** – **Sudarshan Raj Tiwari

Of *tantra, mantra, *and *yantra, *the three conceptual ways of representing the god spirit in tantric practices – *yantra* is in the form of a geometric diagram. Of the many geometric instruments known in *tantra*, the Sriyantra is the most exalted one and represents Tripurasundari, the Goddess of the Universe. Its geometry is shown alongside (Fig.1). The exacting geometry of triple intersections (*marma*) specified for the nine triangles in the core makes its construction extremely complex. *Trivalaya* is the set of three closely drawn concentric circles lying between the outer enclosure formed of three closely drawn squares called the *Chaturastra *and the lotus with sixteen petals called the *sodasadala *on the inside.

**Fig. 1: The Geometry of Sriyantra**

The eleventh stanza of Saundaryalahari, the book of chants composed by Sri Sankaracharya, describes Srichakra as ‘the composition of four *Srikanthas *(Siva-triangles) and five *Sivayuvatis* (Shakti-triangles) gathered to bring forth the nine *Mulaprakritis *(basic triangular manifestations of Nature) as one united around the *bindu *or the origin-point in the center. The eight-petal lotus, sixteen-petal lotus, the triple girdle and the triple lines of enclosures (the Siva-chakras) along with the core make your refuge with the forty-four *Saranakonas *(nooks)’.

* rt’le{Z>Ls07}M lzjo’jltleM k~rle®lk*

* k|leÌfleM zDef]g{jle/lk d”nk|s[ltleM . *

* rt’ZrTjfl/+zå;’bnsnf>lqjno— *

* lq/]vfleM ;fw+{ tj z/0fsf]0ffM kl/0ftfM ..** ** *

– Saundaryalahari.xi (Venkatanathacharya, 1969)

Although visually, the diagram shows only forty-three small triangles making five concentric chakras of fourteen triangles, outer ten triangles, inner ten triangles, eight triangles and one triangle – literally, the house of goddess Sri is said to be made of forty-four places of refuge as the origin and the center point, the *bindu *of Srichakra is also taken as the ultimate all-encompassing abode. Literally, the stanza states that the four Siva Chakra, including the *trivalaya*, frame the five triangulated Shakti Chakra located inside making the Sri Chakra as an integrated whole of the nine. Also, while the central *bindu *is recognized as an additional resting place of Sri, the static emblem is taken as a Siva Chakra also. It may have been this additional ‘Siva Chakra’ construct and its’ rising esoteric importance that led to the discounting of ‘trivalaya’ itself as Siva Chakra in some later practices. But there are also exponents, who while accepting the place of the central *bindu *as an all-pervading chakra, still do not count it as one of the nine, and continue to place *trivalaya *as a chakra on the outside of *Sodasadala*. But however one may count the set of nine, Srichakrais always understood as ‘the form of nine-fold assemblies’ – nine triangles, nine chakras, nine yonis, and nine gateways, and so on.

Since initiation of any practitioner into Tantra is done only and fully under the authority of the master (the Guru) and tantra itself was passed on as experiential praxis, the words and instructions of the different Gurus are prone to be taken as the ultimate truth by the respective practitioner. Thus, varying interpretation of texts and ritual practices with nuanced differences and their acceptance as schools is but natural to this system of knowledge transfer. From the perspective of this research seeking to investigate the geometric relevance of *trivalaya *in the drawing of Sriyantra, such varied practices and practitioner’s interpretations can be added resources potentially providing alternative steps or geometric clues for construction. That the whole purpose of *trivalaya *could have been its instructive geometry itself is reinforced by the fact that the three lines making it are specified so close to each other that it could not have been a visual idea. Also unlike other component chakras, *trivalaya *has not been given an esoteric name and possibly suggests that it has limited use in ritual worship or as a step for esoteric gains.

