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744 If plants can "generally" fertilize themselves without insect aid, simply preferring cross-fertilization through insect agency when they can get it, the abundance or partial absence of insects there is of little consequence in the argument.

The question being purely entomological, and no longer of importance to the botanist, I should feel sorry for having put Mr. Putnam 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 discussion.

February 22, 1877.

To the Editor of the Popular Science Monthly.

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 negatively-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 disingenuous in representing that' Spencer had reference to those technical methods of geometrical construction to which engineers apply the name. But Wright, like you, understood 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 describe 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 conscientiously; and very few have succeeded as well as he did in comprehending thought remote from the channel of their own. The present case illustrates this, for Wright seemed to detect that Spencer had two very different 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-estimated 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 respect, and no unending progress. But such sharp antagonism only serves to make his criticisms all the more instructive. Whatever 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 himself 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 specialists. 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 knowledge 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 question of mathematics, but of logic: and it goes very deep into the theory of logic, too. But, while it does not concern the "mathematical expert," as such, one does not perceive 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 ventured to express himself on the subject, without proper study of what Spencer has said.

You well say that we here "encounter a difficulty which always arises when knowledge outgrows old definitions." Prof. Peirce, in his "Linear Associative Algebra," offers a definition of mathematics, the acceptance of which would not necessarily involve any decision of the question whether that geometry which is not metrical is quantitative or not. Although linear associative algebra is certainly not popular science, 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 definition is too restricted. The sphere of mathematics is here extended, in accordance 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 discoverer of laws, for it is not induction; neither is it the framer of theories, for it is not