574 
[114.] But there remain eight points P,,,, and eight points P,,,, in 
the plane now considered; namely two of each group, on each of the 
four sides of the quadrilateral. There are, therefore, 16 such points ; 
which, with the 12 points p,,,; the 4 points P,,,; the 8 points P,,.; 
the 2 points p.,,; the 4 points P,; and the one point P,, make up (as 
has been said in [109.]) a system of 47 points, given or derived, in any 
one of the fifteen planes 1,,,. f 
It may be remarked that with the initial arrangement [91.] of the 
five given points, the four points 8’c’B,c,, im a new plane II,,,, are 
corners of a square, which has the point = for its centre; and that thus 
the Figure, of the 47 points in such a plane, may be thrown into a 
clear and elegant perspective. 
[115.] As regards the distribution in a plane II,,,, such as the 
Third Typical Plane [34.], it may here be sufficient to observe, that 
besides containing three lines A,,2, namely the sides of a triangular face 
[34.] of one of the five inscribed pyramids [28.], and three points P,, 
which are the corners of that triangle, and serve to determine the plane 
[1.], it contains also forty points P,, which are arranged in groups, as 
follows. Each of the four first groups, of second construction, P2,,,-. - 
P.,4, gives three points to the plane ; the fifth group, Ps, , furnishes only 
one point; and the sixth, seventh, and eighth groups, Po, 6 - - Pays supply 
six, twelve, and nine points, respectively. Of these 40 points P., twenty- 
four are ranged, eight by eight, on the three sides of the triangle, as was 
to be expected from [56.|; and the existence of at least 27 points, P,, Po, 
in a plane II,,,, might thus have been at once foreseen. But we have 
also to consider the éraces, on that plane, of the 52 lines, A,, A,, which 
are not contained therein. Of these lines, it is found that 36 zntersect 
the sides of the triangle, and give therefore no new points. But the six- 
teen other lines intersect the plane, in so many new and distinct points ; 
and thus the total number [109.], of forty-three derived points, Py, P2, in 
a plane II,,,, which contains no given point Py, is made up. 
[116.] Without attempting here to enumerate the cases of collinea- 
tion, in either of the two typical planes I,, we may just remark, that while 
the traces of four of the planes 1, on the typical plane II,,, are the four 
sides, and the traces of four others are the diagonals, of the quadrilateral 
- already mentioned, the trace of a ninth plane 1,, namely azc, on that 
plane I,,,, has been already considered, as the trace aa” of the latter 
on the former; but that the trace of the tenth plane I1,, namely apz, or 
[01100], on 43B,c,C,B,, or on [01111], is a new line, AD’,; which passes 
thus through one point P, and one point P,,,, and also through two points 
P,,., namely (01120) and (01102), and through two points P,,., namely 
(20011) and (20011): being, however, syntypical with the formerly 
considered trace 4a”, and therefore leading to no new harmonie or an- 
harmonic relations. 
a 
