propagation of light was established. Solar shadows, and also lunar, were very valuable in the beginning; and much was obtained from the simple device of a style, so fixed as to cast a shadow corresponding with the diurual rotation to be observed: but the alterations rendered necessary by the annual motion, and impossible to make on that apparatus, rendered the instrument unfit for precise observations. Again, by comparing the length of the shadow cast by a vertical style with the height of the style, the correspond- ing angular distance of the sun from the zenith was com- puted: and a valuable method this was: but the penumbra rendered the accurate measurement of the shadow impos- sible. The difficulty, aggravated by its unequal amount at different distances from the zenith, was partly removed by the use of very large gnomons; but not completely. These imperfections determined astronomers to get rid as soon as possible of the process of gnomonic measurement. Shadows will always be at hand to measure by when better means are wanting: and one application of this instru- 158 POSITIVE PHILOSOPHY.
meut remains in our observatories, — as the basis of the meridian line, regarded as dividing into two equal parts the angle formed by the horizontal shadows of the same length which corresj^ond to the two equivalent parts of the same day. In this case, the j^enumbra is harmless, as it affects the two parts equally; and as for the obli- quity of the sun's motion, that may be mainly got rid of by choosing the period of either solstice, — especially the summer one. It is easy, too, to rectify the observation by the stars.
Proceeding to more exact methods, and, first, with regard to measurement of time, it is clear that the most perfect of all chronoineters is the sky. It seems as if it would be enough, after knowing precisely the latitude of one's obser- vatory, to measure the distance of any star from the zenith, and learn its horary angle, and, as an immediate conse- quence, the time that has elapsed, by resolving the spherical triangle formed by the pole, the zenith, and the star. If a sufficiently wide observation of this kind had been made, and numerical tables formed for certain selected stars, great results might have been obtained from this natural method; but it is insufficient; and it has the defect of making the measure of time depend on that of angles, which is the least perfect of the two, in our day. This method is thei'efore used only in the absence of a better, as in nautical astronomy; and its commonest service is in regulating other chronometers, by a comparison with that of the heavens themselves. Artificial methods of measuring time are therefore indispensable in astronomy..., Every phenomenon which exhibits conmethods tmuous change might serve, m a rough way, to mark time: various chemical processes, or even the beating of our own pulses, might afford a measure, more or less inaccurate: astronomical phenomena are ex- cluded, because they are what Ave want to measure: and we therefore have recourse to physical means, and find weight the best. The ancients tried it in the form of the fiow of liquids; and to water clocks succeeded the hour- glass; but the uncertainty of these led to solids being preferred; and in the form of weight having a vertical descent. By no care, however, could the disturbances MEASUREMENT OF TIME AND OF ANGLES. 159 caused by natural forces be remedied, till Galileo, by liis creation of rational dynamics, suggested the pendulum. Wliether it is or is not correct to assign to rr,, i, Galileo the idea of using the pendulum as a measurer of time, it is certain that his discoveries sug- gested it, and that Huyghens enabled us to use it. He had recourse to the highest principles of science to render this service, and discovered the princij^le of vires vivse, vrhich, besides being scientifically indispensable, afforded to art new means of modifying oscillations without changing the dimensions of the apparatus. Considered as a collection of discoveries for a single aim, Huyghens' treatise De Horologio oscillatorio, is perhaps the most remarkable example of special researches that the history of the human mind has yet exhibited. From that time, the perfecting of astronomical clocks became merely a matter of art. In regard to fixed clocks, two things have to be attended to; — the diminution of friction, by improved methods of sus- pension, and the correction by a compensating apj^aratus of irregularities caused by variations of temjierature. As for portable chronometers, worked by a spiral spring, they are a marvellous invention; but they belong to the pro- vince of art, and not science.
it IS clear that an instrument which would ^£ angles admit of au allowance for minutes and seconds, must be of a size incompatible with minute pre- cision. It must always be that large apparatus must be so affected in its weight and temperature as to be impaired in its accuracy. The large telescopes of modern times are intended to show us stars otherwise invisible: and no one thinks of using them for purposes of precise measurement. It is generally agreed now that instruments for measuring angles should not be more than ten feet in diameter when we are dealing with an entire circle; and they are usually not more than six or seven. The wonder then is how we are to estimate angles to a second, as we do every day, with circles whose size would scarcely indicate minutes. It is done by the concurrent use of three methods, — the eye-piece, the use of the vernier (so called after its inventor), and the repetition of angles.
