PROP. III. THEOR.

IF the first be the same multiple of the second, which the third is of the fourth; and if of the first and third there be taken like multiples, these will be like multiples of the second and fourth, each of each.

Let A the first be the same multiple of B the second, that C the third is of D the fourth; and of A, C, let the same multiples EF, GH be taken: then EF is the same multiple of B, that GH is of D.

F

K+

H

E A B G C

Because EF is the same multiple of A, that GH is of C, there are as many magnitudes in EF equal to A, as are in GH equal to C; let EF be divided into the parts EK, KF, each equal to A, and GH into GL, LH, each equal to C: the number therefore of the magnitudes EK, KF, is equal to the number of the others GL, LH. And because A is the same multiple of B, that C is of D, and that EK is equal to A, and GL to C; therefore EK is the same multiple of B, that GL is of D. For the same reason, KF is the same multiple of B, that LH is of D; and so, if there be more parts in EF, GH, equal to A, C. Because, therefore, the first EK is the same multiple of the second B, which the third GL is of the fourth D, and that the fifth KF is the same multiple of the second B, which the sixth LH is of the fourth D; EF, the first together with the fifth, is the same multiple (V. 2.) of the second B, which GH, the third together with the sixth, is of the fourth D. If, therefore, &c.

PROP. IV. THEOR.

IF the first of four magnitudes have the same ratio to the second which the third has to the fourth; then any like multiples whatever of the first and third have the same ratio to any like multiples of the second and fourth.

Let A the first have to B the second, the same ratio which the third C has to the fourth D; and of A and C let there be taken any like multiples whatever E, F; and of B and D any like multiples whatever G, H: then E has the same ratio to G, which F has to H.

Take of E and F any like multiples whatever K, L, and of G, H, any like multiples K

E A BG M

F

C

D H N

whatever M, N. Then, because E is the L same multiple of A, that F is of C; and of E and F have been taken like multiples K, L; therefore (V.3.) K is the same multiple of A, that L is of C. For the same reason, M is the same multiple of B, that N is of D. And because (hyp.) as A is to B, so is C to D, and of A and C have been taken certain like multiples K, L; and of B and D have been taken certain like multiples M, N; if therefore (V. def. 5.) K be greater than M, L is greater than N: and if equal, equal; if less, less. But K, L are any like multiples whatever of E, F ; and M, N any whatever of G, H: therefore (V. def. 5.) as E: G :: F: H. Therefore, &c.

Cor. Likewise, if the first have the same ratio to the second, which the third has to the fourth, then also any like multiples whatever of the first and third have the same ratio to the second and fourth and in like manner, the first and the third have the same ratio to any like multiples whatever of the second and fourth.

Let A the first have to В the second, the same ratio which the third C has to the fourth D, and of A and C let E and F be any like multiples whatever; then E: B:: F: D.

:

Take of E, F any like multiples whatever K, L, and of B, D any like multiples whatever G, H. Then it may be demonstrated, as before, that K is the same multiple of A, as L is of C: and, because A B C D, and of A and C like multiples K and L have been taken; and of Band D like multiples G and H; therefore, (V. def. 5.) if K be greater than G, L is greater than H; if equal, equal; and if less, less. And K, L are any like multiples of E, F ; and G, H any whatever of B, D: as therefore (V. def. 5.) E:B::F:D: and in the same way the other case is demonstrated.

PROP. V. THEOR.

IF one magnitude be the same multiple of another, which a part taken from the first is of a part taken from the other; the first remainder is the same multiple of the second, that the first magnitude is of the second.

GI

Let the magnitude AB be the same multiple of CD, that AE taken from the first, is of CF taken from the other; the remainder EB is the same multiple of the remainder FD, that the whole AB is of the whole CD.

Take AG the same multiple of FD, that AE is of CF: therefore (V. 1.) AE is the same multiple of CF, that GE is of CD. But (hyp.) AE is the same multiple of CF, that AB is of CD therefore EG is the same multiple of CD that AB is of CD; wherefore (V. ax. 1.) EG is equal to AB. Take from them the common magnitude AE; and the remainder AG is

A

E+

B

D

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.

