[Ill.u.s.tration: FIG. 115.--Supposed capture of the November meteors by Ura.n.u.s.]

174. BREAKING UP A METEOR SWARM.--How closely packed together these meteors were at the time of their annexation to the solar system is unknown, but it is certain that ever since that time the sun has been exerting upon them a tidal influence tending to break up the swarm and distribute its particles around the orbit, as the Perseids are distributed, and, given sufficient time, it will accomplish this, but up to the present the work is only partly done. A certain number of the meteors have gained so much over the slower moving ones as to have made an extra circuit of the orbit and overtaken the rear of the procession, so that there is a thin stream of them extending entirely around the orbit and furnishing in every November a Leonid shower; but by far the larger part of the meteors still cling together, although drawn out into a stream or ribbon, which, though very thin, is so long that it takes some three years to pa.s.s through the perihelion of its...o...b..t. It is only when the earth plunges through this ribbon, as it should in 1899, 1900, 1901, that brilliant Leonid showers can be expected.

175. RELATION OF COMETS AND METEORS.--It appears from the foregoing that meteors and comets move in similar orbits, and we have now to push the a.n.a.logy a little further and note that in some instances at least they move in identically the same orbit, or at least in orbits so like that an appreciable difference between them is hardly to be found. Thus a comet which was discovered and observed early in the year 1866, moves in the same orbit with the Leonid meteors, pa.s.sing its perihelion about ten months ahead of the main body of the meteors. If it were set back in its...o...b..t by ten months' motion, _it would be a part of the meteor swarm_.

Similarly, the Perseid meteors have a comet moving in their orbit actually immersed in the stream of meteor particles, and several other of the more conspicuous star showers have comets attending them.

Perhaps the most remarkable case of this character is that of a shower which comes in the latter part of November from the constellation Andromeda, and which from its a.s.sociation with the comet called Biela (after the name of its discoverer) is frequently referred to as the Bielid shower. This comet, an inconspicuous one moving in an unusually small elliptical orbit, had been observed at various times from 1772 down to 1846 without presenting anything remarkable in its appearance; but about the beginning of the latter year, with very little warning, it broke in two, and for three months the pieces were watched by astronomers moving off, side by side, something more than half as far apart as are the earth and moon. It disappeared, made the circuit of its...o...b..t, and six years later came back, with the fragments nearly ten times as far apart as before, and after a short stay near the earth once more disappeared in the distance, never to be seen again, although the fragments should have returned to perihelion at least half a dozen times since then. In one respect the orbit of the comet was remarkable: it pa.s.sed through the place in which the earth stands on November 27th of each year, so that if the comet were at that particular part of its...o...b..t on any November 27th, a collision between it and the earth would be inevitable. So far as is known, no such collision with the comet has ever occurred, but the Bielid meteors which are strung along its...o...b..t do encounter the earth on that date, in greater or less abundance in different years, and are watched with much interest by the astronomers who look upon them as the final appearance of the _debris_ of a worn-out comet.

