# Popular Science Monthly/Volume 10/February 1877/Distance and Dimensions of the Sun

(1877)
Distance and Dimensions of the Sun by Charles Augustus Young

 DISTANCE AND DIMENSIONS OF THE SUN.

By Professor C. A. YOUNG,

OF DARTMOUTH COLLEGE.

THE problem of finding the distance of the sun is one of the most important and difficult presented by astronomy. Its importance lies in this, that this distance—the radius of the earth's orbit—is the base-line by means of which we measure every other celestial distance, excepting only that of the moon; so that error in this base propagates itself in all directions through all space, affecting with a corresponding proportion of falsehood every measured line—the distance of every star, the radius of every orbit, the diameter of every planet.

Our estimates of the masses of the heavenly bodies also depend upon a knowledge of the sun's distance from the earth. The quantity of matter in a star or planet is determined by calculations whose fundamental data include the distance between the investigated body and some other body whose motion is controlled or modified by it; and this distance generally enters into the computation by its cube, so that any error in it involves a more than threefold error in the resulting mass. An uncertainty of one per cent, in the sun's distance implies an uncertainty of more than three per cent, in every celestial mass and every cosmical force.

Error in this fundamental element propagates itself in time also, as well as in space and mass. That is to say, our calculations of the mutual effects of the planets upon each other's motions depend upon an accurate knowledge of their masses and distances. By these calculations, were our data perfect, we could predict for all futurity, or reproduce for any given epoch of the past, the configurations of the planets and the conditions of their orbits, and many interesting problems in geology and natural history seem to require for their solution just such determinations of the form and position of the earth's orbit in by-gone ages.

Now, the slightest error in the data, though hardly affecting the result for epochs near the present, leads to uncertainty which accumulates with extreme rapidity in the lapse of time; so that even the present uncertainty of the sun's distance, small as it is, renders precarious all conclusions from such computations when the period is extended more than a few hundred thousand years from the present time. If, for instance, we should find as the result of calculation with the received data that two millions of years ago the eccentricity of the earth's orbit was at a maximum, and the perihelion so placed that the sun was nearest during the northern winter (a condition of affairs which it is thought would produce a glacial epoch in the southern hemisphere), it might easily happen that our results would be exactly contrary to the truth, and that the state of affairs indicated did not occur within half a million years of the specified date—and all because in our calculation the sun's distance, or solar parallax by which it is measured, was assumed half of one per cent, too great or too small. In fact, this solar parallax enters into almost every kind of astronomical computations, from those which deal with stellar systems and the constitution of the universe to those which have for their object nothing higher than the prediction of the moon's place as a means of finding the longitude at sea.

Of course, it hardly need be said that its determination is the first step to any knowledge of the dimensions and constitution of the sun itself.

This parallax of the sun is simply the angular semi-diameter of the earth as seen from the sun; or, it may be defined in another way as the angle between the direction of the sun ideally observed from the centre of the earth, and its actual direction as seen from a station where it is just rising above the horizon.

We know with great accuracy the dimensions of the earth. Its mean equatorial radius, according to the latest and most reliable determination (agreeing, however, very closely with previous ones), is 3962.720 English miles [6377.323 kilometres], and the error can hardly amount to more than 11000000 of the whole—perhaps, 200 feet one way or the other. Accordingly, if we know how large the earth looks from any point, or, to speak technically, if we know the parallax of the point, its distance can at once be found by a very easy calculation: it equals simply [206,265[1] X the radius of the earth] ÷ [the parallax in seconds of arc].

Now, in the case of the sun it is very difficult to find the parallax with sufficient precision on account of its smallness—it is less than 9", almost certainly between 8.8" and 8.9". But this tenth of a second of doubtfulness is more than 1100 of the whole, although it is no more than the angle subtended by a single hair at a distance of nearly 800 feet. If we call the parallax 8.86", which is probably very near the truth, the distance of the sun will come out 92,254,000 miles, while a variation of 120 of a second either way will change it nearly half a million of miles.

When a surveyor has to find the distance of an inaccessible object, he lays off a convenient base-line, and from its extremities observes the directions of the object, considering himself very unfortunate if he cannot get a base whose length is at least 110 of the distance to be measured. But the whole diameter of the earth is less than 111000 of the distance of the sun, and the astronomer is in the predicament of a surveyor who, having to measure the distance of an object ten miles off, finds himself restricted to a base of less than five feet, and herein lies the difficulty of the problem.

