# Popular Science Monthly/Volume 64/February 1904/The Predecessors of Copernicus

(1904)
The Predecessors of Copernicus by Edward Singleton Holden

 THE PREDECESSORS OF COPERNICUS.

By EDWARD S. HOLDEN, Sc.D., LL.D.,

LIBRARIAN OF THE U. S. MILITARY ACADEMY.

THE records of the earliest Greek astronomy are very meager. Pythagoras, in the sixth century B. C., held that the heavenly bodies, the earth included, were spheres. Pythagoras is supposed to have known that lunar phases were caused by illumination from the sun; and the curved line separating the bright and dark parts of the moon throughout the month would naturally suggest that it was not a flat disc but a globe. He imagined all the stars to be fixed to a crystal sphere which daily turned round the earth and produced their rising and setting. Each of the seven planets (sun, moon, Mercury, Venus, Mars, Jupiter, Saturn) was attached to a sphere of its own, and their turning made harmonious sound—the music of the spheres. The distances of the several spheres were assigned in accordance with certain laws of music that Pythagoras had himself discovered. The idea of a spherical earth is thus some twenty-five hundred years old.

Philolaus, a Phythagorean of the fifth century B. C., maintained that the earth and all the planets (including the sun) revolved about a central fire. The idea of a moving earth was, therefore, not unfamiliar after his time and Copernicus quotes the Phythagoreans as authorities in the first chapters of his book De Revolutionibus Orbium Cælestium (1543). But the sun was not the central fire in their system, as it is in nature. "This world Pythagoras and his followers asserted to be one of the stars, and they also said that there was another opposite to it, similar to it; and they called that one Anticthona; and he said that both were in one sphere which revolved from east to west, and by this revolution the sun was circled round us; now he was seen, and now he was not seen. And he said that the fire was in the center of these, considering the fire to be a more noble body than the water and than the earth, and giving the noblest center" (Dante, Convito, iii., chap. v.). The Pythagoreans took the sun to be about three times the distance of the moon from the earth.

We know too little of the reasons that led Aristarchus of Samos, in the third century B. C., to hold that the sun was motionless at the center of the celestial sphere and that the earth revolved about him, rotating on her axis as she went. He taught also that the fixed stars are at rest, and measured the sun's apparent diameter, fixing it at half a degree. The little that remains of his writings gives the very highest idea of his originality and practical genius.

The views of Aristarchus on the system of the universe are reported by Archimedes. "The World," he says, "is by the greater part of astronomers called a sphere whose center is the center of the earth and whose radius is the distance from the earth to the sun. But Aristarchus of Samos, in quoting this opinion, refutes it. According to him, the world is very much greater; he supposes the sun to be immovable, as also are the stars, and he believes that the earth turns round the sun as a center, and that the magnitude of the sphere of the fixed stars, whose center is that of the sun, is such that the circumference of the circle described by the earth is in proportion to the distance of the fixed stars as the center of a circle is to its surface." Copernicus himself did not announce and describe his system with the magistral completeness and brevity of these few words. It is clear that we have here not only the view of Aristarchus, but also the opinion of Archimedes. We must assume that this announcement was unknown to Copernicus who reports the misty theories of the Pythagoreans, but makes no mention of Aristarchus in this connection.[1]

Plato (428–347 B. C.) taught that the earth was the center of celestial motions and that the planets and stars revolved about it on eight concentric spheres or circles. He plainly states that the moon shines by the sun's reflected light. Plato was not primarily an astronomer, and in fact held astronomy to be less dignified than the pure geometry that underlaid celestial motions, but his astronomical opinions were always of influence, especially in the orient where he held a high authority. He expressed and enforced the general idea that the heavenly bodies, being perfect in their essence, must necessarily revolve in circles, and with uniform, not variable, motion.

Eudoxus of Cnidus (409–356 B. C.) elaborated the ideas of Plato into a scientific system. By this time the simpler motions of the moon were well known, and to account for them he found three spheres to be necessary. One produced its daily motion of rising and setting, another its monthly motion from west to east, while the third had to do with its motions north and south of the ecliptic. The sun was likewise provided with three spheres, and each of the planets had four (since the planets sometimes appear to 'retrograde' from east to west, though their usual progress is from west to east). The system of Eudoxus thus required twenty-seven spheres; one for the fixed stars, twenty for the planets, six for the sun and moon. It is not probable that Eudoxus and his school regarded these spheres as material crystal shells, but rather as geometrical and abstract vehicles for the resolution of observed mechanical movements into intelligible parts. But the notion of material crystal spheres perpetually recurs in Greek astronomy after his day, and was universally held by the vulgar.

Consider, for an instant, what is involved in the theory of revolving material crystal shells. The stars are at an immense distance, all fixed to a crystal surface, which revolves once in twenty-four hours. The sun is situated on the surface of another shell, but it can not be in one fixed spot on the surface, for we see it rise and set at different points of our horizon at different times of the year. What kind of a crystal shell is it upon which the sun can glide so far and no farther? No wonder that certain medieval writers felt the necessity of imagining two shells for each luminary between which the motion took place with freedom, beyond which there was no passage. What sort of shells are those that correspond to the planets, each of which moves at various rates in varied directions—sometimes eastward, sometimes westward, sometimes north, sometimes south? The details of a scheme like this are literally unthinkable. It must be accepted, if at all, by faith—by a faith founded in phrases.

The ancient astronomers did not, in general, seek knowledge for its own sake. They were either concerned about some practical matter, as the length of the year, the prediction of the seasons and the like; or else sought acquaintance with some aspect of divine or partly divine matter, such as formed the planets and the stars. The science of the middle ages has been summarized in a sentence: 'It was all divination, clairvoyance, unsubjected to our modern exact formulas, seeking in an instant of vision to concentrate a thousand experiences' (Pater). A few of the ancients, Archimedes and Aristarchus, for example, had what we call the modern spirit. Roger Bacon was the first to formulate it. Newton may be taken as its first thorough-going representative, for even Kepler and Galileo were deeply tinged at times with the medieval color.

The Meteorologica and the De Cœlo of Aristotle (384–322 B. C.) were the text-books of the middle ages. The doctrine of material spheres was frankly adopted in these books and in the writings derived from them. The geometric scheme of Eudoxus was transformed into a clumsy mechanism, and its complexity was further increased by the addition of other spheres, so that fifty-six in all were necessary to explain celestial motions. "The glorious philosopher, to whom nature opened her secrets most freely, proved in the second chapter of his De Cœlo, that this world, the earth, is of itself stable and fixed to all eternity. . . . Let it be enough to know, upon his great authority, that this earth is fixed and does not revolve, and that it, with the sea, is the center of the heavens. These heavens revolve round this center continuously even as we see" (Dante, Convito, iii., chap. v.). Until we remember that mechanics was an unknown science to the ancients and in the middle ages, it is almost impossible to conceive how professors could teach, or students accept, a system like Aristotle's that was, in essence, unintelligible. While Cremonini was expounding the De Cœlo in one lecture-room at the University of Padua in 1592, Galileo was teaching the Euclid's Elements in another. It is easier to comprehend how students flocked to listen when a few years later Galileo began his lectures upon astronomy, although by the conditions of his professorship he was only permitted to expound the astronomy of Sacro Bosco.

Aristotle taught that the earth was spherical and gave reasons, good and bad, for his belief. The distance of the sun was fixed by a most ingenious method invented by Aristarchus of Samos (270 B. C.) who concluded that the sun was about 19 times more distant than the moon (it is, in fact, 390 times more distant). Hipparchus determined the moon's distance for himself[2] and took the sun to be 19 times more distant. He did not leave the earth in the central point of the sun's orbit, but shifted that center towards the sixth degree of Gemini by one twenty-fourth of the radius so as to account for observed inequalities in the annual motion. Ptolemy adopted this result without question, and it was accepted by astronomers for twelve centuries. It was not until the time of Kepler that it was proved that the sun must be at least fifty times as far away as the moon. This was one of the consequences of Tycho's accurate observations.

