10. Composition and division. Suppose we have given the propor

in which A and B are any quantities of the same kind, and A' and

B' quantities of the same kind. Let unity be added to both mem

results which are briefly expressed by the theorem, if four quantities are in proportion, they are in proportion by composition; the term composition being employed to express the addition of antecedent and consequent in each ratio.

If we had subtracted unity from both members of [1], we should have found

results which are briefly expressed by the theorem, if four quantities are in proportion, they are in proportion by division; where the term division is employed to express the subtraction of consequent from antecedent in each ratio, this subtraction being conceived to divide. or to separate, the antecedent into parts.

The quotient of [2] divided by [3] is

that is, if four quantities are in proportion, they are in proportion by composition and division.

11. Definition. A continued proportion is a series of equal ratios, as

A: B = A': B' = A" : B" = A": B"" = etc.

12. Let r denote the common value of the ratio in the continued proportion of the preceding article; that is, let

▲ + A' + ▲′′ + A"" + etc. (B + B' + B" + B""' + etc.) r.

A"

= ·

whence

A+ A+ A+ A"" + etc.
B + B' + B" + B""' + etc.

A A'

=r=

= = etc.; B B'

that is, the sum of any number of the antecedents of a continued proportion is to the sum of the corresponding consequents as any antecedent is to its consequent.

If any antecedent and its corresponding consequent be taken with the negative sign, the theorem still holds, provided we read algebraic sum for sum.

In this theorem the quantities A, B, C, etc., must all be quantities of the same kind.

13. If we have any number of proportions, as

a: b = c: d,

a': b'c': d',

a": b" c": d", etc.;

then, writing them in the form,

α

b

a'

=

b'

c'

"

α

C"

d"

1-2 = = etc.,

and multiplying these equations together, we have

that is, if the corresponding terms of two or more proportions are mul tiplied together, the products are in proportion.

If the corresponding terms of the several proportions are equal, that is, if a = a' = a", b = b' = b", etc., then the multiplication of two or more proportions gives

that is, if four numbers are in proportion, like powers of these numberɛ are in proportion.

14. If A, B and Care like quantities of any kind, and if

If A, B and C were numbers, this would be proved, arithmetically, by simply omitting the common factor B in the multiplication of the two fractions; but when they are not numbers we cannot regard B as a factor, or multiplier, and therefore we should proceed more strictly as follows. By the nature of ratio we have

a result usually expressed as follows: the ratio of the first of three quantities to the third is compounded of the ratio of the first to the second and the ratio of the second to the third.

PROPORTIONAL LINES.

PROPOSITION I.-THEOREM.

15. A parallel to the base of a triangle divides the other two sides proportionally.

Let DE be a parallel to the base, BC, of the triangle ABC; then,

AB: AD= AC: AE.

1st. Suppose the lines AB, AD, to have a common measure which is contained, for example, 7 times in AB, and 4 times in AD; so that if AB is divided into 7 parts each equal to the common measure, AD will contain 4 of these parts. Then the ratio of AB to AD is 7: 4 (II. 43); that is

Through the several points of division of AB, draw parallels to the base; then AC will be divided into 7 equal parts (I. 125), of which Hence the ratio of AC to AE is 7: 4; that is,

AE will contain 4.

2d. If AB and AD are incommensurable, suppose one of them, as AD, to be divided into any number n of equal parts; then, AB will contain a certain number m of these parts plus a remainder less than one of these parts. The numerical expression of the ratio AB

m

1

will then be correct within (II. 48). Drawing parallels to

n

n

AD BC, through the several points of division of AB, the line AE will be divided into n equal parts, and the line AC will contain m such parts plus a remainder less than one of the parts. Therefore, the

Since, then, the two ratios always have the same approximate numerical expression, however small the parts into which AD is divided, these ratios must be absolutely equal (II. 49), and we have, as before,

or

AB: AD

AC: AE

16. Corollary I. By division (10), the proportion [1] gives

[1]

Also, if the parallel DE intersect the sides BA and CA produced through A, we find, as in the preceding demonstration,

17. Corollary II. By alternation (7), the preceding proportions give

AB: AC AD: AE,

=

which may both be expressed in one continued proportion,

This proportion is indeed the most general statement of the proposition (15), which may also be expressed as follows: if a straight line is drawn parallel to the base of a triangle, the corresponding segments on the two sides are in a constant ratio.