906 Transactions.— Miscellaneous. 
Now, it is thought necessary for a boy to know, at least, two books of 
Euclid before he ean properly commence to survey with the chain alone, 
and as Euclid, like an ancient philosopher, is speeulative rather than 
practieal in his propositions, the abstract relations of magnitudes are alone 
regarded, and none of the particular or general terms used in calculation 
are at all mentioned. The result is that, after a pupil has mastered the 
two books, he cannot, of himself, discover anything in them relating to 
calculation of areas. And besides, his attention has been directed to the 
demonstration alone, and the essential part of construction is left to his 
 owninvention. Euclid's long proofs and admirable chains of reasoning 
are not put to practice after the books are read, for, in actual questions, the 
proofs are written in the algebraic form. 
Again, it is found that some of Euclid's propositions are made unneces- 
sarily difficult, some of them are self-evident, and none of them expressed 
in modern mathematical language. 
Take, as an example, the fifth in the first book, called the pons asinorum, 
which is a real barrier to many learners, and was at one time the limit to 
geometrical studies for the generality at the universities. Now this propo- 
position can be proved like the fourth by super-position, so that the dullest 
can at once comprehend it; but, as teachers must conform to Euclid and 
Euclid alone, two weeks of school life, allowing two hours a-week for 
geometry, must be spent before the most intelligent boys can comprehend it. 
The name that this fifth proposition has obtained shows the vast number 
who have failed to pursue the study of mathematies owing to their inability 
to see the proof veiled, as it is, in Euclid's drapery of words. 
The eighth proposition is another stumbling-block, owing to the in- 
direct method, although the theorem can be far more easily proved directly, 
and at the same time render the seventh proposition unnecessary. The 
thirteenth proposition really needs no proof, and for this very reason boys 
find great difficulty in writing down or saying Euclid's proof for it; and 
such instances can be multiplied. 
It is nothing new to be aware of the faults in Euclid’s Elements, but 
the range of mathematies and physical science was so limited until the 
eighteenth century, that the time might be spared for Euclid's proofs, as 
there was little more to learn. 
We know what took place when Euclid undertook to teach Ptolemy 
Soter. After that the latter had learned the definitions, many of which 
are more difficult than the things defined—had passed safely through the . 
postulates and axioms, and arrived, as we may suppose, at the pons 
asinorum—the King asked him if there was no easier method. Euclid gave 
the reply so often repeated, * There is no royal road to geometry." I 
