48 Mr Buchanan, On a solar Calorimeter used in Egypt 
is 31 42 inches. The difference between these two circumferences 
is 13‘02 inches. If 44*44 represent 360° of an arc then 13*02 
represents 105*5°. From the common centre of the two circles 
draw to the outer circumference two radii inclined to one another 
at an angle of 105*5°. The construction is then complete. If it 
has been carried out on the sheet of metal from which the actual 
mirror is to be constructed, we first cut out the disc of 7*07 inches 
radius ; we then apply the shears to the point where one of the 
radii cuts the circumference ; we cut along it until we reach the 
inner circumference, we then cut round this circumference along 
an arc of 254*5°, when we arrive at its inner section with the 
second radius, which is then followed until the outer circumference 
is reached. The annular disc, less the sector of 105*5° amplitude, 
which remains, is the metal band which, when bent round until 
its edges abut, forms the 45° mirror. 
Outer Mirror. Through B 3 draw 0 : >,P 3 parallel to OP and on it 
lay off B 3 A 3 = BA = 2 inches. From A 3 as centre, at the distance 
A 3 B 3 describe a circular arc. Join AA 3 and produce the line AA 3 
till it cuts the arc in B 4 . Join B 3 B 4 . B 3 B 4 is the line of section of 
the outer mirror. For, having in view the properties of triangles 
and of parallel lines, it is clear that the lines 0 4 B 4 and B 4 A make 
equal angles with the line B 3 B 4 . But 0 4 B 4 is the direction of the 
incident ray at B 4 ; therefore B 4 A is the direction of the reflected 
ray, and the ray which strikes the outer rim of the mirror is 
reflected upon the upper extremity of the focal line. In the same 
way it is evident that the incident ray 0 3 B 3 , which strikes the 
inner rim of the mirror, is reflected along the line B 3 B and 
falls upon B, the lower extremity of the focal line. Consequently 
parallel rays which strike the mirror in points between B 4 and B 3 
are reflected on AB and strike it in points between A and B 
which are homologous as regards position with the points in the 
mirror between B 4 and B 3 which are struck by the primary rays. 
In the isosceles triangle B 3 A 3 B 4 the angle A 3 is equal to the 
angle A 3 AB, therefore 
therefore 
tan A 3 
b,a 3 
ab 3 
AB 
AB 3 
A* = 111° 48' 
-0*4; 
and the angle at the base 
i=\ (180° — A 3 ) = 34° 6'. 
Further, the base 
B 3 B 4 = 2 x AB cos i = 3 31 inches. 
