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

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.

This leads naturally to two questions. How did Levi ben Gerson become an expert in so many subjects? How did he make a living? Neither question can be answered with any certainty, but we can say a little. For example, given the family of scholars he came from, it is almost certain that he received training in religious matters by family members. There is no evidence where he could have received a scientific training, however, so many experts believe that he was self-taught in these topics. As to his profession, the most common theory is that he made a living from his medical expertise, but other theories suggest that he made money from money lending, or astrology, or that he was of independent means and did not need to earn a living. One might expect that someone expert in so many areas would read the languages of the day with ease but this appears not to be the case. He spoke Provençal but wrote all his works in Hebrew. This may at first look unlikely since only Latin versions of many of his works survive. However, these are translations, the original Hebrew text being lost. He certainly refers to Latin as the "language of Christians," and almost all historians believe that he could not read Latin. Whenever he quotes from Latin works by authors such as Aristotle, he always gives the quotation in Hebrew. Similarly, it is believed that he could not read Arabic, so had to read the works of the Arabic scholars in Hebrew translations. These deductions clearly have some bearing on his education. The only other (almost certain) fact about his life is that he married the sister of Judah Delesfils, whom we mentioned above, so was distantly related to his wife; no record exists of any children from the marriage. We now look at Levi's works.

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 12^th 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 4¹/₂ 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 . 10¹² 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 99^th 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]:-

1. J Samso, Biography in Dictionary of Scientific Biography (New York 1970-1990). 
   http://www.encyclopedia.com/topic/Levi_ben_Gershon.aspx
2. Biography in Encyclopaedia Britannica. 
   http://www.britannica.com/EBchecked/topic/337902/Levi-ben-Gershom

Books:

1. J D Bleich, Providence in the philosophy of Gersonides, Studies in Judaica (Yeshiva Univ. Press, New York, 1973).
2. J Carlebach, Levi Ben Gerson Als Mathematiker (J Lamm, Berlin, 1910).
3. G Dahan (ed.), Gersonide En Son Temps (E Peeters, Louvain-Paris, 1991).
4. R Eisen, Gersonides on Providence, Covenant, and the Chosen People: A Study in Medieval Jewish Philosophy and Biblical Commentary (State University of New York, 1995).
5. G Freudenthal (ed.), Studies on Gersonides : A Fourteenth-Century Jewish Philosopher-Scientist (E J Brill, Leiden, 1992).
6. B R Goldstein, The astronomy of Levi ben Gerson (1288-1344). A critical edition of Chapters 1-20 with translation and commentary (Springer-Verlag, New York, 1985).
7. C H Manekin, The Logic of Gersonides : An Analysis of Selected Doctrines (Kluwer Academic, Dordrecht, 1992).
8. C Touati, La Pensée Philosophique et Théologique de Gersonide (Paris, 1973).

Articles:

