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The sum without end

परिधिः सुसूक्ष्मः · जीवा · आसन्न

The zero essay followed a sign; this one follows a suspicion — that some numbers cannot be finished. It begins with Āryabhaṭa giving π to four places and calling it, with a single loaded word, 'approached'; runs through the sine tables every siddhānta computed; and ends on the Malabar coast around 1400 CE, where the school of Mādhava wrote the circumference of a circle as a sum that never terminates — the series Europe would rediscover and name for Leibniz. The texts carry it in working Sanskrit verse, with an honesty the modern reader should notice: the result is called 'very precise'. Not exact. Precision, the verse knows, is bought term by term, and the sum has no last term.

Every quote is a slice from Jñānakośa, the research database behind this atlas; each links to the e-text it was sliced from. Translations are literal renderings; dates are approximate spans of compilation. Display markup and OCR artifacts of the e-texts are handled as flagged in the database.

Four places, and a confession

आसन्न · the loaded word

Āryabhaṭa's π verse gives 62,832/20,000 — 3.1416 — and plants one word in front of it: āsanna, 'approached'. Not 'is', but 'has come near'. Whether he suspected irrationality or merely honesty of rounding, the tradition read the word and remembered it:

Āryabhaṭīya, Gaṇita 10 · 499 CE

चतुरधिकं शतं अष्टगुणं द्वाषष्टिस्तथा सहस्राणाम् । अयुत द्वयविष्कम्भस्य आसन्नस्वृत्तपरिणाहस् । । २.१० । ।

One hundred increased by four, multiplied by eight, together with sixty-two thousand: this is the near (āsanna) circumference of a circle whose diameter is two ayutas. [(100+4)×8 + 62,000 = 62,832 for a diameter of 20,000 — π ≈ 3.1416.]

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The sine, tabulated

ज्या · the table

The working object of siddhāntic astronomy is the jyā — the half-chord, our sine — kept as a table of twenty-four values. Āryabhaṭa compresses the whole table into one alphabet-coded verse:

Āryabhaṭīya, Gītikā 12 · 499 CE

मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व । घ्लकि किग्र हक्य धकि किच स्ग झश ङ्व क्ल प्त फ छ कला अर्धज्यास् ॥ १.१२ ॥

Makhi (225), bhakhi (224), phakhi (222), dhakhi (219), ṇakhi (215), ñakhi (210), ṅakhi (205), hasjha (199), skaki (191), kiṣga (183), śghaki (174), kighva (164); ghlaki (154), kigra (143), hakya (131), dhaki (119), kica (106), sga (93), jhaśa (79), ṅva (64), kla (50), pta (37), pha (22), cha (7) — [these are] the half-chords, in minutes.

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And the Sūrya-siddhānta canonizes the recursive recipe — each difference computed from the sum of what came before — a table that builds itself:

Sūrya-siddhānta 2.15–17 · c. 400–800 CE

राशिलिप्ताष्टमो भागः प्रथमं ज्यार्धं उच्यते । तत्तद्विभक्तलब्धोनमिश्रितं तद्द्वितीयकम् । । २.१५ । । आद्येनैवं क्रमात्पिण्डान्भक्त्वा *लब्धोनसंयुताः ।(C लब्धोनितैर्युतैः) *खण्डकाः स्युश्चतुर्विंशज्यार्धपिण्डाः क्रमादमी । ।(C खण्डकैस्) २.१६ । । तत्त्वाश्विनो +अङ्काब्धिकृता रूपभूमिधरर्तवः ।(२२४, ४४९, ६९१) खाङ्काष्टौ पञ्चशून्येशा बाणरूपगुणेन्दवः । ।(८९०, ११०५, १३१५) २.१७ । ।

The eighth part of the minutes of a zodiacal sign is called the first half-chord [225′]; that, increased by [itself] less the quotient of itself divided by itself(?), is the second. Dividing the [accumulated] values in order by the first, and adding [each difference] diminished by the quotient — thus arise, in order, the segments: the twenty-four half-chord values. Twenty-five-and-two [225]; nine-four-four [449]; one-seven-six [671]; sky-nine-eight [890]; five-zero-one-one [1105]; five-one-three-one [1315] …

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See: how the table was coded into syllables, in the wanderers essay