176. PERIODIC COMETS.--The Biela comet is a specimen of the type which astronomers call periodic comets--i. e., those which move in small ellipses and have correspondingly short periodic times, so that they return frequently and regularly to perihelion. The comets which accompany the other meteor swarms--Leonids, Perseids, etc.--also belong to this cla.s.s as do some 30 or 40 others which have periodic times less than a century. As has been already indicated, these deviations from the normal parabolic orbit call for some special explanation, and the substance of that explanation is contained in the account of the Leonid meteors and their capture by Ura.n.u.s. Any comet may be thus captured by the attraction of a planet near which it pa.s.ses. It is only necessary that the perturbing action of the planet should result in a diminution of the comet's velocity, for we have already learned that it is this velocity which determines the character of the orbit, and anything less than the velocity appropriate to a parabola must produce an ellipse--i. e., a closed orbit around which the body will revolve time after time in endless succession. We note in Fig. 115 that when the Leonid swarm encountered Ura.n.u.s it pa.s.sed _in front of_ the planet and had its velocity diminished and its...o...b..t changed into an ellipse thereby. It might have pa.s.sed behind Ura.n.u.s, it would have pa.s.sed behind had it come a little later, and the effect would then have been just the opposite. Its velocity would have been increased, its...o...b..t changed to a hyperbola, and it would have left the solar system more rapidly than it came into it, thrust out instead of held in by the disturbing planet. Of such cases we can expect no record to remain, but the captured comet is its own witness to what has happened, and bears imprinted upon its...o...b..t the brand of the planet which slowed down its motion. Thus in Fig. 115 the changed orbit of the meteors has its _aphelion_ (part remotest from the sun) quite close to the orbit of Ura.n.u.s, and one of its nodes, ?, the point in which it cuts through the plane of the ecliptic from north to south side, is also very near to the same orbit. It is these two marks, aphelion and node, which by their position identify Ura.n.u.s as the planet instrumental in capturing the meteor swarm, and the date of the capture is found by working back with their respective periodic times to an epoch at which planet and comet were simultaneously near this node.

Jupiter, by reason of his great ma.s.s, is an especially efficient capturer of comets, and Fig. 116 shows his group of captives, his family of comets as they are sometimes called. The several orbits are marked with the names commonly given to the comets. Frequently this is the name of their discoverer, but often a different system is followed--e. g., the name 1886, IV, means the fourth comet to pa.s.s through perihelion in the year 1886. The other great planets--Saturn, Ura.n.u.s, Neptune--have also their families of captured comets, and according to Schulhof, who does not entirely agree with the common opinion about captured comets, the earth has caught no less than nine of these bodies.

[Ill.u.s.tration: FIG. 116.--Jupiter's family of comets.]

177. COMET GROUPS.--But there is another kind of comet family, or comet group as it is called, which deserves some notice, and which is best exemplified by the Great Comet of 1882 and its relatives. No less than four other comets are known to be traveling in substantially the same orbit with this one, the group consisting of comets 1668, I; 1843, I; 1880, I; 1882, II; 1887, I. The orbit itself is not quite a parabola, but a very elongated ellipse, whose major axis and corresponding periodic time can not be very accurately determined from the available data, but it certainly extends far beyond the orbit of Neptune, and requires not less than 500 years for the comet to complete a revolution in it. It was for a time supposed that some one of the recent comets of this group of five might be a return of the comet of 1668 brought back ahead of time by unknown perturbations. There is still a possibility of this, but it is quite out of the question to suppose that the last four members of the group are anything other than separate and distinct comets moving in practically the same orbit. This common orbit suggests a common origin for the comets, but leaves us to conjecture how they became separated.

The observed orbits of these five comets present some slight discordances among themselves, but if we suppose each comet to move in the average of the observed paths it is a simple matter to fix their several positions at the present time. They have all receded from the sun nearly on line toward the bright star Sirius, and were all of them, at the beginning of the year 1900, standing nearly motionless inside of a s.p.a.ce not bigger than the sun and distant from the sun about 150 radii of the earth's...o...b..t. The great rapidity with which they swept through that part of their orbit near the sun (see -- 162) is being compensated by the present extreme slowness of their motions, so that the comets of 1668 and 1882, whose pa.s.sages through the solar system were separated by an interval of more than two centuries, now stand together near the aphelion of their orbits, separated by a distance only 50 per cent greater than the diameter of the moon's...o...b..t, and they will continue substantially in this position for some two or three centuries to come.

The slowness with which these bodies move when far from the sun is strikingly ill.u.s.trated by an equation of celestial mechanics which for parabolic orbits takes the place of Kepler's Third Law--viz.:

r^3 / T^2 = 178,

where _T_ is the time, in years, required for the comet to move from its perihelion to any remote part of the orbit, whose distance from the sun is represented, in radii of the earth's...o...b..t, by _r_. If the comet of 1668 had moved in a parabola instead of the ellipse supposed above, how many years would have been required to reach its present distance from the sun?

