SINE, COSINE, RADIUS AND ARC
Anyonya kotihathayorabhimatha gunayosthrijeejavayaa hathayo: yogaviyogow syaathaamabhimathagunachaapa yogavivaragunow
The sum of the products of Sin A and Cos B and when angles are exchanged, Sin B and Cos A, gives the Sin of the sum of the angles. Similarly the difference of the above gives the value of the sin of angular difference. Sin (A+B) = Sin A Cos B+Cos A Sin B And Sin (A-B) = Sin A Cos B – Cos A Sin B.
Yadveshta chaapagunatha ccharavargayoga moolaardhamishta dhanurardhaguna: pradishta: jyaanaam nijathriguna vargaviseshamoolam kotisthadoona sahithow thrigunow svabhaanow
Square root, of the square of a chord (R sin _) diminished from squares of radius gives the koti (R cos _). This subtracted from radius gives the (small) arrow of arc. This added to radius is big arrow of the arc…..
PUTHUMANA SOMAYAJI – KARANA PADDHATI 1450
TAYLOR (1685 AD) SERIES OF SINE AND COSINE DISCOVERED BY NILAKANTA
ista-dohkotidhanushoh svasamipasamirate jye dve saavayave nyasya kuryaad unaadhikam dhanuh dvighna talliptikaptaikasarasailasikhindavah nyasyacchedaaya cha mithastatsamskaaravidhitsaya anyasyam atha taam dvighnaam tathaa syam iti samskriti: santha te krtasamskare svagunau dhanusas tayo:
Placing the sine and cosine chords nearest to the arc, whose sine and cosine chords are required, get the arc difference to be subtracted or added. For making the correction, 13,751 should be divided by twice the arc difference in minutes and the quotient is to be placed as the divisor, divide the one (sine or cosine) by this divisor and add to or subtract from the other (cosine or sine) according as the arc difference is to be added or subtracted. Double this result and do as before. Add or subtract the result to or from the first sine or cosine to get the desired sine or cosine chords.
NILAKANTA – TANTRA SANGRAHA 1444 AD
NEWTON GAUSS (1670) INTERPOLATION FORMULA DISCOVERED BY GOVINDASWAMI
gacchad-yata-gunantharavapuryathaishya-disvasanaa cchedaabhyaasa-samuha-kaarmukakrti-praapthath tribhisthaadithah vedaihi sadbhir avaaptam antyagunaje rasyo: kramad antyabhe ganthavaahata-varthamaana-gunajaaccha paatham ekaadibhi:antyad utkramatah kramena vishamai: sankhyaviseshai: khsipedbhankthvaptam, yadi maurvikavidhir ayam makhyah kramad vartate sodhyam vyutkramathaa stathakrthaphlam…..
Mathematicaly this formula is summarised as follows: F(x+nh)=_f(x)+nf(x)+½n(n-1)(_f(x)-_f(x-h) Multiply the difference of the last and the current sine differences by the square of the elemental arc and further mutiply by three. Now divide the result so obtained by four in the first rasi, or by six in the second rasi. The final result thus obtained should be added to the portion of the current sine difference (got by linear proportion). In the last rasi, multiply the linearly promotional part of the current sine differences by the remaining part of the elemental arc and divide by the elemental arc. Now, divide the result by the odd numbers according to the current sine difference, when counted from the end in the reverse order. Add the final result thus obtained to the portion of the current sine difference. These are the rules for computing true sine differences for sines. In the case of versed sines, apply the rules in the reverse order and the above corrections are to be subtracted from the respective differences.
GOVINDASWAMI – COMMENTARY FOR MAHABHASKAREEYA 800 AD
NEWTON’S (1660 AD) POWER SERIES DISCOVERED BY SOMAYAJI
nihatya chapavargena chapam tatthathphalani cha haret samulayugvargaistrijyavargahatai: kramaat chapam phlani chadhodhonyasyoparyupari tyajet jivaptyai, sangraho syaiva vidvan-ityadina krtha: nihathya chapavargena rupam tattatphalani cha hared vimulayugvargaistrijyavargahatai: kramat kintu vyasadalenaiva dvighnenadyam vibhajyataam phalanyadhodha: kramaso nyasyoparyupari tyajet saraptyai, sangraho asyaiva stenastri-tyadinaa krta:
Multiply repeatedly the arc by its square and divide by the square of even numbers increased by that number and then multiplied by the square of radius. Place the arc and result one below the other and subtract each from what is above it. To derive the arc, which are collected, beginning with the expression Vidvan (katapayadi number). Multiply repeatedly, the unit measurement which is the radius, by the square of the arc and divide by the square of even numbers decreased by that number and then multiplied by the square of radius; the first is, however, to be divided by twice the radius. Place the results one below the other and subtract each from the one above it. That is the method to derive the saras, which are collected in the beginning with stena. (This equation is now known as Newton power series.)
