IKS/ Part 3· Mathematics Chapter 8 of 14
Part 3 · Chapter 8

Mathematics

गणितम्

A cord, a peg and a patch of level ground are enough to build a right angle, a square, and a square equal in area to a given circle. From that beginning — the geometry of altar construction — runs an unbroken line to fifteenth-century Kerala, where infinite series for π, sine and cosine were derived, summed with correction terms, and proved.

This chapter follows the line. It is the longest in the series, because there is the most to show.

1Rope geometryशुल्बम्

The word शुल्ब (śulba) means a cord. The Śulba-sūtras are manuals of construction with a cord, and they are the oldest geometry texts in India — some universities now teach the subject under the name “rope geometry”.

The equipment is a peg, a length of cord with marks on it, and level ground. That is enough for a surprising amount.

Constructing a square with cord and pegs, as in the Baudhayana Sulba-sutra An east-west line is laid out, arcs struck from its ends give a perpendicular through the midpoint, and further arcs complete a square. STEP 1 — the base line WE midpoint Stretch the cord east–west between two pegs and mark its midpoint. STEP 2 — the perpendicular Fix the cord's ends at the two pegs; swing arcs from each. The line joining the intersections is perpendicular — and exactly so, not by eye. STEP 3 — close the square Mark equal lengths along both lines and join. The result is a true square, with sides in a stated orientation to the compass points. Every construction in the Śulba-sūtras is of this kind: a procedure, stated as a sequence of physical acts, that produces an exact figure.
There are no proofs in the Greek sense here, and no diagrams in the manuscripts — the text is the diagram, and it is written to be executed rather than contemplated.

2Building a falconश्येनचितिः

Why does anyone need this much geometry? Because the altars were not squares.

More than seventy altar shapes are prescribed for different rites — a tortoise, a chariot wheel, a trough, a rhombus — and the most demanding is the श्येनचिति (śyena-citi), the falcon-shaped altar. It must be built of exactly two hundred bricks of five specified shapes, in a prescribed distribution, to a prescribed total area, and then rebuilt in later layers with the joints offset.

The falcon-shaped altar and its brick count A schematic falcon altar with head, body, two wings and tail, and a table of how many bricks of each of five shapes go into each part, totalling two hundred. head · 14 bricks body 46 left wing right wing — 108 both tail · 32 Schematic. The proportions are those given in the sūtras: the wings are swept back, the tail spread, and the whole is oriented east. Later layers use different brick shapes so that no joint sits above a joint below — a genuine bricklaying constraint, stated as ritual requirement.
Bricks in the first layer of a śyena-citi: five shapes, two hundred bricks
PartShape B1B2B3B4B5Total
Head1—66114
Body30610——46
Wings306216——108
Tail8420——32
Total69725266200
The problem this generates

The altar must have a prescribed area. Its shape is a falcon. Its bricks come in five fixed shapes. It must be rebuilt in successive rites with the area increased by a stated amount while the shape stays the same. Solve that and you have needed: the area of an isosceles triangle and a rhombus, the ability to transform a rectangle into a square of equal area, the ability to scale a figure by a given ratio, and the Pythagorean relation. All of these appear in the Śulba-sūtras, and every one of them is there because of this problem.

3What makes it distinctiveवैशिष्ट्यम्

Constructive, not existential

The characteristic Indian question is “how do I find it?” rather than “does one exist?”. The output of a mathematical investigation is an algorithm — a stated procedure that terminates in the answer. This is why so much of the material translates directly into code.

In verse, and in company

Results are stated as sūtras or ślokas, in metre, with the numbers encoded by the schemes of chapter 6. Mathematics sits inside literature rather than beside it — Bhāskara's Līlāvatī poses its problems as riddles addressed to a student, about bees, peacocks and necklaces.

Continuous and geographically spread

An unbroken line of named authors from at least the fifth century BCE to the seventeenth CE, working from Gandhāra to Bengal and from Kashmir to Kerala. There is no Indian dark age in mathematics.

Proofs exist — as upapatti

The claim that Indian mathematics has no proofs is a half-truth resting on the base texts, which are deliberately terse. The commentaries supply demonstrations: Bhāskara's own Vāsanā-bhāṣya, the Kriyākramakarī on the Līlāvatī, and above all the Yukti-bhāṣā, which is nothing but proofs.

