Pythagorean theorem (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Pythagorean theorem" in English language version.

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Euclid (1956), pp. 351–352 Euclid (1956). The Thirteen Books of Euclid's Elements, Translated from the Text of Heiberg, with Introduction and Commentary. Vol. 1 (Books I and II). Translated by Heath, Thomas L. (Reprint of 2nd (1925) ed.). Dover. On-line text at archive.org
(Loomis 1940) Loomis, Elisha Scott (1940). The Pythagorean Proposition (2nd ed.). Ann Arbor, Michigan: Edwards Brothers. ISBN 9780873530361. Reissued 1968 by the National Council of Teachers of Mathematics. A lower-quality scan was published online by the Education Resources Information Center, ERIC ED037335.
Jan Gullberg (1997). Mathematics: from the birth of numbers. W. W. Norton & Company. p. 435. ISBN 0-393-04002-X.
(Loomis 1940, p. 113, Geometric proof 22 and Figure 123) Loomis, Elisha Scott (1940). The Pythagorean Proposition (2nd ed.). Ann Arbor, Michigan: Edwards Brothers. ISBN 9780873530361. Reissued 1968 by the National Council of Teachers of Mathematics. A lower-quality scan was published online by the Education Resources Information Center, ERIC ED037335.
Ernest Julius Wilczynski; Herbert Ellsworth Slaught (1914). "Theorem 1 and Theorem 2". Plane trigonometry and applications. Allyn and Bacon. p. 85.
Lawrence S. Leff (2005). PreCalculus the Easy Way (7th ed.). Barron's Educational Series. p. 296. ISBN 0-7641-2892-2.
Howard Whitley Eves (1983). "§4.8:...generalization of Pythagorean theorem". Great moments in mathematics (before 1650). Mathematical Association of America. p. 41. ISBN 0-88385-310-8.
For the details of such a construction, see Jennings, George (1997). "Figure 1.32: The generalized Pythagorean theorem". Modern geometry with applications: with 150 figures (3rd ed.). Springer. p. 23. ISBN 0-387-94222-X.
An extensive discussion of the historical evidence is provided in (Euclid 1956, p. 351) page=351 Euclid (1956). The Thirteen Books of Euclid's Elements, Translated from the Text of Heiberg, with Introduction and Commentary. Vol. 1 (Books I and II). Translated by Heath, Thomas L. (Reprint of 2nd (1925) ed.). Dover. On-line text at archive.org
Robert P. Crease (2008). The great equations: breakthroughs in science from Pythagoras to Heisenberg. W W Norton & Co. p. 25. ISBN 978-0-393-06204-5.

