April 26, 1883} 
NATURE 
599 
with a view to sparing the patience of both, I shall 
restrict myself to a few general remarks. In both the 
great branches of mathematics, viz. geometry and algebra, 
new schools have arisen within the last few years. In its 
primary aspect the movement has tended to separate the 
two; geometry has become more purely geometrical in 
its conceptions and methods, algebra more independent 
of geometrical considerations. The geometry of to-day 
is more like the Greek than was that of fifty years ago; 
and yet at the same time they have not only many prin- 
ciples really in common, but many methods which, 
although independent, are strictly analogous. Geometry 
regards its figures, algebra its forms, not as isolated indi- 
viduals, but as associated with others (concomitants, as 
they are called) whose properties characterise those of 
their primitives. The principles of both may be regarded 
as the same, but dual in their application. Geometry, 
again, is dual within itself: points and lines may be so 
viewed that theorems concerning the one give rise to 
analogies concerning the other; the principle the same, 
but dual in its manifestation. In this way we seem to be 
rising to laws which transcend the distinctions between 
the two parts of geometry—between geometry and 
algebra. 
“ Descending a little further into particulars, in another 
way again we seem to be gaining some steps—but as yet 
only a few steps—towards a higher scheme both of 
geometry and algebra. There are a few certain relations 
so elementary in their conception, yet so universal in 
their application, that they seem capable of forming the 
basis of extensive theories: such, for example, in 
geometry, is that of Anharmonic Ratio—a particular kind 
of ratio applicable alike to points and rays, to lines and 
to angles, on which M. Chasles has founded his new and 
classical work on Conic Sections. Such, again, in 
algebra, are those of homogeneity and of symmetry, 
which prove to be not merely improvements in form, but 
actually new powers for progress in the hands of the 
mathematician. The calculus of homogeneous forms has 
marked a new era in the history of algebra ; the theory 
of equations has been transfigured in its light ; mechanics, 
both ordinary and molecular, have been elucidated by it ; 
and the remote applications of the integral calculus have 
felt its ever-extending influence. Under these, as it were, 
new fundamental conceptions, whole theories may be co- 
ordinated, and of these, again, perhaps some coordina- 
tion may one day be contemplated. As another instance 
of this generalisation of principles and of this dual aspect 
of the principles so generalised within almost the present 
generation, it has been discovered, or at all events been 
duly realised, that symbols of operation combine accord- 
ing to definite laws, comprising as a particular case those 
of ordinary number. This fertile idea has, year by year, 
been receiving fuller developments, till it has at last 
assumed the form of a complete calculus.’’ 
We, too, must join our apologies with those of the 
learned speaker for lingering so long upon a favourite 
subject. 
The following is as complete a list of Mr. Spottiswoode’s 
papers as we have been able to make:' they are 
grouped, not according to subjects nor in order of time, 
but as they occur in the several journals in which they 
originally appeared :— 
Phil. Magazine.—(1) On the Equation Q=g (a, 1, y, 2) 
* We trust our readers will pardon our imperfect treatment of these 
papers: we had formed quite a mass of notes—a “‘ rudisindigestaque moles "’— 
but we have had, through circumstances over which we had no control, an 
utterly inadequate periud in which to prune them and shape them into 
comely form. The prefixed numbers are those of the ‘‘ Royal Society's 
Catalogue ”’ and the notes are in most cases derived from the papers them- 
selves. In our haste we have preferred to insert notes to the less familiar 
papers ; the papers read before the Royal and Mathematical Societies are 
without doubt those by which Mr. Spottiswoode’s rank as a mathematician 
has been determined, but these are just the ones that are most familiar to 
students. 
=w+ir+jy+ kz (vol. xxxvi. 1850); this is a theorem 
of considerable importance in the calculus of quaternions, 
and indeed essential for the application of that method 
to geometrical and physical problems. (2) On the Qua- 
ternion Expressions of Coplanarity and Homoconicism 
(z.). (3) On the Geometrical Interpretation of Qua- 
ternions (vol. xxxvii. 1850), the working out on other lines 
of results stated in a previous volume by Prof. Donkin. 
