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[Illustration: SIR ISAAC NEWTON (From the bust by Roubiliac In Trinity College, Cambridge.)]
HISTORY OF ASTRONOMY
M.A., F.R.S., M. INST. C. E.,
(FORMERLY PROFESSOR OF NATURAL PHILOSOPHY, ANDERSON’S COLLEGE, GLASGOW)
AUTHOR OF “THE TRANSIT OF VENUS,” RENDU’S “THEORY OF THE GLACIERS OF SAVOY,” ETC., ETC.
BOOK I. THE GEOMETRICAL PERIOD
1. PRIMITIVE ASTRONOMY AND ASTROLOGY
2. ANCIENT ASTRONOMY–CHINESE AND CHALDAEANS
3. ANCIENT GREEK ASTRONOMY
4. THE REIGN OF EPICYCLES–FROM PTOLEMY TO COPERNICUS
BOOK II. THE DYNAMICAL PERIOD
5. DISCOVERY OF THE TRUE SOLAR SYSTEM–TYCHO BRAHE–KEPLER
6. GALILEO AND THE TELESCOPE–NOTIONS OF GRAVITY BY HORROCKS, ETC.
7. SIR ISAAC NEWTON–LAW OF UNIVERSAL GRAVITATION
8. NEWTON’S SUCCESSORS–HALLEY, EULER, LAGRANGE, LAPLACE, ETC.
9. DISCOVERY OF NEW PLANETS–HERSCHEL, PIAZZI, ADAMS, AND LE VERRIER
BOOK III. OBSERVATION
10. INSTRUMENTS OF PRECISION–SIZE OF THE SOLAR SYSTEM
11. HISTORY OF THE TELESCOPE–SPECTROSCOPE
BOOK IV. THE PHYSICAL PERIOD
12. THE SUN
13. THE MOON AND PLANETS
14. COMETS AND METEORS
15. THE STARS AND NEBULAE
An attempt has been made in these pages to trace the evolution of intellectual thought in the progress of astronomical discovery, and, by recognising the different points of view of the different ages, to give due credit even to the ancients. No one can expect, in a history of astronomy of limited size, to find a treatise on “practical” or on “theoretical astronomy,” nor a complete “descriptive astronomy,” and still less a book on “speculative astronomy.” Something of each of these is essential, however, for tracing the progress of thought and knowledge which it is the object of this History to describe.
The progress of human knowledge is measured by the increased habit of looking at facts from new points of view, as much as by the accumulation of facts. The mental capacity of one age does not seem to differ from that of other ages; but it is the imagination of new points of view that gives a wider scope to that capacity. And this is cumulative, and therefore progressive. Aristotle viewed the solar system as a geometrical problem; Kepler and Newton converted the point of view into a dynamical one. Aristotle’s mental capacity to understand the meaning of facts or to criticise a train of reasoning may have been equal to that of Kepler or Newton, but the point of view was different.
Then, again, new points of view are provided by the invention of new methods in that system of logic which we call mathematics. All that mathematics can do is to assure us that a statement A is equivalent to statements B, C, D, or is one of the facts expressed by the statements B, C, D; so that we may know, if B, C, and D are true, then A is true. To many people our inability to understand all that is contained in statements B, C, and D, without the cumbrous process of a mathematical demonstration, proves the feebleness of the human mind as a logical machine. For it required the new point of view imagined by Newton’s analysis to enable people to see that, so far as planetary orbits are concerned, Kepler’s three laws (B, C, D) were identical with Newton’s law of gravitation (A). No one recognises more than the mathematical astronomer this feebleness of the human intellect, and no one is more conscious of the limitations of the logical process called mathematics, which even now has not solved directly the problem of only three bodies.
These reflections, arising from the writing of this History, go to explain the invariable humility of the great mathematical astronomers. Newton’s comparison of himself to the child on the seashore applies to them all. As each new discovery opens up, it may be, boundless oceans for investigation, for wonder, and for admiration, the great astronomers, refusing to accept mere hypotheses as true, have founded upon these discoveries a science as exact in its observation of facts as in theories. So it is that these men, who have built up the most sure and most solid of all the sciences, refuse to invite others to join them in vain speculation. The writer has, therefore, in this short History, tried to follow that great master, Airy, whose pupil he was, and the key to whose character was exactness and accuracy; and he recognises that Science is impotent except in her own limited sphere.
It has been necessary to curtail many parts of the History in the attempt–perhaps a hopeless one–to lay before the reader in a limited space enough about each age to illustrate its tone and spirit, the ideals of the workers, the gradual addition of new points of view and of new means of investigation.
It would, indeed, be a pleasure to entertain the hope that these pages might, among new recruits, arouse an interest in the greatest of all the sciences, or that those who have handled the theoretical or practical side might be led by them to read in the original some of the classics of astronomy. Many students have much compassion for the schoolboy of to-day, who is not allowed the luxury of learning the art of reasoning from him who still remains pre-eminently its greatest exponent, Euclid. These students pity also the man of to-morrow, who is not to be allowed to read, in the original Latin of the brilliant Kepler, how he was able–by observations taken from a moving platform, the earth, of the directions of a moving object, Mars–to deduce the exact shape of the path of each of these planets, and their actual positions on these paths at any time. Kepler’s masterpiece is one of the most interesting books that was ever written, combining wit, imagination, ingenuity, and certainty.
Lastly, it must be noted that, as a History of England cannot deal with the present Parliament, so also the unfinished researches and untested hypotheses of many well-known astronomers of to-day cannot be included among the records of the History of Astronomy. The writer regrets the necessity that thus arises of leaving without mention the names of many who are now making history in astronomical work.
_August 1st, 1909._
BOOK I. THE GEOMETRICAL PERIOD
1. PRIMITIVE ASTRONOMY AND ASTROLOGY.
The growth of intelligence in the human race has its counterpart in that of the individual, especially in the earliest stages. Intellectual activity and the development of reasoning powers are in both cases based upon the accumulation of experiences, and on the comparison, classification, arrangement, and nomenclature of these experiences. During the infancy of each the succession of events can be watched, but there can be no _a priori_ anticipations. Experience alone, in both cases, leads to the idea of cause and effect as a principle that seems to dominate our present universe, as a rule for predicting the course of events, and as a guide to the choice of a course of action. This idea of cause and effect is the most potent factor in developing the history of the human race, as of the individual.
In no realm of nature is the principle of cause and effect more conspicuous than in astronomy; and we fall into the habit of thinking of its laws as not only being unchangeable in our universe, but necessary to the conception of any universe that might have been substituted in its place. The first inhabitants of the world were compelled to accommodate their acts to the daily and annual alternations of light and darkness and of heat and cold, as much as to the irregular changes of weather, attacks of disease, and the fortune of war. They soon came to regard the influence of the sun, in connection with light and heat, as a cause. This led to a search for other signs in the heavens. If the appearance of a comet was sometimes noted simultaneously with the death of a great ruler, or an eclipse with a scourge of plague, these might well be looked upon as causes in the same sense that the veering or backing of the wind is regarded as a cause of fine or foul weather.
For these reasons we find that the earnest men of all ages have recorded the occurrence of comets, eclipses, new stars, meteor showers, and remarkable conjunctions of the planets, as well as plagues and famines, floods and droughts, wars and the deaths of great rulers. Sometimes they thought they could trace connections which might lead them to say that a comet presaged famine, or an eclipse war.
Even if these men were sometimes led to evolve laws of cause and effect which now seem to us absurd, let us be tolerant, and gratefully acknowledge that these astrologers, when they suggested such “working hypotheses,” were laying the foundations of observation and deduction.
If the ancient Chaldaeans gave to the planetary conjunctions an influence over terrestrial events, let us remember that in our own time people have searched for connection between terrestrial conditions and periods of unusual prevalence of sun spots; while De la Rue, Loewy, and Balfour Stewart thought they found a connection between sun-spot displays and the planetary positions. Thus we find scientific men, even in our own time, responsible for the belief that storms in the Indian Ocean, the fertility of German vines, famines in India, and high or low Nile-floods in Egypt follow the planetary positions.
And, again, the desire to foretell the weather is so laudable that we cannot blame the ancient Greeks for announcing the influence of the moon with as much confidence as it is affirmed in Lord Wolseley’s _Soldier’s Pocket Book_.
Even if the scientific spirit of observation and deduction (astronomy) has sometimes led to erroneous systems for predicting terrestrial events (astrology), we owe to the old astronomer and astrologer alike the deepest gratitude for their diligence in recording astronomical events. For, out of the scanty records which have survived the destructive acts of fire and flood, of monarchs and mobs, we have found much that has helped to a fuller knowledge of the heavenly motions than was possible without these records.
So Hipparchus, about 150 B.C., and Ptolemy a little later, were able to use the observations of Chaldaean astrologers, as well as those of Alexandrian astronomers, and to make some discoveries which have helped the progress of astronomy in all ages. So, also, Mr. Cowell has examined the marks made on the baked bricks used by the Chaldaeans for recording the eclipses of 1062 B.C. and 762 B.C.; and has thereby been enabled, in the last few years, to correct the lunar tables of Hansen, and to find a more accurate value for the secular acceleration of the moon’s longitude and the node of her orbit than any that could be obtained from modern observations made with instruments of the highest precision.
So again, Mr. Hind  was enabled to trace back the period during which Halley’s comet has been a member of the solar system, and to identify it in the Chinese observations of comets as far back as 12 B.C. Cowell and Cromellin extended the date to 240 B.C. In the same way the comet 1861.i. has been traced back in the Chinese records to 617 A.D. 
The theoretical views founded on Newton’s great law of universal gravitation led to the conclusion that the inclination of the earth’s equator to the plane of her orbit (the obliquity of the ecliptic) has been diminishing slowly since prehistoric times; and this fact has been confirmed by Egyptian and Chinese observations on the length of the shadow of a vertical pillar, made thousands of years before the Christian era, in summer and winter.
There are other reasons why we must be tolerant of the crude notions of the ancients. The historian, wishing to give credit wherever it may be due, is met by two difficulties. Firstly, only a few records of very ancient astronomy are extant, and the authenticity of many of these is open to doubt. Secondly, it is very difficult to divest ourselves of present knowledge, and to appreciate the originality of thought required to make the first beginnings.
With regard to the first point, we are generally dependent upon histories written long after the events. The astronomy of Egyptians, Babylonians, and Assyrians is known to us mainly through the Greek historians, and for information about the Chinese we rely upon the researches of travellers and missionaries in comparatively recent times. The testimony of the Greek writers has fortunately been confirmed, and we now have in addition a mass of facts translated from the original sculptures, papyri, and inscribed bricks, dating back thousands of years.