books.google.com

Judith D. Sally; Paul Sally (2007). "Chapter 3: Pythagorean triples". Roots to research: a vertical development of mathematical problems. American Mathematical Society Bookstore. p. 63. ISBN 978-0-8218-4403-8.
Benson, Donald. The Moment of Proof : Mathematical Epiphanies, pp. 172–173 (Oxford University Press, 1999).
(Maor 2007, p. 39) Maor, Eli (2007). The Pythagorean Theorem: A 4,000-Year History. Princeton, New Jersey: Princeton University Press. ISBN 978-0-691-12526-8.
Stephen W. Hawking (2005). God created the integers: the mathematical breakthroughs that changed history. Philadelphia: Running Press Book Publishers. p. 12. ISBN 0-7624-1922-9. This proof first appeared after a computer program was set to check Euclidean proofs.
The proof by Pythagoras probably was not a general one, as the theory of proportions was developed only two centuries after Pythagoras; see (Maor 2007, p. 25) Maor, Eli (2007). The Pythagorean Theorem: A 4,000-Year History. Princeton, New Jersey: Princeton University Press. ISBN 978-0-691-12526-8.
Published in a weekly mathematics column: James A Garfield (1876). "Pons Asinorum". The New England Journal of Education. 3 (14): 161. as noted in William Dunham (1997). The mathematical universe: An alphabetical journey through the great proofs, problems, and personalities. Wiley. p. 96. ISBN 0-471-17661-3. and in A calendar of mathematical dates: April 1, 1876 Archived July 14, 2010, at the Wayback Machine by V. Frederick Rickey
Judith D. Sally; Paul J. Sally Jr. (2007-12-21). "Theorem 2.4 (Converse of the Pythagorean theorem).". Roots to Research. American Mathematical Society. pp. 54–55. ISBN 978-0-8218-4403-8.
Law, Henry (1853). "Corollary 5 of Proposition XLVII (Pythagoras's Theorem)". The Elements of Euclid: with many additional propositions, and explanatory notes, to which is prefixed an introductory essay on logic. John Weale. p. 49.
Shaughan Lavine (1994). Understanding the infinite. Harvard University Press. p. 13. ISBN 0-674-92096-1.
(Heath 1921, Vol I, pp. 65); Hippasus was on a voyage at the time, and his fellows cast him overboard. See James R. Choike (1980). "The pentagram and the discovery of an irrational number". The College Mathematics Journal. 11: 312–316. Heath, Sir Thomas (1921). "The 'Theorem of Pythagoras'". A History of Greek Mathematics (2 Vols.) (Dover Publications, Inc. (1981) ed.). Clarendon Press, Oxford. pp. 144 ff. ISBN 0-486-24073-8.
Jon Orwant; Jarkko Hietaniemi; John Macdonald (1999). "Euclidean distance". Mastering algorithms with Perl. O'Reilly Media, Inc. p. 426. ISBN 1-56592-398-7.
Wentworth, George (2009). Plane Trigonometry and Tables. BiblioBazaar, LLC. p. 116. ISBN 978-1-103-07998-8., Exercises, page 116
Pertti Lounesto (2001). "§7.4 Cross product of two vectors". Clifford algebras and spinors (2nd ed.). Cambridge University Press. p. 96. ISBN 0-521-00551-5.
Francis Begnaud Hildebrand (1992). Methods of applied mathematics (Reprint of Prentice-Hall 1965 2nd ed.). Courier Dover Publications. p. 24. ISBN 0-486-67002-3.
Lawrence S. Leff (2005-05-01). cited work. Barron's Educational Series. p. 326. ISBN 0-7641-2892-2.
Judith D. Sally; Paul Sally (2007-12-21). "Exercise 2.10 (ii)". Roots to Research: A Vertical Development of Mathematical Problems. American Mathematical Soc. p. 62. ISBN 978-0-8218-4403-8.
Claudi Alsina, Roger B. Nelsen: Charming Proofs: A Journey Into Elegant Mathematics. MAA, 2010, ISBN 9780883853481, pp. 77–78 (excerpt, p. 77, at Google Books)
Rajendra Bhatia (1997). Matrix analysis. Springer. p. 21. ISBN 0-387-94846-5.
Ferdinand van der Heijden; Dick de Ridder (2004). Classification, parameter estimation, and state estimation. Wiley. p. 357. ISBN 0-470-09013-8.
Qun Lin; Jiafu Lin (2006). Finite element methods: accuracy and improvement. Elsevier. p. 23. ISBN 7-03-016656-6.
Howard Anton; Chris Rorres (2010). Elementary Linear Algebra: Applications Version (10th ed.). Wiley. p. 336. ISBN 978-0-470-43205-1.
Karen Saxe (2002). "Theorem 1.2". Beginning functional analysis. Springer. p. 7. ISBN 0-387-95224-1.
Douglas, Ronald G. (1998). Banach Algebra Techniques in Operator Theory (2nd ed.). New York, New York: Springer-Verlag New York, Inc. pp. 60–61. ISBN 978-0-387-98377-6.
Eric W. Weisstein (2003). CRC concise encyclopedia of mathematics (2nd ed.). CRC Press. p. 2147. ISBN 1-58488-347-2. The parallel postulate is equivalent to the Equidistance postulate, Playfair axiom, Proclus axiom, the Triangle postulate and the Pythagorean theorem.
