Chapter L: Death of Vidūratha 477 (4)

The author of the Lílávatí has solved the problems, but given no account of the way in which he got the numbers stated by him; if they had been obtained by solution of the above mentioned equation, they would probably have been more accurate than they are. He only lays down an arbitrary rule, that the side of the heptagon is 52055/120000 of the diameter, and that of the nonagon 41081/120000 of the same. Neither of these is very far from the truth. The accurate value of the side of the heptagon lies between 82/182 and 105/242. The side of the nonagon lies between 13/38 and 105/307.

[Sidenote: Commentators on Lílávatí.]

Among the commentators on Lílávatí, Rámakrishna, Gangádhara, and Ranganátha have not attempted any demonstration of the problems in question, and have contented themselves with merely repeating the figures contained in the text. Ganesa confesses that the proof of the sides of the regular pentagon, heptagon and nonagon cannot be shown in a manner similar to that of the triangle, square and octagon.

[Sidenote: The Pentagon.]

But this is untrue of the pentagon; its side can be geometrically found as shown in Euclid Book IV. Prop 11; and the admission of Ganesa serves only to prove, that he was unacquainted with the Sanskrit translation of Euclid which contains a solution of this problem. Ganesa cannot mean only that the side of the pentagon is incommensurable with the diameter; for that is equally true of the triangle, square and octagon, inscribed in a circle.

THE FIGURES.

OF OM (ON OR EN) OF HINDU ONTOLOGY.

I. Mudrá, Madawar, Sphere or Sphaira.
A Symbol of the Universe and Universalia.
A System of the Universal Religion.
ॐकार मुद्रा । अखण्डमण्डलाकारं व्याप्तयेनचराचर स्वरूपं ।

II. The circle O, An Emblem of infinity and Eternity.
A Type of the Catholic Theism of Hindus.
ॐकार वृत्तं । अनाद्यनन्तसनातन व्राह्मधर्म्मस्वरूपञ्च ।

III. The convexity of O. A Type of the Extramundane,
Unknowable and Absolute Supreme Brahma.
Significant of Agnoism and Agnosticism.
ॐवृत्तपरिधेःपरं । अपरिमेयाज्ञेयतुरीयपरव्रह्म स्वरूपं ।

IV. The concavity of O. Emblem of Intramundane
Immensity of knowable Nature and its God Brahma .
And Indicative of Gnosticism and Pantheism.
ॐकारबृत्तमध्यं । प्रज्ञेय निर्गुणब्रह्मतथाज्ञेयब्रह्माण्डस्वरूपं ।

V. The circle with the Central point or Monad.
A Symbol of the Definite and known world and its God.
And signifying the Monotheism of all nations.
ॐबृत्तं सविन्दुमध्यं । सगुणब्रह्मविन्दुब्रह्माण्डवीजमपर तैजस ब्रह्मस्वरूपं ।

1. The circle with the central A, अ Alif or Unit.
Emblematical of the unity of a Personal God.
And the Primary unity of all things in Nature.
And significant of unitarianity or _Advaita matam_.
ॐकार बृत्त केन्द्रस्थअकार सहितं । सोपाधिक एकमेवाद्वितीयमतरूपं ।

2. The two Semicircles of O. Symbolical of Duad or Duality .
In the dualism of Persons in the God-head as _Dvaitam_.
And the Duads of Co-ordinate Principles in Nature.
And signifying the Ditheism of all Dualistic creeds.
ॐ वृत्तार्द्धद्वयं । सोपाधिकेश्वरस्यभेदद्वयं । जीवव्रह्म पुम्प्रकृत्यादि दैतमतरूपं ।

3. The Trisected circle of _Om_. A symbol of the Triad or Trinity.
Indicative of a Triality of Persons in the God-head as _Traitam_.
And the co-ordinate Triples of the Principles in Nature.
And signifying the Tritheism of Trinitarianity.
त्रिभक्त वृत्तं । अउमयुक्तं । त्रिगुणात्मक व्रह्मादित्रिमूर्त्ति । व्याहृति त्रयादि । त्रैतमतरूपं ।

3. (_a_) The Tripartite circle. With the Inscribed Triangle
Euclid (IV. 2).
A symbol of the Holy Trinity (_Trimúrti_ on the three sides).
And the Triangular female emblem of God-mother in the midst.
And Indicating the Materialistic Trinitarianism of Hindus.
ॐ वृत्तान्तर्गत त्रिभुजं । अव्यक्तशक्ति वा प्रकृत्यात्मकत्रिमूर्त्तिरूपं । त्रैतं मतं ।