4Two and a half millenniaपरम्परा

Work or authorPeriod and placeContribution
Vedic texts3000 BCE or earlierThe earliest recorded mathematics: the number system, Pythagorean triples, decimal naming of numbers, the notion of the unbounded
Lagadha — Vedāṅga-jyotiṣac. 1300 BCEA quantitative model of solar and lunar motion; equinoxes and solstices
Śulba-sūtras — Baudhāyana, Āpastamba, Kātyāyana, Mānava800–600 BCEThe earliest geometry texts; √2 and π approximated; exact construction and transformation of squares, rectangles and trapezia
Pāṇini — Aṣṭādhyāyīc. 500 BCE, ŚālāturaAlgorithmic method; context-sensitive rules; recursion. See chapter 5
Piṅgala — Chandaḥ-śāstrac. 300 BCEBinary sequences and conversion both ways; the Meru-prastāra; optimal exponentiation; zero as a symbol
Buddhist worksc. 500 BCE – 500 CEMulti-valued logic; indeterminate and unbounded quantities
Jaina works — Sūrya-prajñapti, Anuyogadvāra-sūtra, Tiloyapaṇṇatti and others200 BCE – 300 CELogarithm-like ideas; very large numbers; exponentiation algorithms; arcs; combinatorics; mensuration; π ≈ √10
Āryabhaṭa — Āryabhaṭīya476–550 CE, KusumapuraSquare and cube root algorithms; place value; the sine table; quadratic and linear indeterminate equations; sums of squares and cubes; spherical trigonometry
Varāhamihira — Pañca-siddhāntikā, Bṛhat-saṃhitā482–565 CE, UjjainFive earlier siddhāntas summarised; sine table; sin²+cos²=1; combinatorics; magic squares
Bhāskara I600–680 CE, SaurashtraInteger solutions of indeterminate equations; a rational approximation to the sine function
Brahmagupta — Brāhmasphuṭa-siddhānta598–668 CE, BhillamālaArithmetic with zero and negatives; algebra; the cyclic-quadrilateral area and diagonal formulae; Pythagorean triples
Vṛhaṅka — Vṛttajātisamuccayac. 600 CEThe Fibonacci numbers, in the context of moric metres
Work or authorPeriod and placeContribution
Mahāvīrācārya — Gaṇita-sāra-saṅgraha800–870 CE, GulbargaA complete standalone mathematics textbook: arithmetic, geometry, algebra; permutations and combinations; series
Śrīdharācārya — Pāṭīgaṇita, Triśatikā870–930 CE, BengalCommercial arithmetic; approximation of roots of non-squares; quadratic equations
Jayadeva10th c. CE or earlierThe cakravāla or cyclic method for second-order indeterminate equations
Śrīpati1019–1066 CE, MaharashtraPlanetary astronomy; Gaṇita-tilaka
Bhāskarācārya (Bhāskara II) — Līlāvatī, Bījagaṇita, Siddhānta-śiromaṇi1114–1185 CEThe canonical textbooks of Indian mathematics; surds; combinatorics; the sine addition formula; indeterminate equations; the instantaneous rate of motion and a form of the mean value theorem
Nārāyaṇa Paṇḍita — Gaṇita-kaumudī1325–1400 CECyclic quadrilaterals; repeated summation of series; a theory and construction of magic squares; combinatorics
Mādhava of Saṅgamagrāma1340–1425 CE, KeralaFounder of the Kerala school. Infinite series for π, sine and cosine, with correction terms
Parameśvara — Dṛggaṇita1360–1460 CE, KeralaIterative techniques; the circumradius formula for a cyclic quadrilateral; a revised observational system
Work or authorPeriod and placeContribution
Nīlakaṇṭha Somayājī — Tantra-saṅgraha, Āryabhaṭīya-bhāṣya1444–1544 CE, KeralaArgues that π is irrational; the basic ideas of calculus; a revised planetary model in which the inner planets orbit the sun
Jyeṣṭhadeva — Yukti-bhāṣā1500–1575 CE, KeralaFull demonstrations of the Mādhava series. Written in Malayalam, and often called the first textbook of calculus
Śaṅkara Vāriyar — Kriyākramakarī1500–1569 CE, KeralaProofs and explanations of the procedures of the Līlāvatī
Gaṇeśa Daivajña — Buddhi-vilāsinī1504 CE, GujaratDemonstrations of the Līlāvatī's results
Kṛṣṇa Daivajña — Bījapallavac. 1600 CE, DelhiDemonstrations of the Bījagaṇita
Munīśvara — Siddhānta-sārvabhauma17th c. CE, VaranasiTrigonometric identities; commentary on the Līlāvatī
Kamalākara — Siddhānta-tattva-viveka1616–1700 CE, VaranasiAddition and subtraction theorems for sine and cosine; multiple-angle formulae
Pāṇini and BNF — the claim, stated carefully

The Aṣṭādhyāyī's rule format bears a real structural resemblance to Backus–Naur Form, the notation used to specify programming-language grammars, and in 1967 Peter Ingerman proposed that BNF be renamed “Pāṇini–Backus Form” in recognition of it. What is established is the resemblance and the acknowledgement. What is not established is historical influence: Backus and Naur did not work from Pāṇini. Stating the claim in the first form is defensible and interesting; stating it in the second is not.

5The Śulba theoremभुजाकोटिकर्णन्यायः

The relation between the sides and the diagonal of a rectangle is stated as a general rule in the Baudhāyana Śulba-sūtra, some centuries before Pythagoras.

