Most of what remains of Joseph Priestley’s laboratory is in the Smithsonian Institution in Washington, D.C. And, of course, it has no business to be there. This apparatus ought to be in Birmingham in England, the centre of the Industrial Revolution, where Priestley did his most splendid work. Why is it here? Because a mob drove Priestley out of Birmingham in 1791.
Priestley’s story is characteristic of another conflict between originality and tradition. In 1761 he had been invited, at the age of twenty-eight, to teach modern languages at one of the dissenting academies (he was a Unitarian) which took the place of universities for those who were not conformists of the Church of England. Within a year, Priestley was inspired by the lectures in science of one of his fellow teachers to begin a book about electricity; and from that he turned to chemical experiments. He also became excited about the American Revolution (he had been encouraged by Benjamin Franklin) and later the French Revolution. And so, on the second anniversary of the storming of the Bastille, the loyal citizens burned down what Priestley described as one of the most carefully assembled laboratories in the world. He went to America, but was not made welcome. Only his intellectual equals appreciated him; when Thomas Jefferson became President, he told Joseph Priestley, ‘Yours is one of the few lives precious to mankind’.
I would like to be able to tell you that the mob that destroyed Priestley’s house in Birmingham shattered the dream of a beautiful, lovable, charming man. Alas, I doubt if that would really be true. I do not think Priestley was very lovable, any more than Paracelsus. I suspect that he was a rather difficult, cold, cantankerous, precise, prim, puritanical man. But the ascent of man is not made by lovable people. It is made by people who have two qualities: an immense integrity, and at least a little genius. Priestley had both.
The discovery that he made was that air is not an elementary substance: that it is composed of several gases and that, among those, oxygen – what he called ‘dephlogisticated air’ – is the one that is essential to the life of animals. Priestley was a good experimenter, and he went forward carefully in several steps. On 1 August 1774 he made some oxygen, and saw to his astonishment how brightly a candle burned in it. In October of that year he went to Paris, where he gave Lavoisier and others news of his finding. But it was not until he himself came back and, on 8 March 1775, put a mouse into oxygen, that he realised how well one breathed in that atmosphere. A day or two after, Priestley wrote a delightful letter in which he said to Franklin: ‘Hitherto only two mice and myself have had the privilege of breathing it’.
Priestley also discovered that the green plants breathe out oxygen in sunlight, and so make a basis for the animals who breathe it in. The next hundred years were to show this is crucial; the animals would not have evolved at all if the plants had not made the oxygen first. But in the 1770s nobody had thought about that.
The discovery of oxygen was given meaning by the clear, revolutionary mind of Antoine Lavoisier (who perished in the French Revolution). Lavoisier repeated an experiment of Priestley’s which is almost a caricature of one of the classical experiments of alchemy which I described at the beginning of this essay. Both men heated the red oxide of mercury, using a burning glass (the burning glass was fashionable just then), in a vessel in which they could see gas being produced, and could collect it. The gas was oxygen. That was the qualitative experiment; but to Lavoisier it was the instant clue to the idea that chemical decomposition can be quantified.
The idea was simple and radical; run the alchemical experiment in both directions, and measure the quantities that are exchanged exactly. First, in the forward direction: burn mercury (so that it absorbs oxygen) and measure the exact quantity of oxygen that is taken up from a closed vessel between the beginning of the burning and the end. Now turn the process into reverse: take the mercuric oxide that has been made, heat it vigorously and expel the oxygen from it again. Mercury is left behind, oxygen flows into the vessel, and the crucial question is: ‘How much?’ Exactly the amount that was taken up before. Suddenly the process is revealed for what it is, a material one of coupling and uncoupling fixed quantities of two substances. Essences, principles, phlogiston, have disappeared. Two concrete elements, mercury and oxygen, have really and demonstrably been put together and taken apart.
It might seem a dizzy hope that we can march from the primitive processes of the first coppersmiths and the magical speculations of the alchemists to the most powerful idea in modern science: the idea of the atoms. Yet the route, the firewalker’s route, is direct. One step remains beyond the notion of chemical elements that Lavoisier quantified, to its expression in atomic terms by the son of a Curnberland hand-loom weaver, John Dalton.
After the fire, the sulphur, the burning mercury, it was inevitable that the climax of the story should take place in the chill damp of Manchester. Here, between 1803 and 1808, a Quaker schoolmaster called John Dalton turned the vague knowledge of chemical combination, brilliantly illuminated as it had been by Lavoisier, suddenly into the precise modern conception of atomic theory. It was a time of marvellous discovery in chemistry – in those five years ten new elements were found; and yet Dalton was not interested in any of that. He was, to tell the truth, a somewhat colourless man. (He was certainly colour-blind, and the genetic defect of confusing red with green that he described in himself was long called ‘Daltonism’.)
Dalton was a man of regular habits, who walked out every Thursday afternoon to play bowls in the countryside. And the things he was interested in were the things of the countryside, the things that still characterise the landscape in Manchester: water, marsh gas, carbon dioxide. Dalton asked himself concrete questions about the way they combine by weight. Why, when water is made of oxygen and hydrogen, do exactly the same amounts always come together to make a given amount of water? Why when carbon dioxide is made, why when methane is made, are there these constancies of weight?
