1. Which empirical geometry should one teach?
- We teach math for its practical purposes. So we should teach empirical geometry. But which one?
- 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.
- So string geometry has an advantage.
Angle
- 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 relative length of a curved arc.
Measure of angle
- When this arc length is measured relative to the circumference, the unit of angle is called a degree with 360 degrees making up the 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.
Conclusion
- Thus, we should teach the geometry of the sulba sutra.
2. What is π? 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.
Height of a tree
- 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 sine
- The origin of the term sine, and sine as a circular function. (Trigonometry is a misnomer.)
- 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.