William Kingdon Clifford

On the Space-Theory of Matter

Riemann has shown that as there are different kinds of lines and surfaces, so there are different kinds of space of three dimensions; and that we can only find out by experience to which of these kinds the space in which we live belongs. In particular, the axioms of plane geometry are true within the limits of experiment on the surface of a sheet of paper, and yet we know that the sheet is really covered with a number of small ridges and furrows, upon which (the total curvature not being zero) these axioms are not true. Similarly, he says although the axioms of solid geometry are true within the limits of experiment for finite portions of our space, yet we have no reason to conclude that they are true for very small portions; and if any help can be got thereby for the explanation of physical phenomena, we may have reason to conclude that they are not true for very small portions of space.

On the Classification of Loci

By a curve we mean a continuous one-dimensional aggregate of any sort of elements, and therefore not merely a curve in the ordinary geometrical sense, but also a singly infinite system of curves, surfaces, complexes, &c., such that one condition is sufficient to determine a finite number of them. The elements may be regarded as determined by k coordinates; and then, if these be connected by k—1 equations of any order, the curve is either the whole aggregate of common solutions of these equations, or, when this breaks up into algebraically distinct parts, the curve is one of these parts. It is thus convenient to employ still further the language of geometry, and to speak of such a curve as the complete or partial intersection of k—1 loci in flat space of k dimensions, or, as we shall sometimes say, in a k-flat. If a certain number, say h, of the equations are linear, it is evidently possible by a linear transformation to make these equations equate h of the coordinates to zero ; it is then convenient to leave these coordinates out of consideration altogether, and only to regard the remaining k—h—1 equations between k—h coordinates. In this case the curve will, therefore, be regarded as a curve in flat space of k—h dimensions. And, in general, when we speak of a curve as in flat space of k dimensions, we mean that it cannot exist in flat space of k—1 dimensions.

On Mr. Spottiswoode's Contact Problems.

The present communication consists of two parts. The first part treats of the contact of conics with a given surface at a given point; this class of questions was first treated by Mr. Spottiswoode in his paper “On the Contact of Conics with Surfaces,” and general formulæ applicable to all such questions were given. The results of that paper are here reproduced with some additions; with the exception of a few collateral theorems, these are all contained in the following Table:— Number of five-point conics through fixed point ........................ = 6 Order of surface formed by five-point conics through fixed axis = 8 Number of six-point conics through fixed axis.............................= 9 Number of seven-point conics ................................................. = 70