दीर्घचतुरश्रस्याक्ष्णया रज्जुः पार्श्वमानी तिर्यङ्मानी च
यत्पृथग्भूते कुरुतस्तदुभयं करोति ॥

dīrgha-caturaśrasyākṣṇayā rajjuḥ pārśvamānī tiryaṅmānī ca yat pṛthag bhūte kurutas tad ubhayaṃ karoti

Baudhāyana Śulba-sūtra 1.48 — “The cord stretched across the diagonal of a rectangle produces both the areas which the side and the breadth produce separately.”

The square on the diagonal equals the sum of the squares on the sides A rectangle with squares erected on its breadth, length and diagonal, illustrating the equality of areas. bhujā — the side koṭi karṇa — the diagonal bhujā² koṭi² karṇa² = bhujā² + koṭi² THE TRIPLES THE TEXT LISTS 3, 4, 55, 12, 13 8, 15, 177, 24, 25 12, 35, 3715, 36, 39 Given as ready-made cord markings for laying out right angles on the ground.
Note the form of the statement: it is about areas produced, not about numbers, and it is stated for the rectangle as the general case with the square following. The listed triples are practical instructions — knot a cord at 3, 4 and 5 units and you have a right angle you can carry to the site.

6√2 to five placesद्विकरणी

The altar work needs the diagonal of a square, so it needs √2. Baudhāyana gives it as a recipe:

प्रमाणं तृतीयेन वर्धयेत् तच्चतुर्थेन आत्मचतुस्त्रिंशोनेन सविशेषः ।

pramāṇaṃ tṛtīyena vardhayet tac caturthena ātma-catustriṃśonena saviśeṣaḥ

Baudhāyana Śulba-sūtra 2.12

Decoding the Sulba approximation to the square root of two The phrase decodes to one plus a third, plus a quarter of that third, less a thirty-fourth of that quarter, giving 577 over 408. pramāṇaṃ tṛtīyena vardhayet “increase the measure by a third” 1 + 1/3 tac caturthena “and by a fourth of that” + 1/(3×4) ātma-catustriṃśonena “less its own thirty-fourth part” − 1/(3×4×34) √2 ≈ 1 + 1/3 + 1/12 − 1/408 = 577/408 = 1.414 215 686 27… saviśeṣaḥ — “with a remainder”, i.e. approximate true value 1.414 213 562 37… the sūtra's value 1.414 215 686 27… Agreement to five decimal places — an error of about 1 part in 700,000. And the text says so: the final word saviśeṣaḥ flags it as approximate.
What the honesty of that last word buys: nobody reading the sūtra can mistake it for an exact value, and the Śulba texts elsewhere distinguish carefully between exact constructions and approximate ones. Where the fractions 1/3, 1/12 and 1/408 come from is not stated, and reconstructing the derivation is an open question with several plausible answers.

7The gnomonशङ्कुः

A vertical rod on level ground — the शङ्कु, śaṅku — is the single most productive instrument in Indian science. Similar triangles turn its shadow into a measuring device, and the same figure does duty in surveying and in astronomy.

The shadow problem and its astronomical application A lamp post and a gnomon give similar triangles that determine the shadow length; the same figure with the sun and earth gives the length of the earth's shadow cone. A C E F B the shadow Triangles FEB and CAB are similar, so the shadow length follows at once: EB = (EF × AE) / (AC − EF) lamp post AC, gnomon EF, shadow EB THE SAME FIGURE, TURNED ON THE SKY sun earth shadow cone Replace the lamp post by half the sun's diameter and the gnomon by half the earth's; AE becomes the sun–earth distance, and EB the length of the earth's shadow — the quantity needed to predict a lunar eclipse. Chapter 9 uses this figure directly.
One proportion, two uses. The transferability is the point: a technique developed for surveying a field is the same technique that gives the eclipse geometry, and the Indian astronomical texts move between the two without remark.

8Squaring a numberवर्गः

Indian arithmetic is taught as procedures, and the procedures are stated in verse. Here is squaring, in three sūtra fragments, applied to 1638.

SūtraInstruction
अन्त्यपदस्य वर्गं कृत्वा
antya-padasya vargaṃ kṛtvā		Square the leading digit and place it in a new row.
द्विगुणं तदेव चान्त्यपदम्
dviguṇaṃ tad eva cāntya-padam		Multiply twice that digit by each remaining digit, placing the products to the right in the same row.
शेषपदैराहन्यात् उत्सार्योत्सार्य
śeṣa-padair āhanyāt utsāryotsārya		Drop the leading digit and repeat with what remains; then add the rows.
Squaring 1638 by the sutra method Four rows of partial products, one per digit, added to give 2683044. 10⁶10⁵10⁴10³10²10¹10⁰ leading digit 1 1² then 2·1·6, 2·1·3, 2·1·8 1 12 6 16 next digit 6 6² then 2·6·3, 2·6·8 36 36 96 next digit 3 3² then 2·3·8 9 48 last digit 8 8², nothing to its right 64 sum the columns, carrying 2683044 1638² = 2 683 044 Four rows, no digit multiplied against itself more than once.
The method is the expansion (a+b+c+d)² = Σaᵢ² + 2Σaᵢaⱼ, organised so that each cross-term is computed exactly once and placed in the right column automatically. Sixteen digit-products become ten. This is what an algorithm designed for mental arithmetic looks like.

