Celebratio Mathematica

The con­tri­bu­tions made by Gil­bert Ames Bliss to math­em­at­ics and to edu­ca­tion­al and sci­entif­ic activ­it­ies in the United States were many and var­ied. The fol­low­ing is a quite in­ad­equate sum­mary of his life and work, with spe­cial ref­er­ence to his math­em­at­ic­al activ­it­ies. Gil­bert Bliss was born in Chica­go on May 9, 1876, and died in In­galls Me­mori­al Hos­pit­al in Har­vey, Illinois, on May 8, 1951. In 1893, one year after the Uni­versity of Chica­go first opened its doors to stu­dents, he en­rolled there as a stu­dent, and was awar­ded the de­grees of B.S. in 1897, M.S. in 1898, and Ph.D. in 1900. He stud­ied at the Uni­versity of Göttin­gen dur­ing the year 1902–1903. His first teach­ing was done as a sub­sti­tute for a mem­ber of the staff of Kala­ma­zoo Col­lege for sev­er­al weeks dur­ing the year 1898–1899. He was an in­struct­or in math­em­at­ics at the Uni­versity of Min­nesota 1900–1902, an as­so­ci­ate at Chica­go 1903–1904, as­sist­ant pro­fess­or at the Uni­versity of Mis­souri 1904–1905, pre­cept­or (\( = \) as­sist­ant pro­fess­or) at Prin­ceton 1905–1908, and at Chica­go was as­so­ci­ate pro­fess­or 1908–1913, pro­fess­or 1913–1933, Mar­tin A. Ry­er­son Dis­tin­guished Ser­vice Pro­fess­or 1933–1941, and chair­man of the de­part­ment 1927–1941. He re­tired from act­ive ser­vice in 1941. In vari­ous sum­mer or au­tumn terms from 1906 to 1911 he gave courses at Wis­con­sin, Chica­go, Prin­ceton, and Har­vard.

Dur­ing his stu­dent days Bliss first fell un­der the in­flu­ence of F. R. Moulton, then a young as­sist­ant who was teach­ing at Chica­go, and his first pub­lished pa­per was en­titled The mo­tion of a heav­enly body in a res­ist­ing me­di­um. The fel­low­ship in as­tro­nomy for which he ap­plied was not gran­ted, and he even­tu­ally de­cided to de­vote him­self to pure math­em­at­ics. His thes­is for the M.S. was en­titled The geodes­ic lines on the an­chor ring, and his thes­is for the Ph.D. bore the same title. The first de­veloped ex­pli­cit for­mu­las for the geodes­ies in terms of el­lipt­ic in­teg­rals and dis­cussed some of their prop­er­ties. In the Ph.D. dis­ser­ta­tion [1] it was shown that the points on the in­ner equat­or of the an­chor ring are of the first kind (in the clas­si­fic­a­tion due to Man­goldt), i.e., each geodes­ic passing through such a point con­tains no con­jug­ate point, while all oth­er points on the an­chor ring are of the second kind, i.e., not of the first kind. Pre­vi­ous in­vest­ig­a­tions by Jac­obi and by Man­goldt had shown that on a sur­face of neg­at­ive curvature all points are of the first kind, while on a closed sur­face of pos­it­ive curvature all points are of the second kind. Dur­ing his time as a gradu­ate stu­dent Bliss made a copy of Bolza’s re­cord of Wei­er­strass’ 1879 course on the cal­cu­lus of vari­ations. This re­cord to­geth­er with Bolza’s in­flu­ence un­doubtedly helped fix in Bliss’ mind the in­terest in the sub­ject which dom­in­ated his re­search. Dur­ing his stay at Min­nesota he stud­ied Kneser’s book on the sub­ject, which was the first prin­ted ex­pos­i­tion of Wei­er­strass’ ideas, and pub­lished a pa­per [2] on the second vari­ation and suf­fi­cient con­di­tions for a min­im­um when one end point is vari­able. A second pa­per [4], tak­ing up the case when both end points are vari­able, was writ­ten dur­ing his stay in Göttin­gen. In both these pa­pers, as in his thes­is, geo­met­ric con­sid­er­a­tions play a prom­in­ent role. There are fre­quent evid­ences of his con­tinu­ing in­terest in geo­metry. In his pa­pers [6], [8], [17], he dis­cussed “Finsler” geo­metry of two di­men­sions, where the arc length is giv­en by an in­teg­ral of the form \[ \int f(x,y,\tau)(x^{\prime\mkern1mu 2} + y^{\prime\mkern1mu 2})^{1/2}dt,\\ \quad \tan \tau = \frac{y^{\prime}}{x^{\prime}}. \] Finsler’s thes­is treat­ing the \( n \)-di­men­sion­al case did not ap­pear un­til 1918. The tensor ana­lys­is of such spaces was de­veloped still later by Élie Cartan. Bliss kept in touch with these ideas, and de­vel­op­ments of them ap­peared in the dis­ser­ta­tions of Taylor [e6], John­son [e14], House­hold­er [e21], and Stokes [e25], which he sug­ges­ted and su­per­vised. Dur­ing the years 1908–1910, im­me­di­ately fol­low­ing the death of Masch­ke, he lec­tured on geo­metry at Chica­go, but when Bolza re­turned to Ger­many in 1910, Bliss seemed glad to make ana­lys­is again the fo­cus of his teach­ing.

