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Book V. is to A, so is D to C. If, then, four magnitudes, &c.
Q. E. D.
PROP. C. THEOR.
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
Take of A and C any equimultiples what-
F is of C; E is the same multiple of B that E a 3. 3. F is of Da; therefore E and F are the same
multiples of B and D: but G and H are equi-
and G, H any equimultiples whatever of B, b 5.def. 5. D. Therefore A is to B 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 preceding case, B is to A as D is to C; and inversely, A is to B as C is to D. Therefore, if the first be the same multiple, &c. Q. E. D.
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.
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 B Wherefore C is the same multiple of D that A is of B.
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.
of the Because A is to B as C is to D; then,
precedinversely, 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;
cB. 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.
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 any other. A and
Book V. of C any multiple whatever F: then, because D is the same
D be greater than F, E is greater than F;
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
PROP. VIII. THEOR.
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.
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 Fig. 1. than BC to D: and D has a greater ra
E tio to BC than unto AB.
If the magnitude which is not the
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
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 K, and, by the conFig. 2,
Fig. 3. 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 with D, is equal to L; therefore EG is great
c! er than L; but FG is not greater than L;
F and ÉG, FG are equi
C multiples of AB, BC,
B and L is a multiple of D; thereforeb AB has
b 7. def. L K HD L K D to D a greater ratio
5. 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.
PROP. IX. THEOR.
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