Editing Gersonides (section)

==Works in mathematics and astronomy/astrology==
Gersonides was the first to make a number of major mathematical and scientific advances, though since he wrote only in Hebrew and few of his writings were translated to other languages, his influence on non-Jewish thought was limited.<ref name="Simonson.2000"/>

Gersonides wrote ''Maaseh Hoshev'' in 1321 dealing with arithmetical operations including extraction of [[Square root|square]] and [[cube root]]s, various algebraic identities, certain sums including sums of consecutive integers, squares, and cubes, binomial coefficients, and simple combinatorial identities. The work is notable for its early use of proof by [[mathematical induction]], and pioneering work in combinatorics.<ref name=induction>{{cite journal|title=Rabbi Levi Ben Gershon and the Origins of Mathematical Induction|first=Nachum L.|last=Rabinovitch|journal=[[Archive for History of Exact Sciences]]|volume=6|number=3|year=1970|pages=237–248|doi=10.1007/BF00327237|jstor=41133303|s2cid=119948133}}</ref>  The title Maaseh Hoshev literally means the Work of the thinker, but it is also a pun on a biblical phrase meaning "clever work".  Maaseh Hoshev is sometimes mistakenly referred to as Sefer Hamispar (The Book of Number), which is an earlier and less sophisticated work by [[Abraham ibn Ezra|Rabbi Abraham ben Meir ibn Ezra]] (1090–1167).  In 1342, Gersonides wrote ''On Sines, Chords and Arcs'', which examined [[trigonometry]], in particular proving the [[sine law]] for plane triangles and giving five-figure [[sine table]]s.<ref name="Simonson.2000">{{cite journal | url=http://web.stonehill.edu/compsci/Shai_papers/MathofLevi.pdf | author=Charles G. Simonson | title=The Mathematics of Levi ben Gershon, the Ralbag | journal=Bekhol Derakhekha Daehu | publisher=Bar-Ilan University Press | volume=10 | pages=5&ndash;21 | date=Winter 2000 }}</ref>

One year later, at the request of the [[bishop of Meaux]], he wrote ''The Harmony of Numbers'' in which he considers a problem of [[Philippe de Vitry]] involving so-called ''harmonic numbers'', which have the form 2<sup>''m''</sup>·3<sup>''n''</sup>. The problem was to characterize all pairs of harmonic numbers differing by 1. Gersonides proved that there are only four such pairs: (1,2), (2,3), (3,4) and (8,9).<ref>{{Cite journal|first=Ivar|last=Peterson|title=Medieval Harmony|url=http://www.maa.org/mathland/mathtrek_1_25_99.html|journal=Ivar Peterson's MathTrek|access-date=2010-12-26|archive-date=2012-08-04|archive-url=https://archive.today/20120804182624/http://www.maa.org/mathland/mathtrek_1_25_99.html|url-status=dead}}</ref>

He is also credited to have invented the [[Jacob's staff]],<ref>{{Cite journal|first=David G. |last=Krehbiel |title=Jacob's Staff |url=http://www.surveyhistory.org/jacob's_staff1.htm |journal=The Ontario Land Surveyor |date=Spring 1990}}</ref> an instrument to measure the angular distance between celestial objects. It is described as consisting {{Quote|…of a staff of 4.5 feet (1.4 m) long and about one inch (2.5 cm) 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.<ref>[http://mathshistory.st-andrews.ac.uk/Biographies/Levi.html Levi ben Gerson]</ref>}}

Gersonides observed a [[solar eclipse]] on March 3, 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 Gersonides 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]]. Some of his 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. 

Gersonides was also the earliest known mathematician to have used the technique of mathematical induction in a systematic and self-conscious fashion and anticipated Galileo's error theory.<ref name=induction/>

The lunar crater [[Rabbi Levi (crater)|Rabbi Levi]] is named after him.

Gersonides believed that [[astrology]] was real, and developed a naturalistic, non-supernatural explanation of how it works. [[Julius Guttman]] explained that for Gersonides, astrology was:{{Quote|founded on the metaphysical doctrine of the dependence of all earthly occurrences upon the heavenly world. The general connection imparted to the prophet by the active intellect is the general order of the astrological constellation. The constellation under which a man is born determines his nature and fate, and constellations as well determine the life span of nations. …The active intellect knows the astrological order, from the most general form of the constellations to their last specification, which in turn contains all of the conditions of occurrence of a particular event. Thus, when a prophet deals with the destiny of a particular person or human group, he receives from the active intellect a knowledge of the order of the constellations, and with sufficient precision to enable him to predict its fate in full detail. …This astrological determinism has only one limitation. The free will of man could shatter the course of action ordained for him by the stars; prophecy could therefore predict the future on the basis of astrological determination only insofar as the free will of man does not break through the determined course of things.<ref>{{Cite book|first=Julius |last=Guttmann |author-link=Julius Guttmann |title=Philosophies of Judaism: The History of Jewish Philosophy from Biblical Times to Franz Rosenzweig |publisher=[[Henry Holt and Company|Holt, Rinehart and Winston]] |location=[[New York City]] |date=1964 |page=217 |oclc=1497829}}</ref>}}

===Estimation of stellar distances and refutation of Ptolemy's model===
Gersonides appears to be the only astronomer before modern times to have surmised that the fixed stars are much further away than the planets. Whereas all other astronomers put the fixed stars on a rotating sphere just beyond the outer planets, Gersonides estimated the distance to the fixed stars to be no less than 159,651,513,380,944 earth radii,<ref>Albert Van Helden, ''Measuring the Universe: Cosmic Dimensions from Aristarchus to Halley'' (Chicago: The University of Chicago Press, 1985), p.&nbsp;40.</ref> or about 100,000 lightyears in modern units.

Using data he collected from his own observations, Gersonides refuted Ptolemy's model in what the notable physicist [[Yuval Ne'eman]] has considered as "one of the most important insights in the history of science, generally missed in telling the story of the transition from epicyclic corrections to the geocentric model to [[Copernican heliocentrism|Copernicus' heliocentric model]]". Ne'eman argued that after Gersonides reviewed Ptolemy's model with its epicycles he realized that it could be checked, by measuring the changes in the apparent brightnesses of Mars and looking for cyclical changes along the conjectured epicycles. These thus ceased being dogma, they were a theory that had to be experimentally verified, "à la Popper". Gersonides developed tools for these measurements, essentially pinholes and the [[camera obscura]].

The results of his observations did not fit Ptolemy's model at all. Concluding that the model was inadequate, Gersonides tried (unsuccessfully) to improve on it. That challenge was finally answered, of course, by [[Nicolaus Copernicus|Copernicus]] and [[Johannes Kepler|Kepler]] three centuries later, but Gersonides was the first to falsify the Alexandrian dogma - the first known instance of modern falsification philosophy. Gersonides also showed that Ptolemy's model for the lunar orbit, though reproducing correctly the evolution of the Moon's position, fails completely in predicting the apparent sizes of the Moon in its motion. Unfortunately, there is no evidence that the findings influenced later generations of astronomers, even though Gersonides' writings were translated and available.<ref>[[Yuval Ne'eman]]: Astronomy in Sefarad [http://wise-obs.tau.ac.il/judaism/sefarad.html]</ref>