178. RELATION OF COMETS TO THE SOLAR SYSTEM.--The orbits of these comets ill.u.s.trate a tendency which is becoming ever more strongly marked.

Because comet orbits are nearly parabolas, it used to be a.s.sumed that they were exactly parabolic, and this carried with it the conclusion that comets have their origin outside the solar system. It may be so, and this view is in some degree supported by the fact that these nearly parabolic orbits of both comets and meteors are tipped at all possible angles to the plane of the ecliptic instead of lying near it as do the orbits of the planets; and by the further fact that, unlike the planets, the comets show no marked tendency to move around their orbits in the direction in which the sun rotates upon his axis. There is, in fact, the utmost confusion among them in this respect, some going one way and some another. The law of the solar system (gravitation) is impressed upon their movements, but its order is not.

But as observations grow more numerous and more precise, and comet orbits are determined with increasing accuracy, there is a steady gain in the number of elliptic orbits at the expense of the parabolic ones, and if comets are of extraneous origin we must admit that a very considerable percentage of them have their velocities slowed down within the solar system, perhaps not so much by the attraction of the planets as by the resistance offered to their motion by meteor particles and swarms along their paths. A striking instance of what may befall a comet in this way is shown in Fig. 117, where the tail of a comet appears sadly distorted and broken by what is presumed to have been a collision with a meteor swarm. A more famous case of impeded motion is offered by the comet which bears the name of Encke. This has a periodic time less than that of any other known comet, and at intervals of forty months comes back to perihelion, each time moving in a little smaller orbit than before, unquestionably on account of some resistance which it has suffered.

[Ill.u.s.tration: FIG. 117.--Brooks's comet, October 21, 1893.--BARNARD.]

179. THE DEVELOPMENT OF A COMET.--We saw in -- 174 that the sun's action upon a meteor swarm tends to break it up into a long stream, and the same tendency to break up is true of comets whose attenuated substance presents scant resistance to this force. According to the mathematical a.n.a.lysis of Roche, if the comet stood still the sun's tidal force would tend first to draw it out on line with the sun, just as the earth's tidal force pulled the moon out of shape (-- 42), and then it would cause the lighter part of the comet's substance to flow away from both ends of this long diameter. This destructive action of the sun is not limited to comets and meteor streams, for it tends to tear the earth and moon to pieces as well; but the densities and the resulting mutual attractions of their parts are far too great to permit this to be accomplished.

As a curiosity of mathematical a.n.a.lysis we may note that a spherical cloud of meteors, or dust particles weighing a gramme each, and placed at the earth's distance from the sun, will be broken up and dissipated by the sun's tidal action if the average distance between the particles exceeds two yards. Now, the earth is far more dense than such a cloud, whose extreme tenuity, however, suggests what we have already learned of the small density of comets, and prepares us in their case for an outflow of particles at both ends of the diameter directed toward the sun. Something of this kind actually occurs, for the tail of a comet streams out on the side opposite to the sun, and in general points away from the sun, as is shown in Fig. 109, and the envelopes and jets rise up toward the sun; but an inspection of Fig. 106 will show that the tail and the envelope are too unlike to be produced by one and the same set of forces.

It was long ago suggested that the sun possibly exerts upon a comet's substance a repelling force in addition to the attracting force which we call gravity. We think naturally in this connection of the repelling force which a charge of electricity exerts upon a similar charge placed on a neighboring body, and we note that if both sun and comet carried a considerable store of electricity upon their surfaces this would furnish just such a repelling force as seems indicated by the phenomena of comets' tails; for the force of gravity would operate between the substance of sun and comet, and on the whole would be the controlling force, while the electric charges would produce a repulsion, relatively feeble for the big particles and strong for the little ones, since an electric charge lies wholly on the surface, while gravity permeates the whole ma.s.s of a body, and the ratio of volume (gravity) to surface (electric charge) increases rapidly with increasing size. The repelling force would thrust back toward the comet those particles which flowed out toward the sun, while it would urge forward those which flowed away from it, thus producing the difference in appearance between tail and envelopes, the latter being regarded from this standpoint as stunted tails strongly curved backward. In recent years the Russian astronomer Bredichin has made a careful study of the shape and positions of comets'

tails and finds that they fit with mathematical precision to the theories of electric repulsion.

