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Celebratio Mathematica

Gilbert Ames Bliss

Gilbert Ames Bliss
May 9, 1876 – May 8, 1951
A Biographical Memoir

by E. J. McShane

Today there is a thickly in­hab­ited part of the South Side of Chica­go where the word “Ken­wood” is still to be seen as the name of an av­en­ue and of a few shops. Eighty years ago this re­gion was the quiet, con­ser­vat­ive sub­urb Ken­wood of the rap­idly grow­ing city Chica­go. In this sub­urb Gil­bert Ames Bliss was born, on May 9, 1876. His fath­er had long been as­so­ci­ated with vari­ous elec­tric­al en­ter­prises, and shortly after 1880 be­came pres­id­ent of one of the early Chica­go Edis­on com­pan­ies. Thus through his child­hood the young Gil­bert Ames heard much about the new and re­volu­tion­ary in­ven­tions of the day, and soon ac­quired what was to be a lifelong in­terest in sci­entif­ic sub­jects.

An­oth­er abid­ing in­terest had its roots in those youth­ful days in Ken­wood. While Gil­bert Ames was still a school­boy his older broth­er had be­come prom­in­ent in bi­cycle ra­cing, which then was far more pop­u­lar and fash­ion­able in the United States than it now is; at one time he held two world’s re­cords in this sport. Nat­ur­ally the whole fam­ily took an in­terest in com­pet­it­ive sports, and Gil­bert Ames re­tained this in­terest all his life. In col­lege he was a mem­ber of the track team, com­pet­ing in the now van­ished event of bi­cycle ra­cing. Later he played ten­nis, and still later he was an en­thu­si­ast­ic golfer un­til age and ill­ness for­bade it.

In gram­mar school and high school he was rather pre­co­cious, without be­ing a prodigy. When he entered the Uni­versity of Chica­go in 1893, with the second en­ter­ing class, he had no strong choice of dir­ec­tion in his stud­ies, but with the study of cal­cu­lus his in­terest in math­em­at­ics was aroused, and in a rather queer way. The in­struct­or was giv­en to sever­ity and sar­casm, and daily brought his stu­dents to wrath or tears ac­cord­ing to sex and dis­pos­i­tion. A few of the stu­dents found that by hard study they could with­stand him, and thus Bliss came to learn cal­cu­lus well and to find it fas­cin­at­ing. Pre­sum­ably the in­struct­or’s con­tri­bu­tion to math­em­at­ics was pos­it­ive, since it is un­likely that a po­ten­tial equal of Bliss was among the oth­ers who fin­ished with an in­cur­able dis­like for math­em­at­ics. Moreover, his re­com­mend­a­tion helped Bliss to ob­tain a schol­ar­ship for the Seni­or Col­lege, which was im­port­ant be­cause his fath­er’s in­creas­ing age and un­cer­tain health, to­geth­er with the de­pres­sion of the nineties, had made the fin­an­cial situ­ation of the fam­ily some­what cramped.

The need of earn­ing a good part of his ex­penses in Seni­or Col­lege was a slight han­di­cap schol­astic­ally, but he ad­jus­ted to it and man­aged to do very well with his stud­ies and to join the track team and the glee club. This last activ­ity was re­lated to his way of earn­ing money; he was a mem­ber of a pro­fes­sion­al man­dolin quar­tet. It must have been a good one, for it was fin­an­cially prof­it­able and Bliss him­self felt that they played well.

Bliss’s first gradu­ate work (1897) was centered around math­em­at­ic­al as­tro­nomy, un­der the in­spir­ing guid­ance of F. R. Moulton. In this year he wrote his first pub­lished pa­per on “The mo­tion of a heav­enly body in a res­ist­ing me­di­um.” Nev­er­the­less, his ap­plic­a­tion for a fel­low­ship was un­suc­cess­ful. This is the sort of mis­take on the part of a fel­low­ship com­mit­tee that seems es­pe­cially ludicrous to those who have nev­er served on a fel­low­ship com­mit­tee. It did not stop Bliss, but it did make him re­con­sider his plans. The res­ult was a re­cog­ni­tion that it was math­em­at­ics that held most at­trac­tion for him, and a de­cision to study for a doc­tor­ate in math­em­at­ics. His later re­search in­dic­ates clearly that he al­ways re­tained a lik­ing for the kind of math­em­at­ics that at least con­ceiv­ably has ap­plic­a­tions.

The lead­ers in math­em­at­ics at the Uni­versity of Chica­go in those days the fam­ous three, E. H. Moore, O. Bolza, and H. Masch­ke. All three of them served as in­spir­ing teach­ers and as fine ex­amples of schol­ars. However, it was Bolza who most in­flu­enced Bliss, both dir­ectly by his lec­tures and in­dir­ectly by let­ting him make a copy of Bolza’s own re­cord of the fam­ous course of lec­tures on the cal­cu­lus of vari­ations giv­en by Wei­er­strass in 1879, and then avail­able only in manuscript form. It was un­der Bolza’s guid­ance that Bliss wrote his doc­tor­al dis­ser­ta­tion, on the geodesics on an an­chor ring.

After re­ceiv­ing his doc­tor­ate (1900), Bliss took up his first reg­u­lar teach­ing po­s­i­tion, at the Uni­versity of Min­nesota. He en­joyed the teach­ing and the stu­dents from the start; but his prin­cip­al sci­entif­ic activ­ity was a self-dis­cip­lined study of a large part of the avail­able lit­er­at­ure on the cal­cu­lus of vari­ations. He felt that he profited so much by this pro­gram of mas­tery of the sub­ject that he con­sist­ently re­com­men­ded it to stu­dents, lightly, as when one of my fel­low-stu­dents ad­mit­ted that he had read only one present­a­tion of a sub­ject, and Bliss ad­vised him to “read two books on it, then you’ll be an ex­pert”; or ser­i­ously, in private con­ver­sa­tion; but al­ways earn­estly.

In 1902 Bliss was able to go to Göttin­gen for a year of fur­ther study. This was in the time of Hil­bert, Klein, and Minkowski. But Göttin­gen offered oth­er ad­vant­ages be­sides the lec­tures and sem­inars of these lead­ers. There was the op­por­tun­ity to be­come ac­quain­ted with the five young math­em­aticians Ab­ra­ham, Carathéodory, Fe­jer, E. Schmidt, and Zer­melo. There was the ex­per­i­ence of speak­ing sev­er­al times in sem­inars and in the Ges­sell­schaft; on this last oc­ca­sion he hor­ri­fied the audi­ence by a slip in lan­guage, an­swer­ing Klein’s blunt ques­tion “Who would be in­ter­ested in this?” with “Wenn Sie et­was von Vari­ation­srech­nung ken­nten…” in­stead of “Wenn man et­was ….” There was the lib­rary, with re­cords of courses by Wei­er­strass, Hil­bert, Som­mer­feld, and Zer­melo. And, too, there was the op­por­tun­ity of com­par­ing the res­ults of Amer­ic­an and Ger­man teach­ing sys­tems.

