The principal use that philosophy has to make of the conception of n-dimensional space is in explaining why the dimensions of real space are three in number. We see from this study that it is the restriction in the number of dimensions which constitutes the fact to be explained. No explanation of why space has more than two dimensions is called for, because that is a mere indeterminacy. But why bodies should be restricted to move in three dimensions is one of the problems which philosophy has to solve.
Another mathematical conception to be studied is that of imaginary quantities. Several illusory accounts of this conception have been given; yet I believe the true account is the most usual. If one man can lift a barrel of flour, how many men can just lift a bushel? The answer one fourth of a man is absurd, because we are dealing with a kind of quantity which does not admit of fractions. But a similar solution in continuous quantity would be correct. So, there is a kind of quantity which admits of no negative values. Now, as the scheme of quantity with negatives is an extension of that of positive quantity, and as the scheme of positive quantity is an extension of that of discrete quantity, so the scheme of imaginary quantity is an extension of that of real quantity. To determine the position of a point upon a plane requires two numbers (like latitude and longitude) and if we choose to use a single letter to denote a position on a plane and choose to call what that letter signifies a quantity, then that quantity is one which can only be expressed by two numbers. Any point on the plane taken arbitrarily is called zero, and any other is called 1. Then, the point which is just as far on the other side of zero is, of course, −1; for the mean of 1 and −1 is 0. Take any three points ABC forming a triangle, and find another point, D, such that the triangle ACD is similar to the triangle ABC. Then, we naturally write (B − A):(C − A) = (C − A):(D − A). Apply this to the case where A is the zero point, B the point, 1, and D the point −1. Then C will be the point at unit distance from A, but at right angles to AB and this point will represent a quantity i such that i2 = −1.
Imaginary quantities are put to two very different uses in mathematics. In some cases, as in the theory of functions, by considering imaginary quantity, and not limiting ourselves to real quantity (which is but a special case of imaginary quantity) we are able to form important generalizations and bind together different doctrines in a manner which leads us to great advances of the most practical kind. In other cases, as in geometry, we use imaginary quantity, because the problems to be solved are too difficult in the case of real quantity. It is very easy to say how many inflexions a curve of a certain description will have, if imaginary inflexions are included, but very difficult if we are restricted to real inflexions. Here imaginaries serve only a temporary purpose, and will one of these days give place to a more perfect doctrine.
To give a single illustration of the generalizing power of imaginaries, take this problem. Two circles have their centres at the distance D, their radii being R1 and R2. Let a straight line be drawn between their two points of intersection, at what distances will the two centres be from this line? Let these distances be x1 and x2, so that x1 + x2 = D. Then the square of the distance from one of the intersections of the circle to the line through their centres will be, by the Pythagorean proposition
These two equations give
Now the two circles may not really intersect at all, and yet x1 and x2 continue to be real. In other words, there is a real line between two imaginary intersections whose distance from the line of centres is
I do not know whether the theory of imaginaries will find any direct application in philosophy or not. But, at any rate, it is needed for the full comprehension of the mathematical doctrine of the absolute. For this purpose we must first explain the mathematical extension of the theory of perspective.2 In the figure, let O be the eye, or centre of projection, let the line afeDc represent the plane3 of projection seen edgewise, and let the line ABCDE represent a natural plane seen edgewise. Any straight line, as OE, being drawn from the eye to any point on the natural plane, will cut the plane of the picture, or plane of projection, in e, the point which represents the point E. The mathematician (sometimes, the artist, too) extends this to the case where C, the natural point, is nearer the eye than the corresponding point of the picture. He also extends the same rule to the case where A, the natural point, and a, the point of the picture, are on opposite sides of the eye. Here is the whole principle of geometrical projection. Suppose now that three points in nature, say P, Q, R, really lie in one straight line. Then the three lines OP, OQ, OR, from these points to the eye, will lie in one plane. This plane will be cut by the plane of the picture in a straight line (because the intersection of any two planes is a straight line). Hence, the points p, q, r, which are the representations in the picture of P, Q, R, also lie in a straight line, and in general every straight line in nature is represented by a straight line in the picture and every straight line in the picture representing a line in a plane not containing the eye represents a straight line. Now according to the doctrine of Euclid, that the sum of the angles of a triangle is 180°, the parts of a natural plane at an infinite distance are also represented by a straight line in the picture, called the vanishing line of that plane. In the figure f is the vanishing line (seen endwise) of the plane ABCDE. Note how the passage from e through f to a corresponds to a passage from E off to infinity and back from infinity on the other side to A. Euclid or no Euclid, the geometer is forced by the principles of perspective to conceive the plane as joined on to itself through infinity. Geometers do not mean that there is any continuity through infinity; such an idea would be absurd. They mean that if a cannon-ball were to move at a continually accelerated rate toward the north so that its perspective representation should move continuously, it would have to pass through infinity and reappear at the south. There would really be a saltus at infinity and not motion proper. Persons absorbed in the study of projective geometry almost come to think there really is in every plane a line at infinity. But those who study the theory of functions regard the parts at infinity as a point. Both views are fictions which severally answer the purposes of the two branches of mathematics in which they are employed.
As I was saying, if the Euclidean geometry be true and the sum of the angles of a triangle equal 180°, it follows that the parts of any plane at infinity are represented by a right line in perspective. If that proposition be not true, still the perspective representation of everything remains exactly the same. Could some power suddenly change the properties of space so that the Euclidean doctrine should cease to be true,—and all things would look exactly as they did before. Only when you came to measure the real differences between objects you would find those distances, especially the long distances, essentially altered. There would be two possible cases. Either, according to Helmholtz’s supposition, you would find you could measure right through what used to be infinity, nothing being infinitely distant; so that a man might walk round space, somewhat as he might walk round our globe, only he would come round walking on the under side of the floor (if he had anything to hold him on); or a man could part his back hair by looking round space, only he would see himself upside down. Or, on the other hand, according to Lobatchewsky’s hypothesis, you would find that all the points of a certain (geometrical, not physical) sphere or ellipsoid about you were at any infinite distance. On the outside