PUZZLE OF ANCIENT MATHEMATICS II: TRUE VEDIC MATHEMATICS

S›ulbasutras I: India and Pythagorean Greece

As every schoolchild knows (or should know), the most important theorem in

geometry is the Theorem of Pythagoras. And yet this theorem is

as much a result in algebra and arithmetic as geometry.

Remarkably, there is no evidence whatsoever that either the

statement or the proof of the theorem was known to the man to

whom this seminal result is credited. The earliest statement of the

theorem, as well as traces of its proof are found in the sulbasμutra

ofBaudhayana. No less interestingly, in his work we find also the

germination of the theorem from its religious-ritualistic roots.

The fact that the ancient Indians knew the so-called Theorem

of Pythagoras was recognized quite early by several European

scholars. Among the first was Thibaut, who left the impression,

even if he didnit explicitly state it, that in geometry the

Pythagoreans were pupils of the Indians. A majority of scholars,

however, did not like the idea and tried to refute it: Their

erefutations, as Seidenberg noted, were no more than haughty

dismissals. An effort was mounted almost at once to manipulate

the chronology so that all Indian scientific knowledge could be

derived from Alexandrian Greece, a campaign that to some extent

still persists. Weber’s assertion regarding the indebtedness of the

›ulbasto Hero of Alexandria, as well as Keith‘s chronological

scheme both now discredited by science should be seen as

parts of that effort. This has continued in the work of recent scholars like David Pingree and his underlings like Kim Plofker.

The moving impulse in all this was the romantic notion of

something they called the Greek miracle. This was a product of

the Romantic Age that exercised a powerful hold on the collective

psyche of nineteenth century Europe. It was a fascination with

everything Greek, and a belief in Greece as the source of all

knowledge. Their heroic resistance to foreign domination, and

their only recent liberation from centuries long subjugation by the

Turks, had also served to burnish the ideal. Poet Byron gave

his life fighting for it. Heinrich Schliemann spent most of his life

and fortune looking for the Homeric Troy and the grave of

Agamemnon of Mycenae. What the idea then meant to the

European intellectual has probably been best expressed by Edith

Hamilton in her admirable book The Greek Way:

“Something had awakened in the minds and spirits of the men there

which was to so influence the world that the slow passage of long

time, of century upon century and the shattering changes they

brought, would be powerless to wear away that deep impress.

Athens had entered upon her brief and magnificent flowering of

genius which so molded the world of mind and of spirit that our

own mind and spirit today are different. We think and feel

differently today because of what a little Greek town did during a

century or two, twenty four centuries ago. … In that black and fierce

world a little center of white hot spiritual energy was at work. A

new civilization had arisen in Athens, unlike all that had gone

before.” (Hamilton 1930: pp 3-4)

As seen by her the world before the rise of Athens was all

engulfed in darkness and sunk in barbarism. This is not ancient

history but modern romance; Hamilton was only describing what

she and others like her wished to believe, and not the world as it

really was. Remarkable indeed as the achievements of classical

Greece are, it was not the seed from which all knowledge sprang.

Also, only a romantic totally ignorant of science could believe

that all the achievements attributed to the Greeks could have

sprung and blossomed forth within a span of a century or two,

with no one before them to prepare the soil. But arch romantics

like Hamilton are not worried about such mundane things.

Though this myth died hard, it had to give way before the

reality of evidence and the inexorable logic of mathematics.

Thibaut was one of the first to make a serious study of the ›ulbas,

and he, as previously noted, left unsaid the unmistakable

impression that Greek geometry was a derivative of the ›ulba.

The factor that seems to have influenced Thibaut most was what

he called the imighty sway of religioni, the vivid trail of the

evolution of mathematical thought from religious and ritualistic

roots that he found in the ›ulbasμutras. As we shall see later, this

trail of evolution of the so-called Theorem of Pythagoras from its

religious roots is still clearly visible in the ›ulba of Baudhayana.

No such roots are to be found anywhere in Greece. And this

has led eminent scholars like Hankel (1874) and Junge (1907; see

also Datta 1932) to deny Pythagoras (see below) credit for the discovery, let

alone the proof, of the famous theorem that goes by his name.

The tradition which attributes the theorem to him seems to have begun

some five centuries after his time. Pythagoras (c. 580 BCE – c.

500 BCE) was an Ionian from Samoa, who founded a religious-philosophic

brotherhood that also cultivated an interest in

mathematics. This could account for the brotherhood’s interest in

the›ulbasalso religious-philosophic in nature pointing to

the line through which Indian geometry might have reached the

Greeks. Unlike the ›ulbas, none of the writings of Pythagoras

have survived. His reputation as a mathematician rests almost

entirely upon his later followers’ habit of indiscriminately

invoking him as the authority in advancing their own ideas. The

Encyclopaedia Britannica records:

Other discoveries often attributed to him (e.g., the incommensurability

of the side and the diagonal of a square, and the

Pythagorean theorem for right triangles) were probably developed

only later by the Pythagorean school. More probably, the bulk of

the intellectual tradition originating with Pythagoras himself

belongs to mystical wisdom rather than scientific scholarship.

(1984, 9: p 827)

All this was known by the end of the last century. It was also

evident by then that the Kalpasμutraliterature (of which the

›ulbasare part) was widely known as Ketubha(Sanskrit

Kaitabha) in the early Buddhist literature (c. 500 BCE), making

it impossible for the ›ulbasto be a derivative of Alexandrian

knowledge. And yet this attachment to the Greek miracle was

such that even so eminent a historian of science as Cantor

iconfesses himself not charmed with the ideai that in geometry,

the Pythagoreans were pupils of the Indians! To his credit,

however, Cantor seems to have later dropped this irrational

objection. This was the situation well into the twentieth century,

reinforced by the reality that most Indologists were (and are)

ignorant of science.

