गणितसारसंग्रह (महावीराचार्य - प्रो. लक्ष्मीचन्द्र जैन सविस्तार हिन्दी अनुवाद सटीक)

५६ गणितसारसंग्रह
प्रति क्रमांक—अ० नं० ६४
(१) सारसंग्रह गणितशास्त्र ।
(२) पत्रसंख्या १४२ प्रतिपत्र १० पंक्तियों—प्रतिपंक्ति २५ अक्षर आकार ५"४ x १६" ।
(३) प्रथमव्यवहार ३७ द्वितीय ७८ तृतीय ९५ चतुर्थ १०४ पञ्चम १११ षष्ठ १३१ सप्तम
१४० अंतिम १४२ ।
(४) प्रारंभ— ८० ॥ श्री जिनाय नमः ॥ श्रीगुरुभ्यो नमः ॥ प्रणिपत्य वर्द्धमानं विद्यानंदं
विशुद्धगुणनिलयं । सूरिं च महावीरं कुर्वे तद्गणितशास्त्रसद्वृत्तिं ॥ १ ॥ अलंघ्यं इत्यादि ।
(५) अंतिम—छत्तीसी टीका ग्रंथसंख्या ३०००५ शुभं भवतु ॥ श्रीरस्तु ॥ शुभं ॥ स्वस्ति श्री
संवत् १६१६ वर्षे कार्तिक सुदि ३ गुरौ श्रीगंधारशुभस्थाने श्रीमदादिजिनचैत्यालये
श्रीमूहलसंघे श्रीसरस्वतीगच्छे श्रीबलात्कारगणे श्रीकुंदकुंदाचार्यान्वये भ० पद्मनंदिदेवास्तत्पट्टे
भ० श्रीदेवेंद्रकीर्तिदेवास्तत्पट्टे भ० श्रीविद्यानंदिदेवास्तत्पट्टे भ० श्री मल्लिभूषणदेवास्तत्पट्टे भ०
श्रीलक्ष्मीचंद्रदेवास्तत्पट्टे भ० श्रीवीरचंद्रदेवास्तत्पट्टे भ० श्रीज्ञानभूषणदेवास्तदन्वये आचार्य-
सुमतिकीर्तेरुपदेशात् श्रीहुंब जातीय सोनी सांतू भार्या बाई हरषाई तयोः पुत्र सोनी देधर
भार्या मरघाई तयोः सुतौ सोनी देवजी क्षीमजी एतेषां मध्ये सोनी देधरकेन इदं शास्त्रं
लिखाप्य दत्तं किंचत् श्रावकैः लिखापितं ॥ छ ॥
आ० वीरत्तभूषणानामिदं ।।
छत्तीसि गणितनि टिका
संवत् १८४२ मिति वैसाख सुदि ११ भट्टारक श्रीवीद्याभूषणइदं गणत छत्तिसी भट्टारक श्री देवेन्द्र-
कीर्तिजीज्यां प्रदत्तं सुभं भूयात् ।
(६) बलात्कार मंदिर कारंजा क्र० ६४ ।
प्रति क्रमांक—अ० नं० ६५
(१) सारसंग्रह गणितशास्त्र—प्रशस्ति मे—षट्त्रिंशतिकागणितशास्त्र ।
(२) पत्र ५३; प्रति पत्र १० पंक्तियाँ; आकार ११" x ४"७५ ।
(३) प्रथम व्यवहार १६; द्वितीय ३४; तृतीय ४०; चतुर्थ ४६; पंचम ५३ ।
(४) प्रारंभ—८०॥ श्रीवीतरागाय नमः ॥ अलंघ्यं त्रिजगत्सारं इत्यादि ।
(५) अन्तिम—(पत्र ५३) घनं ॥ इति सारसंग्रहे गणितशास्त्रे महावीराचार्यस्य कृतौ
वर्गसंकलितादिव्यवहारः पंचमः समाप्तः ॥
संवत् १७२५ वर्षे कार्तिक श्रुदि १० भौमे श्रीमूहलसंघे सरस्वतीगछे बलात्कारगणे
श्रीकुंदकुंदाचार्यान्वये भ० श्रीसकलकीर्त्यन्वये भ० श्रीवादिभूषणदेवास्तत्पट्टे भ० श्रीरामकीर्ति
देवास्तत्पट्टे भ० श्रीपद्मनंदिदेवास्तत्पट्टे भ० श्रीदेवेंद्रकीर्तिगुरूपदेशात् मुनि श्रीश्रुतकीर्ति-
स्तच्छिष्य मुनि श्रीदेवकीर्तिस्तच्छिष्य आचार्य श्रीकल्याणकीर्तिस्तच्छिष्य मुनि श्रीत्रिभुवन-
चंद्रेणेदं षट्त्रिंशतिका गणितशास्त्रं कर्मक्षयार्थे लिखितं ।
(७) प्राप्तिस्थान—बलात्कारगणमंदिर, कारंजा, अ० नं० ६५ ।
(८) स्थिति मध्यम, अक्षर स्पष्ट ।
(९) विशेषता—समास मे टिप्पण; क्वचित् पृष्ठमात्रा ।

गणितसारसंग्रह ५७
नोट—ऐसा प्रतीत होता है मानो यह माधवचंद्र त्रैविद्यदेव का विभिन्न ग्रंथ हो—
१. वर्ग संकलितानयनसूत्रं । २९६-९७ ।
२. घनसंकलितानयनसूत्रं । ३०१-८२ ।
३. एकवारादिसंकलितधनानयनसूत्रं ।
४. सर्वधनानयने सूत्रद्वयं ।
५. उत्तरोत्तरचयभवसंकलितधनानयनसूत्रं ।
६. उभयान्तादागत पुरुषद्वयसंयोगानयनसूत्रं ।
७. वणिक्करस्थितधनानयनसूत्रं ।
८. समुद्रमध्ये—१-२-३ ।
९. छेदोशशेषजातौ करणसूत्रं ।
१०. करणसूत्रत्रयम् ।
११. गुणगुण्यमिश्रे सति गुणगुण्यानयनसूत्रं ।
१२. बाहुकरणानयनसूत्रं ।
१३. व्यासाद्यानयनसूत्रं ।
इति सारसंग्रहे गणितशास्त्रे महावीराचार्यस्य कृतौ वर्गसंकलितादिव्यवहारः पंचमः समाप्तः ।
प्रति क्रमांक—अ० नं० ६२
( १ ) उत्तरछत्तीसी टीका ।
( २ ) पत्र १९; प्रति पत्र १३ पंक्तियाँ; आकार ११" × ४"•७५ ।
( ३ ) आरंभ—ॐ नमः सिद्धेभ्यः ॥ सिद्धेभ्यो निष्ठितार्थेभ्यो इ० ।
( ४ ) अन्तिम — घनः २९२७७१५५८४ ॥ छ ॥
इति श्रीउत्तरछत्तीसी टीका समाप्ता ॥