For the next year he chose to ac­cept E. H. Moore’s of­fer of a one-year as­sist­ant­ship at the Uni­versity of Chica­go, for whose de­part­ment of math­em­at­ics he had a re­spect en­hanced by his year abroad. This was fol­lowed by an en­joy­able year at the Uni­versity of Mis­souri, from which he was at­trac­ted by an in­vit­a­tion to join the Prin­ceton staff. He was one of the “pre­cept­ors” (roughly the equi­val­ent of as­sist­ant pro­fess­ors) ad­ded in 1905 to carry out Woo­drow Wilson’s plan for the re­or­gan­iz­a­tion of edu­ca­tion at Prin­ceton, and so he was a spec­tat­or of the struggles between Wilson and his op­pon­ents that fi­nally forced Wilson to leave the fac­ulty and be­gin a new and (I am told he called it) a “less polit­ic­al” ca­reer.

In the sum­mer of 1908 Masch­ke died, and Bliss was in­vited to the Uni­versity of Chica­go as as­so­ci­ate pro­fess­or to re­place him. From 1908 un­til his re­tire­ment he re­mained on the fac­ulty of the Uni­versity of Chica­go. For two years he gave the ad­vanced courses in geo­metry which Masch­ke would have giv­en. But in 1910 Bolza resigned from the de­part­ment, Wil­czyn­ski was ad­ded to the staff, and Bliss was able to re­turn to his fa­vor­ite field, ana­lys­is in gen­er­al and the cal­cu­lus of vari­ations in par­tic­u­lar.

Prob­ably it was his earli­er ca­reer in sports that pro­duced the sur­prise in­vit­a­tion to ac­com­pany the uni­versity base­ball team to Ja­pan, where it had been in­vited to play against the teams of Waseda and Keio Uni­versit­ies. The Uni­versity gran­ted sev­en months leave, and he went with the team as fac­ulty rep­res­ent­at­ive, and re­turned by go­ing around the world. The trip seems to have had no sci­entif­ic con­nota­tions, but he re­tained a wealth of in­ter­est­ing memor­ies.

In June of 1912 Bliss was mar­ried to Helen Hurd, who also was of a Ken­wood fam­ily. They had two chil­dren, Eliza­beth (Mrs. Rus­sell Wiles), born in 1914, and Gil­bert Ames, Jr., born in 1918.

In 1913 Bliss was pro­moted to a full pro­fess­or­ship. When the United States entered the First World War, the De­part­ment of Math­em­at­ics un­der­took to teach nav­ig­a­tion to men about to enter the nav­al school in Chica­go. Bliss taught about a hun­dred of these. But in the sum­mer of 1918 Os­wald Veblen began press­ing him to come to Ab­er­deen Prov­ing Ground to join the math­em­aticians there in their ur­gent task of de­vis­ing math­em­at­ic­al meth­ods ad­equate for mod­ern ar­til­lery. He was re­luct­ant to do this, feel­ing that his work at the Uni­versity was his best con­tri­bu­tion to the coun­try. But even­tu­ally he was per­suaded to go to Ab­er­deen. It was a good de­cision, for here he made a con­tri­bu­tion of con­sid­er­able and last­ing im­port­ance, which will be de­scribed in a later para­graph. He was still busy writ­ing this for per­man­ent re­cord in Decem­ber, 1918, when he re­ceived a tele­gram that his wife had been stricken in the in­flu­enza epi­dem­ic. She sur­vived his re­turn only a few days.

Two years later he was mar­ried to Olive Hunter, also a nat­ive of Chica­go. They had al­most thirty-one years of a happy life to­geth­er. From 1931 on they made their home in Flossmoor; there are many who will vividly re­mem­ber the charm­ing home and the gra­cious and hos­pit­able host and host­ess.

In 1927 E. H. Moore was past the nor­mal age of re­tire­ment, and wished to resign as head of the de­part­ment. Bliss was ap­poin­ted chair­man, at first without pub­lic an­nounce­ment. From the end of the war un­til his re­tire­ment Moore had tried to hold down the size of the per­man­ent staff in the face of un­pre­ced­en­tedly large en­roll­ments. Bliss took over the chair­man­ship when the en­roll­ments had be­gun a slight de­cline which with the great de­pres­sion turned in­to a rap­id shrink­age. En­lar­ging the de­part­ment was out of the ques­tion; but be­cause Moore had held the size down, and be­cause Bliss was al­ways will­ing to ex­ert him­self to the ut­most for his staff, there were no dis­missals and no re­duc­tions in salary. One im­port­ant and con­spicu­ous me­mori­al to Bliss’s chair­man­ship is Eck­hart Hall at the Uni­versity of Chica­go. The Pres­id­ent of the Uni­versity, Max Ma­son, had ob­tained funds for the new build­ing from Bern­ard Al­bert Eck­hart, Ju­li­us Ros­en­wald, and the Rock­e­feller Found­a­tion. Bliss gave un­stint­ingly of his time and en­ergy in the plan­ning of the build­ing and its fur­nish­ings — I re­call a ses­sion of test­ing chairs for com­fort, and a math­em­at­ic­al con­fer­ence with Bliss that had to be aban­doned after a few minutes be­cause he was ex­hausted — but the res­ults jus­ti­fy the ex­pendit­ure. However, there is an­oth­er me­mori­al to his chair­man­ship in the large group of Chica­go stu­dents of math­em­at­ics spread around the coun­try. Ma­son re­peatedly ex­er­ted pres­sure on the de­part­ment to lim­it its stu­dent body by tight­en­ing re­quire­ments for ad­mis­sion. Bliss op­posed this as a mat­ter of prin­ciple. He felt that some­times bril­liant be­gin­ners fade away, while oth­ers de­vel­op real power after a weak start. Moreover, many who have not the ca­pa­city to take an ad­vanced de­gree be­come bet­ter teach­ers be­cause of study­ing ad­vanced courses. This does not at all mean that Bliss con­doned cheapen­ing of the high­er de­grees. He felt that the time to se­lect care­fully among stu­dents is at the time of ap­plic­a­tion for can­did­acy for a high­er de­gree: “The can­did­ates for high­er de­grees are the ones who take the time, not the listen­ers in lec­ture courses.”

Con­cern­ing the doc­tor­ate, I quote his own words: “It seems to me that there is wide-spread mis­un­der­stand­ing of the sig­ni­fic­ance of doc­tor’s de­grees in math­em­at­ics. The com­ment is of­ten made that the pur­pose of such a de­gree is to train stu­dents for re­search in math­em­at­ics, and that the suc­cess of the de­gree is doubt­ful be­cause most of those who ob­tain it do not af­ter­ward do math­em­at­ic­al re­search. My own feel­ing about our high­er de­grees is quite dif­fer­ent. The real pur­pose of gradu­ate work in math­em­at­ics, or 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.”

In 1933 Bliss was ap­poin­ted to the Mar­tin A. Ry­er­son Dis­tin­guished Ser­vice Pro­fess­or­ship, re­tain­ing the head­ship of the de­part­ment. In 1935 the Uni­versity of Wis­con­sin awar­ded him the hon­or­ary de­gree of Doc­tor of Sci­ence. In 1941, be­ing sixty-five years old, he re­tired, but did not aban­don math­em­at­ic­al activ­ity; his book Math­em­at­ics for Ex­ter­i­or Bal­list­ics was pub­lished dur­ing the war, in 1944, and his great Lec­tures on the Cal­cu­lus of Vari­ations in 1946. However, his health de­clined slowly dur­ing the next sev­er­al years, and rap­idly in 1951. He died in Flossmoor on May 8, 1951, one day be­fore his sev­enty-fifth birth­day.

