and by Art. 43, a: a b: c: с

49. If we have any number of sets of proportionals, and if the corresponding terms be multiplied together, the products are proportionals.

If a b c

d, and p q :: 1: 8,

and u : v :: x: Y,

50. If the same quantities occur in the antecedents of one set of proportionals and the consequents of another set, the resulting proportionals will be reduced.

If a b c d, and b: e :: df,

51. Quantities of the same kind assume different values under constant conditions, and when these different values are compared, the quantities are spoken of as variable, and the proportion of the different values may be expressed by two terms of a proportion instead of four.

Thus if a man travel with a constant velocity (for example 4 miles an hour,) the space travelled over in any one time is to the space travelled over in any other time as the first time is to the second time; and this may be expressed by saying that the space varies as the time, or is as the time.

52. One quantity is said to vary directly as another when the two quantities depend wholly upon each other, in such a manner that if the one be changed the other is changed in the same proportion.

If the altitude of a triangle be invariable, the area varies as the base. For if the base be increased or diminished in any proportion, the area is increased or diminished in the same proportion. (Euc. vi. 1.)

53. One quantity is said to vary inversely as another, when the former cannot be changed in any manner, but the reciprocal of the latter is changed in the same manner.

If the area of a triangle be given the base varies as the perpendicular altitude.

If A, a represent the altitudes, B, b the bases of two triangles, since a triangle is half the rectangle on the same base and of the same altitude, and the triangles are equal, AB = ab. (See Geometry.) Therefore

54. One quantity is said to vary as others jointly, if, when the former is changed in any manner, the product of the others is changed in the same proportion.

The area of a triangle varies as its altitude and base jointly.

Let A, B, a, b be the altitudes and bases of two triangles as before, and S, s the areas; then

S = AB, 8 = ab and S: s AB ab.

55. In the same manner A: a ::

A varies as S directly and B inversely.

56.

The symbol is often used for variation. Thus the above variations may be expressed

57. When the increase or decrease of one quantity depends upon the increase or decrease of two others, and it appears that if either of these latter be constant, the first varies as the other, when they both vary, the first varies as their product.

Thus, if V be the velocity of a body moving uniformly, T the time of motion, and S the space described; if T be constant SV; if V be constant ST; but if neither be constant S∞ TV.

Let s, v, t be any other velocity, space and time; and let X be the space described with the velocity v in the time T: then

S: X :: V: v, because T is the same in both, X: S :: T: t, because v is the same in both. Therefore (Art. 50.)

S: s

:: TV tv; that is, S∞ TV.

(12.) Of Arithmetical Progression.

58. Quantities are said to be in arithmetical progression, when they increase or decrease by a common difference.

Thus 1, 3, 5, 7, 9, &c., where the increase is by the difference 2;

a, a + b, a + 2b, a + 3b, &c. where the increase is by the difference b;

9a+7x, 8a + 6x, 7a + 5x, &c. where the decrease is by the difference a + x;

are in arithmetical progression.

59. To find any term of an arithmetical progression, multiply the difference by the number of the term minus one, and add the product to the first term,

if the progression be an increasing one, or subtract the product, if a decreasing one.

Thus the 10th term of 1, 3, 5, &c. is 1 + 9 × 2:

= 19.

The nth term of a, a + b, a + 2b, &c. is a + n − 1b. The 6th term of 9a +7x, 8a+ 6x, &c. is

60. To find the sum of an arithmetical progression, multiply the sum of the first and last terms by half the number of terms.

Thus the sum of 10 terms of 1, 3, 5, &c. is

or 20 x 10 = 28, or 20 × 5 = s.

Also n terms of a, a + b, a + 2b, &c.

(a + n − 1b) + (a + n − 2 b) + &c.

to a (n terms) = 8;

therefore (2a + n − 1 b) + (2a + n − 1b) + &c.