574 



NATURE 



{Oct. lo, 1889 



'is not felt till we are provided with such a table giving the 

 ■product of two five-figure numbers. As already stated, 

 'the fact that the limit of Laundy's table was only ioo,od3 

 has deprived it of most of its value, for it is obvious that, 

 ^utiless all five-figure numbers can be treated by a uniform 

 ""method, the table could not be conveniently employed in 

 practice. 



Mr. Blater's work consists of the principal table (giving 

 quarter squares up to 200,000), which occupies 200 pages ; 

 a small table of four pages, called the index, to facilitate 

 the use of the table in the extraction of square roots ; and 

 an introduction, &c., of fourteen pages. 



The arrangement of the table (in which the author has 

 followed the plan adopted by Kulik in his table of 1852, 

 already referred to) is somewhat peculiar. The table is 

 first entered {i.e. the required page of the table is found) 

 by means of the last three figures of the number : the 

 table is then entered on this page (or, more correctly, 

 double page), by means of the preceding figures. For 

 example, the quarter square corresponding to 126,993 is 

 found by turning to the double page headed 990. In one 

 of the four columns headed 993 we enter the table at the 

 line 126: from this line we obtain, in the first column, 

 the first four figures of the result, 4031 ; in the column 

 under 993, the next three, 805 ; from the bottom of the 

 column we take the last three figures, 512. The result is 

 therefore given in three parts A, B, C ; A being common 

 to ten numbers (in the same line) beginning with 126, C 

 being common to fifty numbers (in the same column) 

 ending with 993, and B being special to the combination 

 126,993. 



The table is beautifully printed in large antique figures 

 on thick and excellent paper, and is a handsome piece of 

 typography. The author mentions that it was entirely set 

 up by a single compositor at the printing-office of Mr. 

 . Falk, at Mayence, and that it occupied his whole time for 

 eleven months. Besides being admirably printed, the 

 table is no doubt very correct, as a triple calculation was 

 made, and no pains seem to have been spared by Mr. 

 Blater for insuring accuracy. 



The book is dedicated to Mr. Anthony Steinhauser, of 

 Vienna, who has contributed a short historical preface. 

 Mr. Steinhauser, who is himself the author of several 

 logarithmic tables, encouraged Mr. Blater in his work, 

 and rendered him great assistance throughout. The 

 actual calculation occupied eighteen months. 



With respect to the general employment of Mr. Blater's 

 table for the performance of multiplications, it is to be 

 feared that its utility has been jeopardized by the size of 

 page adopted. Anyone who has had occasion to make 

 constant use of tables knows the enormous advantage of 

 the octavo form over the quarto. The book is placed to 

 the left of the computer, and the effort of carrying by 

 the eye a series of figures from the left hand page of a 

 quarto table to the paper — a distance of 18 inches to 

 2 feet — is but ill compensated for by larger figures of 

 fewer pages. Handsome as the book is to look at, it 

 seems to us that the table would have had much more 

 chance of bringing the method into general use if it had 

 been of octavo form. It is greatly to be regretted that 

 it was not printed on 400 octavo instead of 200 quarto 

 pages, which would have been quite possible with the 

 existing arrangement of the table. If this had been done, 

 and if the type had been somewhat smaller, a neat and 

 handy volume might have been produced. 



The mode of entering the table is very insufficiently 

 explained in the introduction. This is unfortunate, as 

 the mode of entry (by the last figures) is so unusual in 

 tables that it should have been explicitly mentioned. 

 Also the translation into English is so very unsatisfactory 

 as to be obscure in places. These, however, are minor 

 blem.ishes which would have but slight influence on the 

 general utility of the table, if only the form were 

 convenient. 



The question of how far the method of quarter 

 squares is likely to come into use is of some interest. 

 Hitherto the method has been very little known, 

 and, so far as we know, it has never been used in 

 practice on any extended scale. The mere fact that it 

 has been so constantly discovered anew is sufficient evi- 

 dence of the slight attention that it has received. Still, 

 there ought to be room for a table that gives, to the last 

 figure, the products of any two five-figure numbers with 

 only two entries. A seven-figure table of logarithms is 

 inadequate for this purpose, for, besides requiring three 

 entries, it only gives the first seven figures of the result. 

 On the other hand, it may be said that in ordinary calcula- 

 tions seven figures are as many as are required, and 

 that logarithms possess the advantage of being equally 

 convenient for divisions and multiplications. It must be 

 admitted that a five-figure quarter- square table is appro- 

 priate to only a limited class of calculations : it applies 

 only to multiplications, and the number of figures in each 

 of the two numbers must not be greater than five. These 

 conditions are of a somewhat special kind. In recent 

 years when heavy multiphcations have been required it 

 has become the custom to make use of Thomas de 

 Colmar's arithmometer ; and probably, at the present 

 time, nearly all systematic work of this character is carried 

 out either by Crelle's tables or by the arithmometer. 



Passing now to the general question of multiplication 

 by means of a table of single entry, we have the two 

 niethods of quarter squares and logarithms, each pos- 

 sessing its special advantages. There is also an older 

 method which passed out of notice with the invention of 

 logarithms. This method was called " prosthaphaeresis," 

 and depended on the formula 



sinrt %v!\b = i[sin!90' — [ii - b)] - sinlgo'' - {a -f b))']. 



A table of natural sines could therefore be used as a 

 multiplication table, four entries being required. This 

 method is due to Wittich, of Breslau, who was assistant 

 for a short time to Tycho Brah^, and it was used by 

 them in their calculations in 1582. It is referred to by 

 Raymarus Ursus, Clavius, and Longomontanus : and it 

 seems to have been used for performing multiplications 

 not only when the numbers occurred as sines but also in 

 the case of ordinary numbers. 



The method of quarter squares depends upon so simple 

 a formula, that it is strange that the first table should not 

 have appeared until 1817. There seems no reason why 

 it should not have been employed before the invention of 

 logarithms, when it would have been a most valuable aid 

 to calculation. The geometrical theorem, which is equi- 

 valent to the algebraical identity {a -\- bf - {a - /')- = ^(ih, 

 on which the method depends, forms Prop. viii. of the 

 second book of Euclid ; and one would think that the 

 application of the geometrical] or algebraical theorem to 

 arithmetic might have been noticed at any time. The 

 actual history of mathematical tables is, however, entirely 

 different from what we might expect it to have been, 

 owing to the wonderfully early invention of logarithms : 

 and it was, in fact, only just about that time that the 

 importance of tables as an aid to general calculation was 

 beginning to be felt. The date of Herwart ab Hohen- 

 burg's great double-entry multiplication table, extending 

 to 1000 X 1000 (the same limit as Crelle's table, and 

 which has never been exceeded) is only four years earlier 

 (1610) than that of Napier's " Canon Mirificus" (1614). 



It is interesting to notice that the method of quarter 

 squares is more closely connected mathematically with the 

 method of prosthaphseresis than with that of logarithms ; 

 in fact, from the formula 



sin a sin b = i {cos((2 — 

 we readily deduce 



ab = l['a + b)^ 



b) - cos{a -f b)] 

 - {a-bf\. 



