**Thanks to Arvind Gupta for sending this paper.**

**A SCHOLAR IN HIS TIME: CONTEMPORARY VIEWS OF KOSAMBI THE MATHEMATICIAN**

**RAMAKRISHNA RAMASWAMY**

University of Hyderabad,
Hyderabad, TS 500 034

“

*Kosambi introduced a new method into historical scholarship, essentially by application of modern mathematics.”*J. D. Bernal [1], who shared some of his interests and much of his politics, summarized the unique talents of DDK [2] in an obituary that appeared in the journal Nature, adding, “*Indians were not themselves historians: they left few documents and never gave dates. One thing the Indians of all periods did leave behind, however, were hoards of coins. [...] By statistical study of the weights of the coins, Kosambi was able to establish the amount of time that had elapsed while they were in circulation . . .*”
The facts of DDK’s
academic life, in brief are as follows. He attended high–school in the US, in
Cambridge, MA, and undergraduate college at Harvard, graduating in 1929. Returning
to India, he then worked as a mathematician at Banaras Hindu University (1930-31),
Aligarh Muslim University (1931-33), Fergusson College, Pune (1933-45), and the
Tata Institute of Fundamental Research (1945-62), after which he held an emeritus
fellowship of the CSIR until his death at the age of 59, in 1966.

Today the significance
of D. D. Kosambi’s mathematical contributions [3–71] tends to be underplayed,
given the impact of his scholarship as historian, and Indologist. His work in
the latter areas has been collected in several volumes [72] and critical commentaries
have appeared over the years [73, 74], but his work in mathematics has not been
compiled and reviewed to the same extent [75, 76, 77, 78]. Indeed, a complete
bibliography is not available in the public domain so far [79]. This asymmetry
is unfortunate since, as commented elsewhere [75], an understanding of Kosambi
the historian can only be enhanced by an appreciation of Kosambi the mathematician
[80].

DDK is known for
several contributions, some of which like the Kosambi-Cartan-Chern (KCC) theory
[81], carry his name, and some like the Karhunen–Loève expansion [37, 39, 82],
that do not. The Kosambi mapping function in genetics [40] continues to be used
to this day [83], but the path geometry that he studied for much of his life
[84] has not found further application. DDK’s final years were mired in controversies,
both personal and professional. His papers on the Riemann hypothesis (RH) [65,
66] brought him a great deal of criticism and not a little ridicule, while his
personal politics put him in direct conflict with Homi Bhabha and the Department
of Atomic Energy. This contributed to his eventual and somewhat ignominious
ouster from employment at the Tata Institute of Fundamental Research. His early
and passionate advocacy of solar energy was practical and based on sound
scientific common sense. In some of his arguments, he seems even somewhat
Gandhian, and although this was a contrary position to hold in the TIFR at that
time, the essential validity of his argument remains to this day [85].

DDK was just about 23
years old when he returned to India and took up a temporary position at Banaras
Hindu University with a BA (summa cum laude) from Harvard. A year later he had
moved to the Aligarh Muslim University where he was appointed in the
Mathematics Department at the suggestion of André Weil [86] who, just about a
year older, was then already well known as a mathematician and as a prodigy,
and who had been invited to the AMU by then Vice Chancellor Syed Ross Masood.

Although Weil did not
last long in Aligarh, his influence on Kosambi was considerable. In addition to
giving him the position and encouraging him on the matter of the Bourbaki prank~~,~~ [86],
Weil helped DDK forge early mathematical links with, among others, T.
Vijayaraghavan [87] and S. Chowla [88]. He undoubtedly influenced his taste in
mathematics, possibly sparking DDK’s interest in the Riemann hypothesis. Weil
would, in the early 1940’s make important contributions to this field [89] although
when DDK turned to it almost thirty years later [65] his efforts were to come a
cropper. Weil spent the summer of 1931 in Europe and upon his return to
Aligarh, he found that not only had his own position been compromised, but the
group of mathematicians that he had put together had also fragmented, with Vijayaraghavan
having moved to take up a Professorship in Dacca [90]. By early 1932, Weil had
returned to Europe, and DDK was to leave Aligarh soon thereafter.

Kosambi started his
independent work in Aligarh, choosing the area of

*path–geometry*, a term he coined, submitting his papers to leading European journals [7, 9, 10]. One that was sent to Mathematische Zeitschrift was also communicated to Elie Cartan who was inspired enough by the result to write a detailed commentary, which included an extract of the correspondence that Kosambi had with him. This was also published in Mathematische Zeitschrift [11] immediately following DDK’s paper in 1933. Along with a later paper by the Chinese mathematician, S. S. Chern, these three works constitute what is now termed the KCC-theory, a topic that has, in recent years, found new applications in physics and biology [81]. Some years later, in 1946, Kosambi tried to have Chern invited to visit India when he was at the TIFR but nothing came of it.
DDK wrote many papers
on path geometry, and in the mid 1940’s summarized his work in a manuscript
that was submitted to Marston Morse at the Institute for Advanced Study in
Princeton. In a letter [91] to Bhabha he says, “The book on Path-Geometry will,
according to a letter from Morse, appear in the Annals of Mathematics Studies,
Princeton.” This book was never published—indeed very few books in this series
were, and efforts to locate a copy of the manuscript in the Morse archives have
proved fruitless [92]. DDK makes reference to a second copy of the manuscript
that he gave to Bhabha, but that copy has not been located either.

The Nobel laureate, C.
V. Raman had visited Aligarh in 1931 as the member of a selection committee,
and although there is no specific record of his having met Kosambi, his
subsequent actions suggest that he quickly gathered, either directly or
indirectly, a very high opinion of DDK. In 1934 when Raman founded the Indian
Academy of Sciences in Bangalore, he elected Vijayaraghavan and Chowla. The very
next year Kosambi was elected to the IASc at the age of 28, when his
mathematical œuvre was slight, and along with others such as P C Mahalanobis
and V V Narlikar. Kosambi was one of the younger of the Founding or Foundation
Fellows (namely those elected in 1934 and 1935). Since the initial election to
the Academy was almost entirely his decision, the estimation that Raman had of
Kosambi’s scholarship or of his potential, must have been considerable. It is
possible that Vijayaraghavan may have played some role in this early
recognition [93], and it is also likely that the award of the first Ramanujan
Prize of the Madras University in 1934 to S. Chandrashekar, S. Chowla and DDK
[94] would have favourably impressed Raman. As it happened, in later years Kosambi
was privately and publicly very critical of Raman’s style of functioning [95].

