Early Indian Mathematical Pilgrims to Cambridge

K. Razi Naqvi Department of Physics, Norwegian University of Science and Technology N-7491 Trondheim, Norway

During the nineteenth and the early part of the next century, the University of Cambridge (UoC) was the Mecca of Mathematcis for students from the British Isles and other parts of the British Empire. UoC concentrated on undergraduate teaching; its examination system, particularly the Mathematical Sciences Tripos (hereafter abbreviated as MathTrip), served as a launch pad for re- munerative and inﬂuential careers in law, church, politics, even medicine, and various adminis- trative organs of the British Raj. Most examiners were not engaged in research, and few examinees dreamt of qualifying as a high wrangler, or of becoming a creative mathematician. The purpose of this article is to scrutinise the performance of the few Indian students who completed one or both parts of MathTrip, the career choices they made after returning to India, and their efforts, if any, towards the diffusion of modern mathematics in Indian schools and colleges. Almost all of the returnees became functionaries of the colossal British bureaucracy. Rejuvenation of Indian mathematics was carried out mostly by other enthusiasts, among whom Muslims are conspicuous by their scarcity.

“Whether good mathematicians, when women and a handful of men from other they die, go to Cambridge, I do not parts of the British Empire were two mi- know. But it is well known that a large nority groups, whose members performed [?] number of men go there when they equally well. I am concerned here essen- are young for the purpose of being con- tially with those who came from the In- verted into senior wranglers and Smith’s dian subcontinent, and even out of this prizemen.” I would have been happier to small group, I will only speak of those who quote this remark if its author, the eccen- passed MathTrip during the period 1898– tric Oliver Heaviside, had used small in- 1909. One of my purposes for selecting stead of large. When these words were writ- this group is to ask what contribution, if ten [1], British men did indeed form the any, they made to mathematical research bulk of the students aspiring to complete and to promotion of mathematical educa- the Mathematical Tripos (for short, Math- tion in India; another is to see what frac- Trip), but most candidates knew that they tion of the graduates had identiﬁably Mus- would not be able to reach the peaks men- lim names. I will subject to closer scrutiny tioned by Heaviside, and did not think of the credentials of two individuals, namely themselves as good-mathematicians-to-be, Inayat Ullah Khan (also Inayatullah Khan; let alone senior wranglers. Quite a few hereafter abbreviated as IUK) and Ziaud-

1 din Ahmad (alternative spellings of the first Both Parts of the Examination are held in June. component: Zia Uddin or Zia-ud-Din or InPart I the namesare in order of merit, the names Zia-ud-din); my motive is to see whether being bracketed in all cases of equality. In Part II. the claims made by their devotees about they are arranged in each Division in alphabetical their academic achievements in Cambridge order. and/or a European university are sup- To paraphrase the above: The list of the ported by such facts as cannot be reason- men who obtained Honours in Part I (Math- ably disputed, and to ask whether it is rea- Trip1) of a given year was grouped in three sonable to call either of them a mathemati- classes (Wranglers, Senior Optimes and Ju- cian or a scientist. nior Optimes), the names in each class being placed according to the marks awarded by the examiners. 1 MainSources Among the wranglers, those placed at the first and second position were called Se- It will suffice to examine two documents nior Wrangler (SW) and the Second Wran- titled Historical Registers, one covering the gler (2W), respectively; after these came period 1753–1910 [2] and the other the fol- the Third Wrangler (3W), Fourth Wrangler lowing decade [3]. The scope of both com- (4W), and so on. Two or more wranglers pendia can be summed up by quoting the who got the same marks were bracketed to- first and the last paragraph of the preface to gether by a closing brace “}”. the second: In the list for Part II (MathTrip2), each Fist paragraph: “The publication of The His- candidate was placed in one of three classes torical Register of the University of Cambridge (First, Second, Third), each class itself being followed upon a decision of the Syndics of the further divided into three divisions (First, Press to lighten the ever-increasing bulk of The Second, Third); the names in each category Cambridge University Calendar by transferring followed alphabetical ordering. Success in the historical information hitherto contained in it Part II is to be seen as a postgraduate quali- to a separate volume. That volume, published in fication. 1917, contained historical records from the earliest times up to and including the year 1910, the fruit of diligent research on the part of its editor, Dr J. 3 AFewExamples R. Tanner.”

