Letter XVII

On the Infinite and on Chronology

The labyrinth and abyss of the infinite is another new domain traversed by Newton, and from him we have the thread by which we may find our way through it.

Descartes again appears as his forerunner in this astonishing innovation; in his geometry he advanced with great strides toward the infinite, but stopped at its edge. Mr. Wallis, around the middle of the last century, was the first to reduce a fraction, by perpetual division, to an infinite series.

Lord Brouncker used this series to square the hyperbola.

Mercator published a demonstration of this quadrature. It was at about this time that Newton, at the age of twenty-three, invented a general method for doing with every curve what had just been attempted with the hyperbola.

This method of everywhere subjecting the infinite to algebraic calculation is called differential calculus, or the calculus of fluxions, and integral calculus. It is the art of counting and measuring exactly that whose very existence one cannot even conceive.

Indeed, would you not think someone was making sport of you if you were told that there are infinitely long lines forming an infinitely small angle?

That a straight line, straight as long as it is finite, becomes an infinite curve by changing its direction infinitely little; that a curve can become infinitely less curved?

That there are squares of infinity, cubes of infinity, and infinities of infinity, the next to last of which is nothing beside the last?

All this, which at first appears the height of unreason, is in fact a triumph of the subtlety and scope of the human mind, and a method for discovering truths hitherto unknown.

This daring edifice is even founded on simple ideas. The task is to measure the diagonal of a square, find the area under a curve, find a square root for a number that has none in ordinary arithmetic.

And, after all, so many orders of infinity should no more affront the imagination than the familiar proposition that one can always draw curves between a circle and a tangent, or the proposition that matter can always be divided. Both truths have long been demonstrated and are no more comprehensible than the rest.

Newton's invention of this famous calculus was long contested. Mr. Leibnitz passed in Germany for the inventor of the differentials Newton calls fluxions, and Bernoulli laid claim to the integral calculus; but the honor of the first discovery remained Newton's, while the others retained the glory of having been able to make the choice between themselves and him seem doubtful.

So it was that Harvey's discovery of the circulation of the blood was disputed, as was Mr. Perrault's discovery of the circulation of sap. Hartsoeker and Leuvenhoeck disputed which of them had first seen the tiny worms of which we are made. The same Hartsoeker disputed with Mr. Huyghens the invention of a new way to calculate the distance of a fixed star. It is still unknown which philosopher solved the problem of the cycloid.

Be that as it may, it was by this geometry of the infinite that Newton attained the most sublime knowledge.

I have yet to speak to you of another work, more within the reach of mankind, but still marked by the creative spirit Newton brought to all his researches: an entirely new chronology. For whatever he undertook, he had to change the ideas accepted by other men. Accustomed to bringing order to chaos, he wanted at least to cast some light on the chaos of ancient fables mingled with history and to settle an uncertain chronology. It is true that there is no family, city, or nation that does not try to push its origins farther back; moreover, the earliest historians are the most careless about recording dates. Books were a thousand times less common than today; consequently, being less exposed to criticism, people could deceive the world with greater impunity. And since facts have plainly been fabricated, it is quite likely that dates have been fabricated too. In general, it seemed to Newton that the world was five hundred years younger than the chronologists say; he bases his idea on the ordinary course of nature and on astronomical observations.

By the course of nature we mean here the duration of each human generation. The Egyptians were the first to use this uncertain way of counting. When they wished to write the beginnings of their history, they counted three hundred and forty-one generations from Menes to Sethon; and, having no fixed dates, they reckoned three generations to a hundred years. Thus they counted eleven thousand three hundred and forty years from the reign of Menes to the reign of Sethon. Before they counted by Olympiads, the Greeks followed the Egyptians' method and even lengthened the duration of generations somewhat, extending each generation to forty years. In this calculation, however, the Egyptians and the Greeks were mistaken. It is quite true that, in the ordinary course of nature, three generations make about a hundred to a hundred and twenty years; but three reigns fall far short of that many years. It is perfectly clear that, in general, people live longer than kings reign. Thus someone wishing to write history without precise dates, who knows that a nation had nine kings, would be very wrong to allow three hundred years for those nine kings. Each generation lasts about thirty-six years; each reign, on average, about twenty. Take the thirty kings of England from William the Conqueror to George I: they reigned six hundred and forty-eight years, which, divided among the thirty kings, gives each a reign of twenty-one and a half years. Sixty-three kings of France reigned, on average, about twenty years each. Such is the ordinary course of nature. Thus the ancients were mistaken when they generally equated the length of reigns with the length of generations; thus they counted too much; thus it is fitting to subtract a little from their reckoning.

