Photo, Mt. Wilson Solar Observatory
An Active Prominence of the Sun, 140,000 Miles High, photographed July 9, 1917.

ASTRONOMY

The Science of the Heavenly Bodies

BY

DAVID TODD

Director Emeritus, Amherst College Observatory

NEW YORK AND LONDON
HARPER & BROTHERS
PUBLISHERS MCMXXII

Copyright 1922
By P. F. Collier & Son Company
MANUFACTURED IN U. S. A.

PREFACE

Sir William Rowan Hamilton, the eminent mathematician of Dublin, has, of all writers ancient and modern, most fittingly characterized the ideal science of astronomy as man's golden chain connecting the heavens to the earth, by which we "learn the language and interpret the oracles of the universe."

The oldest of the sciences, astronomy is also the broadest in its relations to human knowledge and the interests of mankind. Many are the cognate sciences upon which the noble structure of astronomy has been erected: foremost of all, geometry and the higher mathematics, which tell us of motions, magnitudes and distances; physics and chemistry, of the origin, nature, and destinies of planets, sun, and star; meteorology, of the circulation of their atmospheres; geology, of the structure of the moon's surface; mineralogy, of the constitution of meteorites; while, if we attack, even elementally, the fascinating, though perhaps forever unsolvable, problem of life in other worlds, the astronomer must invoke all the resources that his fellow biologists and their many-sided science can afford him.

The progress of astronomy from age to age has been far from uniform—rather by leaps and bounds: from the earliest epoch when man's planet earth was the center about which the stupendous cosmos wheeled, for whom it was created, and for whose edification it was maintained—down to the modern age whose discoveries have ascertained that even our stellar universe, the vast region of the solar domain, is but one of the thousands of island universes that tenant the inconceivable immensities of space.

Such results have been attainable only through the successful construction and operation of monster telescopes that bring to the eye and visualize on photographic plates the faintest of celestial objects which were the despair of astronomers only a few years ago.

But the end is not yet; astronomy to-day is but passing from infancy to youth. And with new and greater telescopes, with new photographic processes of higher sensitivity, with the help of modern invention in overcoming the obstacle of the air—that constant foe of the astronomer—who will presume to set down any limit to the leaps and bounds of astronomy in the future?

So rapid, indeed, has been the progress of astronomy in very recent years that the present is especially favorable for setting forth its salient features; and this book is an attempt to present the wide range of astronomy in readable fashion, as if a story with a definite plot, from its origin with the shepherds of ancient Chaldea down to present-day ascertainment of the actual scale of the universe, and definite measures of the huge volume of supersolar giants among the stars.

David Todd

Amherst College Observatory
November, 1921

CONTENTS

LIST OF ILLUSTRATIONS

CHAPTER I
ASTRONOMY A LIVING SCIENCE

Like life itself we do not know when astronomy began; we cannot conceive a time when it was not. Man of the early stone age must have begun to observe sun, moon, and stars, because all the bodies of the cosmos were there, then as now. With his intellectual birth astronomy was born.

Onward through the childhood of the race he began to think on the things he observed, to make crude records of times and seasons; the Chaldeans and Chinese began each their own system of astronomy, the causes of things and the reasons underlying phenomena began to attract attention, and astronomy was cultivated not for its own sake, but because of its practical utility in supplying the data necessary to accurate astrological prediction. Belief in astrology was universal.

The earth set in the midst of the wonders of the sky was the reason for it all. Clearly the earth was created for humanity; so, too, the heavens were created for the edification of the race. All was subservient to man; naturally all was geocentric, or earth-centered. From the savage who could count only to five, the digits of one hand, civilized man very slowly began to evolve; he noted the progress of the seasons; the old records of eclipses showed Thales, an early Greek, how to predict their happenings, and true science had its birth when man acquired the power to make forecasts that always came true.

Few ancient philosophers were greater than Pythagoras, and his conceptions of the order of the heavens and the shape and motion of the earth were so near the truth that we sometimes wonder how they could have been rejected for twenty centuries. We must remember, however, that man had not yet learned the art of measuring things, and the world could not be brought into subjection to him until he had. To measure he must have tools—instruments; to have instruments he must learn the art of working in metals, and all this took time; it was a slow and in large part imperceptible process; it is not yet finished.

The earliest really sturdy manifestation of astronomical life came with the birth of Greek science, culminating with Aristarchus, Hipparchus and Ptolemy. The last of these great philosophers, realizing that only the art of writing prevents man's knowledge from perishing with him, set down all the astronomical knowledge of that day in one of the three greatest books on astronomy ever written, the Almagest, a name for it derived through the Arabic, and really meaning "the greatest."

The system of earth and heaven seemed as if finished, and the authority of Ptolemy and his Almagest were as Holy Writ for the unfortunate centuries that followed him. With fatal persistence the fundamental error of his system delayed the evolutionary life of the science through all that period.

But man had begun to measure. Geometry had been born and Eratosthenes had indeed measured the size of the earth. Tools in bronze and iron were fashioned closely after the models of tools of stone; astrolabes and armillary spheres were first built on geometric spheres and circles; and science was then laid away for the slumber of the Dark Ages.

Nevertheless, through all this dreary period the life of the youthful astronomical giant was maintained. Time went on, the heavens revolved; sun, moon, and stars kept their appointed places, and Arab and Moor and the savage monarchs of the East were there to observe and record, even if the world-mind was lying fallow, and no genius had been born to inspire anew that direction of human intellect on which the later growth of science and civilization depends. With the growth of the collective mind of mankind, from generation to generation, we note that ordered sequence of events which characterizes the development of astronomy from earliest peoples down to the age of Newton, Herschel, and the present. It is the unfolding of a story as if with a definite plot from the beginning.

Leaving to philosophical writers the great fundamental reason underlying the intellectual lethargy of the Dark Ages, we only note that astronomy and its development suffered with every other department of human activity that concerned the intellectual progress of the race. To knowledge of every sort the medieval spirit was hostile. But with the founding and growth of universities, a new era began. The time was ripe for Copernicus and a new system of the heavens. The discovery of the New World and the revival of learning through the universities added that stimulus and inspiration which marked the transition from the Middle Ages to our modern era, and the life of astronomy, long dormant, was quickened to an extraordinary development.

It fell to the lot of Copernicus to write the second great book on astronomy, "De Revolutionibus Orbium Cœlestium." But the new heliocentric or sun-centered system of Copernicus, while it was the true system bidding fair to replace the false, could not be firmly established except on the basis of accurate observation.

How fortunate was the occurrence of the new star of 1572, that turned the keen intellect of Tycho Brahe toward the heavens! Without the observational labors of Tycho's lifetime, what would the mathematical genius of Kepler have availed in discovery of his laws of motion of the planets?

Historians dwell on the destruction and violent conflicts of certain centuries of the Middle Ages, quite overlooking the constructive work in progress through the entire era. Much of this was of a nature absolutely essential to the new life that was to manifest itself in astronomy. The Arabs had made important improvements in mathematical processes, European artisans had made great advances in the manufacture of glass and in the tools for working in metals.

Then came Galileo with his telescope revealing anew the universe to mankind. It was the north of Italy where the Renaissance was most potent, recalling the vigorous life of ancient Greece. Copernicus had studied here; it was the home of Galileo. Columbus was a Genoese, and the compass which guided him to the Western World was a product of deft Italian artisans whose skill with that of their successors was now available to construct the instruments necessary for further progress in the accurate science of astronomical observation. Even before Copernicus, Johann Müller, better known as Regiomontanus, had imbibed the learning of the Greeks while studying in Italy, and founded an observatory and issued nautical almanacs from Nuremberg, the basis of those by which Columbus was guided over untraversed seas.

About this time, too, the art of printing was invented, and the interrelation of all the movements then in progress led up to a general awakening of the mind of man, and eventually an outburst in science and learning, which has continued to the present day. Naturally it put new life into astronomy, and led directly up from Galileo and his experimental philosophy to Newton and the Principia, the third in the trinity of great astronomical books of all time.

To get to the bottom of things, one must study intimately the history of the intellectual development of Europe through the fifteenth and sixteenth centuries. Many of the western countries were ruled by sovereigns of extraordinary vigor and force of character, and their activities tended strongly toward that firm basis on which the foundations of modern civilization were securely laid.

Contemporaneously with this era, and following on through the seventeenth century, came the measurements of the earth by French geodesists, the construction of greater and greater telescopes and the wonderful discoveries with them by Huygens, Cassini, and many others.

Most important of all was the application of telescopes to the instruments with which angles are measured. Then for the first time man had begun to find out that by accurate measures of the heavenly bodies, their places among the stars, their sizes and distances, he could attain to complete knowledge of them and so conquer the universe.

But he soon realized the insufficiency of the mathematical tools with which he worked—how unsuited they were to the solution of the problem of three bodies (sun, earth, and moon) under the Newtonian law of gravitation, let alone the problem of n-bodies, mutually attracting each the other; and every one perturbing the motion of every other one. So the invention of new mathematical tools was prosecuted by Newton and his rival Leibnitz, who, by the way, showed himself as great a man as mathematician: "taking mathematics," wrote Leibnitz, "from the beginning of the world to the times when Newton lived, what he had done was much the better half." Newton was the greatest of astronomers who, since the revival of learning, had observed the motions of the heavenly bodies and sought to find out why they moved.

Copernicus, Tycho Brahe, Galileo, Kepler, Newton, all are bound together as in a plot. Not one of them can be dissociated from the greatest of all discoveries. But Newton, the greatest of them all, revealed his greatness even more by saying: "If I have seen further than other men, it is because I have been standing on the shoulders of giants." Elsewhere he says: "All this was in the two plague years of 1665 and 1666 [he was then but twenty-four], for in those days I was in the prime of my age for invention, and minded mathematics and philosophy more than at any time since." All school children know these as the years of the plague and the fire; but very few, in school or out, connect these years with two other far-reaching events in the world's history, the invention of the infinitesimal calculus and the discovery of the law of gravitation.

We have passed over the name of Descartes, almost contemporary with Galileo, the founder of modern dynamics, but his initiation of one of the greatest improvements of mathematical method cannot be overlooked. This era was the beginning of the Golden Age of Mathematics that embraced the lives of the versatile Euler, equally at home in dynamics and optics and the lunar theory; of La Grange, author of the elegant "Mécanique Analytique"; and La Place, of the unparalleled "Mécanique Céleste." With them and a fully elaborated calculus Newton's universal law had been extended to all the motions of the cosmos. Even the tides and precession of the equinoxes and Bradley's nutation were accounted for and explained. Mathematical or gravitational astronomy had attained its pinnacle—it seemed to be a finished science: all who were to come after must be but followers.

The culmination of one great period, however, proved to be but the inception of another epoch in the development of the living science.

The greatest observer of all time, with a telescope built by his own hands, had discovered a great planet far beyond the then confines of the solar system. Mathematicians would take care of Uranus, and Herschel was left free to build bigger telescopes still, and study the construction of the stellar universe. Down to his day astronomy had dealt almost wholly with the positions and motions of the celestial bodies—astronomy was a science of where. To inquire what the heavenly bodies are, seemed to Herschel worthy of his keenest attention also. While "a knowledge of the construction of the heavens has always been the ultimate object of my observations," as he said, and his ingenious method of star-gauging was the first practicable attempt to investigate the construction of the sidereal universe, he nevertheless devoted much time to the description of nebulæ and their nature, as well as their distribution in space. He was the founder of double-star astronomy, and his researches on the light of the stars by the simple method of sequences were the inception of the vast fields of stellar photometry and variable stars. The physics of the sun, also, was by no means neglected; and his lifework earned for him the title of father of descriptive astronomy.

