Nicolas Oresme, Grand Master of the high school of Navarre, and the origin of modern science
Nicolas Oresme, Grand Master of the high school of Navarre, and the origin of modern science
Author: Mariano Artigas
Published in: Príncipe de Viana (Science Supplement), year IX, no. 9, Annual Supplement 1989, pp. 297-331.
Date of publication: 1989
1. The high school of Navarre in its first hundred years.
1.1. The foundation of the high school of Navarre.
1.2. The high school of Navarra and the University of Paris.
1.3. The political, ecclesiastical and intellectual environment.
1.4. The beginnings: Jean de Jandun.
1.5. The Oresme era.
1.6. A new era: d'Ailly, Gerson, Clamanges.
1.7. evaluation as a whole.
2. Oresme and the physical school in Paris.
2.1. Science in the 14th century: Oxford.
2.2. The approaches of the Paris School of physics.
2.3. Jean Buridan and his disciples.
2.4. Nicolas Oresme.
3. Oresme's scientific contributions.
3.2. The geometric representation of qualities.
3.3. The law of accelerated motion.
3.4. The fall of the bass.
3.5. The impetus theory.
3.7. Science and astrology.
3.8. The Economics.
3.9. The scientific method.
4. Oresme's place in the history of science.
Since Pierre Duhem's monumental historical work on the development of science from Antiquity to the Renaissance, it is generally acknowledged that the systematic birth of modern experimental science in the 17th century was preceded by a long gestation in which the medievalists played an important part. framework During the 14th century, which is the time frame of our study, the most important work was carried out in Oxford and Paris.
One of the most influential centres of the University of Paris was the high school of Navarre, to which are associated the names of such outstanding figures as Pierre d'Ailly and Jean Gerson at the end of the 14th and beginning of the 15th century. But it is undoubtedly Nicolas Oresme who was the great figure of high school in Navarre in the 14th century. Together with Jean Buridan, Oresme is rightly considered one of the main precursors of modern science. In recent decades, his work has been studied with increasing interest. For us, it also has a special significance because of its close connection with the high school of Navarre, whose importance in France and in Europe as a whole has been too little emphasised to date *(1).
In a recent monograph, Bernard Guenée points out that Jean Buridan was undoubtedly, in the mid-14th century, the best of these excellent Parisian masters... He had many disciples, including, above all, Nicolas Oresme. Let us not deceive ourselves: in the history of Western thought,
Nicolas Oresme is an important name. He has the stature of a Descartes... From 1362 onwards, the favour of King John and then King Charles removed Nicolas Oresme from high school in Navarre, but he had dominated it with his presence for more than fifteen years. There he had studied theology and taught the arts since 1348. He had been at the head of the high school from 1356 to 1361. And it was in that high school of Navarre, still so marked by the very recent presence of such a great spirit, that Pierre d'Ailly went to Paris to do his programs of study, around 1363, 1364 or 1365 *(2).
And, with regard to the Colleges in general and that of Navarre in particular, Guenée points out:
The young man who enrolled at School in the Arts, like the more advanced student, was not abandoned in the big city. He was generally welcomed in one of the numerous Colleges that had been founded in the 13th and 14th centuries, where he was lodged, fed and received teaching. The fundamental solidarity, the dominant medium, was high school. And Pierre d'Ailly's destiny owed much to his host, the high school of Navarre. By her will, Joan of Navarre, wife of Philip the Fair, had founded the high school of Navarre before her death in 1305... she had endowed it with magnanimity. It was to house 70 scholars, 50 to study Arts and 20 Theology... The high school of Navarre soon became one of the most remarkable. Just a few decades after its foundation, two features made it a remarkable institution. On the one hand, it was an intellectual nucleus of exceptional quality, as was sufficiently clear from sample the presence of Nicolas Oresme. On the other... this royal foundation high school was a place of particular devotion to the king. It was the king's confessor who chose the scholars. The king's scholars were particularly attached to the king. Every 25th August, the feast of St. Louis, the feast of the high school was celebrated, often attended by the king himself *(3).
The special relationship between the high school of Navarre and the king had important consequences at the beginning of the 15th century. From 1407, the high school was loyal to the Armagnac party, which led to it being taken over and pillaged in 1418. The buildings were almost ruined, and were restored by King Louis XI in 1464. The high school was then enlarged and the Colleges of Boucourt and Tournay were added. After having played a major role in the life of France for almost five centuries, the high school was suppressed in 1790. It was replaced by the Ecole Centrale des Travaux, which later became known as the Ecole Polytechnique.
We will focus on the period from the foundation of the high school in 1304 to the 1420s. This is the first century of the existence of the high school, during which Nicolas Oresme's activity developed.
1. The high school of Navarre in its first hundred years
1.1. The foundation of the high school of Navarre
Joan of Navarre was the only daughter of Henry I, King of Navarre and Count of Champagne, and Blanche, daughter of Robert, Count of Artois. Having lost her father at a very young age, she was taken to Paris by her mother Blanche, who had her educated at the court of King Philip III. In 1284 King Philip gave her in marriage to his son, the future King Philip IV the Fair, so that Joan became Queen of Navarre and of France. The good understanding between husband and wife lasted until her death.
Joan's will is dated 25 March 1304. It begins with some considerations in which she denotes a lively religious sentiment and perfect understanding with her husband. She then goes on to detail in a long list the amounts of money she leaves to each of the people who served her in any way, mentioning them by name and circumstances. In the next clause she provides for the foundation of a hospital. Then, in the most extensive part of the document, he orders the foundation of the high school of Navarre. He then details the sums he leaves to a long list of religious institutions. The final part is devoted to naming the executors of the will, to whom he leaves broad powers to modify as they see fit the provisions concerning the high school of Navarre. The will concludes with two clauses in which King Philip and his son Louis declare their acceptance of everything established by the queen. Further details are found in an addition dated the last day of March. Also dated 25 March, Queen Joan established the Statute of Foundation of the high school of Navarre, written in Latin, which also concludes with a paragraph in which King Philip and his son Louis express their agreement *(4).
For the establishment of the high school of Navarre, Queen Joan left a building in Paris called the Hotel de Navarre, which was located in the rue de S. André des Arcs. It was one of the eight buildings that the kings of Navarre came to own in Paris. However, the executors of the will judged it preferable to sell it and, with the proceeds, to buy a large plot of land situated in the district where the University was located, almost at the end of the ascent of the Saint Genevieve mountain. They were able to do so by virtue of the broad powers that the queen had left them in her will. The buildings of high school were constructed there. The first stone of the chapel was laid on 2 April 1309, and the buildings were ready for use in 1315, when the high school began to function.
1.2. The high school of Navarra and the University of Paris
In 1677 Jean de Launoy, who was Grand Master of the high school of Navarre, published a documented history of the high school *(5), and Hastings Rashdall published in 1895 a documented study on the Universities of Europe in the Age average, which contains numerous data on the University of Paris and its Colleges *(6). These works make it possible to situate the high school of Navarre in the intellectual environment of its first century of operation.
The Colleges, which still play a role today in English-speaking universities B , originated at the University of Paris. To understand their nature, the social circumstances of the time must be taken into account. The students who came to Paris to study at the University were very young. At School of Arts, a compulsory step to continue on to Law, Medicine or Theology, there were students as young as 14 years old. There was therefore a concern to find a place for foreigners where they could be properly cared for at Education. Undoubtedly, there was freedom in this respect, and during the 13th century various formulas were used, but gradually residency program came to be generalised in a house presided over by one of the Masters of the University. The exceptions were the richest, who lived in their own private accommodation with a private tutor , and the poorest who could not afford to pay the residency program. It is understandable that, under these conditions, the foundation of Colleges, endowed with an income by their founders, was of great importance for students who, while not necessarily poor in the usual sense of the term, lacked the means to pay for a decent accommodation in Paris.
This was Queen Juana's intention when she founded the high school de Navarra. It was to house 70 scholarship students. To this end, she arranged for 2,000 livres tornesas, a large sum for the time, to be set aside from her estate as rent for the high school. Rashdall grade that the foundation of the high school de Navarra was much more splendid than that of the already existing high school de la Sorbonne, although the latter would eventually give its name to the University. The students were divided into three groups: 20 in grammar, 30 in arts and 20 in theology, who received a weekly stipend: grammarians received 4 Parisian salaries, artists 6, and theologians 8. It is not surprising that no place was allocated to students of law and medicine, since these were lucrative professions that did not need special encouragement or assistance. Each group was presided over by a Master of the respective School , who received a stipend double that of his own students.
The Master of the Theologians was at the same time the Grand Master of the high school, and was to be appointed by the Dean and the doctors of the School of Theology of Paris. Queen Joan had entrusted the government and patronage of the high school to this School, but the Bishop of Meaux and the Abbot of St. Denis, of agreement with the freedom left by the Queen to the executors of her will, retained the position of governor for themselves and their successors. They drew up Statutes, the observance of which was sworn by the chapter of masters and scholars on 3 April 1315, and which were approved in 1316 by Pope John XXII. On the death of the two figures, there were complaints about the changes they had made in interpreting the will of the foundress, which led to the intervention of the parliament in 1331 and the change of the degree scroll of governor, which later fell to the king's confessor.
The Masters had to live at high school, and it was their responsibility to see to the complete Education of the students, both intellectually and morally. Each class of students had its own area, and they all met together only for religious functions in the chapel. Once the respective Degree was obtained, the scholarship of the student expired. However, if he wished to continue programs of study at another School, although a new concession was required, he had preference over those applying for entrance at high school. In fact, theological students were often associated with the Master in the governance of the high school, and their consent was often required for the Admissions Office of new students.
The Masters of the Parisian Colleges, in addition to having their own classes, supplemented the public instruction of the students in their respective programs of study with their private instruction at their respective Schools. At high school in Navarre, the Master of the artists had to listen to his students' lessons and instruct them, answer their questions and comment on them, in addition to the texts studied at School, on some book of Logic, Mathematics or Grammar, from agreement with the majority of the students; furthermore, the students had to meet to review the lessons on their return from School to help each other. The high school of Navarre accumulated a good Library Services, since it was established that if there was any money left over, it was to be used to buy books. There were also activities of their own, such as the classic intellectual disputes. At the Sorbonne and at Navarre, the Masters of Theology gave lessons, which seem to have been open to non-resident students, and which counted as regular lessons of the School. Grammarians had all their lessons at the high school, since they did not yet frequent a School, while students of arts and theology had to attend to the classes of their respective Schools . B The instruction at high school was complementary to that given in public, and was an advantage for the students at high school.
These characteristics explain why the Masters of the high school had a B influence on the pupils and why there was a certain esprit de corps among them. There are many testimonies from alumni of the high school in Navarre who express great affection for this institution years later and translate this into donations. Likewise, as mentioned above, the high school came to have its own political significance at the beginning of the 15th century.
In the 15th century, the lessons held at high school became even more important. Around 1404, non-residents began to be admitted to study grammar, and gradually the door was opened to philosophers and theologians as well. In 1459 the multiplication of external students led to disorders that required the intervention of a royal commission, one of whose provisions was to establish that only those who were resident in the high school should eat there. At this time, the classrooms in the Rue du Fouarre, where the public buildings were located, became deserted. The importance of the high school of Navarre grew, until it fell, along with the whole system of Colleges and other organisations, at the time of the Revolution.
1.3. The political, ecclesiastical and intellectual environment
The first century of the high school of Navarre is situated in clearly defined political, ecclesiastical and intellectual coordinates.
Politically, much of this period coincided with the conflicts between France and England known as the Hundred Years' War; these conflicts were projected in the early 15th century into the interior of France, giving rise to internal strife in which the high school of Navarre sided with the Armagnacs, leading to their devastation in 1418. However, between 1330 and 1380, coinciding in part with the reign of Charles V, circumstances in Paris were generally peaceful.
In the ecclesiastical field, the stay of the popes in Avignon from 1309 to 1376 led to an increase in relations between the papacy and France, which partly explains the great influence of the University of Paris on major events in the Church. The Western Schism, between 1378 and 1417, had notable repercussions, such as the departure abroad of important professors from the University of Paris and conciliarist attempts to achieve unity. All this was linked to the development of ideas contrary to the temporal power of the papacy, which played a certain role in the Pope's confrontations with the King of France and with the Emperor already at the beginning of the 14th century.
On the intellectual side, the University of Paris was a centre where nominalism had a special echo, which joined the Aristotelian and Averroist traditions already present since the 13th century, and was the origin of a broad scientific movement of great importance for the later development of modern science.
In all three areas, the main characters are linked to the high school of Navarre, including Jean de Jandun, Nicolas Oresme, Pierre d'Ailly, Jean de Montreuil, Jean Gerson and Nicolas de Clamanges.
1.4. The beginnings: Jean de Jandun
Jean de Jandun was one of the first masters of high school in Navarre, as he was there as Master of the artists in 1315, the same year that the high school was set up. He was then about 25 years old. He took part in the aforementioned meeting of masters and students, at which the Statutes of the high school were sworn in. According to the founding charter, the Master of the Artists was to be an outstanding member of the School. It was during his stay at high school that Jandun composed his commentaries on Aristotle's Physics, Metaphysics, Treatise on the Soul and Books on Heaven. These works, which made him famous, must be the product of his teaching in Navarre.
