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Henry Frederick Baker was born at Cambridge on 3rd July 1866. He spent his whole life there, living to within 4 months of his 90th birthday.
Baker went into residence at St John's College in October 1884 and was, bracketed with 3 others, Senior Wrangler in 1887. Elected Fellow of St John's in 1888, he remained a Fellow without intermission for 68 years. He won a Smith's Prize in 1889 for an essay on the complete system of 148 concomitants of 3 ternary quadrics. Cayley would doubtless propose the subject; Cayley certainly gave advice on several points before the substance of the essay was published. It shows Baker as an expert manipulator of hyperdeterminants and Clebsch-Aronhold symbols, and as one already well read in the works of the German invariantists. It was thus only natural that J H Grace and Alfred Young should later have recourse to Baker when they began to plan their book on the Algebra of Invariants. Baker threw himself wholeheartedly, after his appointment as College Lecturer in 1890, into the work of lecturing and teaching, of coaching for the Tripos and of supervising research. He was soon propounding problems on double theta-functions, and one of his early students was H E Atkins of Leicester, among the Wranglers of 1893 and later to be British chess champion.
Baker twice went to Göttingen to consult with Felix Klein and there met, among other mathematicians, Hurwitz and Burkhardt. Many years later he would delight in recollecting his taking a walk by invitation with Klein and Gordan, and tell how Frau Klein sent him with her husband to search for coloured eggs under the laurels before breakfast on the morning of Easter Day. Baker had, before going into residence as an undergraduate, enjoyed a trip up the Rhine on a cargo boat, but these Göttingen visits were his last journeys abroad. He married in 1893; his wife, a talented violinist, was delicate and 10 years later he was left a widower with 2 young sons. And although freer to travel abroad again by 1928, when both the Royal Society and Cambridge University appointed him a delegate to the International Congress of Mathematicians at Bologna, he was disinclined to go.
The impetus to visit Göttingen came while he was amassing the material of A [letters and numerals in bold refer to the bibliography at the end], his huge volume on Abelian Functions. Not least among its merits is its use, following Cayley, of matrices to display the periods of abelian integrals and to subject them to linear transformations. Matrices were a tool used effectively by Baker on many occasions. He had attended lectures by the originator of matrices and enjoyed telling how he was, as junior member of an audience of three, obliged to sit opposite Cayley and so be constrained to contemplate the ramifications of the algebra upside down as it evolved under Cayley's hand. On Cayley's right sat J W L Glaisher; on Cayley's left A R Forsyth. One day Cayley enunciated the Cayley-Hamilton Theorem, but proved it for 2-rowed matrices only saying, just as he had written when first publishing the theorem in 1857, that he did not think it necessary to undertake the labour of a formal proof in the general case. Forsyth brought one the next day. And Baker not only heard Cayley talk about matrices; he studied what Frobenius had written about them.
In 1899 Cambridge celebrated Sir George Stokes' 50 years tenure of his professorial chair and Baker contributed 1, a paper on the theory of functions of several complex variables; he followed this in 1903 with 2. The theorem that a function of several variables that has no essential singularity at any finite point can be expressed as a quotient of two integral functions is notable, among other reasons, because an attempt that Weierstrass made to prove it was unsuccessful. Poincaré, however, proved the result for periodic functions by using an expression for the real part of an integral periodic function as a multiple integral. In 1 Baker showed that the imaginary part of the same function was also, expressible as a multiple integral, as therefore was the complete function, and, instigated by another paper of Poincaré's published in 1902, gave in 2, using his result of 1, a simpler method of obtaining Poincaré's solution of Weierstrass' problem. Baker was now an analyst of power. But in the same volume as 2 he published 3, wherein he finds a geometrical property of Weddle's quartic surface - the locus, W say, of vertices of quadric cones through 6 points. These points are nodes of W; the join of any point on W to a node only meets W in a single further point. Each node thus affords an operation of period 2 by which to pass from one point of W to another. Baker demonstrates that the 6 operations are commutative, and deduces that the points, other than those on certain curves, of W fall into closed sets of 32. 3 also contains a geometrical explanation of the known birational correspondence between W and a Kummer surface.
