TRANSACTIONS OF SECTION A. 395 
with their extraordinary analytical elegance had the greatest influence. But in his 
works anything geometrical was studiously avoided. Lagrange prided himself that 
there was no figure in his ‘ Mécanique analytique.’ 
The best analytical methods of the Continent were thus introduced into 
England, rapidly assimilated and made the foundation of new theories, so that the 
mathematical activity in this country is now at least as great as it ever has been 
anywhere. 
But whilst analysis, algebra, and with it analytical geometry, made rapid 
progress, pure geometry was not equally fortunate. Here the hold which Euclid 
had long obtained, strengthened, no doubt, by Newton’s example, prevented any 
change in the methods of teaching. 
Most of all, perhaps, solid geometry has suffered, because Euclid’s treatment of 
it is scanty, and it seems almost incredible that a great part of it—the mensuration 
of areas of simple curved surfaces and of volumes of simple solids—is not included 
in ordinary school teaching. The subject is, possibly, mentioned in arithmetic, 
where, under the name of mensuration, a number of rules are given. But the 
justification of these rules is not supplied, except to the student who reaches the 
application of the integral calculus; and what is almost worse is that the general 
relations of points, lines, and planes, in space, is scarcely touched upon, instead 
of being fully impressed on the student’s mind. 
The methods for doing this have long been developed in the new geometry 
which originated in France with Monge. But these have never been thoroughly 
introduced. 
Works written in the German language naturally received even less attention. 
But it was in Germany, at the beginning of the second quarter of this century, 
that geometry received at the hands of several masters an impulse which put the 
subject on an entirely new footing. 
I may mention here especially four men of whom each invented a new method 
and established a new system of geometry. Two of these, Mobius and Pliicker, 
still use algebra, but in perfectly new and original manners, which, although very 
different from each other, have this in common, that in both we have not algebra 
interpreted geometrically, but rather geometry veiled in an algebraic garb. The 
geometrical meaning is never lost sight of. 
But perfectly independent of algebra was the great Steiner, the greatest 
geometrician since the times of Euclid, Appolonius, and Archimedes. In his 
celebrated ‘ Systematische Entwickelungen’ he has laid the foundation of a pure 
geometry, on which a wonderful edifice has since been raised. His treatment 
of the principle of duality, and his methods of generating conics by projective, or 
homographic, rows and pencils which have been extended to curves of all degrees, 
have given to geometrical reasoning a generality never before dreamed of. He is 
in one respect the opposite of Lagrange, hating and despising analysis as much as 
ever Lagrange disliked pure geometry. Steiner started from the geometry of the 
Greeks, Euclid’s elements, and a few other metrical properties he takes for granted ; 
but then he-goes on with essentially modern methods of his own to investigate 
what are now called projective properties of curves and surfaces. 
This metrical foundation Von Staudt changed. In his ‘Geometrie der Lage,’ 
published fifteen years after Steiner’s ‘Entwickelungen,’ he established a most 
remarkable and complete system, into which the notion of a magnitude does not 
enter at all. He shows that projective properties of figures, which have no 
relation whatsoever to measurements, can be established without any mention 
of them. He goes so far as even to give a geometrical definition of a number, in 
its relation to geometry as determining the position of a point, in his theory of what 
he calls ‘ Wiirfe’; and one of the most interesting parts of his work is the purely 
geometrical treatment of imaginary points, lines, and planes. 
In the hands of these men, and since their time, pure geometry has become a 
most important instrument for research, rivalling in power the more or less 
algebraical methods, and surpassing them all in the manner in which they raise 
before the mind’s eye a clear realisation of the forms and figures which are the 
object of the investigation. 
