Thirty-three sūtras in six chapters encode a modern formula compendium – circular and hyperbolic identities, binomial and logarithm laws, differential and integral rules and Eulerian special functions – in classical Sanskrit meters with Pāṇinian terminology.
सार-असार-विवेकः · छन्दोयोजना

सार-असार-विवेकः (Ontological Mapping and Filtration Paradigm)

The primary source document 1 is a comprehensive formula sheet encompassing Trigonometry, Hyperbolic Functions, Algebra, Logarithms, Special Functions (Gamma and Beta), and Differential and Integral Calculus. To codify this text effectively, an ontological filter must be applied to separate universal axioms from transient, specific instances.

रक्षणीयांशः (Criteria for Retention)

The elements selected for verse codification represent invariant laws, geometric identities, and functional properties:

  1. त्रिकोणमितिः (Trigonometric Identities): Generalized sum and difference formulas for sine and cosine, multiple angle formulas, transformation identities, and the foundational Pythagorean-equivalent identity for hyperbolic functions.1
  2. बीजगणितम् (Algebraic Binomial Theorem): The generalized expansion of a binomial polynomial, utilizing combinatorics, abstracting specific cubic expansions as mere corollaries.8
  3. लघुगणकः (Logarithmic Axioms): The translation of multiplicative and divisional functions into additive and subtractive functions.10
  4. चलनकलनम् (Rules of Differential Calculus): The Product Rule, Quotient Rule, and Chain Rule, along with standard derivatives of transcendental functions.12
  5. समाकलनम् (Rules of Integral Calculus): Integration by parts, standard anti-derivatives, and the properties of definite integrals regarding even and odd functions.14
  6. विशेषफलनम् (Eulerian Special Functions): The precise integral definitions of the Gamma and Beta functions.16

त्याज्यांशः (Criteria for Omission)

  1. विशिष्ट-बिन्दवः (Specific Numerical Coordinates): Coordinate pairs from the unit circle, such as the exact numerical values for specific radians.1 These are tabular data points rather than abstract rules and are traditionally mapped into separate numeric tables (ज्या-सारिणी) rather than axiomatic verses.17
  2. स्वतःसिद्ध-नियमाः (Self-evident Arithmetic Derivatives): Formulas representing the derivative of a constant as zero 1, which are self-evident boundary conditions of the differential operator and philosophically understood as unmoving (अचल).
  3. पुनरुक्त-विस्ताराः (Redundant Expansions): Specific instances of expansions, which are automatically satisfied by the codified binomial theorem and do not require independent verse allocation.

छन्दोयोजना-प्रकरण-विभागश्च (Structural Estimation of Chapters)

The codified text is systematically divided into six specialized chapters (प्रकरण), requiring a carefully estimated number of verses to capture the retained concepts without redundancy.

प्रकरणम् Subject Area श्लोक-सङ्ख्या (Estimated Verses)
त्रिकोणमिति-प्रकरणम् Trigonometry ८
अतिपरवलयिक-प्रकरणम् Hyperbolic Functions ३
बीजगणित-प्रकरणम् Algebra & Logarithms ४
अवकलन-प्रकरणम् Differential Calculus १०
समाकलन-प्रकरणम् Integral Calculus ७
विशेषफलन-प्रकरणम् Gamma & Beta Functions २
योगः (Total) Complete Codification ३४
धातुपाठ-शब्दनिर्मितिः

धातुपाठ-शब्दनिर्मितिः (Lexicography and Epistemology)

To encode modern formulas into Sanskrit, foreign names and modern operational concepts are derived from the verbal roots (धातु) outlined in Pāṇini’s धातुपाठ.18 Existing terms found in classical astronomy (ज्योतिष) and mathematics (गणित) are repurposed where historically continuous.21

कर्म-सङ्केताः (Mathematical Operations and Operators)

  • अवकलनम् / चलन-कलनम्: Used for Differentiation. Derived from the root कल् (to count/calculate) with the prefix अव (downward/reduction), capturing the reduction of degree in polynomials.23
  • समाकलनम्: Used for Integration. Derived from सम् (together) + आ (fully) + कल् (count), signifying the summation of infinitesimals.25
  • ज्या, कोटिज्या, स्पर्शज्या: Used for Sine, Cosine, and Tangent. ज्या refers to the bowstring (half-chord), while कोटि refers to the perpendicular complement.26
  • अतिपरवलयिकम्: Used for Hyperbolic. Formed from अति (beyond) and परवलय (parabola).7
  • लघुगणकः / छेदा: Used for Logarithm. From लघु (small) and गण (count), referring to the scaling down of large multiplicative values into additive ones.10

नाम-संस्कृतीकरणम् (Sanskritization of Proper Nouns)

  • उल्लारः (Euler): Phonetically and conceptually adapted. Derived from the prefix उद् (upwards/exalted) and the root लृ (to move or attain), yielding उल्लारः: “He who attains exalted heights”.28
  • नापीरः (Napier): Derived from the root पॄ (to fill/complete) and न (negation) combined with आ, yielding नापीरः: “That which bounds the unbounded” (referencing logarithmic asymptotes).30
  • गामा (Gamma): Derived from the root गम् (to go/extend). The Gamma function extends the factorial into the continuous domain; hence गामा (that which extends).32
  • विटः (Beta): Derived from the root विट् (to sound/divide). The Beta function deals with fractional partitions of the integral space between zero and one.33
  • पाशकलः (Pascal): Derived from पाश (binding/matrix) and कल (calculate), referring to the binding matrix of combinatorial coefficients in the binomial theorem.34
  • फेनमनः (Feynman): Formed from फेन (foam - analogous to quantum vacuum fluctuations) and मन (mind).
  • उद्गारलयतानः (Edgar Leitan): Formed from उद्गार (expression), लय (rhythm), and तान (extension).
## प्रथमं प्रकरणम् – त्रिकोणमिति-प्रकरणम् (Trigonometry) – mixed meters

Sūtras: 1–8 | Theme: Sine, cosine, tangent sums; double, triple, product rules

सूत्रम् 1 (अनुष्टुभ्)

सूत्रम्: $\sin(-\theta) = -\sin(\theta)$, $\cos(-\theta) = \cos(\theta)$

ज्या कोणे ऋणयुक्ते तु
ऋणज्या सम्प्रजायते ।
कोटिज्या ऋणकोणेऽपि
धनरूपा प्रतिष्ठिता ॥ १ ॥
पदच्छेदः · शब्दार्थः · व्याकरणम्

पदच्छेदः: ज्या कोणे ऋण-युक्ते तु / ऋण-ज्या सम्प्रजायते / कोटि-ज्या ऋण-कोणे अपि / धन-रूपा प्रतिष्ठिता //

शब्दार्थः:

ज्या (Sine) कोणे (in angle) ऋण-युक्ते (joined with negative) तु (indeed) / ऋण-ज्या (negative sine) सम्प्रजायते (is produced) / कोटि-ज्या (Cosine) ऋण-कोणे (in negative angle) अपि (even) / धन-रूपा (positive form) प्रतिष्ठिता (remains established) //

व्याकरणम्:

‘सम्प्रजायते’ uses सम् + प्र + जन् + लट्. The verse distinguishes the odd function property of sine from the even function property of cosine.

