enIKSwiki - User contributions [en]

Timekeeping Instruments

2024-01-09T05:44:01Z

Ashish:

Time measurement is an important aspect of astronomical observations. In the 14th century, with the invention of the first mechanical clock, instruments such as sundials, and water clocks gradually faded away from the category of essential astronomical instruments. Unfortunately, we often forget that inventions are not one-day efforts but the gradual development of ideas leads to inventions. Numerous developments have been made in the past to find the accurate measure of time. Initially, people were dependent on nature, they observed periodicity in the movement of the Sun and the moon and took it as the primary basis to set their daily activities, Later on, People started to take an interest in numeration methods, and also observed such unusual activities in the nature, resulting them to take interest in astronomy and mathematics. Their initial efforts included finding the set the systemic units of measurement for the fundamental quantities, length, time, etc, Various units have been adopted, the smallest unit used is ''nimesha'' (time spent in the blinking of the eye)- found its mention in Rigveda, Satapatha Brahmana.

दश च वै सहस्राण्यष्टौ च शतानि संवत्सरस्य मुहूर्ताः। यावन्तो मुहूर्तास्तावन्ति पञ्चदशकृत्वः क्षिप्राणि । यावन्ति क्षिप्राणि तावन्ति पञ्चचदशकृत्व एतर्हीणि । यावन्त्येतर्हीणि तावन्ति पञ्चदशकृत्व इदानीनि। यावन्तीदानीनि तावन्तः पञ्चदशकृत्वः प्राणाः । यावन्तः प्राणास्तावन्तोऽक्तनाः । यावन्तोऽक्तनास्तावन्तो निमेषाः। यावन्तो निमेषास्तावन्तो लोमगर्ताः । यावन्तो लोमगर्तास्तावन्ति स्वेदायनानि। यावन्ति स्वेदायनानि तावन्त एते स्तोका वर्षन्ति ॥ ŚB (12.3.2.5).

Now the question arises as to how and why people got interested in measuring time. There could be many reasons, for example, to measure the exact time for conducting Vedic practices, and other, agricultural practices. The primary source of living was agriculture, and agricultural practices were rain-dependent, which in turn encouraged people to speculate the beginning of the rainy season. Moreover, Celestial navigation is an important part of observational astronomy, Marine navigation was equally important for trade, and many instruments have been developed to precisely measure the time for the same. The earliest known instruments include water clocks which were used for time measurement from the earliest times up to the nineteenth century. Gnomons, mentioned in the Shulbasutra, were initially used to measure the directions, Later on, it was included as a major component of sundials.

'''Water Clock'''

A water clock is an instrument that uses the flow of water through a small aperture as a measure of the interval of time. It can be of three types, a water clock of outflow type was mentioned in the Vedanga Jyotisha, However, the specification of the apparatus has not been defined, those texts in which outflow type water clock is mentioned, incorporate the units of measurement “nali, nalika, nadika, nadi” the instrument is called a Nadika Yantra. As the name suggests nali means reed, tube, of cylindrical shape. The outflow-type water clock consists of a vessel with a hole through which water flows at a constant rate. In addition to Vedāṅga-jyotiṣa, Kauṭilya’s Arthaśāstra, the Śārdūlakarṇāvadāna of the Divyāvadāna and Jyotiṣkaraṇḍaka also describes outflow type instrument, some of them includes their specifications as well.

Later at the beginning of Siddhanta astronomy, Aryabhata describes another type of water clock (sinking bowl), which is called Ghatika Yantra in which time is measured in ghatis, one ghati, one nadi, which is similar to modern 24 minutes, further smaller unit palas – 24 seconds. A vessel of hemispherical shape made of copper and a hole is so formed at the center that a bowl sinks into the larger vessel filled with water exactly 60 times a day, and once every 24 minutes.

