Celebratio Mathematica

One of the im­port­ant con­cepts in­tro­duced in­to math­em­at­ic­al thought by E. H. Moore is that of the “Ex­ten­sion­al At­tain­ab­il­ity” of prop­er­ties (see New Haven Col­loqui­um Lec­tures, p. 53). This concept, apart from its tech­nic­al sig­ni­fic­ance in the the­ory of prop­er­ties of classes, ad­mits of an­oth­er in­ter­pret­a­tion of per­haps broad­er ap­plic­ab­il­ity, one sug­gest­ive of con­quest. So, an in­fant reach­ing out for its playthings might be said to be ex­per­i­ment­ing with the ex­ten­sion­al at­tain­ab­il­ity of sat­is­fac­tion for its de­sires; ex­plorers il­lus­trate the ex­ten­sion­al at­tain­ab­il­ity of man’s con­trol over the globe; oth­er in­stances will oc­cur to the read­er.

The Carus Math­em­at­ic­al Mono­graphs, of which Pro­fess­or Bliss’s book is the first, are in­ten­ded “to con­trib­ute to the dis­sem­in­a­tion of math­em­at­ic­al know­ledge by mak­ing ac­cess­ible at nom­in­al cost a series of ex­pos­it­ory present­a­tions of the best thoughts and keen­est re­searches in pure and ap­plied math­em­at­ics,” “in a man­ner com­pre­hens­ible not only to teach­ers and stu­dents spe­cial­iz­ing in math­em­at­ics, but also to sci­entif­ic work­ers in oth­er fields, and es­pe­cially to the wide circle of thought­ful people who, hav­ing a mod­er­ate ac­quaint­ance with ele­ment­ary math­em­at­ics, wish to ex­tend their know­ledge without pro­longed and crit­ic­al study of the math­em­at­ic­al journ­als and treat­ises.” Is this not an ex­hib­i­tion of faith in the ex­ten­sion­al at­tain­ab­il­ity of a math­em­at­ic­ally in­formed pub­lic? It cer­tainly is most fit­ting that this series of mono­graphs should have been con­ceived by a Chica­go group and that its first num­ber should come from the pen of one of the mem­bers of the De­part­ment of Math­em­at­ics at the Uni­versity of Chica­go.

To Mrs. Mary Hegel­er Carus and to her son Dr. Ed­ward Carus be­longs the hon­or of hav­ing re­cog­nized the im­port­ance of such an un­der­tak­ing and of hav­ing provided the ne­ces­sary means. Through the pub­lic­a­tion of these books, the Open Court Pub­lish­ing Com­pany con­tin­ues its fine ser­vice to math­em­at­ic­al edu­ca­tion in this coun­try.

To Pro­fess­or H. E. Slaught be­longs the cred­it for the in­cep­tion of the idea which gave rise to the mono­graphs and for hav­ing so­li­cit­ously guided it to suc­cess­ful real­iz­a­tion. This series of books will forever be a re­mind­er of his farsighted and in­tel­li­gent de­vo­tion to the cause of math­em­at­ic­al edu­ca­tion and to his skill in lead­ing it on in­to new and sig­ni­fic­ant fields of con­quest.

In how far the wider dis­sem­in­a­tion of know­ledge con­trib­utes to the en­large­ment of its do­main, not many would ven­ture to say. And, wheth­er math­em­at­ics is bet­ter served by mak­ing res­ults long fa­mil­i­ar to the spe­cial­ists ac­cess­ible to a wider group than by the pub­lic­a­tion of new res­ults, is, or should be, a rather fu­tile ques­tion. Neither of these tasks should be al­lowed to be neg­lected. We must con­stantly labor to en­large the found­a­tions if they are to sup­port the cease­lessly ex­pand­ing su­per­struc­ture, without, however, us­ing up so much ma­ter­i­al in the pro­cess that the su­per­struc­ture must suf­fer.

The simile is mani­festly in­ad­equate be­cause it does not al­low for the hu­man ele­ments in­volved. Dif­fu­sion of ideas among a lar­ger group is likely to gen­er­ate curi­os­ity, to lead to the dis­cov­ery of hitherto un­sus­pec­ted con­nec­tions, to sug­gest new ideas. There is no doubt that a much more healthy de­vel­op­ment may be ex­pec­ted if broad­er con­nect­ing av­en­ues are laid out between the fields of pure and ap­plied math­em­at­ics, if the en­gin­eers and the phys­i­cists be­came ac­quain­ted with some of the de­vel­op­ments and some of the res­ults that have been se­cured from ideas that per­haps were first brought to light in their fields of know­ledge. If the Carus mono­graphs con­trib­ute to a re­mov­al of the bar­ri­ers which keep mod­ern math­em­at­ic­al sci­ence isol­ated, they will fully jus­ti­fy the faith of the founders and the hopes of its pro­moters. May the series prove to be a mul­tiple series stretch­ing out its arms in many dir­ec­tions, so as to at­tain the widest pos­sible ex­ten­sion.

