W. H. Davenport Adams

"The contemplation of the works of creation elevates the mind to the admiration of whatever is great and noble, accomplishing the object of all study, which is to inspire the love of truth, of wisdom, of beauty, especially of goodness, the highest beauty, and of that supreme and eternal Mind which contains all truth and wisdom, all beauty and goodness."—Mary Somerville.

T To discern the luminous point which should guide us in the shadows of the infinite, is the gift of genius. The first to discern this point in astronomy, the illustrious Kepler thereby succeeded in formulating those laws, or rather rules, by which the movements of the stars are regulated. How did he succeed? How did he arrive at a goal so much to be desired? By intelligence in full possession of itself. It was by abstracting his thoughts from all systematic conceptions,—the shackles of science; it was by defying the traditional authority which had so long enslaved men's minds; it was by interrogating nature, which leaves all liberty to her interrogator, that Kepler was able to deserve and win the glorious title of "legislator of the heavens,"—a title which we must not, however, understand too literally; it bears witness only to the power of intellect.

Let us attempt, at a modest distance, to proceed like Kepler; let us make astronomy without troubling ourselves concerning astronomers. This is the sole means of seizing the luminous point which should guide our steps.

The movement, in virtue of which every star performs the circuit of heaven in four-and-twenty hours, is incessantly reproduced in a uniform and a constant manner. The acquisition of this first fact, simple as it seems, was a somewhat laborious task, and undoubtedly dates back to a distant antiquity. But now comes another fact, where observation demands the closest mental attention, and which is of a more recent discovery.

To comprehend it clearly, let us first call to mind that the moment when the sun crosses the Equator,—whether to return into the northern hemisphere (at the spring equinox), or into the southern (at the autumn equinox),—is instantaneous. More than one way exists of determining this moment exactly; but here we need not enter upon the subject.

Is the interval of time occupied by the sun in travelling from the spring to the autumn equinox equal to the interval which our luminary requires to pass from the autumn equinox to the vernal?

A singular question, you reply. Who, indeed, would venture to maintain that the number of days, hours, minutes, seconds, was not exactly the same in the one case as in the other?

Well, the period is not the same; and, therefore, merely to propound this question was a masterpiece of genius. For no ordinary intellectual audacity was needed to doubt the reality of the supposed perfect circle which the sun apparently describes—according to the recognised authorities—in its uniform progress around the globe; that globe believed by all the early astronomers to be imperturbably and everlastingly situated in the centre of their thrice-sacred geometrical figure. This dogma being accepted as infallible, there was every evidence that the two intervals of time, which divided the astronomical year into two moieties, would be of equal duration. It did not occur to the mind of any one of the faithful that the sojourn of the sun, in his circular and uniform movement, might be longer or shorter in the northern than in the southern hemisphere.

What, then, was the name of the audacious innovator who ventured upon putting forth so revolutionary a suggestion?

It was Hipparchus. At least it was he who, confidently relying upon his observations, was the first to affirm that the sun remains longer in the northern than in the southern hemispheres; or, more accurately speaking, that its passage from the spring to the autumn equinox occupies 187 days, while from the autumn to the spring equinox the duration of its course is only 178 days 6 hours (nearly). The year of 365-1/4 days—that is, the Egyptian year, which was universally adopted by the ancient astronomers—was thus discovered to be really divided into two unequal portions, although, theoretically, the sun ought to occupy exactly the same space of time in passing from the spring to the autumn, as from the autumn to the spring equinox.

The fact pointed out and attested by Hipparchus had an influence which he never anticipated on the progress of science. In opposition to all the systems previously designed by man, it followed, in the first place, that the movement of the sun, in relation to a mean movement, must sometimes be accelerated, sometimes be retarded; that the solar arc described in a given time would be greater in winter than in summer.

Astronomers who, trammelled by particular theories, were unable and unwilling to accept of any new light, immediately hastened to raise, as is invariably the case with those who defend a bad cause, a subsidiary and damaging question. They asked whether those inequalities of the sun's movement were real, or only apparent; whether they were more than a mere optical phenomenon, arising from the sun's position vis-À-vis to an observer placed on the earth's surface. And they unhesitatingly pronounced in favour of the appearance, and against the reality.

