diff --git "a/validation/1700-1799/euler_rechenkunst02_1740.jsonl" "b/validation/1700-1799/euler_rechenkunst02_1740.jsonl" new file mode 100644--- /dev/null +++ "b/validation/1700-1799/euler_rechenkunst02_1740.jsonl" @@ -0,0 +1,1901 @@ +{"basename":"euler_rechenkunst02_1740","par_idx":0,"date":1740,"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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+{"basename":"euler_rechenkunst02_1740","par_idx":2,"date":1740,"orig":"Einleitung,","norm":"Einleitung,","orig_tok":["Einleitung",","],"norm_tok":["Einleitung",","],"orig_spans":[[0,10],[10,11]],"norm_spans":[[0,10],[10,11]],"alignment":[[0,0],[1,1]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":3.391799926757812} +{"basename":"euler_rechenkunst02_1740","par_idx":3,"date":1740,"orig":"DEr fuͤrnehmſte Theil der Rechen-Kunſt, welcher inſonderheit in dem gemeinen Leben gebrauchet wird, beſtehet darinn, daß man die verſchiedenen Sorten, nach welchen allerley Groͤſſen beſchrieben zu werden pflegen, ſich bekannt mache und die Regeln der Rechen-Kunſt bey denſelben anzubringen wiſſe.","norm":"Der fürnehmste Teil der Rechenkunst, welcher in Sonderheit in dem gemeinen Leben gebraucht wird, bestehet darin, dass man die verschiedenen Sorten, nach welchen allerlei Größen beschrieben zu werden pflegen, sich bekannt mache und die Regeln der Rechenkunst 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+{"basename":"euler_rechenkunst02_1740","par_idx":17,"date":1740,"orig":"Dann wann man die Vergleichung der verſchiedenen Sorten ſchon weiß, ſo iſt genug um zur vollkommenen Erkaͤntniß zu gelangen, daß man die wahre Groͤſſe einer einzigen Sorte beſtimme: in dem daraus zugleich die wahre Groͤſſe der uͤbrigen Sorten erkannt wird.","norm":"Dann wann man die Vergleichung der verschiedenen Sorten schon weiß, so ist genug um zur vollkommenen Erkenntnis zu gelangen, dass man die wahre Größe einer einzigen Sorte bestimme: in dem daraus zugleich die wahre Größe der übrigen Sorten erkannt 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+{"basename":"euler_rechenkunst02_1740","par_idx":19,"date":1740,"orig":"Solche verſchiedenen Sorten werden nun nicht nur bey den Gewichten, ſondern auch bey den meiſten andern Arten von Ausmeſſungen gebraucht; dergleichen ſind die verſchiedenen Sorten von Muͤntzen, von aller Gattungen Maaſſen, die Eintheilung und Abmeſſung der Zeit und andere dergleichen.","norm":"Solche verschiedenen Sorten werden nun nicht nur bei den Gewichten, sondern auch bei den meisten anderen Arten von Ausmessungen gebraucht; dergleichen sind die verschiedenen Sorten von münzen, von aller Gattungen Massen, die Einteilung und Abmessung der Zeit und andere 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+{"basename":"euler_rechenkunst02_1740","par_idx":20,"date":1740,"orig":"Da nun ſo wohl die Eintheilung in verſchiedene Sorten als die Groͤſſe einer Sorte fuͤr ſich ſelbſt etwas willkuͤrliches iſt, und bloß allein auf dem Gutduͤnken derjenigen, welche in einem jeglichen Land zuerſt ſolche verſchiedene Nahmen und Sorten eingefuͤhrt haben, beruhet, ſo iſt leicht zu erachten, daß in verſchiedenen Laͤndern ſo wohl die Eintheilung und Verhaͤltniß der verſchiedenen Sorten, als auch die wahre Groͤſſe an ſich ſelbſt unterſchieden ſeyn muͤſſe.","norm":"Da nun so wohl die Einteilung in verschiedene Sorten als die Größe einer Sorte für sich selbst etwas Willkürliches ist, und bloß allein auf dem Gutdünken derjenigen, welche in einem jeglichen Land zuerst solche verschiedene Namen und Sorten eingeführt haben, beruhet, so ist leicht zu erachten, dass in verschiedenen Ländern sowohl die Einteilung und Verhältnis der verschiedenen Sorten, als auch die wahre Größe an sich selbst Unterschieden sein 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+{"basename":"euler_rechenkunst02_1740","par_idx":200,"date":1740,"orig":"Wann wir nehmlich voraus ſetzen, daß man einen hinlaͤnglichen Begriff von dem Werth eines Copeken hat, ſo wird man einen eben ſo deutlichen Begriff von einer jeglichen anderwerts uͤblichen Muͤntz-Sorte erhalten, wann man erkennt, wieviel dieſelben an Copeken austragen.","norm":"Wann wir nämlich voraussetzen, dass man einen hinlänglichen Begriff von dem Wert eines Kopeken hat, so wird man einen eben so deutlichen Begriff von einer jeglichen anderwärts üblichen Münzsorte erhalten, wann man erkennt, wie viel dieselben an Kopeken 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+{"basename":"euler_rechenkunst02_1740","par_idx":201,"date":1740,"orig":"Weilen aber bey dem Werth der Muͤntzen allenthalben ſehr viel auf dem Gutduͤncken der Menſchen beruhet, ſo kan eine ſolche Vergleichung nicht ſo genau angeſtellet werden; uͤber das macht auch die Handelſchaft hierinn oͤfters groſſe Veraͤnderungen, als worinn die Veraͤnderlichkeit des Wechſel-Courſes beſtehet.","norm":"Weilen aber bei dem Wert der Münzen allenthalben sehr viel auf dem Gutdünken der Menschen beruhet, so kann eine solche Vergleichung nicht so genau angestellt werden; über das macht auch die Handelsschaft hierin öfters große Veränderungen, als worin die Veränderlichkeit des Wechselkurses 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+{"basename":"euler_rechenkunst02_1740","par_idx":202,"date":1740,"orig":"Allhier wird nehmlich bald ein Rubl mit mehr als 50 Stuͤber Hollaͤndiſch Corrent Geld, bald mit weniger gleich geſchaͤtzet; ſo daß in dieſer Vergleichung nichts feſtes geſetzt werden kan.","norm":"Allhier wird nämlich bald ein Rubel mit mehr als 50 Stüber Holländisch Corrent Geld, bald mit weniger gleich geschätzt; so dass in dieser Vergleichung nichts Festes gesetzt werden 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+{"basename":"euler_rechenkunst02_1740","par_idx":203,"date":1740,"orig":"Um aber doch einiger maſſen einen Begriff von der Verhaͤltniß der Muͤntzen unter ſich zu erhalten, ſo erwehlet man hiezu einen mitleren Preiß, welches das Pary genennet wird: und nach ſolchem kan man die verlangte Vergleichung anſtellen.","norm":"Um aber doch einigermaßen einen Begriff von der Verhältnis der Münzen unter sich zu erhalten, so erwählt man hierzu einen mittleren Preis, welches das Pari genannt wird: und nach solchem kann man die verlangte Vergleichung 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+{"basename":"euler_rechenkunst02_1740","par_idx":443,"date":1740,"orig":"Vor allen Dingen iſt nun zu mercken, daß alle Sachen, welche man durch Zahlen ausdruͤckt, entweder nach eintzelen Stuͤcken gezehlet, oder dazu verſchiedene Sorten von Maſſen gebraucht werden.","norm":"Vor allen Dingen ist nun zu merken, dass alle Sachen, welche man durch Zahlen ausdrückt, entweder nach einzeln Stücken gezählt, oder dazu verschiedene Sorten von Massen gebraucht 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+{"basename":"euler_rechenkunst02_1740","par_idx":444,"date":1740,"orig":"Wann nach eintzelen Stuͤcken gezehlet wird, ſo ſind die obbeſchriebenen Operationen hinlaͤnglich, und iſt keine beſondere Anleitung dazu noͤthig.","norm":"Wann nach einzeln Stücken gezählt wird, so sind die obbeschriebenen Operationen hinlänglich, und ist keine besondere Anleitung dazu 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+{"basename":"euler_rechenkunst02_1740","par_idx":445,"date":1740,"orig":"Alſo wann alles Geld nach einerley Muͤntz-Sorten als Copeken gezehlet werden ſollte, ſo wuͤrde es keine Schwierigkeiten haben, verſchiedene Anzahlen von Copeken entweder zu addiren oder von einander zu ſubtrahiren, ingleichem auch eine gegebene Zahl von Copeken zu multipliciren oder zu diuidiren.","norm":"Also wann alles Geld nach einerlei Münzsorten als Kopeken gezählt werden sollte, so würde es keine Schwierigkeiten haben, verschiedene Anzahlen von Kopeken entweder zu addieren oder voneinander zu subtrahieren, ingleichem auch eine gegebene Zahl von Kopeken zu multiplizieren oder zu 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+{"basename":"euler_rechenkunst02_1740","par_idx":446,"date":1740,"orig":"Eine gleiche Bewandnuͤß wuͤrde es auch mit den Gewichten haben, wann man ſich in Ausdruͤckung derſelben nur einerley Sorten als Pfunde bedienen ſollte:","norm":"Eine gleiche Bewandtnis würde es auch mit den Gewichten haben, wann man sich in Ausdrückung derselben nur einerlei Sorten als Pfunde bedienen 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+{"basename":"euler_rechenkunst02_1740","par_idx":447,"date":1740,"orig":"und dieſes iſt von allen Gattungen Maaſſen zu verſtehen, wann beſtaͤndig durch die Unitaͤt 1 einerley Maaß angedeutet wird.","norm":"und dieses ist von allen Gattungen Massen zu verstehen, wann beständig durch die Unität 1 einerlei Maß angedeutet 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+{"basename":"euler_rechenkunst02_1740","par_idx":448,"date":1740,"orig":"Es iſt aber bey allen Rechnungen ſehr gebraͤuchlich, daß einerley Quantitaͤten durch verſchiedene Sorten ausgedruͤckt werden: alſo wird das Geld allhier nach Rubeln, Griven, Copeken und Poluſchken; das Gewicht nach Puden, Pfunden, Lothen und Solomiken; die Zeit nach Jahren, Monathen, Wochen, Tagen, Stunden, Minuten und Secunden beſchrieben.","norm":"Es ist aber bei allen Rechnungen sehr gebräuchlich, dass einerlei Quantitäten durch verschiedene Sorten ausgedrückt werden: also wird das Geld allhier nach Rubeln, griffen, Kopeken und Poluschken; das Gewicht nach Buden, Pfunden, Loten und Solomicken; die Zeit nach Jahren, Monaten, Wochen, Tagen, Stunden, Minuten und Sekunden 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+{"basename":"euler_rechenkunst02_1740","par_idx":457,"date":1740,"orig":"Der Endzweck dieſer Operation aber iſt zweyfach; dann erſtlich erhaͤlt man dadurch, daß die verſchiedenen Benennungen gehoben und auf einerley Nahmen oder Sorten gebracht, und ſolchergeſtalt als ſimple Zahlen tractirt werden koͤnnen.","norm":"Der Endzweck dieser Operation aber ist zweifach; dann erstlich erhält man dadurch, dass die verschiedenen Benennungen gehoben und auf einerlei Namen oder Sorten gebracht, und solchergestalt als simple Zahlen traktiert werden 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+{"basename":"euler_rechenkunst02_1740","par_idx":459,"date":1740,"orig":"Dann wann uns zum Exempel ein Gewicht vorgelegt worden, welches 12 Pud, 10 Pfund, 22 Loth und 1 Solotnick gewogen; ſo koͤnnen dieſe verſchiedenen Sorten auf mehr als eine Weiſe unter einerley Nahmen gebracht werden; dann erſtlich kan ich das gantze Gewicht nur allein nach Puden durch Huͤlfe der Bruͤche ausdrucken, und ſagen, daß dieſes Gewicht halte 12 [Formel] Pud; wobey der Nahme Pud allein vorkommt.","norm":"Dann wann uns zum Exempel ein Gewicht vorgelegt worden, welches 12 Pud, 10 Pfund, 22 Lot und 1 Solotnick gewogen; so können diese verschiedenen Sorten auf mehr als eine Weise unter einerlei Namen gebracht werden; dann erstlich kann ich das ganze Gewicht nur allein nach Buden durch Hilfe der Brüche ausdrücken, und sagen, dass dieses Gewicht halte 12 [Formel] Pud; wobei der Nahme Pud allein 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Gewichts nach Lothen angezeigt werden, da daſſelbe dann halten wird 15702⅓ Loth.","norm":"Drittens kann auch diese Schwere dieses Gewichts nach Loten angezeigt werden, da dasselbe dann halten wird 15702 1/3 Lot.","orig_tok":["Drittens","kan","auch","dieſe","Schwehre","dieſes","Gewichts","nach","Lothen","angezeigt","werden",",","da","daſſelbe","dann","halten","wird","15702⅓","Loth","."],"norm_tok":["Drittens","kann","auch","diese","Schwere","dieses","Gewichts","nach","Loten","angezeigt","werden",",","da","dasselbe","dann","halten","wird","15702","1","/","3"," ","Lot","."],"orig_spans":[[0,8],[9,12],[13,17],[18,23],[24,32],[33,39],[40,48],[49,53],[54,60],[61,70],[71,77],[77,78],[79,81],[82,90],[91,95],[96,102],[103,107],[108,114],[115,119],[119,120]],"norm_spans":[[0,8],[9,13],[14,18],[19,24],[25,32],[33,39],[40,48],[49,53],[54,59],[60,69],[70,76],[76,77],[78,80],[81,89],[90,94],[95,101],[102,106],[107,112],[113,114],[114,115],[115,116],[117,118],[118,121],[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],[null,18],[null,19],[null,20],[null,21],[18,22],[19,23]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":6.889699935913086} +{"basename":"euler_rechenkunst02_1740","par_idx":462,"date":1740,"orig":"Wann man endlich wiſſen will, wieviel Solotnick dieſes Gewicht enthalte, ſo findt man 47107 Solotnick.","norm":"Wann man endlich wissen will, wie viel Solotnick dieses Gewicht enthalte, so findet man 47107 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verlangt dieſes vorgegebene Gewicht in einerley Sorte ausgedruckt zu haben, ſo kan ſolches entweder nach Puden, oder Pfunden oder Lothen oder Solotnicken geſchehen; wann man aber zugleich eine gantze Zahl ohne Bruͤche verlangt, ſo muß ſolches in der kleinſten Sorte, welche vorkommt, nehmlich in Solotnicken geſchehen.","norm":"Wann man also nur verlangt dieses vorgegebene Gewicht in einerlei Sorte ausgedrückt zu haben, so kann solches entweder nach Buden, oder Pfunden oder Loten oder Solotnick geschehen; wann man aber zugleich eine ganze Zahl ohne Brüche verlangt, so muss solches in der kleinsten Sorte, welche vorkommt, nämlich in Solotnick 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+{"basename":"euler_rechenkunst02_1740","par_idx":465,"date":1740,"orig":"Hernach wollen wir doch auch die Regeln vorbringen, vermittelſt welcher eine beſchriebene Quantitaͤt auf einerley Sorte, ſo nicht die kleinſte iſt, gebracht werden kan: als worauf die Wechſel Rechnung beruhet, darinn Muͤntz-Sorten von verſchiedenen Landen unter ſich verglichen, und nach einer beliebigen Art ausgedruͤckt werden.","norm":"Hernach wollen wir doch auch die Regeln vorbringen, vermittels welcher eine beschriebene Quantität auf einerlei Sorte, so nicht die kleinste ist, gebracht werden kann: als worauf die Wechsel Rechnung beruhet, darin Münzsorten von verschiedenen Landen unter sich verglichen, und nach einer beliebigen Art ausgedrückt 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+{"basename":"euler_rechenkunst02_1740","par_idx":466,"date":1740,"orig":"Wann aber eine ſolche Quantitaͤt, welche man nach verſchiedenen Sorten auszuſprechen pflegt, in einerley etwa groͤſſeren oder kleineren Sorten ausgedruͤckt wird, mit oder ohne Bruͤche, ſo lehret die Reduction, wie man daraus hinwiedexum dieſelbe Quantitaͤt nach verſchiedenen Sorten auf gewoͤhnliche Art anzeigen ſoll.","norm":"Wann aber eine solche Quantität, welche man nach verschiedenen Sorten auszusprechen pflegt, in einerlei etwa größeren oder kleineren Sorten ausgedrückt wird, mit oder ohne Brüche, so lehrt die Reduktion, wie man daraus hinwiedexum dieselbe Quantität nach verschiedenen Sorten auf gewöhnliche Art anzeigen 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+{"basename":"euler_rechenkunst02_1740","par_idx":471,"date":1740,"orig":"Nun wiſſen wir aus dem Gebrauch, daß 32 Loth auf ein Pfund gehen, und multipliciren alſo nach Anweiſung der gegebenen Regel die vorgegebene Anzahl Pfund nehmlich 15 mit 32.","norm":"Nun wissen wir aus dem Gebrauch, dass 32 Lot auf ein Pfund gehen, und multiplizieren also nach Anweisung der gegebenen Regel die vorgegebene Anzahl Pfund nämlich 15 mit 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+{"basename":"euler_rechenkunst02_1740","par_idx":472,"date":1740,"orig":"Das Product nehmlich 480 zeigt uns alsdann die verlangte Anzahl von Lothen; und ſagen alſo, daß 480 Loth eben ſo viel iſt als 15 Pfund.","norm":"Das Produkt nämlich 480 zeigt uns alsdann die verlangte Anzahl von Loten; und sagen also, dass 480 Lot ebenso viel ist als 15 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+{"basename":"euler_rechenkunst02_1740","par_idx":473,"date":1740,"orig":"Der Grund dieſer Regel iſt leicht aus dieſem Exempel einzuſehen:","norm":"Der Grund dieser Regel ist leicht aus diesem Exempel einzusehen:","orig_tok":["Der","Grund","dieſer","Regel","iſt","leicht","aus","dieſem","Exempel","einzuſehen",":"],"norm_tok":["Der","Grund","dieser","Regel","ist","leicht","aus","diesem","Exempel","einzusehen",":"],"orig_spans":[[0,3],[4,9],[10,16],[17,22],[23,26],[27,33],[34,37],[38,44],[45,52],[53,63],[63,64]],"norm_spans":[[0,3],[4,9],[10,16],[17,22],[23,26],[27,33],[34,37],[38,44],[45,52],[53,63],[63,64]],"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":4.92579984664917} +{"basename":"euler_rechenkunst02_1740","par_idx":474,"date":1740,"orig":"dann da ein Pfund 32 Loth in ſich enthaͤlt, ſo ſo ſind 2 Pfund ſo viel als 2 mal 32 Loth, und 3 Pfund ſo viel als 3 mal 32 Loth, und 4 Pfund ſo viel als 4 mal 32 Loth und ſo weiter.","norm":"dann da ein Pfund 32 Lot in sich enthält, so so sind 2 Pfund so viel als 2 Mal 32 Lot, und 3 Pfund so viel als 3 Mal 32 Lot, und 4 Pfund so viel als 4 Mal 32 Lot und so 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+{"basename":"euler_rechenkunst02_1740","par_idx":475,"date":1740,"orig":"Hieraus werden alſo 15 Pfund ſo viel ſeyn als 15 mal 32 Loth; woraus folgt, daß man, um zu finden wieviel Loth in 15 Pfunden enthalten ſind, die Anzahl der Pfunden nehmlich 15 mit 32 als der Anzahl der Lothe, welche auf ein Pfund gehen, multipliciren muͤſſe.","norm":"Hieraus werden also 15 Pfund so viel sein als 15 Mal 32 Lot; woraus folgt, dass man, um zu finden wie viel Lot in 15 Pfunden enthalten sind, die Anzahl der Pfunden nämlich 15 mit 32 als der Anzahl der Lote, welche auf ein Pfund gehen, multiplizieren 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+{"basename":"euler_rechenkunst02_1740","par_idx":476,"date":1740,"orig":"Dieſe Regel er��trecket ſich nun gleichergeſtalt auf alle andere Gattungen von groͤſſeren und kleineren Sorten, und koͤnnen vermittelſt derſelben nachfolgende Aufgaben leicht ausgerechnet werden.","norm":"Diese Regel erstrecket sich nun gleichergestalt auf alle andere Gattungen von größeren und kleineren Sorten, und können vermittels derselben nachfolgende Aufgaben leicht ausgerechnet 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solchen Exempeln muss nämlich der völlige Quotient genommen und zu dem nach den Regeln der Division gefundenen Quoto in ganzen Zahlen noch der Bruch, dessen Zähler der übergebliebene Rest der Nenner aber der Divisor ist, hinzugesetzt 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Weilen ein Grad in ſich begreifft 104½ Werſt, ſo muß man 513 durch 104½ dividiren:","norm":"Antw. Weilen ein Grad in sich begreift 1041/2 Wirst, so muss man 513 durch 1041/2 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Erſtlich muͤſſen die Solotnick auf Loth gebracht werden, weilen alſo 3 Solotnick auf 1 Loth gehen, ſo dividire man die zwey Solotnick ſo vorhanden ſind durch 3, der Quotient, der ⅔ ſeyn wird, zeigt an daß 2 Solotnick ſo viel ſind als ⅔ Loth, und alſo haben wir 14 Pfund und 22⅔ Loth, ſtatt des vorgegebenen Gewichts, und alſo nur noch zwey Benennungen oder Sorten nehmlich Pfund und Loth.","norm":"Antw. Erstlich müssen die Solotnick auf Lot gebracht werden, weilen also 3 Solotnick auf 1 Lot gehen, so dividiere man die zwei Solotnick so vorhanden sind durch 3, der Quotient, der 2/3 sein wird, zeigt an dass 2 Solotnick so viel sind als 2/3 Lot, und also haben wir 14 Pfund und 22 2/3 Lot, statt des vorgegebenen Gewichts, und also nur noch zwei Benennungen oder Sorten nämlich Pfund und Lot.","orig_tok":["Antw.","Erſtlich","muͤſſen","die","Solotnick","auf","Loth","gebracht","werden",",","weilen","alſo","3","Solotnick","auf","1","Loth","gehen",",","ſo","dividire","man","die","zwey","Solotnick","ſo","vorhanden","ſind","durch","3",",","der","Quotient",",","der","⅔","ſeyn","wird",",","zeigt","an","daß","2","Solotnick","ſo","viel","ſind","als","⅔","Loth",",","und","alſo","haben","wir","14","Pfund","und","22⅔","Loth",",","ſtatt","des","vorgegebenen","Gewichts",",","und","alſo","nur","noch","zwey","Benennungen","oder","Sorten","nehmlich","Pfund","und","Loth","."],"norm_tok":["Antw",".","Erstlich","müssen","die","Solotnick","auf","Lot","gebracht","werden",",","weilen","also","3","Solotnick","auf","1","Lot","gehen",",","so","dividiere","man","die","zwei","Solotnick","so","vorhanden","sind","durch","3",",","der","Quotient",",","der"," ","2","/","3"," ","sein","wird",",","zeigt","an","dass","2","Solotnick","so","viel","sind","als"," ","2","/","3"," ","Lot",",","und","also","haben","wir","14","Pfund","und","22","2","/","3"," ","Lot",",","statt","des","vorgegebenen","Gewichts",",","und","also","nur","noch","zwei","Benennungen","oder","Sorten","nämlich","Pfund","und","Lot","."],"orig_spans":[[0,5],[6,14],[15,22],[23,26],[27,36],[37,40],[41,45],[46,54],[55,61],[61,62],[63,69],[70,74],[75,76],[77,86],[87,90],[91,92],[93,97],[98,103],[103,104],[105,107],[108,116],[117,120],[121,124],[125,129],[130,139],[140,142],[143,152],[153,157],[158,163],[164,165],[165,166],[167,170],[171,179],[179,180],[181,184],[185,186],[187,191],[192,196],[196,197],[198,203],[204,206],[207,210],[211,212],[213,222],[223,225],[226,230],[231,235],[236,239],[240,241],[242,246],[246,247],[248,251],[252,256],[257,262],[263,266],[267,269],[270,275],[276,279],[280,283],[284,288],[288,289],[290,295],[296,299],[300,312],[313,321],[321,322],[323,326],[327,331],[332,335],[336,340],[341,345],[346,357],[358,362],[363,369],[370,378],[379,384],[385,388],[389,393],[393,394]],"norm_spans":[[0,4],[4,5],[6,14],[15,21],[22,25],[26,35],[36,39],[40,43],[44,52],[53,59],[59,60],[61,67],[68,72],[73,74],[75,84],[85,88],[89,90],[91,94],[95,100],[100,101],[102,104],[105,114],[115,118],[119,122],[123,127],[128,137],[138,140],[141,150],[151,155],[156,161],[162,163],[163,164],[165,168],[169,177],[177,178],[179,182],[183,184],[184,185],[185,186],[186,187],[188,189],[189,193],[194,198],[198,199],[200,205],[206,208],[209,213],[214,215],[216,225],[226,228],[229,233],[234,238],[239,242],[243,244],[244,245],[245,246],[246,247],[248,249],[249,252],[252,253],[254,257],[258,262],[263,268],[269,272],[273,275],[276,281],[282,285],[286,288],[289,290],[290,291],[291,292],[293,294],[294,297],[297,298],[299,304],[305,308],[309,321],[322,330],[330,331],[332,335],[336,340],[341,344],[345,349],[350,354],[355,366],[367,371],[372,378],[379,386],[387,392],[393,396],[397,400],[400,401]],"alignment":[[0,0],[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],[null,36],[null,37],[null,38],[null,39],[null,40],[35,null],[36,41],[37,42],[38,43],[39,44],[40,45],[41,46],[42,47],[43,48],[44,49],[45,50],[46,51],[47,52],[48,null],[49,null],[null,53],[null,54],[null,55],[null,56],[null,57],[null,58],[50,59],[51,60],[52,61],[53,62],[54,63],[55,64],[56,65],[57,66],[58,67],[58,68],[null,69],[null,70],[null,71],[59,72],[60,73],[61,74],[62,75],[63,76],[64,77],[65,78],[66,79],[67,80],[68,81],[69,82],[70,83],[71,84],[72,85],[73,86],[74,87],[75,88],[76,89],[77,90],[78,91]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.619699954986572} +{"basename":"euler_rechenkunst02_1740","par_idx":666,"date":1740,"orig":"Die 22⅔ Loth muͤſſen ferner zu Pfunden gebracht werden, welches geſchicht, wann man 22⅔ dividirt durch 32, da dann der Quotient [Formel] anzeigt, daß 22⅔ Loth ſo viel ſind als [Formel] ℔. Weilen nun 14 Pfund wuͤrcklich vorhanden ſind, ſo haben wir in allem 14 [Formel] Pfund, welches ſo viel iſt als das vorgebene Gewicht 14 Pfund, 22 Loth und 2 Solotnick.","norm":"Die 22 2/3 Lot müssen ferner zu Pfunden gebracht werden, welches geschieht, wann man 22 2/3 dividiert durch 32, da dann der Quotient [Formel] anzeigt, dass 22 2/3 Lot so viel sind als [Formel] ℔. Weilen nun 14 Pfund wirklich vorhanden sind, so haben wir in allem 14 [Formel] Pfund, welches so viel ist als das Vorgeben Gewicht 14 Pfund, 22 Lot und 2 Solotnick.","orig_tok":["Die","22⅔","Loth","muͤſſen","ferner","zu","Pfunden","gebracht","werden",",","welches","geſchicht",",","wann","man","22⅔","dividirt","durch","32",",","da","dann","der","Quotient","[Formel]","anzeigt",",","daß","22⅔","Loth","ſo","viel","ſind","als","[Formel]","℔.","Weilen","nun","14","Pfund","wuͤrcklich","vorhanden","ſind",",","ſo","haben","wir","in","allem","14","[Formel]","Pfund",",","welches","ſo","viel","iſt","als","das","vorgebene","Gewicht","14","Pfund",",","22","Loth","und","2","Solotnick","."],"norm_tok":["Die","22","2","/","3"," ","Lot","müssen","ferner","zu","Pfunden","gebracht","werden",",","welches","geschieht",",","wann","man","22","2","/","3"," ","dividiert","durch","32",",","da","dann","der","Quotient","[","Formel","]","anzeigt",",","dass","22","2","/","3"," 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+{"basename":"euler_rechenkunst02_1740","par_idx":702,"date":1740,"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_rechenkunst02_1740","par_idx":703,"date":1740,"orig":"VII.","norm":"VII.","orig_tok":["VII."],"norm_tok":["VII","."],"orig_spans":[[0,4]],"norm_spans":[[0,3],[3,4]],"alignment":[[0,0],[0,1]],"lang_fastText":"en","lang_py3langid":"en","lang_cld3":"la","orig_lang_de":0,"norm_lmscore":3.952300071716308} +{"basename":"euler_rechenkunst02_1740","par_idx":704,"date":1740,"orig":"Einer hat an Silber nach dem Coͤllniſchen Silber-Gewicht 21¾ Mark, 27⅔ Unzen, 12½ Loth, 18⅝ Quintl. 23⅕ Engliſch und 48 Aeß, und will dieſes Gewicht in Lothen ausgedruͤckt haben.","norm":"Einer hat an Silber nach dem kölnischen Silbergewicht 213 / 4 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sollen.","orig_tok":["Gemeiniglich","wird","zwar","dieſe","Operation","nur","auf","die","kleinſte","Sorte","gezogen",",","und","lehret","nur","die","groͤſſern","Sorten","in","kleinere","verwandeln",";","allein","da","oͤffters","die","Rechnungen","nicht","wenig","abgekuͤrtzet","werden","koͤnnen",",","wann","man","die","verſchiedenen","Sorten","nicht","ſo","wohl","in","die","kleinſte","als","in","eine","andere","verwandelt",",","ſo","haben","wir","allhier","der","Reſolution","eine","groͤſſere","Ausdehnung","gegeben",",","und","darinn","gelehrt",",","wie","vielerley","Sorten","auf","eine","einige","Sorte","gebracht","werden","ſollen","."],"norm_tok":["Gemeiniglich","wird","zwar","diese","Operation","nur","auf","die","kleinste","Sorte","gezogen",",","und","lehrt","nur","die","größeren","Sorten","in","kleinere","verwandeln",";","allein","da","öfters","die","Rechnungen","nicht","wenig","abgekürzt","werden","können",",","wann","man","die","verschiedenen","Sorten","nicht","so","wohl","in","die","kleinste","als","in","eine","andere","verwandelt",",","so","haben","wir","allhier","der","Resolution","eine","größere","Ausdehnung","gegeben",",","und","darin","gelehrt",",","wie","vielerlei","Sorten","auf","eine","einige","Sorte","gebracht","werden","sollen","."],"orig_spans":[[0,12],[13,17],[18,22],[23,28],[29,38],[39,42],[43,46],[47,50],[51,59],[60,65],[66,73],[73,74],[75,78],[79,85],[86,89],[90,93],[94,103],[104,110],[111,113],[114,122],[123,133],[133,134],[135,141],[142,144],[145,153],[154,157],[158,168],[169,174],[175,180],[181,193],[194,200],[201,208],[208,209],[210,214],[215,218],[219,222],[223,236],[237,243],[244,249],[250,252],[253,257],[258,260],[261,264],[265,273],[274,277],[278,280],[281,285],[286,292],[293,303],[303,304],[305,307],[308,313],[314,317],[318,325],[326,329],[330,340],[341,345],[346,355],[356,366],[367,374],[374,375],[376,379],[380,386],[387,394],[394,395],[396,399],[400,409],[410,416],[417,420],[421,425],[426,432],[433,438],[439,447],[448,454],[455,461],[461,462]],"norm_spans":[[0,12],[13,17],[18,22],[23,28],[29,38],[39,42],[43,46],[47,50],[51,59],[60,65],[66,73],[73,74],[75,78],[79,84],[85,88],[89,92],[93,101],[102,108],[109,111],[112,120],[121,131],[131,132],[133,139],[140,142],[143,149],[150,153],[154,164],[165,170],[171,176],[177,186],[187,193],[194,200],[200,201],[202,206],[207,210],[211,214],[215,228],[229,235],[236,241],[242,244],[245,249],[250,252],[253,256],[257,265],[266,269],[270,272],[273,277],[278,284],[285,295],[295,296],[297,299],[300,305],[306,309],[310,317],[318,321],[322,332],[333,337],[338,345],[346,356],[357,364],[364,365],[366,369],[370,375],[376,383],[383,384],[385,388],[389,398],[399,405],[406,409],[410,414],[415,421],[422,427],[428,436],[437,443],[444,450],[450,451]],"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]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.215199947357178} +{"basename":"euler_rechenkunst02_1740","par_idx":711,"date":1740,"orig":"6.)","norm":"6.)","orig_tok":["6.",")"],"norm_tok":["6.",")"],"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":6.128200054168701} +{"basename":"euler_rechenkunst02_1740","par_idx":712,"date":1740,"orig":"Eine Quantitaͤt wird auf gewoͤhnliche Art in verſchiedenen Sorten ausgedruͤckt, wann erſtlich von keiner kleineren Sorte ſo viel oder mehr Stuͤcke vorkommen, als ein Stuͤck von der naͤchſtfolgenden groͤſſeren Sorte ausmachen.","norm":"Eine Quantität wird auf gewöhnliche Art in verschiedenen Sorten ausgedrückt, wann erstlich von keiner kleineren Sorte so viel oder mehr Stücke vorkommen, als ein Stück von der nächstfolgenden größeren Sorte ausmachen.","orig_tok":["Eine","Quantitaͤt","wird","auf","gewoͤhnliche","Art","in","verſchiedenen","Sorten","ausgedruͤckt",",","wann","erſtlich","von","keiner","kleineren","Sorte","ſo","viel","oder","mehr","Stuͤcke","vorkommen",",","als","ein","Stuͤck","von","der","naͤchſtfolgenden","groͤſſeren","Sorte","ausmachen","."],"norm_tok":["Eine","Quantität","wird","auf","gewöhnliche","Art","in","verschiedenen","Sorten","ausgedrückt",",","wann","erstlich","von","keiner","kleineren","Sorte","so","viel","oder","mehr","Stücke","vorkommen",",","als","ein","Stück","von","der","nächstfolgenden","größeren","Sorte","ausmachen","."],"orig_spans":[[0,4],[5,15],[16,20],[21,24],[25,37],[38,41],[42,44],[45,58],[59,65],[66,78],[78,79],[80,84],[85,93],[94,97],[98,104],[105,114],[115,120],[121,123],[124,128],[129,133],[134,138],[139,146],[147,156],[156,157],[158,161],[162,165],[166,172],[173,176],[177,180],[181,197],[198,208],[209,214],[215,224],[224,225]],"norm_spans":[[0,4],[5,14],[15,19],[20,23],[24,35],[36,39],[40,42],[43,56],[57,63],[64,75],[75,76],[77,81],[82,90],[91,94],[95,101],[102,111],[112,117],[118,120],[121,125],[126,130],[131,135],[136,142],[143,152],[152,153],[154,157],[158,161],[162,167],[168,171],[172,175],[176,191],[192,200],[201,206],[207,216],[216,217]],"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":5.259900093078613} +{"basename":"euler_rechenkunst02_1740","par_idx":713,"date":1740,"orig":"Zweytens wird auch erfordert, daß die Anzahl von einer jeglichen Sorte eine gantze Zahl ſey, nur die kleinſte Sorte ausgenommen, bey welcher Brüche vorkommen koͤnnen.","norm":"Zweitens wird auch erfordert, dass die Anzahl von einer jeglichen Sorte eine ganze Zahl sei, nur die kleinste Sorte ausgenommen, bei welcher Brüche vorkommen können.","orig_tok":["Zweytens","wird","auch","erfordert",",","daß","die","Anzahl","von","einer","jeglichen","Sorte","eine","gantze","Zahl","ſey",",","nur","die","kleinſte","Sorte","ausgenommen",",","bey","welcher","Brüche","vorkommen","koͤnnen","."],"norm_tok":["Zweitens","wird","auch","erfordert",",","dass","die","Anzahl","von","einer","jeglichen","Sorte","eine","ganze","Zahl","sei",",","nur","die","kleinste","Sorte","ausgenommen",",","bei","welcher","Brüche","vorkommen","können","."],"orig_spans":[[0,8],[9,13],[14,18],[19,28],[28,29],[30,33],[34,37],[38,44],[45,48],[49,54],[55,64],[65,70],[71,75],[76,82],[83,87],[88,91],[91,92],[93,96],[97,100],[101,109],[110,115],[116,127],[127,128],[129,132],[133,140],[141,147],[148,157],[158,165],[165,166]],"norm_spans":[[0,8],[9,13],[14,18],[19,28],[28,29],[30,34],[35,38],[39,45],[46,49],[50,55],[56,65],[66,71],[72,76],[77,82],[83,87],[88,91],[91,92],[93,96],[97,100],[101,109],[110,115],[116,127],[127,128],[129,132],[133,140],[141,147],[148,157],[158,164],[164,165]],"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":4.861499786376953} +{"basename":"euler_rechenkunst02_1740","par_idx":714,"date":1740,"orig":"Die vielerley Sorten von Muͤntzen, Gewicht, und Maaß ſind nicht nur aus bloſſer Gewohnheit angenommen und in Gebrauch gebracht worden, ſondern die Bequemlichkeit im Zehlen und Rechnen ſcheinet inſonderheit den Alten hiezu Gelegenheit gegeben zu haben.","norm":"Die vielerlei Sorten von münzen, Gewicht, und Maß sind nicht nur aus bloßer Gewohnheit angenommen und in Gebrauch gebracht worden, sondern die Bequemlichkeit im Zählen und Rechnen scheinet in Sonderheit den Alten hierzu Gelegenheit gegeben zu haben.","orig_tok":["Die","vielerley","Sorten","von","Muͤntzen",",","Gewicht",",","und","Maaß","ſind","nicht","nur","aus","bloſſer","Gewohnheit","angenommen","und","in","Gebrauch","gebracht","worden",",","ſondern","die","Bequemlichkeit","im","Zehlen","und","Rechnen","ſcheinet","inſonderheit","den","Alten","hiezu","Gelegenheit","gegeben","zu","haben","."],"norm_tok":["Die","vielerlei","Sorten","von","münzen",",","Gewicht",",","und","Maß","sind","nicht","nur","aus","bloßer","Gewohnheit","angenommen","und","in","Gebrauch","gebracht","worden",",","sondern","die","Bequemlichkeit","im","Zählen","und","Rechnen","scheinet","in","Sonderheit","den","Alten","hierzu","Gelegenheit","gegeben","zu","haben","."],"orig_spans":[[0,3],[4,13],[14,20],[21,24],[25,33],[33,34],[35,42],[42,43],[44,47],[48,52],[53,57],[58,63],[64,67],[68,71],[72,79],[80,90],[91,101],[102,105],[106,108],[109,117],[118,126],[127,133],[133,134],[135,142],[143,146],[147,161],[162,164],[165,171],[172,175],[176,183],[184,192],[193,205],[206,209],[210,215],[216,221],[222,233],[234,241],[242,244],[245,250],[250,251]],"norm_spans":[[0,3],[4,13],[14,20],[21,24],[25,31],[31,32],[33,40],[40,41],[42,45],[46,49],[50,54],[55,60],[61,64],[65,68],[69,75],[76,86],[87,97],[98,101],[102,104],[105,113],[114,122],[123,129],[129,130],[131,138],[139,142],[143,157],[158,160],[161,167],[168,171],[172,179],[180,188],[189,191],[192,202],[203,206],[207,212],[213,219],[220,231],[232,239],[240,242],[243,248],[248,249]],"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],[31,32],[32,33],[33,34],[34,35],[35,36],[36,37],[37,38],[38,39],[39,40]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.571599960327148} +{"basename":"euler_rechenkunst02_1740","par_idx":715,"date":1740,"orig":"Allem Anſehen nach iſt der Endzweck, welchen man bey Einfuͤhrung ſo vielerley Sorten gehabt haben mag, zweyfach geweſen: erſtlich und fuͤrnehmlich im Zehlen und Rechnen ſo viel als moͤglich die Bruͤche zu vermeiden; und zweytens um allzugroſſer Zahlen uͤberhoben zu ſeyn: welches beydes bey dem gemeinen Mann, ſo im Rechnen nicht geuͤbt iſt, kein geringer Vortheil iſt.","norm":"Allem Ansehen nach ist der Endzweck, welchen man bei