This early
recognition, however, stood him in good stead. He published a couple of papers
in the Academy journal, Proceedings of the Indian Academy of Sciences in 1935
(and not again until the 1960’s when, as S. Ducray, he published two more).
Reviews of his papers in other journals began to appear in

*Current Science*, the general science journal started by Raman, in addition to original articles that he chose to publish in this journal as well. Indeed his initial papers on the quantitative approach to numismatics [26, 27, 34, 36] all appeared in*Current Science*.**1. Reviews and Commentaries**

One of the early
references to the work of DDK on numismatics that was brought to the attention
of readers of

*Current Science*was a review in 1941 [96] by K. A. N. (this was probably the well known historian K. A. Nilakantha Sastry) of two papers of DDK’s in the*New Indian Antiquary*[97]. By this time, DDK seems to have been well established as an eminent mathematician. While generally admiring of the work, KAN comments on a number of DDKs characteristics: the use of “hard phrases” in his critique of the methods used by others, his exposure “of the hollowness of much pseudo–expertise that has held the field”, etc. Nevertheless, the review is not uniformly accepting of DDKs conclusions, and KAN does alert the reader to potential areas of inaccuracy. In a charming final paragraph, for instance, he says “Yet, this conclusion hardly tallies with the impressions of the Mauryan epoch gathered from other sources like the inscriptions of Asoka, or the polished stone pillars–-not to speak of Megasthenes and the*Arthasastra*. There are other statements,*obiter dicta*, which may surprise the reader, and even shock him; but there is much, very much in these papers and their method for which he will be grateful”.
The journal

*Mathematical Reviews*(MR) was started in 1940 by the American Mathematical Society as a way for working mathematicians to keep up with the increasing numbers of papers that appeared each year in diverse journals. The practice was (and still is) to have a brief summary of these papers sometimes with commentary, and sometimes without. Indeed, some papers are merely noted or abstracted, and all reviews are signed.
Of DDK’s sixty or so
papers in mathematics, about half were reviewed in MR; these are indicated in
the bibliography [3–71]. The reviewers include R. L. Anderson, R. P. Boas, Jr.,
N. Coburn, J. L. Doob, W. Feller, V. Hlavaty, M. Janet, A. Kawaguchi, J. B.
Kelly, M. S. Knebelman, J. Korevaar, J. Kubilius, R. G. Laha, W. J. LeVeque, A.
Nijenhuis, E. S. Pondiczery (a pseudonym of R. P. Boas Jr), A. Rényi, J. A.
Schouten, E. W. Titt, J. L. Vanderslice, O. Varga, B. Volkmann, A. Wald, and J.
Wolfowitz. Several of these reviews are just summaries of the papers, but some
are serious commentaries on the work of Kosambi, and, significantly, are by
some of the leading contemporaneous mathematicians, probabilists, and statisticians.
Indeed R. P. Boas Jr. who reviewed some of the papers was one of the main
editors of

*Mathematical Reviews*.
It may be pertinent to
note that it is not just DDK’s papers that were published in journals outside
India that were reviewed in

*Mathematical Reviews*; several of the papers published in Indian journals were also commented upon critically. These include the important paper, “Statistics in function space” [39] that was reviewed by the probabalist, J. L. Doob, who went on to become the President of the American Mathematical Society (and who was awarded the National Medal of Science by then President of the United States, Jimmy Carter in 1979).
Although Doob gave a
careful and comprehensive review of the work soon after it was published in
1943, unfortunately neither Karhunen nor Loève who essentially rediscovered
these results [82] were aware either of the paper or of its review, and today
these results go under their names, and Kosambi’s contribution is largely unrecognized.
One important feature of the paper pointed out in the review,

The author discusses statistical problems
connected with continuous stochastic processes whose representative functions

*x(t)*[. . . ] Various mechanical and electrical methods are suggested for combining functions*x(t),*given graphically, as necessitated by this type of statistical approach.
was the idea of a
mechanical or electrical computer. This was to be part of Kosambi’s

*Kosmagraph*project that was in part funded by a grant from the J. R. D. Tata Trust in 1945. It is not clear if a working model was ever successfully constructed, though there is a reference to it in a report he sent to the Tata Trust [80],
The

*Kosmagraph*is finished, and a working model being improved at St. Xavier’s College. The total outlay for workshop charges, electric motors, cathode ray oscillographs, valve tubes etc. would have exceeded the total amount of the Tata grant. But the St. Xavier’s authorities stood the expense of these items, as Fr. Rafael has collaborated in the work. My total expenses from the grant have been a nominal honorarium of Rs. 250/- to K. B. McCabe, the third collaborator; and another of Rs. 50 to Salvador D’Souza, head mechanic at the St. Xavier’s workshop. Both have deserved far more, and the work of McCabe in particular seems to me to be beyond recompense.
A joint paper is being made ready for
publication, though it will be some months before all the points are checked.

The paper alluded to
does not appear to have been published, and no drafts been located among DDK’s
papers. It is also not clear what became of the project; the interest in a
computing machine stayed with Kosambi when he later moved to TIFR, and indeed
was one of purposes of his visit to the USA in 1948-49 [75].

Another of DDK’s
reviewer’s was Abraham Wald (who was later to die in a plane crash in India
when he was visiting the country at the invitation of the Indian government)
who commented, generally favourably, on four of his papers. What is interesting
is that many of the papers were published in journals such as Mathematics
Student and the Journal of the Indian Mathematical Society, both of limited
circulation, and which to this day remain somewhat difficult to locate [98].

It should be mentioned
that most of DDK’s publications in mathematics are independently authored. He
did, however, mentor several students, both formally and informally at the TIFR
in the 1950’s, and among these were S. Raghavachari and U. V. Ramamohana Rao
who are his only coauthors.

**2. The RH papers**

Arguably the most
important as yet unresolved problem in pure mathematics is a hypothesis that
was enunciated in 1857 by the celebrated mathematician, Bernhard Riemann. A
brief introduction to the nature of the mathematical problem [99] is included
here for those who are less acquainted with it, to give some flavour of why it
is interesting and a challenge. (I should also add that at the risk of losing
half the potential readership with each equation [100], it is absolutely
essential that some be retained. For all of mathematics there is no greater
game than to solve the Riemann hypothesis, and to appreciate both what Kosambi
tried, and where he did not succeed, some equations are needed.)

The Riemann Hypothesis
concerns properties of a mathematical function that has been studied for at
least four centuries. This is the zeta function, the sum of inverse powers of
the integers,

the ellipsis
signifying that the sequence does not terminate. When

*z*is equal to zero, then each term is 1, and the sum, namely 1+1+1+1…. becomes infinitely large:
Such infinite sums have
long been of interest: an example that will be familiar to many is the sum that
arises in Zeno’s paradox regarding Achilles and the tortoise. (In a 100 metre
race, the tortoise, which is 100 times slower than Achilles is given, say, a
head-start of 90 metres. In the time that Achilles covers 90 m, the tortoise
covers 90 centimetres and is therefore still ahead. In the time that Achilles
covers the 90 cm, the tortoise goes ahead by 9 millimetres, when Achilles
covers the 9 mm, the tortoise is ahead by a smaller fraction, and so on. So
Achilles would, it seems, never catch up with the tortoise. The resolution of
the paradox is that this infinite sum is actually a finite quantity, and
Achilles wins the race easily [101].)

