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

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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.

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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

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(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

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77

=

=

1.97436 nearly.

39

4 x 32 + 6 × 30 2 × 77

=

=

78

R = 2. 6 x 32 + 4 × 30 Take a 1.97; then a 29 670928, etc.; therefore by the formula 298.683712 R = 1.97 X = 1.9743504 = root very nearly. 298 025568

LOGARITHMS.

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 +≈ be the true root R; then by the binomial theorem,

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

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

b

nar-i

Substitute this value

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

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But since Na + b, we have b = Na", and this being written for b in the last equation, we get,

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

with 1, and over these terms are placed a corresponding series of terms of an arithmetical progression commencing with O, 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,
Geometrical series 1, a*,

2x, 3x, 4x, 5x, 6x, 7x, etc.
a2, a3*, a**, a3*, 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 Loga

rithms.

APPLICATION OF ARITHMETIC TO COMMERCIAL

CALCULATIONS.

PARTNERSHIP.

120. PARTNERSHIP is the method of dividing any quantity into any proposed number of parts, having a given ratio to one another. By it the gains or losses of partners in trade are adjusted, the effects of bankrupts are divided amongst creditors, and contributions are levied.

When two or more partners invest their money together, and gain or lose a certain sum, it is evident that the gain or loss ought not to be divided equally among them all, unless each partner contributed the same sum. Suppose that P contributes 3 times as much as Q, it is evident that P's share of the gain or loss ought to be 3 times as much as the share of Q; hence dividing the gain or loss into 4 equal parts, P must receive or pay 3 of these parts, and Q one of them.

Ex. 1. A ship is to be insured, in which P has ventured £2500; Q, £3500; and R, £4800. The expense of insurance is £495. 10s.; how much must each pay of it?

The entire amount of money risked is £10800, and if the expense of insurance be divided into 10800 equal parts, each of them will express the expense of insurance for one pound of capital; consequently the sum that each must pay will be expressed by

× 2500;

£495. 10s.
10800

× 3500; and

£495. 10s.
10800

X 4800.

£495. 10s. 10800 These results are furnished by the following proportions, in which the first term is the sum of the money risked by all the partners, viz. £10800.

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Ex. 2. Three persons, A, B, C, have a pasture in common, for which they are to pay £30 per annum, into which A put 7 oxen for 3 months, B put 9 oxen for 5 months, and C put 4 oxen for 12 months; how much must each person pay of the rent?

The principle in questions of this kind is, that the same sum should be paid for the keep of one ox for one month or one year by each person. Now since A put in 7 oxen for 3 months, he might have had an equal share of the pasture by putting in 7 oxen x 3, or 21 oxen for 1 month. In like manner, B might have put in 9 oxen x 5, or 45 oxen for 1 month, and C might have put in 4 oxen x 12, or 48 oxen for 1 month. Hence, if we divide £30 into 7 × 3 +9 × 5+ 4 × 12, or 114 equal parts, A must pay 7 x 3 or 21, B must pay 9 x 5 or 45, and C must pay 4 × 12 or 48 of those parts. The three persons A, B, C, must therefore pay respectively,

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or £5. 10s. 6d. P, £11. 16s. 10d. Fr, and £12. 12s. 74d. To.

This question may consequently be resolved in the following manner :

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The reasoning employed above may be conducted in a somewhat different manner; thus, suppose one pound were charged for the pasturage of one ox for a month, it is obvious that A would have to pay 7 pounds for having 7 oxen for 1 month, and consequently £7 × 3, or £21, for having 7 oxen in the pasture for 3 months. In like manner B would have to pay £9 × 5, or £45, for having 9 oxen at pasture for 5 months, and C, £4 x 12, or £48, for having 4 oxen at pasture for 12 months. Hence it is evident that the rent must be divided amongst them in such a manner, that if it be divided into 114 equal parts, A must pay 21, B 45, and C 48 of these parts.

INTEREST.

121. INTEREST is the sum of money paid for the use of other money, and is always estimated at so much for £100 during a year. Thus, if £100 are lent at 4 per cent., it must be understood to mean 4 per cent. per annum, that is, that £4 are paid annually for the use of £100.

Principal is the money lent; the rate per cent. is the interest of £100 for a year; and the amount is the interest and principal together. Simple interest is only the interest of the principal for the whole time it is lent, and compound interest is not only the interest of the principal for the whole time it is lent, but if the interest is not paid at the stated intervals it is considered as principal as soon as it is due, and then the original principal, together with the unpaid interest, forms a new principal, the interest of which becomes due at the next stated time of payment.

Ex. 1. Find the interest of £355. 12s. 6d. for 4 years at 4 per cent. per annum.

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In practice, it is usual to multiply the principal by the rate per cent., and by the number of years; then to divide by 100, as in the margin. If the interest be required for any number of days, we must find the interest for one year, or 365 days, and then by the Rule of Three find the interest for any given number of days

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