when one more than half the number of figures in the root have been obtained, the remaining figures may be found by dividing the last remainder by its corresponding divisor as in contracted division of decimals.

Ex. Find the square root of 10, and also of 17.108 each to 6 places of decimals.

111. The square root of a fraction is equal to the square root of the numerator divided by the square root of the denominator. If the terms of the fraction have not exact square roots, the fraction may be reduced to a decimal, and its root extracted, or if the denominator has no exact root, the terms of the fractions may be multiplied by such a number as will render the denominator a complete square.

Thus :

=

112. The cube root of a number is that number which multiplied twice by itself produces the proposed number. Since 10 1000, 100% = 1000000, etc., the cubes of all numbers between 1 and 10 must consist of 1, 2, or 3 figures; the cubes of all numbers between 10 and 100 must consist of more than 3 but not more than 6 figures, and so on; hence the number whose cube root is to be found is to be divided into periods of three figures each, beginning at the unit's place, and putting a point over it, and a point over every third figure towards the left in integers and the right in decimals. In decimals supply one or two ciphers, if necessary, to complete the period of 3 figures.

113. Let a and b represent the tens and units of the root; then (a+b) is the same as a + b repeated a times and 6 times. But a + b repeated a times is a2+ab, and repeated 6 times is ab+b2; hence a+b repeated a+b times, or (a + b)2 = a2 +2ab+b2. Again

VOL. I.

E

(a + b) is the

that is (a + b)3 + 3 a b2 + b3.

same as a2+2 ab + b2 repeated a times and b times; = (a3 + 2 a2 b + ab2) + (a2 b + 2 a b2 + b3) = a3 +3 aa b This may be put in either of the forms

a3 + (3 a2 + 3 ab + b2) b or a3 + 3 a b (a + b) +63.

From this result the following simple rule for finding the cube root of a number is deduced. Let n be the number, and take a number a whose cube does not exceed n. Find the remainder, take a second number b, such that the remainder may bear the subtraction of the cube of b, and the continued product of thrice a, the second number b, and the sum of a and b. If there be a remainder, consider a + b as the first number, and proceed as before.

3 a2

3a + b
26
За+36

3 ab+b2)

3 a +3 ab + b2.

62 ) 3a2+6 ab + 3 b2

The following mode of forming the successive numbers to be subtracted is the most convenient in practice. Write down 3 times the first number, and three times its square separately, the former one line lower than the other, and to the left of it as in the margin. To 3 a add the second number b, and multiply the sum by b, placing the product below 3 a2 and adding it thereto. The sum 3 a2 + 3 ab+b2 being multiplied by b, produces the entire number to be subtracted. As the part 3 a2 forms a large part of the factor 3 a + 3 ab +b2, it soon becomes available for the determination of the next figure, by using it as a trial divisor. To show how the process may be continued, change a into a + b, and b into c in the arrangement in the margin; then 3 a becomes 3 a +36 and 3 a becomes 3 a2 + 6ab+ 3b2; now to obtain these, we have only to add 2b to the one column, and b2 to the sum of the last two lines of the other column. Then to 3 a + 3b add the next number c; multiply the sum by c, placing the product below 3 a2+6 ab + 3 b2; then adding and multiplying the sum by c, the next entire number to be subtracted will be obtained. In this way the cube root may be extracted with the greatest facility, and when the root cannot be accurately obtained, it may be approximated to, and the work contracted as in the following example.

Ex. 1. Find the cube root of 46656. Here the number of tens is evidently 3, for 303 is less, and 403 greater than 46656, or 33 is ess and 43 greater than the second period 46. Writing 3 times 30, and 3 times the square of 30 on the left, we ask how many times is 2700 contained

90 2700

46656 (30+6
27000

=

36

6 576

19656

96 3276

19656

in the remainder 19656? The quotient is 7, which will be found a unit too much; taking 6 then it is added to 90, and the sum 96 is multiplied by 6. The product 576 is written below 2700 and added thereto; then the sum 3276 is multiplied by 6, and the product being equal to the remainder, the process terminates. In the next example we shall omit the ciphers, and place the figures as they arise in their proper places.

Ex. 2. Find the cube root of 21035 8 to ten places of decimals.

