diff --git "a/validation/1700-1799/euler_rechenkunst01_1738.jsonl" "b/validation/1700-1799/euler_rechenkunst01_1738.jsonl" new file mode 100644--- /dev/null +++ "b/validation/1700-1799/euler_rechenkunst01_1738.jsonl" @@ -0,0 +1,1929 @@ +{"basename":"euler_rechenkunst01_1738","par_idx":0,"date":1738,"orig":"Einleitung zur Rechen-Kunſt zum Gebrauch des GYMNASII bey der Kayſerlichen Academie der Wiſſenſchafften in St. Petersburg.","norm":"Einleitung zur Rechenkunst zum Gebrauch des GYMNASII bei der kaiserlichen Akademie der Wissenschaften in St. 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Rechen-Buͤcher, welche ſo wohl in Teutſchland als anderwerts herausgegeben worden, iſt ſo groß und uͤberhaͤufft, daß manchem dieſe Arbeit hoͤchſt unnoͤthig und uͤberfluͤßig ſcheinen moͤchte.","norm":"Die Anzahl der Rechen-Bücher, welche so wohl in Deutschland als anderwärts herausgegeben worden, ist so groß und überhäuft, dass manchem diese Arbeit höchst unnötig und überflüssig scheinen 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+{"basename":"euler_rechenkunst01_1738","par_idx":7,"date":1738,"orig":"Uber das befinden ſich bey den meiſten auslaͤndiſchen Rechen-Buͤcheren ſolche Maͤngel, welchen man allhier abzuhelffen fuͤr hoͤchſt rathſam hielt.","norm":"Über das befinden sich bei den meisten ausländischen Rechen-Büchern solche Mängel, welchen man allhier abzuhelfen für höchst ratsam 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+{"basename":"euler_rechenkunst01_1738","par_idx":8,"date":1738,"orig":"Dann entweder ſind darinn nichts als die bloſſen Regeln nebſt einer groſſen Anzahl Exempel enthalten; von dem Grunde aber und den Urſachen, worauf die Regeln beruhen, wird nicht die geringſte Meldung gethan: oder dergleichen Anweiſungen gehen zwar auf das wahre Fundament der Rechen-Kunſt, der Vortrag aber iſt ſo beſchaffen, daß ſich nicht leicht andere, als welche ſich an die Mathematiſche Lehrart gewoͤhnet haben, darein finden koͤnnen; und uͤber das pflegt man ſich bey ſolchen Abhandlungen, nicht’ genugſam um die Vortheile und Compendia, wodurch die Fertigkeit und Geſchwindigkeit im Rechnen erlanget wird, zu bekuͤmmern, ſondern begnuͤgt ſich von allem den Grund nur mit kurtzem anzuweiſen.","norm":"Dann entweder sind darin nichts als die bloßen Regeln nebst einer großen Anzahl Exempel enthalten; von dem Grunde aber und den Ursachen, worauf die Regeln beruhen, wird nicht die geringste Meldung getan: oder dergleichen Anweisungen gehen zwar auf das wahre Fundament der Rechenkunst, der Vortrag aber ist so beschaffen, dass sich nicht leicht andere, als welche sich an die mathematische Lehrart gewöhnet haben, darein finden können; und über das pflegt man sich bei solchen Abhandlungen, nicht genügsam um die Vorteile und Compendia, wodurch die Fertigkeit und Geschwindigkeit im Rechnen erlanget wird, zu bekümmern, sondern begnügt sich von allem den Grund nur mit kurzem 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+{"basename":"euler_rechenkunst01_1738","par_idx":9,"date":1738,"orig":"Da nun die Erlernung der Rechen-Kunſt ohne einigen, Grund weder hinreichend iſt alle vorkommende Faͤlle aufzuloͤſen, noch den Verſtand ſchaͤrfet, als dahin die Abſicht inſonderheit gehen ſolte; ſo hat man ſich bemuͤhet in gegenwaͤrtiger Anleitung von allen Regeln und Operationen den Grund ſo vorzutragen, und zu erklaͤren, daß denſelben auch ſolche Leute, welche in gruͤndlichen Abhandlungen noch nicht geuͤbet ſind, einſehen und verſtehen koͤnnen: dabey aber hat man gleichwohl die Regeln und Vortheile, welche im Rechnen zuſtatten kommen koͤnnen, ausfuͤhrlich beſchrieben und mit Exempeln genugſam erlaͤutert.","norm":"Da nun die Erlernung der Rechenkunst ohne einigen, Grund weder hinreichend ist alle vorkommende Fälle aufzulösen, noch den Verstand schärfet, als dahin die Absicht in Sonderheit gehen sollte; so hat man sich bemühet in gegenwärtiger Anleitung von allen Regeln und Operationen den Grund so vorzutragen, und zu erklären, dass denselben auch solche Leute, welche in gründlichen Abhandlungen noch nicht geübt sind, einsehen und verstehen können: dabei aber hat man gleichwohl die Regeln und Vorteile, welche im Rechnen zustatten kommen können, ausführlich beschrieben und mit Exempeln genügsam erläutert.","orig_tok":["Da","nun","die","Erlernung","der","Rechen-Kunſt","ohne","einigen",",","Grund","weder","hinreichend","iſt","alle","vorkommende","Faͤlle","aufzuloͤſen",",","noch","den","Verſtand","ſchaͤrfet",",","als","dahin","die","Abſicht","inſonderheit","gehen","ſolte",";","ſo","hat","man","ſich","bemuͤhet","in","gegenwaͤrtiger","Anleitung","von","allen","Regeln","und","Operationen","den","Grund","ſo","vorzutragen",",","und","zu","erklaͤren",",","daß","denſelben","auch","ſolche","Leute",",","welche","in","gruͤndlichen","Abhandlungen","noch","nicht","geuͤbet","ſind",",","einſehen","und","verſtehen","koͤnnen",":","dabey","aber","hat","man","gleichwohl","die","Regeln","und","Vortheile",",","welche","im","Rechnen","zuſtatten","kommen","koͤnnen",",","ausfuͤhrlich","beſchrieben","und","mit","Exempeln","genugſam","erlaͤutert","."],"norm_tok":["Da","nun","die","Erlernung","der","Rechenkunst","ohne","einigen",",","Grund","weder","hinreichend","ist","alle","vorkommende","Fälle","aufzulösen",",","noch","den","Verstand","schärfet",",","als","dahin","die","Absicht","in","Sonderheit","gehen","sollte",";","so","hat","man","sich","bemühet","in","gegenwärtiger","Anleitung","von","allen","Regeln","und","Operationen","den","Grund","so","vorzutragen",",","und","zu","erklären",",","dass","denselben","auch","solche","Leute",",","welche","in","gründlichen","Abhandlungen","noch","nicht","geübt","sind",",","einsehen","und","verstehen","können",":","dabei","aber","hat","man","gleichwohl","die","Regeln","und","Vorteile",",","welche","im","Rechnen","zustatten","kommen","können",",","ausführlich","beschrieben","und","mit","Exempeln","genügsam","erläutert","."],"orig_spans":[[0,2],[3,6],[7,10],[11,20],[21,24],[25,37],[38,42],[43,50],[50,51],[52,57],[58,63],[64,75],[76,79],[80,84],[85,96],[97,103],[104,115],[115,116],[117,121],[122,125],[126,134],[135,144],[144,145],[146,149],[150,155],[156,159],[160,167],[168,180],[181,186],[187,192],[192,193],[194,196],[197,200],[201,204],[205,209],[210,218],[219,221],[222,236],[237,246],[247,250],[251,256],[257,263],[264,267],[268,279],[280,283],[284,289],[290,292],[293,304],[304,305],[306,309],[310,312],[313,322],[322,323],[324,327],[328,337],[338,342],[343,349],[350,355],[355,356],[357,363],[364,366],[367,379],[380,392],[393,397],[398,403],[404,411],[412,416],[416,417],[418,426],[427,430],[431,440],[441,448],[448,449],[450,455],[456,460],[461,464],[465,468],[469,479],[480,483],[484,490],[491,494],[495,504],[504,505],[506,512],[513,515],[516,523],[524,533],[534,540],[541,548],[548,549],[550,562],[563,574],[575,578],[579,582],[583,591],[592,600],[601,611],[611,612]],"norm_spans":[[0,2],[3,6],[7,10],[11,20],[21,24],[25,36],[37,41],[42,49],[49,50],[51,56],[57,62],[63,74],[75,78],[79,83],[84,95],[96,101],[102,112],[112,113],[114,118],[119,122],[123,131],[132,140],[140,141],[142,145],[146,151],[152,155],[156,163],[164,166],[167,177],[178,183],[184,190],[190,191],[192,194],[195,198],[199,202],[203,207],[208,215],[216,218],[219,232],[233,242],[243,246],[247,252],[253,259],[260,263],[264,275],[276,279],[280,285],[286,288],[289,300],[300,301],[302,305],[306,308],[309,317],[317,318],[319,323],[324,333],[334,338],[339,345],[346,351],[351,352],[353,359],[360,362],[363,374],[375,387],[388,392],[393,398],[399,404],[405,409],[409,410],[411,419],[420,423],[424,433],[434,440],[440,441],[442,447],[448,452],[453,456],[457,460],[461,471],[472,475],[476,482],[483,486],[487,495],[495,496],[497,503],[504,506],[507,514],[515,524],[525,531],[532,538],[538,539],[540,551],[552,563],[564,567],[568,571],[572,580],[581,589],[590,599],[599,600]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[27,28],[28,29],[29,30],[30,31],[31,32],[32,33],[33,34],[34,35],[35,36],[36,37],[37,38],[38,39],[39,40],[40,41],[41,42],[42,43],[43,44],[44,45],[45,46],[46,47],[47,48],[48,49],[49,50],[50,51],[51,52],[52,53],[53,54],[54,55],[55,56],[56,57],[57,58],[58,59],[59,60],[60,61],[61,62],[62,63],[63,64],[64,65],[65,66],[66,67],[67,68],[68,69],[69,70],[70,71],[71,72],[72,73],[73,74],[74,75],[75,76],[76,77],[77,78],[78,79],[79,80],[80,81],[81,82],[82,83],[83,84],[84,85],[85,86],[86,87],[87,88],[88,89],[89,90],[90,91],[91,92],[92,93],[93,94],[94,95],[95,96],[96,97],[97,98]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.694699764251709} +{"basename":"euler_rechenkunst01_1738","par_idx":10,"date":1738,"orig":"Durch dieſe Einrichtung verhofft man alſo dieſen Vortheil zu erlangen, daß die Jugend auſſer der gehoͤrigen Fertigkeit im Rechnen den wahren Grund von einer jeglichen Operation immer vor Augen habe, und dadurch zu gruͤndlichem Rachdencken nach und nach angewoͤhnet werde.","norm":"Durch diese Einrichtung verhofft man also diesen Vorteil zu erlangen, dass die Jugend außer der gehörigen Fertigkeit im Rechnen den wahren Grund von einer jeglichen Operation immer vor Augen habe, und dadurch zu gründlichem Rachdenken nach und nach angewöhnet 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+{"basename":"euler_rechenkunst01_1738","par_idx":12,"date":1738,"orig":"Man hat auch im geringſten nicht zu befuͤrchten, daß die Erlernung der Arithmetic auf dieſe Art ſchwehrer fallen und mehr Zeit erfordern werde, als wann man nur die bloſſen Regeln ohne einigen Grund vortraͤgt:","norm":"Man hat auch im Geringsten nicht zu befürchten, dass die Erlernung der Arithmetik auf diese Art schwerer fallen und mehr Zeit erfordern werde, als wann man nur die bloßen Regeln ohne einigen Grund 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+{"basename":"euler_rechenkunst01_1738","par_idx":38,"date":1738,"orig":"Und deswegen haben wir im Anfang vorausgeſetzet, daß die Arithmetic auſſer der Betrachtung der Zahlen, einige Regeln an die Hand gebe, wodurch die meiſten in dem gewoͤhnlichen Handel und Wandel vorfallenden Rechnungen koͤnnen bewerckſtelliget werden.","norm":"Und deswegen haben wir im Anfang vorausgesetzt, dass die Arithmetik außer der Betrachtung der Zahlen, einige Regeln an die Hand gebe, wodurch die meisten in dem gewöhnlichen Handel und Wandel vorfallenden Rechnungen können bewerkstelliget 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+{"basename":"euler_rechenkunst01_1738","par_idx":46,"date":1738,"orig":"Es iſt nehmlich nicht genug die gedachten Arithmetiſchen Operationen zu verſtehen, ſondern man muß fuͤr einen jeglichen Fall eine Regel wiſſen, welche lehret was fuͤr Operationen gebraucht werden muͤſſen, um dasjenige, was zu wiſſen verlangt wird, zu finden.","norm":"Es ist nämlich nicht genug die gedachten arithmetischen Operationen zu verstehen, sondern man muss für einen jeglichen Fall eine Regel wissen, welche lehrt was für Operationen gebraucht werden müssen, um dasjenige, was zu wissen verlangt wird, zu 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+{"basename":"euler_rechenkunst01_1738","par_idx":47,"date":1738,"orig":"Dieſe Regeln haben nun ihren Grund nicht in der Arithmetic; ſondern ſind aus der allgemeinen Analyſi oder Algebra gelehnet; Als wofuͤr eine jede Art von Aufgaben aus den Umſtaͤnden ſonderbare Regeln hergeleitet werden, durch welcher Huͤlfe man zu richtiger Aufloͤſung gelangen kan.","norm":"Diese Regeln haben nun ihren Grund nicht in der Arithmetik; sondern sind aus der allgemeinen Analysi oder Algebra gelehnt; als wofür eine jede Art von Aufgaben aus den Umständen sonderbare Regeln hergeleitet werden, durch welcher Hilfe man zu richtiger Auflösung gelangen 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+{"basename":"euler_rechenkunst01_1738","par_idx":48,"date":1738,"orig":"Es werden demnach aus der Algebra ſo viel und ſolche Regeln in die Rechenkunſt angenommen, als zu den gewoͤhnlichen Vorfaͤllen auszurechnen noͤthig ſind.","norm":"Es werden demnach aus der Algebra so viel und solche Regeln in die Rechenkunst angenommen, als zu den gewöhnlichen Vorfällen auszurechnen nötig 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+{"basename":"euler_rechenkunst01_1738","par_idx":49,"date":1738,"orig":"Solchergeſtalt ſind in die Arithmetic aufgenommen worden, die Regula Detri Regula Quinque, Regula Aligationis, Regula Falſi etc. als ohne welche ein Rechenmeiſter, welcher in der Algebra nicht geuͤbet iſt, ſchwehrlich fortkommen kan.","norm":"Solchergestalt sind in die Arithmetik aufgenommen worden, die Regula Detri Regula Quinque, Regula Aligationis, Regula Falsi etc. als ohne welche ein Rechenmeister, welcher in der Algebra nicht geübt ist, schwerlich fortkommen kann.","orig_tok":["Solchergeſtalt","ſind","in","die","Arithmetic","aufgenommen","worden",",","die","Regula","Detri","Regula","Quinque",",","Regula","Aligationis",",","Regula","Falſi","etc.","als","ohne","welche","ein","Rechenmeiſter",",","welcher","in","der","Algebra","nicht","geuͤbet","iſt",",","ſchwehrlich","fortkommen","kan","."],"norm_tok":["Solchergestalt","sind","in","die","Arithmetik","aufgenommen","worden",",","die","Regula","Detri","Regula","Quinque",",","Regula","Aligationis",",","Regula","Falsi","etc.","als","ohne","welche","ein","Rechenmeister",",","welcher","in","der","Algebra","nicht","ge��bt","ist",",","schwerlich","fortkommen","kann","."],"orig_spans":[[0,14],[15,19],[20,22],[23,26],[27,37],[38,49],[50,56],[56,57],[58,61],[62,68],[69,74],[75,81],[82,89],[89,90],[91,97],[98,109],[109,110],[111,117],[118,123],[124,128],[129,132],[133,137],[138,144],[145,148],[149,162],[162,163],[164,171],[172,174],[175,178],[179,186],[187,192],[193,200],[201,204],[204,205],[206,217],[218,228],[229,232],[232,233]],"norm_spans":[[0,14],[15,19],[20,22],[23,26],[27,37],[38,49],[50,56],[56,57],[58,61],[62,68],[69,74],[75,81],[82,89],[89,90],[91,97],[98,109],[109,110],[111,117],[118,123],[124,128],[129,132],[133,137],[138,144],[145,148],[149,162],[162,163],[164,171],[172,174],[175,178],[179,186],[187,192],[193,198],[199,202],[202,203],[204,214],[215,225],[226,230],[230,231]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28],[29,29],[30,30],[31,31],[32,32],[33,33],[34,34],[35,35],[36,36],[37,37]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.750699996948242} +{"basename":"euler_rechenkunst01_1738","par_idx":50,"date":1738,"orig":"3)","norm":"3)","orig_tok":["3",")"],"norm_tok":["3",")"],"orig_spans":[[0,1],[1,2]],"norm_spans":[[0,1],[1,2]],"alignment":[[0,0],[1,1]],"lang_fastText":"es","lang_py3langid":"en","lang_cld3":"ja","orig_lang_de":0,"norm_lmscore":2.315399885177612} +{"basename":"euler_rechenkunst01_1738","par_idx":51,"date":1738,"orig":"Wenn viel Stuͤcke von einer Artvorhanden ſind, ſo wird dieſe Vielheit durch eine Zahl angedeutet.","norm":"Wenn viel Stücke von einer Artvorhanden sind, so wird diese Vielheit durch eine Zahl 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und daraus diejenigen Operationen hergeleitet werden, welche zu Vollziehung der im zweyten Theile vorkommenden Regeln noͤthig ſind; ſo muß man ſich vor allen Dingen einen deutlichen Begriff von den Zahlen zu wege bringen.","norm":"Da in dem ersten Teile der Rechenkunst die Natur der Zahlen soll untersuchet, und daraus diejenigen Operationen hergeleitet werden, welche zu Vollziehung der im zweiten Teile vorkommenden Regeln nötig sind; so muss man sich vor allen Dingen einen deutlichen Begriff von den Zahlen zu Wege 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+{"basename":"euler_rechenkunst01_1738","par_idx":56,"date":1738,"orig":"Was aber die Groͤſſe der Zahlen betrifft, ſo wird hier vorausgeſetzet, daß derjenige, welcher die Arithmetic zu lernen geſinnet iſt, von der Groͤſſe einer jeden Zahl einen Begriff habe, und die Worte wiſſe, damit die Zahlen benennet werden.","norm":"Was aber die Größe der Zahlen betrifft, so wird hier vorausgesetzt, dass derjenige, welcher die Arithmetik zu lernen gesinnt ist, von der Größe einer jeden Zahl einen Begriff habe, und die Worte wisse, damit die Zahlen benennet 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+{"basename":"euler_rechenkunst01_1738","par_idx":58,"date":1738,"orig":"Dann auf dieſe Art wird ein Menſch mit Zehlen ſo weit fortfahren koͤnnen, als man verlangt; und wird dabey von der Menge der Stuͤcken, welche eine jede Zahl andeutet, einen deutlichen Begriff erhalten.","norm":"Dann auf diese Art wird ein Mensch mit zählen so weit fortfahren können, als man verlangt; und wird dabei von der Menge der Stücken, welche eine jede Zahl andeutet, einen deutlichen Begriff 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+{"basename":"euler_rechenkunst01_1738","par_idx":63,"date":1738,"orig":"Dieſe Art aber iſt, wenn die Zahlen ſehr groß ſind, einer groſſen Weitlaͤuffigkeit und Undeutlichkeit unterworffen; indem erſtlich lange Zeit und ein groſſer Raum eine groſſe Zahl zu ſchreiben erfordert, und hernach auch, wenn eine ſolche Zahl geſchrieben, ſehr ſchwehr fallen wuͤrde, die Zahl zu erkennen.","norm":"Diese Art aber ist, wenn die Zahlen sehr groß sind, einer großen Weitläufigkeit und Undeutlichkeit unterworfen; indem erstlich lange Zeit und ein großer Raum eine große Zahl zu schreiben erfordert, und hernach auch, wenn eine solche Zahl geschrieben, sehr schwerfallen würde, die Zahl zu 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+{"basename":"euler_rechenkunst01_1738","par_idx":64,"date":1738,"orig":"Nach der Roͤmiſchen Schreibart wird zwar dieſe Weitlaͤuffigkeit und Undeutlichkeit etwas verringert, indem anſtatt fuͤnf Strichen dieſes Zeichen V. anſtatt zehen dieſes Zeichen X. und ſo fort geſchrieben wird; allein da dieſe Art gleichwohl fuͤr groſſe Zahlen noch ziemlich weitlaͤuffig und undeutlich, dabey auch nicht durch feſte Regeln genugſam eingeſchraͤncket iſt, ſo iſt dieſelbe nicht bequem die Arithmetiſchen Operationen darnach einzurichten.","norm":"Nach der römischen Schreibart wird zwar diese Weitläufigkeit und Undeutlichkeit etwas verringert, indem anstatt fünf Strichen dieses Zeichen V. anstatt zehn dieses Zeichen X. und so fort geschrieben wird; allein da diese Art gleichwohl für große Zahlen noch ziemlich weitläufig und undeutlich, dabei auch nicht durch feste Regeln genügsam eingeschränkt ist, so ist dieselbe nicht bequem die arithmetischen Operationen danach 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+{"basename":"euler_rechenkunst01_1738","par_idx":65,"date":1738,"orig":"Noch mehr Schwierigkeiten ſind diejenigen Arten die Zahleu zu ſchreiben unterworfen, in welchen die Buchſtaben des Alphabets zu Bedeutung der Zahlen gebraucht werden; gleichwie vormahls bey den meiſten Voͤlckern geſchehen.","norm":"Noch mehr Schwierigkeiten sind diejenigen Arten die Zahleu zu schreiben unterworfen, in welchen die Buchstaben des Alphabets zu Bedeutung der Zahlen gebraucht werden; gleichwie vormals bei den meisten Völkern 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+{"basename":"euler_rechenkunst01_1738","par_idx":66,"date":1738,"orig":"Vor dieſen Arten hat nun die anjetzo faſt allenthalben gebraͤuchliche Art die Zahlen durch Huͤlffe der zehen angefuͤhrten Zeichen zu ſchreiben, einen ſehr groſſen Vorzug, wie mit mehrerem aus folgendem zu erſehen.","norm":"Vor diesen Arten hat nun die jetzt fast allenthalben gebräuchliche Art die Zahlen durch Hilfe der zehn angeführten Zeichen zu schreiben, einen sehr großen Vorzug, wie mit mehrerem aus folgendem zu 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+{"basename":"euler_rechenkunst01_1738","par_idx":102,"date":1738,"orig":"Wobey die Gewohnheit mit ſich bringt, in Benennung derſelben von der Lincken zu der Rechten fortzugehen.","norm":"Wobei die Gewohnheit mit sich bringt, in Benennung derselben von der Linken zu der Rechten 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+{"basename":"euler_rechenkunst01_1738","par_idx":105,"date":1738,"orig":"Dieſemnach wurde die obige Zahl 7302568 ſo viel ausmachen wie folgt: Sieben Millionen, drey hundert tauſend und zwey tauſend und fuͤnf hundert und ſechzig und acht.","norm":"Diesemnach wurde die obige Zahl 7302568 so viel ausmachen wie folgt: Sieben Millionen, drei hunderttausend und zweitausend und fünfhundert und sechzig und acht.","orig_tok":["Dieſemnach","wurde","die","obige","Zahl","7302568","ſo","viel","ausmachen","wie","folgt",":","Sieben","Millionen",",","drey","hundert","tauſend","und","zwey","tauſend","und","fuͤnf","hundert","und","ſechzig","und","acht","."],"norm_tok":["Diesemnach","wurde","die","obige","Zahl","7302568","so","viel","ausmachen","wie","folgt",":","Sieben","Millionen",",","drei","hunderttausend","und","zweitausend","und","fünfhundert","und","sechzig","und","acht","."],"orig_spans":[[0,10],[11,16],[17,20],[21,26],[27,31],[32,39],[40,42],[43,47],[48,57],[58,61],[62,67],[67,68],[69,75],[76,85],[85,86],[87,91],[92,99],[100,107],[108,111],[112,116],[117,124],[125,128],[129,134],[135,142],[143,146],[147,154],[155,158],[159,163],[163,164]],"norm_spans":[[0,10],[11,16],[17,20],[21,26],[27,31],[32,39],[40,42],[43,47],[48,57],[58,61],[62,67],[67,68],[69,75],[76,85],[85,86],[87,91],[92,106],[107,110],[111,122],[123,126],[127,138],[139,142],[143,150],[151,154],[155,159],[159,160]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,16],[18,17],[19,18],[20,18],[21,19],[22,20],[23,20],[24,21],[25,22],[26,23],[27,24],[28,25]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.131100177764893} +{"basename":"euler_rechenkunst01_1738","par_idx":106,"date":1738,"orig":"Nach der Eigenſchaft unſerer Sprache aber wird dieſe Zahl allſo ausgeſprochen: Sieben Millionen, drey hundert und zwey tauſend, fuͤnf hundert und acht und ſechzig; welche Art von der vorigen nur darinn unterſchieden iſt, daß da oben tauſend zweymahl nach einander vorkommt, hier nur das letztere Mahl geſetzet wird, indem es auf dieſe Art geſetzet auch zugleich zu dem vorhergehenden gehoͤret.","norm":"Nach der Eigenschaft unserer Sprache aber wird diese Zahl also ausgesprochen: Sieben Millionen, dreihundert und zweitausend, fünfhundert und achtundsechzig; welche Art von der vorigen nur darin unterschieden ist, dass da oben tausend zweimal nach einander vorkommt, hier nur das letztere Mal gesetzt wird, indem es auf diese Art gesetzt auch zugleich zu dem vorhergehenden 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Zahlen mit Worten auszuſprechen ſehr genau uͤbereinkomme, indem uns eine jegliche Zahl bey nahe die gewoͤhnlichen Worte, und das in eben der Ordnung in den Mund legt: Welche Gemeinſchaft faſt in allen Sprachen, in einer aber mehr als in der anderen beobachtet wird.","norm":"Aus welchem allen erhellet, dass diese Art die Zahlen zu schreiben mit der gewöhnlichen Art die Zahlen mit Worten auszusprechen sehr genau übereinkomme, indem uns eine jegliche Zahl bei nahe die gewöhnlichen Worte, und das in eben der Ordnung in den Mund legt: Welche Gemeinschaft fast in allen Sprachen, in einer aber mehr als in der anderen beobachtet 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+{"basename":"euler_rechenkunst01_1738","par_idx":112,"date":1738,"orig":"Jn der Teutſchen Sprache iſt nur einige Schwierigkeit, eine Zahl, ſo aus zweyen Characteren beſtehet auszuſprechen, indem die letztere gegen der rechten Hand zu erſt genennet wird.","norm":"In der Deutschen Sprache ist nur einige Schwierigkeit, eine Zahl, so aus zweien Charakter bestehet auszusprechen, indem die letztere gegen der rechten Hand zuerst genannt 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+{"basename":"euler_rechenkunst01_1738","par_idx":113,"date":1738,"orig":"Jſt aber dieſer Character ein 0 ſo wird nur der erſte gegen der lincken Hand mit dem Nahmen, welchen er in der zweyten Stelle hat benennet.","norm":"Ist aber dieser Charakter ein 0 so wird nur der erste gegen der linken Hand mit dem Namen, welchen er in der zweiten Stelle hat 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zehen; 13 dreyzehen; 14 vierzehen und ſo fort bis auf zwantzig.","norm":"Weiter ist 11 elf oder eins und zehn; 12 zwölf oder zwei und zehn; 13 dreizehn; 14 vierzehn und so fort bis auf 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+{"basename":"euler_rechenkunst01_1738","par_idx":117,"date":1738,"orig":"Hat man nun die Ausſprechung zweyer Character begriffen, ſo iſt ſehr leicht alle Zahlen, welche mit drey Characteren geſchrieben werden, auszuſprechen, indem nur erſtlich der erſte von der Lincken nebſt Zuſetzung des Worts hundert genennet, und die zwey folgenden wie gelehret, mit den Worten hinzugeſetzet werden.","norm":"Hat man nun die Aussprechung zweier Charakter begriffen, so ist sehr leicht alle Zahlen, welche mit drei Charakter geschrieben werden, auszusprechen, indem nur erstlich der erste von der Linken nebst Zusetzung des Worts hundert genannt, und die zwei folgenden wie gelehrt, mit den Worten hinzugesetzt 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+{"basename":"euler_rechenkunst01_1738","par_idx":118,"date":1738,"orig":"Allſo iſt 114 hundert und vierzehen; 570 fuͤnf hundert und ſiebenzig; 324 dreyhundert und vier und zwantzig; 208 zwey hundert und acht; 600 ſechs hundert; und ſo fort.","norm":"also ist 114 hundertundvierzehn; 570 fünfhundert und siebzig; 324 dreihundertundvier und zwanzig; 208 zweihundert und acht; 600 sechshundert; und so 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+{"basename":"euler_rechenkunst01_1738","par_idx":168,"date":1738,"orig":"Das fuͤrnehmſte hierinn iſt, daß man die Zahlen, welche kleiner ſind als tauſend, wohl wiſſe zu ſchreiben und auf ihre gehoͤrigen drey Stellen zu ſetzen, denn ſo wohl die Tauſender als Millionen, Billionen ꝛc. durch ſolche Zahlen gezehlet zu werden pflegen.","norm":"Das fürnehmste hierin ist, dass man die Zahlen, welche kleiner sind als tausend, wohl wisse zu schreiben und auf ihre gehörigen drei Stellen zu setzen, denn so wohl die Tausender als Millionen, Billionen etc. durch solche Zahlen gezählt zu werden 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+{"basename":"euler_rechenkunst01_1738","par_idx":169,"date":1738,"orig":"Hernach iſt auch zu beobachten, daß die Millionen, Billionen, Trillionen ꝛc. ſechs Stellen in ihrem Bezirck haben; da dann eine jegliche Art ins beſondere kan geſchrieben werden: wobey nur zu mercken, daß nach den Millionen gegen der rechten noch 6 Stellen, nach den Billionen zwoͤlff Stellen und ſo fort folgen muͤſſen.","norm":"Hernach ist auch zu beobachten, dass die Millionen, Billionen, Trillionen etc. sechs Stellen in ihrem Bezirk haben; da dann eine jegliche Art ins Besondere kann geschrieben werden: wobei nur zu merken, dass nach den Millionen gegen der rechten noch 6 Stellen, nach den Billionen zwölf Stellen und so fort folgen 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+{"basename":"euler_rechenkunst01_1738","par_idx":172,"date":1738,"orig":"11)","norm":"11)","orig_tok":["11",")"],"norm_tok":["11",")"],"orig_spans":[[0,2],[2,3]],"norm_spans":[[0,2],[2,3]],"alignment":[[0,0],[1,1]],"lang_fastText":"es","lang_py3langid":"en","lang_cld3":"ja","orig_lang_de":0,"norm_lmscore":3.216099977493286} +{"basename":"euler_rechenkunst01_1738","par_idx":173,"date":1738,"orig":"Dasjenige, welches bisher iſt erklaͤret worden, nehmlich wie man eine durch Characteres beſchriebene Zahl mit Worten ausſprechen und hinwiederum eine jede Zahl durch ſolche Characteres ſchreiben ſoll, wird die Numeration genennet und pflegt gimeiniglich fuͤr die erſte Arithmet ſche Operation gehalten zu werden.","norm":"Dasjenige, welches bisher ist erkläret worden, nämlich wie man eine durch Characteres beschriebene Zahl mit Worten aussprechen und hinwiederum eine jede Zahl durch solche Characteres schreiben soll, wird die Enumeration genannt und pflegt gemeiniglich für die erste Arithmet sche Operation gehalten zu 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+{"basename":"euler_rechenkunst01_1738","par_idx":174,"date":1738,"orig":"Es iſt willkuͤhrlich was fuͤr Character zu Beſchreibung der Zahlen gebrauchet werden; eine jede Art aber der Zahlen auszudrucken erfordert beſondere Regeln zu den Arithmetiſchen Operationen, welche aus der Beſchaffenheit einer jeglichen Art muͤſſen hergeleitet werden.","norm":"Es ist willkürlich was für Charakter zu Beschreibung der Zahlen gebraucht werden; eine jede Art aber der Zahlen auszudrucken erfordert besondere Regeln zu den arithmetischen Operationen, welche aus der Beschaffenheit einer jeglichen Art müssen hergeleitet werden.","orig_tok":["Es","iſt","willkuͤhrlich","was","fuͤr","Character","zu","Beſchreibung","der","Zahlen","gebrauchet","werden",";","eine","jede","Art","aber","der","Zahlen","auszudrucken","erfordert","beſondere","Regeln","zu","den","Arithmetiſchen","Operationen",",","welche","aus","der","Beſchaffenheit","einer","jeglichen","Art","muͤſſen","hergeleitet","werden","."],"norm_tok":["Es","ist","willkürlich","was","für","Charakter","zu","Beschreibung","der","Zahlen","gebraucht","werden",";","eine","jede","Art","aber","der","Zahlen","auszudrucken","erfordert","besondere","Regeln","zu","den","arithmetischen","Operationen",",","welche","aus","der","Beschaffenheit","einer","jeglichen","Art","müssen","hergeleitet","werden","."],"orig_spans":[[0,2],[3,6],[7,20],[21,24],[25,29],[30,39],[40,42],[43,55],[56,59],[60,66],[67,77],[78,84],[84,85],[86,90],[91,95],[96,99],[100,104],[105,108],[109,115],[116,128],[129,138],[139,148],[149,155],[156,158],[159,162],[163,177],[178,189],[189,190],[191,197],[198,201],[202,205],[206,220],[221,226],[227,236],[237,240],[241,248],[249,260],[261,267],[267,268]],"norm_spans":[[0,2],[3,6],[7,18],[19,22],[23,26],[27,36],[37,39],[40,52],[53,56],[57,63],[64,73],[74,80],[80,81],[82,86],[87,91],[92,95],[96,100],[101,104],[105,111],[112,124],[125,134],[135,144],[145,151],[152,154],[155,158],[159,173],[174,185],[185,186],[187,193],[194,197],[198,201],[202,216],[217,222],[223,232],[233,236],[237,243],[244,255],[256,262],[262,263]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28],[29,29],[30,30],[31,31],[32,32],[33,33],[34,34],[35,35],[36,36],[37,37],[38,38]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.9390997886657715} +{"basename":"euler_rechenkunst01_1738","par_idx":175,"date":1738,"orig":"Wir haben aber bisher genugſam dargethan, daß die gewoͤhnliche Art vermittelſt der zehen Character am allerbeſten mit den Worten, dadurch die Zahlen benennet werden, uͤbereinkommen; wie es dann auf dieſe Art ſehr leicht iſt eine jede durch ſolche Characteres beſchriebene Zahl mit Worten auszuſprechen, und hinwiederum eine mit Worten benennte Zahl zu ſchreiben.","norm":"Wir haben aber bisher genügsam dargetan, dass die gewöhnliche Art vermittels der zehn Charakter am allerbesten mit den Worten, dadurch die Zahlen benennet werden, übereinkommen; wie es dann auf diese Art sehr leicht ist eine jede durch solche Characteres beschriebene Zahl mit Worten auszusprechen, und hinwiederum eine mit Worten benennte Zahl zu 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+{"basename":"euler_rechenkunst01_1738","par_idx":176,"date":1738,"orig":"Da nun die Arithmetiſchen Operationen nach dieſer Art am bequemſten eingerichtet worden; ſo war, ehe man zu den Operationen ſelbſt ſchreiten koͤnnte, unumgaͤnglich noͤthig dieſe Ausdruͤckungs-Art der Zahlen ausfuͤhrlich zu erklaͤren, damit daraus die Regeln fuͤr die Operationen koͤnnten hergeleitet werden.","norm":"Da nun die arithmetischen Operationen nach dieser Art am bequemsten eingerichtet worden; so war, ehe man zu den Operationen selbst schreiten könnte, unumgänglich nötig diese Ausdrückungs-Art der Zahlen ausführlich zu erklären, damit daraus die Regeln für die Operationen könnten hergeleitet 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+{"basename":"euler_rechenkunst01_1738","par_idx":177,"date":1738,"orig":"Dieſe Vorbereitung zu den Arithmetiſchen Operationen wird nun Numeratio oder Notatio genennet, welche lehret eine jegliche Zahl ſchreiben, und wenn eine Zahl geſchrieben, wiederum ausſprechen.","norm":"Diese Vorbereitung zu den arithmetischen Operationen wird nun Numeratio oder Notatio genannt, welche lehrt eine jegliche Zahl schreiben, und wenn eine Zahl geschrieben, wiederum aussprechen.","orig_tok":["Dieſe","Vorbereitung","zu","den","Arithmetiſchen","Operationen","wird","nun","Numeratio","oder","Notatio","genennet",",","welche","lehret","eine","jegliche","Zahl","ſchreiben",",","und","wenn","eine","Zahl","geſchrieben",",","wiederum","ausſprechen","."],"norm_tok":["Diese","Vorbereitung","zu","den","arithmetischen","Operationen","wird","nun","Numeratio","oder","Notatio","genannt",",","welche","lehrt","eine","jegliche","Zahl","schreiben",",","und","wenn","eine","Zahl","geschrieben",",","wiederum","aussprechen","."],"orig_spans":[[0,5],[6,18],[19,21],[22,25],[26,40],[41,52],[53,57],[58,61],[62,71],[72,76],[77,84],[85,93],[93,94],[95,101],[102,108],[109,113],[114,122],[123,127],[128,137],[137,138],[139,142],[143,147],[148,152],[153,157],[158,169],[169,170],[171,179],[180,191],[191,192]],"norm_spans":[[0,5],[6,18],[19,21],[22,25],[26,40],[41,52],[53,57],[58,61],[62,71],[72,76],[77,84],[85,92],[92,93],[94,100],[101,106],[107,111],[112,120],[121,125],[126,135],[135,136],[137,140],[141,145],[146,150],[151,155],[156,167],[167,168],[169,177],[178,189],[189,190]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.115200042724609} +{"basename":"euler_rechenkunst01_1738","par_idx":178,"date":1738,"orig":"Die Numeration kan allſo nicht mit unter die Operationen gezehlet werden, wenn wir durch eine Operation eine beſondere Art verſtehen, aus zweyen oder mehr gegebenen Zahlen eine neue herauszubringen.","norm":"Die Enumeration kann also nicht mit unter die Operationen gezählt werden, wenn wir durch eine Operation eine besondere Art verstehen, aus zweien oder mehr gegebenen Zahlen eine neue herauszubringen.","orig_tok":["Die","Numeration","kan","allſo","nicht","mit","unter","die","Operationen","gezehlet","werden",",","wenn","wir","durch","eine","Operation","eine","beſondere","Art","verſtehen",",","aus","zweyen","oder","mehr","gegebenen","Zahlen","eine","neue","herauszubringen","."],"norm_tok":["Die","Enumeration","kann","also","nicht","mit","unter","die","Operationen","gezählt","werden",",","wenn","wir","durch","eine","Operation","eine","besondere","Art","verstehen",",","aus","zweien","oder","mehr","gegebenen","Zahlen","eine","neue","herauszubringen","."],"orig_spans":[[0,3],[4,14],[15,18],[19,24],[25,30],[31,34],[35,40],[41,44],[45,56],[57,65],[66,72],[72,73],[74,78],[79,82],[83,88],[89,93],[94,103],[104,108],[109,118],[119,122],[123,132],[132,133],[134,137],[138,144],[145,149],[150,154],[155,164],[165,171],[172,176],[177,181],[182,197],[197,198]],"norm_spans":[[0,3],[4,15],[16,20],[21,25],[26,31],[32,35],[36,41],[42,45],[46,57],[58,65],[66,72],[72,73],[74,78],[79,82],[83,88],[89,93],[94,103],[104,108],[109,118],[119,122],[123,132],[132,133],[134,137],[138,144],[145,149],[150,154],[155,164],[165,171],[172,176],[177,181],[182,197],[197,198]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28],[29,29],[30,30],[31,31]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.56059980392456} +{"basename":"euler_rechenkunst01_1738","par_idx":179,"date":1738,"orig":"Da wir nun durch Ausfuͤhrung der Numeration das Fundament zu den A rithmetiſchen Operationen geleget, daraus dieſelben gruͤndlich koͤnnen erklaͤret werden, ſo ſchreiten wir zu dieſen Operationen ſelbſt fort, wenn einige Exempel zur Ubung werden beygebracht ſeyn.","norm":"Da wir nun durch Ausführung der Enumeration das Fundament zu den A rithmetischen Operationen gelegt, daraus dieselben gründlich können erkläret werden, so schreiten wir zu diesen Operationen selbst fort, wenn einige Exempel zur Übung werden beigebracht 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Man hat gefunden, daß der Umkreiß der Erde ſo viel Teutſche Meilen halte als dieſe Zahl 5400 andeutet:","norm":"1. Man hat gefunden, dass der Umkreis der Erde so viel Deutsche Meilen halte als diese Zahl 5400 andeutet:","orig_tok":["1.","Man","hat","gefunden",",","daß","der","Umkreiß","der","Erde","ſo","viel","Teutſche","Meilen","halte","als","dieſe","Zahl","5400","andeutet",":"],"norm_tok":["1.","Man","hat","gefunden",",","dass","der","Umkreis","der","Erde","so","viel","Deutsche","Meilen","halte","als","diese","Zahl","5400","andeutet",":"],"orig_spans":[[0,2],[3,6],[7,10],[11,19],[19,20],[21,24],[25,28],[29,36],[37,40],[41,45],[46,48],[49,53],[54,62],[63,69],[70,75],[76,79],[80,85],[86,90],[91,95],[96,104],[104,105]],"norm_spans":[[0,2],[3,6],[7,10],[11,19],[19,20],[21,25],[26,29],[30,37],[38,41],[42,46],[47,49],[50,54],[55,63],[64,70],[71,76],[77,80],[81,86],[87,91],[92,96],[97,105],[105,106]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.79449987411499} +{"basename":"euler_rechenkunst01_1738","par_idx":182,"date":1738,"orig":"nun fragt man wie groß die Anzahl der Meilen ſey?","norm":"nun fragt man wie groß die Anzahl der Meilen sei?","orig_tok":["nun","fragt","man","wie","groß","die","Anzahl","der","Meilen","ſey","?"],"norm_tok":["nun","fragt","man","wie","groß","die","Anzahl","der","Meilen","sei","?"],"orig_spans":[[0,3],[4,9],[10,13],[14,17],[18,22],[23,26],[27,33],[34,37],[38,44],[45,48],[48,49]],"norm_spans":[[0,3],[4,9],[10,13],[14,17],[18,22],[23,26],[27,33],[34,37],[38,44],[45,48],[48,49]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.4842000007629395} +{"basename":"euler_rechenkunst01_1738","par_idx":183,"date":1738,"orig":"Antw. fuͤnf tauſend vier hundert Teutſche Meilen.","norm":"Antwort fünftausend vierhundert Deutsche Meilen.","orig_tok":["Antw.","fuͤnf","tauſend","vier","hundert","Teutſche","Meilen","."],"norm_tok":["Antwort","fünftausend","vierhundert","Deutsche","Meilen","."],"orig_spans":[[0,5],[6,11],[12,19],[20,24],[25,32],[33,41],[42,48],[48,49]],"norm_spans":[[0,7],[8,19],[20,31],[32,40],[41,47],[47,48]],"alignment":[[0,0],[1,1],[2,1],[3,2],[4,2],[5,3],[6,4],[7,5]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"ga","orig_lang_de":0.667,"norm_lmscore":7.8480000495910645} +{"basename":"euler_rechenkunst01_1738","par_idx":184,"date":1738,"orig":"2. Bibliander hat die Unkoͤſten ausgerechnet, welche der Koͤnig Salomon bey Aufbauung des Tempels zu Jeruſalem aufgewandt, und ſetzt dieſelben auf 13695380050 Kronen; nun wird die Groͤſſe dieſer Zahl gefraget?","norm":"2. Bibliander hat die Unkosten ausgerechnet, welche der König Salomon bei Aufbauung des Tempels zu Jerusalem aufgewandt, und setzt dieselben auf 13695380050 Kronen; nun wird die Größe dieser Zahl gefragt?","orig_tok":["2.","Bibliander","hat","die","Unkoͤſten","ausgerechnet",",","welche","der","Koͤnig","Salomon","bey","Aufbauung","des","Tempels","zu","Jeruſalem","aufgewandt",",","und","ſetzt","dieſelben","auf","13695380050","Kronen",";","nun","wird","die","Groͤſſe","dieſer","Zahl","gefraget","?"],"norm_tok":["2.","Bibliander","hat","die","Unkosten","ausgerechnet",",","welche","der","König","Salomon","bei","Aufbauung","des","Tempels","zu","Jerusalem","aufgewandt",",","und","setzt","dieselben","auf","13695380050","Kronen",";","nun","wird","die","Größe","dieser","Zahl","gefragt","?"],"orig_spans":[[0,2],[3,13],[14,17],[18,21],[22,31],[32,44],[44,45],[46,52],[53,56],[57,63],[64,71],[72,75],[76,85],[86,89],[90,97],[98,100],[101,110],[111,121],[121,122],[123,126],[127,132],[133,142],[143,146],[147,158],[159,165],[165,166],[167,170],[171,175],[176,179],[180,187],[188,194],[195,199],[200,208],[208,209]],"norm_spans":[[0,2],[3,13],[14,17],[18,21],[22,30],[31,43],[43,44],[45,51],[52,55],[56,61],[62,69],[70,73],[74,83],[84,87],[88,95],[96,98],[99,108],[109,119],[119,120],[121,124],[125,130],[131,140],[141,144],[145,156],[157,163],[163,164],[165,168],[169,173],[174,177],[178,183],[184,190],[191,195],[196,203],[203,204]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28],[29,29],[30,30],[31,31],[32,32],[33,33]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.861700057983398} +{"basename":"euler_rechenkunst01_1738","par_idx":185,"date":1738,"orig":"Antw. dreyzehen tauſend ſechs hundert fuͤnf und neunzig Millionen; 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Der Schatz mit welchem ſich der Koͤnig Sardanapalus von Aſſyrien ſelbſt ſoll verbrannt haben, wird auf 145000000000 Goldguͤlden geſchaͤtzet; fragts ſich wie groß dieſer Schatz geweſen?","norm":"4. Der Schatz mit welchem sich der König Sardanapalus von Assyrien selbst soll verbrannt haben, wird auf 145000000000 Goldgülden geschätzt; fragts sich wie groß dieser Schatz gewesen?","orig_tok":["4.","Der","Schatz","mit","welchem","ſich","der","Koͤnig","Sardanapalus","von","Aſſyrien","ſelbſt","ſoll","verbrannt","haben",",","wird","auf","145000000000","Goldguͤlden","geſchaͤtzet",";","fragts","ſich","wie","groß","dieſer","Schatz","geweſen","?"],"norm_tok":["4.","Der","Schatz","mit","welchem","sich","der","König","Sardanapalus","von","Assyrien","selbst","soll","verbrannt","haben",",","wird","auf","145000000000","Goldgülden","geschätzt",";","fragts","sich","wie","groß","dieser","Schatz","gewesen","?"],"orig_spans":[[0,2],[3,6],[7,13],[14,17],[18,25],[26,30],[31,34],[35,41],[42,54],[55,58],[59,67],[68,74],[75,79],[80,89],[90,95],[95,96],[97,101],[102,105],[106,118],[119,130],[131,142],[142,143],[144,150],[151,155],[156,159],[160,164],[165,171],[172,178],[179,186],[186,187]],"norm_spans":[[0,2],[3,6],[7,13],[14,17],[18,25],[26,30],[31,34],[35,40],[41,53],[54,57],[58,66],[67,73],[74,78],[79,88],[89,94],[94,95],[96,100],[101,104],[105,117],[118,128],[129,138],[138,139],[140,146],[147,151],[152,155],[156,160],[161,167],[168,174],[175,182],[182,183]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28],[29,29]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.109799861907959} +{"basename":"euler_rechenkunst01_1738","par_idx":191,"date":1738,"orig":"Antw. hundert und fuͤnf und vierzig tauſend Millionen Goldgulden.","norm":"Antw. hundert und fünfundvierzig tausend Millionen Goldgulden.","orig_tok":["Antw.","hundert","und","fuͤnf","und","vierzig","tauſend","Millionen","Goldgulden","."],"norm_tok":["Antw",".","hundert","und","fünfundvierzig","tausend","Millionen","Goldgulden","."],"orig_spans":[[0,5],[6,13],[14,17],[18,23],[24,27],[28,35],[36,43],[44,53],[54,64],[64,65]],"norm_spans":[[0,4],[4,5],[6,13],[14,17],[18,32],[33,40],[41,50],[51,61],[61,62]],"alignment":[[0,0],[0,1],[1,2],[2,null],[3,null],[4,3],[5,4],[6,5],[7,6],[8,7],[9,8]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.559700012207031} +{"basename":"euler_rechenkunst01_1738","par_idx":192,"date":1738,"orig":"5. Archimedes beweiſet, daß nicht nur alle Sand- Koͤrner auf der gantzen Erde koͤnnen gezehlet werden; ſondern daß man ſo gar eine Zahl anzeigen koͤnne, welche groͤſſer waͤre, als die Anzahl der Sand-Koͤrner, welche den gantzen Raum der Welt bis an die aͤuſſerſten Fixſternen zu erfuͤllen erfordert wuͤrde.","norm":"5. 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+{"basename":"euler_rechenkunst01_1738","par_idx":217,"date":1738,"orig":"ſo iſt noͤthig, daß, wenn in der Addition mehr als 9 Stuͤcke von einer Gattung vorkommen, dieſelben zu hoͤheren Gattungen geſchlagen werden, ſo daß niemahls mehr als 9 Stuͤcke von einer Gattung in Conſideration kommen.","norm":"so ist nötig, dass, wenn in der Addition mehr als 9 Stücke von einer Gattung vorkommen, dieselben zu höheren Gattungen geschlagen werden, so dass niemals mehr als 9 Stücke von einer Gattung in Konsideration 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+{"basename":"euler_rechenkunst01_1738","par_idx":242,"date":1738,"orig":"4)","norm":"4)","orig_tok":["4",")"],"norm_tok":["4",")"],"orig_spans":[[0,1],[1,2]],"norm_spans":[[0,1],[1,2]],"alignment":[[0,0],[1,1]],"lang_fastText":"es","lang_py3langid":"en","lang_cld3":"ja","orig_lang_de":0,"norm_lmscore":2.607399940490722} +{"basename":"euler_rechenkunst01_1738","par_idx":243,"date":1738,"orig":"Wenn zwey oder mehr Zahlen ſollen zuſammen geſetzet oder in eine Summ gebracht werden, ſo wird die Summ gefunden wenn man alle Unitæten zuſammen ſetzet, und denn alle Decades, ferner alle Centenarios, Millenarios und ſo fort.","norm":"Wenn zwei oder mehr Zahlen sollen zusammen gesetzt oder in eine Summe gebracht werden, so wird die Summe gefunden wenn man alle Unitäten zusammen setzet, und denn alle Decades, ferner alle Centenarios, Millenarios und so 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oder geſchrieben werden, ſo koͤnnen niemahls mehr als 9 Stuͤcke von einer Gattung vorkommen; und demnach um die Unitæten zu addiren iſt genug wenn man weiß zu einer jeglichen Zahl eine einfache Zahl hinzuzuſetzen.","norm":"denn wenn die gegebenen Zahlen wie wir voraussetzen, auf die gewöhnliche Art entweder ausgesprochen oder geschrieben werden, so können niemals mehr als 9 Stücke von einer Gattung vorkommen; und demnach um die Unitäten zu addieren ist genug wenn man weiß zu einer jeglichen Zahl eine einfache Zahl hinzuzusetzen.","orig_tok":["denn","wenn","die","gegebenen","Zahlen","wie","wir","vorausſetzen",",","auf","die","gewoͤhnliche","Art","entweder","ausgeſprochen","oder","geſchrieben","werden",",","ſo","koͤnnen","niemahls","mehr","als","9","Stuͤcke","von","einer","Gattung","vorkommen",";","und","demnach","um","die","Unitæten","zu","addiren","iſt","genug","wenn","man","weiß","zu","einer","jeglichen","Zahl","eine","einfache","Zahl","hinzuzuſetzen","."],"norm_tok":["denn","wenn","die","gegebenen","Zahlen","wie","wir","voraussetzen",",","auf","die","gewöhnliche","Art","entweder","ausgesprochen","oder","geschrieben","werden",",","so","können","niemals","mehr","als","9","Stücke","von","einer","Gattung","vorkommen",";","und","demnach","um","die","Unitäten","zu","addieren","ist","genug","wenn","man","weiß","zu","einer","jeglichen","Zahl","eine","einfache","Zahl","hinzuzusetzen","."],"orig_spans":[[0,4],[5,9],[10,13],[14,23],[24,30],[31,34],[35,38],[39,51],[51,52],[53,56],[57,60],[61,73],[74,77],[78,86],[87,100],[101,105],[106,117],[118,124],[124,125],[126,128],[129,136],[137,145],[146,150],[151,154],[155,156],[157,164],[165,168],[169,174],[175,182],[183,192],[192,193],[194,197],[198,205],[206,208],[209,212],[213,221],[222,224],[225,232],[233,236],[237,242],[243,247],[248,251],[252,256],[257,259],[260,265],[266,275],[276,280],[281,285],[286,294],[295,299],[300,313],[313,314]],"norm_spans":[[0,4],[5,9],[10,13],[14,23],[24,30],[31,34],[35,38],[39,51],[51,52],[53,56],[57,60],[61,72],[73,76],[77,85],[86,99],[100,104],[105,116],[117,123],[123,124],[125,127],[128,134],[135,142],[143,147],[148,151],[152,153],[154,160],[161,164],[165,170],[171,178],[179,188],[188,189],[190,193],[194,201],[202,204],[205,208],[209,217],[218,220],[221,229],[230,233],[234,239],[240,244],[245,248],[249,253],[254,256],[257,262],[263,272],[273,277],[278,282],[283,291],[292,296],[297,310],[310,311]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28],[29,29],[30,30],[31,31],[32,32],[33,33],[34,34],[35,35],[36,36],[37,37],[38,38],[39,39],[40,40],[41,41],[42,42],[43,43],[44,44],[45,45],[46,46],[47,47],[48,48],[49,49],[50,50],[51,51]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.964799880981445} +{"basename":"euler_rechenkunst01_1738","par_idx":250,"date":1738,"orig":"Mit den anderen und folgenden Sorten als Decadibus, Centenariis, Millenariis und ſo fort hat es eine gleiche Bewandnuͤß, und wer die Unitæten zuſammen addiren kan, derſelbe kan auf gleiche Weiſe die Decades, Centenarios und folgenden Sorten addiren.","norm":"Mit den anderen und folgenden Sorten als Decadibus, Centenariis, Millenariis und so fort hat es eine gleiche Bewandtnis, und wer die Unitäten zusammen addieren kann, derselbe kann auf gleiche Weise die Decades, Centenarios und folgenden Sorten 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+{"basename":"euler_rechenkunst01_1738","par_idx":251,"date":1738,"orig":"Denn gleichwie 7 Unitæten und 9 Unitæten, zuſammen ſechszehen Unitæten machen; ſo machen auch 7 Decades und 9 Decades zuſammen ſechszehen Decades: und 7 Stuͤcke und 9 Stuͤcke von einerley Sorten, machen zuſammen 16 Stuͤcke von eben der Sorte.","norm":"Denn gleichwie 7 Unitäten und 9 Unitäten, zusammen sechzehn Unitäten machen; so machen auch 7 Decades und 9 Decades zusammen sechzehn Decades: und 7 Stücke und 9 Stücke von einerlei Sorten, machen zusammen 16 Stücke von eben der 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+{"basename":"euler_rechenkunst01_1738","par_idx":252,"date":1738,"orig":"Woraus erhellet, daß verſchiedene Stuͤcke von einer jeglichen Gattung, als Decades, Centenarii, Millenarii und ſofort, eben ſo leicht und auf eben die Art zuſammen geſetzet werden, als die Unitæten.","norm":"Woraus erhellet, dass verschiedene Stücke von einer jeglichen Gattung, als Decades, Centenarii, Millenarii und sofort, eben so leicht und auf eben die Art zusammen gesetzt werden, als die 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+{"basename":"euler_rechenkunst01_1738","par_idx":253,"date":1738,"orig":"Dieſes beſſer zu erlaͤuteren, ſo ſeyen dieſe Zahlen 5326 und 4937 gegeben, derer Summ gefunden werden ſoll.","norm":"Dieses besser zu erläutere, so seien diese Zahlen 5326 und 4937 gegeben, derer Summe gefunden werden 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+{"basename":"euler_rechenkunst01_1738","par_idx":254,"date":1738,"orig":"Nach der gegebenen Anleitung muß nun die Summ erſtlich 6 und 7 das iſt nach der vorigen Tabelle 13 Unitæten enthalten, zweytens 2 und 3 das iſt 5 Decades; drittens 3 und 9 das iſt 12 Centenarios; und viertens 5 und 4 das iſt 9 Millenarios.","norm":"Nach der gegebenen Anleitung muss nun die Summe erstlich 6 und 7 das ist nach der vorigen Tabelle 13 Unitäten enthalten, zweitens 2 und 3 das ist 5 Decades; drittens 3 und 9 das ist 12 Centenarios; und viertens 5 und 4 das ist 9 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+{"basename":"euler_rechenkunst01_1738","par_idx":255,"date":1738,"orig":"Und derohalben kan man mit Gewißheit ſagen, daß die Summ dieſer gegebenen Zahlen ſeye 9 Millenari, 12 Centenarii, 5 Decades und 13 Unitates.","norm":"Und derohalben kann man mit Gewissheit sagen, dass die Summe dieser gegebenen Zahlen sei 9 Millenari, 12 Centenarii, 5 Decades und 13 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+{"basename":"euler_rechenkunst01_1738","par_idx":265,"date":1738,"orig":"Um aber leichter zu ſehen, wieviel eine gewiſſe Anzahl Unitæten, Decades, oder eine gewiſſe Anzahl Decades, Centenarios in ſich begreiffen und ſo weiter; ſo iſt dienlich, daß man die gefundene Summ der Unitæten oder Decadum oder Centenariorum und folgenden Sorten auf die gewoͤhnliche Art ſchreibe und ſehe, ob dieſelbe aus mehr als einem Character beſtehe.","norm":"Um aber leichter zu sehen, wie viel eine gewisse Anzahl Unitäten, Decades, oder eine gewisse Anzahl Decades, Centenarios in sich begreifen und so weiter; so ist dienlich, dass man die gefundene Summe der Unitäten oder Decadum oder Centenariorum und folgenden Sorten auf die gewöhnliche Art schreibe und sehe, ob dieselbe aus mehr als einem Charakter 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+{"basename":"euler_rechenkunst01_1738","par_idx":266,"date":1738,"orig":"Denn beſtehet die Summ von einer Sorte nur aus einem Character, ſo enthaͤlt dieſelbe kein Stuͤck von der folgenden Sorte, ſondern behaͤlt den Nahmen von Unitæten oder Decaden und ſo fort, aus welchen ſie iſt gefunden worden.","norm":"Denn bestehet die Summe von einer Sorte nur aus einem Charakter, so enthält dieselbe kein Stück von der folgenden Sorte, sondern behält den Namen von Unitäten oder Dekaden und so fort, aus welchen sie ist gefunden 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+{"basename":"euler_rechenkunst01_1738","par_idx":267,"date":1738,"orig":"Beſtehet aber die Summ auf dieſe Art geſchrieben aus zwey Characteren, ſo deutet der zur lincken Hand an, wie viel Stuͤck von der folgenden Sorte in dieſer Summ enthalten, welche folglich mit zu der Summ der folgendem Sorte muͤſſen geſchlagen werden.","norm":"Bestehet aber die Summe auf diese Art geschrieben aus zwei Charakter, so deutet der zur linken Hand an, wie viel Stück von der folgenden Sorte in dieser Summe enthalten, welche folglich mit zu der Summe der folgendem Sorte müssen geschlagen 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+{"basename":"euler_rechenkunst01_1738","par_idx":272,"date":1738,"orig":"Die Decades aber von dieſen drey Zahlen zuſammen genommen geben 5 Decades, und zu dieſen die obige eine Decas gethan macht 6 Decades; worinn alſo kein Centenarius enthalten iſt.","norm":"Die Decades aber von diesen drei Zahlen zusammen genommen geben 5 Decades, und zu diesen die obige eine Decas getan macht 6 Decades; worin also kein Centenarius enthalten 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+{"basename":"euler_rechenkunst01_1738","par_idx":273,"date":1738,"orig":"Ferner addire man die 3 und 6 und 2 Centenarios; ſo findet man eilf oder 11 Centenarios; dieſe Summ iſt allſo ſo viel als 1 Centenarius und 1 Millenarius, welcher zu den Millenariis muß hinzugethan werden.","norm":"Ferner addiere man die 3 und 6 und 2 Centenarios; so findet man elf oder 11 Centenarios; diese Summe ist also so viel als 1 Centenarius und 1 Millenarius, welcher zu den Millenariis muss hinzugetan 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+{"basename":"euler_rechenkunst01_1738","par_idx":274,"date":1738,"orig":"Dieſer Millenarius allſo und 2 und 5 und 7 Millenarii machen zuſammen 15 Milenarios: das iſt 5 Millenarii und eine Decas Millenariorum.","norm":"Dieser Millenarius also und 2 und 5 und 7 Millenarii machen zusammen 15 Milenarios: das ist 5 Millenarii und eine Decas 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+{"basename":"euler_rechenkunst01_1738","par_idx":275,"date":1738,"orig":"Alles dieſes zuſammen oder die Summ der drey gegebenen Zahlen iſt derowegen eine Decas Millenariorum und 5 Millenarii, und 1 Centenarius und 6 Decades und 3 Unitæten; welche geſchrieben geben 15163, oder fuͤnfzehen tauſend einhundert und drey und ſechzig.","norm":"Alles dieses zusammen oder die Summe der drei gegebenen Zahlen ist derowegen eine Decas Millenariorum und 5 Millenarii, und 1 Centenarius und 6 Decades und 3 Unitäten; welche geschrieben geben 15163, oder fünfzehntausend einhundertunddrei und 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+{"basename":"euler_rechenkunst01_1738","par_idx":276,"date":1738,"orig":"Solten aber in Addirung einer Sorte hundert oder mehr Stuͤcke herauskommen, ſo enthaͤlt die Summ zehen oder mehr von der folgenden und folglich ein Stuͤck zu der zweyten folgenden Sorte.","norm":"Sollten aber in Addierung einer Sorte hundert oder mehr Stücke herauskommen, so enthält die Summe zehn oder mehr von der folgenden und folglich ein Stück zu der zweiten folgenden 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+{"basename":"euler_rechenkunst01_1738","par_idx":282,"date":1738,"orig":"Alsdenn wird von der rechten Hand der Anfang gemacht und die Unitæten zuſammen addiret; deren Summ wenn ſie kleiner iſt als 10 wird unter die Unitæten unter die Linie geſchrieben, iſt die Summ aber groͤſſer als 9 und enthaͤlt folglich eine oder mehr Decades nebſt etlichen Unitæten, ſo wird nur dieſe Anzahl der Unitæten unter die Linie geſchrieben, die Decades aber bey Add rung der Decadum noch hinzugethan.","norm":"Alsdann wird von der rechten Hand der Anfang gemacht und die Unitäten zusammen addieret; deren Summe wenn sie kleiner ist als 10 wird unter die Unitäten unter die Linie geschrieben, ist die Summe aber größer als 9 und enthält folglich eine oder mehr Decades nebst etlichen Unitäten, so wird nur diese Anzahl der Unitäten unter die Linie geschrieben, die Decades aber bei Add Runge der Decadum noch 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+{"basename":"euler_rechenkunst01_1738","par_idx":283,"date":1738,"orig":"Auf gleiche Art werden auch ferner die Decades addiret und weiter die Centenarii, Millenarii und ſo fort.","norm":"Auf gleiche Art werden auch ferner die Decades addieret und weiter die Centenarii, Millenarii und so 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dieſes alles geſchehen, ſo iſt die Zahl, welche herausgekommen und unter die Linie geſetzet worden, die verlangte Summ der gegebenen Zahlen.","norm":"Wo nun dieses alles geschehen, so ist die Zahl, welche herausgekommen und unter die Linie gesetzt worden, die verlangte Summe der gegebenen 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+{"basename":"euler_rechenkunst01_1738","par_idx":285,"date":1738,"orig":"Die Zahlen, welche addiret werden ſollen, werden deswegen unter einander geſchrieben, damit die Zahlen welche in einer Reihe von oben herab ſiehen, einerley Sorten nehmlich entweder Unitæten oder Decadesoder Centenarii und ſo fort bedeuten, und alſo beſſer ins Geſicht fallen und deſto bequemer addiret werden koͤnnen.","norm":"Die Zahlen, welche addieret werden sollen, werden deswegen untereinander geschrieben, damit die Zahlen welche in einer Reihe von oben herab siechen, einerlei Sorten nämlich entweder Unitäten oder Dekadesoter Centenarii und so fort bedeuten, und also besser ins Gesicht fallen und desto bequemer addieret werden 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+{"basename":"euler_rechenkunst01_1738","par_idx":287,"date":1738,"orig":"Die gantze Operation kan im uͤbrigen durch Exempel am deutlichſten gewieſen werden.","norm":"Die ganze Operation kann im Übrigen durch Exempel am deutlichsten gewiesen werden.","orig_tok":["Die","gantze","Operation","kan","im","uͤbrigen","durch","Exempel","am","deutlichſten","gewieſen","werden","."],"norm_tok":["Die","ganze","Operation","kann","im","Übrigen","durch","Exempel","am","deutlichsten","gewiesen","werden","."],"orig_spans":[[0,3],[4,10],[11,20],[21,24],[25,27],[28,36],[37,42],[43,50],[51,53],[54,66],[67,75],[76,82],[82,83]],"norm_spans":[[0,3],[4,9],[10,19],[20,24],[25,27],[28,35],[36,41],[42,49],[50,52],[53,65],[66,74],[75,81],[81,82]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.770999908447266} +{"basename":"euler_rechenkunst01_1738","par_idx":288,"date":1738,"orig":"Als es ſollen nachfolgende Zahlen 53237; 8729; und 10237 addiret werden; ſo werden dieſe Zahlen unter einander geſchrieben wie folget:","norm":"Als es sollen nachfolgende Zahlen 53237; 8729; und 10237 addieret werden; so werden diese Zahlen untereinander geschrieben wie 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+{"basename":"euler_rechenkunst01_1738","par_idx":289,"date":1738,"orig":"[Formel] da dann die erſte Reihe von oben herab Unitæten, die zweyte Decades, die dritte Centenarios die vierte Millenarios, und die fuͤnfte Decades Millenariorum bedeutet.","norm":"[Formel] da dann die erste Reihe von oben herab Unitäten, die zweite Decades, die dritte Centenarios die vierte Millenarios, und die fünfte Decades Millenariorum 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+{"basename":"euler_rechenkunst01_1738","par_idx":290,"date":1738,"orig":"Nun werden erſtlich die Unitæten addiret und