29 On Johannes Regiomontanus (1549) CR XI, 817–26
29 On Johannes Regiomontanus (1549)
CR XI, 817–26
Oration on the mathematician Johannes Regiomontanus, given by Erasmus Reinhold, of Saalfeld, professor of mathematics, at the proclamation of the degree of Master of Philosophy1
To begin with, I give thanks to the eternal God, Father of our Lord Jesus Christ, the Maker of Heaven and Earth and of His angelic and human Church, together with His Son, our Lord Jesus Christ and His Holy Spirit, for not allowing studies of this subject to perish. And I pray to Him with all my heart that He may guide us and sustain our studies, and assemble among us a following for His Son, our Lord Jesus Christ, who was crucified for us and was resurrected; a following which may delight in His goodness for all eternity and worship Him. It is proper for you all to beg for this with fervent prayers – for in these scholarly assemblies, too, prayers should be mingled with admonitions about knowledge.
But it is the custom occasionally to declaim here the stories of famous men, especially of learned ones, because these stories teach us much about the arts which they rendered illustrious, and because the commemoration of such men is agreeable to worthy minds. Therefore I have now put together an account of Johannes Regiomontanus, partly from his own writings, partly from the oration of that most learned man, Johannes Schöner.2 Although it is rather short and will not fulfil your expectations, it nevertheless needs to be handed on to the young, so that they may come to know and admire this great man, and may strive after these arts whose seeds he sowed. Admittedly the power and sharpness of his intelligence, his erudition and the correctness of his judgement can be seen especially in his writings, but this realisation is limited to the masters. Although this account of his zeal, of the works he published, his struggles with the masters, his travels and
1 In 1549 Erasmus Reinhold (1511–53) was Dean of the arts faculty. This is the oldest extant biography of Regiomontanus, for whose life see E. Zinner, Regiomontanus: His Life and Work, trans. E. Brown, Studies in the History and Philosophy of Mathematics (Amsterdam, New York, Oxford, Tokyo, 1990). For his achievement, see Menso Folkerts, ‘Regiomontanus’ Rôle in the Transmission and Transformation of Greek Mathematics' in Tradition, Transmission, Transformation: Proceedings of Two Conferences on Pre-modern Science Held at the University of Oklahoma, ed. F. Jamil Ragep and Sally R. Ragep, with S. Livesey (Leiden, 1996), pp. 89–113.
2 Johann Schöner (1477–1547) of Nuremberg had access to several works of Regiomontanus and printed them. Schöner's own oration of Regiomontanus, however, on which this oration was based, is no longer extant.
his friendships with the greatest men show that he had many other virtues, it also shows that he had a powerful intellect and an enthusiasm, implanted by divine providence, for rendering illustrious the arts. Indeed, I admire the brilliance of his mind as much as I am delighted by the subtlety of his books. At the same time as his mind flits through the entire nature of things, and measures the heavens and the spaces of the Earth, his body travels through Germany, Italy and Hungary; he learns Greek, and even that peculiar language particular to the geometers, and he compares his ideas with other learned men. He not only wrote much that would be of use for future generations, but he also fashioned machines by his own hand, and he built a printing-press. A slothful nature neither undertakes such a variety of toils, nor can it support it. But whatever trace of his virtues can be shown in such a narrative, I hope that this account will be agreeable to you.
Johannes Regiomontanus was born in a town of Franconia called Königsberg, which because of its closeness is known to most of you. I hear that his parents were worthy people of some means, as things go in our towns, so that they were able to bring up and educate their children in any way they desired. I know that some men born in that family were also known to some of our colleagues. And since it is a matter of fact that dispositions take on some impetus from heavenly causes, I shall also recount the time at which he was born and the position of the stars which shone at his birth. He was born in the year 1436, according to the customary way of counting, on the sixth of June, at the fourth hour and fortieth minute after noon.3 I would wish that the complete positions of the celestial signs at birth were preserved in all histories for those who either excelled by their natural goodness or were of monstrous perversity. Nevertheless, I shall now recount the horoscope and the position of the planets so that the entire position of celestial signs can be the better judged by the eyes. The horoscope was the fifteenth degree of Scorpio, with the sun, the moon, Mars and Mercury together in Gemini – a sign that portends very little wildness, but a moderate and docile nature suitable for discipline; and since the sun and moon and Mars and Mercury came together in it, it was a disposition that was neither dull nor sluggish. Jupiter in Libra and Venus in Cancer increase the goodness of the temperament. I shall not add a longer explanation, but I leave it to
3 It was then customary to use unequal hours (dividing the period between sun-up and sun-down into twelve equal hours): Zinner, Regiomontanus, p. 1. It was unusual to give the exact minute of birth, but it may have had astrological significance.