**The worship of Trivalaya**

In the prevailing *tantric *practices, the *trivalaya, *rarely makes a circuit of veneration, evocation or worship. In the Hayagriva-tradition, it is not even drawn and altogether omitted from Srichakra. In the Ananda-Bhairava-tradition, it is however drawn but not worshipped as a chakra. Here, the circles represent the three worlds (*bhu, bhuvah *and *svar), *the three *gunas *(*sattva, rajas *and *tamas*) and the basal center (*muladhara*) in the living beings (Rao, 1983). In terms of the triad, *muladhara *chakra of the human being is the base in which the three life channels, *ida, pingala, *and *sushumna*, are gathered together in origin and flow out as separate channels. Apparently, all these representations are not for evocation and worship or veneration, but steps in the passage of the tantric. Or, is *trivalaya* symbolizing the principle of fire (*sushumna *in case of the tantrik) encircled by the principle of lunar energy (*chitrina*) and the whole enveloped by the principle of solar energy (*vajrini*) as in human *merudanda*?

It is in Dakshinamurti-tradition, where the worship of *trivalalya *is still conducted in detail, the outer circle represents ‘twentynine *matrikas*, beginning with Kalaratri, Khandita, Gayatri, … etc. and ending with Saraswoti; the middle circle sixteen *matrikas* beginning with Amrita, Aakarshini, … etc.; and the inner circle the sixteen *nitya*-divinities beginning with Kameswori[i]’ (Rao, 1983). The *matrikas* invoked in the outermost circle are the 29 of the 35 sounds of consonants letters of the Sanskrit alphabet – the ones left out are the six consonants generally used to denote the subtle body chakras. Likewise, the sixteen *matrika *invoked in the middle circle represent the sixteen vowels in the Sanskrit alphabets.

Since the mother-like goddesses dispersed all over the various circuits are worshipped in corners (saranakona), the worship ritual practices on *trivalaya *may be interpreted to demand 29 kinks in the outside circle*, *16 kinks on the middle and the inside circle – this is so that the points of worship or contemplation for all the goddesses are geometrically located. For the *grahastha* tantric, like the Rajopadhyaya worshipper at the Dakshinamurti temple, as the worshipper constructs and steps in through each of the Siva and Shakti circuits (to ultimately reach the *bindu *in contemplation), the contemplation of the 29, 16 and 16 mother-like goddesses in *trivalaya* also allows him to symbolically transcend ‘through the three objectives of life, virtuous living (*dharma*), wealth (*artha*), and pleasures (*kama*), also concurrently symbolized by the three circles!

Like in the innermost circle of the *trivalaya, *the same sixteen nitya-divinities, are also worshipped in the *Sodasadala *chakra – in the sixteen niches formed there by the *karnika *circle and the lotus petals. This ritual coincidence appears to have prompted the acceptance of this circle as part of the *trivalaya *chakra itself by some exponents. In this reckoning, the three circles outside the reference circle framing the fourteen-triangles chakra – the circle making the seed pod (*karnika*) of the *Astadala *chakra*,* the circle making the seed pod (karnika) of the *Sodasadala *chakra*, *and a framing circle drawn between the *karnika *and the *Trilokyamohan *chakra –are taken as making the *trivalaya. *Between these two ways of recognizing the *brittatraya, *one circle placed between *sodasadala *and *trilokyamohan *is common. This common circle appears as the critical element in *trivalaya* and its geometry of construction deserves a close analytical scrutiny.

**Potential uses of Sriyantra**

The esoteric use of Sriyantra is made in the worship of goddess Tripurasundari in the Sri-chakra, which describes both the visual pattern (*yantra*) and the sound pattern (*mantra) *of the supreme spirit. The actual process of meditational worship (*puja*)is the symbolic contemplation (*bhavana*) of the spirit by the practitioner-devotee, who as the medium of transaction between the two patterns seeks to achieve a transcendent state of union with the supreme – thus *tantra *is both the process and the product (pattern of contemplative union). In a metaphysical way, the practitioner of *tantra *attains through *tantra *his dissolution with the *bindu, *the ‘invisible first principle, the self-originated seed of Being and consciousness’ (Rawson, 1978) at the center of the Sriyantra.