वृत्तं ताम्रमयं पात्रं कारयेद् दशभिः पलैः |

षडङ्गुलं तदुत्सेधो विस्तारो द्वादशानने ||

तस्याधः कारयेच् छिद्रं पलेनाष्टाङ्गुलेन तु |

इत्य् एतद् घटिकासंज्ञं पलषष्ट्याम्बुपूरणात् ||

स्वेष्टं वान्यद् अहोरात्रे षष्ट्याम्भसि निमज्जति |

ताम्रपात्रम् अधश्छिद्रम् अम्बुयन्त्रं कपालकम् || आर्यभटीय सिद्धान्त||

‘''Get a round (i.e. hemispherical) bowl made with ten palas of copper. Let its height be six aṅgulas and the diameter at the mouth twelve aṅgulas. Get a hole bored at its bottom [so that a gold wire] one pala [in weight] and eight aṅgulas [in length can pass through it]. This is designated as Ghaṭikā -Yantra because it is filled with sixty palas of water”''. After Aryabhata, the sinking bowl-type water is described in several astronomical texts. Lallacharya (8th or 9th century) in ''Sishyadhivriddhitantra'' modified the size of the perforation as,

दशभिः शुल्बस्य पलैः पात्रं कलशार्धसन्निभं घटितम् |

हस्तार्धमुखव्यासं समघटवृत्तं दलोच्छ्रायम् ||

सत्र्यंशमाषकत्रयकृतनलया समवृत्तया हेम्नः |

चतुरङ्गुलया विद्धं मज्जति विमले जले नाड्या ||

''“The bowl, which resembles half a pot (i.e. hemispherical), which is made of ten palas of copper, which is half a cubit (i.e. twelve aṅgulas) in diameter at the mouth and half (i.e. six aṅgulas) as high, which is evenly circular, and which is bored by a uniformly circular needle, made of three and one-third māṣas of gold and of four aṅgulas in length, sinks into clear water in one ghaṭikā (nāḍī)”.''

This description was more accurate than the earlier texts but possibly some trial and error methods have always been involved in measuring the size of the perforation.

'''Gnomons and Sundials'''

Although the use of gnomon was prevalent since the period of Sulbasutra but to find direction. directions. Kautilya in his Arthashastra mentioned the measurement of time by means of the length of the shadow (Chaya- Pramana) as

“''He should divide the day into eight parts as also the night by means of nālikās, or by the measure of the shadow (of the gnomon). (A shadow measuring three pauruṣas, one pauruṣa, (and) four aṅgulas, and the midday when the shadow disappears are the four eighth parts of the day. By them are explained the later (four)”.''

Further, Siddhanta texts AryabhatiyaShiddhanta, BrahmaShutaSiddhanta, Siddhantatatvaviveka, Siddhanta Shekhara describes different varieties of sundials for measuring time using sunlight. Different varieties of sundials include the horizontal dial with a triangular gnomon (Palabhā-yantra), the column dial with a horizontal gnomon (Kaśā-yantra), and the graduated ring in which sunlight enters through an aperture (Cūḍā-yantra). Palabha denotes the length of the shadow thrown at noon by a gnomon of 12 units (aṅgulas) on the days of the equinox. In Jai Singh’s Observatories, two masonry instruments, viz. the Nadivalaya-yantra and the Samrat-yantra are constructed on the principle of Palabha Yantra.

'''Perpetual machines:''' A perpetual motion machine (Ajasra Yantra) is a device that is supposed to perform useful work without any external source of useful energy, along with various astronomical instruments Brahmagupta in his Brahmasphuta Siddhanta (628CE) describes perpetual motion wheel thus,

''“Make a wheel of light timber with uniformly hollow spokes at equal intervals. Fill each spoke up to half with mercury and seal its opening situated in the rim. Set up the wheel so that its axle rests horizontally on two upright supports. Then the mercury runs in some hollow spaces and downwards in some others, as a result of which the wheel rotates automatically forever''”.

In his opinion, if mercury overcomes the inertia, it can be used as the time measuring device ( Akhanda kala- immeasurable cyclic time). Further, in the twelfth century, Bhaskara added that the hollow spokes of the wheel mercury wheel should be ''kinchid vakrah'' like '''''Nandyavarta Flower.'''''

ताश्च '''नद्यावर्तवदेकत''' एव सर्वाः'''किञ्चिद्वक्रा''' योज्याः|.... अत्र युक्तिः|यन्त्रैकभागे रसो ह्यारामूलं प्रविशति| अन्यभागेत्वाराग्रं धावति| तेनाऽऽकृष्टं तत् स्वयं भ्रमतीति|| (सिद्धान्तशिरोमणि-५०)

“Spokes should be slightly curved, all towards the same direction like the petal of the '''''Nandyavarta'' (Tabernaemontana coronaria) flower'''”.

