-Bible The Bible does not deal directly with proper mathematical subjects; however there are some parts that do relate indirectly to different mathematical topics. These are widely discussed by the various commentators on the Bible and Talmud: the ratio of 300:50:30 between the three dimensions of Noah's ark (in the past, a basic ratio in shipbuilding), the mathematical model of a rainbow, the number of 220 sheep and goats sent by Jacob to Esau (220 as the first number of the smallest pair of amicable numbers), the calculations of the visibility of the crescent of the new moon, the total amount and volume of the daily manna , Moses' financial report on the donations for the building of the Tabernacle (mishkan), the commandment of keeping exact measures and balances, chance and probability in relation to the fair division of the land of Israel, lot-drawing to insure the fair division of holy duties and privileges, the curve of "projectile motion" in relation to the unintentional killing of a man by throwing a stone, the surprising distribution of the 12 tribes into two equal groups of six on Mt. Gerizim and Mt. Ebal, and more. The members of the tribe of Issachar were known as "Marei de-Ḥushbena" – the masters of calculations – as their elders specialized in astronomical and calendar calculations. Christian scholars have dealt extensively with Bible mathematics. Among others, an early 18th-century scholar, J.J. Schmidt published an interesting tractate called Biblicus Mathematicus (Zuellichau, 1732) in which many biblical-mathematical subjects are discussed, often based on Jewish sources. In another tractate there is a report on a request to the rabbinical court of Frankfurt to elaborate upon the issue of the geometry of the "Sea of Solomon" in the holy Temple. Many examples of biblical mathematics can be found in Be'er Hetev (Vilna, 1866), a commentary on Leviticus by R. Aryeh Huminer. Sefer Yeẓirah According to ancient Jewish tradition, Sefer Yeẓirah is ascribed to Abraham, as stated at the end of Sefer Yeẓirah. Others ascribe the authorship of the current version of Sefer Yeẓirah to R. Akiba (second century). At its very beginning, "sefar" (arithmetic = the wisdom of mensuration and numbers) is mentioned as one of three dimensions in which the world was created (Kuzari and others). Sefer Yeẓirah deals extensively with permutations and combinations of the 22 letters of the alphabet. The end of Chapter 4 concludes with the statement that the number of permutations of all the 22 letters of the alphabet – (22\! = 1,124,000,727,777,607,680,000) – is a number "which the mouth cannot speak and the ear cannot hear." (It would take 3,564,182,926 people and more than 10,000 years to speak out this number, even at a rate of one number per second.) R. Joseph ben Kalonymos (the elder; mid-13th century), in his commentary to Chapter 2 – erroneously ascribed to R. Abraham ben David (1120–1198) – was one of the earliest to use the decimal system to express large numbers using Hebrew letters. This was noticed by R. elijah of vilna (1720–1797), who indicated that the Hebrew word "<cite>ג״אוח</cite>" which appears in the calculations is actually the result of 144 × 22 = 3,168, written in Hebrew numerals. -Mishnah and Talmud The Mishnah and Talmud, dealing with all aspects of daily life, discuss many mathematical subjects. Yet the main reason for dealing with mathematics was mostly either to bolster the study of the Bible and its commandments or to clarify everyday applications of mathematical methods. As is clear from tractate Avot, the study and transmission of pure mathematical knowledge per se were matters of secondary importance. Nevertheless, for many practical halakhic issues, a considerable body of basic mathematical knowledge is required. The following are a few examples only: the basics of plane geometry in connection with the measuring of the Sabbath eruv boundary (2,000 cubits) over hills and ditches (methods of leveling); elements of knot theory in relation to the Sabbath laws; the layout of family graves; the mathematics of calculating an optimal seeding area without violating the biblical laws of prohibition of sowing with mixed seeds; the number of grapes that are within a circle with a given radius (i.e., Gauss' circle lattice point problem); the mathematics of inheritance leading to geometric progressions and exponential equations; the calculation of square roots leading to irrational numbers; the mensuration of circular and polygonal geometric forms or the proper division of assets among people with different kinds of claims. Another well-known topic is found in tractate Kinnim, the last in the Mishnah order Kodashim. Among other things, it deals with the laws concerning the "confusion of birds," e.g., birds assigned as sin offerings mixed up with those assigned as burnt offerings. Especially the last chapter requires advanced algebra and logic. Among the more recent mathematical commentaries are those by Moshe Koppel of Bar-Ilan University (1998) and B. Engelman of the Nahal Sorek Nuclear Research Center (1992). Knowledge of basic trigonometry and astronomy was needed for the fixing of the new month by reckoning and calculating. The exact methods used by the rabbinical court (bet din) were not made public. It was Maimonides who described appropriate calculations in his "Laws of Sanctification of the New Month" in his Mishneh   Torah. Many of these topics were discussed and elaborated upon by commentators throughout the ages. It has to be remembered that most of these scholars had acquired their knowledge autodidactically and lacked any formal education. From a comment by R. Hai Gaon on Mishnah Kelim (16:1; 24:7) it seems that the Pythagorean writing table (abacus) and "Indian arithmetic" (i.e., numeration system and numerals) were known and in use at this time (Smith 2, 177). Although the Mishnah and Talmud use an approximation of 3 for the better value of 3.1415… for π, it is clear from various discussions in the Talmud, that the amoraim must have been aware of much better values for both π and √2. The Mishnah and Talmud mention a few individuals as having outstanding mathematical knowledge. rabban gamaliel , who used a Heron-type of dioptra to reckon distances; R. Eleazar Ḥisma and R. johanan ben gudgada , whose vast mathematical knowledge was described as enabling them to estimate "the number