My Time in Space, page 12
There are, however, simpler and more basic interpretations, to restore a cutting edge to this soliloquy. The long cliff-sequence, the hundred pages of ‘South’ in Stones of Aran, can be read as a meditation on death. A thought of suicide, suppressed from that book, could find its place in the section called ‘Perdition’s Edge’. Occasionally when we were in wintry mood M and I would visit it – climbing the barren hill to the prehistoric cliff-fort of Dún Aonghasa, skirting its gloomy walls and following the windy clifftop a hundred yards farther west to a projecting angle called Carraig an Smáil, which an island author used to render as the Rock of Perdition. We knew what waves would be thundering against the wall of rock below on such a day, having watched them often and intently on lower shores; they bulge, burst, spout, gape and founder so individually one could give each a proper name – Stag-head, Sloppy Jim, The Walrus, Seven Hills, Hullabaloo – wave after wave, hour after hour, day after day. But from this height all that flux of identity was abstracted by distance into a slow regular hammering. We would lie face down on the almost bare stone of the clifftop and work our way forward until our heads stuck out over the appalling drop like gargoyles from a cathedral’s eaves. But this was the anti-cathedral, the smithy of natural law; all things are on the anvil, they shall be thus and thus. Each blow shook the land and was transmitted to our frail soft bodies. Our minds too were shaken. After a time, immeasurable although perhaps brief, we would withdraw, reshaped, annealed, and take ourselves dizzily homewards, our personalities temporarily in a vanishing perspective.
If such a cliff can be a renewal, it is because it is also a means, invitation and temptation, to end one’s life. Just four and a half seconds, I calculate, would separate – by what sublime panic? – the last step on earth from obliteration of the self. Believing that the right to one’s own death is essential to one’s personhood, however its exercise is subject to individual and communal obligations, I should find it possible to discuss ways and means in a usual tone. To throw myself off the cliffs that I have written up at such length would be fitting, to the point of melodrama. But nobody thereafter would be able to read me unconstrained by a fateful commentary; I would have permanently conditioned the reception of my work, at least until the reading world had forgotten either or both of it and me. Then there are the practicalities. It is a strenuous climb to this point, or would be for one suffering terminal illness, terminal sorrow; this is no death for the dying. Also, the kindly folk of Gort na gCapall would inherit the horrible task of finding and retrieving the ruins of my body. And, even if I got so far, exhibited such life-powers, could I nerve myself to the act? I have imagined it and dreamed it too often. The rollers curve in and are reflected back from the cliff with the regularity of balance-wheels, escapements, ratchets; I am falling into the teeth of the world’s clockwork…. At what should have been the last, or nearly the last, moment, flesh and bone would wrench themselves away from the edge, and I would creep home to shameful continuance of life.
What strange relativity is this, that one sees another’s last years as decline, whereas to the person who treads it with such effort the way is uphill, the ground ever steeper, the breath ever shorter. Finally there is no more ground, no more breath. If one could then come down from that painfully attained summit like a triumphant mountaineer, to admiring toasts and a well-paid lecture tour! But one cannot share this lonely heroism, though it is written into the genes of all mortals, or even savour it in solitude, for clouds of forgetfulness and bewilderment close in. There is not even a view to reward the climber, neither a panorama of the biographical archipelago nor an upwards vision of eternity; consciousness constricts itself into bone-ache.
No, of course, with luck, with love, with forethought, it may not, need not, come to this. Part of my forethought has been this deathly and unrealistic fantasy of the cliff. But if it is not to be the cliff, discussion of it is not matter for this book, and so I leave it. May it go easily, the untwisting of the threads of my life, and of that other life braided into mine! I am very much afraid of it.
* Consider the cathedrals: having lost the élan that lifted up their bulk, become stone again, they shrink and crumple; even their spire, which in former times aimed insolently for the sky, submits to the contamination of heaviness and imitates the modesty of our lassitudes.
CONSTELLATION AND QUESTIONMARK
The Mystic Hexagram
Pascal, before I opened biographies, was chiefly known to me for three pronouncements: his wager on salvation, his confession of cosmic agoraphobia, and his theorem. The first – ‘If God does not exist, one will lose nothing by believing in him, while if he does exist, one will lose everything by not believing’ – is unworthy of both God and human. The second, a mere whisper – ‘Le silence de ces espaces infinis m’effraie’ – still rattles the windows of civilization, and I shall follow its echoes in another essay. The third is this:
If a hexagon is inscribed in a conic, the points of intersection of its opposite sides will lie in a straight line.
