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Book V.

PROP. Ą.: THEOR. If the first of four magnitudes has to the second, the Sec N.

same ratio which the third has to the fourth; then if the firit be greater than the second, the third is also greater than the fourth; and if equal, equal; if less, less.

Take any equimultiples of each of them, as the doubles of each. then by Def. ;th of this Book, if the double of the first be greater than the double of the second, the double of the third is greater than the double of the fourth. but if the first be greater than the second, the double of the first is greater than the double of the fecond. wherefore also the double of the third is greater than the double of the fourth; therefore the third is greater than the fourth. in like manner, if the first be equal to the second, or less than it, the third can be proved to be equal to the fourth, or less than it. Therefore if the first, &c. Q. E, D.

PROP. B. THEOR.

IF

four magnitudes are proportionals, they are propor- see N. tionals also when taken in versely.

If the magnitude A be to B, as C is to D, then also inversely B is to A, as D to C.

Take of B and D any equimultiples what. ever E and F; and of A and C any equimultiples whatever G and H. First, Let E be greater than G, then G is less than E; and because A is to B, as C is to D, and of A and C the first and third, G and H are equimultiples; and of B and D the second and fourth,

G A B E E and F are equimultiples; and that G is less than E, H is also less than F; that is, Fis HC D F

a. s. Def. s. greater than H. if therefore E be greater than G, F is greater than H. in like manner, if E be equal to G, F may be shewn to be equal to H; and if less, less. and E, F are any equimultiples whatever of B and D, and G, H

any

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Book V. any whatever of A and C. Therefore as B is to A, fo is D to C. mIf then four magnitudes, &c. Q. E. D.

PROP. C. THEOR.

See N.

IF the first be the same multiple of the second, or the

same part of it, that the third is of the fourth; the first is to the second, as the third is to the fourth.

Let the first A be the same multiple of B the second, that C the third is of the fourth D. A is to B, as C is to D.

Take of A and Cany equimultiples whatever E and F; and of B and D any equimultiples whatever G and H. then because A is the same multiple of B that C is of D; and that E is the

fame multiple of A, that F is of C; E is the fame A B C D a. 3. 5. multiple of B, that Fis of D’; therefore E and E G FH

F are the same multiples of B and D. but G
and H are equimultiples of B and D; therefore
if Ebe a greater multiple of B, than G is; F is a
greater multiple of D, than H is of D; that is, if
E be greater than G, F is greater than H. in like
inanner, if E be equal to G, or less; F is equal
to H, or less than it. But E, F are equimultiples,
any whatever, of A, C, and G, H any equimul-

tiples whatever of B, D. Therefore A is to B, k. 5. Def.s. as C is to Db.

Next, Let the first A be the same part of the second B, that the third C is of the fourth D. A is to B, as C is to D. for B is the same multiple of A, that D is of C;

wherefore by the preceeding case B is to c. B. 5. A, as D is to C; and inversely A is to B,

as C is to D. Therefore if the first be the
fame multiple, &c. Q. E. D.

A B CD

PROP.

Book V.

PROP. D.

THEOR.

If the first be to the second as the third to the fourth, See N.

and if the first be a multiple, or part of the second; the third is the same multiple, or the same part of the fourth.

b. A. S.

Let A bc to B, as C is to D; and first let A be a multiple of B;
C is the same multiple of D.

Take E equal to A, and whatever multiple
A or E is of B, make F the same multiple of
D. then because A is to B, as C is to D; and
of B the second and D the fourth equimul-
tiples have been taken E and F; A is to E, as
C to F?. but A is equal to E, therefore C is

a. Cor.4.5
equal to Fb. and F is the same multiple of D,
that A is of B. Wherefore C is the same mul- A B C D
tiple of D, that A is of B.

E F
Next, Let the first A be a part of the fe-

See the Fi-
cond B; C the third is the same part of the
fourth D.

preceeding Because A is to B, as C is to D; then, in

page. versely B iso to A, as D to C. but A is a part

c. B. s. of B, therefore B is a multiple of A, and, by the preceeding case, D is the fame multiple of C; that is, C is the same part of D, that A is of B. Therefore if the first, &c. Q. E. D.

gure at the

foot of the

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PRO P. VII.

THE O R.

EQUAL magnitudes have the same ratio to the fame

magnitude; and the same has the same ratio to equal magnitudes.

Let A and B be equal magnitudes, and C any other. A and B have each of them the same ratio to C. and C has the same ratio to each of the magnitudes A and B.

Take of A and B any equimultiples whatever D and E, and of

C any

Book V. C any multiple whatever F. then because D is the fame multiple

w of A, that E is of B, and that A is equal to a. 1. Ax. 5. B; D is equal to E. therefore if D be greater

than F, E is greater than F'; and if equal, e-
qual; if lefs, less. and D, E are any equi-

multiples of A, B, and F is any multiple of 5. Dcf. 5. C. Therefore bas A is to C, so is B to C.

Likewise C has the same ratio to A that it
has to B. for, having made the fame construc- DA
tion, D may in like manner be shewn equal E B
to E. therefore if F be greater than D, it is

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likewise greater than E; and if equal, equal ;
if less, less. and F is any multiple whatever of
C, and D, E, are any equimultiples whatever
of A, B. Therefore C is to A, as C is to Bb.
Therefore equal magnitudes, &c.

Q. E. D.

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S: N.

OF unequal magnitudes the greater has a greater ratia

to the same than the less has. and the same magnitude has a greater ratio to the less than it has to the greater. .

17

Let AB, BC be unequal magnitudes of which AB is the greater,
and let D be any magnitude whatever. AB
has a greater ratio to D than BC to D. and E
D has a greater ratio to BC than unto AB.
If the magnitude which is not the

F+ A
greater of the two AC, CB, be not less
than D, take EF, FG the doubles of AC, C
CB, as in Fig. 1. but if that which is not
the greater of the two AC, CB be less than

GB
D (as in Fig. 2. and 3.) this magnitude can
be multiplied so as to become

L K H D
D, whether it be AC or CB. Let it be
multiplied until it become greater than D,
and let the other be multiplied as often ;
and let EF be the multiple thus taken of
AC, and FG the fame multiple of CB.

therefore

greater than

therefore EF and FG are each of them greater than D. and in Book V. every one of the cases take H the double of D, K its triple, and so on, till the multiple of D be that which first becomes greater than FG. let L be that multiple of D which is first greater than FG, and K the multiple of D which is next less than L.

Then because L is the multiple of D which is the first that becomes greater than FG, the next preceeding multiple K is not greater than FG; that is, FG is not less than K. and since EF is the same multiple of AC, that FG is of CB; FG is the same multiple of CB, that EG is of AB*; wherefore EG and FG are equimultiples of AB a. 1. s. and CB. and it was shewn

E that FG was not less than

EL K, and, by the constructi- FH on, EF is greater than D; therefore the whole EG is greater than K and D A together. but K together

Ft A with D is equal to L; therefore EG is greater GB

CH than L; but FG is not greater than L; and EG, L K HD

GB FG are equimultiples of

I. KD AB, BC, and L is a mul. tiple of D; thereforebAB

b.7. Def. 5 has to D a greater ratio than BC has to D.

Also D has to BC a greater ratio than it has to AB. for, having made the same construction, it may be shewn, in like manner, that L is greater than FG, but that it is not greater than EG. and L is a multiple of D; and FG, EG are equimultiples of CB, AB. Therefore D has to CB a greater ratio b than it has to AB. Wherefore of unequal magnitudes, &c. Q. E. D.

PROP.

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