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Science in Saffron: Skeptical Essays on History of Science


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Challenging Hindu nationalist myths about history of science in ancient India.
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Science in Saron
Science in Saron
Skeptical Essays on History of Science
Meera Nanda
First Edition January 2016
Copyright©ree Essays Collective
All rights reserved
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mechanical, including photocopying, recording or by any information storage or retrieval system,
without the prior written permission of the publisher.
ISBN 978-93-83968-08-4
B-957 Palam Vihar, GURGAON (Haryana) 122 017 India
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Printed and bound by Chaman Oset Printers, New Delhi
In memoriam
Praful Bidwai 1949-2015
Ajita Kamal 1978-2011
dear friends and comrades
Introduction 1
1. Who Discovered the Pythagorean eorem? 19
2. Nothing at Is: Zero’s Fleeting Footsteps 49
3. Genetics, Plastic Surgery and Other Wonders of Ancient
Indian Medicine 93
4. Yoga Scientized: How Swami Vivekananda rewrote
Patanjalis Yoga Sūtra 127
References 181
Index 193
Some years ago, I happened to watch an advertisement for Rajnigandha
paan masala on TV that stuck a nerve with me. is is how it went:
A bespectacled young Indian man in a tweed jacket is sitting in a
classroom in an American campus where a professor is writing some
rather complicated looking mathematical equations on the chalk board.
e young man appears bored; he is looking out of the window and
doodling on his notepad. Speaking in an exaggerated American drawl,
the professor asks how much time the class will need to solve a prob-
lem, and all the European and Chinese-looking students balk at the
task, saying the problem is too tough. Muttering racist-sounding epi-
thets about “you Indians” and “trees,” the professor calls upon the desi.
e Indian student gets up, takes out a small can of paan-masala from
his jacket and puts some in his mouth. He then walks up to the board
and solves the mathematical problem without a moments hesitation.
e American classroom breaks into cheers, and the young man takes a
bow. e image of a packet of Rajnigandha paan masala appears on the
screen with the following voice-over: “जवाब तो हम पहले से जानते थे, सवाल
का इि�ज़ार करना हमारी तहज़ीब ह (We already knew the answer. Waiting
for the question is our culture”). e advertisement ends with a jingle:
मुंह में रजनीगंधा, पैरों पे दुिनया (“With Rajnigandha in your mouth, the
world is at your feet.)”1
1 e advertisement can be viewed at
 d Science in Saron
With some foresight, the ad-agency could have amassed a con-
siderable fortune by selling this slogan (sans the jingle) to the Sangh
Parivar. e young paan-masala consuming fellow could have achieved
lasting fame as the mascot of the new geeky Indian that we are so love
to celebrate.
जवाब तो हम पहले से जानते थे, सवाल का इन्तज़ार करना हमारी तहज़ीब ह
would make an excellent backdrop for any number of “science in the
Vedas” events that the Parivar and its allies like to host. e beauty of
the slogan is that it can capture the spirit of whatever scientic जवाब you
may be in the mood to t into “the Vedas” on any particular day – ro-
botics, nuclear energy, quantum physics, Einstein, theory of evolution,
genetics, consciousness … the list is limited only by your imagination.
Why just the Sangh Parivar, even the Indian Science Congress could
have trotted out the slogan for the more colorful sessions during its an-
nual meet in Mumbai in January 2015!
is book is provoked by the constant assault that on our collective in-
telligence from those who are convinced that “जवाब तो हम पहले से जानते
थे.” But it is more than mere irritation that has motivated me. I believe
that the constant appropriation of modern scientic concepts and theo-
ries for the glory of “the Vedas” is one, if not the, central plank on which
the myth of Hindu supremacy rests. It is thanks to this myth of “sci-
entic Hinduism” that our preeminent national gures, past and pre-
sent, habitually sneer at the “superstitions” of Abrahamic religions. It is
thanks to this myth that we think of ourselves as a “race” endowed with
a special faculty for science. It is thanks to this myth that we go around
the world thumping our chests as “scientic Indians” without whom the
world science and economy would grind to a halt.
Such myths of national exceptionalism and supremacy are danger-
ous. Nothing but evil follows when such myths manage to take hold of
a nations imagination.
It is for this reason, this smug, self-adoring myth of the “Vedas” as
having all the answers – even before scientic questions were even pos-
sible to ask! – must be taken seriously. Each one of its claims must be
Who Discovered the Pythagorean eorem? c   
examined with utmost attention, using the best available evidence that
history of science has to oer. Aer we are done laughing at some of
the utterly outlandish claims, we must get down to the serious business
of analyzing what they are saying in the light of what we know of how
science developed in the modern world and how it diers from other
forms of knowing the world. Time has come for intellectuals to step out
of their ivory towers to challenge the distortion of history of science for
ideological ends.2
Such a response has not been forthcoming, or at least, has not been
proportional to the enormity of the challenge. e scientic commu-
nity in India – whose turf is being encroached upon – has oered only
a deafening silence so far (with rare exceptions who can be counted on
the ngers of one hand). What is even more disheartening is the silence
of Indian historians of science against the blatant encroachment of their
turf.3 Indeed, the silence of academic historians of science is more wor-
risome, as it is symptomatic of postmodernist malaise that continues
to aict the humanities and social sciences in India. How can those
who cannot utter the words modern science without putting them under
contemptuous scare-quotes that question the very distinctiveness and
legitimacy of the enterprise of science, be expected to start demarcat-
ing modern science from Vedic or any other “alternative” knowledge
system? How can those who cannot bear to refer to the mainstream,
global history of science without qualifying it as “colonial” and “Euro-
centric” be expected to turn to the same history for evidence to counter
the priority-claims of our nationalist mythmakers?4
2 Romila apar (2015) distinguishes public intellectuals from technical experts
and ordinary academic scholars by two necessary qualities: Public intellectuals
question authority; and they “defend the primacy of reasoned, logical arguments
in explaining the world around us as well as its past.” p. 2
3 is refers strictly to those historians who specialize in science. History of science
is a relatively specialized sub-set of political and cultural history. Mainstream
historians, archeologists and other concerned intellectuals, to their credit, have
continued to raise their voices against mixing up of myth and science.
4 See for instance Sundar Sarukkai (2014) for a defense of “alternative rationalities”
and the importance of “cultural ownership” of science by Indians. Sarukkai’s insis-
tence that modern conceptions of physics – mass, energy, motion etc. – must be
mapped on to philosophical terms derived from Indian philosophy is no dierent
from the recent intervention by Rajiv Malhotra (2015) to “t modern science into
 d Science in Saron
is book is meant to take on the substantive claims of those who
would saronize modern science. It oers a detailed and through ex-
amination of the priority-claims on behalf of ancient Indian mathema-
ticians and physicians regarding landmark scientic discoveries (the
Pythagorean eorem, zero, genetics and surgery). Such claims have
been a xture of Indian public discourse for a long time, and have been
given a fresh impetus at a variety of high-visibility gatherings over the
last year of so.
e goal of the book is to save the ancient Indian geometers, math-
ematicians, physicians and the unknown artisans-crasmen/women
from both the glorication at the hands of the Hindu Right and the con-
descension at the hands of rationalist fundamentalists who see no value
whatsoever in anything that predates the Scientic Revolution. is can
be done, I believe, by placing their achievements in their own times,
and alongside the achievements of their peers in sister civilizations. A
comparative history, devoid of presentist biases, can bring the true ac-
complishments of our ancestors into a sharper focus – something that
this book tries to do.
Claims to the eect that “it is all in the Vedas” –where “all” includes all
known facts and artifacts of modern science and technology (yes, the
airplanes, too) are not new. Swami Dayananda Sarasvati, the founder
of Arya Samaj, had already proclaimed that as far back as around the
mid-19th century.5 Likewise, claims of there being “perfect harmony”
between the teachings of Hindu shastras and modern science can be
Vedic framework.” Malhotra is a much admired gure in the Hindu Right circles
and has received glowing accolades from Narendra Modi himself. Skepticism
toward the universal metanarrative of science has been declared (Prakash, 1994,
p.1483) a necessary precondition for recovering the voices of the subaltern. I have
defended the universality and objectivity of modern science form its de-construc-
tors in Nanda (2004).
5 e basic idea that motivated Swami Dayananda was this: Because the Vedas are
divinely-inspired “books of true knowledge,” they must contain the basic prin-
ciples of all sciences, and accordingly, every scientic discovery and technological
development of modern times must nd an expression in them. is amounts to
scientization of the Vedas by at. See Arvind Sharma (1989).
Who Discovered the Pythagorean eorem? c   
traced back to the New Dispensation of Keshub Chandra Sen in the late
1800s, and to his more famous protégé, Swami Vivekananda. In his fa-
mous address to the World Parliament of Religions in Chicago in 1893,
Vivekananda proudly proclaimed the latest discoveries of modern sci-
ence to be mere “echoes” of Vedanta philosophy.6
us, the current craze for nding modern science in ancient re-
ligious texts is part and parcel of the history of modernity in India. It
has been the dominant trope for accommodating modern science with
the Hindu belief-system. In the hundred plus years that separate Swami
Dayananda and Swami Vivekananda from us in the 21st century, this
style of accommodating science and Hindu beliefs has become a part
of the common sense of most Indians. It is not considered particularly
right-wing or le-wing, as elements of it can be found among people
and parties of all political persuasions.
While it cuts across political aliations, the eagerness for scientic
legitimation of Hindu dharma is more actively and self-consciously fos-
tered by Hindu nationalists and their allies. Attribution of great scien-
tic discoveries to ancient Hindu rishi-munis has been an integral part
of the indoctrination of swayamsevaks since the very beginnings of the
organized Hindu Right in the early decades of the 20th century.
is explains why every time the Hindu nationalists come to pow-
er, the rst thing they do is to start revising history, with a special place
reserved for the history of science.
During their rst stint from 1998 to 2004, the BJP-led NDA pushed
for introducing degree-courses in astrology, karma-kanda (rituals) and
consciousness studies” of Advaitic variety in colleges and universities.7
6 We explore Swami Vivekananda’s views on science and yoga in the last chapter of
this book.
7 e prestigious Birla Institute of Technology and Science, in collaboration with
Bhaktivedanta Institute now oers M.Phil. and Ph.D. degrees in “consciousness
studies.” is program sells itself as an “equivalent of a graduate program in ‘cog-
nitive studies’ in any Western university.” But it is hard to imagine any respectable
cognitive studies school in the West accepting the fundamental premise that this
program operates with: that consciousness is a pre-existing constituent of matter.
is is simply Advaita by another name. Bhaktivedanta Institute is the “research
wing of the International Society for Krishna Consciousness, aka “Hare Krishnas.
 d Science in Saron
anks to the policies put in place by NDA 1.0, any aspiring astrologer
or priest can get a diploma from public or private institutions that have
been given the status of universities.8
Now that the BJP-led alliance is back in power, revising history
of science is once again on the top of the list of educational “reforms.
NDA 2.0 has lost no time in extending its campaign rhetoric of “In-
dia First” to history of science. Claims of Indias priority in everything
from mathematics, medicine and surgery – to say nothing of nuclear
weapons, spaceships and other Star Trek-style technologies – have been
made by prominent people at prestigious, national-level gatherings.
e ball was set rolling by none other than the Prime Minister in his
inaugural address at Sir H.N. Reliance Foundation Hospital in Mumbai
in October 2014. is was followed by events at the 102nd annual Indian
Science Congress in Mumbai in early January, 2015. Other relatively
high-visibility events where a seamless continuity between modern sci-
ence and ancient sciences and myths was on the agenda include the ex-
hibition in Lalit Kala Academy in New Delhi titled “Cultural Continuity
from Rigveda to Robotics,” and a seminar on Vedic chronology organ-
ized by the Sanskrit department in Delhi University, both in September,
2015. Behind all these high-prole events, there are any numbers of
Shiksha Bachao” (“save our Education”) activists who want this “his-
tory” to become a part of school curricula.
Roughly four kinds of appropriations of modern science for the
glory of Hindu sages-scientists can be discerned:
1. Staking priority-claims for ancient India for landmark discov-
eries in mathematics and medicine. e perennial favorites in
this category are the Pythagorean theorem, algebra and zero in
mathematics. (We will ask “who discovered the Pythagorean
Certicate programs oered by training centers associated with Aurobindo Ash-
ram are recognized by IGNOU, Indira Gandhi National Open University.
8 I have tried to document how educational institutionns set up by prominent
religious gurus and sects include a variety of pseudosciences as a legitimate part
of their curricula. Many of these institutions have been “deemed” as universities.
As “deemed universities” they have been given the authority to set their own cur-
ricula and hand out degrees and diplomas. Many receive state support in the form
of land-grants and tax-breaks. See, Nanda, 2009.
Who Discovered the Pythagorean eorem? c   
eorem?” in Chapter 1, while the next chapter will look at the
hallowed Indian invention of zero as a number).
2. Erasure of lines of demarcation between myth and histori-
cal evidence. is was the Prime Minister’s chosen rhetorical
device at the inaugural address at the Mumbai hospital men-
tioned above. He invoked the elephant-headed god Ganesh as
evidence for plastic surgery, and Karna, a character from the
Mahabharata as evidence for “genetic science.” (We will exam-
ine the history of medicine in chapter 3).
3. Erasure of lines of demarcation between science and certied
pseudosciences like astrology. While this strategy of giving
sheen of respectability to discarded knowledge has not disap-
peared from the public sphere, it has not been openly espoused
from high places lately.
4. A higher kind of pseudoscience that is generated by graing
spiritual concepts like prana (or breath), prakriti or akasha (the
“subtle” material substrate of nature) on to physicists’ concepts
of “energy” and “ether”; karmically determined birth and re-
birth on theories of evolution of species; chakras with actual
neural structures, so on and so forth. (Swami Vivekananda
was the pioneer of this kind of scientization and we will exam-
ine how he re-wrote Patanjali’s Yoga Sutras in a scientistic vein
in the nal chapter of this book).
Why such mental gymnastics? Why this national itch to be crowned
What look like obvious, and even laughable, contortions begin to
make perfect sense when we understand what our saronizers are re-
9 Very similar in spirit to the great eagerness of Indians to have their names
recorded, for most bizarre feats, in the Guinness Book of Records. According to
Vinay Lal, nearly one-tenth of the mail that the Guinness headquarters in London
receives is from Indians. Where else but in India will you nd a “ World Record
Holder Club” whose president has “changed his name from Harparkash Rishi to
Guinness Rishi”? see
 d Science in Saron
ally up to. What is it that they seek to accomplish by their constant and
desperate attempts to claim the stamp of “science” for the worldview
they want to propagate?
We have to understand that the Hindu nationalists are not in the
business of history-writing, even though they may use historical evi-
dence if and when it suits them. No, what they are doing is fabricating
a heritage that we are supposed to kneel before in awe and wonder and
feel special about. While no history is completely free of biases and er-
rors, historians at least try to correct their narratives in the light of bet-
ter evidence. Heritage-makers, on the other hand, thrive on errors and
biases. e torturous logic, the ights of fancy, the mental gyrations are
no circus: ey are the tools of the trade needed to create the myth of
the “scientic Indian,” the bearer of the ancient Hindu heritage which
was scientic – in the sense of Science as We Know it Today, or SaWKiT)
– even before SaWKiT was even born.
e distinction between history and heritage brought out by David
Lowenthal in his well-known book, e Heritage Crusade and the Spoils
of History, is relevant to the Indian situation:
Heritage is not “bad” history. In fact, heritage is not history at all; while it bor-
rows from and enlivens historical study, heritage is not an inquiry into the past,
but a celebration of it; not an eort to know what actually happened, but a pro-
fession of faith in the past tailored to present day purposes…. 10
Heritage is not a testable or even a reasonably plausible account of some past,
but a declaration of faith in that past…. Heritage is not history, even when it
mimics history. It uses historical traces and tells historical tales, but these tales
and traces are stitched into fables that are open neither to critical analysis nor
to comparative scrutiny….. 11
And again:
Heritage is immune to critical reappraisal because it is not erudition but cat-
echism; what counts is not checkable fact but credulous allegiance. Commit-
ment and bonding demand uncritical endorsement and preclude dissenting
voices. …. Prejudiced pride in the past is not a sorry consequence of heritage; it
is its essential purpose. 12
10 Lowenthal, 1998, p. X. emphasis added.
11 Lowenthal, 1998, p. 121. Emphasis added.
12 Lowenthal, 1998, p. 121-122, Emphasis added.
Who Discovered the Pythagorean eorem? c   
e “scientic Vedas” rightfully belong to the “Incredible India!”
campaign which sells Indian heritage primarily to foreign tourists, with
the dierence that the “heritage sites” for the former are not physical
but textual, and the target audience includes Indians rst and foreign-
ers only secondarily. e way the “scientic heritage” is constructed and
sold, however, is turning Indians into tourists to their own history. e
very idea of such a narrative being taught to school children as history
of science is frightening indeed.
It is clear that this enterprise is aimed not at educating but, to use
Lowenthal’s apt words, at creating a “prejudiced pride” in India’s past
through “celebration” and “declaration of faith” in it. Indeed, this is ex-
actly what the heritage-fabricators openly profess.
A case in point: When the Prime Minister Modi invoked Ganesh
from mythology, and Karna from the Mahabharata as “evidence” that
plastic surgery and genetic science existed in ancient India, he explained
his motive for this foray into mythology in the following words:13
We have our own skills. Now, we are not new to medial science…. We can take
pride in the world of medicine. Our nation was great one time. … What I mean
to say is that ours is a country that once had these abilities [for advanced medi-
cine]. We can regain these abilities.
e PM is hardly alone. Indian Firsters routinely claim that by
highlighting the scientic accomplishments of ancient Hindus, they are
actually trying to promote a culture of science and scientic temper.
is is how the argument unfolds: Indians are heirs to a great civili-
zation which promoted reasoned inquiry, which then led to scientic
ideas which are only now being “rediscovered” by modern science. As
the beneciaries of this great civilization, we ought to be inspired by
it, reclaim its scientic spirit and produce world-class science again.
While they would not put it so starkly, even some secular historians of
13 ßgekjk viuk ;s dkS'kY; gS] vc esfMdy lkbal esa ge u, ugha gSaA esfMdy lkbal dh nqfu;k esa
ge xoZ dj ldrs gSa] gekjk ns'k fdlh le; D;k FkkA dgus dk rkRi;Z ;g gS fd ;g oks ns'k gS]
ftlds ikl ;s lkeF;Z jgk FkkA bldks ge fQj dSls nksckjk regain djsaAÞ e complete ad-
dress is available at the PMO website
mumbai/?comment=disable. Translation is mine.
 d Science in Saron
science have bought into this business of promoting “cultural owner-
ship” for the goal of doing good science. 14
Once we see the “science in the Vedas” discourse for what it is – a
fabrication of heritage – three questions arise, which will be examined
in the rest of the Introduction. e rst question has to do with the
relationship between the glorious past and the present state of aairs.
Here we will ask if it is really the case that because we were, presum-
ably, great in sciences once, we will be great again. e other two ques-
tions have to do with how the “scientic” heritage is put together and
made to appear reasonable. Here we will examine two favorite ploys of
heritage-makers, namely, presentism and parochialism. Let us look at
these issues seriatim.
Let us start with the promise of becoming great “again.
We seem to think that by glorifying our ancient knowledge-tradi-
tions, we are providing cultural self-condence to the present and fu-
ture generations of scientists. We seem to think that if we can establish
continuity between ancient and modern modes of inquiry, we will gain
condence in our presumably “innate” acumen to do science.
But the notion of continuity between the science of the antiquity –
not just the sciences of Indian antiquity, but of any ancient civilization
14 is group largely includes those scholars who have accepted multiculturalist and
relativist view of science wherein modern “Western” science is seen as only one
form of science at par with other cultural constructions. is view has become
quite pervasive, especially among feminist and postcolonial scholars of science.
See Harding 2011, for a recent overview.
Multiculturalism in science assumes that all standards of evaluation of evidence
and judgement as to the soundness of a belief are internal to the culture, gender,
social class/caste one is born in, and therefore, when students are exposed to mod-
ern science they are being asked to embrace culturally alien denitions of nature
and standards of judgments. If the students were exposed to science using “their
own” cultural vocabulary, they will become better learners and better scientists.
Sundar Sarukkai (2014) oers a well-articulated statement of this position.
Based upon my experience rst as a young woman trained in microbiology in
India, and now as someone who teaches history of science to science students in
the Indian Institute of Science Education and Research in Mohali, India, I believe
that the cultural relativity of standards of evidence and judgment is overstated.
Who Discovered the Pythagorean eorem? c   
in the world – and modern science is unwarranted and unproductive. It
is unwarranted because it does not acknowledge the break from the tra-
dition that happened with modern science. e science that emerged
aer the Scientic Revolution through the 16th to 18th centuries was a
very dierent enterprise from all earlier attempts to understand nature.
Most historians of science 15 agree on the following revolutionary trans-
formations that marked the birth of modern science:
1. Mathematization of nature, i.e. a growing attempt to describe
natural things and events in mathematical terms which could
be quantied, using increasingly precise tools of measurement
(clocks, compasses, thermometers, barometers and such).
2. Fact-nding experiments in addition to direct observations. In
the hands of early modern scientists (represented by the para-
digmatic gure of Galileo), mathematization of nature was
brought together with controlled experimentation.
3. Development of a mechanistic world picture which tried to
explain the workings of the natural world in nothing but cor-
puscles of matter in motion.
4. An uncommon appreciation of manual work, which led to the
relative lowering of barriers between university-trained natu-
ral philosophers and artisans and crasmen. 16
Undoubtedly, this revolution was made possible by a conuence of
a multitude of earlier achievements of many civilizations – the ancient
Greeks, Christianity, Islam, and through Islam, the contributions of an-
cient and classical India and China. But the new science that emerged
aer the Scientic Revolution was most unlike any of the nature-knowl-
edge traditions that went to into it, including the Greco-Roman, and
Judeo-Christian tradition, that are the direct ancestors of the Western
civilization. While it took on board some elements of mathematical and
observational stock of knowledge from earlier civilizations, modern
science – the SaWKiT – turned the ancient cosmos and ancient meth-
ods of speculative reason upside down, and produced a new concep-
15 At least those historians of science who have not written-o the very idea of a
scientic “revolution” as a Western ploy to project its superiority over all others.