सूत्रम् 2 (उपजाति (११ अक्षराणि))

सूत्रम्: $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$

ज्यायोगसूत्रं शृणु साकमेव
ज्या पूर्वका कोटिरथ द्वितीया ।
कोटी च पूर्वा परज्यायुता च
चिह्नं समानं खलु मध्यभागे ॥ २ ॥
पदच्छेदः · शब्दार्थः · व्याकरणम्

पदच्छेदः: ज्या-योग-सूत्रम् शृणु साकम् एव / ज्या पूर्वका कोटिः अथ द्वितीया / कोटी च पूर्वा पर-ज्या-युता च / चिह्नम् समानम् खलु मध्य-भागे //

शब्दार्थः:

ज्या-योग-सूत्रम् (Formula for the addition of sines) शृणु (listen) साकम् एव (all at once) / ज्या पूर्वका (Sine of the first) कोटिः अथ द्वितीया (and then cosine of the second) / कोटी च पूर्वा (And cosine of the first) पर-ज्या-युता च (joined with the sine of the latter) / चिह्नम् समानम् (The sign is the same) खलु मध्य-भागे (indeed in the middle portion) //

व्याकरणम्:

‘साकम्’ means simultaneously. ‘चिह्नम् समानम्’ dictates that the mathematical operator ($+$ or $-$) between the terms mirrors the operator within the function argument.

सूत्रम् 3 (अनुष्टुभ्)

सूत्रम्: $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$

कोटियोगे द्वयोः कोटी
ज्याद्वयं च प्रयुज्यते ।
योगे हीनं वियोगे च
युक्तं चिह्नं विलोमतः ॥ ३ ॥
पदच्छेदः · शब्दार्थः · व्याकरणम्

पदच्छेदः: कोटि-योगे द्वयोः कोटी / ज्या-द्वयम् च प्रयुज्यते / योगे हीनम् वियोगे च / युक्तम् चिह्नम् विलोमतः //

शब्दार्थः:

कोटि-योगे (In the addition of cosines) द्वयोः कोटी (cosines of both) / ज्या-द्वयम् च (and the two sines) प्रयुज्यते (are employed) / योगे हीनम् (in addition, it is minus) वियोगे च युक्तम् (and in subtraction, it is plus) / चिह्नम् विलोमतः (the sign is inverted) //

व्याकरणम्:

‘विलोमतः’ (inversely) strictly codifies the operational inversion unique to the cosine sum formula.

सूत्रम् 4 (वसन्ततिलका (१४ अक्षराणि))

सूत्रम्: $\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$

स्पर्शज्या-योग-समये लवरूप-भागे
स्पर्शज्या-युग्म-सहितं खलु चिह्न-तुल्यम् ।
हारस्य भाग इह रूप-वियुक्त-घातो
चिह्नं विलोममिति पण्डित-सम्प्रदिष्टम् ॥ ४ ॥
पदच्छेदः · शब्दार्थः · व्याकरणम्

पदच्छेदः: स्पर्शज्या-योग-समये लव-रूप-भागे / स्पर्शज्या-युग्म-सहितम् खलु चिह्न-तुल्यम् / हारस्य भागे इह रूप-वियुक्त-घातः / चिह्नम् विलोमम् इति पण्डित-सम्प्रदिष्टम् //

शब्दार्थः:

स्पर्शज्या-योग-समये (At the time of adding tangents) लव-रूप-भागे (in the numerator part) / स्पर्शज्या-युग्म-सहितम् (joined with the pair of tangents) खलु चिह्न-तुल्यम् (indeed with identical sign) / हारस्य भागे (in the denominator part) इह (here) रूप-वियुक्त-घातः (one minus the product) / चिह्नम् विलोमम् (the sign is inverted) इति पण्डित-सम्प्रदिष्टम् (thus is taught by scholars) //

व्याकरणम्:

लव (numerator) and हार (denominator) are standard terms in Indian mathematics.36 रूप conventionally means the number one in classical algebra.

सूत्रम् 5 (अनुष्टुभ्)

सूत्रम्: $\sin 2A = 2 \sin A \cos A$, $\cos 2A = \cos^2 A - \sin^2 A$

द्विगुणे कोणमाने तु
ज्याकोटी-घात-युग्मकम् ।
कोटि-ज्यायाः फलं तत्र
कोटिवर्गात् ऋणं ज्या-कृत् ॥ ५ ॥
पदच्छेदः · शब्दार्थः · व्याकरणम्

पदच्छेदः: द्वि-गुणे कोण-माने तु / ज्या-कोटी-घात-युग्मकम् / कोटि-ज्यायाः फलम् तत्र / कोटि-वर्गात् ऋणम् ज्या-कृत् //

शब्दार्थः:

द्वि-गुणे कोण-माने तु (In a double angle measure indeed) / ज्या-कोटी-घात-युग्मकम् (the pair of the product of sine and cosine - for sine) / कोटि-ज्यायाः फलम् तत्र (the result of cosine there) / कोटि-वर्गात् ऋणम् ज्या-कृत् (is the square of sine subtracted from the square of cosine) //

व्याकरणम्:

‘कृत्’ is used interchangeably with ‘वर्ग’ to mean square.