While progress and discovery in the earlier fields of astronomy were going on, the initial discoveries in the vast group of small planets were made at the beginning of the nineteenth century. The great Bessel added new life to the science by revolutionizing the methods and instruments of accurate observation, his work culminating in the measure of the distance of 61 Cygni, first of all the stars whose distance from the sun became known.

Wonderful as was this achievement, however, a greater marvel still was announced just before the middle of the century—a new planet far beyond Uranus, whose discovery was made as a direct result of mathematical researches by Adams and Le Verrier, and affording an extraordinary verification of the great Newtonian law. These were the days of great discoveries, and about this time the giant of all the astronomical tools of the century was erected by Lord Rosse, the "Leviathan" reflector with a speculum six feet in diameter, which remained for more than half a century the greatest telescope in the world, and whose epochal discovery of spiral nebulæ has greater significance than we yet know or perhaps even surmise.

The living science was now at the height of a vigorous development, when a revolutionary discovery was announced by Kirchhoff which had been hanging fire nearly half a century—the half century, too, which had witnessed the invention of photography, the steam engine, the railroad, and the telegraph: three simple laws by which the dark absorption lines of a spectrum are interpreted, and the physical and chemical constitution of sun and stars ascertained, no matter what their distance from us.

Huggins in England and Secchi in Italy were quick to apply the discovery to the stars, and Draper and Pickering by masterly organization have photographed and classified the spectra of many hundred thousand stars of both hemispheres, a research of the highest importance which has proved of unique service in studies of stellar movements and the structure of the universe by Eddington and Shapley, Campbell and Kapteyn, with many others who are still engaged in pushing our knowledge far beyond the former confines of the universe.

Few are the branches of astronomy that have not been modified by photography and the spectroscope. It has become a measuring tool of the first order of accuracy; measuring the speed of stars and nebulæ toward and from us; measuring the rotational speed of sun and planets, corona and Saturnian ring; measuring the distances of whole classes of stars from the solar system; measuring afresh even the distance of the sun—the yardstick of our immediate universe; measuring the drift of the sun with his entire family of planets twelve miles every second in the direction of Alpha Lyræ; and discovering and measuring the speed of binary suns too close together for our telescopes, and so making real the astronomy of the invisible.

Impatient of the handicap of a turbulent atmosphere, the living science has sought out mountain tops and there erected telescopes vastly greater than the "Leviathan" of a past century. There the sun in every detail of disk and spectrum is photographed by day, and stars with their spectra and the nebulæ by night. Great streams of stars are discovered and the speed and direction of their drift ascertained. The marvels of the spiral nebulæ are unfolded, their multitudinous forms portrayed and deciphered.

And their distances? And the distances of the still more wonderful clusters? Far, inconceivably far beyond the Milky Way. And are they "island universes"? And can man, the measurer, measure the distance of the "mainland" beyond?

CHAPTER II
THE FIRST ASTRONOMERS

Who were the first astronomers? And who wrote the first treatise on astronomy, oldest of the sciences?

Questions not easy to answer in our day. With the progress of archæological research, or inquiry into the civilization and monuments of early peoples, it becomes certain that man has lived on this planet earth for tens of thousands of years in the past as an intelligent, observing, intellectual being; and it is impossible to assign any time so remote that he did not observe and philosophize upon the firmament above.

We can hardly imagine a people so primitive that they would fail to regard the sun as "Lord of the Day," and therefore all important in the scheme of things terrestrial. Says Anne Bradstreet of the sun in her "Contemplations":

What glory's like to thee?

Soul of this world, this universe's eye,

No wonder some made thee deity.

To the Babylonians belongs the credit of the oldest known work on astronomy. It was written nearly six thousand years ago, about B. C. 3800, by their monarch Sargon the First, King of Agade. Only the merest fragments of this historic treatise have survived, and they indicate the reverence of the Babylonians for the sun. Another work by Sargon is entitled "Omens," which shows the intimate relationship of astronomy to mysticism and superstitious worship at this early date, and which persists even at the present day.

As remotely as B. C. 3000, the sun-god Shamash and his wife Aya are carved upon the historic cylinders of hematite and lapis lazuli, and one of the oldest designs on these cylinders represents the sun-god coming out of the Door of Sunrise, while a porter is opening the Gate of the East. The Semitic religion had as its basis a reverence for the bodies of the sky; and Samson, Hebrew for sun, was probably the sun-god of the Hebrews. The Phœnician deity, Baal, was a sun-god under differing designations; and at the epoch of the Shepherd Kings, about B. C. 1500, during the Hyksos dynasty, the sun-god was represented by a circle or disk with extended rays ending in hands, possibly the precursor of the frequently recurring Egyptian design of the winged disk or winged solar globe. Hittites, Persians, and Assyrians, as well as the Phœnicians, frequently represented the sun-god in similar fashion in their sacred glyphs or carvings.

For a long period in early human history, astronomy and astrology were pretty much the same. We can trace the history of astrology back as far as B. C. 3000 in ancient Babylonia. The motions of the sun, moon, and the five lucid planets of that time indicated the activity of the various gods who influenced human affairs. So the Babylonian priests devised an elaborate system of interpreting the phenomena of the heavens; and attaching the proper significance in human terms to everything that took place in the sky. In Babylonia and Assyria it was the king and his people for whom the prognostications were made out. It was the same in Egypt. Later, about the fifth century B. C., astrology spread through Greece, where astrologers developed the idea of the influence of planets upon individual concerns. Astrology persisted through the Dark Ages, and the great astronomers Copernicus, Tycho, Kepler, Gassendi, and Huygens were all astrologers as well. Milton makes many references to planetary influence, our language has many words with a direct origin in astrology, and in our great cities to-day are many astrologers who prepare individual horoscopes of more than ordinary interest.

It is difficult to assign the antiquity of the Chinese astronomy with any approach to definiteness. Their earliest records appear to have been total eclipses of the sun, going back nearly 2,200 years before the Christian era; and nearly a thousand years earlier the Hindu astronomy sets down a conjunction of all the planets, concerning which, however, there is doubt whether it was actually observed or merely calculated backward. Owing to a colossal misfortune, the burning of all native scientific books by order of the Emperor Tsin-Chi-Hwang-Ti, in B. C. 221, excepting only the volumes relating to agriculture, medicine, and astrology, the Chinese lost a precious mass of astronomical learning, accumulated through the ages. No less an authority than Wells Williams credits them with observing 600 solar eclipses between B. C. 2159 and A. D. 1223, and there must have been some centuries of eclipses observed and recorded anterior to B. C. 2159, as this is the date assigned to the eclipse which came unheralded by the astronomers royal, Hi and Ho, who had become intoxicated and forgot to warn the Court, in accord with their duty. China was thereby exposed to the anger of the gods, and Hi and Ho were executed by his Majesty's command. It is doubtful if there is an earlier record of any celestial phenomenon.

CHAPTER III
PYRAMID, TOMB, AND TEMPLE

Inquiry into the beginnings of astronomy in ancient Egypt reveals most interesting relations of the origins of the science to the life and work and worship of the people. Their astronomers were called the "mystery teachers of heaven"; their monuments indicate a civilization more or less advanced; and their temples were built on astronomical principles and dedicated to purpose of worship. The Egyptian records carry us back many thousands of years, and we find that in Egypt, as in other early civilizations, observation of the heavenly bodies may be embraced in three pretty distinct stages. Awe, fear, wonder and worship were the first. Then came utility: a calendar was necessary to tell men when "to plow and sow, to reap and mow," and a calendar necessitated astronomical observations of some sort. Following this, the third direction required observations of celestial positions and phenomena also, because astrology, in which the potentates of every ancient realm believed, could only thrive as it was based on astronomy.

Sun worship was preeminent in early Egypt as in India, where the primal antithesis between night and day struck terror in the unformed mind of man. In one of the Vedas occurs this significant song to the god of day: "Will the Sun rise again? Will our old friend the Dawn come back again? Will the power of Darkness be conquered by the God of Light?"

Quite different from India, however, is Egypt in matters of record: in India, records in papyrus, but no monuments of very great antiquity; in Egypt, no papyrus, but monuments of exceeding antiquity in abundance. Herodotus and Pliny have told us of the great antiquity of these monuments, even in their own day, and research by archæologist and astronomer has made it certain that the pyramids were built by a race possessing great knowledge of astronomy. Their temples, too, were constructed in strict relation to stars. Not only are the temples, as Edfu and Denderah, of exceeding interest in themselves, but associated with them are often huge monoliths of syenite, obelisks of many hundred tons in weight, which the astronomer recognizes as having served as observation pillars or gnomons. Specimens of these have wandered as far from home as Central Park and the bank of the Thames. But there is an even more remarkable wealth of temple inscriptions, zodiacs especially.

Next to the sun himself was the worship of the Dawn and Sunrise, the great revelations of nature. There were numerous hymns to the still more numerous sun-gods and the powers of sunlight. Ra was the sun-god in his noontide strength; Osiris, the dying sun of sunset. Only two gods were associated with the moon, and for the stars a special goddess, Sesheta. Sacrifices were made at day-break; and the stars that heralded the dawn were the subjects of careful observation by the sacrificial priests, who must therefore have possessed a good knowledge of star places and names, doubtless in belts of stars extending clear around the heavens. These decans, as they were called, are the exact counterparts of the moon stations devised by the Arabians, Indians, and other peoples for a like purpose.

The plane or circle of observation, both in Egypt and India, was always the horizon, whether the sun was observed or moon or stars. So the sun was often worshiped by the ancient Egyptians as the "Lord of the Two Horizons." It is sometimes difficult to keep in mind the fact, in regard to all temples of the ancients, whether in Egypt or elsewhere, that in studying them we must deal with the risings or settings of the heavenly bodies in quite different fashion from that of the astronomer of to-day, who is mainly concerned only with observing them on the meridian. The axis of the temple shows by its direction the place of rising or setting: if the temple faces directly east or west, its amplitude is 0. Now the sun, moon, and planets are, as everyone knows, very erratic as to their amplitudes (i. e., horizon points) of rising and setting; so it must have been the stars that engrossed the attention of the earliest builders of temples. After that, temples were directed to the rising sun, at the equinox or solstices. Then came the necessity of finding out about the inclination or obliquity of the ecliptic, and this is where the gnomon was employed.

At Karnak are many temples of the solstitial order: the wonderful temple of Amen-Ra is so oriented that its axis stands in amplitude 26 degrees north of west, which is the exact amplitude of the sun at Thebes at sunset of the summer solstice. The axis of a lesser temple adjacent points to 26 degrees south of east, which is the exact amplitude of sunrise at the winter solstice. At Gizeh we find the temples oriented, not solstitially, but by the equinoxes, that is, they face due east and west. Peoples who worshiped the sun at the solstice must have begun their year at the solstice; and Sir Norman Lockyer shows how the rise of the Nile, which took place at the summer solstice, dominated not only the industry but the astronomy and religion of Egypt.

