## Some visions of infinity from the past and our times

The great Hindu mathematician and astronomer Bhāskara-II’s work preserves a high-point of Hindu knowledge. His work contains ideas that are often seen as characterizing “modern” scientific understanding i.e. what in the west would seen as starting with Leibniz and Newton going down to our times. About 40 years from the time of Bhāskara’s death around 1185 CE we see the first irruptions of the monstrous Mohammedans in south India, initially spear-headed by the ghāzi sufi-s and then by then by the main army of the Sultanate. The first of these invasions were successfully repulsed by the yādava monarchs. We hear in their inscriptions from around that time titular statements such as:

turuṣka-kopa-pralaya-mahārṇava-magna-medinī-samuddharaṇa-mahā-varāha |
The great boar [the yādava monarch] who lifted up the world sunk by the deluge of the Islamic irruption.

gambhīrābhīra-pracaṇḍa … prarājya-rājya-turuṣkopaplava-medinī-samuddharaṇa-[kartṛ] |
The mighty fierce warrior of the cowherd race who lifted up the kingdom, frontiers, and the world from the Islamic irruption.

Despite their initial successes it appears that the yādava monarchs did not entirely understand the extent of the threat they were facing. This conspired with internal fissures in their polity and about 50 years from that point in time their kingdom came to a cataclysmic end at the hands of the Army of Islam. This marked the drastic cessation of creativity in many of the deeper aspects of Hindu knowledge. Thus, the high-point reached by Bhāskara-II remained that. Though Bhāskara continued to be studied and commented upon the creative expansion of the foundations laid by him mostly ceased in Hindudom other than in the heroics of the nambūtiri-s and their students in the south. This was mainly because of the Vijayanagaran empire reviving Hindudom in the south after overthrowing Mohammedanism in a series of brilliant campaigns. While this provided a refugium for the survival of Hindu knowledge, the spread of ideas and participation across the subcontinent which drove earlier Hindu creativity was no longer possible due to the truncation caused by the horrors of Mohammedanism.

In Bhāskara’s thought we see facets of something which fascinated us since childhood, though we never really understood it in completeness: the idea of infinities. Yet we talk about it for after all life is full of things that we can see and visualize but perhaps hardly “understand”. As the old Hindus would say we experience their pratyakṣa-s but may not realize their parokṣa. Hence, we use Bhāskara’s work as the starting point of some musings on this matter. In Bhāskara’s work we explicitly or implicitly encounter at least two forms of infinity that a typical student of the sciences grapples with by the time he enters the second decade of his life. One of these is the infinity associated with calculus and its counterpart the infinitesimal. The development of the apparatus of calculus had a long history in Hindu mathematics that is not yet fully known. We already see use of what we today call derivatives in the calculation of the motion of the Moon by the great astronomer Mañjula the Bharadvāja in his surviving work the Laghumānasa written around 932 CE. The rationale for this is presented by Bhāskara-II along with other mathematical operations that imply the knowledge of infinity of calculus and limits. This comes out clearly in his commentators like Munīśvara and the nambūtiri-s and their students. A second infinity is an algebraic infinity that Bhāskara lays out in his bīja-gaṇita on the six-fold operations of zero.

vadha-ādau viyat khasya kham khena ghāte kha-hāro bhavet khena bhaktaś ca rāśiḥ | 2.18
In multiplication etc of a number by zero the result is zero, in the multiplication by zero of a number the result is zero [the transitivity of zero multiplication]. The division of a number by zero results in [an infinity called] khahara.

dvi-ghnaṃ tri-hṛt khaṃ kha-hṛtaṃ trayaṃ ca śūnyasya vargaṃ vada me padaṃ ca | 2.19
$0 \times 2; \frac{0}{3}; \frac{3}{0}; 0^2; \sqrt{0}$ tell me [the answers]

asmin vikāraḥ kha-hare na rāśāv api praviṣṭeṣv api niḥsṛteṣu |
bahuṣv api syāl laya-sṛṣṭi-kāle anante acyute bhūta-gaṇeṣu yad-vat || 2.20

In this number khahara there is no change from addition to or subtraction from it of many quantities even as at the time of destruction and emission there is no change in Ananta (Saṃkarṣaṇa) or Acyuta (Vāsudeva) [by the absorption or emission] of material beings.

There are two aspects that are striking in this account of the khahara. The first of these relates to how it is handled in algebra. We encounter this in a simple equation which Bhāskara provides to test the student for this concept:

kaḥ khena vihṛto rāśiḥ koṭyā yukto (a)thavo(ū)nitaḥ |
vargitaḥ svapadenāḍhyaḥ khaguṇo navatir bhavet ||
Which number divided by zero, then increased or reduced by crore, then squared, then increased by its own square root, then multiplied by zero becomes 90.
Now let us unpack this using $\circledcirc$ for kha and $\bowtie$ for khahara of Bhāskara:
$x\\ \frac{x}{\circledcirc}=x\cdot \bowtie \\ x\cdot \bowtie \pm 10^8 = x\cdot \bowtie\\ \left (x\cdot \bowtie \right)^2=x^2 \cdot \bowtie \\ x^2 \cdot \bowtie + \sqrt{x^2 \cdot \bowtie} = (x^2+x) \cdot \bowtie\\ (x^2+x) \cdot \bowtie \cdot \circledcirc = (x^2+x) \cdot 1 = 90\\ x^2+x-90=0\\ \therefore x=9; x=-10$

The positive root is $x=9$ is the preferred answer of the Hindus as rāśi is taken to mean positive quantity. If one observes this operation carefully, the reduction to an ordinary quadratic would not happen if one were to treat the kha and the khahara as the zero and infinity of limits as in calculus. It would blow up to infinity and not leave behind a solvable quadratic residue. This happens only if one strictly adheres to Bhāskara’s prescription regarding khahara. A further discussion of such problems can be found in Avinash Sathaye’s work on Hindu infinities.

This brings us to the second striking aspect of the account of khahara: Its philosophical implication(s), especially given that Bhāskara goes out of his way to use pāñcaratrika vaiṣṇava terminology to explain khahara. Pāñcaratrika tradition uses the concept of large numbers to illustrate the idea of the emission of the universe from the Lakṣmī-Nārāyaṇa continuum. For instance the Lakṣmī-tantra postulates that the whole universe has emerged from a particle that is merely $10^{-18}$ the quantity of the Lakṣmī-Nārāyaṇa continuum. Thus, it tries to illustrate that emission of something as huge as the universe does not make much of a change in the continuum. Bhāskara too uses such huge numbers but he sees them like other good Hindu astronomers of the age as merely large finite numbers. For example he states a hypothesis that the diameter of the universe is $4.69947571\times 10^{16}$ km. He is not sure if the hypothesis is correct and adds that it might merely be the distance that light from the sun has travelled rather than the actual size of the universe (while this is a large number it is still short of the modern estimate of the estimate of the diameter of the visible universe at $8.8 \times 10^{23}$ km). Bhāskara’s concept of khahara is not merely a huge number but a special kind of infinity where operations like addition, subtraction, squaring etc have completely eroded. Thus, it may be visualized as the entity that restores the balance caused by the introduction of zero — a principle of conservation to the number system which is shattered by the “black hole” of zero i.e. kha, viyat, śūnya. This conservation principle almost has a paradox in it because after all the example he uses is of things being emitted and absorbed by Viṣṇu without making any difference to the god. But once one sees the other pole of this system as the abyss of kha it becomes consistent. Moreover, Bhāskara was not the only scientist to see philosophical implications for zero and infinity. Such an allusion is also made by the mathematician Nārāyaṇa paṇḍita as pointed out by Sūryadāsa:

evam etat prasaṃgena sāhityo(u)ktyā nārāyaṇo’api svakṛta-bīje nirūpayāṃ cakāra yathā |
In this way Nārāyaṇa had also defined [note use of periphrastic perfect] this in his own algebra by means of a poetic utterance:

śūnyābhyāsa vaśāt-khatām upagupto rāśiḥ punaḥ svaddhṛto
‘apy āvṛttiṃ punar eva tanmayatayā na prāktanīṃ gacchati |
ātmābhyāsa-vaśād ananyam amalaṃ cid-rūpam ānandaṃ
prāpya brahma-padaṃ na saṃsṛtipathaṃ yogī garīyān ive(a)(i)ti |

Under the operation of multiplication by zero the number vanishes (literally: is concealed by) into zeroness. But again when divided by zero it gets absorbed in that (khahara) and does not return to its previous (finite) state, even as a respectable yogin through the practice of self-realization having attained the unique bliss of pure-consciousness, free from impurities, does not return to the path of union (with matter).

Nārāyaṇa’s analogy is one of yoga. We posit that implicit in this again is a philosophical conservation principle: The destruction of finite number by zero is offset by the its absorption into the khahara. In the sense of a yogin the reduction of this material “mala” to zero (state of amala) is offset by his becoming infinite by oneness with the universal consciousness.

The philosophical interpretations of the algebraic infinity in Bhāskara and Nārāyaṇa may be linked comparable concepts from earlier Hindu tradition. In the context of Bhāskara’s khahara his commentator viloma-kavi Sūryadāsa cites the verse of Bhīṣma from the Mahābhārata regarding the Vāsudeva the supreme god of the pāñcaratrika-s:
yataḥ sarvāṇi bhūtāni bhavanty ādi yugāgame |
yasmiṃś ca pralayaṃ yānti punar eva yuga-kṣaye ||
From whom all entities come into being at the begining of the first yuga and also into whom all go to dissolution at the end of the yuga-cycle.

Further one may also point to a much earlier, famous mantra from the Śukla-yajurveda that contains a comparable concept:
oṃ pūrṇam adaḥ pūrṇam idaṃ pūrṇāt pūrṇam udacyate |
pūrṇasya pūrṇam ādāya pūrṇam evāvaśiṣyate ||
That is the whole; this is whole; from the whole comes out a whole.
When the whole is removed from the whole what is left is also the whole.

We had learned this mantra as the concluding chant of the only upaniṣat of the Śukla-Yajurvedin-s whose recitation our teacher taught. Perhaps we were subliminally influenced by what we had just learned as we wandered to ironically attend a class on partial derivatives in our college. The attendance of that mathematics class was always sparse but that particular teacher, a good Hindu nationalist, was one of the few who had a soft-corner for us; hence, we saw it as an obligation to return the favor by sitting in his class. As we waited for the class to begin we stared out from terrace of the mathematics department into an open patch of untamed wild life that still existed in those days of our youth. That patch often filled us with a remarkable visions of natural selection action. But that day had begun early for us and we lapsed into a hypnogogia: We witnessed two visions. One was somewhat tame: We found ourselves on an endless staircase, with a serially increasing integer on each step. We kept walking up that staircase much like the escalator at the train station next to our workplace. We kept walking and walking but the stairs and the numbers on them never seemed to end. If we looked back we could see the stairs endless leading down to a great dark abyss whose bottom could not be seen. If we looked up the stairs endlessly led to the realm of skyblue light that never seemed closer how much ever we walked. Our legs seemed to hurt but with manly intent we labored on and on with no respite. It was then that we realized that this is how the ordinary infinity of arithmetic manifests — the endless realm of numbers with neither a beginning nor an end. If the old jñānātman Bhāskara compared it to Viṣṇu, the vision that came to us was that of the śaiva-s — of manifestation of their prime god Rudra as the liṅgodbhava-mūrti. Suddenly we snapped out of that vision. A new vision filled our eyes. We saw a stairway in the distance with an oddly distributed size-pattern for the stairs it contained. We looked at it more closely and each stair seemed to be made up of smaller stairs arranged like the whole stairway itself. Now when we looked at each stair of those miniature stairways on a single stair of the stairway we saw more miniature stairways in them. Our vision kept zooming and there was a whole in every whole and whole within it. The physicality of the vision faded and it looked more and more like a plot of a peculiar 2D mathematical function. We raked our minds to think what might be the equation of such a function but it evaded us. Since, after all a mathematics class lay before us, we asked the teacher after it was over if he might have some suggestions. He lent us a Russian book and asked us to see if might help us in our quest.

