Three Scientific Revolutions: How They Transformed Our Conceptions of Reality (5 page)

One can discern one revolutionary feature of his “hypotheses” in the fact that it was “the lack of certitude in the traditional mathematics” that he cites as the justification for attributing motions to the earth despite its opposition to ordinary experience, showing the increasing influence of mathematics in scientific inquiry (an affirmation of Plato over Aristotle later extolled by Galileo). It also shows a new awareness of the tremendous complexity of the problems, along with the required freedom to explore alternative possibilities offered by the new evidence, again rejecting the unwarranted injunctions of the Catholic Church.

Thus began the dismantling of the ancient model of the universe as a celestial sphere composed of an aethereal substance bounded by the sphere of the fixed stars revolving around the stationary central earth. Within this sphere the seven planets, Moon, Mercury, Venus, Sun, Mars, Jupiter, and Saturn, in that ascending order, revolved in heavenly spheres in uniform circular motion at extended distances. Their individual motions were variously attributed to intelligences (Aristotle), to having been created by God in their respective orbits and endowed with their uniform circular motions (Christians), or to their having souls as Kepler originally believed and then replaced by a materialistic “clockwork” system driven by gravity, further emended by Galileo and Newton.

Although Copernicus deserves immense credit for initiating the unraveling considering its consequences, other than exchanging the positions of the earth and the sun he retained most of the explanatory system of the traditional cosmology, such as uniform circular motions. The more radical revisions were by Kepler and Galileo in the seventeenth century and by the telescopic discoveries of Erwin Hubble of the Mount Wilson Observatory in the early twentieth century revealing the existence of billions of additional solar systems, galaxies, and in 1929 the expansion law of the universe.

Kepler was attracted to Copernicus's system—as Copernicus was—because of its harmonious mathematical integration of the astronomical evidence and what Kepler believed would facilitate the discovery of the as yet undetermined exact ratios of the distances, motions, sizes, and of the planetary orbits owing to the sun's influence. Then, when he gained access to Tycho Brahe's naked eye observations (his being the most exact at the time) and the depiction of Mars's orbit noticing its variations in brightness and velocity and that its shape was oval or elliptical, it became apparent that its orbital motion could not be circular and uniform. As he wrote to his friend Johann Fabricius, a clergyman and amateur astronomer in Freesland: “if only the shape were a perfect ellipse all the answers could be found in Archimedes' and Apollonius' works,”
¹⁵
especially the latter's conic sections. Illustrating not only his respect for the evidence, it shows the continued influence of the Hellenistic mathematicians. Indicative of an extraordinary flexibility and openness of mind, Kepler discarded his earlier simplistic explanations for a revised final interpretation that even Tycho Brahe was unable to accept.

Also aware that the sun's emission of light became dimmer with the increased distance, it occurred to him that rather than the planetary-souls producing the motion, perhaps another “force” (
vis
) emanated by the sun analogous to light, “corporeal but immaterial,” could provide a more effective explanation. This hypothesis was reinforced by his reading of William Gilbert's book,
De Magnete
(On the Magnet), in which he described the earth and other planets, including the sun, as having magnetic poles on their opposite axes. Kepler inferred that if, as the planets revolved around the sun, their respective polarities were opposed causing them to be attracted or repelled depending on their alignment, this would explain the shapes and motions of their orbits deviating from circular and uniform motions. This recourse to forces was unusual at the time.

He thus concluded that the two forces, the sun's emanation causing the planets revolution and Gilbert's contrasting magnetic forces, could explain the observed ovoid or elliptical motion of the planets. But it was not until he measured the deviation of Mar's orbit from circular to elliptical as .00427 of the radius, that he was finally convinced that the orbit must be elliptical. It was this replacement of his earlier explanations, consisting of the Platonic solids and souls, with “forces” that enabled him to formulate his astronomical laws. Noted author Arthur Koestler describes the tremendous import of this change:

It would be difficult to overestimate the revolutionary significance of this proposal. For the first time since antiquity, an attempt was made not only to
describe
heavenly motions in geometrical terms, but to assign them a
physical cause
. We have arrived at the point where astronomy and physics meet again, after a divorce which lasted for two thousand years. This reunion . . . produced explosive results. It led to Kepler's three Laws, the pillars on which Newton built the modern universe. (p. 258)