In attempting to appraise the efforts of the beginners we must remember that it was natural to look upon the earth (as all the first astronomers did) as a circular plane, surrounded and bounded by the heaven, which was a solid vault, or hemisphere, with its concavity turned downwards. The stars seemed to be fixed on this vault; the moon, and later the planets, were seen to crawl over it. It was a great step to look on the vault as a hollow sphere carrying the sun too. It must have been difficult to believe that at midday the stars are shining as brightly in the blue sky as they do at night. It must have been difficult to explain how the sun, having set in the west, could get back to rise in the east without being seen _if_ it was always the same sun. It was a great step to suppose the earth to be spherical, and to ascribe the diurnal motions to its rotation. Probably the greatest step ever made in astronomical theory was the placing of the sun, moon, and planets at different distances from the earth instead of having them stuck on the vault of heaven. It was a transition from “flatland” to a space of three dimensions.
Great progress was made when systematic observations began, such as following the motion of the moon and planets among the stars, and the inferred motion of the sun among the stars, by observing their _heliacal risings_–i.e., the times of year when a star would first be seen to rise at sunrise, and when it could last be seen to rise at sunset. The grouping of the stars into constellations and recording their places was a useful observation. The theoretical prediction of eclipses of the sun and moon, and of the motions of the planets among the stars, became later the highest goal in astronomy.
To not one of the above important steps in the progress of astronomy can we assign the author with certainty. Probably many of them were independently taken by Chinese, Indian, Persian, Tartar, Egyptian, Babylonian, Assyrian, Phoenician, and Greek astronomers. And we have not a particle of information about the discoveries, which may have been great, by other peoples–by the Druids, the Mexicans, and the Peruvians, for example.
We do know this, that all nations required to have a calendar. The solar year, the lunar month, and the day were the units, and it is owing to their incommensurability that we find so many calendars proposed and in use at different times. The only object to be attained by comparing the chronologies of ancient races is to fix the actual dates of observations recorded, and this is not a part of a history of astronomy.
In conclusion, let us bear in mind the limited point of view of the ancients when we try to estimate their merit. Let us remember that the first astronomy was of two dimensions; the second astronomy was of three dimensions, but still purely geometrical. Since Kepler’s day we have had a dynamical astronomy.
 Trans. R. S. E., xxiii. 1864, p. 499, _On Sun Spots_, etc., by B. Stewart. Also Trans. R. S. 1860-70. Also Prof. Ernest Brown, in _R. A. S. Monthly Notices_, 1900.
 _R. A. S. Monthly Notices_, Sup.; 1905.
[Illustration: CHALDAEAN BAKED BRICK OR TABLET, _Obverse and reverse sides_, Containing record of solar eclipse, 1062 B.C., used lately by Cowell for rendering the lunar theory more accurate than was possible by finest modern observations. (British Museum collection, No. 35908.)]
 _R. A. S. Monthly Notices_, vol. x., p. 65.
 R. S. E. Proc., vol. x., 1880.
2. ANCIENT ASTRONOMY–THE CHINESE AND CHALDAEANS.
The last section must have made clear the difficulties the way of assigning to the ancient nations their proper place in the development of primitive notions about astronomy. The fact that some alleged observations date back to a period before the Chinese had invented the art of writing leads immediately to the question how far tradition can be trusted.
Our first detailed knowledge was gathered in the far East by travellers, and by the Jesuit priests, and was published in the eighteenth century. The Asiatic Society of Bengal contributed translations of Brahmin literature. The two principal sources of knowledge about Chinese astronomy were supplied, first by Father Souciet, who in 1729 published _Observations Astronomical, Geographical, Chronological, and Physical_, drawn from ancient Chinese books; and later by Father Moyriac-de-Mailla, who in 1777-1785 published _Annals of the Chinese Empire, translated from Tong-Kien-Kang-Mou_.
Bailly, in his _Astronomie Ancienne_ (1781), drew, from these and other sources, the conclusion that all we know of the astronomical learning of the Chinese, Indians, Chaldaeans, Assyrians, and Egyptians is but the remnant of a far more complete astronomy of which no trace can be found.
Delambre, in his _Histoire de l’Astronomie Ancienne_ (1817), ridicules the opinion of Bailly, and considers that the progress made by all of these nations is insignificant.
It will be well now to give an idea of some of the astronomy of the ancients not yet entirely discredited. China and Babylon may be taken as typical examples.
_China_.–It would appear that Fohi, the first emperor, reigned about 2952 B.C., and shortly afterwards Yu-Chi made a sphere to represent the motions of the celestial bodies. It is also mentioned, in the book called Chu-King, supposed to have been written in 2205 B.C., that a similar sphere was made in the time of Yao (2357 B.C.). It is said that the Emperor Chueni (2513 B.C.) saw five planets in conjunction the same day that the sun and moon were in conjunction. This is discussed by Father Martin (MSS. of De Lisle); also by M. Desvignolles (Mem. Acad. Berlin, vol. iii., p. 193), and by M. Kirsch (ditto, vol. v., p. 19), who both found that Mars, Jupiter, Saturn, and Mercury were all between the eleventh and eighteenth degrees of Pisces, all visible together in the evening on February 28th 2446 B.C., while on the same day the sun and moon were in conjunction at 9 a.m., and that on March 1st the moon was in conjunction with the other four planets. But this needs confirmation.
Yao, referred to above, gave instructions to his astronomers to determine the positions of the solstices and equinoxes, and they reported the names of the stars in the places occupied by the sun at these seasons, and in 2285 B.C. he gave them further orders. If this account be true, it shows a knowledge that the vault of heaven is a complete sphere, and that stars are shining at mid-day, although eclipsed by the sun’s brightness.
It is also asserted, in the book called _Chu-King_, that in the time of Yao the year was known to have 365-1/4 days, and that he adopted 365 days and added an intercalary day every four years (as in the Julian Calendar). This may be true or not, but the ancient Chinese certainly seem to have divided the circle into 365 degrees. To learn the length of the year needed only patient observation–a characteristic of the Chinese; but many younger nations got into a terrible mess with their calendar from ignorance of the year’s length.
It is stated that in 2159 B.C. the royal astronomers Hi and Ho failed to predict an eclipse. It probably created great terror, for they were executed in punishment for their neglect. If this account be true, it means that in the twenty-second century B.C. some rule for calculating eclipses was in use. Here, again, patient observation would easily lead to the detection of the eighteen-year cycle known to the Chaldeans as the _Saros_. It consists of 235 lunations, and in that time the pole of the moon’s orbit revolves just once round the pole of the ecliptic, and for this reason the eclipses in one cycle are repeated with very slight modification in the next cycle, and so on for many centuries.
It may be that the neglect of their duties by Hi and Ho, and their punishment, influenced Chinese astronomy; or that the succeeding records have not been available to later scholars; but the fact remains that–although at long intervals observations were made of eclipses, comets, and falling stars, and of the position of the solstices, and of the obliquity of the ecliptic–records become rare, until 776 B.C., when eclipses began to be recorded once more with some approach to continuity. Shortly afterwards notices of comets were added. Biot gave a list of these, and Mr. John Williams, in 1871, published _Observations of Comets from 611 B.C. to 1640 A.D., Extracted from the Chinese Annals_.
With regard to those centuries concerning which we have no astronomical Chinese records, it is fair to state that it is recorded that some centuries before the Christian era, in the reign of Tsin-Chi-Hoang, all the classical and scientific books that could be found were ordered to be destroyed. If true, our loss therefrom is as great as from the burning of the Alexandrian library by the Caliph Omar. He burnt all the books because he held that they must be either consistent or inconsistent with the Koran, and in the one case they were superfluous, in the other case objectionable.
_Chaldaeans_.–Until the last half century historians were accustomed to look back upon the Greeks, who led the world from the fifth to the third century B.C., as the pioneers of art, literature, and science. But the excavations and researches of later years make us more ready to grant that in science as in art the Greeks only developed what they derived from the Egyptians, Babylonians, and Assyrians. The Greek historians said as much, in fact; and modern commentators used to attribute the assertion to undue modesty. Since, however, the records of the libraries have been unearthed it has been recognised that the Babylonians were in no way inferior in the matter of original scientific investigation to other races of the same era.
The Chaldaeans, being the most ancient Babylonians, held the same station and dignity in the State as did the priests in Egypt, and spent all their time in the study of philosophy and astronomy, and the arts of divination and astrology. They held that the world of which we have a conception is an eternal world without any beginning or ending, in which all things are ordered by rules supported by a divine providence, and that the heavenly bodies do not move by chance, nor by their own will, but by the determinate will and appointment of the gods. They recorded these movements, but mainly in the hope of tracing the will of the gods in mundane affairs. Ptolemy (about 130 A.D.) made use of Babylonian eclipses in the eighth century B.C. for improving his solar and lunar tables.
Fragments of a library at Agade have been preserved at Nineveh, from which we learn that the star-charts were even then divided into constellations, which were known by the names which they bear to this day, and that the signs of the zodiac were used for determining the courses of the sun, moon, and of the five planets Mercury, Venus, Mars, Jupiter, and Saturn.
We have records of observations carried on under Asshurbanapal, who sent astronomers to different parts to study celestial phenomena. Here is one:–
To the Director of Observations,–My Lord, his humble servant Nabushum-iddin, Great Astronomer of Nineveh, writes thus: “May Nabu and Marduk be propitious to the Director of these Observations, my Lord. The fifteenth day we observed the Node of the moon, and the moon was eclipsed.”
The Phoenicians are supposed to have used the stars for navigation, but there are no records. The Egyptian priests tried to keep such astronomical knowledge as they possessed to themselves. It is probable that they had arbitrary rules for predicting eclipses. All that was known to the Greeks about Egyptian science is to be found in the writings of Diodorus Siculus. But confirmatory and more authentic facts have been derived from late explorations. Thus we learn from E. B. Knobel about the Jewish calendar dates, on records of land sales in Aramaic papyri at Assuan, translated by Professor A. H. Sayce and A. E. Cowley, (1) that the lunar cycle of nineteen years was used by the Jews in the fifth century B.C. [the present reformed Jewish calendar dating from the fourth century A.D.], a date a “little more than a century after the grandfathers and great-grandfathers of those whose business is recorded had fled into Egypt with Jeremiah” (Sayce); and (2) that the order of intercalation at that time was not dissimilar to that in use at the present day.
Then again, Knobel reminds us of “the most interesting discovery a few years ago by Father Strassmeier of a Babylonian tablet recording a partial lunar eclipse at Babylon in the seventh year of Cambyses, on the fourteenth day of the Jewish month Tammuz.” Ptolemy, in the Almagest (Suntaxis), says it occurred in the seventh year of Cambyses, on the night of the seventeenth and eighteenth of the Egyptian month Phamenoth. Pingre and Oppolzer fix the date July 16th, 533 B.C. Thus are the relations of the chronologies of Jews and Egyptians established by these explorations.