Jandun became even more famous because of his relationship with Marsilius of Padua, author of the famous work Defensor Pacis, which was published in 1324 and was subtitled Against the Usurped Jurisdiction of the Roman Pontiff. Marsilius of Padua was President of the University of Paris in 1312, when Jandun was studying at School of Arts. Along with a critique of the interventions of the ecclesiastical hierarchy in temporal matters, the work contained unorthodox thesis about the spiritual power of the Pope and the superiority of the Council over the Pope. Six propositions taken from this work were condemned by Pope John XXII in 1327. The condemnation also threatened Jean de Jandun; although he did not appear in the book, he was considered by some to be a co-author or translator of it. In fact, in 1328 Jandun left for Germany to seek safety at the court of Ludwig of Bavaria, as did Marsilius, and probably died shortly afterwards.
At Philosophy, Jandun followed an Aristotelian line with an Averroist slant. His impact is undoubted. He is also credited with a work entitled Treatise on the Praises of Paris, which contains comments on the University and its various Schools at the time of the foundation of high school of Navarre *(7). Here are some of the comments he dedicates to the School de Artes:
Beginning with the kind of goods which is the first in honour and dignity, I say that in the villa of villas, in Paris, in the street called du Fouarre, not only are the seven liberal arts taught, but also the most agreeable clarity of all philosophical light, extending the rays of pure truth, illuminates the souls capable of receiving it.... The wonders of divine principles, the secrets of nature, astrology, mathematics, and the salutary resources of moral virtues are there revealed to the eye. Here are gathered together a whole host of wise teachers who teach not only logic, but also all the knowledge that prepares for the higher sciences. Illustrious doctors flourish there, who, with the rapidity of an exercised spirit, go through the mysteries of the lower natures and the heavenly virtues..... There they demonstrate the certain results of an infallible Philosophy and of an indisputable mathematical science, which indicates the marvellous encounters of numbers and figures, whether considered in themselves, or applied to celestial magnitudes, to harmonic sounds and visual rays. O most glorious God, what an idea you have given us of your love for men by giving them the means of knowing the periods which you have fixed for the celestial movements, the distances of the centres, the magnitude of the worlds, the status of the poles, the virtues of the Signs, the order and dispositions of the planets! *(8)
Jandun's commentary on the School of Theology, Rue de la Sorbonne, after the corresponding praise, includes a commentary that contrasts sharply with the one dedicated to the School of Arts:
Nevertheless, although all these men, who profess to be ardent seekers after truth, happen to aim at a single and supreme end, namely the knowledge and the love of the sovereign Trinity, it frequently happens to them (which does not fail to astonish simple people) that they hold opposite opinions about the same conclusions.... The one objects, the other resolves the objection; the one retorts, the other refutes. And, to sum it up in a few words, whatever in the discussion of these problems the one endeavours to hold strongly with a mighty hand, the other studies to turn over with an uplifted arm, except that they confess from the beginning fully their sacred and inviolable acceptance of the articles of faith. What use or advantage does the Catholic religion derive from this exercise? God knows, and these same men will endeavour to make known to anyone who asks them, not for the purpose of mockery but of instruction, in the proper place and at the proper time, the reason for this mode of proceeding *(9).
It is not difficult to suppose that these comments come from a spirit tormented by conflicts, typical of Averroism, between Philosophy and theology. Averroism had been one of the constants of the University of Paris in the 13th century, and Jandun presents himself as a faithful follower of Averroes, although he disagrees with him in astronomy by not admitting eccentrics and epicycles. The harmony between faith and reason, achieved in the great systematisation of Thomas Aquinas, was also compromised by the nominalist current promoted by William of Ockham. Ockham's fate bore some resemblance to that of Jandun, for after being summoned to the papal court of Avignon in 1323 he also sought refuge, in 1326, in the domain of Louis of Bavaria.
But Jandun's comments are of interest for another reason, for they give an insight into the orientation of the School of Arts, centred on logic, Philosophy and the natural sciences. It is tempting to imagine the teachers and students of the fourteenth-century School of Arts immersed in trivial and irrelevant questions which, of course, would have nothing to do with natural science. The reality was, however, very different. In the first place, as Jandun's commentary shows, the School de Artes could also be considered as School of Philosophy; in fact, the chapter he devotes to it refers in his degree scroll to the School de Philosophy or de Artes. Moreover, it is also noted grade that there are numerous references to questions of mathematics and natural sciences. This is not accidental. As Edward Grant notes, the teaching in the School of Arts or Philosophy was strongly oriented towards logic and science and, given that the Arts curriculum was a prerequisite for access to the other Schools, it can be concluded that the scientific orientation was B, more so than at any other time in the history of the higher teaching *(10). Of course, this judgement has to be qualified by the fact that the science in question was not yet what it would become in the 17th century.
Precisely in 1328, when it seems with some probability that Jandun died, Jean Buridan was President of the University of Paris. Buridan was again President in 1340 and, although he held various ecclesiastical benefices, he was always teaching in Paris. He was the inspirational head of the physics school in Paris, which had an influence B not only in France but throughout Europe. His influence on the high school of Navarre was felt through his disciple Nicolas Oresme.
The decades in which Buridan and Oresme exercised their magisterium were relatively peaceful at the University of Paris. The confusions of the 13th century were long gone. In 1210, the bishops of the provincial council of Sens had forbidden the use in Paris of the books of Philosophy natural of Aristotle and his commentators, although Pope Gregory IX ordered in 1231 that those who had incurred in this prohibition be absolved, and in the same year he ordered that the books be revised in order to remove what was erroneous so that what was useful could be studied. In 1255, the works of Aristotle were ordered to be read at the School de Artes. But the disputes provoked by Averroism led to the condemnation of 219 propositions by the bishop of Paris, Stephen Tempier, in 1277, and among them were quite a few of Aristotelian flavour and even some of Thomas Aquinas. In 1325, this decree was revoked insofar as it affected Thomistic doctrine *(11). It is true that there were new condemnations, this time concerning nominalism, around 1340, but they do not seem to have had any major consequences, and something similar happened with the echoes, still lasting after several decades, of the conflict of Jean de Jandun. The era of Buridan and Oresme, which lasted from about 1330 to 1380, meant half a century of peace at the University.
Something similar happened with the status interior of France from the 1360s to 1380s, which coincided with Oresme's period of flourishing. Charles V of France, first as dauphin and governor and from 1364 as king, achieved a B unity, and Oresme was one of his advisors. From 1348 Oresme was in the high school of Navarre, of which he was appointed Grand Master in 1356. It seems that it was his work at that time that attracted the attention of the future king, so that he called on Oresme as partner in economic and cultural matters, and commissioned him to write several commentaries on the works of Aristotle, eventually promoting him to position as bishop of Lisieux, where he stayed from 1377 until his death in 1382. In this peaceful half-century we find the important works of the Physics School of Paris, whose main protagonists, together with Buridan and Oresme, were Albert of Saxony, Henry of Hesse and Marsilius of Inghen.
Shortly before Oresme's death, the status changed abruptly. Charles V died in 1380, and the internal peace of France was disturbed from then until the middle of the 15th century. In 1378 the Western Schism began, which shook the University of Paris to its core: several important masters were in favour of the Pope of Rome, Urban VI, against the Pope of Avignon, and this provoked an exodus which, on the one hand, meant the decline of the University of Paris and, on the other, the spread in Europe of the ideas of Buridan and Oresme. Albert of Saxony had left Paris earlier, and in 1365 he was the first President of the University of Vienna. Because of the Schism, Henry of Hesse went to Germany and then to Vienna, and Marsilius of Inghen left Paris for Heidelberg, where he was the first President of the University. A new epoch began in which the central topic was the internal peace of the Church. At this time, several figures of the high school of Navarre played a very important role in the University of Paris and in the life of the Church.
1.6. A new era: d'Ailly, Gerson, Clamanges
In the following decades, among the great figures of the University of Paris were Pierre d'Ailly, Jean Gerson, Jean de Montreuil and Nicolas de Clamanges, all of them from the high school of Navarre. The first two were also prominent protagonists in the attempts to bring peace to the Church.
Pierre d'Ailly's public activity roughly coincided with the beginning of the Great Schism in 1378. He had joined the high school of Navarre in 1363, almost immediately after Oresme ceased to be Grand Master. In 1368 he was Master of Arts at high school, of which he became Grand Master in 1384, having Gerson and Clamanges as pupils. In 1389 he was Chancellor of the University of Paris. He insisted on the need for a Council to end the division of the Church, and played an important role in the Council of Pisa (1409) and the ecumenical Council of Constance (1414-1418).
Pierre d'Ailly also wrote works related to science. In his Imago mundi of 1410, Christopher Columbus found support on how to travel to the Indies from the west, and the text of this work, annotated by Columbus' own hand, is preserved. In 1411 he finished a pamphlet on the correction of the calendar, and read it in March 1417 at the Council of Constance, preparing for the reform of Gregory XIII. In 1414 he finished his work Concordance of Astronomy with Historical Truth, and was the author of other pamphlets on geography, astrological superstitions and astronomy. He carried out his scientific work in the midst of intense public activity, and was appointed Cardinal in 1411, going to Germany in 1414 as bequest of the Pope. He died on 9 August 1420, leaving generous donations to the high school of Navarre, to hospitals and to the poor.
Jean Le Charlier, called Gerson after the place of his birth, studied from 1377 at the high school in Navarre, where he was taught by Pierre d'Ailly. He was awarded a doctorate in theology in 1394, and in 1395 he succeeded d'Ailly as Chancellor of the University of Paris. He achieved great renown and had an intense public activity, dealing especially with the problem of the unity of the Church, together with d'Ailly. He did not take part in the Council of Pisa, but he did, and with great force, in the Council of Constance.
It was precisely in Constance that he played a role of great importance for the events that afflicted the high school of Navarre in the following years. On 23 November 1407, the Duke of Orleans had been assassinated in Paris at the instigation of the Duke of Burgundy, and Jean sans Peur not only took responsibility for the act, but also justified himself to the king and entrusted his defence to Jean Petit, who publicly defended the legitimacy of the tyrannicide. As Chancellor of the University, Gerson submitted this doctrine to the judgement of the doctors of theology, who condemned nine propositions of Jean Petit. In 1415, at the Council of Constance, Gerson again denounced these ideas and obtained a general condemnation of tyrannicide by the Council Fathers. This general condemnation did not satisfy Gerson and the Armagnacs, and Gerson still strongly expressed his dissatisfaction in his speech of 5 May 1416 in Constance. When the Council ended in 1418, he learned that Jean sans Peur had sworn to get rid of him and that others had order fiercely punished him; consequently, he did not return to Paris, and took the path of exile, going to Austria and then to a Benedictine abbey in Germany. In 1419 he learned that Jean sans Peur had been murdered. He set out on his return to France, but did not reach Paris. He stayed in Lyon, where he lived until his death in 1429. In Lyon he met again his old friend Gérard Machet, confessor to King Charles VII and also to high school of Navarre.
In Paris, the high school of Navarre was clearly in favour of the Armagnacs and against the Bourgognons. Gerson had a discussion with the members of the high school, which led to the bishop of Paris condemning, on 23 February 1414, the statements with which Jean Petit justified the tyrannicide of Jeans sans Peur. This led to reprisals when power was in the hands of the Armagnacs. On 12 June 1418 there was an insurrection in Paris, accompanied by a massacre of prisoners, including several members of the high school of Navarre, which was invaded and sacked. On 14 July Jean sans Peur solemnly entered Paris. On 20 August there was a new massacre, which together with the previous one resulted in some 3,500 dead. Jean sans Peur was assassinated on 10 September 1419, but neither d'Ailly nor Gerson returned to Paris.
The status of the University and the Colleges became lamentable at this time. From 1420 onwards, English rule was established in Paris, and the decline of the University was increasing. The number of graduates decreased considerably issue . Finally, normality returned in 1444, and the Hundred Years' War ended in 1453. The University, which received from the Pope's bequest a new Statute in 1452, slowly recovered.
Like Gerson, Nicolas de Clamanges had been a disciple of d'Ailly at the high school of Navarre. The three of them formed a group committed to ecclesiastical reform and the peace of the Church that had B influence at the University of Paris, in France and throughout the Church. Clamanges entered the high school of Navarre when he was twelve years old, becoming professor of theology in 1386. In 1393 he was President of the University. In 1397 he left the teaching to become private secretary to Benedict XIII, so that he was involved in difficulties when France broke with the Pope of Avignon. Clamanges always sought peace and understanding, and wrote several writings in which he described the circumstances of the time with lamentation; one of them is a letter addressed to the professors of the high school of Navarre in which he justifies his actions.
Unlike d'Ailly and Gerson, Clamanges returned to high school in Navarre after the events of 1418 and continued to teach there. He was buried in the chapel of the high school in 1434. Gilbert Ouy argues that Clamanges was a pioneer of the new Humanism, which was usually confined at that time to Italy, so that the high school of Navarre would have been the main origin of Humanism in Paris.