The Chaldeans and Egyptians held the earth to be a flat disc canopied by the sky—the firmament—and this was the view of the Hebrews. A distinctly Christian theory of the figure of the earth and heavens, drawn from scripture, was formulated by the Egyptian monk and traveler Cosmas Indicopleustes. According to this theory, the earth was a flat parallelogram surrounded by the four seas. "We say, therefore, with Isaiah, that the heaven embracing the universe is a vault; with Job, that it is joined to the earth; and with Moses, that the length of the earth is greater than its breadth." This explanation of appearances was very generally accepted as orthodox, and was held by the common people long after the learned had been convinced of the earth's sphericity by the arguments of Ptolemy and Aristotle. Isidore of Seville in the seventh century, and the Venerable Bede in the eighth, declared for the opinion of Aristotle; Dante in the thirteenth century supported it, and Columbus proved it in the fifteenth. In the sixteenth, Magellan's voyage of circumnavigation settled the vexed question once and for all.

There is in the library of the University of Cambridge, so Dr. Whewell reports, a French poem of the time of Edward the Second (1307–27) illustrated with drawings that show men standing upright on all parts of a spherical earth. By way of illustrating the tendency of heavy bodies towards the earth's center other men are dropping balls into holes bored entirely through the globe and these balls are falling to the earth's midmost point—

That point to which from every part is dragged all heavy substance,

as Virgil explains to Dante in the thirty-fourth canto of the Inferno. The cosmogony of Dante in the Divina Commedia was accepted for centuries by roman Catholics, as Milton's in the Paradise Lost has been adopted by protestants. For Dante the globe of the earth was the center of the world. It was surrounded by nine transparent spheres moved by angels. There was a crystal sphere for the moon, and others for Mercury, Venus, the sun, Mars, Jupiter, Saturn and the fixed stars, and beyond them the Primum Mobile—nine in all. Beyond the outer sphere was the Empyrean—here God sate. Below the earth is hell and here its god—Lucifer—reigned over bad angels. All the discord in the world came from them, even its storms, hail and lightning. The spheres of Eudoxus served as a base to Dante's system, which was adapted, with a poet's license, to a poet's use.[3]

In the De Cœlo, Aristotle lays down certain fundamental principles: The things of which the world is made are all solid bodies, and all have, therefore, three dimensions. The simple elements of nature must also have simple motions. So, indeed, fire and air have their natural motions upwards, water and earth, downwards, both in straight lines. But besides these motions there is also a circular motion, not natural to these elements, although it is a much more complete motion than the rectilinear. For the circle is, in itself, a complete line, which a straight line is not: There must, therefore, be certain things to which complete circular motion is natural: It follows that there must be a certain sort of bodies very different from the four elementary bodies, bodies that are more godlike, that must therefore stand above them: This finer essence was later named by the commentators 'Quinta Essentia'—our quintessence. The heavenly bodies are formed of this; they are spheres endowed with life and activity.

The question of the revolution of the earth in an orbit round the sun is discussed by Aristotle, and he rejects the idea for the reason that such a motion would necessarily produce a corresponding alteration in the place of each and every fixed star. The objection was perfectly valid. If the stars were only a little farther from us than Saturn, as Aristotle believed, a motion of the earth in an orbit would cause each star to move in an apparent parallactic orbit, a miniature copy of that of the earth. No such alteration of place was observable. Hence, said he, the earth did not move. Even the nearest stars are, we now know, twenty thousand times as far from us as Saturn, and it is this fact—which was not finally established till 1837—that explains why the miniature apparent orbits of the stars were not seen by the Greeks or by their successors. They were too minute to be discoverable.

All the important writings of Hipparchus, who lived in the second century B. C., are lost, and the doctrines of this 'most truth-loving and labor-loving man' are known to us only through Ptolemy, his expositor and ardent admirer. Hipparchus was an indefatigable observer, a mathematician of tact and insight, an astronomer of original and profound genius. By his own observations, made at Rhodes (188–127 B. C.), he fixed the positions (the celestial longitudes and latitudes) of the principal fixed stars. Comparing their present places with their past positions as determined by Timocharis and Aristillus, he discovered that backward motion of the equinoctial points which causes the epoch of the sun's passage through the equinox to recur earlier and earlier each year—the precession of the equinoxes—and fixed its probable annual amount. Comparing his own determinations of the date of the vernal equinoctial passage of the sun with those of Aristarchus, he determined the length of the year with accuracy.[4] It is by systematic comparisons of the sort that many of his discoveries were made.

It is very noteworthy that he gives not only his results, but likewise an estimate of their probable errors. His observations of the time of the sun's arriving at a solstice might be erroneous, he says, by about three fourths of a day; at an equinox by about one fourth. Comparisons made in this systematic fashion, and estimates of error of this sort, we are apt to think of as 'modern.' Certainly they are not characteristic of observational astronomy till the eighteenth century, two thousand years after Hipparchus showed the way. Some of his most important researches related to measures of time. What was the length of the year? Were all years of the same length? His observations showed him no difference between one year and another. It is interesting to note how he formulates his conclusions. He does not say that all years are, without doubt, of one and the same length; he asserts simply that the differences, if any, must be very small, so small that his observations are not delicate enough to detect them.

In the year 134 B. C. a new star suddenly appeared in Scorpio, and Hipparchus began the formation of a catalogue of stars visible to him. With such a conspectus of the present state of the sky no new appearances could subsequently occur without detection. His catalogue gave the position and magnitude (brightness) of 1,080 stars for the epoch 128 B. C., and arranged them in the constellation figures that have come down to us only slightly changed.[5] Hipparchus' catalogue stood unique for a thousand years.

An instance of his practical tact as an observer may be quoted. If a straight ruler be held up against the starry sky there will, now and again, be instances where its edge passes through three stars at the same time. Many such cases are recorded by Hipparchus. No one of the three stars can change its situation without detection. A simple observation of the same sort at any subsequent time will at once exhibit any change that may have taken place in the interval between the two observations.

The work of Hipparchus as a theoretical astronomer is as remarkable as his observing skill. The positions of the heavenly bodies are calculated by solving triangles, both plane and spherical. The doctrine of such solutions—trigonometry—was perhaps invented by him; at all events it was greatly developed and improved. Observations give the celestial longitudes and latitudes of planets at the instant of observation. Their positions at past epochs, a month or a year ago, are given by preceding observations of the same sort. Where will Jupiter or Saturn be found in the future—a month or a year hence? It is necessary to invent a geometry of planetary motion that will account for all past and future motions; and this problem was elaborately developed by Hipparchus. We must recollect that his vast activity was exercised under conditions of the most discouraging kind. His best instruments were but rude; all sightings were made with the eye unaided by telescopes; he had only clepsydras (sand or water-clocks) to measure intervals of time; the Greek system of arithmetic in which his calculations were made was cumbrous in the extreme. What he accomplished is little less than astounding.

From his theory of Epicycles Hipparchus was able to construct his tables of the sun and moon. The tables gave the particulars of the motion of these bodies and enabled predictions to be made of coming solar and lunar eclipses. It was sufficient for the purposes of the time to assert that an eclipse would occur on a certain day, about a certain hour of the morning or afternoon, and the tables were adequate to such predictions. His theory was sufficient; it fulfilled all the tests applied to it. The motion of the moon was more complex than that of the sun, but it, too, was reduced to a sufficient order and important discoveries made. The elements of the motions of these bodies were not derived, as we to-day derive them, from continuous observations, but rather from observations made at certain critical times. For the sun the observations were made at the equinoxes and solstices. Six eclipses of the moon sufficed to give him the elements of the lunar orbit and the rate at which they were changing.