With aid from that volume we arrived at the function known as the Devil’s staircase. Our versions of it were slightly different from the traditionally defined one which we learnt off. Let us define various $\textrm{Dscf}(x)$ thus:
$\textrm{Dscfa}(x)=\displaystyle\sum_{n=1}^\infty\dfrac{\lfloor nx\rfloor}{a^n}$; where $a>1$
The classic $\textrm{Dscf}(x)$ which we encountered in books is based on this version where $a=2$,
$\textrm{Dscf2}(x)=\displaystyle\sum_{n=1}^\infty\dfrac{\lfloor nx\rfloor}{2^n}$
We also defined other versions for $a=\phi$ where $\phi$ is the Golden ratio. Finally, we also define a variant version of the function $\textrm{Dscr}(x)$ where we used the round function $\lfloor x \rceil$ which rounds to the nearest whole number instead of the floor function in the above definition. This was what we used in our original definitions because the $\lfloor x \rceil$ was the intuitive way of producing the infinite staircase:
$\textrm{Dscfa}(x)=\displaystyle\sum_{n=1}^\infty\dfrac{\lfloor nx\rceil}{a^n}$; where $a>1$

In figure 1 we illustrate some of these functions which clearly showing the part as a whole concept inherent in these functions. A remarkable result pertaining to one of these functions is,
$\textrm{Dscf2}(\phi)=2+\cfrac{1}{2^{F_0}+\cfrac{1}{2^{F_1}+\cfrac{1}{2^{F_2}+\cdots}}}$

where $F_{0:n}= 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55...$, i.e. the meru-prastha series.

The exploration of $\textrm{Dscf}(x)/\textrm{Dscr}(x)$ led us to another remarkable function we learnt of in an obscure pamphlet of mathematical illustrations (sadly whose name I forget by noted Indian mathematician and politician: Badri Nath Prasad). This function was discovered by the famous mathematician Hermann Minkowski, the teacher of Albert Einstein. Minkowski was among those advocating, what we as heathens would term, a return to “Pythagorean” thinking by emphasizing mathematical beauty as a guiding principle for discovery of new physics. Fittingly, in his study on the geometry of numbers he presented the discovery of the question mark function, $\textrm{?}(x)$, which maps quadratic surds on to rational numbers! Let the number $x$ be defined as a continued fraction,
$x=a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\cdots}}}$ then,

$\textrm{?}(x)=a_0+2\displaystyle\sum_{j=1}^n\dfrac{(-1)^{j+1}}{2^{a_1+a_2+a_3...+a_j}}$ where $n$ is a finite number if $x$ is rational and $n=\infty$ if $x$ is irrational.

Figure 2 shows that $\textrm{?}(x)$ has a structure similar to Devil’s staircases.

When our code plotted out Devil’s staircases and $\textrm{?}(x)$ successfully for the first time our vision had become a pratyakṣa and the apprehension of third type of infinity infinity was under way. This infinity, distinct from that of calculus and algebra, was one which embodied each part of the whole being a whole in itself. Thus, it seemed to be a mathematical reflex of the above mantra from the Śukla-yajurveda. While these mysterious functions were now pratyakṣa on our computer screen, thereby bringing into the apprehension of a lesser mortal like us what was only previously accessible to the mathematician, the question that confronted us was whether such an infinity had a place in the real world.

It immediately caught our attention that we could convert these monotonically increasing functions to periodic or quasi-periodic functions (Such a thing had been independently done by others for $\textrm{?}(x)$). For the Devil’s staircases we centered the functions on 2, which resulted in a saw-tooth function. Thus we get:
$\textrm{Ddiff2}(x)=2\textrm{Dscf2}(x)-4x+1$
$\textrm{Ddifr2}(x)=2\textrm{Dscr2}(x)-4x+1$
We can likewise define attenuated functions like,
$\textrm{Ddiff2},\phi(x)=2\textrm{Dscf}\phi(x)-4x+1$, etc
For Minkowski’s function we can define the periodic version simply as:
$\textrm{?dif}(x)=\textrm{?}(x)-x$
Now when we use these functions to define a polar curve e.g. $r=a\cdot\textrm{Ddiff2}(\theta)$ then we get the below fractal forms of great interest:

Figure 3 $r=a\cdot\textrm{Ddiff2}(\theta); 0 \le \theta le 3\pi$

Figure 4 $r=a\cdot\textrm{Ddifr2}(\theta); 0 \le \theta \le 2\pi$

Figure 5 $r=a\cdot\textrm{?dif}(\theta); 0 \le \theta \le 7$

Figure 6 $r=a\cdot\textrm{Ddiff2},\phi(\theta); 0 \le \theta \le 3\pi$

One notes that these forms bear resemblance to those that appear in nature in jagged leaves and petals (e.g. Dianthus), in inflorescences and petals. Thus, it appears that the biological processes that generate these natural forms are in some way trying to capture the geometry predicated by the nesting of wholes within a whole. However, as on the computer screen the biological forms are by no means totally scale-free facsimiles of the mathematical objects they resemble. The nesting has a certain lower bound beyond which the structure changes. Thus, in a sense they are approximate real world reflections of the Platonic forms of the third type of infinity — that of infinite nesting. While the actual biological process that result in the form might differ from case to case and the mechanistic process might not involve constructing and plotting a $\textrm{Ddiff2}(x)$ or a $\textrm{?dif}(x)$, the forms nevertheless converge to those dictated by such functions. Thus, this type of infinity, even more so than others, might impinge upon the real world. Indeed, perhaps, Yājñavalkya wished to capture the Platonic ideal behind such forms when he recited the above mantra of the Vājasaneyin-s. The realization of this infinity has since then had a deep influence on our way of looking at the world and experiencing beauty.

## Āryabhaṭa and his sine table

Everyone and his son have written about Āryabhaṭa and his sine table. Yet we too do this because sometimes the situation arises where you have to explain things clearly to a layman who might have some education but is unfamiliar with the intricacies of, or in some cases lacks the correct perspective on, Hindu tradition.

siddhānta-pañcaka-vidhāv api dṛg viruddham
auḍhyoparāga-mukha-khecara-cāra-kḷptau |
sūryaḥ svayaṃ kusumapury abhavat kalau tu
bhūgolavit kulapa āryabhaṭābhidhānaḥ ||
When the predictions of the five siddhānta-s and observations of conjunctions, occultations and setting times of planets began to conflict the solar deity himself incarnated in Kusumapuri in the Kali age in the form of the geographer and head professor Āryabhaṭa.

The great astronomer and mathematician Āryabhaṭa-I’s, who was seen by some Hindus as the incarnation of the solar deity, was born in 476 CE in the Aśmaka country (close to modern Maharashtra-Telangana border). He was active at Pāṭaliputra in the golden age of Hindu power during the reign of emperor Budhagupta. He is known to have composed at least two works the Āryabhaṭīya, which was an update of the old Svāyambhuva-siddhānta tradition, and the Āryabhaṭa-siddhānta which was modeled after the old Sūrya siddhānta. We are informed by Bhāskara-I that in his Āryabhaṭīya Āryabhaṭa was following in the footsteps of the great astronomer of antiquity Pārāśarya, who was likely one of the early promulgators of the Svāyambhuva tradition. Āryabhaṭa as head professor is said to have had the following notable students: Lāṭadeva, Niśaṅku, Pāṇḍuraṅgasvamin and Prabhākara. Sadly their works have been lost. Of them Lāṭadeva is recorded in old Hindu scientific tradition as having written several works including on Hellenistic astronomy and likely succeeded Āryabhaṭa as the ācārya of his school.

Thus, Āryabhaṭa’s work is the earliest surviving record of one of the most important Hindu scientific traditions, namely that of Svāyambhuva-s. In the manner of the scientists of old Hindu naturalistic tradition Āryabhaṭa presents the acquisition of scientific knowledge as the attainment of brahmavidyā:
daśa-gītika-sūtram idam bhū-graha-caritam bha-pañjare jñātvā |
graha-bha-gaṇa-paribhramaṇaṃ sa yāti bhittvā param brahma ||
Having known these sūtra-s in the ten verses composed in the gītika meter providing the motions of the Earth and the planets within the celestial sphere [pañjara: marked by the coordinate grid], and having penetrated the orbits of the planets and the stars he attains the supreme brahman.

His clear mention of the movement of the Earth in the celestial sphere along with the other planets in the celestial sphere has been taken as Āryabhaṭa’s discovery of heliocentricity. However, here we are not going into this issue and the real nature of his unique planetary model which sets the old Hindu Svāyambhuva planetary model apart from those of the Greeks. Nevertheless, as a testimony of his astronomical achievements we will merely state the period of sidereal day of the Earth as determined by Āryabhaṭa in modern units: $23^h 56^m 4^{s.}1$, which is practically the modern value.

One of the important features of Āryabhaṭa’s work was his presentation of the old Hindu sine difference table. Āryabhaṭa gives the table using his syllable-numeral equivalence which goes as:
makhi bhaki phakhi dhakhi ṇakhi ñakhi ṅakhi hasjha skaki kiṣga śghaki kighva |
ghlaki kigra hakya dhaki kica sga jhaśa ṅva kla pta pha cha kalā ardha-jyāH ||
225, 224, 222, 219, 215, 210, 205, 199, 191, 183, 174, 164, 154, 143, 131, 119, 106, 93, 79, 65, 51, 37, 22, 7 are the $R\sin(\theta)$ [differences].

