Lalla ( c. 720–790 CE) was an Indian mathematician , astronomer , and astrologer who belonged to a family of astronomers. Lalla was the son of Trivikrama Bhatta and the grandson of Śâmba. He lived in central India, possibly in the Lāṭa region in modern south Gujarat . Lalla was known as being one of the leading Indian astronomers of the eighth century. Only two of his works are currently thought to be extant.
42-470: His best-known work is the Śiṣyadhīvṛddhidatantra ("Treatise which expands the intellect of students"). This text is one of the first major Sanskrit astronomical texts known from the period following the 7th-century works of Brahmagupta and Bhāskara I . It generally treats the same astronomical subject matter and demonstrates the same computational techniques as earlier authors, although there are some significant innovations, such that Lalla’s treatise offers
84-413: A / c + b / d × a / c = a ( d + b ) / cd ; and a / c − b / d × a / c = a ( d − b ) / cd . Brahmagupta then goes on to give the sum of the squares and cubes of the first n integers. 12.20. The sum of
126-567: A compromise between the rival astronomical schools of his predecessors, Āryabhaṭa I and Brahmagupta. It is within the Śiṣyadhīvṛddhidatantra that the earliest known description of perpetual motion is described. The other extant work by Lalla is the Jyotiṣaratnakośa ("Treasury of Jewels"), a treatise on catarchic astrology. This work is one of the earliest known Sanskrit astrological works for determining auspicious and inauspicious times. No edition of this text has ever been published while
168-442: A distance d from the top of a mountain of height m , and then travels in a straight line to a city at a horizontal distance mx from the base of the mountain, travels the same distance as one who descends vertically down the mountain and then travels along the horizontal to the city. Stated geometrically, this says that if a right-angled triangle has a base of length a = mx and altitude of length b = m + d , then
210-401: A formula useful for generating Pythagorean triples : 12.39. The height of a mountain multiplied by a given multiplier is the distance to a city; it is not erased. When it is divided by the multiplier increased by two it is the leap of one of the two who make the same journey. Or, in other words, if d = mx / x + 2 , then a traveller who "leaps" vertically upwards
252-430: A great deal of originality into his revision, adding a considerable amount of new material. The book consists of 24 chapters with 1008 verses in the ārya metre . A good deal of it is astronomy, but it also contains key chapters on mathematics, including algebra, geometry, trigonometry and algorithmics, which are believed to contain new insights due to Brahmagupta himself. Later, Brahmagupta moved to Ujjaini , Avanti ,
294-571: A later commentator, called him Bhillamalacharya , the teacher from Bhillamala. Bhillamala was the capital of the Gurjaradesa , the second-largest kingdom of Western India, comprising southern Rajasthan and northern Gujarat in modern-day India. It was also a centre of learning for mathematics and astronomy. He became an astronomer of the Brahmapaksha school, one of the four major schools of Indian astronomy during this period. He studied
336-536: A lineal descendant in Ujjain, who described Brahmagupta as the ganaka-chakra-chudamani (the gem of the circle of mathematicians). Prithudaka Svamin wrote commentaries on both of his works, rendering difficult verses into simpler language and adding illustrations. Lalla and Bhattotpala in the 8th and 9th centuries wrote commentaries on the Khanda-khadyaka . Further commentaries continued to be written into
378-420: A major centre for astronomy in central India. At the age of 67, he composed his next well-known work Khanda-khādyaka , a practical manual of Indian astronomy in the karana category meant to be used by students. Brahmagupta died in 668 CE, and he is presumed to have died in Ujjain. Brahmagupta composed the following treatises: Brahmagupta's mathematical advances were carried on further by Bhāskara II ,
420-420: A multiplier and increased or diminished by an arbitrary [number]. The product of the first [pair], multiplied by the multiplier, with the product of the last [pair], is the last computed. 18.65. The sum of the thunderbolt products is the first. The additive is equal to the product of the additives. The two square-roots, divided by the additive or the subtractive, are the additive rupas . The key to his solution
