RULE. Always begin with that figure which ftands in the units place of the multiplier, and with it multiply the figure which ftands in the units place of the multiplicand; if their product be less than ten, fet it down underneath its own place of units, and proceed to the next figure of the multiplicand. But if their product be above ten (or tens) then fet down the overplus only (or odd figures, as in addition) and bear (or carry) the faid ten (or tens) in mind, until you have multiplied the next figure of the multiplicand with the fame figure of the multiplier; then to their product add the ten or tens beared in mind, fetting down the overplus of their fum above the tens, as before; and fo proceed in the very fame manner, until all the figures of the multiplicand are multiplied with that figure of the multiplier. 2. When the multiplier is any number between 12 and 20; multiply by the figure in the units place; and as you multiply, add to the product of each fingle figure, that of the multiplicand, which stands next on the right-hand. 3. But when the multiplier confifts of feveral figures, the multiplicand must be multiplied with every fingle figure of the multiplier; always placing the firft figure, or cypher, of every particular product, directly underneath that figure of the multiplier you then multiply with. 4739284 4. If there be a cypher, or cyphers, intermixed with the figures, move for every figure, or cypher, one place toward the left-hand, and take care that every firft figure of the feveral products ftand directly under its refpective multiplier. 5. Cyphers placed at the end of either or both factors, are to be omitted till the laft, product, and then the number of cyphers as are at the end of both must be annexed to it. 6. Any number given, being multiplied by 1, undergoes no alteration; but if by 10, a cypher is to be annexed; if by 100, annex two cyphers; by 1000, annex three, &c. 7. In geometrical progreffions, converging feries, &c, when multiplications have been very operofe, I have frequently added, fubtracted, or divided; or multiplied a product by a smaller, when the former happens to be a multiple of the latter; as I fhall endeavour to explain in the example following. 8496427 874359 By fubtracting the right-hand figure from a cypher,, and each preceding figure from that following. 76467843 42482135 2548928 33985708 - 59474989 67971416 7428927415293 By dividing the multiplicand by 2. But before the learner attempts to perform operations by this method, he ought to be acquainted with divifion. 8. If the multiplier be any number near 100, 1000, 10000, &c. increase the multiplicand by as many cyphers as there are figures in the multiplier; and fubtract the multiplicand from itself thus increased, as often as the multiplier wants units of that by which the multiplicand was increafed. Let 7943628 X 999 7943628000 7943628 7935684372 And 4372845 X 9997 13118535 multiplic. × 3. 43715331465 9. If the multiplier be a repetend of the fame figure, multiply by one of the repeating figures; and the figures of that that product added, as if they had been wrote down in as many products as the multiplier repeated the fame figure, give the product required. 10. When the repeating figure is a high digit, collect the product of as many ones as there are digits in the multiplier, from the multiplicand, according to the rule in the laft contraction; which product being multiplied into the repetend, will give the true product. 784325634 into 7777777. 871472839519374 Products collected for IIIIIII. 7 6100309876635618 Product of 7777777 11. Find the product of the given multiplicand by the likę number of nines, and divide that product by 9; the quotient multiplied by the digit which repeats in the given multiplier, will be the product required. Ex. Let 4538769 be multiplied by 7777777. 4538769 945387685461231 N. B. Divifion must be learned before examples of this kind 5043076162359 be attempted. X 7 35301533136513 D 4 12. When 12. When the multiplier can be parted into periods, which are multiples of one another, the operation may be contracted in the following manneṛ. 13. To multiply by a factor, confifting of as many cyphers between two digits as there are places in the multiplicand, multiply by a fingle digit; and the product by the fecond figure will fall directly to the left-hand of the product by the firft figure; but if the product of the first figure be less than 10, then a cypher must be put down between the two products. 84629 7000003 592403253887 14. The proof of multiplication, is by making the mul tiplicand to be the multiplier; then if the product comes out the fame as before, your work is right. 15. Or by cafting away the nines, which, though not infallible, ferves to confirm the other. Thus, in the laft example, make a cross, and add all the figures, or digits, of the multiplicand together, as units, thus, 8+4+6+2+9=29; caft away the nines, and fet the remainder two on onc fide the crofs. Do the fame with the multiplier 7+3= 10; fet the remainder 1 on the other fide the cross. Do the like by the product, and fet the remainder at top. Lastly, multiply the figures on the fides, and fet the remainder at the bottom, after the nines (if any) are caft away; which must be the fame with the top, if the work is right. QUESTIONS |