Jyotpatti (Sanskrit: "production of chords") is the Indian mathematical tradition of trigonometry — the study of relationships between angles and lengths in triangles and circles, developed primarily for astronomical computation. Indian mathematicians, beginning with Āryabhaṭa in the 5th century CE, defined and tabulated the jya (half-chord, equivalent to the modern sine function) and developed trigonometric methods far earlier and more systematically than their contemporaries elsewhere. The modern terms "sine" and "cosine" derive, through an Arabic mistranslation, from the Sanskrit word jya.
The trigonometric function we call "sine" has an etymology that traces directly to India. The Sanskrit word jya (or jīva) originally meant "bowstring" — a string drawn across a bow, geometrically representing a chord of a circle. Āryabhaṭa used jya to refer to the half-chord (which is our modern sine). Arabic translators rendered jīva as jiba (keeping the sound but losing the meaning, since the word has no meaning in Arabic). Later Latin translators, mistaking jiba for the Arabic word jaib (meaning "bay" or "fold"), translated it as sinus (Latin for bay or fold) — giving us "sine." The route: Sanskrit jya → Arabic jiba → Latin sinus → English sine.
Indian mathematical trigonometry was developed primarily for astronomical computation — calculating the positions of planets, predicting eclipses, and determining the time of day and year. This practical astronomical motivation drove centuries of increasingly precise sine table construction, the development of interpolation methods, and eventually the discovery of infinite series for sine and cosine by Mādhava of Sangamagrama in the 14th century (see that entry) — roughly 200 years before the same series appeared in Europe.
Āryabhaṭa's Āryabhaṭīya (499 CE) contains a table of 24 "jya" values for arcs at intervals of 3.75° (from 3.75° to 90°), expressed in a unit where the radius R = 3438 (the number of arc-minutes in a radian, making the table effectively a table of R·sin(θ)). The table is given as a series of first differences, with a rule for computing the differences using a second-difference formula — an early form of numerical differentiation. Āryabhaṭa also defines the kojya (co-jya, equivalent to cosine) as the jya of the complementary angle.
Indian mathematicians developed a richer set of trigonometric quantities than Greek astronomers used:
The versine (utkramajya) was particularly useful in Indian astronomical calculations and appears prominently in Brahmagupta's interpolation formula.
Brahmagupta's Brahmasphutasiddhanta (628 CE) contains a second-order interpolation formula for computing sine values between tabulated entries — essentially a finite difference method equivalent to what is now called Newton's forward difference formula (or Stirling's formula in another form). This is one of the earliest known examples of higher-order numerical interpolation anywhere in the world, predating Newton's own work on finite differences by over a millennium.
The Kerala mathematician Mādhava of Sangamagrama (c. 1340–1425 CE — see that entry) derived infinite series for sine, cosine, and arctangent — results now attributed in European mathematics to Gregory, Newton, and Leibniz, who discovered them 200–300 years later. Mādhava's sine series is: sin(θ) = θ − θ³/3! + θ⁵/5! − ..., exactly the modern Taylor/Maclaurin series. These results appear in the works of his students and the later Kerala school, including Nilakantha Somayaji's Tantrasangraha (c. 1500 CE).
The Kerala school of mathematics (14th–16th centuries CE), centred on Mādhava and his successors (Parameshvara, Nilakantha, Jyeshthadeva, Achyuta Pisharati), produced a remarkable body of results in trigonometry and infinite series — some of which anticipate European calculus by 200 years. The question of whether these results were transmitted to Europe (via Jesuit missionaries or trade routes) before their independent European discovery remains debated. George Gheverghese Joseph (The Crest of the Peacock) and others have argued for possible transmission; most historians of mathematics currently regard the question as unresolved but lean toward independent discovery in Europe. Either way, the priority of the Kerala school's results is not in scholarly dispute.
Indian trigonometric development was consistently embedded in a larger programme of astronomical table computation — the Karana texts, which were practical computation manuals for astronomers and calendar-makers. This tight coupling between trigonometric mathematics and practical astronomical application drove an unusually sustained tradition of numerical precision: successive Indian astronomers corrected and refined earlier tables, producing, over the span of roughly a millennium (5th–15th centuries CE), sine and cosine computations of accuracy that was not matched in Europe until the 16th century.
The Indian jyotpatti tradition represents one of the most sustained and original mathematical research programmes in the pre-modern world. Its key contributions — the half-chord as the fundamental trigonometric unit, sine tables of high precision, higher-order interpolation, and eventually infinite series for trigonometric functions — were each conceptually significant advances that shaped the subsequent development of both astronomy and mathematics. That these contributions reached Europe partly anonymised (via the Arabic transmission, which often did not explicitly credit Indian sources) is a recurring pattern in the broader history of mathematical knowledge transfer from India to the Islamic world to Europe — one that this Codex is committed to documenting correctly.