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 C as E to H (by part preceding), but C is to G as 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 Ĉ, 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 THE 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 any 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

N

take on the base DE, as often as possible, the parts, DI and IE, equal to AF, and draw BF, BK, and BI.

Therefore, since the right

F K с

D

E

lines AF, FK, KC, DI, and IE are equal (by Const.) but the triangles constructed upon them are of the same altitude, they shall be also equal (by Prop. 38, B. 1.); therefore A such a submultiple as AF is of AC, such is the triangle ABF of ABC, and as often as AF is contained in the base DE, so often is the triangle ABF contained in the triangle DBE, and it can be similarly shown that as often as any other submultiple of AC is contained in DE, so often is an equi-submultiple of ABC contained in DBE; therefore the triangle ABC is to the triangle DBE, as the base AC to the base DE (by Def. 5, B. 5).

A

F

PART 2.-Parallelograms, BA and CF, of the same altitude, are to one another as their bases, BC and CD. For draw BA and AD, and since the triangles BAC, CAD, are of the same altitude, they shall be to one another B as their bases BC and CD (by preceding part); but the parallelograms are double of them (by Prop. 34, B. 1), and therefore the parallelograms are to one another as their bases BC and CD (by Prop. 28, B. 5).

C

COR. 1.-Triangles or parallelograms which have equal altitudes, are to one another as their bases. For, placing the bases in directum, a right line joining the vertices will be parallel to the right line in which the bases are; for letting perpendiculars fall from the vertices on the base, these shall be parallel; and since they are also equal (by Hypoth.), the right lines joining them will be parallel (by Prop. 33, B. 1); and therefore, it can be similarly demonstrated as in the

Proposition, that the triangles or parallelograms are to one another as their bases.

COR. 2.-Triangles, ABC, DEF, and parallelograms, AB and DE, upon equal bases AC and DF, are to one another as their altitudes.

If the triangles be rectangular, it is evident. If not, draw BK and EG parallel to AC and DF, and through C and F draw CK and FG per

pendicular to AC and DF, and join AK and DG.

Therefore since the triangles ABC and AKC are upon the same base, and between the same parallels, ABC shall be equal to AKC (by Prop. 37, B. 1); and DGF is similarly equal to DEF; but if the right lines CK and FG be assumed for the bases of the triangles AKC and DGF, their altitudes AC and DF shall be equal (by Hypoth.), and therefore, they are to each other as their bases CK and FG (by preceding Cor.), therefore ABC and DEF, which are equal to them, are to one another as CK and FG, but these lines are equal to their altitudes (by Prop. 34, B. 1, Def. 3. B. 6), but the parallelograms are double of the triangles, and therefore are also to each other as CK and FG.

If a right line (DE) be drawn parallel to any side (AC) of a triangle (ABC), it shall cut the remaining sides, or the sides produced proportionally.

And if the right line (DE) cut the sides of the triangle, or the sides produced, proportionally; it shall be parallel to the remaining side (AC).

PART 1.-Let DE be be to DB as CE to EB. since the triangles EAD and ECD are upon the sam base ED, and between the same parallels ED and CA

parallel to AC, and AD will For draw AE and DC, and