PROPOSITION VIII.—If there be a proportion, consisting of three or more equal ratios, then either antecedent will be to its consequent, as the sum of all the antecedents is to the sum of all the consequents.

Suppose

abcde:f=g: h=, etc.

Then by comparing the ratio, a: b, first with itself, and afterward with each of the following ratios in succession, we obtain ab = ba,

ad = bc,

af = be,

ah = bg, etc.;

whence, a (b+d+f+h+etc.) =b(a+c+e+g+etc.),

or,

a:b=a+c+e+g+etc.: b+d+f+h+etc.

PROPOSITION IX.-If four quantities are in proportion, the terms of either couplet may be multiplied or divided by any number, and the results will be proportional.

And since the value of a fraction is not changed by multiplying or dividing both of its terms by the same number,

in which n may be either integral or fractional. If n be integral, we have, from (1) and (2),

in which the terms of the given couplets are multiplied. But

PROPOSITION X.-If four quantities are in proportion, either the antecedents or the consequents may be multiplied or divided by any number, and the results in every case will be proportional.

in which n may be either integral or fractional. If n be integral, we have from (2) and (3),

in which the given antecedents and consequents are multiplied.

in which the given antecedents and consequents are divided.

PROPOSITION XI.-If four quantities which are in proportion, be multiplied or divided, term by term, by four other quantities also in proportion, the products, or quotients, taken in order, will be proportional.

multiplying (3) by (4), (ax) (dn) = (by) (cm)

PROPOSITION XII.—If four quantities are in proportion, like powers or roots of the same quantities will be in proportion.

Raising (1) to the nth power, also taking the nth root of the

PROPOSITION XIII.—If three quantities are in continued proportion, the product of the extremes is equal to the square of the

SCHOLIUM. Taking the square root of the last equation, we

b = √ac;

have hence, The mean proportional between two quantities is equal to the square root of their product.

PROPOSITION XIV.-If three quantities are in continued proportion, the first is to the third, as the square of the first is to the square of the second; that is, in the duplicate ratio of the first and second.

PROPOSITION XV.-If four quantities are in continued proportion, the first is to the fourth, as the cube of the first is to the cube of the second; that is, in the triplicate ratio of the first and second.

To show some of the applications of the preceding principles, we give the following problems :

1. Find two numbers, the greater of which shall be to the less as their sum to 42, and as their difference to 6.

Let x = the greater, and y = the less.

√x: y = x + y: 42

x: y=xy: 6

By the conditions,

(1),

(2).

(7) ;

(8);

From (1) and (6), Prop. V, 4:3=x+y: 42

2. Divide the number 14 into two such parts that the quotient of the greater divided by the less, shall be to the quotient of the less divided by the greater, as 16 to 9.

Multiplying terms, Prop. IX, x2 : (14 — x)2 = 16 : 9,

3. There are three numbers in continued proportion; their sum is 52, and the sum of the extremes is to the mean as 10 to 3. Required the numbers.

Three numbers in continued proportion may be represented by x, xy, xy; for we observe that the product of the extremes will then be equal to the square of the mean. Hence,

(1),

x + xy + xy2 = 52
xy2 + x:xy = 10:3 . (2).
y2+1: y = 10:3 ... (3),
y2+1:2y=10:6..

y2 + 2y + 1: y2 — 2y + 1 = 16:4;

by the conditions,

From (2),

or,

by Prop. VII,

(4);

4. The product of two numbers is 112; and the difference of their cubes is to the cube of their difference as 31 to 3. What

are the numbers ?

xy=112

(1),

—

x3- y3: (xy)331: 3.. (2). x2+xy+y2: x2-2xy + y2 = 31:3.. (3); 3xy: (xy)2 = 28: 3.. (4);

336 (xy)2 = 28: 3.. (5);

By the conditions,

From (2), Prop. IX,

by Prop. VI,

by substitution,

whence,

or,

From (1) and (7), we obtain