Jordanus de Nemore (fl. 13th century), also known asJordanus Nemorarius andGiordano of Nemi, was athirteenth-century European mathematician and scientist. The literal translation of Jordanus de Nemore (Giordano of Nemi) would indicate that he was anItalian.[1] He wrote treatises on at least 6 different important mathematical subjects: the science of weights; “algorismi” treatises on practicalarithmetic; pure arithmetic;algebra;geometry; andstereographic projection. Most of these treatises exist in several versions or reworkings from theMiddle Ages. We know nothing about him personally, other than the approximate date of his work.
No biographical details are known about Jordanus de Nemore. Cited in the early manuscripts simply as “Jordanus”, he was later given the sobriquet of “de Nemore” (“of the Forest,” “Forester”) which does not add any firm biographical information. In theRenaissance his name was often given as "Jordanus Nemorarius", an improper form.
An entry in the nineteenth-century manuscript catalogue for theSächsische Landesbibliothek inDresden suggested that Jordanus taught at theUniversity of Toulouse, but the text in question was not written by Jordanus and this possible association is without foundation.[2] A fourteenth-century chronicle of the Order of Preachers by the EnglishmanNicholas Trivet (or Triveth, 1258–1328) suggested that the second master-general of theDominican Order, Jordanus of Saxony (d. 1237) wrote two mathematical texts with titles similar to two by Jordanus de Nemore, but this late suggestion is more likely a confusion on the part of Trivet, rather than any proof of identity. Jordanus of Saxony never uses the name “de Nemore” and is nowhere else credited with mathematical writings – in fact he had lectured in theology at theUniversity of Paris. Likewise the name of Jordanus of Saxony is never found with a mathematical text. This identity, popular among some in the nineteenth and twentieth centuries, has been for the most part abandoned.
It is assumed that Jordanus did work in the first part of the thirteenth century (or even in the late twelfth) since his works are contained in a booklist, theBiblionomia ofRichard de Fournival, compiled between 1246 and 1260.[3]
The medieval “science of weights” (i.e.,mechanics) owes much of its importance to the work of Jordanus. In theElementa super demonstrationem ponderum, he introduces the concept of “positionalgravity” and the use of componentforces.Pierre Duhem (in hisOrigines de la statique, 1905) thought that Jordanus also introduces infinitesimal considerations intostatics in his discussion of "virtual" displacements (this being another interpretation of Duhem) of objects in equilibrium. He proves thelaw of the lever by means of the principle of work. TheDe ratione ponderis also proves the conditions of equilibrium of unequal weights on planes inclined at different angles – long before it was re-established bySimon Stevin (with his clootcrans -- "wreath of spheres" experiment) and later byGalileo.
TheElementa super demonstrationem ponderum seems to be the one work which can definitely be ascribed to Jordanus; and the first of the series. Jordanus took what Joseph Brown has called the "Logician’s Abstract ofOn the Karaston" (a skillful compression of the conclusions ofThābit ibn Qurra’sLiber karastonis) and created a new treatise (7 axioms and 9 propositions) in order to establish a mathematical basis for the four propositions on theRomanbalance called theLiber de canonio. An early commentary on this (which also contains a necessary correction to Proposition 9) is the “Corpus Christi Commentary”.
TheLiber de ponderibus fuses the seven axioms and nine propositions of theElementa to the four propositions of theDe canonio. There are at least two commentary traditions to theLiber de ponderibus which improve some of the demonstrations and better integrate the two sources.
TheDe ratione ponderis is a skillfully corrected and expanded version (45 propositions) of theElementa. This is usually ascribed to Jordanus, but more likely it is the work of an unidentified mathematician because the citations by Jordanus of his other works are deleted.
Related to these treatises is an anonymous set of comments, each of which begins with the words “Aliud commentum” (and thus known as the “Aliud commentum” version). This commentary surpasses all others, especially the commentary on Proposition 1.
There are 5algorismi treatises in this category, examined byGustaf Eneström early in the twentieth century, dealing with practicalarithmetic.
TheCommunis et consuetus (its opening words) appears to be the earliest form of the work, closely related to the much expandedDemonstratio de algorismo. Eneström believed that theCommunis et consuetus was certainly by Jordanus.
The laterDemonstratio de algorismo contains 21 definitions and 34 propositions. This is probably a later version of theCommunis et consuetus, made either by Jordanus himself or by some other thirteenth-century mathematician.
TheTractatus minutiarum onfractions seems to be a second part of theCommunis et consuetus – they are often found together in the manuscripts.
TheDemonstratio de minutiius likewise is linked to theDemonstratio de algorismo, and contains and expands the propositions found in theTractatus minutiarum – again a re-edition of the original text.
TheAlgorismus demonstratus is a spurious attribution although for a long time this item was ascribed to Jordanus. Up until Eneström began to sort out the various treatises, theAlgorismus demonstratus – since it was the only one published (ed.Johannes Schöner, Nuremberg, 1543) – was the heading under which all the treatises were grouped. Eneström thought it highly unlikely, however, that this version was the work of Jordanus since no manuscript ascribes it to him (if they give an author, it is generally a Magister Gernarus, or Gerhardus or Gernandus). The first part of this treatise (also known as theAlgorismus de integris) contains definitions, axioms and 43 propositions. The second part (theAlgorismus de minutiis) contains definitions and 42 propositions. Eneström shows that while different from the algorismi treatises of Jordanus, theAlgorismus demonstratus is still closely related to them.
