Levi ben Gerson

Quick Info

Born
1288
Bagnols now Bagnols sur Cèze, Provence, France
Died
20 April 1344
Avignon, France

Summary
Levi ben Gerson was a French scholar who wrote on trigonometry as well as astronomy and philosophy.

Biography

Levi ben Gerson is known by a large number of different names. Other than Levi ben Gerson, the commonest seem to be Gersonides and Levi ben Gershon. However, others refer to him as Levi ben Gershom, and less commonly as Gersoni, Leo de Bagnols, Leo de Balneolis, Leo Judaeus, Leo Hebraeus , or by the acronym Ralbag. The large number of references attached to this article testify to the large number of studies by historians. Yet little is known of his life and what some historians claim as certain, is sometimes disputed by others. The 'facts' we give here are, therefore, simply those which are believed by the majority. Levi ben Gerson certainly came from a family of outstanding scholars. His father was Gershon ben Solomon, a scholar who wrote Sha'ar ha-Shamayim. Levi states that his grandfather was Levi ha-Kohen, and it is believed that he was his mother's father. Levi was also closely related to Nahmanides (1194-1270) who was a Spanish scholar, a rabbi and Jewish religious leader who is known for his work on philosophy, poetry, and medicine. Levi was a cousin of Judah Delesfils whose grandson was Simon ben Zemah Duran (1361-1444), a Spanish Jewish rabbi who wrote several famous works.

We know that Levi ben Gerson lived in Orange, a town which today is in France but this was not so when he lived there. This is, in fact, quite significant, for the king of France at the time Levi was born was Philip IV, known as Philip the Fair. He expelled the Jews from France in 1306 seizing their property and confiscating all money owed to them, but Levi was not affected. However, even in Provence Jews had a difficult time and Levi wrote in Preface to Milhamot (1329) that the suffering of the Jews:-
... are so intense that they render meditation impossible.
The period 1309-1377 is that of the Roman Catholic papacy in Avignon and these popes had quite good relations with the Jews. Levi certainly had good relations with leading men of the district, both Jewish and Christian. He dedicated one of his works to Pope Clement VI, one of the Avignon popes, and another work was requested by Philip of Vitry, Bishop of Meaux. Yet another of his works was written at the request of a group of Jewish and Christian noblemen. As is already evident from what we have just written, what little information we have about Levi comes from his works. Sadly, these contain very little information about his life. Similarly, there are mentions of him in the writings of others during his lifetime, but again little can be gleaned about his life. For example Isaac de Lattes writes in the Preface to one of his works:-
The great prince, our master Levi ben Gershon, was the author of many valuable works. He wrote a commentary on the Bible and the Talmud; and in all branches of science, especially in logic, physics, metaphysics, mathematics, and medicine, he has no equal on earth.

First we examine his contributions to mathematics. He wrote Art of Calculation (or Art of the Computer) in 1321, which is sometimes incorrectly described as the Book of Number. It deals with arithmetical operations, including extraction of square roots and cube roots. In this work he also looks at the summation of series, permutations and combinations, and basic algebraic identities. He gives formulas for the sum of squares and the sum of cubes of natural numbers as well as studying the binomial coefficients. In proofs, he uses induction making this one of the earliest texts to use this important technique. In [57] and [58], Shai Simonson publishes the problems from this remarkable work by Levi. In fact, it is the Art of Calculation which allows us to give the year of Levi's birth, since he says he finished writing it in 1321, when he was thirty-three years old.

In 1342, at the request of the bishop of Meaux, he wrote The Harmony of Numbers which contains a proof that (1,2), (2,3), (3,4) and (8,9) are the only pairs of consecutive numbers whose only factors are 2 or 3. One year later, he wrote On Sines, Chords and Arcs which examined trigonometry, in particular proving the sine theorem for plane triangles and giving 5 figure sine tables. He calculated his sine tables using Ptolemy's methods and his tables are very accurate. In this work he studied chords, sines, versed sines, cosines but not tangents (which were not in use at this time). Gino Loria suggested that the sine theorem be named after Levi but he was not the first to present the theorem, which was known to Jabir ibn Aflah in the 12th century, but he may have rediscovered it. He also published two geometry books, one being a commentary and introduction to the first five books of Euclid, but not presented axiomatically. The other is the Science of Geometry of which only a fragment has survived. It is interesting to note that Levi was interested in Euclid's parallel postulate and appears to have been part of a lively debate about whether it could be deduced from the other axioms. He proved the parallel postulate with an argument based on an assumption on the convergence or divergence of straight lines that is (as of course it must be) equivalent to the parallel postulate; see [43] for further details.

