In order now to realise what one second of arc really means, let us look at the circle here shown, which is as nearly as possible one-tenth of an inch in diameter—(one-O-tenth of an inch). If we remove this circle to a distance of 28 feet 8 inches it will subtend an angle of one minute, and we shall have to place it at a distance of nearly 1730 feet—almost one-third of a mile—to reduce the angle to one second. But the very nearest to us of the fixed stars, Alpha Centauri, has a parallax of only three-fourths of a second; that is, the distance of the earth from the sun—about 923/4 millions of miles—would appear no wider, seen from the nearest star, than does three-fourths of the above small circle at one-third of a mile distance. To see this circle at all at that distance would require a very good telescope with a power of at least 100, while to see any small part of it and to measure the proportion of that part to the whole would need very brilliant illumination and a large and powerful astronomical telescope.
What is a Million?
But when we have to deal with millions, and even with hundreds and thousands of millions, there is another difficulty—that few people can form any clear conception of what a million is. It has been suggested that in every large school the walls of one room or hall should be devoted to showing a million at one view. For this purpose it would be necessary to have a hundred large sheets of paper each about 4 feet 6 inches square, ruled in quarter inch squares. In each alternate square a round black wafer or circle should be placed a little over-lapping the square, thus leaving an equal amount of white space between the black spots. At each tenth spot a double width should be left so as to separate each hundred spots (10 × 10). Each sheet would then hold ten thousand spots, which would all be distinctly visible from the middle of a room 20 feet wide, each horizontal or vertical row containing a thousand. One hundred such sheets would contain a million spots, and they would occupy a space 450 feet long in one row, or 90 feet long in five rows, so that they would entirely cover the walls of a room, about 30 feet square and 25 feet high, from floor to ceiling, allowing space for doors but not for windows, the hall or gallery being lighted from above. Such a hall would be in the highest degree educational in a country where millions are spoken of so glibly and wasted so recklessly; while no one can really appreciate modern science, dealing as it does with the unimaginably great and little, unless he is enabled to realise by actual vision, and summing up, what a vast number is comprised in one of those millions, which, in modern astronomy and physics, he has to deal with not singly only, but by hundreds and thousands or even by millions. In every considerable town, at all events, a hall or gallery should have a million thus shown upon its walls. It would in no way interfere with the walls being covered when required with maps, or ornamental hangings, or pictures; but when these were removed, the visible and countable million would remain as a permanent lesson to all visitors; and I believe that it would have widespread beneficial effects in almost every department of human thought and action. On a small scale any one can do this for himself by getting a hundred sheets of engineer's paper ruled in small squares, and making the spots very small; and even this would be impressive, but not so much so as on the larger scale.
In order to enable every reader of this volume at once to form some conception of the number of units in a million, I have made an estimate of the number of letters contained in it, and I find them to amount to about 420,000—considerably less than half a million. Try and realise, when reading it, that if every letter were a pound sterling, we waste as many pounds as there are letters in two such volumes whenever we build a battleship.
Having thus obtained some real conception of the immensity of a million, we can better realise what it must be to have every one of the dots above described, or every one of the letters in two such volumes as this lengthened out so as to be each a mile long, and even then we should have reached little more than a hundredth part of the distance from our earth to the sun. When, by careful consideration of these figures, we have even partially realised this enormous distance, we may take the next step, which is, to compare this distance with that of the nearest fixed star. We have seen that the parallax of that star is three-fourths of a second, an amount which implies that the star is 271,400 times as far from us as our sun is. If after seeing what a million is, and knowing that the sun is 923/4 times this distance from us in miles—a distance which itself is almost inconceivable to us—we find that we have to multiply this almost inconceivable distance 271,400 times—more than a quarter of a million times—to reach the nearest of the fixed stars, we shall begin to realise, however imperfectly, how vast is the system of suns around us, and on what a scale of immensity the material universe, which we see so gloriously displayed in the starry heavens and the mysterious galaxy, is constructed.