 These ancient dates are uncertain.
 _R. A. S. Monthly Notices_, vol. lxviii., No. 5, March, 1908.
3. ANCIENT GREEK ASTRONOMY.
We have our information about the earliest Greek astronomy from Herodotus (born 480 B.C.). He put the traditions into writing. Thales (639-546 B.C.) is said to have predicted an eclipse, which caused much alarm, and ended the battle between the Medes and Lydians. Airy fixed the date May 28th, 585 B.C. But other modern astronomers give different dates. Thales went to Egypt to study science, and learnt from its priests the length of the year (which was kept a profound secret!), and the signs of the zodiac, and the positions of the solstices. He held that the sun, moon, and stars are not mere spots on the heavenly vault, but solids; that the moon derives her light from the sun, and that this fact explains her phases; that an eclipse of the moon happens when the earth cuts off the sun’s light from her. He supposed the earth to be flat, and to float upon water. He determined the ratio of the sun’s diameter to its orbit, and apparently made out the diameter correctly as half a degree. He left nothing in writing.
His successors, Anaximander (610-547 B.C.) and Anaximenes (550-475 B.C.), held absurd notions about the sun, moon, and stars, while Heraclitus (540-500 B.C.) supposed that the stars were lighted each night like lamps, and the sun each morning. Parmenides supposed the earth to be a sphere.
Pythagoras (569-470 B.C.) visited Egypt to study science. He deduced his system, in which the earth revolves in an orbit, from fantastic first principles, of which the following are examples: “The circular motion is the most perfect motion,” “Fire is more worthy than earth,” “Ten is the perfect number.” He wrote nothing, but is supposed to have said that the earth, moon, five planets, and fixed stars all revolve round the sun, which itself revolves round an imaginary central fire called the Antichthon. Copernicus in the sixteenth century claimed Pythagoras as the founder of the system which he, Copernicus, revived.
Anaxagoras (born 499 B.C.) studied astronomy in Egypt. He explained the return of the sun to the east each morning by its going under the flat earth in the night. He held that in a solar eclipse the moon hides the sun, and in a lunar eclipse the moon enters the earth’s shadow–both excellent opinions. But he entertained absurd ideas of the vortical motion of the heavens whisking stones into the sky, there to be ignited by the fiery firmament to form stars. He was prosecuted for this unsettling opinion, and for maintaining that the moon is an inhabited earth. He was defended by Pericles (432 B.C.).
Solon dabbled, like many others, in reforms of the calendar. The common year of the Greeks originally had 360 days–twelve months of thirty days. Solon’s year was 354 days. It is obvious that these erroneous years would, before long, remove the summer to January and the winter to July. To prevent this it was customary at regular intervals to intercalate days or months. Meton (432 B.C.) introduced a reform based on the nineteen-year cycle. This is not the same as the Egyptian and Chaldean eclipse cycle called _Saros_ of 223 lunations, or a little over eighteen years. The Metonic cycle is 235 lunations or nineteen years, after which period the sun and moon occupy the same position relative to the stars. It is still used for fixing the date of Easter, the number of the year in Melon’s cycle being the golden number of our prayer-books. Melon’s system divided the 235 lunations into months of thirty days and omitted every sixty-third day. Of the nineteen years, twelve had twelve months and seven had thirteen months.
Callippus (330 B.C.) used a cycle four times as long, 940 lunations, but one day short of Melon’s seventy-six years. This was more correct.
Eudoxus (406-350 B.C.) is said to have travelled with Plato in Egypt. He made astronomical observations in Asia Minor, Sicily, and Italy, and described the starry heavens divided into constellations. His name is connected with a planetary theory which as generally stated sounds most fanciful. He imagined the fixed stars to be on a vault of heaven; and the sun, moon, and planets to be upon similar vaults or spheres, twenty-six revolving spheres in all, the motion of each planet being resolved into its components, and a separate sphere being assigned for each component motion. Callippus (330 B.C.) increased the number to thirty-three. It is now generally accepted that the real existence of these spheres was not suggested, but the idea was only a mathematical conception to facilitate the construction of tables for predicting the places of the heavenly bodies.
Aristotle (384-322 B.C.) summed up the state of astronomical knowledge in his time, and held the earth to be fixed in the centre of the world.
Nicetas, Heraclides, and Ecphantes supposed the earth to revolve on its axis, but to have no orbital motion.
The short epitome so far given illustrates the extraordinary deductive methods adopted by the ancient Greeks. But they went much farther in the same direction. They seem to have been in great difficulty to explain how the earth is supported, just as were those who invented the myth of Atlas, or the Indians with the tortoise. Thales thought that the flat earth floated on water. Anaxagoras thought that, being flat, it would be buoyed up and supported on the air like a kite. Democritus thought it remained fixed, like the donkey between two bundles of hay, because it was equidistant from all parts of the containing sphere, and there was no reason why it should incline one way rather than another. Empedocles attributed its state of rest to centrifugal force by the rapid circular movement of the heavens, as water is stationary in a pail when whirled round by a string. Democritus further supposed that the inclination of the flat earth to the ecliptic was due to the greater weight of the southern parts owing to the exuberant vegetation.
For further references to similar efforts of imagination the reader is referred to Sir George Cornwall Lewis’s _Historical Survey of the Astronomy of the Ancients_; London, 1862. His list of authorities is very complete, but some of his conclusions are doubtful. At p. 113 of that work he records the real opinions of Socrates as set forth by Xenophon; and the reader will, perhaps, sympathise with Socrates in his views on contemporary astronomy:–
With regard to astronomy he [Socrates] considered a knowledge of it desirable to the extent of determining the day of the year or month, and the hour of the night, … but as to learning the courses of the stars, to be occupied with the planets, and to inquire about their distances from the earth, and their orbits, and the causes of their motions, he strongly objected to such a waste of valuable time. He dwelt on the contradictions and conflicting opinions of the physical philosophers, … and, in fine, he held that the speculators on the universe and on the laws of the heavenly bodies were no better than madmen (_Xen. Mem_, i. 1, 11-15).
Plato (born 429 B.C.), the pupil of Socrates, the fellow-student of Euclid, and a follower of Pythagoras, studied science in his travels in Egypt and elsewhere. He was held in so great reverence by all learned men that a problem which he set to the astronomers was the keynote to all astronomical investigation from this date till the time of Kepler in the sixteenth century. He proposed to astronomers _the problem of representing the courses of the planets by circular and uniform motions_.
Systematic observation among the Greeks began with the rise of the Alexandrian school. Aristillus and Timocharis set up instruments and fixed the positions of the zodiacal stars, near to which all the planets in their orbits pass, thus facilitating the determination of planetary motions. Aristarchus (320-250 B.C.) showed that the sun must be at least nineteen times as far off as the moon, which is far short of the mark. He also found the sun’s diameter, correctly, to be half a degree. Eratosthenes (276-196 B.C.) measured the inclination to the equator of the sun’s apparent path in the heavens–i.e., he measured the obliquity of the ecliptic, making it 23 degrees 51′, confirming our knowledge of its continuous diminution during historical times. He measured an arc of meridian, from Alexandria to Syene (Assuan), and found the difference of latitude by the length of a shadow at noon, summer solstice. He deduced the diameter of the earth, 250,000 stadia. Unfortunately, we do not know the length of the stadium he used.
Hipparchus (190-120 B.C.) may be regarded as the founder of observational astronomy. He measured the obliquity of the ecliptic, and agreed with Eratosthenes. He altered the length of the tropical year from 365 days, 6 hours to 365 days, 5 hours, 53 minutes–still four minutes too much. He measured the equation of time and the irregular motion of the sun; and allowed for this in his calculations by supposing that the centre, about which the sun moves uniformly, is situated a little distance from the fixed earth. He called this point the _excentric_. The line from the earth to the “excentric” was called the _line of apses_. A circle having this centre was called the _equant_, and he supposed that a radius drawn to the sun from the excentric passes over equal arcs on the equant in equal times. He then computed tables for predicting the place of the sun.
He proceeded in the same way to compute Lunar tables. Making use of Chaldaean eclipses, he was able to get an accurate value of the moon’s mean motion. [Halley, in 1693, compared this value with his own measurements, and so discovered the acceleration of the moon’s mean motion. This was conclusively established, but could not be explained by the Newtonian theory for quite a long time.] He determined the plane of the moon’s orbit and its inclination to the ecliptic. The motion of this plane round the pole of the ecliptic once in eighteen years complicated the problem. He located the moon’s excentric as he had done the sun’s. He also discovered some of the minor irregularities of the moon’s motion, due, as Newton’s theory proves, to the disturbing action of the sun’s attraction.
In the year 134 B.C. Hipparchus observed a new star. This upset every notion about the permanence of the fixed stars. He then set to work to catalogue all the principal stars so as to know if any others appeared or disappeared. Here his experiences resembled those of several later astronomers, who, when in search of some special object, have been rewarded by a discovery in a totally different direction. On comparing his star positions with those of Timocharis and Aristillus he found no stars that had appeared or disappeared in the interval of 150 years; but he found that all the stars seemed to have changed their places with reference to that point in the heavens where the ecliptic is 90 degrees from the poles of the earth–i.e., the equinox. He found that this could be explained by a motion of the equinox in the direction of the apparent diurnal motion of the stars. This discovery of _precession of the equinoxes_, which takes place at the rate of 52″.1 every year, was necessary for the progress of accurate astronomical observations. It is due to a steady revolution of the earth’s pole round the pole of the ecliptic once in 26,000 years in the opposite direction to the planetary revolutions.
Hipparchus was also the inventor of trigonometry, both plane and spherical. He explained the method of using eclipses for determining the longitude.
In connection with Hipparchus’ great discovery it may be mentioned that modern astronomers have often attempted to fix dates in history by the effects of precession of the equinoxes. (1) At about the date when the Great Pyramid may have been built gamma Draconis was near to the pole, and must have been used as the pole-star. In the north face of the Great Pyramid is the entrance to an inclined passage, and six of the nine pyramids at Gizeh possess the same feature; all the passages being inclined at an angle between 26 degrees and 27 degrees to the horizon and in the plane of the meridian. It also appears that 4,000 years ago–i.e., about 2100 B.C.–an observer at the lower end of the passage would be able to see gamma Draconis, the then pole-star, at its lower culmination. It has been suggested that the passage was made for this purpose. On other grounds the date assigned to the Great Pyramid is 2123 B.C.
(2) The Chaldaeans gave names to constellations now invisible from Babylon which would have been visible in 2000 B.C., at which date it is claimed that these people were studying astronomy.
