case. But let me pause here. We have sufficiently shown our plural without even mentioning Cayley and Sylvester's invariantive algebra; Riemann's theory of a complex variable; the algebra of polar elements; or any of the many others that have sprung or are springing into being.
As for pluralizing the idea of space, that would follow very briefly if only I might talk in terms of the "Ausdehnungslehre." Quaternions, as Professor Tait has said, is content with one flat space; but Grassmann, in a little appendix of only two pages, has shown the ability of his extensive algebra to cope with the modern double plural of the old idea of space. Before this idea had germinated, while therefore there was no real use for the word "spaces," the parsimony of language applied it to mean pieces of space; but in the fullness of time it has received its heritage, and by spaces I mean an aggregate of which the space hypothetically infinite and containing the material universe is but one. A statement in the technical terms of analysis would probably tend very little toward clearing up this matter to one not already familiar with it. Let us, then, use rather the historical method—attack in the light of history.
As an eternal treasure and model to the world the Greeks bequeathed the synthetic science of a space. This is the particular space in which you believe, and are sure you and the stars are inhabiting. You will be glad to know that it has been made a fitting monument to the writer of the greatest classic, and inscribed with the name of Euclid. This Euclidean space is a tridimensional homaloid, and so, in distinction from it, spaces with positive or negative curvature are called non-Euclidean.
Through all the centuries up to the present Euclid's space contained at least the thought-world. The space analyzed in Euclid's "Elements" was supposed to be the only possible form, the only non-contradictory sort of space. And, after more than twenty centuries, it is to a little point in that same book that the new idea attaches itself and sprouts into being. This slender link is one of Euclid's postulates, misplaced in the English editions as the twelfth axiom. As the last of his six αὶτἠυατα (requests) Euclid says: "Let it be granted that if a straight line meet two other straight lines, so as to make the two interior angles on the same side of it, taken together, less than two right angles, these straight lines being continually produced shall at length meet upon that side on which are the angles, which are together less than two right angles." This somewhat complicated so-called axiom is only the converse or inverse of proposition seventeen, that "any two angles of a triangle are together less than two right angles," a theorem readily demonstrated from the preceding postulates and axioms. An inverse is usually exceedingly easy to prove. Then why not remove this inverse from among the postulates, place it after seventeen, and demonstrate it? This obvious way to improve on Euclid suggested itself to nu-
