A Short History of Astronomy (1898)/Chapter 3
THE MIDDLE AGES.
"The lamp burns low, and through the casement bars
Grey morning glimmers feebly."
55. About fourteen centuries elapsed between the publication of the Almagest and the death of Coppernicus (1543), a date which is in astronomy a convenient landmark on the boundary between the Middle Ages and the modern world. In this period, nearly twice as long as that which separated Thales from Ptolemy, almost four times as long as that which has now elapsed since the death of Coppernicus, no astronomical discovery of first-rate importance was made. There were some important advances in mathematics, and the art of observation was improved; but theoretical astronomy made scarcely any progress, and in some respects even went backward, the current doctrines, if in some points slightly more correct than those of Ptolemy, being less intelligently held.
In the Western World we have already seen that there was little to record for nearly five centuries after Ptolemy. After that time ensued an almost total blank, and several more centuries elapsed before there was any appreciable revival of the interest once felt in astronomy.
56. Meanwhile a remarkable development of science had taken place in the East during the 7th century. The descendants of the wild Arabs who had carried the banner of Mahomet over so large a part of the Roman empire, as well as over lands lying farther east, soon began to feel the influence of the civilisation of the peoples whom they had subjugated, and Bagdad, which in the 8th century became the capital of the Caliphs, rapidly developed into a centre of literary and scientific activity. Al Mansur, who reigned from A.D. 754 to 775, was noted as a patron of science, and collected round him learned men both from India and the West. In particular we are told of the arrival at his court in 772 of a scholar from India bearing with him an Indian treatise on astronomy, which was translated into Arabic by order of the Caliph, and remained the standard treatise for nearly half a century. From Al Mansur's time onwards a body of scholars, in the first instance chiefly Syrian Christians, were at work at the court of the Caliphs translating Greek writings, often through the medium of Syriac, into Arabic. The first translations made were of the medical treatises of Hippocrates and Galen; the Aristotelian ideas contained in the latter appear to have stimulated interest in the writings of Aristotle himself, and thus to have enlarged the range of subjects regarded as worthy of study. Astronomy soon followed medicine, and became the favourite science of the Arabians, partly no doubt out of genuine scientific interest, but probably still more for the sake of its practical applications. Certain Mahometan ceremonial observances required a knowledge of the direction of Mecca, and though many worshippers, living anywhere between the Indus and the Straits of Gibraltar, must have satisfied themselves with rough-and-ready solutions of this problem, the assistance which astronomy could give in fixing the true direction was welcome in larger centres of population. The Mahometan calendar, a lunar one, also required some attention in order that fasts and feasts should be kept at the proper times. Moreover the belief in the possibility of predicting the future by means of the stars, which had flourished among the Chaldaeans (chapter i., § 18), but which remained to a great extent in abeyance among the Greeks, now revived rapidly on a congenial oriental soil, and the Caliphs were probably quite as much interested in seeing that the learned men of their courts were proficient in astrology as in astronomy proper.
The first translation of the Almagest was made by order of Al Mansur's successor Harun al Rasid (A.D. 765 or 766–(A.D. 809), the hero of the Arabian Nights. It seems, however, to have been found difficult to translate; fresh attempts were made by Honein ben Ishak (?–873) and by his son Ishak ben Honein (?–910 or 911), and a final version by Tabit ben Korra (836–901) appeared towards the end of the 9th century. Ishak ben Honein translated also a number of other astronomical and mathematical books, so that by the end of the 9th century, after which translations almost ceased, most of the more important Greek books on these subjects, as well as many minor treatises, had been translated. To this activity we owe our knowledge of several books of which the Greek originals have perished.
57. During the period in which the Caliphs lived at Damascus an observatory was erected there, and another on a more magnificent scale was built at Bagdad in 829 by the Caliph Al Mamun. The instruments used were superior both in size and in workmanship to those of the Greeks, though substantially of the same type. The Arab astronomers introduced moreover the excellent practice of making regular and as far as possible nearly continuous observations of the chief heavenly bodies, as well as the custom of noting the positions of known stars at the beginning and end of an eclipse, so as to have afterwards an exact record of the times of their occurrence. So much importance was attached to correct observations that we are told that those of special interest were recorded in formal documents signed on oath by a mixed body of astronomers and lawyers.
Al Mamun ordered Ptolemy's estimate of the size of the earth to be verified by his astronomers. Two separate measurements of a portion of a meridian were made, which, however, agreed so closely with one another and with the erroneous estimate of Ptolemy that they can hardly have been independent and careful measurements, but rather rough verifications of Ptolemy's figures.
