Harappan Mathematics .Solution of a problem

Suvarna Nalapat's Blog (Diagrams are not given in this Blog )

The harappan civilization (Gregory Posshell 2nd revised ed pp 415)had two systems of weights.
1.The hexahedrons –cubical-chert weights of urban phase with ratio of 1:2:4:6:16:32:64.This was the Indus std seen in all Harappan towns including Lothal and Rangapura.
2.Another spheroidal weight system og agate,chert and dolerite in Lothal and Rangapura (where only dolerite and sandstone weights ).The smallest weight of the hexahedron was 4.337 and the next was 8.5733 which was almost near a shekel of Babylon of 837.The truncated spheroids of Lothal had 98.2 gm,156 gm,229.5 gm,271.2 gm,275.2 gm,280 gm,300gm.And 8.5733 was their standardization.
Taking all the hexahedrons of the Harappan sites from 3/2 ;3;6;7;1;18; and 32 and multiplying each with this std number ratio of 1:3/2 :3:6:12:18:32 was obtained.

What I notice is that the 7th number is 32 in each system and thereafter the ratio is thus unified.The 8th is 64 which is 8 X 8 and is a square .In the first system there are 2 and 4 and 16 as sqares before 64 but in the second system 64 is the first sqare number.

Page 236 of the same book reports the occurrence of two types of sling pellets found in Mohenjo Daro near the granary .One was round and spherical ,the size of a marble and the other was more rare and ovoid and had 2.5” length and 1.6” dia .This type occurred at all levels of excavation.They were baked ones and similar pellets were discovered in Sumer ,Turkestan and Indian sites.Sankalia questions the identification of these objects as ammunition by Marshall(Sankalia 1977:68-69).
Page 192 tells of excavations in Bhagaval pura in Kurukshethra district .(Late Harappan period).Two oval shaped structures one having 1.8 by 0.85 cm and another 1.6 by 0 .92 cms was identified from there.In the third structural phase of construction the houses were constructed by baked bricks with sizes:-
1. 1.2 X 1.2 X 18 cms
2. 12 X 12 X 8 cms
3. 29X 22/12 X ½ cms (wedgeshaped)
4. 20 X 20 X8 cms
5. 16 X 12 X 4 cms

Dr Bryan Wells have measured three ceramic glazed pots from Harappa with circumference of 193 cm and height 64 cms ; another with circumference of 269 cms and height 74 cms ; and the third with 262.5 circumference and height 85 cms .The volume of each was measured by him and it was 27.30 litres,55.56 litres and 65.89 liters respectively.Each of the pots had three,six and seven long strokes inscribed and he calculates the value of one long stroke as 9.24 or approximately 10.
That means a pot with three long strokes and circumference of 193 cms and height of 64 cms has a volume of approximately 30 liters ( 27.30 only since a jar is a truncated spheroid and not a perfect sphere.).
The one with 6 long strokes and a circumference of 269 and height 74 has a volume of 60 liters for a perfect sphere and since it is a truncated sphere only 55.56 liters.
The one with a circumference of 262.5 and height of 85 cms and with seven strokes has a volume of 70 litres as a perfect sphere and as a truncated sphere it has only 65.89 liter capacity.

Quoting B.B.Lal, H.J.Winter and S.Rao C.P.A .Vasudevan has given a series of weights from Indus valley and has introduced a problem with Rao’s calculation.(page38-40 The nature of European expansion and the Indus Valley weights .C.P.A.Vasudevan ed.Dr N Mahalingam .International socity for investigation of Ancient civilizations .102.Mount Road ,Guindy ,Madras 600032)It is as follows:-
B.R.Lal:- The weights fall in the progression of 1, 2, 8/3, 8, 16 ,32 upto 12800 .
H.J.Winter of Exeter uty :- In the ratio of 1:2:8/3:4:16:32:64:160:200:320:640.He says the ratios are based on an important number 16,in ancient Indian numerology (Shodasa).Certain others are obtained from it bydoubling or halving.The use of fractional thirds and development of a decimel form of higher numbers is also seen which is interesting.

1, 2, 8/3 ,8, 16,32 and 64 do not form a regular line of progression.The 8/3 stands outside the line formed by doubling or halfing the numbers .
Similarly 160, 200, 320 , 640 ,and 12800 also fail to fall in a line.Some being productions of multiplications of ten (of the figures obtained by doubling).Others are multiplications of 100.Thus arranging them in 4 rows we have
1.The thirds 8/3
2.Figures by doubling . 1 2 4 8 16 32 64 128
3.multiplied by 10 160 320 640
4.Multiplied by 100 200 12800

5.Now we have to fill in the gaps in each line .The complete series then is .

A. Thirds. 1/3 2/3 4/3 8/3………………
B.Primary sreni. 1 2 4 8 16 32 64 1280
C Dasaguna(X 10) 10 20 40 80 160 320 640 1280
D.Sathaguna (X 100) 100 200 400 800 1600 3200 6400 12800

This actually goes on increasing to any number of powers of (10) upto parardham and above.As given in the Veda ( Refer Raagachikitsa .Readworthy publications .Dr Suvarna Nalapat pp and DC Books page )

The progression in two dimensions (Horizontal and vertical )is thus seen.
Horizontal by doubling repeated seven times
Vertical in three steps as division into thirds,multiplication by 10 and by 100.
You can draw this on a sign + which is the KA of Brahmi and the part of swasthika or sarvathobhadra in Indus lipi.

About Rao’s observation and CP.A Vasudevan’s problem about it :-

According to Rao the smallest Indus weight as a unit (From Lothal as 1.8233 gms) the ratio of 2 4 6 8 16 32 64 120 was found and the mean values of other groups were 3.488,5.172,6.896,13.792,27.584 ,55.168 and 137 gms respectively.
He says a second set of weights from Lothal is remarkable for regularity of ratio and the smallest of the series weighed 1.2184 gms and others stand in the ratio 7/2 , 7, 14 ,28 respectively weighing 4.33370,8.5753,18.1650 and 32.3052 gms .
Sri Vasudevan notes :-
1.The unit of 1.8233 gm on multiplication by the given factors will not give the products 3.448, 5.172 , 6.896 etc but if the smallest weight is 17.24 and the factors are 2 3 4 8 16 32 and 80 respectively these values are obtained.(In fact the wikipedia says the smallest weight of Hrappa was 17. 24 so that the ratio is correct).
In that case the resultant figures are accurate to third decimal place ,all but the last (137.90) which ought to be 137.920.

The second unit 1.2184 gm also will not give the figures 4.3370 and the series when multiplied by 7/2. 7, 14 , 28 or any whole numbers .Thus he finds out something wrong in Rao’s smallest weight and ratios observed from it.
If 3 ,6 120 etc were in use another series of progression was in use :-
1 3 6 12 24
10 30 60 120 240
100 300 600 1200 2400

As well as according to Rao’s conclusion a fourth series :-
½ 7/2
1 7 14 28 35 42 49 56 63 70

10 70 140 280 350 420 490 560 630 700

And this elaborate series of weights and measures and progressions which occurred in India at least from 3000 BC (even before that ) is thus a computer type clue to the knowledge systems of all Indian arts ,sciences and philosophy and that is what I am trying to integrate .It is not just a language script alone but a script in which numbers and the whole knowledge system is given as a soothtra ( short formula) for entire world and this they did since there was aprediction that the people and the palm leaf manuscripts and other things are about to be lost in a great deluge and they wanted it to preserve for the posterity so that someone some day will find it and decipher it for entire world.

