Kalpa in Jyotish was not an imaginary quantity of Time!

The vastness of time as well as the skies contrasts so vividly with the delimited and bounded physical space of earth that it could have easily overwhelmed humans into thinking that the universe was a philosophical entity. However, as the expanse was sized up by finiteness of measures, its material and mathematical dimensions got as revealed. Jyotish, as the study of the ‘bodies with light in the sky’, had sought to measure the universe is all its three dimensions – time, space and heavens through the observation of the movements of the objects in the sky. In these motions, time and space seemed to intermingle. We know today that the length of days, nights and seasons, and the length of the year are results of the Earth going round the Sun in an orbit while it rotated in its axis with a tilt. As the earth was rotating on its axis from west to east, everything in the sky appeared to go from east to west on the ecliptic. Jyotish’s overall reference dial for the measurement of time and space is set in the stellar background of the Solar system. Two dials were conceived – one charting the Sun’s movement called Rasi and the other referring to the Moon’s movement called Naxyatra. Both the scales were angular and calibrated on the imaginary great circle of the ecliptic.

The use of two different angular scales with its own system of calibration plotted in the circumference of a circle with units of natural and as apparent motion of the Sun and the Moon appears to have given them great accuracy. However, the angular dimensions based on measurement of arc would only be approximate because arcs are measured only as straight segments and also as the value of **π** is an irrational number. By fifth century, the standard value of **π**, measured as ratio of circumference of a circle to its diameter, was empirically specified as 3.1416 with a circle of diameter 20000 and circumference 62832 by Aryabhatt. Similarly, for general mathematical works, one ninety sixth division of the arc could be taken as straight. The problem was also sought to be resolved taking the reference of a circle with radius of 3438 units, whose measure of circumference 21600 was numerically same as the number of minutes in the whole angle (360×60). A standard table of measures of sets of chords and perpendiculars defining the angle was also in use, with angular intervals specified to one two hundred fifty sixth division of the Rasi angle of 30^{0} greatly increasing the accuracy of calculations[1]. This series of angles (in degrees measure) obtained by halving and quartering of 1R was thus detailed to the interval of 7.03 minutes: 30, 15, 7.5, 3.75, 1.875, 0.937, 0.468, 0.234, and 0.117 (or, 1/256 of 30^{0}). The question, ‘why was it worked to such miniscule divisions?’ may be answered simply as an accuracy call. But it may also have been the result of a quest for a point of approximate convergence of the Rasi scale (worked through bisection of 30^{0}) divisions with those of Naxyatra scale (which was worked by trisection of 40^{0}). This is revealed by the following series (in degrees) obtained on the basis of trisection of 1N angle: 13.333, 4.444, 1.481, 0.493, 0.164, 0.055 (or, 1/243 of 13.33^{0}). Since half of the last term of R series is so close to the last term of the N series, we can infer that circular scale with intervals of 3.5 minute could have been used with accuracy in both the Rasi and Naxyatra computations.

The Naxyatra scale, worked on the trisection of 40^{0} angle in the circle deserves a closer examination here. Since method of trisecting an angle by construction is not known to mathematics, it could not have been an arbitrary theoretical calibration but a very natural unit of measure for the problem of the measurement of time as a product of the motion of the heavenly bodies of Moon and Sun as seen from earth. The primary prompting to the making of a scale with 27 subdivisions would have been the period of 27.32 days that the moon took to complete one round in the field of stars. The nine angles of 40^{0} each that made up the *navayoni *circuit in whole circular angle of 360^{0} almost matched the precession of perigee per year of the moon, which had a measure of 8.84 years and which was related to the occurrence of solar eclipse and the new moon cycle producing it. Various other periodicities of relevance to the exacting measurement of time, as they were discovered in due course, appears to have led the observer to change the Naxyatra divisions from 32 to 28 and finally settle on 27 as the most natural.