16 In two magisterial books, H. Floris Cohen, a Dutch historian of science, has ex-
plored the emergence of distinctively modern science. See Cohen, 1994 and 2010.
 d Science in Saron
tion of the cosmos and the humanity’s place in it. So revolutionary and
sweeping have the changes been that it is oxymoronic to say that any pre-
modern knowledge tradition – be it Hindu, Christian, Islamic, Jewish,
Buddhist, Taoist, animistic – had the answer to the questions asked by
modern scientists. Of course the nature of the natural world (its compo-
sition, the fundamental laws governing its operations) has not changed,
but the conceptual categories, methodological criteria and the aims of
inquiry have undergone such a radical transformation that it is safe
to say with omas Kuhn that the ancients and the modern scientists
practically live in dierent worlds. 17
If one accepts this picture of the birth of modern science, then
the very idea of ancients having the answers that have emerged only
in the last 500 years or so makes no sense. Of course, there are nug-
gets of useful empirical knowledge – the knowledge of useful medic-
inal plants, or organic methods of farming, for example – that can be
incorporated into the modern corpus provided they pass the strin-
gent tests that all empirical claims must go through to be deemed
“scientic.” But beyond that, it is simply vainglorious to claim that
modern science is only repeating what the ancients already knew. 18
Not only is the insistence of continuity between ancient and mod-
ern sciences unwarranted, it is entirely unproductive. e conviction
that we have always-already known everything that is worth knowing,
and that everything we knew is only conrmed – never rejected – by
science, has prevented us from developing an ethos of honest inquiry.
e compulsion to establish harmony with the core of the Vedic world-
view has held back the progress of science in the past, and will continue
to hold us back if we continue to go down this path.
Admitting to being an ignoramus – Latin for “we don’t know” – is
the rst step toward acquiring knowledge. is point has been well-
articulated by Yuval Harari in his inuential book, Sapiens:
17 omas Kuhn’s 1962 masterpiece, e Structure of the Scientic Revolutions has
revolutionized the study of history of science.
18 e classical Indian statement of this sentiment comes from Swami Vivekananda
who in his famous Chicago address insisted that that the discoveries of modern
science are only restating “in a more forcible language…what the Hindu has been
cherishing in his bosom for ages.
Who Discovered the Pythagorean eorem? c   
e Scientic Revolution has not been a revolution of knowledge. It has been
above all a revolution of ignorance. e great discovery that launched the Sci-
entic Revolution was the discovery that humans do not know the answers to
their most important questions…. Even more critically, modern science ac-
cepts that the things we know could be proven wrong as we gain more knowl-
edge. No concept, idea or theory is beyond challenge. 19
Acknowledging that we do not have all the answers, and the an-
swers we do have could well turn out to be all wrong, is what allowed
modern science to emerge and ourish in Europe in the early modern
era, from the 16th to the 18th century. It was not a matter of some spe-
cial “Faustian Spirit” that existed only in the West, but rather a coming
together of theological justications for empiricism, political and mer-
cantile interests, technological breakthroughs, along with a regard for
manual labor that set the stage for the Scientic Revolution.
is process was by no means smooth. ere was resistance from
the Church and the Aristotelian professors who controlled the medieval
universities. Yet eventually, an awareness emerged that the conclusions
of the Greek philosophers (the earth-centered universe, the humoral
theory of disease, Aristotle’s theory of falling objects) and the Bible
(the seven-day Creation, the Great Flood) were incorrect, as they failed
to adequately explain the evidence obtained through systematic and
increasingly precise observations and controlled experiments. Even
though all the pioneers – Copernicus, Vesalius, Galileo, Newton and
later, Darwin – were devout Christians working from within the tra-
ditional medieval view of the world derived from parts of Greek phi-
losophy and the Bible, they managed to set a process in motion which
ended up overturning the inherited framework.
What is even more important is that despite religious resistance,
the scientic revolutionaries were not so compelled by the forces of tra-
dition that they felt forced to “harmonize” their theories and methods
with those prescribed by Aristotle and the Bible: Had that been the case,
the new science would have died in its cradle. e Copernican theory
of sun-centered universe was not absorbed back into the ancient earth-
centered universe of Ptolemy, nor was Darwin’s theory of natural selec-
tion contorted to make it appear as if it was in harmony with the Bible.
19 Yuval Noah Harari, 2011, p. 250-251.
 d Science in Saron
Despite initial condemnation on the part of religious forces, it was the
bastions of tradition that had to capitulate to the force of evidence. (Yes,
there are creationists among fundamentalist Christians who still believe
in the literal truth of the creation story, but they are opposed by the
mainstream of Christianity.) e metaphysical speculations of the early
natural philosophers eventually had to give way to the experimental
method, which involved precise measurement and quantication.
In India, on the other hand, the forces of tradition have managed
to overpower and tame any idea that threatened to challenge the es-
sential Vedic outlook of the primacy of consciousness, or spirit. History
of Indian science abounds in examples of self-censorship by otherwise
ne minds; whenever they perceived a contradiction between the Pura-
nas and the mathematical astronomy of the Siddhantas, for example,
some of our well-known astronomers allowed the Puranas to overrule
the Siddhantas. Disheartening examples include Brahmagupta in the
7th century opposing Aryabhata’s theory of eclipses in favor of Rahu
and Ketu, as well as Yajñ̃eśvara Rode in the 17th century “crushing the
contradictions” that the Copernican astronomy posed to the Puranic
worldview.20 When confronted with conicting arguments, our learned
men did not stand up for what they knew to true and backed by bet-
ter evidence. For the most part, they chose to kneel before the Eternal
Truths of Vedas and Puranas. e forces of conservatism and conformi-
ty have been so deeply entrenched in the system of rituals, social habits,
and beliefs that govern our society, that our learned men did not have
to be hauled up before an Inquisition (as Galileo was) to force them to
renounce what they knew to be true – they did that willingly, on their
own volition.
e same compulsion to let the Vedas and Puranas have the last
word is evident in how the torch-bearers of the Indian Renaissance
co-opted scientic theories of physics and biology. e current crop of
heritage-makers, including the Prime Minister and the academics who
made the Science Congress so memorable are travelling down the road
carved out by two of the most illustrious leader of Indian Renaissance,
Swami Dayananda and Swami Vivekananda. Like the two swamis, they
20 See Christopher Minkowski, 2001; Robert Fox Young, 2003.
Who Discovered the Pythagorean eorem? c   
too are intent on picking out those modern scientic ideas and methods
that they can then fuse with the Vedas and the Puranas.
If history is any guide, the rhetorical illusion of “harmony” be-
tween modern science and traditional views has only served the cause
of the orthodoxy in India. Far from being a source of critical thinking
that accepts that our holy books, our ancestors, and our traditions could
be wrong; far from accepting that the old ways must be given up if they
don’t measure up to best available evidence, this celebration of “har-
mony” has only co-opted science into religious dogmas. is road leads
not to science, but to pseudoscience – whitewashing pet ideas to make
them look as if they are scientic.
Fabrication of heritage is, thus,s a process of domesticating the past,
turning it into stories that serve our purposes today.
Presentism, or anachronism, is how the past is domesticated and
history turned into heritage.
Presentism means simply this: to see the past through the lens of
the present. It has been called the “fallacy of nunc pro tunc” which is
Latin for “now for then.21 In history of science (and intellectual his-
tory more generally), presentism works by simply introducing contem-
porary conceptual categories and aims into the depictions of what the
“scientists” of earlier epochs were trying to do.
Professional historians are taught to recognize this fallacy of pre-
sentism and are trained to avoid it with all their might. “e past is a
foreign country: they do things dierently there” is the mantra of pro-
fessional historians. 22 e objective of history then become to study the
past ideas and practices within their own social-cultural milieu.
21 David Hackett Fischer, 1970, who goes on to add: [the substitution of now for
then is the] “mistaken idea that the proper way to do history is to prune away the
dead branches of the past, and to preserve the green buds and twigs which have
grown into the dark forest of our contemporary world.” P. 135.
22 is is the opening line of e Go-Between, a novel by L. P. Hartley, published in
1953. It is also the title of a well-known book by David Lowenthal, the well-known
historian whose work on heritage we have already referred to.
 d Science in Saron
While historians shun presentism as best as they can, those who
peddle heritage nd it indispensable. e whole purpose of fabricating
a heritage is to infuse the past with present meanings. is requires that
the present be projected back into the past. For our purpose at hand – to
understand how history of science is saronized – we have to under-
stand how conceptual categories available to modern science (genetic
science, quantum physics, nuclear energy and such) are read back into
the minds of our ancestors. In this book, especially in the nal chapter,
we will examine the use of resemblances and parallelisms that are de-
ployed to make such projections look reasonable and plausible.
Presentist history is not just bad history; it is dangerous history as
well. I agree with Eric Hobsbawms observation that “the most usual
ideological abuse of history is based on anachronism rather than lies.
is kind of history, again quoting Hobsbawm:
is the raw material for nationalist or ethnic or fundamentalist ideologies, as
poppies are the raw material for heroin addiction. e past is an essential ele-
ment, perhaps the essential element in these ideologies. If there is no suitable
past, it can always be invented….the past legitimizes. e past gives a more
glorious background to a present that does not have much to celebrate. 23
e other major tool for fabricating a suitable heritage is to cordon o
your own past from the rest of the world. I believe there is an absence of
a serious and honest comparative perspective in the Hindu nationalist
history of science. Or rather, to put a ner point on this statement, the
comparative perspective is not entirely absent from their analysis, but
it is deeply colored by what can only be called a “jagat-guru complex”:
invariably, India appears as the giver of science, but never a taker.
While this kind of history might be tonic for the Indian ego, it hap-
pens to be bad history. It is bad history for the same reason not stepping
outside the boundary of your village limits what you can see and expe-
rience. It is bad history because it does not allow you to ask new and
interesting questions about social and cultural dierences that might
23 Eric Hobsbawm, 1997, p. 7, 5.
Who Discovered the Pythagorean eorem? c   
have made a dierence in the trajectories that science and technology
followed in dierent societies.
What I nd even more distorting about this kind of Indo-centric
historiography is that it fails to see and acknowledge how ideas cross
national and cultural boundaries: circulation of ideas did not have to
wait for the World Wide Web; it has been a part of human history from
the very beginning. I share Joseph Needham’s call for taking what he
calls an ecumenical view of the world:
e dierent civilizations did have scientic interchanges of great importance.
It is surely quite clear by now that in the history of science and technology, the
Old World must thought of as a whole. 24
Once we see the Old World as an interconnected whole, we have no
choice but to see our civilization as one among others bound to them
by mutual exchange of goods, people and ideas. Ideas were not always
radiating from India to the rest of the world, but also coming into India
from the rest of the world. Like every other sister civilization, we were
givers and we were takers, with no monopoly on giving.
As the reader will discover in the rst three chapters, once we get
over our Jagatguru complex and see India as one in the network of civi-
lizations, a newer, more complex appreciation of India’s achievements
begins to take shape.
Before I conclude this introduction, I would like to share with the read-
ers the story of how I came to write this book.
Sometime in January 2015, immediately following the Science
Congress in Mumbai, I received a call from a national weekly magazine
(that shall remain unnamed) to write a piece analyzing the historical
claims that were made at that venue.
As this was an issue that was on my mind anyway, I immediately got
to work. Within a week or so, I sent the magazine not one, but two es-
says – one on Pythagoras and the other on plastic surgery, genetics etc.
For reasons that were never explained, the magazine sat on its hands for
three weeks. Naturally, I withdrew the essays from consideration.
24 Joseph Needham, 1969, p. 16.
 d Science in Saron
If the editor of the magazine is reading these words, please know I
am sincerely grateful to you for not publishing those essays!
I realized I had much more to say on these matters than the mere
two thousand words that I was limited to for the magazine. I decided
to expand my mission and to present an exhaustive analysis of these
issues. I proposed the idea to my friend, Asad Zaidi, the publisher of
ree Essay Collective. He gave me the green light and I went to work
on this book.
e product of my labors is now in your hands.
Who Discovered the Pythagorean eorem? c   
Cha pt er 1
Who Discovered the Pythagorean eorem?
1. Introduction
Poor Pythagoras! at gentle vegetarian1 mystic-mathematician would
have never imagined that over 2,500 years aer his time, hearing his
name would have the same eect on some Indians as showing a red rag
has on a bull!
At the Indian Science Congress earlier this year, Pythagoras and
his theorem were mentioned by many very important persons who
went out of their way to make him look like an imposter basking in
the lime-light that rightfully belongs to us, the brainy Indians. It is not
that Pythagoras doesn’t need to be taken down a notch or two, for the
evidence that he was the original discoverer of the theorem named aer
him is simply not there. But that does not by itself mean that the vacated
pedestal now belongs exclusively to our own Baudhāyana and his fellow
priest-artisans who used ropes to build geometrically complex Vedic
altars. And yet, this is exactly what was clearly and repeatedly asserted
at the Science Congress.
Here is what the Minister of Science and Technology, Dr. Harsh
Vardhan had to say on the matter:
1 For reasons that continue to puzzle historians, Pythagoras, who abstained from
meat-eating, hated beans. e “Pythagorean diet” was bean-free, as well as meat
and sh free. His followers had to swear to follow this diet.
 d Science in Saron
Our scientists discovered the Pythagoras theorem, but we gave its credit to the
Greeks. We all know that we knew bijaganit much before the Arabs, but seless-
ly we allowed it to be called Algebra. …whether related to solar system, medi-
cine, chemistry or earth sciences, we have shared all our knowledge selessly
e Minister was backed by Dr. Gauri Mahulikar, a Sanskrit scholar
from Mumbai University:
In the Śulvasutras, written in 800 BCE, Baudhāyana wrote the geometric for-
mula now known as Pythagoras theorem. It was written by Baudhāyana 300
years before Pythagoras.…2
Between the two of them, the Minister and the Professor proved
a theorem dear to the Indian heart, namely: we are not just brainy, but
big-hearted as well. We are so big-hearted that we let the likes of Py-
thagoras to claim priority for what our own Baudhāyana accomplished.
We are so big-hearted that we selessly give away our intellectual rich-
es – from the geometry of Śulvasūtras to advanced mathematical and
medical concepts – to the rest of the world. Giving is in what we do.
Compared to the rest of the howlers at the Science Congress – the
ancient interplanetary ying machines, the alchemist cows turning
grass into gold, for example3 – the priority-claim for Baudhāyana has at
least one virtue: it is not entirely insane. ere is a substantial nugget of
truth hidden in an Everest of hype.
ere is no doubt that our śulvakaras had indeed mastered the Py-
thagorean conjecture thoroughly and used it in ingenious ways to create
Vedic altars of dierent areas, while conserving the shapes. ey were
the rst to state it unambiguously. But they were neither alone, nor the
rst in having this understanding. e rst recorded evidence for this
conjecture dates back to some 1800 years BCE and it comes from Meso-
2 See
of-ancient-india/article6754106.ece. e priority of ancient priest-crasmen
who composed the Śulvasūtras over Pythagoras has a long history. As early as
1906, Har Bilas Sarda was cheering for Baudhāyana over Pythagoras in his book,
Hindu Superiority, pp. 286-287. More recently, Subhash Kak has claimed that the
geometry of the Vedic altars contains – in a coded form – advanced astrophysi-
cal knowledge such as the exact length of the tropical year and the lunar year, the
distance between the sun and the earth, the distance between the moon and the
earth in lunar diameters. See Kak, 2005.
3 India Today has very helpfully listed these howlers. See http://indiatoday.intoday.
Who Discovered the Pythagorean eorem? c   
potamia, the present day Iraq. e rst proof comes from the Chinese,
preempting the Euclidean proof by a couple of centuries, and the Indian
proof by at least 1000 years. Even though Pythagoras was not the rst to
discover and prove this theorem, it does not diminish his achievement.
He remains an extremely inuential gure not just for history of mathe-
matics, but history of science as well. Pythagoras and his followers were
the “rst theorists to have attempted deliberately to give the knowledge
of nature a quantitative, mathematical foundation.4 Giants of the Scien-
tic Revolution, including Johannes Kepler and Galileo Galilei walked
in the footsteps of Pythagoras.
In this chapter, we will start with a quick refresher on the Pythago-
rean eorem. We will follow this with a straightforward narrative of
the dierent formulations and uses of this theorem, starting with an-
cient Egypt and Mesopotamia, followed by ancient Greece, India and
China. e order is not chronological, and nor does it represent a chain
of transmission. While we have evidence of the Greeks getting their
start in geometry from the Egyptians and the Mesopotamians, it is quite
likely that this conjecture was independently discovered in India and
e idea of following the trail of the Pythagorean eorem from
Mesopotamia to China is simply to place ancient India as one among
other sister civilizations. It is only through a comparative history of the
idea behind this famous conjecture that we will be in a position to judi-
ciously assess India’s contribution.
2. What is the Pythagorean eorem?
Before proceeding any further, let us be clear on what the Pythagorean
eorem is all about. Most of us learnt it in middle or high school, but
it is a good idea to quickly review it.
e theorem simply states that in a right-angle triangle, the square
on the hypotenuse is equal to the sum of the squares on the two sides.
(A hypotenuse, to joggle your memory, dear reader, is the longest side
of a right-angle triangle which also happens to be the side opposite the
right angle).
4 G.E.R. Lloyd, 1970, p. 26.
 d Science in Saron
In gure 1, c is the hypotenuse, while a and b are short and long
sides of the right angle triangle, respectively.
So the theorem simply states the following
c2 = a2 + b2, a relationship that is represented in gure 2.
is theorem seems simple and intuitive. at is why it has been
nominated as a calling-card for the human species to be beamed into
Figure 1
Figure 2
Who Discovered the Pythagorean eorem? c   
the outer space.5 e idea is that any intelligent beings, anywhere in
the universe, would recognize its logic – and even perhaps be moved
by its beauty. Eli Maor reports that in a 2004 “beauty contest” organ-
ized by the journal Physics World, the top winners were Euler’s formula,
Maxwell’s four electromagnetic eld equations, Newton’s second law,
followed by the Pythagorean equation. Not bad for an equation that has
been around for more than 3000 years.6
It is also one of the most frequently used theorems in all of math-
ematics. Algebra and trigonometry make use of the equation. Its most
obvious and practical use is in the building trade, where it is used for
constructing walls perpendicular to the ground, or for constructing
perfect squares or rectangles.
is use follows from the fact that the theorem is reversible which
means that its converse is also true. e converse states that a triangle
whose sides satisfy a² + b² = c² is necessarily right angled. Euclid was the
rst (1.48) to mention and prove this fact. So if we use lengths which
satisfy the relationship, we can be sure that the angle between the short
and the long side of a triangle will have to be right angle.
Any three whole numbers that satisfy the Pythagorean relationship
and yield a right angled triangle are called Pythagorean triples. e most
obvious and the easiest example of these triples is 3, 4, 5. at is to say:
32 + 42 = 52 or
9+16 = 25.
at means that any triangle with sides 3, 4 and 5 will be a right-
angle triangle. As we will see in the rest of this chapter, this method for
building right-angle structures was known to all ancient civilizations,
not just India. is method is still used by carpenters and architects to
get a perfect perpendicular or a perfect square.7
5 In his classic work of science ction, From the Earth to the Moon (1865), Jules Verne
mentions a German mathematician who suggested that a team of scientists go to
Siberia and on its vast plains, set up an enormous, illuminated diagram of Pythago-
rean theorem so that inhabitants of the Moon would see that we are trying to get
in touch. Verne’s un-named mathematician has been identied as Carl Friedrich
Gauss. See Eli Maor, p. 203.
6 Maor, p. xii.
7 If you want to construct a perfect square and you don’t have anything but a tape
measure and a marker try this: draw a straight line roughly 3 units long where you
 d Science in Saron
While all right-angle triangles will bear the relationship described
by c2= a2+b2, not all a and b lengths can be expressed as whole numbers
or as ratios of whole numbers. You can see it for yourself: try calculat-
ing c for a=4 and b=5, or a=7, b= 9. In both cases, you will see that the
c cannot be expressed as a whole number. Actually there are only 16
set of whole numbers below 100 that t into the Pythagorean equation.
ere is one particular number for a and for b that puzzled all an-
cient civilizations that we have records from. at number is one. Imag-
ine a square with side measuring one unit. Now draw a diagonal cutting
the square into two right angle triangles.
e simple question is this: how long is the diagonal?
Let us see:
For a right angle triangle, we know that
c2 = a2+b2
want to locate the corner of the square. On the other side of the corner, draw a line
4 units long, roughly vertical to the rst line. Now use the tape to make sure that
the edges of the two lengths are exactly 5 units apart. e angle between the two
corner lines will be exactly 90 degrees.
Who Discovered the Pythagorean eorem? c   
In this case,
c2= 12 + 12
c2= 2, therefore c=
If you recall your middle-school mathematics, the symbol
stands for square root. Square root of a number is simply a value which,
when multiplied by itself, gives that number.
In the above case, in order to nd how long the hypotenuse is, we
have to nd out square root of two, or in other words, nd out that
number which, when multiplied with itself will produce the number 2.
Try guring out the square root of number 2. You will notice some-
thing strange: you simply cannot express the number as a fraction of
two whole numbers. What you nd is that the decimal fractions of the
number that will give you 2 when multiplied by itself simply go on and
on, without ending and without repeating themselves. For practical
purposes, square root of 2 is taken to be 1.4142136 but the number can
go on forever.
