Parts (a and b) compared among themselves, have the same ratio, which their equi-multiples have, (AC and BD).

For since AC and BD are equi-multiples of a and b, there are as many parts in AC equal to a, as there are in BD equal to b; divide AC into the parts, Am, mn, nC, each equal to a, and BD into parts, Bo, os, sD, equal to b.

Therefore, Am is to Bo, as mn to os, (by Cor. Prop. 14, B. 5), and mn to os as nC to sD, (by same); therefore as one antecedent, Am, is to one consequent, Bo, so are all the antecedents together, or AC, to all the consequents together, or BD, (by Prop. 22, B. 5); but as Am is to Bo, so is a to b, (by Cor. Prop. 14, B. 5), therefore as a is to b, so is AC to BD, (by Prop. 18, B. 5).

COR. 1.--If there be four magnitudes proportional, a to b as c to d, and there be taken equi-submultiples, A, B, of a and b, and also other equi-multiples, C and D, of c and d, A shall be to B as C to D, as is evident from Prop. 28, and Prop. 18, B. 5.

COR. 2. Similarly, if A be to B as C to D, and there be taken equi-submultiples, a and b of A and B, and any equi-submultiples, c and d of C and D, a shall be to b as c to d.

If there be four magnitudes proportional, (a to b as c to d) any multiple (A) of the first will be to the second, as an equimultiple (C) of the third to the fourth.

For such a multiple as A is of a, let B be the same multiple of b, and also D of d, and A will be to B as C to D, (by Cor. 1, Prop. 28, B. 5); therefore A is to b as C to d, (by Prop. 72, B. 5).

If there be four magnitudes proportional (a to b as c to d) the first will be to any multiple (B) of the second, as the third to an equi-multiple (D) of the fourth.

For such a multiple as B is of b, let A be the same of a, and also C of c, A will be to B as C to D, (by Cor. 1, Prop. 28, B. 5), and therefore a is to B as c to D, (by Prop. 26, B. 5).

PROPOSITION XXXI. THEOREM.

If there be four magnitudes proportional, (a to bas c to d) and there be taken equi-multiples (A and C) of the antecedents, and any other equi-multiples (B and D) of the consequents, these multiples will be also proportional.

For since a is to b as c to d, but A and C are equi-multiples of a and c, (by Hypoth.), A will be to bas C to d, (by Prop. 29, B. 5), therefore since B and D are equi-multiples of b and d, (by Hypoth.), A will be to B as C to D, (by Prop. 30, B. 5).

If there be four magnitudes proportional, (a to b as c to d), any equi-multiples (A and C) of the first and third, are together either greater, or equal, or less than the equi-multiples (B and D) of the second and fourth, however multiplied.

For since A and C are equi-multiples of the antecedents, (by Hypoth.), but B and D equi-multiples of the consequents, (by Hypoth.), A shall be to Bas C to D, (by Prop. 31, B. 5); and therefore if A be greater than B, C will be greater than D; and if equal, equal; and if less, less (by Prop. 24, B. 5).

If there be four magnitudes of the same kind proportional (A to B as C to DO), they will be also alternately proportional, (A to C as B to DO).

For if there be taken any equi-submultiples a and b of A and B; as often as a is contained in C, so often is b contained in DO.

For if not, but if it be possible, let a be contained oftener in C than b in DO, and as often as a is contained in C, so often repeat b, and hence make up DY; DY shall be greater than DO (by Prop. 5, B. 5); therefore b is a submultiple of DY, let x be the equisubmultiple of DO, and z also of C.

