12. A party at the distance of 378 miles from the head quarters, receive orders to join their corps in 18 days: what number of miles must they march each day to obey their orders? Ans. 21. 13. The annual revenue of a gentleman being 383301; how much per day is that equivalent to, there being in the year? 365 days Ans. 104/. CONTRACTIONS IN DIVISION.` There are certain contractions in Division, by which the operation in particular cases may be performed in a shorter manner as follows: 1. Divi I. Division by any Small Number, not greater that 12, may be expeditiously performed, by multiplying and subtracting mentally, omitting to set down the work, except only the quotient immediately below the dividend. II. * When Ciphers are annexed to the Divisor; cut off those ciphers from it, and cut off the same number of figures from the right-hand of the dividend; then divide with the remaining figures, as usual. And if there be any thing remaining after this division, place the figures cut off from the dividend to the right of it, and the whole will be the true remainder; otherwise, the figures cut off only will be the remainder. This method is only to avoid a needless repetition of ciphers which would happen in the common way. And the truth of the principle Ans. 3971509 3800 III. When the Divisor is the exact Product of two or more of the small Numbers not greater than 12: Divide by each of those numbers separately, instead of the whole divisor at once. N. B. There are commonly several remainders in working by this rule, one to each division; and to find the true or whole remainder, the same as if the division had been performed all at once, proceed as follows: Multiply the last remainder by the preceding divisor, or last but one, and to the product add the preceding remainder; multiply this sum by the next preceding divisor, and to the product add the next preceding remainder; and so on, till you have gone backward through all the divisors and remainders to the first. As in the example following: principle on which it is founded, is evident; for, cutting off the same number of ciphers, or figures, from each, is the same as dividing each of them by 10, or 100, or 1000, &c. according to the number of ciphers cut off; and it is evident, that as often as the whole divisor is contained in the whole dividend, so often must any part of the former be contained in a like part of the latter. This follows from the second contraction in Multiplication, being only the converse of it; for the half of the third part of any thing, is evidently the same as the sixth part of the whole; and so of any other numbers.-The reason of the method of finding the whole remainder from the several particular ones, will best appear from the nature of Vulgar fractions. Thus, in the first example above, the first remainder being 1, when the divisor is 7, makes this must be added to the second remainder, 6, making 64 to the divisor 8, or to be divided by 8. But 6 = 6 × 7+1 43 43 43 = 7 ; and this divided by 8 IV. Common IV. Common Division may be performed more concisely, by omitting the several products, and setting down only the remainders; namely, multiply the divisor by the quotient figures as before, and, without setting down the product, subtract each figure of it from the dividend, as it is produced; always remembering to carry as many to the next figure as were borrowed before. REDUCTION is the changing of numbers from one name or denomination to another, without altering their value.This is chiefly concerned in reducing money, weights, and measures. When the numbers are to be reduced from a higher name to a lower, it is called Reduction Descending; but when, contrarywise, from a lower name to a higher, it is Reduction Ascending. Before proceeding to the rules and questions of Reduction, it will be proper to set down the usual Tables of money, weights, and measures, which are as follow: of MONEY, WEIGHTS, AND MEASURES. * £ denotes pounds, & shillings, and d denotes pence. denotes 1 farthing, or one quarter of any thing. denotes a halfpenny, or the half of any thing. denotes 3 farthings, or three quarters of any thing. PENCE The The standard for Federal Money of Gold and Silver is 11 parts fine, and 1 part alloy. A Dollar is equal to 48 and 8d in South-Carolina,to 68 in the New-England States and Virginia, to 78 and 6d in NewJersey, Pennsylvania, Delaware, and Maryland, and to 8s in New-York, and North-Carolina. TROY The full weight and value of the English gold and silver coin, is as here below: Half-guinea 0 10.6 The usual value of gold is nearly 4/ an ounce, or 2d a grain; and that of silver is nearly 5s an ounce. Also the value of any quantity of gold, is to the value of the same weight of standard silver, nearly as 15 to 1, or more nearly as 15 and 1-14th to 1. Pure gold, free from mixture with other metals, usually called fine gold, is of so pure a nature, that it will endure the fire "without |