ON RATIOS.

1. RATIO is the relation which one quantity bears to another of the same kind, with respect to magnitude; and the comparison is made by considering how often the one is contained in the other, or how often the one contains the other.

Thus the ratio of A to B is expressed by and the ra

B

tio of B to A by the former of these quantities, or the numerator, is called the antecedent; and the latter, or the denominator, is called the consequent of the ratio. 2. When the antecedent is equal to the consequent, viz. if=1, it is called a ratio of equality if be

B

B

greater than 1, we call it a ratio of greater inequality; and if be less than 1, it is called a ratio of less in

B

equality.

3. The antecedent and consequent are called the terms of the ratio, and the quotient of the two terms is called the index, or exponent of the ratio.

A

Thus, ifm, then m is called the exponent of the

B

ratio of A to B.

4. Compound ratio is made up of two or more ratios, by multiplying their terms and exponents together.

B

AC

=m, and —=n, then-x=mn== so that the

D

Α C
B D

BD'

ratio of AC to BD, is said to be compounded of the ratios of A to B, and c to D.

5. If a ratio be compounded of two equal ratios, it is called a duplicate ratio; if of three equal ratios, it is called a triplicate ratio, &c.

Thus, if

A

-- =m, -=m, then, =m2, hence the ratio

of AC to BD is duplicate of the ratio of A to B, or of C to D.

ratio of ACE to BDF, is triplicate to the ratio of a to B, C to D, or E to F.

6. If the terms of a ratio be prime to each other, no other quantities can be found in the same ratio, but what shall be some multiple thereof.

Let -=

m, and

B

*

=

m, where A and B are prime to

each other, I say c shall be a multiple of A, and D a mul

C
B D'

multiply by D, then c =

DA

B

tiple of B. Fornow it is evident, if в measures DA, it must measure D alone, because A is prime to B: consequently D is some multiple of B, therefore c is some multiple of A.

7. Cor. 1. The like multiples, or the like parts of the terms of a ratio, have the same ratio as the terms themselves. 8. Cor. 2. Numbers that are prime to each other, are the least of all numbers in the same ratio.

9. Having the terms of a ratio given in large numbers that are prime to each other, to find a ratio, nearly equivalent, whose terms are expressed by smaller numbers.

This is performed by reducing the terms of the given ratio into a series, of what are called continued fractions.

b

Thus, let the given ratio be expressed by ; and let a be

1

contained in b, c times, with a remainder a)b(c

d; again let d be contained in a, e times, with a remainder ƒ, and so on, we shall

d)a(e

One number is said to be a multiple of another, when the former contains the latter some exact number of times. Thus, m R is a mul

Hence we deduce the following general rule.

1. Divide the greater term by the less, and that di

visor by the remainder, &c. as in Prop. 1, page 65, Vulgar Fractions. Then, if the antecedent be greater than the consequent, the first quotient divided by 1, gives the first ratio; if less, an unit divided by the first quotient, will express the first ratio.

2. Multiply the terms of the first ratio by the second quotient, and add an unit to the numerator, or denominator, according as the antecedent of the original terms is greater or less than its consequent, and you will have the second ratio.

3. Then, in general, multiply the terms of the ratio last found by the next succeeding quotient, and to the two products add the corresponding terms of the preceding ratio, and you will have the next succeeding ratio, &c.

Example 1. Let it be required to find a series of ratios in less numbers, constantly approaching to the ratio of 314159 to 100000, which is nearly the ratio of the circumference of a circle to its diameter.

the second ratio, being the approximation of Archimedes,

Example 2. Let it be required to find a series of ratios in less numbers, constantly approaching to the

ratio of 7853981633 to 10000000000, which is nearly the ratio of the area of a circle to the square of its dia

ON PROPORTION,

10. Proportion is the equality of ratios.

and =n; then, if m be equal to n,

the ratios are equal; that is, A has the same ratio to B, which c has to D, and the four quantities are said to be

A C

proportional; viz. A : B :: C: D, or B

If m be greater than n, then A has to в a greater ratio than c has to D, and the four quantities are not proportional.

If m be less than n, then A has to B a less ratio than c has to D, and the four quantities are not proportional.