At the close of the third century Pappus of Alexandria (295 A.D.) wrote
on geometry, and one of his theorems, a generalized form of the
Pythagorean proposition, is mentioned in Chapter XVI of this work. Only
two other Greek writers on geometry need be mentioned. Theon of
Alexandria (370 A.D.), the father of the Hypatia who is the heroine of
Charles Kingsley's well-known novel, wrote a commentary on Euclid to
which we are indebted for some historical information. Proclus (410-485
A.D.) also wrote a commentary on Euclid, and much of our information
concerning the first Book of Euclid is due to him.
The East did little for geometry, although contributing considerably to
algebra. The first great Hindu writer was Aryabhatta, who was born in
476 A.D. He gave the very close approximation for [pi], expressed in
modern notation as 3.1416. He also gave rules for finding the volume of
the pyramid and sphere, but they were incorrect, showing that the Greek
mathematics had not yet reached the Ganges. Another Hindu writer,
Brahmagupta (born in 598 A.D.), wrote an encyclopedia of mathematics. He
gave a rule for finding Pythagorean numbers, expressed in modern symbols
as follows:
1/4((_p_^2/_q_) + _q_)^2 = 1/4((_p_^2/_q_) - _q_)^2 + _p_^2.
He also generalized Heron's formula by asserting that the area of an
inscribed quadrilateral of sides _a_, _b_, _c_, _d_, and semiperimeter
_s_, is [sqrt]((_s_ - _a_)(_s_ - _b_)(_s_ - _c_)(_s_ - _d_)).
The Arabs, about the time of the "Arabian Nights Tales" (800 A.D.), did
much for mathematics, translating the Greek authors into their language
and also bringing learning from India. Indeed, it is to them that modern
Europe owed its first knowledge of Euclid. They contributed nothing of
importance to elementary geometry, however.
The greatest of the Arab writers was Mohammed ibn Musa al-Khowarazmi
(820 A.D.). He lived at Bagdad and Damascus. Although chiefly interested
in astronomy, he wrote the first book bearing the name "algebra"
("Al-jabr wa'l-muq[=a]balah," Restoration and Equation), composed an
arithmetic using the Hindu numerals,[20] and paid much attention to
geometry and trigonometry.
Euclid was translated from the Arabic into Latin in the twelfth century,
Greek manuscripts not being then at hand, or being neglected because of
ignorance of the language. The leading translators were Athelhard of
Bath (1120), an English monk; Gherard of Cremona (1160), an Italian
monk; and Johannes Campanus (1250), chaplain to Pope Urban IV.
The greatest European mathematician of the Middle Ages was Leonardo of
Pisa[21] (_ca._ 1170-1250). He was very influential in making the
Hindu-Arabic numerals known in Europe, wrote extensively on algebra, and
was the author of one book on geometry. He contributed nothing to the
elementary theory, however. The first edition of Euclid was printed in
Latin in 1482, the first one in English appearing in 1570.
Public-domain text, read in full here on John Shaqi.
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