Prop. 64. To explain the Rules of Fellowship.

1st. Let the times, for which each partner has been engaged, be the same. It is evident that the gain or loss of each ought to vary as the stock of each; so that the several gains or losses will bear the same ratio to one another, as the several stocks. Hence we must divide the whole gain or loss into a number of parts in these ratios. Now the ratios of the stocks are the same as the ratios of the numbers of pounds in the stocks, or of the numbers of shares; therefore (by Prop. 46) the several shares of gain or loss are found by dividing the whole gain or loss by the sum of these numbers, and mnltiplying the result by each in succession; or by finding the ratio of the whole gain or loss to the whole stock, and multiplying by it each stock.

2nd. Let the times be different. Here it is plain, that the gain or loss ought to vary directly as the amount of stock, when the time is the same; and also directly as the time, when the stock is the same: therefore when both time and stock are altered, the gain or loss will vary as the product of these. Hence the whole gain or loss must be divided into parts, which shall have the same ratios as the products of the units in the several stocks by the units in the several times. So that the several shares of gain or loss will be obtained by dividing the whole gain by the sum of these products, and multiplying by each product separately. If the numbers of shares in the concern be given instead of the amounts of stock, these may be used for the numbers of units, the unit being the amount of a share.

Prop. 65. The product of different powers of the same number is a power of a degree equal to the sum of the degrees of the several powers.

Every power is composed of factors, equal to the number, and in number equal to the degree of the power. Thus in the 3rd, 4th, &c. powers there are 3, 4, &c. factors. Hence if two or more powers be multiplied together, the result is a number of as many factors as there are in the several powers together, i.e. a power of a degree equal to the sum of the degrees of the several powers. Thus if the 3rd and 5th powers of a number be multiplied, the result is a number composed of 8 factors, or a power of the 8th degree. And generally if a denote a number, am X a1 = (a × a × a X &c. to m factors) X (a X a X a X &c. to n factors)

= axaxax &c. to (m + n) factors

am+n.

Prop. 66.-A power of a power is another of a degree equal to the product of the degrees of the two.

If a power of a number be raised to a given power, the result is a number composed by the product of the first power, repeated a number of times, equal to the degree to which it has to be raised; i.e. (by last Prop.) is a power of a degree equal to the sum of the degree of the first power repeated a number of times equal to the degree of the second, or of a degree equal to the product of the two degrees. Thus if the 8th power is to be cubed, the result is the product of three 8th powers, i.e a power of a degree equal to three times 8 or 24. And generally if a be any number, then

(am)nam xam xam x &c. to n factors

=am+m+m+ &c. to n times = am n

Cor. Hence the nth root of the (m n)th power is the mth power; and therefore the mth root of the nth root is the first power or the (mn)th root of the (m n)th power. So that the extraction of two roots in succession is equivalent to the extraction of a root of a degree equal to the product of the degrees of the roots.

Prop. 67.-The power of a product is equal to the product of the powers of the factors.

For in the power each factor will be repeated a number of times equal to the degree of the power; so that the product of all the factors is equal to the product of the powers of the factors.

Cor. Conversely, the root of a product is equal to the product of the roots of all the factors.

Prop. 68.-The square of a number is equal to the sum of the squares of any two parts into which it may be divided, together with twice the product of these parts.

In multiplying one number by another, we may separate the multiplicand and multiplier into any parts we please, then multiply each part of the multiplicand by each part of the multiplier, and add the products. Let then the number, whose square is required, be separated into two parts; the square will be found by multiplying both these parts by each of them, and adding the products. These products will be in order, the square of the 1st part, the product of the two parts, the product of the two parts, the square of the 2nd part. Hence the whole square is equal to the sum of the together with twice the product of the parts.

squares of the two parts

Thus the square of 16 = (124)×(12+4)

= (12+4) × 12 + (12 + 4) × 4

= 12 x 12 + 4 × 12 + 12 × 4+4×4
= 122+2 X 4 X 12 +42

Algebraically expressed the Prop. will stand thus :

(a + b)2 = a2+2ab+b2.

Prop. 69.-The cube of any number is equal to the sum of the cubes of any two parts into which it may be divided, together with three times the sum of the products of the square of each into the other.

The cube of a number is obtained by multiplying the square by the number. Now the square is equal to the sum of the squares of the two parts together with twice the product of the parts. If this be now multiplied by one of the parts, the result will be the cube of the first part, twice the product of the square of the first into the second, and the product of the first into the square of the second. If it be multiplied by the other part, the result will be the product of the square of the first into the second, twice the product of the first into the square of the second, and the cube of the second. Hence adding these results, the cube of the number is equal to the sum of the cubes of the two parts, together with three times the sum of the products of the square of each part into the other. Thus the cube of 16 (122 + 2 × 4 × 12 + 42) × (12 + 4)

=128+2×4× 122+42× 12+122×4+2×42×12+43 123 + 3 x 4 × 122 + 3 × 42 × 12 +43

Algebraically expressed the Prop. will stand thus:

(a + b)3 = a3 + 3 a2 b + 3 a b2 +b3.

