Read Is God a Mathematician? Online
Authors: Mario Livio
Is God a Mathematician? (15 page)
On 15 April 1726 I paid a visit to Sir Isaac at his lodgings in Orbels buildings in Kensington, dined with him and spent the whole day with him, alone…After dinner, the weather being warm, we went into the garden and drank thea, under the shade of some apple trees, only he and myself. Amidst other discourse, he told me he was just in the same situation, as when formerly [in 1666, when Newton returned home from Cambridge because of the plague], the notion of gravitation came into his mind. It was occasion’d by the fall of an apple, as he sat in contemplative mood. Why should that apple always descend perpendicularly to the ground, thought he to himself. Why should it not go sideways or upwards, but constantly to the earth’s centre? Assuredly, the reason is, that the earth draws it. There must be a drawing power in matter: and the sum of the drawing power in the matter of the earth must be in the earth’s centre, not in any side of earth. Therefore does this apple fall perpendicularly, or towards the centre. If matter thus draws matter, it must be in proportion of its quantity. Therefore the apple draws the earth, as well as the earth draws the apple. That there is a power, like that we here call gravity, which extends its self thro’ the universe…This was the birth of those amazing discoverys, whereby he built philosophy on a solid foundation, to the astonishment of all Europe.
Irrespective of whether the mythical event with the apple actually occurred in 1666 or not, the legend sells Newton’s genius and unique depth of analytic thinking rather short. While there is no doubt that Newton had written his first manuscript on the theory of gravity before 1669, he did not need to physically see a falling apple to know that the Earth attracted objects near its surface. Nor could his incredible insight in the formulation of a universal law of gravitation stem from the mere sight of a falling apple. In fact, there are some indica
tions that a few crucial concepts that Newton needed to be able to enunciate a universally acting gravitational force were only conceived as late as 1684–85. An idea of this magnitude is so rare in the annals of science that even someone with a phenomenal mind—such as Newton—had to arrive at it through a long series of intellectual steps.
It may have all started in Newton’s youth, with his less-than-perfect encounter with Euclid’s massive treatise on geometry,
The Elements.
According to Newton’s own testimony, he first “read only the titles of the propositions,” since he found these so easy to understand that he “wondered how any body would amuse themselves to write any demonstrations of them.” The first proposition that actually made him pause and caused him to introduce a few construction lines in the book was the one stating that “in a right triangle the square of the hypothenuse is equal to the squares of the two other sides”—the Pythagorean theorem. Somewhat surprisingly perhaps, even though Newton did read a few books on mathematics while at Trinity College in Cambridge, he did not read many of the works that were already available at this time. Evidently he didn’t need to!
The one book that turned out to be perhaps the most influential in guiding Newton’s mathematical and scientific thought was none other than Descartes’
La Géométrie
. Newton read it in 1664 and re-read it several times, until “by degrees he made himself master of the whole.” The flexibility afforded by the notion of functions and their free variables appeared to open an infinitude of possibilities for Newton. Not only did analytic geometry pave the way for Newton’s founding of calculus, with its associated exploration of functions, their tangents, and their curvatures, but Newton’s inner scientific spirit was truly set ablaze. Gone were the dull constructions with ruler and compass—they were replaced by arbitrary curves that could be represented by algebraic expressions. Then in 1665–66, a horrible plague hit London. When the weekly death toll reached the thousands, the colleges of Cambridge had to close down. Newton was forced to leave school and return to his home in the remote village of Woolsthorpe. There, in the tranquility of the countryside, Newton made his first attempt to prove that the force that held the Moon in its orbit around the
Earth and the Earth’s gravity (the very force that caused apples to fall) were, in fact, one and the same. Newton described those early endeavors in a memorandum written around 1714:
And the same year [1666] I began to think of gravity extending to the orb of the Moon, and having found out how to estimate the force with which [a] globe revolving within a sphere presses the surface of the sphere, from Kepler’s Rule of the periodical times of the Planets being in a sesquialternate proportion of their distances from the centres of their Orbs I deduced that the forces which keep the Planets in their Orbs must [be] reciprocally as the squares of their distances from the centres about which they revolve: and thereby compared the force requisite to keep the Moon in her Orb with the force of gravity at the surface of the earth, and found them answer pretty nearly. All this was in the two plague years of 1665 and 1666, for in those years I was in the prime of my age for invention, and minded Mathematicks and Philosophy more than at any time since.
Newton refers here to his important deduction (from Kepler’s laws of planetary motion) that the gravitational attraction of two spherical bodies varies inversely as the square of the distance between them. In other words, if the distance between the Earth and the Moon were to be tripled, the gravitational force that the Moon would experience would be nine times (three squared) smaller.
For reasons that are not entirely clear, Newton essentially abandoned any serious research on the topics of gravitation and planetary motion until 1679. Then two letters from his archrival Robert Hooke renewed his interest in dynamics in general and in planetary motion in particular. The results of this revived curiosity were quite dramatic—using his previously formulated laws of mechanics, Newton proved Kepler’s second law of planetary motion. Specifically, he showed that as the planet moves in its elliptical orbit about the Sun, the line joining the planet to the Sun sweeps equal areas in equal time intervals (figure 28). He also proved that for “a body revolving in an ellipse…the
law of attraction directed to a focus of the ellipse…is inversely as the square of the distance.” These were important milestones on the road to
Principia
.