58. The careful observations of the Arabs soon shewed the defects in the Greek astronomical tables, and new tables were from time to time issued, based on much the same principles as those in the Almagest, but with changes in such numerical data as the relative sizes of the various circles, the positions of the apogees, and the inclinations of the planes, etc.
To Tabit ben Korra, mentioned above as the translator of the Almagest, belongs the doubtful honour of the discovery of a supposed variation in the amount of the precession (chapter ii., §§ 42, § 50). To account for this he devised a complicated mechanism which produced a certain alteration in the position of the ecliptic, thus introducing a purely imaginary complication, known as the trepidation, which confused and obscured most of the astronomical tables issued during the next five or six centuries.
59. A far greater astronomer than any of those mentioned in the preceding articles was the Arab prince called from his birthplace Al Battani, and better known by the Latinised name Albategnius, who carried on observations from 878 to 918 and died in 929. He tested many of Ptolemy's results by fresh observations, and obtained more accurate values of the obliquity of the ecliptic (chapter i., § 11) and of precession. He wrote also a treatise on astronomy which contained improved tables of the sun and moon, and included his most notable discovery namely—that the direction of the point in the sun's orbit at which it is farthest from the earth (the apogee), or, in other words, the direction of the centre of the eccentric representing the sun's motion (chapter ii., § 39), was not the same as that given in the Almagest; from which change, too great to be attributed to mere errors of observation or calculation, it might fairly be inferred that the apogee was slowly moving, a result which, however, he did not explicitly stale. Albategnius was also a good mathematician, and the author of some notable improvements in methods of calculation.
60. The last of the Bagdad astronomers was Abul Wafa (939 or 940–998), the author of a voluminous treatise on astronomy also known as the Almagest, which contained some new ideas and was written on a different plan from Ptolemy's book, of which it has sometimes been supposed to be a translation. In discussing the theory of the moon Abul Wafa found that, after allowing for the equation of the centre and for the evection, there remained a further irregularity in the moon's motion which was imperceptible at conjunction, opposition, and quadrature, but appreciable at the intermediate points. It is possible that Abul Wafa here detected an inequality rediscovered by Tycho Brahe (chapter v., § 111) and known as the variation, but it is equally likely that he was merely restating Ptolemy's prosneusis (chapter ii., § 48). In either case Abul Wafa's discovery appears to have been entirely ignored by his successors and to have borne no fruit. He also carried further some of the mathematical improvements of his predecessors.
Another nearly contemporary astronomer, commonly known as Ibn Yunos (?–1008), worked at Cairo under the patronage of the Mahometan rulers of Egypt. He published a set of astronomical and mathematical tables, the Hakemite Tables, which remained the standard ones for about two centuries, and he embodied in the same book a number of his own observations as well as an extensive series by earlier Arabian astronomers.
61. About this time astronomy, in common with other branches of knowledge, had made some progress in the Mahometan dominions in Spain and the opposite coast of Africa. A great library and an academy were founded at Cordova about 970, and centres of education and learning were established in rapid succession at Cordova, Toledo, Seville, and Morocco.
The most important work produced by the astronomers of these places was the volume of astronomical tables published under the direction of Arzachel in 1080, and known as the Toletan Tables, because calculated for an observer at Toledo, where Arzachel probably lived. To the same school are due some improvements in instruments and in methods of calculation, and several writings were published in criticism of Ptolemy, without, however, suggesting any improvements on his ideas.
Gradually, however, the Spanish Christians began to drive back their Mahometan neighbours. Cordova and Seville were captured in 1236 and 1248 respectively, and with their fall Arab astronomy disappeared from history.
62. Before we pass on to consider the progress of astronomy in Europe, two more astronomical schools of the East deserve mention, both of which illustrate an extraordinarily rapid growth of scientific interests among barbarous peoples. Hulagu Khan, a grandson of the Mongol conqueror Genghis Khan, captured Bagdad in 1258 and ended the rule of the Caliphs there. Some years before this he had received into favour, partly as a political adviser, the astronomer Nassir Eddin (born in 1201 at Tus in Khorassan), and subsequently provided funds for the establishment of a magnificent observatory at Meraga, near the north-west frontier of modern Persia. Here a number of astronomers worked under the general superintendence of Nassir Eddin. The instruments they used were remarkable for their size and careful construction, and were probably better than any used in Europe in the time of Coppernicus, being surpassed first by those of Tycho Brahe (chapter v.).
