808 Transactions.—Miscellaneous. 
The object of mathematics is calculation, and they become extended just 
in proportion as the method of calculation is improved. 
Now it is worth while to consider the facilities that existed for calcula- 
tion in Euclid’s time, and for many hundred years afterwards. Arithmetic, 
as we now know it, was then in its infancy. In fact, the seventh, eighth, 
and ninth books of Euclid’s Elements are devoted to this important subject ; 
but Euclid has written with such obscurity that his most devoted worship- 
pers do not insist upon our reading these books. The clumsy symbols that 
were employed effectually hindered progress, and no one but a philosopher 
could multiply fractions. It would repay the trouble to work out a few 
sums in the Greek method, which continued to be employed until the so- 
called Arabie symbols and method were introduced from the east. Frac- 
tions, that enter so largely into our arithmetical calculations, have not been 
long properly understood. Killand and Tait, in their preface to Quater- 
nious, give a curious instance of the conceptions entertained of them in the 
sixteenth century. At the present time we can solve all questions by them, 
and thus dispense with the so-called rules of arithmetic. In fact arith- 
metic cannot be taught as a branch of mathematics, unless by the aid of 
fractions, which enable us to keep the whole question before the mind at 
the same time. But Euclid had no such method of expressing the ideas in 
his learned head, and so he expressed them in the best manner he could. 
Where he speaks of multiples we use fractions, and his equality of ratios 
are with us equality of fractions. Duplicate, compound, and triplicate 
ratios lose their learned and formidable appearance when we employ 
fractions. 
It is scarcely credible that Euclid would have devoted so many words to 
prove propositions if he had our concise method of recording results. If, 
for instance, we know that the area of a triangle is half the product of the 
base by the perpendicular, we must see that, when the perpendicular is 
constant, the areas of triangles vary as the length ofthe bases. Yet we 
know what scaffolding Euclid has erected in order to prove this self-evident 
proposition, and his proof is most difficult for learners to fully comprehend. 
As an example of the contrast between the modern mode of proof and 
Euclid's, the nineteenth proposition of the sixth book may be taken. It is 
required to prove that “ similar triangles are to each other in the duplicate 
ratio of their homologous sides." Most of us, in thinking over the proof in 
the Elements, will remember the number of anxious students who could 
not understand the proof, but a little progress in algebra makes them reason 
thus :—Since the triangles a b c and a’ b’ ¢’ are similar, the perpendiculars 
(p and p) on the sides b and b from the vertical angles will divide the 
triangles into others which are respectively similar. Then =< and 