In an effort to put all of these various periodicities of the earth, Moon and Sun together as well as the other bodies in the sky and so as to get a fix on the present moment of time, Jyotish came up with a five-way Luni-Solar definition and quantification on the two angular scales e.g. day and date of the Solar month, time, day and date of the Lunar month (*tithi, * *naxyatra*, and *karana*), and *yoga *(cumulated travel of Moon and Sun). It has now become common knowledge in the Sciences that a point in plane space can be fixed absolutely with reference two other fixed points – a principle of triangulation. To fix a point in the three dimensional universal space, however, we need to triangulate it with three other points making a triangle in a given plane i.e. we need a tetrahedron or four plain triangles. As time is a dimension resulting from the play of three moving points so to speak (Sun, Moon and Earth), we may appreciate the complication behind such pentagonal fix designed by the ancients.

Yet, there must have been no doubt in the minds of the pioneers in Jyotish that time was very much of a human dimension of the universe. As it renewed on and on with the phenomena of eclipse making the point of renewal, the earthly flavor was unmistakable. Tracing the movement of the Moon to further particularize time must have led the ancients to complicated mathematics too as the Moon was going round the earth in a rhythm of its own and with its plane of orbit tilted at a small angle and wobbling to another rhythm! This tilt had brought into play a set of two moving points of intersections on the ecliptic (and named as Rahu and Ketu). And get the point of time of eclipses, when the earth, the Moon and the Sun momentarily come together in a linear alignment only to begin wandering away from each other to repeat the convergence again and so on and the trace and measure of the apparent motion of the Moon and the other eight spheres in the sky on the starry scale seem to have expanded the repeat cycle of some phenomena by several times over. Jyotish, thus appears to have jumped the human scale of time to transcend to the so called God’s scale of time as these long durations became apparent. The idea of Mahayuga had been born as the point of time for the present Kaliyuga was set about 3^{rd} century CE. The Jyotish, as it developed more finer measures of time, it even had to create new and higher level of transcending scale of time, called Kalpa. It is almost as if the measurement of the vastness of time and space led the Jyotish to get philosophical again. This article seeks to show that such quantities of transcending time, whether they be yuga, mahayuga or Kalpa, are not products of imagination but actually measured real quantities of time pertaining in the human fathomed depths of universe!

The variation in length of days and nights along the year, the shortest days in winter, the longest days in summer and the equinoxes in between in spring and fall and the cycle of seasons, and the critical role they played in the nourishment of human life on earth encapsulates the basic human dimension of time and must have been points in time to celebrate life itself. In due course of observation, the Jyotish would have sensed that the location of these particular days were also moving on the scale of seasons in a microscopic measure. It is not clear as to when it was sensed or estimated, but it’s periodicity of 25920 years is reported in both the Greek and Hindu astronomy almost about the same time e.g. as Pythagoras’ Great Year and in Bhrigu Samhita’s birth chart (HORA) rules. This appears almost 2500 years following the Druid debacle at Stonehenge[2]. For the Vedic astronomer, sixty years at this pace of 50 seconds per year made 50 minutes, which equaled 1/432^{nd} part of the whole angle (360^{0}) or a quarter of a *pada* in the Naxyatra scale. Apparently, this new measuring unit of ‘*yuga*’ spanning 60 *samvatsara* was used in the *Brihatsamhita *replacing the ‘*yuga*’ that spanned five *samvatsara* as per *Vedangajyotish [3]. *This change of nature of ‘

*yuga*’ indicates that the measure of precession of equinoxes was already incorporated in the calculation of time by third century CE.

32 Padadevatas: on periphery

[ 28 Naxyatra + 4 Directions N – Soma, E – Surya, S – Yama and

W – Varuna]

