![Jordanus - Sphaerae atque astrorum coelestium ratio, natura, et motus, 1536 - 105369](https://upload.wikimedia.org/wikipedia/commons/3/34/Jordanus_-_Sphaerae_atque_astrorum_coelestium_ratio%2C_natura%2C_et_motus%2C_1536_-_105369.jpg)
Jordanus de Nemore (fl. 13th century), also known as Jordanus Nemorarius and Giordano of Nemi, was a thirteenth-century European mathematician and scientist. The literal translation of Jordanus de Nemore (Giordano of Nemi) would indicate that he was an Italian.
[ Bertrand Gille, ''Les ingénieurs de la Renaissance''.] He wrote treatises on at least 6 different important mathematical subjects: the science of weights; “algorismi” treatises on practical arithmetic; pure arithmetic; algebra; geometry; and
stereographic projection
In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (geometry), plane (the ''projection plane'') perpendicular to ...
. Most of these treatises exist in several versions or reworkings from the Middle Ages. We know nothing about him personally, other than the approximate date of his work.
Life
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 the
Renaissance
The Renaissance ( , ) , from , with the same meanings. is a period in European history marking the transition from the Middle Ages to modernity and covering the 15th and 16th centuries, characterized by an effort to revive and surpass ideas ...
his name was often given as "Jordanus Nemorarius", an improper form.
An entry in the nineteenth-century manuscript catalogue for the
Sächsische Landesbibliothek in
Dresden
Dresden (, ; Upper Saxon: ''Dräsdn''; wen, label=Upper Sorbian, Drježdźany) is the capital city of the German state of Saxony and its second most populous city, after Leipzig. It is the 12th most populous city of Germany, the fourth larg ...
suggested that Jordanus taught at the
University of Toulouse
The University of Toulouse (french: Université de Toulouse) was a university in the French city of Toulouse that was established by papal bull in 1229, making it one of the earliest universities to emerge in Europe. Suppressed during the Frenc ...
, but the text in question was not written by Jordanus and this possible association is without foundation. A fourteenth-century chronicle of the Order of Preachers by the Englishman
Nicholas Trivet
Nicholas Trivet (or Trevet, as he himself wrote) (c. 1258 – c. 1328) was an English Anglo-Norman chronicler.
Life
Trivet was born in Somerset and was the son of Sir Thomas Trevet (died 1283), a judge who came of a Norfolk or Somerset family. ...
(or Triveth, 1258–1328) suggested that the second master-general of the
Dominican Order
The Order of Preachers ( la, Ordo Praedicatorum) abbreviated OP, also known as the Dominicans, is a Catholic mendicant order of Pontifical Right for men founded in Toulouse, France, by the Spanish priest, saint and mystic Dominic of Cal ...
, 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 the
University of Paris
, image_name = Coat of arms of the University of Paris.svg
, image_size = 150px
, caption = Coat of Arms
, latin_name = Universitas magistrorum et scholarium Parisiensis
, motto = ''Hic et ubique terrarum'' (Latin)
, mottoeng = Here and a ...
. 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, the ''Biblionomia'' of
Richard de Fournival
Richard de Fournival or Richart de Fornival (1201 – ?1260) was a medieval philosopher and trouvère perhaps best known for the '' Bestiaire d'amour'' ("The Bestiary of Love").
Life
Richard de Fournival was born in Amiens on October 10, 1201. He ...
, compiled between 1246 and 1260.
Writings
Mechanics: ''scientia de ponderibus'' (the science of weights)
The medieval “science of weights” (i.e.,
mechanics
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...
) owes much of its importance to the work of Jordanus. In the ''Elementa super demonstrationem ponderum'', he introduces the concept of “positional
gravity
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
” and the use of component
forces.
Pierre Duhem
Pierre Maurice Marie Duhem (; 9 June 1861 – 14 September 1916) was a French theoretical physicist who worked on thermodynamics, hydrodynamics, and the theory of elasticity. Duhem was also a historian of science, noted for his work on the Euro ...