Baker's second book B appeared in 1907 and is more geometrical in tone than A. Some geometry in A there is, notably chapter VI; indeed the subject of abelian functions is, in Baker's view, the parent of all systematic algebraic geometry and appeals frequently to geometrical ideas. But in B one is very conscious of the Kummer and Weddle surfaces which, indeed, dominate chapters III and V. As for the book's title let its author speak for himself, 7 years after its publication. In 8 there occurs this passage.
Our ordinary integral calculus is well-nigh powerless when the result of integration is not expressible by algebraic or logarithmic functions. The attempt to extend the possibilities of integration to the case when the function to, be integrated involves the square root of a polynomial of the fourth order, led first, after many efforts, among which Legendre's devotion of forty years was part, to the theory of doubly-periodic functions. To-day this is much simpler than ordinary trigonometry, and, even apart from its applications, it is quite incredible that it should ever again pass from being among the treasures of civilised man. Then, at first in uncouth form, but now clothed with delicate beauty, came the theory of general algebraic integrals, of which the influence is spread far and wide; and with it all that is systematic in the theory of plane curves, and that is associated with the conception of a Riemann surface. After this came the theory of multiply-periodic functions of any number of variables, which, though still very far indeed from being complete, has, I have always felt, a majesty of conception which is unique.
One of the main features of the second part of B is a proof that the most general one-valued multiply-periodic meromorphic function is expressible in terms of theta functions. The proof leans heavily on work of Kronecker on how to define an algebraic construct by systems of equations; but it also uses defective integrals, and before considering Baker's later work as a geometer these should be described in some detail. For they are, as will be seen, close to geometrical concepts; they always occur when there is an algebraic correspondence other than a (1,1) correspondence between two curves, and a curve can often be precisely characterised by some geometrical property when defective integrals occur on it. Baker attached much importance to these integrals which, he contended, were not, in 1907, appreciated as they should have been.
There are, for any algebraic curve C of genus p, integrals that are finite everywhere on the corresponding Riemann surface R; these are linearly dependent on p among them, and this fact can serve to define p - the maximum number of linearly independent integrals. (Throughout the following the single word 'integral' is to be taken as meaning -- everywhere finite integral.') Now R can be subjected to a canonical dissection by 2p closed circuits or cycles
a1 , a2 , ... , ap ; b1 , b2 , ... , bp ;
no cycle meets any other save the one with the same suffix, while aj and bj have, for each j, a single intersection. R remains a connected surface after the 2p cuts are made, but 2p is the maximum number of closed cuts that can be made on R without it falling into separate pieces. This connectivity of R also serves to define p. It is easy to visualise the dissection, starting with p = 1 when R can be a torus, or anchor-ring, generated by the revolution of a circle round a line in its plane that does not meet it. The curves a and b that dissect the torus are, for a, one position of the generating circle and, for b, the path traced, through the revolution, by a point on a. For p > 1 take a sphere with p holes in it and attach, by a short tube, a torus to each hole; then dissect each torus by the above rule. There is no need for the sphere when p = 1 since it can then be shrunk on to the surface of the torus; but if p > 1 this shrinking is prevented by the other p - 1 holes.
The value of an integral round a cycle is a period, so that each integral has 2p periods. The periods of the p basic integrals can be displayed in an array of p rows and 2p columns, each row corresponding to an integral and each column to a cycle of the dissection; the left-hand square half of the array corresponds to the a-cuts, the right-hand half to the b-cuts. Now C may be such that it is possible to select r (< p) of the integrals which are linearly independent and such that there is a canonical dissection of R for which these r integrals all have zero periods except over
a1 , a2 , ... , ar ; b1 , b2 , ... , br ;
that is, the matrix has the shape
It then happens, by a theorem of Poincaré, that there is a complementary system of p - r linearly independent integrals which have zero periods except over
ar+1 , ar+2 , ... , ap ; br+1 , br+2 , ... , bp .
The period matrix thus has the form
It is important to emphasise that there is no linear dependence between the r integrals of one batch and the p - r of the other. The integrals are defective as having less than the, full quota of periods. An elliptic integral, with 2 periods, occurs when r = 1.