But man, says an old adage, is always punished after the manner of his sin. Dogmatic and obstinate authority involved our anti-revolutionary astronomers in fresh complications. Such is the case, too, very frequently, in the domain of theology!

Does the sun—the sun as each of us beholds him—ever change his size? Does he ever shrink in his majestic proportions? Is the magnitude of his broad golden disc ever lessened?

Assuredly this new question, which was not less audacious than its predecessor, did not come—there is sacrilege in the thought—from the conservative areopagus of all ancient doctrines; the learned areopagus, or supreme tribunal, which had erected into a dogma the circular orbit and uniform movement of the sun around the earth, the centre of the universe! It could only have been suggested by some unworthy heterodoxical disturber of men's minds,—his name, alas! has not been handed down to us,—who had dared to look upon the heavens, and learn from their bright and beautiful face, without a master, the A B C of science. This "pestilent heretic" had, probably, remarked one of the commonest phenomena connected with the celestial bodies, which astronomers hitherto had not deigned to notice.

Undoubtedly, dear reader, you will have been more than once impressed by the appearance of the solar orb, when obscured in one of those mists so frequent towards the end of autumn:—

"Cold grew the foggy morn, the day was brief;

Loose on the cherry hung the crimson leaf;

The dew dwelt ever on the herb; the woods

Roared with strong blasts, with mighty showers the floods."

Fig. 66.—"When the glory of the woods is rapidly departing."

Such a day, in this sad season of the year, when the glory of the woods is rapidly departing, and from the swollen streams and dewy pastures the vapours ascend in a dense whirl of clouds, is of frequent occurrence; and on such a day, the solar sphere, as it struggles through the screening mists, seems like the face of the moon at its full, when slightly veiled. Your eye rests upon it without pain. And as observation sharpens your mind, you put to yourself the natural question, Is not the sun farther from the earth at this epoch when it affords us the least heat, than at that period of the year when its vivifying power is greatest? I think it is obvious that, to the unexperienced, such a method of explaining the cold of winter and the heat of summer by the variation in the distance of our great solar luminary would naturally occur.

But the demon of certainty—an excellent demon, whatever the orthodox may say—is present, to stimulate us all. You may have just formed your theories, you may cite your traditional authorities, but these will not satisfy our awakened curiosity. We ask for demonstrations, for irrefragable proofs drawn from the Bible of Nature. We will listen to no oracles but those which are confirmed by the voices of God's second revelation.

Therefore, men required to be assured that the sun was really nearer to us in summer than in winter. For this purpose, it was requisite to make, at the beginning of summer, an observation analogous to that which had been made at the beginning of winter, and afterwards to compare the apparent magnitudes of the solar disc at these two opposite periods of the year.

Behold us, then, at work. You are perfectly tranquil as to the result; for you are persuaded beforehand that the sun must be farther from us in the cold season than in the hot. You regard this as a self-evident truth, like an axiom of Euclid's.

But Nature is a great magician; she contrives the most dramatic surprises for the mind which takes the trouble to interrogate her in all simplicity and without dogmatic pretensions.

What a coup-de-thÉÂtre it was for the observer who first established experimentally that the apparent diameter of the sun is greater in winter than in summer—that we are nearer the sun in the cold season, than in the hot!

On more closely examining a result apparently so paradoxical, man discovered that the angle which subtracts the sun, as seen from the earth,—the visual angle which gives the sun's apparent diameter,—varies necessarily throughout the year. Thus, the semi-diameter, or radius, which on the 24th of June equals 15' 45", will, a month later, have increased one second (15' 46"); on the 2d of August will equal 15' 47"; on the 2d September, 15' 53", and so on. We put the exact measurements before the reader in a tabulated form:—

Length of the Sun's Radius.

We do not trouble the reader with the fractions of a second, which indicate the quantity of the apparent increase of the radius from the end of June to the end of December, and its apparent decrease from the beginning of January to the end of June.

A glance at the above figures shows that the mean of the apparent diameters, all measured at the moment of the sun's passing the meridian, is about half a degree, or 30'; and that—which is sufficiently curious—720 of these mean suns, set one against another, would be required to fill up the contour of a great circle of the celestial sphere. Is it this fact which suggested the idea of dividing the circle into 720/2 = 360°?