A num­ber of pa­pers ap­ply­ing the meth­ods of Wei­er­strass to a vari­ety of prob­lems in cal­cu­lus of vari­ations ap­peared from Bliss’ pen dur­ing the years be­fore the first world war. Some of these were writ­ten in col­lab­or­a­tion with Max Ma­son [7], [9], [11] and one was a joint pa­per with A. L. Un­der­hill [16]. An idea which has be­come ba­sic for much sub­sequent work in the cal­cu­lus of vari­ations ap­peared in the pa­per [18]. This is the con­sid­er­a­tion of the min­im­um prop­er­ties of the second vari­ation. It makes pos­sible simple proofs of the ne­ces­sity of the Jac­obi con­di­tion, and of the ana­log­ous con­di­tion for the more gen­er­al prob­lem of Bolza, and is a guid­ing prin­ciple in the con­struc­tion of suf­fi­ciency proofs. It led to the pa­per [30] on the trans­form­a­tion of Cleb­sch which ex­presses the second vari­ation in the form \[ I_2(\eta) = \int^{x_2}_{x_1} F y^{\prime}_{i}y^{\prime}_{k} (\eta^{\prime}_{i} - \pi_i)(\eta^{\prime}_{k} - \pi_k)\, dx. \] This for­mula is val­id when there ex­ists a con­jug­ate sys­tem of ac­cess­ory ex­tremals whose de­term­in­ant does not van­ish on the in­ter­val \( [x_1, x_2] \) and the vec­tor func­tion \( \pi \) is suit­ably defined in terms of the ad­miss­ible vari­ation \( \eta \) (van­ish­ing at \( x_1 \) and \( x_2 \)) and of the giv­en con­jug­ate sys­tem. It is de­rived very simply with the help of the the­ory of fields of ex­tremals and the for­mula of Wei­er­strass. A simple dir­ect proof is also giv­en. The meth­ods of this pa­per are ba­sic for the vari­ous suf­fi­ciency proofs sub­sequently giv­en for the prob­lems of Lag­range, May­er, and Bolza.

Dur­ing the years 1916–1946 Bliss de­voted much time to im­prov­ing and ex­tend­ing the the­or­ies of the prob­lems of Lag­range, May­er, and Bolza. In this he had the co­oper­a­tion of sev­er­al of his stu­dents, not­ably Hestenes. The res­ults ap­peared in defin­it­ive form in 1946 in the second part of his “Lec­tures” [38] where the prob­lem of Bolza is treated in de­tail. If one com­pares this with earli­er ex­pos­i­tions of these gen­er­al prob­lems — for ex­ample, of the prob­lem of Lag­range in Bolza’s Vor­le­sun­gen, or in Bliss’ pa­per [31] — one is im­me­di­ately im­pressed with the great­er scope of the the­ory, due to weak­en­ing of hy­po­theses both for ne­ces­sary con­di­tions and for suf­fi­cient con­di­tions, and with the sim­pli­fic­a­tions ob­tained in the proofs. The first part of the “Lec­tures” con­tains an un­usu­ally clear present­a­tion of the the­ory of the cal­cu­lus of vari­ations for cases when there are no side con­di­tions.

Al­though Bliss pub­lished only one pa­per on mul­tiple in­teg­rals in the cal­cu­lus of vari­ations [37] he began to dis­cuss the sub­ject in courses and sem­inars in the 1920’s, and con­tin­ued at fre­quent in­ter­vals up to the sum­mer of 1942. Cor­al, Cour­ant, Mc­Shane, Radó, and Smi­ley were among those from out­side the Uni­versity of Chica­go who par­ti­cip­ated in some of these sem­inars. Bliss ex­pounded the sub­ject of mul­tiple in­teg­rals in mi­meo­graphed lec­ture notes [33], [34], where vari­ous im­prove­ments in the the­ory may be found. Some of his ideas were de­veloped in the doc­tor­al dis­ser­ta­tions of Sim­mons [e7], Cor­al [e16], Raab [e18], Cosby [e19], Nord­haus [e23] and Landers [e24]. One of the out­stand­ing prob­lems in this do­main is to find con­di­tions en­sur­ing the ex­ist­ence of a field suit­able for use in a suf­fi­ciency proof.