9Extracting a square rootवर्गमूलम्

Āryabhaṭa gives a digit-by-digit root extraction — the ancestor of the long-division square-root method that was taught in schools until calculators arrived. Its two operative sūtras alternate:

At a varga (square) place

वर्गाद्वर्गे शुद्धे

vargād varge śuddhe

Subtract the square of the digit just found.

At an avarga (non-square) place

भागं हरेत् अवर्गात् नित्यं द्विगुणेन वर्गमूलेन

bhāgaṃ haret avargāt nityaṃ dviguṇena vargamūlena

Divide by twice the root so far; the quotient is the next digit of the root.

Extracting the square root of 19881 Alternating subtraction of squares and division by twice the running root yields 141. MARK THE PLACES ALTERNATELY VARGA (V) AND AVARGA (A), FROM THE RIGHT 19881 VAVAV THE ROOT LINE — ACCUMULATES THE ANSWER 1 4 1 VARGA STEP · subtract the largest square Leading digit 1. The largest square not exceeding it is 1² = 1. Subtract: remainder 0. Write 1 on the root line. AVARGA STEP · divide by twice the root so far Bring down the next digit: 09. Twice the root line is 2×1 = 2. Divide: 09 ÷ 2 = 4, remainder 1. Write 4 on the root line, which now reads 14. VARGA STEP · subtract the square of the last quotient Bring down the next digit: 18. Subtract 4² = 16. Remainder 2. AVARGA STEP 28 ÷ (2×14) = 1, remainder 0. Root line: 141. VARGA STEP · finish Bring down 1; subtract 1² = 1. Remainder 0. √19881 = 141. Exact.
The alternation is the whole design: the odd places consume a square, the even places consume a division by twice the running root. Both sūtras together are three lines of Sanskrit, and they specify a complete algorithm — with a termination condition, and with a remainder that tells you whether the input was a perfect square.

10Roots of non-squaresबक्षालीविधिः

The Bakhshālī manuscript — birch bark, found near Peshawar, usually placed somewhere in 300–600 CE — gives a formula for the root of a number that is not a perfect square. Write N = A² + b, with A² the nearest square below:

The Bakhshali formula for approximate square roots The root of A squared plus b is approximated by A plus b over 2A, minus a correction term. √(A² + b) ≈ A + b 2A − (b / 2A)² 2 (A + b/2A) The first two terms are the obvious linear estimate. The third is a correction that removes most of its error — the same value one gets from a second Newton–Raphson step. WORKED: √41, taking A = 6, b = 5 first estimate 6 + 5/12 = 6.41666… correction − (5/12)² / (2 × 6.41666) = − 0.013528… result 6.403138… true √41 = 6.403124… — five correct figures from one step
The interest is not the numerical accuracy but the structure: an estimate plus an explicit correction term. That pattern — take a first approximation, then write down what to subtract — recurs throughout Indian mathematics, and reappears in section 20 as the correction terms that make Mādhava's series for π converge usefully.

11Sums of squares and cubesवर्गचितिघनः

Āryabhaṭa gives both closed forms in a single verse, using compound words as technical terms:

सैकसगच्छपदानां क्रमात् त्रिसंवर्गितस्य षष्ठोंऽशः ।
वर्गचितिघनः स भवेत् चितिवर्गो घनचितिघनश्च ॥

saika-sagaccha-padānāṃ kramāt trisaṃvargitasya ṣaṣṭhoṃ'śaḥ |
varga-citighanaḥ sa bhavet citivargo ghana-citighanaś ca ||

Āryabhaṭīya, Gaṇitapāda 22

The two summation formulas unpacked from Aryabhata's verse The sum of squares is n times n plus one times two n plus one, over six; the sum of cubes is the square of the sum of the first n numbers. VARGA-CITIGHANA — the sum of squares “the product of three quantities — the number of terms, that plus one, and the same increased by the number of terms — divided by six” 1² + 2² + 3² + … + n² = n (n+1) (2n+1) / 6 note: “(n+1) increased by n” is exactly 2n+1 GHANA-CITIGHANA — the sum of cubes “the square of the sum of the series of natural numbers” 1³ + 2³ + 3³ + … + n³ = [ n(n+1)/2 ]² that is, the square of 1+2+…+n Both results, stated exactly, in two lines of verse — and stated as constructions of the answer rather than as identities to be verified.

