110 CKITICAL NOTICES: frequent halts, and sometimes very long halts, in order to reflect. This is not altogether the author's fault. The truth is that the subject of manifolds is extremely difficult to understand, and still more difficult to explain. The meaning of the word manifold, as defined by its inventor, Eiemann, is so very general, not to say vague and attenuated, that it may be called the ether of mathe- matical conceptions. From some point of view or another almost anything may be regarded as a manifold and resolvable into con- stituents which are also manifolds. Mr. Whitehead might, I think, with advantage have restricted his discussion to the general characteristics of the manifolds which enter into his compared algebras, and he should have illustrated these more copiously with simple and concrete examples. In the third chapter we have an explanation of the principles of " Universal Algebra ". " Universal Algebra," says the author, '" is the name applied to that calculus which symbolises general operations, defined later, which are called Addition and Multi- plication". From this definition it is clear that the word universal, as Mr. Whitehead uses it, must be understood in a somewhat limited sense. We now learn more precisely the particular lines of investigation which the author intends to follow, and the order in which he takes the special algebras to which he limits his discussion. " The Algebra of Symbolic Logic," he says, " is the simplest possible species of its genus and has accordingly the simplest interpretation in the field of de- ductive logic. It is, however, always desirable while developing the symbolism of a calculus to reduce the interpretation to the utmost simplicity consistent with complete generality. Accord- ingly, in discussing the main theory of this algebra, the difficulties peculiar to Symbolic Logic will be avoided by adopting the equally general interpretation which considers merely the intersection or non-intersection of regions of space." Farther on, on the next page, he says : " This spatial interpretation, which also applies to the algebra of Symbolic Logic, will in some form or other apply to every special algebra, in so far as interpretation is possible. This fact is interesting and deserves investigation. The result of it is that a treatise on Universal Algebra is also to some extent a treatise on certain generalised ideas of space." Now, if Mr. Whitehead, in the preceding quotations, only means that appeals to diagrams and to spatial problems are of great utility as particular illustrations of general theorems in Symbolic Logic, my opinion wholly coincides with his. But, if I under- stand him aright, he means much more than this. His words seem to imply that to every valid formula in Symbolic Logic, no matter how abstract the region of thought, and no matter what the elementary symbols may be defined as representing, there always corresponds an analogous formula (symbolically identical) of which the elementary symbols represent spatial magnitudes;
- so that any argument referring to the non-spatial region of