of drops in the sea"; R. zadok , who revealed to the Romans an advanced system of finger-calculation as well as the underlying mathematics of what later came to be known as the "Roman Statyra" (steelyard). R. joshua b. hananiah – called the Escolasticus – was versed in astronomy and mathematics, R. abbahu is mentioned as having calculated the length of the cycle of service of the different tribes in the Holy Temple. The amora Samuel Yarḥina'ah was versed in calendar calculations. This enabled him to calculate the calendar for the Diaspora for more than 60 years in advance. Nevertheless this knowledge was called "simple calculation," as it did not show a deeper understanding of Jewish law proper. In the name of the amora adda a more accurate estimate of the duration of the seasons is reported. The Talmud also mentions a "Kippah shel Ḥesbonot," i.e., a covered place outside Jerusalem serving people visiting the holy city in arranging their financial calculations and transactions. MISHNAT HA-MIDDOT Bible commentators of the Middle Ages mention the existence of a treatise called the Baraita of 49 Rules (Middot). This treatise from the tannaitic era was said to contain geometrical formulas and calculations. It was Abraham ben Solomon, the son of R. Elijah of Vilna (the Gaon of Vilna) in his Rav Pe'alim (Warsaw, 1894), who first drew attention to this treatise, though no existing copy of it was known. In 1864 Moritz Steinschneider found a Hebrew mathematical manuscript, identified it as Baraitat ha-Middot and published it in 1864. A critical edition including the geometrical drawings omitted by Steinschneider was published by the mathematician Hermann Schapira in 1880 as Mishnat ha-Middot. Among others things, Mishnat ha-Middot contains the Pythagorean formula allowing the calculation of the square root of 2 and uses a value of 31/7 for π. Haim Horovitz of Frankfurt tried to prove that this Mishnat ha-Middot was actually part of a Tosefta to the Mishnah tractate Middot (describing the measurements of the Holy Temple). This suggestion was later supported by the discovery of additional fragments and accepted by solomon gandz in his critical edition of the Mishnat ha-Middot. This suggests that the Mishnat ha-Middot is one of the oldest known Hebrew mathematical works. -The Era of the Geonim R. nahshon bar zadok , who headed the yeshivah of sura from 874 to 882, stated that the order of the weekdays on which any particular festival occurs in successive years repeats itself after a cycle of 247 years. Thus he was able to arrange these years and their characteristic dates in 14 tables. This system is known as "Iggul de-Rav Nahshon" (R. Nahshon's cycle). R. Abraham Azulai (1570–1643) explains in Nahshon Gaon's name the concept of "amicable" (or "friendly") number pairs. He also mentions the common belief in the peacemaking powers of these pairs of numbers. This was known to Jacob and explains the number of 220 sheep and goats that Jacob sent to his brother. The suggestion has been made that R. Nahshon Gaon received this information from his contemporary Thabit ibn Qurra (836–901), a Sabbean mathematician in Baghdad famous for his work in amicable numbers. R. hai ben sherira (939–1038), head of the yeshivah of pumbedita , in one of his responsa, explains the use of the Heron-type dioptra used by Rabban Gamaliel. His mathematical description of the various methods is practically identical to the way Heron himself described it in his book, including the accompanying diagram. This gives rise to the conjecture that R. Hai Gaon was familiar with the original source of Heron himself. The gaon who dealt most extensively with mathematical subjects was R. saadiah ben joseph (892–942), head of the yeshivah of Sura. Examples are his commentary on Sefer Yeẓirah and his Sefer ha-Yerushot ("The Book of Inheritances"). The latter is an extensive mathematical-halakhic text showing how to divide an inheritance according to Jewish law. Other figures from this era who commented on mathematical subjects were Rabbenu Hananel ben Ḥushiel , head of the yeshivah of Kairouan (980–1050), in his commentary to the Talmud and R. shabbetai donnolo (913–c. 982), an Italian physician and writer on medicine, in his Taḥkemoni, a commentary on Sefer Yeẓirah. -11th and 12th Centuries FRANCE The tosafists concentrated their literary efforts on the elucidation of the Bible and the Talmud and did not hand down much mathematical work per se. Although they had no formal education in mathematics, and in most cases had no possibility of learning from Greek or Latin sources, they did acquire some basic knowledge in an autodidactic way. Thus they commented on various talmudic discussions involving arithmetic and simple geometry. On the one hand their comments show a basic knowledge of arithmetic, e.g., in proposing several ways to calculate the connection between the basic halakhic unit of a ¼ of a log (revi'it) and the required amount of 40 se'ah of pure water for the ritual bath – the mikveh – yet it seems that some of the tosafists were unaware of the Pythagorean law and did not have a good approximation of π or of irrational numbers like √2 or √5. But evidently they were aware that surveyors used a better value than 3. They also used   a method of applying and proving the Archimedean formula for the calculation of the area of a circle based on the radius and the circumference. It is highly probable that they adopted the method of their Spanish contemporary R. Abraham bar Ḥiyya . His geometrical treatise Ḥibbur ha-Meshiḥah ve-ha-Tishboret ("Treatise on Mensuration and Calculation") was written for the rabbis of southern France. rashi (1140–1205), the forerunner of the tosafists, himself used a very interesting geometric method for calculating the square root, a method which had its sources in the Jerusalem Talmud. One of the tosafists, R. asher ben jehiel (called the "Rosh"; c. 1250–1327), raised the following question: Why does the Talmud, as a book of law, discuss an inaccurate approximation of 3 for the value of π rather than using a better value, which had been known for a long time? The place for such an excursus would be a geometry text and not the Talmud. In reply, he showed that the famous Mishnah in question (Er. 1:6) was not intended to teach a geometrical principle but rather to introduce the halakhic rule that in certain instances this