12Bīja-gaṇitaबीजगणितम्

Algebra's Indian name is बीजगणित — bīja, seed or element, plus gaṇita, the science of calculation: the calculus of elements.

Brahmagupta, in 628 CE, states the sign rules — the first unambiguous statement anywhere that negative quantities are numbers and can be operated on:

The sum of two positives is positive; of two negatives, negative; of a positive and a negative, their difference. … A negative minus zero is negative, a positive minus zero is positive, zero minus zero is zero.

He is thinking of them as debts and fortunes, and the physical interpretation does real work: it makes “subtracting a larger from a smaller” meaningful rather than forbidden, which is precisely the obstacle that kept negative roots out of European algebra until the seventeenth century.

Unknowns by colour

Several unknowns are distinguished by naming them for colours and using the first syllable of each — yāvat-tāvat (“as much as”) for the first, then kālaka (black), nīlaka (blue), pītaka (yellow), lohitaka (red). Symbolic algebra, with a mnemonic naming scheme.

Quadratics

Śrīdhara's rule for the quadratic — multiply through by four times the coefficient of the square, add the square of the linear coefficient, take the root — is the completion of the square, stated as a procedure and quoted by later authors under his name.

Surds and operations

Systematic treatment of irrational quantities: addition, multiplication and rationalisation of expressions in surds, with rules for when two surds can be combined into one.

13Pulveriser and cyclic methodकुट्टकः · चक्रवालम्

Two Indian algorithms for indeterminate equations, both of which were unmatched anywhere else for centuries.

Kuṭṭaka — “the pulveriser”

कुट्टकः

Āryabhaṭa's method for solving ax − by = c in integers. The name describes the procedure: the coefficients are repeatedly pulverised by mutual division — the Euclidean algorithm run in reverse to build up a solution.

It arose from astronomy. If two planets have periods of 4,320,000 and 1,577,917,500 units, when do they next coincide? That is an integer problem, and it needs exactly this.

Cakravāla — “the cyclic method”

चक्रवालम्

For x² − N y² = 1, the equation Europe would later call Pell's. Jayadeva describes it by the tenth century; Bhāskara II perfects it. The method cycles through a sequence of auxiliary equations, each closer to the target, and provably terminates.

It is faster than the continued-fraction method Lagrange proved correct in 1768.

The case N equals 61 Bhaskara II's solution to x squared minus 61 y squared equals one, with the same problem posed by Fermat as a challenge five centuries later. THE HARDEST SMALL CASE — N = 61 x = 1 766 319 049 y = 226 153 980 and 1766319049² − 61 × 226153980² = 1 exactly Bhāskara II, 1150 CE Gives the solution, by the cakravāla, as a worked example — one case among several in the Bījagaṇita. Fermat, 1657 CE Poses N = 61 as a challenge problem to the English mathematicians, precisely because it is so hard.
The smallest solution for N = 61 has ten digits — which is why Fermat chose it. That a twelfth-century Indian text carries the answer, obtained by a general method, is one of the sharpest single facts in the history of mathematics.

14Piṅgala's six operationsप्रत्ययाः

Chapter 6 introduced the light–heavy syllable as a binary digit. Chapter 8 of the Chandaḥ-śāstra then defines six operations on binary sequences, each with a name and an algorithm.

The six combinatorial operations of Pingala Prastara, sankhya, nasta, uddista, lagakriya and adhvayoga, with their modern equivalents. Prastāra · प्रस्तारः Generate all patterns of a given length, in order, as an array. = enumerate all 2ⁿ binary strings of length n Saṅkhyā · सङ्ख्या Find how many rows the array has, without building it. = compute 2ⁿ — and Piṅgala's method is repeated squaring Naṣṭa · नष्टम् Given a row number, produce its pattern directly. = convert an integer to binary Uddiṣṭa · उद्दिष्टम् Given a pattern, produce its row number directly. = convert binary to an integer Lagakriyā · लगक्रिया Count the patterns having exactly k light syllables. = the binomial coefficient ⁿCₖ — computed by the Meru Adhvayoga · अध्वयोगः Find the space the whole array occupies when written out. = a storage-size calculation, so you know how big a floor to clear
The last one is the most disarming: a formula for how much floor space you need if you want to lay out the entire array. Piṅgala is thinking about memory allocation.

Naṣṭa and uddiṣṭa, worked and cross-checked

The nasta and uddista algorithms applied and checked against each other Row 15 converts to the pattern 0111, and the pattern 0111 converts back to row 15. NAṢṬA — row 15 → its pattern Rule: if the number is even, write 1 and halve it; if odd, write 0, add one, and halve. 15 is odd → write 0, (15+1)/2 = 8 8 is even → write 1, 8/2 = 4 4 is even → write 1, 4/2 = 2 2 is even → write 1, 2/2 = 1 0111 four syllables: G L L L UDDIṢṬA — pattern 0111 → its row Rule: scanning from the right, start at 1; for each 1 double; for each 0 double and subtract one. start 1 rightmost 1 → 1 × 2 = 2 next 1 → 2 × 2 = 4 next 1 → 4 × 2 = 8 leftmost 0 → 8 × 2 − 1 = 15 row 15 ⇄
The two algorithms are exact inverses, which is what a pair of conversion routines has to be, and Piṅgala states them as a pair. This is a positional binary numeration system with both directions of conversion specified — around the third century BCE.