Last paragraph: “In order to make this volume In the Introduction to the collected works of as self-contained as possible, indexes to the Tripos G. H. Hardy, we find the following descrip- Lists, to the winners of University Scholarships tions: and Prizes, and to the holders of University ap- Hardy was fourth wrangler in 1898, R. W. H. pointments during the period 1911–20 have been T. Hudson being Senior Wrangler, with J. H. Jeans included in it.” and J. F. Cameron, later Master of Gonville and Caius, bracketed next. He took Part II of the Tri- pos in 1900, being placed in the first division of 2 New Regulations the first class, Jeans being then below him in the second division of the first class. In the same year In 1886, new regulations came into effect [2, he was elected to a Prize Fellowship at Trinity, and p. 549], which are quoted verbatim below: his early ambition was thus fulfilled. Hardy and Jeans, in that order, were awarded Smith’s Prizes Those who have obtained Honours in Part I. in 1901. are admissible to the Examination in Part II. following. In 1899, the top four wranglers were G.

2 Birtwistle (SW), R. P. Paranjpye (2W), S. B. years. In certain respects his position was McLaren, and P. V. Bevan. In Part II of the unique, for he was a link between the older following year, six candidates were placed theoretical physics and the new.” in the First Class: P. V. Bevan, G. Birtwistle, Among the Indians, Paranjpye (1876– G. H. Hardy, and R. P. Paranjpye (all in the 1966) became an administrator in the British First Division), J. H. Jeans and S. B. McLaren Raj, was knighted and served as India’s (both in the Second Division). High Commissioner to Australia during the In 1900, there were 16 wranglers, of period (1944–7). Hoon Balak Ram, whose whom Balak Ram was 4W. There were 26 last name is sometimes spelt as Balakram wranglers in 1905; J. E. Littlewood and (1876–1929) also joined the prestigious In- J. Mercer (both of Trinity College), were dian Civil Service, and was appointed as bracketed as SW, and F. M. Khan (of St. a judge in the Bombay High Court shortly John’s College) was 23W; I have come to before his death. Both Paranjpye and Balak conclude that his full name must have been Ram participated in the revival of mathe- Fazl Muhammad Khan. matics in India [6], and played an active The 1909 list for MathTrip1 has 31 wran- role in the Indian Mathematical Society [7]; glers, the top four of whom are: P. J. a recent book on Catalan numbers men- Daniell, E. H. Neville, L. J. Mordell, W. E. tions the latter’s contribution to number H. Berwick, and C. G. Darwin; on Nr. 14 theory and gives a biographical sketch [8, stands the name of G. S. Chowla, and two p. 69]. S. M. Sulaiman (1886–1941) was scholars are bracketed under Nr. 27: Inayat- knighted in 1929, served as the Chief Jus- ullah and J. R. Stickland. In the second class tice of Allahabad High Court, and spent (Senior Optimes), four names are bracketed the last few years of his life as one of the in the tenth place, among them S. M. Su- three judges of the Federal Court at Delhi, laiman. established in 1937. Sir Shah Muhammad Sulaiman did not lose his interest in science, and spent his spare time in formulat- 4 What Happened to Most of ing a uniﬁed theory of physical phenomena Those Named Above and a non-Einsteinian theory of relativity [9–12]. I reproduce an excerpt from an obit- A few became well known mathematicians uary written by C. V.Raman for Nature [13]: or mathematical physicists. In this group Sulaiman studied mathematics and physics at fall G. H. Hardy (1876–1947), J. H. Jeans Cambridge, taking Part II [sic] of the Mathemati- (1877–1946), J. E. Littlewood (1885–1977) cal Tripos in 1909. During his subsequent career and C. G. Darwin (1887–1962). The careers as a practising barrister and as a judge at Alla- of Hudson (1875–1904), Bevan (1875–1913) habad, he continued to retain a general interest and McLaren (1876–1916) might have been in the progress of physical science. Later in life, just as productive and distinguished if they the stimulus of contact with the University staff at Allahabad and Aligarh led him actively to un- had lived longer. Birtwistle (1877–1929), ex- dertake the study of theoretical physics as a sub- celled as a teacher and author of books [4], sidiary pursuit. as may be judged from the following re- Sir Shah’s high position in public life secured mark in his obituary notice [5]: “It was as for his writings and lectures on scientiﬁc top- a teacher rather than as an investigator that ics the widest publicity in India, as also a sym- Birtwistle was known, and as a teacher that pathetic, though critical, reception from his aca- he played a conspicuous part in Cambridge demic friends and colleagues. His published pa- mathematics, especially during the last ten pers indicate a marked reluctance to accept the

