We cannot go on any longer with this. In spite of it all, the volume on mathematics is full of profound thoughts, and will be very suggestive to those who take up the subject after M. Comte. What deep meaning there is, for example, in the idea that the infinitesimal calculus is a conception analogous to the corpuscular hypothesis in physics; which last M. Comte has always considered as a logical artifice; not an opinion respecting matters of fact. The assimilation, as it seems to us, throws a flood of light on both conceptions; on the physical one still more than the mathematical. We might extract many ideas of similar, though none perhaps of equal, suggestiveness. But mixed with these, what pitiable _niaiseries_! One of his great points is the importance of the "moral and intellectual properties of numbers." He cultivates a superstitious reverence for some of them. The first three are sacred, _les nombres sacres_: One being the type of all Synthesis, Two of all Combination, which he now says _is_ always binary (in his first treatise he only said that we may usefully represent it to ourselves as being so), and Three of all Progression, which not only requires three terms, but as he now maintains, never ought to have any more. To these sacred numbers all our mental operations must be made, as far as possible, to adjust themselves. Next to them, he has a great partiality for the number seven; for these whimsical reasons: "Composed of two progressions followed by a synthesis, or of one progression between two couples, the number seven, coming next after the sum of the three sacred numbers, determines the largest group which we can distinctly imagine. Reciprocally, it marks the limit of the divisions which we can directly conceive in a magnitude of any kind." The number seven, therefore, must be foisted in wherever possible, and among other things, is to be made the basis of numeration, which is hereafter to be septimal instead of decimal: producing all the inconvenience of a change of system, not only without getting rid of, but greatly aggravating, the disadvantages of the existing one. But then, he says, it is absolutely necessary that the basis of numeration should be a prime number. All other people think it absolutely necessary that it should not, and regard the present basis as only objectionable in not being divisible enough. But M. Comte's puerile predilection for prime numbers almost passes belief. His reason is that they are the type of irreductibility: each of them is a kind of ultimate arithmetical fact. This, to any one who knows M. Comte in his later aspects, is amply sufficient. Nothing can exceed his delight in anything which says to the human mind, Thus far shalt thou go and no farther. If prime numbers are precious, doubly prime numbers are doubly so; meaning those which are not only themselves prime numbers, but the number which marks their place in the series of prime numbers is a prime number.