Einführung so vielerlei Sorten gehabt haben mag, zweifach gewesen: erstlich und vornehmlich im Zählen und Rechnen so viel als möglich die Brüche zu vermeiden; und zweitens um allzu großer Zahlen überhoben zu sein: welches beides bei dem gemeinen Mann, so im Rechnen nicht geübt ist, kein geringer Vorteil 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+{"basename":"euler_rechenkunst02_1740","par_idx":716,"date":1740,"orig":"Zu Vermeidung der Bruͤche ſind alſo die kleineren Sorten erdacht und in Gebrauch gebracht worden: dann wann man ſich nnr bey einer jeglichen Ausmeſſung der groͤſſeren Sorten bedienen wollte, ſo wuͤrde man ſo oft auf Bruͤche gerathen, als weniger als ein gantzes Stuͤck von derſelben Sorte vorkommt.","norm":"Zu Vermeidung der Brüche sind also die kleineren Sorten erdacht und in Gebrauch gebracht worden: dann wann man sich nur bei einer jeglichen Ausmessung der größeren Sorten bedienen wollte, so würde man so oft auf Brüche geraten, als weniger als ein ganzes Stück von derselben Sorte 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+{"basename":"euler_rechenkunst02_1740","par_idx":717,"date":1740,"orig":"Als wann man allhier zu Berechnung des Gelds keine andere Sorte oder keinen andern Nahmen als Rubl haͤtte, ſo wuͤrden wenig Rechnungen ohne Bruͤche vollfuͤhret werden koͤnnen.","norm":"Als wann man allhier zu Berechnung des Gelds keine andere Sorte oder keinen anderen Namen als Rubel hätte, so würden wenig Rechnungen ohne Brüche vollführet werden 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+{"basename":"euler_rechenkunst02_1740","par_idx":718,"date":1740,"orig":"Die Bruͤche nun zu vermeiden, ſind die kleineren Sorten als Griwen, Altin, Copeken, Denuſchken und Poluſchken ſehr dienlich, in dem man dadurch, ſo oft kein gantzer Rubl vorkommt, den Werth davon in dieſen kleineren Sorten gemeiniglich ohne Bruͤche anzeigen kan.","norm":"Die Brüche nun zu vermeiden, sind die kleineren Sorten als Griwen, Altin, Kopeken, Denuschken und Poluschken sehr dienlich, in dem man dadurch, so oft kein ganzer Rubel vorkommt, den Wert davon in diesen kleineren Sorten gemeiniglich ohne Brüche anzeigen 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+{"basename":"euler_rechenkunst02_1740","par_idx":719,"date":1740,"orig":"Kan aber ſolches nicht gaͤntzlich ohne Bruͤche geſchehen, ſo gewinnt man dadurch doch ſo viel, daß der Bruch nur zur kleinſten Sorte kommt, und folglich aus weit kleineren Zahlen beſteht.","norm":"Kann aber solches nicht gänzlich ohne Brüche geschehen, so gewinnt man dadurch doch so viel, dass der Bruch nur zur kleinsten Sorte kommt, und folglich aus weit kleineren Zahlen 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+{"basename":"euler_rechenkunst02_1740","par_idx":720,"date":1740,"orig":"Uber das wird auch auf einen ſolchen Bruch, welcher nur Theile von Poluſchken als der kleinſten Sorte enthaͤlt, im Rechnen oͤffters gar nicht geſehen, in Auszahlung des Gelds aber gantz und gar nicht in Acht genommen.","norm":"Über das wird auch auf einen solchen Bruch, welcher nur Teile von Poluschken als der kleinsten Sorte enthält, im Rechnen öfters gar nicht gesehen, in Auszahlung des Gelds aber ganz und gar nicht in Acht 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+{"basename":"euler_rechenkunst02_1740","par_idx":722,"date":1740,"orig":"Dann obgleich noch ein Drittel Poluſchken vorkommt, ſo wird ſolcher niemand groſſe Schwie igkeiten verurſachen.","norm":"Dann obgleich noch ein Drittel Poluschken vorkommt, so wird solcher niemand große Schwie igkeiten 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Bruͤche eingefuͤhret ſind, ſo koͤnnte man auf die Gedancken gerathen, als wann es bequemer ſeyn wuͤrde, wann man ſich nur allein der kleinſten Sorten bey einer jeglichen Rechnung bedienen ſollte, in dem man ſolcher geſtalt ſelten in Bruͤche verfallen wuͤrde, und wann auch dieſes geſchaͤhe, dieſelben ohne groſſe Gefahr verwerfen koͤnnte.","norm":"Da nun die kleineren Sorten zu Vermeidung der Brüche eingeführt sind, so könnte man auf die Gedanken geraten, als wann es bequemer sein würde, wann man sich nur allein der kleinsten Sorten bei einer jeglichen Rechnung bedienen sollte, in dem man solcher Gestalt selten in Brüche verfallen würde, und wann auch dieses geschähe, dieselben ohne große Gefahr verwerfen 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+{"basename":"euler_rechenkunst02_1740","par_idx":724,"date":1740,"orig":"Allein hiebey iſt zu bedencken, daß man in Ausdruͤckung groſſer Summen auf ſehr groſſe Zahlen kommen wuͤrde, welche zu uͤberſehen dem gemeinen Mann nicht weniger ſchwehr fallen wuͤrde.","norm":"Allein hierbei ist zu bedenken, dass man in Ausdrückung großer Summen auf sehr große Zahlen kommen würde, welche zu übersehen dem gemeinen Mann nicht weniger schwerfallen 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+{"basename":"euler_rechenkunst02_1740","par_idx":725,"date":1740,"orig":"Als wann die Rede waͤre von 573648 Poluſchken, ſo doͤrfte dieſe Summ zu begreiffen manchem nicht wenig Schwierigkeiten erwecken, wann man aber anſtatt derſelben ſagt 1434 Rubl und 12 Copeken, ſo wird ſich davon einjeder leicht einen deutlichen Begriff zu machen wiſſen.","norm":"Als wann die Rede wäre von 573648 Poluschken, so dürfte diese Summe zu begreifen manchem nicht wenig Schwierigkeiten erwecken, wann man aber anstatt derselben sagt 1434 Rubel und 12 Kopeken, so wird sich davon einjeder leicht einen deutlichen Begriff zu machen 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+{"basename":"euler_rechenkunst02_1740","par_idx":726,"date":1740,"orig":"Da nun bey den meiſten Rechnungen von Muͤntz, Gewicht und Maaß die verſchiedenen Sorten zu Vermeidung ſo wohl der Bruͤche als allzugroſſer Zahlen eingefuͤhret worden, ſo iſt leicht zu erachten, wie man ſich dieſem Endzweck gemaͤß der verſchiedenen Sorten bedienen muͤſſen.","norm":"Da nun bei den meisten Rechnungen von Münze, Gewicht und Maß die verschiedenen Sorten zu Vermeidung so wohl der Brüche als allzu großer Zahlen eingeführt worden, so ist leicht zu erachten, wie man sich diesem Endzweck gemäß der verschiedenen Sorten bedienen 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+{"basename":"euler_rechenkunst02_1740","par_idx":727,"date":1740,"orig":"Fuͤr das erſte muß man ſich nehmlich huͤten, daß von keiner groͤſſeren Sorte Bruͤche in die Rechnung gebracht werden:","norm":"Für das erste muss man sich nämlich hüten, dass von keiner größeren Sorte Brüche in die Rechnung gebracht 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+{"basename":"euler_rechenkunst02_1740","par_idx":728,"date":1740,"orig":"ſondern wann ſolches geſchieht, muß man die Bruͤche auf die folgenden kleineren Sorten reduciren, bis endlich die Bruͤche entweder gantz und gar verſchwinden, oder nur bey der kleinſten Sorte uͤbrig bleiben.","norm":"sondern wann solches geschieht, muss man die Brüche auf die folgenden kleineren Sorten reduzieren, bis endlich die Brüche entweder ganz und gar verschwinden, oder nur bei der kleinsten Sorte übrig 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+{"basename":"euler_rechenkunst02_1740","par_idx":743,"date":1740,"orig":"Dieſes kan nun fuͤr das erſte am natuͤrlichſten durch die Subtraction geſchehen, in dem man von der vorhandenen Anzahl Stuͤcke der kleinern Sorte ſo viel Stuͤcke abnimmt als ein Stuͤck der groͤſſeren Sorte ausmachen, und dafuͤr ein Stuͤck zu der groͤſſern Sorte ſetzt.","norm":"Dieses kann nun für das erste am natürlichsten durch die Subtraktion geschehen, in dem man von der vorhandenen Anzahl Stücke der kleineren Sorte so viel Stücke abnimmt als ein Stück der größeren Sorte ausmachen, und dafür ein Stück zu der größeren Sorte 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+{"basename":"euler_rechenkunst02_1740","par_idx":744,"date":1740,"orig":"Bleiben, nach dem dieſes geſchehen, noch mehr Stuͤcke von der kleinern Sorte uͤber, als ein Stuͤck der groͤſſern ausmachen, ſo ſubtrahirt man nochmahls eben ſo viel Stuͤck als in der groͤſſern Sorte enthalten, und ſchreibt dafuͤr wiederum ein Stuͤck zur groͤſſeren Sorte.","norm":"Bleiben, nach dem dieses geschehen, noch mehr Stücke von der kleineren Sorte über, als ein Stück der größeren ausmachen, so subtrahiert man nochmals ebenso viel Stück als in der größeren Sorte enthalten, und schreibt dafür wiederum ein Stück zur größeren 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+{"basename":"euler_rechenkunst02_1740","par_idx":749,"date":1740,"orig":"Was aber auf ſolche Art durch die viel mahl wiederholte Subtraction geſchieht, daſſelbe kan kuͤrtzer und auf ein mahl durch die Diviſion bewerckſtelliget werden, in dem die Diviſion nichts anders iſt als eine etliche mahl wiederholte Subtraction; und derohalben kan dieſe Reduction fuͤglicher vermittelſt der Diviſion angeſtellet werden, nach Anweiſung der gegebenen Regel; wovon alſo der Grund hieraus zugleich erhellet.","norm":"Was aber auf solche Art durch die viel Mal wiederholte Subtraktion geschieht, dasselbe kann kürzer und auf ein Mal durch die Division bewerkstelliget werden, in dem die Division nichts anders ist als eine etliche Mal wiederholte Subtraktion; und derohalben kann diese Reduktion füglicher vermittels der Division angestellt werden, nach Anweisung der gegebenen Regel; wovon also der Grund hieraus zugleich erhellet.","orig_tok":["Was","aber","auf","ſolche","Art","durch","die","viel","mahl","wiederholte","Subtraction","geſchieht",",","daſſelbe","kan","kuͤrtzer","und","auf","ein","mahl","durch","die","Diviſion","bewerckſtelliget","werden",",","in","dem","die","Diviſion","nichts","anders","iſt","als","eine","etliche","mahl","wiederholte","Subtraction",";","und","derohalben","kan","dieſe","Reduction","fuͤglicher","vermittelſt","der","Diviſion","angeſtellet","werden",",","nach","Anweiſung","der","gegebenen","Regel",";","wovon","alſo","der","Grund","hieraus","zugleich","erhellet","."],"norm_tok":["Was","aber","auf","solche","Art","durch","die","viel","Mal","wiederholte","Subtraktion","geschieht",",","dasselbe","kann","kürzer","und","auf","ein","Mal","durch","die","Division","bewerkstelliget","werden",",","in","dem","die","Division","nichts","anders","ist","als","eine","etliche","Mal","wiederholte","Subtraktion",";","und","derohalben","kann","diese","Reduktion","füglicher","vermittels","der","Division","angestellt","werden",",","nach","Anweisung","der","gegebenen","Regel",";","wovon","also","der","Grund","hieraus","zugleich","erhellet","."],"orig_spans":[[0,3],[4,8],[9,12],[13,19],[20,23],[24,29],[30,33],[34,38],[39,43],[44,55],[56,67],[68,77],[77,78],[79,87],[88,91],[92,100],[101,104],[105,108],[109,112],[113,117],[118,123],[124,127],[128,136],[137,153],[154,160],[160,161],[162,164],[165,168],[169,172],[173,181],[182,188],[189,195],[196,199],[200,203],[204,208],[209,216],[217,221],[222,233],[234,245],[245,246],[247,250],[251,261],[262,265],[266,271],[272,281],[282,292],[293,304],[305,308],[309,317],[318,329],[330,336],[336,337],[338,342],[343,352],[353,356],[357,366],[367,372],[372,373],[374,379],[380,384],[385,388],[389,394],[395,402],[403,411],[412,420],[420,421]],"norm_spans":[[0,3],[4,8],[9,12],[13,19],[20,23],[24,29],[30,33],[34,38],[39,42],[43,54],[55,66],[67,76],[76,77],[78,86],[87,91],[92,98],[99,102],[103,106],[107,110],[111,114],[115,120],[121,124],[125,133],[134,149],[150,156],[156,157],[158,160],[161,164],[165,168],[169,177],[178,184],[185,191],[192,195],[196,199],[200,204],[205,212],[213,216],[217,228],[229,240],[240,241],[242,245],[246,256],[257,261],[262,267],[268,277],[278,287],[288,298],[299,302],[303,311],[312,322],[323,329],[329,330],[331,335],[336,345],[346,349],[350,359],[360,365],[365,366],[367,372],[373,377],[378,381],[382,387],[388,395],[396,404],[405,413],[413,414]],"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]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.952000141143799} +{"basename":"euler_rechenkunst02_1740","par_idx":750,"date":1740,"orig":"Nehmlich in dem gegebenen Exempel, wann ich die 16 Solotnick durch 3 dividire, ſo weißt mir der Quotus 5, wieviel mahl 3 in 16 enthalten ſeyen oder wieviel mahl man 3 von 16 abziehen koͤnne, der Reſt aber 1 zeiget an, wieviel noch zuruͤck bleibt wann man 3 fuͤnf mahl abgezogen.","norm":"Nämlich in dem gegebenen Exempel, wann ich die 16 Solotnick durch 3 dividiere, so weißt mir der Quotus 5, wie viel Mal 3 in 16 enthalten seien oder wie viel Mal man 3 von 16 abziehen könne, der Rest aber 1 zeiget an, wie viel noch zurückbleibt wann man 3 fünfmal abgezogen.","orig_tok":["Nehmlich","in","dem","gegebenen","Exempel",",","wann","ich","die","16","Solotnick","durch","3","dividire",",","ſo","weißt","mir","der","Quotus","5",",","wieviel","mahl","3","in","16","enthalten","ſeyen","oder","wieviel","mahl","man","3","von","16","abziehen","koͤnne",",","der","Reſt","aber","1","zeiget","an",",","wieviel","noch","zuruͤck","bleibt","wann","man","3","fuͤnf","mahl","abgezogen","."],"norm_tok":["Nämlich","in","dem","gegebenen","Exempel",",","wann","ich","die","16","Solotnick","durch","3","dividiere",",","so","weißt","mir","der","Quotus","5",",","wie","viel","Mal","3","in","16","enthalten","seien","oder","wie","viel","Mal","man","3","von","16","abziehen","könne",",","der","Rest","aber","1","zeiget","an",",","wie","viel","noch","zurückbleibt","wann","man","3","fünfmal","abgezogen","."],"orig_spans":[[0,8],[9,11],[12,15],[16,25],[26,33],[33,34],[35,39],[40,43],[44,47],[48,50],[51,60],[61,66],[67,68],[69,77],[77,78],[79,81],[82,87],[88,91],[92,95],[96,102],[103,104],[104,105],[106,113],[114,118],[119,120],[121,123],[124,126],[127,136],[137,142],[143,147],[148,155],[156,160],[161,164],[165,166],[167,170],[171,173],[174,182],[183,189],[189,190],[191,194],[195,199],[200,204],[205,206],[207,213],[214,216],[216,217],[218,225],[226,230],[231,238],[239,245],[246,250],[251,254],[255,256],[257,262],[263,267],[268,277],[277,278]],"norm_spans":[[0,7],[8,10],[11,14],[15,24],[25,32],[32,33],[34,38],[39,42],[43,46],[47,49],[50,59],[60,65],[66,67],[68,77],[77,78],[79,81],[82,87],[88,91],[92,95],[96,102],[103,104],[104,105],[106,109],[110,114],[115,118],[119,120],[121,123],[124,126],[127,136],[137,142],[143,147],[148,151],[152,156],[157,160],[161,164],[165,166],[167,170],[171,173],[174,182],[183,188],[188,189],[190,193],[194,198],[199,203],[204,205],[206,212],[213,215],[215,216],[217,220],[221,225],[226,230],[231,243],[244,248],[249,252],[253,254],[255,262],[263,272],[272,273]],"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],[22,23],[23,24],[24,25],[25,26],[26,27],[27,28],[28,29],[29,30],[30,31],[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],[46,49],[47,50],[48,51],[49,51],[50,52],[51,53],[52,54],[53,55],[54,55],[55,56],[56,57]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.241600036621094} +{"basename":"euler_rechenkunst02_1740","par_idx":751,"date":1740,"orig":"Da man nun zu den Lothen ſo viel Loth addiren muß als oft man 3 ſubtrahirt hat, ſo weißt der Quotus 5 ſogleich, wieviel Loth in den Solotnicken enthalten und folglich zu den Lothen geſchlagen werden muͤſſen, der Reſt aber 1, weiſet daß noch 1 Solotnick zuruͤck bleibt, und unter dieſem Nahmen bleiben muͤſſe.","norm":"Da man nun zu den Loten so viel Lot addieren muss als oft man 3 subtrahiert hat, so weißt der Quotus 5 sogleich, wie viel Lot in den Solotnick enthalten und folglich zu den Loten geschlagen werden müssen, der Rest aber 1, weiset dass noch 1 Solotnick zurückbleibt, und unter diesem Namen bleiben 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+{"basename":"euler_rechenkunst02_1740","par_idx":752,"date":1740,"orig":"Nach dieſer Regel wird alſo das vorgegebene Gewicht 5 Loth, 16 Solotnick wie folget reducirt werden [Formel] wie nach der vorigen Operation.","norm":"Nach dieser Regel wird also das vorgegebene Gewicht 5 Lot, 16 Solotnick wie folget reduziert werden [Formel] wie nach der vorigen 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+{"basename":"euler_rechenkunst02_1740","par_idx":753,"date":1740,"orig":"Wann demnach von der kleinern Sorte mehr Stuͤcke vorhanden ſind, als 1 Stuͤck der groͤſſeren ausmachen, ſo muß man die vorgelegte Anzahl Stuͤck der kleinern Sorte dividiren durch diejenige Zahl, welche anzeigt, wieviel Stuͤck von der kleinern Sorte ein Stuͤck der groͤſſern ausmachen; wann dieſe Diviſion geſchehen, ſo muß man ſo viel Stuͤck als der Quotient anzeigt zur groͤſſeren Sorten addiren, von der kleinern Sorte aber bleiben ſo viel Stuͤck zuruͤck, als der Reſt ausweiſet.","norm":"Wann demnach von der kleineren Sorte mehr Stücke vorhanden sind, als 1 Stück der größeren ausmachen, so muss man die vorgelegte Anzahl Stück der kleineren Sorte dividieren durch diejenige Zahl, welche anzeigt, wie viel Stück von der kleineren Sorte ein Stück der größeren ausmachen; wann diese Division geschehen, so muss man so viel Stück als der Quotient anzeigt zur größeren Sorten addieren, von der kleineren Sorte aber bleiben so 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+{"basename":"euler_rechenkunst02_1740","par_idx":770,"date":1740,"orig":"Man faͤngt nehmlich bey der kleinſten Sorte an, und wann von derſelben ſo viel oder mehr Stuͤcke vorhanden ſind, als ein Stuͤck der naͤchſtfolgenden groͤſſeren Sorte ausmachen, ſo verrichtet man die Reduction zwiſchen dieſen beyden Sorten wie gelehrt; und erhaͤlt dadurch, daß von der kleinſten Sorten weniger Stuͤcke vorkommen, als 1 Stuͤck von der naͤchſtfolgenden groͤſſeren Sorte ausmachen.","norm":"Man fängt nämlich bei der kleinsten Sorte an, und wann von derselben so viel oder mehr Stücke vorhanden sind, als ein Stück der nächstfolgenden größeren Sorte ausmachen, so verrichtet man die Reduktion zwischen diesen beiden Sorten wie gelehrt; und erhält dadurch, dass von der kleinsten Sorten weniger Stücke vorkommen, als 1 Stück von der nächstfolgenden größeren Sorte 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+{"basename":"euler_rechenkunst02_1740","par_idx":817,"date":1740,"orig":"12 Aeß betrage.","norm":"12 Aß betrage.","orig_tok":["12","Aeß","betrage","."],"norm_tok":["12","Aß","betrage","."],"orig_spans":[[0,2],[3,6],[7,14],[14,15]],"norm_spans":[[0,2],[3,5],[6,13],[13,14]],"alignment":[[0,0],[1,1],[2,2],[3,3]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"hu","orig_lang_de":0.667,"norm_lmscore":7.231299877166748} +{"basename":"euler_rechenkunst02_1740","par_idx":818,"date":1740,"orig":"Dieſe 12 Aeß mit den vorhandenen 13 Aeß zuſammen machen 25 Aeß, und alſo wird obiges Gewicht nach beyden Regeln alſo zu ſtehen kommen, 7 Mrk. 3 Untz, 1 Loth, 2 Quintl, 0 Engl. 25 Aeß.","norm":"Diese 12 Aß mit den vorhandenen 13 Aß zusammen machen 25 Aß, und also wird obiges Gewicht nach beiden Regeln also zu stehen kommen, 7 Mrk. 3 Untz, 1 Lot, 2 Quintl, 0 Engl. 25 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+{"basename":"euler_rechenkunst02_1740","par_idx":822,"date":1740,"orig":"Solte bey dieſer kleineren Sorte noch ein Bruch vorkommen, ſo wird ſolcher in die naͤchſtfolgende kleinere Sorte auf gleiche Art gebracht, bis endlich alle Sorten die kleinſte ausgenommen von Bruͤchen voͤllig befreyet werden.","norm":"Sollte bei dieser kleineren Sorte noch ein Bruch vorkommen, so wird solcher in die nächstfolgende kleinere Sorte auf gleiche Art gebracht, bis endlich alle Sorten die kleinste ausgenommen von Brüchen völlig befreiet 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+{"basename":"euler_rechenkunst02_1740","par_idx":823,"date":1740,"orig":"Eine gegebene Anzahl Stuͤcke von einer groͤſſeren Sorte wird in eine kleinere Sorte verwandelt, wann man dieſelbe Anzahl multiplicirt mit derjenigen Zahl, welche anzeigt, wieviel Stuͤck von der kleineren Sorte ein Stuͤck der groͤſſeren in ſich enthaͤlt, wie wir oben gewieſen haben.","norm":"Eine gegebene Anzahl Stücke von einer größeren Sorte wird in eine kleinere Sorte verwandelt, wann man dieselbe Anzahl multipliziert mit derjenigen Zahl, welche anzeigt, wie viel Stück von der kleineren Sorte ein Stück der größeren in sich enthält, wie wir oben gewiesen 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+{"basename":"euler_rechenkunst02_1740","par_idx":824,"date":1740,"orig":"Da nun dieſe Regel allgemein iſt, ſo wird auch eine gebrochene Anzahl Stuͤck von der groͤſſeren Sorte in die kleinere verwandelt, wann man denſelben Bruch durch den beſchriebenen Multiplicatorem multiplicirt.","norm":"Da nun diese Regel allgemein ist, so wird auch eine gebrochene Anzahl Stück von der größeren Sorte in die kleinere verwandelt, wann man denselben Bruch durch den beschriebenen Multiplikatorrem multipliziert.","orig_tok":["Da","nun","dieſe","Regel","allgemein","iſt",",","ſo","wird","auch","eine","gebrochene","Anzahl","Stuͤck","von","der","groͤſſeren","Sorte","in","die","kleinere","verwandelt",",","wann","man","denſelben","Bruch","durch","den","beſchriebenen","Multiplicatorem","multiplicirt","."],"norm_tok":["Da","nun","diese","Regel","allgemein","ist",",","so","wird","auch","eine","gebrochene","Anzahl","Stück","von","der","größeren","Sorte","in","die","kleinere","verwandelt",",","wann","man","denselben","Bruch","durch","den","beschriebenen","Multiplikatorrem","multipliziert","."],"orig_spans":[[0,2],[3,6],[7,12],[13,18],[19,28],[29,32],[32,33],[34,36],[37,41],[42,46],[47,51],[52,62],[63,69],[70,76],[77,80],[81,84],[85,95],[96,101],[102,104],[105,108],[109,117],[118,128],[128,129],[130,134],[135,138],[139,148],[149,154],[155,160],[161,164],[165,178],[179,194],[195,207],[207,208]],"norm_spans":[[0,2],[3,6],[7,12],[13,18],[19,28],[29,32],[32,33],[34,36],[37,41],[42,46],[47,51],[52,62],[63,69],[70,75],[76,79],[80,83],[84,92],[93,98],[99,101],[102,105],[106,114],[115,125],[125,126],[127,131],[132,135],[136,145],[146,151],[152,157],[158,161],[162,175],[176,192],[193,206],[206,207]],"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":5.33050012588501} +{"basename":"euler_rechenkunst02_1740","par_idx":825,"date":1740,"orig":"Ein Bruch wird aber durch eine jegliche Zahl multipiicirt, wann man den Zehler deſſelben damit multiplicirt; und deswegen muß man den Zehler des Bruchs mit dem gemeldten Multiplicatore multipliciren; den Nenner aber unveraͤndert laſſen.","norm":"Ein Bruch wird aber durch eine jegliche Zahl multipiicirt, wann man den Zähler desselben damit multipliziert; und deswegen muss man den Zähler des Bruchs mit dem gemeldeten Multiplikator multiplizieren; den Nenner aber unverändert lassen.","orig_tok":["Ein","Bruch","wird","aber","durch","eine","jegliche","Zahl","multipiicirt",",","wann","man","den","Zehler","deſſelben","damit","multiplicirt",";","und","deswegen","muß","man","den","Zehler","des","Bruchs","mit","dem","gemeldten","Multiplicatore","multipliciren",";","den","Nenner","aber","unveraͤndert","laſſen","."],"norm_tok":["Ein","Bruch","wird","aber","durch","eine","jegliche","Zahl","multipiicirt",",","wann","man","den","Zähler","desselben","damit","multipliziert",";","und","deswegen","muss","man","den","Zähler","des","Bruchs","mit","dem","gemeldeten","Multiplikator","multiplizieren",";","den","Nenner","aber","unverändert","lassen","."],"orig_spans":[[0,3],[4,9],[10,14],[15,19],[20,25],[26,30],[31,39],[40,44],[45,57],[57,58],[59,63],[64,67],[68,71],[72,78],[79,88],[89,94],[95,107],[107,108],[109,112],[113,121],[122,125],[126,129],[130,133],[134,140],[141,144],[145,151],[152,155],[156,159],[160,169],[170,184],[185,198],[198,199],[200,203],[204,210],[211,215],[216,228],[229,235],[235,236]],"norm_spans":[[0,3],[4,9],[10,14],[15,19],[20,25],[26,30],[31,39],[40,44],[45,57],[57,58],[59,63],[64,67],[68,71],[72,78],[79,88],[89,94],[95,108],[108,109],[110,113],[114,122],[123,127],[128,131],[132,135],[136,142],[143,146],[147,153],[154,157],[158,161],[162,172],[173,186],[187,201],[201,202],[203,206],[207,213],[214,218],[219,230],[231,237],[237,238]],"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":4.984300136566162} +{"basename":"euler_rechenkunst02_1740","par_idx":826,"date":1740,"orig":"Jſt dieſes nun geſchehen, ſo weiſet der herausgekommene Bruch den Werth des vorgegebenen Bruchs in der kleineren Sorte.","norm":"Ist dieses nun geschehen, so weiset der herausgekommene Bruch den Wert des vorgegebenen Bruchs in der kleineren Sorte.","orig_tok":["Jſt","dieſes","nun","geſchehen",",","ſo","weiſet","der","herausgekommene","Bruch","den","Werth","des","vorgegebenen","Bruchs","in","der","kleineren","Sorte","."],"norm_tok":["Ist","dieses","nun","geschehen",",","so","weiset","der","herausgekommene","Bruch","den","Wert","des","vorgegebenen","Bruchs","in","der","kleineren","Sorte","."],"orig_spans":[[0,3],[4,10],[11,14],[15,24],[24,25],[26,28],[29,35],[36,39],[40,55],[56,61],[62,65],[66,71],[72,75],[76,88],[89,95],[96,98],[99,102],[103,112],[113,118],[118,119]],"norm_spans":[[0,3],[4,10],[11,14],[15,24],[24,25],[26,28],[29,35],[36,39],[40,55],[56,61],[62,65],[66,70],[71,74],[75,87],[88,94],[95,97],[98,101],[102,111],[112,117],[117,118]],"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":5.576000213623047} +{"basename":"euler_rechenkunst02_1740","par_idx":827,"date":1740,"orig":"Jſt aber ferner der Zehler dieſes gefundenen Bruchs groͤſſer als der Nenner, ſo muß man den gefundenen Zehler durch den Nenner dividiren, da dann der voͤllige Quotient den Werth des Bruchs in der verlangten kleinern Sorte anzeigt: und dieſes iſt eben diejenige Operation, welche im Satze iſt vorgeſchrieben worden.","norm":"Ist aber ferner der Zähler dieses gefundenen Bruchs größer als der Nenner, so muss man den gefundenen Zähler durch den Nenner dividieren, da dann der völlige Quotient den Wert des Bruchs in der verlangten kleineren Sorte anzeigt: und dieses ist eben diejenige Operation, welche im Satze ist vorgeschrieben 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+{"basename":"euler_rechenkunst02_1740","par_idx":828,"date":1740,"orig":"Auf dieſe Art wird nun ein Bruch aus der groͤſſeren Sorte gehoben, und deſſelben Werth in die folgende kleinere Sorte gebracht; die gantzen Stuͤcke aber, welche bey der groͤſſeren Sorte auſſer dem Bruche vorhanden geweſen, bleiben bey derſelben unveraͤndert.","norm":"Auf diese Art wird nun ein Bruch aus der größeren Sorte gehoben, und desselben Wert in die folgende kleinere Sorte gebracht; die ganzen Stücke aber, welche bei der größeren Sorte außer dem Bruche vorhanden gewesen, bleiben bei derselben unverändert.","orig_tok":["Auf","dieſe","Art","wird","nun","ein","Bruch","aus","der","groͤſſeren","Sorte","gehoben",",","und","deſſelben","Werth","in","die","folgende","kleinere","Sorte","gebracht",";","die","gantzen","Stuͤcke","aber",",","welche","bey","der","groͤſſeren","Sorte","auſſer","dem","Bruche","vorhanden","geweſen",",","bleiben","bey","derſelben","unveraͤndert","."],"norm_tok":["Auf","diese","Art","wird","nun","ein","Bruch","aus","der","größeren","Sorte","gehoben",",","und","desselben","Wert","in","die","folgende","kleinere","Sorte","gebracht",";","die","ganzen","Stücke","aber",",","welche","bei","der","größeren","Sorte","außer","dem","Bruche","vorhanden","gewesen",",","bleiben","bei","derselben","unverändert","."],"orig_spans":[[0,3],[4,9],[10,13],[14,18],[19,22],[23,26],[27,32],[33,36],[37,40],[41,51],[52,57],[58,65],[65,66],[67,70],[71,80],[81,86],[87,89],[90,93],[94,102],[103,111],[112,117],[118,126],[126,127],[128,131],[132,139],[140,147],[148,152],[152,153],[154,160],[161,164],[165,168],[169,179],[180,185],[186,192],[193,196],[197,203],[204,213],[214,221],[221,222],[223,230],[231,234],[235,244],[245,257],[257,258]],"norm_spans":[[0,3],[4,9],[10,13],[14,18],[19,22],[23,26],[27,32],[33,36],[37,40],[41,49],[50,55],[56,63],[63,64],[65,68],[69,78],[79,83],[84,86],[87,90],[91,99],[100,108],[109,114],[115,123],[123,124],[125,128],[129,135],[136,142],[143,147],[147,148],[149,155],[156,159],[160,163],[164,172],[173,178],[179,184],[185,188],[189,195],[196,205],[206,213],[213,214],[215,222],[223,226],[227,236],[237,248],[248,249]],"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":4.908699989318848} +{"basename":"euler_rechenkunst02_1740","par_idx":829,"date":1740,"orig":"Und deswegen wann der Bruch, welcher bey der groͤſſern Sorte vorkommt, groͤſſer iſt als ein gantzes, ſo muͤſſen vorhero daraus die gantzen gezogen, und nur der uͤbrige Bruch, welcher kleiner iſt als ein gantzes, in die folgende kleinere Sorte verwandelt werden.","norm":"Und deswegen wann der Bruch, welcher bei der größeren Sorte vorkommt, größer ist als ein ganzes, so müssen vorher daraus die ganzen gezogen, und nur der übrige Bruch, welcher kleiner ist als ein ganzes, in die folgende kleinere Sorte verwandelt 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+{"basename":"euler_rechenkunst02_1740","par_idx":831,"date":1740,"orig":"Derowegen wann man auch gantze Stuͤcke aus einer groͤſſeren Sorte nehmen und in die kleineren verwandeln wollte, ſo wuͤrde man ſich nur die Arbeit verdoppeln, und nachgehends ſolche nach dem vorigen Satz wiederum auf die groͤſſeren Sorten reduciren muͤſſen.","norm":"Derowegen wann man auch ganze Stücke aus einer größeren Sorte nehmen und in die kleineren verwandeln wollte, so würde man sich nur die Arbeit verdoppeln, und nachgehend solche nach dem vorigen Satz wiederum auf die größeren Sorten reduzieren 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+{"basename":"euler_rechenkunst02_1740","par_idx":832,"date":1740,"orig":"Wann auf ſolche Weiſe der Werth des Bruchs bey der groͤſſern Sorte auf die folgende kleinere Sorte gebracht worden, ſo muß derſelbe zu demjenigen was von dieſer Sorte ſchon allbereit vorhanden iſt addirt werden: findt ſich alsdann bey dieſer kleinern Sorte noch ein Bruch, ſo muß derſelbe auf eben dieſe Art noch weiter auf kleinere Sorten gebracht werden, bis man endlich auf die allerkleinſte Sorte kommt, in welcher Bruͤche geduldet werden.","norm":"Wann auf solche Weise der Wert des Bruchs bei der größeren Sorte auf die folgende kleinere Sorte gebracht worden, so muss derselbe zu demjenigen was von dieser Sorte schon allbereits vorhanden ist addiert werden: findet sich alsdann bei dieser kleineren Sorte noch ein Bruch, so muss derselbe auf eben diese Art noch weiter auf kleinere Sorten gebracht werden, bis man endlich auf die allerkleinste Sorte kommt, in welcher Brüche geduldet 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+{"basename":"euler_rechenkunst02_1740","par_idx":862,"date":1740,"orig":"Jn der Addition muͤſſen immer ſolche Zahlen zuſammen addirt werden, welche ſich auf einerley Unitaͤten beziehen: und dieſes iſt eben diejenige Regel, welche bey der Addition der unbenannten Zahlen im erſten Theil iſt gegeben worden: krafft welcher erſtlich die Unitaͤten und dann die Decades, hernach die Centenarii, Millenarii und ſo fort addirt werden muͤſſen.","norm":"In der Addition müssen immer solche Zahlen zusammen addiert werden, welche sich auf einerlei Unitäten beziehen: und dieses ist eben diejenige Regel, welche bei der Addition der unbenannten Zahlen im ersten Teil ist gegeben worden: Kraft welcher erstlich die Unitäten und dann die Decades, hernach die Centenarii, Millenarii und so fort addiert werden 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+{"basename":"euler_rechenkunst02_1740","par_idx":863,"date":1740,"orig":"Dieſe Regel erſtreckt ſich nun gleichfals auf alle verſchiedene Sorten und Benennungen, welche zu addiren vorkommen koͤnnen, und muͤſſen nach derſelben immer einerley Sorten zuſammen addirt werden.","norm":"Diese Regel erstreckt sich nun gleichfalls auf alle verschiedene Sorten und Benennungen, welche zu addieren vorkommen können, und müssen nach derselben immer einerlei Sorten zusammen addiert 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+{"basename":"euler_rechenkunst02_1740","par_idx":864,"date":1740,"orig":"Als wann verſchiedene Summen Geldes, welche gewoͤhnlicher maſſen in Rublen Griwen und Copeken ausgedruͤckt ſind, vorgegeben werden, ſo addirt man insbeſondere die Copeken, dann die Griwen und endlich die Rublen, und erhaͤlt ſolcher geſtalt die wahre Summ.","norm":"Als wann verschiedene Summen Geldes, welche gewöhnlicher Maßen in Rubeln Griwen und Kopeken ausgedrückt sind, vorgegeben werden, so addiert man insbesondere die Kopeken, dann die Griwen und endlich die Rubeln, und erhält solcher Gestalt die wahre 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+{"basename":"euler_rechenkunst02_1740","par_idx":865,"date":1740,"orig":"Eine jegliche dieſer Additionen geſchieht nun gaͤntzlich wie oben bey der Addition von unbenannten Zahlen gelehret worden:","norm":"Eine jegliche dieser Additionen geschieht nun gänzlich wie oben bei der Addition von unbenannten Zahlen gelehrt 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+{"basename":"euler_rechenkunst02_1740","par_idx":866,"date":1740,"orig":"und werden die Zahlen der Stuͤcke, ſo von einerley Sorten vorkommen nicht anderſt als unbenannte Zahlen addirt.","norm":"und werden die Zahlen der Stücke, so von einerlei Sorten vorkommen nicht anders als unbenannte Zahlen 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+{"basename":"euler_rechenkunst02_1740","par_idx":867,"date":1740,"orig":"Dann gleichwie in unbenannten Zahlen 12 und 5 addirt 17 ausmachen, alſo machen auch 12 Copeken und 5 Copeken zuſammen 17 Copeken: ingleichem 12 Loth und 5 Loth zuſammen 17 Loth, und ſo fort, was auch immer fuͤr Sorten und Benennungen vorkommen, ſo machen allezeit 12 Stuͤck und 5 Stuͤck von einerley Sorte zuſammen 17 Stuͤck von eben derſelben Sorte.","norm":"Dann gleichwie in unbenannten Zahlen 12 und 5 addiert 17 ausmachen, also machen auch 12 Kopeken und 5 Kopeken zusammen 17 Kopeken: ingleichem 12 Lot und 5 Lot zusammen 17 Lot, und so fort, was auch immer für Sorten und Benennungen vorkommen, so machen allezeit 12 Stück und 5 Stück von einerlei Sorte zusammen 17 Stück von eben derselben 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+{"basename":"euler_rechenkunst02_1740","par_idx":871,"date":1740,"orig":"Zu dieſem Ende iſt alſo die angezeigte Manier die verſchiedenen Sorten insbeſondere zu addiren am bequemſten, als wodurch die Summ wiederum in eben denſelben Sorten herauskommt, in welchen dieſelbe nach der angenommenen Regel ausgedruͤckt werden ſoll.","norm":"Zu diesem Ende ist also die angezeigte Manier die verschiedenen Sorten insbesondere zu addieren am bequemsten, als wodurch die Summe wiederum in eben denselben Sorten herauskommt, in welchen dieselbe nach der angenommenen Regel ausgedrückt werden 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+{"basename":"euler_rechenkunst02_1740","par_idx":872,"date":1740,"orig":"Man