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Ganitapada Of Aryabhata I

Ganitapada is a poetic text discussing mathematics written by Aryabhata I (476 CE). Ganitapada in Sanskrit is rendered in arya meter and has thirty three verses, and throws light on the concepts of Algebra and arithmetic.

Ganitapada deals with the following topics:

  1. Decimal number system
  2. Areas and Volumes
  3. Geometrical constructions
  4. Progressions
  5. Linear equations
  6. Trigonometry
  7. Squares and Cubes
  8. Rule of three
  9. Kuttaka (equations of the type): ax + b = cy + d (indeterminate equation of the first degree)

The results arrived at in Ganitapada are relevant even today. Here are some of its details:

Verse 1 – In praise of Brahmadeva, creator of the universe.

Verse 2 – Number System

Ekam ca dasam ca satam ca
Shasram Tvyutniyute thata prayutam
Kotyarbudam ca vruadam
Sthanat sthanam dasgunam syat

Meaning – in the sequence of 1,10, 100, 1000, 10000, 100000, 1000000, 1000000o, 100000000 each number is ten times the previous number.

Verse 3 – Squares and cubes. The figure of four sides in which all four sides are equal and the diagonals are equal is a square. When a number is multiplied three times itself, it is the cube of that number.

Verse 4 and 5 – Obtain the square root of 54756 – Solution 234

Obtain the cube root of 1860876 – solution 123

Verse 6 – Expansion of (x+y)2 and (x+y)3 (3)

Verse 7 – Samaparinahsyardhamviskambharadha vruttafalan.
Tannijmolen hantam ghangolfalam nirvasesham

Meaning – Multiply half the circumference by half of the diameter of a circle and we get the area of a circle.

Verse 8 – Area of a trapezium, triangular pyramid.

Verse 9 – Area of polygon.

Verse 10 – Ratio of circumference of a circle to its diameter, i.e. value of Pi.

Caturadhikam satmastgunam dvastistatha sahastranam
Ayutdvayavishlambhasayasanno vruttaparinah

Meaning – If the diameter of a circle is 20000, its circumference is about 62832. This gives Pi = 3.1416 approximately.

Verse 11 to 14 – Properties of circle, chords

Verse 15 and Verse 16 – Properties of cone.

Verse 17 and 18 – Yascaiv Bhujavargah kotivargasch a karnavargha sa
Vrutte sarsanvargordhjvargah Sa khalu dhamso

Meaning – Sum of squares of two sides of a right angled triangle is equal to the square of its hypotenuse.

Verse 19 to 22 – Progressions, sum to n terms.

Verse 23 to 31 – Number, rule of three; impact of two moving bodies. Profit and loss, interest and investment.

Verse 32 and 33 – Linear equations of the form ax+b = cx+d

Ganitapada shows the status of mathematics in the days of Aryabhata I, around 500 CE.