**Herman Goldstine** was brought up in Chicago where his father was a lawyer. He studied at the University of Chicago where he was awarded a bachelor's degree in mathematics in 1933, graduating Phi Beta Kappa. He remained in Chicago studying for a Master's degree which he received in the following year, then was awarded a doctorate in 1936 for his thesis *Conditions for Minimum of a Functional*. His thesis advisor had been Lawrence Graves and he had also been taught by William Reid. Appointed an assistant to Gilbert Bliss at Chicago in 1936 working with him on the calculus of variations for three years before moving to the University of Michigan in 1939 as an Instructor. In 1941 he married Adele Katz who was an ENIAC (Electronic Numerical Integrator And Computer) programmer; they had a daughter Madlen and a son Jonathan.

The first publication of Goldstine's was *Minimum problems in the functional calculus* which was based on results in his thesis. Lawrence Graves, his thesis advisor, writes that Goldstine first gives:-

... necessary conditions and sufficient conditions, in terms of the first and second differentials, that a real-valued function defined on an open set in a normed linear space should have a minimum. Most of the paper is taken up with the more difficult problem of determining such conditions when the class of admissible points is(1940)required to satisfy an equation of an abstract functional character. Various hypotheses are described under which a multiplier rule can be obtained. Sufficient conditions for a minimum are also given.

There followed *A generalized Pell equation. I* (1940), written jointly with R P McKeon, and *Linear functionals and integrals in abstract spaces *(1942) in which he shows that the Daniell integral and the integral similarly defined by Banach in his addendum to Saks's "Theory of the Integral" are Lebesgue integrals with respect to regular Carathéodory outer measures. In 1942 he published two papers, *The modular space determined by a positive function* (with R W Barnard) and *The calculus of variations in abstract spaces*.

The United States entered World War II following the Japanese attack on the American fleet at Pearl Harbour on 7 December 1941. Goldstine joined the US army and in July 1942, with the rank of lieutenant, he was posted to the Ballistics Research Laboratory at the Aberdeen Proving Ground, Maryland. In [2] Ralph Gomory explains how this came about:-

Bliss had been one of a group, organized by the Princeton mathematician Oswald Veblen, that had worked on ordnance problems for the army during World War I. Bliss had made significant contributions to the theory of ballistics. When Herman was called into the army in July of1942, Bliss wrote to Veblen. Veblen had returned to the army as the chief scientist of the Army Ballistics Research Laboratory in Aberdeen, Maryland. Bliss suggested to Veblen that he should get Herman assigned to the Ballistics Research Laboratory.

However, this led to a degree of confusion. Goldstine was already in the army at Stockton, California, when he received two sets of orders. One set of orders instructed him to proceed to the Far East, while the second set instructed him to report to the Research Laboratory in Aberdeen, Maryland. He spoke to his commanding officer explaining that he had two sets of conflicting orders. The commanding officer asked him which he would prefer and, after being told that he would rather go to the Ballistics Research Laboratory he was told to drive there as quickly as possible and sort out the mess from Aberdeen. This is exactly what Goldstine did and, as Gomory writes [2]:-

It was of course a decision with momentous consequences for the early history of the computer.

Research at the Moore School of Electrical Engineering at the University of Pennsylvania was being carried out using early forms of computers. In particular the School used a Bush analyser, designed by Vannevar Bush, specifically to integrate systems of ordinary differential equations. However John Mauchly and J Presper Eckert Jr at the Moore School had ideas about how a better computer might be built. The main task of the Ballistics Research Laboratory was to produce firing tables which could be used by to direct guns to hit precise targets several miles away. The calculations were tedious and were carried out in Aberdeen by around one hundred women. The Ballistics Research Laboratory were certainly aware of the potential impact that computers would have on this work and they began using the computing facilities at the Moore School of Electrical Engineering. Goldstine was given the task of liaising with the mathematicians at the Moore School. After Mauchly wrote a report towards the end of 1942 detailing the ideas for a better computer that he had been working on with Eckert, Goldstine quickly saw the potential in what was being proposed.

Various committees then considered the proposal before money could be assigned to the project of building the computer, and in April 1943 Goldstine persuaded Veblen, who was the chief scientist at the Ballistics Research Laboratory in Aberdeen, to approve the idea. The army allocated $500,000 to pay for the research on the ENIAC project. The massive computer weighing 30 tons was completed by February 1946. In that year Herman and his wife Adele published the joint paper *The electronic numerical integrator and computer (ENIAC)*. L J Comrie clearly understood the significance of ENIAC when he reviewed the paper. Here is a short extract from the review:-

This is the first technical description of the first electronic calculator. The machine was developed at the Moore School of Electrical Engineering at Philadelphia for the Ballistic Research Laboratory of the Army Ordnance Department at Aberdeen, Maryland, where it is now located. The discarding of moving parts and the use of electronic circuits has, almost overnight as it were, speeded up certain calculations by a factor that handsomely exceeds1000. ... The importance of this pioneer effort cannot be overestimated. It has ushered in a new era in calculations with discrete numbers and has not only swept away the bottleneck of many existing and known forms of computation, but has also opened up new vistas of numerical thought by providing the means of doing what has never yet been attempted because of prohibitive time and cost.