geſagt: 7 und 9 macht 16 und noch 7 dazu macht 23 Unitæten, das iſt 2 Decades, welche zu der zweyten Reihe muͤſſen hinzugethan und deswegen bey dieſer Reihe mit 2 Punckten bemercket werden; die drey Unitæten aber werden unter die Linie auf die erſte Stelle von der rechten Hand, das iſt, auf die Stelle der Unitæten geſchrieben.","norm":"Nun werden erstlich die Unitäten addieret und gesagt: 7 und 9 macht 16 und noch 7 dazu macht 23 Unitäten, das ist 2 Decades, welche zu der zweiten Reihe müssen hinzugetan und deswegen bei dieser Reihe mit 2 Punkten bemerket werden; die drei Unitäten aber werden unter die Linie auf die erste Stelle von der rechten Hand, das ist, auf die Stelle der Unitäten 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+{"basename":"euler_rechenkunst01_1738","par_idx":291,"date":1738,"orig":"Zweytens geht man zu den Decaden und ſagt 3 und 2 macht 5 und noch 3 macht 8 und noch 2, welche durch die 2 Puncten angedeutet worden, macht 10 Decades das iſt, 1 Centenarius und keine Decas; weil nun keine Decas vorhanden, ſo wird unter die Linie auf die zweyte Stelle eine 0 geſchrieben; der 1 Centenarius aber wird durch ein Punct bey der dritten Reihe der Centenariorum angedeutet.","norm":"Zweitens geht man zu den Dekaden und sagt 3 und 2 macht 5 und noch 3 macht 8 und noch 2, welche durch die 2 Punkten angedeutet worden, macht 10 Decades das ist, 1 Centenarius und keine Decas; weil nun keine Decas vorhanden, so wird unter die Linie auf die zweite Stelle eine 0 geschrieben; der 1 Centenarius aber wird durch ein Punkt bei der dritten Reihe der Centenariorum angedeutet.","orig_tok":["Zweytens","geht","man","zu","den","Decaden","und","ſagt","3","und","2","macht","5","und","noch","3","macht","8","und","noch","2",",","welche","durch","die","2","Puncten","angedeutet","worden",",","macht","10","Decades","das","iſt",",","1","Centenarius","und","keine","Decas",";","weil","nun","keine","Decas","vorhanden",",","ſo","wird","unter","die","Linie","auf","die","zweyte","Stelle","eine","0","geſchrieben",";","der","1","Centenarius","aber","wird","durch","ein","Punct","bey","der","dritten","Reihe","der","Centenariorum","angedeutet","."],"norm_tok":["Zweitens","geht","man","zu","den","Dekaden","und","sagt","3","und","2","macht","5","und","noch","3","macht","8","und","noch","2",",","welche","durch","die","2","Punkten","angedeutet","worden",",","macht","10","Decades","das","ist",",","1","Centenarius","und","keine","Decas",";","weil","nun","keine","Decas","vorhanden",",","so","wird","unter","die","Linie","auf","die","zweite","Stelle","eine","0","geschrieben",";","der","1","Centenarius","aber","wird","durch","ein","Punkt","bei","der","dritten","Reihe","der","Centenariorum","angedeutet","."],"orig_spans":[[0,8],[9,13],[14,17],[18,20],[21,24],[25,32],[33,36],[37,41],[42,43],[44,47],[48,49],[50,55],[56,57],[58,61],[62,66],[67,68],[69,74],[75,76],[77,80],[81,85],[86,87],[87,88],[89,95],[96,101],[102,105],[106,107],[108,115],[116,126],[127,133],[133,134],[135,140],[141,143],[144,151],[152,155],[156,159],[159,160],[161,162],[163,174],[175,178],[179,184],[185,190],[190,191],[192,196],[197,200],[201,206],[207,212],[213,222],[222,223],[224,226],[227,231],[232,237],[238,241],[242,247],[248,251],[252,255],[256,262],[263,269],[270,274],[275,276],[277,288],[288,289],[290,293],[294,295],[296,307],[308,312],[313,317],[318,323],[324,327],[328,333],[334,337],[338,341],[342,349],[350,355],[356,359],[360,373],[374,384],[384,385]],"norm_spans":[[0,8],[9,13],[14,17],[18,20],[21,24],[25,32],[33,36],[37,41],[42,43],[44,47],[48,49],[50,55],[56,57],[58,61],[62,66],[67,68],[69,74],[75,76],[77,80],[81,85],[86,87],[87,88],[89,95],[96,101],[102,105],[106,107],[108,115],[116,126],[127,133],[133,134],[135,140],[141,143],[144,151],[152,155],[156,159],[159,160],[161,162],[163,174],[175,178],[179,184],[185,190],[190,191],[192,196],[197,200],[201,206],[207,212],[213,222],[222,223],[224,226],[227,231],[232,237],[238,241],[242,247],[248,251],[252,255],[256,262],[263,269],[270,274],[275,276],[277,288],[288,289],[290,293],[294,295],[296,307],[308,312],[313,317],[318,323],[324,327],[328,333],[334,337],[338,341],[342,349],[350,355],[356,359],[360,373],[374,384],[384,385]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28],[29,29],[30,30],[31,31],[32,32],[33,33],[34,34],[35,35],[36,36],[37,37],[38,38],[39,39],[40,40],[41,41],[42,42],[43,43],[44,44],[45,45],[46,46],[47,47],[48,48],[49,49],[50,50],[51,51],[52,52],[53,53],[54,54],[55,55],[56,56],[57,57],[58,58],[59,59],[60,60],[61,61],[62,62],[63,63],[64,64],[65,65],[66,66],[67,67],[68,68],[69,69],[70,70],[71,71],[72,72],[73,73],[74,74],[75,75],[76,76]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.678899765014648} +{"basename":"euler_rechenkunst01_1738","par_idx":292,"date":1738,"orig":"Drittens ſagt man 2 und 7 macht 9 und noch 2 macht 11 und noch 1 wegen dem Punct macht 12 Centenarios, das iſt 1 Millenarius, welcher durch ein Punct bey der folgenden Reihe angedeutet wird und zwey Centenarii, welche unter die Linie auf die dritte Stelle geſchrieben werden.","norm":"Drittens sagt man 2 und 7 macht 9 und noch 2 macht 11 und noch 1 wegen dem Punkt macht 12 Centenarios, das ist 1 Millenarius, welcher durch ein Punkt bei der folgenden Reihe angedeutet wird und zwei Centenarii, welche unter die Linie auf die dritte Stelle geschrieben 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+{"basename":"euler_rechenkunst01_1738","par_idx":293,"date":1738,"orig":"Viertens machen 3 Millenarii und 8 und noch einer zuſammen 12 Millenarios oder 1 Decadem Millenariorum, ſo durch ein Punct bey dieſer Sorte angezeiget wird, und 2 Millenarios welche unter die Linie auf die vierte Stelle geſchrieben werden.","norm":"Viertens machen 3 Millenarii und 8 und noch einer zusammen 12 Millenarios oder 1 Decadem Millenariorum, so durch ein Punkt bei dieser Sorte angezeigt wird, und 2 Millenarios welche unter die Linie auf die vierte Stelle geschrieben 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der Rest oder dasjenige was überbleibt gefunden; welcher vondieser Eigenschaft seynwird, dass wenn man zu demselben die kleinere Zahl addieret, die größere Zahl herausgebracht 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+{"basename":"euler_rechenkunst01_1738","par_idx":341,"date":1738,"orig":"Wenn nehmlich in einem Sacke eine gewiſſe Anzahl Rubeln befindlich, ſo kan man nicht mehr daraus nehmen, als darinnen iſt; eben ſo viel aber, oder weniger kan wohl daraus genommen werden.","norm":"Wenn nämlich in einem Sacke eine gewisse Anzahl Rubeln befindlich, so kann man nicht mehr daraus nehmen, als darinnen ist; ebenso viel aber, oder weniger kann wohl daraus genommen 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+{"basename":"euler_rechenkunst01_1738","par_idx":342,"date":1738,"orig":"Die Subtraction lehret alſo, wie man finden ſoll, wieviel Rubeln in dem Sacke noch uͤbrig bleiben, wenn aus demſelben eine gewiſſe Summ ausgezehlet worden.","norm":"Die Subtraktion lehrt also, wie man finden soll, wie viel Rubeln in dem Sacke noch übrig bleiben, wenn aus demselben eine gewisse Summe ausgezählt 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+{"basename":"euler_rechenkunst01_1738","par_idx":343,"date":1738,"orig":"Hieraus iſt nun klar daß wenn ſo viel heraus genommen wird, als darinn iſt, nichts im Sacke zuruͤck bleiben werde; wird aber weniger daraus genommen, ſo muß im Sacke noch etwas zuruͤckbleiben, welches der Reſt genennet wird.","norm":"Hieraus ist nun klar dass wenn so viel herausgenommen wird, als darin ist, nichts im Sacke zurückbleiben werde; wird aber weniger daraus genommen, so muss im Sacke noch etwas zurückbleiben, welches der Rest genannt 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+{"basename":"euler_rechenkunst01_1738","par_idx":357,"date":1738,"orig":"Wenn man derohalben die Unitæten der kleineren Zahl von den Unitæten der groͤſſeren Zahl abziehet, gleichergeſtalt auch die Decades von den Decadlbos, die Centenarios von den Centenariis und ſo fort; ſo iſt klar daß dieſe uͤbergebliebenen Unitæten, Decades, Centenarii und ſo fort zuſammen den geſuchten Reſt ausmachen muͤſſen.","norm":"Wenn man derohalben die Unitäten der kleineren Zahl von den Unitäten der größeren Zahl abziehet, gleichergestalt auch die Decades von den Decadlbos, die Centenarios von den Centenariis und so fort; so ist klar dass diese übergebliebenen Unitäten, Decades, Centenarii und so fort zusammen den gesuchten Rest ausmachen 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+{"basename":"euler_rechenkunst01_1738","par_idx":358,"date":1738,"orig":"Dieſe Operation nun ins Werck zurichten, ſo iſt noͤthig, daß man wiſſe die Unitæten von den Unitæten, die Decades von den Decadibus und ſo fort zu ſubtrahiren; welches deswegen zu erlernen ſehr leicht iſt, weilen niemahls mehr als 9 Stuͤcke von einer Gattung vorkommen.","norm":"Diese Operation nun ins Werk zurichten, so ist nötig, dass man wisse die Unitäten von den Unitäten, die Decades von den Decadibus und so fort zu subtrahieren; welches deswegen zu erlernen sehr leicht ist, weilen niemals mehr als 9 Stücke von einer Gattung 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+{"basename":"euler_rechenkunst01_1738","par_idx":400,"date":1738,"orig":"Derowegen wird von der folgenden Sorte der oberen Zahl nehmlich den 6 Decadibus eine Decas weggenommen oder gelehnet, und zu den 4 Unitæten geſchlagen, welches alſo zuſammen 14 Unitæten ausmacht.","norm":"Derowegen wird von der folgenden Sorte der oberen Zahl nämlich den 6 Decadibus eine Decas weggenommen oder gelehnt, und zu den 4 Unitäten geschlagen, welches also zusammen 14 Unitäten 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+{"basename":"euler_rechenkunst01_1738","par_idx":401,"date":1738,"orig":"Nun koͤnnen alſo von dieſen 14 Unitæten die 9 Unitæten abgezogen werden, und bleiben 5 uͤber, welche folglich unter die Linie geſchrieben werden.","norm":"Nun können also von diesen 14 Unitäten die 9 Unitäten abgezogen werden, und bleiben 5 über, welche folglich unter die Linie geschrieben 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+{"basename":"euler_rechenkunst01_1738","par_idx":402,"date":1738,"orig":"Wobey aber zu mercken iſt, daß anjetzo in der oberen Zahl nicht mehr 6, ſondern nur 5 Decades vorhanden, indem eine davon weggenommen worden; welche Verminderung derowegen mit einem Punct angedeutet wird.","norm":"Wobei aber zu merken ist, dass jetzt in der oberen Zahl nicht mehr 6, sondern nur 5 Decades vorhanden, indem eine davon weggenommen worden; welche Verminderung derowegen mit einem Punkt angedeutet wird.","orig_tok":["Wobey","aber","zu","mercken","iſt",",","daß","anjetzo","in","der","oberen","Zahl","nicht","mehr","6",",","ſondern","nur","5","Decades","vorhanden",",","indem","eine","davon","weggenommen","worden",";","welche","Verminderung","derowegen","mit","einem","Punct","angedeutet","wird","."],"norm_tok":["Wobei","aber","zu","merken","ist",",","dass","jetzt","in","der","oberen","Zahl","nicht","mehr","6",",","sondern","nur","5","Decades","vorhanden",",","indem","eine","davon","weggenommen","worden",";","welche","Verminderung","derowegen","mit","einem","Punkt","angedeutet","wird","."],"orig_spans":[[0,5],[6,10],[11,13],[14,21],[22,25],[25,26],[27,30],[31,38],[39,41],[42,45],[46,52],[53,57],[58,63],[64,68],[69,70],[70,71],[72,79],[80,83],[84,85],[86,93],[94,103],[103,104],[105,110],[111,115],[116,121],[122,133],[134,140],[140,141],[142,148],[149,161],[162,171],[172,175],[176,181],[182,187],[188,198],[199,203],[203,204]],"norm_spans":[[0,5],[6,10],[11,13],[14,20],[21,24],[24,25],[26,30],[31,36],[37,39],[40,43],[44,50],[51,55],[56,61],[62,66],[67,68],[68,69],[70,77],[78,81],[82,83],[84,91],[92,101],[101,102],[103,108],[109,113],[114,119],[120,131],[132,138],[138,139],[140,146],[147,159],[160,169],[170,173],[174,179],[180,185],[186,196],[197,201],[201,202]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28],[29,29],[30,30],[31,31],[32,32],[33,33],[34,34],[35,35],[36,36]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.206600189208984} +{"basename":"euler_rechenkunst01_1738","par_idx":403,"date":1738,"orig":"Hierauf ſollten demnach 8 Decades von 5 Decadibus abgezogen werden, welches weil es gleichfals nicht angeht, ſo wird von den 3 Centenariis ein Stuͤck weggenommen, ſo daß nur noch 2 zuruͤckbleiben, welches durch das da zugeſetzte Punct angedeutet wird.","norm":"Hierauf sollten demnach 8 Decades von 5 Decadibus abgezogen werden, welches weil es gleichfalls nicht angeht, so wird von den 3 Centenariis ein Stück weggenommen, so dass nur noch 2 zurückbleiben, welches durch das da zugesetzte Punkt angedeutet wird.","orig_tok":["Hierauf","ſollten","demnach","8","Decades","von","5","Decadibus","abgezogen","werden",",","welches","weil","es","gleichfals","nicht","angeht",",","ſo","wird","von","den","3","Centenariis","ein","Stuͤck","weggenommen",",","ſo","daß","nur","noch","2","zuruͤckbleiben",",","welches","durch","das","da","zugeſetzte","Punct","angedeutet","wird","."],"norm_tok":["Hierauf","sollten","demnach","8","Decades","von","5","Decadibus","abgezogen","werden",",","welches","weil","es","gleichfalls","nicht","angeht",",","so","wird","von","den","3","Centenariis","ein","Stück","weggenommen",",","so","dass","nur","noch","2","zurückbleiben",",","welches","durch","das","da","zugesetzte","Punkt","angedeutet","wird","."],"orig_spans":[[0,7],[8,15],[16,23],[24,25],[26,33],[34,37],[38,39],[40,49],[50,59],[60,66],[66,67],[68,75],[76,80],[81,83],[84,94],[95,100],[101,107],[107,108],[109,111],[112,116],[117,120],[121,124],[125,126],[127,138],[139,142],[143,149],[150,161],[161,162],[163,165],[166,169],[170,173],[174,178],[179,180],[181,195],[195,196],[197,204],[205,210],[211,214],[215,217],[218,228],[229,234],[235,245],[246,250],[250,251]],"norm_spans":[[0,7],[8,15],[16,23],[24,25],[26,33],[34,37],[38,39],[40,49],[50,59],[60,66],[66,67],[68,75],[76,80],[81,83],[84,95],[96,101],[102,108],[108,109],[110,112],[113,117],[118,121],[122,125],[126,127],[128,139],[140,143],[144,149],[150,161],[161,162],[163,165],[166,170],[171,174],[175,179],[180,181],[182,195],[195,196],[197,204],[205,210],[211,214],[215,217],[218,228],[229,234],[235,245],[246,250],[250,251]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28],[29,29],[30,30],[31,31],[32,32],[33,33],[34,34],[35,35],[36,36],[37,37],[38,38],[39,39],[40,40],[41,41],[42,42],[43,43]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.288599967956543} +{"basename":"euler_rechenkunst01_1738","par_idx":404,"date":1738,"orig":"Dieſer Centenarius macht nun 10 Decades, welche mit den 5 ſchon vorhandenen 15 Decades ausmachen.","norm":"Dieser Centenarius macht nun 10 Decades, welche mit den 5 schon vorhandenen 15 Decades 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uͤber, welche unter die Linie in die Stelle der Decaden geſetzet werden.","norm":"Von diesen 15 werden nun die 8 Decades der unteren Zahl abgezogen und bleiben 7 über, welche unter die Linie in die Stelle der Dekaden gesetzt werden.","orig_tok":["Von","dieſen","15","werden","nun","die","8","Decades","der","unteren","Zahl","abgezogen","und","bleiben","7","uͤber",",","welche","unter","die","Linie","in","die","Stelle","der","Decaden","geſetzet","werden","."],"norm_tok":["Von","diesen","15","werden","nun","die","8","Decades","der","unteren","Zahl","abgezogen","und","bleiben","7","über",",","welche","unter","die","Linie","in","die","Stelle","der","Dekaden","gesetzt","werden","."],"orig_spans":[[0,3],[4,10],[11,13],[14,20],[21,24],[25,28],[29,30],[31,38],[39,42],[43,50],[51,55],[56,65],[66,69],[70,77],[78,79],[80,85],[85,86],[87,93],[94,99],[100,103],[104,109],[110,112],[113,116],[117,123],[124,127],[128,135],[136,144],[145,151],[151,152]],"norm_spans":[[0,3],[4,10],[11,13],[14,20],[21,24],[25,28],[29,30],[31,38],[39,42],[43,50],[51,55],[56,65],[66,69],[70,77],[78,79],[80,84],[84,85],[86,92],[93,98],[99,102],[103,108],[109,111],[112,115],[116,122],[123,126],[127,134],[135,142],[143,149],[149,150]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.373199939727783} +{"basename":"euler_rechenkunst01_1738","par_idx":406,"date":1738,"orig":"Ferner haben wir 7 Centenarios von 2 Centenariis abzuziehen, weswegen gleichergeſtalt von den 7 Millenariis ein Stuͤck genommen und zu den 2 Centenariis geſchlagen wird, ſo daß 12 Centenarii herauskommen.","norm":"Ferner haben wir 7 Centenarios von 2 Centenariis abzuziehen, weswegen gleichergestalt von den 7 Millenariis ein Stück genommen und zu den 2 Centenariis geschlagen wird, so dass 12 Centenarii herauskommen.","orig_tok":["Ferner","haben","wir","7","Centenarios","von","2","Centenariis","abzuziehen",",","weswegen","gleichergeſtalt","von","den","7","Millenariis","ein","Stuͤck","genommen","und","zu","den","2","Centenariis","geſchlagen","wird",",","ſo","daß","12","Centenarii","herauskommen","."],"norm_tok":["Ferner","haben","wir","7","Centenarios","von","2","Centenariis","abzuziehen",",","weswegen","gleichergestalt","von","den","7","Millenariis","ein","Stück","genommen","und","zu","den","2","Centenariis","geschlagen","wird",",","so","dass","12","Centenarii","herauskommen","."],"orig_spans":[[0,6],[7,12],[13,16],[17,18],[19,30],[31,34],[35,36],[37,48],[49,59],[59,60],[61,69],[70,85],[86,89],[90,93],[94,95],[96,107],[108,111],[112,118],[119,127],[128,131],[132,134],[135,138],[139,140],[141,152],[153,163],[164,168],[168,169],[170,172],[173,176],[177,179],[180,190],[191,203],[203,204]],"norm_spans":[[0,6],[7,12],[13,16],[17,18],[19,30],[31,34],[35,36],[37,48],[49,59],[59,60],[61,69],[70,85],[86,89],[90,93],[94,95],[96,107],[108,111],[112,117],[118,126],[127,130],[131,133],[134,137],[138,139],[140,151],[152,162],[163,167],[167,168],[169,171],[172,176],[177,179],[180,190],[191,203],[203,204]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28],[29,29],[30,30],[31,31],[32,32]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.497900009155273} +{"basename":"euler_rechenkunst01_1738","par_idx":407,"date":1738,"orig":"Von dieſen ziehet man nun die 7 Centenarios ab, ſo bleiben 5 uͤber, ſo unter die Linie in die dritte Stelle geſetzet werden.","norm":"Von diesen ziehet man nun die 7 Centenarios ab, so bleiben 5 über, so unter die Linie in die dritte Stelle gesetzt werden.","orig_tok":["Von","dieſen","ziehet","man","nun","die","7","Centenarios","ab",",","ſo","bleiben","5","uͤber",",","ſo","unter","die","Linie","in","die","dritte","Stelle","geſetzet","werden","."],"norm_tok":["Von","diesen","ziehet","man","nun","die","7","Centenarios","ab",",","so","bleiben","5","über",",","so","unter","die","Linie","in","die","dritte","Stelle","gesetzt","werden","."],"orig_spans":[[0,3],[4,10],[11,17],[18,21],[22,25],[26,29],[30,31],[32,43],[44,46],[46,47],[48,50],[51,58],[59,60],[61,66],[66,67],[68,70],[71,76],[77,80],[81,86],[87,89],[90,93],[94,100],[101,107],[108,116],[117,123],[123,124]],"norm_spans":[[0,3],[4,10],[11,17],[18,21],[22,25],[26,29],[30,31],[32,43],[44,46],[46,47],[48,50],[51,58],[59,60],[61,65],[65,66],[67,69],[70,75],[76,79],[80,85],[86,88],[89,92],[93,99],[100,106],[107,114],[115,121],[121,122]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.10290002822876} +{"basename":"euler_rechenkunst01_1738","par_idx":408,"date":1738,"orig":"Endlich werden die 5 Millenarii von den 6 oberen abgezogen, und der eine ſo uͤberbleibet unter die Linie geſchrieben, womit die gantze Operation geendigt iſt, und hat alſo dieſen Reſt gefunden 1575.","norm":"Endlich werden die 5 Millenarii von den 6 oberen abgezogen, und der eine so überbleibet unter die Linie geschrieben, womit die ganze Operation geendigt ist, und hat also diesen Rest gefunden 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+{"basename":"euler_rechenkunst01_1738","par_idx":409,"date":1738,"orig":"Wir haben hier bey einer jeden Operation den Grund und das Fundament derſelben beygeſetzet, weswegen die gantze Operation ziemlich weitleiffig ſcheinet, allein wenn die bloſſe Operation beſchrieben wird, ſo wird dieſelbe gantz kurtz.","norm":"Wir haben hier bei einer jeden Operation den Grund und das Fundament derselben beigesetzt, weswegen die ganze Operation ziemlich weitleiffig scheinet, allein wenn die bloße Operation beschrieben wird, so wird dieselbe ganz 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Ferner 8 von 5 kan man nicht, allſo 8 von 15 bleiben 7, und ſetzt ein Punet zu 3. Drittens 7 von 2 kan man nicht, allſo 7 von 12 bleiben 5, und ſetzt ein Punct zu 7. Endlich 5 von 6 bleiben 1.","norm":"Also kann man bei eben diesem Exempel auf folgende Weise den gesuchten Rest gleich finden, wenn man sagt 9 von 4 kann man nicht, deswegen 9 von 14 bleiben 5, und setzt ein Punkt zu 6. Ferner 8 von 5 kann man nicht, also 8 von 15 bleiben 7, und setzt ein Bunt zu 3. Drittens 7 von 2 kann man nicht, also 7 von 12 bleiben 5, und setzt ein Punkt zu 7. 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+{"basename":"euler_rechenkunst01_1738","par_idx":419,"date":1738,"orig":"Um demnach abziehen zukoͤnnen, muß von der zweyten folgenden Sorte nehmlich den Centenariis ein Stuͤck weggenommen werden, und wenn auch von ſolchen nichts vorhanden waͤre, muͤſte ſo gar von den Millenariis ein Stuͤck genommen werden.","norm":"Um demnach abziehen zukönnen, muss von der zweiten folgenden Sorte nämlich den Centenariis ein Stück weggenommen werden, und wenn auch von solchen nichts vorhanden wäre, müsste so gar von den Millenariis ein Stück genommen 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+{"basename":"euler_rechenkunst01_1738","par_idx":420,"date":1738,"orig":"Jn dieſem Exempel aber haben wir 2 Centenarios davon ein Stuͤck genommen, welches durch das hinzugeſetzte Punct angedeutet wird, macht 10 Decades.","norm":"In diesem Exempel aber haben wir 2 Centenarios davon ein Stück genommen, welches durch das hinzugesetzte Punkt angedeutet wird, macht 10 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+{"basename":"euler_rechenkunst01_1738","par_idx":421,"date":1738,"orig":"Da wir nun Decades haben, ſo koͤnnen wir davon ein Stuͤck nehmen, und zu den Unitæten ſchlagen; da dann noch 9 Decaden zuruͤck bleiben, welche man ſich anſtatt der 0 auf der zweyten Stelle der oberen Zahl einbilden muß.","norm":"Da wir nun Decades haben, so können wir davon ein Stück nehmen, und zu den Unitäten schlagen; da dann noch 9 Dekaden zurückbleiben, welche man sich anstatt der 0 auf der zweiten Stelle der oberen Zahl einbilden 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+{"basename":"euler_rechenkunst01_1738","par_idx":422,"date":1738,"orig":"Auf dieſe Weiſe haben wir nun 15 Unitæten, davon die 7 weggenommen bleiben 8 Unitæten uͤber, ſo in den Reſt auf die Stelle der Unitæten geſetzet werden.","norm":"Auf diese Weise haben wir nun 15 Unitäten, davon die 7 weggenommen bleiben 8 Unitäten über, so in den Rest auf die Stelle der Unitäten gesetzt 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+{"basename":"euler_rechenkunst01_1738","par_idx":423,"date":1738,"orig":"Wegen dieſer Operation haben wir nun 2 Decades nicht von 0, ſondern von 9 Decadibus abzuziehen, bleiben allſo 7 uͤbrig, ſo unter die Linie auf die zweite Stelle geſchrieben werden.","norm":"Wegen dieser Operation haben wir nun 2 Decades nicht von 0, sondern von 9 Decadibus abzuziehen, bleiben also 7 übrig, so unter die Linie auf die zweite Stelle geschrieben 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+{"basename":"euler_rechenkunst01_1738","par_idx":424,"date":1738,"orig":"Drittens ſind 8 Centenarii von einem Centenario abzuziehen, welches, weilen es nicht geſchehen kan, wird der eine Millenarius gleich dazu gethan, daß man 11 Centenarios bekommt, davon ſo die 8 Centenarii abgezogen werden, 3 zuruͤck bleiben, und folglich dieſer Reſt 378 herauskommt.","norm":"Drittens sind 8 Centenarii von einem Centenario abzuziehen, welches, weilen es nicht geschehen kann, wird der eine Millenarius gleich dazugetan, dass man 11 Centenarios bekommt, davon so die 8 Centenarii abgezogen werden, 3 zurückbleiben, und folglich dieser Rest 378 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+{"basename":"euler_rechenkunst01_1738","par_idx":425,"date":1738,"orig":"Aus dieſem Exempel ſieht man nun deutlich, daß die obgegebene Regel nicht voͤllig hinlaͤnglich ſey, ſondern oͤffters einen Zuſatz vonnoͤthen haben, wodurch in den Figuren der oberen Zahl, groſſe Veraͤnderungen entſpringen.","norm":"Aus diesem Exempel sieht man nun deutlich, dass die gegebene Regel nicht völlig hinlänglich sei, sondern öfters einen Zusatz vonnöten haben, wodurch in den Figuren der oberen Zahl, große Veränderungen 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+{"basename":"euler_rechenkunst01_1738","par_idx":426,"date":1738,"orig":"Dieſem ſoll allſo durch die nachfolgende Regel abgeholffen werden.","norm":"Diesem soll also durch die nachfolgende Regel abgeholfen werden.","orig_tok":["Dieſem","ſoll","allſo","durch","die","nachfolgende","Regel","abgeholffen","werden","."],"norm_tok":["Diesem","soll","also","durch","die","nachfolgende","Regel","abgeholfen","werden","."],"orig_spans":[[0,6],[7,11],[12,17],[18,23],[24,27],[28,40],[41,46],[47,58],[59,65],[65,66]],"norm_spans":[[0,6],[7,11],[12,16],[17,22],[23,26],[27,39],[40,45],[46,56],[57,63],[63,64]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.283899784088135} +{"basename":"euler_rechenkunst01_1738","par_idx":427,"date":1738,"orig":"7.)","norm":"7.)","orig_tok":["7.",")"],"norm_tok":["7.",")"],"orig_spans":[[0,2],[2,3]],"norm_spans":[[0,2],[2,3]],"alignment":[[0,0],[1,1]],"lang_fastText":"hr","lang_py3langid":"en","lang_cld3":"ja","orig_lang_de":0,"norm_lmscore":5.9008002281188965} +{"basename":"euler_rechenkunst01_1738","par_idx":428,"date":1738,"orig":"Wenn, wie vorher geſetzet worden, die Anzahl der Stuͤcke von einer Sorte in der unteren oder kleineren Zahl groͤſſer iſt, als die Anzahl der Stuͤcke von eben der Sorte in der oberen Zahl, ſo muͤſſen zu dieſen Stuͤcken der oberen Zahl noch 10 Stuͤcke im Sinn hinzugeſetzet werden, da denn die Subtraction wird geſchehen koͤnnen.","norm":"Wenn, wie vorher gesetzt worden, die Anzahl der Stücke von einer Sorte in der unteren oder kleineren Zahl größer ist, als die Anzahl der Stücke von eben der Sorte in der oberen Zahl, so müssen zu diesen Stücken der oberen Zahl noch 10 Stücke im Sinn hinzugesetzt werden, da denn die Subtraktion wird geschehen 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+{"basename":"euler_rechenkunst01_1738","par_idx":437,"date":1738,"orig":"Laſſet uns nun ein Exempel, ſo nach der erſteren Regel iſt berechnet worden, davon wir den Grund ſchon dargethan, vor die Hand nehmen, und uns dabey dieſes jetztgegebenen Grundſatzes bedienen.","norm":"Lasst uns nun ein Exempel, so nach der ersteren Regel ist berechnet worden, davon wir den Grund schon dargetan, vor die Hand nehmen, und uns dabei dieses jetztgegebenen Grundsatzes 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+{"basename":"euler_rechenkunst01_1738","par_idx":438,"date":1738,"orig":"Nehmlich es ſollen 38 von 82 abgezogen werden, welche Zahlen allſo wie folgt zu ſtehen kommen.","norm":"Nämlich es sollen 38 von 82 abgezogen werden, welche Zahlen also wie folgt zu stehen 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+{"basename":"euler_rechenkunst01_1738","par_idx":441,"date":1738,"orig":"Ferner muß ich 3 Decades nur von 7 Decaden abziehen, weilen von 8 ſchon eine Decas iſt weggenommen, und zu den Unitæten geſchlagen worden.","norm":"Ferner muss ich 3 Decades nur von 7 Dekaden abziehen, weilen von 8 schon eine Decas ist weggenommen, und zu den Unitäten geschlagen 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+{"basename":"euler_rechenkunst01_1738","par_idx":442,"date":1738,"orig":"Wenn ich aber Kraft des gegebenen Grundſatzes dieſe beyden Zahlen 3 und 7 um eines vermehre, ſo bekomme ich fuͤr die obere Zahl wiederum 8, wie dieſelbe ſchon wuͤrcklich da ſteht, anſtatt der unteren Zahl 3 aber bekomme ich 4, welche von 8 abgezogen 4 zuruͤck laſſen, eben als wenn ich nach der erſten Regel 3 von 7 ſubtrahiret haͤtte.","norm":"Wenn ich aber Kraft des gegebenen Grundsatzes diese beiden Zahlen 3 und 7 um eines vermehre, so bekomme ich für die obere Zahl wiederum 8, wie dieselbe schon wirklich da steht, anstatt der unteren Zahl 3 aber bekomme ich 4, welche von 8 abgezogen 4 zurücklassen, eben als wenn ich nach der ersten Regel 3 von 7 subtrahieret 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ſage ich wegen dem bey dem 8 ſtehenden Punct 9 von 6 kan ich nicht abziehen, ſage deswegen 9 von 16, und ſetze zu der folgenden unteren Figur 7 ein Punckt, 9 aber von 16 genommen laſſen 7 zuruͤck, welche unter die Linie auf die zweyte Stelle ſchreibe.","norm":"Ferner sage ich wegen dem bei dem 8 stehenden Punkt 9 von 6 kann ich nicht abziehen, sage deswegen 9 von 16, und setze zu der folgenden unteren Figur 7 ein Punkt, 9 aber von 16 genommen lassen 7 zurück, welche unter die Linie auf die zweite Stelle 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+{"basename":"euler_rechenkunst01_1738","par_idx":461,"date":1738,"orig":"Man kan aber auch bey der dritten Subtraction, da oben wuͤrcklich 12 ſtehet, gleich 9 von den 12 abziehen; da dann die gantze Operation ein Ende hat, dadurch man dieſen Reſt gefunden 378, welcher auch auf die vorhergegebene Art iſt heraus gebracht worden.","norm":"Man kann aber auch bei der dritten Subtraktion, da oben wirklich 12 steht, gleich 9 von den 12 abziehen; da dann die ganze Operation ein Ende hat, dadurch man diesen Rest gefunden 378, welcher auch auf die vorhergegeben Art ist herausgebracht 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+{"basename":"euler_rechenkunst01_1738","par_idx":478,"date":1738,"orig":"Auf ſolche Art ſtelle man alſo die Subtraction bey einer jeglichen Sorte an, und ſetze einen jeglichen Reſt auf ſeine gehoͤrige Stelle unter die Linie.","norm":"Auf solche Art stelle man also die Subtraktion bei einer jeglichen Sorte an, und setze einen jeglichen Rest auf seine gehörige Stelle unter die 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+{"basename":"euler_rechenkunst01_1738","par_idx":479,"date":1738,"orig":"Da man denn nach Endigung der gantzen Operation den voͤlligen geſuchten Reſt unter der Linie finden wird.","norm":"Da man denn nach Endigung der ganzen Operation den völligen gesuchten Rest unter der Linie finden 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+{"basename":"euler_rechenkunst01_1738","par_idx":483,"date":1738,"orig":"Allein da, wenn eine Figur in der unteren Zahl groͤſſer iſt, als die obſtehende, die nach der lincken Hand folgende Figur in der unteren Zahl um ein Stuͤck vermehret und allſo veraͤndert werden muß, ſo muß in ſolchem Fall die Operation von der rechten Hand angefangen und nach der lincken fortgeſetzet werden.","norm":"Allein da, wenn eine Figur in der unteren Zahl größer ist, als die stehende, die nach der linken Hand folgende Figur in der unteren Zahl um ein Stück vermehret und also verändert werden muss, so muss in solchem Fall die Operation von der rechten Hand angefangen und nach der Linken fortgesetzt 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+{"basename":"euler_rechenkunst01_1738","par_idx":484,"date":1738,"orig":"Was nun bey Subtrahirung einer jeglichen Sorte uͤberbleibt, wird unter die Linie unter eben dieſe Sorte geſetzet, damit eine jegliche in der Subtraction gefundene Zahl auf ihre gehoͤrige Stelle zu ſtehen komme.","norm":"Was nun bei Subtrahierung einer jeglichen Sorte überbleibt, wird unter die Linie unter eben diese Sorte gesetzt, damit eine jegliche in der Subtraktion gefundene Zahl auf ihre gehörige Stelle zu stehen 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+{"basename":"euler_rechenkunst01_1738","par_idx":485,"date":1738,"orig":"Wo eine Figur der unteren Zahl der obſtehenden gleich iſt und allſo nichts uͤberbleibt, wird eine Ziffer 0 unter die Linie an dieſe Stelle geſchrieben, woferne ſolches nicht zu Ende der Operation geſchieht.","norm":"Wo eine Figur der unteren Zahl der stehenden gleich ist und also nichts überbleibt, wird eine Ziffer 0 unter die Linie an diese Stelle geschrieben, sofern solches nicht zu Ende der Operation 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+{"basename":"euler_rechenkunst01_1738","par_idx":486,"date":1738,"orig":"Denn in ſolchem Falle waͤre es unnoͤthig die 0 zu ſchreiben, weilen die 0 von der lincken Hand anfangs nichts bedeuten, und auch auf die Bedeutung der folgenden Zahlen keinen Einfluß haben.","norm":"Denn in solchem Falle wäre es unnötig die 0 zu schreiben, weilen die 0 von der linken Hand anfangs nichts bedeuten, und auch auf die Bedeutung der folgenden Zahlen keinen Einfluss 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+{"basename":"euler_rechenkunst01_1738","par_idx":487,"date":1738,"orig":"Die gantze Operation wird aber am fuͤglichſten durch einige Exempel erlaͤutert werden.","norm":"Die ganze Operation wird aber am füglichsten durch einige Exempel erläutert werden.","orig_tok":["Die","gantze","Operation","wird","aber","am","fuͤglichſten","durch","einige","Exempel","erlaͤutert","werden","."],"norm_tok":["Die","ganze","Operation","wird","aber","am","füglichsten","durch","einige","Exempel","erläutert","werden","."],"orig_spans":[[0,3],[4,10],[11,20],[21,25],[26,30],[31,33],[34,46],[47,52],[53,59],[60,67],[68,78],[79,85],[85,86]],"norm_spans":[[0,3],[4,9],[10,19],[20,24],[25,29],[30,32],[33,44],[45,50],[51,57],[58,65],[66,75],[76,82],[82,83]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.394999980926514} +{"basename":"euler_rechenkunst01_1738","par_idx":488,"date":1738,"orig":"Als von 273024 ſoll abgezogen werden 65372, welche demnach auf folgende Art geſchrieben werden.","norm":"Als von 273024 soll abgezogen werden 65372, welche demnach auf folgende Art geschrieben 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+{"basename":"euler_rechenkunst01_1738","par_idx":489,"date":1738,"orig":"Reſtirt.","norm":"Restiert.","orig_tok":["Reſtirt","."],"norm_tok":["Restiert","."],"orig_spans":[[0,7],[7,8]],"norm_spans":[[0,8],[8,9]],"alignment":[[0,0],[1,1]],"lang_fastText":"en","lang_py3langid":"vi","lang_cld3":"hu","orig_lang_de":0,"norm_lmscore":5.8302998542785645} +{"basename":"euler_rechenkunst01_1738","par_idx":490,"date":1738,"orig":"[Formel]","norm":"[Formel]","orig_tok":["[Formel]"],"norm_tok":["[","Formel","]"],"orig_spans":[[0,8]],"norm_spans":[[0,1],[1,7],[7,8]],"alignment":[[0,0],[0,1],[0,2]],"lang_fastText":"sh","lang_py3langid":"en","lang_cld3":"cy","orig_lang_de":0,"norm_lmscore":7.932600021362305} +{"basename":"euler_rechenkunst01_1738","par_idx":491,"date":1738,"orig":"Hierauf ſagt man 2 von 4, bleiben 2 ſo unter die Linie geſchrieben werden.","norm":"Hierauf sagt man 2 von 4, bleiben 2 so unter die Linie geschrieben 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Drittens 4 von 0 kan man nicht, ſetzt deswegen zum folgenden 5 ein Punct und ſagt 4 von 10 bleiben 6. Viertens 6 von 3 kan man nicht, ſetzt allſo ein Punct zu der folgenden Figur 6 und ſagt 6 von 13 bleiben 7. Fuͤnftens ſagt man 7 von 7 geht auf, ſchreibet allſo eine 0 unter die Linie.","norm":"Ferner 7 von 2 kann man nicht, setzt deswegen zum folgenden 3 ein Punkt und sagt 7 von 12 bleiben 5. Drittens 4 von 0 kann man nicht, setzt deswegen zum folgenden 5 ein Punkt und sagt 4 von 10 bleiben 6. Viertens 6 von 3 kann man nicht, setzt also ein Punkt zu der folgenden Figur 6 und sagt 6 von 13 bleiben 7. Fünftens sagt man 7 von 7 geht auf, schreibet also eine 0 unter die 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+{"basename":"euler_rechenkunst01_1738","par_idx":545,"date":1738,"orig":"Jn der Multiplication wird nehmlich gelehret, wie man nur allein die Summ zweyer oder mehr Zahlen finden ſoll, welche einander gleich ſind; da ſich die Addition auf die Erfindung der Summ von zweyen oder mehr gegebenen Zahlen, ſo einander auch nicht gleich ſind, erſtrecket.","norm":"In der Multiplikation wird nämlich gelehrt, wie man nur allein die Summe zweier oder mehr Zahlen finden soll, welche einander gleich sind; da sich die Addition auf die Erfindung der Summe von zweien oder mehr gegebenen Zahlen, so einander auch nicht gleich sind, 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+{"basename":"euler_rechenkunst01_1738","par_idx":549,"date":1738,"orig":"wodurch denn gefunden wird, daß 128 dreymahl genommen 384 ausmache.","norm":"wodurch denn gefunden wird, dass 128 dreimal genommen 384 ausmache.","orig_tok":["wodurch","denn","gefunden","wird",",","daß","128","dreymahl","genommen","384","ausmache","."],"norm_tok":["wodurch","denn","gefunden","wird",",","dass","128","dreimal","genommen","384","ausmache","."],"orig_spans":[[0,7],[8,12],[13,21],[22,26],[26,27],[28,31],[32,35],[36,44],[45,53],[54,57],[58,66],[66,67]],"norm_spans":[[0,7],[8,12],[13,21],[22,26],[26,27],[28,32],[33,36],[37,44],[45,53],[54,57],[58,66],[66,67]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":7.067200183868408} +{"basename":"euler_rechenkunst01_1738","par_idx":550,"date":1738,"orig":"Dieſes Exempel kan zwar leicht durch die Regeln der Addition gerechnet 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+{"basename":"euler_rechenkunst01_1738","par_idx":586,"date":1738,"orig":"Dieſe letztere Benennung fuͤhret dieſelbe deswegen, weilen man gemeiniglich von einmahl eins iſt eins anzufangen pflegt.","norm":"Diese letztere Benennung führt dieselbe deswegen, weilen man gemeiniglich von ein mal eins ist eins anzufangen 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+{"basename":"euler_rechenkunst01_1738","par_idx":587,"date":1738,"orig":"Da aber eine jede Zahl mit eins multipliciret oder einmahl genommen in ihrer Groͤſſe unveraͤndert bleibt, ſo haben wir die Multiplication der einfachen Zahlen mit eins nicht beygeſetzet.","norm":"Da aber eine jede Zahl mit eins multiplizieret oder einmal genommen in ihrer Größe unverändert bleibt, so haben wir die Multiplikation der einfachen Zahlen mit eins nicht 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Hiebey iſt auch ferner zu mercken, daß eine jegliche Zahl mit 0 multipliciret nichts ausmache, weilen nichts oder 0, es mag ſo vielmahl genommen werden, als man will, immer nichts bleibt.","norm":"Derowegen pflegt man zu sagen, dass eins nicht multipliziere; also ist einmal 2 zwei, 1 Mal 3 drei und so fort in allen Zahlen welche auch größer sind als 9. 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iſt, ſo iſt nur eine Reihe vorhanden, und folglich das Productum ſo groß als der Multiplicandus ſelbſt.","norm":"Wenn nun der Multiplicator eins ist, so ist nur eine Reihe vorhanden, und folglich das Productum so groß als der Multiplicandus 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+{"basename":"euler_rechenkunst01_1738","par_idx":598,"date":1738,"orig":"Derohalben kan man durch Huͤlfe dieſer Tabelle die Anzahl der Stuͤcke, ſo von einer jeglichen Sorte in einer zuſammengeſetzten Zahl vorhanden ſind, mit einer jeglichen einfachen Zahl multipliciren.","norm":"Derohalben kann man durch Hilfe dieser Tabelle die Anzahl der Stücke, so von einer jeglichen Sorte in einer zusammengesetzten Zahl vorhanden sind, mit einer jeglichen einfachen Zahl 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+{"basename":"euler_rechenkunst01_1738","par_idx":599,"date":1738,"orig":"Wenn aber eine zuſammengeſetzte Zahl durch eine gegebene Zahl multipliciret werden ſoll, ſo wird man das geſuchte Product finden, wenn man einen jeglichen Theil, daraus dieſelbe Zahl beſtehet, mit dieſer vorgegebenen multipliciret, und alle dieſe herausgebrachten Producta zuſammen in eine Summ bringet.","norm":"Wenn aber eine zusammengesetzte Zahl durch eine gegebene Zahl multiplizieret werden soll, so wird man das gesuchte Produkt finden, wenn man einen jeglichen Teil, daraus dieselbe Zahl bestehet, mit dieser vorgegebenen multiplizieret, und alle diese herausgebrachten Producta zusammen in eine Summe 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+{"basename":"euler_rechenkunst01_1738","par_idx":600,"date":1738,"orig":"Dieſes erhellet aus der Addition, als in welcher die Multiplication gegruͤndet iſt, in welcher man um die Summ vieler Zahlen zu finden, alle beſondren Sorten oder Theile, aus welchen dieſelben Zahlen beſtehen, zuſammen thut, da denn alle Summen von allen beſonderen Sorten zuſammen die gantze Summ ausmachen.","norm":"Dieses erhellet aus der Addition, als in welcher die Multiplikation gegründet ist, in welcher man um die Summe vieler Zahlen zu finden, alle besonderen Sorten oder Teile, aus welchen dieselben Zahlen bestehen, zusammen tut, da denn alle Summen von allen besonderen Sorten zusammen die ganze Summe 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+{"basename":"euler_rechenkunst01_1738","par_idx":601,"date":1738,"orig":"Wenn ich zum Exempel die Zahl 237 ſoll mit 4 multipliciren, und dieſes durch die Addition verrichte, indem ich die Zahl 237 vier mahl untereinander ſchreibe und dieſe 4 Zahlen zuſammen addire wie folget: [Formel] ſo nehme ich in der That erſtlich die 7 Unitæten vier mahl, zweytens auch die 3 Decades vier mahl, und drittens auch die 2 Centenarios vier mahl, welche 3 beſonderen Theile vier mahl genommen zuſammen die gantze Zahl vier mahl genommen ausmachen nehmlich 948. Eben dieſes Exempel nun durch die Multiplication auszurechnen, ſo hat man erſtlich zu mercken, daß die gegebene Zahl 237 aus folgenden Theilen beſtehe, nehmlich aus 7 Unitæten, 3 Decaden und 2 Centenariis.","norm":"Wenn ich zum Exempel die Zahl 237 soll mit 4 multiplizieren, und dieses durch die Addition verrichte, indem ich die Zahl 237 viermal untereinander schreibe und diese 4 Zahlen zusammen addiere wie folget: [Formel] so nehme ich in der Tat erstlich die 7 Unitäten viermal, zweitens auch die 3 Decades viermal, und drittens auch die 2 Centenarios viermal, welche 3 besonderen Teile viermal genommen zusammen die ganze Zahl viermal genommen ausmachen nämlich 948. Eben dieses Exempel nun durch die Multiplikation auszurechnen, so hat man erstlich zu merken, dass die gegebene Zahl 237 aus folgenden Teilen bestehe, nämlich aus 7 Unitäten, 3 Dekaden und 2 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+{"basename":"euler_rechenkunst01_1738","par_idx":620,"date":1738,"orig":"Endlich ſagt man 7 mahl 3 Millenarii machen 21 Millenarii dazu die vorigen 4 Millenarii hinzugethan machen 25 Millenarios das iſt 5 Millenarios, ſo in die gehoͤrige Stelle ins Product geſetzet werden, und 2 Decades Millenariorum welche, weilen keine Operation mehr uͤbrig iſt gleichfalls auf ihre gehoͤrige Stelle kommen.","norm":"Endlich sagt man 7 Mal 3 Millenarii machen 21 Millenarii dazu die vorigen 4 Millenarii hinzugetan machen 25 Millenarios das ist 5 Millenarios, so in die gehörige Stelle ins Produkt gesetzt werden, und 2 Decades Millenariorum welche, weilen keine Operation mehr übrig ist gleichfalls auf ihre gehörige Stelle 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+{"basename":"euler_rechenkunst01_1738","par_idx":621,"date":1738,"orig":"Das gantze Product, welches gefunden worden, iſt demnach dieſes 25172, welche Zahl folglich 7 mahl groͤſſer iſt als die vorgegebene 3596.","norm":"Das ganze Produkt, welches gefunden worden, ist demnach dieses 25172, welche Zahl folglich 7 Mal größer ist als die vorgegebene 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+{"basename":"euler_rechenkunst01_1738","par_idx":622,"date":1738,"orig":"Wenn in der Operation, wie ſie in dieſem Exempel iſt gemacht worden, die Nahmen der Sorten ausgelaſſen werden, weilen bey einer jeglichen Sorte die Operation einerley iſt, ſo wird die gantze Operation weit kuͤrtzer.","norm":"Wenn in der Operation, wie sie in diesem Exempel ist gemacht worden, die Namen der Sorten ausgelassen werden, weilen bei einer jeglichen Sorte die Operation einerlei ist, so wird die ganze Operation weit 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+{"basename":"euler_rechenkunst01_1738","par_idx":623,"date":1738,"orig":"Auf ſolche Art wollen wir derohalben folgendes Exempel ausrechnen.","norm":"Auf solche Art wollen wir derohalben folgendes Exempel ausrechnen.","orig_tok":["Auf","ſolche","Art","wollen","wir","derohalben","folgendes","Exempel","ausrechnen","."],"norm_tok":["Auf","solche","Art","wollen","wir","derohalben","folgendes","Exempel","ausrechnen","."],"orig_spans":[[0,3],[4,10],[11,14],[15,21],[22,25],[26,36],[37,46],[47,54],[55,65],[65,66]],"norm_spans":[[0,3],[4,10],[11,14],[15,21],[22,25],[26,36],[37,46],[47,54],[55,65],[65,66]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":6.848800182342529} +{"basename":"euler_rechenkunst01_1738","par_idx":624,"date":1738,"orig":"[Formel] Multiplicandus","norm":"[Formel] 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+{"basename":"euler_rechenkunst01_1738","par_idx":634,"date":1738,"orig":"Achtens ſagt man 5 mahl 9 iſt 45 und die im Sinn behaltenen 6 macht 51, welche gantze Zahl, weilen die Multiplication geendigt, ins Product geſchrieben wird; und auf dieſe Art iſt das unter der Linie | ſtehende Product gefunden worden.","norm":"Achtens sagt man 5 Mal 9 ist 45 und die im Sinn behaltenen 6 macht 51, welche ganze Zahl, weilen die Multiplikation geendigt, ins Produkt geschrieben wird; und auf diese Art ist das unter der Linie | stehende Produkt gefunden 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multipliciren und das Product finden ſoll; und iſt von der gantzen Operation der Grund ausfuͤhrlick erklaret worden.","norm":"Aus welchen Exempeln genügsam zu ersehen ist, wie man eine jegliche Zahl, so groß dieselbe auch immer sein mag, durch eine einfache Zahl multiplizieren und das Produkt finden soll; und ist von der ganzen Operation der Grund ausführlick erklaret 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+{"basename":"euler_rechenkunst01_1738","par_idx":677,"date":1738,"orig":"8.","norm":"8.","orig_tok":["8."],"norm_tok":["8."],"orig_spans":[[0,2]],"norm_spans":[[0,2]],"alignment":[[0,0]],"lang_fastText":"en","lang_py3langid":"en","lang_cld3":"ja","orig_lang_de":0,"norm_lmscore":1.9600000381469722} +{"basename":"euler_rechenkunst01_1738","par_idx":678,"date":1738,"orig":"Wann der Multiplicator eine zuſammengeſetzte Zahl iſt oder aus vielen Figuren beſtehet, ſo muß der Multiplicandus mit einem jeglichen Theil, daraus der Multiplicator beſteht multiplicirt, und darauf alle dieſe gefundenen Producte zuſammen addirt werden, da dann die Summa, welche herauskommt das verlangte Productum ſeyn wird.","norm":"Wann der Multiplicator eine zusammengesetzte Zahl ist oder aus vielen Figuren bestehet, so muss der Multiplicandus mit einem jeglichen Teil, daraus der Multiplicator besteht multipliziert, und darauf alle diese gefundenen Produkte zusammen addiert werden, da dann die Summa, welche herauskommt das verlangte Productum sein 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+{"basename":"euler_rechenkunst01_1738","par_idx":679,"date":1738,"orig":"Wir haben oben gewieſen, daß wann der Multiplicandus aus etlichen Theilen beſteht, ein jeglicher Theil insbeſondere mit dem Multiplicator muͤſſe multiplicirt, und dieſe beſonderen Product zuſammen geſetzet werden, als deren Summ das geſuche Product geben muß.","norm":"Wir haben oben gewiesen, dass wann der Multiplicandus aus etlichen Teilen besteht, ein jeglicher Teil insbesondere mit dem Multiplicator müsse multipliziert, und diese besonderen Produkt zusammen gesetzt werden, als deren Summe das Gesuche Produkt geben 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+{"basename":"euler_rechenkunst01_1738","par_idx":680,"date":1738,"orig":"Da nun der Multiplicandus und der Multiplicator unter ſich verwechſelt, und einer an des anderen Stelle geſetzet werden kan, ſo iſt eben dieſes auch von dem Multiplicator zuverſtehen.","norm":"Da nun der Multiplicandus und der Multiplicator unter sich verwechselt, und einer an des anderen Stelle gesetzt werden kann, so ist eben dieses auch von dem Multiplicator zu 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+{"basename":"euler_rechenkunst01_1738","par_idx":681,"date":1738,"orig":"Derohalben wann der Multiplicator eine zuſammengeſetzte Zahl iſt, oder aus mehr als einer Figur beſtehet, ſo muß der Multiplicandus mit einem jeglichen Theil des Multiplicators multiplicirt und alle dieſe ſonderbaren Producte zuſammen addiret werden:","norm":"Derohalben wann der Multiplicator eine zusammengesetzte Zahl ist, oder aus mehr als einer Figur bestehet, so muss der Multiplicandus mit einem jeglichen Teil des Multiplikators multipliziert und alle diese sonderbaren Produkte zusammen addieret werden:","orig_tok":["Derohalben","wann","der","Multiplicator","eine","zuſammengeſetzte","Zahl","iſt",",","oder","aus","mehr","als","einer","Figur","beſtehet",",","ſo","muß","der","Multiplicandus","mit","einem","jeglichen","Theil","des","Multiplicators","multiplicirt","und","alle","dieſe","ſonderbaren","Producte","zuſammen","addiret","werden",":"],"norm_tok":["Derohalben","wann","der","Multiplicator","eine","zusammengesetzte","Zahl","ist",",","oder","aus","mehr","als","einer","Figur","bestehet",",","so","muss","der","Multiplicandus","mit","einem","jeglichen","Teil","des","Multiplikators","multipliziert","und","alle","diese","sonderbaren","Produkte","zusammen","addieret","werden",":"],"orig_spans":[[0,10],[11,15],[16,19],[20,33],[34,38],[39,55],[56,60],[61,64],[64,65],[66,70],[71,74],[75,79],[80,83],[84,89],[90,95],[96,104],[104,105],[106,108],[109,112],[113,116],[117,131],[132,135],[136,141],[142,151],[152,157],[158,161],[162,176],[177,189],[190,193],[194,198],[199,204],[205,216],[217,225],[226,234],[235,242],[243,249],[249,250]],"norm_spans":[[0,10],[11,15],[16,19],[20,33],[34,38],[39,55],[56,60],[61,64],[64,65],[66,70],[71,74],[75,79],[80,83],[84,89],[90,95],[96,104],[104,105],[106,108],[109,113],[114,117],[118,132],[133,136],[137,142],[143,152],[153,157],[158,161],[162,176],[177,190],[191,194],[195,199],[200,205],[206,217],[218,226],[227,235],[236,244],[245,251],[251,252]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28],[29,29],[30,30],[31,31],[32,32],[33,33],[34,34],[35,35],[36,36]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.5183000564575195} +{"basename":"euler_rechenkunst01_1738","par_idx":682,"date":1738,"orig":"da dann derſelben Summe das geſuchte Product geben wird.","norm":"da dann derselben Summe das gesuchte Produkt geben wird.","orig_tok":["da","dann","derſelben","Summe","das","geſuchte","Product","geben","wird","."],"norm_tok":["da","dann","derselben","Summe","das","gesuchte","Produkt","geben","wird","."],"orig_spans":[[0,2],[3,7],[8,17],[18,23],[24,27],[28,36],[37,44],[45,50],[51,55],[55,56]],"norm_spans":[[0,2],[3,7],[8,17],[18,23],[24,27],[28,36],[37,44],[45,50],[51,55],[55,56]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.44290018081665} +{"basename":"euler_rechenkunst01_1738","par_idx":683,"date":1738,"orig":"Die Theile aber, daraus eine zuſamme geſetzte Zahl beſtehet, ſind die verſchiedenen Sorten als Unitæten, Decades, Centenarii und ſo fort, von deren jeder nicht mehr als 9 Stuͤcke vorhanden ſeyn koͤnnen.","norm":"Die Teile aber, daraus eine zusame gesetzte Zahl bestehet, sind die verschiedenen Sorten als Unitäten, Decades, Centenarii und so fort, von deren jeder nicht mehr als 9 Stücke vorhanden sein 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+{"basename":"euler_rechenkunst01_1738","par_idx":684,"date":1738,"orig":"Derohalben muß der Multiplicandus erſtlich mit ſo viel Unitæten und dann mit ſo viel Decaden, ingleichem mit ſo viel Centenariis, und ſo weiter, als im Multiplicatore befindlich ſind multipliciret und alle heraus gebrachten Producte in eine Summe gebracht werden.","norm":"Derohalben muss der Multiplicandus erstlich mit so viel Unitäten und dann mit so viel Dekaden, ingleichem mit so viel Centenariis, und so weiter, als im Multiplikator befindlich sind multiplizieret und alle heraus gebrachten Produkte in eine Summe gebracht 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dieſes gibt 1421400.","norm":"Drittens multipliziere man die vorgegebene Zahl mit 300, dieses gibt 1421400.","orig_tok":["Drittens","multiplicire","man","die","vorgegebene","Zahl","mit","300",",","dieſes","gibt","1421400","."],"norm_tok":["Drittens","multipliziere","man","die","vorgegebene","Zahl","mit","300",",","dieses","gibt","1421400."],"orig_spans":[[0,8],[9,21],[22,25],[26,29],[30,41],[42,46],[47,50],[51,54],[54,55],[56,62],[63,67],[68,75],[75,76]],"norm_spans":[[0,8],[9,22],[23,26],[27,30],[31,42],[43,47],[48,51],[52,55],[55,56],[57,63],[64,68],[69,77]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,11]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.207300186157227} +{"basename":"euler_rechenkunst01_1738","par_idx":689,"date":1738,"orig":"Nun dieſe drey Producta addire man zuſammen wie folget.","norm":"Nun diese drei Producta addiere man zusammen wie 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werden könne, so pflegt man diese besonderen Produkte gleich so untereinander zu schreiben, dass die Unitäten unter die Unitäten, und alle gleichen Sorten untereinander zu stehen kommen, damit man dieselben gleich zusammen addieren könne, als wie 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+{"basename":"euler_rechenkunst01_1738","par_idx":710,"date":1738,"orig":"Hat man nun auf dieſe Art alle Producte von allen Figuren des Multiplicators gefunden, und auf beſchriebene Art unter einander geſetzet, wird darunter nochmahls eine Linie gezogen, und alle dieſe beſonderen Producte zuſammen addiret, da dann die Summe das geſuchte Product ſeyn wird.","norm":"Hat man nun auf diese Art alle Produkte von allen Figuren des Multiplikators gefunden, und auf beschriebene Art untereinander gesetzt, wird darunter nochmals eine Linie gezogen, und alle diese besonderen Produkte zusammen addieret, da dann die Summe das gesuchte Produkt sein 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+{"basename":"euler_rechenkunst01_1738","par_idx":744,"date":1738,"orig":"Es beſteht nehmlich das Fundament davon hierinn, daß wenn ein Multiplicator ein Factum iſt von zwey Factor bus oder aus der Multiplication zweyer Zahlen mit einander entſprungen, man das wahre Product erhalte, wenn man den Multiplicandum erſtlich mit einem Factore des Multiplicators multiplicire, und was herausgekommen nochmahls mit dem andern Factore multiplicire.","norm":"Es besteht nämlich das Fundament davon hierin, dass wenn ein Multiplicator ein Faktum ist von zwei Faktor bus oder aus der Multiplikation zweier Zahlen miteinander entsprungen, man das wahre Produkt erhalte, wenn man den Multiplicandum erstlich mit einem Factore des Multiplikators multipliziere, und was herausgekommen nochmals mit dem anderen Factore 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Gleichergeſtalt iſt 478000 ſo viel als 478 mahl 1000 oder das Product von dieſen Zahlen.","norm":"Als 230 ist das Produkt von 23 und 10 oder diese beiden Zahlen sind die Factores von 230. 