Oc­ca­sion­ally one hears from a Ph.D. some bit­ter re­mark about the man un­der whom he wrote a dis­ser­ta­tion. I have nev­er heard such a re­mark about Bliss, and am con­fid­ent that none was ever made. He was by nature kind, and his cri­ti­cisms of stu­dents’ work nev­er stung. He en­joyed teas­ing his young­er col­leagues, al­ways with an air of in­no­cence, and nev­er ma­li­ciously. As a small ex­ample, shortly after he be­came head of the Po­lice Com­mis­sion of Flossmoor he sol­emnly handed sev­er­al guests a set of cata­logues of po­lice in­signia and had them search out the largest and most dec­or­at­ive po­lice star lis­ted, quite as ser­i­ously as though he really in­ten­ded to wear one.

Bliss had many math­em­at­ic­al activ­it­ies be­sides those with­in the de­part­ment. In 1916 he was elec­ted to the Na­tion­al Academy of Sci­ences. He also be­came a mem­ber of the Amer­ic­an Philo­soph­ic­al So­ci­ety (1926) and a fel­low of the Amer­ic­an Academy of Arts and Sci­ences (1935). He was pres­id­ent of the Amer­ic­an Math­em­at­ic­al So­ci­ety in 1921 and 1922, when its fin­an­cial situ­ation was dif­fi­cult. With E. R. Hedrick he de­voted a great deal of time and en­ergy to a cam­paign to in­crease mem­ber­ship, res­ult­ing in an in­crease of about fifty per­cent. 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. He was as­so­ci­ate ed­it­or of the An­nals of Math­em­at­ics from 1906 to 1908, and of the Trans­ac­tions of the Amer­ic­an Math­em­at­ic­al So­ci­ety from 1909 to 1916. For many years he was an ed­it­or of the “Carus Math­em­at­ic­al Mono­graphs,” and (after 1929) chair­man of the ed­it­or­i­al com­mit­tee of the “Uni­versity of Chica­go Sci­ence Series.” For twelve years (1924-1936) he served on the Fel­low­ship Board of the Na­tion­al Re­search Coun­cil. This he did con­scien­tiously, al­though he found no pleas­ure in rank­ing people ac­cord­ing to es­tim­ated abil­ity; this was against his nat­ur­al in­clin­a­tion to find what was good in each man and to en­cour­age it. Nev­er­the­less, a stat­ist­ic­al self-eval­u­ation of the Board in­dic­ated clearly that its work had been well done.

There were oth­er activ­it­ies, too. For some years he was a trust­ee of the Teach­ers In­sur­ance and An­nu­ity As­so­ci­ation. In Flossmoor he served as a mem­ber of the Vil­lage Board of Trust­ees and as head of the Po­lice Com­mis­sion. His abil­ity to speak clearly and in­ter­est­ingly caused him to be in­vited of­ten to speak in­form­ally on sci­entif­ic sub­jects.

Look­ing through his list of pub­lic­a­tions, one is struck by the way in which the cal­cu­lus of vari­ations serves as cen­ter of at­trac­tion. There are de­par­tures, but al­ways there is a re­turn. Moreover, a few de­tails of his way of think­ing are clear. He must have visu­al­ized clearly; a curve was a pic­ture in his mind, not a sys­tem of func­tions. He did not seek max­im­al gen­er­al­ity, but pre­ferred to ex­hib­it the prob­lem’s true cen­ter of in­terest with clar­ity. Thus he did not choose to use Le­besgue in­teg­rals in the cal­cu­lus of vari­ations, pre­sum­ably feel­ing that the real in­terest lay in the be­ha­vi­or of fam­il­ies of smooth curves, and that the ex­ten­sion of the the­ory, for ex­ample to all curves of fi­nite length, could be ad­ded af­ter­wards if one wished. He had no tend­ency to join any “Py­thagorean broth­er­hood”; he wrote care­fully so as to be eas­ily in­tel­li­gible to as many read­ers as pos­sible. It was most ap­pro­pri­ate that when the Chauven­et Prize for math­em­at­ic­al ex­pos­i­tion was first awar­ded by the Math­em­at­ic­al As­so­ci­ation of Amer­ica, in 1925, Bliss was the re­cip­i­ent. (The pa­per for which he re­ceived the award was “Al­geb­ra­ic Func­tions and their Di­visors,” lis­ted in the bib­li­o­graphy as [49].)

His doc­tor­al dis­ser­ta­tion [3] con­cerned geodesics on a tor­us. Points on a tor­us, or an­chor ring, can be loc­ated by means of func­tions which de­pend on two angles and are thus doubly peri­od­ic. Bliss found the spe­cif­ic for­mu­las for the geodesics in terms of el­lipt­ic func­tions. He thus could show that through every point there pass geodesics which cross the in­ner equat­or, all such geodesics be­ing free of pairs of con­jug­ate points; and through every point not on the in­ner equat­or there pass geodesics which do not cross that equat­or, and on such geodesics each point has con­jug­ate points. This pa­per is an ad­di­tion to the small col­lec­tion of in­ter­est­ing spe­cial prob­lems whose de­tailed dis­cus­sion is the ground-strat­um for gen­er­al­iz­a­tions.

Pa­pers [2] (writ­ten in Min­nesota) and [4] (writ­ten in Göttin­gen) can be thought of as a pair. In a plane, the shortest curve join­ing a fixed point A to a fixed curve C is a line-seg­ment which is per­pen­dic­u­lar to C at the point of in­ter­sec­tion. There is however an­oth­er con­di­tion; the cen­ter of curvature of C at B must not lie between A and B. If we strengthen the con­di­tion by de­mand­ing that A it­self is not the cen­ter of curvature, the con­di­tions are suf­fi­cient to guar­an­tee that the seg­ment has min­im­um length when com­pared with nearby curves from A to C. Bliss ex­ten­ded this to gen­er­al plane prob­lems in [2], and in [4] gave the first com­plete treat­ment of the case in which both end points are per­mit­ted to vary along fixed curves.

Two fun­da­ment­ally im­port­ant ele­ments of dif­fer­en­tial geo­metry are the ex­pres­sion for the length of a curve and the idea of geodes­ic. But the length is giv­en by an in­teg­ral of the kind stud­ied in the cal­cu­lus of vari­ations, and the ex­tremals of this in­teg­ral are the geodesics. Bliss, be­ing in­ter­ested in geo­metry and in the cal­cu­lus of vari­ations, wondered if the cal­cu­lus of vari­ations could also fur­nish gen­er­al­iz­a­tions of oth­er parts of dif­fer­en­tial geo­metry. He pub­lished three pa­pers [12], [14], [30] con­cerned with ex­tremals in two-di­men­sion­al space, and ob­tained par­tial gen­er­al­iz­a­tions of sev­er­al the­or­ems. These pa­pers now would be said to be on “two-di­men­sion­al Finsler geo­metry.” Finsler treated the n-di­men­sion­al case, but not un­til 1918. Al­though Finsler geo­metry is partly swal­lowed up by tensor ana­lys­is, it still re­tains a meas­ure of in­de­pend­ent ex­ist­ence and is still be­ing stud­ied.