My calculation and conclusion:- I have proof to show that the present Indians (probably all people on the globe)is still following the same mathematics as the ancient Indians followed.Iwill start from the old pre-independent and the present system we follow (which are only two ways of expression of the Harappan system)
4pai=1 anna
16 anna=1Rupee This increase in 1, 2 , 5 ,10 ,20 50 , denominations to 100 Rs 100 and so on.
½ of a Rupee=0.50 =old 8 anna
¼ of a Rupee=0.25 = 4 anna
1/8th of Rupee =0.125=2 anna
1/16th of Rupee=0.0625=1anna=4pai
½ of an anna= o.50=2 pai
¼ of an anna =0.25=1 pai
1/8th of an anna= 0.125= 1/8th 0f 4 pai only .
1/16th of an anna is a pai.
Division of pai=
Half of 1/16 as 1/32
Half of that ,or ¼ of pai =1/64
1/8th of 1/16 or a pai = 1/128
1/16th of 1/16th=1/256 and so on the value decrease at same rate on the opposite number of the expanding numbers .(Both +ve and ---ve numbers used in mathematics.

1/4th of 10 Rs =2.5 Rs
½ of 10 Rs =5
¾ of 10 Rs= 7.5 Rs ( a decimal which is a fraction 7 and a ½)
Then come next as 100 .Rs (Taken as a whole 1)
½ of it =50 Rs
¼ of it 25
3/4th of it =75 and it is not a fraction.So as numbers increase in numismatics fractions and negative numbers become whole numbers.
Then 1000 Rs .Same principle .Like this one can expand to 10 to the power of n numbers which is called the para .10 to the power of 13 is prardha so that of para is 27thpower and beyond.
1/16 for a people who use decimal is 0.0625
1/32=0.03125
1/64=0.0156440625
1/128=0.0078203046875
So 1/128 has a 13 digit decimal just like 10 to the power of 13 or parardha .
From here either you can add 12800 like that or as a number to the power of n to change the position.There is only position change in decimal system.This is what the IVC Harappan weights indicate.They knew all the problems with which the European mathematicians had been toiling with from BC 500 to 16th century(both periods being their contact periods with India.
Now about the problems of the three fractions which we saw earlier
1. The 3/2 fraction used in std hexagon ratios of Harappa
2. The 8/3 used in series 1 2 8/3 .4 8, 16 ,32 ,64 100 .,200 etc in B.R.Lal series
3. The 7/2 fraction as C.P.AVasudevan pointed out in Rao’s series.


3/2 is 1.5 so that the std hexagon ratio is used as 1:1.5:3:6:12:18:32
8/3 is 2.26666…a recurrent decimel.
So the series can be rewritten as 1 2 2.66 4 8 16 32 64 100 200 and so on.
7/2 is 3.5 .
So the entire series including the fractions become :-
1: 1.5: 2 : 2.66: 3.5 4 8 16 32 64 128 and their multiplications by 100
Thus one fraction is between 1 and 2 and two fractions are between 2 and 4 .
It introduce the remaining numbers 3 , 7 and 9 as well as 11 into the series.
1/3 is 0.3333333..recurring decimel1/7 is 0.142857, 142857,142857 …recurring decimel used by Paithamahasidhantha.
1/9 is 0.1111111…a recurrent decimel.
Using 8/3 the number 3 and its multiples are included into the series of 2 4 8 16…
7 is to be included along with the next number 11.(the next to 10 ).1x 7=7 .
11X 7 =77
7/1==7
1/7=The recurrent decimel mentioned above which is the astronomical value .
11X 2=22
223 71 (220+3 and 70+1) contain 294 which include all numbers upto 11 .
223/71 is thus described as a paridhi and its fractional value obtained.
The value of 22/7 is fixed thus as the value of pai.
The pai is the pazhaya kaasu (karshaapanam) as well as the bag in which it is contained or the limit of a purse that can be filled with it (the volume).This is also the same measure of the Kudam or ghatam .Those who measure volume by pai,and by kudam or any vessel use same formula used by astronomers to measure zodiac and stars and by vaasthu people to measure geometry(jyamithy) .Thus algebra and geometry and weight and measures of the various objects from grains to gold was standardized in the same way which shows the accuracy of the scientific mind of Harappans.They experimented with kunni9Abrus precatorious seeds,cowri,grains and liquids and found out the volume of each urn and pot and object they make and the ships and boats they make before they started their trade.So if we are able to see the objects and weights and measures thay made from 7000- 8000 BC from Mehrgarh period ,they must have started the process of scientific thought several millennia before that.
But more remarkable is that they standardized their music also on the same scales.The paramaavadhi(the ultimate limit concept of the human intellect ,as well as that of the universe reached the ParaBrahman concept and the expansion has to be followed by a contraction as in a + and --ve flow of electromagnetis field and this they expressed as their mathematics and astronomy.

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Now go back to historical times and to chapter on the measurements of Koutilya for the gold,for the thulakol,for samavrithathulaakkol,and also for the dronam which is an urn with a truncated spheroidal shape .The system of volumetric analysis that was followed from Indus valley /Harappan to historical ChandragupthaMourya period was the same and continuous will be cognizable.For construction of a equal vritha(samavritha)thula(balance)he uses a 72”or 3 muzham kol length and takes 70 as 9 palam.(72 then is approximately 9.24 to 9.33 or 9 1/3 palam ).
6”=1 palam
3”=1/2 palam 0.5
2”=1/3 palam 0.33 a recurrent decimal
1 ½ “=1/4 palam 0.25
¾ “=1/8 palam 0.125
To make it 10 X 7 divided by 2 =35 palam iron needed.Or roughly 9.33 X 9.33 =87.0389 which if the difference from the 8573 as 85.73 dasamsa is calculated will be 1.20 only.Thus the calculation is a decimel way of doing it .
What Koutalya says about the volume of such Dronam has to be then carefully reexamined.
He describes four types of volumetric measures for Dronam.
1.Aayamaanam contains 200 palam grains.A quarter of it (1/4th)with 50 palam is called a Aadakam and ½ of an aayamaanam is a Prastham with 100 palam grains.1/4th of a prastham is a Naazhi or naazhika containing 25 palam and is also called a Kudavam(kudam/ghatam)or a Kudumba.
That is measuring the jars as cylinders (Para or Parstham has a cylindrical shape)and as truncated spheroids (Kudavam ).
2.The same when we apply to a Vyavaharika Dronam which contain 187 ½ palam grains .47 palam for a ¼ of it, 93 for a ½ and 23 to 24 for 1 ¼ of a quarter .
3.For a Bhaajaneeyam which contains 175 palam grains the same values in order will be 43 ¾ ; 87.5;and 22
(The 87.5 is comparable with the value we get for a squaring of 9.33 as 87.0389. as shown above).This calculation is for measuring of grains for workers as Koutalya says.
4.The drona called Anthapura in the same way contain 162 palam of grains and the ratios will be as follows .40-41; 81 ¼ ; 20-21 .
Another interesting point about the construction of a balance kol is the difference in its length (as 8 ) needing 1 palam increase in the amount of iron needed
6 ‘ length =1 palam
14’= 2
22’= 3
30” = 4
38’= 5
46’= 6
54’= 7
62’= 8
70’= 9
78’= 10

72 ‘ being a samavritha balance it is 9 palam + 1/3rd of 6”length and that is how the approximate value was reached.
Volume of a Naalika /naazhikavatta( a pipe or a cylinder is a nazhi and a vritha is a part of a sphere) and measuring time by it by looking at floatation and sinking by volumetry:-
The two methods of measuring time during day and night by chayamaanam using a sankuchaya or Purushamaanam and by using the floatation and sinking technic of volumetric analysis were present in India .In Bhagavatham Vyasa has described how to construct a naazhikavatta and in Arthasasthra Koutilya has described its construction and in the 4th-5th century AD Varahamihira has described its construction.So that is a continuous tradition in India .It existed till my grandmother’s time (and I have a old Nazhikavatta with me which belonged to her and her mother ).84 “is a vyamam or maaru (when a man holds his arms spread) and with this the rajjumanam(for measuring with rajju or a coir /thread)and khathapourushamaanam(for measuring a depth of a mine )is measured.The Garhapathyadandam or kol for grihastha fire is 4 arakhni or 96 “
A dhanus is 108” for a path and wall etc.
A pourushamaanam or Agnichayas is 96” =8 chaya (when 1/ 18th of the day is over.)
6 kambam is a Brahmadeyam or an Athithisathram.
10 dandam is one Rajjju
That is 1920”
2 rajju= 1 paridesam
3 rajju=1 nivarthanam
42 finger =36”
Therefore 84 viral or finger = 72 “
*(See picture of a thumb or viral with a Navaranellida or size of a paddy seed in the middle )

