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Book V, is to A, so is D to C.

Q. E. D.

If, then, four magnitudes, &c.

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.

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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 C any 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 same multiple of A that

F is of C; E is the same multiple of B that a 3. 5.

F is of Da; therefore E and F are the same
multiples of B and D: but G and H are equi.
multiples of B and D; therefore, if E be a
greater multiple of B than Gis, F is a great.
er multiple of D than H is of D; that is,
if E be greater than G, F is greater than H:
in like manner, 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 equimultiples whatever of B, b 5.def.5. D. Therefore A is to B as C is to Db.

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Book V.

PROP. D.

THEOR.

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

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B

Let A be 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 four h equimultiples have been taken E and F; A is to É as C to Fa: but A is equal to

a Cor.4.5. E, therefore C is equal to Fb: and F is

b A.5. the same multiple of D that A is of B. А Wherefore C is the same multiple of D that A is of B.

E

F Next, Let the first A be a part of the se

See the cond B; C the third is the same part of the

figure at fourth D.

the foot Because A is to B as C is to D; then,

of the inversely, B is c to A as D to C: but A is

ing page: a part of B, therefore B is a multiple of A;

c B. 5. and, by the preceding case, D is the same 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.

preced

PROP. VII. THEOR.

EQUAL magnitudes have the same ratio to the same 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 has the same ratio to each of the magnitudes A and B.

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

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Book V. of C any multiple whatever F: then, because D is the same

multiple of A that E is of B, and that A is a 1.Ax.5. equal to B; D is a equal to E: therefore, if

D be greater than F, E is greater than F;
and if equal, equal; if less, less: and D, E

are any equimultiples of A, B, and F is any b 5. def.5. multiple of C. Therefore b, as 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 same con-

D A
struction, D may in like manner be shown
equal to E: therefore, if F be greater than

E B
D, it is likewise greater than E; and if equal,

C F
equal; if less, less : and F is any multiple
whatever of C, and D, E are any equimulti-
ples whatever of A, B. Therefore Cis to A
as C is to Bb. Therefore, equal magnitudes,
&c. Q. E. D.

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See N.

OF unequal magnitudes, the greater has a greater ratio to the same than the less has; and the same magnitude has a greater ratio to the less, than it has to the greater.

Fig. 1.

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 D has a greater ra

E tio to BC than unto AB.

If the magnitude which is not the greater of the two AC, CB, be not less .

A than D, take EF, FG, the doubles of

F AC, CB, as in Fig. 1. But, if that which is not the greater of the two AC, CB be less than D (as in Fig. 2. and 3.) this magnitude can be multiplied, so as to G B become greater than D, whether it be AC, or CB. Let it be multiplied until L K HD 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 same multiple of CB: therefore EF and FG are each of them greater than

Fig. 3.

D: and in every one of the cases, take H the double of D, K Book V. 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 preceding 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 a; wherefore EG and a 1. 5. FG are equimultiples of AB and CB: and it was shown, that FG was not less than

Fig. 2. K, and, by the con

E struction, EF is great

E er than D; therefore

F the whole EG is greater than K and D together : but K, together

A

A with D, is equal to L; therefore EG is greater than L; but FG is not greater than L;

F and EG, FG are equi

Cmultiples of AB, BC, G B

G B and L is a multiple of D; therefore b AB has

b 7. def. L Κ Η D L KD to D a greater ratio

3. 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 shown, 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.

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Book V.

PROP. IX. THEOR.

See N.

MAGNITUDES which have the same ratio to the same magnitude are equal to one another; and those to which the same magnitude has the same ratio are equal to one another.

Let A, B have each of them the same ratio to C: A is equal to B: for, if they are not equal, one of them is greater than the other; let A be the greater; then, by what was shown in the preceding proposition, there are some equimultiples of A and B, and some multiple of C such, that the multiple of A is greater than the multiple of C, but the multiple of B is not greater than that of C. Let such multiples be taken, and let D, E be the equimultiples of A, B, and F the multiple of C, so that D may be greater than F, and E not greater than F: but, because A is to C as B is to C, and of A, B are taken equimultiples D, E, and of C is taken a

multiple F; and that D is greater than F; a 5. def. E shall also be greater than Fa; but E is

D
5.
not greater than F, which is impossible;

A
A therefore and B are not unequal; that is,
they are equal.

Next, Let C have the same ratio to each
of the magnitudes A and B; A is equal to
B: for, if they are not, one of them is B
greater than the other; let A be the

E
greater; therefore, as was shown in Prop.
8th, there is some multiple F of C, and
some equimultiples E and D, of B and A
such, that F is greater than E, and not greater than D; but be-
cause C is to B, as C is to A, and that F, the multiple of the
first, is greater than E, the multiple of the second; F, the mul-
tiple of the third, is greater than D, the multiple of the fourtha:
but F is not greater than D, which is impossible. Therefore A
is equal to B. Wherefore, magnitudes which, &c. Q. E. D.

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