"itself. 
Fune 4, 1885] 
NATURE 
103 
Long Sight 
I was at school at Rossall, between Fleetwood and Black- 
pool, on the coast of Lancashire. One day, being on the sea- 
wall with Arthur A. Dawson, an Irish boy, we could see the 
Isle of Man as if it were ten miles away, and then to the south 
of the Calf of Man we could distinctly see on the horizon the 
summits of two mountains, which we pronounced must be in 
Treland. Four years later I was staying at Blackpool with my 
mother, when we distinctly saw the same blue mountains just 
appearing above the sea. Being in the Isle of Man later on, 
I was at Port Erie, to the west of Castletown, and saw the same 
summits, and was told they were the mountains of Mourne. 
From there the mountains stood well out of the water, though 
we could not see the rest of the coast. The Mourne Mountains 
are 2798 feet high. They are 125 miles from Blackpool. 
A. SHAW PAGE 
Selsby Vicarage, Gloucestershire, May 28 
Museums 
THE interest which the readers of NATURE in this country 
and in America take in the promotion of museums has induced 
so many of them to inquire of me for a paper recently noticed 
by yourself that, to spare their time and my own, I shall be glad 
if you will enable me to refer inquirers to your advertising 
columns. 
THE AUTHOR OF ‘‘ MUSEUMS OF NATURAL HisToRV ” 
A NEW EXAMPLE OF THE USE OF THE 
INFINITE AND IMAGINARY IN THE SER- 
VICE OF THE FINITE AND REAL 
EOMETERS are wont to speak (it seems to me) some- 
what laxly of “the line at infinity” as if there were 
only one such line in a plane; ina certain but not in the 
most obvious sense this is true—viz. there is but one right 
line of which all the points are at an infinite distance from 
all lines external to them in the finite region of the plane, 
and except these points there are none others having this 
property ; but in the sense that there is but one line 
infinitely distant from all points external to it in the finite 
region, the statement is obviously erroneous, for it need 
only to be mentioned to be at once perceived to be true 
by any tyro in geometry that all rays passing through 
either of the two “circular points at infinity” (Cayley’s 
absolute) are infinitely distant from any external point in 
the finite region ; these two imaginary points may indeed 
without any reference to the circle be defined as the 
points which radiate out in all directions rays infinitely 
distant from the finite region; the “absolute” being, so 
to say, the common depository, 7.e. the crossing points 
of all infinitely distant rays as the “line at infinity” 
is the locus of all infinitely distant points. Similarly in 
space : there is not one infinitely distant plane, “the plane 
at infinity,” but an infinitely infinite number of such 
planes—viz. any plane touching “the circle at infinity” 
(an imaginary circle in the plane at infinity) will at once 
be recognised to be infinitely distant from any external 
point in the finite region, or, as we may say more briefly 
and picturesquely, infinitely distant from the finite region 
It will give greater vivacity to this conception to 
imagine an axis through which pass planes in all directions, 
and to travel in idea this axis round “the circle at infinity ” 
keeping it always tangential thereto ; the complex or corolla 
of planes, so to say, thus formed (infinitely infinite in 
number) contains only planes of infinite distance from 
the finite region ; and “the plane at infinity” is but one 
of them—viz. the one which passes through all the axes 
named, just as the line at infinity in a plane is the line 
which passes through both the centres of infinite distance. 
The infinitely infinite series of infinitely distant planes is 
of course the correlative of the infinitely infinite series of 
infinitely distant points whose locus is the so-called 
“plane at infinity.” 
The above statements have only to be made, to be 
accepted by the geometer, although I do not remember 
seeing them * anywhere explicitly given ; but what I want 
to show is that, although supersensuous abstractions, so 
far from being barren they are capable of immediate 
application to the world of reality, and afford an instan- 
taneous answer to a very simple practical question which 
has only just lately been mooted. The question is this: 
Suppose acd to bea given pyramid, and that perpen- 
diculars are drawn from its four vertices, say 4, A, C, D, 
to a variable plane, then it is easy to show that a certain 
homogeneous quadratic function of 4, B,C, D ccpending 
on the form of the pyramid or relative lengths of its edges 
must be constant, and the question arises, What is this 
constant quadratic function, this quadric in 4, B,C, D, 
expressed in terms of the edges of the pyramid 
exclusively ?+ Just so if we take a triangle, adc, in a plane 
there will be a constant quadratic homogeneous function 
of the distances of its three vertices from a variable line ; 
and it is well known in this case that if 4, B, C are the 
distances the constant quadratic function in question 
will be— 
(2oy (4 — €)(B— €) + (be)? (B— A) (C— A) + 
(ca)?(C — B)(A B). 
But if we had not known this fact it could have been 
found as follows :—Calling the above function 7, when 
A, #£,C all or two of them become infinite the relation 
between the ratios of 4 : 8: C will be such as would 
arise from making “=o; on no other supposition will 
this be the case. 
Now, if we use trilinear co-ordinates with adc as the 
triangle of reference, and take as the co-ordinates of any 
variable point P, the areas aPé, 6 Pc, cPa instead of 
the simple distances of P from a6, dc, ca, then every 
body knows that the line at infinity has for its equation— 
(1) rty+z=0, 
and will easily see that the circle circumscribing @éc has 
for its equation-— 
(2) (ab)xy + (bc (y2)+ (caPrz =o. 
Moreover, when such co-ordinates are employed the 
distances of any line 424+ By + Cz=0 (3) from the 
three vertices are A, 4, C each multiplied by the same 
known quantity. 
If then A,&,C become infinite this line must pass 
through one of the intersections of the line at infinity with 
the circle, or, in other words, the equations (1), (3), (2) 
must be capable of being satisfied simultaneously, and 
accordingly by a well-known algebraical law it follows 
that the determinant to (2) dordered by the coefficients of 
(1) and (3) must vanish. Consequently this determinant 
so bordered will represent the sought-for form /, z.e. the 
constant quadratic function will be represented by— 
e A B G c 
A @ (a6) (ac)? I 
IR NOEAP @ (6c)? I 
Cc (ea) (c6)? e I 
® I I I 
On calculating this determinant it will be found to be the 
function of A — B, B— C, C — A above given, except 
that each term is multiplied by the constant factor — 2, 
which may of course be dispensed with. 
Now let us apply similar or analogous considerations 
to the determination of the constant quadratic function of 
* The statement concerning the circular point-pair at infinity being centres 
of pencils of infinitely distant rays I have since met with somewhere in Dr. 
Salmon’s Conics, but stated in quite a casual manner. It may not be un- 
worthy of notice that just as the distance between any two points in a ray 
passing through either point of the adsodufe in a plane vanishes, so similarly 
vanishes the area of any triangle drawn in ary plane touching “the 
imaginary circle at infinity” in space. : 
+ If Z, m, m. p are the distances of the vertices from the opposite faces, 
and x, y, 2, ¢ from the variable plane, it is well knows that 
>> G — = COS 7/2, n) 
is constant, in fact is unity. 