'''Yasti Yantra (The staff)''': It is basically a representation of the radius of the celestial sphere. To measure time, Brahmagupta describes the use of Yasti Yantra as,

यष्टिव्यासार्धे ऽग्रायष्ट्यग्रान्तरसमज्यया धनुषि ।

घटिका द्वितीयवृत्ते याताः प्रागपरतः शेषाः । । 21 । ।

"On the sphere whose Radius is equal to the length of the yaşți, the distance between the rising (or setting) point and the tip of the yaşți is represented by an arc (dhanus). On the second circle (i.e. the diurnal circle drawn inside the level circle), the ghatikās elapsed or remaining, in the forenoon or afternoon respectively, are measured."

'''Chakra Yantra:''' Chakra Yantra is an instrument resembling a circular hoop, Aryabhata, in his commentary on Surya Siddhanta describes chakra Instruments, similarly, Varahamihira, Brahmagupta, Lalla, Sripati, and Bhaskara also give some information on chakra Yantra.

Lalla explanation of chakra Yantra can be represented as,

वृत्तं कृत्वा फलकं षड्वर्गाकं तथा च षष्ट्यङ्कम् ।

मध्यस्थितावलम्बं मध्यस्थित्या प्रविष्टोष्णम् ।|20||

तदधोलम्बविमुक्तं गृहादि यत्तदुदितं दिनकरांशात् ।

नाड्यः पूर्वकपाले द्युगतास्ताः पश्चिमे द्युदलात् ।। 21 ।।

"Construct a circular board, graduate it with the 360 divisions (of degrees) as well as the 60 divisions (of ghaṭīs), fix a vertical needle at its center, and set it up in such a way that the Sun may lie centrally in its plane. Moreover, according to K. S. Shukla, Lalla must have used the following formula in order to deduce the time elapsed since sunrise using the solar altitude.

Time elapsed = (semi-duration of day x Sun's altitude)/ (Sun's meridian altitude)

The other instruments are generally variations of '''Chakra Yantra, Kapala, and Kartari Yantra,''' Kapala is the horizontally placed semicircular plate, axis pointing upwards, diameter in the north-south directions, and arc to the east or to the west where the shadow happens to fall. These instruments are used to measure time from sunrise to sunset.

'''Phalaka Yantra (type of Quadrant):'''Phalaka Yantra was invented by Bhaskaracharya, a simple instrument used to measure time after sunrise. It consists of a vertical gnomon and a rectangular board. The shadow of the gnomon is used to determine the time, and markings on the board help interpret the position of the sun or other celestial bodies. Yantraprakasha (1428CE), a text by Ramachandra Vajpeyin mentions the '''Kachayantra''' “glass instrument” to measure time, It description is neither based on European source, nor on an Islamic prototype, (Sarma 2014)

काचस्य पात्र्यौ वदने युते सृजेत्|

ऊर्ध्वं भृतां वालुकयावधारयेत्||

घट्यां तुतत्पातत एव वालुका-

धरोर्ध्वगा स्वेततरा पुणस् तथा||

“Make two glass ampoules joined mouth to mouth. The upper one should be filled with sand. Owing to its trickling down, after one ghati all the sand will be below, and the upper [ ampoule becomes] white Do the same again.”

'''Astrolabes'''

Another important instrument of the medieval period is the '''Astrolabe''', it is a multifunctional instrument, that contains two- a dimensional projection of the heavens. The rete is the perforated open plate containing the ecliptic and a star map with pointers indicating the positions of some bright stars. Beneath the rete are to be found a series of plates or discs which are specific to a particular terrestrial latitude and display stereographic projections of local horizon, equal altitude circles. Mahendra Suri called it ‘Yantraraja’ the king of astronomical instruments. It is not known precisely when the astrolabe was invented or by whom, but astrolabes were introduced in India in the eleventh century most probably by al-Biruni. Mahendra suri, a Jaina monk wrote a Sanskrit manual on astrolabe entitled Yantra Raja under the reign of Sultan Firuz Shah in 1370. Along with these instruments, various instruments have been constructed even after the invention of the first mechanical clock, In some places traditional methods are still in use, particularly for ritual purposes.

'''References'''

Ohashi, Y., 1994. Astronomical instruments in classical Siddhantas. ''Indian journal of history of science'', ''29''(2), pp.155-313.

Plofker, K., 2011. The mathematics of measuring time: astronomical timekeeping and the sinking-bowl water-clock in India. ''Mathematical Time Capsules: Historical Modules for the Mathematics Classroom'', ''77'', pp.55-62.