3 ideas of the newer physics as expounded by the Govt of India Scholar at St John College, Cam- leading authorities on the subject. They largely bridge IOR/L/PJ/6/705, File 40 13 Jan 1905 consist of attempts to explain the facts of the newer physics on the basis of classical or semi- I was not able to find much about the ac- classical ideas aided by special hypotheses. It tivities of F. M. Khan after his graduation. A could scarcely be hoped that work on such lines few lines in a 1937 compilation of the wor- would find general acceptance. thies and notables affiliated with the State About the wrangler Chowla: I have of Hyderabad indicate that he earned his not been able to find out much more than living as an administrator [14]: that he co-authored a book on elementary Fazl Muhammad Khan, Khan, M.A. plane trigonometry. As for Fazl Muham- Muhammadan, Sunni, Official. Born in May mad Khan, I can only tell you that he held a 1882 at Hoshyarpur, Punjab. Was formerly in high office in Hyderabad state, but in order the Punjab Educational Service. Entered His to convince you that the paucity of the in- Exalted Highness the Nizam’s service on lath formation is not due to lack of effort on my December 1928. Is Director of Public Instruc- part, I reproduce here (apart from showing tion. some words in bold font) what the daily London Standard wrote (June 14, 1905, p. 4): FROM OUR CORRESPONDENT in CAM- BRIDGE. June 13. There was a large assembly 5 The Strange Case of 27W/1909 of students at the Senate House this morning to hear the Mathematical Honours list read. There A candidate named Inayat Ullah Khan (of are two Senior Wranglers this year, both of Trinity Christ’s College) is stated to have passed College—Mr. J. E. Littlewood and Mr. J. Mercer— the following examinations: who are bracketed equal. There is only one woman wrangler [sic], Miss Ethel May Newbold, 1. Natural Sciences Tripos (Part I) in 1911, placed a former Tunbridge Wells High School scholar, in the Third Class [3, p. 122]; who is equal to the twenty-sixth. It is interesting 2. Oriental Language Tripos in 1911, placed in to note that the mens list includes eight oarsmen, the First Class [3, p. 170]; among whom is Littlewood,four footballers, three minor cricketers, two lacrosse players, one hockey 3. Mechanical Sciences Tripos (Part I) in 1912, player (Priestley being a well-known player), one placed in the Second Class [3, p. 183]. boxer (Khan, a Hindu), one runner (Coggin), and Greenhalgh, who represented his University in Let us note first that the names in each the Inter-University chess match. [This is fol- list are arranged alphabetically, and one lowed by the official list.] can tell that our candidate is under the According to a notification in the India letter “I”, because in the Natural Sciences Record Office (http://archive.is/Ojiu# list the candidates before and after him are selection-713.0-717.11): “Correct name “Hill, J. C.” and “Jordan, J. H.”, and that of the scholarship holder Fazl Muham- Inayatullah (27W/1909) has now become mad Khan IOR/L/PJ/6/623, File 32 18 Dec Inayat Ullah; all this makes sense if one un- 1902”. In the same selection, one also finds derstands how Muslim names are written two other entries (both of which probably in Urdu, and how they are handled/inter- refer to the same scholar): preted by native Urdu speakers. British bu- Application of Mr H M Khan, a Govt reaucrats in those days were evidently more of India scholar, for an advance of £20 conversant with these details, and under- IOR/L/PJ/6/702, File 2952 30 Nov 1904 stood that the formula “First Name-Middle Scholarship allowance of Mr F M Khan, Name(s)-Surname” could not be applied

