The Legacy of Mr. Carr, A Gift for the Gifted

Amitabha Sen

Browsing in the Chicago Public Library one lazy Sunday afternoon, I wandered into the mathematics section, which was deserted due to lack of interest in the subject. I quickly surveyed the untouched shelves to see if there was anything interesting. Here "The Collected Papers of G.H. Hardy" was wedged between "Powers, Roots, and Reciprocals from 0.0001 to 1.0000" and "Logarithm and Trigonometric Tables", books rendered useless today by pocket calculators. There, interestingly enough, was Gauss' famous "Disquisitiones Arithmeticae" and nearby, a book on Diophantus of Alexandria, the Greek exponent of algebra. Then turning the corner, past "Remarks on the Foundations of Mathematics" by Ludwig Wittgenstein, the sight of "Formulas and Theorems in Pure Mathematics" by George Shoobridge Carr caught me completely by surprise. What was an obscure handbook of great curiosity in the annals of mathematics doing here? The book was first published in England in 1880 under the title of "Synopsis of Results Pure Mathematics". (Click here for an online version*)

Back in 1903, in a small town of Kumbakonam, 160 miles southwest of Madras (Chennai), a boy of sixteen had got hold of a copy of this book and used it as a primary source of knowledge about advanced mathematics. Several years later, a letter written in longhand had arrived at the offices of the Madras Port Trust:

9th February 1912
I understand there is a clerkship vacant in your office, and I beg to apply for the same. I have passed the Matriculation Examination and studied up to the F.A. but was prevented from pursuing my studies owing to several untoward circumstances. I have, however been devoting all my time to Mathematics and developing the subject. I can say I am quite confident I can do justice to my work if I am appointed to the post. I therefore beg to request that you will be good enough to confer the appointment on me. I beg to remain,
Your most obedient Servant,
S. Ramanujan"

A letter of recommendation attached to the letter said:

"I can strongly recommend the applicant. He is a young man of quite exceptional capacity in Mathematics and especially in work relating to numbers. He has a natural aptitude for computation and is very quick at figure work. Though he has no experience of statistical work I am confident that he can pick up the details in a very short time.
E. W. Middlemast
Ag. Principal and Professor of Mathematics
The Presidency College"

From the age of sixteen to twenty five, Srinivasa Ramanujan had taught himself mathematics using Carr's Synopsis as a guide. He tried unsuccessfully to have other mathematicians in Madras to take an interest in his discoveries but his talents did not go unnoticed. He would often pose problems and solve them in the recently founded Journal of the Indian Mathematical Society. His single-minded devotion to mathematics, neglecting other subjects, caused him to fail his exams, cost him his admission to a university education and subsequently made it difficult for him to get a regular job. In 1905, suffering from depression and in financial difficulty, he ran away from home to Vishakhapatnam. Later in 1912, after having being married off early in the old Hindu tradition, he finally got a job at the Madras Port Trust as a clerk.

While he worked as a clerk, several reputable officials in Madras were trying to help him. Among them was R. Ramachandra Rao, Collector of Nellore, mathematics professors Edward Ross of Madras Christian College and Edward Middlemast of Presidency College, Charles Griffith, a professor of civil engineering at the Madras College of Engineering, and Sir Gilbert Walker, Director General of Indian Observatories. Sir Walker, himself a mathematician from Cambridge, is known to be the "father" of monsoon studies and global weather forecasting.

The English officials exchanged letters trying to decide how best to nurture his talents. One letter to the Chairman of the Port Trust said "You have in your office as an accountant on Rs 25 a young man named S. Ramanujan who is a most remarkable mathematician. He may be a very poor accountant, but I hope you will see that he is kept happily employed until something can be done to make use of his extraordinary gifts... If there is any real genius in him he will have to be provided with money for books and with leisure, but until I hear from home, I don't feel sure it is worth spending much time and money on him". Another said, "... I don't like the suggestion that access to a library would ruin any genius; it savours of the middle ages..."

