ON MATHEMATICAL TABLES. 21 



parts of all numbers from 43 to 434 (note that near the commencement of 

 the table, viz. from diff. 434 to diff. 340, the proportional parts are only 

 given for every other difference in some tables ; whether a table gives the 

 proportional parts of all the differences or not is generally noted in § 4), 

 ISeveral scven-fignre tables extend to 108,000 ; and for the last 8000 the dif- 

 fei'ences decrease from 434 to 403. Tables in which «=60 often accompany 

 canons of trigonometrical functions that give the results for every minute, for 

 convenience of interpolating for seconds; such must be sought from the 

 descriptions of trigonometrical tables in § 3, arts. 10 and 15, and in § 4 ; 

 we have also seen tables for which rt=30, where the functions are tabulated 

 for every two minutes or two seconds. 



There are several tables to which proportional parts of the differences to 

 hundredths (viz. in which «=:100) are attached, e.g. Gray (§ 3, art. 19), 

 FiLiPowsKi (§ 4), and Pineto (§ 3, art. 13) ; but the ranges of the differences 

 arc generally so small that it is not worth while giving references. In 

 PiNETO, for instance, the range of the differences is only from 4295 to 4343 

 (in this work multiples are given, the last two figures being separated by a 

 comma). 



The only separate table of proportional parts, properly so called, that we 

 have seen, is 



Bremiker, 1843 (' Tafel der Proportionaltheile '). Proportional parts to 

 hundredths (viz. miiltiples from 1 to 100, with the last figure omitted, and 

 the last but one corrected) of all numbers from 70 to 099. A very useful 

 table, chiefly intended for use in interpolating for the sixth and seventh figures 

 in logarithmic calculations. 



T. III. of ScnEON (§ 4) (which is there called an Interpolation Table) is a 

 large table of proportional parts. 



It is to be noticed that all multiplication tables are, or rather can be used 

 as proportional-part tables. A table of multiples, with the last figure omitted, 

 and the last but one corrected (which can be done at sight), is a proportional- 

 part table to tenths ; and if the last two figures are omitted, and the last 

 remaining figure corrected, to hundredths (see therefore § 3, arts. 1 and 3). 



It is proper here to allude to slide-rules and other mechanical appliances 

 for working proportions &c. A card intended to do the work of a very large 

 slide-rule is described in § 4 (Eveeett) ; and some information and references 

 about slide-rules of different shapes M'ill be found in a paper " On a New 

 Proportion Table," by Prof. Everett, in the Phil. Mag. for Nov. 1806. 



The following are references to works described in § 4 : — 



T(djJes of Fro2>ortio7ial Parts.— iiir J. Mooke, 1681 [T. II.]; DucoM, 1820, 

 T. XX.; Lynn, 1827, T. Z; Callet, 1853 [T. YIII.]; Schkon, 1860, 

 T. Ill, 



Art. 3. Tables of Quarter Squares. 



Tables of quarter squares have for their object to facilitate the performance 

 of multiplications ; and the principle on which their utility depends is con- 

 tained in the formula 



ab = i(a+hy-l{a-hy-, 



so that with such a table to multiply two numbers we subtract the quarter 

 square of the difference from that of their sum ; the multiplication is there- 

 fore replaced by an addition, a subtraction, two single entries of the tables, 

 and a final subtraction — a very considerable saving if the numbers be high. 

 The work is more than with a product table, where a double entry gives the 

 result at once ; but the quarter squares occupy much less space, and can 