Nassir Eddin and his assistants translated or commented on nearly all the more important available Greek writings on astronomy and allied subjects, including Euclid's Elements, several books by Archimedes, and the Almagest. Nassir Eddin also wrote an abstract of astronomy, marked by some little originality, and a treatise on geometry. He does hot appear to have accepted the authority of Ptolemy without question, and objected in particular to the use of the equant (chapter ii., § 51), which he replaced by a new combination of spheres. Many of these treatises had for a long time a great reputation in the East, and became in their turn the subject-matter of commentary.
But the great work of the Meraga astronomers, which occupied them 12 years, was the issue of a revised set of astronomical tables, based on the Hakemite Tables of Ibn Yunos (§ 60), and called in honour of their patron the Ilkhanic Tables. They contained not only the usual tables for computing the motions of the planets, etc., but also a star catalogue, based to some extent on new observations.
An important result of the observations of fixed stars made at Meraga was that the precession (chapter ii., § 42) was fixed at 51", or within about 1" of its true value. Nassir Eddin also discussed the supposed trepidation (§ 58), but seems to have been a little doubtful of its reality. He died in 1273, soon after his patron, and with him the Meraga School came to an end as rapidly as it was formed.
63. Nearly two centuries later Ulugh Begh (born in 1394), a grandson of the savage Tartar Tamerlane, developed a great personal interest in astronomy, and built about 1420 an observatory at Samarcand (in the present Russian Turkestan), where he worked with assistants. He published fresh tables of the planets, etc., but his most important work was a star catalogue, embracing nearly the same stars as that of Ptolemy, but observed afresh. This was probably the first substantially independent catalogue made since Hipparchus. The places of the stars were given with unusual precision, the minutes as well as the degrees of celestial longitude and latitude being recorded; and although a comparison with modern observation shews that there were usually errors of several minutes, it is probable that the instruments used were extremely good. Ulugh Begh was murdered by his son in 1449, and with him Tartar astronomy ceased.
64. No great original idea can be attributed to any of the Arab and other astronomers whose work we have sketched. They had, however, a remarkable aptitude for absorbing foreign ideas, and carrying them slightly further. They were patient and accurate observers, and skilful calculators. We owe to them a long series of observations, and the invention or introduction of several important improvements in mathematical methods. Among the most important of their services to mathematics, and hence to astronomy, must be counted the introduction, from India, of our present system of writing numbers, by which the value of a numeral is altered by its position, and fresh symbols are not wanted, as in the clumsy Greek and Roman systems, for higher numbers. An immense simplification was thereby introduced into arithmetical work. More important than the actual original contributions of the Arabs to astronomy was the service that they performed in keeping alive interest in the science and preserving the discoveries of their Greek predecessors.
Some curious relics of the time when the Arabs were the great masters in astronomy have been preserved in astronomical language. Thus we have derived from them, usually in very corrupt forms, the current names of many individual stars, e.g. Aldebaran, Altair, Betelgeux, Rigel, Vega (the constellations being mostly known by Latin translations of the Greek names), and some common astronomical terms such as zenith and nadir (the invisible point on the celestial sphere opposite the zenith); while at least one such word, almanack, has passed into common language.
65. In Europe the period of confusion following the break-up of the Roman empire and preceding the definite formation of feudal Europe is almost a blank as regards astronomy, or indeed any other natural science. The best intellects that were not absorbed in practical life were occupied with theology. A few men, such as the Venerable Bede (672–735), living for the most part in secluded monasteries, were noted for their learning, which included in general some portions of mathematics and astronomy; none were noted for their additions to scientific knowledge. Some advance was made by Charlemagne (742–814), who, in addition to introducing something like order into his extensive dominions, made energetic attempts to develop education and learning. In 782 he summoned to his court our learned, countryman Alcuin (735–804) to give instruction in astronomy, arithmetic, and rhetoric, as well as in other subjects, and invited other scholars to join him, forming thus a kind of Academy of which Alcuin was the head.
Charlemagne not only founded a higher school at his own court, but was also successful in urging the ecclesiastical authorities in all parts of his dominions to do the same. In these schools were taught the seven liberal arts, divided into the so-called trivium (grammar, rhetoric, and dialectic) and quadrivium, which included astronomy in addition to arithmetic, geometry, and music.