approximate value of 3 should be used. After his escape from Germany to Spain, R. Asher ben Jehiel asked one of his disciples – the astronomer R. Isaac ben joseph israeli – to elaborate on the Pythagorean law and other geometric principles. It was in response to this request by R. Asher ben Jehiel that he wrote his famous trigonometric-astronomic treatise Yesod Olam. SPAIN AND PORTUGAL In Spain and Portugal Jewish intellectual and scientific growth continued to flourish. There was a lively exchange of mathematical knowledge between Jews and Arabs, and many Arabic mathematical ideas are reflected in Jewish literature. Likewise Jews contributed much to the Arabic corpus of scientific knowledge. Thus one finds much more elaborated mathematical ideas in both talmudic literature proper and original mathematical works. Among the best-known mathematical figures of the 11th and 12th centuries was the above-mentioned Abraham bar Ḥiyya. Until the publication of Ḥibbur ha-Meshiḥah veha-Tishboret in 1910, this mathematical magnum opus was known only in manuscript. This text is probably the earliest post-talmudic mathematical text per se. The author states that the reason he compiled the discourse was the lack of knowledge of geometry among the Jews of southern France. As he writes in his introduction, the text was meant to serve as a textbook for judges who had to deal with legal issues concerning the surveying and measuring of fields. Towards the end Bar Ḥiyya provides an interesting demonstration – using a model built from concentric circles of thin rope – of the Archimedean formula for the calculation of the circle's area based on its radius (r) and its circumference (C) (½∙c∙r), without actually using the value of π. It is this very demonstration that is used by the tosafists in their Talmud commentary. Bar Ḥiyya gives the fair approximation of 1.4143 (1⅖ + 1/10) for the square root of 2 and 3.141593 for π. He was also among the first to introduce to Europe the complete solution of quadratic equations. This fine textbook was translated by Plato of Tivoli in 1145, just a few years after the death of Bar Ḥiyya, under the title Liber Embadorum. Other works by Bar Ḥiyya dealing with mathematics are Ẓurat ha-Areẓ ("Form of the Earth"), a basic introduction to spherical trigonometry and astronomy, and Sefer ha-Ibbur ("Book of Intercalation"). Based on a statement by Maimonides, it seems that he knew this text. Another work of his is Yesodot ha-Tevunah u-Migdal ha-Emunah ("The Foundations of Understanding and the Tower of Faith"), an encyclopedia on arithmetic, geometry, optics, astronomy, and music. Parts of it were translated into Latin by the Hebraists Sebastian Münster (1488–1552) and his disciple Erasmus Oswald Schreckenfuchs (1511–1575). ABRAHAM B. MEIR IBN EZRA ibn ezra (1092–1167), a tosafist, was one of the most prolific commentators on the Bible, who at the same time also wrote extensively on mathematics. The best known of his mathematical contributions are Sefer ha-Shem ("Book of the Holy Name"), Sefer ha-Eḥad ("Book of the Number 1"), Sefer Keli ha-Neḥoshet ("Book of the Copper Instrument (i.e., the astrolabe)"), and Sefer ha-Mispar ("Book of Numbers"). (Some 100 years later, Jacob b. Machir ibn Tibbon, known as Profatius Judaeus, invented an improved version of the astrolabe, known as the quadrant.) Ibn Ezra also translated into Hebrew the commentary of al-Biruni on al-Khwarizmi's tables. In the introduction to this work he gives an historical account on the involvement of Jews in the introduction of Indian mathematics to the Arabic world. Ibn Ezra is one of the earliest Hebrew writers to introduce the "0," which he called galgal (wheel). In his Sefer ha-Mispar Ibn Ezra presents many exercises, which also appeared more than 250 years later in the Sefer ha-Mispar of Elijah Mizraḥi . In addition, Ibn Ezra deals with the mathematics of inheritance and with the history of PI and mentions perfect numbers. He is also famous for his "prisoner problem," a mathematical puzzle first presented by josephus flavius in his Jewish War and which is often used in introductory courses in mathematical programming. Many scientific articles have been written on Ibn Ezra's mathematical writings. MAIMONIDES Aside from a short tractate on calendar calculations, the works of Maimonides (1135–1204) are primarily nonmathematical. Yet in his Mishnah commentary Maimonides mentions that the ratio between the circumference of a circle and its diameter (π) cannot be expressed as a ratio of two natural numbers, and that this fact is not due to a lack of our understanding but is in the very nature of this number. He further states that there is no possibility to know the exact value of this ratio, that mathematicians have written various treatises on this subject, and that the approximation used by scientists is 22/7. Yet in his halakhic Mishneh Torah Maimonides requires the use of the old talmudic (and Babylonian) approximation of 3. A similar statement by Maimonides relates to two other irrational numbers, namely √2 and √5000. It is interesting to note that in the Western world it was Lambert (1728–1777)   who first proved the irrationality of π in 1761. Much has been written in rabbinic literature about Maimonides' geometrical explanation of the mishnayyot in Kilayim 3:1 and 5:5. In his Moreh Nevukhim (Guide of the Perplexed), Maimonides mentions the difficulty in imagining the concept of the hyperbola and its asymptote, i.e., a curved line and a straight one, constantly approaching one another ad infinitum, without ever meeting. This subject – including the detailed explanations of the Jewish writers – was later elaborated by S. Motot, a 14th-century Jewish mathematician, and by Francesco Barozzi in his Admirandum Illud Geometricum Problema, Tredicim Modis Demonstratum (Venice, 1585). -13th and 14th Centuries LEVI BEN GERSHOM levi ben gershom (Ralbag/Gersonides; 1288–ca. 1344) was probably the most advanced Hebrew mathematician of his generation. Widely known for his biblical commentaries, he dealt with all the three branches of Arabic mathematics: arithmetic, geometry, and trigonometry. An extensive corpus of research about Gersonides has come into being (spearheaded by Bernhard Goldstein of Yale University). A comprehensive bibliography on Gersonides has been published by Menachem Kellner of Haifa University (1992). Gersonides