Of course, it would be hopeless to attempt this problem by direct observations, such as answer perfectly in the case of the moon, whose distance is only thirty times the earth's diameter. In her case, observations taken from stations widely separated in latitude, like Berlin and the Cape of Good Hope, or Washington and Santiago, determine her parallax and distance with very satisfactory precision; but if observations of the same accuracy could be made upon the sun (which is not the case, since its heat disturbs the adjustments of an instrument), they would only show the parallax to be somewhere between 8" and 10", its distance between 126,000,000 and 82,000,000 miles.

Astronomers, therefore, have been driven to employ indirect methods based on various principles: some on observations of the nearer planets, some on calculations founded upon the irregularities—the so called perturbations—of lunar and planetary movements, and some upon observations of the velocity of light. Indeed, before the Christian era, Aristarchus of Samos had devised a method so ingenious and pretty in theory that it really deserved success, and would have attained it were the necessary observations susceptible of sufficient accuracy. Hipparchus also devised another founded on observations of lunar eclipses, which also failed for much the same reasons as the plan of Aristarchus.

The idea of Aristarchus was to observe carefully the number of hours between new moon and the first quarter, and also between the quarter and the full. The first interval should be shorter than the second, and the difference would determine how many times the distance of the sun from the earth exceeds that of the moon, as will be clear from the accompanying figure. The moon reaches its quarter, or appears as a half-moon, when it arrives at the point Q, where the lines drawn from it to the sun and earth are perpendicular to each other. Since the angle H E Q = E S Q, it will follow that H Q is the same fraction of H E as Q E is of E S; so that, if H Q can be found, we shall at once have the ratio of Q E and E S. Aristarchus thought he had ascertained that the first quarter of the month (from N to Q) was about 12 hours shorter than the second, from which he computed the sun to be about 19 times as distant as the moon. The difficulty

Fig. 1.

lies in the impossibility of determining the precise moment when the disk of the moon is an exact semicircle. The real difference between the first and second quarters is really not quite 36 minutes, and the sun's distance is about 400 times the moon's.

The different methods upon which our present knowledge of the sun's distance depends may be classified as follows:

1. Observations upon the planet Mars near opposition, in two distinct ways:
(a) Observations of the planet's declination made from stations widely separated in latitude.
(b) Observations from a single station of the planet's right ascension when near the eastern and western horizons—known as Flamsteed's method.
2. Observations of Venus at or near inferior conjunctions:
(a) Observations of her distance from small stars measured at stations widely different in latitude.
(b) Observations of the transits of the planet: 1. By noting the duration of the transit at widely-separated stations; 2. By noting the true Greenwich time of contact of the planet with the sun's limb; 3. By measuring the distance of the planet from the sun's limb with suitable micrometric apparatus; 4. By photographing the transit, and subsequently measuring the pictures.
3. By observing the oppositions of the nearer asteroids in the same manner as those of Mars.
4. By means of the so-called parallactic inequality of the moon.
5. By means of the monthly equation of the sun's motion.
6. By means of the perturbations of the planets, which furnish us the means of computing the ratios between the masses of the planets and the sun, and consequently their distances—known as Leverrier's method.
7. By measuring the velocity of light, and combining the result (a) with equation of light between the earth and sun, and (b) with the constant of aberration.

Our scope and limits do not, of course, require or allow any exhaustive discussion of these different methods and their results, but some of them will repay a few moments' consideration:

The first three methods are all based upon the same general idea, that of finding the actual distance of one of the nearer planets by serving its displacement in the sky as seen from remote points on the earth. The relative distances of the planets are easily found in several different ways,[2] and are known with very great accuracy—the possible error hardly reaching the ten-thousandth in even the most unfavorable cases. In other words, we are able to draw for any moment an exceedingly accurate map of the solar system—the only question being as to the scale. Of course, the determination of any line in the map will fix this scale; and for this purpose one line is as good as another, so that the measurement of the distance from the earth to the planet Mars, for instance, will settle all the dimensions of the system.

Fig. 2.

The figure illustrates the method of observation. Suppose two observers, situated one near the north pole of the earth, the other near the south. Looking at the planet, the northern observer will see it at N (in the upper figure), while the other will see it at S, farther north in the sky. If the northern observer sees it as at A (in the lower part of the figure), the southern will at the same time see it as at B; and, by measuring carefully at each station the apparent distance of the planet from several of the little stars (a, b, c) which appear in the field of view, the amount of the displacement can be accurately ascertained. The figure is drawn to scale. The circle E being taken to represent the size of the earth as seen from Mars when nearest us, the black disk represents the apparent size of the planet on the same scale, and the distance between the points N and S, in either figure A or B, represents, on the same scale also, the displacement which would be produced in the planet's position by a transference of the observer from Washington to Santiago, or vice versa.