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+{"basename":"euler_rechenkunst01_1738","par_idx":750,"date":1738,"orig":"Als wann man ſoll 5339 mit 24600 multipliciren, ſo multipliciret man nur mit 246, welche man alſo unter die Zahl 5339 ſchreibt.","norm":"Als wann man soll 5339 mit 24600 multiplizieren, so multiplizieret man nur mit 246, welche man also unter die Zahl 5339 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+{"basename":"euler_rechenkunst01_1738","par_idx":751,"date":1738,"orig":"Damit man aber die Nullen nicht vergeſſe, kan man dieſelben gleichwohl zum Multiplicatore hinzuſetzen, bey der Multiplication aber hat man auf dieſelben nicht zu ſehen; ſondern ſchreibt dieſelben nur zum gefundenen Product, wie aus beyſtehender Operation zu ſehen.","norm":"Damit man aber die Nullen nicht vergesse, kann man dieselben gleichwohl zum Multiplikator hinzusetzen, bei der Multiplikation aber hat man auf dieselben nicht zu sehen; sondern schreibt dieselben nur zum gefundenen Produkt, wie aus beistehender Operation zu 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+{"basename":"euler_rechenkunst01_1738","par_idx":756,"date":1738,"orig":"Man kan auch der Deutlichkeit halben die Nullen gegen der rechten Hand vorſchieſſend zum Multiplicando ſetzen, damit man nach geendigter Operation gleich ſehe, wieviel Nullen man zum Product zu ſetzen habe, wie aus folgender Operation zu ſehen.","norm":"Man kann auch der Deutlichkeit halben die Nullen gegen der rechten Hand vorschießend zum Multiplicando setzen, damit man nach geendigter Operation gleich sehe, wie viel Nullen man zum Produkt zu setzen habe, wie aus folgender Operation zu 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+{"basename":"euler_rechenkunst01_1738","par_idx":759,"date":1738,"orig":"Wann nun ſo wohl der Multiplicandus als Multiplicator ſich mit Nullen endigen, ſo kan die Operation aus dieſen beyden Faͤllen angeſtellet werden.","norm":"Wann nun so wohl der Multiplicandus als Multiplicator sich mit Nullen endigen, so kann die Operation aus diesen beiden Fällen angestellt 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+{"basename":"euler_rechenkunst01_1738","par_idx":760,"date":1738,"orig":"Als wann man 1987000 mit 3700 multipliciren ſollte, ſo multiplicire man erſtlich 1987000 mit 37 und zum Product ſchreibe man zwey Nullen.","norm":"Als wann man 1987000 mit 3700 multiplizieren sollte, so multipliziere man erstlich 1987000 mit 37 und zum Produkt schreibe man zwei 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+{"basename":"euler_rechenkunst01_1738","par_idx":761,"date":1738,"orig":"Um aber 1987000 mit 37 zu multipliciren, ſo multiplicirt man nach der gegebenen Regel 1987 mit 37 und an das Product hangt man drey Nullen.","norm":"Um aber 1987000 mit 37 zu multiplizieren, so multipliziert man nach der gegebenen Regel 1987 mit 37 und an das Produkt hangt man drei 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+{"basename":"euler_rechenkunst01_1738","par_idx":762,"date":1738,"orig":"Derowegen um die beyden vorgegebenen Zahlen mit einander zu multipliciren, ſo multiplicirt man nur, nachdem man die Nullen beyderſeits zu Ende weggeworfen, 1987 mit 37 und ſchreibt zum gefundenen Product fuͤnf Nullen, nehmlich ſo viel als man weggeworfen.","norm":"Derowegen um die beiden vorgegebenen Zahlen miteinander zu multiplizieren, so multipliziert man nur, nachdem man die Nullen beiderseits zu Ende weggeworfen, 1987 mit 37 und schreibt zum gefundenen Produkt fünf Nullen, nämlich so viel als man 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+{"basename":"euler_rechenkunst01_1738","par_idx":804,"date":1738,"orig":"Dann wann durch die Addition zwey Zahlen in eine Summ gebracht werden, ſo lehret die Subtraction, wie man, wann die Summ und eine derſelben beyden Zahlen gegeben ſind, die andere Zahl finden ſoll.","norm":"Dann wann durch die Addition zwei Zahlen in eine Summe gebracht werden, so lehrt die Subtraktion, wie man, wann die Summe und eine derselben beiden Zahlen gegeben sind, die andere Zahl finden 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+{"basename":"euler_rechenkunst01_1738","par_idx":818,"date":1738,"orig":"Da nun der Quotus oder quotient dieſes anzeiget ſo iſt derſelbe die Zahl, welche geſucht wird, und um welche zu finden die Regeln der Diuiſion gegeben werden muͤſſen.","norm":"Da nun der Quotus oder Quotient dieses anzeiget so ist derselbe die Zahl, welche gesucht wird, und um welche zu finden die Regeln der Division gegeben werden 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+{"basename":"euler_rechenkunst01_1738","par_idx":819,"date":1738,"orig":"Wie wir nun vorher gewieſen, ſo iſt der Quotus eine Zahl, welche mit dem Diuiſor multiplicirt im Product den Dividendum gibt, weswegen in der Diuiſion der Quotus das iſt eine ſolche Zahl geſucht wird, welche, wann ſie mit dem Diuiſore multiplicirt wird, den Diuidendum herausbringt.","norm":"Wie wir nun vorher gewiesen, so ist der Quotus eine Zahl, welche mit dem Divisor multipliziert im Produkt den Dividendum gibt, weswegen in der Division der Quotus das ist eine solche Zahl gesucht wird, welche, wann sie mit dem Divisor multipliziert wird, den Dividenden 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+{"basename":"euler_rechenkunst01_1738","par_idx":820,"date":1738,"orig":"Wann man alſo fragt, wie viel mal 12 in 180 enthalten ſey, oder wann, wie man zu reden pflegt 180 durch 12 diuidirt werden ſollen, ſo iſt 180 der Diuidendus und 12 der Diuiſor.","norm":"Wann man also fragt, wie viel Mal 12 in 180 enthalten sei, oder wann, wie man zu reden pflegt 180 durch 12 dividiert werden sollen, so ist 180 der Dividend und 12 der 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+{"basename":"euler_rechenkunst01_1738","par_idx":821,"date":1738,"orig":"Die Zahl aber, welche geſucht wird oder der Quotus zeigt an, wie viel mal 12 in 180 enthalten ſey, und iſt ſo beſchaffen, daß derſelbe 12 mal genommen 180 ausmacht.","norm":"Die Zahl aber, welche gesucht wird oder der Quotus zeigt an, wie viel Mal 12 in 180 enthalten sei, und ist so beschaffen, dass derselbe 12 Mal genommen 180 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+{"basename":"euler_rechenkunst01_1738","par_idx":825,"date":1738,"orig":"Dieſe Summ aber iſt ſo groß, daß wann man dieſelbe 12 mal nimmt, 1728 herauskommen muß.","norm":"Diese Summe aber ist so groß, dass wann man dieselbe 12 Mal nimmt, 1728 herauskommen 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+{"basename":"euler_rechenkunst01_1738","par_idx":827,"date":1738,"orig":"Dieſes Exempel gehoͤrt derohalben zur Diuiſion und iſt 1728 der Diuidendus, 12 der Diuiſor, der Quotus aber, ſo durch die Diuiſion gefunden werden muß, zeigt an wieviel eine Perſon bekommen wird.","norm":"Dieses Exempel gehört derohalben zur Division und ist 1728 der Dividend, 12 der Divisor, der Quotus aber, so durch die Division gefunden werden muss, zeigt an wie viel eine Person bekommen 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+{"basename":"euler_rechenkunst01_1738","par_idx":828,"date":1738,"orig":"Nachdem man allſo dieſes Exempel auf dieſe Art unterſuchet hat, ſo iſt nicht nur klar, daß daſſelbe in die Diuiſion lauffe, ſondern auch, was fuͤr Zahlen fuͤr den Diuidendum und Diuiſorem angenommen werden muͤſſen.","norm":"Nachdem man also dieses Exempel auf diese Art untersuchet hat, so ist nicht nur klar, dass dasselbe in die Division laufe, sondern auch, was für Zahlen für den Dividenden und Divisor angenommen werden 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+{"basename":"euler_rechenkunst01_1738","par_idx":836,"date":1738,"orig":"Wenn aber eine Zahl von der anderen ſubtrahirt werden ſoll, ſo muß jene kleiner ſeyn als dieſe, ſonſten kan der Reſt mit den gewoͤhnlichen Zahlen, die uns noch allein bekannt ſind, nicht angedeutet werden.","norm":"Wenn aber eine Zahl von der anderen subtrahiert werden soll, so muss jene kleiner sein als diese, sonsten kann der Rest mit den gewöhnlichen Zahlen, die uns noch allein bekannt sind, nicht angedeutet 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+{"basename":"euler_rechenkunst01_1738","par_idx":843,"date":1738,"orig":"Jn den Exempeln von der erſten Art muß alſo der Diuidendus ſo beſchaffen ſeyn, daß derſelbe wuͤrcklich ein Factum ſey, davon der eine Factor der Diuiſor ſelbſt iſt.","norm":"In den Exempeln von der ersten Art muss also der Dividend so beschaffen sein, dass derselbe wirklich ein Faktum sei, davon der eine Faktor der Divisor selbst 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+{"basename":"euler_rechenkunst01_1738","par_idx":844,"date":1738,"orig":"Ein ſolches Exempel iſt wann 182 durch 13 diuidiret werden ſoll, dann da iſt der Quotus 14, und 182 entſpringt, wann man 13 mit 14 multiplicirt.","norm":"Ein solches Exempel ist wann 182 durch 13 dividieret werden soll, dann da ist der Quotus 14, und 182 entspringt, wann man 13 mit 14 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+{"basename":"euler_rechenkunst01_1738","par_idx":845,"date":1738,"orig":"Von ſolchen Exempeln ſagt man, daß ſich der Diuidendus wuͤrcklich durch den Diuiſorem diuidiren laſſe; alſo laͤßt ſich72 durch 8 diuidiren, dann 8 mahl 9 gibt 72.","norm":"Von solchen Exempeln sagt man, dass sich der Dividend wirklich durch den Divisor dividieren lasse; also lässt sich72 durch 8 dividieren, dann 8 Mal 9 gibt 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+{"basename":"euler_rechenkunst01_1738","par_idx":846,"date":1738,"orig":"Ein Exempel ſo zur anderen Art gehoͤret iſt, wann13 durch 3 diuidiret werden ſoll.","norm":"Ein Exempel so zur anderen Art gehöret ist, wann13 durch 3 dividieret werden 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alſo iſt der wahre Quotus groͤſſer als 4 und kleiner als 5, und kan alſo durch keine gantze Zahl angegeben werden.","norm":"Dann man kann keine ganze Zahl angeben, welche mit 3 multipliziert 13 ausmache; dann 3 mit 4 multipliziert gibt 12, und 3 mit 5 multipliziert 15; also ist der wahre Quotus größer als 4 und kleiner als 5, und kann also durch keine ganze Zahl angegeben 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+{"basename":"euler_rechenkunst01_1738","par_idx":848,"date":1738,"orig":"Derohalben weilen hier noch nicht der Ort iſt von Bruͤchen zu handeln, ſo muß man ſich begnuͤgen anſtatt des Quoti die naͤchſte Zahl anzugeben, und dabey zu mercken, wieviel dieſelbe fehle.","norm":"Derohalben weilen hier noch nicht der Ort ist von Brüchen zu handeln, so muss man sich begnügen anstatt des Quoti die nächste Zahl anzugeben, und dabei zu merken, wie viel dieselbe 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+{"basename":"euler_rechenkunst01_1738","par_idx":849,"date":1738,"orig":"Als in dem Exempel da 13 durch 3 diuidiret werden ſoll, ſo kan man ſagen daß 4 der Quotus ſey, aber nicht vollkommen, dann 4 mahl 3 macht nur 12 nicht 13, und iſt alſo 1 der Unterſcheid.","norm":"Als in dem Exempel da 13 durch 3 dividieret werden soll, so kann man sagen dass 4 der Quotus sei, aber nicht vollkommen, dann 4 Mal 3 macht nur 12 nicht 13, und ist also 1 der 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+{"basename":"euler_rechenkunst01_1738","par_idx":850,"date":1738,"orig":"Dieſer Unterſcheid iſt demnach der Reſt, welcher bey einer ſolchen Diuiſion zuruͤck bleibt.","norm":"Dieser Unterschied ist demnach der Rest, welcher bei einer solchen Division 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allezeit die naͤchſt kleinere Zahl fuͤr den Quotum zu nehmen, deswegen wird in dieſem Exempel 8 der Quotus ſeyn, weil aber 8 mahl 12 nur 96 macht, welche Zahl um 5 kleiner iſt als die gegebene 101, ſo iſt der Reſt 5. Jn ſolchen Exempeln iſt derowegen der angegebene Quotus ſo beſchaffen, daß wann man denſelben mit dem Diuiſore multiplicirt und zum Product den Reſt addirt, der Diuidendus herauskomme.","norm":"Ingleichem, wann 101 durch 12 dividieret werden soll, so sieht man dass 12 mehr als achtmal in 101 begriffen sein, aber weniger als 9 Mal; nun pflegt man allezeit die nächst kleinere Zahl für den Quotum zu nehmen, deswegen wird in diesem Exempel 8 der Quotus sein, weil aber 8 Mal 12 nur 96 macht, welche Zahl um 5 kleiner ist als die gegebene 101, so ist der Rest 5. In solchen Exempeln ist derowegen der angegebene Quotus so beschaffen, dass wann man denselben mit dem Divisor multipliziert und zum Produkt den Rest addiert, der Dividend 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+{"basename":"euler_rechenkunst01_1738","par_idx":861,"date":1738,"orig":"Hiebey iſt nun hauptſaͤchlich noͤthig den Unterſcheid zu bemercken zwiſchen denjenigen Exempeln, in welchen der wahre Quotus kan angegeben werden, und denjenigen, in welchen ein Reſt zuruͤck bleibt.","norm":"Hierbei ist nun hauptsächlich nötig den Unterschied zu bemerken zwischen denjenigen Exempeln, in welchen der wahre Quotus kann angegeben werden, und denjenigen, in welchen ein Rest 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+{"basename":"euler_rechenkunst01_1738","par_idx":862,"date":1738,"orig":"Was die Exempel der erſten Art anbetrifft da der wahre Quotus angegeben werden kan, dieſelben ſind aus der bey der Multiplication gegebenen Tabelle leicht zu erkennen, wann man nehmlich dieſelbe Tabelle dem Gedaͤchtniß wohl eingepraͤgt hat.","norm":"Was die Exempel der ersten Art anbetrifft da der wahre Quotus angegeben werden kann, dieselben sind aus der bei der Multiplikation gegebenen Tabelle leicht zu erkennen, wann man nämlich dieselbe Tabelle dem Gedächtnis wohl eingeprägt 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+{"basename":"euler_rechenkunst01_1738","par_idx":864,"date":1738,"orig":"Wir wollen aber dem ungeacht folgende Tabelle bey fuͤgen.","norm":"Wir wollen aber dem ungeachtet folgende Tabelle bei Fügen.","orig_tok":["Wir","wollen","aber","dem","ungeacht","folgende","Tabelle","bey","fuͤgen","."],"norm_tok":["Wir","wollen","aber","dem","ungeachtet","folgende","Tabelle","bei","Fügen","."],"orig_spans":[[0,3],[4,10],[11,15],[16,19],[20,28],[29,37],[38,45],[46,49],[50,56],[56,57]],"norm_spans":[[0,3],[4,10],[11,15],[16,19],[20,30],[31,39],[40,47],[48,51],[52,57],[57,58]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":6.947700023651123} +{"basename":"euler_rechenkunst01_1738","par_idx":865,"date":1738,"orig":"Aus dieſer Tabelle ſieht man alſo alle diejenige Faͤlle, in welchen ſo wohl der Diuiſor als der wahre Quotus einfache Zahlen oder kleiner ſind als 10.","norm":"Aus dieser 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+{"basename":"euler_rechenkunst01_1738","par_idx":866,"date":1738,"orig":"Und wer dieſe Tabelle wohl erlernet hat, derſelbe wird bey einem jeglichen vorkommenden Fall, der in dieſer Tabelle begriffen iſt, den wahren Quotum gleich ſagen koͤnnen.","norm":"Und wer diese Tabelle wohl erlernet hat, derselbe wird bei einem jeglichen vorkommenden Fall, der in dieser Tabelle begriffen ist, den wahren Quotum gleich sagen 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+{"basename":"euler_rechenkunst01_1738","par_idx":867,"date":1738,"orig":"Wann zum Exempel die Frage iſt wieviel mahl 7 in 56 enthalten ſey, ſo weißt derſelbe gleich, daß es 8 mahl ſey.","norm":"Wann zum Exempel die Frage ist wie viel Mal 7 in 56 enthalten sei, so weißt derselbe gleich, dass es 8 Mal 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+{"basename":"euler_rechenkunst01_1738","par_idx":868,"date":1738,"orig":"Wir haben aber in dieſer Tabelle diejenigen Faͤlle ausgelaſſen, in welchen der Diuiſor 1 iſt.","norm":"Wir haben aber in dieser Tabelle diejenigen Fälle ausgelassen, in welchen der Divisor 1 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anzeigt.","norm":"Dann 1 ist in einer jeglichen Zahl so viel Mal begriffen, als dieselbe Zahl selbst anzeigt.","orig_tok":["Dann","1","iſt","in","einer","jeglichen","Zahl","ſo","viel","mahl","begriffen",",","als","dieſelbe","Zahl","ſelbſt","anzeigt","."],"norm_tok":["Dann","1","ist","in","einer","jeglichen","Zahl","so","viel","Mal","begriffen",",","als","dieselbe","Zahl","selbst","anzeigt","."],"orig_spans":[[0,4],[5,6],[7,10],[11,13],[14,19],[20,29],[30,34],[35,37],[38,42],[43,47],[48,57],[57,58],[59,62],[63,71],[72,76],[77,83],[84,91],[91,92]],"norm_spans":[[0,4],[5,6],[7,10],[11,13],[14,19],[20,29],[30,34],[35,37],[38,42],[43,46],[47,56],[56,57],[58,61],[62,70],[71,75],[76,82],[83,90],[90,91]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.942999839782715} +{"basename":"euler_rechenkunst01_1738","par_idx":870,"date":1738,"orig":"Das iſt wann der Diuiſor 1 iſt, ſo iſt der Quotus allezeit dem Diuidendo gleich.","norm":"Das ist wann der Divisor 1 ist, so ist der Quotus allezeit dem Dividenden 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muß, ſo iſt klar, daß wann der Diuiſor 1 iſt, der Quotus dem Diuidendo gleich ſeyn muͤſſe.","norm":"Dieses sieht man aus der Multiplikation, dann weilen der Quotus mit dem Divisor multipliziert den Dividenden herausbringen muss, so ist klar, dass wann der Divisor 1 ist, der Quotus dem Dividenden gleich sein 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+{"basename":"euler_rechenkunst01_1738","par_idx":872,"date":1738,"orig":"Alſo wann zum Exempel23 durch 1 diuidirt werden ſoll, ſo iſt der Quotus 23, dann 23 mahl 1 macht 23. Daher pflegt man zu ſagen, daß eins nicht diuidire, weilen der Diuidendus ſelbſt den Quotum anzeigt.","norm":"Also wann zum Exempel23 durch 1 dividiert werden soll, so ist der Quotus 23, dann 23 Mal 1 macht 23. Daher pflegt man zu sagen, dass eins nicht dividiere, weilen der Dividend selbst den Quotum 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+{"basename":"euler_rechenkunst01_1738","par_idx":873,"date":1738,"orig":"Ferner erhellet auch, daß wann der Diuiſor dem Diuidendo gleich iſt, der Quotus allezeit 1 ſeyn muͤſſe, dann eine jegliche Zahl iſt in ſich ſelber ein mahl enthalten.","norm":"Ferner erhellet auch, dass wann der Divisor dem Dividenden gleich ist, der Quotus allezeit 1 sein müsse, dann eine jegliche Zahl ist in sich selber ein Mal 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+{"basename":"euler_rechenkunst01_1738","par_idx":874,"date":1738,"orig":"Endlich waͤre auch anzu mercken, daß wann der Diuiſor 0 iſt, der Quotus unendlich groß ſey: allein weil dieſer Fall bey gemeinen Diuiſionen nicht vorkommt, ſo iſt nicht noͤthig einem Anfaͤnger etwas von dem unendlichen vorzutragen.","norm":"Endlich wäre auch anzu merken, dass wann der Divisor 0 ist, der Quotus unendlich groß sei: allein weil dieser Fall bei gemeinen Divisionen nicht vorkommt, so ist nicht nötig einem Anfänger etwas von dem unendlichen 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+{"basename":"euler_rechenkunst01_1738","par_idx":875,"date":1738,"orig":"Wir ſchreiten derohalben fort zu den Exempeln der anderen Art, in welchen der wahre Quotus nicht kan in gantzen Zahlen angegeben werden, und bey welchen man ſich begnuͤgt den naͤchſten Quotum anzuzeigen, nebſt dem uͤberbleibenden Reſt.","norm":"Wir schreiten derohalben fort zu den Exempeln der anderen Art, in welchen der wahre Quotus nicht kann in ganzen Zahlen angegeben werden, und bei welchen man sich begnügt den nächsten Quotum anzuzeigen, nebst dem überbleibenden 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+{"basename":"euler_rechenkunst01_1738","par_idx":876,"date":1738,"orig":"Man ſieht nehmlich aus der vorigen Tabelle, daß die Zahlen in den zweyten Reihen von oben herab nicht in der Ordnung fortgehen, ſondern daß zwiſchen denſelben immer eine oder mehr Zahlen begriffen ſind.","norm":"Man sieht nämlich aus der vorigen Tabelle, dass die Zahlen in den zweiten Reihen von oben herab nicht in der Ordnung fortgehen, sondern dass zwischen denselben immer eine oder mehr Zahlen begriffen 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+{"basename":"euler_rechenkunst01_1738","par_idx":877,"date":1738,"orig":"Wann demnach eine ſolche Zahl, welche nicht in der Tabelle ſteht, ſondern zwiſchen dieſelben Zahlen hineingehoͤret, durch eine einfache Zahl diuidirt werden ſoll, ſo kan der wahre Quotus nicht gegeben werden, ſondern man muß die naͤchſt kleinere Zahl darfuͤr nehmen und den ruͤckſtehenden Reſt dabey anzeigen.","norm":"Wann demnach eine solche Zahl, welche nicht in der Tabelle steht, sondern zwischen dieselben Zahlen hineingehöret, durch eine einfache Zahl dividiert werden soll, so kann der wahre Quotus nicht gegeben werden, sondern man muss die nächst kleinere Zahl dafür nehmen und den rückstehenden Rest dabei anzeigen.","orig_tok":["Wann","demnach","eine","ſolche","Zahl",",","welche","nicht","in","der","Tabelle","ſteht",",","ſondern","zwiſchen","dieſelben","Zahlen","hineingehoͤret",",","durch","eine","einfache","Zahl","diuidirt","werden","ſoll",",","ſo","kan","der","wahre","Quotus","nicht","gegeben","werden",",","ſondern","man","muß","die","naͤchſt","kleinere","Zahl","darfuͤr","nehmen","und","den","ruͤckſtehenden","Reſt","dabey","anzeigen","."],"norm_tok":["Wann","demnach","eine","solche","Zahl",",","welche","nicht","in","der","Tabelle","steht",",","sondern","zwischen","dieselben","Zahlen","hineingehöret",",","durch","eine","einfache","Zahl","dividiert","werden","soll",",","so","kann","der","wahre","Quotus","nicht","gegeben","werden",",","sondern","man","muss","die","nächst","kleinere","Zahl","dafür","nehmen","und","den","rückstehenden","Rest","dabei","anzeigen","."],"orig_spans":[[0,4],[5,12],[13,17],[18,24],[25,29],[29,30],[31,37],[38,43],[44,46],[47,50],[51,58],[59,64],[64,65],[66,73],[74,82],[83,92],[93,99],[100,114],[114,115],[116,121],[122,126],[127,135],[136,140],[141,149],[150,156],[157,161],[161,162],[163,165],[166,169],[170,173],[174,179],[180,186],[187,192],[193,200],[201,207],[207,208],[209,216],[217,220],[221,224],[225,228],[229,236],[237,245],[246,250],[251,258],[259,265],[266,269],[270,273],[274,288],[289,293],[294,299],[300,308],[308,309]],"norm_spans":[[0,4],[5,12],[13,17],[18,24],[25,29],[29,30],[31,37],[38,43],[44,46],[47,50],[51,58],[59,64],[64,65],[66,73],[74,82],[83,92],[93,99],[100,113],[113,114],[115,120],[121,125],[126,134],[135,139],[140,149],[150,156],[157,161],[161,162],[163,165],[166,170],[171,174],[175,180],[181,187],[188,193],[194,201],[202,208],[208,209],[210,217],[218,221],[222,226],[227,230],[231,237],[238,246],[247,251],[252,257],[258,264],[265,268],[269,272],[273,286],[287,291],[292,297],[298,306],[306,307]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28],[29,29],[30,30],[31,31],[32,32],[33,33],[34,34],[35,35],[36,36],[37,37],[38,38],[39,39],[40,40],[41,41],[42,42],[43,43],[44,44],[45,45],[46,46],[47,47],[48,48],[49,49],[50,50],[51,51]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.42579984664917} +{"basename":"euler_rechenkunst01_1738","par_idx":878,"date":1738,"orig":"Dieſes geſchieht nun alſo, man ſucht in demjenigen Theil der Tabelle, in welchem der gegebene Diuiſor voraus ſteht, in der zweyten Reihen die dem Diuidendo naͤchſt kleinere Zahl, und zieht dieſelbe von dem Diuidendo ab, da dann der Reſt den zuruͤckbleibenden Reſt der Diuiſion anzeigt.","norm":"Dieses geschieht nun also, man sucht in demjenigen Teil der Tabelle, in welchem der gegebene Divisor voraus steht, in der zweiten Reihen die dem Dividenden nächst kleinere Zahl, und zieht dieselbe von dem Dividenden ab, da dann der Rest den zurückbleibenden Rest der Division 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+{"basename":"euler_rechenkunst01_1738","par_idx":879,"date":1738,"orig":"Die Zahl aber in der dritten Reihe, welche dabey ſteht gibt den Quotum.","norm":"Die Zahl aber in der dritten Reihe, welche dabei steht gibt den Quotum.","orig_tok":["Die","Zahl","aber","in","der","dritten","Reihe",",","welche","dabey","ſteht","gibt","den","Quotum","."],"norm_tok":["Die","Zahl","aber","in","der","dritten","Reihe",",","welche","dabei","steht","gibt","den","Quotum","."],"orig_spans":[[0,3],[4,8],[9,13],[14,16],[17,20],[21,28],[29,34],[34,35],[36,42],[43,48],[49,54],[55,59],[60,63],[64,70],[70,71]],"norm_spans":[[0,3],[4,8],[9,13],[14,16],[17,20],[21,28],[29,34],[34,35],[36,42],[43,48],[49,54],[55,59],[60,63],[64,70],[70,71]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":6.35230016708374} +{"basename":"euler_rechenkunst01_1738","par_idx":880,"date":1738,"orig":"Als wann die Frage iſt wieviel mahl 7 in 38 enthalten ſeyn, oder wann 38 durch 7 ſoll diu dirt werden, ſo ſieht man in demjenigen Theil da 7 in der erſten Reihe ſteht, daß 35, darinn ſieben 5 mahl enthalten iſt, die naͤchſt kleinere Zahl ſey als 38, und iſt der Reſt 3, ſo uͤberbleibt wann 35 von 38 abgezogen wird.","norm":"Als wann die Frage ist wie viel Mal 7 in 38 enthalten sein, oder wann 38 durch 7 soll diu dirt werden, so sieht man in demjenigen Teil da 7 in der ersten Reihe steht, dass 35, darin sieben 5 Mal enthalten ist, die nächst kleinere Zahl sei als 38, und ist der Rest 3, so überbleibt wann 35 von 38 abgezogen wird.","orig_tok":["Als","wann","die","Frage","iſt","wieviel","mahl","7","in","38","enthalten","ſeyn",",","oder","wann","38","durch","7","ſoll","diu","dirt","werden",",","ſo","ſieht","man","in","demjenigen","Theil","da","7","in","der","erſten","Reihe","ſteht",",","daß","35",",","darinn","ſieben","5","mahl","enthalten","iſt",",","die","naͤchſt","kleinere","Zahl","ſey","als","38",",","und","iſt","der","Reſt","3",",","ſo","uͤberbleibt","wann","35","von","38","abgezogen","wird","."],"norm_tok":["Als","wann","die","Frage","ist","wie","viel","Mal","7","in","38","enthalten","sein",",","oder","wann","38","durch","7","soll","diu","dirt","werden",",","so","sieht","man","in","demjenigen","Teil","da","7","in","der","ersten","Reihe","steht",",","dass","35",",","darin","sieben","5","Mal","enthalten","ist",",","die","nächst","kleinere","Zahl","sei","als","38",",","und","ist","der","Rest","3",",","so","überbleibt","wann","35","von","38","abgezogen","wird","."],"orig_spans":[[0,3],[4,8],[9,12],[13,18],[19,22],[23,30],[31,35],[36,37],[38,40],[41,43],[44,53],[54,58],[58,59],[60,64],[65,69],[70,72],[73,78],[79,80],[81,85],[86,89],[90,94],[95,101],[101,102],[103,105],[106,111],[112,115],[116,118],[119,129],[130,135],[136,138],[139,140],[141,143],[144,147],[148,154],[155,160],[161,166],[166,167],[168,171],[172,174],[174,175],[176,182],[183,189],[190,191],[192,196],[197,206],[207,210],[210,211],[212,215],[216,223],[224,232],[233,237],[238,241],[242,245],[246,248],[248,249],[250,253],[254,257],[258,261],[262,266],[267,268],[268,269],[270,272],[273,284],[285,289],[290,292],[293,296],[297,299],[300,309],[310,314],[314,315]],"norm_spans":[[0,3],[4,8],[9,12],[13,18],[19,22],[23,26],[27,31],[32,35],[36,37],[38,40],[41,43],[44,53],[54,58],[58,59],[60,64],[65,69],[70,72],[73,78],[79,80],[81,85],[86,89],[90,94],[95,101],[101,102],[103,105],[106,111],[112,115],[116,118],[119,129],[130,134],[135,137],[138,139],[140,142],[143,146],[147,153],[154,159],[160,165],[165,166],[167,171],[172,174],[174,175],[176,181],[182,188],[189,190],[191,194],[195,204],[205,208],[208,209],[210,213],[214,220],[221,229],[230,234],[235,238],[239,242],[243,245],[245,246],[247,250],[251,254],[255,258],[259,263],[264,265],[265,266],[267,269],[270,280],[281,285],[286,288],[289,292],[293,295],[296,305],[306,310],[310,311]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[20,21],[21,22],[22,23],[23,24],[24,25],[25,26],[26,27],[27,28],[28,29],[29,30],[30,31],[31,32],[32,33],[33,34],[34,35],[35,36],[36,37],[37,38],[38,39],[39,40],[40,41],[41,42],[42,43],[43,44],[44,45],[45,46],[46,47],[47,48],[48,49],[49,50],[50,51],[51,52],[52,53],[53,54],[54,55],[55,56],[56,57],[57,58],[58,59],[59,60],[60,61],[61,62],[62,63],[63,64],[64,65],[65,66],[66,67],[67,68],[68,69],[69,70]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.17609977722168} 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Wann man obige Tabelle wohl im Gedaͤchtniß hat, ſo ſieht man gleich wieviel mahl man den Diuiſorem nehmen muͤſſe, daß die naͤchſt kleinere Zahl als der Diuidendus iſt herauskomme.","norm":"Derohalben ist der Quotus 5 und der Rest 3, wann38 durch 7 dividiert wird; dann 5 Mal 7 ist 35 und dazu der Rest 3 getan macht 38. Wann man obige Tabelle wohl im Gedächtnis hat, so sieht man gleich wie viel Mal man den Divisor nehmen müsse, dass die nächst kleinere Zahl als der Dividend ist 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+{"basename":"euler_rechenkunst01_1738","par_idx":899,"date":1738,"orig":"Was nun in dieſem Exempel von den Rubl. iſt geſagt worden, verſteht ſich gleichermaſſen bey einer jeglichen Benennung, welche der Diuidendus fuͤhrt.","norm":"Was nun in diesem Exempel von den Rubel . ist gesagt worden, versteht sich gleichermaßen bei einer jeglichen Benennung, welche der Dividend 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+{"basename":"euler_rechenkunst01_1738","par_idx":900,"date":1738,"orig":"Und iſt alſo hieraus genugſam klar, daß der Quotus und Reſt eben den Nahmen fuͤhren muͤſſen, welchen der Diuidendus hatte; weswegen man alſo um ſo viel weniger zu zweiflen hat, was die Benennungen als Decaden, Centanarios und ſo fort betrifft.","norm":"Und ist also hieraus genügsam klar, dass der Quotus und Rest eben den Namen führen müssen, welchen der Dividend hatte; weswegen man also um so viel weniger zu zweifeln hat, was die Benennungen als Dekaden, Centanarios und so fort 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+{"basename":"euler_rechenkunst01_1738","par_idx":902,"date":1738,"orig":"Alſo geben 69 Decades durch 8 Unitæten Diuidirt, 8 Decades fuͤr den Quotum und 5 Decades fuͤr den Reſt.","norm":"Also geben 69 Decades durch 8 Unitäten dividiert, 8 Decades für den Quotum und 5 Decades für den 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+{"basename":"euler_rechenkunst01_1738","par_idx":905,"date":1738,"orig":"Als wann 2400 durch 4 diuidirt werden ſollen, ſo ſage ich 2400 iſt ſo viel als 24 Centenarii, und diuidire alſo 24 Centenarios durch 4, und finde 6 Centenarios fuͤr den Quotum ohne Reſt.","norm":"Als wann 2400 durch 4 dividiert werden sollen, so sage ich 2400 ist so viel als 24 Centenarii, und dividiere also 24 Centenarios durch 4, und finde 6 Centenarios für den Quotum ohne Rest.","orig_tok":["Als","wann","2400","durch","4","diuidirt","werden","ſollen",",","ſo","ſage","ich","2400","iſt","ſo","viel","als","24","Centenarii",",","und","diuidire","alſo","24","Centenarios","durch","4",",","und","finde","6","Centenarios","fuͤr","den","Quotum","ohne","Reſt","."],"norm_tok":["Als","wann","2400","durch","4","dividiert","werden","sollen",",","so","sage","ich","2400","ist","so","viel","als","24","Centenarii",",","und","dividiere","also","24","Centenarios","durch","4",",","und","finde","6","Centenarios","für","den","Quotum","ohne","Rest","."],"orig_spans":[[0,3],[4,8],[9,13],[14,19],[20,21],[22,30],[31,37],[38,44],[44,45],[46,48],[49,53],[54,57],[58,62],[63,66],[67,69],[70,74],[75,78],[79,81],[82,92],[92,93],[94,97],[98,106],[107,111],[112,114],[115,126],[127,132],[133,134],[134,135],[136,139],[140,145],[146,147],[148,159],[160,164],[165,168],[169,175],[176,180],[181,185],[185,186]],"norm_spans":[[0,3],[4,8],[9,13],[14,19],[20,21],[22,31],[32,38],[39,45],[45,46],[47,49],[50,54],[55,58],[59,63],[64,67],[68,70],[71,75],[76,79],[80,82],[83,93],[93,94],[95,98],[99,108],[109,113],[114,116],[117,128],[129,134],[135,136],[136,137],[138,141],[142,147],[148,149],[150,161],[162,165],[166,169],[170,176],[177,181],[182,186],[186,187]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28],[29,29],[30,30],[31,31],[32,32],[33,33],[34,34],[35,35],[36,36],[37,37]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.884300231933594} 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Wann aber 46000 das iſt 46 Millenarii durch 7 diuidirt werden ſollen, ſo wird der Quotus ſeyn 6 Millenarii das iſt 6000, wobey 4 Millenarii reſtiren das iſt 4000 Unitæten, welche aber weiter durch 7 diuidirt werden koͤnnen, wovon im folgenden weiter gehandelt werden wird.","norm":"Ich sage deshalb dass der gesuchte Quotus sei 600. Wann aber 46000 das ist 46 Millenarii durch 7 dividiert werden sollen, so wird der Quotus sein 6 Millenarii das ist 6000, wobei 4 Millenarii restieren das ist 4000 Unitäten, welche aber weiter durch 7 dividiert werden können, wovon im folgenden weiter gehandelt werden 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+{"basename":"euler_rechenkunst01_1738","par_idx":907,"date":1738,"orig":"6)","norm":"6)","orig_tok":["6",")"],"norm_tok":["6",")"],"orig_spans":[[0,1],[1,2]],"norm_spans":[[0,1],[1,2]],"alignment":[[0,0],[1,1]],"lang_fastText":"es","lang_py3langid":"en","lang_cld3":"ja","orig_lang_de":0,"norm_lmscore":2.728699922561645} +{"basename":"euler_rechenkunst01_1738","par_idx":908,"date":1738,"orig":"Wann eine zuſammen geſetzte Zahl ſo groß dieſelbe immer ſeyn mag, durch eine einfache Zahl diuidirt werden ſoll, ſo muß man alle Theile derſelben, das iſt alle beſonderen Sorten, aus welchen dieſelbe Zahl beſtehet, durch den Diuiſorem diuidiren, wobey der Anfang von den groͤſten Sorten gemacht werden muß.","norm":"Wann eine zusammen gesetzte Zahl so groß dieselbe immer sein mag, durch eine einfache Zahl dividiert werden soll, so muss man alle Teile derselben, das ist alle besonderen Sorten, aus welchen dieselbe Zahl bestehet, durch den Divisor dividieren, wobei der Anfang von den größten Sorten gemacht werden 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+{"basename":"euler_rechenkunst01_1738","par_idx":909,"date":1738,"orig":"Der Reſt aber welcher bey einer jeglichen Sorte uͤber bleibt, wird in die folgende geringere Sorte verwandelt und zu derſelbigen Sorte hinzugeſetzt, und alſo mit der Diuiſion bis zu den Unitæten als der kleinſten Sarte fortgefahren:","norm":"Der Rest aber welcher bei einer jeglichen Sorte überbleibt, wird in die folgende geringere Sorte verwandelt und zu derselbigen Sorte hinzugesetzt, und also mit der Division bis zu den Unitäten als der kleinsten Sarte 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+{"basename":"euler_rechenkunst01_1738","par_idx":910,"date":1738,"orig":"da dann alle dieſe beſonderen Quoti zuſammen den geſuchten Quotum ausmachen, und was bey der letzten Diuiſion uͤbrig bleibt, iſt der ruͤckſtehende Reſt.","norm":"da dann alle diese besonderen Quoti zusammen den gesuchten Quotum ausmachen, und was bei der letzten Division übrig bleibt, ist der rückstehende 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+{"basename":"euler_rechenkunst01_1738","par_idx":911,"date":1738,"orig":"Gleich wie in der Multiplication das verlangte Product gefunden wird, wann man alle Theile des Multiplicandi mit dem Multiplicatore multiplicirt, und alle dieſe beſonderen Product zuſammen addirt; alſo findet man auch in der Diuiſion den geſuchten Quotum, wann man alle Theile des Diuidendi durch den Diuiſorem diuidirt und alle dieſe beſonderen Quotos zuſammen addirt.","norm":"Gleich wie in der Multiplikation das verlangte Produkt gefunden wird, wann man alle Teile des Multiplicandi mit dem Multiplikator multipliziert, und alle diese besonderen Produkt zusammen addiert; also findet man auch in der Division den gesuchten Quotum, wann man alle Teile des Dividenden durch den Divisor dividiert und alle diese besonderen Quotos zusammen 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+{"basename":"euler_rechenkunst01_1738","par_idx":912,"date":1738,"orig":"Dann da in der Diuiſion die Frage iſt, wieviel mahl der Diuiſor in dem Diuidendo enthalten ſey, ſo wird man dieſe geſuchte Zahl oder den Quotum anzeigen koͤnnen, wann man weiß, wieviel mahl der Diuiſor in einem jeglichen Theil des Diuidendi enthalten iſt, dann alle dieſe beſonderen Quoti zuſammen geben den gantzen geſuchten Quotum.","norm":"Dann da in der Division die Frage ist, wie viel Mal der Divisor in dem Dividenden enthalten sei, so wird man diese gesuchte Zahl oder den Quotum anzeigen können, wann man weiß, wie viel Mal der Divisor in einem jeglichen Teil des Dividenden enthalten ist, dann alle diese besonderen Quoti zusammen geben den ganzen gesuchten 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+{"basename":"euler_rechenkunst01_1738","par_idx":913,"date":1738,"orig":"Als wann zum Exempel 6903 durch 3 diuidirt werden ſoll, ſo ſind die Theile des Diuidendi 6 Millenarii, 9 Centenariii und 3 Unitæten.","norm":"Als wann zum Exempel 6903 durch 3 dividiert werden soll, so sind die Teile des Dividenden 6 Millenarii, 9 Centenariii und 3 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+{"basename":"euler_rechenkunst01_1738","par_idx":914,"date":1738,"orig":"Der erſte Theil nehmlich 6 Millenarii durch 3 diuidirt geben 2 Millenarios fur den Quotum.","norm":"Der erste Teil nämlich 6 Millenarii durch 3 dividiert geben 2 Millenarios für den 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im Quoto.","norm":"Der zweite Teil 9 Centenarii durch 3 dividiert geben 3 Centenarios im Quoto, und endlich 3 Unitäten durch 3 dividiert geben 1 Unität im 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+{"basename":"euler_rechenkunst01_1738","par_idx":916,"date":1738,"orig":"Alle dieſe Quoti zuſammen ſind nur 2 Millenarii 3 Centenarii und 1 Unitæt das iſt 2301 und dieſe Zahl iſt der geſuchte Quotus, welcher herauskommt wann 6903 durch 3 diuidirt wird, und bleibt kein Reſt zuruͤck.","norm":"Alle diese Quoti zusammen sind nur 2 Millenarii 3 Centenarii und 1 Unität das ist 2301 und diese Zahl ist der gesuchte Quotus, welcher herauskommt wann 6903 durch 3 dividiert wird, und bleibt kein Rest 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+{"basename":"euler_rechenkunst01_1738","par_idx":917,"date":1738,"orig":"Jn dieſem Exempel hat ſich zwar ein jeglich Theil des Diuidendi durch den Diuiſorem ohne Reſt diuidiren laſſen; allein aus demſelben iſt gleich wohl leicht zu ſchlieſſen, wie man ſich zu verhalten habe, wann bey dieſen beſonderen Diuiſionen etwas zuruͤck bleiben ſollte.","norm":"In diesem Exempel hat sich zwar ein jeglich Teil des Dividenden durch den Divisor ohne Rest dividieren lassen; allein aus demselben ist gleich wohl leicht zu schließen, wie man sich zu verhalten habe, wann bei diesen besonderen Divisionen etwas zurückbleiben 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+{"basename":"euler_rechenkunst01_1738","par_idx":918,"date":1738,"orig":"Dann da der Reſt, welcher in der Diuiſion eines Theils oder einer Sorte des Diuidendi durch den Diuiſorem zuruͤck bleibet, noch nicht diuidirt worden iſt, in dem man noch nicht gefunden, wieviel mahl der Diuiſor darinn enthalten iſt, ſo muß derſelbe Reſt in die folgende kleinere Sorte verwandelt, und zu derſelben geſetzet, und darauf dieſes zuſammen durch den Diuiſorem getheilet werden.","norm":"Dann da der Rest, welcher in der Division eines Teils oder einer Sorte des Dividenden durch den Divisor zurückbleibt, noch nicht dividiert worden ist, in dem man noch nicht gefunden, wie viel Mal der Divisor darin enthalten ist, so muss derselbe Rest in die folgende kleinere Sorte verwandelt, und zu derselben gesetzt, und darauf dieses zusammen durch den Divisor geteilt 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+{"basename":"euler_rechenkunst01_1738","par_idx":919,"date":1738,"orig":"Auf dieſe Art muß man alſo in der Diuiſion von den groͤſſeren Sorten des Diuidendi zu den kleineren fortfahren, biß man zu den Unitæten kommt, und wann dabey ein Reſt zuruͤck bleibt, ſo iſt derſelbe auch der wuͤrckliche Reſt, welcher nebſt dem Quoto muß angezeiget werden.","norm":"Auf diese Art muss man also in der Division von den größeren Sorten des Dividenden zu den kleineren fortfahren, bis man zu den Unitäten kommt, und wann dabei ein Rest zurückbleibt, so ist derselbe auch der wirkliche Rest, welcher nebst dem Quoto muss angezeigt 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+{"basename":"euler_rechenkunst01_1738","par_idx":920,"date":1738,"orig":"Als wann die Zahl 8359 durch 6 diuidirt werden ſoll, ſo muß von den 8 Millenariis, als der groͤſten Sorte des Diuidendi der Anfang gemacht werden.","norm":"Als wann die Zahl 8359 durch 6 dividiert werden soll, so muss von den 8 Millenariis, als der größten Sorte des Dividenden der Anfang gemacht 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+{"basename":"euler_rechenkunst01_1738","par_idx":922,"date":1738,"orig":"Damit nun dieſes geſchehen koͤnne, ſo werden daraus Centenarii gemacht, wodurch man alſo 20 Centenarios bekommt, hiezu aber die 3 Centenarii, welche im Diuidendo wuͤrcklich vorhanden ſind gethan machen 23 Centenarios; dieſe alſo durch 6 diuidirt geben 3 Centenarios fuͤr den Quotum, und 5 Centenarii bleiben fuͤr den Reſt.","norm":"Damit nun dieses geschehen könne, so werden daraus Centenarii gemacht, wodurch man also 20 Centenarios bekommt, hierzu aber die 3 Centenarii, welche im Dividenden wirklich vorhanden sind getan machen 23 Centenarios; diese also durch 6 dividiert geben 3 Centenarios für den Quotum, und 5 Centenarii bleiben für den 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+{"basename":"euler_rechenkunst01_1738","par_idx":923,"date":1738,"orig":"Dieſe 5 Centenarios verwandelt man nun in Decades das gibt 50 Decades, weilen aber 5 Decades im Diuidendo wuͤrcklich vorhanden ſind, ſo hat man 55 Decades durch 6 zu diuidiren, dieſe geben demnach 9 Decades zum Quoto und bleibt 1 Decas zuruͤck.","norm":"Diese 5 Centenarios verwandelt man nun in Decades das gibt 50 Decades, weilen aber 5 Decades im Dividenden wirklich vorhanden sind, so hat man 55 Decades durch 6 zu dividieren, diese geben demnach 9 Decades zum Quoto und bleibt 1 Decas 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+{"basename":"euler_rechenkunst01_1738","par_idx":924,"date":1738,"orig":"Dieſe 1 Decas macht 10 Unitæten, welche mit den 9 Unitæten des Diuidendi 19 Unitæten ausmachen, dieſe durch 6 diuidirt geben 3 Unitæten zum Quoto und 1 Unitæt bleibt als Reſt.","norm":"Diese 1 Decas macht 10 Unitäten, welche mit den 9 Unitäten des Dividenden 19 Unitäten ausmachen, diese durch 6 dividiert geben 3 Unitäten zum Quoto und 1 Unität bleibt als 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+{"basename":"euler_rechenkunst01_1738","par_idx":925,"date":1738,"orig":"Weilen nun die Unitæten nicht weiter in kleinere Sorten verwandelt werden koͤnnen, ſo bleibt alſo die 1 Unitæt wuͤrcklich zuruͤck und kan nicht getheilet werden.","norm":"Weilen nun die Unitäten nicht weiter in kleinere Sorten verwandelt werden können, so bleibt also die 1 Unität wirklich zurück und kann nicht geteilt 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+{"basename":"euler_rechenkunst01_1738","par_idx":933,"date":1738,"orig":"Hierauf ſagt man 6 in23 iſt 3 mahl enthalten und blejben 5 zuruͤck, die 3 ſchreibt man unter die Linie nach der vorhergehenden Zahl der Reſt 5 aber uͤber das 3, welcher mit den 5 Decaden des Diuidendi 55 ausmacht.","norm":"Hierauf sagt man 6 in23 ist 3 Mal enthalten und bleiben 5 zurück, die 3 schreibt man unter die Linie nach der vorhergehenden Zahl der Rest 5 aber über das 3, welcher mit den 5 Dekaden des Dividenden 55 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+{"basename":"euler_rechenkunst01_1738","par_idx":934,"date":1738,"orig":"Man ſagt alſo ferner 6 in55 iſt 9 mahl enthalten und bleibt 1 uͤber; man ſchreibt alſo 9 unter die Linie und den Reſt 1 uͤber die Decaden nehmlich uͤber 5. 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+{"basename":"euler_rechenkunst01_1738","par_idx":936,"date":1738,"orig":"Auf ſolche Art ſind nun folgende Exempel ausgerechnet worden.","norm":"Auf solche Art sind nun folgende Exempel ausgerechnet worden.","orig_tok":["Auf","ſolche","Art","ſind","nun","folgende","Exempel","ausgerechnet","worden","."],"norm_tok":["Auf","solche","Art","sind","nun","folgende","Exempel","ausgerechnet","worden","."],"orig_spans":[[0,3],[4,10],[11,14],[15,19],[20,23],[24,32],[33,40],[41,53],[54,60],[60,61]],"norm_spans":[[0,3],[4,10],[11,14],[15,19],[20,23],[24,32],[33,40],[41,53],[54,60],[60,61]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.948500156402588} +{"basename":"euler_rechenkunst01_1738","par_idx":937,"date":1738,"orig":"[Formel]","norm":"[Formel]","orig_tok":["[Formel]"],"norm_tok":["[","Formel","]"],"orig_spans":[[0,8]],"norm_spans":[[0,1],[1,7],[7,8]],"alignment":[[0,0],[0,1],[0,2]],"lang_fastText":"sh","lang_py3langid":"en","lang_cld3":"cy","orig_lang_de":0,"norm_lmscore":7.932600021362305} +{"basename":"euler_rechenkunst01_1738","par_idx":938,"date":1738,"orig":"Bey dem erſten dieſer Exempel iſt zu erinneren, daß weilen die erſte Figur von der lincken des Diuidendi nehmlich 1 kleiner iſt als der Diuiſor und alſo eine 0 in den Quotum gegen der lincken geſetzt werden muͤßte, welche keine Bedeutung hat, ſo wird dieſes 1 gleich zur folgenden Sorte gethan, welches 13 ausmacht und dabey die Diuiſion angefangen.","norm":"Bei dem ersten dieser Exempel ist zu erinnern, dass weilen die erste Figur von der Linken des Dividenden nämlich 1 kleiner ist als der Divisor und also eine 0 in den Quotum gegen der Linken gesetzt werden müsste, 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+{"basename":"euler_rechenkunst01_1738","par_idx":939,"date":1738,"orig":"Eine gleiche Bewandnuͤß hat es auch mit dem anderen Exempel, in welchem man gleich34 durch 8 zu diuidiren anfaͤngt.","norm":"Eine gleiche Bewandtnis hat es auch mit dem anderen Exempel, in welchem man gleich34 durch 8 zu dividieren 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+{"basename":"euler_rechenkunst01_1738","par_idx":940,"date":1738,"orig":"Wann aber mitten oder zum Ende des Quoti eine 0 kommt, ſo muß dieſelbe nothwendig geſchrieben werden, damit eine jede Figur auf ihre gehoͤrige Stelle komme.","norm":"Wann aber mitten oder zum Ende des Quoti eine 0 kommt, so muss dieselbe notwendig geschrieben werden, damit eine jede Figur auf ihre gehörige Stelle 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+{"basename":"euler_rechenkunst01_1738","par_idx":941,"date":1738,"orig":"Dieſer Fall kommt im erſten Exempel vor, welches auf folgende Weiſe operirt wird: 4 in13 iſt 3 mahl enthalten und bleibt 1 uͤber, ſchreibt 3 unter| die Linie und 1 uͤber das 3 im Diuidendo.","norm":"Dieser Fall kommt im ersten Exempel vor, welches auf folgende Weise operiert wird: 4 in13 ist 3 Mal enthalten und bleibt 1 über, schreibt 3 unter | die Linie und 1 über das 3 im 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+{"basename":"euler_rechenkunst01_1738","par_idx":942,"date":1738,"orig":"Ferner ſagt man 4 in16 iſt 4 mahl enthalten und bleibt nichts uͤber, ſchreibt alſo 4 unter die Linie, und weil kein Reſt vorhanden, ſagt man 4 in 2 iſt kein mahl enthalten, ſetzt alſo 0 in den Quotum, und weilen die 2 wuͤrcklich der Reſt ſind, ſo nimmt man dieſelben gleich mit der folgenden 8 zuſammen, das gibt 28 darinn 4, 7 mahl begriffen iſt, und kein Reſt zuruͤck bleibt; ſo daß alſo der Quotus iſt 3407.","norm":"Ferner sagt man 4 in16 ist 4 Mal enthalten und bleibt nichts über, schreibt also 4 unter die Linie, und weil kein Rest vorhanden, sagt man 4 in 2 ist kein Mal enthalten, setzt also 0 in den Quotum, und weilen die 2 wirklich der Rest sind, so nimmt man dieselben gleich mit der folgenden 8 zusammen, das gibt 28 darin 4, 7 Mal begriffen ist, und kein Rest zurückbleibt; so dass also der Quotus ist 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+{"basename":"euler_rechenkunst01_1738","par_idx":945,"date":1738,"orig":"Dann man hat nur noͤthig von dem Diuiſore die Cyphren, und von dem Diuidendo auch eben ſo viel Figuren von der rechten Hand weg zu ſchmeiſſen, und ſo dann dieſen herausgekommenen Diuidendum durch den einfachen Diuiſorem zu diuidiren, da man dann den wahren Quotum bekommen wird.","norm":"Dann man hat nur nötig von dem Divisor die Ziffern, und von dem Dividenden auch ebenso viel Figuren von der rechten Hand weg zu schmeißen, und so dann diesen herausgekommenen Dividenden durch den einfachen Divisor zu dividieren, da man dann den wahren Quotum bekommen 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+{"basename":"euler_rechenkunst01_1738","par_idx":946,"date":1738,"orig":"Zu dem Reſt aber, der uͤberbleibt, muß man die von dem Diuidendo abgeſchnittenen Figuren von der rechten Hand hinzuſetzen, ſo wird man den wahren Reſt haben.","norm":"Zu dem Rest aber, der überbleibt, muss man die von dem Dividenden abgeschnittenen Figuren von der rechten Hand hinzusetzen, so wird man den wahren Rest 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+{"basename":"euler_rechenkunst01_1738","par_idx":947,"date":1738,"orig":"Um dieſe Operation deutlicher vorzuſtellen ſo laſſt uns dieſe Zahl 156327 durch 700 diuidiren.","norm":"Um diese Operation deutlicher vorzustellen so lasst uns diese Zahl 156327 durch 700 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folgt.","norm":"Wir schmeißen also von 700 die zwei Ziffern und von dem Dividenden 156327 die zwei letzten Figuren 27 weg, und dividieren 1563 durch 7 wie folgt.","orig_tok":["Wir","ſchmeiſſen","alſo","von","700","die","zwey","Cyphren","und","von","dem","Diuidendo","156327","die","zwey","letſten","Figuren","27","weg",",","und","diuidiren","1563","durch","7","wie","folgt","."],"norm_tok":["Wir","schmeißen","also","von","700","die","zwei","Ziffern","und","von","dem","Dividenden","156327","die","zwei","letzten","Figuren","27","weg",",","und","dividieren","1563","durch","7","wie","folgt","."],"orig_spans":[[0,3],[4,14],[15,19],[20,23],[24,27],[28,31],[32,36],[37,44],[45,48],[49,52],[53,56],[57,66],[67,73],[74,77],[78,82],[83,90],[91,98],[99,101],[102,105],[105,106],[107,110],[111,120],[121,125],[126,131],[132,133],[134,137],[138,143],[143,144]],"norm_spans":[[0,3],[4,13],[14,18],[19,22],[23,26],[27,30],[31,35],[36,43],[44,47],[48,51],[52,55],[56,66],[67,73],[74,77],[78,82],[83,90],[91,98],[99,101],[102,105],[105,106],[107,110],[111,121],[122,126],[127,132],[133,134],[135,138],[139,144],[144,145]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.997099876403809} +{"basename":"euler_rechenkunst01_1738","par_idx":949,"date":1738,"orig":"[Formel]","norm":"[Formel]","orig_tok":["[Formel]"],"norm_tok":["[","Formel","]"],"orig_spans":[[0,8]],"norm_spans":[[0,1],[1,7],[7,8]],"alignment":[[0,0],[0,1],[0,2]],"lang_fastText":"sh","lang_py3langid":"en","lang_cld3":"cy","orig_lang_de":0,"norm_lmscore":7.932600021362305} +{"basename":"euler_rechenkunst01_1738","par_idx":950,"date":1738,"orig":"Auf dieſe Art haben wir alſo fuͤr den geſuchten Quotum 223 gefunden.","norm":"Auf diese Art haben wir also für den gesuchten Quotum 223 gefunden.","orig_tok":["Auf","dieſe","Art","haben","wir","alſo","fuͤr","den","geſuchten","Quotum","223","gefunden","."],"norm_tok":["Auf","diese","Art","haben","wir","also","für","den","gesuchten","Quotum","223","gefunden","."],"orig_spans":[[0,3],[4,9],[10,13],[14,19],[20,23],[24,28],[29,33],[34,37],[38,47],[48,54],[55,58],[59,67],[67,68]],"norm_spans":[[0,3],[4,9],[10,13],[14,19],[20,23],[24,28],[29,32],[33,36],[37,46],[47,53],[54,57],[58,66],[66,67]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.705100059509277} +{"basename":"euler_rechenkunst01_1738","par_idx":951,"date":1738,"orig":"Der Reſt aber iſt nicht 2 ſondern 227; indem zu dem gefundenen Reſt 2 die abgeſchnittenen zwey Figuren 27 von dem Diuidendo ſind angehaͤngt worden.","norm":"Der Rest aber ist nicht 2 sondern 227; indem zu dem gefundenen Rest 2 die abgeschnittenen zwei Figuren 27 von dem Dividenden sind angehängt 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+{"basename":"euler_rechenkunst01_1738","par_idx":952,"date":1738,"orig":"Wann man alſo die Zahl 156327 durch 700 diuidirt ſo kommt fuͤr den Quotum heraus 223, fuͤr den Reſt aber 227, wovon die Wahrheit gleich erhellet, wann man den Quotum 223 mit dem Diuiſore 700 multiplicirt und zum Product 227 hinzuthut, da dann der vorgegebene Diuidendus 156327 herauskommt.","norm":"Wann man also die Zahl 156327 durch 700 dividiert so kommt für den Quotum heraus 223, für den Rest aber 227, wovon die Wahrheit gleich erhellet, wann man den Quotum 223 mit dem Divisor 700 multipliziert und zum Produkt 227 hinzutut, da dann der vorgegebene Dividend 156327 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+{"basename":"euler_rechenkunst01_1738","par_idx":953,"date":1738,"orig":"Der Grund aber von dieſer Operation beſtehet darinn; daß man immer einerley Quotum findet, wann man den Diuiſorem und den Diuidendum beyde mit einerley Zahl multiplicirt.","norm":"Der Grund aber von dieser Operation bestehet darin; dass man immer einerlei Quotum findet, wann man den Divisor und den Dividenden beide mit einerlei Zahl 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+{"basename":"euler_rechenkunst01_1738","par_idx":954,"date":1738,"orig":"Als wann man den Diuidendum und den Diuiſorem beyde mit 10 oder mit 100 oder mit 1000 oder mit einer jeglichen anderen beliebten Zahl multiplicirt, ſo wird man immer eben denſelben Quotum finden, der herauskommt wann man den bloſſen Diuidendum durch den bloſſen Diuiſorem diuidirt.","norm":"Als wann man den Dividenden und den Divisor beide mit 10 oder mit 100 oder mit 1000 oder mit einer jeglichen anderen beliebten Zahl multipliziert, so wird man immer eben denselben Quotum finden, der herauskommt wann man den bloßen Dividenden durch den bloßen Divisor 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+{"basename":"euler_rechenkunst01_1738","par_idx":955,"date":1738,"orig":"Dann da der Quotus mit dem Diuiſore multiplicirt den Diuidendum herausbringt, ſo muß eben der Quotus mit einem 10 mahl groͤſſeren Diuiſore multiplicirt einen 10 mahl groͤſſeren Diuidendum, mit einem 100 mahl groͤſſeren Diuiſore aber einen 100 mahl groͤſſeren Diuidendum und ſo fort herfuͤrbringen, wie aus der Multiplication genugſam bekannt iſt.","norm":"Dann da der Quotus mit dem Divisor multipliziert den Dividenden herausbringt, so muss eben der Quotus mit einem 10 Mal größeren Divisor multipliziert einen 10 Mal größeren Dividenden, mit einem 100 Mal größeren Divisor aber einen 100 Mal größeren Dividenden und so fort hervorbringen, wie aus der Multiplikation genügsam bekannt 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+{"basename":"euler_rechenkunst01_1738","par_idx":956,"date":1738,"orig":"Da nun in dem gegebenen Exempel 1563 durch 7 diuidirt 223 fuͤr den Quotum gibt, 2 aber fuͤr den Reſt, ſo muß 100 mahl 1563 das iſt 156300 durch 100 mahl 7 das iſt durch 700 diuidirt eben den Quotum nehmlich 223 geben.","norm":"Da nun in dem gegebenen Exempel 1563 durch 7 dividiert 223 für den Quotum gibt, 2 aber für den Rest, so muss 100 Mal 1563 das ist 156300 durch 100 Mal 7 das ist durch 700 dividiert eben den Quotum nämlich 223 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+{"basename":"euler_rechenkunst01_1738","par_idx":957,"date":1738,"orig":"Der Reſt aber, welcher ein Theil des Diuidendi iſt, ſo ſich nicht weiter durch den Diuiſorem diuidiren laͤſſt, wird folglich auch 100 mahl groͤſſer und alſo 200 ſeyn.","norm":"Der Rest aber, welcher ein Teil des Dividenden ist, so sich nicht weiter durch den Divisor dividieren lässt, wird folglich auch 100 Mal größer und also 200 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+{"basename":"euler_rechenkunst01_1738","par_idx":958,"date":1738,"orig":"Derowegen wann man 156300 durch 700 diuidirt ſo wird der Quotus 223 ſeyn, der Reſt aber 200.","norm":"Derowegen wann man 156300 durch 700 dividiert so wird der Quotus 223 sein, der Rest aber 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ſich dieſe 27 durch den Diuiſorem nicht diuidiren laſſen, ſo kommen dieſe 27, wann man 156327 durch 700 diuidirt, noch mit zu dem Reſt, ſo daß in dieſem Fall der Quotus 223 bleibt, der Reſt aber um 27 groͤſſer und folglich 227 ſeyn wird.","norm":"Da nun 156327 nur um 27 größer ist als 156300 und sich diese 27 durch den Divisor nicht dividieren lassen, so kommen diese 27, wann man 156327 durch 700 dividiert, noch mit zu dem Rest, so dass in diesem Fall der Quotus 223 bleibt, der Rest aber um 27 größer und folglich 227 sein 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+{"basename":"euler_rechenkunst01_1738","par_idx":972,"date":1738,"orig":"Hierauf ſieht man wieviel mahl der Diuiſor in dieſer abgeſchnittenen Zahl enthalten iſt, und dieſe Anzahl gibt die erſte Figur von der lincken Hand in den Quotum.","norm":"Hierauf sieht man wie viel Mal der Divisor in dieser abgeschnittenen Zahl enthalten ist, und diese Anzahl gibt die erste Figur von der linken Hand in den 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+{"basename":"euler_rechenkunst01_1738","par_idx":979,"date":1738,"orig":"Nur beſtehet der Unterſcheid darinn, daß mit einem einfachen Diuiſore viel Operationen im Sinne vollbracht werden koͤnnen, welche bey einem zuſammen geſetzten Diuiſore wuͤrcklich auf dem Papier geſchehen muͤſſen.","norm":"Nur bestehet der Unterschied darin, dass mit einem einfachen Divisor viel Operationen im Sinne vollbracht werden können, welche bei einem zusammen gesetzten Divisor wirklich auf dem Papier geschehen 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+{"basename":"euler_rechenkunst01_1738","par_idx":980,"date":1738,"orig":"Als wann man bey einem einfachen Diuiſore eine jegliche Figur des Quoti mit dem Diuiſore multiplicirt, und das Product von dem gehoͤrigen Theil des Diuidendi abzieht, ſo geſchieht beydes im Kopf, welche beyden Operationen aber, wann der Diuiſor eine groſſe Zahl iſt, auf dem Papier wuͤrcklich berechnet werden muͤſſen.","norm":"Als wann man bei einem einfachen Divisor eine jegliche Figur des Quoti mit dem Divisor multipliziert, und das Produkt von dem gehörigen Teil des Dividenden abzieht, so geschieht beides im Kopf, welche beiden Operationen aber, wann der Divisor eine große Zahl ist, auf dem Papier wirklich berechnet werden 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+{"basename":"euler_rechenkunst01_1738","par_idx":981,"date":1738,"orig":"Dieſes wird nun deutlicher aus dem folgenden Exempel zu ſehen ſeyn, in welchem wir 178093 durch 23 diuidiren wollen.","norm":"Dieses wird nun deutlicher aus dem folgenden Exempel zu sehen sein, in welchem wir 178093 durch 23 dividieren 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+{"basename":"euler_rechenkunst01_1738","par_idx":982,"date":1738,"orig":"Dieſes Exempel pflegt nun erſtlich ſolcher geſtalt geſchrieben zu werden.","norm":"Dieses Exempel pflegt nun erstlich solcher Gestalt geschrieben zu werden.","orig_tok":["Dieſes","Exempel","pflegt","nun","erſtlich","ſolcher","geſtalt","geſchrieben","zu","werden","."],"norm_tok":["Dieses","Exempel","pflegt","nun","erstlich","solcher","Gestalt","geschrieben","zu","werden","."],"orig_spans":[[0,6],[7,14],[15,21],[22,25],[26,34],[35,42],[43,50],[51,62],[63,65],[66,72],[72,73]],"norm_spans":[[0,6],[7,14],[15,21],[22,25],[26,34],[35,42],[43,50],[51,62],[63,65],[66,72],[72,73]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":6.793600082397461} +{"basename":"euler_rechenkunst01_1738","par_idx":983,"date":1738,"orig":"[Formel]","norm":"[Formel]","orig_tok":["[Formel]"],"norm_tok":["[","Formel","]"],"orig_spans":[[0,8]],"norm_spans":[[0,1],[1,7],[7,8]],"alignment":[[0,0],[0,1],[0,2]],"lang_fastText":"sh","lang_py3langid":"en","lang_cld3":"cy","orig_lang_de":0,"norm_lmscore":7.932600021362305} +{"basename":"euler_rechenkunst01_1738","par_idx":984,"date":1738,"orig":"Wann man nun den Diuiſor als eine einfache Zahl anſieht, und die Diuiſion auf die vorher gelehrte Art anſtellen will, ſo muß man anfanglich die 3 erſten Figuren des Diuidendi nehmlich 178 zuſammen nehmen, und dieſelben durch 23 diuidiren, weilen die erſte nehmlich 1 und die zwey erſten 17 allein kleiner ſind als der Diuiſor, und ſich alſo durch denſelben nicht diuidiren laſſen.","norm":"Wann man nun den Divisor als eine einfache Zahl ansieht, und die Division auf die vorher gelehrte Art anstellen will, so muss man anfanglich die 3 ersten Figuren des Dividenden 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+{"basename":"euler_rechenkunst01_1738","par_idx":985,"date":1738,"orig":"Derowegen muß man ſuchen, wieviel mahl 23 in 178 begriffen iſt, und was uͤberbleibt; welches fuͤr den Anfang durch das probiren geſchehen muß, ehe wir darzu einige Regeln geben konnen.","norm":"Derowegen muss man suchen, wie viel Mal 23 in 178 begriffen ist, und was überbleibt; welches für den Anfang durch das Probieren geschehen muss, ehe wir dazu einige Regeln geben 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+{"basename":"euler_rechenkunst01_1738","par_idx":986,"date":1738,"orig":"Nun aber iſt leicht zu ſehen daß 23 in 178 nicht mehr als 7 mahl enthalten iſt, weilen 8 mahl 23 ſchon 184 das iſt mehr als 178 ausmacht.","norm":"Nun aber ist leicht zu sehen dass 23 in 178 nicht mehr als 7 Mal enthalten ist, weilen 8 Mal 23 schon 184 das ist mehr als 178 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+{"basename":"euler_rechenkunst01_1738","par_idx":988,"date":1738,"orig":"Nun 23 mahl 7 Millenarii macht 161 Millenarios, welche von den 178 Millenariis abgezogen 17 Millenarios zuruͤck laſſen; dieſe Subtraction wird nun wuͤrcklich auf dem Papier verrichtet.","norm":"Nun 23 Mal 7 Millenarii macht 161 Millenarios, welche von den 178 Millenariis abgezogen 17 Millenarios zurücklassen; diese Subtraktion wird nun wirklich auf dem Papier 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+{"basename":"euler_rechenkunst01_1738","par_idx":995,"date":1738,"orig":"Dieſe 4 ſind nun Decades und kommen in den Quotum auf die Stelle der Decaden, 23 mahl 4 Decaden aber machen 92 Decaden, welche von den 99 abgezogen 7 Decaden zuruͤck laſſen.","norm":"Diese 4 sind nun Decades und kommen in den Quotum auf die Stelle der Dekaden, 23 Mal 4 Dekaden aber machen 92 Dekaden, welche von den 99 abgezogen 7 Dekaden 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+{"basename":"euler_rechenkunst01_1738","par_idx":996,"date":1738,"orig":"Dieſe 7 Decaden machen endlich 70 Unitæten, welche mit den 3 Unitæten des Diuidendi 73 Unitæten betragen; oder man hat nur noͤthig zu den uͤbergebliebenen 7 die 3 hinzuzuſchreiben.","norm":"Diese 7 Dekaden machen endlich 70 Unitäten, welche mit den 3 Unitäten des Dividenden 73 Unitäten betragen; oder man hat nur nötig zu den übergebliebenen 7 die 3 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+{"basename":"euler_rechenkunst01_1738","par_idx":997,"date":1738,"orig":"Jn 73 iſt endlich 23 nur 3 mahl enthalten, welche 3 Unitæten ſind, und alſo im Quoto auf die letzte Stelle geſchrieben werden muͤſſen.","norm":"In 73 ist endlich 23 nur 3 Mal enthalten, welche 3 Unitäten sind, und also im Quoto auf die letzte Stelle geschrieben werden 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+{"basename":"euler_rechenkunst01_1738","par_idx":998,"date":1738,"orig":"Weilen aber 3 mahl 23 nur 69 ausmacht, ſo muͤſſen dieſe 69 von den 73 abgezogen werden, da dann der Reſt 4 der wahre Reſt iſt, welcher in dieſer Diuiſion zuruͤck bleibt; ſo daß alſo der gefundene Quotus iſt 7743 