Professor Chrystal has remarked that mathematical works should be read backwards as well as forwards, i.e., advanced notions of mathematics often throw light on elementary problems. While this "back-tracking" is frequently brought out in analysis, it is seldom emphasised in connection with plane and solid geometry.After graduating, Franklin spent a year at the U.S. Army Proving Grounds in Aberdeen, Maryland. This was to have a very considerable influence on Franklin, mainly because he met and became friendly with Norbert Wiener. In  Wiener writes:-
I shared a roughly boarded compartment in the civilian barracks with Philip Franklin ... and with Bennington Gill of the College of the City of New York. ... Franklin and Gill, nineteen and thus considerably younger than I, were my particular cronies. When we were not working on the noisy hand-computing machines, which we knew as "crashers," we were playing bridge together after hours using the same computing machines to record our scores. Sometimes we played chess or a newly devised three-handed variant of it on a board made of a piece of jump-screen, or risked the dangers of burning smokeless powder or TNT. We went swimming together in the tepid, brackish waters of Chesapeake Bay, or took walks in the woods amid a flora that was too southern to be familiar to us. I remember the pawpaws and their exotic tropical manner of growing their fruits directly from the tree trunk. Whatever we did, we always talked mathematics. Much of our talk led to no immediate research. I remember some half-baked ideas about the geometry of Pfaffians ... I cannot remember all the other subjects we discussed, but I am sure that this opportunity to live for a protracted period with mathematics and mathematicians greatly contributed to the devotion of all of us to our science.While he was in the Ordnance Department of the Aberdeen Proving Grounds, Franklin submitted Calculation of the Complex Zeros of the Function P(z) Complementary to the Incomplete Gamma Function to the Annals of Mathematics; it was published in September 1919. Franklin had already presented the paper on 26 April 1919 at a meeting of the American Mathematical Society held at New York City. From the Aberdeen Proving Grounds, he went to Princeton University to undertake doctoral studies. He was awarded his Ph.D. in 1921 for a thesis The Four Color Problem written under Oswald Veblen's supervision. On completing his doctorate Franklin remained at Princeton where he was an Instructor in Mathematics during 1921-22. While at Princeton, Franklin submitted a number of papers on widely different topics which were published in 1921 and 1922. He published: Generalized Conjugate Matrices (1921) and (with Oswald Veblen) On Matrices Whose Elements Are Integers (1921) both in the Annals of Mathematics; An Arithmetical Perpetual Calendar (1921) and On Curves Whose Evolutes are Similar Curves (1921) in the American Mathematical Monthly; The Meaning of Rotation in the Special Theory of Relativity (1922) in the Proceedings of the National Academy of Sciences of the United States of America; and The Four Color Problem (1922) in the American Journal of Mathematics. This last mentioned paper contained results from his doctoral thesis. In it he listed six properties which must be satisfied by a map with a minimal number of regions which cannot be four-coloured. His two main results were the first proof that all planar graphs with at most 25 vertices can be four-coloured and that there exists a map with 42 regions which can be four-coloured yet it satisfies all six properties given for a graph which cannot be four-coloured.
After his year as an instructor at Princeton, in 1922 Franklin went to Harvard University where he was appointed a Benjamin Peirce Instructor. We have not yet mentioned one important outcome of Franklin's time at the Aberdeen Proving Grounds. His friendship with Norbert Wiener led to his meeting Wiener's sister Constance who, like Franklin, was born in 1898. They married on 14 June 1924 and had three children, David, Janet (who married Vaclav E Benes), and Hope (who married Barrett O'Neill). In 1924 Franklin was appointed Instructor in Mathematics at the Massachusetts Institute of Technology. He was to remain at MIT, being promoted to assistant professor in 1925. Then, during 1927-28 he held a Guggenheim Fellowship, before being promoted to associate professor in 1930 and full professor seven years later. He spent 1935-36 undertaking research at the Institute for Advanced Study at Princeton and undertook foreign travel. In 1948 he lectured on mathematical physics at Harvard University.
We have already looked at the titles of some of Franklin's papers up to 1922. Let us record a few later highlights. In 1926, in the paper Approximations to Topological Transformations written in collaboration with his brother-in-law Norbert Wiener, it is proved that every continuous one-to-one mapping of a two-dimensional region of finite connectivity may be approximated as closely as required by an analytic one-to-one mapping. In 1928 he described an orthonormal basis for L([0,1]) consisting of continuous functions. This is now known as "Franklin's system". In A Step-Polygon of a Denumerable Infinity of Sides which Bounds No Finite Area (1933), written in collaboration with Jesse Douglas, the authors gave an explicit construction for functions which had been shown to exist by Jesse Douglas in his solution of Plateau's problem in his paper published in 1931 (for which he received a Fields Medal). In A Six Color Problem (1934) Franklin gave a 12-vertex cubic graph whose embedding into the Klein bottle provides the only known counter-example to the Heawood conjecture.