In this manner we have found the cube root to ten or eleven places of decimals with comparatively little trouble. When the contraction is commenced it is only necessary to cut off one figure from the right of the middle column, and two from the right of the left column; because in this way three figures or a period is struck off from each column, the period on the right-hand column not being annexed to the right of it.

114. The cube root of a fraction may be found by reducing it to a decimal, or by multiplying both numerator and denominator by such a number as will render the denominator a cube number.

115. Let N be the number whose root is to be extracted, and n the index of the root; then assume a root a whose nth power, a", is as near to the given number as convenient, and let R represent the true root of the number.

Then will

(n + 1) a" + (n − 1) N

Ex. 1. Find the cube root of 21035.8.

= root nearly.*

Here n = 3, and if we assume a = 28, which is rather too great, since the cube of 28 is 21952 = a3, we have

21952 + 21035.8 x 2

= 28

21952 x 2 + 21035.8

=

Ex. 2. Find the fifth root of 30.

27.6049, true to the last figure

Here n = 5, and if we assume a = 2, then a = 32, which is a little

too great; hence,

116. The principle of logarithms is essentially arithmetical, depending on the relation subsisting between the corresponding terms of an arithmetical and a geometrical progression, which is this, viz. :—If any number of terms are arranged in a geometrical progression commencing

* Let N = a + b, where b may be either additive or subtractive, and let a + x be the true root R; then by the binomial theorem,

N = a + b = (a + x)" = a" + na"-1 x +

n (n − 1)

1.2

Now if x be small, we may neglect all the terms in the denominator except the b

first; hence, for a first approximation, we get x =

nar-1'

Substitute this value

for x in the denominator above, and take two terms of it; then will

This may be put in the form of a proportion, and easily recollected; thus, (n + 1) a" + (n − 1) N : (n − 1) a2 + (n + 1) N:: a: R.

with 1, and over these terms are placed a corresponding series of terms of an arithmetical progression commencing with 0, it will be found that the sum of any two of the numbers in the upper line will constitute the number in that line which corresponds to the product of the two numbers in the lower line, and that the difference of any two of the upper line will be the number standing over that number in the lower line, which is equal to the quotient arising from dividing the greater number by the less. Thus let the

Arithmetical series be
Geometrical series

0, 1, 2, 3, 4, 5, 6, 7, etc.
1, 2, 4, 8, 16, 32, 64, 128, etc.

Here 8 x 16 = 128, and if the numbers 3 and 4 placed over 8 and 16 be added together, the sum is 7, which is the number in the upper line standing over the product 128. In like manner, if 64 be divided by 4, the quotient is 16, and if the number 2 placed over 4 be subtracted from 6, the number standing over 64, the difference 4 is the number standing over the quotient 16.

117. It is a further property of two such progressions, that if we double any one of the terms in the upper line, it will give the number standing over the square of the corresponding number in the lower line, and three times that number will give the term standing over the cube of the corresponding number in the lower line, and so on. The same property extends to the extraction of the square, or cube, or any other root. Thus the square root of 64 is 8, and the number standing over 64 is 6, and, dividing this by 2, gives 3, which is the number standing over 8, the square root of 64; or, if divided by 3, we have 2, the number standing over 4, the cube root of 64.

118. We have here given two of the most simple series for the sake of illustration, but with them we can only deal with the numbers belonging to these series, while in the more general form, viz.,

Arithmetical series 0, x, 2x, 3x, 4 x, 5x, 6x, 7x, etc.
Geometrical series 1, a*, a2, a3*, a13, a33, a**, a13, etc.

we can include every number, integral, decimal, or mixed of both, from O to any extent required in the upper series, and in the lower every number, integral, decimal, or both, from 1 to any extent. The upper line constitutes a series of numbers which are termed the logarithms of the corresponding numbers in the lower line, and we hence obtain cur first general idea and definition of a logarithm, viz. :— The logarithm of a number is that index of the power of a given radix or base which is equal to that number.

119. It appears, then, that logarithms are strictly of arithmetical origin, but it would be laborious, if possible, to prove by Arithmetic, unassisted by Algebra, that it is possible, by different powers of any given radix a, to express every intervening number and fraction between Ŏ and one hundred or a thousand millions, and to supply at the same time the means of computing them. For this reason we shall defer the more extended development of the principle and properties of logarithms to its proper place in the Algebra, and the rules for applying them, with the description of the Tables to the Introduction in the Volume of Logarithms.