For the first part of this obituary see this link

Baker was, as all researchers ought to be, always eager to find things out; he wanted to know. Many were the searching interrogatories to which his pupils were subjected. Let one, whereof the occasion is of no importance, serve as an instance. On 26/4/37 he wrote thus.

1. I want to know whether Room has any better method than Dixon's. I cannot find out at present. Grace gives J.L.M.S. 11 an a posteriori verification of the number of constants.

2. I want to know how Dixon's method fits into the usual Abelian thy (my Abelian Fctns. p. 390). It appears to be special for curves of order n - 1 and equivalent to saying that the general contact curve of order n - 1 is

3. I want to know how the contact curve of order n + 1 on p. 414 of Enriques-Campedelli (Alg. Surf.) fits into the general theory.

4. Now you have found the linear forms for x⁴ + y⁴ + z⁴ = 0. I should like to know what you get.

I have never properly read Cayley's paper on "Polyzomal curves". I believe you have?

But Baker did not rest content with mere discovery; he wanted an aesthetically satisfying explanation, and the choice between different explanations he would determine by this criterion. This was not a habit only of his later years, it was a lifelong characteristic. In the preface to A he had written:

An endeavour has been made to point out what are conceived to be the most artistic ways of formally developing the theory regarded as complete.

He would remark on the long process that began with the struggle of the early 19th century French Dynamiker with the axes of permanent rotation of a rigid body, on how these were then recognised as normals of a set of confocal quadrics, which normals, in their turn, were seen to constitute a tetrahedral complex. Each stage, he would insist, was an advance, but the advance was not complete until Corrado Segre announced that the lines of a tetrahedral complex were just the intersections, of the [3] in which they lay, with those planes that meet 3 given lines in a [4]. Another achievement of Segre's was to geometrise Kronecker's results on a pencil of singular quadratic forms. The difficulty of Kronecker's algebraic reduction is notorious but Segre, by equating a singular form to zero and interpreting the equation as a cone in [n], where n + 1 is the number of variables in the form, considers the locus of vertices (which need not be points merely, but spaces of larger dimension) of the cones. As Baker says in 10 :

To the geometrically minded, to be able to state a geometrical criterion for all the quadrics of the pencil to be singular, of a specified degree, will appeal as a consummation of Kronecker's remarkable work.

A figure might be widely known and have been described by eminent investigators, but yet still be awaiting the best explanation of its raison d'être. Just after Baker won his Fellowship there was established a birational correspondence between the Weddle surface W and the Kummer surface; it was a consequence of relations between double theta functions. But has it, in essence, anything to do with theta functions? W is the locus of points whereat all quadrics through 6 given points have a common tangent line. Baker, in 3, recalls that the quadrics through 5 points represent the space sections of a Segre cubic primal in [4]; those through 6 points therefore represent the sections through a given point P of the primal. The correspondence is an immediate consequence of the above definition of W and the fact that the apparent contour of the primal from P is a Kummer surface.

The striking elucidations by geometry of phenomena that sprang from other branches of mathematics, the sudden perception in a figure of some intrinsic incandescence, these had a profound effect on and a singular fascination for Baker: as though he were being led to recognise the verities of things sub specie aeternitatis. Herein may well have lain the chief reason for his turning to geometry. He so decided, not in the flush of youthful enthusiasm but in the ripening wisdom of mature years. His high appreciation of analysis remained: Riemann he almost worshipped, so impressed was he with the deep insight of his ideas; the work of Weierstrass and Poincaré he knew as few others could know; yet he chose geometry. Amplissima est et pulcherrima scientia figurarum. So he inscribed the title page of the first volume of Principles of Geometry, and so he believed.

The preceding sentences have indicated that a knowledge of geometry in higher space may be necessary for a proper appreciation of geometry in ordinary space, and Baker's main preoccupation in writing F was to publicise this fact; therein Segre's generation of the tetrahedral complex appears on p. 33 while the Kummer surface is found as an apparent contour on p. 156 and closely scrutinised in chapter VII. Cayley, as long ago as 1846, said, after remarking that Desargues' figure in a plane is a projection of the 10 edges and 10 vertices of a pentahedron, that it was only reasonable to expect, by analogy, a simplification of geometry in space by using figures in higher space. Klein, in 1872, explained how geometries in spaces of different dimensions could be equivalent; a geometry does not depend primarily on the ambient space but on the group of self-transformations of the figure. In Cambridge the situation was fully appreciated by H W Richmond, who had exploited it in papers written about 1900. Yet, at home, there seemed no other awareness and Baker would be pondering again the question of his exordium (7): why is it so often the case that the early history in England of a department of Pure Mathematics is a history of importation? He began to unload his cargo.