Be­cause the cal­cu­lus of vari­ations deals with fam­il­ies of curves sat­is­fy­ing the Euler-Lag­range dif­fer­en­tial equa­tions and sat­is­fy­ing some sort of end-con­di­tion, it has use for ex­ist­ence the­or­ems of con­sid­er­able strength. Bliss found the need of these the­or­ems and of the­or­ems on im­pli­cit func­tions. He wrote sev­er­al pa­pers [5], [8], [22], [39] and gave a sys­tem­at­ic present­a­tion of res­ults in the Prin­ceton Col­loqui­um lec­tures [24]. It is char­ac­ter­ist­ic of Bliss’s work that the the­ory is de­veloped in all the gen­er­al­ity called for by the ap­plic­a­tions; that the gen­er­al­iz­a­tion in­to the realm of the Le­besgue meth­ods was es­chewed, pre­sum­ably be­cause the ap­plic­a­tions did not call for it; and that the meth­ods in­ven­ted and the style of writ­ing made the sub­ject eas­ily and pleas­antly ac­cess­ible.

The im­pli­cit func­tions the­or­em has, as one of its ap­plic­a­tions in the cal­cu­lus of vari­ations, the use of de­term­in­ing the para­met­ers which se­lect from a giv­en fam­ily of curves (ex­tremals) that par­tic­u­lar curve which passes through a giv­en point. Un­der some con­di­tions, for ex­ample, when all the ex­tremals are tan­gent to one fixed curve, the prob­lem re­quires solv­ing a sys­tem of equa­tions near a sin­gu­lar point. If the func­tions in­volved are (real) ana­lyt­ic, the “pre­par­a­tion the­or­em” of Wei­er­strass is an im­port­ant tool. Bliss ex­ten­ded this to the gen­er­al­ity needed, and gave an el­eg­ant demon­stra­tion [19], [23],

From the study of sin­gu­lar points of ana­lyt­ic trans­form­a­tions to the study of al­geb­ra­ic curves is a nat­ur­al step. Bliss, hav­ing made sig­ni­fic­ant con­tri­bu­tions to the one, be­came in­ter­ested in the oth­er [46], [47], [49], [66]. Since he liked to think in geo­met­ric im­ages, he of course stud­ied the curves from the geo­met­ric and ana­lyt­ic point of view. The more re­cent de­vel­op­ments have shown that al­geb­ra­ic meth­ods yield strong res­ults in this field. Nev­er­the­less, Bliss’s work on the geo­met­ric the­ory left that as­pect of the sub­ject in a much more com­plete and co­her­ent state than it had pre­vi­ously at­tained.

Meas­ured by util­it­ari­an stand­ards there can be no doubt that Bliss’s out­stand­ing con­tri­bu­tion was in bal­list­ics; and this con­tri­bu­tion is by no means trivi­al even from a pure math­em­at­ic­al point of view. A mod­ern fir­ing table has two es­sen­tial parts. Both refer to a giv­en com­bin­a­tion of gun, pro­jectile, and charge. One gives the el­ev­a­tion ne­ces­sary to at­tain a de­sired range un­der “nor­mal con­di­tions”; the oth­er gives the change in this el­ev­a­tion needed to cor­rect for the way in which con­di­tions at time of fir­ing dif­fer from stand­ard. These in­clude ef­fects of wind, of non-stand­ard dens­ity, of non-stand­ard pro­jectile-weight, and oth­ers. At the be­gin­ning of the First World War all the na­tions in­volved were com­put­ing tra­ject­or­ies by the meth­od of Si­acci, based on ap­prox­im­a­tions ad­equate for pro­jectile trav­el­ing in level, flat tra­ject­or­ies. But guns were be­ing used at long ranges and high el­ev­a­tions, and the Si­acci the­ory was no longer ad­equate. “Fudge-factors” were in­tro­duced to make ad hoc cor­rec­tions, but these were neither the­or­et­ic­ally sound nor use­fully ac­cur­ate. F. R. Moulton re­placed these out­grown devices by a meth­od of nu­mer­ic­al in­teg­ra­tion sim­il­ar to that used in com­put­ing or­bits, and cap­able of very high ac­cur­acy.

This took care of the prob­lem of the tra­ject­or­ies un­der “nor­mal con­di­tions” so well that it con­tin­ues to be used, un­changed ex­cept in de­tail, even with the elec­tron­ic com­puters of today. However, Moulton’s treat­ment of the cor­rec­tions was less sat­is­fact­ory both the­or­et­ic­ally and com­pu­ta­tion­ally. The change of range due to wind (or to non-stand­ard dens­ity, or tem­per­at­ure) is an ex­ample of a “func­tion­al”; it is a func­tion which is not de­term­ined by a single num­ber, the “in­de­pend­ent vari­able,” nor in­deed by any fi­nite set of num­bers, but is de­term­ined only when we know the en­tire course of an­oth­er func­tion, namely, the wind at each alti­tude. When the Si­acci meth­od was stand­ard, there was no ser­i­ous at­tempt to dis­cuss this dif­fi­cult de­pend­ence; some crude over-sim­pli­fic­a­tion was used, such as re­pla­cing the ac­tu­al vari­able wind by a con­stant, agree­ing with the ac­tu­al wind at two thirds of the max­im­um alti­tude reached by the pro­jectile. It was pos­sible to de­vise meth­ods ap­plic­able to the tra­ject­or­ies com­puted by Moulton’s nu­mer­ic­al in­teg­ra­tion pro­cess, but these meth­ods needed both math­em­at­ic­al jus­ti­fic­a­tion and com­pu­ta­tion­al sim­pli­fic­a­tion.

Bliss’s work in ana­lys­is had provided him with just the ap­pro­pri­ate back­ground for this prob­lem. He had been led to study func­tion­als, and pub­lished one pa­per [32] on the sub­ject, and had guided the writ­ing of three doc­tor­al dis­ser­ta­tions ( C. A. Fisc­her, 1912; Miss Le Stur­geon, 1917; I. A. Barnett, 1918) on the sub­ject. Thus he could con­trib­ute quickly to the work of the group, and soon provided both the math­em­at­ic­al found­a­tion and the com­pu­ta­tion­al pro­ced­ure.

To handle the math­em­at­ic­al found­a­tion, he de­vised a nu­mer­ic­al meas­ure, or norm, to spe­cify the “size” of a dis­turb­ance; and he proved that the ef­fect of a dis­turb­ance can be ap­prox­im­ated by a lin­ear es­tim­ate close enough so that for small dis­turb­ances, the er­ror is an ar­bit­rar­ily small per­cent of the norm of the dis­turb­ance, no mat­ter what the shape of the dis­turb­ance (e.g., the pat­tern of winds) might be [44], [43], [66]. This lin­ear es­tim­ate is the “dif­fer­en­tial ef­fect” of the dis­turb­ance, and is ex­actly what is used in ser­vice. It is not dif­fi­cult to find the sys­tem of dif­fer­en­tial equa­tions which the dif­fer­en­tial ef­fect must sat­is­fy; in fact, these were known be­fore Bliss gave the proof that the dif­fer­en­tial ef­fect really ex­ists.

