Above Title page of “Acta Eruditorium Anno, 1684,” Leipzig, 1684, in which appeared Gottfried Wilhelm von Leibniz’s paper on his discover of the differential calculus, “Nova Methodus pro Maximus et Minimus
Curiously, Newton was very reticent about discussing what he called the method of “fluxions.” He apparently discovered the Fundamental Theorem between 1664 and 1666, but showed his work only in bits and pieces to a handful of people. Mathematicians had not yet realised that publishing, not hoarding secrets, was the surest route to progress.
When Leibniz started thinking about rates of change and infinite sums in the 1670s, he surely had second-hand information about what Newton claimed he could do: namely, that he could compute infinite sums, areas, arc lengths, and so on. Sometime between 1673 and 1675, Leibniz unlocked the Fundamental Theorem, too. At this point he contacted Newton, to find out exactly what Newton knew, and proposed a sort of exchange of information: you tell me this, and I will tell you that.
Newton wrote back very cautiously, sending Leibniz only two letters. In the second, he divulged the Fundamental Theorem of Calculus—but concealed it in an indecipherable anagram. It is obvious that Newton did not want to share his discovery with Leibniz—he only wanted to be able to prove that he had gotten it first, in case Leibniz should later claim it as his own.
UNFORTUNATELY, THAT IS EXACTLY what then happened. Leibniz published his version of calculus in 1684 in a book called Nova methodus, while Newton, amazingly, waited until 1704 before publishing his first account of the method of fluxions. An extremely bitter dispute ensued over who should be known as the discoverer of calculus. English mathematicians supported Newton, while continental scholars mostly sided with Leibniz. Each side accused the other of plagiarism.
The consensus of modern historians is that they both were wrong, and they both were right. There was no plagiarism on either side, and both men independently made the same discovery. Newton undoubtedly was aware of the Fundamental Theorem first, but as I have said before, it does no good to discover America and then keep it to yourself. Leibniz was the first to tell the world about calculus. Partly for that reason, and partly because Leibniz’s notation was simpler, the notation we use today is almost entirely due to Leibniz. No one today talks about “fluxions” and “fluents”—the words died with Newton.
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† Here I have to confess that an undergraduate student can only succeed in a relatively small number of cases. However, beginning in the 1800s and continuing to the present day, the more difficult cases, such as rectifying an ellipse or a lemniscate, led to deep and beautiful new theories.
11
of apples, legends … and comets newton’s laws
Ask most people what they know about Isaac Newton, and there is a good chance that they will tell you about an apple falling from a tree. According to legend, Newton was inspired to formulate his universal law of gravitational attraction by witnessing the fall of an apple, and realizing that the same force that explained its motion could also explain the motion of the planets. In some more recent embellishments, perhaps Newton was inspired by being hit on the head by the apple.
Here is an equally unverifiable counterlegend, which first appeared in print in 1858, in a delightful English journal called Notes & Queries. According to a contributor named “W.”, Karl Friedrich Gauss—the leading mathematician of the day—dismissed the legend as follows: “The history of the apple is too absurd. Undoubtedly, the occurrence was something of this sort. There comes to Newton a stupid importunate man, who asks him how he hit upon his great discovery. When Newton had convinced himself what a noodle he had to do with, and wanted to get rid of the man, he told him that an apple fell on his nose; and this made the matter quite clear to the man, and he went away satisfied.”
What is one to make of such legends? In reality, there is more substance to the apple story than Gauss realized (if the quote from him is authentic). The story is attested by two sources, one of whom was the famous French writer Voltaire, who heard it personally from Newton’s niece. Hardly the sort of “stupid importunate man” that Gauss envisioned! One might think of the story as a highly encoded version of what actually happened. If you do not know the code, then you end up with the cartoonish story that Gauss so vigorously objected to.