Looking into the question of temple orientation in other countries, as China, for example, Lockyer finds that the most important temple of that country, the Temple of the Sun at Peking, is oriented to the winter solstice; and Stonehenge, as has long been known, is oriented to sunrise at the summer solstice.

In like fashion the rising and setting of many stars were utilized by the Egyptians, in both temple and pyramid; and no astronomer who has ever seen these ancient structures and studied their orientations can doubt that they were built by astronomers for use by astronomers of that day. The priests were the astronomers, and the temples had a deep religious significance, with a ceremony of exceeding magnificence wherever observations of heavenly bodies were undertaken, whether of sun or stars.

Hindu and Persian astronomy must be passed over very briefly. Interesting as their systems are historically, there were few, if any, original contributions of importance, and the Indian treatises bear strong evidence of Greek origin.

CHAPTER IV
ORIGIN OF GREEK ASTRONOMY

While the Greeks laid the foundations of modern scientific astronomy, they were not as a whole observers: rather philosophers, we should say. The later representatives of the Greek School, however, saw the necessity of observation as a basis of true induction; and they discovered that real progress was not possible unless their speculative ideas were sufficiently developed and made definite by the aid of geometry, so that they became capable of detailed comparison with observation. This was the necessary and ultimate test with them, and the same is true to-day. The early Greek philosophers were, however, mainly interested, not in observations, but in guessing the causes of phenomena.

Thales of Miletus, founder of the Ionian School, introduced the system of Egyptian astronomy into Greece, about the end of the seventh century B. C. He is universally known as the first astronomer who ever predicted a total eclipse of the sun that happened when he said it would: the eclipse of B. C. 585. This he did by means of the Chaldean eclipse cycle of 18 years known as the Saros.

Aristarchus of Samos was the first and most eminent of the Alexandrian astronomers, and his treatise "On the Magnitudes and Distances of the Sun and Moon" is still extant. This method of ascertaining how many times farther the sun is than the moon is very simple, and geometrically exact. Unfortunately it is impossible, even to-day, to observe with accuracy the precise time when the moon "quarters," (an observation essential to his method), because the moon's terminal, or line between day and night, is not a straight line as required by theory, but a jagged one. By his observation, the sun was only twenty times farther away than the moon, a distance which we know to be nearly twenty times too small.

His views regarding other astronomical questions were right, although they found little favor among contemporaries. Not only was the earth spherical, he said, but it rotated on its axis and also traveled round the sun. Aristarchus was, indeed, the true originator of the modern doctrine of motions in the solar system, and not Copernicus, seventeen centuries later; but Seleucus appears to have been his only follower in these very advanced conceptions. Aristarchus made out the apparent diameters of sun and moon as practically equal to one another, and inferred correctly that their real diameters are in proportion to their distances from the earth. Also he estimated, from observations during an eclipse of the moon, that the moon's diameter is about one-third that of the earth. Aristarchus appears to have been one of the clearest and most accurate thinkers among the ancient astronomers; even his views concerning the distances of the stars were in accord with the fact that they are immeasurably distant as compared with the distances of the sun, moon, and planets.

Practically contemporary with Aristarchus were Timocharis and Aristillus, who were excellent observers, and left records of position of sun and planets which were exceedingly useful to their successors, Hipparchus and Ptolemy in particular. Indeed their observations of star positions were such that, in a way, they deserve the fame of having made the first catalogue, rather than Hipparchus, to whom is universally accorded that honor.

Spherical astronomy had its origin with the Alexandrian school, many famous geometers, and in particular Euclid, pointing the way. Spherics, or the doctrine of the sphere, was the subject of numerous treatises, and the foundations were securely laid for that department of astronomical research which was absolutely essential to farther advance. The artisans of that day began to build rude mechanical adaptations of the geometric conceptions as concrete constructions in wood and metal, and it became the epoch of the origin of astrolabes and armillary spheres.

CHAPTER V
MEASURING THE EARTH—ERATOSTHENES

All told, the Greek philosophers were probably the keenest minds that ever inhabited the planet, and we cannot suppose them so stupid as to reject the doctrine of a spherical earth. In fact so certain were they that the earth's true figure is a sphere that Eratosthenes in the third century B. C. made the first measure of the dimensions of the terrestrial sphere by a method geometrically exact.

At Syene in Upper Egypt the sun at the summer solstice was known to pass through the zenith at noon, whereas at Alexandria Eratosthenes estimated its distance as seven degrees from the zenith at the same time. This difference being about one-fiftieth of the entire circumference of a meridian, Eratosthenes correctly inferred that the distance between Alexandria and Syene must be one-fiftieth of the earth's circumference. So he measured the distance between the two and found it 5,000 stadia. This figured out the size of the earth with a percentage of error surprisingly small when we consider the rough means with which Eratosthenes measured the sun's zenith distance and the distance between the two stations.

Greatest of all the Greek astronomers and one of the greatest in the history of the science was Hipparchus who had an observatory at Rhodes in the middle of the second century B. C. His activities covered every department of astronomy; he made extensive series of observations which he diligently compared with those handed down to him by the earlier astronomers, especially Aristillus and Timocharis. This enabled him to ascertain the motion of the equinoxial points, and his value of the constant of precession of the equinoxes is exceedingly accurate for a first determination.

In 134 B. C. a new star blazed out in the constellation Scorpio, and this set Hipparchus at work on a catalogue of the brighter stars of the firmament, a monumental work of true scientific conception, because it would enable the astronomers of future generations to ascertain what changes, if any, were taking place in the stellar universe. There were 1,080 stars in his catalogue, and he referred their positions to the ecliptic and the equinoxes. Also he originated the present system of stellar magnitudes or orders of brightness, and his catalogue was in use as a standard for many centuries.

Hipparchus was a great mathematician as well, and he devoted himself to the improvement of the method of applying numerical calculations to geometrical figures: trigonometry, both plane and spherical, that is; and by some authorities he is regarded as the inventor of original methods in trigonometry. The system of spheres of Eudoxus did not satisfy him, so he devised a method of representing the paths of the heavenly bodies by perfectly uniform motion in circles. There is slight evidence that Apollonius of Perga may have been the originator of the system, but it was reserved for Hipparchus to work it out in final form. This enabled him to ascertain the varying length of the seasons, and he fixed the true length of the year as 365¼ days. He had almost equal success in dealing with the irregularities of the moon's motion, although the problem is much more complicated. The distance and size of the moon, by the method of Aristarchus, were improved by him, and he worked out, for the distance of the sun, 1,200 radii of the earth—a classic for many centuries.

Hipparchus devoted much attention to eclipses of both sun and moon, and we owe to him the first elucidation of the subject of parallax, or the effect of difference of position of an observer on the earth's surface as affecting the apparent projection of the moon against the sun when a solar eclipse takes place; whereas an eclipse of the moon is unaffected by parallax and can be seen at the same time by observers everywhere, no matter what their location on the earth. Indeed, with all that Hipparchus achieved, we need not be surprised that astronomy was regarded as a finished science, and made practically no progress whatever for centuries after his time.

Then came Claudius Ptolemæus, generally known as Ptolemy, the last great name in Greek astronomy. He lived in Alexandria about the middle of the second century A. D. and wrote many minor astronomical and astrological treatises, also works on geography and optics, in the last of which the atmospheric refraction of rays of light from the heavenly bodies, apparently elevating them toward the zenith, is first dealt with in true form.

CHAPTER VI
PTOLEMY AND HIS GREAT BOOK

Ptolemy was an observer of the heavens, though not of the highest order; but he had all the work of his predecessors, best of all Hipparchus, to build upon. Ptolemy's greatest work was the "Megale Syntaxis," generally known as the Almagest. It forms a nearly complete compendium of the ancient astronomy, and although it embodies much error, because built on a wrong theory, the Almagest nevertheless is competent to follow the motions of all the bodies in the sky with a close approach to accuracy, even at the present day. This marvelous work written at this critical epoch became as authoritative as the philosophy of Aristotle, and for many centuries it was the last word in the science. The old astrology held full sway, and the Ptolemaic theory of the universe supplied everything necessary: further progress, indeed, was deemed impossible.

The Almagest comprises in all thirteen books, the first two of which deal with the simpler observations of the celestial sphere, its own motion and the apparent motions of sun, moon, and planets upon it. He discusses, too, the postulates of his system and exhibits great skill as an original geometer and mathematician. In the third book he takes up the length of the year, and in the fourth book similarly the moon and the length of the month. Here his mathematical powers are at their best, and he made a discovery of an inequality in the moon's motion known as the evection. Book five describes the construction and use of the astrolabe, a combination of graduated circles with which Ptolemy made most of his observations. In the sixth book he follows mainly Hipparchus in dealing with eclipses of sun and moon. In the seventh and eighth books he discusses the motion of the equinox, and embodies a catalogue of 1,028 stars, substantially as in Hipparchus. The five remaining books of the Almagest deal with the planetary motions, and are the most important of all of Ptolemy's original contributions to astronomy. Ptolemy's fundamental doctrines were that the heavens are spherical in form, all the heavenly motions being in circles. In his view, the earth too is spherical, and it is located at the center of the universe, being only a point, as it were, in comparison. All was founded on mere appearance combined with the philosophical notion that the circle being the only perfect curve, all motions of heavenly bodies must take place in earth-centered circles. For fourteen or fifteen centuries this false theory persisted, on the authority of Ptolemy and the Almagest, rendering progress toward the development of the true theory impossible.

Ptolemy correctly argued that the earth itself is a sphere that is curved from east to west, and from north to south as well, clinching his argument, as we do to-day, by the visibility of objects at sea, the lower portions of which are at first concealed from our view by the curved surface of the water which intervenes. To Ptolemy also the earth is at the center of the celestial sphere, and it has no motion of translation from that point; but his argument fails to prove this. Truth and error, indeed, are so deftly intermingled that one is led to wonder why the keen intelligence of this great philosopher permitted him to reject the simple doctrine of the earth's rotation on its axis. But if we reflect that there was then no science of natural philosophy or physics proper, and that the age was wholly undeveloped along the lines of practical mechanics, we shall see why the astronomers of Ptolemy's time and subsequent centuries were content to accept the doctrines of the heavens as formulated by him.

When it came to explaining the movements of the "wandering stars," or planets, as we term them, the Ptolemaic theory was very happy in so far as accuracy was concerned, but very unhappy when it had to account for the actual mechanics of the cosmos in space. Sun and moon were the only bodies that went steadily onward, easterly: whereas all the others, Mercury, Venus, Mars, Jupiter, Saturn, although they moved easterly most of the time, nevertheless would at intervals slow down to stationary points, where for a time they did not move at all, and then actually go backward to the west, or retrograde, then become stationary again, finally resuming their regular onward motion to the east.

To help out of this difficulty, the worst possible mechanical scheme was invented, that known as the epicycle. Each of the five planets was supposed to have a fictitious "double," which traveled eastward with uniformity, attached to the end of a huge but mechanically impossible bar. The earth-centered circle in which this traveled round was called the "deferent." What this bar was made of, what stresses it would be subjected to, or what its size would have to be in order to keep from breaking—none of these questions seems to have agitated the ancient and medieval astronomers, any more than the flat-earth astronomy of the Hindu is troubled by the necessity of something to hold up the tortoise that holds up the elephant that holds up the earth.