12 Adityas in middle ring

1 Brahma in center

Since the Jyotish thinkers had linked renewal of time in human scale through the eclipse, the recurrence cycle of an identified specific eclipse or its periodicity was used as markers in time. In this count, the period of 25920 years taken by the precession of equinoxes to complete the whole angle of 360^{0}, is divided into 32 parts of 810 years each and each new cycle starting with every 32^{nd} recurrence of the eclipse. So, in the square diagram called Vastupurushamandala, these 32 divisions (*pada*) in the periphery are each assigned to one deva (god). And with Brahma in its center, the count of gods in the mandala becomes 33 and Vastupurushamandala itself becomes a personification of the existential world with its 33 gods. With the 12 Adityas resident in between and in the middle ring, the universe was pictured as a square mandala of 45 Gods, held in space in a web of Luni-Solar time! Clearly the cycle of the Great Year was known long before the precession of equinoxes had been measured and incorporated in the *Suryasiddhanta *of *Varahamihira *by 247CE. From the account of Sumatitantra, a Jyotish treatise compiled in 7^{th} century in Kathmandu Valley, that the knowledge and mathematics of *Suryasiddhanta *was got from proxy of the Sun God himself towards the end of Satyayuga, we can rationally infer that Satyayuga and its measure as a duration of 1728000 years had been computed as *Suryasiddhanta *was firmed up. Clearly, the enormous precision in measure of time cycle achieved in the conceptualization of the module of sixty samvatsara as yuga in human scale had led to the firming up of the huge new ‘godly’ time frame. How was this transcendence of the scale of time of the mortals to that of the immortals computed? Details suggest that the transcendence is set through the equivalence of one day of the Gods to one year of men, 360 such days making one year of the Gods and 12000 such years making their era called Mahayuga. The Mahayuga is equal to one cycle of Satyayuga (1728000 Solar years), followed by Tretayuga (1296000 Solar years), Dvaperyuga (864000 Solar years) and Kaliyuga (432000 Solar years) and so lasts for 4320000 Solar years in total.

It can be seen through simple calculation that in reference to the circular angle of 360^{0}, while one Yuga of humans of 60 years duration was 1/432^{nd} part of it, with a value of one Solar year assigned to one second, 120^{0} will make 432000 Solar years or one Kaliyuga. It seems that the total angular precession of equinoxes in 60 years i.e. 50 minutes expressed in seconds is itself 3000 yielding the higher value scale automatically. What was the causative factor for such a change of scale then? We know today that apart from the periodicity of daily, yearly and precession movement of earth, the inclined axis of earth itself has a periodicity of 41000 years and this would have caused the need to think of a larger cycle of time. A further level of transcendence is set through the equivalence of one day of Brahma to 1000 Mahayuga era of Gods. This is reiterated in Bhagavatgita 8.17:

सहस्रयुगपर्यन्तमहर्यद्ब्रह्मणो विदुः ।

रात्रिं युगसहस्रान्तां तेऽहोरात्रविदो जनाः ।। १७ ।। Bhagavadgita 8.17

(Tr. The *yogi *who knows that a day of Brahma lasts for thousand yugas, each a sum of the set of four yugas, and the night as long, has understood the reality of time.)

But the same duration for a day of Brahma is already told in the following hymn from the Rigveda, and this must mean that such long duration of time was already possible to be computed much before its detailing in *Suryasiddhanta*:

चत्वारि श्रृङ्गा त्रयो अस्य पादा द्वे शीर्षे सप्त हस्तासो अस्य ।

त्रिधा बद्धो वृषभो रोरवीति महो देवो मत्यीँ आ विवेश ।। ३ ।। Rigveda 4.58.3

(Tr. This god of fire sacrifice has 4 horns, three feet, two heads and seven hands. The great god making loud sound like a roaring bull held by ropes from three directions enters the arena of the mortals.)

The symbolism in the second part of the hymn, the tripartite roping in of time and space finds repeated in other myths too. For example, Brahmanda as the universe of human existence and the chariot of Brahma, is said to be in three wheels incessantly moving on the track of the ecliptic with all objects in it in motion relative to each other and the chariot itself making no translation. The radiant nature of dispersal of the essence of time and space and its tripartite anchoring to the ecliptic circle tell that the circular scales have three points of references at 120^{0} to each other in the space. We can speculate that the Vedics may have observed some changes in previously known cycles and allowed for a hidden cycle of 40000 years periodicity somewhere in the motion of the Sun and the Moon to accommodate the observed nuances!