(31) On a Geometrical Theorem (7é., 1850), viz. if three 
cones of the second order, having a common vortex, 
intersect one another two and two, the nine lines of inter- 
section (three being selected from each pair of cones) 
will lie on a cone of the fourth order. (7) On a Problem 
in Combinational Analysis (vol. iii. 1852) connected with 
the 15-girl Problem and a more general form of it, the 
solution of which turns upon certain determinants. (41) 
On some Experiments on Successive Polarisation of 
Light made by Sir C. Wheatstone (vol. xli. 1871); the 
introduction of instrumental means for converting the 
plane of polarisation of the ordinary apparatus into 
successive, or, as it is more commonly called, circular 
polarisation, and the explanation of the phenomena 
thence arising, constitutes the main purpose of the 
communication. See also Proc. of R. Inst., vol. vi. 1872. 
1875 (a) on a Revolving Polariscope; 1882 (4) on a 
Separator and Shunt for Alternate Currents. 
Camb. and Dub. Math. Fournal—(4) On certain Geo- 
metrical Theorems (vol. vi. 1851). This is an anonymous 
article which gives simple algebraical demonstrations of 
certain of Steiner’s Theorems in the Systematische Ent- 
wickelung, and also of some relations given by M. Chasles 
in his “Apergu.’’ (9) On Certain Theorems in the Cal- 
culus of Operations (vol. viii. 1853); an extension of 
theorems by Boole (P&zl. Trans., 1844), relating to the 
operation symbol Dax", and by Carmichael relating to 
BS 
the symbol vy = 1% ie + x, a 
ax, “AX, 
which the order of the Variables by which the Symbols of 
Differentiation are Multiplied is not the same as that of 
the Variables with respect to which the Differentiations 
are to be performed; (2) in which the Variables by 
which the Symbols of Differentiation are Multiplied are 
any linear Function of the Given Variables; (10) on 
Certain Geometrical Theorems (26. 1853); two Ele- 
mentary Theorems in anharmonics proved by aid of deter- 
minants. (11) On the Curvature of Curves in Space 
(vol. ix. 1854) ; on this M. Chasles (‘‘ Rapports,”” p. 162) 
remarks : “‘M. W. Spottiswoode est parvenu a la méme 
expression dans une Note... ” ze. to the expression— 
7 x 4 
1 = cosec $) = + = == ee 
P p Pis P P14 : 
Quarterly Journal of Mathematics—(15) Note on 
Axes of Equilibrium (vol. i. 1857). The axes (Mobius, 
““Statik’’) possess the property of allowing the body to 
be turned about them, the forces retaining their directions 
in space without a disturbance of equilibrium. The 
paper is an application of formule given by Rodrigues to 
a proof of the property. (16) Ona Theorem in Statics 
(26. 1857) is a proof of the following, due to Mobius 
(“Statik’’): “ If there be any forces in equilibrium, and 
a series of pyramids be constructed having for one edge 
a common line, and for their opposite edges the lines 
which represent the forces, in both magnitude and direc- 
tion, respectively, the algebraical sum of the volumes of 
the pyramids will vanish. It is of this M. Chasles 
(“ Rapport,” p. 59) writes: “ M. W. Spottiswoode, 4 qui 
toutes les ressources des nouvelles théories de l’analyse 
sont si familiéres, s’est plu a les appliquer 4 la démon- 
stration de cette proposition (¢.e. Mobius’s) et d’un autre 
passage du traité de statique de Mobius, sur les axes de 
lV équilibre.” (23) On Petzval’s Asymptotic Method of Solv- 
ing Differential Equations (vol. vy. 1862). Also ina somewhat 
different form in 4rzt, Assoc. Report (part ii.), 1861. (29) 
. . . to the cases (1) in 
