The **Mishnat ha-Middot** (
Hebrew: מִשְׁנַת הַמִּדּוֹת,
lit. 'Treatise of Measures') is the earliest known
Hebrew
treatise on
geometry, composed of 49 *
mishnayot* in six chapters. Scholars have dated the work to either the
Mishnaic period or the
early Islamic era.

Moritz Steinschneider dated the *Mishnat ha-Middot* to between 800 and 1200 CE.^{
[1]} Sarfatti and Langermann have advanced Steinschneider's claim of
Arabic influence on the work's
terminology, and date the text to the early ninth century.^{
[2]}^{
[3]}

On the other hand,
Hermann Schapira 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.^{
[4]}
Solomon Gandz conjectured that the text was compiled no later than 150 CE (possibly by
Rabbi Nehemiah) and intended to be a part of the
Mishnah, but was excluded from its final canonical edition because the work was regarded as too
secular.^{
[5]} The content resembles both the work of
Hero of Alexandria (c. 100 CE) and that of
al-Khwārizmī (c. 800 CE) and the proponents of the earlier dating therefore see the *Mishnat ha-Middot* linking
Greek and
Islamic mathematics.^{
[6]}

The *Mishnat ha-Middot* was discovered in MS 36 of the
Munich Library by Moritz Steinschneider in 1862.^{
[1]} The manuscript, copied in
Constantinople in 1480, goes as far as the end of Chapter V. According to the
colophon, the copyist believed the text to be complete.^{
[7]} Steinschneider published the work in 1864, in honour of the seventieth birthday of
Leopold Zunz.^{
[8]} The text was edited and published again by mathematician Hermann Schapira in 1880.^{
[4]}

After the discovery by
Otto Neugebauer of a
genizah-fragment in the
Bodleian Library 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.^{
[7]}

Although primarily a practical work, the *Mishnat ha-Middot* attempts to define terms and explain both geometric application and theory.^{
[9]} The book begins with a discussion that defines "aspects" for the different kinds of
plane figures (
quadrilateral,
triangle,
circle, and
segment of a circle) in Chapter I (§1–5), and with the basic principles of measurement of
areas (§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 (§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.^{
[10]} The text concludes with a discussion of the proportions of the
Tabernacle in Chapter VI.^{
[11]}^{
[12]}

The treatise argues against the common belief that the
Tanakh defines the
geometric ratio
π as being exactly equal to 3 and defines it as
31⁄7 instead.^{
[5]} The book arrives at this approximation by calculating the
area of a circle according to the formulae

- and .
^{ [11]}^{: II§3, V§3 }

- ^
^{a}^{b}Steinschneider, Moritz, ed. (1864).*Mischnat ha-Middot, die erste Geometrische Schrift in Hebräischer Sprache, nest Epilog der Geometrie des Abr. ben Chija*(in Hebrew and German). Berlin. **^**Sarfatti, Gad B. (1993). "Mishnat ha-Middot". In Ben-Shammai, H. (ed.).*Ḥiqrei Ever ve-Arav [Festschrift Joshua Blau]*(in Hebrew). Tel Aviv and Jerusalem. p. 463.**^**Langermann, Y. Tzvi (2002). "On the Beginnings of Hebrew Scientific Literature and on Studying History through "Maqbiloṯ" (Parallels)".*Aleph*. Indiana University Press.**2**(2): 169–189. doi: 10.2979/ALE.2002.-.2.169. JSTOR 40385478. S2CID 170928770.- ^
^{a}^{b}Schapira, Hermann, ed. (1880). "Mischnath Ha-Middoth".*Zeitschrift für Mathematik und Physik*(in Hebrew and German). Leipzig. - ^
^{a}^{b}Gandz, Solomon (January 1936). "The Sources of Al-Khowārizmī's Algebra".*Osiris*. University of Chicago Press.**1**: 263–277. doi: 10.1086/368426. JSTOR 301610. S2CID 60770737. **^**Gandz, Solomon (1938–1939). "Studies in Hebrew Mathematics and Astronomy".*Proceedings of the American Academy for Jewish Research*. American Academy for Jewish Research.**9**: 5–50. doi: 10.2307/3622087. JSTOR 3622087.- ^
^{a}^{b}Scheiber, Sándor (1974). "Prague manuscript of*Mishnat ha-Middot*".*Hebrew Union College Annual*.**45**: 191–196. ISSN 0360-9049. JSTOR 23506854. **^**Thomson, William (November 1933). "Review: The Mishnat ha-Middot by Solomon Gandz".*Isis*. University of Chicago Press.**20**(1): 274–280. doi: 10.1086/346775. JSTOR 224893.**^**Levey, Martin (June 1955). "Solomon Gandz, 1884–1954".*Isis*. University of Chicago Press.**46**(2): 107–110. doi: 10.1086/348405. JSTOR 227124. S2CID 143232106.**^**Neuenschwander, Erwin (1988). "Reflections on the Sources of Arabic Geometry".*Sudhoffs Archiv*. Franz Steiner Verlag.**72**(2): 160–169. JSTOR 20777187.- ^
^{a}^{b}Gandz, Solomon, ed. (1932).*The Mishnat ha-Middot, the First Hebrew Geometry of about 150 C. E., and the Geometry of Muhammad Ibn Musa Al-Khowarizmi, the first Arabic Geometry (c. 820), Representing the Arabic Version of the Mishnat ha-Middot*. Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik A. Vol. 2. Translated by Gandz, Solomon. Berlin: Springer. **^**Sarfatti, Gad B. (1974). "Some remarks about the Prague manuscript of*Mishnat ha-Middot*".*Hebrew Union College Annual*.**45**: 197–204. ISSN 0360-9049. JSTOR 23506855.

- MS Heb. c. 18, Catalogue of the Genizah Fragments in the Bodleian Libraries.