ſchreibt deswegen die gegebenen Quantitaͤten, welche addirt werden ſollen, ſolcher geſtalt unter einander, daß immer die gleichen Sorten oder Benennungen unter einander zu ��tehen kommen, und addirt von einer jeglichen Sorte alle Zahlen, welche davon vorkommen und ſchreibt die Summ unter denſelbigen Nahmen.","norm":"Man schreibt deswegen die gegebenen Quantitäten, welche addiert werden sollen, solcher Gestalt untereinander, dass immer die gleichen Sorten oder Benennungen untereinander zu stehen kommen, und addiert von einer jeglichen Sorte alle Zahlen, welche davon vorkommen und schreibt die Summe unter denselbigen 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+{"basename":"euler_rechenkunst02_1740","par_idx":876,"date":1740,"orig":"Hernach wird unter die ſolcher geſtalt geſchriebenen Quantitaͤten, welche addirt werden ſollen, eine Linie gezogen, und alle verſchiedenen Sorten insbeſondere addirt und die gefundenen Summen unter eben dieſelben Sorten geſchrieben.","norm":"Hernach wird unter die solcher Gestalt geschriebenen Quantitäten, welche addiert werden sollen, eine Linie gezogen, und alle verschiedenen Sorten insbesondere addiert und die gefundenen Summen unter eben dieselben Sorten 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+{"basename":"euler_rechenkunst02_1740","par_idx":878,"date":1740,"orig":"ſo daß die gefundene Summ ſeyn wird 14 ℔, 28 Loth, 3 Quintl.","norm":"so dass die gefundene Summe sein wird 14 ℔, 28 Lot, 3 Quintl.","orig_tok":["ſo","daß","die","gefundene","Summ","ſeyn","wird","14","℔",",","28","Loth",",","3","Quintl","."],"norm_tok":["so","dass","die","gefundene","Summe","sein","wird","14","℔",",","28","Lot",",","3","Quintl","."],"orig_spans":[[0,2],[3,6],[7,10],[11,20],[21,25],[26,30],[31,35],[36,38],[39,40],[40,41],[42,44],[45,49],[49,50],[51,52],[53,59],[59,60]],"norm_spans":[[0,2],[3,7],[8,11],[12,21],[22,27],[28,32],[33,37],[38,40],[41,42],[42,43],[44,46],[47,50],[50,51],[52,53],[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],[10,10],[11,11],[12,12],[13,13],[14,14],[15,15]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"az","orig_lang_de":0.667,"norm_lmscore":6.107699871063232} +{"basename":"euler_rechenkunst02_1740","par_idx":879,"date":1740,"orig":"Daß dieſes aber die wahre Summ ſey, daran iſt im geringſten nicht zu zweifeln, weilen auf dieſe Art von jeglicher Sorte die Summ richtig genommen worden.","norm":"Dass dieses aber die wahre Summe sei, daran ist im Geringsten nicht zu zweifeln, weilen auf diese Art von jeglicher Sorte die Summe richtig genommen 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+{"basename":"euler_rechenkunst02_1740","par_idx":880,"date":1740,"orig":"Gleicher geſtalt ſind auch folgende Exempel addirt worden [Formel]","norm":"Gleicher Gestalt sind auch folgende Exempel addiert worden [Formel]","orig_tok":["Gleicher","geſtalt","ſind","auch","folgende","Exempel","addirt","worden","[Formel]"],"norm_tok":["Gleicher","Gestalt","sind","auch","folgende","Exempel","addiert","worden","[","Formel","]"],"orig_spans":[[0,8],[9,16],[17,21],[22,26],[27,35],[36,43],[44,50],[51,57],[58,66]],"norm_spans":[[0,8],[9,16],[17,21],[22,26],[27,35],[36,43],[44,51],[52,58],[59,60],[60,66],[66,67]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[8,9],[8,10]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.990200042724609} +{"basename":"euler_rechenkunst02_1740","par_idx":881,"date":1740,"orig":"[Formel] Dieſe Exempel ſind ſo beſchaffen, daß die gefundene Summ ſchon den vorgeſchriebenen Regeln gemaͤß iſt, und keiner 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+{"basename":"euler_rechenkunst02_1740","par_idx":882,"date":1740,"orig":"Wann aber dieſes nicht geſchieht, ſondern von den kleineren Sorten ſo viel oder mehr Stuͤcke in der Addition gefunden werden, als ein Stuͤck der naͤchſtfolgenden groͤſſeren Sorte ausmachen, ſo kan die auf ſolche Art gefundene Summ hernach nach den Regeln der Reduction in die gehoͤrige Form gebracht werden.","norm":"Wann aber dieses nicht geschieht, sondern von den kleineren Sorten so viel oder mehr Stücke in der Addition gefunden werden, als ein Stück der nächstfolgenden größeren Sorte ausmachen, so kann die auf solche Art gefundene Summe hernach nach den Regeln der Reduktion in die gehörige Form gebracht 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+{"basename":"euler_rechenkunst02_1740","par_idx":885,"date":1740,"orig":"Dieſes iſt zwar ſchon die wahre Summ der vorgegebenen 4 Gewichte: weilen aber in derſelben von allen kleinern Sorten mehr Stuͤcke vorkommen als ein Stuͤck der naͤchſtfolgenden groͤſſeren Sorte ausmachen, ſo muß noch die vorher beſchriebene Reduction angeſtellt werden: wodurch die Summ in gehoͤriger Form alſo ausgedruͤckt werden wird:","norm":"Dieses ist zwar schon die wahre Summe der vorgegebenen 4 Gewichte: weilen aber in derselben von allen kleineren Sorten mehr Stücke vorkommen als ein Stück der nächstfolgenden größeren Sorte ausmachen, so muss noch die vorher beschriebene Reduktion angestellt werden: wodurch die Summe in gehöriger Form also ausgedrückt werden 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+{"basename":"euler_rechenkunst02_1740","par_idx":886,"date":1740,"orig":"Summa 153 ℔, 7 ℥, 3 ℨ, 1 ℈, 7 gr.","norm":"Summa 153 ℔, 7 ℥, 3 ℨ, 1 ℈, 7 gr.","orig_tok":["Summa","153","℔",",","7","℥",",","3","ℨ",",","1","℈",",","7","gr","."],"norm_tok":["Summa","153","℔",",","7","℥",",","3","ℨ",",","1","℈",",","7","gr","."],"orig_spans":[[0,5],[6,9],[10,11],[11,12],[13,14],[15,16],[16,17],[18,19],[20,21],[21,22],[23,24],[25,26],[26,27],[28,29],[30,32],[32,33]],"norm_spans":[[0,5],[6,9],[10,11],[11,12],[13,14],[15,16],[16,17],[18,19],[20,21],[21,22],[23,24],[25,26],[26,27],[28,29],[30,32],[32,33]],"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]],"lang_fastText":"de","lang_py3langid":"qu","lang_cld3":"fi","orig_lang_de":0.333,"norm_lmscore":5.222899913787842} +{"basename":"euler_rechenkunst02_1740","par_idx":887,"date":1740,"orig":"Dieſe Reduction kan aber ſogleich der Addition ſelbſt ſo einverleibet werden, daß man 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+{"basename":"euler_rechenkunst02_1740","par_idx":888,"date":1740,"orig":"2.)","norm":"2.)","orig_tok":["2.",")"],"norm_tok":["2.",")"],"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":5.8993000984191895} +{"basename":"euler_rechenkunst02_1740","par_idx":889,"date":1740,"orig":"Wann nun die Addition nach der vorher beſchriebenen Regel angeſtellet wird, ſo muß man den Anfang zu addiren von der kleinſten Sorte machen, und von derſelben zu den groͤſſeren Sorten fortſchreiten.","norm":"Wann nun die Addition nach der vorher beschriebenen Regel angestellt wird, so muss man den Anfang zu addieren von der kleinsten Sorte machen, und von derselben zu den größeren Sorten 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+{"basename":"euler_rechenkunst02_1740","par_idx":890,"date":1740,"orig":"Kommen nun durch die Addition von einer kleineren Sorte weniger Stuͤcke heraus, als ein Stuͤck der naͤchſtfolgenden groͤſſeren Sorte ausmachen, ſo ſchreibt man ſogleich die gefundene Summ unter die Linie auf ihre gehoͤrige Stelle.","norm":"Kommen nun durch die Addition von einer kleineren Sorte weniger Stücke heraus, als ein Stück der nächstfolgenden größeren Sorte ausmachen, so schreibt man sogleich die gefundene Summe unter die Linie auf ihre gehörige 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immer in derjenigen Form heraus, in welcher man ſolche zu verlangen pflegt.","norm":"Nach der vorher gegebenen Regel wird zwar die verlangte Summe immer richtig gefunden, allein dieselbe kommt nicht immer in derjenigen Form heraus, in welcher man solche zu verlangen 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+{"basename":"euler_rechenkunst02_1740","par_idx":894,"date":1740,"orig":"Es geſchieht nehmlich gemeiniglich daß von den kleineren Sorten mehr Stuͤcke herauskommen als ein Stuͤck von der groͤſſeren folgenden Sorte ausmachen; welches derjenigen Regel, nach welcher alle aus vielerley Sorten beſtehende Quantitaͤten ausgedruͤckt werden ſollen, zu wieder iſt.","norm":"Es geschieht nämlich gemeiniglich dass von den kleineren Sorten mehr Stücke herauskommen als ein Stück von der größeren folgenden Sorte ausmachen; welches derjenigen Regel, nach welcher alle aus vielerlei Sorten bestehende Quantitäten ausgedrückt werden sollen, zu wieder 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+{"basename":"euler_rechenkunst02_1740","par_idx":895,"date":1740,"orig":"Derohalben um dieſer Regel ein Genuͤgen zu leiſten, muß man entweder die auf vorher gehende Art gefundene Summ durch die Reduction in die verlangte Form bringen, oder die Reduction ſelbſt ſo gleich mit der Additions- Arbeit verknuͤpfen; davon das letztere mit weit geringererer Muͤhe geſchehen kan.","norm":"Derohalben um dieser Regel ein Genügen zu leisten, muss man entweder die auf vorher gehende Art gefundene Summe durch die Reduktion in die verlangte Form bringen, oder die Reduktion selbst so gleich mit der Additions-Arbeit verknüpfen; davon das letztere mit weit geringerer Mühe geschehen 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+{"basename":"euler_rechenkunst02_1740","par_idx":896,"date":1740,"orig":"Zu dieſem Ende bedienet man ſich alſo der allhier beſchriebenen Regel, nach welcher ſogleich bey Addirung einer jeglichen kleineren Sorte die noͤthige Reduction zugleich angeſtellet wird.","norm":"Zu diesem Ende bedienet man sich also der allhier beschriebenen Regel, nach welcher sogleich bei Addierung einer jeglichen kleineren Sorte die nötige Reduktion zugleich angestellt 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+{"basename":"euler_rechenkunst02_1740","par_idx":897,"date":1740,"orig":"Da nun mit der Reduction immer von den kleinſten Sorten der Anfang gemacht werden muß, ſo muß auch in der Addition von den kleinſten Sorten der Anfang gemacht werden, damit man immer, ſo oft die Reduction noͤthig gefunden wird dieſelbe ſo fort anbringen koͤnnen.","norm":"Da nun mit der Reduktion immer von den kleinsten Sorten der Anfang gemacht werden muss, so muss auch in der Addition von den kleinsten Sorten der Anfang gemacht werden, damit man immer, so oft die Reduktion nötig gefunden wird dieselbe so fort anbringen können.","orig_tok":["Da","nun","mit","der","Reduction","immer","von","den","kleinſten","Sorten","der","Anfang","gemacht","werden","muß",",","ſo","muß","auch","in","der","Addition","von","den","kleinſten","Sorten","der","Anfang","gemacht","werden",",","damit","man","immer",",","ſo","oft","die","Reduction","noͤthig","gefunden","wird","dieſelbe","ſo","fort","anbringen","koͤnnen","."],"norm_tok":["Da","nun","mit","der","Reduktion","immer","von","den","kleinsten","Sorten","der","Anfang","gemacht","werden","muss",",","so","muss","auch","in","der","Addition","von","den","kleinsten","Sorten","der","Anfang","gemacht","werden",",","damit","man","immer",",","so","oft","die","Reduktion","nötig","gefunden","wird","dieselbe","so","fort","anbringen","können","."],"orig_spans":[[0,2],[3,6],[7,10],[11,14],[15,24],[25,30],[31,34],[35,38],[39,48],[49,55],[56,59],[60,66],[67,74],[75,81],[82,85],[85,86],[87,89],[90,93],[94,98],[99,101],[102,105],[106,114],[115,118],[119,122],[123,132],[133,139],[140,143],[144,150],[151,158],[159,165],[165,166],[167,172],[173,176],[177,182],[182,183],[184,186],[187,190],[191,194],[195,204],[205,212],[213,221],[222,226],[227,235],[236,238],[239,243],[244,253],[254,261],[261,262]],"norm_spans":[[0,2],[3,6],[7,10],[11,14],[15,24],[25,30],[31,34],[35,38],[39,48],[49,55],[56,59],[60,66],[67,74],[75,81],[82,86],[86,87],[88,90],[91,95],[96,100],[101,103],[104,107],[108,116],[117,120],[121,124],[125,134],[135,141],[142,145],[146,152],[153,160],[161,167],[167,168],[169,174],[175,178],[179,184],[184,185],[186,188],[189,192],[193,196],[197,206],[207,212],[213,221],[222,226],[227,235],[236,238],[239,243],[244,253],[254,260],[260,261]],"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]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.142300128936768} +{"basename":"euler_rechenkunst02_1740","par_idx":898,"date":1740,"orig":"Dieſe gedoppelte Operation geſchiehet nun folgender geſtalt:","norm":"Diese gedoppelte Operation geschieht nun folgender Gestalt:","orig_tok":["Dieſe","gedoppelte","Operation","geſchiehet","nun","folgender","geſtalt",":"],"norm_tok":["Diese","gedoppelte","Operation","geschieht","nun","folgender","Gestalt",":"],"orig_spans":[[0,5],[6,16],[17,26],[27,37],[38,41],[42,51],[52,59],[59,60]],"norm_spans":[[0,5],[6,16],[17,26],[27,36],[37,40],[41,50],[51,58],[58,59]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7]],"lang_fastText":"de","lang_py3langid":"nl","lang_cld3":"fy","orig_lang_de":0.333,"norm_lmscore":6.480299949645996} +{"basename":"euler_rechenkunst02_1740","par_idx":899,"date":1740,"orig":"Man addirt zuſammen alle Stuͤcke ſo von der kleinſten Sorte vorhanden ſind, und wann dieſe Summ davon kleiner iſt, als ein Stuͤck von der naͤchſtfolgenden groͤſſeren Sorte, ſo ſchreibet man dieſelbe, weilen keine Reduction von noͤthen in die Summ.","norm":"Mann addiert zusammen alle Stücke so von der kleinsten Sorte vorhanden sind, und wann diese Summe davon kleiner ist, als ein Stück von der nächstfolgenden größeren Sorte, so schreibet man dieselbe, weilen keine Reduktion vonnöten in die 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+{"basename":"euler_rechenkunst02_1740","par_idx":902,"date":1740,"orig":"man dividirt nehmlich die herausgebrachte Summ der kleineren Sorte durch diejenige Zahl, welche anzeigt, wieviel Stuͤck von der kleineren Sorte ein Stuͤck der groͤſſeren ausmachen, und ſchreibt nur den in dieſer Diviſion zuruͤck gebliebenen Reſt in die geſuchte Summ unter den Nahmen der kleinſten Sorte; den gefundenen Quotienten aber, welcher ſo viel Stuͤcke der groͤſſeren Sorte anzeigt, addirt man mit zu den vorgegebenen Stuͤcken von der groͤſſeren Sorte.","norm":"man dividiert nämlich die herausgebrachte Summe der kleineren Sorte durch diejenige Zahl, welche anzeigt, wie viel Stück von der kleineren Sorte ein Stück der größeren ausmachen, und schreibt nur den in dieser Division zurück gebliebenen Rest in die gesuchte Summe unter den Namen der kleinsten Sorte; den gefundenen Quotienten aber, welcher so viel Stücke der größeren Sorte anzeigt, addiert man mit zu den vorgegebenen Stücken von der größeren 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+{"basename":"euler_rechenkunst02_1740","par_idx":907,"date":1740,"orig":"Antw. Erſtlich muͤſſen dieſe 4 gegebenen Summen Geld gehoͤriger maſſen unter einander geſchrieben werden, wie folgt.","norm":"Antw. Erstlich müssen diese 4 gegebenen Summen Geld gehöriger maßen untereinander geschrieben werden, wie 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+{"basename":"euler_rechenkunst02_1740","par_idx":980,"date":1740,"orig":"Dieſe Regel aber findet nur ſtatt, wann von einer jeglichen Sorte in der groͤſſeren Quantitaͤt mehr Stuͤcke vorhanden ſind, als in der kleineren; dann wo dieſes nicht geſchieht, ſo muß man ſich in der folgenden Regel Raths erhohlen.","norm":"Diese Regel aber findet nur statt, wann von einer jeglichen Sorte in der größeren Quantität mehr Stücke vorhanden sind, als in der kleineren; dann wo dieses nicht geschieht, so muss man sich in der folgenden Regel Rats 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+{"basename":"euler_rechenkunst02_1740","par_idx":981,"date":1740,"orig":"Die Subtraction lehret, wie man eine kleinere Quantitaͤt von einer groͤſſeren abziehen, und dasjenige anzeigen ſoll, welches uͤberbleibt, wann man die kleinere Quantitaͤt von der groͤſſeren weg genommen hat.","norm":"Die Subtraktion lehrt, wie man eine kleinere Quantität von einer größeren abziehen, und dasjenige anzeigen soll, welches überbleibt, wann man die kleinere Quantität von der größeren weg genommen 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+{"basename":"euler_rechenkunst02_1740","par_idx":982,"date":1740,"orig":"Von dieſer Operation haben wir ſchon im erſteren Theile die noͤthigen Regeln gegeben, wann die zwey vorgelegten Quantitaͤten ſo wohl gantze als gebrochene und vermiſchte oder aus gantzen und Bruͤchen zuſammen geſetzte Zahlen ſind.","norm":"Von dieser Operation haben wir schon im ersteren Teile die nötigen Regeln gegeben, wann die zwei vorgelegten Quantitäten so wohl ganze als gebrochene und vermischte oder aus ganzen und Brüchen zusammen gesetzte Zahlen 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+{"basename":"euler_rechenkunst02_1740","par_idx":983,"date":1740,"orig":"Da wir aber anjetzo ſolche Quantitaͤten vorhaben, welche aus verſchiedenen Sorten beſtehen, ſo bleibt zwar das Fundament der Subtraction einerley, allein die Application muß auf dieſen Fall insbeſondere eingerichtet werden.","norm":"Da wir aber jetzt solche Quantitäten vorhaben, welche aus verschiedenen Sorten bestehen, so bleibt zwar das Fundament der Subtraktion einerlei, allein die Applikation muss auf diesen Fall insbesondere eingerichtet 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+{"basename":"euler_rechenkunst02_1740","par_idx":984,"date":1740,"orig":"Wann aber wie wir allhier geſetzt haben, von allen Sorten in der groͤſſeren Quantitaͤt mehr Stuͤcke vorhanden ſeyn als in der kleineren, ſo hat man zur Subtraction keine beſondere Anleitung noͤthig; ſondern ſubtrahirt nur eine jegliche Sorte der kleineren Zahl von einer jeden gleichen Sorte der groͤſſeren Zahl, und ſetzt alle dieſe gefundene Reſte zuſammen, welche den voͤlligen geſuchten Reſt austragen werden.","norm":"Wann aber wie wir allhier gesetzt haben, von allen Sorten in der größeren Quantität mehr Stücke vorhanden sein als in der kleineren, so hat man zur Subtraktion keine besondere Anleitung nötig; sondern subtrahiert nur eine jegliche Sorte der kleineren Zahl von einer jeden gleichen Sorte der größeren Zahl, und setzt alle diese gefundene Reste zusammen, welche den völligen gesuchten Rest austragen 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Erſtlich iſt klar, daß die Antwort durch die Subtraction gefunden werde; und man zu dieſem Ende die ausgezahlte Summ von derjenigen, welche ſich anfaͤnglich in der Caſſa befunden, abziehen muͤſſe, welches folgender geſtalt geſchieht.","norm":"Antw. Erstlich ist klar, dass die Antwort durch die Subtraktion gefunden werde; und man zu diesem Ende die ausgezahlte Summe von derjenigen, welche sich anfänglich in der Kassa befunden, abziehen müsse, welches folgender Gestalt 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+{"basename":"euler_rechenkunst02_1740","par_idx":991,"date":1740,"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_rechenkunst02_1740","par_idx":992,"date":1740,"orig":"Man ſchreibt nehmlich wie in der Subtraction mit unbenannten Zahlen die kleinere Zahl unter die groͤſſere und unterſtreicht dieſelben mit einer Linie.","norm":"Man schreibt nämlich wie in der Subtraktion mit unbenannten Zahlen die kleinere Zahl unter die größere und unterstreicht dieselben mit einer 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+{"basename":"euler_rechenkunst02_1740","par_idx":993,"date":1740,"orig":"Hernach ſubtrahirt man die 25 Cop. von den 57 Cop. und ſchreibt die reſtirenden 32 Cop. unter die Linie; hierauf ſubtrahirt man gleicher geſtalt die 2340 Rubl von den 5736 Rubl, und ſchreibt die reſtirenden 3396 Rubl ebenfals unter die Linie.","norm":"Hernach subtrahiert man die 25 Cop. von den 57 Cop. und schreibt die restierenden 32 Cop. unter die Linie; hierauf subtrahiert man gleicher Gestalt die 2340 Rubel von den 5736 Rubel, und schreibt die restierenden 3396 Rubel ebenfalls unter die 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+{"basename":"euler_rechenkunst02_1740","par_idx":994,"date":1740,"orig":"Wann nun dieſes geſchehen, ſo ſieht man, daß nach geſchehener Auszahlung der Kaufmann noch 3396 Rubl, 32 Cop. in ſeiner Caſſa behaͤlt.","norm":"Wann nun dieses geschehen, so sieht man, dass nach geschehener Auszahlung der Kaufmann noch 3396 Rubel, 32 Cop. in seiner Kassa 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+{"basename":"euler_rechenkunst02_1740","par_idx":995,"date":1740,"orig":"Dann wann man ſich vorſtellt, daß der Kaufmann in ſeinem Koſten erſtlich 5736 Rubl und noch uͤber das in einem Beutel 57 Cop. hat, ſo kan man ſich die Auszahlung dergeſtalt vorſtellen, daß der Kaufmann erſtlich aus dem Beutel 25 Cop. und dann aus dem Kaſten die 2340 Rubl auszahlt, wodurch die erforderte Bezahlung voͤllig erſtattet wird.","norm":"Dann wann man sich vorstellt, dass der Kaufmann in seinem Kosten erstlich 5736 Rubel und noch über das in einem Beutel 57 Cop. hat, so kann man sich die Auszahlung dergestalt vorstellen, dass der Kaufmann erstlich aus dem Beutel 25 Cop. und dann aus dem Kasten die 2340 Rubel auszahlt, wodurch die erforderte Bezahlung völlig erstattet 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+{"basename":"euler_rechenkunst02_1740","par_idx":1017,"date":1740,"orig":"4.)","norm":"4.)","orig_tok":["4.",")"],"norm_tok":["4.",")"],"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":6.027200222015381} +{"basename":"euler_rechenkunst02_1740","par_idx":1018,"date":1740,"orig":"Wann in der groͤſſeren Quantitaͤt, von welcher die kleinere abgezogen werden ſoll, von einer Sorte weniger Stuͤcke vorhanden ſind, als von eben der Sorte in der kleineren Quantitaͤt, und alſo die Subtraction nach der vorigen Regel nicht verrichtet werden kan; ſo muß man von der naͤchſtfolgenden groͤſſeren Sorte aus der groͤſſeren Quantitaͤt ein Stuͤck entlehnen und daſſelbe nach ſeinem Werth zur kleineren Sorte ſchlagen, da dann die Subtraction wird geſchehen koͤnnen.","norm":"Wann in der größeren Quantität, von welcher die kleinere abgezogen werden soll, von einer Sorte weniger Stücke vorhanden sind, als von eben der Sorte in der kleineren Quantität, und also die Subtraktion nach der vorigen Regel nicht verrichtet werden kann; so muss man von der nächstfolgenden größeren Sorte aus der größeren Quantität ein Stück entlehnen und dasselbe nach seinem Wert zur kleineren Sorte schlagen, da dann die Subtraktion wird geschehen 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kleineren.","orig_tok":["Dem","ungeachtet","aber","kan","es","geſchehen",",","daß","in","der","groͤſſeren","Quantitaͤt","von","den","kleineren","Sorten","weniger","Stuͤcke","vorhanden","ſind",",","als","von","eben","denſelben","Sorten","in","der","kleineren","."],"norm_tok":["Dem","ungeachtet","aber","kann","es","geschehen",",","dass","in","der","größeren","Quantität","von","den","kleineren","Sorten","weniger","Stücke","vorhanden","sind",",","als","von","eben","denselben","Sorten","in","der","kleineren","."],"orig_spans":[[0,3],[4,14],[15,19],[20,23],[24,26],[27,36],[36,37],[38,41],[42,44],[45,48],[49,59],[60,70],[71,74],[75,78],[79,88],[89,95],[96,103],[104,111],[112,121],[122,126],[126,127],[128,131],[132,135],[136,140],[141,150],[151,157],[158,160],[161,164],[165,174],[174,175]],"norm_spans":[[0,3],[4,14],[15,19],[20,24],[25,27],[28,37],[37,38],[39,43],[44,46],[47,50],[51,59],[60,69],[70,73],[74,77],[78,87],[88,94],[95,102],[103,109],[110,119],[120,124],[124,125],[126,129],[130,133],[134,138],[139,148],[149,155],[156,158],[159,162],[163,172],[172,173]],"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":4.238100051879883} 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+{"basename":"euler_rechenkunst02_1740","par_idx":1023,"date":1740,"orig":"Wann derohalben geſchieht, daß von einer kleineren Sorte eine groͤſſere Zahl von einer kleineren abgezogen werden ſoll, ſo kan dieſes nicht unmittelbar geſchehen, ſondern man muß dazu die folgende groͤſſere Sorte zu Huͤlfe nehmen, nach der im Satze beſchriebenen Regel.","norm":"Wann derohalben geschieht, dass von einer kleineren Sorte eine größere Zahl von einer kleineren abgezogen werden soll, so kann dieses nicht unmittelbar geschehen, sondern man muss dazu die folgende größere Sorte zu Hilfe nehmen, nach der im Satze beschriebenen 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+{"basename":"euler_rechenkunst02_1740","par_idx":1024,"date":1740,"orig":"Dieſe Regel aber beruhet auf eben demjenigen Grund, als die in der gemeinen Subtraction mit gantzen Zahlen vorgeſchriebene Regel, wann eine groͤſſere Anzahl von Unitaͤten, oder Decaden, oder Centurien und dergleichen von einer kleineren abgezogen werden ſoll.","norm":"Diese Regel aber beruhet auf eben demjenigen Grund, als die in der gemeinen Subtraktion mit ganzen Zahlen vorgeschriebene Regel, wann eine größere Anzahl von Unitäten, oder Dekaden, oder Centuriones und dergleichen von einer kleineren abgezogen werden 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+{"basename":"euler_rechenkunst02_1740","par_idx":1025,"date":1740,"orig":"Gleichwie nun in dieſem Falle ein Stuͤck von der naͤchſt groͤſſeren Sorte genommen und ſeinem Werthe noch zur kleineren Sorte geſchlagen werden muß, gleichergeſtalt muß man auch in unſerem gegenwaͤrtigen Falle verfahren; dann die verſchiedenen Arten als Unitaͤten, Decades, Centuriae, und ſo fort, ſind ebenfals nichts anders als verſchiedene Sorten der Quantitaͤten.","norm":"Gleichwie nun in diesem Falle ein Stück von der nächst größeren Sorte genommen und seinem Werte noch zur kleineren Sorte geschlagen werden muss, gleichergestalt muss man auch in unserem gegenwärtigen Falle verfahren; dann die verschiedenen Arten als Unitäten, Decades, Centuriae, und so fort, sind ebenfalls nichts anders als verschiedene Sorten der 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+{"basename":"euler_rechenkunst02_1740","par_idx":1026,"date":1740,"orig":"Derohalben wird ſo wohl die gegebene Regel mehr erlaͤutert als der Grund davon deutlich dargethan werden, wann wir davon ein Exempel anfuͤhren.","norm":"Derohalben wird so wohl die gegebene Regel mehr erläutert als der Grund davon deutlich dargetan werden, wann wir davon ein Exempel anführen.","orig_tok":["Derohalben","wird","ſo","wohl","die","gegebene","Regel","mehr","erlaͤutert","als","der","Grund","davon","deutlich","dargethan","werden",",","wann","wir","davon","ein","Exempel","anfuͤhren","."],"norm_tok":["Derohalben","wird","so","wohl","die","gegebene","Regel","mehr","erläutert","als","der","Grund","davon","deutlich","dargetan","werden",",","wann","wir","davon","ein","Exempel","anführen","."],"orig_spans":[[0,10],[11,15],[16,18],[19,23],[24,27],[28,36],[37,42],[43,47],[48,58],[59,62],[63,66],[67,72],[73,78],[79,87],[88,97],[98,104],[104,105],[106,110],[111,114],[115,120],[121,124],[125,132],[133,142],[142,143]],"norm_spans":[[0,10],[11,15],[16,18],[19,23],[24,27],[28,36],[37,42],[43,47],[48,57],[58,61],[62,65],[66,71],[72,77],[78,86],[87,95],[96,102],[102,103],[104,108],[109,112],[113,118],[119,122],[123,130],[131,139],[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],[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":5.872499942779541} +{"basename":"euler_rechenkunst02_1740","par_idx":1027,"date":1740,"orig":"Wir wollen demnach ſetzen, man ſoll nach Hollaͤndiſchen Gelde 125 fl. 15 Stuͤber 13 ₰ von 231 fl. 9 St. 8 ₰ ſubtrahiren, dieſe zwey Summen werden erſtlich unter einander geſchrieben wie folgt.","norm":"Wir wollen demnach setzen, man soll nach Holländischen Gelde 125 fl. 15 Stüber 13 ₰ von 231 fl. 9 St. 8 ₰ subtrahieren, diese zwei Summen werden erstlich untereinander geschrieben wie folgt.","orig_tok":["Wir","wollen","demnach","ſetzen",",","man","ſoll","nach","Hollaͤndiſchen","Gelde","125","fl.","15","Stuͤber","13","₰","von","231","fl.","9","St.","8","₰","ſubtrahiren",",","dieſe","zwey","Summen","werden","erſtlich","unter","einander","geſchrieben","wie","folgt","."],"norm_tok":["Wir","wollen","demnach","setzen",",","man","soll","nach","Holländischen","Gelde","125","fl",".","15","Stüber","13","₰","von","231","fl",".","9","St.","8","₰","subtrahieren",",","diese","zwei","Summen","werden","erstlich","untereinander","geschrieben","wie","folgt","."],"orig_spans":[[0,3],[4,10],[11,18],[19,25],[25,26],[27,30],[31,35],[36,40],[41,55],[56,61],[62,65],[66,69],[70,72],[73,80],[81,83],[84,85],[86,89],[90,93],[94,97],[98,99],[100,103],[104,105],[106,107],[108,119],[119,120],[121,126],[127,131],[132,138],[139,145],[146,154],[155,160],[161,169],[170,181],[182,185],[186,191],[191,192]],"norm_spans":[[0,3],[4,10],[11,18],[19,25],[25,26],[27,30],[31,35],[36,40],[41,54],[55,60],[61,64],[65,67],[67,68],[69,71],[72,78],[79,81],[82,83],[84,87],[88,91],[92,94],[94,95],[96,97],[98,101],[102,103],[104,105],[106,118],[118,119],[120,125],[126,130],[131,137],[138,144],[145,153],[154,167],[168,179],[180,183],[184,189],[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],[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],[31,32],[32,33],[33,34],[34,35],[35,36]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.74459981918335} 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sind.","orig_tok":["Wir","fangen","demnach","die","Subtraction","von","der","kleinſten","Sorte",",","nehmlich","den","Pfenningen","an",";","da","13","₰","abgezogen","werden","ſollen",",","oben","aber","nur","8","₰","vorhanden","ſind","."],"norm_tok":["Wir","fangen","demnach","die","Subtraktion","von","der","kleinsten","Sorte",",","nämlich","den","Pfenningen","an",";","da","13","₰","abgezogen","werden","sollen",",","oben","aber","nur","8","₰","vorhanden","sind","."],"orig_spans":[[0,3],[4,10],[11,18],[19,22],[23,34],[35,38],[39,42],[43,52],[53,58],[58,59],[60,68],[69,72],[73,83],[84,86],[86,87],[88,90],[91,93],[94,95],[96,105],[106,112],[113,119],[119,120],[121,125],[126,130],[131,134],[135,136],[137,138],[139,148],[149,153],[153,154]],"norm_spans":[[0,3],[4,10],[11,18],[19,22],[23,34],[35,38],[39,42],[43,52],[53,58],[58,59],[60,67],[68,71],[72,82],[83,85],[85,86],[87,89],[90,92],[93,94],[95,104],[105,111],[112,118],[118,119],[120,124],[125,129],[130,133],[134,135],[136,137],[138,147],[148,152],[152,153]],"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":4.954400062561035} +{"basename":"euler_rechenkunst02_1740","par_idx":1030,"date":1740,"orig":"Weilen nun dieſes nicht geſchehen kan, ſo nehmen wir von den 9 St. einen Stuͤber weg, und verwechſeln denſelben in Pfenning, welcher folglich 16 ₰ austraͤgt, dieſe 16 ₰ ſchlagen wir zu den 8 ₰, und bekommen 24 ₰, von welchen wir die 13 ₰ abziehen, da dann 11 ₰ uͤberbleiben Nun ſchreiten wir zu den Stuͤbern fort, und muͤſſen 15 Stuͤber nicht von 9 Stuͤbern, ſondern nur von 8 St. weilen wir ſchon 1 Stuͤber zu dem ₰ geſchlagen, abziehen, welches wiederum nicht geſchehen kan.","norm":"Weilen nun dieses nicht geschehen kann, so nehmen wir von den 9 St. einen Stüber weg, und verwechseln denselben in Pfennig, welcher folglich 16 ₰ austrägt, diese 16 ₰ schlagen wir zu den 8 ₰, und bekommen 24 ₰, von welchen wir die 13 ₰ abziehen, da dann 11 ₰ überbleiben nun schreiten wir zu den Stübern fort, und müssen 15 Stüber nicht von 9 Stübern, sondern nur von 8 St. weilen wir schon 1 Stüber zu dem ₰ geschlagen, abziehen, welches wiederum nicht geschehen 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+{"basename":"euler_rechenkunst02_1740","par_idx":1031,"date":1740,"orig":"Derowegen nehmen wir von den vorhandenen 231 fl. 1 Gulden weg, welcher 20 Stuͤber betraͤgt, und dieſe thun wir zu den 8 Stuͤbern, da wir dann 28 Stuͤber bekommen, wovon die 15 St. abgezogen 13 St. zuruͤck laſſen.","norm":"Derowegen nehmen wir von den vorhandenen 231 fl. 1 Gulden weg, welcher 20 Stüber beträgt, und diese tun wir zu den 8 Stübern, da wir dann 28 Stüber bekommen, wovon die 15 St. abgezogen 13 St. zurücklassen.","orig_tok":["Derowegen","nehmen","wir","von","den","vorhandenen","231","fl.","1","Gulden","weg",",","welcher","20","Stuͤber","betraͤgt",",","und","dieſe","thun","wir","zu","den","8","Stuͤbern",",","da","wir","dann","28","Stuͤber","bekommen",",","wovon","die","15","St.","abgezogen","13","St.","zuruͤck","laſſen","."],"norm_tok":["Derowegen","nehmen","wir","von","den","vorhandenen","231","fl",".","1","Gulden","weg",",","welcher","20","Stüber","beträgt",",","und","diese","tun","wir","zu","den","8","Stübern",",","da","wir","dann","28","Stüber","bekommen",",","wovon","die","15","St.","abgezogen","13","St.","zurücklassen","."],"orig_spans":[[0,9],[10,16],[17,20],[21,24],[25,28],[29,40],[41,44],[45,48],[49,50],[51,57],[58,61],[61,62],[63,70],[71,73],[74,81],[82,90],[90,91],[92,95],[96,101],[102,106],[107,110],[111,113],[114,117],[118,119],[120,128],[128,129],[130,132],[133,136],[137,141],[142,144],[145,152],[153,161],[161,162],[163,168],[169,172],[173,175],[176,179],[180,189],[190,192],[193,196],[197,204],[205,211],[211,212]],"norm_spans":[[0,9],[10,16],[17,20],[21,24],[25,28],[29,40],[41,44],[45,47],[47,48],[49,50],[51,57],[58,61],[61,62],[63,70],[71,73],[74,80],[81,88],[88,89],[90,93],[94,99],[100,103],[104,107],[108,110],[111,114],[115,116],[117,124],[124,125],[126,128],[129,132],[133,137],[138,140],[141,147],[148,156],[156,157],[158,163],[164,167],[168,170],[171,174],[175,184],[185,187],[188,191],[192,204],[204,205]],"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],[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],[41,41],[42,42]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.035999774932861} +{"basename":"euler_rechenkunst02_1740","par_idx":1032,"date":1740,"orig":"Wann dies geſchehen, ſo ſubtrahiren wir endlich die 125 fl. von 230 fl. weilen ſchon 1 fl. in Stuͤber verwechſelt worden, da dann 105 fl. uͤberbleiben, ſo daß der voͤllige Reſt ſeyn wird 105 fl. 13 St. 11 ₰.","norm":"Wann dies geschehen, so subtrahieren wir endlich die 125 fl. von 230 fl. weilen schon 1 fl. in Stüber verwechselt worden, da dann 105 fl. überbleiben, so dass der völlige Rest sein wird 105 fl. 13 St. 11 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+{"basename":"euler_rechenkunst02_1740","par_idx":1033,"date":1740,"orig":"Daß nun dieſes der wahre Reſt ſey, kan durch die Addition leicht erwieſen werden, weilen immer wann man den in der Subtraction gefundenen Reſt zur kleineren Zahl addirt, die groͤſſere Zahl heraus kommen muß.","norm":"Dass nun dieses der wahre Rest sei, kann durch die Addition leicht erwiesen werden, weilen immer wann man den in der Subtraktion gefundenen Rest zur kleineren Zahl addiert, die größere Zahl herauskommen 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+{"basename":"euler_rechenkunst02_1740","par_idx":1034,"date":1740,"orig":"Wir wollen demnach dieſe Probe zu mehrerem Beweißtum hieher ſetzen.","norm":"Wir wollen demnach diese Probe zu mehrerem