Stringing the bow – a backgrounder on Indian trigonometry and the concept of jyā

Having calculated the ratio of the circumference and diameter of a circle to any accuracy we want with the Madhava series, we move onto the question of calculating ज्या/jyā or the half-chord of any arc. ज्या is the Indian counterpart of the modern sine function, and a backgrounder on how Indians thought of it, and the entire subject of what we call “trigonometry” would be interesting. We have seen a brief summary of this in an earlier post, but this would be a good time for a more detailed picture.

Bows, strings and arrows

Think of an arc of a circle as a bow – depicted in red in the figure. The chord of the arc represents the bowstring, and is therefore called समस्तज्या – full bowstring – and the half-chord (green in the figure) correspondingly becomes अर्धज्या, or just ज्या. Āryabhaṭa is often fastidious about calling the half-chord अर्धज्या, but later mathematicians merely say ज्या. The radius of the circle (purple) represents half of the bowstring in stretched position. The कोटीज्या or कोटिः (in yellow) is the horizontal difference in positions between the normal and stretched bowstrings. The difference of radius and कोटीज्या is the arrow or शरः (in blue) placed on the bow. The शरः is not commonly seen in modern trigonometry, but turns out to be quite useful in Indian Trigonometry.

A more formal picture

Consider the circle centered on O, oriented with East upwards, as is the convention. The North South diameter is NOS, and the East West diameter EOW. Consider Arc EA of the circle. The ज्या of this arc is segment AJ, and the corresponding कोटीज्या (or कोटिः) is AK. EJ is the शरः.

OE/OA/ON/OS are all equal to the radius (व्यासार्धः, also called कर्णः in this context). कर्णः means radius in this context, but can also mean diagonal, or hypotenuse of a right triangle. OA fulfils all these meanings at the same time.

Using the notation \(EA = s \text{,} \quad \theta = \frac{s}{R} \), we get:

\(\begin{align*}
jyā(s) &= AJ = R.sin(\theta) \\
koṭi(s) &= AK = R.cos(\theta) \\
śara(s) &= EJ = R.versine(\theta) = R.(1-cos(\theta)) \\
\end{align*}\)

Quite evidently, we can derive the ज्याकोटिकर्णन्यायः \( jyā^2 + koṭi^2 = karṇa^2 \), by noting that \( AJ^2 + AK^2 = OA^2 \).

In modern trigonometry, we talk of sines and cosines of angles (arcs divided by the radius), whereas ज्या etc. are functions of arcs. Again, the ज्या etc. are lengths, not ratios as the sine and cosine functions are. Therefore, ज्या etc. are dependent on the actual radius/circumference of the circle, which can make life complicated if we use them as is. Therefore, a convention of all circles having radius 21600′ was adopted.

These names and conventions were first used by Āryabhaṭa, and adopted by later Indian mathematicians.

Building a table of jyā

Āryabhaṭa's first step towards being able to calculate the jyā of any arc was to build a table of a fixed number of them. This construction later became standard among all of his followers, thanks to his ingenious method of calculating those tabular jyās. Āryabhaṭa divided the NE quadrant of the circle into 24 arcs of equal length, EC₁, C₁C₂, … C₂₂C₂₃, C₂₃N. The respective jyās are C₁B₁, C₂B₂, … C₂₃B₂₃ and their respective koṭis are C₁K₁, C₂K₂, … C₂₃K₂₃. How are these calculated?

There is the hard, but accurate way. Through geometrical reasoning, Āryabhaṭa proved that \(jyā(30) = \frac{r}{2}\), \(jyā(90) = r \) and \( jyā(\frac{s}{2}) = \frac{1}{2}. \sqrt{jyā(s)^2 + śara(s)^2}\). Using these and the ज्याकोटिकर्णन्यायः all of tabular jyā can be calculated accurately. However, a whole lot of square roots need to be calculated, which limits practical calculation accuracy. This problem sets the stage for Āryabhaṭa's ingenious method.

The Āryabhaṭa Recurrence Relation

प्रथमाच्चापज्यार्धाद्यैरूनं खण्डितं द्वितीयार्धम् ।
तत्प्रथमज्यार्धांशैस्तैस्तैरूनानि शेषाणि ।।
From the first jyā. subtract its ratio with the first jya (ie itself, therefore 1) to get the second difference
Further differences are computed from the previous by subtracting the ratio of the previous sine with the first

Each of the equal arcs EC₁, C₁C₂, … C₂₂C₂₃, C₂₃N has length equal to 21600/(4*24) =225, which was one reason exactly 24 divisions were chosen rather than 23 or 25. Using a small arc approximation, Āryabhaṭa set jyā(225′) = 225′. Further jyās are calculated using recurrence relations.