66. In the 10th century the fame of the Arab learning began slowly to spread through Spain into other parts of Europe, and the immense learning of Gerbert, the most famous scholar of the century, who occupied the papal chair as Sylvester II. from 999 to 1003, was attributed in large part to the time which he spent in Spain, either in or near the Moorish dominions. He was an ardent student, indefatigable in collecting and reading rare books, and was especially interested in mathematics and astronomy. His skill in making astrolabes (chapter ii., § 49) and other instruments was such that he was popularly supposed to have acquired his powers by selling his soul to the Evil One. Other scholars shewed a similar interest in Arabic learning, but it was not till the lapse of another century that the Mahometan influence became important.
At the beginning of the 12th century began a series of translations from Arabic into Latin of scientific and philosophic treatises, partly original works of the Arabs, partly Arabic translations of the Greek books. One of the most active of the translators was Plato of Tivoli, who studied Arabic in Spain about 1116, and translated Albategnius's Astronomy (§ 59), as well as other astronomical books. At about the same time Euclid's Elements, among other books, was translated by Athelard of Bath. Gherardo of Cremona (1114–1187) was even more industrious, and is said to have made translations of about 70 scientific treatises, including the Almagest, and the Toletan Tables of Arzachel (§ 61). The beginning of the 13th century was marked by the foundation of several Universities, and at that of Naples (founded in 1224) the Emperor Frederick II., who had come into contact with the Mahometan learning in Sicily, gathered together a number of scholars whom he directed to make a fresh series of translations from the Arabic.
Aristotle's writings on logic had been preserved in Latin translations from classical times, and were already much esteemed by the scholars of the nth and 12th centuries. His other writings were first met with in Arabic versions, and were translated into Latin during the end of the 12th and during the 13th centuries; in one or two cases translations were also made from the original Greek. The influence of Aristotle over mediæval thought, already considerable, soon became almost supreme, and his works were by many scholars regarded with a reverence equal to or greater than that felt for the Christian Fathers.
Western knowledge of Arab astronomy was very much increased by the activity of Alfonso X. of Leon and Castile (1223–1284), who collected at Toledo, a recent conquest from the Arabs, a body of scholars, Jews and Christians, who calculated under his general superintendence a set of new astronomical tables to supersede the Toletan Tables. These Alfonsine Tables were published in 1252, on the day of Alfonso's accession, and spread rapidly through Europe. They embodied no new ideas, but several numerical data, notably the length of the year, were given with greater accuracy than before. To Alfonso is due also the publication of the Libros del Saber a voluminous encyclopaedia of the astronomical knowledge of the time, which, though compiled largely from Arab sources, was not, as has sometimes been thought, a mere collection of translations. One of the curiosities in this book is a diagram representing Mercury's orbit as an ellipse, the earth being in the centre (cf. chapter vii., § 140), this being probably the first trace of the idea of representing the celestial motions by means of curves other than circles.
67. To the 13th century belong also several of the great scholars, such as Albertus Magnus, Roger Bacon, and Cecco d'Ascoli (from whom Dante learnt), who took all knowledge for their province. Roger Bacon, who was born in Somersetshire about 1214 and died about 1294, wrote three principal books, called respectively the Opus Majus, Opus Minus, and Opus Tertium, which contained not only treatises on most existing branches of knowledge, but also some extremely interesting discussions of their relative importance and of the right method for the advancement of learning. He inveighs warmly against excessive adherence to authority, especially to that of Aristotle, whose books he wishes burnt, and speaks strongly of the importance of experiment and of mathematical reasoning in scientific inquiries. He evidently had a good knowledge of optics and has been supposed to have been acquainted with the telescope, a supposition which we can hardly regard as confirmed by his story that the invention was known to Caesar who when about to invade Britain surveyed the new country from the opposite shores of Gaul with a telescope!
Another famous book of this period was written by the Yorkshireman John Halifax or Holywood, better known by his Latinised name Sacrobosco, who was for some time a well-known teacher of mathematics at Paris, where he died about 1256. His Sphaera Mundi was an elementary treatise on the easier parts of current astronomy, dealing in fact with little but the more obvious results of the daily motion of the celestial sphere. It enjoyed immense popularity for three or four centuries, and was frequently re-edited, translated, and commented on: it was one of the very first astronomical books ever printed; 25 editions appeared between 1472 and the end of the century, and 40 more by the middle of the 17th century.