If p = 3 the presence of a defective integral causes either r or p - r to be 1; whenever a non-singular plane quartic Q possesses defective integrals one has to be elliptic. This happens when it is possible to choose the triangle of reference XYZ so that only even powers of one of the co-ordinates, say of x, occur in the equation of Q. The cubic curve which is the first polar of X then splits into YZ and a conic for which X and YZ are pole and polar, and Q has 4 bitangents, concurrent at X, whose contacts constitute the set of its intersections with the conic. A general plane quartic does not have any 4 of its bitangents concurrent.
When Q has an elliptic integral it also has, by Poincaré's theorem, a set of 2 integrals, linearly independent of the elliptic integral and of each other, with 4 periods. For some quartics these 2 integrals can be further broken down, being linearly dependent on 2 elliptic integrals; such a curve has all its integrals linear combinations of 3 elliptic integrals. The possession of defective integrals amplifies geometrical properties; the more defective integrals there are the more interesting is the curve likely to be, and the plane quartics with 3 elliptic integrals are interesting indeed. One of them has 21 elliptic integrals from which a linearly independent set of 3 can be chosen in 14 ways; this curve admits a group of 168 self-collineations and the 14 ways of choosing independent elliptic integrals answer to the 14 octahedral sub-groups. Baker evaluates a set of elliptic integrals for this curve and finds their periods (B p. 266; note also 12). Another curve of great interest in this regard admits a group of 96 self -collineations; its equation can be reduced, in one way only, to x4 + y4 + z4 = 0 and Baker gives (B p. 260) the diagram of a dissection of its 4-sheeted Riemann surface and, on p. 261, evaluates the 3 elliptic integrals. This book indeed has the merit of not confining itself to the general theory; many particular examples are provided to illustrate this theory and throw light upon its difficulties. On p. 272 there appears a curve with p = 9 which has 3 systems of defective integrals corresponding to the partition 2 + 2 + 5 of 9, and this simply in consequence of its being a plane section of a hyperelliptic surface. The book closes with a description of a curve H possessing 5 linearly independent elliptic integrals. H is the curve of contact of a Weddle surface W with the tangent cone from one of its nodes N and had in fact been encountered in a different guise by Georges Humbert in 1894 and its elliptic integrals recorded. Each node N' of W other than N is the vertex of a cubic cone whose generators each meet H in 2 points other than N', and the elliptic integral on a plane section of the cone is one of those on H; thus each integral is associated with a (2,1) correspondence between H and an elliptic curve. Baker gives these geometrical properties of H, as well as some of its projection on to a plane from N, with explicit forms for the elliptic integrals whether on H or on this projection.
Poincaré's theorem on complementary batches of defective integrals dates from 1886; there is a proof, amounting to little more than a piece of matrix algebra, on p. 240 of B. Later, in 1916, Rosati gave a geometrical proof, taking the 2p numbers of a row of the period matrix as homogeneous co-ordinates (A positive integer n enclosed in square brackets denotes a linear space of n dimensions) in [2p - 1]. This Baker took over and gave, with acknowledgment and manifest approbation, at the close of chapter 1 of H.
There is, however, some awkwardness at the end of Rosati's proof, it being essential to verify that there is no linear dependence among the two sets of defective integrals that have emerged or, in the geometrical setting of Rosati's work, that a certain two linear spaces do not meet. Matters might have rested there, but Baker took them up again in 11. This seems to have been prompted by the appearance of Zariski's book in 1935; much work had by then been done in America on Riemann matrices, of which the period matrix is a special instance, and A A Albert had published a purely algebraic proof of Poincaré's theorem (and of the analogous theorem for any Riemann matrix). It seems permissible, reading between the lines of 11, to suspect that Baker was displeased that the American mathematicians had not explicitly alluded to his proof; be that as it may, he is at pains to emphasise the reliance of his argument on matrix theory, and adds an explanation of how his proof of 1907 can be shaped so as to dispense with the a posteriori verification of linear independence at the close of Rosati's proof.