• आचार्य श्रीकल्याणकीर्तिस्तच्छिष्य मुनि श्रीत्रिभुवनचंद्रेणेदं गणितशास्त्रं लिखितं ॥
  उजलो पाषाण सुतारी गज १ समचोरस मण ४८ पालेवो पाषाण गज १ मण ६० खारो
  पाषाण गज १ मण ४० ।
  ( ५ ) प्राप्तिस्थान —अ० नं० ६२ ।
  ( ६ ) स्थिति उत्तम, अक्षर स्पष्ट ।
  ( ७ ) क्वचित् टिप्पण ।
  प्रति क्रमांक—अ० नं० ६६
  ( २ ) पत्र १५; प्रतिपत्र १४ पंक्तियाँ; आकार ११"•५ × ५"
  ( ३ ) * ब्रह्म जसवंताख्येन स्वपरपठनार्थं स्वहस्तेन लिखितं ।
  ( ५ ) अ० नं० ६६ ।
  प्रति क्रमांक—अ० नं० ६०
  ( २ ) पत्र २०; प्रतिपत्र ११ पंक्तियाँ; आकार १२" × ५"•५ ।
  ( ५ ) अ० नं० ६० ।

५८ गणितसारसंग्रह
प्रति क्रमांक - अ० नं० ६१
( २ ) पत्र १८; प्रतिपत्र १४ पंक्तियाँ; आकार १०"०५ x ६" ।
( ५ ) अ० नं० ६१ ।
गणितसारसंग्रह
प्रतक्रमांक ६३ = अ, प्र० क्र० ६५ = ब, प्र० क्र० ६४ = स
अर्थबोधक टिप्पण
श्लोक १-१ अलङ्घ्यम्—अ मिथ्यादृष्टिभिः । ब मिथ्यादृष्टिभिः लङ्घयितुम् अशक्यमित्यर्थः ।
स आप्ताभासागम्यम् अतल्लक्ष्यमस्ति । स त्रिजगत्सारम्—निरावरणत्वादनन्यसाधारणत्वाच्च लोकत्रयसारम्,
त्रिजगद्वन्द्याराध्यमित्यर्थः । अ अनन्तचतुष्टयम् अनन्तज्ञान-दर्शन-सुख-वीर्यचतुष्टयम् । स तस्मै महावीराय
वर्धमानस्वामिने । स जिनेन्द्राय—एकदेशेन कर्मारातीन् जयन्तीति जिना असंयतसम्यग्दृष्ट्यादयस्तेषामिन्द्रः
स्वामी, तस्मै नमः । अ तायिने-धर्मोपदेशकत्वेन भव्यत्राणाय ।
श्लोक १-२ अ जि [जै]नेन्द्रेण—जिनो देवता येषां ते जैनाः, तेषामिन्द्रः, तेन । पक्षे—
जिनेन्द्रस्यायं सम्बन्धी जैनेन्द्रः तेन वा । जिन एव जैनः, स एव इन्द्रः प्रधानो यत्र संख्याज्ञानप्रदीपे सः,
तेन । स जैनेन्द्रेण—जिनप्रणीतेन । स संख्याज्ञानप्रदीपेन—गणितशास्त्रज्योतिषा । स महात्विषा—
बहुप्रकाशेन । स सर्वम्—षड्भव्यसमुदायस्वरूपम् । अ तम्—महावीरम्, पक्षे संख्याज्ञानप्रदीपम् ।
श्लोक १-३ स प्रीणितः—तर्पितः । स प्राणिसस्यौघः विनेयजनस्य संघातः । अ निरीतिः—
निर्गता ईतयः अतिवृष्ट्यनावृष्टिमूषक-शलभ-शुक-स्वचक्र-परचक्रलक्षणाः यस्मात् असौ निरीतिः । अ
निरवग्रहः—निर्गतोऽवग्रहः शत्रुः यस्मात् यत्र वा सः, व्यथा—वर्षाविघातरहितः । स श्रीमता—लक्ष्मी-
मता । अ अमोघवर्षेण—सफलवृष्ट्या, पक्षे सत्यस्वरूपोपदेशवृष्ट्या । स सफलसद्धर्मोपदेशामृतवृष्ट्या ।
अ स्वेष्टहितैषिणा—स्वस्य इष्टं स्वेष्टम्, तच्च तद्धितं च स्वेष्टहितम्, तदिच्छतीति स्वेष्टहितैषी तेन । वा
स्वस्य इष्टाः स्वेष्टाः, तान् प्रति हितम् इच्छतीति स्वेष्टहितैषी, तेन । स स्वेष्टहितमिच्छता ।
श्लोक १-४ अ चित्तवृत्तिद्विषिर्भुजी [जि]—शुक्लध्यानाग्नौ । स भस्मसात् भावम्—भस्मस्वरूपम् ।
अ ईयुः—गच्छन्ति स्म । अ ते—आगमप्रसिद्धाः काम-क्रोधादिशत्रवः । अ अवन्ध्यकोपाः [पः]—
सफलकोपाः इत्यर्थः ।
श्लोक १-५ स वशीकुर्वन्—स्वाधीनं विदधत् । स नानुवशः—अन्याधीनो न भवति । स परैः—
एकान्तवादिभिः । अभिभूतः—अ पराभूतः । स तिरस्कृतः । स प्रभुः—जगदाराध्यः । स अपूर्वमकर-
ध्वजः—अभिनवमीनकेतनः ।
श्लोक १-६ अ विक्रम-क्रमाक्रान्त-चक्रीचक्र-कृतक्रियः—विक्रमक्रमेण पराक्रमसंतत्या आक्रान्ताः
ते च ते चक्रिणश्च, तेषां चक्रं समूहः, तेन कृतक्रिया सेवा यस्यासौ तथोक्तः । पक्षे चक्रं सेनाऽस्ति येषां ते
चक्रिणः, शेषं पूर्ववत् । अ चक्रिकाभञ्जनः—संसारचक्रभञ्जनः, पक्षे—परचक्रभञ्जनः । अ अञ्जसा—
परमार्थेन ।
श्लोक १-७ अ विद्यानद्यधिष्ठानः—विद्या द्वादशाङ्गलक्षणाः पक्षे—द्वासप्ततिकलालक्षणास्ता एव
नद्यः तासाम् अधिष्ठानम् आश्रयः यः सः । स मर्यादावज्रवेदिकः—मर्यादैव वज्रवेदिका यस्य सः । अ
रत्नगर्भः—रत्नानि सम्यग्दर्शनादीनि, पक्षे—स्वादीनि, गर्भे ते यस्य सो [यस्यासौ] । ब रत्नानि सम्यग्दर्शना-
दीनि, पक्षे—हस्त्यश्वादीनि गर्भे ते यस्यासौ तथोक्तः । अ यथाख्यातचारित्र [त्र] जलधिः—क्षायिक-
चारित्र्य [त्र] जलधिः, पक्षे—यथाख्यातं प्रवृद्धैर्यथोक्तम्, तच्चतच्चारित्र्यं [त्रं] आचरणं च ।