The prac­tic­al side of the prob­lem was to de­vise a meth­od of hand­ling these equa­tions so that giv­en a vari­ety of dis­turb­ances, the range-ef­fect of each dis­turb­ance could be quickly com­puted. Bliss did this [40], [41], [42] by in­tro­du­cing an­oth­er sys­tem of dif­fer­en­tial equa­tions re­lated to the equa­tions for the dif­fer­en­tial ef­fect, and called the “ad­joint sys­tem.” For each un­dis­turbed tra­ject­ory a single solu­tion of the ad­joint sys­tem is found, de­term­ined by a cer­tain known set of val­ues at the end of the tra­ject­ory. This one solu­tion brings us al­most to the end of the prob­lem. For, giv­en any spe­cif­ic dis­turb­ance func­tion, we need only mul­tiply it in­to the solu­tion of the ad­joint sys­tem and per­form a nu­mer­ic­al quad­rat­ure (say, by Simpson’s rule) to ob­tain the range ef­fect of the dis­turb­ance. As com­pared with the best of pre­ced­ing meth­ods, this device saves about three quar­ters of the work. It con­tin­ued in use, with at most small amend­ments, through all of the Second World War. The ad­vent of high-speed com­put­ing ma­chines took away some of its pre-em­in­ence, but it is not likely that such a con­veni­ent meth­od will be per­man­ently shelved; in some modi­fic­a­tion, it will prob­ably be an aux­il­i­ary in any com­put­ing pro­gram in­volving ef­fects of small changes in the data.

After his war work, Bliss re­turned to ana­lys­is, centered again on the cal­cu­lus of vari­ations. He did not con­trib­ute to the new the­ory be­gun by Morse, nor to the dir­ect-meth­ods the­ory of Ton­elli (al­though he en­cour­aged at least one stu­dent to work in that field). The in­verse prob­lem re­ceived some at­ten­tion from him; in 1908 he had pub­lished a pa­per [17] on the sub­ject, and he dir­ec­ted three gradu­ate stu­dents (D. R. Dav­is, L. La Paz, N. A. Mo­scov­itch) in writ­ing dis­ser­ta­tions on this prob­lem. Like­wise, he gave some at­ten­tion to mul­tiple in­teg­ral prob­lems. Al­though he pub­lished only one pa­per on the sub­ject [86], he con­duc­ted sev­er­al courses and sem­inars on the sub­ject, and pub­lished mi­meo­graphed notes.

However, the cen­ter of his re­search was con­sist­ently the single-in­teg­ral prob­lem of the cal­cu­lus of vari­ations. Of the vari­ous forms of such prob­lems he chose the prob­lem of Bolza, since it most read­ily spe­cial­ized down to in­clude the oth­er forms. The res­ult of his work is his fine book Lec­tures on the Cal­cu­lus of Vari­ations [64], pub­lished in 1946, after his re­tire­ment. Be­sides his own work this con­tains the ad­vances made by oth­er work­ers in the field. But so much of this oth­er work was done by men who had stud­ied un­der Bliss and re­ceived their in­spir­a­tion from him that the book is to an un­usu­al de­gree a monu­ment to Bliss him­self.

If it were ne­ces­sary to pick out one of his con­tri­bu­tions as a spe­cial ex­hib­it, the choice al­most cer­tainly should be his treat­ment of the Jac­obi con­di­tion [35]. This had been handled, since the middle of the nine­teenth cen­tury, by an ana­lyt­ic device called the trans­form­a­tion of the second vari­ation; but for the simplest prob­lems this was cum­ber­some, and for the more com­plic­ated prob­lems it was hope­lessly un­wieldy. A geo­met­ric sub­sti­tute due to Kneser was el­eg­ant for the simplest prob­lems, but did not seem to ex­tend to more in­volved ones. Bliss re­marked that the second vari­ation of an in­teg­ral is it­self giv­en by an in­teg­ral of the same type as we star­ted with, and if it is nev­er neg­at­ive then the identic­ally-van­ish­ing vari­ation gives it its least value, 0. From this the de­sired res­ults fol­low quite read­ily. Moreover, the device is equally ap­plic­able to the more com­plic­ated prob­lems, such as the Lag­range and Bolza prob­lems. The prin­cip­al pur­pose of the trans­form­a­tion of the second vari­ation is at­tained without the trans­form­a­tion. But even the sec­ond­ary res­ults of the trans­form­a­tion are not lost in the pro­cess. For by use of the Bliss tech­nique the second vari­ation can be trans­formed in­to the form de­sired by earli­er in­vest­ig­at­ors, without any of the cum­ber­some ana­lyt­ic ma­chinery of pre­vi­ous meth­ods. As an in­dic­a­tion of how “right” Bliss’s meth­od is, less than four years after its pub­lic­a­tion it was re­ferred to, in a pa­per pub­lished by an­oth­er math­em­atician, as the “clas­sic­al” meth­od.

This was one of his many con­tri­bu­tions to what might be called a pro­gram of sav­ing the cal­cu­lus of vari­ations from death by ele­phant­ias­is. The life of a math­em­at­ic­al sci­ence comes from its in­tel­lec­tu­al at­tract­ive­ness. In the past it has happened that some branch of math­em­at­ics has be­come bulky by the pil­ing up of minu­ti­ae and the long-win­ded dis­cus­sion of in­tric­ate and of­ten un­in­ter­est­ing prob­lems by meth­ods stretched out bey­ond their do­main of ap­pro­pri­ate­ness. Such branches nat­ur­ally lose all ap­peal, and be­come senile un­less re­ju­ven­ated by new ideas and re-think­ing that suc­ceed in at­tain­ing the prin­cip­al res­ults (and new ones, too) more read­ily and more beau­ti­fully. In the early twen­ti­eth cen­tury the cal­cu­lus of vari­ations was in danger of los­ing its ap­peal be­cause of mount­ing com­plex­ity. How much Bliss con­trib­uted to its res­cue, as well as to its ad­vance­ment, can be seen by any­one who will com­pare the com­pact­ness and gen­er­al­ity of the the­ory in the Lec­tures on the Cal­cu­lus of Vari­ations with the older pa­pers on the same sub­ject. It is a worthy monu­ment.

Works

[1]G. Bliss: “The mo­tion of a heav­enly body in a res­ist­ing me­di­um,” Pop. As­tron. 6 (1898), pp. 20–​29.

[2]G. A. Bliss: “The second vari­ation of a def­in­ite in­teg­ral when one end-point is vari­able,” Trans. Amer. Math. Soc. 3 : 1 (1902), pp. 132–​141. MR 1500591 JFM 33.​0385.​01

[3]G. A. Bliss: “The geodes­ic lines on the an­chor ring,” Ann. of Math. (2) 4 : 1 (October 1902), pp. 1–​21. MR 1502291 JFM 33.​0670.​02

[4]G. A. Bliss: “Jac­obi’s cri­terion when both end-points are vari­able,” Math. Ann. 58 : 1–​2 (1903), pp. 70–​80. MR 1511229 JFM 34.​0402.​01

[5]G. A. Bliss: “An ex­ist­ence the­or­em for a dif­fer­en­tial equa­tion of the second or­der, with an ap­plic­a­tion to the Cal­cu­lus of Vari­ations,” Trans. Amer. Math. Soc. 5 : 2 (1904), pp. 113–​125. MR 1500665 JFM 35.​0340.​01

[6]G. A. Bliss: “Suf­fi­cient con­di­tion for a min­im­um with re­spect to one-sided vari­ations,” Trans. Amer. Math. Soc. 5 : 4 (1904), pp. 477–​492. MR 1500686 JFM 35.​0370.​02