Among the members of the high school of Navarre who were particularly influential at that time was also Jean de Montreuil, who was financial secretary to King Charles V and has been described as the first humanist in France. A disciple of d'Ailly, he died during the sacking of the high school of Navarre in 1418.
It is no exaggeration to say that the high school of Navarre exerted an influence B on the intellectual, political and religious life of France during the first century of its existence. In the programs of study on that period, references to the high school are very frequent, and there is unanimous recognition of its importance. José Yanguas y Miranda, writing about the death of Queen Joan of Navarre in 1305, notes that
The Queen was buried in the convent of Saint Francis in Paris, in whose city she had founded the famous high school called de los Navarros *(12).
José María Lacarra mentions it in these terms:
Finally, we will remember, as a work of the court, but of the French court, the famous high school of Navarre, founded in Paris in 1304 by Queen Jeanne, to house 70 poor students, and in order to have more masters and doctors in the field of theology. It was the first establishment of its kind at the University of Paris *(13).
From agreement with Gilbert Ouy, it seems that the relevance of the high school of Navarre is even greater than it is already recognised. George Sarton, in his monumental History of Science, states that the high school of Navarre played a fundamental role not only in the University of Paris but in the Kingdom of France, being the first real foundation of that subject and perhaps the most famous of all French Colleges, and certainly the most important of them at the time of Oresme *(14). This assessment does not only apply to humanism, as Ouy does. As far as science is concerned, the impact of Navarre's high school reached a universal level, as is evident from a closer look at the figure and work of Nicolas Oresme.
2. Oresme and the physical school in Paris
2.1. Science in the 14th century: Oxford
The Universities of Paris and Oxford were during the 14th century the two main centres for the formulation of concepts that were to have a major influence on the birth of modern science. The two universities were closely linked, which explains why it is sometimes difficult to determine whether a particular development took place first in Oxford or in Paris. In any case, Oxford's orientation was more towards logic, although there was no lack of programs of study on natural science. In the 15th century, the logical orientation of Oxford also influenced Paris, so much so that Pierre Duhem came to consider this fact as a cause of the decline of the scientific programs of study in Paris.
Among the pinocheros of Oxonian science, Robert Grosseteste (+1253) and Roger Bacon (+1292) stand out. Grosseteste probably studied Arts at Oxford, but went to Paris to study Theology, reaching his doctorate around 1214. He was later one of the first chancellors of Oxford University. His works include commentaries on Aristotle's Physics and Posterior Analytics, as well as various treatises on cosmological, geometrical and optical questions. He made interesting experiments on the problems of optics. Besides contributing to the introduction of Aristotle to the West, one of his main merits is that he was Bacon's teacher.
Roger Bacon studied at the University of Paris, where he was one of the first to use Aristotle's works on natural science around 1237. He returned to Oxford around 1247 to devote himself to research, carrying out an extensive experimental work over the next twenty years. His impact was not only in terms of the concrete results he achieved, for example in the field of optics, but also in the great emphasis he placed on the use of mathematics and experimentation as a means for the progress of science.
The tradition of Grosseteste and Bacon was continued in the 14th century by Richard Swineshead, John Dumbleton and Thomas Bradwardine, whose work lasted until about the middle of the century, and William Heytesbury, who died around 1373. They are a clear sample of the interest of the 14th century University in questions of logic and natural science. Of particular note is Bradwardine's mathematical representation of motion *(15) and Heytesbury's velocity theorem average , also known as the Merton College theorem, which played an important role in the formulation of Galileo's law of the fall of gravities and thus in the foundation of modern physics.
Pierre Duhem insisted on the superiority of Paris over Oxford, claiming that there were few and uncertain data about the work done at Oxford. In any case, it is clear that there were close links between the two universities and that extensive efforts were made at both, which had a decisive influence on the development of natural science.
2.2. The approaches of the Paris School of Physics
The influence of Aristotle in the Age average was subject to great variations. It would not be correct to imagine that era as having been subject to a universal and unconditional adherence to the doctrines of the Stagirite. For one thing, their reception in the West was rather late, and took place mainly in the 13th century. Moreover, as recalled above, that century was marked by a distrust of the possibility of harmonising Aristotelianism with Catholic theology, and there were various prohibitions and condemnations in this respect. The difficulties were even greater because Aristotelianism was presented together with Averroist interpretations.
The condemnation of 1277 had a great impact. It particularly emphasised divine omnipotence as opposed to Aristotelian necessitarianism. Among the 219 propositions condemned, 28 explicitly referred to natural science and its philosophical implications, and had notable repercussions for the subsequent development of science, so much so that Pierre Duhem went so far as to assert that, if a specific date were to be given for the origin of modern science, it would be 7 March 1277, the day on which Bishop Tempier's decree is dated. In any case, the consideration of some of the propositions condemned by Tempier will help to assess important aspects for the development of science.
Proposition 6 stated that "when all the celestial bodies return to the same point, which happens every 36,000 years, the same effects will return as now exist". Stanley Jaki has devoted an entire, extensively documented work to show that science encountered successive abortions in ancient cultures due, in large part, to pantheistic ideas, often associated with an organic and cyclical universe in which everything would be governed by an inscrutable destiny *(16). The idea of eternally recurring cycles is also found in the Greek world. Jaki sample the importance of the Christian idea of creation in the birth of modern science. This idea definitively broke the old pantheistic paradigms and instilled a strong confidence in the rationality of the universe, created by an infinitely intelligent God staff , and in the cognitive capacity of man, created by God in his image and likeness, so that the Christian cultural matrix had a decisive impact on the birth of science in Christian Europe. This fact, moreover, has been emphasised by researchers of very different ideological tendencies.
This cultural matrix was taking shape in the Age average, and the condemnation of 1277 concretely embodied some of its basic features. For example, proposition 34 stated "that the first cause (i.e. God) could not make more than one world". This conclusion, which plays an important role in Aristotelian physics, was unacceptable for the Christian faith in an all-powerful creator God who created the world freely. The same was true of proposition 49, according to which "God could not move the heavens with a rectilinear motion, for in that case he would leave a void". In 92 it was stated that "the heavenly bodies move due to an internal principle, which is the soul". By rejecting these and many similar propositions which implied a single world, reduced to the solar system and the visible stars, composed of incorruptible stars which necessarily moved in a circular motion and in which there was no void, the door was opened to new speculations which were of decisive importance in the birth of modern science.
William of Ockham's nominalism seemed to go from agreement with the condemnation of 1277, since it stressed divine omnipotence and seemed to cast doubt on the possibility of a natural science capable of reaching conclusive demonstrations. In fact, the impact of nominalism at the University of Paris was B. But it also met with resistance and condemnation: between 1339 and 1346 there were two reprobations of nominalism by the School des Arts de Paris and two others by the papal curia *(17). The Paris School of Physics is often considered as a branch of nominalism. It is, however, a very mitigated nominalism, which does not accept Ockham's extreme thesis which, in fact, is incompatible with a valid science, since Ockham does not admit the existence of a natural order cognizable by reason. Jean Buridan and Nicolas Oresme admitted the value of natural science and rational theology, and for this reason they were far removed from nominalist scepticism. This distancing enabled them to tackle in depth the scientific problems, the approach to which requires the admission of the existence of a natural order goal and of the human capacity to know it.
The approaches of the Paris School of physics have an originality that may escape the superficial observer. This is due, in part, to the method followed in the exhibition of the ideas. A large part of the works of the School are commentaries on the works of Aristotle, from agreement in accordance with the custom of the time. However, although Aristotle is treated with respect by the commentators, there are many discrepancies, which become substantial and extensive on many basic points. This led to a singular approach : on the one hand, interest in Aristotle led to the admission of the existence of a true science of nature and stimulated the formulation of problems, but at the same time many of Aristotle's ideas were clearly and harshly criticised as an obstacle to the development of science.
So, when it is claimed that Aristotelianism was an obstacle to science, this may perhaps be valid if one thinks of the 17th century Aristotelians, with whom Galileo was confronted. But in the physics school in Paris in the 14th century, 300 years earlier, the study of Aristotle was carried out in a very different critical spirit. Although there were not yet sufficient experimental concepts and means at their disposal, they worked with a genuinely scientific orientation that gave rise to the development of scientific concepts of great relevance.
2.3. Jean Buridan and his disciples
The soul of the physical school of Paris was Jean Buridan. He was born at the end of the 13th century in Béthune. In addition to the documents in which he appears as President of the University of Paris in 1328 and 1340, there is another one dated 12 July 1358 in which his name appears signature, and which is the last documentary reference letter of him. He wrote a commentary on the logic of William of Ockham, and numerous commentaries on the works of Aristotle, not only on physics but also on metaphysics, ethics and politics. Not many originals are available, but there are notes by his Germanic disciples, which testify to Buridan's influence in the new universities of central Europe.
Although he is often placed in a nominalist line, Buridan's nominalism is certainly very mitigated; for example, he explicitly admits the value of causality and its use to prove the existence of God, which is incompatible with a genuinely nominalist position.
Buridan's main merit in the scientific field consists in his theory of motion and impetus, which is the impulse statement to a projectile when it is launched, and applies also to the fall of bodies under the action of gravity. Although he was not the first to propose this notion, he developed it very effectively, applied it to various problems, and his ideas continued through his disciples and played a decisive role in the formulation of the basic concepts of the science of Galileo and Newton. His works were widely studied in Europe during the 16th century, which accounts for his influence on the gestation of modern science. Although they are mainly commentaries on Aristotle, they are critical commentaries which, according to the custom of the time, are entitled Questions, in which many problems and criticisms are raised, and ideas that are really incompatible with Aristotle's mechanics are proposed. This was the path which, in fact, made it possible to take advantage of Aristotle's philosophical depth while overcoming the limitations of his scientific approaches.
In addition to Oresme, Buridan's disciples included Albert of Saxony, Henry of Hesse, Marsilius of Inghen and Themon.
Albert of Saxony or Helmstaedt, who was also called Albertutius to distinguish him from Albert the Great, was born about 1316 in leave Saxony, and acted as a transmitter of Parisian ideas to Vienna. He obtained the Degree master's degree in Paris in 1351, and at that university he held various offices in the English nation; by that name was designated one of the four nations or groups into which the students were organised, and on the occasion of the Hundred Years' War against England it was taken by the Germanics. In 1353 it was President of the University. It is recorded that in 1362 he was still in Paris. In the following years he was one of the promoters of the new University of Vienna, of which he was the first President in 1365. He died in 1390, while bishop of Halberstadt. He wrote six works on logic, five of which were published in print in the 15th century, when the printing press was already available, which is a sign of his influence. He wrote commentaries on Aristotle's physical works, which were also widely circulated, as well as on ethics and Economics. In physics he used Buridan's ideas and proposed several developments of his own, for example about the centre of gravity and uniformly accelerated motion, and admitted the rotation of the earth. In addition to the influence of his works, he also influenced posterity through Henry of Hesse and Marsilius of Inghen.
Henry of Hesse, of Langenstein or of Hainbuch was born around 1325. It is recorded that in 1363 he was in Paris and that he was a doctor of theology, also in Paris, in 1376. Before 1381 he was vice-chancellor of the Sorbonne. Due to the difficulties created by the Great Schism, he left Paris in 1383, as he was a supporter of the Roman pope Urban VI, while the king of France adhered to the pope of Avignon, Clement VII. After passing through Germany he went to the University of Vienna, and it seems likely that it was he who obtained from the pope the bull founding that university in 1384. He died in Vienna on 11 February 1397. His writings are very numerous. He seems to have played an important role in the development of mathematics in Vienna and, through there, in Germany. He dealt especially with astronomical questions, helping to discredit astrological beliefs; in this respect he followed in the footsteps of Oresme, whose disciple he may be regarded as to some extent. He rejected the Aristotelian theory of the different nature of the celestial and sublunar worlds. Together with Oresme, he contributed to the diffusion of the scientific mentality that was painstakingly making its way. In addition to other scientific contributions, he was the author of books on spirituality which were very successful, and he was active in the unity of the Church in the same conciliarist line as the leading personalities of the University of Paris.
Marsilius of Inghen was born in Holland around 1340. In 1362 he gave his first lesson as a teacher in Paris. He was President of the University in 1367 and 1371, and held various offices in the English nation. He was a very popular teacher. In 1376 he accompanied Pope Gregory XI to Rome, and was there when in 1378 Urban VI was elected, of whom he was a supporter during the Schism. For this reason he had to leave Paris. He was the first President of the University of Heidelberg, founded in 1386 by Pope Urban VI, and was re-elected to that position six times. He died in Heidelberg on 20 August 1396. Like Albert of Saxony and Henry of Hesse, he basically followed the scientific line of Buridan and Oresme. He composed commentaries on Peter Lombard's Sentences and on various works of Aristotle.
Themon, called the son of the Jew, who was in his youth in Westphalia, was also in the same line. Converted to Christianity, he finished the programs of study at the School of Arts in Paris in 1349, and at that University he was a famous teacher with B influence over his pupils. After a stay in Germany, he returned to Paris in 1353, and held offices in the English nation between that year and 1361. He is missing from data from the latter date. His writings have great similarities with those of Albert of Saxony. Many ideas of both are later found in Leonardo da Vinci and other later scientists.