Simultaneously with the discovery of the variations of the solar charioteer, it was ascertained that the moments of the sun's passage of the meridian—moments which measure the 365 different positions occupied by the sun in the 365 days of the year—are not separated by equal intervals, or that equal intervals of time do not correspond to the equal angular displacements,—in fine, that the maximum and minimum of the sun's angular velocity coincide with the maximum and minimum of its apparent diameter. Now, remember that the extreme points where the sun experiences its maximum and minimum angular displacement are named, according to PtolemÆus, the former the perigee, the latter the apogee; or, if we follow Copernicus, the former the perihelion, and the latter the aphelion.

The aggregate of these facts was known to the ancients; but the manner in which it was sought to explain them merits notice as a specimen of blind attachment to a preconceived system.

PtolemÆus, the organ of the dictatorial astronomy of antiquity, declares, ex cathedrÂ, that "the inequalities of the sun's movements are only apparent; that they are simply the effects of the position and of the arrangements of the circles in which these movements are accomplished; and that, in this apparent disorder of the phenomena (pe?? t?? ?p???????? t?? fa??????? ?ta??a?), nothing really occurs contrary to their actual immobility (t? ??t? p?f??e s?a??e?? ??d?? ????t???? a?t?? t?? ??d??t?t??)."

Now, according to this dogmatic immutability, the straight lines, or radii, which proceed from the revolving star to the centre of the circle, would describe "equal angles in equal times." This is exactly the contrary of the result obtained, as we have seen, by careful observation.

But this difficulty no more embarrassed the great pontiff of astronomy than a conscientious scruple would perplex the author of a theological dogma. Listen to him:—

"The true cause of these apparent irregularities is explained by two very simple hypotheses. Either the one or the other would account for the phenomena. In fact, if we suppose the movement to occur in a circle described around the centre of the world, and in the plane of the ecliptic, so that the point whence we are looking corresponds with this centre, we must admit either that the planets make their movements equal in non-concentric circles, or that, if these circles are concentric, it is not simply in these circles that they move, but in others, called epicycles, carried through the concentric."[73]

Examine Fig. 67. Here A B G D represent the ecliptic, E its centre, and A E G its diameter; Z H T K is the epicycle, in which the planet moves uniformly around the centre A, while the epicycle uniformly traverses the circle A B G D. Now, suppose that the star has arrived at H; it would appear to an observer at E to be more advanced by the uniform movement of all the arc A H; if it be at K, it would appear, on the contrary, to be less advanced by all the arc A K. At Z the star would appear more distant, and at T, nearer than if it were at A.

To explain the other phenomena, such as the stations and retrocessions of the planets, recourse was again had to the epicycles or deferred eccentric circles. By multiplying these it was possible to account for all the angular inequalities in the movements of a planet. It is of importance to note this point, in order to show how very dangerous it is to trust absolutely to mathematics in our search after the truth; that science which, by the certainty of its demonstrations, nourishes our intellectual pride, and may, therefore, occasionally lull the mind into a false security. The theory of epicycles, from a mathematical point of view, was irreproachable, and it sufficiently accounted for the facts which threatened to overthrow the dogma of circular orbits and uniform planetary movement.

But by degrees, as observations grew more accurate and comprehensive, these and other theories, however fine in appearance,—teres atque rotundus,—gradually disappeared, if fundamentally erroneous. By the invention of micrometers, we were enabled to measure more exactly than had formerly been possible the variations of diameter or the modifications of distance, and afterwards to compare them with the changes of velocity. From this comparison it results that the latter are not greater than is compatible with the alterations of distance indicated by the variations of diameter; in a word, that the hypothesis of epicycles is decidedly insufficient to account for all the inequalities detected by careful investigation.

Kepler was the first to break the charm which had held captive the mind of astronomers, including even Copernicus and Tycho BrahÉ. PtolemÆus had considered the mean positions of the stars to be real. Kepler, strong in his researches, declared that they were but a factitious mode of calculation by which the true positions might be ascertained; that the mean movement is simply an artifice representing the star's place, if no inequality existed; in fine, that we must take the movements as they are in nature,—the true movements, given by observation,—and not the mean movements, deduced from an erroneous hypothesis.