गणितसारसंग्रह ५९
श्लोक १-८ स देवस्य—स जिनस्य । स शासनम् अनेकान्तरूपं वर्धताम् ।
श्लोक १-९ स लौकिके—वृद्धिव्यवहारादौ । अ वैदिके—आगमे । स सामायिके—प्रतिक्रमणादौ ।
अ यः—यः कश्चित् व्यापारः प्रवृत्तिः तत्र सर्वत्र संख्यानं गणितम् उपयुज्यते उपयोगी भवति ।
श्लोक १-१० अ अर्थशास्त्रे — जीवादिपदार्थे ।
श्लोक १-११ अ प्रस्तुतम्—कथितम् । अ पुरा— पूर्वम् ।
श्लोक १-१२ अ ग्रहचारेषु—संक्रमणेपु । ब सूर्यादिसंक्रमणेपु । स ग्रहणे—चन्द्र-सूर्योपरागे ।
अ ग्रहसंयुतौ—ग्रहयुद्धे । अ त्रिप्रश्ने—त्रयः प्रश्नाः नष्ट-मुष्टि-चिन्तारूपाः यत्र तत् त्रिप्रश्नम्, होराशास्त्र-
मित्यर्थः, तस्मिन् । स अथवा त्रयो धातुं-मूँलं-जीवँविषयाः प्रश्नाः यत्र तत् त्रिप्रश्नम् । प्रश्नव्याकरणाय
सद्भावकेवलज्ञानहोरादिशास्त्रम् । स चन्द्रवृत्तौ—चन्द्रचारे । ब omits बुध्यन्ते (श्लोक १४) ।
ब omits—यात्राद्याः (श्लोक १५) ।
श्लोक १-१३ अ परिक्षेपः—परिधियः ।
श्लोक १-१४ अ उत्कराः—समूहाः । अ बुध्यन्ते—ज्ञायन्ते ।
श्लोक १-१५ अ तत्र—श्रेणीबद्धादिषु जीवानाम् । अ संस्थानम्—समचतुरस्रादि । अ अष्ट-
गुणादयः—अणिमादयः । अ यात्राद्याः—गति । अ संहितायाश्च—संधिप्रतिष्ठाग्रन्थो वा ।
श्लोक १-१७ अ गुरुपर्वतः—गुरुपरिपाटीभ्यः ।
श्लोक १-२०—अ कलासवर्णसरुलठत्पाटीनसंकुले—कीदृग्विधे सारसंग्रहवारिधौ । कलासवर्णाः
भिन्नप्रत्युत्पन्नादयः ते एव लुठत्पाटीनास्तेषां संकटे संकोचस्थाने ।
श्लोक १-२१ अ प्रकीर्णक—अ तृतीयव्यवहारः । अ महाग्राहे—मत्स्यविशेषः । अ मिश्रक—
अ वृद्धिव्यवहारादि ।
श्लोक १-२२ अ क्षेत्रविस्तीर्णपाताले—त्रिभुज-चतुर्भुजादिक्षेत्राणि एव विस्तीर्णपातालानि यत्र स
तस्मिन् । अ खाताख्यसिकताकुले—खाताख्यम् एव सिकताः ताभिः आकुले । अ करणस्कन्धसंबन्धच्छाया-
बेलाविराजिते—करणस्कन्धेन करणसूत्रसमूहेन संबन्धो यस्याः सा करणस्कन्धसंबन्धा, सा चासौ छाया-
गणितं (?) करणस्कन्धसंबन्धच्छाया, सा एव वेला, तया विराजिता तस्मिन् ।
श्लोक १-२३ अ गुणसंपूर्णैः—लघुकरणाद्यष्टगुणसंपूर्णैः । करणोपायैः—अ करणानुपयोगोपायैः सूत्रैः ।
श्लोक १-२४ अ यत्—यस्मा

६० गणितसारसंग्रह
श्लोक १-२६ अ अतः—अणोतः । तस्मात्—त्रसरेणोः । शिरोरुहः—( भवन्ति ) ।
श्लोक १-२७ अ लिक्षा—लिक्षाप्रमाणस्कन्धः । सः—स तिलः । अष्टगुणानि—अष्टगुणानि भवन्ति
त्रसरेष्वाद्यङ्गुलान्तानि ।
श्लोक १-२८ अ प्रमाणम्—प्रमाणाङ्गुलम् ।
श्लोक १-२९ अ तिर्यक्पादः—पादस्य अङ्गुष्ठकनिष्ठापर्यन्त भाग तिर्यक्पादः । तिर्यक्पादद्वयं
वितस्तिः । ब तिर्यग्पादः—omits.
श्लोक १-३१ अ परिभाषा—अनियमेन नियमंकारिणी परिभाषा ।
श्लोक १-३२ अ अणुरण्वन्तरम्—मन्दगतिमाश्रितः सन्, शीघ्रगतिमाश्रितश्चेत् चतुर्दशरज्जुम्
अतिक्रामति । समयः—प्रोक्तः । असंख्यैः—जघन्ययुक्तासंख्यैः । ब असंख्यैः—omits, लोके—
omits (?)
श्लोक १-३३ अ स्तोक इति मानम् । तेषाम्—लवानाम् । सार्धाष्टात्रिंशता—३८ १/२ ।
श्लोक १-३४ अ पक्षः—भवेत् ।
श्लोक १-३५ अ तैः—ऋतुभिः । वत्सरो संवत्सरः ।
श्लोक १-३६ अ तत्र—धान्यमाने । चतस्रः—षोडशिकः । कुडवः—सहस्रैश्च त्रिभिः षड्भिः
शतैश्च व्रीहिभिः समैः । यः संपूर्णो भवेत् सोऽयं कुडवः परिभाष्यते ॥ लोके पवाळु ८। प्रस्थः—लोके
पाली ८। ब प्रस्थः—omits.
श्लोक १-३८ अ सेयं प्रवर्तिका । ताः खार्याः [ यैः ] । तस्याः प्रवर्तिकायाः ।
श्लोक १-३९ अ गण्डकैः—कस्तुंबुरुभिः, लोके धाना, धरणे-धरणद्वयम् ।
श्लोक १-४० अ धान्यद्वयेन—लोके धानाद्वयेन ब कुस्तुंबरद्वयेन । अत्र—रजतपरिकर्मणि ।
श्लोक १-४१ अ पुराणाम्—कर्षान् । रूप्ये—रजत—परिभाषायां मागधदेशव्यवहारमाश्रित्य ।
श्लोक १-४२ अ कल—कलेति नाम भवेत् ।
श्लोक १-४३ अ अस्मात्—द्रक्षूणात् । सतेरं—सतेराख्यं मानं भवति । ब लोहे—लोह-
परिभाषायाम् ।
श्लोक १-४४ अ 'प्रचक्षते' अन्तस्य 'अत्' आदेशो भवति ।
श्लोक १-४५ अ ब वस्त्राभरण-कटादीनाम् ।
श्लोक १-४६ ब अत्र—परिकर्मणि ।
श्लोक १-४८ अ भिन्नानि—यथा गुणाकारभिन्नः भागहारभिन्नः कृतिभिन्नः प्रत्येकभिन्नः इति
परं योज्यम् ।
ब तच्च—'विद्या कलासवर्णस्य' इति वा पाठः ।
श्लोक १-४९ ब हृतः शून्येन भक्तः सन् । खवधादिः—शून्यस्य भजन-गुणन-वर्गमूलादिः ।
योज्यरूपकम्—योज्यराशिसमानम् ।
स शून्येन ताडितो गुणितो राशिः खं शून्यं स्यात् । स राशिः शून्येन हतः [ हृतः ] भक्तः ।
शून्येन युतः सहितः । शून्येन हीनो रहितोऽपि अविकारी विकारवान् न भवति तदवस्थ एव—
खवधादिः ख शून्यस्य वधो गुणनं खं शून्यं स्यात् । आदिशब्देन भजन-वर्ग-घन-तन्मूलानि गृह्यं ।
श्लोक १-५० ब घाते गुणने । विवरं—महाराशौ स्वल्पराशिमपनीयावशिष्टशेषो विवरमित्युच्यते ।