So how do we understand this? As per Hindu terminology if the arc is equivalent to the bow then the chord is equivalent to the bowstring (jyā). Hence, the sine can be seen as half a jyā (ardhajyā) as Āryabhaṭa terms it in the above sūtra. For simplicity call Āryabhaṭa’s ardhajyā a function named jyā and represent it thus in modern notation:
Let the function jyā $(\theta)$ be defined as,
jyā $(\theta)=R\sin(\theta)$, where $R=3438$.
Why the number 3438? Āryabhaṭa conceives a circle whose circumference is divided into $360 \times 60=21600$ parts. Now the radius of this circle given Āryabhaṭa’s $\pi \approx 3.1416$ will be,
$\dfrac{21600}{3.1416}=3437.73 \approx 3438 \Rightarrow 1\;radian$
Thus, Āryabhaṭa’s R value is for the first time a radian-like concept was used in trigonometry. Now, Āryabhaṭa divides his quadrant into 24 parts; thus, his minimal angle is $\theta_1=\frac{\pi}{48}$. To see why he chose this value of $\theta_1$ note the following:

$\dfrac{\pi}{48} \times 3438 \approx 225$

jyā $(\dfrac{\pi}{48})=3438\sin(\dfrac{\pi}{48})=3438\times0.06540 \approx 225$

$\therefore$ jyā $(\dfrac{\pi}{48})\Big/R\cdot\dfrac{\pi}{48} \approx 1$; actual value 0.99982

Thus, Āryabhaṭa’s value of $\theta_1$ is chosen such that the angle and its jyā are practically the same (Figure 1). This value as a proxy for $\displaystyle \lim_{\theta \to 0}\frac{\sin(\theta)}{\theta}=1$ continued to be used in the subsequent development of Hindu calculus. The $\theta$ and jyā $(\theta)$ being nearly the same at this value allows linear interpolation for intermediate values.

Figure 1

Now the rest of his table is in the form of differences. So to get a jyā $(\theta_n)$ we have to do the following:
jyā $(\theta_2)=225+224=449$; jyā $(\theta_3)=225+224+222=671$
The table below compares Āryabhaṭa’s jyā $(\theta)$ values to the exact modern values.

## Euler and Ramanujan: primes, near integers and cakravāla

Mathematician Watson who worked on the famed notebooks said regarding some of Srinivasa Ramanujan’s equations: “a thrill which is indistinguishable from the thrill which I feel when I enter the Sagrestia Nuovo of the Capella Medici and see before me the austere beauty of the four statues representing “Day,” “Night,” “Evening,” and “Dawn” which Michelangelo has set over the tomb of Giuliano de’Medici and Lorenzo de ‘Medici.” [Quoted by computer scientists Borwein, Borwein and Bailey in their great paper on Ramanujan and $\pi$]. This very thrill is something that a mere mortal like us can also experience when gazing at the monuments of the great men like Euler, Gauss and Ramanujan. This indeed is the basis of this note.

In 1772 the blind Euler discovered a remarkable parabola: $y=x^2+x+41$. For $x=0,1,2...39$ the value of $y$ is a prime number. Thus we see 41-1 primes consecutively generated by it:
41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 853, 911, 971, 1033, 1097, 1163, 1231, 1301, 1373, 1447, 1523, 1601

Interestingly this parabola is one of a set of special parabolas with deeper connections elsewhere. One notices that it can be written as:
$y=(x+\frac{1}{2})^2+\frac{163}{4}$
Now the following other parabolas of such a form exist:
$y=(x+\frac{1}{2})^2+\frac{67}{4}$
$y=(x+\frac{1}{2})^2+\frac{43}{4}$
$y=(x+\frac{1}{2})^2+\frac{19}{4}$
$y=(x+\frac{1}{2})^2+\frac{11}{4}$
All these parabolas can also be written as:
$y=x^2+x+n; n \Rightarrow 41, 17, 11, 5, 3$
They all generate $n-1=40, 16, 10, 4, 2$ primes consecutively for $x=0:n-2$. Moreover they are increasingly productive in terms of prime-production. For each value of $n=3...41$ we tabulate below $4n-1$ and the number of primes produced by these equations for $x=0:99$

One notices that the set culminates in Euler’s original parabola, which is rather remarkably rich in its prime-generative capacity. The $4n-1$ numbers also have several remarkable properties. Consider the following cubic equations whose non-cube coefficients relate to the number -2:
$x^3-2x^2+2x-2 = 0; \; x=1.543689012$
$x^3-2x-2 = 0; \; x=1.769292354$
$x^3-2x^2-2 = 0; \; x=2.359304085$
$x^3-2x^2-2x-2 = 0; \; x=2.91963956$
$x^3-6x^2+4x-2 = 0; \; x=5.318628217$
Their roots form a peculiar figure on the complex plane (Figure 1) with the real roots (shown next to each equation) falling successively on the positive x-axis.

Figure 1

Now for each of these roots if we compute: $y=x^{24}-24$ successively we interestingly get $y \approx e^{\pi\sqrt{k}}$ where $k=4n-1$ from the above Euler parabolas. Even more strikingly these values of $e^{\pi\sqrt{k}}$ are approximations of $b=a^3+744$ where $a$ is a multiple of 32. These striking relationships are tabulated below. Notably, the largest $k$ exhibits the relationship $\frac{163}{\log(163)}=31.999998$ which is almost the integer 32.

Further as $k$ gets bigger the values of $e^{\pi\sqrt{k}}$ and either approximation get closer and strikingly the values become almost integers. The last expression for $k=163$ also leads to the below expression which is a very close near integer:
$\left(\dfrac{\log\left(640320^3+744\right)}{\pi}\right)^2=163.0000000000000000000000000000232$

This is also the basis of the famous and good approximation of $\pi$:
$\pi \approx \dfrac{\log(640320^3+744)}{\sqrt{163}}$
It is correct to 30 decimal places and is greater $\pi$ by $2.23\times 10^{-31}$. Mysteriously, the largest $k=163$ is associated with another near integer relationships involving $\pi$ and $e$:
$163*(\pi-e)=68.99966$

The near integer values of $e^{\pi\sqrt{k}}$ for the larger of these $k$ brings us to the remarkable discovery of several such near integers for other $e^{\pi\sqrt{n}}$ where $n$ is a positive integer by the 26 year old Srinivasa Ramanujan. To look closer at these we computed the values of $e^{\pi\sqrt{n}}$ for $n=1:1000$ and looked for those values. Figure 2 shows the $\log(|e^{\pi\sqrt{n}}-nint|)$, where $nint$ is the nearest integer. A table of these values along with the difference from $nint$ and also the $a^3+744$ = $b$ values and differences (last two columns) are provided.

Figure 2

Figure 3 shows the distribution of the $e^{\pi\sqrt{n}}-nint$. We notice a striking peak of 27 at $-.005. Otherwise with a mean of 10 and standard deviation of 3.52 the distribution of differences is fairly uniform. Hence, this production of near integers by $y=e^{\pi\sqrt{n}}$ does not seem to be by chance alone ($p=6.93 \times 10^{-7}$).

Figure 3

In 16 of these cases (tabulated below) $e^{\pi\sqrt{n}}-\left \lceil e^{\pi\sqrt{n}} \right \rceil<.005$ similar to the near integers of the 43, 67, 163 system we saw above.

Of these three (43, 67, 163) belong to the above system. A further $n=326$ is the double of 163 but it does not show the $a^3+744$ = $b$ approximation. Three numbers are multiples of 29: $58=2\times29$; $232=2^3\times29$; $522=2\times3^2\times 29$. $386= 2\times 193; 772=2^2\times 193$ feature 193. Then we have 37 and its double 74. Of these the Ramanujan set of $n=22,37,58$ show a relationship similar to those seen from the above set with 744: $a^2 \pm 104=b$ where $b$ is the integer approximation of $e^{\pi\sqrt{n}}$:
$e^{\pi\sqrt{22}} \approx 1584^2 - 104=2508952$
$e^{\pi\sqrt{37}} \approx 14112^2 + 104=199148648$
$e^{\pi\sqrt{58}} \approx 156816^2 - 104=24591257752$

But beyond these I am not aware of any other such simple formulae that produce the remaining approximating integer values. The link to 29 however lies at the heart of a class of remarkable approximations of $\pi$ discovered by Ramanujan of the type:
$\pi \approx \dfrac{12 \log(5 + \sqrt{29}) - \log(64)}{\sqrt{58}}$

There is some connection between this and our previous article. The explosive values emerging from the solution of cakravāla equation are analogous by to the “special” near integer approaches of $y=e^{\pi\sqrt{n}}$. Indeed, deeper links beyond the superficial comparison exist between the cakravāla-type equations and these systems, which were discovered by Ramanujan through the approximations of $\pi$ like the one provided above. One such link works thus: Consider the cakravāla indeterminate equation,
$x^2=ay^2+1$
Consider a second indeterminate equation in $(p,q)$ for same $a$,
$aq^2=p^2+4$
Now if $a\mod 8 \equiv 5$ then the following holds,
$\left(\dfrac{p+q\sqrt{a}}{2}\right)^6=x+y\sqrt{a}$
We see that $a=29=3\times 8+5$ fulfills the above criterion. Hence,
$(p=5, q=1)$; by cakravāla we get $(x=9801, y=1820)$, thus:
$\left(\dfrac{5+1\cdot\sqrt{29}}{2}\right)^6=9801+1820\sqrt{29}$
Thus, we can write an approximation of $\pi$ in terms of the cakravāla result for 29 as:
$\pi \approx \dfrac{\log\left(64\left(9801+1820\sqrt{29}\right)^2\right)}{\sqrt{58}}$

Finally, while not a near integer there is one involving the Golden ratio $\phi$ in the same genre:
$e^{\pi\sqrt{190}} \approx \left(\sqrt{2}\phi^3\left(3+\sqrt{10}\right)\right)^{12}+24$; thus, we get.
$\pi \approx \dfrac{\left(6\log(2)+36\log(\phi)+12\log\left(3+\sqrt{10}\right)\right)}{\sqrt{190}}$

Other than the famous $B^3$ paper mentioned above while writing this we came across formulae of the Ramanujan type discovered by Tito Piezas with new cakravāla connections. One may also study the related formulae of the Chudnovsky brothers. All of these show that what’s discussed here is merely the tip of the iceberg. Much of the deeper connections regarding these systems lie outside our very meager mathematical knowledge, hence we stop here. Briefly, the $k=4n-1$ values of the above Euler parabolas are precisely the numbers which Gauss first connected to some deep mathematics. Thus, the mysterious connections presented here are not coincidental. It stems from some very deep modern results which started with Carl Gauss’ discovery of a function of the complex variable known as the j-function. It has wended its way via Gauss’s student Dedekind’s $\eta$ function and through Ramanujan’s discovery of the above-discussed near integers and other things from his “lost notebook”. Finally in our times it has led to something called the Monstrous Moonshine which lies outside the perimeter of our mathematical education. We are simply recording some of what we have explored for ourselves (well-known to mathematicians) for their astounding nature. Fittingly, some physicists hold that these might have deep connections to the very nature of existence.

## Early Hindu mathematics and the exploration of some second degree indeterminate equations

The following is merely a record of our exploration as a non-mathematician/non-computer scientist of a remarkable (at least to us) class of numerical relationships.