So how much will be 84 inches?
84X 84/72= 7126/72 = 0.9 and 646/648 almost 10
That 84 “was thus taken as a maaru or a mans outstretched arms on either side (the diameter )and his height as the vertical to that horizontal line making a golam or sphere corresponding to the cosmic sphere .
Chaya 8 pourushamanam = 96” or 1/18th of day over
6 pourushamaanam chaya =72”=1/14th of day over
4 pourushamaanam =48”=1/8th part over
2 pourushamaaanm =24”=1/6th part over
1 pourushamaanm = 1”=1/4th over
8”chaya means 3/10 part over.
4”= 3/8th over
0 “= noon
(That is from 8 pourushamaana the 8th is the 0 where no shadow is there.)Now this is in reverse order for the rest of day till sunset.
At night it is better to use the nazhikavatta since no sunlight and shadow seen especially for seafarers.
¼ nimisham =1 trudi
2 thrudi=1 lavam
2 lavam=1 nimesham
5 nimesham=1 kashta
30 kashta =1 kala
40 kala =1 nazhika
Thus time and space have been unified by a naazhika.
Take 4 maasham gold and with it make a 4”needle(salaaka) and with it make a even opening on the base of a ardhakunda (hemispherical thin pot)made of Thamram .Then put the pot in a bigger vessel with water .The water will enter through the small opening drp by drop.And when the pot sinks by the water one is measuring the volume it contains and the time taken for its sinking as a nazhika and the problem of volume of a cylinder and a sphere is solved.This was the secret which Archemedes heard and tried to figure out and finally did figure out .But the water clock of Greek and this water clock of Indians are not the same .the water clock of Greeks though used the floatation principle was not measuring the volume of water needed to sink a vessel of a particular capacity and the time needed for it.

2 naazhika =1 muhoortahm
15 muhurtham= I day or 1 night (for months of Chaithram and Aswayujam as samarathra or Vishuvam)
Then every 3 muhurtha decrease and increase and day and night length increase or decrease as it is towards aphelion or perihelion .
Thus Indian volumetry extended to comparison of volume of a vessel,a ship,of a sphere and cylinder and to cosmos and human body as a Gola /naalika combination with vertical and horizontal measures .The spheres of celestial bodies,of human body and of vessels were done with same precision and measurement and at a scale difference only.This was applied not only to geography,astrophysics and aesthetics and mundane measurements of vasthu and vessels but also to music in the same manner .Thus the mathematics itself shows the originality of Indian scientists and the secret of Indian history is in this .Those who just repeat that India received science from Greece and from Rome and from 16th century Europe has to do some more homework and correct their views from such solid evidences.
COMPARISON OF HARAPPAN WEIGHTS IN GRAMS WITH NO OF KUNNI (GUNJA SEEDS) AND SOUTH INDIAN NAMES
Harappan wts gms Binary unit Decimal unit Average of the 21 wts per unit No of kunni seed Binary and decimal Tamil/southern names
0.571 1 .87 .87 8 2 panam ida
1.770 2 .89 16 4 panam ida
2.285 2.5 .86 20 1 kazhanju
3.434 4 .86 32 1 varaku nellida
4.337 5 .87 40 2 kazhanju
6.829 8 .85 64 2 varakunellida
8.575 10 .86 80 1 kaasu/karshapanam
13.731 16 .86 128 4 varakanellida
18.165 20 .91 160 2 kaasu
27.405 32 .86 256 8 varakanellida
33.05 40 .83 320 1 palam
54.359 64 .85 512 16 varakanellida
136.02 160 .85 1280 4 palam
174.5 200 .87 1600 5 palam
271.33 320 .85 2560 1 ser/cher
546.7 640 .85 5120 2 ser/cher
1417.5 1600 .86 12800 1 veesam
2781.4 3200 .84 25600 1 thada
5556 6400 .87 51200 2 thada
6603 8000 .83 64000 1 thulam
10875 12800 .85 102400 1 madanku
Total 28456 Total 33525 .85

Conversion of kunni(gunja)weights to fractions and their breakups .
Fractions Names of fractions Binary/decimal as in previous table of Kunnimani Break up of fraction
1/8 arakkaal 8 ½ X ¼
1/16 Veesam/makaaaani 16
1/20 Oru Maa 20
1/32 Ara veesam 32 ½ X 1/16
1/40 Ara maa 40
1/64 kal veesam 64
1/80 kaani 80
1/128 Araikkal veesam 128 ½ X ¼ X 1/16
1/160 araikaani 160 ½ X 1/80
1/256 Munthriye keel kaal 256 1/320 +1/320 X ¼
1/320 Munthry 320
1/512 Keel aria arakkal 512 1/320 X 5/8
1/1280 Keel kaal 1280 1/320 X 1/4
1/1600 Keel nanma 1600 1/320 X 4/20
1/2560 Keel kaal nama 2560 1/320 X ¼ X 4/20
1/5120 Keel veesam 5120 1/320 X 1/16
1/12800 Keel aria maa 12800 1/320X 1/40
1/25600 Keel kaani 25600 1/320 X 1/80
1/51200 Keel araikkani 51200 1/320 X 1/160
1/64000 Keel nanma ara maa 64000 1/320 X 4/20 X 1/40
1/102400 Keel munthri 102400 1/320 X 1/320
Note the sandwiching of binary and decimal system and making sreni or series with it and measuring everything by such simplified ways which existed in BC 3500 and to till date as a continuous tradition and now we have to compare these mathematical genius of ancestors with our modern mathematical science of the west .Therefore the history of western mathematics from BC 500 given in a nutshell.
(comparison of two sizes of Kunnimani with manjadi, and seed of Ilanji and ricegrains.Note the shapes ,size length,bredth differ and the number of these filled in the same jar will be different).