the learned to consider, who see what threatens the disposition of life. But the signs of disposition and character will not be discovered to be absurd. The seeds of his nature were sufficiently propitious, and his disposition was docile and keen; nevertheless, without divine inspiration he could not have thoroughly grasped so many and such varied arts, learnt so many languages, supported simultaneously so many toils in studying, teaching, fashioning machines, observing the positions of the stars, writing most learned explanations, disclosing so many mazes, publishing his own tables and those of others, and also Ephemerides, especially in the course of such a short life, if God had not kindled and assisted the fire of his mind and heart. In that way, in His great goodness, He is wont continually to encourage and help masters in every field in order to restore and protect the arts that are necessary for life. For truly the preservation of arts and sciences among humankind is the favour of the eternal God, who is the fount of wisdom. Therefore let us give thanks to Him, who repeatedly restores the light of the sciences, and let us pray that He will not let it be extinguished. However, He wants some diligence and zeal in us to be added to His generosity; and He does not want His gifts cast away with disgust and contempt, as it is expressed in a more weighty argument in the parable about the traders. And it is not incongruous that Plato says in book four of the Laws: ‘That the god and opportunity steer all human matters, it is more civilised to agree that art has to follow these as a third’ [709b].4 That is, a god and opportunity rule all affairs, but it is nevertheless more civil to grant that art is to be added, as on a ship: even if a storm has the upper hand, nevertheless the oars should not be idle. In this way let us also accept the divine gifts with respect, delight in them assiduously, and protect them for the glory of God and for the common good, as much as we can with our mediocre diligence, as it is written. This is demanded of us, in order that we be faithful, and imitate Regiomontanus in this matter. When God had given him the powers of the intellect and the body necessary for learning, an excellent teacher and books as companions for his studies, he did not neglect these opportunities; he did not, like many others, prefer pleasures or money, but attached himself to these studies with all the powers of his intellect and his mind. He searched everywhere for learned men by whose wisdom he could be instructed, and in turn he adorned the arts by his works as much as he could. Let us therefore give thanks to God, who does not allow these His gifts to perish utterly, and let us praise the will, effort and assiduity of
4 The quotation is slightly shortened to omit the word ‘chance’.
Regiomontanus. In order that you may better behold his virtues, you should know his ways.
After studying grammar in his home-town, he came, still a boy, to the University of Leipzig, where he seized eagerly upon dialectic and the elements of the science of celestial motions. But burning, by disposition, with the desire to learn, he needed guides in this study as well as companions in this most beautiful toil, and he heard that Vienna was a particular home to this philosophy. Therefore he travelled to Austria when still a youth. There he encountered the Bavarian Peuerbach, a most ingenious man, in the prime of his age and reputation.5 The latter valued Regiomontanus, when he had examined the power of the youth's nature and his will, and became not only his encourager, but also his guide to the most sacred sanctuary of these arts. When I think of their companionship, studies and struggles I also see in my mind's eye the ancient schools by whose diligence these arts were propagated. Thus the Egyptian priests taught Theletes, Meton and Eudoxus, and Eudoxus taught Plato. In that way it is said that Atlas taught this knowledge to Orion; indeed, as I believe, in that way those long-lived men in the first ages of the world, Adam, Noah, Shem and Joseph, passed on this knowledge to coming generations, and its possession was preserved in Egypt from the time of Joseph to the age of Ptolemy, and somewhat longer, for over two thousand years. For such a long time God bestowed upon this knowledge a stable and fixed abode, among the changes of the various monarchies. But then the wisdom of the principal monarchs was such that they knew that they ought to wage war against the enemies and not against the arts, and in the midst of wars they protected the abodes of erudition and the assemblies of the learned. Now, oh grief, that knowledge roams without protectors and without a fixed abode, so that one must fear the ruin of these arts. But let us pray to God the Maker not to allow His gifts to perish.
