Bas E to F (by Hypoth.), and therefore E has a greater ratio to F, than the same E has to D (by Prop. 19, B. 5), therefore Dis greater than F (by Prop. 17, B. 5).

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 there be taken any equi-submultiples (a, b, and d) of the first and second in the first series, and of the first in the last series (of A, B, and D); and also any equi-submultiples (c, e, and f): of the third in the first series, and of the second and third in the next series (of C, E, and F), these equi-submultiples of the quantities in the first series, and of these in the second series, shall be also perturbate proportionals (a to b as e to f, and b to c as d to e).

For since A is to B as E is to F (by Hypoth.), and a and b are taken equi-submultiples of A and B, and e and f equi-submultiples of E and F, a will be to bas e to f (by Cor. 2, Prop. 28, B. 5).

And since B is to C as D to E (by Hypoth.), and b and d are equi-submultiples of B and D, and also c and e equi-submultiples of C and E (by Hypoth.), b will be to c as d to e (by Cor. 2, Prop. 27, B. 5).

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 there be taken any equi-submultiples (a and d) of the first in each series (A and D), and any other equi-submultiples (c and f) of the last in each series (C and F), and if the part (a) of the first in the first series, be greater than the part (c) of the last in the same series, the part (d) of the first in the second series will be greater than the part (f) of the last.

For take b,

which is the same submultiple of B that a is of A, and also e which is the same submultiple of E that f is of F; there will be three magnitudes, a, b, c, and three others, d, e, f, if taken two by two are proportional, but perturbate (by Prop. 36, B. 5); and therefore if a be greater than e, d will be greater than f (by Prop. 35, 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, but perturbate; they will be also ex æquali in the same ratio.

First let there be three magnitudes, A, B, and CO, and three others, D, E, F; if A be to B as E to F, and B to CO as D to E, A will 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 also in F. For if not, but, 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 let CY be made up; CY shall be greater than CO (by Prop. 5, B. 5). Therefore a is a submultiple of CY; let x be an equi-submultiple of CO, and also z of F. Therefore since x and a are equi-submultiples of CO and CY, but CO is less than CY, x shall be less than a (by Ax. 3, B. 5); and since z is an equi-submultiple of F, and x of CO (by Const.), but as often as d is contained in F, so often is x in CO (by Const.), d is contained in F as often as z, a submultiple of F, is contained in the same F, therefore d is either less than z or equal to it (by Cor. 3, Prop. 7, B. 5). Therefore there are taken equi-submultiples, a and d, of A and D, and also other equi-submultiples, x and z of CO and F, and a is greater than x, but d is not greater than z, which is impossible (by Prop. 37, B. 5). Therefore no submultiple of D is contained oftener in

F, than an equi-submultiple of A is contained in Co; and it can be similarly demonstrated that no submultiple of A is contained oftener in CO, than an equi-submultiple of D is contained in F, therefore A is to CO as D to F.

Now let there be four magnitudes, A, B, C, G, and four others, D, E, F, H; and let A be to B as F to H; but B to C as E to F, and C to Gas D to E; A will be also to G, as D to H. For since there are three magnitudes, A, B, C, and three others, E, F, H, which, if taken two by two are proportional, but perturbate, A will be to Cas E to H (by part preceding), but C is to Gas D to E (by Hypoth.), therefore A will be to G, as D to H (by part preceding). And similarly, if there be any number of magnitudes given.

PROPOSITION XXXIX. THEOREM.

If there be three magnitudes proportional, (A to B as B to C), and three others proportional, (D to E as E to F); and there be, in the first series, the first to the last, (A to C), as the first to the last, (D to F), in the next series; the first will be also to the second, in the first series, (A to B), as the first to the second in the next series, (D to E).

For if not, but if it be possible, let one of them, A, have a less ratio to B, than D has to E; and since B is to C as A to B (by Hypoth.). and E to F as D to E (by Hypoth.), B also has a less ratio to C, than E has to F, (by Prop. 19, B. 5); therefore a submultiple of B can be taken, which is oftener contained in C, than an equi-submultiple of E is contained in F, (by Def. 7, B. 5): let them be b and e; and take equi-submultiples, a and d, of A and D, and since they are assumed equi-submultiples, a and b of A and B, a will be to b as A to B, (by Prop. 28, B. 5), and similarly d is to e as D to E, (by same), therefore a has a less ratio to b than d has to e, (by Hypoth.), and therefore there can be taken

a submultiple of a, which is contained oftener in b, than an equi-submultiple of d is contained in e, (by Def. 7, B. 5); let them be also assumed x and z; and since x is contained oftener in b than z in d, but b is contained oftener in C than d in F, x is also contained oftener in C than z in F, (by Cor. Prop. 10, B. 5), but x and z are equi-submultiples of a and d, and a and d are equi-submultiples of A and D, and therefore x and z are equi-submultiples of A and D, (by Prop. 7, B. 5); therefore as often as x is contained in C, so often also is z contained in F, (by Hypoth. and Def. 5, B. 5); but x was shown to be oftener contained in C, than z in F, which is absurd. Therefore A has not a less ratio to B, than D has to E; and it can be similarly demonstrated, that D has not a less ratio to E, than A has to B; therefore A is to B, as D to E.

END OF FIFTH BOOK.

SIXTH BOOK.

DEFINITIONS.

1. Similar rectilineal figures, are those which have the angles in each respectively equal, and the sides about the equal angles proportional.

2. A right line is said to be cut in extreme and mean ratio, when, as the whole is to the greater segment, so is the greater to the less.

3. The altitude of any figure, is a right line drawn from the vertex perpendicular to the base.

4. A parallelogram described upon a right line, is said to be applied to that line.

5. A parallelogram described upon a part of any right line, is said to be applied to that line deficient by a parallelogram; namely, by that parallelogram which is described upon the remaining part.

6. When a given right line is produced, the parallelogram described upon the whole is said to be applied to the given right line, exceeding by a parallelogram; namely, by that parallelogram which is described upon the produced part.

PROPOSITION I. THEOREM.

Triangles (ABC, DBE) and parallelograms, (BA and CF), which have the same altitude, are to one another as their bases.

PART 1. Let the base of the triangle ABC be divided into any number of equal parts, AF, FK, and KC, and