This declaration of principles met, at the time, with the hostility of all astronomers of any reputation, but it has become the starting-point of the discovery of the laws on which the whole edifice of astronomy reposes. Had Kepler, however, been left to depend entirely on his own resources, he might, perhaps, have never completed his task. A fortunate circumstance brought him an unexpected ally. Tycho, having taken refuge in Bohemia, sent for the young astronomer (Kepler then was but twenty-nine years old), to assist him in the composition of the "Rudolphine Tables."[74]

"This," says Kepler, "was a providential interposition. I repaired to Bohemia early in the year 1600, in the hope of learning the correction of the eccentricities of the planets. Perceiving that Tycho made use of a mixed system (which made Mercury and Venus revolve around the sun, and all these planets, with their companions, around the earth), I asked his permission to follow out my own ideas. It was the will of Providence again, that we should occupy ourselves with Mars. My whole attention, therefore, was directed to this planet: and it is through the movements of Mars we must obtain our insight into the secrets of astronomy, or remain ignorant of them for ever (ex Martis motibus omnino necesse est nos in cognitionem astronomiÆ arcanorum venire aut ea perpetuo nescire)."[75]

Why this preference given to Mars? In the first place, because, among all the planets then known, it was Mars which, in its movement round the sun, departed most from the circle; next, its orbit approaches nearest to the earth's; the earth is very near to Mars when she passes between that planet and the sun,—that is to say, when she is in opposition, while she retires from it triple the distance when in conjunction,—that is, when the sun is between her and Mars. Hence arise certain variations of aspect, particularly adapted to make manifest the form of the orbit, and the law of the real movement of the "red planet, Mars." As for the other planets, as far as they were then known, their orbits differ so little from the circle, that the nature of the curve which they describe in reality would never have been exactly recognised by any inexperienced star-gazer.

For these reasons Kepler regarded as providential the choice he had been led to make of Mars at the outset of his astronomical career. Before the close of 1601, Tycho died, bequeathing to his young fellow-worker a treasury of observation. Thenceforth Kepler undertook to finish without assistance the famous Rudolphine Tables. They cost him five-and-twenty years of assiduous labour. Looking upon Tycho's observations, because of their exactness, as "a gift from the Divine Goodness," he employed them, in the first place, as a test of the old hypotheses of planetary orbits and movements. Let us do our best to grasp the range and bearing of this part of his work.

In the system of Copernicus, which Kepler ardently adopted, the earth revolves around the sun. Now, observation having shown that the sun remains seven or eight days longer in the northern than in the southern signs of the Zodiac, we must of necessity admit that the sun, instead of being situated in the centre of the terrestrial orbit, occupies a point outside that centre, in such a manner that the earth must sometimes be nearer to, and sometimes farther from, the sun. The distance by which it departs from the centre of its orbit, which Copernicus, like the ancients, supposed to be circular, is called its eccentricity.

Astronomers were long preoccupied with the idea of seeking in this eccentricity a point where the movements should appear equal. This point was the centre of the equant,—a name given to the eccentric circle described from the point of equality or from the centre of the mean movements.

Now, let us recall the principal condition of the problem which Kepler had undertaken to solve. This condition required that the straight line drawn from the centre of our globe to the centre of the sun,—in a word, that the vector radius, as it is called, should describe around the sun certain angles, whose variability should agree with the results of observation.

Starting from this point, Kepler found that, for certain positions of Mars (in the aphelion and perihelion, corresponding to the minimum and maximum of velocity), the centre of the orbit, always supposing it to be circular, divided into two equal parts (or bisected) the total eccentricity: in other words, that it exactly occupied the middle between the centre of the eccentric and the equant of PtolemÆus; but it did not appear to him necessary to bisect it in other positions, intermediate between those of the aphelion and the perihelion. He established that the differences in longitude amounted to eight or nine minutes. Now, observations so exact as those of Tycho were altogether incompatible with such great error.[76] Therefore, the geometrical hypothesis which gave these errors was false; the orbit of Mars could not be a circle, and to save these eight or nine minutes, furnished by observation but in disaccord with theory, it would be needful to recommence all the calculations of astronomy. This conclusion, not less legitimate than daring, supplied Kepler with the first decisive step in the task he had undertaken.

