36 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Mathematics is usually spoken of as a deductive science, but it is also 

 inductive. Klein divides the mathematicians into three classes, the logi- 

 cians, formulists, and iutuitionists. The intuitionists lay particular stress 

 on geometrical intuition, — not in pure geometry only, but in all branches 

 of mathematics. " What Benjamin Peirce calls geometrizing a matl>e- 

 matical question," says Klein, " seems to express the same idea." Each 

 class is useful in its own sphere, but hereafter the sphere of the iutui- 

 tionist will be vastly extended. 



In mathematics and physics all analytical work can be put into geo- 

 metrical shape, and so expressed that one or two pages will not only 

 indicate but explain the work of a volume and help to recall its contents. 

 In Analytic Geometry, as ordinarily treated, the Geometry is the object, 

 the Algebra the means. In cartographic representation the Geometry 

 not only explains itself, but explains the Algebra also. 



Algebraical Formula3 should also be tabulated by themselves for some 

 purposes. Tables of Integrals should be made as complete and system- 

 atic as possible, and form the basis for a treatment of the Integral 

 Calculus. 



The higher branches of mathematics lend themselves as readily to 

 cartographic representation as the simpler branches. 



The purely analytical methods of much of the mathematics of the 

 nineteenth century make it comprehensible only to a mathematician, and 

 only to him when he begins at the first chapter. 



I had occasion once to look into a question of mathematical physics 

 that was not so elementary as to be found in an engineer's pocket-book ; 

 and so I consulted a treatise on the special branch in which I was 

 working. From the index and table of contents, I had no trouble in 

 finding the place, but did not know what the letters stood for. 1 read 

 the chapter through without learning, and then read the chapter before 

 it. There were the same signs, and I was referred back to chapter after 

 chapter, then to another book on general physics, whicli I read hall 

 through, and then to a book on mathematics, with no better result. 

 Then I took up one of the books on physics again, and began at the 

 Preface. I found nothing about the signs, but much about the Cambridge 

 Tripos. I had not been connected with any university for several years, 

 and I thought a Tripos was a thing to set a theodolite on. I looked in 

 the dictionary, and found that it was a writer of Latin verses on the back 

 of a slip of paper containing the names of bachelors who were highest in 

 the list. That explained everything. The Latin verses were the end and 

 motive of the whole system. The books were not meant to be understood 