mentions both Abraham Ibn Ezra and Abraham bar Ḥiyya as sources from which he derived some of his knowledge. His arithmetical works – Ma'aseh Ḥoshev ("The Practice of Arithmetic") and "De numeris harmonicis" have been studied since the publications of Ma'aseh Ḥoshev by R. Joseph Carlebach of Hamburg, and some years later, at the beginning of the 20th century, by Gerson Lange. A hitherto missing part of problems of Ma'aseh Ḥoshev has been published by S. Simonson of Stonehill College. In this treatise Gersonides deals with arithmetic, algebra, and combinatorics. The short tractate on harmonic numbers was written as a response to an inquiry by Philip of Vitry, the bishop of Maux, shortly before Gersonides passed away. Another mathematical work is a commentary on Euclid's Elements. Gersonides' text on trigonometry, De sinibus, chordis et arcubus (originally written in Hebrew but immediately translated into Latin) is a commentary on the relevant chapters of Ptolemy's Almagest and was originally part of Gersonides' major work Milḥamot Adonai ("The Book of the Wars of the Lord"), part V, ch. 1. It was omitted in the printed Venice edition of 1560. In it, Gersonides presents a proof of the theorem of sines. He had arranged for a translation into Latin which is still extant. Carlebach showed that Gersonides was the inventor of the "cross-staff" (sometimes called "bacculus" or "Jacob's staff"), a simple yet powerful surveying device which allowed nautical and astronomical measurements. Carlebach even reconstructed a model of this instrument following Gersonides' description. The Jacob's staff was in use until the 17th century. IMMANUEL BEN JACOB BONFILS Another Jewish mathematician of this era is immanuel ben jacob bonfils of Tarascon (1300–1377), a contemporary of R. Levi b. Gershom. He is known as Ba'al Kenafayim after his astronomical tables published under the name of Shesh Kenafayim ("Six Wings"). Bonfils taught astronomy and mathematics at Orange for some time. He also was one of the forerunners of exponential calculus, some 150 years before its adoption in Europe, as is evident from his Derekh Ḥilluk. A great number of his many mathematical and astronomical works are still in MS. A special volume on the history of science in the Middle Ages by G. Sarton was called Six Wings after Bonfils' astronomical tables. Important findings relating to Jewish mathematics in the 12–14th centuries have been contributed by various researchers, among them G. Freudenthal, G. Safatti, T. Levy, and D. Zeilberger. -15th Century MOSES BEN ABARAHAM PROVENCAL Moses ben Abaraham provencal (1503–1575) was considered one of the greatest talmudists and most illustrious scholars of Italian Jewry in the Renaissance period. For many decades he was rabbi of the Italian community of Mantua, which therefore became a center of talmudic study. His mathematical knowledge is evident from his Be'ur Inyan Shenei Kavvim. In his Guide of the Perplexed Maimonides mentions the concept of the asymptote, a straight line constantly approaching a curved line without ever touching it, referring to the Conics of Appolonius. Provencal wrote a four-page Hebrew explanation of this subject, which was added to the Sabionetta (1553) edition of the Guide of the Perplexed. This kuntres (pamphlet), which became famous, was translated into Italian by Joseph Shalit (Mantua, 1550) and was included in the well-known volume on the concept of the asymptote by Franceso Barocius (Venice, 1586). The latter also contains geometric explanations by other Jewish commentators on the Guide of the Perplexed. The subject itself became a major topic of rabbinical mathematics and was discussed in rabbinical literature from the 14th to the 19th centuries. It has been suggested that Provencal was familiar with Simon Motot's book on algebra. In his pamphlet, Provencal includes an explanation of the Greek concept of the "mean and extreme proportion" (the "golden section") and proofs related to the connection between the lengths of the sides of a regular hexagon and a regular decagon, both inscribed to the same circle. This concept, described by Euclid and mentioned by joseph albo in his Sefer ha-Ikkarim, inspired many discussions in the rabbinical literature, mainly because of the lack of knowledge of the proper definition of the Greek concept of "mean and extreme proportion." Provencal specifically refers the reader to the source in Euclid's Elements. MORDECAI COMTINO Mordecai Comtino (1420–d. before 1487) was the teacher of R. Elijah ben Abraham Mizraḥi . He was on friendly terms with the Karaites and was the teacher of two of their leaders, caleb afendopolo and Elijah bashyazi . His literary output includes Sefer ha-Ḥeshbon ve-ha-Middot on arithmetic and geometry and commentaries on Abraham Ibn Ezra's Sefer ha-Eḥad, Yesod Mora, and Sefer ha-Shem, in which various mathematical subjects are discussed. His Sefer ha-Ḥeshbon ve-ha-Middot was known only in manuscript until   a careful analysis and partial translation was published by Moritz Silberberg of Schrimm in 1905. The plan of this treatise follows a statement of the Greek Nichomachus of Gerasa regarding the logical order of basic mathematical subjects. Following Ibn Ezra, Comtino introduces the full decimal numeration, including the "0," which again he called "galgal" (the wheel). After introducing standard subjects in the first part, on arithmetic, Comtino deals with the measurement and division of plane figures and then proceeds to calculation of volumes of geometrical bodies and their parts. He also provides a detailed vocabulary of the different scientific terms. A special addition is the collection of problems part of which are borrowed from Ibn Ezra. Mizraḥi in his Sefer ha-Mispar (see below) drew upon some of the problems presented by Comtino. Besides this, Comtino also dealt with the construction of astronomical instruments. ELIJAH BEN ABRAHAM MIZRAḤI Mizraḥi (Re'em; c. 1450–1526) is known primarily from his famous supercommentary to Rashi's Bible commentary. Mizraḥi's mathematical works are Sefer ha-Mispar (Constantinople, 1534), on arithmetic, and a commentary on Ptolemy's Almagest (no longer extant). The former book became a standard text for the study of arithmetic. It deals with whole numbers, fractions, and mixed numbers, with the extraction of the square and cube roots, proportions, and arithmetical and