Infinitesimals in Geometry and Sir Isaac Newton’s Chronology

The labyrinth and abyss of infinity is also a new course Sir Isaac Newton has gone through, and we are obliged to him for the clue, by whose assistance we are enabled to trace its various windings.

Sir Isaac Newton has also explored the labyrinth and abyss of infinity. We are grateful to him for the guiding thread that lets us follow the labyrinth’s many twists and turns.

Descartes got the start of him also in this astonishing invention. He advanced with mighty steps in his geometry, and was arrived at the very borders of infinity, but went no farther. Dr. Wallis, about the middle of the last century, was the first who reduced a fraction by a perpetual division to an infinite series.

Descartes had also beaten him to this astonishing invention. He made great strides in geometry and reached the very edge of infinity, but went no further. Around the middle of the last century, Dr. Wallis was the first to turn a fraction into an infinite series. He did so through repeated division.

The Lord Brouncker employed this series to square the hyperbola.

Lord Brouncker used this series to calculate the area under the hyperbola.

Mercator published a demonstration of this quadrature; much about which time Sir Isaac Newton, being then twenty-three years of age, had invented a general method, to perform on all geometrical curves what had just before been tried on the hyperbola.

Mercator published a proof of this quadrature. Around that time, Sir Isaac Newton, then twenty-three years old, had invented a general method for all geometrical curves. His method did what had just been tried on the hyperbola.

It is to this method of subjecting everywhere infinity to algebraical calculations, that the name is given of differential calculations or of fluxions and integral calculation. It is the art of numbering and measuring exactly a thing whose existence cannot be conceived.

The method of applying algebraic calculations to infinity in every case is called differential calculus, or fluxions, and integral calculus. It is the art of counting and measuring exactly something whose existence we cannot conceive.

And, indeed, would you not imagine that a man laughed at you who should declare that there are lines infinitely great which form an angle infinitely little?

And really, wouldn't you think someone was mocking you if he made this claim? Infinitely long lines can form an infinitely small angle.

That a right line, which is a right line so long as it is finite, by changing infinitely little its direction, becomes an infinite curve; and that a curve may become infinitely less than another curve?

That a straight line, which stays straight as long as it is finite, becomes an infinite curve through an infinitely small change in direction? And that one curve may become infinitely smaller than another?

That there are infinite squares, infinite cubes, and infinites of infinites, all greater than one another, and the last but one of which is nothing in comparison of the last?

That there are infinite squares, infinite cubes, and infinities of infinities, each greater than the one before it? And that the next-to-last infinity is nothing compared with the last?

All these things, which at first appear to be the utmost excess of frenzy, are in reality an effort of the subtlety and extent of the human mind, and the art of finding truths which till then had been unknown.

All these things at first seem like the height of madness. But they actually show the human mind’s subtlety and range, and its skill in discovering previously unknown truths.

This so bold edifice is even founded on simple ideas. The business is to measure the diagonal of a square, to give the area of a curve, to find the square root of a number, which has none in common arithmetic. After all, the imagination ought not to be startled any more at so many orders of infinites than at the so well-known proposition, viz., that curve lines may always be made to pass between a circle and a tangent; or at that other, namely, that matter is divisible in infinitum. These two truths have been demonstrated many years, and are no less incomprehensible than the things we have been speaking of.

Even this bold structure rests on simple ideas. The task is to measure a square’s diagonal or find the area of a curve. Or it is to find the square root of a number that has none in ordinary arithmetic. After all, so many orders of infinity should be no more startling than a familiar proposition. Curved lines can always be drawn between a circle and a tangent. Nor should these orders of infinity be more startling than another proposition: matter is divisible without end. Both truths were proved many years ago. They are no less beyond our understanding than the things we have been discussing.

For many years the invention of this famous calculation was denied to Sir Isaac Newton. In Germany Mr. Leibnitz was considered as the inventor of the differences or moments, called fluxions, and Mr. Bernouilli claimed the integral calculus. However, Sir Isaac is now thought to have first made the discovery, and the other two have the glory of having once made the world doubt whether it was to be ascribed to him or them. Thus some contested with Dr. Harvey the invention of the circulation of the blood, as others disputed with Mr. Perrault that of the circulation of the sap.

For many years, Sir Isaac Newton was denied credit for inventing this famous method of calculation. In Germany, Mr. Leibnitz was considered the inventor of differences or moments, called fluxions. Mr. Bernouilli claimed to have invented integral calculus. However, Sir Isaac is now thought to have made the discovery first. The other two have the glory of having once made the world question who deserved credit: Sir Isaac or themselves. Similarly, some challenged Dr. Harvey’s claim to have discovered the circulation of the blood. Others disputed Mr. Perrault’s claim to have discovered the circulation of sap.