Greece to Babylonia

Then in 1928, the great historian of science Otto Neugebauer

found that the so-called Pythagorean triples, i.e., whole

numbers of the form (3, 4, 5) which satisfy the equation a2 + b2 =

c2 were known to the Old-Babylonians before 1700 BCE, who

were also seen to have been no mean arithmeticians. Since they

could not have derived their knowledge from the Greeks,

Neugebauer rather hastily concluded that the Greeks in turn had

borrowed the Theorem of Pythagoras from the Babylonians. He

wrote in 1937:

What is called Pythagorean in the Greek tradition had better be

called Babylonian. (Neugebauer 1945: p 41)

And that seemed to settle the problem of Greek indebtedness

to India for good. But the problem was only beginning, for a

closer examination of the two mathematics shows that it is not

possible to derive Pythagorean Greek geometry from the entirely

computational arithmetic of Old-Babylonia. Thus the roots of

geometry are nowhere to be found in Babylonia. Then there is the

problem of time o a vast gap of some 1400 years o after the

Old-Babylonia of 1700 BCE when we have no mathematical

records at all from Babylonia, until we start picking up the

threads again during the Seleucid period around 300 BCE. The

problem however is not primarily one of time, but of content and

spirit. Greek and Old-Babylonian mathematics are as

fundamental1y mutually incommensurable as the circle and the

square. But the picture changes at once if we include the ›ulba. As

Return to Vedic mathematcs

Seidenberg noted:

… if one includes the Vedic mathematics, one will get quite a

different perspective on ancient mathematics.

The main issue is the origin of geometric algebra. The Sulvasutras

have geometric algebra … Greece and India have a common

heritage that cannot have derived from Old-Babylonia, i.e., Old-

Babylonia of about 1700 BCE. (Seidenberg 1978: p 318)

What then is there in the ›ulbasthat made scholars like

Cantor, Thibaut, Burk and Seidenberg look to them for the roots

of Greek geometry?

It is connected with the directions found in

he Kalpasμutras for the design of altars for various sacrificial rites.

In the words of Datta, the foremost modern student of the ›ulbas:

The ›ulbas, or as they are commonly known at present amongst

oriental scholars, the ›ulba-sutras, are manuals for the construction

of altars which are necessary in connexion with the sacrifices of the

Vedic Hindus. They are sections of the Kalpa-sutras, more

particularly of the Shrauta-sutras, which form one of the six

Vedangas (or Members of the Veda) and deal specially with

rituals. Each Shrauta-sutra seems to have had its own ›ulbasection.

So there were, very likely, several such works in ancient times.

At present we know, however, of only seven ›ulba-sutras … (Datta

1993: 1)

Of these seven, however, only three, those of Baudhayana,

Apastamba and Katyayana are of first importance. Even among

these, the ›ulba of Baudhayana is the largest and very probably

also the oldest. The other four ›ulbas, namely, the Manava, the

Maitrayaƒa, the Varaha and the Vidula are of lesser importance.

As regards their connection with the Vedas, the ›ulbasof

Baudhayana and Apastamba belong to the Taittir∂ya Samhita or

the so-called Black Yajurveda; that of Katyayana to the

Vajasaneya a recension or the so-called White Yajurveda.This also

suggests that both Baudhayana and Apastamba were in all

likelihood southerners, for the Taittir∂yawas the recension of the

Yajurvedaused almost exclusively in the south. Since the ›ulbas

are part of Kalpasμutraswhich are works concerned primarily

with ritual, it is entirely natural that they should be connected

with the Yajurveda. Unlike the Rigveda, the Yajurvedais

primarily a book of sacrifice and ritual.

As Datta observes:

It was perhaps primarily in connexion with the construction of

sacrificial altars of proper size and shape that the problems of

geometry and also of arithmetic and algebra presented themselves,

and were studied in ancient India, just as the study of astronomy is

known to have begun and developed out of the necessity of fixing

the proper time for the sacrifice. At any rate from the ›ulba-sutras,

we get a glimpse of the knowledge of geometry that the Vedic

Hindus had. Incidentally, they furnish us with a few other subjects

of much mathematical interest. (Datta 1993: 2)

We can thus see that the origins of mathematics o of

arithmetic as well as geometry can be traced to the ›ulbas.

Seidenberg calls these the two great traditions of mathematics.

And as he perceptively noted, the first of these, namely the

arithmetic or the computational, led to the mathematics of Old-

Babylonia and Egypt; while the other, involving geometric

constructions and geometric algebra, and the concept of the proof

led to Greek mathematics.

Incidentally, scholars are entirely

wrong in claiming that there are no proofs before Euclid; indeed

proofs are found in the ›ulba. After reproducing a proof relating

to the area of a trapezoid given by Apastamba, Seidenberg (below)

observed:

Many writers who refer to the Sulvasutras say that there are no

proofs there. We can only suppose that these writers have not

bothered to examine the work. (Seidenberg 1962)

Interesting as it is to the mathematician, the origins of

mathematical proof need not detain us here. Also, there are

mathematical examples that are given purely as a matter of

illustration or erudition, having no immediate applications

whatsoever. Proofs however are stated simply as a matter of

course, and not endowed with the formalisms, later to attain

perfection at the hands of Euclid. The ›ulbastherefore are much

more than an altar builderis manual; they are in fact

mathematical texts that display considerable virtuosity on the part

of its authors, and the oldest comprehensive works of their kind

found anywhere in the world. The Baudhayana ›ulbain

particular is a work of great beauty and perfection.

To return to the Theorem of Pythagoras, in its geometric form

it is a theorem about triangles and rectangles. The Greeks state it

as a theorem about triangles while the Indians know it as a

theorem about rectangles. But the difference is trivial and

mathematically the two are equivalent. (The earliest Babylonian

and Chinese versions also speak of the width and the diagonal,

which bespeaks Indian influence.)

Indian texts, i.e., the ›ulbasof Baudhayana, Apastamba and

Katyayana, state two cases: for the square as well as the case of

the general rectangle (of unequal sides). But the order in which

they are stated in the three ›ulbasis different, and this fact is of

prime historical importance. Baudhayana states the theorem for

the square first, and then its general rectangular form; Apastamba

andKatyayana follow the reverse course. They both state the

general case first and then note the special case as a corollary.

This indicates that by the time of the latter two, the

power and importance of the general form of the theorem was

fully recognized, while Baudhayana was still treating the results

in the order in which they were discovered. The general form of

the theorem is stated by Baudhayana as follows:

The diagonal of a rectangle produces both (areas) which its length

and breadth produce separately. (Baudhayana ›ulbasμutra1.48;

cited in Datta 1932: p 104)

He however first derives the theorem for the special case of the

square:

The diagonal of a square produces an area twice as much as itself.