This linear independence of two batches of defective integrals came to have, for Baker, another aspect. He encountered, in the study of algebraic correspondences between two curves, precisely two such batches, on one of the curves, whose independence he strongly suspected but was only able to conjecture. The conjecture was, he showed in 13, equivalent to a lemma of Severi (1905) which was well-known both because of its implications in the theory of Picard integrals on a surface and because there was some hesitation, and not only on Baker's part, in accepting Severi's proof as valid. But just before Baker published 13 Hodge, who saw it in manuscript, succeeded in showing that Baker's conjecture was indeed true. There is an incident relating to this conjecture that is not generally known. In the summer of 1929 Baker was lecturing on these topics and put the conjecture to his class. A few days later he asked for our proof, and showed no little chagrin and disappointment that none of us had found one. Had he tried himself? And, if he had, did he guess that he would have to admit failure 7 years later? And, if he did guess so, would he still have felt entitled to, demand success of his class? Perhaps. He had watched T G Room writing his paper on the double-ten with the pride of a chief architect observing the specification of a new cathedral. He was about to communicate J G Semple's paper, on Cremona Transformations in [4], and H S M Coxeter's, on Polytopes with regular-prismatic vertex figures, to successive volumes of the Philosophical Transactions. These essays were hardly such as to cause him to minimise the capabilities of his pupils. Baker rejoiced in their achievements and they gave him ample cause; a Smith's Prize was won by one of them every year from 1927 to 1933, a run of 7 consecutive prizes: few Cambridge teachers can have had so rewarding an experience. Yet these were the years, immediately subsequent to 1926, that saw legions of researchers and a spate of papers on quantum theory. A new pupil, eager to learn geometry, was always warmly welcomed and Baker's satisfaction was deepened when geometry was chosen in conscious preference to the allurements of the new physical discoveries. But he warned newcomers that discoveries in geometry were unlikely to win recognition whereas "if you go and discover a comet you can write a letter to The Times about it." These same years too were the great years of the Baker tea-party: Saturdays at 4.15 during term: a gathering sui generis if ever any gathering was. Any attempt to describe it would be an attempt to communicate the incommunicable. There one listened to the exposition of the embryonic Smith's Prize essays. Most of us were young, but H W Richmond and F S Macaulay aided Baker to leaven the assembly with the senatorial dignity of years.
The momentum of Poincaré's theorem has thrown this biographical notice forward, and one must now revert to the years following 1907. Here it should be said that Baker edited the 4 volumes of Sylvester's papers, totalling more than 2,800 pages; a task spanning the years 1904-12.
In 1910 appeared 4 wherein the theorem of the double-six is proved, without using the cubic surface through the lines, by purely projective arguments that use only lines and quadrics. It is a paper on which its author set some value since, in his candidature for the Lowndean professorship in December 1913, he singles it out, from among the 43 papers he had by then published, as "evidence of the fact that I have cultivated the constructive methods of the old-established geometry." From 1910, too, dates 5, a paper of over 50 pages concerned with the cubic surface. Most of the treatment here is algebraic; Baker pleads that this is "of advantage to readers not familiar with the matter" and adds that a geometrical treatment is much more attractive, an avowal indicative of his growing predilection for geometry. Early in 1912 came 6, exploiting the fact that every algebraic curve on a cubic surface is expressible as a linear combination of 7 curves, namely one twisted cubic and the lines of one half of a double-six. This has, in its final form, an elementary aspect, but it would not have attained this form unless work of Burnside on groups of linear substitutions had recently become available, while the use of a base of 7 curves stems from Severi's work on algebraic surfaces - a topic that made 1912 a significant year for Baker. The International Congress of Mathematicians met at Cambridge; Baker was one of its organising committee of six and presided at the first meeting of the geometry section on 23rd August. The brief address he then gave is a terse summary of the topics which he speaks of in 7, his masterly Presidential Address, read on 12th December, to the London Mathematical Society. This was hailed by a German reviewer as diese anziehend geschriebenen Darstellung der geschichtlichen Entwicklung der neueren Theorie der algebraischen Oberflächen, and so say all of us. It is a splendid document, the ripeness of the subject for exposition combining most felicitously with the erudition and authority of the expositor. Neither the writing nor the reading of it but would he the better for the prelusion of 4 months earlier when the Italian geometers whose work he was extolling were sitting among his audience. At that congress Baker met, for the only time, Castelnuovo, Enriques and Severi.