With the newly recalibrated angular scale, completion of a period of Mahayuga would require three full circles and one 120^{0} segment, making the zero point shift by 120^{0} just as hinted by allusion to the bull held by ropes in three directions. So where was the zero mark in the circular Naxyatra scale at the start of the present Kaliyuga? This is what the Vedic thinkers were most probably referring to in the following *sloka*:

त्रीणि शतात्री सहस्राण्यग्निं त्रिशंच्चदेवा नवचास पर्यन ।

औक्षन्घृतैरस्तृणान्वर्हिरस्मा आदिध्होतारंन्यसादयन्त ।। (Rigveda 3.9.9)

(Tr. 3339 devas are welcoming Agni in worship, they had bathed him in Ghee, laid a Kusa grass seat and had him seated as the receiver of the sacrifice.)

We have seen in the Vastupurushamandala representation of universe as a square Luni-Solar model of time and space that these 32 divisions (*pada*) in the periphery are each assigned to one deva (god) and with Brahma in its center, the count of *deva *was 33, also referred as 33 *kotideva, *particularly when referring to the * *periphery of universe as circumference of a circle. The *kotidevata *have been named as 12 Adityas, 8 Basus, 11 Rudras, 2 Aswini Kumars, the god-names clearly indicating it as a Vedic period detail. The term ‘*koti*’ has been used in Sanskrit with various contextual meanings such as (i) a number equal to 100 lakhs or one Krore, (ii) Tip or edge of an arm like sword, (iii) Tip of the arrow, (iv) class, level or rank, and (v) base of a right-angled triangle. The application of the first meaning of Koti as count of gods in the universe has led to the belief that there are 330 million gods in Hindu pantheon. But taking the fifth meaning of the word ‘*Koti*’ as used in *suryasiddhanta* and other mathematics (*Jyotpatti*) literature, it states a measure of the arc angle in circular scales in terms of the base segment of a right triangle. In the above stanza, the number of ‘*deva*’ noted refer to the angle indicated by the 3339 segments of the base radius of a circle divided into 3438 ‘Koti’ divisions (as per the reference table also attributed to Bhasker), one minute yielding one segment on the circumference. On this basis, the following computation can be made to pin-point the start of present Kaliyuga.

According to *panchanga *calendar, the year 2022 CE is year 5123 in Post-Kali era (Pant, 2022) and the position of Sun at start of Daxinayana this year was at 65^{0}21’11”(read from Pant, 2022 pp.34). We can also see from the calendar that it uses the point at start of Mesh Rasi or start of Aswini Naxyatra as the general reference.

According to Varahamihira, on the base year of compilation of Suryasiddhanta, the Sun at the start of Daxinayana was at start of Karkat Rasi, i.e. 90^{0}. Thus the total precession between the year and 2022 CE is 88729” and using the per year precession rate of 50”, we get the time lapsed as 1774.58 – So we can say that Suryasiddhanta was compiled in the year 247 CE or 3348 Post-Kali year. This computation shows that the result of precession makes the summer solstice points move anti-clockwise

Using the same rate of precession of 50”, the angular location of Sun at the Summer Solstice at the start year of Kaliyuga, would have moved 5123 x 50 or 256150 secs further anti-clockwise from the position of Summer Solstice in year 2022 CE (65^{0}21’11”) or 235271 secs. The position in relation to the zero point at Mesh Rasi (or also starting point of Aswini Naxyatra) is thus 491421 sec or 136^{0}30’21”.

According to the above hymn, the angular position of the Sun (as 3339 Kotijya[4] on the radius of circle of 3438 divisions) would be further anti-clockwise by Cos^{-1 }(3339/3438) or 13^{0}47’.

Since in the then system, beginning of a new year was taken at the vernal equinox, the zero point in the Naxyatra scale at the start of the present Kaliyuga would be at 240^{0}17’ anti-clockwise of starting point of Aswini. We can therefore conclude that statement 3.9.9 in Rigveda is actually showing that the base Naxyatra for the present era is Mula, which as a word also means the base, starting or zero point of a scale.