Dur­ing the 1920’s Bliss also lec­tured on bound­ary value prob­lems as­so­ci­ated with the cal­cu­lus of vari­ations, and on ap­plic­a­tions to quantum mech­an­ics and re­lativ­ity. Some of his work on bound­ary value prob­lems ap­peared in the pa­pers [29], [28], [32], [36]. For the sake of the greatest sym­metry and gen­er­al­ity it is con­veni­ent to study dif­fer­en­tial sys­tems of the form \begin{align} \label{one} & \frac{dy}{dt}=[A(x)+ \lambda B(x)]y,\\ & My(a) + Ny(b)=0. \notag \end{align} (Here cap­it­al let­ters are used to de­note square matrices of or­der \( n, y \) and \( z \) de­note matrices of one column, and the trans­pose of \( A \), for ex­ample, is de­noted by \( A^{\prime} \).) The ad­joint sys­tem to \eqref{one} is defined to be \begin{align*} & \frac{dz}{dt} = -z^{\prime} [A+\lambda B],\\ & z^{\prime}(a)P+z^{\prime}(b)Q=0, \end{align*} where \( MP - NQ = 0 \). The sys­tem \eqref{one} is said to be self-ad­joint if it is equi­val­ent to its ad­joint un­der a trans­form­a­tion \( z = T(x)y \) where the mat­rix \( T \) is nonsin­gu­lar on the in­ter­val \( [a, b] \). In the pa­per [36], a self-ad­joint sys­tem \eqref{one} was defined to be def­in­ite in case: (a) the mat­rix \( T^{\prime}B \) is sym­met­ric and pos­it­ive semi­def­in­ite, and (b) the sys­tem \eqref{one} has no non­trivi­al solu­tions for which \( B_y = 0 \). The bound­ary value prob­lems arising from prob­lems of Bolza sat­is­fy­ing cer­tain mild re­stric­tions are of this type. The defin­i­tion of “def­in­itely self-ad­joint” giv­en in the earli­er pa­per [28] ex­cluded many such prob­lems. However, in both pa­pers prop­er­ties of the sys­tem are de­rived which are like those which hold for the Fred­holm equa­tion with real sym­met­ric ker­nel. Stu­dent theses re­lated to the sub­ject of bound­ary value prob­lems in­clude those of Miss Stark [e8], Bam­forth [e10], Cope [e11], Hick­son [e15], Miss Jack­son [e12], Hu [e17], and Miss Wig­gin [e20]. The in­verse prob­lem of the cal­cu­lus of vari­ations drew some at­ten­tion from Bliss, al­though it was not in the main line of his in­terest. (See the pa­per [10]. Among his stu­dents Dav­is [e9], La Paz [e13], and Mo­scov­itch [e22] con­trib­uted to the dis­cus­sion of the prob­lem.

Cal­cu­lus of vari­ations the­ory re­quires the use of ex­ist­ence the­or­ems for im­pli­cit func­tions and dif­fer­en­tial equa­tions which yield more in­form­a­tion than those com­monly giv­en in the text­books and treat­ises on ana­lys­is. Hence Bliss was led to write a num­ber of pa­pers on the sub­ject [4], [5], [14], [19] and his Col­loqui­um lec­tures de­livered be­fore the Amer­ic­an Math­em­at­ic­al So­ci­ety in 1909 [15] were de­voted to these and re­lated top­ics. The ana­lys­is of sin­gu­lar points for trans­form­a­tions of the plane was one of the top­ics treated in the Col­loqui­um. This work grew out of spe­cial cases arising in the cal­cu­lus of vari­ations. The dis­ser­ta­tion of Lovitt [e2] treated some cases for trans­form­a­tions of three-space. The trans­form­a­tions of the plane con­sidered in the Col­loqui­um are real ana­lyt­ic ones, and real branches of plane ana­lyt­ic curves play a prom­in­ent role. This led to the two pa­pers on the factor­ing of power series [12], [13], and later to the study of al­geb­ra­ic curves [25], [26], [35]. At the date of these stud­ies the geo­met­ric proofs for the the­or­ems on the re­duc­tion of the sin­gu­lar­it­ies of al­geb­ra­ic curves were quite un­sat­is­fact­ory. In re­cent years mod­ern al­gebra has provided the means for a very ab­stract and gen­er­al treat­ment of the the­ory of al­geb­ra­ic curves. Bliss gave a clear treat­ment from the view­point of ana­lys­is of the case when the base field is the com­plex num­ber field. This phase of his work in­dic­ates that he was not al­ways in­ter­ested in the max­im­um ab­stract­ness, al­though he al­ways sought for sim­pli­city, clar­ity, and com­pre­hens­ive­ness in his math­em­at­ic­al ex­pos­i­tion.