Throughout the summer of 1803 Dalton worked at the question. He wrote: ‘An enquiry into the relative weights of the ultimate particles is, as far as I know, entirely new. I have lately been prosecuting this enquiry with remarkable success.’ And he thereby realised that the answer must be, Yes, the old-fashioned Greek atomic theory is true. But the atom is not just an abstraction; in a physical sense, it has a weight which characterises this element or that element. The atoms of one element (Dalton called them ‘ultimate or elementary particles’) are all alike, and are different from the atoms of another element; and one way in which they exhibit the difference between them is physically, as a difference in weight. ‘I should apprehend there are a considerable number of what may properly be called elementary particles, which can never be metamorphosed one into another.’
In 1805 Dalton published for the first time his conception of atomic theory, and it went like this. If a minimum quantity of carbon, an atom, combines to make carbon dioxide, it does so invariably with a prescribed quantity of oxygen – two atoms of oxygen.
If water is then constructed from the two atoms of oxygen, each combined with the necessary quantity of hydrogen, it will be one molecule of water from one oxygen atom and one molecule of water from the other.
The weights are right: the weight of oxygen that produces one unit of carbon dioxide will produce two units of water. Now are the weights right for a compound that has no oxygen in it – for marsh gas or methane, in which carbon combines directly with hydrogen? Yes, exactly. If you remove the two oxygen atoms from the single carbon dioxide molecule, and from the two water molecules, then the material balance is precise: you have the right quantities of hydrogen and carbon to make methane.
The weighed quantities of different elements that combine with one another express, by their constancy, an underlying scheme of combination between their atoms.
It is the exact arithmetic of the atoms which makes of chemical theory the foundation of modern atomic theory. That is the first profound lesson that comes out of all this multitude of speculation about gold and copper and alchemy, unti
l it reaches its climax in Dalton.
The other lesson makes a point about scientific method. Dalton was a man of regular habits. For fifty-seven years he walked out of Manchester every day; he measured the rainfall, the temperature – a singularly monotonous enterprise in this climate. Of all that mass of data, nothing whatever came. But of the one searching, almost childlike question about the weights that enter the construction of these simple molecules – out of that came modern atomic theory. That is the essence of science: ask an impertinent question, and you are on the way to the pertinent answer.
CHAPTER FIVE
THE MUSIC OF THE SPHERES
Mathematics is in many ways the most elaborated and sophisticated of the sciences – or so it seems to me, as a mathematician. So I find both a special pleasure and constraint in describing the progress of mathematics, because it has been part of so much human speculation: a ladder for mystical as well as rational thought in the intellectual ascent of man. However, there are some concepts that any account of mathematics should include: the logical idea of proof, the empirical idea of exact laws of nature (of space particularly), the emergence of the concept of operations, and the movement in mathematics from a static to a dynamic description of nature. They form the theme of this essay.
Even very primitive peoples have a number system; they may not count much beyond four, but they know that two of any thing plus two of the same thing makes four, not just sometimes but always. From that fundamental step, many cultures have built their own number systems, usually as a written language with similar conventions. The Babylonians, the Mayans, and the people of India, for example, invented essentially the same way of writing large numbers as a sequence of digits that we use, although they lived far apart in space and in time.
So there is no place and no moment in history where I could stand and say ‘Arithmetic begins here, now’. People have been counting, as they have been talking, in every culture. Arithmetic, like language, begins in legend. But mathematics in our sense, reasoning with numbers, is another matter. And it is to look for the origin of that, at the hinge of legend and history, that I went sailing to the island of Samos.
In legendary times Samos was a centre of the Greek worship of Hera, the Queen of Heaven, the lawful (and jealous) wife of Zeus. What remains of her temple, the Heraion, dates from the sixth century before Christ. At that time there was born on Samos, about 580 BC, the first genius and the founder of Greek mathematics, Pythagoras. During his lifetime the island was taken over by the tyrant, Polycrates. There is a tradition that before Pythagoras fled, he taught for a while in hiding in a small white cave in the mountains which is still shown to the credulous.
Samos is a magical island. The air is full of sea and trees and music. Other Greek islands will do as a setting for The Tempest, but for me this is Prospero’s island, the shore where the scholar turned magician. Perhaps Pythagoras was a kind of magician to his followers, because he taught them that nature is commanded by numbers. There is a harmony in nature, he said, a unity in her variety, and it has a language: numbers are the language of nature.
Pythagoras found a basic relation between musical harmony and mathematics. The story of his discovery survives only in garbled form, like a folk tale. But what he discovered was precise. A single stretched string vibrating as a whole produces a ground note. The notes that sound harmonious with it are produced by dividing the string into an exact number of parts: into exactly two parts, into exactly three parts, into exactly four parts, and so on. If the still point on the string, the node, does not come at one of these exact points, the sound is discordant.
