198 
MR. REEVE IN REPLY TO MR. GLOAG. 
It appears, however, to me, that Mr. Gloag entertains a some- 
what exaggerated idea with respect to the amount of dilatation 
necessary to constitute lateral expansion ; since, in reference to 
my experiments in clay and gutta percha, he inquires, if I com- 
pared the mould in clay (taken during action) with that in gutta 
percha (taken during repose); observing, that it was an “oppor- 
tunity of silencing all argument,” and inquiring, “ What better 
experiment could possibly be instituted I” 
I answer, that the total amount of expansion that could have 
taken place under the most favourable circumstances would have 
been too minute to have been ascertained by such rude means ; 
since, from the moist and yielding nature of the clay upon the 
withdrawal of the horse’s foot, the breadth of the mould would 
have been affected to a degree equivalent to, if not greater, than 
the expansion itself. 
We may easily arrive at an approximate to the expansion 
which takes place in a foot 5 inches wide, and having a concave 
sole half-an-inch high, when that sole descends to the depth of 
one-tenth of an inch. 
In a transverse section of a hoof of the above dimensions from 
J which the frog had been extracted, thereby 
leaving an aperture 1 inch wide, we per- 
ceive that the edge of the sole (a, a) be- 
comes the hypothenuse of a right angled 
triangle on each side of the frog, the base of 
which is 2 inches, and perpendicular \ inch 
in measurement. 
Now, the sum of the increase which takes place in each base 
upon the descent of the sole, or hypothenuse, to the depth of the 
one-tenth of an inch, would, together with the amount of expan- 
sion which simultaneously takes place in the frog and commissures, 
give a very fair approximate to the actual degree of expansion. 
By a descent of the sole one-tenth of an inch, the perpendicular 
would be reduced from five inches to four inches, and our question 
is, how much has each base increased thereby? 
Now, by the 47th problem of the first book of Euclid, the sum 
of the squares of the base and perpendiculars equals the square of 
the hypothenuse, therefore (2* + ,5 2 ) or 4,25 would represent the 
square of the line indicated by the edge of the sole ; and it there- 
fore follows, when the sole has descended so as to reduce the 
perpendicular to ,4 of an inch, that the formula (4,25 — ,16) 
or 2,022 inches would be the breadth of the base, which being 
doubled (for there is a triangle on each side) gives 4,044, or an 
excess of ,044 of an inch, which is the expansion caused by the 
descent of the sole; viz., a little more than the one twenty-fifth part 
