Āryabhaṭīya/Chapter 02
ब्रह्मकुशशिबुधभृगुरविकुजगुरुकोणभगणान् नमस्कृत्य ।
आर्यभटस्तु इह निगदति कुसुमपुरे अभ्यर्चितं ज्ञानम् ।। २.१ ।।
Having bowed with reverence to Brahma, Earth, Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn and the asterisms, Aryabhata sets forth here the knowledge honoured at Kusumapura.
Āryabhaṭīya/Chapter 02
एकं दश च शतं च सहस्रं अयुत नियुते तथा प्रयुतम् ।
कोटि अर्बुदं च वृन्दं स्थानात्स्थानं दशगुणं स्यात् ।। २.२ ।।
Eka (units place), dasa (tens place), Sata (hundreds place), sahasra (thousands place), ayuta (ten thousands place), niyuta (hundred thousands place), prayuta (millions place), koti (ten millions place), arbuda (hundred millions place), and vrnda (thousand millions place) are, respectively, from place to place, each ten times the preceding.
Āryabhaṭīya/Chapter 02
वर्गस्सम चतुरश्रस्फलं च सदृश द्वयस्य संवर्गस् ।
सदृश त्रयसंवर्गस्घनस्तथा द्वादशाश्रिस् स्यात् ।। २.३ ।।
(a-b) An equilateral quadrilateral with equal diagonals and also the area thereof are called 'square'. The product of two equal quantities is also 'square'. (c-d) The continued product of three equals as also the (rectangular) solid having twelve (equal) edges is called a ;cube;
Āryabhaṭīya/Chapter 02
भागं हरेतवर्गान्नित्यं द्विगुणेन वर्गमूलेन ।
वर्गात्वर्गे शुद्धे लब्धं स्थानान्तरे मूलम् ।। २.४ ।।
(Having subtracted the greatest possible square from the last odd place and then having written down the square root of the number subtracted in the line of the square root) always divide the even place (standing on the right) by twice the square root. Then, having subtracted the square (of the quotient) from the odd place (standing on the right), set down the quotient at the next place (i.e., on the right of the number already written in the line of the square root). This is the square root. (Repeat the process if there are still digits on the right).
Āryabhaṭīya/Chapter 02
अघनात् भजेत् द्वितीयात् त्रिगुणेन घनस्य मूलवर्गेण ।
वर्गस् त्रिपूर्वगुणितस्शोध्यस् प्रथमात्घनस्च घनात् ।। २.५ ।।
(Having subtracted the greatest possible cube from the last cube place and then having written down the cube root of the number subtracted in the line of the cube root), divide the second non-cube place (standing onthe right of the last cube place) by thrice the square of the cube root (already obtained); (then) subtract from the first non-cube place (standing on the right of the second non-cube place) the square of the quotient multiplied by thrice the previous (cube root); and (then subtract) the cube (of the quotient) from the cube place (standing on the right of the first non-cube place) (and write down the quotient on the right of the previous cube root in the line of the cube root, and treat this as the new cube root. Repeat the process if there are still digits on the right).
Āryabhaṭīya/Chapter 02
त्रिभुजस्य फलशरीरं समदलकोटीभुजा अर्धसंवर्गस् ।
ऊर्ध्वभुजातद्संवर्ग अर्धं सस्घनस् षषश्रिसिति ।। २.६ ।।
{a-b) The product of the perpendicular (dropped from the vertex on the base) and half the base gives the measure of the area of a triangle. (c-d) Half the product of that area (of the triangular base) and the height is the volume of a six-edged solid.
Āryabhaṭīya/Chapter 02
समपरिणाहस्य अर्धं विष्कम्भ अर्धहतं एव वृत्तफलम् ।
तद्निजमूलेन हतं घनगोलफलं निरवशेषम् ।। २.७ ।।
(a-b) Half of the circumference, multiplied by the semi-diameter certainly gives the area of a circle. (c-d) That area (of the diametral section) multiplied by its own square root gives the exact volume of a sphere.
Āryabhaṭīya/Chapter 02
आयामगुणे पार्श्वे तद्योगहृते स्वपातलेखे ।
विस्तरयोग अर्धगुणे ज्ञेयं क्षेत्रफलं आयामे ।। २.८ ।।
(Severally) multiply the base and the face (of the trapezium) by the height, and divide (each product) by the sum of the base and the face; the results are the lengths of the perpendiculars on the base and the face (from the point of intersection of the diagonals). The results obtained by multiplying half the sum of the base and the face by the height is to be known as the area (of the trapezium).