(Baudhayana ›ulbasμutra1.48; cited in Datta 1932: p 104-5)

As previously observed, Apastamba and Katyayana reverse

the course. Specifically, Apastamba and Katyayana state the

* The earliest Old-Babylonian version also speaks of the ewidthi and the

ediagonali. For the Chinese also it is a theorem on rectangles. The ancient Choupei

states: iMake the breadth 3, the length 4. The king-yu, that is, the way that

joins the corners is 5. Take the halves of the rectangle around the outside. There

will be (left) a kuu.i

general case first (Apastamba ›ulbasμutra1.4; Katyayana

›ulbasμutra2.11), and state the result for the square as a corollary

(Apastamba ›ulbasμutra1.5; Katyayana ›ulbasμutra2.12).

No less important is the fact that the roots of its discovery

are still preserved in the form of the catura‹ra-‹yenacit, one of

the oldest known of the Vedic altars (Datta, 1934). Figure 1

below reproduces part of this ancient altar. This makes it selfevident

howBaudhayana (or someone before him) must have

seen the theorem for the special case of the square.

When we examine the figure, it becomes apparent that the

square on the diagonal is composed of four triangles, while that

on a side has only two. Baudhayana does not claim to have

discovered the result, but he may well have. In stating it he does

not use the term iti ‹rμuyate, meaning eso we heari. As a Vedic

priest it was not for him much more than a useful detail in the

construction of altars. In any event, the spirit of the age did not

place the same high value on the priority of publications that we

now do. But he certainly merits recognition as the first known of

theworldis great mathematicians.

The really interesting question then is to know if knowledge

of the theorem can be traced to the Rigvedaitself. Seidenberg

sought to find some evidence for it in Rigvedic hymn X.90, the

famousPurusaSμukta. We cannot agree with his reading of it; nor

can we find any evidence that the Rigvedaknows the Theorem of

Pythagoras, though knowledge of the other great problem of

geometric algebra o the canonical circling of the square o is

implied, if only indirectly. The Rigvedaknows the spoked wheel,

and there is an ingenious construction of it given in the ›ulbaof

Baudhayana based on successive circling of concentric squares.

One is therefore led to conclude that the Rigvedatoo knows it.

Harappan archaeology also establishes that the Pythagorean

theorem was known and applied as early as 3000 BCE. Several

massive structures and lengthy streets are laid out using perfect

perpendicular and parallel lines. One may draw good parallel and

perpendicular lines that are a few feet long through visual

inspection and experience. But planning and building structures

that are hundreds of meters long demands knowledge of

mathematics.

Thus, objective needs of Harappan architecture

leave no doubt at all that geometry, including the so-called

Pythagorean Theorem was known to Harappan architects.

All evidence therefore points to the Theorem of Pythagoras

as having been known in the Sutra literature going back almost

to 3000 BCE, as we shall see later. Datta has noted that traces of

Indian terminology are to be found in the works of Greeks like

Democritos (c. 440 BCE). Further, the gnomon also links

Pythagorean geometry to India. The Greek geometry thus greatly

resembles the Indian both in spirit and content. Therefore until a

convincing alternative source is found, we are justified in holding

Greek geometry to be a derivative of the ›ulba. (The gnomon is

an L-shaped figure obtained by removing a square from a larger

square having a common vortex.)

There is ample evidence showing that India and Greece had

long been in contact o going back well before the coming of

Alexander. Thus, the Greeks would have been familiar with the

Vedic mathematics, not the ›ulbas themselves perhaps which

were quite ancient by then, but through various commentaries

that were then current.

culminating in the work of Seidenberg, the layer of

theSμutra literature containing the works of Baudhayana,

A‹valayana, Apastamba and Katyayana, must be dated to no later

than 2100 BCE. And this is an absolute lower limit that derives

from the fact that the mathematics of Old-Babylonia (before 1700

BCE) and the Egyptian Middle Kingdom (2050 to 1800 BCE)

both derive from the ›ulbas.

Nor does the story end here. It should further be noted that

considerable time must have elapsed before the religious-ritual

form of the geometric algebra found in the ›ulbaof Baudhayana

et. al. evolved into the purely secular arithmetic methods found in

the Old-Babylonia and Egypt. Recognizing this fact, Seidenberg

required a date ifar back of 1700 BCE for the mathematics of

the›ulba. This mathematically determined date is supported also

by ancient astronomical records.

But returning to ancient mathematics, once we recognize the

›ulbasas the source of both Egyptian and Old-Babylonian

mathematics, it then becomes possible to fill otherwise

inexplicable gaps found in them. Specifically, the origins of

several significant results from Egyptian and Old-Babylonian

records can now be traced to the ›ulbas.

One of the most

interesting of these is the use of the so-called unit fractions, i.e.,

approximations to irrational numbers with terms of the form 1/n,

where n is a whole number. The following unit fraction

approximation appears frequently (Seidenberg, 1978; Datta 1934).

√2= 1 + 1/3 + 1/(3.4) – 1/(3.4.34)

Such unit fractions in Egyptian (and Babylonian) mathematics

are justly famous. They appear also in the ›ulbas. From this

Cantor stated that the Indians got their mathematics from the

Egyptians. (Remember others claimed they got their geometry

and astronomy from the Greeks, following Alexander’s invasion.

There is of course no record of any Egyptian Pharaoh invading

India.) But this claim is contradicted by the fact its origins are

nowhere to be found in Egypt. Where then did they come from?

The roots again are found only in the ›ulbas, where this unit

fraction approximation arises naturally as the outgrowth of the

problem of squaring the circle. Like the Theorem of Pythagoras,

the square-circle equivalence (approximate) is one of the seminal

problems of ancient mathematics. Further, the value of pi=

3.16049 used by Ahmes of Egypt (c. 1550 BCE) is exactly the same

as the one given in the relatively late Manava ›ulbasμutra, and,

what is more, is obtained in precisely the same fashion, i.e., as

4.(8/9) 2. That is to say, the Manava ›ulbaand Ahmes of Egypt

both use the exact same approximation (in modern notation, Datta

1932):

Pi = 3.16049 = 4.(8/9)2

With two specific examples agreeing down to the most

minute detail, the probability of its being a coincidence can be

dismissed. In the problem of the circle-square equivalence, we

see therefore the roots for going over into arithmetic methods.

