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A II I T H M E T I C K. confequentjy we ftiail have fix uncertain places in the to- tions, will throw a light upon the matter, and unfold the tal product toward the right, and alfo one uncertain place reafon of the rule. Which takehs follows. more on account of the uncertain carriage from the co- Multiply 18.634375 into 9-S75, and limit the produft lumn in which the right-hand figure of the lalt particular to four decimal places. product Hands. By the rule. By the other method. The reafon of Rule 2. is alfo obvious. For in Ex. 2. 18.634375 18.634375 by making the finite faftor the multiplier, we have four 578.9 A 9-875 particular produ&s, in each of which the right-hand figure is uncertain; and fo we have four uncertain places 1677093 1677093 75 in the total produft, and one uncertain place more arifing 149075 149075 000 I from the uncertain carriage. 130440625 3°44 The carriage in foroe cafes may affed feveral columns '931 71875 on the left, and thereby render fo many more figures uncertain. 184.0143-f184.0144I53125 The fiureft way therefore to determine the certain B places in the product of approximates, is by a fecond In working by this rule, you invert the multiplier, and operation, giving the approximates contrary ligns; for place 9, the units figure, under 3, the fourth place of then, fo far as the two products agree, the figures are decimals, becaufe the produdt is limited to four decicertaim The fecond operations of the two former ex- mal places; then multiply, faying, 9X3=127, and 6 carried from the right makes 33, fcc. In multiplying amples, follow. Ex. :. Ex. 2. by 8, fay, 8 X4= 32, and 3 of carriage makes 35 ; fo fet 5 under 3 ; and proceed in like manner -to multiply 245.117+ • .210527the figures on the left. The right-hand figure of the .253— 2.875 product is defective, as wanting the carriage from the 1 columns cut off on the right by the line A B. The fi23535 1052635 gures expreffing the fum of the columns fo cut off, are 1225585 M73689 735351 1684216 fo many uncertain places of the product, when the fac421054 tors are approximate, and on that account to be rejected as ufelefs. The figures* moreover, on the right of 86.526301 .605265125 the line A B, flrow how far the operation is contracted, Xn Ex. 1. 865 is certain, and all the othen figures or how much labour is faved in working by the rule uncertain. In Ex. 2. .60526 are the only certain places. If there be no units in the multiplier, in this cafe fet Becaufe the multiplication of decimals that confift of the right-hand figure of the inverted multiplier under many places, proves, in the way hitherto pra&ifed, a te- that figure of the multiplicand, below which it would dious operation, we fhall here explain a method where- have flood had there been units. by decimals of this fort, whether finite or approximate, Ex. Multiply .825 by .825* limiting the product to may be multiplied expeditioufly, and at the fame time three decimal places. By the rule. The common way at large. have the decimal places in the product limited to any .825, .Say number propofed. This may be effected by the fol528. .825 lowing Rule. Under the multiplicand place the multiplier 64a <r'*25 inverted, fo that its units place may Hand under that 16 16I501 place of the multiplicand to which you propofe to limit 4 66o(o the rrroduS:; then multiply the right-hand figure of the multiplier into that figme of the multiplicand which Hands .680+ .68o|62 5 direflly over it, taking in the carriage from the right, and go on to multiply it into all the other figures on the The decimal places of the faClors may either be releft. Proceed in like manner with every other figure of tained at full-length, or turned into approximates before the multiplier, placing the right-hand figures of all the you begin to multiply. particular prod& ds direftly under other. The total pro- Ex. Multiply 25.8490.13^25 by 42 92235, limiting, places. duft will be approximate, and the right-hand figure un- the product to two decimal 25.849013635. certain. To make this rule more eafily underfiood, the readeir 53*79-24 may look back to the multiplication of integers ; where: 10339& it was obfetved, that infiesd of beginning with the right5169 hand figure of the multiplier, we may begin with the 2326 left, and Hill have a juft product, provided the righttffo. hand figure of every particular prodnff be placed di5 reflly under the multiplying figure. Now, the working, an example, both in this manner, and alfo by the rule,, LI LO.764and comparing the fteps and refuks of the two operaWe