CORRESP ONDENCE. 



609 



CORRESPONDENCE. 



SPENCER'S CLASSIFICATION OF THE 

 ABSTRACT SCIENCES. 

 To the Editor of the Popular Science Monthly. 



I AM a great admirer of Herbert Spen- 

 cer, and especially of his wonderful 

 "Answers to Criticisms " in your journal. 

 When he seems entirely caught and inwoven 

 by his adversaries, with one blow of his 

 trenchant blade he cuts the net, and is free. 

 He is one of the highest of living au- 

 thorities, and I read with deep attention 

 his two editions of " The Classification of 

 the Sciences," being particularly interested 

 in Table I., " The Abstract Sciences." All 

 of it but two divisions he devotes ta mathe- 

 matics as exactly equivalent to quantitative 

 relations; still, at the present day, it seems 

 an untenable cramping of mathematics to 

 define it as the science of quantity. 



_ A candid note in Mr. Spencer's first 

 edition shows that it was not till after he 

 had actually drawn up this table that he 

 became aware of one of the most impor- 

 tant points in the question to be solved. 



It is a note to his first great division 

 of mathematics, and says : " I was igno- 

 rant of the existence of this as a separate 

 division of mathematics, until it was de- 

 scribed to me by Mr. Hirst, whom I have 

 also to thank for pointing out the omission 

 of the subdivision 4 Kinematics.' It was 

 only when seeking to affiliate and define 

 1 Descriptive Geometry ' that I reached the 

 conclusion that there is a negatively-quan- 

 titative mathematics as well as a positively- 

 quantitative mathematics." 



All this confession is omitted in the 

 second edition, where, however, the much 

 superior expression "Geometry of Posi- 

 tion "is substituted in the table for " De- 

 scriptive Geometry," which latter was very 

 apt to be misleading, especially to engi- 

 neers, from its technical sense, in which 

 sense, of course, Spencer did not mean it. 



Now let us try to explain, in few words, 

 what the problem was that Hirst so unex- 

 pectedly put before Spencer's mind, that 

 you may judge whether "seeking to affili- 

 ate " it to a scheme already drawn up was 

 a proper mental condition in which to deal 

 with a question so important, so subtile, so 

 profound. 



Geometry, as the abstract science of 

 space, naturally resolves itself into two 

 great divisions, geometry of measurement 

 and geometry of position geometry quan- 

 titative or metrical, and geometry morpho-, 

 logical or positional. 



As an example of the first, we may take 

 vol. x 39 



the most ordinary illustration, that of 

 equivalent triangles. Any two triangles 

 having the same base, and their vertices in 

 a line parallel to that base, will be of equal 

 or "equivalent"" superficial magnitude. 

 Although the sum of the three sides of the 

 one triangle might be a thousand times as 

 great as the sum of the three sides of the 

 other, they will contain the same number 

 of square inches or square feet. This is a 

 metrical or quantitative proposition ; but, 

 on the other hand, many propositions are 

 known which are purely descriptive or 

 morphological. Take the one, perhaps, best 

 known, the celebrated hexagram. 



In any circle join any six points of the 

 circumference by consecutive straight lines 

 in any order: the intersections of the three 

 pairs of opposite sides are in a straight line. 

 Or, take any two straight lines in a plane, 

 and draw at random other straight lines 

 traversing in a zigzag fashion between them, 

 so as to obtain a twisted hexagon or sort 

 of cat's-cradle figure : if you consider the 

 six lines so drawn symmetrically in couples, 

 then, no matter how the points have been 

 selected on the given lines, the three points 

 through which these three couples of lines 

 respectively pass will lie all in one and the 

 same straight line. So great an authority 

 as Prof. Sylvester has stated that this prop- 

 osition " refers solely to position, and nei- 

 ther invokes nor involves the idea of quan- 

 tity or magnitude." Take another : If any 

 pencil of four rays is cut by a transversal, 

 any anharmonic ratio of the four points of 

 intersection is constant for all positions of 

 the transversal. 



Now, Carnot in his splendid " Geometry 

 of Position," and many before and after 

 him, have laid open a whole world of truths 

 of this kind, truths undeniably geometrical 

 in their nature, but founded on the primi- 

 tive idea of position, and bringing in any 

 idea of quantity only incidentally and after- 

 ward. Now, this was evidently a branch of 

 mathematics, but, having made his scheme 

 mathematics only coextensive with quanti- 

 tative relations, Herbert Spencer must force 

 this under the quantitative rubric, and thus 

 was betrayed into error. Seeing that it was 

 not really positively quantitative, he could 

 only call it negatively quantitative, but in 

 doing this entirely misrepresents it. In 

 Table I. he has, under " Abstract Science: " 



"Laws of Relations 



that are Quantitative (Mathematics). 