The circumference as an endless sum

परिधिः सुसूक्ष्मः · the series

Then the leap. In the Karaṇapaddhati of Putumana Somayāji — a working manual of the Kerala school — the circumference is no longer a value at all. It is a procedure without a final step: four times the diameter, divided by the odd numbers in turn, signs alternating. This is π/4 = 1 − 1/3 + 1/5 − 1/7 …, the series Europe would find in 1673 and name for Leibniz; Mādhava of Saṅgamagrāma had it around 1400:

Karaṇapaddhati 6.1 · c. 16th–18th c. CE

व्यासाच्चतुर्भाद् बहुशः पृथकुस्थात् त्रिपञ्चसप्ताद्ययुगाहृतानि । व्यासे चतुर्भे क्रमशस्त्वृणं स्वं कुर्यात् तदा स्यात् परिधिः सुसूक्ष्मः ॥ १ ॥

From the diameter multiplied by four, set down separately many times and divided [in turn] by the odd numbers three, five, seven and so on — subtracting and adding [these] in order to four times the diameter — then the circumference comes out very precise.

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Read the last word again: susūkṣmaḥ, 'very precise'. The verse does not claim the sum ends — it tells you the answer sharpens as long as you keep dividing. That is an infinite series, stated as such, in a Sanskrit karaṇa manual.

Faster: the school keeps a family

विमूलघनैः · the family

The alternating series converges miserably — and the very next verse shows the school knew it, and did what analysts do: transform the series. Divide instead by the odd cubes less their roots, start from three diameters, and the sum closes in far faster:

Karaṇapaddhati 6.2 · c. 16th–18th c. CE

व्यासाद् वनसङ्गुणितात् पृथगातंत्र्याधयुग्विमूलघनैः । त्रिगुणव्यासे स्वमृणं क्रमशः कृत्वापि परिधिरानेयः ॥ २ ॥

Or: from the diameter multiplied [by four], divided separately by the cubes-less-their-roots of the odd numbers beginning with three, adding and subtracting in order to three times the diameter — the circumference may [also] be obtained.

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π = 3 + 4/(3³−3) − 4/(5³−5) + … One series is a curiosity; a family of series, chosen for convergence, is a method — the heart of what the Kerala school possessed.

The sine, in Mādhava's house

तन्त्रसंग्रह · the school

The school's flagship treatise, Nīlakaṇṭha's Tantrasaṅgraha (1501), works the sine with the same self-building recursions the old siddhāntas used — here the doubling relations among the tabulated jyās:

Tantrasaṅgraha 2 (jyā section) · 1501 CE

यदि स्थादाद्यजीवेयं द्वितीयज्यादलं भवेत्‌ । द्वितीयज्यापि सा नित्य॑ चतुर्थज्यादलं मवेत्‌ ॥ ३२४ ॥

If this be the first sine, it is half the second sine; and the second sine likewise is always half the fourth sine.

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The explicit Mādhava attributions — the jīve-paraspara sine formula, the correction terms bearing his name — live in the school's commentarial layer (the Yuktidīpikā, the Malayalam Yuktibhāṣā), which the database records as a testimonium debt, exactly as it records Garga's lost comet-course. What the pages here carry verbatim is enough: the series, the acceleration, and the school still computing.

The sum ends only when you stop

From āsanna to susūkṣma, the thread holds one idea: some numbers are not values but pursuits. Āryabhaṭa flags it in a single participle; the sine tables chase it recursively; and the Kerala school states it in full — an unending alternating sum, plus the transformations that make it usable. Whether any of this reached Europe before Newton and Leibniz remains unproven and much argued; what needs no argument is the date-stamped Sanskrit record itself: the series were here first, in working verse, labeled honestly as precise-not-exact.

Each quote carries its full citation, edition, revision-pinned source link and confidence grading in Jñānakośa — batch kerala-series-pass1 (which opened the Tantrasaṅgraha and Karaṇapaddhati as sources) with the standing trigonometry passes. The circle in tonight's sky — any of them — still refuses to hand over its circumference exactly. The Kerala verses tell you how close you are allowed to get: as close as you have patience for.

The series

Sanskrit texts on the sky and the sciences around it, one thread per essay — readable in any order.