However, he is best known for textbooks he published on calculus, differential equations, complex variable and Fourier series. In particular he wrote Differential equations for electrical engineers (1933), Treatise on advanced calculus (1940), The four color problem (1941), Methods of advanced calculus (1944), Fourier methods (1949), Differential and integral calculus (1953), Functions of a complex variable (1958) and Compact calculus (1963).
Extracts from reviews of each of these papers can he seen at THIS LINK which give a good flavour of the different styles of book that Franklin wrote, all with great success.
Let us here quote from Richard Courant's review of A Treatise on Advanced Calculus :-
Rigour, whatever this word may mean, was one of the great mathematical achievements of the nineteenth century. Only gradually has this tendency penetrated into textbooks. The first great work of this kind, Jordan's "Cours d'analyse," was followed by many others, of which Hardy's "Pure Mathematics" seems to be the foremost in English. Franklin's book is an admirable attempt on a much broader scale to combine rigour with completeness in a volume of modest size.Ernst Hellinger, reviewing the same textbook by Franklin, writes :-
This book is an extraordinarily satisfactory addition to the literature of advanced calculus. This text is indeed a treatise which covers completely the infinitesimal calculus and includes much prerequisite algebra and analysis (and most other concepts) that are needed for geometric and physical applications. The theorems and proofs are given with utmost precision and completeness, and the reader will never have need of another text for the material that is treated here.Four years after publishing this rigorous text, Franklin published another text Methods of advanced calculus on similar material but now with very different students in mind. Morris Marden contrasts the two different style books in :-
The two books are in some respects different as to content, but in all respects different as to point of view. ... While the first book was a sort of "cours d'Analyse," the new book ... has the purpose of serving the needs of prospective engineers, physicists, and others who may regard mathematics primarily as a tool. ... the new volume may be regarded as one of the best textbooks now available for any advanced calculus course which is intended to be a terminal course in mathematics for engineers, physicists and the like.In addition to this impressive collection of books, Franklin was editor of the Journal of Mathematics and Physics from 1929. In 1943 he was honoured by his old College when the College of the City of New York awarded him their Townsend Harris Medal. The Medal was presented on 13 November at a dinner held in the Hotel Roosevelt. The citation reads:-
At various times a member of the faculties of Princeton, Harvard and the Institute for Advanced Study, and now professor of mathematics in the Massachusetts Institute of Technology, your colleagues consider you one of the outstanding mathematicians in the United States. As an editor of 'The Journal of Mathematics and Physics' and the author of studies in algebra and calculus you have made important contributions to scholarship.As a summary of Franklin's contributions we quote Dirk Struik writing in :-
Phil was an even-tempered, mild, humorous man, "of almost Mr Chips proportions", as Dean Harrison said in 1965 at his funeral service. His Princeton Ph.D. thesis of 1922 was in the Veblen topology field and a contribution to the four-colour problem. Coming to Harvard first then to MIT he brought a new field to Cambridge. "Franklin", Marshall Stone writes, "gave us [Harvard] our first systematic introduction to topology." In the MIT Journal of 1933-34 he extended his studies to the six-colour problem for one-sided surfaces. He was well-versed in many fields of geometry, algebra and analysis. In 1936 he lectured before the American Mathematical Society on transcendental numbers. In the early 30s he published with Moore a set of papers on algebraic Pfaffians. Since Franklin brought topology to MIT in his "analysis situs" form, and Wiener in its "point-set-Lebesgue" form, we see that it came to the Institute through two brothers-in-law. I have always had the feeling that living in the shadow, so to speak, of his overwhelming brother-in-law, cramped his style. At any rate, he devoted much of his time to the writing of eight excellent textbooks ...We should mention Franklin's service to the mathematical community through serving on the Board of Governors of the Mathematical Association of America during 1940-42 and again during 1954-56. He was chairman of the Nominating Committee of the Association in 1949-50. He also served on the Council of the American Academy of Arts and Sciences in 1947-50. In 1950 he was one of a 3-man team asked "to report on ways of discovering and encouraging latent scientific talent in New England."
In June 1964 he retired but continued to teach part time. He died unexpectedly while recovering from surgery. He was buried in Pawlet, Vermont.
Article by: J J O'Connor and E F Robertson