The potential use of Sriyantra as a tool for concentration derives out of its asymmetry along its North-South axis, which provides two points at the central area – one, the *bindu *atthe center of the circle and another, the center point of the central triangle. From the side of creation, the first principle’s first act of projection consists in splitting itself into male and female; the points making a *visarga*-mandala (Khanna, 1979) in the central region. The points of this *visarga*(:) mandala are separated by about two units on the radius measured at 24 units and present a oscillating field for concentration on the asymmetry or power of the *yantra*.

The potential use of Sriyantra as a template tool in *Jyotish** [ii]*, the study of time,can be inferred by putting two facts together – the facts that the yantra of goddess Tripurasundari mirrors the universe (

*Bhairavayamala Tantra*) and that the universe itself is expressed in three forms –

*desharupa, uvayarupa and kalarupa*or as existential space, the heavens or sky space and time. According to Chandrahari (2012), ‘Sriyantra is Jyotish chakra itself in a nine fold geometric representation. As a measure, the unit Sri Yoni is 40

^{0}and nine of such Yonis constitute the Nava-Yoni Chakra.’ From available literature, the interrelated proportioning, placing and pairing of the triangles and their placing and pairing, it can be inferred that Sriyantra is an orthocentric mapping of the naturally regulated cyclic motion of the nine planets of our universe (the solar system) in a equilibrium state. It charts the renewal cycles of Lunar-solar time using Naxyatra scale for the upper half and Rasi scale on the lower half of the circle. Based on study of these aspects of universe as time, I have shown how geometrically perfect Sriyantra (Tiwari, 2008)[iii] can be drawn substituting Naxyatra and Rasi based angular specifications for the chords and perpendiculars sets as traditionally prescribed. It will be seen from the analysis of drawing methods that the ability to draw the

*yoni*or 40

^{0}is critical in establishing the basic

*trikona*chakra!

The most authoritative instructions for drawing Sriyantra available today come from the commentary of Laxmidhara, who is thought to have belonged to early 14^{th} century CE. He describes two methods in practice then – *samharakrama* (outside in construction or absorption mode) practiced by Kaulachara, and *shristhikrama* (inside out construction or creation mode) in Vamachara practice.

**Fig. 2: Astara – the Navayoni chakra – in shristhikrama.**

The *sristhikrama *instructions of construction steps are so lacking in exacting measures that correct drawing appears left to chance or some unstated knowledge of exact proportioning and placing of the apex angle of the first *trikona *itself. Similar accurate visual judgment and purposeful acuity are required for proper sizing of second and third triangles so that the eight outer triangles of the *astara *chakra correctly point to the cardinal and corner directions. A close scrutiny of the above figure (Fig.2) will evidence that if the *astara *is somehow correctly drawn, then the two *dasara *and the *chaturdasara *can be sequentially drawn correctly from within the information contained in the *astara. *

This directional property of the *astara *also forecasts the spirit of the Siva chakra *astadala. *Since the three yantras of *bindu, trikona, *and *astara *make the central *Navayoni *chakra[i], which conceptually summarizes the whole Sriyantra itself, *shristhikrama *method clearly presupposes some intellectual understanding of the requisite geometry of the *trikona. *It can be easily observed that the *trikona *triangle is equivalent to the topmost Shakti triangle, which has been named *Ketu *after its *jyotish *function as descending node of moon, is also critical to the total structure of Sriyantra.

The *Samharakrama* instruction starts with drawing an east-west line called *brahmasutra *and dividing it into 72 parts. A circle measuring 45 parts in diameter is then drawn in the mid-point of the axis. Then *chaturastra *and *trivalaya *are drawn within 4 parts on the ends of *brahmasutra *and the *astadala *contained within four and a half parts measure outside the central circle and the *sodasadala *within further five parts around the *astadala* circle. The drawing of the central composition of nine triangles is instructed through the specification of perpendiculars and chords[ii] for vertices of all triangles and a sequential process of drawing using the 24 triple intersections as check points.

