## Āryabhaṭa and his sine table

Everyone and his son have written about Āryabhaṭa and his sine table. Yet we too do this because sometimes the situation arises where you have to explain things clearly to a layman who might have some education but is unfamiliar with the intricacies of, or in some cases lacks the correct perspective on, Hindu tradition.

siddhānta-pañcaka-vidhāv api dṛg viruddham
auḍhyoparāga-mukha-khecara-cāra-kḷptau |
sūryaḥ svayaṃ kusumapury abhavat kalau tu
bhūgolavit kulapa āryabhaṭābhidhānaḥ ||
When the predictions of the five siddhānta-s and observations of conjunctions, occultations and setting times of planets began to conflict the solar deity himself incarnated in Kusumapuri in the Kali age in the form of the geographer and head professor Āryabhaṭa.

The great astronomer and mathematician Āryabhaṭa-I’s, who was seen by some Hindus as the incarnation of the solar deity, was born in 476 CE in the Aśmaka country (close to modern Maharashtra-Telangana border). He was active at Pāṭaliputra in the golden age of Hindu power during the reign of emperor Budhagupta. He is known to have composed at least two works the Āryabhaṭīya, which was an update of the old Svāyambhuva-siddhānta tradition, and the Āryabhaṭa-siddhānta which was modeled after the old Sūrya siddhānta. We are informed by Bhāskara-I that in his Āryabhaṭīya Āryabhaṭa was following in the footsteps of the great astronomer of antiquity Pārāśarya, who was likely one of the early promulgators of the Svāyambhuva tradition. Āryabhaṭa as head professor is said to have had the following notable students: Lāṭadeva, Niśaṅku, Pāṇḍuraṅgasvamin and Prabhākara. Sadly their works have been lost. Of them Lāṭadeva is recorded in old Hindu scientific tradition as having written several works including on Hellenistic astronomy and likely succeeded Āryabhaṭa as the ācārya of his school.

Thus, Āryabhaṭa’s work is the earliest surviving record of one of the most important Hindu scientific traditions, namely that of Svāyambhuva-s. In the manner of the scientists of old Hindu naturalistic tradition Āryabhaṭa presents the acquisition of scientific knowledge as the attainment of brahmavidyā:
daśa-gītika-sūtram idam bhū-graha-caritam bha-pañjare jñātvā |
graha-bha-gaṇa-paribhramaṇaṃ sa yāti bhittvā param brahma ||
Having known these sūtra-s in the ten verses composed in the gītika meter providing the motions of the Earth and the planets within the celestial sphere [pañjara: marked by the coordinate grid], and having penetrated the orbits of the planets and the stars he attains the supreme brahman.

His clear mention of the movement of the Earth in the celestial sphere along with the other planets in the celestial sphere has been taken as Āryabhaṭa’s discovery of heliocentricity. However, here we are not going into this issue and the real nature of his unique planetary model which sets the old Hindu Svāyambhuva planetary model apart from those of the Greeks. Nevertheless, as a testimony of his astronomical achievements we will merely state the period of sidereal day of the Earth as determined by Āryabhaṭa in modern units: $23^h 56^m 4^{s.}1$, which is practically the modern value.

One of the important features of Āryabhaṭa’s work was his presentation of the old Hindu sine difference table. Āryabhaṭa gives the table using his syllable-numeral equivalence which goes as:
makhi bhaki phakhi dhakhi ṇakhi ñakhi ṅakhi hasjha skaki kiṣga śghaki kighva |
ghlaki kigra hakya dhaki kica sga jhaśa ṅva kla pta pha cha kalā ardha-jyāH ||
225, 224, 222, 219, 215, 210, 205, 199, 191, 183, 174, 164, 154, 143, 131, 119, 106, 93, 79, 65, 51, 37, 22, 7 are the $R\sin(\theta)$ [differences].

So how do we understand this? As per Hindu terminology if the arc is equivalent to the bow then the chord is equivalent to the bowstring (jyā). Hence, the sine can be seen as half a jyā (ardhajyā) as Āryabhaṭa terms it in the above sūtra. For simplicity call Āryabhaṭa’s ardhajyā a function named jyā and represent it thus in modern notation:
Let the function jyā $(\theta)$ be defined as,
jyā $(\theta)=R\sin(\theta)$, where $R=3438$.
Why the number 3438? Āryabhaṭa conceives a circle whose circumference is divided into $360 \times 60=21600$ parts. Now the radius of this circle given Āryabhaṭa’s $\pi \approx 3.1416$ will be,
$\dfrac{21600}{3.1416}=3437.73 \approx 3438 \Rightarrow 1\;radian$
Thus, Āryabhaṭa’s R value is for the first time a radian-like concept was used in trigonometry. Now, Āryabhaṭa divides his quadrant into 24 parts; thus, his minimal angle is $\theta_1=\frac{\pi}{48}$. To see why he chose this value of $\theta_1$ note the following:

$\dfrac{\pi}{48} \times 3438 \approx 225$

jyā $(\dfrac{\pi}{48})=3438\sin(\dfrac{\pi}{48})=3438\times0.06540 \approx 225$

$\therefore$ jyā $(\dfrac{\pi}{48})\Big/R\cdot\dfrac{\pi}{48} \approx 1$; actual value 0.99982

Thus, Āryabhaṭa’s value of $\theta_1$ is chosen such that the angle and its jyā are practically the same (Figure 1). This value as a proxy for $\displaystyle \lim_{\theta \to 0}\frac{\sin(\theta)}{\theta}=1$ continued to be used in the subsequent development of Hindu calculus. The $\theta$ and jyā $(\theta)$ being nearly the same at this value allows linear interpolation for intermediate values.

Figure 1

Now the rest of his table is in the form of differences. So to get a jyā $(\theta_n)$ we have to do the following:
jyā $(\theta_2)=225+224=449$; jyā $(\theta_3)=225+224+222=671$
The table below compares Āryabhaṭa’s jyā $(\theta)$ values to the exact modern values.