The figures

Of Om (On or En) of Hindu Ontology.

I. Mudra, Madawar, Sphere or Sphaira.

 A Symbol of the Universe and Universalia.

 A System of the Universal Religion.   ~~

II. The circle O, An Emblem of infinity and Eternity.

 A Type of the Catholic Theism of Hindus.   ~~

III. The convexity of O. A Type of the Extramundane,

 Unknowable and Absolute Supreme Brahma.

 Significant of Agnoism and Agnosticism.   ~~

IV. The concavity of O. Emblem of Intramundane

 Immensity of knowable Nature and its God Brahma.

 And Indicative of Gnosticism and Pantheism.   ~~

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.   ~~

1.  The circle with the central A, 
~~ 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 Advaila matam.

 ~~

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.

 ~~

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.

 ~~

3. (a) The Tripartite circle. With the Inscribed Triangle Euclid (IV. 2).

A symbol of the Holy Trinity (Trimurti on the three sides).

And the Triangular female emblem of God-mother in the midst.

And Indicating the Materialistic Trinitarianism of Hindus.

 ~~

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.

 ~~

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.

 ~~

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.

 ~~

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 Lilavati 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.

 ~~

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 Murti.

 ~~

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.

 ~~

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.

 ~~

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 Suryadasa's commentary on Lilavati.  ~~

"For the heptagon  ~~: describe a circle, and an equilateral heptagon in it, then a line being drawn between the  ~~ extremities of any two sides—at pleasure, and three lines from the centre of the circle  ~~ to the angles indicated by those extremities  ~~, an unequal quadrilateral  ~~ is formed. The greater sides and the least diagonal

 ~~ thereof are equal to the semi diameter  ~~

The value of the greater diagonal, which is assumed arbitrarily, is the chord of the arc  ~~ encompassing the two sides. Its arrow  ~~ being deduced in the manner before directed, is the side of a small rectangular triangle  ~~

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

These numbers are given from the copy of Suryadasa's commentary on the Lilavati in the library of the As. Society. There are two obvious errors in them, probably of the copyist  ~~; viz. 22,579 should be 22.581, and 21997604 should be 2199797604.

Note to Fig. 9.

To inscribe a nonagon in a circle,  ~~ i. e, to divide it into nine parts. "A circle being described as before, inscribe a triangle  ~~ in it. Thus the circle is divided into three parts. Three equal chords  ~~ being drawn in each of these portions, a nonagon is thus inscribed in it  ~~; and three oblongs  ~~ are formed within the same; of which the base is equal to the side of the (inscribed) triangle  ~~ Then two perpendiculars  ~~ being drawn in the oblong, it is divided into three portions, the first and last of which are triangles  ~~; and the intermediate one is a tetragon.  ~~ The base in each of them is a third part of the side of the inscribed triangle  ~~(?). It is the upright (of a rectangular triangle)  ~~; the perpendicular is its side; and the square root of the sum of their squares  ~~ is the hypotenuse  ~~:, and is the side of the nonagon  ~~.

To find the perpendicular  ~~; put an assumed chord  ~~ equal to half the chord  ~~ of the (inscribed) tetragon; find its arrow in the manner aforesaid, and subtract that from the arrow of the chord  ~~ of the (inscribed) triangle, the remainder is the perpendicular.  ~~ Thus the perpendicular  ~~ comes out 21,989: it is the side of a rectangular triangle. The third part of the inscribed  ~~ triangle is 34,641: it is the upright.  ~~ The square root of the sum of their squares  ~~ is 41,031: and is the side of the inscribed nonagon."  ~~