THE TAXONOMY OF ALGEBRAIC SURFACES 
R. P. BAKER 
The problems of taxonomy are problems of order. Any dis- 
crete set can be arranged in linear order but it does not follow 
that any linear order is satisfactory. The separation of natural 
neighbors may be inevitable. Examples are Linnaeus’ botanical 
classification, or the arrangement of logical classes {ahcd, abed, 
) where a natural arrangement applies in general surfaces 
of connectivity greater than one, or w-dimensional space. Any 
number of interrelations of a discrete finite set can be indicated 
by a three dimensional model where the elements are points and 
the relations say colored lines, as in a Cayley color group ab- 
stracted from a surface. 
When the set of elements is infinite the only apparent arrange- 
ment is in a space of n dimensions. For the cubic surfaces every 
real point in a space of nineteen dimensions corresponds to a real 
cubic. 
To condense this two fundamental schemes are used, either a 
classification by projections (real or complex) or by a birational 
reduction. 
The literature of the cubic surface contains three different pro- 
jective attacks. For the general form’ Cayley ^ gave a four 
parameter form (Imnp) having in general twenty-seven lines 
rational in {Imnp). Schlafli ^ reduced the set to a four parameter 
double trihehedral form. Rodenburg ^ took the reduction to sum 
of five cubes and classified by the coefficients. 
The comparison of these forms is rendered difficult by the fact 
that one of two algebraid equations (neither of which has been 
explicitly written) is encountered: a quintic for the pentahedron 
of the five cubes, discussed by Clebsch ^ and the well known equa- 
tion of order 27 for the lines. 
The groups are in general of order 120 and 51840 respectively. 
By a theorem of Jordan the adjunction of the quintic roots is 
ineffective as to the resolution of the line equation, and the ad- 
1 Phil. Trans., 1869. 
2 Phil. Trans., 1863. 
3 Math. Ann., 14. 
4CrelIe, 59. 