Alexander R. Pruss (2006). The principle of sufficient reason: a reassessment. Cambridge University Press. p. 11. ISBN 0-521-85959-X. We could include...the parallel postulate and derive the Pythagorean theorem. Or we could instead make the Pythagorean theorem among the other axioms and derive the parallel postulate.
Stephen W. Hawking (2005). cited work. Running Press. p. 4. ISBN 0-7624-1922-9.
Barrett O'Neill (2006). "Exercise 4". Elementary Differential Geometry (2nd ed.). Academic Press. p. 441. ISBN 0-12-088735-5.
Saul Stahl (1993). "Theorem 8.3". The Poincaré half-plane: a gateway to modern geometry. Jones & Bartlett Learning. p. 122. ISBN 0-86720-298-X.
Jane Gilman (1995). "Hyperbolic triangles". Two-generator discrete subgroups of PSL(2,R). American Mathematical Society Bookstore. ISBN 0-8218-0361-1.
Neugebauer 1969, p. 36. Neugebauer, Otto (1969). The exact sciences in antiquity (2nd ed.). Courier Dover Publications. ISBN 0-486-22332-9. OCLC 638685764.
Neugebauer 1969: p. 36 "In other words it was known during the whole duration of Babylonian mathematics that the sum of the squares on the lengths of the sides of a right triangle equals the square of the length of the hypotenuse." Neugebauer, Otto (1969). The exact sciences in antiquity (2nd ed.). Courier Dover Publications. ISBN 0-486-22332-9. OCLC 638685764.
Kim Plofker (2009). Mathematics in India. Princeton University Press. pp. 17–18. ISBN 978-0-691-12067-6.
Carl Benjamin Boyer; Uta C. Merzbach (2011). "China and India". A history of mathematics (3rd ed.). Wiley. p. 229. ISBN 978-0470525487. Quote: [In Sulba-sutras,] we find rules for the construction of right angles by means of triples of cords the lengths of which form Pythagorean triages, such as 3, 4, and 5, or 5, 12, and 13, or 8, 15, and 17, or 12, 35, and 37. Although Mesopotamian influence in the Sulvasũtras is not unlikely, we know of no conclusive evidence for or against this. Aspastamba knew that the square on the diagonal of a rectangle is equal to the sum of the squares on the two adjacent sides. Less easily explained is another rule given by Apastamba – one that strongly resembles some of the geometric algebra in Book II of Euclid's Elements. (...)
"Introduction and books 1,2". The University Press. March 25, 1908 – via Google Books.
(Heath 1921, Vol I, p. 144): "Though this is the proposition universally associated by tradition with the name of Pythagoras, no really trustworthy evidence exists that it was actually discovered by him. The comparatively late writers who attribute it to him add the story that he sacrificed an ox to celebrate his discovery." Heath, Sir Thomas (1921). "The 'Theorem of Pythagoras'". A History of Greek Mathematics (2 Vols.) (Dover Publications, Inc. (1981) ed.). Clarendon Press, Oxford. pp. 144 ff. ISBN 0-486-24073-8.
An extensive discussion of the historical evidence is provided in (Euclid 1956, p. 351) page=351 Euclid (1956). The Thirteen Books of Euclid's Elements, Translated from the Text of Heiberg, with Introduction and Commentary. Vol. 1 (Books I and II). Translated by Heath, Thomas L. (Reprint of 2nd (1925) ed.). Dover. On-line text at archive.org
Asger Aaboe (1997). Episodes from the early history of mathematics. Mathematical Association of America. p. 51. ISBN 0-88385-613-1. ...it is not until Euclid that we find a logical sequence of general theorems with proper proofs.
This work is a compilation of 246 problems, some of which survived the book burning of 213 BC, and was put in final form before 100 AD. It was extensively commented upon by Liu Hui in 263 AD. Philip D. Straffin Jr. (2004). "Liu Hui and the first golden age of Chinese mathematics". In Marlow Anderson; Victor J. Katz; Robin J. Wilson (eds.). Sherlock Holmes in Babylon: and other tales of mathematical history. Mathematical Association of America. pp. 69 ff. ISBN 0-88385-546-1. See particularly §3: Nine chapters on the mathematical art, pp. 71 ff.
Kangshen Shen; John N. Crossley; Anthony Wah-Cheung Lun (1999). The nine chapters on the mathematical art: companion and commentary. Oxford University Press. p. 488. ISBN 0-19-853936-3.
In particular, Li Jimin; see Centaurus, Volume 39. Copenhagen: Munksgaard. 1997. pp. 193, 205.
Chen, Cheng-Yih (1996). "§3.3.4 Chén Zǐ's formula and the Chóng-Chã method; Figure 40". Early Chinese work in natural science: a re-examination of the physics of motion, acoustics, astronomy and scientific thoughts. Hong Kong University Press. p. 142. ISBN 962-209-385-X.
Wen-tsün Wu (2008). "The Gougu theorem". Selected works of Wen-tsün Wu. World Scientific. p. 158. ISBN 978-981-279-107-8.