An equation like $x^2-x-1=0$ can be solved to obtain specific solutions as: $x=\frac{1+\sqrt{5}}{2},\;\frac{1-\sqrt{5}}{2}$. However, if we have an equation like then $437x+592y=298374$ it is termed indeterminate as it has no unique set of solutions. Rather the values of the variables satisfying it define a curve in a space with as many dimensions as the number of variables, which in the above case is a straight line on a plane. However, if we constrain the solutions to being integers we get specific, though sometimes infinite solutions. This constraint essentially reduces the problem to finding the points on the curve that lie on a fixed lattice. Further, if we add the constraint that the solutions should be positive then we can come down to a limited set of solutions, like for the above equation of the line we have $(x=126, y=411)$ as lying on the integer lattice in the positive quadrant. Such integral solutions to indeterminate equations are referred to in western and thereby in modern mathematics as Diophantine equations after the yavana mathematician Diophantos.

However, Hindu tradition has one of the deepest histories of the analysis of such indeterminate equations. The earliest studies go back to the ritual vedī-s of the Yajurveda and they find mention in the problems of the śulbasūtra-s. This suggests that the Hindu interest in them arose in the context of ritual. Similarly, Proclus, Diophantos and Archimedes before him were likely building on a Pythagorean base, suggesting that such problems might have existed in the ancestral ritual tradition of both the yavana-s and ārya-s. Already in the Yajurveda tradition we encounter such indeterminate equations of the linear form and also those resulting in the bhujā-koṭi-karṇa (known in the west as Pythagorean) triples (see below). It is also likely that the adhvaryu-s used an equation of the form $x^2=2y^2+1$ to obtain the solutions $(x=577, y=408)$ to obtain the value of $\sqrt{2} \approx 1+\frac{1}{3}+\frac{1}{3}\times\frac{1}{4}-\frac{1}{3}\times \frac{1}{4} \times \frac{1}{34} = \frac{577}{408}$ correct to 5 decimal places. That this was the case is supported by the commentary of the great Nīlakaṇṭha somayājin who notes that the factor 17 in the denominator of the śulbasūtra value comes from $17^2=2 \times 12^2+1$. In the post-Vedic period the analysis of indeterminate equations was advanced further. We learn in the teachings of: 1) Āryabhaṭa of the kuṭṭaka algorithm for linear indeterminate equations. 2) Brahmagupta and his successors Jayadeva and Bhāskara-II of the algorithms to solve certain degree 2 indeterminate equations. 3) Nārāyaṇa a general summary of these equations and their applications (his text the Gaṇita-kaumudi).

Of the degree 2 indeterminate equations, integral solutions of those of the form $x^2=ay^2+1$ where $a$ is an integer were of greatest interest to the Hindus. As we saw above such an equation with $a=2$ might have been used by seers of the Yajurveda to approximate $\sqrt{2}$. Looking at this equation some things become immediately clear. For $(x=1, y=0)$ we get a trivial solution which is true for all $a$. If $a$ is a perfect square then $ay^2+1$ will never be a perfect square so we have no solutions beyond the trivial one. If $a$ is not a perfect square we could potentially have infinite solutions because at some point $ay^2+1$ is free to become a perfect square. This can be easily understood geometrically: The equation $x^2-ay^2=1$ defines a family of hyperbolas (Figure 1). Integral solutions will occur where ever the hyperbola passes through the integer lattice. From the equation we can deduce that one of the two asymptotes of the hyperbola is $y=\frac{x}{\sqrt{a}}$. If $a$ is a perfect square then the asymptote will pass through points on the integer lattice but beyond the trivial case the hyperbola will never pass through the points on the lattice. However, if $a$ is not a perfect square then $\sqrt{a}$ will be irrational. Hence, as the hyperbola converges towards it asymptote, each time it passes through a lattice point $\frac{x}{y}$ will be a closer and closer approximation of $\sqrt{a}$.

Figure 1

For the particular case $x^2=2y^2+1$ there is a special significance that links arithmetic to geometry. The sum of integers is given by the formula presented by sage Śākapūṇi in Vedic antiquity:
$1+2+3...n \Rightarrow \sum \limits_1^n i =\dfrac{n(n+1)}{2}$
By laying the integers as equidistant points on successive rows we get an equilateral triangle lattice with the sum up to the nth integer completing the next equilateral triangle (Figure 2). Hence these sums are known as triangular numbers.

Figure 2

Now consider the sum of odd integers:
$1+3+5+7...(2n-1) \Rightarrow \sum \limits_1^n (2i-1)=n^2$
Laying out the nth odd number $2n-1$, with each new addition bracketing the previous one we get a square lattice (Figure 3). Hence, their sum is $n^2$. Thus these sums are known as square numbers.

Figure 3

Now a question which arises is whether there are any triangular numbers that are also square numbers. Such a question might be germane to a neo-ritualist who wishes to make a triangular vedī with the same number of lattice points as a square vedī. This is not entirely out of place because square numbers arose first in the vaidika system of square altars and the triangular numbers had a significance for the Pythagoreans (also see earlier note). In any case, the answer to this question is obtained thus: Let $r$ be the number to which we desire to sum ( i.e. $1+2+3...+r$) to get the triangular number. Let $s$ be the number which we desire to be side of the square number (i.e $1+3+5+...(2s-1)=s^2$). Then we have:
$\dfrac{r(r+1)}{2}=s^2\\ r^2+r=2s^2\\ \therefore 4r^2+4r=2(4s^2)\\ \therefore 4r^2+4r+1=2(4s^2)+1\\ \therefore (2r+1)^2=2(2s)^2+1\\ let \; x=2r+1 \; and \; y=2s \\ \therefore x^2=2y^2+1\\ \therefore r=\dfrac{x-1}{2} \; and \; s=\dfrac{y}{2}$

Thus the integral solutions of the above equation give us the triangular numbers which are also square numbers. As one can notice $2y^2+1$ will always be an odd number; hence, for the solutions $x$ will always be an odd number and $y$ will always be even [an odd square will be $(2n-1)^2= 4n^2-4n+1$; Now $2y^2+1=4n^2-4n+1$; thus we have $y^2=2n^2-2n$; hence $y$ is always even]. Thus, for each solution of the above indeterminate equation we get the equivalent triangular and square numbers. Interestingly, each successive approximation of $\sqrt{2}$ yield the point where triangular and square numbers are equivalent. Illustrated in Figures 2 and 3 are the cases where the triangular and square numbers become equal till $r=49, s=35)$. Below is a table of the first 11 such values of $(x,y)$ and corresponding $(r,s)$.

As can be seen above these numbers can quickly large. A further question is can we get such solutions more generally for the above equation where $a$ takes values other than 2. The answer to this lies in one of the greatest discoveries of the Hindu mind. For the rest of this note we shall restrict ourselves to the first $(x,y)$ that satisfies the equation, though the method discussed can be used to get further values too.

Our encounter with this equation goes back to our youth when our father had taken us to a bookshop. There a book on Hindu mathematics caught our sight and in it we encountered this equation $x^2=ay^2+1$ for the first time. We also noted in that the book that Brahmagupta had mentioned that one could be considered as having some mettle in mathematics if one solved the equation for $a=83, a=92$. Interestingly, when these equations were first being explored in the West (1600s) the Frenchman Fermat too posed one of them as a major challenge to rival English mathematicians and others. Not being of extensive means we were not in position of buying many books those days and had to leave it in the shop and return home. Nevertheless the problem stuck in our mind and we started exploring it with paper and pen. We were able obtain the first solutions for numbers $\leq10$ but soon realized that 83 and 92 were beyond our means, plainly showing us that we had no mathematical talents. However, the book usefully informed us that there was an algorithm developed by the Hindu mathematicians/scientists from Brahmagupta onwards culminating in Jayadeva, which was known as the cakravāla that allowed one to solve any such equation. Commonly one finds it attributed to Bhāskara-II but it is already mentioned in the commentary called Sundarī of Udayadivākara from 1073 CE where he says it was achieved by Jayadeva before him. Hence, it was already in place long before Bhāskara-II, who when providing it merely says its is known or has been recited as the cakravāla (cakravālam idaṃ jaguḥ |).

However, the time was too brief at the time of our first encounter for us to understand the cakravāla. Hence, a little later when we had access to a computer for the first time we wrote a simple program to carry out a brute force search for all the first solutions to the equation till $a=100$. We thus found the answers for $a=83: (x=82, y=9)$ and $a=92: (x=1151, y=120)$. However, since our search was carried out only to a maximum of $10^5$ (1 lakṣa) we had several holes. This indicated a surprising fact that for some values of $a$ $(x,y)$ could suddenly take large values among otherwise pedestrian values. This meant we had to have a better way of finding them. Hence, we got back to study the cakravāla algorithm and implement it having found a book describing it with a richer relative. In doing so we realized that the way it was presented in the sources we used was not the best for implementation on a computer. Hence, we had to implement it in the form of a systematic search which catches all corner cases that cannot be handled by the “simple” presentation we usually see for this algorithm. Below is a presentation of such an implementation of the function $cv(a)=(x,y)$ which would output the first non-trivial solutions of the equation (Hindus termed the solutions $(x,y)$ as antya and ādi: Brahmagupta or as jyeṣṭa and hrasva: Jayadeva and Bhāskara-II)

Do until $val=(x,y)$ is obtained
$p_n=\lfloor \sqrt{a} \rfloor \Rightarrow \lfloor x \rfloor$ is the floor function
if $p_n==\sqrt{a}$ then $val=(x=1,y=0)$; end execution
write array $j=(0,1,-1,2,-2,...(p_n-1),(-pn+1))$
$n=1$
Do until $val=(x,y)$ is obtained
$p_n=\lfloor \sqrt{a} \rfloor$
$p_n=p_n+j[n]$
$q_n=1$
$m_n=p_n^2-a$
$x_n=-p_n\; mod\; |m_n|$
if $x_n==0$ then $x_n= \lfloor\sqrt{a}\rfloor+j[n]$
Do until $val=(x,y)$ is obtained
$p_{n+1}=\dfrac{p_n x_n+a q_n}{|m_n|}$
$q_{n+1}=\dfrac{p_n+x_n q_n}{|m_n|}$
$m_{n+1}=\dfrac{x_n^2-a}{m_n}$
$x_n= -x_n\; mod\; |m_{n+1}|$
if $x_n==0$ then $x_n= \lfloor\sqrt{a}\rfloor+j[n]$
$p_n=p_{n+1}$
$q_n=q_{n+1}$
$m_n=m_{n+1}$
if $q_n-\lfloor q_n\rfloor>0$ then break this loop; $n=n+1$; restart from outer loop with next value in array j $(j[n+1])$
if $m_n==1$ then $val=(x=p_n, y=q_n)$; end execution
return(val)

If one instructs a computer to do the above procedure in any language it understands one can unerringly get the first non-trivial solutions of the equation under consideration. The implementation has three nested loops. The outer loop catches the cases where $a$ is a perfect square and has only trivial solutions. The innermost loop is the core cakravāla procedure where the search iteratively proceeds until $m_n == 1$. As the first approach it starts with an initial $p_n$, i.e. $p_0=\lfloor \sqrt{a} \rfloor$. This is what you typically see in examples illustrating the procedure. However, one quickly realizes that with $a=14$ that does not work even though it has a relatively simple solution $(x=15, y=4)$. Here, we need to start with $p_n=4$ instead of $p_n=3$ because the latter results in fractional $q_n$ (note it has to always be integer for procedure). Bhāskara-II seems indicates that one might need take an alternative starting point with the ‘athavā’: guṇavarge prakṛtya +ūne .athavā +alpaṃ śeṣakaṃ yathā | (guṇa-varga $\Rightarrow p_0^2$ i.e. square of starting $p_n$; prakṛtya $\Rightarrow a$). But when we go to $a=21$ we see that both $p_0=4,5$ fail. Hence, we need to go to $p_0=3$ to get the solution without hitting a fractional $q_n$. It is for this issue that we need to incorporate the second loop where we search for the appropriate $p_0$ systematically starting from $\lfloor\sqrt{a} \rfloor$ and moving up and down from it by 1, 2 so on. Thus, we have a simple implementation of the cakravāla on a computer that anyone with even limited mathematical capacity like us can achieve. Since by this procedure we always reach the solution, i.e. the algorithm always halts, we have not implemented Brahmagupta and Bhāskara-II’s shortcuts.