Numbers such as these were given the name “alogon” by the Greeks
which means “unsayable or inexpressible. We call them irrational num-
Irrational numbers were known to all the ancient civilizations that
are examined in this chapter. All of them tried to represent these num-
bers by using rough approximations. Only among the Greeks, however,
it led to a crisis of spiritual dimensions. We will shortly explain why, and
what they did about it. But we have to start our story from the begin-
ning in Egypt and Mesopotamia.
3. Egypt and Mesopotamia
If anyone can take credit for being the rst to gure out the Pythagorean
eorem, they have to be the unknown and unnamed builders, land-
surveyors, accountants and scribes of ancient Egypt and Mesopotamia
(the land we know as Iraq today) sometime between 2000 to 1700 BCE.
Just as ancient India had its śulvakaras who used a length of rope
to map out altar designs, ancient Egypt had its harpedonaptai, the “rope
stretchers”. If Herodotus, the Greek historian who lived in the h cen-
 d Science in Saron
tury BCE is to be trusted, these rope-stretchers were surveyors sent out
by the pharaohs to measure the farm land for tax purposes every time
the river Nile would ood and change the existing boundaries. ey are
rightly considered the true fathers of geometry, which literally means
measurement (metery) of earth (geo): they were the land surveyors sent
out by the pharaohs to measure the land for taxation purposes every-
time the river Nile would ood and change the existing boundaries.
One would think that a civilization that built the Great Pyramids8
would have mastered the right-angle rule and much-much more. In-
deed, it has been claimed by Martin Bernal in his well-known book, e
Black Athena, that the Greeks learned their sciences and mathematics
from Egypt, with its roots in Black Africa. is is not the right forum
to resolve this huge controversy, but Bernal’s claims regarding the ad-
vanced state of mathematics and astronomy in Egypt have been chal-
lenged, and are no longer held to be credible by most historians.9
e two main mathematical papyri – the Ahmes Papyrus (also
called the Rhind Papyrus) that dates back to 1650 BCE and the so-called
Moscow Mathematical Papyrus that contains text written some 1850
BCE – don’t make any reference to this theorem. While both these pa-
pyri contain geometrical problems like calculating the areas of squares,
volume of cylinders (for the jars they stored grain in), circumference
and areas of circles, the familiar Pythagorean relation is not there. Yet
it is hard to imagine how the pyramid makers could have laid the foun-
dations of the square base of pyramid without the familiar 3, 4, 5 rule
described in the previous section.
A more recent nd has thrown new light on this issue: the so-called
Cairo Mathematical Papyrus, which was unearthed in 1938 and con-
tains materials dating back to 300 BCE shows that the Egyptians of this,
much later era, did know that a triangle with sides 3,4,5 is right-angled,
8 e best known of them, the Great Pyramid at Gizeh, built around 2600 BCE was
the largest building of the ancient world. It rose 481 feet above the ground, with
four sides inclined at an angle of 51 degrees with the ground. Its base was a perfect
square with an area of 13 acres – equal to the combined base areas of all the major
cathedrals in all of Europe. Some 400,000 workers labored on it for 30 years. Bur-
ton, 2011, p. 58.
9 See the important paper by Robert Palter (1993) titled, ‘Black Athena, Afro-cen-
trism and the History of Science’.
Who Discovered the Pythagorean eorem? c   
as are triangles with sides 5, 12, 13 and 20,21,29. is papyrus contains
40 problems of mathematical nature, out of which 9 deal with the Py-
thagorean relationship between the three sides of a right triangle.10
We may never get the complete story of Egyptian mathematics, as
the ancient Egyptians wrote their texts on scrolls made out of at strips
of pith of the papyrus reeds that grew abundantly in the marshes and
wetlands of the region. e problem with papyrus is that it is perishable.
But the Mesopotamian civilization that grew not too far away from
Egypt on the fertile land between the rivers Tigris and Euphrates in
modern-day Iraq is a whole dierent story in so far historical records
go. e clever Sumerians, Assyrians and Babylonians who successively
ruled this land have le us a huge library of their literary and mathe-
matical works chiseled on clay tablets which were dried in the sun (and
oen baked in accidental res) and are practically indestructible.
As in Egypt, the Mesopotamian mathematics and geometry grew
out of administrative needs of the highly centralized state. Temples of
local gods and goddesses also needed to keep accounts of the gis and
donations. is led to the ourishing of many scribe-training schools
where men (they seem to be all men) learned how to write and do ele-
mentary arithmetic. Fortunately for historians, the Mesopotamian peo-
ple chose a non-degradable material – wet clay that their rivers brought
in plenty – to write upon. ey used a reed with an edge – quite like our
kalam – that could make wedge-shaped marks on the clay. ese tablets
were then dried in the sun which made them practically indestructi-
ble.11 Literally thousands of these clay tablets have been recovered and
deciphered, including the famous Flood Tablet which tells the story of
a great ood, very similar to the Biblical story of the ood and Noah’s
A small fraction of the tablets recovered from schools for scribes
contain numerical symbols which were painstakingly deciphered by
Professor Otto Neugebauer at Brown University, USA in the 1930s. It is
now well-established that the Babylonian people had developed a pretty
10 Burton, 2011, p. 78.
11 Clay tablets were also recyclable: if a scribe made an error, he could simply knead
his tablet into a ball and make a fresh tablet out of it.
 d Science in Saron
ingenious system that allowed them to use just two symbols – a wedge
for the number one and a hook-shaped symbol for the number ten – to
represent and manipulate any number, however large. ey could do
that because they had gured out what is called place value, in which the
value of a number changes with the position it occupies. What is more,
they also started using a symbol indicating empty space – a forerunner
of zero. (Place-value and zero will be examined in the next chapter).
But what is of special interest to us are two tablets which have an
iconic status in history of mathematics, namely, Plimpton 322 and a
tablet called YBC7289 housed in Columbia and Yale universities, re-
spectively. ese tablets reveal that the Mesopotamians knew how to
gure out Pythagorean triples, and could also calculate square roots.
Some historians conjecture that Plimpton might even be the rst record
of trigonometry anywhere in the world.12
Wikipedia provides a very good description of Plimpton 322:
Plimpton 322 is partly broken clay tablet, approximately 13cm wide, 9cm tall,
and 2cm thick. New York publisher George Arthur Plimpton purchased the
tablet from an archaeological dealer, Edgar J. Banks, in about 1922, and be-
queathed it with the rest of his collection to Columbia University in the mid-
1930s. e tablet came from Senkereh, a site in southern Iraq corresponding
to the ancient city of Larsa. e tablet is believed to have been written about
1800 BC, based in part on the style of handwriting used for its cuneiform script.
A line-drawing of Plimpton 322 (Figure 4a) and a transcript of cu-
neiform numerals into modern numbers (Figure 4b) are given below.
What is written on it that makes it so important? It has four columns of
numbers and it appears that there was a h column on the le which
broken o. e rst column from the right is simply a column of serial
numbers, from 1-15, while the other three columns contain 15 numbers
written in Cuneiform script.
What do these columns of numbers mean? is tablet was rst de-
ciphered by Otto Neugebauer and his colleague Alfred Sachs in 1945.
Without going into details which can now be found in any standard
text book of history of mathematics, they concluded that “the num-
bers b and d in the second and third columns (from right to le) are
12 “If the missing part of the tablet shows up in the future…. Plimpton 322 will go
down as history’s rst trigonometric table.” Eli Maor, 2007, p. 11.
Who Discovered the Pythagorean eorem? c   
Width Diagonal
1:59:00:15 1:59 2:49 1
1:56:56:58:14:50:06:15 56:07 1:20:25 2
1:55:07:41:15:33:45 1:16:41 1:50:49 3
1:53:10:29:32:52:16 3:31:49 5:09:01 4
1:48:54:01:40 1:05 1:37 5
1:47:06:41:40 5:19 8:01 6
1:43:11:56:28:26:40 38:11 59:01 7
1:41:33:45:14:03:45 13:19 20:49 8
1:38:33:36:36 8:01 12:49 9
1:35:10:02:28:27:24:26 1:22:41 2:16:01 10
1:33:45 45 1:15 11
1:29:21:54:02:15 27:59 48:49 12
1:27:00:03:45 2:41 4:49 13
1:25:48:51:35:06:40 29:31 53:49 14
1:23:13:46:40 56 1:46 15
Figure 4a. Line drawing of Plimpton 322. Source: Eleanor Robson
Figure 4b. Transcription of the Plimpton 322 tablet using modern digits. Source
Clark University, Department of Mathematics and Computer Science http://¬)
 d Science in Saron
Pythagorean numbers, this means that they are integer solutions to
d2=b2+l2 where d and b stand for the diagonal and the leg of the triangle
respectively.”13 To use modern terminology, the numbers tabulated in
Plimpton 322 are Pythagorean triples, which as dened in section 2, are
whole numbers that fulll the Pythagorean relation, a2+b2=c2.
In other words, Plimpton 322 is the work of some unknown Bab-
ylonian mathematician, or a teacher or a scribe trying to nd sets of
whole numbers which will automatically generate a right angle. What is
most striking is that some of the triples listed in the tablet are simply too
large for a random, hit-and-trial discovery.14 ere are many guesses as
to how they managed to get these values, but nothing denite can be
said about their method.
e second tablet that has received great amount of scrutiny is
called YBC 7289, making it the tablet number 7289 in the Yale Babylo-
nian Collection. e tablet dates from the old Babylonian period of the
Hammurabi dynasty, roughly 1800-1600 BCE.
is celebrated tablet shows a tilted square with two diagonals,
with some marks engraved along one side and under the horizontal
diagonal. A line-drawing of the tablet and a sketch in which the cunei-
form numerals are written in modern numbers is given below (Figures
5a and 5b on the next page):
e number on the top of the horizontal diagonal when translated
from the base-60 of Mesopotamians to our modern 10-based numerals,
gives us this number: 1.414213, which is none other than square root
of 2, accurate to the nearest one hundred thousandth. e number be-
low the horizontal diagonal is what we get on multiplying the 1.414213
with the length of the side (30) which, in modern numbers comes to
42.426389. is tablet is interpreted as showing that the Mesopotami-
ans knew how to calculate the square root of a number to a remarkable
ese two tablets are the rst evidence we have of the knowledge of
what we today call Pythagorean eorem.
13 Otto Neugebauer, p. 37.
14 For example, row 4 has the following triples 12,709 (the short side), 18,541 the
hypotenuse, and 13,500 the third side of a right angle triangle. See Katz, p. 20.
Who Discovered the Pythagorean eorem? c   
Figure 5b. YBC 7289 transcribed into modern numerals. Source: McTutor History of
Mathematics Archives at
Figure 5a. Line-drawing of the Yale tablet, YBC 7289.
Source: Mathematical Association of America.
 d Science in Saron
4. Pythagoras, the Pythagoreans and Euclid
Pythagoras (about 569 BC-about 475 BC) is perhaps the most misun-
derstood of all gures that have come down through history. We all
know him as the man who gave us the theorem that – rightly or wrongly
– bears his name. But for Pythagoras and his followers, this theorem
was not a formula for doubling the square or building precise perpen-
diculars, as it was for all other civilizations of that time. It is a safe bet
that neither Pythagoras nor his followers ever lied a length of rope,
got down on their knees to measure and build anything, for that kind of
work was seen t only for the slaves.
e real – and path-breaking – contribution of Pythagoras was the
fundamental idea that nature can be understood through mathematics.
He was the rst to imagine the cosmos as an ordered and harmonious
whole, whose laws could be understood by understanding the ratios
and proportions between the constituents. It was this tradition that was
embraced by Plato, and through Plato became a part of Western Chris-
tianity, and later became a fundamental belief of the Scientic Revolu-
tion expressed eloquently by Galileo: “e Book of Nature is written in
the language of mathematics.
It is well-recognized that Pythagoras himself was not the original
discoverer of the relationship between three sides of a right-angled tri-
angle. Greek accounts written by his contemporaries are very clear that
Pythagoras got the idea from the Mesopotamians and perhaps Egyp-
tians, among whom he spent many years as a young man. e words
of Sir omas Heath, the well-known historian of Greek mathematics,
written as long ago as 1921, are apt:
ough this is the proposition universally associated by tradition with the
name of Pythagoras, no really trustworthy evidence exists that it was actually
discovered by him.15
15 Heath, 1921, p. 144. Our esteemed Minister and the Professor were really tilting
at windmills. Greeks have always admitted that they learned their geometry from
Egyptians and Mesopotamians. All serious historians of mathematics would agree
with Sir Heath’s words.
Who Discovered the Pythagorean eorem? c   
Neither is there any clear-cut evidence that Pythagoras or his fol-
lowers oered a proof of the theorem. ose who attribute the proof to
Pythagoras cite as evidence stories about him sacricing a number of
oxen when he proved the theorem. Apparently the story about oxen be-
ing sacriced comes from a writer by the name of Apollodorus. But as
omas Heath has argued, the passage from Apollodorus does mention
the sacrice without mentioning which theorem was being celebrated.
e sacrice story has been challenged on the grounds of the Pythago-
reans’ strictures against animal sacrices and meat-eating.16
e rst Greek proof of the theorem appears in Euclid’s classic of
geometry called Elements, which was written at least three centuries af-
ter Pythagoras. Euclid (around 365 BCE-275 BCE) provides not one,
but two proofs of this theorem – theorem 42 in Book I, and theorem 31
of the Book VI. Nowhere does Euclid attribute the proofs to Pythago-
Why then did this theorem get Pythagoras’ name? No one knows
for sure. It is possible that Greeks were following a tradition of attribut-
ing new ideas to well-recognized sages – a practice that is very common
in Indian scientic and spiritual literature as well. Pythagoras, aer all,
was no ordinary man: he had a semi-divine status among his followers.
While he did not discover it or prove it, this equation played a most
dramatic – one can say, catastrophic – role in Pythagoras’ condence in
mathematics and numbers. To understand the catastrophe, one has to
understand the fundamental place numbers and ratios occupied in the
Pythagorean view of the world.
Pythagoras was a mystic-mathematician, a cross between “Einstein
and Mrs. Eddy” to use Bertrand Russell’s words.18 Or one can say that he
was a mystic with a mathematical bent of mind. He saw contemplation
of mathematical proportions and ratios as the highest form of medita-
tion that can bring the mind in tune with the Ultimate Reality that he
16 Heath, 1921, pp. 144-145.
17 e rst proof, I:42, is generally attributed to Eudoxus, who was a student of Plato,
while the second proof is attributed to Euclid himself. See Eli Maor, chapter 3.
18 Mrs. Mary Baker Eddy founded a spiritualist movement called Christian Science
in 1879. e quotation is from Russell’s well-known History of Western Philosophy,
p. 31.
 d Science in Saron
believed existed independently of material stu. What is more, he be-
lieved that mathematical knowledge can purify the soul and free it from
the cycles of rebirth. (Yes, his spiritual beliefs overlapped with the belief
system prevalent in India. More on this below).
Pythagoras was born in 571 BCE (which makes him a rough con-
temporary of Gautam Buddha in India and Confucius in China) on the
island of Samos in the Aegean Sea, just o the coast of modern-day
Turkey. He spent many years of his youth in Egypt and later in Mesopo-
tamia. In both places, he immersed himself in the spiritual and math-
ematical traditions of the host cultures. ere is no evidence that he
travelled as far east as India, but there is a strong possibility that he
picked up the belief in immortality of the soul and its reincarnation
from Hindu teachers who were probably present in the courts of Per-
sian kings before Alexander opened a direct line between India and
Greece when he came as far as the Indus river in 326 BCE. It was his
belief in reincarnation that led him to oppose eating meat and stick to
a bean-free vegetarian diet – a dietary practice which is as un-Greek
today, as it was then. Like the Hindus, he believed in purication of the
soul through contemplation of the Ultimate Reality in order to break
the chain of rebirth – except that for him, mathematics was the form
that the contemplation of the Ultimate took.19
Could he not have picked up the geometry of Baudhāyana and oth-
er śulvakaras as well who are estimated to have lived anywhere between
800-300 BCE? It is entirely possible, although the Greek historians of
that time have le no record of it. e same historians, on the other
hand, have le meticulous records of what he learned from Mesopota-
mians and Egyptians.
But wherever Pythagoras learned this theorem from, it played a
unique role in his philosophy. It led to the discovery of irrational num-
19 For pre-Alexandrian contacts between Indians and Greeks, see McEvilley, 2002,
ch. 1. e possibility of Pythagoras learning his beliefs in immortality and rebirth
of the soul from Indian philosophers is accepted by many scholars. See Kahn
2001 and McEvilley,2002 for example. One of the many stories that are told about
Pythagoras is that he once stopped a man from beating a dog by telling him that
he recognized the dog as an old friend, reincarnated. His followers believed that
Pythagoras could recall many of his earlier births and that was one reason they
treated him as a divine man.
Who Discovered the Pythagorean eorem? c   
bers (see section 2) which led to a great spiritual crisis for himself and
his followers. To understand why a mathematical result would lead to a
spiritual crisis, some background is needed.
While we don’t have any evidence for Pythagoras discovering
the Pythagorean eorem, his role in discovering the laws of musical
sounds is well-attested. It appears that one day as he was walking past
a blacksmith’s workshop, he was intrigued by the sounds coming from
within. So he went in to investigate and found that the longer the sheets
of metal that were being hit by the blacksmiths hammer, the lower was
the pitch of the sound. When he came back home, he experimented
with bells and water-lled jars and observed the same relationship: the
more massive an object that is being struck or plucked, the lower the
pitch of the sound it produces. He experimented with strings and ob-
served that the pitch of the sound is inversely proportional to the length
of the string that is vibrating. He gured out that if a string is plucked
at a ratio of 2:1 it produces an octave, 3:2 produces a h, 4:3 a fourth.
is was a pivotal discovery – of far greater importance to Pythag-
oras than the famous theorem he is known for. It made him realize that
human experience of something as subjective as music could be under-
stood in terms of numerical ratios: the quality of what pleases the ear
was determined by the ratios of the lengths that were vibrating. is
was the rst successful reduction of quality to quantity, and the rst step
towards mathematization of human experience.20
e realization that what produces music are certain numerical ra-
tios led Pythagoras to derive a general law: that the ultimate stu out of
which all things are made are numbers. Understand the numbers and
their ratios and you have understood the Ultimate Reality that lies be-
hind all phenomena, which you can only see in your mind, not through
your senses. If all is number – and numbers rule all – then obviously,
we should be able to express that number either as whole numbers (in-
tegers like 2, 6, 144 etc.) or as fractions of whole numbers (for example,
half can be written as one divided by two).
Given how central numbers and numerical ratios were to Pythag-
oras’s view of the unseen reality which humans could access through
20 is interpretation is from Arthur Koestler’s well-known book, e Sleepwalkers.
 d Science in Saron
mathematics, one can understand that the discovery that square root of
two cannot be expressed as either a whole number or a fraction of two
whole numbers would lead to an unprecedented crisis.
is discovery was a direct result of the Pythagorean eorem.
Here is what happened: having understood the right-angled triangle
relationship (i.e., a2+b2=c2) either Pythagoras himself or one of his stu-
dents tried use it to calculate the diagonal of a square whose side is one
unit. ey discovered that they simply can’t get to a denite number
that would terminate somewhere. In other words, they realized that
some lengths cannot be expressed as a number. is shattered their fun-
damental belief that all is number and the ratio of numbers can explain
the order of the cosmos.
e legend has it that Pythagoras swore his followers to complete
secrecy regarding this awful discovery: they were never to disclose the
existence of irrational numbers to anyone. One unfortunate follower by
the name of Hippasus who broke the vow of secrecy was pushed to his
death from a boat into the Mediterranean Sea – so the story goes.
e crisis led to further developments in Greek mathematics. To
begin with, it led to a split between geometry and arithmetic. For Py-
thagoras, all numbers had shapes. But irrational numbers could not be
expressed in shapes. e existence of irrationality was proven later by
Aristotle and Euclid.
To conclude this section: yes, Pythagoras was not the original dis-
cover of this theorem. But he put it to a dierent use than it was any-
where else. e truly important discovery of Pythagoras was not the
famous theorem, but the laws of music and the existence of irrational
5. Śulvasūtras
We now come to the central theme inspired by the Minister and the
Professor mentioned earlier. To recapitulate, they asserted that what
the world knows as the Pythagorean eorem should be renamed aer
Baudhāyana who discovered it in 800 BCE, which is nearly 200 years
before Pythagoras was even born.
Who Discovered the Pythagorean eorem? c   
As we have already seen, this claim is factually incorrect: there is
a great amount of evidence chiseled into the Mesopotamian clay that
proves that Pythagoras was already outdone before even Baudhāyana
was born! But if we let go of this madness for who came rst, we will
see that Baudhāyana and his colleagues who lived and worked some-
where between 800 to 500 BCE (or between 600-200 BCE, according to
some estimates21) were extremely creative artisans-geometers in their
own right. eir accomplishments don’t need to be judged from the Py-
thagorean or the Greek lens.
What are these Śulvasūtras that we keep hearing about? Who com-
posed them? When? Why? ese are some of the questions we will try
to answer in this section.
As mentioned earlier, śulva means a cord or string, while sutras are
short, poetic sentences that are easy to memorize. ese “sutras of the
cord” are a part of the kalpa-texts that make up one of the six Vedan-
gas, or limbs of the Vedas, each dealing with a specialized topic ranging
from grammar to astronomy. e kalpa literature specializes in ritual
matters, including building of re altars, or vedis, some of them very
intricate in shapes and sizes.
e most succinct denition is provided by George ibaut, the
German philologist who rst translated these sutras:
e class of writings, commonly called Śulvasūtras means the “sutras of the
cord”. Śulvasūtras is the name given to those portions or supplements of the
Kalpasūtras which treat of the measurement and construction of dierent ve-
dis, or altars, the word śulva referring to the cords which were employed for
those measurements. I may remark at once that the sutras themselves don’t
make use of the word śulva; a cord is regularly called by them rajju (rope).22
Out of four extant texts, the two most important are those by
Baudhāyana and Āpastamba. Next to nothing is known about these
men, but “most likely they were not just scribes but also priest-cras-
men, performing a multitude of tasks, including construction of the
21 Agathe Keller (2012) dates Baudhāyana to 600 BCE.
22 ibaut 1992[1875], p. 417. George ibaut is an interesting gure in Indology.