The first equation is Newton’s Second Law of Motion, the second is Newton’s Law of Universal Gravitation. In both equations, F represents a force. The symbol a represents the acceleration of an object with mass m. In the law of universal gravitation, F is specifically the gravitational force between masses m and M, while r represents the distance between the objects. G is the universal gravitational constant, 6.672 × 10-8 cm3 g-1 sec-2.
Was there an apple tree? Yes. It was located at Isaac Newton’s homestead in Woolsthorpe, England. He had lived there until 1661, when he went to Trinity College at Cambridge, and most importantly he returned there in 1665, when the last major outbreak of plague struck England. For close to two years Newton remained in his rural sanctuary. Those were the two years during which he developed the basics of calculus and began thinking about planetary motions. Newton wrote, “In those days I was in the prime of my age for invention and minded Mathematics and Philosophy more than at any time since.”
However, Newton did not need the fall of an apple for inspiration. Gauss was right about that. Newton was surely inspired by the problem itself, which already had centuries of history behind it. Did the Moon, the Sun, and the planets require some sort of external agency to make them move? If so, what was it? Aristotle had argued that heavenly bodies were made of different stuff than Earth, and that their natural motion was circular. Kepler thought that a propulsive force was needed to keep the planets in their orbits. Descartes more or less agreed; in his elaborate theory, the universe was composed of vortices that swept the planets along in their orbits. It is only natural that Newton, as a young scholar, would have been passionately interested in one of the leading scientific debates of the day. He worked out a system in which apples are subject to the same forces as planets. (This is why the apple is important! It refutes Aristotle.) And remarkably, the planets are in free fall at all times; they require no propulsion. (This refutes Descartes and Kepler.)
Opposite The first reflecting telescope, made by Issac Newton in 1668, stands by his manuscript of Principia Mathematica.
While the apple story has some merit, it fails to explain how Newton convinced the rest of the scientific world that his theory was correct. His masterpiece, The Mathematical Principles of Natural Philosophy (often called the Principia after its Latin title), set out to do for physics exactly what Euclid had done for geometry. At the very outset Newton stated three axioms: three laws of motion that all material objects obey, whether they be apples or moons. Later he added the law of universal gravitation, which quantifies how objects attract one another through gravity. From these principles alone, he proved that planets orbiting the Sun obey Kepler’s three laws.
In fact, Kepler’s first law is probably the main reason Newton wrote the Principia. Several other physicists—notably Newton’s rival Robert Hooke, the architect Christopher Wren, and the Dutch physicist Christian Huygens—had also arrived at “Newton’s” law of universal gravitation by the early 1680s. But they had been unable to show that the law causes planets to orbit in ellipses; they could only account for the mathematically much simpler case of circular orbits. In 1684, Newton’s friend Edmund Halley asked Newton if he could prove that planets had elliptical orbits. Newton said that he could, and Halley cajoled him into putting his argument into print. The result, three years later, was much more than the solution of one problem; it was the blueprint for all future physics books.
Halley, who paid for part of the printing costs out of his own pocket, was eventually rewarded for his efforts in a very unique way. Newton’s theory applies to comets, as well as to apples and planets. (In fact, Newton himself emphasized this point.) Because comets follow elliptical orbits, they
must return over and over again. Halley realized that one comet in particular had been seen repeatedly at roughly 75-year-intervals: in 1456, 1531, 1606, and 1682. Thus he predicted, correctly, that it would return in 1758 (long after his own death). It has continued to return every 75 to 76 years ever since then, and is now known as Halley’s Comet.
NEWTON’S FIRST LAW states that a moving object will continue moving in a straight line forever, unless some external force stops it or changes its path. This seems quite surprising at first: after all, golf balls don’t keep going forever, and planets don’t move in straight lines. In both cases, the reason is that there are external forces acting on the object. In the case of the golf ball, the forces are gravity, wind resistance (while the ball is in the air), and friction with the ground after it lands. In the case of planets, the hidden force is the Sun’s gravity.