Āryabhaṭīya/Chapter 02
सर्वेषां क्षेत्राणां प्रसाध्य पार्श्वे फलं तदभ्यासस् ।
परिधेस् षष्भागज्या विष्कम्भ अर्धेन सा तुल्या ।। २.९ ।।
(a-b) In the case of all the plane figures, one should determine the adjacent sides (of the rectangle into which that figure can be transformed) and find the area by taking their product. (c-d) The chord of one-sixth of the circumference (of a circle) is equal to the radius.
Āryabhaṭīya/Chapter 02
चतुरधिकं शतं अष्टगुणं द्वाषष्टिस्तथा सहस्राणाम् ।
अयुत द्वयविष्कम्भस्य आसन्नस्वृत्तपरिणाहस् ।। २.१० ।।
100 plus 4, multiplied by 8, and added to 62,000 : this is the nearly approximate measure of the circumference of a circle whose diameter is 20,000.
Āryabhaṭīya/Chapter 02
समवृत्तपरिधिपादं छिन्द्यात् त्रिभुजात् चतुर्भुजात्च एव ।
समचापज्या अर्धानि तु विष्कम्भ अर्धे यथा इष्टानि ।। २.११ ।।
Divide a quadrant of the circumferenc of a circle (into as many parts as desired). Then, from (right) triangles and quadrilaterals, one can find as many Rsines of equal arcs as one likes, for any given radius.
Āryabhaṭīya/Chapter 02
प्रथमात्चापज्या अर्धात्यैरूनं खण्डितं द्वितीय अर्धम् ।
तद् प्रथमज्या अर्धांशैस्तैस्तैसूनानि शेषाणि ।। २.१२ ।।
The first Rsine divided by itself and then diminished by the quotient gives the second Rsine-difference. The same first Rsine diminished by the quotients obtained by dividing each of the preceding Rsines by the first Rsine gives the remaining Rsine-differences.
Āryabhaṭīya/Chapter 02
वृत्तं भ्रमेण साध्यं च चतुर्भुजं च कर्णाभ्याम् ।
साध्या जलेन समभूरध ऊर्ध्वं लम्बकेन एव ।। २.१३ ।।
A circle should be constructed by means of a pair of compasses; a triangle and a quadrilateral by means of the two hypotenuses (Karna). The level of ground should be tested by means of water; and verticality by means of a plumb.
Āryabhaṭīya/Chapter 02
शङ्कोस्प्रमाणवर्गं छायावर्गेण संयुतं कृत्वा ।
यत्तस्य वर्गमूलं विष्कम्भ अर्धं स्ववृत्तस्य ।। २.१४ ।।
Add the square of the height of the gnomon to the square of its shadow. The square root of that sum is the semi-diameter of the circle of shadow.
Āryabhaṭīya/Chapter 02
शङ्कुगुणं शङ्कुभुजाविवरं शङ्कुभुजयोर्विशेषहृतम् ।
यत्लब्धं सा छाया ज्ञेया शङ्कोस्स्वमूलात्हि ।। २.१५ ।।
Multiply the distance between the gnomon and the lamp-post (the latter being regarded as base) by the height of the gnomon and divide (the product) by the difference between (the heights of) the lamp-post (base) and the gnomon. The quotient (thus obtained) should be known as the length of the shadow measured from the foot of the gnomon.
Āryabhaṭīya/Chapter 02
छायागुणितं छायाअग्रविवरं ऊनेन भाजितं कोटी ।
शङ्कुगुणा कोटी सा छायाभक्ता भुजा भवति ।। २.१६ ।।
(When there are two gnomons of equal height in the same direction from the lamp-post), multiply the distance between the tips of the shadows (of the two gnomons) by the (larger or shorter) shadow and divide by the larger shadow diminished by the shorter one; the result is the upright (i.e., the distance of the tip of the larger or shorter shadow from the foot of the lamp-post). The upright multiplied by the height of the gnomon and divided by the (larger or shorter) shadow gives the base (i.e., the height of the lamp-post).
Āryabhaṭīya/Chapter 02
यस्च एव भुजावर्गस्कोटीवर्गस्च कर्णवर्गस्सस् ।
वृत्ते शरसंवर्गस् अर्धज्यावर्गस्सस्खलु धनुषोस् ।। २.१७ ।।
(In a right-angled triangle) the square of the base plus the square of the upright is the square of the hypotenuse. In a circle (when a chord divides it into two arcs), the product of the arrows of the two arcs is certainJy equal to the square of half the chord.