The first modern attempt to determine the sun's parallax was made by this method in 1670, when the French Academy of Sciences sent Richer to Cayenne to observe the opposition of Mars, while Cassini (who proposed the expedition), Roemer, and Picard, observed it from different stations in France. When the results came to be compared, however, it was found that the planet's displacement was imperceptible by their existing means of observation: from this they inferred that the planet's parallax could not exceed half a minute of arc, and that the sun's could not be more than 10”.

In 1752 Lacaille at the Cape of Good Hope made similar observations, and their comparison with corresponding observations in Europe showed that instruments had so far improved as to make the displacement quite sensible. He fixed the sun's parallax at 10”, corresponding to a distance of about 82,000,000 miles.

In more recent times the method has been frequently applied, and with results on the whole satisfactory. In 1849-'52 Lieutenant Gilliss was sent by the United States Government to Santiago, in Chili, to observe both Mars and Venus in connection with northern observatories. In 1862 a still more extended campaign was organized, in which a great number of observatories in both hemispheres participated. Prof. Newcomb's careful reduction of the work puts the resulting parallax at 8.855”. The method can be used to the best advantage, of course, when at the time of opposition the planet is near its perihelion and the earth near its aphelion; these favorable oppositions occur about once in fifteen years, and the one which is next to occur, in September, 1877, is so exceptionally advantageous that already somewhat extensive preparations are on foot to secure its careful and general observation.

In observations of this sort upon Mars or the asteroids, the position and displacement of the planet, as seen from different stations, are determined by comparing it with neighboring stars. When Venus, however, is nearest us, she can be observed only by day, so that in her case star comparisons are as a general thing out of the question. But occasionally at her inferior conjunction she passes directly across the disk of the sun, and her parallactic displacement from different stations can then be determined by making any such observations as will enable the computer to ascertain accurately her apparent distance and direction from the sun's centre at some given moment. Gregory in 1663 first pointed out the utility of such observations for ascertaining the parallax, but it was not until some fifteen years later that the attention of astronomers was secured to the subject by Halley, who discussed the matter thoroughly, and showed how the problem might be solved with accuracy by observations such as were entirely practicable even by the instruments and with the knowledge then at command. In 1761 and 1769 two transits occurred which were observed in all accessible quarters of the globe by expeditions sent out by the different governments. From different sets of these observations variously combined by different computers, values of the solar parallax were obtained ranging all the way from 7.5" to 9.2". A general discussion of all the materials afforded by the two transits was first made by Encke in 1822, and he obtained, as the most probable result, the value 8."5776, which from that time for more than thirty years was accepted by all astronomers as the best attainable approximation to the truth. In order to harmonize the results, however, he thought himself obliged to reject the observations of several stations. In 1854 Hansen, in publishing some of his results respecting the motion of the moon, announced that Encke's value of the solar parallax could not be reconciled with his investigations; within the next six or seven years several independent researches by other astronomers confirmed his conclusions, and the most recent recomputations show that the observations rejected by Encke are as trustworthy as any, and that the errors of observation were so considerable in 1769 that nothing more can be fairly deduced from that transit than that the solar parallax is probably somewhere between 8.7" and 8.9".

The method of observation then used consisted simply in noting the moment when the limb of the planet came in contact with that of the sun—an observation which is attended with much more difficulty and uncertainty than would at first be supposed. The difficulties depend in part upon the imperfections of optical instruments and the human eye, partly upon the essential nature of light, leading to what is known as diffraction, and partly upon the action of the planet's atmosphere. The two first named causes produce what is called irradiation, and operate to make the apparent diameter of the planet, as seen on the solar disk, smaller than it really is—smaller, too, by an amount which varies with the size of the telescope, the perfection of its lenses, and the tint and brightness of the sun's image. The edge of the planet's image is also rendered slightly hazy and indistinct.