While the first of these laws was anticipated in Kepler's early work,
Commentaries on the Movement of Mars
, it was in his
Astronomia Nova
(New Astronomy) published in 1609 (the year Galileo began his telescopic observations), acclaimed as “the first modern treatise on astronomy” that, having read Gilbert's book, he offered his first two planetary laws: (1) “that the planets travel round the sun not in circles but in elliptical orbits,” and (2) “that a planet moves in its orbit not at uniform speed but in such a manner that a line drawn from the planet to the sun sweeps over equal areas in equal times” (p. 313).

Yet despite these striking innovations he still had not determined the exact ratios of the velocities of the planets' orbits to their distances from the sun, which was his original motivation in studying astronomy. Initially he assumed the simplest ratio, that the velocity diminished inversely with its distance from the sun, but in the
Dioptrice
, published in 1611, believing that the intensity of light lessened inversely with the
square
of the distance, he attributed this also to the sun's “gravity,” a term he had introduced in the
Nova
as the sun's emanation causing the orbital motions.

There he declared that “Gravity is the mutual bodily tendency between cognate [i.e., material ] bodies towards unity or contact . . . each approaching the other in proportion to the other's mass.”
¹⁶
Attributing this gravitational force to all physical objects such as the earth and its moon, he also inferred that it is the earth's attractive force that prevents its seas from rising to the moon. While Galileo rejected this explanation on the basis that it involved an “occult force,” it became a crucial principle in Newton's celestial mechanics.

Then in the
Harmonice Mundi
(World Harmony), published in 1619, based on Tycho Brahe's measurements comparing the ratios of the periods of the planets with their distances from the sun, he deduced that “
The squares of the periodic times are to each other as the cubes of the mean distances.
”
¹⁷
That is, the period of the revolutions vary with the 3/2
^th
power of their distances: “
it is certain . . . that the ratio which exists between the periodic times of any two planets is precisely the ratio of the 3/2th power of the mean distances, i.e., of the spheres themselves
. . . .”
¹⁸
It was this crucial ratio that would provide Newton with the key to his universal law of gravitation.

What an extraordinary achievement! With these laws one could finally cast aside the spiritual or heavenly nature of the universe, along with all the past fabricated devises such as celestial spheres, epicycles, eccentrics, equants, and souls (though not God as the initial cause) previously used to explain the movements and dimensions of the planetary orbits. In fact, replacing the conception of a celestial universe with a mechanistic one seems to have been his ultimate intention, as he wrote to his friend Herwart von Hohenburg, the Catholic chancellor of Bavaria who asked Kepler, along with other astronomers, for his opinion on certain astronomical problems,

My aim is to show that the heavenly machine is not a kind of divine, live being, but a kind of clockwork . . . insofar as nearly all the manifold motions of the clock are caused by a most simple . . . weight. And I also show how these physical causes are to be given numerical and geometrical expression.
¹⁹

Though this was his vision, in his written works he was unable to present it in such a clear and convincing manner that it lacked the impact it should have had, thus his reputation was largely overshadowed by Galileo and others in the seventeenth century.

Yet his legacy did include a final significant achievement, namely, the creation of the
Rudolphine Tables.
As I wrote previously:

In 1614 John Napier had published a much praised work,
Merifici Logarithmorum Canonis Descriptio
[Different Descriptions of Logarithmic Canons], containing logarithmic tables that facilitated astronomical calculations, but had not shown how they had been computed. Knowing of its popularity but limited explication, in the years 1621–1622 Kepler wrote a work that contained not only logarithmic tables along with instructions for their use, but also considerable planetary data and a star catalogue comprising over a thousand fixed stars. Published five years later and entitled the
Tabula Rudophinæ
in honor of his deceased patron Rudolph II, it served for over a century as the basis of astronomical calculations and predictions.
²⁰

I should not leave this discussion without correcting what may have been a false impression that Kepler's intellectual development was a natural, direct, and smooth process and that his personal life was a prosperous one. Nothing could be further from the truth. Not only was his remarkable intellectual development arduous, conflicted, and at times regressive, despite his achievements, his personal life was fraught with tragedy indicative of the times. His first wife and their child died as did three other children from his second marriage; he was beset by constant financial problems due to the Crown withholding his earnings, and he had to face the awful charge of witchcraft brought against his aged mother, a charge that was eventually withdrawn owing to her unwavering insistence on her innocence, even under threat of torture and death, and assisted by Kepler's devoted defense supported by friends at court.