Āryabhaṭīya/Chapter 02
ग्रासऊने द्वे वृत्ते ग्रासगुणे भाजयेत्पृथक्त्वेन ।
ग्रासऊनयोगलब्धौ संपातशरौ परस्परतस् ।। २.१८ ।।
When one circle intersects another circle) multiply the diameters of the two circles each diminished by the erosion, by the erosion and divide (each result) by the sum of the diameters of the two circles after each has been diminished by the erosion: then are obtained the arrows of the arcs (of the two circles) intercepted in each other.
Āryabhaṭīya/Chapter 02
इष्टं वि एकं दलितं सपूर्वं उत्तरगुणं समुखं मध्यम् ।
इष्टगुणितं इष्टधनं तु अथ वा आदिअन्तं पद अर्धहतम् ।। २.१९ ।।
Diminish the given number of terms by one, then divide by two, then increase by the number of the preceding terms (if any), then multiply by the common difference, and then increase by the first term of the (whole) series : the result is the arithmetic mean (of the given number of terms). This multiplied by the given number of terms is the sum of the given terms. Alternatively, multiply the sum of the first and last terms (of the series or partial series which is to be summed up) by half the number of terms.
Āryabhaṭīya/Chapter 02
गच्छस् अष्टौत्तरगुणितात् द्विगुणऽदिउत्तरविशेषवर्गयुतात् ।
मूलं द्विगुणऽदिऊनं स्वौत्तरभजितं सरूपार्धम् ।। २.२० ।।
The number of terms (is obtained as follows): Multiply (the sum of the series) by eight and by the common difference, increase that by the square of the difference between twice the first term and the common difference, and then take the square root; then subtract twice the first term, then divide by the common difference, then add one (to the quotient), and then divide by two.
Āryabhaṭīya/Chapter 02
एकौत्तरऽदिउपचितेस्गच्चऽदि एकौत्तर त्रिसंवर्गस् ।
षष्भक्तस्सस्चितिघनस्स एकपदघनस्विमूलस्वा ।। २.२१ ।।
Of the series (upaciti) which has one for the first term and one for the common difference, take three terms in continuation, of which the first is equal to the given number of terms, and find their continued product. That (product), or the number of terms plus one subtracted from the cube of that, divided by 6, gives the citighana.
Āryabhaṭīya/Chapter 02
स एकसगच्छपदानां क्रमात् त्रिसंवर्गितस्य षष्ठसंशस् ।
वर्गचितिघनस्सस् भवेत्चितिवर्गस्घनचितिघनस्च ।। २.२२ ।।
The continued product of the three quantities, viz., the number of terms plus one, the same increased by the number of terms, and the number of terms, when divided by 6 gives the sum of the series of squares of natural numbers (vargacitighana). The square of the sum of the series of natural numbers (citi) gives the sum of the series of cubes of natural numbers (ghanacitighana).
Āryabhaṭīya/Chapter 02
सम्पर्कस्य हि वर्गात् विशोधयेतेव वर्गसम्पर्कम् ।
यत्तस्य भवति अर्धं विद्यात्गुणकारसंवर्गम् ।। २.२३ ।।
From the square of the sum of the two factors subtract the sum of their squares. One-half of that (difference) should be known as the product of the two factors.
Āryabhaṭīya/Chapter 02
द्विकृतिगुणात्संवर्गात् द्विअन्तरवर्गेण संयुतात्मूलम् ।
अन्तरयुक्तं हीनं तद्गुणकार द्वयं दलितम् ।। २.२४ ।।
Multiply the product by four, then add the square of the difference of the two (quantities), and then take the square root. (Set down this square root in two places). (In one place) increase it by the difference (of the two quantities), and (in the other place) decrease it by the same. The results thus obtained, when divided by two, give the two factors (of the given product).
Āryabhaṭīya/Chapter 02
मूलफलं सफलं कालमूलगुणं अर्धमूलकृतियुक्तम् ।
तद्मूलं मूल अर्धऊनं कालहृतं स्वमूलफलम् ।। २.२५ ।।
Multiply the interest on the principal plus the interest on that interest by the time and by the principal; (then) add the square of half the principal; (then) take the square root (then) subtract half the principal; and (then) divide by the time : the result is the interest on the principal.
