24 EEroRT — 1873. 



preface) the Bmaller multi])lication table is not added ; squares and cubes only 

 arc given in the other small table. 



Centnerschwer, lb25. [T. I.] A table of quarter squares to 20,000 ; viz. 



-r, is tabulated from .r=l to .r=20,000, the fraction I, -which occurs -when 

 -± 



X is odd, being omitted. The last two figures of the quarter square, whicli 



only depend on the last two figures of the number, are given once for all 



on two slips bound up to face pp. 2 & 41. 



TuU rules are given as to how to use the table as a table of squares ; and 

 three small tables are added, by means of which the square of any number 

 of five figures can be found tolerably easily. The arguments are printed 

 in red. 



[T. II.] Square roots of numbers from 1 to 1000 to six places. 



There is a long and full introduction prefixed. 



In his prefiice Centnerschwer states that after his work was in the press, 

 lie received from C'relle a table, by J. A. P. Burger, entitled " Tafeln zur 

 Erleichtcrung in Eechnungen," Karlsruhe, 1817, in wbich the author claims 

 to be inventor of the method, while Centnerschwer states it was known to 

 LuDOLF (1690), and even Euclid. That Ludolf was the inventor of the 

 method is true ; and there is attached to his work a table of squares to 

 100,000 (see Ludolf, § 3, art. 4). 



The full title of Biirger's work, which we have not been successful in ob- 

 taining a sight of, is (after Hogg) as follows : — " Tafeln zur Erleichtcrung in 

 llechnungen fiir den allgemeinen Gebrauch eingerichtet. Deren ausserst ein- 

 fach gegebene Ilcgeln, nach Avelchen man das Product zweier Zahlen ohne Mul- 

 tiplication finden, auch sie sehr vortheilhaft bei Ausziehung der Quadrat- iind 

 Cubicwurzel anwendcn kann, sich auf den binomischen Lehrsatz griinden. 

 Nebst Anhaug iiber meine im vorigen Jahr erschienene Paralleltheorie. 

 Carlsruhe, 1817. 4to." The book last referred to was entitled "VoUstiindige 

 Thcorie der Parallellinien &c. Carlsruhe, 1817 ; 2nd edit. 1821," as given 

 by Hogg under Elcmentar-Geometrie. 



Merpaut, 1832. The premeire partle gives the arWmome (/. e. quarter 

 square) of all numbers from 1 to 40,000, so arranged that the first three 

 figures of the argument are sought at the head of the table, the fourth figure 

 at the head of one of the vertical columns, in which, in the line with the final 

 (fifth) figure in the left-hand column, is given the quarter square required. 

 The quarter squares are printed in groups of three figures, the second group 

 being under the first, &c. A specimen of this table is given by LArNnr 

 (1850, p. V of his Introduction). 



The deuxibne ]partie gives the reciprocals of all numbers from 1 to 10,000 

 to nine figures. 



The author seems not to have been aware of the existence of any of the 

 previous works on the subject of quarter squares. 



Laundy, 1S5G. Quarter squares of all numbers from unity to 100,000, 

 the fraction |, which occurs when the number is odd, being, as usual, omitted. 

 The arrangement is rs in a seven-figure logarithm table ; viz. the first four 

 figures are found in the left-hand column, and the fifth in the top row ; the 

 three or four figures common to the block of figures are also separated as in 

 logarithmic tables, and the change in the fourth or fifth figure is denoted by 

 an asterisk prefixed to all the quarter squares affected : at the extreme left 

 of each page is a column of corresponding degrees, minutes, and seconds 

 (thus, corresponding to 43510 we have 12° 5' 10" = 43510"). At the bottom 

 of the page arc differences (contracted by the omission of the last two figures) 