Thus the origins even of arithmetic methods, so characteristic of

the mathematics of Babylon and Egypt, are again traceable to the

›ulbas.We can therefore conclude that attempts at squaring the

circle led to the arithmetic methods that lie at the heart of Old-

Babylonian mathematics of 1700 BCE (and Egypt four centuries

earlier). It is therefore beyond question that their mathematics is

also a derivative of the ›ulba. Thus Vedic mathematics, or, more

exactly, the mathematics of the ›ulbasμutrasmust have been

known in India no later than 2100 BCE. Also as already noted,

many of the structures and cities of the Indus Valley presuppose

considerable knowledge of geometry nearly a thousand years

before Old-Babylonia and the Egyptian Middle Kingdom.

A question that arises is the following: Can we reasonably

conclude that Vedic mathematics was the source of all the

technical knowledge of the ancient world, from Egypt to China?

The Egyptian Middle Kingdom certainly owed its mathematics to

theSμulbas, as did Old-Babylonia. But how about Sumeria or

early Egypt? Or China? Did the Egyptians seek the help of some

Vedic priests in their constructions?

To these questions, the answer at this time has to be a

qualified affirmative. There is at least one other connection, quite

apart from unit fractions and the value of pi. Trapezoidal figures

shaped like Vedic altars are found on Egyptian monuments of all

times. Two of them, taken from Histoire de liart Egyptian

diapres le monuments are reproduced in Figure 2. (They are

originally given by Cantor. References are to the book by Rajaram and Frawley.) As Seidenberg observes:

It is not merely the trapezoidal shapes which impress us, but

their subdivisions. The subdivision of the first figure occurs

in the Sulvasutras, and the second calls to mind the

computation of the area [and its proof] in the Apastamba

Sulvasutra.If these figures occurred on Indian monuments,

we could understand the Indian interest in them: all the hopes

of the Indian for health and wealth were tied up in the

trapezoid. (Seidenberg 1962: p.519)

Trapezoids are common in Vedic altars. But how are we to

account for their presence on Egyptian monuments? Particularly

when they would have been quite cumbersome to construct and

draw using the earithmetici techniques so favored by the

Egyptians (and the Old-Babylonians)? Now we know that durinng

Astronomy, Alpha Draconis

Astronomical references in the work of Ashvalayana (not

Discussed here) establishes this date far back of 1700 BCE to

be around 3100 to 2600 BCE o the time when the star

Alpha Draconis (also called Thuban in the constellation Dragon)

was the pole star, a fact noted in the Satapatha Brahmana also.

This receives additional support from other astronomical

observations in the ancient literature.

This is also in accord withIndian tradition which places Ashvalayana about five generations

removed from the Mahabharata War c. 3100 BCE). Baudhayana also belongs

to the same period, perhaps a generation earlier. Further, since the

Sutra literature presupposes the existence of the Vedas, it follows

that the four Vedas must have been in existence by 3000 BCE.

Thus, the tradition of Vyasa as the editor of the four Vedas, as

well as his date of the Mahabharata War (c.3102 BCE) are in

agreement with ancient mathematics and astronomical data.

Further, we have also found connections between the so-called

Step Pyramid or the mastababuilt c. 2650 BCE by Djoser

(c. 2686 to c. 2613 BCE) and the smashana-cit altar described by

Baudhayana. This Step Pyramid was the forerunner of all the

future pyramids of Egypt. The smashana-citaltar (i.e., cemetery

shaped altar), as its name itself clearly indicates was connected

with Vedic funerary rituals. Since all Egyptian pyramids were

erected to serve as mausoleums, the connection is not only

mathematical but also of religion and ritual.

            These issues are discussed in more fully in the book Vedic Aryans and the Origins of Civilization by Rajaram and Frawley, along with relevant illustrations.

S›ulbasutras I: India and Pythagorean Greece

As every schoolchild knows, the most important theorem in

geometry is the Theorem of Pythagoras. And yet this theorem is

as much a result in algebra and arithmetic as geometry.

Remarkably, there is no evidence whatsoever that either the

statement or the proof of the theorem was known to the man to

whom this seminal result is credited. The earliest statement of the

theorem, as well as traces of its proof are found in the sulbasμutra

ofBaudhayana. No less interestingly, in his work we find also the

germination of the theorem from its religious-ritualistic roots.

The fact that the ancient Indians knew the so-called Theorem

of Pythagoras was recognized quite early by several European

scholars. Among the first was Thibaut, who left the impression,

even if he didnit explicitly state it, that in geometry the

Pythagoreans were pupils of the Indians. A majority of scholars,

however, did not like the idea and tried to refute it: Their

erefutations, as Seidenberg noted, were no more than haughty

dismissals. An effort was mounted almost at once to manipulate

the chronology so that all Indian scientific knowledge could be

derived from Alexandrian Greece, a campaign that to some extent

still persists. Weber’s assertion regarding the indebtedness of the

›ulbasto Hero of Alexandria, as well as Keith‘s chronological

scheme both now discredited by science should be seen as

parts of that effort. This has continued in the work of recent scholars like David Pingree and his underlings like Kim Plofker.

The moving impulse in all this was the romantic notion of

something they called the Greek miracle. This was a product of

the Romantic Age that exercised a powerful hold on the collective

psyche of nineteenth century Europe. It was a fascination with

everything Greek, and a belief in Greece as the source of all

knowledge. Their heroic resistance to foreign domination, and

their only recent liberation from centuries long subjugation by the

Turks, had also served to burnish the ideal. Poet Byron gave

his life fighting for it. Heinrich Schliemann spent most of his life

and fortune looking for the Homeric Troy and the grave of

Agamemnon of Mycenae. What the idea then meant to the

European intellectual has probably been best expressed by Edith

Hamilton in her admirable book The Greek Way:

“Something had awakened in the minds and spirits of the men there

which was to so influence the world that the slow passage of long

time, of century upon century and the shattering changes they

brought, would be powerless to wear away that deep impress.

Athens had entered upon her brief and magnificent flowering of

genius which so molded the world of mind and of spirit that our

own mind and spirit today are different. We think and feel

differently today because of what a little Greek town did during a

century or two, twenty four centuries ago. … In that black and fierce

world a little center of white hot spiritual energy was at work. A

new civilization had arisen in Athens, unlike all that had gone

before.” (Hamilton 1930: pp 3-4)

As seen by her the world before the rise of Athens was all

engulfed in darkness and sunk in barbarism. This is not ancient

history but modern romance; Hamilton was only describing what

she and others like her wished to believe, and not the world as it

really was. Remarkable indeed as the achievements of classical

Greece are, it was not the seed from which all knowledge sprang.