सूत्रम् 6 (शालिनी (१५ अक्षराणि))

सूत्रम्: $\sin 3A = 3 \sin A - 4 \sin^3 A$, $\cos 3A = 4 \cos^3 A - 3 \cos A$

त्रिगुणिते खलु कोणे ज्या त्रिघ्नी ज्या विहीना
चतुर-गुणित-ज्यायाः घनरूपेण नूनम् ।
चतुर-गुणित-कोटी-घनतः त्र्याहत-कोटिः
अपहृत-परिमाणा कोटि-रूपं त्रिभागे ॥ ६ ॥
पदच्छेदः · शब्दार्थः · व्याकरणम्

पदच्छेदः: त्रि-गुणिते खलु कोणे ज्या त्रि-घ्नी ज्या विहीना / चतुर-गुणित-ज्यायाः घन-रूपेण नूनम् / चतुर-गुणित-कोटी-घनतः त्रि-आहत-कोटिः / अपहृत-परिमाणा कोटि-रूपम् त्रि-भागे //

शब्दार्थः:

त्रि-गुणिते खलु कोणे (In a tripled angle indeed) ज्या (sine) त्रि-घ्नी ज्या (sine multiplied by three) विहीना (minus) / चतुर-गुणित-ज्यायाः घन-रूपेण (four times the cube of sine) नूनम् (certainly) / चतुर-गुणित-कोटी-घनतः (from four times the cube of cosine) त्रि-आहत-कोटिः (cosine multiplied by three) / अपहृत-परिमाणा (measure subtracted) कोटि-रूपम् (is the form of cosine) त्रि-भागे (in the triple part) //

व्याकरणम्:

घ्नी and आहत signify multiplication. घन indicates the mathematical cube.

सूत्रम् 7 (अनुष्टुभ्)

सूत्रम्: $\sin A + \sin B = 2 \sin(\frac{A+B}{2}) \cos(\frac{A-B}{2})$

ज्यायोगे द्वे गुणे ज्या स्याद्
अर्ध-योग-प्रमाणतः ।
कोटि-रूपेण सङ्गुण्या
अर्ध-भेद-प्रमाणतः ॥ ७ ॥
पदच्छेदः · शब्दार्थः · व्याकरणम्

पदच्छेदः: ज्या-योगे द्वे गुणे ज्या स्याद् / अर्ध-योग-प्रमाणतः / कोटि-रूपेण सङ्गुण्या / अर्ध-भेद-प्रमाणतः //

शब्दार्थः:

ज्या-योगे (In the sum of sines) द्वे गुणे (two times) ज्या स्याद् (sine should be) / अर्ध-योग-प्रमाणतः (of the measure of half the sum) / कोटि-रूपेण सङ्गुण्या (multiplied by the cosine form) / अर्ध-भेद-प्रमाणतः (of the measure of half the difference) //

व्याकरणम्:

‘सङ्गुण्या’ uses सम् + गुण् + ण्यत् indicating multiplication.

सूत्रम् 8 (अनुष्टुभ्)

सूत्रम्: $\sin A \cos B = \frac{1}{2}$

ज्या-कोटी-गुणने नूनम्
अर्ध-भागो विधीयते ।
योग-भेद-प्रमाणानां
ज्या-योगस्य फलं स्मृतम् ॥ ८ ॥
पदच्छेदः · शब्दार्थः · व्याकरणम्

पदच्छेदः: ज्या-कोटी-गुणने नूनम् / अर्ध-भागः विधीयते / योग-भेद-प्रमाणानाम् / ज्या-योगस्य फलम् स्मृतम् //

शब्दार्थः:

ज्या-कोटी-गुणने नूनम् (In the multiplication of sine and cosine certainly) / अर्ध-भागः विधीयते (a half portion is prescribed) / योग-भेद-प्रमाणानाम् (of the measures of sum and difference) / ज्या-योगस्य फलम् स्मृतम् (the result of the sum of sines is remembered) //

व्याकरणम्:

The genitive ‘प्रमाणानाम्’ connects to the parameters of the newly formed sine functions.

—

द्वितीयं प्रकरणम् - अतिपरवलयिक-प्रकरणम् (Hyperbolic Formulations)

This section translates the definitions and primary identity of hyperbolic trigonometry, relying on the exponential base $e$.1

## द्वितीयं प्रकरणम् – अतिपरवलयिक-प्रकरणम् (Hyperbolic functions) – mixed meters

Sūtras: 9–10 | Theme: Definitions and fundamental identity

सूत्रम् 9 (उपजाति)

सूत्रम्: $\sinh A = \frac{e^A - e^{-A}}{2}$, $\cosh A = \frac{e^A + e^{-A}}{2}$

उल्लार-घाते धन-रूप-युक्ते
ऋणात्मक-घ्नेन विहीनरूपे ।
अर्धीकृते सा अतिज्या प्रदिष्टा
योगे तयोः कोटि-अतिज्या-रूपम् ॥ ९ ॥
पदच्छेदः · शब्दार्थः · व्याकरणम्

पदच्छेदः: उल्लार-घाते धन-रूप-युक्ते / ऋण-आत्मक-घ्नेन विहीन-रूपे / अर्धीकृते सा अति-ज्या प्रदिष्टा / योगे तयोः कोटि-अतिज्या-रूपम् //

शब्दार्थः:

उल्लार-घाते (In the power of Euler’s number $e$) धन-रूप-युक्ते (joined with a positive form) / ऋण-आत्मक-घ्नेन (by the negative power) विहीन-रूपे (in the subtracted form) / अर्धीकृते (when halved) सा अति-ज्या प्रदिष्टा (that is taught as hyperbolic sine) / योगे तयोः (in the addition of those two) कोटि-अतिज्या-रूपम् (is the form of hyperbolic cosine) //

व्याकरणम्:

उल्लार (Euler) acts as the base.28 ‘अतिज्या’ is a metrically convenient abbreviation for अतिपरवलयिक-ज्या. ‘अर्धीकृते’ uses the च्वि suffix indicating transformation into a half.

सूत्रम् 10 (अनुष्टुभ्)

सूत्रम्: $\cosh^2 x - \sinh^2 x = 1$

अतिपरवलये कोटि-
वर्गान्तरित-ज्याकृतिः ।
सर्वदा एकरूपैव
सूत्रमेतद् प्रकीर्तितम् ॥ १० ॥
पदच्छेदः · शब्दार्थः · व्याकरणम्

पदच्छेदः: अतिपरवलये कोटि-वर्ग-ान्तरित-ज्या-कृतिः / सर्वदा एक-रूपा एव / सूत्रम् एतद् प्रकीर्तितम् //

शब्दार्थः:

अतिपरवलये (In the hyperbola) कोटि-वर्ग (the square of the cosine) अन्तरित (differenced by) ज्या-कृतिः (the square of the sine) / सर्वदा एक-रूपा एव (is always equal to the form of one) / सूत्रम् एतद् प्रकीर्तितम् (this formula is proclaimed) //

व्याकरणम्:

‘कृतिः’ and ‘वर्ग’ act as synonyms for mathematical squaring. The prefix ‘अतिपरवलये’ specifies that this rule applies exclusively to hyperbolic functions.7

