244 Proceedings of Royal Society of Edinburgh. [sess. 
The Theory of Ortkogonants in the Historical Order of 
its development up to 1832. By Thomas Muir, 
LL.D. 
(Read May 19, 1902.) 
The special form of determinant to which we have now come is 
connected with a problem in coordinate geometry — the problem 
of transformation from one set of rectangular axes to another set 
having the same origin. The actual appearance of determinants 
in any of the attempts to solve the geometrical problem did not 
take place until comparatively late in its history, probably because 
the connection between the two subjects was less patent than in 
other cases, the problem when transformed into algebraical language 
being not a mere matter of elimination of unknowns from a set of 
linear equations. The earlier portion of the history of orthogonal 
substitution, although of considerable interest, is thus not sufficiently 
germane to our subject to warrant detailed treatment of it. For 
those interested in this earlier portion it will suffice to give the 
following chronologically arranged list of papers from 1770 to 
1840:— 
1748. Euler. Introductio in Analysin Infinitorum. Tomi duo. 
Lausannae et Genevae (v. ii. Appendix de Super- 
ficiebus*). 
1770. Euler. Problema algebraicum ob affectiones prorsus 
singulares memorabile. Novi Commentarii Acad . 
Petrop., xv. p. 75; or, Commentationes Aritli. Collectae , 
i. p. 427. 
1772. Laplace. Eecherches sur le calcul integral et sur le 
systeme du monde. Hist, de Vacad. roy. des sciences 
(Paris), 2 e partie, pp. 267-376. 
1773. Lagrange. Nouvelle solution du probleme du mouvement 
de rotation d’un corps de figure quelconque qui n’esfc 
anime par aucune force acceldratrice. Nouv. mem. de 
Vacad. roy. . . . (Berlin.), pp. 85-120. 
* Or in Labey’s French Translation, ii. pp. 370-378. 
