Apaus.— Elements of Mathematics. 807 
think I may venture to say that there are excellent navigators, surveyors, 
and engineers, who could not prove this proposition in Euclid's manner ; 
and if there was no shorter method we should not have attained to our 
present knowledge of mathematics. 
For the second step in geometry, namely, to commence the study of 
trigonometry, it is usual to learn six books of Euclid. This means four 
years at least of careful teaching in a secondary school; for, as our primary 
schools are, as a rule, of a most inferior description, nearly all the elemen- 
tary work must be done in the secondary schools. Four years are spent 
before the pupil enters on the study of mathematies in such a form as to 
induce him to pursue the study after leaving school Now, after these 
books are mastered, the pupil finds that he has not learned the language of 
trigonometry, nor the method of the proofs. 
It is as if he had learned Latin in order to speak French. He will have 
acquired such terms as invertendo, convertendo, ex aequali, and ex «quo, 
duplicate and triplicate ratios; but not a word of sines and cosines, nor 
even the relative values, in general terms, of the sides of a triangle to each 
other, nor of the side of a regular figure inscribed in a circle to the diameter. 
The abstract proofs of Euclid confuse rather than clear his understanding, 
when he has to calculate areas and distances ‘by general or by concrete 
values. The natural order of instruction is the concrete first, then the 
general, and last of all the abstract; but in teaching mathematies we 
reverse the order. Let anyone who knows Euclid’s Elements, as now read 
commence Plane Trigonometry, and I feel sure he will be astounded at the 
little preparation he has made. He will find that the work of the modern 
mathematieian is with definite values, and with every variety of new pro- 
blems, of which the construction must be found, and moreover that tho 
proof is in the algebraic form, quite regardless of Euclid's language. 
When it is considered that every branch of mathematies has its own 
language or terminology—and this is the real diffieulty in trigonometry, 
conie sections, mechanics, hydrostatics, and every other subject—what is 
more reasonable than to dispense with Euclid's language, which does not 
apply to our modern methods of calculation ? If we had a text book, with 
the single definite object of preparing pupils to enter on the study of 
trigonometry, there would be a great deal of valuable time saved. Nor is 
there any fear of speculative geometry dying out, as those with only a taste 
for mathematics are too prone to it, but in the greater mathematicians it 
amounts almost to a disease. Mathematical proofs cannot be otherwise 
than rigid, whether we use the modern or the ancient method. But to 
solve a problem by the analytic method, and then write out the proof in the 
synthetic is exactly what Macaulay charges Samuel Johnson with doing, 
namely, first writing in English and then translating into Johusonese. 