(3) In the Odyssey, Calypso directs Odysseus, in accordance with Phoenician rules for navigating the Mediterranean, to keep the Great Bear “ever on the left as he traversed the deep” when sailing from the pillars of Hercules (Gibraltar) to Corfu. Yet such a course taken now would land the traveller in Africa. Odysseus is said in his voyage in springtime to have seen the Pleiades and Arcturus setting late, which seemed to early commentators a proof of Homer’s inaccuracy. Likewise Homer, both in the _Odyssey_  (v. 272-5) and in the _Iliad_ (xviii. 489), asserts that the Great Bear never set in those latitudes. Now it has been found that the precession of the equinoxes explains all these puzzles; shows that in springtime on the Mediterranean the Bear was just above the horizon, near the sea but not touching it, between 750 B.C. and 1000 B.C.; and fixes the date of the poems, thus confirming other evidence, and establishing Homer’s character for accuracy. 
(4) The orientation of Egyptian temples and Druidical stones is such that possibly they were so placed as to assist in the observation of the heliacal risings  of certain stars. If the star were known, this would give an approximate date. Up to the present the results of these investigations are far from being conclusive.
Ptolemy (130 A.D.) wrote the Suntaxis, or Almagest, which includes a cyclopedia of astronomy, containing a summary of knowledge at that date. We have no evidence beyond his own statement that he was a practical observer. He theorised on the planetary motions, and held that the earth is fixed in the centre of the universe. He adopted the excentric and equant of Hipparchus to explain the unequal motions of the sun and moon. He adopted the epicycles and deferents which had been used by Apollonius and others to explain the retrograde motions of the planets. We, who know that the earth revolves round the sun once in a year, can understand that the apparent motion of a planet is only its motion relative to the earth. If, then, we suppose the earth fixed and the sun to revolve round it once a year, and the planets each in its own period, it is only necessary to impose upon each of these an additional _annual_ motion to enable us to represent truly the apparent motions. This way of looking at the apparent motions shows why each planet, when nearest to the earth, seems to move for a time in a retrograde direction. The attempts of Ptolemy and others of his time to explain the retrograde motion in this way were only approximate. Let us suppose each planet to have a bar with one end centred at the earth. If at the other end of the bar one end of a shorter bar is pivotted, having the planet at its other end, then the planet is given an annual motion in the secondary circle (the epicycle), whose centre revolves round the earth on the primary circle (the _deferent_), at a uniform rate round the excentric. Ptolemy supposed the centres of the epicycles of Mercury and Venus to be on a bar passing through the sun, and to be between the earth and the sun. The centres of the epicycles of Mars, Jupiter, and Saturn were supposed to be further away than the sun. Mercury and Venus were supposed to revolve in their epicycles in their own periodic times and in the deferent round the earth in a year. The major planets were supposed to revolve in the deferent round the earth in their own periodic times, and in their epicycles once in a year.
It did not occur to Ptolemy to place the centres of the epicycles of Mercury and Venus at the sun, and to extend the same system to the major planets. Something of this sort had been proposed by the Egyptians (we are told by Cicero and others), and was accepted by Tycho Brahe; and was as true a representation of the relative motions in the solar system as when we suppose the sun to be fixed and the earth to revolve.
The cumbrous system advocated by Ptolemy answered its purpose, enabling him to predict astronomical events approximately. He improved the lunar theory considerably, and discovered minor inequalities which could be allowed for by the addition of new epicycles. We may look upon these epicycles of Apollonius, and the excentric of Hipparchus, as the responses of these astronomers to the demand of Plato for uniform circular motions. Their use became more and more confirmed, until the seventeenth century, when the accurate observations of Tycho Brahe enabled Kepler to abolish these purely geometrical makeshifts, and to substitute a system in which the sun became physically its controller.
 _Phil. Mag_., vol. xxiv., pp. 481-4.
Plaeiadas t’ esoronte kai ophe duonta bootaen ‘Arkton th’ aen kai amaxan epiklaesin kaleousin, ‘Ae t’ autou strephetai kai t’ Oriona dokeuei, Oin d’ammoros esti loetron Okeanoio.
“The Pleiades and Bootes that setteth late, and the Bear, which they likewise call the Wain, which turneth ever in one place, and keepeth watch upon Orion, and alone hath no part in the baths of the ocean.”
 See Pearson in the Camb. Phil. Soc. Proc., vol. iv., pt. ii., p. 93, on whose authority the above statements are made.
 See p. 6 for definition.
4. THE REIGN OF EPICYCLES–FROM PTOLEMY TO COPERNICUS.
After Ptolemy had published his book there seemed to be nothing more to do for the solar system except to go on observing and finding more and more accurate values for the constants involved–viz., the periods of revolution, the diameter of the deferent, and its ratio to that of the epicycle, the distance of the excentric from the centre of the deferent, and the position of the line of apses, besides the inclination and position of the plane of the planet’s orbit. The only object ever aimed at in those days was to prepare tables for predicting the places of the planets. It was not a mechanical problem; there was no notion of a governing law of forces.
From this time onwards all interest in astronomy seemed, in Europe at least, to sink to a low ebb. When the Caliph Omar, in the middle of the seventh century, burnt the library of Alexandria, which had been the centre of intellectual progress, that centre migrated to Baghdad, and the Arabs became the leaders of science and philosophy. In astronomy they made careful observations. In the middle of the ninth century Albategnius, a Syrian prince, improved the value of excentricity of the sun’s orbit, observed the motion of the moon’s apse, and thought he detected a smaller progression of the sun’s apse. His tables were much more accurate than Ptolemy’s. Abul Wefa, in the tenth century, seems to have discovered the moon’s “variation.” Meanwhile the Moors were leaders of science in the west, and Arzachel of Toledo improved the solar tables very much. Ulugh Begh, grandson of the great Tamerlane the Tartar, built a fine observatory at Samarcand in the fifteenth century, and made a great catalogue of stars, the first since the time of Hipparchus.
At the close of the fifteenth century King Alphonso of Spain employed computers to produce the Alphonsine Tables (1488 A.D.), Purbach translated Ptolemy’s book, and observations were carried out in Germany by Muller, known as Regiomontanus, and Waltherus.
Nicolai Copernicus, a Sclav, was born in 1473 at Thorn, in Polish Prussia. He studied at Cracow and in Italy. He was a priest, and settled at Frauenberg. He did not undertake continuous observations, but devoted himself to simplifying the planetary systems and devising means for more accurately predicting the positions of the sun, moon, and planets. He had no idea of framing a solar system on a dynamical basis. His great object was to increase the accuracy of the calculations and the tables. The results of his cogitations were printed just before his death in an interesting book, _De Revolutionibus Orbium Celestium_. It is only by careful reading of this book that the true position of Copernicus can be realised. He noticed that Nicetas and others had ascribed the apparent diurnal rotation of the heavens to a real daily rotation of the earth about its axis, in the opposite direction to the apparent motion of the stars. Also in the writings of Martianus Capella he learnt that the Egyptians had supposed Mercury and Venus to revolve round the sun, and to be carried with him in his annual motion round the earth. He noticed that the same supposition, if extended to Mars, Jupiter, and Saturn, would explain easily why they, and especially Mars, seem so much brighter in opposition. For Mars would then be a great deal nearer to the earth than at other times. It would also explain the retrograde motion of planets when in opposition.
We must here notice that at this stage Copernicus was actually confronted with the system accepted later by Tycho Brahe, with the earth fixed. But he now recalled and accepted the views of Pythagoras and others, according to which the sun is fixed and the earth revolves; and it must be noted that, geometrically, there is no difference of any sort between the Egyptian or Tychonic system and that of Pythagoras as revived by Copernicus, except that on the latter theory the stars ought to seem to move when the earth changes its position–a test which failed completely with the rough means of observation then available. The radical defect of all solar systems previous to the time of Kepler (1609 A.D.) was the slavish yielding to Plato’s dictum demanding uniform circular motion for the planets, and the consequent evolution of the epicycle, which was fatal to any conception of a dynamical theory.
Copernicus could not sever himself from this obnoxious tradition. It is true that neither the Pythagorean nor the Egypto-Tychonic system required epicycles for explaining retrograde motion, as the Ptolemaic theory did. Furthermore, either system could use the excentric of Hipparchus to explain the irregular motion known as the equation of the centre. But Copernicus remarked that he could also use an epicycle for this purpose, or that he could use both an excentric and an epicycle for each planet, and so bring theory still closer into accord with observation. And this he proceeded to do. Moreover, observers had found irregularities in the moon’s motion, due, as we now know, to the disturbing attraction of the sun. To correct for these irregularities Copernicus introduced epicycle on epicycle in the lunar orbit.
This is in its main features the system propounded by Copernicus. But attention must, to state the case fully, be drawn to two points to be found in his first and sixth books respectively. The first point relates to the seasons, and it shows a strange ignorance of the laws of rotating bodies. To use the words of Delambre, in drawing attention to the strange conception,
he imagined that the earth, revolving round the sun, ought always to show to it the same face; the contrary phenomena surprised him: to explain them he invented a third motion, and added it to the two real motions (rotation and orbital revolution). By this third motion the earth, he held, made a revolution on itself and on the poles of the ecliptic once a year…. Copernicus did not know that motion in a straight line is the natural motion, and that motion in a curve is the resultant of several movements. He believed, with Aristotle, that circular motion was the natural one.
Copernicus made this rotation of the earth’s axis about the pole of the ecliptic retrograde (i.e., opposite to the orbital revolution), and by making it perform more than one complete revolution in a year, the added part being 1/26000 of the whole, he was able to include the precession of the equinoxes in his explanation of the seasons. His explanation of the seasons is given on leaf 10 of his book (the pages of this book are not all numbered, only alternate pages, or leaves).
In his sixth book he discusses the inclination of the planetary orbits to the ecliptic. In regard to this the theory of Copernicus is unique; and it will be best to explain this in the words of Grant in his great work. He says:–
Copernicus, as we have already remarked, did not attack the principle of the epicyclical theory: he merely sought to make it more simple by placing the centre of the earth’s orbit in the centre of the universe. This was the point to which the motions of the planets were referred, for the planes of their orbits were made to pass through it, and their points of least and greatest velocities were also determined with reference to it. By this arrangement the sun was situate mathematically near the centre of the planetary system, but he did not appear to have any physical connexion with the planets as the centre of their motions.
According to Copernicus’ sixth book, the planes of the planetary orbits do not pass through the sun, and the lines of apses do not pass through to the sun.
