Summary of day 3

1. Which empirical geometry should one teach?

• Thus, the compass box geometry and sulba sutra geometry are both empirical geometry, but the compass box has no instrument in it with which to measure curved lines.

• Because one cannot measure curved lines, one defines an angle as a pair of rays. But this does not explain why a semi-circular protractor is needed to measure angles. It does not explain why the size of the protractor does not matter in measuring angles.

• This can be explained only by using string geometry, and defining an angle as the length of a curved arc.

• When this arc length is measured relative to the circumference, the unit of angle is called a degree with 360 degrees making up the semi-circle.

• When the arc length is measured by taking the radius of the circle as the unit of length, the unit of angle is called a radian.

• Angles so defined can be greater than 360 degrees and also negative.

• Thus, we should teach the geometry of the sulba sutra.
  

2. What is pi? And how to calculate it?

• Once a measure of curved lines is available, we can define pi as the ratio of the circumference of a circle to its diameter.

• Method 1: This can be calculated empirically by drawing a large circle on the ground, using a string.

• Method 2: This can be calculated by the octagon doubling method which involves cutting the corners and approximating a circle by a polygon.

• Method 3: it can be calculated by using a computer program to generate random numbers.

3. The second “Pythagorean calculation” vs the theorem.

• The Manava sulba sutra version uses square roots to calculate the diagonal using the square root algorithm. Recalling the notion of surd.

• But we also need to calculate the sides given one side and an angle in a right angled triangle.

• For example, we need it to calculate the height of a tree.

• But first we must learn how to measure real life angles.

• Then we must learn how to calculate sine values.

• The origin of the term sine, and sine as a circular function.

• Sine values for easy cases.

• Why have we fallen behind by 1500 years. Aryabhata's 24 sine values.

• Interpolating sine values.

• Calculating the height of a tree.

• Calculating the height of a mountain.

• Calculating the size of the earth.