160 POSITIVE PHILOSOPHY.
It was long before it occurred to astronomers to use their lenses for any other purpose than the discovery of new objects: but at last it occurred to them to replace the ancient transoms and modern sights by an eye-piece which should secure the advantages without the inaccuracies of a large instrument. Morin first made this iise of a lens. Auzout followed with his invention of the reticle; and, a century after, Dollond gave us a power of absolute pre- cision by his invention of the achi'omatic object-glass. Vernier proposed in 1631 to divide intervals into parts much moi'e minute than could be marked. He enabled us to ascertain angles, within half a minute, or circles divided only into sixth-parts of a degree. The j^recision obtainable by his simple apparatus is indefinite, being limited only by our difficulty in detecting the coincidence of the line of the vernier with that of the limb. The union of the third method with these two gives us the perfection we have attained. It is strange that we should have been so long in perceiving that, the imperfection of angular instruments having nothing to do with the dimensions of the angle to be measured, we should gain much by increasing, in fixed proportions, the magnitude of the angles, which is equivalent to diminishing the imperfection of the instruinent. The repetition of angles served every purpose immediately, with I'egard to terrestrial objects, on account of the steadi- ness of the point of view; but there was the difficulty, with regard to the heavenly bodies, of their perpetual change of place. Borda applied himself to measure the distance from the zenith of the stars when they crossed the meridian; and the star then remains sensibly at the same distance from the zenith long enough to allow the oi)eration of the multiplication of the angle. By these means, angular instruments are matched with horary in regard to precision. They require from the observer a diligent patience in applying all the minute precautions and rectifications which experience has proved to be indis- pensable to the fullest use of these instruments.
Then, we have Roemer's meridional eye-glass, which fixes the instant of the passage of a star over the meridian. The plane of the meridian is made in this case purely geome- trical, by being described by the optic axis of a simjjle eye- DIFFICULTIES FROM REFRACTIOX. 161 glass, properly disposed; ■which is enough when all we wan. to know is the precise moment of the star's passage. Then there are the micrometrical instruments, by which we mea- sure the diameters of stars, and, generally all small an- gular intervals. These are the material instruments of observation,- — horary and angular. We must now advert to the intellectual means, — that is, to the corrections which astronomers must apply to the results exhibited by their instruments. There would be little use in perfecting our instruments, if refraction and j^arallax introduced as much error into our observations as we had got rid of by the im- jjrovement of our apparatus.
The corrections required are of two kinds. „.. The first relate to the errors caused by the corrections position of the observer, — the ordinary re- fraction and parallax. No deep astronomical knowledge is required for the correction of these. The second class, arising from the same cause, since they proceed from the observer being on a moving planet, are founded on primary astronomical theories: they are the annual parallax, the precession of the equinoxes, aberration, and nutation. Our business now is with the first and most important class.
SECTION II.
REFRACTION.
The light which comes to us from any star t,,,.
ri.PTT'fi.OLion must be more or less turned aside by the action of the terrestrial atmosphere. We must estimate the amount of this deviation before our observations can answer any theoretical purpose. The star is, by this re- fraction, made to appear too near the zenith, while left in the same vertical plane. Only at the zenith is the error absent, while it increases as the star descends to the horizon. This error, primarily affecting distances from the zenith, must affect, indirectly, all other astronomical measure- ments, except azimuths: but it would be easy to calculate them, if we once knew the law of diminution and increase of refraction at different distances from the zenith. Philo- I. M 162 POSITIVE PHILOSOPHY.
sopliers have tried the logical way and the empirical, and have ended by combining the two.