In 1920 appeared 9, of which Figure 2 is the well-known frontispiece of F; it supersedes 4 wherein the discussion, repeated at the opening of 9, began with 8 lines in [3] whose relation appears artificial because they are not there recorded as being the projection of a completely natural set of 8 lines in [4]. Though F was not to appear for another 5 years it was already being drafted; meanwhile, as harbingers thereof, C, D, E, were duly issued. Nor was it only Baker who so carefully prepared the ground: the lectures and the advocacy of F P White helped to sow in Cambridge the seed for the approaching harvest. Then the volume came: 1925. For all its idiosyncrasies and, one must declare, its anfractuosities of style it is a glorious book, a cornucopia crammed with riches. The cramming is prodigious. A reviewer wrote:

The book naturally suffers from the compression and is not one to he read in an arm-chair; indeed, any one page will furnish matter for several hours' cogitation by the ordinary mortal. But it is a fascinating study, and British mathematicians may well be proud of such a splendid mine of geometrical lore as is to be found in the four volumes of Principles of Geometry.

Not only is every word of this true, but every phrase is an instance of the Englishman's talent for understatement.

A book of such immense influence cannot be adequately summarised here, yet a fair fraction of this notice must be given to it. Perhaps the best course is to give a full account of a certain equivalence, signalised by Klein and exploited by Segre, between geometries in different spaces, and then indicate a few of the many uses to which Baker puts it.

It may help to describe a metrical example before the projective generalisation. Project a sphere S stereographically from its north pole N onto its equatorial plane h. All circles of S become circles in h, save that circles through N become lines. If, then, we regard lines in h as special cases of circles, any circle on either S or h corresponds to a circle on the other. All this is visually obvious when attention is fixed on real points. But points with complex numbers for their co-ordinates are not visible; there are generating lines lying on S, two through each point. Such a pair of generators has a real intersection (the only real point on either) and spans a real plane (the tangent plane of S at their intersection). The generators i and j at N span the tangent plane at N; this plane is parallel to h so that the intersections I, J of h with i, j are at "infinity"; moreover they are "conjugate imaginaries." Not only so: both i and j meet (at conjugate imaginary points) every real plane section of S, and indeed I and J are Poncelet's pair of points common to every circle in h. A line in h is amplified to a circle by adding the line IJ.

Plane geometry does not demand that the absolute points I, J be either imaginary or at infinity, as they are in the Euclidean plane. So any non-singular quadric S can be projected stereographically from any point N of S onto any plane n not through N; the generators i, j of S at N meet h in points I, J which can serve as absolute points in the plane geometry. Then all plane sections of S are projected into circles while every circle of h so arises from a section of S. Chapter II of F opens thus.

Hart's Theorem, for circles in a plane, or for section of a quadric.

Given three lines in a plane, there are four circles touching them; these circles, we know, are all touched by another circle, the nine-points circle (Feuerbach's theorem; see Vol. II). In other words, given three lines, we can add to them a circle such that the four, these lines and the circle, are all touched by four other circles.

In the present chapter we show how, given any three circles in a plane, we can add to them another circle, which we call the Hart circle, such that the four circles are all touched by four other circles (Hart, Quart. J. of Maths., IV (1861), p. 260).

The three original circles are in fact touched by eight other circles, as we shall prove. There are fourteen ways of choosing, from these eight, four circles which all touch another circle. In six of these ways, the four circles chosen have a common orthogonal circle; and the four circles consisting of the original circles, and their Hart circle, have also a common orthogonal circle.

We have shown that circles in a plane may be regarded as projections of plane sections of a quadric. We prove the results enunciated as theorems for such plane sections. This appears greatly to increase the interest and clearness of the matter.