Newton’s second law says that the force on an object equals the rate of change of its momentum. In the language of calculus, we would say that:
recalling that d/dt denotes the rate of change and mv (where m is the mass of the object and v is its velocity) denotes the momentum. In most applications the mass of the object does not change, and in this case Newton’s second law becomes F = ma (i.e., force equals mass times acceleration), a formula that is today memorized by every beginning physics student.
Newton’s third law, “for every action there is an equal and opposite reaction,” is somewhat less often used by physicists than the first two, but it explains, for example, why a rocket works. The action of propelling exhaust out of the rocket’s nozzles creates a reaction: the acceleration of the rocket in the opposite direction.
Collectively, these three laws explain how all forces affect the motion of all solid bodies. On the other hand, Newton’s law of gravitation pertains to one force only, the force of gravity. It states that the gravitational attraction between any two objects, one of mass M and the other of mass m, is:
The denominator r2 indicates that the strength of the gravitational force is inversely proportional to the square of the planet’s distance (r) from the Sun. (This is the part of the formula that Hooke, Wren, and Huygens had already guessed.) The minus sign and the vector (read as “r-hat”) indicates that the direction of the force is toward the Sun. In other words, Kepler and Descartes were wrong. There is no force pushing the planets forward in their orbits, only a gravitational force pulling them to the side (that is, toward the Sun).
Newton’s truly novel accomplishment was his ability, using calculus,‡ to combine the law of gravitation with his laws of motion to set up—and then solve—equations describing a planet’s orbit. Together, his physical insight and his mathematical tools ushered in a new era of celestial dynamics, when the motion of planets—and eventually, rockets and spacecraft—could be predicted and controlled, rather than merely observed.
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‡ It is often claimed that Newton deliberately avoided the use of calculus in the Principia, rewriting all the proofs in terms of Euclidean geometry. It is true that he avoids the notation of calculus, but his work is fully imbued with the ideas of calculus.
12
the great explorer euler’s theorems
In 1988, the magazine Mathematical Intelligencer organized a poll to determine the most beautiful mathematical theorems in history. Amazingly, four of the top five theorems on the list were proved by the same man: Leonhard Euler. Even more remarkably, it is easy to come up with a half-dozen more theorems by Euler that could have made the list. In fact, Euler authored more than 800 articles and about 50 books and memoirs. The Academy of Sciences in St. Petersburg (where he spent the last 17 years of his life) was unable to keep up with his output, and continued publishing articles by Euler for half a century after his death!
Leonhard Euler was born in Basel, Switzerland, in 1707, and remained proud of his native country and town throughout his life, even though he never again set foot in Basel after age twenty. He had the good fortune to come of age in an era when mathematics was beginning to turn from a scholarly pursuit into a profession. England’s Royal Society had been founded in 1660, and the French Academy of Sciences soon afterward, in 1666. Gottfried Wilhelm Leibniz, after returning from France, persuaded King Frederick I to establish the Prussian Academy of Sciences in 1700. By the early 1720s, when Tsar Peter I of Russia was building his new capital at St. Petersburg, an academy of science was almost de rigueur for a royal court. Monarchs had begun to realize that mathematicians and scientists could play an important role in building their countries’ infrastructure and military prowess.
The number e = 2.718281828459045… is the base of the natural logarithm function and the second most ubiquitous constant in mathematics, after π. The letter i represents the imaginary unit, . The functions cos, and sin are the cosine function, and sine function respectively.
In 1724, Peter I founded the Russian Academy of Sciences and invited a number of foreign scientists to move to his raw new capital. At that time, the opportunities for a mathematician in Switzerland were limited, so Euler seized the opportunity. During his first period in Russia, from 1727 to 1741, Euler’s reputation throughout Europe rose rapidly. In 1735, he stunned the world of mathematics by evaluating an apparently simple infinite sum, that no one had been able to crack:
demonstrating that it is equal to:
Euler’s argument is explained in masterful fashion in George Pólya’s 1954 book Mathematics and Plausible Reasoning, Part I. For readers who know their trigonometry and have some knowledge of infinite series, it is one of the best expositions of how a mathematical genius thinks (both Pólya and Euler).