geometrical problems. In his lengthy introduction he describes the importance of the study of mathematics as a bridge between the different sciences. Sefer ha-Mispar is based on Ibn Ezra's Sefer ha-Mispar and the mathematical work of his teacher Mordecai Comtino (see above). A Latin abridgment by Sebastian Muenster was published by his disciple Schreckenfuchs (Basel, 1546). At the end of the 19th century an in-depth description of Sefer ha-Mispar (Die Arithmetik des Elija Misrachi) was published by Gustav Wertheim (Frankfurt, 1893). An excerpt and analysis of those of his mathematical problems related to physics (Ueber physikalische Aufgaben by Elia Misrachi) was prepared by E. Wiedeman in 1910. MORDECAI B. ABRAHAM FINZI finzi (c. 1407–1476), a banker and mathematician, was known mainly for his mathematical and astronomical works, which included Luḥot, tables on the length of days (published by Abraham Conat, Mantua, c. 1479), and an astronomical work entitled Netiv Ḥokhmah (unpublished). He translated into Hebrew the Algebra of the Arab mathematician Abu Kamil Soga (c. 850–930). In 1934, a young Jewish mathematician by the name of Joseph Weinberg, submitted a critical edition of Finzi's translation as a thesis to the University of Munich. (Weinberg later was murdered by the Nazis.) An English translation of Finzi's commentary on Abu Kamil's Algebra was published in 1966 by Martin Levey. Finzi also translated into Hebrew various works on astronomy and geometry, wrote commentaries on some of them, described and explained recently invented astronomical instruments, and wrote treatises on grammar and mnemonics. ABRAHAM BEN SAMUEL ZACUTO zacuto (1452–c. 1515) was known as a talmudic scholar, historian, mathematician, and astronomer. He was appointed professor of astronomy and mathematics at the University of Salamanca. His famous almanacs and astronomical tables became a principal base for Portuguese navigators. Thus, Columbus, whom he met in Salamanca, was able to garner important information before his famous expedition. Using the tables of Zacuto, Columbus was able to predict an eclipse of the moon and so save the lives of his men by demonstrating to hostile natives that he could shut out the light of the sun and moon. R. levi ben habib , a contemporary of Zacuto, remarks in one of his responsa that Abraham Zacuto wrote a commentary on the Talmud, from which he quotes a small geometrical explanation. Zacuto's well-known Sefer Yuḥasin contains several references to mathematics. -16th and 17th Centuries david b. solomon gans The astronomer, mathematician, and historian david b. solomon gans (1541–1613) was raised and educated in the home of R. moses isserles , the Rema. He also belonged to the circle of judah loew ben bezalel (the Maharal). Gans was in close contact with Tycho Brahe and Johann Kepler. Besides his history, Ẓemaḥ David, Gans compiled an astronomical-mathematical textbook called Neḥmad ve-Na'im (Jesnitz, 1743), an extract of which appeared in Prague in 1612 under the name of Magen David. In the preface to Neḥmad ve-Na'im he presents an abridged history of the transmission of mathematics and astronomy among the Jews, based on Jewish sources. The main part is devoted to pre-Copernican celestial mechanics, whereas toward the end he introduces basic instrumentation as well as trigonometry and its applications in daily life, enriching the text with many contemporary examples. MENAHEM ZION PORTO (RAFA) Porto was an Italian rabbi born in Trieste toward the end of the 16th century; he died in Padua around 1660. He was an excellent mathematician and astronomer. His works were highly praised by Andrea Argoli and extolled in Italian sonnets by Tomaso Ercaloni and Benedetto Luzzatto. In 1641 Gaspard Scüppius, editor of the Mercurius Quadralinguis, recommended Porto, in terms that were very complimentary to the rabbi, to johannes buxtorf (the younger), with whom Porto later carried on an active correspondence. Among other works, Porto published a "Hand-book for the Merchant" (Over la-Soḥer, Venice, 1627), a compendium of basic arithmetic and many examples of business calculations for merchants. He also published a two-volume treatise of close to 400 pages – <cite>םימעה יניעל םכתניבו םתמכח</cite> – Porto Astronomico – dealing with trigonometry and astronomy. JOSEPH SOLOMON ROFE DELMEDIGO delmedigo (YaSHaR; 1591–1655) studied mathematics, mechanics, and astronomy under Galileo at Padua. His major work, Elim (Amsterdam, 1629), is a classic compendium of 16th-century mathematics,   physics and astronomy, and scientific instruments. He describes his use of Galileo's telescope to observe the planet Mars. Delmedigo displays a profound knowledge of the Greek, Renaissance, and contemporary literature dealing with mathematics and physics. The first part of Elim contains mathematical discourses dealing with both classical problems of Greek mathematics like the solution of the famous Alexander problem leading to Diophantine equations, the trisection of angles using the Conchoid of Nichomedes, the calculation of the octagon, and post-Renaissance mathematics like the solution of the cubic equations or the "squaring" of a circle, and the 355/113 and Ludolphine approximations to π. In Mayan Ganim Delmedigo deals mainly with spherical trigonometry, with the proof and application of the law of sines, the history of trigonometry, the prosthaphaeresis (trigonometric formulas for the conversion of a product of functions into a sum or a difference), and the law of tangents of the sine function. Gevurat Adonai deals mainly with astronomy but has some mathematics as well. Another part, Ma'yan Ḥatum, is devoted mainly to the discussion of physical problems and paradoxes – mostly from Aristotle's Mechanical Problems – as well as to topics from early 17th-century classical physics. JAIR ḤAYYIM BACHARACH bacharach (1638–1702), rabbi of Worms, had a keen interest in mathematics. His first volume of responsa, Ḥut ha-Shani ("Scarlet Thread," Frankfurt, 1679. contains a lengthy responsum (§98) dealing first with talmudic metrology and then, in the second part, with many mathematical subjects. His main source is Gevurot Adonai by Joseph Solomon Delmedigo (see above). He mentions Hero's formula for extracting the square root, the (Roman) system of finger calculation, the approximation of a circle's circumference by polygons, mathematical