Thus, armed with the cakravāla in a second or so we obtained the values of the first $(x,y)$ for all $a=1:100$. In the process we blazed through Bhāskara-II’s famous illustration of $a=61$ which yields big values: $(x=1766319049, y=226153980)$. While big, to put it in perspective the jyeṣṭa is still only about $\frac{1}{3}$ number of people alive on earth. As we tried to go past the $a=100$ barrier towards $a=200$ we hit a snag as our numbers ran beyond our language’s standard numbers kept as double-precision floating-point (64 bits ~ 15:17 digits). So we had re-implement the algorithm using multiple precision floating-point numbers. This slowed it down a bit but was not too bad given the cakravāla’s inherent speed. Now we could obtain a table of all first $(x, y)$ all the way to $a=1000$. Now, we are not talking just big but truly large numbers that are a delight to the Hindu eye. In this range we reach the maximum $(x=16421658242965910275055840472270471049,\\ y=638728478116949861246791167518480580)$ for $a=661$ interestingly 600 ahead of Bhāskara-II’s $a=61$. To put is perspective that is close to the diameter of the universe measured in Hydrogen atom diameters. At $a=421$ we get $(x=3879474045914926879468217167061449,\\ y=189073995951839020880499780706260)$ which is roughly the diameter of an apple (6.27 cm) in Planck’s lengths. We provide this table for the reader’s enjoyment as such information does not seem to be easily available. The first two columns give the jyeṣṭa and hrasva. The next two columns $(n,m)$ give the number of iterations of the second and total number of iterations of the third loop that were needed until the solution was obtained. The first non-trivial $x$ values are also plotted for $a=1:1000$ (Figure 4).

Figure 4

Is there any pattern to these values? As noted above $a=14,21$ need us to search beyond the $p_0=\lfloor \sqrt{a} \rfloor$. Indeed, multiples of 7 seem have a higher propensity for this problem because we encounter a similar issues with 14, 21, 28, 70, 77, 98, 112, 126, 133, 140, 154, 161, 189, 238, 252, 259, 266, 273…: Out of the 142 multiples of 7 that $a$ assumes between 1 and 1000, we need multiple rounds of exploration $(n)$ for 79 of them. Indeed the highest values of $n=21$ are attained for $a= 966=7 \times 138$ and $a=987= 7 \times 141$.

For the actual values of $(x,y)$ there are some patterns that can be easily proved: When $a=k^2-1$, i.e. $a$ is one less than a perfect square the solution is $(x=k, y=1)$. For $a=k^2+1$ we can easily show that $(x=2k^2+1, y=2k)$. When $a=k^2 \pm 2$ it is obvious that $(x=k^2 \pm 1, y=k)$. Similarly, when $a=4k^2 \pm 4$ then $(x=2k^2 \pm 1, y=k)$. Slightly more subtle ones are: when $a=(3k)^2 \pm 3$ then $(x=6k^2 \pm 1, y=2k)$; when $a=9k^2 \pm 6$ then $(x=3k^2 \pm 1, y=k)$; when $a=25k^2 \pm 10$ then $(x=5k^2 \pm 1, y=k)$. Such may be termed regular categories of $(a:x,y)$

It also becomes apparent (Figure 4, Table) that for several prime values we get explosive eruptions of the $(x,y)$ values. The famous explosive values like $a=61, 109, 181, 397, 409, 421, 661, 673, 769, 937$ are all primes. Now if a prime happens to fall in one of the above categories where $a$ can be regularly defined in terms of a perfect square then we get get pedestrian values despite it being a prime (e.g. $a=47 \Rightarrow k^2-2$ category or $a=101 \Rightarrow k^2+1$ category). However, double their value despite not being a prime has an explosive value if it escapes the above categories; e.g. for $a=94=47 \times 2$, $(x=2143295, y=221064)$ and for $a= 202 = 101 \times 2$, $(x=19731763, y=1388322)$. Similarly, when take the case of 17 we see that $17 \Rightarrow k^2+1$; $17 \times 2 =34 \Rightarrow k^2-2$; $17 \times 3 =51 \Rightarrow k^2+2$; $17 \times 4 =68 \Rightarrow 4k^2+4$. Thus, the prime 17 is suppressed over 4 successive multiples before it explodes finally at $a=85$ with $(x=285769, y=30996)$. Thus, more generally we see explosions whenever a prime or its multiples escapes the regular categories. Among the primes, a famous set of primes (studied intensely from Fermat, Lagrange, Gauss to this date) that are of the form: $p=r^2+ns^2$ (typically written as $p=x^2+ny^2$, where $n=1,3$) tend to be particularly explosive if they are not suppressed by belonging one of the above regular categories. For example the above listed famous explosive values (e.g 61, 109, 181 etc) can be expressed as both $p=r^2+s^2$ and $p=r^2+3s^2$. Thus by Fermat’s theorems these primes are simultaneously: $p\;mod\;4 \equiv 1$ and $p \;mod\; 3 \equiv 1$.

Figure 5

We can now understand what we see when we look closely at the log-scale plot of $x$ against $a$ (Figure 5). We see “cycloid” like arcs between successive $a$ which are perfect squares. These peaks of these arcs are more pronounced for such even $a$. These arcs arise from the regular categories. The “central” part of these arcs are interrupted by eruptions arising from unsuppressed primes and their multiples. As $a$ increases the explosions get larger and larger, and as the distance between successive perfect squares increases there are more unsuppressed values which erupt. The distribution of values of $x$ show an interesting statistical property. Though the median value (lower dotted line in Figure 4, 5) is rather pedestrian the mean value is huge (upper dotted line). This illustrates how the minority explosive values have a much greater influence on the cumulative value of the system — a pattern often seen in natural systems where a few “hubs” dominate a distribution. This remarkable system made easy by the cakravāla illustrates the beauty of this method of old Hindu mathematics. In the past many have said that the Hindus were unlikely to have known of a proof for the cakravāla. We do not think so; rather it is likely they had a proof of some kind for why it always works and with it had some grasp of its consequences for the theory of numbers.

In the final part of this note we shall consider another second degree indeterminate equation, the bhujā-koṭi-karṇa nyāya: $x^2+y^2=z^2$. Its integral solutions where $(x,y,z \neq 0)$ are the famous b-k-k triples (Pythagorean triples in the west), which were studied by the Hindus since the days of the Veda. This likely has a deep history among the ārya-s and other IE people, which the white indological hot air is keen to ascribe to borrowing from the Mesopotamians (a discerning Hindu will see through the foundations of Mesopotamian boosterism that is common in the Abrahamosphere). In any case, the properties of its solutions are well-know to lay people with even the slightest mathematical interest but we are reiterating them here as another illustration of how even a simple indeterminate equations can produce complex patterns. whose. We encounter its solutions explicitly in the śulbasūtra-s and they continue to be discussed in subsequent Hindu mathematics. For instance the great sage Baudhāyana says in his śulbasūtra:
dīrgha-catur-aśrasya+akṣṇyā-rajjuḥ pārśva-mānī tiryañ-mānī ca yat pṛthag bhūte kurutas tad ubhayaṃ karoti |
The hypotenuse (akṣṇyā-rajju) i.e. diagonal of a rectangle produces
[the area] that is equal to both the areas [produced] separately by the horizontal and vertical sides.
Among the [sides showing above relationship] are those obtained in this form: (3,4,[5]); (5,12,[13]); (8,15,[17]); (7,24,25); (12,35,[37]); (15,36,[39]).

The first of the triples is the most primitive of triples and sets the properties for all the other solutions: 1) $xy\; mod \; 12 \equiv 0$: the product of the two legs of the triple is always divisible by 12. 2) $xyz \; mod \; 60 \equiv 0$: the product of all three of them is always divisible by 60. Now of the forms listed in the sūtra, the first 5 are illustrations of primitives i.e. are integers with greatest common divisor 1 (i.e. they are coprime). The 6th is merely a illustration of how further trivial triples can be obtained by merely scaling a primitive in this case $(5,12,13) \times 3$. However, triples of sūtra are likely chosen to illustrate another point:
$z^2-y^2=(z-y)(z+y)=x^2$
Three of them are of the form $z-y=1$, two are $z-y=2$ and the odd one out is $z-y=3$. Since the adhvaryu uses difference of squares formula $z^2-y^2=(z-y)(z+y)$ in the layout of the citi (3 sutra-s later in Baudhāyana) this was likely used to obtain the illustrated triples by setting $(z-y)=1,2,3$. With $z-y=1$ we get $x^2=z+y$ which easily yields the most primitive triple $(5+4=3^2)$ and further $(12+13=5^2), (24+25=7^2)$ etc.

More generally all primitive triples can be obtained using the formula in the complex plane:
$x+iy=(m+in)^2$ where $(m,n)$ are coprime and of opposite parity (i.e. one is even and the other odd).
Then the hypotenuse is $z=Mod\left(\left(m+in\right)^2\right)=m^2+n^2$. Thus one can see that in the triples $z$ and one of the legs would be odd and the other even. If we compute the primitive triples and plot them on the xy plane (Figure 6) we see that they present a complex pattern.

Figure 6

There are two aspects to the pattern: A relatively trivial aspect is the arrangement of the points on parabolic arcs which stems from the parabolic form of the generative formulae. Indeed, if one plots the points as $(x,y,z)$ in 3D, one can see that they lie on the parabolic sections cutting the surface of the generative cone of the conic (Figure 7).

