ON MATHEMATICAL TABLES. 33 



" In ail}' three consecutive vulgar fractions in the table, if the numerators of 

 the extremes and the denominators be added together, the sum will form the 

 numerator and denominator of a fraction equal to the mean." That this is 

 the case with all fractions, ranged in order, whose numerators and denomi- 

 nators are integers less than given integers, is a theorem discovered by Cauchy 

 and published by him in his ' Exercices.' 



It has been thought worth while to describe Goodwyn's works at some 

 length, as they are almost unique of their kind, and are rarely to be met 

 with. 



De Morgan states that " Mr. Goodwyn's manuscripts, an enormous mass 

 of similar calculations, came into the possession of Dr. Olinthus Gregory, 

 and were purchased by the Eoyal Society at the sale of his books in 1842."' 

 There is no mention of them, however, in the lloyal Society's Catalogue of 

 MSS. ; and nothing is known of them at the Society. They may possibly be 

 brought to light in the rearrangement of the manuscripts consequent upon the 

 approaching change of rooms. 



Art. 7. Tables of Mecijirocals. 



The most extensive table is 



Oakes, 1865. Reciprocals from 1 to 100,000. This table gives seven figures 

 of the reciprocal, and is arranged as in tables of seven-figure logarithms ; viz. 

 the first four figures are found in the column at the left-hand side of the page, 

 the fifth figures run along the top line, and the sixth and seventh are inter- 

 polated for by proportional parts. The reciprocal of a number of five figures 

 is therefore taken out at once, and the process of taking out a recipi'ocal is 

 exactly similar to that of taking out a logarithm. 



From 10,000 to 22,500 the differences and proportional parts (being 

 numerous) are placed on the lower half of the page, the differences being 

 also placed at the side of each line ; but above 22,500 the differences and 

 proportional parts are placed at the side of the page as in tables of logarithms. 

 The figures have heads and tails ; and the change in the third figure of the 

 reciprocal is made evident by prefixing an asterisk to the succeeding numbers 

 in the line. The table is the result of an original calculation, and was con- 

 structed by means of the obvious theorem that the difference of two recipro- 

 cals, divided by the difference of the corresponding numbers, is the reciprocal 

 of the product of those numbers. The reciprocals of the highei' numbers, 

 however, were calculated by differences, which difterences were found by 

 logarithms. Various checks were applied ; and the whole was virtually re- 

 computed on the Arithmometer of M. Thomas de Colmar. The significant 

 figures of the reciprocals alone are tabulated, decimal points and ciphers 

 being omitted, for the same reason that characteristics are left out in loga- 

 rithmic tables. 



In T. I. of Baelow (§ 4) reciprocals are given of numbers from 1 to 10,000 ; 

 and this table also appears in the stereotype reprint of 1840 (see § 3, art. 4) : 

 the latter is the most generally used table of reciprocals, and is of sufficient 

 extent for most purposes ; it is also reputed to be very accurate, and is perhaps 

 free from error. 



It must be added that Goodwyn's ' Table of Circles,' and ' Tabular Series,' 

 &c., 1823 (§ 3, art. 6), give reciprocals of numbers less than 1024 complete; 

 viz. the whole period is given, even where it exceeds a thousand figures. 



See also the reference to Gauss, vol. ii., near the beginning of the last 

 article (§ 3, art. 6). 



As most nearly connected with a table of reciprocals (it gives not only 

 1873. D 



