188 Proceedings of the Eo>/al Tri'^h Academy. 



dimensions, wherever n variables are used, we are logically dealing with n 

 dimensions. All Applied Mathematics is ideally a form of « -dimensional 

 geometry ; e.g., in Dynamics of a particle, we assume that every point is 

 weighted with a mass ra, and with velocities u, t-, vj, and thus at any instant 

 has at least seven coordinates. Again, the temperature of a body is a new 

 coordinate ; temperatures form a continuous ideal series, not intuitible in space, 

 though capable of being put into one-to-one correspondence with the points on 

 a line (as, e.g., in the thermometer). In Thermodynamics the relations between 

 pressure, temperature, and volume of a gas are actually representable as a 

 three-dimensional geometry, of which actual space is only symbolic* Facts 

 like these have kept alive the belief brought into prominence by Locke and 

 Descartes, that the so-called secondary qualities of bodies are reducible to 

 the primary or geometrical.f The only reduction possible is, however, a 

 one-to-one correspondence between the variations in secondary or intensive 

 qualities and the points in space or moments in time. This is a purely logical 

 idea and far more philosophical and true to experience than the counter- 

 doctrine still current in theories of the ultimate constitution of matter, 

 that the relation between these two kinds of qualities can be reduced to one 

 of identity. The secondary qualities &.j:q functions of the primary; but the 

 primary qualities are likewise functions of the secondary. 



VI. 



The question whether the premisses of mathematics are or are not 

 hypothetical is one of great interest and difficulty. The mere fact that some 

 mathematicians and thinkers believe that they are hypothetical would seem 

 to prove that this is the case ; because, if the premisses of mathematics are 

 given to the mind as absolutely existing objects or relations, the question 

 could never be raised. The two opposite views may be called the ' Absolutist ' 

 and the ' Hypothetical ' respectively. The ' Absolutist ' ^•iew has always been 

 popular with the Intellectualists, like Plato and Spinoza; and this was 

 consistent, because for them the intelligible as such is real— for them the 

 logical is the true and objective. The apodictic certainty of mathematics was 

 often, as by Descartes, Leibniz, and Kant, taken as the ideal type of perfect 



♦Thermodynamics, it may be mentioned, presents the cnrious case of a ' non-Euclidean space,' 

 in which at least one of the dimensions (temperature) is believed to have a last term (absolute zero) ; 

 and it is not known whether there is or is not a last term in the other direction. The non-Euclidean 

 nature of series in Applied Mathematics cannot be escaped except by asserting that temperatures, 

 electrical charges, masses, and so forth, are actually identical with spatial points or volumes— an 

 obvious absurdity. 



t The Electric Theory of Matter is remarkable as being the first attempt to correlate all secondary 

 qualities (as well as Solidity, treated by Locke as primary) with one intensive or secondary quality. 



