So vast are the distances of the stars that all attempts of the early astronomers to ascertain them necessarily proved futile. This led many astronomers after Copernicus to reject his doctrine of the earth's motion round the sun, so that they clung rather to the Ptolemaic view that the earth was without motion and was the center about which all the celestial motions took place. The geometry of stellar distances was perfectly understood, and many were the attempts made to find the parallaxes and distances of the stars; but the art of instrument making had not yet advanced to a stage where astronomers had the mechanisms that were absolutely necessary to measure very small angles.
About 1835, Bessel undertook the work of determining stellar parallax in earnest. His instrument was the heliometer, originally designed for measuring the sun's diameter; but as modified for parallax work it is the most accurate of all angle-measuring instruments that the astronomers employ. The star that he selected was 61 Cygni, not a bright star, of the sixth magnitude only, but its large proper motion suggested that it might be one of those nearest to us. He measured with the heliometer, at opposite seasons of the year, the distance of 61 Cygni from another and very small star in the same field of view, and thus determined the relative parallax of the two stars. The assumption was made that the very faint star was very much more distant than the bright one, and this assumption will usually turn out to be sound. Bessel got 0".35 for his parallax of 61 Cygni, and Struve by applying the same method to Alpha Lyræ, about the same time, got 0".25 for the parallax of that star.
These classic researches of Bessel and Struve are the most important in the history of star distances, because they were the first to prove that stellar parallax, although minute, could nevertheless be actually measured. About the same time success was achieved in another quarter, and Henderson, the British astronomer at the Cape of Good Hope, found a parallax of nearly a whole second for the bright star Alpha Centauri.
Although the parallaxes of many hundreds of stars have been measured since, and the parallaxes of other thousands of stars estimated, the measured parallax of Alpha Centauri, as later investigated by Elkin and Sir David Gill, and found to be 0".75, is the largest known parallax, and therefore Alpha Centauri is our nearest neighbor among the stars, so far as we yet know. This star is a binary system and the light of the two components together is about the same as that of Capella (Alpha Aurigæ). But it is never visible from this part of the world, being in 60 degrees of south declination: one might just glimpse it near the southern horizon from Key West.
How the distances of the stars are found is not difficult to explain, although the method of doing it involves a good deal of complication, interesting to the practical astronomer only. Recall the method of getting the moon's distance from the earth: it was done by measuring her displacement among the stars as seen from two widely separated observatories, as near the ends of a diameter of the earth as convenient. This is the base line, and the angle which a radius of the earth as seen from the center of the moon fills, or subtends, is the moon's parallax.
So near is the moon that this angle is almost an entire degree, and therefore not at all difficult to measure. But if we go to the distance of even Alpha Centauri, the nearest of the stars, our earth shrinks to invisibility; so that we must seek a longer base line. Fortunately there is one, but although its length is 25,000 times the earth's diameter, it is only just long enough to make the star distances measurable. We found that the sun's distance from the earth was 93 million miles; the diameter of the earth's orbit is therefore double that amount. Now conceive the diameter of the earth replaced by the diameter of the earth's orbit: by our motion round the sun we are transported from one extremity of this diameter to the opposite one in six month's time; so we may measure the displacement of a star from these two extremities, and half this displacement will be the star's parallax, often called the annual parallax because a year is consumed in traversing its period. And it is this very minute angle which Bessel and Struve were the first to measure with certainty, and which Henderson found to be in the case of Alpha Centauri the largest yet known.
Evidently the earth by its motion round the sun makes every star describe, a little parallactic ellipse; the nearer the star is the larger this ellipse will be, and the farther the star the smaller: if the star  were at an infinite distance, its ellipse would become a point, that is, if we imagine ourselves occupying the position of the star, even the vast orbit of the earth, 186 million miles across, would shrink to invisibility or become a mathematical point.
Measurement of stellar parallax is one of many problems of exceeding difficulty that confront the practical astronomer. But the actual research nowadays is greatly simplified by photography, which enables the astronomer to select times when the air is not only clear, but very steady for making the exposures. Development and measurement of the plates can then be done at any time. Pritchard of Oxford, England, was among the earliest to appreciate the advantages of photography in parallax work, and Schlesinger, Mitchell, Miller, Slocum and Van Maanen, with many others in this country, have zealously prosecuted it.
How shall we intelligently express the vast distances at which the stars are removed from us? Of course we can use miles, and pile up the millions upon millions by adding on ciphers, but that fails to give much notion of the star's distance. Let us try with Alpha Centauri: its parallax of 0".75 means that it is 275,000 times farther from the sun than the earth is. Multiplying this out, we get 25 trillion miles, that is, 25 millions of million miles—an inconceivable number, and an unthinkable distance.