180. COMET TAILS.--According to Bredichin, a comet's tail is formed by something like the following process: In the head of the comet itself a certain part of its matter is broken up into fine bits, single molecules perhaps, which, as they no longer cling together, may be described as in the condition of vapor. By the repellent action of both sun and comet these molecules are cast out from the head of the comet and stream away in the direction opposite to the sun with different velocities, the heavy ones slowly and the light ones faster, much as particles of smoke stream away from a smokestack, making for the comet a tail which like a trail of smoke is composed of constantly changing particles. The result of this process is shown in Fig. 118, where the positions of the comet in its...o...b..t on successive days are marked by the Roman numerals, and the broken lines represent the paths of molecules _m^{I}_, _m^{II}_, _m^{III}_, etc., expelled from it on their several dates and traveling thereafter in orbits determined by the combined effect of the sun's attraction, the sun's repulsion, and the comet's repulsion. The comet's attraction (gravity) is too small to be taken into account. The line drawn upward from _VI_ represents the positions of these molecules on the sixth day, and shows that all of them are arranged in a tail pointing nearly away from the sun. A similar construction for the other dates gives the corresponding positions of the tail, always pointing away from the sun.

[Ill.u.s.tration: FIG. 118.--Formation of a comet's tail.]

Only the lightest kind of molecules--e. g., hydrogen--could drift away from the comet so rapidly as is here shown. The heavier ones, such as carbon and iron, would be repelled as strongly by the electric forces, but they would be more strongly pulled back by the gravitative forces, thus producing a much slower separation between them and the head of the comet. Construct a figure such as the above, in which the molecules shall recede from the comet only one eighth as fast as in Fig. 118, and note what a different position it gives to the comet's tail. Instead of pointing directly away from the sun, it will be bent strongly to one side, as is the large plume-shaped tail of the Donati comet shown in Fig. 101. But observe that this comet has also a nearly straight tail, like the theoretical one of Fig. 118. We have here two distinct types of comet tails, and according to Bredichin there is still another but unusual type, even more strongly bent to one side of the line joining comet and sun, and appearing quite short and stubby. The existence of these three types, and their peculiarities of shape and position, are all satisfactorily accounted for by the supposition that they are made of different materials. The relative molecular weights of hydrogen, some of the hydrocarbons, and iron, are such that tails composed of these molecules would behave just as do the actual tails observed and cla.s.sified into these three types. The spectroscope shows that these materials--hydrogen, hydrocarbons, and iron--are present in comets, and leaves little room for doubt of the essential soundness of Bredichin's theory.

181. DISINTEGRATION OF COMETS.--We must regard the tail as waste matter cast off from the comet's head, and although the amount of this matter is very small, it must in some measure diminish the comet's ma.s.s. This process is, of course, most active at the time of perihelion pa.s.sage, and if the comet returns to perihelion time after time, as the periodic ones which move in elliptic orbits must do, this waste of material may become a serious matter, leading ultimately to the comet's destruction.

It is significant in this connection that the periodic comets are all small and inconspicuous, not one of them showing a tail of any considerable dimensions, and it appears probable that they are far advanced along the road which, in the case of Biela's comet, led to its disintegration. Their fragments are in part strewn through the solar system, making some small fraction of its cloud of cosmic dust, and in part they have been carried away from the sun and scattered throughout the universe along hyperbolic orbits impressed upon them at the time they left the comet.