The harmonic series, is the sum of inverses
of the integers,

and this also diverges or becomes
infinite (the → in the equation below signifies
“tends to”). In contrast, when

*z*= 2, the sum is a finite number,
Clearly the value of
the zeta function,

depends on the value
of

*z,*and Riemann was interested in its “zeroes”, namely those values of*z*when
In order to study the general properties of
such a function, it is necessary to consider all possible values for

*z*, in particular when*z*is a*complex*number, namely of the form*z = x + iy*where*x*is the real part and*y*the imaginary part, and . It turns out then that the*ζ*function can take values that are either positive or negative depending on the value of*z*, or equivalently, on the values of*x*and*y*. When*y=*0*and**x*is a negative even integer, namely -2, -4, -6 and so on, the function takes the value 0: these are termed “trivial” zeroes since the function can be shown to vanish through a straightforward procedure [102].
The

*ζ*function has in addition an infinite number of “nontrivial” zeros, and Riemann’s hypothesis is that for*all*of these,*x*(namely the real part of*z*) has the value 1/2. In the complex plane, these zeroes therefore all lie on the so–called “critical” line,*x*= 1/2. While being simple enough to state, it remains unproven to this day. Because of connections between the zeta function and prime numbers, a proof of the RH would have significant implications for the distribution of prime numbers, and via this, for much of mathematics. An alternate measure of its importance can be gauged from the fact that it is one of the so–called Millennium Prize problems for which the Clay Mathematical Institute has announced a grand cash award in the past decade.
DDK’s mathematical
reputation suffered greatly as a result of two papers he published in the
Journal of the Indian Society of Agricultural Statistics [63, 67]; in one of
them, he claimed a result that was essentially a proof of the RH.
Notwithstanding its name, the journal does publish serious mathematics,
particularly in the area of probability. Although obscure and highly
specialized, the journal may not have been as inappropriate for the papers as
might appear since the methods suggested by Kosambi were probabilistic.
However, it is not clear that the journal had a proper peer—review process in
place whereby submitted articles would, prior to publication, be examined by
experts in the field. The lack of appropriate reviewing was a real deficiency,
more so for a claim of this magnitude and the charge remains that DDK chose to
publish the papers in JISAS to be able to pass off a doubtful “proof”.

Both papers were
reviewed subsequently in MR, one by W. J. LeVeque, a number theorist who
eventually became Executive Director of the American Mathematical Society. His
critique of “An application of stochastic convergence” [63] goes straight to
the point, that the claim made by DDK

*is a result which easily implies the Riemann hypothesis.*However, since the proof is probabilistic in nature, there are major problems that he identifies.*Of the two proofs given for the crucial Lemma 1.2, the reviewer does not understand the first, which seems to involve more ‘hand-waving’ than is customarily accepted even in proofs of theorems less significant than the present one. The second proof appears to be erroneous*. The review concludes*The reviewer is unable either to accept this proof or to refute it conclusively. The author must replace verbal descriptions, qualitative comparisons and intuition by precise definitions, equations and inequalities, and rigorous reasoning, if he is to claim to have proved a theorem of the magnitude of the Riemann hypothesis.*
The kindest analysis
of these works of DDK comes from the Hungarian mathematician, A. Rényi who says
in a posthumous review of the paper “Statistical methods in number theory” [67]
that

The late author tried in the last 10 years of
his life to prove the Riemann hypothesis by probabilistic methods. Though he
did not succeed in this, he has formulated the following highly interesting
conjecture on prime numbers.

Rényi, who had been
sent both this and the earlier papers [63, 64] prior to publication, goes on to
say that

*Neither in this paper nor in his previous paper [Proc. Nat. Acad. Sci. U.S.A. 49 (1963), 20–23; MR0146168 (26 #3690)] did the author succeed in proving his hypothesis, nor in deducing from it the Riemann hypothesis*. The PNAS paper [64] was reviewed by J. B. Kelley who states, after summarizing the main result, that*The exposition is rather sketchy; in particular, the reviewer could not follow the proof of the crucial Lemma 4.*
Either because of the
timing of the review or because he may have appreciated the valiant attempts of
DDK to prove the Riemann hypothesis by an unusual route, Rényi, concludes the
review by saying that at that point in time (1968)

*one does not have enough knowledge of the fine structure of the distribution of primes to prove or disprove the author’s conjecture. The problem seems to be even more difficult than the problem of the validity of the Riemann hypothesis. As a matter of fact, no obvious method exists to prove the author’s hypothesis even under the assumption of the Riemann hypothesis. Nevertheless, the conjecture is worthy of study in its own right, and the reviewer proposes to call it “the Kosambi hypothesis” in commemoration of the enthusiastic efforts of the late author.*
Rényi’s suggestion has
not found favour. The probabilistic approach has inherent limitations, as the
physicist Michael Berry points out [103]. Indeed, as these reviews suggest, the
rigour emphasised by DDK in his early years had deserted him. What is somewhat
surprising is that there are elementary errors in these papers that become
evident even with a fairly cursory examination, and which could have been
detected by an alert referee. The fact that IJSAS published this paper with the
errors added to the feeling that DDK deliberately chose the journal to avoid
qualified peer review. These papers essentially destroyed DDK’s mathematical
reputation.

Given the ongoing
interest in the RH, only in part increased by its inclusion as a Millennium
Prize problem, there are a number of popular books [104] that summarize the
approaches to proving it. Not surprisingly, the work of DDK is not mentioned,
although Berry remarks [103] that

*his idea for proving RH based on showing that a certain function is nonsingular off the line, is ingenious.*Andrew Odlyzko, another mathematician who has worked extensively on the RH says [105] that he*was really intrigued by these approaches, but after a while decided that it would take some clever insights far beyond what [he] could think of to accomplish anything rigorous in this area*. Among Odlyzko’s major contributions to a study of the RH is the computation of a large number of zeros (several million of them, in fact) to fairly high precision; for*all*of these, the real part equals 1/2. As an experimental mathematician he has a good insight into the approach suggested by DDK, adding,*In summary, I think it is a pity that Kosambi did not see the flaws in his arguments and published this paper, but the basic idea is an interesting one, and certainly worth exploring. I would be surprised, but not shocked, if somebody clever managed to do something with it.***3. Bhabha and DDK**

DDK joined the newly
formed Tata Institute of Fundamental Research on June 16, 1945. His
appointment, which was for an initial period of five years was decided at the
first meeting of the provisional council of TIFR.

The initial
correspondence between Bhabha and DDK, although formal, was extremely cordial
[74]. In 1946, when Bhabha traveled to England, he appointed DDK Acting
Director, leaving him in charge of the fledgling institute. This was a position
of considerable responsibility, and one that DDK clearly enjoyed, and in a long
letter [91] written on 8th July he writes “About building up a School of
Mathematics in India, we also think alike; but, as you are fully aware, we have
to get people trained in a considerable number of branches for which there are
no real specialists in this country.”

The relationship also
grew warm, especially since they had to plan the Institute together, concerning
themselves with details regarding land acquisition, equipping the laboratories,
hiring staff, planning for the future. That same year DDK was elected Fellow of
the Indian National Science Academy, and the next, in 1947, was awarded the
Bhabha Prize (named for Bhabha’s father, Jehangir Hormusji Bhabha). He was also
chosen President of the Mathematics section of the 34th Indian Science Congress
that was held in Delhi in December 1947 [47] with the active support of Bhabha
who also realized that this would bring DDK into contact with Nehru. Kosambi’s
mathematical and statistical expertise was also greatly appreciated—a number of
colleagues, Bhabha among them, acknowledge his advice and help explicitly in
their scientific publications.