But at the end of this bar is jointed or swiveled another shorter bar, to the revolving end of which is attached the actual planet itself; and the second bar, by swinging once round the end of the primary advancing bar, would account for the backward or retrograde motion of the planet as seen in the sky. For every new irregularity that was found, in the motion of Mars, for instance, a new and additional bar was requisitioned, until interplanetary space was hopelessly filled with revolving bars, each producing one of the epicycles, some large, some small, that were needed to take up the vagaries of the several planets.

The Arabic astronomers who kept the science alive through the Middle Ages added epicycle to epicycle, until there was every justification for Milton's verses descriptive of the sphere:

With Centric and Eccentric scribbled o'er,

Cycle and Epicycle, Orb in Orb.

CHAPTER VII
ASTRONOMY OF THE MIDDLE AGES

With the fall of Alexandria and the victory of Mohammed throughout the West, and a consequent decline in learning, supremacy in science passed to the East and centered round the caliphs of Bagdad in the seventh and eighth centuries. They were interested in astronomy only as a practical, and to them useful, science, in adjusting the complicated lunar calendar of the Mohammedans, in ascertaining the true direction of Mecca which every Mohammedan must know, and in the revival of astrology, to which the Greeks had not attached any particular significance.

Harun al-Rashid ordered the Almagest and many other Greek works translated, of which the modern world would otherwise no doubt never have heard, as the Greek originals are not extant.

Splendid observatories were built at Damascus and Bagdad, and fine instruments patterned after Greek models were continuously used in observing. The Arab astronomers, although they had no clocks, were nevertheless so fully impressed with the importance of time that they added extreme value to their observations of eclipses, for example, by setting down the altitudes of sun or stars at the same time. On very important occasions the records were certified on oath by a body of barristers and astronomers conjointly—a precedent which fortunately has never been followed.

About the middle of the ninth century, the Caliph Al-Mamun directed his astronomers to revise the Greek measures of the earth's dimensions, and they had less reverence for the Almagest than existed in later centuries: indeed, Tabit ben Korra invented and applied to the tables of the Almagest a theoretical fluctuation in the position of the ecliptic which he called "trepidation," which brought sad confusion into astronomical tables for many succeeding centuries.

Albategnius was another Arab prince whose record in astronomy in the ninth and tenth centuries was perhaps the best: the Ptolemaic values of the precession of the equinoxes and of the obliquity of the ecliptic were improved by new observations, and his excellence as mathematician enabled him to make permanent improvements in the astronomical application of trigonometry.

Abul Wefa was the last of the Bagdad astronomers in the latter half of the tenth century, and his great treatise on astronomy known as the Almagest is sometimes confused with Ptolemy's work. Following him was Ibn Yunos of Cairo, whose labors culminated in the famous Hakemite Tables, which became the standard in mathematical and astronomical computations for several centuries.

Mohammedan astronomy thrived, too, in Spain and northern Africa. Arzachel of Toledo published the Toledan Tables, and his pupils made improvements in instruments and the methods of calculation. The Giralda was built by the Moors in Seville in 1196, the first astronomical observatory on the continent of Europe; but within the next half century both Seville and Cordova became Christian again, and Arab astronomy was at an end.

Through many centuries, however, the science had been kept alive, even if no great original advances had been achieved; and Arab activities have modified our language very materially, adding many such words as almanac, zenith, and radii, and a wealth of star names, as Aldebaran, Rigel, Betelgeuse, Vega, and so on.

Meanwhile, other schools of astronomy had developed in the East, one at Meraga near the modern Persia, where Nassir Eddin, the astronomer of Hulagu Khan, grandson of the Mongol emperor Genghis Khan, built and used large and carefully constructed instruments, translated all the Greek treatises on astronomy, and published a laborious work known as the Ilkhanic Tables, based on the Hakemite Tables of Ibn Yunos.

More important still was the Tartar school of astronomy under Ulugh Beg, a grandson of Tamerlane, who built an observatory at Samarcand in 1420, published new tables of the planets, and made with his excellent instruments the observations for a new catalogue of stars, the first since Hipparchus, the star places being recorded with great precision.

The European astronomy of the Middle Ages amounted to very little besides translation from the Arabic authors into Latin, with commentaries. Astronomers under the patronage of Alfonso X of Leon and Castile published in 1252 the Alfonsine Tables, which superseded the Toledan tables and were accepted everywhere throughout Europe. Alfonso published also the "Libros del Saber," perhaps the first of all astronomical cyclopedias, in which is said to occur the earliest diagram representing a planetary orbit as an ellipse: Mercury's supposed path round the earth as a center.

Purbach of Vienna about the middle of the 15th century began his "Epitome of Astronomy" based on the "Almagest" of Ptolemy, which was finished by his collaborator Regiomontanus, who was an expert in mathematics and published a treatise on trigonometry with the first table of sines calculated for every minute from 0° to 90°, a most helpful contribution to theoretical astronomy.

Regiomontanus had a very picturesque career, finally taking up his residence in Nuremberg, where a wealthy citizen named Walther became his patron, pupil, and collaborator. The artisans of the city were set at work on astronomical instruments of the greatest accuracy, and the comet of 1472 was the first to be observed and studied in true scientific fashion. Regiomontanus was very progressive and the invention of the new art of printing gave him an opportunity to publish Purbach's treatise, which went through several editions and doubtless had much to do in promoting dissatisfaction with the ancient Ptolemaic system, and was thus most significant in preparing a background for the coming of the new Copernican order.

The Nuremberg presses popularized astronomy in other important ways, issuing almanacs, the first precursors of our astronomical Ephemerides. Regiomontanus was practical as well, and invented a new method of getting a ship's position at sea, with tables so accurate that they superseded all others in the great voyages of discovery, and it is probable that they were employed by Columbus in his discovery of the American continent. Regiomontanus had died several years earlier, in 1475 at Rome, where he had gone by invitation of the Pope to effect a reformation in the calendar. He was only forty, and his patron Walther kept on with excellent observations, the first probably to be corrected for the effect of atmospheric refraction, although its influence had been known since Ptolemy. The Nuremberg School lasted for nearly two centuries.

Nearly contemporary with Regiomontanus were Fracastoro and Peter Apian, whose original observations on comets are worthy of mention because they first noticed that the tails of these bodies always point away from the sun. Leonardo da Vinci was the first to give the true explanation of earth-shine on the moon, and similarly the moon-illumination of the earth; and this no doubt had great weight in disposing of the popular notion of an essential difference of nature between the earth and celestial bodies—all of which helped to prepare the way for Copernicus and the great revolution in astronomical thought.

CHAPTER VIII
COPERNICUS AND THE NEW ERA

Throughout the Middle Ages the progress of astronomy was held back by a combination of untoward circumstances. A prolonged reaction from the heights attained by the Greek philosophers was to be expected. The uprising of the Mohammedan world, and the savage conquerors in the East did not produce conditions favorable to the origin and development of great ideas.

At the birth of Copernicus, however, in 1473, the time was ripening for fundamental changes from the ancient system, the error of which had helped to hold back the development of the science for centuries. The fifteenth century was most fruitful in a general quickening of intelligence, the invention of printing had much to do with this, as it spread a knowledge of the Greek writers, and led to conflict of authorities. Even Aristotle and Ptolemy were not entirely in harmony, yet each was held inviolate. It was the age of the Reformation, too, and near the end of the century the discovery of America exerted a powerful stimulus in the advance of thought.

Copernicus searched the works of the ancient writers and philosophers, and embodied in this new order such of their ideas as commended themselves in the elaboration of his own system.

Pythagoras alone and his philosophy looked in the true direction. Many believe that he taught that the sun, not the earth, is at the center of our solar system; but his views were mingled with the speculative philosophy of the Greeks, and none of his writings, barring a few meager fragments, have come down to our modern age.

To many philosophers, through all these long centuries, the true theory of the celestial motions must have been obvious, but their views were not formulated, nor have they been preserved in writing. So the fact remains that Copernicus alone first proved the truth of the system which is recognized to-day. This he did in his great treatise entitled "De Revolutionibus Orbium Cœlestium," the first printed copy of which was dramatically delivered to him on his deathbed, in May, 1543. The seventy years of his life were largely devoted to the preparation of this work, which necessitated many observations as well as intricate calculations based upon them. Being a canon in the church, he naturally hesitated about publishing his revolutionary views, his friend Rheticus first doing this for him in outline in 1540.

So simple are the great principles that they may be embodied in very few words; what appears to us as the daily revolution of the heavens is not a real motion, but only an apparent one; that is, the heavens are at rest, while the earth itself is in motion, turning round an axis which passes through its center. And the second proposition is that the earth is simply one of the six known planets; and they all revolve round the sun as the true center. The solar system, therefore, is "heliocentric," or sun-centered, not "geocentric" or earth-centered, as taught by the Ptolemaic theory.

Copernicus demonstrates clearly how his system explains the retrograde motion of the planets and their stationary points, no matter whether they are within the orbit of the earth, as Mercury and Venus, or outside of it, as Mars, Jupiter, and Saturn. His system provides also the means of ascertaining with accuracy the proportions of the solar system, or the relative distances of the planets from the sun and from each other. In this respect also his system possessed a vast advantage over that of Ptolemy, and the planetary distances which Copernicus computed are very close approximations to the measures of the present day.

Reinhold revised the calculations of Copernicus and prepared the "Tabulæ Prutenicæ," based on the "De Revolutionibus," which proved far superior to the Alfonsine Tables, and were only supplanted by the Rudolphine Tables of Kepler. On the whole we may regard the lifework of Copernicus as fundamentally the most significant in the history and progress of astronomy.

CHAPTER IX
TYCHO, THE GREAT OBSERVER

Clear as Copernicus had made the demonstration of the truth of his new system, it nevertheless failed of immediate and universal acceptance. The Ptolemaic system was too strongly intrenched, and the motions of all the bodies in the sky were too well represented by it. Accurate observations were greatly needed, and the Landgrave William IV. of Hesse built the Cassel Observatory, which made a new catalogue of stars, and introduced the use of clocks to carry on the time as measured by the uniform motion of the celestial sphere. Three years after the death of Copernicus, Tycho Brahe was born, and when he was 30 the King of Denmark built for him the famous observatory of Uraniborg, where the great astronomer passed nearly a quarter of a century in critically observing the positions of the stars and planets. Tycho was celebrated as a designer and constructor of new types of astronomical instruments, and he printed a large volume of these designs, which form the basis of many in use at the present day. Unfortunately for the genius of Tycho and the significance of his work, the invention of the telescope had not yet been made, so that his observations had not the modern degree of accuracy. Nevertheless, they were destined to play a most important part in the progress of astronomy.

Tycho was sadly in error in his rejection of the Copernican system, although his reasons, in his day, seemed unanswerable. If the outer planets were displaced among the stars by the annual motion of the earth round the sun, he argued, then the fixed stars must be similarly displaced—unless indeed they be at such vast distances that their motions would be too slight to be visible. Of course we know now that this is really true, and that no instruments that Tycho was able to build could possibly have detected the motions, the effects of which we now recognize in the case of the nearer fixed stars in their annual, or parallactic, orbits.

The remarkably accurate instruments devised by Tycho Brahe and employed by him in improving the observations of the positions of the heavenly bodies were no doubt built after descriptions of astrolabes such as Hipparchus used, as described by Ptolemy. In his "Astronomiæ Instauratæ Mechanica" we find illustrations and descriptions of many of them.

