by Andrew Brown
Adding to his unsettled mood that summer were two guests at Brookwatson – his American second cousin, Virginia Crawford, and her mother, Aunt Sarah. Although Bernal had originally thought Virginia rather plain, he grew to like her spirit and even allowed her into his private study, where she persuaded him to tidy his books. With Kevin and Gigi, they took trips in the horse-drawn milk wagon to visit local places of interest. One of these was Handley’s Mill, where they watched the process of producing tweed from wool. Desmond’s attention was caught by an emaciated young girl of twelve, whose job it was to load the raw wool into the carding machine: she had ‘a pallid face shining with grease out of which shone two deep eyes looking at us…with an indescribable expression that had misery in it and yet pride’. He was also moved by the contrast of the noisy, dusty, slavish conditions of the mill and the joyous freedom of the surrounding countryside. At the hand-looms, there were rows of bent-backed men, sitting on narrow benches, pedalling fiercely, and pulling threads back and forth with each hand, while ‘behind each man was a window that looked out on the gurgling river below and away over the sunbathed country to the blue mountains beyond’.56 He concluded that Ireland would be better without such cottage industries.
By the end of the summer, Desmond was spending every minute of the day with Virginia, causing his sister to become quite jealous. As Virginia was packing to leave, he finally confided his feelings to her and there was a histrionic build-up to their first kiss. After a period of sobbing, Virginia yielded to his request. The kiss was emunctory rather than erotic because he noticed that she had not blown her nose. Desmond did not feel emotionally involved – in fact, he imagined himself to be a character from a novel. He thought he should pledge his love to Virginia, and suggested that they should become engaged. She accepted, and they spent the next day kissing more passionately in his laboratory, where they were almost caught by Biddy, the housemaid, who instead found them ‘intent over a copy of Bell’s Solid Geometry’.57 Bernal thought that Biddy suspected something.
When he returned to Emmanuel College in October, he was advised by his college tutor to switch to Natural Sciences in the light of his disappointing showing in the Part I of the Mathematics tripos. The science curriculum – physics, chemistry, geology and mineralogy – was a far better match for his broad interests. He soon began to regain some of his self-confidence through verbal sparring with his contemporaries. His diary for 27th October was triumphal after a commanding display to Dickinson and other members of CUSS: ‘I surpassed myself in arriving at rapid and novel conclusions. I shall never be afraid of philosophers again, it takes them years to arrive at a truth that I take a minute to reach and it is just as liable to be quite wrong.’58 He wondered if his mind was capable of originality or ‘merely an undigested mass of Einstein and Freud with a top dressing of Wilde and Shaw’.
The originality of Bernal’s mind soon emerged, fertilized by the work of an Irish genius – not Wilde or Shaw, but Sir William Hamilton. Hamilton* was perhaps the most remarkable Irish prodigy of the nineteenth century: he knew at least a dozen languages including Greek, Latin, Hebrew, Persian and Hindustani, and at the age of eighteen was hailed by the Royal Astronomer of Ireland as the first mathematician of the age.59 His greatest contribution to mathematics was his investigation of complex numbers, culminating in his invention of Quaternions in 1843. Complex or imaginary numbers involve the concept of the square root of minus one, , usually represented by the symbol, i. Real numbers, such as the familiar integers 1, 2, 3, etc., can be located on a number line running from left to right, with negative numbers being to the left of the central point, 0, and positive numbers to the right. The product of multiplying any two real numbers that have the same sign (+ or –) is always positive (if we ignore zero). The square of any real number is therefore always positive and lies on the number line to the right of 0. The square root of a negative number cannot be expressed as a real number, and cannot be located on the real number line running from west to east on a page. Complex numbers can take the form a + bi, where a and b are real numbers and i is the imaginary unit defined as: i2 = –1. It had been accepted since the late seventeenth century that a complex number, such as z = a + bi, can be represented on a plane surface by locating its real part a along the west–east number line or axis, and the imaginary part b along an orthogonal north–south axis. The complex number will then appear as a point (a, b) in the coordinate plane.