He was born in Germany in 1848 and later moved to England to work with Max
Muller. In 1875 he became a professor of Sanskrit at Benares Sanskrit College.
It was here that he produced his studies on the Śulvasūtras. But his real claim to
fame was his work on mimamsa texts. See Keller, 2012, pp. 261-262.
 d Science in Saron
vedis, maintaining agni and instructing worshippers on appropriate
choice of sacrices and altars.23
If it was the need for repeated measurements of land in the ood-
zones of rivers that gave birth to geometry in Egypt and Mesopotamia,
it was the need for precision in Vedic rituals that gave birth to geometry
in India. In order for the Vedic yagnas to bear fruit, they had to be car-
ried out precisely according to the guidelines laid out in the Brahmana
texts of Yajurveda: the mantras had to be recited just so, the sacricial
animal quartered exactly at specic vertebra, the altar (vedi) for the sac-
rice had to be constructed exactly following the prescribed shapes and
sizes. us, ritual has been recognized as the source of sciences and in-
deed, by some, of all civilization.24 Let us see how the need for exactness
in ritual led to advances in geometry in ancient India.
To begin with, the shape of the altar was decided by the goal of the
yagna. For example those who desired to go to heaven were required to
construct a falcon (syena in Sanskrit)-shaped vedi because as Taittirīya
Sahitā explained: “the falcon is the best yer among the birds; and
thus he (the sacricer) having become a falcon himself ies up to the
heavenly world.25 For those seeking food, the altar should be in the
shape of a trough (called drona-cit), while those seeking victory over the
enemy were to build an altar in the shape of a rathachakra or a wheel.
What is geometrically challenging about these requirements is this:
• To use ibaut’s words: “every one of these altars had to be
constructed out of ve layers of bricks… every layer was to
consist of 200 bricks [arranged in such a manner] that in all
ve layers, one brick was never lying upon another brick of the
same size and form.
• If this wasn’t challenging enough, the area of every altar, what-
ever its shape – falcon with curved wings, wheel, tortoise,
23 George G. Joseph, p. 327.
24 A. Seidenberg, 1962, proposes that civilization itself has its origin in rituals. We
will discuss the contribution of the ritual horse sacrice (Ashvamedha yagna) in
understanding equine anatomy in ancient India in chapter 3.
25 ibaut, p. 419.
Who Discovered the Pythagorean eorem? c   
trough etc. – had to be equal to 71/2 square purusha, where a
purusha is the height of a man with uplied arms.26
• ere was yet another challenge: every-time the sacrice was
carried out aer the rst construction and consecration, the
area had to be increased by one square purusha, until one
comes to the one-hundred-and-a-half-fold altar. As Seiden-
berg explains, “the sacricer is [symbolically] climbing a lad-
der, his sacricial rank being determined by, or determining,
the area.27
• Here comes the most daunting challenge of all: while the area
had to be increased by one square purusha at each subsequent
construction, the relative proportions of the single parts had to
remain unchanged. In other words, area was to be increased
while preserving the shape of the altar.
• ere is another twist to altar-making which shows the deep
roots of the varna order: If the yajman, or the host of the yag-
na, was a Brahmin, he was required to set up the sacred re at
eight units east of the household re, if a prince, eleven and the
Vaisya twelve.28
Clearly, constructing such altars was no mere “carpentry problem”,
to use Seidenberg’s words, that could be solved with a few “carpenter’s
rules”.29 e technical problems were not trivial, for as ibaut puts it:
Squares had to be found which would be equal to two or more given squares,
or equal to the dierence of two given squares; oblongs had to be turned into
squares and squares into oblongs; triangles had to be constructed equal to given
squares and oblongs and so on….[Even for the most ordinary of vedis] care had
to be taken that the sides really stood at right angles, for would the āhavaniya
re have carried up the oerings of the sacricer to the gods if its hearth had
26 It is not entirely clear how this man of one-purusha height is chosen. Is he any
average sized man, or the yajman hosting the yagna?
27 Seidenberg, 1962, p. 491.
28 Kim Ploer, p. 24. Ploer calls these units “double-paces where a pace equals 15
angulas”. An angula or digit is said to be equal to 14 grains of millet.
29 Seidenberg is right in poking fun at those Hellenophiles who treat any tradition
of geometry that does not justify itself through a Euclidean deduction as merely
 d Science in Saron
not the shape of a perfect square?... [there were also occasions when] a square
had to be turned into a circle of the same area.30
e most important arsenal in the mental tool-kit of the altar-mak-
ers was what we call Pythagorean eorem.
Baudhāyana gave a very close approximation to this theorem,
even though he used four-sided right-angled structures rather than the
right-angled triangle that we are familiar with. Here are three sutras
(1.9-1.13) from Baudhāyana Śulvasūtras which capture the essence of
this theorem, one for the diagonal of a square and another for the diago-
nal of an oblong or rectangle, followed by Pythagorean triples:
1. “e cord which is stretched across in the diagonal of a square (sama-catu-
rasra) produces an area of double the size.
at is: the square of the diagonal of a square is twice as large as
the area of the square.
2. a. “e cord stretched on the diagonal of an oblong (dirgha chaturasra) pro-
duces both areas which the cords forming the longer and the shorter side of an
oblong produce separately.
at is: the square of the diagonal of an oblong is equal to the
square on both of its sides. is is an unambiguous statement of
the Pythagorean theorem.
2. b. “is (2a) is seen in those oblongs the sides of which are 3 and 4, 12 and 5,
15 and 8, 7 and 24, 12 and 35, 15 and 36.
Here, Baudhāyana is enumerating ve Pythagorean triangles,
that is, right-angled triangles whose sides will yield a hypot-
enuse, which when squared will yield twice the area of the two
sides which have the dimensions described in 2a. All three sides
of the resulting triangles can be expressed in whole numbers.31
e numbers in 2b are none other than our old friends, the Pythag-
orean triples. We encountered them rst on Plimpton 322 which dates
back at least a thousand years before Baudhāyana. e Pythagoreans
not only knew about the triples, but had actually worked out a formula
30 ibaut, pp. 420-421.
31 ibaut, p. 422-424.
Who Discovered the Pythagorean eorem? c   
for nding these triples.32 So we can say that Baudhāyana was no less
than his contemporaries, but he was not ahead of them either.
Once these insights were acquired, it became easy to conduct many
operations required for altar construction. us, doubling the area of a
square became a breeze: all you had to do was to gure out the diagonal
of the existing square and construct a square on it. Or you could easily
triple the size of a square by building an oblong on the diagonal of the
second square obtained by doubling the rst square.
We also nd these principles at work in the construction of a vedi
for the soma ritual described by Āpastamba. If one follows Āpastambas
instructions described by ibaut, it becomes obvious that the altar-
makers were using cords and pegs in the ratio of what we would call
Pythagorean triples (5, 12, 13) to construct the east and west side of the
vedi at right angles on the axis of the vedi running through the center.33
ere is lot more to these sutras than just the rst enunciation of
Pythagoras theorem. Of special interest is the discovery of a procedure
for calculating the square roots. e need for calculating the square
roots emerged for the same “irrationality” that so bothered the Pythag-
oreans. e problem is that the diagonal of any square is incommensu-
rable with the length of the sides. is creates a problem for someone
who is trying to calculate the diagonal of a square, knowing its sides.
We nd in Baudhāyana an approximate method of nding square roots,
and using this method we get a fairly accurate square root of two to the
h decimal place.34 Here again, our śulvakaras were in good company:
the Yale tablet shows the Babylonians knew how to solve the square root
of 2 problem, and the Greeks nearly had a mental breakdown over it!
We now come to the controversial matter of proof. For a long time,
the mathematical traditions of ancient India and China have been put
down as merely “carpenter’s rules” which lack proof, while the only
32 e Pythagoreans gured out formulas for calculating triples for an odd number
and an even number. ese formulas were later given a proof by Euclid. See Katz,
p. 38-39 for details.
33 ibaut, pp. 424-426.
34 See ibaut, pp. 430-431 and Joseph, pp. 334 -336 for details. ibaut provides
useful explanations of how śulvakaras could square a circle, build a falcon shaped
altar and other complex altars.
 d Science in Saron
valid model of proof that is admitted is that modeled on Euclid that
proceeds through deductions from rst principles. It is true that the au-
thors of Śulvasūtras only meant to convey, in short memorable sutras,
how to construct the altars. As a result, they did not try to explain how
they arrived at their methods. But that does not mean that the later In-
dian commentators on these sutras did not feel the need to “remove
confusion and doubts regarding the validity of their results and proce-
dures; and to obtain consent of the community of mathematicians.35
e Greeks were not the only ones to feel the itch to justify their theo-
rems, albeit the deductive method of proof was unique to them.
Even though Baudhāyana and other śulvakaras don’t provide a
proof, later texts do. e rst Indian proof of the insights regarding
right angle and diagonals was provided by Bhaskara who lived in the
12th century.
6. “Was Pythagoras Chinese?”: the Kou-Ku theorem
Sometime in the 6th century BCE when Pythagoras and his followers
were working out their number-based cosmology in islands around the
Aegean Sea, when Śulvasūtras were being composed in India, the Chi-
nese, too, had gured out the Pythagorean theorem. Not only that, they
had also given an elegant proof for it. Later they would call it kou-ku
theorem, which is sometimes also referred to as gou-gu theorem.
e rst reference and proof of this theorem appears in the oldest
mathematical text known in China. It is called Chou Pei Suan Ching
which translates into e Arithmetical Classic of the Gnomon and the
Circular Paths of Heaven. Just as in the case of Śulvasūtras, the exact
date of this text is not known. To quote from Frank Swetz and T.I.Kao,
authors of Was Pythagoras Chinese:
While the exact date of its origin is controversial, with estimates ranging as far
back as 1100 BCE, it can generally be accepted on the basis of astronomical
evidence that much of the material in the book was from the time of Confucius,
the sixth century BCE and its contents would reect the mathematical knowl-
edge accumulated in China until that time.36
35 Srinivas 2008, p. 1833.
36 Swetz and Kao, 1977, p. 14.
Who Discovered the Pythagorean eorem? c   
Chou Pei is largely devoted to using the gnomon to measure the
length of the shadow of the sun.37 But the rst part is devoted to the
properties of right-angle triangles. is part consists of a dialogue be-
tween Chou Kung (the ruler of Chou) and a wise man by the name of
Shang Kao who “knows the art of numbering”. Chou Kung wants to
know how the astronomers could have “established the degrees of the
celestial spheres?” He is puzzled because as he says, “there are no steps
by which one may ascent to heavens, and the earth is not measurable by
a footrule. I should like to ask you what is the origin of these numbers?”
Shang Kao explains that the art of numbering originates from “the
circle and the square. e circle is derived from the square and square
from a rectangle.” What follows is a statement of what would later be
given the name of kou-ku theorem:
let us cut a rectangle diagonally and make the width (kou) 3 units, and the
length (ku) 4 units. e diagonal (ching) between the two corners will then be
5 units long.38
is statement is immediately followed with a proof:
aer drawing a square on this diagonal, circumscribe it by half-rectangles like
that which has been le outside, so as to form a square plate. us the four
outer half-rectangles of width 3, length 4 and diagonal 5, together make two
rectangles (of area 24); then, when this is subtracted from the square plate of
area 49, the remainder is of area 25. is process is called piling up the rectan-
gles (chi chu).
e methods used by Yu the Great39 in governing the world were derived from
these numbers.
Chou Kung exclaimed “great indeed is the art of numbering. I would like to ask
about the Tao of the use of right-angle triangle.
37 Gnomon is a primitive form of a sun-dial. Mesopotamians are known to have
used it, the Greeks are known to have borrowed it from Mesopotamians. Indian
astronomers knew it as shanku.
38 Notice the familiar triples 3, 4, 5.
39 According to Needham (1959, p. 23), “the legendary Yu was the patron saint of
hydraulic engineers and all those concerned with water-control, irrigation and
conservancy. Epigraphic evidence from the later Han, when the Chou Pei had
taken its present form, shows us, in reliefs on the walls of the Wu Liang tomb-
shrines the legendary culture-heroes Fu-Hsi and Nu-Kua holding squares and
compasses. e reference to Yu here undoubtedly indicates the ancient need for
mensuration and applied mathematics.
 d Science in Saron
Aer Shang Kao explains the “Tao of the use of right-angle
triangle,40 the dialogue ends with Kao declaring:
40 Which is nothing more than rules for using a T-square.
Figure 6. Hsuan-u is considered the earliest proof of Pythagoras eorem, dat-
ing back to around 600 BCE.
Who Discovered the Pythagorean eorem? c   
He who understands the earth is a wise man and he who understands the heav-
ens is a sage. Knowledge is derived from a straight line. e straight line is
derived from the right angle. And the combination of the right angle with num-
bers is what guides and rules ten thousand things.
Chou Kung exclaimed: “Excellent indeed.41
is dialogue is accompanied by a diagram (gure 6 on page 44).
is is what is called hsuan-thu and is considered one of the earliest
and most elegant proofs of the hundreds of proofs of the Pythagorean
theorem that exist today.42
is proof is relevant to the Indian story. As mentioned in the pre-
vious section, Śulvasūtras did not provide any proof of the theorem and
the rst Indian proof appears in the work of Bhaskara in the 12th cen-
Some historians believe that Bhaskara’s proof is inuenced by this
ancient Chinese proof. is was rst pointed out by Joseph Needham
who writes:
Liu Hui [see below] called this gure ‘the diagram giving the relations between
the hypotenuse and the sum and dierence of the other two sides, whereby one
can nd the unknown from the known.’ In the time of Liu and Chao, it was
colored, the small central square being yellow and the surrounding rectangles
red. e same proof is given by the Indian Bhaskara in the +12th century.
e Hsuan thu proof of the Pythagoras theorem given in the +3rd century
commentary of Chao Chun-Ching on the Chou Pei is reproduced exactly by
Bhaskara in +12the century. It does not occur anywhere else.43
is proof is oen confused with Pythagorean proof. But this proof
shows an arithmetical-algebraic style of the Chinese which was totally
alien to the Greek geometry which abstracted ideal forms from num-
bers. As Needham puts it, the classic passage from Chou Pei quoted
…shows the Chinese arithmetical-algebraic mind at work from the earliest
times, apparently not concerned with abstract geometry independent of con-
41 e complete dialogue can be found in Swetz and Kao, pp.14-16, and also in
42 For a simple explanation of this proof that even those without much mathemati-
cal aptitude (including myself) can understand, see
43 Needham and Wang Ling, 1959, p. 96 and p. 147. is position is supported by
Swetz and Kao, p.40, and also nds support from Victor Katz, pp. 240-241.
 d Science in Saron
crete numbers, and consisting of theorems and propositions capable of proof,
given only certain fundamental postulates at the outset. Numbers might be un-
known, or they might not be any particular numbers, but numbers there had
to be. In the Chinese approach, geometrical gures acted as a means for trans-
mutation whereby numerical relations were generalized into algebraic forms.44
is theorem shows up again in the 9th chapter of the most well-
known ancient classics of mathematics in China, called Chiu Chang
Suan Shu, which translates as Nine Chapters on the Mathematical Art.
is work was composed in the Han dynasty (3rd C. BCE). e version
that survives to the present is a commentary by Liu Hui in 250 CE. Liu
Hui has the same iconic stature in China as Aryabhata has in India.
e ninth chapter of the book is titled “Kou-ku” which is an elabo-
ration, in algebraic terms, of the properties of right-angle triangles rst
described in Chou Pei.
Why kou-ku or gou-gu? To cite Swetz and Kao, in a right angle, the
short side adjacent to the right angle is called kou or gou (or “leg”). e
longer side adjacent to the right angle is called ku or gu (or “thigh”).
e side opposite to the right angle (the hypotenuse) is called hsien (or
is chapter contains some kou-ku problems that are famous
around the world for their elegance and the delicately drawn sketches
that accompany them. ese include the so-called “broken bamboo
problem” and “the reed in the pond problem. Both of these problems, it
is claimed, found their way into medieval Indian and European math-
ematics texts. e “reed in the pond” problem appears in Bhaskara and
the “broken bamboo” in the 9th century Sanskrit classic Ganit Sara by
One thing that the Chinese and the Indian geometers shared – and
what set them apart from the Greeks – was that geometry never got
44 Needham and Wang Ling, 1959, pp. 23-24.
45 Swetz and Kao, pp. 26-28.
46 Swetz and Kao, pp. 32-33 for the “reed in the pond” problem, pp. 44-45 for the
“broken bamboo” problem. See also Needham, p. 147.
Not having training in mathematics, I am not in a position to render my inde-
pendent judgement. But there are striking similarities between Li Hui’s (250 CE)
reed problem and Bhaskara’s (12th century) Lotus problem. See Swetz and Kao for
detailed statement of the problem in both cases.
Who Discovered the Pythagorean eorem? c   
linked to spiritual and/or philosophical questions, as it did in Greece.
It remained more of an art used for practical matters. But that does not
mean that their ideas were not supported by arguments and spatial ma-
nipulation (as in the Hsuan-thu proof). Just because these proofs did
not follow Euclid’s axiomatic-deductive methods, does not make them
any less persuasive.
7. Conclusion
is chapter has followed the trail of the so-called Pythagoras eo-
rem through centuries, crisscrossing the islands on the Aegean Sea,
and traveling through the river valleys of the Nile, the Tigris and the
Euphrates, the Indus and the Ganges, and the Yellow River. We have
looked at the archeological evidence le behind on Mesopotamian clay
tablets and on Egyptian and Chinese scrolls. We have examined the
writings of the Greeks and the sutras of our own altar-makers. We have
wondered at the achievements of the ancient land-surveyors, builders
and mathematicians.
Having undertaken this journey, we are in a better position to an-
swer the question: “Who discovered the Pythagorean eorem?” e
answer is: the geometric relationship described by this theorem was
discovered independently in many ancient civilizations. e likely ex-
planation is that the knowledge of the relationship between sides of a
right-angle triangle emerged out of practical problems that all civiliza-
tions necessarily face, namely, land measurement and construction of
buildings – buildings as intricate as the Vedic re altars, as grand as the
Pyramids, as functional as the Chinese dams and bridges, or as humble
as simple dwellings with walls perpendicular to the oor.
Where is India in this picture? Indian śulvakaras were one among
the many in the ancient world who hit upon the central insight con-
tained in the Pythagorean eorem: they were neither the leaders,
nor laggards, but simply one among their peers in other ancient civi-
lizations. Our Baudhāyana need not displace their Pythagoras, as they
were not running a race. ey were simply going about their business,
Baudhāyana and his colleagues concentrating on the sacred geometry
 d Science in Saron
of re altars, Pythagoras and his followers worrying about the ratios and
proportions that underlie the cosmos.
It is merely an accident of history, undoubtedly fed by the Euro-
centric and Hellenophilic biases of Western historians, that the insight
contained in the theorem got associated with the name of Pythagoras.
Apart from shoring up pride in our civilization, nothing much is to be
gained by insisting on a name change. e correct response to Euro-
centrism is not Indo-centrism of the kind that was on full display at the
Mumbai Science Congress. e correct response is to stop playing the
game of one-upmanship altogether.
e itch to be e First is unproductive for many reasons. For one,
it turns evolution of science into a competitive sport, history of science
into a matter of score- keeping and the historians of science into ref-
erees and judges who hand out trophies to the winner. e far greater
damage, however, is inicted on the integrity of ancient sciences and
their practitioners whose own priorities and methods get squeezed into
the narrow connes of the Greek achievements.
If we really want to honor Baudhāyana and other śulvakaras,
a far more sincere and meaningful tribute would be to understand
their achievements in the totality of their own context, including the
ingenious methods they employed for solving complex architectural
problems. Viewing ancient Indian geometry purely, or even primarily,
through the lens of Pythagoras actually does a disservice to Baudhāyana,
for there is lot more to Śulvasūtras than this one theorem.
It is high time we freed ourselves from our xation on Pythagoras.
Let him Rest in Peace.
Nothing at Is: Zero’s Fleeting Footsteps c   
Cha pt er 2
Nothing at Is: Zero’s Fleeting Footsteps1
जब ज̣ीरो िदया मेरे भारत ने,
भारत ने, मेरे भारत ने,
दुिनया को तब िगनती आई।
तारो ं की भाषा भारत ने,
दुिनया को पहले िसखलाई।
देता ना दशमलव भारत तो,
यूँ चाँद पे जाना मुिकल था।
धरती और चाँद की दूरी का,
अंदाज़ लगाना मुिकल था।2
1. e question
Indians of a certain age grew up humming this ditty from the 1970
Hindi lm, Purab aur Paschim. Judging by the hits it gets on YouTube, it
seems as if the song continues to tug at Indian heartstrings even today,
almost half-a-century aer it was composed.
e lyrics resonate because they arm what is already imprinted
in our national psyche. It is drilled into our heads by school teachers,
religious teachers, family elders, Bollywood lms, and politicians, that
1 e title of this chapter is inspired by books by Robert Kaplan (1999) and Lam
Lay-Yong (2004).
2 e movie’s title, Purab aur Paschim translates into “East and West.” Here is a
rough translation of the lyrics: Only aer my Bharat (India) gave zero, could the
rest of the world learn how to count; my Bharat rst taught the world the language
of the stars. Had Bharat not given the decimal system to the world, it would have
been dicult to reach the moon, or even to calculate the distance between the
earth and the moon.
 d Science in Saron
zero and the decimal system are Indias unique and revolutionary con-
tributions to world mathematics and science. Countless books and es-
says are written by Indian mathematicians and historians that declare
zero to be an “entirely Hindu” contribution to world mathematics.3 e
sentiment expressed in the song – that if it were not for the genius of an-
cient Bharat, the world wouldn’t have known how to count – is widely
shared. If there is one “fact” in history of science that is assumed to be
true beyond any reasonable doubt, it is the Indian origin of zero.