His theory of the planets was not so complete, for there was no sufficient body of ancient observations to be compared with those which he himself accumulated with so much diligence. Considering the data at his disposition and the use made of them, the work of Hipparchus is of the first order. Astronomers of all ages are agreed that he was 'one of the most extraordinary men of antiquity; the very greatest in the sciences that require a combination of observation with geometry' (Delambre).

His expositor, Ptolemy of Alexandria, was primarily a geometer and made few original observations. The Almagest is, in essence, a restatement of the theories of Hipparchus with additions, not all of which are improvements. It begins by laying down certain postulates: The earth is spherical and a mere point in respect of the heavens; its circumference is 180,000 stadia; the heavens are likewise spherical and revolve about the earth, which is in the center and has no motion. So far he is in agreement with Aristotle. Where he differs, astronomers who succeeded him followed the Almagest while philosophers were more apt to take Aristotle as authority.

Ptolemy's theory of the moon's motion led him to important discoveries, which need not be described here. It is mentioned because it also contained a contradiction of the precise sort that is best suited to lead to further discoveries, and because this contradiction was passed over and entirely neglected by him and by his successors for centuries. His theory gave the position of the moon with satisfactory accuracy. It was, in so far, presumably true. It assumed that at times the moon was twice as far from the earth as at others. If this were true the moon's apparent diameter should sometimes have been twice as great as at other times. But no such variation was observed. The necessary conclusion: Hence the theory can not possibly be true—was not drawn by Ptolemy. The instance is significant; it marks a radical difference between the modern attitude and that of the ancients in matters of physical science. Ptolemy and his successors really held two antagonistic theories of the moon's motion and distance at the same time. Each theory satisfied the conditions of part of the problem. They did not seek for a unique theory. This was not done until the time of Kepler, whose whole life was spent in searching for the physical causes of observed phenomena, and who was not content with mere analytic devices by which the phenomena could be predicted. He sought for these, but he looked deeper and further.

All but a few of the greatest of the ancients regarded a physical problem in the light of a riddle to which an answer was required. Any plausible answer would do. The fixed belief that there was one answer and could be only one did not arise till quite modern times. Modern science is a search for such unique solutions. Most of ancient science was a search for an hypothesis to account for a set of observed facts.

A page from Ptolemy's note-book may be transcribed. He was seeking the position of the bright star Regulus: "In the second year of Antoninus, the ninth day of Pharmauthi, the sun being near setting, the last division of Taurus being on the meridian (that is 5½ equinoctial hours after noon) the moon was in three degrees of Pisces by her distance from the sun (which was 92° 8'); and half an hour after, the sun having set, and the quarter of Gemini on the meridian, Regulus appeared, by the other circle of the astrolabe, 57½ degrees to the eastward of the moon in longitude." The position of the sun was known from the day of the year by the solar tables; the moon, at 5½ hours, was 92° 8' east of the sun; the moon's motion in half an hour was also known from the tables, and hence her position at 6 hours was determined; Regulus was at that time 57½ degrees east of the moon, and its place was thus fixed with respect to the sun. A modern note-book would give the year and day, and would record that Regulus crossed the meridian at a certain hour, minute, second and decimal of a second by the clock. The correction of the clock would be given as so many seconds and hundredths of a second. The sum of the clock-time and the correction is the position of the star. In Ptolemy's case it was known to half a degree (two minutes of time). A modern observation gives it with an error not above one tenth of a second; that is with an accuracy about 1,200 times greater.

Ptolemy's Almagest is, in essence, 'modern' in respect of the fact that its theories are designed to give quantitative results and are presented as general bases for special calculations. With a certain set of observations as data the desired results could be worked out in numbers. Tables for calculating future positions of the planets were also given and in Ptolemy's time the actual positions were fairly well represented by the predictions. As time went on, more accurate observations with better instruments, were made. The observed places of the planets did not agree with the predictions. The ingenuity of his disciples in the middle ages was taxed to improve the theory, and the tables of Ptolemy were supplanted in turn by the Hakemite tables of Ibn Yunus (about A. D. 1000), the Toledan tables of Arzachel (1080), the Alphonsine tables of Alphonso the Wise (1252) and others. Finally, in the first half of the sixteenth century it became evident that Ptolemy's theory was itself gravely at fault. It was the fortune of Copernicus to open a new way to scientific thought—to lay down a new theory of the world.

The details of the long history thus sketched out are only interesting to astronomers.[6] We are here concerned with the main outlines alone.

The Arabian school of astronomers added nothing to the theory of Ptolemy. They transmitted the text of the Almagest to the west accompanied by intelligent comment and almost without criticism except in the cases of Alpetragins and Geber. The Arab observations were very numerous, and resulted in fixing new and much more accurate values of the constant of precession, the length of the year, the obliquity of the ecliptic, the eccentricity of the sun's orbit and the motion of its apogee. Their arithmetic was the clumsy sexagesimal arithmetic of the Greeks, until in the eleventh century the Hindu decimal system began to make its way in Egypt, Spain and Europe. Geometry is not indebted to the Arabs for any marked advances. On the other hand, trigonometry was greatly improved.

As observers the astronomers of the Arab school had great merit. They grasped the need for continuous observations, whereas the Greeks in general had contented themselves with making observations at certain critical times only—at the solstices and equinoxes, for instance. The Arabs were the first to assign the exact time at which any phenomenon occurred—a fundamental datum. They measured the altitude of the sun at the beginning and ending of solar eclipses, for example, in order that the time might be known. The calculation of a spherical triangle enabled the instants of beginning and ending to be accurately assigned. The Greeks never employed this device and the times of phenomena recorded by them are seldom known with any accuracy. Indeed Ptolemy has no formula by which to calculate the time when the sun's altitude is given, and it is noteworthy that the Arab device was not known in Europe until 1457, when Purbach used it for the first time. Yet it was employed at Bagdad at the solar eclipse of A. D. 829, six hundred 3ears earlier. Even the times of phenomena recorded by Tycho Brahe in 1600 are seldom known so close as a quarter of an hour. Short intervals of time were measured by the Arabs by counting the beats of pendulums.

A few of the greatest Arabians are named in what follows. Albategnius was an Arab prince of Syria who flourished at the end of the ninth century of our era. His observations were made at Aracte (Eachah) in Mesopotamia and at Antioch, between the years 878 and 918. After studying the Syntaxis of Ptolemy he set himself to correct the errors of its catalogue of stars by observations of his own, made with apparatus fashioned after Ptolemy's descriptions. It appears that some of his instruments could be read to single minutes (1′) and were divided possibly to 2′ (or it may be to 6′). He detected the change of position of the sun's apogee, determined the obliquity of the ecliptic, the length of the year, the precession-constant (54″), observed and calculated solar and lunar eclipses and computed new tables of the planetary motions, although he did not seek to improve Ptolemy's planetary theory. He was original and inventive as an observer; a sound mathematician; an expert and careful computer; and he introduced marked improvements in the methods of calculation. He holds the very first rank in the Arabian school. In his trigonometry he substituted sines for chords; reduced the calculation of spherical right triangles to four cases; and possessed a general rule for the solution of oblique spherical triangles in the cases (I.) given a, b, c, required A; (II.) given a, b, C, required c; was acquainted with the doctrine of tangents and cotangents, though he made no useful application of it; and seems to have known something of secants and cosecants. To understand his exact merits as an observer, it would be necessary to go into details that have no place here.

