mer quantities have to each other, or that the four quantities are proportionals. In like manner, by observing one quantity to be triple, quadruple, or any multiple of another, we acquire ideas of other ratios; and thus we obtain ideas of all ratios where the greater quantity is a multiple of the less.

9. The comparison between two quantities is made by considering how often one quantity contains the other, or how of ten one quantity is contained in the other. Thus, in comparing 6 with 3, we observe that 6 has a certain magnitude with respect to 3, for 6 contains 3 twice; and in comparing 6 with 2, we observe that 6 has a different relative magnitude, for 6 contains 2 three times.

10. The quantities must be of the same kind, else they cannot be compared together, and therefore no judgment of their equality or inequality can be formed. Thus, 2 hours and 3 yards cannot be compared together, because they are quanti ties of a different nature.

A

11. The ratio of A to B is expressed by, or by two points

placed between them, as A: B. The former quantity A is called the antecedent of the ratio, and the latter B the consequent.

12. The antecedent and consequent are called the terms of the ratio, and the quotient of the two terms is called the mea

A sure, index, or exponent of the ratio. Thus, ifm, then m

is called the measure, &c. of the ratio of A to B.

13. One ratio is greater than another when its antecedent is a greater multiple, part, or parts of its consequent than the antecedent of the other ratio is of its consequent. Thus, the ratio 7: 4 is greater than the ratio 8: 5, because 4 and reduced to a common denominator are 35 and 32, and 3 is greater than 32 by 20

3

14. If the antecedents of any ratios be multiplied together, and also the consequents, a new ratio results, which is said to be compounded of the former ratios. Thus, if A: B and C : D be two ratios, then AC BD is said to be compounded of the two ratios A: B and C ; D,

15. If a ratio be compounded of two equal ratios, it is called a duplicate ratio; if of three equal ratios, it is called a tripli

that is, the ratio of AC to BD is duplicate of the ratio of A to

m3, that is, the ratio of ACE to BDF is triplicate of the ratio of

A to B, or of C to D, or of E to F.

16. The ratio of A to B is said to be one third of the ratio of

A3 to B3, and the ratio of Am to Bm is said to be an mth part of the ratio of A to B.

17. Let the first ratio be a : 1, then a : 1, a3 : 1, an 1, are twice, thrice,

n times the first ratio a : 1. n, the index of a, shows what multiple or part of the ratio a" : 1 the first ratio is. For this reason the indices 1, 2, are called measures of the ratios a1: 1, a2: 1, a3 : 1, an: 1.

18. Proportion is an equality of ratios.

Thus, let =m, and. ee

A
B

3,. n

n; then, if m=n, the two ra

tios are equal, that is, A has the same ratio to B which C has to D.

If m be greater than n, then A has to B 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.

A proportion is thus expressed, A: B::C: D, or A: B

= C:D, or

A

B

C
D'

and is read A is to B as C is to D.

19. Hence four quantities are proportional when the first contains the second as often as the third contains the fourth.

NOTE. It will not always happen that the first quantity contains the second exactly, or the third contains the fourth; but the criterion of proportion is complete if the two fractions which consist of the terms of the two ratios be equal; that is,

8

4

2. Hence 10:5::8:4. Again, 3:4::9:12, for

The following is a general definition of proportion, whether the terms of the two ratios be commensurable or incommensurable.

20. Four quantities are proportional when any multiple of the first contains the second as often as the same multiple of the third contains the fourth.

Let A, B, C, D be four quantities, and m any number, and

let ma

mA mc ; then A: B::C: D. Let A=2, B3, C4,

B D

21. The terms A and D are called extremes, and the terms B and C are called means.

22. In any proportion the two antecedents, or the two consequents, are sometimes called homologous terms; and each antecedent with its consequent are called analogous terms.

23. If the sum or difference of two numbers be multiplied by any number, the product is equal to the sum or difference of the separate products of the first two numbers multiplied by the third.

Let A, B, C, be three numbers; (B+C) A=AB+AC, and (B-C) A-AB-AC.

For the product AB is the same as each unit in B repeated A times, and the product AC is the same as each unit in C repeated A times; therefore the sum of the products AB+AC is equal to the units contained in B+C repeated A times, or AB +AC is equal to the sum of the numbers B and C multiplied by A.

Again, for the same reason, the difference between the products AB and AC, or AB-AC, must be equal to the difference of the units contained in B and C, or in B- C, repeated A times; that is, AB. -AC is equal to the difference of the numbers B and C multiplied by A.

24. Cor. 1. Hence a number which measures (or divides) any two numbers will measure their sum and difference. Thus, A measures AB and AC, and also AB+AC and AB-AC.

25. Cor. 2. Hence it is manifest that the first part of the proposition may be extended to more than two numbers, or that AB+AC+AD+ &c. = A (B+C+D+ &c.)

26. If four quantities be proportional, the product of the extremes is equal to the product of the means.

If A: B::C: D, then AD-BC.

27. If the first quantity be to the second as the second to the third, the product of the extremes will be equal to the square of the mean.

If AB:: B: C, then AC=Bo.

28. Cor. B✔ AC, that is, a geometrical mean proportional between two quantities is equal to the square root of their product.

29. If any three terms of a proportion be given, the fourth term may be found.

Let x be the unknown term, and let A: B:: C:x, then Ax

NOTE. This article contains the demonstration of the Rule of Three in Arithmetic.

30. Equimultiples of any quantities have the same ratio to one another which the quantities have; and like parts of any quantities have the same ratio to one another which the quantities have.

Let A and B be any quantities, and m any number; mA: mB:: A: B, and 1⁄2A : B::A : B.

31. If four quantities he proportional, according as the first quantity is greater than, equal to, or less than the second, the third quantity is greater than, equal to, or less than the fourth.

=

Let A B C : D, then AD : BC (26), .. if A be greater than B, then C is greater than D, if A=B, then C=D, and if A be less than B, then C is less than D.

32. If four quantities be proportional, according as the first quantity is greater than, equal to, or less than the third, the second quantity is greater than, equal to, or less than the fourth.

:

Let A B C : D, then AD BC (26), .. if A be greater than C, then B is greater than D, if A-C, then B=D, and if A be less than C, then B is less than D.

33. If the product of two quantities be equal to the product of two other quantities, these four quantities may be turned into a proportion by making the terms of one product the means, and the terms of the other product the extremes.

A C

Let AD-BC, then

that is, A: B:: C: D.

B

34. Quantities which have the same ratio to the same quantities are proportional.

Let AB::C: D, and C :D :: E: F, then

35. If four quantities be proportional, they are also proportional when taken inversely.

36. If four quantities be proportional, they are also proportional when taken alternately.