In 1948, when DDK was
to go to the US for a year’s visit, to Chicago and Princeton, Bhabha threw a
party for him at his residence in Malabar Hill. This visit was in fact largely
arranged by Bhabha, and among other things, DDK was to investigate the
possibility of getting a computing machine for the new institute [91] as well
as to attract new faculty, K. Chandrasekharan and S. Minakshisundaram in
particular. On this trip, he pursued all aspects of his wide–ranging interests,
visiting Einstein and von Neumann in Princeton, Norbert Wiener in Boston, as
well as the historian, A. L. Basham in London. In Chicago, he was visiting
Professor at the University, where he gave a course of 36 lectures on tensor
analysis. This was a special interest of his: he had been invited to the
editorial board of the Hokkaido University journal, Tensor (New Series), and
indeed an article of his had been translated into Japanese already in 1939 by
the same journal [25].

In the event,
Chandrasekharan joined the TIFR in 1950 or so, and shortly thereafter, so did
K. G. Ramanathan, who had obtained his Ph. D. at Princeton. They were to play a
much more influential role in shaping the TIFR School of Mathematics. In the
next few years, though, the cracks in the relationship between Bhabha and DDK
surfaced, first in regard to students and then gradually, with regard to
details such as his attendance in office and other aspects of his working.

The spiral downwards,
though, began in 1959 with the publication of the JISAS paper [63], and the
subsequent grand obsession with a probabilistic proof of the Riemann
hypothesis. His differences became more pronounced with Bhabha who relied more
and more on Chandrasekharan’s opinion and estimation of DDK’s work. The

*coup de grace*was a letter signed by four of the mathematicians at TIFR stating that Kosambi had become an embarrassment to the Institute with his claim of the proof of the RH and of Fermat’s Last Theorem [106] that was being broadcast internationally.
There were other differences
with Bhabha which were of a political nature, but these differences were
already present in 1946 when Bhabha invited DDK to join TIFR. The unpublished
(and largely unknown) essay ‘An Introduction To Lectures On Dialectical
Materialism’ relates to a set of 15 lectures given by Kosambi to the citizenry
of Pune in 1943. Later, when he gave a set of lectures on Statistics at TIFR
the notes conclude with an appreciation of Lenin [107]. Indeed, Bhabha
facilitated DDK’s visits to the Soviet Union and China, and it is not possible
that DDK’s views were hidden under a bushel until the early 1960’s.

In July 1960 DDK gave
a talk to the Rotary Club of Poona on “Atomic Energy for India”. This essay [108]
is an unabashed advocacy of solar power over atomic power, mirroring in a sense
his ideological conflict with the DAE. Half a century later, many of these
issues remain current and the arguments remain valid, as for example the
following observation.

It seems to me that research on the
utilization of solar radiation, where the fuel costs nothing at all, would be
of immense benefit to India, whether or not atomic energy is used. But by
research is not meant the writing of a few papers, sending favoured delegates
to international conferences and pocketing of considerable research grants by
those who can persuade complaisant politicians to sanction crores of the
taxpayers’ money. Our research has to be translated into use.

There is more in these
essays on solar energy that merits attention even today such as his observations
on energy storage and distribution, and on environmental issues [108].
Eventually matters came to such a pass as to cause the DAE to not renew DDK’s
contract. As already pointed out, the RH papers had caused a serious blow to
Kosambi’s mathematical reputation and while this was made out as the proximate
cause for his dismissal from TIFR, trouble had been brewing for some time. The
letters between Bhabha and DDK grew increasingly formal, bureaucratic, and
strained. There was a distinct difference in styles, and the iconoclastic
Kosambi was hardly one to fit into the DAE mould.

**4. Pseudonyms and Aliases**

DDK was responsible
for the first mention of Bourbaki in the mathematics literature in his
publication [4] in the Proceedings of the Academy of Sciences, UP, in 1931,
although the obscurity of the journal has resulted in the article receiving
less attention than it deserved, even from a purely historical point of view.
André Weil had suggested a prank, that he ascribe a theorem to a nonexistent
Russian mathematician, in order to put down an older colleague in Aligarh who
was giving the young Kosambi a difficult time. There is not much more than a
paragraph in Weil’s autobiography [86] on this episode, so the circumstances
surrounding the event are difficult to reconstruct. Nevertheless, this

*parodic note passed off as a serious contribution to a provincial journal*is not entirely facetious.
It was not until
December 1934 that the Bourbaki idea acquired more momentum [109, 110], when
Weil along with Henri Cartan, Claude Chevalley, Jean Delsarte, Jean Dieudonné,
and René de Possel, decided

*... to define for 25 years the syllabus for the certificate in differential and integral calculus by writing, collectively, a treatise on analysis. Of course, this treatise will be as modern as possible*. The book [111] would eventually appear in 1938, authored by the group that now called themselves Nicolas Bourbaki [112]; they then went on to write many more (and extremely influential) volumes. An Indian connection remained: when Boas mentioned (in the Britannica Book of the Year) that Bourbaki was a collective pseudonym, he got an indignant letter of protest, from Bourbaki, writing*from his ashram in the Himalayas*[113]. It should also be noted that Kosambi cites D. Bourbaki [4] who is allegedly of Russian extraction, while the first name eventually adopted by the Bourbaki collective [114] is Nicolas, who is of Greek descent.
Aliases were used by
DDK on several occasions although he did not use them extensively enough to warrant
a distinction between his “aesthetic” or pseudonymous writing and that
published under his own name. S. Ducray was merely the last nom de plume in a
series, although by far the most elaborate. His first article in the magazine
of Fergusson College was signed off as ‘Ahriman’ [115]. Subsequently he wrote
an expository article on the Raman effect as ‘Indian Scientist’ [116], and a
note as ‘Vidyarthi’ [117]: this was almost surely his nod to William Sealy
Gosset, the chemist and statistician who, as ‘Student’ invented the t-test in
statistics.

It is difficult to
discern what led him to use the pseudonym S. Ducray. The alleged etymology is
that Bonzo, the Kosambi family dog in the 1960’s was quite plump, and DDK
affectionately called him

*Dukker*, namely ‘pig’ in Marathi. This evolved into Ducray, a name that sounds vaguely French, with the forename being the Sanskrit for dog, namely*Svana*. The choice of such a name remains enigmatic, and while it may have been prompted initially by his anger with the establishment—to date Kosambi is among the very few persons to have had their appointment terminated by the Department of Atomic Energy—there is enough to suggest that there may be more to the use of this alias than pique.
DDK published four
articles as S. Ducray, two in the Journal of the University of Bombay [65, 66]
and two in the

*Proceedings of the Indian Academy of Sciences*[68, 69]. The latter two were in fact communicated by C. V. Raman. While this may have been a formal device employed by the journal, it is highly unlikely that Raman knew of the masquerade. Had Raman known, it is also highly unlikely that he would have permitted such subterfuge in a journal of his Academy. These two papers were serious enough as works of mathematics, as were the other two Ducray papers that were submitted to the Journal of the University of Bombay. Indeed, two of these four papers were reviewed in*Mathematical Reviews*. All the four articles show a strong connection to DDK, acknowledging him in one and quoting a private communication from Paul Erdös in another, in addition, of course, to citing his related papers written as D. D. Kosambi.
These papers continued
the prime obsession that DDK showed in his last years. Regrettably, the
manuscript of his book [71] that was mailed to the publishers a short time
before his death has never been retrieved. If nothing else, it would have
provided some clues as to how he hoped to use probability theory in this arena.