Ibn Yunus, the scion of a noble family, was the astronomer-royal of the Fatimite Caliphs of Cairo, where he constructed the Hakemite tables, in 1008, from his own observations. Comparing his own observations with those determined by Hipparchus or Ptolemy, he obtained accurate values of the changes that had supervened. They were accurate for two reasons: In the first place, the modern observation was very near to the truth; and in the second, the annual change was better determined the greater the interval of elapsed years. Albategnius and Ibn Yunus were 800 years after Ptolemy, while Ptolemy was but 263 years after Hipparchus, and Hipparchus but two centuries after Timocharis. The divisors increased with the lapse of time.[7]

"At Nishapur lived and died (early in the twelfth century) Omar Khayyam busied in winning knowledge of every kind, and especially in astronomy, wherein he attained to high preeminence. When Malik Shah determined to reform the calendar, Omar was one of the eight learned men required to do it; the result was the Jalili era, 'a computation of time,' says Gibbon, 'which surpassess the Julian and approaches the accuracy of the Gregorian style.' He is also the author of astronomical tables, and of a treatise on algebra" (Fitzgerald).

It is interesting to note that the Bagdad astronomers observed an eclipse of the sun by its reflection in water. The obliquity of the ecliptic for the year 1000 Ibn Yunus found to be 23° 33' (the true value is 23° 34' 16"). The latitude of Cairo he determined to be 30° (the true value is 30° 2'). Like all the Arabs he adopted the theories of the Almagest without change;[8] but his observations were materially better than Ptolemy's and his numerical results were, consequently, much more accurate. What is said of Ibn Yunus is, in general, true of the whole school of Arab and Moorish astronomers.

Ibn Yunus was acquainted with the Indian numerals 1, 2, 3, 4, 5, 6, 7, 8, 9, and used them occasionally in place of the clumsy Greek system, and he also introduced tangents and secants into trigonometry, as well as auxiliary angles (which latter were not used in Europe till the eighteenth century), but he continued to calculate triangles by formulæ involving sines only. Abul-Wafa of Bagdad (940–948) gave the formulæ relating to tangents and cotangents, and also to secants and cosecants, and even calculated tables of tangents; though he also stopped short of useful applications that were well within his reach. The science of trigonometry was, however, built up by Arabs, and the way was prepared for Vieta, who is the founder of the accepted doctrine. Abul-Wafa is the discoverer of the third inequality of the moon—the variation. Observing at a time when the first and second inequalities (discovered by Hipparchus and Ptolemy) had no effect, he noticed that the moon was a degree and a quarter from her calculated place. "Hence," he says, "I perceived that this inequality exists independently of the two first." This discovery remained unknown in Europe for six centuries until Tycho Brahe independently came to the same result.

Alhazen was an Arabian mathematician and astronomer of the eleventh century who is noteworthy for his treatment of physical problems, especially that of refraction. Ptolemy had experimented on the refraction of glass and of water and had made out the law that the angle of refraction is a fixed submultiple of the angle of incidence (r=1/mi). This was denied by Alhazen, but the true law was not discovered till the time of Willebrod Snell in 1621, who found the relation sine r=1/m⋅sine i, where m has a different value for each different substance. Alhazen's 'Optics' treats of the anatomy of the eye, and of vision, and has several propositions relating to the physiology of seeing, and it remained the standard work until the time of Roger Bacon and Vitello (thirteenth century).

This is the place to describe the system by which Ptolemy explained the world. It will be sufficient to explain the two main problems that any system of astronomy was bound to consider, and to leave details to one side. These two chief problems were: (1) How to account for the rising and setting of the sun, moon, stars and planets—how to explain the general diurnal motion of all celestial bodies; (2) how to explain the motions of the planets among the stars. These motions are, in general, towards the east—but are varied by occasional westward motions, and interrupted by periods of no motion at the 'stations.' As we have seen, Ptolemy declared the earth to be a sphere fixed in the center of the heavens. The sphere of the fixed stars was at an immense distance, so that the earth was a mere point in respect of the distance of the stars and the stars revolved about the earth. All the observed phenomena of the rising and setting of the stars are satisfactorily explained in this way. Ptolemy perfectly understood that they could also be explained by the hypothesis of a rotating earth, but he concluded that it was easier to attribute motion to bodies like the stars which seem to be of the nature of fire, than to the solid earth. The sun, moon and planets share in the diurnal motion of the stars. It will be seen that no mechanical conception of the diurnal motion is attainable in this way without the assumption of crystal spheres. Ptolemy sought an analytic device by which calculations of phenomena could be made, not a physical explanation based on mechanical laws.

The problems relating to the motions of the heavenly bodies are more complex and must be considered somewhat in detail. It is necessary to describe the observed phenomena for each body separately, and to adopt a system which will explain every phenomenon and appearance in turn.

The Moon.—The facts of observation, familiar to us all, are that the new moon sets in the west about sunset, and that on every succeeding night the moon sets at a later hour. It, therefore, moves to the east among the stars from night to night, which can readily be verified by observation. If the moon is near the stars of Orion on one night, it will be found many degrees to the east of them on the night following. It sets later and later every night throughout the month. If it is in the same longitude as the stars of Orion on any one day, it will be again in that longitude about a month (27 days) later (more exactly, 27d 7h 43m 11s.5). It has moved through the whole circuit of the heavens, 360°, in 27 days. Ptolemy explained the phenomena, as we explain them to-day, by asserting that the moon revolves in an orbit, about the earth as a center, making a complete revolution among the stars (from one star back to the same star again) in 27 days. Its motion among the stars is always forward—always from west to east.

The Sun.—The observed phenomena with regard to the sun are of the same nature. If the sun rises at the same time as the bright star Sirius on a particular day of the year, on the next day it will rise later than Sirius. It has, therefore, moved a certain distance (about one degree) eastwardly during that day. On the next following day it will have moved about two degrees east of Sirius and will rise correspondingly later; and so on for each succeeding day. After 180 days (six months) the sun will have moved about 180 degrees to the east of Sirius. Sirius will be visible on the meridian at midnight, (when the sun is 180° away from the meridian). At the end of 365 days (more exactly, 365.2564 days) the sun will have moved eastward through 360° and will again rise at the same moment as Sirius. The sun, then, appears to move eastwardly among the stars (from one star back to the same star again) once in 36514 days. At different times of the year it is among different groups of stars, and it is for this reason, therefore, that we see different groups of stars at different seasons of the year. Orion is visible in the winter skies, Scorpio in the summer, because Orion and Scorpio are 180 degrees apart in longitude. Ptolemy's explanation of all these phenomena is that the sun moves about the earth in a circular orbit at such a rate as to make a complete revolution in 36514 days. The explanation of Copernicus is that the earth revolves about the sun in the same period. It is to be noted that either of these explanations will completely account for all the observed phenomena.

New moon occurs when the earth, moon and sun are in a straight line. At sunset the new moon is seen in the west. After 27 days the moon has made one circuit among the stars, moving from west to east. But in those 27 days the sun has likewise moved eastwardly, about 27 degrees. The moon, then, has to make one circuit and a little more in order to be again in the line joining the earth and sun, in order to be again 'new.' The time from one new moon to the next—the lunar month—is about 2912 days (more exactly 29d 12h 55m 2s.9) for this reason. Just as there is a difference between the moon's sidereal and synodic period, so there is a corresponding difference between the sidereal and tropical year, because the equinoctial points are in motion relatively to the stars.

The Superior Planets.—In the system of Ptolemy, Mars, Jupiter and Saturn were supposed to be further from the earth than the sun—to be above it—and they were, therefore, called superior planets; while Mercury and Venus were called inferior planets. The facts of observations for one of the superior planets, for Mars, for example, are as follows. If on any day Mars rises at the same time as Sirius, on the next day it will rise a little later, and so on. The planet, therefore, moves eastwardly among the stars. It continues its motion so that at the end of 687 days (1.88 years) the planet again rises at the same time as Sirius. It has therefore made a complete circuit of the sky (from one star back to the same star again) in a little less than two years. Its orbit was supposed by Ptolemy to be a circle (the deferent) about the earth like the sun's orbit. In like manner Jupiter makes a revolution in 4,333 days (11.86 years), and Saturn in 10,759 days (29.46 years). Such are the general motions of the three superior planets; but there are irregularities in their motions that must be accounted for.