Also, only a romantic totally ignorant of science could believe

that all the achievements attributed to the Greeks could have

sprung and blossomed forth within a span of a century or two,

with no one before them to prepare the soil. But arch romantics

like Hamilton are not worried about such mundane things.

Though this myth died hard, it had to give way before the

reality of evidence and the inexorable logic of mathematics.

Thibaut was one of the first to make a serious study of the ›ulbas,

and he, as previously noted, left unsaid the unmistakable

impression that Greek geometry was a derivative of the ›ulba.

The factor that seems to have influenced Thibaut most was what

he called the imighty sway of religioni, the vivid trail of the

evolution of mathematical thought from religious and ritualistic

roots that he found in the ›ulbasμutras. As we shall see later, this

trail of evolution of the so-called Theorem of Pythagoras from its

religious roots is still clearly visible in the ›ulba of Baudhayana.

No such roots are to be found anywhere in Greece. And this

has led eminent scholars like Hankel (1874) and Junge (1907; see

alsoDatta 1932) to deny Pythagoras credit for the discovery, let

alone the proof, of the famous theorem that goes by his name.

The tradition which attributes the theorem to him seems to have begun

some five centuries after his time. Pythagoras (c. 580 BCE – c.

500 BCE) was an Ionian from Samoa, who founded a religious-philosophic

brotherhood that also cultivated an interest in

mathematics. This could account for the brotherhood’s interest in

the›ulbasalso religious-philosophic in nature pointing to

the line through which Indian geometry might have reached the

Greeks. Unlike the ›ulbas, none of the writings of Pythagoras

have survived. His reputation as a mathematician rests almost

entirely upon his later followers’ habit of indiscriminately

invoking him as the authority in advancing their own ideas. The

Encyclopaedia Britannica records:

Other discoveries often attributed to him (e.g., the incommensurability

of the side and the diagonal of a square, and the

Pythagorean theorem for right triangles) were probably developed

only later by the Pythagorean school. More probably, the bulk of

the intellectual tradition originating with Pythagoras himself

belongs to mystical wisdom rather than scientific scholarship.

(1984, 9: p 827)

All this was known by the end of the last century. It was also

evident by then that the Kalpasμutraliterature (of which the

›ulbasare part) was widely known as Ketubha(Sanskrit

Kaitabha) in the early Buddhist literature (c. 500 BCE), making

it impossible for the ›ulbasto be a derivative of Alexandrian

knowledge. And yet this attachment to the Greek miracle was

such that even so eminent a historian of science as Cantor

iconfesses himself not charmed with the ideai that in geometry,

the Pythagoreans were pupils of the Indians! To his credit,

however, Cantor seems to have later dropped this irrational

objection. This was the situation well into the twentieth century,

reinforced by the reality that most Indologists were (and are)

ignorant of science.

Greece to Babylonia

Then in 1928, the great historian of science Otto Neugebauer

found that the so-called Pythagorean triples, i.e., whole

numbers of the form (3, 4, 5) which satisfy the equation a2 + b2 =

c2 were known to the Old-Babylonians before 1700 BCE, who

were also seen to have been no mean arithmeticians. Since they

could not have derived their knowledge from the Greeks,

Neugebauer rather hastily concluded that the Greeks in turn had

borrowed the Theorem of Pythagoras from the Babylonians. He

wrote in 1937:

What is called Pythagorean in the Greek tradition had better be

called Babylonian. (Neugebauer 1945: p 41)

And that seemed to settle the problem of Greek indebtedness

to India for good. But the problem was only beginning, for a

closer examination of the two mathematics shows that it is not

possible to derive Pythagorean Greek geometry from the entirely

computational arithmetic of Old-Babylonia. Thus the roots of

geometry are nowhere to be found in Babylonia. Then there is the

problem of time o a vast gap of some 1400 years o after the

Old-Babylonia of 1700 BCE when we have no mathematical

records at all from Babylonia, until we start picking up the

threads again during the Seleucid period around 300 BCE. The

problem however is not primarily one of time, but of content and

spirit. Greek and Old-Babylonian mathematics are as

fundamental1y mutually incommensurable as the circle and the

square. But the picture changes at once if we include the ›ulba. As

Return to Vedic mathematcs

Seidenberg noted:

… if one includes the Vedic mathematics, one will get quite a

different perspective on ancient mathematics.

The main issue is the origin of geometric algebra. The Sulvasutras

have geometric algebra … Greece and India have a common

heritage that cannot have derived from Old-Babylonia, i.e., Old-

Babylonia of about 1700 BCE. (Seidenberg 1978: p 318)

What then is there in the ›ulbasthat made scholars like

Cantor, Thibaut, Burk and Seidenberg look to them for the roots

of Greek geometry?

It is connected with the directions found in

T

the Kalpasμutras for the design of altars for various sacrificial rites.

In the words of Datta, the foremost modern student of the ›ulbas:

The ›ulbas, or as they are commonly known at present amongst

oriental scholars, the ›ulba-sutras, are manuals for the construction

of altars which are necessary in connexion with the sacrifices of the

Vedic Hindus. They are sections of the Kalpa-sutras, more

particularly of the Shrauta-sutras, which form one of the six

Vedangas (or Members of the Veda) and deal specially with

rituals. Each Shrauta-sutra seems to have had its own ›ulbasection.

So there were, very likely, several such works in ancient times.

At present we know, however, of only seven ›ulba-sutras … (Datta

1993: 1)

Of these seven, however, only three, those of Baudhayana,

Apastamba and Katyayana are of first importance. Even among

these, the ›ulba of Baudhayana is the largest and very probably

also the oldest. The other four ›ulbas, namely, the Manava, the

Maitrayaƒa, the Varaha and the Vidula are of lesser importance.

As regards their connection with the Vedas, the ›ulbasof

Baudhayana and Apastamba belong to the Taittir∂yaSamhitaor

the so-called Black Yajurveda; that of Katyayana to the

Vajasaneyarecension or the so-called White Yajurveda.This also

suggests that both Baudhayana and Apastamba were in all

likelihood southerners, for the Taittir∂yawas the recension of the

Yajurvedaused almost exclusively in the south. Since the ›ulbas

are part of Kalpasμutraswhich are works concerned primarily

with ritual, it is entirely natural that they should be connected

with the Yajurveda. Unlike the Rigveda, the Yajurvedais

primarily a book of sacrifice and ritual. As Datta observes:

It was perhaps primarily in connexion with the construction of

sacrificial altars of proper size and shape that the problems of

geometry and also of arithmetic and algebra presented themselves,

and were studied in ancient India, just as the study of astronomy is

known to have begun and developed out of the necessity of fixing

the proper time for the sacrifice. At any rate from the ›ulba-sutras,

we get a glimpse of the knowledge of geometry that the Vedic

Bhibutibhushan Datta, the foremost expert on the Sulbas.