If our atmosphere were homogeneous, the refraction of light would be uniform and calculable. But our atmo- sphere is composed of strata; and the consequent refrac- tions are excessively unequal, and increasing as the light penetrates a denser stratum, so that its passage constitutes a curve of the last degree of complication. Even this would be calculable, with more or less pains, if we knew the law of variation of these atmosj^heric densities: but we do not and cannot know that law. We have no exact knowledge of the laws of temperature, and cannot estimate atmo- spheric changes, either as to number or degree: and all mathematical processes founded on laws of pressure, etc., may be good as exercises, but are of no value in estimating refraction. As to the empirical method, if the refraction remained always constant at the same height, we might construct tables; and, by extending our observations, and instituting various comparisons, we might hope to obtain such a mass of materials as would afford us some certain results. This is what astronomers have, in fact, patiently and laboriously done, by the help of the improved instru- ments we have spoken of. They have used whatever geo- metrical help they could make applicable: but the resiilts are discouraging enough. There is nothing like uniformity in the results: for the changes in the atmosphere are beyond our calculation and measurement. We study the barometer, the thermometer, and the hygrometer, at the right moment; we can learn from them only the changes taking place on the spot in which we are; and our tables of refraction vary as our observatories, and even in one observatory at diffe- rent times. Delambre found differences of four or five minutes between one day and another, after taking all imaginable pains. All that we can do is to confine our observations to the nearest possible approach to the zenith, and to place no reliance on what we attempt near the horizon. By doing this, we shall find our astronomical ob- servations less affected by the unmanageable difficulties of refraction than might be anticipated.
DIFFICULTIES FROM PARALLAX. 163 SECTION III.
PARALLAX.
The difficulty of the parallaxes can be dealt p • 11 - with much more easily and satisfactorily than that of the refractions. Observations of the heavenly bodies made in different places could not be exactly com- pared without a reference, in idea, to those which would be made from an imag-inary observatory, situated in the middle of the earth, which is besides the true centre of apparent diurnal motions. This correction, which is called the parallax, is analogous to that which is constantly made in measurements of the earth's surface, under the more logical name of reduction to the centre of the station.
The effect of the parallax, like that of refraction, is upon the distance of stars from the zenith alone, leaving the star in the same vertical plane, and placing it too far fi'om the zenith, instead of too near, as in the case of re- fraction. In this instance too, as in the other, though not according to the same law, the deviation increases as the star descends to the horizon. In like manner, too, there must be secondary modifications for all the other astro- nomical quantities, except with regard to the azimuths. The rectification is easy in comparison with the other case, from the absence of the hopeless difficulties caused by our ill-understood atmosphere. The similar course of the two difficulties, producing counteracting effects, has, we may observe, relaxed the attention of astronomers to the facts of refraction and parallax, by partly concealing their in- fluence on actual observations.
The parallax does not, like refraction, affect all the stars alike, but, on the contrary, affects all unequally, and each according to its position. It is insensible with regard to all which lie outside the limits of our system, on account of their immense distance; and it varies extremely within our system, from the horizontal parallax of Uranus, which can never reach a half-second, to that of the moon, which may at times exceed a degree. Here lies the radical distinction, in astronomical calculations, between the theory of paral- 164 POSITIVE PHILOSOPHY.
laxes and that of refractions. The determination of ques- tions of parallaxes does not wholly depend, like that of re- fraction, on methods of observation in astronomy, but is truly a portion of science. Depending as it does, ultimately, on the estimate of the distances of the stars from the earth, it pertains to celestial geometry, through the necessity of knowing the law of motion of each star. Thus, it consti- tutes a part of the science itself; though, in the absence of direct knowledge of the distances of stars, an empirical method of determining the coefficients, analogous to that employed in the case of refraction, may be adopted. The method which will suffice is to choose a place and time which will show the proposed star passing the meridian very near the zenith: then to measure, for several consecu- tive days, its polar distance, so as to know pretty nearly the amount of this distance at any moment of the process: and this being laid down, then to calculate, for this instant, according to the horary angle and its two sides, the true distance from the star to the zenith, when it is considerably remote from it, without being too near the horizon (say from 75° to 80°): and then, the comparison of this distance with that which is actually observed at the moment, will evidently disclose the corresponding parallax, and therefore the horizontal parallax, provided the due correc- tion for the refraction has been made. This is the method by which it is most easily established that the parallax of all the stars is absolutely insensible.