We now look at some of Levi's contributions to astronomy. One of these is associated with astronomical instruments. He invented Jacob's staff, an instrument to measure the angular distance between celestial objects. We should note that the term 'Jacob's staff' was not used by Levi but rather by his Christian contemporaries; he used a Hebrew name which translates as 'Revealer of Profundities'. It is described as consisting:-
... of a staff of $4{1\over2}$ feet long and about one inch wide, with six or seven perforated tablets which could slide along the staff, each tablet being an integral fraction of the staff length to facilitate calculation, used to measure the distance between stars or planets, and the altitudes and diameters of the Sun, Moon and stars.
This was far from his only contribution to improvements in astronomical instruments. In [23] Bernard Goldstein looks at Levi's various innovations in instrument design. A striking example is the design of a transversal scale for reading fifteenths of degrees on the graduated outer circle of an astrolabe. We note that, remarkably, it was around 250 years later that Tycho Brahe used a similar transversal scale on his great mural quadrant. Goldstein also, in an appendix to [23], examines Levi's transversal scale for the Jacob staff. We note that while Levi's method for constructing the scale is theoretically correct, it requires making measurements that seem extremely difficult, so perhaps the theory was never put into practice.

These instruments are described in the astronomical part of Levi's major work The Wars of the Lord which took him twelve years to compose beginning in 1317 and finishing on 24 November 1328. The work is divided into six books, with the fifth of these dealing with astronomy. The astronomy part was translated into Latin at the request of Pope Clement VI in 1340 but this translation includes later revisions of the work by Levi. We will look later at some of the other books but for the moment we make some comments on Book V which alone contains 136 sections. The Hebrew text, with translation and commentary, of the first 20 sections is published in [8]. H W Guggenheimer writes in a review:-
This is the first edition of any of Levi's astronomical writings in Hebrew; as the editor points out, because of the author's critical attitude towards Ptolemy his work did not find many readers who were willing to acknowledge that they had read it (which was translated into Latin at the papal court of Avignon). These first 20 sections contain a general philosophical introduction, Levi ben Gerson's trigonometry, a description of the construction and use of an instrument invented by him (the Jacob staff) for exact determination of angular differences, remarks about the exact construction of the astrolabe and improvements invented by him, followed by practical and theoretical methods for the determination of the meridian and the position of stars and, finally, a proof that Ptolemy's theory does not account for the facts - neither for the moon (which in an epicycle motion would have to be seen from both its sides) nor for the planets, and a preliminary exploration of alternative (excentric circle) theories.
So Levi's passion for accurate astronomical instruments came from a desire to make sure that any proposed theoretical model was actually in accord with observation, and the more accurate the observation, the better a theory could be put to the test. It was a bold move by Levi to reject Ptolemy's system and certainly shows his strength of character that he was prepared to argue against such a deeply held standard model of the universe. However, he realised that Ptolemy's model does not match the facts. Most importantly, although various tinkerings with epicycles and other corrective devices could always be used to correct the position of the heavenly objects, it still left the apparent size of the objects far from that which was observed. For example, according to Ptolemy's model the size of Mars should vary by a factor of 6 but Levi observed only a factor of 2 in the size of the planet. Levi observed a solar eclipse in 1337. After he had observed this event he proposed a new theory of the Sun, which he proceeded to test by further observations. Another eclipse observed by Levi was the eclipse of the Moon on 3 October 1335. He described a geometrical model for the motion of the Moon and made other astronomical observations of the Moon, Sun and planets using a camera obscura. José Luis Mancha, reviewing [25] writes:-
Faced with the conflict between physical and mathematical accounts of celestial phenomena, [Levi] rejected (against Aristotelian tradition) the confinement of astronomy within the limits of a merely predictive theory, and intended (against most of the Ptolemaic tradition) to construct a true mathematical representation - not only a possible one - of the heavens and the motions of the heavenly bodies, being able to satisfy at the same time the requirements of observation and natural philosophy.
Having explained that Levi rejected Ptolemy's model, we must make it clear that the model which Levi gives is of a similar type consisting of 48 spheres. Some of these spheres are concentric with the Earth, while others are not. Levi considers that these solid spheres are separated by layers of fluid that lubricate their motions, each sphere being moved by an immaterial intelligence. He gives an iterative calculation for the size of the fluid layer and is led to a calculation of the size of the universe which differs very significantly from Ptolemy's. For example, for Ptolemy the distance to the sphere of fixed stars is 20,000 Earth radii, while Levi calculates it to be $160 \times 10^{12}$ Earth radii. Of course these beliefs were well wide of the truth such as his belief that the Milky Way was on the sphere of the fixed stars and shines by the reflected light of the Sun. His lunar model, however, is very interesting for he was able to eliminate epicycles and come up with a model which agreed with observation much better than that of Ptolemy, see [28] and [29]. The 99th chapter of Book V consists of astronomical tables made for the meridian of Orange in 1320.

