ON MATHEMATICAL TABLES. 25 



and proportional parts. The figures are very clear ; and there is a full intro- 

 duction, with explanations of the use, &c. of the tables. 



Mr. Laundy was induced to construct his tabic by Prof. Sjdvester's paper 

 in vol. iv. of the 'Assurance Magazine,' referred to above ; and a description 

 of the mode of construction &c. of the table (most of which is also incor- 

 porated in the introduction to it) is given in vol. vi. of the ' Assurance 

 Magazine.' 



Art. 4. Tables of Squares, Cubes, Square roots, and Cube roots. 



Tables of squares (or square roots of square numbers) are of nearly as 

 great antiquity as multiplication tables, and would, wc think, be found to be 

 rather common in early manuscripts on arithmetic. They are, as a role, but 

 slightly noticed in histories of the subject (see references in § 3, art. 1), partly 

 because the latter are very meagre, and very many manuscripts remain still 

 unexamined, and partly because it is rather the province of a history to de- 

 scribe the improvement of processes. The perfection of the methods of ex- 

 tracting the square root of numbers not complete squares, however, occupies 

 a conspicuous place. 



In the MSS. Gg. ii. 33 of the Cambridge University Library, are two frag- 

 ments, one of Theodorus Meletiniotes, the second of Isaac Argyrus (botb much of 

 the same date, time of John Palaeologus, 1360) (concerning the first, see Vin- 

 cent, Manuscrit de la Bibliotheque Imperiale, xix. pt. 2. p. 6). The fragment 

 is a portion of the first book, and contains rules and small tables for multi- 

 plication, fractional computation &c. 



The tract of Isaac Argyrus is entitled " tov 'Apyvpou evpeffis rwy Terpayw- 

 riKiJii}' TrXevpdJi' tiLv fxt) prjrwi' uptdjjiiiijy. 



At the end there is a table of the square roots of all integral numbers from 

 1 to 120, in sexagesimal notation. The table is prepared as if for three 

 places of sexagesimals ; but usually two only are perfect. Errors (probably 

 due to the copyist) are frequent. Before the table is a description of the 

 method of its use, including an explanation of the method of proportional 

 parts. 



De Morgan speaks of two early (printed) tables in Pacioli's ' Summa,' 

 1494, and by Cosmo Bartoli, 1564, extending respectively to the squares of 

 100 and 661. The tables which we have examined are described below; but 

 there are several of some extent, which De Morgan refers to, that we have not 

 seen, viz. : — Guldinus, 1635, squares and cubes to those of 10,000 ; AY. Hunt, 

 1687, squares to that of 10,000 ; and J. P. Biichner's ' Tabula Radicum,' 

 Nuremberg, 1701, which gives squares and cubes up to 'that of 12,000 (full 

 title given in Eogg). Lambeet (Introd. ad Suppl. &c. 1798) says that 

 Biichner's table is " j^lena errorum." Eogg gives the title " Bobert, K. W., 

 Tafeln der Quadratzahlen aller natiii-lichen Zahlen von 1-25,200 ; der Kubik- 

 zahlenvon 1-1200; der Quadrat- u.Cubicwurzeln von 1-1000. Neu berechnet, 

 Leipzig, 1812 ;" and the title occurs in the Eoy. Soc. Lib. Cat. (though the 

 book is not to be found in the Library). De Morgan mentions " Schiert, 

 'Tafeln,' &c. Eohn om Eheim, 1827," as giving squares to 10,000, which is 

 no doubt a misprint for " Schiereck, J. F., Tafeln aller Quadrate von 1 bis 

 10,000. 4to. Koln am Ithein, 1827," which occurs in the Babbage Catalogue, 

 and also in Eogg. From the title of another work of Schiercck's given in 

 the former catalogue, it appears that the table of squares also appeared as an 

 appendix to his ' Handbuch fiir Geometer,' published in the same year. 



De Morgan speaks of Ludolf's ' Tetragonometria,' 1090, which gives 

 squares up to that of 100,000, " as being the largest in existence, and very 