Although reviewed in
MR, the papers had serious shortcomings. J. Kubilius who himself worked in the
area of probabilistic number theory says of ‘Probability and prime numbers’
[68] that

*The reviewer could not follow the proof of the cardinal Lemma 3.*The paper “Normal Sequences” [66] was comprehensively reviewed by B. Volkmann who pointed out a number of inaccuracies and misprints.
One of DDK’s earlier
papers had been reviewed in

*Mathematical Reviews*by E. S. Pondiczery: this was the editor Ralph Boas Jr’s pseudonym, a fanciful ‘slavic’ spelling of Pondicherry. The name, which Boas used even when writing serious mathematics, was apparently concocted for its initials, ESP, and was to have been used for writing an article debunking extra–sensory perception. Boas had a well–developed sense of the ludic and was one of the authors of the brilliant article “A Contribution to the Mathematical Theory of Big Game Hunting” that was published in the American Mathematical Monthly under the (collective) pseudonym H. W. O. Pétard [118]. Both Boas and Kosambi were publicly dismissive of extra–sensory perception, and in 1958 DDK in collaboration with U. V. R. Rao authored an article analysing the statistical defects underlying parapsychological experiments [60]. This paper was subsequently commented upon by A. W. Joseph [119] who pointed out an error in analysis as well as in the conclusions, ending with*The above comments do not detract from the valuable experiments in card–shuffling made by the authors, but it is suggested that there is little weight left in their criticism of the ESP investigations.*Perhaps it was these connections that inspired Kosambi when he was to later adopt the Ducray alias.**5. Concluding Remarks**

History may not have
been particularly kind to Kosambi the mathematician, but in his lifetime DDK
was appreciated for his scholarship and intelligence [120] early in his career
and by his peers. The manner in which Kosambi was viewed by his
contemporaries—many of who were more distinguished than him and had a more
significant impact in mathematics—is revealing. From 1930 to 1958 or so, DDK
enjoyed the respect and admiration of a large professional circle. As has been
noted earlier [75], his contributions in areas such as ancient Indian history,
Sanskrit epigraphy, Indology, as well as his writings of a political and
pacific nature grew both in volume and in substance in the 1940’s and 1950’s,
overshadowing his mathematics, although the constancy of his work in the area
remained. His wide scholarship and his ability to integrate different strands
of thought gave him a large and dispersed audience, although his temperament
and his politics were also well known and not as widely appreciated.

One important
recognition that was accorded him, in part due to his being at the TIFR and the
association with Bhabha, but also for his work and his mathematical antecedents
[121], was his appointment as a member, in 1950, of the Interim Executive
Committee of the International Mathematical Union, to serve along with Harald
Bohr, Lars Ahlfors, Karol Borsuk, Maurice Fréchet, William Hodge, A. N. Kolmogorov
and Marston Morse. One of the tasks of this rather distinguished group was to
choose Fields medalists, and DDK served on this committee for two years.

It is thus noteworthy
that in a period that spans three decades, Kosambi was mathematically productive,
prolific, original, and was taken seriously by the scientific establishment in
the country, as his elections to the Fellowships of the Indian Academy of
Sciences and the Indian National Science Academy and the Presidency of the
Mathematics section of the 34th Indian Science Congress in 1947, among other
distinctions, testify. His papers appeared in leading journals of the world,
and were communicated by or reviewed by some of the leading mathematicians of
the time. And that this happened while his reputation in a diametrically
different field was also burgeoning can only be seen as evidence of a complex
but nevertheless Promethean intellect.

**Acknowledgment**

I have greatly
benefited from conversations and/or correspondence with Michael Berry, S. G. Dani,
Meera Kosambi, Mrs. Marston Morse, Rajaram Nityananda and Andrew Odlyzko. The
TIFR archives have been very helpful in providing copies of the correspondence
between Kosambi and Bhabha, and Kapilanjan Krishan, Rahim Rajan and Mudit
Trivedi have helped me obtain copies of articles by DDK that proved to be the
most difficult to locate. The main effort of putting together the collected
mathematical works of DDK was completed at the University of Tokyo in January
2010, and their hospitality is gratefully acknowledged.

**References**

[1]
J. D. Bernal, Nature

**211**, 1024 (1966).
[2]
For convenience, in this essay I will refer to Professor Damodar
Dharmananda Kosambi as
DDK or just Kosambi. Other abbreviations used
frequently in this essay are MR (Mathematical Reviews), JISAS (Journal of the
Indian Society for Agricultural Statistics), RH (Riemann hypothesis), TIFR
(Tata Institute of Fundamental Research). Journal names are given in full, and
the MR reference number will help locate the reviews of the pertinent papers.

[3]
D. D. Kosambi, “Precessions of an elliptical orbit”, Indian Journal of
Physics

**5**, 359–64 (1930)
[4]
D. D. Kosambi, “On a generalization of the second theorem of Bourbaki”,
Bulletin of the
Academy of Sciences, U. P.

**1**, 145–47 (1931)
[5]
D. D. Kosambi, “Modern differential geometries”, Indian Journal of
Physics

**7**, 159–64 (1932)
[6]
D. D. Kosambi, “On differential equations with the group property”,
Journal of the Indian
Mathematical Society

**19**, 215–19 (1932)
[7] D. D. Kosambi, “Geometrie
differentielle et calcul des variations”, Rendiconti della Reale Accademia
Nazionale dei Lincei

**16**, 410–15 (1932)
[8]
D. D. Kosambi, “On the existence of a metric and the inverse variational
problem”, Bulletin of the Academy of Sciences, U. P.

**2**, 17–28 (1932)
[9]
D. D. Kosambi, “Affin-geometrische Grundlagen der Einheitlichen
Feld–theorie”, Sitzungsberichten der Preussische Akademie der Wissenschaften,
Physikalisch-mathematische klasse

**28**, 342–45 (1932)
[10]
D. D. Kosambi, “Parallelism and path-spaces”, Mathematische Zeitschrift

**37**, 608–18 (1933); MR1545422.
[11]
D. D. Kosambi, “Observations sur le memoire precedent”, Mathematische
Zeitschrift

**37**, 619– 22 (1933); MR1545423.
[12]
D. D. Kosambi, “The problem of differential invariants”, Journal of the
Indian Mathematical Society

**20**, 185–88 (1933)
[13]
D. D. Kosambi, “The classification of integers”, Journal of the
University of Bombay

**2**, 18–20 (1933)
[14]
D. D. Kosambi, “Collineations in path-space”, Journal of the Indian
Mathematical Society

**1**, 68–72 (1934)
[15]
D. D. Kosambi, “Continuous groups and two theorems of Euler”, The
Mathematics Student

**2**, 94–100 (1934)
[16]
D. D. Kosambi, “The maximum modulus theorem”, Journal of the University
of Bombay

**3**, 11–12 (1934)
[17]
D. D. Kosambi, “Homogeneous metrics”, Proceedings of the Indian Academy
of Sciences

**1**, 952–54 (1935)
[18]
D. D. Kosambi, “An affine calculus of variations”, Proceedings of the
Indian Academy of Sciences