Beweißtum hierhersetzen.","orig_tok":["Wir","wollen","demnach","dieſe","Probe","zu","mehrerem","Beweißtum","hieher","ſetzen","."],"norm_tok":["Wir","wollen","demnach","diese","Probe","zu","mehrerem","Beweißtum","hierhersetzen","."],"orig_spans":[[0,3],[4,10],[11,18],[19,24],[25,30],[31,33],[34,42],[43,52],[53,59],[60,66],[66,67]],"norm_spans":[[0,3],[4,10],[11,18],[19,24],[25,30],[31,33],[34,42],[43,52],[53,66],[66,67]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,8],[10,9]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.757699966430664} +{"basename":"euler_rechenkunst02_1740","par_idx":1035,"date":1740,"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_rechenkunst02_1740","par_idx":1036,"date":1740,"orig":"Nach dieſer Operation, welche unmittelbar in der Natur der Sach gegruͤndet iſt, haben wir die naͤchſtfolgende Sorte der oberen und groͤſſeren Quantitaͤt um ein Stuͤck kleiner betrachtet, weilen ſchon 1 Stuͤck davon in die kleineren Sorten verwechſelt worden.","norm":"Nach dieser Operation, welche unmittelbar in der Natur der Sache gegründet ist, haben wir die nächstfolgende Sorte der oberen und größeren Quantität um ein Stück kleiner betrachtet, weilen schon 1 Stück davon in die kleineren Sorten verwechselt 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+{"basename":"euler_rechenkunst02_1740","par_idx":1037,"date":1740,"orig":"Weilen aber in der Subtraction einerley herauskommt ob man die obere Zahl uͤm eins kleiner oder die untere um eins groͤſſer macht, wie wir in der Subtraction mit unbenannten Zahlen gewieſen haben, ſo kan man ſich auch allhier dieſes Vortheils bedienen, und die Subtraction folgender geſtalt anſtellen.","norm":"Weilen aber in der Subtraktion einerlei herauskommt ob man die obere Zahl um eins kleiner oder die untere um eins größer macht, wie wir in der Subtraktion mit unbenannten Zahlen gewiesen haben, so kann man sich auch allhier dieses Vorteils bedienen, und die Subtraktion folgender Gestalt 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+{"basename":"euler_rechenkunst02_1740","par_idx":1038,"date":1740,"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_rechenkunst02_1740","par_idx":1039,"date":1740,"orig":"Man ſage alſo: 13 ₰ von 8 ₰ kan ich nicht abziehen, deswegen nehme ich 1 Stuͤber, ſo 16 ₰ austraͤgt, dieſe 16 ₰ zu den vorhandenen 8 ₰ gethan machen 24 ₰, davon 13 ₰ abgezogen bleiben 11 ₰. Weilen nun 1 Stuͤber iſt gelehnet worden, ſo mache ich die Anzahl der Stuͤber in der unteren Zahl um 1 groͤſſer, welches durch ein Punct angedeutet werden kan, und ſage 16 St. von 9 St. kan ich nicht abziehen, nehme deswegen dazu 1 fl. und ſetze ſogleich 1 fl. zu den folgenden 125 fl. in der unteren Zahl.","norm":"Man sage also: 13 ₰ von 8 ₰ kann ich nicht abziehen, deswegen nehme ich 1 Stüber, so 16 ₰ austrägt, diese 16 ₰ zu den vorhandenen 8 ₰ getan machen 24 ₰, davon 13 ₰ abgezogen bleiben 11 ₰. Weilen nun 1 Stüber ist gelehnt worden, so mache ich die Anzahl der Stüber in der unteren Zahl um 1 größer, welches durch ein Punkt angedeutet werden kann, und sage 16 St. von 9 St. kann ich nicht abziehen, nehme deswegen dazu 1 fl. und setze sogleich 1 fl. zu den folgenden 125 fl. in der unteren 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über.","orig_tok":["Dieſer","fl.","betraͤgt","20","St.","welche","mit","den","9","St.","zuſammen","machen","29","St.","davon","16","Stuͤber","abgezogen",",","bleiben","13","St.","uͤber","."],"norm_tok":["Dieser","fl",".","beträgt","20","St.","welche","mit","den","9","St.","zusammen","machen","29","St.","davon","16","Stüber","abgezogen",",","bleiben","13","St.","über","."],"orig_spans":[[0,6],[7,10],[11,19],[20,22],[23,26],[27,33],[34,37],[38,41],[42,43],[44,47],[48,56],[57,63],[64,66],[67,70],[71,76],[77,79],[80,87],[88,97],[97,98],[99,106],[107,109],[110,113],[114,119],[119,120]],"norm_spans":[[0,6],[7,9],[9,10],[11,18],[19,21],[22,25],[26,32],[33,36],[37,40],[41,42],[43,46],[47,55],[56,62],[63,65],[66,69],[70,75],[76,78],[79,85],[86,95],[95,96],[97,104],[105,107],[108,111],[112,116],[116,117]],"alignment":[[0,0],[1,1],[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]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.380300045013428} +{"basename":"euler_rechenkunst02_1740","par_idx":1041,"date":1740,"orig":"Endlich ſubtrahire ich 126 fl. von 231 fl. ſo bleiben 105 fl; und wird alſo der Reſt gefunden wie vorher.","norm":"Endlich subtrahiere ich 126 fl. von 231 fl. so bleiben 105 fl; und wird also der Rest gefunden wie 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+{"basename":"euler_rechenkunst02_1740","par_idx":1042,"date":1740,"orig":"Wann in ſolchen Faͤllen ein Stuͤck von der naͤchſtfolgenden groͤſſeren Sorte abgenommen, und in die kleinere Sorte verwechſelt werden muß, ſo pflegt dieſes zwar im Sinn verrichtet zu werden, man kan ſich dabey aber einiger Vortheile bedienen, wodurch oͤfters dieſe Operation weit leichter gemachet wird.","norm":"Wann in solchen Fällen ein Stück von der nächstfolgenden größeren Sorte abgenommen, und in die kleinere Sorte verwechselt werden muss, so pflegt dieses zwar im Sinn verrichtet zu werden, man kann sich dabei aber einiger Vorteile bedienen, wodurch öfters diese Operation weit leichter gemacht 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+{"basename":"euler_rechenkunst02_1740","par_idx":1043,"date":1740,"orig":"Nehmlich anſtatt, daß man, wie allhier geſchehen iſt, das Stuͤck von der groͤſſeren Sorte in die kleinere verwechſelt, und den Betrag davon zu den vorhandenen Stuͤcken von der kleineren Sorte in der oberen Quantitaͤt addirt, und alsdann die Anzahl der Stuͤcke von eben dieſer Sorte in der unteren Quantitaͤt ſubtrahirt: ſo kan man entweder ſogleich die untere Zahl von dem Betrag des genommenen Stuͤcks der groͤſſeren Sorte ſubtrahiren und den Reſt zur oberen Zahl addiren: oder man kan auch die obere Zahl von der unteren Zahl ſubtrahiren, und den Reft nochmalen von dem Werth des abgenommenen Stuͤckes der groͤſſeren Sorte ſubtrahiren.","norm":"Nämlich anstatt, dass man, wie allhier geschehen ist, das Stück von der größeren Sorte in die kleinere verwechselt, und den Betrag davon zu den vorhandenen Stücken von der kleineren Sorte in der oberen Quantität addiert, und alsdann die Anzahl der Stücke von eben dieser Sorte in der unteren Quantität subtrahiert: so kann man entweder sogleich die untere Zahl von dem Betrag des genommenen Stücks der größeren Sorte subtrahieren und den Rest zur oberen Zahl addieren: oder man kann auch die obere Zahl von der unteren Zahl subtrahieren, und den Refft nochmal von dem Wert des abgenommenen Stückes der größeren Sorte 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+{"basename":"euler_rechenkunst02_1740","par_idx":1044,"date":1740,"orig":"Dann durch dieſe beyden Wege wird einerley gefunden, oͤfters aber iſt bald dieſer bald jener bequemer um dieſe Subtraction bloß allein im Sinne zu verrichten.","norm":"Dann durch diese beiden Wege wird einerlei gefunden, öfters aber ist bald dieser bald jener bequemer um diese Subtraktion bloß allein im Sinne zu 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+{"basename":"euler_rechenkunst02_1740","par_idx":1045,"date":1740,"orig":"Jn welchen Faͤllen aber der eine oder der andere Weg einen groͤſſeren Vortheil bringe, wird ein jeder durch eine geringe Ubung bald ſelbſt einſehen.","norm":"In welchen Fällen aber der eine oder der andere Weg einen größeren Vorteil bringe, wird ein jeder durch eine geringe Übung bald selbst einsehen.","orig_tok":["Jn","welchen","Faͤllen","aber","der","eine","oder","der","andere","Weg","einen","groͤſſeren","Vortheil","bringe",",","wird","ein","jeder","durch","eine","geringe","Ubung","bald","ſelbſt","einſehen","."],"norm_tok":["In","welchen","Fällen","aber","der","eine","oder","der","andere","Weg","einen","größeren","Vorteil","bringe",",","wird","ein","jeder","durch","eine","geringe","Übung","bald","selbst","einsehen","."],"orig_spans":[[0,2],[3,10],[11,18],[19,23],[24,27],[28,32],[33,37],[38,41],[42,48],[49,52],[53,58],[59,69],[70,78],[79,85],[85,86],[87,91],[92,95],[96,101],[102,107],[108,112],[113,120],[121,126],[127,131],[132,138],[139,147],[147,148]],"norm_spans":[[0,2],[3,10],[11,17],[18,22],[23,26],[27,31],[32,36],[37,40],[41,47],[48,51],[52,57],[58,66],[67,74],[75,81],[81,82],[83,87],[88,91],[92,97],[98,103],[104,108],[109,116],[117,122],[123,127],[128,134],[135,143],[143,144]],"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":4.926000118255615} +{"basename":"euler_rechenkunst02_1740","par_idx":1046,"date":1740,"orig":"Jnzwiſchen koͤnnen wir ſo viel melden, daß wann die untere Zahl um viel groͤſſer iſt als die obere, alsdann der erſtere der zweyen gewieſenen Vortheile die Operation leichter mache; hingegen aber der andere Vortheil beſſere Dienſte leiſte, wann die obere Zahl nur um ſehr wenig kleiner iſt als die untere.","norm":"Inzwischen können wir so viel melden, dass wann die untere Zahl um viel größer ist als die obere, alsdann der erstere der zweien gewiesenen Vorteile die Operation leichter mache; hingegen aber der andere Vorteil bessere Dienste leiste, wann die obere Zahl nur um sehr wenig kleiner ist als die untere.","orig_tok":["Jnzwiſchen","koͤnnen","wir","ſo","viel","melden",",","daß","wann","die","untere","Zahl","um","viel","groͤſſer","iſt","als","die","obere",",","alsdann","der","erſtere","der","zweyen","gewieſenen","Vortheile","die","Operation","leichter","mache",";","hingegen","aber","der","andere","Vortheil","beſſere","Dienſte","leiſte",",","wann","die","obere","Zahl","nur","um","ſehr","wenig","kleiner","iſt","als","die","untere","."],"norm_tok":["Inzwischen","können","wir","so","viel","melden",",","dass","wann","die","untere","Zahl","um","viel","größer","ist","als","die","obere",",","alsdann","der","erstere","der","zweien","gewiesenen","Vorteile","die","Operation","leichter","mache",";","hingegen","aber","der","andere","Vorteil","bessere","Dienste","leiste",",","wann","die","obere","Zahl","nur","um","sehr","wenig","kleiner","ist","als","die","untere","."],"orig_spans":[[0,10],[11,18],[19,22],[23,25],[26,30],[31,37],[37,38],[39,42],[43,47],[48,51],[52,58],[59,63],[64,66],[67,71],[72,80],[81,84],[85,88],[89,92],[93,98],[98,99],[100,107],[108,111],[112,119],[120,123],[124,130],[131,141],[142,151],[152,155],[156,165],[166,174],[175,180],[180,181],[182,190],[191,195],[196,199],[200,206],[207,215],[216,223],[224,231],[232,238],[238,239],[240,244],[245,248],[249,254],[255,259],[260,263],[264,266],[267,271],[272,277],[278,285],[286,289],[290,293],[294,297],[298,304],[304,305]],"norm_spans":[[0,10],[11,17],[18,21],[22,24],[25,29],[30,36],[36,37],[38,42],[43,47],[48,51],[52,58],[59,63],[64,66],[67,71],[72,78],[79,82],[83,86],[87,90],[91,96],[96,97],[98,105],[106,109],[110,117],[118,121],[122,128],[129,139],[140,148],[149,152],[153,162],[163,171],[172,177],[177,178],[179,187],[188,192],[193,196],[197,203],[204,211],[212,219],[220,227],[228,234],[234,235],[236,240],[241,244],[245,250],[251,255],[256,259],[260,262],[263,267],[268,273],[274,281],[282,285],[286,289],[290,293],[294,300],[300,301]],"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]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.577499866485596} +{"basename":"euler_rechenkunst02_1740","par_idx":1047,"date":1740,"orig":"Wir wollen dieſe beyden Wege aber im folgenden Exempel deutlicher beſchreiben und den Gebrauch davon anzeigen.","norm":"Wir wollen diese beiden Wege aber im folgenden Exempel deutlicher beschreiben und den Gebrauch davon 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+{"basename":"euler_rechenkunst02_1740","par_idx":1048,"date":1740,"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_rechenkunst02_1740","par_idx":1049,"date":1740,"orig":"Dieſes Exempel wollen wir aber erſtlich nach der im Satze beſchriebenen Art berechnen, damit man die Ubereinſtimmung des natuͤrlichen Wegs mit den angezeigten Vortheilen deſto deutlicher einſehe.","norm":"Dieses Exempel wollen wir aber erstlich nach der im Satze beschriebenen Art berechnen, damit man die Übereinstimmung des natürlichen Wegs mit den angezeigten Vorteilen desto deutlicher 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+{"basename":"euler_rechenkunst02_1740","par_idx":1050,"date":1740,"orig":"Jch ſage alſo 18 gr. von 5 gr. kan ich nicht, lehne alſo einen Serupel, welchen durch ein Punckt bey den ℈ in der unteren Quantitaͤt andeute.","norm":"Ich sage also 18 gr. von 5 gr. kann ich nicht, lehne also einen Serupel, welchen durch ein Punkt bei den ℈ in der unteren Quantität 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+{"basename":"euler_rechenkunst02_1740","par_idx":1059,"date":1740,"orig":"Nun wollen wir eben dieſes Exempel durch die angewieſenen Vortheile berechnen.","norm":"Nun wollen wir eben dieses Exempel durch die angewiesenen Vorteile berechnen.","orig_tok":["Nun","wollen","wir","eben","dieſes","Exempel","durch","die","angewieſenen","Vortheile","berechnen","."],"norm_tok":["Nun","wollen","wir","eben","dieses","Exempel","durch","die","angewiesenen","Vorteile","berechnen","."],"orig_spans":[[0,3],[4,10],[11,14],[15,19],[20,26],[27,34],[35,40],[41,44],[45,57],[58,67],[68,77],[77,78]],"norm_spans":[[0,3],[4,10],[11,14],[15,19],[20,26],[27,34],[35,40],[41,44],[45,57],[58,66],[67,76],[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":5.9781999588012695} +{"basename":"euler_rechenkunst02_1740","par_idx":1060,"date":1740,"orig":"[Formel] Nehmlich weilen ich 18 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+{"basename":"euler_rechenkunst02_1740","par_idx":1061,"date":1740,"orig":"Zweytens ſage ich 2 und 1 ℈ machen 3 ℈, welchen von den obſtehenden 2 ℈ nicht abgezogen werden koͤnnen, lehne allſo 1 ℨ welche 3 ℈ betraͤgt, und ſage nach dem erſteren Vortheil 3 ℈ von 1 ℨ oder 3 ℈ bleibt nichts uͤber, dieſes zu den 2 ℈ gethan gibt 2 ℈ fuͤr den Reſt: oder nach dem zweyten Vortheil ſage ich die oberen 2 ℈ von den unteren 3 ℈ abgezogen bleibt 1 ℈ uͤber, dieſer von 1 ℨ oder 3 ℈ abgezogen laͤßt 2 ℈ fuͤr den Reſt.","norm":"Zweitens sage ich 2 und 1 ℈ machen 3 ℈, welchen von den stehenden 2 ℈ nicht abgezogen werden können, lehne also 1 ℨ welche 3 ℈ beträgt, und sage nach dem ersteren Vorteil 3 ℈ von 1 ℨ oder 3 ℈ bleibt nichts über, dieses zu den 2 ℈ getan gibt 2 ℈ für den Rest: oder nach dem zweiten Vorteil sage ich die oberen 2 ℈ von den unteren 3 ℈ abgezogen bleibt 1 ℈ über, dieser von 1 ℨ oder 3 ℈ abgezogen lässt 2 ℈ für den 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+{"basename":"euler_rechenkunst02_1740","par_idx":1062,"date":1740,"orig":"Drittens ſollen wir 7 ℨ von 1 ℨ abziehen, welches weilen es nicht geſchehen kan, ſo lehnen wir 1 ℥, die betraͤgt 8 ℨ; und ſagen nach dem erſteren Weg 7 ℨ von 1 ℥ oder 8 ℨ bleiben 1 ℨ und 1 ℨ machen 2 ℨ fuͤr den Reſt; oder nach der anderen Art 1 ℨ von 7 ℨ bleiben 6 ℨ, dieſe von 1 ℥ oder 8 ℨ abgezogen, bleiben 2 ℨ fuͤr den Reſt.","norm":"Drittens sollen wir 7 ℨ von 1 ℨ abziehen, welches weilen es nicht geschehen kann, so lehnen wir 1 ℥, die beträgt 8 ℨ; und sagen nach dem ersteren Weg 7 ℨ von 1 ℥ oder 8 ℨ bleiben 1 ℨ und 1 ℨ machen 2 ℨ für den Rest; oder nach der anderen Art 1 ℨ von 7 ℨ bleiben 6 ℨ, diese von 1 ℥ oder 8 ℨ abgezogen, bleiben 2 ℨ für den 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+{"basename":"euler_rechenkunst02_1740","par_idx":1108,"date":1740,"orig":"Nehmlich wann man die folgende groͤſſere Sorte durch den Multiplicatorem multiplicirt hat, ſo addirt man zum Product die aus der vorigen Multiplication entſprungenen Stuͤcke von dieſer Sorte, und ſucht wiederum wieviel gantze Stuͤcke von der folgenden groͤſſeren Sorte in dieſer Summ enthalten ſind, welche wiederum zur folgenden Multiplication aufbehalten werden, in das Product aber werden ſo viel Stuͤcke geſchrieben, als nach geſchehener Reduction von dieſer Sorte uͤberſchieſſen: und ſolcher geſtalt verfaͤhrt man, bis man zur groͤſten Sorte komt, welche man multiplicirt und das Product ohne weitere Reduction hinſchreibt.","norm":"Nämlich wann man die folgende größere Sorte durch den Multiplikatorrem multipliziert hat, so addiert man zum Produkt die aus der vorigen Multiplikation entsprungenen Stücke von dieser Sorte, und sucht wiederum wie viel ganze Stücke von der folgenden größeren Sorte in dieser Summe enthalten sind, welche wiederum zur folgenden Multiplikation aufbehalten werden, in das Produkt aber werden so viel Stücke geschrieben, als nach geschehener Reduktion von dieser Sorte überschießen: und solcher Gestalt verfährt man, bis man zur größten Sorte kommt, welche man multipliziert und das Produkt ohne weitere Reduktion hinschreibt.","orig_tok":["Nehmlich","wann","man","die","folgende","groͤſſere","Sorte","durch","den","Multiplicatorem","multiplicirt","hat",",","ſo","addirt","man","zum","Product","die","aus","der","vorigen","Multiplication","entſprungenen","Stuͤcke","von","dieſer","Sorte",",","und","ſucht","wiederum","wieviel","gantze","Stuͤcke","von","der","folgenden","groͤſſeren","Sorte","in","dieſer","Summ","enthalten","ſind",",","welche","wiederum","zur","folgenden","Multiplication","aufbehalten","werden",",","in","das","Product","aber","werden","ſo","viel","Stuͤcke","geſchrieben",",","als","nach","geſchehener","Reduction","von","dieſer","Sorte","uͤberſchieſſen",":","und","ſolcher","geſtalt","verfaͤhrt","man",",","bis","man","zur","groͤſten","Sorte","komt",",","welche","man","multiplicirt","und","das","Product","ohne","weitere","Reduction","hinſchreibt","."],"norm_tok":["Nämlich","wann","man","die","folgende","größere","Sorte","durch","den","Multiplikatorrem","multipliziert","hat",",","so","addiert","man","zum","Produkt","die","aus","der","vorigen","Multiplikation","entsprungenen","Stücke","von","dieser","Sorte",",","und","sucht","wiederum","wie","viel","ganze","Stücke","von","der","folgenden","größeren","Sorte","in","dieser","Summe","enthalten","sind",",","welche","wiederum","zur","folgenden","Multiplikation","aufbehalten","werden",",","in","das","Produkt","aber","werden","so","viel","Stücke","geschrieben",",","als","nach","geschehener","Reduktion","von","dieser","Sorte","überschießen",":","und","solcher","Gestalt","verfährt","man",",","bis","man","zur","größten","Sorte","kommt",",","welche","man","multipliziert","und","das","Produkt","ohne","weitere","Reduktion","hinschreibt","."],"orig_spans":[[0,8],[9,13],[14,17],[18,21],[22,30],[31,40],[41,46],[47,52],[53,56],[57,72],[73,85],[86,89],[89,90],[91,93],[94,100],[101,104],[105,108],[109,116],[117,120],[121,124],[125,128],[129,136],[137,151],[152,165],[166,173],[174,177],[178,184],[185,190],[190,191],[192,195],[196,201],[202,210],[211,218],[219,225],[226,233],[234,237],[238,241],[242,251],[252,262],[263,268],[269,271],[272,278],[279,283],[284,293],[294,298],[298,299],[300,306],[307,315],[316,319],[320,329],[330,344],[345,356],[357,363],[363,364],[365,367],[368,371],[372,379],[380,384],[385,391],[392,394],[395,399],[400,407],[408,419],[419,420],[421,424],[425,429],[430,441],[442,451],[452,455],[456,462],[463,468],[469,483],[483,484],[485,488],[489,496],[497,504],[505,514],[515,518],[518,519],[520,523],[524,527],[528,531],[532,540],[541,546],[547,551],[551,552],[553,559],[560,563],[564,576],[577,580],[581,584],[585,592],[593,597],[598,605],[606,615],[616,627],[627,628]],"norm_spans":[[0,7],[8,12],[13,16],[17,20],[21,29],[30,37],[38,43],[44,49],[50,53],[54,70],[71,84],[85,88],[88,89],[90,92],[93,100],[101,104],[105,108],[109,116],[117,120],[121,124],[125,128],[129,136],[137,151],[152,165],[166,172],[173,176],[177,183],[184,189],[189,190],[191,194],[195,200],[201,209],[210,213],[214,218],[219,224],[225,231],[232,235],[236,239],[240,249],[250,258],[259,264],[265,267],[268,274],[275,280],[281,290],[291,295],[295,296],[297,303],[304,312],[313,316],[317,326],[327,341],[342,353],[354,360],[360,361],[362,364],[365,368],[369,376],[377,381],[382,388],[389,391],[392,396],[397,403],[404,415],[415,416],[417,420],[421,425],[426,437],[438,447],[448,451],[452,458],[459,464],[465,477],[477,478],[479,482],[483,490],[491,498],[499,507],[508,511],[511,512],[513,516],[517,520],[521,524],[525,532],[533,538],[539,544],[544,545],[546,552],[553,556],[557,570],[571,574],[575,578],[579,586],[587,591],[592,599],[600,609],[610,621],[621,622]],"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],[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]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.543399810791016} 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+{"basename":"euler_rechenkunst02_1740","par_idx":1113,"date":1740,"orig":"Alſo kan man nicht fragen, wieviel heraus komme, wann man 100 Rubl mit 10 Rubl multiplicirt; dann wann man antworten ſollte, daß 1000 Rubl herauskaͤmen, ſo wuͤrde dieſes die gehoͤrige Antwort ſeyn, wann gefragt worden waͤre, wieviel 100 Rubl 10 mahl, aber nicht 10 Rubl mahl genommen ausmachten.","norm":"Also kann man nicht fragen, wie viel herauskomme, wann man 100 Rubel mit 10 Rubel multipliziert; dann wann man antworten sollte, dass 1000 Rubel herauskämen, so würde dieses die gehörige Antwort sein, wann gefragt worden wäre, wie viel 100 Rubel 10 Mal, aber nicht 10 Rubel Mal genommen 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+{"basename":"euler_rechenkunst02_1740","par_idx":1116,"date":1740,"orig":"Als da 8 mit 3 multiplicirt 24 ausmachen, ſo geben auch 8 Rubl mit 3 multiplicirt 24 Rubl, und 8 Pfund mit 3 multiplicirt 24 Pfund und ſo fort: weswegen zur Multiplication der benannten Zahlen keine andere Regeln erfordert werden, als zur Multiplication der unbenannten Zahlen gegeben worden ſind.","norm":"Als da 8 mit 3 multipliziert 24 ausmachen, so geben auch 8 Rubel mit 3 multipliziert 24 Rubel, und 8 Pfund mit 3 multipliziert 24 Pfund und so fort: weswegen zur Multiplikation der benannten Zahlen keine andere Regeln erfordert werden, als zur Multiplikation der unbenannten Zahlen gegeben worden 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+{"basename":"euler_rechenkunst02_1740","par_idx":1119,"date":1740,"orig":"dann wann derſelbe ein Bruch waͤre, ſo muß man zu einer ſolchen Multiplication noch die Diviſion zu Huͤlfe nehmen, weswegen wir allhier nur die Multiplication mit gantzen Zahlen fuͤr uns nehmen, und die mit Bruͤchen ins folgende Capitel verſparen.","norm":"dann wann derselbe ein Bruch wäre, so muss man zu einer solchen Multiplikation noch die Division zu Hilfe nehmen, weswegen wir allhier nur die Multiplikation mit ganzen Zahlen für uns nehmen, und die mit Brüchen ins folgende Kapitel 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+{"basename":"euler_rechenkunst02_1740","par_idx":1120,"date":1740,"orig":"Um alſo eine aus vielevley Sorten beſtehende Quantitaͤt durch einen vorgeſchriebenen Multiplicator zu multipliciren, ſo darf man nur eine jegliche Sorte durch den Multiplicatorem insbeſondere multipliciren; da dann alle dieſe Producte zuſammen das voͤllige verlangte Product geben werden.","norm":"Um also eine aus vielevley Sorten bestehende Quantität durch einen vorgeschriebenen Multiplicator zu multiplizieren, so darf man nur eine jegliche Sorte durch den Multiplikatorrem insbesondere multiplizieren; da dann alle diese Produkte zusammen das völlige verlangte Produkt geben 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+{"basename":"euler_rechenkunst02_1740","par_idx":1121,"date":1740,"orig":"Als wann man dieſes Gewicht 9 Pud, 24 ℔, 15 Loth 8 mahl nehmen oder durch 8 multipliciren ſoll, ſo wird man das verlangte Product folgender geſtalt durch Multiplicirung einer jeden Sorte mit 8 finden.","norm":"Als wann man dieses Gewicht 9 Pud, 24 ℔, 15 Lot 8 Mal nehmen oder durch 8 multiplizieren soll, so wird man das verlangte Produkt folgender Gestalt durch Multiplizierung einer jeden Sorte mit 8 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+{"basename":"euler_rechenkunst02_1740","par_idx":1122,"date":1740,"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_rechenkunst02_1740","par_idx":1123,"date":1740,"orig":"Allein da man dergleichen Ausdruͤckungen in ſolcher Form verlangt, daß vom keiner kleineren Sorte ſo viel oder mehr Stuͤcke vorkommen als ein Stuͤck von der naͤchſtfolgenden groͤſſeren Sorte ausmachen; ſo muß bey einem auf dieſe Art gefundenen Product nach die gehoͤrige Reduction angeſtellet werden.","norm":"Allein da man dergleichen Ausdrückungen in solcher Form verlangt, dass vom keiner kleineren Sorte so viel oder mehr Stücke vorkommen als ein Stück von der nächstfolgenden größeren Sorte ausmachen; so muss bei einem auf diese Art gefundenen Produkt nach die 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+{"basename":"euler_rechenkunst02_1740","par_idx":1124,"date":1740,"orig":"Damit man aber nicht noͤthig habe zwey mahl zu operiren, und das Product in der gehoͤrigen Form ſogleich finde, ſo haben wir in der im Satze beſchriebenen Regel die Multiplications und Reductions Operationen mit einander verknuͤpfet.","norm":"Damit man aber nicht nötig habe zweimal zu operieren, und das Produkt in der gehörigen Form sogleich finde, so haben wir in der im Satze beschriebenen Regel die Multiplikation und Reduktion Operationen miteinander 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+{"basename":"euler_rechenkunst02_1740","par_idx":1125,"date":1740,"orig":"Weilen demnach mit der Reduction der Anfang immer von der kleinſten Sorte gemacht werden muß, ſo iſt auch noͤthig zu dieſem Ende die Multiplication von der kleinſten Sorte anzufangen.","norm":"Weilen demnach mit der Reduktion der Anfang immer von der kleinsten Sorte gemacht werden muss, so ist auch nötig zu diesem Ende die Multiplikation von der kleinsten Sorte 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+{"basename":"euler_rechenkunst02_1740","par_idx":1126,"date":1740,"orig":"Wann nun die kleinſte Sorte multiplicirt worden, ſo muß man ſehen, ob das gefundene Product mehr Stuͤcke anzeige, als ein Stuͤck der naͤchſtfolgenden groͤſſeren Sorte ausmachen oder nicht: befindet ſich das letztere, ſo darf man nur ſogleich das Product hinſchreiben; ſind aber im Product ein oder mehr Stuͤcke von der naͤchſtfolgenden groͤſſeren Sorte enthalten, ſo muͤſſen ſolche durch die Reduction daraus gezogen und nachgehends zum Product, welches aus der folgenden Sorte entſpringen wird, gethan werden; unter dem Nahmen aber einer jeglichen Sorte werden nur ſo viel Stuͤcke ins Product geſchrieben als in der Reduction uͤbergeſchoſſen.","norm":"Wann nun die kleinste Sorte multipliziert worden, so muss man sehen, ob das gefundene Produkt mehr Stücke anzeige, als ein Stück der nächstfolgenden größeren Sorte ausmachen oder nicht: befindet sich das letztere, so darf man nur sogleich das Produkt hinschreiben; sind aber im Produkt ein oder mehr Stücke von der nächstfolgenden größeren Sorte enthalten, so müssen solche durch die Reduktion daraus gezogen und nachgehend zum Produkt, welches aus der folgenden Sorte entspringen wird, getan werden; unter dem Namen aber einer jeglichen Sorte werden nur so viel Stücke ins Produkt geschrieben als in der Reduktion 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Weilen im Apotheker Gewicht noch zwey hoͤhere Sorten als Drachmen uͤblich ſind, nehmlich Unzen und Pfund und man im Product auf dieſe hoͤheren Sorten kommen wird, ſo betrachtet man dieſelben als wann ſie im Multiplicando ſchon wuͤrcklich vorhanden, ihre Stellen aber ledig waͤren.","norm":"Antw. 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+{"basename":"euler_rechenkunst02_1740","par_idx":1183,"date":1740,"orig":"[Formel] Nehmlich wann man das Product nach der vorher beſchriebenen Art ausrechnet und keine groͤſſere Sorten in Betrachtung ziehet als im Multiplicando vorhanden geweſen ſind, ſo findt man 586 ℨ, 2 ℨ, — gr.","norm":"[Formel] Nämlich wann man das Produkt nach der vorher beschriebenen Art ausrechnet und keine größere Sorten in Betrachtung ziehet als im Multiplicando vorhanden gewesen sind, so findet man 586 ℨ, 2 ℨ, — 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+{"basename":"euler_rechenkunst02_1740","par_idx":1188,"date":1740,"orig":"Jngleichem wann der Multiplicator zweymal ſo groß iſt als die benannte Anzahl Stuͤcke, welche von der kleineren Sorte ein Stuͤcke der groͤſſeren ausmachen, ſo darf man nur die kleinere Sorte mit zwey multipliciren, und dem Product den Nahmen der groͤſſeren Sorte beylegen.","norm":"Ingleichem wann der Multiplicator zweimal so groß ist als die benannte Anzahl Stücke, welche von der kleineren Sorte ein Stücke der größeren ausmachen, so darf man nur die kleinere Sorte mit zwei multiplizieren, und dem Produkt den Namen der größeren Sorte 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+{"basename":"euler_rechenkunst02_1740","par_idx":1189,"date":1740,"orig":"Eben dieſes Vortheils kan man ſich bedienen wann der Multiplicator drey, vier oder mehrmalen groͤſſer iſt als die gedachte Zahl, welche anzeint wie viel Stuͤcke der kleineren Sorte ein Stuͤck der groͤſſeren ausmachen.","norm":"Eben dieses Vorteils kann man sich bedienen wann der Multiplicator drei, vier oder mehrmals größer ist als die gedachte Zahl, welche anzeint wie viel Stücke der kleineren Sorte ein Stück der größeren 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+{"basename":"euler_rechenkunst02_1740","par_idx":1190,"date":1740,"orig":"Jn welchen Faͤllen dieſer gemeldte Vortheil ſtatt finde, iſt aus der gegebenen Beſchreibung leicht abzunehmen, der Vortheil aber beſtehet an und fuͤr ſich ſelbſt darinn, daß man die ſonſt nach der Multiplication noͤthige Reduction zur folgenden groͤſſeren Sorte zugleich mit anſtellet.","norm":"In welchen Fällen dieser gemeldete Vorteil stattfinde, ist aus der gegebenen Beschreibung leicht abzunehmen, der Vorteil aber bestehet an und für sich selbst darin, dass man die sonst nach der Multiplikation nötige Reduktion zur folgenden größeren Sorte zugleich mit 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+{"basename":"euler_rechenkunst02_1740","par_idx":1191,"date":1740,"orig":"Als wann man zum Exempel 6 ℨ mit 8 multipliciren ſollte, ſo wuͤrden nach dem gewoͤhnlichen Wege 48 ℨ heraus kommen, welche weilen 8 ℨ eine Unze ausmachen, durch 8 dividirt 6 ℥ geben.","norm":"Als wann man zum Exempel 6 ℨ mit 8 multiplizieren sollte, so würden nach dem gewöhnlichen Wege 48 ℨ herauskommen, welche weilen 8 ℨ eine Unze ausmachen, durch 8 dividiert 6 ℥ geben.","orig_tok":["Als","wann","man","zum","Exempel","6","ℨ","mit","8","multipliciren","ſollte",",","ſo","wuͤrden","nach","dem","gewoͤhnlichen","Wege","48","ℨ","heraus","kommen",",","welche","weilen","8","ℨ","eine","Unze","ausmachen",",","durch","8","dividirt","6","℥","geben","."],"norm_tok":["Als","wann","man","zum","Exempel","6","ℨ","mit","8","multiplizieren","sollte",",","so","würden","nach","dem","gewöhnlichen","Wege","48","ℨ","herauskommen",",","welche","weilen","8","ℨ","eine","Unze","ausmachen",",","durch","8","dividiert","6","℥","geben","."],"orig_spans":[[0,3],[4,8],[9,12],[13,16],[17,24],[25,26],[27,28],[29,32],[33,34],[35,48],[49,55],[55,56],[57,59],[60,67],[68,72],[73,76],[77,90],[91,95],[96,98],[99,100],[101,107],[108,114],[114,115],[116,122],[123,129],[130,131],[132,133],[134,138],[139,143],[144,153],[153,154],[155,160],[161,162],[163,171],[172,173],[174,175],[176,181],[181,182]],"norm_spans":[[0,3],[4,8],[9,12],[13,16],[17,24],[25,26],[27,28],[29,32],[33,34],[35,49],[50,56],[56,57],[58,60],[61,67],[68,72],[73,76],[77,89],[90,94],[95,97],[98,99],[100,112],[112,113],[114,120],[121,127],[128,129],[130,131],[132,136],[137,141],[142,151],[151,152],[153,158],[159,160],[161,170],[171,172],[173,174],[175,180],[180,181]],"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],[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]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.744800090789795} +{"basename":"euler_rechenkunst02_1740","par_idx":1192,"date":1740,"orig":"Da man nun erſtlich durch 8 multipliciren, und als dann das Product wiederum durch 8 dividiren muß, ſo iſt klar, daß wiederum das erſte heraus kommen muͤſſe.","norm":"Da man nun erstlich durch 8 multiplizieren, und als dann das Produkt wiederum durch 8 dividieren muss, so ist klar, dass wiederum das erste herauskommen 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+{"basename":"euler_rechenkunst02_1740","par_idx":1194,"date":1740,"orig":"Gleichergeſtalt, da 3 Solotnick ein Loth machen, ſo gibt 1 Solotnick 3 mal genommen 1 Loth und 2 Solotnick mit 3 multiplicirt 2 Loth.","norm":"Gleichergestalt, da 3 Solotnick ein Lot machen, so gibt 1 Solotnick 3 Mal genommen 1 Lot und 2 Solotnick mit 3 multipliziert 2 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+{"basename":"euler_rechenkunst02_1740","par_idx":1195,"date":1740,"orig":"Und weilen 96 Solotnick ein Pfund machen, ſo iſt ſehr leicht eine jegliche gegebene Anzahl Solotnick mit 96 zu multipliciren, in dem des Product eben ſo viel Pfund anzeigen muß, als der Multiplicandus Solotnick gehalten hat.","norm":"Und weilen 96 Solotnick ein Pfund machen, so ist sehr leicht eine jegliche gegebene Anzahl Solotnick mit 96 zu multiplizieren, in dem des Produkt ebenso viel Pfund anzeigen muss, als der Multiplicandus Solotnick gehalten 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+{"basename":"euler_rechenkunst02_1740","par_idx":1196,"date":1740,"orig":"Alſo werden 15 Solotnick mit 96 multiplicirt 15 Pfund geben und ſo ſort.","norm":"Also werden 15 Solotnick mit 96 multipliziert 15 Pfund geben und so sort.","orig_tok":["Alſo","werden","15","Solotnick","mit","96","multiplicirt","15","Pfund","geben","und","ſo","ſort","."],"norm_tok":["Also","werden","15","Solotnick","mit","96","multipliziert","15","Pfund","geben","und","so","sort","."],"orig_spans":[[0,4],[5,11],[12,14],[15,24],[25,28],[29,31],[32,44],[45,47],[48,53],[54,59],[60,63],[64,66],[67,71],[71,72]],"norm_spans":[[0,4],[5,11],[12,14],[15,24],[25,28],[29,31],[32,45],[46,48],[49,54],[55,60],[61,64],[65,67],[68,72],[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],[11,11],[12,12],[13,13]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":7.456600189208984} +{"basename":"euler_rechenkunst02_1740","par_idx":1197,"date":1740,"orig":"Hieraus erhellet ferner, daß 8 Copeken mit 10 multiplicirt 8 Griven, und 3 Griven mit 10 multiplicirt 3 Rubl. geben muͤſſen, weilen 10 Copeken 1 Griven und 10 Griven 1 Rubl. ausmachen.","norm":"Hieraus erhellet ferner, dass 8 Kopeken mit 10 multipliziert 8 griffen, und 3 griffen mit 10 multipliziert 3 Rubel . geben müssen, weilen 10 Kopeken 1 griffen und 10 griffen 1 Rubel . ausmachen.","orig_tok":["Hieraus","erhellet","ferner",",","daß","8","Copeken","mit","10","multiplicirt","8","Griven",",","und","3","Griven","mit","10","multiplicirt","3","Rubl",".","geben","muͤſſen",",","weilen","10","Copeken","1","Griven","und","10","Griven","1","Rubl",".","ausmachen","."],"norm_tok":["Hieraus","erhellet","ferner",",","dass","8","Kopeken","mit","10","multipliziert","8","griffen",",","und","3","griffen","mit","10","multipliziert","3","Rubel",".","geben","müssen",",","weilen","10","Kopeken","1","griffen","und","10","griffen","1","Rubel",".","ausmachen","."],"orig_spans":[[0,7],[8,16],[17,23],[23,24],[25,28],[29,30],[31,38],[39,42],[43,45],[46,58],[59,60],[61,67],[67,68],[69,72],[73,74],[75,81],[82,85],[86,88],[89,101],[102,103],[104,108],[108,109],[110,115],[116,123],[123,124],[125,131],[132,134],[135,142],[143,144],[145,151],[152,155],[156,158],[159,165],[166,167],[168,172],[172,173],[174,183],[183,184]],"norm_spans":[[0,7],[8,16],[17,23],[23,24],[25,29],[30,31],[32,39],[40,43],[44,46],[47,60],[61,62],[63,70],[70,71],[72,75],[76,77],[78,85],[86,89],[90,92],[93,106],[107,108],[109,114],[115,116],[117,122],[123,129],[129,130],[131,137],[138,140],[141,148],[149,150],[151,158],[159,162],[163,165],[166,173],[174,175],[176,181],[182,183],[184,193],[193,194]],"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":5.152699947357178} +{"basename":"euler_rechenkunst02_1740","par_idx":1198,"date":1740,"orig":"Weilen ein Hollaͤndiſcher Stuͤber 16 ₰ halt, ſo muͤſſen auch zum Exempel 9 ₰ mit 16 multiplicirt 9 St. und 14 ₰ mit 16 multiplicirt 14 St. hervor bringen, welches alles fuͤr ſich ſo deutlich iſt, daß wir keiner weiteren Ausfuͤhrung vonnoͤthen haben.","norm":"Weilen ein Holländischer Stüber 16 ₰ halte, so müssen auch zum Exempel 9 ₰ mit 16 multipliziert 9 St. und 14 ₰ mit 16 multipliziert 14 St. hervorbringen, welches alles für sich so deutlich ist, dass wir keiner weiteren Ausführung vonnöten haben.","orig_tok":["Weilen","ein","Hollaͤndiſcher","Stuͤber","16","₰","halt",",","ſo","muͤſſen","auch","zum","Exempel","9","₰","mit","16","multiplicirt","9","St.","und","14","₰","mit","16","multiplicirt","14","St.","hervor","bringen",",","welches","alles","fuͤr","ſich","ſo","deutlich","iſt",",","daß","wir","keiner","weiteren","Ausfuͤhrung","vonnoͤthen","haben","."],"norm_tok":["Weilen","ein","Holländischer","Stüber","16","₰","halte",",","so","müssen","auch","zum","Exempel","9","₰","mit","16","multipliziert","9","St.","und","14","₰","mit","16","multipliziert","14","St.","hervorbringen",",","welches","alles","für","sich","so","deutlich","ist",",","dass","wir","keiner","weiteren","Ausführung","vonnöten","haben","."],"orig_spans":[[0,6],[7,10],[11,25],[26,33],[34,36],[37,38],[39,43],[43,44],[45,47],[48,55],[56,60],[61,64],[65,72],[73,74],[75,76],[77,80],[81,83],[84,96],[97,98],[99,102],[103,106],[107,109],[110,111],[112,115],[116,118],[119,131],[132,134],[135,138],[139,145],[146,153],[153,154],[155,162],[163,168],[169,173],[174,178],[179,181],[182,190],[191,194],[194,195],[196,199],[200,203],[204,210],[211,219],[220,231],[232,242],[243,248],[248,249]],"norm_spans":[[0,6],[7,10],[11,24],[25,31],[32,34],[35,36],[37,42],[42,43],[44,46],[47,53],[54,58],[59,62],[63,70],[71,72],[73,74],[75,78],[79,81],[82,95],[96,97],[98,101],[102,105],[106,108],[109,110],[111,114],[115,117],[118,131],[132,134],[135,138],[139,152],[152,153],[154,161],[162,167],[168,171],[172,176],[177,179],[180,188],[189,192],[192,193],[194,198],[199,202],[203,209],[210,218],[219,229],[230,238],[239,244],[244,245]],"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],[29,28],[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]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.611599922180176} +{"basename":"euler_rechenkunst02_1740","par_idx":1199,"date":1740,"orig":"Hierauf beruhet ferner der Grund der andern gemeldten Vortheile, wann nehmlich der Multiplicator zwey oder drey oder mehr mal groͤſſer iſt, als ſolcher im vorigen Falle iſt angenommen worden.","norm":"Hierauf beruhet ferner der Grund der anderen gemeldeten Vorteile, wann nämlich der Multiplicator zwei oder drei oder mehr Mal größer ist, als solcher im vorigen Falle ist angenommen 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+{"basename":"euler_rechenkunst02_1740","par_idx":1200,"date":1740,"orig":"Dann da ein zweymal ſo groſſer Multiplicator ein zweymal ſo groſſes Product hervorbringt, wann nehmlich der Multiplicandus einerley iſt, ſo muß auch leicht ſeyn durch einen Multiplicatorem zu multipliciren, welcher 2, 3, oder mehrmal groͤſſer iſt als derjenige vom obigen falle.","norm":"Dann da ein zweimal so großer Multiplicator ein zweimal so großes Produkt hervorbringt, wann nämlich der Multiplicandus einerlei ist, so muss auch leicht sein durch einen Multiplikatorrem zu multiplizieren, welcher 2, 3, oder mehrmals größer ist als derjenige vom obigen falle.","orig_tok":["Dann","da","ein","zweymal","ſo","groſſer","Multiplicator","ein","zweymal","ſo","groſſes","Product","hervorbringt",",","wann","nehmlich","der","Multiplicandus","einerley","iſt",",","ſo","muß","auch","leicht","ſeyn","durch","einen","Multiplicatorem","zu","multipliciren",",","welcher","2",",","3",",","oder","mehrmal","groͤſſer","iſt","als","derjenige","vom","obigen","falle","."],"norm_tok":["Dann","da","ein","zweimal","so","großer","Multiplicator","ein","zweimal","so","großes","Produkt","hervorbringt",",","wann","nämlich","der","Multiplicandus","einerlei","ist",",","so","muss","auch","leicht","sein","durch","einen","Multiplikatorrem","zu","multiplizieren",",","welcher","2",",","3",",","oder","mehrmals","größer","ist","als","derjenige","vom","obigen","falle","."],"orig_spans":[[0,4],[5,7],[8,11],[12,19],[20,22],[23,30],[31,44],[45,48],[49,56],[57,59],[60,67],[68,75],[76,88],[88,89],[90,94],[95,103],[104,107],[108,122],[123,131],[132,135],[135,136],[137,139],[140,143],[144,148],[149,155],[156,160],[161,166],[167,172],[173,188],[189,191],[192,205],[205,206],[207,214],[215,216],[216,217],[218,219],[219,220],[221,225],[226,233],[234,242],[243,246],[247,250],[251,260],[261,264],[265,271],[272,277],[277,278]],"norm_spans":[[0,4],[5,7],[8,11],[12,19],[20,22],[23,29],[30,43],[44,47],[48,55],[56,58],[59,65],[66,73],[74,86],[86,87],[88,92],[93,100],[101,104],[105,119],[120,128],[129,132],[132,133],[134,136],[137,141],[142,146],[147,153],[154,158],[159,164],[165,170],[171,187],[188,190],[191,205],[205,206],[207,214],[215,216],[216,217],[218,219],[219,220],[221,225],[226,234],[235,241],[242,245],[246,249],[250,259],[260,263],[264,270],[271,276],[276,277]],"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":4.9257001876831055} +{"basename":"euler_rechenkunst02_1740","par_idx":1201,"date":1740,"orig":"Alſo wann man 4 ℨ mit 16 multipliciren ſoll, ſo muß das Product zweymal ſo groß ſeyn als dasjenige welches herauskommen wuͤrde, wann man 4 ℨ mit 8 multiplicirte.","norm":"Also wann man 4 ℨ mit 16 multiplizieren soll, so muss das Produkt zweimal so groß sein als dasjenige welches herauskommen würde, wann man 4 ℨ mit 8 multiplizierte.","orig_tok":["Alſo","wann","man","4","ℨ","mit","16","multipliciren","ſoll",",","ſo","muß","das","Product","zweymal","ſo","groß","ſeyn","als","dasjenige","welches","herauskommen","wuͤrde",",","wann","man","4","ℨ","mit","8","multiplicirte","."],"norm_tok":["Also","wann","man","4","ℨ","mit","16","multiplizieren","soll",",","so","muss","das","Produkt","zweimal","so","groß","sein","als","dasjenige","welches","herauskommen","würde",",","wann","man","4","ℨ","mit","8","multiplizierte","."],"orig_spans":[[0,4],[5,9],[10,13],[14,15],[16,17],[18,21],[22,24],[25,38],[39,43],[43,44],[45,47],[48,51],[52,55],[56,63],[64,71],[72,74],[75,79],[80,84],[85,88],[89,98],[99,106],[107,119],[120,126],[126,127],[128,132],[133,136],[137,138],[139,140],[141,144],[145,146],[147,160],[160,161]],"norm_spans":[[0,4],[5,9],[10,13],[14,15],[16,17],[18,21],[22,24],[25,39],[40,44],[44,45],[46,48],[49,53],[54,57],[58,65],[66,73],[74,76],[77,81],[82,86],[87,90],[91,100],[101,108],[109,121],[122,127],[127,128],[129,133],[134,137],[138,139],[140,141],[142,145],[146,147],[148,162],[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],[28,28],[29,29],[30,30],[31,31]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":3.821500062942505} +{"basename":"euler_rechenkunst02_1740","par_idx":1202,"date":1740,"orig":"Da nun 4 ℨ mit 8 multiplicirt 4 ℥ geben, ſo werden 4 ℨ mit 16 oder mit 2 mal 8 multiplicirt 8 ℥ das iſt 2 mal 4 ℥ geben.","norm":"Da nun 4 ℨ mit 8 multipliziert 4 ℥ geben, so werden 4 ℨ mit 16 oder mit 2 Mal 8 multipliziert 8 ℥ das ist 2 Mal 4 ℥ geben.","orig_tok":["Da","nun","4","ℨ","mit","8","multiplicirt","4","℥","geben",",","ſo","werden","4","ℨ","mit","16","oder","mit","2","mal","8","multiplicirt","8","℥","das","iſt","2","mal","4","℥","geben","."],"norm_tok":["Da","nun","4","ℨ","mit","8","multipliziert","4","℥","geben",",","so","werden","4","ℨ","mit","16","oder","mit","2","Mal","8","multipliziert","8","℥","das","ist","2","Mal","4","℥","geben","."],"orig_spans":[[0,2],[3,6],[7,8],[9,10],[11,14],[15,16],[17,29],[30,31],[32,33],[34,39],[39,40],[41,43],[44,50],[51,52],[53,54],[55,58],[59,61],[62,66],[67,70],[71,72],[73,76],[77,78],[79,91],[92,93],[94,95],[96,99],[100,103],[104,105],[106,109],[110,111],[112,113],[114,119],[119,120]],"norm_spans":[[0,2],[3,6],[7,8],[9,10],[11,14],[15,16],[17,30],[31,32],[33,34],[35,40],[40,41],[42,44],[45,51],[52,53],[54,55],[56,59],[60,62],[63,67],[68,71],[72,73],[74,77],[78,79],[80,93],[94,95],[96,97],[98,101],[102,105],[106,107],[108,111],[112,113],[114,115],[116,121],[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],[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":3.80019998550415} +{"basename":"euler_rechenkunst02_1740","par_idx":1203,"date":1740,"orig":"Jngleichem da 24 Stunden auf einen Tag gerechnet werden, ſo muͤſſen zum Exempel 15 Stunden mit 24 multiplicirt 15 Tage geben.","norm":"Ingleichem da 24 Stunden auf einen Tag gerechnet werden, so müssen zum Exempel 15 Stunden mit 24 multipliziert 15 Tage 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+{"basename":"euler_rechenkunst02_1740","par_idx":1204,"date":1740,"orig":"Dahero wann 15 Stunden mit 2 mal 24 das iſt mit 48 multiplicirt werden, ſo muͤſſen 2 mal 15 das iſt 30 Tage herauskommen: ſollte aber der Multiplicator 3 mal 24 oder 72 ſeyn, ſo wuͤrden 3 mal 15 das iſt 45 Tage ins Product kommen.","norm":"Daher wann 15 Stunden mit 2 Mal 24 das ist mit 48 multipliziert werden, so müssen 2 Mal 15 das ist 30 Tage herauskommen: sollte aber der Multiplicator 3 Mal 24 oder 72 sein, so würden 3 Mal 15 das ist 45 Tage ins Produkt 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+{"basename":"euler_rechenkunst02_1740","par_idx":1206,"date":1740,"orig":"Da nun 20 St. einen fl. ausmachen, und folglich 17 St. mit 20 multiplicirt 17 fl. betragen, ſo muͤſſen dieſelben mit 100 multiplicirt 5 mal 17 fl. geben, weilen 100 fuͤnf mal groͤſſer iſt als 20; derohalben darf man nur die 17 St. mit 5 multipliciren und im Product anſtatt der St. den Nahmen fl. ſetzen; wodurch das Product 85 fl. ſeyn wird.","norm":"Da nun 20 St. einen fl. ausmachen, und folglich 17 St. mit 20 multipliziert 17 fl. betragen, so müssen dieselben mit 100 multipliziert 5 Mal 17 fl. geben, weilen 100 fünfmal größer ist als 20; derohalben darf man nur die 17 St. mit 5 multiplizieren und im Produkt anstatt der St. den Namen fl. setzen; wodurch das Produkt 85 fl. sein 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+{"basename":"euler_rechenkunst02_1740","par_idx":1208,"date":1740,"orig":"Da wir nehmlich gewieſen haben, daß ein zweyfacher Multiplicator ein zweyfaches Product, ein dreyfacher ein dreyfaches und ſofort hervorbringe; ſo kan man daraus, wann der Multiplicator eine groſſe Zahl iſt die Multiplication in zwey oder mehr Operationen zertheilen; wodurch oͤfters die Multiplication leichter und geſchwinder verrichtet werden kan, als auf die gewoͤhnliche Art.","norm":"Da wir nämlich gewiesen haben, dass ein zweifacher Multiplicator ein zweifaches Produkt, ein dreifacher ein dreifaches und sofort hervorbringe; so kann man daraus, wann der Multiplicator eine große Zahl ist die Multiplikation in zwei oder mehr Operationen zerteilen; wodurch öfters die Multiplikation leichter und geschwinder verrichtet werden kann, als auf die gewöhnliche 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Derowegen kan man erſtlich die vorgegebene Quantitaͤt durch 4 multipliciren, und als dann das gefundene Product noch mal mit 3 multipliciren; da dann eben ſo viel heraus kommen wird, als wann man ſo gleich mit 12 multiplicirt haͤtte.","norm":"Als wann eine vorgegebene so wohl unbenannte als benannte Zahl durch 12 multipliziert werden soll, so ist zu merken, dass 12 so viel ist als 3 Mal 4, und folglich der Multiplicator 12 ein dreimal größeres Produkt geben müsse, als der Multiplicator 4. Derowegen kann man erstlich die vorgegebene Quantität durch 4 multiplizieren, und als dann das gefundene Produkt noch mal mit 3 multiplizieren; da dann ebenso viel herauskommen wird, als wann man so gleich mit 12 multipliziert 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Eben dieſes Product wird auch herauskommen, wann man erſtlich mit 3 und hernach mit 4 multiplicirt.","norm":"Dieses Produkt ist nun gleich demjenigen, welches würde gefunden worden sein, wann man sogleich mit 12 multipliziert hätte, und das deswegen, weilen 12 so viel ist als 3 Mal 4. 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+{"basename":"euler_rechenkunst02_1740","par_idx":1213,"date":1740,"orig":"Ferner da auch 12 ſo viel iſt als 2 mahl 6, ſo koͤnte man das erſte mahl mit 2 und das andere mahl mit 6 multipliciren, oder umgekehrt das erſte mahl mit 6 und das andere mahl mit 2, wie folgt.","norm":"Ferner da auch 12 so viel ist als 2 Mal 6, so könnte man das erste Mal mit 2 und das andere Mal mit 6 multiplizieren, oder umgekehrt das erste Mal mit 6 und das andere Mal mit 2, wie 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+{"basename":"euler_rechenkunst02_1740","par_idx":1214,"date":1740,"orig":"[Formel] Solche Zertheilungen des Multiplicatoris in zwey Factores oder Multiplicatores ſind nun zwar an und fuͤr ſich ſelbſt vortheilhaft, weilen mit kleineren Zahlen leichter zu multipliciren iſt als mit groſſen: inzwiſchen aber wird hinwiederum die Arbeit groͤſſer, weilen man zwey Multiplicationen anſtellen muß.","norm":"[Formel] Solche Zerteilungen des Multiplicatoris in zwei Factores oder Multiplikators sind nun zwar an und für sich selbst vorteilhaft, weilen mit kleineren Zahlen leichter zu multiplizieren ist als mit großen: inzwischen aber wird hinwiederum die Arbeit größer, weilen man zwei Multiplikationen anstellen 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+{"basename":"euler_rechenkunst02_1740","par_idx":1215,"date":1740,"orig":"Dem ungeacht aber behaͤlt dennoch dieſe Zertheilung einen mercklichen Nutzen, wann man ſich im Rechnen ſchon eine ſolche Fertigkeit erworben hat, daß man mit kleinen Zahlen gleichſam im Sinn die Multiplication verrichten kan.","norm":"Dem ungeachtet aber behält dennoch diese Zerteilung einen merklichen Nutzen, wann man sich im Rechnen schon eine solche Fertigkeit erworben hat, dass man mit kleinen Zahlen gleichsam im Sinn die Multiplikation verrichten 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+{"basename":"euler_rechenkunst02_1740","par_idx":1216,"date":1740,"orig":"Jn dieſem Falle erhaͤlt man auch oͤfters einen Vortheil, wann man den Multiplicatorem in drey oder mehr Factores zertheilet.","norm":"In diesem Falle erhält man auch öfters einen Vorteil, wann man den Multiplikatorrem in drei oder mehr Factores 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+{"basename":"euler_rechenkunst02_1740","par_idx":1217,"date":1740,"orig":"Als wann man durch 210 multipliciren ſollte, ſo koͤnte man erſtlich durch 7 und dann durch 30 multipliciren, weilen 210 ſo viel iſt als 7 mahl 30. Wann aber durch 30 zu multipliciren noch ſchwehr faͤllt, ſo kan man den Multiplicatorem 30 noch in 5 und 6 vertheilen, weilen 30 ſo viel iſt als 5 mahl 6.","norm":"Als wann man durch 210 multiplizieren sollte, so könnte man erstlich durch 7 und dann durch 30 multiplizieren, weilen 210 so viel ist als 7 Mal 30. Wann aber durch 30 zu multiplizieren noch schwerfällt, so kann man den Multiplikatorrem 30 noch in 5 und 6 verteilen, weilen 30 so viel ist als 5 Mal 6.","orig_tok":["Als","wann","man","durch","210","multipliciren","ſollte",",","ſo","koͤnte","man","erſtlich","durch","7","und","dann","durch","30","multipliciren",",","weilen","210","ſo","viel","iſt","als","7","mahl","30.","Wann","aber","durch","30","zu","multipliciren","noch","ſchwehr","faͤllt",",","ſo","kan","man","den","Multiplicatorem","30","noch","in","5","und","6","vertheilen",",","weilen","30","ſo","viel","iſt","als","5","mahl","6","."],"norm_tok":["Als","wann","man","durch","210","multiplizieren","sollte",",","so","könnte","man","erstlich","durch","7","und","dann","durch","30","multiplizieren",",","weilen","210","so","viel","ist","als","7","Mal","30.","Wann","aber","durch","30","zu","multiplizieren","noch","schwerfällt",",","so","kann","man","den","Multiplikatorrem","30","noch","in","5","und","6","verteilen",",","weilen","30","so","viel","ist","als","5","Mal","6."],"orig_spans":[[0,3],[4,8],[9,12],[13,18],[19,22],[23,36],[37,43],[43,44],[45,47],[48,54],[55,58],[59,67],[68,73],[74,75],[76,79],[80,84],[85,90],[91,93],[94,107],[107,108],[109,115],[116,119],[120,122],[123,127],[128,131],[132,135],[136,137],[138,142],[143,146],[147,151],[152,156],[157,162],[163,165],[166,168],[169,182],[183,187],[188,195],[196,202],[202,203],[204,206],[207,210],[211,214],[215,218],[219,234],[235,237],[238,242],[243,245],[246,247],[248,251],[252,253],[254,264],[264,265],[266,272],[273,275],[276,278],[279,283],[284,287],[288,291],[292,293],[294,298],[299,300],[300,301]],"norm_spans":[[0,3],[4,8],[9,12],[13,18],[19,22],[23,37],[38,44],[44,45],[46,48],[49,55],[56,59],[60,68],[69,74],[75,76],[77,80],[81,85],[86,91],[92,94],[95,109],[109,110],[111,117],[118,121],[122,124],[125,129],[130,133],[134,137],[138,139],[140,143],[144,147],[148,152],[153,157],[158,163],[164,166],[167,169],[170,184],[185,189],[190,201],[201,202],[203,205],[206,210],[211,214],[215,218],[219,235],[236,238],[239,243],[244,246],[247,248],[249,252],[253,254],[255,264],[264,265],[266,272],[273,275],[276,278],[279,283],[284,287],[288,291],[292,293],[294,297],[298,300]],"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],[37,36],[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],[61,59]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.583099842071533} +{"basename":"euler_rechenkunst02_1740","par_idx":1218,"date":1740,"orig":"Und alſo kan die Multiplication durch 210 dergeſtalt in drey Multiplicationen vertheilet werden, daß man erſtlich durch 7, hernach durch 6 und drittens durch 5 multiplicire, weilen 210 ſo viel iſt als 7 mahl 6 mahl 5.","norm":"Und also kann die Multiplikation durch 210 dergestalt in drei Multiplikationen verteilet werden, dass man erstlich durch 7, hernach durch 6 und drittens durch 5 multipliziere, weilen 210 so viel ist als 7 Mal 6 Mal 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+{"basename":"euler_rechenkunst02_1740","par_idx":1219,"date":1740,"orig":"Wir wollen davon ein Exempel geben: es ſollen 21 L. Sterl. 14 ß. 5 ₰ mit 210 multiplicirt werden; wovon die Operation alſo wird zu ſtehen kommen.","norm":"Wir wollen davon ein Exempel geben: Es sollen 21 L. Sterl . 14 ß . 5 ₰ mit 210 multipliziert werden; wovon die Operation also wird zu stehen 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+{"basename":"euler_rechenkunst02_1740","par_idx":1220,"date":1740,"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_rechenkunst02_1740","par_idx":1221,"date":1740,"orig":"Dieſer Vortheil findet nun Statt, wann ſich der Multiplicator in zwey oder mehr kleine Factores zertheilen laͤſt, mit welchen man leicht und bequem multipliciren kan.","norm":"Dieser Vorteil findet nun statt, wann sich der Multiplicator in zwei oder mehr kleine Factores zerteilen lässt, mit welchen man leicht und bequem multiplizieren 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+{"basename":"euler_rechenkunst02_1740","par_idx":1222,"date":1740,"orig":"Da ſich nun dieſes nicht bey allen vorkommenden Multiplicationen bewerckſtelligen laͤſt, ſo kan man ſich auch dieſes Vortheils nicht allzeit bedienen.","norm":"Da sich nun dieses nicht bei allen vorkommenden Multiplikationen bewerkstelligen lässt, so kann man sich auch dieses Vorteils nicht allzeit bedienen.","orig_tok":["Da","ſich","nun","dieſes","nicht","bey","allen","vorkommenden","Multiplicationen","bewerckſtelligen","laͤſt",",","ſo","kan","man","ſich","auch","dieſes","Vortheils","nicht","allzeit","bedienen","."],"norm_tok":["Da","sich","nun","dieses","nicht","bei","allen","vorkommenden","Multiplikationen","bewerkstelligen","lässt",",","so","kann","man","sich","auch","dieses","Vorteils","nicht","allzeit","bedienen","."],"orig_spans":[[0,2],[3,7],[8,11],[12,18],[19,24],[25,28],[29,34],[35,47],[48,64],[65,81],[82,87],[87,88],[89,91],[92,95],[96,99],[100,104],[105,109],[110,116],[117,126],[127,132],[133,140],[141,149],[149,150]],"norm_spans":[[0,2],[3,7],[8,11],[12,18],[19,24],[25,28],[29,34],[35,47],[48,64],[65,80],[81,86],[86,87],[88,90],[91,95],[96,99],[100,104],[105,109],[110,116],[117,125],[126,131],[132,139],[140,148],[148,149]],"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":4.5690999031066895} +{"basename":"euler_rechenkunst02_1740","par_idx":1223,"date":1740,"orig":"Man kan aber in ſolchen Faͤllen einen anderen Vortheil zu Huͤlfe nehmen, welcher aus nachfolgendem Grunde flieſſet.","norm":"Man kann aber in solchen Fällen einen anderen Vorteil zu Hilfe nehmen, welcher aus nachfolgendem Grunde 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+{"basename":"euler_rechenkunst02_1740","par_idx":1224,"date":1740,"orig":"Wann man den Multiplicatorem in zwey Theile vertheilet, welche zuſammen addirt denſelben wiederum hervor bringen, ſo kan man den Multiplicandum durch jeden Theil insbeſondere multipliciren, und die beyden Product zuſammen addiren; da dann dieſe Summ dem geſuchten Product gleich ſeyn wird.","norm":"Wann man den Multiplikatorrem in zwei Teile verteilet, welche zusammen addiert denselben wiederum hervorbringen, so kann man den Multiplicandum durch jeden Teil insbesondere multiplizieren, und die beiden Produkt zusammen addieren; da dann diese Summe dem gesuchten Produkt gleich sein 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+{"basename":"euler_rechenkunst02_1740","par_idx":1236,"date":1740,"orig":"Als wann man durch 37 multipliciren ſollte, ſo koͤnte man darfuͤr dieſe Theile annehmen 36 und 1, und jenen in dieſe ſeine beyde Factores 6 und 6 reſolviren.","norm":"Als wann man durch 37 multiplizieren sollte, so könnte man dafür diese Teile annehmen 36 und 1, und jenen in diese seine beide Factores 6 und 6 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+{"basename":"euler_rechenkunst02_1740","par_idx":1237,"date":1740,"orig":"Jn dieſem Falle muͤßte man alſo den Multiplicandum erſtlich mit 6 und das Product nochmalen durch 6 multipliciren, und zum letzten Product den Multiplicandum durch 1 multiplicirt das iſt den Multiplicandum ſelbſt addiren.","norm":"In diesem Falle müsste man also den Multiplicandum erstlich mit 6 und das Produkt nochmal durch 6 multiplizieren, und zum letzten Produkt den Multiplicandum durch 1 multipliziert das ist den Multiplicandum selbst 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+{"basename":"euler_rechenkunst02_1740","par_idx":1238,"date":1740,"orig":"Laſt uns alſo nach dieſer Art, uachfolgendes Apotheker Gewicht durch 37 multipliciren.","norm":"Last uns also nach dieser Art, uachfolgendes Apotheker Gewicht durch 37 multiplizieren.","orig_tok":["Laſt","uns","alſo","nach","dieſer","Art",",","uachfolgendes","Apotheker","Gewicht","durch","37","multipliciren","."],"norm_tok":["Last","uns","also","nach","dieser","Art",",","uachfolgendes","Apotheker","Gewicht","durch","37","multiplizieren","."],"orig_spans":[[0,4],[5,8],[9,13],[14,18],[19,25],[26,29],[29,30],[31,44],[45,54],[55,62],[63,68],[69,71],[72,85],[85,86]],"norm_spans":[[0,4],[5,8],[9,13],[14,18],[19,25],[26,29],[29,30],[31,44],[45,54],[55,62],[63,68],[69,71],[72,86],[86,87]],"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":8.257399559020996} +{"basename":"euler_rechenkunst02_1740","par_idx":1239,"date":1740,"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_rechenkunst02_1740","par_idx":1240,"date":1740,"orig":"Eben dieſes Product wuͤrde man gefunden haben, wann man das vorgegebene Gewicht nach der gewoͤhnlichen Art multiplicirt haͤtte durch 37.","norm":"Eben dieses Produkt würde man gefunden haben, wann man das vorgegebene Gewicht nach der gewöhnlichen Art multipliziert hätte durch 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+{"basename":"euler_rechenkunst02_1740","par_idx":1241,"date":1740,"orig":"Wir wollen noch ein Exempel geben in welchem mit 157 multiplicirt werden ſoll; dieſe Zahl laͤſt ſich nun nicht in Factores zertheilen, deswegen muß man ſuchen dieſelbe in zwey ſolche Theile zu zerſchneiden, deren einer fuͤr ſich klein genug, der andere aber ſich in bequeme Factores zertheilen laſſe.","norm":"Wir wollen noch ein Exempel geben in welchem mit 157 multipliziert werden soll; diese Zahl lässt sich nun nicht in Factores zerteilen, deswegen muss man suchen dieselbe in zwei solche Teile zu zerschneiden, deren einer für sich klein genug, der andere aber sich in bequeme Factores zerteilen 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Mühe kosten würde, als die ganze Operation nach der gewöhnlichen 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+{"basename":"euler_rechenkunst02_1740","par_idx":1279,"date":1740,"orig":"Dieſe Summ dividirt man ferner durch den Diviſorem, ſchreibt den Quotum mit dem Nahmen der Sorte in den geſuchten Quotient, und verwechſelt den Reſt in die kleinere folgende Sorte, welche man zuſamt demjenigen, was von dieſer Sorte im Dividendo da iſt, ferner durch den Diviſorem dividirt.","norm":"Diese Summe dividiert man ferner durch den Divisorem, schreibt den Quotum mit dem Namen der Sorte in den gesuchten Quotient, und verwechselt den Rest in die kleinere folgende Sorte, welche man samt demjenigen, was von dieser Sorte im Dividendo da ist, ferner durch den Divisorem 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+{"basename":"euler_rechenkunst02_1740","par_idx":1280,"date":1740,"orig":"Solcher geſtalt verfaͤhrt man bis zur kleinſten Sorte, und was in der letzten Diviſion uͤbrig bleibt, daſſelbe ſchreibt man in Form eines Bruchs in den Quotienten.","norm":"Solcher Gestalt verfährt man bis zur kleinsten Sorte, und was in der letzten Division übrig bleibt, dasselbe schreibt man in Form eines Bruchs in den 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+{"basename":"euler_rechenkunst02_1740","par_idx":1281,"date":1740,"orig":"Jn der Diviſion wird immer eine ſolche Quantitaͤt geſucht, welche mit dem Diviſore multiplicirt den Dividendum wiederum hervorbringet.","norm":"In der Division wird immer eine solche Quantität gesucht, welche mit dem Divisore multipliziert den Dividendum wiederum 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+{"basename":"euler_rechenkunst02_1740","par_idx":1282,"date":1740,"orig":"Es iſt alſo der Dividendus nicht anderſt anzuſehen als ein Product, welches aus der Multiplication des Quoti mit dem Diviſore entſpringt.","norm":"Es ist also der Dividendus nicht anders anzusehen als ein Produkt, welches aus der Multiplikation des Quoti mit dem Divisore 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+{"basename":"euler_rechenkunst02_1740","par_idx":1283,"date":1740,"orig":"Weilen nun in der Multiplication der Multiplicator allzeit eine unbenannte Zahl ſeyn muß.","norm":"Weilen nun in der Multiplikation der Multiplicator allzeit eine unbenannte Zahl sein muss.","orig_tok":["Weilen","nun","in","der","Multiplication","der","Multiplicator","allzeit","eine","unbenannte","Zahl","ſeyn","muß","."],"norm_tok":["Weilen","nun","in","der","Multiplikation","der","Multiplicator","allzeit","eine","unbenannte","Zahl","sein","muss","."],"orig_spans":[[0,6],[7,10],[11,13],[14,17],[18,32],[33,36],[37,50],[51,58],[59,63],[64,74],[75,79],[80,84],[85,88],[88,89]],"norm_spans":[[0,6],[7,10],[11,13],[14,17],[18,32],[33,36],[37,50],[51,58],[59,63],[64,74],[75,79],[80,84],[85,89],[89,90]],"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":5.184999942779541} +{"basename":"euler_rechenkunst02_1740","par_idx":1284,"date":1740,"orig":"Das Product aber dem Nahmen nach dem Multiplicando aͤhnlich gefunden wird:","norm":"Das Produkt aber dem Namen nach dem Multiplicando ähnlich gefunden wird:","orig_tok":["Das","Product","aber","dem","Nahmen","nach","dem","Multiplicando","aͤhnlich","gefunden","wird",":"],"norm_tok":["Das","Produkt","aber","dem","Namen","nach","dem","Multiplicando","ähnlich","gefunden","wird",":"],"orig_spans":[[0,3],[4,11],[12,16],[17,20],[21,27],[28,32],[33,36],[37,50],[51,59],[60,68],[69,73],[73,74]],"norm_spans":[[0,3],[4,11],[12,16],[17,20],[21,26],[27,31],[32,35],[36,49],[50,57],[58,66],[67,71],[71,72]],"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":7.04040002822876} +{"basename":"euler_rechenkunst02_1740","par_idx":1285,"date":1740,"orig":"ſo muß auch in der Diviſion entweder der Diviſor 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entſtehen zweyerley Arten der Diviſion mit benannten Zahlen: in der erſteren iſt der Diviſor eine unbenannte Zahl und folglich der Quotus eine benannte Zahl; welche dem Nahmen nach dem Dividendo aͤhnlich ſeyn muß.","norm":"Daher entstehen zweierlei Arten der Division mit benannten Zahlen: in der ersteren ist der Divisor eine unbenannte Zahl und folglich der Quotus eine benannte Zahl; welche dem Namen nach dem Dividendo ähnlich sein 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+{"basename":"euler_rechenkunst02_1740","par_idx":1290,"date":1740,"orig":"das iſt, wann der Diviſor 2 iſt, ſo wird die Helfte der Dividendi geſuchet, iſt der Diviſor 3, der Drittel, iſt der Diviſor 4, der Viertel und ſo fort; und demnach muß der Quotus dem Nahmen nach dem Dividendo aͤhnlich ſeyn.","norm":"das ist, wann der Divisor 2 ist, so wird die Hälfte der Dividendi gesucht, ist der Divisor 3, der Drittel, ist der Divisor 4, der Viertel und so fort; und demnach muss der Quotus dem Namen nach dem Dividendo ähnlich 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Weilen der Kaufmann den vierten Theil verkauft hat, ſo muß man ſuchen, wieviel der vierte Theil der gantzen Partie austrage, das iſt man muß die gantze Partie durch 4 dividiren, da dann der Quotus das geſuchte anzeigen wird.","norm":"Antw. 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Weilen die Sonne immer gleich geſchwind zu lauffen angenommen wird, ſo wird der taͤgliche Weg, welchen die Sonne zuruͤck legt der 365ſte Theil ſeyn von den 12 Himmliſchen Zeichen.","norm":"Antw. 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+{"basename":"euler_rechenkunst02_1740","par_idx":1344,"date":1740,"orig":"VII.","norm":"VII.","orig_tok":["VII","."],"norm_tok":["VII","."],"orig_spans":[[0,3],[3,4]],"norm_spans":[[0,3],[3,4]],"alignment":[[0,0],[1,1]],"lang_fastText":"en","lang_py3langid":"en","lang_cld3":"la","orig_lang_de":0,"norm_lmscore":3.952300071716308} +{"basename":"euler_rechenkunst02_1740","par_idx":1345,"date":1740,"orig":"Ein Stuͤck Gold wiegt 5 ℳ, 6 ℥, 9 Engl. 20¾ Aß, daſſelbe ſoll in 16 gleiche Theile getheilt werden, wie ſchwehr muß ein Theil ſeyn?","norm":"Ein Stück Gold wiegt 5 M, 6 ℥, 9 Engl. 203/4 Aß, dasselbe soll in 16 gleiche Teile geteilt werden, wie schwer muss ein Teil 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+{"basename":"euler_rechenkunst02_1740","par_idx":1346,"date":1740,"orig":"Antw.","norm":"Antw.","orig_tok":["Antw","."],"norm_tok":["Antw","."],"orig_spans":[[0,4],[4,5]],"norm_spans":[[0,4],[4,5]],"alignment":[[0,0],[1,1]],"lang_fastText":"pl","lang_py3langid":"de","lang_cld3":"sr","orig_lang_de":0.333,"norm_lmscore":3.781899929046631} +{"basename":"euler_rechenkunst02_1740","par_idx":1347,"date":1740,"orig":"Hier iſt klar, daß das gegebene Gewicht des Stuͤcks Goldes durch 16 dividirt werden muͤſſe, da dann der Quotient das Gewicht eines Theiles anzeigen wird.","norm":"Hier ist klar, dass das gegebene Gewicht des Stücks Goldes durch 16 dividiert werden müsse, da dann der Quotient das Gewicht eines Teiles anzeigen 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+{"basename":"euler_rechenkunst02_1740","par_idx":1348,"date":1740,"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_rechenkunst02_1740","par_idx":1349,"date":1740,"orig":"In dieſem und den vorhergehenden Exempeln haben wir immer bey der Multiplication, wodurch eine groͤſſere Sorte in eine kleinere reſolvirt wird, zugleich die Stuͤcke von der kleineren Sorte mit addirt, wie aus Anſehung der Operationen leicht zu ſehen.","norm":"In diesem und den vorhergehenden Exempeln haben wir immer bei der Multiplikation, wodurch eine größere Sorte in eine kleinere resolviert wird, zugleich die Stücke von der kleineren Sorte mit addiert, wie aus Ansehung der Operationen leicht zu 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+{"basename":"euler_rechenkunst02_1740","par_idx":1350,"date":1740,"orig":"Wir koͤnnten hier bey der Diviſion auch dergleichen Vortheile anzeigen, wie bey der Multiplication, allein da der Nutzen davon von keiner oder doch ſehr geringer Wichtigkeit iſt; ſo haben wir nicht noͤthig, uns dabey aufzuhalten.","norm":"Wir könnten hier bei der Division auch dergleichen Vorteile anzeigen, wie bei der Multiplikation, allein da der Nutzen davon von keiner oder doch sehr geringer Wichtigkeit ist; so haben wir nicht nötig, uns dabei 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iſt angefuͤhret worden betrifft, ſo iſt wohl zu mercken, daß dieſelbe bey der Diviſion gantz und gar nicht Statt finde; weswegen man ſich darvor, um nicht in Jrrthum zu verfallen, wohl fuͤrzuſehen hat.","norm":"Was aber die andere Zerteilung in Teile, welche bei der Multiplikation ist angeführt worden betrifft, so ist wohl zu merken, dass dieselbe bei der Division ganz und gar nicht statt finde; weswegen man sich davor, um nicht in Irrtum zu verfallen, wohl vorzusehen 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+{"basename":"euler_rechenkunst02_1740","par_idx":1364,"date":1740,"orig":"Jn der vorhergehenden Art haben wir nun geſetzet, daß der Diviſor eine unbenannte Zahl ſey, da dann der Quotus eine benannte Zahl worden iſt.","norm":"In der vorhergehenden Art haben wir nun gesetzt, dass der Divisor eine unbenannte Zahl sei, da dann der Quotus eine benannte Zahl worden 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+{"basename":"euler_rechenkunst02_1740","par_idx":1365,"date":1740,"orig":"Nunmehro nehmen wir aber fuͤr den Diviſorem eine benannte Zahl an, und muß folglich fuͤr den Quotum eine unbenannte Zahl gefunden werden.","norm":"Nunmehr nehmen wir aber für den Divisorem eine benannte Zahl an, und muss folglich für den Quotum eine unbenannte Zahl gefunden 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+{"basename":"euler_rechenkunst02_1740","par_idx":1366,"date":1740,"orig":"Jn dieſem Falle ſind demnach der Diviſor und der Dividendus benannte Zahlen von einerley Art, das iſt ſolche welche entweder gleiche Nahmen fuͤhren, oder doch ſolche verſchiedene Nahmen, welche unter ſich verglichen werden koͤnnen.","norm":"In diesem Falle sind demnach der Divisor und der Dividendus benannte Zahlen von einerlei Art, das ist solche welche entweder gleiche Namen führen, oder doch solche verschiedene Namen, welche unter sich verglichen werden können.","orig_tok":["Jn","dieſem","Falle","ſind","demnach","der","Diviſor","und","der","Dividendus","benannte","Zahlen","von","einerley","Art",",","das","iſt","ſolche","welche","entweder","gleiche","Nahmen","fuͤhren",",","oder","doch","ſolche","verſchiedene","Nahmen",",","welche","unter","ſich","verglichen","werden","koͤnnen","."],"norm_tok":["In","diesem","Falle","sind","demnach","der","Divisor","und","der","Dividendus","benannte","Zahlen","von","einerlei","Art",",","das","ist","solche","welche","entweder","gleiche","Namen","führen",",","oder","doch","solche","verschiedene","Namen",",","welche","unter","sich","verglichen","werden","können","."],"orig_spans":[[0,2],[3,9],[10,15],[16,20],[21,28],[29,32],[33,40],[41,44],[45,48],[49,59],[60,68],[69,75],[76,79],[80,88],[89,92],[92,93],[94,97],[98,101],[102,108],[109,115],[116,124],[125,132],[133,139],[140,147],[147,148],[149,153],[154,158],[159,165],[166,178],[179,185],[185,186],[187,193],[194,199],[200,204],[205,215],[216,222],[223,230],[230,231]],"norm_spans":[[0,2],[3,9],[10,15],[16,20],[21,28],[29,32],[33,40],[41,44],[45,48],[49,59],[60,68],[69,75],[76,79],[80,88],[89,92],[92,93],[94,97],[98,101],[102,108],[109,115],[116,124],[125,132],[133,138],[139,145],[145,146],[147,151],[152,156],[157,163],[164,176],[177,182],[182,183],[184,190],[191,196],[197,201],[202,212],[213,219],[220,226],[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]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.605500221252441} 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+{"basename":"euler_rechenkunst02_1740","par_idx":1368,"date":1740,"orig":"Dann wann man zum Exempel 12 Rubl. durch 4 Rubl. theilen ſoll, ſo iſt dieſes nichts anders, als man ſoll anzeigen, wieviel mal 4 Rubl. in 12 Rubl. enthalten ſeyen; oder wieviel malen 4 Rubl. genommen werden muͤſſen, daß 12 Rubl. heraus kommen.","norm":"Dann wann man zum Exempel 12 Rubel . durch 4 Rubel . teilen soll, so ist dieses nichts anders, als man soll anzeigen, wie viel Mal 4 Rubel . in 12 Rubel . enthalten seien; oder wie viel malen 4 Rubel . genommen werden müssen, dass 12 Rubel . 