Here the second term is 14 times the first; therefore the fourth must be 14 times the third.

Ex. 2. If a person travel 1800 miles in 7 days of 16 hours each, in how many days of 12 hours each will he travel the same distance?

As the fourth term is to be days, the third must be 7 days; but if it require 7 days of 16 hours each to travel the given, or any other distance, it will require a greater number of days of only 12 hours each to accomplish the same distance; therefore the quantities must be placed in the following manner :

9 days and of a day of 12 hours each; therefore he would travel the distance in 9 days 4 hours. The distance, 1800 miles, has not been employed, being a superfluous quantity.

Ex. 3. If the provisions of a garrison will serve 2500 men for 73 days, how long will they last if the garrison be reinforced by 750 men ?

Here the fourth term is evidently less than 73 days, because the garrison is increased from 2500 to 3250 men; hence, 3250 men is the first term, 2500 men is the second, and 73 days is the third term of the proportion.

3250 men : 2500 men :: 73 days

103. Questions frequently arise in which five quantities are given to find a sixth, or seven quantities to find an eighth, and so on. In such cases it becomes necessary to repeat the process already adverted to in the Rule of Three, or to combine two or more proportions so as to reduce them all to a single proportion, and then if any three of the terms of the reduced proportion be known, the fourth can be found as before.

Ex. 1. If the expenses of 7 persons for 3 months be 70 guineas, what will be the expenses of 10 persons for 12 months at the same rate?

Here we must find what the expenses of 10 persons for 3 months would be, and this is done by the Rule of Three in the following

manner:

7 persons 10 persons :: 70 guineas: 100 guineas .... (A). Now, since 10 persons expend 100 guineas in 3 months, we must next inquire how much they will expend in 12 months; and, by the same rule, we get

3 months 12 months: 100 guineas: 400 guineas (B). Instead of repeating the operation in the Rule of Three, the terms of

the proportion (A) may be multiplied by the corresponding terms of the proportion (B), and the products will still be proportional (96). Hence 7 × 3: 10 x 12 :: 70 x 100 100 × 400;

but as the ratio of the third term to the fourth will not be altered by dividing each of its terms by 100, it is obvious that the operation for finding the fourth term (100 guineas) in (A) is superfluous, and may be dispensed with entirely. The statement of the terms and the operation will then be as follows:

7 persons: 10 persons :: 70 guineas;

3 months 12 months,

:

10 x 10 x 4 = 400 guineas, the expense re

Ex. 2. If the carriage of 30 tons, through 36 miles, cost £12. 10s., what weight ought to be carried 48 miles for £6. 13s. 4d.

Here weight is required, and therefore 30 tons will stand in the third term; then it is obvious that the weight which will be carried through 36 miles for £6. 13s. 4d. will be less than the weight carried through the same distance for £12. 10s. On this account £12. 10s., or 3000 pence, will be the first term, and 67. 13s. 4d., or 1600 pence, the second term of the first proportion. Again, whatever be the weight carried through 36 miles, for 3000d. : 1600d. £6. 13s. 4d., a less weight 48 miles : 36 miles, will be carried through 48 miles for the same sum; hence, on this account, 48 miles is to be the first term, and 36 miles the second term of the second proportion. Then multiply

30 x 1600 × 36

:: 30 tons

16 x 36

=

48

hence

3000 X 48

16 x 3

=

= 4 × 3 =

4

12 tons.

the corresponding terms of these two ratios, the reduced proportion is 3000 x 48: 1600 × 36 :: 30 tons: 12 tons.

104. If one or more of the terms be fractions or decimals, the result will be obtained in the same manner, provided the principles of the multiplication and division of fractions or decimals be carefully observed.

INVOLUTION.

nth

105. By multiplying a number by itself one, two, three, or (n−1) times successively, we obtain the second, third, fourth, or that number; hence a power of a number is the successive multiplications by itself. Thus 3 × 3 second power of 3; and 5 x 5 x 5 = 125, the of 5. In a similar manner the square, the cube, and generally any power

powers of number arising from 9, is the square or cube, or third power

=

2 2 4 3 3

of a fraction or a decimal is found by multiplication. So X

=

2 the square of and 025 × 025 000625, the square of 025, and so on. 3'

These operations are denoted by means of Indices, or small figures placed on the right of the numbers a little above the line; thus, 22 = 2 × 2 = 4, 33 3 x 3 x 3 = 27, and 25 32, where the index

or exponent denotes the number of factors employed.

This method of notation furnishes some important conclusions; for, since 52 = 5 × 5 and 5* = 5 × 5 × 5 × 5; therefore we have,

5' x 5o (5 x 5 x 5 x 5) X (5 × 5) = 5° = 5*+2;
5* 5% =

consequently powers of the same number are multiplied by adding their indices and divided by subtractiny the index of the divisor from that of the dividend.

Thus, since 11 5"=5*X 5' x 53.