—

तृतीयं प्रकरणम् - बीजगणित-लघुगणक-प्रकरणम् (Algebra & Logarithms)

This chapter encapsulates combinatorial expansions and the transformation properties of logarithms.1

## तृतीयं प्रकरणम् – बीजगणित-लघुगणक-प्रकरणम् (Algebra and logarithms) – mixed meters

Sūtras: 11–14 | Theme: Binomial, cubes, Napier laws

सूत्रम् 11 (आर्या (मात्रा-छन्दः))

सूत्रम्: $(x+y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k$

द्विपदस्य घातविस्तृतिः मेरुप्रस्तार-पाशकल-युक्ता ।
आद्यस्य हीनघातोऽन्त्यस्य च वृद्ध्या समं भवेत् क्रमशः ॥ ११ ॥
पदच्छेदः · शब्दार्थः · व्याकरणम्

पदच्छेदः: द्वि-पदस्य घात-विस्तृतिः मेरु-प्रस्तार-पाशकल-युक्ता / आद्यस्य हीन-घातः अन्त्यस्य च वृद्ध्या समम् भवेत् क्रमशः //

शब्दार्थः:

द्वि-पदस्य घात-विस्तृतिः (The power expansion of a binomial) मेरु-प्रस्तार-पाशकल-युक्ता (is joined with the Pascal coefficients of the Meru-Prastāra) / आद्यस्य हीन-घातः (the descending power of the first variable) अन्त्यस्य च वृद्ध्या (and with the increasing power of the last) समम् भवेत् क्रमशः (becomes equal sequentially) //

व्याकरणम्:

The translation leverages the ancient Indian combinatorial structure ‘मेरु-प्रस्तारः’ (the geometric mountain), combined with ‘पाशकल’ (Pascal).34 The verse explicitly dictates the descending polynomial degree of $x$ and the ascending degree of $y$.

सूत्रम् 12 (अनुष्टुभ्)

सूत्रम्: $x^3 \pm y^3 = (x \pm y)(x^2 \mp xy + y^2)$

घनानां योग-भेदौ तु
मूल-योग-वियोगकैः ।
वर्ग-योग-विपरीत-
घात-घ्नैश्च प्रकीर्तितौ ॥ १२ ॥
पदच्छेदः · शब्दार्थः · व्याकरणम्

पदच्छेदः: घनानाम् योग-भेदौ तु / मूल-योग-वियोगकैः / वर्ग-योग-विपरीत-घात-घ्नैः च प्रकीर्तितौ //

शब्दार्थः:

घनानाम् योग-भेदौ तु (The sum and difference of cubes indeed) / मूल-योग-वियोगकैः (by the sum and difference of the roots) / वर्ग-योग-विपरीत-घात-घ्नैः च (and multiplied by the sum of squares and the inverse product) प्रकीर्तितौ (are proclaimed) //

व्याकरणम्:

‘विपरीत-घात’ perfectly encapsulates the $\mp xy$ term, ensuring the sign opposite to the binomial root factor is applied.

सूत्रम् 13 (अनुष्टुभ्)

सूत्रम्: $\log(ab) = \log a + \log b$, $\log(a/b) = \log a - \log b$

लघुगणके पदघातो
योगतुल्यः प्रजायते ।
भागो वियोगतुल्यः स्याद्
इति नापीरनिश्चयः ॥ १३ ॥
पदच्छेदः · शब्दार्थः · व्याकरणम्

पदच्छेदः: लघु-गणके पद-घातः / योग-तुल्यः प्रजायते / भागः वियोग-तुल्यः स्याद् / इति नापीर-निश्चयः //

शब्दार्थः:

लघु-गणके (In the logarithm) पद-घातः (the product of terms) योग-तुल्यः प्रजायते (is born equal to their sum) / भागः वियोग-तुल्यः स्याद् (division becomes equal to their difference) / इति नापीर-निश्चयः (thus is Napier’s certainty) //

व्याकरणम्:

The reference to ‘नापीर’ attributes the law to John Napier, fulfilling the criteria for Sanskritizing the names of modern mathematical progenitors through grammatical derivation.30

सूत्रम् 14 (अनुष्टुभ्)

सूत्रम्: $\log 1 = 0$, $a = e^{\log a}$

एकस्य लघुगणकं
शून्यमेव सदैव हि ।
उल्लारे लघु-घातेन
मूल-सङ्ख्या प्रजायते ॥ १४ ॥
पदच्छेदः · शब्दार्थः · व्याकरणम्

पदच्छेदः: एकस्य लघु-गणकम् / शून्यम् एव सदैव हि / उल्लारे लघु-घातेन / मूल-सङ्ख्या प्रजायते //

शब्दार्थः:

एकस्य लघु-गणकम् (The logarithm of one) शून्यम् एव सदैव हि (is always zero indeed) / उल्लारे (In Euler’s number) लघु-घातेन (by the logarithmic power) मूल-सङ्ख्या प्रजायते (the original number is produced) //

व्याकरणम्:

‘उल्लार’ explicitly references the base $e$.29

—

चतुर्थं प्रकरणम् - अवकलन-प्रकरणम् (Differential Calculus)

Encoding the core operative rules of Leibniz and Newton requires leveraging the Sanskrit concept of अवकलन (differentiation, literally “downward counting” or “reduction”).12

## चतुर्थं प्रकरणम् – अवकलन-प्रकरणम् (Differential calculus) – mixed meters

Sūtras: 15–23 | Theme: Product, quotient, chain rules and standard derivatives

सूत्रम् 15 (उपजाति (११ अक्षराणि))

सूत्रम्: $\frac{d}{dx}[u \cdot v] = u \frac{dv}{dx} + v \frac{du}{dx}$

घाते फलानां खलु भेदकाले
पूर्वं स्थिरं चान्तिम-भेदकञ्च ।
अन्त्यं स्थिरं पूर्व-विभेदकञ्च
फलं तयोर्योगसमं प्रदिष्टम् ॥ १५ ॥
पदच्छेदः · शब्दार्थः · व्याकरणम्

पदच्छेदः: घाते फलानाम् खलु भेद-काले / पूर्वम् स्थिरम् च अन्तिम-भेदकम् च / अन्त्यम् स्थिरम् पूर्व-विभेदकम् च / फलम् तयोः योग-समम् प्रदिष्टम् //

शब्दार्थः:

घाते फलानाम् (In the product of functions) खलु भेद-काले (indeed during the time of differentiation) / पूर्वम् स्थिरम् (the first is constant) च अन्तिम-भेदकम् च (and the derivative of the last) / अन्त्यम् स्थिरम् (the last is constant) पूर्व-विभेदकम् च (and the derivative of the first) / फलम् तयोः योग-समम् प्रदिष्टम् (the result is taught as the sum of those two) //

व्याकरणम्:

A structurally precise transcription of the Leibniz product rule. ‘भेदक’ denotes the differential operator applied to a given function.

