Indian Discoveries and Inventions – Part I

indian discoveries and inventions – Part I : WE CLAIM THESE ARE INDIAN DISCOVERIES

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DISCOVERY AND USE OF ZERO

Gaayathre shadsankhyaamardhe
apaneethe dvayanke avasishtasthrayastheshu roopamapaneeya
dvayankaadha: soonyam sthaapyam

In gayatri chandas, one pada has six letters. When this number is made half, it becomes three (i.e the pada can be divided into two). Remove one from three and make it half to get one. Remove one from it, thus gets the zero (Soonya).

— PINGALACHARYA IN CHANDA SASTRA 200 B.C.

CALCULATIONS WITH ZERO

Vikaaramaayaanthi dhanarunakhaani na soonya samyoga viyogathasthu soonyaaddhi suddham swamrunam kshayam swam vadhaadinaa kham khaharam vibhakthaa:

Nothing happens (to the number) when a positive or negative number is added with 0. When +ve and -ve numbers are subtracted from 0, the +ve number becomes negative and -ve number becomes +ve. When multiplied with 0, the values of both +ve and -ve numbers become 0, when divided by 0, it becomes infinity (khahara).

— SRIPATI IN SIDDHANTHA SEKHARA 1039 AD

Yathaa ekarekhaa sathasthaane satham dasasthane dasaiam chaikasthaane yathaa cha ekathvepi sthree mathaa cha uchyathe duhithaa svasaa cha ithi

In the unit place the digit has the same value, in 10th place, 10 times the value and in 100th place 100 times the value, is given.

— VYASA BHASHAYA TO YOGA SUTRA 650 AD

DISCOVERY OF PLACE VALUES – II

Yathaachaikaapi rekha sthaananyathvena nivisamaanaika dasa satha sahasraadi sabda prathyaya bhedhamanubhavathi

One and the same numerical sign when occupying different places is conceived as measuring 1, 10, 100, 1000 etc.

— SANKARACHARYA VEDANTA SUTRA BHASHAYA

KNOWLEDGE ON INFINITY

Asmin vikara khahare na raasaavapi praveshteshvapi ni: srutheshu bahushvapi syaallaya srushtikaalenanthe chyuthe bhoothaganeshu yaddhath

Nothing happens to the (huge number) infinity, when any number enters (added) or leaves (subtrated) the infinity. During pralaya many things get dissolved in Mahavishnu and after pralaya, during srushti all those things get out of him. This happens without affecting the lord himself. Like that, whatever number is added to infinity or whatever is subtracted from it, the infinity remains unchanged.

— BRAHMAGUPTHA IN BRAHMASPHUTA SIDDHANTA 600 AD
— BHAKARACHARYA II – BEEJAGANITA 1148 AD

USE OF AVERAGE VALUES

Ganayithva visthaaram bahushusthaneshu thadyuthirbhaayyaa sthaanakamithyaa samamithirevam dairgye cha vedhe cha

(For length, breadth and depth) the measurements should be taken at many places and the sum should be divided by the number of times (places) the measurement is taken.

— BHASKARACHARYA II IN LILAVATI 1150 AD

USE OF FRACTIONS

Drammaardha thrilavadvayasya sumathe paadathrayam yadbhaveth that panchaamsaka shoda saamsa charana: sampraarthithenaa- rthinaa datto yenavaraatakaa: kathi kadaryenarpithastena me broohithvam yadi vetsi vatsaganitha jaathim prabhagaabhidhaam

One man has given to a beggar fraction of 1 dramma (a unit of money). That fraction is one fourth of the one sixth of one fifth of the three fourth of the two third of the half of a dramma. Then tell how much kowdi (a unit fraction of the amount dramma) was given to the beggar?

— BHASKARACHARYA I – ARYABHATEEYA BHASHAYA 628 AD

USE OF RATIO AND PROPORTION

Ashtow daanthaa sthryo damyaa ithi gaava: prakeerthi thaa: ekaagrasya sahasrasya kathi daanthaa: katheetharai:

(Out of 11 cattle) Eight are tamed and 3 are to be tamed and (how many are) to be tamed) if the number of cows is 1001 ?