15The Meru-prastāraमेरुप्रस्तारः

To answer lagakriyā — how many patterns of n syllables have exactly k light ones — Piṅgala gives a construction: a triangle of numbers, each the sum of the two above it, named for the world-mountain Meru because of its shape.

The Meru-prastara, a triangle of binomial coefficients Rows of the triangle from 1 to the sixth row, each entry the sum of the two above it. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 3 + 3 = 6 — each entry is the sum of the two above ROW n n = 1n = 2n = 3n = 4n = 5n = 6 Row 4 reads 1 4 6 4 1: of the sixteen four-syllable patterns, one has no light syllable, four have one, six have two, four have three, one has four. Total 16 = 2⁴ — the saṅkhyā. Blaise Pascal published the same triangle in 1665. It is called Pascal's triangle in modern books; in India it is the Meru-prastāra, and it is roughly eighteen centuries older.
Perfumes, and a real application

Varāhamihira's Bṛhat-saṃhitā (c. 550 CE) computes the number of perfumes obtainable by choosing four ingredients from sixteen: 1,820. That is 16C4, and it is correct. The combinatorics is not being displayed for its own sake — it is being used to organise a manufacturing problem.

16Magic squaresभद्रगणितम्

The Indian tradition has a name for the study of magic squares — भद्रगणित, bhadra-gaṇita — and a speciality within it: the sarvatobhadra or pan-diagonal square, which is much harder to construct.

An ordinary magic square compared with a pan-diagonal one Both have rows, columns and main diagonals summing to 34, but only the second has all broken diagonals summing to 34 as well. ORDINARY MAGIC SQUARE · magic sum 34 12 3 6 13 14 5 4 11 7 16 9 2 1 10 15 8 rows ✓ columns ✓ main diagonals ✓ broken diagonal 3 + 4 + 2 + 1 = 10 ✗ PAN-DIAGONAL · SARVATOBHADRA · magic sum 34 10 3 13 8 5 16 2 11 4 9 7 14 15 6 12 1 rows ✓ columns ✓ main diagonals ✓ every broken diagonal too: 3 + 2 + 14 + 15 = 34 ✓ Also: every 2×2 block of adjacent cells sums to 34. Think of it as a torus Roll the square into a cylinder, then join the ends. Now every diagonal is a closed diagonal, and pan-diagonality is simply the statement that all of them sum alike. There are 384 pan-diagonal 4×4 squares on 1…16, all equivalent. A 4×4 square appears in an eleventh-century Jaina inscription at Khajuraho, and another dated 1480 CE in the Gwalior fort. Various 3×3 squares are attributed to the astronomer Garga around 100 BCE; Nāgārjuna's Kakṣapuṭa (c. 100 CE) gives a construction method; Nārāyaṇa Paṇḍita's Gaṇita-kaumudī (14th c.) gives a full theory. The first chapter of Ramanujan's notebooks is on magic squares.

Nāgārjuna's construction, in one line of verse

A generic pan-diagonal square, encoded in kaṭapayādi (chapter 6):

अर्क इन्दुनिधा नारी तेन लग्न विनाशनम्

arka indunidhā nārī tena lagna vināśanam

Decoding Nagarjuna's magic square mnemonic The verse decodes to sixteen entries, half literal and half complements of n; setting n to fifty gives a pan-diagonal square with magic sum one hundred. 1 · DECODED DIGITS 0 1 0 8 0 9 0 2 6 0 3 0 4 0 7 0 Eight cells carry a value; eight are marked 0. 2 · THE TEMPLATE — zeros become complements n−3 1 n−6 8 n−7 9 n−4 2 6 n−8 3 n−1 4 n−2 7 n−9 n is free. Choose it and the square appears. 3 · SET n = 50 47 1 44 8 43 9 46 2 6 42 3 49 4 48 7 41 CHECK rows 100 ✓ columns 100 ✓ diagonals 100 ✓ broken diags: 1+46+49+4 = 100 ✓ pan-diagonal magic sum = 2n = 100
A parametrised family of pan-diagonal magic squares, memorised as one line of verse. Choose n, decode, and you have a square with magic sum 2n — the construction, not just an instance.

17Jyā, and the word “sine”ज्या

The trigonometric functions used today are Indian in origin, and the English word “sine” is a chain of accidents that can be traced back link by link.