For example, the actual motion of the planet Jupiter among the stars for the year 1897 is as follows: Beginning on October 28, 1897, the planet's motion is eastwards until January 22, 1898; here it turns and moves westwards until May 28; here, again, it turns and moves eastwards and its direct motion continues for the rest of its period of nearly twelve years. Ptolemy accounted for the irregularities of motion just described by supposing that Jupiter revolved in a small circular orbit—the epicycle—once in 365 days, while, at the same time, the center of the epicycle moved along the circumference of the deferent circle, making a complete revolution in about twelve years.

As time elapses the center of Jupiter's epicycle will move onwards on the deferent while Jupiter will move onwards in its epicycle. The combination of these two motions will produce a direct motion of the planet. After Jupiter has moved through a quarter of a circumference on its epicycle the planet will appear to the observer on the earth to move in a retrograde direction, because it will move to the right or left on its epicycle faster than the center of the epicycle moves to the left or right. Hence the planet will appear to an observer to be moving in a retrograde direction—east to west. The epicyclic motion combined with the motion of the epicycle forwards along the deferent will produce first the retrograde and (in the last quadrant of the epicycle) again the direct motion of the planet in the sky. By taking the diameter of the epicycle of an appropriate size all the circumstances of the apparent motion of Jupiter can be represented. The explanation given by Ptolemy is complete and satisfactory. The general motion of the planet around the sky in twelve years is explained by the motion along the deferent. Its retrogradations and stations are explained by the combination of its epicyclic motion with its general motion. A like explanation serves for the other superior planets, Mars and Saturn.

The Inferior Planets.—The inferior planets, Mercury and Venus, appear sometimes east of the sun, sometimes west of it, but are never very far distant from the sun. We see them at sunset and sunrise as the morning and evening stars, Hesperus and Phosphorus, always in the sun's vicinity. Ptolemy explained their apparent motions completely and accurately by supposing that the centers of their epicycles revolved round the circumferences of their deferents in 36514 days; and that Mercury revolved round the circumference of its epicycle in 88 days, Venus round the circumference of its epicycle in 225 days. The sizes of the epicycles were chosen to correspond to the amount of each planet's greatest elongation from the sun. In the foregoing summary explanation only the main phenomena are described and explained. Irregularities in the moon's motion were explained by supposing that the earth did not lie at the center of the moon's orbit, but to one side; and other irregularities in the motions of the sun and planets were explained in a similar way. All motions took place in circles; the circle was the only 'perfect' curve. But the circles were eccentrics; the earth did not lie at their centers.

The periods of revolution of the planets were known to Ptolemy, but he knew little of their distances and nothing of their actual dimensions. The moon, he knew, shone by reflected light from the sun and he explained the lunar phases in this way, as is done to-day. The planets he supposed to shine by their own light, just as the fixed stars do. Astronomy to-day asserts that the planets, like the moon, shine by reflected light, and that the fixed stars are suns situated at immense distances.

Ptolemy solved the problem of the universe by solving the problem of the motion of each planet separately and by annexing each solution to the others. He never sought, it seems, for a single law governing all the cases. But such a law is patent. The radii of the epicycles of the superior planets are always parallel to the line joining the earth and the sun. The deferents of Mercury and Venus were really identical with the sun's orbit. It would seem that these very obvious laws could not escape a geometer of the caliber of Ptolemy. It appears that he never attempted the generalization; nor did his successors till the time of Copernicus. Each case was treated separately. When each was solved the explanation was complete. It required fourteen hundred years to make a generalization which is, in reality, simple, almost obvious.

Ptolemy's explanation of the system of the world accounted for all the facts known to him. As time went on, those assiduous observers, the Arabians, discovered other irregularities in the lunar and planetary motions unknown to Ptolemy. Every new irregularity required a new epicycle to explain it and in time the commentators of Ptolemy had added cycle on cycle, orb on orb until more than sixty spheres were necessary. The system lost its simplicity as more and more facts had to be explained and became a tangle of single instances, a web of particularities. It was never refuted. It broke of its own weight. The heliocentric hypothesis of Copernicus explained all these matters so simply, so convincingly, that it was soon adopted by all competent persons who examined it. The simplicity of a hypothesis is, of course, no evidence of its truth. Many modern theories are complex to a degree, but this is no proof that they are not true.

A layman seldom understands the attitude of a man of science towards 'theories,' as they are often half-contemptuously termed. Theory is popularly used as a synonym of opinion. 'His theory' is thought of as merely 'his opinion.' When, let us ask, is a science perfect? It is perfect when the circumstances of a phenomenon that is to occur in the future can be as accurately predicted now, as they can subsequently be observed when the actual phenomenon occurs. The 'theory' of transits of Venus over the sun's disc is practically perfect. We can predict the conditions of the next transit in A. D. 2004 almost as well as the astronomers of that day can observe it. The theory of Neptune's motion is so well known that the position of the planet in 1999 can be now predicted almost as accurately as it can be observed in that year. The theory of the circulation of the sap in plants is, on the other hand, far from perfect. We understand its general laws very well, but it is quite impossible to predict the circumstances for any particular plant in any particular season. The theories of hail, of lightning, of auroras and many others are in the same state.

It is obvious that if our own methods and instruments of observation are greatly improved at any particular epoch the science to which they belong will cease to be perfect even if it were a perfect science in the first instance. Tycho Brahe observed the longitudes of the stars by the naked eye. It is impossible, as we now know, to fix a longitude by such observations within one minute of arc (1′). This depends on the very constitution of the eye. When the telescope was invented and provided with a micrometer it became possible to fix star-places to within about one second of are (1″). Tycho's observing science, perfect in his day—incapable of further improvement—was no more than a rude approximation to Bradley, Astronomer-Royal of England in 1750. Bradley's tests were at least sixty times more delicate (1′ = 60″). Examples of this sort show how theories are held. Certain tests are now available—tests of a certain delicacy. When phenomena can be predicted beforehand as well as they can be subsequently observed, science is perfect up to that point. Increase the delicacy of the tests and a new standard is set up. Wave-motion was pretty well understood at the end of the nineteenth century until the X-rays came and refused, at first, to be reflected, refracted or polarized.

We in our day have learned a patient tolerance of opinion; wait, these theories that seem so baseless may, perhaps, come to something, as others have done in the past. To what especial and peculiar merit do we owe this acquired virtue of tolerant patience? It is owed solely to the experience of centuries. We have so often seen the impossible become the plausible, and at last the proved and the practical. Can we justly expect that our frame of mind—the strict result of centuries of experience—should have been the attitude of the doctors of the middle ages? Galileo was a great physicist: would not even he require time to accept our modern cobweb theory of the constitution of matter with its ether, molecules, atoms, electrified and non-electrified half-atoms, ions, dissociation, radio-activity and the like? Centuries of experience have taught us to hold theories lightly even while we are using them for present interpretations of phenomena. What physicist doubts that our present theory of electricity needs a thoroughgoing revision? And yet, who fails to use it where it can serve even a temporary purpose, foreseeing all the while new interpretations in the future?

The fundamental necessity in studies like the present is to realize the state of mind of our heroes and of the communities in which they lived. The only data are the words of the books they have left us.

How to interpret their words in their sense is the central difficulty; it is often most misleading to interpret them in our own. 'Do unto others as you would that they should do to you' is a golden rule that has been given in nearly the same words by Aristotle, by Christ and by Confucius; yet by 'others' Christ meant all men; Aristotle meant all the free born men of Greece, not their slaves; and Confucius meant the virtuous among his countrymen and excluded all wicked men and all foreign barbarians. If we consider what was meant by the words 'citizen,' 'honor,' 'duty,' in ancient Rome; in the later Roman Empire; in Constantinople; in the free towns of Italy; in the England of the middle ages; we shall understand the snares that lie latent in words which at first glance seem obvious in meaning.

