A Mathematical and Philosphical Dictionary
, a very eminent astronomer and mathematician, was born at Wiel, in the county of Wirtemberg, in 1571. He was the disciple of Mæstlinus, a learned mathematician and astronomer, of whom he learned those sciences, and became afterwards professor of them to three successive emperors, viz. Matthias, Rudolphus, and Ferdinand the 2d.
To this sagacious philosopher we owe the first discovery of the great laws of the planetary motions, viz. that the planets describe areas that are always proportional to the times; that they move in elliptical orbits, having the sun in one focus; and that the squares of their periodic times, are proportional to the cubes of their mean dislances; which are now generally known by the name of Kepler's Laws. But as this great man stands as it were at the head of the modern reformed astronomy, he is highly deserving of a pretty large account, which we shall extract chiefly from the words of that great mathematician Mr. Maclaurin.
Kepler had a particular passion for finding analogies and harmonies in nature, after the manner of the Pythagoreans and Platonists; and to this disposition we owe such valuable discoveries, as are more than sufficient to excuse his conceits. Three things, he tells us, he anxiously sought to find out the reason of, from his early youth; viz, Why the planets were 6 in number? Why the dimensions of their orbits were such as Copernicus had deseribed from observations? And what was the analogy or law of their revolutions? He sought for the reasons of the two first of these, in the properties of numbers and plane figures, without success. But at length reflecting, that while the plane regular sigures may be infinite in number, the regular solids are only five, as Euclid had long ago demonstrated: he imagined, that certain mysteries in nature might correspond with this remarkable limitation inherent in the essences of things; and the rather, as he found that the Pythagoreans had made great use of those sive regular solids in their philosophy. He therefore endeavoured to find some relation between the dimensions of these solids and the intervals of the planetary spheres; thus, imagining that a cube, inscribed in the sphere of Saturn, would touch by its six planes the sphere of Jupiter; and that the other four regular solids in like manner fitted the intervals that are between the spheres of the other planets: he became persuaded that this was the true reason why the primary planets were precisely six in number, and that the author of the world had determined their distances from the sun, the centre of the system, from a regard to this analogy. Being thus possessed, as he thought, of the grand secret of the Pythagoreans, and greatly pleased with his discovery, he published it in 1596, under the title of Mysterium Cosmographicum; and was for some time so charmed with it, that he said he would not give up the honour of having invented what was contained in that book, for the electorate of Saxony.
Kepler sent a copy of this book to Tycho Brahe, who did not approve of those abstract speculations concerning the system of the world, but wrote to Kepler, first to lay a solid foundation in observations, and then, by ascending from them, to endeavour to come at the causes of things. Tycho however, pleased with his genius, was very desirous of having Kepler with him to assist him in his labours: and having settled, under the protection of the emperor, in Bohemia, where he passed the last years of his life, after having left his native country on some ill usage, he prevailed upon Kepler to leave the university os Gratz, and remove into Bohemia, with his family and library, in the year 1600. But Tycho dying the next year, the arranging the observations devolved upon Kepler, and from that time he had the title of Mathematician to the Emperor all his life, and gained continually more and more reputation by his works. The emperor Rudolph ordered him to finish the tables of Tycho Brahe, which were to be called the Rudolphine Tables. Kepler applied diligently to the work: but unhappy are those learned men who depend upon the good-humour of the intendants of the finances; the treasurers were so ill-affected towards our author, that he could not publish these tables till 1627. He died at Ratisbon, in 1630, where he was soliciting the payment of the arrears of his pension.
Kepler made many important discoveries from Tycho's observations, as well as his own. He found, that astronomers had erred, from the first rise of the science, in ascribing always circular orbits and uniform motions to the planets; that, on the contrary, each of them moves in an ellipsis which has one of its foci in the sun: that the motion of each is really unequable, and varies so, that a ray supposed to be always drawn from the planet to the sun describes equal areas in equal times.
It was some years later before he discovered the analogy there is between the distances of the several planets from the sun, and the periods in which they complete their revolutions. He easily saw, that the higher planets not only moved in greater circles, but also more slowly than the nearer ones; so that, on a double account, their periodic times were greater. Saturn, for example, revolves at the distance from the sun 9 1/2 times greater than the earth's distance from it; and the circle described by Saturn is in the same proportion: but as the earth revolves in one year, so, if their velocities were equal, Saturn ought to revolve in 9 years and a half; whereas the periodic time of Saturn is about 29 years. The periodic times of the planets increase, therefore, in a greater proportion than their distances from the sun: but yet not in so great a proportion as the squares of those distances; for if that were the law of the motions, (the square of 9 1/2 being 90 1/4), the periodic time of Saturn ought to be above 90 years. A mean proportion between that of the distances of the planets, and that of the squares of those distances, is the true proportion of the periodic times; as the mean between 9 1/2 and its square 90 1/4, gives the periodic time of Saturn in years. Kepler, after having committed several mistakes in determining this analogy, hit upon it at last, May the 15, 1618; for he is so particular as to mention the precise day when he found that “The squares of the periodie times were always in the same proportion as the cubes of their mean distances from the sun.”