Since the number 72 resolves as 2^{3}x3^{2}, the *brahmasutra *line would need to be bisected three times over and trisected twice over to get the unit part. In Fig 3, top right quadrant (obtained by halving the axis once) shows the sides of the square of 36 units halved twice over yielding the measure of 27 units. The bisecting lines also make a 3-4-5 right triangle with base 27 and height 36, and thus, a hypotenuse of 45 units. The construction lines also halve the hypotenuse so that the circle of 45 unit diameter is consequently drawn as required. The same construction lines can be used to draw a circle of radius 27 units required to define the *astadala* chakra. The *sodasadala *is defined bythe circle inscribing the square that prescribes the circle of 45 unit diameter – this circle has the point (22.5, 22.5) on its circumference and so a radius of 2 or 31.8 units, which has been rounded as 32 in the instructions. Another circle drawn through the identified point (27, 22.5)[iii] will have a radius of 35.14 and makes the outer circle of the *trivalaya *chakra. Likewise, in the lower left quadrant, two points – one at 1/3 (12 units) and another at ¾ (27 units) from the axis – defining the gateway outline of *chaturastra *has been shown. The three squares of *chaturastra *are drawn by the lines that trisect the 36^{th} unit. The *samharakrama *steps for drawing the four outlying Siva chakras, thus, presuppose the knowledge of basic mathematical operations like drawing a square, bisecting and trisecting line segments, drawing 3-4-5 right triangle as well as inscribing and circumscribing squares and prescribing circles.

**Fig. 3: Dividing the Brahmasutra to inform drawing of the four Siva chakras.**

Since the Shakti chakras inside the reference circle are drawn such that the upper half is measured in Naxyatra scale (night sky) and the lower half in Rasi scale (day sky), all Shakti triangles with apices down and bases on the upper part (e.g. Ketu, Rahu, Sani, Sukra and Guru) are accurately specified in angles expressed in Naxyatra scale. Likewise, the Siva triangles with apices up and bases on the lower half (e.g. Surya, Chandra, Mangal and Budha) are accurately specified in angles expressed in Rasi scale. Tiwari (2008) has shown that angles measuring 50^{0} (for Ketu), 16.66^{0} (for Sani), 166.66^{0} and 136.66^{0} (for Guru) and 15^{0} (for Mangal) and 45^{0} (for Surya) truly specify the triangles named in brackets. It is interesting to note that 1/3^{rd} of angle for Ketu is the angle for Sani just as 1/3^{rd} of angle for Surya is the angle for Mangal. It is also obvious that angles of measure 166.66^{0} and 136.66^{0} required for Guru triangle can be constructed by adding 150^{0} and 120^{0} respectively on the angle 16.66^{0}. Both these angular additions, like angles 15^{0} and 45^{0} can be drawn through simple angular bisection constructs. Therefore we can conclude that knowledge of accurately drawing angle 50^{0} and its 1/3^{rd} hold the clue to a perfect drawing of Sriyantra.