Hartsocher and Leuwenhoek disputed with each other the honour of having first seen the vermiculi of which mankind are formed. This Hartsocher also contested with Huygens the invention of a new method of calculating the distance of a fixed star. It is not yet known to what philosopher we owe the invention of the cycloid.

Hartsocher and Leuwenhoek each claimed the honor of being the first to see the vermiculi from which humans are formed. Hartsocher also disputed Huygens’s claim to have invented a new method for calculating the distance of a fixed star. We still do not know which philosopher invented the cycloid.

Be this as it will, it is by the help of this geometry of infinites that Sir Isaac Newton attained to the most sublime discoveries. I am now to speak of another work, which, though more adapted to the capacity of the human mind, does nevertheless display some marks of that creative genius with which Sir Isaac Newton was informed in all his researches. The work I mean is a chronology of a new kind, for what province soever he undertook he was sure to change the ideas and opinions received by the rest of men.

Be that as it may, Sir Isaac Newton used this geometry of infinites to make the most remarkable discoveries. I will now discuss another work, though one more suited to the limits of the human mind. Even so, the work shows some signs of the creative genius that guided Sir Isaac Newton in all his research. I mean a new kind of chronology. For whatever field he took up, he was sure to change the ideas and opinions everyone else accepted.

Accustomed to unravel and disentangle chaos, he was resolved to convey at least some light into that of the fables of antiquity which are blended and confounded with history, and fix an uncertain chronology. It is true that there is no family, city, or nation, but endeavours to remove its original as far backward as possible. Besides, the first historians were the most negligent in setting down the eras: books were infinitely less common than they are at this time, and, consequently, authors being not so obnoxious to censure, they therefore imposed upon the world with greater impunity; and, as it is evident that these have related a great number of fictitious particulars, it is probable enough that they also gave us several false eras.

Used to untangling chaos, he was determined to shed at least some light on ancient myths mixed up with history. He also aimed to put an uncertain chronology in order. It is true that every family, city, and nation tries to trace its origins as far back as possible. Besides, the earliest historians were the most careless about recording dates. Books were far less common than they are today, so authors faced less criticism. As a result, authors could deceive the world with less fear of consequences. And since those historians clearly reported many invented details, it is quite likely they also gave us several false dates.

It appeared in general to Sir Isaac that the world was five hundred years younger than chronologers declare it to be. He grounds his opinion on the ordinary course of Nature, and on the observations which astronomers have made.

In general, Sir Isaac believed the world was five hundred years younger than experts in chronology claimed. He based his view on the usual course of nature and on astronomers’ observations.

By the course of Nature we here understand the time that every generation of men lives upon the earth. The Egyptians first employed this vague and uncertain method of calculating when they began to write the beginning of their history. These computed three hundred and forty-one generations from Menes to Sethon; and, having no fixed era, they supposed three generations to consist of a hundred years. In this manner they computed eleven thousand three hundred and forty years from Menes’s reign to that of Sethon.

By the course of Nature, we mean the length of time each generation lives on earth. The Egyptians first used this vague and uncertain way of measuring time when they began writing their earliest history. They counted 341 generations from Menes to Sethon. Since they had no fixed starting point for dating events, they assumed that three generations lasted a hundred years. By this calculation, they placed 11,340 years between the reigns of Menes and Sethon.

The Greeks before they counted by Olympiads followed the method of the Egyptians, and even gave a little more extent to generations, making each to consist of forty years.

Before counting time by Olympiads, the Greeks used the Egyptians’ method. They even made generations a little longer, counting each as forty years.

Now, here, both the Egyptians and the Greeks made an erroneous computation. It is true, indeed, that, according to the usual course of Nature, three generations last about a hundred and twenty years; but three reigns are far from taking up so many. It is very evident that mankind in general live longer than kings are found to reign, so that an author who should write a history in which there were no dates fixed, and should know that nine kings had reigned over a nation; such an historian would commit a great error should he allow three hundred years to these nine monarchs. Every generation takes about thirty-six years; every reign is, one with the other, about twenty. Thirty kings of England have swayed the sceptre from William the Conqueror to George I., the years of whose reigns added together amount to six hundred and forty-eight years; which, being divided equally among the thirty kings, give to every one a reign of twenty-one years and a half very near. Sixty-three kings of France have sat upon the throne; these have, one with another, reigned about twenty years each. This is the usual course of Nature. The ancients, therefore, were mistaken when they supposed the durations in general of reigns to equal that of generations. They, therefore, allowed too great a number of years, and consequently some years must be subtracted from their computation.