But is this really true? Is it really irrefutably established that ancient
India was the original source of the number zero and the decimal num-
bering system that is the foundation of modern mathematics?
is is the question that this chapter seeks to answer.
2. Two principles and a hypothesis
Two principles will guide this chapter.
e rst is a commitment to comparative history. Comparative
history simply means studying ideas, institutions, and historical pro-
cesses across dierent socio-cultural systems separated by space and/or
time. Histories that can be compared can exist within the same nation
(the structure of family in Kerala and Punjab, for example), across na-
tions (historical trajectories of economic growth in China and India, for
example) or across civilizations (the philosophies of nature in ancient
Greece and India at a comparable time period, for example). Compara-
tive history can produce valuable insights for identifying events/pro-
cesses/institutions that could have set a society on a certain trajectory
3 e classic in this genre is Bibhutibhusan Datta and Avadesh Singh’s History
of Hindu Mathematics published in 1938. is strain of “out of India” writing
continues in academic circles, including elite science and engineering institu-
tions, see Datta, 2002, for example. Prestigious scientic institutions routinely host
ideologues with scientic credentials who make hyperbolic claims about Hindu
origins of zero (and just about every landmark in history of science) backed by
nothing more than Sanskrit shlokas, randomly selected and idiosyncratically
interpreted. For a recent example, see the lecture by Dr. N. Gopalakrishnan, the
founder-director the Indian Institute of Scientic Heritage at IIT-Madras recently.
(e lecture can be seen at
Books like Dr. Priyadarshi’s Zero is not the Only Story continue to gain traction in
the public sphere, including places like IIT-Kanpur.
Nothing at Is: Zero’s Fleeting Footsteps c   
which may dier from other societies in direction or in pace. Only by
comparison can historians get to understand whether the develop-
ments in question are unique to a society, or are part of a much broader
Cross-civilizational comparisons are of special relevance in history
of science, especially when scientic achievements are appropriated for
nationalistic purposes and cast in the following form: “ the civilization
Y was the rst to come up with the idea X.” Replace Y with “ancient
Indian mathematicians/scientists-sages” and X with zero and decimal
system, or with any other notable scientic discovery, and we get exact-
ly the kind of history of science that is being propagated in India today.
e very idea of being “the rst” is an inherently comparative idea:
unless there are others engaged in a similar project, how can there be
a “rst”? Can there be a winner in a marathon without there being a
marathon? To crown any civilization Y the “rst” to come up with X,
requires us to see what Y’s sister-civilizations A, B, or C were doing in
the domain of X.
It is true that there has been a long tradition of history-writing that
simply assumed that the European civilization – with its birth in ancient
Greece and its coming of age in Christianity – gave birth to everything
of value. is kind of history suers from a serious bout of Hellenophil-
4 According to Karl Marx, “Events strikingly similar but occurring in a dierent
historical milieu, lead to completely dierent results. By studying each of these
evolutions separately, and then comparing them, it is easier to nd the key to
the understanding of this phenomenon; but it is never possible to arrive at this un-
derstanding by using the passe-partout [ master-key] of some universal-historical
theory whose great virtue is to stand above history.” Quoted from Raymond Grew,
1980, p. 766. Marc Bloch, another master of comparative history, has emphasized
the importance of the comparative method as a means of systematically gather-
ing evidence to test the validity of explanations of historical changes. As William
Sewell (1967,p.208-209) writes in his exposition of Bloch’s work, “if an historian
attributes the appearance of phenomenon A in one society to the existence of con-
dition B, he can check this hypothesis by trying to nd other societies where A oc-
curs without B, or vice versa. If he nds no case which contradict his hypothesis,
his condence in its validity will increase…. If he nds contradictory evidence, he
will either reject the hypothesis outright or reformulate and rene it so as to take
into account the contradictory evidence and then subject it again to comparative
testing.” Clearly, historical comparisons serve a function that is very similar the
role of experiments in natural sciences.
 d Science in Saron
ia, which David Pingree has described as “a kind of madness” caused by
the delusion that the only real sciences are those that began in Greece
and then spread to the rest of the world. On this account, Europe is the
sole Giver, while the rest of the world is merely a passive Receiver.5
Eurocentric history is deeply awed and condescending to the rest
of the world. Yet, an uncritical and self-celebratory Indo-centric history
is nothing but a mirror image of uncritical and self-celebratory Euro-
centrism: Both are equally illogical and equally chauvinistic. What we
instead need is an appreciation that independent mathematical and sci-
entic developments can cross cultural barriers, take root and blossom
in many dierent kinds of soils.6
e second principle that will guide this chapter is that history of
even as abstract a subject as mathematics cannot be understood without
paying close attention to the “counter-culture” of mathematics, that is,
the practical manipulation of numbers using movable counters, be they
clay tokens, pebbles, sticks, or beads on an abacus. Most histories of
mathematics tend to focus exclusively on the written number-records
while completely ignoring number-manipulations using counters.
Moreover, most histories of mathematics are histories of “pure” math-
ematics which, in the ancient world, was invariably intertwined with
the quest for the divine (the Pythagorean number mysticism, construc-
tion of altars and temples for ritual purposes, for example. See Chapter
1). What these standard histories overlook is the history of practices
– counting, computing, measuring, weighing – that ordinary people in
5 David Pingree, 1992, p. 555. See also George G. Joseph’s important book, Crest of
the Peacock, for a critique of what he calls “classical Eurocentrism” in history of
While Eurocentrism is obviously problematic, anti-Eurocentric histories oen
have the feel of ogging a dying horse. anks to the sustained post-colonial
critiques and equally sustained self-critiques by European and North-American
scholars, history of science has moved away from an uncritical Hellenophilia of
the earlier generations, and has made signicant strides toward a more inclusive
and context-sensitive historiography. Even a cursory glance at the educational
curricula in the US schools and universities will show that Europe is by no means
treated as the center of the known universe. Today, it is not Eurocentrism, but
romantic multiculturalism that poses a greater danger to the study of history of
science because of its potential to foster ethnocentrism and nativism.
6 is formulation is from Joseph, 2011, p. 12
Nothing at Is: Zero’s Fleeting Footsteps c   
all cultures developed for taking care of their ordinary material needs.
Simple practices like keeping track of sheep in a herd, trading, collect-
ing taxes, and building houses, bridges, dams, temples etc. all required
some methods for manipulating numbers. Once we pay attention to
such practices, a whole another dimension is added to the historical
search for the origin of zero.7
Once these two principles are applied to available evidence, two
new ways of understanding the evolution of zero, and India’s role in it,
begin to emerge:
a. When the Indian evidence is placed alongside the evidence from
other civilizations of comparable development, it becomes clear that
the Indian contributions were neither unique, nor without precedent.
Evidence presented in this chapter will show that the Hindu-Arabic sys-
tem of numeration involves no principle that was not already familiar
to India’s sister civilizations. All the elements that went into the crea-
tion of zero – counting by powers of ten, decimal place-value, and the
concept of empty space in the decimal ranking – were well-known for
many centuries in many diverse cultures before they all came together
in India in the form that we use today.8 Indian mathematicians further
developed zero as a number like any other number which could be used
in arithmetical operations. All of these are substantial contributions for
which India is justly admired. But the claims of India being the one and
only civilization to arrive at zero as a mathematical concept are simply
not substantiated by the historical evidence taken in its totality.
7 e idea of “counter-culture” and the distinction between “number manipulation
and “number recording” are from the important work of Reviel Netz (2002). Netz
makes a powerful case for writing history of numeracy and even literacy from
the perspective of computational practices: “e rule is that across cultures, and
especially in early cultures, the record and manipulation of numerical symbols
precede and predominate over the record and manipulation of verbal symbols….
In other words, in early cultures, numeracy drives literacy, rather than the other way
around,” p. 323. A similar point has been argued by Sal Restivo (1992). Both see
the practices of counting and manipulating numbers as more fundamental than the
practices of recording numbers in texts.
8 ere is a substantial literature on Tamil and Sinhalese numerals. See Georges
Ifrah, 2000, for details. We will, however, focus only on the Brahmi-Nagari nu-
merals, as they are the ancestors of the modern Hindu-Arabic numerals.
 d Science in Saron
b. secondly, when the sources of evidence are widened beyond
metaphysical speculations to include every-day, practical counting and
computing practices of ordinary people, a new window opens up which
faces East of India, to China and South-East Asia. e window- opener
is the renowned historian of Chinese science, Joseph Needham (aptly
described as “the man who loved China”) who, along with his Chinese
colleagues produced the monumental multi-volume work Science and
Civilization in China which is considered a landmark in comparative
history of science. In the third volume of this massive work, Needham
and his co-author, Wang Ling, propose a Chinese origin of zero as a
number, which they say travelled from China, through Southeast Asia
to India where it acquired the familiar form that the whole world uses
today. In their own words:
… the written symbol for nil value, emptiness, sunya, i.e., the zero, is an Indian
garland thrown around the nothingness of the vacant space on the Han count-
ing boards.9
A very similar thesis has been put forward in recent years by Lam
Lay Yong, a renowned historian of mathematics based in National Uni-
versity of Singapore. In her various writings which culminated in her
book titled Fleeting Footsteps, Lam has argued that “the Hindu-Arabic
numeral system had its origin in the Chinese rod-numeral system.10
Lam supports Needham’s thesis that it is through Southeast Asia, “where
the eastern zone of Hindu culture met the southern zone of the culture
of the Chinese” that the practice of using empty space in the process of
counting and doing basic arithmetic crossed over from China into In-
dia where it acquired the shape that the whole world is familiar with.11
9 Needham and Wang Ling, 1959, p. 148. Simon Winchester’s recent biography of
Needham is titled e Man who Loved China.
Joseph Needham (1900-1995) was a British biochemist who became fascinated
with the Chinese language which he learnt from his Chinese students at Cam-
bridge University. (He would marry one of these students much later in life aer
his rst wife passed away). He spent extended periods of time in China, travelling
and studying.
10 Lam Lay Yong and Ang Tian Se, 2004: 170. Professor Lam was the recipient in
2002 of the Kenneth O. May medal, the highest honor in history of mathematics.
11 Needham and Wang Ling, 1959, p. 148.
Nothing at Is: Zero’s Fleeting Footsteps c   
e Needham-Lam thesis will be examined in greater details in this
chapter. By bringing China into the story of zero, my intention is not to
start another Indo-Chinese race. Instead, my motivation is to de-na-
tionalize the way we write history of science in India. Needhams well-
known work on Chinese mathematics has been around since the 1960s,
while Lam Lay Yong has been publishing her work since the 1990s
to great acclaim by professional historians. While this thesis is gain-
ing wide acceptance and is making its way into well-known textbooks,
we in India have remained oblivious to it.12 Most Indian historians of
mathematics (with the exception of George Joseph’s Crest of the Pea-
cock) that do touch upon India-China cultural exchanges start with the
assumption that India was the Giver and China was the grateful Receiv-
er of not just Buddhism, but of everything else of value in mathematics
and sciences.13 Does this assumption of unilateral ow of mathematical
ideas from India to China, Southeast Asia and later, through the Arab
mathematicians, to the rest of the world hold up against the best avail-
able historical evidence? e chapter will try to answer this question.
Before we get into the historical details, some clarity about the
mathematical concepts of decimal base, place value and zero is called
3. e preliminaries: numbers, decimal, place-value and zero
It is quite common in Indocentric histories to nd zero indiscrimi-
nately grouped with decimal counting and decimal place value – and
all three given an Indian birth-certicate. e problem is that the three
concepts did not evolve together: presence of decimal counting systems
don’t necessarily imply the knowledge of decimal place value, albeit the
knowledge of place value was a pre-requisite for the evolution of zero.
12 See Victor Katz’s (2009) well-known text book on history of mathematics.
13 For example, R. C. Gupta’s (2011) paper on Indian contributions to Chinese math-
ematics starts with this totally unhistorical statement: “the countries in East Asia
received with Buddhism not only their religion but practically the whole of their
civilization and culture.” p. 33, emphasis added. Such a stance clearly devalues
centuries of pre-Buddhist achievements of ancient China.
 d Science in Saron
Unless we un-bundle them and understand each on its own terms, we
simply can’t understand their evolution.
To begin with, the word “number” is more complicated than we
give it credit for. Numbers obviously count things, but they are not the
things themselves; you can pick up two cups, or two of anything, but
you can’t pick up the number “two. “Two” is an abstraction, an idea in
our heads. It does not exist by itself.14
ere are two ways all cultures seem to have for expressing num-
bers. e rst is through “number words” which can be spoken or writ-
ten in the local language. For example, a Sanskrit speaker would use the
word शतम् (shatam) to mean 100, an Arabic speaker would call it ةئم/
ةئام (mia) while a Greek would call it Hekaton.
e other way to represent numbers is through the use of symbols.
e symbols can be of two kinds. e rst kind are what are called “nu-
merals” which are simply marks representing numbers. Our own Hin-
du-Arabic 1,2,3… 9 and 0 are the most obvious example of numerals.
But the markings that are used for representing numbers have var-
ied through history and cultures. For example, ancient Babylonians
represented the number one hundred using Cuneiform symbols, while
Greeks would write the Greek letter ϱ (“ rho”) to write their
number-word Hekaton. We don’t know what numeral the com-
posers of the Vedas would have used to represent shatam or any other
Sanskrit number-word, because we don’t have any written records of
Sanskrit numerals.15 e rst numerals we nd in India are written in
Kharosthi and Brahmi and date back only to 300 BCE. In the modern
world, however, peoples of all countries throughout the world under-
stand and use the Hindu-Arabic number-symbols, or numerals – the
familiar 1,2,3…. Even though dierent cultures continue to use number
words in their own local language (ek, do, teen in Devanagari; ena, dio,
tria in Greek), the Hindu-Arabic numerals are now universally under-
stood and used. (We will look at the evolution of Hindu numerals from
Brahmi later).
14 See Ian Stewart, 2007, p. 9.
15 e numerals used in Sanskrit today are Devanagari numerals which date back to
the Gupta period (200-550 CE).
Nothing at Is: Zero’s Fleeting Footsteps c   
e second kind of symbols use words which symbolize number
words. Referred to as Bhūta-sankhyā in medieval Sanskrit mathemati-
cal texts, this way of representing numbers is also called “object num-
bers” or “concrete numbers”. Rather than use number word or numer-
als, this system makes it possible to express any number by the name
of whatever object – real or mythical – that routinely occurs in that
number. us, the number two can be represented by all the Sanskrit
words for “eyes” because eyes naturally come in pairs. e symbols can
also come from religious texts and ritual practices: thus the word “agni”
can stand for number three, as there are three ritual res; “anga” can
stand for the number six, as there are six limbs of the Vedas. Alberuni,
writing around the turn of the rst 1000 years aer the Common Era,
describes this system thus:
… [for] each number, Hindu astronomers have appropriated quite a great
quantity of words. Hence if one word does not suit the metre, you may easily
exchange for a synonym which suits. Brahmagupta says: ‘if you want to write
one, express it by everything that is unique, as the earth, the moon; two, by
everything that is double e.g., black and white; three by everything which is
threefold; the naught by heaven, the twelve by the names of the Sun.16
is unique way of recording numbers arose out of the compul-
sion to write mathematical and astronomical ideas in verse so that they
could be easily memorized. Our mathematically-minded poets faced a
problem; it wasn’t easy to use number-words in verse all the time. ey
needed synonyms which would sound better and be easy to remember.
ese terse sutras were committed to memory, while the guru directly
explained the full meaning to the students. Commentaries in prose were
written to expound on the meaning of the symbols and the sutras.17
e point is this: you can record, versify, and memorize number-
words and concrete symbols, but you cannot compute with them. (Try
16 Alberuni in Sachau 1971, p. 177.
17 ere is a vast literature on the preference for orality and the inuence it had on
development of sciences in India. For mathematics, see Yano, 2006, Ploer, 2009,
Filliozat, 2004. Ifrah believes that this system is unique to India (p. 409) and also
provides an extensive list of number symbols (p. 499).
 d Science in Saron
adding “chakshu-akaash-agni” to “ashvin-anga-pitamaha, or, even
better, try multiplication or division!).18
Let us turn to the
term “decimal”. According to the Oxford Eng-
lish Dictionary, the word decimal simply means: “a system of numbers
and arithmetic based on the number ten, tenth parts, and powers of
ten.” is simply means counting by power of tens, or bundling
by tens.
Counting by tens is the result of the fact that human beings have
ten ngers. Earliest records show that numbers beyond one and two
evolved by addition – three by adding two and one; four as two and
two, and so on. As commerce and cras developed, the need for larger
numbers grew. is led to the bundling of numbers rst in ves as in
, and later the base 10. With 10 as the base, larger numbers could be
constructed by addition or subtraction (12 is 10 plus 2, 20 is 10 plus 10,
and 19 as 20 minus 1 etc.) and by multiplication (20 is 2 times 10 etc.).
Some civilizations (the ancient Mayans) used 20 as the base (ngers
and toes), while for reasons that are not clear, ancient Mesopotamians
used 60 as the base. As we will see in the next section, base 10 or deci-
mal system of counting, did not originate in India: it is simply the most
common method of counting and cuts across civilizations.
You can have a decimal system of counting without a zero, but you
cannot have decimal place value without having a symbol for an empty
place or what we today call zero. In other words, the existence of deci-
mal counting itself does not constitute evidence for zero; but a decimal
place value does. While 10-based or decimal counting is almost univer-
sal and has been around from the very beginnings of civilization, deci-
mal place value is another story altogether.
What is place value? Place value, also called positional notation,
has been described as “one of the most fertile inventions of humanity,
comparable to the invention of the alphabet which replaced thousands
of picture-signs.19 In the place-value method of writing numbers, the
position of a number symbol determines its value. Consider the num-
ber 211 written in our modern decimal notation: the numeral 1 has the
18 Hint: the numbers are 302, and 162, as the number symbols are read from right to
19 Otto Neugebauer, 1962, p. 5.
Nothing at Is: Zero’s Fleeting Footsteps c   
value of one if it occupies the rst place from the right, but the same 1
stands for a ten as it moves one place to the le. Similarly, 2 is not simply
the sum of one and one, but has the value of two-hundred. If the order
changes, the value changes; for example, 112 is a very dierent number
from 211.
Consider a larger number: 4567. If we stop to think about it, this
number is actually made up of the following: 4x1000 + 5x100 + 6x10 +
7x1. In other words, every position from right to le is a multiple of 10
(unit, 1 is one tenth of 10, tens (10x1), hundred (10x10, or 102), thou-
sand (100x10, or 103) and so on to millions, billions, trillions, etc. If we
accept this rule, then instead of explicitly spelling out the powers – four
thousand, ve hundred, sixty seven, we simple assume that in this case,
4 is to be multiplied by 103, 5 by 102 and so on.
e beauty of place value – and the reason it is considered revo-
lutionary – is that it allows you to write any number, however large or
small, with just a few numerals. Using the modern decimal system, nine
digits (1 to 9) and a zero are sucient to write any number without
having to invent new symbols for each digit of a number, or for each
multiple of the base 10.
If the place value notation did not exist, separate symbols would be
required for writing 10, 20,30, … 90 and for 200, 300 … 900. Let us take
an example. We know from the existing evidence (which will be exam-
ined in more details in later sections), that in India place-value nota-
tions rst made their appearance around the time of Asoka around 300
BCE, while the ancient Greeks never developed it at all. us, someone
living before the Asokan era in India would write the number 456 (for
example) in Brahmi by using a symbol for 400, followed by a separate
symbol for 50, followed by a symbol for six. Similarly, his Greek coun-
terpart would write the same number as υνϛ, where these letters from
the Greek alphabet stand for 400, 50 and 6 respectively. To contrast, in
a positional value notation (in our example 456), the numeral 4 would
stand for 400 (4 units at the 100th position), ve would stand for 50
(ve units at 10th position) and six for 6 units. In other words, the order
in which the numerals are written or spoken would automatically indi-
cate whether they represented a thousand, hundred etc. In this system,
 d Science in Saron
the work of “power words” – special words or signs indicating numeri-
cal rank (thousand, hundred, tens etc.) – is simply transferred to the
places any of the rst 9 numerals occupy.
We can now understand fully what Neugebauer meant when he
said that the invention of place value notation is analogous to the crea-
tion of the alphabet. Inventing place value meant that the thousands
of separate symbols that were needed to represent individual numbers
became obsolete, just as the alphabet rendered hieroglyphics obsolete.
Instead of memorizing large number of symbols, just the nine digits
(1 to 9) plus a sign for empty space – the familiar, somewhat oval-ish
empty circle we call zero – are enough to write any number however
big or small.
What does place value notation have to do with zero?
e answer: Zero was born out of place value notation. To be more
precise, place value is necessary for the evolution of zero as a numeral,
but it is not sucient, for you can have place value without a zero if
you have number words that are larger than the base. For example, you
could write or say 2004 as “two thousand and four”, without using a
zero. But if you are writing in numerals, you cannot write 2004 without
indicating that there is nothing under tens and hundreds – and zero is
what indicates the absence of any number, or the presence of nothing.
Without some way of indicating nothing, the numerals 2 and 4 could
well mean 24 or 204. As Georges Ifrah put it in his well-known book,
e Universal History of Numbers:
In any numeral system using the rule of position, there comes a point where a
special sign is needed to represent units that are missing from the number to
be represented… It became clear in the long run that nothing had to be rep-
resented by something. e something that means nothing, or rather the sign
that signies the absence of units in a given order of magnitude is …[ what we
call zero.]20
As the above example shows, the philosophers and scribes who
used number words could get by without having a special numeral that
indicated nothing. But it is also important to note that the need for zero
was not obvious to those who practiced everyday mathematics in their
daily lives either. As Alfred North Whitehead put it, “the point about
20 Georges Ifrah, 2000, pp. 149-150. Emphasis in the original.
Nothing at Is: Zero’s Fleeting Footsteps c   
zero is that we do not need to use it in the operations of daily life. No
one goes to buy zero sh. Charles Seife, whose book this quote is taken
from, goes on to add, “you never need to keep track of zero sheep or
tally your zero children. Instead of ‘we have zero bananas,’ the grocer
says, “we have no bananas. We dont have to have a number to express
the lack of something.”21
It is only when numbers are written as numerals or as number-
symbols in a positional order, does the need for a zero emerge. In other
words, when 10-based numerals began to be arranged according to
their rank, the symbol for zero became necessary.