Since there was the greatest friendship between him and Peuerbach, and an assiduous comparison of ideas for about a decade, they shed light on many obscure matters and, as they say, the two beginning together (synte dyō archomenō) they observed the motion and detected that the motion of Mars did not correspond to the tables, but that there was an error of about two degrees. Therefore they began the correction of the tables. By chance
5 Georg Peuerbach (1423–61) attended the University of Vienna, obtaining his BA in 1448 and MA in 1453. He lectured on astronomy and poetry at Vienna: see Zinner, Regiomontanus, p. 8. Regiomontanus matriculated at the University of Vienna in 1450.
Bessarion, a Greek, who excelled in knowledge, was at the time ambassador to the emperor Frederick, since he was a cardinal.6 When he had inquired much about the knowledge of Ptolemy from Peuerbach, and had admired his skill in expounding, he asked him to compose a résumé of the book entitled Megalē Syntaxis (the Almagest). Regiomontanus completed that work that was interrupted by the death of Peuerbach, because – as he himself relates – the dying Peuerbach had asked him on the point of death. In this matter it is much to be marvelled at that at the time they were able, without the Greek sources, to pluck the meaning from those Arabic translations. This could in no way have been done if the demonstrations had not been obvious and easily penetrable to them because of their great practice in that entire circle of arts. But nevertheless later Regiomontanus corrected that work in Rome, when he had learnt Greek and compared it with Theon. There survives a letter by Regiomontanus full of honesty, in which he deplores the death of his teacher and proclaims his genius, saying that he preferred him to all the mathematicians of his day.7
The desire to go to Italy struck both Peuerbach and Regiomontanus, in order to listen to learned men in Italy – for Bianchini's renown8 was being praised – as well as to apply themselves to the Greek language, in which they knew the sources of these arts to be contained, and they wanted to read Ptolemy speaking his own language. And Bessarion encouraged them to travel with him. Regiomontanus also relates that he was much aggrieved that he could not examine the sources when, as a youth, he saw two most ingenious men, Peuerbach and Franciscus Mantuanus,9 at a loss in the translations of the geography of Ptolemy because of their ignorance of Greek. There were no teachers for the Greek language, and because of that he decided at some time to travel to Italy. The youth took it ill that there were so many errors in the translation of the description of the Danube. For when the text had ektropē, which is passage, where another stream flows into it, such as the Inn flowing into the Danube at Passau, the translator
6 See note 4, oration 12, p. 108.
7 See Regiomontanus, Rudimenta Astronomica Alfagani. Item Albategnius Astronomus Peritissimus de Motu Stellarum … Item Oratio Introductoria in Omnes Scientias Mathematicas (Nuremberg: J. Petreius, 1537). This contains a preface by Melanchthon addressed to the Senate of Nuremberg.
8 Giovanni Bianchini had no formal university training and was in the service of Nicolò d'Este at Ferrara. He published several large astronomical tables, which were used by Peuerbach and Regiomontanus to provide the Ephemerides.
9 Franciscus Mantuanus: Regiomontanus is here probably referring to Francesco Berlinghieri, who was an orator at the Este court in Mantua. For Berlinghieri's translation of Ptolemy's Geographica, see Jerry Brotton, Trading Territories: Mapping the Early Modern World (London, 1997), pp. 87–99.
had translated it as divisions in many places, as though the other rivers were arms of the Danube. The description of the Po is obscured by a similar wandering of the mind.
However, Peuerbach's untimely death cut short his plan for this journey. After his death, sorrow also influenced Regiomontanus to leave Vienna more easily. He therefore accompanied Bessarion who was returning in Italy, when he was not far from his thirtieth year. In Rome Regiomontanus often met with Trebizond. He also explored the libraries, which spurred him even more to give attention to the Greek language. Therefore, burning with this desire, he copied many things by his own hand, and made himself familiar with the books of Ptolemy. Then he also wrote the dialogue against the Theorica of Gerard of Cremona.10 Some time later he went to Ferrara, where Blanchinus and Theodorus Gaza were at the time, and Guarinus taught there in Greek.11 There he accomplished with zeal an excellent work in Greek, in more than a year; and he studied so much that he was also able to write verse in Greek, which is still extant, and to busy himself with the Greek philosophers without difficulty. There he already began to compare the commentaries of Ptolemy and Theon. Some time later he left Ferrara and went to the University of Padua, where, upon request by the students of philosophy, he gave a public lecture on Alfraganus.12 The introductory address given there still exists, in which he not only extols the subjects, but also, with great brilliance, praises the mathematicians, who were famous in Italy at the time, and whom he had attached to himself by friendship because of their fellowship of studies and virtues. When Alfraganus' lecture had finished, and Bessarion had already returned from Greece, Regiomontanus betook himself to Venice, in the year 1464. There he composed the book on triangles with great diligence, and the refutations of the quadrature of the circle, which13 Nicolas Cusanus was believed to have demonstrated.14 Then he returned to Rome, where he copied some
10 The Theorica Planetarum is usually attributed to Gerard of Cremona (1114–87). For Regiomontanus' criticism of it, see his Contra Cremonensia in Planetarum Theroicas Deliramenta … Disputationes, which underwent several editions since 1483.
