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familiar to modern savages.

With these facts it is convenient to begin, taking them in the order in which they most readily present themselves to any ordinary observer.

3. The sun is daily seen to rise in the eastern part of the sky, to travel across the sky, to reach its highest position in the south in the middle of the day, then to sink, and finally to set in the western part of the sky. But its daily path across the sky is not always the same: the points of the horizon at which it rises and sets, its height in the sky at midday, and the time from sunrise to sunset, all go through a series of changes, which are accompanied by changes in the weather, in vegetation, etc.; and we are thus able to recognise the existence of the seasons, and their recurrence after a certain interval of time which is known as a year.

4. But while the sun always appears as a bright circular disc, the next most conspicuous of the heavenly bodies, the moon, undergoes changes of form which readily strike the observer, and are at once seen to take place in a regular order and at about the same intervals of time. A little more care, however, is necessary in order to observe the connection between the form of the moon and her position in the sky with respect to the sun. Thus when the moon is first visible soon after sunset near the place where the sun has set, her form is a thin crescent (cf. fig. 11 on p. 31), the hollow side being turned away from the sun, and she sets soon after the sun. Next night the moon is farther from the sun, the crescent is thicker, and she sets later; and so on, until after rather less than a week from the first appearance of the crescent, she appears as a semicircular disc, with the flat side turned away from the sun. The semicircle enlarges, and after another week has grown into a complete disc; the moon is now nearly in the opposite direction to the sun, and therefore rises about at sunset and sets about at sunrise. She then begins to approach the sun on the other side, rising before it and setting in the daytime; her size again diminishes, until after another week she is again semicircular, the flat side being still turned away from the sun, but being now turned towards the west instead of towards the east. The semicircle then becomes a gradually diminishing crescent, and the time of rising approaches the time of sunrise, until the moon becomes altogether invisible. After two or three nights the new moon reappears, and the whole series of changes is repeated. The different forms thus assumed by the moon are now known as her phases; the time occupied by this series of changes, the month, would naturally suggest itself as a convenient measure of time; and the day, month, and year would thus form the basis of a rough system of time-measurement.

5. From a few observations of the stars it could also clearly be seen that they too, like the sun and moon, changed their positions in the sky, those towards the east being seen to rise, and those towards the west to sink and finally set, while others moved across the sky from east to west, and those in a certain northern part of the sky, though also in motion, were never seen either to rise or set. Although anything like a complete classification of the stars belongs to a more advanced stage of the subject, a few star groups could easily be recognised, and their position in the sky could be used as a rough means of measuring time at night, just as the position of the sun to indicate the time of day.

6. To these rudimentary notions important additions were made when rather more careful and prolonged observations became possible, and some little thought was devoted to their interpretation.

Several peoples who reached a high stage of civilisation at an early period claim to have made important progress in astronomy. Greek traditions assign considerable astronomical knowledge to Egyptian priests who lived some thousands of years B.C., and some of the peculiarities of the pyramids which were built at some such period are at any rate plausibly interpreted as evidence of pretty accurate astronomical observations; Chinese records describe observations supposed to have been made in the 25th century B.C.; some of the Indian sacred books refer to astronomical knowledge acquired several centuries before this time; and the first observations of the Chaldaean priests of Babylon have been attributed to times not much later.

On the other hand, the earliest recorded astronomical observation the authenticity of which may be accepted without scruple belongs only to the 8th century B.C.

For the purposes of this book it is not worth while to make any attempt to disentangle from the mass of doubtful tradition and conjectural interpretation of inscriptions, bearing on this early astronomy, the few facts which lie embedded therein; and we may proceed at once to give some account of the astronomical knowledge, other than that already dealt with, which is discovered in the possession of the earliest really historical astronomersβ€”the Greeksβ€”at the beginning of their scientific history, leaving it an open question what portions of it were derived from Egyptians, Chaldaeans, their own ancestors, or other sources.