4. e evidence
In popular discourse, the Indian origin of zero has become an article of
faith: it has acquired the status of an established fact which is beyond
any doubt. For professional historians of mathematics, however, the In-
dian origin story remains a puzzle, with many unsolved elements. As
Kim Ploer, the author of the well-received Mathematics in India puts
e Indian development of place value decimal system … is such a famous
achievement that it would be very gratifying to have a detailed record of it. …
Exactly how and when the Indian decimal place value system rst developed,
and how and when a zero symbol was incorporated into it, remains mysteri-
In this section, we will examine in details the evidence that is of-
fered for India’s priority-claims on zero. As explained in the previous
section, history of zero cannot be understood without understanding
the history of decimal place value. Our examination of the Indian case
will start with decimals and decimal place values, and gradually move
towards the emergence of zero. We will, as promised, juxtapose the
21 Charles Seife, 2000, p. 8. Emphasis added.
22 Ploer, 2009, pp. 44, 47. Even Datta and Singh, one of the earliest advocates of
the exclusively Hindu origin of zero and place value admit that there are holes in
the evidence: “between the nds of Mohenjodaro and the inscription of Asoka,
there is a gap of 2,700 years of more,” p. 20. ey also acknowledge that answers to
questions regarding who? Where? when? of the invention of place value are “not
known,” p. 49.
 d Science in Saron
Indian evidence against evidence from sister Euro-Asian civilizations,
and we will pay attention to everyday methods of counting and com-
puting, in addition to Sanskrit texts.
4.1 Antiquity of the decimal system in India
It is oen implied that the decimal system is an Indian invention. Oen
the credit for this achievement is ascribed to the inherently scientic
nature of Sanskrit language. Statements like these from an eminent his-
torian of Indian science, B.A. Subbrayappa, that “Indians’ invention of
the decimal system, especially zero, has paved the way for today’s IT
revolution,are the stu of everyday discourse in India.23 As we shall see,
ancient Indians were in no way the rst, or the only, creators of the decimal
system of counting.
ere is, of course, no doubt that as far back as we can go, Indians
have used a decimal or a 10-based counting method. e g Veda and
Yajurveda provide ample evidence that by the early Vedic times, a regu-
larized decimal system of number counting was well established. Kim
Ploer has usefully provided English translations of shlokas from the
Vedic corpus which give a good idea of how the power of ten was used.
Two representative examples are quoted below.
You Agni, are the lord of all [oerings], you are the distributor of thousands,
hundreds, tens [of good things]. g Veda, 2.1.8
Come Indra, with twenty, thirty, forty horses; come with y horses yoked to
your chariot, with sixty, seventy to drink the soma; Come carried by eighty,
ninety and a hundred horses. g Veda, 2.18:5-6
By the middle-Vedic period, one nds number words for much
larger powers of ten. A verse from Yajurveda (7.2.20) for example, oers
praise to numbers which range from one, two … to ayuta (ten thou-
sand), niyuta (hundred thousand), prayata (one million), arbuda (ten
million), nyarbuda (hundred million), samudra (billion) madhya (ten
billion), anta (hundred billion) parardha (trillion).24
23 ‘India invented decimal system’ e Hindu, Feb. 17, 2007. Available at http://www.
24 Ploer, 2009, p. 13-16. is penchant for large numbers is a hallmark of Indian
mathematical texts and we will return to this issue below.
Nothing at Is: Zero’s Fleeting Footsteps c   
All of this is well-established and beyond any doubt. However,
counting in bundles of tens was so widely practiced, it could almost
be considered universal. For example, of the 307 number systems of
Native American peoples investigated in an anthropological study car-
ried out in the early decades of the 20th century, 146 were found to be
decimal and 106 of them used the base 5 or 20.25 Moreover, as Georges
Ifrah points out, counting by tens is shared by all members of the family
of Indo-European languages. e rule in the Indo-European language-
family is this: “the numbers from 1 to 9 and each of the powers of ten
(100, 1000, 10,000 etc.) has a separate name, while all other numbers
being expressed analytical combinations of these names.26 Given the
near universal use of decimal counting, it would have been surprising if
ancient Indians had been innocent of it.
While the Indian case for ten-based numeration rests upon textual
evidence, archeological evidence from Greece and China clearly shows
decimal system being used for practical purposes.
e Greek evidence is found engraved on the walls of a tunnel in
the island of Samos, which was constructed around 550 BCE to bring
water from a spring outside the capital city. Modern archaeological ex-
cavation has revealed that the tunnel was dug by two teams who started
from the opposite side and met in the middle. e numbers engraved
on the walls read “10, 20, 30…. 200” from the south entrance and
“10,20, 30… 300” from the north entrance, and were used to keep track
of the distance dug.27
As far as China is concerned, Joseph Needhams judgment that
“there was never a time when the Chinese did not have a decimal place
value” is only partially correct.28 e Chinese number system as written
was based on powers of ten, but it was not place value. However, the
Chinese performed their computations using counting rods, which was
25 Dirk Struik, 1987, p. 10.
26 Georges Ifrah, 2000, p. 31.
27 Victor Katz, pp. 34-35.
28 Needham, pp. 12-13. While the antiquity of decimal is uncontested, Needham
exaggerates the antiquity of place value in Chinese numerals, which when written
down, used special characters for powers of ten, hundred etc. In a true positional
system, the position of the number itself will show if it is a multiple of thousand,
hundred etc. without using any special characters for ten, hundred …etc.
 d Science in Saron
a decimal place value system, conceptually identical with the modern
“Hindu Arabic” numerals which we use today. (We will look at the issue
of place value in the next section).
e earliest evidence of Chinese number-system comes from the
so-called “oracle bones” inscribed with royal records of divinations
written on bones and tortoise shells dating back to 1500 BCE (See Fig-
ure 1) ese bones contain numerical records of tribute received, ani-
mals hunted, number of animals sacriced, counts of days, months, and
other miscellaneous quantities related to divination.29
e oracle bones are mathematically important because they show
an advanced numeral system, which allowed any number, however
large, to be expressed by the use of nine unit signs, along with a select-
ed number of “power-signs” for representing powers of tens, twenties,
29 Farmers found these bones in their elds in Henan Province at the end of the 19th
century. Initially, they were thought to be “dragon” bones with medicinal value.
Fortunately, they were rescued before they could be powdered and sold as medi-
cine. Many more bones carrying similar inscriptions have been found through the
last century.
Figure 1. Oracle bones from the Shang Dynasty in China (c. 1800-1200 BCE)
Nothing at Is: Zero’s Fleeting Footsteps c   
hundreds, thousands etc. e standard number system used today in
China is a direct descendant of the ancient Shang system.30
In light of this evidence, claims about the Indian invention of the
decimal system must be re-evaluated.
4.2 Antiquity of decimal place value in India
As discussed above, the invention of place value is a necessary stage of
mathematical development that needs to take place if an empty space
in the order of numbers is to make an appearance. It is therefore under-
standable why Indo-centric histories lay priority claim to it nding it in
Sanskrit texts written as early as 200 CE, becoming fully operational in
Aryabhatas work, around 500 CE.31
e historical evidence is complicated by the fact that we have two
distinct number systems evolving at the same time – Brahmi numer-
als which lack place value, and the Sanskrit number-symbols, or bhūta-
sankhyā, which do show place-value from early centuries of the Com-
mon Era. Let us examine the evidence, starting with Brahmi numerals.
e oldest written script from the Indian subcontinent is that
found on the yet un-deciphered Harrapan seals, but the oldest deci-
phered script is Brahmi that dates back to around 4th century BCE.
30 is description of the oracle bones is from Joseph, 2011, pp. 199-200, and Chri-
somalis, 2010, pp. 260-261.
31 For a non-specialist take on this issue, see
Figure 2. Brahmi Numerals
 d Science in Saron
e general consensus is that the Brahmi script was formalized at about
the time of the Mauryan emperor, Ashoka. It was devised to give writ-
ten expression to the spoken language of the region, called Prakrit. e
earliest inscriptions in Brahmi can be found on the rock edicts installed
by Ashoka, around the middle of the third century BCE. It is in these
rock edicts we get the rst glimpse of how numbers were written in
Brahmi. Numerals 1,4 and 6 are found in various Ashokan inscriptions,
while numbers 2, 4, 6, 7 and 9 in the Nana Ghat inscriptions about a
century later; and the 2, 3, 4, 5, 6, 7, and 9 in the Nasik caves of the 1st
or 2nd century. Figure 2 (at page 65) is a composite of Brahmi numerals
obtained from sites all over the subcontinent, including Nepal.
e familiar Nagari numerals descended from Brahmi numerals
sometime in the Gupta period, and gradually evolved into the “Hindu-
Arabic” numerals the whole world uses today (Fig. 3).32
32 See Ifrah for an exhaustive treatment of Brahmi numerals, including speculations
about their origin, pp. 367-399.
Figure 3. Evolution of modern numerals from Brahmi
Nothing at Is: Zero’s Fleeting Footsteps c   
ere is a consensus among historians that Brahmi numerals did
not have a concept of place value and did not have a symbol for zero. Nu-
merals inscribed into the wall of Nanaghat cave clearly show a number
which has been deciphered as 24,400 (Fig. 4). It is written using special
symbols for 20,000, followed by another symbol for 4000 and 400. If
these numerals had followed place-value notation it would have been
written with only the numerals for 2 and 4 followed by two zeros.33
None of the Brahmi inscriptions with numerals discovered so far
show any sign of place value.34 Place value suddenly begins to make an
appearance sometime around the sixth century, starting with the dates
written on copper land-grants.35 Gradually one begins to see numbers
written without special symbols indicating power or rank; the posi-
33 See Ifrah, p. 399, for how the number 24,000 would appear in Brahmi if Brahmi
were a place-value number system.
34 See Ifrah, pp. 397-398 for a compilation of various Brahmi inscriptions.
35 e authenticity of some of these copper-plates has been challenged. See Datta
and Singh for details.
Figure 4. A Pencil rubbing of Nanaghat Cave Inscription (second century BCE). e num-
ber 24,000 is represented by three marks occupying positions 4, 5, 6 from the le hand
corner on the bottom line. Source: Hindu-Arabic Numerals, by David Eugene Smith and
Louis Charles Karpinski, 1911. e Project Gutenberg EBook.
 d Science in Saron
tion of numeral itself begins to indicate what power of ten they carried.
Zero, initially a dot, begins to make an appearance around this time, the
rst incontrovertible proof appearing in a Gwalior temple in the year
876. (More on this in section 4.4).
ere is a gap of about 900 years between Brahmi to Nagari place-
value numerals. ere are all kinds of wild guesses about what caused
this crucial transition, but most of them are just that – guesses. Even
those scholars (like Ifrah, to take a prominent example) who rmly and
fervently believe that Indians alone invented zero without any outside
inuence, are unable to oer any clues to how this transition took place,
what could have caused it, and how it spread all over the subcontinent
and beyond to Southeast Asian lands.
Brahmi inscriptions disappoint us in our search for zero. However,
Sanskrit scripts using bhūta-sankhyā (see section 3) indicate a knowl-
edge of place value. We see this method of notation in use from Sanskrit
texts starting around mid-third century CE, to around the 18th century.
e way Sanskrit number-symbols, or bhūta-sankhyā, were used
is exemplied by the Yavana-jataka, or “Greek horoscopy”, of Sphujid-
havja, which is a versied form of a translated Greek work on astrology.
is text places the “wise king Sphujidhavja in the year Vishnu/hook-
sign/moon” which translates into numerals one (moon), nine (hook
sign) and one (the deity Vishnu) giving us the year 191 of the Saka era
beginning in 78 CE. (e year corresponds to 269 or 270 CE.) More
mathematical examples can be cited from Surya Siddhanta, an early 6th
century text, and the 14th century writings of Madhava.36
is manner of writing numbers shows one thing; the order in
which the symbols were written or recited determined their value, with
a proviso that the least signicant number came rst, followed by high-
er powers. e order itself indicated power, and there was no need to
use power-words. us Sphujidhavja could write 191 using only three
e other textual evidence that is routinely cited to support the
idea that place-value was known to Indians as early as the 5th century
36 For Yavana-jataka, see Ploer 2009, p. 47; for a verse from Surya Siddhanta, see
Ifrah, p. 411 and for Madhava, see Pingree 2003, p. 49.
Nothing at Is: Zero’s Fleeting Footsteps c   
CE is a commentary on a verse of Patanjali’s Yoga Sūtras [3.13] which
reads as follows:
Just as a line in the hundreds place means a hundred, in the tens place ten, and
one in the ones place, so one and the same woman is called a mother, daughter
and sister.
e author of this commentary, as Ploer rightly points out, clear-
ly expected his audiences to be familiar with the concept of numeri-
cal symbols representing dierent powers of ten depending upon their
e textual evidence is not in question. Concrete number system
is a place value system. However cumbersome and full of ambiguities it
was, there is no doubt that the order of the symbols alone determined
their value.
What is in question is whether India was the rst to combine dec-
imal powers of ten with place value, as is routinely claimed. e use
of concrete symbols (bhuta) for numbers is denitely unique to India.
But was the practice of reading the value of a number from its position
unique to India? Another important question has to do with the practi-
cal limitations of bhūta-sankhyā method of enumeration. Using sym-
bols to represent numbers was a wonderful device for generating verses
that rhymed and could be easily memorized, but it was most imprac-
tical for actual computations. (Try your hand at adding, subtracting,
multiplying or dividing these two numbers: Vishnu-hook sign-moon
and ashvin-anga-pitamaha, and you will see the problem). e follow-
ing observation by Stephen Chrisomalis, author of a recent book on
comparative history of numerical notations, is right on the mark:
e bhūta-sankhyā system is suggestive of positionality, but does not constitute
a system of graphic numeral signs, nor should its use be taken to imply the
widespread use of decimal positional numerals in Indian manuscripts.38
Beyond its extensive use in recording dates and years in land-deeds,
and for recording the nal results of computations, bhūta-sankhyā sys-
tem did not nd much use. ere is no evidence to indicate that Brahmi
numerals were much in use for practical purposes either. What then,
37 Ploer, 2009, p. 46. Similar analogies are also recorded in the Buddhist literature,
dating back to rst century CE.
38 Chrisomalis, 2010, p. 195.
 d Science in Saron
were the computational practices needed for commerce, account-keep-
ing, tax-collection and myriad other uses of numbers in everyday life?
We do hear of dust-boards used for computations, but we have no clue
what the numerals looked like, what the rules of computation were, or
who used these boards.39
Let us now put the development of place-value in India in a com-
parative perspective.
It is well-documented that as far back as 1800 BCE, the Mesopota-
mian cultures were using a base- 60, or sexagesimal, place-value system
to write any number, however large, using just two symbols (a hook for
1 and a wedge for 10). It is also well-established that the Mayan people
(of what we today call South America) had developed a base-20, or vi-
gesimal, place-value system. But as the modern numerals have followed
a base-10, or decimal system, we will exclude these outliers. at leaves
us with the Greeks and the Chinese. Both civilizations had a decimal
number system that used a hybrid form of place-value. What is more,
both have le us evidence of well-developed technologies of practical
computations – abacus in the case of Greco-Roman civilization, and
counting rods in the case of China.40
It is well known that the Greeks thought of numbers in geometrical
patterns and, consequently, remained largely preoccupied with geom-
etry. It has been well documented elsewhere how the Greeks adapted
Egyptian numbers to their own purposes, and gradually came to adopt
what is called the Ionian system of numeration around sixth century
BCE. is system was alphabetical: the rst nine letters of the Greek
alphabet were associated with numbers 1 to 9, the next nine alphabets
39 e best description that I have come across is from Datta and Singh: “For the
calculations involved in ganita, the use of some writing material was essential. e
calculations were performed on board with a chalk, or on sand (dhuli) spread on
the ground or on a board. us the terms pati-ganita (“science of calculation on
the board”) or dhuli-karma (“dust-work”) came to be used for higher mathemat-
ics. Later on, the section dealing with algebra was given the name bīja-gaita.”
1938, p. 8. ey provide no further details. Robert Kaplan also refers to these
sand-boards and conjectures that zero was the empty space le behind in the
sand when a Greco-Roman style “counter” – most likely a rounded pebble – was
40 e Chinese abacus, which is still in use, is very dierent from the ancient Greco-
Roman abacus.
Nothing at Is: Zero’s Fleeting Footsteps c   
represented multiples of 10 (10, 20….90), while the last batch of alpha-
bets (which included three archaic alphabets) stood for the rst nine
integral multiples of 100 (100, 200, 300…900). To take an example, 654
could be written as χνδ, where χ stands for 600, ν for 50 and δ for four.
e system is not positional. Yet, as Carl Boyer pointed out, “that the
Greeks had such a principle more or less in mind, is evident not only in
the repeated use of symbols from α through ϴ for units and thousands,
but also in the fact that the symbols are arranged in order of magnitude,
from the smallest on the right to the largest on the le.41
If we to turn to the “counter culture” of Greeks and Romans – lit-
erally, counters which could be pebbles to clay shards being moved
around on counting boards – we nd a positional decimal system rm-
ly in place, with spaces le empty, signifying what we today call zero.
Archeologists have recovered actual abacuses and counting tables
going as far back as the third century BCE Greece. Some thirty abacuses
have so far been found in the region around the Aegean Sea, includ-
ing Greece and what is now Turkey. ese counting devices are simple
structures, consisting of a at surface on which lines are marked be-
tween which counters are moved. e most famous abacus is the Table
of Salamis, dating back to 5th BCE (see Figures 5 and 6), and the most
famous image of a money- counter using a counting board is from the
Darius vase, dating back 350 BCE (see gure 7).42
We must include these early calculators for this reason: they oper-
ated on the principle of positional value. In other words, the pebbles/
counters changed value according to the position they occupied. e
basic operation was as follows:43 counters move between lines, based
upon simple equivalences between numbers. Five times ten is y, and
therefore ve counters on the ten-line are equivalent to one counter on
the y line; likewise, two counters on the y line can be replaced with
one counter on the one hundred-line. Suppose you have four counters
on the ten-line and one counter on the y-line. Let us suppose you
want to add ten. You add a single counter on the ten-line. Now that you
41 Merzbach and Boyer, 2011, p. 54.
42 See Ifrah, pp. 200-211, Kaplan, pp. 23-24.
43 From Reviel Netz, p. 326.
 d Science in Saron
Figure 5. e Salamis Tablet, 300 BCE
Source: Ancient Computers,
Figure 6. Roman hand abacus, mapped on to the Salamis Tablet.
Source: Ancient Computers,
Nothing at Is: Zero’s Fleeting Footsteps c   
Figure 7. Details of the table abacus from vase painting, "e War Council of Darius” c. 340-
320 BCE. Source: Computer History Museum at
 d Science in Saron
have ve counters on the ten-line, you are allowed to remove all ve and
add one counter to the y-line, and so on.
Did these counters have a zero? ey surely had empty spaces.
As in the example above, the board was a dynamic space, constantly
changing as counters moved from one line to another. But the empty
space was not given a numerical sign, as most of this computation was
done manually and the nal number recorded in words which did not
need a zero.
is procedure for counting simply assumes that the value of the
same counter depends on which line it sits on. is assumption and
the method of counting using the counting boards must be widespread
enough for the Athenian law giver Solon (550 BCE) to have compared
“a tyrant’s favorite to a counter whose value depends upon the whim of
the tyrant pushing it from column to column.” e same words were
repeated by historian Polybius (200 BCE) with some extra elaboration:
the courtiers who surround the king are exactly like counters on the lines of
a counting board. For depending upon the will of the reckoner, they may be
valued either at no more than an obol, or else at a whole talent.44
is bears striking resemblance in logic – if not in the imagery
– with the 5th century commentary on the Yoga Sūtra cited above. To
remind ourselves: “Just as a line in the hundreds place means a hun-
dred, in the tens place ten, and one in the ones place, so one and the
same woman is called a mother, daughter and sister.” e Greek sources
are dated many centuries before the Indian reference. at itself proves
nothing, except that Indians were neither the only ones, nor the rst
ones, to be familiar with the idea that a number can take on dierent
values, depending upon the position.
But we have evidence from much closer home – China – where
decimal place value was already widespread by 400 BCE. As the evi-
dence from oracle bones shows, writing numbers in powers of tens has
very ancient roots in China (just as it has in India). But a distinct use of
decimal place value – where the position of a number decides its value,
complete with empty space indicating absence of any numeral – was
44 Quoted from Kaplan, p. 22. Talent and obol are names of the Greek currency, with
30,000 obols to a talent.
Nothing at Is: Zero’s Fleeting Footsteps c   
already a common, everyday practice in China 400 years before the rst
millennium of the Common Era.