That is the opening of Chapter II, but the technique has already been applied to much benefit in Chapter I. One must record, writing for an Edinburgh society of which Baker was glad to be an honorary member, his handling of Wallace's theorem (F p. 18). Wallace, who was appointed to the Edinburgh chair in 1819, is usually credited with having been the first to prove, about 1806, that the 4 circles which circumscribe the triangles formed by omitting, in turn, each side of a quadrilateral are concurrent. Baker shows this to be equivalent to a theorem about Möbius tetrahedra - two tetrahedra both inscribed and circumscribed to one another; this theorem in space involves only points, lines and planes. And many additions to and extensions of Wallace's theorem are established.

It is not possible, by linear transformation in a plane, to turn lines into circles; but it is possible by inversion. In inversive geometry the lines and circles form a closed family. Klein showed, in the Erlanger Programm of 1872, that inversive geometry in a plane is equivalent to projective geometry on a quadric; this is because, the lines and circles in 7) answering to the plane sections of S, it is precisely the plane sections of S that must form a closed family in a geometry equivalent to the inversive geometry in h so that S must, as a surface, be unaltered and its plane sections permuted among themselves. Inversion in a circle g of h transposes in pairs those points of h which do not lie on g while leaving every point of g invariant: g is the projection from N of a section G of S and the inversion answers to a projectivity in space that transposes in pairs those points of S that do not lie on G while leaving every point of G invariant. This is the projectivity that transposes every pair of points on S whose join passes through the pole O of the plane of G. Hence the projection of the curve of intersection of S with any cone whose vertex is O is its own inverse in g. When this cone is a plane the projection is a circle orthogonal to g. When it is a quadric cone the situation is fully explained (p. 96) in Chapter III of F, which opens thus.

In this chapter we obtain some properties of a plane curve by projection of a curve which lies in space of three dimensions. The plane curve is one which meets an arbitrary line in four points, and has two double points, or points where the curve crosses itself. The curve in space is the curve of intersection of two quadric surfaces. The matter is dealt with in more detail than is required by its difficulty, because the theory is a model for the subsequent theory of the Cyclide, a quartic surface in three dimensions, regarded as the projection of the intersection of two quadrics of fourfold space. (Chap. VI, below.)

For this equivalence between inversion in a plane and geometry on a quadric in space is only an instance, for n = 3, of the equivalence between inversion in [n - 1] and geometry on a quadric in [n]. Klein pointed this out and said that it would, for n = 4 elucidate the properties of cyclides and anallagmatic surfaces. The cyclides had not long to wait for the elucidation. In 1884 Segre published a 130-page paper which is one of the landmarks of descriptive geometry and gives one to understand why Baker spoke, in 10, of Segre's power of fashioning a new world from the bare suggestions of others. After an introduction wherein due acknowledgments are made Segre breaks new ground and, acting on Klein's hint, obtains 78 different types of cyclide as projections of the surface of intersection of two quadrics in [4]. This, with the self-inversions and the generations of a cyclide as an envelope of spheres, is the subject matter of Chapter VI, although Baker, in contrast to Segre's wholly descriptive and synthetic argument, gives a fair amount of algebra. He used to say, years after 1925, that the chapter was too analytical and perhaps, when he wrote it, he was of the same mind, in thinking that algebra would be a help "to readers not familiar with the matter", that he was when he wrote 5. Yet one cannot, whatever his frame of mind, see Baker forgoing the allusions to Maxwell on pp. 193 and 198.

The volumes G and H are quite distinct from C, D, E, F. Baker's intention had been to publish a large volume on the theory of algebraic curves and surfaces which would include a topological treatment of the subject in addition to what has now appeared in G and H; but this third part of the volume was not near enough to completion when Baker began to feel, as a man nearing 70 and tiring, that the compilation of the first two-thirds was perhaps as much as he could confidently hope to accomplish. When he retired from his professorship a dinner was given in his honour in St John's College on Saturday, 6th June 1936. On that day the Edinburgh Mathematical Society was meeting in St Andrews and their congratulatory telegram was put into Baker's hand as he entered college.

In 1941 Baker sent the manuscript of I to the press. He was glad to have done this book and set some store by its logical framework, claiming to start absolutely from scratch with no foundations of Euclid's results or of propositions from "sequels" to Euclid, doing the geometry of circles ab initio. He said modestly when it was finished that he did not think highly of many parts of it, but that it might serve as a model for someone some day to make a less imperfect book. It contains an abundance of examples, and a skilled draughtsman with sufficient leisure could fill a large portfolio with handsome diagrams.

