Apams.—Elements of Mathematics, 305 
They confessed their inability to comprehend how that acumen of intellect 
could lead to the summum bonum— 
“ Which could distinguish and divide 
A hair 'twixt south and south-west side.” 
It was fully believed by the supporters of the old system, that such a 
change would banish learning from the earth, and that, like Orpheus, it 
would be torn to pieces by the Bacchanalians. But as no such consum- 
mation took place—but, on the contrary, that literature, and science, and 
art flourished with renewed vigour—the denunciations ceased, and similar 
changes in the school system were adopted in other countries. England, 
since that time, has been slowly making changes. Greek and Latin verses 
are not now composed with the same assiduity as formerly. The rules of 
the Latin Grammar are written in English ; the as in presenti is not so 
commonly learned, and many other changes have been grudgingly made 
with the view of cutting off the non-essential, and thus affording more time 
for the pupils to study physical science and mathematics. 
The English have not followed the example of the French in dis- 
carding Euclid’s Elements, but, on the contrary, they have made it a 
standard text-book for those who intend to study the higher mathematics. 
And an admirable book it is, as it contains the summary of what was 
known of mathematics during that brilliant period of history, when 
Ptolemy Soter ruled Egypt; and not only so, but geometry cannot be 
known without the proof of Euclid’s propositions. But, that it should 
be absolutely necessary to use Euclid's own words in the proofs, now 
that the range of mathematics is so much wider and the aim so different, 
is open to grave doubt. 
The difficulty a pupil experiences in entering on the study of geometry 
is great enough, by his having for the first time to form conceptions of 
quantities of two dimensions without adding any unnecessary obstacle. 
The actual work he ought to accomplish is quite enough without leading 
him to it by a most circuitous route. This work may be thus arranged :— 
First, to calculate areas from line measurements. 
Second, to calculate areas and distances by means of measured lines 
and angles. 
The third step takes in the additional element of force, but the object 
is still to determine some point or points in space. This embraces the 
usual mathematical course for secondary schools, namely—mensuration, 
trigonometry, and mixed mathematics. 
It is evident that, in order to learn the first, mathematically, the pupil 
must solve a great many geometrical problems, understand the principles 
on which constructions are obtained, and*aequire the method of calculation. 
But as all this leads to a definite object, he learns only what he shall 
absolutely require, and of which he must constantly make use. 
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