In ac­cord­ance with the gen­er­al aim set for the series, the volume now un­der re­view has the pur­pose of bring­ing the gen­er­al meth­ods of the cal­cu­lus of vari­ations with­in the reach of a lar­ger pub­lic than can be ex­pec­ted to mas­ter all the re­quire­ments which a sys­tem­at­ic study of the sub­ject would re­quire. And to the re­view­er it seems that the book suc­ceeds ad­mir­ably in this pur­pose. Wheth­er this judg­ment is cor­rect can only be es­tim­ated by the fu­ture his­tor­i­an.

The pro­cess is in­duct­ive. In an in­tro­duct­ory chapter of six­teen pages, the cal­cu­lus of vari­ations is presen­ted to the read­er in a semi­his­tor­ic­al way, chiefly by means of an ana­logy with the prob­lem of find­ing the max­ima and min­ima of func­tions, and through some of its il­lus­trat­ive prob­lems. Three of these, all be­long­ing to the “simplest prob­lem of the cal­cu­lus of vari­ations,” viz. the prob­lems of the shortest dis­tance, of the bra­chis­to­chrone, and of the sur­face of re­volu­tion of min­im­um area, form the sub­ject mat­ter of chapters II, III, and IV, re­spect­ively, a total of 111 pages. The fifth chapter treats the min­im­iz­ing of the in­teg­ral \( \int f (x, y,y^{\prime})\,dx \). Then fol­low a list of ref­er­ences, notes, and in­dex.

“The au­thor as­sumes,” so the warn­ing on the jack­et reads, “that the read­er has an ac­quaint­ance with the ele­ment­ary prin­ciples of the Dif­fer­en­tial and In­teg­ral Cal­cu­lus.” Even with this as­sump­tion as to the amount of pre­par­a­tion of his read­ers, the au­thor must have been in doubt many times as to wheth­er or not to pre­sup­pose know­ledge of a par­tic­u­lar fact. In some in­stances he ap­par­ently con­cluded that a re­mind­er was the only thing ne­ces­sary. As such we have to con­sider for in­stance the clause on page 21 con­cern­ing the de­riv­at­ive of an in­teg­ral with re­spect to its up­per lim­it, and the brief para­graph on the cyc­loid on page 52. Would not ref­er­ences to fuller treat­ment of such ques­tions be use­ful in these places?

Each of the prob­lems to which sep­ar­ate chapters are de­voted re­ceives a more com­plete treat­ment than they do when used as ex­amples fol­low­ing the de­vel­op­ment of the gen­er­al the­ory, a treat­ment moreover which util­izes to the full the more re­cent work done in con­nec­tion with the clas­sic­al the­ory. In pre­par­a­tion for the de­riv­a­tion of the Euler equa­tion in the shortest dis­tance prob­lem, the lemma, usu­ally at­trib­uted to Du Bois-Rey­mond, is proved in a very simple man­ner, al­though per­haps a little too clev­erly for the un­soph­ist­ic­ated read­er. The Hil­bert in­de­pend­ent in­teg­ral, the concept of a field and re­lated ideas and the­or­ems are in­tro­duced in each of chapters II, III, IV, in the spe­cial forms which they take for the prob­lems there treated. The usu­al meth­ods of the suf­fi­ciency proofs are ex­em­pli­fied in a sim­il­ar man­ner, as are also the cases in which one or two en­d­points are vari­able on ar­bit­rary curves. In the dis­cus­sion of these lat­ter ques­tions the au­thor is not sat­is­fied to treat merely first or­der con­di­tions, but gives a com­plete ac­count of the the­ory of the fo­cal point, which re­ceives in each of the spe­cial cases a more or less simple geo­met­ric­al in­ter­pret­a­tion. The proof of the ex­ist­ence of a unique ex­tremal through two giv­en points and the con­struc­tion of a field in the bra­chis­to­chrone prob­lem are car­ried out by draw­ing in a very skill­ful way upon well known prop­er­ties of the cyc­loid. In con­nec­tion with the same prob­lem we are in­tro­duced to the en­vel­ope the­or­em and to the geo­met­ric proof of Jac­obi’s the­or­em which de­pends upon it.

