Fune 4, 1885] 
NATURE 
105 
that the cosine of the angle between zéc, aéd, the faces 
of a tetrahedron, will be the determinant— 
e ab ad I 
ab e ba? I 
ca cb ca I 
I I I 
divided by sixteen times the product of the faces adc, 
abd. 
Or, again, if a4, cd be any two non-intersecting edges 
of the tetrahedron, + 2a@4.cd cos (a6, cd) ought to be 
equal to— 
ac aad I 
6c ba? I 
I I e 
and as a matter of fact the cosine between (aé, cd) is 
ad*?+bce—ac —bd*, 
2ab : 
Again, if ac, def are any two triangles in space of 5, 4, 
or 3 dimensions the product of their areas into the cosine 
of their inclination will be a numerical multiple of the 
bordered determinant of the group aéc in regard to def, 
and if they lie in the same plane their product itself will 
be that numerical multiple. 
Similarly for two groups of four points lying in one 
space (as ex gr. that in which we ééve, move, and have 
our being +) the product of their bordered self-determinants 
will be equal to the bordered determinant of either group 
in respect of the other, because their #zveaus coincide, 
and if we take two groups of five points each in ordinary 
space it again follows from the theorem that the bordered 
determinant between them must vanish, a statement 
which when the two groups coincide reverts to Cayley’s 
condition concerning the mutual squared distances of five 
points in ordinary space. 
Finally, there can be little doubt, I think, of the truth of 
the following theorem dealing with determinants (but un- 
bordered) t of which the general theorem we have been 
considering which deals with bordered determinants must 
needs be a corollary. 
By P:: QO where P, Q are two groups of 7 points each, 
let us understand the determinant formed by taking the 
cosines of the angles which the 7° lines connecting P and 
Q subtend at a point O equidistant, in space of the 
necessary number of dimensions, from each of the 2 
given points, and let ?’, QO’ mean the groups ? and Q 
augmented by the addition of O to each of them, the 
theorem is that— 
cos (P’, Q') = 
equal to 
JE (0) R 
V(P:P)(Q:0Q)° 
* Obviously therefore we can express the squared shortest distance between 
two non-intersecting edges of a tetrahedron as a rational function of the 
squares of all six. The formula in the text is well known and easily proved 
for the case of a4c¢d being in a plane, which is enough to show that it must 
be true universally, for if we make BCD rotate about AC, the projection 
of C upon BD does not move, and consequently A C into the cosine of A C, 
B D is invariable. 
+ It would perhaps be more correct to say “‘ which has its being in ys.’ 
t From which it follows that every algebraical theorem regarding square 
matrices expressed in the umbral notation is immediately convertible into a 
proposition in universal geometry ; the umbrz cease to be mere abstractions, 
and acquire a local Aabitation and a name as points in extension. 
§ “P: P isin fact the factorial of 2 divided by the »th power of the dis- 
tance of O from each point in P into the content of (what I call) the plasm 
(of order x) denoted by ?’. 
A plasm of the order 1, 2, 3 means a rectilinear segment, a triangle, a 
tetrahedron—whence it is easy to deduce and define in exact terms the mean- 
ing of a plasm of any order as a figure bounded by plasms of the order next 
below its own. Thesquared content ofa triangle is equal to the sum of the 
3 Squared contents of its projections on mutually perpendicular planes in 
ordinary space : but also to the sum of the 6 squared contents of its projec- 
tions on 6 such planes in extension of 4 dimensions and so en—and in general 
the square of the content of a Alaswz denoted by x points is similarly re- 
a(n —1)...(#-—i+1) 5 
— . ; such squares in extension of 
soluble into a sum of 
I. o-e 2 
( +2— 1) dimensions; as these squared contents are all expressible imme- 
diately by Cayley’s theorem in terms of squared distances, the above state- 
ment gives rise to a far from self-evident theorem in determinants. What I 
Thus for the case of 7 equal to 2 if O is the centre of 
the sphere passing through a, 4, c,d, we ought to find the 
cosine of the angle between the arcs a6, cd equal to 
cosac coscd 
cos 6c cosébd 
divided by a square root of 
cosaa cosadb 
coséa cos bé 
into a square root of 
cos ¢c coscad 
cosdc¢ cosdd 
z.e. equal to 
cos ac. cos 6d — cos ad cos bc 
sinad.sincad 
as is the case. 