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+{"basename":"euler_rechenkunst02_1740","par_idx":1369,"date":1740,"orig":"Solches geſchieht nun in 3 malen und dahero ſagt man, daß 4 Rubl. in 12 Rubl.","norm":"Solches geschieht nun in 3 malen und daher sagt man, dass 4 Rubel . in 12 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Laſſt uns vor allen Dingen beyde Quantitaͤten in ℔ reſolviren.","norm":"Antw. 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+{"basename":"euler_rechenkunst02_1740","par_idx":1427,"date":1740,"orig":"Jm vorigen Satze haben wir zwar zu dieſem Ende die kleinſte Sorte erwehlet, welche in entwederem von beyden vorkommt, welches hauptſaͤchlich um Bruͤche zu vermeiden geſchehen iſt, in dem darinn ein nicht geringer Vortheil ſtecket, wann man gantze Zahlen ſtatt Bruͤche durch einander zu dividiren hat.","norm":"Im vorigen Satze haben wir zwar zu diesem Ende die kleinste Sorte erwählt, welche in entwederem von beiden vorkommt, welches hauptsächlich um Brüche zu vermeiden geschehen ist, in dem darin ein nicht geringer Vorteil stecket, wann man ganze Zahlen statt Brüche durcheinander zu dividieren 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+{"basename":"euler_rechenkunst02_1740","par_idx":1428,"date":1740,"orig":"Allein da es auch ſehr leicht iſt die Diviſion in Bruͤchen zu bewerckſtelligen wann nur der Diviſor eine gantze Zahl iſt, ſo iſt unnoͤthig den Diviſorem in kleinere Sorten zu verwandeln als darinn wuͤrcklich vorkommen.","norm":"Allein da es auch sehr leicht ist die Division in Brüchen zu bewerkstelligen wann nur der Divisor eine ganze Zahl ist, so ist unnötig den Divisorem in kleinere Sorten zu verwandeln als darin wirklich 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+{"basename":"euler_rechenkunst02_1740","par_idx":1429,"date":1740,"orig":"Jm Dividendo hat man alſo darauf nicht zu ſehen, ob derſelbe mit Bruͤchen verknuͤpfet iſt oder nicht, wann nur der Diviſor nur eine einige Sorte enthaͤlt und dabey eine gantze Zahl iſt, ſo darf man nur den gantzen Dividendum auf eben diejenige Sorte reſolviren; und nicht darauf ſehen ob Bruͤche darinn vorkommen oder nicht.","norm":"Im Dividendo hat man also darauf nicht zu sehen, ob derselbe mit Brüchen verknüpfet ist oder nicht, wann nur der Divisor nur eine einige Sorte enthält und dabei eine ganze Zahl ist, so darf man nur den ganzen Dividendum auf eben diejenige Sorte resolvieren; und nicht darauf sehen ob Brüche darin vorkommen oder 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+{"basename":"euler_rechenkunst02_1740","par_idx":1430,"date":1740,"orig":"Wann nun dieſes geſchehen, ſo dividirt man nach der allgemeinen Regel, den Dividendum durch den Diviſorem als unbenannte Zahlen.","norm":"Wann nun dieses geschehen, so dividiert man nach der allgemeinen Regel, den Dividendum durch den Divisorem als unbenannte 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+{"basename":"euler_rechenkunst02_1740","par_idx":1431,"date":1740,"orig":"Beſtehet aber der Diviſor aus etlichen Sorten, ſo bringt man denſelben auf den Nahmen der kleinſten Sorte welche darinnen vorkommt; auf eben denſelbigen Nahmen aber bringt man auch den Dividendum, wann auch gleich darinn noch kleinere Sorten vorhanden ſeyn ſollten.","norm":"Bestehet aber der Divisor aus etlichen Sorten, so bringt man denselben auf den Namen der kleinsten Sorte welche darinnen vorkommt; auf eben denselbigen Namen aber bringt man auch den Dividendum, wann auch gleich darin noch kleinere Sorten vorhanden sein 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+{"basename":"euler_rechenkunst02_1740","par_idx":1432,"date":1740,"orig":"Die Haupt- Regel hiebey iſt, daß man die Zahl des Diviſoris ohne Noth nicht vergroͤſſere, ſondern ſo klein behalte als immer moͤglich iſt, wann ſolche nur eine gantze Zahl bleibet.","norm":"Die Haupt-Regel hierbei ist, dass man die Zahl des Divisoris ohne Not nicht vergrößere, sondern so klein behalte als immer möglich ist, wann solche nur eine ganze Zahl 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Weilen allhier der Diviſor 1 fl. iſt, ſo muß man auch den Dividendum auf den Nahmen fl. reduciren, bey welcher Operation man von der kleinſten Sorte anfaͤngt, bis man gefunden den vievielten Theil eines Guldens die vorhandenen kleineren Sorten 5 St. 12 ₰ austragen, welches alſo geſchieht.","norm":"Antw. 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+{"basename":"euler_rechenkunst02_1740","par_idx":1474,"date":1740,"orig":"Die Abſicht dieſes Satzes gehet wieder dahin, daß man ſo viel als moͤglich groſſe Zahlen und auch Bruͤche vermeide.","norm":"Die Absicht dieses Satzes geht wieder dahin, dass man so viel als möglich große Zahlen und auch Brüche 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+{"basename":"euler_rechenkunst02_1740","par_idx":1475,"date":1740,"orig":"Gleichwie wir nun im vorigen Satze angezeigt haben, daß man den Diviſorem auf die kleinſte gantze Zahl bringen, und den Dividendum in eben diejenige Sorte reſolviren ſoll, alſo kehren wir hier die Sache um und reſolviren den Dividendum in einer ſolche Sorte, in welcher derſelbe durch die kleinſte gantze Zahl ausgedruͤckt wird.","norm":"Gleichwie wir nun im vorigen Satze angezeigt haben, dass man den Divisorem auf die kleinste ganze Zahl bringen, und den Dividendum in eben diejenige Sorte resolvieren soll, also kehren wir hier die Sache um und resolvieren den Dividendum in einer solche Sorte, in welcher derselbe durch die kleinste ganze Zahl ausgedrückt 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+{"basename":"euler_rechenkunst02_1740","par_idx":1476,"date":1740,"orig":"Jn beyden Faͤllen iſt aber der Vortheil bey nahe einerley, und iſt faſt gleich leicht eine gebrochene oder vermiſchte Zahl durch eine gantze, oder umgekehrt eine gantze Zahl durch eine gebrochene oder vermiſchte zu dividiren; wie aus den oben gegebenen Regeln der Diviſion mit Bruͤchen genugſam erhellet.","norm":"In beiden Fällen ist aber der Vorteil bei nahe einerlei, und ist fast gleich leicht eine gebrochene oder vermischte Zahl durch eine ganze, oder umgekehrt eine ganze Zahl durch eine gebrochene oder vermischte zu dividieren; wie aus den oben gegebenen Regeln der Division mit Brüchen genügsam 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+{"basename":"euler_rechenkunst02_1740","par_idx":1477,"date":1740,"orig":"Dieſes wird aber deutlicher dargethan durch die im Satze gemeldte Verſetzung des Diviſoris und Dividendi unter ſich, welche, weilen ſie uns die Natur der Diviſion gruͤndlicher vor die Augen leget wohl verdienet mit groͤſſerer Aufmerckſamkeit betrachtet zu werden.","norm":"Dieses wird aber deutlicher dargetan durch die im Satze gemeldete Versetzung des Divisoris und Dividendi unter sich, welche, weilen sie uns die Natur der Division gründlicher vor die Augen leget wohl verdienet mit größerer Aufmerksamkeit betrachtet zu 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+{"basename":"euler_rechenkunst02_1740","par_idx":1478,"date":1740,"orig":"Wir haben nehmlich geſagt, daß man um den Quotum zu finden, welcher aus der Diviſion des Dividendi durch den Diviſorem entſpringet, die Operation umkehren und den Diviſorem durch den Dividendum dividiren koͤnne; als dann aber dieſen gefundenen Quotum, nach dem ſolcher in die Form eines einzelen Bruchs gebracht worden, wiederum umkehren, und den Zehler an des Nenners, den Nenner aber an des Zehlers Stelle ſetzen muͤſſe.","norm":"Wir haben nämlich gesagt, dass man um den Quotum zu finden, welcher aus der Division des Dividendi durch den Divisorem entspringet, die Operation umkehren und den Divisorem durch den Dividendum dividieren könne; als dann aber diesen gefundenen Quotum, nach dem solcher in die Form eines einzeln Bruchs gebracht worden, wiederum umkehren, und den Zähler an des Nenners, den Nenner aber an des Zählers Stelle setzen 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+{"basename":"euler_rechenkunst02_1740","par_idx":1479,"date":1740,"orig":"Daß aber auf dieſe Art der wahre Quotus entſpringe, kan aus demjenigen, was oben von der Natur der Bruͤche erwieſen worden, leicht dargethan werden.","norm":"Dass aber auf diese Art der wahre Quotus entspringe, kann aus demjenigen, was oben von der Natur der Brüche erwiesen worden, leicht dargetan 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+{"basename":"euler_rechenkunst02_1740","par_idx":1480,"date":1740,"orig":"Dann da ein Bruch nichts anders anzeiget, als den Quotum, welcher herauskommt, wann man den Zehler durch den Nenner dividirt, ſo kan hinwiederum der aus einer Diviſion entſtehende Quotus durch einen Bruch ausgedruͤckt werden, deſſen Zehler der Dividendus, der Nenner aber der Diviſor iſt.","norm":"Dann da ein Bruch nichts anders anzeiget, als den Quotum, welcher herauskommt, wann man den Zähler durch den Nenner dividiert, so kann hinwiederum der aus einer Division entstehende Quotus durch einen Bruch ausgedrückt werden, dessen Zähler der Dividendus, der Nenner aber der Divisor 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+{"basename":"euler_rechenkunst02_1740","par_idx":1481,"date":1740,"orig":"Wann wir nun auf die verkehrte Art den Diviſorem durch den Dividendum dividiren, und den gefundenen Quotum in die Form eines einzelen Bruchs bringen, ſo erhalten wir einen Bruch deſſen Zehler der Diviſor der Nenner aber der Dividendus ſeyn wird.","norm":"Wann wir nun auf die verkehrte Art den Divisorem durch den Dividendum dividieren, und den gefundenen Quotum in die Form eines einzeln Bruchs bringen, so erhalten wir einen Bruch dessen Zähler der Divisor der Nenner aber der Dividendus sein 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+{"basename":"euler_rechenkunst02_1740","par_idx":1482,"date":1740,"orig":"Wann wir nun ferner dieſen Bruch wiederum umkehren, den Zehler und Nenner nehmlich unter ſich verwechſeln, ſo erhalten wir einen Bruch deſſen Zehler der Dividendus, der Nenner aber der Diviſor ſeyn wird; und iſt folglich dieſer Bruch der wahre Quotus, welcher herauskommt, wann man den Dividendum durch den Diviſorem dividirt.","norm":"Wann wir nun ferner diesen Bruch wiederum umkehren, den Zähler und Nenner nämlich unter sich verwechseln, so erhalten wir einen Bruch dessen Zähler der Dividendus, der Nenner aber der Divisor sein wird; und ist folglich dieser Bruch der wahre Quotus, welcher herauskommt, wann man den Dividendum durch den Divisorem 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+{"basename":"euler_rechenkunst02_1740","par_idx":1483,"date":1740,"orig":"Der Vortheil, welcher aus dieſer Betrachtung entſpringt, beſtehet darinn, daß wann man mit geringerer Muͤhe den Diviſorem durch den Dividendum dividiren kan, man dieſe Diviſion verrichte, und alsdann den gefundenen Quotum nach der gegebenen Vorſchrifft umkehre.","norm":"Der Vorteil, welcher aus dieser Betrachtung entspringt, bestehet darin, dass wann man mit geringerer Mühe den Divisorem durch den Dividendum dividieren kann, man diese Division verrichte, und alsdann den gefundenen Quotum nach der gegebenen Vorschrift 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+{"basename":"euler_rechenkunst02_1740","par_idx":1484,"date":1740,"orig":"Als wann man ſollte 3 durch 15 dividiren, ſo dividire ich 15 durch 3 und kehre den Quotum 5 oder in Bruchs Form [Formel] um, und bekomme alſo ⅕, welches der Quotus iſt, wann 3 durch 15 dividirt wird.","norm":"Als wann man sollte 3 durch 15 dividieren, so dividiere ich 15 durch 3 und kehre den Quotum 5 oder in Bruchs Form [Formel] um, und bekomme also 1/5 , welches der Quotus ist, wann 3 durch 15 dividiert 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+{"basename":"euler_rechenkunst02_1740","par_idx":1485,"date":1740,"orig":"Wann man ferner 6 durch 9 dividiren ſollte, ſo kan man 9 durch 6 dividiren, und den Quotum 1½ in Form eines einzelen Bruchs [Formel] umkehren, da dann ⅔ den geſuchten Quotum gibt.","norm":"Wann man ferner 6 durch 9 dividieren sollte, so kann man 9 durch 6 dividieren, und den Quotum 11/2 in Form eines einzeln Bruchs [Formel] umkehren, da dann 2/3 den gesuchten Quotum gibt.","orig_tok":["Wann","man","ferner","6","durch","9","dividiren","ſollte",",","ſo","kan","man","9","durch","6","dividiren",",","und","den","Quotum","1½","in","Form","eines","einzelen","Bruchs","[Formel]","umkehren",",","da","dann","⅔","den","geſuchten","Quotum","gibt","."],"norm_tok":["Wann","man","ferner","6","durch","9","dividieren","sollte",",","so","kann","man","9","durch","6","dividieren",",","und","den","Quotum","11","/","2","in","Form","eines","einzeln","Bruchs","[","Formel","]","umkehren",",","da","dann"," ","2","/","3"," 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+{"basename":"euler_rechenkunst02_1740","par_idx":1487,"date":1740,"orig":"So oft ſich nun ſolches findet, ſo darf man nur die Diviſion umkehren, und nach denſelbigen Regeln den Diviſorem durch den Dividendum dividiren: und wann dieſes geſchehen den gefundenen Quotum, nachdem man denſelben in die Form eines einzelen Bruchs gebracht hat, umkehren.","norm":"So oft sich nun solches findet, so darf man nur die Division umkehren, und nach denselbigen Regeln den Divisorem durch den Dividendum dividieren: und wann dieses geschehen den gefundenen Quotum, nachdem man denselben in die Form eines einzeln Bruchs gebracht hat, 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+{"basename":"euler_rechenkunst02_1740","par_idx":1523,"date":1740,"orig":"Aus allen dieſen verſchiedenen Arten iſt nun dienlich diejenige auszuleſen, nach welcher der Quotus mit der leichteſten Muͤhe gefunden werden kan; in welcher Wahl nicht alle Rechner uͤbereinſtimmen werden.","norm":"Aus allen diesen verschiedenen Arten ist nun dienlich diejenige auszulesen, nach welcher der Quotus mit der leichtesten Mühe gefunden werden kann; in welcher Wahl nicht alle Rechner übereinstimmen 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+{"basename":"euler_rechenkunst02_1740","par_idx":1524,"date":1740,"orig":"Dann diejenigen, welche nicht gerne mit Bruͤchen umgehen, werden die erſt angefuͤhrte Art den uͤbrigen weit vorziehen, in welcher beydes der Diviſor und Dividendus auf die kleinſte vorhandene Sorte reducirt werden.","norm":"Dann diejenigen, welche nicht gerne mit Brüchen umgehen, werden die erst angeführte Art den übrigen weit vorziehen, in welcher beides der Divisor und Dividendus auf die kleinste vorhandene Sorte reduziert werden.","orig_tok":["Dann","diejenigen",",","welche","nicht","gerne","mit","Bruͤchen","umgehen",",","werden","die","erſt","angefuͤhrte","Art","den","uͤbrigen","weit","vorziehen",",","in","welcher","beydes","der","Diviſor","und","Dividendus","auf","die","kleinſte","vorhandene","Sorte","reducirt","werden","."],"norm_tok":["Dann","diejenigen",",","welche","nicht","gerne","mit","Brüchen","umgehen",",","werden","die","erst","angeführte","Art","den","übrigen","weit","vorziehen",",","in","welcher","beides","der","Divisor","und","Dividendus","auf","die","kleinste","vorhandene","Sorte","reduziert","werden","."],"orig_spans":[[0,4],[5,15],[15,16],[17,23],[24,29],[30,35],[36,39],[40,48],[49,56],[56,57],[58,64],[65,68],[69,73],[74,85],[86,89],[90,93],[94,102],[103,107],[108,117],[117,118],[119,121],[122,129],[130,136],[137,140],[141,148],[149,152],[153,163],[164,167],[168,171],[172,180],[181,191],[192,197],[198,206],[207,213],[213,214]],"norm_spans":[[0,4],[5,15],[15,16],[17,23],[24,29],[30,35],[36,39],[40,47],[48,55],[55,56],[57,63],[64,67],[68,72],[73,83],[84,87],[88,91],[92,99],[100,104],[105,114],[114,115],[116,118],[119,126],[127,133],[134,137],[138,145],[146,149],[150,160],[161,164],[165,168],[169,177],[178,188],[189,194],[195,204],[205,211],[211,212]],"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]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.362599849700928} +{"basename":"euler_rechenkunst02_1740","par_idx":1525,"date":1740,"orig":"Auf dieſe Art aber, welche zwar wegen der Vermeidung der Bruͤche ihre beſondere Vortheile hat, kommt man oͤfters auch auf ſehr groſſe Zahlen; wer demnach lieber mit gebrochenen als allzugroſſen gantzen Zahlen rechnet, derſelbe wird die zwey letzteren Arten der erſten oͤfters vorziehen.","norm":"Auf diese Art aber, welche zwar wegen der Vermeidung der Brüche ihre besondere Vorteile hat, kommt man öfters auch auf sehr große Zahlen; wer demnach lieber mit gebrochenen als allzu großen ganzen Zahlen rechnet, derselbe wird die zwei letzteren Arten der ersten öfters 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+{"basename":"euler_rechenkunst02_1740","par_idx":1526,"date":1740,"orig":"Dieſes beruhet nun hauptſaͤchlich auf dem Genie des Rechners, welcher durch eine geringe Muͤhe die fuͤr ſich vortheilhafteſte Art zu dividiren in einem jeglichen Fall bald wird ausfinden koͤnnen.","norm":"Dieses beruhet nun hauptsächlich auf dem Genie des Rechners, welcher durch eine geringe Mühe die für sich vorteilhafteste Art zu dividieren in einem jeglichen Fall bald wird ausfinden 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+{"basename":"euler_rechenkunst02_1740","par_idx":1527,"date":1740,"orig":"An und fuͤr ſich ſelbſten pflegen zwar dergleichen Diviſions-Exempel, in welchen ſo wohl der Diviſor als Dividendus benannte Zahlen ſind, ſehr ſelten vorzukommen, weswegen man auch in den meiſten Rechen-Buͤchern dieſe Art der Diviſion unberuͤhret antrifft.","norm":"An und für sich selbst pflegen zwar dergleichen Divisionsexempel, in welchen so wohl der Divisor als Dividendus benannte Zahlen sind, sehr selten vorzukommen, weswegen man auch in den meisten Rechenbüchern diese Art der Division unberührt 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+{"basename":"euler_rechenkunst02_1740","par_idx":1528,"date":1740,"orig":"Dem ungeacht aber iſt dieſe Art nicht nur von ſehr groſſem Nutzen, ſondern iſt ſo gar das Fundament der Regula de Tri mit benannten Zahlen; und enthaͤlt den fuͤrnehmſten Theil der gantzen Operation in ſich.","norm":"Dem ungeachtet aber ist diese Art nicht nur von sehr großem Nutzen, sondern ist so gar das Fundament der Regula de Tri mit benannten Zahlen; und enthält den fürnehmsten Teil der ganzen Operation in 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+{"basename":"euler_rechenkunst02_1740","par_idx":1529,"date":1740,"orig":"Dahero geſchieht es, daß diejenigen, welche dieſe Diviſion in den Arithmetiſchen Operationen uͤberſprungen haben, hernach in der Regula de Tri entweder dieſe Operation allererſt beſchreiben, und zur Ubung bringen muͤſſen; oder aber dieſelbe in die ſo genannte Jtaliaͤniſche Practicam einhuͤllen.","norm":"Daher geschieht es, dass diejenigen, welche diese Division in den arithmetischen Operationen übersprungen haben, hernach in der Regula de Tri entweder diese Operation allererst beschreiben, und zur Übung bringen müssen; oder aber dieselbe in die so genannte Italienische Practicam 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+{"basename":"euler_rechenkunst02_1740","par_idx":1530,"date":1740,"orig":"Weilen nun unſer Endzweck in dieſem Wercke iſt, die Arithmetiſchen Operationen nicht nur aus ihrem Grunde zu beſchreiben, ſondern auch die fuͤrnehmſten Vortheile, deren man ſich dabey bedienen kan, anzuzeigen, als worinn die gantze gemeldete Welſche Practic beſtehet; ſo haben wir auch dieſe Art der Diviſion, zugleich mit der folgenden Art der Multiplication allhier bey den Operationen auszufuͤhren fuͤr noͤthig befunden; damit wir hernach in der Regula de Tri und uͤbrigen der Arithmetic einverleibten Regeln nicht allererſt noͤthig haben die bey den Operationen dienlichen Vortheile zu beſchreiben.","norm":"Weilen nun unser Endzweck in diesem Werke ist, die arithmetischen Operationen nicht nur aus ihrem Grunde zu beschreiben, sondern auch die fürnehmsten Vorteile, deren man sich dabei bedienen kann, anzuzeigen, als worin die ganze gemeldete Welsche Praktik bestehet; so haben wir auch diese Art der Division, zugleich mit der folgenden Art der Multiplikation allhier bei den Operationen auszuführen für nötig befunden; damit wir hernach in der Regula de Tri und übrigen der Arithmetik einverleibten Regeln nicht allererst nötig haben die bei den Operationen dienlichen Vorteile zu 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V.","norm":"Cap V.","orig_tok":["Cap.","V."],"norm_tok":["Cap","V."],"orig_spans":[[0,4],[5,7]],"norm_spans":[[0,3],[4,6]],"alignment":[[0,0],[1,1]],"lang_fastText":"en","lang_py3langid":"en","lang_cld3":"ca","orig_lang_de":0,"norm_lmscore":6.413300037384033} +{"basename":"euler_rechenkunst02_1740","par_idx":1532,"date":1740,"orig":"Von der Multiplication und Diviſion benannter Zahlen durch Bruͤche.","norm":"Von der Multiplikation und Division benannter Zahlen durch 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benannte Zahl, aus ſo viel Sorten dieſelbe auch immer beſteht, multiplicirt, wann man dieſelbe erſtlich durch den Zehler des Bruchs multiplicirt, und hernach das herausgebrachte Product durch den Nenner deſſelben Bruchs dividirt:","norm":"Durch einen Bruch wird eine benannte Zahl, aus so viel Sorten dieselbe auch immer besteht, multipliziert, wann man dieselbe erstlich durch den Zähler des Bruchs multipliziert, und hernach das herausgebrachte Produkt durch den Nenner desselben Bruchs dividiert:","orig_tok":["DUrch","einen","Bruch","wird","eine","benannte","Zahl",",","aus","ſo","viel","Sorten","dieſelbe","auch","immer","beſteht",",","multiplicirt",",","wann","man","dieſelbe","erſtlich","durch","den","Zehler","des","Bruchs","multiplicirt",",","und","hernach","das","herausgebrachte","Product","durch","den","Nenner","deſſelben","Bruchs","dividirt",":"],"norm_tok":["Durch","einen","Bruch","wird","eine","benannte","Zahl",",","aus","so","viel","Sorten","dieselbe","auch","immer","besteht",",","multipliziert",",","wann","man","dieselbe","erstlich","durch","den","Zähler","des","Bruchs","multipliziert",",","und","hernach","das","herausgebrachte","Produkt","durch","den","Nenner","desselben","Bruchs","dividiert",":"],"orig_spans":[[0,5],[6,11],[12,17],[18,22],[23,27],[28,36],[37,41],[41,42],[43,46],[47,49],[50,54],[55,61],[62,70],[71,75],[76,81],[82,89],[89,90],[91,103],[103,104],[105,109],[110,113],[114,122],[123,131],[132,137],[138,141],[142,148],[149,152],[153,159],[160,172],[172,173],[174,177],[178,185],[186,189],[190,205],[206,213],[214,219],[220,223],[224,230],[231,240],[241,247],[248,256],[256,257]],"norm_spans":[[0,5],[6,11],[12,17],[18,22],[23,27],[28,36],[37,41],[41,42],[43,46],[47,49],[50,54],[55,61],[62,70],[71,75],[76,81],[82,89],[89,90],[91,104],[104,105],[106,110],[111,114],[115,123],[124,132],[133,138],[139,142],[143,149],[150,153],[154,160],[161,174],[174,175],[176,179],[180,187],[188,191],[192,207],[208,215],[216,221],[222,225],[226,232],[233,242],[243,249],[250,259],[259,260]],"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":4.538899898529053} 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Gestalt man durch Brüche multiplizieren müsse.","orig_tok":["Wir","haben","ſchon","oben","bey","den","Bruͤchen","gewieſen",",","welcher","geſtalt","man","durch","Bruͤche","multipliciren","muͤſſe","."],"norm_tok":["Wir","haben","schon","oben","bei","den","Brüchen","gewiesen",",","welcher","Gestalt","man","durch","Brüche","multiplizieren","müsse","."],"orig_spans":[[0,3],[4,9],[10,15],[16,20],[21,24],[25,28],[29,37],[38,46],[46,47],[48,55],[56,63],[64,67],[68,73],[74,81],[82,95],[96,102],[102,103]],"norm_spans":[[0,3],[4,9],[10,15],[16,20],[21,24],[25,28],[29,36],[37,45],[45,46],[47,54],[55,62],[63,66],[67,72],[73,79],[80,94],[95,100],[100,101]],"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]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"lb","orig_lang_de":0.667,"norm_lmscore":5.283899784088135} +{"basename":"euler_rechenkunst02_1740","par_idx":1537,"date":1740,"orig":"Wir haben zwar dort hauptſaͤchlich Bruͤche mit Bruͤchen multipliciren gelehret, und darfuͤr dieſe Regel gegeben, daß man von den Bruͤchen, welche mit einander multiplicirt werden ſollen, erſtlich die Zehler und dann die Nenner durch einander multipliciren, und das erſtere Product fuͤr den Zehler, das letztere aber fuͤr den Nenner des geſuchten Products annehmen muͤſſe.","norm":"Wir haben zwar dort hauptsächlich Brüche mit Brüchen multiplizieren gelehrt, und dafür diese Regel gegeben, dass man von den Brüchen, welche miteinander multipliziert werden sollen, erstlich die Zähler und dann die Nenner durcheinander multiplizieren, und das erstere Produkt für den Zähler, das letztere aber für den Nenner des gesuchten Produkts annehmen müsse.","orig_tok":["Wir","haben","zwar","dort","hauptſaͤchlich","Bruͤche","mit","Bruͤchen","multipliciren","gelehret",",","und","darfuͤr","dieſe","Regel","gegeben",",","daß","man","von","den","Bruͤchen",",","welche","mit","einander","multiplicirt","werden","ſollen",",","erſtlich","die","Zehler","und","dann","die","Nenner","durch","einander","multipliciren",",","und","das","erſtere","Product","fuͤr","den","Zehler",",","das","letztere","aber","fuͤr","den","Nenner","des","geſuchten","Products","annehmen","muͤſſe","."],"norm_tok":["Wir","haben","zwar","dort","hauptsächlich","Brüche","mit","Brüchen","multiplizieren","gelehrt",",","und","dafür","diese","Regel","gegeben",",","dass","man","von","den","Brüchen",",","welche","miteinander","multipliziert","werden","sollen",",","erstlich","die","Zähler","und","dann","die","Nenner","durcheinander","multiplizieren",",","und","das","erstere","Produkt","für","den","Zähler",",","das","letztere","aber","für","den","Nenner","des","gesuchten","Produkts","annehmen","müsse","."],"orig_spans":[[0,3],[4,9],[10,14],[15,19],[20,34],[35,42],[43,46],[47,55],[56,69],[70,78],[78,79],[80,83],[84,91],[92,97],[98,103],[104,111],[111,112],[113,116],[117,120],[121,124],[125,128],[129,137],[137,138],[139,145],[146,149],[150,158],[159,171],[172,178],[179,185],[185,186],[187,195],[196,199],[200,206],[207,210],[211,215],[216,219],[220,226],[227,232],[233,241],[242,255],[255,256],[257,260],[261,264],[265,272],[273,280],[281,285],[286,289],[290,296],[296,297],[298,301],[302,310],[311,315],[316,320],[321,324],[325,331],[332,335],[336,345],[346,354],[355,363],[364,370],[370,371]],"norm_spans":[[0,3],[4,9],[10,14],[15,19],[20,33],[34,40],[41,44],[45,52],[53,67],[68,75],[75,76],[77,80],[81,86],[87,92],[93,98],[99,106],[106,107],[108,112],[113,116],[117,120],[121,124],[125,132],[132,133],[134,140],[141,152],[153,166],[167,173],[174,180],[180,181],[182,190],[191,194],[195,201],[202,205],[206,210],[211,214],[215,221],[222,235],[236,250],[250,251],[252,255],[256,259],[260,267],[268,275],[276,279],[280,283],[284,290],[290,291],[292,295],[296,304],[305,309],[310,313],[314,317],[318,324],[325,328],[329,338],[339,347],[348,356],[357,362],[362,363]],"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],[25,24],[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],[39,37],[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]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.023399829864502} +{"basename":"euler_rechenkunst02_1740","par_idx":1538,"date":1740,"orig":"Ob nun gleich hier nur von Bruͤchen die Rede iſt, ſo erſtrecket ſich dennoch dieſe Regel auch auf ſolche Faͤlle, in welchen entwedere Quantitaͤt von denen ſo durch einander multiplicirt werden ſollen eine gantze Zahl iſt; dann eine gantze Zahl kan immer in Form eines Bruchs vorgeſtellet werden, wann man dieſelbe als den Zehler und 1 als den Nenner betrachtet.","norm":"Ob nun gleich hier nur von Brüchen die Rede ist, so erstrecket sich dennoch diese Regel auch auf solche Fälle, in welchen entweder Quantität von denen so durcheinander multipliziert werden sollen eine ganze Zahl ist; dann eine ganze Zahl kann immer in Form eines Bruchs vorgestellt werden, wann man dieselbe als den Zähler und 1 als den Nenner betrachtet.","orig_tok":["Ob","nun","gleich","hier","nur","von","Bruͤchen","die","Rede","iſt",",","ſo","erſtrecket","ſich","dennoch","dieſe","Regel","auch","auf","ſolche","Faͤlle",",","in","welchen","entwedere","Quantitaͤt","von","denen","ſo","durch","einander","multiplicirt","werden","ſollen","eine","gantze","Zahl","iſt",";","dann","eine","gantze","Zahl","kan","immer","in","Form","eines","Bruchs","vorgeſtellet","werden",",","wann","man","dieſelbe","als","den","Zehler","und","1","als","den","Nenner","betrachtet","."],"norm_tok":["Ob","nun","gleich","hier","nur","von","Brüchen","die","Rede","ist",",","so","erstrecket","sich","dennoch","diese","Regel","auch","auf","solche","Fälle",",","in","welchen","entweder","Quantität","von","denen","so","durcheinander","multipliziert","werden","sollen","eine","ganze","Zahl","ist",";","dann","eine","ganze","Zahl","kann","immer","in","Form","eines","Bruchs","vorgestellt","werden",",","wann","man","dieselbe","als","den","Zähler","und","1","als","den","Nenner","betrachtet","."],"orig_spans":[[0,2],[3,6],[7,13],[14,18],[19,22],[23,26],[27,35],[36,39],[40,44],[45,48],[48,49],[50,52],[53,63],[64,68],[69,76],[77,82],[83,88],[89,93],[94,97],[98,104],[105,111],[111,112],[113,115],[116,123],[124,133],[134,144],[145,148],[149,154],[155,157],[158,163],[164,172],[173,185],[186,192],[193,199],[200,204],[205,211],[212,216],[217,220],[220,221],[222,226],[227,231],[232,238],[239,243],[244,247],[248,253],[254,256],[257,261],[262,267],[268,274],[275,287],[288,294],[294,295],[296,300],[301,304],[305,313],[314,317],[318,321],[322,328],[329,332],[333,334],[335,338],[339,342],[343,349],[350,360],[360,361]],"norm_spans":[[0,2],[3,6],[7,13],[14,18],[19,22],[23,26],[27,34],[35,38],[39,43],[44,47],[47,48],[49,51],[52,62],[63,67],[68,75],[76,81],[82,87],[88,92],[93,96],[97,103],[104,109],[109,110],[111,113],[114,121],[122,130],[131,140],[141,144],[145,150],[151,153],[154,167],[168,181],[182,188],[189,195],[196,200],[201,206],[207,211],[212,215],[215,216],[217,221],[222,226],[227,232],[233,237],[238,242],[243,248],[249,251],[252,256],[257,262],[263,269],[270,281],[282,288],[288,289],[290,294],[295,298],[299,307],[308,311],[312,315],[316,322],[323,326],[327,328],[329,332],[333,336],[337,343],[344,354],[354,355]],"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],[30,29],[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],[63,62],[64,63]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.339499950408935} +{"basename":"euler_rechenkunst02_1740","par_idx":1539,"date":1740,"orig":"Wann demnach eine gantze Zahl mit einem Bruche multiplicirt werden ſoll, ſo multiplicirt man dieſelbe mit dem Zehler des Bruches, und ſchreibt unter das Product den Nenner deſſelben Bruchs in Bruchs-Form: ſo daß fuͤr das verlangte Product ein Bruch gefunden wird, deſſen Zehler das Product iſt aus dem Zehler des Bruchs und der gantzen Zahl, welche mit einander multiplicirt werden ſollen: der Nenner aber kommt mit dem Nenner des Bruchs dadurch multiplicirt werden ſoll uͤberein.","norm":"Wann demnach eine ganze Zahl mit einem Bruche multipliziert werden soll, so multipliziert man dieselbe mit dem Zähler des Bruches, und schreibt unter das Produkt den Nenner desselben Bruchs in Bruchsform: so dass für das verlangte Produkt ein Bruch gefunden wird, dessen Zähler das Produkt ist aus dem Zähler des