Buridan undoubtedly deserves credit for having been the head of the Paris School of physics, whose ideas were elaborated and transmitted throughout Europe thanks to the work of the Germanic masters mentioned above. But, in a certain sense, Nicolas Oresme is the centre of this school, because of the breadth of his work, the originality of his ideas and the great influence he exerted on the intellectual, political and ecclesiastical spheres.
Sarton called Oresme one of the greatest scientists of the 14th century; one of the greatest mathematicians, mechanics and economists of the Age average; and one of the founders of French scientific language and French prose in general.
Nicole Oresme was born around 1325 near Caen in Normandy. He entered the high school of Navarre in 1348 to study Theology, which implies that he was already a Master of Arts; on the list of 29 November 1348 he appears, on the high school of Navarre, as Master of the Norman nation. He must have obtained the doctorate in Theology in 1356 at the latest, as this was a requirement to become Grand Master of the high school of Navarre, position which he held from 4th October 1356. At that time he must have written many of his Latin treatises, which attracted the attention of the royal family, with whom he was closely associated from then on. He had to abandon his appointment as archdeacon of Bayeux in 1361, because the Parliament of Paris ruled that position was not compatible with his position as Grand Master of high school of Navarre. He was appointed canon of Rouen on 23 November 1962, although it seems that he continued to teach at the University of Paris. On 18 March 1964 he was Dean of the Cathedral Chapter of Rouen. From 1369 to 1377 he composed his commentaries on Aristotle's Ethics, Politics and De Caelo; in addition to their philosophical and scientific interest, these were translations into French, the first complete versions of Aristotle's works in a modern language , commissioned by King Charles V. On 3 August 1377, on the king's initiative, he was appointed bishop of Lisieux, position which he held until his death on 11 July 1382 *(18).
Oresme's activity was multifaceted. A profound connoisseur of Philosophy and theology, interested in scientific questions to which he made notable personal contributions, advisor to the king, author of economic programs of study whose ideas served to stabilise the French Economics , influential ecclesiastic, and innovator of the French language in both its ordinary and scientific usage, he displayed a balanced personality which enabled him to work successfully in all the above-mentioned fields and which earned him the royal confidence, even though he openly fought the king's astrological interests in writing. This balance can also be seen in his writings, sample , where he shows himself to be a man of his time, who was subject to the intellectual limitations of his cultural environment, but who had the necessary firmness to decisively depart from traditional ideas, however deep-rooted they might be, whenever he felt it necessary to do so.
Oresme's works have been the subject of extensive programs of study since the beginning of the 20th century. Among the pioneers of these works are P. Duhem, H. Wieleitner and L. Thorndike. Some have been published in recent decades, in whole or in part, in annotated editions *(19). With some exceptions, as in the case of the commentary on Aristotle's De caelo which ends by indicating that it was composed in 1377, it is not easy to establish the dates on which they were written. There are four writings devoted to combating astrology, three on mathematics and eleven on physical matters, in addition to his treatise on coinage, and the translation and commentary on Aristotle's De caelo *(20).
3. Oresme's scientific contributions
Given the multifaceted nature of Oresme's work, it seems appropriate to classify his contributions to the development of science from agreement with the different disciplines. The areas to which he contributed original works are mathematics, mathematical physics, cosmology, scientific method, the fight against astrology and Economics. These subjects are dealt with simultaneously in several works and there are, of course, important relationships between them *(21).
Oresme's mathematical contributions are mainly to be found in his works De proportionibus proportionum, Quaestiones super geometriam Euclidis and Algoritmus proportionum *(22).
At the basis of these contributions is the study of the use of fractional numbers as instructions and exponents of algebraic relations. Oresme uses the term proportio as a ratio, fraction or simple proportion (ratio ), and as proportionality or equality of two ratios, fractions or simple proportions. The title De proportionibus proportionum refers to simple ratios raised to an exponent which, in turn, is a simple proportion: if the exponent is rational, the expression will also be rational, and it will be irrational otherwise. For example, 4/1 raised to 1/3 or 2/3 are irrational expressions; in Oresme's terminology , the former is a part of the rational expression 4/1 because it is smaller than it and the exponent is a fraction whose numerator is unity, while the latter is a part of 4/1 because, being also smaller than it, the exponent contains a numerator and denominator that are minimal and distinct from unity. These considerations are applied by Oresme when he studies the concepts of commensurability and incommensurability, which play an important role in his attacks on astrology.
The treatise Algoritmus proportionum is dedicated to Philip of Vitri, Petrarch's friend, while he was bishop of Meaux, which allows us to place its composition between 1351 and 1361, therefore at the time of the high school of Navarre. Philip was a well-known musician and Oresme begins his dedication with a touch of humorous politeness, as he writes that he would call his friend Pythagoras if it were possible to believe in the doctrine of the transmigration of souls. After establishing the nomenclature he will use, Oresme sets out the rules for multiplying and dividing proportions in which integer or fractional exponents are found. He first establishes the rules for multiplying or dividing rational expressions, and then analyses the expressions in which fractional exponents are found: he studies how to formulate irrational expressions more appropriately, then sets out the rules for multiplying or dividing a rational expression and an irrational one, and then studies the general rules for performing these operations with two irrational expressions.
According to Grant, this is the first known attempt to systematically study operational rules for this subject of mathematical expressions. Oresme's terminology refers to addition and subtraction, probably because he is thinking of the operations to be carried out with exponents in order to multiply and divide expressions raised to any power. On the basis of the available documents, Grant formulates the conjecture, which he considers perfectly plausible, that Oresme created a new type of mathematical treatise.
The problem of the proportion of proportions, in the sense already mentioned, is recurrent in the 14th century. Grant comments that, in Oresme's treatise De proportionibus proportionum, we seem to find for the first time an explicit study of irrational exponents.
Moreover, Oresme formulated in the same treatise a proposition about mathematical probability which, according to Grant, may be one of the first formal propositions of probability theory, and perhaps the first in an absolute sense.
Finally, in his treatise Questiones super geometriam Euclidis, written perhaps around 1350, at the time of high school of Navarre, Oresme studies the topic of infinite mathematical series, to which much effort was devoted in the 14th century. In the first half of the century, in Oxford, Swineshead, Dumbleton and Heytesbury worked extensively on this topic . Oresme's analysis contains elements of great value.
In addition to the topics mentioned above, another mathematical contribution of Oresme, and probably the most important, was the introduction of coordinates to represent graphically the variation of qualities. Because of the great importance of this issue in its application to physics, it seems appropriate to consider it among the contributions to mathematical physics.
3.2. Geometric representation of qualities
Oresme's contributions to physics include those to cosmology or the study of the universe, which will be considered elsewhere section. Limiting attention for the moment to more specific aspects of mathematical physics, Oresme's contributions mainly cover three issues: the graphical representation of qualities, the application of that representation to the study of uniformly accelerated motion, and considerations on the vacuum. The study of these topics can be found in the works Tractatus de configurationibus qualitatum et motuum, De proportionibus proportionum and Le livre du ciel et du monde *(23).
Marshall Clagett states that the Tractatus de configurationibus qualitatum et motuum was probably written by Oresme in the 1350s, during his stay at high school in Navarre. This treatise contains 93 chapters divided into three parts. In the first, the doctrine of geometrical representation is formulated and applied to qualities, and it is broadly suggested how it can be used to explain many physical and even psychological phenomena.
This is, of course, a first-order topic with regard to the actual formulation of a mathematical science of nature. Although philosophical doctrines according to which numbers and geometry played a central role in the explanation of nature had already been proposed since antiquity, it was not until the 14th century that such ideas had been applied in many concrete areas. One of the most important aspects for the birth of the new science was undoubtedly the formulation of quantitative explanations of qualitative phenomena, and this required adequate means.
It was not until the 17th century that mathematical methods, such as differential calculus, which would revolutionise scientific methods, were achieved, but in the meantime two important steps were taken which prepared for future developments. One was conceptual, and concerned the possibility of studying mathematically those aspects of nature that did not seem to be related to the quantitative, i.e. qualities. The other was mathematical, and consisted of providing instruments that, although still to be improved, greatly expanded the effective possibilities of mathematical physics. In both respects, Oresme made important contributions.
In addition to Oresme, others worked along these lines. Prominent among them were the Oxonians, and especially William Heytesbury. Robert Grosseteste had already insisted on the fundamental importance of mathematics in the study of physical phenomena and had applied geometry to optics, thus promoting the scientific orientation of Oxford. There, motion was studied quantitatively, a distinction was made between kinematics and dynamics, the concept of velocity was analysed and the law of uniformly accelerated motion was basically established, formulating the famous velocity theorem average of Merton College. In these aspects, the priority probably belongs to the Oxford authors, although Oresme also dealt with them and it is not easy to establish the relationship between the Oxford work and the Paris work. As already noted, the exchange between the two universities was very wide and similar problems were often studied at both.
However, Oresme has undoubted primacy in two central aspects: the breadth of problems to which he applies mathematical approaches and the use of coordinates for the graphical representation of variations in qualities and motions. For this reason, Duhem stated categorically, in the title of a section of his magnum opus, that Oresme was the inventor of analytical geometry *(24). This assertion seems risky and, in fact, Duhem was more moderate in detailing this question, but he rightly concluded that, at least, it is difficult to deny that Oresme took the first steps in this direction.
At the beginning of his treatise on the representation of qualities, Oresme presents his work as a development of his own ideas and of those who before him have dealt with this subject of programs of study, without claiming absolute originality. Oresme's basic idea is that any quality that can acquire successively different intensities can be represented by a straight line raised vertically above each point of the subject affected by this quality. On a horizontal line the extension of the body on which the quality is studied is represented, and at each point of this line a vertical line is raised whose height is proportional to the intensity of the quality. This results in a geometrical figure that financial aid helps us to easily understand the characteristics of the phenomenon being studied, since, as Oresme reminds us when dealing with this question, our knowledge relies on the senses and is aided by the imagination through resource .
The thought that guide to Oresme is that everything that can be measured can be imagined in the manner of a continuous quantity, such as lines and surfaces. Therefore, the intensities that can be acquired successively can be imagined by means of a straight line raised vertically above each point of the subject they affect, so that the measure of these lines will provide the measure of the intensities. The qualities studied can be those that are usually considered as qualities, such as colour, but also others that are not usually understood under this concept, such as speed. The latter is particularly important for the representation of movement.
Oresme expressly states that the representation he proposes extends universally to every conceivable intensity, both in terms of types of qualities and subjects, which may or may not be sentient; and, as has been pointed out, it expressly refers also to psychological phenomena. Stanley Jaki notes in this respect that this is an effort to apply the quantitative method, characteristic of experimental science, to an enormously broad field of phenomena; it is not only a partial contribution to specific problems, however important they may be, but it is also necessary to underline the mentality involved in these approaches and what this mentality means for the consolidation of the quantitative method of modern science *(25).
This use of rectangular coordinates is an original contribution by Oresme, as is his application to the mathematical study of qualities and, in particular, to the study of motion. For this reason, Hugo Dingler attributed a decisive importance to Oresme as the author of a conceptual revolution which made it possible, for the first time in history, to study motion according to the requirements of mathematical physics. Dingler compared Oresme's ideas with those of the Greek tradition, which tried to explain motion in terms of something fixed and constant; according to Dingler, the mathematical representation of the variations of qualities, proposal by Oresme, would have been a decisive step that made it possible to refer motion to a temporal framework , which was indispensable for the formulation of dynamics. Dingler concluded that Oresme's work was a novelty, unthinkable for the Greeks, which provided the basis for all exact natural science, and which was the gateway to later developments: the fact that this gateway seems modest when viewed in the light of current knowledge does not prevent it from being considered as a fundamental progress, perhaps more meritorious for its character of source or origin which inaugurates a new way of conceptualising physics *(26)
In the simplest case envisaged by Oresme, which refers to the representation of a quality by a vertical proportional to its intensity, the points on the horizontal axis represent the points affected by the quality. A geometric figure is thus obtained whose properties correspond to those of the quality under study. In this way, linear qualities are represented by flat surfaces. Oresme studies several cases that refer to qualities that he calls uniformly diffuse, represented by right triangles or trapezoids, and to diffusely diffuse qualities, which cover a great variety of linear qualities. He notes that different plane figures can be constructed which, if they are proportional, serve to represent the same quality. This is equivalent to saying, like grade Duhem, that the unit of intensity can be chosen arbitrarily, so that an infinite number of equivalent figures can correspond to the same horizontal axis, and can be passed from one to another by operations in which the intensities represented on the vertical axis of coordinates are multiplied by an arbitrary issue .
The study extends to the geometrical properties of the six possible types of simple configurations, but Oresme also studies compound configurations, which admit 62 species. Although he does not give an algebraic formulation of their geometric representation, if we translate his considerations into algebraic language, we obtain, in the case of two dimensions, the equation of the straight line. There is therefore justification for considering Oresme as the precursor, if not the inventor, of analytic geometry.
