exultation to doubt, from doubt to confessed impotence, yet still without abandoning the old assumption of confident strength; and all his turnings and windings were now laid bare in one of the most trenchant pieces of controversial writing ever penned. Wallis's _Elenchus geometriae Hobbianae_, published in 1655 about three months after the _De corpore_, contained also an elaborate criticism of Hobbes's whole attempt to relay the foundations of mathematical science in its place within the general body of reasoned knowledge--a criticism which, if it failed to allow for the merit of the conception, exposed only too effectually the utter inadequacy of the result. Taking up mathematics when not only his mind was already formed but his thoughts were crystallizing into a philosophical system, Hobbes had, in fact, never put himself to school and sought to work up gradually to the best knowledge of the time, but had been more anxious from the first to become himself an innovator with whatever insufficient means. The consequence was that, when not spending himself in vain attempts to solve the impossible problems that have always waylaid the fancy of self-sufficient beginners, he took an interest only in the elements of geometry, and never had any notion of the full scope of mathematical science, undergoing as it then was (and not least at the hands of Wallis) the extraordinary development which made it before the end of the century the potent instrument of physical discovery which it became in the hands of Newton. He was even unable, in dealing with the elementary conceptions of geometry, to work out with any consistency the few original thoughts he had, and thus became the easy sport of Wallis. At his advanced age, however, and with the sense he had of his powers, he was not likely to be brought to a better mind by so insulting an opponent. He did indeed, before allowing an English translation of the _De corpore_ (_E.W._ i.) to appear in 1656, take care to remove some of the worst mistakes exposed by Wallis, and, while leaving out all the references to Vindex, now profess to make, in altered form, a series of mere "attempts" at quadrature; but he was far from yielding the ground to the enemy. With the translation,[14] in the spring of 1656, he had ready _Six Lessons to the Professors of Mathematics, one of Geometry, the other of Astronomy, in the University of Oxford_ (_E.W._ vii. 181-356), in which, after reasserting his view of the principles of geometry in opposition to Euclid's, he proceeded to repel Wallis's objections with no lack of dialectical skill, and with an unreserve equal to Wallis's own. He did not scruple, in the ardour of conflict, even to maintain positions that he had resigned in the translation, and he was not afraid to assume the offensive by a counter criticism of three of Wallis's works then published. When he had thus disposed