But for the pure mathematician there are no infinitesimals, and all statements in which they seem to occur must be expressible as limits of what happens to finite quantities. To take our particular case: We must be able to say of a small finite quadrilateral that it is approximately a parallelogram, if we are to be able to assign a meaning to the statement that an infinitesimal quadrilateral may be accurately a parallelogram. The case is exactly analogous to velocity in elementary kinematics: we can assign a meaning to velocity only because we can measure finite distances and times, and so form the conception of the limit of their quotient. It is not wholly clear how we are to satisfy this requirement in the case of Weyl's theory. I think, however, that there is not the slightest reason to suppose that it cannot be satisfied. Let "" mean " form a parallelogram." We are supposed to have also , , etc., but not etc. Also if we have and , we are to have . But if we take "" to mean ", , , form an approximate parallelogram," we cannot (if there[Pg 105] is any way of specifying a degree of approximation) argue from and to . Now if we assume, as Weyl does, that lengths at a given point are comparable, we can perhaps give the necessary definitions. We shall have to take , not , as our fundamental relation, since the distance between any two points is finite, and it is assumed that no finite quadrilateral can be accurately a parallelogram. Or perhaps we shall have to go a step further, and take as fundamental a relation of eight points, say meaning " is more nearly a parallelogram than " We shall then say that, given any four points, , , , , it is possible to find points , nearer to and respectively than and are, such that Further, we can say that, if , , , are sufficiently near together, and then the ratio of to can be made to approach zero as a limit by diminishing the size of in a purely ordinal sense. (Ordinal relations among points, as we saw earlier, are presupposed in the theory of relativity.) It is highly probable that the above process can be simplified. It is, however, of no importance in itself; its only purpose is to show that the derivatives required can be correctly defined, and that, however the mathematical treatment may confine itself to infinitesimals, relations between points whose distances are finite must be presupposed if the infinitesimal calculus is to be applicable.