und der Reſt 4.","norm":"Weilen aber 3 Mal 23 nur 69 ausmacht, so müssen diese 69 von den 73 abgezogen werden, da dann der Rest 4 der wahre Rest ist, welcher in dieser Division zurückbleibt; so dass also der gefundene Quotus ist 7743 und der Rest 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+{"basename":"euler_rechenkunst01_1738","par_idx":999,"date":1738,"orig":"Aus dieſem Exempel ſind nun die Operationen leicht zu erſehen, welche bey dergleichen Diuiſionen vorgenommen werden muͤſſen.","norm":"Aus diesem Exempel sind nun die Operationen leicht zu ersehen, welche bei dergleichen Divisionen vorgenommen werden 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+{"basename":"euler_rechenkunst01_1738","par_idx":1000,"date":1738,"orig":"Um dieſelben aber mit deſtoweniger Muͤhe anzuſtellen wollen wir nachfolgende Regeln an die Hand geben, welcher Grund aus dem angefuͤhrten leicht folget.","norm":"Um dieselben aber mit desto weniger Mühe anzustellen wollen wir nachfolgende Regeln an die Hand geben, welcher Grund aus dem angeführten leicht 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+{"basename":"euler_rechenkunst01_1738","par_idx":1001,"date":1738,"orig":"I. Wann erſtlich die Frage iſt, wieviel mahl der Diuiſor in einem jeglichen Theil des Diuidendi enthalten iſt, durch welche Operation wie in dem vorigen Exempel zu ſehen ein jeglicher Theil des Quoti gefunden wird, ſo iſt zu wiſſen, daß der Diuiſor auf das hoͤchſte 9 mahl darinn enthalten ſeyn koͤnne, weilen durch eine ſolche Operation eine Zahl in den Quotum kommt, welche nicht groͤſſer ſeyn kan als 9. Derowegen wuͤrde man auch mit dem probieren nicht viel Zeit verlieren, wann man den Diuiſorem mit allen einfachen Zahlen multipliciren wollte, damit man ſo gleich ſehen koͤnnte, welches Product am naͤchſten komme.","norm":"I. Wann erstlich die Frage ist, wie viel Mal der Divisor in einem jeglichen Teil des Dividenden enthalten ist, durch welche Operation wie in dem vorigen Exempel zu sehen ein jeglicher Teil des Quoti gefunden wird, so ist zu wissen, dass der Divisor auf das höchste 9 Mal darin enthalten sein könne, weilen durch eine solche Operation eine Zahl in den Quotum kommt, welche nicht größer sein kann als 9. Derowegen würde man auch mit dem probieren nicht viel Zeit verlieren, wann man den Divisor mit allen einfachen Zahlen multiplizieren wollte, damit man so gleich sehen könnte, welches Produkt am nächsten 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+{"basename":"euler_rechenkunst01_1738","par_idx":1005,"date":1738,"orig":"Hernach ſtellt man ſich die erſte Figur um eins groͤſſer vor, und ſieht wiederum, wieviel mahl dieſer Diuiſor in derſelben Zahl enthalten ſey.","norm":"Hernach stellt man sich die erste Figur um eins größer vor, und sieht wiederum, wie viel Mal dieser Divisor in derselben Zahl enthalten 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+{"basename":"euler_rechenkunst01_1738","par_idx":1008,"date":1738,"orig":"Mit dieſem Quoto probirt man nun die Operation, und wann derſelbe noch entweder zu groß oder zu klein gefunden wird, ſo muß man es mit einem kleineren oder groͤſſeren probiren.","norm":"Mit diesem Quoto probiert man nun die Operation, und wann derselbe noch entweder zu groß oder zu klein gefunden wird, so muss man es mit einem kleineren oder größeren 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+{"basename":"euler_rechenkunst01_1738","par_idx":1009,"date":1738,"orig":"Als in dem vorhergehenden Exempel, da die Frage war, wieviel mahl 23 in 178 enthalten ſey, ſo diuidire man erſtlich 178 durch 20 oder 17 durch 2, und dann 178 durch 30 oder 17 durch 3.","norm":"Als in dem vorhergehenden Exempel, da die Frage war, wie viel Mal 23 in 178 enthalten sei, so dividiere man erstlich 178 durch 20 oder 17 durch 2, und dann 178 durch 30 oder 17 durch 3.","orig_tok":["Als","in","dem","vorhergehenden","Exempel",",","da","die","Frage","war",",","wieviel","mahl","23","in","178","enthalten","ſey",",","ſo","diuidire","man","erſtlich","178","durch","20","oder","17","durch","2",",","und","dann","178","durch","30","oder","17","durch","3","."],"norm_tok":["Als","in","dem","vorhergehenden","Exempel",",","da","die","Frage","war",",","wie","viel","Mal","23","in","178","enthalten","sei",",","so","dividiere","man","erstlich","178","durch","20","oder","17","durch","2",",","und","dann","178","durch","30","oder","17","durch","3."],"orig_spans":[[0,3],[4,6],[7,10],[11,25],[26,33],[33,34],[35,37],[38,41],[42,47],[48,51],[51,52],[53,60],[61,65],[66,68],[69,71],[72,75],[76,85],[86,89],[89,90],[91,93],[94,102],[103,106],[107,115],[116,119],[120,125],[126,128],[129,133],[134,136],[137,142],[143,144],[144,145],[146,149],[150,154],[155,158],[159,164],[165,167],[168,172],[173,175],[176,181],[182,183],[183,184]],"norm_spans":[[0,3],[4,6],[7,10],[11,25],[26,33],[33,34],[35,37],[38,41],[42,47],[48,51],[51,52],[53,56],[57,61],[62,65],[66,68],[69,71],[72,75],[76,85],[86,89],[89,90],[91,93],[94,103],[104,107],[108,116],[117,120],[121,126],[127,129],[130,134],[135,137],[138,143],[144,145],[145,146],[147,150],[151,155],[156,159],[160,165],[166,168],[169,173],[174,176],[177,182],[183,185]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[20,21],[21,22],[22,23],[23,24],[24,25],[25,26],[26,27],[27,28],[28,29],[29,30],[30,31],[31,32],[32,33],[33,34],[34,35],[35,36],[36,37],[37,38],[38,39],[39,40],[40,40]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.918799877166748} +{"basename":"euler_rechenkunst01_1738","par_idx":1010,"date":1738,"orig":"Es wird alſo fuͤr den Diuiſor 20 der Quotus 8 ſeyn; fuͤr den Diuiſor30 aber 5. Weilen nun der wahre Diuiſor 23 dem erſteren Diuiſori naher kommt, ſo muß auch der wahre Quotus dem 8 naͤher ſeyn als dem 5, wie er dann auch 7 iſt gefunden worden.","norm":"Es wird also für den Divisor 20 der Quotus 8 sein; für den Diuisor30 aber 5. Weilen nun der wahre Divisor 23 dem ersteren Diuisori naher kommt, so muss auch der wahre Quotus dem 8 näher sein als dem 5, wie er dann auch 7 ist gefunden worden.","orig_tok":["Es","wird","alſo","fuͤr","den","Diuiſor","20","der","Quotus","8","ſeyn",";","fuͤr","den","Diuiſor30","aber","5.","Weilen","nun","der","wahre","Diuiſor","23","dem","erſteren","Diuiſori","naher","kommt",",","ſo","muß","auch","der","wahre","Quotus","dem","8","naͤher","ſeyn","als","dem","5",",","wie","er","dann","auch","7","iſt","gefunden","worden","."],"norm_tok":["Es","wird","also","für","den","Divisor","20","der","Quotus","8","sein",";","für","den","Diuisor30","aber","5.","Weilen","nun","der","wahre","Divisor","23","dem","ersteren","Diuisori","naher","kommt",",","so","muss","auch","der","wahre","Quotus","dem","8","näher","sein","als","dem","5",",","wie","er","dann","auch","7","ist","gefunden","worden","."],"orig_spans":[[0,2],[3,7],[8,12],[13,17],[18,21],[22,29],[30,32],[33,36],[37,43],[44,45],[46,50],[50,51],[52,56],[57,60],[61,70],[71,75],[76,78],[79,85],[86,89],[90,93],[94,99],[100,107],[108,110],[111,114],[115,123],[124,132],[133,138],[139,144],[144,145],[146,148],[149,152],[153,157],[158,161],[162,167],[168,174],[175,178],[179,180],[181,187],[188,192],[193,196],[197,200],[201,202],[202,203],[204,207],[208,210],[211,215],[216,220],[221,222],[223,226],[227,235],[236,242],[242,243]],"norm_spans":[[0,2],[3,7],[8,12],[13,16],[17,20],[21,28],[29,31],[32,35],[36,42],[43,44],[45,49],[49,50],[51,54],[55,58],[59,68],[69,73],[74,76],[77,83],[84,87],[88,91],[92,97],[98,105],[106,108],[109,112],[113,121],[122,130],[131,136],[137,142],[142,143],[144,146],[147,151],[152,156],[157,160],[161,166],[167,173],[174,177],[178,179],[180,185],[186,190],[191,194],[195,198],[199,200],[200,201],[202,205],[206,208],[209,213],[214,218],[219,220],[221,224],[225,233],[234,240],[240,241]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28],[29,29],[30,30],[31,31],[32,32],[33,33],[34,34],[35,35],[36,36],[37,37],[38,38],[39,39],[40,40],[41,41],[42,42],[43,43],[44,44],[45,45],[46,46],[47,47],[48,48],[49,49],[50,50],[51,51]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.217599868774414} +{"basename":"euler_rechenkunst01_1738","par_idx":1011,"date":1738,"orig":"Kommt aber der Diuiſor einem von den zweyen, welche angenommen werden, gar um viel naher als dem anderen, ſo hat man auch nur mit dem naͤheren allein zu probiren, und zwar mit dieſer Vorſichtigkeit, daß wann der kleinere naͤher kommt der Quotus bisweilen nur um ein Unitæt zu groß, im andren Fall aber zu klein herauskomme.","norm":"Kommt aber der Divisor einem von den zweien, welche angenommen werden, gar um viel naher als dem anderen, so hat man auch nur mit dem näheren allein zu probieren, und zwar mit dieser Vorsichtigkeit, dass wann der kleinere näher kommt der Quotus bisweilen nur um ein Unität zu groß, im anderen Fall aber zu klein herauskomme.","orig_tok":["Kommt","aber","der","Diuiſor","einem","von","den","zweyen",",","welche","angenommen","werden",",","gar","um","viel","naher","als","dem","anderen",",","ſo","hat","man","auch","nur","mit","dem","naͤheren","allein","zu","probiren",",","und","zwar","mit","dieſer","Vorſichtigkeit",",","daß","wann","der","kleinere","naͤher","kommt","der","Quotus","bisweilen","nur","um","ein","Unitæt","zu","groß",",","im","andren","Fall","aber","zu","klein","herauskomme","."],"norm_tok":["Kommt","aber","der","Divisor","einem","von","den","zweien",",","welche","angenommen","werden",",","gar","um","viel","naher","als","dem","anderen",",","so","hat","man","auch","nur","mit","dem","näheren","allein","zu","probieren",",","und","zwar","mit","dieser","Vorsichtigkeit",",","dass","wann","der","kleinere","näher","kommt","der","Quotus","bisweilen","nur","um","ein","Unität","zu","groß",",","im","anderen","Fall","aber","zu","klein","herauskomme","."],"orig_spans":[[0,5],[6,10],[11,14],[15,22],[23,28],[29,32],[33,36],[37,43],[43,44],[45,51],[52,62],[63,69],[69,70],[71,74],[75,77],[78,82],[83,88],[89,92],[93,96],[97,104],[104,105],[106,108],[109,112],[113,116],[117,121],[122,125],[126,129],[130,133],[134,142],[143,149],[150,152],[153,161],[161,162],[163,166],[167,171],[172,175],[176,182],[183,197],[197,198],[199,202],[203,207],[208,211],[212,220],[221,227],[228,233],[234,237],[238,244],[245,254],[255,258],[259,261],[262,265],[266,272],[273,275],[276,280],[280,281],[282,284],[285,291],[292,296],[297,301],[302,304],[305,310],[311,322],[322,323]],"norm_spans":[[0,5],[6,10],[11,14],[15,22],[23,28],[29,32],[33,36],[37,43],[43,44],[45,51],[52,62],[63,69],[69,70],[71,74],[75,77],[78,82],[83,88],[89,92],[93,96],[97,104],[104,105],[106,108],[109,112],[113,116],[117,121],[122,125],[126,129],[130,133],[134,141],[142,148],[149,151],[152,161],[161,162],[163,166],[167,171],[172,175],[176,182],[183,197],[197,198],[199,203],[204,208],[209,212],[213,221],[222,227],[228,233],[234,237],[238,244],[245,254],[255,258],[259,261],[262,265],[266,272],[273,275],[276,280],[280,281],[282,284],[285,292],[293,297],[298,302],[303,305],[306,311],[312,323],[323,324]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28],[29,29],[30,30],[31,31],[32,32],[33,33],[34,34],[35,35],[36,36],[37,37],[38,38],[39,39],[40,40],[41,41],[42,42],[43,43],[44,44],[45,45],[46,46],[47,47],[48,48],[49,49],[50,50],[51,51],[52,52],[53,53],[54,54],[55,55],[56,56],[57,57],[58,58],[59,59],[60,60],[61,61],[62,62]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.8730998039245605} +{"basename":"euler_rechenkunst01_1738","par_idx":1012,"date":1738,"orig":"Auf dieſe Art nimmt man alſo anſtatt des wahren Diuiſoris ſolche an, welche aus einer einfachen Zahl mit daran gehaͤngten Cyphren beſtehen, mit welchen die Diuiſion oder vielmehr nur die Findung des Quoti, indem der Reſt nicht von noͤthen, nach dem vorhergehenden Punckt eben ſo leicht als mit einfachen Zahl bewerckſtelliget wird.","norm":"Auf diese Art nimmt man also anstatt des wahren Divisors solche an, welche aus einer einfachen Zahl mit daran gehängten Ziffern bestehen, mit welchen die Division oder vielmehr nur die Findung des Quoti, indem der Rest nicht vonnöten, nach dem vorhergehenden Punkt eben so leicht als mit einfachen Zahl bewerkstelliget wird.","orig_tok":["Auf","dieſe","Art","nimmt","man","alſo","anſtatt","des","wahren","Diuiſoris","ſolche","an",",","welche","aus","einer","einfachen","Zahl","mit","daran","gehaͤngten","Cyphren","beſtehen",",","mit","welchen","die","Diuiſion","oder","vielmehr","nur","die","Findung","des","Quoti",",","indem","der","Reſt","nicht","von","noͤthen",",","nach","dem","vorhergehenden","Punckt","eben","ſo","leicht","als","mit","einfachen","Zahl","bewerckſtelliget","wird","."],"norm_tok":["Auf","diese","Art","nimmt","man","also","anstatt","des","wahren","Divisors","solche","an",",","welche","aus","einer","einfachen","Zahl","mit","daran","gehängten","Ziffern","bestehen",",","mit","welchen","die","Division","oder","vielmehr","nur","die","Findung","des","Quoti",",","indem","der","Rest","nicht","vonnöten",",","nach","dem","vorhergehenden","Punkt","eben","so","leicht","als","mit","einfachen","Zahl","bewerkstelliget","wird","."],"orig_spans":[[0,3],[4,9],[10,13],[14,19],[20,23],[24,28],[29,36],[37,40],[41,47],[48,57],[58,64],[65,67],[67,68],[69,75],[76,79],[80,85],[86,95],[96,100],[101,104],[105,110],[111,121],[122,129],[130,138],[138,139],[140,143],[144,151],[152,155],[156,164],[165,169],[170,178],[179,182],[183,186],[187,194],[195,198],[199,204],[204,205],[206,211],[212,215],[216,220],[221,226],[227,230],[231,238],[238,239],[240,244],[245,248],[249,263],[264,270],[271,275],[276,278],[279,285],[286,289],[290,293],[294,303],[304,308],[309,325],[326,330],[330,331]],"norm_spans":[[0,3],[4,9],[10,13],[14,19],[20,23],[24,28],[29,36],[37,40],[41,47],[48,56],[57,63],[64,66],[66,67],[68,74],[75,78],[79,84],[85,94],[95,99],[100,103],[104,109],[110,119],[120,127],[128,136],[136,137],[138,141],[142,149],[150,153],[154,162],[163,167],[168,176],[177,180],[181,184],[185,192],[193,196],[197,202],[202,203],[204,209],[210,213],[214,218],[219,224],[225,233],[233,234],[235,239],[240,243],[244,258],[259,264],[265,269],[270,272],[273,279],[280,283],[284,287],[288,297],[298,302],[303,318],[319,323],[323,324]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28],[29,29],[30,30],[31,31],[32,32],[33,33],[34,34],[35,35],[36,36],[37,37],[38,38],[39,39],[40,40],[41,40],[42,41],[43,42],[44,43],[45,44],[46,45],[47,46],[48,47],[49,48],[50,49],[51,50],[52,51],[53,52],[54,53],[55,54],[56,55]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.035999774932861} +{"basename":"euler_rechenkunst01_1738","par_idx":1013,"date":1738,"orig":"Als wann die Frage iſt, wieviel mahl 319 in 1268 enthalten ſey, ſo ſehe ich nur wieviel mahl 300 darinn enthalten ſey, und probire nicht ein mahl mit 400, weilen 319 jener Zahl weit naͤher kommt als dieſer.","norm":"Als wann die Frage ist, wie viel Mal 319 in 1268 enthalten sei, so sehe ich nur wie viel Mal 300 darin enthalten sei, und probiere nicht ein Mal mit 400, weilen 319 jener Zahl weit näher kommt als dieser.","orig_tok":["Als","wann","die","Frage","iſt",",","wieviel","mahl","319","in","1268","enthalten","ſey",",","ſo","ſehe","ich","nur","wieviel","mahl","300","darinn","enthalten","ſey",",","und","probire","nicht","ein","mahl","mit","400",",","weilen","319","jener","Zahl","weit","naͤher","kommt","als","dieſer","."],"norm_tok":["Als","wann","die","Frage","ist",",","wie","viel","Mal","319","in","1268","enthalten","sei",",","so","sehe","ich","nur","wie","viel","Mal","300","darin","enthalten","sei",",","und","probiere","nicht","ein","Mal","mit","400",",","weilen","319","jener","Zahl","weit","näher","kommt","als","dieser","."],"orig_spans":[[0,3],[4,8],[9,12],[13,18],[19,22],[22,23],[24,31],[32,36],[37,40],[41,43],[44,48],[49,58],[59,62],[62,63],[64,66],[67,71],[72,75],[76,79],[80,87],[88,92],[93,96],[97,103],[104,113],[114,117],[117,118],[119,122],[123,130],[131,136],[137,140],[141,145],[146,149],[150,153],[153,154],[155,161],[162,165],[166,171],[172,176],[177,181],[182,188],[189,194],[195,198],[199,205],[205,206]],"norm_spans":[[0,3],[4,8],[9,12],[13,18],[19,22],[22,23],[24,27],[28,32],[33,36],[37,40],[41,43],[44,48],[49,58],[59,62],[62,63],[64,66],[67,71],[72,75],[76,79],[80,83],[84,88],[89,92],[93,96],[97,102],[103,112],[113,116],[116,117],[118,121],[122,130],[131,136],[137,140],[141,144],[145,148],[149,152],[152,153],[154,160],[161,164],[165,170],[171,175],[176,180],[181,186],[187,192],[193,196],[197,203],[203,204]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[18,20],[19,21],[20,22],[21,23],[22,24],[23,25],[24,26],[25,27],[26,28],[27,29],[28,30],[29,31],[30,32],[31,33],[32,34],[33,35],[34,36],[35,37],[36,38],[37,39],[38,40],[39,41],[40,42],[41,43],[42,44]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.9959001541137695} +{"basename":"euler_rechenkunst01_1738","par_idx":1014,"date":1738,"orig":"Um aber zu finden wieviel mahl 300 in 1268 enthalten ſey, ſo darf man nur ſehen wieviel mahl 3 in 12 begriffen ſey, welches 4 mahl iſt, alſo wird der Quotus 4 ſeyn, oder auf das hoͤchſte nur 3. Wann aber geſucht wird, wieviel mahl 2976 in 15873 enthalten ſey, ſo bediene man ſich nur des Diuiſoris 3000 allein, und diuidire alſo15 durch 3, ſo |wird der Quotus 5 der wahre Quotus ſeyn.","norm":"Um aber zu finden wie viel Mal 300 in 1268 enthalten sei, so darf man nur sehen wie viel Mal 3 in 12 begriffen sei, welches 4 Mal ist, also wird der Quotus 4 sein, oder auf das höchste nur 3. Wann aber gesucht wird, wie viel Mal 2976 in 15873 enthalten sei, so bediene man sich nur des Divisors 3000 allein, und dividiere also15 durch 3, so | wird der Quotus 5 der wahre Quotus 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+{"basename":"euler_rechenkunst01_1738","par_idx":1015,"date":1738,"orig":"Vermittelſt dieſer Anleitung wird man nun leicht’ finden koͤnnen, wieviel mahl ein jeglicher vorgegebener Diuiſor in einem jeden Theil des Diuidendi enthalten ſey, und wird alſo die Figuren, aus welchen der Quotus beſteht, finden koͤnnen.","norm":"Vermittels dieser Anleitung wird man nun leicht finden können, wie viel Mal ein jeglicher vorgegebener Divisor in einem jeden Teil des Dividenden enthalten sei, und wird also die Figuren, aus welchen der Quotus besteht, finden 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+{"basename":"euler_rechenkunst01_1738","par_idx":1016,"date":1738,"orig":"Durch eine fleißige Ubung aber wird man ſich dieſe Arbeit ſehr erleichtern.","norm":"Durch eine fleißige Übung aber wird man sich diese Arbeit sehr erleichtern.","orig_tok":["Durch","eine","fleißige","Ubung","aber","wird","man","ſich","dieſe","Arbeit","ſehr","erleichtern","."],"norm_tok":["Durch","eine","fleißige","Übung","aber","wird","man","sich","diese","Arbeit","sehr","erleichtern","."],"orig_spans":[[0,5],[6,10],[11,19],[20,25],[26,30],[31,35],[36,39],[40,44],[45,50],[51,57],[58,62],[63,74],[74,75]],"norm_spans":[[0,5],[6,10],[11,19],[20,25],[26,30],[31,35],[36,39],[40,44],[45,50],[51,57],[58,62],[63,74],[74,75]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.292200088500977} +{"basename":"euler_rechenkunst01_1738","par_idx":1017,"date":1738,"orig":"II. Weilen es aber auf dieſe Art geſchehen kan, daß man den Quotum um eins entweder zu groß oder zu klein angenommen, ſo kan man dieſes auf folgende Art leicht innen werden, und alſo corrigiren.","norm":"II. Weilen es aber auf diese Art geschehen kann, dass man den Quotum um eins entweder zu groß oder zu klein angenommen, so kann man dieses auf folgende Art leicht innen werden, und also 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+{"basename":"euler_rechenkunst01_1738","par_idx":1019,"date":1738,"orig":"Jſt aber dieſes Product kleiner, ſo daß die Subtraction geſchehen kan, der Reſt aber, der uͤberbleibt ſo groß oder groͤſſer als der Diuiſor, ſo iſt dieſes eine Anzeige, daß man den Quotum zu klein angenommen, und denſelben alſo um eins groͤſſer annehmen muͤſſe.","norm":"Ist aber dieses Produkt kleiner, so dass die Subtraktion geschehen kann, der Rest aber, der überbleibt so groß oder größer als der Divisor, so ist dieses eine Anzeige, dass man den Quotum zu klein angenommen, und denselben also um eins größer annehmen 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Figur des Diuidendi an, und ſucht wiederum wieviel mahl der Diuiſor in dieſer Zahl enthalten iſt; welche Zahl die zweyte Figur des Quoti gibt, und mit dieſer multiplicirt man wieder den Diuiſorem, ſubtrahirt das Product von jener Zahl und haͤngt an den Reſt die folgende Figur des Diuidendi.","norm":"An den Rest hängt man die nach dem Abschnitt folgende Figur des Dividenden an, und sucht wiederum wie viel Mal der Divisor in dieser Zahl enthalten ist; welche Zahl die zweite Figur des Quoti gibt, und mit dieser multipliziert man wieder den Divisor, subtrahiert das Produkt von jener Zahl und hängt an den Rest die folgende Figur des 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+{"basename":"euler_rechenkunst01_1738","par_idx":1028,"date":1738,"orig":"Was bey der letzten Subtraction zuruͤckbleibt, iſt der Reſt, ſo bey der Diuiſion noch uͤbrig iſt.","norm":"Was bei der letzten Subtraktion zurückbleibt, ist der Rest, so bei der Division noch übrig 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die folgende Zahl des Diuidendi 9 dazu ſchreibt ſo hat man nur 9, in welcher Zahl der Diuiſor kein mahl begriffen iſt; derowegen ſchreibt man 0 in den Quotum, und da 0 mahl251 auch 0 ausmacht, und 0 von 9 ſubtrahirt 9 zuruͤck laͤßt, ſo iſt unnoͤthig dieſe Operation hinzuſchreiben, ſondern man betrachtet gleich dieſe 9 als den Reſt, und ſchreibt dazu die folgende Figur 7.","norm":"Wann man nun nach der Regel die folgende Zahl des Dividenden 9 dazuschreibt so hat man nur 9, in welcher Zahl der Divisor kein Mal begriffen ist; derowegen schreibt man 0 in den Quotum, und da 0 mahl251 auch 0 ausmacht, und 0 von 9 subtrahiert 9 zurücklässt, so ist unnötig diese Operation hinzuschreiben, sondern man betrachtet gleich diese 9 als den Rest, und schreibt dazu die folgende Figur 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00 des Diuidendi; und da 0 mit dem Diuiſor multiplicirt 0 gibt, und 0 von 0 ſubtrahirt 0 zuruͤcklaͤßt, ſo wird der Reſt 0 ſeyn, und der Quotus 309000.","norm":"Die folgende Figur des Dividenden 0 herabgeschrieben, gibt in Quotum eine 0, ingleichem auch die zwei letzten 00 des Dividenden; und da 0 mit dem Divisor multipliziert 0 gibt, und 0 von 0 subtrahiert 0 zurücklässt, so wird der Rest 0 sein, und der Quotus 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+{"basename":"euler_rechenkunst01_1738","par_idx":1104,"date":1738,"orig":"Jn dieſen Faͤllen iſt demnach der daſelbſt gefundene Quotus nicht der wahre Quotus:","norm":"In diesen Fällen ist demnach der daselbst gefundene Quotus nicht der wahre Quotus:","orig_tok":["Jn","dieſen","Faͤllen","iſt","demnach","der","daſelbſt","gefundene","Quotus","nicht","der","wahre","Quotus",":"],"norm_tok":["In","diesen","Fällen","ist","demnach","der","daselbst","gefundene","Quotus","nicht","der","wahre","Quotus",":"],"orig_spans":[[0,2],[3,9],[10,17],[18,21],[22,29],[30,33],[34,42],[43,52],[53,59],[60,65],[66,69],[70,75],[76,82],[82,83]],"norm_spans":[[0,2],[3,9],[10,16],[17,20],[21,28],[29,32],[33,41],[42,51],[52,58],[59,64],[65,68],[69,74],[75,81],[81,82]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.349299907684326} +{"basename":"euler_rechenkunst01_1738","par_idx":1105,"date":1738,"orig":"ſondern dazu muß noch der herauskommende Reſt in Betrachtung gezogen werden, um einen hinlanglichen Begriff zu haben, wieviel mahl der Diuiſor in dem Diuidendo enthalten ſey.","norm":"sondern dazu muss noch der herauskommende Rest in Betrachtung gezogen werden, um einen Hinlanglichen Begriff zu haben, wie viel Mal der Divisor in dem Dividenden enthalten 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+{"basename":"euler_rechenkunst01_1738","par_idx":1106,"date":1738,"orig":"Als wann 17 durch 5 getheilet werden ſoll, ſo finden wir durch die daſelbſt gegebenen Regeln, daß der Quotus 3, und dazu noch ein Reſt nehmlich 2 ſey.","norm":"Als wann 17 durch 5 geteilt werden soll, so finden wir durch die daselbst gegebenen Regeln, dass der Quotus 3, und dazu noch ein Rest nämlich 2 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+{"basename":"euler_rechenkunst01_1738","par_idx":1107,"date":1738,"orig":"Hieraus erhellet, daß 5 in 17 mehr als 3 mahl enthalten, und folglich der wahre Quotus groͤſſer als 3 ſeyn muͤſſe.","norm":"Hieraus erhellet, dass 5 in 17 mehr als 3 Mal enthalten, und folglich der wahre Quotus größer als 3 sein 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+{"basename":"euler_rechenkunst01_1738","par_idx":1108,"date":1738,"orig":"Dann da 5 in 15 drey mahl enthalten iſt, ſo muß 5 in 17 nothwendig mehr als 3 mahl begriffen ſeyn.","norm":"Dann da 5 in 15 dreimal enthalten ist, so muss 5 in 17 notwendig mehr als 3 Mal begriffen 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+{"basename":"euler_rechenkunst01_1738","par_idx":1109,"date":1738,"orig":"Dennoch aber iſt 5 in 17 auch nicht gar 4 mahl enthalten, weilen 20 diejenige Zahl iſt, welche 5 vier mahl in ſich begreifft.","norm":"Dennoch aber ist 5 in 17 auch nicht gar 4 Mal enthalten, weilen 20 diejenige Zahl ist, welche 5 viermal in sich 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nun ſehr bequem auf die angefuͤhrte Art, da der Diuidendus uͤber den Diuiſor geſetzet und eine Linie dazwiſchen gezogen wird.","norm":"Dieses geschieht nun sehr bequem auf die angeführte Art, da der Dividend über den Divisor gesetzt und eine Linie dazwischen gezogen 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+{"basename":"euler_rechenkunst01_1738","par_idx":1122,"date":1738,"orig":"Dann auf dieſe Weiſe erkennt man zugleich den Urſprung und Werth eines Beuches; in dem auf dieſe Art der Quotus angedeutet wird, welcher herauskommt, wann man die obere Zahl durch die untere diuidirt.","norm":"Dann auf diese Weise erkennt man zugleich den Ursprung und Wert eines Beuche; in dem auf diese Art der Quotus angedeutet wird, welcher herauskommt, wann man die obere Zahl durch die untere 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+{"basename":"euler_rechenkunst01_1738","par_idx":1123,"date":1738,"orig":"Jn dem vorher gegebenen Exempel, da17 durch 5 ſollte diuidiret werden, wird alſo der Quotus, welcher ein Bruch iſt, auf dieſe Art angezeiget [Formel] .","norm":"In dem vorher gegebenen Exempel, da17 durch 5 sollte dividieret werden, wird also der Quotus, welcher ein Bruch ist, auf diese Art angezeigt 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+{"basename":"euler_rechenkunst01_1738","par_idx":1124,"date":1738,"orig":"Durch dieſe Schreib- Art wird demnach ein Bruch ausgedrucket, und daraus erkennt man zugleich, was daſſelbe fuͤr ein Bruch ſey; nehmlich [Formel] iſt ein Bruch, und deutet an wieviel mahl 5 in 17 enthalten ſey, oder dieſer Bruch iſt der wahre Quotus, der berauskommt, wann man 17 durch 5 diuidirt.","norm":"Durch diese Schreib-Art wird demnach ein Bruch ausgedruckt, und daraus erkennt man zugleich, was dasselbe für ein Bruch sei; nämlich [Formel] ist ein Bruch, und deutet an wie viel Mal 5 in 17 enthalten sei, oder dieser Bruch ist der wahre Quotus, der berauskommt, wann man 17 durch 5 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Und durch dieſe Schreib-Art [Formel] wird der Quotus angedeutet, welcher herauskommt, wann man 5 durch 3 diuidirt.","norm":"Gleichergestalt wann 8 durch 7 geteilt werden soll, so ist der Quotus keine ganze Zahl, sondern ein Bruch und wird also geschrieben [Formel]. Und durch diese Schreibart [Formel] wird der Quotus angedeutet, welcher herauskommt, wann man 5 durch 3 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+{"basename":"euler_rechenkunst01_1738","par_idx":1128,"date":1738,"orig":"Dieſe Art ſich einen Begriff von dem Werth eines Bruchs zu machen, ſcheinet zwar von der vorigen unterſchieden zu ſeyn, kommt aber in der That mit derſelben ſehr genau uͤberein.","norm":"Diese Art sich einen Begriff von dem Wert eines Bruchs zu machen, scheinet zwar von der vorigen Unterschieden zu sein, kommt aber in der Tat mit derselben sehr genau 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+{"basename":"euler_rechenkunst01_1738","par_idx":1129,"date":1738,"orig":"Wann nehmlich dieſer Bruch [Formel] vorgegeben iſt, ſo deutet derſelbe nach der erſten Art den Quotum an, welcher herauskommt, wann mann 7 durch 4 diuidirt.","norm":"Wann nämlich dieser Bruch [Formel] vorgegeben ist, so deutet derselbe nach der ersten Art den Quotum an, welcher herauskommt, wann Mann 7 durch 4 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+{"basename":"euler_rechenkunst01_1738","par_idx":1130,"date":1738,"orig":"Nach dieſer Art aber ſagen wir, daß wann ein gantzes in 4 gleiche Theile getheilet wird, der Bruch 7 dergleichen Theile andeute und in ſich begreiffe.","norm":"Nach dieser Art aber sagen wir, dass wann ein ganzes in 4 gleiche Teile geteilt wird, der Bruch 7 dergleichen Teile andeute und in sich 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+{"basename":"euler_rechenkunst01_1738","par_idx":1131,"date":1738,"orig":"Die Ubereinſtimmung aber dieſer zwey verſchiedenen Arten den Werth eines Bruchs zu beſchreiben kan auf dieſe Weiſe gewiefen werden.","norm":"Die Übereinstimmung aber dieser zwei verschiedenen Arten den Wert eines Bruchs zu beschreiben kann auf diese Weise gewiesen 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+{"basename":"euler_rechenkunst01_1738","par_idx":1132,"date":1738,"orig":"Da [Formel] den Quotum andeutet, der herauskommt, wann man 7 durch 4 diuidirt, ſo wird dadurch der vierte Theil von 7 angezeiget, dann 7 durch 4 diuidiren iſt nichts anders als den vierten Theil von 7 finden.","norm":"Da [Formel] den Quotum andeutet, der herauskommt, wann man 7 durch 4 dividiert, so wird dadurch der vierte Teil von 7 angezeigt, dann 7 durch 4 dividieren ist nichts anders als den vierten Teil von 7 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+{"basename":"euler_rechenkunst01_1738","par_idx":1133,"date":1738,"orig":"Woraus erhellet daß ein jeglicher Bruch nichts anders bedeute, als den ſo vielten Theil der obſtehenden Zahl, als die unten ſtehende ausweiſet; welches wieder eine neue Art iſt ſich den Werth eines Bruchs vorzuſtellen.","norm":"Woraus erhellet dass ein jeglicher Bruch nichts anders bedeute, als den so vielten Teil der stehenden Zahl, als die unten stehende ausweiset; welches wieder eine neue Art ist sich den Wert eines Bruchs 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+{"basename":"euler_rechenkunst01_1738","par_idx":1134,"date":1738,"orig":"Weilen nun um bey dem gegebenen Exempel von [Formel] zu bleiben, 7 ſieben mahl groͤſſer iſt als 1, ſo muß folglich auch der vierte Theil von 7, ſieben mahl groͤſſer ſeyn als der vierte Theil von 1. Wann demnach 1 in 4 gleiche Theile getheilet wird, ſo iſt einer derſelben der vierte Theil von 1, und alſo [Formel] ſieben mahl groͤſſer als ein ſolcher Theil.","norm":"Weilen nun um bei dem gegebenen Exempel von [Formel] zu bleiben, 7 siebenmal größer ist als 1, so muss folglich auch der vierte Teil von 7, siebenmal größer sein als der vierte Teil von 1. Wann demnach 1 in 4 gleiche Teile geteilt wird, so ist einer derselben der vierte Teil von 1, und also [Formel] siebenmal größer als ein solcher 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+{"basename":"euler_rechenkunst01_1738","par_idx":1150,"date":1738,"orig":"Gleichergeſtalt heiſſt dieſer Bruch [Formel] fuͤnfzehn acht und zwantzigſte Theil, und dieſer ¾ drey vierte Theil.","norm":"Gleichergestalt heißt dieser Bruch [Formel] fünfzehn acht und zwanzigste Teil, und dieser ¾ drei vierte 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Ist aber der Zähler g Öser als der Nenner, so ist auch der Inhalt des Bruchs größer als 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+{"basename":"euler_rechenkunst01_1738","par_idx":1157,"date":1738,"orig":"Ein Bruch aber, da der Zehler dem Nenner gleich iſt, haͤlt juſt ein gantzes.","norm":"Ein Bruch aber, da der Zähler dem Nenner gleich ist, hält just ein 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vorgeſtellt, leicht erweiſen.","norm":"Die Wahrheit dieses, was hier ist vorgebracht worden, lässt sich aus den beiden Arten nach denen wir uns die Brüche vorgestellt, leicht 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+{"basename":"euler_rechenkunst01_1738","par_idx":1159,"date":1738,"orig":"Dann da nach der erſten Art ein Bruch den wahren Quotum anzeiget, welcher herauskommt, wann man die obere Zahl durch die untere diuidirt:","norm":"Dann danach der ersten Art ein Bruch den wahren Quotum anzeiget, welcher herauskommt, wann man die obere Zahl durch die untere 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+{"basename":"euler_rechenkunst01_1738","par_idx":1160,"date":1738,"orig":"ſo iſt klar, daß wann die obere Zahl kleiner iſt als die untere, dieſe in jener nicht ein mahl, ſonderen weniger mahl darinnen enthalten ſey; weswegen in ſolchem Fall der Quotus das iſt der Jnhalt des Bruchs kleiner als 1 ſeyn muß.","norm":"so ist klar, dass wann die obere Zahl kleiner ist als die untere, diese in jener nicht ein Mal, sondern weniger Mal darinnen enthalten sei; weswegen in solchem Fall der Quotus das ist der Inhalt des Bruchs kleiner als 1 sein muss.","orig_tok":["ſo","iſt","klar",",","daß","wann","die","obere","Zahl","kleiner","iſt","als","die","untere",",","dieſe","in","jener","nicht","ein","mahl",",","ſonderen","weniger","mahl","darinnen","enthalten","ſey",";","weswegen","in","ſolchem","Fall","der","Quotus","das","iſt","der","Jnhalt","des","Bruchs","kleiner","als","1","ſeyn","muß","."],"norm_tok":["so","ist","klar",",","dass","wann","die","obere","Zahl","kleiner","ist","als","die","untere",",","diese","in","jener","nicht","ein","Mal",",","sondern","weniger","Mal","darinnen","enthalten","sei",";","weswegen","in","solchem","Fall","der","Quotus","das","ist","der","Inhalt","des","Bruchs","kleiner","als","1","sein","muss","."],"orig_spans":[[0,2],[3,6],[7,11],[11,12],[13,16],[17,21],[22,25],[26,31],[32,36],[37,44],[45,48],[49,52],[53,56],[57,63],[63,64],[65,70],[71,73],[74,79],[80,85],[86,89],[90,94],[94,95],[96,104],[105,112],[113,117],[118,126],[127,136],[137,140],[140,141],[142,150],[151,153],[154,161],[162,166],[167,170],[171,177],[178,181],[182,185],[186,189],[190,196],[197,200],[201,207],[208,215],[216,219],[220,221],[222,226],[227,230],[230,231]],"norm_spans":[[0,2],[3,6],[7,11],[11,12],[13,17],[18,22],[23,26],[27,32],[33,37],[38,45],[46,49],[50,53],[54,57],[58,64],[64,65],[66,71],[72,74],[75,80],[81,86],[87,90],[91,94],[94,95],[96,103],[104,111],[112,115],[116,124],[125,134],[135,138],[138,139],[140,148],[149,151],[152,159],[160,164],[165,168],[169,175],[176,179],[180,183],[184,187],[188,194],[195,198],[199,205],[206,213],[214,217],[218,219],[220,224],[225,229],[229,230]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28],[29,29],[30,30],[31,31],[32,32],[33,33],[34,34],[35,35],[36,36],[37,37],[38,38],[39,39],[40,40],[41,41],[42,42],[43,43],[44,44],[45,45],[46,46]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.533899784088135} +{"basename":"euler_rechenkunst01_1738","par_idx":1161,"date":1738,"orig":"Als [Formel] deutet den Quotum an, welcher herauskommt, wann man 3 durch 7 diuidirt.","norm":"Als [Formel] deutet den Quotum an, welcher herauskommt, wann man 3 durch 7 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das iſt mehr als 3; dennoch aber iſt 7 in 3 mehr als kein mahl oder 0 mahl enthalten, dann 0 mahl 7 macht 0, das iſt weniger als 3. Hieraus folget alſo daß dieſer Bruch [Formel] oder der wahre Quotus, ſo herauskommt, wann 3 durch 7 diuidirt wird, kleiner ſey als 1, und doch groͤſſer als nichts.","norm":"Nun aber ist 7 in drei nicht ein Mal enthalten, dann 1 Mal 7 macht 7 das ist mehr als 3; dennoch aber ist 7 in 3 mehr als kein Mal oder 0 Mal enthalten, dann 0 Mal 7 macht 0, das ist weniger als 3. Hieraus folget also dass dieser Bruch [Formel] oder der wahre Quotus, so herauskommt, wann 3 durch 7 dividiert wird, kleiner sei als 1, und doch größer als nichts.","orig_tok":["Nun","aber","iſt","7","in","drey","nicht","ein","mahl","enthalten",",","dann","1","mahl","7","macht","7","das","iſt","mehr","als","3",";","dennoch","aber","iſt","7","in","3","mehr","als","kein","mahl","oder","0","mahl","enthalten",",","dann","0","mahl","7","macht","0",",","das","iſt","weniger","als","3.","Hieraus","folget","alſo","daß","dieſer","Bruch","[Formel]","oder","der","wahre","Quotus",",","ſo","herauskommt",",","wann","3","durch","7","diuidirt","wird",",","kleiner","ſey","als","1",",","und","doch","groͤſſer","als","nichts","."],"norm_tok":["Nun","aber","ist","7","in","drei","nicht","ein","Mal","enthalten",",","dann","1","Mal","7","macht","7","das","ist","mehr","als","3",";","dennoch","aber","ist","7","in","3","mehr","als","kein","Mal","oder","0","Mal","enthalten",",","dann","0","Mal","7","macht","0",",","das","ist","weniger","als","3.","Hieraus","folget","also","dass","dieser","Bruch","[","Formel","]","oder","der","wahre","Quotus",",","so","herauskommt",",","wann","3","durch","7","dividiert","wird",",","kleiner","sei","als","1",",","und","doch","größer","als","nichts","."],"orig_spans":[[0,3],[4,8],[9,12],[13,14],[15,17],[18,22],[23,28],[29,32],[33,37],[38,47],[47,48],[49,53],[54,55],[56,60],[61,62],[63,68],[69,70],[71,74],[75,78],[79,83],[84,87],[88,89],[89,90],[91,98],[99,103],[104,107],[108,109],[110,112],[113,114],[115,119],[120,123],[124,128],[129,133],[134,138],[139,140],[141,145],[146,155],[155,156],[157,161],[162,163],[164,168],[169,170],[171,176],[177,178],[178,179],[180,183],[184,187],[188,195],[196,199],[200,202],[203,210],[211,217],[218,222],[223,226],[227,233],[234,239],[240,248],[249,253],[254,257],[258,263],[264,270],[270,271],[272,274],[275,286],[286,287],[288,292],[293,294],[295,300],[301,302],[303,311],[312,316],[316,317],[318,325],[326,329],[330,333],[334,335],[335,336],[337,340],[341,345],[346,354],[355,358],[359,365],[365,366]],"norm_spans":[[0,3],[4,8],[9,12],[13,14],[15,17],[18,22],[23,28],[29,32],[33,36],[37,46],[46,47],[48,52],[53,54],[55,58],[59,60],[61,66],[67,68],[69,72],[73,76],[77,81],[82,85],[86,87],[87,88],[89,96],[97,101],[102,105],[106,107],[108,110],[111,112],[113,117],[118,121],[122,126],[127,130],[131,135],[136,137],[138,141],[142,151],[151,152],[153,157],[158,159],[160,163],[164,165],[166,171],[172,173],[173,174],[175,178],[179,182],[183,190],[191,194],[195,197],[198,205],[206,212],[213,217],[218,222],[223,229],[230,235],[236,237],[237,243],[243,244],[245,249],[250,253],[254,259],[260,266],[266,267],[268,270],[271,282],[282,283],[284,288],[289,290],[291,296],[297,298],[299,308],[309,313],[313,314],[315,322],[323,326],[327,330],[331,332],[332,333],[334,337],[338,342],[343,349],[350,353],[354,360],[360,361]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28],[29,29],[30,30],[31,31],[32,32],[33,33],[34,34],[35,35],[36,36],[37,37],[38,38],[39,39],[40,40],[41,41],[42,42],[43,43],[44,44],[45,45],[46,46],[47,47],[48,48],[49,49],[50,50],[51,51],[52,52],[53,53],[54,54],[55,55],[56,56],[56,57],[56,58],[57,59],[58,60],[59,61],[60,62],[61,63],[62,64],[63,65],[64,66],[65,67],[66,68],[67,69],[68,70],[69,71],[70,72],[71,73],[72,74],[73,75],[74,76],[75,77],[76,78],[77,79],[78,80],[79,81],[80,82],[81,83],[82,84]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.251100063323975} 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+{"basename":"euler_rechenkunst01_1738","par_idx":1164,"date":1738,"orig":"Alſo iſt [Formel] groͤſſer als 1, dann wann ich 7 durch 5 diuidire, ſo kommt in Quotum 1 und bleibt noch 2 uͤber, weswegen der wahre Quotus das iſt der Werth des Bruchs [Formel] groͤſſer ſeyn muß als 1. Jngleichem gibt es auch Bruͤche welche groͤſſer ſind als 2, 3, 4, und ſo fort; als [Formel] iſt groͤſſer als 3, und [Formel] groͤſſer als 4, wie aus der Diuiſion erhellet.","norm":"Also ist [Formel] größer als 1, dann wann ich 7 durch 5 dividiere, so kommt in Quotum 1 und bleibt noch 2 über, weswegen der wahre Quotus das ist der Wert des Bruchs [Formel] größer sein muss als 1. Ingleichem gibt es auch Brüche welche größer sind als 2, 3, 4, und so fort; als [Formel] ist größer als 3, und [Formel] größer als 4, wie aus der Division erhellet.","orig_tok":["Alſo","iſt","[Formel]","groͤſſer","als","1",",","dann","wann","ich","7","durch","5","diuidire",",","ſo","kommt","in","Quotum","1","und","bleibt","noch","2","uͤber",",","weswegen","der","wahre","Quotus","das","iſt","der","Werth","des","Bruchs","[Formel]","groͤſſer","ſeyn","muß","als","1.","Jngleichem","gibt","es","auch","Bruͤche","welche","groͤſſer","ſind","als","2",",","3",",","4",",","und","ſo","fort",";","als","[Formel]","iſt","groͤſſer","als","3",",","und","[Formel]","groͤſſer","als","4",",","wie","aus","der","Diuiſion","erhellet","."],"norm_tok":["Also","ist","[","Formel","]","größer","als","1",",","dann","wann","ich","7","durch","5","dividiere",",","so","kommt","in","Quotum","1","und","bleibt","noch","2","über",",","weswegen","der","wahre","Quotus","das","ist","der","Wert","des","Bruchs","[","Formel","]","größer","sein","muss","als","1.","Ingleichem","gibt","es","auch","Brüche","welche","größer","sind","als","2",",","3",",","4",",","und","so","fort",";","als","[","Formel","]","ist","größer","als","3",",","und","[","Formel","]","größer","als","4",",","wie","aus","der","Division","erhellet","."],"orig_spans":[[0,4],[5,8],[9,17],[18,26],[27,30],[31,32],[32,33],[34,38],[39,43],[44,47],[48,49],[50,55],[56,57],[58,66],[66,67],[68,70],[71,76],[77,79],[80,86],[87,88],[89,92],[93,99],[100,104],[105,106],[107,112],[112,113],[114,122],[123,126],[127,132],[133,139],[140,143],[144,147],[148,151],[152,157],[158,161],[162,168],[169,177],[178,186],[187,191],[192,195],[196,199],[200,202],[203,213],[214,218],[219,221],[222,226],[227,234],[235,241],[242,250],[251,255],[256,259],[260,261],[261,262],[263,264],[264,265],[266,267],[267,268],[269,272],[273,275],[276,280],[280,281],[282,285],[286,294],[295,298],[299,307],[308,311],[312,313],[313,314],[315,318],[319,327],[328,336],[337,340],[341,342],[342,343],[344,347],[348,351],[352,355],[356,364],[365,373],[373,374]],"norm_spans":[[0,4],[5,8],[9,10],[10,16],[16,17],[18,24],[25,28],[29,30],[30,31],[32,36],[37,41],[42,45],[46,47],[48,53],[54,55],[56,65],[65,66],[67,69],[70,75],[76,78],[79,85],[86,87],[88,91],[92,98],[99,103],[104,105],[106,110],[110,111],[112,120],[121,124],[125,130],[131,137],[138,141],[142,145],[146,149],[150,154],[155,158],[159,165],[166,167],[167,173],[173,174],[175,181],[182,186],[187,191],[192,195],[196,198],[199,209],[210,214],[215,217],[218,222],[223,229],[230,236],[237,243],[244,248],[249,252],[253,254],[254,255],[256,257],[257,258],[259,260],[260,261],[262,265],[266,268],[269,273],[273,274],[275,278],[279,280],[280,286],[286,287],[288,291],[292,298],[299,302],[303,304],[304,305],[306,309],[310,311],[311,317],[317,318],[319,325],[326,329],[330,331],[331,332],[333,336],[337,340],[341,344],[345,353],[354,362],[362,363]],"alignment":[[0,0],[1,1],[2,2],[2,3],[2,4],[3,5],[4,6],[5,7],[6,8],[7,9],[8,10],[9,11],[10,12],[11,13],[12,14],[13,15],[14,16],[15,17],[16,18],[17,19],[18,20],[19,21],[20,22],[21,23],[22,24],[23,25],[24,26],[25,27],[26,28],[27,29],[28,30],[29,31],[30,32],[31,33],[32,34],[33,35],[34,36],[35,37],[36,38],[36,39],[36,40],[37,41],[38,42],[39,43],[40,44],[41,45],[42,46],[43,47],[44,48],[45,49],[46,50],[47,51],[48,52],[49,53],[50,54],[51,55],[52,56],[53,57],[54,58],[55,59],[56,60],[57,61],[58,62],[59,63],[60,64],[61,65],[62,66],[62,67],[62,68],[63,69],[64,70],[65,71],[66,72],[67,73],[68,74],[69,75],[69,76],[69,77],[70,78],[71,79],[72,80],[73,81],[74,82],[75,83],[76,84],[77,85],[78,86],[79,87]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.005899906158447} +{"basename":"euler_rechenkunst01_1738","par_idx":1165,"date":1738,"orig":"Daß aber ein Bruch, in welchem der Zehler dem Nenner gleich iſt, juſt 1 ausmache, laͤſſt ſich hieraus auch leicht erſehen.","norm":"Dass aber ein Bruch, in welchem der Zähler dem Nenner gleich ist, just 1 ausmache, lässt sich hieraus auch leicht 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+{"basename":"euler_rechenkunst01_1738","par_idx":1166,"date":1738,"orig":"Dann da die obere Zahl der unteren gleich iſt, ſo iſt dieſe in jener juſt ein mahl enthalten, und alſo der wahre Quotus 1. Nehmlich [Formel] iſt ſo viel als 1:","norm":"Dann da die obere Zahl der unteren gleich ist, so ist diese in jener just ein Mal enthalten, und also der wahre Quotus 1. Nämlich [Formel] ist so viel als 1:","orig_tok":["Dann","da","die","obere","Zahl","der","unteren","gleich","iſt",",","ſo","iſt","dieſe","in","jener","juſt","ein","mahl","enthalten",",","und","alſo","der","wahre","Quotus","1.","Nehmlich","[Formel]","iſt","ſo","viel","als","1",":"],"norm_tok":["Dann","da","die","obere","Zahl","der","unteren","gleich","ist",",","so","ist","diese","in","jener","just","ein","Mal","enthalten",",","und","also","der","wahre","Quotus","1.","Nämlich","[","Formel","]","ist","so","viel","als","1",":"],"orig_spans":[[0,4],[5,7],[8,11],[12,17],[18,22],[23,26],[27,34],[35,41],[42,45],[45,46],[47,49],[50,53],[54,59],[60,62],[63,68],[69,73],[74,77],[78,82],[83,92],[92,93],[94,97],[98,102],[103,106],[107,112],[113,119],[120,122],[123,131],[132,140],[141,144],[145,147],[148,152],[153,156],[157,158],[158,159]],"norm_spans":[[0,4],[5,7],[8,11],[12,17],[18,22],[23,26],[27,34],[35,41],[42,45],[45,46],[47,49],[50,53],[54,59],[60,62],[63,68],[69,73],[74,77],[78,81],[82,91],[91,92],[93,96],[97,101],[102,105],[106,111],[112,118],[119,121],[122,129],[130,131],[131,137],[137,138],[139,142],[143,145],[146,150],[151,154],[155,156],[156,157]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[27,28],[27,29],[28,30],[29,31],[30,32],[31,33],[32,34],[33,35]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.138700008392334} +{"basename":"euler_rechenkunst01_1738","par_idx":1167,"date":1738,"orig":"dann [Formel] iſt der Quotus ſo herauskommt, wann man 4 durch 4 diuidirt; dieſer Quotus aber iſt 1 ohne Reſt, und alſo iſt [Formel] ſo viel als 1. Gleicher maſſen gibt es auch Bruͤche, welche 2, 3, oder eine andere gantze Zahl ausmachen: alſo iſt [Formel] ſo viel als 2; [Formel] ſo viel als 3. Dergleichen Bruͤche aber ſind eigentlich keine Bruͤche, in dem ihr Werth durch gantze Zahlen angegeben werden kan.","norm":"dann [Formel] ist der Quotus so herauskommt, wann man 4 durch 4 dividiert; dieser Quotus aber ist 1 ohne Rest, und also ist [Formel] so viel als 1. Gleichermaßen gibt es auch Brüche, welche 2, 3, oder eine andere ganze Zahl ausmachen: also ist [Formel] so viel als 2; [Formel] so viel als 3. Dergleichen Brüche aber sind eigentlich keine Brüche, in dem ihr Wert durch ganze Zahlen angegeben werden 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+{"basename":"euler_rechenkunst01_1738","par_idx":1168,"date":1738,"orig":"Weilen aber doch die Schreib-Art die Geſtalt eines Bruchs hat, ſo werden ſolche Bruͤche Schein-Bruͤche oder ſcheinbare Bruͤche genennt, und werden in der Bruͤche-Rechnung auch gebraucht.","norm":"Weilen aber doch die Schreibart die Gestalt eines Bruchs hat, so werden solche Brüche Scheinbrüche oder scheinbare Brüche genannt, und werden in der Brüche-Rechnung auch 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+{"basename":"euler_rechenkunst01_1738","par_idx":1172,"date":1738,"orig":"Alles dieſes aber, was wir aus unſerem erſten Begriff der Bruͤchen hergeleitet, folget gleicher maſſen auch aus dem anderen und noch leichter.","norm":"Alles dieses aber, was wir aus unserem ersten Begriff der Brüchen hergeleitet, folget gleichermaßen auch aus dem anderen und noch 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+{"basename":"euler_rechenkunst01_1738","par_idx":1173,"date":1738,"orig":"Dann da man ein gantzes in ſo viel Theile theilt, als der Nenner eines Bruchs anzeiget, und der Bruch ſelbſt alsdann dergleichen Theile ſo viel enthaͤlt, als der Zehler anweiſt; ſo iſt klar daß wann der Zehler dem Nenner gleich iſt, alsdann der Bruch eben ſo viel Theile enthalte, als ein gantzes ausmachen, und folglich ſelbſt ein gantzes betrage.","norm":"Dann da man ein ganzes in so viel Teile teilt, als der Nenner eines Bruchs anzeiget, und der Bruch selbst alsdann dergleichen Teile so viel enthält, als der Zähler anweist; so ist klar dass wann der Zähler dem Nenner gleich ist, alsdann der Bruch ebenso viel Teile enthalte, als ein ganzes Ausmachen, und folglich selbst ein ganzes 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+{"basename":"euler_rechenkunst01_1738","par_idx":1174,"date":1738,"orig":"Und weilen ferner ein gantzes ſo viel Theile haͤlt, als der Nenner ausweiſt, ſo muß ein Bruch, deſſen Zehler kleiner iſt als der Nenner, kleiner als 1, und ein Bruch deſſen Zehier groſſer iſt als der Nenner groͤſſer als 1 ſeyn.","norm":"Und weilen ferner ein ganzes so viel Teile hält, als der Nenner ausweist, so muss ein Bruch, dessen Zähler kleiner ist als der Nenner, kleiner als 1, und ein Bruch dessen Zehier großer ist als der Nenner größer als 1 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+{"basename":"euler_rechenkunst01_1738","par_idx":1175,"date":1738,"orig":"Dann in jenem Fall ſind weniger Theile, in dieſem aber mehr enthalten als zu einem gantzen erfordert werden.","norm":"Dann in jenem Fall sind weniger Teile, in diesem aber mehr enthalten als zu einem ganzen erfordert 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+{"basename":"euler_rechenkunst01_1738","par_idx":1180,"date":1738,"orig":"Nun um zu wiſſen wieviel gantze in dieſem Bruch enthalten ſind, und auſſer denſelben was fuͤr ein Bruch, ſo diuidirt man den Zehler 20 durch den Nenner 3; da dann in Quotum 6 gantze kommen und noch 2 fuͤr den Reſt zuruͤck bleiben.","norm":"Nun um zu wissen wie viel ganze in diesem Bruch enthalten sind, und außer denselben was für ein Bruch, so dividiert man den Zähler 20 durch den Nenner 3; da dann in Quotum 6 ganze kommen und noch 2 für den Rest 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+{"basename":"euler_rechenkunst01_1738","par_idx":1181,"date":1738,"orig":"Dieſer Reſt 2 gibt nun den Zehler des Bruchs, deſſen Nenner iſt 3 nehmlich ⅔.","norm":"Dieser Rest 2 gibt nun den Zähler des Bruchs, dessen Nenner ist 3 nämlich 2/3 .","orig_tok":["Dieſer","Reſt","2","gibt","nun","den","Zehler","des","Bruchs",",","deſſen","Nenner","iſt","3","nehmlich","⅔","."],"norm_tok":["Dieser","Rest","2","gibt","nun","den","Zähler","des","Bruchs",",","dessen","Nenner","ist","3","nämlich"," 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man dass der vorgelegte Bruch [Formel] so viel sei als 6 ganze nebst 2/3 , welche ganze Zahl nebst dem Bruch also geschrieben zu werden pflegt 6 2/3 ; da der Bruch hinter die ganze Zahl gesetzt wird, und heißt eine solche Ausdrückung eine ganze Zahl nebst einem Bruch.","orig_tok":["Hierauf","ſagt","man","daß","der","vorgelegte","Bruch","[Formel]","ſo","viel","ſey","als","6","gantze","nebſt","⅔",",","welche","gantze","Zahl","nebſt","dem","Bruch","alſo","geſchrieben","zu","werden","pflegt","6⅔",";","da","der","Bruch","hinter","die","gantze","Zahl","geſetzt","wird",",","und","heiſſt","eine","ſolche","Ausdruͤckung","eine","gantze","Zahl","nebſt","einem","Bruch","."],"norm_tok":["Hierauf","sagt","man","dass","der","vorgelegte","Bruch","[","Formel","]","so","viel","sei","als","6","ganze","nebst"," 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+{"basename":"euler_rechenkunst01_1738","par_idx":1183,"date":1738,"orig":"Alſo iſt [Formel] eben ſo viel als 6⅔; gleichergeſtalt iſt [Formel] ſo viel als 4 [Formel] ; und [Formel] ſo viel als 4 [Formel] .","norm":"Also ist [Formel] ebenso viel als 6 2/3 ; gleichergestalt ist [Formel] so viel als 4 [Formel]; und [Formel] so viel als 4 [Formel].","orig_tok":["Alſo","iſt","[Formel]","eben","ſo","viel","als","6⅔",";","gleichergeſtalt","iſt","[Formel]","ſo","viel","als","4","[Formel]",";","und","[Formel]","ſo","viel","als","4","[Formel]","."],"norm_tok":["Also","ist","[","Formel","]","ebenso","viel","als","6","2","/","3",";","gleichergestalt","ist","[","Formel","]","so","viel","als","4","[","Formel","]",";","und","[","Formel","]","so","viel","als","4","[","Formel","]","."],"orig_spans":[[0,4],[5,8],[9,17],[18,22],[23,25],[26,30],[31,34],[35,37],[37,38],[39,54],[55,58],[59,67],[68,70],[71,75],[76,79],[80,81],[82,90],[91,92],[93,96],[97,105],[106,108],[109,113],[114,117],[118,119],[120,128],[129,130]],"norm_spans":[[0,4],[5,8],[9,10],[10,16],[16,17],[18,24],[25,29],[30,33],[34,35],[36,37],[37,38],[38,39],[40,41],[42,57],[58,61],[62,63],[63,69],[69,70],[71,73],[74,78],[79,82],[83,84],[85,86],[86,92],[92,93],[93,94],[95,98],[99,100],[100,106],[106,107],[108,110],[111,115],[116,119],[120,121],[122,123],[123,129],[129,130],[130,131]],"alignment":[[0,0],[1,1],[2,2],[2,3],[2,4],[3,5],[4,5],[5,6],[6,7],[7,8],[null,9],[null,10],[null,11],[8,12],[9,13],[10,14],[11,15],[11,16],[11,17],[12,18],[13,19],[14,20],[15,21],[16,22],[16,23],[16,24],[17,25],[18,26],[19,27],[19,28],[19,29],[20,30],[21,31],[22,32],[23,33],[24,34],[24,35],[24,36],[25,37]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"ca","orig_lang_de":0.667,"norm_lmscore":3.635499954223633} +{"basename":"euler_rechenkunst01_1738","par_idx":1184,"date":1738,"orig":"Dann wann man 33 durch ‒ diuidirt ſo kommen in Quotum 4 gantze, und in Reſt 5 woraus der angehaͤngte Bruch [Formel] entſtehet.","norm":"Dann wann man 33 durch ‒ dividiert so kommen in Quotum 4 ganze, und in Rest 5 woraus der angehängte Bruch [Formel] 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+{"basename":"euler_rechenkunst01_1738","par_idx":1185,"date":1738,"orig":"Diuidirt man aber 51 durch 11 ſo kommen in Quotum 4 gantze und reſtiren noch 7 daher der Bruch [Formel] entſpringet.","norm":"Dividiert man aber 51 durch 11 so kommen in Quotum 4 ganze und restieren noch 7 daher der Bruch [Formel] 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+{"basename":"euler_rechenkunst01_1738","par_idx":1186,"date":1738,"orig":"Auf dieſe Art erkennt man alſo gleich, wieviel gantze in einem Bruche enthalten ſind, und was fuͤr ein Bruch noch auſſer denſelben dazu gehoͤre.","norm":"Auf diese Art erkennt man also gleich, wie viel ganze in einem Bruche enthalten sind, und was für ein Bruch noch außer denselben dazugehöre.","orig_tok":["Auf","dieſe","Art","erkennt","man","alſo","gleich",",","wieviel","gantze","in","einem","Bruche","enthalten","ſind",",","und","was","fuͤr","ein","Bruch","noch","auſſer","denſelben","dazu","gehoͤre","."],"norm_tok":["Auf","diese","Art","erkennt","man","also","gleich",",","wie","viel","ganze","in","einem","Bruche","enthalten","sind",",","und","was","für","ein","Bruch","noch","außer","denselben","dazugehöre","."],"orig_spans":[[0,3],[4,9],[10,13],[14,21],[22,25],[26,30],[31,37],[37,38],[39,46],[47,53],[54,56],[57,62],[63,69],[70,79],[80,84],[84,85],[86,89],[90,93],[94,98],[99,102],[103,108],[109,113],[114,120],[121,130],[131,135],[136,143],[143,144]],"norm_spans":[[0,3],[4,9],[10,13],[14,21],[22,25],[26,30],[31,37],[37,38],[39,42],[43,47],[48,53],[54,56],[57,62],[63,69],[70,79],[80,84],[84,85],[86,89],[90,93],[94,97],[98,101],[102,107],[108,112],[113,118],[119,128],[129,139],[139,140]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18],[18,19],[19,20],[20,21],[21,22],[22,23],[23,24],[24,25],[25,25],[26,26]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.885000228881836} +{"basename":"euler_rechenkunst01_1738","par_idx":1187,"date":1738,"orig":"Man findet nemlich eine gantze Zahl nebſt einem Bruche, welche zuſammen eben ſo viel ausmachen, als der vorgelegte Bruch.","norm":"Man findet nämlich eine ganze Zahl nebst einem Bruche, welche zusammen ebenso viel ausmachen, als der vorgelegte 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+{"basename":"euler_rechenkunst01_1738","par_idx":1188,"date":1738,"orig":"Durch dieſe Operation erhaͤlt man alſo einen deutlichern Begriff von einem Bruch, indem man erkennet, wieviel derſelbe gantze und nebſt denſelben noch was fuͤr einen Brnch in ſich begreiffe.","norm":"Durch diese Operation erhält man also einen deutlicheren Begriff von einem Bruch, indem man erkennt, wie viel derselbe ganze und nebst denselben noch was für einen Brnch in sich 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+{"basename":"euler_rechenkunst01_1738","par_idx":1189,"date":1738,"orig":"Es iſt aber klar, daß dieſer angehaͤngte Bruch allezeit kleiner ſeyn als ein gantzes, dann ſein Zehler iſt der aus der Diuiſion entſprungene Reſt, welcher allezeit kleiner iſt als der Theiler, ſo zum Nenner gemacht wird.","norm":"Es ist aber klar, dass dieser angehängte Bruch allezeit kleiner sein als ein ganzes, dann sein Zähler ist der aus der Division entsprungene Rest, welcher allezeit kleiner ist als der Teiler, so zum Nenner gemacht 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+{"basename":"euler_rechenkunst01_1738","par_idx":1197,"date":1738,"orig":"Jn ſolchem Fall iſt alſo der wahre Quotus nichts anders als der gefundene Quotus in gantzen Zahlen nebſt dem Bruch deſſen Zehler der Reſt, der Nenner aber der Theiler oder des vorigen Bruchs Nenner ſelbſt iſt.","norm":"In solchem Fall ist also der wahre Quotus nichts anders als der gefundene Quotus in ganzen Zahlen nebst dem Bruch dessen Zähler der Rest, der Nenner aber der Teiler oder des vorigen Bruchs Nenner selbst 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+{"basename":"euler_rechenkunst01_1738","par_idx":1198,"date":1738,"orig":"Da alſo ein jeder Bruch nichts anders iſt, als der voͤllige Quotus, der herauskommt, wann man den Zehler durch den Nenner diuidirt, ſo iſt derſelbe auch gleich dem auf beſchriebene Art durch die Diuiſion gefundenen voͤlligen Quoto; nehmlich der durch die Diuiſion fuͤr den Quotum gefundenen gantzen Zahl, nebſt dem Bruch deſſen Zehler der zuruͤck gebliebene Reſt, der Nenner aber eben des vorigen Bruchs Nenner iſt.","norm":"Da also ein jeder Bruch nichts anders ist, als der völlige Quotus, der herauskommt, wann man den Zähler durch den Nenner dividiert, so ist derselbe auch gleich dem auf beschriebene Art durch die Division gefundenen völligen Quoto; nämlich der durch die Division für den Quotum gefundenen ganzen Zahl, nebst dem Bruch dessen Zähler der zurück gebliebene Rest, der Nenner aber eben des vorigen Bruchs Nenner 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+{"basename":"euler_rechenkunst01_1738","par_idx":1199,"date":1738,"orig":"Dieſes iſt demnach der Grund der gegebenen Regul, durch welche man einen Bruch, der groͤſſer iſt als 1, jn eine gantze Zahl nebſt einem Bruch verwandelt.","norm":"Dieses ist demnach der Grund der gegebenen Regel, durch welche man einen Bruch, der größer ist als 1, ihn eine ganze Zahl nebst einem Bruch 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+{"basename":"euler_rechenkunst01_1738","par_idx":1204,"date":1738,"orig":"Beyde Operationen haben ihren groſſen Nutzen:","norm":"Beide Operationen haben ihren großen