This is not the place to relate all the essays and miscarriages through which this man of genius passed before finally completing his discovery of the rules that bear his name. But we may put before the reader the construction which led to them.

On a sheet of paper let us mark down by a point (Fig. 68) the place occupied by the earth in relation to the sun.[77] From this point o, we draw a right line terminating at a, the sun's noon-day position (for example, on the 1st of January); the succeeding lines shall touch upon a´ a´´, which the sun occupies successively after the same interval of time (twenty-four hours, or the exact duration of the earth's rotation on its axis);—and let us continue after this mode until the sun has accomplished, by its own proper movement from west to east, the whole circuit of the heavens, traversing 360 degrees in the space of a year. If we ascribe to the radius o a a certain length, corresponding to a definite solar diameter, the lengths of all the others, corresponding to the variations of the same diameter, will depend upon that of the first, which, for facility of calculation, we suppose to be divided into one thousand parts.

After having thus allotted to each straight line its approximate length, let us join their extremities by a curve. What do we see before us? A geometrical figure widely different from a circle, for the diameters (i.e., the straight lines passing through the centre) are far from being equal. The figure is an ellipse.

If now we pass from the appearance to the reality, o will be the sun, and a a´ a´´, m m´ will indicate the terrestrial orbit, or the points of the curve successively occupied by the earth in movement. The moveable straight lines, free at one extremity, and at the other attached to the centre of the sun, are called the Vector heliocentric radii. By the help of this construction, you see that the point occupied by the sun is beyond or without the centre; this eccentric point is the focus of the ellipse, and the distance from this focus to the centre, its eccentricity. The extremity of the major axis, the nearest to the focus, is the perihelion, and its farthest extremity the aphelion. The difference of the angles formed by the vector radii indicate the inequality of the movements: to the greatest angle, the perihelion, corresponds the maximum of velocity (a a´ a´´), just as to the smallest, or aphelion, corresponds the minimum (m m´); the other angles mark the velocities intermediary between these two extremes. We have thus before us a series of triangles with their apices at the focus of the ellipse, and their bases on the contour of the curve.

But these latter are not sufficient for the mind, whose principal function lies in seeking unity among the variety of phenomena.

In what way are the variations of distance connected with the variations of velocity? What is the simplest expression of their relationship? These are questions which naturally presented themselves to Kepler's inquiring intellect. By dint of immeasurable patience, and recommencing more than once the same toil, this great astronomer discovered that the variable arc traversed by the earth (or, in appearance, the sun), in four-and-twenty hours, multiplied by one half the corresponding vector radius, is a constant quantity: is the product which, as elementary geometry teaches, gives the surface of a triangle. And, in fact, look at the matter carefully: the vector radii form triangles whose base is the arc traversed in the same interval of time, and whose apices rest upon the centre of the sun (or, in appearance, the observer, or the centre of the earth).

To fix these ideas thoroughly in our minds,—and a superficial knowledge is worse than useless,—let us imagine to ourselves a man holding horizontally extended a tube of a certain length, capable, like a telescope, of being lengthened or shortened at pleasure; and let us fancy him pivoting upon himself, in such a manner that he sweeps, every minute, exactly the same area or same quantity of surface, while varying perpetually the swiftness of movement and the length of the tube; this "ideal man" will have solved the problem whose solution is inscribed, in ineffaceable letters, on the machinery of our globe; he will describe around him an ellipse, of which he himself occupies one of the foci.

By this method of investigation and deduction, Kepler succeeded in breaking up the traditionary authority of the circle and of uniform movement. He broke it up for ever by two of his celebrated laws, which may be rendered in the following terms:—

1st, The orbit of the earth, as well as the curves described by the other planets, are ellipses, one of whose foci is represented by the sun;

2d, The heliocentric vector radius of a planet describes around the sun areas equal with the times; or, in other words, the surfaces described by the vector radii, in equal times, are also equal.

The ancients had looked for equality in the movements of planets traversing the circumferences of circles: they were mistaken. It is true this equality exists; only, not where they supposed. If they had sought it in the surfaces described by the vector radii, they would have anticipated Kepler's discovery of the laws which govern our world.

But their astronomical dogmas prevented them from seeing the path which led to this great discovery.

Hence we may conclude that Dogma is an evil thing.