SPECULATIVE SCIENCE. 159 



Again : 



Whatever we may say of the utility of such investigations, one thing is cer- 

 tain they are perfectly harmless. At the very worst they can do no more 



injury to scientific conceptions than the careless author of an elementary algebra 

 will do his pupil by loading an hypothetical baker's wagon with more loaves of 

 bread than the baker could get into it. If Judge Stallo had taken up a book on 

 algebra, found a problem the answer to which required five thousand loaves 

 of bread to be carried by a single baker, and had devoted sixty-two pages to an 

 elaborate statistical and mechanical proof that no wagon could possibly hold 

 that number of loaves, his criticisms would have been as valuable and perti- 

 nent as those which he devotes to his imaginary school of pangeometry. 



After reading these passages I am sorely perplexed. When Pro- 

 fessor 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 Clif- 

 ford, 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 pan- 

 geometrical speculations respecting the necessity of assuming the ex- 

 istence of a fourth dimension for the purpose of explaining certain 

 optic and magnetic phenomena. There are mathematicians and phys- 

 icists in Europe excellent mathematicians and physicists, too who 

 maintain that space must have at least four dimensions, because with- 

 out it a reconciliation of Avogadro's law with the first proposition of 

 the atomo-mechanical theory is impossible. According to them, experi- 

 ence 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 Xewcomb 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-tivo pages to the attempt at proving 

 that space has in fact but three dimensions, is a pitiful misrepresen- 

 tation, akin to the statement that I am the defender of the propositions 

 of the atomo-mechanical theory. In my two chapters on transcendent- 

 al 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-f old extended multiple, 



