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Again, let c have a greater ratio to B, than it has to A: I fay, B is less than A.

For if it be not lefs, it is either equal to it or greater. But B is not equal to A, for then c [by 7. 5.] would have the fame ratio to A as to B. But it has not: Therefore A is not equal to B, nor is B greater than A; for then [by 8. 5.] would have a less ratio to B than to A. But it has not: Therefore B is not greater than A: But it has been proved not to be equal to it: Confequently B will be less than A:

Therefore that magnitude of those magnitudes which have a ratio to the fame magnitude, is the greater which has the greater ratio, and that magnitude to which the fame magnitude has the greater ratio, is the leffer.

PROP. XI. THEOR.

Thofe ratios that are the fame to the fame ratio, are the fame between themselves.

For let A be to B, as c is to D, and as c is to D, fo is E to F: I fay, as A is to B, fo is E to F.

For take G, H, K equimultiples of A, C, E, and any other equimultiples L, M, N of B, D, F.

Then because it is as A to B, fo is C to D, and G, H, are equimulti

ples of A,C, and L, M any other equimultiplesof B, D: If G exceeds L, [by 5: def. 5.] H will exceed M; if, the one be equal, the other will be equal; if lefs, lefs. Again, because it is as c to D, fo

is E to F, and GABL HCDM KEFN H, K are taken equimultiples of C, E; alfo M, N any other equimultiples of D, F. If H exceeds M, K will exceed N; If equal, equal; but if lefs, lefs. But if H.ex

ceeds

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ceeds M, & will exceed L: If it be equal, equal; if lefs, less. Wherefore if G exceeds L, K will exceed N; if equal, equal; if lefs, lefs. And G, K are equimultiples of A, E, and L, N any others of B, F: Therefore [by 5. def. 5.] as A is to B, fo will E be to F.

Wherefore those ratios that are the fame to the fame ratio, are the fame between themselves. Which was to be demonftrated.

PROP. XII. THEOR.

If any magnitudes foever be proportional; as one of the antecedents is to its confequent, so will all the antecedents together be to all the confequents toge

ther Y.

Let any magnitudes foever A, B, C, D, E, F be proportional; and let A be to B, as c is to D, and as E is to ř: I fay, as A is to B, fo is A, C, E, to B, D, F.

For take the equimultiples G, H, K of A, C, E, and any other equimultiples L, M, N of B, D, F.

Then because as A is to B, fo is c to D, and E to F: And there are taken the equimultiples G, H, K of A, C, E, and any other equimultiples L, M, N of B, D, F: If G exceeds L [by 5. def. 5.] will exceed M, and K will exceed N: If the one

GHKACE BDFLMN be equal, the other will be equal; if lefs, lefs. Wherefore also if G exceeds 1; G, H, K will exceed L, M, N; if equal, equal; and if lefs, lefs. And G, and G, H, K are equimultiples of A and A, C, E: Because [by 1. 5.] if there be any number of magnitudes foever equimultiples of an equal number of magnitudes, each the fame multiple of each, the fame muluple that one of the magnitudes is of one of the other, will all the magnitudes be of all the others. By the fame reafon L and L, M, N are equimultiples of B, and B, D, F:

B, D, F,: Therefore [by 5. def. 5.] as A is to B, fo is a, ċ, E, to B, D, F.

Wherefore if any magnitudes foever be proportional: as one of the Antecedents is to its confequent, fo will all the antecedents together be to all the confequents together. Which was to be demonftrated.

y The twelfth propofition of the feventh book of Euclid anfwers to this in numbers.

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If there be fix magnitudes and the first magnitude bas the fame ratio to the fecond, as the third to the fourth, but the third to the fourth has a greater ratio than the fifth to the fixth; then will the first have a greater ratio to the fecond, than the fifth bas to the fixth.

For let the firft of fix magnitude A, have the same ratio to the second B, that the third c, has to the fourth D; and let the third c have a greater ratio to the fourth D, than the fifth E has to the fixth F: I fay the firft A will have a greater ratio to the second B, than the fifth E has to the fixth F.