गणितसारसंग्रह ६१
स ऋणयोः—ऋणरूपराश्योः । धनयोः—धनरूपराश्योः । भाजने—भागहारे । फलम्—गुणित-
फलम् । तु—पुनः ।—adds चेयमंकसंहतिः ।—adds illustrations to explain rules
on 50 ( stanza ).
श्लोक १—५१ स योगः—संयोजनम् । शोध्यम्—अपनेयम् ।
श्लोक १—५२— ब मूले—वर्गमूले । स्वर्णे—धनऋणे स्याताम् । Adds two stanzas
after 52. Printed in text at No. 69–70.
लघुकरणोहापोहानालस्यग्रहणधारणोपायैः ।
व्यक्तिकराङ्कविशिष्टैः गणकोष्टाभिर्गुणैर्ज्ञेयः ॥ १ ॥
इति संज्ञा समासेन भाषिता मुनिपुंगवैः ।
विस्तरेणागमाद् वेद्यं वक्ष्यन्थे यदितः परम् ॥ २ ॥
तत्पदम्—ऋणरूपवर्गराशेर्मूलं कथं भवेत् इत्याशङ्कायाम् इदमाह—ऋणराशिः निजऋणवर्गो न
भवेत्, किंतु धनरूपेण वर्गो भवेत् । तस्मात् ऋणराशेः सकाशात् मूलं न भवेत्, किंतु धनराशेः सकाशात्
ऋणराशेर्मूलं स्यात् ।
स धनराशेः ऋणराशेश्च वर्गो धनं भवति । Adds illustrations to explain rules
on 52 ( stanza ).
श्लोक १—५८ अ ऋतुर्जीवो—षड् जीवाः । कुमारवदनम्—कार्तिक [ केय ] वदनम् । ब
कुमारवदनम्—कार्तिकेयवदनम् ।
श्लोक १—६९ ब शीघ्रगुणन-भजनादिलक्षणं लघुकरणम् । अनेन प्रकारेण गुणनादौ कृते
सतीप्सितं लब्धं स्यादिति पूर्वमेव परिज्ञानलक्षणः ऊहः । इत्थं गुणनादौ कृते सतीप्सितं लब्धं न स्यादिति
पूर्वमेव परिज्ञानलक्षणः अपोहः । गुणनादिक्रियायां मन्दभावरहित्यलक्षणमनालस्यम् । कथितार्थलक्षणं
ग्रहणम् । कथितार्थस्य कालान्तरेऽप्यविस्मरणलक्षणा धारणा । सूत्रोक्तगुणनादिकमाधारं कृत्वा स्वबुद्ध्या
प्रकारान्तरगुणनादिविचारलक्षणः उपायः । अंकं व्यक्तं स्थापयित्वा गुणनादिकरणलक्षणो व्यक्तिकरांकः ।
इत्यष्टभिर्गुणै गणितज्ञो भवेदिति ज्ञेयः । इति ।
श्लोक २-१ अ ( १ ) येन राशिना गुण्यस्य भागो भवेत् तेन गुण्यं भङ्क्त्वा गुणकारं गुणयित्वा
स्थापनालक्षणो राशिखण्डः । येन राशिना गुणगुणकारस्य भागो भवेत् तेन गुणकारं भङ्क्त्वा गुण्यं गुणयित्वा
स्थापनालक्षणोऽर्धखण्डः । गुण्य-गुणकारो [ रौ ] अभेदयित्वा स्थापनालक्षणः तत्स्थः । इति त्रिप्रकारैः
स्थितगुण्य-गुणकारराशियुगलं कवाटसंधाणक्रमेण विन्यस्य । ( २ ) राशेरादितः आरभ्यान्तपर्यन्तं गुणनलक्षणेन
अनुलोममार्गेण । ( ३ ) राशेरन्ततः आरभ्यादिपर्यन्तं गुणनलक्षणेन विलोममार्गेण च गुण्यराशि गुणकार-
राशिना गुणयेत् । ( ४ ) 'गुणयेत् गुणेन गुण्यं कवाटसंधिक्रमेण संस्थाप्य' इति पाठान्तर—पादद्वयम् ।
( ५ ) गुण्यगुणकारं यथा व १४४ गुण्यं = प्रत्येक पञ्चानि गुणकार इति = ८ ; २४
४८
११५२ राशिखण्ड