1. M Curtze, Die Abhandlung des Levi ben Gerson ueber Trigonometrie und den Jacobstab. Bibliotheca Mathematica 12 (1898), 97-112.
2. K Chemla and S Pahaut, Remarques sur les ouvrages mathematiques de Gersonide, in G Freudenthal (ed.), Studies on Gersonides : A Fourteenth-Century Jewish Philosopher-Scientist (E J Brill, Leiden, 1992), 97-112.
3. P H Espenshade, A text on trigonometry by Levi ben Gerson (1288-1344), Math. Teacher 60 (1967), 628-637.
4. S Feldman, Gersonides' Proofs for the Creation of the Universe, Proc. Amer. Acad. Jewish Research (1967), 113-37.
5. S Feldman, Levi ben Gershom, in D H Frank and O Leaman (eds.), History of Jewish Philosophy (Routledge Univ Press, 2003), 379-398.
6. G Freudenthal, Épistemologie, astronomie et astrologie chez Gersonide, Revue des études juives 146 (3-4) (1987), 357-365.
7. R Glasner, The Early Stages in the Evolution of Gersonides' The Wars of the Lord, The Jewish Quarterly Review (1996), 1-46.
8. R Glasner, Levi ben Gershom and the study of Ibn Rushd in the fourteenth century, The Jewish Quarterly Review 86 (1-2) (1995), 51-90.
9. R Glasner, Gersonides on simple and composite movements, Stud. Hist. Philos. Sci. B Stud. Hist. Philos. Modern Phys. 28 (4) (1997), 545-584.
10. R Glasner, Gersonide, le pseudo-Tusi, et le postulat des parallèles. Les mathématiques en Hébreu et leurs sources arabes, Arabic Sci. Philos. 2 (1) (1992), 39-82.
11. B R Goldstein, Preliminary Remarks on Levi ben Gerson's Contributions to Astronomy, Proc. Israel Acad. Sci. Humanities 3 (1969), 239-254.
12. B R Goldstein, Levi ben Gerson's analysis of precession, Journal for the History of Astronomy 6 (1975), 31-34.
13. B R Goldstein, Levi ben Gerson : On instrumental errors and the transversal scale, Journal for the History of Astronomy 8 (1977), 102-112.
14. B R Goldstein, Astronomical and astrological themes in the philosophical works of Levi ben Gerson, Archives internationales d'histoire des sciences 26 (1976), 221-224.
15. B R Goldstein, The physical astronomy of Levi ben Gerson, Perspect. Sci. 5 (1) (1997), 1-30.
16. B R Goldstein, Levi ben Gerson's theory of planetary distances, Centaurus 29 (4) (1986), 272-313.
17. B R Goldstein, The astronomy of Levi ben Gerson (1288-1344), Studies in the History of Mathematics and Physical Sciences 11 (New York-Berlin, 1985).
18. B R Goldstein, Levi ben Gerson's preliminary lunar model, Centaurus 18 (1973-74), 275-288.
19. B R Goldstein, Levi ben Gerson's lunar model, Centaurus 16 (4) (1971-72), 257-284.
20. B R Goldstein, Levi ben Gerson's analysis of precession, J. Hist. Astronom. 6 (1975), 31-41.
21. B R Goldstein, Levi ben Gerson's preliminary lunar model, in Theory and observation in ancient and medieval astronomy (London, 1985), 94-107.
22. B R Goldstein, Levi ben Gerson's contributions to astronomy, in G Freudenthal (ed.), Studies on Gersonides : A Fourteenth-Century Jewish Philosopher-Scientist (E J Brill, Leiden, 1992), 3-19.
23. B R Goldstein, Levi Ben Gerson's Astrology in Historical Perspective, in G Dahan (ed.), Gersonide En Son Temps (E Peeters, Louvain-Paris, 1991), 287-300.
24. B R Goldstein and D Pingree, Levi Ben Gerson's Prognostication for the Conjunction of 1345, Trans. Amer. Philos. Soc. 80 (1990), 1-60.
25. J Guttman, Levi ben Gerson, in Philosophies of Judaism (1964), 214-215.
26. I Husik, Studies in Gersonides, Jewish Quarterly Review 7 (1916-1917), 553-594.
27. I Husik, Studies in Gersonides, Jewish Quarterly Review 8 (1917-1918), 113-156.
28. I Husik, Studies in Gersonides, Jewish Quarterly Review 8 (1917-1918), 231-268.
29. M Kellner, R Levi Ben Gerson : A Bibliographical Essay, Studies in Bibliography and Booklore 12 (1979), 13-23.
30. M Kellner, An annotated list of writings by and about R Levi ben Gershom, in G Freudenthal (ed.), Studies on Gersonides : A Fourteenth-Century Jewish Philosopher-Scientist (E J Brill, Leiden, 1992), 367-414.
31. S Klein-Braslavy, Gersonides on the Magnet and the Heat of the Sun, in G Freudenthal (ed.), Studies on Gersonides (E J Brill, Leiden, 1992), 267-284.
32. S Klein-Braslavy, Gersonides and Astrology, in S Feldman (ed.), Levi Ben Gershom : The Wars of the Lord (Jewish Publication Society, New York, 1999), 506-519.
33. T Levy, Gersonide, le Pseudo-Tusi et le Postulat des Paralleles, Arabic Science and Philosophy 2 (1992), 39-82.
34. J L Mancha, The Provençal version of Levi ben Gerson's tables for eclipses, Arch. Internat. Hist. Sci. 48 (141) (1998), 268-352.
35. J L Mancha, Heuristic reasoning: approximation procedures in Levi ben Gerson's astronomy, Arch. Hist. Exact Sci. 52 (1) (1998), 13-50.
36. J L Mancha, Levi ben Gerson's astronomical work: chronology and Christian context, Sci. Context 10 (3) (1997), 471-493.
37. C H Manekin, Conservative Tendencies in Gersonides' Religious Philosophy, in D H Frank and O Leaman (eds.), The Cambridge Companion to Medieval Jewish Philosophy (Cambridge University Press, 2003), 304-344.
38. N Mindel, Rabbi Levi Ben Gershon (1288-1344)
    http://www.chabad.org/library/article_cdo/aid/111901/jewish/Rabbi-Levi-Ben-Gershon.htm
39. J D North, Levi's astronomical tables, Journal for the History of Astronomy 7 (1976), 212-213.
40. N L Rabinovitch, Rabbi Levi Ben Gershon and the Origins of Mathematical Induction, Arch. Hist. Exact Sci. 6 (1970), 237-248.
41. B A Rosenfeld, Les démonstrations du 5-me postulat d'Euclide chez Ibn al-Haithm et Léon Gersonide (Russian), Istor.-Mat. Issled. 11 (1958), 733-742.
42. B A Rosenfeld, Notes aux textes d'Ibn al-Haitham et Léon Gersonide (Russian), Istor.-Mat. Issled. 11 (1958), 777-782.
43. T Rudavsky, Gersonides, in E N Zalta (ed.), The Stanford Encyclopedia of Philosophy (2008).
    http://plato.stanford.edu/entries/gersonides/
44. T Rudavsky, Divine Omniscience and Future Contingents in Gersonides, J. Hist. Philos. 21 (1983), 513-536.
45. T Rudavsky, Creation, Time and Infinity in Gersonides, J. Hist. Philos. 26 (1) (1988), 25-44.
46. M Seligsohn, K Kohler and I Broydé, Levi ben Gershon (JewishEncyclopedia.com, 2002).
    http://www.jewishencyclopedia.com/view.jsp?artid=247&letter=L
47. S Simonson, The missing problems of Gersonides - a critical edition. I, Historia Math. 27 (3) (2000), 243-302.
48. S Simonson, The missing problems of Gersonides - a critical edition. II, Historia Math. 27 (4) (2000), 384-431.
49. S Simonson, Mathematical gems of Levi ben Gershon. Mathematics Teacher (2000).
50. S Simonson, The mathematics of Levi ben Gershon, Derakhekha Daehu 10 (2000).