Despite spending some time on considering Levi's mathematical and astronomical contributions, we have not yet mentioned the bulk of his work which was on philosophy and religious studies; he wrote many complex Biblical commentaries. It is important to realise that Levi's scientific work was often in conflict with Judaism. He leaves us in no doubt as to his views on such conflicts writing in the Preface to The Wars of the Lord:-
The Law cannot prevent us from considering to be true that which our reason urges us to believe.
We promised to say a little more about The Wars of the Lord. In fact the six books of this work are designed to examine six philosophical questions: the immortality of the soul; prophecy; God's omniscience; divine providence; the nature of the celestial spheres; and the eternity of matter. These six questions are examined, one in each of the six books, beginning with a discussion of the views of earlier philosophers, in particular Aristotle. In his discussion of the immortality of the soul, Levi examines the material intellect, which he believes is born with man but only has potential existence, but acquired intellect reaches an effective existence by acquired ideas and conceptions. The acquired intellect, Levi argues, does not cease to exist with the death of the body, so achieving immortality. On prophecy, Levi considers whether it is possible and argues strongly that it is. For anyone like Levi who believes in astrology, such a conclusion is inevitable. The topic of the third book, God's omniscience, was a subject debated at length by philosophers and Levi gives his own views [56]:-
The sublime thought of God, he says, embraces all the cosmic laws which regulate the evolutions of nature, the general influences exercised by the celestial bodies on the sublunary world, and the specific essences with which matter is invested; but sublunary events, the multifarious details of the phenomenal world, are hidden from His spirit. Not to know these details, however, is not imperfection, because in knowing the universal conditions of things, He knows that which is essential, and consequently good, in the individual.
On divine providence, Levi reaches the conclusion that some are protected and guided by general providence while others are under an individual providence. We have looked in some detail at Book V, Levi's astronomical theories, so it remains to comment on Book VI which discusses creation and miracles. On creation, he argues against the 'ex nihilo' theory, arguing instead that 'inert undetermined matter, devoid of form and attribute' had existed for all time. This was really potential matter brought into form and being in an instant act of creation by God. His argument regarding miracles is interesting, for he claims that they exist but are neither due to God nor to prophets. They are the result, he argues, of the active intellect. But miracles have their own limitations: they cannot alter the laws of nature; they cannot produce a contradiction; and they cannot affect the motion of the celestial spheres.

Let us end this biography by quoting from Tamar Rudavsky's conclusion [53]:-
Gersonides' philosophical ideas went against the grain of traditional Jewish thought. Gersonides reflects the following characteristics: first, his writings demonstrate a fundamental interplay and harmony between astrological and theological beliefs. It is clear that the appeal of astrology lay in the fact that it offered useful information, while it looked and operated like a science. ... Gersonides believed that life on earth had a meaning, and that terrestrial events had an order. Astrology was a means of ascertaining that meaning. Gersonides' views on prophecy, providence, free-will and evil reflected ingredients of this philosophical determinism. Whereas his commentaries occupied a central place in Jewish theology, his philosophical work was rejected. .... Only in recent years has Gersonides received his rightful place in the history of philosophy. As scholars have rediscovered his thought and have made his corpus available to a modern audience, Gersonides is once again appreciated as an insightful, ruthlessly consistent philosopher.

References (show)

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Written by J J O'Connor and E F Robertson
Last Update July 2009