Raju, C.K., 2007. ''Cultural foundations of mathematics: The nature of mathematical proof and the transmission of the calculus from India to Europe in the 16th c. CE''. Pearson Education India

Sarma, S.R. and Knudsen, T.L., 2011. The Archaic and the Exotic: Studies in the History of Indian Astronomical Instruments. ''Aestimatio: Sources and Studies in the History of Science'', ''8'', pp.121-127.

Authored By: Mahima Pandey, Ashish Karn

Meru Prastara

2024-01-09T05:43:10Z

Ashish:

Meru Prastara is basically a triangular array of numbers (arithmetic triangle), consisting of binomial coefficients often known as Pascal's Triangle. The term Meru in ‘Meru Prastara’ stands for Mount Meru in Sanskrit literature. Thus, Meru Prastara is “steps of Mount Meru”. The mention of Meru Prastara was initially found in the ancient text on Sanskrit prosody called Chandaḥśhastra, which was written by Pingalacharya around 200- 300 BCE. The last chapter of the book contains a mathematical treatment of Sanskrit metres. which offers the notion of triangular tabulation of numbers.

This triangle tabulation of numbers has travelled a long way before came into light as Pascal’s triangle. Initially, around 200 BCE Indian text, the method of construction of a triangle was mentioned. Later, in China (Yang Hui's triangle (c. 1238-1298), in Italy "Tartaglia's Triangle," named after Italian algebraist Tartaglia, born Niccolò Fontana (1499-1557), and Pascal's Triangle, which came into existence in Europe after the publication of Pascal's book Traité du Triangle arithmétique in 1665.

Construction of Meru Prastara

Pingala, in Chandahshastra discussed some algorithms for arranging Sanskrit syllables Laghu(L)- short syllables (अ),i(इ),u(उ),ṛ(ऋ),ḷ(लृ) and Guru(G)- long syllables ā(आ),ī(ई),ū(ऊ),ṝ(ऋृ),e(ए), ai(ऐ),o(ओ),au(औ)) which forms a basis for writing Vedic metre. Sanskrit Metres may be broadly divided into three categories. First, according to the number of syllables in the metre (varnavrttas) and other, number of syllabic instants or matras (morae) - time measure (matravṛttas) and third ganas - groups of syllables (Singh, 2001). In the concluding verse of Chandaḥśāstra, a cryptic statement is given, which provides the notion for the construction of Varna Meru(Meru Prastara), Pingala lists six Pratyayas (problems and answers) for Varnavrttas in the Chandahshastra. One of the problems is Prastara Pratyaya (algorithm), which offered an account for the binary representation of syllabic metres. La-Ga- kriya Pratyaya elaborates on the number of combinations generated by taking "n" syllables at a time. The letters "L" and "G" stand for the laghus and gurus in a specific metre. Pingala's aphorism “परेपूर्णम्” (6.34), परेपूर्णमिति (6.35), this is too compressed and cryptic can be understood only with the help of commentary, In the tenth century, Halayudha, an Indian mathematician elaborated that statement in his text Mṛtasañjīvani, a commentary on Chandaḥśāstra and derived a triangular table called Meru Prastara, The explanation as follows,

उपरिष्टादेकं चतुरस्त्रकोष्ठं लिखित्वा तस्याधस्ता दुभयतोऽर्धनिष्क्रान्तं कोष्ठद्वयं लिखेत्| तस्याप्यधस्तात्त्रयं तस्याप्यधस्तात्त्रयं यावद्भिमतं स्थानमिति मेरुप्रस्तारः| तस्यप्रथमे कोष्ठे एकसंख्यां व्यवस्थाप्य लक्षणमिदंप्रवर्तयेत| तत्र परे कोष्ठे यद्वृत्तसंख्याजातं तत् पूर्वकोष्ठयोःपूर्णंनिवेशयेत्| तत्रोभयोः कोष्ठकयोरेकैकमङ्कं दद्यात् मध्ये कोष्ठे तु परकोष्ठद्वयाङ्कमेकी- कृत्य पूर्णं निवेशयेदिति पूर्णशब्दार्थः | चतुर्थ्यां पङ्क्तावपि पर्यन्तकोष्ठयो- रेकैकमेवस्थापयेत् |मध्यमकोष्ठ योस्तु परकोष्ठद्वयाअङ्कमेकी कृत्य पूर्णं त्रिसन्ख्यारूपं स्थापयेत्|…….