4. The Four Quadrants of the circle of _Om_ or a square.
Emblematical of the Tetrad of the Divinity.
And the co-ordinate Quadruples of Things.
And signifying the Quaternity of certain creeds.
ॐकारवृत्तपादाः । अउमविन्दुयुक्ताः । विश्वादितुरीय ब्रह्म । वा सप्रकृतित्रिमूर्त्तिरूपं ।

5. The Pentagon Inscribed in the circle. Denoting the Pentad.
The Angular Points A. B. C. D. & E. Meeting at the Centre O
(Euclid IV. 11).
Indicative of the Quintuple Hypostases of the Deity.
The Quintessence and the Five fold co-ordinates of Elementary
bodies.
ॐ वृत्तस्य पञ्चभुजं । अउमविन्दुनादयुक्तं । पञ्चकोश पञ्चप्राण पञ्चभूतमयं ।

6. The Hexagon in the Circle. Significant of the Hexad.
The Angular Points A.B.C.D.E.F. Meeting at the centre
(Euclid IV. 15).
Denotative of the sextuple Evolutions of the Monad O.
And Indicative of the Six Internal and External Organs of sense.
वृत्तस्य षड्भुजं । ॐ नमो षोढ़ारूपं । अउम॰ ৺नमषडिन्द्रिय षड्रस षडङ्गादिस्वरूपं ।

7. The Heptagon. Inscribed in the circle O. Indicates the Heptad.
The Angular Points A. B. C. D. E. F. G. Meeting at the centre O.
According to the Process of Lílávatí mentioned below.
Indicates the septuple Hypostases of Divine Essence, _viz_;
The Five External senses, mind and intellect. (Anquetil).
And the seven fold co-ordinate bodies in creation, _viz_;
the seven Worlds, seven Planets, seven Continents and Oceans.
ॐवृत्तस्य सप्तभुजं । अउम॰ ৺ नमःयुक्तं । प्रज्ञादि आत्मारूपं । सप्तभुवनमयविराजरूपं ।
सप्त ग्रह सप्त द्वीप सप्त समुद्रञ्च ।

8. The Octagon (A. B. C. D. E. F. G. H). Inscribed in the circle O.
By Bisection of the Quadrants (in Figure 4).
Indicative of the Octad or Octuple states of Spirit and Body.
_viz_, the five Vital airs or the five external or five
Internal senses with the Mind, Intellect and consciousness (Chittam).
All forms of the Spirit.
And the eight material forms of Earth &c., treated of in the Ashta
Múrti.
ॐ वृत्तस्याष्टभुजं । ॐ नमोयुक्तं अउओम॰ ৺ नमः । अष्ट बुद्धीन्द्रिय तथा ईश्वराष्टमूर्तिरूपं ।

9. The Nonagon A, B, C, D, E, F, G, H, I. Inscribed in the circle O.
By Trisection of the three sections of a Tripartite circle.
Symbolical of a _nonad_ or nine fold nature of the Deity.
And the nine doors or organs of Animal bodies.
ॐ वृत्तस्य नव भुजं । त्रिभुजस्यत्रिभक्तभुजत्रयं । ओन्नमो शान्ति युक्तं । अउओम॰ ৺ नमः शान्ति ।
आत्मनः नवधारूपाणि । तथाजीवदेहस्य नवद्वारानिच । आत्मनः नव रूपाणि यथा—
अ-जाग्रत् बैश्वानरः । उ-स्वप्नजतैजनः । म-सुसुप्त प्रज्ञा । ओ-तुरीयपरात्पर व्रह्म ।
৺ नाद समष्टि सूत्रात्मा । ॰ विन्दुव्यष्टि हिरण्य गर्भः । नमः स्थूलविश्वः । शक्ति विश्वरूपः ।
शान्तिविराजरूपं । सर्ब्बमोङ्कारस्य नव रेखभिः दर्शितं । यथा ॐ

10. The Decagon in a Circle. Emblematical of the Decad.
The Decagon A, B, C, D, E, F, G, H, I, J. By Bisection of the
Pentagon.
Significant of the Five Internal and five External Senses.
And the Ten Directions of space. All filled by Divine Spirit.
ॐवृत्तस्य दशभुजं । वृत्तस्य पञ्च भुजंद्विर्बिभक्तं । दशदिग्व्यापि ईश्वरात्माज्ञापकं ।
ॐन्नमः शक्तिःयुक्तं । यथा अ, उ, ओ, म, ৺, ॰, नमः, शान्तिः, ।
वा अ, उ, ओ, म, ৺, ॰, नमः शान्तिः, शान्तिः, शान्तिः ।

These figures might be multiplied _ad infinitum_, as there is no
limit of created things and the attributes of the Creator; but as
neither Infinity nor Immensity is comprehensible by the limited
understanding of man, the Yogi takes some definite ideas and
determinate objects for his meditation, as he is directed by the
Natural Religion of mankind.

