After reading these passages I am sorely perplexed. When Professor Newcomb penned them he had before him my extracts (in a note to page 211 of my book) from the Exeter address of Professor Sylvester, embodying a reference to the speculations of Professor Clifford, and another independent citation from Clifford's writings on page 213. And, being himself a writer on geometry of more than three dimensions, he can hardly have been ignorant of the many other pangeometrical speculations respecting the necessity of assuming the existence of a fourth dimension for the purpose of explaining certain optic and magnetic phenomena. There are mathematicians and physicists in Europe—excellent mathematicians and physicists, too—who maintain that space must have at least four dimensions, because without it a reconciliation of Avogadro's law with the first proposition of the atomo-mechanical theory is impossible. According to them, experience shows that matter has not only extension but also intension, which directly evidences the actual existence of a fourth dimension in space. Among those who advocate views like this is Professor Ernst Mach, in Prague. How, in the face of all this, Professor Newcomb could have the hardihood to assure his readers that no mathematician has ever pretended that space has more than three dimensions, I am at a loss to understand.
But it is 1 not worth while to quarrel with him on this head; for his statement, that I devote sixty-two pages to the attempt at proving that space has in fact but three dimensions, is a pitiful misrepresentation, akin to the statement that I am the defender of the propositions of the atomo-mechanical theory. In my two chapters on transcendental geometry there is not a page, not even a line, devoted to such an undertaking. I discuss two main questions: first, whether or not it is true, as Lobatschewsky, Riemann, and Helmholtz assert, that space is a real thing, an object of direct sensation whose "properties," such as the number of its dimensions and the form or degree of its inherent curvature, are to be ascertained by observation and experiment—by telescopic observation, for instance; and, secondly, whether or not the empirical possibility and character of several kinds of space can be deduced a priori from the concept of an n-fold extended multiple,