Celebratio Mathematica

This re­view might very prop­erly be­gin with the quo­ta­tion of the first para­graph of the au­thor’s pre­face which ex­plains the ori­gin and pur­pose of these mono­graphs. “This book is the first of a series of mono­graphs on math­em­at­ic­al sub­jects which are to be pub­lished un­der the aus­pices of the Math­em­at­ic­al As­so­ci­ation of Amer­ica and whose pub­lic­a­tion has been made pos­sible by a very gen­er­ous gift to the As­so­ci­ation by Mrs. Mary Hegel­er Carus as trust­ee of the Ed­ward C. Hegel­er Trust Fund. The pur­pose of the mono­graphs is to make the es­sen­tial fea­tures of vari­ous math­em­at­ic­al the­or­ies ac­cess­ible and at­tract­ive to as many per­sons as pos­sible who have an in­terest in math­em­at­ics but who may not be spe­cial­ists in the par­tic­u­lar the­ory presen­ted, a pur­pose which Mrs. Carus has very ap­pro­pri­ately de­scribed to be the ‘dif­fu­sion of math­em­at­ic­al and form­al thought as con­trib­ut­ory to ex­act know­ledge and clear think­ing not only for math­em­aticians and teach­ers of math­em­at­ics but also for oth­er sci­ent­ists and the pub­lic at large.’ ”

The prin­cip­al aim of this re­view is to give an ap­prais­al of this work as to the ex­tent to which it has ful­filled the pur­poses of these mono­graphs, leav­ing to oth­ers to give a de­tailed de­scrip­tion and cri­ti­cism of the sub­jects dis­cussed. I might say at the out­set that the pub­lic­a­tion com­mit­tee could have made no wiser choice for the sub­ject of the first mono­graph than that of the cal­cu­lus of vari­ations. The beau­ti­ful geo­met­ric­al and mech­an­ic­al prop­er­ties of curves and sur­faces which un­fold them­selves so nat­ur­ally in this sub­ject are sure to ar­rest the at­ten­tion of the read­er who is even only mildly in­ter­ested in math­em­at­ics. Coupled with this in­her­ent beauty of the sub­ject, we have here an ex­pos­i­tion by a man whose abil­ity both as a teach­er and schol­ar is of such out­stand­ing char­ac­ter as to in­sure the suc­cess of any math­em­at­ic­al work, let alone a sub­ject with which he has such an in­tim­ate ac­quaint­ance and in which he has such a deep in­terest. Those of us who have al­ways re­garded math­em­at­ics merely as a sci­ence find in these pages an artist­ic struc­ture and a beauty of tech­nique that lead us to think that math­em­at­ics in this coun­try is at last tak­ing on the form of an art also.

The book opens with an in­tro­duct­ory chapter giv­ing a state­ment and his­tor­ic­al sketch of some of the typ­ic­al prob­lems of the cal­cu­lus of vari­ations. This chapter is so clear and vivid that it must surely arouse the in­terest and curi­os­ity of the read­er. For those who have no ac­quaint­ance with this sub­ject it may not be amiss at this place to give a brief state­ment of one of the most in­ter­est­ing of these prob­lems, viz., the bra­chis­to­chrone prob­lem which was first pro­posed by John Bernoulli in 1696. For some reas­on a rivalry had sprung up between him and his older broth­er James and this prob­lem was pro­posed as a sort of a chal­lenge. It is re­quired to pick out of all the pos­sible paths join­ing two points in a ver­tic­al plane that one down which a particle will fall in the shortest time. It is clear that this prob­lem is one which is quite dis­tinct from the max­ima and min­ima prob­lems solved in the dif­fer­en­tial cal­cu­lus. In fact we have here to find not merely a value, or val­ues, of a vari­able which will give an as­signed func­tion the largest or smal­lest value, but a whole curve or func­tion which, when sub­sti­tuted in the in­teg­rand of a cer­tain def­in­ite in­teg­ral, will give the lat­ter a value smal­ler than would be ob­tained by the sub­sti­tu­tion of any oth­er func­tion. The broth­ers both solved the prob­lem in 1697; and they found much to their sur­prise that the curve down which a particle will fall in the least time is the same curve which will be gen­er­ated by a particle on the rim of a wheel as the wheel rolls along a straight line, and more sur­pris­ing still it is the same curve on which a particle start­ing at rest will fall to the low­est point in the same time no mat­ter at what point of the curve the particle is star­ted. In oth­er words, a pen­du­lum con­strained to move along a cyc­loid, as this curve is called, would have a peri­od in­de­pend­ent of the arc of swing, a prop­erty not en­joyed by the so-called simple pen­du­lum. We can feel with John Bernoulli when he says (see page 54), “With justice we ad­mire Huy­gens be­cause he first dis­covered that a heavy particle falls on a cyc­loid in the same time al­ways, no mat­ter what the start­ing point may be. But you will be pet­ri­fied with as­ton­ish­ment when I say. that ex­actly this same cyc­loid, the tau­to­chrone of Huy­gens, is the bra­chis­to­chrone which we are seek­ing.”