There ought also to exist analogous theorems applicable 
to non-equi-numerous point groups depending in some 
way upon the minors of a corresponding rectangular 
matrix.* J. J. SYLVESTER 
New College, Oxford, April 1885 
GRESHAM COLLEGE 
ete question of what is to be done with one of the 
greatest of existing London abuses, Gresham College, 
has again come up in connection with a letter from a 
“Londoner” in the Zzmes. The 7izmes, in a somewhat 
incomplete leader, animadverts strongly on the abuse, 
and urges its prompt remedying. Surely when the fact 
that London has no university in the true sense is 
attracting so much attention and the movement to supply 
the want is so powerful, it is absurd to allow the funds 
to be worse than wasted which represent the wreck 
of those which were originally intended for the main- 
tenance of a real institution of this class. There were 
once 20,000 students at Gresham College, and when 
London does have a university, as it must. have some 
time, even Gresham College will be without razson d’étre. 
“Topographically,” the Zzmes says, “the lecture-rooms 
are off the track of students. None of the apparatus of 
systematic instruction, in the way of examinations, 
accompanies the courses. Provision does not exist, 
have here termed f/asms might with more exactitude be termed Arofo- 
plasnis, as being the elements into which all other figures are capable of 
being resolved. 
* It may be objected that the theorems of the text applied in their full 
generality beyond the limits of empirical space cease to affirm a relation 
between two different things and therefore lose their efficacy as such and 
become mere definztions of the meaning of the inclination of two figures in 
supersensible space. To meet this objection it is sufficient to give a general 
method for determining algebraically the projection of a point in space of # 
dimensions on the niveau of » points where v is any number not greater than 
2; this it is easy to see may be effected as follows :— 
(a) I observe that the niveau of any » given points in a space of z dimen- 
sions may be expressed in Cartesian co-ordinates by means of equating to 
zero each of 7 — 42 +1 independent minors of a rectangular matrix containing 
2 +x columns and u + 1 lines, the formation of which is too obvious to need 
stating in detail. 
(8) In order to project orthogonally a point whose % co-ordinates in a space 
of 2 dimensions are x’, y’, . . . 2’ upon aniveau (of the (# — 1)th order) passing 
through given points defined by theequation dx + By+...C2+ 
we have only to write r—2:y—y': nse easter vate 
combining the(# — 1) equations contained in this proportion with the 
equation, the resulting values of x, y,... determine the projection of the 
given point on the given niveau. 
If now v points are given in a space of % dimensions and the projection is 
required of a given point upon their niveau we may proceed as follows :— 
(x) Find the % — v + 1 equations which define the niveau. 
(2) On each of the niveaus of the (# — 1)th order which correspond thereto 
respectively find the orthogonal projections of the given point. 
(5) Through these x — v + 1 projections of the given points and the given 
point itself draw a niveau which will be defined by (% + 1) — (2 — v+ 2), 
7.€. v — I equations. 
Finally, combining these with the z — v + 1 original equations we have # 
equations in all, and these will serve to determine the » co-ordinates of the 
projection required. 7‘ 
This method is not always the most compendious, but is always sufficient, 
and enables us to attach a definite meaning to the inclination of two spaces of 
any the same order to one another: thus ex. g7., the content of the pro- 
jection of abcd on efgh divided by the content of aécd itself is the 
cosine of the inclination of the niveaus abcd, e/g /, and the projections of 
the several points a, 6,c,d on e/g h (say a’, 6’, c,d’) being found by the 
preceding method, the content of the tetrahedron a’ é'c’ d' (and therefore 
the inclination of the two niveaus) is a known quantity. 
