Proposition 13.

Theorem.-If two proportions have in each ratio an antecedent of one the same as a consequent of the other, the other terms are in proportion, antecedent remaining antecedent, and consequent, consequent.

C=mD (IV. 6), and if B=nE, F-nC; therefore F=mnD

Corollary. From this we see that the ratio of A to E is the product of the ratios of A to B and of B to E; that is, if

Theorem.-Equal quantities have the same ratio to the same quantity, and quantities which have the same ratio to the same quantity, are equal to each other.

Let A and B be equal magnitudes, and C another.

let Am C, then (IV. 6) b=m C. Hence

Proposition 15.

Theorem.—If two proportions have one ratio in each the sume, the remaining terms are in proportion.

Theorem.—If any number of quantities be in proportion, any antecedent is to its consequent, as the sum of all the antecedents is to the sum of all the consequents.

If

A:

B:: C: D:: E. F, etc.,

then A B A+ C+E, etc. : B+D+F, etc.

Let A = mB, then (IV. 6) C=mD and E=mF, etc.
Adding these, we have

A+ C+E, etc. -m(B+D+F), etc.,

therefore A : B :: A+ C+E+etc. : B+D+F+etc.

BOOK V.

SIMILAR POLYGONS.-MEASUREMENT OF

POLYGONS.

DEFINITIONS.

1. Similar rectilineal figures are those which have their several angles equal, each to each, and the sides about the equal angles proportional.

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

3. The altitude of a triangle is the straight line drawn from its vertex perpendicular to the base,

or the base produced.

As any side of a triangle may be considered the base, a triangle may have three altitudes. The altitude of a parallelogram is the perpen

dicular distance between either pair of parallel sides.

4. The homologous sides of similar rectilineal figures are

A

D

those which are adjacent the equal angles; in triangles they are those which are opposite the equal angles. Thus, if AD, B= E, and C= F, AB and DE are homologous sides, as also AC and DF, and BC and EF. The corresponding parts of two figures are called homologous whether they be lines or angles.

B

Proposition 1.

CE

Theorem.-Rectangles of equal altitude are proportional to

their bases.

There are two cases: 1. Where the bases are commensurable. 2. Where the bases are incommensurable.*

1. Let AB and CD be two rectangles having equal altitudes, AE and CF;

*Quantities are commensurable when they exactly contain the same thus, two lines respectively 7 and 4 feet long are commensurable, but two lines respectively 7 and 4 feet long are incommensurable in feet.

unit;

Lay off the unit of measure, which we take less than CD, on EB; at least one point of division, as F, will fall between C and D; draw FG parallel to AE.

Then, according to Case 1,

But

AF AB:: EF: EB.

AB AC EB: ED;

(IV. 13), AF: AC:: EF: ED.

But EF is less than ED, therefore AF is less than AC, which is impossible. Therefore, no other line but EC can be a fourth proportional to AB, AC, and EB.

AB AC:: EB : EC.

Corollary 1.-Parallelograms of equal altitude are proportional to their bases.

For any parallelogram is equivalent to a rectangle having the same base and altitude (I. 33).

Corollary 2.-Triangles of equal altitude are proportional to their bases.

For a triangle is half a parallelogram of the same base and altitude (I. 35, Cor.).