When the war ended Baker was near 80, and the fell sergeant had made his preliminary foray. On 20/7/46 Baker wrote:

Did you ever hear that it was said of - that at a certain stage, smitten by conscience at his unanswered letters, he bought a basket to put them in? I tie mine up with string and they make a pile on the table. Yours has been at the top for a long time.

I have some excuse. I am training for a Valetudinarian, not having ever got back to the vigour I enjoyed before I was laid abed on 6th May 1944. I go short walks, most often in the garden, and, in those terms, am quite well. If you carry out your plan of coming to Cambridge this summer I shall be very glad to see you; but shall not be able to go right round the Farm. I have missed 2 harvests (with this one), to my great regret.

What had laid him abed on 6th May 1944 was pleurisy, and he never walked round the university farm again. The walk had been for years a constant solace: winter and summer, week-days and Sundays, wet or fine, he made the round and his inability to resume it was one of the deep sorrows that tinged his latter days. But, though he might have finished walking, he had not yet finished work. By the middle of September 1945 he had completed the manuscript of J; although he only then assembled it in its final form the subject was clearly in his mind when he wrote 6, and indeed he had been intrigued by it ever since those colloquies with Burkhardt in the distant days at Göttingen. The tract describes the geometry, in [4], of a group of 25920 linear transformations, and its first consequence was J A Todd's using the geometry to decompose the group into its conjugate classes. In the letter, already quoted, of 20/7/46 Baker said:

My short screed, in which I use the "synthemes", is in type, with the proofs revised, as a Cambridge-Tract. It will give me much pleasure when I am able to send you a copy. It has inspired Todd, who was kind enough to read the proofs, to write a very remarkable paper on groups - a great satisfaction to me.

Todd's paper has proved to be only the first in these last 10 years of at least 10 papers by various authors that would never have been written but for the tract. Moreover it, in its turn, provoked Baker's still insatiable curiosity; by June 1947 he was inquiring how to classify the transformations not as products of projections as Todd, following the tract, did, but by their sets of invariant points. He still wanted to know, and this piece of information he did eventually receive, but only in the closing months of his life when he could no longer see to read it. In 1947 however he was still reading: he read with close attention Hodge and Pedoe's Method of Algebraic Geometry, noting particularly the manner in which they introduced co-ordinates. And at the very end of 1948 he wrote (23/12/48)

I have spent some time of late in looking carefully through (B) Segre's recent "Modern Geometry, Vol. 1", which led me to read van der Waerden's "Algebra, Vol. 1", and the two led me to read about ideals in the old book, Dirichlet's edition of Dedekind's "Theory of Numbers", about a third of which is taken up with the Theory of Ideals.

Afterwards he took up the Hodge and Pedoe book again and, in consequence, at the age of 84, wrote 14 and communicated it to the Royal Society on 6th October 1950. This was his last paper, 15 being a brief pendant to it whereby the long procession of impressive works falls quietly to its close with a diagram depicting basic propositions of projective geometry in a plane. The last word lies, after all, with "the constructive methods of the old-established geometry." In minimis maxima.

By the end of 1950 his eyesight was but fitful, and he had to suffer the deprivation of not being able to read on duller days. The cloud of affliction darkened as the last years passed heavily by. In January 1953, he paid the bitter price sometimes exacted for longevity, being predeceased by his younger son. The following December Mrs Baker had to leave home. Pneumonia had hastened the decline of a memory that had begun to fail, and it was essential that she be moved to a nursing home. Baker, though practically confined to his study, was able to make one or two visits by taxi, but it soon became necessary to move her away from Cambridge. The partnership of 40 years was sundered and they never saw one another again. But while these inexorable events could enforce physical separation they could not remove her from his thoughts, and he wrote almost daily. That he felt unable to visit her was a sign of his own awareness of failing strength and, refuse as he might, and for as long as possible, to admit it, his growing realisation that she would not improve (Mrs Baker died on 17th December 1956.) was a potent influence of his own decline. Yet in April 1954, when a colloquium was held in Cambridge, several of his old pupils called to pay their homage and there was no trace in his conversation or demeanour of any resentment or distress. In January 1956 came an onset of phlebitis, while his eyesight by now was poor indeed and, though able to distinguish furniture and large objects, he was quite unable to read. He retained his equanimity throughout and his spirit was still unconquered and undaunted when the end, came peacefully on 17th March.