Figure 7

The intricacy of their actual pattern arises from the interplay of this more obvious feature with the distribution of coprimes with opposite parity.  One specific aspect of this arrangement is the presence of certain $(x,y)$ pairs that differ by 1, e.g. (3,4), i.e. triples with legs differing by 1. Rather interestingly these pairs are related to a specific case of the above considered indeterminate equations: $x^2=8y^2+1$. The successive solutions of this equation are shown in the below table:

From the $y_n$ as above values (i.e. the y column) we get:
$p_n=y_n+y_{n+1}\\ a_n=\left \lfloor\dfrac{p_n}{2} \right \rfloor \\ b_n=p_n-a_n\\ c_n=y_{n+1}-y_n$
Now $(a_n,b_n,c_n)$ are the triples whose legs differ by 1

A further striking relationship captures a more general aspect of the distribution of triples: We first order all triples by increasing magnitude of their hypotenuse. $z_i$ will be the ith hypotenuse and $N(z)$ will be the number of triples with $z. Then we see that:
$\lim \limits_{z_i \to \infty} \dfrac{N(z)}{z_i} =\dfrac{1}{2\pi}$
This convergence is illustrated in Figure 8.  Thus, remarkably, as is typical of mathematics, we start with the elementary geometry of right triangles which are part of the rectilinear figures of the vaidika ritual, pass on to conics, intersect with the object of the cakravāla, and end up with a nice appearance of that quintessential number $\pi$.  Thus, one appreciates Gauss’s famous statement: “ἀεὶ ὁ θεὸς ἀριθμετἰζεῖ (the god always arithmeticizes)”.

Figure 8

## The Rāmāyaṇa and a para-rāmāyaṇa in numbers-I: epic as religion

This note may be read as part of our studies on the Rāmāyaṇa and para-Rāmāyaṇa-s of which an earlier part is presented here.

A study of the epic in Indo-European tradition suggests that there were two registers of the old Indo-European religion. While today both of them survive together with any vigor only among the Hindus, until not too long ago these registers showed some survival even among their Iranian cousins. From these it is apparent the first register is the “high religion” which manifests as śrauta and smārta performance. Among the ārya-s this further evolved into other manifestations as seen in the tantra-s of the sectarian traditions. Nevertheless, the Vedic base remained the model for most of these later developments. On the other hand the lay manifestation of religion was by the medium of the epic or itihāsa-s in India. Their religious value elsewhere in the Indo-European world was apparent in Greece. Indeed, in the classical Greco-Roman confluence the last attempt of reviving the religion by emperor Julian, which was being swept away by the “Typhonic” evil of the preta-moha, involved a focus on the religious facet of the Homeric epics.

In both India and Greece there are two epics, which have numerous parallels in their motifs, and resonate even in their overall themes. However, in India each has a distinct character. The Rāmāyaṇa is what might be termed “the universal epic of ideals.” The Mahābhārata is on the other hand our national epic, the epic of the first ārya nation in India, the foundation on which the modern Hindu nation rests. The Iranians have a comparable national epic in the form of the Kśathāya-nāmag and its precursors but apparently lack the universal epic. Among the Greeks to an extent the Iliad probably played a national role but tended towards the universal in the later phase. It was the universal epic, the Rāmāyaṇa, which was the vehicle of the ārya-dharma beyond boundaries of Jambudvīpa. In its role as the foundation of the “lay religion” it was remarkably tenacious and withstood the assault of the other Abrahamistic evil in the form the marūnmāda in Indonesia. It also served as a means of preserving the ārya-dharma in both India and in the east against the assault of the Aryan counter-religions promulgated by the naked-one and the ground-toucher. Indeed, in India the powerful force of the itihāsa-s was realized by successors of both these heterodox promulgators, who either attacked the itihāsa-s or tried to have people not attend their exposition.

The remainder of this note we shall look at the Rāmāyaṇa via numbers, which was part of my self-discovery of its key religious facet. Most importantly, it reveals something about the deep layers of the ārya-dharma and its evolution over time. Before we get started, a few caveats should be stated upfront: The texts I am using are the so-called “critical editions” of the Rāmāyaṇa and Mahābhārata for the first para-Rāmāyaṇa, the Rāmopākhyana of Mārkaṇḍeya. These critical editions have their faults but are available in electronic form and are thus amenable to semi-automatic text analysis by regular expression searches. Almost all of these analysis were performed by means of such. The Heidelberg system has a very sophisticated text-parsing mechanism for several Sanskrit works but I did not use it except for one word search (inspired by an interlocutor on Twitter), which will be discussed as part of another note, as it was not quite compatible with my command line pipeline. So my system could have some deficiencies but manual checking of the results shows that it is largely correct and the magnitudes should be taken as genuinely representative. In general for this activity you need to have a good knowledge of the various names of the gods, characters and weapons used in the text. Although not a paṇḍita, being a brāhmaṇa, I believe that I have a level of command of this as a reasonable representative of my varṇa should, so the results might be taken as generally reliable. Finally, I am aware that in white indological circles some work in this direction has been done by the likes of Brockington. However, I did not consult his papers as I wanted to have my own unbiased experience of the data and conclusions from it. More generally, wherever there is tractable data I believe that an educated man should analyze it himself rather than wholly relying on hearsay of others.

First we shall look at the gross features of the Rāmāyaṇa (Figure 1):

Figure 1

-The text has seven kāṇḍa-s, which are composed of multiple sarga-s, which in turn are composed of śloka-s. The Ayodhyā, Yuddha and Uttara have much more than median number of sarga-s and śloka-s.
-However, it is notable that except for Yuddha the other kāṇḍa-s have a nearly constant median śloka count per sarga (~24-27). This was the likely count maintained by Pracetas and his son Vālmīki the original composers of the Rāmāyaṇa for a typical kāṇḍa, probably aiming to be around 25 śloka-s. The Yuddha in contrast is longer both in terms of number of sarga-s and also the number of śloka-s per sarga. Clearly, this is a distinct composition suggesting that a different style was adopted on purpose for the military narratives typical of Indo-European epics. Unlike the median, the mean śloka count per sarga is higher with anomalies for both the Yuddha and Sundara. We shall take a closer look at this in Figure 2

Figure 2.

-Here we see the actual frequency distribution of the sarga length across the Rāmāyaṇa and per kāṇḍa in śloka-s: Here the differences are more apparent.
-The first three kāṇḍa-s are “tighter” in distribution with modal sarga length close to the median length. The Kiṣkindhā shows some divergence in the form of a fat tail with several sarga-s in of great length (40-70 śloka-s).
-The Sundara is most unusual in having a bimodal distribution with short sarga-s peaking less than 20 in length and longer ones peaking around 35. This pattern suggests a deliberate compositional shift perhaps reflecting the peculiar nature of the Sundara as an avenue for display of poetic beauty.
-The Yuddha is clearly distinct with the general peak and median length being shifted to being between 30 and 40. There is also a sizable fraction of very long sarga-s above 40 going all the way to well over 80. This again emphasizes the distinctness of the battle narratives where the long recitations perhaps appealed to the war-like ancient ārya audience who might have been in similar battles in their own lives.
-Finally, Uttara shows a typical median distribution of sarga length with a major fraction of sarga-s distributed around this value. However, it is distinct in showing a bimodality with two peaks one with length between 10-20 śloka-s and another with length between 40-45 śloka-s. This suggests a certain composite character with the shorter sarga-s probably representing the several short narratives included in it and the long ones relating to battle-sequences comparable to the Yuddha.

Now coming to the core issue of religion we shall look at the frequency of occurrence of the gods in the Rāmāyaṇa (Figure 3)

Figure 3

-It is apparent that Indra is literally the leader of the gods. He occurs nearly twice as frequently as the next contender Prajāpati or Brahmā. He is the standard for all comparisons and the hero of the Rāmāyaṇa is frequently likened to him. Indeed, there is a the tacit understanding that Indra used his māyā to take the form of a man in order to slay Rāvaṇa. This is suggested by Mandodarī’s lament upon her husband’s death:

atha vā rāma-rūpeṇa vāsavaḥ svayam āgataḥ |
māyāṃ tava vināśāya vidhāyāpratitarkitām || R 6.99.10

Or indeed Indra himself appeared in the form of Rāma,
for ruining and slaying you using impenetrable illusion.

Thus, it is hinted that Indra, who right in the Ṛgveda is famous for his māyā, uses it to kill the rakṣas.

Now again, though the core kāṇḍa narrative itself mentions Rāma taking the weapons of Viṣṇu from Agastya, in the preamble it is mentioned that they were the weapons of Indra himself.

agastyavacanāc caiva jagrāhaindraṃ śarāsanam |
khaḍgaṃ ca paramaprītas tūṇī cākṣaya-sāyakau || R 1.1.34c

At Agastya’s words Rāma verily took up Indra’s bow,
sword and the excellent inexhaustible quiver.

Of course the grand finale of the Yuddhakāṇḍa has Rāma ride the chariot of Indra steered by Mātali himself and using Indra-s weapons:

sahasrākṣeṇa kākutstha ratho ‘yaṃ vijayāya te |
dattas tava mahāsattva śrīmāñ śatrunibarhaṇaḥ ||

O descendant of Kakutstha, the slayer of foes, one of great strength and opulence, the thousand-eyed Indra has given for your victory this chariot.

idam aindraṃ mahaccāpaṃ kavacaṃ cāgni-saṃnibham |
śarāś cādityasaṃkāśāḥ śaktiś ca vimalā śitāḥ ||R 6.90.9-6.90.10

[He has also given] this great bow of Indra and his armor which glow like fire,
as also these arrows blazing like the sun and this bright sharp spear.

Finally, to slay Rāvaṇa he is said to use the missile made by Brahmā. But even here it is a mighty missile made by Brahmā in the manner of Tvaṣṭṛ in the Veda for Indra to conquer the three worlds:

brahmaṇā nirmitaṃ pūrvam indrārtham amitaujasā |
dattaṃ surapateḥ pūrvaṃ triloka-jayakāṅkṣiṇaḥ || R 6.97.5c

[The missile] was formerly made by the god Brahmā of immeasurable might for the sake of Indra. It was given to the lord of the gods [Indra] when he formerly sought to conquer the three worlds.

The missile itself has characteristics that are clearly suggestive of the vajra of Indra:
ratha-nāgāśva-vṛndānāṃ bhedanaṃ kṣiprakāriṇam |R 6.97.8c”
The swift acting [missile] was the smasher of [entire] troops of chariots, elephants and horses.
dvārāṇāṃ parighāṇāṃ ca girīṇām api bhedanam |R 6.97.9”
It was capable of smashing its way through through bar-reinforced doors and also mountains”

Tellingly it is described as “vajrasāram” (imbued with the essence of the vajra), and “yama-rūpam” (of the form of Yama). The latter epithet directly recalls the the first person statement of Indra in the 10th maṇḍala of the Ṛgveda where he says that he wields a missile that is like Yama himself.

The deployment of this missile by Rāma on Rāvaṇa is again thus described thus:

sa vajra iva durdharṣo vajrabāhu-visarjitaḥ |
kṛtānta iva cāvāryo nyapatad rāvaṇorasi ||

The missile, difficult to defend against like the vajra hurled by the arm of Indra, unstoppable like the causer of death (Yama), hit Rāvaṇa on his chest.

Thus struck Rāvaṇa fell:

gatāsur bhīmavegas tu nairṛtendro mahādyutiḥ |
papāta syandanād bhūmau vṛtro vajrahato yathā || R 6.97.021

His life-breath having departed the lord of the nairṛta-s of fierce speed and great luster fell from his battle-car to the ground like Vṛtra struck by the vajra.

