Oct, lo, 1889] 



NATURE 



575 



by expanding the sines and cosines in ascending powers 

 of their arguments and equating the terms of two dimen- 

 sions. 



The method of quarter squares enables us to multiply 

 together two numbers of n figures each if we have a table 

 extending to 2 X 10". If the latter only extends to 10" 

 three entries are required, and the final result has to be 

 doubled whenever the sum of the numbers exceeds io« (as 

 in Laundy's table). If we consider the question of the 

 multiplication of two numbers of 71 figures each by means 

 of a table extending only to 10", the same process being 

 employed in all cases, it appears that three entries are 

 necessary, and that it would be better to tabulate half 

 squares, using the formula 



ab^la^ + W - \ia-bY-, 



In tabulating the half squares the fraction \ would be 

 thrown off, so that if a and b- were both uneven, unity 

 would have to be added to the result. 



It would, however, we think, if the table is not to 

 go beyond 10", be more convenient to employ a table 

 of triangular numbers. The «th triangular number is 

 \n{n X i), and if we are provided with a table extending 

 to 10" we may multiply any two numbers not exceeding 

 10" by means of the formula 



ab = h(ci - i)a + hK^ + i) - l{a - b - i)(a - b) ; 



or, as we may write it 



ab = T(a - i) + T{b) - T(a - i - b), 



T(n) denoting the nth triangular number.^ 



Thus, to multiply two numbers we subtract unity from 

 the larger number, and enter the table with the larger 

 number so diminished, with the smaller number, and 

 with the difference of these two numbers. For example, 

 to multiply 5289 and 2156, we add the tabular results 

 corresponding to 5288 and 2156, and subtract from this 

 sum the tabular. result corresponding to 3132. 



The mode of construction of a table of triangular 

 numbers is almost the simplest possible, the numbers 

 being formed by adding to zero the natural numbers 

 I, 2, 3, . . . . e.g., 



0+ I = I, I + 2 - 3, 3 + 3 = 6, 6+4=10, 10+5 = 15, 



and so on. It may be noticed also that any two consecu- 

 tive triangular numbers are the most nearly equal parts 

 into which a square of points can be divided by a line 

 parallel to the diagonal. For example, in the square of 

 16 points, the two most nearly equal triangular parts are, 

 1+2+3 = 6^ and 1+2+3 + 4=10. This is shown 

 in the following diagram : — 



\ 



■ ■ ■ ■ 



■ ■ ■ ■ ~^ 



Whether the square contains an even or an uneven 

 number of points, the diagonal, which is in the middle, 

 has to be given to one of the two parts, which there- 

 fore necessarily differ by the number of points it con- 

 tains. In the square «^, the two consecutive triangular 

 numbers which form it are l/i{n — i) and pi{n + i), dif- 

 fering, as they should, by n, the number of points in the 



' It is interesting to compare the two formula; which involve half s:iuares 

 and triangular numbers respectively. In the former case we tabulate a dis- 

 continuous function, and in the use of the formula a unit has sometimes to be 

 arbitrarily added. Jn the latter case we tabulate a continuous function, and 

 the formula always holds good (the larger of the arguments beiog always 

 reduced by unity). One formula depends en squares, «■* ; the other on 

 factorials of the second order, «(« — i). 



diagonal. Viewing the same matter from a slightly 

 different point of view, we see that any two consecutive 

 triangular numbers always make a square, e.g., 



1 + 3 = 4, 3+6 = 9, 6 + 10=16, &c. 



It is interesting to exhibit by means of a diagram the 

 manner in which the rectangle representing the product 

 ab is derived from the three triangular numbers corre- 

 sponding to a — I, b, a — I— b. As an example, the mode 

 of formation of the product 8x4 is shown below, the 

 triangular number corresponding to 7 being represented 

 by dots and the triangular number corresponding to 4 by 

 stars : — 



The dots above the line form the triangular number 

 corresponding to 7 — 4 = 3.^ 



It is not suggested that the method just described by 

 means of triangular numbers is comparable to that of 

 quarter squares. It is certainly better to double the 

 extent of the table and have but two entries. Still, it is 

 interesting to note how readily a table of triangular num- 

 bers extending only to 10" is available for the perform- 

 ance of multiplications of two //-figure numbers. So far as 

 we know, only one extended table of triangular numbers 

 has ever been published. This table, which gives the 

 value of in{?t-\- i) from « = 1 to n = 20,000, was pub- 

 lished at the Hague, by E. de Joncourt, in 1762, under the 

 title" De Natura et Pritclaro Usu Simplicissimae Speciei 

 Numerorum Trigonalium." The book is a small and 

 handsomely printed volume of 267 pages, 224 of which 

 are occupied by the table. 



In tabulating quarter squares, the fraction \ which oc- 

 curs when the square is uneven is omitted. If we denote 

 by qsq« the tabulated quarter square of 7t, we have, 

 therefore — 



qsq (2«) = it^, 



qsq (2« + 1) = n'^ + n. 



A table of quarter squares may be formed by 

 adding to zero the numbers 1, i, 2, 2, 3, 3, ... . e.g. 

 0+1 = 1, 1 + 1=2, 2 + 2 = 4, 4+2 = 6, 6 + 3 = 9, 

 9+3= 12, and so on. Its construction, therefore, is 



' We might of course alsa perform the multiplication thus ; — 



■ ■ 



■ I 



■ ■■■■■■« 



corresponding to the formula 



abulia) + 1(J)-i)-1{a- 6). 



But if unity is sr.btracted from the smaller, instead of from the larger, 

 number, slightly higher numbers are involved in the process. 