The concept of Mathematical discoveries of Europe at present accepted by modern science:-
1.Measurement of a sphere Archemedes 287-212 BC
2. Fraction of Egypt exist from 1650 BC scroll of ancient Egypt but it was a clumpsy aliquot fraction and if 2/5 is to be written they had to write for the second fraction 1/6 + 1/30 (not 1/5 + 1/5).
2/5= 1/5+ 1/6+ 1/30
Babylonians followed system of whole numbers each unit divided to 60 parts called the second minute prts and this continue with 3rd and 4th minute parts.This was used to tell time ,and is used still.Hour into 60 mts ,minute into 60 sec like that.The second is then divided to fractions of decimals rather than 3rds and 4ths.60 has many divisors and many fractions terminate in it.
½ 1/3 ¼ 1/5 1/6 1/7 1/8 1/9 .
Of these ½ ¼ 1/5 1/8 has a terminating destination.
1/3,1/6,1/7 1/9 has recurring representation. 1/3 = 0.333333………ad infinitum
Using 6 ,60,600 etc the only fraction that has no terminating representation is 1/7 .
(In India 5/8 is written as 0.625 . And 8/3 = 2.6666666666…..and tabulation of whole number 2 as acomplete revolution and taking 6 as aconvenient number for time calculation is easy. Please do see the chapter on weights and measures of Harappan and IVC in 3500 BC -7000 BC )
3.Quadratic equation includes square of the unknown.Quadratus is latin word for square which is a Chathurbhuja in Sanskrit . Chathura in Indian regional languages.Babylonian clay tablet show the following statement followed by a question..”The area of a square added to side of square comes to 0.75.What is the side of the square ?”
How this is worked out in an Indian context I will write here.
Area (Called Kshethraphala) + side of square (called Mukham or face) =0.75
Expressed as formula Ksa + Mu = 0.75
If Mukham is one ,
Half of it will be 0.5
Multiply that half with itself to get a sqauare it will be 0.5 X 0.5= 0.25
Then if you add this square 0.25 of the mukham with the already known 0.75 you get mukham as 1 (in modern terms the co-efficient of X ).This 1 is the square of 1.
Substract the 0.5 which you multiplied .Then 0.5 is the side of the square.
Or it is a way of finding the sqareroot and also to find the faces of a structure with several sides. Both Babylonians and Indians knew this.
4.The construction of a pyramid with top cut off was known to Egypt and to American people .
A solid like a cone is a pyramid.It has slopes uniformly from base to a point at top in Egyptian pyramid.If a piece is cut at top that is a frustrum.1850 Moscow papyrus show Egyptians knew this though their pyramids were not having frustrum.It says ,If aTruncated pyramid of height 6 and square base of side 4 on base and 2 at top ,square the 4 and get 16; multiply th e4 and 2 and you get 8;square the 2 and result is 4.
Now add the 16, the 8 and the 4 and you get 28.
Take 1/3rd of 6 we get 2 .Multiply 28 and 2 and we get 56.
This give formula of volume as:-
1/3 X 6 (4 square + 2 X 4 = 2 square )=56
If we take height as h sqare top of side r and square base of side R the formula for volume is
1/3 h ( R2 +Rr +r2 )
Diagram Truncated pyramid



(Now see the BC 3500- 7000 old Harappan /IVC measures of weights and its shape as both cubes and spheroids before you proceed and compare the numbers used ).Because we are now passing on to the circumference of a sphere .
5.Value of Pai and the ratio of circumference of a circle to its diameter.
Now considered as a discovery of Archimedes in 3rd century BC .Though both Indians and Egyptians knew it.
The Pai is now considered as derived from the Greek letter P pronounced Pie.(Actually from Paridhi a Sanskrit word which is written as Pa by Mathematicians and the letter in old Brahmi and in Sanskrit is similar to the symbol that we use even now.).

Early Egyptian value was 4 X(8/9)2=3.16 close to 3.14.
Bible 1 kings 7 verse 23 gives an approximate value.
Archemedes drew several polygons inside and outside circles with more and more sides and was able to close in on value of pie. With polygon of 96 sides the pai is between 223/71 and 22/7.Zu Chongzhi in 5th century used more accurate 355/113.Madhava by trigonometry demonstrated a seris or sreni for it which continue forever as :-
π/4 = 1-1/3 +1/5-1/7 +……By this he calculated pai to 11 decimal place.
With computers in 1949 ENIAC calculated to 2037 decimal place taking 70 hrs to do it.Modern computers upto a million decimal place is calculated. Pai occur both in pure and applied mathematics .
(In India this was used for musical circles as well which will be seen and described from the weights of the Harapappan and IVC periods and the musical scales and mela followed even today in the southern Parts of the country and how this is derived and applied to saama veda of Jaimini and Kouthuma traditions.)
The quadrature of the parabola or the area evaluation between a curve and a chord (which is a Danush or a bow and a chaapa/Jya in Jyamithy -from which term geometry originated in Europe - in Indian languages and seen frequently in Harapppan IVC script with sign of a bow )was attributed to Archimedes of 287-212 BC.He found a method to add infinitely many numbers.A parabola is a conic . A classic example is a human female breast.If a straight line is drawn crossing it near its tip ,the shaded area between that line and the curve is the area between the curve and a chord (The chaapa/Dhanus) but it is more like a nipple on the breast with one triangle in the area leaving two gaps on either side .One has to draw two small triangles in the 2 gaps then there are 4 gaps left This will repeat indefinitely .The area of all these small triangles approach the area required.Thus by a method of exhaustion one has to find the area (Like the milk of the breast measured by its enjoyment by a child).An infinite succession of triangles exhaust the area between the curve and line like the infinite succession of generations exhaust the nourishing food of the earth mother as a Kaamadhenu.)
Greeks distrusted infinite process.(Zenos paradox).Archemedes proved that it is possible to add infinitely many terms and obtain something finite.He showed that this sum of infinitely many fractions :1/1+ ¼ +1/16 +1/64 +1/256+……has a finite value 4/3 .The children are taught that this is the first recorded example of a summation of an infinite series,a key part of mathematics .But that is not true as we see from evidence of Harappa and IVC civilizations which are several millennia before Archemedes.
Another question and answer credited to him is the sandreckoner .How to estimate the size and volume of universe and size and volume and number of grains of sand needed to fill that universe ?How many grains of sand will fill the universe and what is thus the volume of universe/This was the question given to Archemedes .Archemedes called numbers upto 10 to the power of 8 as first order numbers.From that starting point he took successive multiples of this new unit.That is 10 to the power of 16 is the second order.This continued upto a myriad myriadth order which starts at 10 to the power of eight ,the whole to the power of 10 to the power of eight . This was known to ancient Harappans who found volume of grains to be put in vessels and how the volumes of celestaial objects and stars fir into the spacetime infinitely in same way but on astronomical proportions and the Veda starts the parardha or 10 to power of 13 in dasamsa .