Such was the theory advanced by Copernicus: The earth moves in an epicycle, on a deferent whose centre is a little distance from the sun. The planets move in a similar way on epicycles, but their deferents have no geometrical or physical relation to the sun. The moon moves on an epicycle centred on a second epicycle, itself centred on a deferent, excentric to the earth. The earth’s axis rotates about the pole of the ecliptic, making one revolution and a twenty-six thousandth part of a revolution in the sidereal year, in the opposite direction to its orbital motion.
In view of this fanciful structure it must be noted, in fairness to Copernicus, that he repeatedly states that the reader is not obliged to accept his system as showing the real motions; that it does not matter whether they be true, even approximately, or not, so long as they enable us to compute tables from which the places of the planets among the stars can be predicted. He says that whoever is not satisfied with this explanation must be contented by being told that “mathematics are for mathematicians” (Mathematicis mathematica scribuntur).
At the same time he expresses his conviction over and over again that the earth is in motion. It is with him a pious belief, just as it was with Pythagoras and his school and with Aristarchus. “But” (as Dreyer says in his most interesting book, _Tycho Brahe_) “proofs of the physical truth of his system Copernicus had given none, and could give none,” any more than Pythagoras or Aristarchus.
There was nothing so startlingly simple in his system as to lead the cautious astronomer to accept it, as there was in the later Keplerian system; and the absence of parallax in the stars seemed to condemn his system, which had no physical basis to recommend it, and no simplification at all over the Egypto-Tychonic system, to which Copernicus himself drew attention. It has been necessary to devote perhaps undue space to the interesting work of Copernicus, because by a curious chance his name has become so widely known. He has been spoken of very generally as the founder of the solar system that is now accepted. This seems unfair, and on reading over what has been written about him at different times it will be noticed that the astronomers–those who have evidently read his great book–are very cautious in the words with which they eulogise him, and refrain from attributing to him the foundation of our solar system, which is entirely due to Kepler. It is only the more popular writers who give the idea that a revolution had been effected when Pythagoras’ system was revived, and when Copernicus supported his view that the earth moves and is not fixed.
It may be easy to explain the association of the name of Copernicus with the Keplerian system. But the time has long passed when the historian can support in any way this popular error, which was started not by astronomers acquainted with Kepler’s work, but by those who desired to put the Church in the wrong by extolling Copernicus.
Copernicus dreaded much the abuse he expected to receive from philosophers for opposing the authority of Aristotle, who had declared that the earth was fixed. So he sought and obtained the support of the Church, dedicating his great work to Pope Paul III. in a lengthy explanatory epistle. The Bishop of Cracow set up a memorial tablet in his honour.
Copernicus was the most refined exponent, and almost the last representative, of the Epicyclical School. As has been already stated, his successor, Tycho Brahe, supported the same use of epicycles and excentrics as Copernicus, though he held the earth to be fixed. But Tycho Brahe was eminently a practical observer, and took little part in theory; and his observations formed so essential a portion of the system of Kepler that it is only fair to include his name among these who laid the foundations of the solar system which we accept to-day.
In now taking leave of the system of epicycles let it be remarked that it has been held up to ridicule more than it deserves. On reading Airy’s account of epicycles, in the beautifully clear language of his _Six Lectures on Astronomy_, the impression is made that the jointed bars there spoken of for describing the circles were supposed to be real. This is no more the case than that the spheres of Eudoxus and Callippus were supposed to be real. Both were introduced only to illustrate the mathematical conception upon which the solar, planetary, and lunar tables were constructed. The epicycles represented nothing more nor less than the first terms in the Fourier series, which in the last century has become a basis of such calculations, both in astronomy and physics generally.
[Illustration: “QUADRANS MURALIS SIVE TICHONICUS.” With portrait of Tycho Brahe, instruments, etc., painted on the wall; showing assistants using the sight, watching the clock, and recording. (From the author’s copy of the _Astronomiae Instauratae Mechanica._)]
 For definition see p. 22.
 For definition see p. 18.
 For definition see p. 18.
 In his great book Copernicus says: “The movement of the heavenly bodies is uniform, circular, perpetual, or else composed of circular movements.” In this he proclaimed himself a follower of Pythagoras (see p. 14), as also when he says: “The world is spherical because the sphere is, of all figures, the most perfect” (Delambre, _Ast. Mod. Hist_., pp. 86, 87).
 Kepler tells us that Tycho Brahe was pleased with this device, and adapted it to his own system.
 _Hist. Ast._, vol. i., p. 354.
 _Hist. of Phys. Ast._, p. vii.
 “Est enim Astronomi proprium, historiam motuum coelestium diligenti et artificiosa observatione colligere. Deinde causas earundem, seu hypotheses, cum veras assequi nulla ratione possit … Neque enim necesse est, eas hypotheses esse veras, imo ne verisimiles quidem, sed sufficit hoc usum, si calculum observationibus congruentem exhibeant.”
BOOK II. THE DYNAMICAL PERIOD
5. DISCOVERY OF THE TRUE SOLAR SYSTEM–TYCHO BRAHE–KEPLER.
During the period of the intellectual and aesthetic revival, at the beginning of the sixteenth century, the “spirit of the age” was fostered by the invention of printing, by the downfall of the Byzantine Empire, and the scattering of Greek fugitives, carrying the treasures of literature through Western Europe, by the works of Raphael and Michael Angelo, by the Reformation, and by the extension of the known world through the voyages of Spaniards and Portuguese. During that period there came to the front the founder of accurate observational astronomy. Tycho Brahe, a Dane, born in 1546 of noble parents, was the most distinguished, diligent, and accurate observer of the heavens since the days of Hipparchus, 1,700 years before.
Tycho was devoted entirely to his science from childhood, and the opposition of his parents only stimulated him in his efforts to overcome difficulties. He soon grasped the hopelessness of the old deductive methods of reasoning, and decided that no theories ought to be indulged in until preparations had been made by the accumulation of accurate observations. We may claim for him the title of founder of the inductive method.
For a complete life of this great man the reader is referred to Dreyer’s _Tycho Brahe_, Edinburgh, 1890, containing a complete bibliography. The present notice must be limited to noting the work done, and the qualities of character which enabled him to attain his scientific aims, and which have been conspicuous in many of his successors.
He studied in Germany, but King Frederick of Denmark, appreciating his great talents, invited him to carry out his life’s work in that country. He granted to him the island of Hveen, gave him a pension, and made him a canon of the Cathedral of Roskilde. On that island Tycho Brahe built the splendid observatory which he called Uraniborg, and, later, a second one for his assistants and students, called Stjerneborg. These he fitted up with the most perfect instruments, and never lost a chance of adding to his stock of careful observations.
The account of all these instruments and observations, printed at his own press on the island, was published by Tycho Brahe himself, and the admirable and numerous engravings bear witness to the excellence of design and the stability of his instruments.
His mechanical skill was very great, and in his workmanship he was satisfied with nothing but the best. He recognised the importance of rigidity in the instruments, and, whereas these had generally been made of wood, he designed them in metal. His instruments included armillae like those which had been used in Alexandria, and other armillae designed by himself–sextants, mural quadrants, large celestial globes and various instruments for special purposes. He lived before the days of telescopes and accurate clocks. He invented the method of sub-dividing the degrees on the arc of an instrument by transversals somewhat in the way that Pedro Nunez had proposed.
He originated the true system of observation and reduction of observations, recognising the fact that the best instrument in the world is not perfect; and with each of his instruments he set to work to find out the errors of graduation and the errors of mounting, the necessary correction being applied to each observation.
When he wanted to point his instrument exactly to a star he was confronted with precisely the same difficulty as is met in gunnery and rifle-shooting. The sights and the object aimed at cannot be in focus together, and a great deal depends on the form of sight. Tycho Brahe invented, and applied to the pointers of his instruments, an aperture-sight of variable area, like the iris diaphragm used now in photography. This enabled him to get the best result with stars of different brightness. The telescope not having been invented, he could not use a telescopic-sight as we now do in gunnery. This not only removes the difficulty of focussing, but makes the minimum visible angle smaller. Helmholtz has defined the minimum angle measurable with the naked eye as being one minute of arc. In view of this it is simply marvellous that, when the positions of Tycho’s standard stars are compared with the best modern catalogues, his probable error in right ascension is only +/- 24″, 1, and in declination only +/- 25″, 9.
Clocks of a sort had been made, but Tycho Brahe found them so unreliable that he seldom used them, and many of his position-measurements were made by measuring the angular distances from known stars.
Taking into consideration the absence of either a telescope or a clock, and reading his account of the labour he bestowed upon each observation, we must all agree that Kepler, who inherited these observations in MS., was justified, under the conditions then existing, in declaring that there was no hope of anyone ever improving upon them.
In the year 1572, on November 11th, Tycho discovered in Cassiopeia a new star of great brilliance, and continued to observe it until the end of January, 1573. So incredible to him was such an event that he refused to believe his own eyes until he got others to confirm what he saw. He made accurate observations of its distance from the nine principal stars in Casseiopeia, and proved that it had no measurable parallax. Later he employed the same method with the comets of 1577, 1580, 1582, 1585, 1590, 1593, and 1596, and proved that they too had no measurable parallax and must be very distant.
The startling discovery that stars are not necessarily permanent, that new stars may appear, and possibly that old ones may disappear, had upon him exactly the same effect that a similar occurrence had upon Hipparchus 1,700 years before. He felt it his duty to catalogue all the principal stars, so that there should be no mistake in the future. During the construction of his catalogue of 1,000 stars he prepared and used accurate tables of refraction deduced from his own observations. Thus he eliminated (so far as naked eye observations required) the effect of atmospheric refraction which makes the altitude of a star seem greater than it really is.
Tycho Brahe was able to correct the lunar theory by his observations. Copernicus had introduced two epicycles on the lunar orbit in the hope of obtaining a better accordance between theory and observation; and he was not too ambitious, as his desire was to get the tables accurate to ten minutes. Tycho Brahe found that the tables of Copernicus were in error as much as two degrees. He re-discovered the inequality called “variation” by observing the moon in all phases–a thing which had not been attended to. [It is remarkable that in the nineteenth century Sir George Airy established an altazimuth at Greenwich Observatory with this special object, to get observations of the moon in all phases.] He also discovered other lunar equalities, and wanted to add another epicycle to the moon’s orbit, but he feared that these would soon become unmanageable if further observations showed more new inequalities.
But, as it turned out, the most fruitful work of Tycho Brahe was on the motions of the planets, and especially of the planet Mars, for it was by an examination of these results that Kepler was led to the discovery of his immortal laws.
After the death of King Frederick the observatories of Tycho Brahe were not supported. The gigantic power and industry displayed by this determined man were accompanied, as often happens, by an overbearing manner, intolerant of obstacles. This led to friction, and eventually the observatories were dismantled, and Tycho Brahe was received by the Emperor Rudolph II., who placed a house in Prague at his disposal. Here he worked for a few years, with Kepler as one of his assistants, and he died in the year 1601.