Oresme extends his study to figures of higher dimensions. What has been said about linear qualities, which give rise to representations by means of flat figures, can be extended to superficial qualities. In the context of these considerations, Oresme even refers to a fourth dimension that would allow the representation used for linear and superficial qualities to be extended to bodily qualities. He clearly notes the imaginary nature of this dimension, but opens the way to a mathematical work which, although it does not immediately translate the properties of bodies, is a useful tool for their scientific study.
3.3. The law of accelerated motion
It has already been pointed out that Oresme refers to the representation of any subject of qualities. But of particular interest is the application of his ideas to the study of the motion of bodies.
The consolidation of a scientific discipline requires that some basic laws about the scope of its objects become established. This was fulfilled in the first branch of physics to achieve a rigorous formulation in the 17th century, mechanics. And among the laws of mechanics that were formulated in its early days, the law of accelerated motion undoubtedly occupies a prominent place.
The formulation of this law is generally attributed to Galileo. However, there are clear precedents in the 14th century, both in Oxford and in Paris. Duhem entitled a section of his work with some significant words: How Nicolas Oresme established the law of uniformly varied motion *(27). Oresme's explanations, which Duhem analysed in the aforementioned section, constitute parts II and III of the Tractatus de configurationibus qualitatum et motuum, and in them he applies to this problem the ideas already presented and contained in part I.
Duhem points out in detail that Oresme's work , in this case, is based on principles that are found at the same time by Albert of Saxony, and that analogous formulations are found at Oxford: they are what are called the speed theorem average of Merton College. Moreover: in dealing with this question, Oresme refers back to the ancients, and these were probably the Oxford authors, stating that he only intends to establish order and clarity. And he certainly succeeds in doing so, thanks to the use of his geometrical representations, which are not used in the Oxford works. This point is important, since the physical demonstrations of the 17th century will continue to be based to a large extent on geometrical methods, which sample underlines the importance of Oresme's contribution. The most rigorous demonstrations were only obtained three centuries after Oresme, when infinitesimal calculus became available and the concept of instantaneous velocity could be defined.
From agreement with the method used by Oresme for the graphic representation of the qualities, the measure of a certain linear quality is given by the area of the respective figure, or by the volume if it is a superficial quality. It follows that if a linear quality is subject to a uniformly uniform change, its measure is equivalent to that which would result if it only affected the subject with the value it has at the midpoint. When the quality in question is the velocity in a uniformly accelerated movement, we obtain a geometric representation of the velocity theorem average of Merton College, according to which the space covered by an animated mobile in uniformly accelerated movement is equivalent to that which the same mobile would cover in a uniform movement whose velocity is equal to that which it has at the mean instant of the time elapsed during the accelerated movement.
The Oxford texts expressly mention the concept of instantaneous velocity, and define uniformly accelerated motion as that in which, in any equal part of time, an equal increment of velocity is acquired. The reference letter to any part of time is important, since it could be the case that the same spaces were traversed in equal times with non-uniform velocities; this point was carefully emphasised by Galileo when he formulated, three centuries later, the law of accelerated motion; but he did not have to add anything to what had already been expounded by the medieval authors.
The geometrical formulation of the velocity theorem average, proposal by Oresme, was the clearest demonstration of those proposed at the time, since it allows us to understand in an intuitive and imaginative way, thanks to the representation by means of coordinates, concepts that would otherwise escape immediate comprehension. Indeed, to define the concept of instantaneous velocity, the concept of the space that would be covered if the movement were uniform is used, but it is precisely the case in which it is not. For the purposes of calculation, the assimilation of accelerated motion to uniform motion, although formulated by other authors using arithmetical procedures, is clearer if Oresme's geometrical representation is used.
This formulation assumes, in the simplest case, that the mobile starts from a state of rest. Time is represented on the horizontal coordinate axis and velocity on the vertical axis, so that at each instant there is a vertical line representing the velocity at that instant. Since the velocity increments are assumed to be uniform, the line joining the velocities is a straight line forming a right triangle with the time and velocity axes, and the space travelled is given by the area of that triangle, which is equal to half the product of the base times the height, i.e. half the total time spent in the movement multiplied by the final velocity reached. Since this is a uniformly accelerated movement, the final velocity is equal to the constant acceleration multiplied by the time, which gives the well-known formula according to which the space travelled is equal to one half of the acceleration multiplied by the square of the time. Geometrically, if we draw the vertical line that represents the velocity at the instant that is in the middle of the time invested in the movement, it is easy to show that the area of the primitive triangle, which represents the space travelled, is equal to the area of the rectangle that has the same base, that is to say the time invested, and as height the velocity at the middle instant of the journey. The abstract mathematical formula of the velocity theorem average is thus intuitively proved.
This theorem, with its arithmetical proofs and with Oresme's geometrical test , was widely disseminated and known in Europe during the 14th and 15th centuries and, from the end of the 15th century, printed editions were available, and these ideas were especially widespread in Italy. It is not easy to prove which path led Galileo in his formulation of the law of free fall of bodies. But two facts are clear. First, Galileo himself, when describing his experiments, states that his goal is to prove that the acceleration of falling bodies follows the law expressed in the velocity theorem average. Moreover, he expounds this theorem by placing it as the first proposition used to support the new mechanics, and the test he provides is not only strikingly similar to that of Oresme, but is even accompanied by the same geometrical figure *(28).
All this does not in the least diminish Galileo's merit as the founder of modern mechanics. In a balanced way, Grant argues that Galileo's contributions have been exaggerated, because they were developed between the 17th and 19th centuries in almost total ignorance of the medieval works. These have been brought to light only in the 20th century, through painstaking research in which medieval manuscripts and early printed editions have been carefully studied; Pierre Duhem was the pioneer of this research, which has revolutionised the interpretation of the birth of modern science. However, this does not detract from Galileo's importance. The medieval contributions, with all their importance, are set in a context that was still to evolve into systematic formulations integrated into an original science, and in this respect Galileo plays an indisputable role of the first order. Current knowledge of the medieval works shows that the age average was not a time disinterested in science, and that there were then first-rate contributions which directly prepared the birth of classical physics; but the elaboration of the new physics is still the work of geniuses among whom Galileo occupies an eminent place *(29).
The law of accelerated motion, considered in the previous section , falls under kinematics, since it refers to motion without considering the physical causes of motion. The kinematics of Oxford and Paris was accompanied by dynamic explanations, in which the causes of motion were taken into account. Albert of Saxony explicitly referred to the relationship between the fall of bodies and the law of accelerated motion. His ideas were widely disseminated through editions of his work that were printed in Pavia and Venice, and are reproduced verbatim in publications from the late 15th century and the first third of the 16th century. It is not surprising, therefore, that Leonardo da Vinci and Domingo de Soto affirmed that the fall of a grave is a uniformly accelerated movement. If this had been combined with the law formulated already in the 14th century and with Oresme's geometrical representation, the law of the fall of the gravius would have been obtained. As Duhem points out, before 1370 it had been glimpsed that this fall is a uniformly accelerated motion and the law of such a motion had been explicitly formulated subject, but more than a century and a half had to pass before the two discoveries were unequivocally related. That relationship is to be found in the writings of Domingo de Soto. Until then, the two ideas were handed down from generation to generation and remained separate *(30).
The problem of the motion of free falling bodies had been widely discussed since antiquity, and with reference letter to observable phenomena. It is not, therefore, an original 17th century question. For example, Jean Buridan examined and criticised the arguments of Aristotle and Averroes, referring to the facts of experience; he affirmed that, together with natural gravity, there is an impetus that is acquired with the same development of the movement and, although he does not state it expressly, it is easy to interpret that he relates the acceleration of the falling body to the time elapsed and not only to the space travelled *(31).
Although Oresme did not develop a theory of his own in this respect, he did offer some reflections related to the conceptual progress of the problem. He did so by studying a very different problem, that of the plurality of worlds. According to the Platonic tradition, bodies would tend to unite with those of a similar nature, whatever world they were in. This explanation clashed with the reasoning of Aristotle, who opposed the existence of other worlds because he claimed that, if they existed, bodies would not have a natural place; he argued that at the centre of the celestial orbits there must be a fixed body, which was the earth, so that weight determined that bodies would move towards a natural place that was totally fixed. In contrast, Oresme relativised the concepts of weight and lightness, stating that these are qualities by virtue of which heavy bodies tend to be in the midst of light bodies, without it being possible to establish a stationary place towards which they tend. In this perspective, gravity was conceived as a property by which heavier bodies tend towards the centre of the masses of subject, without admitting the existence of an absolute direction of gravity anywhere in space.
These ideas also had important repercussions in the field of cosmology, since they made it possible to dispense with a fixed and immobile Earth as the centre of the universe. The relativisation of the forces of attraction, proposal by Oresme, was a step forward in the arduous work leading up to Newtonian dynamics. The theory of impetus played a central role in this progress.
One of the main contributions of the Paris School of physics to the development of modern science was undoubtedly the theory of impetus. It was formulated by Jean Buridan. It was not entirely new, as the idea is found in Hipparchus in the 2nd century BC, in John Philopon, Greek commentator of Aristotle's Physics in the 6th century, and in Avicenna in the 11th century.
According to the Aristotelian conception, natural motions were caused by an intrinsic principle, while violent motions required an external cause whose efficiency lasted only as long as it was at contact with the mover. This seems to agree with a great many facts of experience, but it is hardly reconcilable with what happens in the accelerated fall of bodies and with the motion of projectiles. Indeed, when a body is thrown, it continues to move even though it is no longer at contact with the agent cause of the throw. Particularly practical cases were the motion of a stone thrown upwards and the flight of an arrow. Aristotle tried to explain these phenomena by referring to the motion that the agent cause communicates to the air, so that this motion would be transmitted to the air that the projectile successively passes through. This problem was related to the Aristotelian denial of the vacuum; it was argued that, if there were a vacuum, the motion of bodies in it would be instantaneous because they would meet no resistance, and the projectiles could not move because the cause that would keep them in motion could not act.
Philopon opposed the Aristotelian explanation and argued on the basis of various experiences. He claimed that the agent cause imparts a capacity of motion to the projectile; this would be a transitory quality, which would diminish due to the natural tendencies of the projectile and the resistance it encounters in its motion. Avicenna held a similar idea, but asserted that the motion would continue indefinitely if the projectile encountered no obstacles and, furthermore, that a body of greater weight would travel a proportionately greater distance. These ideas were taken up in the early 14th century by François de Marchia, who claimed that the agent cause produced in the mobile an accidental quality that had a limited duration; however, many contemporaries accepted the Aristotelian explanation.
Buridan developed the theory of the force impressed on the mobile, which he called impetus. Anneliese Maier has argued that there was no dependence between this theory and the precedents of Philopon and Avicenna. In any case, the theory of impetus became, in the 14th century, a widespread explanation that was applied by Buridan and his disciples to problems of great interest *(32).
The impetus is conceived by Buridan as a force capable of moving the body in the direction in which it was launched by the agent. It will be greater the faster the mover moves the mobile and the more subject the mobile contains, so that if two bodies are launched with the same speed, the denser and heavier one will receive a greater force and its movement, which will tend to diminish due to the natural tendency of the body and the resistance it encounters, will last longer. Applying these ideas to the free fall of bodies, Buridan asserted that, at first, the body only moves under the effect of gravity, but that gravity gives it an impetus which is added to gravity and grows progressively as the movement becomes more rapid, which explains the accelerated character of the fall.
Moreover, Buridan argues that the existence of the impetus makes it unnecessary to admit, as was common in the cosmology of the time, that the celestial spheres are moved by intelligent beings. Indeed, according to Buridan, it can be said that God, in creating the universe, impressed on each sphere an impetus which from then on causes its movement without it being necessary to admit any other action, since, as was then admitted, in the celestial bodies there would be no natural inclination towards another subject of movement and there would be no resistance that would oppose their movement. The impetus is conceived by Buridan as a permanent quality, not a transitory one, which only diminishes by virtue of the natural motion of the mobile and the resistance it encounters in its motion.
These ideas are an anticipation of the concept of quantity of motion, which results from the product of mass and velocity, and of the notion of inertia. It would be an exaggeration to consider them as a simple anticipation of the concepts of classical physics, since they were put forward in a context that still had to evolve considerably. But they constitute an important step which, in fact, had an influence B, as the documentary programs of study shows. Even the reference letter to the celestial spheres, which might seem anecdotal, is of undoubted scientific interest, since it means that the same physical laws apply to the celestial bodies and to the sublunar world, and precisely one of the main obstacles to the progress of the new science was the radical distinction between these two spheres.
The theory of impetus is typical of the Paris School of physics. Oresme admits it and applies it. For example, in his commentary on Book I of Aristotle's De caelo, besides referring twice to his references to impetus in the commentary on the Physics, he mentions impetus as an accidental quality which is found in all movement, whether natural or violent, when the velocity increases, and which is the cause of the movement of thrown projectiles; and in his commentary on Book II he explains the theory in greater detail.
Oresme formulated ideas of great interest for the development of the new science by studying the universe as a whole, mainly in Le livre du ciel et du monde, commenting on the ideas of Aristotle, and in the Tractatus de commensurabilitate vel incommensurabilitate motuum celi *(33).