Par­tic­u­larly valu­able is the present­a­tion in the fourth chapter of the res­ults ob­tained by Sin­clair and MacNeish in the caten­ary prob­lem, not here­to­fore avail­able in a con­nec­ted form, and the clear dis­cus­sion of the re­la­tion between the caten­ary and the Gold­schmidt straight line solu­tions of the prob­lem of the sur­face of re­volu­tion of min­im­um area.

The fi­nal chapter, more than any of the oth­ers, is of the usu­al text­book char­ac­ter. The read­er who has fol­lowed the dis­cus­sions of the three spe­cial prob­lems in the pre­ced­ing chapters should be well pre­pared to un­der­stand now the gen­er­al the­ory as here presen­ted. He meets again, in the form of gen­er­al the­or­ems, state­ments with whose gen­er­al char­ac­ter he has had an op­por­tun­ity to be­come ac­quain­ted. But the chapter con­tains more than these gen­er­al­iz­a­tions of the res­ults pre­vi­ously ob­tained for spe­cial cases. It is here that we learn for the first time of the dis­tinc­tion between weak and strong min­ima, of the Wei­er­strass \( E \)-func­tion, of Le­gendre’s con­di­tion, of the Wei­er­strass–Erd­mann corner point con­di­tion, etc., which for vari­ous reas­ons have not been brought for­ward be­fore. It is fur­ther­more made clear that the geo­met­ric­al treat­ment of the con­jug­ate point con­di­tion is not suf­fi­cient in all cases. This leads to a dis­cus­sion of the second vari­ation by the el­eg­ant meth­od which the au­thor in­tro­duced some years ago and which has since been ap­plied by sev­er­al of his pu­pils to a vari­ety of more gen­er­al prob­lems, but a sys­tem­at­ic ex­pos­i­tion of which has not hitherto been avail­able out­side the journ­als. Moreover, the read­er who merely wants to get the res­ults will find an ac­cur­ate state­ment of sets of ne­ces­sary and of suf­fi­cient con­di­tions for the cases both of fixed and of vari­able en­d­points. The chapter closes in the same key in which the first one opened, viz., with his­tor­ic­al re­marks.

Enough has been said to make clear that in the judg­ment of the re­view­er, Pro­fess­or Bliss has car­ried through a dif­fi­cult task with re­mark­able suc­cess. We have here a new type of math­em­at­ic­al book. It is not a text­book, neither is it ad­dressed to the spe­cial­ist, ac­tu­al or in spe. It is in­ten­ded for that very in­def­in­ite group, the in­tel­li­gent gen­er­al pub­lic. This group surely in­cludes teach­ers of math­em­at­ics, even those who may have taken a course in the cal­cu­lus of vari­ations! Cer­tainly en­gin­eers who ex­pect to do more than routine work, phys­i­cists and chem­ists and oth­er work­ers in nat­ur­al sci­ence can profit largely from fa­mil­i­ar­iz­ing them­selves with the lead­ing ideas set forth in this book. To many it should be of im­me­di­ate use­ful­ness.

But to write on an ad­vanced math­em­at­ic­al top­ic for a pub­lic of such mixed and un­cer­tain qual­i­fic­a­tions is much like lec­tur­ing to a ra­dio audi­ence: one nev­er can tell wheth­er there is any real catch­ing on. In his se­lec­tion of ma­ter­i­al the au­thor had to trust to his judg­ment; the style of treat­ment is in a large meas­ure de­term­ined by this choice. Now one can read­ily con­ceive of oth­er se­lec­tions which might have been made. A treat­ment of an iso­peri­met­ric prob­lem, men­tion at least of some prob­lems lead­ing to in­teg­rals with more un­known func­tions, and to ex­trema of mul­tiple in­teg­rals would seem to me to be ne­ces­sary in or­der to give an ad­equate pic­ture of the broad scope of the cal­cu­lus of vari­ations. This re­mark however amounts to little else than to ask­ing for “more of the same kind”; for I should not want any of these top­ics to re­place any­thing that is in the book now. And this in­deed is my chief cri­ti­cism; the book is too short. It is a fine be­gin­ning; but it should be con­tin­ued.

This then is the re­mark with which I wish to close this re­view: Let the series of Carus Mono­graphs, for which I an­ti­cip­ate a long and suc­cess­ful ca­reer, de­vel­op a semi-peri­od­ic char­ac­ter which will make it pos­sible to carry for­ward to oth­er parts of the field the dis­cus­sion of the cal­cu­lus of vari­ations which Pro­fess­or Bliss’s volume has so ex­cel­lently ini­ti­ated.