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.

### The standard circle

The standard circle in Indian trigonometry has circumference 21600′ (=360*60), which is its arc-length in minutes. The corresponding radius can be calculated by the circumference-diameter relation to be *approximately* 3438′. All calculations of jyā and others happen assuming this standard circle. When a true result is required, it is scaled to the real size of the circle using the rule of three (त्रैराशिकम्) which we have seen before.

The standard circle is divided as

Divided Into | |

1 Standard circle | 12 Rashi/राशिः (divisions) |

1 Rashi | 30 Amsha/अंशः (degrees) |

1 Amsha | 60 Kalā / कला (minutes) |

1 Kalā | 60 Vikalā / विकला (seconds) |

It can be seen that 1 Standard circle = 12*30*60 = 21600 कला (minutes). Beyond Vikalā, further subdivisions into sixtieths as thirds, fourths and so on can be made if necessary.

For more accurate estimates of radius, the encoding श्रीरुद्रः श्रीधरः श्रेष्ठो देवो विश्वस्थली भृगुः corresponds to 3437,44,48,22,29,22,22 , or 3437 minutes 44 seconds, 48 thirds, 22 fourths, 29 fifths, 22 sixths and 22 sevenths.

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

### How did the name “sine” originate?

Surprisingly enough, the modern term “sine” itself derives from ज्या, in spite of the lack of superficial similarity. Indian texts used any synonym of bowstring for ज्या, including मोर्वी, शिञ्जिनी, and जीवा. When Indian texts were translated into Arabic, जीवा became *jiba*, written in Arabic as جب . European translators misinterpreted this as the Arabic word *jayb* (pocket, bay, inlet), and translated it into Latin as *sinus *(bay). Thus, through a rather circuitous route, ज्या became sine.

## 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,** EC_{1}, C_{1}C_{2}, … C_{22}C_{23}, C_{23}N. The respective jyās are C_{1}B_{1}, C_{2}B_{2}, … C_{23}B_{23} and their respective koṭis are C_{1}K_{1}, C_{2}K_{2}, … C_{23}K_{23}. 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 differenceFurther differences are computed from the previous by subtracting the ratio of the previous sine with the first*

Each of the equal arcs EC_{1}, C_{1}C_{2}, … C_{22}C_{23}, C_{23}N 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 EC_{1}, C_{1}C_{2}, … C_{22}C_{23}, C_{23}N be denoted as B_{1}, B_{2}, … B_{24}. We denote their differences as K_{1}, K_{2}, … K_{24}, such that K_{1} = B_{1}, K_{2} = B_{2} – B_{1}, K_{3} = B_{3} – B_{2} 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 K_{1} = B_{1} = 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 A_{1}, A_{2}, A_{3}, A_{4 }. 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 WA_{2}, and read out (or interpolate) its tabular jyā, to get A_{2}B_{2}, the jyā of EA_{2}. For A_{3}, we use WA_{3} to calculate A_{3}B_{3} with a negative sign, and for A_{4}, we use the complementary arc EA_{4} to calculate A_{4}B_{4}, 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.

I like your blog; interesting. You mention the synonyms of ज्या … मोर्वी, शिञ्जिनी, and जीवा. I can see ज्या and जीवा in the Surya Siddhanta and in the Aryabhatiya … but in which texts does one find मोर्वी and शिञ्जिनी?