Since a constructional method of trisecting angles is still unknown to mathematics, the *Jyotish *mathematics was apparently forced to imagine and construct the circular scale of Naxyatra with 27 divisions to chart the simplest of the motions of the Moon, the cycle (sidereal month) of 27.32 days it took to complete one round in the field of stars. The angle such as Yoni (40^{0} as 1/9 of whole circular angle of 360^{0}) also occurred naturally as the precession of perigee per year in the anomalistic cycle (*chandra mandoccha bhagan)* of 8.84 years. Considering the fact that no solar eclipse can occur outside of perigee, the linkage of Yoni angle with *Ketu *(descending node) and cyclical time can be guessed. Naxyatra (800” as 1/3 of Yoni or 1/27 of 360^{0}) then naturally measured moon’s one day’s journey in the ecliptic. Thus these measures and scales are not arbitrary but actual mapping of naturally occurring reality of our universe of time resulting from the unique combination of revolution and rotation of the heavenly bodies with light (the *navagraha *with sun, moon and visible planets included) in the starry field. Indeed, just like Kepler’s or Einstein’s formulations, these could be mathematical laws and scales describing real patterns of movements occurring in nature or *mulaprakriti.* Actually, *Jyotish* further quartered the measure of Naxyatra to arrive at a *pada*, which would equal 1/108^{th} part of the whole circular angle because the same factor also measured the auspicious ratio of the diameter of the Sun to its distance from earth or the ratio of the diameter of the Moon to its distance from earth. Of course, the natural positioning of the earth vis-à-vis Sun and Moon meant that the apparent diameter of both the celestial bodies measured the same (31 minutes 50 seconds) on average for the observer on earth and directly impacted the phenomena of eclipses so critical to the concept of cyclical eternal time contextualized in the natural laws of rotations and revolutions, axial and plane tilts, and other aspects of the earth, moon, and sun. But for the Naxyatra scale, it was also magical that one quarter of the *pada *of the Naxyatra, i.e. 1/432^{nd} part of 360^{0} (or, 50 minutes) also measured the precession of equinoxes in 60 years, for which the *Jyotish* has given sixty different names (Dwivedi, 1977) for the *samvatsaras*[iv] and provided the tally for *adisamvatsara, *the period taken for the precession to complete 360^{0}at 25920 solar years. Magically too, it could also provide a scale of measure of the moon’s cycle of eclipse producing nodes (*chandra pata vagana*) of Rahu and Ketu of 6794.51 Solar days/18.60 Solar years duration (*Sumatitantra** [v]*), for the apparent radius of Sun/Moon Tan

^{-1}(1/216) averaged 15’54”, a figure close to 18.6×50” or 15’30”.

**The Geometry of Trivalaya**

It is interesting to note that while only *Daxinamurti *subsect of *Kaulachara *continues to ritually worship the *trivalaya *chakra and while all of *Kaulachara *are required to worship on Srichakra drawn using *Samharakrama *method only, only this method gives instructions for drawing all the four Siva chakra including the *trivalaya. *Actually many Vamachara practitioners take *chaturdasara *Shakti chakra as the outermost element of Sriyantra and hold all the four outer Siva chakras as just perceived patterns and hence redundant in ritual worship. Some exponents of *shristhikrama* even take that the drawing of the four Siva chakras on the outside of the *chaturdasara* are only provided for the mental exercise of the learner (*‘shisya-buddhi-vikasha’*). Still others expand on the principle of inseparability or oneness of Siva and Shakti to draw one to one correspondence between Siva and Shakti chakras in an inside out order, e.g. *bindu *and *trikona, astadala *and *astara, sodasadala *and the two *dasara, *and between *chaturastra *and *chaturdasara*. But since Sriyantra is a composition of nine chakras – four Siva and five Shakti in nature – a one to one, and Siva to Shakti, correspondence is problematic. So we find such oddity as proposing correspondence of two *dasara *Shakti chakras to a single Siva chakra of *sodasadala* and at the same time the omission of *trivalaya *altogether as the *bindu *is taken in as a Siva chakra. Since *bindu *is located at the center and *trivalaya *second from outside originally, this order of correspondence between Siva and Shakti chakras could hardly be expected to reveal any constructional sequence relationship between them. Similarly, the esoteric belief that the outermost *chaturastra *chakra sums the *bindu, trikona, astadala, *and *sodasadala *chakras could be only partially verified in its drawing stepse.g. fixing the center, the eight directions (cardinal and corner axes), and thirty-two intermediate directions obtained by bisecting the eight angles twice (See Fig. 1 and 3). And the relationship of *chaturastra *with the *trikona *is unsubstantiated – a critical failure in that the *trikona *forms part of all the four other *Shakti *chakras, the *astaara, *the inner *dasaara, *the outer *dasaara, *and the *chaturdasaara. *Since *trivalaya *is the only Siva chakra left to account for, we have a perfectly good reason to seek this missing information in there – both in its *samharakrama *construction and in its *daxinamurti *esotery*. *It appears that the urge to keep this critical information closely guarded may have led the masters to keep the procedure of *trivalaya *construction obscure and even encourage omission of esoteric usage. Just as the practitioner of *Dakshinamurti* tradition of *Samharakrama *worship of Sriyantra seeks sequential absorption into Siva chakras in the outside-in order and thus get enabled for absorption through the Shakti chakras to ultimately get absorbed into the *bindu, *we may hypothesize that the mathematical/trigonometric information contained in the outer chakra diagrams critically enable the construction of the inner chakras.

