52 Mr Buchanan, On a solar Calorimeter used in Egypt 
Abscissae are to be measured on a line at right angles to OP, 
and they are positive when measured to the right. 
J oin BB 1 ; and through B L draw a line 0 1 P 1 parallel to OP, and 
on it lay off B X A X = BA. 
Join A A 1 and produce it to a point B 2 , so that A 1 B 2 — AB. 
Join j B 1 B 2 ; then B 1 B 2 is the line which represents in section 
the innermost mirror. 
Through B 2 draw a straight line 0 2 P 2 parallel to OP, and on it 
lay off the length B 2 A 2 = BA. 
Join AA 2 and produce it to a point B 3 , so that A 2 B 3 = AB. 
Join B 2 B 3 ; then B 2 B 3 is the line which represents in section the 
second mirror of the series. 
Through B 3 draw 0 3 P 3 parallel to OP, and on it lay off 
B 3 A 3 = BA. 
Join AA 3 and produce it to a point B 4 , so that A 3 B 4 = AB. 
Join B 3 B 4 ; then B 3 B 4 is the line which represents in section 
the third mirror in the series ; and so on. 
It is evident from the properties of parallel lines that the angle 
which the incident ray makes with the outer extremity of any one 
of these lines is equal to the angle made with it by the line 
connecting that point with the upper extremity of the focal line. 
Therefore all the rays parallel to the axis which strike the outer 
extremity of a line of section are reflected upon A, the upper 
extremity of the focal line. In the same way all the rays parallel 
to the axis which strike the inner extremity of a line of section 
are reflected upon B the lower extremity of the focal line. Con- 
sequently all the rays parallel to the axis which fall upon inter- 
mediate points in the line of section are reflected upon the corre- 
sponding points between A and B on the focal line. Therefore 
all the rays parallel to the axis which strike the reflector are 
reflected and condensed on the focal line AB. 
If the graphic construction is effected on the natural scale, all 
the measurements, both linear and angular, can be taken from it 
directly with sufficient exactness to enable the reflector to be con- 
structed. On the other hand, the geometrical construction is 
so simple that there is no difficulty in arriving at all the values 
by calculation. It will be apparent from the diagram (Fig. 6) 
that most of the elements of each section are contained in an 
isosceles triangle A n B n B n+1 . In it the angle A at the apex of 
one triangle is derived from the data of the previous triangle. 
The angle of inclination to the axis of the mirror is 
i = i (180° - A), 
and the length of the base or width of the mirror is 
m = 2 AB cos i. 
