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

xiii
It is interesting to note that this general formula was discovered in
Europe as late as in 1634 by Herigone (Smith's History of
Mathematics Vol. II). We may also recall here that the number 7
which occurs in Saptabhaṅgī provides a simple example in the
theory of Permutations and Combinations. A layman can verify
that he can form seven and only seven different combinations of
three distinct objects. Jainas have been using mathematics freely in
their sacred literature from very remote antiquity. The above
example supports this fact.
Problem ( VI 220 ) : O friend, tell me quickly how many
varieties there may be, owing to variation in combination of a
single-string necklace made up of diamonds, sapphires, emeralds,
corals and pearls ?
Problem ( VI 287 ) : What is that quantity which when divided
by 7, ( then ) multiplied by 3, ( then ) squared, ( then ) increased by
5, ( then ) divided by 3/5, ( then ) halved and ( then ) reduced to its
square root, happens to be 59.
Note the sheer devilry of it !
In chapters VII and VIII problems on mensuration are treated.
Some of the formulas used are noted here :
(1) The Pythargorean formula for the sides of a right angled
triangle is a² = b² + c² where a is the hypotenuse.
(2). Area of △ ABC is
√[s (s - a) (s - b) (s - c)] where 2 s = a + b + c.
(3). The area and the diagonals of a quadrilateral ABCD are :
√[(s - a) (s - b) (s - c) (s - d)] where 2 s = a + b + c + d ;
√[((ac + bd) (ab + cd)) / (ad + bc)] ; √[((ac + bd) (ad + bc)) / (ab + cd)].
It is unfortunate that both Mahāvīrāchārya and his predecessor
Brahmagupta made the common mistake of not mentioning the fact
that these formulas hold for cyclic quadrilaterals only.
( 4 ). π = 3 or √10.
( 5 ). The circumference of an ellipse whose major and minor
axes are of lengths 2 a and 2 b is √(24 b² + 16 a²) which reduces to
2π a √(1 - ⅗ e²) where e is the eccentricity. It is difficult to imagine,