When Kepler saw, according to better observations, that his disposition of the five regular solids among the planetary spheres, was not agreeable to the intervals between their orbits, he endeavoured to discover other schemes of harmony. For this purpose, he compared the motions of the same planet at its greatest and least distances, and of the different planets in their several orbits, as they would appear viewed from the sun; and here he fancied that he found a similitude to the divisions of the octave in music. These were the dreams of this ingenious man, which he was so fond of, that, hearing of the discovery of four new planets (the satellites of Jupiter) by Galileo, he owns that his first reflections were from a concern how he could save his favourite scheme, which was threatened by this addition to the number of the planets. The same attachment led him into a wrong judgment concerning the sphere of the fixed stars: for being obliged, by his doctrine, to allow a vast superiority to the sun in the universe, he restrains the fixed stars within very narrow limits. Nor did he consider them as suns, placed in the centres of their several systems, having planets revolving round them; as the other followers of Copernicus have concluded them to be, from their having light in themselves, from their immense distances, and from the analogy of nature. Not contented with these harmonies, which he had learned from the observations of Tycho, he gave himself the liberty to imagine several other analogies, that have no foundation in nature, and are overthrown by the best observations. Thus from the opinions of Kepler, though most justly admired, we are taught the danger of espousing principles, or hypotheses, borrowed from abstract sciences, and of applying them, with such freedom, to natural enquiries.
A more recent instance of this fondness, for discovering analogies between matters of abstract speculation, and the constitution of nature, we find in Huygens, one of the greatest geometricians and astronomers any age has produced: when he had discovered that satellite of Saturn, which from him is still called the Huygenian satellite, this, with our moon, and the four satellites of Jupiter, completed the number of six secondary planets then discovered in the system; and because the number of primary planets was also six, and this number is called by mathematicians a perfect number (being equal to the sum of its aliquot parts, 1, 2, 3,) Huygens was hence induced to believe that the number of the planets was complete, and that it was in vain to look for any more. This is not mentioned to lessen the credit of this great man, who never perhaps reasoned in such a manner on any other occasion; but only to shew, by another instance, how ill-grounded reasonings of this kind have always proved. For, not long after, the celebrated Cassini discovered four more satellites about Saturn, not to mention the two more that have lately been discovered to that planet by Dr. Herschel, with another new primary planet and its two satellites, besides many others, of both sorts, as yet unknown, which possibly may belong to our system. The same Cassini having found that the analogy, discovered by Kepler, between the periodic times and the distances from the centre, takes place in the lesser systems of Jupiter and Saturn, as well as in the great solar system; his observations overturned that groundless analogy which had been imagined between the number of the planets, both primary and secondary, and the number six: but established, at the same time, that harmony in their motions, which will afterwards appear to flow from one real principle extended over the universe.
But to return to Kepler; his great sagacity, and continual meditations on the planetary motions, suggested to him some views of the true principles from which these motions flow. In his preface to the Commentaries concerning the planet Mars, he speaks of gravity as of a power that was mutual between bodies, and tells us, that the earth and moon tend towards each other, and would meet in a point, so many times nearer to the earth than to the moon, as the earth is greater than the moon, if their motions did not hinder it. He adds, that the tides arise from the gravity of the waters towards the moon. But not having notions sufficiently just of the laws of motion, it seems he was not able to make the best use of these thoughts; nor does it appear that he adhered to them steadily, since in his Epitome of Astronomy, published many years after, he proposes a physical account of the planetary motions, derived from dif ferent principles.
He supposes, in that treatise, that the motion of the sun on his axis, is preserved by some inherent vital principle; that a certain virtue, or immaterial image of the sun, is diffused with his rays into the ambient spaces, and, revolving with the body of the sun on his axis, takes hold of the planets, and carries them along with it in the same direction; like as a loadstone turned round near a magnetic needle, makes it turn round at the same time. The planet, according to him, by its inertia, endeavours to continue in its place, and the action of the sun's image and this inertia are in a perpetual struggle. He adds, that this action of the sun, like his light, decreases as the distance increases; and therefore moves the same planet with greater celerity when nearer the sun, than at a greater distance. To account for the planet's approaching towards the sun as it descends from the aphelion to the perihelion, and receding from the sun while it ascends to the aphelion again, he supposes that the sun attracts one part of each planet, and repels the opposite part; and that the part attracted is turned towards the sun in the descent, and the other towards the sun in the ascent. By suppositions of this kind, he endeavoured to account for all the other varieties of the celestial motions.
But, now that the laws of motion are better known than in Kepler's time, it is easy to shew the fallacy of every part of this account of the planetary motions. The planet does not endeavour to stop in consequence of its inertia, but to persevere in its motion in a right line. An attractive force makes it descend from the aphelion to the perihelion in a curve concave towards the sun: but the repelling force, which he supposed to begin at the perihelion, would cause it to ascend in a figure convex towards the sun. There will be occasion to shew afterwards, from Sir Isaac Newton, how an attraction or gravitation towards the sun, alone produces the effects, which, according to Kepler, required both an attractive and repelling force; and that the virtue which he ascribed to the sun's image, propagated into the planetary regions, is unnecessary, as it could be of no use for this effect, though it were admitted. For now his own prophecy, with which he concludes his book, is verisied; where he tells us, that “the discovery of such things was reserved for the succeeding ages, when the author of nature would be pleased to reveal these mysteries.”