PUTHUMANA SOMAYAJI – KARANAPADDHATI (1450 AD)
VOLUMES OF CONES
Samakhaatha phalathryamasai: soochikhathe phalam bhavathi
The one third of the volume of the uniform cylinder is the volume of the cone.
Pardhirbhitthilagrasya raasesthrimsathkara: kila anthakonasthithasyaapi thithithulyakara: sakhe bahishkona sthithasyaapi panchaghnanava sammitha: theshaa ma chakshva me kshipram ghanahasthaath pruthak pruthak
Friend, the food grains are kept at a circumference of 30 cubit in the floor, outside corner of the room, inside corner and side of the wall. Find out the volume of the grain if the height is 45 cubit.
BHASKARA II LILAVATI 1114 AD
LHUILER’S (1782 AD) FORMULA DISCOVERED BY SOMAYAJI
Doshnamdvayordvayor ghaatayutaanaam tisraanaam vadhaat ekaikonetarattraikyam catushkavadhabhajitam Iabdha mulena yadvrttam vishkambhaardhena nirmitam sarvam caturbhujakshetram tasminneva tisthtahathe
The three sums of the product of sides, taken two at a time are to be multiplied together and divided by the product of the sums of the sides taken three at a time and diminished by the fourth. If a circle is drawn with the square root of this quantity as radius, the whole quadrilateral will be situated inside it.
PARAMESWARA COMMENTARY FOR LILAVATI (1360 AD)
GREGORY’S (1632 AD) SERIES FOR INVERSE TANGENT DISCOVERED BY MADHAVA CHARYA
istajya-trijyayorghathath kotyaptam prathamam phalam jyavargam gunakam kritva kotivargam cha haarakam pratha maadiphalebhyo atha neya phalakrtir muhu: eka-tryaady-ojasankhyabhirabhakteshveteshv anukramaat ojanam samyutesthyaktva yugmayogam dhanur bhavet doh-kotyor alpameveha kalpaniyam iha smrtam labdhinam avasanam syanna thathaapi muhu: krte
Obtain the first result of multiplying the jya (R sine _) by the trijya (radius) and dividing the product by koti (R cos _). Multiply this result by the square of the jya and divide the square by the koti. Thus we obtain a second result a sequence of the further results by repeatedly multiply by the square of the jya and dividing by the square of the koti. Divide the terms of the sequence in order by the odd numbers 1,3,5,…; after this, add all the odd terms and subtract from them all the even terms (without disturbing the order of the terms). Thus is obtained the dhanus whose two elements are the given jya and koti. (Here the smaller of the two elements should be taken as the jya, since other wise the series obtained will be non finite) (use of Tangent)
MADHAVA YUKTI BHASHA? (1350 AD)
DE MOIVRE’S (1650 AD) APPROXIMATION DISCOVERED BY MADHAVA CHARYA
Asmat sukshmataroanyo vilikhyate kashcanapi samskara: ante samasankhyadalavarga saiko guna:, sa eva puna: yugagunito rupayuta: samasankhyadalahato bhaved haara: trisaradivisa mashankhyaharanat param etad eve va karyam
A correction for cirumference still more precise is being stated here. The multiplier is the square of half the even integer increased by unity. This multiplier multiplied by 4, then increased by unity and then multiplied by half the even integer is the divisor. This correction may be applied after the division by odd integers,3, 5, etc. i.e Circumference = 4D (1-1/3+1/5-1/7….. + ..-1/n(½(n+1)2+1___((½(n+1)2 x 4 +1) (½(n+1))
MADHAVA KRIYA KRAMAKARI (1350 AD)
DE MOIVRE’S (1650 AD) APPROXIMATION
yatsankhyaaatra harane krte nivrtta hrtis tu jamitaya tasya urdhvagatasyas samasankhya taddalam guno ante syat tadvargai rupahato haaro vyasabdhighatata: pragvat tasyam aptam svamrne krte dhane sodhanan cha karaniyam sukhma: paridhi: sa syat bahukrtvo haranato atisukshmas cha
………. Let the process stop at a certain stage, giving rise to a finite sum, multiply four times the diameter by half the even integer subsequent to the last odd integer used as divisor and then divide by the square of the integer increased by unity. The result is the correction to be added to or subtracted from finite sum. The choice of addition or subtraction is depending on sign of the last term in the sum. The final result is the circumference determined more accurately than by taking a large number of terms:
MADHAVA YUKTIBHASHA? (1350 AD)
Aaveshtamaanamatha thaani dalapravruthyaa yadvrutthamathra harijam kshithijam thadaahu: yasmin bhaveth samudayasthamayo akhilaanaam praachyaam kramaadaparadisyudu khecharaanaam
The great circle which goes round them, dividing each of them into two equal parts, is called harija or kshitija. This in modern astronomy is horizon. This is the circle on which rising and setting of stars and planets take place towards east and west respectively.
VATESWARA SIDDHANTA 880 AD