Within a few months of joining the Port Trust, Ramanujan mailed a memorable letter to G.H. Hardy, a mathematician at Cambridge University:

"Dear Sir,
I beg to introduce myself to you as a clerk in the Accounts Department of the Port Trust Office at Madras on a salary of only £ 20 per annum. I am now about 23 years of age. I have had no University education but I have undergone the ordinary school course. After leaving school I have been employing the spare time at my disposal to work at Mathematics. I have not trodden through the conventional regular course which is followed in the University course, but I am striking out a new path for myself."

The letter ended with a list of some 52 mathematical formulas written in the concise style of Carr's Synopsis.

In later writings, Hardy dismissed Carr's Synopsis as being unremarkable. But Hardy was scarcely able to appreciate how the book might have been a rich source of material for a self-taught student. Anyone who has faced the difficulties of getting access to scientific literature will be able to appreciate the value of a book like Carr's. Here, in one place, was a comprehensive summary of basic mathematical knowledge known to the West. It also contained a very substantial cross-referenced index to European papers on pure mathematics of the nineteenth century, although this may not have been of much value to Ramanujan sitting in Madras.

A small quotation placed casually at the end of Carr's book sums up its usefulness:

"There is an immense amount of knowledge lying scattered at the present day, and almost useless from the difficulty of finding it when wanted - Professor J. D. Everett"

The value of the book comes from its method of organization. There are 6165 results covering algebra, theory of equations, plane trigonometry, spherical trigonometry, elementary geometry, geometrical conics, differential calculus, integral calculus, calculus of variations, differential equations, calculus of finite differences, plane coordinate geometry, and solid coordinate geometry.

Many of the results are at a high school level but some are fairly advanced and complicated. The book states a formula or a theorem but gives only a sketch of proof, so the reader is forced to work out the details. As Carr put it, in his nineteenth century pedagogical diction:

"Let them be read once, but recalled often. The difference in the effect upon the mind between reading a mathematical demonstration, and originating one wholly or partly, is very great. It may be compared to the difference between the pleasure experienced, and interest aroused, when in the one case a traveller is passively conducted through the roads of a novel and unexplored country, and in the other case he discovers the roads for himself with the assistance of a map."

Professor Hardy had grudgingly conceded that it is "a book written with some real scholarship and with a style and individuality of its own". In reality, the book is an elaborate crib sheet, such as one would create to prepare for a difficult exam.

Indeed, the book originated as a study guide for the Cambridge Tripos exams. During the nineteenth century, mathematical education at British Universities was largely oriented towards preparing students for the Tripos examinations, which involved solving tricky problems within a set time. The exams originated in 1748, sometime after Isaac Newton's tenure at Cambridge, and the name Tripos came from the three-legged stool on which the student sat to argue his points. Those who placed in the first class were known as Wranglers, and were assured of a successful future in any profession. Those who placed in the second and third classes were known as Senior and Junior Optimes. A market for study guides and coaches naturally arose under these circumstances. George Shoobridge Carr was a Tripos coach. The Synopsis came from his coaching notes, refined meticulously over a period of fourteen years.

Ramanujan of course was not studying for the Tripos. He had the book of answers. He needed to discover the questions. In mathematics, asking the right questions is as important as getting the right answers. Many formulas in mathematics are curiosities. The formulas for calculating the value of p are curiosities these days, but in ancient times they were useful, for instance, in astronomy. The Hindu mathematician Aryabhata (c. 510 AD) gave one formula:

"Add 4 to 100, multiply by 8, add 62,000, and you have for a diameter of 2 ayutas (10,000) the approximate value of the circumference."

Ramanujan also gave many formulas for p (pi) (my favourite: take the fourth root of 97.5 - 1/11) but of course the importance of his formulas was in the method he used to arrive at them.

When the Cambridge professors first looked at Ramanujan's formulas, they had mixed feelings, as they were not certain whether to take the author seriously. Hardy's reaction went something like this: " formulas (1) to (4) are harder than they look... (5) and (6) I could do, but with difficulty... (7) I have done myself... (8) is a formula by Laplace which was proved by Jacobi... (9) is in a paper by Rogers in 1907... (10) and (11) are not true... (12) to (15) defeated me completely ..."