It is an interesting fact that Tycho Brahe had a firm conviction that mundane events could be predicted by astrology, and that this belief was supported by his own predictions.
It has already been stated that Tycho Brahe maintained that observation must precede theory. He did not accept the Copernican theory that the earth moves, but for a working hypothesis he used a modification of an old Egyptian theory, mathematically identical with that of Copernicus, but not involving a stellar parallax. He says (_De Mundi_, etc.) that
the Ptolemean system was too complicated, and the new one which that great man Copernicus had proposed, following in the footsteps of Aristarchus of Samos, though there was nothing in it contrary to mathematical principles, was in opposition to those of physics, as the heavy and sluggish earth is unfit to move, and the system is even opposed to the authority of Scripture. The absence of annual parallax further involves an incredible distance between the outermost planet and the fixed stars.
We are bound to admit that in the circumstances of the case, so long as there was no question of dynamical forces connecting the members of the solar system, his reasoning, as we should expect from such a man, is practical and sound. It is not surprising, then, that astronomers generally did not readily accept the views of Copernicus, that Luther (Luther’s _Tischreden_, pp. 22, 60) derided him in his usual pithy manner, that Melancthon (_Initia doctrinae physicae_) said that Scripture, and also science, are against the earth’s motion; and that the men of science whose opinion was asked for by the cardinals (who wished to know whether Galileo was right or wrong) looked upon Copernicus as a weaver of fanciful theories.
Johann Kepler is the name of the man whose place, as is generally agreed, would have been the most difficult to fill among all those who have contributed to the advance of astronomical knowledge. He was born at Wiel, in the Duchy of Wurtemberg, in 1571. He held an appointment at Gratz, in Styria, and went to join Tycho Brahe in Prague, and to assist in reducing his observations. These came into his possession when Tycho Brahe died, the Emperor Rudolph entrusting to him the preparation of new tables (called the Rudolphine tables) founded on the new and accurate observations. He had the most profound respect for the knowledge, skill, determination, and perseverance of the man who had reaped such a harvest of most accurate data; and though Tycho hardly recognised the transcendent genius of the man who was working as his assistant, and although there were disagreements between them, Kepler held to his post, sustained by the conviction that, with these observations to test any theory, he would be in a position to settle for ever the problem of the solar system.
[Illustration: PORTRAIT OF JOHANNES KEPLER. By F. Wanderer, from Reitlinger’s “Johannes Kepler” (original in Strassburg).]
It has seemed to many that Plato’s demand for uniform circular motion (linear or angular) was responsible for a loss to astronomy of good work during fifteen hundred years, for a hundred ill-considered speculative cosmogonies, for dissatisfaction, amounting to disgust, with these _a priori_ guesses, and for the relegation of the science to less intellectual races than Greeks and other Europeans. Nobody seemed to dare to depart from this fetish of uniform angular motion and circular orbits until the insight, boldness, and independence of Johann Kepler opened up a new world of thought and of intellectual delight.
While at work on the Rudolphine tables he used the old epicycles and deferents and excentrics, but he could not make theory agree with observation. His instincts told him that these apologists for uniform motion were a fraud; and he proved it to himself by trying every possible variation of the elements and finding them fail. The number of hypotheses which he examined and rejected was almost incredible (for example, that the planets turn round centres at a little distance from the sun, that the epicycles have centres at a little distance from the deferent, and so on). He says that, after using all these devices to make theory agree with Tycho’s observations, he still found errors amounting to eight minutes of a degree. Then he said boldly that it was impossible that so good an observer as Tycho could have made a mistake of eight minutes, and added: “Out of these eight minutes we will construct a new theory that will explain the motions of all the planets.” And he did it, with elliptic orbits having the sun in a focus of each.
It is often difficult to define the boundaries between fancies, imagination, hypothesis, and sound theory. This extraordinary genius was a master in all these modes of attacking a problem. His analogy between the spaces occupied by the five regular solids and the distances of the planets from the sun, which filled him with so much delight, was a display of pure fancy. His demonstration of the three fundamental laws of planetary motion was the most strict and complete theory that had ever been attempted.
It has been often suggested that the revival by Copernicus of the notion of a moving earth was a help to Kepler. No one who reads Kepler’s great book could hold such an opinion for a moment. In fact, the excellence of Copernicus’s book helped to prolong the life of the epicyclical theories in opposition to Kepler’s teaching.
All of the best theories were compared by him with observation. These were the Ptolemaic, the Copernican, and the Tychonic. The two latter placed all of the planetary orbits concentric with one another, the sun being placed a little away from their common centre, and having no apparent relation to them, and being actually outside the planes in which they move. Kepler’s first great discovery was that the planes of all the orbits pass through the sun; his second was that the line of apses of each planet passes through the sun; both were contradictory to the Copernican theory.
He proceeds cautiously with his propositions until he arrives at his great laws, and he concludes his book by comparing observations of Mars, of all dates, with his theory.
His first law states that the planets describe ellipses with the sun at a focus of each ellipse.
His second law (a far more difficult one to prove) states that a line drawn from a planet to the sun sweeps over equal areas in equal times. These two laws were published in his great work, _Astronomia Nova, sen. Physica Coelestis tradita commentariis de Motibus Stelloe; Martis_, Prague, 1609.
It took him nine years more to discover his third law, that the squares of the periodic times are proportional to the cubes of the mean distances from the sun.
These three laws contain implicitly the law of universal gravitation. They are simply an alternative way of expressing that law in dealing with planets, not particles. Only, the power of the greatest human intellect is so utterly feeble that the meaning of the words in Kepler’s three laws could not be understood until expounded by the logic of Newton’s dynamics.
The joy with which Kepler contemplated the final demonstration of these laws, the evolution of which had occupied twenty years, can hardly be imagined by us. He has given some idea of it in a passage in his work on _Harmonics_, which is not now quoted, only lest someone might say it was egotistical–a term which is simply grotesque when applied to such a man with such a life’s work accomplished.
The whole book, _Astronomia Nova_, is a pleasure to read; the mass of observations that are used, and the ingenuity of the propositions, contrast strongly with the loose and imperfectly supported explanations of all his predecessors; and the indulgent reader will excuse the devotion of a few lines to an example of the ingenuity and beauty of his methods.
It may seem a hopeless task to find out the true paths of Mars and the earth (at that time when their shape even was not known) from the observations giving only the relative direction from night to night. Now, Kepler had twenty years of observations of Mars to deal with. This enabled him to use a new method, to find the earth’s orbit. Observe the date at any time when Mars is in opposition. The earth’s position E at that date gives the longitude of Mars M. His period is 687 days. Now choose dates before and after the principal date at intervals of 687 days and its multiples. Mars is in each case in the same position. Now for any date when Mars is at M and the earth at E3 the date of the year gives the angle E3SM. And the observation of Tycho gives the direction of Mars compared with the sun, SE3M. So all the angles of the triangle SEM in any of these positions of E are known, and also the ratios of SE1, SE2, SE3, SE4 to SM and to each other.
For the orbit of Mars observations were chosen at intervals of a year, when the earth was always in the same place.
But Kepler saw much farther than the geometrical facts. He realised that the orbits are followed owing to a force directed to the sun; and he guessed that this is the same force as the gravity that makes a stone fall. He saw the difficulty of gravitation acting through the void space. He compared universal gravitation to magnetism, and speaks of the work of Gilbert of Colchester. (Gilbert’s book, _De Mundo Nostro Sublunari, Philosophia Nova_, Amstelodami, 1651, containing similar views, was published forty-eight years after Gilbert’s death, and forty-two years after Kepler’s book and reference. His book _De Magnete_ was published in 1600.)
A few of Kepler’s views on gravitation, extracted from the Introduction to his _Astronomia Nova_, may now be mentioned:–
1. Every body at rest remains at rest if outside the attractive power of other bodies.
2. Gravity is a property of masses mutually attracting in such manner that the earth attracts a stone much more than a stone attracts the earth.
3. Bodies are attracted to the earth’s centre, not because it is the centre of the universe, but because it is the centre of the attracting particles of the earth.
4. If the earth be not round (but spheroidal?), then bodies at different latitudes will not be attracted to its centre, but to different points in the neighbourhood of that centre.
5. If the earth and moon were not retained in their orbits by vital force (_aut alia aligua aequipollenti_), the earth and moon would come together.
6. If the earth were to cease to attract its waters, the oceans would all rise and flow to the moon.
7. He attributes the tides to lunar attraction. Kepler had been appointed Imperial Astronomer with a handsome salary (on paper), a fraction of which was doled out to him very irregularly. He was led to miserable makeshifts to earn enough to keep his family from starvation; and proceeded to Ratisbon in 1630 to represent his claims to the Diet. He arrived worn out and debilitated; he failed in his appeal, and died from fever, contracted under, and fed upon, disappointment and exhaustion. Those were not the days when men could adopt as a profession the “research of endowment.”
Before taking leave of Kepler, who was by no means a man of one idea, it ought to be here recorded that he was the first to suggest that a telescope made with both lenses convex (not a Galilean telescope) can have cross wires in the focus, for use as a pointer to fix accurately the positions of stars. An Englishman, Gascoigne, was the first to use this in practice.
From the all too brief epitome here given of Kepler’s greatest book, it must be obvious that he had at that time some inkling of the meaning of his laws–universal gravitation. From that moment the idea of universal gravitation was in the air, and hints and guesses were thrown out by many; and in time the law of gravitation would doubtless have been discovered, though probably not by the work of one man, even if Newton had not lived. But, if Kepler had not lived, who else could have discovered his laws?
 When the writer visited M. D’Arrest, the astronomer, at Copenhagen, in 1872, he was presented by D’Arrest with one of several bricks collected from the ruins of Uraniborg. This was one of his most cherished possessions until, on returning home after a prolonged absence on astronomical work, he found that his treasure had been tidied away from his study.
 An ellipse is one of the plane, sections of a cone. It is an oval curve, which may be drawn by fixing two pins in a sheet of paper at S and H, fastening a string, SPH, to the two pins, and stretching it with a pencil point at P, and moving the pencil point, while the string is kept taut, to trace the oval ellipse, APB. S and H are the _foci_. Kepler found the sun to be in one focus, say S. AB is the _major axis_. DE is the _minor axis_. C is the _centre_. The direction of AB is the _line of apses_. The ratio of CS to CA is the _excentricity_. The position of the planet at A is the _perihelion_ (nearest to the sun). The position of the planet at B is the _aphelion_ (farthest from the sun). The angle ASP is the _anomaly_ when the planet is at P. CA or a line drawn from S to D is the _mean distance_ of the planet from the sun.
