grown up under the name of "Descriptive Geometry." But where is the evidence that he was ignorant of these theorems? He certainly does not say that he was made acquainted with them by Mr. Hirst, but simply that he was first informed by him that they had been grouped into "a separate division of mathematics." Why he did not know of this is readily explained, as the title Descriptive Geometry had never been adopted in England for the subject to which it had been applied, from Monge, to Reye on the Continent; and its modern restricted use was very naturally known only to professed mathematicians. What Prof. Hirst put before Mr. Spencer was, therefore, not any new mathematical problems or principles which he found it necessary as an afterthought to thrust into a previously-formed mathematical philosophy, but only the recognized differentiation of a certain mathematical province.
As for the non-quantitative mathematics, we fail to see that Mr. Halsted gets up much of a difference with Spencer. Mr. Halsted thinks that the "Geometry of Position" does not involve the notion of quantity, and Mr. Spencer thinks the same. But the experts of "Harvard" and of "Johns Hopkins" are squarely at issue on this point. After making his case against Mr. Spencer on a false interpretation of what he said, Mr. Wright admitted that, perhaps, after all, he did not mean that—possibly, instead of a branch of the engineer's art, Spencer was referring to "certain propositions in the higher geometry concerning the relations of position and direction in points and lines." But he opens a battery of sarcasms upon the idea of non-quantitative mathematics, and says of these geometrical propositions that they "cannot be made to stand alone, or independently of dimensional properties." Spencer was thus attacked by a skilled mathematician a dozen years ago for taking substantially the same ground that Mr. Halsted now advocates.
In regard to the terminology of the subject, Mr. Halsted encounters the difficulty which always arises when knowledge outgrows old definitions. No doubt, if positional geometry is non-quantitative, and is still a branch of mathematics, we should have a new definition of mathematics; but it is much easier to discredit the old one than to replace it by a better. Why does Mr. Halsted continue to apply the term geometry, which, by its very structure and etymology, implies measure and quantity, to that which has no quantity? Mr. Spencer evidently saw the difficulty; but, rather than attempt to redefine mathematical science, he preferred the alternative of marking off the newly-recognized province by a title that excluded the element of quantity—that is, he called it negatively quantitative. Mr. Halsted does not like this term. Speaking of a certain proposition given as an illustration by Spencer, he says: "It is not 'a negatively quantitative proposition,' as Spencer asserts in his note. It is, primarily, not quantitative at all." But what does Mr. Halsted suppose Mr. Spencer means by "negatively quantitative," unless he means not quantitative at all, or the denial and exclusion of quantity? Let us observe exactly whit Spencer says: "In explanation of the term 'negatively quantitative,' it will be sufficient to instance the proposition that certain three lines will meet in a point, as a negatively-quantitative proposition, since it asserts the absence of any quantity of space between their intersections. Similarly, the assertion that certain three points would always fall in a straight line is 'negatively quantitative,' since the conception of a straight line implies the negation of any lateral quantity or deviation." The italics are ours, but the statement is sufficiently explicit. The absence or negation of quantity is as strong an expression as could be used for no quantity at all, or that which Spencer calls negatively quantitative. Mr. Spencer designates the "Geometry of Position" as of this kind, and yet Mr. Halsted imputes to him the error of ranging it under and trying to make it depend upon quantity.
Mr. Halsted reports that, in his last bulletin, Cayley stands opposed to Spencer's views. It is to be* hoped that he understands him; but what is his relation to Wright and Halsted?
And now, apologizing to our readers for introducing this remote discussion, and passing it off under the head of popular science, we call upon the heirs and repre-
