पञ्चसिद्धान्तिका (आचार्य वराहमिहिर - सूर्य, रोमक, पौलिश, वासिष्ठ एवं पितामह सिद्धान्त)

IV.2 IV. THREE PROBLEMS 77 mentioned in the succeeding verses, because it can be shown that the author derives the sines from a correct formula, (not dependent on this wrong ratio of the diameter to the circumference), based on 120' as the radius of the circle. If he had depended on the wrong value, the first tabular sine, i.e. sin 3° 45' would be 7' 54", (being the 96th part of the circumference, where the sine is indistin- guishable from the arc), and not the correct 7' 51" as given by the author. Taking the diameter as 4°, and thereby the maximum sine (i.e. the radius) as 120', is arbitrary. In general, the Siddhāntas give the maximum sine, 3438', as arrived at from taking the circumfer- ence as 360° or 21600'. The Vākyakaraṇa makes it 43°. In actual work, the sines enter only as a ratio to the maximum sine, and therefore no harm, will result by taking these different maximum sines. TS and NP have not understood the meaning of the second half of the verse, and mis-interpret aṁśacatuṣkam as quadrant. व्यासार्ध[स्य] कृतिर्ध्रुवसंज्ञिका कृतांशस्ततः स मेषस्य | ध्रुवकरणी मेषोना द्वयोस्तु राश्योः पदं ज्याः स्युः ॥ २ ॥ 2. The square of the radius, (i.e. 14,400), is called dhruva (karaṇī), (literally, ‘Fixed Irrational’). The fourth part of it, (i.e. 3600), is the karaṇī (Irrational) related to the first sign, (or 30°). Dhruvakaraṇī minus the karaṇī of Meṣa, (i.e. 14,400 − 3600 = 10,800), is the karaṇī of two signs, (or 60°). The square root of a karaṇī is the tabular sine. Being square of tabular sines given in minutes, the karaṇīs are squares of minutes, which is their peculiarity as given by the author, though this is not mentioned explicitly. The other well-known characteristic of a karaṇī, viz. irrationality, is found in all karaṇīs except 14,400 and 3600, though the author calls these also karaṇīs in a general way. In modern terminology the word sine used in connection with the angle is defined thus: [Figure: Right-angled triangle ABC with right angle at C and angle marked at B] Fig. IV. 1-a In the right angled triangle, (fig. 1-a), sine ∠ B = AC/AB, or sine ∠ A = BC/AB, i.e. as the ratio of the opposite side to the hypotenuse. In tabulating the sines, the hypotenuse is taken as unity, and the ratio expressed as a decimal fraction. 2a. A.B. कृते ध्रुव०; C.D. कृतिध्रुव० c. B1.3. ये योना; B3. येषोना b. A.B. ०ज्ञिता. A.B. कृताशाःस्ततः. A.B. सशेषस्य d. A. दयोस्तु; B. दयो सु