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some twenty centuries ago) that the illumination of the Moon by the Sun affords a means of estimating the Sun's distance.

If m M' M (fig. 5) represent the Moon's path about the Earth E, and s be the place of the Sun, we know that the Moon is half full when near M. But clearly it is not when the Moon has reached the point м, such that M Em is a right angle, that she is exactly half full, but when she is at the point M', such that E M's is a right angle. If, then, we can only determine the arc m M', or find out how soon after new moon the Moon appears exactly half full, we can tell in what proportion

FIG. 5.

MM'

E m

the distance of the Sun exceeds the Moon's distance; for in that case we have the angle M'E S as well as the right angle at M', and thus the shape of the triangle E M's is assigned, and with it the proportion of E S to E M', which is what we require.

Let us pause to notice the ingenuity of this method. The point to be determined is, in reality, the distance between M and M', or the angle M E M', which is the same as the angle at s. In other words, instead of estimating the angle which the Earth's radius subtends as seen from s, this plan requires that we should determine the angle which the Moon's distance subtends as seen from s;-a much easier problem, first,

because the latter angle is 60 times as great as the former, and secondly, because the necessary observations can be made at one terrestrial station.

Ingenious as the plan was, however, it was totally inadequate to meet the real (but as yet unknown) difficulty of the problem. Aristarchus estimated the Sun's distance ES at nineteen times the Moon's, or (roughly) at a twentieth of its true value.

However, we should perhaps regard the estimate by Aristarchus as corresponding to those modern estimates of certain stellar distances, regarding which astronomers only say that they do not fall short of a

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certain value, without claiming to know how far they may exceed it.

The next plan of attack was devised by Hipparchus.

Let м (fig. 6) represent the Moon just entering the shadow of the Earth E, s being the Sun. It is clear that if the Sun were just as large as the Earth, the shadow's width m m' would be exactly equal to the Earth's diameter. If the Sun were less than the Earth, the shadow at m m' would be wider than this; and if the Sun were greater than the Earth, the shadow at m m' would be narrower than the Earth's diameter. Hipparchus reasoned that if m m' is known, then by

с

combining this measure with our knowledge of the Moon's distance and the Sun's apparent diameter, we can determine the Sun's distance.*

This method, like the former, was exceedingly ingenious, because it promised to enable a single observer, by merely timing the duration of a lunar eclipse, to solve a problem which, attacked directly, requires very delicate observations, made at stations very far apart.

Again, however, the as yet unknown vastness of the Sun's distance foiled the ingenuity of astronomers. We now know that the plan just described is utterly inadequate; and we can readily understand how it was that Hipparchus and Ptolemy (who followed him in applying the method) arrived at a measure of the Sun's distance which lay no nearer to the truth than the determination made by Aristarchus of Samos.

Thus it came to pass that until the time of Tycho Brahe the received estimate of the Sun's distance was no greater than five millions of miles; nor is it too much to say that the methods applied by Aristarchus and Hipparchus might equally well have given any result whatever, from a million miles to infinity. In other words, the limits of error by these methods, and with the means available to ancient astronomers, actually exceed the quantity to be determined.

We come now, however, to the methods belonging

* We can determine at once the angle included between e m and e' m'; it is easily seen that the angle subtended by the Sun's semi-diameter exceeds this angle by twice the Sun's horizontal parallax.

to modern astronomy. Before dealing with them it will be convenient to indicate the quantity whichinstead of the distance-is the direct object of the researches to be described. Of course the distance is what astronomers really require; but this distance is determined (at least as far as direct surveying methods are concerned) by the measurement of the angle between lines directed towards the Sun's centre from different parts of the Earth. For convenience, one of the

FIG. 7.

points is taken to be at the Earth's centre as E (fig. 7). Now, if E s' represent a line directed from E towards the Sun's centre, e s" a line directed to the Sun from a point e on the Earth's surface, so placed that e s" is an horizon-line (that is, square to E e), then the angle between the lines E s' and e s" is called the Sun's horizontal parallax,* and this is the quantity which astronomers set themselves to determine in the first place. Of course, the distance of the Sun becomes known so soon as this angle is determined; and

As the Earth is not a perfect sphere, horizontal parallax is different in different places. Further, the Earth's path is eccentric, and so there is a variation depending on her position in her orbit. To secure uniformity, the results obtained by astronomers are always referred to the horizontal parallax of the Sun at his mean distance and for a place on the Earth's equator-or the mean equatorial horizontal solar parallax, as it is called. It may perhaps be useful to remind the reader that this expression means merely the apparent length of half the greater diameter of the Earth's disc as seen from the Sun (at his mean distance).

throughout the remainder of this chapter, besides mentioning the parallax deduced by each method, I shall always mention the corresponding distance.

Six several methods have been devised, each at least as ingenious as the methods of Aristarchus and Hipparchus, and each requiring an exactness of observation which would have seemed to the old astronomers altogether hopeless of attainment.

The first two methods to be described are intimately associated with Kepler's laws of the planetary distances.

So long as no known law associated the distances of the planets from the Sun, it did not seem advisable to attempt to measure the distance of any planet from the Earth as a preliminary to determining the Sun's distance; for further observations were required in order to determine what relation the latter distance bore to the former. But so soon as Kepler proved that the distances and periods of the planets are associated by a simple law, it seemed a promising course to attack instead of the Sun-some planet which approaches us within a less distance.

Let us consider, for example, the case of the planet Mars, in order that we may judge how much is to be gained by the course suggested. In doing this we are still following the actual order of events, for the first determination of the Sun's distance by modern astronomers, and with modern instrumental means, was founded on observations made upon the Planet of War. In fig 8, the orbits of Mars (M) and the Earth (E)

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