Nutzen:","orig_tok":["Beyde","Operationen","haben","ihren","groſſen","Nutzen",":"],"norm_tok":["Beide","Operationen","haben","ihren","großen","Nutzen",":"],"orig_spans":[[0,5],[6,17],[18,23],[24,29],[30,37],[38,44],[44,45]],"norm_spans":[[0,5],[6,17],[18,23],[24,29],[30,36],[37,43],[43,44]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.569200038909912} +{"basename":"euler_rechenkunst01_1738","par_idx":1205,"date":1738,"orig":"denn durch die erſte erhaͤlt man, wie ſchon gemeldt, einen deutlichern Begriff von dem Jnhalt oder Werth eines Bruchs, die andere aber iſt in denen folgenden Operationen mit dem Bruͤchen hoͤchſt noͤthig, da um dieſelben zu bewerckſtelligen gemeiniglich eine gantze Zahl nebſt angehaͤngtem Bruche in einen einzelen Bruch verwandelt werden muß.","norm":"denn durch die erste erhält man, wie schon gemeldet, einen deutlicheren Begriff von dem Inhalt oder Wert eines Bruchs, die andere aber ist in denen folgenden Operationen mit dem Brüchen höchst nötig, da um dieselben zu bewerkstelligen gemeiniglich eine ganze Zahl nebst angehängtem Bruche in einen einzeln Bruch verwandelt werden 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+{"basename":"euler_rechenkunst01_1738","par_idx":1206,"date":1738,"orig":"Die gegebene Regel verhaͤlt ſich nun alſo:","norm":"Die gegebene Regel verhält sich nun also:","orig_tok":["Die","gegebene","Regel","verhaͤlt","ſich","nun","alſo",":"],"norm_tok":["Die","gegebene","Regel","verhält","sich","nun","also",":"],"orig_spans":[[0,3],[4,12],[13,18],[19,27],[28,32],[33,36],[37,41],[41,42]],"norm_spans":[[0,3],[4,12],[13,18],[19,26],[27,31],[32,35],[36,40],[40,41]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"tr","orig_lang_de":0.667,"norm_lmscore":5.925600051879883} +{"basename":"euler_rechenkunst01_1738","par_idx":1207,"date":1738,"orig":"es ſey gegeben 7⅔ nehmlich eine gantze Zahl 7 nebſt dem Bruch ⅔; woraus ein einzeler Bruch gemacht werden ſoll.","norm":"es sei gegeben 7 2/3 nämlich eine ganze Zahl 7 nebst dem Bruch 2/3 ; woraus ein Einzeller Bruch gemacht werden 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Product 21 thut man 2, ſo bekommt man 23 fuͤr den Zehler des geſuchten Bruchs, deſſen Nenner iſt 3, nehmlich [Formel] .","norm":"Man multipliziert also 7 mit 3, und zum Produkt 21 tut man 2, so bekommt man 23 für den Zähler des gesuchten Bruchs, dessen Nenner ist 3, nämlich 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+{"basename":"euler_rechenkunst01_1738","par_idx":1209,"date":1738,"orig":"Daß nun dieſer Bruch [Formel] eben ſo viel ſey als 7⅔, erhaͤlt aus dem vorigen Satz’ dadurch [Formel] in 7⅔ verwandelt wird.","norm":"Dass nun dieser Bruch [Formel] ebenso viel sei als 7 2/3 , erhält aus dem vorigen Satz dadurch [Formel] in 7 2/3 verwandelt wird.","orig_tok":["Daß","nun","dieſer","Bruch","[Formel]","eben","ſo","viel","ſey","als","7⅔",",","erhaͤlt","aus","dem","vorigen","Satz’","dadurch","[Formel]","in","7⅔","verwandelt","wird","."],"norm_tok":["Dass","nun","dieser","Bruch","[","Formel","]","ebenso","viel","sei","als","7","2","/","3",",","erhält","aus","dem","vorigen","Satz","dadurch","[","Formel","]","in","7","2","/","3"," 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gantze Zahl der Quotus in gantzen Zahlen, wie derſelbe in der Diuiſion iſt gefunden worden; der der Zehler des Bruchs aber der Reſt iſt.","norm":"Eine jede Zahl nebst angehängtem Bruche kann angesehen werden, als ein aus der Division entsprungener wahrer Quotus, da der Nenner des angehängten Bruchs der Divisor, die ganze Zahl der Quotus in ganzen Zahlen, wie derselbe in der Division ist gefunden worden; der der Zähler des Bruchs aber der Rest 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+{"basename":"euler_rechenkunst01_1738","par_idx":1226,"date":1738,"orig":"Wann nun ferner eben dieſes Bruchs ⅔ Zehler und Nenner durch 3 multiplicirt wird, ſo hat man [Formel] ; welcher wiederum ſo viel iſt als ⅔.","norm":"Wann nun ferner eben dieses Bruchs 2/3 Zähler und Nenner durch 3 multipliziert wird, so hat man [Formel]; welcher wiederum so viel ist als 2/3 .","orig_tok":["Wann","nun","ferner","eben","dieſes","Bruchs","⅔","Zehler","und","Nenner","durch","3","multiplicirt","wird",",","ſo","hat","man","[Formel]",";","welcher","wiederum","ſo","viel","iſt","als","⅔","."],"norm_tok":["Wann","nun","ferner","eben","dieses","Bruchs"," ","2","/","3"," ","Zähler","und","Nenner","durch","3","multipliziert","wird",",","so","hat","man","[","Formel","]",";","welcher","wiederum","so","viel","ist","als"," 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+{"basename":"euler_rechenkunst01_1738","par_idx":1227,"date":1738,"orig":"Wann man alſo fortfaͤhrt durch 4, 5, 6 und ſo fort an zu multipliciren, ſo kommen folgende Bruͤche heraus [Formel] , [Formel] , [Formel] , und ſo weiter fort; welche alle eben ſo viel halten als ⅔.","norm":"Wann man also fortfährt durch 4, 5, 6 und so fort an zu multiplizieren, so kommen folgende Brüche heraus [Formel], [Formel], [Formel], und so weiter fort; welche alle ebenso viel halten als 2/3 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+{"basename":"euler_rechenkunst01_1738","par_idx":1228,"date":1738,"orig":"Gleichergeſtalt ſind auch alle folgenden Bruͤche ½, [Formel] , [Formel] , [Formel] , [Formel] , [Formel] , und ſo fort einander gleich, und iſt ein jeglicher davon ſo viel als ein halbes.","norm":"Gleichergestalt sind auch alle folgenden Brüche ½, [Formel], [Formel], [Formel], [Formel], [Formel], und so fort einander gleich, und ist ein jeglicher davon so viel als ein 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+{"basename":"euler_rechenkunst01_1738","par_idx":1229,"date":1738,"orig":"Es kan allſo eben derſelbige Bruch auf unendlich vielerley Arten vorgeſtellet werden, indem wenn ſo wohl der Zehler als Nenner durch eine jegliche Zahl multiplicirt wird; ein Bruch heraus kommt, der dem vorigen gleich iſt.","norm":"Es kann also eben derselbige Bruch auf unendlich vielerlei Arten vorgestellt werden, indem wenn so wohl der Zähler als Nenner durch eine jegliche Zahl multipliziert wird; ein Bruch herauskommt, der dem vorigen gleich 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+{"basename":"euler_rechenkunst01_1738","par_idx":1230,"date":1738,"orig":"Auf dieſe Weiſe aber nehmlich durch das multipliciren erhalt man allzeit Bruͤche, welche aus groͤſſeren Zahlen beſtehen als der vorgelegte.","norm":"Auf diese Weise aber nämlich durch das Multiplizieren erhalte man allzeit Brüche, welche aus größeren Zahlen bestehen als der 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+{"basename":"euler_rechenkunst01_1738","par_idx":1231,"date":1738,"orig":"Es iſt aber klar, daß man hinwiederum aus dieſen aus groſſen Zahlen beſtehenden Bruͤchen diejenigen muͤſſe finden koͤnnen, welche aus kleinern Zahlen beſtehen, und aus welchen jene durch die Multiplication entſtanden ſind.","norm":"Es ist aber klar, dass man hinwiederum aus diesen aus großen Zahlen bestehenden Brüchen diejenigen müsse finden können, welche aus kleineren Zahlen bestehen, und aus welchen jene durch die Multiplikation entstanden 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+{"basename":"euler_rechenkunst01_1738","par_idx":1232,"date":1738,"orig":"Dieſes geſchicht nun durch die Diuiſion, da beydes der Nenner und Zehler durch eine beliebige Zahl diuidirt wird, wann nehmlich die Diuiſion angeht.","norm":"Dieses geschieht nun durch die Division, da beides der Nenner und Zähler durch eine beliebige Zahl dividiert wird, wann nämlich die Division 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+{"basename":"euler_rechenkunst01_1738","par_idx":1233,"date":1738,"orig":"Dann gleich wie aus dieſem Bruche ⅗; wenn oben und unten durch 7 multiplicirt wird, dieſer [Formel] entſpringt, ſo erhaͤlt man hinwiederum aus dieſem Bruche [Formel] den vorigen ⅗, wann man beydes den Nenner und Zehler durch 7 diuidirt.","norm":"Dann gleich wie aus diesem Bruche 3/5 ; wenn oben und unten durch 7 multipliziert wird, dieser [Formel] entspringt, so erhält man hinwiederum aus diesem Bruche [Formel] den vorigen 3/5 , wann man beides den Nenner und Zähler durch 7 dividiert.","orig_tok":["Dann","gleich","wie","aus","dieſem","Bruche","⅗",";","wenn","oben","und","unten","durch","7","multiplicirt","wird",",","dieſer","[Formel]","entſpringt",",","ſo","erhaͤlt","man","hinwiederum","aus","dieſem","Bruche","[Formel]","den","vorigen","⅗",",","wann","man","beydes","den","Nenner","und","Zehler","durch","7","diuidirt","."],"norm_tok":["Dann","gleich","wie","aus","diesem","Bruche"," 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+{"basename":"euler_rechenkunst01_1738","par_idx":1234,"date":1738,"orig":"Aus dieſen zweyerley Arten einen Bruch in andere Formen zu verwandeln, ſieht man nun, daß man Bruͤche angeben koͤnne, welche ſo wohl aus aroͤſſeren als kleineren Zahlen, als ein vorgegebener Bruch iſt, beſtehen, und demſelben dennoch dem Werth nach gleich ſind; Deren jenes vermittelſt der Multiplication, dieſes aber durch die Diuiſion geſchicht.","norm":"Aus diesen zweierlei Arten einen Bruch in andere Formen zu verwandeln, sieht man nun, dass man Brüche angeben könne, welche so wohl aus arösseren als kleineren Zahlen, als ein vorgegebener Bruch ist, bestehen, und demselben dennoch dem Wert nach gleich sind; deren jenes vermittels der Multiplikation, dieses aber durch die Division 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+{"basename":"euler_rechenkunst01_1738","par_idx":1235,"date":1738,"orig":"Hiebey aber iſt zum voraus zu erinnern, daß man dieſe beyden Operationen der Multiplication und Diuiſion nicht mit der eigentlichen Multiplication und Diuiſion der Bruͤche confundire; dann auf die jetzt beſchriebene Art, wird ein Bruch nur in eine andere Geſtalt gebracht, ohne ſeinen Werth zu veraͤnderen; Wann aber ein Bruch entweder multiplicirt oder diuidirt werden ſoll, ſo ſuchet man einen Bruch, welcher entweder groͤſſer oder kleiner ſeyn ſoll als der vorgelegte; ſo daß dieſe Operationen, welche zu den Speciebus der Bruͤche gehoͤren, von der hier beſchriebenen Verwandlung gaͤntzlich unterſchieden ſind.","norm":"Hierbei aber ist zum voraus zu erinnern, dass man diese beiden Operationen der Multiplikation und Division nicht mit der eigentlichen Multiplikation und Division der Brüche konfundiere; dann auf die jetzt beschriebene Art, wird ein Bruch nur in eine andere Gestalt gebracht, ohne seinen Wert zu verändere; wann aber ein Bruch entweder multipliziert oder dividiert werden soll, so suchet man einen Bruch, welcher entweder größer oder kleiner sein soll als der vorgelegte; so dass diese Operationen, welche zu den Speciebus der Brüche gehören, von der hier beschriebenen Verwandlung gänzlich unterschieden 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+{"basename":"euler_rechenkunst01_1738","par_idx":1237,"date":1738,"orig":"Ein jeglicher Bruch zeiget demnach an, wieviel mahl der Nenner im Zehler enthalten ſey.","norm":"Ein jeglicher Bruch zeiget demnach an, wie viel Mal der Nenner im Zähler enthalten sei.","orig_tok":["Ein","jeglicher","Bruch","zeiget","demnach","an",",","wieviel","mahl","der","Nenner","im","Zehler","enthalten","ſey","."],"norm_tok":["Ein","jeglicher","Bruch","zeiget","demnach","an",",","wie","viel","Mal","der","Nenner","im","Zähler","enthalten","sei","."],"orig_spans":[[0,3],[4,13],[14,19],[20,26],[27,34],[35,37],[37,38],[39,46],[47,51],[52,55],[56,62],[63,65],[66,72],[73,82],[83,86],[86,87]],"norm_spans":[[0,3],[4,13],[14,19],[20,26],[27,34],[35,37],[37,38],[39,42],[43,47],[48,51],[52,55],[56,62],[63,65],[66,72],[73,82],[83,86],[86,87]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.047100067138672} +{"basename":"euler_rechenkunst01_1738","par_idx":1238,"date":1738,"orig":"Es iſt aber klar, daß ſo viel mahl bey einem Bruche der Nenner im Zehler enthalten iſt, eben ſo vielmal der doppelte Nenner im doppelten Zehler enthalten ſey; und folglich auch eben ſo viel mal der halbe Nenner im halben Zehler: woraus dann erhellet, daß wann man beydes den Nenner und Zehler eines Bruchs durch 2 entweder multiplicirt oder diuidirt, der hieraus entſtehende Bruch eben ſo viel betrage als der vorgegebene.","norm":"Es ist aber klar, dass so viel Mal bei einem Bruche der Nenner im Zähler enthalten ist, ebenso vielmal der doppelte Nenner im doppelten Zähler enthalten sei; und folglich auch ebenso viel Mal der halbe Nenner im halben Zähler: woraus dann erhellet, dass wann man beides den Nenner und Zähler eines Bruchs durch 2 entweder multipliziert oder dividiert, der hieraus entstehende Bruch ebenso viel betrage als der vorgegebene.","orig_tok":["Es","iſt","aber","klar",",","daß","ſo","viel","mahl","bey","einem","Bruche","der","Nenner","im","Zehler","enthalten","iſt",",","eben","ſo","vielmal","der","doppelte","Nenner","im","doppelten","Zehler","enthalten","ſey",";","und","folglich","auch","eben","ſo","viel","mal","der","halbe","Nenner","im","halben","Zehler",":","woraus","dann","erhellet",",","daß","wann","man","beydes","den","Nenner","und","Zehler","eines","Bruchs","durch","2","entweder","multiplicirt","oder","diuidirt",",","der","hieraus","entſtehende","Bruch","eben","ſo","viel","betrage","als","der","vorgegebene","."],"norm_tok":["Es","ist","aber","klar",",","dass","so","viel","Mal","bei","einem","Bruche","der","Nenner","im","Zähler","enthalten","ist",",","ebenso","vielmal","der","doppelte","Nenner","im","doppelten","Zähler","enthalten","sei",";","und","folglich","auch","ebenso","viel","Mal","der","halbe","Nenner","im","halben","Zähler",":","woraus","dann","erhellet",",","dass","wann","man","beides","den","Nenner","und","Zähler","eines","Bruchs","durch","2","entweder","multipliziert","oder","dividiert",",","der","hieraus","entstehende","Bruch","ebenso","viel","betrage","als","der","vorgegebene","."],"orig_spans":[[0,2],[3,6],[7,11],[12,16],[16,17],[18,21],[22,24],[25,29],[30,34],[35,38],[39,44],[45,51],[52,55],[56,62],[63,65],[66,72],[73,82],[83,86],[86,87],[88,92],[93,95],[96,103],[104,107],[108,116],[117,123],[124,126],[127,136],[137,143],[144,153],[154,157],[157,158],[159,162],[163,171],[172,176],[177,181],[182,184],[185,189],[190,193],[194,197],[198,203],[204,210],[211,213],[214,220],[221,227],[227,228],[229,235],[236,240],[241,249],[249,250],[251,254],[255,259],[260,263],[264,270],[271,274],[275,281],[282,285],[286,292],[293,298],[299,305],[306,311],[312,313],[314,322],[323,335],[336,340],[341,349],[349,350],[351,354],[355,362],[363,374],[375,380],[381,385],[386,388],[389,393],[394,401],[402,405],[406,409],[410,421],[421,422]],"norm_spans":[[0,2],[3,6],[7,11],[12,16],[16,17],[18,22],[23,25],[26,30],[31,34],[35,38],[39,44],[45,51],[52,55],[56,62],[63,65],[66,72],[73,82],[83,86],[86,87],[88,94],[95,102],[103,106],[107,115],[116,122],[123,125],[126,135],[136,142],[143,152],[153,156],[156,157],[158,161],[162,170],[171,175],[176,182],[183,187],[188,191],[192,195],[196,201],[202,208],[209,211],[212,218],[219,225],[225,226],[227,233],[234,238],[239,247],[247,248],[249,253],[254,258],[259,262],[263,269],[270,273],[274,280],[281,284],[285,291],[292,297],[298,304],[305,310],[311,312],[313,321],[322,335],[336,340],[341,350],[350,351],[352,355],[356,363],[364,375],[376,381],[382,388],[389,393],[394,401],[402,405],[406,409],[410,421],[421,422]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,19],[21,20],[22,21],[23,22],[24,23],[25,24],[26,25],[27,26],[28,27],[29,28],[30,29],[31,30],[32,31],[33,32],[34,33],[35,33],[36,34],[37,35],[38,36],[39,37],[40,38],[41,39],[42,40],[43,41],[44,42],[45,43],[46,44],[47,45],[48,46],[49,47],[50,48],[51,49],[52,50],[53,51],[54,52],[55,53],[56,54],[57,55],[58,56],[59,57],[60,58],[61,59],[62,60],[63,61],[64,62],[65,63],[66,64],[67,65],[68,66],[69,67],[70,68],[71,68],[72,69],[73,70],[74,71],[75,72],[76,73],[77,74]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.208899974822998} +{"basename":"euler_rechenkunst01_1738","par_idx":1239,"date":1738,"orig":"Gleich wie man nun leicht ſieht, daß was hier von der Zahl 2 geſagt worden, ſeine Richtigkeit hat; ſo laͤßt ſich eben daſſelbe von der Zahl 3, 4, und ſo gar von einer jeglichen Zahl begreiffen.","norm":"Gleich wie man nun leicht sieht, dass was hier von der Zahl 2 gesagt worden, seine Richtigkeit hat; so lässt sich eben dasselbe von der Zahl 3, 4, und so gar von einer jeglichen Zahl begreifen.","orig_tok":["Gleich","wie","man","nun","leicht","ſieht",",","daß","was","hier","von","der","Zahl","2","geſagt","worden",",","ſeine","Richtigkeit","hat",";","ſo","laͤßt","ſich","eben","daſſelbe","von","der","Zahl","3",",","4",",","und","ſo","gar","von","einer","jeglichen","Zahl","begreiffen","."],"norm_tok":["Gleich","wie","man","nun","leicht","sieht",",","dass","was","hier","von","der","Zahl","2","gesagt","worden",",","seine","Richtigkeit","hat",";","so","lässt","sich","eben","dasselbe","von","der","Zahl","3",",","4",",","und","so","gar","von","einer","jeglichen","Zahl","begreifen","."],"orig_spans":[[0,6],[7,10],[11,14],[15,18],[19,25],[26,31],[31,32],[33,36],[37,40],[41,45],[46,49],[50,53],[54,58],[59,60],[61,67],[68,74],[74,75],[76,81],[82,93],[94,97],[97,98],[99,101],[102,107],[108,112],[113,117],[118,126],[127,130],[131,134],[135,139],[140,141],[141,142],[143,144],[144,145],[146,149],[150,152],[153,156],[157,160],[161,166],[167,176],[177,181],[182,192],[192,193]],"norm_spans":[[0,6],[7,10],[11,14],[15,18],[19,25],[26,31],[31,32],[33,37],[38,41],[42,46],[47,50],[51,54],[55,59],[60,61],[62,68],[69,75],[75,76],[77,82],[83,94],[95,98],[98,99],[100,102],[103,108],[109,113],[114,118],[119,127],[128,131],[132,135],[136,140],[141,142],[142,143],[144,145],[145,146],[147,150],[151,153],[154,157],[158,161],[162,167],[168,177],[178,182],[183,192],[192,193]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28],[29,29],[30,30],[31,31],[32,32],[33,33],[34,34],[35,35],[36,36],[37,37],[38,38],[39,39],[40,40],[41,41]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.446499824523926} +{"basename":"euler_rechenkunst01_1738","par_idx":1240,"date":1738,"orig":"Hieraus folget nun der vorgebrachte Satz, daß ein Bruch an ſeinem Werth nichts verliere, wann gleich beydes der Zehler und Nenner durch eine jegliche beliebige Zahl entweder multiplicirt oder diuidirt werden.","norm":"Hieraus folget nun der vorgebrachte Satz, dass ein Bruch an seinem Wert nichts verliere, wann gleich beides der Zähler und Nenner durch eine jegliche beliebige Zahl entweder multipliziert oder dividiert werden.","orig_tok":["Hieraus","folget","nun","der","vorgebrachte","Satz",",","daß","ein","Bruch","an","ſeinem","Werth","nichts","verliere",",","wann","gleich","beydes","der","Zehler","und","Nenner","durch","eine","jegliche","beliebige","Zahl","entweder","multiplicirt","oder","diuidirt","werden","."],"norm_tok":["Hieraus","folget","nun","der","vorgebrachte","Satz",",","dass","ein","Bruch","an","seinem","Wert","nichts","verliere",",","wann","gleich","beides","der","Zähler","und","Nenner","durch","eine","jegliche","beliebige","Zahl","entweder","multipliziert","oder","dividiert","werden","."],"orig_spans":[[0,7],[8,14],[15,18],[19,22],[23,35],[36,40],[40,41],[42,45],[46,49],[50,55],[56,58],[59,65],[66,71],[72,78],[79,87],[87,88],[89,93],[94,100],[101,107],[108,111],[112,118],[119,122],[123,129],[130,135],[136,140],[141,149],[150,159],[160,164],[165,173],[174,186],[187,191],[192,200],[201,207],[207,208]],"norm_spans":[[0,7],[8,14],[15,18],[19,22],[23,35],[36,40],[40,41],[42,46],[47,50],[51,56],[57,59],[60,66],[67,71],[72,78],[79,87],[87,88],[89,93],[94,100],[101,107],[108,111],[112,118],[119,122],[123,129],[130,135],[136,140],[141,149],[150,159],[160,164],[165,173],[174,187],[188,192],[193,202],[203,209],[209,210]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28],[29,29],[30,30],[31,31],[32,32],[33,33]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.92140007019043} +{"basename":"euler_rechenkunst01_1738","par_idx":1241,"date":1738,"orig":"Zu fernerer Erlaͤuterung dieſer Operation durch die Multiplication koͤnnen folgende Exempel dienen.","norm":"Zu fernerer Erläuterung dieser Operation durch die Multiplikation können folgende Exempel dienen.","orig_tok":["Zu","fernerer","Erlaͤuterung","dieſer","Operation","durch","die","Multiplication","koͤnnen","folgende","Exempel","dienen","."],"norm_tok":["Zu","fernerer","Erläuterung","dieser","Operation","durch","die","Multiplikation","können","folgende","Exempel","dienen","."],"orig_spans":[[0,2],[3,11],[12,24],[25,31],[32,41],[42,47],[48,51],[52,66],[67,74],[75,83],[84,91],[92,98],[98,99]],"norm_spans":[[0,2],[3,11],[12,23],[24,30],[31,40],[41,46],[47,50],[51,65],[66,72],[73,81],[82,89],[90,96],[96,97]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.857800006866455} +{"basename":"euler_rechenkunst01_1738","par_idx":1242,"date":1738,"orig":"iſt ſo viel als [Formel] oder [Formel] oder [Formel] . Jmgleichen 2⅓ iſt ſo viel als 2 [Formel] oder 2 [Formel] , weilen [Formel] und [Formel] ſo viel ſind als ⅓ und die gantze Zahl 2 bey allen einerley iſt.","norm":"ist so viel als [Formel] oder [Formel] oder [Formel]. Im gleichen 2 1/3 ist so viel als 2 [Formel] oder 2 [Formel], weilen [Formel] und [Formel] so viel sind als 1/3 und die ganze Zahl 2 bei allen einerlei ist.","orig_tok":["iſt","ſo","viel","als","[Formel]","oder","[Formel]","oder","[Formel]",".","Jmgleichen","2⅓","iſt","ſo","viel","als","2","[Formel]","oder","2","[Formel]",",","weilen","[Formel]","und","[Formel]","ſo","viel","ſind","als","⅓","und","die","gantze","Zahl","2","bey","allen","einerley","iſt","."],"norm_tok":["ist","so","viel","als","[","Formel","]","oder","[","Formel","]","oder","[","Formel","]",".","Im","gleichen","2","1","/","3"," ","ist","so","viel","als","2","[","Formel","]","oder","2","[","Formel","]",",","weilen","[","Formel","]","und","[","Formel","]","so","viel","sind","als"," ","1","/","3"," ","und","die","ganze","Zahl","2","bei","allen","einerlei","ist","."],"orig_spans":[[0,3],[4,6],[7,11],[12,15],[16,24],[25,29],[30,38],[39,43],[44,52],[53,54],[55,65],[66,68],[69,72],[73,75],[76,80],[81,84],[85,86],[87,95],[96,100],[101,102],[103,111],[112,113],[114,120],[121,129],[130,133],[134,142],[143,145],[146,150],[151,155],[156,159],[160,161],[162,165],[166,169],[170,176],[177,181],[182,183],[184,187],[188,193],[194,202],[203,206],[206,207]],"norm_spans":[[0,3],[4,6],[7,11],[12,15],[16,17],[17,23],[23,24],[25,29],[30,31],[31,37],[37,38],[39,43],[44,45],[45,51],[51,52],[52,53],[54,56],[57,65],[66,67],[68,69],[69,70],[70,71],[72,73],[73,76],[77,79],[80,84],[85,88],[89,90],[91,92],[92,98],[98,99],[100,104],[105,106],[107,108],[108,114],[114,115],[115,116],[117,123],[124,125],[125,131],[131,132],[133,136],[137,138],[138,144],[144,145],[146,148],[149,153],[154,158],[159,162],[163,164],[164,165],[165,166],[166,167],[168,169],[169,172],[173,176],[177,182],[183,187],[188,189],[190,193],[194,199],[200,208],[209,212],[212,213]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[4,5],[4,6],[5,7],[6,8],[6,9],[6,10],[7,11],[8,12],[8,13],[8,14],[9,15],[10,16],[10,17],[11,18],[null,19],[null,20],[null,21],[null,22],[12,23],[13,24],[14,25],[15,26],[16,27],[17,28],[17,29],[17,30],[18,31],[19,32],[20,33],[20,34],[20,35],[21,36],[22,37],[23,38],[23,39],[23,40],[24,41],[25,42],[25,43],[25,44],[26,45],[27,46],[28,47],[29,48],[30,null],[31,null],[null,49],[null,50],[null,51],[null,52],[null,53],[null,54],[32,55],[33,56],[34,57],[35,58],[36,59],[37,60],[38,61],[39,62],[40,63]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":3.885299921035766} +{"basename":"euler_rechenkunst01_1738","par_idx":1243,"date":1738,"orig":"Gleicher Geſtalt iſt 3 ſo viel als [Formel] , item als [Formel] , item als [Formel] , und ſo fort; dann 3 iſt ſo viel als [Formel] , wann man nun oben und unten durch 2 oder 3, oder 4 multiplicirt, ſo kommen [Formel] , [Formel] , und [Formel] heraus, welche Bruͤche folglich ſo viel ſind als 3. Hieraus ſieht man nun, daß man eine jegliche gantze Zahl in eine Bruchs Form verwandeln kan, von einem beliebigen Nenner, als wann man einen Bruche verlangte, der ſo viel iſt als 5 und deſſen Nenner 6 ſeyn ſoll, ſo hat man [Formel] .","norm":"Gleicher Gestalt ist 3 so viel als [Formel], item als [Formel], item als [Formel], und so fort; dann 3 ist so viel als [Formel], wann man nun oben und unten durch 2 oder 3, oder 4 multipliziert, so kommen [Formel], [Formel], und [Formel] heraus, welche Brüche folglich so viel sind als 3. Hieraus sieht man nun, dass man eine jegliche ganze Zahl in eine Bruchs Form verwandeln kann, von einem beliebigen Nenner, als wann man einen Bruche verlangte, der so viel ist als 5 und dessen Nenner 6 sein soll, so hat man 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+{"basename":"euler_rechenkunst01_1738","par_idx":1244,"date":1738,"orig":"Um Exempel von dieſer Operation durch die Diuiſion anzufuͤbren, ſo muß man ſolche Bruͤche nehmen, deren Nenner und Zehler ſich durch eine Zahl theilen laſſen, welches nicht bey allen angeht.","norm":"Um Exempel von dieser Operation durch die Division anzufübren, so muss man solche Brüche nehmen, deren Nenner und Zähler sich durch eine Zahl teilen lassen, welches nicht bei allen 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+{"basename":"euler_rechenkunst01_1738","par_idx":1245,"date":1738,"orig":"Dahero obgleich die Multiplication bey allen Bruͤchen Statt findet, ſo kan doch die Diuiſion nur bey ſolchen angebracht werden, in welchen der Zehler und Nenner ſich durch eine gemeine Zahl theilen laſſen.","norm":"Daher obgleich die Multiplikation bei allen Brüchen statt findet, so kann doch die Division nur bei solchen angebracht werden, in welchen der Zähler und Nenner sich durch eine gemeine Zahl teilen 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+{"basename":"euler_rechenkunst01_1738","par_idx":1251,"date":1738,"orig":"Ferner laſſen ſich auch hier beyde Zahlen wiederum durch 3 Theilen, da dann heraus komt ⅗, welcher Bruch folglich ſo viel iſt als [Formel] , und aus dieſem auf einmal haͤtte koͤnnen heraus gebracht werden, wann man geſehen haͤtte, daß ſich beyde Zahlen 36 und 60 durch 12 theilen laſſen.","norm":"Ferner lassen sich auch hier beide Zahlen wiederum durch 3 Teilen, da dann herauskommt 3/5 , welcher Bruch folglich so viel ist als [Formel], und aus diesem auf einmal hätte können heraus gebracht werden, wann man gesehen hätte, dass sich beide Zahlen 36 und 60 durch 12 teilen 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+{"basename":"euler_rechenkunst01_1738","par_idx":1252,"date":1738,"orig":"Dann wann den Zehler und Nenner dieſes Bruchs [Formel] durch 12 diuidirt, ſo kommen ⅗ heraus.","norm":"Dann wann den Zähler und Nenner dieses Bruchs [Formel] durch 12 dividiert, so kommen 3/5 heraus.","orig_tok":["Dann","wann","den","Zehler","und","Nenner","dieſes","Bruchs","[Formel]","durch","12","diuidirt",",","ſo","kommen","⅗","heraus","."],"norm_tok":["Dann","wann","den","Zähler","und","Nenner","dieses","Bruchs","[","Formel","]","durch","12","dividiert",",","so","kommen"," ","3","/","5"," 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laſſen, und ferner was dieſer Theiler fuͤr eine Zahl iſt, ſo wollen wir in folgenden Saͤtzen dazu Anleitung geben.","norm":"Weilen nun bei dieser Operation, welche durch die Division geschieht, und dadurch ein Bruch in kleinere Zahlen gebracht wird, vor allen Dingen zu wissen nötig ist, ob sich beide Zahlen eines Bruchs durch eine gemeine Zahl teilen lassen, und ferner was dieser Teiler für eine Zahl ist, so wollen wir in folgenden Sätzen dazu Anleitung 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Durch 2 laſſen ſich alle diejenigen Zahlen theilen, deren letzte Figur nach der rechten Hand ſich durch 2 theilen laͤßt.","norm":"1. Durch 2 lassen sich alle diejenigen Zahlen teilen, deren letzte Figur nach der rechten Hand sich durch 2 teilen lässt.","orig_tok":["1.","Durch","2","laſſen","ſich","alle","diejenigen","Zahlen","theilen",",","deren","letzte","Figur","nach","der","rechten","Hand","ſich","durch","2","theilen","laͤßt","."],"norm_tok":["1.","Durch","2","lassen","sich","alle","diejenigen","Zahlen","teilen",",","deren","letzte","Figur","nach","der","rechten","Hand","sich","durch","2","teilen","lässt","."],"orig_spans":[[0,2],[3,8],[9,10],[11,17],[18,22],[23,27],[28,38],[39,45],[46,53],[53,54],[55,60],[61,67],[68,73],[74,78],[79,82],[83,90],[91,95],[96,100],[101,106],[107,108],[109,116],[117,122],[122,123]],"norm_spans":[[0,2],[3,8],[9,10],[11,17],[18,22],[23,27],[28,38],[39,45],[46,52],[52,53],[54,59],[60,66],[67,72],[73,77],[78,81],[82,89],[90,94],[95,99],[100,105],[106,107],[108,114],[115,120],[120,121]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.602099895477295} +{"basename":"euler_rechenkunst01_1738","par_idx":1257,"date":1738,"orig":"2. Durch 4 laͤßt ſich eine Zahl theilen, wann ſich die zwey letſten Zahlen gegen der rechten durch 4 theilen laſſen.","norm":"2. Durch 4 lässt sich eine Zahl teilen, wann sich die zwei letzten Zahlen gegen der rechten durch 4 teilen lassen.","orig_tok":["2.","Durch","4","laͤßt","ſich","eine","Zahl","theilen",",","wann","ſich","die","zwey","letſten","Zahlen","gegen","der","rechten","durch","4","theilen","laſſen","."],"norm_tok":["2.","Durch","4","lässt","sich","eine","Zahl","teilen",",","wann","sich","die","zwei","letzten","Zahlen","gegen","der","rechten","durch","4","teilen","lassen","."],"orig_spans":[[0,2],[3,8],[9,10],[11,16],[17,21],[22,26],[27,31],[32,39],[39,40],[41,45],[46,50],[51,54],[55,59],[60,67],[68,74],[75,80],[81,84],[85,92],[93,98],[99,100],[101,108],[109,115],[115,116]],"norm_spans":[[0,2],[3,8],[9,10],[11,16],[17,21],[22,26],[27,31],[32,38],[38,39],[40,44],[45,49],[50,53],[54,58],[59,66],[67,73],[74,79],[80,83],[84,91],[92,97],[98,99],[100,106],[107,113],[113,114]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.6971001625061035} +{"basename":"euler_rechenkunst01_1738","par_idx":1258,"date":1738,"orig":"3. Durch 8 laͤſſt ſich eine Zahl theilen, wann die drey letzten Zahlen gegen der Rechten durch 8 getheilt werden koͤnnen.","norm":"3. Durch 8 lässt sich eine Zahl teilen, wann die drei letzten Zahlen gegen der Rechten durch 8 geteilt werden können.","orig_tok":["3.","Durch","8","laͤſſt","ſich","eine","Zahl","theilen",",","wann","die","drey","letzten","Zahlen","gegen","der","Rechten","durch","8","getheilt","werden","koͤnnen","."],"norm_tok":["3.","Durch","8","lässt","sich","eine","Zahl","teilen",",","wann","die","drei","letzten","Zahlen","gegen","der","Rechten","durch","8","geteilt","werden","können","."],"orig_spans":[[0,2],[3,8],[9,10],[11,17],[18,22],[23,27],[28,32],[33,40],[40,41],[42,46],[47,50],[51,55],[56,63],[64,70],[71,76],[77,80],[81,88],[89,94],[95,96],[97,105],[106,112],[113,120],[120,121]],"norm_spans":[[0,2],[3,8],[9,10],[11,16],[17,21],[22,26],[27,31],[32,38],[38,39],[40,44],[45,48],[49,53],[54,61],[62,68],[69,74],[75,78],[79,86],[87,92],[93,94],[95,102],[103,109],[110,116],[116,117]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.091599941253662} +{"basename":"euler_rechenkunst01_1738","par_idx":1259,"date":1738,"orig":"4. Durch 5 laͤſſt ſich eine Zahl theilen, wann die letſte Figur nach der Rechten entweder 5 iſt oder 0.","norm":"4. Durch 5 lässt sich eine Zahl teilen, wann die letzte Figur nach der Rechten entweder 5 ist oder 0.","orig_tok":["4.","Durch","5","laͤſſt","ſich","eine","Zahl","theilen",",","wann","die","letſte","Figur","nach","der","Rechten","entweder","5","iſt","oder","0","."],"norm_tok":["4.","Durch","5","lässt","sich","eine","Zahl","teilen",",","wann","die","letzte","Figur","nach","der","Rechten","entweder","5","ist","oder","0."],"orig_spans":[[0,2],[3,8],[9,10],[11,17],[18,22],[23,27],[28,32],[33,40],[40,41],[42,46],[47,50],[51,57],[58,63],[64,68],[69,72],[73,80],[81,89],[90,91],[92,95],[96,100],[101,102],[102,103]],"norm_spans":[[0,2],[3,8],[9,10],[11,16],[17,21],[22,26],[27,31],[32,38],[38,39],[40,44],[45,48],[49,55],[56,61],[62,66],[67,70],[71,78],[79,87],[88,89],[90,93],[94,98],[99,101]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,20]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.428999900817871} +{"basename":"euler_rechenkunst01_1738","par_idx":1260,"date":1738,"orig":"5. Durch 10 laſſen ſich keine anderen Zahlen theilen, als deren letzte Figur nach der Rechten 0 iſt.","norm":"5. Durch 10 lassen sich keine anderen Zahlen teilen, als deren letzte Figur nach der Rechten 0 ist.","orig_tok":["5.","Durch","10","laſſen","ſich","keine","anderen","Zahlen","theilen",",","als","deren","letzte","Figur","nach","der","Rechten","0","iſt","."],"norm_tok":["5.","Durch","10","lassen","sich","keine","anderen","Zahlen","teilen",",","als","deren","letzte","Figur","nach","der","Rechten","0","ist","."],"orig_spans":[[0,2],[3,8],[9,11],[12,18],[19,23],[24,29],[30,37],[38,44],[45,52],[52,53],[54,57],[58,63],[64,70],[71,76],[77,81],[82,85],[86,93],[94,95],[96,99],[99,100]],"norm_spans":[[0,2],[3,8],[9,11],[12,18],[19,23],[24,29],[30,37],[38,44],[45,51],[51,52],[53,56],[57,62],[63,69],[70,75],[76,80],[81,84],[85,92],[93,94],[95,98],[98,99]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.555799961090088} +{"basename":"euler_rechenkunst01_1738","par_idx":1261,"date":1738,"orig":"6. Durch 3 laͤſſt ſich eine Zahl theilen, wann ſich die Summ von allen Figuren, aus welchen die Zahl beſteht, durch 3 theilen laͤſſt.","norm":"6. Durch 3 lässt sich eine Zahl teilen, wann sich die Summe von allen Figuren, aus welchen die Zahl besteht, durch 3 teilen lässt.","orig_tok":["6.","Durch","3","laͤſſt","ſich","eine","Zahl","theilen",",","wann","ſich","die","Summ","von","allen","Figuren",",","aus","welchen","die","Zahl","beſteht",",","durch","3","theilen","laͤſſt","."],"norm_tok":["6.","Durch","3","lässt","sich","eine","Zahl","teilen",",","wann","sich","die","Summe","von","allen","Figuren",",","aus","welchen","die","Zahl","besteht",",","durch","3","teilen","lässt","."],"orig_spans":[[0,2],[3,8],[9,10],[11,17],[18,22],[23,27],[28,32],[33,40],[40,41],[42,46],[47,51],[52,55],[56,60],[61,64],[65,70],[71,78],[78,79],[80,83],[84,91],[92,95],[96,100],[101,108],[108,109],[110,115],[116,117],[118,125],[126,132],[132,133]],"norm_spans":[[0,2],[3,8],[9,10],[11,16],[17,21],[22,26],[27,31],[32,38],[38,39],[40,44],[45,49],[50,53],[54,59],[60,63],[64,69],[70,77],[77,78],[79,82],[83,90],[91,94],[95,99],[100,107],[107,108],[109,114],[115,116],[117,123],[124,129],[129,130]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":3.9565999507904053} +{"basename":"euler_rechenkunst01_1738","par_idx":1262,"date":1738,"orig":"7. Durch 9 laͤſſt ſich eine Zahl theilen, wann ſich gleichfals die Summ aller Figuren durch 9 theilen laͤſſt.","norm":"7. Durch 9 lässt sich eine Zahl teilen, wann sich gleichfalls die Summe aller Figuren durch 9 teilen lässt.","orig_tok":["7.","Durch","9","laͤſſt","ſich","eine","Zahl","theilen",",","wann","ſich","gleichfals","die","Summ","aller","Figuren","durch","9","theilen","laͤſſt","."],"norm_tok":["7.","Durch","9","lässt","sich","eine","Zahl","teilen",",","wann","sich","gleichfalls","die","Summe","aller","Figuren","durch","9","teilen","lässt","."],"orig_spans":[[0,2],[3,8],[9,10],[11,17],[18,22],[23,27],[28,32],[33,40],[40,41],[42,46],[47,51],[52,62],[63,66],[67,71],[72,77],[78,85],[86,91],[92,93],[94,101],[102,108],[108,109]],"norm_spans":[[0,2],[3,8],[9,10],[11,16],[17,21],[22,26],[27,31],[32,38],[38,39],[40,44],[45,49],[50,61],[62,65],[66,71],[72,77],[78,85],[86,91],[92,93],[94,100],[101,106],[106,107]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.001999855041504} +{"basename":"euler_rechenkunst01_1738","par_idx":1263,"date":1738,"orig":"8. Durch 6 laſſen ſich alle diejenigen Zahlen theilen, welche zugleich durch 2 und durch 3 getheilt werden koͤnnen.","norm":"8. Durch 6 lassen sich alle diejenigen Zahlen teilen, welche zugleich durch 2 und durch 3 geteilt werden 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Hierauf beruhen auch die 4te und 5te Regel; dann die Decades, Centenarii, Millenarii und folgende laſſen ſich fuͤr ſich durch 5 und durch 10 Theilen:","norm":"Wann sich demnach auch die Unitäten durch 2 teilen lassen, so lässt sich auch die ganze Zahl durch 2 teilen; dieses aber geschieht, wann sich die letzte Figur nach der rechten Hand durch 2 teilen lässt, oder wann dieselbe ist entweder 0 oder 2, 4, 6, 8. Hierauf beruhen auch die 4te und 5te Regel; dann die Decades, Centenarii, Millenarii und folgende lassen sich für sich durch 5 und durch 10 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+{"basename":"euler_rechenkunst01_1738","par_idx":1270,"date":1738,"orig":"Nun aber enthaͤlt die letzte Figur von der rechten die Unitæten; und folglich laͤſt ſich eine Zahl durch 5 oder 10 theilen, wann ſich die letzte Figur dadurch theilen laͤſt, das iſt fuͤr den erſteren Fall nehmlich 5 wann die letzte Figur entweder 0 oder 5 iſt, im anderen Fall fuͤr 10 aber wann die letzte Figur 0 iſt.","norm":"Nun aber enthält die letzte Figur von der rechten die Unitäten; und folglich lässt sich eine Zahl durch 5 oder 10 teilen, wann sich die letzte Figur dadurch teilen lässt, das ist für den ersteren Fall nämlich 5 wann die letzte Figur entweder 0 oder 5 ist, im anderen Fall für 10 aber wann die letzte Figur 0 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+{"basename":"euler_rechenkunst01_1738","par_idx":1271,"date":1738,"orig":"Die zweyte Regel zu beweiſen ſo iſt zu mercken, daß ſich alle Centenarii, Millenarii, und ſo fort durch 4 theilen laſſen; wann ſich demnach die Decades zuſammt den Unitæten auch durch 4 theilen laſſen, ſo wird die gantze Zahl durch 4 koͤnnen getheilet werden.","norm":"Die zweite Regel zu beweisen so ist zu merken, dass sich alle Centenarii, Millenarii, und so fort durch 4 teilen lassen; wann sich demnach die Decades zusamt den Unitäten auch durch 4 teilen lassen, so wird die ganze Zahl durch 4 können geteilt 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+{"basename":"euler_rechenkunst01_1738","par_idx":1272,"date":1738,"orig":"Die zwey letzteren Figuren aber nach der rechten Hand enthalten die Decades und Unitates, und folglich kommt die gantze Sache darauf an, ob ſich dieſe zwey Zahlen, oder vielmehr die Zahl, welche dadurch angedeutet wird, durch 4 theilen laͤſt; alſo laͤſt ſich 1736 durch 4 theilen, weil 36 dadurch getheilet werden kan.","norm":"Die zwei letzteren Figuren aber nach der rechten Hand enthalten die Decades und Unitates, und folglich kommt die ganze Sache darauf an, ob sich diese zwei Zahlen, oder vielmehr die Zahl, welche dadurch angedeutet wird, durch 4 teilen lässt; also lässt sich 1736 durch 4 teilen, weil 36 dadurch geteilt werden 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+{"basename":"euler_rechenkunst01_1738","par_idx":1280,"date":1738,"orig":"Bey groſſen Zahlen wann die Summ der Figuren ſelbſt wieder groß wird, und aus etlichen Figuren beſteht, ſo kan dieſer Vortheil wieder angebracht, und die Summ dieſer Figuren ſelbſt unterſuchet werden.","norm":"Bei großen Zahlen wann die Summe der Figuren selbst wieder groß wird, und aus etlichen Figuren besteht, so kann dieser Vorteil wieder angebracht, und die Summe dieser Figuren selbst untersuchet 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+{"basename":"euler_rechenkunst01_1738","par_idx":1281,"date":1738,"orig":"Als wann gefragt wuͤrde, ob ſich dieſe Zahl 5 9 8 7 6 2 5 7 9 8 6 3 4 durch 3 oder 9 theilen laſſe, ſo addire man alle Figuren zuſammen, da dann 79 herauskommt; dieſer Zahl Figuren zuſammen machen nun ferner 16, und weil dieſe Zahl noch aus zwey Figuren beſteht, ſo addire man dieſelben noch mahls zuſammen, da dann 7 herauskommt.","norm":"Als wann gefragt würde, ob sich diese Zahl 5 9 8 7 6 2 5 7 9 8 6 3 4 durch 3 oder 9 teilen lasse, so addiere man alle Figuren zusammen, da dann 79 herauskommt; dieser Zahl Figuren zusammen machen nun ferner 16, und weil diese Zahl noch aus zwei Figuren besteht, so addiere man dieselben noch Mals zusammen, da dann 7 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+{"basename":"euler_rechenkunst01_1738","par_idx":1282,"date":1738,"orig":"Woraus erhellet, daß wann die vorgegebene Zahl durch 3 oder 9 diuidirt werden ſollte, eben ſo viel uͤberbleiben wuͤrde, als wann 7 dadurch getheilet wurde, nehmlich im erſtern Fall 1, im letzten 7.","norm":"Woraus erhellet, dass wann die vorgegebene Zahl durch 3 oder 9 dividiert werden sollte, ebenso viel überbleiben würde, als wann 7 dadurch geteilt wurde, nämlich im ersteren Fall 1, im letzten 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+{"basename":"euler_rechenkunst01_1738","par_idx":1283,"date":1738,"orig":"Die 8te Regel folget aus der erſten und ſechſten, dann wann ſich eine Zahl in zwey und zugleich auch in drey gleiche Theile zertheilen laͤſt, ſo laͤſt ſich dieſelbe auch in 6 gleiche Theile theilen.","norm":"Die 8te Regel folget aus der ersten und sechsten, dann wann sich eine Zahl in zwei und zugleich auch in drei gleiche Teile zerteilen lässt, so lässt sich dieselbe auch in 6 gleiche Teile 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+{"basename":"euler_rechenkunst01_1738","par_idx":1284,"date":1738,"orig":"Endlich iſt zu mercken, daß man durch alle dieſe Regeln nicht nur erkennt, ob ſich eine Zahl durch eine ſolche vorgeſchriebene theilen laſſe oder nicht, ſondern auch wieviel im letzteren Fall uͤbrig bleibe, wie aus dem letzt-angebrachten Exempel von 3 und 9 zu erſehen, obgleich dieſes zu unſerem jetzigen Vorhaben nicht dienet, in anderen Faͤllen aber dennoch von groſſem Vortheil ſeyn kan.","norm":"Endlich ist zu merken, dass man durch alle diese Regeln nicht nur erkennt, ob sich eine Zahl durch eine solche vorgeschriebene teilen lasse oder nicht, sondern auch wie viel im letzteren Fall übrig bleibe, wie aus dem letzt-angebrachten Exempel von 3 und 9 zu ersehen, obgleich dieses zu unserem jetzigen Vorhaben nicht dient, in anderen Fällen aber dennoch von großem Vorteil sein 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+{"basename":"euler_rechenkunst01_1738","par_idx":1286,"date":1738,"orig":"Dann zu Erkennung der Bruͤche tragt ſehr viel bey, wann die Zahlen daraus derſelbe beſteht ſo klein ſind als moͤglich; und iſt alſo die Verkleinerung der Bruͤche zu deutlicherem Begriff derſelben hoͤchſt nuͤtzlich.","norm":"Dann zu Erkennung der Brüche tragt sehr viel bei, wann die Zahlen daraus derselbe besteht so klein sind als möglich; und ist also die Verkleinerung der Brüche zu deutlicherem Begriff derselben höchst nützlich.","orig_tok":["Dann","zu","Erkennung","der","Bruͤche","tragt","ſehr","viel","bey",",","wann","die","Zahlen","daraus","derſelbe","beſteht","ſo","klein","ſind","als","moͤglich",";","und","iſt","alſo","die","Verkleinerung","der","Bruͤche","zu","deutlicherem","Begriff","derſelben","hoͤchſt","nuͤtzlich","."],"norm_tok":["Dann","zu","Erkennung","der","Brüche","tragt","sehr","viel","bei",",","wann","die","Zahlen","daraus","derselbe","besteht","so","klein","sind","als","möglich",";","und","ist","also","die","Verkleinerung","der","Brüche","zu","deutlicherem","Begriff","derselben","höchst","nützlich","."],"orig_spans":[[0,4],[5,7],[8,17],[18,21],[22,29],[30,35],[36,40],[41,45],[46,49],[49,50],[51,55],[56,59],[60,66],[67,73],[74,82],[83,90],[91,93],[94,99],[100,104],[105,108],[109,117],[117,118],[119,122],[123,126],[127,131],[132,135],[136,149],[150,153],[154,161],[162,164],[165,177],[178,185],[186,195],[196,203],[204,213],[213,214]],"norm_spans":[[0,4],[5,7],[8,17],[18,21],[22,28],[29,34],[35,39],[40,44],[45,48],[48,49],[50,54],[55,58],[59,65],[66,72],[73,81],[82,89],[90,92],[93,98],[99,103],[104,107],[108,115],[115,116],[117,120],[121,124],[125,129],[130,133],[134,147],[148,151],[152,158],[159,161],[162,174],[175,182],[183,192],[193,199],[200,208],[208,209]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28],[29,29],[30,30],[31,31],[32,32],[33,33],[34,34],[35,35]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.6809000968933105} +{"basename":"euler_rechenkunst01_1738","par_idx":1287,"date":1738,"orig":"Derowegen wird nicht undienlich ſeyn einige Exempel vorzubringen, in welchen Bruͤche vermittelſt der gegebenen Regeln in leichtere verwandelt werden.","norm":"Derowegen wird nicht undienlich sein einige Exempel vorzubringen, in welchen Brüche vermittels der gegebenen Regeln in leichtere verwandelt 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+{"basename":"euler_rechenkunst01_1738","par_idx":1288,"date":1738,"orig":"1. Es ſey uns dieſer Bruch [Formel] vorgeleget, in welchem wir nach der erſten Regel ſehen, daß ſich beyde Zahlen durch 2 theilen laſſen, weilen die letzten Figuren derſelben 2 und 6 dadurch getheilt werden koͤnnen: wann wir derohalben den Zehler und Nenner durch 2 diuidiren, ſo kommt dieſer Bruch heraus [Formel] , welcher dem vorgelegten gleich iſt.","norm":"1. Es sei uns dieser Bruch [Formel] vorgelegt, in welchem wir nach der ersten Regel sehen, dass sich beide Zahlen durch 2 teilen lassen, weilen die letzten Figuren derselben 2 und 6 dadurch geteilt werden können: wann wir derohalben den Zähler und Nenner durch 2 dividieren, so kommt dieser Bruch heraus [Formel], welcher dem vorgelegten gleich ist.","orig_tok":["1.","Es","ſey","uns","dieſer","Bruch","[Formel]","vorgeleget",",","in","welchem","wir","nach","der","erſten","Regel","ſehen",",","daß","ſich","beyde","Zahlen","durch","2","theilen","laſſen",",","weilen","die","letzten","Figuren","derſelben","2","und","6","dadurch","getheilt","werden","koͤnnen",":","wann","wir","derohalben","den","Zehler","und","Nenner","durch","2","diuidiren",",","ſo","kommt","dieſer","Bruch","heraus","[Formel]",",","welcher","dem","vorgelegten","gleich","iſt","."],"norm_tok":["1.","Es","sei","uns","dieser","Bruch","[","Formel","]","vorgelegt",",","in","welchem","wir","nach","der","ersten","Regel","sehen",",","dass","sich","beide","Zahlen","durch","2","teilen","lassen",",","weilen","die","letzten","Figuren","derselben","2","und","6","dadurch","geteilt","werden","können",":","wann","wir","derohalben","den","Zähler","und","Nenner","durch","2","dividieren",",","so","kommt","dieser","Bruch","heraus","[","Formel","]",",","welcher","dem","vorgelegten","gleich","ist","."],"orig_spans":[[0,2],[3,5],[6,9],[10,13],[14,20],[21,26],[27,35],[36,46],[46,47],[48,50],[51,58],[59,62],[63,67],[68,71],[72,78],[79,84],[85,90],[90,91],[92,95],[96,100],[101,106],[107,113],[114,119],[120,121],[122,129],[130,136],[136,137],[138,144],[145,148],[149,156],[157,164],[165,174],[175,176],[177,180],[181,182],[183,190],[191,199],[200,206],[207,214],[214,215],[216,220],[221,224],[225,235],[236,239],[240,246],[247,250],[251,257],[258,263],[264,265],[266,275],[275,276],[277,279],[280,285],[286,292],[293,298],[299,305],[306,314],[315,316],[317,324],[325,328],[329,340],[341,347],[348,351],[351,352]],"norm_spans":[[0,2],[3,5],[6,9],[10,13],[14,20],[21,26],[27,28],[28,34],[34,35],[36,45],[45,46],[47,49],[50,57],[58,61],[62,66],[67,70],[71,77],[78,83],[84,89],[89,90],[91,95],[96,100],[101,106],[107,113],[114,119],[120,121],[122,128],[129,135],[135,136],[137,143],[144,147],[148,155],[156,163],[164,173],[174,175],[176,179],[180,181],[182,189],[190,197],[198,204],[205,211],[211,212],[213,217],[218,221],[222,232],[233,236],[237,243],[244,247],[248,254],[255,260],[261,262],[263,273],[273,274],[275,277],[278,283],[284,290],[291,296],[297,303],[304,305],[305,311],[311,312],[312,313],[314,321],[322,325],[326,337],[338,344],[345,348],[348,349]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[6,7],[6,8],[7,9],[8,10],[9,11],[10,12],[11,13],[12,14],[13,15],[14,16],[15,17],[16,18],[17,19],[18,20],[19,21],[20,22],[21,23],[22,24],[23,25],[24,26],[25,27],[26,28],[27,29],[28,30],[29,31],[30,32],[31,33],[32,34],[33,35],[34,36],[35,37],[36,38],[37,39],[38,40],[39,41],[40,42],[41,43],[42,44],[43,45],[44,46],[45,47],[46,48],[47,49],[48,50],[49,51],[50,52],[51,53],[52,54],[53,55],[54,56],[55,57],[56,58],[56,59],[56,60],[57,61],[58,62],[59,63],[60,64],[61,65],[62,66],[63,67]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.577700138092041} +{"basename":"euler_rechenkunst01_1738","par_idx":1289,"date":1738,"orig":"2. Wenn dieſer Bruch [Formel] vorkame, ſo ſaͤhe man nach der zweyten Regel gleich, daß beyde Zahlen ſich durch 4 theilen laſſen, weilen die zwey letzteren Figuren davon nehmlich 68 und 32 dadurch theilbar ſind.","norm":"2. Wenn dieser Bruch [Formel] vorkam, so sähe man nach der zweiten Regel gleich, dass beide Zahlen sich durch 4 teilen lassen, weilen die zwei letzteren Figuren davon nämlich 68 und 32 dadurch teilbar 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+{"basename":"euler_rechenkunst01_1738","par_idx":1290,"date":1738,"orig":"Ja man kan hier ſo gar die dritte Regel anbringen und ſehen, daß ſich beyden Zahlen durch 8 theilen laſſen, weilen die drey letzten Figuren, nehmlich 368 und 032, das iſt32 durch 8 theilbar ſind.","norm":"Ja man kann hier so gar die dritte Regel anbringen und sehen, dass sich beiden Zahlen durch 8 teilen lassen, weilen die drei letzten Figuren, nämlich 368 und 032, das ist32 durch 8 teilbar 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+{"basename":"euler_rechenkunst01_1738","par_idx":1291,"date":1738,"orig":"Wann man demnach durch 8 diuidirt, ſo wird der vorgelegte Bruch in dieſen [Formel] verwandelt.","norm":"Wann man demnach durch 8 dividiert, so wird der vorgelegte Bruch in diesen [Formel] 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habe ſich viel Muͤhe fuͤr die 2te und 3te Regel zu geben, indem der Gebrauch der erſten beyde in ſich begreifft; als im vorgegebenen Bruche [Formel] kan genug ſeyn, wann man ſieht, daß ſich beyde Zahlen durch 2 theilen laſſen, wodurch alſo dieſer Bruch [Formel] herauskommt; bey welchem man ſieht, daß beyde Zahlen ſich nochmahls durch 2 theilen laſſen, da man dann [Formel] bekommt.","norm":"Hierbei aber ist zu erinnern, dass man nicht nötig habe sich viel Mühe für die 2te und 3te Regel zu geben, indem der Gebrauch der ersten beide in sich begreift; als im vorgegebenen Bruche [Formel] kann genug sein, wann man sieht, dass sich beide Zahlen durch 2 teilen lassen, wodurch also dieser Bruch [Formel] herauskommt; bei welchem man sieht, dass beide Zahlen sich nochmals durch 2 teilen lassen, da man dann [Formel] 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Wann dieſer Bruch vorkaͤme [Formel] ſo ſehe man nach der fuͤnfften Regel gleich, daß beyde Zahlen durch 10 theilbar ſind, weswegen nach verrichteter Diuiſion durch 10 dieſer Bruch [Formel] herauskommt.","norm":"3. Wann dieser Bruch vorkäme [Formel] so sehe man nach der fünften Regel gleich, dass beide Zahlen durch 10 teilbar sind, weswegen nach verrichteter Division durch 10 dieser Bruch [Formel] herauskommt.","orig_tok":["3.","Wann","dieſer","Bruch","vorkaͤme","[Formel]","ſo","ſehe","man","nach","der","fuͤnfften","Regel","gleich",",","daß","beyde","Zahlen","durch","10","theilbar","ſind",",","weswegen","nach","verrichteter","Diuiſion","durch","10","dieſer","Bruch","[Formel]","herauskommt","."],"norm_tok":["3.","Wann","dieser","Bruch","vorkäme","[","Formel","]","so","sehe","man","nach","der","fünften","Regel","gleich",",","dass","beide","Zahlen","durch","10","teilbar","sind",",","weswegen","nach","verrichteter","Division","durch","10","dieser","Bruch","[","Formel","]","herauskommt","."],"orig_spans":[[0,2],[3,7],[8,14],[15,20],[21,29],[30,38],[39,41],[42,46],[47,50],[51,55],[56,59],[60,69],[70,75],[76,82],[82,83],[84,87],[88,93],[94,100],[101,106],[107,109],[110,118],[119,123],[123,124],[125,133],[134,138],[139,151],[152,160],[161,166],[167,169],[170,176],[177,182],[183,191],[192,203],[203,204]],"norm_spans":[[0,2],[3,7],[8,14],[15,20],[21,28],[29,30],[30,36],[36,37],[38,40],[41,45],[46,49],[50,54],[55,58],[59,66],[67,72],[73,79],[79,80],[81,85],[86,91],[92,98],[99,104],[105,107],[108,115],[116,120],[120,121],[122,130],[131,135],[136,148],[149,157],[158,163],[164,166],[167,173],[174,179],[180,181],[181,187],[187,188],[189,200],[200,201]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[5,6],[5,7],[6,8],[7,9],[8,10],[9,11],[10,12],[11,13],[12,14],[13,15],[14,16],[15,17],[16,18],[17,19],[18,20],[19,21],[20,22],[21,23],[22,24],[23,25],[24,26],[25,27],[26,28],[27,29],[28,30],[29,31],[30,32],[31,33],[31,34],[31,35],[32,36],[33,37]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.909800052642822} +{"basename":"euler_rechenkunst01_1738","par_idx":1295,"date":1738,"orig":"Bey dieſem Bruche kan ferner die vierte Regel Statt finden, weilen die obere Zahl ſich mit 5, die untere aber mit 0 endigt; daher beyde durch 5 theilbar ſind.","norm":"Bei diesem Bruche kann ferner die vierte Regel statt finden, weilen die obere Zahl sich mit 5, die untere aber mit 0 endigt; daher beide durch 5 teilbar sind.","orig_tok":["Bey","dieſem","Bruche","kan","ferner","die","vierte","Regel","Statt","finden",",","weilen","die","obere","Zahl","ſich","mit","5",",","die","untere","aber","mit","0","endigt",";","daher","beyde","durch","5","theilbar","ſind","."],"norm_tok":["Bei","diesem","Bruche","kann","ferner","die","vierte","Regel","statt","finden",",","weilen","die","obere","Zahl","sich","mit","5",",","die","untere","aber","mit","0","endigt",";","daher","beide","durch","5","teilbar","sind","."],"orig_spans":[[0,3],[4,10],[11,17],[18,21],[22,28],[29,32],[33,39],[40,45],[46,51],[52,58],[58,59],[60,66],[67,70],[71,76],[77,81],[82,86],[87,90],[91,92],[92,93],[94,97],[98,104],[105,109],[110,113],[114,115],[116,122],[122,123],[124,129],[130,135],[136,141],[142,143],[144,152],[153,157],[157,158]],"norm_spans":[[0,3],[4,10],[11,17],[18,22],[23,29],[30,33],[34,40],[41,46],[47,52],[53,59],[59,60],[61,67],[68,71],[72,77],[78,82],[83,87],[88,91],[92,93],[93,94],[95,98],[99,105],[106,110],[111,114],[115,116],[117,123],[123,124],[125,130],[131,136],[137,142],[143,144],[145,152],[153,157],[157,158]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28],[29,29],[30,30],[31,31],[32,32]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.8084001541137695} +{"basename":"euler_rechenkunst01_1738","par_idx":1296,"date":1738,"orig":"Wann man nun beyde Zahlen durch 5 diuidirt, ſo bekommt man dieſen Bruch [Formel] , welcher eben ſo viel haͤlt als der vorgelegte [Formel] . 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Es ſey uns dieſer Bruch [Formel] vorgegeben durch kleinere Zahlen auszudrucken, weilen nun beyde Zahlen zu gleich weder durch 2 noch 5 noch 10 getheilt werden koͤnnen, ſo wollen wir ſehen, ob nicht beyde durch 3 oder 9 theilbar ſind, welches nach der ſechſten und ſiebenten Regel geſchieht, wann man die Figuren ſo wohl des Zehlers als Nenners zuſammen addirt.","norm":"4. Es sei uns dieser Bruch [Formel] vorgegeben durch kleinere Zahlen auszudrucken, weilen nun beide Zahlen zugleich weder durch 2 noch 5 noch 10 geteilt werden können, so wollen wir sehen, ob nicht beide durch 3 oder 9 teilbar sind, welches nach der sechsten und siebenten Regel geschieht, wann man die Figuren so wohl des Zählers als Nenners zusammen 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+{"basename":"euler_rechenkunst01_1738","par_idx":1300,"date":1738,"orig":"Ob aber dieſe Regeln gleich einen groſſen Vortheil in Verkleinerung der Bruͤche haben, ſo kan man dennoch vermittelſt derſelben nur ſehen, ob beyde Zahlen durch 2, 3, 5 oder 10 theilbar ſind, und folglich dadurch dergleichen Bruͤche nicht in kleinere Zahlen bringen, bey welchen dieſe Regel n nicht ſtatt finden.","norm":"Ob aber diese Regeln gleich einen großen Vorteil in Verkleinerung der Brüche haben, so kann man dennoch vermittels derselben nur sehen, ob beide Zahlen durch 2, 3, 5 oder 10 teilbar sind, und folglich dadurch dergleichen Brüche nicht in kleinere Zahlen bringen, bei welchen diese Regel n nicht 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+{"basename":"euler_rechenkunst01_1738","par_idx":1317,"date":1738,"orig":"Solche Zahlen aber, welche auſſer der Unitæt keinen gemeinen Theiler haben, werden unter ſich untheilbare Zahlen genennt, ſolche ſind 7 und 9; item 15 und 28. Wann derohalben ein Bruch ſo beſchaffen iſt, daß der Zehler und Nenner unter ſich untheilbare Zahlen ſind, ſo kan derſelbe nicht durch kleinere Zahlen ausgedruͤckt werden; dergleichen Bruͤche pflegen unaufhebliche Bruͤche genennt zu werden, weilen ſie ſich durch die Diuiſion nicht in kleinere Zahlen bringen laſſen, welche Operation das Aufheben der Bruͤche genennt zu werden pflegt.","norm":"Solche Zahlen aber, welche außer der Unität keinen gemeinen Teiler haben, werden unter sich unteilbare Zahlen genannt, solche sind 7 und 9; item 15 und 28. Wann derohalben ein Bruch so beschaffen ist, dass der Zähler und Nenner unter sich unteilbare Zahlen sind, so kann derselbe nicht durch kleinere Zahlen ausgedrückt werden; dergleichen Brüche pflegen unaufhebliche Brüche genannt zu werden, weilen sie sich durch die Division nicht in kleinere Zahlen bringen lassen, welche Operation das Aufheben der Brüche genannt zu werden 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+{"basename":"euler_rechenkunst01_1738","par_idx":1318,"date":1738,"orig":"Wann aber der Zehler und Nenner eines Bruchs unter ſich theilbare Zahlen ſind, ſo kan der Bruch durch den gemeinen Theiler aufgehoben das iſt durch kleinere Zahlen ausgedruͤckt werden, weswegen auch ſolche Bruͤche aufhebliche Bruͤche genennt werden.","norm":"Wann aber der Zähler und Nenner eines Bruchs unter sich teilbare Zahlen sind, so kann der Bruch durch den gemeinen Teiler aufgehoben das ist durch kleinere Zahlen ausgedrückt werden, weswegen auch solche Brüche aufhebliche Brüche genannt 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+{"basename":"euler_rechenkunst01_1738","par_idx":1319,"date":1738,"orig":"Um nun die aufheblichen Bruͤche zu erkennen, und dieſelben in kleinere Zahlen zu bringen, ſo haben wir die beſchriebene Regel vorgebracht, vermittelſt welcher man nicht nur von zweyen gegebenen Zahlen einen gemeinen Theiler wann ſie nehmlich unter ſich theilbar ſind, ſondern ſo gar den groͤſten gemeinen Theiler finden kan.","norm":"Um nun die Aufheblichen Brüche zu erkennen, und dieselben in kleinere Zahlen zu bringen, so haben wir die beschriebene Regel vorgebracht, vermittels welcher man nicht nur von zweien gegebenen Zahlen einen gemeinen Teiler wann sie nämlich unter sich teilbar sind, sondern so gar den größten gemeinen Teiler finden 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+{"basename":"euler_rechenkunst01_1738","par_idx":1320,"date":1738,"orig":"Dadurch erhaͤlt man aber dieſen Vortheil, daß man ſogleich alle aufhebliche Bruͤche durch den groͤſten gemeinen Theiler in die kleinſten moͤglichen Zahlen bringet, und in unaufhebliche verwandelt, von welchen man verſichert ſeyn kan, daß ſie alsdann durch keine kleinere Zahlen weiter ausgedruͤckt werden koͤnnen.","norm":"Dadurch erhält man aber diesen Vorteil, dass man sogleich alle aufhebliche Brüche durch den größten gemeinen Teiler in die kleinsten möglichen Zahlen bringt, und in unaufhebliche verwandelt, von welchen man versichert sein kann, dass sie alsdann durch keine kleinere Zahlen weiter ausgedrückt werden 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[Formel]","norm":"Weiter durch 66 dividiere man 252. 