The Mishnat ha-Middot ( he, מִשְׁנַת הַמִּדּוֹת, 'Treatise of Measures') is the earliest known
Hebrew
Hebrew (; ; ) is a Northwest Semitic language of the Afroasiatic language family. Historically, it is one of the spoken languages of the Israelites and their longest-surviving descendants, the Jews and Samaritans. It was largely preserved ...
treatise
A treatise is a formal and systematic written discourse on some subject, generally longer and treating it in greater depth than an essay, and more concerned with investigating or exposing the principles of the subject and its conclusions."Treat ...
on
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 ...
, composed of 49 ''
mishnayot
The Mishnah or the Mishna (; he, מִשְׁנָה, "study by repetition", from the verb ''shanah'' , or "to study and review", also "secondary") is the first major written collection of the Jewish oral traditions which is known as the Oral Torah ...
Moritz Steinschneider
Moritz Steinschneider (30 March 1816, Prostějov, Moravia, Austrian Empire – 24 January 1907, Berlin) was a Moravian bibliographer and Orientalist. He received his early instruction in Hebrew from his father, Jacob Steinschneider ( 1782; ...
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 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 ...
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 word, or multi-wor ...
, and date the text to the early ninth century.
On the other hand,
Hermann Schapira
Zvi Hermann Schapira ( he, צבי הרמן שפירא; 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 ...
argued that the treatise dates from an earlier era, most likely the Mishnaic period, as its mathematical terminology differs from that of the Hebrew mathematicians of the Arab period. Solomon Gandz 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 (; he, מִשְׁנָה, "study by repetition", from the verb ''shanah'' , or "to study and review", also "secondary") is the first major written collection of the Jewish oral traditions which is known as the Oral Torah ...
, but was excluded from its final canonical edition because the work was regarded as too
secular
Secularity, also the secular or secularness (from Latin ''saeculum'', "worldly" or "of a generation"), is the state of being unrelated or neutral in regards to religion. Anything that does not have an explicit reference to religion, either negativ ...
. The content resembles both the work of
Hero of Alexandria
Hero of Alexandria (; grc-gre, Ἥρων ὁ Ἀλεξανδρεύς, ''Heron ho Alexandreus'', also known as Heron of Alexandria ; 60 AD) was a Greece, Greek mathematician and engineer who was active in his native city of Alexandria, Roman Egy ...
(c. ) and that of
al-Khwārizmī
Muḥammad ibn Mūsā al-Khwārizmī ( ar, محمد بن موسى الخوارزمي, Muḥammad ibn Musā al-Khwārazmi; ), or al-Khwarizmi, was a Persian polymath from Khwarazm, who produced vastly influential works in mathematics, astronomy ...
(c. ) and the proponents of the earlier dating therefore see the ''Mishnat ha-Middot'' linking
Greek
Greek may refer to:
Greece
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 ...
and
Islamic mathematics
Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built on Greek mathematics (Euclid, Archimedes, Apollonius) and Indian mathematics (Aryabhata, Brahmagupta). Important progress was made, such as full ...
.
Modern history
The ''Mishnat ha-Middot'' was discovered in MS 36 of the Munich Library by Moritz Steinschneider in 1862. The manuscript, copied in
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 ( he, יום טוב צונץ—''Yom Tov Tzuntz'', yi, ליפמן צונץ—''Lipmann Zunz''; 10 August 1794 – 17 March 1886) was the founder of academic Judaic Studies (''Wissenschaft des Judentums''), the critical investigation ...
. 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 anti ...
of a
genizah
A genizah (; , also ''geniza''; plural: ''genizot'' 'h''or ''genizahs'') is a storage area in a Jewish synagogue or cemetery designated for the temporary storage of worn-out Hebrew-language books and papers on religious topics prior to proper ceme ...
-fragment in the
Bodleian Library
The Bodleian Library () is the main research library of the University of Oxford, and is one of the oldest libraries in Europe. It derives its name from its founder, Sir Thomas Bodley. With over 13 million printed items, it is the second- ...
containing Chapter VI, Solomon Gandz 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 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
A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type.
A plane shape or plane figure is constrained to lie o ...
s (
quadrilateral
In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...
,
triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, an ...
,
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
, and segment of a circle) in Chapter I (§1–5), and with the basic principles of measurement of
area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape
A shape or figure is a graphics, graphical representation of an obje ...
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 occupied 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). The de ...
(§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 ( he, מִשְׁכַּן, mīškān, residence, dwelling place), also known as the Tent of the Congregation ( he, link=no, אֹהֶל מוֹעֵד, ’ōhel mō‘ēḏ, also Tent of Meeting, etc.), ...
in Chapter VI.
The treatise argues against the common belief that the
geometric
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 ca ...
ratio π as being exactly equal to 3 and defines it as 3 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 origi ...