It is a serious inconvenience in this method, that all the uncertainty of the case of refraction is introduced into that of parallax. In regard to a body whose parallax is very great, as the moon, the uncertainty is of small consequence; but in regard to the sun, or other distant body, an error of one-third, or even one-half, in the value of its horizontal parallax, might be occasioned. The method is absolutely inapplicable to the remotest of our planets; and not only to Uranus, but to Saturn and Jui:)iter. The rational method must be resorted to, in the case of these. The empirical method has been mentioned here from the philo- sophical interest which attaches to the fact that, w]) to a certain point, the true distances of stars from the earth, at least in proportion to its radius, may be ascertained by ob- CATALOGUE OF STARS. 165 servations made iu one place; a thing which appears, at first sight, geometrically impossible.
SECTION IV.
CATALOGUE OF STARS.
trary to custom, to the Catalogiie of stars, s,t^i-s " which I think should be reckoned among our necessary means of observation in astronomy. This catalogue is a mathematical table of directions by which we find the different stars. Such a determination is a basis of direct knowledge in regard to Sidereal astronomy: while, in regard to our own system, it is simply a valuable means of observation, which supplies us with terms of comparison indispensable for the study of the interior movements of the system. Such has been the essential use of catalogues of stars, from Hipparchus, who began them, to this day. — In order to fulfil their purposes, these catalogues should contain the greatest possible number of stars, spread over every region of the sky. Asti'onomers have done their duty well; for it is a settled habit with them to determine, as far as they can, the co-ordinates of every new star which they observe; and thus our catalogues ai'e very voluminous, and for ever augmenting. Our business here is not with the system of classification and nomenclature adopted in these catalogues. The nomenclature, bearing as it does the marks of the primitive theological state of astronomy, might be easily replaced by one of a methodical character, — the objects to be classified being of the simplest nature, and the distinctions being, in fact, only those of position. But it is this very simplicity which prevents the need from being felt as it would among more complex elements, — useful as a rational system would no doubt be in finding and assigning the places of stars. The change will be made in time, no doubt, and the need is not urgent. Stars are not known by their names, for astronomical purposes, but by their desci-iptions; and the classification and nomencla- ture in the catalogue, resulting from the fundamental divi- 166 POSITIVE PHILOSOPHY.
sion of the circle, are as perfect as jiossible; and all else is of little importance. I would only ask that we should cease to speak of the magnitudes of stars, as marking their rank, and substitute the word hrightness, in order to avoid all risk of supposing stars to be large or small in proportion to their brightness or dimness. The word brigJitness would be a simple declaration of the fact, without judging the causes, which we are far from understanding.
By viewing these methods as I have brought them to- gether, we may trace the progress of the science from its earliest days. With regard to angular measurement, for instance, the ancients observed with exactness a degree at the utmost; Tycho Brahe carried up the precision to a minute, and the moderns to a second; — a perfection so recent, that observations which lie more than a century behind our time are considered, from their want of pre- cision, inadmissible in the formation of astronomical theories.
My object has been, chiefly, to show the harmony which exists among these different methods of observation; a harmony which, while it tends to perfect them all, up to a certain jioint, still restricts them all, by making each a limit to the rest. No improvement in horary or angular instruments, for instance, could carry us far, while our knowledge of refraction remains as imperfect as it is. But there is no reason to suppose that we have approached the limits imposed liy the conditions of the subject.
167 CHAPTER III.
GEOMETRICAL PHENOMENA OF THE HEAVENLY BODIES.
SECTION I. STATICAL PHENOMENA.