11 Theodore Gaza (1400–76) taught Greek at the University of Ferrara between 1447 and 1449. For Guarinus, see note 3, oration 28, p. 230.
12 Oratio Iohannis de Monteregio quam habuit ipse Patavij in praelectione Alfragani (Wittenberg: V. Creutzer, 1549), with this oration by Reinhold attached.
13 Reading quam for quas.
14 Nicholas Cusanus' (1401–64) attempt to square the circle appears in the De Quadratura Circuli (1450), a copy of which was sent by Cusanus to Peuerbach, who in turn indicated corrections to Regiomontanus. See his De Triangulis Omnimodis. Libri quinque … Accesserunt huc Incalce Deque Recti ac Curvi Commensuratione … (Nuremberg: J. Petreius, 1533).
Greek and Latin books by his own hand – for he drew the shapes of the letters with great skill and ease – and had some of them copied at great expense. There he also had the most vehement disputes with Trebizond,15 whose reveries in criticising Theon he refuted ingenuously.
After having travelled through Italy he returned to Vienna, where he began to teach, but was summoned by King Matthias and by the bishop of Esztergom, and was given great rewards by both.16 The king also decreed for him an annual pension of 200 Hungarian gold coins. There he also constructed instruments for observing the motions of the heavens for both the king and the bishop. For the bishop had great learning in these subjects. And although no other prince of that time was more warlike, the king was nevertheless not uneducated in the knowledge of the celestial motions. It is said that he was also saved from a very serious illness by the counsel of a mathematician. For when the doctors found no cause for the illness in the humours, and noticed only that the faintness of the heart was strange, a mathematician advised that this feebleness of the vital powers was because of an eclipse which had threatened the king's life, and that the cause was not to be sought in the humours,17 but the powers of the heart could only be assisted by pleasant things. Thus the king recovered.
At that time Regiomontanus also composed for the bishop the tables of directions. Although he had taken over their foundations from others, a great part of the work is nevertheless his own. And this imparting of works is proper and most worthy of praise – for preceding generations to pass on the seeds of the arts to following generations, and for the following generations to preserve them faithfully and render them illustrious.
At the same time the king of Hungary, Matthias, waged war against the Bohemians, a fact that troubled the peace of many regions nearby. Some of the Bohemians invited Casimir, the king of Poland, some attempted to surrender the kingship to Albert Duke of Saxony, and others invited Matthias, who, since he was the most spirited, preceded the
15 In 1451, George of Trebizond (1395–1484) was commissioned by Pope Nicholas V to produce a Latin translation of Ptolemy's Almagest from a Greek manuscript borrowed from Bessarion. Trebizond criticised the corrections made by Theon of Alexandria in his edition, for which see John Monfasani, Collectanea Trapezuntina: Texts, Documents and Bibliographies of George of Trebizond (Binghamton, NY, 1984), pp. 671–87. Bessarion objected to Trebizond's criticism and defended Theon's commentary on Ptolemy's, Almagest.
16 Matthias Corvinus I (1443–90), King of Hungary, book-collector and patron for scholars and artists.Johannes Vitez, Archbishop of Gran (Esztergom) was also a patron of arts.
17 Reading humoribus for honoribus.
others.18 Since, therefore, Hungary, Austria and all the regions nearby were in great commotion, Regiomontanus looked for a quiet place for his studies and, in 1471, he betook himself to Nuremberg. He himself said that he would like to spend the rest of his life in that town, because there were craftsmen there who were capable of constructing machines necessary for this art, and, furthermore, because he could easily send letters from there to other regions by merchants. For, when he was already famously learned in the subject, had examined the judgements of others, and equipped himself for making mathematics illustrious and promoting it as much as he could, he determined to edit also the ancient writers, and to publish his interpretations.