In comparing the view-point of different ages with our own we continually meet with surprises. The uncritical attitude of the men of the thirteenth century towards miracles and wonders is little less than astounding to us. Our thought seems to be ages in advance of theirs. On the other hand, we often meet with an insight that has what we call the distinctly modern note. An instance from literature will illustrate:

A man's character is his fate

is a sentence that one would assign to Taine or to Stendhal in the nineteenth century, if one did not know it to have been written by Heracleitus in the fifth century before Christ. In like manner, some of the scientific processes of Hipparchus, Archimedes and Roger Bacon are so 'modern' as to bring a glow of delighted wonder when they are met with. Their failure to draw certain conclusions that seem almost obvious to us is equally astonishing. A formal explanation of the differences and of the resemblances of ancient ages with our own may be had somewhat as follows. We may suppose that a completely developed man of our day has educated his sympathies and intelligence to have outlets in a certain large number of directions—let us say, in the directions

A, B, C, D, E, F, G, H, I, J, K, L, M, N, 0, P, Q, R, S, T, U, V, W, X, Y, Z.

It is possible, however, that some few of these outlets are absent, or nearly closed, E and O for instance. The men of the eighteenth century may be supposed to have had fewer outlets, and those of the thirteenth still fewer; but the intensity and refinement of their sympathies in certain directions may not have been less but greater than ours. The feeling of the thirteenth century for religion, and of the sixteenth for art, for example, were not only different in intensity, but very different in quality from our own. When we make a formal comparison of our age with that of St. Thomas Aquinas and of Newton the table might stand thus:

A, B, C, D, –, F, G, H, I, J, K, L, M, N, –, P, Q, R, X, Y, Ztwentieth century
a, b, c, d, e, f, g, h, i, –, –, –, m, nthirteenth century
a, i, –, –, –, –, g, h, i, j, k, l, m, n, o, p, q, reighteenth century

If in a comparison of the thirteenth century with our own the discourse is upon the matters A, B, C and D we may find their insights, a, b, c, d, singularly like our own. The case may be the same for the matters G, H, I compared with g, h, i. But if, by chance, we are comparing their insight e with our absence of insight or our X, Y, Z, with the blanks in their experience, we are astonished at the difference of outlook. This formal and unimaginative illustration may not be quite useless in clarifying one's thought upon a matter easy to describe in words and exceedingly difficult to realize. It is essential to admit the presence of blanks in the experience of past centuries; and also the presence of insights upon fundamental matters astonishingly different in intensity and in quality from our own. The experience of the thirteenth century was handed onwards to succeeding ages; it could be understood by the ages near to it; words continued to mean in the fourteenth very nearly what they meant in the preceding century. But as ideas changed, the signs for ideas changed with them; and we must be constantly on our guard lest we unthinkingly admit an old form as if it had the new meaning.

Consider, for example, what astrology meant to Roger Bacon and what it means to us. He had no difficulty in reconciling the fateful influence of the stars with a scheme of salvation for men possessed of free-will. Words had different meanings to him and to us. His mind was conscious of no conflict between his religion and his science. His religion—that of the thirteenth century—is in absolute conflict with our science—that of the twentieth. Let his one example stand as a type of many that might be brought forward.

In what follows we shall study the words of Roger Bacon, the highest product of the thirteenth century.[9] His Opus Majus has recently (1897) been admirably edited by Bridges. Bacon has there expressed himself fully; and his century can be understood by implications. For this reason—to recreate, as it were, the background upon which the figure of Copernicus is projected, I have set down a few sentences. The paragraphs chosen relate chiefly to science, in which Bacon was far advanced, but enough is given of his views of philosophy, theology and morals to assist our judgment of his time. These extracts show what was possible to a man of the thirteenth century; and Bacon did not stand alone. He is the representative of a spirit that was active and widespread. It was creative; and it formed the scientific thought of succeeding centuries. Extracts from the summary of Bridges follow:

Opus Majus.—The four general causes of human ignorance—(1) subjection to unworthy authority; (2) habit; (3) popular prejudice; (4) false conceit of our own wisdom. Popular prejudice is a potent cause of error—pearls should not be cast before swine. Aristotle, wisest of philosophers, was not perfect. Avicenna sees where Aristotle erred; Averroes corrects Avicenna. Errors are found among the fathers of the church. By the habit of discussing received opinion we cease to be its slaves. The best Greek thought was not known to the Latin fathers. The early church made no use of Greek science, and the same is true of Bacon's own day, though without a good excuse.

The Connection of Philosophy with Theology.—Reason comes from God, therefore philosophy is divine. It is not an invention of heathen nations. The business of philosophy is to furnish a criterion of knowledge. All speculative philosophy has moral philosophy for its end and aim.

The Study of Language.—The quality of one language can never be perfectly reproduced in another; Latin altogether lacks many necessary words; a translator must not only be perfectly acquainted with his subject, but also with the two languages with which he deals; the translators of Aristotle have not fulfilled this condition; errors remain in the Vulgate; Hebrew, Chaldean, Latin, Greek (and Arabic) should be studied.

Mathematical Science is the key to all sciences; astronomy depends on mathematics; things terrestrial no less, since they are governed by things celestial.

Rays issuing in infinite number from a point in every direction find their termination on the hollow surface of a sphere. Light and other forces propagate themselves in this way. By the foregoing principles and others akin to them all natural actions are to be explained—as, for example, eclipses. (The moon and planets are self-luminous.) Refraction is discussed. The varying heat of different regions of the earth at difl'erent seasons is explained.

The emanations from the stars affect not merely climate but character; implanting in the new-born child dispositions to good and evil: though freewill, God's grace, temptations of the devil, or education may modify these innate tendencies. (Temperament is itself a result of the influence of the stars.)

The theory of (radiations of force) may be applied to the tides. These evidently depend on the moon. It is still unexplained why tides occur in the hemisphere averted from the moon.

On geometrical grounds the shape of the universe may be inferred to be spherical. The water, air and the fire surrounding the earth concentrically are of similar form (the sphere of fire is neither luminous nor visible).

In a body falling towards the earth's center a strain is involved; from this strain, heat results; experiment shows this to be a fact.

The Application of Mathematics to Sacred Subjects.—Astronomy shows the insignificance of the earth as compared with the heavens. The smallest of the stars is larger than the earth. Chronology is dependent upon astronomy. The lunar periods are discussed.

The rainbow is produced by the solar rays striking on the raindrops and being refracted or reflected thence.

One degree of latitude is 56 miles. The semi-diameter of the earth is 3,250 miles. The distance of the starry sphere (according to Alfraganus) is 130,715,000 miles. The distances of the planets are given—Saturn's as 65,357,500 miles (it is in fact 886,000,000) and the moon's 208,541 (it is in fact 238,840).

There are 1,022 fixed stars catalogued. The largest are 107 times as large as the earth; the smallest 18 times. Besides these there are infinite numbers of other stars.

Astrology as it relates to church government is discussed. Christianity is associated with the conjunctions of the planets Jupiter and Mercury; the extinction of religions is related to the conjunctions of Jupiter and Saturn with the moon.

The correction of the Church Calendar is discussed. At the present time its errors are so great as to attract the ridicule of Jewish and Arab astronomers.

Geography is discussed at length. The space of ocean separating Spain and India is inconsiderable; the seven climates of Ptolemy.

Astrology.—The sun may be said to beget no less than the parent. Each planet has an influence. Special parts of the body are affected by different constellations. The bearing of this on medical art is obvious. The comet of 1264 was due to Mars, and it was related to the European wars of that year.