Thus a two-dimensional plane can be represented by complex numbers; Hamilton’s ambition was to develop an algebraic system that would account for three-dimensional space, since this would have great utility in physics and mechanics. If one imagines the west–east number line as an axle for a cycle wheel, while there will be spokes running in the north–south line, there are many spokes lying out of the page, as it were, representing alternative planes all perpendicular to the original west–east axis. After fifteen years of trying to devise a method, Hamilton had a momentary flash of inspiration and realized that a four-dimensional algebra would provide the necessary tools for the job – the quaternion. A quaternion consists of an ordered set of four terms, a + bi + cj + dk, three of which involve imaginary parts defined by the relationships: i2 = j2 = k2 = ijk = –1.
The quaternion system expresses all lines in space by a distinct square root of minus one, and if the three symbols, i, k, and j represent mutually perpendicular square roots of minus one, they can serve as a mathematical device to rotate a line through ninety degrees. The fourth term in the quaternion allows the treatment of lines not mutually perpendicular. Hamilton extended his ideas in a book, Lectures on Quaternions, which was incomprehensible even to many mathematicians, but was notable for the first use of the term vector to describe quantities such as force, velocity or acceleration that have direction as well as magnitude. He established that vectors when multiplied together as vector products, are anti-commutative: that is to say, the product of vector a multiplied by vector b will have exactly the opposite direction to the product of vector b times vector a (written as a × b = –{b × a}).
As we have seen, as a result of Grace’s lectures Bernal was particularly imbued with the concepts of symmetry, and now he was studying crystals in geology and mineralogy naturally applied his mind to their three-dimensional symmetries. Geometers divide symmetry into two broad classes: point groups or space groups. Point-group symmetry depends on one unique point in a symmetrical figure that is unique and not repeated elsewhere in the pattern – think of the bulls-eye on a dartboard. Space groups consist of symmetrical patterns that lack a unique point; they are created by the repetition of a certain element in one, two or three dimensions. An example of a one-dimensional space group would be a row of identical columns or a single file of soldiers of the same height. The pattern results from translation or the shifting of the motif by regular intervals along a line. If the repetition of the pattern is then extended in two directions (e.g. multiple rows of soldiers standing at attention), a planar pattern of two-dimensional symmetry is obtained. Other common examples of planar patterns include tiles on a roof or patterns printed on fabric. Such planar patterns can be generated from the elemental motif of one soldier or one tile, by a variety of operations such as translation, rotation about an axis perpendicular to the plane or the use of a mirror to produce reflections. For any motif, there are only seventeen symmetry variations, or ways of manipulating it into different, two-dimensional, repeating patterns.60
When one comes to three-dimensional symmetry, the permutations are of course much more complicated. As mentioned in Chapter 1, the nineteenth-century geometers had recognized that there were thirty-two possible crystal shapes, and that the facets shown on the exterior of a crystal were an embodiment of the internal planes created by the regular arrangement of molecules within the crystal. The operations of translation, rotation and reflection still apply to three-dimensional space groups, and there are the additional complexities of glide planes and screw axes. All these operations are
applied to bring the crystal into congruence with itself: that is to say, if an observer leaves the room while a crystal undergoes one of these symmetry operations, the crystal would look exactly the same when he returns to the room. As a simple example, consider a six-sided, unsharpened pencil. If it is cut in half and a mirror held perpendicular to the cut end, the whole pencil will be restored by the mirror image. Now imagine the whole pencil being rotated around an axis passing down the centre of the lead. This is a six-fold axis of rotation because the pencil comes into congruence with itself every sixty degrees of turn. In crystals, there are two-, three-, four- and six-fold axes of symmetry,61 but not five-fold axes so commonly seen in the petal arrangements of blossoms, as well as in many starfish.
The basic building block or motif that is repeated to build up the three-dimensional structure of a crystal is called the unit cell. That there are 230 space groups, or different symmetric arrangements of unit cells, in crystal lattices was first shown by a German mathematician, Artur Schoenflies, and a Russian crystallographer, E.S. Federov, in 1891. Employing different methods, they reached this final total after several years of independent work.62 It seems that Bernal became seriously interested in this complex subject in the spring of 1921. His diary entry for 4th April contains the following statement: ‘An idea has blossomed in my mind on the connection between the atomic structure of crystals and their elastic and thermal constants. I attempted to attack the problem mathematically with a certain success.’ The idea, which he developed over the next few months, was to treat the unit cell of a crystal lattice as three vectors, a, b, c that intersect at a common point or origin, and then to develop the symmetrical possibilities using Hamiltonian quaternions. This was a completely original approach to the daunting set of 230 space groups that had previously been solved only in a qualitative geometrical way, which allowed one crystal structure to be distinguished from another. Bernal’s aim was to develop a quantitative theory so that every crystal structure could be represented by a formula and the analysis of that structure could be reduced to solving a number of equations.
