Studying mathematics with our father was not exactly an easy-going experience; nevertheless, it was the source of many a spark that inspired fruitful explorations and life-lessons. We recount one such thread here, and reflect on how our personal education matched up to history. When we were a kid, and learning about the area and perimeter of plane figures — we used to do it with a graph paper — our father told us that was not as our school lessons claimed but a “never-ending” number. He pointed to us that was just like which repeated after a certain run of numbers. This inculcated in us a life-long fascination, to the degree our meager mathematical capacity allowed, for both these types of numbers and their deeper significance. Some time thereafter our parents took us to watch a series of documentaries that were screened at a nearby auditorium on the evolution of man and various intellectual developments in science and mathematics in the Occident. In one of them, the presenter mentioned that the yavanācārya of yore, Archimedes, had arrived at the approximation for by inscribing polygons in a circle. This sparked a great a excitement in us for we were then fascinated by construction of regular polygons . On returning home from the screening, we quickly got back to that geometric activity realizing that the very first construction we learned in life, that of a regular hexagon, yielded (Figure 1).

That was hardly anything to write home about, but even as people used to think that “developmental ontology recapitulates phylogeny”, this realization recapitulated the very beginnings of the human knowledge of . This value was used for the crudest of constructions in the Vedic ritual going back to its ancient roots. It has been suggested that this knowledge is encapsulated in a cryptic manner with a peculiar play on the name of the god Trita (meaning 3rd):

indro yad vajrī dhṛṣamāṇo andhasā bhinad valasya paridhīṃr iva tritaḥ |

When the vajra-wielding Indra invigorated by the soma draught, split the [fortification] perimeter of Vala (the dānava’s name is used in a possible play on term for a circle), even as Trita [had done].

The allusion to the god Trita in the simile here is unusual and is evidently an allusion to his breaking out a well (an enclosure with a circular section); this strengthens the idea that a word play on the circular perimeter being split up in 3 by the diameter was exploited by Savya Āṅgirasa. This crude approximation continued to be used in sthūla-vyavahāra by the Hindus and the Nirgrantha-s till the medieval period (e.g. in the nagna text Tiloyasāra: vāso tiguṇo parihi |; the Prakrit corresponds to the Sanskrit vyāso triguṇaḥ paridhiḥ |: ). We also hear that the Jews used the same value in building a religious structure in their early history. While that is a lot of words to expend on this crudest of approximations, one could say that it at least gets you to of the real thing. Around that time our father had introduced us to the radian measure of an angle and informed us that it was the natural one, for after all the number 360 for the degree measure was an arbitrary one coming from an approximation of the year. This most elementary of constructions, the hexagon, gave us an indelible visual feel for the radian for after all if 3 got us to of then the radian should be roughly . More importantly, it informed us that this unit is best understood in multiples of and that the interior angle of the hexagon should be . In terms of history we had caught up with the emergence of the germ of this concept in Āryabhaṭa.

The aftermath of the above apprehension led us to doing a few more constructions and origami folding that led us to a somewhat more interesting realization from an aesthetically pleasing construction of a regular dodecagon which goes thus (Figure 2):

1) Draw a starting square and an equilateral triangle facing inwards on each of its sides (relates to the basic origami construction of an equilateral triangle from a square paper as indicated by Sundara Rao more than 100 years ago).

2) The inward-facing vertices of these equilateral triangles will define a new square orthogonal to the original square.

3) The 4 midpoints of the sides of this new square and the intersections of the equilateral triangles help define the sides of a regular dodecagon — in effect arising from a geometric multiplication of . Thus, this dodecagon is inscribed in the inner square.

4) Notably, this construction by itself helps define two tiles, a isosceles triangle (violet) and an equilateral triangle (green). Using these tiles both the inner square and the dodecagon can be completely tiled thus (Figure 2): and . This means that the area of the inscribed dodecagon is the area of the inner square.

5) A corollary to the above is that if a unit circle were inscribed in the inner square then square will have area 4 and the dodecagon will have area 3.