The planet's atmosphere also causes its disk to be surrounded by a narrow ring of light, which becomes visible long before the planet touches the sun, and at the moment of internal contact produces an appearance of which the accompanying figure is intended to give an

Fig. 4.

idea though on an exaggerated scale. The planet moves so slowly as. to occupy more than twenty minutes in crossing the sun's limb; so that, even if the planet's edge were perfectly sharp and definite, and the sun's limb undistorted, it would be very difficult to determine the precise second at which contact occurs; but as things are, observers, with precisely similar telescopes, and side by side, often differ from each other five or six seconds; and where the telescopes are not similar the differences and uncertainties are much greater. The contact observations of the last transit in 1874 do not appear to be much more accordant than those of 1769, notwithstanding the great improvement in telescopes; and astronomers at present are pretty much agreed that such observations can be of little value in removing the remaining uncertainty of the solar parallax, and are disposed to put more reliance upon the micrometric and photographic methods, which are free from these peculiar difficulties, though of course beset with others; which, however, it is hoped will prove less formidable.

The micrometric method requires the use of a peculiar instrument known as the heliometer, an instrument common only in Germany, and requiring much skill and practice in its use in order to obtain with it accurate measures. At the late transit a single English party, two or three of the Russian parties, and all five of the German, were equipped with these instruments, and at some of the stations extensive series of measures were made. None of the results, however, have appeared as yet, so that it is impossible to say how greatly, if at all, this method will have the advantage in precision over the contact observations.

The Americans and French placed their main reliance upon the photographic method, while the English and Germans also provided for its use to a certain extent. The great advantage of this method is that it makes it possible to perform the necessary measurements, upon whose accuracy everything depends, at leisure after the transit, without hurry, and with all possible precautions. The field-work consists merely in obtaining as many and as good pictures as possible. The only objection to the method lies in the difficulty of obtaining good pictures, i. e., pictures free from distortion, and so distinct and sharp as to bear high magnifying power in the microscopic apparatus used for their measurement. It is necessary also that the exact scale of the pictures, or the number of seconds of arc to the linear inch, be known, as well as the precise Greenwich time at which each picture is taken, and it is also extremely desirable that the orientation of the picture should be accurately determined, that is, the north and south, east and west points of the solar image on the finished plate. There has been a good deal of anxiety lest the image, however accurate and sharp when first produced, should alter in course of time through the contraction of the collodion film on the glass plate, but the experiments of Rutherfurd, Huggins, and Paschen, seem to show that this danger is imaginary; that if a plate is properly prepared the collodion film never creeps at all, but remains firmly attached to the glass. It requires but a very trifling amount of distortion or inaccuracy of the image to render it useless. The uncertainty in our present knowledge of the sun's parallax is so small that it would only involve an error of about one-quarter of a second in the calculated position of Venus on the sun's disk as seen from any station at any given time during the transit, and this would be about 12000 of an inch on a four-inch picture of the sun. Unless, then, the picture is so distinct and free from distortion that the relative positions of Venus and the sun's centre can be determined from it within 12000 of an inch, it is worthless as a means of correcting the received determination of the parallax.

But it is to be noted that any mere enlargement or diminution of the diameter of sun or planet will do no harm, provided it is alike all around the circumference of the disk, since the measurement is not from the edge of Venus to the edge of the sun, but between their centres. Photographic determinations of contact, on the contrary (such as Janssen and some of the English parties attempted by a peculiar and complicated apparatus), are affected with all the uncertainties of the old-fashioned observations of the eye alone, and with others in addition; so that, astronomically considered, they are entirely worthless, although interesting from a chemical and physical point of view.

Two essentially different lines of proceeding were adopted, at the last transit, in the photographic observations. The English and Germans attached a camera to the eye-end of an ordinary telescope, which was pointed directly at the sun; the image formed at the focus of the telescope was enlarged to the proper size by a combination of lenses in the camera; and a small plate of glass ruled with squares was placed at the focus of the telescope and photographed with the sun's image, furnishing a set of reference-lines, which give the means of detecting and allowing for any distortion caused by the enlarging lenses.