Therefore, because x and b are equi-submultiples of DO and DY, (by Const.), but DO is less than DY, x shall be less than b, (by Ax. 3, B 5), and since z and x are equi-submultiples of C and DO, (by Const,), z will be to x as C to DO, (by Prop. 28, B. 5); but C is to Do as A to B, (by Hypoth.), and therefore z is to x as A to B, (by Prop. 18, B. 5); but a and b are equi-submultiples of A and B, and therefore a is to b as A to B, (by Prop. 28, B. 5), therefore z is to x as a to b, (by Prop. 18, B. 5), but x is less than b, and therefore z is less than a (by Prop. 23, B. 5), therefore there is given a submultiple of z, which is oftener contained in C, than an equi-submultlple of a is contained in the same C, (by Prop. 8, B. 5); but z and x are equi-submultiples of C and DO, (by Const.), and therefore as often as any submultiple of z is contained in C, so often is an equi-submultiple of x contained in DO, (by Prop. 7, B. 5); therefore there is a submultiple of x, which is oftener contained in DO, than an equi-submultiple of a is contained in C; but x repeated as often as a is contained in C, makes up DO, (by Const.); therefore there is not a submultiple of x which is oftener contained in DO, than an equi-submultiple of a is contained in C, (by Cor. 2, Prop. 7, B. 5), but it is also given, which is absurd.

Therefore there is not any submultiple of A contained in C, oftener than an equi-submultiple of B is contained in DO; and it can be similarly demonstrated that there is no submultiple of B oftener contained in DO, than an equi-submultiple of A is contained in B; therefore A is to C as B to DO, (by Def. 5, B. 5).

If there be any number of magnitudes, and an equal number of others, which if taken two by two are in the same ratio, and if their proportion be ordinate; they shall be ex æquali in the same ratio.

First, let there be three magnitudes, A, B, and CO, and three others D, E, and F; if A be to B as D to E, and B to CO as E to F, A shall be to CO as D to F. For if there be taken any equi-submultiples, a and d, of A and D, as often as a is contained in CO, so often is d contained in F.

For if not, if it be possible, let d be contained oftener in F than a in Co; and as often as d is contained in F, so often repeat a, and hence make up CY; CY shall be greater than CO, (by Prop. 5, B. 5); a is therefore a submultiple of CY, let x be the equisubmultiple of CO, as also z of F.

Therefore since x and a are equi-submultiples of CO and CY, (by Const.), but CO is less than CY, x will be less than a, (by Ax. 3, B. 5), and since B is to CO as E to F, (by Hypoth.), CO shall be to B as F to E; but x and z are equi-submultiples of CO and F, (by Const.), and therefore x is to B as z to E, (by Prop. 26, B. 5); therefore as often as any submultiple of x is contained in B, so often is an equi-submultiple of z contained in E; but x is less than a, and therefore there is given a submultiple of x, which is oftener contained in B, than an equi-submultiple of a is contained in the same B, (by Prop. 8, B. 5); and therefore oftener than an equi-submultiple of d is contained in E, (by Hypoth. and Def. 5, B. 5); therefore there

is a submultiple of z given, which is oftener contained in E, than an equi-submultiple of d is contained in E, therefore z is less than d, (by Prop, 9, B. 5); but z repeated as often as d is contained in F, makes up F, (by Const.); and therefore z is either greater than d or equal to it, (by Cor. 3, Prop. 7, B. 5), but it was shown to be less than it, which is absurd.

Therefore there is not any submultiple of D contained oftener in F, than an equi-submultiple of A is contained in CO; and it can be similarly demonstrated that there is no submultiple of A contained often in CO, than an equi-submultiple of D is contained in F; therefore A is to C as D to F, (by Def. 5, B. 5).

Now let there be four magnitudes, A, B, C, G, and four others, D, E, F, H; and let A be to B, as D to E, and B to C as E to F, and C to G as F to A, A shall be also to G as D to H.

For since there are three magnitudes, A, B, C, and three others, D, E, F, which if taken two by two, are in the same ratio, A will be to Cas D to F, (by preceding part), but C is to Gas F to H, (by Hypoth.), and therefore A shall be to G as D to H (by preceding part); and similarly, whatever be the number of magnitudes given.

If there be three magnitudes (A, B, C), and three others (D, E, F), which if taken two by two are proportional, but perturbate (A to B as E to F, and B to C as D to E), and if in the first series, the first magnitude be greater than the third, the first in the next series shall be also greater than the third.

For since D is to E as B to C (by Hypoth.), E will be to D as C to B (by Prop. 20, B. 5), but A is greater than C (by Hypoth.), therefore A has a greater ratio to B, than C has to the same B (by Prop. 16, B. 5), and therefore a greater ratio than E has to D (by Cor. Prop. 19, B. 5): but A is to