सूत्रम् 16 (वसन्ततिलका (१४ अक्षराणि))

सूत्रम्: $\frac{d}{dx}[\frac{u}{v}] = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}$

भागे हरः स्थिरतरो लव\-भेद-निघ्नो
हीनं लवेन हर\-भेदक-ताडितेन ।
सर्वं हृतं हर\-कृतेन फलावकालः
सूत्रं पुरातनविदैः कथितं स्फुटं तत् ॥ १६ ॥
पदच्छेदः · शब्दार्थः · व्याकरणम्

पदच्छेदः: भागे हरः स्थिर-तरः लव-भेद-निघ्नः / हीनम् लवेन हर-भेदक-ताडितेन / सर्वम् हृतम् हर-कृतेन फल-अवकालः / सूत्रम् पुरातन-विदैः कथितम् स्फुटम् तत् //

शब्दार्थः:

भागे (In quotient) हरः स्थिर-तरः (the denominator is constant) लव-भेद-निघ्नः (multiplied by the derivative of the numerator) / हीनम् लवेन (minus the numerator) हर-भेदक-ताडितेन (multiplied by the derivative of the denominator) / सर्वम् हृतम् हर-कृतेन (all is divided by the square of the denominator) फल-अवकालः (is the differentiation of the function) / सूत्रम् पुरातन-विदैः कथितम् स्फुटम् तत् (that clear formula is told by the knowers of old) //

व्याकरणम्:

Utilizing traditional Indian fractional vocabulary–हर for denominator and लव for numerator.36 The terms निघ्न and ताडित both mean “struck” or “multiplied by”.

सूत्रम् 17 (अनुष्टुभ्)

सूत्रम्: $\frac{d}{dx}[f(g(x))] = f’(g(x)) \cdot g’(x)$

फलानां फलने भेदो
बाह्यभेदोऽन्तरस्थितैः ।
अन्तःस्थ-भेद-गुणितः
शृङ्खला-नियमः स्मृतः ॥ १७ ॥
पदच्छेदः · शब्दार्थः · व्याकरणम्

पदच्छेदः: फलानाम् फलने भेदः / बाह्य-भेदः अन्तर-स्थितैः / अन्तःस्थ-भेद-गुणितः / शृङ्खला-नियमः स्मृतः //

शब्दार्थः:

फलानाम् फलने (In a function of functions) भेदः (the derivative) / बाह्य-भेदः अन्तर-स्थितैः (is the outer derivative with the inner situated within it) / अन्तःस्थ-भेद-गुणितः (multiplied by the derivative of the inner) / शृङ्खला-नियमः स्मृतः (this is known as the Chain Rule) //

व्याकरणम्:

‘शृङ्खला’ is the exact Sanskrit equivalent for “chain”. The text captures the recursive nested nature of composite function differentiation.12

सूत्रम् 18 (अनुष्टुभ्)

सूत्रम्: $\frac{d}{dx}(\sin ax) = a \cos ax$, $\frac{d}{dx}(\cos ax) = -a \sin ax$

ज्या-भेदः कोटि-रूपः स्यात्
गुणक-गुणितस्तथा ।
कोटि-भेदो ऋणज्या स्यात्
अङ्क-ताडित-रूपकः ॥ १८ ॥
पदच्छेदः · शब्दार्थः · व्याकरणम्

पदच्छेदः: ज्या-भेदः कोटि-रूपः स्यात् / गुणक-गुणितः तथा / कोटि-भेदः ऋण-ज्या स्यात् / अङ्क-ताडित-रूपकः //

शब्दार्थः:

ज्या-भेदः (The derivative of sine) कोटि-रूपः स्यात् (becomes the form of cosine) / गुणक-गुणितः तथा (and multiplied by the coefficient) / कोटि-भेदः ऋण-ज्या स्यात् (the derivative of cosine becomes negative sine) / अङ्क-ताडित-रूपकः (in the form multiplied by the constant) //

व्याकरणम्:

The chain rule application of multiplying by ‘$a$’ (the inner constant) is handled by ‘गुणक-गुणितः’.

सूत्रम् 19 (अनुष्टुभ्)

सूत्रम्: $\frac{d}{dx}(\tan ax) = a \sec^2 ax$

स्पर्शज्या-भेदने नूनं
छेदिका-वर्ग उच्यते ।
चर-सङ्ख्या-गुणाङ्केन
गुणितं तत् फलं स्मृतम् ॥ १९ ॥
पदच्छेदः · शब्दार्थः · व्याकरणम्

पदच्छेदः: स्पर्शज्या-भेदने नूनम् / छेदिका-वर्गः उच्यते / चर-सङ्ख्या-गुण-अङ्केन / गुणितम् तत् फलम् स्मृतम् //

शब्दार्थः:

स्पर्शज्या-भेदने नूनम् (In the differentiation of tangent certainly) / छेदिका-वर्गः उच्यते (the square of secant is spoken of) / चर-सङ्ख्या-गुण-अङ्केन (by the multiplier coefficient of the variable) / गुणितम् तत् फलम् स्मृतम् (multiplied, that result is remembered) //

व्याकरणम्:

‘छेदिका’ represents the secant line crossing the circle.39

सूत्रम् 20 (अनुष्टुभ्)

सूत्रम्: $\frac{d}{dx}(\sin^{-1}x) = \frac{1}{\sqrt{1-x^2}}$

विलोम-ज्या-प्रभेदे तु
रूपं वर्ग-विहीनकम् ।
मूल-रूपेण यद्-दृष्टं
तस्य व्युत्क्रम-रूपकम् ॥ २० ॥
पदच्छेदः · शब्दार्थः · व्याकरणम्

पदच्छेदः: विलोम-ज्या-प्रभेदे तु / रूपम् वर्ग-विहीनकम् / मूल-रूपेण यद्-दृष्टम् / तस्य व्युत्क्रम-रूपकम् //

शब्दार्थः:

विलोम-ज्या-प्रभेदे तु (In the derivative of the inverse sine indeed) / रूपम् वर्ग-विहीनकम् (one minus the square) / मूल-रूपेण यद्-दृष्टम् (that which is seen in the form of a square root) / तस्य व्युत्क्रम-रूपकम् (is the reciprocal form of that) //

व्याकरणम्:

‘व्युत्क्रम’ implies the reciprocal ($1 / x$), effectively codifying the fractional denominator geometry.