Prop. 70.-A power of a fraction is the fraction formed by raising the numerator and denominator to the required power.

For the product of fractions is obtained by multiplying the numerators for a new numerator, and the denominators for a new denominator. When therefore all the numerators are the same, and all the denominators, the numerator and denominator of the power will be the power of the numerator and denominator of the original fraction.

Cor. 1. A root of a fraction is the fraction formed by extracting the root of the numerator and denominator. For if this fraction be raised to the power indicated by the degree of the root, it will become equal to the given fraction.

Cor. 2. Hence the square and cube root of decimals may be found by extracting the root of the number, considered as integral, and marking off as decimals one-half, or one-third as many figures, as there are decimals in the given number, which must therefore be some multiple of 2 or 3.

Prop. 71.-To prove the Rule for pointing in extraction of

the square of

the Square Root.

10 is 100

100 is 10000

the square of 1000 is 1000000, and so on,

it appears that the square root of a number

of 1 or 2 digits will consist of 1 digit;

of 3 or 4 digits will consist of 2 digits;

of 5 or 6 digits will consist of 3 digits, and so on.

Hence if every alternate figure be marked by a point, the number of points will be the number of digits in the square root.

Again, since the square of a decimal contains twice the number of decimal places, which the decimal contains, therefore if a point be placed over every alternate figure in a given decimal, the number of points will shew the number of decimal places in the square root; but there must always be an even number of decimals in the given number, that the extraction of the square root may be possible.

It is of no consequence with which figure the pointing is commenced, but it is usual to begin with the units' figure, and to point every alternate figure right and left.

Prop. 72.--To prove the Rule for pointing in extraction of

of 4 and not more than 6 digits consists of 2 digits;

of 7 and not more than 9 digits consists of 3 digits, and so on. Hence if every third figure be marked by a point, the number of points will shew the number of digits in the cube root.

Again, since the cube of a decimal contains three times as many decimal places as the decimal contains, therefore every decimal, which is a cube, must have a number of decimal places divisible by 3, and if a point be placed over every third figure, the number of points will shew the number of decimal places in the root.

It is usual to commence the pointing with the units' figure, and to continue it right and left, pointing every third figure.

Prop. 73.-To prove and explain the Rule for the extraction of the Square Root.

Let N be the given number of 3 or 4 digits, whose nearest square root therefore will contain 2 digits, which call a and b. Then the integral part of the root will be denoted by 10 a + b. Let the number be pointed by the Rule for pointing, thus being divided into periods from point to point, the figures as far as the first point counting from the left being reckoned as the first period. Let c be the greatest number, whose square does not exceed the first period in N,which is the number of hundreds in N; then (c+1)2 is greater than the number of hundreds in N, and 100 (c + 1)2 or

2

{10 (c+1)} is greater than N. Hence a cannot be greater than c; for if it could be equal to c + 1, then since {10 (c+1)}* is greater than N,

2

the square of a part of the root would be greater than the given number, which is the square of the whole root. Nor can a be less than c; for then 10 a would be less than 10 c, and 10 a + b would also (b being a number of units less than 10) be less than 10 c, and therefore (10 a + b)2, which is the greatest square in N, would be less than (10 c)2, i.e. would be less than the greatest square number of hundreds in N, which is manifestly impossible. Hence a is equal to c, or the first digit in the root is the greatest number whose square does not exceed the first period. We have now to find b. Let x be the difference between 10 a + b and the complete root of N; x is of course a fraction.

Then

N = ( 10 a + b + x)2 = (10 a)2 + 2 × 10 a (b + x) + (b + x)2

fraction, the integral quotient arising from the division of N - (10 a)2 by 2 X 10 a will be the second digit. But as x may be very nearly equal to 1, and b may be as large as 9, while a may be as small as 1, the value of this expression may be very nearly equal to 6, or the quotient may be larger by 5, than the second digit. But from the second form given of the expression b it appears that the error decreases or increases, as ', and therefore the ratio of the quotient to a, decreases or increases. And the error is larger as x, and therefore the remainder from the division, is larger. Also if a be greater than 4, or b be less than 4, the error cannot exceed 1, ie. if the first digit be greater than 4, or the quotient less than 5. But practice will soon make the detection of the error easy.

α