Optics.—The organs of vision, psychical and cerebral; the organs of the sensitive soul are in the brain; threefold division of the brain; the heart is the seat of life; the brain first receives impressions; the nerves; the anatomy of the eye; its humors; the function of vision; vision is the result of radiation; vision is not completed in the eye but in the brain; matter is infinitely divisible; as many divisions can be made in a grain of millet as in the diameter of the world; theory of color; conditions of vision; time is required for the propagation of light; double images; radiations from the object and from the eye; perception; the Milky Way is a multitude of small stars clustered; shooting stars are probably bodies of small magnitudes seen by (persistent vision); phases of the moon; the surface of the heavens is spherical; illusions respecting relative motion; twinkling of stars; animals pass through a train of mental processes akin to syllogistic reasoning, though they can not put it into a logical figure; they have a storehouse of mental impressions; can generalize and draw conclusions.

On Reflected Vision.—The angles of incidence and reflection are equal whether the mirror be plane or spherical; mirrors; illusions; color; refraction; by refraction great wonders may be wrought; small things may be made to seem great, distant things near.

Moral Philosophy.—Civic morality; personal morality; we must pursue our steady course, not diverted from it by the varying blasts of opinion; proof of the truth of the Christian religion; revelation is necessary; it is not enough for the reason to be convinced in this matter—the heart must be stirred. If we are made one with God and Christ, to what greater good can we aspire?

The thirteenth century is memorable by the appearance of three great men, Roger Bacon, Albertus Magnus and St. Thomas Aquinas. Albert was born in Suabia in 1193, the descendant of a celebrated and powerful family. He may be reckoned as the best product of the middle ages. He studied in Padua and Bologna, taught at Cologne and Paris (1245) and returned to Cologne. He became provincial of the Dominicans in Germany in 1254, and was Bishop of Ratisbon (in 1260) till he resigned about 1263. He was the friend of kings and popes. His great service to the church was a systematic presentation of the philosophy of Aristotle with a full accompaniment of Arab commentary. Among his contemporaries he was known as Doctor Universalis, and, in the history of the world, is especially famous for his works on physical science. Like all the learned men of his time he was supposed by the vulgar to practise magic and, as a matter of course, he sought the philosopher's stone. It was even currently believed that he paid the large debt of his bishopric of Ratisbon with transmuted gold. The flowers that he grew in the winter time, which the wondering townsfolk called magical, were in all likelihood the product of the first hot-house constructed in Europe. An edition of his works in twenty-one volumes was first published in 1651. This complete collection shows, in the first place, that he was thoroughly familiar with all the learning of the Arabs up to his own time. He was to the west what Avicenna was to the east, an encyclopedia of all knowledge. His philosophy is, of course, Aristotle's, elucidated by the schools of Arabia and Spain. His works on physical science are, in a large degree, mere reproductions of Greek treatises, but they are nearly everywhere enriched by original observations. In his treatise De Animalibus (Vol. VI. of his works) he begins by a study of the spinal column, calls the sponges the lowest forms of animal life, improves on the zoological classification of his forerunners, and includes good descriptions of the fauna of the Arctic, and at the same time admits the legendary monsters of the Bestiaires of the tenth century, the barnacle-goose, anser arboreus, for example.

His botany is said to be full of errors. He quotes Pliny's facts relating to the fecundation of the date-palm, and correctly explains them, it is believed for the first time. The term affinity is first used in his chemical writings. There was no branch of knowledge that he did not treat, from mineralogy to magnetism, and it is noteworthy that he describes the magnet as in use by navigators in his time.[10]

It is not necesary in this place to speak of the achievement of St. Thomas Aquinas, the pupil of Albertus. His work does not lie in the field of physics, but in the universe of man. His Summa treats more than five hundred questions, but only one section refers to the phenomena of the material world. One remark may be made on the activity of the thirteenth century. Every one of the distinctly 'modern' problems was propounded in that age. Few were solved; but substantially all of them were stated. When a problem is clearly stated it is at least half solved.

The most noted figure in the generation preceding Copernicus was Regiomontanus, who is always thought of in company with his colleagues Purbach and Walther. They formed part of a group of German and Italian astronomers, calculators and teachers, no one of whom made any signal advance, but all of whom were well instructed in the fashion of the time. There are many names of men forgotten to-day among this group; but, on the other hand, the faint beginnings of a critical spirit are here and there to be noted. The Almagest was not always taken as infallible; observation began to be accepted as a test of theory. Dominicus of Bologna, the teacher of Copernicus, is a marked example of the new spirit.

George Purbach (or Beurbach, from his native village), professor of astronomy in the University of Vienna, was born in 1433 (died 1461), and studied at Vienna and in Italy. He was a votary of the old astronomy, and his chief work, Theoricæ Novæ Planetarum (1460), is a development of the doctrine of crystalline spheres. At the same time he was an ardent student of Ptolemy. The epicycles of Ptolemy were a geometrical conception; the crystalline spheres of Eudoxus and Purbach a crude cosmological idea; they could not be reconciled with nature. In so far Purbach was on the wrong road. He saw, however, the necessity for further observations of the planets and for accurate tables to replace those constructed for Alphonso the Wise, which no longer served to predict eclipses or to account for the configurations of the planets.[11] Errors of a couple of hours in the predicted time of a lunar eclipse were noted, and Mars was two degrees away from its calculated place.

In the work of observation and calculation he gained an invaluable coadjutor in Johann Müller, of Königsberg, a village of Franconia, one of his pupils. Müller called himself, after the fashion of the time, Johannes de Monteregio, but is known to us as Regiomontanus. Together they studied the works of Ptolemy, and together they observed the planets with the best instruments they could construct, though their apparatus was much inferior to that of the Arabs. Like all men of their time they were believers in judicial astrology, and their tables were arranged to meet the wants of this pseudo-science. At the same time astronomy benefited by their investigations, which began to be based on actual observation of the sky.

The Papal Nuncio in Vienna was then Cardinal Bessarion, once Bishop of Nicaea in the Greek, now high in power in the Latin Church and a friend of learning. Purbach's enthusiasm for the works of Ptolemy was shared by the cardinal, and they planned a new edition of his writings. For such an edition it was necessary to collect Greek manuscripts. After the death of Purbach (1461) Regiomontanus went to Italy in the cardinal's suite for this purpose (1462). Here he remained some seven years, collecting manuscripts, mastering the Greek language, studying the sciences, and writing his treatise on trigonometry. His text of Ptolemy was printed at Basel in 1538 and was used by Copernicus.

In 1471 he was settled in Nuremberg near the printing presses that had been installed a few years earlier. Here he had the fortune to meet a wealthy amateur, Bernhard Walther (1430–1504), who built an observatory for their joint use, and aided him in his publication of various writings, his own and Purbach's. The Ephemerides of Regiomontanus made him famous. They were the nautical almanacs of those days, and were used by Columbus and Vasco da Gama in their voyages of discovery. He is also the inventor of the method of lunar distances for determining the longitude at sea. He was invited to Rome by the Pope in 1475 to reform the calendar and there died in 1476.

There is a legend that Regiomontanus was assassinated by the sons of George of Trebizond, the first translator of the Almagest of Ptolemy from the Greek, because of strictures passed upon it. The legend is probably not true, but it is, perhaps, worth repeating, as it was credited at the time and casts a light on the age. During his short life Regiomontanus accomplished much, and gave promise of more. In particular he greatly improved the doctrine of trigonometry. Purbach and himself were the very first Europeans to utilize the discoveries of the Arabs in this science. As every astronomical calculation depends upon the solution of spherical triangles, the tables of sines and tangents computed by Regiomontanus were of fundamental importance, since they gave numerical values of these trigonometric functions calculated once for all, and saved the computer endless special reckonings.

It is difficult for us to conceive the state of science in those days. The school-boy problem: given a, b, c, in a spherical triangle, to find A, B, C, was considered operose by Regiomontanus and his friends, although the solution had been reached long before, by Albategnius. Blanchini, a contemporary of note, sends him the following equations for solution:

${\displaystyle x:y=5:8;x+y=xy}$.