The Americans and French, on the other hand, preferred to make the picture of full size, without the intervention of any enlarging lens: as this requires an object-glass with a focal length of thirty or forty feet, which could not be easily pointed at the sun, a plan proposed first, I believe, by M. Laussedat, but also independently by our own Prof. Winlock, was adopted. The telescope is placed horizontal, and the rays are reflected into the object-glass by a plane mirror suitably mounted. The French used mirrors of silvered glass, and took their pictures (about two and a half inches in diameter) by the old daguerreotype process on silvered plates of copper, in order to avoid the risk of collodion-contraction. With the silvered mirror the time of exposure is so short that no clock-work is required. The Americans used unsilvered mirrors, in order to avoid any distorting action of the sun's rays upon the form of the mirror. This, of course, made the light feebler, and the time of exposure longer, so that a clock-work movement of the mirror was needed to keep the image from changing its place on the plate during the exposure, which, however, never exceeded half a second. The American pictures were taken by the ordinary wet process on glass, and were about four inches in diameter. Just in front of the sensitive plate, at a distance of about one-eighth of an inch, was placed a reticle, or a plate of glass ruled in squares, and between this and the collodion-plate hung a fine silver wire suspending a plumb-bob. Thus the finished negative was marked into squares, and also bore the image of the plumb-line, which, of course, indicated precisely the direction of the vertical. The Americans also placed the photographic telescope exactly in line with a meridian instrument, and so determined, with the extremest precision, the direction in which it was pointed. Knowing this, and the time at which any picture was taken, it becomes possible, with the help of the plumb-line image, to determine precisely the orientation of the picture—an advantage possessed by the American pictures alone, and making their value nearly twice as great as otherwise it would have been.

The following figure is a representation of one of the American photographs reduced about one-half. V is the image of Venus, which on the actual plate is about one-seventh of an inch in diameter; a a' is the image of the plumb-line. The centre of the reticle is marked by the little cross, and the word "China," written on the reticle-plate with a diamond—and, of course, copied on the photograph—indicates that it is one of the Peking pictures. Its number in the series is given in the right-hand upper corner. About 90 such pictures were obtained at Peking during the transit, and about 350 at all the eight American stations, the work being much interfered with by unfavorable weather at most of them. If we add those obtained by the French, Germans, and English, the total number available reaches nearly 1,200, according to the best estimates.

Fig. 5.

After the pictures are made and safely brought home, they have next to be measured—i. e., the distance (and in the American pictures the direction also) between the centre of Venus and the centre of the sun must be determined in each picture. This is an exceedingly delicate and tedious operation, rendered more difficult by the fact that the image of the sun is never truly circular; but, even supposing the instrument to be perfect in all its adjustments, is somewhat distorted by the effect of atmospheric refraction; so that the true position of the sun's centre with reference to the squares of the reticle is determined only by an intricate calculation from measurements made with a microscopic apparatus on a great number of points suitably chosen on the circumference of the image. The final result of the measurement would come out something in this form: Peking, No. 32, time, 14h 08’ 20.2” (Greenwich mean time); Venus north of sun's centre, 735.32”; east of centre, 441.63”; distance from centre of sun, 857.75”. (The numbers given are only imaginary.) It is this process of measurement which has required so long a time since the transit, and is not yet completed. When it is finished, and the results published in the form indicated, then will come the work of combining all the data thus obtained at all the stations, and from them deducing the true value of the solar parallax. Since, however, another transit is to occur so soon (in 1882), it is not unlikely that astronomers may defer the final grand combination until the observations of that transit also are ready to be included. It is very confidently hoped by most of those who have studied the subject that the remaining uncertainty in the sun's distance will be greatly reduced as the result of this work; and yet there are some grounds for anxiety lest the photographic data prove as intractable and inconsistent as those derived from contact observations. Time only can positively settle the question.

One of the best methods of determining the solar parallax is based upon the careful observation of the motions of the moon. It will be recollected that the first suspicion as to the correctness of the then received distance of the sun was raised in 1854 by Hansen's announcement that the moon's parallactic inequality led to a smaller value than that deduced from the transit of Venus—a conclusion corroborated by Leverrier four years later. It seems at first sight strange, but it is true, as Laplace long since pointed out, that the skillful astronomer, by merely watching the movements of our satellite, and without leaving his observatory, can obtain the solution of problems which, attacked by other methods, require tedious and expensive expeditions to remote corners of the earth. Our scope and object do not require us to enter into detail respecting this lunar method of finding the sun's parallax; it must suffice to say that the disturbing action of the sun makes the interval from new moon to full a little longer than that from full to new; and this difference depends upon the ratio between the diameter of the moon's orbit and the distance of the sun in such a manner that, if the inequality is accurately observed, the ratio can be calculated. Since we know the distance of the moon, this will give that of the sun. The results obtained in this way, according to the most recent investigations, fix the solar parallax between 8.83" and 8.92".