problems from the Sefer ha-Mispar of R. Elijah Mizraḥi (see above), the famous Alexander problem from R. Joseph Solomon Delmedigo, and others. In his second volume of responsa Ḥavvat Ya'ir, Bacharach deals at length with the Euclidean concept of "mean and extreme proportion" (the golden section) mentioned by R. Joseph Albo in his Sefer ha-Ikkarim (Responsa §111). MOSES HEFETZ GENTILI In both his Ḥanukkat ha-Bayit (Venice, 1696), on the architecture of the Temple, and Melekhet Maḥshevet, a commentary on the Torah (Venice, 1710), Gentili (1663–1711) presents material that reflects mathematical thinking. Referring to the weekly portion of Noah, he describes Descartes' mathematical model of the rainbow and in his commentary to the weekly Torah portion Ma'asei, he describes Tartaglia's model for the motion of a projectile. In Ḥanukkat ha-Bayit he makes extensive use of the "Pythagorean theorem" in discussing the structure of the altar. ELIJAH BEN SOLOMON ZALMAN OF VILNA Already in his very early life elijah ben solomon zalman (the Vilna Gaon; 1685–1779) showed great interest in the study of mathematics and astronomy as an aid to furthering and deepening the study of Jewish law. In his halakhic commentary on the Shulḥan Arukh he added a great number of notes which disclose his profound mathematical and astronomical knowledge. He encouraged his students to translate basic mathematical texts into Hebrew, and even wrote a small and very concise tractate on arithmetic, geometry, and trigonometry with an introduction to basic astronomy (Ayil Meshulash, 1834). An analysis and description of Ayil Meshulash was published by Elias Fink (Eliah Wilna und sein elementar-geometrisches Compendium, Frankfurt, 1903). Fresh interest in this compendium was aroused with the publication of a new edition with a modern Hebrew commentary. One of his students, baruch schick , a rabbi and physician, published various Hebrew texts on astronomy and medicine as well as a Hebrew translation of the first six books of Euclid's Elements. RAPHAEL LEVI OF HANOVER raphael levi hannover (1685–1779) showed his mathematical talent already as a child, when studying at the Jewish orphanage. One day, upon returning from his studies, observing the construction of the new royal stables in Hanover, he was able to prevent a serious engineering mistake. This drew the attention of Leibniz, the famous mathematician, who after meeting the young Raphael Levi offered to pay part of his tuition and to tutor him privately in mathematics and astronomy. In the last paragraph of his Tekhunat ha-Shamayim (Amsterdam, 1756) on geometry, trigonometry, astronomy, and calendar-making, Levi describes and strongly supports the Copernican system. In an addendum, Raphael Levi's disciple Moses Titkin elaborates a few difficult talmudic passages connected to mathematics. In addition to the Hebrew works, Raphael Levi also invented a system of using logarithms in currency conversions and wrote two mathematical compendia in German: (1) Wechsel-Tabellen Tractaetgen (Hanover, 1746), tables for currency conversion, and (2) Vorbericht vom Gebrauch der neuerfundenen Logarithmischen Wechsel-Tabellen (Hanover, 1747), a preliminary report on the use of the newly invented logarithmic currency conversion method. This latter text contains a good deal of advanced exercises and numerical examples. He also left a Hebrew manuscript of the first part of an introduction to algebra. Among many other individuals who dealt with mathematical subjects were R. yom tov lipmann heller Wallerstein (1579–1654), a disciple of the Maharal, who used his extensive knowledge of mathematics and astronomy throughout the whole of his commentary Tosefot Yom Tov to the Mishnah. R. David Nieto of London (1654–1728), in his Kuzari Sheni, devoted a whole chapter to the explanation of some of the geometrical issues discussed in the Talmud and the commentary of the tosafists and many others. -18th Century The expansion of general knowledge caused a shift in the content of rabbinical mathematics, from basic arithmetic and geometry only to a much broader scope of talmudic subjects related to mathematics. This was made possible by the availability of the Hebrew texts of Abraham Ibn Ezra, Elijah   Mizraḥi, Joseph Solomon Delmedigo, and others mentioned above. Topics like cubic equations, arithmetic and geometric progressions, logarithms, spherical trigonometry (for calculations not connected directly to astronomy), the use of trigonometric tables, logarithms, and methods similar to calculus – like analysis of functions for maxima and minima – were used. This brought forth a great number of texts containing mathematical excursus. At the beginning of the 18th century R. Samuel Schotten mentions in his Kos ha-Yeshu'ot (Frankfurt, 1711) his plan to publish a collection of talmudic-mathematical essays. His basic knowledge in mathematics enabled him to give several approbations to Hebrew astronomical texts. Some years later jonathan of ruzhany published his Yeshu'ah be-Yisrael, a commentary on the laws of kiddush ha-ḥodesh (concerning the blessing of the New Moon) in Maimonides' Mishneh Torah (Frankfurt, 1720). This tractate has an interesting appendix on some halakhot requiring basic mathematical knowledge. In the same year, Jonathan published a compendium of three astronomical works which naturally contain various mathematical elaborations. Some years later R. Emanuel Hai Rikki (1680–1744) published his Ḥoshev Maḥashavot (Amsterdam, 1727), an interesting halakhic-mathematical discourse of 70 short chapters based on an inquiry concerning the measurements of a mikveh (ritual bath). Within these discussions he elaborates the subject of measuring the circumference of a circle, the calculation of √2, and the concepts of asymptotes, and shows how to calculate and to graph two curved functions, each approaching the other without ever touching it. R. jonah landsofer 's Me'il Ẓedakah (Prague, 1757) contains a responsum (§28) – an answer to an inquiry from a "learned man well versed in geometry" – elaborating the geometrical aspects of the altar in the holy Temple as discussed in the Talmud. This subject is often discussed in talmudic literature. In his answer, Landsofer shows his profound knowledge of the relevant texts and also provides a proof of the Pythagorean Theorem. This responsum follows the previous one (§27) dealing