Thus, to the ancient ārya audience this recitation would have immediately evoked the imagery of the Ṛgveda, where Indra’s heroic deeds in battle are praised in the ritual.
In conclusion, this makes it is clear that the original Rāmāyaṇa was composed in a setting where the aindra flavor of the ārya-dharma was the still the main expression of the religion. It is indeed likely that that it was tacitly implied that Rāma was a manifestation of Indra in human form to kill Rāvaṇa.

Now what about the rest of the Vaidika pantheon. Was it like the late Vedic age or the saṃhitā-s themselves?

-We see considerable prominence for Sūrya, Vāyu, Viṣṇu, Yama, Rudra in addition the Prajāpati/Brahmmā. However, the Aśvin-s, the Marut-s, the distinct āditya-s are not prominent. Agni has a moderate presence although primarily in the sense of poetic similes. This suggests that period of composition while still marked by Aindra dominance was one which was probably positionally distinct and temporally much later than the saṃhitā period. Of the prominent deities the indistinct solar deity suggests the rise of the new Indic solar cult with links to the older Āditya system but certainly very distinct in its manifestation with parallels to those seen in the Iranian world.

The prominence of Vāyu is related to his association with Indra in battle against the dānava-s, a feature which was prominent in both the Veda and the para-Vedic tradition. The latter is partly reflected in the Rāmāyaṇa and also relates to the importance of his son, Hanumat in the epic. We should mention here that in counting Vāyu we have almost entirely avoided including the incidental occurrence of his name as a epithet of Hanumat. A similar situation accounts in part of the prominence of Viṣṇu; however, his story has more which will be further discussed below. If Indra is identified with Rāma, and the role of Vāyu is taken by Hanumat, then it is rather obvious that the place of Viṣṇu is taken by Lakṣmaṇa. The Rāmāyaṇa makes this obvious in the statement:

vikramiṣyati rakṣaḥsu bhartā te saha-lakṣmaṇaḥ|
yathā śatruṣu śatrughno viṣṇunā saha vāsavaḥ || 6.024.029c

Your husband [Rāma] with invade the rakṣas with his brother Lakṣmaṇa even as the foe-killing Indra against his foes along with Viṣṇu.
or:

sa dadarśa tato rāmaṃ tiṣṭhantam aparājitam |
lakṣmaṇena saha bhrātrā viṣṇunā vāsavaṃ yathā || R 6.87.9

He then saw the undefeated Rāma standing with his brother Lakṣmaṇa like Indra with Viṣṇu.

Like in the Veda the most frequently referred act of Viṣṇu are three world-conquering strides suggesting that this old motif was still of great importance in the age of the Rāmāyaṇa rather than later elements like his incarnations or battles with certain demons. His weapon, the cakra is frequently mentioned, unlike in the Veda, where other gods are described as wielding it but not Viṣṇu. This suggests that the Rāmāyaṇa marks a stage after the saṃhitā period where the cakra became established as the favored weapon of Viṣṇu. However, it does preserve the memory of Indra’s cakra mentioned in the śruti in R 1.26.5. Notably, Viṣṇu is mentioned as killing the demon Naraka in a conflict which was perhaps coupled with Indra’s battle with Śambara:
śambaro devarājena narako viṣṇunā yathā | R 6.57.7

Thus is appears possible that this exploit of Viṣṇu was transferred to his avatāra Kṛṣṇa in a later retelling of the legend. Indeed, the whole Kārṣṇī retelling has Viṣṇu only thinly veiled by the Yadu hero.

-Of the other gods, Garuḍa and Kubera despite having a presence in the Veda are not prominent there beyond specific rituals. Nevertheless, even there, there is an under-current that they had a role of some note in household rituals. Their importance clearly comes out in the Rāmāyaṇa. In particular it is clear that the whole epic has a frame that tries to highlight the might of Rāvaṇa as the expense of Kubera, implying that he was an important deity of the time. He is named as one of the great regal gods along with kings Varuṇa and Yama and his greatness is repeatedly mentioned. This importance of Kubera, as we have seen before has a strong para-Rāmāyaṇa tradition too as laid out in the Rāmopākhyāna. Notably, in that relatively short text he is 3rd most frequently mentioned deity (Figure 4) suggesting that his importance was visible throughout the whole early phase of the Rāmāyaṇa tradition.

Figure 4

His importance is also implied by his airplane the Puṣpaka playing a notable role in the epic. His son Nalakūbara is also seen as cursing Rāvaṇa resulting in the protection of Sitā’s chastity upon her abduction. Kubera is also described as providing a secret missile to Lakṣmaṇa in his dream that allowed him to counter the Yama weapon of Meghanāda in their final encounter.

lakṣmaṇo ‘py ādade bāṇam anyaṃ bhīma-parākramaḥ |
kubereṇa svayaṃ svapne yad dattam amitātmanā ||

Lakṣmaṇa of fierce valor also deployed another missile, which given [to him] by the incomparable Kubera himself in a dream.

When the two missiles collided a great explosion is said to have taken place with a fire breaking out as they neutralized each other – in a sense implying that Kubera is no less than the god of death in his might.

-Yama in the Ṛgveda is strictly associated with the context of the funerary and ancestor rituals. However, there is again the under-current in the other saṃhitā-s that he was an important deity in regular existence as the god of death. This role of his in the Rāmāyaṇa is rather prominent and both in terms of numbers and the way he is referred to as a great king suggests that he was an important god in the ārya-dharma of the time. The death-dealing rod of Yama and entering his abode are common similes.

-Prajāpati: This deity is hardly present in the core clan-specific works of Ṛgveda – he is mentioned only twice outside of maṇḍala-10. But in maṇḍala-10 he has already risen to being the supreme deity in certain sūkta-s. He is conceived as both the overlord deity as well as the protogonic “golden-egg”. Now this would suggest that he was a late-emerging deity, probably specifically in the Indic setting after the ārya-s had left their ancestral steppe regions. However, we do not think this is the case. Comparisons with protogonic deities in the Greek realm suggest that such a deity predated the Greco-Aryan split. Rather we posit that he was not a key protogonic deity of the normative Indo-European pantheonic system but was the focus of one of several Indo-European cults outside the standard polytheism. Some deities who were part of the standard polytheism were also foci of such extra-normative cults but others like Prajāpati were solely cultic to start with. In both India and Greece the proponents of such protogonic deities started acquiring great prestige and religious centrality. In India this is reflected in the late Ṛgveda of the maṇḍala-10 and the brāhmaṇa-s where we witness the meteoric rise of Prajāpati. In the process of his rise he began to eat into the dominance of Indra, the head deity of the standard IE model.

In the itihāsa-s his ectype Brahman is likewise prominent as the head of the pantheon, though he is already beginning to face competition from the radiations from the cultic foci around Skanda, Rudra and Viṣṇu. What we see in the Rāmāyaṇa is that he is without any close competitor the second most frequently mentioned deity (Figure 3). His prominence in this itihāsa seems to be similar to what we see in the brāhmaṇa-s: As a deity at the head of the pantheon Brahman shares the position with Indra, but his prominence is clearly eating into that of Indra. This suggests two possible scenarios: 1) He was already a prominent figure from the very beginning of the Rāmāyaṇa tradition and his “power-sharing” with Indra is reflective of the parallel scenario in the brāhmaṇa-s were he had already risen to the highest rank. Thus this would imply that both aindra and prājāpatya memes were already active as the epic was being composed. 2) The Rāmāyaṇa as proposed above was primarily an aindra epic and Brahman secondarily encroached on Indra’s share in an independent replay of what happened in the brāhmaṇa-s.

On historical grounds we favor the second scenario. A comparison of the nāstika productions of the ground-toucher and the naked-one’s cults clearly indicate that at their time the prājāpatya strand of the religion was primarily among brāhmaṇa “intellectuals”. This intellectual link continued to later times when we see mathematical and scientific authors like Āryabhaṭa and Brahmagupta invoke Brahman as their deity (contrast with older scientific tradition in the Caraka-saṃhitā where Indra is dominant). The rest of the people in large part seem to have still followed the aindra religion until pretty late in Indian history with some competition from the other cultic foci mentioned above. This is indicated by the fact that the two nāstika teachers accepted this aindra mainstream as their background and mention the prājāpatya tradition primarily in the context of their brāhmaṇa rivals. Notably, in the first of the many Rāmāyaṇa of the jaina-s, the Paumacariyaṃ, Vimalasūri explicit calls out the stupidity of the āstika versions on grounds of their denigration of the great god Indra. This historical background would imply that the prājāpatya-s first rose as a dominant force inside the Vedic intellectual circles. The mark of this rise was first seen in the brāhmaṇa texts. Then as the prājāpatya-s “conquered” the intellectual landscape they extended their influence to more “secular” intellectual activities such as the itihāsa-s and mathematics/science. This was when Brahman came to prominence in the Rāmāyaṇa tradition. However, by the time the purāṇa-s started taking shape in their extant form, the other cultic sectarian foci had radiated enough to catch up and supersede Brahman. Of the old cultic foci, Skanda after an initial rise faded away. In contrast, Viṣṇu and Rudra came up to Brahman and soon overtook him to the point that despite the three of them being acknowledge as a trinity Brahman sunk to the “junior” position of the trinity. In part the tale of him having no temples might reflect the inability of the intellectual-centered Prājāpatya system to capitalize on the rising āgama-dharma, despite an early attempt hinted by the Atharvaveda pariśiṣṭa-s.

So what do the numbers from the text tell us? First looking at the Rāmopākhyāna we find that Brahman/Prajāpati has gone ahead of Indra (Figure 4). It was created by an author(s) who were clearly Prājāpatya and did not see any need to emphasize or maintain the position of Indra beyond what was absolutely unavoidable. What this tells us is that the Rāmāyaṇa tradition passed through a distinct phase after its original composition where Prajāpati had become important in it and it was in this phase that the fork leading to the Rāmopākhyāna was created. More tellingly, this proposal is supported when we look at the by kāṇḍa counts of key deities (Figure 5: shown as percentage of verses featuring particular deva). Here we see that Brahman has a peculiar distribution that is distinct from that of Indra and Vāyu. While the latter two show clear kāṇḍa-specific differences, they are more uniformly distributed across the kāṇḍa-s. In contrast the occurrences of Brahman show a significantly higher occurrence in the Bāla and Uttara kāṇḍa-s while being greatly under-represented in the rest. We know that both these kāṇḍa-s were clearly subject to reshaping after the core epic was composed because they try to explain things which were not clear elsewhere in the epic (e.g. the origin of the heroes and villains of the text). This together with the above observation clinches the case for the second of the above proposals: after the original epic in an aindra form was composed the Prājāpatya-s refashioned it by primarily redacting the first and last kāṇḍa-s.

Figure 5

-Viṣṇu again: Two other major deities show a similar of kāṇḍa-wise pattern of distribution as Brahman: Viṣṇu and Rudra. Importantly, they are minor in their presence in the Rāmopākhyana (Figure 4). This suggests that the vaiṣṇava and śaiva redaction occurred later than the forking of the Rāmopākhyāna and acted in manner very similar to the prājāpatya action before them. That they were also directly in conflict with each other is suggested by the fight between Rudra and Viṣṇu which is inserted into the bāla-kāṇḍa. Another key point is that the vaiṣṇava material show no strong hints of the avatāra doctrine nor the early pāṅcarātrika tradition which is strong in the Mahābhārata. This suggests that the vaiṣṇava redaction comes from an early stream of the sect that underwent further evolution by time of the redaction of the Bhārata.