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6 der groͤſte gemeine Theiler von den zwey vorgegebenen Zahlen iſt.","norm":"Endlich hat man 12 durch 6 zu teilen, welche Division, weilen sie ohne Rest aufgeht, anzeigt dass 6 der größte gemeine Teiler von den zwei vorgegebenen Zahlen 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+{"basename":"euler_rechenkunst01_1738","par_idx":1332,"date":1738,"orig":"Wann alſo dieſer Bruch [Formel] waͤre vorgelegt worden, ſo koͤnnte man denſelben durch 6 aufheben und in dieſen Bruch [Formel] verwandeln, welcher unaufheblich und nicht mehr durch kleinere Zahlen ausgedruͤckt werden kan.","norm":"Wann also dieser Bruch [Formel] wäre vorgelegt worden, so könnte man denselben durch 6 aufheben und in diesen Bruch [Formel] verwandeln, welcher unaufheblich und nicht mehr durch kleinere Zahlen ausgedrückt werden 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dadurch die Unitæt als der groͤſte gemeine Theiler gefunden wird, wie aus folgendem Exempel, da dieſe Zahlen 36 und 151 gegeben ſind zu erſehen.","norm":"Wären aber die Zahlen unter sich unteilbar, so weiset auch dasselbe diese Operation, dadurch die Unität als der größte gemeine Teiler gefunden wird, wie aus folgendem Exempel, da diese Zahlen 36 und 151 gegeben sind zu ersehen.","orig_tok":["Waͤren","aber","die","Zahlen","unter","ſich","untheilbar",",","ſo","weiſet","auch","daſſelbe","dieſe","Operation",",","dadurch","die","Unitæt","als","der","groͤſte","gemeine","Theiler","gefunden","wird",",","wie","aus","folgendem","Exempel",",","da","dieſe","Zahlen","36","und","151","gegeben","ſind","zu","erſehen","."],"norm_tok":["Wären","aber","die","Zahlen","unter","sich","unteilbar",",","so","weiset","auch","dasselbe","diese","Operation",",","dadurch","die","Unität","als","der","größte","gemeine","Teiler","gefunden","wird",",","wie","aus","folgendem","Exempel",",","da","diese","Zahlen","36","und","151","gegeben","sind","zu","ersehen","."],"orig_spans":[[0,6],[7,11],[12,15],[16,22],[23,28],[29,33],[34,44],[44,45],[46,48],[49,55],[56,60],[61,69],[70,75],[76,85],[85,86],[87,94],[95,98],[99,105],[106,109],[110,113],[114,121],[122,129],[130,137],[138,146],[147,151],[151,152],[153,156],[157,160],[161,170],[171,178],[178,179],[180,182],[183,188],[189,195],[196,198],[199,202],[203,206],[207,214],[215,219],[220,222],[223,230],[230,231]],"norm_spans":[[0,5],[6,10],[11,14],[15,21],[22,27],[28,32],[33,42],[42,43],[44,46],[47,53],[54,58],[59,67],[68,73],[74,83],[83,84],[85,92],[93,96],[97,103],[104,107],[108,111],[112,118],[119,126],[127,133],[134,142],[143,147],[147,148],[149,152],[153,156],[157,166],[167,174],[174,175],[176,178],[179,184],[185,191],[192,194],[195,198],[199,202],[203,210],[211,215],[216,218],[219,226],[226,227]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28],[29,29],[30,30],[31,31],[32,32],[33,33],[34,34],[35,35],[36,36],[37,37],[38,38],[39,39],[40,40],[41,41]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.4593000411987305} +{"basename":"euler_rechenkunst01_1738","par_idx":1337,"date":1738,"orig":"[Formel]","norm":"[Formel]","orig_tok":["[Formel]"],"norm_tok":["[","Formel","]"],"orig_spans":[[0,8]],"norm_spans":[[0,1],[1,7],[7,8]],"alignment":[[0,0],[0,1],[0,2]],"lang_fastText":"sh","lang_py3langid":"en","lang_cld3":"cy","orig_lang_de":0,"norm_lmscore":7.932600021362305} +{"basename":"euler_rechenkunst01_1738","par_idx":1338,"date":1738,"orig":"Damit wir aber endlich auf den Grund dieſer Operation kommen, ſo iſt vor allen Dingen zu mercken, daß wann zwey Zahlen einen gemeinen Theiler haben, alsdenn auch die Differenz derſelben Zahlen durch eben denſelben Theiler getheilt werden koͤnne: ingleichem auch die Differenz zwiſchen der einen und dem doppelten oder dreyfachen oder unter einem anderen vielfachen der anderen Zahl.","norm":"Damit wir aber endlich auf den Grund dieser Operation kommen, so ist vor allen Dingen zu merken, dass wann zwei Zahlen einen gemeinen Teiler haben, alsdann auch die Differenz derselben Zahlen durch eben denselben Teiler geteilt werden könne: ingleichem auch die Differenz zwischen der einen und dem doppelten oder dreifachen oder unter einem anderen vielfachen der anderen 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+{"basename":"euler_rechenkunst01_1738","par_idx":1339,"date":1738,"orig":"Nun aber wann die groͤſſere Zahl durch die kleinere diuidirt wird, ſo iſt der Reſt nichts anders als die Differenz zwiſchen der groͤſſeren Zahl und einem multiplo der kleineren.","norm":"Nun aber wann die größere Zahl durch die kleinere dividiert wird, so ist der Rest nichts anders als die Differenz zwischen der größeren Zahl und einem multiplo der 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+{"basename":"euler_rechenkunst01_1738","par_idx":1340,"date":1738,"orig":"Derohalben muß ein gemeiner Theiler zweyer Zahlen auch den Reſt theilen, welcher in der Diuiſion der groͤſſeren Zahl durch die kleinere zuruͤckbleibt.","norm":"Derohalben muss ein gemeiner Teiler zweier Zahlen auch den Rest teilen, welcher in der Division der größeren Zahl durch die kleinere zurückbleibt.","orig_tok":["Derohalben","muß","ein","gemeiner","Theiler","zweyer","Zahlen","auch","den","Reſt","theilen",",","welcher","in","der","Diuiſion","der","groͤſſeren","Zahl","durch","die","kleinere","zuruͤckbleibt","."],"norm_tok":["Derohalben","muss","ein","gemeiner","Teiler","zweier","Zahlen","auch","den","Rest","teilen",",","welcher","in","der","Division","der","größeren","Zahl","durch","die","kleinere","zurückbleibt","."],"orig_spans":[[0,10],[11,14],[15,18],[19,27],[28,35],[36,42],[43,49],[50,54],[55,58],[59,63],[64,71],[71,72],[73,80],[81,83],[84,87],[88,96],[97,100],[101,111],[112,116],[117,122],[123,126],[127,135],[136,149],[149,150]],"norm_spans":[[0,10],[11,15],[16,19],[20,28],[29,35],[36,42],[43,49],[50,54],[55,58],[59,63],[64,70],[70,71],[72,79],[80,82],[83,86],[87,95],[96,99],[100,108],[109,113],[114,119],[120,123],[124,132],[133,145],[145,146]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.201099872589111} +{"basename":"euler_rechenkunst01_1738","par_idx":1341,"date":1738,"orig":"Solcher Geſtalt wird ein jeder gemeiner Theiler der zwey gegebenen Zahlen zugleich ein gemeiner Theiler ſeyn des Diuiſoris und des Reſts.","norm":"Solcher Gestalt wird ein jeder gemeiner Teiler der zwei gegebenen Zahlen zugleich ein gemeiner Teiler sein des Divisors und des 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+{"basename":"euler_rechenkunst01_1738","par_idx":1342,"date":1738,"orig":"Auf gleiche Weiſe wann der vorige Diuiſor durch den Reſt getheilt wird, ſo wird wiederum einjeder gemeiner Theiler der zwey Anfangs vorgegebenen Zahlen den Diuiſor und Reſt dieſer letzten Diuiſion theilen, und ſo weiter fort bey allen folgenden Diuiſionen.","norm":"Auf gleiche Weise wann der vorige Divisor durch den Rest geteilt wird, so wird wiederum einjeder gemeiner Teiler der zwei Anfangs vorgegebenen Zahlen den Divisor und Rest dieser letzten Division teilen, und so weiter fort bei allen folgenden 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+{"basename":"euler_rechenkunst01_1738","par_idx":1343,"date":1738,"orig":"Wann man endlich allſo zu einer Diuiſion kommt, welche ohne Reſt aufgeht, ſo haben auch der Diuidendus und Diuiſor dieſer letzten Diuiſion eben die gemeinen Theiler, welche die beyden Anfangs gegebenen Zahlen unter ſich haben.","norm":"Wann man endlich also zu einer Division kommt, welche ohne Rest aufgeht, so haben auch der Dividend und Divisor dieser letzten Division eben die gemeinen Teiler, welche die beiden Anfangs gegebenen Zahlen unter sich 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+{"basename":"euler_rechenkunst01_1738","par_idx":1344,"date":1738,"orig":"Weilen aber dieſe letzte Diuiſion ohne Reſt aufgeht, ſo iſt der Diuiſor nicht nur ein gemeiner Theiler des Diuiſoris ſelbſt und des Diuidendi ſondern auch der groͤſte gemeine Theiler: woraus dann folgt, daß dieſer letzte Diuiſor auch der groͤſte gemeine Theiler beyder vorgegebenen Zahlen ſeyn muͤſſe.","norm":"Weilen aber diese letzte Division ohne Rest aufgeht, so ist der Divisor nicht nur ein gemeiner Teiler des Divisors selbst und des Dividenden sondern auch der größte gemeine Teiler: woraus dann folgt, dass dieser letzte Divisor auch der größte gemeine Teiler beider vorgegebenen Zahlen sein müsse.","orig_tok":["Weilen","aber","dieſe","letzte","Diuiſion","ohne","Reſt","aufgeht",",","ſo","iſt","der","Diuiſor","nicht","nur","ein","gemeiner","Theiler","des","Diuiſoris","ſelbſt","und","des","Diuidendi","ſondern","auch","der","groͤſte","gemeine","Theiler",":","woraus","dann","folgt",",","daß","dieſer","letzte","Diuiſor","auch","der","groͤſte","gemeine","Theiler","beyder","vorgegebenen","Zahlen","ſeyn","muͤſſe","."],"norm_tok":["Weilen","aber","diese","letzte","Division","ohne","Rest","aufgeht",",","so","ist","der","Divisor","nicht","nur","ein","gemeiner","Teiler","des","Divisors","selbst","und","des","Dividenden","sondern","auch","der","größte","gemeine","Teiler",":","woraus","dann","folgt",",","dass","dieser","letzte","Divisor","auch","der","größte","gemeine","Teiler","beider","vorgegebenen","Zahlen","sein","müsse","."],"orig_spans":[[0,6],[7,11],[12,17],[18,24],[25,33],[34,38],[39,43],[44,51],[51,52],[53,55],[56,59],[60,63],[64,71],[72,77],[78,81],[82,85],[86,94],[95,102],[103,106],[107,116],[117,123],[124,127],[128,131],[132,141],[142,149],[150,154],[155,158],[159,166],[167,174],[175,182],[182,183],[184,190],[191,195],[196,201],[201,202],[203,206],[207,213],[214,220],[221,228],[229,233],[234,237],[238,245],[246,253],[254,261],[262,268],[269,281],[282,288],[289,293],[294,300],[300,301]],"norm_spans":[[0,6],[7,11],[12,17],[18,24],[25,33],[34,38],[39,43],[44,51],[51,52],[53,55],[56,59],[60,63],[64,71],[72,77],[78,81],[82,85],[86,94],[95,101],[102,105],[106,114],[115,121],[122,125],[126,129],[130,140],[141,148],[149,153],[154,157],[158,164],[165,172],[173,179],[179,180],[181,187],[188,192],[193,198],[198,199],[200,204],[205,211],[212,218],[219,226],[227,231],[232,235],[236,242],[243,250],[251,257],[258,264],[265,277],[278,284],[285,289],[290,295],[295,296]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28],[29,29],[30,30],[31,31],[32,32],[33,33],[34,34],[35,35],[36,36],[37,37],[38,38],[39,39],[40,40],[41,41],[42,42],[43,43],[44,44],[45,45],[46,46],[47,47],[48,48],[49,49]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":3.934099912643432} +{"basename":"euler_rechenkunst01_1738","par_idx":1345,"date":1738,"orig":"Dieſes iſt alſo der Grund der erklaͤrten Regel, durch welche der groͤſte gemeine Theiler zweyer Zahlen gefunden werden kan, davon der Nutzen in Verkleinerung oder Aufhebung der Bruͤche zwar ſchon einiger maſſen angefuͤhrt worden iſt, dennoch aber zu groͤſſerem Gebrauch im folgenden Satz ausgefuͤhrt werden ſoll.","norm":"Dieses ist also der Grund der erklärten Regel, durch welche der größte gemeine Teiler zweier Zahlen gefunden werden kann, davon der Nutzen in Verkleinerung oder Aufhebung der Brüche zwar schon einigermaßen angeführt worden ist, dennoch aber zu größerem Gebrauch im folgenden Satz ausgeführt werden soll.","orig_tok":["Dieſes","iſt","alſo","der","Grund","der","erklaͤrten","Regel",",","durch","welche","der","groͤſte","gemeine","Theiler","zweyer","Zahlen","gefunden","werden","kan",",","davon","der","Nutzen","in","Verkleinerung","oder","Aufhebung","der","Bruͤche","zwar","ſchon","einiger","maſſen","angefuͤhrt","worden","iſt",",","dennoch","aber","zu","groͤſſerem","Gebrauch","im","folgenden","Satz","ausgefuͤhrt","werden","ſoll","."],"norm_tok":["Dieses","ist","also","der","Grund","der","erklärten","Regel",",","durch","welche","der","größte","gemeine","Teiler","zweier","Zahlen","gefunden","werden","kann",",","davon","der","Nutzen","in","Verkleinerung","oder","Aufhebung","der","Brüche","zwar","schon","einigermaßen","angeführt","worden","ist",",","dennoch","aber","zu","größerem","Gebrauch","im","folgenden","Satz","ausgeführt","werden","soll","."],"orig_spans":[[0,6],[7,10],[11,15],[16,19],[20,25],[26,29],[30,40],[41,46],[46,47],[48,53],[54,60],[61,64],[65,72],[73,80],[81,88],[89,95],[96,102],[103,111],[112,118],[119,122],[122,123],[124,129],[130,133],[134,140],[141,143],[144,157],[158,162],[163,172],[173,176],[177,184],[185,189],[190,195],[196,203],[204,210],[211,221],[222,228],[229,232],[232,233],[234,241],[242,246],[247,249],[250,260],[261,269],[270,272],[273,282],[283,287],[288,299],[300,306],[307,311],[311,312]],"norm_spans":[[0,6],[7,10],[11,15],[16,19],[20,25],[26,29],[30,39],[40,45],[45,46],[47,52],[53,59],[60,63],[64,70],[71,78],[79,85],[86,92],[93,99],[100,108],[109,115],[116,120],[120,121],[122,127],[128,131],[132,138],[139,141],[142,155],[156,160],[161,170],[171,174],[175,181],[182,186],[187,192],[193,205],[206,215],[216,222],[223,226],[226,227],[228,235],[236,240],[241,243],[244,252],[253,261],[262,264],[265,274],[275,279],[280,290],[291,297],[298,302],[302,303]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28],[29,29],[30,30],[31,31],[32,32],[33,32],[34,33],[35,34],[36,35],[37,36],[38,37],[39,38],[40,39],[41,40],[42,41],[43,42],[44,43],[45,44],[46,45],[47,46],[48,47],[49,48]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.710299968719482} +{"basename":"euler_rechenkunst01_1738","par_idx":1346,"date":1738,"orig":"11.)","norm":"11.)","orig_tok":["11.",")"],"norm_tok":["11.",")"],"orig_spans":[[0,3],[3,4]],"norm_spans":[[0,3],[3,4]],"alignment":[[0,0],[1,1]],"lang_fastText":"hr","lang_py3langid":"en","lang_cld3":"ja","orig_lang_de":0,"norm_lmscore":5.8007001876831055} +{"basename":"euler_rechenkunst01_1738","par_idx":1347,"date":1738,"orig":"Um von einem vorgegebenen Bruche zu urtheilen, ob derſelbe durch kleinere Zahlen ausgedruckt werden koͤnne oder nicht, ſo muß man von dem Zehler und Nenner deſſelben den groͤſten gemeineu Theiler ſuchen.","norm":"Um von einem vorgegebenen Bruche zu urteilen, ob derselbe durch kleinere Zahlen ausgedrückt werden könne oder nicht, so muss man von dem Zähler und Nenner desselben den größten gemeineu Teiler 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+{"basename":"euler_rechenkunst01_1738","par_idx":1348,"date":1738,"orig":"Findet man nun 1 fuͤr den groͤſten gemeinen Theiler, ſo iſt daſſelbe ein Anzeigen, daß der Bruch durch kleinere Zahlen nicht ausgedruͤckt werden koͤnne:","norm":"Findet man nun 1 für den größten gemeinen Teiler, so ist dasselbe ein Anzeigen, dass der Bruch durch kleinere Zahlen nicht ausgedrückt werden 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+{"basename":"euler_rechenkunst01_1738","par_idx":1349,"date":1738,"orig":"Kommt aber ein anderer groͤſſerer gemeiner Theiler heraus, ſo kan der vorgegebene Bruch in kleinere Zahlen gebracht werden, wann man nehmlich den Zehler und Nenner des gegebenen Bruchs durch den gefundenen groͤſten gemeinen Theiler diuidirt, wobey noch dieſes zu mercken iſt, daß der Bruch, welchen man auf dieſe Weiſe erhaͤlt, nicht weiter verkleinert oder aufgehoben werden koͤnne, und dadurch folglich der vorgelegte Bruch in den kleineſten Zahlen ausgedruͤckt werde.","norm":"Kommt aber ein anderer größerer gemeiner Teiler heraus, so kann der vorgegebene Bruch in kleinere Zahlen gebracht werden, wann man nämlich den Zähler und Nenner des gegebenen Bruchs durch den gefundenen größten gemeinen Teiler dividiert, wobei noch dieses zu merken ist, dass der Bruch, welchen man auf diese Weise erhält, nicht weiter verkleinert oder aufgehoben werden könne, und dadurch folglich der vorgelegte Bruch in den kleinsten Zahlen 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+{"basename":"euler_rechenkunst01_1738","par_idx":1352,"date":1738,"orig":"Derowegen iſt die hier gegebene Regel, durch welche man lernet einen Bruch in den kleinſten moͤglichen Zahlen vorzuſtellen, von ſehr groſſem Nutzen; indem man durch Huͤlfe derſelben einen Bruch entweder ſicher in die kleineſten Zahlen bringen, oder wo eine ſolche Aufhebung nicht Statt findet, verſichert ſeyn kan, daß der vorgelegte Bruch unaufheblich ſey, und durch kleinere Zahlen unmoͤglich vorgeſtellt werden koͤnne.","norm":"Derowegen ist die hier gegebene Regel, durch welche man lernet einen Bruch in den kleinsten möglichen Zahlen vorzustellen, von sehr großem Nutzen; indem man durch Hilfe derselben einen Bruch entweder sicher in die kleinsten Zahlen bringen, oder wo eine solche Aufhebung nicht statt findet, versichert sein kann, dass der vorgelegte Bruch unaufheblich sei, und durch kleinere Zahlen unmöglich vorgestellt werden 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+{"basename":"euler_rechenkunst01_1738","par_idx":1353,"date":1738,"orig":"Dieſe Verwandlung in die leichteſte Form geſchicht nun durch die Ausfindung des groͤſten gemeinen Theilers der beyden Zahlen des Bruchs nehmlich des Zehlers und Nenners, wozu in vorigen Satze genugſame Anleitung gegeben worden iſt.","norm":"Diese Verwandlung in die leichteste Form geschieht nun durch die Ausfindung des größten gemeinen Teilers der beiden Zahlen des Bruchs nämlich des Zählers und Nenners, wozu in vorigen Satze genugsame Anleitung gegeben worden 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sind.","orig_tok":["Erſtlich","ſieht","man",",","weil","der","Zehler","groͤſſer","iſt","als","der","Nenner",",","daß","in","dieſem","Bruche","ein","oder","etliche","gantze","enthalten","ſind",",","weswegen","vor","allen","Dingen","dienlich","ſeyn","wird","zu","ſuchen","wieviel","gantze","vorhanden","ſind",",","weilen","man","alsdann","ſchon","einen","deutlicheren","Begriff","von","dem","Werthe","deſſelben","erhaͤlt",",","als","wann","die","gantzen","mit","im","Bruche","eingewickelt","ſind","."],"norm_tok":["Erstlich","sieht","man",",","weil","der","Zähler","größer","ist","als","der","Nenner",",","dass","in","diesem","Bruche","ein","oder","etliche","ganze","enthalten","sind",",","weswegen","vor","allen","Dingen","dienlich","sein","wird","zu","suchen","wie","viel","ganze","vorhanden","sind",",","weilen","man","alsdann","schon","einen","deutlicheren","Begriff","von","dem","Werte","desselben","erhält",",","als","wann","die","ganzen","mit","im","Bruche","eingewickelt","sind","."],"orig_spans":[[0,8],[9,14],[15,18],[18,19],[20,24],[25,28],[29,35],[36,44],[45,48],[49,52],[53,56],[57,63],[63,64],[65,68],[69,71],[72,78],[79,85],[86,89],[90,94],[95,102],[103,109],[110,119],[120,124],[124,125],[126,134],[135,138],[139,144],[145,151],[152,160],[161,165],[166,170],[171,173],[174,180],[181,188],[189,195],[196,205],[206,210],[210,211],[212,218],[219,222],[223,230],[231,236],[237,242],[243,255],[256,263],[264,267],[268,271],[272,278],[279,288],[289,296],[296,297],[298,301],[302,306],[307,310],[311,318],[319,322],[323,325],[326,332],[333,345],[346,350],[350,351]],"norm_spans":[[0,8],[9,14],[15,18],[18,19],[20,24],[25,28],[29,35],[36,42],[43,46],[47,50],[51,54],[55,61],[61,62],[63,67],[68,70],[71,77],[78,84],[85,88],[89,93],[94,101],[102,107],[108,117],[118,122],[122,123],[124,132],[133,136],[137,142],[143,149],[150,158],[159,163],[164,168],[169,171],[172,178],[179,182],[183,187],[188,193],[194,203],[204,208],[208,209],[210,216],[217,220],[221,228],[229,234],[235,240],[241,253],[254,261],[262,265],[266,269],[270,275],[276,285],[286,292],[292,293],[294,297],[298,302],[303,306],[307,313],[314,317],[318,320],[321,327],[328,340],[341,345],[345,346]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28],[29,29],[30,30],[31,31],[32,32],[33,33],[33,34],[34,35],[35,36],[36,37],[37,38],[38,39],[39,40],[40,41],[41,42],[42,43],[43,44],[44,45],[45,46],[46,47],[47,48],[48,49],[49,50],[50,51],[51,52],[52,53],[53,54],[54,55],[55,56],[56,57],[57,58],[58,59],[59,60],[60,61]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.869900226593018} +{"basename":"euler_rechenkunst01_1738","par_idx":1372,"date":1738,"orig":"Um nun dieſes zu finden, ſo hat man nach dem ſechſten Satz den Zehler durch den Nenner zu diuidiren wie folgt.","norm":"Um nun dieses zu finden, so hat man nach dem sechsten Satz den Zähler durch den Nenner zu dividieren wie 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+{"basename":"euler_rechenkunst01_1738","par_idx":1373,"date":1738,"orig":"[Formel]","norm":"[Formel]","orig_tok":["[Formel]"],"norm_tok":["[","Formel","]"],"orig_spans":[[0,8]],"norm_spans":[[0,1],[1,7],[7,8]],"alignment":[[0,0],[0,1],[0,2]],"lang_fastText":"sh","lang_py3langid":"en","lang_cld3":"cy","orig_lang_de":0,"norm_lmscore":7.932600021362305} +{"basename":"euler_rechenkunst01_1738","par_idx":1374,"date":1738,"orig":"Alſo ſieht man ſchon, daß der vorgegebene Bruch in dieſe Form 5 [Formel] gebracht werde, welche ſchon leichter zu begreiffen iſt als die vorgelegte.","norm":"Also sieht man schon, dass der vorgegebene Bruch in diese Form 5 [Formel] gebracht werde, welche schon leichter zu begreifen ist als die 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+{"basename":"euler_rechenkunst01_1738","par_idx":1375,"date":1738,"orig":"Ferner hat man zu ſehen, ob der Bruch [Formel] nicht durch kleinere Zahlen ausgedruͤcket werden koͤnne, welches geſchieht wann man den groͤſten gemeinen Theiler des Zehlers und Nenners ſuchet, ſolcher geſtalt.","norm":"Ferner hat man zu sehen, ob der Bruch [Formel] nicht durch kleinere Zahlen ausgedrückt werden könne, welches geschieht wann man den größten gemeinen Teiler des Zählers und Nenners suchet, solcher Gestalt.","orig_tok":["Ferner","hat","man","zu","ſehen",",","ob","der","Bruch","[Formel]","nicht","durch","kleinere","Zahlen","ausgedruͤcket","werden","koͤnne",",","welches","geſchieht","wann","man","den","groͤſten","gemeinen","Theiler","des","Zehlers","und","Nenners","ſuchet",",","ſolcher","geſtalt","."],"norm_tok":["Ferner","hat","man","zu","sehen",",","ob","der","Bruch","[","Formel","]","nicht","durch","kleinere","Zahlen","ausgedrückt","werden","könne",",","welches","geschieht","wann","man","den","größten","gemeinen","Teiler","des","Zählers","und","Nenners","suchet",",","solcher","Gestalt","."],"orig_spans":[[0,6],[7,10],[11,14],[15,17],[18,23],[23,24],[25,27],[28,31],[32,37],[38,46],[47,52],[53,58],[59,67],[68,74],[75,88],[89,95],[96,102],[102,103],[104,111],[112,121],[122,126],[127,130],[131,134],[135,143],[144,152],[153,160],[161,164],[165,172],[173,176],[177,184],[185,191],[191,192],[193,200],[201,208],[208,209]],"norm_spans":[[0,6],[7,10],[11,14],[15,17],[18,23],[23,24],[25,27],[28,31],[32,37],[38,39],[39,45],[45,46],[47,52],[53,58],[59,67],[68,74],[75,86],[87,93],[94,99],[99,100],[101,108],[109,118],[119,123],[124,127],[128,131],[132,139],[140,148],[149,155],[156,159],[160,167],[168,171],[172,179],[180,186],[186,187],[188,195],[196,203],[203,204]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[9,10],[9,11],[10,12],[11,13],[12,14],[13,15],[14,16],[15,17],[16,18],[17,19],[18,20],[19,21],[20,22],[21,23],[22,24],[23,25],[24,26],[25,27],[26,28],[27,29],[28,30],[29,31],[30,32],[31,33],[32,34],[33,35],[34,36]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.360000133514404} +{"basename":"euler_rechenkunst01_1738","par_idx":1376,"date":1738,"orig":"[Formel]","norm":"[Formel]","orig_tok":["[Formel]"],"norm_tok":["[","Formel","]"],"orig_spans":[[0,8]],"norm_spans":[[0,1],[1,7],[7,8]],"alignment":[[0,0],[0,1],[0,2]],"lang_fastText":"sh","lang_py3langid":"en","lang_cld3":"cy","orig_lang_de":0,"norm_lmscore":7.932600021362305} +{"basename":"euler_rechenkunst01_1738","par_idx":1377,"date":1738,"orig":"Demnach iſt 174 der groͤſte gemeine Theiler, wann man nun den gefundenen Bruch [Formel] dadurch aufhebt ſo bekommt man dieſen [Formel] . Derowegen iſt der im Anfang gegebenen Bruch [Formel] ſo viel als 5 [Formel] das iſt ſo viel als fuͤnf gantze und drey Zehntel eines gantzen.","norm":"Demnach ist 174 der größte gemeine Teiler, wann man nun den gefundenen Bruch [Formel] dadurch aufhebt so bekommt man diesen [Formel]. Derowegen ist der im Anfang gegebenen Bruch [Formel] so viel als 5 [Formel] das ist so viel als fünf ganze und drei Zehntel eines ganzen.","orig_tok":["Demnach","iſt","174","der","groͤſte","gemeine","Theiler",",","wann","man","nun","den","gefundenen","Bruch","[Formel]","dadurch","aufhebt","ſo","bekommt","man","dieſen","[Formel]",".","Derowegen","iſt","der","im","Anfang","gegebenen","Bruch","[Formel]","ſo","viel","als","5","[Formel]","das","iſt","ſo","viel","als","fuͤnf","gantze","und","drey","Zehntel","eines","gantzen","."],"norm_tok":["Demnach","ist","174","der","größte","gemeine","Teiler",",","wann","man","nun","den","gefundenen","Bruch","[","Formel","]","dadurch","aufhebt","so","bekommt","man","diesen","[","Formel","]",".","Derowegen","ist","der","im","Anfang","gegebenen","Bruch","[","Formel","]","so","viel","als","5","[","Formel","]","das","ist","so","viel","als","fünf","ganze","und","drei","Zehntel","eines","ganzen","."],"orig_spans":[[0,7],[8,11],[12,15],[16,19],[20,27],[28,35],[36,43],[43,44],[45,49],[50,53],[54,57],[58,61],[62,72],[73,78],[79,87],[88,95],[96,103],[104,106],[107,114],[115,118],[119,125],[126,134],[135,136],[137,146],[147,150],[151,154],[155,157],[158,164],[165,174],[175,180],[181,189],[190,192],[193,197],[198,201],[202,203],[204,212],[213,216],[217,220],[221,223],[224,228],[229,232],[233,238],[239,245],[246,249],[250,254],[255,262],[263,268],[269,276],[276,277]],"norm_spans":[[0,7],[8,11],[12,15],[16,19],[20,26],[27,34],[35,41],[41,42],[43,47],[48,51],[52,55],[56,59],[60,70],[71,76],[77,78],[78,84],[84,85],[86,93],[94,101],[102,104],[105,112],[113,116],[117,123],[124,125],[125,131],[131,132],[132,133],[134,143],[144,147],[148,151],[152,154],[155,161],[162,171],[172,177],[178,179],[179,185],[185,186],[187,189],[190,194],[195,198],[199,200],[201,202],[202,208],[208,209],[210,213],[214,217],[218,220],[221,225],[226,229],[230,234],[235,240],[241,244],[245,249],[250,257],[258,263],[264,270],[270,271]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[14,15],[14,16],[15,17],[16,18],[17,19],[18,20],[19,21],[20,22],[21,23],[21,24],[21,25],[22,26],[23,27],[24,28],[25,29],[26,30],[27,31],[28,32],[29,33],[30,34],[30,35],[30,36],[31,37],[32,38],[33,39],[34,40],[35,41],[35,42],[35,43],[36,44],[37,45],[38,46],[39,47],[40,48],[41,49],[42,50],[43,51],[44,52],[45,53],[46,54],[47,55],[48,56]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.156000137329102} +{"basename":"euler_rechenkunst01_1738","par_idx":1378,"date":1738,"orig":"Man kan aber auch gleich den groͤſten gemeinen Theiler des Zehlers und Nenners des gegebenen Bruchs ſuchen, alſo [Formel]","norm":"Man kann aber auch gleich den größten gemeinen Teiler des Zählers und Nenners des gegebenen Bruchs suchen, also [Formel]","orig_tok":["Man","kan","aber","auch","gleich","den","groͤſten","gemeinen","Theiler","des","Zehlers","und","Nenners","des","gegebenen","Bruchs","ſuchen",",","alſo","[Formel]"],"norm_tok":["Man","kann","aber","auch","gleich","den","größten","gemeinen","Teiler","des","Zählers","und","Nenners","des","gegebenen","Bruchs","suchen",",","also","[","Formel","]"],"orig_spans":[[0,3],[4,7],[8,12],[13,17],[18,24],[25,28],[29,37],[38,46],[47,54],[55,58],[59,66],[67,70],[71,78],[79,82],[83,92],[93,99],[100,106],[106,107],[108,112],[113,121]],"norm_spans":[[0,3],[4,8],[9,13],[14,18],[19,25],[26,29],[30,37],[38,46],[47,53],[54,57],[58,65],[66,69],[70,77],[78,81],[82,91],[92,98],[99,105],[105,106],[107,111],[112,113],[113,119],[119,120]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[19,20],[19,21]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.699100017547607} +{"basename":"euler_rechenkunst01_1738","par_idx":1379,"date":1738,"orig":"Weilen nun 174 der groͤſte gemeine Theiler iſt, ſo wird durch die Diuiſion der gegebene Bruch in dieſe Form [Formel] gebracht ſo daß [Formel] eben ſo viel iſt als [Formel] . Da aber der Bruch [Formel] mehr iſt als 1, ſo wird derſelbe, wann man den Zehler durch den Nenner wuͤrcklich diuidirt in dieſe Form 5 [Formel] verwandelt wie vorher.","norm":"Weilen nun 174 der größte gemeine Teiler ist, so wird durch die Division der gegebene Bruch in diese Form [Formel] gebracht so dass [Formel] ebenso viel ist als [Formel]. Da aber der Bruch [Formel] mehr ist als 1, so wird derselbe, wann man den Zähler durch den Nenner wirklich dividiert in diese Form 5 [Formel] verwandelt wie 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Wann aber zu einer gantzen Zahl eine gantze Zahl ſammt einem Bruche addirt werden ſoll, ſo darf man nur erſtlich die gantzen Zahlen addiren, und zu der herausgekommenen Zahl noch den Bruch, wie im vorigen Falle.","norm":"Als wann dieser Bruch [Formel] zu dieser Zahl 107 addiert werden soll, so wird die Summe sein 107 [Formel]. Wann aber zu einer ganzen Zahl eine ganze Zahl samt einem Bruche addiert werden soll, so darf man nur erstlich die ganzen Zahlen addieren, und zu der herausgekommenen Zahl noch den Bruch, wie im vorigen Falle.","orig_tok":["Als","wann","dieſer","Bruch","[Formel]","zu","dieſer","Zahl","107","addirt","werden","ſoll",",","ſo","wird","die","Summ","ſeyn","107","[Formel]",".","Wann","aber","zu","einer","gantzen","Zahl","eine","gantze","Zahl","ſammt","einem","Bruche","addirt","werden","ſoll",",","ſo","darf","man","nur","erſtlich","die","gantzen","Zahlen","addiren",",","und","zu","der","herausgekommenen","Zahl","noch","den","Bruch",",","wie","im","vorigen","Falle","."],"norm_tok":["Als","wann","dieser","Bruch","[","Formel","]","zu","dieser","Zahl","107","addiert","werden","soll",",","so","wird","die","Summe","sein","107","[","Formel","]",".","Wann","aber","zu","einer","ganzen","Zahl","eine","ganze","Zahl","samt","einem","Bruche","addiert","werden","soll",",","so","darf","man","nur","erstlich","die","ganzen","Zahlen","addieren",",","und","zu","der","herausgekommenen","Zahl","noch","den","Bruch",",","wie","im","vorigen","Falle","."],"orig_spans":[[0,3],[4,8],[9,15],[16,21],[22,30],[31,33],[34,40],[41,45],[46,49],[50,56],[57,63],[64,68],[68,69],[70,72],[73,77],[78,81],[82,86],[87,91],[92,95],[96,104],[105,106],[107,111],[112,116],[117,119],[120,125],[126,133],[134,138],[139,143],[144,150],[151,155],[156,161],[162,167],[168,174],[175,181],[182,188],[189,193],[193,194],[195,197],[198,202],[203,206],[207,210],[211,219],[220,223],[224,231],[232,238],[239,246],[246,247],[248,251],[252,254],[255,258],[259,275],[276,280],[281,285],[286,289],[290,295],[295,296],[297,300],[301,303],[304,311],[312,317],[317,318]],"norm_spans":[[0,3],[4,8],[9,15],[16,21],[22,23],[23,29],[29,30],[31,33],[34,40],[41,45],[46,49],[50,57],[58,64],[65,69],[69,70],[71,73],[74,78],[79,82],[83,88],[89,93],[94,97],[98,99],[99,105],[105,106],[106,107],[108,112],[113,117],[118,120],[121,126],[127,133],[134,138],[139,143],[144,149],[150,154],[155,159],[160,165],[166,172],[173,180],[181,187],[188,192],[192,193],[194,196],[197,201],[202,205],[206,209],[210,218],[219,222],[223,229],[230,236],[237,245],[245,246],[247,250],[251,253],[254,257],[258,274],[275,279],[280,284],[285,288],[289,294],[294,295],[296,299],[300,302],[303,310],[311,316],[316,317]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[4,5],[4,6],[5,7],[6,8],[7,9],[8,10],[9,11],[10,12],[11,13],[12,14],[13,15],[14,16],[15,17],[16,18],[17,19],[18,20],[19,21],[19,22],[19,23],[20,24],[21,25],[22,26],[23,27],[24,28],[25,29],[26,30],[27,31],[28,32],[29,33],[30,34],[31,35],[32,36],[33,37],[34,38],[35,39],[36,40],[37,41],[38,42],[39,43],[40,44],[41,45],[42,46],[43,47],[44,48],[45,49],[46,50],[47,51],[48,52],[49,53],[50,54],[51,55],[52,56],[53,57],[54,58],[55,59],[56,60],[57,61],[58,62],[59,63],[60,64]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.178400039672852} +{"basename":"euler_rechenkunst01_1738","par_idx":1394,"date":1738,"orig":"Als wann zu 17 addirt werden ſoll 9 [Formel] ſo wird die Summ ſeyn 26 [Formel] . Wann nun hinwiederum von 26 [Formel] ſollte 17 ſubtrahirt werden, ſo ſieht man aus dem vorigen Exempel, daß der Reſt 9 [Formel] ſeyn muͤſſe: dieſer Reſt aber wird gefunden, wann man 17 von 26 ſubtrahirt, und zum uͤbergebliebenen nehmlich 9 den Bruch [Formel] hinzuſetzt.","norm":"Als wann zu 17 addiert werden soll 9 [Formel] so wird die Summe sein 26 [Formel]. 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+{"basename":"euler_rechenkunst01_1738","par_idx":1399,"date":1738,"orig":"Hieraus ſieht man alſo, daß die groͤſte Schwierigkeit bey der Addition und Subtraction mit gebrocheuen Zahlen nur auf den Bruͤchen allein beruhe, und wann gantze Zahlen mit den Bruͤchen verknuͤpfet ſind, dadurch die Operation nicht ſchwehrer gemacht werde.","norm":"Hieraus sieht man also, dass die größte Schwierigkeit bei der Addition und Subtraktion mit gebrocheuen Zahlen nur auf den Brüchen allein beruhe, und wann ganze Zahlen mit den Brüchen verknüpfet sind, dadurch die Operation nicht schwerer gemacht 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+{"basename":"euler_rechenkunst01_1738","par_idx":1406,"date":1738,"orig":"Derowegen dieſe Bruͤche zuſammen addiren iſt nichts anders als finden wieviel dergleichen Theile alle insgeſammt enthalten.","norm":"Derowegen diese Brüche zusammen addieren ist nichts anders als finden wie viel dergleichen Teile alle insgesamt 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+{"basename":"euler_rechenkunst01_1738","par_idx":1407,"date":1738,"orig":"Wann man alſo alle Zehler zuſammen addirt, ſo weiſet die Summ, wieviel dergleichen Theile alle Bruͤche insgeſammt ausmachen.","norm":"Wann man also alle Zähler zusammen addiert, so weiset die Summe, wie viel dergleichen Teile alle Brüche insgesamt 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+{"basename":"euler_rechenkunst01_1738","par_idx":1408,"date":1738,"orig":"Da nun dieſes ſolche Theile ſind als der gemeine Nenner der gegebenen Bruͤche anzeigt, ſo iſt die Summ derſelben Bruͤche ein Bruch, deſſen Nenner der gemeine Nenner, der Zehler aber die Summ der Zehler iſt.","norm":"Da nun dieses solche Teile sind als der gemeine Nenner der gegebenen Brüche anzeigt, so ist die Summe derselben Brüche ein Bruch, dessen Nenner der gemeine Nenner, der Zähler aber die Summe der Zähler 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+{"basename":"euler_rechenkunst01_1738","par_idx":1409,"date":1738,"orig":"Als wann zum Exempel dieſe Bruͤche [Formel] , [Formel] , und [Formel] zuſammen addirt werden ſollten, ſo ſieht man daß ein jeder Bruch einerley, nehmlich fuͤnf und zwanzigſte, Theile eines gantzen andeute, dergleichen der erſte 2, der andere 4 und der dritte 7 enthaͤlt.","norm":"Als wann zum Exempel diese Brüche [Formel], [Formel], und [Formel] zusammen addiert werden sollten, so sieht man dass ein jeder Bruch einerlei, nämlich fünf und zwanzigste, Teile eines ganzen andeute, dergleichen der erste 2, der andere 4 und der dritte 7 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+{"basename":"euler_rechenkunst01_1738","par_idx":1410,"date":1738,"orig":"Alle drey zuſammen alſo machen 12 fuͤnf und zwanzigſte Theile eines gantzen aus, welche alſo [Formel] geſchrieben werden, und folglich iſt dieſer Bruch [Formel] , deſſen Nenner dem gemeinen Nenner der gegebenen Bruͤche, der Zehler aber der Summ der Zehler gleich iſt, die geſuchte Summ der gegebenen Bruͤche [Formel] , [Formel] und [Formel] . Hieraus erhellet nun daß die Summ zweyer oder mehr gegebenen Bruͤche, welche gleiche Nenner baben ein Bruch ſey, deſſen Nenner der vorige gemeine Nenner, der Zehler aber die Summ der Zehler der gegebenen Bruͤche iſt.","norm":"Alle drei zusammen also machen 12 fünf und zwanzigste Teile eines ganzen aus, welche also [Formel] geschrieben werden, und folglich ist dieser Bruch [Formel], dessen Nenner dem gemeinen Nenner der gegebenen Brüche, der Zähler aber der Summe der Zähler gleich ist, die gesuchte Summe der gegebenen Brüche [Formel], [Formel] und [Formel]. Hieraus erhellet nun dass die Summe zweier oder mehr gegebenen Brüche, welche gleiche Nenner haben ein Bruch sei, dessen Nenner der vorige gemeine Nenner, der Zähler aber die Summe der Zähler der gegebenen Brüche 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+{"basename":"euler_rechenkunst01_1738","par_idx":1434,"date":1738,"orig":"Soll aber von einer gantzen Zahl nebſt einem Bruche eine andere gantze Zahl nebſt einem Bruche, deſſen Nenner des vorigen Bruchs Nenner gleich iſt, ſubtrahirt werden, ſo wird der Bruch der kleineren Zahl von dem Bruche der groͤſſeren, und die gantze kleinere Zahl von der gantzen groͤſſeren ſubtrahirt, wann der Bruch der groͤſſern Zahl groͤſſer iſt als der Bruch der kleineren Zahl.","norm":"Soll aber von einer ganzen Zahl nebst einem Bruche eine andere ganze Zahl nebst einem Bruche, dessen Nenner des vorigen Bruchs Nenner gleich ist, subtrahiert werden, so wird der Bruch der kleineren Zahl von dem Bruche der größeren, und die ganze kleinere Zahl von der ganzen größeren subtrahiert, wann der Bruch der größeren Zahl größer ist als der Bruch der kleineren 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+{"basename":"euler_rechenkunst01_1738","par_idx":1437,"date":1738,"orig":"Wann man nun den kleineren Bruch vom groͤſſeren ſubtrahiren will, ſo zieht man die kleinere Anzahl ſolcher Theile von der groͤſſeren ab, das iſt, man ſubtrahirt den kleineren Zehler vom groͤſſeren, und unter den Reſt als den Zehler ſchreibt man den gemeinen Nenner.","norm":"Wann man nun den kleineren Bruch vom größeren subtrahieren will, so zieht man die kleinere Anzahl solcher Teile von der größeren ab, das ist, man subtrahiert den kleineren Zähler vom größeren, und unter den Rest als den Zähler schreibt man den gemeinen 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Laſſt uns nun zwey aus gantzen und Bruͤchen zuſammen geſetzte Zahlen betrachten, ſo iſt diejenige Zahl die groͤſſere, in welcher die gantze Zahl groͤſſer iſt, wann nehmlich die Bruͤche kleiner ſind als ein gantzes: allſo iſt 4⅓ mehr als 3⅔, obgleich der Bruch der kleineren groͤſſer iſt als der Bruch der groͤſſeren.","norm":"Hierbei ist nun zu merken, welches aus der Natur der Brüche von selbst folgt, dass von zweien Brüchen, welche gleiche Nenner haben, derjenige der größere ist, welcher den größeren Zähler hat; sind also die Zähler einander gleich, so sind auch die Brüche einander gleich und folglich der Rest nichts als 2/3 von 2/3 bleibt 0. Lasst uns nun zwei aus ganzen und Brüchen zusammen gesetzte Zahlen betrachten, so ist diejenige Zahl die größere, in welcher die ganze Zahl größer ist, wann nämlich die Brüche kleiner sind als ein ganzes: also ist 4 1/3 mehr als 3 2/3 , obgleich der Bruch der kleineren größer ist als der Bruch der 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+{"basename":"euler_rechenkunst01_1738","par_idx":1460,"date":1738,"orig":"Die gegebene Regel eine ſolche gemeine theilbare Zahl zu finden iſt leicht zu begreiffen, dann wann man die gegebenen Zahlen mit einander multiplicirt, ſo laͤſt ſich hinwiederum das Product durch eine jegliche der gegebenen Zahlen theilen und iſt folglich davon eine gemeine theilbare Zahl.","norm":"Die gegebene Regel eine solche gemeine teilbare Zahl zu finden ist leicht zu begreifen, dann wann man die gegebenen Zahlen miteinander multipliziert, so lässt sich hinwiederum das Produkt durch eine jegliche der gegebenen Zahlen teilen und ist folglich davon eine gemeine teilbare 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+{"basename":"euler_rechenkunst01_1738","par_idx":1461,"date":1738,"orig":"Ferner iſt auch klar, daß wann ſich eine Zahl durch die gegebenen Zahlen theilen laͤſt, auch das doppelte, dreyfache, und ſo fort, dieſelbe theilbare Zahl mit einer jeglichen beliebigen Zahl multiplicirt ſich dadurch theilen laſſe; dann eine jegliche Zahl, welche ſich durch die gemeine theilbare Zahl theilen laͤſt, laͤſt ſich auch durch die gegebenen Zahlen theilen.","norm":"Ferner ist auch klar, dass wann sich eine Zahl durch die gegebenen Zahlen teilen lässt, auch das doppelte, dreifache, und so fort, dieselbe teilbare Zahl mit einer jeglichen beliebigen Zahl multipliziert sich dadurch teilen lasse; dann eine jegliche Zahl, welche sich durch die gemeine teilbare Zahl teilen lässt, lässt sich auch durch die gegebenen Zahlen 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Ferner ſind auch alle Zahlen, welche ſich durch 30 theilen laſſen, gemeine theilbare Zahlen von 2, 3, und 5; dieſe werden gefunden, wann man 30 mit einer beliebigen Zahl multiplicirt; als da ſind 60, 90, 120, 150 und ſo weiter.","norm":"Als bei den gegebenen Zahlen 2, 3, 5 multipliziert man nach der Regel erstlich 2 mit 3 und das Produkt 6 noch mit 5, so ist 30 das Produkt von 2, 3, und 5, und folglich eine gemeine teilbare Zahl von 2, 3 und 5. Ferner sind auch alle Zahlen, welche sich durch 30 teilen lassen, gemeine teilbare Zahlen von 2, 3, und 5; diese werden gefunden, wann man 30 mit einer beliebigen Zahl multipliziert; als da sind 60, 90, 120, 150 und so 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Hernach multiplicirt man die gefundene Zahl 70 mit 9, ſo iſt das Product 630 eine gemeine theilbare Zahl von 70 und 9 und folglich auch von 2, 5, 7, und 9. Endlich multiplicirt man 630 mit 11, ſo iſt das Product 6930 eine gemeine theilbare Zahl von 2, 5, 7, 9, und 11, dergleichen verlanget worden.","norm":"Ferner sucht man von 10 und 7 eine gemeine teilbare Zahl, in dem man 10 mit 7 multipliziert; so ist 70 schon eine gemeine teilbare Zahl von 2, 5 und 7. Hernach multipliziert man die gefundene Zahl 70 mit 9, so ist das Produkt 630 eine gemeine teilbare Zahl von 70 und 9 und folglich auch von 2, 5, 7, und 9. Endlich multipliziert man 630 mit 11, so ist das Produkt 6930 eine gemeine teilbare Zahl von 2, 5, 7, 9, und 11, dergleichen verlanget 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+{"basename":"euler_rechenkunst01_1738","par_idx":1468,"date":1738,"orig":"Ob wir nun gleich im folgenden dazu die gehoͤrige Regel geben werden, ſo wollen wir doch hier ein Exempel von einem ſolchen Falle vorbringen, damit man ſich davon zum voraus einen Begriff machen koͤnne.","norm":"Ob wir nun gleich im folgenden dazu die gehörige Regel geben werden, so wollen wir doch hier ein Exempel von einem solchen Falle vorbringen, damit man sich davon zum voraus einen Begriff machen 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+{"basename":"euler_rechenkunst01_1738","par_idx":1469,"date":1738,"orig":"Wann alſo von dieſen Zahlen 2, 4, 6, 9 eine gemeine theilbare Zahl geſucht werden ſollte; ſo nehme man erſtlich 2 mit 4; davon ſieht man, daß 4 eine gemeine theilbare Zahl iſt, welche kleiner iſt, als die, ſo durch die gegebene Regel gefunden wird, nehmlich 8.","norm":"Wann also von diesen Zahlen 2, 4, 6, 9 eine gemeine teilbare Zahl gesucht werden sollte; so nehme man erstlich 2 mit 4; davon sieht man, dass 4 eine gemeine teilbare Zahl ist, welche kleiner ist, als die, so durch die gegebene Regel gefunden wird, nämlich 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+{"basename":"euler_rechenkunst01_1738","par_idx":1470,"date":1738,"orig":"Man nehme alſo nicht 8 ſondern 4 und dazu 6, und ſuche von 4 und 6 eine gemeine theilbare Zahl; welche nach der Regel 24 ſeyn wuͤrde, man ſieht aber daß ſich auch12 durch 4 und 6 theilen laſſe; welche Zahl man alſo der anderen billich vorzieht.","norm":"Man nehme also nicht 8 sondern 4 und dazu 6, und suche von 4 und 6 eine gemeine teilbare Zahl; welche nach der Regel 24 sein würde, man sieht aber dass sich auch12 durch 4 und 6 teilen lasse; welche Zahl man also der anderen billig 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+{"basename":"euler_rechenkunst01_1738","par_idx":1480,"date":1738,"orig":"Wann man alſo das Product durch den groͤſten gemeinen Theiler diuidirt, ſo muß der Quotus die kleinſte gemeine theilbare Zahl ſeyn von den zwey gegebenen Zahlen, ſo moͤglich iſt.","norm":"Wann man also das Produkt durch den größten gemeinen Teiler dividiert, so muss der Quotus die kleinste gemeine teilbare Zahl sein von den zwei gegebenen Zahlen, so möglich 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+{"basename":"euler_rechenkunst01_1738","par_idx":1481,"date":1738,"orig":"Wie aber der groͤſte gemeine Theiler zweyer Zahlen gefunden werden ſoll, iſt ſchon oben gelehret worden, und vermittelſt deſſelben kan man alſo allezeit zweyer gegebenen Zahlen kleinſte gemeine theilbare Zahl ausfinden.","norm":"Wie aber der größte gemeine Teiler zweier Zahlen gefunden werden soll, ist schon oben gelehrt worden, und vermittels desselben kann man also allezeit zweier gegebenen Zahlen kleinste gemeine teilbare Zahl 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+{"basename":"euler_rechenkunst01_1738","par_idx":1482,"date":1738,"orig":"Es iſt aber gleich viel ob man das Product der zwey gegebenen Zahlen durch den groͤſten gemeinen Theiler diuidirt, oder ob man vor der Multiplication die eine Zahl durch den groͤſten gemeinen Theiler diuidirt, und hernach durch den gefundenen Quotum die andere Zahl multiplicirt.","norm":"Es ist aber gleich viel ob man das Produkt der zwei gegebenen Zahlen durch den größten gemeinen Teiler dividiert, oder ob man vor der Multiplikation die eine Zahl durch den größten gemeinen Teiler dividiert, und hernach durch den gefundenen Quotum die andere Zahl 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+{"basename":"euler_rechenkunst01_1738","par_idx":1489,"date":1738,"orig":"Hernach multiplicirt man man durch das Kreutz eine jede Zahl mit dem Quoto der anderen, da dann beyderſeits 45 herauskommt, welches die kleinſte gemeine theilbare Zahl der beyden gegebenen iſt.","norm":"Hernach multipliziert man man durch das Kreuz eine jede Zahl mit dem Quoto der anderen, da dann beiderseits 45 herauskommt, welches die kleinste gemeine teilbare Zahl der beiden gegebenen 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+{"basename":"euler_rechenkunst01_1738","par_idx":1490,"date":1738,"orig":"Ob aber gleich von dieſen beyden Operationen eine allein genug waͤre, ſo iſt gleichwohl dieſe doppelte Operation nicht gaͤntzlich als unnuͤtz zu verwerfen:","norm":"Ob aber gleich von diesen beiden Operationen eine allein genug wäre, so ist gleichwohl diese doppelte Operation nicht gänzlich als unnütz zu 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+{"basename":"euler_rechenkunst01_1738","par_idx":1491,"date":1738,"orig":"dann da durch beyde Multiplicationen ein Product herauskommen muß, ſo dienet dieſe Operation zugleich als eine Probe, daß man ſich im Rechnen nicht geirret; indem wann nicht einerley Zahl gefunden werden ſollte, daſſelbe ein gewiſſes Zeichen eines Fehlers ſeyn wuͤrde.","norm":"dann da durch beide Multiplikationen ein Produkt herauskommen muss, so dient diese Operation zugleich als eine Probe, dass man sich im Rechnen nicht geirrt; indem wann nicht einerlei Zahl gefunden werden sollte, dasselbe ein gewisses Zeichen eines Fehlers sein 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+{"basename":"euler_rechenkunst01_1738","par_idx":1497,"date":1738,"orig":"[Formel]","norm":"[Formel]","orig_tok":["[Formel]"],"norm_tok":["[","Formel","]"],"orig_spans":[[0,8]],"norm_spans":[[0,1],[1,7],[7,8]],"alignment":[[0,0],[0,1],[0,2]],"lang_fastText":"sh","lang_py3langid":"en","lang_cld3":"cy","orig_lang_de":0,"norm_lmscore":7.932600021362305} +{"basename":"euler_rechenkunst01_1738","par_idx":1498,"date":1738,"orig":"Hieraus ſieht man alſo, daß wann ſich von den zweyen gegebenen Zahlen die groͤſſere durch die kleinere theilen laͤſt, ſo dann die groͤſſere Zahl ſelbſt die kleinſte gemeine theilbare Zahl ſey; in welchen Faͤllen man alſo nicht einmahl noͤthig hat die vorgeſchriebenen Operationen anzuſtellen.","norm":"Hieraus sieht man also, dass wann sich von den zweien gegebenen Zahlen die größere durch die kleinere Teilen lässt, so dann die größere Zahl selbst die kleinste gemeine teilbare Zahl sei; in welchen Fällen man also nicht einmal nötig hat die vorgeschriebenen 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+{"basename":"euler_rechenkunst01_1738","par_idx":1499,"date":1738,"orig":"Wer nun von zweyen gegebenen Zahlen die kleinſte gemeine theilbare Zahl finden kan, derſelbe iſt zugleich im Stande von ſo viel Zahlen als vorgegeben ſeyn moͤchten, die kleinſte gemeine theilbare Zahl zu finden.","norm":"Wer nun von zweien gegebenen Zahlen die kleinste gemeine teilbare Zahl finden kann, derselbe ist zugleich im Stande von so viel Zahlen als vorgegeben sein möchten, die kleinste gemeine teilbare Zahl zu 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+{"basename":"euler_rechenkunst01_1738","par_idx":1500,"date":1738,"orig":"Dann von den vorgegebenen Zahlen nimmt man zwey nach Belieben, und ſucht davon die kleinſte gemeine theilbare Zahl; welche in die Stelle derſelben zweyen Zahlen geſetzt werden kan, ſo daß auf ſolche Weiſe die Anzahl der gegebenen Zahlen um eine kleiner wird.","norm":"Dann von den vorgegebenen Zahlen nimmt man zwei nach Belieben, und sucht davon die kleinste gemeine teilbare Zahl; welche in die Stelle derselben zweien Zahlen gesetzt werden kann, so dass auf solche Weise die Anzahl der gegebenen Zahlen um eine kleiner 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+{"basename":"euler_rechenkunst01_1738","par_idx":1501,"date":1738,"orig":"Ferner nimmt man wiederum nach Belieben zwey Zahlen, und ſucht davon die kleinſte gemeine theilbare Zahl, und ſetzt dieſelbe an die Stelle derſelben zweyen Zahlen, ſo daß die Anzahl der Zahlen wiederum um eine vermindert wird.","norm":"Ferner nimmt man wiederum nach Belieben zwei Zahlen, und sucht davon die kleinste gemeine teilbare Zahl, und setzt dieselbe an die Stelle derselben zweien Zahlen, so dass die Anzahl der Zahlen wiederum um eine vermindert 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+{"basename":"euler_rechenkunst01_1738","par_idx":1502,"date":1738,"orig":"Solchergeſtalt faͤhrt man alſo fort, bis man alle gegebenen Zahlen auf zwey gebracht hat, deren kleinſte gemeine theilbare Zahl zugleich die kleinſte gemeine theilbare Zahl von allen vorgegebenen Zahlen iſt.","norm":"Solchergestalt fährt man also fort, bis man alle gegebenen Zahlen auf zwei gebracht hat, deren kleinste gemeine teilbare Zahl zugleich die kleinste gemeine teilbare Zahl von allen vorgegebenen Zahlen ist.","orig_tok":["Solchergeſtalt","faͤhrt","man","alſo","fort",",","bis","man","alle","gegebenen","Zahlen","auf","zwey","gebracht","hat",",","deren","kleinſte","gemeine","theilbare","Zahl","zugleich","die","kleinſte","gemeine","theilbare","Zahl","von","allen","vorgegebenen","Zahlen","iſt","."],"norm_tok":["Solchergestalt","fährt","man","also","fort",",","bis","man","alle","gegebenen","Zahlen","auf","zwei","gebracht","hat",",","deren","kleinste","gemeine","teilbare","Zahl","zugleich","die","kleinste","gemeine","teilbare","Zahl","von","allen","vorgegebenen","Zahlen","ist","."],"orig_spans":[[0,14],[15,21],[22,25],[26,30],[31,35],[35,36],[37,40],[41,44],[45,49],[50,59],[60,66],[67,70],[71,75],[76,84],[85,88],[88,89],[90,95],[96,104],[105,112],[113,122],[123,127],[128,136],[137,140],[141,149],[150,157],[158,167],[168,172],[173,176],[177,182],[183,195],[196,202],[203,206],[206,207]],"norm_spans":[[0,14],[15,20],[21,24],[25,29],[30,34],[34,35],[36,39],[40,43],[44,48],[49,58],[59,65],[66,69],[70,74],[75,83],[84,87],[87,88],[89,94],[95,103],[104,111],[112,120],[121,125],[126,134],[135,138],[139,147],[148,155],[156,164],[165,169],[170,173],[174,179],[180,192],[193,199],[200,203],[203,204]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28],[29,29],[30,30],[31,31],[32,32]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.152699947357178} +{"basename":"euler_rechenkunst01_1738","par_idx":1503,"date":1738,"orig":"Dieſe Regel iſt von der im Satze gegebenen nur darinn unterſchieden, daß man nach jener immer die letztgefundene kleinſte theilbare Zahl mit einer neuen Zahl zuſammen nimmt, und davon die kleinſte gemeine theilbare Zahl ſucht:","norm":"Diese Regel ist von der im Satze gegebenen nur darin unterschieden, dass man nach jener immer die letztgefundene kleinste teilbare Zahl mit einer neuen Zahl zusammen nimmt, und davon die kleinste gemeine teilbare Zahl sucht:","orig_tok":["Dieſe","Regel","iſt","von","der","im","Satze","gegebenen","nur","darinn","unterſchieden",",","daß","man","nach","jener","immer","die","letztgefundene","kleinſte","theilbare","Zahl","mit","einer","neuen","Zahl","zuſammen","nimmt",",","und","davon","die","kleinſte","gemeine","theilbare","Zahl","ſucht",":"],"norm_tok":["Diese","Regel","ist","von","der","im","Satze","gegebenen","nur","darin","unterschieden",",","dass","man","nach","jener","immer","die","letztgefundene","kleinste","teilbare","Zahl","mit","einer","neuen","Zahl","zusammen","nimmt",",","und","davon","die","kleinste","gemeine","teilbare","Zahl","sucht",":"],"orig_spans":[[0,5],[6,11],[12,15],[16,19],[20,23],[24,26],[27,32],[33,42],[43,46],[47,53],[54,67],[67,68],[69,72],[73,76],[77,81],[82,87],[88,93],[94,97],[98,112],[113,121],[122,131],[132,136],[137,140],[141,146],[147,152],[153,157],[158,166],[167,172],[172,173],[174,177],[178,183],[184,187],[188,196],[197,204],[205,214],[215,219],[220,225],[225,226]],"norm_spans":[[0,5],[6,11],[12,15],[16,19],[20,23],[24,26],[27,32],[33,42],[43,46],[47,52],[53,66],[66,67],[68,72],[73,76],[77,81],[82,87],[88,93],[94,97],[98,112],[113,121],[122,130],[131,135],[136,139],[140,145],[146,151],[152,156],[157,165],[166,171],[171,172],[173,176],[177,182],[183,186],[187,195],[196,203],[204,212],[213,217],[218,223],[223,224]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28],[29,29],[30,30],[31,31],[32,32],[33,33],[34,34],[35,35],[36,36],[37,37]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.897799968719482} +{"basename":"euler_rechenkunst01_1738","par_idx":1504,"date":1738,"orig":"nach dieſer Regel aber man nach Belieben zwey Zahlen nehmen kan; welche noch nicht in Betrachtung gezogen worden ſind.","norm":"nach dieser Regel aber man nach Belieben zwei Zahlen nehmen kann; welche noch nicht in Betrachtung gezogen worden 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+{"basename":"euler_rechenkunst01_1738","par_idx":1505,"date":1738,"orig":"Dieſe Freyheit der letzteren Regel iſt aber nicht ohne Nutzen; dann da kan man immer ſolche zwey Zahlen ausleſen, davon man am leichteſten die kleinſte gemeine theilbare Zahl ausfinden kan, dergleichen ſind ſolche zwey Zahlen, davon die groͤſſere ſich durch die kleinere theilen laͤſt, dann da iſt die groͤſſere Zahl ſelbſt die kleinſte gemeine theilbare Zahl; wie ſchon gemeldet worden iſt.","norm":"Diese Freiheit der letzteren Regel ist aber nicht ohne Nutzen; dann da kann man immer solche zwei Zahlen auslesen, davon man am leichtesten die kleinste gemeine teilbare Zahl ausfinden kann, dergleichen sind solche zwei Zahlen, davon die größere sich durch die kleinere Teilen lässt, dann da ist die größere Zahl selbst die kleinste gemeine teilbare Zahl; wie schon gemeldet worden 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+{"basename":"euler_rechenkunst01_1738","par_idx":1506,"date":1738,"orig":"Oder man nimmt auch zwey ſolche Zahlen davon der groͤſte gemeine Theiler ſchon bekannt iſt, und iſt alſo der Muͤhe uͤberhoden ſich der vorgegebenen Operation zu bedienen.","norm":"Oder man nimmt auch zwei solche Zahlen davon der größte gemeine Teiler schon bekannt ist, und ist also der Mühe Überhoden sich der vorgegebenen Operation zu bedienen.","orig_tok":["Oder","man","nimmt","auch","zwey","ſolche","Zahlen","davon","der","groͤſte","gemeine","Theiler","ſchon","bekannt","iſt",",","und","iſt","alſo","der","Muͤhe","uͤberhoden","ſich","der","vorgegebenen","Operation","zu","bedienen","."],"norm_tok":["Oder","man","nimmt","auch","zwei","solche","Zahlen","davon","der","größte","gemeine","Teiler","schon","bekannt","ist",",","und","ist","also","der","Mühe","Überhoden","sich","der","vorgegebenen","Operation","zu","bedienen","."],"orig_spans":[[0,4],[5,8],[9,14],[15,19],[20,24],[25,31],[32,38],[39,44],[45,48],[49,56],[57,64],[65,72],[73,78],[79,86],[87,90],[90,91],[92,95],[96,99],[100,104],[105,108],[109,114],[115,125],[126,130],[131,134],[135,147],[148,157],[158,160],[161,169],[169,170]],"norm_spans":[[0,4],[5,8],[9,14],[15,19],[20,24],[25,31],[32,38],[39,44],[45,48],[49,55],[56,63],[64,70],[71,76],[77,84],[85,88],[88,89],[90,93],[94,97],[98,102],[103,106],[107,111],[112,121],[122,126],[127,130],[131,143],[144,153],[154,156],[157,165],[165,166]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":6.001999855041504} +{"basename":"euler_rechenkunst01_1738","par_idx":1507,"date":1738,"orig":"Durch ſolche Handgriffe aber, welche bey dieſer Regel angebracht werden koͤnnen, kan die gantze Operation ungemein abgekuͤrtzet werden; inſonderheit wann man ſich durch eine fleißige Ubung darinn feſtgeſetzt hat.","norm":"Durch solche Handgriffe aber, welche bei dieser Regel angebracht werden können, kann die ganze Operation ungemein abgekürzt werden; in Sonderheit wann man sich durch eine fleißige Übung darin festgesetzt hat.","orig_tok":["Durch","ſolche","Handgriffe","aber",",","welche","bey","dieſer","Regel","angebracht","werden","koͤnnen",",","kan","die","gantze","Operation","ungemein","abgekuͤrtzet","werden",";","inſonderheit","wann","man","ſich","durch","eine","fleißige","Ubung","darinn","feſtgeſetzt","hat","."],"norm_tok":["Durch","solche","Handgriffe","aber",",","welche","bei","dieser","Regel","angebracht","werden","können",",","kann","die","ganze","Operation","ungemein","abgekürzt","werden",";","in","Sonderheit","wann","man","sich","durch","eine","fleißige","Übung","darin","festgesetzt","hat","."],"orig_spans":[[0,5],[6,12],[13,23],[24,28],[28,29],[30,36],[37,40],[41,47],[48,53],[54,64],[65,71],[72,79],[79,80],[81,84],[85,88],[89,95],[96,105],[106,114],[115,127],[128,134],[134,135],[136,148],[149,153],[154,157],[158,162],[163,168],[169,173],[174,182],[183,188],[189,195],[196,207],[208,211],[211,212]],"norm_spans":[[0,5],[6,12],[13,23],[24,28],[28,29],[30,36],[37,40],[41,47],[48,53],[54,64],[65,71],[72,78],[78,79],[80,84],[85,88],[89,94],[95,104],[105,113],[114,123],[124,130],[130,131],[132,134],[135,145],[146,150],[151,154],[155,159],[160,165],[166,170],[171,179],[180,185],[186,191],[192,203],[204,207],[207,208]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[21,22],[22,23],[23,24],[24,25],[25,26],[26,27],[27,28],[28,29],[29,30],[30,31],[31,32],[32,33]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.187699794769287} +{"basename":"euler_rechenkunst01_1738","par_idx":1508,"date":1738,"orig":"Wir wollen aber den Gebrauch dieſer Regel durch einige Exempel deutlicher erklaͤren.","norm":"Wir wollen aber den Gebrauch dieser Regel durch einige Exempel deutlicher erklären.","orig_tok":["Wir","wollen","aber","den","Gebrauch","dieſer","Regel","durch","einige","Exempel","deutlicher","erklaͤren","."],"norm_tok":["Wir","wollen","aber","den","Gebrauch","dieser","Regel","durch","einige","Exempel","deutlicher","erklären","."],"orig_spans":[[0,3],[4,10],[11,15],[16,19],[20,28],[29,35],[36,41],[42,47],[48,54],[55,62],[63,73],[74,83],[83,84]],"norm_spans":[[0,3],[4,10],[11,15],[16,19],[20,28],[29,35],[36,41],[42,47],[48,54],[55,62],[63,73],[74,82],[82,83]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.22130012512207} +{"basename":"euler_rechenkunst01_1738","par_idx":1509,"date":1738,"orig":"Es