548 Proceedings of the Eoyal Society 
smoothness in the narrative by doing away with the necessity of 
frequent harkings back, but it will also be of material importance 
to investigators who may wish to find out what has already been 
done in advancing any particular department of the subject. To 
fihis end, also, each new result as it appears will be numbered in 
Eoman figures ; and if the same result be obtained in a different 
way, or be generalised, by a subsequent worker, it will be marked 
among the contributions of the latter with the same Eoman figures, 
followed by an Arabic numeral. Thus the theorem regarding the 
effect of the transposition of two rows of a determinant will be 
found under Vandermonde, marked with the number xi., and the 
information intended thus to be conveyed is that in the order of 
discovery the said theorem was the eleventh noteworthy result 
obtained: while the mark xi. 2, which occurs under Laplace, is 
meant to show that the theorem was not then heard of for the first 
time, but that Laplace contributed something additional to our 
knowledge of it. In this way any reader who will take the trouble 
to look up the sequence xi., xi. 2, xi. 3, &c., may be certain, it is 
hoped, of obtaining the full history of the theorem in question. 
The early foreshadowings of a new domain of science, and tenta- 
tive gropings at a theory of it, are so difficult for the historian to 
represent without either conveying too much or too little, that the 
only satisfactory way of dealing with a subject in its earliest stages 
seems to be to reproduce the exact words of the authors where 
essential parts of the theory are concerned. This I have resolved 
to do, although to some it may have the effect of rendering the 
account at the commencement somewhat dry and forbidding. 
No author, so far as I am aware, has preceded me in the task I 
have chosen. Sketches of the history have appeared in a number 
of text-books of the subject, notably in Gunther’s Lehrbuch der 
Determinanten-Theorie fur Studirende (2 te Aufl. xii. 209 pp., 
Erlangen, 1877), which contains a considerable quantity of detail. 
The early history has been very carefully dealt with by E. J. 
Studnicka, in a memoir published in the Abhandlungen der honigl. 
bohm. Gesellschaft der Wissenschaften , 6 Eolge, viii. 40 pp. (24th 
March 1876), and entitled “A. L. Cauchy als formaler Begriinder 
der Determinanten-Theorie. Eine literarisch-historische Studie.” 
There is also an academic thesis ( Teorin for Determinant-Kalkylen, 