cam.ac.uk

dpmms.cam.ac.uk

For an extended discussion of this generalization, see, for example, Willie W. Wong Archived 2009-12-29 at the Wayback Machine 2002, A generalized n-dimensional Pythagorean theorem.

clarku.edu

aleph0.clarku.edu

"Euclid's Elements, Book I, Proposition 47". See also a web page version using Java applets by Prof. David E. Joyce, Clark University.
Euclid's Elements, Book I, Proposition 48 From D.E. Joyce's web page at Clark University

colgate.edu

math.colgate.edu

Lantz, David. "Garfield's proof of the Pythagorean Theorem". Math.Colgate.edu. Archived from the original on 2013-08-28. Retrieved 2018-01-14.

cut-the-knot.org

Alexander Bogomolny. "Cut-the-knot.org: Pythagorean theorem and its many proofs, Proof #4". Cut the Knot. Retrieved 4 November 2010.
Alexander Bogomolny. "Cut-the-knot.org: Pythagorean theorem and its many proofs, Proof #3". Cut the Knot. Retrieved 4 November 2010.
Alexander Bogomolny. "Pythagorean theorem, proof number 10". Cut the Knot. Retrieved 27 February 2010.
Bogomolny, Alexander. "Pythagorean Theorem". Interactive Mathematics Miscellany and Puzzles. Alexander Bogomolny. Archived from the original on 2010-07-06. Retrieved 2010-05-09.
Alexander Bogomolny, Pythagorean Theorem for the Reciprocals,https://www.cut-the-knot.org/pythagoras/PTForReciprocals.shtml

doi.org

Mike Staring (1996). "The Pythagorean proposition: A proof by means of calculus". Mathematics Magazine. 69 (1). Mathematical Association of America: 45–46. doi:10.2307/2691395. JSTOR 2691395.
Bruce C. Berndt (1988). "Ramanujan – 100 years old (fashioned) or 100 years new (fangled)?". The Mathematical Intelligencer. 10 (3): 24–31. doi:10.1007/BF03026638. S2CID 123311054.
Kurt Von Fritz (Apr 1945). "The Discovery of Incommensurability by Hippasus of Metapontum". Annals of Mathematics. Second Series. 46 (2): 242–264. doi:10.2307/1969021. JSTOR 1969021.
WS Massey (Dec 1983). "Cross products of vectors in higher-dimensional Euclidean spaces" (PDF). The American Mathematical Monthly. 90 (10). Mathematical Association of America: 697–701. doi:10.2307/2323537. JSTOR 2323537. S2CID 43318100. Archived from the original (PDF) on 2021-02-26.
Aydin Sayili (Mar 1960). "Thâbit ibn Qurra's Generalization of the Pythagorean Theorem". Isis. 51 (1): 35–37. doi:10.1086/348837. JSTOR 227603. S2CID 119868978.
Donald R Conant & William A Beyer (Mar 1974). "Generalized Pythagorean Theorem". The American Mathematical Monthly. 81 (3). Mathematical Association of America: 262–265. doi:10.2307/2319528. JSTOR 2319528.
Victor Pambuccian (December 2010). "Maria Teresa Calapso's Hyperbolic Pythagorean Theorem". The Mathematical Intelligencer. 32 (4): 2. doi:10.1007/s00283-010-9169-0.
Friberg, Jöran (1981). "Methods and traditions of Babylonian mathematics: Plimpton 322, Pythagorean triples, and the Babylonian triangle parameter equations". Historia Mathematica. 8: 277–318. doi:10.1016/0315-0860(81)90069-0.: p. 306 "Although Plimpton 322 is a unique text of its kind, there are several other known texts testifying that the Pythagorean theorem was well known to the mathematicians of the Old Babylonian period."
Robson, Eleanor (2001). "Neither Sherlock Holmes nor Babylon: a reassessment of Plimpton 322". Historia Mathematica. 28 (3): 167–206. doi:10.1006/hmat.2001.2317.