68. The European writers of the Middle Ages whom we have hitherto mentioned, with the exception of Alfonso and his assistants, had contented themselves with collecting and rearranging such portions of the astronomical knowledge of the Greeks and Arabs as they could master; there were no serious attempts at making progress, and no observations of importance were made. A new school, however, grew up in Germany during the 15th century which succeeded in making some additions to knowledge, not in themselves of first-rate importance, but significant of the greater independence that was beginning to inspire scientific work. George Purbach, born in 1423, became in 1450 professor of astronomy and mathematics at the University of Vienna, which had soon after its foundation (1365) become a centre for these subjects. He there began an Epitome of Astronomy based on the Almagest, and also a Latin version of Ptolemy's planetary theory, intended partly as a supplement to Sacrobosco's textbook, from which this part of the subject had been omitted, but in part also as a treatise of a higher order; but he was hindered in both undertakings by the badness of the only available versions of the Almagest—Latin translations which had been made not directly from the Greek, but through the medium at any rate of Arabic and very possibly of Syriac as well (cf. § 56), and which consequently swarmed with mistakes. He was assisted in this work by his more famous pupil John Müller of Königsberg (in Franconia), hence known as Regiomontanus, who was attracted to Vienna at the age of 16 (1452) by Purbach's reputation. The two astronomers made some observations, and were strengthened in their conviction of the necessity of astronomical reforms by the serious inaccuracies which they discovered in the Alfonsine Tables, now two centuries old; an eclipse of the moon, for example, occurring an hour late and Mars being seen 2° from its calculated place. Purbach and Regiomontanus were invited to Rome by one of the Cardinals, largely with a view to studying a copy of the Almagest contained among the Greek manuscripts which since the fall of Constantinople (1453) had come into Italy in considerable numbers, and they were on the point of starting when the elder man suddenly died (1461).
Regiomontanus, who decided on going notwithstanding Purbach's death, was altogether seven years in Italy; he there acquired a good knowledge of Greek, which he had already begun to study in Vienna, and was thus able to read the Almagest and other treatises in the original; he completed Purbach's Epitome of Astronomy, made some observations, lectured, wrote a mathematical treatise of considerable merit, and finally returned to Vienna in 1468 with originals or copies of several important Greek manuscripts. He was for a short time professor there, but then accepted an invitation from the King of Hungary to arrange a valuable collection of Greek manuscripts. The king, however, soon turned his attention from Greek to fighting, and Regiomontanus moved once more, settling this time in Nürnberg, then one of the most flourishing cities in Germany, a special attraction of which was that one of the early printing presses was established there. The Nürnberg citizens received Regiomontanus with great honour, and one rich man in particular, Bernard Walther (1430-1504), not only supplied him with funds, but, though an older man, became his pupil and worked with him. The skilled artisans of Nürnberg were employed in constructing astronomical instruments of an accuracy hitherto unknown in Europe, though probably still inferior to those of Nassir Eddin and Ulugh Begh (§§ 62, 63). A number of observations were made, among the most interesting being those of the comet of 1472, the first comet which appears to have been regarded as a subject for scientific study rather than for superstitious terror. Regiomontanus recognised at once the importance for his work of the new invention of printing, and, finding probably that the existing presses were unable to meet the special requirements of astronomy, started a printing press of his own. Here he brought out in 1472 or 1473 an edition of Purbach's book on planetary theory, which soon became popular and was frequently reprinted. This book indicates clearly the discrepancy already being felt between the views of Aristotle and those of Ptolemy. Aristotle's original view was that sun, moon, the five planets, and the fixed stars were attached respectively to eight spheres, one inside the other; and that the outer one, which contained the fixed stars, by its revolution was the primary cause of the apparent daily motion of all the celestial bodies. The discovery of precession required on the part of those who carried on the Aristotelian tradition the addition of another sphere. According to this scheme, which was probably due to some of the translators or commentators at Bagdad (§ 56), the fixed stars were on a sphere, often called the firmament, and outside this was a ninth sphere, known as the primum mobile, which moved all the others; another sphere was added by Tabit ben Korra to account for trepidation (§ 58), and accepted by Alfonso and his school; an eleventh sphere was added towards the end of the Middle Ages to account for the supposed changes in the obliquity of the ecliptic. A few writers invented a larger number. Outside these spheres mediaeval thought usually placed the Empyrean or Heaven. The accompanying diagram illustrates the whole arrangement.
