Note to Figure 7.
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Valmiki and Mitra, Viharilala
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.
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.