Hindus had. Incidentally, they furnish us with a few other subjects

of much mathematical interest. (Datta 1993: 2)

We can thus see that the origins of mathematics o of

arithmetic as well as geometry can be traced to the ›ulbas.

Seidenberg calls these the two great traditions of mathematics.

And as he perceptively noted, the first of these, namely the

arithmetic or the computational, led to the mathematics of Old-

Babylonia and Egypt; while the other, involving geometric

constructions and geometric algebra, and the concept of the proof

led to Greek mathematics.

Incidentally, scholars are entirely

wrong in claiming that there are no proofs before Euclid; indeed

proofs are found in the ›ulba. After reproducing a proof relating

to the area of a trapezoid given by Apastamba, Seidenberg

commented:

Many writers who refer to the Sulvasutras say that there are no

proofs there. We can only suppose that these writers have not

bothered to examine the work. (Seidenberg 1962)

Interesting as it is to the mathematician, the origins of

mathematical proof need not detain us here. Also, there are

mathematical examples that are given purely as a matter of

illustration or erudition, having no immediate applications

whatsoever. Proofs however are stated simply as a matter of

course, and not endowed with the formalisms, later to attain

perfection at the hands of Euclid. The ›ulbastherefore are much

more than an altar builderis manual; they are in fact

mathematical texts that display considerable virtuosity on the part

of its authors, and the oldest comprehensive works of their kind

found anywhere in the world. The Baudhayana ›ulbain

particular is a work of great beauty and perfection.

To return to the Theorem of Pythagoras, in its geometric form

it is a theorem about triangles and rectangles. The Greeks state it

as a theorem about triangles while the Indians know it as a

theorem about rectangles. But the difference is trivial and

mathematically the two are equivalent. (The earliest Babylonian

and Chinese versions also speak of the width and the diagonal,

which bespeaks Indian influence.)

Indian texts, i.e., the ›ulbasof Baudhayana, Apastamba and

Katyayana, state two cases: for the square as well as the case of

the general rectangle (of unequal sides). But the order in which

they are stated in the three ›ulbasis different, and this fact is of

prime historical importance. Baudhayana states the theorem for

the square first, and then its general rectangular form; Apastamba

andKatyayana follow the reverse course. They both state the

general case first and then note the special case as a corollary.

This indicates that by the time of the latter two, the

power and importance of the general form of the theorem was

fully recognized, while Baudhayana was still treating the results

in the order in which they were discovered. The general form of

the theorem is stated by Baudhayana as follows:

The diagonal of a rectangle produces both (areas) which its length

and breadth produce separately. (Baudhayana ›ulbasμutra1.48;

cited in Datta 1932: p 104)

He however first derives the theorem for the special case of the

square:

The diagonal of a square produces an area twice as much as itself.

(Baudhayana ›ulbasμutra1.48; cited in Datta 1932: p 104-5)

As previously observed, Apastamba and Katyayana reverse

the course. Specifically, Apastamba and Katyayana state the

* The earliest Old-Babylonian version also speaks of the ewidthi and the

ediagonali. For the Chinese also it is a theorem on rectangles. The ancient Choupei

states: iMake the breadth 3, the length 4. The king-yu, that is, the way that

joins the corners is 5. Take the halves of the rectangle around the outside. There

will be (left) a kuu.i

general case first (Apastamba ›ulbasμutra1.4; Katyayana

›ulbasμutra2.11), and state the result for the square as a corollary

(Apastamba ›ulbasμutra1.5; Katyayana ›ulbasμutra2.12).

No less important is the fact that the roots of its discovery

are still preserved in the form of the caturasra-‹yenacit, one of

the oldest known of the Vedic altars (Datta, 1934). Figure 1

below reproduces part of this ancient altar. This makes it selfevident

howBaudhayana (or someone before him) must have

seen the theorem for the special case of the square.

When we examine the figure, it becomes apparent that the

square on the diagonal is composed of four triangles, while that

on a side has only two. Baudhayana does not claim to have

discovered the result, but he may well have. In stating it he does

not use the term iti shruyate, meaning so we hear. As a Vedic

priest it was not for him much more than a useful detail in the

construction of altars. In any event, the spirit of the age did not

place the same high value on the priority of publications that we

now do. But he certainly merits recognition as the first known of

theworldis great mathematicians.

The really interesting question then is to know if knowledge

of the theorem can be traced to the Rigvedaitself. Seidenberg

sought to find some evidence for it in Rigvedic hymn X.90, the

famousPurusaSμukta. We cannot agree with his reading of it; nor

can we find any evidence that the Rigvedaknows the Theorem of

Pythagoras, though knowledge of the other great problem of

geometric algebra o the canonical circling of the square o is

implied, if only indirectly. The Rigvedaknows the spoked wheel,

and there is an ingenious construction of it given in the ›ulbaof

Baudhayana based on successive circling of concentric squares.

One is therefore led to conclude that the Rigveda too knows it.

Harappan archaeology also establishes that the Pythagorean

theorem was known and applied as early as 3000 BCE. Several

massive structures and lengthy streets are laid out using perfect

perpendicular and parallel lines. One may draw good parallel and

perpendicular lines that are a few feet long through visual

inspection and experience. But planning and building structures

that are hundreds of meters long demands knowledge of

mathematics.

Thus, objective needs of Harappan architecture (see above)

leave no doubt at all that geometry, including the so-called

Pythagorean Theorem was known to Harappan architects.

All evidence therefore points to the Theorem of Pythagoras

as having been known in the Sutra literature going back almost

to 3000 BCE, as we shall see later. Datta has noted that traces of

Indian terminology are to be found in the works of Greeks like

Democritos (c. 440 BCE). Further, the gnomon also links

Pythagorean geometry to India. The Greek geometry thus greatly

resembles the Indian both in spirit and content. Therefore until a

convincing alternative source is found, we are justified in holding

Greek geometry to be a derivative of the ›ulba. (The gnomon is

an L-shaped figure obtained by removing a square from a larger

square having a common vortex.)