A colleague of Hardy, John Littlewood's reaction was more of a roller coaster: "... the stuff about primes is wrong... I have a vague theory as to how his mistakes have come about... I still can't believe he knows any function theory... (d) is still wrong, of course, rather a howler... my hopes now are that he has made important discoveries about continued fractions and elliptic functions", and then finally ending with "... I can believe that he's at least a Jacobi."

Many years later, Littlewood would observe that every positive number was one of Ramanujan's personal friends and would ask, rhetorically, what would have happened if he had come in touch with Leonhard Euler, the eighteenth century Swiss prodigy. Of Euler it was said that he calculated without effort, just as men breathe, as eagles sustain themselves in the air.

Ramanujan died at the age of thirty-three, leaving behind a large body of unpublished notes. While arrangements were being made to publish his works, Hardy realized that he did not have a good portrait photograph to go along with it. Subsequently, the astrophysicist S. Chandrasekhar, went to Madras to look for a suitable photograph.

4 August 1937
"Dear Hardy,
I remember your telling me that when Ramanujan's Collected papers were being edited, it was your original intention to include a portrait of Ramanujan, but eventually you abandoned the plan as no good photograph was available. So, when I was in India last summer, I made an effort to find a reasonably good photograph. I met Mrs. Ramanujan - his wife... I am not in a position to judge how true it is, but I have on Mrs. Ramanujan's authority that the rather worried look Ramanujan has in the picture was extremely frequent during his last year...
S. Chandrasekhar"
15 December 1937
"Dear Chandrasekhar,
...I was very glad indeed to have the photograph, which seems to me an extremely good one. He looks rather ill (and no doubt was very ill): but he looks all over the genius he was....
G.H. Hardy"

Both Carr and Hardy had a strong influence on Ramanujan's intellectual growth. Both took the Tripos, Carr a Senior Optime and Hardy a Wrangler. Carr dedicated his life to compiling old knowledge and Hardy, the abler mind, dedicated his to producing new mathematics. Yet at the end of the day, it seems Carr's gift to Ramanujan of teaching him the art of how-to-know is an achievement that matches or exceeds Hardy's lessons on what-to-know.

Today, almost a hundred years after Ramanujan picked up a copy of Carr, there is renewed interest in concise formulas and ways of doing fast calculations, driven by the inexorable needs of software and digital technology. As the body of knowledge about these methods continues to accumulate, their organization in a useful form like Carr's Synopsis has immense economic value, more so than any book by Hardy.

Meanwhile, mathematics departments the world over continue to take crank-letters seriously (many, until recently, claiming to prove Fermat's last theorem), in case they come from yet another lone student of George Shoobridge Carr.

* Inserted in 2007.

References related to Ramanujan:

  • The Man Who knew Infinity; Robert Kanigel, Simon Schuster (New York, 1991)

  • Ramanujan, Letters and Commentary; Bruce C. Berndt and Robert A. Rankin, American Mathematical Society and London Mathematical Society (1995)

  • Formulas and Theorems in Pure Mathematics; George S. Carr, 2nd Edition, Chelsea Publishing (New York 1970); first edition published in 1880 in London and Cambridge

  • References related to ancient Indian mathematics

  • History of Hindu Mathematics, A Source Book Bibhutibhushan Dutta and Avadhesh Narayan Singh (1962, Asia Publishing House)

  • Alberuni's India; Edward C. Sachau, S. Chand Publishers (New Delhi 1964, original London 1888)

  • References related to popular writings on number theory and its history

  • Higher arithmetic; H Davenport, Dover.

  • Fermat's Enigma; Simon Singh, Anchor Books (1997)

  • Pencil sketch of Ramanujan - Amitabha Sen

    Amitabha Sen is a Chicago based former mathematical physicist now with a Swiss investment bank, UBS Warburg Dillon Read

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