NOTE TO FIGURE 7.

Solution of the Problem of inscribing a heptagon in a circle, or dividing the circle into seven equal parts. According to Súryadása’s commentary on Lílávatí. वृत्तान्तर्गत सप्तभुजाङ्कनं सूर्य्यदासमतं ।

“For the heptagon समवाहुक सप्तभुजः describe a circle, and an equilateral heptagon in it, then a line being drawn between the भुजाग्ररेखा extremities of any two sides—at pleasure, and three lines from the centre of the circle वृत्तान्तर्गत केन्द्रं to the angles indicated by those extremities भुजाग्रचिन्हित कोणं, an unequal quadrilateral विषम चतुर्भुजं is formed. The greater sides and the least diagonal क्षुद्रतरकर्णं thereof are equal to the semi-diameter व्यसार्द्धतुल्यं”. The value of the greater diagonal, which is assumed arbitrarily, is the chord of the arc चापस्य पूर्णज्या encompassing the two sides. Its arrow शरः being deduced in the manner before directed, is the side of a small rectangular triangle एकजात्य त्रिभुजं ।

Thus the greater diagonal बृहत्तरकर्णं, being arbitrarily assumed to be 93,804, is the chord sought इष्टज्याः; its arrow found in the manner directed is 22,579; this is the side, and half the base or chord जीवारूप भूम्यर्द्ध कोटि is the upright 46,902; their squares are 509711241 and 21997604; the square root of the sum of which is the side वर्गमूलं of the heptagon or 52,055 योगमूलं ।

These numbers are given from the copy of Súryadása’s commentary on the Lílávatí in the library of the As. Society. There are two obvious errors in them, probably of the copyist लिपिकर प्रमादः; _viz._ 22,579 should be 22.581, and 21997604 should be 2199797604.

NOTE TO FIG. 9.

To inscribe a nonagon in a circle, वृत्तान्तर्गत नवभुजं i. e., to divide it into nine parts. “A circle being described as before, inscribe a triangle वृत्तान्तर्गत त्रिभुजं in it. Thus the circle is divided into three parts. Three equal chords समान पूर्णज्या being drawn in each of these portions, a nonagon is thus inscribed in it वृत्तस्य नव भुजक्षेत्रं; and three oblongs वृत्तस्य चतुर्भुज क्षेत्रं are formed within the same; of which the base is equal to the side of the (inscribed) triangle भूमिवृत्तस्य त्रिभुजभुजतुल्यं । Then two perpendiculars लम्बपातद्वयं being drawn in the oblong, it is divided into three portions, the first and last of which are triangles परपूर्ब्बांश त्रिभुजं; and the intermediate one is a tetragon. मध्यांश चतुर्भुजं । The base in each of them is a third part of the side of the inscribed triangle त्रिभुजवाहोस्त्रितीयांशः(?). It is the upright (of a rectangular triangle) जात्यत्रिभुजकोटि; the perpendicular is its side; and the square root of the sum of their squares भुजकोटिवर्गयोगमूलं is the hypotenuse कर्णः, and is the side of the nonagon नवभुजं.

“To find the perpendicular लम्बं, put an assumed chord कल्पितज्या equal to half the chord पूर्णज्यार्द्धं of the (inscribed) tetragon; find its arrow in the manner aforesaid, and subtract that from the arrow of the chord तत्रिभुजस्यशर of the (inscribed) triangle, the remainder is the perpendicular. लम्बपरिमाणं । Thus the perpendicular लम्बं comes out 21,989: it is the side of a rectangular triangle. The third part of the inscribed एकजात्य त्रिभुजं triangle is 34,641: it is the upright. कोटि । The square root of the sum of their squares वर्गयोग मूलं is 41,031: and is the side of the inscribed nonagon.” वृत्तस्यनवभुजं ।

YOGA VÁSISHTHA.