Special case.

36. Cor. 1. The ratio of two whole numbers is not altered by multiplying both of them by the same number.

Multiples of one magnitude.

38. THEOREM 10. The ratio of the mth multiple of any magnitude to the nth multiple of the same magnitude is equal to the ratio of the number m to the number n.

Take p times each antecedent, and q times each consequent, and compare the order of size of the two pairs of multiples

and these two multiples of A are in the same order of size

as the pair of numbers

pm, qn,

which is the second pair of multiples above.

[9 (11)

Hence the two pairs of multiples above are in the same order of size whatever the whole numbers p and q are. Therefore, by the definition of equal ratios,

MA: NA=m: n.

Another statement of 38.

39. Cor. If two magnitudes have a

common measure

which is contained m times in the first magnitude and n times in the second, then the ratio of the two magnitudes is equal to the ratio of the number m to the number n.

Equivalent multiples.

40. THEOREM 11. If the ratio of one magnitude to another is equal to the ratio of the number m to the number n, then the nth multiple of the first magnitude is equivalent to the mth multiple of the second.

Take the nth multiple of each antecedent and the mth multiple of each consequent; then, from the hypothesis, the pairs of multiples

are in the same order of size; but the members of the latter pair are equivalent; therefore the members of the former pair are equivalent.

41. Cor. I. If the ratio of two magnitudes is greater than the ratio of two whole numbers m and n, then the nth multiple of the first magnitude is greater than the mth multiple of the second.

[Show that the first pair of multiples above are then in descending order (16).]

Combined statement.

42. Cor. 2. According as A: B >=< m : n,

PROPORTION

PROPERTIES OF A PROPORTION

44. Definition. A proportion is a statement of the equality of two ratios, as A: B = X : Y.

These four magnitudes are said to form a proportion, of which A and Y are the extremes, and B and X the means; and Y is called the fourth proportional to the three terms A, B, and X. The proportion is sometimes read thus: A is to B as X is to Y.

The next three theorems are concerned with the establishment of certain general "rules of inference," by which, from a given proportion, certain other proportions can be at once derived. They are the Rules of Equi-multiplication, Alternation, and Composition.

Equi-multiples of homologous terms.

45. THEOREM 12. If two ratios are equal, and if any like multiples of the antecedents are taken, and also any like multiples of the consequents, then the multiple of the first antecedent is to the multiple of the first consequent as the multiple of the second antecedent is to the multiple of the second consequent. Given

to prove

A: B = X: Y;

MA:nB = mx: nY.

To compare the latter two ratios, take the pth multiple of each antecedent, and the qth multiple of each consequent, and compare the order of size of the two pairs of resulting multiples

According to 9 (9), these may be written in the form

Now these two pairs of multiples are in the same order of size, because the ratios B and X: Y are equal.

Therefore the former pairs of multiples are in the same order of size, whatever whole numbers Ρ and q may be.

NOTE. This corollary may be stated in words as follows:

If four magnitudes form a proportion, and if the first is any multiple, or part, or multiple of a part, of the second, then the third is the like multiple, or part, or multiple of a part, of the fourth.

Rule of alternation.

47. THEOREM 13. If four magnitudes of the same kind form a proportion, then the first is to the third as the second is to the fourth.

Let A, B, C, D be four magnitudes of the same kind such that

To prove

A: B = C: D.

A: CB: D.

Since the ratio of two magnitudes equals the ratio of their like multiples,

hence

MA: MB = nC : nD.

[35

Therefore, by comparison of homologous terms in equal ratios, the two pairs

and

mA, по

mB, nD

are in the same order of size.

[33

Now m and n are any whole numbers; hence, by definition

Rule of composition.

48. THEOREM 14. If four magnitudes form a proportion, then the sum of the first and second is to the second as the sum of the third and fourth is to the fourth.

Given

to prove

A:BX: Y;

A+ BB = X + Y : Y.

In order to compare the latter two ratios take any like multiples of the antecedents, and any like multiples of the consequents; and then compare the order of size in the two resulting pairs

The order of the first pair of multiples is not altered by subtracting mB from each; and the order of the second pair is not altered by subtracting my from each.

Therefore the above pairs of multiples are in the same order, respectively, as the pairs

Now, from the hypothesis, these are in the same order of size; hence the above pairs are in the same order of size. Next, let n be not greater than m.

Then the pairs of multiples in question are evidently both in descending order.

are always in the same order of size whatever m and n are.

The complete statement and proof are left to the student.