There is still another lunar method, mentioned in the synopsis; but its results are much less reliable—subject, that is, to a much larger probable error, though not at all contradictory to those just given.

But the method by which ultimately we shall obtain the most accurate determination of the dimensions of our system is that proposed by Leverrier, making use of the secular perturbations produced by the earth upon her neighboring planets, especially in causing the motions of their nodes and perihelia. These motions are very slow, but continuous; and hence, as time goes on, they will become known with ever-increasing accuracy. If they were known with absolute precision, they would enable us to compute, with absolute precision also, the ratio between the masses of the sun and earth, and from this ratio we can calculate[3] the distance of the sun by either of two or three different methods.

As matters stand at present, the majority of astronomers would probably consider that these secular perturbations are not yet known with an exactness sufficient to render this method superior to the others that have been named—perhaps as yet not even their rival. Leverrier, on the other hand, himself puts such confidence in it that he declined to sanction or coöperate in the operations for observing the recent transit of Venus, considering all labor and expense in that direction as merely so much waste.

But, however the case may be now, there is no question that as time goes on, and our knowledge of the planetary motions becomes more minutely precise, this method will become continually and cumulatively more exact, until finally, and not many centuries hence, it will supersede all the others that have been described. The parallax of the sun, determined by Leverrier in this method, in 1872, comes out 8.86".

The last of the methods mentioned in the synopsis given on page 405 is interesting as an example of the manner in which the sciences are mutually connected and dependent. Before the experiments of Fizeau in 1849, and of Foucault a few years later, our knowledge of the velocity of light depended on our knowledge of the dimensions of the earth's orbit: it had been found by astronomical observations upon the eclipses of Jupiter's satellites that light occupied a little more than 16 minutes in crossing the orbit of the earth, or about 8 minutes in coming from the sun; and hence, supposing the sun's distance to be 95,600,000 miles, as was long believed, the velocity of light must be about 192,000 miles per second; thus optics was indebted to astronomy for this fundamental element. But when Foucault in 1862 announced that, according to his unquestionably accurate experiments, the velocity of light could not be much more than 186,000 miles per second, the obligation was returned, and the suspicions as to the received value of the sun's parallax, which had been raised by the lunar researches of Hansen and Leverrier, were changed into certainty. The most recent experimental determinations of the velocity of light by Cornu in 1873-'74 fix the solar parallax between 8.80" and 8.85", according as we use Peters's "constant of aberration" or Delambre's value of the "equation of light," which is the name given to the time required for light to traverse the interval between the sun and the earth.

Collecting all the evidence at present attainable, it would seem that the solar parallax cannot differ much from 8.86", though it may be as much as 0.04" greater or smaller; this would correspond, as has

already been said, to a distance of 92,250,000 miles, with a probable error of about one-half per cent., or 450,000 miles.

But, though the distance can thus easily be stated in figures, it is not so easy to give any real idea of a space so enormous; it is quite beyond our power of conception. If one were to try to walk such a, distance, supposing even that he could walk 4 miles an hour, and keep it up for 10 hours every day, it would take 6812 years to make a single million of miles, and more than 6,300 years to traverse the whole.

If some celestial railway could be imagined, the journey to the sun, even if our trains ran 60 miles an hour, day and night and without a stop, would require over 175 years. Sensation, even, would not travel so far in a human lifetime. To borrow the curious illustration of Prof. Mendenhall, if we could imagine an infant with an arm long enough to enable him to touch the sun and burn himself, he would die of old age before the pain could reach him, since, according to the experiments of Helmholtz and others, a nervous shock is communicated only at the rate of about 100 feet per second, or 1,637 miles a day, and would need more than 150 years to make the journey. Sound would do it in about 14 years if it could be transmitted through celestial space, and a cannon-ball in about 9, if it were to move uniformly with the same speed as when it left the muzzle of the gun. If the earth could be suddenly stopped in her orbit, and allowed to fall unobstructed toward the sun under the accelerating influence of his attraction, she would reach the central fire in about four months. I have said if she could be stopped, but such is the compass of her orbit that, to make its circuit in a year, she has to move nearly 19 miles a second, or more than fifty times faster than the swiftest rifle-ball; and in moving 20 miles her path deviates from perfect straightness by less than one-eighth of an inch. And yet, over all the circumference of this tremendous orbit, the sun exercises his dominion, and every pulsation of his surface receives its response from the subject earth.

