Bibliotheca Polyglotta

PROPOSITION 19

第十九題其系為轉理

If, as a whole is to a whole, so is a part subtracted to a part subtracted, the remainder will also be to the remainder as whole to whole. If a:b=c:d (c<a and d<b), then (a-c)=(b-d).

兩幾何。各截取一分。其所截取之比例。與兩全之比例等。則分餘之比例。與兩全之比例亦等。

For, as the whole AB is to the whole CD, so let the part AE subtracted be to the part CF subtracted; I say that the remainder EB will also be to the remainder FD as the whole AB to the whole CD.

For since, as AB is to CD, so is AE to CF, alternately also, as BA is to AE, so is DC to CF. [V. 16] And, since the magnitudes are proportional componendo, they will also be proportional separando, [V. 17] that is, as BE is to EA, so is DF to CF, and, alternately, as BE is to DF, so is EA to FC. [V. 16] But, as AE is to CF, so by hypothesis is the whole AB to the whole CD. Therefore also the remainder EB will be to the remainder FD as the whole AB is to the whole CD. [V. 11]

Therefore etc.

.

[PORISM.
From this it is manifest that, if magnitudes be proportional componendo, they will also be proportional convertendo.] Q. E. D.

PROPOSITION 20

第二十題　三支

If there be three magnitudes, and others equal to them in multitude, which taken two and two are in the same ratio, and if ex aequali the first be greater than the third, the fourth will also be greater than the sixth; if equal, equal; and, if less, less. If a:b=d:e and b:c=e:f, then, accordingly as a>=<c, also d>=<f.

有三幾何。又有三幾何。相為連比例。而第一幾何大於第三。則第四亦大於第六。第一或等、或小於第三。則第四亦等、亦小、於第六。

Let there be three magnitudes A, B, C, and others D, E, F equal to them in multitude, which taken two and two are in the same ratio, so that, as A is to B, so is D to E, and as B is to C, so is E to F; and let A be greater than C ex aequali; I say that D will also be greater than F; if A is equal to C, equal; and, if less, less.

For, since A is greater than C, and B is some other magnitude, and the greater has to the same a greater ratio than the less has, [V. 8] therefore A has to B a greater ratio than C has to B. But, as A is to B, so is D to E, and, as C is to B, inversely, so is F to E; therefore D has also to E a greater ratio than F has to E. [V. 13] But, of magnitudes which have a ratio to the same, that which has a greater ratio is greater; [V. 10] therefore D is greater than F. Similarly we can prove that, if A be equal to C, D will also be equal to F; and if less, less.

Therefore etc. Q. E. D.

PROPOSITION 21.

第二十一題　三支

If there be three magnitudes, and others equal to them in multitude, which taken two and two together are in the same ratio, and the proportion of them be perturbed, then, if ex aequali the first magnitude is greater than the third, the fourth will also be greater than the sixth; if equal, equal; and if less, less. If a:b=e:f and b:c=d:e, then, accordingly as a>=<c, also d>=<f.

有三幾何。又有三幾何。相為連比例而錯。以平理推之。若第一幾何大於第三。則第四亦大於第六。若第一或等、或小、於第三。則第四亦等、亦小、於第六。

Let there be three magnitudes A, B, C, and others D, E, F equal to them in multitude, which taken two and two are in the same ratio, and let the proportion of them be perturbed, so that, as A is to B, so is E to F, and, as B is to C, so is D to E, and let A be greater than C ex aequali; I say that D will also be greater than F; if A is equal to C, equal; and if less, less.

For, since A is greater than C, and B is some other magnitude, therefore A has to B a greater ratio than C has to B. [V. 8] But, as A is to B, so is E to F, and, as C is to B, inversely, so is E to D. Therefore also E has to F a greater ratio than E has to D. [V. 13] But that to which the same has a greater ratio is less; [V. 10] therefore F is less than D; therefore D is greater than F. Similarly we can prove that, if A be equal to C, D will also be equal to F; and if less, less.

Therefore etc. Q. E. D.

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