Thus, it indicated that we would need a polygon of twice the number of sides to get the same approximation of via its area as that of an inscribed polygon which gives the same from its perimeter. Hence, perimeter of the inscribed polygon is better than the area to obtain an approximation of . Further, these exercises taught us something notable: If the Yajurvedic tradition had used polygon inscription (likely it did not) then it would have required a decagon to get something close to its values the real value. While Baudhāyana or Kātyāyana are not explicit about it, Āpastamba is clear that these conversions are approximate. Squeezing out additional digits beyond that point is a process of diminishing returns; however, in the ancient world the Maitrāyaṇīya school of the Ādhvaryava tradition and the Egyptians achieved similar success reaching close to of the real value. If one wanted to achieve such a level of approximation with the polygon method you would need to inscribe to get almost exactly the value of the Maitrāyaṇīya tradition. While old Archimedes is said to have labored with a 96-side polygon to reach his , Āryabhaṭa would have needed something like 360-polygon to get his value. This made us suspect that it was unlikely he used polygon inscription and instead had a trick inherited from the non-polygonal methods typical of the old Hindu quadrature of the circle [Footnote 1]. On the other hand Archimedes’ early triumph undoubtedly rode on the quadrature of the circle achieved by the Platonic school.

From our father’s instruction we were reasonably conversant with basic trigonometry before it had been taught in school and he used that as base to introduce us to the basics of calculus in the form of limits. He told us that we could use our polygon inscription to informally understand the limit . With this in hand, he told us that we could, if we really understood it, prove the formulae of the perimeter and area of a circle as the limiting case of the -sided polygon. From our earliest education in mensuration we had been puzzled by how that mysterious number appeared in these formulae — we understood quite easily how the formulae of rectilinear figures like rectangles and triangles had been derived but this “correction factor” for the circle had been an open question for us. Hence, we were keen to figure this out using our newly acquired knowledge of limits. Being of only modest mathematical ability it took us a few days until we arrived at the proof with limits but we were then satisfied beyond words by the experience of putting down the below:

From Figure 3 we can write the perimeter of an inscribed polygon in a unit circle as:

Similarly we can write its area as:

By taking the above limit we get

While this gave us the formulae for the perimeter and the area of a circle, the actual value of was still a challenge and progress on that front had to wait for other developments. Around the same time, our fascination with the other conics was growing, mainly as an offshoot of our concomitant interest in astronomy. Armed with the high-precision German-made templates we had received from our father we began studying these conics closely. We soon realized that the circle was at one end of the continuum of ellipses and the parabola the end. The hyperbolae lay beyond that end almost as if the ellipse had wrapped around infinity and its two apices had folded back towards each other. It also struck us right away that the method of limits we had used to derive the area and perimeter of a circle could not applied to these other conics. Informally (i.e. by squeezing a circle perpendicular to one of the diameters while preserving area), we could figure out that the area of an ellipse should be where are its semimajor and semiminor axes. We also got the idea of “area under a curve” intuitively; however, it was unclear how the formulae for perimeters of these other conics could be derived. We had seen formulae for them in tables of functions we had at home [Footnote 2]. However, the tables stated that the multiple formulae it offered perimeter of the ellipse were approximate:

The first two formulae were attributed in the tables to Johannes Kepler, who had reason to calculate this as he studied ellipses in course of his monumental work on planetary orbits. While I have confirmed him as the source of the first formula, it is not clear if he was the first to propose the second one. The third formula was proposed by Leonhard Euler. In Hindu tradition, to our knowledge, the perimeter of an ellipse (āyata-vṛtta) was treated for the first time by Mahāvīra in this Gaṇita-sāra-saṃgraha. He gives a formula that goes thus:

vyasa-krtiḥ ṣadguṇitā dvi-saṅ-guṇāyama-krtiyutā padam paridhiḥ |

vyāsa-catur-bhāga-guṇaś cāyata-vṛttasya sukṣma-phalam ||

Six times the square of the minor axis plus the square of twice its major axis; the root of this gives the perimeter. That multiplied by one fourth of its minor axis is the high precision area of the ellipse.

In modern usage the perimeter will be: