676 Proceedings of the Royal Society of Edinburgh. [Sess. 
XLI. — The Theory of General Determinants in the Historical 
Order of Development up to 1860. By Thomas Muir, LL.D. 
(MS. received June 15, 1908. Read July 13, 1908.) 
My last communication in reference to the history of general determinants 
dealt with the period 1844-1852 ( Proc . Roy. Soc. Edin., xxv. pp. 908-947). 
The present paper continues the history up to 1860, but in addition contains 
an account of five writings belonging to previous periods, namely, by 
Bianchi (1839), Chelini (1840), Terquem (1846), Hermite (1849), Salmon 
(1852). 
Bianchi, G. (1839, January). 
[Sopra l’analisi lineare per la risoluzione dei problemi di primo grado. 
Mem. della Soc. ital. delle Sci., xxii. pp. 184-227.] 
Bianchi’s knowledge of previous work on simultaneous linear equations 
must have been slight — confined, probably, to an acquaintance with Cramer’s 
rule and with Cauchy’s so-called “ symbolical ” solution as given in the Cours 
d’ Analyse of 1821 : unless this were so, he would scarcely have referred to 
the methods given in such text-books as Ruffini’s Elementary Algebra and 
Euler’s Elements. One is thus prepared to find little new in his conscien- 
tiously laboured monograph, consisting of an introduction of five pages, a 
section of twenty -seven pages on the solution of a set of n equations with 
n unknowns, and a section of twelve pages on n equations with fewer 
unknowns. The main interest lies in the first fifteen pages (pp. 189-204) 
of the earlier section, these being devoted to establishing the validity of 
Cramer’s rule. The procedure consists in eliminating one and the same 
unknown between the first equation and each of the other equations of the 
set, then in treating in the same way the set of n — 1 equations thus derived, 
and so on until a single equation = D results. As negligible factors are 
not struck out in the course of the work, the discovery of the law of 
formation of the coefficients in the successive sets of equations is made un- 
necessarily difficult, and N and D are obtained in unwieldy forms. Thus, 
in the case of the six equations 
a 1 z 1 + b x x 2 + . . . . + = Sj \ 
a 2 x i + b 2 x 2 + ....+ f 2 x 6 = s 2 > 
