Thribdhyankaagninabha schandraisthri bhaanaa shtayugaashtabhi: vedaagni baanakhaaschaicha khakhaabhraa bhrarasai: kramaath baaneshu nakha baanai schadvidvi nandeshu saagarai: kuraamadasavedaischa vruthhavyaase samaahathe khakhakhaabhraarka sambhakthe labhyanthe kramasobhujaa: vrutthaantha sthraya poorvaanaam navaasraantham pruthak pruthak
For cyclic equilateral triangle, cyclic square, cyclic equilateral pentagon,…. to cyclic equilateral nonagon, (cyclic figures having 3 to 9 sides with equal side measurements) their sides can be calculated respectively when diameter is multiplied separetely with 103923 (triangle) 84854 (quadrilateral) 70534 (pentagon), 60000 (hexagon) 52055 (septagon) 45922 (octagon) and 41031 (nonagon) and divided by 120000, the value will be the measurements of the sides of cyclic equilateral triangles to cyclic equilateral nonagon. Bhaskaracharya has given the example: If 2000 is the diameter of circle, equilateral geometrical figures inscribed inside that circle will have sides as follows:
Geometrical figure Bhaskara’s value Modern value
Triangle 1732 + .05 1732.043
Square 1414 + .021 1414.211
Pentagon 1175 + .056 1175.5619
Hexagon 1000 + .00 999.996
Septagon 867 + .58 867.5799
Octagon 765 + .36 765.3636
Nonagon 683 + .85 683.85
BHASKARA II – LILAVATI 1114 AD
CIRCLE – VALUE OF Π
Chathuradhikam sathatmashtagunam dvaashashtisthathaa sahasraanaam ayuthadvya vishkambasyaasannoo vruthhaparinaaha:
When 100 increased by 4 multiplied by 8 and added to 62,000 gives an approximate value for the circumference of a circle having diameter 20,000 units.
ARYABHATA I ARYABHATEEYA 499 AD
Ashtadvaadasa shadkaa: vishkambhasthathvatho mayaa drushtaa: theshaam samavrutthaanaam parithiphalam me pruthak broohi
Diameter of 3 circles are correctly seen by me to be 8, 12 and 6 units respectively. Tell me separately the circumference and areas of the circles.
BHASKARACHARYA I – 628 AD
Vyaasaath vanasangunithaath pruthagaaptam thryaadyayugvimoola ghanai: thrigunavygaase svamrunam kramasa: kruthvaapi paridhiraaneyu:
Multiply the diameter of a circle with 4 and keep it at different places and divide each with the odd numbers beginning from 3, 5, 7,… as their cubes subtracted by the same value. Repeat this and add/subtract alternatively the results to three times the diameter of the circle to get the circumference with the highest degree of accuracy. This theorem can be mathematically represented as follows:
Circumference = 3D+4D/(33-3)-4D/(53-5)+4D/73 7)-..
Vargairyujaam vaa dvigunairnirekair vargeekruthair varji thayugma vargai: vyaasam cha chadghnam vibhajeth phalam svam vyaase thrinighne paridhi sthadaasyaath
Six times the diameter is divided separetely by the square of twice the square of even integers 2,4,6…. minus one, diminished by the squares of even integers themselves. The sum of the resulting quotient by thrice the diameter is the circumference.
This can be mathematically written as follows:Circumference =
3D+6D([1/2×22-1]2-22) + ([1/2×42-1]2-42)+[(1/2×62-1)2-62])+….
PUTHUMANA SOMAYAJI – KARANAPADDHATI 1450 AD
AREA OF CIRCLE AND SPHERE
Vrutthakshethre paridhigunitha vyaasapaada: phalam thath kshunnam vedairupari paritha:kandukasyeva jaalam golasyaivam thadapi cha phalam prushtajam vyaasanighnam shadbhirbhaktham bhavathi niyatham golagarbhe ghanaakhyam
When circumference is multiplied with diameter and that result divided by 4, that will give the area of a circle. This when multiplied with 4 gives the surface area of the globe which is like surface of a ball. This when multiplied with diameter and divided by 6 gives the volume of the sphere of globe.
Mathematically it can be written as 2_r x 2r/4 =_r2
BHASKARACHARYA II – LILAVATI – 1114 A.D
NEWTON GAUSS (1670AD) BACKWARD
INTERPOLATION DISCOVERED BY VATESWARACHARYA
Dhanushaaptha bhuktha jeevaghaathe labdham saroopakam dalitham labdaghna vivarahatham cha samsodhya niyogya vikalajyaa
In modern mathematical form this interpolation formula can be written as f(x) = f(xi)+ (x-xi)1/h Df(xi-h) + (x-xi)1/h. (x-xi+h)1/h. D2f(xi-h)½.
VATESWARA VATESWARA SIDDHANTA 904 AD
ARC AND CHORD
Svalpachaapaacchaghanashashta bhaagatho vistaraardhakruthir- bhaktha varjitham sishtachaapamihasinjanee bhaveth thadyuth o_alpaka guno_asakruthdhanu:
The chord of an arc of a circle is obtained from the result of the cube of the length of the arc divided by six times the cube of radius and subtracted from the arc. This can be mathematically presented as follows: Chord (R Sine _) = s – (s3 / 6r3). Here length of the arc s is in angular dimensions, r is the radius and _ is the angle of the arc.
PUTHUMANA SOMAYAJI – KARANA PADDHATHI – 1450 AD
Paridhe: shadbhaagajyaa vishkambhaardhena saa thulyaa
The chord of one sixth of circumference is equal to the radius of that circle.
ARYABHATTA I – ARYABHATEEYA 499 AD
LENGTH OF ARC – CHORD
Vyaasaabdhighaathayuthamourvikayaa vibhaktho jeevangghri panchagunitha: paridhesthuvarga: labdhonithaath paridhivarga chathurtha bhaagaadaapte pade vruthidalaath pathithedhanu: syaath.
One fourth of five times the chord multiplied with square of circumference divided by four times the diameter added with the chord. This value is subtracted from one fourth of the square of circumference. Square root of this is taken and subtracted from half of the circumference to get the arc.
BHASKARA II – LILAVATI 1114 AD
ARC AND ARROW
Jyaavyaasayogaanthara ghaathamoolam vyaasasthadoono dalitha: sara: syaath vyaasaaccharonaacchara sangunaa cha moolam dvinighnam bhavatheeha geevaa yeevaardhavarge sarabhaktha yukthe vyaasapramaanam pravadanthi vrutthe
When the sum and differences of diameter and the chord are multiplied, and their square root is taken and if half of that is subtracted from the diameter, the arrow is obtained. The difference of diameter and the arrow multiplied with the arrow, twice the square root of that value gives the chord. The square of half the chord divided by arrow and added with arrow gives the diameter of the circle.
BHASKARA II – LILAVATI 1114 AD
NEWTON’S INFINITE GP CONVERGENT SERIES
DISCOVERED BY NILAKANTA SOMAYAJI
Evam yasthuthya ccheda paramabhaaga paramaparyayaa ananthaayaa api samyoga: thasya ananthaanaam api kalpyamaanasya yogasyaaddhyaavayavina: parasparama cchedaad ekonacchedaa mamsa saadhyam sarvathraapi samaanam eva…
Thus the sum of an infinite series, whose later terms (after the first) are got by diminishing the preceding or by the same divisor, is always equal to the first term divided by one less than the common mutual divisor.
NILKANTA ARYABHATEEYA BHASHAYA 1444