Apaus.—JHlemenis of Mathematics. 309 
also ^ — E . Multiply these equal fractions and a 4 or 23 — Eo 
Thus the areas of similar triangles vary as the squares of the idus 
| opposite the equal angles. 
The books of Euclid are read by pupils, as commanded, but as soon as 
they read the chapter on ratio in the algebra they adopt the algebraic 
method, because the mind always takes the shortest route to a conclusion, 
and this appears to be the reason that self-evident propositions present so 
much diffieulty to beginners. 
The great fault in Euclid is that the pupil is not allowed to know as 
much about the proposition as the instructor. 
The process by which Euclid arrived at his proof had been known to 
philosophers at least from the time of Plato. It is called the analytic 
method in geometrical researches, for by this method the problem is 
supposed to be solved, and then by comparing the magnitudes under this 
new aspect, and observing the relation between those given and those 
sought, a way to solve the problem is discovered. This is algebra as 
applied to mathematics, and it is the grand method of invention. All the 
progress made in mathematics during the eighteenth century is owing to it. 
Pascal and Roberval made use of it; but when they had solved the 
problem they wrote out the demonstration in Euclid’s synthetic method, 
and designedly concealed their method of invention. : 
Bir Isaae Newton did the same thing, not that he desired to hide the 
method by which he arrived at the solution, for even with this aid mathe- 
maties are not very easy, but he considered a proof was unfit for publica- 
tion unless given after the manner of Euclid, and clothed as far as possible 
in his language. The following is a quotation from his work on F'luxions :— 
** Postquam area alicujus curve ita (analytically) reperta. est et constructa, 
induganda est demonstratio constructionis ut omisso quatenus fieri potest calculo 
algebraico theorema fiat concinnum et elegans ac lumen publicum sustinere 
valeat. 
It appears from this that in such a question as the following taken from 
Todhunter's Mechanics—‘ Find the centre of equal like parallel forces 
acting at seven of the angular points of a cube "—that it is not sufficient 
to determine the point, which is easily done by supposing the forces to act 
at all the angular points; but Newton considered such a method unfit to see 
the light, and that the position of the point must be shown by geometrical 
construction ; and this is often so difficult that the exercise may well com- 
mend itself to the great mathematicians as a kind of mental gymnastics. 
But for a teacher who is anxious to`conduct a pupil in the study of mathe- 
maties to appoint from whence he can see their value and their beauty, it is 
most injudicious to perplex the learner's mind with intricate questions which 