Astronomical observations seem to offer our philosopher still greater help: he appears stronger fighting on his own ground.

You know that the earth, besides its annual motion, which carries it around the sun from west to east in the space of a year, undergoes another remarkable revolution, suspected rather than known until recent times. Its poles move very slowly backward from east to west, so that from one day to the next their position does not correspond precisely to the same points in the sky. This difference, imperceptible in a year, becomes considerable in time; and after seventy-two years the difference is found to be one degree, that is, one three-hundred-and-sixtieth part of the entire sky. Thus, after seventy-two years, the colure of the spring equinox, which passes through one fixed star, corresponds to another fixed star one degree away from the first. That is why the sun, instead of being in the region of the sky where Aries was in Hipparchus's time, is found to correspond to the region where Pisces is, and Gemini occupies the place where Taurus then stood. All the signs have changed places; yet we still retain the ancients' way of speaking: we say that the sun is in Aries in spring, by the same indulgence with which we say the sun moves around us.

Hipparchus was the first among the Greeks to notice certain changes in the constellations relative to the equinoxes, or rather to learn of them from the Egyptians. Philosophers attributed this motion to the stars, for at that time they were far from imagining such a revolution of the earth: they believed it immobile in every respect. They therefore devised a heaven to which they attached all the stars, and gave this heaven a motion of its own carrying it toward the east while all the stars seemed to make their daily journey from east to west. To this error they added a second, far more consequential one: they believed that the supposed heaven of fixed stars advanced eastward one degree in a hundred years. Thus they were mistaken in their astronomical reckoning as well as in their physical system. An astronomer then might have said, for example: “In the time of a certain observer the spring equinox was in such-and-such a sign, by such-and-such a star; it has moved two degrees between that observer's time and ours; now two degrees amount to two hundred years, so that observer lived two hundred years before me.” An astronomer reasoning in this way would certainly have been mistaken by about fifty years. That is why the ancients, doubly deceived, made their great year of the world, that is, the revolution of the entire heavens, about thirty-six thousand years long. But the moderns know that this imaginary revolution of the heaven of stars is nothing other than the revolution of the earth's poles, which takes twenty-five thousand nine hundred years. It is worth noting in passing that Newton, in determining the shape of the earth, very successfully explained the reason for this revolution.

With all this established, what remains in order to settle the chronology is to see by which star the colure of the equinoxes crosses the ecliptic today in spring, and to find out whether some ancient writer has told us at what point the ecliptic was crossed in his time by the same colure of the equinoxes.

Clement of Alexandria reports that Chiron, who took part in the expedition of the Argonauts, observed the constellations at the time of that famous expedition and placed the spring equinox in the middle of Aries, the autumn equinox in the middle of Libra, our summer solstice in the middle of Cancer, and the winter solstice in the middle of Capricorn.

Long after the expedition of the Argonauts, and one year before the Peloponnesian War, Meton observed that the point of the summer solstice passed through the eighth degree of Cancer.

Now each sign of the zodiac has thirty degrees. In Chiron's time the solstice was halfway through the sign, that is, at the fifteenth degree; one year before the Peloponnesian War it was at the eighth: it had therefore moved backward seven degrees. One degree amounts to seventy-two years: therefore, from the beginning of the Peloponnesian War to the expedition of the Argonauts, there are only seven times seventy-two years, which makes five hundred and four years, not seven hundred years, as the Greeks said. Thus, comparing the present state of the sky with its state then, we see that the expedition of the Argonauts must be placed about nine hundred years before Jesus Christ, and not about fourteen hundred years; consequently the world is about five hundred years younger than was thought. In this way all the epochs are brought closer together, and everything happened later than is said. I do not know whether this ingenious system will meet with great success, or whether people will resolve, on the strength of these ideas, to reform the chronology of the world; perhaps the learned would find it too much to grant one man the honor of having perfected physics, geometry, and history at once: it would be a sort of monarchy that pride can scarcely endure. Indeed, while the partisans of vortices and grooved matter were attacking demonstrated gravitation, the Reverend Father Souciet and Mr. Fréret were writing against Newton's chronology before it was printed.