6.6th century BC Greece and Southern Italy had the Pythagoreans who thought everything is number. They discovered earth is a sphere ,earth is not the center of universe,musical harmony depends on ratio of numbers (But was it actually they who discovered it or the Indus valley/Harappan people is what I am trying to prove).The European scientists say that no one is able to decipher the saying of Pythagoreans that everything is numbers.They probably meant all could be explained through a number language or code of numbers and that the seen world is only an illusion and the unseen world of numbers and laws of nature is the real world (as conjunctured by modern scientists).The important fact I find about Pythagoreans is that they had learnt that the area of land(Geographical measurement)and of weights of corn can be extended to a very abstract subject. But who taught them that? That is what I am speaking about when I speak of the global history of Mathematics. Who knew that everything on known objects,measureable objects could be extended to unmanifested abstract as well on an astronomical scale and who practiced this philosophy except the Harappans /IVC people and their wide trade and commerce agents ? And how the Harappan series of weights and measures were taught to the Greeks and Romans through these merchant class people (The phoenicians) in BC 500?
The Pythagoras theorem of a rightangle triangle that its square on hypotenuse is equal to sum of the squares on other two sides was known to Babylonians from BC 2000 and to Harappans from BC 6000-7000.
7 Pythagoreans were taught only part of mathematics ,not the whole.Because they thought everything can be explained in terms of whole numbers and their ratios.(Fractions).But this was not true. Because square root of 2 is not a ratio of two whole numbers but an irrational numbers (Akarani in Sanskrit).This is called proof by contradiction (Which in Indian terms is called Nethi ,Nethi .It is not this,not this .An exclusion principle ).You first assume that square root of 2 is a rational number and attempt to prove it and get a contradiction as proof saying that it is not a rational number but is irrational.(Karani means that which can be measured and akarani is that which is not possible to achieve).When this was taught to the Pythagoreans (whoever was that unfortunate teacher)they thought he was impudent and in their wrath drowned him in the Aegean sea. So by 5th century the Indian /Phoenicians had become cautious of teaching others certain secrets or ideas ,we can assume.(They found that these new people are different from Babylonians and Egyptians and Americans who accepted teachers and their knowledge for betterment of society ).
I will explain a little bit more of what the unknown teacher was trying to teach the Pythagoreans and lost his life for it.The greeks problems of doubling the cube,trisecting an angle,sqaring a circle involve constructions using a straight edge and compasses.What lenghths are constructed by these instruments?There are essentially five operations .One can join two points with the straight edge. Draw a circle with a given center through a given point.Find the intersection of two straight lines.Can find intersection of a line and circle.Can find intersection of two circles.All these operations were known to Harappans from the evidence of their symbols and weights and measure series. If a problem involves a length which cannot be built up from 1 by + ,---, √÷ and X ,then it cannot be constructed. Doubling cube with exactly twice the volume of given cube,and trisecting certain angles with these instruments alone is impossible according to Pierre Wantzel (1814-1848).The Harappans were trisecting an angle of 90 degrees to three angles of 30.Suppose we want to construct an angle of 10 degree we are trisecting a 30 degree.Then we have to construct a length of sin 10 degree.This number is a solution of a cubic equation that cannot be broken to simpler equations..The numbers that are not algebraic are the transcendental number.Both √ and ∏ are transcendental. The Pythagorean Greeks could not accept an akarani or a transcendental number.
8 What is a perfect number?
A number is perfect if it equals the sum of its proper divisors.In 6th century AD St Augustine thought perfect numbers are mystically superior to others.He wrote:- 6 is a perfect number.God created world in 6 days.Because six is perfect. 6 and 28 are the first two perfect numbers.Divisors of 6 are 1 ,2 and 3 .
6= 1 +2+3
Divisors of 28 are 1,2,4,7 and 14
28= 1+2+4+7+14
The next perfect numbers are 496 and 8128 and upto this was known to 13th century Arabs .The next 3 with a set of three incorrect numbers were added by the Arab Ibn Fallus.The even perfect numbers are easily found with formula of Euclid .to the power of n-1 ( 2 to the power of1 -1)provided the term inside bracket is a prime number.Upto 10 to the power of 300 ( 1 followed by three hundred zeroes).During vedic period upto parardham (13 zeroes after 1 )was thus studied.
There is no odd perfect number.If at all a odd perfect number exists it should have at least nine prime factors. ( Therefore to make a perfect solar system and a perfect understanding of it sitting on the earth as a central point of the observer who makes his observations , nine graham or nine prime factors are needed. The Indians thus added sun,moon ,and the two nodes (or 1/2s of earths orbits) ,to Mercury,Mars,Jupiter,venus and Saturn )
9.A regular polygon has equal angles and equal sides.examples are equilateral trangles with 60 degree angles,the square or a samachathura,all angles being 90 degree,then a pentagon,hexagon and so on.Construction of these is possible with only 2 instruments -a staright edge and compasses.This is part of school mathematics only.This was known to Pythagoreans (taught by someone to them ).Octagon or a Ashtadalapadma was known to Harappans/IVC and to Moscow Papyrus period Egyptians (much later than Harappans).Heptagons and Nonagons(with 7 sides and 9 sides)is that which is measured with 7 days,7 swaras ,nine graham and 9 rasa in musical aesthetics and in astronomy.That is more higher grade mathematics which was not taught to Pythagoreans by their teachers ,probably because they had killed him before that.That is why Archemedes had to wait for more centuries to acquire a knowledge of it.What we call Platonic solids are the regular polygons.
Tetrahedron hs four triangular faces.Octahedron has eight triangular faces.Cube has 6 square faces.Dodecahedron has 12 pentagonal faces.Icosahedron has 20 triangular faces.The golden ratio and the golden law of the Phoenicians or the Phaneesa(king of the Phani ) was taught to Pythagoreans by their teacher Phoenician and it is written with symbol that resembles a Naamam on the forehead of south Indians as the hood of a serpent with a U and a central line dipping in (The letter Zha in Malaylalam is pronounced as in pazhaya,pazham signifying an ancient race of fame )And its value is 1.618.This is a pentagon,pentagram,fivepointed star,Penrose tilling and the ratio of successive terms of Fibonacci series ,and several natural patterns have this golden rule .The musical aesthetics is also by this .This was known to Harappan and IVC people in 5000 BC but in Europe only in 1707-1783 by Euler this was identified .The rule of connecting vertice,face and edge of a solid is shown in table below where v is vertex,f is face and e is edge
Solid V F E
Tetrahedron 4 4 6
Cube 8 6 12
Octahedron 6 8 12
Dodecahedron 20 12 30
Icosahedron 12 20 30