Blind harpist, Egypt, 1579-1293 BC
As we shift the node along the string, we recognise the notes that are harmonious when we reach the prescribed points. Begin with the whole string: this is the ground note. Move the node to the midpoint: this is the octave above it. Move the node to a point one third of the way along: this is the fifth above that. Move it to a point one fourth along: this is the fourth, another octave above. And if you move the node to a point one fifth of the way along, this (which Pythagoras did not reach) is the major third above that.
Pythagoras had found that the chords which sound pleasing to the ear – the western ear – correspond to exact divisions of the string by whole numbers. To the Pythagoreans that discovery had a mystic force. The agreement between nature and number was so cogent that it persuaded them that not only the sounds of nature, but all her characteristic dimensions, must be simple numbers that express harmonies. For example, Pythagoras or his followers believed that we should be able to calculate the orbits of the heavenly bodies (which the Greeks pictured as carried round the earth on crystal spheres) by relating them to the musical intervals. They felt that all the regularities in nature are musical; the movements of the heavens were, for them, the music of the spheres.
These ideas gave Pythagoras the status of a seer in philosophy, almost a religious leader, whose followers formed a secret and perhaps revolutionary sect. It is likely that many of the later followers of Pythagoras were slaves; they believed in the transmigration of souls, which may have been their way of hoping for a happier life after death.
I have been speaking of the language of numbers, that is arithmetic, but my last example was the heavenly spheres, which are geometrical shapes. The transition is not accidental. Nature presents us with shapes: a wave, a crystal, the human body, and it is we who have to sense and find the numerical relations in them. Pythagoras was a pioneer in linking geometry with numbers, and since it is also my choice among the branches of mathematics, it is fitting to watch what he did.
Pythagoras had proved that the world of sound is governed by exact numbers. He went on to prove that the same thing is true of the world of vision. That is an extraordinary achievement. I look about me; here I am, in this marvellous, coloured landscape of Greece, among the wild natural forms, the Orphic dells, the sea. Where under this beautiful chaos can there lie a simple, numerical structure?
The question forces us back to the most primitive constants in our perception of natural laws. To answer well, it is clear that we must begin from universals of experience. There are two experiences on which our visual world is based: that gravity is vertical, and that the horizon stands at right angles to it. And it is that conjunction, those cross-wires in the visual field, which fixes the nature of the right angle; so that if I were to turn this right angle of experience (the direction of ‘down’ and the direction of sideways’) four times, back I come to the cross of gravity and the horizon. The right angle is defined by this fourfold operation, and is distinguished by it from any other arbitrary angle.
In the world of vision, then, in the vertical picture plane that our eyes present to us, a right angle is defined by its fourfold rotation back on itself. The same definition holds also in the horizontal world of experience, in which in fact we move. Consider that world, the world of the flat earth and the map and the points of the compass. Here I am looking across the straits from Samos to Asia Minor, due south. I take a triangular tile as a pointer and I set it pointing there, south. (I have made the pointer in the shape of a right-angled triangle, because I shall want to put its four rotations side by side.) If I turn that triangular tile through a right angle, it points due west. If I now turn it through a second right angle, it points due north. And if I now turn it through a third right angle, it points due east. Finally, the fourth and last turn will take it due south again, pointing to Asia Minor, in the direction in which it began.
Not only the natural world as we experience it, but the world as we construct it is built on that relation. It has been so since the time that the Babylonians built the Hanging Gardens, and earlier, since the time that the Egyptians built the pyramids. These cultures already knew in a practical sense that there is a builder’s set square in which the numerical relations dictate and make the right angle. The Babylonians knew many, perhaps hundreds of formulae for this by 2000 BC. The Indians and the Egyptians knew som
e. The Egyptians, it seems, almost always used a set square with the sides of the triangle made of three, four, and five units. It was not until 550 BC or thereabouts that Pythagoras raised this knowledge out of the world of empirical fact into the world of what we should now call proof. That is, he asked the question, ‘How do such numbers that make up these builder’s triangles flow from the fact that a right angle is what you turn four times to point the same way?’
His proof, we think, ran something like this. (It is not the proof that stands in the school books.) The four leading points – south, west, north, east – of the triangles that form the cross of the compass are the corners of a square. I slide the four triangles so that the long side of each ends at the leading point of a neighbour. Now I have constructed a square on the longest side of the right-angled triangles – on the hypotenuse. Just so that we should know what is part of the enclosed area and what is not, I will fill in the small inner square area that has now been uncovered with an additional tile. (I use tiles because many tile patterns, in Rome, in the Orient, from now on derive from this kind of wedding of mathematical relation to thought about nature.)
Now we have a square on the hypotenuse, and we can of course relate that by calculation to the squares on the two shorter sides. But that would miss the natural structure and inwardness of the figure. We do not need any calculation. A small game, such as children and mathematicians play, will reveal more than calculation. Transpose two triangles to new positions, thus. Move the triangle that pointed south so that its longest side lies along the longest side of the triangle that pointed north. And move the triangle that pointed east so that its longest side lies along the longest side of the triangle that pointed west.
The Ascent of Man Page 10