The quantities treated in the theories of heat and electricity are numerous and complicated as regards their dimensions. Thermal capacity has the dimensions *ML*^{-3}, thermal conductivity, *ML*^{-1}*T*^{-1}. In Magnetism the dimensions of the strength of pole are *M*^{1/2}*L*^{3/2}*T*^{-1}, the dimensions of field-intensity are *M*^{1/2}*L*^{-1/2}*T*^{-1}, and the intensity of magnetisation has the same dimensions. In the science of electricity physicists have to deal with numerous kinds of quantity, and their dimensions are different too in the electro-static and the electro-magnetic systems. Thus electro-motive force has the dimensions *M*^{1/2}*L*^{1/2}*T*^{-1}, in the former, and *M*^{1/2}*L*^{3/2}*T*^{-2} in the latter system. Capacity simply depends upon length in electro-statics, but upon *L*^{-1}*T*^{2} in electro-magnetics. It is worthy of particular notice that electrical quantities have simple dimensions when expressed in terms of density instead of mass. The instances now given are sufficient to show the difficulty of conceiving and following out the relations of the quantities treated in physical science without a systematic method of calculating and exhibiting their dimensions. It is only in quite recent years that clear ideas about these quantities have been attained. Half a century ago probably no one but Fourier could have explained what he meant by temperature or capacity for heat. The notion of measuring electricity had hardly been entertained. Besides affording us a clear view of the complex relations of physical quantities, this theory is specially useful in two ways. Firstly, it affords a test of the correctness of mathematical reasoning. According to the *Principle of Homogeneity*, all the quantities *added* together, and equated in any equation, must have the same dimensions. Hence if, on estimating the dimensions of the terms in any equation, they be not homogeneous, some blunder must have been committed. It is impossible to add a force to a velocity, or a mass to a momentum. Even if the numerical values of the two members of a non-homogeneous equation were equal, this would be accidental, and any alteration in the physical units would produce inequality and disclose the falsity of the law expressed in the equation.