 The ruled logarithmic paper we now use was not then to be had by going into a stationer’s shop. Else he would have accomplished this in five minutes.
6. GALILEO AND THE TELESCOPE–NOTIONS OF GRAVITY BY HORROCKS, ETC.
It is now necessary to leave the subject of dynamical astronomy for a short time in order to give some account of work in a different direction originated by a contemporary of Kepler’s, his senior in fact by seven years. Galileo Galilei was born at Pisa in 1564. The most scientific part of his work dealt with terrestrial dynamics; but one of those fortunate chances which happen only to really great men put him in the way of originating a new branch of astronomy.
The laws of motion had not been correctly defined. The only man of Galileo’s time who seems to have worked successfully in the same direction as himself was that Admirable Crichton of the Italians, Leonardo da Vinci. Galileo cleared the ground. It had always been noticed that things tend to come to rest; a ball rolled on the ground, a boat moved on the water, a shot fired in the air. Galileo realised that in all of these cases a resisting force acts to stop the motion, and he was the first to arrive at the not very obvious law that the motion of a body will never stop, nor vary its speed, nor change its direction, except by the action of some force.
It is not very obvious that a light body and a heavy one fall at the same speed (except for the resistance of the air). Galileo proved this on paper, but to convince the world he had to experiment from the leaning tower of Pisa.
At an early age he discovered the principle of isochronism of the pendulum, which, in the hands of Huyghens in the middle of the seventeenth century, led to the invention of the pendulum clock, perhaps the most valuable astronomical instrument ever produced.
These and other discoveries in dynamics may seem very obvious now; but it is often the most every-day matters which have been found to elude the inquiries of ordinary minds, and it required a high order of intellect to unravel the truth and discard the stupid maxims scattered through the works of Aristotle and accepted on his authority. A blind worship of scientific authorities has often delayed the progress of human knowledge, just as too much “instruction” of a youth often ruins his “education.” Grant, in his history of Physical Astronomy, has well said that “the sagacity and skill which Galileo displays in resolving the phenomena of motion into their constituent elements, and hence deriving the original principles involved in them, will ever assure to him a distinguished place among those who have extended the domains of science.”
But it was work of a different kind that established Galileo’s popular reputation. In 1609 Galileo heard that a Dutch spectacle-maker had combined a pair of lenses so as to magnify distant objects. Working on this hint, he solved the same problem, first on paper and then in practice. So he came to make one of the first telescopes ever used in astronomy. No sooner had he turned it on the heavenly bodies than he was rewarded by such a shower of startling discoveries as forthwith made his name the best known in Europe. He found curious irregular black spots on the sun, revolving round it in twenty-seven days; hills and valleys on the moon; the planets showing discs of sensible size, not points like the fixed stars; Venus showing phases according to her position in relation to the sun; Jupiter accompanied by four moons; Saturn with appendages that he could not explain, but unlike the other planets; the Milky Way composed of a multitude of separate stars.
His fame flew over Europe like magic, and his discoveries were much discussed–and there were many who refused to believe. Cosmo de Medici induced him to migrate to Florence to carry on his observations. He was received by Paul V., the Pope, at Rome, to whom he explained his discoveries.
He thought that these discoveries proved the truth of the Copernican theory of the Earth’s motion; and he urged this view on friends and foes alike. Although in frequent correspondence with Kepler, he never alluded to the New Astronomy, and wrote to him extolling the virtue of epicycles. He loved to argue, never shirked an encounter with any number of disputants, and laughed as he broke down their arguments.
Through some strange course of events, not easy to follow, the Copernican theory, whose birth was welcomed by the Church, had now been taken up by certain anti-clerical agitators, and was opposed by the cardinals as well as by the dignitaries of the Reformed Church. Galileo–a good Catholic–got mixed up in these discussions, although on excellent terms with the Pope and his entourage. At last it came about that Galileo was summoned to appear at Rome, where he was charged with holding and teaching heretical opinions about the movement of the earth; and he then solemnly abjured these opinions. There has been much exaggeration and misstatement about his trial and punishment, and for a long time there was a great deal of bitterness shown on both sides. But the general verdict of the present day seems to be that, although Galileo himself was treated with consideration, the hostility of the Church to the views of Copernicus placed it in opposition also to the true Keplerian system, and this led to unprofitable controversies. From the time of Galileo onwards, for some time, opponents of religion included the theory of the Earth’s motion in their disputations, not so much for the love, or knowledge, of astronomy, as for the pleasure of putting the Church in the wrong. This created a great deal of bitterness and intolerance on both sides. Among the sufferers was Giordano Bruno, a learned speculative philosopher, who was condemned to be burnt at the stake.
Galileo died on Christmas Day, 1642–the day of Newton’s birth. The further consideration of the grand field of discovery opened out by Galileo with his telescopes must be now postponed, to avoid discontinuity in the history of the intellectual development of this period, which lay in the direction of dynamical, or physical, astronomy.
Until the time of Kepler no one seems to have conceived the idea of universal physical forces controlling terrestrial phenomena, and equally applicable to the heavenly bodies. The grand discovery by Kepler of the true relationship of the Sun to the Planets, and the telescopic discoveries of Galileo and of those who followed him, spread a spirit of inquiry and philosophic thought throughout Europe, and once more did astronomy rise in estimation; and the irresistible logic of its mathematical process of reasoning soon placed it in the position it has ever since occupied as the foremost of the exact sciences.
The practical application of this process of reasoning was enormously facilitated by the invention of logarithms by Napier. He was born at Merchistoun, near Edinburgh, in 1550, and died in 1617. By this system the tedious arithmetical operations necessary in astronomical calculations, especially those dealing with the trigonometrical functions of angles, were so much simplified that Laplace declared that by this invention the life-work of an astronomer was doubled.
Jeremiah Horrocks (born 1619, died 1641) was an ardent admirer of Tycho Brahe and Kepler, and was able to improve the Rudolphine tables so much that he foretold a transit of Venus, in 1639, which these tables failed to indicate, and was the only observer of it. His life was short, but he accomplished a great deal, and rightly ascribed the lunar inequality called _evection_ to variations in the value of the eccentricity and in the direction of the line of apses, at the same time correctly assigning _the disturbing force of the Sun_ as the cause. He discovered the errors in Jupiter’s calculated place, due to what we now know as the long inequality of Jupiter and Saturn, and measured with considerable accuracy the acceleration at that date of Jupiter’s mean motion, and indicated the retardation of Saturn’s mean motion.
Horrocks’ investigations, so far as they could be collected, were published posthumously in 1672, and seldom, if ever, has a man who lived only twenty-two years originated so much scientific knowledge.
At this period British science received a lasting impetus by the wise initiation of a much-abused man, Charles II., who founded the Royal Society of London, and also the Royal Observatory of Greeenwich, where he established Flamsteed as first Astronomer Royal, especially for lunar and stellar observations likely to be useful for navigation. At the same time the French Academy and the Paris Observatory were founded. All this within fourteen years, 1662-1675.
Meanwhile gravitation in general terms was being discussed by Hooke, Wren, Halley, and many others. All of these men felt a repugnance to accept the idea of a force acting across the empty void of space. Descartes (1596-1650) proposed an ethereal medium whirling round the sun with the planets, and having local whirls revolving with the satellites. As Delambre and Grant have said, this fiction only retarded the progress of pure science. It had no sort of relation to the more modern, but equally misleading, “nebular hypothesis.” While many were talking and guessing, a giant mind was needed at this stage to make things clear.
7. SIR ISAAC NEWTON–LAW OF UNIVERSAL GRAVITATION.
We now reach the period which is the culminating point of interest in the history of dynamical astronomy. Isaac Newton was born in 1642. Pemberton states that Newton, having quitted Cambridge to avoid the plague, was residing at Wolsthorpe, in Lincolnshire, where he had been born; that he was sitting one day in the garden, reflecting upon the force which prevents a planet from flying off at a tangent and which draws it to the sun, and upon the force which draws the moon to the earth; and that he saw in the case of the planets that the sun’s force must clearly be unequal at different distances, for the pull out of the tangential line in a minute is less for Jupiter than for Mars. He then saw that the pull of the earth on the moon would be less than for a nearer object. It is said that while thus meditating he saw an apple fall from a tree to the ground, and that this fact suggested the questions: Is the force that pulled that apple from the tree the same as the force which draws the moon to the earth? Does the attraction for both of them follow the same law as to distance as is given by the planetary motions round the sun? It has been stated that in this way the first conception of universal gravitation arose.
Quite the most important event in the whole history of physical astronomy was the publication, in 1687, of Newton’s _Principia (Philosophiae Naturalis Principia Mathematica)_. In this great work Newton started from the beginning of things, the laws of motion, and carried his argument, step by step, into every branch of physical astronomy; giving the physical meaning of Kepler’s three laws, and explaining, or indicating the explanation of, all the known heavenly motions and their irregularities; showing that all of these were included in his simple statement about the law of universal gravitation; and proceeding to deduce from that law new irregularities in the motions of the moon which had never been noticed, and to discover the oblate figure of the earth and the cause of the tides. These investigations occupied the best part of his life; but he wrote the whole of his great book in fifteen months.
Having developed and enunciated the true laws of motion, he was able to show that Kepler’s second law (that equal areas are described by the line from the planet to the sun in equal times) was only another way of saying that the centripetal force on a planet is always directed to the sun. Also that Kepler’s first law (elliptic orbits with the sun in one focus) was only another way of saying that the force urging a planet to the sun varies inversely as the square of the distance. Also (if these two be granted) it follows that Kepler’s third law is only another way of saying that the sun’s force on different planets (besides depending as above on distance) is proportional to their masses.
Having further proved the, for that day, wonderful proposition that, with the law of inverse squares, the attraction by the separate particles of a sphere of uniform density (or one composed of concentric spherical shells, each of uniform density) acts as if the whole mass were collected at the centre, he was able to express the meaning of Kepler’s laws in propositions which have been summarised as follows:–
The law of universal gravitation.–_Every particle of matter in the universe attracts every other particle with a force varying inversely as the square of the distance between them, and directly as the product of the masses of the two particles_.
But Newton did not commit himself to the law until he had answered that question about the apple; and the above proposition now enabled him to deal with the Moon and the apple. Gravity makes a stone fall 16.1 feet in a second. The moon is 60 times farther from the earth’s centre than the stone, so it ought to be drawn out of a straight course through 16.1 feet in a minute. Newton found the distance through which she is actually drawn as a fraction of the earth’s diameter. But when he first examined this matter he proceeded to use a wrong diameter for the earth, and he found a serious discrepancy. This, for a time, seemed to condemn his theory, and regretfully he laid that part of his work aside. Fortunately, before Newton wrote the _Principia_ the French astronomer Picard made a new and correct measure of an arc of the meridian, from which he obtained an accurate value of the earth’s diameter. Newton applied this value, and found, to his great joy, that when the distance of the moon is 60 times the radius of the earth she is attracted out of the straight course 16.1 feet per minute, and that the force acting on a stone or an apple follows the same law as the force acting upon the heavenly bodies.