On Infinity and Chronology

The maze and the bottomless depths of infinity are another new field that Newton explored. He gave us the thread that lets us find our way through it.

Descartes was once again a forerunner of this amazing discovery. He took great strides toward infinity in his geometry, but stopped at the edge. Around the middle of the last century, Mr. Wallis was the first to turn a fraction, by dividing it without end, into an infinite series.

Lord Brouncker used this series to find the area of the hyperbola.

Mercator published a proof of how to find this area. At about the same time, Newton, at the age of twenty-three, had invented a general method for doing with every curve what had just been tried with the hyperbola.

This method of bringing infinity everywhere within the reach of algebraic calculation is called differential calculus, or the calculus of fluxions, and integral calculus. It is the art of counting and measuring exactly what we cannot even imagine existing.

Really, wouldn't you think someone was making fun of you if they told you that infinitely long lines form an infinitely small angle?

Or that a straight line which remains straight as long as it is finite can change direction by an infinitely small amount and become an infinite curve? Or that one curve can become infinitely less curved?

Or that there are squares of infinity, cubes of infinity, and infinities of infinity, in which the next-to-last is nothing beside the last?

All this seems at first to be completely unreasonable. In fact, it shows how subtle and far-reaching the human mind can be. It is a way of finding truths that were unknown until then.

This daring structure is even based on simple ideas. The problems are to measure the diagonal of a square, to find the area bounded by a curve, and to find a square root for a number that has none in ordinary arithmetic.

After all, these many kinds of infinity should be no more shocking to the imagination than the familiar claim that you can always draw curves between a circle and a line tangent to it. Or than the claim that matter can always be divided. Both truths were proved long ago, and they are no easier to understand than the rest.

For a long time, people disputed Newton's claim to have invented this famous calculus. Mr. Leibnitz was regarded in Germany as the inventor of the differences that Newton calls fluxions, and Bernoulli claimed the integral calculus. But Newton kept the honor of having made the first discovery. The others kept the credit for having given people reason to wonder whether the discovery was theirs instead of his.

In the same way, people challenged Harvey's discovery of the circulation of the blood, and Mr. Perrault's discovery of the circulation of sap. Hartsoeker and Leuvenhoeck disputed which of them first saw the tiny worms we are made of. The same Hartsoeker challenged Mr. Huyghens's claim to have invented a new way of calculating the distance of a fixed star. We still do not know which philosopher solved the problem of the cycloid.

Whatever the case, this geometry of infinity enabled Newton to reach his most remarkable insights.

I still have to tell you about another work of his that is easier for most people to understand. Yet it too shows the inventive mind that Newton brought to all his research: an entirely new chronology. In everything he undertook, he had to change the ideas other people had accepted. Used to bringing order to chaos, he wanted at least to shed some light on the chaos of ancient stories mixed up with history, and to establish a chronology that had been uncertain. It is true that every family, city, and nation tries to push its origins further back. Also, the earliest historians are the least careful about recording dates. Books were a thousand times less common than they are now. So people faced less criticism and could deceive the world more easily. Since events have plainly been invented, it is quite likely that dates have been invented too. In general, Newton thought the world was five hundred years younger than the experts in chronology say. He based this view on the ordinary course of nature and on astronomical observations.

By the course of nature here I mean the length of a human generation. The Egyptians were the first to use this uncertain way of counting. When they wanted to write the beginnings of their history, they counted three hundred and forty-one generations from Menes to Sethon. As they had no fixed dates, they gave three generations a hundred years. In this way, they counted eleven thousand three hundred and forty years from the reign of Menes to the reign of Sethon. Before the Greeks counted time by Olympiads, they followed the Egyptian method. They even made generations a little longer, stretching each one to forty years. But both the Egyptians and the Greeks got this calculation wrong. It is quite true that in the ordinary course of nature, three generations last about a hundred to a hundred and twenty years. But three reigns come nowhere near lasting so long. It is very clear that, in general, people live longer than kings reign. So anyone writing a history without exact dates who knows that a nation had nine kings would be very wrong to give those nine kings three hundred years. A generation lasts about thirty-six years; a reign about twenty, on average. Take the thirty kings of England from William the Conqueror to George I. They reigned for six hundred and forty-eight years. Divided among the thirty kings, that gives each one twenty-one and a half years on the throne. Sixty-three kings of France reigned for about twenty years each, on average. This is the ordinary course of nature. So the ancients were wrong when they generally made reigns as long as generations. They counted too many years, and it makes sense to reduce their totals a little.