**2**, 333–35 (1935)
[19]
D. D. Kosambi, “Systems of differential equations of the second order”,
Quarterly Journal of Mathematics (Oxford)

**6**, 1–12 (1935)
[20]
D. D. Kosambi, “Differential geometry of the Laplace equation”, Journal
of the Indian Mathematical Society

**2**, 141–43 (1936)
[21]
D. D. Kosambi, “Path-spaces of higher order”, Quarterly Journal of
Mathematics (Oxford)

**7**, 97–104 (1936)
[22]
D. D. Kosambi, “Path-geometry and cosmogony”, Quarterly Journal of
Mathematics (Oxford) 7, 290–93 (1936)

[23]
D. D. Kosambi, “Les metriques homogenes dans les espaces cosmogoniques”,
Comptes rendus de l’Académie des Sciences

**206**, 1086–88 (1938)
[24]
D. D. Kosambi, “Les espaces des paths generalises qu’on peut associer
avec un espace de Finsler”, Comptes rendus de l’Académie des Sciences

**206**, 1538–41 (1938)
[25]
D. D. Kosambi, “The tensor analysis of partial differential equations”,
Journal of the Indian Mathematical Society

**3**, 249–53 (1939); MR0001882 (1,313f) Reviewer: E. W. Titt. Tensor, 2, 36–39 (1939); MR0001075 (1,176c) Reviewer: A. Kawaguchi.
[26]
D. D. Kosambi, “A statistical study of the weights of the old Indian
punch-marked coins”, Current Science

**9**, 312–14 (1940)
[27]
D. D. Kosambi, “On the weights of old Indian punch-marked coins”, Current
Science

**9**, 410–11 (1940)
[28]
D. D. Kosambi, “Path-equations admitting the Lorentz group”, Journal of
the London Mathematical Society

**15**, 86–91 (1940); MR0002258 (2,21f) Reviewer: J. L. Vanderslice.
[29]
D. D. Kosambi, “The concept of isotropy in generalized path-spaces”,
Journal of the Indian Mathematical Society

**4**, 80–88 (1940); MR0003125 (2,166g) Reviewer: J. L. Vanderslice.
[30]
D. D. Kosambi, “A note on frequency distribution in series”, The
Mathematics Student

**8**, 151–55 (1940); MR0005390 (3,147h).
[31]
D. D. Kosambi, “A bivariate extension of Fisher’s Z–test”, Current
Science

**10**, 191–92 (1941); MR0005589 (3,175h) Reviewer: A. Wald.
[32]
D. D. Kosambi, “Correlation and time series”, Current Science

**10**, 372–74 (1941); MR0005590 (3,175i) Reviewer: A. Wald.
[33]
D. D. Kosambi, “Path-equations admitting the Lorentz group–II”, Journal
of the Indian Math
ematical Society

**5**, 62–72 (1941); MR0005713 (3,192g) Reviewer: J. L. Vanderslice.
[34]
D. D. Kosambi, “On the origin and development of silver coinage in
India”, Current Science

**10**, 395–400 (1941)
[35]
D. D. Kosambi, “On the zeros and closure of orthogonal functions”,
Journal of the Indian
Mathematical Society

**6**, 16–24 (1942); MR0006770 (4,39d) Reviewer: E. S. Pondiczery.
[36]
D. D. Kosambi, “The effect of circulation upon the weight of metallic
currency”, Current
Science

**11**, 227–31 (1942)
[37]
D. D. Kosambi, “A test of significance for multiple observations”,
Current Science

**11**, 271–74 (1942); MR0007235 (4,107b) Reviewer: A. Wald.
[38]
D. D. Kosambi, “On valid tests of linguistic hypotheses”, New Indian
Antiquary

**5**, 21—24 (1942); MR0007247 (4,109a) Reviewer: A. Wald.
[39]
D. D. Kosambi, “Statistics in function space”, Journal of the Indian
Mathematical Society

**7**, 76–88 (1943); MR0009816 (5,207c) Reviewer: J. L. Doob.
[40]
D. D. Kosambi, “The estimation of map distance from recombination
values”, Annals of Eugenics

**12**, 172–75 (1944)
[41]
D. D. Kosambi, “Direct derivation of Balmer spectra”, Current Science

**13**, 71–72 (1944)
[42]
D. D. Kosambi, “The geometric method in mathematical statistics”,
American Mathematical
Monthly

**51**, 382–89 (1944); MR0010937 (6,91c) Reviewer: R. L. Anderson.
[43]
D. D. Kosambi, “Parallelism in the tensor analysis of partial
differential equations”, Bulletin of the American Mathematical Society

**51**, 293–96 (1945); MR0011793 (6,217e) Reviewer: J. L. Vanderslice.
[44]
D. D. Kosambi, “The law of large numbers”, The Mathematics Student

**14**, 14–19 (1946); MR0023471 (9,360i) Reviewer: W. Feller.
[45]
D. D. Kosambi, “Sur la differentiation covariante”, Comptes rendus de
l’Académie des Sciences

**222**, 211–13 (1946); MR0015274 (7,396b) Reviewer: J. L. Vanderslice.
[46]
D. D. Kosambi, “An extension of the least–squares method for statistical
estimation”, Annals
of Eugenics

**18**, 257–61 (1947); MR0021290 (9,49d) Reviewer: J. Wolfowitz.
[47]
D. D. Kosambi, “Possible Applications of the Functional Calculus”,
Proceedings of the 34th
Indian Science Congress. Part II: Presidential
Addresses, 1–13 (1947)

[48] D. D. Kosambi, “Les invariants
differentiels d’un tenseur covariant a deux indices”, Comptes rendus de l’Académie
des Sciences

**225**, 790–92 (1947); MR0022433 (9,207b) Reviewer: N. Coburn.
[49]
D. D. Kosambi, “Systems of partial differential equations of the second
order”, Quarterly
Journal of Mathematics (Oxford)

**19**, 204–19 (1948); MR0028514 (10,458d) Reviewer: M. Janet.
[50]
D. D. Kosambi, “Characteristic properties of series distributions”,
Proceedings of the National
Institute of Science of India

**15**, 109–13 (1949); MR0030731 (11,42h) Reviewer: J. L. Doob.
[51]
D. D. Kosambi, “Lie rings in path-space”, Proceedings of the National
Academy of Sciences (USA)

**35**, 389–94 (1949); MR0030807 (11,56a) Reviewer: O. Varga.
[52]
D. D. Kosambi, “The differential invariants of a two-index tensor”,
Bulletin of the American Mathematical Society

**55**, 90–94 (1949); MR0028653 (10,480b) Reviewer: V. Hlavaty ́.
[53]
D. D. Kosambi, “Series expansions of continuous groups”, Quarterly
Journal of Mathematics (Oxford, 2)

**2**, 244–57 (1951); MR0045732 (13,624b) Reviewer: M. S. Knebelman.
[54]
D. D. Kosambi and S. Raghavachari, “Seasonal variations in the Indian
birth–rate”, Annals of Eugenics

**16**, 165–92 (1951); MR0046135 (13,691b) Reviewer: R. P. Boas, Jr.
[55]
D. D. Kosambi, “Path-spaces admitting collineations”, Quarterly Journal
of Mathematics (Oxford, 2)