462-399: A negative is positive; a zero divided by zero is zero; a positive divided by a negative is negative; a negative divided by a positive is [also] negative. 18.35. A negative or a positive divided by zero has that [zero] as its divisor, or zero divided by a negative or a positive [has that negative or positive as its divisor]. The square of a negative or positive is positive; [the square] of zero
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#1732773125508504-553: A number in its own right, rather than as simply a placeholder digit in representing another number as was done by the Babylonians or as a symbol for lack of quantity as was done by Ptolemy and the Romans . In chapter eighteen of his Brāhmasphuṭasiddhānta , Brahmagupta describes operations on negative numbers. He first describes addition and subtraction, 18.30. [The sum] of two positives is positives, of two negatives negative; of
546-413: A positive and a negative [the sum] is their difference; if they are equal it is zero. The sum of a negative and zero is negative, [that] of a positive and zero positives, [and that] of two zeros zero. [...] 18.32. A negative minus zero is negative, a positive [minus zero] is positive; zero [minus zero] is zero. When a positive is to be subtracted from a negative or a negative from a positive, then it
588-464: A recurrence relation for generating solutions to certain instances of Diophantine equations of the second degree such as Nx + 1 = y (called Pell's equation ) by using the Euclidean algorithm . The Euclidean algorithm was known to him as the "pulverizer" since it breaks numbers down into ever smaller pieces. The nature of squares: 18.64. [Put down] twice the square-root of a given square by
630-520: A theoretical treatise, and the Khandakhadyaka ("edible bite", dated 665), a more practical text. In 628 CE, Brahmagupta first described gravity as an attractive force, and used the term "gurutvākarṣaṇam (गुरुत्वाकर्षणम्)" in Sanskrit to describe it. He is also credited with the first clear description of the quadratic formula (the solution of the quadratic equation) in his main work,
672-404: Is his formula for cyclic quadrilaterals . Given the lengths of the sides of any cyclic quadrilateral, Brahmagupta gave an approximate and an exact formula for the figure's area, 12.21. The approximate area is the product of the halves of the sums of the sides and opposite sides of a triangle and a quadrilateral. The accurate [area] is the square root from the product of the halves of the sums of
714-401: Is in four metrical lines called pāda s. Unlike the majority of meters employed in classical Sanskrit, the āryā meter is based on the number of mātrā s ( morae ) per pāda . A short syllable counts for one mātrā , and a long syllable (that is, one containing a long vowel, or a short vowel followed by two consonants) counts for two mātrā s. It is believed that āryā meter
756-529: Is not known and it is possible that both Greek and Indian syncopation may be derived from a common Babylonian source. The four fundamental operations (addition, subtraction, multiplication, and division) were known to many cultures before Brahmagupta. This current system is based on the Hindu–Arabic numeral system and first appeared in the Brāhmasphuṭasiddhānta . Brahmagupta describes multiplication in
798-408: Is to be added. He goes on to describe multiplication, 18.33. The product of a negative and a positive is negative, of two negatives positive, and of positives positive; the product of zero and a negative, of zero and a positive, or of two zeros is zero. But his description of division by zero differs from our modern understanding: 18.34. A positive divided by a positive or a negative divided by
840-486: Is zero. That of which [the square] is the square is [its] square root. Here Brahmagupta states that 0 / 0 = 0 and as for the question of a / 0 where a ≠ 0 he did not commit himself. His rules for arithmetic on negative numbers and zero are quite close to the modern understanding, except that in modern mathematics division by zero is left undefined . In chapter twelve of his Brāhmasphuṭasiddhānta , Brahmagupta provides
882-631: The Brāhma-sphuṭa-siddhānta . Brahmagupta, according to his own statement, was born in 598 CE. Born in Bhillamāla in Gurjaradesa (modern Bhinmal in Rajasthan , India) during the reign of the Chavda dynasty ruler Vyagrahamukha. He was the son of Jishnugupta and was a Hindu by religion, in particular, a Shaivite . He lived and worked there for a good part of his life. Prithudaka Svamin ,