By observing the slight changes in the sun's apparent diameter, we find that its distance varies somewhat at different times of the year, about 3,000,000 miles in all; and minute investigation shows that the earth's orbit is almost an exact ellipse, whose nearest point to the sun, or perihelion, is passed by the earth about the 1st of January, at which time she is 90,750,000 miles distant.

The distance of the sun being once known, its dimensions are easily ascertained—at least, within certain narrow limits of accuracy. The angular semi-diameter of the sun when at the mean distance is almost exactly 962", the uncertainty not exceeding 12000 of the whole. The result of twelve years' observations at Greenwich (1836 to 1847) gives 961.82", and other determinations oscillate around the value first mentioned, which is that adopted in the "American Nautical Almanac." Taking the distance as 92,250,000 miles, this makes the sun's diameter 860,500; and the probable error of this quantity, depending as it does both on the error of the measured diameter and of the distance, is some 4,000 or 5,000 miles; in other words, the chances are strong that the actual diameter is between 855,000 and 865,000 miles.

Measurements made by the same person, however, and with the same instrument, but at different times, sometimes differ enough to raise a suspicion that the diameter is slightly variable, which would be nothing surprising considering the nature of the solar surface. There is no sensible difference between the equatorial and polar diameters, the rotation of the sun on its axis not being sufficiently rapid to make the polar compression (which must, of course, necessarily result from the rotation) marked enough to be perceived by our present means of observation.

It is not easy to obtain any real conception of the vastness of this enormous sphere. Its diameter is 108.7 times that of the earth, and its circumference proportional, so that the traveler who could make the circuit of the world in 80 days would need nearly 24 years for his journey around the sun. Since the surfaces of spheres vary as the squares, and bulks as the cubes, of their diameters, it follows that the sun's surface is nearly 12,000 times, and its volume, or bulk, more than 1,280,000 times, greater than that of the earth. If the earth be represented by one of the little three-inch globes common in school apparatus, the sun on the same scale will be more than 27 feet in diameter, and its distance nearly 3,000 feet. Imagine the sun to be hollowed out and the earth placed in the centre of the shell thus formed, it would be like a sky to us, and the moon would have scope for all her motions far within the inclosing surface; indeed, since she is only 240,000 miles away, while the sun's radius is more than 430,000, there would be room for a second satellite 190,000 miles beyond her.

The mass of the sun, or quantity of matter contained in it, can also be computed when we know its distance, and comes out 325,600 times as great as the earth. The calculation may be made either by means of the proportion given in the note to page 413, or by comparing the attracting force of the sun upon the earth, as indicated by the curvature of her orbit (about 0.119 inch per second), with the distance a body at the surface of the earth falls in the same time under the action of gravity, a quantity which has been determined with great accuracy by experiments with the pendulum. Of course, the fact that the sun produces its effect upon the earth at a distance of 92,250,000 miles, while a falling body at the level of the sea is only about 4,000 miles from the centre of the attraction which produces its motion, must also enter into the reckoning.[4] This mass, if we express it in pounds or tons, is too enormous to be conceived: it is 2 octillions of tons—that is, 2 with 27 ciphers annexed; it is nearly 750 times as great as the combined masses of all the planets and satellites of the solar system—and Jupiter alone is more than 300 times as massive as the earth. The sun's attractive power is such that it dominates all surrounding space, even to the fixed stars, so that a body at the distance of our nearest stellar neighbor, α Centauri, which is more than 200,000 times remoter than the sun, could free itself from the solar attraction only by darting away with a velocity of more than 300 feet per second, or over 200 miles an hour; unless animated by a greater velocity than this, it would move around the sun in a closed orbit—an ellipse of some shape, or a circle, with a period of revolution which, in the smallest possible orbit, would be about 31,600,000 years, and if the orbit were circular, would be nearly 90,000,000. We say it would revolve thus—that is, of course, unless intercepted or diverted from its course by the influence of some other sun, as it probably would be. And we may notice here that in many cases certainly, and in most cases probably, the stars are flying through space at a far swifter rate, with velocities of many miles per second.

If we calculate the force of gravity at the sun's surface, which is easily done by dividing its mass, 325,600, by the square of 10834 (the number of times the sun's diameter exceeds the earth's), we find it to be 2712 times as great as on the earth; a man who on the earth would weigh 150 pounds, would there weigh nearly two tons; and, even if the footing were good, would be unable to stir. A body which at the earth falls a little more than 16 feet in a second would there fall 443. A pendulum which here swings once a second would there oscillate more than five times as rapidly, like the balance-wheel of a watch—quivering rather than swinging.

