26 REPORT—1883. 
As regards geometry, I have already spoken of the invention of the 
Cartesian coordinates (1637). This gave to geometers the whole series 
of geometric curves of higher order than the conic sections: curves of the 
third order, or cubic curves; curves of the fourth order, or quartic curves ; 
and so on indefinitely. The first fruits of it were Newton’s ‘Hnumeratio 
linearum tertii ordinis,’ and the extremely interesting investigations of 
Maclaurin as to corresponding points on a cubic curve. This was at once 
enough to show that the new theory of cubic curves was a theory quite 
as beautiful and far more extensive than that of conics. And I must 
here refer to Euler’s remark in the paper ‘Sur une contradiction appa- 
rente dans la théorie des courbes planes’ (Berlin Memoirs, 1748), in 
regard to the nine points of intersection of two cubic curves (viz. that 
when eight of the points are given the ninth point is thereby completely 
determined) : this is not only a fundamental theorem in cubic curves 
(including in itself Pascal’s theorem of the hexagon inscribed in a conic), 
but it introduces into plane geometry a new notion—that of the point- 
system, or system of the points of intersection of two curves. 
A theory derived from the conic, that of polar reciprocals, led to the 
general notion of geometrical duality—viz. that in plane geometry the 
point and the line are correlative figures; and founded on this we have 
Pliicker’s great work, the ‘Theorie der algebraischen Curven’ (Bonn, 
1839), in which he establishes the relation which exists between the order 
and class of a curve and the number of its different point- and line- 
singularities (Pliicker’s six equations). It thus appears that the true 
division of curves is not a division according to order only, but according 
to order and class, and that the curves of a given order and class © 
are again to be divided into families according to their singularities: 
this is not a mere subdivision, but is really a widening of the field of 
investigation; each such family of curves is in itself a subject as wide 
as the totality of the curves of a given order might previously have 
appeared. 
We wnite families by considering together the curves of a given 
Geschlecht, or deficiency ; and in reference to what I shall have to say on 
the Abelian functions, I must speak of this notion introduced into 
geometry by Riemann in the memoir ‘ Theorie der Abel’schen Functionen,’ 
Crelle, t. 54 (1857). For a curve of a given order, reckoning cusps 
as double points, the deficiency is equal to the greatest number 
(x — 1) (w — 2) of the double points which a curve of that order can 
have, less the number of double points which the curve actually has. 
Thus a conic, a cubic with one double point, a quartic with three double 
points, &c., are all curves of the deficiency 0; the general cubic is a curve, 
and the most simple curve, of the deficiency 1; the general quartic is a 
curve of deficiency 3; and soon. The deficiency is usually represented 
by the letter py. Riemann considers the general question of the rational 
transformation of a plane curve: viz. here the coordinates, assumed to 
be homogeneous or trilinear, are replaced by any rational and integral 
