write the Multiplier, in an inverted or retrograde Order; and, in multiplying, begin always with that Fi-. gure of the Multiplicand, which stands directly over that of the Multiplier which you are going to multiply by; remembering to add in the Increase, or Carriage, that would arife from the two next Figures, (were they to be multiplied) that are to the right Hand of that Figure, which you begin with in the Multiplicand; alfo remembering to let the first Figure of each particular Product stand directly one under the other. 18. Example 1. Let it be required to multiply 54.321711 by 3.12321, and have only four Places of Decimals in the Product. Here we put the Units Place of the Multiplier 3 under the fourth Place of the Multiplier 7: In the second Multiplication, because 7 × 1 = 7 is nearer to carrying 1 than o, we add in one, and, in like Manner, in the other Products: 54.321711 3.12321 54321711 108 643422 162965133 10864 3422 54321711 1629651 33 169.65811101231 By comparing the foregoing Contraction with the Operation here worked at large, the Reafon of that ¥ 3 Method Method will plainly appear; for the Figures to the right Hand of the Line are omitted in that fhort Method, and the Operation inverted, the laft particular Product here being the first in that. 19. Example 2. Multiply 231.312 by 21.32, and have only 3 Places of Decimals in the Product. The common Way 231.3121 21.32 4626242 6939363 2313121 4626242 69394 4626 4931.574 4931.5731972 20. Example 3. Multiply 432.12 by 0, 785, and have only the whole Numbers in the Product. 21. Multiplication of Decimals may also be contracted, without inverting the Multiplier, by the following Rule, viz. The Multiplier and Multiplicand being placed as in their natural Order, from the Number of Decimal Places in both the Factors, deduct the Number of Decimal Places that you intend to keep in the Product; and then cut off as many Figures as remain from the Multiplicand, counting from the right Hand towards the Left; but, if there is not a fufficient Number of Figures in the Multi N. B. As fome Figures of the Product are omitted in thefe Contractions, the Products cannot be proved by cafting Out the Nines, Multiplicand, cut off the Defect from the Multiplier. Then multiply the Figures to the left of the Line of Separation, by the firft Figure (to the right Hand) of the Multiplier, remembering to add the Carriage that would arife from the Multiplication of two Figures to the right Hand (which must be alfo done in the Multiplication by the other Figures) and fet down the Product. Secondly, in Multiplying by the fecond Figure take in one Figure more of the Multiplicand; and in this Manner proceed, each Time taking in one Figure more of the Multiplicand, and writing the Units Place of each particular Product directly under the Units Place of the fuperior or preceding Product. Note. In Order to prevent forgetting what Figures of the Multiplicand and Multiplier we are at any Time come to, it may be useful to dot as we proceed. An Example, or two, will make this intelligible. 22. Example 1. Multiply 54.321711 by 3.12321, and have only four Places of Decimals in the Proz duct. 4 Here are 11 Decimal Places in both the Multiplicand and Multiplier, and we are only to have in the Product, we have 11-47 Places to be cut off in the Multiplicand. 54.321711 3.12321 5 = 5 x I 109 = 54 + 2, + 1 Carriage 1630=543 × 3, +1 carried 108645432 x 2 5432254321x1,+1 for nearest Car. 1629651 =543217 x 3 169.6581 23. Example 2. Multiply 43.212 by 0.785, and have only the Integers in the Product. Y 4 Here Here being 6 Places to be cut off, and only 5 Figures in the Multiplicand, all the five must be cut off, and one Figure from the Multiplier. 0143.21 .78 ox8+3 carried 3 33 Whoever compares this Method of contracting Decimal Multiplications with that before delivered, will plainly fee, that they are in Effect the fame. We have given both Methods, that the Learner make 'Ufe of that which he likes best. " 24. I CHA P. V. DIVISION of DECIMALS. N Divifion of Decimals, take the Divifor and Dividend as whole Numbers, and divide as has been already taught in Divifion of whole Numbers; but, if there are not fo many Decimal Places in the Dividend as there are in the Divifor, that Defect is firft to be fupplied by annexing o's to the right Hand of the Dividend; in like Manner, if the Divifor, confidered as a whole Number, cannot be taken once out of the Dividend, alfo confidered as a whole Number, we must firft add as many Cyphers (o's) to the right Hand of the Dividend, as will make the Dividend, taken as a Whole, greater than (or at least equal to) the Divifor. And the annexing thefe o's does not alter the Value of the Dividend, as appears by Notation of Decimals; but only prepares it for the Operation. Having found how many Times the Divifor is con tained in the Dividend, both confidered as whole Numbers, we must now fee, how many of the Places in the Quotient must be Decimal Parts; and this we do by the following Rule, viz. Mark off in the Quotient as many Places of Figures for the Decimal Part, as as there are Decimal Places in the Dividend more than in the Divifor. The Reafon of which will easily appear; for, by Multiplication of Decimals, the Decimal Places in the Multiplicand plus the Decimal Places in the Multiplier, are equal to the Decimal Places in the Product; and, by the Proof of Divifion, the Divifor, multiplied by the Quotient, is equal to the Dividend; therefore, the Decimal Places in the Divifor plus the Decimal Places in the Quotient are equal to the Decimal Places in the Diyidend; hence, by fubtracting the Decimal Places in the Divifor from both Sides of this Equation, we have the Decimal Places in the Quotient equal to the Decimal Places in the Dividend minus the Number of Decimal Places in the Divifor. N. B. When it happens that there are not fo many Places of Figures in the Quotient, found by the Divifion, as there must be Decimal Places in the Quotient, that Deficiency must be fupplied by placing Cyphers to the right Hand of the Figures. Note. If, after all the Figures in the Dividend have been taken down in the Operation, there be a Remainder, we may continue the Divifion by adding Cyphers, each Cypher added giving one Decimal Place more in the Quotient. A few Examples will better explain this, than more Words. 25. Example 1. Divide 763.2417 by 33.17. 6634. 9984 9951 36. 3317 3317 |