Baker visited Edinburgh in the first decade of the century, his contemporary F W Dyson being Astronomer Royal for Scotland. An appointment for lunch took him up Blackford Hill, then utterly innocent of suburban contiguity, on a cold winter day in deep snow. The recollection of the stark frigidity of his first climb up that eminence was counterpoised by the warmth of his attachment to it, presided over as it was, from the appointment of Dyson in 1905 till the death of Greaves in 1955, by a personal friend. R A Sampson, who succeeded Dyson in 1910, was 2 years junior to Baker at Cambridge and emulated his friend by winning a Smith's Prize and a St John's Fellowship; this was one of Baker's closest friendships and Sampson was Baker's best man in 1893. But Baker's ties to Scotland were also to other places than Blackford Hill - and other friends than Astronomers Royal. He was an honorary LL.D. of Edinburgh University; he stayed several times with the Whittakers in George Square and had known E T Whittaker and H W Turnbull since their Cambridge days. Turnbull's name is a reminder that Baker's thoughts, when they turned to Scotland, as like as not dwelt longest on the old grey city of St Andrews. He gave courses of lectures there at two colloquia, those of 1926 and 1930. The writer of this notice happened to pay his first call at Walcott in June 1926, and Baker was preparing to leave for St Andrews with Mrs Baker and their daughter in a few days. Twelve years after, on a walk round the Farm, he harked back to this first call and said, with emotion, "Never, never in my life have I enjoyed anything so much as my first visit to St Andrews." At that colloquium H F Baker and H W Richmond used to walk together along the Scores. Those remaining few who beheld that promenade speak about it still, and though the writer did not see it he is able to record it and thereby preserve it from the encroachment of oblivion. Let it be there that we look our last on him, the Cambridge professor savouring the most enjoyable of his too rare holidays, on the meridian of life and at the zenith of happiness, gazing along the West Sands and across the bright estuary to the outline of the Angus hills.

W L Edge

Baker wrote 10 books, all published by the Cambridge University Press. The dates of the editions are as follows :

A. Abelian Functions 1897

B. Multiply Periodic Functions 1907

Principles of Geometry

C. Foundations 1922, 1929

D. Plane Geometry 1922, 1930

E. Solid Geometry 1923

F. Higher Geometry 1925

G. Theory of Curves 1933

H. Algebraic Surfaces 1933

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I. Introduction to Plane Geometry 1943

J. A locus with 25920 linear self-transformations 1946

The following list of Baker's papers does not include any not referred to in this notice. The list given by Professor Hodge (Biographical Memoirs of Fellows of the Royal Society, Vol. 2) includes 97 papers: a fact that gives some indication of how far, short the above notice falls of an adequate assessment of Baker's work.

1. On the theory of functions of several complex variables.
Trans. Camb. Phil. Soc. 18 (1899), 408.

2. On functions of several variables.
Proc. London Math. Soc.,(2) 1 (1904), 14.

3. Elementary note on the Weddle quartic surface.
Proc. London Math. Soc. (2) 1 (1904), 247.

4. A geometrical proof of the theorem of a double-six of straight lines.
Proc. Royal Soc. (A) 84 (1911), 597.

5. Notes on the theory of the cubic surface.
Proc. London Math. Soc. (2) 9 (1910), 145.

6. On the curves which lie on a cubic surface.
Proc. London Math. Soc. (2) 11 (1912), 285.

7. On some recent advances in the theory of algebraic surfaces.
Proc. London Math. Soc. (2) 12 (1913), 1.

8. The place of pure mathematics.
Nature 92 (1914), 69.

9. On a proof of the theorem of a double-six of lines by projection from four dimensions.
Proc. Camb. Phil. Soc. 20 (1920), 133.

10. Corrado Segre.
Journal London Math. Soc. 1 (1926), 263.

11. On Poincaré's theorem for defective integrals on a Riemann surface.
Journal London Math. Soc. 10 (1935), 281.

12. Note introductory to the study of Klein's group of order 168.
Proc. Camb. Phil. Soc. 31 (1935), 468.

13. On the proof of a lemma enunciated by Severi.
Proc. Camb. Phil. Soc. 32 (1936), 253.

14. On non-commutative algebra, and the foundations of projective geometry.
Proc. Royal Soc. (A) 205 (1951), 178.

15. Note on the foundations of projective geometry.
Proc. Camb. Phil. Soc. 48 (1952), 363.

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