CORRESP ONDENCE. 



61: 



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 ac- 

 quainted with tbem by Mr. Hirst, but sim- 

 ply 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 adopt- 

 ed 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 be- 

 fore Mr. Spencer was, therefore, not any 

 new mathematical problems or principles 

 which he found it necessary as an after- 

 thought to thrust into a previously-formed 

 mathematical philosophy, but only the rec- 

 ognized differentiation of a certain mathe- 

 matical 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 fabe interpretation of what 

 he said, Mr. Wright admitted that, perhaps, 

 after all, he did not mean that possibly, in- 

 stead of a branch of the engineer's art, 

 Spencer was referring to " certain proposi- 

 tions in the higher geometry concerning the 

 relations of position and direction in points 

 and lines." But he opens a battery of sar- 

 casms 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 diffi- 

 culty 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 I Mr. Spen- 

 cer evidently saw the difficulty; but, rather 

 than attempt to redefine mathematical sci- 

 ence, he preferred the alternative of mark- 

 ing off the newly-recognized province by 

 a title that excluded the element of quan- 

 tity that is, he called it negatively quan- 

 titative. Mr. Halsted docs not like this 

 term. Speaking of a certain proposition 

 given as an illustration by Spencer, he 

 says : " It is not ' a negatively quantita- 

 tive proposition,' as Spencer asserts in his 

 note. It is, primarily, not quantitative 

 at all." But what does Mr. Halsted sup- 

 pose Mr. Spencer means by " negatively 

 quantitative," unless he means not quanti- 

 tative at all, or the denial and exclusion of 

 quantity ? Let us observe exactlv whit Spen- 

 cer 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 nega- 

 tively-quantitative proposition, since it as- 

 serts 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 quan- 

 tity 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 Po- 

 sition " as of this kind, and yet Mr. Hal- 

 sted imputes to him the error of ranging it 

 under and trying to make it depend upon 

 quantity. 



Mr. Halsted reports that, in his last bul- 

 letin, Cayley stands opposed to Spencer's 

 views. It is to be* hoped that he under- 

 stands 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- 