६२ गणितसारसंग्रह
(६) गुणकारं ८ अस्य भाग ४, अनेन गुण्यं गुणित चेत् ४ | ५ | ७ | ६
२ | १/१ | १/४ | १/२
(७) व = वश [स] ति । (८) ता = तामरसं । (९) प = पदमानि । (१०) विनष्टो एकः
येभ्यस्तेष्विकाम् । (११) मणयः । (१२) खर इति षड् जीव । (१३) राशिना गुण्यलब्धम् उपरिनन-
भागे स्थाप्यमधः तेनैव गुणकारं गुणयित्वा स्थापना* ।
श्लोक २-७ अ विषनिधिः = जलनिधिः ।
श्लोक २...अ पुरुषः — जीवो इत्यर्थः ।
श्लोक २-९ अ [ खरः-] “सत्यसंघः खरो ज्ञेयः खरोऽपि पुरुषो मतः” इत्यभिधानात् ।
श्लोक २-१० अ तत्-राशिम् ।
श्लोक २-११ अ पञ्चषट्कं च — आदौ ७ पञ्चषट्कं ६६६६६ षट्त्रिकं ३३३३३३ तत् भिन्नं
लिखितम् — ३३३३३३६६६६६७ ।
श्लोक २-१५ अ त्रयः — सान्तः त्रयःशब्दोऽयम् ।
श्लोक २-१७ अ हिमांश्वग्र — हिमांशु अग्रे [ रग्रे ] येषां तानि, हिमांश्वग्राणि च तानि रन्ध्राणि च
तत्तथोक्तानि, तैः । कण्ठिका — कण्ठभूषणम् । ब एकरूपम् — एकस्याभिधानं ग्रन्थान्तरे ।
श्लोक २-१८ की उत्थानिका — च परमागमप्रतिपादितकरणानुयोगे ग्रह-नक्षत्र-प्रकीर्णक-तारादि-
गणनामभिधानं करणमित्युच्यते, तस्य सूत्रम्, सूचयति संक्षेपेणार्थं सूचयति इति सूत्रं तत्तथोक्तम् ।
श्लोक २-१९ अ प्रतिलोमपथेन — विलोममार्गेण भाव्यम् — अंकानां वामतो गतिः, तेन अन्ततः
आरभ्य भाव्यम् । विधाय — अपवर्तनविधिं विधाय । तयोः — भाज्य-भागहारराश्योः । स उपस्थितं
भाज्यराशिं अधःस्थितेन भागहारेणानन्तां आरभ्यादिपर्यन्त भजनलक्षणेन प्रतिलोमपथेन भजेत् ।
यदि तयोर्भाज्य-भागहारयोः सदृशापवर्तनविधिः समानराशिना भाज्य-भागहाराद्यपवर्तनलक्षणविधानं संभवति
तर्हितं कृत्वा भजेत ।
श्लोक २-२० अ अंशो भागः । नुः नरस्य । —भागहारस्य भाग (?) द्वौ वा चत्वारो वा तेषु एकभागेन
भाज्यं भाजयेत्, द्वितीयभागेन भाज्यं भाजयेत्, तृतीयभागेन भाज्यं भाजयेत्, चतुर्थभागेन भाज्यं
भाजयेत् । अपवर्तनविधिः । एकशतयुतम् — एकेनाधिकं शतम् एकशतम् ।
श्लोक २-२६ अ त्रिदशसहस्री — त्रिभिः गुणिता दश त्रिदश, त्रिदशानां सहस्राणां समाहारः
त्रिदशसहस्री । हाटकानि — कनकानि ।
श्लोक २-२९ अ घातो वर्ग ६४ स्यात् । स्वेष्टोनयुतद्वयस्य — समानौ द्वौ राशी विन्यस्य ८।८
स्वेष्टोन-युत ६।१० तयोर्घातः ६० स्वेष्ट २ कृती ४ युक्तः ६४ वर्गः स्यात् । सेष्टकृतिः — इष्टकृतिसहितः ।
एकादि — एकादि द्विचयेष्टगच्छानां | ८ | युतिः संकलनं रूपेणोणो [नो] गच्छः दलितः प्रचयताडितो
मिश्रः प्रभवेण पदाम्यस्तः इति सूत्रेण | २ | वर्गो भयेत् ६४। इति घनं ८।
| १ |
श्लोक २-३० अ द्विस्यानप्रभृतीनाम् — षट्पंचाशत् द्विशत (२५६) इति त्रिस्थानान्तं वर्गे ।

• यह ज्ञात नहीं होता कि इनका सम्बन्ध किस-किस श्लोक से है ।
  † ( णान्ततः ? )

गणितसारसंग्रह ६३
षड्वर्गः ३६। पंचाशतवर्गः २५००। द्विशतवर्गः ४००००। सर्ववर्गसंयोगः ४२५३६। द्विशत-षट्पंचांषड्-
[ ०शद् ] घातः ११२००। पंचाशत्-षड्घातः ३००। तद्विगुणः २२४००।६००। तेन विमिश्रितः सर्व-
वर्गसंयोगः ६५५३६। तेषाम्—द्विप्रभृतिकल्पितस्थानानाम् । क्रमघातेन—द्विस्थानप्रभृतिराशीनाम्
अन्त्यस्थानं शेषस्थानैर्गुणयित्वा, पुनः शेषान्यस्थानं शेषस्थानैर्गुणयित्वा, तेन क्रमेण प्रथमस्थानपर्यन्त
गुणनक्षण क्रमघातः। तेन पुनः द्विस्थानप्रभृतीनां राशीनाम्, इत्यभिप्रायेण वर्गरचनां स्फुटयति ।
| ४ | द्विवर्ग ४ त्रिवर्ग ९ चतुर्वर्ग १६ तत्संयोगः २९ तेषां क्रमघातः द्विकत्रिकमिश्रेण चतुष्कं
| ३ | गुणयेत् २०। द्विकेन त्रिकं गुणयित्वा मिश्रितः सन् २६। द्विगुणो ५२। अनेन
| २ | मिश्रितेन वर्गः ८१।
श्लोक २-३१ अ कृत्वान्त्यकृतिम्-कृत्वा ७५ अन्त्यकृतिं ४९'५ अन्त्यं द्विगुणमुत्सार्य | ४३'५ / १४ | शेष
५ पदैर्हन्यात् | ४९'५ / ७० | शेषानुत्सार्य | ४९ X ५ / ७० | कृत्वा तस्यकृतिं | ४९२५ / ७० | लब्धः ५६२५ इति सर्वत्र
| ७ | X | ५ | कर्तव्यः द्वयंकानां वर्गकोष्ठः। पंचांकानां वर्गकोष्ठरचना
| ४ | ९ | ० | ५ |
| | ७ | २ | |
| ६ | X | ५ | X | ५ | X | ३ | X | ६ |
| ६ | ६ | ४ | ३ | २ | ० | ० | ६ | ६ | लब्धवर्गाः
| ६ | २ | ५ | ३ | ६ | ६ | ९ | ३ | | ४२९४९६७२९६॥ उ० १०
| | ५ | २ | ५ | ० | ३ | | | |
| | | | ३ | | | | | |
स अयमर्थः—अन्त्यराशिं वर्गे कृत्वा पुनरन्त्यराशिं द्विगुणं कृत्वा पुरो गमयित्वा शेषस्थानैर्गुणयेत् ।
शेषस्थानानि पुरो गमयित्वा पूर्वंकथितक्रिया कर्तव्या ।

परिशिष्ट-६
[ Reprinted from the First Edition. ]
PREFACE
Soon after I was appointed Professor of Sanskrit and Compara-
tive Philology in the Presidency College at Madras, and in that
capacity took charge of the office of the Curator of the Goverment
Oriental Manuscripts Library, the late Mr. G. H. Stuart, who was
then the Director of Public Instruction, asked me to find out if in
the Manuscripts Library in my charge there was any work of
value capable of throwing new light on the history of Hindu
mathematics, and to publish it, if found, with an English trans-
lation and with such notes as were necessary for the elucidation
of its contents. Accordingly the mathematical manuscripts in the
Library were examined with this object in view; and the examina-
tion revealed the existence of three incomplete manuscripts of
Mahāvīrācārya's Gaṇita-sāra-saṅgraha. A cursory persual of these
manuscripts made the value of this work evident in relation to the
history of Hindu Mathematics. The late Mr. G. H. Stuart's interest
in working out this history was so great that, when the existence of
the manuscripts and the historical value of the work were brought to
his notice, he at once urged me to try to procure other manuscripts
and to do all else that was necessary for its proper publication.
He gave me much advice and encouragement in the early stages of
my endeavour to publish it; and I can well guess how it would have
gladdened his heart to see the work published in the form he
desired. It has been to me a source of very keen regret that it did
not please Providence to allow him to live long enough to enable
me to enhance the value of the publication by means of his continued
guidance and advice; and my consolation now is that it is something
to have been able to carry out what he with scholarly delight
imposed upon me as a duty.
Of the three manuscripts found in the library one is written
on paper in Grantha characters, and contains the first five chapters
of the work with a running commentary in Sanskrit; it has been
denoted here by the letter P. The remaining two are palm-leaf