For because the ratio of c to Dis greater than the ratio of E to F, there are [by 7. def. 5.] fome equimultiples of c, E, and others of D, F ; where the multiple of c exceeds the multiple of D, but the multiple of E does not

exceed the multi-MABN GCDK HEFL ple of F. Také

thefe equimultiples. And let G, H be equimultiples of C, E, and K, L other equimultiples of D, F, fuch that G exceeds K, but H does not exceed L: and the fame multiple that G is of c, the fame let M be of A; and the fame that K is of D, let N be of B.

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Book V. Then becaufe A is to в, as is to D, and there are taken the equimultiples M, G of A, C, and other equimultiples N, K of B, D: [by 5. def. 5.] If м exceeds N, G will exceed K if м be equal to N, G will be equal to K: if lefs, lefs: But G exceeds K; therefore м will exceed N : But H does not exceed L ; and M, H are equimultiples of A, E, and N, L fome other equimultiples of B, F: Therefore [by 7 def. 5] A will have a greater ratio to в than

E has to F.

If therefore the first of fix magnitudes has the fame ratio to the fecond, as the third to the fourth; and the third to the fourth has a greater ratio than the fifth to the fixth : then the first to the fecond will have a greater ratio than the fifth has to the fixth. Which was to be demonftrated.

PROP. XIV. THEOR.

If the first of four magnitudes has the fame ratio to the Second, as the third has to the fourth; and the first be greater than the third: then will the fecond be greater than the fourth. If the first be equal to the third, the fecond will be equal to the fourth. If the first be less than the third, the fecond will be less than the fourth.

For let the first magnitude A have the fame ratio to the fecond B, as the third c has to the fourth D, and let a be greater than c I fay B is alfo greater than D.

For becaufe A is greater than c, and B is any other. magnitude; [by 8. 5.1 A will have a greater ratio to B,

A

than c has to B. But as A is to B, fo is c to D: Therefore [by 13. 5.] the ratio of c to D, will be greater than the ratio of c to B. But [by 10. 5] that magnitude to which the fame magnitude Thas the greater ratio, is the leffer magnitude wherefore D is less than B and accordingly B will be greater than D.

After the like manner we demonftraté if A be equal to C, that B is equal to D and if a be less than C, B is lefs Dthan D.

Note,

Note, If A be equal to C, B will be equal to D. Follows [by 9.5.] for here A is to B, as C, that is A to D; B fhall be equal to D. If A be less than C, B fhall be lefs than D, for c is greater than A; and here c is to D, as A to B; D will be greater than в by the firft cafe, wherefore B is less than D.

Therefore if the first magnitude has the fame ratio to the fecond, as the third has to the fourth, and the first be greater than the third: then will the fecond be greater than the fourth; if the first be equal to the third, the second will be equal to the fourth: and if the first be less than the third, the fecond will be less than the fourth. Which was to be demonftrated.

PROP. XV. THEOR. Magnitudes have the fame ratio to one another which their equimultiples have*.

For let A B be the fame multiple of C, that D E is off I fay c is to F, as A B is to D E.

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For because A B is the fame multiple of C, that DE is of F: there will be as many magnitudes in A & equal to c, as there are in D E, equal to F. Di-A vide A B into magnitudes equal to c, which let be AG, GH, HB; and divide DE into the magnitudes D K, KL, LE each equal to F. Then the multitude of AG, GH, HB will be equal to the multiude of D K, KL, LE. And because A G, GH, H B are equal to one another, as also D K, KL, L E equal to one another: [by 7. 5.] AG will be to H DK, as GH is to KL, and as HB is to LE: But [by 12. 5.] as one of the antecedents is to one of the confequents, fo will all the antecedents be to all the confequents. Therefore AC is to D B C as A B is to D E. But AG is equal

K

L

EF

to C, and DK equal to F. Therefore as c is to F, fo will AB be to DE.

Wherefore magnitudes have the fame ratio to one an other which their equimultiples have. Which was to be demonftrated.

z The feventeenth propofition of the feventh book answers to this propofition.

Q 2

PROP.

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