Ever since I came across it, in my first year at Cambridge, it has been my paradigm of mathematical beauty. When I go now to refresh my memory of its proof in a little textbook of projective geometry that has followed me around since then, I find a hole from which the relevant diagram has been cut out, reminding me that I used it (in those pre-photocopier days) to illustrate an article I wrote for a college magazine. I suspect that my jejune reflections on abstract art, maths and Eastern calligraphy would not have been accepted for publication had I not also been the editor; but one point from the essay comes in handy here. The geometrical air of some modernist art (I was thinking of my hero Mondrian, and Gabo and Hepworth among others) might lead one to suppose that the beauty of such works is in some way dependent on the beauty of mathematics; however, descriptions of them as geometrical configurations would not be of much interest to a mathematician. Conversely, the diagram exemplifying a theorem may or may not be beautiful, but the beauty of the theorem itself derives from invisible considerations such as the richness of connections it reveals among different mathematical ideas, its ‘seriousness’. I borrowed this term from G.H. Hardy’s A Mathematician’s Apology, that high-minded extension of the Bloomsbury ethos to mathematics. Hardy further analyzes a theorem’s ‘seriousness’ as dependent upon its ‘generality’ or relevance to a wide but well-discriminated body of ideas, and its ‘depth’, by which he seems to mean something like its explanatory power in the hierarchy of concepts. Nowadays I can recognize that my mathematical preferences are influenced by less purist factors as well: in this instance by the association of the theorem with Pascal’s fast-burning lifestory, and indeed with the sound of his name, which promises both Gallic grace and passionate intellectuality.
Figs. 1–3. Cross sections of the cone: (1) the ellipse; (2) the parabola; (3) the hyperbola. Only the central portion of the cone is shown; a complete cone has two ‘branches’, each of which extends to infinity. In (3) the cross-section cuts both branches, and so the hyperbola also has two branches.
In the language of geometry a hexagon need not be the sort of honeycomb-cell shape we call by that name in daily life; any six points joined into a circuit by six lines constitute a hexagon, even if the lines cross one another. (I will say simply ‘line’, instead of specifying each time a straight line.) A conic (or conic section) is any of the curves obtained by taking a cross-section of a cone (Figs. 1–3). The circle is the simplest of them, and the others are familiar presences to those who read by night: the ellipse of light cast on the ceiling from the circular aperture in the top of a tilted lampshade, and the lovely and evocative shape it projects onto a nearby wall, elegantly rounded below and fading away upwards with the grace of an angel’s wing until clipped by the ceiling – a parabola if the two limbs of the illuminated area would eventually become parallel, part of a hyperbola if they would diverge forever. Pascal’s theorem, then, is a powerful generalization, covering any of the endless variety of hexagons that can be inscribed in a conic of any type or size. A watertight proof of it, starting from the basic definitions and axioms of lines and their intersections, is lengthy and involved. But one can almost see that it must be true, by looking at a few examples, starting with a very simple case: Take six points evenly distributed around a circle, A, B, C, D, E and F, and join them up in this order: ADBFCE and so back to A (Fig. 4). Calling AD the first side, the side opposite to it is the fourth, FC, and they intersect in X. The second side is DB and it intersects the fifth, CE, in Y. Finally the third side, BF, intersects the sixth, EA, in Z. Then, the diagram being perfectly symmetrical, it is obvious that X, Y and Z lie on a straight line.
Now join up the same six points in another hexagon thus: ADEFCB and back to A (Fig. 5). This time, to find the intersection of opposite sides DE and CB, for instance, we have to extend them. But, once again, from symmetry, the three crucial points (as I will call them) lie on a line. And a few experiments with less symmetrical hexagons will convince one that the theorem is certainly true for circles (Fig. 6).
Next, imagine any such diagram of a hexagon in a circle being constructed out of strings stretched across that aperture in the top of a lampshade. The doctored lampshade will throw a shadow diagram onto a wall or ceiling, showing a hexagon of some sort inscribed in a conic of some sort – and, since the shadows of straight lines are themselves straight, the crucial points in that diagram will lie in a line (Fig. 7). Surely enough examples have been exhibited to convince reason that Pascal’s theorem is a general truth? Proving it – proving that there are no exceptions – is of course another story. (Pascal did so at the age of sixteen.) A complication arises if two opposite sides are parallel, but they can be considered to meet at infinity; if all three crucial points are infinitely far away, one can take them to lie on a conceptual ‘line at infinity’ – or one can ignore such anomalous cases, infinity being one of the more easily ignored aspects of the world.
Figs. 4–9. Instances of Pascal’s Theorem.
I have deliberately surrounded this theorem with images of lamps projecting pools of light and of shapes projecting shadows. Pascal’s is one of many theorems that survive projection; the conic may have its shape altered (a circle may become a parabola, for instance, as in Fig. 8), the sides of the hexagon may become longer or shorter and its angles larger or smaller, but the crucial points will still lie on a line. But most of the theorems in Euclid do not remain true under projection. The theorem that the two tangents from a point to a circle are equal, for example; project a diagram of it obliquely, and the circle will become an ellipse and the two tangents unequal. In general, distances and angles are changed by projection, but Pascal’s theorem makes no mention of them; it belongs to a stratum of geometry more basic (deeper, in Hardy’s sense) than the geometry of distances and angles.