Greek astronomy worked with the chord of an arc. Indian astronomers worked instead with the half-chord, which is what we call the sine — and this is the more useful quantity, because it is what appears in the right triangle. The half-chord was called ज्या (jyā) or jīvā, the bowstring, because a chord across an arc looks like a strung bow.

Jya and kotijya, and the etymological chain to the word sine A circle with an arc, its half-chord jya and the complementary kotijya, and the sequence jya to jiva to jiba to jayb to sinus to sine. θ jyā = R sin θ koṭijyā = R cos θ R the arc Full chord = 2 R sin θ. The Indian innovation is to tabulate the half of it — because that is the quantity that actually enters a calculation. jyā · jīvāSanskrit — “bowstring” the half-chord jībaArabic transliteration a meaningless borrowed sound jaybmisread — Arabic script omits short vowels; jayb = “fold, pocket, bay” sinusLatin, 12th century a literal translation of “fold, bay” sineEnglish What survived the journey The mathematics came through intact. The meaning of the name did not: “bowstring” became “pocket” by way of an unvocalised script, and then got faithfully translated as such. koṭijyā, by the same route, is “cosine”.
Every time you write sin θ you are using a word that means “bay” because a twelfth-century Latin translator correctly rendered an Arabic word that was itself a misreading of a Sanskrit one.

18Āryabhaṭa's sine tableज्यार्धानि

Āryabhaṭa gives twenty-four sine values at intervals of 3° 45′, and — more remarkably — a rule for generating them that is a difference equation.

प्रथमाच्चापज्यार्धाद् यैरूनं खण्डितं द्वितीयार्धम् ।
तत्प्रथमज्यार्धांशैस्तैस्तैरूनानि शेषाणि ॥

Āryabhaṭīya, Gaṇitapāda 12

The recursion generating Aryabhata's sine differences Each sine difference is the previous one reduced by the running total divided by the first difference. THE RULE Let R₁ … R₂₄ be the twenty-four sines and δ₁ … δ₂₄ their successive differences, with δ₁ = R₁. δn+1 = δn − (δ₁ + δ₂ + … + δn ) / R₁ = δn − Rn / R₁ since the differences sum to the sine itself What this is, in modern terms The second difference of the sine is proportional to the sine itself, with a negative constant: Δ²R ≈ − (θ step)² · R — the discrete form of d²(sin x)/dx² = − sin x. Delambre, 1817 “The method is curious: it indicates a method of calculating the table of sines by means of their second differences … Here then is a method which the Indians possessed, and which is found neither among the Greeks nor among the Arabs.”
Delambre — Napoleon's astronomer, and no partisan of Indian science — is pointing at exactly the right thing. A table built by recursion from a differential relation is a different kind of object from a table built by interpolating measurements.

19The pursuit of πपरिधिमानम्

Successive Indian approximations to pi From the Sulba-sutras' one correct place to Madhava's eleven, with the method used at each stage. CORRECT DECIMAL PLACES Śulba-sūtras · c. 800 BCE3.0888… 1 geometrical, from altar constructions Jaina texts · c. 500 BCE√10 = 3.1623… 1 geometrical Āryabhaṭa · 499 CE62832 / 20000 = 3.1416 4 polygon doubling, to 4 × 2⁸ = 1024 sides Bhāskara II · 1150 CE3927 / 1250 = 3.1416 4 polygon doubling Mādhava · c. 1375 CE3.14159265359… 11 infinite series with end correction — a different kind of method entirely Ramanujan · 1914 CEmodular equations series later used to compute 10⁷ places and beyond The jump at Mādhava is the important one. Everything before it is polygon approximation, which converges slowly and can be pushed only so far by hand. A convergent infinite series is a change of method, not an improvement of the old one — and Nīlakaṇṭha, a century later, argues that no finite ratio can ever be exact.
Nīlakaṇṭha on irrationality

In the Āryabhaṭīya-bhāṣya, Nīlakaṇṭha asks why the ratio is always given as approximate, and answers that if a measure divides the diameter without remainder it cannot divide the circumference without remainder, and conversely — so however far one goes, only an approximate value can be stated. That is not a proof of irrationality, and he does not claim it is. It is a clear statement of what would need to be true, made in the fifteenth century.

20The Kerala schoolकेरलगणितम्

Between about 1350 and 1600, a lineage of mathematicians in a small area of Kerala derived infinite series for π, sine and cosine, developed correction terms to accelerate their convergence, and wrote out the demonstrations.

