The San Francisco call. [volume] (San Francisco [Calif.]) 1895-1913, 1895-04-28

each section of solid is plane. so each
section of a fourth dimensiona object must
be a solid. Each section of a cube is
square, SO each section of the four square
must be a cube. Each section of sphere
is circular plate or slice, so each section
of the corresponding object in fourth
space must be a sphere.
Suppose something falls across the Line
mans line of motion, he is completely
stopped in that direction. If something
also crosses his path on the other side he
would be completely stopped, for he would
have no motion known by which he might
60 round the object, as the Flatlander
would. So too, suppose the Flatlander sur
rounded by ring or square, escape would
be prevented entirely, for he could have
no notion of surmounting the obstacle
as the Spacelander would at once pro-
coed to do. So if a three -dimensional
creature should be placed inside shell or
closed room it would have no notion of
how to escape without penetrating the
walls of the shell or room; while four
dimensional creature would Immediately
start out upon new direction and without
the necessity of breaking the walls would
come out and settle down into space on
the outside with as much ease as man
can climb fence or a bird fly over a hill.
Now this looks unreasonable, does it not?
But are all your conceptions reasonable or
real? Ever since you began the study of
algebra you have been considering i'm
aginary quantities, unreal quantities.
Such are the indicated even roots of all
negative quantities. The square root of
1 is an example. There is no rational
number that you can picture even in your
thought which multiplied by itself will
produce i.
Again, ask one of the children in the
first, second or third grade to subtract
Z from 3, he will undoubtedly answer
that the larger of two mum bers cannot be
taken from the less. The child is right
from his point of view. The algebraic sub
traction OF from leaves 4; but the child
has not been taught the theory of mega
tive quantities, and in his mind there ex-
lists no number which, added to Z, pro-
duces B. We say that -7x-3=+21, but we
cannot form realistic conception of such
an operation. 1t is impossible to think of
u quantities taken 3 times and giving
81, a conceivable number. Yet we find that
this algebraical law invented by mathe
maticians, that the products of quantities
with like signs are plus quantities, is law
that produces correct results. The realistic
conception of the operation is impossible
for us. Yet we accept the law and use it
with never a doubt as to its correctness.
This Is necessary to the development Of
the highly useful science of algebra. So,
also the admission of a fourth incon-
celvable but possible direction is alb
solutely necessary for a profound
knowledge OF modern analytic geometry.
Take the equation of a circle, I2+y2=a2.
This is the equation of the bounding curve
OF the circle. Using three co-ordinates and
three directions, represented by y and z.
We can readily obtain the equation
I2fy2+ z2=d2, the equation of the sphere,
not true of points inside the sphere,
but OF points upon its surface, the
bounding surface of a solid. Now, shall
analytic geometry stop bere? Are we
allowed TO use but three co-ordinates and
three directions? If so, who shall stop us?
Every student knows that by the conside-
ration OF higher algebra many of the prob
lens, principles and truths of the eleInen.
tary part of the subject are made palmer.
Also by the consideration of the geometry
of space, that concerning the plane is
made more clear. Let us then take
the equation G2ty9 t@2 +at2=a2, u repre-
senting a new direction at right angles
to the other three, We cannot picture
such a direction, but we can assume it.
Our equation then represents the bound
ing solids of a fourth dimensional object
Such an equation is just as capable
of mathematical treatment. Why should
it not represent something, have some geo.
metric meaning? And if by the consider
alton of fourth dimensional objects we can
make the geometry of space more clear, it
is then the mathematician duty to use
the fourth dimensional assumption.
Given the equations of two lines, we cam
by considering them simultaneous equa-
tions solve them and determine their
points of intersection. Given the equa-
lions of two surfaces, we can eliminate one
variable and obtain the equation of their
curve of intersection. Given the equations
of the bounding solids of two fourth di-
mensional objects, we can eliminate one