Here, both the Egyptians and the Greeks made a mistake in their calculations. It is true that, in the usual course of nature, three generations last about a hundred and twenty years. But three reigns last far less time. Clearly, people generally live longer than kings reign. Suppose a historian were writing a history without fixed dates and knew that nine kings had ruled a nation. The historian would make a serious mistake by assigning three hundred years to those nine monarchs. Each generation lasts about thirty-six years. Each reign lasts about twenty years on average. Thirty kings of England ruled from William the Conqueror to George I. Their reigns together total six hundred and forty-eight years. Dividing that total equally among the thirty kings gives each a reign of nearly twenty-one and a half years. Sixty-three kings of France have occupied the throne, reigning about twenty years each on average. This is the usual course of nature. The ancients were therefore mistaken in assuming that reigns generally lasted as long as generations. They consequently counted too many years, so some years must be subtracted from their calculations.

Astronomical observations seem to have lent a still greater assistance to our philosopher. He appears to us stronger when he fights upon his own ground.

Astronomical observations seem to have helped our philosopher even more. He seems stronger to us when he fights on his own ground.

You know that the earth, besides its annual motion which carries it round the sun from west to east in the space of a year, has also a singular revolution which was quite unknown till within these late years. Its poles have a very slow retrograde motion from east to west, whence it happens that their position every day does not correspond exactly with the same point of the heavens. This difference, which is so insensible in a year, becomes pretty considerable in time; and in threescore and twelve years the difference is found to be of one degree, that is to say, the three hundred and sixtieth part of the circumference of the whole heaven. Thus after seventy-two years the colure of the vernal equinox which passed through a fixed star, corresponds with another fixed star. Hence it is that the sun, instead of being in that part of the heavens in which the Ram was situated in the time of Hipparchus, is found to correspond with that part of the heavens in which the Bull was situated; and the Twins are placed where the Bull then stood. All the signs have changed their situation, and yet we still retain the same manner of speaking as the ancients did. In this age we say that the sun is in the Ram in the spring, from the same principle of condescension that we say that the sun turns round.

You know that the earth travels around the sun from west to east once a year. Besides this annual motion, the earth has another unusual rotation, unknown until recent years. Its poles move very slowly backward, from east to west. As a result, each day their position shifts slightly relative to the heavens. This shift is barely noticeable over a year, but becomes quite considerable over time. In seventy-two years, it amounts to one degree, or one three-hundred-and-sixtieth of the full circle of the heavens. Thus, after seventy-two years, the colure of the vernal equinox, a line that passed through one fixed star, aligns with another. As a result, the sun no longer appears in the part of the heavens occupied by the Ram in Hipparchus’s time. Instead, the sun appears in the part occupied by the Bull, and the Twins stand where the Bull once stood. All the signs have changed position, yet we still speak as the ancients did. Today we say that the sun is in the Ram in spring. We make the same concession to common speech when we say that the sun moves around the earth.

Hipparchus was the first among the Greeks who observed some change in the constellations with regard to the equinoxes, or rather who learnt it from the Egyptians. Philosophers ascribed this motion to the stars; for in those ages people were far from imagining such a revolution in the earth, which was supposed to be immovable in every respect. They therefore created a heaven in which they fixed the several stars, and gave this heaven a particular motion by which it was carried towards the east, whilst that all the stars seemed to perform their diurnal revolution from east to west. To this error they added a second of much greater consequence, by imagining that the pretended heaven of the fixed stars advanced one degree eastward every hundred years. In this manner they were no less mistaken in their astronomical calculation than in their system of natural philosophy. As for instance, an astronomer in that age would have said that the vernal equinox was in the time of such and such an observation, in such a sign, and in such a star. It has advanced two degrees of each since the time that observation was made to the present. Now two degrees are equivalent to two hundred years; consequently the astronomer who made that observation lived just so many years before me. It is certain that an astronomer who had argued in this manner would have mistook just fifty-four years; hence it is that the ancients, who were doubly deceived, made their great year of the world, that is, the revolution of the whole heavens, to consist of thirty-six thousand years. But the moderns are sensible that this imaginary revolution of the heaven of the stars is nothing else than the revolution of the poles of the earth, which is performed in twenty-five thousand nine hundred years. It may be proper to observe transiently in this place, that Sir Isaac, by determining the figure of the earth, has very happily explained the cause of this revolution.