This treatise onarithmetic contains over 400 propositions divided into ten books. There are three versions or editions in manuscript form, the second one with different or expanded proofs than found in the first, and a number of propositions added at the end; the third version inserts the added propositions into their logical position in the text, and again changed some of the proofs. Jordanus’ aim was to write a complete summary of arithmetic, similar to whatEuclid had done forgeometry.[4]
Jordanus collected and organized the whole field of arithmetic, based both on Euclid’s work and on that ofBoethius. Definitions, axioms and postulates lead to propositions with proofs which are somewhat sketchy at times, leaving the reader to complete the argument. Here also Jordanus uses letters to represent numbers, but numerical examples, of the type found in theDe numeris datis, are not given.[5]
The editor of this treatise onalgebra, Barnabas Hughes, has found two sets of manuscripts for this text, one containing 95 propositions, the other, 113. As well some of the common propositions have different proofs. There are also 4 digests or revisions in manuscript form.
Jordanus’De numeris datis was the first treatise in advanced algebra composed in Western Europe, building on elementary algebra provided in twelfth-century translations fromArabic sources. It anticipates by 350 years the introduction of algebraic analysis byFrançois Viète intoRenaissancemathematics. Jordanus used a system similar to that of Viète (although couched on non-symbolic terms) of formulating the equation (setting out the problem in terms of what is known and of what is to be found), of transforming the initial given equation into a solution, and the introduction of specific numbers that fulfil the conditions set by the problem.
This is medievalgeometry at its best. It contains propositions on such topics as the ratios of sides and angles of triangles; the division of straight lines, triangles, and quadrangles under different conditions; the ratio of arcs and plane segments in the same or in different circles; trisecting an angle; the area of triangles given the length of the sides; squaring the circle.
Again there are two versions of this text: the shorter and presumably first edition (theLiber philotegni Iordani de Nemore) and a longer version (Liber de triangulis Iordani) which divides the text into books, re-arranges and expands book 2, and adds propositions 4-12 to 4-28. This latter set of 17 propositions also circulated separately. While the longer version may not be by Jordanus, it was certainly complete by the end of the thirteenth century.
This treatise of five propositions deals with various aspects ofstereographic projection (used in planisphericastrolabes). The first and historically the most important proposition proves for all cases that circles on the surface of a sphere when projected stereographically on a plane remain circles (or a circle of infinite radius, i.e., a straight line). While this property was known long before Jordanus, it had never been proved.
There are three versions of the treatise: the basic text, a second version with an introduction and a much expanded text, and a third, only slightly expanded. The introduction is sometimes found with version 1 and 3, but it was obviously written by someone else.
TheDe proportionibus (onratios), theIsoperimetra (on figures with equal perimeters),[6] theDemonstrationes pro astrolapsu (onastrolabe engraving), and thePre-exercitamina (“a short introductory exercise”?) are dubiously ascribed to Jordanus. A number of other texts including aLiber de speculis and aCompositum astrolabii are spurious ascriptions.[7]
The book "Eresia Pura", byAdriano Petta is a fiction, in italian, based on historical research, around the life of Jordanus de Nemore.[8]
Most of Jordanus' works have been published in critical editions in the twentieth century.[9]
1. Mechanics: The three main treatises and the “Aliud commentum” version (Latin and English) are published inThe Medieval Science of Weights, ed. Ernest A. Moody and Marshall Clagett (Madison: University of Wisconsin Press, 1952). The commentaries are also found in Joseph E. Brown, “The ‘Scientia de ponderibus’ in the Later Middle Ages,” PhD. Dissertation, University of Wisconsin, 1967. TheLiber de ponderibus and the “Aliud commentum” version were published byPetrus Apianus (= Peter Bienewitz) in Nuremberg, 1533; and theDe ratione ponderis was published byNicolò Tartaglia in Venice, 1565.
2. TheAlgorismi treatises: The articles by Gustaf Eneström, which contain the Latin text of the introductions, definitions and propositions, but only some of the proofs, were published inBiblioteca Mathematica, ser 3, vol. 7 (1906–07), 24-37; 8 (1907–08), 135-153; 13 (1912–13), 289-332; 14 (1913–14) 41-54 and 99-149.
3. Arithmetic (theDe elementis arithmetice artis): Jacques Lefèvre d’Étaples (1455–1536) published a version (with his own demonstrations and comments) in Paris in 1496; this was reprinted Paris, 1514. The modern edition is: H. L. L. Busard,Jordanus de Nemore, De elementis arithmetice artis. A Medieval Treatise on Number Theory (Stuttgart: Franz Steiner Verlag, 1991), 2 parts.
4. Algebra (De numeris data): The text was published in the 19th century, but a critical edition now exists: Jordanus de Nemore,De numeris datis, ed. Barnabas B. Hughes (Berkeley: University of California Press, 1981).
5. Geometry: "De triangulis" was first published by M.Curtze in "Mittheilungen des Copernicusvereins für Wissenschaft und Kunst" Heft VI - Thorn, 1887. See in Kujawsko-Pomorska Digital Library:http://kpbc.umk.pl/dlibra/docmetadata?id=39881. More recently, theLiber philotegni Iordani and theLiber de triangulis Iordani have been critically edited and translated in: Marshall Clagett,Archimedes in the Middle Ages (Philadelphia: American Philosophical Society, 1984), 5: 196-293 and 346-477, which is much improved over Curtze's edition.
6. Stereographic projection: The text of version 3 of theDemonstratio de plana spera and the introduction were published in the sixteenth century – Basel, 1536 and Venice, 1558. All versions are edited and translated in: Ron B. Thomson,Jordanus de Nemore and the Mathematics of Astrolabes: De Plana Spera (Toronto: Pontifical Institute of Mediaeval Studies, 1978).
