[...]... square-root is, at this point: 25 The remainder of the division of 22 by 5 is set down in the place of the previously written digits: av 2 v 5 The quotient is the root in the next place When subtracting the square from the square The quotient is 5 The next place being a square-place, one subtracts the square of 5 The square-root found is 25 B .2 av 2 A − a2 1 02 − 2ab10 − b2 is computed v 5 − 52 0 a.10... one has made the square of the last term’ (a.1 02 )2 + 2a1 02 (b.10 + c) + (b.10 )2 + 2b10.c 5 6 5 0 − 52 = 25 When adding this value to the partial square found according to its power of ten, and placing it: 1 5 6 2 (a.1 02 )2 + 2a1 02 (b.10 + c) + (b10 )2 + 2b10.c + c2 5 5 The process ends here as there are no more digits The square obtained is: 15 625 (a.1 02 + b.10 + c )2 10 Supplements A .2. 2 Cubing No extensive... the ‘non-square’ place Twice the partial square root is 2 × 2 = 4 One performs the following division: 22 2 =5+ 4 4 Extracting the squareroot of A = (a.10 + b )2 b is computed as the quotient of the division of the two higher digits by a2 Then A − a2 1 02 − 2ab10 is set down a.10 + b is the partial square-root extracted 5 is the quotient, it is the second digit of the square-root to be extracted The partial... to the respective decimal places of each digit: 2 5 0 0 6 3 2 − Erasing the last digit 2 5 0 0 3 2 − Considering that the number to cube is b  12 Supplements Table 2: Cubing 63 Cubing digit the next As 33 = 27 , the disposition would be: 2 5 0 0 a3 103 + 3a2 1 02 b 3a10.b2 + b3 + 3 2 − 2 7 Adding according to the respective places of each digit: 2 As there are no more digits the process ends here The. .. with the remaining digits As 3 × 62 × 3 = 324 , the disposition would be: 2 (a.10)3 + 3a2 1 02 b 6 3 1 6 − − − 3 2 4 − − Adding according to the respective decimal places of each digit: 2 Computing successively the product of 3 times the last digit with the square of the following digits 4 8 6 4 − 3 − As 3 × 6 × 32 = 1 62, the disposition would be: 2 4 (a.10)3 + 3a10.b2 3a2 1 02 b + 6 3 8 4 − − 1 6 2 −... than the first digit (Or smaller than a two-digit/three digit-number if the first digit of the number whose root is extracted does not fall on a place whose power of ten is a cube.) Step 2 Subtract the cube from the first digit (or from the two-digit/three-digit number) Replace the digit (resp two-digit/three-digit number) by the difference The cube-root of the subtractor is the first digit of the cube-root... Step 2 Computing the successive products of 2 times the last digit with the remaining digits (2ab.103 and 2ac.1 02 ); Step 3 Adding the successive products, according to their respective powers of 10 to the partial square (a2 104 + 2ab.103 + 2ac.1 02 ); Step 4 Erasing the last digit, and “shifting” Then starting the process again, until no more digits of the initial number are left (Reiterating the process... fraction Dividing the cube of the numerator by the cube of the denominator: 25 0047 5 12 = 488 + 191 5 12 , which corresponds to the last column set down as a result: 488 191 5 12 B BAB .2. 4-5 B.1 Extracting square-roots B.1.1 Square and non-square places The procedure of square root extraction rests upon a categorization of the places ¯ of the decimal place-value system (defined in AB .2. 2) Aryabhata distinguishes... by the cube of the denominator: 25 0047 5 12 = 488 + 191 5 12 , which corresponds to the last column set down as a result: 488 191 5 12 B BAB .2. 4-5 B.1 Extracting square-roots B.1.1 Square and non-square places The procedure of square root extraction rests upon a categorization of the places ¯ of the decimal place-value system (defined in AB .2. 2) Aryabhata distinguishes square (varga) and non-square (a-varga)... concern directly the “Pythagoras Theorem”6 it is closely related to it Let us look at Figure 1 page 2 again The area of the square in the middle can be seen as the square of the diagonal of the rectangle (c2 ) But we can also consider the area of the first drawn square This is equal to the square of the sum of the two adjoining sides of the rectangle ((a + b )2 ) Now if we cut off the areas of the four triangles . in Germany Vol. 1/SN 30: ISBN 10: 3-7 64 3-7 29 1-5 e-ISBN: 3-7 64 3-7 59 2- 2 ISBN 13: 97 8-3 -7 64 3-7 29 1-0 Vol. 2/ SN 31: ISBN 10: 3-7 64 3-7 29 2- 3 e-ISBN: 3-7 64 3-7 59 3-0 ISBN 13: 97 8-3 -7 64 3-7 29 2- 7 Set SN 30/31:. 100 N .2 Twoarrowsandtheirhalf-chord 101 O BAB .2. 18 105 Contents vii P BAB .2. 1 9 -2 2 106 P.1 Ab .2. 19 107 P .2 Ab .2. 20:Thenumberofterms 111 P.3 Ab .2. 21:Progressivesumsofnaturalnumbers 111 P.4 Ab .2. 22: Sumofsquaresandcubes. Measureunits 22 2 3.1 Unitsoflength 22 2 3 .2 Measuresofweight 22 3 3.3 Coins 22 3 3.4 Timeunits 22 4 3.5 Subdivisionsofacircle 22 4 4 Namesofplanets,constellations,zodiacsigns 22 4 5 Daysoftheweek 22 5 6 Godsandmythologicalfigures