In 1913 Baker, a widower for 10 years, married again and brought his bride to North Walsham where he owned a house. He now built the house in Storey's Way, then a private road with gates at either end, and during the building travelled to North Walsham for the week-ends in term. Mrs Baker had been brought up in Norfolk, of which both she and her husband were very fond, and they enjoyed long bicycle rides. They named the new house Walcott, after a hamlet near the Norfolk coast east of North Walsham; it is a house that has meant home to them and much to many people, some of whom will call to mind Belloc's apostrophe:
Stand thou for ever among human Houses,
House of the Resurrection, House of Birth;
House of the rooted hearts and long carouses,
Stand, and be famous over all the Earth.
But there were no carouses, long or short, at Walcott, and its fame is sure without them. For it is the house where so many pupils were received on so many occasions: where they paid their first tentative call to petition to commence research under the Master: where they received their formidable reading-lists: where they were encouraged in their difficulties and guardedly congratulated on their surmounting them: where they came, established mathematicians, from different continents, to pay their calls. And on the more social visits they were welcomed by a hostess whose staid graciousness will be long remembered.
Baker, while preparing to move into his new home, was appointed to the Lowndean Professorship of Astronomy and Geometry. His claims were very strong, for his publications were voluminous, his erudition profound, and he was now an established teacher of long experience. But some electors might insist on choosing a professor with a knowledge of practical astronomy, and it looks as if they emphatically did. Elections to professorships at Cambridge were presided over by the Vice-Chancellor, who issued the notice of the election on the day the electors met. But Baker's appointment did not conform to this routine - the notice was signed not by the Vice-Chancellor but by the Chancellor, and it was dated not 22nd December 1913, the day that the electors met, but 5th January, 1914. This could be explained by the Vice-Chancellor declining to use a casting vote. Baker's pupils could fittingly make a pilgrimage to Terling Place, Witham, where the Chancellor signed that notice for, in addition to making 5th January 1914 a red-letter day in Baker's life, the appointment has been of immense benefit to the study of geometry at Cambridge and elsewhere. To commemorate our benefactors is a pious duty incumbent on us all, and all who have studied under Baker are eternally indebted to Lord Rayleigh and those who helped to promote Baker to his chair.
These years of the Presidential Address, the second marriage and the appointment to the chair mark a watershed in Baker's life. War supervened, and thereafter Baker was set on the path that he henceforward travelled. His choice of geometry as the dominating intellectual passion of his life is very striking, for he was widely learned and could have been eminent in other branches. He was, after all, of the 19th century and his contemporaries were not prone to specialisation. He lectured on dynamical astronomy almost throughout his tenure of the chair and wrote papers on it as well as on hydrodynamics. In his obituary notice of Poincaré he is every bit as enthusiastic and as well-informed on Poincaré's writings on celestial mechanics as he is on those on automorphic functions and multiply-periodic functions. He proposed the Principle of Least Action for a Smith's Prize essay. Natural Philosophy and its spectacular achievements always captivated him; so close was his long friendship with C T R Wilson that he felt an almost proprietary interest in cloud chambers; he was a friend of P A M Dirac for nigh 30 years and proud that his friendship was reciprocated. But pure mathematics was supreme and, to quote from 8 again, he said publicly
Pure mathematics is not the rival, even less is it the handmaid, of other branches of science. Properly pursued, it is the essence and soul of them all. It is not for them; they are for it; and its results are for all time.
Baker did not leave analysis completely: far from it. The volume G stands to refute any such view. Chapter VII, though perhaps too compressed for so intricate a matter, is concerned with work of Kronecker and Dedekind on the relation between everywhere finite integrals and integral functions. This had been treated-long before in chapter IV of A, and Baker used to say in his old age that something had still to emerge from this arithmetic theory. But he did turn from analysis to geometry. It has been said, and Baker knew of its being said, and by a very eminent mathematician, that Baker thereby made an error of judgment. But Baker, while acknowledging the implied compliment to his early work, did not think so.
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