Bibliotheca Polyglotta

PROPOSITION 9.

PROBL. 1 PROPOS. 9

第九題

From a given straight line to cut off a prescribed part.

ADATA recta linea imperatam partem auferre.

一直線。求截所取之分。

Let AB be the given straight line; thus it is required to cut off from AB a prescribed part.

IMPERETVR, ut ex linea AB, auferamus partem tertiam.

法曰。甲乙直線。求截取三分之一。

Let the third part be that prescribed. Let a straight line AC be drawn through from A containing with AB any angle; let a point D be taken at random on AC, and let DE, EC be made equal to AD. [I. 3] Let BC be joined, and through D let DF be drawn parallel to it. [I. 31]

Ex A, ducatur recta AC, utcunque faciens angulum CAB; et ex AC, abscindantur tot partes aequales cuiuslibet magnitudinis, quota pars detrahenda est ex AB, ut in proposito exemplo tres AD, DE, EF. Deinde ex F, ad B, recta ducatur FB, cui per D, parallela agatur DG.

先從甲、任作一甲丙線、為丙甲乙角。次從甲向丙。任作所命分之平度。如甲丁、丁戊、戊己、為三分也。次作己乙直線。末作丁庚線。與己乙平行。卽甲庚為甲乙三分之一。

Then, since FD has been drawn parallel to BC, one of the sides of the triangle ABC, therefore, proportionally, as CD is to DA, so is BF to FA. [VI. 2] But CD is double of DA; therefore BF is also double of FA; therefore BA is triple of AF.

Dico AG, esse partem tertiam imperatam rectae AB. Nam cum in triangulo ABF, lateri FB, parallela sit recta DG;¹ erit ut FD, ad DA, ita BG, ad GA. ² Componendo igitur , ut FA, ad DA, ita BA, erit ad GA: sed FA, ipsius AD, est tripla, ex constructione. Igitur et BA, ipsis AG, erit tripla, ideoque AG, tertia pars erit ipsius AB, quae imperabatur.

1. 2. sexti. 2. 18. quinti.

論曰。甲乙己角形內之丁庚線。旣與乙己邊平行。卽己丁與丁甲之比例。若乙庚與庚甲也。本篇二合之。己甲與甲丁。若乙甲與庚甲也。五卷十八而甲丁旣為己甲三分之一。卽庚甲亦為乙甲三分之一也。

Therefore from the given straight line AB the prescribed third part AF has been cut off. Q. E. F.

A data ergo recta linea imperatam partem abstulimus. Quod faciendum erat.

SCHOLIUM. QVOD si ex AB, auferenda sit pars non aliquota, sed quae plures aliquotas non efficientes unam complectatur, nimirum quae contineat quatuor undecimas ipsius AB, sumendae erunt ex AC, undecim partes aquales usque ad D, punctum, ex quo ad B recta ducatur DB; et huic parallela EF, ex E, termino quatuor partium. Nam AF, erit pars imperata. Erit enim rursus ut DA, ad AE, ita BA, ad AF. Quare, et convertendo ut AE, ad AD, ita AF, ad AB: Est autem AE, pars continens quatuor undecimas ipsius AD, ex constructione. Igitur et AF, eadem pars erit rectae AB. Quod est propositum. Non aliter detrahetur ex AB, pars complectens quotcunque partes ipsius aliquotas non facientes unam.

注曰。甲乙線。欲截取十一分之四。先作甲丙線、為丙甲乙角。從甲向丙。任平分十一分、至丁。次作丁乙線。末從甲取四分、得戊。作戊己線。與丁乙平行。卽甲己為十一分甲乙之四。何者。依上論、丁甲與戊甲之比例。若乙甲與己甲也。反之。甲戊與甲丁。若甲己與甲乙也。五卷四甲戊為十一分甲丁之四。則甲己亦十一分甲乙之四矣。依此可推不盡分之數。蓋四不為十一之盡分故。

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