Introduction

Many different ways of representing a three-dimensional unit vector or axis have been developed over the centuries, due not only to the requirements of different disciplines (Astronomy, Geodesy, Geology, Geophysics, Mathematics, …) but also to diverse needs within a discipline: in Geology, for example, there appear to be five or six systems in current use. In this book, we shall use either polar coordinates or the corresponding direction cosines for all purposes of statistical analysis. The following sub-section (§2.2) defines several of the coordinate systems and gives the mathematical relationship of each to polar coordinates.

Later chapters of this book, concerned with statistical analysis, abound with words and phrases which have particular meanings in Statistics, and, possibly, rather different meanings in other areas. A good example of this is the word “sample”, which for our purposes is loosely taken to mean a collection of measurements of a particular characteristic, but which has a general scientific meaning of an observational or sampling unit (e.g. a drill-core specimen on which a single measurement may be made). §2.3 gives definitions of a number of such words and phrases.

Spherical coordinate systems

The type of data we shall be dealing with will be either directed lines or undirected lines. For the former, the measurements will be unit vectors, such as the direction of magnetisation of a rock specimen, or the direction of palaeocurrent flow. For the latter, which we shall term axes (cf. §2.3), the line measured might be the normal to a fracture plane, and so have no sense (direction) unless this is ascribed on some other basis.

Let us now turn our thoughts beyond the earth and its atmosphere to the phenomena which may properly be described as astronomical. We see a procession of objects moving ceaselessly across the sky—the sun by day, the moon and stars by night. These all appear to cross the sky from east to west, because the rotation of the earth, from which we view the spectacle, causes us to move continually from west to east.

The most conspicuous phenomenon is of course the daily motion of the sun across the heavens, producing the alternations of light and darkness, heat and cold, which we describe as day and night. The rising and setting of the moon and its passage across the sky are only one degree less conspicuous, and must have been not only noticed, but also familiar, since the days when human beings first appeared on the earth.

The sun shews no changes either of shape or brightness, except when our own atmosphere dims its light, but the moon continually varies in both respects. Every month it goes through the complete cycle of changes, which we call its “phases”. It begins as a thin crescent of light, which we describe as the new moon. This increases in size until after about a week we have the semicircle of light we call half moon, and then a week later the complete circle we call full moon.

We know that the moon always looks about the same size in the sky and from this we can conclude that it is always at about the same distance from the earth. And we can measure the distance in the same way as we measure the distance of an inaccessible mountain peak, or the height of an aeroplane.

When an aeroplane is up in the air, people who are standing at different points must look in different directions to see it. If it is directly overhead for one man, it will not be directly overhead for another man a mile away, and its height can be calculated simply by noticing how far its position appears to be out of the vertical for the second man. Using this method, astronomers find that the distance of the moon varies between the limits of 221,462 miles and 252,710 miles, the average distance being 238,857 miles. Thus, in round numbers, we may think of the moon as being a quarter of a million miles away.

At such a distance, we can hardly expect to see much detail with our unaided eyes. Indeed, as we watch the moon sailing through the night sky, we can detect nothing on its surface beyond a variety of light and dark patches, which, with a bit of imagination, we can make into the man in the moon with his bundle of sticks, or an old woman reading a book, or—as the Chinese prefer to think—a jumping hare.

Every year for more than a century, the Royal Institution has invited, some man of science to deliver a course of lectures at Christmastide in a style “adapted to a juvenile auditory”. In practice, this rather quaint phrase means that the lecturer will be confronted with an eager and critical audience, ranging in respect of age from under eight to over eighty, and in respect of scientific knowledge from the aforesaid child under eight to staid professors of science and venerable Fellows of the Royal Society, each of whom will expect the lecturer to say something that will interest him.

The present book contains the substance of what I said when I was honoured with an invitation of this kind for the Christmas season 1933—4, fortified in places with what I have said on other slightly more serious occasions, both at the Royal Institution and elsewhere.

It is a pleasure to acknowledge many courtesies and return thanks for much valuable help. I am indebted to Sir T. L. Heath for permission to borrow largely from his Greek Astronomy and other books; to many Institutions, Publishers and private individuals for the loan of negatives, prints, blocks, etc., and permission to reproduce these in my book—detailed acknowledgment is made in the List of Illustrations.