गणितसारसंग्रह ६५
manuscripts in Kanarese characters, one of them containing, like P,
the first five chapters, and the other the seventh chapter dealing
with the geometrical measurement of areas. In both these manu-
scripts there is to be found, in addition to the Sanskrit text of the
original work, a brief statement in the Kanarese language of the
figures relating to the various illustrative problems as also of the
answers to those same problems. Owing to the common charact-
eristics of these manuscripts and also owing to their not overlapping
one another in respect of their contents, it has been thought advisable
to look upon them as one manuscript and denote them by K. Another
manuscript, denoted by M, belongs to the Government Oriental
Library at Mysore, and was received on loan from Mr. A Mahadeva
Sastri, B. A., the Curator of that institution. This manuscript is a
transcription on paper in Kanarese characters of an original palm-
leaf manuscript belonging-to a Jaina Pandit, and contains the whole
of the work with a short commentary in the Kanarese language
by one Vallabha, .who claims to be the author of also a Telugu
commentary on the same work. Althought incorrect in many places,
it proved to be of great value on account of its being complete and
containing the Kanarese commentary; and my thanks are specially
due to Mr. A. Mahadeva Sastri for his leaving it sufficiently long at my
disposal. A fifth manuscript, denoted by B, is a transcription on paper
in Kanarese characters of a palm-leaf manuscript found in a Jaina
monastery at Mudbidri in South Canara, and was obtained through
the kind effort of Mr. R. Krishnamacharyar, M. A., he Sub-assistant
Inspector of Sanskrit Schools in Madras, and Mr. U. B. Venkatara-
manaiya of Mudbidri. This manuscript also contains the whole
work, and gives, like K, in Kanarese a brief statement of the pro-
blems and their answers. The endeavour to secure more manuscripts
having proved fruitless, the work has had to be brought out with
the aid of these five manucripts; and owing to the technical charac-
ter of the work and its elliptical and often riddle-like language and
the inaccuracy of the manuscripts, the labour involved in bringing
it out with the translation and the requisite notes has been heavy
and trying. There is, however, the satisfaction that all this labour
has been bestowed on a worthy work of considerable historical
value.

६६ गणितसारसंग्रह
It is a fortunate circumstance about the Gaṇita-sāra-saṅgraha
that the time when its author Mahāvīrācārya lived may be made
out with fair accuracy. In the very first chapter of the work, we have,
immediately after the two introductory stanzas of salutation to
Jina Mahāvīra, six stanzas describing the greatness of a king, whose
name is said to have been Cakrikā-bhañjana, and who appears to have
been commonly known by the title of Amōghavarṣa Nṛpatuṅga; and
in the last of these six stanzas there is a benediction wishing progre-
ssive prosperity to the rule of this king. The results of modern
Indian epigraphical research show that this king Amōghavarṣa
Nṛpatuṅga reigned from A. D. 814 or 815 to A. D. 877 or 878.* Since
it appears probable that the author of the Gaṇita-sāra-saṅgraha
was in some way attached to the court of this Rāṣṭrakūṭa king
Amōghavarṣa Nṛpatuṅga, we may consider the work to belong to
the middle of the ninth century of the Christian era. It is now gene-
rally accepted that, among well-known early Indian mathematicians
Āryabhaṭa lived in the fifth, Varāhamihira in the sixth, Brahma-
gupta in the seventh and Bhāskarācārya in the twelfth century of
the Christian era; and chronologically, therefore, Mahāvīrācārya
comes between Brahmagupta and Bhāskarācārya. This in itself is a
point of historical noteworthiness; and the further fact that the
author of the Gaṇita-sāra-saṅgraha belonged to the Kanarese speak-
ing portion of South India in his days and was a Jaina in religion
is calculated to give an additional importance to the historical value
of his work. Like the other mathematicians mentioned above,
Mahāvīrācārya was not primarily an astronomer, although he knew
well and has himself remarked about the usefulness of mathematics
for the study of astronomy. The study of mathematics seems to have
been popular among Jaina scholars; it forms, in fact, one of their
four Anuyōgas or auxiliary sciences indirectly serviceable for the
attainment of the salvation of soul-liberation known as mōkṣa.
A comparison of the Gaṇita-sāra-saṅgraha with the correspon-
ding portions in the Brahmasphuṭa-siddhānta of Brahmagupta is

• Vide Nilgund Inscription of the time of Amoghavarṣa I, A. D. 866; edited
  by J. F. Fleet, PH. D., C. I. E., in Epigraphia Indica, Vol. VI, pp. 98--108.

गणितसारसंग्रह ६७
calculated to lead to the conclusion that, in all probability,
Mahāvīrācārya was familiar with the work of Brahmagupta and
endeavoured to improve upon it to the extent to which the scope
of his Gaṇita-sāra-saṅgraha permitted such improvement. Mahāvīrā-
chārya’s classification of arithmetical operations is simpler, his
rules are fuller and he gives a large number of examples for
illustration and exercise. Pṛthūdaksvāmin, the well-known commen-
tator on the Brahmasphuṭa-siddhānta, could not have been chronolo-
gically far removed form Mahāvīrācārya, and the similarity of some
of the examples given by the former with some of those of the latter
naturally arrests attention. In any case it cannot be wrong to believe,
that, at the time, when Mahāvīrācārya wrote his Gaṇita-sāra-saṅgraha,
Brahmagupta must have been widely recognized as a writer of
authority in the field of Hindu astronomy and mathematics. Whether
Bhāskarācārya was at all acquainted with the Gaṇita-sāra-saṅgraha
of Mahāvīrācārya, it is not quite easy to say. Since neither
Bhāskarācārya nor any of his known commentators seem to quote
from him or mention him by name, the natural conclusion appears
to be that Bhāskarācārya’s Siddhānta-śirōmaṇi, including his
Līlāvatī and Bījagaṇita, was intended to be an improvement in the
main upon the Brahmasphuṭa-siddhānta of Brahmagupta. The fact
that Mahāvīrācārya was a Jaina might have prevented Bhāskarācārya
from taking note of him; or it may be that the Jaina mathematician’s
fame had not spread far to the north in the twelfth century of the
Christian era. His work, however, seems to have been widely known
and appreciated in Southern India. So early as in the course
of the eleventh century and perhaps under the stimulating
influence of the enlightened rule of Rājarājanarēndra of
Rajahmundry, it was translated into Telugu in verse by Pāvulūri
Mallana; and some manuscripts of this Telugu translation are now
to be found in the Government Oriental Manuscripts Library here
at Madras. It appeared to me that to draw suitable attention to
the historical value of Mahāvīrācārya’s Gaṇita-sāra-saṅgraha, I could
not do better than seek the help of Dr. David Eugene Smith of the
Columbia University of New York, whose knowledge of the history
of mathematics in the West and in the East is known to be wide