with the area calculations of various shapes, all of the size of a "lense." In 1794, David Pivani published Zikhron Yosef, a textbook on arithmetic, geometry, and plane and spherical trigonometry. In the introduction he explains some of the geometrical aspects underlying Maimonides' commentary to the Mishnah Kila'im 5:5. Three years later, David Friesenhausen published his Kelil ha-Ḥeshbon (Berlin, 1797), in collaboration with the Jewish Freeschool in Berlin. This text, an introduction to algebra, contains a variety of challenging examples and exercises. Among the more interesting topics are arithmetical and geometrical series, cubic roots, and Lambert's law concerning the brightness of an illuminated surface. Towards the end of the 18th century and in connection with the social emancipation and the resulting assimilation of the Jews, Jewish mathematics began to develop into two main streams: the traditional Jewish talmud scholar who used mathematical knowledge mainly for the purpose of expounding religious subjects and the new type of a modern mathematician of Jewish origin who pursues mathematics as an academic profession. -19th Century The study of mathematics in the 19th century is widely characterized by the efforts of the haskalah movement to introduce general secular education among the Jews. It was one of the goals of the Haskalah to show that the Jewish people too have a basic mathematical tradition. Therefore one finds regular contributions on mathematics in periodicals like Sulamith in German, Ha-Me'assef, Ha-Ẓefirah, and Ha-Carmel in Hebrew – dealing with the study of mathematics in the framework of talmudic studies or dealing with mathematical problems as such. One of the first texts of the 19th century was Beirurei ha-Middot by Tovia Segal of Horoshitz (Prague, 1807), on the geometry of Sabbath distances and diameters of levitical cities. This text contains an introduction to geometry and trigonometry using logarithms. At the beginning of the 19th century Meyer Hirsch published a textbook/collection of exercises on geometry, Sammlung Geometrischer Aufgaben (Berlin, 1809). This collection of problems was used in Germany for almost a century and contains a formula proposed by the 15th-century Simeon ben Ẓemaḥ Duran . A little later (1828–31) michael creizenach , a teacher at the Frankfurt Philantrophin Reform school, published a series of textbooks on descriptive geometry, algebra, and technical geometry. In the first quarter of this century the first Hebrew article on "binary numbers" (a basic concept in computer engineering and digital electronics) was written by R. zechariah jolles . (The author passed away two years before George Boole published his paper taking up again the concept of "binary numbers" in 1854.) Jolles' paper is basically a translation and elaboration of Leibniz's famous paper on the same subject. It is included, among other mathematical writings, in his Ha-Torah ve-ha-Ḥokhmah (Vilna, 1913). In 1834 Ḥayyim Selig Slonimsky published his Mosdei Ḥokhmah, an introduction to mathematics. A second edition of David Friesensohn's Kelil ha-Ḥeshbon was reprinted in Zolkiev in 1835. In 1845 the famous leopold kronecker (baptized 1863) began his brilliant career as mathematician after receiving his Ph.D. In 1856 the great historian of mathematics Moritz Cantor published his fundamental paper "Ueber die Einfuehrung unserer gegenwaertigen Ziffern in Europa" ("On the Introduction of Our Present Numerals in Europe"). Later on he published his monumental Vorlesungen ueber Geschichte der Mathematik ("Lectures on the History of Mathematics"), which is considered as marking the beginning of the modern history of mathematics. In keeping with the spirit of the Haskalah, the works of Joseph Solomon Delmedigo were reprinted (Odessa, 1865. as was Shevilei de-Raki'a by Eliah Hochheim (Warsaw, 1863). The editor of this tractate, Baruch Lowenstein, added a monograph of his own, Bikkurei ha-Limmudiot ("Firstlings of Mathematics"), discourses on various historical topics in Jewish mathematics. About the same time Ẓevi ha-Cohen Rabinowitz published his fine series of Hebrew texts on popular experimental physics, Yesodei   Ḥokhmat ha-Teva (Warsaw, 1865), including special parts on mathematics and a short bibliography of Hebrew mathematical works. In those years yom tov lipman lipkin – a son of the famous R. israel lipkin Salanter, the founder of the musar movement – invented the "Lipkin linkage," a mechanical device to transform circular motion into linear motion, and became a famous mathematician who contributed mathematical problems to the Ha-Ẓefirah periodical. In the last quarter of the 19th century the first modern systematic texts on Jewish mathematics appeared: baruch zuckermann 's Das Mathematische im Talmud (Breslau, 1878; see J. Szechtman , "Notes on Dr. Zuckerman's 'Introduction to his Mathematical Concepts in the Talmud,'" in: Scripta Mathematica, vol. 25 (1960), pp 49–62). The most important work in this field is moritz steinschneider 's series of articles on Jewish mathematics, published between 1893 and 1898 in Ennestroem's Bibliotheca Mathematica. Part of these articles (covering the 9th–16th centuries) were reprinted as Mathematik bei den Juden in 1964. Additional relevant information can be found in Steinschneider's contributions to the famous Realencyclopaedie by Ersch and Gruber. This was made possible with the opening of German and Italian libraries, allowing the study of ancient Hebrew mathematical manuscripts and books. In 1879 hermann schapira , at that time still a student in Heidelberg, edited and published a German translation of the Mishnat ha-Middot, discovered by Steinschneider in 1864. Towards the end of the 19th century Gustav Wertheim published Elemente der Zahlentheorie (Leipzig, 1887) and some years later an interesting monograph on the mathematics of Elijah Mizraḥi (Die Mathematik des Elia Misrachi, Frankfurt, 1893). A year later Israel Michel Rabbinowits of Paris published a Hebrew introduction to the Talmud containing an interesting appendix on the extraction of square roots, based on Heron's algorithm, as well as a discussion of negative and irrational numbers (Mavole-Talmud, Vilna, 1894). (Shimon Bollag (2nd ed.) -20th Century Jewish mathematicians continued to make major contributions throughout the 20th century and into the 21st, as is evidenced by