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#1732773125508924-552: The mahāyuga in the traditional way, following the Brāhmapakṣa school of Brahmagupta . Brahmagupta This is an accepted version of this page Brahmagupta ( c. 598 – c. 668 CE ) was an Indian mathematician and astronomer . He is the author of two early works on mathematics and astronomy : the Brāhmasphuṭasiddhānta (BSS, "correctly established doctrine of Brahma ", dated 628),
966-799: The 12th century. A few decades after the death of Brahmagupta, Sindh came under the Arab Caliphate in 712 CE. Expeditions were sent into Gurjaradesa (" Al-Baylaman in Jurz ", as per Arab historians). The kingdom of Bhillamala seems to have been annihilated but Ujjain repulsed the attacks . The court of Caliph Al-Mansur (754–775) received an embassy from Sindh, including an astrologer called Kanaka, who brought (possibly memorised) astronomical texts, including those of Brahmagupta. Brahmagupta's texts were translated into Arabic by Muḥammad ibn Ibrāhīm al-Fazārī , an astronomer in Al-Mansur's court, under
1008-488: The Indians". In the Brāhmasphuṭasiddhānta , four methods for multiplication were described, including gomūtrikā , which is said to be close to the present day methods. In the beginning of chapter twelve of his Brāhmasphuṭasiddhānta , entitled "Calculation", he also details operations on fractions. The reader is expected to know the basic arithmetic operations as far as taking the square root, although he explains how to find
1050-416: The constants c and e . The solution given is equivalent to x = e − c / b − d . He further gave two equivalent solutions to the general quadratic equation 18.44. 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. [The result is]
1092-413: The cube and cube-root of an integer and later gives rules facilitating the computation of squares and square roots. He then gives rules for dealing with five types of combinations of fractions: a / c + b / c ; a / c × b / d ; a / 1 + b / d ;
1134-492: The desired variable must first be isolated, and then the equation must be divided by the desired variable's coefficient . In particular, he recommended using "the pulverizer" to solve equations with multiple unknowns. 18.51. Subtract the colors different from the first color. [The remainder] divided by the first [color's coefficient] is the measure of the first. [Terms] two by two [are] considered [when reduced to] similar divisors, [and so on] repeatedly. If there are many [colors],
1176-432: The first n natural numbers as n ( n + 1)(2 n + 1) / 6 and the sum of the cubes of the first n natural numbers as ( n ( n + 1) / 2 ) . Brahmagupta's Brahmasphuṭasiddhānta is the first book that provides rules for arithmetic manipulations that apply to zero and to negative numbers . The Brāhmasphuṭasiddhānta is the earliest known text to treat zero as
1218-466: The five traditional Siddhantas on Indian astronomy as well as the work of other astronomers including Aryabhata I , Latadeva, Pradyumna, Varahamihira , Simha, Srisena, Vijayanandin and Vishnuchandra. In the year 628, at the age of 30, he composed the Brāhmasphuṭasiddhānta ("improved treatise of Brahma") which is believed to be a revised version of the received Siddhanta of the Brahmapaksha school of astronomy. Scholars state that he incorporated
1260-476: The following way: The multiplicand is repeated like a string for cattle, as often as there are integrant portions in the multiplier and is repeatedly multiplied by them and the products are added together. It is multiplication. Or the multiplicand is repeated as many times as there are component parts in the multiplier. Indian arithmetic was known in Medieval Europe as modus Indorum meaning "method of
1302-401: The known manuscripts are incomplete. In his work, Lalla drew on his predecessors Brahmagupta , and Bhāskara I . In turn, he influenced later generations of astronomers, including Āryabhaṭa II , Śrīpati , Vaṭeśvara , and Bhāskara II (who later wrote a commentary on the Śiṣyadhīvṛddhidatantra ). He followed the Āryapakṣa or the school of Āryabhaṭa (continued by Bhāskara I), but divided