६८ गणितसारसंग्रह
and comprehensive, and who on the occasion when he met me in
person at Madras showed great interest in the contemplated
publication of the Gaṇita-sāra-saṅgraha and thereafter read a paper
on that work at the Fourth International Congress of Mathe-
maticians held at Rome in April 1908. Accordingly I requested him
to write an introduction to this edition of the Gaṇita-sāra-saṅgraha,
given in brief outline what he considers to be its value in building
up the history of Hindu mathematics. My thanks as well as the
thanks of all those who may as scholars become interested in this
publication are therefore due to him for his kindness in having
readily complied with my request; and I feel no doubt that his
introduction will be read with great appreciation.
Since the origin of the decimal system of notation and of the
conception and symbolic representation of zero are considered to
be important questions connected with the history of
Hindu mathematics, it is well to point out here that in the
Gaṇita-sārasaṅgraha twenty four notational places are mentioned,
commencing with the units place and ending with the place called
mahākṣōbha, and that the value of each succeeding place is taken
to be ten times the value of the immediately preceding place.
Although certain words forming the names of certain things are
utilized in this work to represent various numerical figures, still in
the numeration of of numbers with the aid of such words
the decimal system of notation is almost invariably followed.
If we took the words moon, eye, fire and sky to represent respectively
1, 2, 3 and 0, as their Sanskrit equivalents are understood in this
work, then, for instance, fire-sky-moon-eye would denote the number
2103, and moon-eye-sky-fire would denote 3021, since these nominal
numerals denoting numbers are generally repeated in order from
the units place upwards. This combination of nominal numerals
and the decimal system of notation has been adopted obviously for
the sake of securing metrical convenience and avoiding at the same
time cumbrous ways of mentioning numerical expressions; and it
may well be taken for granted that for the use of such nominal
numerals as well as the decimal system of notation Mahāvīrācārya
was indebted to his predecessors. The decimal system of notation is

गणितसारसंग्रह ६९
distinctly described by Āryabhaṭa, and there is evidence in his
writings to show that he was familiar with nominal numerals. Even
in his brief mnemonic method of representing numbers by certain
combinations of the consonants and vowels found in the Sanskrit
language, the decimal system of notation is taken for granted; and
ordinarily 19 notational places are provided for therein. Similarly
in Brahmagupta's writings also there is evidence to show that he
was acquainted with the use of nominal numerals and the decimal
system of notation. Both Āryabhaṭa and Brahmagupta claim that
their astronomical works are related to the Brahma-siddhānta; and
in a work of this name, which is said to form a part of what is
called Śākalya-saṁhitā and of which a manuscript copy is to be
found in the Government Oriental Manuscripts Library here,
numbers are expressed mainly by nominal numerals used in accordance
with the decimal system of notation. It is not of course meant to
convey that this work is necessarily the same as what was known
to Ārayabhaṭa and Brahmagupta; and the fact of its using nominal
numerals and the decimal system of notation is mentioned here for
nothing more than what it may be worth.
It is generally recognized that the origin of the conception of
zero is primarily due to the invention and practical utilization of a
system of notation wherein the several numerical figures used have
place-values apart from what is called their intrinsic value. In
writing out a number according to such a sytem of notation, any
notational place may be left empty when no figure with an intrinsic
value is wanted there. It is probable that owing to this very reason
the Sanskrit word śūnya, meaning 'empty', came to denote the zero;
and when it is borne in mind that the English word 'cipher' is derived
from an Arabic word having the same meaning as the Sanskrit śūnya,
we may safely arrive at the conclusion that in this country the
conception of the zero came naturally in the wake of the decimal
system of notation : and so early as in the fifth century of the
Christian era, Āryabhaṭa is known to have been fully aware of this
valuable mathematical conception. And in regard to the question
of a symbol to represent this conception, it is well worth
bearing in mind that operations with the zero cannot be

७० गणितसारसंग्रह
carried on–not to say cannot be even thought of easily–without a
symbol of some sort to represent it. Mahāvīrācārya gives, in the
very first chapter of his Gaṇita-sāra-saṅgraha, the results of the
operations of addition, subtraction, multiplication and division
carried on in relation to the zero quantity; and although he is wrong
in saying that a quantity, when divided by zero, remains unaltered,
and should have said, like Bhāskarācārya, that the quotient in such
a case is infinity, still the very mention of operations in relation to
zero is enough to show that Mahāvīrācārya must have been aware
of some symbolic representation of the zero quantity. Since
Brahmagupta, who must have lived at least 150 years before
Mahāvīrācārya, mentions in his work the results of operations in
relation to the zero quantity, it is not unreasonable to suppose that
before his time the zero must have had a symbol to represent it in
written calculations. That even Āryabhaṭa knew such a symbol is
not at all improbable. It is worthy of note in this connection that in
enumerating the nominal numerals in the first chapter of his work,
Mahāvīrācārya mentions the names denoting the nine figures from 1 to
9, and then gives in the end the names denoting zero, calling all the
ten by the name of saṅkhyā : and from this fact also, the inference
may well be drawn that the zero had a symbol, and that it was well
known that with the aid of the ten digits and the decimal system of
notation numerical quantites of all values may be definitely and
accurately expressed. What this known zero-symbol was, is, however,
a different question.
The labour and attention bestowed upon the study and trans-
lation and annotation of the Gaṇita-sāra-saṅgraha have made it
clear to me that I was justified in thinking that its publication might
prove useful in elucidating the condition of mathematical studies as
they flourished in South India among the Jainas in the ninth century
of the Christian era; and it has been to me a source of no small
satisfaction to feel that in bringing out this work in this form,
I have not wasted my time and thought on an, unprofitable
undertaking. The value of the work is undoubtedly more historical
than mathematical. But it cannot be denied that the step by step
construction of the history of Hindu culture is a worthy endeavour,

गणितसारसंग्रह ७१
and that even the most insignificant labourer in the field of such
an endeavour deserves to be looked upon as a useful worker.
Although the editing of the Gaṇita-sāra-saṅgraha has been to me a
labour of love and duty, it has often been felt to be heavy and taxing;
and I, therefore, consider that I am specially bound to acknowledge
with gratitude the help which I have received in relation to it. In
the early stage, when conning and collating and interpreting the
manuscripts was the chief work to be done, Mr. M. B. Varadaraja
Aiyangar, B. A., B. L., who is an Advocate of the Chief Court at
Bangalore, co-operated with me and gave me an amount of aid for
which I now offer him my thanks. Mr. K. Krishnaswami Aiyangar,
B. A.; of the Madras Christian College, has also rendered consider-
able assistance in this manner; and to him also I offer my thanks.
Latterly I have had to consult on a few occasions Mr. P. V. Seshu
Aiyar, B. A., L. T., Professor of Mathematical Physics in the Presidency
College here, in trying to explain the rationale of some of the rules
given in the work; and I am much obliged to him for his ready
willingness in allowing me thus to take advantage of his expert
knowledge of mathematics. My thanks are, I have to say in
conclusion, very particularly due to Mr. P. Varadacharya, B. A.,
Librarian of the Government Oriental Manuscripts Library at Madras,
but for whose zealous and steady co-operation with me throughout
and careful and continued attention to details, it would indeed
have been much harder for me to bring out this edition of the
Gaṇit-sāra-saṅgraha.
$\left. {l} February 1912,
Madras. \right}$ M. RANGACHARYA.