THE plienomeua of our solar system divide _ themselves into two classes, — the Statical ^A^^r,l,^T.^n ^ and the Dynamical. Ihe first class compre- hends the circumstances of the star itself, independent of its motions; as its distance, magnitude, form, atmosphere, etc.: the other comprehends the facts of its displacements, and the mathematical considerations belonging to its diffe- rent positions. According to the usual analogy, the first is independent of the second; while the second could have no existence without the first. The Statical phenomena would exist if the system was immovable: while the dynamical are wholly determined by the statical conditions.
The first thing necessary to be known about any heavenly body is its distance from tanctL^*"^""' the earth: and the diflficulty of obtaining this ground for further observations is extremely great, — the smallnessof the base of our triangle, and the immensity of the distance of the planet, rendering all accuracty hopeless in very many cases. Towards the middle of the last century, when it was desired to determine the horizontal parallax of the moon, — the most manageable of the heavenly bodies, — Lacaille went to the Cape of Good Hope, and Lalande to Berlin, to observe its distance at the same moment from the zenith, — that moment being appointed,— as the middle of an anticipated eclipse. The stations were so chosen as to afford a pretty accurate knowledge of the extent of the line of the base, — which was about as long a one as our globe could afford. The observations of the two distances 168 POSITIVE PHILOSOPHY.
of the moon from the zenith must thus afford the necessary data for the resolution of the triangle which must give the distance sought: and thus we have obtained a very exact knowledge of the moon's distance, which, at its mean, is about sixty terrestrial diameters, and about which we are sure that we cannot be mistaken to the extent of more than twelve miles. The same method might serve to give us, though with much less precision, the distance of Venus and even Mars, if the observation was made when they were nearest to the earth; but it becomes too uncertain with regard to the sun. It would leave an uncertainty of at least an eighth, or about twelve milHons of miles. Of course, it is of no avail with regard to yet more distant bodies.
The method used by astronomers under this difficulty is to measure, first, distances for which our small terrestrial bases will serve; and on these, according to their related jihenomena, to erect other calculations; thus making of the first a basis for the support of new estimates. Aristarchus of Samos conceived of an ingenious method of discovering the distance of the sun through that of the moon; but the uncertainty about seizing the exact moment of the quad- rature of the moon introduced fatal inaccuracy into the calculation. Halley's method, by means of the passage of Mercury and Venus over the sun, is more circuitous, and suitable only to an advanced state of geometrical science; but it is far more accurate, and the only one now admissible, for determining the parallax of those planets and of the sun, and therefore the distance of the sun from the earth, through the differences in the transit observable at two very distant stations. By this method, we can estimate, within a hundredth part, the distance of the sun from the earth. This distance being ascertained, we have it for a basis for other calculations. We have only to observe the angular distance from the sun to the pi'oposed body, at two periods separated by six mouths, — that is, from opposite points of the earth's orbit. This gives us an immense triangle, the base of which is twice the length of the dis- tance of the earth from the sun: and thus it is that our knowledge of the earth's motion has helped us to a base twenty-four thousand times longer than the longest that can be conceived on our own globe. It is true, the j^lanet PLANETARY DISTANCES. 169 observed will have changed its place in the interval; but the remoter planets, — which alone are in question here, — move very slowly; — Saturn's circuit, for instance, occupying thirty years; and our times of observation being practically reducible to a shorter time than six months, — even to two or one, with regard to those planets of our system which move more rapidly; while the slower ones may be con- sidered almost stationary, during such short periods of time; and again, allowance can be made for this small change of place, according to the geometrical theory of its proper motion. It is in this way that astronomers have attained to their knowledge of the positions of the remotest bodies of our system. The numbers by which we express their relations to the distance of the earth from the sun, are now certain to the third decimal at least.
The vast increase of the basis of observation afforded us by our knowledge of the earth's movement is clearly the greatest that we can attain. If we have cleared the bounds of our globe, we certainly cannot go beyond its orbit. Great as this distance appears to us, it vanishes when we want to ascertain the distances of stars outside our system. All measurement is here so out of the question that the most we can do is to fix a limit within which they cer- tainly are not, — saying, for instance, that the nearest star is at least two hundi-ed thousand times more remote than the sun, or ten thousand times further off than the remotest planet of our system; which is quite sufficient to establish the independence of our system.