One is a polar astrolabe, mounted somewhat as a modern equatorial telescope is, and the meridian circle is adjustable so that it can be used in any place, no matter what its latitude might be. There is a graduated equatorial ring at right angles to the polar axis, so that the astrolabe could be used for making observations outside the meridian as well as on it. This equatorial circle slides through grooves, and is furnished with movable sights, and a plumb line from the zenith or highest point of the meridian circle makes it possible to give the necessary adjustment in the vertical. Screws for adjustment at the bottom are provided, just as in our modern instruments, and two observers were necessary, taking their sights simultaneously; unless, as in one type of the instrument, a clock, or some sort of measure of time, was employed.

Another early type of instrument is called by Tycho the ecliptic astrolabe (Armillæ Zodiacales, or the Zodiacal Rings). It resembles the equatorial astrolabe somewhat, but has a second ring inclined to the equatorial one at an angle equal to the obliquity of the ecliptic. In observing, the equatorial ring was revolved round till the ecliptic ring came into coincidence with the plane of the ecliptic in the sky. Then the observation of a star's longitude and latitude, as referred to the ecliptic plane, could be made, quite as well as that of right ascension and declination on the equatorial plane. But it was necessary to work quickly, as the adjustment on the ecliptic would soon disappear and have to be renewed.

Tycho is often called the father of the science of astronomical observation, because of the improvements in design and construction of the instruments he used. His largest instrument was a mural quadrant, a quarter-circle of copper, turning parallel to the north-and-south face of a wall, its axis turning on a bearing fixed in the wall. The radius of this quadrant was nine feet, and it was graduated or divided so as to read the very small angle of ten seconds of arc—an extraordinary degree of precision for his day.

Tycho built also a very large alt-azimuth quadrant, of six feet radius. Its operation was very much as if his mural quadrant could be swung round in azimuth. At several of the great observatories of the present day, as Greenwich and Washington, there are instruments of a similar type, but much more accurate, because the mechanical work in brass and steel is executed by tools that are essentially perfect, and besides this the power of the telescope is superadded to give absolute direction, or pointing on the object under observation.

Excellent clocks are necessary for precise observation with such an instrument; but neither Tycho Brahe, nor Hevelius was provided with such accessories. Hevelius did not avail himself of the telescope as an aid to precision of observation, claiming that pinhole sights gave him more accurate results. It was a dispute concerning this question that Halley was sent over from London to Danzig to arbitrate.

There could be but one way to decide; the telescope with its added power magnifies any displacement of the instrument, and thereby enables the observer to point his instrument more exactly. So he can detect smaller errors and differences of direction than he can without it. And what is of great importance in more modern astronomy, the telescope makes it possible to observe accurately the position of objects so faint that they are wholly invisible to the naked eye.

CHAPTER X
KEPLER, THE GREAT CALCULATOR

Most fortunate it was for the later development of astronomical theory that Tycho Brahe not only was a practical or observational astronomer of the highest order, but that he confined himself studiously for years to observations of the places of the planets. Of Mars he accumulated an especially long and accurate series, and among those who assisted him in his work was a young and brilliant pupil named Johann Kepler.

Strongly impressed with the truth of the Copernican System, Kepler was free to reject the erroneous compromise system devised by Tycho Brahe, and soon after Tycho's death Kepler addressed himself seriously to the great problem that no one had ever attempted to solve, viz: to find out what the laws of motion of the planets round the sun really are. Of course he took the fullest advantage of all that Ptolemy and Copernicus had done before him, and he had in addition the splendid observations of Tycho Brahe as a basis to work upon.

Copernicus, while he had effected the tremendous advance of substituting the sun for the earth as the center of motion, nevertheless clung to the erroneous notion of Ptolemy that all the bodies of the sky must perforce move at uniform speeds, and in circular curves, the circle being the only "perfect curve." Kepler was not long in finding out that this could not be so, and he found it out because Tycho Brahe's observations were much more accurate than any that Copernicus had employed.

Naturally he attempted the nearest planet first, and that was Mars—the planet that Tycho had assigned to him for research. How fortunate that the orbit of Mars was the one, of all the planets, to show practically the greatest divergence from the ancient conditions of uniform motion in a perfectly circular orbit! Had the orbit of Mars chanced to be as nearly circular as is that of Venus, Kepler might well have been driven to abandon his search for the true curve of planetary motion.

However, the facts of the cosmos were on his side, but the calculations essential in testing his various hypotheses were of the most tedious nature, because logarithms were not yet known in his day. His first discovery was that the orbit of Mars is certainly not a circle, but oval or elliptic in figure. And the sun, he soon found, could not be in the center of the ellipse, so he made a series of trial calculations with the sun located in one of the foci of the ellipse instead.

Then he found he could make his calculated places of Mars agree quite perfectly with Tycho Brahe's observed positions, if only he gave up the other ancient requisite of perfectly uniform motion. On doing this, it soon appeared that Mars, when in perihelion, or nearest the sun, always moved swiftest, while at its greatest distance from the sun, or aphelion, its orbital velocity was slowest.

Kepler did not busy himself to inquire why these revolutionary discoveries of his were as they were; he simply went on making enough trials on Mars, and then on the other planets in turn, to satisfy himself that all the planetary orbits are elliptical, not circular in form, and are so located in space that the center of the sun is at one of the two foci of each orbit. This is known as Kepler's first law of planetary motion.

The second one did not come quite so easy; it concerned the variable speed with which the planet moves at every point of the orbit. We must remember how handicapped he was in solving this problem: only the geometry of Euclid to work with, and none of the refinements of the higher mathematics of a later day. But he finally found a very simple relation which represented the velocity of the planet everywhere in its orbit. It was this: if we calculate the area swept, or passed over, by the planet's radius vector (that is, the line joining its center to the sun's center) during a week's time near perihelion, and then calculate the similar area for a week near aphelion, or indeed for a week when Mars is in any intermediate part of its orbit, we shall find that these areas are all equal to each other. So Kepler formulated his second great law of planetary motion very simply: the radius vector of any planet describes, or sweeps over, equal areas in equal times. And he found this was true for all the planets.

But the real genius of the great mathematician was shown in the discovery of his third law, which is more complex and even more significant than the other two—a law connecting the distances of the planets from the sun with their periods of revolution about the sun. This cost Kepler many additional years of close calculation, and the resulting law, his third law of planetary motion is this: The cubes of the mean or average distances of the planets from the sun are proportional to the squares of their times of revolution around him.

So Kepler had not only disposed of the sacred theories of motion of the planets held by the ancients as inviolable, but he had demonstrated the truth of a great law which bound all the bodies of the solar system together. So accurately and completely did these three laws account for all the motions, that the science of astronomy seemed as if finished; and no matter how far in the future a time might be assigned, Kepler's laws provided the means of calculating the planet's position for that epoch as accurately as it would be possible to observe it. Kepler paused here, and he died in 1630.

CHAPTER XI
GALILEO, THE GREAT EXPERIMENTER

The fifteenth and sixteenth centuries, containing the lives and work of Copernicus, Tycho, Galileo, Kepler, Huygens, Halley, and Newton, were a veritable Golden Age of astronomy. All these men were truly great and original investigators.

None had a career more picturesque and popular than did Galileo. Born a few years earlier and dying a few years later than Kepler, the work of each of these two great astronomers was wholly independent of the other and in entirely different fields. Kepler was discovering the laws of planetary motion, while Galileo was laying the secure foundations of the new science of dynamics, in particular the laws of falling bodies, that was necessary before Kepler's laws could be fully understood. When only eighteen Galileo's keen power of observation led to his discovery of the laws of pendulum motion, suggested by the oscillation to and fro of a lamp in the cathedral of Pisa.

The world-famous leaning tower of this place, where he was born, served as a physical laboratory from the top of which he dropped various objects, and thus was led to formulate the laws of falling bodies. He proved that Aristotle was all wrong in saying that a heavy body must fall swifter in proportion to its weight than a lighter one. These and other discoveries rendered him unpopular with his associates, who christened him the "Wrangler."

The new system of Copernicus appealed to him; and when he, first of all men, turned a telescope on the heavenly bodies, there was Venus with phases like those of the moon, and Jupiter with satellites traveling about it—a Copernican system in miniature. Nothing could have happened that would have provided a better demonstration of the truth of the new system and the falsity of the old. His marvelous discoveries caused the greatest excitement—consternation even, among the anti-Copernicans. Galileo published the "Sidereus Nuncius," with many observations and drawings of the moon, which he showed to be a body not wholly dissimilar to the earth: this, too, was obviously of great moment in corroboration of the Copernican order and in contradiction to the Ptolemaic, which maintained sharp lines of demarcation between things terrestrial and things celestial.

His telescopes, small as they were, revealed to him anomalous appearances on both sides of the planet Saturn which he called ansæ, or handles. But their subsequent disappearance was unaccountable to him, and later observers, who kept on guessing ineffectively till Huygens, nearly a half century after, showed that the true nature of the appendage was a ring. Spots on the sun were frequently observed by Galileo and led to bitter controversies. He proved, however, that they were objects on the sun itself, not outside it, and by noticing their repeated transits across the sun's disk, he showed that the sun turned round on his axis in a little less than a month—another analogy to the like motion of the earth on the Copernican plan.

Galileo's appointment in 1610 as "First Philosopher and Mathematician" to the Grand Duke of Tuscany gave him abundant time for the pursuit of original investigations and the preparation of books and pamphlets. His first visit to Rome the year following was the occasion of a reception with great honor by many cardinals and others of high rank. His lack of sympathy with others whose views differed from his, and his naturally controversial spirit, had begun to lead him headlong into controversies with the Jesuits and the church, which culminated in his censure by the authorities of the church and persecution by the Inquisition.

In 1618 three comets appeared, and Galileo was again in controversial hot water with the Jesuits. But it led to the publication five years later of "Il Saggiatore" (The Assayer), of no great scientific value, but only a brilliant bit of controversial literature dedicated to the newly elevated Pope, Urban VIII. Later he wrote through several years a great treatise, more or less controversial in character, entitled a "Dialogue on the Two Chief Systems of the World" between three speakers, and extending through four successive days. Simplicio argues for the Aristotelians, Salviati for the Copernicans, while Sagredo does his best to be neutral. It will always be a very readable book, and we are fortunate to have a recent translation by Professor Crew of Evanston.

Here we find the first suggestion of the modern method of getting stellar parallaxes, the relative parallax, that is, of two stars in the same field—a method not put into service till Bessel's time, two centuries later. But the most important chapters of the "Dialogue" deal with Galileo's investigations of the laws of motion of bodies in general, which he applied to the problem of the earth's motion. In this he really anticipated Newton in the first of his three laws of motion, and in a subsequent work, dealing with the theory of projectiles, he reaches substantially the results of Newton's second law of motion, although he gave no general statement of the principle. Nevertheless, in the epoch where his life was lived and his work done, his telescopic discoveries, combined with his dynamic researches in untrodden fields, resulted in the complete and final overthrow of the ancient system of error, and the secure establishment of the Copernican system beyond further question and discussion. Only then could the science of astronomy proceed unhampered to the fullest development by the master minds of succeeding centuries.

CHAPTER XII
AFTER THE GREAT MASTERS

Following Kepler and Galileo was a half century of great astronomical progress along many lines laid out by the work of the great masters. The telescope seemed only a toy, but its improvement in size and quality showed almost inconceivable possibilities of celestial discoveries.