“First write one square cell at the top. Below it, write two cells, extending halfway on both sides. Below that three, below that again, four, until the desired number of places (are obtained.) Begin by writing 1 in the first cell. In the other cells, put down the sum of the numbers of the two cells above it. ⋅⋅⋅”

According to Halayudha, In the triangular array of numbers, the second row represents the expansion of a meter with one syllable, and the third row, the expansion of the metre with two syllables, and so on. It expresses the basic rule for the construction of Meru Prastara, the number of metrical forms of r Gurus (or Laghus) in the Prastara of metre of n syllables is the binomial coefficient nCr. In terms of the binomial coefficients, the recursive relation is as follows: nCr = n-1Cr-1 + n-1Cr.

The triangular array of numbers can be represented in the form of binomial sequence (L+G)n. The first row of the triangular array represents (L+G)0, similarly, the second row represents (L+G)1, and so on.

(L+G)0 = 1

(L+G)1 = 1G +1L

(L+G)2 = 1G2 +2GL +1 L3

(L+G)3 = 1 G3 + 3G2L + 3 GL2 +1 L3

(G+L) n = 1 Gn + n Gn-1L .…... + n.GLn-1 +1.Ln. This technique of expanding expression that has been raised to some finite powers, of summation of laghus and gurus, also leads to the developments of the binomial theorem.

The Agni Purana also mentions Meru Prastara and states Pingala’s rule for its formation. The rule is also given by Virahanka in Vrittajatisamucchaya, Trivikarma and Bhatta Narayana ( the commentators of Vrttaratnakara of Kedara) also in the work of Prakrta Pingala, Moreover, Gopala (c.1135)in his commentary on Vrittajatisamucchaya have illustrated the rule with figures of Meru Prastara. La-Ga-Kriya has also been described by Acharya Bharata in his Natyashastra. Another form of figures of numbers which also follows the same recursive formula, called Khand Meru (in Lilavati )and Suci Prastara has been given by Varahamihira, Janasrya, Virahanka, Jayadeva, Jaykirti, Kedara, Trivikarma, Hemachanda.

'''Mathematics of Meru Prastara'''

Meru Prastara illustrates many concepts of elementary mathematics as well as higher-level mathematics, the number which forms a triangle are all natural numbers, The first row of Meru Prastara represents 1, and the next row represents the sum of two values from the previous row, and this forms Triangular numbers, numbers associated with a triangular array of dots. The nth number of the sequence can be represented by T(n) = n(n+1)/2 (Subramanium, K.B.,1994). The resulting triangular sequence is 1,3,6,10,15,21,28,36…so forth. The sum of the first n triangular number is (n (n+1)( n+2))/6 which represents the nth Tetrahedral number (Asiru, M.A.,2008). The sum of two values from the previous row in the Meru Prastara gives Tetrahedral numbers through the figurate numbers (Willson, R., 2015). The resulting tetrahedral sequence of numbers is 1,4,10,20,35,56…....so forth. Moreover, this triangular array of numbers also depicts powers of 2 and powers of 11.

The diagonal number of Meru Prastara denotes Fibonacci numbers. This popular sequence comprised of 0, 1, 1, 2, 3, 5, 8,13….so on, the recursive relation Un =Un-1+ Un-2 yields the nth term in the series. Although the name L.Fibonacci(1140) is attached to the series of numbers, the sequence was well known in India prior to him, in the process of expansion of matravrttas. In the expansion of matra vṛttas, the number of variations of one mora is one, the number of variations of two morae is 2, 3 morae are 3, 4 morae are 5, and so on, which ultimately gives such a pattern of numbers. According to Pingala’s rule, miśrau ca (“and the two mixed”) is also meant for the expansion of matravrttas, The rule for the formation of numbers of variation of matravrttas was continued to be given in the later works on metrics, and various revised series have been mentioned in the works of mathematicians, including Abhinavgupta, Achrya Bharata, Acharya Virahanka, Acharya Gopala, Acharya Hemachandra, Bhatt Narayana. Acharya Hemachandra Suri(1150CE) gives the rule of formation of matra vrittas which is अङ्कान्त्योपान्त्य योगः परे परेमात्रा मात्राणाम्| stating that “the sum of the last and the last but one number [of variations] is [that] of the matra-vrtta coming next”. In response to his own rule of variation of matra vṛttas, Hemachandra proposed the "statement 1,2,3,5,8,13,21,34 and in this way forward". Thus, It can be seen that the so-called Fibonacci numbers resulted from the variations of matravṛttas in the Indian subcontinent. While in Europe, Fibonacci’s Liber Abaci in 1202 describes the sequence in his book to explain the Indian number system and its mathematics which he learned from Arabian teachers.