their extremely high representation among the winners of major awards: 27% for the Fields Medal (the "Nobel Prize of Mathematics") and 40% for the Wolf Prize. Of those still active around the outset of the 20th century mention may be made of rudolf otto sigismund lipschitz (1832–1903), whose contributions to mathematics and physical mathematics were mostly in the theory of numbers, the computation of variations, progressive series, and the theory of potential and analytic mechanics. With the French mathematician Augustin Louis Cauchy (1789–1857), he proved the theorem of prime importance in differential calculus and equations concerning the existing solutions to the equation dy/dx = f (x, y). herman minkowski (1864–1909) is entitled to nearly all the credit for creating the geometry of numbers. He was one of the earliest mathematicians to realize the significance of cantor 's theory of sets at a time when this theory was not appreciated by most mathematicians. The later work of Minkowski was inspired by einstein 's special theory of relativity which was first published in 1905. He produced the four-dimensional formulation of relativity which has given rise to the term "Minkowski space." james joseph sylvester (1814–1897) dominated the development of the theories of algebraic and differential invariants, and many of the technical terms now in use were coined by him. In Italy vito volterra (1860–1940) wrote numerous papers on partial differential equations, integral equations, calculus of variations, elasticity, and topology, and initiated the subjects of functionals and mathematical biology. tullio levi-civita (1873–1942) developed the absolute differential calculus, which was the essential mathematical tool required by Einstein for his development (in 1916. of the general theory of relativity. Levi-Civita's most important contribution in this field was the theory of "parallel displacement." He also produced significant papers on relativity, analytical dynamics, hydrodynamics, and systems of partial differential equations. Two outstanding French mathematicians were jacques salomon hadamard (1865–1963), who produced important work in analysis, number theory, differential geometry, calculus of variations, functional analysis, partial differential equations, and hydrodynamics, and inspired research among successive generations of mathematicians, and laurent schwartz (1915–2002), whose work broadened the scope of calculus and brought Paul Dirac's ideas of "delta functions" in quantum mechanics within the scope of rigorous mathematics. For this work he was awarded the Fields Medal in 1950. Another winner of the Fields Medal was Paul joseph cohen (1934– ), for his fundamental work on the foundations of set theory. Cohen used a technique called "forcing" to prove the independence in set theory of the axiom of choice and of the generalized continuum hypothesis. felix hausdorff (1868–1942) was also an authority on set theory and its applications to sets of points and real analysis. His textbook Mengenlehre (Leipzig, 1935. is recognized as one of the great classics of set theory. The depth and simplicity of his research into fundamental problems was a source of inspiration in the rapid development of modern mathematics. johann ludwig von neumann (1903–1957) sought to develop the subject of quantum mechanics as a mathematical discipline, which led him to research in Hilbert space and the initiation of continuous geometry. In addition, Von Neumann made important contributions to measure theory, ergodic theory, continuous groups, topology, classical mechanics, hydrodynamic turbulence, and shock wave, and was a pioneer of game theory. issai schur (1875–1941) specialized in the theory of numbers, particularly with regard to finite groups and their representations. He is widely known as the author of "Schur's lemma," which states that the only operators that commute with a unitary irreducible representation are the scalar multiples of the identity operator. Schur is also credited with extending the finite group theory to compact groups, and is noted for his work in the representation theory of the rotation   group. André Weil (1906–1998) contributed widely to many branches of mathematics, including the theory of numbers, algebraic geometry, and group theory. norbert wiener (1894–1964) invented the science of cybernetics. As a mathematician, Wiener's main innovation was to develop a mathematics based upon imprecise terms reflecting the irregularities of the physical world. He sought to reduce these random movements to a minimum in order to bring them into harmony. During World War II, he applied his concepts to work connected with antiaircraft defense, and this led to advances in radar, high-speed electric computation, the automatic factory, and a new science he created called cybernetics, a word he coined from the Greek word for "steersman," meaning the study of control. This followed his attempt as a mathematician to find the basis of the communication of information, and of the control of a system based on such communication. Wiener suggested the use of cybernetics in diagnostic procedures and indicated the similarity between certain types of nervous pathology and servomechanism (goal-directed machines such as guns which correct their own fixing malfunctioning). See also benoit mandelbrot and robert aumann . -BIBLIOGRAPHY: R. Aumann and M. Mashler, "Game Theoretic Analysis of a Bankruptcy Problem from the Talmud," in: J. Economic Theory, 36 (1985), 195–213; W. Feldman, Rabbinical Mathematics … Astronomy (1931; repr. 1978); S. Gandz, Studies in Hebrew Astronomy and Mathematics (1971); M. Littman, Approaching Infinity, Selected Mathematical writings of R. Shlomo of Chelme (1989); G.B. Sarfatti, Mathematical Terminology in Hebrew Scientific Literature of the Middle Ages (Heb., 1968); M. Steinschneider, Die Mathematik bei den Juden (1964); N.E. Rabinovitch, Probability and Statistical Inference in Ancient and Medieval Jewish Literature (1973); J. Rosenberg, "Some Examples of Mathematical Analysis Applied to Talmud Study," in: Mathematical Analysis Applied to Talmud Study, at: ; B. Tsaban and D. Graber, Mathematics in Jewish Sources, at: <> . WEBSITES: ; <> (for mathematics in Israel); <> (for mathematicians persecuted by the Nazis).

Encyclopedia Judaica. 1971.

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