There is ample evidence showing that India and Greece had

long been in contact o going back well before the coming of

Alexander. Thus, the Greeks would have been familiar with the

Vedic mathematics, not the ›ulbas themselves perhaps, which

were quite ancient by then, but through various commentaries

that were then current.

culminating in the work of Seidenberg, the layer of

the Sutra literature containing the works of Baudhayana,

A‹valayana, Apastamba and Katyayana, must be dated to no later

than 2100 BCE. And this is an absolute lower limit that derives

from the fact that the mathematics of Old-Babylonia (before 1700

BCE) and the Egyptian Middle Kingdom (2050 to 1800 BCE)

both derive from the ›ulbas.

Nor does the story end here. It should further be noted that

considerable time must have elapsed before the religious-ritual

form of the geometric algebra found in the ›ulbaof Baudhayana

et. al. evolved into the purely secular arithmetic methods found in

the Old-Babylonia and Egypt. Recognizing this fact, Seidenberg

required a date ifar back of 1700 BCE for the mathematics of

the›ulba. This mathematically determined date is supported also

by ancient astronomical records.

But returning to ancient mathematics, once we recognize the

›ulbasas the source of both Egyptian and Old-Babylonian

mathematics, it then becomes possible to fill otherwise

inexplicable gaps found in them. Specifically, the origins of

several significant results from Egyptian and Old-Babylonian

records can now be traced to the ›ulbas.

One of the most

interesting of these is the use of the so-called unit fractions, i.e.,

approximations to irrational numbers with terms of the form 1/n,

where n is a whole number. The following unit fraction

approximation appears frequently (Seidenberg, 1978; Datta 1934).

√2= 1 + 1/3 + 1/(3.4) – 1/(3.4.34)

Such unit fractions in Egyptian (and Babylonian) mathematics

are justly famous. They appear also in the ›ulbas. From this

Cantor stated that the Indians got their mathematics from the

Egyptians. (Remember others claimed they got their geometry

and astronomy from the Greeks, following Alexander’s invasion.

There is of course no record of any Egyptian Pharaoh invading

India.) But this claim is contradicted by the fact its origins are

nowhere to be found in Egypt. Where then did they come from?

The roots again are found only in the ›ulbas, where this unit

fraction approximation arises naturally as the outgrowth of the

problem of squaring the circle. Like the Theorem of Pythagoras,

the square-circle equivalence (approximate) is one of the seminal

problems of ancient mathematics. Further, the value of pi=

3.16049 used by Ahmes of Egypt (c. 1550 BCE) is exactly the same

as the one given in the relatively late Manava ›ulbasμutra, and,

what is more remarkable, is obtained in precisely the same fashion, i.e., as

4.(8/9)(8/9). That is to say, the Manava ›ulbaand Ahmes of Egypt

both use the exact same approximation (in modern notation, Datta

1932):

Pi = 3.16049 = 4.(8/9)2

With two specific examples agreeing down to the most

minute detail, the probability of its being a coincidence can be

dismissed. In the problem of the circle-square equivalence, we

see therefore the roots for going over into arithmetic methods.

Thus the origins even of arithmetic methods, so characteristic of

the mathematics of Babylon and Egypt, are again traceable to the

›ulbas.We can therefore conclude that attempts at squaring the

circle led to the arithmetic methods that lie at the heart of Old-

Babylonian mathematics of 1700 BCE (and Egypt four centuries

earlier). It is therefore beyond question that their mathematics is

also a derivative of the ›ulba. Thus Vedic mathematics, or, more

exactly, the mathematics of the ›ulbasμutrasmust have been

known in India no later than 2100 BCE. Also as already noted,

many of the structures and cities of the Indus Valley presuppose

considerable knowledge of geometry nearly a thousand years

before Old-Babylonia and the Egyptian Middle Kingdom.

A question that arises is the following: Can we reasonably

conclude that Vedic mathematics was the source of all the

technical knowledge of the ancient world, from Egypt to China?

The Egyptian Middle Kingdom certainly owed its mathematics to

theSμulbas, as did Old-Babylonia. But how about Sumeria or

early Egypt? Or China? Did the Egyptians seek the help of some

Vedic priests in their constructions?

To these questions, the answer at this time has to be a

qualified affirmative. There is at least one other connection, quite

apart from unit fractions and the value of pi. Trapezoidal figures

shaped like Vedic altars are found on Egyptian monuments of all

times. Two of them, taken from Histoire de liart Egyptian

diapres le monuments are reproduced in Figure 2. (They are

originally given by Cantor.) As Seidenberg notes:

It is not merely the trapezoidal shapes which impress us, but

their subdivisions. The subdivision of the first figure occurs

in the Sulvasutras, and the second calls to mind the

computation of the area [and its proof] in the Apastamba

Sulvasutra.If these figures occurred on Indian monuments,

we could understand the Indian interest in them: all the hopes

of the Indian for health and wealth were tied up in the

trapezoid. (Seidenberg 1962: p.519)

Trapezoids are common in Vedic altars. But how are we to

account for their presence on Egyptian monuments? Particularly

when they would have been quite cumbersome to construct and

draw using the earithmetici techniques so favored by the

Egyptians (and the Old-Babylonians)? Now we know that durinng

Astronomy, Alpha Draconis

Astronomical references in the work of Ashvalayana (not

Discussed here) establishes this date far back of 1700 BCE to

be around 3100 to 2600 BCE o the time when the star

Alpha Draconis (also called Thuban in the constellation Dragon)

was the pole star, a fact noted in the Satapatha Brahmana also.

This receives additional support from other astronomical

observations in the ancient literature.

This is also in accord withIndian tradition which places Ashvalayana about five generations

removed from the Mahabharata War c. 3100 BCE). Baudhayana also belongs

to the same period, perhaps a generation earlier. Further, since the

Sutra literature presupposes the existence of the Vedas, it follows

that the four Vedas must have been in existence by 3000 BCE.

Thus, the tradition of Vyasa as the editor of the four Vedas, as

well as his date of the Mahabharata War (c.3102 BCE) are in

agreement with ancient mathematics and astronomical data.