From the *Samharakrama *construction data and process given above, we can see that all the information needed to draw all of the four Siva chakras are revealed (Fig. 3) when a particular sequence of bisecting and trisecting of the *brahmasutra *is followed to divide it into 72 parts. Here, in Fig 4, the bisections and trisections of a square are done in traditional drawing method using just a compass and a straight edge (Tiwari, 2009).

The sequential unveiling of the data is almost magical and this has been shown above how we can sequentially draw the reference circle of radius 22.5 units, the circle of radius 27 units for Siva chakra *astadala, *the circle of radius 31.8 from a square prescribing the reference circle to contain the Siva chakra *sodasadala, *and the circle of radius 35.14 making the outer circle of the Siva chakra *trivalaya*! We have seen above how a nearby point in a circle with radius of 35 units can makes 50^{0} at the center and help draw the base of *ketu *triangle of the *chaturdasara *chakra*.*By requiring *trivalaya *of three lines, the practitioner is clearly instructed to draw another circle with radius somewhere in between 35.14 and 35. It would seem that the *trivalaya *is drawn by trisecting the space in between 35 and 35.14 so as to yield a circle of radius 35.04, which can be used indirectly to get angle 16.66^{0} and thus, a trisection of 50^{0}! This angle drawn at the center defines the triangle of *Sani*. We can thus conclude that the data from bisecting and trisecting the *brahmasutra *define all the four Siva chakras.

**Fig. 4. Bisecting and Trisecting lines using straight edges on square** * *

We have seen earlier that two different sets of three circles are recognized as *trivalaya *by practitioners from different sects – one is with all the circles closely drawn between *chaturastra *and *sodasadala *(Ri= 35, Rm= 35.04, and Ro= 35.14), and another made of *karnika *circles of *astadala *(Ri= 27)*, sodasadala *(Rm= 31.8) and the outermost circle (Ro= 35.14). For the adept and the knower, the common circle (Ro= 35.14) should have been enough to lay bare the significance and mentally draw the other two nearby (Rm= 35.04, and Ri= 35) and get the angles 50^{0} and 16.66^{0} to accurately cast the *Shakti *chakras. Thus we can conclude that geometrically the *trivalaya *chakra holds critical information that enables and ensures accuracy of construction of all of the five Shakti triangles.

In both the sets of circles making *trivalaya, *the outermost circle where the twenty nine consonant sound matrikas are worshipped is the same for all sects. Additionally, other points of interest in examining the esotery of *trivalaya *in *daxinamurti *traditions, are – (1) its equivalence to the basal center of *muladhara *chakraof the human being (tantric practitioner), where the life channels of *ida*, *pingala*, and *sushumna *are gathered together and, (2) from where the practitioner herds up the spine the three life principles of fire, moon and sun unto the *sahasradala *as part of *tantra. *As the practitioner goes up the chakras, the remaining six consonants sounds, *vam*, *lam*, *ram*, *yam*, *hum* and *aum* get connected. The middle circle will allow the practitioner to meditate on the sixteen vowels sounds (identified to points in head and face such as ears, eyes, nose, mouth, lips, tongue, and etc. fire principle) and connect to all of the 51 *matrikas*. Esoterically therefore, the instruction to worship of twenty nine *matrikas *takes the practitioner to the thirty five *matrikas – *a parallel to the hint of circle of *matra *Ri= 35 in geometrical construct! And in the worship of sixteen *nityas, *the practitioner seeks dissolution with the moon principle. Given the nature of *tantra, *only the adept *dixit *could experience verification, if you will!