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1344-459: The length, c , of its hypotenuse is given by c = m (1 + x ) − d . And, indeed, elementary algebraic manipulation shows that a + b = c whenever d has the value stated. Also, if m and x are rational, so are d , a , b and c . A Pythagorean triple can therefore be obtained from a , b and c by multiplying each of them by the least common multiple of their denominators . Brahmagupta went on to give
1386-438: The middle [number]. 18.45. Whatever is the square-root of the rupas multiplied by the square [and] increased by the square of half the unknown, diminished that by half the unknown [and] divide [the remainder] by its square. [The result is] the unknown. which are, respectively, solutions for the equation ax + bx = c equivalent to, and He went on to solve systems of simultaneous indeterminate equations stating that
1428-543: The names Sindhind and Arakhand . An immediate outcome was the spread of the decimal number system used in the texts. The mathematician Al-Khwarizmi (800–850 CE) wrote a text called al-Jam wal-tafriq bi hisal-al-Hind (Addition and Subtraction in Indian Arithmetic), which was translated into Latin in the 13th century as Algorithmi de numero indorum . Through these texts, the decimal number system and Brahmagupta's algorithms for arithmetic have spread throughout
1470-489: The pulverizer [is to be used]. Like the algebra of Diophantus , the algebra of Brahmagupta was syncopated. Addition was indicated by placing the numbers side by side, subtraction by placing a dot over the subtrahend, and division by placing the divisor below the dividend, similar to our notation but without the bar. Multiplication, evolution, and unknown quantities were represented by abbreviations of appropriate terms. The extent of Greek influence on this syncopation , if any,
1512-515: The sides diminished by [each] side of the quadrilateral. So given the lengths p , q , r and s of a cyclic quadrilateral, the approximate area is p + r / 2 · q + s / 2 while, letting t = p + q + r + s / 2 , the exact area is Arya metre Āryā meter is a meter used in Sanskrit , Prakrit and Marathi verses. A verse in āryā metre
1554-446: The solution of the general linear equation in chapter eighteen of Brahmasphuṭasiddhānta , 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 the square and the unknown are to be subtracted. which is a solution for the equation bx + c = dx + e where rupas refers to
1596-401: The squares is that [sum] multiplied by twice the [number of] step[s] increased by one [and] divided by three. The sum of the cubes is the square of that [sum] Piles of these with identical balls [can also be computed]. Here Brahmagupta found the result in terms of the sum of the first n integers, rather than in terms of n as is the modern practice. He gives the sum of the squares of
1638-401: The world. Al-Khwarizmi also wrote his own version of Sindhind , drawing on Al-Fazari's version and incorporating Ptolemaic elements. Indian astronomic material circulated widely for centuries, even making its way into medieval Latin texts. The historian of science George Sarton called Brahmagupta "one of the greatest scientists of his race and the greatest of his time." Brahmagupta gave
1680-522: Was able to find integral solutions to Pell's equation through a series of equations of the form x − Ny = k i . Brahmagupta was not able to apply his solution uniformly for all possible values of N , rather he was only able to show that if x − Ny = k has an integer solution for k = ±1, ±2, or ±4, then x − Ny = 1 has a solution. The solution of the general Pell's equation would have to wait for Bhāskara II in c. 1150 CE . Brahmagupta's most famous result in geometry
1722-485: Was taken from the gāthā meter of Prakrit. Āryā metre is common in Jain Prakrit texts and hence considered as favourite metre of early authors of Jainism . The earlier form of the āryā metre is called old gīti , which occurs in a some very early Prakrit and Pāli texts. The basic āryā verse has 12, 18, 12 and 15 mātrā s in the first, second, third, and fourth pāda s respectively. An example
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1764-430: Was the identity, which is a generalisation of an identity that was discovered by Diophantus , Using his identity and the fact that if ( x 1 , y 1 ) and ( x 2 , y 2 ) are solutions to the equations x − Ny = k 1 and x − Ny = k 2 , respectively, then ( x 1 x 2 + Ny 1 y 2 , x 1 y 2 + x 2 y 1 ) is a solution to x − Ny = k 1 k 2 , he
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