PROP. V. THEOR.

If one magnitude be the same multiple of another, which a magnitude taken from the first is of a magnitude taken from the 'other; the remainder shall be the same multiple of the remainder, that the whole is of the whole.

that AB is of CD; Therefore EG is the same multiple of CD that AB is of CD; wherefore EG is equal to AB. (1. Ax. 5.) Take from them the common magnitude AE; the remainder AG is equal to the remainder EB. Wherefore, since AE is the same multiple of CF, that AG is of FD, and that AG is equal to EB; therefore AE is the same multiple of CF, that EB is of FD: But AE is the same multiple of CF, that AB is of CD; therefore EB is the same multiple of FD, that AB is of CD. Therefore, if one magnitude, &c. Q. E. D.

PROP. VI. THEOR.

If two magnitudes be equimultiples of two others, and if equimultiples of these be taken from the first two, the remainders are either equal to these others, or equimultiples of them.

Let the two magnitudes AB, CD be equimultiples of the two E, F, and AG, CH taken from the first two be equimultiples of the same E, F; the remainders GB, HD are either equal to E, F, or equimultiples of them.

First, Let GB be equal to E; HD is equal to F: Make CK equal to F; and

because AG is the H
same multiple of E,
that CH is of F, and
that GB is equal to

B

E, and CK to F; therefore AB is the same multiple of E, that KH is of F. But AB, by the hypothesis, is the same multiple of E that CD is of F; therefore KH is the same multiple of F, that CD is of F; wherefore KH is

equal CD: (1. Ax. 5.) Take away the common magnitude CH, then the remainder KC is equal to the remainder HD: But KC is equal to F; HD therefore is equal to F.

K

G. FL

But let GB be a multiple of E; then HD is the same multiple of F: Make CK the same multiple of F, that GB is of E: And because AG is the same multiple of E, that CH is of F; and GB the same multiple of E, that CK is of F; therefore AB is the same multiple of E, that KH is of F: (2. 5.) But AB is the same multiple of E, that CD is of F; therefore KH is

B D

the same multiple of F, that CD is of it: wherefore KH is equal to CD: (1. Ax. 5.) Take away CH from both; therefore the remainder KC is equal to the remainder HD: And because GB is the same multiple of E,

that KC is of F, and that KC is equal to HD; therefore HD is the same multiple of F, that GB is of E: If therefore two magnitudes, &c. Q. E. D.

PROP. A. THEOR.

If the first of four magnitudes has to the second the same ratio which the third has to the fourth; then, if the first 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. 5th 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 second;

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 proportionals also when taken inversely.

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. 3) 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 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 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.

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 mul- is of the fourth D: A is to B as C is tiple of B the second, that C the third to D

FH

Take of A and C any equimultiples whatever E and F; and of B & 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

PROP. D. THEOR.

If 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,

PROP. VII. THEOR.

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

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 shewn 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, É 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. E. D.

PROP. VIII. THEOR.

Of unequal magnitudes, the greater has the 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.

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 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.

which is the first that becomes greater Then, because Lis the multiple of 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.