The principal series of the Kerala school The arctangent, sine and cosine series attributed to Madhava, with the correction term for pi. THE ARCTANGENT SERIES — and π as its special case π/4 = 1 − 1/3 + 1/5 − 1/7 + 1/9 − … discovered in Europe by Gregory and Leibniz, c. 1670 Mādhava states it, and states that it converges far too slowly to be used as it stands. THE CORRECTION TERM — the part that makes it usable Truncate after n terms and add a rational correction whose form Mādhava specifies. With it, a few dozen terms give more accuracy than millions of terms of the raw series — which is how the eleven-place value of chapter 6 was obtained. Recognising that a slowly convergent series needs an end correction, and finding one, is a more advanced act than finding the series. SINE sin x = x − x³/3! + x⁵/5! − … Newton, 1669 COSINE cos x = 1 − x²/2! + x⁴/4! − … Newton, 1669
The series are given in verse, with the coefficients encoded in the systems of chapter 6, and are attributed by later Kerala authors to Mādhava by name. The Yukti-bhāṣā of Jyeṣṭhadeva then does what no earlier Indian text had done at this length: it sets out the derivations in full, in Malayalam prose, including the summation of powers that a modern reader recognises as an integration.
Two claims to keep apart

Established: the series above were obtained in Kerala, with demonstrations, before their European discovery. This is not disputed by historians of mathematics.

Not established: that this work reached Europe and influenced Newton or Leibniz. There is a live and serious argument for transmission — Jesuit missionaries were in Kerala, collecting astronomical material, in the relevant period — but the documentary chain has not been closed. The honest statement is that transmission is possible and unproven. The Kerala achievement does not depend on it.

21Self-checkपरीक्षा

Check your reading

1 · Why did altar construction generate so much geometry?

Area preservation under change of shape, and scaling a figure by a given ratio, are exactly the problems the Śulba-sūtras solve.

2 · What does the word saviśeṣaḥ do at the end of Baudhāyana's √2 rule?

The Śulba texts distinguish carefully between exact constructions and approximations, and label the latter. This matters when assessing claims about the tradition's rigour.

3 · Naṣṭa and uddiṣṭa together amount to —

Row 15 → 0111 by naṣṭa; 0111 → row 15 by uddiṣṭa. A positional binary numeration with both conversions specified, around the third century BCE.

4 · What made Fermat's 1657 challenge — solve x² − 61y² = 1 — a hard one?

x = 1,766,319,049 and y = 226,153,980. The cakravāla finds it; trial and error does not.

5 · What is the strongest correct statement about the Kerala series and European calculus?

Both of the other two overstate what is known — in opposite directions. The first has no documentary chain; the second dismisses a serious and live scholarly argument.

Questions worth arguing about

Does Indian mathematics have proofs?

It depends what counts. If a proof must be a deductive derivation from stated axioms in the Euclidean manner, then mostly no — that programme was not attempted. If a proof is a demonstration that convinces a competent reader why a procedure works, then yes, abundantly: the upapatti tradition, the Yukti-bhāṣā above all, is exactly this. The interesting question is not which tradition had proofs but why the two developed such different conceptions of what a justification is for.

Why did the Kerala school not continue?

Nobody knows, and the honest answer includes several partial ones: the work was carried by a small lineage in a small region; it was written partly in Malayalam and partly in a verse idiom that limited its readership; the political and economic disruption of the coast from the sixteenth century onward was severe; and the mathematics had no institutional home comparable to a university. Any of these could be decisive, and there is no consensus.

Is the constructive style a limitation or an advantage?

Both, at different moments. It makes the tradition superb at algorithms and at problems that can be computed, and it produced results — the cakravāla, the sine recursion, the series with corrections — that a purely existential approach would not have reached. It is weaker at the questions that need a proof of impossibility, which is a genre the Greek tradition owned. Notably, the two styles are now much closer than they were: computational mathematics has made the constructive question respectable again.

22Glossaryशब्दकोशः

IASTDevanāgarīSense
bhadra-gaṇitaभद्रगणितThe study of magic squares.
bhujā, koṭi, karṇaभुजा, कोटि, कर्णThe two sides and the hypotenuse of a right triangle.
bīja-gaṇitaबीजगणितAlgebra: “the calculus of elements”.
cakravālaचक्रवालThe cyclic method for x² − Ny² = 1.
citiचितिA pile or layer; hence a summed series.
jyā, koṭijyāज्या, कोटिज्याSine and cosine — literally bowstring and its complement.
kuṭṭakaकुट्टक“Pulveriser”: the algorithm for linear indeterminate equations.
lagakriyāलगक्रियाCounting patterns with a given number of light syllables; binomial coefficients.
meru-prastāraमेरुप्रस्तारThe triangle of binomial coefficients.
naṣṭa / uddiṣṭaनष्ट / उद्दिष्टIndex-to-pattern and pattern-to-index conversion.
prastāraप्रस्तारThe systematic array of all patterns of a given length.
śaṅkuशङ्कुA gnomon: a vertical rod whose shadow is measured.
sarvatobhadraसर्वतोभद्रA pan-diagonal magic square.
śulbaशुल्बA cord; hence the geometry of cord construction.
upapattiउपपत्तिDemonstration; the justification of a procedure.
varga / ghanaवर्ग / घनSquare and cube; also the alternating place-classes in root extraction.
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