Hevelius of Danzig took up the study of the moon, and his "Selenographia" was finely illustrated by plates which he not only drew but engraved himself. Lunar names of mountains, plains, and craters we owe very largely to him. Also he published among other works two on comets, the second of which was published in 1668 and called the "Cometographia," the first detailed account of all the comets observed and recorded to date.

Many were the telescopes turned on the planet Saturn, and every variety of guess was made as to the actual shape and physical nature of the weird appendages discovered by Galileo. The true solution was finally reached by Huygens, whose mechanical genius had enabled him to grind and polish larger and better lenses than his contemporaries; in 1659 he published the "Systema Saturnium" interpreting the ring and the cause of its various configurations, and the first discovery of a Saturnian satellite is due to him.

Gascoigne in England about 1640 was the first to make the important application of the micrometer to enhance the accuracy of measurement of small angles in the telescopic field; an invention made and applied independently many years later by Huygens in Holland and Auzout and Picard in France, where the instrument was first regularly employed as an accessory in the work of an observatory.

Another Englishman, Jeremiah Horrocks, was the first observer of a transit of Venus over the disk of the sun, in 1639. Horrocks was possessed of great ability in calculational astronomy also. This was about the time of the invention of the pendulum clock by Huygens, which in conjunction with the later invention of the transit instrument by Roemer wrought a revolution in the exacting art of practical astronomy. This was because it enabled the time to be carried along continuously, and the revolution of the earth could be utilized in making precise measures of the position of sun, moon, and stars. Louis XIV had just founded the new Observatory at Paris in 1668, and Picard was the first to establish regular time-observations there.

Huygens followed up the motion of the pendulum in theory as well as practice in his "Horologium Oscillatorium" (1673), showing the way to measure the force of gravity, and his study of circular motion showed the fundamental necessity of some force directed toward the center in planetary motions.

The doctrine of the sphericity of the earth being no longer in doubt, the great advance in accuracy of astronomical observation indicated to Willebrord Snell in Holland the best way to measure an arc of meridian by triangulation. Picard repeated the measurements near Paris with even greater accuracy, and his results were of the utmost significance to Newton in establishing his law of gravitation.

Domenico Cassini, an industrious observer, voluminous writer, and a strong personality, devised telescopes of great size, discovered four Saturnian satellites and the main division in the ring of Saturn, determined the rotation periods of Mars and Jupiter, and prepared tables of the eclipses of Jupiter's satellites. At his suggestion Richer undertook an expedition to Cayenne in latitude 5 degrees north, where it was found that the intensity of gravity was less than at Paris, and his clock therefore lost time, thus indicating that the earth was not a perfect sphere as had been thought, but a spheroid instead.

The planet Mars passed a near opposition, and Richer's observations of it from Cayenne, when combined with those of Cassini and others in France, gave a new value of the sun's parallax and distance, really the first actual measurement worth the name in the history of astronomy.

To close this era of signal advance in astronomy we may cite a discovery by Roemer of the first order: no less than that of the velocity of transmission of light through space. At the instigation of Picard, Roemer in studying the motions of Jupiter's satellites found that the intervals between eclipses grew less and less as Jupiter and the earth approached each other, and greater and greater than the average as the two planets separated farther and farther. Roemer correctly attributed this difference to the progressive motion of light and a rough value of its velocity was calculated, though not accepted by astronomers generally for more than a century.

Why the laws of Kepler should be true, Kepler himself was unable to say. Nor could anyone else in that day answer these questions: (1) The planets move in orbits that are elliptical not circular—why should they move in an imperfect curve, rather than the perfect one in which it had always been taught that they moved? (2) Why should our planet vary its velocity at all, and travel now fast, now slow; especially why should the speed so vary that the line of varying length, joining the planet to the sun, always passes over areas proportional to the time of describing them? And (3) Why should there be any definite relation of the distances of planets from the sun to their times of revolution about him? Why should it be exactly as the cube of one to the square of the other?

We must remember that the Copernican system itself was not yet, in the beginning of the seventeenth century, accepted universally; and the great minds of that period were most concerned in overturning the erroneous theory of Ptolemy.

The next step in logical order was to find a basic explanation of the planetary motions, and Descartes and his theory of vortices are worthy of mention, among many unsuccessful attempts in this direction. Descartes was a brilliant French philosopher and mathematician, but his hypothesis of a multitude of whirlpools in the ether, while ingenious in theory, was too vague and indefinite to account for the planetary motions with any approach to the precision with which the laws of Kepler represented them.

Another great astronomer whose labors helped immensely in preparing the way for the signal discoveries that were soon to come was Huygens, a man of versatility as natural philosopher, mechanician, and astronomical observer. Huygens was born thirteen years before the death of Galileo, and to the discovery of the laws of motion by the latter Huygens added researches on the laws of action of centrifugal forces. Neither of them, however, appeared to see the immediate bearing on the great general problem of celestial motions in its true light, and it was reserved for another generation, and an astronomer of another country, to make the one fundamental discovery that should explain the whole by a single simple law.

CHAPTER XIII
NEWTON AND MOTION

"How is it that you are able to make these great discoveries?" was once asked of Sir Isaac Newton, facile princeps of all philosophers, and the discoverer of the great law of universal gravitation.

"By perpetually thinking about them," was Newton's terse and illuminating reply. He had set for himself the definite problem of Kepler's laws: why is it that they are true, and is there not some single, general law that will embody all the circumstances of the planetary motions?

Newton was born in 1643, the year after the death of Galileo. He had a thorough training in the mathematics of his day, and addressed himself first to an investigation and definite formulation of the general laws of motion, which he found to be three in number, and which he was able to put in very simple terms. The first one is: Any body, once it is set in motion, will continue to move forward in a straight line with a uniform velocity forever, provided it is acted upon by no force whatever. In other words, a state of motion is as natural as a state of rest (rest in relation to things everywhere adjacent) in which we find all things in general.

Here on earth where gravity itself pulls all objects downward toward the earth, and where resistance of the air tends to hold a moving body back and bring it to rest, and where friction from contact with whatever material substance may be in its path is perpetually tending to neutralize all motion—with all three of these forces always at work to stop a moving body, the truth of this first and fundamental law of motion was not apparent on the surface.

Till Galileo's time everyone had made the mistake of supposing that some force or other must be acting continually on every moving body to keep it in motion. Ptolemy, Copernicus, Kepler, Leonardo da Vinci—all failed to see the truth of this law which Newton developed in the immortal Principia. And at the present day it is not always easy to accept at first, although the progress of mechanical science, by reducing friction and resistance, has produced machines in which motion of large masses may be kept up indefinitely with the application of only the merest minimum of force.

Once a planet is set in motion round the sun, it would go on forever through frictionless, non-resistant space; but there must be a central force, as Huygens saw clearly, to hold it in its orbit. Otherwise it would at any moment take the direction of a tangent to the orbit. Here is where Newton's second law of motion comes in, and he formulated it with great definiteness. When any force acts on a moving body, its deviation from a straight line will be in the direction of the force applied and proportional to that force.

In accord with this law, Newton first began to inquire whether the force of attraction here on earth, which everyone commonly recognizes as gravity, drawing all things down toward the center of the earth, might not extend upward indefinitely. It is found in operation on the summits of mountain peaks, and the clouds above them and the rain falling from them are obviously drawn downward by the same force. May it not extend outward into space, even as far as the moon?

This was an audacious question, but Newton not only asked, but tried to answer it in the year 1665, when he was only twenty-three. On the surface of the earth this attraction is strong enough to draw a falling body downward through a vertical space of sixteen feet in a second of time. What ought it to be at the distance of the moon. The distance of the moon in Newton's time was better known in terms of the earth's size than was the size of the earth itself: the earth's radius was known to be one-sixtieth of the moon's distance, but the earth's diameter was thought to be something under 7,000 miles, so that Newton's first calculations were most disappointing, and he laid them aside for nearly twenty years.

Meanwhile the French astronomers led by Picard had measured the earth anew, and showed it to be nearly 8,000 miles in diameter. As soon as Newton learned of this, he revised his calculations, and found that by the law of the inverse square the moon, in one second, should fall away from a tangent to its orbit one thirty-six hundredth of sixteen feet.

This accorded exactly with his original supposition that the earth's attraction extended to the moon. So he concluded that the force which makes a stone fall, or an apple, as the story goes, is the same force that holds the moon in its orbit, and that this force diminishes in the exact proportion that the square of the distance from the earth's center increases. The moon, indeed, becomes a falling body; only, as Kingdon Clifford puts it: "She is going so fast and is so far off that she falls quite around to the other side of the earth, instead of hitting it; and so goes on forever."

NICHOLAS COPERNICUS

GALILEO GALILEI

JOHANN KEPLER

SIR ISAAC NEWTON

Newton goes on in the Principia to explain the extension of gravitation to the other bodies of the solar system beyond the earth and moon. Clearly the same gravitation that holds the moon in its orbit round the earth, must extend outward from the sun also, and hold all the planets in their orbits centered about him. Newton demonstrates by calculation based on Kepler's third law that (1) the forces drawing the planets toward the sun are inversely as the squares of their mean distances from him; and (2) if the force be constantly directed toward the sun, the radius vector in an elliptic orbit must pass over equal areas in equal times.

CHAPTER XIV
NEWTON AND GRAVITATION

So all of Kepler's laws could be embodied in a single law of gravitation toward a central body, whose force of attraction decreases outward in exact proportion as the square of the distance increases.

Only one farther step had to be taken, and this the most complicated of all: he must make all the bodies of the sky conform to his third law of motion. This is: Action and reaction are equal, or the mutual actions of any two bodies are always equal and oppositely directed. There must be mutual attractions everywhere: earth for sun as well as sun for earth, moon for sun and sun for moon, earth for Venus and Venus for earth, Jupiter for Saturn and Saturn for Jupiter, and so on.

The motions of the planets in the undisturbed ellipses of Kepler must be impossible. As observations of the planets became more accurate, it was found that they really did fail to move in exact accord with Kepler's laws unmodified. Newton was unable, with the imperfect processes of the mathematics of his day to ascertain whether the deviations then known could be accounted for by his law of gravitation; but he nevertheless formulated the law with entire precision, as follows:

Every particle of matter in the universe attracts every other particle with a force exactly proportioned to the product of their masses, and inversely as the square of the distance between their centers.

The centuries of astronomical research since Newton's day, however, have verified the great law with the utmost exactness. Practically every irregularity of lunar and planetary motion is accounted for; indeed, the intricacies of the problems involved, and the nicety of their solution, have led to the invention of new mathematical processes adequate to the difficulties encountered.

And about the middle of the last century, when Uranus departed from the path laid out for it by the mathematical astronomers, its orbital deviations were made the basis of an investigation which soon led to the assignment of the position where a great planet could be found that would account for the unexplained irregularities of the motion of Uranus. And the immediate discovery of this planet, Neptune, became the most striking verification of the Newtonian law that the solar system could possibly afford.

The astronomers of still later days investigating the statelier motions of stellar systems find the Newtonian law regnant everywhere among the stars where our most powerful telescopes have as yet reached. So that Newton's law is known as the law of Universal Gravitation, and its author is everywhere held as the greatest scientist of the ages.