Upon the ur­ging of Veblen in 1918 Bliss went to Ab­er­deen as a sci­entif­ic ex­pert in the Range Fir­ing Sec­tion. His work on the dif­fer­en­tial cor­rec­tions of tra­ject­or­ies [20], [21], [22] made pos­sible enorm­ous sav­ings in the time re­quired to com­pute the range cor­rec­tions which are ne­ces­sary in al­low­ing for the ro­ta­tion of the earth, ef­fects of wind, and vari­ations in the dens­ity of the air, powder charge, etc. His meth­ods con­tin­ued to be used dur­ing World War II. While the ad­vent of high speed com­put­ing ma­chines has led to some changes in meth­ods, the ba­sic the­or­et­ic­al ideas re­main the same. These ba­sic ideas were ex­pounded in two pa­pers in Trans. Amer. Math. Soc. [23], [24], and later in a book [3]. Bliss’ in­terest in func­tion­al ana­lys­is was not con­fined to this peri­od. In earli­er years the dis­ser­ta­tions of Fisc­her [e1], Lam­son [e3], Le Stour­geon, [e4] and Barnett [e5] were re­lated to this field.

Bliss was known the world around as one of the lead­ing au­thor­it­ies on the cal­cu­lus of vari­ations, al­though he did not con­trib­ute to the new dir­ec­tions of study opened up by Ton­elli and by Morse. He was primar­ily in­ter­ested in math­em­at­ic­al re­search, but he heeded also the call of oth­er du­ties. His broad in­terest in math­em­at­ics was evid­enced, for ex­ample, by his reg­u­lar at­tend­ance at the meet­ings of the Math­em­at­ic­al Club of the Uni­versity of Chica­go, where he con­trib­uted com­ments and ques­tions on a wide vari­ety of top­ics. He felt the duty of ex­pos­i­tion, and wrote the first of the series of Carus Mono­graphs, for which he also served as a mem­ber of the ed­it­or­i­al com­mit­tee. In math­em­at­ic­al pub­lic­a­tion he took a middle ground between those who pour forth un­di­ges­ted ideas and those who in­sist on ex­cess­ive pol­ish­ing. In 1925 he was the re­cip­i­ent of the first award of the Chauven­et Prize by the Math­em­at­ic­al As­so­ci­ation of Amer­ica for his pa­per Al­geb­ra­ic func­tions and their di­visors [27]. Bliss was an as­so­ci­ate ed­it­or of the An­nals of Math­em­at­ics from 1906 to 1908, and of the Trans­ac­tions of the Amer­ic­an Math­em­at­ic­al So­ci­ety from 1909 to 1916. He was chair­man of the Ed­it­or­i­al Com­mit­tee of the Uni­versity of Chica­go Sci­ence Series from 1929 un­til his re­tire­ment. From 1924 to 1936 he served on the Fel­low­ship Board of the Na­tion­al Re­search Coun­cil. He was a trust­ee of the Teach­ers In­sur­ance and An­nu­ity As­so­ci­ation for sev­er­al years. A wide vari­ety of or­gan­iz­a­tions called on him to make in­form­al ad­dresses, and as a res­ult he gave quite a num­ber of such talks on math­em­at­ics and re­lated top­ics.