(in his ''Origines de la statique'', 1905) thought that Jordanus also introduces infinitesimal considerations into
statics
Statics is the branch of classical mechanics that is concerned with the analysis of force and torque (also called moment) acting on physical systems that do not experience an acceleration (''a''=0), but rather, are in static equilibrium with ...
in his discussion of "virtual" displacements (this being another interpretation of Duhem) of objects in equilibrium. He proves the
law of the lever
A lever is a simple machine consisting of a beam or rigid rod pivoted at a fixed hinge, or ''fulcrum''. A lever is a rigid body capable of rotating on a point on itself. On the basis of the locations of fulcrum, load and effort, the lever is divi ...
by means of the principle of work. The ''De ratione ponderis'' also proves the conditions of equilibrium of unequal weights on planes inclined at different angles – long before it was re-established by
Simon Stevin
Simon Stevin (; 1548–1620), sometimes called Stevinus, was a Flemish mathematician, scientist and music theorist. He made various contributions in many areas of science and engineering, both theoretical and practical. He also translated vario ...
(with his clootcrans -- "wreath of spheres" experiment) and later by
Galileo
Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He was ...
.
The ''Elementa 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 of ''On the Karaston''" (a skillful compression of the conclusions of
Thābit ibn Qurra
Thābit ibn Qurra (full name: , ar, أبو الحسن ثابت بن قرة بن زهرون الحراني الصابئ, la, Thebit/Thebith/Tebit); 826 or 836 – February 19, 901, was a mathematician, physician, astronomer, and translator who ...
’s ''Liber karastonis'') and created a new treatise (7 axioms and 9 propositions) in order to establish a mathematical basis for the four propositions on the
Roman
Roman or Romans most often refers to:
*Rome, the capital city of Italy
*Ancient Rome, Roman civilization from 8th century BC to 5th century AD
*Roman people, the people of ancient Rome
*''Epistle to the Romans'', shortened to ''Romans'', a letter ...
balance
Balance or balancing may refer to:
Common meanings
* Balance (ability) in biomechanics
* Balance (accounting)
* Balance or weighing scale
* Balance as in equality or equilibrium
Arts and entertainment Film
* ''Balance'' (1983 film), a Bulgarian ...
called the ''Liber de canonio''. An early commentary on this (which also contains a necessary correction to Proposition 9) is the “Corpus Christi Commentary”.
The ''Liber de ponderibus'' fuses the seven axioms and nine propositions of the ''Elementa'' to the four propositions of the ''De canonio''. There are at least two commentary traditions to the ''Liber de ponderibus'' which improve some of the demonstrations and better integrate the two sources.
The ''De ratione ponderis'' is a skillfully corrected and expanded version (45 propositions) of the ''Elementa''. 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.
''Algorismi'' treatises
There are 5
algorism
Algorism is the technique of performing basic arithmetic by writing numbers in place value form and applying a set of memorized rules and mathematical table, facts to the digits. One who practices algorism is known as an algorist. This positiona ...
i treatises in this category, examined by
Gustaf Eneström
Gustaf Hjalmar Eneström (5 September 1852 – 10 June 1923) was a Swedish mathematician, statistician and historian of mathematics known for introducing the Eneström index, which is used to identify Euler's writings. Most historical scholars re ...
early in the twentieth century, dealing with practical
arithmetic
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
.
The ''Communis et consuetus'' (its opening words) appears to be the earliest form of the work, closely related to the much expanded ''Demonstratio de algorismo''. Eneström believed that the ''Communis et consuetus'' was certainly by Jordanus.
The later ''Demonstratio de algorismo'' contains 21 definitions and 34 propositions. This is probably a later version of the ''Communis et consuetus'', made either by Jordanus himself or by some other thirteenth-century mathematician.
The ''Tractatus minutiarum'' on
fractions
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
seems to be a second part of the ''Communis et consuetus'' – they are often found together in the manuscripts.
The ''Demonstratio de minutiius'' likewise is linked to the ''Demonstratio de algorismo'', and contains and expands the propositions found in the ''Tractatus minutiarum'' – again a re-edition of the original text.
The ''Algorismus 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, the ''Algorismus demonstratus'' – since it was the only one published (ed.
Johannes Schöner
Johannes Schöner (16 January 1477, in Karlstadt am Main – 16 January 1547, in the Free Imperial City of Nuremberg) (aka, Johann Schönner, Johann Schoener, Jean Schönner, Joan Schoenerus) was a renowned and respected German polymath. It i ...
, 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 the ''Algorismus de integris'') contains definitions, axioms and 43 propositions. The second part (the ''Algorismus de minutiis'') contains definitions and 42 propositions. Eneström shows that while different from the algorismi treatises of Jordanus, the ''Algorismus demonstratus'' is still closely related to them.
