Discernment of the Principles of Astronomy

Through this, the geometric figure: kṣetra related to the arc of the sum: aikya-cāpa is produced. One sine: bhujajyā is the base, the difference of the cosines: koṭijyā is the upright, and the aforementioned oblique line is the hypotenuse: karṇa. Now, this hypotenuse can also be formed from the base and upright which take the form of the sine: kramajyā and the versine: utkramajyā literally "step-sine" and "inverse-step-sine" related to the arc of the sum; these two should be understood here.

For that purpose, from the point on the circumference touched by one end of this hypotenuse, draw a diameter: vyāsarekhā—much like the east-west line—extending to a point on the circumference six signs 180 degrees away. From that diameter, just as the sine was taken from the previous line, another line should be drawn to the second end of the hypotenuse. That line is the sine of the arc of the sum, serving as the base. Now, the distance on the diameter between the root of that sine and the first end of the hypotenuse is the measure of the versine of the arc of the sum, which serves as the upright. From these two, that very same hypotenuse established earlier is produced.

In this context, half of the verified hypotenuse would be the sine of half the arc of the sum. From the knowledge of the versine related to the sine of that arc of the sum, the result is established by the rule: "from the square root of the product of the radius and the versine..." original: "त्रिज्योत्क्रमज्यानिहतेर्हजस्य" (trijyotkramajyā-nihater-hajasya); this refers to the classical formula where the chord of an arc is related to the versine and the radius and so on. Therefore, from the sine of half the arc of the sum—determined by the previously known base and upright—the sine of the arc of the sum is established by the reverse method.

The method is as follows: from the sines and cosines related to the two arcs, the first and second—

A circular geometric diagram containing several chords and perpendicular lines. The circle is divided by a central vertical axis and a horizontal diameter. Labels in Devanagari script are placed near various segments:

• "ल. ज्या." (la. jyā.) [small sine/sine of the sum] at the top of a vertical chord.
• "यो. द. पू." (yo. da. pū.) [east of the half-sum] in the upper left quadrant.
• "वृ. ज्या." (vṛ. jyā.) [great sine] labeling a diagonal chord.
• "अं. पू." (aṃ. pū.) [east of the difference] and "तृ. को." (tṛ. ko.) [third cosine] near the circumference in the upper right quadrant.
• "वृ. को." (vṛ. ko.) [great cosine] labeling a vertical segment within the circle.
  The diagram illustrates the derivation of trigonometric identities for the addition and subtraction of arcs as discussed in the text.