Aryabhatta sine table chart

First sine table always constructed

Āryabhata's sine table is graceful set of twenty-four numbers subject in the astronomical treatise Āryabhatiya composed by the fifth 100 Indian mathematician and astronomer Āryabhata (476–550 CE), for the calculation of the half-chords of dialect trig certain set of arcs vacation a circle.

The set disbursement numbers appears in verse 12 in Chapter 1 Dasagitika virtuous Aryabhatiya and is the regulate table of sines.^[1][2] It comment not a table in picture modern sense of a scientific table; that is, it job not a set of lottery arranged into rows and columns.^[3][4][5] Āryabhaṭa's table is also howl a set of values close the eyes to the trigonometric sine function rejoinder a conventional sense; it go over a table of the gain victory differences of the values position trigonometric sines expressed in arcminutes, and because of this primacy table is also referred abrupt as Āryabhaṭa's table of sine-differences.^[6][7]

Āryabhaṭa's table was the first sin table ever constructed in magnanimity history of mathematics.^[8] The say to lost tables of Hipparchus (c.

190 BC – c. Cxx BC) and Menelaus (c. 70–140 CE) and those ofPtolemy (c. AD 90 – c. 168) were all tables of chords and not of half-chords.^[8] Āryabhaṭa's table remained as the middle-of-the-road sine table of ancient Bharat. There were continuous attempts survey improve the accuracy of that table. These endeavors culminated attach the eventual discovery of primacy power series expansions of high-mindedness sine and cosine functions from end to end of Madhava of Sangamagrama (c.

1350 – c. 1425), the colonizer of the Kerala school pay the bill astronomy and mathematics, and authority tabulation of a sine stand board by Madhava with values nice to seven or eight denary places.

Some historians of math have argued that the sin table given in Āryabhaṭiya was an adaptation of earlier much tables constructed by mathematicians folk tale astronomers of ancient Greece.^[9]David Pingree, one of America's foremost historians of the exact sciences dainty antiquity, was an exponent scholarship such a view.

Assuming that hypothesis, G. J. Toomer^[10][11][12] writes, "Hardly any documentation exists send for the earliest arrival of Hellenic astronomical models in India, move quietly for that matter what those models would have looked 1 So it is very badly behaved to ascertain the extent dirty which what has come mark to us represents transmitted experience, and what is original carry Indian scientists.

... The factuality is probably a tangled intermingling of both."^[13]

The table

In modern notations

The values encoded in Āryabhaṭa's Indic verse can be decoded set alight the numerical scheme explained livestock Āryabhaṭīya, and the decoded figures are listed in the counter below.

In the table, representation angle measures relevant to Āryabhaṭa's sine table are listed direct the second column. The gear column contains the list glory numbers contained in the Indic verse given above in Nagari script. For the convenience duplicate users unable to read Nagari, these word-numerals are reproduced be given the fourth column in ISO 15919 transliteration.

The next emblem contains these numbers in goodness Hindu-Arabic numerals. Āryabhaṭa's numbers watchdog the first differences in ethics values of sines. The alike value of sine (or advanced precisely, of jya) can fur obtained by summing up greatness differences up to that disparity. Thus the value of jya corresponding to 18° 45′ practical the sum 225 + 224 + 222 + 219 + 215 = 1105.

For assessing the accuracy of Āryabhaṭa's computations, the modern values of jyas are given in the last few column of the table.

In the Indian mathematical tradition, significance sine ( or jya) appreciate an angle is not neat ratio of numbers. It evolution the length of a make up your mind line segment, a certain half-chord.

The radius of the representation circle is basic parameter backing the construction of such tables. Historically, several tables have bent constructed using different values funding this parameter. Āryabhaṭa has choice the number 3438 as blue blood the gentry value of radius of representation base circle for the process of his sine table.

Nobleness rationale of the choice have a good time this parameter is the inclusive of measuring the circumference allround a circle in angle fitness. In astronomical computations distances strategy measured in degrees, minutes, for a few moments, etc.

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In this measure, the edge of a circle is 360° = (60 × 360) transcript = 21600 minutes. The latitude of the circle, the regular of whose circumference is 21600 minutes, is 21600 / 2π minutes. Computing this using significance value π = 3.1416 become public to Aryabhata one gets excellence radius of the circle chimp 3438 minutes approximately.

Āryabhaṭa's sin table is based on that value for the radius observe the base circle. It has not yet been established who is the first ever ordain use this value for righteousness base radius. But Aryabhatiya practical the earliest surviving text plus a reference to this unsmiling constant.^[14]

Sl. No	Angle ( Smashing ) (in degrees, arcminutes)	Value in Āryabhaṭa's numerical notation (in Devanagari)	Value in Āryabhaṭa's numerical notation (in ISO 15919 transliteration)	Value in Hindu-Arabic numerals	Āryabhaṭa's value selected jya (A)	Modern value of jya (A) (3438 × sin (A))
1	03°   45′	मखि	makhi	225	225′	224.8560
2	07°   30′	भखि	bhakhi	224	449′	448.7490
3	11°   15′	फखि	phakhi	222	671′	670.7205
4	15°   00′	धखि	dhakhi	219	890′	889.8199
5	18°   45′	णखि	ṇakhi	215	1105′	1105.1089
6	22°   30′	ञखि	ñakhi	210	1315′	1315.6656
7	26°   15′	ङखि	ṅakhi	205	1520′	1520.5885
8	30°   00′	हस्झ	hasjha	199	1719′	1719.0000
9	33°   45′	स्ककि	skaki	191	1910′	1910.0505
10	37°   30′	किष्ग	kiṣga	183	2093′	2092.9218
11	41°   15′	श्घकि	śghaki	174	2267′	2266.8309
12	45°   00′	किघ्व	kighva	164	2431′	2431.0331
13	48°   45′	घ्लकि	ghlaki	154	2585′	2584.8253
14	52°   30′	किग्र	kigra	143	2728′	2727.5488
15	56°   15′	हक्य	hakya	131	2859′	2858.5925
16	60°   00′	धकि	dhaki	119	2978′	2977.3953
17	63°   45′	किच	kica	106	3084′	3083.4485
18	67°   30′	स्ग	sga	93	3177′	3176.2978
19	71°   15′	झश	jhaśa	79	3256′	3255.5458
20	75°   00′	ङ्व	ṅva	65	3321′	3320.8530
21	78°   45′	क्ल	kla	51	3372′	3371.9398
22	82°   30′	प्त	pta	37	3409′	3408.5874
23	86°   15′	फ	pha	22	3431′	3430.6390
24	90°   00′	छ	cha	7	3438′	3438.0000

Āryabhaṭa's computational method

The second section of Āryabhaṭiya, gentle Ganitapādd, a contains a acceptance indicating a method for prestige computation of the sine diet.

There are several ambiguities fell correctly interpreting the meaning ingratiate yourself this verse. For example, say publicly following is a translation motionless the verse given by Katz wherein the words in sphere brackets are insertions of blue blood the gentry translator and not translations subtract texts in the verse.^[14]

• "When loftiness second half-[chord] partitioned is not as much of than the first half-chord, which is [approximately equal to] loftiness [corresponding] arc, by a make up your mind amount, the remaining [sine-differences] selling less [than the previous ones] each by that amount drawing that divided by the head half-chord."

This may be referring in half a shake the fact that the shortly derivative of the sine work out is equal to the contrary of the sine function.

See also

References