Beyond the mediaeval walls of Cambridge, a springtide of socialism seemed about to engulf the British Isles in April 1921. The post-war economic boom was over and unemployment began to rise rapidly. Coal miners were on strike after refusing to accept wage cuts, and were trying to persuade the transport workers to join them in a Triple Alliance. Bernal followed events by reading the London newspapers everyday at Brookwatson, and felt that a negotiated settlement ‘seems impossible. The crash must come. Yet like butterflies we sport in the sunshine.’63 He was certainly enjoying the rustic pleasures of Ireland, spending idyllic afternoons in High Wood – ‘a wood of dreams. Green moss with anenomies and hyacinths and primroses, big ivy grown trees and saplings circled with honeysuckle’ – when not engrossed in his work. On 9th April the newspapers announced that the Triple Alliance would strike and army reserves were called in. Bernal thought that civil conflict on the mainland was almost inevitable, and in turn might force the withdrawal of auxiliary troops from Ireland. He remained worried though by ‘the Government’s deep laid plans [that] almost make me despair of victory and that means another century of slavery’. Not that he was intending to join the Volunteers or IRA, but would continue to watch and wait. Two days later, the papers reported negotiations and compromises that angered the armchair socialist: ‘it’s all going to fizzle out, bluff on both sides. I am furious.’64 The next day he went to a dance in Nenagh, where there were ‘tales of raids by the auxiliaries and atrocities of the “Shinnies”’. Surrounded by mostly Protestant young farmers, Bernal held his tongue ‘out of prudence’.
As the level of violence in Ireland escalated, it appears that Desmond also decided to restrain his political comments at home. He had become accustomed to quarrelling with Cuddie, which he did not enjoy, but recent criticisms from his mother and his sister, Gigi, had stung him. His mother, annoyed that he would spend all day reading or playing instead of helping around the farm, accused him of selfishness. He increased her annoyance by replying that his selfishness was a deliberate policy. Privately he did express some remorse, noting in his diary that selfishness seemed to be his one constant trait and that he needed to look over his morals. He decided to write to his cousin Virginia and break off their engagement. Gigi, a rebellious and breezy character, thought him conceited. She had little time for his personal introspections and when he showed her his diary, she ‘rocked with laughter’ over the passages that he had taken the most trouble over. Desmond conceded that ‘I am a joke especially on paper, but if I don’t take myself seriously, who will?’65 Desmond and Gigi never remained at odds for long, and he also enjoyed an affectionate relationship with Gofty, their youngest brother, who was now at school in Cambridge. Desmond would spend Sundays entertaining Gofty in his rooms or in a punt on the river.
Bernal now decided that physics was going to be his subject because ‘my mind delights in taking readings. Chemistry is too full of smells and Geology of names.’66 He would soon be faced with examinations again, this time in Natural Sciences. First came the geology examination in which he did ‘rather badly’ and then a physics paper he spoiled, but at last a triumph: ‘A beautiful chemistry paper, I walked over it, trampled over it and jumped on it.’67 He remained in Cambridge during the summer of 1921 and made the acquaintance of the crystallographer, Arthur Hutchinson, whom he described as ‘a very gentlemanly don at Pemmer [Pembroke College]’. Hutchinson encouraged him to continue with four subjects on the basis of his examination results, to Bernal’s delight. He continued to read extensively and in July came across a paper by W.L. Bragg on the arrangement of atoms in crystals. Bragg had just become Professor of Physics at Manchester University; he had shared the 1915 Nobel Prize for Physics with his father, W.H. Bragg, for the development of X-ray crystallography. In this paper,68 Bragg presented some empirical results on the inter-atomic distances in simple crystals, and then discussed these findings in the light of theories of atomic structure proposed independently by two American chemists, G.N. Lewis and I. Langmuir. This was the first Bernal had heard of Langmuir’s theory, and he felt exultant after reading Bragg’s paper. Two days later, he spent ten hours studying the ideas of Langmuir and found ‘the scope of the theory leaves me dazed. A slight imagination and plenty of donkey-work, and chemistry will emerge as a branch of physics. I had not dared hope that the clue would come into my hands so soon.’69 The following week he found time between his other activities to read more of Langmuir’s ‘beautiful theory of atoms with wobbly tails and fat, insoluble heads’. He felt satisfied that ‘in this glorious new world of science, there may be some paths left for me to walk’.70
He left Cambridge in mid-August 1921 for a truncated summer vacation in Ireland and took books on vector analysis with him. On 2nd September, ‘the morning I spent rapt in geometry and complex vector algebra. It is all new and my method of attack is both haphazard and roundabout; however, I gained a few positions and consolidated them. It was with great reluctance that I left my new set of points to go to help with the threshing.’71 By the next week there had been substantial progress: ‘I have cleared up simple space lattices pretty thoroughly and I have fifty pages of manuscript to copy out. Not so bad for a week’s very interrupted work.’72 His private research continued during the Michaelmas Term, and by November he felt emboldened to make some remarks about vector analysis after a lecture by the physicist, Charles Darwin, on X-ray crystallography.