Furthermore, he suffered a final tragedy when, on a trip to Ratisbon, Bavaria, in an effort to regain 11,818 florins owed him by the Crown, he became fatally ill and died three days later on November 15, 1630, apart from his family and alone. As a further misfortune, he was buried in the cemetery of Saint Peter outside of Ratisbon but his actual gravesite is unknown due to the ravages of numerous successive battles there. However, these humiliations are somewhat mitigated by a fine statue of him beside Tycho Brahe on the hill overlooking the city of Prague honoring their outstanding contributions to the creation of modern classical science, especially Kepler's.

Having limited myself to the major contributors to the development of modern classical science, I shall turn now to the outstanding achievements of Galileo comprising his extraordinary, iconoclastic telescopic observations, his experimental discovery of the mathematical law of free fall and proof of parabolic motion, his eloquent endorsement of mathematics as “the language of nature,” and his continued dismantling of the competing Aristotelian or Scholastic cosmological system in favor of a mechanistic worldview based on mathematical computations.

It was overhearing a lecture by the court mathematician Ostilio Ricci on Euclid that lured Galileo to the study of mathematics and mechanics. Continuing his tutelage under Ricci studying the works of Eudoxus and Archimedes, Ricci soon recognized Galileo's exceptional mathematical ability and encouraged him to study statics and hydrostatics. This led to Galileo's first scientific publication,
La Bilancetta
(The Little Balance), in 1586, written in Italian that introduced modifications in the Westphal balance enhancing measurements of specific gravity and specific weights.

His next book,
De Motu
(On Motion), described his research leading to his criticism of Aristotle' s theory of terrestrial motions, especially the latter's common-sense law that free-falling objects accelerate in proportion to their weights. In 1602 he published a book on
Mechanics
continuing his incline plane experiments begun in
De Motu
to determine their exact rate of free fall. Like Kepler he tried to determine the ratio of the sizes and speeds of the planetary orbits in relation to the sun, but unlike Kepler was unable to arrive at the correct proportion due to his adherence to circular orbits and his rejection of Kepler's ellipses. But in experiments with pendulums he discovered that when the duration of their swings was equal or isochronal, this was due not to their weights, as generally believed, but to their lengths. This showed Galileo's extensive curiosity about natural phenomena and his usual willingness to challenge traditional beliefs and authority, along with a realization of the importance of experiments in testing what seemed obvious common-sense truths. Still, as the acclaimed historian of science James Gleick points out, as science progresses one discovers that these earlier discoveries are approximate.

The regularity Galileo saw is only an approximation. The changing angle of the Bob's motion creates a slight nonlinearity in the equations. At low amplitudes, the error is almost nonexistent. But it is there, and it is measurable even in the experiment as crude as the one Galileo describes.
²¹

Continuing his incline plane experiments, two years later he confirmed that as the ball rolls from rest down the incline its acceleration in
equal times
traverses distances proportional to the odd numbers beginning with one: 1, 3, 5, 7, 9, etc. In addition, he found that the
square roots
of the
successive sums
of the odd numbers gave the successive times of descent: for example, the sum of 1 and 3 = 4 whose square root is 2; the sum of 1, 3, and 5 equals 9 whose square root is 3, etc. In turn, these numbers squared indicate the ratios of the increases of acceleration during the fall. As he later wrote to Fra Paolo Sarpi whom he greatly admired, “he had found a proof for the square law, the odd number rule, and other things he had long been asserting, if granted the assumption that velocità are proportional to distances from rest.”
²²
These early experimental discoveries were important because they would become the basis of his second most important book,
Dialogues Concerning Two New Sciences
, published shortly before his death.