266 
POPULAE SCIENCE EEYIEW. 
after one epoch at which an eclipse (or more than one) of 
either kind has taken place, an interval of nearly six months 
will elapse before another eclipse (or set of eclipses) can take 
place. 
Let SEs' and mEm' (^fig. 2) represent the apparent paths of 
the sun and moon on the celestial sphere. They are, of course, 
great circles of the sphere, and are here supposed to be seen 
sideways, the two nodes being both seen at e, one between e 
and the eye, the other beyond e. The arcs sm, s'm' contain 
each rather more than 5°. Now, suppose that on the same 
scale, the small lines s^m^ and s'^m'* represent the greatest dis- 
tance which can separate the centres of the sun and moon, so 
that there may be a solar eclipse (mere contact, of course, in 
such a case).* Then it is clear, that if conjunction takes place 
when the moon is on either the nearer or the farther arc repre- 
sented by m’m'‘, there will be an eclipse. Let us open out the 
circle mEm' until it is seen as a circle, and let one of the two 
arcs be opened out into the arc AC (m’ moving along the 
line mU, m'‘ along the line m'‘c). The arc Ac is found in this 
w^ay (or rather from the calculations which result from this con- 
struction) to be at the outside 37° 12'. It varies of course with 
the varying distances of the sun and moon from the earth, which 
variations obviously affect the construction indicated in fig. 1. 
Its mean value is 33° 56'. As there are two such arcs, we see 
that whereas the whole circumference of the ecliptic contains 
360°, the portions along which eclipses are possible contain on 
an average only 67° 52'. Reducing both arcs to minutes, we 
obtain the numbers 21,600 and 4,072. It follows that, on an 
average of any very great number of lunations, there are for 
every 21,600 conjunctions of the sun and moon, 4,072 at which 
the sun is eclipsed. 
Next let us do the like for the moon. We take s^m^, s^m^, 
equal to the greatest distance at which the moon’s centre can 
be removed from the centre of the earth’s shadow, in order that 
there may be a lunar eclipse.f Thence we obtain the arc dg, 
whose average value we find to be 21° 47'; and as there are 
two such arcs, we find that along the whole ecliptic of 360° 
there are on an average only 43° 34', along which a lunar 
eclipse can take place. Thus, for every 21,600 lunations, there 
are on an average 2,614 lunar eclipses. 
In all, there are, for every 21,600 lunations, 6,686 eclipses, 
solar or lunar. 
* The angle ni' e m' in fig. 1 corresponds to the angle subtended by 
at E in fig. 2 — not to the angle ndE s'^ as actually seen, but on the supposition 
that E m* and e s^ are foreshortened radii of the sphere A s D s'. 
t This is the distance subtended at E (fig. 1) by the section of the um- 
bral cone opposite m m. 