-Rudra: In the Rāmāyaṇa has his characteristic features of being dark-throated, three-eyed, with braided locks (Kapardin), having a bull as his banner/vehicle, holding a great bow, having Umā for this wife and displaying great ferocity. His destruction of the Tripura-s is frequently mentioned. Additionally, his slaying of Andhaka gets multiple references. These references frequently come in kāṇḍa-s 2-6 suggesting that they are indeed the ancient similes involving the deeds of Rudra. E.g.

sa papāta kharo bhūmau dahyamānaḥ śarāgninā |
rudreṇaiva vinirdagdhaḥ śvetāraṇye yathāndhakaḥ || R 3.29.27

He, Khara, fell to the ground being burnt by the fire of the missile even as Andhaka [fell] burnt down by Rudra in the White Forest.

Most of these features have direct or indirect reference in the Veda, often going back to the oldest layers. However, we do not hear of his exploits made prominent in the purāṇa-s like the killing of Jalandhara or Śaṅkhacūda. Thus Rudra in the Rāmāyaṇa has not changed in any notable way from his Vedic form.

-Finally one may note that in this Kāṇḍa-wise distribution Kubera is mostly uniform across kāṇḍa except for the uttara – paralleling Vāyu to an extent. This we believe suggests his ancient and intrinsic importance to the text with the Uttara merely serving as a receptacle for lore relating to him and Vāyu.

In conclusion, we can say with some confidence that the great Rāmāyaṇa of sage Vālmīki was originally an epic encapsulating the popular register of the Indo-European religion as manifest among the Indians – the ārya-dharma. Its heroes were set in the mold of the great deities Indra (Rāma and Vālin), Vāyu (Hanumat), Viṣṇu (Lakṣmaṇa), Kubera (perhaps some of the Kuberian element transferred to Vibhīṣaṇa), the opaque popular Āditya (Sugrīva), with simile-linkages to Rudra and the Maruts (encompassed in Hanumat). Despite the later sectarian redactions starting from the prājāpatya-s casting it in different light, it retained this ancient religious spirit of the ārya-dharma. It was this that erupted forth like the great ape Hanumat to animate the Hindus in their life and death struggle against the unadulterated evil of Mohammedanism when they seemed all but lost. That is why a memorial to the epic should be built at Ayodhyā after destruction of all marūnmatta elements in the holy city.

## The great faceless man

In the late Yajurvaidika upaniṣat, the Śvetāśvatara, which is the foundational text of the śaiva-śāsana, the god Rudra is described thus:

na tasya pratimā asti yasya nāma mahad yaśaḥ।
There is no one who his equal, whose name [itself] is great fame.

This sentence has also been taken to organically imply something else among Hindus too: Statues are not made of the great people – their name itself is great fame. Keeping with this we mostly do not have statues of many of the great figures of Hindu tradition. For instance, we do not know how Bodhāyana or Āpastamba or Āśvalāyana or Paippalāda looked, though we take their names on a daily basis. So also with great men even closer to our times, like say Vācaspati Miśrā. Now, speaking of other heroes, like Rāma Aikṣvākava, whose name might be almost taken daily in some of our households, we have some kind of a description in the opening of the Rāmāyaṇa:

ikṣvāku-vaṃśa-prabhavo rāmo nāma janaiḥ śrutaḥ |
niyatātmā mahāvīryo dyutimān dhṛtimān vaśī ||

Born in the Ikṣvāku clan, he is known among men by the name of Rāma. He is self-controlled, of great manliness, radiant, resolute, and has his senses under control.

buddhimān nītimān vāgmī śrīmāñ śatru-nibarhaṇaḥ |
vipulāṃso mahābāhuḥ kambugrīvo mahāhanuḥ ||

He is intelligent, politically astute, eloquent, opulent, and an extirpator of foes. He is broad-shouldered, of mighty arms, with a conch-like neck, and strong-jawed.

ājānubāhuḥ suśirāḥ sulalāṭaḥ suvikramaḥ ||

His chest is broad, he is a great archer, his collar-bones are well-concealed, and is a suppressor of foes. With arms reaching up to his knees, with a good head, shapely forehead and good gait.

samaḥ sama-vibhaktāṅgaḥ snigdha-varṇaḥ pratāpavān |
pīnavakṣā viśālākṣo lakṣmīvāñ śubha-lakṣaṇaḥ || 1.1.8-11

His body is well-proportioned, he is of smooth complexion and mighty. His chest is rounded, his eyes large, he is prosperous and with auspicious marks.

Similar accounts might be found elsewhere in the Rāmāyaṇa too. One thing which comes out of this account is that it is fairly generic for a mighty kṣatriya except for one specific, unusual feature namely “ājānubāhuḥ” – i.e. that his arms reached down to his knees, which might have been a peculiar characteristic of the man himself. Thus, while one can build a generic image of emperor Rāma as a mighty kṣatriya, we can still say we do not know how he *exactly* looked. Now, this is in part keeping with a the broader issue we have discussed earlier, namely the iconic depictions of deities among the early Hindus. As we argued before such existed but were not prominent and were perhaps “primitive” keeping with the archaeological evidence from several early Indian sites. In this sense the Indian iconography mirrored the primitivism of the early Greek iconography.

This is in stark contrast to Egypt where their great Pharaohs are known more from their portraits rather than epic narratives. When we see the images of the lordly Pharaohs, while stylized, there is clearly an element of individuality behind them. Over the ages of its heathen existence, in addition to statues, Egypt developed an even more realistic portraiture in other media. It is conceivable that this Egyptian tradition of portraiture spread through West Asia and then Europe influencing other cultures, first Semitic and then Indo-European. Thus, we see it emerge first among the Hittites to some extent and then eventually among the yavana-s (here collectively Greek and Macedonian) and romāka-s. Thus, by the time of the Macedonian invasion of India we we see a vigorous tradition of realistic royal portraiture on their coins, medallions and mosaic work that has moved far away from their ancestral primitivism. Early Indian coinage was abundant in symbolism and even primitive iconography of Hindu deities but not royal portraits. However, in the years following the Macedonian attack we see an emergent tradition of such portraiture both on coins and in the form of larger icons as seen in the case of Aśoka the mighty Maurya. This trend would suggest that the impulse for portraiture eventually reached the Indian world only via the Macedonians. Likewise, there have also been plausible suggestions that the movement of yavana-s eastwards along with contacts with the now fully portrait-using Hindus sparked the emergence realistic portraiture among the cīna-s along side the unification by the Chin. This manifest burst upon the scene on a truly cīna scale in the funerary statuary of the Chin conquerors.

However, it does remain a fact that the explosive spread of *extant* portraiture mostly post-dates the Macedonian invasion, which could imply a causal link between the two. Of this most traces have been erased outside the peninsular tip by the evil hand of the Army of Islam. In Nepal however some early specimens survive such as the image of king Jayavarma-deva from śaka 105= 185 CE. In a 3 century time window from that point we also see traces portraiture among the Śuṅga-s, Andhra-s, and Iranic rulers (Kuṣāṇa, Pahālava, Śaka) in India. In discovering and describing the image of king Jayavarman of Nepal, Tara-ananda Mishra has given an excellent account of the evolution of portraiture including several points we have independently arrived at. Other than royal portraiture there appears to have been a vigorous tradition of the imagery of persons endowing religious images and constructions. Moreover, śaiva teachers of both the mantra-mārga and atimārga, extraordinary śaiva and vaiṣṇava devotees, and siddha-s were also prominent objects of whole-body portraits. Statues of such were once common throughout India but have been mostly damaged or demolished in the north. Thus, in south India we can still see statues of emperor Kṛśṇadeva or Govinda Dīkṣita but we do not have the original image of the great Bhoja-deva or Lalitāditya.

Yet, despite all of this, on the whole the the majority of our great figures have not survived in portraiture, perhaps indicating a real Hindu tendency for the statement: na tasya pratimā asti yasya nāma mahad yaśaḥ। . We too seem to be resonant with this Hindu tendency and take it in a broad sense as was possibly understood by many of our greats of the past. In large part we believe that a man’s visual image or for that matter several details of his life should not matter at all. All that should matter are the words he leaves behind – do you find something in them or not – that’s all. Indeed, this is how it is for Āryabhaṭa-I or Bhāskara or Nīlakaṇṭha Somayājin. They are their words not their portraits or even biographies. Now one may ask: “Have you not said there is great value in studying the biographies of past intellectuals?” I still hold that view but have always held that not all of their biography really matters. Indeed most Hindu notables have left behind colophonic biographies that only stress the needful in the best case scenario. For instance, they are not shy of revealing their youthful genius, like bhaṭṭa Jayanta recording his advanced grammatical research as a kid or mathematician-astronomer Gaṇeśa recording his discoveries like the hyperbolic approximation of trigonometric functions in his teens or Vaṭeśvara his discovery of mathematical recipes for astronomy in his twenties.

This contrasts what we observed among the mleccha-s today and its back-flow to the Hindus. The person, his appearance and certain incidental aspects of his biography (e.g. ones sex, sexual escapades or sexual orientation) seem to matter a lot. Indeed, the appearance of a person has a serious correlation to his/her intellectual influence especially in a lokābhimukha sense. There is a drive to get people who look acceptable in a mleccha sense as the face of even an intellectual matter, like physics. I was amazed by how certain bhārata-s known to me converged to a similar style of appearance after hitting the talking circuit in the mleccha world. More than a person’s intellectual substance what matters is that they lie at one extreme of the bell-curve in terms of the mleccha-defined (partly universal) measure of appearance, appeal, etc. Almost as though out of guilt, the mleccha-s would then balance it with a few chosen tokens from the other extreme of the normal distribution (e.g. the diseased, the handicapped and the rare ethnicities) and pepper it with elements of what matters in the mleccha facade: e.g. aberrant sexuality. Finally, in the biographic sense, often what the person holds on matters well-beyond his/her expertise is of great importance in his/her projection as an icon: s/he better mouth liberal platitudes along the lines of how they are out to save the world, save the disadvantaged, and other feel-good messages quite removed from reality and sometimes even biology all while looking ‘cool’ at the same time. This would have even been half-understandable where it not for the concomitant insistence of the mleccha system being “equal opportunity”.

In face of this facade we think great father Manu had a point – the brāhmaṇa intellectuals are best off being low-key in such dimensions – the faceless intellectual whose biographic peculiarities matter little. In conclusion, perhaps keeping with the Hindu tepidity towards portraiture, we subscribe to the view that it is better for a man’s words than his portraits to linger on after Vaivasvata takes him away – indeed what use is it to be like the mysterious brāhmaṇa from what is today Afghanistan, the fragment of whose portrait is placed for auction in a mleccha market.