The Mishnat ha-Middot (, 'Treatise of Measures') is the earliest known
Hebrew
Hebrew (; ''ʿÎbrit'') is a Northwest Semitic languages, Northwest Semitic language within the Afroasiatic languages, Afroasiatic language family. A regional dialect of the Canaanite languages, it was natively spoken by the Israelites and ...
treatise
A treatise is a Formality, formal and systematic written discourse on some subject concerned with investigating or exposing the main principles of the subject and its conclusions."mwod:treatise, Treatise." Merriam-Webster Online Dictionary. Acc ...
on
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, composed of 49 ''
mishnayot
The Mishnah or the Mishna (; , from the verb ''šānā'', "to study and review", also "secondary") is the first written collection of the Jewish oral traditions that are known as the Oral Torah. Having been collected in the 3rd century CE, it is ...
'' in six chapters. Scholars have dated the work to either the
Mishnaic period
''Tannaim'' ( Amoraic Hebrew: תנאים "repeaters", "teachers", singular ''tanna'' , borrowed from Aramaic) were the rabbinic sages whose views are recorded in the Mishnah, from approximately 10–220 CE. The period of the Tannaim, also refe ...
Moritz Steinschneider
Moritz Steinschneider (; 30 March 1816 – 24 January 1907) was a Moravian bibliographer and Orientalist, and an important figure in Jewish studies and Jewish history. He is credited as having invented the term ''antisemitism.''
Education
Mo ...
dated the ''Mishnat ha-Middot'' to between 800 and 1200 CE. Sarfatti and Langermann have advanced Steinschneider's claim of
Arabic
Arabic (, , or , ) is a Central Semitic languages, Central Semitic language of the Afroasiatic languages, Afroasiatic language family spoken primarily in the Arab world. The International Organization for Standardization (ISO) assigns lang ...
influence on the work's
terminology
Terminology is a group of specialized words and respective meanings in a particular field, and also the study of such terms and their use; the latter meaning is also known as terminology science. A ''term'' is a word, Compound (linguistics), com ...
, and date the text to the early ninth century.
On the other hand,
Hermann Schapira
Zvi Hermann Schapira (; 1840–1898), or Hermann Hirsch Schapira, was a Lithuanian rabbi, mathematician at the University of Heidelberg, and Zionist. He was the first to suggest founding a Jewish National Fund for the purchase of land in Palestin ...
argued that the treatise dates from an earlier era, most likely the
Mishnaic period
''Tannaim'' ( Amoraic Hebrew: תנאים "repeaters", "teachers", singular ''tanna'' , borrowed from Aramaic) were the rabbinic sages whose views are recorded in the Mishnah, from approximately 10–220 CE. The period of the Tannaim, also refe ...
Solomon Gandz
Solomon Gandz (2 February 1883, Tarnobrzeg, Austria-Hungary – 30 March 1954) was a historian of science.
Gandz published on the history of mathematics and astronomy in medieval Jewish and Islamic civilizations.
From 1915 to 1919, Gandz was ...
conjectured that the text was compiled no later than (possibly by
Rabbi Nehemiah
Rabbi Nehemiah was a rabbi who lived circa 150 AD (fourth generation of tannaim).
He was one of the great students of Rabbi Akiva, and one of the rabbis who received semicha from R' Judah ben Baba
The Talmud equated R' Nechemiah with Rabbi Ne ...
) and intended to be a part of the
Mishnah
The Mishnah or the Mishna (; , from the verb ''šānā'', "to study and review", also "secondary") is the first written collection of the Jewish oral traditions that are known as the Oral Torah. Having been collected in the 3rd century CE, it is ...
, but was excluded from its final canonical edition because the work was regarded as too
secular
Secularity, also the secular or secularness (from Latin , or or ), is the state of being unrelated or neutral in regards to religion. The origins of secularity can be traced to the Bible itself. The concept was fleshed out through Christian hi ...
. The content resembles both the work of
Hero of Alexandria
Hero of Alexandria (; , , also known as Heron of Alexandria ; probably 1st or 2nd century AD) was a Greek mathematician and engineer who was active in Alexandria in Egypt during the Roman era. He has been described as the greatest experimental ...
(c. ) and that of
al-Khwārizmī
Muhammad ibn Musa al-Khwarizmi , or simply al-Khwarizmi, was a mathematician active during the Islamic Golden Age, who produced Arabic-language works in mathematics, astronomy, and geography. Around 820, he worked at the House of Wisdom in B ...
(c. ) and the proponents of the earlier dating therefore see the ''Mishnat ha-Middot'' linking
Greek
Greek may refer to:
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group
*Greek language, a branch of the Indo-European language family
**Proto-Greek language, the assumed last common ancestor of all kno ...
and
Islamic mathematics
Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built upon syntheses of Greek mathematics (Euclid, Archimedes, Apollonius) and Indian mathematics (Aryabhata, Brahmagupta). Important developments of ...
.
Modern history
The ''Mishnat ha-Middot'' was discovered in MS 36 of the Munich Library by Moritz Steinschneider in 1862. The manuscript, copied in
Constantinople
Constantinople (#Names of Constantinople, see other names) was a historical city located on the Bosporus that served as the capital of the Roman Empire, Roman, Byzantine Empire, Byzantine, Latin Empire, Latin, and Ottoman Empire, Ottoman empire ...
in 1480, goes as far as the end of Chapter V. According to the colophon, the copyist believed the text to be complete. Steinschneider published the work in 1864, in honour of the seventieth birthday of
Leopold Zunz
Leopold Zunz (—''Yom Tov Tzuntz'', —''Lipmann Zunz''; 10 August 1794 – 17 March 1886) was the founder of academic Judaic Studies ('' Wissenschaft des Judentums''), the critical investigation of Jewish literature, hymnology and ritual. Nah ...
. The text was edited and published again by mathematician Hermann Schapira in 1880.
After the discovery by
Otto Neugebauer
Otto Eduard Neugebauer (May 26, 1899 – February 19, 1990) was an Austrian-American mathematician and historian of science who became known for his research on the history of astronomy and the other exact sciences as they were practiced in an ...
of a
genizah
A genizah (; , also ''geniza''; plural: ''genizot'' 'h''or ''genizahs'') is a storage area in a Judaism, Jewish synagogue or cemetery designated for the temporary storage of worn-out Hebrew-language books and papers on religious topics prior t ...
-fragment in the
Bodleian Library
The Bodleian Library () is the main research library of the University of Oxford. Founded in 1602 by Sir Thomas Bodley, it is one of the oldest libraries in Europe. With over 13 million printed items, it is the second-largest library in ...
containing Chapter VI,
Solomon Gandz
Solomon Gandz (2 February 1883, Tarnobrzeg, Austria-Hungary – 30 March 1954) was a historian of science.
Gandz published on the history of mathematics and astronomy in medieval Jewish and Islamic civilizations.
From 1915 to 1919, Gandz was ...
published a complete version of the ''Mishnat ha-Middot'' in 1932, accompanied by a thorough philological analysis. A third manuscript of the work was found among uncatalogued material in the Archives of the
Jewish Museum of Prague
The Jewish Museum in Prague () is a museum of Jewish heritage in the Czech Republic and one of the most visited museums in Prague. Its collection of Judaica is one of the largest in the world, about 40,000 objects, 100,000 books, and a copious a ...
in 1965.
Contents
Although primarily a practical work, the ''Mishnat ha-Middot'' attempts to define terms and explain both geometric application and theory. The book begins with a discussion that defines "aspects" for the different kinds of
plane figure
Plane most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
* Plane (mathematics), generalizations of a geometrical plane
Plane or planes may also refer to:
Biology
* Plane ...
s (
quadrilateral
In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''l ...
,
triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
,
circle
A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
, and segment of a circle) in Chapter I (§1–5), and with the basic principles of measurement of
area
Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...
s (§6–9). In Chapter II, the work introduces concise rules for the measurement of plane figures (§1–4), as well as a few problems in the calculation of
volume
Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...
(§5–12). In Chapters III–V, the ''Mishnat ha-Middot'' explains again in detail the measurement of the four types of plane figures, with reference to numerical examples. The text concludes with a discussion of the proportions of the
Tabernacle
According to the Hebrew Bible, the tabernacle (), also known as the Tent of the Congregation (, also Tent of Meeting), was the portable earthly dwelling of God used by the Israelites from the Exodus until the conquest of Canaan. Moses was instru ...
in Chapter VI.
The treatise argues against the common belief that the
geometric
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
ratio π as being exactly equal to 3 and defines it as instead. The book arrives at this approximation by calculating the
area of a circle
In geometry, the area enclosed by a circle of radius is . Here, the Greek letter represents the constant ratio of the circumference of any circle to its diameter, approximately equal to 3.14159.
One method of deriving this formula, which ori ...