It has already been mentioned that, according to Oresme, there is no reason to assert that there should be a fixed place of attraction at the centre of the universe. Thomas Kuhn points out the importance of this conception in the path of the new physics:
Nicolas of Oresme was a complete critic of the main Aristotelian argument for the uniqueness of the earth. Aristotle claimed that if there were two earths in space (and when the earth becomes a planet there are six "earths"), they would both fall towards the centre of the universe to unite into one, because the earth naturally tends to occupy the centre of space. This demonstration, says Oresme, is invalid, for it presupposes a theory of motion that has not been proved. Perhaps the earth does not naturally tend towards the centre, but towards other fragments of the earth nearby. Our earth has a centre, and perhaps it is towards it, irrespective of its position in the universe, that all freely abandoned stones are directed. According to this Orsmian theory, the natural motion of a body is governed, not by the position it occupies in an absolute Aristotelian space, but by its relative position, with respect to other fragments of subject. This thesis represents something of a prerequisite for the new cosmologies of the sixteenth and seventeenth centuries; cosmologies in which the earth had lost its characteristics of uniqueness and centrality. Similar theories in several respects are common in the texts of Copernicus, Galileo, Descartes and Newton *(34).
Oresme also discussed the immobility of the Earth. In the second book of his Treatise on Heaven, Aristotle claimed that the Earth is immobile at the centre of the World. In his commentary on this work, Oresme analyses the reasons for the possible daily rotation of the Earth. His thesis is that it cannot be proved, either by experience or by reasoning, that the sky moves with daily rotation and the Earth does not.
First of all, Oresme refers to optical relativity, stating that local motion can only be sensibly perceived when a body changes its position with respect to others. A person on a ship, if he sees only another ship moving in the same way as his own, will have the sensation that both ships are at rest; even if his ship is moving and the other is still, he will think that it is the other ship that is moving. Consequently, it cannot be decided whether it is the sky that rotates daily while the earth is still or whether it is the other way round: in both cases we would have the sensation that we are still and that it is the heavens that are moving, since we would perceive the same phenomena. This reasoning of Oresme's later played an important role in the hands of Copernicus and Galileo.
Oresme then examines other classical objections to the rotation of the earth, such as the one he attributes to Ptolemy: if the earth moves from west to east, just as an arrow thrown vertically from a ship would not fall at its starting point but outside the ship, so a stone thrown upwards from the earth would fall towards the west, contrary to experience. After stating that this is the strongest experimental argument, Oresme retorts that these objects could also move eastwards together with the air and the mass of the terrestrial world, and even resorts to an imaginary experiment in which he uses assumptions admitted by Aristotle. These subject arguments are also widely used by Galileo when he defends the Copernican system.
Oresme concludes that no experience can show that Heaven moves with a daily rotational motion and the earth does not.
He then examines seven arguments of other types. Of particular interest are the last three, in which the objections to the motion of the earth are taken from texts from the Sacred Scripture, where the motion of the sun, the stillness of the earth, and the fact that the sun stood still in the time of Joshua are mentioned. These are reasons that were used in the discussions on the motion of the Earth in the time of Galileo, and it is interesting to note that Nicolas Oresme, an orthodox ecclesiastic, had already examined them three centuries earlier and concluded, with complete firmness and peace, that they do not constitute a real objection to the motion of the Earth. Oresme says in this connection that in these passages Scripture conforms to the usual way of speaking, as it does in many other places. This is a clear sample that, when such arguments were later made against Galileo, they were not authentic Catholic doctrine; indeed, Galileo himself expressly noted this in his 1615 letter to the Grand Duchess Christina. It is also interesting to note that Oresme's interpretation of Joshua's text is clearer than Galileo's, since Oresme simply refers to the character of the scriptural text, while Galileo complicates the topic by alluding to the Sun's control over all celestial movements.
Moreover, Oresme emphasises that all the phenomena explained by astronomy can also be explained by admitting that the Earth is animated by a daily rotational motion. Moreover, he claims that this explains the observable phenomena in a simpler way, analyses it in detail and reproaches Aristotle who, in dealing with this topic, resorts to many obscure reasons: Oresme recalls that Aristotle himself appeals to the simplicity of nature and admits that God and nature do nothing in vain, and that, therefore, no causes other than the necessary ones should be admitted.
Duhem stated that:
When Copernicus, in his book On the Revolutions of the Celestial Orbs, will again take up the hypothesis of terrestrial rotation, what he will say in favour of this hypothesis will be far from having the breadth and clarity of Nicolas Oresme's speech *(35).
However, it is not easy to determine what Oresme thought about the reality of the earth's motion. Despite his arguments in favour of it, and despite the fact that at the beginning of his commentary to the De caelo he states that it is possible and perhaps even physically necessary for the earth to move, and then alludes to the possibility that, over the course of time, the place of sunrise and sunset changes, as is said in an Egyptian tradition, he does not claim that the earth actually moves. What he intends by his arguments is not to defend the movement of the earth as a real physical hypothesis. But his reasons in favour of that hypothesis and his refutation of the objections against it are quite clear and leave the ground prepared for further developments.
Oresme's contribution has an additional merit, since, in this case, he defends personal ideas that were not shared by other authors of the Paris School of physics. Not even Buridan, although he could do so by being faithful to his own ideas, aligned himself with Oresme on this point. Albert of Saxony refers that one of his teachers affirms the impossibility of refuting the hypothesis of the Earth's motion; it is obvious that he is referring to Oresme, for he takes up his argumentation, but does not admit it. Later, Pierre d'Ailly will admit the conclusion of Albert of Saxony. It is clear that Oresme defended his arguments alone.
Oresme's thinking is equally clear on the possible plurality of worlds, topic which played a central role in the formulation of the new physics. Aristotle argued forcefully for the uniqueness of the world, which was closely related to his theory of the natural place and his conception of the perfect and eternal world, composed of the substances of the sublunar world in which the elements are mixed with the ingested and incorruptible celestial substances whose subject is different. The thesis of the unicity of the world was one of those condemned by the Bishop of Paris in 1277, since it was presented as an absolutely necessary conclusion that seemed to limit the divine omnipotence. Nevertheless, teachers such as John of Jandun and Albert of Saxony continued to admit the Aristotelian thesis ; although they introduced a corrective, stating that God could create other worlds, they said that this would be a miracle outside the laws of physics.
In his commentary on the De caelo, Oresme sets out Aristotle's argumentation and criticises it. He then lists three possible ways of imagining several worlds. First, it is possible for these worlds to succeed one another in time; Oresme clearly states that God almighty has been able to create and annihilate several worlds. Secondly, one could be within another; Oresme says that this is not so, but that it is not, however, a contradiction, for the apparent magnitudes are relative, and it does not appear, either from reason or experience, that it is a contradiction. Thirdly, and this is the most important case, one world can be found outside another; Oresme remarks that this is what Aristotle reproves, but adds that his reasons are not convincing.
To this purpose, Oresme states that the notions of high and low are relative, not absolute, since they refer to the mutual order between bodies without recourse to an immobile place; and that heavy things tend to come together, without this being due to anything to do with place. These observations imply the withdrawal of the Aristotelian theory of natural place referring to the immobile Earth at the centre of the world.
Moreover, Oresme explicitly appeals to the omnipotence of God, who can create out of nothing, or make the last heaven have another figure, or even make the void exist. These affirmations also affect important points of the Aristotelian conception, and even a basic aspect that conditions it: the Aristotelian idea of a unique, perfect and eternal world that enjoys total necessity. In this context, Oresme takes important steps in the direction favourable to the new physics, since sample the possibility that the Earth should not be considered as the immobile centre of a single world, and opens the way to the unification of celestial and sublunar physics. The stars would no longer be incorruptible bodies endowed with necessary qualities. Nor would the elements be endowed with a single natural motion, as Aristotelian physics demanded. These ideas represent progress along lines that would later be adopted by Nicholas of Cusa, Leonardo da Vinci and Copernicus.
From the same reflection arises, as has been incidentally noted, another factor that breaks with Aristotle's physics and prepares Newton's: the possibility of a physical vacuum. Oresme affirms that the vacuum can be made by God, and in developing his thought he comes to formulate an idea that constitutes an explicit anticipation of the concept of space in Newtonian physics. Indeed, Oresme refers to the existence, beyond the heavens, of an incorporeal, empty space, of a different nature from corporeal space, just as the duration called eternity is of a different nature from temporal duration, even if the latter were perpetual; such a space would be infinite and indivisible, it would be the immensity of God and God himself, just as his eternity is infinite and indivisible and God himself; we cannot comprehend these realities, he adds, because our understanding depends on our senses which are corporeal.
Oresme's ideas about empty space were not entirely new, for they had precedents in the Stoics and John Philopon. But they differed from the ancient conceptions. In addition to relating it to divine immensity, Oresme asserts that local motion is conceived in relation to space which is imagined to be immobile. In this way, Oresme proposes a conceptual framework which is not Exempt of difficulties and which will later influence the Cambridge Platonists, such as Henry More, and will decisively condition Newton's ideas.
Another cosmological problem that Oresme studied with great interest was that of the commensurability of the movements of the stars. His goal was to discredit astrology.
In his treatise De proportionibus proportionum, Oresme devoted the first three chapters to the commensurability between proportions, and in the fourth chapter he applied his considerations to terrestrial and celestial motions. In Ad pauca respicientes he studied the circular motion, which was attributed to the stars, stopping in a special way to determine the points at which the celestial bodies meet in coincidence or at civil service examination; if the motions are commensurable, the exact time that elapses between these events could be calculated, something that will not be possible if the stars move with speeds that are incommensurable.
A basic proposition in Oresme's study is that any two quantities have a high probability of being incommensurable. Applying it to celestial motions, Oresme tries to show that the configurations of the stars do not repeat themselves, from which he concludes that astrological predictions are false. Later, when revising his writings, he ended up writing a new one which is expressly dedicated to these subjects, graduate De commensurabilitate vel incommensurabilitate motuum celi. There he first considered bodies moving with commensurable velocities along concentric circles, then he considered incommensurable motions, and in the third part he presented a dialogue in which Apollo and his Muses preside over the discussion of topic, in which he seeks to discredit astrological predictions by means of certain arguments taken from mathematics *(36). As Grant points out, although Oresme was not the first to discuss these problems, he was nevertheless unequalled in the development and force of the mathematical arguments, so much so that he exerted a great influence on those who later discussed these questions, as was the case with Henry of Hesse, Marsilius of Inghen and Pierre d'Ailly.
Oresme wrote still other works against astrology: one in Latin, entitled Contra divinatores horoscopios, and one in French with the title degree scroll Le Traité des divinacions, in order to be understood by a wider public and especially by the high nobility of France.
This insistence can be explained by the king's fondness for astrology and the influence Oresme had with the king. He might have expected his arguments to have an effect, but apparently they did not. In his history of Charles V, R. Delachenal states that:
In any case, this fondness persisted, and the arguments of good sense, later accumulated with invincible force by Nicolas Oresme, had no effect on the king who, even when he was heir to the throne, had his astrologer *(37).
Basically, Oresme argued that if celestial movements are incommensurable, as they seem to be, regular cycles and accurate predictions based on the conjunctions and oppositions of the stars are unthinkable. Although the physical reasoning is not conclusive, it is clear that Oresme was determined to fight astrology with all his might. This is consistent with the vigorous scientific bent of Oresme, who knew well what had been written on topic, so that he could quote all subject authors and arguments, and was thoroughly employed in the task of exposing the sophistry and tricks of the astrologers.
The importance of these works can hardly be exaggerated. From antiquity, celestial phenomena were given decisive importance as causes of terrestrial events. These ideas permeated Aristotle's cosmology and appeared to be fully scientific. It must be borne in mind that, for a mentality forged on the basis of the data of ordinary experience, it is natural to admit that celestial phenomena can have such an important impact as that of the sun on the earth, which is certainly of enormous magnitude. It was not, therefore, always a matter of mere superstition, but of a mentality in which it was logical to establish a hierarchy of causes, in which the stars played an essential role.
Even Thomas Aquinas shares this view. Certainly, as with the whole Christian tradition, he rejects outright the direct influence of celestial events on man's free actions and contingent futures. But in addition to these essential caveats, which undoubtedly contributed to the withdrawal of astrology, something else was needed to combat it: namely, the progress of particular scientific knowledge and scientific mentality. For these reasons, it does not seem fair to reproach Oresme, as some authors have done, for still admitting some subject influence of celestial events on terrestrial events. Sarton states that Oresme was in this topic far ahead of his contemporaries, that perhaps for this reason his influence was not immediately great although it was eventually felt, and that it is difficult to overdo the praise he deserves for having gone as far as was possible in his time, displaying great courage and moderation *(38). Oresme had an equally clear attitude towards magic and superstition.
