Why planets should move in ellipses is easy to show in mathematical terms; leaving mathematics aside, one may visualize the mechanism as a tug-of-war between gravity and centrifugal force. If the string to which the revolving stone is attached be made of elastic material, one can imagine it alternately expanding and contracting, thus making the stone's orbit an oval. * Or one can visualize the process as follows: as the planet approaches the sun, its speed increases. It shoots past the sun, but as it does so, the clutching hand of gravity swings it round – as a running child grabbing at a maypole is swung around it – so that it now continues in the opposite direction. If its velocity on the approach-run had been exactly the amount required to prevent it from falling into the sun, it would continue in a circle. But as it was slightly greater, the receding run will carry it into an elongated path, which the planet pursues at slackening speed in the teeth of the sun's attraction, as it were, gradually curving inward; until, after passing the aphelion, the curve again approaches the sun and the whole cycle starts again.
____________________
*
The analogy between elastic resistance and gravitational force is, of course, quite wrong, but it may help to get the "feel" of the elliptic orbit.
The "eccentricity" of the ellipse is the amount by which it deviates from the circle. The eccentricities of the planets are small, owing to the common origin of the solar system which makes their tangential velocities almost precisely balance gravity.
But all this was as yet merely conjecture; and the days of purely speculative hypotheses were past. It was wild conjecture to postulate that the moon was constantly "falling" towards the earth, like a projectile, or like the famous apple in the garden at Woolsthorpe – in other words, that the earth's attraction reached as far as the moon, the sun's attraction as far as the planets, and that interstellar space was indeed "filled" or "charged" with gravity. To transform a wild guess into scientific theory, Newton had to provide rigorous mathematical proof.
This means, he had to calculate: (a) the centrifugal force of the moon; 4 (b) the gravitational force which the earth was supposed to exert on the moon; and (c) he had to show that the interaction of these two forces produced a theoretical orbit which agreed with the moon's observed orbit.
In order to carry out this operation, he must first of all know at what rate the earth's gravity diminished with distance. The apple fell from the tree at a known acceleration of approximately ten yards added speed per second; but what would be the acceleration of the distant moon towards the earth? In other words, he had to discover the Law of Gravity – that the force diminishes with the square of distance. In the second place, he had to know the exact value of the moon's distance. Thirdly, he had to decide whether it was legitimate to treat two huge globes like the earth and the moon in an abstract manner, as if their whole mass were concentrated in a single central point. Lastly, to reduce the mathematical difficulties, the moon's orbit had to be treated as if it were a circle instead of an ellipse.
As a result of all these difficulties, Newton's first calculations only agreed "pretty nearly" with the facts; and that was not good enough. For nearly twenty years he dropped the whole issue.
During these twenty years, Jean Picard's expedition to Cayenne produced much improved data on the earth's diameter and its distance from the moon; Newton himself developed his own brand of infinitesimal calculus, the indispensable mathematical tool for attacking the problem; and the Halley-Hooke-Wren trio kept fitting together further bits of the puzzle. The orchestra had now reached the stage where whole groups of instruments could be discerned running through certain passages; only the rap of the conductor's baton was needed to make everything fall into place.
In 1686, goaded on by Halley, Newton arrived at his ultimate synthesis. He computed the force of the earth's attraction on the moon, and showed that this, combined with the moon's own centrifugal force, satisfied the moon's observed motions. Next he computed the sun's attraction on the planets, and demonstrated that the orbit produced by a force of attraction which diminished with the square of distance was a Keplerian ellipse with the sun in one focus; and conversely, that an elliptic orbit required a gravitational force obeying the inverse square ratio. Kepler's Third Law relating the duration of the planets' periods to their mean distances from the sun became a cornerstone of the system; and the Second Law – equal areas being swept out in equal times – was now shown to hold for any central orbit. Comets were shown to move either in very elongated ellipses or in parabolas, receding into the infinity of space. Newton further proved that any object above the earth's surface behaved as if the whole mass of the earth were concentrated in its centre; which made it possible to treat all heavenly bodies as if they were mathematical points. Lastly, all observable motion in the universe was reduced to four basic laws: the Law of Inertia; the Law of Acceleration under an impressed force; the Law of Reciprocal Action and Reaction; and the Law of Gravity.
The miracle was completed; the fragments had all flown together in this reversed explosion and were fused into a smooth, compact, innocent-looking body; and had Donne still been alive, he could have reversed his lament into a triumphant: "'Tis all in one piece, all coherence now."