But it is not through the tail only that the disintegrating process is worked out. While Biela's comet is perhaps the most striking instance in which the head has broken up, it is by no means the only one. The Great Comet of 1882 cast off a considerable number of fragments which moved away as independent though small comets and other more recent comets have been seen to do the same. An even more striking phenomenon was the gradual breaking up of the nucleus of the same comet, 1882, II, into a half dozen nuclei arranged in line like beads upon a string, and pointing along the axis of the tail. See Fig. 119, which shows the series of changes observed in the head of this comet.

182. COMETS AND THE SPECTROSCOPE.--The spectrum presented by comets was long a puzzle, and still retains something of that character, although much progress has been made toward an understanding of it. In general it consists of two quite distinct parts--first, a faint background of continuous spectrum due to ordinary sunlight reflected from the comet; and, second, superposed upon this, three bright bands like the carbon band shown at the middle of Fig. 48, only not so sharply defined. These bands make a discontinuous spectrum quite similar to that given off by compounds of hydrogen and carbon, and of course indicate that a part of the comet's light originates in the body itself, which must therefore be incandescent, or at least must contain some incandescent portions.

[Ill.u.s.tration: FIG. 119.--The head of the Great Comet of 1882.--WINLOCK.]

By heating hydrocarbons in our laboratories until they become incandescent, something like the comet spectrum may be artificially produced, but the best approximation to it is obtained by pa.s.sing a disruptive electrical discharge through a tube in which fragments of meteors have been placed. A flash of lightning is a disruptive electrical discharge upon a grand scale. Now, meteors and electric phenomena have been independently brought to our notice in connection with comets, and with this suggestion it is easy to frame a general idea of the physical condition of these objects--for example, a cloud of meteors of different sizes so loosely cl.u.s.tered that the average density of the swarm is very low indeed; the several particles in motion relative to each other, as well as to the sun, and disturbed in that motion by the sun's tidal action. Each particle carries its own electric charge, which may be of higher or lower tension than that of its neighbor, and is ready to leap across the intervening gap whenever two particles approach each other. To these conditions add the inductive effect of the sun's electric charge, which tends to produce a particular and artificial distribution of electricity among the comet's particles, and we may expect to find an endless succession of sparks, tiny lightning flashes, springing from one particle to another, most frequent and most vivid when the comet is near the sun, but never strong enough to be separately visible. Their number is, however, great enough to make the comet in part self-luminous with three kinds of light--i. e., the three bright bands of its spectrum, whose wave lengths show in the comet the same elements and compounds of the elements--carbon, hydrogen, and oxygen--which chemical a.n.a.lysis finds in the fallen meteor. It is not to be supposed that these are the only chemical elements in the comet, as they certainly are not the only ones in the meteor. They are the easy ones to detect under ordinary circ.u.mstances, but in special cases, like that of the Great Comet of 1882, whose near approach to the sun rendered its whole substance incandescent, the spectrum glows with additional bright lines of sodium, iron, etc.

183. COLLISIONS.--A question sometimes asked, What would be the effect of a collision between the earth and a comet? finds its answer in the results reached in the preceding sections. There would be a star shower, more or less brilliant according to the number and size of the pieces which made up the comet's head. If these were like the remains of the Biela comet, the shower might even be a very tame one; but a collision with a great comet would certainly produce a brilliant meteoric display if its head came in contact with the earth. If the comet were built of small pieces whose individual weights did not exceed a few ounces or pounds, the earth's atmosphere would prove a perfect shield against their attacks, reducing the pieces to harmless dust before they could reach the ground, and leaving the earth uninjured by the encounter, although the comet might suffer sadly from it. But big stones in the comet, meteors too ma.s.sive to be consumed in their flight through the air, might work a very different effect, and by their bombardment play sad havoc with parts of the earth's surface, although any such result as the wrecking of the earth, or the destruction of all life upon it, does not seem probable. The 40 meteors of -- 169 may stand for a collision with a small comet. Consult the Bible (Joshua x, 11) for an example of what might happen with a larger one.