The works of this celebrated author are many and valuable; as,
1. His Cosmographical Mystery, in 1596.
2. Optical Astronomy, in 1604.
3. Account of a New Star in Sagittarius, 1605.
4. New Astronomy; or, Celestial Physics, in Commentaries on the planet Mars.
5. Dissertations; with the Nuncius Siderius of Galileo, 1610.
6. New Gauging of Wine Casks, 1615. Said to be written on occasion of an erroneous measurement of the wine at his marriage by the revenue officer.
7. New Ephemerides, from 1617 to 1620.
8. Copernican System, three first books of the, 1618.
9. Harmony of the World; and three books of Comets, 1619.
10. Cosmographical Mystery, 2d edit. with Notes, 1621.
11. Copernican Astronomy; the three last books, 1622.
12. Logarithms, 1624; and the Supplement, in 1625.
13. His Astronomical Tables, called the Rudolphine Tables, in honour of the emperor Rudolphus, his great and learned patron, in 1627.
14. Epitome of the Copernican Astronomy, 1635.
Beside these, he wrote several pieces on various other branches, as Chronology, Geometry of Solids, Trigonometry, and an excellent treatise of Dioptrics, for that time.
Kepler's Laws, are those laws of the planetary motions discovered by Kepler. These discoveries in the mundane system, are commonly accounted two, viz. 1st, That the planets describe about the sun, areas that are proportional to the times in which they are described, namely, by a line connecting the sun and planet; and 2d, That the squares of the times of revolution, are as the cubes of the mean distances of the planets from the sun. Kepler discovered also that the orbits of the planets are elliptical.
These discoveries of Kepler, however, were only found out by many trials, in searching among a great number of astronomical observations and revolutions, what rules and laws were found to obtain. On the other hand, Newton has demonstrated, a priori, all these laws, shewing that they must obtain in the mundane system, from the laws of gravitation and centripetal force; viz, the first of these laws resulting f<*>om a centripetal force urging the planets towards the sun, and the 2d, from the centripetal force being in an inverse ratio of the square of the distance. And the elliptic form of the orbits, from a projectile force regulated by a centripetal one.
Kepler's Problem, is the determining the true from the mean anomaly of a planet, or the determining its place, in its elliptic orbit, answering to any given time; and so named from the celebrated astronomer Kepler, who first proposed it. See Anomaly.
The general state of the problem is this: To find the position of a right line, which, passing through one of the foci of an ellipsis, shall cut off an area which shall be in any given proportion to the whole area of the ellipsis; which results from this property, that such a line sweeps areas that are proportional to the times.
Many solutions have been given of this problem, some direct and geometrical, others not: viz, by Kepler, Bulliald, Ward, Newton, Keill, Machin, &c. See Newton's Princip. lib. 1. prop. 31, Keill's Astron. Lect. 23, Philos. Trans. abr. vol. 8. pa. 73, &c.
In the last of these places, Mr. Machin observes, that many attempts have been made at different times, but with no great success, towards the solution of the problem proposed by Kepler: To divide the area of a semici<*>cle into given parts, by a line drawn from a given point in the diameter, in order to find an universal rule for the motion of a body in an elliptic orbit. For among the several methods offered, some are only true in speculation, but are really of no service; others are not different from his own, which he judged improper. And as to the rest, they are all so limited and consined to particular conditions and circumstances, as still to leave the problem in general untouched. To be more particular; it is evident, that all constructions by mechanical curves are seeming solutions only, but in reality unapplicable; that the roots of infinite series are, on account of their known limitations in all respects, so far from being sufficient rules, that they serve for little more than exercises in a method of calculation. And then, as to the universal method, which proceeds by a continued correction of the errors of a false position, it is no method of solution at all in itself; because, unless there be some antecedent rule or hypothesis to begin the operation (as suppose that of an uniform motion about the upper focus, for the orbit of a planet; or that of a motion in a parabola for the perihelion part of the orbit of a comet, or some other such), it would be impossible to proceed one step in it. But as no general rule has ever yet been laid down, to assist this method, so as to make it always operate, it is the same in effect as if there were no method at all. And accordingly in experience it is found, that there is no rule now subsisting but what is absolutely useless in the elliptic orbits of comets; for in such cases there is no other way to proceed but that which was used by Kepler: to compute a table for some part of the orbit, and in it examine if the time to which the place is required, will fall out any where in that part. So that, upon the whole, it appears evident, that this problem, contrary to the received opinion, has never yet been advanced one step towards its true solution.
Mr. Machin then proceeds to give his own solution of this problem, which is particularly necessary in orbits of a great excentricity; and he illustrates his method by examples for the orbits of Venus, of Mercury, of the comet of the year 1682, and of the great comet of the year 1680, sufficiently shewing the universality of the method.