by a Table of Single Entry. 341 



The connexion between prosthaphaeresis and quarter squares, 

 and between the generalization of the prosthaphaeresis formula, 

 as given in § 4, and the formula for the product of n quan- 

 tities as a sum of nth powers, viz. that the algebraical formulas 

 may be obtained by equating terms of the same order in the 

 expansions of the trigonometrical formula, is noteworthy ; but, 

 as mentioned in § l,the trigonometrical formula only requires 

 one and the same table (a table of sines), however many quan- 

 tities have to be multiplied together, while the algebraical for- 

 mula requires a table of squares to multiply two numbers, a 

 table of cubes to multiply three, and a table of nth powers to 

 multiply n numbers ; and to multiply n numbers, 2 n ~ 1 entries 

 would be required. T\ r e might, however, multiply two numbers 

 by means of a table of quarter squares, and then multiply their 

 product by the third number and so on (which would also re- 

 quire 2 n_1 entries) : but we should only obtain the exact 

 value of the result if all the products were included within 

 the limits of the arguments of table, i. e. if the sum of the 

 largest multiplier and the product of the other factors be within 

 these limits. Practically, however, the sine table would only 

 contain a certain number of decimal places ; and if we assume 

 the sine table to be perfect so as to render any number of multi- 

 plications possible, we ought at the same time to assume the 

 quarter-square table to be extended ad libitum; so that the theo- 

 retical advantage of the trigonometrical formula, as only re- 

 quiring one table, is more apparent than real, if we admit the 

 repeated use of the quarter-square table. 



As just remarked, a table of sines, like a table of logarithms, 

 is onlv available for obtaining results to a certain number of 

 figures ; and the superiority of the quarter-square method, 

 when only numbers within the range of the table have to be 

 multiplied together, is very decided. 



§ 9. The title of Herwart ab Hohenburg's multiplication 

 table is " Tabulae arithmeticae TrpoaOafyaipeaews universales, 

 quarum subsidio numerus quilibet, ex multiplicatione produ- 

 cendus, per solam aclditionem ; et quotiens quilibet, e divi- 

 sione eliciendus, per solam subtractionem, sine taediosa & 

 lubrica Multiplicationis, atque Divisionis operatione, etiam ab 

 eo, qui Arithmetices non admodum sit gnarus, exacte, celeriter 

 & nullo negotio invenitur. E museo Ioannis Georgii Herwart 

 ab Hohenburg .... Monachii Bavariarum. Anno Christi, 

 m.dc.x." The book is a very large and thick folio, containing 

 a multiplication table up to 1000 x 1000, the thousand mul- 

 tiples of any one number being given on the same page ; and 

 there is an introduction of seven pages, in which the use of 

 the table in multiplying numbers containing more than three 