CHAPTER XIII

THE FIXED STARS

184. THE CONSTELLATIONS.--In the earlier chapters the student has learned to distinguish between wandering stars (planets) and those fixed luminaries which remain year after year in the same constellation, shining for the most part with unvarying brilliancy, and presenting the most perfect known image of immutability. Homer and Job and prehistoric man saw Orion and the Pleiades much as we see them to-day, although the precession, by changing their relation to the pole of the heavens, has altered their risings and settings, and it may be that their l.u.s.ter has changed in some degree as they grew old with the pa.s.sing centuries.

[Ill.u.s.tration: FIG. 120.--Ill.u.s.trating the division of the sky into constellations.]

The division of the sky into constellations dates back to the most primitive times, long before the Christian era, and the crooked and irregular boundaries of these constellations, shown by the dotted lines in Fig. 120, such as no modern astronomer would devise, are an inheritance from antiquity, confounded and made worse in its descent to our day. The boundaries a.s.signed to constellations near the south pole are much more smooth and regular, since this part of the sky, invisible to the peoples from whom we inherit, was not studied and mapped until more modern times. The old traditions a.s.sociated with each constellation a figure, often drawn from cla.s.sical mythology, which was supposed to be suggested by the grouping of the stars: thus Ursa Major is a great bear, stalking across the sky, with the handle of the Dipper for his tail; Leo is a lion; Ca.s.siopeia, a lady in a chair; Andromeda, a maiden chained to a rock, etc.; but for the most part the resemblances are far-fetched and quite too fanciful to be followed by the ordinary eye.

185. THE NUMBER OF STARS.--"As numerous as the stars of heaven" is a familiar figure of speech for expressing the idea of countless number, but as applied to the visible stars of the sky the words convey quite a wrong impression, for, under ordinary circ.u.mstances, in a clear sky every star to be seen may be counted in the course of a few hours, since they do not exceed 3,000 or 4,000, the exact number depending upon atmospheric conditions and the keenness of the individual eye. Test your own vision by counting the stars of the Pleiades. Six are easily seen, and you may possibly find as many as ten or twelve; but however many are seen, there will be a vague impression of more just beyond the limit of visibility, and doubtless this impression is partly responsible for the popular exaggeration of the number of the stars. In fact, much more than half of what we call starlight comes from stars which are separately too small to be seen, but whose number is so great as to more than make up for their individual faintness.

The Milky Way is just such a cloud of faint stars, and the student who can obtain access to a small telescope, or even an opera gla.s.s, should not fail to turn it toward the Milky Way and see for himself how that vague stream of light breaks up into shining points, each an independent star. These faint stars, which are found in every part of the sky as well as in the Milky Way, are usually called _telescopic_, in recognition of the fact that they can be seen only in the telescope, while the other brighter ones are known as _lucid stars_.

186. MAGNITUDES.--The telescopic stars show among themselves an even greater range of brightness than do the lucid ones, and the system of magnitudes (-- 9) has accordingly been extended to include them, the faintest star visible in the greatest telescope of the present time being of the sixteenth or seventeenth magnitude, while, as we have already learned, stars on the dividing line between the telescopic and the lucid ones are of the sixth magnitude. To compare the amount of light received from the stars with that from the planets, and particularly from the sun and moon, it has been found necessary to prolong the scale of magnitudes backward into the negative numbers, and we speak of the sun as having a stellar magnitude represented by the number -26.5. The full moon's stellar magnitude is -12, and the planets range from -3 (Venus) to +8 (Neptune). Even a very few of the stars are so bright that negative magnitudes must be used to represent their true relation to the fainter ones. Sirius, for example, the brightest of the fixed stars, is of the -1 magnitude, and such stars as Arcturus and Vega are of the 0 magnitude.