Moreover, the elements and patterns of Meru Prastara provide some ideas of Permutations and combinations, which are prerequisites for modern combinatorics and computation, Indian prosody texts not only describe Meru Prastara, It also explore numerous recursive algorithms, computation of Binomial coefficients, the application of repeated partial sums of sequences and the formula for summing a geometric series and many more.
[[File:Meru Prastara and Binomial coefficients .png|thumb|Meru Prastara and Binomial coeffiecients ]]

'''References'''

Bag, A.K.(1966). Binomial theorem in ancient India. Indian Journal of History of Science, 1(1), pp.68-74.

Brown, C.P., 1869. Sanskrit prosody and numerical symbols explained. Trübner & Company.

Dvivedi, K. D., Singh, S., (2019). किंगल िृ त छन्त्ि:सत्रू म: The Prosody of Pingala - A Treatise of Vedic and Sanskrit Metrics with Applications of Vedic Mathematics.Vishvavidyalaya Prakashan, Varanasi.

Dvivedi, P., 1936. and 1942, Ganita Kaumudi of Nârãyana, pts I & II.

Dvivedi, P., 1942. The Ganita-kaumudi. ''Part II''.

Dvivedi, S. (1897) Brihatsamhita– of Varahamihira (c. 6th century): Ed, with the Vivrti of Bhattotpala (c.967), by, 2 vols., Varanasi. Ed, with the Vivrti of Bhat.

Sharma, A. (2007). Chandaḥśāstra of Piṅgala with Sanskrit commentary Mṛtasañjīvanī of Halāyudha, Ananta Sharma, Parimal Publications, Delhi

Shah, J., 2013. A history of Pingala’s combinatorics. Ganita Bhāratī, pp.35-1.

Singh, A.N., 1936. On the use of series in Hindu mathematics. ''Osiris'', ''1'', pp.606-628.Singh, P., 1985. The so-called Fibonacci numbers in ancient and medieval India. ''Historia Mathematica'', ''12''(3), pp.229-244.

Subramaniam K.B. (1994) A generalization of Euler’s reproductive properties of triangular numbers, Int. J. Math. Educ. Sci. Technol. 25, pp. 767–769

Upādhyāya, B. ed., 1966. Agnipurana of Naharsi Vedavyasa. Chowkhamba Sanskrit Series Office pp 487.

Velankar, H.D., (1961). Chando’nuśāsana by Hemacandrasūri. A comprehensive treatise of Sanskrit, Prakrit and Apabhraṁśa Prosody.

Velankar, H.D., (1962). Vrttajatisamuccaya or Kaisitthachanda (in Prakrta) of Virahanka, with commentary of Gopala. Rajasthan Prachya Vidya Pratisthana, Jodhpur.

Velankar, H.D., (1963). Chando'nuśāsana of Hemacandrasūri Chando'nusasana of Hemacandrasuri. Journal of the American Oriental Society, 83(2).

Virahāṅka, Gopāla and Velankar, H.D.,(1962). Vṛttajātisamuccaya: saṭīka. Rājasthāna Prācyavidyā Pratiṣṭhāna.

Authored By: Mahima Pandey, Ashish Karn

Meru Prastara

2023-11-07T11:06:06Z

Ashish:

Timekeeping Instruments

2023-11-07T08:33:31Z

Ashish:

Timekeeping Instruments

2023-11-07T07:23:48Z

Ashish:

Timekeeping Instruments

2023-11-06T08:57:47Z

Ashish: Created page with " Authored by . Published by,"

Authored by .
Published by,

Meru Prastara

2023-11-06T08:11:57Z

Ashish:

File:Binomial coefficients.png

2023-11-06T08:09:13Z

Ashish:

Meru Prastara and Binomial coefficients

Meru Prastara

2023-10-19T06:32:14Z

Ashish:

Meru Prastara is basically a triangular array of numbers (arithmetic triangle), consisting of binomial coefficients often known as Pascal's Triangle. The term Meru in ‘Meru Prastara’ stands for Mount Meru in Sanskrit literature. Thus, Meru Prastara is “steps of Mount Meru”. The mention of Meru Prastara was initially found in the ancient text on Sanskrit prosody called Chandaḥśstra, which was written by Pingalacharya around 200- 300 BCE. The last chapter of the book contains a mathematical treatment of Sanskrit metres. which offers the notion of triangular tabulation of numbers.