For that purpose he fitted out a printing-house and set up a list – which still exists19 – of the books he was about to publish, and he began at a fortunate moment with the publication of the Ephemerides covering thirty years. The work was so pleasing at the time that single copies of it sold for twelve Hungarian gold coins, and they were desired eagerly by men of all nations. No Italian, Frenchman, Hungarian, Pole or German had ever seen a similar calendar. The Venetians also brought it to Greece. You know that it is also the will of God that the years are divided, and that learned distinctions are a great pleasure to all those in their right mind. Therefore we should abhor the ravings of the Cyclopes, who either hate or scorn the study of these arts.
The expense of setting up the printing-house was borne mainly by Bernhard Walter20 – the members of whose family are not unknown in these regions – who had very great means as well as cherishing the study of the knowledge of the heavens. He also attracted the good will of many other citizens, so that they began to acquaint themselves with, and come to know, these subjects. These ‘nurseries’ of the subject were first spread in Nuremberg. Therefore up to now the study of these arts flourishes more
18 There seems to be some confusion of names here: Vladislav II (1456–1516), who was elected King of Bohemia in 1471, was the eldest son of Casimir IV Jagiello and brother of Sigismund I of Poland. Territorial agreements were reached in 1478 and 1479 after a protracted military conflict between Vladislav II and Matthias Corvinus I. Albrecht, Duke of Saxony (1443–1500) had married the daughter of Georg, King of Bohemia, whom he tried to succeed (in vain) after the latter's death in 1471.
19 For a description of the trade-list, see Zinner, Regiomontanus, pp. 112–17; and for a photograph of a surviving copy, see L. Jardine, Worldly Goods (London 1996), p. 348.
20 Bernard Walter (d. 1504), a citizen of Nuremberg, was a student of Regiomontanus and had built an observatory in his own house. He purchased books and instruments of Regiomontanus after the latter's death, but there seems to be no positive proof that he financially assisted Regiomontanus' printing enterprise.
in that city than in most universities. And by decree of the most wise senate there is a certain income for the arts, so that they introduce the young to the knowledge of them. In recent times the learned and most worthy man Johannes Schöner had encouraged many to study. You can see what a great thing Regiomontanus had set in motion.
He established new translations of the Almagest and the Geography of Ptolemy, and he also provided verifications for his corrections. In passing he also examined and disclosed other workings of nature. He researched how high vapours are raised up above the earth, and he discovered with admirable dexterity that they are no further away than 12 German miles.
We read that at the time two comets burned in two consecutive years, namely in 1471 and again in 1472. Such drought ensued that the courses of rivers and springs dried up. They also tell expressly that fires in many towns ensued. Also, the Hungarians and the Poles fought among themselves about Bohemia; and the Turkish tyrant Muhammad occupied Walachia in Europe and the kingdom of Trebizond in Asia,21 to leave out much else. Since Regiomontanus knew that comets do not often burn in two consecutive years, and was impressed by the magnitude of the events that had followed, he published an admirable work about the true location and size of comets.22
At the same time there was a man in Erfurt who was very learned in these subjects, Master Christianus.23 How much Regiomontanus set store by him is apparent from his letter, in which he writes that working on the publication of useful writers, he would therefore like to move to Nuremberg, so that he would not be far away from him, since he desired greatly to be helped by him, and to make use of his judgement and criticism in many matters. I praise the openness of the most excellent master, that he was a stranger to disparagement and envy towards the teachers of his art, and I believe that he sought in earnest for companions in this most beautiful toil and for the criticism of a great master.
The edition of the Ephemerides was particularly popular in Rome, and learned men, who knew his intelligence and erudition, desired to recall him
21 For the struggle between Hungarians and Poles around 1472, see note 18 above. Turks occupied Walachia in Europe in 1393 after the battle of Kossowo (1389) and Trebizond, the last Greek state, fell in 1461.