This sequence of small academic successes did a lot to restore Bernal’s self-confidence. He continued to devour books on science, history, art, and politics, and to be a prominent figure in the Union, the Heretics and the Emmanuel Science Club. His readings had by now led him to embrace communism and he was one of the leading figures of that faction in the CUSS.73 His range and brilliance were encapsulated in his new nickname of ‘Sage’. The name came not from one of Bernal’s fellow students, who had seen him surrounded by obscure tomes i
n the library, but from Dora Grey, who had witnessed his mesmeric performances at the Heretics and CUSS. She lived in Cambridge with her brother, and worked in a second-hand bookshop owned by C.K. Ogden, the creator and president of the Heretics. On May Day 1921, Bernal went to a Labour Party rally addressed by Hugh Dalton, the economist, who was standing for parliament. Bernal found himself ‘packed in a crowd of well dressed proletarians… [to] listen to insipidities and party lies for two hours’.74 He had the luck to run into Dora Grey, and they went for tea in Allen Hutt’s room in Downing College. Hutt was a good friend of Bernal’s from the first-year mathematics course and was also a communist active in CUSS. Bernal would sometimes join Hutt at meetings of a small communist group, the Spillikins,75 along with Maurice Dobb and Ivor Montagu. The quaint name was adopted to avoid the unwelcome interest that the ‘Young Communist League’ might have earned from Scotland Yard’s Special Branch.
Hutt knew Dora Grey through the Cambridge Trades Council and introduced her to Bernal.76 The sobriquet of ‘Sage’ secretly pleased Bernal; she in return was known as ‘Goblin’. On that long May evening, Bernal and Dickinson idly amused themselves by sliding Dora from one end of Hutt’s table to the other. Bernal thought she was ‘a funny little thing but she has a fine mind’.77 Whatever her attributes, ‘Grey’s shop’ became a regular rendezvous for Bernal, Hutt and Dickinson. At the end of the month, Bernal had to take the first-year science exams, and Grey was on hand to cheer him up, after the first paper seemed to go badly. By now he was convinced that he was in love with her, but she and Hutt were already lovers. Bernal admired them as a thoroughly modern pair, disillusioned with the conventions of life, and with no expectations of the future. Free love had spread from Bloomsbury to Cambridge and Freudianism, in Bernal’s phrase, was ‘the new religion’.78 To emphasize the free nature of their association, Hutt challenged Grey to fall in love with Bernal. Bernal on hearing this idea was ‘almost alarmed by the prospects it opened up’ and viewed it as a dangerous experiment, unlikely to succeed. On 22nd July, the day he spent ten hours reading Langmuir’s atomic theory, Bernal invited Grey to his room for supper. She sat in his armchair talking of her affair with Hutt, while Sage sat on a cushion at her feet, growing more and more impressed with her independent attitude. As both of them had been encouraged to experiment by Hutt, they exchanged ‘a white hot kiss that drew the fire from our souls’ before she left to go to a socialist meeting. An ecstatic Sage ‘walked light footed to Hutt’s’ to give the news. He then resumed work on the atomic theory until 2 am, signing off ‘a record day, a new theory of the universe and new love though really they are both second-hand’.79