RECIPROCAL

, in Arithmetic, &c, is the quotient arising by dividing 1 by any number or quantity. So, the Reciprocal of 2 is 1/2; of 3 is 1/3, and of a is 1/a, &c. Hence, the Reciprocal of a vulgar fraction is found, by barely making the numerator and the denominator mutually change places: so the Reciprocal of 1/2 is 2/1 or 2; of 2/3, is 3/2; of a/b, is b/a, &c. Hence also, any quantity being multiplied by its Reciprocal, the product is always equal to unity or 1 : so , and , and . |

Of the preceding Table, the use is evidently to shorten arithmetical calculations, and will appear eminently great to those mathematicians and others who are frequently concerned in such kinds of computations. The structure of the Table is evident; the first column contains the natural series of numbers from 1 to 1000, the 2d the Reciprocals. These Reciprocals (which are no other than the decimal values of the quotients resulting from the division of unity or 1 by each of the several numbers from 1 to 1000) are not only useful in shewing by inspection the quotient when the dividend is unity, but are also applied with much advantage in turning many divi- sions into multiplications, which are much easier performed, and are done by multiplying the Reciprocal of the divisor (as found in the Table) by the dividend, for the quotient; they will also apply to good purpose in summing the terms of many converging series.

The Reciprocals are carried on to 7 places of decimals (for the column of Reciprocals must be accounted all decimal figures, although they have not the decimal point placed before them, which is omitted to save room), each being set down to the nearest figure in the last place, that is, when the next figure beyond the last set down in the Table came out a 5 or more, the last | figure was increased by 1, otherwise not; excepting in the repetends which occurred among the Reciprocals, where the real last figure is always set down; the Reciprocals, which in the Table consist of less than seven figures, are those which terminate, and are complete within that number; such as .5 the Reciprocal of 2, .25 the Reciprocal of 4, &c.

Reciprocal Figures, in Geometry, are such as have the antecedents and conse- quents of the same ratio in both figures. So, in the two rectangles BE and BD, if AB: DC :: BC : AE, then those rectangles are reciprocal figures; and are also equal.

Reciprocal Proportion, is when, in four quantities, the two latter terms have the Reciprocal ratio of the two former, or are proportional to the Reciprocals of them. Thus, 24, 15, 5, 8 form a Reciprocal proportion, because .

Reciprocal Ratio, of any quantity, is the ratio of the Reciprocal of the quantity.

RECIPROCALLY. One quantity is Reciprocally as another, when the one is greater in proportion as the other is less; or when the one is proportional to the Reciprocal of the other. So a is Reciprocally as b, when a is always proportional to 1/b. Like as in the mechanic powers, to perform any effect, the less the power is, the greater must be the time of performing it; or, as it is said, what is gained in power, is lost in time. So that, if p denote any power or agent, and t the time of its performing any given service; then p is as 1/t, and t is as 1/p; that is, p and t are Reciprocally proportionals to each other.