**3**, 1–11 (1952); MR0047387 (13,870d) Reviewer: O. Varga.
[56]
D. D. Kosambi, “Path-geometry and continuous groups”, Quarterly Journal
of Mathematics (Oxford, 2)

**3**, 307–20 (1952); MR0051562 (14,498g) Reviewer: A. Nijenhuis.
[57]
S. Raghavachari and D. D. Kosambi, “Seasonal variations in the Indian
death–rate”, Annals of Human Genetics

**19**, 100–19 (1954)
[58]
D. D. Kosambi, “The metric in path-space”, Tensor (New Series)

**3**, 67–74 (1954); MR0061869 (15,898a) Reviewer: J. A. Schouten.
[59]
D. D. Kosambi, “Classical Tauberian theorems”, Journal of the Indian
Society of Agricultural Statistics

**10**, 141–49 (1958); MR0118997 (22 #9766) Reviewer: J. Korevaar.
[60]
D. D. Kosambi and U. V. R. Rao, “The efficiency of randomization by
card–shuffling”, Journal of the Royal Statistics Society

**121**, 223–33 (1958)
[61]
D. D. Kosambi, “The method of least–squares”, Journal of the Indian
Society of Agricultural Statistics

**11**, 49–57 (1959); MR0114265 (22 #5089) Reviewer: R. G. Laha.
[62]
D. D. Kosambi, “The method of least–squares. (In Chinese.)”, Advancement
in Mathematics

**3**, 485–491 (1957); MR0100960 (20 #7385).
[63]
D. D. Kosambi, “An application of stochastic convergence”, Journal of the
Indian Society of Agricultural Statistics

**11**, 58–72 (1959); MR0122792 (23 #A126) Reviewer: W. J. LeVeque.
[64]
D. D. Kosambi, “The sampling distribution of primes”, Proceedings of the
National Academy of Sciences (USA)

**49**, 20–23 (1963); MR0146168 (26 #3690) Reviewer: J. B. Kelly.
[65]
D. D. Kosambi (as S. Ducray), “A note on prime numbers”, Journal of the
University of Bombay

**31**, 1–4 (1962)
[66]
D. D. Kosambi, (as S. Ducray), “Normal Sequences”, Journal of the
University of Bombay

**32**, 49–53 (1963); MR0197433 (33 #5598) Reviewer: B. Volkmann
[67]
D. D. Kosambi, “Statistical methods in number theory”, Journal of the
Indian Society of Agricultural Statistics

**16**, 126–35 (1964). MR0217024 (36 #119) Reviewer: A. Rényi.
[68]
D. D. Kosambi (as S. Ducray), “Probability and prime numbers”,
Proceedings of the Indian Academy of Sciences

**60**, 159–64 (1964); MR0179148 (31 #3399) Reviewer: J. Kubilius.
[69]
D. D. Kosambi (as S. Ducray)“The sequence of primes”, Proceedings of the
Indian Academy of Sciences

**62**, 145–49 (1965)
[70]
D. D. Kosambi, “Numismatics as a Science”, Scientific American, February
1966, pages 102–11.

[71]
D. D. Kosambi, “Prime Numbers”, Monograph completed a few days before the
author’s death;
untraced.

[72]
“The Oxford India Kosambi”, Compiled, edited and introduced by Brajadulal
Chattopadhyaya (Oxford University Press, New Delhi, 2009); “Combined Methods in
Indology & Other Writings: Collected Essays”. D. D. Kosambi, Compiled, edited
and introduced by Brajadulal Chattopadhyaya, Oxford University Press, New
Delhi, 2005; “Indian Numismatics”, D. D. Kosambi (Orient Longman, Hyderabad,
1981); D. D. Kosambi, Steps in Science (Prof. D. D. Kosambi Commemoration
Volume) Popular Prakashan Bombay, 1974; D. D. Kosambi, Exasperating Essays,
Peoples Publishing House, 1957.

[73]
‘The many careers of D. D. Kosambi: Critical essays’, edited by D. N. Jha
(Leftword, Delhi,
2011); ‘Damodar Dharmanand Kosambi’ (in Hindi) edited by R.
S. Sharma (SAHMAT, New
Delhi 2010).

[74]
Meera Kosambi, ed., “Unsettling the Past: Unknown Aspects and Scholarly
Assessments of D.
D. Kosambi ”, (Permanent Black, New Delhi, 2012).

[75]
R. Ramaswamy, ‘Integrating Mathematics and History: The scholarship of D.
D. Kosambi’,
Economic and Political Weekly,

**47**, 58–62 (2012). Reproduced in [74].
[76]
S. G. Dani, “Kosambi, the Mathematician”, Resonance journal of Science
Education, June 2011, pp. 514–28. This issue of the journal is dedicated to D.
D. Kosambi and contains several articles that discuss the scientific
contributions of DDK as well as two essays on his life and
historical work.

[77]
R. Narasimha, “Kosambi and Proper Orthogonal Decomposition”, Resonance journal
of Science Education, June 2011, pp. 574–81.

[78]
C. K. Raju, “Kosambi the Mathematician”, Economic and Political Weekly,

**54**, 38 (2009).
[79]
References [3–71] are a complete set of the papers of DDK that are of a
mathematical nature.
The list has been compiled in part from incomplete
sources in the biography by Chintamani Deshmukh as well as web listings. In
addition to the papers listed above, several of his essays relate to scientific
issues, but these are not included here.

[80]
R. Ramaswamy, ed., “D. D. Kosambi: Selected Works in Mathematics and
Statistics”, (Springer Verlag), forthcoming.

[81]
P. Antonelli, R. Ingarden and M. Matsumoto, “The Theory of Sprays and Finsler
Spaces with Applications in Physics and Biology”, (Kluwer Academic Publishers,
Amsterdam, 1993).

[82]
K. Karhunen, “Über lineare Methoden in der Wahrscheinlichkeitsrechnung”,
Ann. Acad. Sci.
Fennicae. Ser. A. I. Math.-Phys.

**37**, 1–79 (1947); M. Loève, “Fonctions aleatoires de seconde ordre”, C. R. Acad. Sci.**220**, 295 (1945) and related papers.
[83]
K. K. Vinod, “Kosambi and the genetic mapping function”, Resonance
journal of Science
Education, June 2011, pp. 540–50.

[84]
Starting with [10] in 1933, DDK wrote a series of papers on path spaces,
the last being [58] in
1954.

[85]
“Atomic Energy for India”, the text of a talk by DDK to the Rotary Club
of Poona, on July 25,
1960 was published in the posthumous volume, “Science,
Society, and Peace”, (The Academy
of Political and Social Studies, Pune, 1967,
reprinted by People’s Publishing House, 1995)).

[86]
A. Weil, “The apprenticeship of a mathematician”, (Birkhäuser, Basel, 1992).

[87]
Tirukkannapuram Vijayaraghavan (1902–1955) was a Founding Fellow of the IASc,
being
elected in 1934. He did his Ph. D. under the supervision of G. H. Hardy
in Cambridge. From Dacca he moved to Waltair, and eventually became the
founding director of the Ramanujan Institute of Mathematics in Madras.