Of the three possible usage of Sriyantra – a tool for concentration of mind, a tool for worship and a tool of Jyotish computations, and the three ways the universe finds expression in human experience – an existential space, an eternal time and a moving heavens (sky), we have considered here universe as time and Sriyantra as potential tool of Jyotish computation – specifically exploring the *trivalaya *Siva chakra for relevant linkage between its esotery and *geometry. *We can see that three scales of measuring time appears to have been used, e.g., *agni* scale for measuring the precession of equinoxes (principle of fire, 432 divisions of the sixty *Samvatsara *years each), *rasi *scale for measuring the movement of the Sun (principle of sun, 360 divisions of one day each), and *naxyatra *scale for measuring the movement of the Moon (principle of moon, 108 divisions of 200 minutes each). Similarly, the three cycles of the moon (*chandra bhagan – *synodic lunar month*, chandra mandoccha bhagan – *anomalistic cycle*, *and *chandra pata bhagan – Rahu-Ketu *cycle) could each be mapped by one circle of the *trivritta *outside in. The worship of the sixteen *nityas *remembers the sixteen phases of the moon* – *the *sodasadala *is drawn showing a gap equal to the half circle petal to show the day time. The *astadala *appears to trace the solar directions on the Rasi scale. I would further hypothesize that as many as five circular scales and three squared scales for measuring eternal time are hidden in the four Siva chakras of Sriyantra!

# Works Cited

Dwivedi, K. P. (1977). *Jyotishsar.* Mumbai: Kalyan.

Khanna, M. (1979). *Yantra.* London: Thames and Hudson .

Pant, N. R. (2040 BS). *Trikonamiti (Jyotpatti).* Kathmandu: Royal Nepal Academy.

Rao, S. R. (1983). *The Tantra of Sri Chakra.* Bangalore: Sharada Prakashana.

Rawson, P. (1978). *The Art of Tantra.* London: Thames and Hudson.

Tiwari, S. R. (2009). *The Temples of Nepal Valley.* Lalitpur: Himal Books.

Venkatanathacharya, V. (1969). *Saundaryalahari of Sri Sankaracharya (with commentary of Laxmidhara).* Mysore: University of Mysore.

[i] It has been said that Navayoni as a concept of Sriyantra comes from Vedic thoughts (*Atharva Veda *x, v. 31-4).

[ii] The system of using *jya *(chord) and *kotijya *(perpendicular) to specify angles was used in ancient Hindu trigonometry for ease of practice. The dimensions given here are as detailed by Kaivalyasrama based on standard tables referring to a circle of radius 24 and angular intervals of 3.75^{0}. This results in drawing with just some level of visual accuracy. For greater accuracy, standard tables based on circle with radius of 3438 units and angular intervals of 7.03 minutes (1/256^{th} of 1 Rasi) were in use (Pant, 2040 BS).

[iii] This point subtends angle 50.19^{0}(or, tan^{-1} 27/22.5) at the center. Since tan^{-1}26.81/22.5 is 50^{0} and tan^{-1 }22.5/ 26.81= 40^{0}, and both make right triangles with hypotenuse= 35, we can infer that these points in the circumference of circle of radius 35 will produce angles 50^{0 }and 40^{0} at the center. The 50^{0} angle at the center defines the key triangle of Ketu in *chaturdasara*!

[iv] Named *Prabhavah, Vibhavah, …, Krodhanah, and Kshyaya, *the list can be found in any standard *Jyotish *texts.

[v] This is a 7^{th} century *Jyotish *document written in Kathmandu Valley and is based on *Suryasiddhanta.*

[i] The names represent the phases of the moon e.g. Kameswori, Bhagamalini, Nityaklinna, Bherunda, Vanhivasini, …, Nitya Nitya, Nilapataka, Vijaya, Sarvamangala, Jvalamalini, Chitra and Adya Nitya.

[ii] Jyotish literally stands for the heavenly objects with light – the triad of *agni-surya-soma *or constellations, Sun and Moon – and is the ancient mathematics of luni-solar time.

[iii] For this publication, visit my web site – http://www.kailashkut.com/wp-content/uploads/2017/06/Tiwaris-Method-of-Drawing-Sriyantra.doc