Let us leave the earth, in which we have burrowed for long enough, and turn our thoughts, and our eyes, upwards.

We all know what we may expect to see—the sun, the blue sky, and possibly some clouds, by day; stars, with perhaps the moon and one or more planets, by night. We see these objects by light which has travelled to us through the earth's atmosphere, and if we see them clearly, it is because the atmosphere is transparent—it presents no barrier to the passage of rays of light.

Perhaps we are so accustomed to this fact that we merely take it for granted. Or perhaps we think of the atmosphere as something too flimsy and ethereal ever to stop the passage of rays of light. Yet we know exactly how much atmosphere there is, for the ordinary domestic barometer is weighing it for us all the time. When the barometer needle points to 30, there is as much substance in the atmosphere over our heads as there is in a layer of mercury 30 inches thick. This again is the same amount as there would be in a layer of lead about 36 inches thick, for mercury is heavier than an equal volume of lead in the ratio of about six to five. To visualise the weight of the atmosphere above us, we may think of ourselves as covered up with 144 blankets of lead, each a quarter of an inch in thickness.

These are restless days in which everyone travels who can. The more fortunate of us may have travelled outside Europe to other continents—perhaps even round the world—and seen strange sights and scenery on our travels. And now we are starting out to take the longest journey in the whole universe. We shall travelor pretend to travel—so far through space that our earth will look like less than the tiniest of motes in a sunbeam, and so far through time that the whole of human history will shrink to a tick of the clock, and a man's whole life to something less than the twinkling of an eye.

As we travel through space, we shall try to draw a picture of the universe as it now is—vast spaces of unthinkable extent and terrifying desolation, redeemed from utter emptiness only at rare intervals by small particles of cold lifeless matter, and at still rarer intervals by those vivid balls of flaming gas we call stars. Most of these stars are solitary wanderers through space, although here and there we may perhaps find a star giving warmth and light to a family of encircling planets. Yet few of these are at all likely to resemble our own earth; the majority will be so different that we shall hardly be able to describe their scenery, or imagine their physical condition.

We all know now that our sun is a very ordinary star, but it took men a long time to discover this. Perhaps this is not surprising, for certainly it does not look much like an ordinary star to us. The reason is, of course, that it is enormously nearer than any of the other stars.

We have seen how the ancients imagined the earth to be the fixed centre of the universe, round which everything else moved. The stars merely formed a background of light, against which they could map out the motions of the sun, moon and planets. They thought of the stars as attached to the inside of a hollow sphere, which turned round over the earth much as a telescope dome turns round over the floor of a telescope, or “as one might turn a cap round on one's head”. And although a few of the more philosophical of the Greeks gave reasons for thinking that the earth moved round the sun, they had no means of making their opinions or arguments known to a wide circle of people, so that these were forgotten as the world gradually became submerged in the intellectual darkness of the Middle Ages. Then, in 1543, a Polish monk, Copernicus, advanced views and arguments which were very similar to those which Aristarchus of Samos had propounded 1800 years earlier, although the extent to which he was indebted to his Greek predecessors is not clear.

There are nine planets circling round the sun, of which of course the earth is one. Of the other eight, five have been known from pre-historic times, while the remaining three—the three farthest from the sun—are comparatively recent discoveries.

The row of models exhibited in fig. 60 shew how greatly these nine planets differ in size. Those which are nearest to, and farthest away from, the sun are the smallest, while the middle members, Jupiter and Saturn, are the largest. Jupiter, the central member, is largest of all, with a diameter of nearly 90,000 miles, and a volume 1300 times that of the earth. Jupiter stands in the same proportion to the earth as a football to a marble, while on the same scale Mars would be hardly larger than a pea.

If we wish to complete our model by placing the objects shewn in fig. 60 at their proper distances, the nearest planet, Mercury, must describe an orbit which is not quite circular, but is such that, even at its nearest approach to the sun, the planet would be 20 feet away. The earth must keep at a distance of 50 feet from the sun, while Pluto, the farthest planet of all, must describe an orbit nearly half a mile in radius.

We see that the solar system consists mainly of empty space, and yet the emptiness of the solar system is as nothing compared to the emptiness of space itself.