This triangle tabulation of numbers has travelled a long way before came into light as Pascal’s triangle. Initially, around 200 BCE Indian text, the method of construction of a triangle was mentioned. Later, in China (Yang Hui's triangle (c. 1238-1298), in Italy "Tartaglia's Triangle," named after Italian algebraist Tartaglia, born Niccolò Fontana (1499-1557), and Pascal's Triangle, which came into existence in Europe after the publication of Pascal's book Traité du Triangle arithmétique in 1665.

Construction of Meru Prastara

Pingala, in Chandahshastra discussed some algorithms for arranging Sanskrit syllables Laghu(L)- short syllables (अ),i(इ),u(उ),ṛ(ऋ),ḷ(लृ) and Guru(G)- long syllables ā(आ),ī(ई),ū(ऊ),ṝ(ऋृ),e(ए), ai(ऐ),o(ओ),au(औ)) which forms a basis for writing Vedic metre. Sanskrit Metres may be broadly divided into three categories. First, according to the number of syllables in the metre (varnavrttas) and other, number of syllabic instants or matras (morae) - time measure (matravṛttas) and third ganas - groups of syllables (Singh, 2001). In the concluding verse of Chandaḥśāstra, a cryptic statement is given, which provides the notion for the construction of Varna Meru(Meru Prastara), Pingala lists six Pratyayas (problems and answers) for Varnavrttas in the Chandahshastra. One of the problems is Prastara Pratyaya (algorithm), which offered an account for the binary representation of syllabic metres. La-Ga- kriya Pratyaya elaborates on the number of combinations generated by taking "n" syllables at a time. The letters "L" and "G" stand for the laghus and gurus in a specific metre. Pingala's aphorism “परेपूर्णम्” (6.34), परेपूर्णमिति (6.35), this is too compressed and cryptic can be understood only with the help of commentary, In the tenth century, Halayudha, an Indian mathematician elaborated that statement in his text Mṛtasañjīvani, a commentary on Chandaḥśāstra and derived a triangular table called Meru Prastara, The explanation as follows,

उपरिष्टादेकं चतुरस्त्रकोष्ठं लिखित्वा तस्याधस्ता दुभयतोऽर्धनिष्क्रान्तं कोष्ठद्वयं लिखेत्| तस्याप्यधस्तात्त्रयं तस्याप्यधस्तात्त्रयं यावद्भिमतं स्थानमिति मेरुप्रस्तारः| तस्यप्रथमे कोष्ठे एकसंख्यां व्यवस्थाप्य लक्षणमिदंप्रवर्तयेत| तत्र परे कोष्ठे यद्वृत्तसंख्याजातं तत् पूर्वकोष्ठयोःपूर्णंनिवेशयेत्| तत्रोभयोः कोष्ठकयोरेकैकमङ्कं दद्यात् मध्ये कोष्ठे तु परकोष्ठद्वयाङ्कमेकी- कृत्य पूर्णं निवेशयेदिति पूर्णशब्दार्थः | चतुर्थ्यां पङ्क्तावपि पर्यन्तकोष्ठयो- रेकैकमेवस्थापयेत् |मध्यमकोष्ठ योस्तु परकोष्ठद्वयाअङ्कमेकी कृत्य पूर्णं त्रिसन्ख्यारूपं स्थापयेत्|…….

“First write one square cell at the top. Below it, write two cells, extending halfway on both sides. Below that three, below that again, four, until the desired number of places (are obtained.) Begin by writing 1 in the first cell. In the other cells, put down the sum of the numbers of the two cells above it. ⋅⋅⋅”

According to Halayudha, In the triangular array of numbers, the second row represents the expansion of a meter with one syllable, and the third row, the expansion of the metre with two syllables, and so on. It expresses the basic rule for the construction of Meru Prastara, the number of metrical forms of r Gurus (or Laghus) in the Prastara of metre of n syllables is the binomial coefficient nCr. In terms of the binomial coefficients, the recursive relation is as follows: nCr = n-1Cr-1 + n-1Cr.

The triangular array of numbers can be represented in the form of binomial sequence (L+G)n. The first row of the triangular array represents (L+G)0, similarly, the second row represents (L+G)1, and so on.