Suppose the entire solar system to shrink so that the orbit of Neptune, sixty times 93 million miles in diameter, would be a circle the size of the dot over this letter i. On the same scale the sun itself, although nearly a million miles in diameter, could  not be seen with the most powerful microscope in existence; and on the same scale also we should have to have a circle ten feet in diameter, if the solar system were imagined at its center and Alpha Centauri in its circumference.
So astronomers do not often use the mile as a yardstick of stellar distance, any more than we state the distance from London to San Francisco in feet or inches. By convention of astronomers, the average distance between the centers of sun and earth, or 93 million miles, is the accepted unit of measure in the solar system. So the adopted unit of stellar distance is the distance traveled by a wave of light in a year's time: and this unit is technically called the light-year. This unit of distance, or stellar yardstick, as we may call it, is nearly 6 millions of million miles in length. Alpha Centauri, then, is four and one-third light-years distant, and 61 Cygni seven and one-fifth light-years away.
For convenience in their calculations most astronomers now use a longer unit called the parsec, first suggested by Turner. Its length is equal to the distance of a star whose parallax is one second of arc; that is, one parsec is equal to about three and a quarter light-years. Or the light-year is equal to 0.31 parsec. Also the parsec is equal to 206,000 astronomical units, or about 19 millions of million miles.
We have, then four distinct methods of stating the distance of a star: Sirius, for example, has a parallax of 0".38 or its distance is two and two-thirds parsecs, or eight and a half light-years, or 50 millions of million miles. It is the angle of parallax which is always found first by actual measurement and from this the three other estimates of distance are calculated.
So difficult and delicate is the determination of a stellar distance that only a few hundred parallaxes have been ascertained in the past century. The distance of the same star has been many times measured by different astronomers, with much seeming duplication of effort. Comprehensive campaigns for determining star parallaxes in large numbers have been undertaken in a few instances, particularly at the suggestion of Kapteyn, the eminent astronomer of Groningen, Holland. His catalogue of star parallaxes is the most complete and accurate yet published, and is the standard in all statistical investigations of the stars.
That we find relatively large parallaxes for some of the fainter stars, and almost no measurable parallax for some of the very bright stars is one of the riddles of the stellar universe. We may instance Arcturus, in the northern hemisphere and Canopus in the southern; the latter almost as bright as Sirius. Dr. Elkin and the late Sir David Gill determined exhaustively the parallax of Canopus, and found it very minute, only 0".03, making its distance in excess of a hundred light-years. The stupendous brilliancy of this star is apparent if we remember that the intensity of its light must vary inversely as the square of the distance; so that if Canopus were to be brought as near us as even 61 Cygni is, it would be a hundredfold brighter than Sirius, the brightest of all the stars of the firmament.
In researches upon the distribution of the more distant stars, the method of measuring parallaxes of individual stars fails completely, and the secular parallax, or parallactic motion of the stars is employed instead. By parallactic motion is meant the apparent displacement in consequence of the solar motion which is now known with great accuracy, and amounts to 19.5 kilometers per second. Even in a single year, then, the sun's motion is twice the diameter of the earth's orbit, so that in a hundred or more years, a much longer base line is available than in the usual type of observations for stellar parallax. If we ascertain the parallactic motion of a group of stars, then we can find their average distance. It is found, for example, that the mean parallax of stars of the sixth magnitude is 0".014. Also the mean distances of stars thrown into classes according to their spectral type have been investigated by Boss, Kapteyn, Campbell and others. The complete intermingling of the two great star streams has been proved, too, by using the magnitude of the proper motions to measure the average distances of both streams. These come out essentially the same, so that the streaming cannot be due to mere chance relation in the line of sight.
Most unexpected and highly important is the discovery that the peculiar behavior of certain lines in the spectrum leads to a fixed relation between a star's spectrum and its absolute magnitude, which provides a new and very effective method of ascertaining stellar distances. By absolute magnitudes are meant the magnitudes the stars would appear to have if they were all at the same standard distance from the earth.
Very satisfactory estimates of the distance of exceedingly remote objects have been made within recent years by this indirect method, which is especially applicable to spiral nebulæ and globular clusters. The absolute magnitude of a star is inferred from the relative intensities of certain lines in its spectrum, so that the observed apparent magnitude at once enables us to calculate the distance of the star. Adams and Joy have recently determined the luminosities and parallaxes of 500 stars by this spectroscopic method. Of these stars 360 have had their parallaxes previously measured; and the average difference between the spectroscopic and the trigonometric values of the parallax is only the very small angle 0".0037, a highly satisfactory verification.
An indirect method, but a very simple one, and of the greatest value because it provides the key to stellar distances with the least possible calculation, and we can ascertain also the distances of whole classes of stars too remote to be ascertained in any other way at present known.
The problem of spectroscopic determinations of luminosity and parallax has been investigated at Mount Wilson with great thoroughness from all sides, the separate investigations checking each other. A definitive scale for the spectroscopic determination of absolute magnitudes has now been established, and the parallaxes and absolute magnitudes have already been derived for about 1,800 stars.