MECHANICS

  1. Indirect verification of the law of gravity:
    1. Explanation of tides.
      Both Kepler and Galileo proposed different explanations of the tides. Kepler in his New Astronomy proposed an explanation of the tides that is similar to Newton's: the tides is a result of the motions of the waters "towards the regions where the moon stands in the zenith." Later he in his book the Somnium he explained that tides are caused, not by the attraction of the moon alone, but of the moon and the sun together. Galileo rejected Kepler's explanation of the tides and proposed his own theory, which he believed to be "conclusive physical proof" of the heliocentric system; he proposed that the tides were a direct consequence of the earth's conbined motion of daily rotation on its axis and of the annual revolution about the sun which cause the sea to move at a different speed from the land. Newton rejected Galileo's explanation, and proposed a theory of tides that, like Kepler's theory, was caused by gravitational attraction of the moon on the earth's ocean waters. The waters of the ocean on the side of the earth toward the moon is pulled out a little, since it is closer to the moon than the earth as a whole. For the same reason the earth is pulled a little away from the oceans on the opposite side from the moon. Since the earth daily rotates, there are thus two tides per day at any location on the ocean. But they are not of equal heights because the plane of the orbit of the moon lies close to the plane of the ecliptic, with respect to which the axis of the earth is tilted. Tides are also complicated by the land masses and by the variations in shore and the ocean floor. The sun also contributes to the occurrence of the tides, but the sun is less than half as effective as the moon in causing the tidal bulges. However, the sun contributes to the highest high tides which occur about twice a month, when the sun, the earth an the moon are all position on the same line.

    2. Explanation of the orbits of comets.
      In Newton's day and long before in antiquity and in the middle ages, the appearance of comets were dreaded and were interpreted as signs of disaster. Newton showed that comets were nothing but parts of the solar system, moving in highly elliptical orbits and visible only when near the sun. Comets were clouds of material with very little mass and very large volumes and were made visible by the reflected light from the sun. The great comet of 1682, whose path was carefully watched by Edmund Halley, Newton determined that it had a period of recurrence of approximately 75 years; although an eccentric member of the solar system, Newton showed that it obeyed the laws of motion, the law of gravitation, and Kepler's laws of planetary motion. Its return in 1756 and twice since, after covering a long elliptical path carrying it out far beyond the last planet, has been hailed as significant symbol of the triumph of Newtonian science.

    3. Explanation of local variations in the acceleration due to gravity.
      The acceleration due to gravity, g, is not an universal constant; the value of g varies from place to place on the earth. Changes of altitude alone produces the change: g is smaller at the top of Pike's Peak than at its base, smaller on the 100th floor of the Empire State Building than at street level, for the same reason as the centripetal acceleration of the moon is less than the acceleration of a free falling stone on the earth. The formula, derived above, , where r is the distance from the center of the earth, shows how g varies with altitude. Since w = mg, the weight of the body would also vary with altitude. Also the value of g is not the same every on the surface of the earth at sea level; the earth is not perfect sphere, but more nearly a spheroid, flattened at the poles and bulging somewhat around the equator. Its equatorial radius is about 13 miles greater than it polar radius, and this is one of the reasons that value of g is 0.5% larger at the poles than at the equator, 9.83223 m/sec2 at the North Pole and 9.78049 m/sec2 at the equator at sea level. An object of mass 1 kg will weigh 9.79 nt at Key West and 9.80 nt at Baltimore. Other local variation in g have been observed, that has significance in field geology. Masses of rock in high mountains affect the value of g in the vicinity as well as the direction of free fall. The attraction of large masses of rock in the mountain on a body free falling near them pulls them to the body sideways; a plumb line near a mountain does not point exactly toward the center of the earth. Also hidden deposits of lighter material, like oil or gas, can also be detected by measuring local variations in g.

    4. Explanation of the perturbations of orbits.
      Using Newton's laws of motion and the law of gravitation the orbits of the planets can be exactly calculated. But small deviations from the perfect calculated elliptical orbits, called perturbations of the orbits, have been observed. Most of these can be accounted for by the gravitational attraction of nearby planets.