The motions of sun, moon and the five vagabond stars had been the main problem of cosmology since the days of the Babylonians. Now that they were all shown to follow the same simple laws, the solar system was recognized as an integrated unit. The rapid progress of astronomy and astrophysics soon led to the further realization that this unit was merely a subdivision of a larger one: our galaxy of millions of stars of roughly the same nature as our sun, some of them, no doubt, also surrounded by planets; and that our galaxy again was merely one among other galaxies and nebulae in various stages of their evolution, yet all governed by the same universal set of laws.
But these developments no longer concern us. With the publication of Newton's Principia in 1687 A.D., cosmology became a disciplined science; and at this point our narrative of man's changing vision of the universe must end. The wild dance of shadows thrown by the stars on the wall of Plato's cave was settling into a decorous and sedate Victorian waltz. All mysteries seemed to have been banished from the universe, and divinity reduced to the part of a constitutional monarch, who is kept in existence for reasons of decorum, but without real necessity and without influence on the course of affairs.
It remains to discuss some implications of the story.
CHRONOLOGICAL TABLE TO PARTS FOUR AND FIVE
TYCHO DE BRAHE
GALILEO
KEPLER
A.D. 1546
Born on 14 Dec., at Knudstrup.
" 1559
Studies at Copenhagen, Germany
to 1572
and Switzerland.
A.D. 1564 Born on 15 Feb., at Pisa.
A.D. 1571 Born on 16 May, at
Weil-der-Stadt.
" 1572
Appearance of "Tycho's Nova".
" 1576
Receives the island of Hveen.
Left in care of grand-parents.
" 1581
Matriculates at University of Pisa.
"Put to hard work in the
country".
" 1584
Works at Hveen.
Enters theological Seminary.
" 1589
Appointed Lecturer on Mathematics, University of Pisa.
Professor of Mathematics, Padua University.
Matriculates at Tuebingen
University.
" 1592
" 1593
Teacher of Mathematics,
Provincial School in Gratz.
" 1597
Leaves Hveen.
Writes pro-Copernican letter to Kepler.
Publishes Mysterium Cosmo-
graphicum.
" 1599
Appointed Imperial Mathematicus to Rudolph II.
Exiled from Gratz; school closed down.
" 1600-01
Collaboration Tycho-Kepler.
Kepler at Benatek and Prague.
" 1601
Dies on 13 Oct., Prague.
Appointed Tycho's successor.
GALILEO
KEPLER
" 1609
Publishes Astronomia Nova (First and Second Law).
" 1610
Telescopic discoveries. The Star Messenger. Appointed "Chief Mathematician and Philosopher"
at the Court of Cosmo II de Medici.
Conversation with the Star Messenger.
" 1611
Triumphal visit to Rome.
Dioptrice.
" 1612
Writes Letters on Sunspots.
Death of Rudolph; departure for Linz. Excommunication.
" 1613
Writes Letter to Castelli.
" 1614
Caccini preaches against Galileans.
" 1615
Lorini denounces Galileans. Galileo in Rome. Theory of the tides.
Proceedings against mother start.
" 1616
Copernicus' book banned "until corrected".
Galileo instructed to abandon it.
" 1618
Start of dispute on comets.
Outbreak of Thirty Years War.
" 1619
Harmonice Mundi published (Third Law).
" 1620
Copernicus' book, with minor corrections, again permissible reading.
Mother arrested.
" 1621
Mother acquitted; dies. Publication of Epitome completed.
" 1623
Barberini becomes Urban VIII.
Il Saggiatore published.
" 1625
Starts writing Dialogue.
Printing of Rudolphine Tables begun.
Siege of Linz. Destruction of printing press.
Departure for Ulm.
Printing of Tables completed.
Erratic travels. Obtains post with Wallenstein at Sagan.
" 1630
Dialogue completed. Negotiations about the imprimatur.
Work on Somnium. Last journey to Ratisbon.
Dies on 15 Nov.
" 1632
Dialogue published and banned. Galileo ordered to Rome.
" 1633
Trial of Galileo.
" 1637
Goes blind in both eyes.
" 1638
Two New Sciences published in Leyden.
" 1642
Dies at Arcetri, on 8 Jan.
EPILOGUE
Me thinks there be not impossibilities enough in Religion for an active faith.
SIR THOMAS BROWNE
1. The Pitfalls of Mental Evolution
WE are in the habit of visualizing man's political and social history as a wild zig-zag which alternates between progress and disaster, but the history of science as a steady, cumulative process, represented by a continuously rising curve, where each epoch adds some new item of knowledge to the legacy of the past, making the temple of science grow brick by brick to ever greater height. Or alternately, we think in terms of "organic" growth from the magic-ridden, myth-addicted infancy of civilization through various stages of adolescence, to detached, rational maturity.