22 De Cometae Magnitudine, Longitudine ac de Loco eius Vero Problemata (Nuremberg, 1544).
23 Christian Roder was a professor of mathematics at the University of Erfurt. Regiomontanus' letters to him are edited in M. Curtze, Urkunden zur Geschichte der Mathematik in Mittelalter und der Renaissance (2 vols., Leipzig, 1902), vol. 1, pp. 324–36.
to Italy, so that they could profit from his teaching. These encouraged Pope Sixtus to invite him to Rome, and as a justification it was put forth that there would be deliberations on the emendation of the calendar. Regiomontanus bore it ill that he had to tear himself away from the printing-house and the edition of Ptolemy, and he had a presentiment of his death, and, as they relate it, he complained that he undertook the journey unwillingly; nevertheless the pope's authority won. Also a reason had been proposed which moved a man who desired to serve the public good. For you know how useful it is to have a computation of the year that is properly ordered and corrected. He did not want to be absent from such a great and useful venture. It is also possible that he shrank less from the journey, because he looked forward to conversations with the learned, and the libraries attracted him. He returned to Rome therefore, where he died in the year 1476.24 That was the year in which the king of Hungary, Matthias, defeated Muhammad in a great battle at Smederevo, in which Muhammad is also said to have been wounded.
Who would not grieve that a longer life was not given to such a man of genius, especially as he was undertaking the most useful things? For he was cut off when he had only entered his forty-first year. The rumour is that poison was given to him by the sons of Trebizond, because he had criticised the latter's translation of Ptolemy and refuted the errors in his commentary and his criticism of Theon.25 His horoscope not only announced a man who would die a violent death (biothanaton), but also one who was upright and combative. Therefore perhaps the liberty of his tongue was harmful to him. But whatever way he was cut off, it was an untimely death and one that even now is to be mourned, because we lack those works which he was yet to begin. It should not be disregarded that to such erudition and his other virtues he added piety towards God, and in order to worship Him by true recognition and invocation he copied the entire New Testament in Greek most elegantly by his own hand; Schöner has shown me that book along with his other memorials. And it is fitting that such great men, who everywhere see the marks of divinity engraved upon the nature of things and consider them attentively for the sake of knowledge, are greatly moved by these tokens of evidence. And they declare much more firmly that this world is made and ruled by
24 Reading 1476 for 1470.
25 The rumour of poisoning was first mentioned in 1482 by Jacobus Sentinus: Zinner, Regiomontanus, p. 152.
an eternal constructing mind than do other men who are strangers to this study of contemplating so many and such excellent marks of God in nature. Indeed, I often shudder throughout my body to think that when God shows most truly by so much evidence that He is nearby, and that He is the observer and judge of all our actions, our hearts nevertheless feebly fear Him and do not burn with trust in Him. Wherever you turn in this entire nature of things, whether you contemplate the shape of the heavens, their motions and harmony with the elements, or observe wonderful skill in the construction of our body or the fertility of the earth, the order in all things cries out that there is an eternal constructing mind. And it is to be recognised as it discloses itself in the Church. Since Regiomontanus did that, let us consider that he now sees the idea of the entire world in God Himself, and the causes of the way26 in which everything is made, that he beholds in person the heavenly bodies, and that he converses most pleasantly with the first fathers about the invention of their arts and their observations, as with fellow philosophers (symphilosophois), and worships God the Creator.
The library of Regiomontanus was bought by Walter, the citizen of Nuremberg of whom I spoke earlier; when he had died, the senate took charge of preserving his effects. That most learned man, Schöner, published some useful writings from among those; but we would have many other, and by far more useful, writings, had the author not been cut off before his time by a tragic destiny.
I have surveyed briefly the history of the most learned man Regiomontanus, which is beneficial for students to know. For although natural excellence cannot be transferred to others, nevertheless others can in some way imitate will-power and assiduity. First, however, let us give thanks to God when thinking about this history, for preserving the arts, and for rousing some worthy minds to study and adorn them; and let us pray to Him that He may not allow such studies to be exhausted now. Then let us also praise in gratitude the memory of Regiomontanus himself, and let us strive to profit from his teaching. Let us also imitate, as much as we can, his studies and assiduity, and let us furthermore love and esteem the arts which he adorned. Let us not reckon that the wonderful sight of the motions of the heavens is put before the eyes of men in vain, or that such a knowledge is useless, but let us truly realise that the order of the motions and its knowledge itself are admonitions from God, and that this knowledge is profitable for
26 Reading quo for hoc.
the life of men in many ways, as it is shown clearly in measuring, in the computation of the year and in the knowledge of the position of countries. I have spoken.