[7]G. A. Bliss: “The ex­ter­i­or and in­teri­or of a plane curve,” Bull. Amer. Math. Soc. 10 : 8 (1904), pp. 398–​404. MR 1558133 JFM 35.​0506.​01

[8]G. A. Bliss: “The solu­tions of dif­fer­en­tial equa­tions of the first or­der as func­tions of their ini­tial val­ues,” Ann. of Math. (2) 6 : 2 (January 1905), pp. 49–​68. MR 1503546 JFM 36.​0389.​01

[9]G. A. Bliss and M. Ma­son: “A prob­lem of the Cal­cu­lus of Vari­ations in which the in­teg­rand is dis­con­tinu­ous,” Trans. Amer. Math. Soc. 7 : 2 (1906), pp. 325–​336. MR 1500752 JFM 37.​0402.​01

[10]G. A. Bliss: “A proof of the fun­da­ment­al the­or­em of ana­lys­is sit­us,” Bull. Amer. Math. Soc. 12 : 7 (1906), pp. 336–​341. MR 1558345 JFM 37.​0495.​02

[11]G. A. Bliss: “Book re­view: The the­ory of func­tions of real vari­ables, by James Pier­pont,” Bull. Amer. Math. Soc. 13 : 3 (1906), pp. 119–​130. MR 1558414

[12]G. A. Bliss: “A gen­er­al­iz­a­tion of the no­tion of angle,” Trans. Amer. Math. Soc. 7 : 2 (1906), pp. 184–​196. MR 1500741 JFM 37.​0490.​01

[13]G. A. Bliss: “The con­struc­tion of a field of ex­tremals about a giv­en point,” Bull. Amer. Math. Soc. 13 : 7 (1907), pp. 321–​324. MR 1558473 JFM 38.​0408.​02

[14]G. A. Bliss: “A new form of the simplest prob­lem of the Cal­cu­lus of Vari­ations,” Trans. Amer. Math. Soc. 8 : 3 (1907), pp. 405–​414. MR 1500795 JFM 38.​0408.​01

[15]G. A. Bliss: “A meth­od of de­riv­ing Euler’s equa­tion in the Cal­cu­lus of Vari­ations,” Amer. Math. Monthly 15 : 3 (March 1908), pp. 47–​54. MR 1516998 JFM 40.​1031.​08

[16]M. Ma­son and G. A. Bliss: “The prop­er­ties of curves in space which min­im­ize a def­in­ite in­teg­ral,” Trans. Amer. Math. Soc. 9 : 4 (1908), pp. 440–​466. MR 1500821 JFM 39.​0441.​01

[17]G. A. Bliss: “On the in­verse prob­lem of the Cal­cu­lus of Vari­ations,” Ann. of Math. (2) 9 : 3 (April 1908), pp. 127–​140. MR 1502361 JFM 39.​0445.​01

[18]G. A. Bliss and M. Ma­son: “Fields of ex­tremals in space,” Trans. Amer. Math. Soc. 11 : 3 (1910), pp. 325–​340. MR 1500866 JFM 41.​0437.​01

[19]G. A. Bliss: “A new proof of Wei­er­strass’s the­or­em con­cern­ing the fac­tor­iz­a­tion of a power series,” Bull. Amer. Math. Soc. 16 : 7 (1910), pp. 356–​359. MR 1558920 JFM 41.​0286.​03

[20]G. A. Bliss: “Book re­view: An in­tro­duc­tion to the study of in­teg­ral equa­tions, by Maxime Bôcher,” Bull. Amer. Math. Soc. 16 : 4 (1910), pp. 207–​213. MR 1558889

[21]G. A. Bliss: “The func­tion concept and the fun­da­ment­al no­tions of the Cal­cu­lus,” pp. 263–​304 in Mono­graphs on top­ics of mod­ern math­em­at­ics rel­ev­ant to the ele­ment­ary field. Edi­ted by J. W. A. Young. Long­mans Green and Co., 1911. Zbl 0067.​03101

[22]G. A. Bliss: “A new proof of the ex­ist­ence the­or­em for im­pli­cit func­tions,” Bull. Amer. Math. Soc. 18 : 4 (1912), pp. 175–​179. MR 1559180 JFM 43.​0483.​04

[23]G. A. Bliss: “A gen­er­al­iz­a­tion of Wei­er­strass’ pre­par­a­tion the­or­em for a power series in sev­er­al vari­ables,” Trans. Amer. Math. Soc. 13 : 2 (April 1912), pp. 133–​145. MR 1500910 JFM 43.​0504.​02

[24]G. A. Bliss: “Fun­da­ment­al ex­ist­ence the­or­ems,” pp. i–​ii,1–​107 in The Prin­ceton Col­loqui­um (Prin­ceton Uni­versity, 15–17 Septem­ber 1909), part 1. Amer­ic­an Math­em­at­ic­al So­ci­ety (New York), 1913. JFM 44.​0365.​01

[25]G. A. Bliss and F. B. Wiley: “A meth­od of sub­divid­ing the in­teri­or of a simply closed rec­ti­fi­able curve with an ap­plic­a­tion to Cauchy’s the­or­em,” Bull. Sci. Lab. Den­ison Univ. 17 (1914), pp. 375–​389.

[26]G. A. Bliss: “The Wei­er­strass \( E \)-func­tion for prob­lems of the Cal­cu­lus of Vari­ations in space,” Trans. Amer. Math. Soc. 15 : 4 (October 1914), pp. 369–​378. MR 1500985 JFM 45.​0604.​03

[27]G. A. Bliss and A. L. Un­der­hill: “The min­im­um of a def­in­ite in­teg­ral for uni­lat­er­al vari­ations in space,” Trans. Amer. Math. Soc. 15 : 3 (1914), pp. 291–​310. MR 1500981 JFM 45.​0605.​03

[28]G. A. Bliss: “A note on sym­met­ric matrices,” Ann. of Math. (2) 16 : 1–​4 (1914–1915), pp. 43–​44. MR 1502486 JFM 45.​0262.​05

[29]G. A. Bliss: “A sub­sti­tute for Duhamel’s the­or­em,” Ann. of Math. (2) 16 : 1–​4 (1914–1915), pp. 45–​49. MR 1502487 JFM 45.​0452.​02

[30]G. A. Bliss: “Gen­er­al­iz­a­tions of geodes­ic curvature and a the­or­em of Gauss con­cern­ing geodes­ic tri­angles,” Amer. J. Math. 37 : 1 (January 1915), pp. 1–​18. MR 1507894 JFM 45.​0859.​02

[31]G. A. Bliss: “Book re­view: Leçons sur les fonc­tions des lignes, by Vito Vol­terra,” Bull. Amer. Math. Soc. 21 : 7 (1915), pp. 345–​355. MR 1559651

[32]G. A. Bliss: “A note on func­tions of lines,” Proc. Natl. Acad. Sci. USA 1 : 3 (March 1915), pp. 173–​177. JFM 45.​0550.​01

[33]G. A. Bliss: “A note on the prob­lem of Lag­range in the Cal­cu­lus of Vari­ations,” Bull. Amer. Math. Soc. 22 : 5 (1916), pp. 220–​225. MR 1559765 JFM 45.​0605.​01

