28 KEPORT — 1873. 



each containing two vertical rows of numbers, the one corresponding to W, 

 iukI the other to N'; the lines are numbered 0,1,2. . . .49 (and on the next 

 double page 50 ... . 99). If, then, we wish to find the cube of 79217, we take 

 the figures 49711300i31 from column 792, lino 17, and add the last four 

 figures 1313 (which conclude the cube of 9217 in the same line) ; so that 

 the cube required is 497113061311313. Certain figures, common to the 

 whole or part of a column, are printed at the top, and the change in the 

 column is denoted by an asterisk. This is the largest table of cubes in ex- 

 istence, and (excei>t Ludolf, Avhich is of the same extent) is also the largest 

 table of squares. The printing is clear, and the book not bulky ; so that the 

 table can be readily used. At the end are eleven subsidiary tables. T. 1 

 (Perioden (jcmder Sumnienden) consists of columns marked 4, 6, 8 .... 48 at 

 the top, and 96, 94 .... 52 at the bottom, each containing the " complete 

 period " of the number in question ; thus for 42 we have 42, 84, 26, 68, 10, 

 &c. (these numbers being the last two figm'cs of a series of terms in arith- 

 metical progression, 42 being the common difference) ; and these are given 

 till the period is completed, i. e. till 42 occurs again. This may be at the end 

 of 25 or 50 additions ; if the former, the periods are given commencing 

 with 1 , 2, 3 (as well as 0) ; if the latter, with 1 or 2 only, as the case may 

 be; the periods for .r and 100 — x are of course the same, only in reverse 

 order. Tlao use of the table as a means of verifying the table of squares 

 is obvious. 



T. 2. Primes which are the sum of two squares (these being given also) 

 up to 10,529. 



T. 3. Odd numbers which are the difference of two cubes (these being 

 given also) to 12,097. 



T. 4. Odd numbers which are the sum of two cubes (these being given also) 

 to 18,907. 



T. 5-9. Four-figure additive and subtractive congriicnt endings for numbers 

 ending in 3 and 7, or 1 and 9, etc. : the more detailed description of these 

 tables belongs to the theory of numbers, which will form a j^art of a subse- 

 quent Report. 



T. 10. The 1044 four-figure endings for squares, and the figures in which 

 the corresponding numbers must end. 



T. 11. First hundred multiples of tt and tt"^ to twelve places. There is 

 appended to the tables a vcrj- full description of their object and use. 



Bruno, Faa de, 1809. T. I. of this work (pp. 28) contains squares of 

 numbers from 0-000 to 12-000, at intervals of -001 to four places (stereo- 

 typed), intended for use in connexion with the method of least squares. 



The following are references to § 4 : — 



Tables of Sqiiairs and Cuhes, or hoth Squares and Cubes. — ScnuLZE, 1778 

 [T. IX.] and [T. X.] ; Hutton, 1781 [T. II.] and [T. III.] ; Vega, 1797, 

 VoL II. T. IV. ; Lambert, 1798, T. XXXV. and XXXVI. ; Barlow, 1814, 

 T. I. ; Schmidt, 1821 [T. V.] (with subsidiary tables) ; Hantschl, 1827, 

 T. VIII. ; *Salomon, 1827, T. I. ; Geuson, 1832, T. II. and III. ; Hulsse's 

 Vega, 1840, T. IX. C. ; Trotter, 1841 [T. VI.] ; Muller, 1844 [T. III.] ; 

 MiNsiNGER, 1845 [T. II.] ; KonLER, 1848, T. V. and VI. ; Willich, 1853, 

 T. XXI. ; Beardmore, 1862, T. 35 ; IIankine, 1866, T. I. and II. ; 

 Wackeebarth, 1867, T. VI. ; PARKHrRST, 1871, T. XXVI. and XXXII., 

 and XXXIV. (multiples of squares); Peters, 1871 [T. VI.]. See also 

 Taylor, 17S0 [T. IV.] (§ 3, art. 9). 



Tables of hiquarc Roots and Cube Boots.— I) odsos, 1747, T. XIX. ; 

 ScHULZK, 1778 [T. XI.] and [T. XII.]; Maseres, 1795 (two tables); 