This somewhat lengthy preliminary discussion is thought necessary in order that my readers may form some idea of the enormous difficulty of obtaining any measurement whatever of such distances. I now propose to point out what the special difficulties are, and how they have been overcome; and thus I hope to be able to satisfy them that the figures astronomers give us of the distances of the stars are in no way mere guesses or probabilities, but are real measurements which, within certain not very wide limits of error, may be trusted as giving us correct ideas of the magnitude of the visible universe.
Measurement of Stellar Distances
The fundamental difficulty of this measurement is, of course, that the distances are so vast that the longest available base-line, the diameter of the earth's orbit, only subtends an angle of little more than a second from the nearest star, while for all the rest it is less than one second and often only a small fraction of it. But this difficulty, great as it is, is rendered far greater by the fact that there is no fixed point in the heavens from which to measure, since many of the stars are known to be in motion, and all are believed to be so in varying degrees, while the sun itself is now known to be moving among the stars at a rate which is not yet accurately determined, but in a direction which is fairly well known. As the various motions of the earth while passing round the sun, though extremely complex, are very accurately known, it was first attempted to determine the changed position of stars by observations, many times repeated at six months' intervals, of the moment of their passage over the meridian and their distance from the zenith; and then by allowing for all the known motions of the earth, such as precession of the equinoxes and nutation of the earth's axis, as well as for refraction and for the aberration of light, to determine what residual effect was due to the difference of position from which the star was viewed; and a result was thus obtained in several cases, though almost always a larger one than has been found by later observations and by better methods. These earlier observations, however perfect the instruments and however skilful the observer, are liable to errors which it seems impossible to avoid. The instruments themselves are subject in all their parts to expansion and contraction by changes of temperature; and when these changes are sudden, one part of the instrument may be affected more than another, and this will often lead to minute errors which may seriously affect the amount to be measured when that is so small. Another source of error is due to atmospheric refraction, which is subject to changes both from hour to hour and at different seasons. But perhaps most important of all are minute changes in level of the foundations of the instruments even when they are carried down to solid rock. Both changes of temperature and changes of moisture of the soil produce minute alterations of level; while earth-tremors and slow movements of elevation or depression are now known to be very frequent. Owing to all these causes, actual measurements of differences of position at different times of the year, amounting to small fractions of a second, are found to be too uncertain for the determination of such minute angles with the required accuracy.
But there is another method which avoids almost all these sources of error, and this is now generally preferred and adopted for these measurements. It is, that of measuring the distance between two stars situated apparently very near each other, one of which has large proper motion, while the other has none which is measurable. The proper motions of the stars was first suspected by Halley in 1717, from finding that several stars, whose places had been given by Hipparchus, 130 B.C., were not in the positions where they now ought to be; and other observations by the old astronomers, especially those of occultations of stars by the moon, led to the same result. Since the time of Halley very accurate observations of the stars have been made, and in many cases it is found that they move perceptibly from year to year, while others move so slowly that it is only after forty or fifty years that the motion can be detected. The greatest proper motions yet determined amount to between 7" and 8" in a year, while other stars require twenty, or even fifty or a hundred years to show an equal amount of displacement. At first it was thought that the brightest stars would have the largest proper motion, because it was supposed they were nearest to us, but it was soon found that many small and quite inconspicuous stars moved as rapidly as the most brilliant, while in many very bright stars no proper motion at all can be detected. That which moves most rapidly is a small star of less than the sixth magnitude.
It is a matter of common observation that the motion of things at a distance cannot be perceived so well as when near, even though the speed may be the same. If a man is seen on the top of a hill several miles off, we have to observe him closely for some time before we can be sure whether he is walking or standing still. But objects so enormously distant as we now know that the stars are, may be moving at the rate of many miles in a second and yet require years of observation to detect any movement at all.