Page images
PDF
EPUB
[ocr errors]

D, M together. But D, M to

FI Τ Α gether are equal to n; wherefore FH exceeds N; but K does not exceed n; and FH, K

E se equimultiples of A B, C; and n is any other multiple of D: Therefore [by 7. def. 5.] B A B has a greater racio to D,

+G than c has to D.

I say moreover that the ratio of D to c is greater than the ratio of D to A B.

For the same construction re- K H C D L MN maining, we demonstrate in the like manner, that n exceeds K, but does not exceed F H. But N is a multiple of D, and FH, K any other equimultiples of A B, C. Therefore [by 7. def. 5.) the ratio of d to c is greater than the ratio of D to A B.

Now let A E be greater than E B: Then [by 4. def.:5.] the leffer magnitude E B being multiplied, will at length become greater than D. Let it de multiplied, and let G H the multiple of E B be greater than d: And make F G the fame multiple of a E, and K of c, as Gh is of E B. Then by the like reason as before, we demonstrate that FH, K are equimultiples of AB, c. And likewise take n a multiple of D, [in the first place] greater than FG: Therefore again F G is not less than M.

But G H is greater than D. Therefore the whole F H exceeds d and m together, that is N. But K does not exceed n, because F G, which is greater than HG, that is, than K, does not exceed n. And after the like manner as before we finish the demonstration.

Therefore the greater of (two) unequil magnitudes has a greater ratio to the same magnitude than the leller ; and the same magnitude to the lesser of (two] unequal magnitudes has a greater ratio than it has to the greater of those (two] magnitudes. Which was to be demonstrated,

PROP.

[ocr errors]

Ic

PROP. IX. THEOR. Magnitudes that have the same ratio to the same magnitude, are equal to one another ; and those mag,

; nitudes to which the same magnitude has the same ratio, are also equal to one another.

For let each of the magnitudes A and B have the same satio to the same magnitude c: I say, a is equal to B.

For if it were not equal, A and B would not [by 8. 5.] have the same ratio to c.

But it has. Therefore A is equal to B. 1

Again, let c have the same ratio both to
A, B : I say, A is equal to B.

For if it were not equal, c would not have
the same ratio both to A and B. But it has :

Therefore A is equal to B. A

Wherefore magnitudes that have the same

ratio to the same magnitude, are equal to one another : and those magnitudes to which the same magnitude has the same ratio, are also equal to one ano ther. Which was to be demonstrated

PROP. X. THEOR.
That magnitude of those magnitudes which have a

ratio to the same magnitude, is the greater which
has the greater ratio ; and that magnitude to
which the same magnitude has the greater ratio,
is the lesser.

For let a have a greater ratio to c, than B has to c; I say, A is greater than B.

For if it be not greater, it is either equal

to it, or less than it. But A is not equal to B

B, for then [by 7.5.] A and B would each
have the same ratio to c. But they bave
not the same ratio. Therefore A is not
equal to B. Nor is A less than B, for then

[by 8. 5.) A would have a less ratio toc А

c.

a

than B has to c. But it has not a less :

Wherefore A is not less than B: it is there

fore greater.

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

For if it be not less, it is either equal to it or greater. But B is not equal to A, for then c [by 7. 5.] would have the same 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.] c 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: Consequently 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 same magnitude has the greater ratio, is the lesser.

P R OP. XI. T H E O R. Those ratios that are the same to the same ratio, are

the same 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 say, as A is to B, so is E to F. For take G, H, K equimultiples of A, C, E,

and

any ether equimultiples L, M, N of B, D, F.

Then because it is as A to B, so is c to D, and G, H, are equimultiples 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 less, less. Again, because it is as c to D, so is e to F, and GABL HCDM KÉÉN E ,

KEFN H, K are taken equimultiples of C, E ; alfo M, N any other equimultiples of D, F. If exceeds M, K will exceed N ; If equal, equal ; but if less, less. But if H.exCeeds m, Ġ will exceed L: If it be equal, equal; if less, less. Wherefore if G exceeds L, K will exceed N; if equal, equal ; if less; less. And 6, K. are equimultiples of

ceeds

A, E, and l, N any others of B; F: Therefore [by ść def. 5.] as A is to B; so will E be to F.

Wherefore those ratios that are the same to the same ratio, are the same between themselves. Which was to be demonstrated.

PROP. XII. THEOR. If any magnitudes foever be proportional ; as one of

the antecedents is to its consequent, so will all the antecedents together be to all the consequents togen ther y.

Let any magnitudes sơever Å, B, C, D, E, F be proportional; and let a be to B, as c is to D, and as e is to $ : I say, 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 6, D, F.

Then because as a is to B; so 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 1 [by 5. def. 5.] i will exceed Mj and K will

exceed N: If the one GH KACE BDFLMN be equal, the other will be equal; if less, less. Wherefore also if g exceeds I ; G, H, K will exceed L, M, N; if equal; equal; and if less, less. 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 same multiple 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;

[ocr errors]

B, D, F,: Therefore [by si def. 5.) as A is to ß, so is A,
C, E, to B, D, É.

Wherefore if any magnitudes soever be proportional : as one of the Antecedents is to its confequent, so will all the antecedents together be to all the consequents together. Which was to be demonstrated.

y The twelfth propofition of the seventh book of Euclid answers to this in numbers.

[ocr errors]

PROP. XIII. THEOR.
If there be fix magnitudes and the first magnitude baś

the same ratio to the second, 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 second; than the
fifth bas to the fixtb.

For let the first of fix magnitude A, have the same ratio to the second By 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 say the first a will have a greater ratio to the second B, than the fifth É has to the sixth F.

For because the ratio of c to D is greater than the ratio of E to F, there are (by 7. def. 5.] some 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. Take these equimultiples. And let G, H be equimultiples of C, E, and K, L other equimultiples of D, F, such that G exceeds K, but H does not exceed L : and the same multiple that G is of c, the same let m be of a ; and the same that K is of D, let n be of B.

« PreviousContinue »