of Edinhurgli, Session 1883-84. 
703 
2. On the Computation of Eecurring Functions, by the Aid 
of Chain-Fractions. By Edward Sang, LL.D. 
3. On Extensions of Euclid I. 47. By A. H. Anglin, Esq. 
In Euclid I. 47, it is proved that if regular tetragons be described 
on the sides of a right-angled triangle, that described on the hypo- 
tenuse is equal to the sum of those on the sides containing the 
right angle. This proposition, as is shown in Euclid VI. 31, is 
only a particular case of a more general one ; and the object of this 
Paper is to establish the corresponding result in the case of regular 
trigons, pentagons, hexagons, and generally regular figures of any 
number of sides, and finally, in the case of any similar rectilineal 
figures, without the use of ratio and proportioyi. 
1. The case of regular trigons or equilateral' triangles admits of 
an easy and independent proof, somewhat like that of squares as 
given in Euclid ; but the following general proposition will enable 
us to establish the case of regular polygons of any number of sides, 
including of course these two particular cases : — If isosceles triangles 
of the same vertical angle he described on the sides of a right-angled 
triangle as bases, that one on the hypotenuse is ecpted to the sum of 
the other two. (A.) 
Let G, H, P (fig, 1) be the vertices of isosceles triangles of the 
same vertical angle described on the sides of the right-angled triangle 
CAB; then if D, E, F be 
the middle points of the 
sides, GD and PE will meet 
at F, since the line drawn 
through the middle point 
of the side of a triangle 
parallel to the base bisects 
the other side. Through H 
draw HK parallel to GFD, 
or perpendicular to CB. 
Join FK and GK, and pro- 
duce them to meet BG and 
AB in M and L respectively. Then shall GKL be perpendicular 
to AB. 
