Proceedings. 
591 
tion (actually drawn by some writers of repute) that since the first three 
terms in a series of algebraic powers may have a geometrical import, the 
succeeding terms should also in some way have the same. Of course, as 
an exercise of pure logic, such discussions may he as instructive as the 
developments of any other imaginary values of a variable; hut when the 
products of such pure or abstract logic are supposed to have a real concrete 
existence they evidence fatuity and illustrate absurdity. If logic be the 
science of “ necessary conclusions,” it must as certainly conduct us to neces¬ 
sary falsehood as to necessary truth, accordingly as our postulates are 
faulty or sound. 
Reference was made in this connection to the somewhat astounding 
statement of Professor Peter G. Tait (in his “ Lectures on Recent Advances 
in Physical Science,” published in 1876) that our Solar system might be 
gradually passing into “ curvature of space,” where a fourth-dimension 
change of form will be necessarily evolved. Such arrant nonsense could 
be accounted for only on the supposition that the incongruous algebraic 
square and cube naturally suggested cubic contents of the fourth degree 
and even of still higher powers in grandly ascending orders, and that the 
abstract and imaginary had been hopelessly confounded with the concrete 
* and the real. While it is true that algebraic analysis may be successfully 
applied to geometrical relations, the converse is utterly false; and the 
two departments of mathematical logic are as radically distinct as is the 
science of space dimension from the “ science of pure time.” The speaker 
insisted, in conclusion, that mathematicians, above all others, should culti¬ 
vate not only the finest perspicacity of concept, but the most vigorous 
accuracy of expression. 
[Mr. Taylor added that after writing out his present criticism he had 
just discovered (almost accidentally) that Judge Stallo, in his “ Concepts 
and Theories of Modern Physics,” published in 1882, incidentally remarked 
in a foot-note to chapter 14 (on Riemann’s Dissertation) that such an ex¬ 
pression as x square or x cube assumes algebraic quantity to have an in¬ 
herent geometrical import.] 
Mr. Marcus Baker made a communication on Averages. 