4 to the names of Indian Muslims (Muham- well-to-do and not given to loafing about, madans, as they would have said). Con- became immersed in Urdu and Persian po- temporary Muslims living in the West must etry, having osmosised it at their elder’s conform to the formula willy-nilly. knee(s). It does not come as a great surprise As far as I can tell, IUK made no notable (to me) that, just one year after MathTrip1, contribution to mathematics after his re- IUK passed the Oriental Languages Tripos turn to India, and his time and talents were with a First Class. spent mostly on establishing the Khaksar Excellence in both classics and mathe- movement. Chapter 4 of Cantwell Smith’s matics may be rare but is not totally un- Modern Islam in India [15] is devoted to a de- known. Cambridge itself has produced scription of this movement. such examples. My favourite in this list is Rev. Thomas Jarrett (1805–1882), of St. Since the present article is concerned Catherine’s College, primarily because I only with the academic attainments of the share his objections to the Romanization early Indo-Cantabrigian graduates, I would of Urdu and other Indian languages [16]. like to add a few comments about the three He secured seventh position (First Class) in examinations passed by IUK after he be- Classical Tripos (ClasTrip) and was thirty- came a wrangler. It is not inconceivable fourth Wrangler in the same year (1827). that IUK as well as Sulaiman had contem- Jarrett made an extensive study of algebraic plated taking MathTrip2 (see below), and notation, which he discussed in an article that both abandoned the ambition after the [17] and much more fully in the form of a results of MathTrip1 were announced. The monograph [18]. He was professor of Ara- former, but not the latter, would have been bic in the University of Cambridge from entitled to sit for MathTrip2, but being closer 1831 until 1854 and Regius Professor of He- to the bottom rather than the top of the list brew from 1854 until 1882, while in 1875 of wranglers might have acted as a deter- he edited the Sanskrit text of the Tale of rent. Nala with vocabulary [19]. Other exam- Let us recall Raman’s words: “Sulaiman ples: In 1835, Henry Goulburn (1813–43) studied mathematics and physics at Cam- was 2W and at the top of the First Class in bridge, taking Part II of the Mathematical ClasTrip; also, Alfred Barry (1826–1910) was Tripos in 1909.” Raman might have been 4W and came seventh in the First Class list misled by a remark with which Sulaiman of ClasTrip in the same year (1848). opened the preface to Unified Theory of Phys- I have no knowledge of the require- ical Phenomena [9]: “When I was studying ments for obtaining a particular class in for the Mathematical Tripos, Part II, in 1909, the Natural Science Tripos (NST) and Me- at Cambridge, a new theory of gravitation, chanical Science Tripos (MST) examinations dependent not on external attraction as is taken by IUK, but I am aware of a de- so far supposed but on internal action of a bate concerning the standard of NST in the matter occurred to me.” It may have been 1860s, and feel that a few lines from the con- caused by a simple lapse of memory, but cluding paragraph of one letter [20] are just one cannot rule out the possibility that Su- as relevant for any later decade: laiman was indeed hoping to be a wrangler in 1909, and preparing for MathTrip2 while I may remark that at the time when special credit-marks were awarded in that Tripos waiting for the results of MathTrip1. [NST], no one could go in for the examination Most Muslims of the period prior to the who had not already taken honours, either in partition of India, if they happened to be mathematics or classics. When we find men