Astronomical observations seem to give our philosopher even more help. He looks stronger when he is working in his own field.

You know that the earth moves around the sun from west to east over the course of a year. It also makes another unusual movement, one that people only suspected rather than understood until recent times. Its poles move very slowly backward, from east to west, so that from one day to the next their positions do not line up with exactly the same points in the sky. This difference is too small to notice in one year, but it grows over time. After seventy-two years, the difference amounts to one degree, or one three-hundred-and-sixtieth of the entire sky. Thus after seventy-two years, the circle through the poles and the spring equinox, which once passed through one fixed star, lines up with another fixed star one degree away. That is why the sun no longer lines up with the part of the sky where Aries stood in the time of Hipparchus. Instead it lines up with the part where Pisces stands, while Gemini has taken the place where Taurus stood then. All the signs have changed places. Yet we still speak as the ancients did. We say that the sun is in Aries in spring, just as we go along with saying that the sun turns.

Hipparchus was the first among the Greeks to notice some changes in where the constellations stood in relation to the equinoxes—or rather, to learn of them from the Egyptians. The philosophers attributed this motion to the stars. At that time, no one was close to imagining such a movement of the earth. They thought it stood still in every way. So they invented a sphere to which they attached all the stars. They gave that sphere a special movement that carried it eastward, while all the stars appeared to make their daily journey from east to west. To this mistake they added a second, much more important one: they thought their supposed sphere of fixed stars moved eastward by one degree every hundred years. So they were wrong in their astronomical calculations as well as in their physical theory. For example, an astronomer then might have said: “In the time of such-and-such an observer, the spring equinox was in a particular sign, at a particular star. It has moved two degrees between his time and ours. Two degrees represent two hundred years, so this observer lived two hundred years before me.” An astronomer reasoning this way would certainly have been wrong by about fifty years. These two mistakes led the ancients to make their great year of the world, the time for the entire sky to complete its revolution, about thirty-six thousand years long. But modern observers know that this imaginary revolution of the sphere of stars is really just the movement of the earth's poles, which takes twenty-five thousand nine hundred years. It is worth mentioning here that, by establishing the shape of the earth, Newton gave a very good explanation for this movement.

Given all this, to fix the chronology we need to see which star marks the point where the circle through the poles and the equinoxes crosses the ecliptic in spring today. We also need to know whether any ancient writer has told us where that same circle crossed the ecliptic in his own time.

Clement of Alexandria reports that Chiron, who took part in the Argonauts' expedition, observed the constellations during that famous expedition. He placed the spring equinox in the middle of Aries, the autumn equinox in the middle of Libra, our summer solstice in the middle of Cancer, and the winter solstice in the middle of Capricorn.

Long after the Argonauts' expedition, and one year before the Peloponnesian War, Meton observed that the point of the summer solstice passed through the eighth degree of Cancer.

Each sign of the zodiac has thirty degrees. In Chiron's time the solstice was halfway through the sign, at the fifteenth degree. One year before the Peloponnesian War, it was at the eighth degree. So it had moved back seven degrees. One degree represents seventy-two years. Therefore, between the beginning of the Peloponnesian War and the Argonauts' expedition there were only seven times seventy-two years, or five hundred and four years, not seven hundred as the Greeks said. By comparing the sky today with the sky then, we can see that the Argonauts' expedition should be placed about nine hundred years before Jesus Christ, not about fourteen hundred years before. As a result, the world is about five hundred years younger than people thought. This brings every date closer to us: everything happened later than people say. I do not know whether this clever theory will catch on, or whether people will agree to revise the chronology of the world on the basis of these ideas. Perhaps scholars would think it too much to give one man the honor of improving physics, geometry, and history all at once. That would be a kind of monarchy that their pride could hardly accept. Indeed, while supporters of vortices and grooved matter were attacking the proven theory of gravitation, Reverend Father Souciet and Mr. Fréret were writing against Newton's chronology before it had even been printed.