3. Add this last found number and the uppermost line together, and if their sum be the same as that found by the first addition, the sum is right. 4. Add 8635, 2194, 7421, 5063, 2196, and 1245 to gether. Ans. 26754 5. Add the sum of the former excesses, it is plain this last excess must be equal to the excess of nines contained in the total sum of all these numbers; the parts being equal to the whole. This rule was first given by Dr. WALLIS, in his Arithmetic, published A. D. 1657, and is a very simple easy method; though it is liable to this inconvenience, that a wrong operation may sometimes appear to be right; for if we change the places of any two figures in the sum, it will still be the same; but then a true sum will always appear to be true by this proof; and to make a false one appear true, there must be at least two errors, and these opposite to each other; and if there be more than two errors, they must balance among themselves: but the chance against this particular circumstance is so great, that we may pretty safely trust to this proof. 5. Add 246034, 298765, 47321, 58653, 64218, 5376; 9821, and 340 together. Ans. 730528. 6. Add 562163, 21964, 56321, 18536, 4340, 279, and $3 together. Ans. 663686. 7. How many shillings are there in a crown, a guinea, a moidore, and a six and thirty? Ans. 89. 8. How many days are there in the twelve calendar months? Ans. 365. 9. How many days are there from the 19th day of April, 1774, to the 27th day of November, 1775, both days exclu sive ? Ans. 586. SIMPLE SUBTRACTION. Simple Subtraction teacheth to take a less number from a greater of the same denomination, and thereby, shews the difference or remainder. The less number, or that which is to be subtracted, is called the subtrahend; the other, the minuend; and the number that is found by the operation, the remainder or difference. RULE.* 1. Place the less number under the greater, so that units may stand under units, tens under tens, &c. and draw at line under them. 2. Begin *DEMON. I. When all the figures of the less number are *less than their correspondent figures in the greater, the difference of the figures in the several like places must altogether make the true difference sought; because as the sum of the parts is equal to the whole, so must the sum of the differences of all the similar parts be equal to the difference of the whole. 2. When 2. Begin at the right hand, and take each figure in the lower line from the figure above it, and set down the re mainder. 3. If the lower figure is greater than that above it, add ten to the upper figure; from which figure, so increased, take the lower, and set down the remainder, carrying one to the next lower figure; with which proceed as before, and so on till the whole is finished. Method of PROOF. Add the remainder to the less number, and if the sum is equal to the greater, the work is right. 2. When any figure of the greater number is less than its correspondent figure in the less, the ten, which is added by the rule, is the value of an unit in the next higher place, by the nature of notation; and the one that is added to the next place of the less· number is to diminish the correspondent place of the greater accordingly; which is only taking from one place and adding as much to another, whereby the total is never changed. And by this means the greater number is resolved into such parts, as are each greater than, or equal to, the similar parts of the less and the difference of the corresponding figures, taken together, will evidently make up the difference of the whole. Q. E. D. The truth of the method of proof is evident: for the difference of two numbers, added to the less, is manifeftly equal to the greater. 4. From 2637804 take 2376982. 5. From 3762162 take 826541. 6. From 78213606 take 27821890. Ans. 260822. Ans. 2935621. Ans. 50391716. 7. The Arabian method of notation was first known in England about the year 1150: how long was it thence to the year 1776 ? Ans. 626 years. 8. Sir Isaac Newton was born in the year 1642, and died in 1727 how old was he at the time of his decease? Ans. 85 years. SIMPLE MULTIPLICATION, Simple Multiplication is a compendious method of addition, and teacheth to find the amount of any given number of one denomination, by repeating it any proposed number of times. The number to be multiplied is called the multiplicand. The number you multiply by is called the multiplier. The number found from the operation is called the product. Both the multiplier and multiplicand are, in general, called terms or factors. USE of the Table in MULTIPLICATION. Find the multiplier in the left-hand column, and the multiplicand in the uppermost line; and the product is in the common angle of meeting, or against the multiplier, and under the multiplicand. USE of the Table in DIVISION. Find the divisor in the left-hand column, and the dividend in the same line; then the quotient will be over the dividend, at the top of the column. RULE.* 1. Place the multiplier under the multiplicand, so that units may stand under units, tens under tens, &c. and draw a line under them. 2. Begin * DEMON. I. When the multiplier is a single digit, it is plain that we find the product; for by multiplying every figure, that is, every part of the multiplicand, we multiply the whole; and writing down the products that are less than ten, or the excess of tens, in the places of the figures multiplied, and carrying the num ber of tens to the product of the next place, is only gathering together the similar parts of the respective products, and is, therefore, the same thing, in effect, as though we wrote down the multiplicand as often as the multiplier expresses, and added them together for the sum of every column is the product of the figures in the place of that column; and these products, collected together, are evidently equal to the whole required product. 2. If the multiplier is a number made up of more than one digit. After we have found the product of the multiplicand by the first figure of the multiplier, as above, we suppose the multiplier divided into parts, and find, after the same manner, the product of the multiplicand by the second figure of the multiplier; but as the figure we are multiplying by stands in the place of tens; |