maadhavaachaarya ki jya saarani

keraleeya ganit sampradaaya ke ganitjnya tatha khagolashaastri maadhavaachaarya ne chaudahaveen shataabdi mein vibhinn konon ke jya ke maanon ki ek saarani nirmit ki thi. is saarani mein chaubees konon ke jya ke maan diye gaye hain. jin konon ke jya ke maan diye gaye hain ve hain:

3.75°, 7.50°, 11.25°, ..., tatha 90.00° (arthaat 3.75°a= 1/24 samakon ke poornaank gunak).

yeh saarani ek sanskrut shlok ke roop mein hai jismein sankhyaatmak maanon ko katapayaadi paddhati ka upayog karke niroopit kiya gaya hai.

isse sambandhit maadhav ke mool kaarya praapt naheen hote hain kintu neelakanth somayaaji (1444–1544) ke 'aaryabhateeyabhaashya' tatha shankar variyaar (circa. 1500-1560) dvaara rachit tantrasangrah ki 'yuktideepika/laghuvrutti' naamak teeka mein bhi hain.^[1]

nimnaankit shlok mein maadhav ki jya saarani dikhaayi gayi hai. jo chandrakaant raajoo dvaara likhit kalcharal faaundeshans of maithametiks naamak pustak se liya gaya hai.^[1]

shreshthan naam varishthaanaan himaadrirvedabhaavan:.

tapano bhaanusooktajnyao madhyaman viddhi dohanan..

dhigaajyo naashanan kashtan chhatrabhogaashayaambika.

mrigaahaaro nareshoऽyan veeroranajayotsuk:..

moolan vishuddhan naalasya gaaneshu virla naraa:.

ashuddhiguptaachorashree: shankukarno nageshvar:..

tanujo garbhajo mitran shreemaanatr sukhi sakhe!.

shashi raatrau himaahaaro vegalp: pathi sindhur:..

chhaayaalayo gajo neelo nirmalo naasti satkule.

raatrau darpanamabhraaङagan naagastuङaganakho bali..

dheero yuva kathaalol: poojyo naareejarairbhag:.

kanyaagaare naagavalli devo vishvasthali bhrugu:..

tatparaadiklaantaastu mahaajya maadhavoditaa:.

svasvapoorvavishuddhe tu shishtaastatkhandamaurvikaa:.. 2.9.5

maadhav dvaara diye gaye maanon ki vyaakhya karne wala chitr

maadhav dvaara diye gaye maanon ko samajhne ke liye maana koi kon A hai. O kendra vaale tatha ikaai trijya ke ek vrutt ki kalpana keejiye. maana vrutt ka chaap PQ kendra O par A kon banaata hai. Q se OP par QR lamb kheenchiye. rekhaakhand RQ ka maan hi Sin A ka maan hoga.

udaaharan ke liye maana A ka maan 22.50° hai. sin 22.50° ka aadhunik maan 0.382683432363 hai tatha,

0.382683432363 radians = 180 / π × 0.382683432363 degrees = 21.926145564094 degrees.

tatha

21.926145564094 degree = 1315 aarkamint 34 aarkasekend 07 ka saanthavaaain aarkasekend.

katapayaadi paddhati mein ankon ko ulate kram mein likha gaya hai. aur 22.50° ke sangat jo maan diya gaya hai vah hai : 70435131.

kisi kon A ke liye, maana

${\displaystyle \angle POS=m{\text{ arcminutes, }}s{\text{ arcseconds, }}t{\text{ sixtieths of an arcsecond}}}$

to

${\displaystyle {\begin{aligned}\sin(A)&=RQ\\&={\text{length of arc }}PS\\&=\angle POS{\text{ in radians}}\\&={\frac {\pi }{180\times 60}}\left(m+{\frac {s}{60}}+{\frac {t}{60\times 60}}\right).\end{aligned}}}$

saarani ki pratyek pankti aath ank deti hai. maana kon A ke sangat ank (baaeain se daahine ki taraf padhiye) is prakaar hain-

${\displaystyle d_{1}\quad d_{2}\quad d_{3}\quad d_{4}\quad d_{5}\quad d_{6}\quad d_{7}\quad d_{8}}$

to katapayaadi pranaali ke anusaar

${\displaystyle {\begin{aligned}m&=d_{8}\times 1000+d_{7}\times 100+d_{6}\times 10+d_{5}\\s&=d_{4}\times 10+d_{3}\\t&=d_{2}\times 10+d_{1}\end{aligned}}}$

nimnalikhit saarani mein maadhavaachaarya ke jya maanon ki shuddhata ki tulana sangat aadhunik maanon ke saath ki gayi hai.