ed.gov

eric.ed.gov

(Loomis 1940) Loomis, Elisha Scott (1940). The Pythagorean Proposition (2nd ed.). Ann Arbor, Michigan: Edwards Brothers. ISBN 9780873530361. Reissued 1968 by the National Council of Teachers of Mathematics. A lower-quality scan was published online by the Education Resources Information Center, ERIC ED037335.
(Loomis 1940, p. 113, Geometric proof 22 and Figure 123) Loomis, Elisha Scott (1940). The Pythagorean Proposition (2nd ed.). Ann Arbor, Michigan: Edwards Brothers. ISBN 9780873530361. Reissued 1968 by the National Council of Teachers of Mathematics. A lower-quality scan was published online by the Education Resources Information Center, ERIC ED037335.

jstor.org

Mike Staring (1996). "The Pythagorean proposition: A proof by means of calculus". Mathematics Magazine. 69 (1). Mathematical Association of America: 45–46. doi:10.2307/2691395. JSTOR 2691395.
Kurt Von Fritz (Apr 1945). "The Discovery of Incommensurability by Hippasus of Metapontum". Annals of Mathematics. Second Series. 46 (2): 242–264. doi:10.2307/1969021. JSTOR 1969021.
WS Massey (Dec 1983). "Cross products of vectors in higher-dimensional Euclidean spaces" (PDF). The American Mathematical Monthly. 90 (10). Mathematical Association of America: 697–701. doi:10.2307/2323537. JSTOR 2323537. S2CID 43318100. Archived from the original (PDF) on 2021-02-26.
Aydin Sayili (Mar 1960). "Thâbit ibn Qurra's Generalization of the Pythagorean Theorem". Isis. 51 (1): 35–37. doi:10.1086/348837. JSTOR 227603. S2CID 119868978.
Donald R Conant & William A Beyer (Mar 1974). "Generalized Pythagorean Theorem". The American Mathematical Monthly. 81 (3). Mathematical Association of America: 262–265. doi:10.2307/2319528. JSTOR 2319528.

maa.org

Published in a weekly mathematics column: James A Garfield (1876). "Pons Asinorum". The New England Journal of Education. 3 (14): 161. as noted in William Dunham (1997). The mathematical universe: An alphabetical journey through the great proofs, problems, and personalities. Wiley. p. 96. ISBN 0-471-17661-3. and in A calendar of mathematical dates: April 1, 1876 Archived July 14, 2010, at the Wayback Machine by V. Frederick Rickey

ox.ac.uk

ora.ox.ac.uk

Robson, Eleanor (2001). "Neither Sherlock Holmes nor Babylon: a reassessment of Plimpton 322". Historia Mathematica. 28 (3): 167–206. doi:10.1006/hmat.2001.2317.

researchgate.net

Friberg, Jöran (1981). "Methods and traditions of Babylonian mathematics: Plimpton 322, Pythagorean triples, and the Babylonian triangle parameter equations". Historia Mathematica. 8: 277–318. doi:10.1016/0315-0860(81)90069-0.: p. 306 "Although Plimpton 322 is a unique text of its kind, there are several other known texts testifying that the Pythagorean theorem was well known to the mathematicians of the Old Babylonian period."

ruc.dk

akira.ruc.dk

Høyrup, Jens. "Pythagorean 'Rule' and 'Theorem' – Mirror of the Relation Between Babylonian and Greek Mathematics". In Renger, Johannes (ed.). Babylon: Focus mesopotamischer Geschichte, Wiege früher Gelehrsamkeit, Mythos in der Moderne. 2. Internationales Colloquium der Deutschen Orient-Gesellschaft 24.–26. März 1998 in Berlin (PDF). Berlin: Deutsche Orient-Gesellschaft / Saarbrücken: SDV Saarbrücker Druckerei und Verlag. pp. 393–407., p. 406, "To judge from this evidence alone it is therefore likely that the Pythagorean rule was discovered within the lay surveyors’ environment, possibly as a spin-off from the problem treated in Db₂-146, somewhere between 2300 and 1825 BC." (Db₂-146 is an Old Babylonian clay tablet from Eshnunna concerning the computation of the sides of a rectangle given its area and diagonal.)