Figure 28
Principia
Halley came to visit Newton in Cambridge in the spring or summer of 1684. For some time Halley had been discussing Kepler’s laws of planetary motion with Hooke and with the renowned architect Christopher Wren (1632–1723). At these coffeehouse conversations, both Hooke and Wren claimed to have deduced the inverse-square law of gravity some years earlier, but both were also unable to construct a complete mathematical theory out of this deduction. Halley decided to ask Newton the crucial question: Did he know what would be the shape of the orbit of a planet acted upon by an attractive force varying as an inverse-square law? To his astonishment, Newton answered that he had proved some years earlier that the orbit would be an ellipse. The mathematician Abraham de Moivre (1667–1754) tells the story in a memorandum (from which a page is shown in figure 29):
In 1684 D
r
Halley came to visit him [Newton] at Cambridge, after they had been some time together, the D
r
asked him what he thought the curve would be that would be described by the
planets supposing the force of attraction towards the sun to be reciprocal to the square of their distance from it. S
r
Isaac replied immediately that it would be an Ellipsis [ellipse], the Doctor struck with joy and amazement asked him how he knew it, why saith he [Newton] I have calculated it, whereupon D
r
Halley asked him for his calculation without any farther delay, S
r
Isaac looked among his papers but could not find it, but he promised him to renew it and send it.
Halley indeed came to visit Newton again in November 1684. Between the two visits Newton worked frantically. De Moivre gives us a brief description:
Figure 29
Sr
Isaac in order to make good his promise fell to work again but he could not come to that conclusion w
ch
he thought he had before examined with care, however he attempted a new way which tho
u
longer than the first, brought him again to his former conclusion, then he examined carefully what might be the reason why the calculation he had undertaken before did not prove right, &…he made both his calculations agree together.
This dry summary does not even begin to tell us what Newton had actually accomplished in the few months between Halley’s two visits. He wrote an entire treatise,
De Motu Corporum in Gyrum
(
The Motion of Revolving Bodies
), in which he proved most aspects of bodies moving in circular or elliptical orbits, proved all of Kepler’s laws, and even solved for the motion of a particle moving in a resisting medium (such as air). Halley was overwhelmed. To his satisfaction, he at least managed to persuade Newton to publish all of these staggering discoveries—
Principia
was finally about to happen.
At first, Newton had thought of the book as being nothing but a somewhat expanded and more detailed version of his treatise
De Motu
. As he started working, however, he realized that some topics required further thought. Two points in particular continued to disturb Newton. One was the following: Newton originally formulated his law of gravitational attraction as if the Sun, Earth, and planets were mathematical point masses, without any dimensions. He of course knew this not to be true, and therefore he regarded his results as only approximate when applied to the solar system. Some even speculate that he abandoned again his pursuit of the topic of gravity after 1679 because of his dissatisfaction with this state of affairs. The situation was even worse with respect to the force on the apple. There, clearly the parts of the Earth that are directly underneath the apple are at a much shorter distance to it than the parts that are on the other side of the Earth. How was one to calculate the net attraction? The astronomer Herbert Hall Turner (1861–1930) described Newton’s mental struggle in an article that appeared in the London
Times
on March 19, 1927:
At that time the general idea of an attraction varying as the inverse square of the distance occurred to him, but he saw grave difficulties in its complete application of which lesser minds were unconscious. The most important of these he did not overcome until 1685…It was that of linking up the attraction of the earth on a body so far away as the moon with its attraction on the apple close to its surface. In the former case the various particles composing the earth (to which individually Newton hoped to extend his law, thus making it universal) are at distances from the moon not greatly different either in magnitude or direction; but their distances from the apple differ conspicuously in both size and direction. How are the separate attractions in the latter case to be added together or combined into a single resultant? And in what “centre of gravity,” if any, may they be concentrated?
The breakthrough finally came in the spring of 1685. Newton managed to prove an essential theorem: For two spherical bodies, “the whole force with which one of these spheres attracts the other will be inversely proportional to the square of the distance of the centres.” That is, spherical bodies gravitationally act as if they were point masses concentrated at their centers. The importance of this beautiful proof was emphasized by the mathematician James Whitbread Lee Glaisher (1848–1928). In his address at the bicentenary celebration (in 1887) of Newton’s
Principia,
Glaisher said:
No sooner had Newton proved this superb theorem—and we know from his own words that he had no expectation of so beautiful a result till it emerged from his mathematical investigation—than all the mechanism of the universe at once lay spread before him…How different must these propositions have seemed to Newton’s eyes when he realised that these results, which he had believed to be only approximately true when applied to the solar system, were really exact!…We can imagine the effect of this sudden transition from approximation to exactitude in stimulating Newton’s mind to still greater
efforts. It was now in his power to apply mathematical analysis with absolute precision to the actual problem of astronomy.