152. In multiplication and division it is useless to retain more places of decimals in the result than were certainly correct in the multiplier, &c., which gave that result. Suppose, for example, that 9·98 and 8·96 are distances in inches which have been measured correctly to two places of decimals, that is, within half a hundredth of an inch each way. The real value of that which we call 9·98 may be any where between 9·975 and 9·985, and that of 8·96 may be any where between 8·955 and 8·965. The product, therefore, of the numbers which represent the correct distances will lie between 9·975 × 8·955 and 9·985 × 8·965, that is, taking three decimal places in the products, between 89·326 and 89·516. The product of the actual numbers given is 89·4208. It appears, then, that in this case no more than the whole number 89 can be depended upon in the product, or, at most, the first place of decimals. The reason is, that the error made in measuring 8·96, though only in the third place of decimals, is in the multiplication increased at least 9·975, or nearly 10 times; and therefore affects the second place. The following simple rule will enable us to judge how far a product is to be depended upon. Let _a_ be the multiplier, and _b_ the multiplicand; if these be true only to the first decimal place, the product is within (_a_ + _b_)/20[19] of the truth; if to two decimal places, within (_a_ + _b_)/200; if to three, within (_a_ + _b_)/2000; and so on. Thus, in the above example, we have 9·98 and 8·96, which are true to two decimal places: their sum divided by 200 is ·0947, and their product is 89·4208, which is therefore within ·0947 of the truth. If, in fact, we increase and diminish 89·4208 by ·0947, we get 89·5155 and 89·3261, which are very nearly the limits found within which the product must lie. We see, then, that we cannot in this case depend upon the first place of decimals, as (151) an error of ·05 cannot exist if this place be correct; and here is a possible error of ·09 and upwards. It is hardly necessary to say, that if the numbers given be exact, their product is exact also, and that this article applies where the numbers given are correct only to a certain number of decimal places. The rule is: Take half the sum of the multiplier and multiplicand, remove the decimal point as many places to the left as there are correct places of decimals in either the multiplier or multiplicand; the result is the quantity within which the product can be depended upon. In division, the rule is: Proceed as in the last rule, putting the dividend and divisor in place of the multiplier and multiplicand, and divide by the _square_ of the divisor; the quotient will be the quantity within which the division of the first dividend and divisor may be depended upon. Thus, if 17·324 be divided by 53·809, both being correct to the third place, their half sum will be 35·566, which, by the last rule, is made ·035566, and is to