7. If an observer looks at the stars on any clear night he sees an apparently innumerable1 host of them, which seem to lie on a portion of a spherical surface, of which he is the centre. This spherical surface is commonly spoken of as the sky, and is known to astronomy as the celestial sphere. The visible part of this sphere is bounded by the earth, so that only half can be seen at once; but only the slightest effort of the imagination is required to think of the other half as lying below the earth, and containing other stars, as well as the sun. This sphere appears to the observer to be very large, though he is incapable of forming any precise estimate of its size.2

Most of us at the present day have been taught in childhood that the stars are at different distances, and that this sphere has in consequence no real existence. The early peoples had no knowledge of this, and for them the celestial sphere really existed, and was often thought to be a solid sphere of crystal.

Fig. 1.β€”The celestial sphere.

Moreover modern astronomers, as well as ancient, find it convenient for very many purposes to make use of this sphere, though it has no material existence, as a means of representing the directions in which the heavenly bodies are seen and their motions. For all that direct observation can tell us about the position of such an object as a star is its direction; its distance can only be ascertained by indirect methods, if at all. If we draw a sphere, and suppose the observer’s eye placed at its centre O (fig. 1), and then draw a straight line from O to a star S, meeting the surface of the sphere in the point s; then the star appears exactly in the same position as if it were at s, nor would its apparent position be changed if it were placed at any other point, such as Sβ€² or Sβ€³, on this same line. When we speak, therefore, of a star as being at a point s on the celestial sphere, all that we mean is that it is in the same direction as the point s, or, in other words, that it is situated somewhere on the straight line through O and S. The advantages of this method of representing the position of a star become evident when we wish to compare the positions of several stars. The difference of direction of two stars is the angle between the lines drawn from the eye to the stars; e.g., if the stars are R, S, it is the angle R O S. Similarly the difference of direction of another pair of stars, P, Q, is the angle P O Q. The two stars P and Q appear nearer together than do R and S, or farther apart, according as the angle P O Q is less or greater than the angle R O S. But if we represent the stars by the corresponding points p, q, r, s on the celestial sphere, then (by an obvious property of the sphere) the angle P O Q (which is the same as p O q) is less or greater than the angle R O S (or r O s) according as the arc joining p q on the sphere is less or greater than the arc joining r s, and in the same proportion; if, for example, the angle R O S is twice as great as the angle P O Q, so also is the arc p q twice as great as the arc r s. We may therefore, in all questions relating only to the directions of the stars, replace the angle between the directions of two stars by the arc joining the corresponding points on the celestial sphere, or, in other words, by the distance between these points on the celestial sphere. But such arcs on a sphere are easier both to estimate by eye and to treat geometrically than angles, and the use of the celestial sphere is therefore of great value, apart from its historical origin. It is important to note that this apparent distance of two stars, i.e. their distance from one another on the celestial sphere, is an entirely different thing from their actual distance from one another in space. In the figure, for example, Q is actually much nearer to S than it is to P, but the apparent distance measured by the arc q s is several times greater than q p. The apparent distance of two points on the celestial sphere is measured numerically by the angle between the lines joining the eye to the two points, expressed in degrees, minutes, and seconds.3

We might of course agree to regard the celestial sphere as of a particular size, and then express the distance between two points on it in miles, feet, or inches; but it is practically very inconvenient to do so. To say, as some people occasionally do, that the distance between two stars is so many feet is meaningless, unless the supposed size of the celestial sphere is given at the same time.

It has already been pointed out that the observer is always at the centre of the celestial sphere; this remains true even if he moves to another place. A sphere has, however, only one centre, and therefore if the sphere remains fixed the observer cannot move about and yet always remain at the centre. The old astronomers met this difficulty by supposing that the celestial sphere was so large that any possible motion of the observer would be insignificant in comparison with the radius of the sphere and could be neglected. It is often more convenientβ€”when we are using the sphere as a mere geometrical device for representing the position of the starsβ€”to regard the sphere as moving with the observer, so that he always

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