In fact, we have seen that this progress was neither "continuous" nor "organic". The philosophy of nature evolved by occasional leaps and bounds alternating with delusional pursuits, culs-de-sac, regressions, periods of blindness and amnesia. The great discoveries which determined its course were sometimes the unexpected by-products of a chase after quite different hares. At other times, the process of discovery consisted merely in the cleaning away of the rubbish that blocked the path, or in the rearranging of existing items of knowledge in a different pattern. The mad clockwork of epicycles was kept going for two thousand years; and Europe knew less geometry in the fifteenth century than in Archimedes' time.
If progress had been continuous and organic, all that we know, for instance, about the theory of numbers, or analytical geometry, should have been discovered within a few generations after Euclid. For this development did not depend on technological advances or the taming of nature: the whole corpus of mathematics is potentially there in the ten billion neurons of the computing machine inside the human skull. Yet the brain is supposed to have remained anatomically stable for something like a hundred thousand years. The jerky and basically irrational progress of knowledge is probably related to the fact that evolution had endowed homo sapiens with an organ which he was unable to put to proper use. Neurologists have estimated that even at the present stage we are only using two or three per cent of the potentialities of its built-in "circuits". The history of discovery is, from this point of view, one of random penetrations into the uncharted Arabias in the convolutions of the human brain.
This is a very curious paradox indeed. The senses and organs of all species evolve (via mutation and selection as we suppose), according to adaptative needs; and novelties in anatomical structure are by and large determined by those needs. Nature meets its customers' requirements by providing longer necks to graze off the top of trees, harder hooves and teeth to cope with the coarse grass of the drying steppes; by shrinking the smellbrain and enlarging the visual cortex of birds, arboreals, and bipeds as they slowly raise their heads above ground. But it is entirely unprecedented that nature should endow a species with an extremely complex luxury organ far exceeding its actual and immediate needs, which the species will take millennia to learn to put to proper use – if it ever does. Evolution is supposed to cater for adaptative demands; in this case the goods delivered anticipated the demand by a time-stretch of geological magnitude. The habits and learning potentialities of all species are fixed within the narrow limits which the structure of its nervous system and organs permits; those of homo sapiens seem unlimited precisely because the possible uses of that evolutionary novelty in his skull were quite out of proportion with the demands of his natural environment.
Since evolutionary genetics is unable to account for the fact that a biologically more or less stable race should mentally evolve from cave-dwellers to spacemen, we can only conclude that the term "mental evolution" is more than a metaphor; and that it refers to a process in which some factors operate to which as yet we have not got a clue. All we know is that mental evolution cannot be understood either as a cumulative, linear process, or as a case of "organic growth" comparable to the maturing of the individual; and that it would perhaps be better to consider it in the light of biological evolution, of which it is a continuation.
It would indeed seem more expedient to treat the his
tory of thought in terms borrowed from biology (even if they can yield no more than analogies) than in terms of an arithmetical progression. "Intellectual progress" has, as it were, linear associations – a continuous curve, a steadily rising water level; whereas "evolution" is known to be a wasteful, fumbling process characterized by sudden mutations of unknown cause, by the slow grinding of selection, and by the dead-ends of overspecialization and rigid inadaptability. "Progress" can by definition never go wrong; evolution constantly does; and so does the evolution of ideas, including those of "exact science". New ideas are thrown up spontaneously like mutations; the vast majority of them are useless crank theories, the equivalent of biological freaks without survival-value. There is a constant struggle for survival between competing theories in every branch of the history of thought. The process of "natural selection", too, has its equivalent in mental evolution: among the multitude of new concepts which emerge only those survive which are well adapted to the period's intellectual milieu. A new theoretical concept will live or die according to whether it can come to terms with this environment; its survival value depends on its capacity to yield results. When we call ideas "fertile" or "sterile", we are unconsciously guided by biological analogy. The struggle between the Ptolemaic, Tychonic and Copernican systems, or between the Cartesian and Newtonian views of gravity, was decided by those criteria. Moreover, we find in the history of ideas mutations which do not seem to correspond to any obvious need, and at first sight appear as mere playful whimsies – such as Appollonius' work on conic sections, or the non-Euclidian geometries, whose practical value became apparent only later. Conversely, there are organs which have lost their purpose and are yet carried over as an evolutionary legacy: modern science is full of appendices and rudimentary monkey-tails.