Since the sun's volume is 1,280,000 times that of the earth, while its mass is only 325,600 times as great, it follows at once that the sun's average density (found by dividing the mass by the volume) is only about one-quarter that of the earth. This is a fact of the utmost importance in its bearing upon the constitution of this body. As we shall see hereafter, we know that certain heavy metals, with which we are familiar on the earth, enter largely into the composition of the

sun, so that, if the principal portion of the solar mass were either solid or liquid, its mean density ought to be at least as great as the earth's, especially since the enormous force of solar gravity would tend most powerfully to compress the materials. The low density can only he accounted for on the supposition, which seems fairly to accord also with all other facts, that the sun is mainly a ball of gas, or vapor, powerfully condensed, of course, in the central portion by the super-incumbent weight, but prevented from liquefaction by an exceedly high temperature. And, on the other hand, it could be safely predicted on physical principles that so huge a ball of fiery vapor, exposed to the cold of space, would present precisely such phenomena as we find by observation of the solar surface and surroundings.

to M and the earth to E', observe the planet's elongation from the sun, i. e., the angle M' E' S. Now, since we know the periodic times of both the earth and planet, we shall know both the angle M S M' moved over by the planet in one hundred days, and also E S E described in the same time by the earth, the difference is M' S E', called by some

Fig. 3.

writers the synodic angle. We have, therefore, in the triangle M' S E', the angle at E' measured, and the angle M' S E' known as stated above; this of course gives the third angle at M', and hence we know the shape of the triangle, and by the ordinary processes of trigonometry can find the relative values of its three sides.

1. This number 206,265 is the length of the radius of a circle expressed in seconds of its circumference. A ball one foot in actual diameter would have an apparent diameter of one second at a distance of 206,265 feet, or a little more than 39 miles. If its apparent diameter were 10", its distance would, of course, be only 110 as great.
2. One method of determining the relative distances of a planet and the sun from each other and from the earth is the following, known since the days of Hipparchus: First, observe the date when the planet comes to its opposition—i. e., when sun, earth, and planet, are in line, as in the figure, where the planet and earth are represented by M and E Next, after a known number of days, say one hundred, when the planet has advanced
3. One method of proceeding is as follows: Let M be the mass of the sun, and m that of the earth; let R be the distance of the sun from the earth, and r that of the moon; finally, let T be the number of days in a sidereal year, and t the number in a sidereal month. Then, by elementary astronomy,

${\displaystyle \scriptstyle M:m={\tfrac {R^{3}}{T^{2}}}:{\tfrac {r^{3}}{t^{2}}};}$ whence ${\displaystyle \scriptstyle {R^{3}}={r^{3}}\left({\tfrac {T^{2}}{t^{2}}}\right)\left({\tfrac {M}{m}}\right)}$ ;

or, in words, the cube of the sun's distance equals the cube of the moon's distance, multiplied by the square of the number of sidereal months in a year, and by the ratio between the masses of the sun and earth. It is to be noted, however, that T and t are the periods of the earth and moon, as they would be if wholly undisturbed in their motions, and hence differ slightly from the periods actually observed—the differences are small, but somewhat troublesome to calculate with precision.

4. The calculation of the sun's mass, from the data given, proceeds as follows: Let M = the sun's mass, and m that of the earth; R = the distance from the earth to the sun, and r the mean radius of the earth; T, the length of the sidereal year, reduced to seconds; and 12 g the distance a body falls in a second at the earth's surface. Now, the distance the earth falls toward the sun in a second, or the curvature of her orbit in a second, is equal to ${\displaystyle \scriptstyle {\tfrac {2\pi ^{2}R}{T^{2}}}}$ (about 0.119 inch). Hence, by the law of gravitation, 12 g : ${\displaystyle \scriptstyle {\tfrac {2\pi ^{2}R}{T^{2}}}={\tfrac {m}{r^{2}}}:{\tfrac {M}{R^{2}}}}$, whence, ${\displaystyle \scriptstyle M=m\left({\tfrac {4\pi ^{2}R^{3}}{T^{2}r^{2}g}}\right)}$ In this formula make it = 3.14159 ; R, 92,250,000 miles ; T = 31,558,149.3 seconds ; r = 3,956.179 miles ; and g = 0.0061035 mile (16.113 feet), and we shall. get the result given in the text, viz., M = 325,600 m.