Further, we have also found connections between the so-called

Step Pyramid or the mastababuilt c. 2650 BCE by Djoser

(c. 2686 to c. 2613 BCE) and the smashana-cit altar described by

Baudhayana. This Step Pyramid was the forerunner of all the

future pyramids of Egypt. The smashana-citaltar (i.e., cemetery

shaped altar), as its name itself clearly indicates was connected

with Vedic funerary rituals. Since all Egyptian pyramids were

erected to serve as mausoleums, the connection is not only

mathematical but also of religion and ritual.

            These issues are discussed in more fully in the book Vedic Aryans and the Origins of Civilization by Rajaram and Frawley (above) along with relevant illustrations.

PUZZLE OF ANCIENT MATHEMATICS, INDIA AND THE WORLD I

PUZZLE OF ANCIENT MATHEMATICS, INDIA AND THE WORLD I

There is now a revision of history of mathematics, recognizing Indian contributions in both geometry and algebra.

NavaratnaRajaram

In our study of Indian contributions to the mathematics of the ancient world, we are faced with two hurdles. First, extravagant claims in the name of Vedic Mathematics that have no support in the Vedic literature. The famous book Vedic Mathematics by Swamy Bharati Krishna Tirtha (compiled by his disciples) is a modern text by an accomplished mathematician (at a basic level) and a capable Vedic scholar. It may have some pedagogical value in teaching basic to intermediate mathematics, but is of no historical significance. Its contents cannot be traced to the Atharva Veda or any Vedic  texts as claimed by its proponents. [His book Vedic Metaphysics is more sober and useful but not as well known as it should be.]

The second hurdle is the attempt by some Western scholars and their followers to deny all Indian contributions, attributing them to foreign sources like the Greeks and the Chinese. Its perverse limit was reached probably by David Pingree in his creation of a pseudo-Greek text based on Indian sources which he later claimed to be the source of Indian mathematics. A prime example is the book on Indian mathematics by Pingree student Kim Plofker. But ancient authors including Greeks and Arabs make no such claims and freely acknowledge their indebtedness to other sources.

But a careful examination of the Bakshali Manuscript should have put an end these absurdities, but for reasons known only to herself Kim Plofker seems to take no note of its conents. Pingree too seemed to have been unaware of it when he created his pseudo-Greek source.

It is generally recognized that the so-called Arabic numerals, now used worldwide is a misnomer and they are of Indian origin. Arabs themselves called them Hindi numerals. The Italian mathematician Leonardo of Pisa (better known as Fibanocci) was among the first to introduce them in Europe as well as the decimal place value system (also from India). He states that he got it from the Arabs, then ruling Spain but also notes that the Arabs got them from the Indians. The Fibonacci numbers were also known to the Indians, centuries before they were discovered by Europeans.

More than 500 years before Fibonacci, we have more substantial evidence from Syria suggesting that as early as the seventh century, knowledgeable scholars recognized Indian mathematics to be superior to the Greek.Writing in 662 CE, Servius Sebokht (575-667 CE)., the Bishop of Qinnesrin in North Syria observed:

I will omit all discussion of the science of the Hindus [Indians], discoveries more ingenious than those of the Greeks and the Babylonians:

He notes in particular their valuable method of calculation [the decimal system]; their computing that surpasses description. I wish only to say that this computation is done by means of nine signs. If those who believe because they know Greek, that they have

reached the limits of science should know these things, they would be convinced that there are also others (than the Greeks) who know something.

Aryabhata appears to have been the earliest to use the decimal system with zero extensively.

The origin of the modern decimal-based place value notation can be traced to the Aryabhatiya (c. 500), which states sthānātsthānaṁdaśaguṇaṁsyāt ” from place to place each is ten times the preceding

Even though we take the decimal system as the standard, it is only a special case of the place value system, to base 10. Now binary systems (base 2) are widely used in computations, especially in computer science. Wilhem Leibnitz in the 17th century is usually credited with the discovery of the binary system, but he was preceded by others going back centuries.

The Chandaḥśāstra (c. 400 BCE or earlier), presents the first known description of a binary numeral system in connection with the systematic enumeration of meters with fixed patterns of short and long syllables.The discussion of the combinatorics of meter corresponds to the binomial theorem. Halāyudha’s commentary includes a presentation of Pascal’s triangle (called meruprastāra). Pingala’s work also includes material related to the Fibonacci numbers, called mātrāmeru.[7]

Use of zero is sometimes ascribed to Pingala due to his discussion of binary numbers, usually represented using 0 and 1 in modern discussion, but Pingala used light (laghu) and heavy (guru) rather than 0 and 1 to describe syllables. As Pingala’s system ranks binary patterns starting at one (four short syllables—binary “0000”—is the first pattern), the nth pattern corresponds to the binary representation of n-1 (with increasing positional values).

Pingala is credited with using binary numbers in the form of short and long syllables (the latter equal in length to two short syllables). Pingala used the Sanskrit word śūnya explicitly to refer to zero. This and its derivatives are still used in Indian languages and its derivatives.

Also the modern numeral notation from 0 to 9 are found explicitly presented in the 2000 year old Bakshali Manuscript.

Large numbers in ancient Indian literature

Greeks and the Indians approached mathematics in different spirit. Greeks are better known for geometry, while Indians, while not ignoring geometry were focused more on the representation and manipulation of numbers, excellent at working with large numbers that seemed to beyond the capability of the Greeks . Evidence for this is available from the Mahabharata, in describing the armies engaged in the war. We are told that 18 akshauhanis (or armies) battled it out at Kurukshetra battlefield.

An akshauhini (Sanskrit: अक्षौहिणी akṣauhiṇī) is described in the Mahabharata as a battle formation (or army) consisting of approximately 5 lac. warriors (500 K or half a million) i.e. 21,870 chariots (Sanskrit ratha); 21,870 elephants (Sanskrit gaja); 65,610 horses (Sanskrit turaga) and 109,350 infantry (Sanskrit padasainyam) as per the Mahabharata (AdiParva 2.15-23).[1][2] (thus the total number of humans, warriors in akshauhini is equal to 218,700). The ratio is 1 chariot : 1 elephant : 3 cavalry : 5 infantry soldiers. In each of these large number groups (65,610, etc.), the digits add up to 18.

            It is not our point here that such large armies actually took part in the Mahabharata War. But only ancient Indians were comfortable with large numbers and knew how to handle them. In this they were helped by a vastly superior number system based on place-values that remains the world standard even today, nearly 2000 years later if not more.