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

IV. THREE PROBLEMS 79 Incidentally, we shall derive the karaṇī and sine of one and a half signs, i.e. 45°, mentioned in verse 4. Let A₁BD be equal to 45°. ∠ A₁ = 45°, and ∠ C is a right angle. ∴ A₁C = C₁B. But, A₁B² = A₁C₁² + C₁B² = 2 A₁C₁². ∴ A₁C₁² = 120²/2 = 14400/2 = 7200 = the karaṇī of one and a half signs as mentioned in the text. Its root, 84'51", is sine 45°, agreeing with what is given in the tables. शेषेष्विष्टेषु धनु-र्द्विगु(णं) पदात् प्रोज्झ्य शेषगुणहीना [त्] । [व्यासस्याऽर्धार्द्धर्गं] द्विगुणकरण्यां समायोज्यम् ॥ ३ ॥ तत्पादोऽभिमता [स्याद्] ध्रुवा तदूनाऽवशेषपिण्डस्य । ध्रुवकरणीदलमध्यर्धसंज्ञमन्योऽत्र विधिरुक्तः ॥ ४ ॥ इष्टांशद्विगुणोनत्रिभज्ययोना त्रयस्य चापज्या । षष्टिगुणा सा करणी तया ध्रुवोनाऽवशेषस्य ॥ ५ ॥ 3-5. The other tabular sines, (i.e. sine 3° 45′, sin 7° 30′ etc. other than the four mentioned of the total 24) are formed successively in the following manner: Let the angle or arc for which the sine is required be θ. I. Sin²θ = ¼[sin²2θ + {120 − sin(90° − 2θ)}²] II. Sin²θ = 60 × {120′ − sin (90° − 2θ)}, where the sines are in minutes etc. Of the 24 sines, the karaṇī of the nth sine = 14400 − the karaṇī of the (24 − n)th sine. 7200 is the karaṇī of one and a half signs, i.e. 45°. Thus, as karaṇīs 8, 12, and 16 are known, those of their halves etc. and (24 − halves) etc. can be found successively. Thus all the sines from 1 to 24 can be found. Of the two formulae, the first is suited to geometrical representation, and the second to computation. Example 1. Given the 8th karaṇī (i.e. of 30°) 3600 and its sine 60′, the 16th karaṇī (i.e. of 60°) 10800, and its sine 103′ 55″.33, find the 4th and 20th karaṇīs and sines, using each of the two formulae. The desired sine is the 4th, i.e. of 4 × 3° 45′ = 15°. 2θ = 30°, 90° − 2θ = 60°. 3a. A. धनुर्द्वि. A.B.C.D. ॰गुणपदा॰ 4a. A. तपदो; B. पदो; C.D. त[स्य] पदो. b. A. पदायोज्य; B. पदायोज्य्; C.D. पदायॊग॰ A.B.C.D. ॰भिमतज्या A.C.D. गुणहीना; B. गुणाहीना b. A. तदुना. B. ॰विशेषे c. A.B. तृव्यासपादार्द्धाद्धर्गं; (B. om तु; B2. ०र्गं); d. A. र्द्धसंज्ञा; B1.2. र्द्धसंज्ञां; B3. र्द्धं संज्ञा C.D. [त्रिज्या तदर्धवर्गौ] द्वि C.D. संज्ञकोऽन्योऽत्र. A. विधिनुक्तः d. A. कारथो; B. कारयो 5a. A.C. इच्छांशद्विगुणेन (C. ॰णोन) A. समायोज्यं; B. समाप्रोज्यन्त; b. B3. त्रयंस A.B. वायज्या C. द्विगुणज्यार्धस्य संयोज्यः; D. द्विगुण [र्ध] करणी c. A. स कारणी; B. स करणी समायोज्यः d. B. षपा. A.B. ॰नामशेषस्य