In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among
the three sides of a right triangle (right-angled triangle). In terms of areas, it states:
In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is
equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right
angle).
The theorem can be written as an equation relating the lengths of the sides a, b and c, often called the
Pythagorean equation:[1]
where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.
These two formulations show two fundamental aspects of this theorem: it is both a statement about areas and
about lengths. Tobias Dantzig refers to these as areal and metric interpretations.[2][3] Some proofs of the theorem
are based on one interpretation, some upon the other. Thus, Pythagoras' theorem stands with one foot in
geometry and the other in algebra, a connection made clear originally by Descartes in his work La Géométrie,
and extending today into other branches of mathematics.[4]
The Pythagorean theorem has been modified to apply outside its original domain. A number of these
generalizations are described below, including extension to many-dimensional Euclidean spaces, to spaces that
are not Euclidean, to objects that are not right triangles, and indeed, to objects that are not triangles at all, but n-
dimensional solids.
The Pythagorean theorem is named after the Greek mathematician Pythagoras, who by tradition is credited with
its discovery and proof,[5][6] although it is often argued that knowledge of the theorem predates him. (There is
much evidence that Babylonian mathematicians understood the formula, although there is little surviving
evidence that they fitted it into a mathematical framework.[7]) “[To the Egyptians and Babylonians] mathematics
provided practical tools in the form of "recipes" designed for specific calculations. Pythagoras, on the other
hand, was one of the first to grasp numbers as abstract entities that exist in their own right.”[8] In addition to a
separate section devoted to the history of Pythagoras' theorem, historical asides and sources are found in many
of the other subsections.
The Pythagorean theorem has attracted interest outside mathematics as a symbol of mathematical abstruseness,
mystique, or intellectual power. The article ends with a section on pop references to the theorem.
Bhaskara[1] (Marathi: भास्कर, Kannada: ಭಾಸ್ಕರಾಚಾರ್ಯ) (1114–1185), also known as Bhaskara II and
Bhaskara Achārya ("Bhaskara the teacher"), was an Indian mathematician and an astronomer. He was born
near Bijjada Bida which is in present day Bijapur district, Karnataka, India. Bhaskara was the head of an
astronomical observatory at Ujjain, the leading mathematical center of ancient India. His predecessors in this
post had included both the noted Indian mathematicians Brahmagupta and Varahamihira. He lived in the
Sahyadri region.[1]
Bhaskara and his works represent a significant contribution to mathematical and astronomical knowledge in the
12th century. He has been called the greatest mathematician of medieval India.[2] His main work was the
Siddhanta Siromani which is divided in to four parts called Lilavati , Bijaganita, Grahaganita and Goladhyaya.
[3]
Siddhanta Siromani is Sanskrit for "Crown of treatises".[4] The English translations of four titles are "Dealing
with Arithmetic", Algebra, "Mathematics of the planets" and Sphere respectively.
Bhaskara's work on calculus predates Newton and Leibniz by half a millenium.[5][6] He is particularly known in
the discovery of the principles of differential calculus and its application to astronomical problems and
computations. While Newton and Leibniz have been credited with differential and integral calculus, there is
strong evidence to suggest that Bhaskara was a pioneer in some of the principles of differential calculus. He was
perhaps the first to conceive the differential coefficient and differential calculus.[7]
Mathematics
Some of Bhaskara's contributions to mathematics include the following:
A proof of the Pythagorean theorem by calculating the same area in two different ways and then
canceling out terms to get a² + b² = c².
In Lilavati, solutions of quadratic, cubic and quartic indeterminate equations.
Solutions of indeterminate quadratic equations (of the type ax² + b = y²).
Integer solutions of linear and quadratic indeterminate equations (Kuttaka). The rules he gives are (in
effect) the same as those given by the Renaissance European mathematicians of the 17th century
A cyclic Chakravala method for solving indeterminate equations of the form ax² + bx + c = y. The
solution to this equation was traditionally attributed to William Brouncker in 1657, though his method
was more difficult than the chakravala method.
The first general method for finding the solutions of the problem x² − ny² = 1 (so-called "Pell's
equation") was given by Bhaskara II.[10]
Solutions of Diophantine equations of the second order, such as 61x² + 1 = y². This very equation was
posed as a problem in 1657 by the French mathematician Pierre de Fermat, but its solution was unknown
in Europe until the time of Euler in the 18th century.
Solved quadratic equations with more than one unknown, and found negative and irrational solutions.
Preliminary concept of mathematical analysis.
Preliminary concept of infinitesimal calculus, along with notable contributions towards integral calculus.
Conceived differential calculus, after discovering the derivative and differential coefficient.
Stated Rolle's theorem, a special case of one of the most important theorems in analysis, the mean value
theorem. Traces of the general mean value theorem are also found in his works.
Calculated the derivatives of trigonometric functions and formulae. (See Calculus section below.)
In Siddhanta Shiromani, Bhaskara developed spherical trigonometry along with a number of other
trigonometric results. (See Trigonometry section below.)