This law is not true for the two Kepler solids and for solid with a hole.
Note that cube and octahedron has same edge 12 ,and their values of vertex and edges are reversed and the same is true for dodecahedron and icosahedron.
Doubling a cube to get volume of the altar of Apollo was not possible for the Greeks.
The problem they faced was;-Suppose wach side of altar is k units .Then volume is kXk X k = k 3 .If it is doubled then k3=2
Therefore k must be 3√2 .The cube root of 2.
10.Apart from problem of doubling cubes the Greeks had two other problems which they could not prove for centuries.One was trisecting the angle and the other to construct a square of equal area to a circle.
The squaring the circle ,one has to construct a line of length pai ,given a line of length 1.This is to be done with just a straight edge and compasses.
The area of sqare and circle has to be equal.suppose the radius is 1 unit.Area is pai X 1 to the power of 2.If equivalent square has side X units the area is X to the power of 2.Then find a length such that X to the power of 2= pai .The length was not the problem but the pai was.To draw a line of pai ,one had to use moving parts,or curves that could not be drawn exactly such as spirals.It took 2000 years for Europe all the three problems of the Greeks,their ancestors.But the Harappans had proved them several millennia before the Grreks.
The Zeno’s paradox of 490-430 BC of the dichotomy of the tortoise and Achilles is an example.By the time Achilles run 10 paces tortoise is one pace ahead and the field that they have to cross was peculiar.Before one reach other side you have to get halfway across.Before one reach halfway you must get quarter point and travel an infinite number of smaller distances so that it becomes impossible to get across at all .
This cut the infinite space to infinitely many numbers to have a finite sum.What Archemedes showed with it became important part of Western mathematics only in 17th century .But this was known to Harappans and it was one such person who taught this possible and impossible and was killed by the hasty disciples.Because the dichotomy in Zenos paradox is
½+ ¼ + 1/8 +1/!6+………=1
(Refer to Indus valley weights )
Achilis and tortoise (Rather feet and tortoise or korma )or rabbit and tortoise story of India :-
100 +10+1+1/10+1/100+….111 1/9
Which means after running 111 and 1/9 paces there is a possibility to catch up with the tortoise .The Chandra as a fast runner and the stars as a slow runner in the chandrasidhantha of Indian astronomers is this. See the dasamsa divisions and calculation of Harappan and IVC people again and you can see that they knew their sreni of weights of the finite types was actually to measure the infinite .It is possible to accomplish infinitely many tasks in a finite period of time .But it is also proving that any notion of movement or saayana universe and prakrithy as many objects is an illusion and that to know the immeasurable was impossible.And whatever there is measurable is expressed by their system of weights and measures and the numbers was known to Harappans as prakrithy ,time and that beyond prakrithy and measureable timespace as Brahman.
In mathematics as Plato correctly said there is only discovery and no invention.You can invent only technology ,not truth as an abstract principle.That is why he said “In mathematics men think the thoughts of Gods”.In more precise words mathematics in its abstract idea and thought is an absolute truth which is used by lesser individuals for mundane purposes.
11.The light cone of Einstein is now known to all scientists.It was drawn as a musical instrument of Shiva in India. It is a double cone cut by a plane.When the plane is horizontal it is a circle.When slightly tilted it is a ellipse.The plane parallel to the side of cone is a parabola,and when plane is steeply tilted it is a hyperbola.
If we want to define a cone without involving a threedimensional cone,on a piece of paper :-Suppose a point or bindu( Nadabindu or F) that is fixed and a line (Rekha d that is fixed) and a variable bindu moves so that the ratio is XF:X d is a constant.Then the X is moving in a conic.This ratio is called eccentricity and the fixed point is the focus or focal point .It is with this knowledge one can understand that the orbit of earth is an ellipse with the sun as a focus.This was not accepted by the Greeks as all of us know .But Indians knew this .They knew the eccentricity as 1/90th as we see from their astronomical calculatons from vedic times and from Indus valley and Harappan evidence.Since a wheel seen in an angle is an ellipse they thought of the Discus or chakra of Vishnu and Kaalachakra (Time of wheel )like this .If it is thrown in air it cuts a parabola as its path .This trajectory they saw as the reflecting surface of sunlight from the parabola.For long range navigation systems to guide ships the intersecting hyperbolas has to be used.So being the seafarers who traveled across the world the Phoenicians or kings of the Phani or naaga kings of India had employed this light cones and its tilts and the evidence of it is there in Harappan sites.
12.Fifth postulate of Euclid.
Gives properties of parallel lines. It states : That,if a straight line falling on two straight lines make the interior angles on the same side less than two right angles ,the two straight lines,if produced indefinitely ,meet on that side on which are the angles less than the two right angles.
This is needed to prove the proposition 29 which prove wellknown results about alternate and corresponding angles and also for proving Pythogorus theorem and many other standard results ogf geometry.
But there is a alternative possibility.The sum of angles could be less than 180 or greater than 180 unlike the pair of alternate angles with sum as 180 degrees.
The fundamental theorem of Arithmetic is that every whole number is written as a product of prime numbers .It is possible to factorize any whole number until one is left with prime numbers which by definition cannot be factorised any further.For one number there is only one possible factorization.
12= 2X 2 X 3
35=5 X 7
1001=7X11X13
The analogy with chemistry ,prime numbers are like atomic particles and cannot be split up and every other number can be expressed in terms of them.
The number of prime numbers are always infinite. There will be always a +1 after the last number we discover.
13 Hipparchus (190-120 BC) is credited with Trigonometry by western mathematicians and every one know that it is not true from the ancient archeological sites all over the world.Trigonometry taught in high schools is based on three functions ,the sine,cosine and a tangent and the original trigonometric function was the chord function.If the equal sides of a triangle have length of 1 unit,then the chord function theta gives the third side of triangle .The chord and sine functions are converted with doubling and halving .
Chord ø =2 sin(1/2 ø)
Sin ø = 1.2 chord (2 ø)
The oldest surviving table of sine and cosine is seen in Almagest of Ptolemy ,a work of astronomy which he learned from India (After Alexanders journey to India).The table is numerically starting with results like chord 60 degree =1 chord 90 degrees = √2
And uses formulae for chord (A+B) and Chord A-B)and chords of angles are found for every ½ degree to an accuracy of upto 6 decimal places.This familiar method of sine ,cosine and tangent introduced by Indian mathematicians and navigators are even now used and methods are the same but name of Hipparchus and Ptolemy is attached to the discovery.Extension of a number less than zero is the fractions and –ve numbers .It is important to recognize that sine,cosine balancing depend upon negative numbers and function of a revolving celestial body around earth or any other focal point also depend upon this.But none of the Greek mathematicians believed in negative solutions to equations.Diophantus rejected such equations categorically.But Indians had accepted it from prevedic pre-Harappan times and the use of finding negative roots of quadratic equations.
But negative numbers are an essential part of mathematics as modern mathematicians know very well.Minus times minus equals a plus and this is even applied in the energy equations of the present .
14.Claudius Ptolemy(83-161 AD) misunderstood the earthcentered universe from the method of teaching of the Indians .To suppose that the earth is the center of the universe as the observer sits on it and watches the surrounding moving objects in the sky is something needed for any astronomer to start learning astronomy .If a student mistakes it for an earthcentered theory of universe put forth by the teacher and does not move forward to the next lesson it is only a inadequacy of student not of teacher.A planet move in a small circle around earth (say the moon ).The center of this moon is earth which is moving in a bigger circle.The earth is moving around the sun.The sun which is the centre of solar system is moving in a very big circle than entire solar system put together.Such a system of earth,graham and nakshathra was accurate and used as guide for navigation by Indians ,for astronomical calculations,meteriological observations,for religious and harvest festivals ,calendar setting etc and recorded evidence of it is there from the sea trade of Harappans and their relics and weights and measures and the pictographs and ideograms they used .Between the positive and negative numbers is the number zero which is a prominent ideological symbol of India.This was invented by Pre-Harappan mathematicians of India.
Nicholas Copernicus had to arrive (1473-1543 AD) for renaissance of astronomy in the west .It took nearly 1300 years for European mathematics to revive and that too after another contact with India .
15.Al-Khwarizmi in his book of shifting and balancing gives a comprehensive guide to solving quadratic eqations .He desribes 6 types of quadratic equations.With an equation you shift terms on left side to right and vice versaThen balance terms on either side.His book was a work on Indian mathematics and was named in Arabic kitab wa al jabr wa al muqabalah.From this the middle words when said quickly(Al jabr)was pronounced as algebra and that is how Europeans called it later on.There was no need to write an X for unknown in Indian mathematics .The equation was written in a word and not in a number.Each word carried a number but.(The present computer language is based on this).But Al –Kwarizmi also did not use negative numbers and treated them in a different way.