Newton's Principia may be regarded as the culminating research of the inductive method, and further outline of its contents is desirable. It is divided into three books following certain introductory sections. The first book treats of the problems of moving bodies, the solutions being worked out generally and not with special reference to astronomy. The second book deals with the motion of bodies through resistant media, as fluids, and has very little significance in astronomy. The third book is the all important one, and applies his general principles to the case of the actual solar system, providing a full explanation of the motions of all the bodies of the system known in his day. Anyone who critically reads the Principia of Newton will be forced to conclude that its author was a genius in the highest sense of the word. The elegance and thoroughness of the demonstrations, and the completeness of application of the law of gravitation are especially impressive.

The universality of his new law was the feature to which he gave particular attention. It was clear to him that the gravitation of a planet, although it acted as if wholly concentrated at the center, was nevertheless resident in every one of the particles of which the planet is composed. Indeed, his universal law was so formulated as to make every particle attract every other particle; and an investigation known as the Cavendish experiment—a research of great delicacy of manipulation—not only proves this, but leads also to a measurement of the earth's mean density, from which we can calculate approximately how much the earth actually weighs.

Another way to attack the same problem is by measuring the attraction of mountains, as Maskelyne, Astronomer Royal of Scotland did on Mount Schehallien in Scotland, which was selected because of its sheer isolation. The attraction of the mountain deflected the plumb-lines by measurable amounts, the volume of the mountain was carefully ascertained by surveys, and geologists found out what rocks composed it. So the weight of the entire mountain became pretty well known, and combining this with the observed deflection, an independent value of the earth's weight was found.

Still other methods have been applied to this question, and as an average it is found that the materials composing the earth are about five and a half times as heavy as water, and the total weight of the earth is something like six sextillions of tons.

What is the true shape of the earth? And does the earth's turning round on its axis affect this shape? Newton saw the answer to these questions in his law of gravitation. A spherical figure followed as a matter of course from the mutual attraction of all materials composing the earth, providing it was at rest, or did not turn round on its axis. But rotation bulges it at the equator and draws it in at the poles, by an amount which calculation shows to be in exact agreement with the amount ascertained by actual measurement of the earth itself.

Another curious effect, not at first apparent, was that all bodies carried from high latitudes toward the equator would get lighter and lighter, in consequence of the centrifugal force of rotation. This was unexpectedly demonstrated by Richer when the French Academy sent him south to observe Mars in 1672. His clock had been regulated exactly in Paris, and he soon found that it lost time when set up at Cayenne. The amount of loss was found by observation, and it was exactly equal to the calculated effect that the reduction of gravity by centrifugal action should produce.

Also Newton saw that his law of gravitation would afford an explanation of the rise and fall of the tides. The water on the side of the earth toward the moon, being nearer to the moon, would be more strongly attracted toward it, and therefore raised in a tide. And the water on the farther side of the earth away from the moon, being at a greater distance than the earth itself, the moon would attract the earth more strongly than this mass of water, tending therefore to draw the earth away from the water, and so raising at the same time a high tide on the side of the earth away from the moon. As the earth turns round on its axis, therefore, two tidal waves continually follow each other at intervals of about twelve hours.

The sun, too, joins its gravitating force with that of the moon, raising tides nearly half as high as those which the moon produces, because the sun's vaster mass makes up in large part for its much greater distance. At first and third quarters of the moon, the sun acts against the moon, and the difference of their tide-producing forces gives us "neap tides"; while at new moon and full, sun and moon act together, and produce the maximum effect known as "spring tides."

Newton passed on to explain, by the action of gravitation also, the precession of the equinoxes, a phenomenon of the sky discovered by Hipparchus, who pretty well ascertained its amount, although no reason for it had ever been assigned. The plane of the earth's equator extended to the celestial sphere marks out the celestial equator, and the two opposite points where it intersects the plane of the ecliptic, or the earth's path round the sun, are called the equinoctial points, or simply the equinoxes. And precession of the equinoxes is the motion of these points westward or backward, about 50 seconds each year, so that a complete revolution round the ecliptic would take place in about 26,000 years.

Newton saw clearly how to explain this: it is simply due to the attraction of the sun's gravitation upon the protuberant bulge around the earth's equator, acting in conjunction with the earth's rotation on its axis, the effect being very similar to that often seen in a spinning top, or in a gyroscope. The moon moving near the ecliptic produces a precessional effect, as also do the planets to a very slight degree; and the observed value of precession is the same as that calculated from gravitation, to a high degree of precision.

Newton died in 1727, too early to have witnessed that complete and triumphant verification of his law which ultimately has accounted for practically every inequality in the planetary motions caused by their mutual attractions. The problems involved are far beyond the complexity of those which the mathematical astronomer has to deal with, and the mathematicians of France deserve the highest credit for improving the processes of their science so that obstacles which appeared insuperable were one after another overcome.

Newton's method of dealing with these problems was mainly geometric, and the insufficiency of this method was apparent. Only when the French mathematicians began to apply the higher methods of algebra was progress toward the ultimate goal assured. D'Alembert and Clairaut for a time were foremost in these researches, but their places were soon taken by Lagrange, who wrote the "Mécanique Analytique," and Laplace, whose "Mécanique Céleste" is the most celebrated work of all. In large part these works are the basis of the researches of subsequent mathematical astronomers who, strictly speaking, cannot as yet be said to have arrived at a complete and rigorous solution of all the problems which the mutual attractions of all the bodies of the solar system have originated.

It may well be that even the mathematics of the present day are incompetent to this purpose. When the brilliant genius of Sir William Hamilton invented quaternion analysis and showed the marvelous facility with which it solved the intricate problems of physics, there was the expectation that its application to the higher problems of mathematical astronomy might effect still greater advances; but nothing in that direction has so far eventuated. Some astronomers look for the invention of new functions with numerical tables bearing perhaps somewhat the relation to present tables of logarithms, sines, tangents, and so on, that these tables do to the simple multiplication table of Pythagoras.

CHAPTER XV
AFTER NEWTON

We have said that practically all the motions in the solar system have been accounted for by the Newtonian law of gravitation. It will be of interest to inquire into the instances that lead to qualification of this absolute statement.

One relates to the planet Mercury, whose orbit or path round the sun is the most elliptical of all the planetary orbits. This will be explained a little later.

The moon has given the mathematical astronomers more trouble than any other of the celestial bodies, for one reason because it is nearest to us and very minute deviations in its motion are therefore detectible. Halley it was who ascertained two centuries ago that the moon's motion round the earth was not uniform, but subject to a slight acceleration which greatly puzzled Lagrange and Laplace, because they had proved exactly this sort of thing to be impossible, unless indeed the body in question should be acted on by some other force than gravitation. But Laplace finally traced the cause to the secular or very slow reduction in the eccentricity of the earth's own orbit. The sun's action on the moon was indeed progressively changing from century to century in such manner as to accelerate the moon's own motion in its orbit round the earth.

Adams, the eminent English astronomer, revised the calculations of Laplace, and found the effect in question only half as great as Laplace had done; and for years a great mathematical battle was on between the greatest of astronomical experts in this field of research. Adams, in conjunction with Delaunay, the greatest of the French mathematicians a half century ago, won the battle in so far as the mathematical calculations were concerned; but the moon continues to the present day her slight and perplexing deviation, as if perhaps our standard time-keeper, the earth, by its rotation round its axis, were itself subject to variation. Although many investigations have been made of the uniformity of the earth's rotation, no such irregularity has been detected, and this unexplained variation of the moon's motion is one of the unsolved problems of the gravitational astronomer of to-day.

But we are passing over the most impressive of all the earlier researches of Lagrange and Laplace, which concerned the exceedingly slow changes, technically called the secular variations of the elements of the planetary orbits. These elements are geometrical relations which indicate the form of the orbit, the size of the orbit, and its position in space; and it was found that none of these relations or quantities are constant in amount or direction, but that all, with but one exception, are subject to very slow, or secular, change, or oscillation.

This question assumed an alarming significance at an early day, particularly as it affected the eccentricity of the earth's orbit round the sun. Should it be possible for this element to go on increasing for indefinite ages, clearly the earth's orbit would become more and more elliptical, and the sun would come nearer and nearer at perihelion, and the earth would drift farther and farther from the sun at aphelion, until the extremes of temperature would bring all forms of life on the earth to an end. The refined and powerful analysis of Lagrange, however, soon allayed the fears of humanity by accounting for these slow progressive changes as merely part of the regular system of mere oscillations, in entire accord with the operation of the law of gravitation; and extending throughout the entire planetary system. Indeed, the periods of these oscillations were so vast that none of them were shorter than 50,000 years, while they ranged up to two million years in length—"great clocks of eternity which beat ages as ours beat seconds."

About a century ago, an eminent lecturer on astronomy told his audience that the problem of weighing the planets might readily be one that would seem wholly impossible to solve. To measure their sizes and distances might well be done, but actually to ascertain how many tons they weigh—never!

Yet if a planet is fortunate enough to have one satellite or more, the astronomer's method of weighing the planet is exceedingly simple; and all the major planets have satellites except the two interior ones, Mercury and Venus. As the satellite travels round its primary, just as the moon does round the earth, two elements of its orbit need to be ascertained, and only two. First, the mean distance of the satellite from its primary, and second the time of revolution round it.

Now it is simply a case of applying Kepler's third law. First take the cube of the satellite's distance and divide it by the square of the time of revolution. Similarly take the cube of the planet's distance from the sun and divide by the square of the planet's time of revolution round him. The proportion, then, of the first quotient to the second shows the relation of the mass (that is the weight) of the planet to that of the sun. In the case of Jupiter, we should find it to be 1,050, in that of Saturn 3,500, and so on.

The range of planetary masses, in fact, is very curious, and is doubtless of much significance in the cosmogony, with which we deal later. If we consider the sun and his eight planets, the mass or weight of each of the nine bodies far exceeds the combined mass of all the others which are lighter than itself.

To illustrate: suppose we take as our unit of weight the one-billionth part of the sun's weight; then the planets in the order of their masses will be Mercury, Mars, Venus, Earth, Uranus, Neptune, Saturn, and Jupiter. According to their relative masses, then, Mercury being a five-millionth part the weight of the sun will be represented by 200; similarly Venus, a four hundred and twenty-five thousandth part by 2,350, and so on. Then we have

Curious and interesting it is that Saturn is nearly three times as heavy as the six lighter planets taken together, Jupiter between two and three times heavier than all the other planets combined, while the sun's mass is 750 times that of all the great planets of his system rolled into one.

All the foregoing masses, except those of Mercury and Venus, are pretty accurately known because they were found by the satellite method just indicated. Mercury's mass is found by its disturbing effects on Encke's comet whenever it approaches very near. The mass of Venus is ascertained by the perturbations in the orbital motion of the earth. In such cases the Newtonian law of gravitation forms the basis of the intricate and tedious calculations necessary to find out the mass by this indirect method.

Its inferiority to the satellite method was strikingly shown at the Observatory in Washington soon after the satellites of Mars were discovered in 1877. The inaccurate mass of that planet, as previously known by months of computation based upon years and years of observation, was immediately discarded in favor of the new mass derived from the distance and period of the outer satellite by only a few minutes' calculation.