Alongside the written number ideograms (which date back to
1500 BCE oracle bones) the Chinese had their “counter-culture” rooted
in practice: their “counters” were counting rods which were moved on
any at surface marked into successive powers of tens. ese rods were
not a mere accounting device (as the Grecian abacuses, above) but were
used for all basic arithmetical operations and eventually also for solving
algebraic equations. If the Chinese had transferred their rod-numerals
and the mathematical operations based upon them into writing, the re-
sult would be identical to our modern numeration and mathematical
operations like multiplication, division, root extraction etc.45
A very brief introduction to counting rods will be useful at this
point. We will use Lam and Ang’s Fleeting Footsteps as our guide here.
e rods were in use as far back as 400 BCE (the Warring States
era). e earliest physical rods unearthed by archaeologists go back to
around 170 BCE. Coins and pottery bearing rod-numeral signs have
been dated to around 400 BCE. Records as far back as 202 BCE de-
scribe the rst Han emperor as boasting that he alone knew “how to
plan campaigns with counting rods in his tent.46 ese rods were basi-
cally short sticks about 14cm (5.5 inches) in length, made mostly of
bamboo, but also of wood, bone, horn, iron or even ivory or jade (which
only the very rich could aord). ey were carried (all 271 of them)
in a small hexagonal pouch, much like we carry electronic calculators
or our smart phones today. Bags containing bundles of counting stick
have been found in skeletal remains dating back to the last few centuries
before the Common Era.
Who used them? Practically everybody from traders, travellers,
monks to government ocials, mathematicians and astronomers. In
other words, whenever and wherever computation was required, the
sticks came out of their bags and were spread on a mat, table top, oor
45 is is what Lam Lay-Yong has claimed in her work e Fleeting Footsteps, p. 10
and passim. e Chinese replaced the counting rods with the abacus around 12th
century or so, which Yong believes set them back, as the step-by-step thinking that
rod-numerals required was replaced with rote-learning.
46 Needham and Ling, p. 71.
 d Science in Saron
or any at surface. Evidence shows that during the Tang Dynasty (618-
907), civil and military ocials carried their bags of sticks wherever
they went. e computations carried out with the sticks were written
down on bamboo strips and on paper by the early centuries of the Com-
mon Era.47 Since counting with rods was a practical skill which every-
one was supposed to be familiar with, early mathematical texts (such
as the 3rd century CE Nine Chapters on the Mathematical Arts, and e
Mathematical Classic of Zhou Gnomon that we referred to in the last
chapter) don’t elaborate on how to use them. But a 4th century book
attributed to a Master(Zi) Sun titled Sun zi Suanjing (the Mathemati-
cal Classic of Master Sun), provides details of how computation was to
be carried out with rods. is book was later included in the set of ten
mathematical classics put together during the Tang Dynasty that all as-
piring state ocials had to study in order to pass the entrance exams.
In the early centuries of the Common Era, rod numeral computations
spread to Japan, Korea, Vietnam and other areas in the South-East in-
uenced by both India and China.
How were the rods used? e method is simple and ingenuous.
e rst nine numerals were formed using the rods in the follow-
ing two arrangements: one in which the rods are vertical (zong) and the
other in which the rods are horizontal (heng)
47 Paper was invented by the Chinese around 100 CE, though some archaeological
ndings put the date further back by a century or two.
Figure 8: Counting rods placed vertically, zong, top row; Rods arranged horizontally,
heng, bottom row.
Nothing at Is: Zero’s Fleeting Footsteps c   
To write numbers greater than 10, the rods were set up in columns.
e right-most column was for units, the next one for tens, the next for
hundreds, and so on. A blank column meant no rods were to be placed
there, meaning what we mean today when we write a zero. e Chinese
called the empty space in rod-numerals as kong, , which means emp-
ty, just as Hindus called an empty space śunya. (More on zero in Chi-
nese numerals in 4.4). To make it easier to read the columns, zongs and
hengs were alternated: vertical arrangement of rods (zong) was used in
the unit column, the hundreds column and ten thousand column and
so on, while the horizontal (heng) arrangement was used in tens, thou-
sand, hundred thousand.48 Here are some illustrative examples:49
e columns could be extended in both directions, with columns to
the right of the units column containing negative numbers which were
represented by rods of a dierent color. e rods were used for addi-
tion, subtraction, multiplication and division, the rules for which are
laid out in Sun Zi’s book, translated and explained in Fleeting Footsteps.
In fact, in her lecture when she was awarded the Kenneth May medal
for her distinguished career, Lam Lay Yong took the audience step-by-
step through the steps for multiplication and division that the great 9th
century Muslim mathematician and astronomer al-Khwarizmi uses, to
48 e following formula from Sun zi sums up the arrangement: “the units are
vertical and the tens horizontal, the hundreds stand and the thousands prostrate,
thousands and tens look alike, and so do ten thousand and hundreds.” Lam and
Ang, p. 47.
49 All examples are from MacTutor website.
1234 would be
45698 would be
60390 would be
 d Science in Saron
show that his method is identical to the method that Sun Zi lays out in
his classic text. 50
What interests us are the following similarities between the Chi-
nese rod-numerals and the modern “Hindu-Arabic” numerals. (ey
could not be more dierent in how they look, but looks are deceptive):
ere is an exact correspondence between rod numerals and the
Hindu-Arabic (i.e. modern) decimal place values numerals – to use
Lams words “the two are conceptually identical”.51 In both systems, only
nine numbers and a sign for an empty space are all that is needed to
write any number, however large or small. In both systems, the numeri-
cal values of the digits are built into their positions, going in ascending
power of ten from right to le. Anyone with just the bare-bones infor-
mation supplied above will have no choice but to read this as 60390.
e rod-numeral system is the rst decimal place-value number
system that we have evidence for. All other ancient place-value nota-
tions known to us (the Babylonian and the Mayan) were not decimal.
Even though Brahmi was decimal, we have already established that
Brahmi numerals, which are almost the exact contemporaries of rod
numerals, did not use place value. For this reason, Lam points out, cor-
rectly it seems, that “Brahmi could not have been the conceptual pre-
cursor of Hindu Arabic system”, while fully accepting that the shape of
Hindu-Arabic numerals did evolve from Brahmi via Devanagari.52 (We
will return to the rod-numerals in section 4.4).
50 According to the Encyclopaedia Britannica, Muḥammad ibn Mūsā al-Khwārizmī
(bornc.780 – diedc.850),was a Muslim mathematician and astronomer whose
major works introduced Hindu-Arabic numeralsand the concepts ofalgebrainto
European mathematics. Latinized versions of his name and of his most famous
book title live on in the termsalgorithmandalgebra.
51 Lam, pp. 172, 173.
52 Lam, p. 177.
Nothing at Is: Zero’s Fleeting Footsteps c   
4.3 Antiquity of large numbers in India
e Hindus, the Buddhists and the Jains are well known for using ex-
travagantly large numbers in their cosmological speculations. A verse
from the Yajurveda that oers prayers to numbers that go up to a tril-
lion has been cited above (section 4.1). e Buddhist text, Lalitavistara
which was written around 300 CE tells the story of Buddha who has
been challenged to recite the names of all powers of ten beyond a koti
(i.e., 10 million), each rank being a hundred times greater than the pre-
vious one. e Buddha successfully recites all the names, going up to
the 421st power of ten – that is, one followed by 421 zeros. Many other
examples of breathtakingly large numbers have been documented.53
No one has been able to explain this strange penchant for immense
numbers. ese numbers were obviously not obtained by any kind of
physical measurements, nor did they refer to what exactly was being
counted. Such ights of imagination were obviously of no use in every-
day mathematics. As Sal Restivo points out:
[immense cosmological numbers of the Hindus]... are means for transcending
experience, used for the purpose of mystication, or to convey the notion that
some thing or being is impressive, they are symbols in a mathematical rhetoric
designed to awe the listener into a religious posture… e social roots of this
distinctive mathematical system lie in the particularly exalted status of Indian
religious specialists.54
We could have let the matters rest there. However, many notable
historians see the Indic penchant for large numbers not as a source of
mystication, but as a source of mathematical genius, which led to the
origin of place value and the invention of zero. Ifrah summarizes this
position thus:
e early passion which Indian civilization had for high numbers was a sig-
nicant factor contributing to the discovery of the place-value system, and not
only oered the Indians the incentive to go beyond the “calculable” physical
world, but also led to an understanding (much earlier than in our [western]
civilization) of the notion of mathematical innity itself.55
In other words, creativity in mathematics is ascribed largely (if
not solely) to experience-transcending speculations. e actual math-
53 Kaplan, pp. 37-40; Ifrah, pp. 421-426.
54 Restivo, 1992, p. 49. Emphasis in the original.
55 Ifrah, p. 421.
 d Science in Saron
ematical practices of the calculable world around us are not taken into
But what is the connection between these enormous numbers and
discovery of place value and eventually zero? How are the two related?
For some the connection is obvious and needs no further evidence
or elaboration. Example of this faith-based history comes from the well-
known History of Hindu Mathematics in which the authors, Datta and
Singh, repeatedly inform the reader that “While the Greeks had no ter-
minology for denominations above the myriad (104), and the Romans
above the mille (103), the ancient Hindus dealt freely with no less than
eighteen denominations. … e numeral language of no other nation
is as scientic and perfect as that of the Hindus.” From this they simply
surmise that “even at a remote period, the Hindus must have possessed
a well-developed system of numerical symbols, and again that all these
large numbers “would have been impossible unless arithmetic had at-
tained a considerable degree of progress…56
More recently, in his well-known work Crest of the Peacock, George
Gheverghese Joseph has made a similar argument. Aer citing large
numbers from Yajurveda and Ramayana, and comparing India favora-
bly against Greeks for stopping at a woefully small 104, he points out
what is obviously true: that “the Vedic Indians were quite at home with
very large numbers. However, he goes on to conclude, like Datta and
Singh, that this must have led to the development of place value:
e early use of such large numbers eventually led to the adoption of a series
of names for the successive powers of ten. e importance of these number-
names in the evolution of decimal place value notation cannot be exaggerated.
e word-numeral system, later replaced by alphabetic notation, was the logi-
cal outcome of proceeding by the multiples of ten….57
is argument fails to convince. For one, decimal system – that
is, counting by power of tens – does not itself imply place value. As
explained earlier, counting by the powers of tens can happily carry on
without inventing a system in which the same number acquires a dier-
ent value depending upon where it is placed. Yes, there are verses where
56 Datta and Singh, 1939, pp. 9, 20, 36. ey believe that the existence of large num-
bers also “proves” that “Hindus invented the Brahmi number system” (p. 36).
57 Joseph, 2011, pp. 340-341.
Nothing at Is: Zero’s Fleeting Footsteps c   
number-symbols (bhutas) are used alone and their value is understood
by the sequence in which they are uttered (see section 2). But bhūta-
sankhyā was used only in the verse portions of mathematical texts. e
prose commentaries that accompanied these verses used number words
to express large numbers which did not need a place value system of
notation. us what we will today write as 3045 would be expressed as
tri (three) sahastra (thousand), and chatvaarimshat (four times ten),
pancha (ve), where sahastra and chatvaarimshat are power-words.58
e second aw in this argument has to do with factually incorrect
historical details. If we place Indian evidence in a comparative perspec-
tive, we can clearly see that it was neither the rst nor the only civiliza-
tion that was comfortable with large numbers. We will again bring in
the Greeks and the Chinese.
To begin with, it is simply incorrect that the Greeks could not han-
dle numbers larger than 10,000. Even a cursory familiarity with Greek
numerals would show that they could write any number, however large,
using their alphabetical numerals. It is true that they did not have spe-
cial names (or alphabets) for numbers larger than a myriad (10,000)
which they represented with the Greek letter M (pronounced mu).
But that is hardly the end of the story: a myriad was simply the begin-
ning of a new count which they represented by writing the number of
myriads above M. For example, the number 71,750, 000 was written as
αΜ͵ζροε; 2,056,839,184 becomes βΜκʹ, αΜ͵εχπγ, ͵θρπδ and so
ose who continue to glibly put our ancestors ahead of all others
must pay attention to the well known work by Archimedes (287-212
BCE) called the Sand Reckoner in which the great Greek mathematician
and engineer teaches Gelon, the king of Syracuse, how to nd out how
many grains of sand there are in the universe. He provides names for
increasingly larger orders of a myriad-myriad (that is 108) all the way
to 10 to the power of 80,000 million million! Archimedes accomplishes
58 See Filliozat, 2004 for more on the distinction between how numbers are ex-
pressed in verse and commentary.
59 e example of large numbers and their Greek notations are from Katz, p. 34 and
from the Greek Number Convertor available at
greek/utilities/greeknumberconverter.htm. For more details, consult Boyer, 1944.
 d Science in Saron
this task without the use of a zero, as he uses number-words for the
various orders of 108.60
As Robert Kaplan has pointed out, there are striking similarities
between Archimedes’ method and the story about Buddha that is told
in Lalitavistara described at the beginning of this section. According to
Kaplan, there are structural similarities between the two accounts, in-
cluding even the mention of poppy seeds. Kaplan admits that “clues are
thin on the ground”, but he posits the possibility of Greek inuence on
the evolution of Indian numerals, including the sign of zero which he
believed came from the empty place le behind on Indian sand-boards
when Greek-style pebble-counting spread into India.61 If the clues are
as thin as Kaplan believes them to be, it is better to withhold judgement
and simply admit that question of transmission is perhaps un-answer-
able at this stage.
Turning now to China, we nd that the humble counting rods were
capable of not just expressing any number, however large. ey were ca-
pable of carrying out basic arithmetical manipulations with large num-
bers as well. e following example excerpted here from Lam’s work
will suce:
In Sun Zi suanjing is found the following: Multiply 708,588 by 531, 441 to ob-
tain 376,572,715, 308. When this is divided among 354,294 persons, each per-
son gets 1,062, 882.62
Sun Zi suanjing, remember, is a 4th century text that describes
the rules and methods of carrying out mathematical operations us-
ing counting-rods. It goes without saying that the above problem was
solved using counting rods.
Once again, India cannot rightfully claim to be ahead of other civi-
lizations of comparable age and development when it comes to comfort
with very large numbers. What is most important to note is that the
comparative perspective shows that facility with naming large numbers
is neither necessary nor sucient for the development of decimal place
value and zero. If that were the case, Greek mathematicians, especially
Archimedes, would also have hit upon the idea. On the other hand, we
60 Kaplan, 1999, chapter 3.
61 Kaplan, 1999, chapter 4.
62 Lam and Ang, 2004, p. 14.
Nothing at Is: Zero’s Fleeting Footsteps c   
have seen that both Greek and Chinese “counter-cultures” – hands-on,
everyday calculations – were literally born with decimal place-value,
without which there was no need to represent an empty space in a non-
metaphysical, computational sense. So perhaps, zero was born in the
streets, far away from the ashrams, academies, or mandarin schools
where learned philosophers thought deep thoughts about the void or
4.4. e emergence of zero
e following facts are well-established about the emergence of zero in
the sub-continent and its cultural sphere in South-East Asia:
• Śunya-bindu as a numeral represented initially by a dot begins
to dierentiate from the metaphysical concept of śunya as void
or nothingness sometime around 600 CE – which is also the
time when Brahmi-derived, non-place value numerals begin
to give way to place value numerals.63
• e earliest surviving and unquestioned evidence of śunya-
bindu as a numeral comes not from India, but from Cam-
bodia. It comes from an inscription from a stone pillar
which in part says “the Chaka era reached year 605 on the
h day of the waning moon.” e ‘0’ in 605 is represent-
ed by a dot. As we know that the Chaka era began in the
year 78 A.D., the date of this zero is 683, nearly two centu-
ries before the rst zero shows up in India. (See plate 1)
 e Cambodian inscription was documented rst by a
French scholar George Codes in 1931.e site where the pillar
stood was plundered by the Khmer Rouge and no one knew
what became of it. It was re-discovered – in a storage shed near
the great temple of Angkor Wat – in 2013 by Amir Aczel, an
American-Israeli mathematician and a historian of science. 64
 e Cambodian zero is not a uke. Similar inscriptions
63 According to Chrisomalis (2010, p. 196) śunya-bindu was rst used in Subhandu’s
poetical work, Vasavadatta, written around the 6th century.
64 Aczel has written about this discovery in many forums (apart from a book). See
‘How I rediscovered the oldest zero in history’, available at e Crux, an online
 d Science in Saron
with a dot for a zero are found in Sumatra and Banka islands
of Indonesia, dated 683 and 686 CE respectively. ere are
many more inscriptions, too numerous to list here, from other
South-East Asian lands, especially the present day Malaysia
and Indonesia.65 e implications of the fact that zero shows
up rst in South-East Asia before it makes its appearance in
India have not been fully absorbed by Indian historians, as we
will see in the next section.
• e rst rock engravings in India that indicate the use of zero
in numbers that use decimal place value date back to the sec-
ond half of the 9th century. e most well known is the in-
scription from the Chaturbhujatemple, a rock temple dedi-
cated to Vishnu, near the city of Gwalior. (see plate 2) Inside
the temple (which is no longer used for worship), next to the
murti of the deity, there is an inscription dated year 933 in the
Vikram calendar (which translates into 876 CE). e inscrip-
tion is about a gi of land, measuring 270 x187 hastas, to the
temple. is land was to be turned into a ower garden, from
which 50 garlands were to be oered to the deity everyday.
What makes this inscription a milestone in the history of
mathematics is that the numbers 933, 270, and 50 are written
in Nagari numerals using place-value and a small empty circle
representing zero. is is the rst undisputed evidence of the
use of zero in a number found in India. (see plate 3)
• Two other pieces of contested e vidence are still cited as evidence
for Indian priority over decimal place value numerals with
zero. e rst piece of evidence is a set of copper plates bear-
ing inscriptions about land-grants dating from 594 to 972 CE,
and they are sometimes oered as evidence that zero and place
value were known to us much before the Gwalior inscription.
However, the authenticity of the plates has been questioned.66
 e other piece of evidence is the famous Bakshali manu-
magazine. See also his narrative of the discovery ‘e Origin of the Number’, at
the website of the Smithsonian.
65 See Needham and Ling, 1959, p. 11. An exhaustive list is provided by Ifrah.
66 See Ifrah, pp. 400-402, Datta and Singh, 1938.
Nothing at Is: Zero’s Fleeting Footsteps c   
script found in 1881 in the village called Bakshali in the north-
western region in modern-day Peshawar, Pakistan. e part-
ly-rotted birch-bark manuscript contains problems involving
basic arithmetic, and clearly uses a dot in place-value numer-
als. Augustus F. R. Hoernle, the Indian-born Indologist of Ger-
man descent who rst studied the text, dated the work to the
3rd or the 4th century CE. But that date has been questioned by
later historians, notably by Takao Hayashi in 1995 who places
the mathematics contained in the text to be as late as 7th cen-
tury. If Hayashi is right – as claimed by a general consensus
among scholars – then the earlier date for zero in decimal
place value is ruled out.67
• Once the Classical or Siddhantic period of astronomy and
mathematics begins, the rest of the story has a clear narra-
tive which has the feel of an o-told-tale. Aryabhata, whose
famous work, Aryabhatiya, was written in 510 CE, created his
own (rather cumbersome) alphabetic numeral system which
nobody followed aer he died. Ifrah succinctly describes what
happened aer Aryabhata:
Varhamihira (c. 575) who in his major work Panchasiddhantika, men-
tioned the use of zero in mathematical operations, as did Bhaskara in 629
in his commentary on Aryabhatiya. In 628 in Brahmaguptasiddhanta,
Brahmagupta dened zero as the result of subtraction between of a num-
ber by itself (a- a=0) and described its properties in the following terms:
‘when zero is added to a number or subtracted from a number, the num-
ber remains unchanged, and a number multiplied by zero becomes zero.
…. [And thus] modern algebra was born and the mathematician had thus
formulated the basic rules… this brilliant civilization opened the way to…
development of mathematics and exact sciences.68
67 And yet one nds a scholar of the caliber of Joseph who seems unable to let go of
the earlier date for this manuscript. Aer repeatedly endorsing Hayashi, Joseph
continues to use Bakshali as “substantial piece of evidence, aer Jaina mathemat-
ics, to bridge the long gap between the Śulvasūtras of the Vedic period and the
mathematics of the classical period which began around 500 CE.” p. 358. Clearly,
if the manuscript is dated aer 7th century, this statement is incorrect.
68 Ifrah, p. 439. Ifrah has an entire chapter titled ‘Dictionary of the Numerical Sym-
bolism of Indian Civilization’ where he expands upon these ideas. e interested
reader is advised to consult this dictionary.
 d Science in Saron
Aer all this, the standard story-line is simple: India’s generous gi
spread to all corners of the world. Arab mathematicians picked up nu-
merals from India and transmitted them to Europe. Buddhist monks
from India took the Hindu numerals, complete with place-value and
the symbol for zero, with them to China. Because the decimal numer-
als with a zero were so much more convenient than any other numeral
system for actually manipulating numbers, the entire world discarded
their old numbers and adopted the Hindu-Arabic numerals. anks to
us, the world learned how to count.
We must now do what we have done throughout this chapter; we
must look at the Indian evidence in a comparative perspective.
Even though there were possibilities inherent in the Greco-Roman
abacuses, place-value and zero did not take root in that culture. But
what happened in China is a dierent matter entirely. In China place-
value and blank spaces on counting-boards indicating that a particular
rank (unit, hundred, thousand…) had no number had become a part of
commonsense What is interesting is that in China the walls that sepa-
rated the “specialists” and the “street” were breached to some extent; the
same method was being used by high and low, by the Mandarins and
the learned monks as well as the illiterate farmer or the trader. Because
the “learned” were not separated by high walls of status (at least in this
practical technique), the method of counting sticks became codied in
texts like those of Sun Zi’s and became a part of the scientic tradition.
e Chinese had a name for the empty spaces on their counting
boards: they called them “kong”, , which is exactly how Indians used
the word “śunya. Later on, the exact date is not known, the notation
for “ling” () meaning “last small raindrops aer a storm” was used
to represent a zero.69 So without a doubt, “a strictly decimal positional
system” with a “kong” for an empty space rst appears in China, at least
four centuries before the Common Era. 70
69 Needham and Ling, p. 16.
70 Quoted from Ifrah, p. 279.
Nothing at Is: Zero’s Fleeting Footsteps c   
5. e Chinese origin of zero: Lam-Needham thesis
We come back to the question we started out with: Is it really established
beyond reasonable doubt that ancient India was the original source of
the number zero and the decimal numbering system that is the founda-
tion of modern mathematics?
e answer can only be in the negative.
