in shillings; but if it be not an even part, take such parts as will amount to the whole, and add the quotients together for the whole price. The same principles are to be observed when the price is shillings, &c. If the price is pounds, shillings, &c., the number expressing the quantity may be first multiplied by the number expressing pounds, and the remaining part obtained as before. IV. When there are parts of a yard, ton, &c. in the given sum, the price of the whole number may be obtained, and then the price of the part added to it. Example. What will 54 yards of cloth cost at 4 s. 6 d. a yard? When two or more men trade together, each advancing a certain sum, (to be continued for an equal time,) called stock, or capital, it is evident that each one's share of the gain or loss, must be in proportion to the stock which he puts in. They Three merchants, A, B, and C, entered into trade togethA advanced $2000, B $3000, and C $5000. gained $3000. What is each man's share of the gain? er. The whole stock is $10,000. Had one man put in this whole sum, the whole gain $3000 would have been his; but as this gain is to be divided among three men, it is evident that each one's share of the united stock must bear the same proportion to his gain, that the whole stock does to the whole gain. Therefore $10000 3000 :: 2000 10000 3000 :: 3000 10000 3000 :: 5000 : to A's share of the gain. By performing the operations of the above statements, we obtain the answers. $600 A's share. 900 B's share. 1500 C's share. Proof $3000 The above propositions may be stated in words, thus :— The whole stock to the whole gain :: each man's share of the stock to his share of the gain. It is evident, that the sum of the several shares of the gain will be equal to the whole gain. By adding the shares of the gain together, the work may be proved. If each one's share of the stock had been the same, the shares of the gain would have been the same; to obtain which we should only have had to divide the gain by the number of shares. We may reduce the question to this in the present case, by dividing the whole stock $10000 into 10 partial stocks of $1000 each. The gain of each one of these will evidently be the 10th part of the whole gain; then by multiplying this part by 2, 3, and 5, which have the same relation to each other as the several shares of the whole stock, we shall obtain each man's share. The gain of $1000-$300, and 300 × 2= $600, A's share; $300 X 3=900, B's share; $300 × 5= $1500 C's share. When the stocks of the partners are continued in trade an equal time, as in the preceding case, it is called Single Fellowship. When the stocks are continued in trade unequal times it is called Double Fellowship. A and B enter into partnership. A advanced 600 dollars for 6 months and B 1200 for 12 months. They gained $300. What was each one's share of the gain? If the two shares of stock had been continued in trade the same time, it is evident, that B's share of gain could have been only double that of A. But as $1200 for 12 months is equal to 12 times 12 hundred for one month, and 6 hundred dollars for 6 months is equal to 6 times 6 hun dred for one month, A can have only 400 whole gain. Then 3600 of the A's part is $75 B's 225 The amount put in must be taken in connection with the length of time in which it is employed. Hence the RULE. Multiply each man's share of the capital by the time it is employed: then as the whole stock is to the whole gain, so is each man's share of stock to his share of the gain. SECTION XXIV. LOSS AND GAIN. If I buy 4 yards of cloth for ten dollars, and sell it at two dollars a yard, it is evident that I lose a part of the price. Questions in business often arise, on subjects connected with loss and gain in trade, and also the amount per cent. either lost or gained. If it were asked, how much per cent. was lost on the cloth above, I can ascertain by saying that as $10 lost $2, $1 must lose part of 2, which will of course be of $1 - 20 cents. = Then the loss will be 20 per cent. If it be required to find the loss or gain in laying out $500, when the loss or gain is a certain per cent., say 15; I have only to apply the principles before explained, and say that if $1 gains or loses 15 cents, $500 will gain or lose 500 times as many 7500. As the principles of proportion already explained are all which are required to perform these operations, the rule does not merit further notice in this place. = SECTION XXV. INVOLUTION AND EVOLUTION. These terms are sufficiently defined in the corresponding section of Part I. Any number whatever may be a root, and can be raised to the second, or any given power, by continued multiplication. 2×24 is the 2d power or square of 2. The following Table contains the powers of numbers as far as nine. 6 36 216 1296 7776 46656 279936 1679616 10077696 749343 2401 16807 117649 823543 5764801 40353607 8 64512 4096 32768 262144 2097152 16777216 134217728 981 729 6561 59049 531441 4782969 43046721 387420489 Every number must have a root, that is, a number which, being multiplied into itself once, or a certain number of times, will produce it. But this is not always easily expressed by figures, without having recourse to fractions. The least root which is a whole number is 1. The square of 1 X 1= 1 has one figure less than the numbers in the factors; the cube of 1 X1 X1 = 1 is less by two than the facThe least root consisting of two figures is 10; and 10 × 10=100 has one figure less than the factors. The cube of 10 1000 has two less. The greatest root consisting of one figure is 9. Its square, 81, contains just the number of figures found in its factors; its cube, 729, is just equal to the factors. tors. The greatest root consisting of two figures, is 99; and its square, 9801, contains just the number of figures in the fac 9th power. tors. Hence it will be seen that the second power can have no more than double the figures of its root, and in no case but one less; and that the third power can never have more than three times the number of figures of its root, and in no instance more than two less. Then, to ascertain the number of figures in any required root, distinguish the given sum into parts or periods, by dots, putting into each period the number denoted by the index of the root required. If the 3d root is required, put 3 figures in a period; if the 4th, put in 4, and so on. SECTION XXVI. SQUARE ROOT. The square root of any number, is such a number as multiplied into itself will produce that number. The square root of 144 is of course 12, because 12 × 12=144. If 100 pieces of paper 1 foot square are to be placed in a square form, each side must measure 10 feet, because 10 × 10 100. The following example will exhibit the operation, and the reasons on which it is founded. 66 = If 529 feet of boards be laid down in a square form, what will be the length of the sides of the square ? or, in other words, what is the square root of 529 ? From what was shown in the last Section, we know that the root must consist of two figures, inasmuch as 529 consists of two periods. Now to understand the method of ascertaining these two figures, it may be well to consider how the square root, consisting of two figures, is formed. For this purpose we will take the = |