**Gilbert Bliss**'s parents were George Harrison Bliss and Mary Maria Gilbert. George Bliss was the president of the Chicago Edison Company which, by 1907, supplied all of Chicago's electricity. Bliss was born in the suburbs of Chicago into a wealthy family. However when he was seventeen years old and about to attend College, the United States, which had been in economic depression for six years, plunged into crisis. The level of the gold reserve in the federal Treasury was believed vital to assure redemption of government obligations. On 21 April 1893 the reserve fell below what was generally believed to be the necessary level and the result was panic as investors tried to change their assets into gold. Many businesses and financial institutions failed as prices dropped, and the Chicago Edison Company hit hard times. A period of severe economic depression followed that continued for over three years. Bliss's family were no longer able to support him through College so he had to find a way to support himself. He did so in two ways, first by winning a scholarship and second by making money by being a member of a professional mandolin group.

Bliss was one of the first American mathematicians to complete his studies in the United States before travelling to Europe. He entered the University of Chicago in 1893 and received his B.S. is 1897. He then began his graduate studies at Chicago in mathematical astronomy and his first publication was in that field. However mathematics was his real love and, in 1898, he began his doctoral studies working on the calculus of variations. His interest in the calculus of variations came through two sources, firstly from lecture notes of Weierstrass's 1879 course, of which he had a copy, and secondly from the inspiring lectures by Bolza which Bliss attended. Bliss received his doctorate in 1900 for a dissertation *The Geodesic Lines on the Anchor Ring* which was supervised by Bolza. Then he was appointed as an instructor at the University of Minnesota in 1900. He left Minnesota in 1902 to spend a year in Göttingen where he interacted with Klein, Hilbert, Minkowski, Zermelo, Schmidt, Max Abraham and Carathéodory. His fellow American Max Mason was a doctoral student at Göttingen during the year Bliss spent there. Bliss published two papers in 1902: one in the *Annals of Mathematics* was based on his doctoral dissertation and had the same title *The geodesic lines on the anchor ring* while the second in the *Transactions of the American Mathematical Society* was titled *The second variation of a definite integral when one end-point is variable.*

Returning to the United States, Bliss was appointed to the University of Chicago in 1903, then in 1904 he was appointed as an assistant professor at the University of Missouri. Two further papers by him on the calculus of variations appeared in 1904, both in the *Transactions of the American Mathematical Society.* They were *An existence theorem for a differential equation of the second order, with an application to the calculus of variations* and *Sufficient condition for a minimum with respect to one-sided variations.* At Missouri his Head of Department was Hedrick but after a year he was offered a post at Princeton which he accepted, remaining there until 1908. At Princeton Bliss joined a strong group of young mathematicians including Eisenhart, Veblen, and Robert Moore. While at Princeton he was also an associate editor of the *Annals of Mathematics.* Bliss was appointed as an associate professor at the University of Chicago on the death of Maschke and he remained at Chicago until he retired. At this time he became an editor of the *Transactions of the American Mathematical Society* and continued in this rol until 1916. He was chairman of the Mathematics Department at Chicago from 1927 until he retired in 1941. We should remark however, that Duren in [5] was a graduate student in Chicago in the late 1920s and describes this period as the:-

... down cycle for mathematics at Chicago.

Bliss married Helen Hurd in 1912 and they had two children before Helen tragically died in the influenza epidemic of 1918. This epidemic, often called the Spanish Influenza Epidemic, was the most severe outbreak of influenza during the 20^{th} century. It seems to have originated in Funston, Kansas, in early March 1918 and led to about 550,000 people dying in the United States while world wide around 30 million died. Bliss and his two children survived.

He worked on ballistics during World War I and designed new firing tables for artillery, publishing his work in two papers in the *Journal of U.S. Artillery* in 1919. His book *Mathematics for Exterior Ballistics* (1944) was based on this work and published to make the methods he devised during World War I available during World War II. W E Milne writes in a review:-

This book treats exterior ballistics strictly as a problem in particle dynamics, that is, rotational effects are ignored and resistance is assumed to act along the tangent to the trajectory. The field of ballistics, thus restricted, is presented with exceptional clearness and thoroughness and a surprising amount of information is packed into a little book of only128pages.

He lectured on navigation to about 100 students at the University of Chicago as part of the World War I effort. Also in 1918 he joined Veblen in the Range Firing Section at the Aberdeen Proving Ground, a military weapons testing site established in 1917 in Harford county northeastern Maryland. There he very effectively applied methods from the calculus of variations to solve problems relating to correcting missile trajectories for the effects of wind, changes in air density, rotation of the Earth and other perturbations. Bliss married again in 1920, his second wife being Olive Hunter; there were no children from this marriage.

Bliss's main work was on the calculus of variations and he produced a major book, *Lectures on the Calculus of Variations* , on the topic in 1946. As a consequence of Bliss's results a substantial simplification of the transformation theories of Clebsch and Weierstrass was achieved. O Frink writes in a review:-

This is a sound, thorough and up-to-date text on the single integral problems of the calculus of variations, based on courses given by the author at the University of Chicago. ... The book starts with a typical simple problem, the non-parametric problem in3-space with fixed end-points. Chapter I takes up necessary conditions, chapter II sufficient conditions for this case. Tables show various ways the necessary conditions of Euler, Weierstrass, Legendre and Jacobi may be strengthened to give sufficient conditions. ... Part II presents the first comprehensive treatment of the problem of Bolza, a very general type of problem with side conditions and variable end-points. The form in which the results are presented here is that preferred by Bliss himself. The problem has now reached a state of completeness and simplicity which allows it to be presented in a manner like that used for the older problems of part I. ... The theory here presented marks the culmination of the modern phase of development of the calculus of variations, begun by Weierstrass and continued by Hilbert, Bolza and Bliss. In this treatment the subject is studied rigorously, with no emphasis on the formal manipulation of symbols. On the other hand the subject is taken as an end in itself, and not as a mere adjunct of mechanics.

Bliss also studied singularities of real transformations in the plane. During the last 50 years of his life Bliss played a major role in mathematics in the United States and he was elected to the National Academy of Sciences (United States) in 1916. He was deeply involved in the American Mathematical Society being the Colloquium Lecturer in 1909, the Vice President in 1911, and its President from 1921 to 1922. He received many awards for his work including the first Chauvenet Prize in 1925 from the Mathematical Association of America for his article on *Algebraic functions and their divisors.* On this topic he published the book *Algebraic functions* in 1933 which was reprinted by *Dover Publications* in 1963.

Bliss's interests outside mathematics are described in [3]:-

In order to earn money necessary for his college expenses he became a member of a student professional mandolin quartet. He has always been interested in sport and beginning with bicycle racing in student days he has successively taken up tennis, racquets, and golf.

Also outside mathematics was his work heading [6]:-

... a government commission to construct precise rules for assigning to states seats in the U.S. House of Representatives, under the vague constitutional provisions for proportional representation.

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

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