सूत्रम् 21 (अनुष्टुभ्)

सूत्रम्: $\frac{d}{dx}(e^{ax}) = a e^{ax}$, $\frac{d}{dx}(a^x) = a^x \log a$

उल्लारे भेदने रूपं
तदेव गुणकाहतम् ।
अन्य-मूले तु तद्-रूपं
लघुगणक-ताडितम् ॥ २१ ॥
पदच्छेदः · शब्दार्थः · व्याकरणम्

पदच्छेदः: उल्लारे भेदने रूपम् / तद् एव गुणक-आहतम् / अन्य-मूले तु तद्-रूपम् / लघु-गणक-ताडितम् //

शब्दार्थः:

उल्लारे भेदने (In the differentiation of Euler’s $e^x$) रूपम् तद् एव (the form remains exactly that) गुणक-आहतम् (multiplied by the coefficient) / अन्य-मूले तु (but in another base $a^x$) तद्-रूपम् (that form) / लघु-गणक-ताडितम् (is multiplied by the logarithm) //

व्याकरणम्:

Differentiates the unique property of the natural exponential from general exponentials.

सूत्रम् 22 (अनुष्टुभ्)

सूत्रम्: $\frac{d}{dx}(\log x) = \frac{1}{x}$, $\frac{d}{dx}(\log f(x)) = \frac{f’(x)}{f(x)}$

लघुगणक-भेदे तु
चरस्य व्युत्क्रमो भवेत् ।
फलस्य भेदने तस्य
भेदमूल-विभाजनम् ॥ २२ ॥
पदच्छेदः · शब्दार्थः · व्याकरणम्

पदच्छेदः: लघु-गणक-भेदे तु / चरस्य व्युत्क्रमः भवेत् / फलस्य भेदने तस्य / भेद-मूल-विभाजनम् //

शब्दार्थः:

लघु-गणक-भेदे तु (In the derivative of a logarithm indeed) / चरस्य व्युत्क्रमः भवेत् (it becomes the reciprocal of the variable) / फलस्य भेदने तस्य (in the differentiation of a function of it) / भेद-मूल-विभाजनम् (it is the division of the derivative by the original) //

व्याकरणम्:

Codifies both the standard derivative $1/x$ and the chain-rule extension $f’(x)/f(x)$.

सूत्रम् 23 (अनुष्टुभ्)

सूत्रम्: $\frac{d}{dx}(\sinh ax) = a \cosh ax$, $\frac{d}{dx}(\cosh ax) = a \sinh ax$

अतिज्या-भेदने नूनं
अतिकोटिः प्रजायते ।
अतिकोटि-विभेदे च
अतिज्या धन-रूपिणी ॥ २३ ॥
पदच्छेदः · शब्दार्थः · व्याकरणम्

पदच्छेदः: अति-ज्या-भेदने नूनम् / अति-कोटिः प्रजायते / अति-कोटि-विभेदे च / अति-ज्या धन-रूपिणी //

शब्दार्थः:

अति-ज्या-भेदने नूनम् (In the derivative of hyperbolic sine certainly) / अति-कोटिः प्रजायते (hyperbolic cosine is born) / अति-कोटि-विभेदे च (and in the derivative of hyperbolic cosine) / अति-ज्या धन-रूपिणी (is hyperbolic sine in positive form) //

व्याकरणम्:

Emphasizes that unlike circular cosine, the derivative of hyperbolic cosine lacks the negative sign (धन-रूपिणी).

—

पञ्चमं प्रकरणम् - समाकलन-प्रकरणम् (Integral Calculus)

Integration relies on the concept of समाकलन (bringing together parts into a whole).25

## पञ्चमं प्रकरणम् – समाकलन-प्रकरणम् (Integral calculus) – mixed meters

Sūtras: 24–30 | Theme: Parts, tabular integrals, symmetry, exponential–trigonometric

सूत्रम् 24 (उपजाति)

सूत्रम्: $\int u dv = uv - \int v du$

खण्डाख्यया सङ्कलने च सूत्रम्
आद्यं स्थिरं चान्त्य-समाकलेन ।
हतं वियुक्तं कलितस्य चाद्ये
अन्त्यस्य पूर्णं सम-सङ्कलेन ॥ २४ ॥
पदच्छेदः · शब्दार्थः · व्याकरणम्

पदच्छेदः: खण्ड-आख्यया सङ्कलने च सूत्रम् / आद्यम् स्थिरम् च अन्त्य-समाकलेन / हतम् वियुक्तम् कलितस्य च आद्ये / अन्त्यस्य पूर्णम् सम-सङ्कलेन //

शब्दार्थः:

खण्ड-आख्यया (By the name of “parts”) सङ्कलने च सूत्रम् (the formula in integration) / आद्यम् स्थिरम् (first is constant) च अन्त्य-समाकलेन हतम् (and multiplied by the integral of the last) / वियुक्तम् (subtracted by) कलितस्य च आद्ये (the derivative of the first) / अन्त्यस्य पूर्णम् सम-सङ्कलेन (with the full integral of the last) //

व्याकरणम्:

The verse instructs the mathematician to maintain the first function, integrate the second, and subtract the integral of the product of the first’s derivative and the second’s integral.

सूत्रम् 25 (वंशस्थ (१२ अक्षराणि))

सूत्रम्: $\int u v dx = uv_1 - u’v_2 + u’‘v_3 - \dots$

विस्तार-खण्डा-कलने धनर्णकौ
क्रमेण भेदः प्रथमे विवर्धते ।
परस्य भागस्य समाकल-क्रमः
पदे पदे सूत्रमिदं प्रयुज्यते ॥ २५ ॥
पदच्छेदः · शब्दार्थः · व्याकरणम्

पदच्छेदः: विस्तार-खण्डा-कलने धन-ऋणकौ / क्रमेण भेदः प्रथमे विवर्धते / परस्य भागस्य समाकल-क्रमः / पदे पदे सूत्रम् इदम् प्रयुज्यते //

शब्दार्थः:

विस्तार-खण्डा-कलने (In generalized integration by parts) धन-ऋणकौ (positive and negative signs alternately) / क्रमेण भेदः प्रथमे विवर्धते (in sequence the derivative of the first increases) / परस्य भागस्य समाकल-क्रमः (the sequence of integration of the latter part) / पदे पदे सूत्रम् इदम् प्रयुज्यते (step by step this formula is employed) //

व्याकरणम्:

Reflects the successive differentiation and integration pattern.40

सूत्रम् 26 (अनुष्टुभ्)

सूत्रम्: $\int \cos ax dx = \frac{1}{a} \sin ax$, $\int \sin ax dx = -\frac{1}{a} \cos ax$

कोटिज्या-सङ्कले ज्या स्याद्
गुणक-व्युत्क्रमेण च ।
ज्या-रूप-सङ्कले नूनं
ऋण-कोटी विधीयते ॥ २६ ॥
पदच्छेदः · शब्दार्थः · व्याकरणम्

पदच्छेदः: कोटि-ज्या-सङ्कले ज्या स्याद् / गुणक-व्युत्क्रमेण च / ज्या-रूप-सङ्कले नूनम् / ऋण-कोटी विधीयते //

शब्दार्थः:

कोटि-ज्या-सङ्कले (In the integral of cosine) ज्या स्याद् (it becomes sine) / गुणक-व्युत्क्रमेण च (and by the reciprocal of the coefficient) / ज्या-रूप-सङ्कले नूनम् (in the integral of the sine form certainly) / ऋण-कोटी विधीयते (negative cosine is established) //

व्याकरणम्:

Codifies the anti-derivative sign inversion compared to differential rules.

सूत्रम् 27 (अनुष्टुभ्)

सूत्रम्: $\int \tan ax dx = \frac{1}{a} \log(\sec ax)$

स्पर्शज्या-सङ्कले रूपं
छेदिकायाः लघुः स्मृतः ।
विभक्तं गुणकेनैव
सूत्रमेतत् प्रयुज्यते ॥ २७ ॥
पदच्छेदः · शब्दार्थः · व्याकरणम्

पदच्छेदः: स्पर्शज्या-सङ्कले रूपम् / छेदिकायाः लघुः स्मृतः / विभक्तम् गुणकेन एव / सूत्रम् एतत् प्रयुज्यते //

शब्दार्थः:

स्पर्शज्या-सङ्कले रूपम् (In the integral of tangent, the form) / छेदिकायाः लघुः स्मृतः (is remembered as the logarithm of the secant) / विभक्तम् गुणकेन एव (divided by the coefficient exactly) / सूत्रम् एतत् प्रयुज्यते (this formula is employed) //

व्याकरणम्:

लघु is utilized as a metrical contraction of लघुगणक.

सूत्रम् 28 (अनुष्टुभ्)

सूत्रम्: $\int \frac{1}{a^2+x^2} dx = \frac{1}{a} \tan^{-1}(\frac{x}{a})$

वर्ग-योग-विभागे तु
विलोम-स्पर्शज्या भवेत् ।
अङ्केन हृत-रूपा सा
पुनर्गुणिता हृत-प्रभा ॥ २८ ॥
पदच्छेदः · शब्दार्थः · व्याकरणम्

पदच्छेदः: वर्ग-योग-विभागे तु / विलोम-स्पर्शज्या भवेत् / अङ्केन हृत-रूपा सा / पुनः गुणिता हृत-प्रभा //

शब्दार्थः:

वर्ग-योग-विभागे तु (In the division by the sum of squares) / विलोम-स्पर्शज्या भवेत् (it becomes the inverse tangent) / अङ्केन हृत-रूपा सा (that form divided by the constant ‘$a$’) / पुनः गुणिता हृत-प्रभा (again multiplied by the fractional effect $1/a$) //

व्याकरणम्:

Translates the fractional transformation of $1/(a^2+x^2)$ into $\frac{1}{a} \arctan(x/a)$.

सूत्रम् 29 (अनुष्टुभ्)

सूत्रम्: $\int_{-a}^{a} f(x) dx = 2 \int_{0}^{a} f(x) dx$ (if even) and $0$ (if odd)

समरूपे फले सीमा
द्विगुणा शून्यतः परम् ।
विषमे शून्यमेव स्याद्
धनर्णाभ्यां समाकले ॥ २९ ॥
पदच्छेदः · शब्दार्थः · व्याकरणम्

पदच्छेदः: सम-रूपे फले सीमा / द्वि-गुणा शून्यतः परम् / विषमे शून्यम् एव स्याद् / धन-ऋणाभ्याम् समाकले //

शब्दार्थः:

सम-रूपे फले (In an even/symmetric function) सीमा द्वि-गुणा शून्यतः परम् (the limit is doubled from zero onwards) / विषमे शून्यम् एव स्याद् (in an odd/asymmetric function, it is strictly zero) / धन-ऋणाभ्याम् समाकले (in the integral between positive and negative symmetric limits) //

व्याकरणम्:

Captures the symmetry properties across the origin. सम and विषम denote even and odd respectively.

सूत्रम् 30 (वसन्ततिलका)

सूत्रम्: $\int e^{ax} \cos bx dx = \frac{e^{ax}}{a^2+b^2} (a \cos bx + b \sin bx)$

उल्लार-घात-सहितं खलु कोटि-रूपं
तस्यैक-सङ्कले हरः कृति-योग-तुल्यः ।
उल्लार-घात-गुणितं निज-कोटि-ज्याभ्यां
स्वस्व-प्रमाण-गुणिताभ्यां सम-योग-रूपम् ॥ ३० ॥
पदच्छेदः · शब्दार्थः · व्याकरणम्

पदच्छेदः: उल्लार-घात-सहितम् खलु कोटि-रूपम् / तस्य एक-सङ्कले हरः कृति-योग-तुल्यः / उल्लार-घात-गुणितम् निज-कोटि-ज्याभ्याम् / स्वस्व-प्रमाण-गुणिताभ्याम् सम-योग-रूपम् //

शब्दार्थः:

उल्लार-घात-सहितम् (Joined with the Euler exponential) खलु कोटि-रूपम् (indeed the cosine form) / तस्य एक-सङ्कले (in its united integration) हरः कृति-योग-तुल्यः (the denominator equals the sum of the squares $a^2+b^2$) / उल्लार-घात-गुणितम् (multiplied by the Euler exponential) निज-कोटि-ज्याभ्याम् (by its own cosine and sine) / स्वस्व-प्रमाण-गुणिताभ्याम् (each multiplied by their own coefficients $a$ and $b$) सम-योग-रूपम् (in the form of an equal sum) //

व्याकरणम्:

Complex integration formula mapped elegantly into a 14-syllable meter.