[34]G. A. Bliss: “Book re­view: Top­ics in the the­ory of func­tions of sev­er­al com­plex vari­ables, by Wil­li­am Fogg Os­good,” Bull. Amer. Math. Soc. 23 : 1 (1916), pp. 35–​44. MR 1559859

[35]G. A. Bliss: “Jac­obi’s con­di­tion for prob­lems of the Cal­cu­lus of Vari­ations in para­met­ric form,” Trans. Amer. Math. Soc. 17 : 2 (April 1916), pp. 195–​206. MR 1501037 JFM 46.​0758.​03

[36]G. A. Bliss: “A ne­ces­sary and suf­fi­cient con­di­tion for the ex­ist­ence of a Stieltjes in­teg­ral,” Proc. Natl. Acad. Sci. USA 3 : 11 (November 1917), pp. 633–​637. JFM 46.​1458.​03

[37]G. A. Bliss: “In­teg­rals of Le­besgue,” Bull. Amer. Math. Soc. 24 : 1 (1917), pp. 1–​47. MR 1560001 JFM 46.​0383.​01

[38]G. A. Bliss: “The prob­lem of May­er with vari­able end points,” Trans. Amer. Math. Soc. 19 : 3 (1918), pp. 305–​314. MR 1501104 JFM 46.​0758.​04

[39]G. A. Bliss: “Solu­tions of dif­fer­en­tial equa­tions as func­tions of the con­stants of in­teg­ra­tion,” Bull. Amer. Math. Soc. 25 : 1 (1918), pp. 15–​26. MR 1560139 JFM 47.​0940.​01

[40]G. A. Bliss: “A meth­od of com­put­ing dif­fer­en­tial cor­rec­tions for a tra­ject­ory,” J. U.S. Ar­til­lery 51 (1919), pp. 445–​449. Cor­rec­ted copy of earli­er print­ing in vol. 50 (pp. 455–460).

[41]G. A. Bliss: “The use of ad­joint sys­tems in the prob­lem of dif­fer­en­tial cor­rec­tions for a tra­ject­ory,” J. U.S. Ar­til­lery 51 (1919), pp. 296–​311.

[42]G. A. Bliss: Dif­fer­en­tial cor­rec­tions for anti-air­craft guns. Technical report, Ab­er­deen Prov­ing Ground, 1919.

[43]G. A. Bliss: “Func­tions of lines in bal­list­ics,” Trans. Amer. Math. Soc. 21 : 2 (1920), pp. 93–​106. MR 1501138 JFM 47.​0382.​02

[44]G. A. Bliss: “Dif­fer­en­tial equa­tions con­tain­ing ar­bit­rary func­tions,” Trans. Amer. Math. Soc. 21 : 2 (1920), pp. 79–​92. MR 1501137

[45]G. A. Bliss: “Some re­cent de­vel­op­ments in the Cal­cu­lus of Vari­ations,” Bull. Amer. Math. Soc. 26 : 8 (1920), pp. 343–​361. MR 1560316 JFM 47.​0471.​04

[46]G. A. Bliss: “Bira­tion­al trans­form­a­tions sim­pli­fy­ing sin­gu­lar­it­ies of al­geb­ra­ic curves,” Trans. Amer. Math. Soc. 24 : 4 (1922), pp. 274–​285. MR 1501226 JFM 50.​0266.​02

[47]G. A. Bliss: “The re­duc­tion of sin­gu­lar­it­ies of plane curves by bira­tion­al trans­form­a­tion,” Bull. Amer. Math. Soc. 29 : 4 (1923), pp. 161–​183. Pres­id­en­tial ad­dress de­livered be­fore the Amer­ic­an Math­em­at­ic­al So­ci­ety, Decem­ber 28, 1922. MR 1560693 JFM 49.​0264.​01

[48]G. A. Bliss: “Book re­view: Vor­le­sun­gen über Dif­fer­en­tial­geo­met­rie und geo­met­rische Grundla­gen von Ein­stein’s Re­lativ­itäts­the­or­ie, vol. 1: Ele­ment­are Dif­fer­en­tial­geo­met­rie, by W. Blasch­ke,” Bull. Amer. Math. Soc. 29 : 7 (1923), pp. 322–​325. MR 1560739

[49]G. A. Bliss: “Al­geb­ra­ic func­tions and their di­visors,” Ann. of Math. (2) 26 : 1–​2 (September–October 1924), pp. 95–​124. MR 1502680 JFM 50.​0699.​03

[50]G. A. Bliss: “A bound­ary value prob­lem in the Cal­cu­lus of Vari­ations,” Bull. Amer. Math. Soc. 32 : 4 (1926), pp. 317–​331. MR 1561219 JFM 52.​0509.​02

[51]G. A. Bliss: “A bound­ary value prob­lem for a sys­tem of or­din­ary lin­ear dif­fer­en­tial equa­tions of the first or­der,” Trans. Amer. Math. Soc. 28 : 4 (1926), pp. 561–​584. MR 1501366 JFM 52.​0453.​13

[52]G. A. Bliss: “Book re­view: Ele­menti del cal­colo delle variazioni, by G. Vivanti,” Bull. Amer. Math. Soc. 32 : 4 (1926), pp. 392–​393. MR 1561233

[53]G. A. Bliss: “Con­tri­bu­tions that have been made by pure sci­ence to the ad­vance­ment of en­gin­eer­ing and in­dustry. Math­em­at­ics.,” Sci. Mon. 24 (1927), pp. 308–​319. JFM 52.​0036.​11

[54]G. A. Bliss: “Book re­view: Cal­cu­lus of Vari­ations, by A. R. For­syth,” Bull. Amer. Math. Soc. 34 : 4 (1928), pp. 512–​514. MR 1561597

[55]G. Bliss: “The trans­form­a­tion of Cleb­sch in the Cal­cu­lus of Vari­ations,” pp. 589–​603 in Pro­ceed­ings of the In­ter­na­tion­al Math­em­at­ic­al Con­gress (Toronto, 11–16 Au­gust 1924), vol. 1. Edi­ted by J. C. Fields. Uni­versity of Toronto Press, 1928. JFM 54.​0532.​03

[56]G. A. Bliss: “An in­teg­ral in­equal­ity,” J. Lon­don Math. Soc. S1-5 : 1 (1930), pp. 40–​46. MR 1574997 JFM 56.​0434.​02

[57]G. A. Bliss: “The prob­lem of Lag­range in the Cal­cu­lus of Vari­ations,” Amer. J. Math. 52 : 4 (October 1930), pp. 673–​744. MR 1506783 JFM 56.​0435.​01

[58]Con­tri­bu­tions to the Cal­cu­lus of Vari­ations, 1930. Edi­ted by G. A. Bliss and L. M. Graves. Uni­versity of Chica­go Press, 1931. Zbl 0003.​40002

[59]G. A. Bliss and I. J. Schoen­berg: “On sep­ar­a­tion, com­par­is­on and os­cil­la­tion the­or­ems for self-ad­joint sys­tems of lin­ear second or­der dif­fer­en­tial equa­tions,” Amer. J. Math. 53 : 4 (October 1931), pp. 781–​800. MR 1506854 JFM 57.​0528.​01 Zbl 0003.​25702

[60]G. A. Bliss: “The Cal­cu­lus of Vari­ations and the quantum the­ory,” Bull. Amer. Math. Soc. 38 : 4 (1932), pp. 201–​224. MR 1562365 JFM 58.​0530.​01 Zbl 0004.​15601

[61]G. A. Bliss: “The prob­lem of Bolza in the Cal­cu­lus of Vari­ations,” Ann. of Math. (2) 33 : 2 (April 1932), pp. 261–​274. MR 1503050 JFM 58.​0535.​02 Zbl 0004.​15502

[62]G. A. Bliss and I. J. Schoen­berg: “On the de­riv­a­tion of ne­ces­sary con­di­tions for the prob­lem of Bolza,” Bull. Amer. Math. Soc. 38 : 12 (1932), pp. 858–​864. MR 1562533 JFM 58.​0535.​01 Zbl 0006.​25902

[63]Top­ics of the Cal­cu­lus of Vari­ations. Edi­ted by G. Bliss. Uni­versity of Chica­go, 1932. Prin­cip­ally a re­port on lec­tures by E. J. Mc­Shane and Richard Cour­ant.

