ON MATHEMATICAL TABLES. 29 



Vkoa, 1797, Vol. ir. T. IV. ; Hantscul, 1827, T. YIII. ; *Salomon, 1827, 

 T. I.; GutisoN,11832, T. IV. and V.; Hulsse's Vega, 1840, T. VIII.; 

 Trottek, 1841 [T. VI.] ; Minsinger, 1845 [T. II.] ; Koiilek, 1848, T. VII. ; 

 WiLLicn, 1853, T. XXI. ; Bearduore, 1862, T. 35 ; *ScnLOMiLCH [18G5?] ; 

 Eankine, 1866, T. I, A ; Waceerbarth, 1867, T. VII. Sec also Centner- 

 scnwER, 1825 [T. II.] (§ 3, art. 3). And for Squares (for method of least 

 squares), Muller, 1844 [T. III.]. 



Endings of Squares. — (Three-figure endings') Lambert, 1798, T. IV. 



Art. 5. Tables of Poivers higher than Cubes. 



We know of no work coutaiuing powers of numbers (except squares and 

 cubes) only. Both Hutton, 1781, and Barlow, 1814, give the first ten 

 powers of the first hundred numbers ; but we have seen no more extensive 

 table of this kind. Shanes (§ 4) gives every twelfth power of 2 as far as 2"-' ; 

 and, according to De Morgan, John Hill's 'Arithmetic,' 1745, has all powers 

 of 2 up to 2"^ Tables of compound intei-est are, in fact, merely power tables, 



as the amount of <£M at the end of n years at r per cent, is M( 1 + | . In 



"' ^ \ 100/ 



interest tables r has usually values from 1 to 8 or 10 at intervals of | or | 

 for ditfercnt periods of years ; but they could not be of much use, except for 

 the purpose for which they are calculated. 



A good table of powers is still a desideratum, as the need for it is often 

 felt in mathematical calculations. Very many functions are expansible in an 

 ascending (convergent) scries of the form A^-{- A^x + A^.v^ -\- &c., and a de- 

 scending series (generally semiconvergont) of the form B^4-Bj.i-~^4-B2.r~- + 

 itc. The former is usually very convenient for calculation when x is small, 

 and the latter when x is large ; but between the two, for values of :c included 

 between certain limits above unitj', there will be an interval where neither 

 series is suitable — the ascending series because the terms x, x"^,. . . . (x >1) 

 increase so fast that n must be taken very large (t. e. very many terms must 

 be included) before A„ is so small that A„.r" can be neglected, and the de- 

 scending series because it begins to diverge before it has yielded as many 

 decimals as are required. For these intermediate values the former series 

 (if there is no continued fraction available) must be used ; and then the terms 

 begin by increasing, often so rapidly, if x bo moderately large, that it may be 

 necessary to calculate some of them to fifteen or twenty figures to obtain a 

 correct value for the function to only seven or eight decimals. In these 

 cases, so long as ten figures only are wanted, logarithms are employed ; but 

 when more are required recourse must be had to simple arithmetic ; and it is 

 then that a power table is so much needed. Mr. J. "W. L. Glaisher has had 

 formed in duplicate a table giving the first twelve powers of the first thousand 

 numbers, which, after the calculation has been made independently a third 

 time, will be stereotyped and published, probably in the course of 1873 ; it is 

 hoped that it will help to make the tabulation of mathematical functions 

 somewhat less laborious and difficult. 



The following tables on the subject of this article are described in § 4 : — 

 Tables of Poivers higher than Cubes. — Dodson, 1747, T. XXI. (powers of 2) 

 and T. XXII. ; Schulze, 1778 [T. VIIL] ; Htjtton, 1781 [T. IV.] ; Vega, 

 1797, Vol. II. T. II. (powers of" 2, 3, and 5); Vega, 1797, Vol. II. T. IV. ; 

 Lambert, 1798, T. VIT.-IX. (powers of 2, 3, and 5) and T. XL. ; Barlow, 

 1814, T. II. and III. ; Hijlsse's Vega, 1840, T. VI. (powers of 2, 3, 5) 

 and T. IX. A, B, D, E ; Kohler, 1848, T, II. (powers of 2, 3, and 5) and 

 T. IV. ; Shanks, 1853 (powers of 2 to 2"') ; Beakdmoee, 1862, T. 35 ; 