To appreciate the importance of Oresme's attitude to astrology and its impact on the birth of modern science, it is useful to consider the acceptance in ancient cultures of doctrines which, without being directly superstitious in the popular sense, presupposed a view of the universe that made a rigorous scientific approach impossible. Such was the case with the doctrine of the Great Year, which was supposed to occur at regular cyclical intervals that would take place, it was generally accepted, every 36,000 years, when the celestial bodies would meet in an identical spatial configuration that would be repeated cyclically. Since it was generally accepted that celestial bodies exerted a decisive causality on terrestrial events, it was concluded that the events of human history would repeat themselves exactly. This subject of ideas was widespread, in one form or another, in the main cultures of antiquity, and affected thinkers of the most varied tendencies. Reference has already been made to the work of Stanley Jaki, who has emphasised the importance of Christianity as a factor which, by admitting the creation of the universe and affirming the irreversible sense of history as it results from the Incarnation, contributed decisively to the overcoming of the above ideas and, therefore, to the spread of a vision of nature and history which had a decisive importance in the birth of modern science *(39).
In this sense, Oresme's work is of great importance in relation to the spread of scientific thinking. Regardless of the value of his physical-mathematical reasoning, he made an important contribution to the discrediting of astrological ideas. Oresme applied his mathematical considerations about the commensurability of motions and about probabilities to the motions of the stars, concluding that, with great probability, the motions of the stars are incommensurable and that, consequently, there is no scientific basis for accurately predicting the coincidences of the stars. If the celestial configurations do not repeat themselves exactly, the basis of astrology fails and the concept of the Great Year with its repetitive cycles is unacceptable.
Proposition 6, of the 219 condemned by the Bishop of Paris in 1277, specifically referred to this idea, stating that when all the celestial bodies have returned to the same point, which happens every 36,000 years, the same effects now observed will be repeated. The idea of a universe in which all events, including human acts and the people who perform them, repeat themselves in an identical cyclical fashion, clashed head-on with the Christian view of history as a single, meaningful development . But it also clashed with the rational basis necessary for the development of science, since it implied that all events depended on a kind of blind force or fatal destiny that was beyond any possible rational research , and fostered a state of mind contrary to the search for the concrete laws that govern natural phenomena. In this sense, the fact that reliable predictions about the movements of the stars can be made today does not detract from the validity of Oresme's basic idea. Moreover, in the modern perspective of a universe composed of billions of galaxies in motion, each of which is composed of billions of stars, the prospect of a cyclical repetition of the configurations of the stars appears even more implausible than could have been thought in Oresme's time.
Oresme also made important contributions in the field of Economics. His treatise on the first invention of coins was written in two successive Latin versions, between 1355 and 1358, thus in his time at high school in Navarre. The French version was printed around 1477, and the Latin version was also printed several times in the 16th and 17th centuries. It is acknowledged to be the first monographic treatise devoted to these subjects, which makes Oresme one of the pioneers of economic science. It is a coherent synthesis that marks a milestone in economic writing.
In his treatise, Oresme was sample well versed in what had been said so far, both in medieval and ancient times. He translated into French and commented on Aristotle's Economics , and his familiarity with Aristotle's thought contributed to the depth of his approach. His study includes penetrating considerations of the exchange of money, the origin and utility of currency, and the theory of value, which is based not only on scarcity but on intrinsic qualities. Usury and feudal abuses are criticised, and the economic rights of the community vis-à-vis the prince are strongly asserted: the power of the prince is held to derive from God but only insofar as he acts for the common good, and these ideas are applied to monetary policy.
Of particular importance is his description of the qualities of good money, his insistence on the need for monetary stability and programs of study on the relationship between money and precious metals. It is interesting to note that Oresme's treatise is not an abstract study, as it is written with practical problems in mind, especially the serious economic difficulties that France was going through when the treatise was written. Indeed, it is to Oresme's credit that B made a decisive contribution to overcoming this economic crisis. In his history of Charles V of France, R. Delachenal states that Oresme's great influence on Charles, when he was prince and after he became king, inspired and determined the monetary policy of the whole reign, mainly through the Treatise on currency *(40).
We find an example of this influence in 1360. These were the years when John II was out of the kingdom, as a prisoner of the English, on several occasions, and the dauphin Charles had to take over the government of the kingdom. On 5 December 1360, as Delachenal reports, John II
gave a library porter, which is one of the most important of his reign and was one of the best executed. It established a whole system of aid for the payment of royal tribute. He created a new and strong currency, which was also a stable currency, since during the last years of John II's life, as well as throughout the reign of his son, there were no changes in the handling of gold and silver other than those imposed and legitimised by the variations in the price of precious metals. The return to the principles and prescriptions of the great library porter of December 1955 was certainly due to the influence of the dauphin and marked the triumph of Nicole Oresme's ideas, which inspired and directed Charles V's monetary policy *(41).
The consolidation of France during the reign of Charles V was partly an effect of monetary policy, in which Oresme played a decisive role.
Oresme did not write systematically about scientific method. But his contributions to particular problems, which have been outlined in the previous sections, have a remarkable impact on the evolution of the mentality that would eventually lead to the birth of modern science. There was a need for a proper idea of the method to be followed in the research of natural phenomena and, in fact, this task was not an easy one. For this reason, the partial works that, in a practical way, contributed to the finding and consolidation of the scientific method are of special importance.
Among Oresme's contributions to this progress was his mathematical treatment of qualities, which, as already mentioned, responded to a mentality according to which mathematical study could even be extended to some areas of psychic phenomena, and was accompanied by concrete instruments, among which the geometrical representation of the variations of qualities stands out. This is topic of prime importance in the gestation of modern science.
Another important contribution along the same lines was the application of the same physical ideas to the explanation of celestial and terrestrial phenomena. In this major aspect it took more time for a complete clarification, but Oresme's reflections in considering the problems concerning the centre of the world, the natural motions of the elements, the shape of the celestial orbits, the motion of the earth and the possible plurality of worlds represented a basic break with the commonly accepted Aristotelian ideas, which implied the essential diversity of the celestial and sublunar worlds.
It should also be mentioned that in his reasoning, Oresme often uses the epistemological criterion according to which a physical explanation is acceptable if it can account for observable phenomena. Since Greek antiquity, the realism of astronomical explanations has been debated, and there was a school of thought according to which such explanations did not necessarily correspond to real physical Structures , since it was in any case possible to formulate alternative explanations that were equally satisfactory in order to save the phenomena. Evidently, this could give rise to a fictionalist or instrumentalist epistemology which, if carried to its ultimate consequences, would end up having a negative impact on scientific realism and, therefore, on the progress of science. But in Oresme's hands, this approach was used in a balanced way and served to show the hypothetical character of some important assumptions thesis , as has been pointed out in analysing Oresme's reasoning on the motion of the earth.
All this relates to Oresme's attitude to Aristotle. It is certainly a deeply respectful attitude. But that respect does not imply an unconditional acceptance of Aristotle's basic thesis . On the contrary, as has been pointed out, Oresme rejects some of them outright and raises doubts about others. This is certainly an attitude found in other authors of the time, and is partly conditioned by the difficulty of reconciling certain aspects of Aristotelian thought with the Christian faith. However, unlike the currents which, at the same time and with similar motivations, insisted on a fideistic orientation in which the capacity of reason was undervalued, in Oresme we find a vigorous confidence in the rational possibilities which, precisely, are used by him to critically analyse the solutions of Aristotelian physics. Oresme adopts a realist epistemology that is shared by the other authors of the Paris School of physics, and this allows him to accept Aristotle's general idea of rationality, which was of great importance in the process that led to classical physics, while departing from Aristotelian physics whenever it was necessary to do so. Oresme's attitude is in line with the scientific tradition, and has nothing to do with the uncritical acceptance of authority in the philosophical or scientific field.
Christian convictions about the contingency of nature also played an essential role in Oresme's scientific approach and in the birth of the new physics. They led him to abandon the Aristotelian perspective, which imposed on physical phenomena a necessity arbitrarily identified with conceptions that had nothing to do with the scientific method, and also to criticise the mythical and astrological ideas that prevented a rational study of natural laws.
4. Oresme's place in the history of science
Pierre Duhem was unequivocal in his claim that Oresme deserves a first-rate place in the history of science. Here is one of his most emphatic statements on the subject:
Later we will be able to show that Nicole Oresme was a forerunner of Copernicus; indeed, he argued that it was more plausible to suppose that the heavens are motionless and that the earth is animated by a daily motion of rotation than to follow the opposite hypothesis. But Oresme was not only the precursor of Copernicus. He was also the precursor of Descartes and the precursor of Galileo; he invented analytical geometry; he established the law of the spaces that a mobile travels in accelerated motion *(42).
Duhem's enthusiasm for Oresme is understandable. Duhem was an excellent physicist who was interested in the history of his science, which was marked by the cliché, admitted without much discussion, that the birth of modern science was due to a revolution carried out outside and against a medieval tradition that was not only disinterested in science, but even hostile to it. In the course of his research, Duhem discovered, to his great surprise, that this cliché was unfounded. He came across important medieval works that decisively prepared for the advent of modern physics, and devoted enormous effort to the research of the original manuscripts, which led to the result essay and publication of his monumental programs of study on topic *(43).
Duhem's work resulted in a B increase of interest in the history of science in the Middle Ages average. Some later researchers, such as Anneliese Maier, came to conclusions that were not as enthusiastic as Duhem's. But others, such as Konstanty Michalski, have argued that it is to be attributed to Duhem's ideas. But others, such as Konstanty Michalski, have argued that the ideas that presided over the birth of modern physics, which emerged in the course of the 14th century, must be attributed an even more profound impact than Duhem assumed *(44).
It is not easy for this topic to reach a unanimously accepted conclusion. In 1941, Dana B. Durand presented a balance sheet of the research so far and adopted a less enthusiastic interpretation than those of Duhem, Dingler and Michalski *(45), although he limited his personal considerations to the topic of Oresme's anticipations of Descartes. In more recent times, there has been a proliferation of programs of study on this topic. One of the most opposed positions to the recognition of the continuity between the medieval and the moderns is represented by Alexander Koyré, who stated that:
despite appearances to the contrary, appearances of historical continuity on which Caverni and Duhem in particular have insisted, classical physics, arising from the thought of Bruno, Galileo and Descartes, does not in fact continue the medieval physics of the "Parisian precursors of Galileo"; it is situated, from entrance, on a different plane, on a plane which we would like to call Archimedesian. Indeed, the precursor and master of classical physics is not Buridan or Nicolas d'Oresme, but Archimedes *(46).
Stanley Jaki disagrees with Koyré's evaluation *(47). He notes that the assessment of the historical impact of ideas cannot claim to be accurate enough to generate a unanimous agreement , since it is virtually impossible to determine with precision the true origin of ideas and approaches. Moreover, Oresme's work must necessarily appear inadequate when compared to Galileo and, in general, to the protagonists of the new science of the 17th century: Jaki uses as an analogy the difference between the use of wheels at the dawn of civilisation and their current application in the most sophisticated aeronautical engineering, and comments that the end product of a long development may appear to be something totally different from what existed at the beginning. In this sense, there is every reason to stress the novelty of Galileo's science. But this is compatible with the recognition of the geniuses who, a long time ago, made the birth of the new science possible.
Having made these clarifications, Jaki states that inconsistencies can be detected in Koyré's rejection of the importance of Buridan and Oresme's century for the progress of science. Koyré, a convinced advocate of the importance of conceptual elements in scientific progress, shuns positivist interpretations and stresses that the real innovation of Galileo's science consisted in a new mentality with respect to the research of nature. This perspective is logical and coherent. But a novelty only makes sense with reference letter to the old, which, in this case, was the Aristotelian worldview with all its metaphysical and theological ramifications (pantheistic, Jaki stresses). average In this sense, one finds an undeniable historical fact: that there was a collective mentality, based on the Christian faith, which clashed with some basic aspects of Aristotelian thought and introduced modifications that, in the long run, had an influence B on the approaches that led to modern science.
In Jaki's words:
Aristotle had been subject to criticism long before the medieval period, but the pantheistic necessitarianism of his synthesis had never received a broad and effective challenge before Christianity developed a pervasive cultural matrix during the Age average. Therefore, if the primacy of a mental perspective must be defended over the mere facts of sensible evidence when analysing the development of science, then the faith of the Age average in a transcendent, provident, rational, transcendent Creator staff will be of enormous importance. It was a faith shared enthusiastically by the majority of those who acted as teachers in a culture with truly new features. It was a faith that engendered a sense of confidence, finality, and orientation with regard to the fundamentals of man's place in the universe. It was a faith fully conscious of the rights of reason *(48).
Jaki analyses Oresme's commentary on Aristotle's De caelo and highlights the relationship between Oresme's scientific contributions, which have already been discussed above, with his criticisms of Aristotle, and the relationship between these criticisms and Oresme's Christian convictions. The rejection of some basic aspects of the Aristotelian universe, such as the spherical figure derived from the perfection of the prime mover or the impossibility of the void, was based on the recognition of God's omnipotence, which was arbitrarily limited in Aristotle's worldview. The same is true of the eternity of the world and of celestial movements, which contradicted the temporal duration of the universe as affirmed by the Christian faith. Oresme expressly noted that Aristotle's position was based on the assertion that creation out of nothing is impossible and, on such an important point, he expressed his intention to examine the Aristotelian arguments closely, which he did. The result was that, moved by a primarily theological interest, he unravelled misunderstandings that impeded the progress of science and formulated new concepts that were stimulating to that progress.