ſoll von dieſen Zahlen 4, 5, 6, 9, 10, 16 die kleinſte gemeine theilbare Zahl gefunden werden.","norm":"Es soll von diesen Zahlen 4, 5, 6, 9, 10, 16 die kleinste gemeine teilbare Zahl gefunden 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+{"basename":"euler_rechenkunst01_1738","par_idx":1510,"date":1738,"orig":"Hier kan man zu erſt dieſe Zahlen 4 und 16 annehmen, weil ſich16 durch 4 theilen laͤſt, und folglich davon 16 die kleinſte gemeine theilbare Zahl iſt.","norm":"Hier kann man zuerst diese Zahlen 4 und 16 annehmen, weil sich16 durch 4 teilen lässt, und folglich davon 16 die kleinste gemeine teilbare Zahl 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+{"basename":"euler_rechenkunst01_1738","par_idx":1511,"date":1738,"orig":"Anſtatt dieſer beyden Zahlen 4 und 16 ſetzt man alſo nur 16, und hat folglich nur noch dieſe Zahlen, 5, 6, 9, 10, 16 davon die kleinſte gemeine theilbare Zahl geſucht werden ſoll.","norm":"Anstatt dieser beiden Zahlen 4 und 16 setzt man also nur 16, und hat folglich nur noch diese Zahlen, 5, 6, 9, 10, 16 davon die kleinste gemeine teilbare Zahl gesucht werden 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+{"basename":"euler_rechenkunst01_1738","par_idx":1535,"date":1738,"orig":"Hernach diuidirt man dieſen gemeinen Nenner durch einen jeglichen Nenner der gegebenen Bruͤche, und mit den Quotis multiplicirt man die dahin gehoͤrigen Zehler, ſo geben dieſe Producte die Zehler der geſuchten Bruͤche.","norm":"Hernach dividiert man diesen gemeinen Nenner durch einen jeglichen Nenner der gegebenen Brüche, und mit den Quotis multipliziert man die dahin gehörigen Zähler, so geben diese Produkte die Zähler der gesuchten 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+{"basename":"euler_rechenkunst01_1738","par_idx":1536,"date":1738,"orig":"Auf dieſe Art verwandelt man alſo die gegebenen Bruͤche in andere, welche den gegebenen dem Werthe nach gleich ſind, und dabey gleiche Nenner haben.","norm":"Auf diese Art verwandelt man also die gegebenen Brüche in andere, welche den gegebenen dem Werte nach gleich sind, und dabei gleiche Nenner 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+{"basename":"euler_rechenkunst01_1738","par_idx":1538,"date":1738,"orig":"Derowegen kan man allezeit einen Bruch in einen anderen verwandeln, deſſen Nenner gegeben iſt, wann ſich nur dieſer Nenner durch jenen theilen laͤſt.","norm":"Derowegen kann man allezeit einen Bruch in einen anderen verwandeln, dessen Nenner gegeben ist, wann sich nur dieser Nenner durch jenen teilen 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+{"basename":"euler_rechenkunst01_1738","par_idx":1539,"date":1738,"orig":"Als dieſer Bruch ¾ kan in einen anderen verwandelt werden, deſſen Nenner 12 iſt, weilen ſich12 durch 4 theilen laͤſt, und nehmlich 3 fuͤr den Quotum gibt.","norm":"Als dieser Bruch ¾ kann in einen anderen verwandelt werden, dessen Nenner 12 ist, weilen sich12 durch 4 teilen lässt, und nämlich 3 für den Quotum 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+{"basename":"euler_rechenkunst01_1738","par_idx":1540,"date":1738,"orig":"Weilen nun der neue Nenner 3 mahl ſo groß iſt als der alte, ſo muß auch der neue Zehler 3 mahl groͤſſer ſeyn als der alte, und derowegen wird der neue Bruch gefunden werden [Formel] . Wann ferner dieſer Bruch [Formel] in einen anderen verwandelt werden ſoll, deſſen Nenner 50 ſey, ſo diuidirt man 50 durch den vorigen Nenner, und mit dem Quoto 5 multiplicirt man den vorigen Zehler 7, ſo gibt das Product 35 den Zehler des neuen Bruchs; weswegen alſo der verwandelte Bruch [Formel] ſeyn wird, welcher auch wie leicht zu ſehen dem vorigen Bruche [Formel] gleich iſt:","norm":"Weilen nun der neue Nenner 3 Mal so groß ist als der alte, so muss auch der neue Zähler 3 Mal größer sein als der alte, und derowegen wird der neue Bruch gefunden werden [Formel]. Wann ferner dieser Bruch [Formel] in einen anderen verwandelt werden soll, dessen Nenner 50 sei, so dividiert man 50 durch den vorigen Nenner, und mit dem Quoto 5 multipliziert man den vorigen Zähler 7, so gibt das Produkt 35 den Zähler des neuen Bruchs; weswegen also der verwandelte Bruch [Formel] sein wird, welcher auch wie leicht zu sehen dem vorigen Bruche [Formel] gleich 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+{"basename":"euler_rechenkunst01_1738","par_idx":1541,"date":1738,"orig":"dann wann dieſes Bruchs Nenner nnd Zehler mit 5 multiplicirt wird, ſo kommt dieſer [Formel] heraus.","norm":"dann wann dieses Bruchs Nenner und Zähler mit 5 multipliziert wird, so kommt dieser [Formel] 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Bruch in eine andere Form gebracht werden ſoll, davon der Nenner gegeben iſt, doch ſo, daß ſich derſelbe durch den Nenner des vorgegebenen Bruchs theilen laſſe, ſo kan der neue Bruch auf dieſe Art ſehr leicht gefunden werden.","norm":"Wann also ein Bruch in eine andere Form gebracht werden soll, davon der Nenner gegeben ist, doch so, dass sich derselbe durch den Nenner des vorgegebenen Bruchs teilen lasse, so kann der neue Bruch auf diese Art sehr leicht gefunden 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+{"basename":"euler_rechenkunst01_1738","par_idx":1553,"date":1738,"orig":"Dieſer Vortheil aber wird erhalten, wann man fuͤr den gemeinen Nenner der geſuchten Bruͤche die kleinſte gemeine theilbare Zahl der gegebenen Nenner annimmt.","norm":"Dieser Vorteil aber wird erhalten, wann man für den gemeinen Nenner der gesuchten Brüche die kleinste gemeine teilbare Zahl der gegebenen Nenner 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+{"basename":"euler_rechenkunst01_1738","par_idx":1554,"date":1738,"orig":"Derowegen hat man bey der Reduction der Bruͤche zu gleichen Nenneren dieſe Regel in acht zu nehmen.","norm":"Derowegen hat man bei der Reduktion der Brüche zu gleichen Nennern diese Regel in Acht zu nehmen.","orig_tok":["Derowegen","hat","man","bey","der","Reduction","der","Bruͤche","zu","gleichen","Nenneren","dieſe","Regel","in","acht","zu","nehmen","."],"norm_tok":["Derowegen","hat","man","bei","der","Reduktion","der","Brüche","zu","gleichen","Nennern","diese","Regel","in","Acht","zu","nehmen","."],"orig_spans":[[0,9],[10,13],[14,17],[18,21],[22,25],[26,35],[36,39],[40,47],[48,50],[51,59],[60,68],[69,74],[75,80],[81,83],[84,88],[89,91],[92,98],[98,99]],"norm_spans":[[0,9],[10,13],[14,17],[18,21],[22,25],[26,35],[36,39],[40,46],[47,49],[50,58],[59,66],[67,72],[73,78],[79,81],[82,86],[87,89],[90,96],[96,97]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.7515997886657715} +{"basename":"euler_rechenkunst01_1738","par_idx":1555,"date":1738,"orig":"Erſtlich ſucht man die kleinſte gemeine theilbare Zahl aller gegebenen Nenner; und ſetzt dieſelbe fuͤr den gemeinen Nenner der geſuchten Bruͤche.","norm":"Erstlich sucht man die kleinste gemeine teilbare Zahl aller gegebenen Nenner; und setzt dieselbe für den gemeinen Nenner der gesuchten Brüche.","orig_tok":["Erſtlich","ſucht","man","die","kleinſte","gemeine","theilbare","Zahl","aller","gegebenen","Nenner",";","und","ſetzt","dieſelbe","fuͤr","den","gemeinen","Nenner","der","geſuchten","Bruͤche","."],"norm_tok":["Erstlich","sucht","man","die","kleinste","gemeine","teilbare","Zahl","aller","gegebenen","Nenner",";","und","setzt","dieselbe","für","den","gemeinen","Nenner","der","gesuchten","Brüche","."],"orig_spans":[[0,8],[9,14],[15,18],[19,22],[23,31],[32,39],[40,49],[50,54],[55,60],[61,70],[71,77],[77,78],[79,82],[83,88],[89,97],[98,102],[103,106],[107,115],[116,122],[123,126],[127,136],[137,144],[144,145]],"norm_spans":[[0,8],[9,14],[15,18],[19,22],[23,31],[32,39],[40,48],[49,53],[54,59],[60,69],[70,76],[76,77],[78,81],[82,87],[88,96],[97,100],[101,104],[105,113],[114,120],[121,124],[125,134],[135,141],[141,142]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.966700077056885} +{"basename":"euler_rechenkunst01_1738","par_idx":1556,"date":1738,"orig":"Hernach um die gehoͤrigen Zehler zu finden, ſo diuidirt man dieſen gemeinen Nenner durch den Nenner eines jeglichen gegebenen Bruchs, und mit dem Quoto multiplicirt man den Zehler deſſelben Bruchs ſo hat man den geſuchten Zehler.","norm":"Hernach um die gehörigen Zähler zu finden, so dividiert man diesen gemeinen Nenner durch den Nenner eines jeglichen gegebenen Bruchs, und mit dem Quoto multipliziert man den Zähler desselben Bruchs so hat man den gesuchten Zähler.","orig_tok":["Hernach","um","die","gehoͤrigen","Zehler","zu","finden",",","ſo","diuidirt","man","dieſen","gemeinen","Nenner","durch","den","Nenner","eines","jeglichen","gegebenen","Bruchs",",","und","mit","dem","Quoto","multiplicirt","man","den","Zehler","deſſelben","Bruchs","ſo","hat","man","den","geſuchten","Zehler","."],"norm_tok":["Hernach","um","die","gehörigen","Zähler","zu","finden",",","so","dividiert","man","diesen","gemeinen","Nenner","durch","den","Nenner","eines","jeglichen","gegebenen","Bruchs",",","und","mit","dem","Quoto","multipliziert","man","den","Zähler","desselben","Bruchs","so","hat","man","den","gesuchten","Zähler","."],"orig_spans":[[0,7],[8,10],[11,14],[15,25],[26,32],[33,35],[36,42],[42,43],[44,46],[47,55],[56,59],[60,66],[67,75],[76,82],[83,88],[89,92],[93,99],[100,105],[106,115],[116,125],[126,132],[132,133],[134,137],[138,141],[142,145],[146,151],[152,164],[165,168],[169,172],[173,179],[180,189],[190,196],[197,199],[200,203],[204,207],[208,211],[212,221],[222,228],[228,229]],"norm_spans":[[0,7],[8,10],[11,14],[15,24],[25,31],[32,34],[35,41],[41,42],[43,45],[46,55],[56,59],[60,66],[67,75],[76,82],[83,88],[89,92],[93,99],[100,105],[106,115],[116,125],[126,132],[132,133],[134,137],[138,141],[142,145],[146,151],[152,165],[166,169],[170,173],[174,180],[181,190],[191,197],[198,200],[201,204],[205,208],[209,212],[213,222],[223,229],[229,230]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28],[29,29],[30,30],[31,31],[32,32],[33,33],[34,34],[35,35],[36,36],[37,37],[38,38]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.027500152587891} +{"basename":"euler_rechenkunst01_1738","par_idx":1557,"date":1738,"orig":"Dieſe gantze Operation aber wird durch folgende Exempel mehr erlaͤutert werden.","norm":"Diese ganze Operation aber wird durch folgende Exempel mehr erläutert werden.","orig_tok":["Dieſe","gantze","Operation","aber","wird","durch","folgende","Exempel","mehr","erlaͤutert","werden","."],"norm_tok":["Diese","ganze","Operation","aber","wird","durch","folgende","Exempel","mehr","erläutert","werden","."],"orig_spans":[[0,5],[6,12],[13,22],[23,27],[28,32],[33,38],[39,47],[48,55],[56,60],[61,71],[72,78],[78,79]],"norm_spans":[[0,5],[6,11],[12,21],[22,26],[27,31],[32,37],[38,46],[47,54],[55,59],[60,69],[70,76],[76,77]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.785999774932861} +{"basename":"euler_rechenkunst01_1738","par_idx":1558,"date":1738,"orig":"Erſtlich ſollen dieſe Bruͤche [Formel] , [Formel] , [Formel] und [Formel] zu gleichen Nenneren gebracht oder in andere verwandelt werden, welche gleiche Nenner haben.","norm":"Erstlich sollen diese Brüche [Formel], [Formel], [Formel] und [Formel] zu gleichen Nennern gebracht oder in andere verwandelt werden, welche gleiche Nenner haben.","orig_tok":["Erſtlich","ſollen","dieſe","Bruͤche","[Formel]",",","[Formel]",",","[Formel]","und","[Formel]","zu","gleichen","Nenneren","gebracht","oder","in","andere","verwandelt","werden",",","welche","gleiche","Nenner","haben","."],"norm_tok":["Erstlich","sollen","diese","Brüche","[","Formel","]",",","[","Formel","]",",","[","Formel","]","und","[","Formel","]","zu","gleichen","Nennern","gebracht","oder","in","andere","verwandelt","werden",",","welche","gleiche","Nenner","haben","."],"orig_spans":[[0,8],[9,15],[16,21],[22,29],[30,38],[39,40],[41,49],[50,51],[52,60],[61,64],[65,73],[74,76],[77,85],[86,94],[95,103],[104,108],[109,111],[112,118],[119,129],[130,136],[136,137],[138,144],[145,152],[153,159],[160,165],[165,166]],"norm_spans":[[0,8],[9,15],[16,21],[22,28],[29,30],[30,36],[36,37],[37,38],[39,40],[40,46],[46,47],[47,48],[49,50],[50,56],[56,57],[58,61],[62,63],[63,69],[69,70],[71,73],[74,82],[83,90],[91,99],[100,104],[105,107],[108,114],[115,125],[126,132],[132,133],[134,140],[141,148],[149,155],[156,161],[161,162]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[4,5],[4,6],[5,7],[6,8],[6,9],[6,10],[7,11],[8,12],[8,13],[8,14],[9,15],[10,16],[10,17],[10,18],[11,19],[12,20],[13,21],[14,22],[15,23],[16,24],[17,25],[18,26],[19,27],[20,28],[21,29],[22,30],[23,31],[24,32],[25,33]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.335400104522705} +{"basename":"euler_rechenkunst01_1738","par_idx":1559,"date":1738,"orig":"Man ſuche alſo fuͤr allen Dingen die kleinſte gemeine theilbare Zahl der gegebenen Nenner [Formel] wie vorher gelchret worden, welche 420 iſt.","norm":"Man suche also für allen Dingen die kleinste gemeine teilbare Zahl der gegebenen Nenner [Formel] wie vorher gelchret worden, welche 420 ist.","orig_tok":["Man","ſuche","alſo","fuͤr","allen","Dingen","die","kleinſte","gemeine","theilbare","Zahl","der","gegebenen","Nenner","[Formel]","wie","vorher","gelchret","worden",",","welche","420","iſt","."],"norm_tok":["Man","suche","also","für","allen","Dingen","die","kleinste","gemeine","teilbare","Zahl","der","gegebenen","Nenner","[","Formel","]","wie","vorher","gelchret","worden",",","welche","420","ist","."],"orig_spans":[[0,3],[4,9],[10,14],[15,19],[20,25],[26,32],[33,36],[37,45],[46,53],[54,63],[64,68],[69,72],[73,82],[83,89],[90,98],[99,102],[103,109],[110,118],[119,125],[125,126],[127,133],[134,137],[138,141],[141,142]],"norm_spans":[[0,3],[4,9],[10,14],[15,18],[19,24],[25,31],[32,35],[36,44],[45,52],[53,61],[62,66],[67,70],[71,80],[81,87],[88,89],[89,95],[95,96],[97,100],[101,107],[108,116],[117,123],[123,124],[125,131],[132,135],[136,139],[139,140]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[14,15],[14,16],[15,17],[16,18],[17,19],[18,20],[19,21],[20,22],[21,23],[22,24],[23,25]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":6.644899845123291} +{"basename":"euler_rechenkunst01_1738","par_idx":1560,"date":1738,"orig":"Dieſe Zahl wird nun fuͤr den gemeiner Nenner der geſuchten Bruͤche angenommen; dieſe Bruͤche ſelbſt aber werden auf folgende Weiſe gefunden.","norm":"Diese Zahl wird nun für den gemeiner Nenner der gesuchten Brüche angenommen; diese Brüche selbst aber werden auf folgende Weise gefunden.","orig_tok":["Dieſe","Zahl","wird","nun","fuͤr","den","gemeiner","Nenner","der","geſuchten","Bruͤche","angenommen",";","dieſe","Bruͤche","ſelbſt","aber","werden","auf","folgende","Weiſe","gefunden","."],"norm_tok":["Diese","Zahl","wird","nun","für","den","gemeiner","Nenner","der","gesuchten","Brüche","angenommen",";","diese","Brüche","selbst","aber","werden","auf","folgende","Weise","gefunden","."],"orig_spans":[[0,5],[6,10],[11,15],[16,19],[20,24],[25,28],[29,37],[38,44],[45,48],[49,58],[59,66],[67,77],[77,78],[79,84],[85,92],[93,99],[100,104],[105,111],[112,115],[116,124],[125,130],[131,139],[139,140]],"norm_spans":[[0,5],[6,10],[11,15],[16,19],[20,23],[24,27],[28,36],[37,43],[44,47],[48,57],[58,64],[65,75],[75,76],[77,82],[83,89],[90,96],[97,101],[102,108],[109,112],[113,121],[122,127],[128,136],[136,137]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.5} +{"basename":"euler_rechenkunst01_1738","par_idx":1561,"date":1738,"orig":"[Formel]","norm":"[Formel]","orig_tok":["[Formel]"],"norm_tok":["[","Formel","]"],"orig_spans":[[0,8]],"norm_spans":[[0,1],[1,7],[7,8]],"alignment":[[0,0],[0,1],[0,2]],"lang_fastText":"sh","lang_py3langid":"en","lang_cld3":"cy","orig_lang_de":0,"norm_lmscore":7.932600021362305} +{"basename":"euler_rechenkunst01_1738","par_idx":1562,"date":1738,"orig":"Nehmlich nach dem man die querſtriche der gegebenen Bruͤche fortgezogen, ſo wird unter einen jeglichen der gemeine Nenner 420 geſchrieben; hernach diuidirt man dieſen gemeinen Nenner durch einen jeglichen Nenner der gegebenen Bruͤche; und ſetzt die Quotos weiter zur rechten; als 420 durch 12 diuidirt gibt 35, und 420 durch 15 gibt 28, und 420 durch 20 gibt 21, und 420 durch 21 gibt 20. Endlich multiplicirt man dieſe Quotos mit den gegenuͤber ſtehenden Zehlern der gegebenen Bruͤche und ſchreibt die Product in die Stellen der Zehler der geſuchten Bruͤche.","norm":"Nämlich nach dem man die Querstriche der gegebenen Brüche fortgezogen, so wird unter einen jeglichen der gemeine Nenner 420 geschrieben; hernach dividiert man diesen gemeinen Nenner durch einen jeglichen Nenner der gegebenen Brüche; und setzt die Quotos weiter zur rechten; als 420 durch 12 dividiert gibt 35, und 420 durch 15 gibt 28, und 420 durch 20 gibt 21, und 420 durch 21 gibt 20. Endlich multipliziert man diese Quotos mit den gegenüber stehenden Zählern der gegebenen Brüche und schreibt die Produkt in die Stellen der Zähler der gesuchten Brüche.","orig_tok":["Nehmlich","nach","dem","man","die","querſtriche","der","gegebenen","Bruͤche","fortgezogen",",","ſo","wird","unter","einen","jeglichen","der","gemeine","Nenner","420","geſchrieben",";","hernach","diuidirt","man","dieſen","gemeinen","Nenner","durch","einen","jeglichen","Nenner","der","gegebenen","Bruͤche",";","und","ſetzt","die","Quotos","weiter","zur","rechten",";","als","420","durch","12","diuidirt","gibt","35",",","und","420","durch","15","gibt","28",",","und","420","durch","20","gibt","21",",","und","420","durch","21","gibt","20.","Endlich","multiplicirt","man","dieſe","Quotos","mit","den","gegenuͤber","ſtehenden","Zehlern","der","gegebenen","Bruͤche","und","ſchreibt","die","Product","in","die","Stellen","der","Zehler","der","geſuchten","Bruͤche","."],"norm_tok":["Nämlich","nach","dem","man","die","Querstriche","der","gegebenen","Brüche","fortgezogen",",","so","wird","unter","einen","jeglichen","der","gemeine","Nenner","420","geschrieben",";","hernach","dividiert","man","diesen","gemeinen","Nenner","durch","einen","jeglichen","Nenner","der","gegebenen","Brüche",";","und","setzt","die","Quotos","weiter","zur","rechten",";","als","420","durch","12","dividiert","gibt","35",",","und","420","durch","15","gibt","28",",","und","420","durch","20","gibt","21",",","und","420","durch","21","gibt","20.","Endlich","multipliziert","man","diese","Quotos","mit","den","gegenüber","stehenden","Zählern","der","gegebenen","Brüche","und","schreibt","die","Produkt","in","die","Stellen","der","Zähler","der","gesuchten","Brüche","."],"orig_spans":[[0,8],[9,13],[14,17],[18,21],[22,25],[26,37],[38,41],[42,51],[52,59],[60,71],[71,72],[73,75],[76,80],[81,86],[87,92],[93,102],[103,106],[107,114],[115,121],[122,125],[126,137],[137,138],[139,146],[147,155],[156,159],[160,166],[167,175],[176,182],[183,188],[189,194],[195,204],[205,211],[212,215],[216,225],[226,233],[233,234],[235,238],[239,244],[245,248],[249,255],[256,262],[263,266],[267,274],[274,275],[276,279],[280,283],[284,289],[290,292],[293,301],[302,306],[307,309],[309,310],[311,314],[315,318],[319,324],[325,327],[328,332],[333,335],[335,336],[337,340],[341,344],[345,350],[351,353],[354,358],[359,361],[361,362],[363,366],[367,370],[371,376],[377,379],[380,384],[385,388],[389,396],[397,409],[410,413],[414,419],[420,426],[427,430],[431,434],[435,445],[446,455],[456,463],[464,467],[468,477],[478,485],[486,489],[490,498],[499,502],[503,510],[511,513],[514,517],[518,525],[526,529],[530,536],[537,540],[541,550],[551,558],[558,559]],"norm_spans":[[0,7],[8,12],[13,16],[17,20],[21,24],[25,36],[37,40],[41,50],[51,57],[58,69],[69,70],[71,73],[74,78],[79,84],[85,90],[91,100],[101,104],[105,112],[113,119],[120,123],[124,135],[135,136],[137,144],[145,154],[155,158],[159,165],[166,174],[175,181],[182,187],[188,193],[194,203],[204,210],[211,214],[215,224],[225,231],[231,232],[233,236],[237,242],[243,246],[247,253],[254,260],[261,264],[265,272],[272,273],[274,277],[278,281],[282,287],[288,290],[291,300],[301,305],[306,308],[308,309],[310,313],[314,317],[318,323],[324,326],[327,331],[332,334],[334,335],[336,339],[340,343],[344,349],[350,352],[353,357],[358,360],[360,361],[362,365],[366,369],[370,375],[376,378],[379,383],[384,387],[388,395],[396,409],[410,413],[414,419],[420,426],[427,430],[431,434],[435,444],[445,454],[455,462],[463,466],[467,476],[477,483],[484,487],[488,496],[497,500],[501,508],[509,511],[512,515],[516,523],[524,527],[528,534],[535,538],[539,548],[549,555],[555,556]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28],[29,29],[30,30],[31,31],[32,32],[33,33],[34,34],[35,35],[36,36],[37,37],[38,38],[39,39],[40,40],[41,41],[42,42],[43,43],[44,44],[45,45],[46,46],[47,47],[48,48],[49,49],[50,50],[51,51],[52,52],[53,53],[54,54],[55,55],[56,56],[57,57],[58,58],[59,59],[60,60],[61,61],[62,62],[63,63],[64,64],[65,65],[66,66],[67,67],[68,68],[69,69],[70,70],[71,71],[72,72],[73,73],[74,74],[75,75],[76,76],[77,77],[78,78],[79,79],[80,80],[81,81],[82,82],[83,83],[84,84],[85,85],[86,86],[87,87],[88,88],[89,89],[90,90],[91,91],[92,92],[93,93],[94,94],[95,95],[96,96],[97,97]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.120200157165527} +{"basename":"euler_rechenkunst01_1738","par_idx":1563,"date":1738,"orig":"Als 5 mahl 35 gibt 175, und 8 mahl 28 gibt 224 und ſo fort.","norm":"Als 5 Mal 35 gibt 175, und 8 Mal 28 gibt 224 und so fort.","orig_tok":["Als","5","mahl","35","gibt","175",",","und","8","mahl","28","gibt","224","und","ſo","fort","."],"norm_tok":["Als","5","Mal","35","gibt","175",",","und","8","Mal","28","gibt","224","und","so","fort","."],"orig_spans":[[0,3],[4,5],[6,10],[11,13],[14,18],[19,22],[22,23],[24,27],[28,29],[30,34],[35,37],[38,42],[43,46],[47,50],[51,53],[54,58],[58,59]],"norm_spans":[[0,3],[4,5],[6,9],[10,12],[13,17],[18,21],[21,22],[23,26],[27,28],[29,32],[33,35],[36,40],[41,44],[45,48],[49,51],[52,56],[56,57]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.935500144958496} +{"basename":"euler_rechenkunst01_1738","par_idx":1564,"date":1738,"orig":"Wann dieſes geſchehen, ſo hat man die verlangten Bruͤche von einerley Nenner zur Seite der gegebenen, welche durch einen Strich von einander abgeſondert werden.","norm":"Wann dieses geschehen, so hat man die verlangten Brüche von einerlei Nenner zur Seite der gegebenen, welche durch einen Strich voneinander abgesondert werden.","orig_tok":["Wann","dieſes","geſchehen",",","ſo","hat","man","die","verlangten","Bruͤche","von","einerley","Nenner","zur","Seite","der","gegebenen",",","welche","durch","einen","Strich","von","einander","abgeſondert","werden","."],"norm_tok":["Wann","dieses","geschehen",",","so","hat","man","die","verlangten","Brüche","von","einerlei","Nenner","zur","Seite","der","gegebenen",",","welche","durch","einen","Strich","voneinander","abgesondert","werden","."],"orig_spans":[[0,4],[5,11],[12,21],[21,22],[23,25],[26,29],[30,33],[34,37],[38,48],[49,56],[57,60],[61,69],[70,76],[77,80],[81,86],[87,90],[91,100],[100,101],[102,108],[109,114],[115,120],[121,127],[128,131],[132,140],[141,152],[153,159],[159,160]],"norm_spans":[[0,4],[5,11],[12,21],[21,22],[23,25],[26,29],[30,33],[34,37],[38,48],[49,55],[56,59],[60,68],[69,75],[76,79],[80,85],[86,89],[90,99],[99,100],[101,107],[108,113],[114,119],[120,126],[127,138],[139,150],[151,157],[157,158]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,22],[24,23],[25,24],[26,25]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.065599918365478} +{"basename":"euler_rechenkunst01_1738","par_idx":1565,"date":1738,"orig":"Die Figur der Operation kan ein jeder nach ſeinem Gutbefinden aͤndern, und um der Kuͤrtze willen, ſo wohl die Quotos gar weg laſſen, als auch den gemeinen Nenner nur ein mahl oben a part ſetzen.","norm":"Die Figur der Operation kann ein jeder nach seinem Gutbefinden ändern, und um der Kürze willen, sowohl die Quotos gar weglassen, als auch den gemeinen Nenner nur ein Mal oben a part setzen.","orig_tok":["Die","Figur","der","Operation","kan","ein","jeder","nach","ſeinem","Gutbefinden","aͤndern",",","und","um","der","Kuͤrtze","willen",",","ſo","wohl","die","Quotos","gar","weg","laſſen",",","als","auch","den","gemeinen","Nenner","nur","ein","mahl","oben","a","part","ſetzen","."],"norm_tok":["Die","Figur","der","Operation","kann","ein","jeder","nach","seinem","Gutbefinden","ändern",",","und","um","der","Kürze","willen",",","sowohl","die","Quotos","gar","weglassen",",","als","auch","den","gemeinen","Nenner","nur","ein","Mal","oben","a","part","setzen","."],"orig_spans":[[0,3],[4,9],[10,13],[14,23],[24,27],[28,31],[32,37],[38,42],[43,49],[50,61],[62,69],[69,70],[71,74],[75,77],[78,81],[82,89],[90,96],[96,97],[98,100],[101,105],[106,109],[110,116],[117,120],[121,124],[125,131],[131,132],[133,136],[137,141],[142,145],[146,154],[155,161],[162,165],[166,169],[170,174],[175,179],[180,181],[182,186],[187,193],[193,194]],"norm_spans":[[0,3],[4,9],[10,13],[14,23],[24,28],[29,32],[33,38],[39,43],[44,50],[51,62],[63,69],[69,70],[71,74],[75,77],[78,81],[82,87],[88,94],[94,95],[96,102],[103,106],[107,113],[114,117],[118,127],[127,128],[129,132],[133,137],[138,141],[142,150],[151,157],[158,161],[162,165],[166,169],[170,174],[175,176],[177,181],[182,188],[188,189]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,18],[20,19],[21,20],[22,21],[23,22],[24,22],[25,23],[26,24],[27,25],[28,26],[29,27],[30,28],[31,29],[32,30],[33,31],[34,32],[35,33],[36,34],[37,35],[38,36]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.917900085449219} +{"basename":"euler_rechenkunst01_1738","par_idx":1566,"date":1738,"orig":"Wann dieſe Bruͤche ½, ⅔, ¾, ⅘, ⅚, [Formel] , ⅞, [Formel] , [Formel] zu gleichen Nenneren gebracht werden ſollen; ſo wird erſtlich die kleinſte gemeine theilbare Zahl von allen Nenneren geſucht, und dafuͤr 2520 gefunden; hernach aber die Operation folgender geſtalt verrichtet.","norm":"Wann diese Brüche ½, 2/3 , ¾, ⅘, ⅚, [Formel], ⅞, [Formel], [Formel] zu gleichen Nennern gebracht werden sollen; so wird erstlich die kleinste gemeine teilbare Zahl von allen Nennern gesucht, und dafür 2520 gefunden; hernach aber die Operation folgender Gestalt verrichtet.","orig_tok":["Wann","dieſe","Bruͤche","½",",","⅔",",","¾",",","⅘",",","⅚",",","[Formel]",",","⅞",",","[Formel]",",","[Formel]","zu","gleichen","Nenneren","gebracht","werden","ſollen",";","ſo","wird","erſtlich","die","kleinſte","gemeine","theilbare","Zahl","von","allen","Nenneren","geſucht",",","und","dafuͤr","2520","gefunden",";","hernach","aber","die","Operation","folgender","geſtalt","verrichtet","."],"norm_tok":["Wann","diese","Brüche","½",","," 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+{"basename":"euler_rechenkunst01_1738","par_idx":1567,"date":1738,"orig":"Ein Exempel von Bruͤchen, ſo aus groͤſſeren Zahlen beſtehen, koͤnnen dieſe [Formel] , [Formel] , [Formel] geben, welche da die kleinſte theilbare Zahl der Nenner iſt 1260, wie folget zu gleichen Nenneren gebracht werden.","norm":"Ein Exempel von Brüchen, so aus größeren Zahlen bestehen, können diese [Formel], [Formel], [Formel] geben, welche da die kleinste teilbare Zahl der Nenner ist 1260, wie folget zu gleichen Nennern gebracht 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+{"basename":"euler_rechenkunst01_1738","par_idx":1572,"date":1738,"orig":"Nehmlich bey der Addition werden die Zehler der gefundenen Bruͤche zuſammen addirt, und unter die Summ als einen Zeyler der gemeine Nenner geſchrieben; welcher Bruch die Summ der Bruͤche anzeiget.","norm":"Nämlich bei der Addition werden die Zähler der gefundenen Brüche zusammen addiert, und unter die Summe als einen Zeile der gemeine Nenner geschrieben; welcher Bruch die Summe der Brüche 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+{"basename":"euler_rechenkunst01_1738","par_idx":1573,"date":1738,"orig":"Jſt nun dieſer Bruch groͤſſer als ein gantzes, ſo werden die gantzen daraus gezogen, und ſo noch gantze Zahlen zu addiren da ſind, mit zu derſelben Summ geſchlagen.","norm":"Ist nun dieser Bruch größer als ein ganzes, so werden die ganzen daraus gezogen, und so noch ganze Zahlen zu addieren da sind, mit zu derselben Summe 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+{"basename":"euler_rechenkunst01_1738","par_idx":1574,"date":1738,"orig":"Jn der Subtraction aber wird der Zehler des unteren Bruchs von dem Zehler des oberen Bruchs ſubtrahirt, wofern derſelbe kleiner iſt; ſollte der untere Zehler aber groͤſſer ſeyn, ſo wird der obere Bruch um ein gantzes vermehret und ſo dann die Subtraction vollzogen.","norm":"In der Subtraktion aber wird der Zähler des unteren Bruchs von dem Zähler des oberen Bruchs subtrahiert, sofern derselbe kleiner ist; sollte der untere Zähler aber größer sein, so wird der obere Bruch um ein ganzes vermehret und so dann die Subtraktion 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+{"basename":"euler_rechenkunst01_1738","par_idx":1638,"date":1738,"orig":"Alſo wird das Product von 2 und ⅓ oder zwey mahl ⅓ ſeyn ⅔, und 2 mahl [Formel] wird geben [Formel] oder 1 [Formel] . Gleicher geſtalt, wann ein Bruch mit 3 oder 4 oder einer anderen Zahl multiplicirt werden ſoll, ſo geſchieht dieſe Multiplication, wann man den gegebenen Bruch drey mahl oder 4 mahl oder ſo viel mahl als der Multiplicator anzeiget ſetzt, und dieſe Bruͤche zuſammen addirt.","norm":"Also wird das Produkt von 2 und 1/3 oder zweimal 1/3 sein 2/3 , und 2 Mal [Formel] wird geben [Formel] oder 1 [Formel]. Gleicher Gestalt, wann ein Bruch mit 3 oder 4 oder einer anderen Zahl multipliziert werden soll, so geschieht diese Multiplikation, wann man den gegebenen Bruch dreimal oder 4 Mal oder so viel Mal als der Multiplicator anzeiget setzt, und diese Brüche zusammen addiert.","orig_tok":["Alſo","wird","das","Product","von","2","und","⅓","oder","zwey","mahl","⅓","ſeyn","⅔",",","und","2","mahl","[Formel]","wird","geben","[Formel]","oder","1","[Formel]",".","Gleicher","geſtalt",",","wann","ein","Bruch","mit","3","oder","4","oder","einer","anderen","Zahl","multiplicirt","werden","ſoll",",","ſo","geſchieht","dieſe","Multiplication",",","wann","man","den","gegebenen","Bruch","drey","mahl","oder","4","mahl","oder","ſo","viel","mahl","als","der","Multiplicator","anzeiget","ſetzt",",","und","dieſe","Bruͤche","zuſammen","addirt","."],"norm_tok":["Also","wird","das","Produkt","von","2","und"," ","1","/","3"," ","oder","zweimal"," ","1","/","3"," ","sein"," 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+{"basename":"euler_rechenkunst01_1738","par_idx":1639,"date":1738,"orig":"Weilen nun dieſe Bruͤche einander voͤllig gleich ſind, ſo addirt man nur die Zehler, das iſt man multiplicirt den Zehler des gegebenen Bruchs mit 3, 4 oder einer anderen Zahl ſo gegeben iſt.","norm":"Weilen nun diese Brüche einander völlig gleich sind, so addiert man nur die Zähler, das ist man multipliziert den Zähler des gegebenen Bruchs mit 3, 4 oder einer anderen Zahl so gegeben 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+{"basename":"euler_rechenkunst01_1738","par_idx":1640,"date":1738,"orig":"Hieraus erhellet nun, daß wann ein Bruch mit einer gegebenen Zahl multiplicirt werden ſoll, das Product gefunden werde, wann man nur den Zehler mit der gegebenen Zahl multiplicirt, den Nenner aber unveraͤndert laͤſt.","norm":"Hieraus erhellet nun, dass wann ein Bruch mit einer gegebenen Zahl multipliziert werden soll, das Produkt gefunden werde, wann man nur den Zähler mit der gegebenen Zahl multipliziert, den Nenner aber unverändert 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+{"basename":"euler_rechenkunst01_1738","par_idx":1662,"date":1738,"orig":"Jſt aber die gantze Zahl mit welcher ein Bruch multiplicirt wird, zwey mahl ſo groß als der Nenner, ſo iſt auch das Product zwey mahl ſo groß als der Zehler; und ſo viel mahl dieſelbe Zahl, mit welcher ein Bruch multiplicirt wird, groͤſſer iſt als der Nenner des Bruchs, eben ſo viel mahl wird auch das Product groͤſſer ſeyn als der Zehler deſſelben Bruchs.","norm":"Ist aber die ganze Zahl mit welcher ein Bruch multipliziert wird, zweimal so groß als der Nenner, so ist auch das Produkt zweimal so groß als der Zähler; und so viel Mal dieselbe Zahl, mit welcher ein Bruch multipliziert wird, größer ist als der Nenner des Bruchs, ebenso viel Mal wird auch das Produkt größer sein als der Zähler desselben 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+{"basename":"euler_rechenkunst01_1738","par_idx":1668,"date":1738,"orig":"Ob nun gleich in ſolchen Faͤllen eben dieſes Product auch durch die vorige Regel gefunden wird, ſo muß doch dabey eine Multiplication und Diuiſion gebraucht werden, welche beyden Operationen nach dieſer Regel nicht noͤthig ſind, in dem man nur den bloſſen Zehler fuͤr das Product hinſchreiben darf.","norm":"Ob nun gleich in solchen Fällen eben dieses Produkt auch durch die vorige Regel gefunden wird, so muss doch dabei eine Multiplikation und Division gebraucht werden, welche beiden Operationen nach dieser Regel nicht nötig sind, in dem man nur den bloßen Zähler für das Produkt hinschreiben 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+{"basename":"euler_rechenkunst01_1738","par_idx":1670,"date":1738,"orig":"Dieſe Regel aber wird uns im folgenden hauptſaͤchlich dazu dienen, daß man wiſſe, mit was fuͤr einer Zahl man einen Bruch multipliciren muͤſſe, damit das Product eine gantze Zahl werde.","norm":"Diese Regel aber wird uns im folgenden hauptsächlich dazu dienen, dass man wisse, mit was für einer Zahl man einen Bruch multiplizieren müsse, damit das Produkt eine ganze Zahl 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+{"basename":"euler_rechenkunst01_1738","par_idx":1671,"date":1738,"orig":"Nehmlich man ſieht hieraus, daß man einen Bruch, damit das Product eine gantze Zahl werde, mit ſeinem Nenner multipliciren muͤſſe, dann da wird das Product dem Zehler deſſelben Bruchs gleich ſeyn.","norm":"Nämlich man sieht hieraus, dass man einen Bruch, damit das Produkt eine ganze Zahl werde, mit seinem Nenner multiplizieren müsse, dann da wird das Produkt dem Zähler desselben Bruchs gleich 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+{"basename":"euler_rechenkunst01_1738","par_idx":1674,"date":1738,"orig":"Derowegen iſt klar, daß ſo viel mahl diejenige Zahl, mit welcher ein Bruch multiplicirt werden ſoll, groͤſſer iſt als der Nenner, als dann das Product eben ſo viel mahl groͤſſer ſeyn werde als der Zehler deſſelben Bruchs.","norm":"Derowegen ist klar, dass so viel Mal diejenige Zahl, mit welcher ein Bruch multipliziert werden soll, größer ist als der Nenner, als dann das Produkt ebenso viel Mal größer sein werde als der Zähler desselben 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Gleicher geſtalt wann ⅔ mit 18 multiplicirt werden ſoll, ſo ſieht man, daß der Multiplicator 18 ſechs mahl groͤſſer iſt als der Nenner 3; deswegen wird das Product auch ſechs mahl groͤſſer als der Zehler 2, und folglich 12 ſeyn.","norm":"Also wann [Formel] mit 24 multiplizieret wird, so ist das Produkt 14; dann weilen hier 24 zweimal so groß ist als der Nenner 12, so muss das Produkt 14 zweimal so groß sein als der Zähler 7. 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+{"basename":"euler_rechenkunst01_1738","par_idx":1676,"date":1738,"orig":"Ob es aber gleich unendlich viel Zahlen gibt, welche mit einem Bruche multiplicirt gantze Zahlen hervorbringen, ſo wird dennoch am vortheilhaffteſten ſeyn ſich nur allein des Nenners ſelbſt zu bedienen, weilen auf dieſe Art das kleinſte gantze Product herauskommt, und ohne einige Operation geſunden wird.","norm":"Ob es aber gleich unendlich viel Zahlen gibt, welche mit einem Bruche multipliziert ganze Zahlen hervorbringen, so wird dennoch am vorteilhaftesten sein sich nur allein des Nenners selbst zu bedienen, weilen auf diese Art das kleinste ganze Produkt herauskommt, und ohne einige Operation gesunden 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+{"basename":"euler_rechenkunst01_1738","par_idx":1681,"date":1738,"orig":"Nach dieſer Regel iſt alſo ſehr leicht zwey oder mehr Bruͤche mit einander zu multipliciren, indem dieſe Operation bloß in der Multiplication der Zehler und Nenner der gegebenen Bruͤche beſteht; weswegen die Multiplication der Bruͤche weit leichter faͤllt als die Addition und Subtraction, als zu welchen erfordert wird die Bruͤche vorher zu gleichen Nenneren zu bringen, welche bey der Multiplication nicht von noͤthem iſt.","norm":"Nach dieser Regel ist also sehr leicht zwei oder mehr Brüche miteinander zu multiplizieren, indem diese Operation bloß in der Multiplikation der Zähler und Nenner der gegebenen Brüche besteht; weswegen die Multiplikation der Brüche weit leichter fällt als die Addition und Subtraktion, als zu welchen erfordert wird die Brüche vorher zu gleichen Nennern zu bringen, welche bei der Multiplikation nicht von nöthem 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+{"basename":"euler_rechenkunst01_1738","par_idx":1682,"date":1738,"orig":"Der Grund dieſer Regel aber beruhet auf den zwey vorhergehenden Saͤtzen.","norm":"Der Grund dieser Regel aber beruhet auf den zwei vorhergehenden Sätzen.","orig_tok":["Der","Grund","dieſer","Regel","aber","beruhet","auf","den","zwey","vorhergehenden","Saͤtzen","."],"norm_tok":["Der","Grund","dieser","Regel","aber","beruhet","auf","den","zwei","vorhergehenden","Sätzen","."],"orig_spans":[[0,3],[4,9],[10,16],[17,22],[23,27],[28,35],[36,39],[40,43],[44,48],[49,63],[64,71],[71,72]],"norm_spans":[[0,3],[4,9],[10,16],[17,22],[23,27],[28,35],[36,39],[40,43],[44,48],[49,63],[64,70],[70,71]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.913300037384033} +{"basename":"euler_rechenkunst01_1738","par_idx":1683,"date":1738,"orig":"Dann nach dem erſten wird ein Bruch mit einer jeglichen Zahl multiplicirt, wann man nur den Zehler mit derſelben multiplicirt, den Nenner aber unveraͤndert laͤſt.","norm":"Dann nach dem ersten wird ein Bruch mit einer jeglichen Zahl multipliziert, wann man nur den Zähler mit derselben multipliziert, den Nenner aber unverändert lässt.","orig_tok":["Dann","nach","dem","erſten","wird","ein","Bruch","mit","einer","jeglichen","Zahl","multiplicirt",",","wann","man","nur","den","Zehler","mit","derſelben","multiplicirt",",","den","Nenner","aber","unveraͤndert","laͤſt","."],"norm_tok":["Dann","nach","dem","ersten","wird","ein","Bruch","mit","einer","jeglichen","Zahl","multipliziert",",","wann","man","nur","den","Zähler","mit","derselben","multipliziert",",","den","Nenner","aber","unverändert","lässt","."],"orig_spans":[[0,4],[5,9],[10,13],[14,20],[21,25],[26,29],[30,35],[36,39],[40,45],[46,55],[56,60],[61,73],[73,74],[75,79],[80,83],[84,87],[88,91],[92,98],[99,102],[103,112],[113,125],[125,126],[127,130],[131,137],[138,142],[143,155],[156,161],[161,162]],"norm_spans":[[0,4],[5,9],[10,13],[14,20],[21,25],[26,29],[30,35],[36,39],[40,45],[46,55],[56,60],[61,74],[74,75],[76,80],[81,84],[85,88],[89,92],[93,99],[100,103],[104,113],[114,127],[127,128],[129,132],[133,139],[140,144],[145,156],[157,162],[162,163]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.404900074005127} +{"basename":"euler_rechenkunst01_1738","par_idx":1684,"date":1738,"orig":"Ob aber gleich dieſe Regel nur zu Multiplication der Bruͤche mit gantzen Zahlen iſt gebraucht worden, ſo gilt dieſelbe dennoch auch, wann Bruͤche mit Bruͤchen multiplicirt werden ſollen; wie wir ſchon oben angemercket haben.","norm":"Ob aber gleich diese Regel nur zu Multiplikation der Brüche mit ganzen Zahlen ist gebraucht worden, so gilt dieselbe dennoch auch, wann Brüche mit Brüchen multipliziert werden sollen; wie wir schon oben angemerkt haben.","orig_tok":["Ob","aber","gleich","dieſe","Regel","nur","zu","Multiplication","der","Bruͤche","mit","gantzen","Zahlen","iſt","gebraucht","worden",",","ſo","gilt","dieſelbe","dennoch","auch",",","wann","Bruͤche","mit","Bruͤchen","multiplicirt","werden","ſollen",";","wie","wir","ſchon","oben","angemercket","haben","."],"norm_tok":["Ob","aber","gleich","diese","Regel","nur","zu","Multiplikation","der","Brüche","mit","ganzen","Zahlen","ist","gebraucht","worden",",","so","gilt","dieselbe","dennoch","auch",",","wann","Brüche","mit","Brüchen","multipliziert","werden","sollen",";","wie","wir","schon","oben","angemerkt","haben","."],"orig_spans":[[0,2],[3,7],[8,14],[15,20],[21,26],[27,30],[31,33],[34,48],[49,52],[53,60],[61,64],[65,72],[73,79],[80,83],[84,93],[94,100],[100,101],[102,104],[105,109],[110,118],[119,126],[127,131],[131,132],[133,137],[138,145],[146,149],[150,158],[159,171],[172,178],[179,185],[185,186],[187,190],[191,194],[195,200],[201,205],[206,217],[218,223],[223,224]],"norm_spans":[[0,2],[3,7],[8,14],[15,20],[21,26],[27,30],[31,33],[34,48],[49,52],[53,59],[60,63],[64,70],[71,77],[78,81],[82,91],[92,98],[98,99],[100,102],[103,107],[108,116],[117,124],[125,129],[129,130],[131,135],[136,142],[143,146],[147,154],[155,168],[169,175],[176,182],[182,183],[184,187],[188,191],[192,197],[198,202],[203,212],[213,218],[218,219]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28],[29,29],[30,30],[31,31],[32,32],[33,33],[34,34],[35,35],[36,36],[37,37]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.543300151824951} +{"basename":"euler_rechenkunst01_1738","par_idx":1685,"date":1738,"orig":"Wann man aber nach dieſer Regel den Zehler eines Bruchs mit einem anderen Bruche multiplicirt, um den Zehler des Products zu bekommen, ſo faͤllt man in dieſe Schwierigkeit, daß der Zehler des Products gemeiniglich eine gebrochene Zahl wird, und folglich von dem Werthe eines ſolchen Products kein deutlicher Begriff formirt werden kan.","norm":"Wann man aber nach dieser Regel den Zähler eines Bruchs mit einem anderen Bruche multipliziert, um den Zähler des Produkts zu bekommen, so fällt man in diese Schwierigkeit, dass der Zähler des Produkts gemeiniglich eine gebrochene Zahl wird, und folglich von dem Werte eines solchen Produkts kein deutlicher Begriff formiert werden 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+{"basename":"euler_rechenkunst01_1738","par_idx":1686,"date":1738,"orig":"Alſo wann man [Formel] mit [Formel] multipliciren ſoll, ſo muß man nach dieſer Regel den Zehler 7 mit dem Bruche [Formel] multipliciren, um den Zehler des Products zu bekommen, welcher alſo [Formel] ſeyn wird, der Nenner aber des Products bleibt 12. Alſo iſt in dieſem Exempel das Product ein Bruch deſſen Zehler [Formel] und Nenner 12 iſt.","norm":"Also wann man [Formel] mit [Formel] multiplizieren soll, so muss man nach dieser Regel den Zähler 7 mit dem Bruche [Formel] multiplizieren, um den Zähler des Produkts zu bekommen, welcher also [Formel] sein wird, der Nenner aber des Produkts bleibt 12. Also ist in diesem Exempel das Produkt ein Bruch dessen Zähler [Formel] und Nenner 12 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sind:","orig_tok":["Weilen","wir","aber","von","keinen","anderen","Bruͤchen","bisher","Meldung","gethan",",","als","von","ſolchen",",","deren","Zehler","und","Nenner","gantze","Zahlen","ſind",":"],"norm_tok":["Weilen","wir","aber","von","keinen","anderen","Brüchen","bisher","Meldung","getan",",","als","von","solchen",",","deren","Zähler","und","Nenner","ganze","Zahlen","sind",":"],"orig_spans":[[0,6],[7,10],[11,15],[16,19],[20,26],[27,34],[35,43],[44,50],[51,58],[59,65],[65,66],[67,70],[71,74],[75,82],[82,83],[84,89],[90,96],[97,100],[101,107],[108,114],[115,121],[122,126],[126,127]],"norm_spans":[[0,6],[7,10],[11,15],[16,19],[20,26],[27,34],[35,42],[43,49],[50,57],[58,63],[63,64],[65,68],[69,72],[73,80],[80,81],[82,87],[88,94],[95,98],[99,105],[106,111],[112,118],[119,123],[123,124]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.835700035095215} +{"basename":"euler_rechenkunst01_1738","par_idx":1688,"date":1738,"orig":"ſo muͤſſen wir ſehen, ob wir einen ſolchen uneigentlichen Bruch nicht in einen anderen verwandeln koͤnnen, deſſen Zehler und Nenner gantze Zahlen ſind.","norm":"so müssen wir sehen, ob wir einen solchen uneigentlichen Bruch nicht in einen anderen verwandeln können, dessen Zähler und Nenner ganze Zahlen sind.","orig_tok":["ſo","muͤſſen","wir","ſehen",",","ob","wir","einen","ſolchen","uneigentlichen","Bruch","nicht","in","einen","anderen","verwandeln","koͤnnen",",","deſſen","Zehler","und","Nenner","gantze","Zahlen","ſind","."],"norm_tok":["so","müssen","wir","sehen",",","ob","wir","einen","solchen","uneigentlichen","Bruch","nicht","in","einen","anderen","verwandeln","können",",","dessen","Zähler","und","Nenner","ganze","Zahlen","sind","."],"orig_spans":[[0,2],[3,10],[11,14],[15,20],[20,21],[22,24],[25,28],[29,34],[35,42],[43,57],[58,63],[64,69],[70,72],[73,78],[79,86],[87,97],[98,105],[105,106],[107,113],[114,120],[121,124],[125,131],[132,138],[139,145],[146,150],[150,151]],"norm_spans":[[0,2],[3,9],[10,13],[14,19],[19,20],[21,23],[24,27],[28,33],[34,41],[42,56],[57,62],[63,68],[69,71],[72,77],[78,85],[86,96],[97,103],[103,104],[105,111],[112,118],[119,122],[123,129],[130,135],[136,142],[143,147],[147,148]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.168399810791016} +{"basename":"euler_rechenkunst01_1738","par_idx":1689,"date":1738,"orig":"Dieſes aber kan durch Huͤlfe des vorigen Satzes bewerckſtelliget werden; dann da ein Bruch ſeinem Werthe nach unveraͤndert bleibt, wann man beydes Zehler und Nenner durch eine jegliche beliebige Zahl multiplicirt:","norm":"Dieses aber kann durch Hilfe des vorigen Satzes bewerkstelliget werden; dann da ein Bruch seinem Werte nach unverändert bleibt, wann man beides Zähler und Nenner durch eine jegliche beliebige Zahl multipliziert:","orig_tok":["Dieſes","aber","kan","durch","Huͤlfe","des","vorigen","Satzes","bewerckſtelliget","werden",";","dann","da","ein","Bruch","ſeinem","Werthe","nach","unveraͤndert","bleibt",",","wann","man","beydes","Zehler","und","Nenner","durch","eine","jegliche","beliebige","Zahl","multiplicirt",":"],"norm_tok":["Dieses","aber","kann","durch","Hilfe","des","vorigen","Satzes","bewerkstelliget","werden",";","dann","da","ein","Bruch","seinem","Werte","nach","unverändert","bleibt",",","wann","man","beides","Zähler","und","Nenner","durch","eine","jegliche","beliebige","Zahl","multipliziert",":"],"orig_spans":[[0,6],[7,11],[12,15],[16,21],[22,28],[29,32],[33,40],[41,47],[48,64],[65,71],[71,72],[73,77],[78,80],[81,84],[85,90],[91,97],[98,104],[105,109],[110,122],[123,129],[129,130],[131,135],[136,139],[140,146],[147,153],[154,157],[158,164],[165,170],[171,175],[176,184],[185,194],[195,199],[200,212],[212,213]],"norm_spans":[[0,6],[7,11],[12,16],[17,22],[23,28],[29,32],[33,40],[41,47],[48,63],[64,70],[70,71],[72,76],[77,79],[80,83],[84,89],[90,96],[97,102],[103,107],[108,119],[120,126],[126,127],[128,132],[133,136],[137,143],[144,150],[151,154],[155,161],[162,167],[168,172],[173,181],[182,191],[192,196],[197,210],[210,211]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28],[29,29],[30,30],[31,31],[32,32],[33,33]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.499100208282471} +{"basename":"euler_rechenkunst01_1738","par_idx":1690,"date":1738,"orig":"ſo muͤſſen wir hier nur eine ſolche Zahl ſuchen, mit welcher wann Zehler und Nenner multiplicirt werden, gantze Zahlen herauskommen.","norm":"so müssen wir hier nur eine solche Zahl suchen, mit welcher wann Zähler und Nenner multipliziert werden, ganze Zahlen herauskommen.","orig_tok":["ſo","muͤſſen","wir","hier","nur","eine","ſolche","Zahl","ſuchen",",","mit","welcher","wann","Zehler","und","Nenner","multiplicirt","werden",",","gantze","Zahlen","herauskommen","."],"norm_tok":["so","müssen","wir","hier","nur","eine","solche","Zahl","suchen",",","mit","welcher","wann","Zähler","und","Nenner","multipliziert","werden",",","ganze","Zahlen","herauskommen","."],"orig_spans":[[0,2],[3,10],[11,14],[15,19],[20,23],[24,28],[29,35],[36,40],[41,47],[47,48],[49,52],[53,60],[61,65],[66,72],[73,76],[77,83],[84,96],[97,103],[103,104],[105,111],[112,118],[119,131],[131,132]],"norm_spans":[[0,2],[3,9],[10,13],[14,18],[19,22],[23,27],[28,34],[35,39],[40,46],[46,47],[48,51],[52,59],[60,64],[65,71],[72,75],[76,82],[83,96],[97,103],[103,104],[105,110],[111,117],[118,130],[130,131]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.683300018310547} +{"basename":"euler_rechenkunst01_1738","par_idx":1691,"date":1738,"orig":"Wann alſo der Zehler eines Bruchs ſelbſt eine gebrochene Zahl iſt, ſo darf man nur mit dem Nenner dieſer gebrochenen Zahl beydes den Zehler und Nenner des vorgelegten Bruchs multipliciren.","norm":"Wann also der Zähler eines Bruchs selbst eine gebrochene Zahl ist, so darf man nur mit dem Nenner dieser gebrochenen Zahl beides den Zähler und Nenner des vorgelegten Bruchs multiplizieren.","orig_tok":["Wann","alſo","der","Zehler","eines","Bruchs","ſelbſt","eine","gebrochene","Zahl","iſt",",","ſo","darf","man","nur","mit","dem","Nenner","dieſer","gebrochenen","Zahl","beydes","den","Zehler","und","Nenner","des","vorgelegten","Bruchs","multipliciren","."],"norm_tok":["Wann","also","der","Zähler","eines","Bruchs","selbst","eine","gebrochene","Zahl","ist",",","so","darf","man","nur","mit","dem","Nenner","dieser","gebrochenen","Zahl","beides","den","Zähler","und","Nenner","des","vorgelegten","Bruchs","multiplizieren","."],"orig_spans":[[0,4],[5,9],[10,13],[14,20],[21,26],[27,33],[34,40],[41,45],[46,56],[57,61],[62,65],[65,66],[67,69],[70,74],[75,78],[79,82],[83,86],[87,90],[91,97],[98,104],[105,116],[117,121],[122,128],[129,132],[133,139],[140,143],[144,150],[151,154],[155,166],[167,173],[174,187],[187,188]],"norm_spans":[[0,4],[5,9],[10,13],[14,20],[21,26],[27,33],[34,40],[41,45],[46,56],[57,61],[62,65],[65,66],[67,69],[70,74],[75,78],[79,82],[83,86],[87,90],[91,97],[98,104],[105,116],[117,121],[122,128],[129,132],[133,139],[140,143],[144,150],[151,154],[155,166],[167,173],[174,188],[188,189]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28],[29,29],[30,30],[31,31]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":3.814300060272217} +{"basename":"euler_rechenkunst01_1738","par_idx":1692,"date":1738,"orig":"Als im gegebenen Exempel war das Product ein Bruch deſſen Zehler [Formel] und Nenner 12 iſt; um nun dieſen Bruch in eine gewoͤhnliche Form zu bringen, ſo multiplicire man Zehler und Nenner mit 9; daher wird nun ein anderer Bruch entſpringen, deſſen Zehler 35 und Nenner 108 ſeyn wird; und welcher dem vorigen voͤllig gleich iſt.","norm":"Als im gegebenen Exempel war das Produkt ein Bruch dessen Zähler [Formel] und Nenner 12 ist; um nun diesen Bruch in eine gewöhnliche Form zu bringen, so multipliziere man Zähler und Nenner mit 9; daher wird nun ein anderer Bruch entspringen, dessen Zähler 35 und Nenner 108 sein wird; und welcher dem vorigen völlig gleich ist.","orig_tok":["Als","im","gegebenen","Exempel","war","das","Product","ein","Bruch","deſſen","Zehler","[Formel]","und","Nenner","12","iſt",";","um","nun","dieſen","Bruch","in","eine","gewoͤhnliche","Form","zu","bringen",",","ſo","multiplicire","man","Zehler","und","Nenner","mit","9",";","daher","wird","nun","ein","anderer","Bruch","entſpringen",",","deſſen","Zehler","35","und","Nenner","108","ſeyn","wird",";","und","welcher","dem","vorigen","voͤllig","gleich","iſt","."],"norm_tok":["Als","im","gegebenen","Exempel","war","das","Produkt","ein","Bruch","dessen","Zähler","[","Formel","]","und","Nenner","12","ist",";","um","nun","diesen","Bruch","in","eine","gewöhnliche","Form","zu","bringen",",","so","multipliziere","man","Zähler","und","Nenner","mit","9",";","daher","wird","nun","ein","anderer","Bruch","entspringen",",","dessen","Zähler","35","und","Nenner","108","sein","wird",";","und","welcher","dem","vorigen","völlig","gleich","ist","."],"orig_spans":[[0,3],[4,6],[7,16],[17,24],[25,28],[29,32],[33,40],[41,44],[45,50],[51,57],[58,64],[65,73],[74,77],[78,84],[85,87],[88,91],[91,92],[93,95],[96,99],[100,106],[107,112],[113,115],[116,120],[121,133],[134,138],[139,141],[142,149],[149,150],[151,153],[154,166],[167,170],[171,177],[178,181],[182,188],[189,192],[193,194],[194,195],[196,201],[202,206],[207,210],[211,214],[215,222],[223,228],[229,240],[240,241],[242,248],[249,255],[256,258],[259,262],[263,269],[270,273],[274,278],[279,283],[283,284],[285,288],[289,296],[297,300],[301,308],[309,316],[317,323],[324,327],[327,328]],"norm_spans":[[0,3],[4,6],[7,16],[17,24],[25,28],[29,32],[33,40],[41,44],[45,50],[51,57],[58,64],[65,66],[66,72],[72,73],[74,77],[78,84],[85,87],[88,91],[91,92],[93,95],[96,99],[100,106],[107,112],[113,115],[116,120],[121,132],[133,137],[138,140],[141,148],[148,149],[150,152],[153,166],[167,170],[171,177],[178,181],[182,188],[189,192],[193,194],[194,195],[196,201],[202,206],[207,210],[211,214],[215,222],[223,228],[229,240],[240,241],[242,248],[249,255],[256,258],[259,262],[263,269],[270,273],[274,278],[279,283],[283,284],[285,288],[289,296],[297,300],[301,308],[309,315],[316,322],[323,326],[326,327]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[11,12],[11,13],[12,14],[13,15],[14,16],[15,17],[16,18],[17,19],[18,20],[19,21],[20,22],[21,23],[22,24],[23,25],[24,26],[25,27],[26,28],[27,29],[28,30],[29,31],[30,32],[31,33],[32,34],[33,35],[34,36],[35,37],[36,38],[37,39],[38,40],[39,41],[40,42],[41,43],[42,44],[43,45],[44,46],[45,47],[46,48],[47,49],[48,50],[49,51],[50,52],[51,53],[52,54],[53,55],[54,56],[55,57],[56,58],[57,59],[58,60],[59,61],[60,62],[61,63]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.575900077819824} +{"basename":"euler_rechenkunst01_1738","par_idx":1693,"date":1738,"orig":"Wann derohalben [Formel] mit [Formel] multipliciret werden ſoll, ſo wird das Product [Formel] ſeyn, welches auch ſehr ſchoͤn mit der gegebenen Regel uͤbereinſtimmt; dann hier iſt 35 das Product der Zehler 7 und 5, und 108 das Product von den Nennern 12 und 9; woraus der Grund der gegebenen Regel ſchon einiger maſſen erhellet.","norm":"Wann derohalben [Formel] mit [Formel] multiplizieret werden soll, so wird das Produkt [Formel] sein, welches auch sehr schön mit der gegebenen Regel übereinstimmt; dann hier ist 35 das Produkt der Zähler 7 und 5, und 108 das Produkt von den Nennern 12 und 9; woraus der Grund der gegebenen Regel schon einigermaßen erhellet.","orig_tok":["Wann","derohalben","[Formel]","mit","[Formel]","multipliciret","werden","ſoll",",","ſo","wird","das","Product","[Formel]","ſeyn",",","welches","auch","ſehr","ſchoͤn","mit","der","gegebenen","Regel","uͤbereinſtimmt",";","dann","hier","iſt","35","das","Product","der","Zehler","7","und","5",",","und","108","das","Product","von","den","Nennern","12","und","9",";","woraus","der","Grund","der","gegebenen","Regel","ſchon","einiger","maſſen","erhellet","."],"norm_tok":["Wann","derohalben","[","Formel","]","mit","[","Formel","]","multiplizieret","werden","soll",",","so","wird","das","Produkt","[","Formel","]","sein",",","welches","auch","sehr","schön","mit","der","gegebenen","Regel","übereinstimmt",";","dann","hier","ist","35","das","Produkt","der","Zähler","7","und","5",",","und","108","das","Produkt","von","den","Nennern","12","und","9",";","woraus","der","Grund","der","gegebenen","Regel","schon","einigermaßen","erhellet","."],"orig_spans":[[0,4],[5,15],[16,24],[25,28],[29,37],[38,51],[52,58],[59,63],[63,64],[65,67],[68,72],[73,76],[77,84],[85,93],[94,98],[98,99],[100,107],[108,112],[113,117],[118,124],[125,128],[129,132],[133,142],[143,148],[149,163],[163,164],[165,169],[170,174],[175,178],[179,181],[182,185],[186,193],[194,197],[198,204],[205,206],[207,210],[211,212],[212,213],[214,217],[218,221],[222,225],[226,233],[234,237],[238,241],[242,249],[250,252],[253,256],[257,258],[258,259],[260,266],[267,270],[271,276],[277,280],[281,290],[291,296],[297,302],[303,310],[311,317],[318,326],[326,327]],"norm_spans":[[0,4],[5,15],[16,17],[17,23],[23,24],[25,28],[29,30],[30,36],[36,37],[38,52],[53,59],[60,64],[64,65],[66,68],[69,73],[74,77],[78,85],[86,87],[87,93],[93,94],[95,99],[99,100],[101,108],[109,113],[114,118],[119,124],[125,128],[129,132],[133,142],[143,148],[149,162],[162,163],[164,168],[169,173],[174,177],[178,180],[181,184],[185,192],[193,196],[197,203],[204,205],[206,209],[210,211],[211,212],[213,216],[217,220],[221,224],[225,232],[233,236],[237,240],[241,248],[249,251],[252,255],[256,257],[257,258],[259,265],[266,269],[270,275],[276,279],[280,289],[290,295],[296,301],[302,314],[315,323],[323,324]],"alignment":[[0,0],[1,1],[2,2],[2,3],[2,4],[3,5],[4,6],[4,7],[4,8],[5,9],[6,10],[7,11],[8,12],[9,13],[10,14],[11,15],[12,16],[13,17],[13,18],[13,19],[14,20],[15,21],[16,22],[17,23],[18,24],[19,25],[20,26],[21,27],[22,28],[23,29],[24,30],[25,31],[26,32],[27,33],[28,34],[29,35],[30,36],[31,37],[32,38],[33,39],[34,40],[35,41],[36,42],[37,43],[38,44],[39,45],[40,46],[41,47],[42,48],[43,49],[44,50],[45,51],[46,52],[47,53],[48,54],[49,55],[50,56],[51,57],[52,58],[53,59],[54,60],[55,61],[56,62],[57,62],[58,63],[59,64]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.593400001525879} +{"basename":"euler_rechenkunst01_1738","par_idx":1694,"date":1738,"orig":"Um aber den Grund vollkommen anzuzeigen, ſo laſt uns zwey Bruͤche betrachten, davon der erſte durch den anderen multipliciret werden ſoll; dieſes geſchieht nun wann man den Zehler des erſten mit dem anderen Bruche multiplicirt, den Nenner aber unveraͤndert laͤſt.","norm":"Um aber den Grund vollkommen anzuzeigen, so lässt uns zwei Brüche betrachten, davon der erste durch den anderen multiplizieret werden soll; dieses geschieht nun wann man den Zähler des ersten mit dem anderen Bruche multipliziert, den Nenner aber unverändert 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+{"basename":"euler_rechenkunst01_1738","par_idx":1695,"date":1738,"orig":"Alſo wird das Product ein Bruch ſeyn, deſſen Nenner dem Nenner des erſten Bruchs gleich iſt, der Zehler aber wird fuͤr ſich ein Bruch ſeyn, deſſen Nenner dem Nenner des anderen gegebenen Bruchs gleich, der Zehler aber das Product aus beyden Zehlern iſt.","norm":"Also wird das Produkt ein Bruch sein, dessen Nenner dem Nenner des ersten Bruchs gleich ist, der Zähler aber wird für sich ein Bruch sein, dessen Nenner dem Nenner des anderen gegebenen Bruchs gleich, der Zähler aber das Produkt aus beiden Zählern 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+{"basename":"euler_rechenkunst01_1738","par_idx":1699,"date":1738,"orig":"Wann eine gantze Zahl oder ein Bruch mit ½ multipliciret werden ſoll, ſo wird nichts anders gefragt, als daß man die Helfte derſelben Zahl oder deſſelben Bruchs finden ſoll.","norm":"Wann eine ganze Zahl oder ein Bruch mit ½ multiplizieret werden soll, so wird nichts anders gefragt, als dass man die Hälfte derselben Zahl oder desselben Bruchs finden 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+{"basename":"euler_rechenkunst01_1738","par_idx":1700,"date":1738,"orig":"Dann gleich wie das doppelte oder dreyfache von einer Zahl finden nichts anders iſt als dieſelbe Zahl mit 2 oder mit 3 multipliciren; ſo iſt auch die Helfte von einer Zahl finden nichts anders als dieſelbe Zahl mit ½ multipliciren.","norm":"Dann gleich wie das doppelte oder dreifache von einer Zahl finden nichts anders ist als dieselbe Zahl mit 2 oder mit 3 multiplizieren; so ist auch die Hälfte von einer Zahl finden nichts anders als dieselbe Zahl mit ½ 