5 who have taken high degrees in mathematics of study of Research Students, and instituting getting First Class in Natural Sciences ten or the new Degree of Ph.D. But inasmuch as no eleven months afterwards, with special credit- candidate was presented for this Degree until marks in no less than ﬁve subjects, what is the 1921, the new regulations do not properly come inference? Is it to be supposed that the intel- within the scope of this volume. lect even of a High Wrangler can master science at or about the rate of one subject every The full name of Z. U. Ahmad appears two months? in the list of those who were awarded the following scholarship [2, p. 279]:

ISAAC NEWTON STUDENTSHIPS 6 Advanced/Research Students In 1890 Frank McClean, B.A. 1859, M.A. 1863, of Trinity College, offered about £12,500 for the A certain Mr Z. U. Ahmad took Math- founding of Studentships in Astronomy and Trip2 in 1903 as an “Advanced Student” [2, Physical Optics in memory of Sir Isaac New- p. 584]. This term was applied to students ton. The offer was accepted and Regulations falling in a special category [21, p. 5]: approved by Grace of the Senate, 5 March 1891. Value, £200 a year. Graduates of other Universities who com- ply with the following requirements may be The complete list of the awardees for admitted to the University of Cambridge as the period (1891–1910) appears below the “Advanced Students,” and are thereby placed above text. Let us note three names: J. H. on a footing different from that of ordinary Jeans (1900), Camaji Navroji Cama (1903), undergraduates. They may for one or more and Zia Uddin Ahmad (1904); hereafter I Terms pursue courses of advanced study or re- will refer to the last named as ZUA. search, literary or scientific, under the direc- Who was C. N. Cama? The answer is tion of the University teachers, without follow- riveting, but I will only mention here the ing the usual courses of study and examina- fact that he was an Indian. There seems to tions required for the degrees of the Univer- be only one full-length biography of ZUA sity. They may become members of certain Col- [22], written in Urdu and not easily acces- leges without fulfilling the same conditions as sible. Those who cannot read Urdu are re- are imposed on junior students, and their col- ferred to the relevant Wikipedia entry [23], legiate status is assimilated to that of students to the biographical information provided in who have taken their first degree in the Univer- the web pages of Aligarh Muslim Univer- sity. sity [24], and to a book chapter [25]. Here For an updated account (relative to the I will focus only on ZUA’s postgraduate ca- foregoing), let us recall the following pas- reer. After receiving a D.Sc. from Allahabad sage [3, p. 49]: University in 1901, he proceeded to Cam- From 1897 to 1913 Research Students were bridge for further studies. Contrary to what known as ’Advanced Students.’ Of these there is asserted in [23, 24], ZUA could not have were two kinds, one proceeding to the Degree become a wrangler, simply because he did of B.A. or LL.B. by means of Tripos Examina- not sit for the MathTrip1 examination; the tions and the other by means of Certificates of same sources also claim (incorrectly) that he Research. From 1913 to 1920 Research Students was the first Indian to get an Isaac Newton proceeded to the Degree of B.A. or LL.B. by Studentship. means of Certificates of Research only. One may wonder why ZUA, who re- In May 1920 new regulations were ap- ceived a D.Sc. from Allahabad, enrolled for proved governing the admission and course MathTrip2 as an Advanced Student. This