Bruchs und der ganzen Zahl, welche miteinander multipliziert werden sollen: Der Nenner aber kommt mit dem Nenner des Bruchs dadurch multipliziert werden soll überein.","orig_tok":["Wann","demnach","eine","gantze","Zahl","mit","einem","Bruche","multiplicirt","werden","ſoll",",","ſo","multiplicirt","man","dieſelbe","mit","dem","Zehler","des","Bruches",",","und","ſchreibt","unter","das","Product","den","Nenner","deſſelben","Bruchs","in","Bruchs-Form",":","ſo","daß","fuͤr","das","verlangte","Product","ein","Bruch","gefunden","wird",",","deſſen","Zehler","das","Product","iſt","aus","dem","Zehler","des","Bruchs","und","der","gantzen","Zahl",",","welche","mit","einander","multiplicirt","werden","ſollen",":","der","Nenner","aber","kommt","mit","dem","Nenner","des","Bruchs","dadurch","multiplicirt","werden","ſoll","uͤberein","."],"norm_tok":["Wann","demnach","eine","ganze","Zahl","mit","einem","Bruche","multipliziert","werden","soll",",","so","multipliziert","man","dieselbe","mit","dem","Zähler","des","Bruches",",","und","schreibt","unter","das","Produkt","den","Nenner","desselben","Bruchs","in","Bruchsform",":","so","dass","für","das","verlangte","Produkt","ein","Bruch","gefunden","wird",",","dessen","Zähler","das","Produkt","ist","aus","dem","Zähler","des","Bruchs","und","der","ganzen","Zahl",",","welche","miteinander","multipliziert","werden","sollen",":","Der","Nenner","aber","kommt","mit","dem","Nenner","des","Bruchs","dadurch","multipliziert","werden","soll","überein","."],"orig_spans":[[0,4],[5,12],[13,17],[18,24],[25,29],[30,33],[34,39],[40,46],[47,59],[60,66],[67,71],[71,72],[73,75],[76,88],[89,92],[93,101],[102,105],[106,109],[110,116],[117,120],[121,128],[128,129],[130,133],[134,142],[143,148],[149,152],[153,160],[161,164],[165,171],[172,181],[182,188],[189,191],[192,203],[203,204],[205,207],[208,211],[212,216],[217,220],[221,230],[231,238],[239,242],[243,248],[249,257],[258,262],[262,263],[264,270],[271,277],[278,281],[282,289],[290,293],[294,297],[298,301],[302,308],[309,312],[313,319],[320,323],[324,327],[328,335],[336,340],[340,341],[342,348],[349,352],[353,361],[362,374],[375,381],[382,388],[388,389],[390,393],[394,400],[401,405],[406,411],[412,415],[416,419],[420,426],[427,430],[431,437],[438,445],[446,458],[459,465],[466,470],[471,479],[479,480]],"norm_spans":[[0,4],[5,12],[13,17],[18,23],[24,28],[29,32],[33,38],[39,45],[46,59],[60,66],[67,71],[71,72],[73,75],[76,89],[90,93],[94,102],[103,106],[107,110],[111,117],[118,121],[122,129],[129,130],[131,134],[135,143],[144,149],[150,153],[154,161],[162,165],[166,172],[173,182],[183,189],[190,192],[193,203],[203,204],[205,207],[208,212],[213,216],[217,220],[221,230],[231,238],[239,242],[243,248],[249,257],[258,262],[262,263],[264,270],[271,277],[278,281],[282,289],[290,293],[294,297],[298,301],[302,308],[309,312],[313,319],[320,323],[324,327],[328,334],[335,339],[339,340],[341,347],[348,359],[360,373],[374,380],[381,387],[387,388],[389,392],[393,399],[400,404],[405,410],[411,414],[415,418],[419,425],[426,429],[430,436],[437,444],[445,458],[459,465],[466,470],[471,478],[478,479]],"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],[62,61],[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]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":3.892100095748901} +{"basename":"euler_rechenkunst02_1740","par_idx":1540,"date":1740,"orig":"Weilen nun der Werth eines Bruchs gefunden wird, wann man den Zehler durch den Nenner wuͤrcklich dividirt, ſo wird auch eine jegliche Zahl durch einen Bruch multiplicirt, wann man dieſelbe erſtlich mit dem Zehler des Bruchs multiplicirt, und was herausgekommen, durch den Nenner dividirt.","norm":"Weilen nun der Wert eines Bruchs gefunden wird, wann man den Zähler durch den Nenner wirklich dividiert, so wird auch eine jegliche Zahl durch einen Bruch multipliziert, wann man dieselbe erstlich mit dem Zähler des Bruchs multipliziert, und was herausgekommen, durch den Nenner 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+{"basename":"euler_rechenkunst02_1740","par_idx":1541,"date":1740,"orig":"Dieſe Regel iſt auch allgemein und erſtrecket nicht nur auf gantze Zahlen, welche mit Bruͤchen multiplicirt ſollen, ſondern auf aller Gattung Quantitaͤten, was ſolche auch immer fuͤr Nahmen fuͤhren.","norm":"Diese Regel ist auch allgemein und erstrecket nicht nur auf ganze Zahlen, welche mit Brüchen multipliziert sollen, sondern auf aller Gattung Quantitäten, was solche auch immer für Namen 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+{"basename":"euler_rechenkunst02_1740","par_idx":1542,"date":1740,"orig":"Alles dieſes wird aber deutlicher werden, wann wir erſtlich Statt des Multiplicatoris ſolche Bruͤche annehmen, deren Zehler 1 iſt; und zeigen, daß durch einen ſolchen Bruch multiplicirt wird, wann man durch den Nenner deſſelben multiplicirt.","norm":"Alles dieses wird aber deutlicher werden, wann wir erstlich statt des Multiplicatoris solche Brüche annehmen, deren Zähler 1 ist; und zeigen, dass durch einen solchen Bruch multipliziert wird, wann man durch den Nenner desselben 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+{"basename":"euler_rechenkunst02_1740","par_idx":1543,"date":1740,"orig":"Dann mit ½ multipliciren iſt nichts anders als die Helfte von einer Sach nehmen, und folglich ſo viel als durch 2 dividiren: gleichergeſtalt wird eine Zahl durch ⅓ multiplicirt, wann man dieſelbe durch 3 dividirt und alſo wird die Multiplication durch einen Bruch deſſen Zehler 1 iſt allzeit in eine bloſſe Diviſion verwandelt.","norm":"Dann mit ½ multiplizieren ist nichts anders als die Hälfte von einer Sache nehmen, und folglich so viel als durch 2 dividieren: gleichergestalt wird eine Zahl durch 1/3 multipliziert, wann man dieselbe durch 3 dividiert und also wird die Multiplikation durch einen Bruch dessen Zähler 1 ist allzeit in eine bloße Division 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+{"basename":"euler_rechenkunst02_1740","par_idx":1544,"date":1740,"orig":"Wann nun dieſes ſeine Richtigkeit hat, ſo folgt daraus ſehr leicht wie man durch einen Bruch deſſen Zehler nicht 1 iſt dividiren muͤſſe: wann man dazu den bey der Multiplication oben angefuͤhrten Vortheil in Erwegung ziehet; da wir gewieſen haben, daß wann ſich der Multiplicator in zwey Factores zertheilen laͤſt, man erſtlich die Multiplication durch einen Factorem anſtellen, und das gefundene Product noch mahl durch den anderen Factorem multipliciren koͤnne.","norm":"Wann nun dieses seine Richtigkeit hat, so folgt daraus sehr leicht wie man durch einen Bruch dessen Zähler nicht 1 ist dividieren müsse: wann man dazu den bei der Multiplikation oben angeführten Vorteil in Erwägung ziehet; da wir gewiesen haben, dass wann sich der Multiplicator in zwei Factores zerteilen lässt, man erstlich die Multiplikation durch einen Factorem anstellen, und das gefundene Produkt noch mal durch den anderen Factorem 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+{"basename":"euler_rechenkunst02_1740","par_idx":1545,"date":1740,"orig":"Weilen ſich nun ein jeglicher Bruch deſſen Zehler nicht 1 iſt in zwey Factores zertheilen laͤſt, davon einer eine gantze Zahl und dem Zehler des Bruchs gleich iſt, der andere aber ein Bruch iſt, deſſen Zehler 1 der Nenner aber dem Nenner deſſelben Bruchs gleich iſt, ſo wird durch einen ſolchen Bruch, deſſen Zehler nicht 1 iſt multiplicirt werden, wann man erſtlich mit dem Zehler des Bruchs multiplicirt, und was herausgekommen, durch den Nenner dividirt.","norm":"Weilen sich nun ein jeglicher Bruch dessen Zähler nicht 1 ist in zwei Factores zerteilen lässt, davon einer eine ganze Zahl und dem Zähler des Bruchs gleich ist, der andere aber ein Bruch ist, dessen Zähler 1 der Nenner aber dem Nenner desselben Bruchs gleich ist, so wird durch einen solchen Bruch, dessen Zähler nicht 1 ist multipliziert werden, wann man erstlich mit dem Zähler des Bruchs multipliziert, und was herausgekommen, durch den Nenner 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+{"basename":"euler_rechenkunst02_1740","par_idx":1546,"date":1740,"orig":"Wann man zum Exempel mit [Formel] multipliciren ſolte, ſo iſt erſtlich zu mercken, daß [Formel] ſo viel ſey als 7 mahl [Formel] . Derohalben multipliciet man erſtlich mit 7, und was herausgekommen noch mit [Formel] . Weilen nun mit [Formel] multipliciren nichts anders iſt als durch 12 dividiren, ſo folgt daraus, daß eine Zahl durch [Formel] multiplicirt werde, wann man dieſelbe erſtlich mit 7 multiplicirt, und hernach das Product durch 12 dividirt.","norm":"Wann man zum Exempel mit [Formel] multiplizieren sollte, so ist erstlich zu merken, dass [Formel] so viel sei als 7 Mal [Formel]. Derohalben multipliciet man erstlich mit 7, und was herausgekommen noch mit [Formel]. Weilen nun mit [Formel] multiplizieren nichts anders ist als durch 12 dividieren, so folgt daraus, dass eine Zahl durch [Formel] multipliziert werde, wann man dieselbe erstlich mit 7 multipliziert, und hernach das Produkt durch 12 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Weilen mit [Formel] multipliciren nichts anders iſt als durch 12 dividiren, ſo muß man die Jahrs-Zeit durch 12 dividiren.","norm":"Antw . 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Wann gefragt wird, wieviel ⅔ von einer Quantitaͤt austragen, ſo iſts eben ſo viel, als wan man dieſelbe Quantitaͤt mit ⅔ multipliciren ſoll.","norm":"Antw. 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[Formel]","norm":"Anstatt aber durch 15 zu dividieren, so kann man 15 in seine zwei Factores 3 und 5 verteilen und erstlich durch 3 und hernach durch 5 dividieren, welche beiden Divisionen leichter fallen werden, als die einzelne Division durch 15. 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Weilen allhier der Multiplicator 5¾ eine aus gantzen und Bruͤchen vermiſchte Zahl iſt, ſo kan man denſelben um die gegebene Regel anzubringen in die Form eines einzelen Bruchs bringen, da man dan [Formel] bekommt.","norm":"Antw. 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verlangte Produkt anzeigen 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+{"basename":"euler_rechenkunst02_1740","par_idx":1583,"date":1740,"orig":"Wir haben ſchon mehr als ein mahl erwieſen, daß wann der Multiplicator aus verſchiedenen Theilen beſtehet, das Product gefunden werde, wann man den Multiplicandum insbeſondere durch einen jeglichen Theil des Multiplicatoris multiplicirt, und alle dieſe beſonderen Producte zuſammen addirt: und dieſes findet ſo wohl ſtatt, wann der Multiplicator wuͤrcklich aus verſchiedenen Theilen zuſammen geſetzt iſt, als wann derſelbe nur in den Gedancken in etliche Theile zertheilet wird.","norm":"Wir haben schon mehr als ein Mal erwiesen, dass wann der Multiplicator aus verschiedenen Teilen bestehet, das Produkt gefunden werde, wann man den Multiplicandum insbesondere durch einen jeglichen Teil des Multiplicatoris multipliziert, und alle diese besonderen Produkte zusammen addiert: und dieses findet so wohl statt, wann der Multiplicator wirklich aus verschiedenen Teilen zusammen gesetzt ist, als wann derselbe nur in den 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+{"basename":"euler_rechenkunst02_1740","par_idx":1586,"date":1740,"orig":"Hierdurch erhaͤlt man nun ſehr wichtige Vortheile in der Operation:","norm":"Hierdurch erhält man nun sehr wichtige Vorteile in der Operation:","orig_tok":["Hierdurch","erhaͤlt","man","nun","ſehr","wichtige","Vortheile","in","der","Operation",":"],"norm_tok":["Hierdurch","erhält","man","nun","sehr","wichtige","Vorteile","in","der","Operation",":"],"orig_spans":[[0,9],[10,17],[18,21],[22,25],[26,30],[31,39],[40,49],[50,52],[53,56],[57,66],[66,67]],"norm_spans":[[0,9],[10,16],[17,20],[21,24],[25,29],[30,38],[39,47],[48,50],[51,54],[55,64],[64,65]],"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":4.908699989318848} +{"basename":"euler_rechenkunst02_1740","par_idx":1587,"date":1740,"orig":"dann erſtlich hat man nicht noͤthig die gantze Zahl des Multiplicatoris in die Form des damit verknuͤpften Bruchs zu bringen: und dadurch vermeidet man auch hernach die oͤfters ſehr groſſen und beſchwehrlichen Zahlen, welche wann man den Multiplicatorem in einen einzelen Bruch bringt in den Zehler kommen und wird man folglich der Multiplication mit ſolchen groſſen Zahlen uͤberhoben.","norm":"dann erstlich hat man nicht nötig die ganze Zahl des Multiplicatoris in die Form des damit verknüpften Bruchs zu bringen: und dadurch vermeidet man auch hernach die öfters sehr großen und beschwerlichen Zahlen, welche wann man den Multiplikatorrem in einen einzeln Bruch bringt in den Zähler kommen und wird man folglich der Multiplikation mit solchen großen Zahlen überhoben.","orig_tok":["dann","erſtlich","hat","man","nicht","noͤthig","die","gantze","Zahl","des","Multiplicatoris","in","die","Form","des","damit","verknuͤpften","Bruchs","zu","bringen",":","und","dadurch","vermeidet","man","auch","hernach","die","oͤfters","ſehr","groſſen","und","beſchwehrlichen","Zahlen",",","welche","wann","man","den","Multiplicatorem","in","einen","einzelen","Bruch","bringt","in","den","Zehler","kommen","und","wird","man","folglich","der","Multiplication","mit","ſolchen","groſſen","Zahlen","uͤberhoben","."],"norm_tok":["dann","erstlich","hat","man","nicht","nötig","die","ganze","Zahl","des","Multiplicatoris","in","die","Form","des","damit","verknüpften","Bruchs","zu","bringen",":","und","dadurch","vermeidet","man","auch","hernach","die","öfters","sehr","großen","und","beschwerlichen","Zahlen",",","welche","wann","man","den","Multiplikatorrem","in","einen","einzeln","Bruch","bringt","in","den","Zähler","kommen","und","wird","man","folglich","der","Multiplikation","mit","solchen","großen","Zahlen","überhoben","."],"orig_spans":[[0,4],[5,13],[14,17],[18,21],[22,27],[28,35],[36,39],[40,46],[47,51],[52,55],[56,71],[72,74],[75,78],[79,83],[84,87],[88,93],[94,106],[107,113],[114,116],[117,124],[124,125],[126,129],[130,137],[138,147],[148,151],[152,156],[157,164],[165,168],[169,176],[177,181],[182,189],[190,193],[194,209],[210,216],[216,217],[218,224],[225,229],[230,233],[234,237],[238,253],[254,256],[257,262],[263,271],[272,277],[278,284],[285,287],[288,291],[292,298],[299,305],[306,309],[310,314],[315,318],[319,327],[328,331],[332,346],[347,350],[351,358],[359,366],[367,373],[374,384],[384,385]],"norm_spans":[[0,4],[5,13],[14,17],[18,21],[22,27],[28,33],[34,37],[38,43],[44,48],[49,52],[53,68],[69,71],[72,75],[76,80],[81,84],[85,90],[91,102],[103,109],[110,112],[113,120],[120,121],[122,125],[126,133],[134,143],[144,147],[148,152],[153,160],[161,164],[165,171],[172,176],[177,183],[184,187],[188,202],[203,209],[209,210],[211,217],[218,222],[223,226],[227,230],[231,247],[248,250],[251,256],[257,264],[265,270],[271,277],[278,280],[281,284],[285,291],[292,298],[299,302],[303,307],[308,311],[312,320],[321,324],[325,339],[340,343],[344,351],[352,358],[359,365],[366,375],[375,376]],"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]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.739099979400635} +{"basename":"euler_rechenkunst02_1740","par_idx":1588,"date":1740,"orig":"Ob man aber gleich durch die hier beſchriebene Art zwey Multiplicationen zu verrichten und die Producte zuſammen zu addiren genoͤthiget iſt, ſo uͤberwiegen doch die gemeldten Vortheile dieſen Zuwachs der Arbeit meiſtentheils.","norm":"Ob man aber gleich durch die hier beschriebene Art zwei Multiplikationen zu verrichten und die Produkte zusammenzuaddieren genötigt ist, so überwiegen doch die gemeldeten Vorteile diesen Zuwachs der Arbeit 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+{"basename":"euler_rechenkunst02_1740","par_idx":1589,"date":1740,"orig":"Um ſo viel groͤſſer wird aber der Nutzen noch werden, wann wir hernach noch einige beſondere Vortheile mit Bruͤchen zu multipliciren anzeigen werden.","norm":"Um so viel größer wird aber der Nutzen noch werden, wann wir hernach noch einige besondere Vorteile mit Brüchen zu multiplizieren anzeigen 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+{"basename":"euler_rechenkunst02_1740","par_idx":1590,"date":1740,"orig":"Wir wollen inzwiſchen dieſe Art mit einem aus gantzen und gebrochenen Zahlen zuſammen geſetzten Multiplicatore zu multipliciren durch einige Exempel erlaͤutern.","norm":"Wir wollen inzwischen diese Art mit einem aus ganzen und gebrochenen Zahlen zusammen gesetzten Multiplikator zu multiplizieren durch einige Exempel 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Sterl. 13 ß. 8 ₰, welche mit 4⅓ multiplicirt werden ſoll?","norm":"Man hat an Englischem Gelde diese Summe 209 L. Sterl . 13 ß . 8 ₰, welche mit 4 1/3 multipliziert werden soll?","orig_tok":["Man","hat","an","Engliſchem","Gelde","dieſe","Summ","209","L.","Sterl",".","13","ß",".","8","₰",",","welche","mit","4⅓","multiplicirt","werden","ſoll","?"],"norm_tok":["Man","hat","an","Englischem","Gelde","diese","Summe","209","L.","Sterl",".","13","ß",".","8","₰",",","welche","mit","4","1","/","3"," ","multipliziert","werden","soll","?"],"orig_spans":[[0,3],[4,7],[8,10],[11,21],[22,27],[28,33],[34,38],[39,42],[43,45],[46,51],[51,52],[53,55],[56,57],[57,58],[59,60],[61,62],[62,63],[64,70],[71,74],[75,77],[78,90],[91,97],[98,102],[102,103]],"norm_spans":[[0,3],[4,7],[8,10],[11,21],[22,27],[28,33],[34,39],[40,43],[44,46],[47,52],[53,54],[55,57],[58,59],[60,61],[62,63],[64,65],[65,66],[67,73],[74,77],[78,79],[80,81],[81,82],[82,83],[84,85],[85,98],[99,105],[106,110],[110,111]],"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],[null,20],[null,21],[null,22],[null,23],[20,24],[21,25],[22,26],[23,27]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":5.779300212860107} +{"basename":"euler_rechenkunst02_1740","par_idx":1593,"date":1740,"orig":"Antw. Weilen der Multiplicator 4⅓ iſt, ſo multiplicirt man erſtlich die vorgelegte Summ mit 4, hernach mit ⅓, oder welches gleichviel iſt, man dividirt dieſe Summ durch 3 und addirt den Quotum zum vorigen Product: wie aus der Operation zu erſehen.","norm":"Antw. Weilen der Multiplicator 4⅓ ist, so multipliziert man erstlich die vorgelegte Summe mit 4, hernach mit 1/3 , oder welches gleichviel ist, man dividiert diese Summe durch 3 und addiert den Quotum zum vorigen Produkt: wie aus der Operation zu ersehen.","orig_tok":["Antw.","Weilen","der","Multiplicator","4⅓","iſt",",","ſo","multiplicirt","man","erſtlich","die","vorgelegte","Summ","mit","4",",","hernach","mit","⅓",",","oder","welches","gleichviel","iſt",",","man","dividirt","dieſe","Summ","durch","3","und","addirt","den","Quotum","zum","vorigen","Product",":","wie","aus","der","Operation","zu","erſehen","."],"norm_tok":["Antw",".","Weilen","der","Multiplicator","4⅓","ist",",","so","multipliziert","man","erstlich","die","vorgelegte","Summe","mit","4",",","hernach","mit"," ","1","/","3",",","oder","welches","gleichviel","ist",",","man","dividiert","diese","Summe","durch","3","und","addiert","den","Quotum","zum","vorigen","Produkt",":","wie","aus","der","Operation","zu","ersehen","."],"orig_spans":[[0,5],[6,12],[13,16],[17,30],[31,33],[34,37],[37,38],[39,41],[42,54],[55,58],[59,67],[68,71],[72,82],[83,87],[88,91],[92,93],[93,94],[95,102],[103,106],[107,108],[108,109],[110,114],[115,122],[123,133],[134,137],[137,138],[139,142],[143,151],[152,157],[158,162],[163,168],[169,170],[171,174],[175,181],[182,185],[186,192],[193,196],[197,204],[205,212],[212,213],[214,217],[218,221],[222,225],[226,235],[236,238],[239,246],[246,247]],"norm_spans":[[0,4],[4,5],[6,12],[13,16],[17,30],[31,33],[34,37],[37,38],[39,41],[42,55],[56,59],[60,68],[69,72],[73,83],[84,89],[90,93],[94,95],[95,96],[97,104],[105,108],[109,110],[110,111],[111,112],[112,113],[114,115],[116,120],[121,128],[129,139],[140,143],[143,144],[145,148],[149,158],[159,164],[165,170],[171,176],[177,178],[179,182],[183,190],[191,194],[195,201],[202,205],[206,213],[214,221],[221,222],[223,226],[227,230],[231,234],[235,244],[245,247],[248,255],[255,256]],"alignment":[[0,0],[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],[null,20],[null,21],[null,22],[null,23],[19,null],[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]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.810100078582764} +{"basename":"euler_rechenkunst02_1740","par_idx":1594,"date":1740,"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_rechenkunst02_1740","par_idx":1595,"date":1740,"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_rechenkunst02_1740","par_idx":1596,"date":1740,"orig":"Es ſollen nach dem 5ten Exempel des vorigen Satzes wiederum 574 L. Sterl. 15 ß. durch 5¾ multiplicirt werden?","norm":"Es sollen nach dem 5ten Exempel des vorigen Satzes wiederum 574 L. 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Weilen dieſes Gewicht mit 6 und ⅖ multiplicirt werden muß, und es gleich gilt, mit welchem Theil man zu erſt multiplicirt, ſo wollen wir erſtlich mit ⅖ multipliciren, damit in der Operation die 2 Product, welche zuſammen addirt werden ſollen, unmittelbar untereinander zu ſtehen kommen.","norm":"Antw. Weilen dieses Gewicht mit 6 und 2/5 multipliziert werden muss, und es gleich gilt, mit welchem Teil man zuerst multipliziert, so wollen wir erstlich mit 2/5 multiplizieren, damit in der Operation die 2 Produkt, welche zusammen addiert werden sollen, unmittelbar untereinander zu stehen kommen.","orig_tok":["Antw.","Weilen","dieſes","Gewicht","mit","6","und","⅖","multiplicirt","werden","muß",",","und","es","gleich","gilt",",","mit","welchem","Theil","man","zu","erſt","multiplicirt",",","ſo","wollen","wir","erſtlich","mit","⅖","multipliciren",",","damit","in","der","Operation","die","2","Product",",","welche","zuſammen","addirt","werden","ſollen",",","unmittelbar","untereinander","zu","ſtehen","kommen","."],"norm_tok":["Antw",".","Weilen","dieses","Gewicht","mit","6","und"," ","2","/","5"," ","multipliziert","werden","muss",",","und","es","gleich","gilt",",","mit","welchem","Teil","man","zuerst","multipliziert",",","so","wollen","wir","erstlich","mit"," ","2","/","5"," 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Weilen hier alle Zahlen durch welche ſo wohl multiplicirt als dividirt werden ſoll, ſo groß ſind, daß dieſe Operationen bloß allein im Sinne nicht verrichtet werden koͤnnen: ſo kan dieſes Exempel auf nachfolgende Art zu Papier gebracht werden.","norm":"Antw. 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+{"basename":"euler_rechenkunst02_1740","par_idx":1627,"date":1740,"orig":"Dann erſtlich weilen 64 ſo viel iſt als 4 mahl 4 mahl 4, ſo kan man den Multiplicandum drey mahl durch 4 multipliciren; hernach anſtatt mit 13 zu multipliciren, ſo zergliedere man 13 alſo, 4 mahl 3 und noch 1, das iſt man multiplicire erſtlich den Multiplicandum durch 4, welches ſchon vorher geſchehen, und dieſes 4 fache noch mahl durch 3, zum Product addire man den Multiplicandum ſelbſt, ſo bekommt man das 13 fache, welches noch mit 25 dividirt werden muß, weilen nun 25 ſich in dieſe Factores 5 mahl 5 reſolviren laͤßt, ſo dividire man zwey mahl, nehmlich erſtlich mit 5 und den Quotum noch mahl mit 5. Endlich zu dieſem letzteren Quoto addire man das erſte Product von 64. [Formel]","norm":"Dann erstlich weilen 64 so viel ist als 4 Mal 4 Mal 4, so kann man den Multiplicandum dreimal durch 4 multiplizieren; hernach anstatt mit 13 zu multiplizieren, so zergliedere man 13 also, 4 Mal 3 und noch 1, das ist man multipliziere erstlich den Multiplicandum durch 4, welches schon vorher geschehen, und dieses 4-fache noch mal durch 3, zum Produkt addiere man den Multiplicandum selbst, so bekommt man das 13-fache, welches noch mit 25 dividiert werden muss, weilen nun 25 sich in diese Factores 5 Mal 5 resolvieren lässt, so dividiere man zweimal, nämlich erstlich mit 5 und den Quotum noch mal mit 5. Endlich zu diesem letzteren Quoto addiere man das erste Produkt von 64. 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+{"basename":"euler_rechenkunst02_1740","par_idx":1628,"date":1740,"orig":"Wann aber ſolche Vortheile angebracht werden, ſo iſt dabey inſonderheit zu beobachten, daß man ſich vor allen Dingen die Zergliederung der Multiplicatorum und Diviſorum deutlich bemercke; hernach alle Operationen ordentlich verrichte und bey einer jeglichen anzeige, warum ſolche geſchehen, damit man die gantze Vertheilung der Operation immer vor Augen behalte, und ſich nicht confundire, welcher Behutſamkeit ſich ein jeder auf eine ihm bequeme Art bedienen kan.","norm":"Wann aber solche Vorteile angebracht werden, so ist dabei in Sonderheit zu beobachten, dass man sich vor allen Dingen die Zergliederung der Multiplicatorum und Divisorum deutlich bemerke; hernach alle Operationen ordentlich verrichte und bei einer jeglichen anzeige, warum solche geschehen, damit man die ganze Verteilung der Operation immer vor Augen behalte, und sich nicht konfundiere, welcher Behutsamkeit sich ein jeder auf eine ihm bequeme Art 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+{"basename":"euler_rechenkunst02_1740","par_idx":1633,"date":1740,"orig":"da dann das letzte Product dasjenige iſt, welches verlanget wird.","norm":"da dann das letzte Produkt dasjenige ist, welches verlanget wird.","orig_tok":["da","dann","das","letzte","Product","dasjenige","iſt",",","welches","verlanget","wird","."],"norm_tok":["da","dann","das","letzte","Produkt","dasjenige","ist",",","welches","verlanget","wird","."],"orig_spans":[[0,2],[3,7],[8,11],[12,18],[19,26],[27,36],[37,40],[40,41],[42,49],[50,59],[60,64],[64,65]],"norm_spans":[[0,2],[3,7],[8,11],[12,18],[19,26],[27,36],[37,40],[40,41],[42,49],[50,59],[60,64],[64,65]],"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":5.263199806213379} +{"basename":"euler_rechenkunst02_1740","par_idx":1634,"date":1740,"orig":"Zertheilet man aber den Multiplicatorem in Theile, ſo multiplicirt man den 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+{"basename":"euler_rechenkunst02_1740","par_idx":1636,"date":1740,"orig":"Hat man aber den Diviſorem in bequeme Factores reſolviren koͤnnen, ſo dividirt man den Dividendum erſtlich durch den erſten Factorem hernach den gefundenen Quotum durch den andern Factorem, dieſen zweyten Quotum ferner durch den dritten Factorem, und ſo fort bis man durch alle Factores dividirt hat:","norm":"Hat man aber den Divisorem in bequeme Factores resolvieren können, so dividiert man den Dividendum erstlich durch den ersten Factorem hernach den gefundenen Quotum durch den anderen Factorem, diesen zweiten Quotum ferner durch den dritten Factorem, und so fort bis man durch alle Factores dividiert 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+{"basename":"euler_rechenkunst02_1740","par_idx":1637,"date":1740,"orig":"da dann der letzte Quotus der geſuchte ſeyn wird.","norm":"da dann der letzte Quotus der gesuchte sein wird.","orig_tok":["da","dann","der","letzte","Quotus","der","geſuchte","ſeyn","wird","."],"norm_tok":["da","dann","der","letzte","Quotus","der","gesuchte","sein","wird","."],"orig_spans":[[0,2],[3,7],[8,11],[12,18],[19,25],[26,29],[30,38],[39,43],[44,48],[48,49]],"norm_spans":[[0,2],[3,7],[8,11],[12,18],[19,25],[26,29],[30,38],[39,43],[44,48],[48,49]],"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":5.070700168609619} +{"basename":"euler_rechenkunst02_1740","par_idx":1638,"date":1740,"orig":"Durch dieſe Erleichterung der Multiplication und Diviſion wird aber der Vortheil um ſo viel groͤſſer, wann der Multiplicandus in zweyen verſchiedenen Multiplicationen durch einerley Zahl multiplicirt werden ſoll, oder wann man ein ſchon gefundenes Product zur folgenden Multiplication zu Huͤlfe nehmen kan.","norm":"Durch diese Erleichterung der Multiplikation und Division wird aber der Vorteil um so viel größer, wann der Multiplicandus in zweien verschiedenen Multiplikationen durch einerlei Zahl multipliziert werden soll, oder wann man ein schon gefundenes Produkt zur folgenden Multiplikation zu Hilfe nehmen 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Producte zuſammen addirt geben das verlangte Product.","norm":"Wann der Multiplicator ein Bruch ist, dessen Zähler größer ist als 1, und man folglich nach der ersten Regel durch den Zähler multiplizieren, und durch den Nenner dividieren müsste, so kann die Zerteilung eines solchen Multiplicatoris in zwei oder mehr Teile einen großen Vorteil schaffen wann erstlich die Teile 1 zum Zähler haben, und über das ein Teil in dem anderen etliche Mal enthalten ist: dann wann in solchem Falle durch den größten Teil multipliziert worden, so werden aus diesem Produkt die Produkte für die folgenden Teile durch die Division leicht gefunden; und alle diese Produkte zusammen addiert geben das verlangte 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+{"basename":"euler_rechenkunst02_1740","par_idx":1652,"date":1740,"orig":"Wir haben ſchon etliche mahl von der Zertheilung des Multiplicatoris in Theile und wie nach ſolchen Theilen die Multiplication angeſtellet werden ſoll, Meldung gethan: dieſelbe aber bringet nirgend einen ſo groſſen Vortheil, als wann der Multiplicator ein Bruch iſt, durch welchen die Multiplication ſonſten nach der erſten Regel beſchwehrlich ſeyn wuͤrde.","norm":"Wir haben schon etliche Mal von der Zerteilung des Multiplicatoris in Teile und wie nach solchen Teilen die Multiplikation angestellt werden soll, Meldung getan: Dieselbe aber bringt nirgend einen so großen Vorteil, als wann der Multiplicator ein Bruch ist, durch welchen die Multiplikation sonsten nach der ersten Regel beschwerlich sein 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+{"basename":"euler_rechenkunst02_1740","par_idx":1653,"date":1740,"orig":"Ja der Vortheil, welcher in dieſer Zertheilung des Multiplicatoris, wann derſelbe eine gebrochene Zahl iſt, ſteckt, iſt ſo groß, daß darinn allein faſt die gantze ſo genannte Jtaliaͤniſche Practic enthalten iſt: weswegen dieſer Vortheil mit beſonderer Aufmerckſamkeit abgehandelt zu werden verdienet.","norm":"Ja der Vorteil, welcher in dieser Zerteilung des Multiplicatoris, wann derselbe eine gebrochene Zahl ist, steckt, ist so groß, dass darin allein fast die ganze so genannte italienische Praktik enthalten ist: weswegen dieser Vorteil mit besonderer Aufmerksamkeit abgehandelt zu werden 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+{"basename":"euler_rechenkunst02_1740","par_idx":1654,"date":1740,"orig":"Wir haben aus dem vorhergehenden ſchon genugſam erſehen, daß es ſehr beſchwehrlich iſt benannte aus vielerley Sorten beſtehende Zahlen durch gantze Zahlen ſo wohl zu multipliciren als zu dividiren, und daß man einen nicht geringen Vortheil erhalte, wann man durch kleinere Zahlen operiren koͤnne, obgleich die Anzahl der Operationen dadurch vermehret wird.","norm":"Wir haben aus dem vorhergehenden schon genügsam ersehen, dass es sehr beschwerlich ist benannte aus vielerlei Sorten bestehende Zahlen durch ganze Zahlen so wohl zu multiplizieren als zu dividieren, und dass man einen nicht geringen Vorteil erhalte, wann man durch kleinere Zahlen operieren könne, obgleich die Anzahl der Operationen dadurch vermehret 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+{"basename":"euler_rechenkunst02_1740","par_idx":1656,"date":1740,"orig":"Hiezu ſind zwar ſchon im vorigen einige Vortheile angezeigt worden, welche Statt finden, wann man den Zehler und den Nenner des Bruchs, durch welchen multiplicirt werden ſoll, in bequeme Factores reſolviren kan, da dann ſo wohl die Multiplication durch den Zehler als die Diviſion durch den Nenner erleichtert wird.","norm":"Hierzu sind zwar schon im vorigen einige Vorteile angezeigt worden, welche statt finden, wann man den Zähler und den Nenner des Bruchs, durch welchen multipliziert werden soll, in bequeme Factores resolvieren kann, da dann so wohl die Multiplikation durch den Zähler als die Division durch den Nenner erleichtert 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+{"basename":"euler_rechenkunst02_1740","par_idx":1657,"date":1740,"orig":"Allein dieſes Vortheils kan man ſich erſtlich nicht allzeit bedienen; und hernach erleichtert derſelbe die Arbeit bey weitem nicht ſo ſehr, als dieſer, von welchem allhier die Rede iſt.","norm":"Allein dieses Vorteils kann man sich erstlich nicht allzeit bedienen; und hernach erleichtert derselbe die Arbeit bei weitem nicht so sehr, als dieser, von welchem allhier die Rede 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+{"basename":"euler_rechenkunst02_1740","par_idx":1658,"date":1740,"orig":"Der Grund dieſes Vortheils beſtehet nun darinn, daß man denjenigen Bruch, durch welchen multiplicirt werden ſoll, in zwey oder mehr Theile zertheile; den Multiplicandum durch einen jeglichen Theil insbeſondere multiplicire, und alle dieſe Producte zuſammen addire; von welcher Operation die Richtigkeit ſchon zur Gnuͤge iſt dargethan worden.","norm":"Der Grund dieses Vorteils bestehet nun darin, dass man denjenigen Bruch, durch welchen multipliziert werden soll, in zwei oder mehr Teile zerteile; den Multiplicandum durch einen jeglichen Teil insbesondere multipliziere, und alle diese Produkte zusammen addiere; von welcher Operation die Richtigkeit schon zur Genüge ist dargetan 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+{"basename":"euler_rechenkunst02_1740","par_idx":1659,"date":1740,"orig":"Es kan aber eine ſolche Zertheilung, inſonderheit wann der Zehler eine groſſe Zahl iſt, auf mancherley Art geſchehen; weswegen man hauptſaͤchlich dahin zu ſehen hat, daß man die bequemſte und vortheilhafteſte Zertheilung erwehle.","norm":"Es kann aber eine solche Zerteilung, in Sonderheit wann der Zähler eine große Zahl ist, auf mancherlei Art geschehen; weswegen man hauptsächlich dahin zu sehen hat, dass man die bequemste und vorteilhafteste Zerteilung 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+{"basename":"euler_rechenkunst02_1740","par_idx":1660,"date":1740,"orig":"Dahero muͤſſen die beſonderen Bruͤche, in welche der Multiplicator zergliedert wird, ſo beſchaffen ſeyn, daß man durch dieſelben mit leichter Muͤhe multipliciren koͤnne.","norm":"Daher müssen die besonderen Brüche, in welche der Multiplicator zergliedert wird, so beschaffen sein, dass man durch dieselben mit leichter Mühe multiplizieren 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+{"basename":"euler_rechenkunst02_1740","par_idx":1661,"date":1740,"orig":"Es kan aber durch einen Bruch leicht multiplicirt werden, wann der Zehler deſſelben 1 iſt, weilen man in dieſem Falle nur durch den Nenner zu dividiren hat: und dieſe Operation wird noch um ſo viel leichter, je kleiner der Nenner des Bruchs