"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 quantities, 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 will meet in a point" is not "a negatively-quantitative proposition," as Spencer asserts in his note. It is primarily not quantitative at all, but positional; and, secondarily, 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 coördinates; 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 single positive straight line. The whole question 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 University, 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 Memoir 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 reassured, and venture to think upon the questions they raise, without the sense of being crushed to the dust by the weight of authority.
This is not the first time that Mr. Spencer's note, or, as our contributor calls it, his "confession," has been attacked by mathematicians, and in such a way as to admonish 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 present instance, has exposed him to misunderstanding and misrepresentation. Mr. Halsted recognizes that, by "Descriptive Geometry," Mr. Spencer did not mean those technical methods of geometrical construction 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 passage 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 insinuated 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 particular 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 somewhat 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