Instead of writing X 2 -5X = 6
He would write X2 +5X =6 .This book had no discoveries but collected methodically the different ways of mathematics in an algebraic form and not in a geometric way.
16.when we use a geometric method of solving cubic equations we have to use intersection of certain shapes.An intersecting parabola is a sign of fish (seen in IVC Harapan and in the Tamil Pandha signs).It is also a sign of art,eros and kaama .Omar khayyam in 9-10th centuries concerned with love and wine ,also systematically classified cubic equations and how conic curves could be used to solve problems approaching them in algebraic way.
In 1202 Leonardo Fibonacci applied the sequence of 1 pair,2 pairs,3 pairs,5 pairs 8 pairs breeding successively in a rabbit population and by 5th month 13 pairs will be there.
Start 1 pair =2
1st month first pair breeds thus two pairs
2nd month first pair breeds again.the second pair is immature .so 3 pairs
3rd month 1st pair and 1st offspring breeds =5 pairs
4th month first pair and first two pairs of offsprings breeds =8 pairs
5th month 5+8=13 pairs
6th month 8+13=21
7th month 13+21=34
The sequence of 1 2 3 5 8 13 21 34 with successive ratios
3/2 5/3 8/5 13/8 21/13 …tends to the golden ratio of population growth in Fibonacci series.
17.A painting or a picture represents a three dimensional object on a two dimensional surface.It is the mathematics of perspective that give its depth .In a one-point perspective there is one set of parallel lines which meets a single vanishing point.This perspective is usefl for painting or drawinga corridor seen from one end. In a twopointed perspective there are two sets of parallels and hence two vanishing points eg a floor with square tiles .The tiled floor paintings of renaissance(literally rebirth)was giving an impression that distant objects were far away.A threepoint perspective has three vanishing points.That type of geometry is needed to draw objects in depth and that type of perspective is needed to appreciate it deeply.Universe and time has a four dimensional perspective and culture and human wisdom has a multidimensional perspective.
The solution of a quartic equation is containing a term of the 4th power.In a quadratic equation the highest power is X 2 and in cubic equation it is X 3.One step from the cubic,one reach the quartic and the X 4.If we collect everything on the left side of = sign the general quatic equation is
Ax4+Bx3+Cx2 +Dx+E= 0 (Ferrari 1522-1565)
18.The method of mathematical induction ( Francesco Maurolico 1494-1575)was for proof of theorems about whole numbers.If one can prove that n=1 and if youcan extend that proof from n to n+1 then you have proved it for all n.Though this logic was used by Indians, Arab and Greeks in Europe this was made known by Maurolico.
19.Galileo Galilei (1564-1642)
How a body falls under gravity? If it falls from rest the distance it falls vary with the square of time.When it falls from a height the speed is not uniform as it picks up speed while falling.First second it fall 5m ,first 2 sec 20 m ,in first 3 sec 45 m and so on.But no mention of mass of body is made here by him.The rate of fall and speed is unaffected by its mass.When a body is thrown its trajectory is parabola.The movement of a pendulum to and fro in th same time, regardless of its angle through which it swings.He supported Copernicus suncentered universe and had opposition from Catholic church .
The second law of Kepler that a planet sweeps out equal areas in equal times in an elliptical orbit around the sun was behind Galileos discovery.The third law of Kepler is that cube of distance of planet from sun is proportional to square of length of its year.Thus for earth with 365 days ,the distance is 150million KMs for him.The cube of 150 divided by square of 365 =cube of 108 divided by square of 225 .And this was for Venus,the nearest planet to earth with 225 days as year and 108 million Km as distance fromsun.That is if we calculate the year of a celestial body we can calculate its distance from sun as well as from the point of observation,the earth.
The important thing is that the years of earth,the years of venus,Saturn,Jupiter,mars and Mercury were known to Indians and the distance of earth to sun they called the Raahumaanm (Measurement of the Raahu or Ahorathra) and determined it not as 150 but as varying from 150-153 showing that they were more accurate in their calculation than Keplar.
In 1609 when Galilio visualized moon with his telescope ha wasinterested in the heights and depths that reveal an earthlike moon and in the play of light and shadow raher than mapping the moon and he was disproving the Aristotelian view that heaven was perfect .Moon had craters and peaks like earth and was imperfect like earth, he tried to prove.This he was quoting (and influenced by) Alexandrean philosopher John Philoponus,a Christian thinker who lived 1000 yrs before Galileo.He was challenging not only Aristotle but also church view that heavens are a better place . A perfect place.He said whatever is there is here also and both are alike and this was not against what the ancient Indians thought .The timescale only differed and all other things being similar for men,Pithru and deva etc.Both John and Galileo were considered heretics by church.
20.Logarithms as calculating aids to arithmetic was the contribution of Napier to western science.It reduce multiplication to addition and division to substraction.Such elaborate tables were prepared by astronomers of India for easy calculation of large numbers ,fractions with several decimal positions etc.Looking at such charts they could add logarithms and find out antilogarithms .Briggs published Napier logarithms and in it each number is converted to powers of 10 or dasamsa of Indians.That is logarithm of 100 is 2 as 10 to the power of 2 is 2.Logarithm of 1000 is 10 to the power of 3 and so on. So parardham or 10000000000000 is 10 to the power of 13 as the vedic rishi states.Logarithms of numbers in betweeninteger powers is also found.Many quantities are measured in logarithmic scales.Intensity of sound in decibels in terms of logarithm of air pressure is an example.
21 .The five platonic solids were all convex.With concave solids 4 more can be added making nine. These were added by Kepler mathematically .These regular pentagons,with dodecahedrons,and five sided pyramids on each face of it making small stellated dodecahedrons and a different great stellated dodecahedron. The small stellated dodecahedron was drawn on floor of St Marks cathedral in Venice.This was by Paolo Uccello in 1397-1475 one century before Kepler.Great stellated dodecahedron appeared in 1568 in a book of geometric drawings by Wenzel Jamintzer before Kepler was born.Later on Loius Poinsot described a great icosahedron.Kepler said solar system fitted precisely into a nest of platonic solids.Mercury fits into an octahedron .Venus into an icosahedron. His model of universe was based on this.Modelling universe on geometric shape of panchabhootha and building them as different models of vasthu is known in India and the cubic and spheroidal weights of Harappa were some units which were following the rules.They also modeled it on an algebraic model as equations and compared the two with the observations made by each generation of teachers as the chronologies of teachers indicate.
22.Before calculating machines of Pascal and Leibnitze.before Napier ever thought of a logarithmic table the human brains had made their brains into calculating instruments.Now we are familiar with computrs to do calculations for us.We have to just think of the history of mathematic calculations and computing that occurred for several millennia for us to find out such a technological device and that this technology was built on the brains of all those ancestors and teachers who thought and discovered old ideas again and again and did research on it.Every computer we use and every knowledge we possess we are indebted to known and unknown ancestors of human race on either side of the globe irrespective of race or class and we are the inheritors f all their wisdom.No race is superior to the other simply by technological tools alone unless supported by sound,logical thinking to back up their ideas .In other words human brain is always a step ahead than the most modern computer or tool it had invented.
Therefore analytic geometry is important since it establishes the relation between geometric curves and algebraic equations.Indian philosophers,samkhya physicians practicing ayurveda ,astronomers ,mathematicians were very systematic in this integration.In the west,Rene Descartes ,a philosopher,physicist,physiologist,mathematician developed analytical geometrical ideas systematically.Modern western philosophy begins with his cogito ergo sum or I think therefore I am.
The theory of probability that began with the gambling problem was solved by Fermat and Pascal .This question influenced Pascal to systematically approach the triangle of numbers of probability which existed millennia before his triangles appeared.In China it was Yanghui’s triangle.In Iran as Khayyam’s triangle and in Italy as Tartaglia’s triangle. In India it existed as prasthara of the chandas,of swara ,and as a part of easily finding out an arithematic or algebraic triangle of probabilities of prediction among Chathurangam players,business people,among astrologers etc and among grammarians and poets and thanthric and musicians to elaborate and teach their respective subjects.It had to be used as a probability distribution as in Ashtakavargaprasthara or in gayatry chandas prasthara etc. Thus it is a binomial distribution or a probability distribution for the number of successes in a series of experiments a scientist or a gambler performs.Pascal enlisted the theory of probability to support religious belief in God He said :-God is or is not.At the far end of an infinite distance ,a coin is being spun which will come down heads or tails .How will you wager. You must wager.It is not optional.There is an infinitely happy life to gain and a chance of gain agaist a finite number of chances of loss,and what you stake is finite.It is no use saying that whether we gain is uncertain ,while what we risk is certain.
