744 



THE POPULAR SCIENCE MONTHLY. 



If plants can "generally" fertilize them- 

 selves without insect aid, simply preferring 

 cross -fertilization through insect agency 

 when they can get it, the abundance or par- 

 tial absence of insects there is of little con- 

 sequence in the argument. 



The question being purely entomological, 

 and no longer of importance to the botanist, 

 I should feel sorry for having put Mr. Put- 

 nam to the trouble of writing his letter, only 

 that I know facts are always welcome to the 

 lover of science, though they may have no 

 immediate bearing on questions under dis- 

 cussion. Thomas Meehan. 



Germantown, Pa., February 22, 18TT. 



THE MATHEMATICAL CONTEOVEESY. 

 To the Editor of the Popular Science Monthly. 



Sir: You will doubtless be gratified to 

 find that the premise upon which you have 

 rested a charge of disingenuousness against 

 a gentleman no longer living is mistaken. 

 The case is this : You find that the nega- 

 tively-quantitative geometry of Spencer's 

 first edition of his " Classification of the 

 Sciences " must have been that branch of 

 mathematics which has grown up under the 

 name of " Descriptive Geometry ; " and you 

 find the late Mr. Chauncey Wright disin- 

 genuous in representing that' Spencer had 

 reference to those technical methods of geo- 

 metrical construction to which engineers ap- 

 ply the name. But Wright, like you, under- 

 stood Spencer to refer to Monge's descriptive 

 geometry ; and it was just that which he 

 characterized as a mathematical art having 

 no place among the abstract sciences. 



Permit me to add that I used to talk 

 with Wright, daily, while he was writing the 

 article in which this matter is discussed ; 

 and I declare that nothing could less de- 

 scribe his method than to say that he " was 

 hunting through Spencer's various books in 

 search of flaws." On the contrary, no critic 

 ever studied his author more conscientious- 

 ly ; and very few have succeeded as well as 

 he did in comprehending thought remote 

 from the channel of their own. The pres- 

 ent case illustrates this, for Wright seemed 

 to detect that Spencer had two very differ- 

 ent things confounded together in his mind, 

 viz., descriptive geometry and positional 

 geometry. The second edition of Spencer's 

 book makes it pretty clear that this is so ; 

 for some of his warm disciples maintain 

 that he still means the former, while to the 

 mathematician his present words describe 

 with tolerable accuracy the latter. 



No doubt, Wright greatly under-esti- 

 mated the importance of Herbert Spencer's 

 philosophy. This was natural, because he 

 found in Spencer's fundamental doctrine of 

 the universality of evolution a proposition 

 radically opposed to his own theory that 



there is only an ebb and flow, in this re- 

 spect, and no unending progress. But such 

 sharp antagonism only serves to make his 

 criticisms all the more instructive. What- 

 ever there may be of extravagance in the 

 claims which are made for Spencer will be 

 overthrown in the course of the discussion 

 which is sure to go on, and which he him- 

 self would be among the very last of men 

 to deprecate. It would be strange, indeed, 

 if it were to turn out that an encyclopedic 

 system of philosophy had been produced, 

 so perfect in its details as to satisfy special- 

 ists. But disputation clears the philosophic 

 air, and can only serve to bring into the 

 light and to sharpen the outline of all that 

 is to abide in Spencer's system. In this 

 point of view, I cannot agree with you that 

 Mr. Spencer's distinguished candor has done 

 him any harm, or has postponed the knowl- 

 edge of the truth for which he is striving. 



Mr. Wright occupied a position opposed 

 to that of most modern mathematicians, in 

 maintaining that positional geometry is not 

 quantitative. This, however, is not a ques- 

 tion of mathematics, but of logic : and it 

 goes very deep into the theory of logic, too. 

 But, while it does not concern the " mathe- 

 matical expert," as such, one does not per- 

 ceive that Mr. Spencer has proved himself 

 so supremely the master of the philosophy 

 of mathematics that we need be greatly 

 anxious lest Mr. Cayley should have vent- 

 ured to express himself on the subject, with- 

 out proper study of what Spencer has said. 



You well say that we here " encounter 

 a difficulty which always arises when knowl- 

 edge outgrows old definitions." Prof. 

 Peirce, in his "Linear Associative Algebra," 

 offers a definition of mathematics, the ac- 

 ceptance of which would not necessarily in- 

 volve any decision of the question whether 

 that geometry which is not metrical is quan- 

 titative or not. Although linear associa- 

 tive algebra is certainly not popular sci- 

 ence, perhaps you will find his remarks of 

 sufficient general interest for insertion. He 

 says: 



" Mathematics is the science which draws 

 necessary conclusions. 



" This definition of mathematics is wider 

 than that which is ordinarily given, and by 

 which its range is limited to quantitative 

 research. The ordinary definition, like those 

 of other sciences, is objective ; whereas this 

 is subjective. Recent investigations, of 

 which quaternions is the most noteworthy 

 instance, make it manifest that the old defi- 

 nition is too restricted. The sphere of 

 mathematics is here extended, in accord- 

 ance with the derivation of its name, to all 

 demonstrative research ; so as to include 

 all knowledge strictly capable of dogmatic 

 teaching. Mathematics is not the discov- 

 erer of laws, for it is not induction ; neither 

 is it the framer of theories, for it is not hy- 