6 apparent paradox can be resolved by con- 7 Concluding Remarks sulting the calendar of Allahabad Univer- sity [26], and noting that the D.Sc. degree Securing a high position in the MathTrip ex- was awarded on the basis of performance aminations indicated possession of excep- in three annual examinations, and the syl- tional mathematical skills, but it gave no in- labus of the course was approximately on a dication of a successful candidate’s poten- par with that for a bachelor’s degree (with tial as an original mathematician. Here I honours) in mathematics in a British uni- would like to refer the reader to the argu- versity. The Urdu biography [22] states ments eloquently expressed by G. H. Hardy that he presented a paper on the Problem in his presidential address to the Mathemat- of Three Bodies in the 1904 meeting of the ical Association, whose text was printed British Association (BA, for short), held in and reprinted in the Mathematical Gazette Cambridge. This claim is not borne out by [28, 29] and subsequently in his collected the published account of the meeting [27], papers [30, pp. 527–537]. The text of the ad- which gives only the title of ZUA’s pre- dress should be read and re-read by anyone sentation, namely ‘On the use of divergent who does not understand the difference be- series in astronomy’. The statement that tween mere mathematical callisthenics and the meeting was presided by “Mr Belford, creative flair; here it would be enough to the minister of England (àAJÊÆK@ QK Pð )” is state that, in Hardy’s view [30, p. 538], also manifestly wrong, and more culpable “very difficult honours examinations, like than a case of mistaken identity. Henry Bal- the Mathematical Tripos or Oxford Greats, four, who was president of the Anthropol- were fundamentally vicious and should be ogy Section of BA in 1904, delivered his abolished.” Hardy’s efforts to reform the presidential address on Thursday, 18 Au- Tripos resulted in changing the system from gust, whereas ZUA presented his work in 1910 onwards. the Mathematics Section on the following Indian universities inherited the educa- day. The British prime minister in 1904 tional system developed in England, but was Arthur Balfour. I leave the problem only a diluted and emaciated version of it, of tracking the source of the “three-body- whose principal object was to churn out problem rumour” to such readers as are functionaries dedicated to the running of adept at debunking of myths. The title page the Empire. When one looks at accounts of the Urdu biography also indicates that of the rejuvenation of Indian mathematics ZUA was the holder of a Ph.D. from “Got- [6, 31], one is struck by the facts that the tinjen” (áj J KñÃ ). I havenot beenable to find early Indo-Cantabrigians played a compar- . atively minor role and that Muslims distin- any research work by ZUA (based on his guished themselves only through their ab- BA presentation, or on some other investi- stinence from the activity. gation), nor any hint as to the title, let alone the contents, of his Göttingen dissertation, whose existence is yet to be demonstrated.

References

[1] Oliver Heaviside. The principle of least action. Lagrange’s equations. Nature, 67(1735):297– 303, 1903.

7 [2] J. R. Tanner (editor). The Historical Register of the University of Cambridge: Being a Supplement to the Calendar, with a Record of University Ofﬁces, Honours and distinctions to the Year 1910. Cambridge University Press, Cambridge, 1917.

[3] G. V. C. (editor). The Historical Register of the University of Cambridge. Supplement, 1911–20. Cambridge University Press, Cambridge, 1922.

[4] Jaume Navarro. Teaching quantum physics in Cambridge : George Birtwistle and his two textbooks. In Research and Pedagogy : a History of Quantum Physics Through its Textbooks (edited by Massimiliano Badino and Jaume Navarro), pages 227–243. Open Access Edition, Berlin https://www.edition-open-access.de/studies/2/10/index.html, 2012.

[5] Anonymous. George Birtwistle (Obituary). Nature, 123(3110):881, 1929.

[6] Joseph W. Dauben and Rohit Parikh. Beginnings of modern mathematics in India. Current Science, 99(3):15–37, 2010.

[7] Anonymous. The Succession List of the Ofﬁce-Bearers of the Indian Mathematical Society. Available at: http://www.indianmathsociety.org.in/officebearers.htm.

[8] Thomas Koshy. Catalan Numbers with Applications. Oxford University Press, Inc., New York, 2009.

[9] S. M. Sulaiman. Uniﬁed Theory of Physical Phenomena. Self-published, Allahabad, 1933 (available at: https://archive.org/details/in.ernet.dli.2015.221147).

