Congrats, Jack! Reminiscences and Appreciation

July 10, 2001

John Todd "devoted at least eighty of his ninety years to the cause of mathematics and to its applications," said Philip Davis, an invited speaker at the Caltech conference held in May to honor Todd on his 90th birthday. (Photo by Sarah Emery Bunn, Caltech)
Philip J. Davis

A two-day conference, Numerical Analysis, Linear Algebra and Computations, was held at the California Institute of Technology, May 16-17, to celebrate the 90th birthday of John Todd. Philip Davis, whose association with Todd dates back to the time just after World War II, looked back on those early days of computing and numerical analysis in the first invited talk at the conference. What follows is an adapted version of that talk.

We gathered in Pasadena to honor a nonagenarian of international reputation who has devoted at least eighty of his ninety years to the cause of mathematics and to its applications. There is an old proverb that asks: "Who is honored?" The answer given is: "He who honors other people." So I was greatly honored by the opportunity to pay tribute to John Todd. It is by no means an easy matter to select out the high points of a long profes-sional career, but for this occasion it seemed important that I make the attempt.

Jack Todd was born in 1911. He received his BSc at Queens College Belfast, Northern Ireland, in 1931. During World War II, he was with the Mine Design Department in Portsmouth, England, and later with the Admiralty Computing Service in London. In 1939, just before the war began, Jack and Olga Taussky were married. In 1946, he gave his first course in numerical mathematics at King's College London. In 1949, John Curtiss hired him to head the Computation Laboratory at the National Bureau of Standards (now NIST) in Washington, and from 1954 to 1957 he was chief of the Numerical Analysis Section. He moved to Caltech as a professor of mathematics in 1957 and has been there since that time.

If you check the Web site for this conference,* you will find a complete list of Jack's scientific work, which spans algebra, analysis, computation, special functions, history, biography, and many other topics. You will also find an interesting account of how, late in World War II, he and another British officer were the first to occupy the buildings of the Mathematisches Forschungsinstitut in Oberwohlfach, and how their occupation saved the institute from being destroyed. Anyone who has profited from a stay at Oberwohlfach owes a belated debt of gratitude to Jack.

One can speak generically about a person, and by that I mean provide the standard contents of a curriculum vitae-material that is now often webified. But I had in mind to say something personal: how Jack Todd impacted my life. And for this reason, this piece contains a bit of self-revelation. Olga Taussky-Todd subtitled her autobiography "the truth, nothing but the truth, but not all the truth" [1]. And here I must do likewise.

I received my bachelor's degree in mathematics in the middle of World War II. Shortly thereafter, I was inducted into the Air Force, placed on reserve status, and given a position at NACA, the precursor of NASA, at Langley Field in Virginia. My job was in NACA's Aircraft Loads Division, which studied dynamic loads on the components of fighter aircraft during a variety of maneuvers. I was partially a computer---working with the raw data provided by flight instrumentation, accelerometers, and pressure gauges---and partially an interpreter of what I had computed. My colleagues and I worked with slide rules, planimeters, nomograms, and various electromechanical calculators (Marchants, Friedens, and so forth). And we had one other mathematical aid. Even as Jack Todd reported in his History of Computation [2] how, in his war work in the British Mine Design Department, he had used the (American) WPA tables of special functions, so also did we make substantial use of these and other tables computed some years before.

Reciprocally, we got numerous reports from U.K. laboratories (all stamped "confidential" or classified at even higher levels). These reports were circulated widely, and I recall reading one on eigenvalue computation authored by Olga Taussky. It must have dealt with the Gershgorin circle theorem.

How difficult, how tedious, how time-consuming it was in those days to solve a second-order linear differential equation with constant coefficients but with a graphically given right-hand side that represented the pilot's action. My first published paper reduced, computationally speaking, to this---a task that I would guess is now performed routinely in nanoseconds.

Although I had had no college courses in numerical methods, I didn't come into my job at NACA totally green. In high school I had studied (with no deep understanding, I can assure you) a thin book by David Gibb entitled Interpolation and Numerical Integration (1915), which derived from E.T. Whittaker's Mathematical Laboratory at the University of Edinburgh. (Incidentally, a full-page picture of Whittaker can be found at the beginning of Nash's collection of articles in which Jack's History appeared.)

The war over, I returned to graduate school and did a thesis in pure mathematics under the supervision of Ralph Boas. The subject: uniqueness theorems for infinite interpolatory systems as applied to entire analytic functions of exponential type. This was, by the way, a subject in which Edmund Whittaker's son, J.T. Whittaker, had specialized, producing a Cambridge monograph.

After several years of working on contracts and grants for Stefan Bergman, I accepted a position in the Numerical Analysis Section of NBS in Washington. I recall the snooty disdain, the lifted eyebrows of my contemporaries when they heard I'd accepted a job at a government laboratory. The established wisdom in those days---and it lingers on---was that the only allowable career for a mathematician was to prove theorems in a university environment. What served as significant credentiation for the place, as I began to consider the possibility, was the fact that John Curtiss was then head of mathematics at NBS, that he had written his thesis under J.L. Walsh, and that I had co-authored several papers with Walsh.

