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

**PROGRESSION OF THE TYPE 12 + 22 + 32 + 42 + ….**

*Sapthaanaam ashtaanaam saptadasaanaam chathurbhu jaaschithaya: ekavidyaanaam vaachyam padastharaasthaa hi vargaakhyaa: *

There are (three pyramidal) piles on square bases having 7, 8 and 17 layers which are also squares. Say the number of units there in.

*BHASKARACHARYA I – ARYABHATEEYA BHASHYA 628 AD*

**PROGRESSION OF THE TYPE 13 + 23 + 33 + 43 + **

*Chathurasraghanaschithaya: panchachathurnavastharaa vinirdesyaa: ekaavaghatithaasthaa: samachathura sreshtakaa: kramasa:*

There are three pyramidal piles having 5, 4 and 9 cuboidal layers. They are cuboidal bricks (of unit dimension) with one brick in the topmost layer. Find the number of bricks used in them.

*BHASKARACHARYA I – ARYABHATEEYA BHASHYA 628 AD*

**PROGRESSION OF THE TYPE _n + _n2 + _n3 + _n 4**

*Sankalithakruthighanaanaam sankalithasamaasamaanaam me kathaya shannaam sakhe padaanaam ganayithvaa yadivijaanaasi*

Friend, if you know, then say after calculation (i) the sum of successive sum of 6 natural numbers (ii) the sum of the squares of the first 6 natural numbers and (iii) the sum of the cubes of first 6 natural numbers.

*SREEDHARACHARYA – IN PATIGANITHA 900 AD*

**FIRST DEGREE INDETERMINATE EQUATION**

*Mudgaanaam kudavaa: saptha labhyanthe navabhi: pane: panena kudavasyaardham thandulaanaamavaapyathe thatha: panathrayam saardham gruheethvaa_asu vaningmama thandulaanaam prayacchaamsa mudgaanaam cha dvisangunam *

7 kudavas (unit of measurement) of mudga are obtained for 9 panas and ½ kudava of rice is obtained for one pana. Then O! merchant take 3½ panas and quickly give me one part of rice and two parts of mudga.

*SREEDHARACHARYA – IN PATIGANITHA 900 AD*

**FIRST ORDER EQUATION – I**

*Ye nirjaraa dinadinaardha thrutheeya shashtai: sampoorayanthi pruthak pruthakeva mukthaa: vaapeem yadaa yugapadeva sakhe vimukthaasthe kenavaasaralavena thadaa vadaasu*

By opening 4 inlets separately, one pond gets filled respectively within 1, ½, 1/3, and 1/6 days. If all the four inlets are opened together, how much time (in fraction of the day) is required to fill the pond ?

*BHASKARACHARYA II – IN LILVATI 1114 AD*

**FIRST ORDER EQUATION – II**

*Nava gulikaa saptha (cha) roopakasamaasthrayaanaam (thu) gulikaanaam thrayodasaanaam cha roopakaanaam thadaa kim gulikaa moolyam *

If 9 gulika and 7 rupaka are equal to 3 gulika and 13 rupaka, what is the price of one gulika? (the answer can be determined through the same method followed above)

**SREEDHARACHARYA PATIGANITHA 990 AD**

**EQUATIONS OF HIGHER ORDER- I**

Vaanarakulathribhaga: svathryamsa samanvi1tha: sara: prayayow moolam cha pipaasathi dvow choothathale sthithow seshow

One third of a troop of monkey with one third of itself has gone to the tank; the square root of the whole troop is afflicted with thirst, and the remaining 2 monkeys are sitting under the mango tree. What is the total number of monkeys? 1/3 a + 1/9 a + _a + 2 = a.

**REEDHARACHARYA – PATIGANITHA 990 AD**

**EQUATIONS OF HIGHER ORDER- II**

*Bale maralakula mooladalaani saptha theere vilaasabhara manthara gaanyapasyam kurvancha keleekalaham kalahamsayugmam sesham jale vada maraalakula pramaanam *

I saw that one half of 7 times of the square root of the total number of swans were slowly moving away in the river. Remaining 2 are playing in water. What is the number of total swans? (equation: 7/2 _a+2=a)

BHASKARACHARYA – LILAVATI 1114 AD

**PYTHAGORUS THEOREM DISCOVERED BY BOUDHAYANA**

*Samachathurasrasyakshnayaa rajju dvishtavathim bhoomim karothi *

The diagonal of a square produces double the area of the square.

*Deerghachathurasrasyakshnayaarajju: paarsvamaani thiryanmaani cha yatpruthakbhoothe kuruthasthadubhayam karoti *

Areas produced separately by the length and breadth of rectangle together equal to the area of the (square) produced by the diagonal.

**BOUDHAYANA BOUDHAYANA SULBASUTRA 700 BC**

**EXPLANATION OF BINOMIAL THEOREM**

If a three syllablic Madhya Chanda based on guru and lakhu sounds were followed, then variation of guru and lakhu sound will be on the following pattern:

*3 guru sound occur once, 2 guru and 1 lakhu occur thrice, 1 guru and 2 lakhu sounds occur thrice, 3 lakhu occur once. **The equation can be derived easily. If guru is g and lakhu is 1 then, *

**(g+1)3 = g3+3g21+3g12+l3.**

This equation is the same as (x+y)3. Similarly for finding the pratishta Chanda, in the Chanda sastra of Pingalacharya, the following equation can be indirectly applied in this form: (g+1)4 which is expanded as

**g4+4g31+4g212+4g13+14 **

*I.e 4 guru sound occur once, 3 guru and 1 lakhu occur four times, 2 guru and 2 lakhu occur four times, 1 guru and 3 lakhu occur four times and 4 lakhu occur once.*

PINGALACHARYA – CHANDASASTRA 200 BC

**GEOMETRY IN SULBASUTRA-II**

Thaasaam trika chathushkayordvaadasikapanchikayo: panchadasikaashti kayo: saaptikachathurimsathikayo: dvaadasika panchathrimsathikayo: panchadasikashad- thrimsikayo: ithyethaasoopalabdhi:

Hypotenuse in rectangles having sides 3 and 4 (= 5), 12 and 5 (= 13), 15 and 8 (= 17), 7 and 24 (= 25), 12 and 35 (= 37) and 15 and 36 (= 39) (I.49).

**BOUDHAYANA BOUDHAYANA SULBASUTRA 700 BC**

**ANGULAR DIMENSIONS**

*Angagunavedahuthaasaa: kalikaa vikalaa: samudrajaladhaya: svalpajalakhaashtasasi dhruthisasina: kalikaa: saraagnayo vikalaa: thrijyaakruthivarashta navathribhuvo visve jinaamsajyaa. *

*Thribhujasya phalasareeram samadalakoti bhujaardha samvarga: *

The area of a triangle is the product of the prependicular and half the base.

**ARYABHATTA I ARYABHATEEYA 499 AD**

*Karnasthrayodasa syaath panchadasaanyo mahee drisapthaiva vishamasthri bhujasya sakhe phalasankhyaa kaa bhavedasya *

What is the area of a scalene traingle in which one lateral side is 13 units, other 15 unit and the base is 14 units.

*Ashtaadasakocchrayovamso vaathena paathithomoolaath shadgathvaavasow pathithaasthribhujam kruthvaa kva bhaghna: syaath *

A bamboo of beight 18 cubits fell by the wind, it falls at a distance of 6 cubits from the root, thus forming a right triangle, where is the break?

**BHASKARA I COMMENTARY TO ARYABHATEEYA 628 AD**