Goldstine had been promoted from Instructor to Assistant Professor at the University of Michigan in 1942 and continued to hold this post while undertaking army work at the Ballistics Research Laboratory in Maryland. In 1946, with World War II over, Goldstine left the army and returned to his university position. However, a chance meeting with von Neumann on a railway platform in Aberdeen, Maryland, in the summer of 1944 proved highly significant to his future career. Goldstine started to tell von Neumann about the computing project he was working, in particular that ENIAC could carry out 333 multiplications per second, and then [3]:-

... the whole atmosphere of our conversation changed from one of relaxed good humour to one more like the oral examination for the doctor's degree in mathematics

A project to produce an improved computer began with Goldstine, von Neumann, Mauchly and Eckert all involved. However Mauchly and Eckert left the Moore School in October 1946 and started up the Electronic Control Company which received an order from Northrop Aircraft Company to build the Binary Automatic Computer (BINAC). Goldstine was appointed Associate Project Director of the Electronic Computer Project at the Institute for Advanced Study, Princeton University, and with von Neumann worked on their own ideas for a new computer. They produced a series of reports on the EDVAC (Electronic Differential Variable Computer) which changed the whole concept of computers. There was *Preliminary Discussion of the Logical Design of an Electronic Computing Instrument *(1947) by Arthur Bucks, John von Neumann and Herman Goldstine. Richard Hamming begins a review:-

This report gives a preliminary discussion of a high-speed electronic digital computing machine. The machine is to make full use of the flexible and compact coding of problems which is possible when orders as well as numbers are stored in the high speed memory and can be operated on and modified according to the progress of the computation.

In the same year Goldstine and von Neumann published *Planning and Coding of Problems for an Electronic Computing Instrument* which discusses the general philosophy of coding problems for a large scale digital computers, then the same two authors published *Numerical inverting of matrices of high order*, again in 1947. In the last mentioned paper they state that they:-

... wish to determine the precision and stability of a(long)computation with respect to the round off errors.

In 1948 they continued to produce further reports on their plans writing in one:-

We wish to develop here methods that will permit us to use the coded sequence of a problem, when that problem occurs as part of a more complicated one, as a single entity, as a whole, and avoid the need for recoding it each time when it occurs as a part in a new context, i.e., in a new problem. The importance of being able to do this is very great. It is likely to have a decisive influence on the ease and the efficiency with which a computing automat of the type that we contemplate will be operable.... We envisage that a properly organized automatic, high speed establishment will include an extensive collection of such subroutines, of lengths ranging from about15 - 20words upwards.

Goldstine and von Neumann also produced papers using computation to attack mathematical question, such as their application to number theory in *A numerical study of a conjecture of Kummer* (1953). Goldstine's applications of computing was to a wide variety of problems, for example he published *Calculation of plane cavity flows past curved obstacles* (1954) jointly with G Birkhoff and E H Zarantonello:-

This paper describes the first systematic calculations ever carried out on cavity and jet flows past curved obstacles, comprising altogether fifty symmetric plane flows past convex and concave bodies, under several different conditions of streamline detachment. The computations were performed on a high speed digital computer...

In 1955 a joint work by Goldstine and von Neumann *Blast wave calculation* was published as well as *On the stability of two superposed compressible fluids* which was joint work by Goldstine and J Gillis.

Goldstine became Project Director in 1954. His social relations with von Neumann are described in [2]:-

Many evenings Johnny[von Neumann]would entertain. Usually a few of us, maybe my wife and me. We would just sit around, and he might not even sit in the same room. He had a little study that opened off of the living room, and he would just sit in there sometimes. He would listen, and if something interested him, he would interrupt. Otherwise he would work away.

After von Neumann's death in 1957, Goldstine left Princeton and in 1958 was appointed research administrator at IBM. He did not undertake research solely on computers and their applications, however, for he published a series of three papers on *Hilbert space with non-associative scalars* (1962, 1964, 1966). Goldstine's wife Adele died in 1964 and two years later he married Ellen Watson. He remained working with IBM in a number of leading roles until he retired in 1984. He was appointed director of scientific development for data processing at IBM in 1965, then an IBM Fellow in 1969 and also a consultant to the director of research.

Goldstine wrote a number of outstanding books. These include *The computer from Pascal to von Neumann* (1972), and *A history of numerical analysis from the *16*th through the *19*th century* (1977), described as:-

... a definitive text which is a must for the library of any serious numerical analyst.

*A history of the calculus of variations from the *17*th through the *19*th century* was published in 1980. L G Chambers wrote that:-

... this book is a substantial piece of mathematical scholarship. The exposition is good and the author has clarified the argument of, and has indicated the connection between, many of the fundamental works to which he refers with the result that the book gives a better understanding of the calculus of variations than do many modern texts. ... the book deserves the highest recommendation.

Goldstine received many honours for his outstanding contributions to the development of the computer and its applications to mathematics. For example he received the Harry H Goode Memorial Award in 1979, and the National Medal of Science in 1983 from President Ronald Regan for:-

... fundamental contributions to development of the digital computer, computer programming and numerical analysis.

He was admitted to the Hall of Fame of the Army Ordnance Department in 1997 and also received the IEEE Pioneer Award. He was elected to the National Academy of Science, the American Academy of Arts and Sciences, and the American Philosophical Society. This last mentioned Society appointed him their Executive Director after he retired from IBM in 1984, a position he held until 1997. On 27 May 1998 IBM Research announced that it was to honour:-

... Dr Herman H Goldstine by renaming a major postdoctoral fellowship for the mathematician.

He died in his home of Parkinson's disease at the age of 90.

**Article by:** *J J O'Connor* and *E F Robertson*