These spheres, which were almost entirely fanciful and in no serious way even professed to account for the details of the celestial motions, are of course quite different from the circles known as deferents and epicycles, which Hipparchus and Ptolemy used. These were mere geometrical abstractions, which enabled the planetary motions to be represented with tolerable accuracy. Each planet moved freely in space, its motion being represented or described (not controlled) by a particular geometrical arrangement of circles. Purbach suggested a compromise by hollowing out Aristotle's crystal spheres till there was room for Ptolemy's epicycles inside!
From the new Nürnberg press were issued also a succession of almanacks which, like those of to-day, gave the public useful information about moveable feasts, the phases of the moon, eclipses, etc.; and, in addition, a volume of less popular Ephemerides, with astronomical information of a fuller and more exact character for a period of about 30 years. This contained, among other things, astronomical data for finding latitude and longitude at sea, for which Regiomontanus had invented a new method.
The superiority of these tables over any others available was such that they were used on several of the great voyages of discovery of this period, probably by Columbus himself on his first voyage to America.
In 1475 Regiomontanus was invited to Rome by the Pope to assist in a reform of the calendar, but died there the next year at the early age of forty.
Walther carried on his friend's work and took a number of good observations; he was the first to make any successful attempt to allow for the atmospheric refraction of which Ptolemy had probably had some knowledge (chapter ii., § 46); to him is due also the practice of obtaining the position of the sun by comparison with Venus instead of with the moon (chapter ii., § 39), the much slower motion of the planet rendering greater accuracy possible.
After Walther's death other observers of less merit carried on the work, and a Nürnberg astronomical school of some kind lasted into the 17th century.
69. A few minor discoveries in astronomy belong to this or to a slightly later period and may conveniently be dealt with here.
Lionardo da Vinci (1452–1519), who was not only a great painter and sculptor, but also an anatomist, engineer, mechanician, physicist, and mathematician, was the first to explain correctly the dim illumination seen over the rest of the surface of the moon when the bright part is only a thin crescent. He pointed out that when the moon was nearly new the half of the earth which was then illuminated by the sun was turned nearly directly towards the moon, and that the moon was in consequence illuminated slightly by this earthshine, just as we are by moonshine. The explanation is interesting in itself, and was also of some value as shewing an analogy between the earth and moon which tended to break down the supposed barrier between terrestrial and celestial bodies (chapter vi., § 119).
Jerome Fracastor (1483-1543) and Peter Apian (1495-1552), two voluminous writers on astronomy, made observations of comets of some interest, both noticing that a comet's tail continually points away from the sun, as the comet changes its position, a fact which has been used in modern times to throw some light on the structure of comets (chapter xiii., § 304).
Peter Nonius (1492-1577) deserves mention on account of the knowledge of twilight which he possessed; several problems as to the duration of twilight, its variation in different latitudes, etc., were correctly solved by him; but otherwise his numerous books are of no great interest.
A new determination of the size of the earth, the first since the time of the Caliph Al Mamun (§ 57), was made about 1528 by the French doctor John Fernel (1497-1558), who arrived at a result the error in which (less than 1 per cent.) was far less than could reasonably have been expected from the rough methods employed.
The life of Regiomontanus overlapped that of Coppernicus by three years; the four writers last named were nearly his contemporaries; and we may therefore be said to have come to the end of the comparatively stationary period dealt with in this chapter.
- The data as to Indian astronomy are so uncertain, and the evidence of any important original contributions is so slight, that I have not thought it worth while to enter into the subject in any detail. The chief Indian treatises, including the one referred to in the text, bear strong marks of having been based on Greek writings.
- He introduced into trigonometry the use of sines, and made also some little use of tangents, without apparently realising their importance: he also used some new formulæ for the solution of spherical triangles.
- A prolonged but indecisive controversy has been carried on, chiefly by French scholars, with regard to the relations of Ptolemy, Abul Wafa, and Tycho in this matter.
- For example, the practice of treating the trigonometrical functions as algebraic quantities to be manipulated by formulæ, not merely as geometrical lines.
- Any one who has not realised this may do so by performing with Roman numerals the simple operation of multiplying by itself a number such as mdcccxcviii.
- On trigonometry. He reintroduced the sine, which had been forgotten; and made some use of the tangent, but like Albategnius (§ 59 n.) did not realise its importance, and thus remained behind Ibn Yunos and Abul Wafa. An important contribution to mathematics was a table of sines calculated for every minute from 0° to 90°.
- That of "lunar distances."
- He did not invent the measuring instrument called the vernier, often attributed to him, but something quite different and of very inferior value.