Āryabhaṭīya/Chapter 02
त्रैराशिकफलराशिं तं अथ इच्छाराशिना हतं कृत्वा ।
लब्धं प्रमाणभजितं तस्मातिच्छाफलं इदं स्यात् ।। २.२६ ।।
In the rule of three, multiply the 'fruit' (phala) by the 'requisition' (iccha) and divide the resulting product by the 'argument' (pramana). Then is obtained the 'fruit corresponding to the requisition' (icchaphala).
Āryabhaṭīya/Chapter 02
छेदास्परस्परहतास् भवन्ति गुणकारभागहाराणाम् ।
छेदगुणं सछेदं परस्परं तत्सवर्णत्वम् ।। २.२७ ।।
(a-b) .The numerators and denominators of the multipliers and divisors should be multiplied by one another. (c-d) Multiply the numerator as also the denominator of each fraction by the denominator of the other fraction; then the (given) fractions are reduced to a common denominator.
Āryabhaṭīya/Chapter 02
गुणकारास्भागहरास्भागहरास्ते भवन्ति गुणकारास् ।
यस्क्षेपस्ससपचयसपचयस्क्षेपस्च विपरीते ।। २.२८ ।।
In the method of inversion multipliers become divisors and divisors become multipliers, additive becomes subtractive and subtractive becomes additive.
Āryabhaṭīya/Chapter 02
राशिऊनं राशिऊनं गच्छधनं पिण्डितं पृथक्त्वेन ।
वि एकेन पदेन हृतं सर्वधनं तत् भवति एवम् ।। २.२९ ।।
The sums of all (combinations of) the (unknown) quantities except one (which are given) separately should be added together; and the sum should be written down separately and divided by the number of (unknown) quantities less one : the quotient thus obtained is certainly the total of all the (unknown) quantities, (This total severally diminished by the given sums gives the various unknown quantities).
Āryabhaṭīya/Chapter 02
गुलिकाअन्तरेण विभजेत् द्वयोस्पुरुषयोस्तु रूपकविशेषम् ।
लब्धं गुलिकामूल्यं यदि अर्थकृतं भवति तुल्यम् ।। २.३० ।।
Divide the difference between the rupakas with the two persons by the difference between their gulikas. The quotient is the value of one gulika, if the possessions of the two persons are of equal value.
Āryabhaṭīya/Chapter 02
भक्ते विलोमविवरे गतियोगेन अनुलोमविवरे द्वौ ।
गतिअन्तरेण लब्धौ द्वियोगकालौ अतीताइष्यौ ।। २.३१ ।।
Divide the distance between the two bodies moving in the opposite directions by the sum of their speeds, and the distance between the two bodies moving in the same direction by the difference of their speeds; the two quotients will give the time elapsed since the two bodies met or to elapse before they will meet.
Āryabhaṭīya/Chapter 02
अधिकाग्रभागहारं छिन्द्यातूनाग्रभागहारेण ।
शेषपरस्परभक्तं मतिगुणं अग्रान्तरे क्षिप्तम् ।। २.३२ ।।
Divide the divisor corresponding to the greater remainder by the divisor corresponding to the smaller remainder. (Discard the quotient). Divide the remainder obtained (and the divisor) by one another (until the number of quotients of the mutual division is even and the final remainder is small enough). Multiply the final remainder by an optional number and to the product obtained add the difference of the remainders (corresponding to the greater and smaller divisors; then divide this sum by the last divisor of the mutual division. The optional number is to be so chosen that this division is exact. Now place the quotients of the mutual division one below the other in a column; below them write the optional number and underneath it the quotient just obtained.
Āryabhaṭīya/Chapter 02
अधसुपरिगुणितं अन्त्ययुजूनाग्रछेदभाजिते शेषम् ।
अधिकाग्रछेदगुणं द्विछेदाग्रं अधिकाग्रयुतम् ।। २.३३ ।।
Then reduce the chain of numbers which have been written down one below the other, as follows): Multiply by the last but one number (in the bottom) the number just above it and then add the number just below it (and then discard the lower number).. (Repeat this process until there are only two pumbers in the chain). Divide (the upper number) by the divisor corresponding to the smaller remainder, then multiply the remainder obtained by the divisor corresponding to the greater remainder, and then add the greater remainder: the result is the dvicchedagra . (i.e., the number answering to the two divisors). (This is also the remainder corresponding to the divisor equal to the product of the two divisors).