THEOREM XLV.

If a straight line be divided into two equal parts, and also into two unequal parts, the rectangle contained by the two unequal parts together with the square of the line between the points of division, will be equivalent to the square on one half the line.

Let AB be a line bisected in C, and divided into two unequal parts in D.

=

We see by inspection that AD AC+ CD, and BD -AC-CD; therefore by (Th. 38), we have

AD × BD = AC2 — CD2.

By adding CD to each of these equals, we obtain
AD × BD+CD3 = AC2

Hence the theorem.

BOOK II.

PROPORTION.

DEFINITIONS AND EXPLANATIONS.

THE word Proportion, in its common meaning, denotes that general relation or symmetry existing between the different parts of an object which renders it agreeable to our taste, and conformable to our ideas of beauty or utility; but in a mathematical sense,

1. Proportion is the numerical relation which one quantity bears to another of the same kind.

As the magnitudes compared must be of the same kind, proportion in geometry can be only that of a line to a line, a surface to a surface, an angle to an angle, or a volume to a volume.

2. Ratio is a term by which the number which measures the proportion between two magnitudes is designated, and is the quotient obtained by dividing the one

B

by the other. Thus, the ratio of A to B is or A: B,

A'

in which A is called the antecedent, and B the consequent. If, therefore, the magnitude A be assumed as the unit or standard, this quotient is the numerical value of B expressed in terms of this unit.

It is to be remarked that this principle lies at the foundation of the method of representing quantities by numbers. For example, when we say that a body weighs twenty-five pounds, it is implied that the weight of this body has been compared, directly or indirectly, with that of the standard, one pound. And so of geometrical

magnitudes; when a line, a surface, or a volume is said to be fifteen linear, superficial, or cubical feet, it is understood that it has been referred to its particular unit, and found to contain it fifteen times; that is, fifteen is the ratio of the unit to the magnitude.

When two magnitudes are referred to the same unit, the ratio of the numbers expressing them will be the ratio of the magnitudes themselves.

Thus, if A and B have a common unit, a, which is contained in A, m times, and in B, n times, then A = ma

3. A Proportion is a formal statement of the equality of two ratios.

Thus, if we have the four magnitudes A, B, C and D,

B D

=

such that
A C'

this relation is expressed by the pro

portion A: B:: C: D, or A: BC: D, the first of which is read, A is to B as Cis to D; and the second, the ratio of A to B is equal to that of C to D.

4. The Terms of a proportion are the magnitudes, or more properly the representatives of the magnitudes compared.

5. The Extremes of a proport on are its first and fourth terms.

6. The Means of a proportion are its second and third terms.

7. A Couplet consists of the two terms of a ratio. The

first and second terms of a proportion are called the first couplet, and the third and fourth terms are called the second couplet.

8. The Antecedents of a proportion are its first and third terms.

9. The Consequents of a proportion are its second and fourth terms.

In expressing the equality of ratios in the form of a proportion, we may make the denominators the antecedents, and the numerators the consequents, or the reverse, without affecting the relation between the magnitudes. It is, however, a matter of some little importance to the beginner to adopt a uniform rule for writing the terms of the ratios in the proportion; and we shall always, unless otherwise stated, make the denominators of the ratios the antecedents, and the numerators the consequents.*

10. Equimultiples of magnitudes are the products arising from multiplying the magnitudes by the same number. Thus, the products, Am and Bm, are equimultiples of A and B.

11. A Mean Proportional between two magnitudes is a magnitude which will form with the two a proportion, when it is made a consequent in the first ratio, and an antecedent in the second. Thus, if we have three magnitudes A, B, and C, such that A : B :: B: C, B is a mean proportional between A and C.

12. Two magnitudes are reciprocally, or inversely proportional when, in undergoing changes in value, one is multiplied and the other is divided by the same number. Thus, if A and B be two magnitudes, so related that when

* For discussion of the two methods of expressing Ratio, see University Algebra.

13. A Proportion is taken inversely when the antecedents are made the consequents and the consequents the antecedents.

14. A Proportion is taken alternately, or by alternation, when the antecedents are made one couplet and the consequents the other.

15. Mutually Equiangular Polygons have the same number of angles, those of the one equal to those of the others, each to each, and the angles like placed.

16. Similar Polygons are such as are mutually equiangular, and have the sides about the equal angles, taken in the same order, proportional.

17. Homologous Angles in similar polygons are those which are equal and like placed; and

18. The Homologous Sides are those which are like disposed about the homologous angles.

THEOREM I.

If the first and second of four magnitudes are equal, and also the third and fourth, the four magnitudes may form a proportion.

Let A, B, C, and D represent four magnitudes, such that A = B and C = D; we are to prove that A : B :: C: D.

Now, by hypothesis, A is equal to B, and their ratio is therefore 1; and since, by hypothesis, C is equal to D, their ratio is also 1.

Hence, the ratio of A to B is equal to that of C to D; and, (by Def. 3),

A B C D.

Therefore, four magnitudes which are equal, two and two, constitute a proportion.