PROP. VI. THEOR.

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

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

K

A

C

H+

!!

B D E

First, let GB be equal to E; HD is equal to F. Make CK equal to F; and because AG is the same multiple of E, that CH is of F, and that GB is equal to E, and CK to F; therefore AB is the same multiple of E, that KH is of F. But (hyp.) AB 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 (V. ax. 1.) KH is equal to CD. Take away CH, then the remainder KC is equal to the remainder HD: but KC is equal to F; HD therefore is equal to F.

G

K

CL

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. Then 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 (V. 2.) AB is the same multiple of E, that KH is of F. But AB is the same multiple of E, that CD is of F: therefore KH is the same multiple of F, that CD is of it: wherefore (V. ax. 1.) KH is equal to CD. 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.

PROP. A. THEOR.

B D E P

IF four magnitudes be proportional; 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 like multiples of each of them, as the doubles of each;

then (V. def. 5.) 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 (V. ax. 3.) than the double of the second; wherefore also the double of the third is greater (V. def. 5.) than the double of the fourth; therefore the third is greater (V. ax. 4.) 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 four magnitudes, &c.

PROP. B. THEOR.

Ir four magnitudes be proportionals, they are proportionals also, when taken inversely.

G A B E

H C D F

If A : B :: C: D; then also inversely, B: A :: D: C. Take of B and D any like multiples E and F; and of A and C any like multiples G and H. First, let E be greater than G; and, because A: B:: C: D, and of A and C, the first and third, G and Hare like multiples; and of B and D, the second and fourth, E and F are like multiples; and that G is less than E, H is also (V. 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 like any multiples whatever of B and D, and G, H any whatever of A and C: therefore, as B: A :: D: C. If, then, four magnitudes, &c.

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 the second, that C the third is of D the fourth; then A: B::C: D.

Take of A and C any like multiples whatever E and F; and of B and D any like multiples whatever G and H. Then, because A is the same multiple of A B that C is of D; and that E is the same multiple of E A, as F is of C; E is the same multiple (V. 3.) of B, that F is of D: therefore E and F are the same multiples of B and D. But G and H are like multiples of B and D; therefore, if E be a greater multiple of B, than G is, F is a greater multiple of D,

PH

B C D
GF

than H is; 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 like multiples, any whatever, of A, C,

and G, H any like multiples whatever of B, D. (V. def. 5.) A: B:: C: D.

Therefore

Next, let the first A be the same part of the second B, that the third C is of the fourth D; A : B :: C : D. For B is the same multiple of A, that D is of C: wherefore, by the preceding case, B: A:: D: C; and inversely | (V. B.) A: B:: C: D. If, therefore, &c.

PROP. D. THEOR.

A B C D

IF the first be to the second as the third to the fourth, and if the first be a multiple, or a part of the second; the third is the same multiple, or part of the fourth. Let A: B:: C: 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.

A B C D E F

Then because A: B::C: D; and of B the second, and D the fourth like multiples E and F have been taken; then, (V. 4. cor.) A: E::C: F. But A is equal to E; therefore C (V. A.) is equal to F 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 (see last figure for prop. C.) be a part of the second B; C the third is the same part of the fourth D.

Because A: B:: C: D; then, inversely, (V. B.) B: A:: D: 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. If, therefore, &c.

PROP. VII. THEOR.

EQUAL magnitudes have the same ratio to the same magnitude of the same kind: and the same has the same ratio to equal magnitudes of the same kind.

Let A and B be equal magnitudes, and C any other of the same kind: A and B have each 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 like multiples whatever D and E, and of C any multiple whatever F. Then, because D is the same multiple of A, that E is of B, and that A is equal to B; D is equal (V. ax. 1.) to E: therefore, if D be greater than F, E is greater than F; if equal, D equal; and if less, less. But D, E are any like E B multiples of A, B, and F is any multiple of C. Therefore (V. def. 5.) as A: C:: B: C.

Likewise C has the same ratio to A, that it has to

A

C