_Om_ the object of Yoga meditation, being already described in sections IX. &c. of this article as symbolical of Divine nature, and its different divisions as emblematical of the eternal attributes or hypostases of the Self—same Unity, they are as shown before, represented by the component letters of that mystic syllable, and meditated upon by the mental arithmetic of the speculative theosophist, the vedántist and yogi. But as the majority of people of grosser understandings are more dependant on ocular and sensible symbolism than abstract idealism, the Tantras have purposely contrived many a figure and diagram (_Mudrás_ and _Chakras_) for their guidance, of which we will give a few below with their geometrical names and notations. It will appear from the diagrams described hereafter that _Om_ the symbol of Brahman the Universal _Sat_ or existence, serves to show us as a chart of the world, or representation of the cranium, everything existing in the physical and intellectual world, which is expressed by the word Om (ॐ शब्द सर्ब्बार्थ​​ वाचकः), in its different divisions and partitions for our meditation and contemplation. The pious and religious spiritualist may employ them in Divine contemplation, but the majority are at liberty to use them in the meditation of every other subject which comes to be comprised within the compass of their thought, in the groups of significations which the letters are said to convey. Hence the Yoga of old, meant only an intense application of the mind to all subjects of thought and knowledge. Thus the end of our Yoga philosophy is not only the abstruse meditation of Divine attributes, but the mental reflection of every thing besides. XIX. MATHEMATICAL INVESTIGATION INTO THE DIAGRAMS OF OM. [Sidenote: Correctness of the Diagrams.] We have seen from the diagrams given in the following section, that the Tántrika formulists have spared no pains to divide the great circle of the Universe, filled by the omnipresence of Brahma and represented by the figure _om_, into several parts for the purpose of meditating His different hypostases, and contemplation o£ the various orders of creation. We are now to inquire as to whether these several divisions of a mathematical circle of 360 degrees are geometrically correct, or mere arbitrary partitions made by ignorant priests for their own amusement and deception of their proselytes. [Sidenote: The Heptagon and Nonagon.] Now for instance, the problem of inscribing a heptagon or a nonagon in a circle will at once startle a student of Euclid as altogether impossible, and identical with that which was celebrated among Greek geometricians as the problem of the trisection of the angle. If treated algebraically, it leads to a cubic equation with three real roots, the arithmetical value of which can be found only approximately. [Sidenote: The Lílávatí’s solution.]