In the decade beginning in about 1948, NBS was surely one of the principal places in the world studying and investigating numerical methods, taking into account the potentialities of the new computing machines. The National Physical Laboratory in England was another such place. When I arrived at NBS, I found several senior mathematicians: Jack and Olga Todd, Milton Abramowitz, Irene Stegun, Churchill Eisenhart, Ida Rhodes, Ted Motzkin, Ky Fan. Among my contemporaries were Alan Hoffman, Morris Newman, Karl Goldberg, Henry Antosiewicz, and a bit later, Philip Rabinowitz, Walter Gautschi, Peter Henrici, John Rice, Marvin Marcus, Emilie Haynsworth, Joan Rosenblatt, Marvin Zelen.

There was a steady stream of visitors, often from abroad, all mathematicians of the first class with a deep interest in computation. I can cite Eduard Stiefel, Helmut Wielandt, Alexander Ostrowski, and J.M. Synge. The division had an advisory committee among whose members Mina Rees and Marc Kac played an active role in suggesting names of visitors to invite.

The factors and forces, the people that influence a career, are often difficult to perceive at the time. Many of us, I'm sure, have had this experience: Someone knocks on your door, and a person you do not recognize enters. You look perplexed. He or she then says, "You don't remember me, Professor, but I took your course twenty-five years ago, and it really turned my life around." Embarrassed, you say, "Why yes, of course. Come in; sit down and tell me what you've been doing all this while."

For four years---from 1952 to 1957---Jack Todd was my boss. And I use the word "boss" in the "weak topology," for what he did was great. Over and above certain contractual obligations, which paid for my bread, he left me alone. He left me alone to interact with the great group that was assembled around him and then to do my own thing.

And I did interact with many of the people just mentioned. A number became very good friends: Phil Rabinowitz, Emilie Haynsworth, Alexander Ostrowski. An interest in matrix theory developed (sparked by Jack and Olga, for matrix theory was perhaps the strong suit of the group). I left NBS because I wanted to write, and found by hard experience that a government agency was by no means the ideal place to write.

In the mid-fifties, computers and computation laboratories were beginning to appear at universities where previously only a single, heavy, rusty, dusty, adding machine might have been found in the office of the professor of astronomy. Jack conceived the idea of running a training program for future directors of academic computing labs. He received NSF support for the program, which was held in 1957, with great success. As an inheritance from Jack, I repeated his formula in 1959 and was equally gratified by the results.

Mathematics is probably as old as civilization itself; it has always played a role in the ways in which people deal with one another, with discovery, with the arrangement and incorporation of experience into a coherent, interpretable, occasionally predictable system. Numerical analysis, which is probably where mathematics began three or four millennia ago, has itself changed mightily in Jack Todd's and all of our lifetimes. How many of the tools of yesteryear have become quaint and obsolete! In 1911, numerical analysis was an accumulation of recipes and advice: how to deal with squared paper and what checks to make. It is amusing to read David Gibb's words from 1915:

"Each desk [in the Mathematical Laboratory at the University of Edinburgh] is equipped with a copy of Barlow's Tables, a copy of Crelle's Tables which gives at sight the product of any two numbers less than 1000. For the neat and methodical arrangement of the work computing paper is essential. . . . It will be found conducive to speed and accuracy if, instead of taking down a number one digit at a time, the computer takes it down two digits at a time."

Numerical analysis is now a full-blown theoretical subject that also incorporates the accumulated wisdom of hundreds of thousands of experimental runs, all of whose ideas have diffused and filtered into packages of scientific computation.

Paraphrasing John Milton, we can assert: "They also serve who only make experimental runs." I'm sure that Cleve Moler, himself a student of Jack Todd, would agree with me that in the practical sense the very best numerical analysis is now to be found in Matlab and other such packages, and not on the pages of textbooks.

Our lives are now increasingly mathematized, and there is every indication that this tendency will continue unabated for the foreseeable future. To the average person, these mathematizations are not visible: They are hidden in age-old practices that we take for granted, or they lie buried deep in computer programs and chips.

Mathematics is the backbone of much that is new, surprising, utilitarian, aesthetic, and occasionally regrettable about our contemporary world. Take, for example, modern communications and the Web. The Web site for this conference contains much information about the venue and agenda of the conference. It contains much detail about, even many photos of, the man whose career we are celebrating. I can recommend it to all here.

The organizing committee could have set up a celebratory chat-site conference, attended by hundreds. No airplane or hotel bookings would be necessary, no jet lag. No cancellations or other irritations.

But this would not have done at all. For mathematics, even through the latest communication schemes, is at a loss to summarize an individual, or the career of an individual, or the impact of the individual on other individuals or on the larger social and scientific world, and to do it in a specified number of bytes. Yes, using mathematics, we can make a complete genetic analysis; but mathematics is powerless to describe fully what the eye sees or the ear hears.

And that is why we gathered at Caltech---to learn, and judge, and praise, and come away refreshed and built up. And we came together to pay tribute to a man who has devoted his life to the furtherance of our profession.

[1] O. Taussky-Todd, An autobiographical essay: The truth, nothing but the truth, but not all the truth, in Mathematical People, D.J. Albers and G.L Alexanderson, eds., Birkhauser, Boston, 1985.

[2] J. Todd, The prehistory and early history of computation at the NBS, in A History of Scientific Computation, S.G. Nash, ed., Addison Wesley, 1990, 251-268.

See and

Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island and can be reached at

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