(L+G)0 = 1

(L+G)1 = 1G +1L

(L+G)2 = 1G2 +2GL +1 L3

(L+G)3 = 1 G3 + 3G2L + 3 GL2 +1 L3

(G+L) n = 1 Gn + n Gn-1L .…... + n.GLn-1 +1.Ln. This technique of expanding expression that has been raised to some finite powers, of summation of laghus and gurus, also leads to the developments of the binomial theorem.

The Agni Purana also mentions Meru Prastara and states Pingala’s rule for its formation. The rule is also given by Virahanka in Vrittajatisamucchaya, Trivikarma and Bhatta Narayana ( the commentators of Vrttaratnakara of Kedara) also in the work of Prakrta Pingala, Moreover, Gopala (c.1135)in his commentary on Vrittajatisamucchaya have illustrated the rule with figures of Meru Prastara. La-Ga-Kriya has also been described by Acharya Bharata in his Natyashastra. Another form of figures of numbers which also follows the same recursive formula, called Khand Meru (in Lilavati )and Suci Prastara has been given by Varahamihira, Janasrya, Virahanka, Jayadeva, Jaykirti, Kedara, Trivikarma, Hemachanda.

'''Mathematics of Meru Prastara'''

Meru Prastara illustrates many concepts of elementary mathematics as well as higher-level mathematics, the number which forms a triangle are all natural numbers, The first row of Meru Prastara represents 1, and the next row represents the sum of two values from the previous row, and this forms Triangular numbers, numbers associated with a triangular array of dots. The nth number of the sequence can be represented by T(n) = n(n+1)/2 (Subramanium, K.B.,1994). The resulting triangular sequence is 1,3,6,10,15,21,28,36…so forth. The sum of the first n triangular number is (n (n+1)( n+2))/6 which represents the nth Tetrahedral number (Asiru, M.A.,2008). The sum of two values from the previous row in the Meru Prastara gives Tetrahedral numbers through the figurate numbers (Willson, R., 2015). The resulting tetrahedral sequence of numbers is 1,4,10,20,35,56…....so forth. Moreover, this triangular array of numbers also depicts powers of 2 and powers of 11.

The diagonal number of Meru Prastara denotes Fibonacci numbers. This popular sequence comprised of 0, 1, 1, 2, 3, 5, 8,13….so on, the recursive relation Un =Un-1+ Un-2 yields the nth term in the series. Although the name L.Fibonacci(1140) is attached to the series of numbers, the sequence was well known in India prior to him, in the process of expansion of matravrttas. In the expansion of matra vṛttas, the number of variations of one mora is one, the number of variations of two morae is 2, 3 morae are 3, 4 morae are 5, and so on, which ultimately gives such a pattern of numbers. According to Pingala’s rule, miśrau ca (“and the two mixed”) is also meant for the expansion of matravrttas, The rule for the formation of numbers of variation of matravrttas was continued to be given in the later works on metrics, and various revised series have been mentioned in the works of mathematicians, including Abhinavgupta, Achrya Bharata, Acharya Virahanka, Acharya Gopala, Acharya Hemachandra, Bhatt Narayana. Acharya Hemachandra Suri(1150CE) gives the rule of formation of matra vrittas which is अङ्कान्त्योपान्त्य योगः परे परेमात्रा मात्राणाम्| stating that “the sum of the last and the last but one number [of variations] is [that] of the matra-vrtta coming next”. In response to his own rule of variation of matra vṛttas, Hemachandra proposed the "statement 1,2,3,5,8,13,21,34 and in this way forward". Thus, It can be seen that the so-called Fibonacci numbers resulted from the variations of matravṛttas in the Indian subcontinent. While in Europe, Fibonacci’s Liber Abaci in 1202 describes the sequence in his book to explain the Indian number system and its mathematics which he learned from Arabian teachers.

Moreover, the elements and patterns of Meru Prastara provide some ideas of Permutations and combinations, which are prerequisites for modern combinatorics and computation, Indian prosody texts not only describe Meru Prastara, It also explore numerous recursive algorithms, computation of Binomial coefficients, the application of repeated partial sums of sequences and the formula for summing a geometric series and many more.
[[File:Meru Prastara and Binomial coefficients .png|thumb|Meru Prastara and Binomial coeffiecients ]]

Authored by .
Published by,

File:Meru Prastara and Binomial coefficients .png

2023-10-19T06:29:50Z

Ashish:

Meru Prastara displays that every row represents the binomial expansion.

Meru Prastara

2023-10-17T06:25:20Z

Ashish: Created page with " Authored by . Published by,"

Authored by .
Published by,