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THE POPULAR SCIENCE MONTHLY. 



"Negatively: the terms of the relations 

 being definitely-related sets of positions in 

 space, and the facts predicated being the 

 absence of certain quantities (' Geometry 

 of Position')." 



Now, we contend that there is naturally 

 nothing negative about the matter, and to 

 call it negative is unfairly to wrest it from 

 its proper simplicity in order to force it 

 under a preconceived classification. The 

 primitive and natural idea of position is 

 of any portion of space, as distinct from 

 space in general, and does not depend at 

 all upon any quantitative relations, either 

 positive or negative. But, after this, if we 

 wish to define any position with reference 

 to any other definite known position, we 

 use quantities, coordinates, and by this 

 means we can, by using only positive quan- 

 tities, e. g., a positive straight line and a 

 positive angle, accurately refer any one 

 point in any plane to any other point in 

 the same plane. 



So " the proposition that certain three 

 lines vvill meet in a point " is not " a nega- 

 tively-quantitative proposition," as Spencer 

 asserts in his note. It is primarily not 

 quantitative at all, but positional ; and, sec- 

 ondarily, if one wishes to look at it in a 

 quantitative light, it is then very positively 

 quantitative, since it asserts that the three 

 lines will run together on a point which 

 may be exactly fixed by positive quantities 

 its polar coordinates ; or, having the 

 point fixed by the intersection of either two 

 of the lines, it asserts, directionally, that 

 the third line must go directly through that 

 point. In the same way, the assertion that 

 "certain three points will always fall in 

 a straight line " is primarily an assertion 

 of relative position, in which the relation 

 is defined in the simplest manner by a sin- 

 gle positive straight line. The whole ques- 

 tion is this : Is not position as simple and 

 primitive an idea as quantity ? and is not 

 Spencer in error when he gives its abstract 

 science no separate place, but ranges it 

 under, and tries to make it depend upon, 

 quantity? 



George Bruce Halsted, A. B., 

 Mathematical Fellow of Johns Hopkins Univer- 

 sity, late Mathematical Fellow of Princeton 

 College, Intercollegiate Prizeman. 



P. S. Since the above was in print, I 

 have noticed that Arthur Cayley holds views 

 on this subject very much opposed to those 

 of Mr. Spencer. (See Cayley's " Sixth Me- 

 moir on Quantics," in the " Philosophical 

 Transactions.") G. B. H. 



It has been remarked of Mr. Herbert 

 Spencer that he does not stand well with 

 the experts men trained in specialties, and 

 who know their subjects at first hand, and 

 through and through. This is thought to 



be a formidable charge, and it would be 

 formidable if it were true, and the experts 

 agreed among themselves. But when they 

 coincide in nothing but in differing from 

 Mr. Spencer, we may be moderately reas- 

 sured, and venture to think upon the ques- 

 tions they raise, without the sense of being 

 crushed to the dust by the weight of au- 

 thority. 



This is not the first time that Mr. Spen- 

 cer's note, or, as our contributor calls it, 

 his " confession," has been attacked by 

 mathematicians, and in such a way as to ad- 

 monish him that, as this world is constituted, 

 it is not always wisest to be very candid. 

 It has ever been a rule with him carefully 

 to acknowledge the aid he has received 

 from others a practice which, as in the pres- 

 ent instance, has exposed him to misunder- 

 standing and misrepresentation. Mr. Hal- 

 sted recognizes that, by " Descriptive Ge- 

 ometry," Mr. Spencer did not mean those 

 technical methods of geometrical construc- 

 tion to which engineers apply the name; 

 yet no less a mathematical expert than Mr. 

 Chauncey Wright the pride of Cambridge, 

 and whose biography we are soon to have 

 attacked hi in a dozen years ago, in the 

 North American Review, on the very pas- 

 sage here dealt with by Mr. Halsted, but on 

 the opposite ground that such was Mr. 

 Spencer's meaning of Descriptive Geometry. 

 And having assumed that Spencer meant a 

 mathematical art which he was trying to 

 classify as abstract science, Wright insinu- 

 ated that by his acknowledgment to Hirst 

 he was ignorant even of this. It was a 

 disingenuous piece of work. Mr. Wright 

 was then hunting through Spencer's various 

 books in search of flaws to work up into a 

 sensational article, and he was not very par- 

 ticular how he did it, so he could make a 

 telling point. As his note was liable to 

 such misconstruction, Mr. Spencer very 

 naturally withdrew it in a second edition, 

 and substituted for the title first used one 

 less liable to be misunderstood. 



And now has not Mr. Halsted also some- 

 what misapprehended this memorable note? 

 If Mr. Spencer was not referring to the art 

 of Descriptive Geometry, as Mr. Halsted 

 admits he was not, then he must have been 

 referring to the system of theorems in the 

 science of pure mathematics which has 