That is reward for hope in God is infinite and worldly gains are finite .
23.Suppose we are taking the 15 Thithi of the moon from Prathama to panchadasi or 1 to 15. They are arranged in seven days of a week in groups of three .If we arrange them such that no shall walk together twice in 7 days (We know only three thithi can be there in an 24 hour day).Now give the 15 days each a letter from albhabet /or a name .Say A to O.The sequence will be
Sun Mon Tues Wednes Thurs Frid Saturd
afk abe bcf efi cek egm kmd
bgl cdg deh ghk dfl fhn lne
chm hil ijm lma gio ikb oah
din jkn klo noc hja jlc aci
ego mof nag bdj mnb oad fgj
The formula here is ½(15—1) =7
½(14)=7
14/2=7
14X 2=28
The 14 worlds and the 28 stars of India was trying to do this geometries in which there are finitely many points. 35 different arrangements are seen in the table the double of which is 70. Making such grids of magic squares in Leela or Gnan Choupad was prevalent in scholarly discourses and Ramanujan was an expert in it. Numbers in magical arrangement is the basis of many yanthras like Sriramachakra .The Panchadasi or Lopamudrasuthra of India is called Lossu (Lo for Lopamudra and su for suthra) in China.Thomas Kirkman (1806-1895) defined it as the schoolgirl problem where 15 schoolgirls were arranged in three for 7 days to school as said here. In this n is an odd multiple of three ,arranging them in triplets in m lists ,so that no pair of numbers occur more than once in same triple .
If m is ½(n-1) or less this is possible.That is both fraction and negative number is here.
24. George Boole (1815-1864) tried to set up a system to codify logical argument as a form of algebra.This is Boolean algebra .This uses symbolic language in designing of computer and relay circuits .It also explains our neurological circuits and cosmic circuits.When I tried to decipher Harappan script I thought this as a logical way of transmiting knowledge from BC 4000 or 5000 years to AD 2009 or beyond by our ancestral brain neurochannels as computerized codified systematic way of teaching exactly what they wanted to convey through the highly technical and complex subject Mathematics and musical notation.
25.Reimann hypothesis is about zeros of an infinite series.
The Reimann zeta function is defined by ,
ζ( s)= 1/1 s +½ s+ ⅓ s + ¼ s +
zeta (1) is infinite, ζ(2) =π2/6, ζ(4)=π4/90, ζ(6)=π6/945
whenever ζ(s)=0 the real part of s ,if +ve is ½.Riemann hypothesis is more important than Goldbach and Germats theorems as its solution give answers for many other unsolved problems.
26 Maxwell summarized electricity and magnetism .These fields are propgated through spacetime at speed of light. The speed was 186000 miles/sec ,the same as speed of light .The electromagnetic field and light wave was same .When Paithamahasidhantha takes 366 days (183 +- 3 ) to 180+- 3 ,this is indicated and the Raasimandala(mandal ais a field of energy) itself is the spacetime in which light travels creating such a field as it travels all around .Light consists of electromagnetic waves .Our nervous conduction also is in electromagnetic fields.
27 .The countability of fractions and uncountability of real numbers ( George Canter 1845-1918).the series is ----3----2—1---0,1,2,3 , and so on.Only if we include zero and the negative numbers we get the integers .So,that was why Harappan mathematics of India was using zero and fractions.They knew integers .
The integers are countable.For that one has to list it as 0 1 --1 2 --2 3 --3 and so on.That is why the Harappans gave fractions and negative numbers in their weights and measures.Every natural number is an integer 17 is 17/1
Two sets of natural numbers and integers as a pair each will be
. 1 2 3 4 5 6 7 ….
. 0 1 --1 2 --2 3 --3 …
The infinite list on top including all natural numbers and the bottom including all integers.The two lists are paired together so that as many numbers are in top list will be in bottom list too.Therefore bottom list of integers also is countable as natural numbers.But this is an infinity too and hence uncountable too.
If we take 6 as a number of columns horizontally and vertically ( as 6 seasons and 6 chakras of body from Moolaadhara to Agna) and count fractions in a systematically arranged pattern counting across the diagonal or karma (which also means the ears/sruthy in Sanskrit) the list will be in 36 columns with lot of repetitions .These are the countable +ve fractions with an infinity that is natural and repetitive and it is the basis of Melakartha raaga scale also.
1 2 3 4 5 6
1 1/1 2/1 3/1 4/1 5/1 6/1
2 ½ 2/2 3/2 4/2 5/2 6/2
3 1/3 2/3 3/3 4/3 5/3 6/3
4 1/4 2/4 3/4 4/4 5/4 6/4
5 1/5 2/5 3/5 4/5 5/5 6/5
6 1/6 2/6 3/6 4/6 5/6 6/6
The series here is 1/1 ½ 2/1 3/1 2/2 2/3 ¼ 2/3 3/2 4/1 and so on.
Or 1 ½ 2 3 1 0.66666 ¼ 0.6666 1.5
1 05 2 3 1 0.66 0. 25 0.66 1.5
Thus infinitely recurring and repeating natural and integer numbers are there and the Hrappans used it in daily life.When Cantor first recognized them and named these as infinity א0 he was called a charlatan,a renegade and a corruptor of youth and he died as a mental wreck (though he was not drowned as the Pythagoreans drowned their Guru) due indirect attacks and personality assassinations.Treating infinity as if it is a number was not accepted (1845-1918).But he had also said that the real numbers are ultimately uncountable too. There are different levels of infinity.Infinity of real numbers are greater infinity of whole numbers.
Natural numbers 1 2 3 4 countable
Integers and fractions also countable.
Real numbers are numbers with decimal expansion (on which Indians were experts).The expansion may either terminate or continue forever.one can list the countable numbers as below.
1st 3.294759
2nd 5.268370
3rd 8.371541
4th 0.387928
Take a diagonal from this decimal. Starting from 2 in the first number the diagonal is 2619 .Change each of these numbers by alternately adding and subtracting a number from them
2 to 3 ,6 to 5 , 1 to 2 ,9 to 8 .The new number 3528.This number cannot appear in the list .So real numbers are not countable.Any attempt to count a real number will leave one out.The infinity of real numbers is strictly greater than infinity of natural numbers.This is written as 2 to the power of א0.
To apply this to the squaring of circle, it is impossible to square a circle because it is not a solution to an algebraic equation and cannot be done in a Euclidean space.But it can be done in a non-Euclidean space which was the space of our Harappans/Indian rishis.pai is transcendental for them and this property of pai was known to west only with Ferdinand von Lindemann.(1852-1939).Euclid was dealing with geometry alone but the Harappana were doing something more than that with their mathematics.
Cantor hypothesized that infinity of natural numbers and of real numbers is the same so that there is no contradictin in it.Godel also said this is consistent and contradictory. This is the same infinity of Brahman and prakrithi –The ekam and anekam of ancient Indians .Iam finding a correlation coefficient between the western and eastern science here. When data seem to suggest a connection between two quantities,ideas,cultures is it a pure chance or is there something beyond chance there?In Medicine we try to find out a correlation between aetilogy of cancer and meat eating or tobacco smoking. In psychology we search for connection between extraversion and schizophrenia.In sociology researchers try to find out connection between social class and longevity and so on.The investigator collect and tabulate data and draw graphs to show relation between the two.Is there a definite pattern?.Is the pattern due to random fluctuations even when there is a definite pattern ?
That is why I am searching each and every part of our knowledge systems (Philosophy,medicine,biology,chemistry,physics,arts and science of ancient and new worlds and of India and Europe ,astrophysics and mathematics and languages of India with the history of anthropology and science and development of human consciousness and its development in relation to civilizations in each.The pattern I get, prove that it is not any random fluctuation but real and that India had been the Guru of the west right from the beginning .But this I do not apply to superiority of Indian race but to the presence of the rich agroeconomy,industry,trade and searoutes which they developed and this was at least from 7000 BC means the civilization existed even before that .).
There is a story of an argument about mathematics between the tortoise and Achilles. The argument goes on forever.Whatever proof the Achilles brings about deduction of a triangle with two sides which is isocles according to Achilles but has one side measuring 12.7 cms and the other is 5 inch long.But isocles triangle has to have two equal sides.So the tortoise traps Achilles into a infinite regress of implication.The argument goes on for ever in science in every era. Where does the formal logical deduction and the common sense reasoning begin and the other ends ?This puzzle was raised by Charles Lutwidge Dodgson ,author of An elementary treatise on determinants better known as Lewis Carroll (Author of Alice in wonderland).It is the slow moving tortoise on which this entire logically constructed universe moves so fast is the one who always wins in the end.Queen Victoria( 1819-1901 reigned 1837-1901) asked him to send his second book when she read Alice in wonderland but was disappointed to get An elementary treatise .
When we take larger numbers proportion of primes decrease in a logarithmic pattern
Number n Number of primes ח(n) Ratio of decrease ח(n)/n
10 4 0.4
100 25 0.25
1000 168 0.168
1000000 78498 0.078
1000000000 50 million 0.05
1000000000000 38 billion 0.038
Value of ח(n)1n n/n is 1.08. for n =1 million
1.04 for n=1 billion.
The ratio is therefore tending to one.Only in late 19th century this was proved independently by Hardamard(1865-1963)and Poussin(1866-1962).The proof of this theorem is very difficult and advanced and brings in many notions unconnected with primes and logarithms.Erdos (1913-19996)found an elemenry proof for it.In it the proof is still difficult but it does not involve advanced mathematics.This prime number theorem which is difficult to prove was known to Harappans as their weights and measures and other artifacts and culture show.