Euler continued to move from triumph to triumph. His book Mechanica, published in 1736, took mechanics out of the realm of Euclidean geometry, where Newton had awkwardly placed it, and rephrased it in the much more appropriate language of calculus. In 1738, Euler won the Grand Prix de Paris competition for the first time.* During this period, Euler introduced notation that is used by all mathematicians today: e for the base of the natural logarithms, i for √–1 and f(x) for functions.
Unfortunately, after Empress Anna died in 1740, a backlash set in against the foreigners whom Peter had invited to “cut a window through to Europe.” Euler found his situation increasingly untenable, and accepted an invitation from Emperor Frederick II of Prussia to join the academy of sciences in Berlin.
Opposite Hand-colored engraving of Swiss mathematician Euler, ca. 1770.
If Euler in St. Petersburg was an up-and-coming star, Euler in Berlin was a mature scientist at the height of his power. Among other things, during this period he resuscitated Fermat’s number theory and re-proved most of the things that Fermat had claimed to prove (except, of course, the Last Theorem). He worked out how Newton’s laws of motion apply to fluids, deriving what are still known as Euler’s equations of hydrodynamics. He published books on topics ranging from calculus to naval science. And he worked on some of the ongoing puzzles of astrophysics.
Although Euler prospered in Berlin, the one person he never managed to impress was his employer. The mercurial monarch, Frederick II, was greatly attracted to pomp and culture, particularly French culture, and he could not abide the stolid Swiss mathematician in his court. Though he conceded that Euler was “useful,” Frederick compared him to a Doric column, “anything but elegant.” On another occasion, he wrote to Voltaire, “We have here a great Cyclops of mathematics,” referring unkindly to the fact that Euler had lost the vision in his right eye. Euler longed to become president of the Berlin Academy of Sciences, but it was clear that under Frederick II this would never happen, so in 1766 he accepted Empress Catherine II’s invitation to return to Russia.
During the last years of his life Euler’s work continued unabated, even though a failed cataract operation in 1771 left him nearly blind. He gradually withdrew from the St. Petersburg Academy because of internal politics. However, in 1783, the last year of Euler’s life, Princess Ekaterina Dashkova took over the directorsh
ip of the Academy and insisted on making her entrance with him. When she realized that the seat next to hers was taken, she wrote, “I therefore turned to Mr. Euler and told him to sit down where he thought fit, for any place he occupied would always be the first.” Euler had finally found a place where he was appreciated.
TWO HUNDRED YEARS LATER, mathematicians still appreciate him. Let us take a look at the four theorems of Euler that the readers of Mathematical Intelligencer rated in the top five of all time: eiπ + 1 = 0 (or eiπ = –1) Surely one of the most paradoxical statements in mathematics, it is often written in the former way because this allegedly “unifies” the five most important constants of mathematics: 0, 1, π, e, and i.
Above Title page of the first edition of Leonhard Euler’s Methodus Inveniendi Lineas Curvas, 1744, on the creation of the calculus of variations.
Here is what the “most beautiful equation in history” really means. Probably the most important function in calculus is the exponential function exp(x), because it is the only function that is its own derivative and its own integral. The name is apt because the values of the function are all powers of the number exp(1). For instance, exp(2) = (exp 1)2, exp(3) = (exp 1)3, and so forth. To save space, we can call the number exp(1) “e”, as Euler did. Then the seemingly nonsensical number eπ (e multiplied by itself π times, whatever that means) can be defined as exp(π). Because the function exp(x) is defined by calculus it can be computed using calculus. Thus we can determine that exp(π) = 23.1406…
The Universe in Zero Words Page 8