semanticscholar.org

api.semanticscholar.org

Bruce C. Berndt (1988). "Ramanujan – 100 years old (fashioned) or 100 years new (fangled)?". The Mathematical Intelligencer. 10 (3): 24–31. doi:10.1007/BF03026638. S2CID 123311054.
WS Massey (Dec 1983). "Cross products of vectors in higher-dimensional Euclidean spaces" (PDF). The American Mathematical Monthly. 90 (10). Mathematical Association of America: 697–701. doi:10.2307/2323537. JSTOR 2323537. S2CID 43318100. Archived from the original (PDF) on 2021-02-26.
Aydin Sayili (Mar 1960). "Thâbit ibn Qurra's Generalization of the Pythagorean Theorem". Isis. 51 (1): 35–37. doi:10.1086/348837. JSTOR 227603. S2CID 119868978.

pdfs.semanticscholar.org

WS Massey (Dec 1983). "Cross products of vectors in higher-dimensional Euclidean spaces" (PDF). The American Mathematical Monthly. 90 (10). Mathematical Association of America: 697–701. doi:10.2307/2323537. JSTOR 2323537. S2CID 43318100. Archived from the original (PDF) on 2021-02-26.

slu.edu

See for example Pythagorean theorem by shear mapping Archived 2016-10-14 at the Wayback Machine, Saint Louis University website Java applet

stanford.edu

plato.stanford.edu

Huffman, Carl (23 February 2005). "Pythagoras". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy (Winter 2018 Edition)., "It should now be clear that decisions about sources are crucial in addressing the question of whether Pythagoras was a mathematician and scientist. The view of Pythagoras's cosmos sketched in the first five paragraphs of this section, according to which he was neither a mathematician nor a scientist, remains the consensus."

usma.edu

math.usma.edu

Published in a weekly mathematics column: James A Garfield (1876). "Pons Asinorum". The New England Journal of Education. 3 (14): 161. as noted in William Dunham (1997). The mathematical universe: An alphabetical journey through the great proofs, problems, and personalities. Wiley. p. 96. ISBN 0-471-17661-3. and in A calendar of mathematical dates: April 1, 1876 Archived July 14, 2010, at the Wayback Machine by V. Frederick Rickey

utexas.edu

farside.ph.utexas.edu

Heiberg, J.L. "Euclid's Elements of Geometry" (PDF). pp. 46–47.

cs.utexas.edu

Dijkstra, Edsger W. (September 7, 1986). "On the theorem of Pythagoras". EWD975. E. W. Dijkstra Archive.

web.archive.org

See for example Pythagorean theorem by shear mapping Archived 2016-10-14 at the Wayback Machine, Saint Louis University website Java applet
Published in a weekly mathematics column: James A Garfield (1876). "Pons Asinorum". The New England Journal of Education. 3 (14): 161. as noted in William Dunham (1997). The mathematical universe: An alphabetical journey through the great proofs, problems, and personalities. Wiley. p. 96. ISBN 0-471-17661-3. and in A calendar of mathematical dates: April 1, 1876 Archived July 14, 2010, at the Wayback Machine by V. Frederick Rickey
Lantz, David. "Garfield's proof of the Pythagorean Theorem". Math.Colgate.edu. Archived from the original on 2013-08-28. Retrieved 2018-01-14.
Bogomolny, Alexander. "Pythagorean Theorem". Interactive Mathematics Miscellany and Puzzles. Alexander Bogomolny. Archived from the original on 2010-07-06. Retrieved 2010-05-09.
WS Massey (Dec 1983). "Cross products of vectors in higher-dimensional Euclidean spaces" (PDF). The American Mathematical Monthly. 90 (10). Mathematical Association of America: 697–701. doi:10.2307/2323537. JSTOR 2323537. S2CID 43318100. Archived from the original (PDF) on 2021-02-26.
For an extended discussion of this generalization, see, for example, Willie W. Wong Archived 2009-12-29 at the Wayback Machine 2002, A generalized n-dimensional Pythagorean theorem.

worldcat.org

search.worldcat.org

Neugebauer 1969, p. 36. Neugebauer, Otto (1969). The exact sciences in antiquity (2nd ed.). Courier Dover Publications. ISBN 0-486-22332-9. OCLC 638685764.
Neugebauer 1969: p. 36 "In other words it was known during the whole duration of Babylonian mathematics that the sum of the squares on the lengths of the sides of a right triangle equals the square of the length of the hypotenuse." Neugebauer, Otto (1969). The exact sciences in antiquity (2nd ed.). Courier Dover Publications. ISBN 0-486-22332-9. OCLC 638685764.