Let the jyā of EC₁, C₁C₂, … C₂₂C₂₃, C₂₃N be denoted as B₁, B₂, … B₂₄. We denote their differences as K₁, K₂, … K₂₄, such that K₁ = B₁, K₂ = B₂ – B₁, K₃ = B₃ – B₂ etc. The Āryabhaṭa recurrence relation is

\(
K_2 = K_1 – \frac{B_1}{B_1} \\
K_n = K_{n-1} – \frac{B_{n-1}}{B_1}
\)

Since we already know K₁ = B₁ = 225′, this lets us calculate the full table of jyās. Āryabhaṭa calls the K_n as खण्डज्या and the B_n as पिण्डज्या. Calculating the K_n leads to the verse of all 24 खण्डज्या (encoded in his peculiar encoding)

मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व |
घ्लकि किग्र हक्य धकि किच स्ग झश ङ्व क्ल प्त फ छ कलार्धज्याः ||

S.No	Angle	खण्डज्या	पिण्डज्या	3438′ * Modern Sine
1	03° 45′	225	225′	224.8560
2	07°   30′	224	449′	448.7490
3	11°   15′	222	671′	670.7205
4	15°   00′	219	890′	889.8199
5	18°   45′	215	1105′	1105.1089
6	22°   30′	210	1315′	1315.6656
7	26°   15′	205	1520′	1520.5885
8	30°   00′	199	1719′	1719.0000
9	33°   45′	191	1910′	1910.0505
10	37°   30′	183	2093′	2092.9218
11	41°   15′	174	2267′	2266.8309
12	45°   00′	164	2431′	2431.0331
13	48°   45′	154	2585′	2584.8253
14	52°   30′	143	2728′	2727.5488
15	56°   15′	131	2859′	2858.5925
16	60°   00′	119	2978′	2977.3953
17	63°   45′	106	3084′	3083.4485
18	67°   30′	93	3177′	3176.2978
19	71°   15′	79	3256′	3255.5458
20	75°   00′	65	3321′	3320.8530
21	78°   45′	51	3372′	3371.9398
22	82°   30′	37	3409′	3408.5874
23	86°   15′	22	3431′	3430.6390
24	90°   00′	7	3438′	3438.0000

These are accurate to the minute (= 1/21600 of the circumference), which is remarkable for something that's that simple to remember. Incidentally, the Āryabhaṭa recurrence can be considered a discrete approximation of the modern relation \( \frac{d}{d\theta}sin \theta = – sin \theta\).

Reversing this table gives the values of koṭi for the same arcs, since koṭi(s) = jyā(90-s). Using śara(s) = R- koṭi(s) lets us calculate those as well.

Calculation of jyā at points between the points of the table is done through simple linear interpolation. Thus, we're home and dry to a decent accuracy for all three – jyā, koṭi, and śara, with nothing more than basic arithmetic!

Going beyond 90 degrees

Consider the four points A₁, A₂, A₃, A₄ . By convention, we measure angles in Indian astronomy from the East point (first point of Aries, or मेषादिः), and therefore the points on the circle are marked E, N, W, S, respectively. For a point in the NW quadrant (ie, arc greater than quarter but less than half the circumference), we take the complementary arc WA₂, and read out (or interpolate) its tabular jyā, to get A₂B₂, the jyā of EA₂. For A₃, we use WA₃ to calculate A₃B₃ with a negative sign, and for A₄, we use the complementary arc EA₄ to calculate A₄B₄, again with a negative sign. This corresponds to the modern relationships:

\(\begin{align*}
sin(90+\theta) &= cos(\theta) = sin(90-\theta) \\
sin(180+\theta) &= -sin(\theta) \\
sin(270+\theta) &= -cos(\theta) = -sin(90-\theta) \\
\end{align*}\)

Getting even better

Is this the end of the story then? Not quite. While these values were “good enough” for a long time, errors tended to accumulate, and just as we saw with calculating circumference, the need for more accurate values was felt. Also, linear interpolation is clearly not quite good enough for a non-linear function, as we would instantly note in a modern context.

The Kerala School came up with solutions for both of these.

• First, they discovered much improved recurrence relations, and a proper infinite series for these functions which improved accuracy much further.
• Secondly, they found a much more accurate interpolation, using the जीवे परस्परन्यायः relation for sin(A+B).

All this will be explored in later articles.