be divided by the square of 53·809, or, which will do as well for our purpose, the square of 50, or 2500. The result is something less than ·00002, so that the quotient of 17·324 and 53·809 can be depended on to four places of decimals. [19] These are not quite correct, but sufficiently so for every practical purpose. 153. It is required to multiply two decimal fractions together, so as to retain in the product only a given number of decimal places, and dispense with the trouble of finding the rest. First, it is evident that we may write the figures of any multiplier in a contrary order (for example, 4321 instead of 1234), provided that in the operation we move each line one place to the right instead of to the left, as in the following example: 2221 2221 1234 4321 ---- ---- 8884 2221 6663 4442 4442 6663 2221 8884 ------- ------- 2740714 2740714 Suppose now we wish to multiply 348·8414 by 51·30742, reserving only four decimal places in the product. If we reverse the multiplier, and proceed in the manner just pointed out, we have the following: 3488414 2470315 | ---------+ 17442070 | 3488414| 1046524|2 24418|898 1395|3656 69|76828 ----------+------ 17898·1522|23188 Cut off, by a vertical line, the first four places of decimals, and the columns which produced them. It is plain that in forming our abbreviated rule, we have to consider only, I. all that is on the left of the vertical line; II. all that is carried from the first column on the right of the line. On looking at the first column to the left of the line, we see 4, 4, 8, 5, 9, of which the first 4 comes from 4 × 1′,[20] the second 4 from 1 × 3′, the 8 from 8 × 7′, the 5 from 8 × 4′, and the 9 from 4 × 2′. If, then, we arrange the multiplicand and the reversed multiplier thus, 3488414 2470315