[10] S. M. Sulaiman. The mathematical theory of a new relativity. Proceedings of the Academy of Sciences, U.P., India, 4(1):1–36, 1934.

[11] S. M. Sulaiman. The mathematical theory of a new relativity. Proceedings of the Academy of Sciences, U.P., India, 4(3):217–261, 1934/5.

[12] S. M. Sulaiman. The mathematical theory of a new relativity. Proceedings of the Academy of Sciences, U.P., India, 5(2):123–170, 1935.

[13] C. V. Raman. Sir Shah Muhammad Sulaiman (Obituary). Nature, 148(3751):336–337, 1941.

[14] Anonymous. Hyderabad State List of Leading Ofﬁcials, Nobles and Personages. Hyderabad Res- idency Government Press, Hyderabad, 1937 (available at: https://archive.org/details/ listofleadingoff030666mbp).

[15] W. Cantwell Smith. Modern Islam in India. Minerva Book Shop, Lahore, 1943.

[16] Monier Williams. Original Papers Illustrating the History of the Application of the Roman Alphabet to the Languages of India. Longman, Brown, Green, Longmans, and Roberts, London, 1859.

[17] Thomas Jarrett. On algebraic notation [read november 12, 1827]. Transactions of the Cambridge Philosophical Society, 3(1):65–104, 1830.

[18] Thomas Jarrett. An Essay on Algebraic Development. J. & J. J. Deightons, Cambridge, 1831.

[19] Thomas Jarrett. Nalopakhynam, or, The Tale of Nala. University Press, Cambridge, 1875.

8 [20] Common Sense (pseudonym). The Natural Sciences Tripos, Cambridge. The Lancet, 85(2160):66, 1865.

[21] Donald MacAlister. Advanced Study and Research in the University of Cambridge; A Guide for Students. Cambridge University Press, Cambridge, 1896.

[22] Muh. ammad Am¯ın Zubayr¯ı. Zi¯a-´e-H. ay¯at. D¯ın Muh. ammad¯ıPress, Karachi, undated. [23] Anonymous. Ziauddin Ahmad. https://en.wikipedia.org/wiki/Ziauddin_Ahmad. Online; accessed 8 October 2018.

[24] Anonymous. Sir Ziauddin Hall. https://www.amu.ac.in/amucentre.jsp?did=10093&lid= Home. Online; accessed 8 October 2018.

[25] Ather H. Siddiqi and Syed Zillur Rahman. Aligarh Muslim University: Development and Progress of Science Teaching and Research, 1847–1947”. In History of Science, Philosophy and Culture in Indian Civilization (Geral Editor: D. P. Chattopadhyaya). Volume 15, Part 4: Science and Modern India: An Institutional History, c. 1784–1947 (edited by Uma Das Gupta), pages 747–755. Pearson Longman, Delhi, 2011.

[26] Anonymous. University of Allahabad: Calendar for the Year 1902–1903. Pioneer Press, Alla- habad, 1902.

[27] Z. U. Ahmad. *On the use of divergent series in astronomy. In Report on the Seventy-Fourth Meeting of the British Association for the Advancement of Science Held at Cambridge in August 1904, pages 464–464. John Murray, London, 1905. The following comments (including the square brackets) appear at the head of the Report: [An asterisk * indicates that the title only is given. The mark † indicates the same, but with a reference to the Journal or Newspaper in which it is published in extenso.].

[28] G. H. Hardy. The case against the Mathematical Tripos. The Mathematical Gazette, 13(181):61– 71, 1926.

[29] G. H. Hardy. The case against the Mathematical Tripos. The Mathematical Gazette, 32(300):134– 145, 1948.

[30] G. H. Hardy. Collected Papers of G. H. Hardy Vol. 7. Clarendon Press, Oxford, 1979.

[31] Priyadaranjan Ray and S. N. Sen. The Cultural Heritage of India: Science and Technology Volume VI. Ramakrishna Mission, Institute of Culture, Calcutta, 1986.