— BHASKARACHARYA I – ARYABHATEEYA BHASHAYA 628 AD

PERMUTATIONS AND COMBINATION – I

Katukathiktha kashaayaamla lavana madhurai: sakhe rasai: shadbhi: vidadhaathi soopakaaro vyanchanamaachakshva kathibhedam

Friend, a cook prepared varieties of food with 6 savours: pungent, bitter, astringent, acid, saline and sweet. Say what is the possible number of varieties of food that can be made with these savours.

— SRIDHARACHARYA IN PATIGANITA 990 AD

PERMUTATIONS AND COMBINATION – II

Paasankusaahi damarooka kapaala soolai: khadvangasakthi sara chaapayuthairbhavanthi anyonya hastha kalithai: kathi moorthibhedaa: sambho haririva gadaari saroja sankachakrai:

Pasa, ankusa, serpant, damaru, kapala, soola, khatvanga, sakti, chapa, sara with these (ten) items how many permutations and combinations are possible for Lord Siva. Similarly with the four items, sanku, chakra, gadha and padma holding in the hands, how many combinations are possible for Lord Vishnu?

— BHASKARACHARYA II IN LILAVATI 1114 AD

PARTNERSHIP AND SHARES

Samavaayakaasthu vanija: panchaikaikottharaadhi mooladhanaa: laabha: sahasra sankhyo vada kasmai thathra kim deyam

Five partners collaborate in a business. The capital invested by them are (in the ratio) one and the same number increasing successively by one (i.e 1,2,3,4, & 5) respectively. Profit that accrued amounts to 1000. Say what should be given to whom.

— BHASKARACHARYA I – IN ARYABHATEEYA BHASHYA 628 AD

LOANS AND INTERESTS

Kutumbaarthamasakthena gruheetham vyaadhithena vaa upaplava nimittham cha vidyaathaapalkrutham thath kanyaavaivahikam chaiva prethakaaryeshu yathkrutham ethath sarvam pradaathavyam kutumbena krutham prabho

Loans are taken for meeting the expenditure connected with economic problems due to family burden, health problems, treatment, education, expenditure during accident, marriage of daughter, for performing rituals connected with the demise of the family members, etc.

— VISHNUSMRUTHI 100 BC

INTEREST CALCULATION

Maasena sathasya phalam panchaiko bhavyakerdhamaya vruttho lekhakapaado varshe panchaadika navasatheemisram

The rate of interest being 5% per month, the commission of surety 1% per month, fee for accountant ½% and charges of the scribe 1/4% per month, certain sum amounts to 905 a year. Find the capital, the interest and the shares of the surety?

— SRIDHARACHARYA IN PATIGANITA 990 AD

RULES OF CHARGING INTEREST

Atha utthamarna: adhamarnaadyathaa datthamartham gruhneeyaath dvikam thrikam chathushkam panchakam cha satham prathimaasam

The loans can be given and taken between borrower and lender. Generally charged interest rates are 2, 3, 4, or 5% per month.

Sa paadapanaa dharmyaa maasavruddhi: panassathasya panchapanaa vyaavaharikee

Reasonable (dharmic) rate of interest is 1.25% per month (i.e 15% per annum) on the transactions with common man for non commercial purposes. But for commercial purposes (for making profit out of it) interest rate can be 5% per month.

— VISHNU SMRUTHI 100 B.C

RULES OF BODIES IN MOTION

Bhakthe vilomavivare gathiyogenaanulomavivare dvow gathyantharena labdow dviyogakaalaavatheethaishyow

Whenever two bodies are travelling in the opposite directions, the distance between them is to be divided by the sum of their speeds. If they move in the same direction, the distance is to be divided by the difference of their speeds. This gives the time required for meeting of the bodies or the time elapsed after meeting of the moving bodies.

— ARYABHATA I – ARYABHATEEYA 499 AD

Ekow naa yojananyashtow yaathyanyo yojanadvayam yojanaanaan satham panthaa: sangama: kva gamaagame

One man travels at 8 yojana speed per day. Another travels at 2 yojana per day, starting simultaneously from the same place. After reaching the destination, the first man comes back. If the length of the track is 100 yojana. Say where is the meeting place of the two? (One going forward and the other traveller returning).

— SREEDHARACHARYA PATIGANITHA 990 AD