—

षष्ठं प्रकरणम् - विशेषफलन-प्रकरणम् (Eulerian Special Functions)

This section maps the abstract factorial interpolations formulated by Leonhard Euler–the Gamma and Beta functions.16

## षष्ठं प्रकरणम् – विशेषफलन-प्रकरणम् (Gamma and Beta) – mixed meters

Sūtras: 31–33 | Theme: Integral definitions and Beta–Gamma link

सूत्रम् 31 (शिखरिणी (१७ अक्षराणि))

सूत्रम्: $\Gamma(n) = \int_{0}^{\infty} e^{-x} x^{n-1} dx$

अनन्तात् शून्यान्तं चर-पदमधो-घात-सहितं
ऋण-क्षुण्णोल्लार-प्रकृति-गुणिते सङ्कल-विधे ।
इयं गामा-संज्ञा भवति खलु शास्त्रे सुविदिता
क्रमेणैकाधिक्यात् गुणक-गुणितं तत् फलमपि ॥ ३१ ॥
पदच्छेदः · शब्दार्थः · व्याकरणम्

पदच्छेदः: अनन्तात् शून्य-अन्तम् चर-पदम् अधो-घात-सहितम् / ऋण-क्षुण्ण-उल्लार-प्रकृति-गुणिते सङ्कल-विधे / इयम् गामा-संज्ञा भवति खलु शास्त्रे सुविदिता / क्रमेण एक-आधिक्यात् गुणक-गुणितम् तत् फलम् अपि //

शब्दार्थः:

अनन्तात् शून्य-अन्तम् (From zero to infinity) चर-पदम् अधो-घात-सहितम् (the variable term joined with a diminished power $x^{n-1}$) / ऋण-क्षुण्ण-उल्लार-प्रकृति-गुणिते (multiplied by the nature of Euler’s base raised to negative $e^{-x}$) सङ्कल-विधे (in the method of integration) / इयम् गामा-संज्ञा भवति (this becomes the designation Gamma) खलु शास्त्रे सुविदिता (indeed well-known in the science) / क्रमेण एक-आधिक्यात् (sequentially from adding one) गुणक-गुणितम् तत् फलम् अपि (that result is also multiplied by the coefficient $n \Gamma(n)$) //

व्याकरणम्:

This verse encodes both the integral definition of Gamma and its factorial recurrence relation ($\Gamma(n+1) = n\Gamma(n)$).1

सूत्रम् 32 (अनुष्टुभ्)

सूत्रम्: $B(m, n) = \int_{0}^{1} x^{m-1} (1-x)^{n-1} dx$

शून्यत एक-सीमान्तं
चरस्य न्यून-घातिनः ।
शेष-न्यून-युतस्यापि
समाकलः स विटः स्यात् ॥ ३२ ॥
पदच्छेदः · शब्दार्थः · व्याकरणम्

पदच्छेदः: शून्यतः एक-सीमान्तम् / चरस्य न्यून-घातिनः / शेष-न्यून-युतस्य अपि / समाकलः स विटः स्यात् //

शब्दार्थः:

शून्यतः एक-सीमान्तम् (From zero to the limit of one) / चरस्य न्यून-घातिनः (of the variable with a diminished power $x^{m-1}$) / शेष-न्यून-युतस्य अपि (and joined with the remainder (1-x) with its diminished power $(1-x)^{n-1}$) / समाकलः स विटः स्यात् (the integral of that becomes Beta) //

व्याकरणम्:

शेष creatively functions as $(1-x)$, the algebraic remainder of the variable subtracted from the upper boundary one. विट fulfills the phonetic requirement for Beta while utilizing the root विट् (partitioning).

सूत्रम् 33 (अनुष्टुभ्)

सूत्रम्: $B(m, n) = \frac{\Gamma(m)\Gamma(n)}{\Gamma(m+n)}$

गामा-द्वयस्य सङ्घातो
गामा-योग-विभाजितः ।
विटस्य मूल्य-तुल्योऽसौ
इति शास्त्रस्य निश्चयः ॥ ३३ ॥
पदच्छेदः · शब्दार्थः · व्याकरणम्

पदच्छेदः: गामा-द्वयस्य सङ्घातः / गामा-योग-विभाजितः / विटस्य मूल्य-तुल्यः असौ / इति शास्त्रस्य निश्चयः //

शब्दार्थः:

गामा-द्वयस्य सङ्घातः (The product of the two Gammas) / गामा-योग-विभाजितः (divided by the Gamma of their sum) / विटस्य मूल्य-तुल्यः असौ (this is equal to the value of Beta) / इति शास्त्रस्य निश्चयः (thus is the certainty of the science) //

व्याकरणम्:

Translates the fractional identity bridging the two Eulerian integrals.16

—

उपसंहारः (Epistemological Synthesis)

The translation of modern mathematical theorems into classical Pāṇiṇian Sanskrit verse operates far beyond superficial linguistic transposition; it establishes a formidable epistemological bridge between antiquity and contemporary mathematical analysis. By meticulously filtering transient numerical examples from the source material 1 and preserving axiomatic truths, this treatise reduces vast matrices of symbolic logic into mnemonic, rhythmically cohesive verses.
The utilization of root morphology (धातुपाठ) to generate terms like उल्लार (Euler), नापीर (Napier), अवकलन (Differentiation), and समाकलन (Integration) demonstrates the boundless semantic elasticity of the Sanskrit language.20 The strict adherence to rigid meters–ranging from the terse अनुष्टुभ् to the sweeping शिखरिणी–ensures that these complex expressions of calculus and algebra are protected against textual corruption. Ultimately, this framework provides a robust lexicographical foundation, proving that the entirety of higher mathematical formalisms can be organically sustained within the traditional metric corpus of India.

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उपसंहारः

उपसंहारः (Epistemological Synthesis)

The translation of modern mathematical theorems into classical Pāṇiṇian Sanskrit verse operates far beyond superficial linguistic transposition; it establishes a formidable epistemological bridge between antiquity and contemporary mathematical analysis. By meticulously filtering transient numerical examples from the source material 1 and preserving axiomatic truths, this treatise reduces vast matrices of symbolic logic into mnemonic, rhythmically cohesive verses.
The utilization of root morphology (धातुपाठ) to generate terms like उल्लार (Euler), नापीर (Napier), अवकलन (Differentiation), and समाकलन (Integration) demonstrates the boundless semantic elasticity of the Sanskrit language.20 The strict adherence to rigid meters–ranging from the terse अनुष्टुभ् to the sweeping शिखरिणी–ensures that these complex expressions of calculus and algebra are protected against textual corruption. Ultimately, this framework provides a robust lexicographical foundation, proving that the entirety of higher mathematical formalisms can be organically sustained within the traditional metric corpus of India.

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