Hipparchus was the first Greek to observe some change in the constellations’ positions relative to the equinoxes. Or rather, he learned of the change from the Egyptians. Philosophers attributed this motion to the stars. In those days, people were far from imagining such a movement in the earth, which they thought was completely motionless. So they invented a heaven in which they placed the stars. They gave this heaven its own eastward motion, while all the stars seemed to move daily from east to west. To this error they added a second, far more significant one. They imagined that this supposed heaven of fixed stars moved one degree eastward every hundred years. Their astronomical calculations were therefore just as mistaken as their understanding of nature. For example, an astronomer of that time would have reasoned as follows. At the time of a particular observation, the spring equinox was in a particular sign and at a particular star. Since that observation, the equinox has advanced two degrees relative to each. Now, two degrees equal two hundred years. So the astronomer who made that observation lived exactly two hundred years before me. An astronomer reasoning this way would certainly have been wrong by exactly fifty-four years. These two errors led the ancients to assign thirty-six thousand years to their great year of the world. By this they meant one complete revolution of the heavens. But modern thinkers recognize that this imaginary revolution of the stars’ heaven is simply the revolution of the earth’s poles. That revolution takes twenty-five thousand nine hundred years. It is worth noting in passing that Sir Isaac, by determining the earth’s shape, very successfully explained what causes this revolution.

All this being laid down, the only thing remaining to settle chronology is to see through what star the colure of the equinoxes passes, and where it intersects at this time the ecliptic in the spring; and to discover whether some ancient writer does not tell us in what point the ecliptic was intersected in his time, by the same colure of the equinoxes.

With all this established, only one task remains to settle the chronology. We must identify the star through which the colure of the equinoxes passes. We must also locate where this colure currently crosses the ecliptic in spring. Then we must see whether some ancient writer tells us where the same colure crossed the ecliptic in his time.

Clemens Alexandrinus informs us, that Chiron, who went with the Argonauts, observed the constellations at the time of that famous expedition, and fixed the vernal equinox to the middle of the Ram; the autumnal equinox to the middle of Libra; our summer solstice to the middle of Cancer, and our winter solstice to the middle of Capricorn.

Clemens Alexandrinus tells us that Chiron accompanied the Argonauts and observed the constellations during that famous expedition. Chiron placed the spring equinox in the middle of the Ram and the autumn equinox in the middle of Libra. He placed our summer solstice in the middle of Cancer and our winter solstice in the middle of Capricorn.

A long time after the expedition of the Argonauts, and a year before the Peloponnesian war, Methon observed that the point of the summer solstice passed through the eighth degree of Cancer.

Long after the expedition of the Argonauts, and a year before the Peloponnesian war, Methon made an observation. He observed that the summer solstice point passed through the eighth degree of Cancer.

Now every sign of the zodiac contains thirty degrees. In Chiron’s time, the solstice was arrived at the middle of the sign, that is to say to the fifteenth degree. A year before the Peloponnesian war it was at the eighth, and therefore it had retarded seven degrees. A degree is equivalent to seventy-two years; consequently, from the beginning of the Peloponnesian war to the expedition of the Argonauts, there is no more than an interval of seven times seventy-two years, which make five hundred and four years, and not seven hundred years as the Greeks computed. Thus in comparing the position of the heavens at this time with their position in that age, we find that the expedition of the Argonauts ought to be placed about nine hundred years before Christ, and not about fourteen hundred; and consequently that the world is not so old by five hundred years as it was generally supposed to be. By this calculation all the eras are drawn nearer, and the several events are found to have happened later than is computed. I do not know whether this ingenious system will be favourably received; and whether these notions will prevail so far with the learned, as to prompt them to reform the chronology of the world. Perhaps these gentlemen would think it too great a condescension to allow one and the same man the glory of having improved natural philosophy, geometry, and history. This would be a kind of universal monarchy, with which the principle of self-love that is in man will scarce suffer him to indulge his fellow-creature; and, indeed, at the same time that some very great philosophers attacked Sir Isaac Newton’s attractive principle, others fell upon his chronological system. Time that should discover to which of these the victory is due, may perhaps only leave the dispute still more undetermined.

Each sign of the zodiac contains thirty degrees. In Chiron’s time, the solstice fell in the middle of the sign, at the fifteenth degree. A year before the Peloponnesian war, the solstice was at the eighth degree. So it had moved backward seven degrees. One degree equals seventy-two years. Therefore, the interval between the beginning of the Peloponnesian war and the expedition of the Argonauts is only seven times seventy-two years. That makes five hundred and four years, not seven hundred years as the Greeks calculated. Comparing the positions of the heavens in those two periods places the expedition about nine hundred years before Christ. It should not be placed about fourteen hundred years before Christ. Consequently, the world is five hundred years younger than people generally supposed. This calculation brings all the eras closer to the present. It places the various events later than the accepted calculations do. I do not know whether this ingenious system will be well received. Nor do I know whether these ideas will persuade scholars to revise the chronology of the world. Perhaps these gentlemen would consider it beneath them to grant one man the glory of advancing natural philosophy, geometry, and history. Such glory would amount to a kind of universal monarchy. Human self-love will hardly allow anyone to grant a fellow human being such power. Indeed, while some very great philosophers attacked Sir Isaac Newton’s principle of attraction, others attacked his chronological system. Time should reveal which side deserves victory. Yet time may perhaps only leave the dispute even less settled.