A star rises at Venice at 3h 25m, and transits at 7h 38m, after midnight; required its longitude and latitude: is a problem addressed to Blanchini, in return. The Arabs five centuries earlier would have found these questions easy. Regiomontanus was, nevertheless, the most accomplished man of science in Europe. The ancients determined the longitude of a planet somewhat as follows: The difference of longitude between the planet and the moon was measured (A.) and next the difference of longitude between the moon and the sun (B). The longitude of the sun was calculated from the solar tables (C). The sum of A, B and C gave the planet's longitude. In Walther's observatory the angular distances of the planet from known stars were measured and the required longitude and latitude of the planet were calculated, by the formulae of spherical trigonometry, from the known longitudes and latitudes of the stars. The gain in precision was considerable, and the observations could be made on any clear night, whether the moon was or was not above the horizon.

Walther survived his friend for many years and carried on the observations which they had begun together. It was in their observatory that clocks (not pendulum-clocks) were first employed to measure short intervals of time and that observations were first corrected for terrestrial refraction. A star seen through the atmosphere appears higher above the horizon than if the atmosphere were absent. Its apparent position must then be corrected for refraction in order to obtain its true place. At an altitude greater than 45° the correction is less than 1′, which was inappreciable before the day of the telescope; but near the horizon the correction is large (the line of sight passing through the deepest layer of atmosphere there) and must he taken account of, even with rude observing apparatus. Refraction had been studied by Ptolemy and more deeply by Alhazen and Roger Bacon. Twilight, the scattering of the rays of the sun from the particles of dust and the like in the upper atmosphere, was investigated by Peter Nonius (1492–1577) a voluminous writer on astronomical matters.

All that was known in astronomy was familiar to Regiomontanus, and during his seven years' residence in Italy his relations were with the best instructed savants of Rome, who were then concerned with projects for improving and correcting the calendar. When Copernicus went to Italy in 1496 the best traditions of all Europe had spread throughout its universities and he was, therefore, familiar with all that his predecessors had accomplished.

A passage from the 'Principles of Astronomy' of Gemma Frisius (died 1558) is worth translation, since it fixes an important date and describes methods of determining longitudes and latitudes which are used to-day. He says: "People are beginning to make use of little clocks that are called watches. They are not too heavy to be carried about; they will run nearly twenty-four hours, and even longer if you aid them a bit; they afford a very easy method of determining longitude. Before starting on a journey, set your watch carefully to the local time of the country you are leaving; take pains that the watch doesn't stop on the road; when you have gone twenty leagues, for instance, determine the local time of the place where you are, with an astrolabe; compare this with the time by your watch, and you will have the difference of longitude." The latitude of the place can be had by measuring the altitude of the pole-star. Watches, which were invented about 1525, varied several minutes a day, and the portable astrolabes of the time could hardly give the altitude so close as 10′; but the methods were correct, and are those to-day employed in using the chronometer and the sextant.

Mention must be made of Peter Bienewitz, otherwise Peter Apianus (born 1495, died 1552), who expounded the Ptolemaic system in a great volume—Astronomicum Cæsareum (1540). Apianus was the first to observe the sun through colored glasses. The astronomers of Bagdad had observed an eclipse, when the sun was low, by its reflection in water, and Reinhold had proposed to project the solar image on a card in a camera obscura, a method which was used by the astronomers of Galileo's time. His best contribution to astronomy was the discovery that the tails of comets are generally directed away from the sun, a remark independently made by Fracastor.

Comets in his day were usually supposed to be atmospheric phenomena. Why this connection between them and the sun? Why should the sun, and not the earth, control their forms? The comet of 1472 had been studied by Regiomontanus and its course among the stars traced. This was the very first occasion upon which a comet had been treated as a celestial body like another. How could an object of the sort circulate among material crystal spheres? Questions of this kind were in men's minds; the observations upon which their solutions must depend were a-making; sufficient progress in mathematics had already been made; the time for a recasting of the accepted theory of the world was at hand.

Crystalline spheres were the basis of the theory of Fracastor. To explain the motions of the heavenly bodies he employed sixty-three spheres whose motions were linked one with another like wheel-work. His doctrine is that: All motions take place in circles; uniform motions are the most probable; each planet always remains at a constant distance from the earth; the changes in their observed brilliancy depend not on changes of distance, but on differences in the earth's atmosphere, or in the density of the crystal spheres; the Primum Mobile moves uniformly and always will do so unless God the Creator intervenes by a special act; spheres are of various kinds—conductors, anti-conductors, circling, anticircling, countervailing; sixty-three of them will explain the world; ten orbs belong to Saturn, eleven to Jupiter, nine to Mars, four to the sun, eleven to Venus, eleven to Mercury, seven to the moon. The system of Fracastor is not only complex, but mechanically impossible. It represents the worst aspect of the doctrine which Copernicus was to overthrow and it is interesting as almost the last exposition of its sort, and especially because Fracastor was a contemporary of Copernicus and died in the same year (1543).

1. Copernicus had access to certain large collections of books and the catalogues of these collections exist to-day. I have had them searched and the works of Archimedes are not there mentioned.
2. He fixed the greatest distance of the moon at 78, the least at 67, semi-diameters of the earth. The mean distance is, in fact, 60. The distance of the sun, according to Hipparchus, was 1,300 semi-diameters. It is really about 23,000.
3. The upper regions of Paradise contained the narrow-minded monks of the middle age as well as the great saints. The wisest and most virtuous heathens, like Virgil, were in Limbo—which, it has been remarked, contained the 'best society.' Outcasts from all religions, and sinners of all sorts, were in Hell.
4. Hipparchus fixed the length of the tropical year at 365 days 5 hours 55 minutes. Its true length is 365 days 5 hours 48 minutes 45.51 seconds (1900).
5. Our constellation figures are those designed by Albrecht Dürer on the star maps of Stoeffler and Heinfogel from the descriptions of Ptolemy.
6. They are given in clear form in various encyclopedias and other books of reference. Perhaps Berry's Short History of Astronomy (1899) will best serve the purpose of the general reader. Gyldén's Die Grundlehren der Astronomie (1877) develops the mathematical bases of ancient astronomy in an elementary form. Delambre's Histoire de l'Astronomie is still the best general history.
7. It may be remarked, in passing, that the foregoing explains how it is that Copernicus and Kepler had such accurate values of the periods of revolution of the different planets. Hipparchus noted, for example, that Mars was in conjunction with a certain star—Sirius for instance, on a certain day. Tycho, 1700 years later, observed that Mars was again in conjunction with Sirius on a certain day, at a certain hour. In the seventeen centuries that had elapsed, Mars had made about 860 revolutions. The interval of time between the two epochs, divided by the number of revolutions, gave the time of revolution with great exactness. On the other hand, the distance of Mars from the sun was only roughly known, even to Kepler. Of the dimensions of the planets nothing was known until their apparent angular diameters had been measured with the telescope. Anaxagoras held that the sun was about the same size as the Peloponesus.
8. It is to be noted, however, that the theories of Ptolemy, as understood by the Arabs, made some of the crystal spheres of the planets clash; and that Ptolemy's place for Mercury was consequently changed arbitrarily to allow room for its motion! This is not a change of theory; but it illustrates how slavishly the doctrine of spheres was followed by some of its votaries.
9. See a paper on Roger Bacon in The Popular Science Monthly, January, 1902.
10. It was introduced into Europe by Flavio Gioja, according to common report, about 1302.
11. This great collection of tables was calculated in the middle of the thirteenth century by Arabian and Jewish astronomers, under the patronage of the king, on the system of Ptolemy with some changes. The Libros del Saber were an encyclopedia of all the astronomical knowledge of the time and are invaluable to the historian.