With regard to the theory of impetus, Jaki stresses that, for Aristotle, the process implied in the notion of impetus was a theological, metaphysical and physical impossibility, and in that order. Oresme was aware of the theological sense of some of his basic criticisms. For example, he refers on several occasions to the list of propositions condemned in 1277 by the bishop of Paris; Jaki notes that these references to a document that had even been partially revoked in 1325, half a century before the essay of Oresme's commentary to the De caelo, indicate that the document in question was, for Oresme, the expression of some essential features of the Christian mentality, at least in its fundamental lines.
Today, the importance of Oresme's scientific contributions is often recognised. Thus, Edward Grant sample forcefully states that Galileo was anticipated by his medieval predecessors, including Oresme, in fundamental concepts and theorems, so that he goes so far as to ask where Galileo's originality lies *(49). Of course, it is not difficult to answer this question. But the fact that it is asked at all sample shows the extent to which the medieval works were really relevant. And René Dugas, in his study of mechanics in the 17th century, clearly exposes the importance of the medieval background *(50).
Dugas refers first of all to the Oxford School, pointing out the decisive influence of Robert Grosseteste, the true head of a school in which the scientific programs of study flourished and achieved appreciable results, which Dugas briefly describes. He then devotes a section to the theory of impetus, developed by Jean Buridan, and he overcomes the undervaluation of it in the work of A. Maier; in the same line as Jaki, he affirms that, although Buridan's concepts are still far from modern ones, they have nevertheless rendered an immense service to the progress of science. He continues with another section dedicated to Oresme, in which he refers to his teachings in the high school of Navarre between 1348 and 1362, and summarises some of his main scientific contributions, adopting an interpretation that is neither as enthusiastic as that of Duhem nor as critical as that of Maier. Other sections devoted to Albert of Saxony and Marsilus of Inghen complete the reference letter to the medieval antecedents of classical mechanics, affirming that the classical scientific revolution was not a sudden revolution, but rather the result of an evolution in which the mentioned authors played an appreciable role.
Thomas Kuhn's analysis of the Copernican revolution has already been alluded to above. Referring to the scholastic critique of Aristotle, Kuhn states that:
the very ardor with which Aristotle's texts were studied ensured that inconsistencies in his doctrine or his demonstrations were quickly detected; inconsistencies that often became the basis for new creative achievements. Medieval scholars had barely glimpsed the astronomical and cosmological novelties that their successors of the 16th and 17th centuries would bring to the table. However, they expanded the field of Aristotelian logic, discovered errors in its reasoning and rejected a good issue of its explanations because of their mismatch with the evidence provided by experience. At the same time, they forged a good issue of concepts and tools that proved essential for the future scientific achievements of men like Copernicus or Galileo *(51).
Kuhn goes on to state that Oresme's commentary on Aristotle's De caelo contains important anticipations of Copernican thought. He goes on to add that
In Buridan's writings, perhaps for the first time, we see the attempt to unite heaven and earth under the same set of laws, an idea that will be expanded and deepened by his student, Nicolas d'Oresme... To conceive of heaven as a terrestrial mechanism, as a piece of clockwork, is tantamount to shattering the absolute dichotomy between the sublunar and supralunar regions *(52).
After pointing out the decisive importance of the impetus theory for the formulation of Newtonian physics, Kuhn sets out a general evaluation which, precisely because it does not show any special sympathy for the medieval people and their beliefs, has a special force:
From a modern point of view, the scientific activity of the average age was incredibly inefficient. Yet how else could science have been reborn in the West? The centuries during which scholasticism reigned are those in which the tradition of ancient science and Philosophy was simultaneously reconstructed, assimilated and put to test. As their weaknesses were discovered, they immediately became the focus of the first operational research in the modern world. All the new scientific theories of the sixteenth and seventeenth centuries have their origin in the shreds of Aristotle's thought torn apart by scholastic criticism. Most of these theories also contain key concepts created by scholastic science. Even more important than such concepts is the stance of mind that modern scientists have inherited from their medieval predecessors: an unbounded faith in the power of human reason to solve the problems of nature. As Whitehead has remarked, "the faith in the possibilities of science, engendered prior to the development of modern scientific theory, is an unconscious derivative of medieval theology" *(53).
Despite the diversity of assessments, which is not surprising when it comes to the historical evolution of concepts over centuries, and which depends in part on the different perspectives adopted, it seems possible to affirm that the work of the Medievalists played an important role in the evolution of the thinking that led to the birth of classical science, and that among this work, Nicolas Oresme's work occupies a preferential place.
This work is part of a project directed by Professor Angel Luis González, graduate La influencia de Navarra en los orígenes del pensamiento moderno, and is carried out with the financial aid of the Government of Navarra.
B. Guenée, Entre l'Eglise et l'Etat, Gallimard, Paris 1987, p. 133-134.
These documents are collected in: E. de Boulay,Historia Universitatis Parisiensis, Paris 1668, volume IV, p. 74-85.
J. de Launoy, Regii Navarrae Gymnasii Parisiensis Historia, 2 vols, Paris 1677.
H. Rashdall, The Universities of Europe in the Middle Ages, Oxford University Press 1958, volume I: Salerno-Bologna-Paris (reprint of the 1936 edition).
With the degree scroll Traité des louanges de Paris, its text is collected in: Le Roux de Lincy - L.M. Tisserand,Histoire générale de Paris. Paris et ses historiens aux XIVe et XVe siècles, Imprimerie Impériale, Paris 1867, p. 33 ff. P. 35-45, which constitute the first part, are devoted to the University.
E. Grant, Physical Science in the Middle Ages, Cambridge University Press 1977, p. 21.
The respective documents can be found in H. Denifle, Chartularium Universitatis Parisiensis, Paris 1899 (reprinted by Culture et Civilisation, Brussels 1964), volume I, p. 70 (n. 11), 143-144 (n. 86 and 87), 277-279 (n. 246), 543-555 (n. 473) and 280-281 (n. 838).
J. Yanguas y Miranda, Historia compendiada del Reino de Navarra, Imprenta de Ignacio Ramón Baroja, San Sebastián 1832, p. 156.
J.M. Lacarra, Historia del Reino de Navarra en la Edad average, published by the Caja de Ahorros de Navarra, Pamplona 1976, p. 350-351.
G. Sarton, Introduction to the History of Science. Volume III: Science and Learning in the Fourteenth Century, p. 108, 476 and 1405.
H. Lamar Crosby (publisher), Thomas of Bradwardine: His Tractatus de Proportionibus; Its Significance for the Development of Mathematical Physics, University of Wisconsin Press, Madison 1955.
S.L. Jaki, Science and Creation. From eternal cycles to an oscillating universe, Scottish Academic Press, Edinburgh and London 1974.
H. Denifle, Chartularium Universitatis Parisiensis, volume II, Paris 1891, p. 485-486 (n. 1023), 505-507 (n. 1042), 576-587 (n. 1124) and 587-590 (n. 1125).
A study on the person and works of Oresme can be found in: F. Meunier, Essai sur la vie et les ouvrages de Nicole Oresme, thesis de Paris, 1857.
The commentary on Aristotle's De caelo was edited in French, accompanied by notes and a critical study of Oresme's works, by: A.D. Menut - A.J. Denomy, Maistre Nicole Oresme: Le Livre du ciel et du monde. Text and commentary. Mediaeval Studies, III (1941), p. 185-280; IV (1942), p. 159-297; V (1943), p. 167-333. This work was published in book form by the University of Wisconsin Press, Madison 1968. Also published: H.L.L. Busard,Nicole Oresme: Quaestiones super geometriam Euclidis, Brill, Leiden 1961; E. Grant, Part I of Nicole Oresme's Algorismus proportionum, Isis, LVI (1965), p. 327-341 (with English translation and notes); E. Grant, Nicole Oresme: De proportionibus proportionum and Ad pauca respicientes, University of Wisconsin Press, Madison 1966 (with English translation and annotations); M. Clagett, Nicole Oresme and the Medieval Geometry of Qualities and Motions, A Treatise on the Uniformity and Difformity of Intensities known as Tractatus de configurationibus qualitatum et motuum. University of Wisconsin Press, Madison 1968 (with English translation and commentary); E. Grant, Nicole Oresme and the Kinematics of Circular Motion: Tractatus de commensurabilitate vel incommensurabilitate motuum celi, University of Wisconsin Press, Madison-Milwaukee-London 1971 (with English translation and commentary).
A study of Oresme's works is included in Menut's work mentioned in the previous quotation , which was completed by Menut in two programs of study published in: Mediaeval Studies, XXVIII (1966), p. 279-299 and XXXI (1969), p. 346-347.
The most salient excerpts from Oresme's works that make reference letter to the sciences, accompanied by commentaries, are collected in: E. Grant (publisher),A Source Book in Medieval Science, Harvard University Press, Cambridge, Massachusetts, 1974.
A selection of Oresme's texts on mathematics can be found in Grant's work cited in grade above, p. 130-135 and 150-159.
In the work cited at grade 21, the respective text selections can be found on pp. 243-253 and 306-312. Duhem studied these topics extensively in: P. Duhem, Le Système du Monde, 10 volumes, Hermann, Paris 1913-1917 and 1954-1959, especially in volume VII, p. 534-569 and 582-587, and in volume VIII, p. 214-225 and 296-308. The treatise on qualities edited by Marshall Clagett with the degree scroll indicated in the text (see grade 19), is graduate by Duhem Tractatus de figuratione potentiarum et mensurarum difformitatum. Moreover, in his commentary on Aristotle's Politics, Oresme refers to an earlier treatise graduate De difformitate qualitatum where he had studied the mathematical treatment of qualities, and it seems that the treatise that was published with the degree scroll De latitudinibus formarums would be a summary of Oresme's ideas made by a disciple.
P. Duhem, Le Système du Monde, cited above, volume VII, p. 534.
H. Dingler, Ueber die Stellung von Nicolas Oresme in der Geschichte der Wissenschaften, Philosophisches Jahrbuch, XLV (1932), p. 58-64.
P. Duhem, Le Système du Monde, cited, volume VII, p. 550. Oresme's explanation, together with comments by Clagett, can be found in: E. Grant, A Source Book in Medieval Science, cited, p. 243-253.
Galileo Galilei, Discorsi e dimostrazioni matematiche intorno à due nuove Scienze, 1638, giornata terza, theorema I, propositio I. In Opere, G. Barbèra, Firenze 1968, volume VIII, p. 208-209.
E. Grant, Physical Science in the Middle Ages, cited, p. 58-59. Oresme's explanation can be found in: E. Grant, A Source Book in Medieval Science, cited, p. 243-253.
P. Duhem, Le Système du Monde, cited above, volume VIII, p. 296-297.
Buridan's text, with comments by Clagett, can be found in: E Grant, A Source Book in Medieval Science, cited, p. 280-284.
A good summary of the issue can be found at: A.C. Crombie. Historia de la ciencia: de San Agustín a Galileo, Alianza, Madrid 1974, volume 2, p. 67-75. Buridan's texts with commentary by Clagett can be found in: E. Grant,A Source Book in Medieval Science, cited, p. 275-284.
Selections from Oresme's texts on these subjects, with commentary by E. Grant, can be found in: E. Grant,A Source Book in Medieval Science, cited, p. 503-510, 529-539 and 547-554.
T.S. Kuhn, The Copernican Revolution. La astronomía planetaria en el development del pensamiento occidental, Ariel, Barcelona 1978, p. 161-162.
P. Duhem, Le Système du Monde, cited above, volume IX, p. 341.
A selection of the text, with comments by E. Grant, can be found in: E. Grant, A Source Book in Medieval Science, cited, p. 529-539.
R, Delachenal, Histoire de Charles V, Picard, 5 volumes, Paris 1909-1931: volume II, p. 367.
G. Sarton, Introduction to the History of Science, cited, volume III, p. 1491.
S.L. Jaki, Science and Creation, quoted in The Road of Science and the Ways to God, University of Chicago Press, Chicago 1978. See grade 16 with the corresponding text.
R. Delachenal, Histoire de Charles V, quoted, volume I, p. 15.
P. Duhem, Le Système du Monde, cited above, volume VII, p. 534.
An excellent study on Duhem is: S.L. Jaki, Uneasy Genius: The Life and Work of Pierre Duhem, Nijhoff, Dordrecht 1987.
K. Michalski, La physique nouvelle et les différents courants philosophiques au XIVe siècle, Imprimerie de l'Université, Krakow 1928, p. 207.
D.B. Durand, Nicole Oresme and the Mediaeval Origins of Modern Science, Speculum, XVI (1941), p. 167-185.
A Koyré, programs of study galileanos, Siglo veintiuno, Madrid 1980, p. 5-6.
E. Grant, Physical Science in the Middle Ages, cited, p. 57-59.
R. Dugas, La Mécanique au XVIIe siècle. Des antécédents scolastiques a la pensée classique, Editions du Griffon, Neuchatel 1954.