subtract from (10) the member. From the remainder (1), with (3) the first figure of the following member added, is formed a dividend (13), which I divide by the found figure doubled (6), the quotient (2) will be the second figure of the root; which being multiplied first into the divisor, and then into itself, and the sum of the product, taken so, however, that the latter be removed one place to the right (124), I take away this from the dividend (13), increased by o, the remaining figure of the second member. To the remainder 6, I add 4, the first figure of the third two, and so a new dividend (64) is produced, which being divided by 64, twice the root already found, gives 1, the third of the required root; this being then multiplied into itself and the products added up, I subtract the sum (641) from the dividend, increased by the addition of the other figure of the third member, and in this way we must proceed to whatever length the operation may be carried.

If after the last subtraction there be a remainder, it shows that the given number was not a square; however by adding to it decimal cyphers, the operation can be continued to any extent thought desirable.

If there be any decimal places in the number, for the root of which we are searching, their number divided by two, will show how many should be in the root. The reason of this appears from chap. iv.

The reason of the mode of proceeding is quite clear from what has been stated. For as a divisor I employed 6, the double of the found figure, because, from the formation of the square as it has been explained, I knew that double the rectangle of that figure, multiplied into the following one, comprised the dividend; consequently, if this were divided by the double of one factor (3), that the other factor (2), that is, the next figure of the root, could be obtained. So likewise I have formed a subtrahend from double the rectangle of the quotient and the divisor and the square of the quotient added together, because I found that those two rectangles and the square were contained in that order in the remainder and the following member from which the subtraction was made, and so the evolution of the power is easily effected from its involution or formation.

CHAP. VII.

CONCERNING THE INVOLUTION AND EVOLUTION OF THE CUBE.

THE root multiplied into the square produces the cube. To prepare the way for the analysis we should, as has been done in the former chapter, begin with the composition of the power. In the production then of the cube from a binomial root, the first member of the root, in the first place, meets with its own square, whence results the cube of the first figure; secondly, double the rectangle of the members, whence double the solid of the square of the first figure multiplied into the other; thirdly, the square of the other member, whence the solid produced from the first figure and the square of the second. In the same way, when the multiplication takes place by the second member, there arises the solid of the second figure and of the square of the first; in the second place double the solid of the first figure and of the square of the second; in the third place the cube of the second member.

Therefore the cube produced from the binomial root contains the cubes of the two members and six solids, that is to say, three made from the square of each member, multiplied into the other.

The reasoning being continued according to the analogy of the preceding chapter, it will follow, that if, as the square should be divided into twos, the cube resulting from any root be distributed into threes, that the three, or member first from the left, contains the cube of the figure first on the left, and also the excess, if there be any, of three solids of the square of the same, multiplied into the second; that the first place of the second contains the said solids and the excess of the three solids of the square of the second figure, multiplied into the first; that the second place contains the same three solids and the excess of the cube of the second figure; and that the third is occupied by the said cube and the excess of the three solids produced from the square of the preceding figures, multiplied into the third; and that the solids just mentioned fill the first place of the third member, and so on. From this we shall easily derive the following manner of extracting the cube root.

Beginning from the right, I divide, by means of points, the resolvend (80621568) into threes, except the last member, which can be less. I then take the greatest cube (64) contained in the first member towards the left, and having written down its root (4) for the first figure of the sought root, I annex to the remainder (16) the next figure (6) of the resolvend, whence results a dividend (166), which I divide by 48, thrice the square of the figure which has been found: the quotient (3) is the second figure of the root. I multiply this first into the divisor, secondly its square into three times the first figure, and thirdly itself into itself twice. The products then being collected in this way, that the second be set down one place to the right of the first, the

144

third one place to the right of the second. 108 I subtract

64

48)16621

15507

5547)1114568 1114568

27

it from the dividend increased by the ad80.621.568(432 dition of the two remaining figures of the second member. In this way, however prolonged the operation, a dividend will always result from the remainder, with the addition of the first figure of the following member, and a divisor from three times the square of the figures of the root already found, and a subducend from the figure last found, the square of the same multiplied into three times the preceding figures, lastly its cube, and these collected in the manner set forth. If the resolvend be not a cube, by adding decimals to the remainder you can carry its exhaustion to infinity.

The root should have a third part of the decimal places of the resolvend.

N.B.-Synthetical operations can be examined by means of analytical, and analytical by means of synthetical; so if either number, being subtracted from the sums of two numbers, the other remains, the addition has been rightly performed; and vice versa, subtraction is proved to be right when the sum of the subtrahend and remainder is equal to the greater number. So if the quotient multiplied into the divisor produce the dividend, or the root multiplied into itself produce the resolvend, it is a proof that the division or evolution has been correct.

ARITHMETIC.

PART II.

CHAP. I.

ON FRACTIONS.

IT has been before mentioned that division is signified by setting down the dividend with the divisor under it, and separated from it by a line drawn between them. Quotients of this kind are called broken numbers, or fractions, because the upper number, called also the numerator, is divided or broken into parts, the denomination of which is fixed by the lower, which is therefore called the denominator. For instance, in the fraction, 2 is the dividend or numerator, 4 the divisor or denominator, and the fraction indicates the quotient which arises from 2 divided by 4, that is the fourth of any two things whatever, or two-fourths of one, for they mean the same.

N.B.-It is clear that numbers which denote decimal parts, and which are commonly called decimal fractions, can be expressed as vulgar fractions, if the denominator be written beneath. For instance, 25 is equivalent to 004 is equivalent to, &c., which we must either do, or understand to be done, as often as those are to be reduced to vulgar fractions, or conversely these are to be reduced into those, or any other operation is to take place equally affecting both fractions, decimal and vulgar.

CHAP. II.

OF ADDITION AND SUBTRACTION OF FRACTIONS.

1. IF fractions, whose sum or difference is sought, have the same denominator, the sum or difference of the nu

merators should be taken, and the common denominator written under, and this will be the answer.

2. If they be not of the same denomination let them be reduced to the same denomination. The denominators multiplied into each other will give a new denominator, but the numerator of each fraction multiplied into the denominators of the others will give a numerator of a new fraction of equal value. Then the new fractions should be treated

as above.

3. If an integer is to be added to a fraction or subtracted from it, or vice versa, it should be reduced to a fraction of the same denomination as the given one; that is, it is to be multiplied into the given denominator, and that denominator to be placed under it.

In the first place, it should be explained why fractions should be reduced to the same denomination before we treat them; and it is on this account, that numbers enumerating heterogeneous things cannot be added together, or subtracted from each other. For instance, if I wish to add three pence to two shillings, the sum will not be 5 shillings, or 5 pence, nor can it be ascertained before that the things mentioned be brought to the same sort, by using 24 pence instead of 2 shillings, to which if I add 3 pence, there results a sum of 27 pence; for the same reason, if I have to add 2 thirds and 3 fourths, I do not write down 5 parts either thirds or fourths, but, instead of them I employ 8 twelfths and 9 twelfths, the sum of which is 17 twelfths.

Secondly. I wish to show that fractions after such re