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+{"basename":"euler_rechenkunst01_1738","par_idx":1708,"date":1738,"orig":"Allein dieſer Nahme iſt aus der Multiplication mit gantzen Zahlen hergenommen worden, und wird allhier bey den Bruͤchen nur in Anſehung der Operation bey behalten.","norm":"Allein dieser Nahme ist aus der Multiplikation mit ganzen Zahlen hergenommen worden, und wird allhier bei den Brüchen nur in Ansehung der Operation 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+{"basename":"euler_rechenkunst01_1738","par_idx":1711,"date":1738,"orig":"Dieſes iſt aber nur allein von Bruͤchen zu verſtehen, welche kleiner ſind als ein gantzes, dann wann eine Zahl mit einem Bruche der groͤſſer iſt als 1 multipliciret wird, ſo wird das Product auch groͤſſer als dieſelbe Zahl: als wann man 7 mit [Formel] multiplicirt, ſo kommt [Formel] das iſt 10½ heraus, und alſo mehr als 7. Ferner iſt hier auch, wie bey der Multiplication mit gantzen Zahlen, zu beobachten daß wann zwey Bruͤche mit einander multiplicirt werden ſollen, es gleichviel ſey, welcher mit dem anderen multiplicirt werde.","norm":"Dieses ist aber nur allein von Brüchen zu verstehen, welche kleiner sind als ein ganzes, dann wann eine Zahl mit einem Bruche der größer ist als 1 multiplizieret wird, so wird das Produkt auch größer als dieselbe Zahl: als wann man 7 mit [Formel] multipliziert, so kommt [Formel] das ist 101/2 heraus, und also mehr als 7. Ferner ist hier auch, wie bei der Multiplikation mit ganzen Zahlen, zu beobachten dass wann zwei Brüche miteinander multipliziert werden sollen, es gleichviel sei, welcher mit dem anderen multipliziert 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+{"basename":"euler_rechenkunst01_1738","par_idx":1712,"date":1738,"orig":"Alſo ⅗ mit ⅔ multipliciren, iſt eben ſo viel als ⅔ mit ⅗ multipliciren, dann in beyden Faͤllen iſt das Product [Formel] oder ⅖; demnach iſt ⅔ von ⅗ eben ſo viel als ⅗ von ⅔.","norm":"Also 3/5 mit 2/3 multiplizieren, ist ebenso viel als 2/3 mit 3/5 multiplizieren, dann in beiden Fällen ist das Produkt [Formel] oder 2/5 ; demnach ist 2/3 von 3/5 ebenso viel als 3/5 von 2/3 .","orig_tok":["Alſo","⅗","mit","⅔","multipliciren",",","iſt","eben","ſo","viel","als","⅔","mit","⅗","multipliciren",",","dann","in","beyden","Faͤllen","iſt","das","Product","[Formel]","oder","⅖",";","demnach","iſt","⅔","von","⅗","eben","ſo","viel","als","⅗","von","⅔","."],"norm_tok":["Also"," ","3","/","5"," ","mit"," ","2","/","3"," ","multiplizieren",",","ist","ebenso","viel","als"," ","2","/","3"," ","mit"," ","3","/","5"," ","multiplizieren",",","dann","in","beiden","Fällen","ist","das","Produkt","[","Formel","]","oder"," 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+{"basename":"euler_rechenkunst01_1738","par_idx":1713,"date":1738,"orig":"Und gleicher geſtalt iſt die Helfte von 6 eben ſo viel als 6 mahl ½ das iſt 3.","norm":"Und gleicher Gestalt ist die Hälfte von 6 ebenso viel als 6 Mal ½ das ist 3.","orig_tok":["Und","gleicher","geſtalt","iſt","die","Helfte","von","6","eben","ſo","viel","als","6","mahl","½","das","iſt","3","."],"norm_tok":["Und","gleicher","Gestalt","ist","die","Hälfte","von","6","ebenso","viel","als","6","Mal","½","das","ist","3."],"orig_spans":[[0,3],[4,12],[13,20],[21,24],[25,28],[29,35],[36,39],[40,41],[42,46],[47,49],[50,54],[55,58],[59,60],[61,65],[66,67],[68,71],[72,75],[76,77],[77,78]],"norm_spans":[[0,3],[4,12],[13,20],[21,24],[25,28],[29,35],[36,39],[40,41],[42,48],[49,53],[54,57],[58,59],[60,63],[64,65],[66,69],[70,73],[74,76]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,8],[10,9],[11,10],[12,11],[13,12],[14,13],[15,14],[16,15],[17,16],[18,16]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.65339994430542} +{"basename":"euler_rechenkunst01_1738","par_idx":1714,"date":1738,"orig":"Aus dieſer Operation aber, durch welche wir zwey Bruͤche mit einander multipliciren gelehret, koͤnnen leicht 3 und auch mehr Bruͤche mit einander multiplicirt werden.","norm":"Aus dieser Operation aber, durch welche wir zwei Brüche miteinander multiplizieren gelehrt, können leicht 3 und auch mehr Brüche miteinander multipliziert 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+{"basename":"euler_rechenkunst01_1738","par_idx":1715,"date":1738,"orig":"Dann man multiplicirt erſtlich zwey Bruͤche mit einander, und dann ferner dieſes Product mit dem dritten Bruch, und was herauskommt mit dem vierten Bruche, und ſo weiter bis man mit allen gegebenen Bruͤchen multiplicirt.","norm":"Dann man multipliziert erstlich zwei Brüche miteinander, und dann ferner dieses Produkt mit dem dritten Bruch, und was herauskommt mit dem vierten Bruche, und so weiter bis man mit allen gegebenen Brüchen 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+{"basename":"euler_rechenkunst01_1738","par_idx":1716,"date":1738,"orig":"Hieraus ſieht man aber leicht, daß das letzt gefundene Product herauskomme, wann man alle Zehler, und dann auch alle Nenner, mit einander multiplicirt.","norm":"Hieraus sieht man aber leicht, dass das letzt gefundene Produkt herauskomme, wann man alle Zähler, und dann auch alle Nenner, miteinander 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+{"basename":"euler_rechenkunst01_1738","par_idx":1718,"date":1738,"orig":"Dieſes Product [Formel] oder welches gleichviel iſt ⅕ wird auch gefunden, wann man je nur zwey Bruͤche mit einander multiplicirt; als ½ mit ⅔ multiplicirt, gibt [Formel] das iſt ⅓; ferner dieſes Product ⅓ mit ¾ multiplicirt, gibt [Formel] das iſt ¼; und dieſes Product noch mit ⅘ multiplicirt, gibt [Formel] das iſt ⅕ wie vorher; ſo daß alſo ⅕ das Product iſt, wann man alle dieſe Bruͤche mit einander multiplicirt; ½, ⅔, ¾, und ⅘.","norm":"Dieses Produkt [Formel] oder welches gleichviel ist 1/5 wird auch gefunden, wann man je nur zwei Brüche miteinander multipliziert; als ½ mit 2/3 multipliziert, gibt [Formel] das ist 1/3 ; ferner dieses Produkt 1/3 mit ¾ multipliziert, gibt [Formel] das ist ¼; und dieses Produkt noch mit 4/5 multipliziert, gibt [Formel] das ist 1/5 wie vorher; so dass also 1/5 das Produkt ist, wann man alle diese Brüche miteinander multipliziert; ½, 2/3 , ¾, und 4/5 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+{"basename":"euler_rechenkunst01_1738","par_idx":1719,"date":1738,"orig":"Hieraus koͤnnen wir nun dieſe Frage beantworten:","norm":"Hieraus können wir nun diese Frage beantworten:","orig_tok":["Hieraus","koͤnnen","wir","nun","dieſe","Frage","beantworten",":"],"norm_tok":["Hieraus","können","wir","nun","diese","Frage","beantworten",":"],"orig_spans":[[0,7],[8,15],[16,19],[20,23],[24,29],[30,35],[36,47],[47,48]],"norm_spans":[[0,7],[8,14],[15,18],[19,22],[23,28],[29,34],[35,46],[46,47]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":3.558500051498413} +{"basename":"euler_rechenkunst01_1738","par_idx":1720,"date":1738,"orig":"Es ſind vier Perſonen, die erſte hat 560 Rubl, die zweyte hat ¾ mahl ſo viel als die erſte, die dritte hat ⅖ mahl ſo viel als die zweyte, und der vierten Vermoͤgen iſt [Formel] von demjenigen, was die dritte hat.","norm":"Es sind vier Personen, die erste hat 560 Rubel, die zweite hat ¾ Mal so viel als die erste, die dritte hat 2/5 Mal so viel als die zweite, und der vierten Vermögen ist [Formel] von demjenigen, was die dritte hat.","orig_tok":["Es","ſind","vier","Perſonen",",","die","erſte","hat","560","Rubl",",","die","zweyte","hat","¾","mahl","ſo","viel","als","die","erſte",",","die","dritte","hat","⅖","mahl","ſo","viel","als","die","zweyte",",","und","der","vierten","Vermoͤgen","iſt","[Formel]","von","demjenigen",",","was","die","dritte","hat","."],"norm_tok":["Es","sind","vier","Personen",",","die","erste","hat","560","Rubel",",","die","zweite","hat","¾","Mal","so","viel","als","die","erste",",","die","dritte","hat"," ","2","/","5"," 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ſo wird das Vermoͤgen der zweyten gefundenen, wann man 560 mit ¾ multiplicirt, da dann [Formel] oder 420 herauskommt.","norm":"Weilen die zweite ¾ Mal so viel hat als die erste, deren Vermögen ist 560 Rubel . so wird das Vermögen der zweiten gefundenen, wann man 560 mit ¾ multipliziert, da dann [Formel] oder 420 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nur das Vermoͤgen der vierten Perſon, allein zu wiſſen verlanget haͤtte, ſo wuͤrde daſſelbe daraus gefunden werden, daß daſſelbe iſt [Formel] von ⅖ von ¾ von 560 Rubl.","norm":"Wann man aber nur das Vermögen der vierten Person, allein zu wissen verlanget hätte, so würde dasselbe daraus gefunden werden, dass dasselbe ist [Formel] von 2/5 von ¾ von 560 Rubel.","orig_tok":["Wann","man","aber","nur","das","Vermoͤgen","der","vierten","Perſon",",","allein","zu","wiſſen","verlanget","haͤtte",",","ſo","wuͤrde","daſſelbe","daraus","gefunden","werden",",","daß","daſſelbe","iſt","[Formel]","von","⅖","von","¾","von","560","Rubl","."],"norm_tok":["Wann","man","aber","nur","das","Vermögen","der","vierten","Person",",","allein","zu","wissen","verlanget","hätte",",","so","würde","dasselbe","daraus","gefunden","werden",",","dass","dasselbe","ist","[","Formel","]","von"," ","2","/","5"," 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+{"basename":"euler_rechenkunst01_1738","par_idx":1728,"date":1738,"orig":"Aus dieſen Exempeln erhellet nun, daß oͤfters nach der gegebenen Regel ein Product gefunden werde, welches hernach entweder durch eine gantze Zahl, oder durch einen leichteren Bruch, als gefunden worden, koͤnne ausgedruckt werden.","norm":"Aus diesen Exempeln erhellet nun, dass öfters nach der gegebenen Regel ein Produkt gefunden werde, welches hernach entweder durch eine ganze Zahl, oder durch einen leichteren Bruch, als gefunden worden, könne ausgedrückt 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+{"basename":"euler_rechenkunst01_1738","par_idx":1732,"date":1738,"orig":"4)","norm":"4)","orig_tok":["4",")"],"norm_tok":["4",")"],"orig_spans":[[0,1],[1,2]],"norm_spans":[[0,1],[1,2]],"alignment":[[0,0],[1,1]],"lang_fastText":"es","lang_py3langid":"en","lang_cld3":"ja","orig_lang_de":0,"norm_lmscore":2.607399940490722} +{"basename":"euler_rechenkunst01_1738","par_idx":1733,"date":1738,"orig":"Damit man, wann zwey oder mehr Bruͤche mit einander multiplicirt werden ſollen, gleich das geſuchte Product in den kleinſten moͤglichen Zahlen ausgedruͤckt bekomme, ſo muß man ſehen, ob irgend ein Zehler mit einem Nenner einen gemeinen Theiler habe, und alsdann beyde durch ihren groͤſten gemeinen Theiler diuidiren, und die Quotos an derſelben Stelle ſetzen.","norm":"Damit man, wann zwei oder mehr Brüche miteinander multipliziert werden sollen, gleich das gesuchte Produkt in den kleinsten möglichen Zahlen ausgedrückt bekomme, so muss man sehen, ob irgendein Zähler mit einem Nenner einen gemeinen Teiler habe, und alsdann beide durch ihren größten gemeinen Teiler dividieren, und die Quotos an derselben Stelle 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+{"basename":"euler_rechenkunst01_1738","par_idx":1734,"date":1738,"orig":"Auf dieſe Art verfaͤhrt man mit einem jeglichen Zehler und Nenner, und wann man alle ſo viel moͤglich gegen einander aufgehoben, ſo multiplicirt man nach der vorigen Regel die Zehler und Nenner, oder vielmehr die Zahlen, welche nach geſchehener Aufhebung an derſelben Stelle geſetzt worden ſind, mit einander, und bekommt alſo auf dieſe Art das geſuchte Product in den kleinſten moͤglichen Zahlen ausgedruͤckt.","norm":"Auf diese Art verfährt man mit einem jeglichen Zähler und Nenner, und wann man alle so viel möglich gegen einander aufgehoben, so multipliziert man nach der vorigen Regel die Zähler und Nenner, oder vielmehr die Zahlen, welche nach geschehener Aufhebung an derselben Stelle gesetzt worden sind, miteinander, und bekommt also auf diese Art das gesuchte Produkt in den kleinsten möglichen Zahlen 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+{"basename":"euler_rechenkunst01_1738","par_idx":1735,"date":1738,"orig":"Weilen nach der vorigen Regel zwey und auch mehr Bruͤche mit einander multiplicirt werden, wann man erſtlich alle Zehler und dann auch alle Nenner mit einander multiplicirt, ſo iſt ein jeglicher Zehler ein Factor oder Theiler des Zehlers des Products, und gleicher geſtalt ein jeglicher Nenner ein Factor oder Theiler des Nenners des Products.","norm":"Weilen nach der vorigen Regel zwei und auch mehr Brüche miteinander multipliziert werden, wann man erstlich alle Zähler und dann auch alle Nenner miteinander multipliziert, so ist ein jeglicher Zähler ein Faktor oder Teiler des Zählers des Produkts, und gleicher Gestalt ein jeglicher Nenner ein Faktor oder Teiler des Nenners des 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+{"basename":"euler_rechenkunst01_1738","par_idx":1736,"date":1738,"orig":"Wann derohalben irgend ein Zehler mit irgend einem Nenner einen gemeinen Theiler hat, ſo werden ſich auch des Products Zehler und Nenner durch eben denſelben Theiler theilen und folglich in kleinere Zahlen bringen laſſen.","norm":"Wann derohalben irgendein Zähler mit irgendeinem Nenner einen gemeinen Teiler hat, so werden sich auch des Produkts Zähler und Nenner durch eben denselben Teiler teilen und folglich in kleinere Zahlen bringen 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+{"basename":"euler_rechenkunst01_1738","par_idx":1737,"date":1738,"orig":"Wann man derohalben noch vor der Multiplication derſelben Zehler und Nenner durch ihren gemeinen Theiler theilet, und die Quotienten an derſelben Stelle ſetzt, ſo iſt es eben ſo viel als wann man nach geſchehener Multiplication den Zehler und Nenner des Products durch denſelben gemeinen Theiler diuidirte.","norm":"Wann man derohalben noch vor der Multiplikation derselben Zähler und Nenner durch ihren gemeinen Teiler teilet, und die Quotienten an derselben Stelle setzt, so ist es ebenso viel als wann man nach geschehener Multiplikation den Zähler und Nenner des Produkts durch denselben gemeinen Teiler 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+{"basename":"euler_rechenkunst01_1738","par_idx":1738,"date":1738,"orig":"Durch eine ſolche Aufhebung alſo, da ein Zehler und Nenner durch einen gemeinen Theiler diuidirt werden, erhaͤlt man das Product zugleich in kleineren Zahlen ausgedruͤckt, und hat hernach derſelben Reduction nicht mehr vonnoͤthen.","norm":"Durch eine solche Aufhebung also, da ein Zähler und Nenner durch einen gemeinen Teiler dividiert werden, erhält man das Produkt zugleich in kleineren Zahlen ausgedrückt, und hat hernach derselben Reduktion nicht mehr 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+{"basename":"euler_rechenkunst01_1738","par_idx":1739,"date":1738,"orig":"Woraus erhellet, daß wann man vor der Multiplication einen jeglichen Zehler gegen einen jeglichen Nenner betrachtet, und dieſelben durch ihren groͤſten gemeinen Theiler gegen einander aufhebt, alsdann das Product in den kleinſten moͤglichen Zahlen ausgedruͤckt gefunden werde.","norm":"Woraus erhellet, dass wann man vor der Multiplikation einen jeglichen Zähler gegen einen jeglichen Nenner betrachtet, und dieselben durch ihren größten gemeinen Teiler gegen einander aufhebt, alsdann das Produkt in den kleinsten möglichen Zahlen ausgedrückt gefunden 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+{"basename":"euler_rechenkunst01_1738","par_idx":1742,"date":1738,"orig":"Wann ſich endlich kein Zehler mehr gegen einem Nenner aufheben laͤſt, ſo ſchreitet man zu der Multiplication, da man dann an ſtatt der ausgeſtrichenen Zehler und Nenner, die an derſelben Stelle geſetzten Zahlen multiplicirt.","norm":"Wann sich endlich kein Zähler mehr gegen einem Nenner aufheben lässt, so schreitet man zu der Multiplikation, da man dann anstatt der ausgestrichenen Zähler und Nenner, die an derselben Stelle gesetzten Zahlen 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eine gantze Zahl mit einem Bruche multiplicirt werden, ſo findet der gewieſene Vortheil gleicher maſſen ſtatt, indem die gantze Zahl als ein Zehler angeſehen werden kan, deſſen Nenner 1 iſt; gleich wie wir ſchon oben angemercket, daß zum Exempel [Formel] fuͤr 6 geſchrieben werden koͤnne.","norm":"Soll eine ganze Zahl mit einem Bruche multipliziert werden, so findet der gewiesene Vorteil Maßen statt, indem die ganze Zahl als ein Zähler angesehen werden kann, dessen Nenner 1 ist; gleich wie wir schon oben angemerkt, dass zum Exempel [Formel] für 6 geschrieben werden 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+{"basename":"euler_rechenkunst01_1738","par_idx":1762,"date":1738,"orig":"[Formel]","norm":"[Formel]","orig_tok":["[Formel]"],"norm_tok":["[","Formel","]"],"orig_spans":[[0,8]],"norm_spans":[[0,1],[1,7],[7,8]],"alignment":[[0,0],[0,1],[0,2]],"lang_fastText":"sh","lang_py3langid":"en","lang_cld3":"cy","orig_lang_de":0,"norm_lmscore":7.932600021362305} +{"basename":"euler_rechenkunst01_1738","par_idx":1763,"date":1738,"orig":"Sollen aber mehr als 2 Bruͤche mit einander multiplicirt werden, ſo wird dieſer Vortheil gleicher geſtalt angebracht, wie aus folgenden Exempeln zu ſehen.","norm":"Sollen aber mehr als 2 Brüche miteinander multipliziert werden, so wird dieser Vorteil gleicher Gestalt angebracht, wie aus folgenden Exempeln zu 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+{"basename":"euler_rechenkunst01_1738","par_idx":1764,"date":1738,"orig":"[Formel] Ferner [Formel] Gleicher geſtalt [Formel]","norm":"[Formel] Ferner [Formel] Gleicher Gestalt [Formel]","orig_tok":["[Formel]","Ferner","[Formel]","Gleicher","geſtalt","[Formel]"],"norm_tok":["[","Formel","]","Ferner","[","Formel","]","Gleicher","Gestalt","[","Formel","]"],"orig_spans":[[0,8],[9,15],[16,24],[25,33],[34,41],[42,50]],"norm_spans":[[0,1],[1,7],[7,8],[9,15],[16,17],[17,23],[23,24],[25,33],[34,41],[42,43],[43,49],[49,50]],"alignment":[[0,0],[0,1],[0,2],[1,3],[2,4],[2,5],[2,6],[3,7],[4,8],[5,9],[5,10],[5,11]],"lang_fastText":"sh","lang_py3langid":"de","lang_cld3":"cy","orig_lang_de":0.333,"norm_lmscore":5.515399932861328} +{"basename":"euler_rechenkunst01_1738","par_idx":1765,"date":1738,"orig":"Und alſo| wird in |allen dergleichen Exempeln verfahren.","norm":"Und also | wird in | allen dergleichen Exempeln 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Zahlen, welche mit einander multiplicirt werden ſollten, keine einzelen Bruͤche, ſondern aus gantzen und Bruͤchen zuſammen geſetzt ſind, ſo kan man entweder dieſelben in die Form einzeler Bruͤche bringen, wie oben iſt gelehret worden, und als dann die Multiplication wie vorher vollziehen.","norm":"Wann die Zahlen, welche miteinander multipliziert werden sollten, keine einzeln Brüche, sondern aus ganzen und Brüchen zusammen gesetzt sind, so kann man entweder dieselben in die Form Einzeller Brüche bringen, wie oben ist gelehrt worden, und als dann die Multiplikation wie vorher 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+{"basename":"euler_rechenkunst01_1738","par_idx":1772,"date":1738,"orig":"Die erſte Art beſteht nun darinn, daß man die aus gantzen und Bruͤchen zuſammen geſetzten Zahlen in die Form einzeler Bruͤche bringt, und die Multiplication nebſt denen Vortheilen, wie im vorigen Satze gelehret worden, verrichtet.","norm":"Die erste Art besteht nun darin, dass man die aus ganzen und Brüchen zusammen gesetzten Zahlen in die Form Einzeller Brüche bringt, und die Multiplikation nebst denen Vorteilen, wie im vorigen Satze gelehrt worden, 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+{"basename":"euler_rechenkunst01_1738","par_idx":1785,"date":1738,"orig":"Als es ſollen 2½, 3⅓, 5⅖, mit einander multipliciret werden, ſo wird das Product folgender geſtalt gefunden.","norm":"Als es sollen 21/2, 3 1/3 , 5 2/5 , miteinander multiplizieret werden, so wird das Produkt folgender Gestalt 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+{"basename":"euler_rechenkunst01_1738","par_idx":1805,"date":1738,"orig":"Wann zwey Bruͤche gleiche Nenner haben, ſo ſieht man erſtlich leicht, welcher groͤſſer iſt als der andere:","norm":"Wann zwei Brüche gleiche Nenner haben, so sieht man erstlich leicht, welcher größer ist als der 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+{"basename":"euler_rechenkunst01_1738","par_idx":1806,"date":1738,"orig":"dann derjenige Bruch deſſen Zehler groͤſſer iſt, derſelbe iſt auch der groͤſſere.","norm":"dann derjenige Bruch dessen Zähler größer ist, derselbe ist auch der größere.","orig_tok":["dann","derjenige","Bruch","deſſen","Zehler","groͤſſer","iſt",",","derſelbe","iſt","auch","der","groͤſſere","."],"norm_tok":["dann","derjenige","Bruch","dessen","Zähler","größer","ist",",","derselbe","ist","auch","der","größere","."],"orig_spans":[[0,4],[5,14],[15,20],[21,27],[28,34],[35,43],[44,47],[47,48],[49,57],[58,61],[62,66],[67,70],[71,80],[80,81]],"norm_spans":[[0,4],[5,14],[15,20],[21,27],[28,34],[35,41],[42,45],[45,46],[47,55],[56,59],[60,64],[65,68],[69,76],[76,77]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13]],"lang_fastText":"de","lang_py3langid":"lb","lang_cld3":"fr","orig_lang_de":0.333,"norm_lmscore":6.582600116729736} 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+{"basename":"euler_rechenkunst01_1738","par_idx":1810,"date":1738,"orig":"Weilen nun in der Diuiſion nichts anders geſucht wird, als wieviel mahl eine Zahl groͤſſer ſey als die andere, und die Zahl welche anzeiget, wieviel mahl die eine groͤſſer iſt als die andere, der Quotus genennet wird:","norm":"Weilen nun in der Division nichts anders gesucht wird, als wie viel Mal eine Zahl größer sei als die andere, und die Zahl welche anzeiget, wie viel Mal die eine größer ist als die andere, der Quotus genannt 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+{"basename":"euler_rechenkunst01_1738","par_idx":1811,"date":1738,"orig":"ſo iſt auch einen Bruch durch einen anderen diuidiren nichts anders, als finden, wieviel mahl einer groͤſſer iſt als der andere, welches durch den Quotum angezeiget wird.","norm":"so ist auch einen Bruch durch einen anderen dividieren nichts anders, als finden, wie viel Mal einer größer ist als der andere, welches durch den Quotum angezeigt 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+{"basename":"euler_rechenkunst01_1738","par_idx":1812,"date":1738,"orig":"Da nun alſo von zweyen Bruͤchen, welche gleiche Nenner haben, der eine um ſo viel mahl groͤſſer iſt als der andere, um ſo viel mahl deſſelben Zehler groͤſſer iſt als der Zehler dieſes:","norm":"Da nun also von zweien Brüchen, welche gleiche Nenner haben, der eine um so viel Mal größer ist als der andere, um so viel Mal desselben Zähler größer ist als der Zähler 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+{"basename":"euler_rechenkunst01_1738","par_idx":1827,"date":1738,"orig":"Wann zwey Bruͤche von ungleichen Nennern zu gleichen Benennungen gebracht werden ſollen, ſo multiplicirt man eines jeden Bruchs Zehler und Nenner mit dem Nenner des andern:","norm":"Wann zwei Brüche von ungleichen Nennern zu gleichen Benennungen gebracht werden sollen, so multipliziert man eines jeden Bruchs Zähler und Nenner mit dem Nenner des 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+{"basename":"euler_rechenkunst01_1738","par_idx":1828,"date":1738,"orig":"wie oben ſchon gelehret worden.","norm":"wie oben schon gelehrt worden.","orig_tok":["wie","oben","ſchon","gelehret","worden","."],"norm_tok":["wie","oben","schon","gelehrt","worden","."],"orig_spans":[[0,3],[4,8],[9,14],[15,23],[24,30],[30,31]],"norm_spans":[[0,3],[4,8],[9,14],[15,22],[23,29],[29,30]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.7941999435424805} +{"basename":"euler_rechenkunst01_1738","par_idx":1829,"date":1738,"orig":"Derohalben wann auf dieſe Art zwey Bruͤche, davon einer durch den anderen diuid rt werden ſoll, zu gleichen Benennungen gebracht werden, ſo wird fuͤr den Diuidendum ein Bruch herauskommen, deſſen Zehler der vorige Zehler, mit dem Nenner des Diuiſoris multiplicirt, ſeyn wird.","norm":"Derohalben wann auf diese Art zwei Brüche, davon einer durch den anderen divid rt werden soll, zu 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+{"basename":"euler_rechenkunst01_1738","par_idx":1832,"date":1738,"orig":"Wann aber alſo die vorgegebenen Bruͤche zu gleichen Benennungen gebracht worden, ſo wird der geſuchte Quotus, nach dem vorigen Satze, ein Bruch ſeyn, deſſen Zehler der Zehler des Diuidendi, der Nenner aber der Zehler des Diuiſoris iſt.","norm":"Wann aber also die vorgegebenen Brüche zu gleichen Benennungen gebracht worden, so wird der gesuchte Quotus, nach dem vorigen Satze, ein Bruch sein, dessen Zähler der Zähler des Dividenden, der Nenner aber der Zähler des Divisors 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+{"basename":"euler_rechenkunst01_1738","par_idx":1834,"date":1738,"orig":"Von zweyen gegebenen Bruchen, deren einer durch den anderen diuidirt werden ſoll, wird der Quotus ein Bruch ſeyn, deſſen Zehler das Product aus dem Zehler des Diuidendi und dem Nenner des Diuiſoris, der Nenner aber das Product aus dem Zehler des Diuiſoris und dem Nenner des Diuidendi iſt.","norm":"Von zweien gegebenen Brüchen, deren einer durch den anderen dividiert werden soll, wird der Quotus ein Bruch sein, dessen Zähler das Produkt aus dem Zähler des Dividenden und dem Nenner des Divisors, der Nenner aber das Produkt aus dem Zähler des Divisors und dem Nenner des Dividenden 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+{"basename":"euler_rechenkunst01_1738","par_idx":1835,"date":1738,"orig":"Wann alſo nach dieſer Regel ⅝ durch ⅔ diuidirt werden ſoll, ſo iſt ⅝ der Diuidendus und ⅔ der Diuiſor; demnach gibt 5 mit 3 multiplicirt den Zehler des Quoti, und 8 mit 2 multiplicirt, das iſt 16, den Nenner deſſelben, ſo daß folglich der Quotus [Formel] ſeyn wird.","norm":"Wann also nach dieser Regel 5/8 durch 2/3 dividiert werden soll, so ist 5/8 der Dividend und 2/3 der Divisor; demnach gibt 5 mit 3 multipliziert den Zähler des Quoti, und 8 mit 2 multipliziert, das ist 16, den Nenner desselben, so dass folglich der Quotus [Formel] sein 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+{"basename":"euler_rechenkunst01_1738","par_idx":1836,"date":1738,"orig":"Dieſe Operation pflegt nun folgender Geſtalt vorgeſtellet zu werden.","norm":"Diese Operation pflegt nun folgender Gestalt vorgestellt zu werden.","orig_tok":["Dieſe","Operation","pflegt","nun","folgender","Geſtalt","vorgeſtellet","zu","werden","."],"norm_tok":["Diese","Operation","pflegt","nun","folgender","Gestalt","vorgestellt","zu","werden","."],"orig_spans":[[0,5],[6,15],[16,22],[23,26],[27,36],[37,44],[45,57],[58,60],[61,67],[67,68]],"norm_spans":[[0,5],[6,15],[16,22],[23,26],[27,36],[37,44],[45,56],[57,59],[60,66],[66,67]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":7.712900161743164} +{"basename":"euler_rechenkunst01_1738","par_idx":1837,"date":1738,"orig":"[Formel]","norm":"[Formel]","orig_tok":["[Formel]"],"norm_tok":["[","Formel","]"],"orig_spans":[[0,8]],"norm_spans":[[0,1],[1,7],[7,8]],"alignment":[[0,0],[0,1],[0,2]],"lang_fastText":"sh","lang_py3langid":"en","lang_cld3":"cy","orig_lang_de":0,"norm_lmscore":7.932600021362305} +{"basename":"euler_rechenkunst01_1738","par_idx":1838,"date":1738,"orig":"Nachdem man nehmlich den Diuiſorem vor den Diuidendum geſchrieben, ſo zieht man vom Zehler des Diuidendi zum Nenner des Diuiſoris, und auch vom Nenner des Diuidendi zum Zehler des Diuiſoris, gerade Linien, welche ſich durchſchneiden und ein Creutz vorſtellen werden.","norm":"Nachdem man nämlich den Divisor vor den Dividenden geschrieben, so zieht man vom Zähler des Dividenden zum Nenner des Divisors, und auch vom Nenner des Dividenden zum Zähler des Divisors, gerade Linien, welche sich durchschneiden und ein Kreuz vorstellen 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+{"basename":"euler_rechenkunst01_1738","par_idx":1839,"date":1738,"orig":"Hierauf multiplicirt man nach Anleitung dieſes Creutzes den Zehler des Diuidendi 5 mit dem Nenner des Diuiſoris 3, ſo gibt das Product 15 den Zehler des Quoti.","norm":"Hierauf multipliziert man nach Anleitung dieses Kreuzes den Zähler des Dividenden 5 mit dem Nenner des Divisors 3, so gibt das Produkt 15 den Zähler des 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+{"basename":"euler_rechenkunst01_1738","par_idx":1840,"date":1738,"orig":"Hernach multiplicirt man den Nenner des Diuidendi 8 mit dem Zehler des Diuiſoris 2, ſo gibt das Product 16 den Nenner des Quoti, ſo daß folglich der Quotus [Formel] ſeyn wird.","norm":"Hernach multipliziert man den Nenner des Dividenden 8 mit dem Zähler des Divisors 2, so gibt das Produkt 16 den Nenner des Quoti, so dass folglich der Quotus [Formel] sein 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+{"basename":"euler_rechenkunst01_1738","par_idx":1841,"date":1738,"orig":"Um aber von dieſer Operation deſto gewiſſer zu ſeyn, ſo kan man die vorgegebenen Bruͤche erſtlich zu gleichen Benennungen bringen, da man dann anſtatt ⅔ und ⅝ dieſe Bruͤche bekommt [Formel] und [Formel] ; davon dieſer durch jenen nach dem erſten Satz diuidirt [Formel] fuͤr den Quotum gibt.","norm":"Um aber von dieser Operation desto gewisser zu sein, so kann man die vorgegebenen Brüche erstlich zu gleichen Benennungen bringen, da man dann anstatt 2/3 und 5/8 diese Brüche bekommt [Formel] und [Formel]; davon dieser durch jenen nach dem ersten Satz dividiert [Formel] für den Quotum 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+{"basename":"euler_rechenkunst01_1738","par_idx":1842,"date":1738,"orig":"Man kan ſich auch die gantze Operation folgender geſtalt vorſtellen.","norm":"Man kann sich auch die ganze Operation folgender Gestalt vorstellen.","orig_tok":["Man","kan","ſich","auch","die","gantze","Operation","folgender","geſtalt","vorſtellen","."],"norm_tok":["Man","kann","sich","auch","die","ganze","Operation","folgender","Gestalt","vorstellen","."],"orig_spans":[[0,3],[4,7],[8,12],[13,17],[18,21],[22,28],[29,38],[39,48],[49,56],[57,67],[67,68]],"norm_spans":[[0,3],[4,8],[9,13],[14,18],[19,22],[23,28],[29,38],[39,48],[49,56],[57,67],[67,68]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.411600112915039} +{"basename":"euler_rechenkunst01_1738","par_idx":1843,"date":1738,"orig":"Wann ⅝ durch ⅔ diuidirt werden ſollen, ſo kan ſogleich der Quotus auf ſolche Art [Formel] ausgedruͤckt 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+{"basename":"euler_rechenkunst01_1738","par_idx":1844,"date":1738,"orig":"Um aber dieſe ungewoͤhnliche Bruchs- Form in die gewoͤhnliche Form zu bringen, und dieſen Bruch in einen anderen zu verwandeln, deſſen Zehler und Nenner gantze Zahlen ſind, ſo multiplicire man den Zehler und Nenner beyde durch 8; da dann dieſer Bruch [Formel] herauskomt, weilen ⅝ mit 8 multiplicirt 5 gibt, und ⅔ mit 8 multiplicirt [Formel] .","norm":"Um aber diese ungewöhnliche Bruchs-Form in die gewöhnliche Form zu bringen, und diesen Bruch in einen anderen zu verwandeln, dessen Zähler und Nenner ganze Zahlen sind, so multipliziere man den Zähler und Nenner beide durch 8; da dann dieser Bruch [Formel] herauskommt, weilen 5/8 mit 8 multipliziert 5 gibt, und 2/3 mit 8 multipliziert 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+{"basename":"euler_rechenkunst01_1738","par_idx":1847,"date":1738,"orig":"Zu fernerer Gewißheit kan man ſich auch der bey der Diuiſion in gantzen Zahlen angebrachten Probe bedienen.","norm":"Zu fernerer Gewissheit kann man sich auch der bei der Division in ganzen Zahlen angebrachten Probe 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Wann ſich aber der Zehler durch die gantze Zahl wuͤrcklich theilen laͤßt, ſo darf man nur den Zehler theilen, und den Nenner unveraͤndert laſſen, als [Formel] durch 2 diuidirt gibt [Formel] ; dann nach der vorigen Operation komt [Formel] heraus, welches eben ſo viel iſt als [Formel] weilen ſich Zehler und Nenner durch 2 theilen laſſen.","norm":"Also 5/6 durch 3 dividiert gibt [Formel]; und [Formel] durch 5 dividiert gibt [Formel]. 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+{"basename":"euler_rechenkunst01_1738","par_idx":1861,"date":1738,"orig":"Gleicher geſtalt wann [Formel] durch 8 diuidirt werden ſollen, ſo wird der Quotus [Formel] ſeyn.","norm":"Gleicher Gestalt wann [Formel] durch 8 dividiert werden sollen, so wird der Quotus [Formel] 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derohalben ein Bruch, durch eine gantze Zahl getheilet werden ſoll, ſo ſieht man erſtlich, ob ſich der Zehler dadurch theilen laſſe, in welchem Fall man den Zehler dadurch wuͤrcklich diuidirt, den Nenner aber unveraͤndert laͤſſt, und alſo den Quotum bekommt.","norm":"Wann derohalben ein Bruch, durch eine ganze Zahl geteilt werden soll, so sieht man erstlich, ob sich der Zähler dadurch teilen lasse, in welchem Fall man den Zähler dadurch wirklich dividiert, den Nenner aber unverändert lässt, und also den Quotum 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+{"basename":"euler_rechenkunst01_1738","par_idx":1863,"date":1738,"orig":"Laͤſſt ſich aber der Zehler durch die gantze Zahl nicht theilen, ſo laͤſſt man den Zehler unveraͤndert, und moltiplicirt den Nenner mit der gantzen Zahl, ſo hat man den Quotum.","norm":"Lässt sich aber der Zähler durch die ganze Zahl nicht teilen, so lässt man den Zähler unverändert, und moltiplicirt den Nenner mit der ganzen Zahl, so hat man den 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+{"basename":"euler_rechenkunst01_1738","par_idx":1870,"date":1738,"orig":"Wann dieſer Bruch [Formel] durch ⅖ diuidirt werden ſoll, ſo wird der Quotus [Formel] ſeyn, wie folgende Operation weiſet.","norm":"Wann dieser Bruch [Formel] durch 2/5 dividiert werden soll, so wird der Quotus [Formel] sein, wie folgende Operation weiset.","orig_tok":["Wann","dieſer","Bruch","[Formel]","durch","⅖","diuidirt","werden","ſoll",",","ſo","wird","der","Quotus","[Formel]","ſeyn",",","wie","folgende","Operation","weiſet","."],"norm_tok":["Wann","dieser","Bruch","[","Formel","]","durch"," ","2","/","5"," 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+{"basename":"euler_rechenkunst01_1738","par_idx":1873,"date":1738,"orig":"Dann hieraus ſieht man daß der Quotus herauskommt, wann man des Diuidendi Zehler durch den Zehler des Diuiſoris, und den Nenner des Diuidendi durch den Nenner des Diuiſoris diuidirt.","norm":"Dann hieraus sieht man dass der Quotus herauskommt, wann man des Dividenden Zähler durch den Zähler des Divisors, und den Nenner des Dividenden durch den Nenner des Divisors dividiert.","orig_tok":["Dann","hieraus","ſieht","man","daß","der","Quotus","herauskommt",",","wann","man","des","Diuidendi","Zehler","durch","den","Zehler","des","Diuiſoris",",","und","den","Nenner","des","Diuidendi","durch","den","Nenner","des","Diuiſoris","diuidirt","."],"norm_tok":["Dann","hieraus","sieht","man","dass","der","Quotus","herauskommt",",","wann","man","des","Dividenden","Zähler","durch","den","Zähler","des","Divisors",",","und","den","Nenner","des","Dividenden","durch","den","Nenner","des","Divisors","dividiert","."],"orig_spans":[[0,4],[5,12],[13,18],[19,22],[23,26],[27,30],[31,37],[38,49],[49,50],[51,55],[56,59],[60,63],[64,73],[74,80],[81,86],[87,90],[91,97],[98,101],[102,111],[111,112],[113,116],[117,120],[121,127],[128,131],[132,141],[142,147],[148,151],[152,158],[159,162],[163,172],[173,181],[181,182]],"norm_spans":[[0,4],[5,12],[13,18],[19,22],[23,27],[28,31],[32,38],[39,50],[50,51],[52,56],[57,60],[61,64],[65,75],[76,82],[83,88],[89,92],[93,99],[100,103],[104,112],[112,113],[114,117],[118,121],[122,128],[129,132],[133,143],[144,149],[150,153],[154,160],[161,164],[165,173],[174,183],[183,184]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28],[29,29],[30,30],[31,31]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.006899833679199} 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+{"basename":"euler_rechenkunst01_1738","par_idx":1875,"date":1738,"orig":"Der Grund hievon ergibt ſich am beſten aus der Probe durch die Multiplication; dann da ſieht man deutlich, daß wann man den gefundenen Quotum mit dem Diuiſore multiplicirt, der Diuidendus herauskomme.","norm":"Der Grund hiervon ergibt sich am besten aus der Probe durch die Multiplikation; dann da sieht man deutlich, dass wann man den gefundenen Quotum mit dem Divisor multipliziert, der Dividend 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+{"basename":"euler_rechenkunst01_1738","par_idx":1876,"date":1738,"orig":"Jn welchen Faͤllen aber die Operation abgekuͤrtzet werden koͤnne, wird aus folgendem Satze zu erſehen ſeyn.","norm":"In welchen Fällen aber die Operation abgekürzt werden könne, wird aus folgendem Satze zu ersehen 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+{"basename":"euler_rechenkunst01_1738","par_idx":1879,"date":1738,"orig":"Wann nun ſolcher geſtalt die Diuiſion in eine Multiplication iſt verwandelt worden, ſo kan man auch dabey alle diejenigen Vortheile anbringen, welche im vorigen Cap. bey der Multiplication ſind gelehret worden, wodurch gleichfals die Operation ſo kan abgekuͤrtzet werden, daß man gleich den Quotum in ſeiner kleinſten Form bekommt, und darnach keiner weiteren Reduction mehr bedarf.","norm":"Wann nun solcher Gestalt die Division in eine Multiplikation ist verwandelt worden, so kann man auch dabei alle diejenigen Vorteile anbringen, welche im vorigen Cap. bei der Multiplikation sind gelehrt worden, wodurch gleichfalls die Operation so kann abgekürzt werden, dass man gleich den Quotum in seiner kleinsten Form bekommt, und danach keiner weiteren Reduktion mehr 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Von dieſer Umkehrung der Bruͤche iſt uͤberhaupt anzumercken, daß wann ein Bruch kleiner iſt als ein gantzes, als dann der umgekehrte Bruch groͤſſer ſey als ein gantzes; und hinwiederum, wann der Bruch groͤſſer iſt als 1, ſo iſt der umgekehrte kleiner als 1.","norm":"Ein Bruch wird umgekehrt, wann man den Zähler an des Nenners Stelle, und den Nenner an des Zählers Stelle, setzet; also wann 5/8 umgekehrt werden, so bekommt man [Formel]. Von dieser Umkehrung der Brüche ist überhaupt anzumerken, dass wann ein Bruch kleiner ist als ein ganzes, als dann der umgekehrte Bruch größer sei als ein ganzes; und hinwiederum, wann der Bruch größer ist als 1, so ist der umgekehrte kleiner als 1.","orig_tok":["Ein","Bruch","wird","umgekehret",",","wann","man","den","Zehler","an","des","Nenners","Stelle",",","und","den","Nenner","an","des","Zehlers","Stelle",",","ſetzet",";","alſo","wann","⅝","umgekehret","werden",",","ſo","bekommt","man","[Formel]",".","Von","dieſer","Umkehrung","der","Bruͤche","iſt","uͤberhaupt","anzumercken",",","daß","wann","ein","Bruch","kleiner","iſt","als","ein","gantzes",",","als","dann","der","umgekehrte","Bruch","groͤſſer","ſey","als","ein","gantzes",";","und","hinwiederum",",","wann","der","Bruch","groͤſſer","iſt","als","1",",","ſo","iſt","der","umgekehrte","kleiner","als","1","."],"norm_tok":["Ein","Bruch","wird","umgekehrt",",","wann","man","den","Zähler","an","des","Nenners","Stelle",",","und","den","Nenner","an","des","Zählers","Stelle",",","setzet",";","also","wann"," ","5","/","8"," ","umgekehrt","werden",",","so","bekommt","man","[","Formel","]",".","Von","dieser","Umkehrung","der","Brüche","ist","überhaupt","anzumerken",",","dass","wann","ein","Bruch","kleiner","ist","als","ein","ganzes",",","als","dann","der","umgekehrte","Bruch","größer","sei","als","ein","ganzes",";","und","hinwiederum",",","wann","der","Bruch","größer","ist","als","1",",","so","ist","der","umgekehrte","kleiner","als","1."],"orig_spans":[[0,3],[4,9],[10,14],[15,25],[25,26],[27,31],[32,35],[36,39],[40,46],[47,49],[50,53],[54,61],[62,68],[68,69],[70,73],[74,77],[78,84],[85,87],[88,91],[92,99],[100,106],[106,107],[108,114],[114,115],[116,120],[121,125],[126,127],[128,138],[139,145],[145,146],[147,149],[150,157],[158,161],[162,170],[171,172],[173,176],[177,183],[184,193],[194,197],[198,205],[206,209],[210,220],[221,232],[232,233],[234,237],[238,242],[243,246],[247,252],[253,260],[261,264],[265,268],[269,272],[273,280],[280,281],[282,285],[286,290],[291,294],[295,305],[306,311],[312,320],[321,324],[325,328],[329,332],[333,340],[340,341],[342,345],[346,357],[357,358],[359,363],[364,367],[368,373],[374,382],[383,386],[387,390],[391,392],[392,393],[394,396],[397,400],[401,404],[405,415],[416,423],[424,427],[428,429],[429,430]],"norm_spans":[[0,3],[4,9],[10,14],[15,24],[24,25],[26,30],[31,34],[35,38],[39,45],[46,48],[49,52],[53,60],[61,67],[67,68],[69,72],[73,76],[77,83],[84,86],[87,90],[91,98],[99,105],[105,106],[107,113],[113,114],[115,119],[120,124],[125,126],[126,127],[127,128],[128,129],[130,131],[131,140],[141,147],[147,148],[149,151],[152,159],[160,163],[164,165],[165,171],[171,172],[172,173],[174,177],[178,184],[185,194],[195,198],[199,205],[206,209],[210,219],[220,230],[230,231],[232,236],[237,241],[242,245],[246,251],[252,259],[260,263],[264,267],[268,271],[272,278],[278,279],[280,283],[284,288],[289,292],[293,303],[304,309],[310,316],[317,320],[321,324],[325,328],[329,335],[335,336],[337,340],[341,352],[352,353],[354,358],[359,362],[363,368],[369,375],[376,379],[380,383],[384,385],[385,386],[387,389],[390,393],[394,397],[398,408],[409,416],[417,420],[421,423]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,null],[27,null],[null,26],[null,27],[null,28],[null,29],[null,30],[null,31],[28,32],[29,33],[30,34],[31,35],[32,36],[33,37],[33,38],[33,39],[34,40],[35,41],[36,42],[37,43],[38,44],[39,45],[40,46],[41,47],[42,48],[43,49],[44,50],[45,51],[46,52],[47,53],[48,54],[49,55],[50,56],[51,57],[52,58],[53,59],[54,60],[55,61],[56,62],[57,63],[58,64],[59,65],[60,66],[61,67],[62,68],[63,69],[64,70],[65,71],[66,72],[67,73],[68,74],[69,75],[70,76],[71,77],[72,78],[73,79],[74,80],[75,81],[76,82],[77,83],[78,84],[79,85],[80,86],[81,87],[82,88],[83,88]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.083099842071533} +{"basename":"euler_rechenkunst01_1738","par_idx":1881,"date":1738,"orig":"Der Grund davon iſt klar; dann wann in einem Bruche der Zehler kleiner iſt als der Nenner, ſo iſt auch der Bruch kleiner als 1; und wann der Zehler groͤſſer iſt als der Nenner, ſo iſt der Bruch groͤſſer als 1. Ferner iſt auch zu mercken, daß zwey ſolche Bruͤche, deren Zehler und Nenner, wechſels weiſe einander gleich ſind, wann ſie mit einander multiplicirt werden, allezeit ein gantzes herausbringen; alſo [Formel] mit ¾ multiplicirt gibt 1, und 8 mit ⅛, das iſt [Formel] mit ⅛ multiplicirt gibt auch 1. Weilen nun wie oben gelehret, ein Bruch durch einen anderen Bruch diuidirt wird, wann man den Zehler des Diuidendi mit dem Nenner des Diuiſoris; ingleichen auch den Nenner des Diuidendi mit dem Zehler des Diuiſoris multiplicirt, und dann jenes Product fuͤr den Zehler des Quoti dieſes aber fuͤr den Nenner ſetzt:","norm":"Der Grund davon ist klar; dann wann in einem Bruche der Zähler kleiner ist als der Nenner, so ist auch der Bruch kleiner als 1; und wann der Zähler größer ist als der Nenner, so ist der Bruch größer als 1. Ferner ist auch zu merken, dass zwei solche Brüche, deren Zähler und Nenner, Wechsels weise einander gleich sind, wann sie miteinander multipliziert werden, allezeit ein ganzes herausbringen; also [Formel] mit ¾ multipliziert gibt 1, und 8 mit 1/8 , das ist [Formel] mit 1/8 multipliziert gibt auch 1. Weilen nun wie oben gelehrt, ein Bruch durch einen anderen Bruch dividiert wird, wann man den Zähler des Dividenden mit dem Nenner des Divisors; ingleichen auch den Nenner des Dividenden mit dem Zähler des Divisors multipliziert, und dann jenes Produkt für den Zähler des Quoti dieses aber für den Nenner setzt:","orig_tok":["Der","Grund","davon","iſt","klar",";","dann","wann","in","einem","Bruche","der","Zehler","kleiner","iſt","als","der","Nenner",",","ſo","iſt","auch","der","Bruch","kleiner","als","1",";","und","wann","der","Zehler","groͤſſer","iſt","als","der","Nenner",",","ſo","iſt","der","Bruch","groͤſſer","als","1.","Ferner","iſt","auch","zu","mercken",",","daß","zwey","ſolche","Bruͤche",",","deren","Zehler","und","Nenner",",","wechſels","weiſe","einander","gleich","ſind",",","wann","ſie","mit","einander","multiplicirt","werden",",","allezeit","ein","gantzes","herausbringen",";","alſo","[Formel]","mit","¾","multiplicirt","gibt","1",",","und","8","mit","⅛",",","das","iſt","[Formel]","mit","⅛","multiplicirt","gibt","auch","1.","Weilen","nun","wie","oben","gelehret",",","ein","Bruch","durch","einen","anderen","Bruch","diuidirt","wird",",","wann","man","den","Zehler","des","Diuidendi","mit","dem","Nenner","des","Diuiſoris",";","ingleichen","auch","den","Nenner","des","Diuidendi","mit","dem","Zehler","des","Diuiſoris","multiplicirt",",","und","dann","jenes","Product","fuͤr","den","Zehler","des","Quoti","dieſes","aber","fuͤr","den","Nenner","ſetzt",":"],"norm_tok":["Der","Grund","davon","ist","klar",";","dann","wann","in","einem","Bruche","der","Zähler","kleiner","ist","als","der","Nenner",",","so","ist","auch","der","Bruch","kleiner","als","1",";","und","wann","der","Zähler","größer","ist","als","der","Nenner",",","so","ist","der","Bruch","größer","als","1.","Ferner","ist","auch","zu","merken",",","dass","zwei","solche","Brüche",",","deren","Zähler","und","Nenner",",","Wechsels","weise","einander","gleich","sind",",","wann","sie","miteinander","multipliziert","werden",",","allezeit","ein","ganzes","herausbringen",";","also","[","Formel","]","mit","¾","multipliziert","gibt","1",",","und","8","mit"," 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+{"basename":"euler_rechenkunst01_1738","par_idx":1882,"date":1738,"orig":"ſo ſieht man leicht daß eben dieſer Quotus herauskommen werde, wann man den Diuiſorem umkehret, und damit den Diuidendum multiplicirt; als wann [Formel] durch ⅘ diuidirt werden ſoll, ſo wird nach der erſteren Regel der Quotus folgender geſtalt gefunden.","norm":"so sieht man leicht dass eben dieser Quotus herauskommen werde, wann man den Divisor umkehret, und damit den Dividenden multipliziert; als wann [Formel] durch 4/5 dividiert werden soll, so wird nach der ersteren Regel der Quotus folgender Gestalt gefunden.","orig_tok":["ſo","ſieht","man","leicht","daß","eben","dieſer","Quotus","herauskommen","werde",",","wann","man","den","Diuiſorem","umkehret",",","und","damit","den","Diuidendum","multiplicirt",";","als","wann","[Formel]","durch","⅘","diuidirt","werden","ſoll",",","ſo","wird","nach","der","erſteren","Regel","der","Quotus","folgender","geſtalt","gefunden","."],"norm_tok":["so","sieht","man","leicht","dass","eben","dieser","Quotus","herauskommen","werde",",","wann","man","den","Divisor","umkehret",",","und","damit","den","Dividenden","multipliziert",";","als","wann","[","Formel","]","durch"," ","4","/","5"," 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+{"basename":"euler_rechenkunst01_1738","par_idx":1883,"date":1738,"orig":"[Formel]","norm":"[Formel]","orig_tok":["[Formel]"],"norm_tok":["[","Formel","]"],"orig_spans":[[0,8]],"norm_spans":[[0,1],[1,7],[7,8]],"alignment":[[0,0],[0,1],[0,2]],"lang_fastText":"sh","lang_py3langid":"en","lang_cld3":"cy","orig_lang_de":0,"norm_lmscore":7.932600021362305} +{"basename":"euler_rechenkunst01_1738","par_idx":1884,"date":1738,"orig":"Nach der jetzo gegebenen Regel aber wird eben dieſer Quotus durch die Multiplication des Diuidendi mit dem ungekehrten Diniſore alſo gefunden.","norm":"Nach der jetzt gegebenen Regel aber wird eben dieser Quotus durch die Multiplikation des Dividenden mit dem ungekehrten Dinisore also gefunden.","orig_tok":["Nach","der","jetzo","gegebenen","Regel","aber","wird","eben","dieſer","Quotus","durch","die","Multiplication","des","Diuidendi","mit","dem","ungekehrten","Diniſore","alſo","gefunden","."],"norm_tok":["Nach","der","jetzt","gegebenen","Regel","aber","wird","eben","dieser","Quotus","durch","die","Multiplikation","des","Dividenden","mit","dem","ungekehrten","Dinisore","also","gefunden","."],"orig_spans":[[0,4],[5,8],[9,14],[15,24],[25,30],[31,35],[36,40],[41,45],[46,52],[53,59],[60,65],[66,69],[70,84],[85,88],[89,98],[99,102],[103,106],[107,118],[119,127],[128,132],[133,141],[141,142]],"norm_spans":[[0,4],[5,8],[9,14],[15,24],[25,30],[31,35],[36,40],[41,45],[46,52],[53,59],[60,65],[66,69],[70,84],[85,88],[89,99],[100,103],[104,107],[108,119],[120,128],[129,133],[134,142],[142,143]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":6.326700210571289} +{"basename":"euler_rechenkunst01_1738","par_idx":1885,"date":1738,"orig":"[Formel]","norm":"[Formel]","orig_tok":["[Formel]"],"norm_tok":["[","Formel","]"],"orig_spans":[[0,8]],"norm_spans":[[0,1],[1,7],[7,8]],"alignment":[[0,0],[0,1],[0,2]],"lang_fastText":"sh","lang_py3langid":"en","lang_cld3":"cy","orig_lang_de":0,"norm_lmscore":7.932600021362305} +{"basename":"euler_rechenkunst01_1738","par_idx":1886,"date":1738,"orig":"Hieraus erhellet nun eine ſchoͤne Verwandſchaft zwiſchen der Diuiſion und Multiplication, nehmlich daß durch ⅘ diuidiren eben ſo viel ſey als mit [Formel] multipliciren; und allezeit, daß durch einen jeglichen Bruch diuidiren eben ſo viel ſey als mit demſelben umgekehrten Bruche multipliciren.","norm":"Hieraus erhellet nun eine schöne Verwandtschaft zwischen der Division und Multiplikation, nämlich dass durch 4/5 dividieren ebenso viel sei als mit [Formel] multiplizieren; und allezeit, dass durch einen jeglichen Bruch dividieren ebenso viel sei als mit demselben umgekehrten Bruche multiplizieren.","orig_tok":["Hieraus","erhellet","nun","eine","ſchoͤne","Verwandſchaft","zwiſchen","der","Diuiſion","und","Multiplication",",","nehmlich","daß","durch","⅘","diuidiren","eben","ſo","viel","ſey","als","mit","[Formel]","multipliciren",";","und","allezeit",",","daß","durch","einen","jeglichen","Bruch","diuidiren","eben","ſo","viel","ſey","als","mit","demſelben","umgekehrten","Bruche","multipliciren","."],"norm_tok":["Hieraus","erhellet","nun","eine","schöne","Verwandtschaft","zwischen","der","Division","und","Multiplikation",",","nämlich","dass","durch"," ","4","/","5"," 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+{"basename":"euler_rechenkunst01_1738","par_idx":1888,"date":1738,"orig":"Dieſe Verwandlung der Diuiſion in eine Multiplication ſcheinet zwar die Operation nicht leichter zu machen, weilen auf beyde Arten einerley Multiplicationen verrichtet werden muͤſſen; allein da wir oben bey der Multiplication einige Vortheile anzubringen gelehret, dadurch das Product gleich in ſeiner kleinſten Form herausgebracht wird, ſo koͤnnen bey der Diuiſion, nachdem dieſelbe in eine Multiplication iſt verwandelt worden, eben dieſelben Vortheile angebracht werden, wodurch man der Muͤhe uͤberhoben wird fuͤr die Diuiſion beſondere Vortheile ſo wohl anzuzeigen als zu erlernen.","norm":"Diese Verwandlung der Division in eine Multiplikation scheinet zwar die Operation nicht leichter zu machen, weilen auf beide Arten einerlei Multiplikationen verrichtet werden müssen; allein da wir oben bei der Multiplikation einige Vorteile anzubringen gelehrt, dadurch das Produkt gleich in seiner kleinsten Form herausgebracht wird, so können bei der Division, nachdem dieselbe in eine Multiplikation ist verwandelt worden, eben dieselben Vorteile angebracht werden, wodurch man der Mühe überhoben wird für die Division besondere Vorteile so wohl anzuzeigen als zu 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+{"basename":"euler_rechenkunst01_1738","par_idx":1889,"date":1738,"orig":"Dieſe Vortheile bey der Multiplication beſtehen aber darinn, daß man von zwey Bruͤchen, welche mit einander multiplicirt werden ſollen, je einen Zehler gegen einem Nenner aufhebt, welches durch einen gemeinen Theiler geſchieht; da man an die Stelle derſelben Zahlen, die gefundenen Quotos ſetzet.","norm":"Diese Vorteile bei der Multiplikation bestehen aber darin, dass man von zwei Brüchen, welche miteinander multipliziert werden sollen, je einen Zähler gegen einem Nenner aufhebt, welches durch einen gemeinen Teiler geschieht; da man an die Stelle derselben Zahlen, die gefundenen Quotos 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+{"basename":"euler_rechenkunst01_1738","par_idx":1890,"date":1738,"orig":"Eben dieſes Vortheils kan man ſich alſo bey der Diuiſion bedienen, nachdem dieſelbe in eine Multiplication iſt verwandelt worden; wie aus folgenden Exempeln mit mehrerem erhellen wird.","norm":"Eben dieses Vorteils kann man sich also bei der Division bedienen, nachdem dieselbe in eine Multiplikation ist verwandelt worden; wie aus folgenden Exempeln mit mehrerem erhellen 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+{"basename":"euler_rechenkunst01_1738","par_idx":1891,"date":1738,"orig":"I.","norm":"I.","orig_tok":["I","."],"norm_tok":["I."],"orig_spans":[[0,1],[1,2]],"norm_spans":[[0,2]],"alignment":[[0,0],[1,0]],"lang_fastText":"de","lang_py3langid":"en","lang_cld3":"mg","orig_lang_de":0.333,"norm_lmscore":2.600199937820434} +{"basename":"euler_rechenkunst01_1738","par_idx":1892,"date":1738,"orig":"Wann ⅝ durch ¾ diuidirt werden ſollen, ſo wird die Diuiſion folgender geſtalt verrichtet und der Quotus gefunden werden.","norm":"Wann 5/8 durch ¾ dividiert werden sollen, so wird die Division folgender Gestalt verrichtet und der Quotus gefunden werden.","orig_tok":["Wann","⅝","durch","¾","diuidirt","werden","ſollen",",","ſo","wird","die","Diuiſion","folgender","geſtalt","verrichtet","und","der","Quotus","gefunden","werden","."],"norm_tok":["Wann"," ","5","/","8"," 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+{"basename":"euler_rechenkunst01_1738","par_idx":1893,"date":1738,"orig":"[Formel]","norm":"[Formel]","orig_tok":["[Formel]"],"norm_tok":["[","Formel","]"],"orig_spans":[[0,8]],"norm_spans":[[0,1],[1,7],[7,8]],"alignment":[[0,0],[0,1],[0,2]],"lang_fastText":"sh","lang_py3langid":"en","lang_cld3":"cy","orig_lang_de":0,"norm_lmscore":7.932600021362305} +{"basename":"euler_rechenkunst01_1738","par_idx":1894,"date":1738,"orig":"Nehmlich anſtatt des Diuiſoris ¾ ſetzt man [Formel] , damit man ⅝ mit [Formel] zu multipliciren habe.","norm":"Nämlich anstatt des Divisors ¾ setzt man [Formel], damit man 5/8 mit [Formel] zu multiplizieren habe.","orig_tok":["Nehmlich","anſtatt","des","Diuiſoris","¾","ſetzt","man","[Formel]",",","damit","man","⅝","mit","[Formel]","zu","multipliciren","habe","."],"norm_tok":["Nämlich","anstatt","des","Divisors","¾","setzt","man","[","Formel","]",",","damit","man"," ","5","/","8"," 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Hernach multiplicirt man 1 mit 5, und 3 mit 2, ſo kommt der geſuchte Quotus in ſeiner kleinſten Form nehmlich ⅚ heraus.","norm":"Alsdann sieht man, dass 4 gegen 8 durch 4 können aufgehoben werden, und setzt man also 1 anstatt 4 und 2 anstatt 8. Hernach multipliziert man 1 mit 5, und 3 mit 2, so kommt der gesuchte Quotus in seiner kleinsten Form nämlich 5/6 heraus.","orig_tok":["Alsdann","ſieht","man",",","daß","4","gegen","8","durch","4","koͤnnen","aufgehoben","werden",",","und","ſetzt","man","alſo","1","anſtatt","4","und","2","anſtatt","8.","Hernach","multiplicirt","man","1","mit","5",",","und","3","mit","2",",","ſo","kommt","der","geſuchte","Quotus","in","ſeiner","kleinſten","Form","nehmlich","⅚","heraus","."],"norm_tok":["Alsdann","sieht","man",",","dass","4","gegen","8","durch","4","können","aufgehoben","werden",",","und","setzt","man","also","1","anstatt","4","und","2","anstatt","8.","Hernach","multipliziert","man","1","mit","5",",","und","3","mit","2",",","so","kommt","der","gesuchte","Quotus","in","seiner","kleinsten","Form","nämlich"," ","5","/","6"," ","heraus","."],"orig_spans":[[0,7],[8,13],[14,17],[17,18],[19,22],[23,24],[25,30],[31,32],[33,38],[39,40],[41,48],[49,59],[60,66],[66,67],[68,71],[72,77],[78,81],[82,86],[87,88],[89,96],[97,98],[99,102],[103,104],[105,112],[113,115],[116,123],[124,136],[137,140],[141,142],[143,146],[147,148],[148,149],[150,153],[154,155],[156,159],[160,161],[161,162],[163,165],[166,171],[172,175],[176,184],[185,191],[192,194],[195,201],[202,211],[212,216],[217,225],[226,227],[228,234],[234,235]],"norm_spans":[[0,7],[8,13],[14,17],[17,18],[19,23],[24,25],[26,31],[32,33],[34,39],[40,41],[42,48],[49,59],[60,66],[66,67],[68,71],[72,77],[78,81],[82,86],[87,88],[89,96],[97,98],[99,102],[103,104],[105,112],[113,115],[116,123],[124,137],[138,141],[142,143],[144,147],[148,149],[149,150],[151,154],[155,156],[157,160],[161,162],[162,163],[164,166],[167,172],[173,176],[177,185],[186,192],[193,195],[196,202],[203,212],[213,217],[218,225],[226,227],[227,228],[228,229],[229,230],[231,232],[232,238],[238,239]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15],[16,16],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26],[27,27],[28,28],[29,29],[30,30],[31,31],[32,32],[33,33],[34,34],[35,35],[36,36],[37,37],[38,38],[39,39],[40,40],[41,41],[42,42],[43,43],[44,44],[45,45],[46,46],[47,null],[48,null],[null,47],[null,48],[null,49],[null,50],[null,51],[null,52],[49,53]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.720200061798096} +{"basename":"euler_rechenkunst01_1738","par_idx":1896,"date":1738,"orig":"II.","norm":"II.","orig_tok":["II","."],"norm_tok":["II."],"orig_spans":[[0,2],[2,3]],"norm_spans":[[0,3]],"alignment":[[0,0],[1,0]],"lang_fastText":"en","lang_py3langid":"en","lang_cld3":"xh","orig_lang_de":0,"norm_lmscore":1.316900014877319} +{"basename":"euler_rechenkunst01_1738","par_idx":1897,"date":1738,"orig":"Sollen [Formel] durch [Formel] diuidirt werden.","norm":"Sollen [Formel] durch [Formel] dividiert werden.","orig_tok":["Sollen","[Formel]","durch","[Formel]","diuidirt","werden","."],"norm_tok":["Sollen","[","Formel","]","durch","[","Formel","]","dividiert","werden","."],"orig_spans":[[0,6],[7,15],[16,21],[22,30],[31,39],[40,46],[46,47]],"norm_spans":[[0,6],[7,8],[8,14],[14,15],[16,21],[22,23],[23,29],[29,30],[31,40],[41,47],[47,48]],"alignment":[[0,0],[1,1],[1,2],[1,3],[2,4],[3,5],[3,6],[3,7],[4,8],[5,9],[6,10]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.172200202941894} +{"basename":"euler_rechenkunst01_1738","par_idx":1898,"date":1738,"orig":"Der Quotus wird alſo folgender geſtalt gefunden.","norm":"Der Quotus wird also folgender Gestalt gefunden.","orig_tok":["Der","Quotus","wird","alſo","folgender","geſtalt","gefunden","."],"norm_tok":["Der","Quotus","wird","also","folgender","Gestalt","gefunden","."],"orig_spans":[[0,3],[4,10],[11,15],[16,20],[21,30],[31,38],[39,47],[47,48]],"norm_spans":[[0,3],[4,10],[11,15],[16,20],[21,30],[31,38],[39,47],[47,48]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":6.832099914550781} +{"basename":"euler_rechenkunst01_1738","par_idx":1899,"date":1738,"orig":"[Formel]","norm":"[Formel]","orig_tok":["[Formel]"],"norm_tok":["[","Formel","]"],"orig_spans":[[0,8]],"norm_spans":[[0,1],[1,7],[7,8]],"alignment":[[0,0],[0,1],[0,2]],"lang_fastText":"sh","lang_py3langid":"en","lang_cld3":"cy","orig_lang_de":0,"norm_lmscore":7.932600021362305} +{"basename":"euler_rechenkunst01_1738","par_idx":1900,"date":1738,"orig":"Hier laſſen ſich 9 und 45 durch 9 theilen, deswegen ſchreibt man 1 an ſtatt 9, und 5 an ſtatt 45. 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