Frustratingly, it is not known how Pascal proved his theorem. The result was announced without proof in 1640 in his first published work, a leaflet which is hardly more than an advertisement for a proposed treatise, circulated among the scientists of the day including Huygens, who forwarded it immediately to Descartes. After some basic definitions Pascal states two lemmas, of which the first is his theorem, for the case of a circle. (It is not in the trenchant and symmetrical form he later gave the result, and the hexagon is not explicitly mentioned, but the result is equivalent to what we now call Pascal’s Theorem.) From these he proposes to show that the same is true of any conic section, and thence deduce all the elements of conies, including the subject’s basic theorem, due to his elder contemporary Desargues, to whom he almost over-modestly acknowledges his debt: ‘I owe the little I have discovered on this topic to his writings, and … I have tried to imitate his method, so far as was possible for me.’ There was a certain rivalry between Descartes’ algebraic approach to geometry and Desargues’ more intuitive and visual methods, and perhaps this made Descartes read too hastily, for he mistakenly jumped to the conclusion that the youngster had not done more than imitate Desargues; later on he came to appreciate Pascal’s genius, and was especially interested in his experiments on vacuums or ‘la Vide’.
From the correspondence of Pascal’s mathematical mentor Père Mersenne later in 1640 we learn that Pascal deduced four hundred corollaries from his theorem, covering the field of Apollonius’s treatise, the ancient classic on conies, and that the work was to be published in the following year. Desargues himself followed its progress, and referred in his own writings to ‘this great proposition called la Pascale’. However, although Pascal published work on other branches of mathematics (and designed and built the first calculating machine), the promised Traité des coniques never appeared. When Fermat, with whom he had corresponded on and effectively co-founded the theory of probability, wrote to him in 1660 suggesting a meeting, he replied with expressions of esteem for ‘Europe’s greatest geometer’ (‘geometer’ here meaning ‘mathematician’), but sickness and his increasingly demanding religiosity reduced him to the saddening admission, ‘I would not take two steps for geometry … I am engaged in studies so far from that turn of mind I scarcely remember what it is about.’ We are all the losers by Pascal’s miscalculated wager on God.
After his early death Pascal’s manuscripts on conies were sent by his family to Leibniz, a great admirer, whose own achievement in founding what we now call analysis or calculus owed much to Pascal. Leibniz sorted them out into five treatises, had a copy made of the first, and returned them with a letter summarizing their contents, concluding by stating that ‘This work is in fit state to be printed, and there is no question that it merits it.’ Unforgivably, the family did not act on this advice. The manuscripts were seemingly deposited with Pascal’s other papers at Saint-Germain-des-Prés, and are now lost. One feels like demanding that all the archives in which they could possibly be lying suspend all other business while they turn themselves inside-out like sacks to find them. As it is we have Leibniz’s copy of the first treatise and his letter, from which it appears that the second treatise contained the proof of the theorem Pascal himself called the ‘Hexagramme mystique’, the consequences of which were elaborated in the subsequent treatises. The penetrating luminosity of the first treatise gives one a heightened sense of what is lost.
In the leaflet of 1640 the theorem is first stated for a circle, and then extended to the general conic. It seems clear that Pascal’s method of doing this was to be by projection. In the first treatise he gives his definition of a cone, which amounts to this: if a line is drawn through a point on a circle and another fixed point not in the plane of the circle, then as the first point is allowed to move around the circle, the line sweeps out the surface of a cone (Fig. 10). And he deduces the properties of the various conic sections from those of the circle, by imagining that the eye is placed at the vertex of the cone, so that the conic is the image of the circle; i.e. points of the conic coincide visually with those of the circle. (This eye is highly idealized; it sees in all directions.) Leibniz, it appears from his private notes, was surprised and seduced by this ‘optical method of cast shadows’ and wondered if it could not ‘transcend the cone and rise to higher considerations’, that is to a general projective geometry not limited to lines and conics, which of course has come to pass.
In its origins, then, projective geometry relates to the theory of perspective and to the Enlightenment’s fascination with the new tools of vision, the telescope, the microscope, the camera obscura, and so to the ocular imagery pervading the thought of that time, including Descartes’ foundational doctrine of ‘clear and distinct ideas’ and even his conception of the relationship of soul to body. Later in his life Pascal composed an emblematic ‘device’ for himself, showing an eye surrounded by a crown of thorns. In the daytime world of fact and history, this commemorated ‘the Miracle of the Thorn’, an important event in the turbulent history of Port Royale, the Jansenist abbey with which his family was closely connected: a precious relic, a fragment of a spine from Christ’s crown of thorns, had been lent to the abbey, and Pascal’s niece had been cured of a dreadful fistula or ulcer of a tear-gland by application of the reliquary containing it; the occurrence was investigated by doctors of the Sorbonne and officially declared miraculous. But one could point to other connections, not figuring in the Cartesian realm of clear and distinct ideas, between this discomforting image of the eye surrounded by thorns, and that of the eye at the vertex of a cone, at the origin of the generators of form in general and of the mystic hexagram in particular.