Arithmetic: The ''De elementis arismetice artis''
![Jordanus - Demonstrationes in Arithmetica - 1354431](https://upload.wikimedia.org/wikipedia/commons/7/75/Jordanus_-_Demonstrationes_in_Arithmetica_-_1354431.jpg)
This treatise on
arithmetic
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
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 what
Euclid
Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
had done for
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
.
Jordanus collected and organized the whole field of arithmetic, based both on Euclid’s work and on that of
Boethius
Anicius Manlius Severinus Boethius, commonly known as Boethius (; Latin: ''Boetius''; 480 – 524 AD), was a Roman senator, consul, ''magister officiorum'', historian, and philosopher of the Early Middle Ages. He was a central figure in the tr ...
. 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 the ''De numeris datis'', are not given.
Algebra: The ''De numeris datis''
The editor of this treatise on
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
, 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 from
Arabic
Arabic (, ' ; , ' or ) is a Semitic languages, Semitic language spoken primarily across the Arab world.Semitic languages: an international handbook / edited by Stefan Weninger; in collaboration with Geoffrey Khan, Michael P. Streck, Janet C ...
sources. It anticipates by 350 years the introduction of algebraic analysis by
François Viète
François Viète, Seigneur de la Bigotière ( la, Franciscus Vieta; 1540 – 23 February 1603), commonly know by his mononym, Vieta, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to i ...
into
Renaissance
The Renaissance ( , ) , from , with the same meanings. is a period in European history marking the transition from the Middle Ages to modernity and covering the 15th and 16th centuries, characterized by an effort to revive and surpass ideas ...
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
. 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.
Geometry: ''Liber philotegni'' and the ''De triangulis''
This is medieval
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
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 (the ''Liber 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.
Stereographic projection: ''Demonstratio de plana spera''
This treatise of five propositions deals with various aspects of
stereographic projection
In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (geometry), plane (the ''projection plane'') perpendicular to ...
(used in planispheric
astrolabes
An astrolabe ( grc, ἀστρολάβος ; ar, ٱلأَسْطُرلاب ; persian, ستارهیاب ) is an ancient astronomical instrument that was a handheld model of the universe. Its various functions also make it an elaborate inclin ...
). 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.
Dubious and spurious works
The ''De proportionibus'' (on
ratios
In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
), the ''Isoperimetra'' (on figures with equal perimeters), the ''Demonstrationes pro astrolapsu'' (on
astrolabe
An astrolabe ( grc, ἀστρολάβος ; ar, ٱلأَسْطُرلاب ; persian, ستارهیاب ) is an ancient astronomical instrument that was a handheld model of the universe. Its various functions also make it an elaborate inclin ...
engraving), and the ''Pre-exercitamina'' (“a short introductory exercise”?) are dubiously ascribed to Jordanus. A number of other texts including a ''Liber de speculis'' and a ''Compositum astrolabii'' are spurious ascriptions.
Historical fiction
The book "Eresia Pura", by
Adriano Petta is a fiction, in italian, based on historical research, around the life of Jordanus de Nemore.
Editions of Jordanus’ works
Most of Jordanus' works have been published in critical editions in the twentieth century.
[A discussion of the various texts, and a list of the manuscripts and printed editions (to 1976), are found in Thomson, “Jordanus de Nemore: Opera,” 97-144.]
1. Mechanics: The three main treatises and the “Aliud commentum” version (Latin and English) are published in ''The 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. The ''Liber de ponderibus'' and the “Aliud commentum” version were published by
Petrus Apianus
Petrus Apianus (April 16, 1495 – April 21, 1552), also known as Peter Apian, Peter Bennewitz, and Peter Bienewitz, was a German humanist, known for his works in mathematics, astronomy and cartography. His work on "cosmography", the field that de ...
(= Peter Bienewitz) in Nuremberg, 1533; and the ''De ratione ponderis'' was published by
Nicolò Tartaglia in Venice, 1565.
2. The ''Algorismi'' 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 in ''Biblioteca 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 (the ''De 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, the ''Liber philotegni Iordani'' and the ''Liber 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 the ''Demonstratio 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).
Notes
External links
*
* Nemorarius Jordanus (1553
''De ponderibus propositiones XIII''- digital facsimile from the
Linda Hall Library
The Linda Hall Library is a privately endowed American library of science, engineering and technology located in Kansas City, Missouri, sitting "majestically on a urban arboretum." It is the "largest independently funded public library of scien ...
*
{{DEFAULTSORT:Jordanus De Nemore
13th-century Italian scientists
Medieval European mathematics
13th-century Italian mathematicians
13th-century Latin writers
13th-century Italian writers