In light of the fact that a decimal place-value, conceptually identi-
cal with modern Hindu-Arabic numerals, was fully functional in China
around the time when non-place value Brahmi numerals were barely
emerging in India, some skepticism – and some humility – is warrant-
ed. is means that we must pay serious attention to the lines of pos-
sible transmission from China, through South-East Asia to the Indian
sub-continent, that Needham and his Chinese colleagues hypothesized
in 1959, and that Lam has argued for more recently.
So far, Indian historians have simply assumed a one-way transmis-
sion of mathematical ideas from India to China. But if we look close-
ly, the transmission was always two-way, with at least as much com-
ing from China into India as the other way around. Moreover, if we
look past the monks carrying Buddhist wisdom to also include Indian
merchants hazarding the mountain passes through Tibet and sailing
through the Bay of Bengal, past South East Asian islands to reach Chi-
na, it becomes entirely plausible that they could have brought the Chi-
nese way of counting with them. e beauty of this conjecture is that it
can solve two puzzles in the biography of zero:
1. e rst puzzle is the gap of nearly 900 years between Brahmi
non-place decimals, to Nagari place value decimals numerals
complete with a śunya-bindu. (see section 4.2).
2. e other puzzle has to do with why zero appears in Cambo-
dia and Indonesian islands before it shows up in Gwalior? (see
section 4.4).
It is customary to date Indo-Chinese contact when China estab-
lished an embassy in the court of the Guptas.71 But that is not entirely
true. We know from the account le behind by Zhan Qian, who rst
71 Joseph, p. 304.
 d Science in Saron
explored the lands beyond the Western frontier of China in 138 BCE,
that bamboo and cotton from southwestern provinces of China that
were being supplied by Indian caravans were being sold as far west as
Bactria, the land that straddles today’s Afghanistan, Pakistan and Ta-
jikistan. China was familiar enough to nd a reference in the Mahab-
harata. Moreover, Indians were not the only travellers: Pre-Islamic Ar-
abs, Greeks, Persians and Central Asians have been travelling the many
“silk routes” since 130 BCE, when the Chinese opened their western
Likewise, it is not at all clear that South East Asian countries were
purely Hindu kingdoms, as is oen argued to explain away the puzzle
that zero appears in Cambodia and Indonesian islands before it appears
in Gwalior. e fact is that even those parts of South-East Asia that
were under the cultural and political inuence of Hinduism – includ-
ing Cambodia, parts of Vietnam, ailand, Laos and Burma – were in
constant contact with China. To quote Prabodh C. Bagchi, the author of
a book on India-China relations:
For over a thousand years, the entire Indo-Chinese peninsula and the islands
of the Indian archipelago were for all practical purposes a Greater India. In-
dian colonizers had set up ourishing kingdoms. Indian culture permeated the
people of the country. Regular lines of communication by sea connected these
kingdoms with India on the one hand, and with China on the other.73
is bi-cultural nature of Southeast Asia lies at the heart of Need-
hams conjecture cited earlier (section 2) that the symbol for zero is “an
Indian garland thrown around the nothingness of the vacant space on
the Han counting boards”. How Needham elaborates this conjecture de-
serves to be reproduced in full:
We are free to consider the possibility (or even the probability) that the written
zero symbol, the more reliable calculations it permitted really originated in the
eastern zone of Hindu culture where it met the southern zone of the culture of the
Chinese. What ideographic stimulus it could have received at that interface?
Could it have adopted an encircled vacancy from the empty blanks le for zeros
on the Chinese counting boards? e essential point is that the Chinese had
possessed, long before the time of time of Sun Tzu Suan Ching (late +3rd centu-
ry) a fundamentally decimal place-value system. It may be then that the ‘empti-
72 Bagchi, p. 7.
73 Bagchi, p. 25.
Nothing at Is: Zero’s Fleeting Footsteps c   
ness’ of Taoist mysticism, no less than the “void” of Indian philosophy, contrib-
uted to the invention of symbol for śunya, the zero. It would seem, indeed, that
the ndings of the rst appearance of zero in dated inscriptions on the borderline
of the Indian and the Chinese culture-areas can hardly be a coincidence.74
is thesis, as we have seen, has received a second wind from the
writings of Lam Lay Yong. Lam endorses Needham’s conjecture, and
strongly argues for the Chinese origin of zero, but her evidence comes
more from the conceptual identity between the Chinese and the “Hin-
du-Arabic” mathematical procedures described in three Arabic texts.75
Lams work opens a fresh line of inquiry, namely, the Arab-Chinese cul-
tural exchanges which go back to pre-Islamic era and were intensied
aer the birth of Islam. (As Prophet Mohammad advised his followers,
“seek knowledge as far as China”). e direct contact between Chinese
and Islamic mathematicians and scientists is too complicated to be dis-
cussed here.
Contemporary historians are split over the relevance of the Chi-
nese evidence for understanding the evolution of Indian mathematical
ideas. Some like Ifrah and Chrisomalis dismiss the very idea of zero as
anything but purely Indian, while others like Joseph and Ploer advise
against such summary dismissal. More balanced is the opinion of Victor
Katz, author of a well-respected textbook on the history of mathemat-
ics, who looks on the invention of zero as a multi-step process by which
India and China built upon each other’s ideas. Katz's words deserve to
be quoted at length, because they happen to exactly coincide with the
surmise of this essay:
It has been suggested that the true origins of the system in India come from the
Chinese counting boards. e counting board was a portable object. Certainly
Chinese traders [and Buddhist seekers as well] who visited India carried these
along. In fact, since Southeast Asia is the border between Hindu culture and
Chinese inuence, it may have well been in that area where the interchange
took place. What may have happened is that the Indians were impressed with
the idea of using only nine symbols. But they naturally took for their symbols
the ones they had already been using. ey then improved upon the Chinese
74 Needham and Wang, pp. 11-12, emphasis added. e quote about the “garland”
appears on p. 148.
75 ese texts are: “a Latin translation of al-Khwarizmi’s work on arithmetic, al-
Uqlidisi’s Kitab al-Fusul  al-Hisab al-Hindi, and Kushyar ibn Labban’s Kitab 
usual Hisab al-Hind. Lam, p. 178.
 d Science in Saron
system for counting rods, for using exactly the same symbols for each place
value, rather than alternating two types of symbols [horizontal and vertical].
And because they needed to be able to write numbers in some form, rather than
just have them on the counting board, they were forced to use a symbol, the dot
and later the circle to represent the blank column of the counting board. If this
theory is correct, it is somewhat ironic that the Indian scientists then returned
the favor and brought the new system back to China in the 8th century.76
If Katz is right – as he seems to be, in light of the material we have
discussed at length in these pages – then it should help explain one
more puzzle: why are there no written records in India which men-
tion Chinese way of computation, while there are plenty of translated
“Brahmin” texts in astronomy in China? e answer seems to be that
the counting rods and the methods of using them were not written in
books, but learnt through practice. It will be futile to look for evidence
in learned texts. e evidence lies in the counter-culture of traders, trav-
ellers and even monks who, to use Needham’s words, “had exchanged
metaphysics for mathematics”.
6. Concluding remarks
is chapter has tried to present the history of zero in a new key. A
comparative, practice-centered perspective is what sets this account
apart from the traditional histories of zero that dominate the historical
scholarship, and saturate the public sphere, in India.
I hope that the evidence provided in this chapter will encourage the
readers to look beyond national or civilizational boundaries to develop
a deeper understanding of how ideas evolve through a give-and-take
between civilizations, and how civilizations build upon ideas and prac-
tices that travel back-and-forth across trade routes, pilgrimage circuits
and political relations.
I hope, moreover, that this chapter succeeds in planting in the
readers’ minds a seed of doubt about “purity” of ideas. ere is noth-
ing “pure” about “pure mathematics,” for there is a constant interplay
between practices and concepts. Neither is there anything in history
of science that is “purely” Indian, or “purely” European, Chinese, or
76 Katz, p. 235.
Genetics, Plastic Surgery and Other Wonders of Ancient Indian Medicine c   
Islamic. History of science is a wonderful example of history of inter-
civilizational exchange of ideas. Conning it within nationalistic frame-
works can only lead to a tunnel vision, and there is no reason why we
should accept such a limitations on our ability to see the wider vistas
that encompass the whole world.
 d Science in Saron
 d Science in Saron
Genetics, Plastic Surgery and Other Wonders of Ancient Indian Medicine c   
Cha pt er 3
Genetics, Plastic Surgery and Other Wonders
of Ancient Indian Medicine
1. Introduction
As the title suggests, this chapter is about medical knowledge in ancient
India. But it is more than that. It also proposes a plan for combating
pseudohistory of science – a plan that has the potential to turn the ma-
nia for mythic history into an opportunity for learning.
Even a cursory look at news headlines will show that we are in-
undated these days with myths of our civilization’s singular greatness.
A narrative of Indic, or dharmic, exceptionalism is under construction
which celebrates its spiritual and scientic riches. Not unlike American
exceptionalism, Indic exceptionalism seeks to universalize itself, both
at home and around the world.1
e myth of Indic exceptionalism is a myth wrapped in and
around myths taken straight out of the Mahabharata, the Ramayana
and the many Puranas, the traditional storehouses of mythology. e
1 Dharmic civilization is understood as the civilization that is native to the land of
India. It subsumes Hinduism, Buddhism, Jainism and Sikhism. Its distinctive set
of assumptions regarding “divinity, the cosmos and humanity” are seen as oering
“an Indian challenge to Western Universalism,” as the subtitle of a recent book by
Rajiv Malhotra (2011) would have it. e point to note is that the Indic/dharmic
tradition by denition excludes those Indian religious traditions with roots in the
Judeo-Christian and Islamic traditions.
 d Science in Saron
 d Science in Saron
new myth-makers appropriate popular myths from this rich tradition,
evacuate religious or spiritual meanings out of them, and retell them as
if they are literally true accounts of scientic and technological achieve-
ments. e much beloved gods and goddesses that are imprinted in
the collective psyche of Indian people remain – but now they serve the
earthly ambitions of men and women.
A myth, according to the Oxford English Dictionary, is “a tradi-
tional story, especially one concerning the early history of a people or
explaining a natural or social phenomenon, and typically involving su-
pernatural beings or events: ancient Celtic myths. Myth also means, ac-
cording to OED again, A widely held but false belief or idea.
Both meanings of myth are at work in the public sphere in India
today, with one important dierence: rather than see myths for what
they are – “traditional stories…..involving supernatural beings,” or as
“widely held false beliefs” – they are being served up as legitimate evi-
dence of scientic achievements. Like fundamentalists everywhere who
insist upon reading religious texts as literal accounts of the creation and
evolution of the universe, in India too, the miraculous prowess of su-
pernatural beings is being interpreted as if they provide a literally true
account of the achievements of ancient “scientists” and “engineers.
is chapter will oer a creative way we can turn this asco into a
teaching moment. e basic idea is simple: whenever our political lead-
ers dish out myths and call them “science,” we should take it upon our-
selves to learn some real history of real science in the specic domain
in question.2 Aer we are done laughing at the absurdity of the tall-
tales we are told, we should get down to the more sober task of educat-
ing ourselves with the actual history of science in India as a part of the
global history of medicine, science and technology. is self-education
requires that we arm ourselves with the best, the most reliable evidence
available and approach it with a critical, or a scientic, spirit – that is,
be willing to rethink our preconceived ideas in the light of compelling evi-
dence.3 is is what this chapter intends to do for history of medicine
2 I think of it as my “lemonade model,” inspired by the old proverb, “when world
gives you lemons, make lemonade.
3 Here Garrett Fagan, a critic of pseudo-archeology, is right on the mark: “a basic
characteristic of genuine [as opposed to pseudo-] archeology, of whatever theoret-
Genetics, Plastic Surgery and Other Wonders of Ancient Indian Medicine c   
(as the previous two chapters have tried to do for two landmarks in
Such an exercise, carried out with rigor, honesty and a sturdy re-
spect for historical evidence can yield rich dividends. Its usefulness for
countering ideologically-driven pseudohistory of science is obvious.
Less obvious, but perhaps more important, is how a dose of real his-
tory can save the ancient physicians, crasmen-mathematicians from
becoming civilizational icons (as in the Indocentric discourse), or from
becoming totally invisible (as in the Eurocentric discourse). Under-
standing how the ancients grappled with the natural world armed with
nothing more than their faculty to reason and the evidence of their
senses, can save them from both glorication and condescension at the
hands of their 21st century inheritors.
2. Mythologizing medicine
Scholarly study of myths has come a long way from the 19th centu-
ry understanding of myths as proto-scientic explanations of nature.
roughout the 20th century, as scientic understanding of the natural
world made progress, “ the physical world was conceded to [modern]
science,” as Robert Segal, a leading theorist of myth put it, and myths
were no longer seen as competing with science as explanations of na-
ture; they were instead reconceived as symbolic narratives about the
place of human beings in the world, their unconscious fears and fan-
tasies, their sense of right or wrong.4 As Sudhir Kakar, the pre-eminent
interpreter of the “inner world” of Indians puts it, “myths… are individ-
ual psychology projected onto the outside world… myths can be read as
a kind of collective historical conscience, instructions from the vener-
able ancestors on ‘right’ or ‘wrong,’ which serves to bind the members
of a group to each other.5
ical bent, is the maintenance of conceptual exibility – a willingness to re-examine
favored conclusions in the face of… countervailing evidence, and to change those
conclusions accordingly. It is not unreasonable to brand such an intellectual stance
as broadly scientic insofar as it accepts the capacity of the data to reshape inter-
pretations” (emphasis added), Fagan, 2006, p.25
4 Robert Segal, 2006, pp. 341-342.
5 Sudhir Kakar, 1981, p. 4.
 d Science in Saron
 d Science in Saron
In India of the 21st century we seem to be stuck in the 19th century:
Myths continue to crop up in history of science as if they are literally
true accounts of the physical world, or as literally true descriptions of
technological artifacts. Existence of ancient Vedic-era space-ships ca-
pable of inter-galactic travel, the existence of nuclear weapons in the
time of the Mahabharata and other such fantastic tales continue to be
asserted by learned men and women in academic forums.
It is in this context that when the Prime Minister of India used my-
thology as evidence for the existence of advanced knowledge of genet-
ics and surgery in ancient India, it made news not just in India, but
around the world. One could not but read Mr. Modi’s words as giving
ocial blessings to the mythication of science that has been going on
in the country for a long time, but which seems to intensied under his
Speaking at the inauguration ceremony Sir H.N. Reliance Founda-
tion Hospital and Research Center in Mumbai on October 25, 2014,
Modi invoked familiar Hindu myths to exhort the audience to take
pride in the medical achievements of our ancestors. e Hindi text of
his speech is available on the ocial website of the Prime Minister’s Of-
ce. Excerpts in English translation are reproduced here.
Karna in the Mahabharata, Modi suggested, could well have been
a medical rst; a baby born in-vitro. is is what he said: “We can feel
proud of what our country achieved in medical science at one point of
time. We all read about Karna in the Mahabharata. If we think a little
more, we realize that the Mahabharata says Karna was not born from
his mother’s womb. is means that genetic science was present at that
time. at is why Karna could be born outside his mother’s womb.
Next, Modi invoked Lord Ganesh in the context of plastic surgery.
“We worship Lord Ganesh. ere must have been some plastic surgeon
at that time who got an elephant’s head on the body of a human being,
and began the practice of plastic surgery.”
e PM stopped at Ganesh. But following this line of thinking,
many more medical rsts can be claimed. Aer all, we worship Hanu-
man, and so there must have been biophysicist who could make this
member of higher primates y. We worship gods and goddesses with
Genetics, Plastic Surgery and Other Wonders of Ancient Indian Medicine c   
any number of fully-functional arms and heads and so there must have
been neurosurgeons way back then. So on and so forth. T h e
point is that if we are not going to respect any boundary between myth
and science, then history of science simply collapses into mythology.
Myths interpreted literally come to serve not just as evidence for rudi-
mentary or proto-science, but for the most cutting-edge sciences that
we have today.
No doubt this bit of myth-making at the hospital was done with
best intentions of encouraging pursuit of science. As Modi explained,
“What I mean to say is that we are the country which had these capabili-
ties. We need to regain these.
One could well complain that we are making too much of these
remarks. Aer all, don’t all politicians, from the Le and the Right, go
into a grandstanding mode time to time? is is what politicians do.
But Modi, as is well-known, is a product of the shakha culture of
the RSS. Having joined the local shakha when he was barely eight-years
old, the RSS “[has done] the most to shape him and his worldview, and
to advance his political ambitions,” to quote from Vinod Jose’s bio-
graphical essay on the rise of Narendra Modi.6 Fables about “scientic”
achievements of our Hindu forefathers are as natural in the RSS cul-
ture as water is for sh. With the RSS in an unprecedented position of
power, there is every reason to fear that this mythology will nd a place
in textbooks. is is one very good reason why we must take the PM’s
pronouncements seriously.7
6 Vinod Jose, e Caravan, March 2012.
7 All signs are pointing to a massive push for the Saronization of education. Earlier
this year, the Ministry of Human Resource Development began its consulta-
tive process for a New Education Policy. It has invited input from grassroots
movements regarding 33 topics related to school and higher education posted
on its website e RSS is a major player in the consultative
process. According to the Deccan Herald, “Amid these initiatives and plans of
the government, the Rashtriya Swayamsevak Sangh’s (RSS) education wing is
silently working to assist the government formulate the new policy. A Shiksha Niti
Aayog (education policy commission), set up under the leadership of controver-
sial educationist and former RSS pracharak Dinanath Batra, is holding parallel,
nationwide deliberations to get suggestions from the “right-minded” citizens of
the country. It has plans to hold at least 500 seminars across the country to “make
people aware of the drawbacks of the current education system and get vital
 d Science in Saron
 d Science in Saron
is mythic history of medicine has implications for health policy
as well. Under the Modi government, AYUSH, the government body
that oversees traditional medical systems, has been elevated to a full-
edged ministry with an annual budget of 1,200 crore rupees. Even
though the number of randomized control trials for Ayurveda can be
counted on the “ngers of one hand,” and even though homeopathy has
been proven multiple times to be utterly ineective in rigorous double-
blind trials, resources are going to be diverted to these medical tradi-
tions which are more aptly described as alternatives to medicine, rather
than as alternative medicine.8
e situation is ripe to put “the plan” into action, that is, turn every
mystication into an opportunity to educate ourselves in real history of
real science. Following the PMs mystication, the plan calls for look-
ing up our ancient medical to nd out what they actually have to say
regarding “genetic science” and surgery. When we call them “scientic,
what do we mean? If we really had made such advances in medicine in
the past, why did we stop? Why has Ayurveda not made any real pro-
gress beyond whatever was put down in Charaka and Sushruta samhi-
tas composed in the early centuries of the Common Era?
In this chapter, we will examine these issues in more details. We
will rst look into the question of “genetic science” in Charaka Samhita.
e next section will examine the question of plastic surgery, focusing
on the method of nose reconstruction in Sushruta Samhita. We will fol-
low it up with a comparative history of anatomy where we will address
the question why, despite the promising start in anatomy and surgery,
we fell behind sister civilizations.
But we will start with a brief discussion of the dangers of anach-
ronistic or “presentist” history. Delving into this problem with history
suggestions from them on how to make it relevant for the country. http://www.nitely-not.html
To understand why the leadership of Dinanath Batra should worry us, here is a
gem from his book, Bharatiya Shiksha kaa Swarup: “Charaka explained blood
circulation in 300 BC, while the credit is given to William Harvey.” p. 50. Batra
provides no evidence to back this astounding claim.
8 See Rukmini Shrinivasan, “Medicine Wars,e Hindu, April 26, 2015. AYUSH
stands for Ayurveda, Yoga, Unani medicine, Siddha and homeopathy. e phrase
“alternatives to medicine” was suggested by my friend, Vijayan.
Genetics, Plastic Surgery and Other Wonders of Ancient Indian Medicine c   
writing may seem like a digression, but its relevance to the issue at hand
will soon become evident.
3. Why anachronism is bad history of science
One of the rst things all historians are taught to avoid is the “sin” of
writing “Whig history”, which consists of giving anachronistic or “pre-
sentist” accounts of the past.9 Anachronistic history is simply reading
the past in the vocabulary derived from our present knowledge, beliefs,
or values. It is “unhistorical history writing” that “studies the past with
one eye to the present, to use Buttereld’s famous words. Put another
way, it uses now as the prism through which it views then. Historians
of science are especially wary of presentism for the potential it has to
distort what scientists in the past were trying to achieve. e presentist
distortion in history of science comes when historians “cast a particular
theory, now deemed correct, as proven right from the start,” or to put
it another way, when they cast the “scientists” of earlier eras as working
with the same conceptual and methodological framework as scientists
e opposite of anachronistic history is the diachronic, or contex-
tual, history of ideas in which the historian tries to become an observer
in the past, not just of the past; in which the historian takes a y-on-the-
wall approach to writing history. is requires that the historian must
9 e term “Whig history” was made famous by Herbert Buttereld’s 1931 classic
titled e Whig Interpretation of History. By Whig history Buttereld was referring
to the habit of British liberals to read the political history of Britain as one long
continuous and inevitable march toward parliamentary democracy. is way of
history writing worked by reading the contemporary political philosophy of liber-
alism back into the minds of actors in the past, who in reality may have had totally
dierent motives and meanings for their actions.
10 e quotation is from Douglas Allchin, 2004, p. 182. Strictly speaking, there were
no “scientists” before the term was coined by William Whewell in 1834 to describe
the students of the knowledge of the material world collectively. By “scientist” he
meant an analogue to “artist”, as the term that could provide linguistic unity to
those studying the various branches of the sciences. But, of course, human beings
have been studying the material world from the very beginning of history. e
correct name for pre-modern students of nature is “natural philosophers”. See Syd-
ney Ross, 1962. See also