INTRODUCTION
BY
DAVID EUGENE SMITH
PROFESSOR OF MATHEMATICS IN TEACHERS' COLLEGE,
COLUMBIA UNIVERSITY, NEW YORK.
We have so long been accustomed to think of Pāṭaliputra on
the Ganges and of Ujjain over towards the Western Coast of India
as the ancient habitats of Hindu mathematics, that we experience
a kind of surprise at the idea that other centres equally important
existed among the multitude of cities of that great empire. In the
same way we have known for a century, chiefly through the labours
of such scholars as Colebrooke and Taylor, the works of Āryabhaṭa,
Brahmagupta, and Bhāskara, and have come to feel that to these
men alone are due the noteworthy contributions to be found in
native Hindu mathematics. Of course a little reflection shows this
conclusion to be an incorrect one. Other great schools, particularly of
astronomy, did exist, and other scholars taught and wrote and added
their quota, small or large, to make up the sum total. It has, however,
been a little discouraging that native scholars under the English
supremacy have done so little to bring to light the ancient mathe-
matical material known to exist and to make it known to the Western
world. This neglect has not certainly been owing to the absence of
material, for Sanskrit mathematical manuscripts are known, as are
also Persian, Arabic, Chinese, and Japanese; and many of these are
well worth translating from the historical standpoint. It has rather
been owing to the fact that it is hard to find a man with the requisite
scholarship, who can afford to give his time to what is necessarily
a labour of love.
It is a pleasure to know that such a man has at last appeared
and that, thanks to his profound scholarship and great perseverance,

गणितसारसंग्रह ७३
we are now receiving new light upon the subject of Oriental
mathematics, as known in another part of India and at a time
about midway between that of Āryabhaṭa and Bhāskara, and two
centuries later than Brahmagupta. The learned scholar, Professor
M. Raṅgācārya of Madras, some years ago became interested in
the work of Mahāvīrācārya, and has now completed its translation,
thus making the mathematical world his perpetual debtor; and I
esteem it a high honour to be requested to write an introduction
to so noteworthy a work.
Mahāvīrācārya appears to have lived in the court of an old
Rāṣṭrakūṭa monarch, who ruled probably over much of what is now
the kingdom of Mysore and other Kanarese tracts, and whose name
is given as Amōghavarṣa Nṛpatuṅga. He is known to have ascended
the throne in the first half of the ninth century A. D., so that we
may roughly fix the date of the treatise in question as about 850.
The work itself consists, as will be seen, of nine chapters like
the Bīja-gaṇita of Bhāskara; it has one more chapter than the
Kuṭṭaka of Brahmagupta. There is, however, no significance in
this number, for the chapters are not at all parallel, although certain
of the topics of Brahmagupta's Gaṇita and Bhāskara's Līlāvatī are
included in the Gaṇita-Sāra-Saṅgraha.
In considering the work, the reader naturally repeats to
himself the great questions that are so often raised:—How much of
this Hindu treatment is original? What evidences are there here
of Greek influence? What relation was there between the great
mathematical centres of India? What is the distinctive feature, if
any, of the Hindu algebraic theory?
Such questions are not new. Davis and Strachey, Colebrooke
and Taylor, all raised similar ones a century ago, and they are by
no means satisfactorily answered even yet. Nevertheless, we are
making good progress towards their satisfactory solution in
the not too distant future. The past century has seen several
Chinese and Japanese mathematical works made more or less
familiar to the West; and the more important Arab treatises are now
quite satisfactorily known. Various editions of Bhāskara have
appeared in India; and in general the great treatises of the Orient

७४ गणितसारसंग्रह
have begun to be subjected to critical study. It would be strange,
therefore, if we were not in a position to weigh up, with more
certainty than before, the claims of the Hindu algebra. Certainly
the persevering work of Professor Raṅgācārya has made this more
possible than ever before.
As to the relation between the East and the West, we should
now be in a position to say rather definitely that there is no evidence
of any considerable influence of Greek algebra upon that of India.
The two subjects were radically different. It is true that Diophantus
lived about two centuries before the first Āryabhaṭa, that the paths
of trade were open from the West to the East, and that the itinerant
scholar undoubtedly carried learning from place to place. But the
spirit of Diophantus, showing itself in a dawning symbolism and in
a peculiar type of equation, is not seen at all in the works of the
East. None of his problems, not a trace of his symbolism, and not
a bit of his phraseology appear in the works of any Indian
writer on algebra. On the contrary, the Hindu works have a style
and a range of topics peculiarly their own. Their problems lack the
cold, clear, geometric precision of the West ; they are clothed in that
poetic language which distinguishes the East, and they relate to
subjects that find no place in the scientific books of the Greeks. With
perhaps the single exception of Metrodorus, it is only when we come
to the puzzle problems doubtfully attributed to Alcuin that we find
anything in the West which resembles, even in a slight degree, the
work of Alcuin's Indian contemporary, the author of this treatise.
It therefore seems only fair to say that, although some know-
ledge of the scientific work of any one nation would, even in those
remote times, naturally have been carried to other peoples by some
wandering savant, we have nothing in the writings of the Hindu
algebraists to show any direct influence of the West upon their
problems or their theories.
When we come to the question of the relation between the different
sections of the East, however, we meet with more difficulty. What
were the relations, for example, between the school of Pāṭaliputra,
where Āryabhaṭa wrote, and that of Ujjain, where both Brahmagupta
and Bhāskara lived and taught ? And what was the relation of each

गणितसारसंग्रह ७५
of these to the school down in South India, which produced this
notable treatise of Mahāvīrācārya ? And, a still more interesting
question is, what can we say of the influence exerted on China by
Hindu scholars, or vice versa? When we find one set of early inscrip-
tions, those at Nānā Ghāt, using the first three Chinese numerals,
and another of about the same period using the later forms of Mesopo-
tamia, we feel that both China and the West may have influenced
Hindu science. When, on the other hand, we consider the problems
of the great trio of Chinese algebraists of the thirteenth century,
Ch'in Chiushang, Li Yeh, and Chu Shih-chieh, we feel that
Hindu algebra must have had no small influence upon the North
of Asia, although it must be said that in point of theory the Chinese
of that period naturally surpassed the earlier writers of India.
The answer to the questions as to the relation between the
schools of India cannot yet be easily given. At first it would seem
a simple matter to compare the treatises of the three or four great
algebraists and to note the similarities and differences. When this
is done, however, the result seems to be that the works of Brahma-
gupta, Mahāvīrācārya, and Bhāskara may be described as similar
in spirit but entirely different in detail. For example, all of these
writers treat of the areas of polygones, but Mahāvīrācārya is the
only one to make any point of those that are re-entrant. All of them
touch upon the area of a segment of a circle, but all give different
rules. The so-called janya operation (page 209) is akin to work
found in Brahmagupta, and yet none of the problems is the same.
The shadow problems, primitive cases of trigonometry and gnomo-
nics, suggest a similarity among these three great writers, and yet
those of Mahāvīrācārya are much better than the one to be found in
either Brahmagupta or Bhāskara, and no questions are duplicated.
In the way of similarity, both Brahmagupta and Mahāvīrācārya
give the formula for the area of a quadrilateral,
√((s - a)(s - b)(s - c)(s - d))
—but neither one observes that it holds only for a cyclic
figure. A few problems also show some similarity such as that
of the broken tree, the one about the anchorites, and the