=

5' X 5' x 5' ÷ 5 = 625 × 625 X 125

The 12th power of 5 is 5' x 5' X 5'

is 5a × 53 × 53 × 53

=

=

=

the cube of 5 or (5*)3, or it the fourth power of 5 or (53); hence (53)* (5)3 = 512; and, therefore, the power of a power of a number is expressed by multiplying the index or exponent by the degree of the power. Hence, 9° 91X2X3 9 × 92 × 93 9 x 81 x 729 531441, and the same result will be obtained if either of the following indicated operations be performed :

=

=

=

=

=

=

=

25 =

The operation of evolution is in

106. A root of a number is such a number as, being multiplied by itself one, two, three, or (n-1) times, produces the given number; and the operation by which the root is obtained is termed Evolution. Thus the second or square root of 25 is 5, since 5 × 5 5o; the third, or cube root of 27 is 3, for 33 3 x 3 x 3 27, and the square root of 0625 is 25, since 25 × 25 = ・0625. Also the cube root of 8 2 2 2 2 8 is because X X 27 3' 3 3 3 27 dicated by the sign, accompanied with a small figure in the opening. The sign is called the radical sign, and the figure in its opening indicates the particular root to be determined. In the case of the square root the figure 2 in the radical sign is always omitted; hence the square root of 25 is expressed by 25, the cube, or third root of 27, by 3/27, and so on. Again, by involution (82) 3 = 82X3 86; therefore, conversely, the cube, or third root of 8° is 82, where the index 2 is found by dividing 6, the index of the power, by 3 the index of the root. generally the n root of a power of a number is that power of the number whose index is the nth part of the index of the proposed power. Thus the square root of 2° is 23; the cube root of 26 is 2 and the nth root of 2" is 2.

=

Hence

Suppose, now, we take a number with a fractional index, as 8, and treat the fraction exactly as it were an integer; then since 8a × 83 83+3 =3 8°, we have 8 × 8 – 81+1 must necessarily produce 8; hence / 8 = will represent the second or square root.

= S1

=

83 = 83++++ 83, and the

Hence

8, we have in like manner

8; but 38 × 38 × 38

= 8 hence 38 = ; index characterizes the third or cube root; and so on.

(11)* = 11*×+ = (11*)* or (11)*; hence

; and also

11 = √11 =√√11; (11)* = 111×+×+ = 111×3 = 11°×+ = 1331* = √ 1331.

EXTRACTION OF THE SQUARE ROOT.

=

107. The square root of a number is that number which multiplied by itself produces the proposed number. There are many numbers whose square roots cannot be determined exactly, as 5, 7, 10, etc., but they may always be found to any degree of accuracy by means of decimals. And since 102 100, 100 = 10000, etc., it follows that the squares of all numbers between 1 and 10 must consist of 1 or 2 figures; the squares of all numbers between 10 and 100 must consist of more than 2, but not more than 4 figures; and the squares of all numbers between 100 and 1000 must consist of more than 4, but not more than 6 figures, etc.; hence, if a number be proposed to find its square root, we must place a point over every second figure, beginning at the unit's place; thus dividing the number into periods of two figures each, excepting the last period, which may consist of either one or two figures as the case may be. There will be just as many figures in the root as there are periods. With respect to a number composed of integers and decimals, the points will necessarily fall above the second, fourth, sixth, etc. decimals, counting from the unit's place, which we must do in all cases, and when there is no unit, as in the case of decimals, its place must be supplied with a cipher, and a point placed over it; hence the number of decimals must be one of the even numbers, 2, 4, 6, 8, etc. A cipher must be added to the right of an odd number of decimals to complete the period.

=

108. Let the several parts of a number be denoted by a, b, c; then (a+b+c) is the same as a + b + c repeated a times, b times, and c times. But a+b+c repeated a times is a2 + ab + ac, b times = ab+b2 + bc, and c times = ac + bc + c2; hence by adding all together we get the square of a+b+c expressed in either of the following forms :

(a+b+c)2=a2+(2 a+b)b,
+(2a+2b+c) c

or (a+b+c)2=c2+2c (a+b)

+b2+2ab.

+ a2

109. From the latter of these forms the rule for squaring will be, square each part, and multiply all that precede by twice that part; and a reverse rule for extracting the square root immediately presents itself. Let n denote the given number, and take a number a whose square does not exceed n. Find the remainder; take a second number

b, such that the remainder will bear the subtraction of the square of b and twice b multiplied by the preceding part a. If there be a remainder, take a third number c and find whether the second remainder will allow of the subtraction of the square of c and twice e multiplied by a+b. Proceed in this manner till the process terminate, or until the root be obtained to the nearest unit. The first form, however, affords an easy process for forming the numbers to be subtracted. For we have only to double the sum of all the parts which have been obtained, add the new part, and multiply the sum by the new part. Thus after a is subtracted; (2a + b) × b is the next subtrahend; {2 (a + b) + c} x c is the third, and so on.

root of 273529. Begin

273529 (500+20 + 3 250000

23529
20400

3129

3129

or 523

Ex. Let it be required to extract the square at the unit's place and point off the periods. as already directed; then since there are 3 periods, the first figure of the root will be in the place of hundreds. This first figure will be 5, because 500 is less and 6002 is greater than 270000. Subtracting 250000 from the number leaves the remainder 23529. Let 2 tens or 20 be the next part; then by the processes given above, the number to be subtracted will be 20o +2×20×500, or (2× 500+20) × 20, viz., 20400. Lastly, take 3 as the next part of the root; then, as before, 32+2 × 3 × 520, or (2 × 520 + 3) × 3, gives 3129, which, being exactly equal to the second remainder, shows that the root is = 523. It will be covenient to arrange the operation of forming the numbers to be subtracted on the left of the successive remainders, as in the following example:

Ex. Find the square root of 293764.

In the first of these, the numbers are written at length, but in the second the ciphers on the right are omitted, and each period is annexed to the remainder as it is wanted. The parts of the root a, b, c may be any whatever; but when they are confined to the nearest number of hundreds, tens, and units, the parts 2a, and 2a +26 which form so large a portion of the factors 2a+b, 2a + 2b + c soon become available for the determination of the succeeding parts. Thus 1000 is contained in 43764 more than 40 but less than 50 times. In this way the successive figures of the root can be written in the places which they occupy in the decimal scale, as in the second mode of operation.

110. When the proposed number has not an exact square root, it may be obtained to any degree of accuracy by means of decimals, and