Let us begin with the conservation of momentum and of energy (or mass). Here we start from a proposition of pure mathematics. To explain this proposition will require certain preliminaries. It will be remembered that we had: We put: And we write for the minor of in this determinant, divided by . Also: which = 0 if and =1 if . The next step is the definition of the "three-index symbols," which are:[Pg 85] We can now define the tensor which Einstein uses for his law of gravitation. It is , where: summed for all values of and from 1 to 4. Einstein takes as the law of gravitation in empty space. For the moment, we are not concerned with the law of gravitation, but with certain identities. We put: Further, there is a rule for raising or lowering suffixes in any tensor, of which an illustration is: so that— Generalizing the notion of the "divergence" of a vector, we obtain a general definition of the divergence of any tensor. Taking a tensor of the form for purposes of illustration, its "divergence" has four components: where: and similarly for , etc. These definitions have been given in order to enunciate the proposition:[29] [Pg 86]which Eddington calls "the fundamental theorem of mechanics." In order to see the use made of this proposition, we need to introduce the "material energy-tensor," defined as: where is the "proper density" of the matter concerned—i.e. its density relative to axes moving with the matter. From this, by the usual rule for lowering a suffix, we obtain a tensor . The principles of the conservation of mass and momentum are contained in the statement that the divergence of vanishes. This suggests the identification of with , whose divergence vanishes identically—apart from a numerical factor, which, for convenience, is taken as . Thus Eddington puts: which is the law of gravitation for continuous matter. It has been necessary to make the above excursion into mathematical regions in order to be able to understand the observations which succeed to the above in Eddington's exposition (op. cit. p. 119). He says: [Pg 87] "Appeal is now made to a Principle of Identification. Our deductive theory starts with the interval ..., from which the tensor is immediately obtained. By pure mathematics we derive other tensors.... These constitute our world-building material; and the aim of the deductive theory is to construct from this a world which functions in the same way as the known physical world. If we succeed, mass, momentum, stress, etc., must be the vulgar names for certain analytical quantities in the deductive theory; and it is this stage of naming the analytical tensors which is reached in (54·3). If the theory provides a tensor , which behaves in exactly the same way as the tensor summarizing the mass, momentum and stress of matter is observed to behave, it is difficult to see how anything more could be required of it."