The universality claimed for the law–if not by Newton, at least by his commentators–was bold, and warranted only by the large number of cases in which Newton had found it to apply. Its universality has been under test ever since, and so far it has stood the test. There has often been a suspicion of a doubt, when some inequality of motion in the heavenly bodies has, for a time, foiled the astronomers in their attempts to explain it. But improved mathematical methods have always succeeded in the end, and so the seeming doubt has been converted into a surer conviction of the universality of the law.
Having once established the law, Newton proceeded to trace some of its consequences. He saw that the figure of the earth depends partly on the mutual gravitation of its parts, and partly on the centrifugal tendency due to the earth’s rotation, and that these should cause a flattening of the poles. He invented a mathematical method which he used for computing the ratio of the polar to the equatorial diameter.
He then noticed that the consequent bulging of matter at the equator would be attracted by the moon unequally, the nearest parts being most attracted; and so the moon would tend to tilt the earth when in some parts of her orbit; and the sun would do this to a less extent, because of its great distance. Then he proved that the effect ought to be a rotation of the earth’s axis over a conical surface in space, exactly as the axis of a top describes a cone, if the top has a sharp point, and is set spinning and displaced from the vertical. He actually calculated the amount; and so he explained the cause of the precession of the equinoxes discovered by Hipparchus about 150 B.C.
One of his grandest discoveries was a method of weighing the heavenly bodies by their action on each other. By means of this principle he was able to compare the mass of the sun with the masses of those planets that have moons, and also to compare the mass of our moon with the mass of the earth.
Thus Newton, after having established his great principle, devoted his splendid intellect to the calculation of its consequences. He proved that if a body be projected with any velocity in free space, subject only to a central force, varying inversely as the square of the distance, the body must revolve in a curve which may be any one of the sections of a cone–a circle, ellipse, parabola, or hyperbola; and he found that those comets of which he had observations move in parabolae round the Sun, and are thus subject to the universal law.
Newton realised that, while planets and satellites are chiefly controlled by the central body about which they revolve, the new law must involve irregularities, due to their mutual action–such, in fact, as Horrocks had indicated. He determined to put this to a test in the case of the moon, and to calculate the sun’s effect, from its mass compared with that of the earth, and from its distance. He proved that the average effect upon the plane of the orbit would be to cause the line in which it cuts the plane of the ecliptic (i.e., the line of nodes) to revolve in the ecliptic once in about nineteen years. This had been a known fact from the earliest ages. He also concluded that the line of apses would revolve in the plane of the lunar orbit also in about nineteen years; but the observed period is only ten years. For a long time this was the one weak point in the Newtonian theory. It was not till 1747 that Clairaut reconciled this with the theory, and showed why Newton’s calculation was not exact.
Newton proceeded to explain the other inequalities recognised by Tycho Brahe and older observers, and to calculate their maximum amounts as indicated by his theory. He further discovered from his calculations two new inequalities, one of the apogee, the other of the nodes, and assigned the maximum value. Grant has shown the values of some of these as given by observation in the tables of Meyer and more modern tables, and has compared them with the values assigned by Newton from his theory; and the comparison is very remarkable.
Newton. Modern Tables. degrees ‘ ” degrees ‘ ” Mean monthly motion of Apses 1.31.28 3.4.0 Mean annual motion of nodes 19.18.1,23 19.21.22,50 Mean value of “variation” 36.10 35.47 Annual equation 11.51 11.14 Inequality of mean motion of apogee 19.43 22.17 Inequality of mean motion of nodes 9.24 9.0
The only serious discrepancy is the first, which has been already mentioned. Considering that some of these perturbations had never been discovered, that the cause of none of them had ever been known, and that he exhibited his results, if he did not also make the discoveries, by the synthetic methods of geometry, it is simply marvellous that he reached to such a degree of accuracy. He invented the infinitesimal calculus which is more suited for such calculations, but had he expressed his results in that language he would have been unintelligible to many.
Newton’s method of calculating the precession of the equinoxes, already referred to, is as beautiful as anything in the _Principia_. He had already proved the regression of the nodes of a satellite moving in an orbit inclined to the ecliptic. He now said that the nodes of a ring of satellites revolving round the earth’s equator would consequently all regress. And if joined into a solid ring its node would regress; and it would do so, only more slowly, if encumbered by the spherical part of the earth’s mass. Therefore the axis of the equatorial belt of the earth must revolve round the pole of the ecliptic. Then he set to work and found the amount due to the moon and that due to the sun, and so he solved the mystery of 2,000 years.
When Newton applied his law of gravitation to an explanation of the tides he started a new field for the application of mathematics to physical problems; and there can be little doubt that, if he could have been furnished with complete tidal observations from different parts of the world, his extraordinary powers of analysis would have enabled him to reach a satisfactory theory. He certainly opened up many mines full of intellectual gems; and his successors have never ceased in their explorations. This has led to improved mathematical methods, which, combined with the greater accuracy of observation, have rendered physical astronomy of to-day the most exact of the sciences.
Laplace only expressed the universal opinion of posterity when he said that to the _Principia_ is assured “a pre-eminence above all the other productions of the human intellect.”
The name of Flamsteed, First Astronomer Royal, must here be mentioned as having supplied Newton with the accurate data required for completing the theory.
The name of Edmund Halley, Second Astronomer Royal, must ever be held in repute, not only for his own discoveries, but for the part he played in urging Newton to commit to writing, and present to the Royal Society, the results of his investigations. But for his friendly insistence it is possible that the _Principia_ would never have been written; and but for his generosity in supplying the means the Royal Society could not have published the book.
[Illustration: DEATH MASK OF SIR ISAAC NEWTON. Photographed specially for this work from the original, by kind permission of the Royal Society, London.]
Sir Isaac Newton died in 1727, at the age of eighty-five. His body lay in state in the Jerusalem Chamber, and was buried in Westminster Abbey.
 The writer inherited from his father (Professor J. D. Forbes) a small box containing a bit of wood and a slip of paper, which had been presented to him by Sir David Brewster. On the paper Sir David had written these words: “If there be any truth in the story that Newton was led to the theory of gravitation by the fall of an apple, this bit of wood is probably a piece of the apple tree from which Newton saw the apple fall. When I was on a pilgrimage to the house in which Newton was born, I cut it off an ancient apple tree growing in his garden.” When lecturing in Glasgow, about 1875, the writer showed it to his audience. The next morning, when removing his property from the lecture table, he found that his precious relic had been stolen. It would be interesting to know who has got it now!
 It must be noted that these words, in which the laws of gravitation are always summarised in histories and text-books, do not appear in the _Principia_; but, though they must have been composed by some early commentator, it does not appear that their origin has been traced. Nor does it appear that Newton ever extended the law beyond the Solar System, and probably his caution would have led him to avoid any statement of the kind until it should be proved.
With this exception the above statement of the law of universal gravitation contains nothing that is not to be found in the _Principia_; and the nearest approach to that statement occurs in the Seventh Proposition of Book III.:–
Prop.: That gravitation occurs in all bodies, and that it is proportional to the quantity of matter in each.
Cor. I.: The total attraction of gravitation on a planet arises, and is composed, out of the attraction on the separate parts.
Cor. II.: The attraction on separate equal particles of a body is reciprocally as the square of the distance from the particles.
 It is said that, when working out this final result, the probability of its confirming that part of his theory which he had reluctantly abandoned years before excited him so keenly that he was forced to hand over his calculations to a friend, to be completed by him.
8. NEWTON’S SUCCESSORS–HALLEY, EULER, LAGRANGE, LAPLACE, ETC.
Edmund Halley succeeded Flamsteed as Second Astronomer Royal in 1721. Although he did not contribute directly to the mathematical proofs of Newton’s theory, yet his name is closely associated with some of its greatest successes.
He was the first to detect the acceleration of the moon’s mean motion. Hipparchus, having compared his own observations with those of more ancient astronomers, supplied an accurate value of the moon’s mean motion in his time. Halley similarly deduced a value for modern times, and found it sensibly greater. He announced this in 1693, but it was not until 1749 that Dunthorne used modern lunar tables to compute a lunar eclipse observed in Babylon 721 B.C., another at Alexandria 201 B.C., a solar eclipse observed by Theon 360 A.D., and two later ones up to the tenth century. He found that to explain these eclipses Halley’s suggestion must be adopted, the acceleration being 10″ in one century. In 1757 Lalande again fixed it at 10.”
The Paris Academy, in 1770, offered their prize for an investigation to see if this could be explained by the theory of gravitation. Euler won the prize, but failed to explain the effect, and said: “It appears to be established by indisputable evidence that the secular inequality of the moon’s mean motion cannot be produced by the forces of gravitation.”
The same subject was again proposed for a prize which was shared by Lagrange  and Euler, neither finding a solution, while the latter asserted the existence of a resisting medium in space.
Again, in 1774, the Academy submitted the same subject, a third time, for the prize; and again Lagrange failed to detect a cause in gravitation.
Laplace  now took the matter in hand. He tried the effect of a non-instantaneous action of gravity, to no purpose. But in 1787 he gave the true explanation. The principal effect of the sun on the moon’s orbit is to diminish the earth’s influence, thus lengthening the period to a new value generally taken as constant. But Laplace’s calculations showed the new value to depend upon the excentricity of the earth’s orbit, which, according; to theory, has a periodical variation of enormous period, and has been continually diminishing for thousands of years. Thus the solar influence has been diminishing, and the moon’s mean motion increased. Laplace computed the amount at 10″ in one century, agreeing with observation. (Later on Adams showed that Laplace’s calculation was wrong, and that the value he found was too large; so, part of the acceleration is now attributed by some astronomers to a lengthening of the day by tidal friction.)
Another contribution by Halley to the verification of Newton’s law was made when he went to St. Helena to catalogue the southern stars. He measured the change in length of the second’s pendulum in different latitudes due to the changes in gravity foretold by Newton.
Furthermore, he discovered the long inequality of Jupiter and Saturn, whose period is 929 years. For an investigation of this also the Academy of Sciences offered their prize. This led Euler to write a valuable essay disclosing a new method of computing perturbations, called the instantaneous ellipse with variable elements. The method was much developed by Lagrange.
But again it was Laplace who solved the problem of the inequalities of