In the next three chapters this and two oth­er prob­lems are taken up in great de­tail with a view to ex­plain­ing and il­lus­trat­ing the mod­ern meth­ods of the cal­cu­lus of vari­ations. These mod­ern meth­ods of study­ing this sub­ject are here presen­ted for the first time in a form which it is pos­sible for the in­tel­li­gent lay­man with a thor­ough train­ing in ad­vanced cal­cu­lus to un­der­stand. I do not mean to im­ply that the read­er will al­ways find it easy to fol­low every state­ment at a first glance. In fact I doubt if any one but a spe­cial­ist could read everything in the book that way. But I am sure that it is pos­sible for a read­er with a good train­ing in ele­ment­ary math­em­at­ics to veri­fy every state­ment made in this book, provided he has suf­fi­cient pa­tience and con­cen­tra­tion. If we re­call that math­em­aticians had been work­ing on this sub­ject for over 300 years be­fore it was fi­nally put on a rig­or­ous found­a­tion by Wei­er­strass, we can see why a be­gin­ner should not ex­pect to un­der­stand the cal­cu­lus of vari­ations in all its as­pects at a first read­ing. In fact, I would sug­gest to the be­gin­ner to omit on a first read­ing all the dif­fi­cult proofs in the book and come back to them only after he has seen the un­der­ly­ing gen­er­al ideas of the sub­ject. This, by the way, is I think the best and most ef­fi­cient meth­od of read­ing all math­em­at­ics, even tech­nic­al art­icles. But, in spite of the in­her­ent dif­fi­culty of the sub­ject, Pro­fess­or Bliss has giv­en us an ex­pos­i­tion which I think will be con­sidered a mod­el for many years to come. Whenev­er he needs to use some res­ult from an ad­vanced por­tion of math­em­at­ics, as for ex­ample the the­ory of im­pli­cit func­tions, he either sub­sti­tutes for it a clear geo­met­ric proof (see for ex­ample the proof that a cyc­loid can al­ways be drawn through two points, page 56) or he gives a proof us­ing noth­ing more than what one should know after a thor­ough course in cal­cu­lus.

The book ends with a chapter on the gen­er­al the­ory and we have to mar­vel that any one can in the space of fifty pages give such a clear and com­plete treat­ment of what is known as the simplest prob­lem of the cal­cu­lus of vari­ations. Yet we ac­tu­ally find that the ne­ces­sary con­di­tions as well as the suf­fi­cient con­di­tions for a min­im­um or max­im­um are stated and proved with all the rig­or that one could de­sire. This chapter should be ex­cel­lent read­ing for one who has already had some ac­quaint­ance with the cal­cu­lus of vari­ations, and even the spe­cial­ist will find here a form which is not lack­ing in nov­elty. The chapter closes with an in­ter­est­ing his­tor­ic­al ac­count of the whole sub­ject. It may be seen from this brief de­scrip­tion that this is an in­ter­est­ing and use­ful book for a large group of read­ers whose math­em­at­ic­al equip­ment var­ies all the way from a course in the dif­fer­en­tial and in­teg­ral cal­cu­lus to one in the cal­cu­lus of vari­ations. In fact; I should think it would make an ex­cel­lent in­tro­duc­tion for a course in this sub­ject if not as a text, at least for col­lat­er­al read­ing.