iſt.","norm":"Es kann aber durch einen Bruch leicht multipliziert werden, wann der Zähler desselben 1 ist, weilen man in diesem Falle nur durch den Nenner zu dividieren hat: und diese Operation wird noch um so viel leichter, je kleiner der Nenner des Bruchs 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+{"basename":"euler_rechenkunst02_1740","par_idx":1665,"date":1740,"orig":"dann in dieſem Fall kan derjenige Vortheil angebracht werden, welcher von der Reſolution eines Multiplicatoris in Factores oben iſt beſchrieben worden.","norm":"dann in diesem Fall kann derjenige Vorteil angebracht werden, welcher von der Resolution eines Multiplicatoris in Factores oben ist beschrieben 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+{"basename":"euler_rechenkunst02_1740","par_idx":1667,"date":1740,"orig":"Dann weilen ſich12 durch 3 theilen laͤſt, ſo iſt [Formel] ſo viel als ⅓ mit ¼ multiplicirt, und wird folglich der Multiplicandus durch [Formel] multiplicirt, wann man das Product, welches aus der Multiplication durch ⅓ entſtanden, noch durch ¼ multiplicirt, das iſt durch 4 dividirt.","norm":"Dann weilen sich12 durch 3 teilen lässt, so ist [Formel] so viel als 1/3 mit ¼ multipliziert, und wird folglich der Multiplicandus durch [Formel] multipliziert, wann man das Produkt, welches aus der Multiplikation durch 1/3 entstanden, noch durch ¼ multipliziert, das ist durch 4 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ſchreibt.","norm":"Erstlich wird ein Bruch in Teile zerteilet, wann man den Zähler desselben in verschiedene Teile zerteilet, und unter jeden Teil den Nenner unverändert schreibt.","orig_tok":["Erſtlich","wird","ein","Bruch","in","Theile","zertheilet",",","wann","man","den","Zehler","deſſelben","in","verſchiedene","Theile","zertheilet",",","und","unter","jeden","Theil","den","Nenner","unveraͤndert","ſchreibt","."],"norm_tok":["Erstlich","wird","ein","Bruch","in","Teile","zerteilet",",","wann","man","den","Zähler","desselben","in","verschiedene","Teile","zerteilet",",","und","unter","jeden","Teil","den","Nenner","unverändert","schreibt","."],"orig_spans":[[0,8],[9,13],[14,17],[18,23],[24,26],[27,33],[34,44],[44,45],[46,50],[51,54],[55,58],[59,65],[66,75],[76,78],[79,91],[92,98],[99,109],[109,110],[111,114],[115,120],[121,126],[127,132],[133,136],[137,143],[144,156],[157,165],[165,166]],"norm_spans":[[0,8],[9,13],[14,17],[18,23],[24,26],[27,32],[33,42],[42,43],[44,48],[49,52],[53,56],[57,63],[64,73],[74,76],[77,89],[90,95],[96,105],[105,106],[107,110],[111,116],[117,122],[123,127],[128,131],[132,138],[139,150],[151,159],[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],[17,17],[18,18],[19,19],[20,20],[21,21],[22,22],[23,23],[24,24],[25,25],[26,26]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.977200031280518} +{"basename":"euler_rechenkunst02_1740","par_idx":1671,"date":1740,"orig":"Alſo wann dieſer Bruch [Formel] zertheilet werden ſollte, ſo kan man den Zehler 7 in dieſe Theile 4 und 3 zertheilen, aus welchen dieſe zwey Theile des Bruchs [Formel] und [Formel] das iſt ⅓ und ¼ entſpringen, oder man koͤnnte auch 7 in dieſe Theile 6 und 1 und daraus den Bruch [Formel] in dieſe Theile [Formel] und [Formel] das iſt ½ und [Formel] zertheilen.","norm":"Also wann dieser Bruch [Formel] zerteilet werden sollte, so kann man den Zähler 7 in diese Teile 4 und 3 zerteilen, aus welchen diese zwei Teile des Bruchs [Formel] und [Formel] das ist 1/3 und ¼ entspringen, oder man könnte auch 7 in diese Teile 6 und 1 und daraus den Bruch [Formel] in diese Teile [Formel] und [Formel] das ist ½ und [Formel] 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+{"basename":"euler_rechenkunst02_1740","par_idx":1672,"date":1740,"orig":"Zweytens muß man ſich bemuͤhen, daß man zu allererſt von dem Zehler einen ſolchen Theil nehme, durch welchen ſich der Nenner theilen laſſe: dann dadurch erhaͤlt man ſogleich einen Theil des Bruchs deſſen Zehler 1, der Nenner aber kleiner iſt als vorher.","norm":"Zweitens muss man sich bemühen, dass man zuallererst von dem Zähler einen solchen Teil nehme, durch welchen sich der Nenner teilen lasse: dann dadurch erhält man sogleich einen Teil des Bruchs dessen Zähler 1, der Nenner aber kleiner ist als vorher.","orig_tok":["Zweytens","muß","man","ſich","bemuͤhen",",","daß","man","zu","allererſt","von","dem","Zehler","einen","ſolchen","Theil","nehme",",","durch","welchen","ſich","der","Nenner","theilen","laſſe",":","dann","dadurch","erhaͤlt","man","ſogleich","einen","Theil","des","Bruchs","deſſen","Zehler","1",",","der","Nenner","aber","kleiner","iſt","als","vorher","."],"norm_tok":["Zweitens","muss","man","sich","bemühen",",","dass","man","zuallererst","von","dem","Zähler","einen","solchen","Teil","nehme",",","durch","welchen","sich","der","Nenner","teilen","lasse",":","dann","dadurch","erhält","man","sogleich","einen","Teil","des","Bruchs","dessen","Zähler","1",",","der","Nenner","aber","kleiner","ist","als","vorher","."],"orig_spans":[[0,8],[9,12],[13,16],[17,21],[22,30],[30,31],[32,35],[36,39],[40,42],[43,52],[53,56],[57,60],[61,67],[68,73],[74,81],[82,87],[88,93],[93,94],[95,100],[101,108],[109,113],[114,117],[118,124],[125,132],[133,138],[138,139],[140,144],[145,152],[153,160],[161,164],[165,173],[174,179],[180,185],[186,189],[190,196],[197,203],[204,210],[211,212],[212,213],[214,217],[218,224],[225,229],[230,237],[238,241],[242,245],[246,252],[252,253]],"norm_spans":[[0,8],[9,13],[14,17],[18,22],[23,30],[30,31],[32,36],[37,40],[41,52],[53,56],[57,60],[61,67],[68,73],[74,81],[82,86],[87,92],[92,93],[94,99],[100,107],[108,112],[113,116],[117,123],[124,130],[131,136],[136,137],[138,142],[143,150],[151,157],[158,161],[162,170],[171,176],[177,181],[182,185],[186,192],[193,199],[200,206],[207,208],[208,209],[210,213],[214,220],[221,225],[226,233],[234,237],[238,241],[242,248],[248,249]],"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],[18,17],[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]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.436100006103516} +{"basename":"euler_rechenkunst02_1740","par_idx":1673,"date":1740,"orig":"Dieſer Vortheil aber wird um ſo viel groͤſſer, wann man aus dem Zehler den groͤſten Theil abſchneidet, durch welchen ſich der Nenner theilen laͤßt.","norm":"Dieser Vorteil aber wird um so viel größer, wann man aus dem Zähler den größten Teil abschneidet, durch welchen sich der Nenner teilen lässt.","orig_tok":["Dieſer","Vortheil","aber","wird","um","ſo","viel","groͤſſer",",","wann","man","aus","dem","Zehler","den","groͤſten","Theil","abſchneidet",",","durch","welchen","ſich","der","Nenner","theilen","laͤßt","."],"norm_tok":["Dieser","Vorteil","aber","wird","um","so","viel","größer",",","wann","man","aus","dem","Zähler","den","größten","Teil","abschneidet",",","durch","welchen","sich","der","Nenner","teilen","lässt","."],"orig_spans":[[0,6],[7,15],[16,20],[21,25],[26,28],[29,31],[32,36],[37,45],[45,46],[47,51],[52,55],[56,59],[60,63],[64,70],[71,74],[75,83],[84,89],[90,101],[101,102],[103,108],[109,116],[117,121],[122,125],[126,132],[133,140],[141,146],[146,147]],"norm_spans":[[0,6],[7,14],[15,19],[20,24],[25,27],[28,30],[31,35],[36,42],[42,43],[44,48],[49,52],[53,56],[57,60],[61,67],[68,71],[72,79],[80,84],[85,96],[96,97],[98,103],[104,111],[112,116],[117,120],[121,127],[128,134],[135,140],[140,141]],"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]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.455699920654297} +{"basename":"euler_rechenkunst02_1740","par_idx":1674,"date":1740,"orig":"Alſo wann man durch dieſen Bruch [Formel] multipliciren ſollte, ſo nehme man von dem Zehler 11 den Theil 8, als die groͤſte Zahl, ſo kleiner iſt als 11, und durch welche ſich der Nenner 24 theilen laͤßt: derowegen zertheilet man 11 in dieſe Theile 8 und 3, aus welchen dieſe Theile des Bruchs [Formel] und [Formel] das iſt ⅓ und ⅛ entſtehen werden, durch welche leicht zu multipliciren iſt.","norm":"Also wann man durch diesen Bruch [Formel] multiplizieren sollte, so nehme man von dem Zähler 11 den Teil 8, als die größte Zahl, so kleiner ist als 11, und durch welche sich der Nenner 24 teilen lässt: derowegen zerteilet man 11 in diese Teile 8 und 3, aus welchen diese Teile des Bruchs [Formel] und [Formel] das ist 1/3 und ⅛ entstehen werden, durch welche leicht zu multiplizieren ist.","orig_tok":["Alſo","wann","man","durch","dieſen","Bruch","[Formel]","multipliciren","ſollte",",","ſo","nehme","man","von","dem","Zehler","11","den","Theil","8",",","als","die","groͤſte","Zahl",",","ſo","kleiner","iſt","als","11",",","und","durch","welche","ſich","der","Nenner","24","theilen","laͤßt",":","derowegen","zertheilet","man","11","in","dieſe","Theile","8","und","3",",","aus","welchen","dieſe","Theile","des","Bruchs","[Formel]","und","[Formel]","das","iſt","⅓","und","⅛","entſtehen","werden",",","durch","welche","leicht","zu","multipliciren","iſt","."],"norm_tok":["Also","wann","man","durch","diesen","Bruch","[","Formel","]","multiplizieren","sollte",",","so","nehme","man","von","dem","Zähler","11","den","Teil","8",",","als","die","größte","Zahl",",","so","kleiner","ist","als","11",",","und","durch","welche","sich","der","Nenner","24","teilen","lässt",":","derowegen","zerteilet","man","11","in","diese","Teile","8","und","3",",","aus","welchen","diese","Teile","des","Bruchs","[","Formel","]","und","[","Formel","]","das","ist"," 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+{"basename":"euler_rechenkunst02_1740","par_idx":1675,"date":1740,"orig":"Dieſe Zertheilung aber findet nur Platz, wann der Nenner eine zuſammen geſetzte oder ſolche Zahl iſt, welche ſich durch andere kleinere Zahlen theilen laͤßt, und dabey ſolche Theile hat, welche kleiner ſind als der Zehler des Bruchs.","norm":"Diese Zerteilung aber findet nur Platz, wann der Nenner eine zusammen gesetzte oder solche Zahl ist, welche sich durch andere kleinere Zahlen teilen lässt, und dabei solche Teile hat, welche kleiner sind als der Zähler des 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+{"basename":"euler_rechenkunst02_1740","par_idx":1676,"date":1740,"orig":"Wie aber eine ſolche Zertheilung anzuſtellen ſey, wann der Nenner ſich durch keine Zahl ſo kleiner iſt als der Zehler theilen laͤſt, wollen wir hernach melden.","norm":"Wie aber eine solche Zerteilung anzustellen sei, wann der Nenner sich durch keine Zahl so kleiner ist als der Zähler teilen lässt, wollen wir hernach 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+{"basename":"euler_rechenkunst02_1740","par_idx":1677,"date":1740,"orig":"Drittens wann man den Bruch, durch welchen multiplicirt werden ſoll, ſchon in zwey ſolche Theile zertheilet hat, davon einer zum Zehler 1 zum Nenner aber eine Zahl ſo klein genug iſt, hat, ſo muß man den anderen Theil betrachten, und wann deſſelben Zehler nicht 1 iſt, denſelben nach der vorigen Art ferner in zwey Theile zertheilen, davon einer die Unitaͤt zum Zehler bekomme; den anderen Theil aber wann deſſelben Zehler noch nicht 1 iſt noch ferner zertheilen, bis man lauter ſolche Bruͤche fuͤr die geſuchten Theile bekomme, deren Zehler 1 iſt.","norm":"Drittens wann man den Bruch, durch welchen multipliziert werden soll, schon in zwei solche Teile zerteilet hat, davon einer zum Zähler 1 zum Nenner aber eine Zahl so klein genug ist, hat, so muss man den anderen Teil betrachten, und wann desselben Zähler nicht 1 ist, denselben nach der vorigen Art ferner in zwei Teile zerteilen, davon einer die Unität zum Zähler bekomme; den anderen Teil aber wann desselben Zähler noch nicht 1 ist noch ferner zerteilen, bis man lauter solche Brüche für die gesuchten Teile bekomme, deren Zähler 1 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Davon der letztere ferner in dieſe [Formel] und [Formel] zertheilet wird oder ⅙ und [Formel] ; ſo daß dieſer Bruch [Formel] ſich in dieſe 3 Theile ½ und ⅙ und [Formel] zertheilet; durch welche ſehr leicht multiplicirt wird.","norm":"Als wann dieser Bruch [Formel] vorkommt, so zerteile man erstlich 17 in diese 2 Teile, 12 und 5, weilen sich der Nenner 24 durch 12 teilen lässt, daher entspringen diese 2 Brüche ½ und [Formel]. Davon der letztere ferner in diese [Formel] und [Formel] zerteilet wird oder ⅙ und [Formel]; so dass dieser Bruch [Formel] sich in diese 3 Teile ½ und 1/6 und [Formel] zerteilet; durch welche sehr leicht multipliziert 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+{"basename":"euler_rechenkunst02_1740","par_idx":1679,"date":1740,"orig":"Dann erſtlich dividirt man den Multiplicandum durch 2, ſo bekommt man die Helfte: dieſe Helfte dividirt man ferner durch 3, ſo bekommt man den Sechstel, weilen 6 ſo viel iſt als 2 mahl 3, und endlich den Sechstel dividirt man durch 4, ſo bekommt man den 24ſtel.","norm":"Dann erstlich dividiert man den Multiplicandum durch 2, so bekommt man die Hälfte: Diese Hälfte dividiert man ferner durch 3, so bekommt man den Sechstel, weilen 6 so viel ist als 2 Mal 3, und endlich den Sechstel dividiert man durch 4, so bekommt man den 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+{"basename":"euler_rechenkunst02_1740","par_idx":1680,"date":1740,"orig":"Viertens wann der Nenner des Bruchs, welcher zertheilet werden ſoll, entweder gar keine oder doch keine kleinere Theile hat als der Zehler, ſo verwandele man denſelben in eine andere Form, in dem man den Zehler und Nenner durch eine beliebige Zahl multiplicirt; am dienlichſten aber iſt beyde anfaͤnglich nur mit 2 zu multipliciren, damit man nicht ohne Noth auf allzugroſſe Zahlen komme.","norm":"Viertens wann der Nenner des Bruchs, welcher zerteilet werden soll, entweder gar keine oder doch keine kleinere Teile hat als der Zähler, so verwandele man denselben in eine andere Form, in dem man den Zähler und Nenner durch eine beliebige Zahl multipliziert; am dienlichsten aber ist beide anfänglich nur mit 2 zu multiplizieren, damit man nicht ohne Not auf allzu große Zahlen 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+{"basename":"euler_rechenkunst02_1740","par_idx":1681,"date":1740,"orig":"Wann aber noch keine bequeme Zertheilung ſollte vorgenommen werden koͤnnen, alsdann kan man, anſtatt mit 2, mit 3 oder 4 oder eine groͤſſere Zahl beydes Zehler und Nenner multipliciren.","norm":"Wann aber noch keine bequeme Zerteilung sollte vorgenommen werden können, alsdann kann man, anstatt mit 2, mit 3 oder 4 oder eine größere Zahl beides Zähler und Nenner 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+{"basename":"euler_rechenkunst02_1740","par_idx":1682,"date":1740,"orig":"Als wann dieſer Bruch [Formel] vorgelegt waͤre, weilen 7 keine Theiler hat, ſo multiplicire man oben und unten mit 2; da kommt dieſer Bruch [Formel] welcher ſich in dieſe Bruͤche [Formel] und [Formel] oder ½ und [Formel] zertheilet.","norm":"Als wann dieser Bruch [Formel] vorgelegt wäre, weilen 7 keine Teiler hat, so multipliziere man oben und unten mit 2; da kommt dieser Bruch [Formel] welcher sich in diese Brüche [Formel] und [Formel] oder ½ und [Formel] 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+{"basename":"euler_rechenkunst02_1740","par_idx":1683,"date":1740,"orig":"Gleicher geſtalt [Formel] wann oben und unten mit 2 multiplicirt wird, gibt [Formel] , und daher entſtehen dieſe Theile [Formel] und [Formel] das iſt ½ und [Formel] : davon der letztere Bruch in dieſe [Formel] und [Formel] oder [Formel] und [Formel] zergliedert wird: und iſt folglich [Formel] ſo viel als ½ und [Formel] und [Formel] . Dieſe Verwandlung des vorgelegten Bruchs durch 2 findet aber nur Platz, wann der Bruch groͤſſer iſt als ½, iſt derſelbe aber kleiner als ½ doch aber groͤſſer als ⅓, ſo multiplicire man oben und unten mit 3. Jſt aber derſelbe kleiner als ⅓ doch aber groͤſſer als ¼, ſo multiplicire man oben und unten mit 4, und ſo weiter.","norm":"Gleicher Gestalt [Formel] wann oben und unten mit 2 multipliziert wird, gibt [Formel], und daher entstehen diese Teile [Formel] und [Formel] das ist ½ und [Formel]: davon der letztere Bruch in diese [Formel] und [Formel] oder [Formel] und [Formel] zergliedert wird: und ist folglich [Formel] so viel als ½ und [Formel] und [Formel]. Diese Verwandlung des vorgelegten Bruchs durch 2 findet aber nur Platz, wann der Bruch größer ist als ½, ist derselbe aber kleiner als ½ doch aber größer als 1/3 , so multipliziere man oben und unten mit 3. Ist aber derselbe kleiner als 1/3 doch aber größer als ¼, so multipliziere man oben und unten mit 4, und so 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Hat man nun mit ¼ multiplicirt, ſo dividire man dieſes Product durch 29 ſo bekommt man den 116ſten Theil, dieſer aber mit 2 multiplicirt gibt den 58ſten Theil, weilen [Formel] ſo viel iſt als 2 mahl [Formel] . Aus dieſen Regeln wird nun leicht ſeyn einen jeglichen vorkommenden Bruch in bequeme Theile zu zertheilen, durch welche die Multiplication vortheilhaft angeſtellet werden kan.","norm":"Als wann dieser Bruch vorkommt [Formel], weilen derselbe kleiner ist als 1/3 größer aber als ¼, welches daraus erhellet, weilen 8 in 29 mehr als 3 Mal, doch weniger als 4 Mal enthalten ist; so multipliziere man oben und unten mit 4, kommt [Formel] das ist [Formel] und [Formel] oder ¼ und [Formel], der letzter Bruch [Formel] aber zerteilet sich in [Formel] und [Formel], also dass [Formel] so viel ist als ¼ und [Formel] und [Formel]. Hat man nun mit ¼ multipliziert, so dividiere man dieses Produkt durch 29 so bekommt man den 116sten Teil, dieser aber mit 2 multipliziert gibt den 58sten Teil, weilen [Formel] so viel ist als 2 Mal [Formel]. Aus diesen Regeln wird nun leicht sein einen jeglichen vorkommenden Bruch in bequeme Teile zu zerteilen, durch welche die Multiplikation vorteilhaft angestellt werden 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Weilen 3 keine Theiler hat, und der Bruch ⅔ groͤſſer iſt als ½, ſo multiplicire man oben und unten mit 2, ſo wird der Multiplicator [Formel] das iſt [Formel] und ⅙ oder ½+ [Formel] . Derowegen dividirt man erſtlich den Multiplicandum durch 2, und was herauskommt nochmahls durch 3, und addirt beyde Quotos zuſammen.","norm":"Antw. Weilen 3 keine Teiler hat, und der Bruch 2/3 größer ist als ½, so multipliziere man oben und unten mit 2, so wird der Multiplicator [Formel] das ist [Formel] und 1/6 oder 1/2+ [Formel]. 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Dahero geſchieht die Multiplication wie folgt.","norm":"Der Multiplicator [Formel] zerteilet sich nach der gegebenen Regeln in diese 2 Brüche 1/2+ [Formel] oder 1/2+ [Formel]. Daher geschieht die Multiplikation wie folgt.","orig_tok":["Der","Multiplicator","[Formel]","zertheilet","ſich","nach","der","gegebenen","Regeln","in","dieſe","2","Bruͤche","½+","[Formel]","oder","½+","[Formel]",".","Dahero","geſchieht","die","Multiplication","wie","folgt","."],"norm_tok":["Der","Multiplicator","[","Formel","]","zerteilet","sich","nach","der","gegebenen","Regeln","in","diese","2","Brüche","1","/","2","+","[","Formel","]","oder","1","/","2","+","[","Formel","]",".","Daher","geschieht","die","Multiplikation","wie","folgt","."],"orig_spans":[[0,3],[4,17],[18,26],[27,37],[38,42],[43,47],[48,51],[52,61],[62,68],[69,71],[72,77],[78,79],[80,87],[88,90],[91,99],[100,104],[105,107],[108,116],[117,118],[119,125],[126,135],[136,139],[140,154],[155,158],[159,164],[164,165]],"norm_spans":[[0,3],[4,17],[18,19],[19,25],[25,26],[27,36],[37,41],[42,46],[47,50],[51,60],[61,67],[68,70],[71,76],[77,78],[79,85],[86,87],[87,88],[88,89],[89,90],[91,92],[92,98],[98,99],[100,104],[105,106],[106,107],[107,108],[108,109],[110,111],[111,117],[117,118],[118,119],[120,125],[126,135],[136,139],[140,154],[155,158],[159,164],[164,165]],"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,null],[null,15],[null,16],[null,17],[null,18],[14,19],[14,20],[14,21],[15,22],[null,23],[null,24],[null,25],[null,26],[16,null],[17,27],[17,28],[17,29],[18,30],[19,31],[20,32],[21,33],[22,34],[23,35],[24,36],[25,37]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":4.201700210571289} +{"basename":"euler_rechenkunst02_1740","par_idx":1711,"date":1740,"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_rechenkunst02_1740","par_idx":1712,"date":1740,"orig":"IV.","norm":"IV.","orig_tok":["IV","."],"norm_tok":["IV."],"orig_spans":[[0,2],[2,3]],"norm_spans":[[0,3]],"alignment":[[0,0],[1,0]],"lang_fastText":"en","lang_py3langid":"en","lang_cld3":"zu","orig_lang_de":0,"norm_lmscore":2.057800054550171} +{"basename":"euler_rechenkunst02_1740","par_idx":1713,"date":1740,"orig":"Nach Engliſchem Gelde ſoll dieſe Summ 5720 L. Sterl. 15 ß.","norm":"Nach Englischem Gelde soll diese Summe 5720 L. 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Man theilet alſo die gegebene Summ erſtlich durch 2, was herauskommt durch 3, und dieſen Quotum durch 4, und addirt alle 3 Quotos zuſammen: wie aus folgender Berechnung, ſo nach dieſer Zertheilung eingerichtet iſt, zu erſehen.","norm":"Der Bruch [Formel] zerteilet sich erstlich in diese zwei [Formel] und [Formel] oder 1/2+ [Formel] dieser andere Bruch [Formel] aber in [Formel] und [Formel] oder ⅙+ [Formel], so dass der gegebene Multiplicator [Formel] sich in diese drei Brüche zerteilet 1/2+ 1/6 + [Formel] oder in 1/2+ [Formel] + [Formel]. 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Dieſer letztere Theil [Formel] aber ferner in dieſe [Formel] + [Formel] oder [Formel] + [Formel] ſo daß unſer Multiplicator ſeyn wird ½+ [Formel] + [Formel] : woraus folgende Operation entſpringt.","norm":"Dieser Multiplicator [Formel] zerteilet sich erstlich in diese 2 Teile [Formel] + [Formel] oder 1/2+ [Formel]. 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+{"basename":"euler_rechenkunst02_1740","par_idx":1753,"date":1740,"orig":"Dieſer Multiplicator [Formel] aber reſolvirt ſich in dieſe Theile [Formel] + [Formel] + [Formel] oder ⅛+ [Formel] + [Formel] woraus folgende Operation entſtehet.","norm":"Dieser Multiplicator [Formel] aber resolviert sich in diese Teile [Formel] + [Formel] + [Formel] oder ⅛+ [Formel] + [Formel] woraus folgende Operation 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+{"basename":"euler_rechenkunst02_1740","par_idx":1756,"date":1740,"orig":"Und wann auch ſolches geſchehen koͤnnte, ſo wuͤrde doch die Ausfindung eines ſolchen Vortheils mehr Zeit und Muͤhe koſten, als wann man das Exempel nach der gewoͤhnlichen Art ausrechnen wollte.","norm":"Und wann auch solches geschehen könnte, so würde doch die Ausfindung eines solchen Vorteils mehr Zeit und Mühe kosten, als wann man das Exempel nach der gewöhnlichen Art ausrechnen 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+{"basename":"euler_rechenkunst02_1740","par_idx":1759,"date":1740,"orig":"Dieſe Regeln dienen demnach haͤuptſaͤchlich dazu, um den Anfaͤngern mit einiger Muͤhe die Vortheile beyzubringen, damit ſie hernach dieſelben ohne Regeln mit leichter Muͤhe bey allen Gelegenheiten ſelbſt geſchwind finden koͤnnen.","norm":"Diese Regeln dienen demnach häuptsächlich dazu, um den Anfängern mit einiger Mühe die Vorteile beizubringen, damit sie hernach dieselben ohne Regeln mit leichter Mühe bei allen Gelegenheiten selbst geschwind finden 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+{"basename":"euler_rechenkunst02_1740","par_idx":1760,"date":1740,"orig":"4.)","norm":"4.)","orig_tok":["4.",")"],"norm_tok":["4.",")"],"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":6.027200222015381} +{"basename":"euler_rechenkunst02_1740","par_idx":1761,"date":1740,"orig":"Man kan auf oͤfters mit nicht geringem Vortheile einen Bruch, durch welchen multiplicirt werden ſoll, als einen Reſt anſehen, welcher herauskommt, wann man einen kleineren Bruch von einem groͤſſeren ſubtrahirt.","norm":"Man kann auf öfters mit nicht geringem Vorteile einen Bruch, durch welchen multipliziert werden soll, als einen Rest ansehen, welcher herauskommt, wann man einen kleineren Bruch von einem größeren 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+{"basename":"euler_rechenkunst02_1740","par_idx":1762,"date":1740,"orig":"Jn ſolchem Falle multiplicirt man den Multiplicandum erſtlich durch den groͤſſeren Bruch, hernach durch den kleineren, und ſubtrahirt das letztere Product von dem erſteren, ſo bekommt man das verlangte Facit.","norm":"In solchem Falle multipliziert man den Multiplicandum erstlich durch den größeren Bruch, hernach durch den kleineren, und subtrahiert das letztere Produkt von dem ersteren, so bekommt man das verlangte 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+{"basename":"euler_rechenkunst02_1740","par_idx":1763,"date":1740,"orig":"Um aber hiedurch einigen Vortheil zu erlangen, ſo muͤſſen die beyden Bruͤche, aus deren Subtraction der vorgegebene Multiplicator entſpringt, ſo beſchaffen ſeyn, daß man mit denſelben leicht multipliciren kan.","norm":"Um aber hierdurch einigen Vorteil zu erlangen, so müssen die beiden Brüche, aus deren Subtraktion der vorgegebene Multiplicator entspringt, so beschaffen sein, dass man mit denselben leicht multiplizieren 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+{"basename":"euler_rechenkunst02_1740","par_idx":1766,"date":1740,"orig":"Alſo bedeutet 8 — 5 ſo viel als 8 weniger 5, das iſt die Differenz oder der Reſt, welcher uͤberbleibt, wann man 5 von 8 ſubtrahirt.","norm":"Also bedeutet 8 — 5 so viel als 8 weniger 5, das ist die Differenz oder der Rest, welcher überbleibt, wann man 5 von 8 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+{"basename":"euler_rechenkunst02_1740","par_idx":1768,"date":1740,"orig":"Allhier wollen wir nun diejenigen Vortheile anzeigen, welche man erhalten kan, wann man einen gebrochenen Multiplicatorem als eine aus der Subtraction entſtandene Differenz anſieht, und auf ſolche Art durch dieſes Zeichen — andeutet.","norm":"Allhier wollen wir nun diejenigen Vorteile anzeigen, welche man erhalten kann, wann man einen gebrochenen Multiplikatorrem als eine aus der Subtraktion entstandene Differenz ansieht, und auf solche Art durch dieses Zeichen — 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+{"basename":"euler_rechenkunst02_1740","par_idx":1771,"date":1740,"orig":"Der Grund hievon beruhet darauf: wann man den Multiplicandum durch die groͤſſere Zahl multiplicirt hat, ſo hat man denſelben durch eine allzugroſſe Zahl multiplicirt, in dem man denſelben durch die Differenz zwiſchen der groͤſſeren und kleineren Zahl multipliciren ſollte.","norm":"Der Grund hiervon beruhet darauf: wann man den Multiplicandum durch die größere Zahl multipliziert hat, so hat man denselben durch eine allzu große Zahl multipliziert, in dem man denselben durch die Differenz zwischen der größeren und kleineren Zahl multiplizieren 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+{"basename":"euler_rechenkunst02_1740","par_idx":1772,"date":1740,"orig":"Wann wir aber ferner ſehen, um wieviel die groͤſſere Zahl der Differenz zu groß oder groͤſſer als der gegebene Multiplicator iſt, ſo finden wir daß ſolches die kleinere Zahl anzeige; wann wir alſo den Multiplicandum durch die kleinere Zahl multipliciren, und dieſes Product von dem vorigen ſubtrahiren, ſo nehmen wir accurat eben ſo viel davon hinweg als das erſtere Product zu groß war, und finden alſo das geſuchte Product.","norm":"Wann wir aber ferner sehen, um wie viel die größere Zahl der Differenz zu groß oder größer als der gegebene Multiplicator ist, so finden wir dass solches die kleinere Zahl anzeige; wann wir also den Multiplicandum durch die kleinere Zahl multiplizieren, und dieses Produkt von dem vorigen subtrahieren, so nehmen wir akkurat ebenso viel davon hinweg als das erstere Produkt zu groß war, und finden also das gesuchte 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+{"basename":"euler_rechenkunst02_1740","par_idx":1773,"date":1740,"orig":"Dieſer Schluß weiſet ſich aber deutlicher durch Exempel:","norm":"Dieser Schluss weiset sich aber deutlicher durch Exempel:","orig_tok":["Dieſer","Schluß","weiſet","ſich","aber","deutlicher","durch","Exempel",":"],"norm_tok":["Dieser","Schluss","weiset","sich","aber","deutlicher","durch","Exempel",":"],"orig_spans":[[0,6],[7,13],[14,20],[21,25],[26,30],[31,41],[42,47],[48,55],[55,56]],"norm_spans":[[0,6],[7,14],[15,21],[22,26],[27,31],[32,42],[43,48],[49,56],[56,57]],"alignment":[[0,0],[1,1],[2,2],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8]],"lang_fastText":"de","lang_py3langid":"de","lang_cld3":"de","orig_lang_de":1,"norm_lmscore":6.946000099182129} +{"basename":"euler_rechenkunst02_1740","par_idx":1774,"date":1740,"orig":"wir wollen demnach ſetzen, man ſoll10 durch 4 multipliciren, man betrachte aber 4 als die Differenz zwiſchen 7 und 3, und ſoll folglich10 durch 7 — 3 das iſt 7 weniger 3 multipliciren.","norm":"wir wollen demnach setzen, man soll10 durch 4 multiplizieren, man betrachte aber 4 als die Differenz zwischen 7 und 3, und soll folglich10 durch 7 — 3 das ist 7 weniger 3 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+{"basename":"euler_rechenkunst02_1740","par_idx":1775,"date":1740,"orig":"Da nun 7 — 3 ſo viel iſt als 4, ſo wird auch 2 mahl 7 weniger 2 mahl 3 ſo viel ſeyn als 2 mahl 4, und 3 mahl 7 weniger 3 mahl 3 ſo viel als 3 mahl 4 und folglich 10 mahl 7 weniger 10 mahl 3 ſo viel als 10 mahl 4. Hieraus erhellet nun, daß wann man 10 mit 7 und auch mit 3 multiplicirt und das kleinere Product von dem groͤſſeren ſubtrahirt, eben ſo viel herauskommen muͤſſe, als wann man 10 mit 7 — 3 das iſt mit 4 multiplicirt haͤtte; in beyden Faͤllen kommt nehmlich 4 heraus.","norm":"Da nun 7 — 3 so viel ist als 4, so wird auch 2 Mal 7 weniger 2 Mal 3 so viel sein als 2 Mal 4, und 3 Mal 7 weniger 3 Mal 3 so viel als 3 Mal 4 und folglich 10 Mal 7 weniger 10 Mal 3 so viel als 10 Mal 4. Hieraus erhellet nun, dass wann man 10 mit 7 und auch mit 3 multipliziert und das kleinere Produkt von dem größeren subtrahiert, ebenso viel herauskommen müsse, als wann man 10 mit 7 — 3 das ist mit 4 multipliziert hätte; in beiden Fällen kommt nämlich 4 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+{"basename":"euler_rechenkunst02_1740","par_idx":1786,"date":1740,"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_rechenkunst02_1740","par_idx":1787,"date":1740,"orig":"Wir wollen uns aber bey dergleichen Vortheilen nicht laͤnger aufhalten, ſondern zu unſerem Endzwecke fortſchreiten, und zeigen, wann und wie ein Bruch in eine ſolche Differenz, durch welche leicht multiplicirt werden kan, verwandelt werden koͤnne.","norm":"Wir wollen uns aber bei dergleichen Vorteilen nicht länger aufhalten, sondern zu unserem Endzwecke fortschreiten, und zeigen, wann und wie ein Bruch in eine solche Differenz, durch welche leicht multipliziert werden kann, verwandelt werden 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+{"basename":"euler_rechenkunst02_1740","par_idx":1800,"date":1740,"orig":"Wann alſo dieſe Zahl 156234 durch ⅚ multipliciret werden ſoll, weilen ⅚ ſo viel iſt als 1 — ⅙, ſo ſubtrahirt man von derſelben Zahl 1 mahl genommen, das iſt von derſelben Zahl ſelbſt ihren Sechstel: alſo [Formel]","norm":"Wann also diese Zahl 156234 durch 5/6 multiplizieret werden soll, weilen 5/6 so viel ist als 1 — 1/6 , so subtrahiert man von derselben Zahl 1 Mal genommen, das ist von derselben Zahl selbst ihren Sechstel: also [Formel]","orig_tok":["Wann","alſo","dieſe","Zahl","156234","durch","⅚","multipliciret","werden","ſoll",",","weilen","⅚","ſo","viel","iſt","als","1","—","⅙",",","ſo","ſubtrahirt","man","von","derſelben","Zahl","1","mahl","genommen",",","das","iſt","von","derſelben","Zahl","ſelbſt","ihren","Sechstel",":","alſo","[Formel]"],"norm_tok":["Wann","also","diese","Zahl","156234","durch"," ","5","/","6"," ","multiplizieret","werden","soll",",","weilen"," ","5","/","6"," 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+{"basename":"euler_rechenkunst02_1740","par_idx":1846,"date":1740,"orig":"Daß ſich die Diviſion durch Bruͤche in eine Multiplication verwandeln laſſe, iſt ſchon im vorige Theile bey den Operationen der Bruͤche klar dargethan worden, und bedarf alſo anjetzo keines neuen Beweiſes.","norm":"Dass sich die Division durch Brüche in eine Multiplikation verwandeln lasse, ist schon im vorige Teile bei den Operationen der Brüche klar dargetan worden, und bedarf also jetzt keines neuen 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+{"basename":"euler_rechenkunst02_1740","par_idx":1847,"date":1740,"orig":"Es iſt demnach vorallen Dingen zu merken, daß wann der Diviſor ein ſolcher Bruch iſt, deſſen Zehler 1 iſt, die Diviſion in eine Multiplication durch gantze Zahlen verwandelt werden.","norm":"Es ist demnach vorallen Dingen zu merken, dass wann der Divisor ein solcher Bruch ist, dessen Zähler 1 ist, die Division in eine Multiplikation durch ganze Zahlen verwandelt 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+{"basename":"euler_rechenkunst02_1740","par_idx":1848,"date":1740,"orig":"Alſo iſt durch ½ dividiren eben ſo viel als mit 2 multipliciren, und durch ⅓ dividiren nichts anders als mit 3 multipliciren und ſo fort.","norm":"Also ist durch ½ dividieren ebenso viel als mit 2 multiplizieren, und durch 1/3 dividieren nichts anders als mit 3 multiplizieren und so fort.","orig_tok":["Alſo","iſt","durch","½","dividiren","eben","ſo","viel","als","mit","2","multipliciren",",","und","durch","⅓","dividiren","nichts","anders","als","mit","3","multipliciren","und","ſo","fort","."],"norm_tok":["Also","ist","durch","½","dividieren","ebenso","viel","als","mit","2","multiplizieren",",","und","durch"," ","1","/","3"," 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+{"basename":"euler_rechenkunst02_1740","par_idx":1899,"date":1740,"orig":"Dahero man ſich nicht wundern muß, daß wir dieſe Arithmetiſchen Operationen mit benannten Zahlen weitlaͤuffiger abgehandelt haben, als ſonſten zu geſchehen pflegt.","norm":"Daher man sich nicht wundern muss, dass wir diese arithmetischen Operationen mit benannten Zahlen weitläufiger abgehandelt haben, als sonsten zu geschehen 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+{"basename":"euler_rechenkunst02_1740","par_idx":1900,"date":1740,"orig":"Da wir aber hier die meiſten Vortheile im Rechnen als an ihrem gehoͤrigen Orte angefuͤhret haben, ſo werden die bey der Arithmetic vorkommenden verſchiedenen Regeln deſto leichter und kuͤrtzer abgehandelt werden koͤnnen.","norm":"Da wir aber hier die meisten Vorteile im Rechnen als an ihrem gehörigen Orte angeführt haben, so werden die bei der Arithmetik vorkommenden verschiedenen Regeln desto leichter und kürzer abgehandelt werden 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