[88]
Sarvadaman Chowla (1907–1995) moved to the US in 1947 after a career at
Delhi, Banaras, Waltair and Lahore in undivided India. A student of J. E. Littlewood,
Chowla was a number theorist.

[89]
In 1940, Weil was in military prison in Bonne-Nouvelle for refusing to
take part in the war as a conscientious objector (since his true dharma was the
pursuit of mathematics and not war, he said) when he proved an analogue of the
Riemann hypothesis (for the zeta function of curves over finite fields). He did
discuss the Riemann hypothesis with T. Vijayaraghavan, who is supposed to have
said that if he could have six months—undisturbed and undistracted—in a prison,
he could have a crack at solving the RH. See Ref. [86], and M. Raynaud, ‘André
Weil and the Foundations of Algebraic Geometry’, Notices of the AMS,

**46**, 864 (1999).
[90]
The historical spellings of city names have been retained where it seemed
appropriate.

[91]
The Kosambi–Bhabha correspondence has been made available through the
kind courtesy of the TIFR archives. There are a large number of letters that
are presently being catalogued and
edited. Some have been reproduced in [74].

[92]
The most recent effort was in 2010, when Louise J. (Mrs Marston) Morse
was nearly 100 years
old. She was kind enough to have the Morse archives
searched, but was not able to locate this
manuscript or any reference to it.

[93]
Of the 110 or so Founding Fellows, about two thirds were from the south
of India or worked
there and Raman might have had greater familiarity with
their work or their reputation.

[94]
The University of Madras announced the Ramanujan Memorial Prize for “the
best thesis based on original contributions submitted by an Indian (or one
domiciled in India) on some definite branch of mathematics, applied or pure” in
1933. The prize was awarded in 1934, as reported
in Nature

**135**, 28–28 (1935).
[95]
“A Chapter in the history of Indian science”, an unpublished essay by DDK
is a damning
indictment of Raman’s role in suppressing creativity in Indian
science.

[96]
K. A. N., “Metrology of Punch-Marked Coins”, Current Science

**7**, 345–6 (1941). This might have been the historian of South India, K. A. Nilakantha Sastry (R. Thapar, private communication).
[97]
D. D. Kosambi, ‘A Note on two hoards of punch marked coins found at
Taxila’, New Indian
Antiquary

**3**, 156–57 (1940) and ‘On the study and metrology of silver punch marked coins’, New Indian Antiquary**4**, 1–35 and 49–76 (1941).
[98]
The existence of a complete archive of these journals at the University
of Tokyo has proved to
be an invaluable resource.

[99]
H. M. Edwards, “Riemann’s Zeta Function”, (Academic Press, New York,
1974).

[100] S. Hawking, ‘A
brief history of time’, (Bantam Books, 2011).

[101] See, e.g. The
Stanford Encyclopedia of Philosophy, for a discussion of this very classic
paradox, http://plato.stanford.edu/entries/paradox-zeno/#AchTor

[102] These results
need elementary but unfamiliar methods of complex analysis that can be found in
standard textbooks. There are a number of websites on the internet that give a
reasonable introduction to the mathematics.

[103] M. Berry,
private communication.

[104] M. du Sautoy,
“The Music of the Primes”, (Harper Perennial, London, 2004); K. Sabbagh, “Dr.
Riemann’s Zeros”, (Atlantic Books, NY, 2003); J. Derbyshire, “Prime Obsession:
Bernhard Riemann and the Greatest Unsolved Problem in Mathematics”, (Plume
Books, New York, 2004); D. Rockmore, “Stalking the Riemann Hypothesis”,
(Vintage, New York, 2005)

[105] A. Odlyzko,
private communication.

[106] In addition to
Chandrashekaran and Ramanathan, the other two signatories were C. S. Seshadri
and M. S. Narasimhan. Fermat’s last theorem, that the equation

*xn + yn = zn*has no solutions with integer*x, y, z*if*n*is larger than 2 was proved conclusively only in 1995 by Andrew Wiles (see Simon Singh, “Fermat’s Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem”, (Anchor Books, 1998)). Kosambi could not have had the proof he claimed in his letters to eminent mathematicians such as Carl Siegel (who was at the Institute for Advanced Study, Princeton) and others.
[107] D. D. Kosambi,
‘Lectures on Statistics’, unpublished. This essay concludes,

*But if I go any further into his achievements, I shall be preaching Bolshevism in the sacral portals of Bombay House and so must stop here.*
[108] This is one of three essays on solar
energy that were first reprinted in “Science, Society and Peace”, (The Academy
of Political and Social Studies, Pune, 1986), as well as now in [74].

[109] L. Beaulieu,
‘Bourbaki’s Art of Memory’, Osiris,

**14**, 291 (1999).
[111] N. Bourbaki, ‘Éléments
de Mathématique, Book 1: Théorie des ensembles: Fascicule de Resultats ’,
(Paris, Hermann, 1939).

[112] The history of
the Bourbaki collective has been written about extensively by Maurice Mashaal, Bourbaki:
a secret society of mathematicians, (American Mathematical Society, Providence,
2006) as well as others, including Liliane Beaulieu [114]. While the eventual
name chosen by the group was Nicolas, in the original Kosambi paper, it is D.
Bourbaki. Whether the initial was for Damodar, or whether this refers to
another of the Bourbaki scions remains a mystery.

[113] R. P. Boas Jr.,
‘Bourbaki and me’, Math. Intelligencer,

**8**, 84 (1986).
[114] L. Beaulieu,
‘Nicolas Bourbaki: History and Legend, 1934-1956’, (Springer Verlag, Berlin 2006).

[115] D. D. Kosambi,
“The function of leadership in a mass movement; The Cawnpore Road”, Fergusson
College Magazine, pp. 1–7 (1939).

*Ahriman*is the destructive spirit in Zoroastrian mythology.
[116] D. D. Kosambi,
“The Raman effect”, Peoples Age, July 1945.

[117] D. D. Kosambi
(with Miss Sushila Gokhale), “Progress in the production and consumption of
textile goods in India”, Journal of the Indian Merchants’ Chamber (Bombay),
January, pp. 11–15 (1943).

[118] H. W. O. Pétard,
‘A contribution to the mathematical theory of big game hunting’, American Mathematical
Monthly,

**45**, 446 (1938).
[119] A. W. Joseph, ‘A
note on the paper by D. D. Kosambi and U. V. Ramamohan Rao on “The efficiency
of randomization by card–shuffling”, Journal of the Royal Society of
Statistics,

**122**, 373–74 (1959).
[120] In ‘Artless innocents and ivory-tower sophisticates: Some personalities
on the Indian mathematical scene’, Current Science ~~recalls~~recounts a conversation with André
Weil in 1966 or 1967, when he (Weil) says of DDK, “...

**85**, 526–537 (2003), M. S. Raghunathan*Let me tell you this: he was one of the finest intellects to come out of your country.”*In his autobiography [86] André Weil recalls:*I appointed Kosambi for the following year. He was a young man with an original turn of mind, fresh from Harvard where he had begun to take an interest in differential geometry. I had met him in Benares (now Varanasi) where he had found a temporary position.*
[121] G. D. Birkhoff,
‘Mathematics at Harvard in the 1940’s’, Proceedings of the American Philosophical
Society,

**137**, 268–272 (1993)
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