In weighing the planets, astronomers always use the sun as the unit. What then is the sun's own weight? Obviously the law of gravitation answers this question, if we compare the sun's attraction with the earth's at equal distances. First we conceive of the sun's mass as if all compressed into a globe the size of the earth, and calculate how far a body at the surface of this globe would fall in one second. The relation of this number to 16.1 feet, the distance a body falls in one second on the actual earth, is about 330,000, which is therefore the number of times the sun's weight exceeds that of the earth.

A word may be added regarding the force of gravitation and what it really is. As a matter of fact Newton did not concern himself in the least with this inquiry, and says so very definitely. What he did was to discover the law according to which gravitation acts everywhere throughout the solar system. And although many physicists have endeavored to find out what gravitation really is, its cause is not yet known. In some manner as yet mysterious it acts instantaneously over distances great and small alike, and no substance has been found which, if we interpose it between two bodies, has in any degree the effect of interrupting their gravitational tendency toward each other.

While the Newtonian law of gravitation has been accepted as true because it explained and accounted for all the motions of the heavenly bodies, even including such motions of the stars as have been subjected to observation, astronomers have for a long time recognized that quite possibly the law might not be absolutely exact in a mathematical sense, and that deviations from it would surely make their appearance in time.

A crude instance of this was suggested about a century ago, when the planet Uranus was found to be deviating from the path marked out for it by Bouvard's tables based on the Newtonian law; and the theory was advocated by many astronomers that this law, while operant at the medium distances from the sun where the planets within Jupiter and Saturn travel, could not be expected to hold absolutely true at the vast distance of Uranus and beyond. The discovery of Neptune in 1846, however, put an end to all such speculation, and has universally been regarded as an extraordinary verification of the law, as indeed it is.

When, however, Le Verrier investigated the orbit of Mercury he found an excess of motion in the perihelion point of the planet's orbit which neither he nor subsequent investigators have been able to account for by Newtonian gravitation, pure and simple. If Newton's theory is absolutely true, the excess motion of Mercury's perihelion remains a mystery.

Only one theory has been advanced to account for this discrepancy, and that is the Einstein theory of gravitation. This ingenious speculation was first propounded in comprehensive form nearly fifteen years ago, and its author has developed from it mathematical formulæ which appear to yield results even more precise than those based on the Newtonian theory.

In expressing the difference between the law of gravitation and his own conception, Einstein says: "Imagine the earth removed, and in its place suspended a box as big as a moon or a whole house and inside a man naturally floating in the center, there being no force whatever pulling him. Imagine, further, this box being, by a rope or other contrivance, suddenly jerked to one side, which is scientifically termed 'difform motion,' as opposed to 'uniform motion.' The person would then naturally reach bottom on the opposite side. The result would consequently be the same as if he obeyed Newton's law of gravitation, while, in fact, there is no gravitation exerted whatever, which proves that difform motion will in every case produce the same effects as gravitation…. The term relativity refers to time and space. According to Galileo and Newton, time and space were absolute entities, and the moving systems of the universe were dependent on this absolute time and space. On this conception was built the science of mechanics. The resulting formulas sufficed for all motions of a slow nature; it was found, however, that they would not conform to the rapid motions apparent in electrodynamics…. Briefly the theory of special relativity discards absolute time and space, and makes them in every instance relative to moving systems. By this theory all phenomena in electrodynamics, as well as mechanics, hitherto irreducible by the old formulæ, were satisfactorily explained."

Natural phenomena, then, involving gravitation and inertia, as in the planetary motions, and electro-magnetic phenomena, including the motion of light, are to be regarded as interrelated, and not independent of one another. And the Einstein theory would appear to have received a striking verification in both these fields. On this theory the Newtonian dynamics fails when the velocities concerned are a near approach to that of light. The Newtonian theory, then, is not to be considered as wrong, but in the light of a first approximation. Applying the new theory to the case of the motion of Mercury's perihelion, it is found to account for the excess quite exactly.

On the electro-magnetic side, including also the motion of light, a total eclipse of the sun affords an especially favorable occasion for applying the critical test, whether a huge mass like the sun would or would not deflect toward itself the rays of light from stars passing close to the edge of its disk, or limb. A total eclipse of exceptional duration occurred on May 29, 1919, and the two eclipse parties sent out by the Royal Society of London and the Royal Astronomical Society were equipped especially with apparatus for making this test. Their stations were one on the east coast of Brazil and the other on the west coast of Africa.

Accurate calculation beforehand showed just where the sun would be among the stars at the time of the eclipse; so that star plates of this region were taken in England before the expeditions went out. Then, during the total eclipse, the same regions were photographed with the eclipsed sun and the corona projected against them. To make doubly sure, the stars were a third time photographed some weeks after the eclipse, when the sun had moved away from that particular region.

Measuring up the three sets of plates, it was found that an appreciable deflection of the light of the stars nearest alongside the sun actually exists; and the amount of it is such as to afford a fair though not absolutely exact verification of the theory. The observed deflection may of course be due to other causes, but the English astronomers generally regard the near verification as a triumph for the Einstein theory. Astronomers are already beginning preparations for a repetition of the eclipse programme with all possible refinement of observation, when the next total eclipse of the sun occurs, September 20, 1922, visible in Australia and the islands of the Indian Ocean.

A third test of the theory is perhaps more critical than either of the others, and this necessitates a displacement of spectral lines in a gravitational field toward the red end of the spectrum; but the experts who have so far made measures for detecting such displacement disagree as to its actual existence. The work of St. John at Mt. Wilson is unfavorable to the theory, as is that of Evershed of Kodiakanal, who has made repeated tests on the spectrum of Venus, as well as in the cyanogen bands of the sun.

The enthusiastic advocates of the Einstein theory hold that, as Newton proved the three laws of Kepler to be special cases of his general law, so the "universal relativity theory" will enable eventually the Newtonian law to be deduced from the Einstein theory. "This is the way we go on in science, as in everything else," wrote Sir George Airy, Astronomer Royal; "we have to make out that something is true; then we find out under certain circumstances that it is not quite true; and then we have to consider and find out how the departure can be explained." Meanwhile, the prudent person keeps the open mind.

CHAPTER XVI
HALLEY AND HIS COMET

Halley is one of the most picturesque characters in all astronomical history. Next to Newton himself he was most intimately concerned in giving the Newtonian law to the world.

Edmund Halley was born (1656) in stirring times. Charles I. had just been executed, and it was the era of Cromwell's Lord Protectorate and the wars with Spain and Holland. Then followed (1660) the promising but profligate Charles II. (who nevertheless founded at Greenwich the greatest of all observatories when Halley was nineteen), the frightful ravages of the Black Plague, the tyrannies of James II., and the Revolution of 1688—all in the early manhood of Halley, whose scientific life and works marched with much of the vigor of the contending personalities of state.

The telescope had been invented a half century earlier, and Galileo's discoveries of Jupiter's moons and the phases of Venus had firmly established the sun-centered theory of Copernicus.

The sun's distance, though, was known but crudely; and why the stars seemed to have no yearly orbits of their own corresponding to that of the earth was a puzzle. Newton was well advanced toward his supreme discovery of the law of universal gravitation; and the authority of Kepler taught that comets travel helter-skelter through space in straight lines past the earth, a perpetual menace to humanity.

"Ugly monsters," that comets always were to the ancient world, the medieval church perpetuated this misconception so vigorously that even now these harmless, gauzy visitors from interstellar space possess a certain "wizard hold upon our imagination." This entertaining phase of the subject is excellently treated in President Andrew D. White's "History of the Doctrine of Comets," in the Papers of the American Historical Association. Halley's brilliant comet at its earlier apparitions had been no exception.

Halley's father was a wealthy London soap maker, who took great pride in the growing intellectuality of his son. Graduating at Queen's College, Oxford, the latter began his astronomical labors at twenty by publishing a work on planetary orbits; and the next year he voyaged to St. Helena to catalogue the stars of the southern firmament, to measure the force of terrestrial gravity, and observe a transit of Mercury over the disk of the sun.

While clouds seriously interfered with his observations on that lonely isle, what he saw of the transit led to his invention of "Halley's method," which, as applied to the transit of Venus, though not till long after his death, helped greatly in the accurate determination of the sun's distance from the earth. Halley's researches on the proper motions of the stars of both hemispheres soon made him famous, and it was said of him, "If any star gets displaced on the globe, Halley will presently find it out."

His return to London and election to the Royal Society (of which he was many years secretary) added much to his fame, and he was commissioned by the society to visit Danzig and arbitrate an astronomical controversy between Hooke and Hevelius, both his seniors by a generation.

On the continent he associated with other great astronomers, especially Cassini, who had already found three Saturnian moons; and it was then he observed the great comet of 1680, which led up to the most famous event of Halley's life.

The seerlike Seneca may almost be said to have predicted the advent of Halley, when he wrote ("Quaestiones Naturales," vii): "Some day there will arise a man who will demonstrate in what region of the heavens comets pursue their way; why they travel apart from the planets; and what their sizes and constitution are. Then posterity will be amazed that simple things of this sort were not explained before."

To Newton it appeared probable that cometary voyagers through space might have orbits of their own; and he proved that the comet of 1680 never swerved from such a path. As it could nowhere approach within the moon's orbit, clearly threats of its wrecking the earth and punishing its inhabitants ought to frighten no more.

Halley then became intensely interested in comets, and gathered whatever data concerning the paths of all these bodies he could find. His first great discovery was that the comets seen in 1531 by Apian, and in 1607 by Kepler, traveled round the sun in identical paths with one he had himself observed in 1682. A still earlier appearance of Halley's comet (1456) seems to have given rise to a popular and long-reiterated myth of a papal bull excommunicating "the Devil, the Turk, and the Comet."

No longer room for doubt: so certain was Halley that all three were one and the same comet, completing the round of its orbit in about seventy-six years, that he fearlessly predicted that it would be seen again in 1758 or 1759. And with equal confidence he might have foretold its return in 1835 and 1910; for all three predictions have come true to the letter.

Halley's span of existence did not permit his living to see even the first of these now historic verifications. But we in our day may emphatically term the epoch of the third verified return Annus Halleianus.

Says Turner, Halley's successor in the Savilian chair at Oxford to-day: "There can be no more complete or more sensational proof of a scientific law, than to predict events by means of it. Halley was deservedly the first to perform this great service for Newton's Law of Gravitation, and he would have rejoiced to think how conspicuous a part England was to play in the subsequent prediction of the existence of Neptune."

Halley rose rapidly among the chief astronomical figures of his day. But he had little veneration for mere authority, and the significant veering of his religious views toward heterodoxy was for years an obstacle to his advance.

Still Halley the astronomer was great enough to question any contemporary dicta that seemed to rest on authority alone. Everyone called the stars "fixed" stars; but Halley doubting this, made the first discovery of a star's individual motion—proper motion, as astronomers say. To-day, two hundred years after, every star is considered to be in motion, and astronomers are ascertaining their real motions in the celestial spaces to a nicety undreamed of by even the exacting Halley.

The moon, of priceless service to the early navigator, was regarded by all astronomers as endowed with an average rate of motion round the earth that did not vary from age to age. But Halley questioned this too; and on comparing with the ancient value from Chaldean eclipses, he made another discovery—the secular acceleration of the moon's mean motion, as it is technically termed. This was a colossal discovery in celestial dynamics; and the reason underlying it lay hidden in Newton's law for yet another century, till the keener mathematics of Laplace detected its true origin.