If the discovery of the actual value of **π** as well as the mathematical relation that the length of the circumference of a circle was 2 π times the radius of the circle both came later, ‘what could have led them to make a very close approximation of this magical circle of 3438 units radius by as early as fifteenth century BCE when *Vedangajyotish *is thought to have come into use?’ is a question worth pondering. It appears to me that efforts to use subdivisions of the Naxyatra scale to work on the large base year of four Mahayugas or 17280000 Solar years and other measured cycles of revolutions (*bhagana*) of Moon, its nodes and visible planets in the solar system could have easily revealed the approximate data. It can be seen that a further quartering of 1/432^{nd} division yields the 1/1728^{th} of the circular angle and 1/3456^{th} division was just a step away to discovery that the circumference of circle with such divisions of the chord would be about 21600 units and allowed approximation of the value of **π** to 3.125! It would appear that fixing of the measure of Mahayuga had paved the way for the level of accuracy achieved early on. And as the estimated accuracy increased to 3.1416, the measure of the length of a day of Brahma (Kalpa) was revised downwards from 1000 Mahayugas to 71×14 Mahayugas in the later documents(such as *Visnupurana 1.3.16-22*)*, *the smaller number 994 is clearly the factor 3.125/3.1416 of 1000. Visnupurana details this new corrected duration of the day of Brahma in clear numerical terms:

ब्रह्मणो दिवसे ब्रह्मन्मनवस्तु चतुर्दश ।

भवन्ति परिमाणं च तेषां कालकृतं श्रृणु ।। १६ …

चतुर्युगाणां संख्याता साधिकाह्येकसप्ततिः ।

मन्वन्तरं मनोः कालः सुरादीनां च सत्तम ।। १८ ।।

अष्टौ शत सहस्राणि दिव्यवा संख्यया स्मृतम् ।

व्दिपञ्चाशत्तथान्यानि सहस्राण्यधिकानि तु ।। १९ ।।

त्रिंशत्कोट्यस्तु सम्पूर्णाः संख्याताः संख्यया व्दिज ।

सप्तषष्टिस्तथान्यानि नियुतानि महामुने ।। २० ।।

विंशतिस्तु सहस्राणि कालोऽयमधिकं विना ।

मन्वन्तरस्य सङ्खयेवं मानुवँर्वत्सरैव्दिँज ।। २१ ।।

चतुर्दशगुणो ह्येष कालो ब्राह्ममहः स्मृतम् ।

ब्रह्मो नैमित्तिको नाम तस्यान्ते प्रतिसञ्चरः ।। २२ ।।

(Visnupurana 1.3.16 –22)

(Tr. A day of Brahma is made of 14 Manvas, listen to the subdivision of this time. A time period lasting 71 times the number of years of the four-yugas make a Manva interval– the time of men and gods. The count of godly years in a Manva interval is remembered as 852000 and counted in men’s years Manva interval is 306720000 and not more than this. Fourteen times this duration makes a day of Brahma …)

Clearly, the duration of the day of Brahma was revised downwards from 4320000000 years in Rig Veda knowledge to 4294080000 (= 306720000 x 14) years as the accuracy of computations of the circumference increased in later periods. One year of the gods is as long as 360 years of men or one year of men is equal to one day of the gods. Kalpa, one day of Brahma, thus, was not a philosophical idea but a measurable quantity, verifiable through observations on the circumference, a quantity affected by the value of 𝛑. As a matter of fact, we have seen that all the three levels of time (Yuga, Mahayuga and Kalpa) are all based on observations and measurement on the circumference of the circle of Rasi and Naxyatra scales applied on the ecliptic! A consequence of this would have been the replacement of the square Vastupurushamandala of the Vedic times by the circular one of the Sriyantra.

[1] The table of perpendiculars and bases of the circle with radius of 3438 units is attributed to Bhasker (1150CE) but the idea of the mathematics was obviously there since much earlier.

[2] Druids appear as the first community of faith that suffered for lack of this measure. They seem to have deserted the Stonehenge about 3000BC after around 400 years of use as the magical 40^{0} horizon angle of Sun had visibly failed in the morning of Summer Solstice day there.

[3] From the data given in *Suryasiddhanta*, we can see that *Vedangajyotish* theory was worked about 1431BCE.

[4] Chandra Hari has taken the number as 3339 year of the Kali era and deduced it as the year 238CE (Chandra Hari, 32 (3), 1997).