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Brahmagupta theorem states that,
If a cyclic quadrilateral is orthodiagonal (i.e., has perpendicular diagonals), then the perpendicular to a side from the point of intersection of the diagonals always bisects the opposite side.”
Geometrically, this theorem means that, in a cyclic quadrilateral ABCD, diagonals AC and BD are perpendicular to each other. The intersection of AC and BD is M. Drop the perpendicular from M to the line BC, calling the intersection point E. Let F be the intersection of the line EM and the side AD. Then, according to the theorem, F is the midpoint of side AD.
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According to him,zero is a number that is obtained, when a number is subtracted from itself. He also mentioned some properties of zero, where positive numbers are termed as fortunes and negative numbers are termed as debt.
Diminish by the middle number the square-root of the rupas multiplied by four times the square and increased by the square of the middle number; divide the remainder by twice the square.”
A formula stated by him, for the computation of values of the sine table, having common interval (h) in the underlying base table as 900 minutes or 15 degrees, is given below.
In modern notation, the formula is denoted as sphuta-bhogyakhanda = (Dr + Dr-1)⁄2 ± t|Dr – Dr-1|⁄2, where ± is introduced according to Dr
He was the first who propose an interpolation formula using second-order difference. His Sanskrit verses on this formula were found in the Khandakadyaka work of Brahmagupta. Today, the Brahmagupta interpolation formula is known as Newton- Stirling interpolation formula. In his book, he termed the...
Brahmagupta illustrated the construction of several figures with arbitrary sides. He tried to construct figures such as isosceles triangles, scalene triangles, rectangles, isosceles trapezoids, isosceles trapezoids with three equal sides, and scalene cyclic quadrilateral, mainly, with the help of...
Brahmagupta also tried to approximate the value of π and in stanza 40 of his book he mentioned,
He also discussed rational triangles. A rational triangle with the rational area and sides a, b, c, are of the form:
A major portion of Brahmagupta’s work was dedicated to the study of geometry. One of his theorem about triangles states that,
Brahmagupta further extended his theory and claimed that, The square-root of the sum of the two products of the sides and opposite sides of a non-unequal quadrilateral is the diagonal. The square of the diagonal is diminished by the square of half the sum of the base and the top; the square-root ...
Brahmagupta in chapter 12, entitled “Calculation”, of his book, proposed a formula that was useful in generating Pythagorean triplets. He mentioned,
the sum of, a series of cubes and a series of squares for the first n natural numbers as follows:
"Five hundred drammas were loaned at an unknown rate of interest, The interest on the money for four months was loaned to another at the same rate of interest and amounted in ten mounths to 78 drammas. Give the rate of interest.”
He proposed a method of multiplication, “gomutrika”, in his book “Brahmasphutasiddhanta”. The title of this method was translated by Ifrah as, “Like the trajectory of cow’s urine”. In the 12th chapter of his book, he also tried to explain the rules of simplifying five types of combinations of fra...
‘Durkeamynarda’ (672), ‘Khandakhadyaka’ (665), ‘Brahmasphutasiddhanta’ (628) and ‘Cadamakela’ (624). The ‘Brahmasphutasiddhanta’ meaning the ‘Doctrine of Brahamagupta’ is one of his well-known works. The ‘Brahmasphutasiddhanta’ consists of 25 chapters. In the first 10 chapters, topics covered ar...
and conjunctions of the planets with the fixed stars. The rest of the 15 chapters are focused more on mathematical concepts. One of his fellow mathematicians gave him the title of ‘Ganita Chakra Chudamani’ which when translated meant, ‘The gem of the circle of mathematicians’. It is believed, tha...
He was a great mathematician and astronomer. Brahamgupta considered himself to be an astrologer, maybe because his father Jisnugupta was an astrologer but today, he is mostly remembered for his significant works in the field of mathematics. He was born in 598 CE in Bhinmal, a city in Rajasthan. H...
Brahmagupta’s formula for the cyclic quadrilaterals is regarded as his most famous discovery in geometry. Given the sides of a cyclic quadrilateral, he provided an approximate and exact formula for the area of the cyclic quadrilateral. He mentioned,
Although Brahmagupta thought of himself as an astronomer who did some mathematics, he is now mainly remembered for his contributions to mathematics. He was honoured by the title given to him by a fellow scientist ‘Ganita Chakra Chudamani’ which is translated as ‘The gem of the circle of mathemati...
Brahmagupta gave the solution of general linear equations in chapter 18 of his book and wrote,The difference between rupas, when inverted and divided by the difference of the coefficients of the unknowns, is the unknown in the equation. The rupas are subtracted on the side below that from which t...
Brahmagupta studied this equation 1000 years before Pell’s birth. Pell’s equation is of form, nx² + 1 = y², which can also be written as y² – nx² = 1, where ‘n’ is an integer and we have to solve it for (x, y) integer solutions. Brahmagupta also provided a lemma, in which he stated that if (a, b)...
proposed some methods to solve equations of the type ax + by = c. According to Majumdar, Brahmgupta used continued fractions to solve such equations. He also tried to solve quadratic equations of the type ax² + c = y² and ax² – c = y². For example, for the equation 8x² + 1 = y² he obtained the so...
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