This article aims first at showing that Russell's general doctrine according to which all mathematics is
deducible ‘by logical principles from logical principles' does not require a preliminary reduction of all mathematics
to arithmetic. In the Principles, mechanics (part VII), geometry (part VI), analysis (part IV-V), magnitude theory
(part III) are to be all directly derived from the theory of relations, without being first reduced to arithmetic
(part II). The epistemological importance of this point cannot be overestimated: Russell's logicism does not only
contain the claim that mathematics is no more than logic, it also contains the claim that the differences between
the various mathematical sciences can be logically justified – and thus, that, contrary to the arithmeticisation
stance, analysis, geometry, mechanics are not merely outgrowths of arithmetic.
The second aim of this article is to set out the neglected Russellian theory of quantity. The topic is obviously
linked with the first, since the mere existence of a doctrine of magnitude, in a work dated from 1903, is a sign
of a distrust vis-à-vis the arithmeticisation program. After having showed that, despite the works of Cantor,
Dedekind and Weierstrass, many mathematicians at the end of the XIXth Century elaborated various axiomatic
theories of the magnitude, I will try to define the peculiarity of the Russellian approach. I will lay stress on the
continuity of the logicist's thought on this point: Whitehead, in the Principia, deepens and generalizes the first
Russellian 1903 theory.