He was elec­ted to the Na­tion­al Academy of Sci­ences in 1916, to the Amer­ic­an Philo­soph­ic­al So­ci­ety in 1926, and was made a Fel­low of the Amer­ic­an Academy of Arts and Sci­ences in 1935. Dur­ing his term (1921 and 1922) as pres­id­ent of the Amer­ic­an Math­em­at­ic­al So­ci­ety, he con­duc­ted a cam­paign for new mem­bers, in co­oper­a­tion with E. R. Hedrick, Chair­man of the Mem­ber­ship Com­mit­tee. As a res­ult the mem­ber­ship of the So­ci­ety in­creased from 770 to 1127 dur­ing this bi­en­ni­um. In 1930 he was Vice Pres­id­ent and Chair­man of Sec­tion A of the Amer­ic­an As­so­ci­ation for the Ad­vance­ment of Sci­ence. He was also a mem­ber of the Math­em­at­ic­al As­so­ci­ation of Amer­ica, the Illinois Academy of Sci­ence, the Lon­don Math­em­at­ic­al So­ci­ety, the Deutsche Math­em­at­ische Ver­ein, and the Cir­colo Matem­atico di Palermo. In 1935 the Uni­versity of Wis­con­sin awar­ded him the hon­or­ary de­gree of Doc­tor of Sci­ence.

Bliss played a cent­ral role in the plan­ning of Eck­hart Hall, which houses the De­part­ment of Math­em­at­ics at Chica­go, and he closely su­per­vised its erec­tion. It is very largely due to his care and in­sight that this build­ing fills so well the needs of the De­part­ment, and it will stand as a beau­ti­ful and en­dur­ing me­mori­al to his ef­forts. He ac­cu­mu­lated a rather sub­stan­tial per­son­al lib­rary of math­em­at­ic­al books and peri­od­ic­als, which he presen­ted to the De­part­ment of Math­em­at­ics in 1950 for the use of the staff. This “Gil­bert Ames Bliss Lib­rary” is now housed in the Fac­ulty Con­fer­ence Room in Eck­hart Hall. It in­cludes an un­usu­ally com­plete col­lec­tion of works on the cal­cu­lus of vari­ations.

Bliss be­lieved whole­heartedly in the im­port­ance of teach­ing as an ac­com­pani­ment of math­em­at­ic­al re­search. All those who ex­per­i­enced the stim­u­la­tion of his courses re­tain a high ad­mir­a­tion for him as very nearly the ideal sci­entif­ic teach­er. His judg­ment was highly re­garded, and his coun­sel was sought from many quar­ters, by oth­er uni­versit­ies as well as his own, and by oth­er or­gan­iz­a­tions be­sides the math­em­at­ic­al ones. He was con­ser­vat­ive in tend­ency, and was not one to rush in­to new do­mains of math­em­at­ics or in­to new pro­jects in war time. In his view fun­da­ment­al math­em­at­ic­al re­search and teach­ing were activ­it­ies too im­port­ant to be in­ter­rup­ted ex­cept for very strong reas­ons. At­tempts to re­strict stu­dent en­roll­ments by high ad­mis­sion re­quire­ments seemed to him un­wise, since some poorly trained stu­dents de­vel­op real in­tel­lec­tu­al power, and some bril­liant ones gradu­ally fade. With re­spect to the value of the Ph.D. train­ing, he stated:

The real pur­pose of gradu­ate work in math­em­at­ics, or in any oth­er sub­ject, is to train the stu­dent to re­cog­nize what men call the truth, and to give him what is usu­ally his first ex­per­i­ence in search­ing out the truth in some spe­cial field and re­cord­ing his im­pres­sions. Such a train­ing is in­valu­able for teach­ing, or busi­ness, or whatever activ­ity may claim the stu­dent’s fu­ture in­terest.

Gil­bert Bliss was mar­ried to Helen Hurd in 1912. Their chil­dren are Eliza­beth (Mrs. Rus­sell Wiles), born in 1914, and Ames, born in 1918. His wife was stricken and died in the in­flu­enza epi­dem­ic of Decem­ber, 1918. In 1920 he mar­ried Olive Hunter, who sur­vives him. The Blisses had a sum­mer home in Flossmoor, Illinois, for many years, and be­gin­ning in 1931 they made their year round res­id­ence in Flossmoor, where Gil­bert was at one time a mem­ber of the Vil­lage Board of Trust­ees and head of the Po­lice Com­mis­sion. They were ex­cep­tion­ally friendly in man­ner and in spir­it, with a high sense of hu­mor, and en­joyed en­ter­tain­ing stu­dents, fac­ulty col­leagues, and friends in their home. Golf at one of the nearby coun­try clubs was a fa­vor­ite re­cre­ation un­til de­clin­ing health forced its dis­con­tinu­ance. Bliss was in­ter­ested in com­pet­it­ive sports throughout his life, and in his earli­er years par­ti­cip­ated act­ively in bi­cycle ra­cing, ten­nis, and rac­quets.

His in­flu­ence on math­em­at­ics and math­em­aticians was wide­spread and deep, and his con­tri­bu­tions will be long re­membered.