The relation of these magnitudes to each other has been so chosen that a star of any one magnitude is very approximately 2.5 times as bright as one of the next fainter magnitude, and this ratio furnishes a convenient method of comparing the amount of light received from different stars.

Thus the brightness of Venus is 2.5 2.5 times that of Sirius. The full moon is 2.5^{9} times as bright as Venus, etc.; only it should be observed that the number 2.5 is not exactly the value of the _light ratio_ between two consecutive magnitudes. Strictly this ratio is the 100^{1/5} = 2.5119+, so that to be entirely accurate we must say that a difference of five magnitudes gives a hundredfold difference of brightness. In mathematical symbols, if _B_ represents the ratio of brightness (quant.i.ty of light) of two stars whose magnitudes are _m_ and _n_, then

B = (100)^{(m-n)/5}

How much brighter is an ordinary first-magnitude star, such as Aldebaran or Spica, than a star just visible to the naked eye? How many of the faintest stars visible in a great telescope would be required to make one star just visible to the unaided eye? How many full moons must be put in the sky in order to give an illumination as bright as daylight?

How large a part of the visible hemisphere would they occupy?

187. CLa.s.sIFICATION BY MAGNITUDES.--The brightness of all the lucid stars has been carefully measured with an instrument (photometer) designed for that special purpose, and the following table shows, according to the Harvard Photometry, the number of stars in the whole sky, from pole to pole, which are brighter than the several magnitudes named in the table:

The number of stars brighter than magnitude 1.0 is 11 " " " " " 2.0 " 39 " " " " " 3.0 " 142 " " " " " 4.0 " 463 " " " " " 5.0 " 1,483 " " " " " 6.0 " 4,326

It must not be inferred from this table that there are in the whole sky only 4,326 stars visible to the naked eye. The actual number is probably 50 or 60 per cent greater than this, and the normal human eye sees stars as faint as the magnitude 6.4 or 6.5, the discordance between this number and the previous statement, that the sixth magnitude is the limit of the naked-eye vision, having been introduced in the attempt to make precise and accurate a cla.s.sification into magnitudes which was at first only rough and approximate. This same striving after accuracy leads to the introduction of fractional numbers to represent gradations of brightness intermediate between whole magnitudes. Thus of the 2,843 stars included between the fifth and sixth magnitudes a certain proportion are said to be of the 5.1 magnitude, 5.2 magnitude, and so on to the 5.9 magnitude, even hundredths of a magnitude being sometimes employed.

We have found the number of stars included between the fifth and sixth magnitudes by subtracting from the last number of the preceding table the number immediately preceding it, and similarly we may find the number included between each other pair of consecutive magnitudes, as follows:

Magnitude 0 1 2 3 4 5 6 Number of stars 11 28 103 321 1,020 2,843 4 3^{m} 12 36 108 324 972 2,916

In the last line each number after the first is found by multiplying the preceding one by 3, and the approximate agreement of each such number with that printed above it shows that on the whole, as far as the table goes, the fainter stars are approximately three times as numerous as those a magnitude brighter.

The magnitudes of the telescopic stars have not yet been measured completely, and their exact number is unknown; but if we apply our principle of a threefold increase for each successive magnitude, we shall find for the fainter stars--those of the tenth and twelfth magnitudes--prodigious numbers which run up into the millions, and even these are probably too small, since down to the ninth or tenth magnitude it is certain that the number of the telescopic stars increases from magnitude to magnitude in more than a threefold ratio. This is balanced in some degree by the less rapid increase which is known to exist in magnitudes still fainter; and applying our formula without regard to these variations in the rate of increase, we obtain as a rude approximation to the total number of stars down to the fifteenth magnitude, 86,000,000. The Herschels, father and son, actually counted the number of stars visible in nearly 8,000 sample regions of the sky, and, inferring the character of the whole sky from these samples, we find it to contain 58,500,000 stars; but the magnitude of the faintest star visible in their telescope, and included in their count, is rather uncertain.