[64]G. Bliss: Vari­ation­srech­nung. Edi­ted by F. Schwank. B. G. Teub­n­er (Leipzig and Ber­lin), 1932. JFM 58.​0529.​08 Zbl 0003.​34803

[65]G. A. Bliss: “Ern­est Ju­li­us Wil­czyn­ski,” Sci­ence 76 : 1971 (October 1932), pp. 316–​317. JFM 58.​0995.​07

[66]G. A. Bliss: Al­geb­ra­ic func­tions. AMS Col­loqui­um Pub­lic­a­tions 16. Amer­ic­an Math­em­at­ic­al So­ci­ety (New York), 1933. JFM 59.​0384.​03 Zbl 0008.​21004

[67]G. A. Bliss: The Cal­cu­lus of Vari­ations, mul­tiple in­teg­rals. Uni­versity of Chica­go, 1933. MR 2936825

[68]Con­tri­bu­tions to the Cal­cu­lus of Vari­ations, 1931–32. Edi­ted by G. A. Bliss and L. M. Graves. Uni­versity of Chica­go Press, 1933. Zbl 0006.​40401

[69]G. A. Bliss and M. R. Hestenes: “Suf­fi­cient con­di­tions for a prob­lem of May­er in the Cal­cu­lus of Vari­ations,” Trans. Amer. Math. Soc. 35 : 1 (1933), pp. 305–​326. MR 1501685 JFM 59.​0497.​02 Zbl 0006.​25903

[70]G. A. Bliss: “Eliakim Hast­ings Moore,” Bull. Amer. Math. Soc. 39 : 11 (1933), pp. 831–​838. MR 1562740 JFM 59.​0038.​02

[71]G. A. Bliss: “Eliakim Hast­ings Moore,” Univ. Rec. 19 (1933), pp. 130–​134.

[72]G. A. Bliss: “Math­em­at­ic­al in­ter­pret­a­tions of geo­met­ric­al and phys­ic­al phe­nom­ena,” Amer. Math. Monthly 40 : 8 (October 1933), pp. 472–​480. Re­prin­ted as Chapter 8 of A math­em­atician ex­plains by M. I. Logs­don (Uni­versity of Chica­go Press, 1936) pp. 146–158. MR 1522896 JFM 59.​0060.​01

[73]G. A. Bliss: “The sci­entif­ic work of Eliakim Hast­ings Moore,” Bull. Amer. Math. Soc. 40 : 7 (1934), pp. 501–​514. MR 1562892

[74]G. A. Bliss: The Cal­cu­lus of Vari­ations in three space. Uni­versity of Chica­go, 1934. Lec­ture notes, Spring 1934.

[75]G. A. Bliss: The prob­lem of Bolza in the Cal­cu­lus of Vari­ations. Uni­versity of Chica­go, 1935. Lec­ture notes, Winter 1935.

[76]G. A. Bliss: “Book re­view: ‘In­equal­it­ies’, by Hardy, Lit­tle­wood, and Pólya,” Sci­ence 81 (1935), pp. 565–​566.

[77]G. A. Bliss: “The evol­u­tion of prob­lems of the Cal­cu­lus of Vari­ations,” Amer. Math. Monthly 43 : 10 (December 1936), pp. 598–​609. MR 1523791 Zbl 0015.​35702

[78]G. A. Bliss and L. E. Dick­son: “Bio­graph­ic­al mem­oir of Eliakim Hast­ings Moore, 1862–1932,” Proc. Natl. Acad. Sci. USA 17 (1936), pp. 83–​102.

[79]Con­tri­bu­tions to the Cal­cu­lus of Vari­ations, 1933–37. Edi­ted by G. A. Bliss, L. M. Graves, and W. T. Re­id. Uni­versity of Chica­go Press, 1937.

[80]G. A. Bliss and L. E. Dick­son: “Her­bert Ell­s­worth Slaught,” Sci­ence 86 (1937), pp. 72–​73.

[81]G. A. Bliss: “Her­bert Ell­s­worth Slaught — In me­mori­am,” Bull. Amer. Math. Soc. 43 : 9 (1937), pp. 595–​597. MR 1563596

[82]G. A. Bliss: “Her­bert Ell­s­worth Slaught — teach­er and friend,” Amer. Math. Monthly 45 : 1 (January 1938), pp. 5–​10. MR 1524156 JFM 64.​0023.​14

[83]G. A. Bliss: “Nor­mal­ity and ab­nor­mal­ity in the Cal­cu­lus of Vari­ations,” Trans. Amer. Math. Soc. 43 : 3 (1938), pp. 365–​376. MR 1501950 JFM 64.​0510.​01 Zbl 0019.​12303

[84]G. A. Bliss: “Def­in­itely self-ad­joint bound­ary value prob­lems,” Trans. Amer. Math. Soc. 44 : 3 (1938), pp. 413–​428. MR 1501974 JFM 64.​0438.​02 Zbl 0020.​03204

[85]Con­tri­bu­tions to the Cal­cu­lus of Vari­ations, 1938–41. Edi­ted by G. A. Bliss, M. R. Hestenes, and W. T. Re­id. Uni­versity of Chica­go Press, 1942.

[86]G. A. Bliss: “The Cal­cu­lus of Vari­ations for mul­tiple in­teg­rals,” Amer. Math. Monthly 49 : 2 (February 1942), pp. 77–​89. MR 0006022 Zbl 0063.​00455

[87] G. A. Bliss: “Oskar Bolza,” Science 97 : 2509 (1943), pp. 108–​109. article

[88]G. A. Bliss: Math­em­at­ics for ex­ter­i­or bal­list­ics. John Wiley and Sons (New York), 1944. MR 0010480 Zbl 0063.​00457

[89]G. A. Bliss: “Os­kar Bolza — in me­mori­am,” Bull. Amer. Math. Soc. 50 : 7 (1944), pp. 478–​489. MR 1564626

[90]G. A. Bliss: Lec­tures on the cal­cu­lus of vari­ations. Uni­versity of Chica­go Press (Chica­go, IL), 1946. MR 0017881 Zbl 0063.​00459

[91]G. A. Bliss: Cal­cu­lus of Vari­ations, 6th edition. Carus Math­em­at­ic­al Mono­graphs 1. The Math­em­at­ic­al As­so­ci­ation of Amer­ica (Wash­ing­ton, DC), 1971. Zbl 0317.​49001