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PROP. B. THEOR.

IF four magnitudes be proportionals they are poportionals also when taken inversely."

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 whatever 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, E and F are equimultiples; and that G is less than E, H is also (5. def. 5.) less than F; that is, F is 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 shown to be equal to H; and, if less, less; and E, F are any equimultiples whatever of B and D, and G, H any whatever of A and C; therefore as B is to A, so is D to C. If then, four magnitudes, &c. Q. E. D.

PROP. C. THEOR.

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H

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A B

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D

F

Ir 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 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 F is of D (3. 5.); therefore E and F are the same multiples of B and D: but G and H are equimultiples of B and D: therefore, if E be 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 manner, if E be equal to G, or less; F is equal to H, or less than it. But E, F are any equimultiples whatever of A, C, and G, H any equimultiples whatever of B, D. Therefore A is to B, as C is to D (5. def.)

* See Notes.

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E

B C D

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F H

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 (B. 5.) A is to B, as C is to D. Therefore, if the first be the same multiple, &c. Q. E. D.

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PROP. D. THEOR.

Ir the first be to the second as the third to the 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 fourth equimultiples have been taken E and F; A is to E as C to F (Cor. 4. 5.): but A is equal to E, therefore C is equal to F (A. 5): and F is the same multiple of D, that A is of B. Wherefore C is the same multiple of D, that A is of B.

Next, Let the first A be a part of the second B; C the third is the same part of the fourth† D.

Because A is to B, as C is to D; then inversely, B is (B. 5.) to A, as D to C: but A is a part of B, therefore B is a multiple of A; 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.

PROP. VII. THEOR.

A B

C D

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F

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 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 multiple whatever F; then, because D is the same multi

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ple of A, that E is of B, and that A is equal to B; D is (1. Ax. 5.) 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 multiple of C. Therefore (5. def. 5.) 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 construction, D may in like manner be shown equal to E: therefore, if F be greater than D, it is 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 B (5. def. 5.). Therefore equal magnitudes, &c. Q.

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E. D.

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PROP. VIII. THEOR.

Or 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 than BC to D; and D has a greater ratio to BC than to AB.

If the magnitude which is not the greater of the two AC, CB, be not less than D, take EF, FG, the doubles of 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 become greater than 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 same multiple of CB; therefore EF and FG are each of them greater than D: and in every one of the

Fig. 1.

E

A

F

L

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H

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cases, take H the double 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.

* See Note.

Fig. 2.

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F

A

Fig. 3.

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A

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 (1. 5.): wherefore EG and FG are equimultiples of AB and CB: and it was shown, that FG was not less than K, and, by the construction, EF is greater than D; therefore the whole EG is greater than K and D together; but, K together with D, is equal to L; therefore EG is greater than L; but FG is not greater than L; and EG, FG are equimultiples of AB, BC, and L is a multilple of D; therefore (7. def. 5.) AB has to D a greater ra tio 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

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B

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K D

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 (7. def. 5.) 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 be not equal, one of them is greater than the other; let A be the greater; then, by what was shown in the preceding

* See Note.

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; E shall also be greater than F (5. def. 5.): but E is not greater than F, which is impossible; A therefore and B are not unequal; that is, they are equal.

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B

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Next, let C have the same ratio to each of the magnitudes A and B; A is equal to B: for if they be not, one of them is greater than the other; let A be the 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 because 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 multiple of the third, is greater than D, the multiple of the fourth (5. def. 5.) : but F is not greater than D, which is impossible. Therefore A is equal to B. Wherefore, magnitudes which, &c. Q. E. D.

PROP. X. THEOR.

THAT magnitude which has a greater ratio than another has unto the same magnitude, is the greater of the two: and that magnitude, to which the same has a greater ratio than it has unto another magnitude is the lesser of the two.*

Let A have to C a greater ratio than B has to C: A is greater than B: for, because A has a greater ratio to C, than B has to C, there are (7. def. 5.) 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 it let them

* See Note.

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