Bibliotheca Polyglotta

If a magnitude be the same multiple of a magnitude that a part subtracted is of a part subtracted, the remainder will also be the same multiple of the remainder that the whole is of the whole.

SI magnitudo magnitudinis aeque fuerit multiplex, atque ablata ablatae : Etiam reliqua reliquae ita multiplex erit, ut tota totius.

大小兩幾何。此全所倍於彼全。若此全截取之分、所倍於彼全截取之分。則此全之分餘、所倍於彼全之分餘。亦如之。

For let the magnitude AB be the same multiple of the magnitude CD that the part AE subtracted is of the part CF subtracted; I say that the remainder EB is also the same multiple of the remainder FD that the whole AB is of the whole CD.

Ita multiplex fit tota AB, totius CD, ut est multiplex AE, ablata ablate CF : Dico reliquam EB, ita esse multipticem reliquae CD, ut est tota AB, totius CD.

解曰。甲乙大㡬何。丙丁小㡬何。甲乙所倍于丙丁。若甲乙之截分甲戊、所倍于丙丁之截分、丙己。題言甲戊之分餘、戊乙、所倍于丙己之分餘，己丁。亦如其數。

For, whatever multiple AE is of CF, let EB be made that multiple of CG.

Ponatur enim EB, ita multiplex cuiuspiam magnitudinis ut delicet ipsius GC, ut est AE, ipsus CF.

論曰。試作一他幾何。為庚丙。令戊乙之倍庚丙。若甲戊之倍丙己也。本卷界說增

Then, since AE is the same multiple of CF that EB is of GC, therefore AE is the same multiple of CF that AB is of GF. [V. 1] But, by the assumption, AE is the same multiple of CF that AB is of CD. Therefore AB is the same multiple of each of the magnitudes GF, CD; therefore GF is equal to CD. Let CF be subtracted from each; therefore the remainder GC is equal to the remainder FD. And, since AE is the same multiple of CF that EB is of GC, and GC is equal to DF, therefore AE is the same multiple of CF that EB is of FD. But, by hypothesis, AE is the same multiple of CF that AB is of CD; therefore EB is the same multiple of FD that AB is of CD. That is, the remainder EB will be the same multiple of the remainder FD that the whole AB is of the whole CD.

Quoniam igitur AE, EB, aeque sunt multiplices ipsarum CF, GC,¹ erit tota AB, totius GF, ita multiplex, ut AE, ipsus CF, hoc est, omnes omnium, ut una unius. Sed tam multiplex etiam ponitur AB, ipsius CD, quam est multiplex AE, ipsius CF. Igitur AB, tam est multiplex ipsius GF, quam multiplex est ipsius CD;² atque idcirco aequales sunt GF, CD. Ablata igitur communi CF, aequales erunt GC, FD. Tam multiplex igitur erit EB, ipsius FD, quam multiplex est ipsius GC. Sed ita multiplex posita fuit EB, ipsius GC, ut AE, ipsius CF, hoc est, ut tota AB, totius CD. Quare tam multiplex est reliqua EB, reliquae FD, quam est tota AB, totius CD: quod est propositum.

1. 1. quinti. 2. 6. pron.

甲戊、戊乙、之倍丙己、庚丙。其數等。卽其兩幷、甲乙之倍庚己。亦若甲戊之倍丙己也。本篇一而甲乙之倍丙丁。元若甲戊之倍丙己。則丙丁與庚己等也。次每減同用之丙己。卽庚丙與己丁、亦等。而戊乙之倍己丁。亦若戊乙之倍庚丙矣。夫戊乙之倍庚丙。旣若甲戊之倍丙己。則戊乙、為甲戊之分餘。所倍於己丁、為丙己之分餘者。亦若甲乙之倍丙丁也。

Therefore etc. Q. E. D.

ALITER. Sit ita multiplex tota AB, totius CD, ut ablata AE, ablatae CF. Dico reliquam EB, reliquae FD, esse sic multiplicem, ut est tota totius. Posita enim GA, ita multiplici ipsius FD, ut est AE, ipsius CF, vel ut tota AB, totius CD: quoniam AE, GA, aeque multiplices sunt ipsarum CE, FD,³ erit tota GE, sic multiplex totius CD, ut AE, ipsius CF: Sed ita quoque multiplex est AB, eiusdem CD, ut AE, ipsius C F, ex hypothesi. Aeque multiplices sunt igitur GE, AB, ipsius CD;⁴ atque adeo inter se aequales. Quare dempta communi AE, aequalet erunt GA, EB: Ideoque aequemultiplices ipsius FD; cum GA, sit multiplex posita ipsius FD: Atqui ita est multiplex posita GA, ipsius FD, ut AB, ipsius CD. Igitur et EB, reliqua sic erit multiplex ipsius FD, reliquae, ut AB, tota totius CD; quod est propositum. Si magnitudo itaque magnitudinis aeque fuerit multiplex, etc. Quod erat demonstrandum.

3. 1. quinti. 4. 6. pron.

又論曰。試作一他幾何、為庚甲。令庚甲之倍己丁。若甲戊之倍丙己。本卷界說二十卽其兩幷、庚戊之倍丙丁。亦若甲戊之倍丙己也。本篇一而甲乙之倍丙丁。元若甲戊之倍丙己。是庚戊與甲乙等矣。次每減同用之甲戊。卽庚甲與戊乙等也。而庚甲之倍己丁。若甲乙之倍丙丁也。則戊乙之倍己丁。亦若甲乙之倍丙丁也。

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