    5. Discovery of previously unknown planets.
      But the paths of some of the more remote planets have exhibited perturbations that can not be explained by the attraction by planets already known. The attempt to account for these perturbations has lead to the discovery of previously unknown planets. This happened in the early 19th century, but the story begins in 1781, when William Herschel, a professional musician and amateur astronomer, was systematically searching the skies with a 6.2 inch aperture reflecting telescope of his own making. Near the constellation of Gemini he found a small object which moved eastward nearly along the elliptic. Continued observation verified that it was a planet, which was later name Uranus. This was the first discovery of a planet in man's written history. As a result, Herschel became a professional astronomer and an amateur musician by his appointment as Royal Astronomer to King George III. Continued observations of Uranus showed increasing departures from its orbit calculated from Newtonian mechanics. These perturbations reached 2' of arc in 1844 and could no longer be ignored. Two young men, J. C. Adams in England and U. J. J. Leverrier in France, began to work on the problem independently and unknown to each other. Each postulated the existence of another planet as yet unobserved in an orbit beyond the orbit of Uranus. This unobserved planet would pull Uranus ahead of its calculated position as Uranus approached it and would pull Uranus behind its calculated position as Uranus receded from it. The problem for Adams and Leverrier was to calculate the position of the yet unobserved planet from Newton's theory and the observed departure of Uranus from predicted position. Adams completed his calculations in October, 1845, and sent his result to G. B. Airy, the English Astronomer Royal. For reasons which are still not understandable, Airy did not assign a very high priority to Adam's request to look for the planet. Meanwhile in August, 1846, Leverrier had also solved the problem without knowing of Adam's work, sent his results to the observatory at Berlin. J. G. Galle found the new planet on September 23, 1846 within a half hour after beginning his search and within a degree of the calculated position. The new planet was called Neptune. The spectacular achievement of finding of a previously unsuspected planet purely by mathematics gave immediate and world-wide acceptance of Newton's theories.

      Neptune, in turn, was observed carefully and in time perturbations of its calculated orbit and those of Uranus, that were larger than counted be accounted for by known forces, were observed. An arduous 25 year search began which ended with the discovery of Pluto in 1930, which was announced on the double anniversary of Herschel's discovery of Uranus, and of the birthday of Perceival Lowell, whose calculations had led to the search and who had founded the observatory in Arizona at which the discovery was made. Another astronomer, W. H. Pickering, had made independent calculations and predictions of Pluto's position as far back as 1909, and had initiated a telescopic search for the planet at Mount Wilson Observatory in California. Nothing was found, but after the discovery at the Lowell Observatory in 1930, the old Mount Wilson photographs were re-examined and showed that Pluto could have been discovered in 1919, if its image had not fallen directly on a small flaw in the photographic emulsion. But the attempt to find the planet to account for a problem with the orbit of the planet Mercury were unsuccessful. In the latter days of the 19th century, the long axis of Mercury's elliptical orbit was observed to turning slowly in space. Most of this "advance of perihelion" of Mercury, as it called, could be explained on Newtonian grounds as due to the attractions of Venus, Earth and the other planets; but about 40'' of arc of rotation each 100 years remained unexplained. The previously successful Newtonian methods were applied and a new planet was confidently predicted called Vulcan within the orbit of Mercury. After a long search it became apparent by the end of 19th century that there was no such planet. In 1915 Albert Einstein proposed a new theory of gravitation, called the general theory of relativity. The theoretical astronomer, Karl Schwartzschild, proposed that the new theory could account for the 40'' of arc per century and that there was no necessity for a new planet Vulcan. This failure of the Newtonian theory of gravitation was not because it was wrong, but because of a limitation of that theory that Einstein's theory removed.

    6. Explanation of the precession of the equinoxes.
      The precession of the equinoxes, which was known to Hipparchus, who died around 125 B.C., is the slow shift of the seasons with respect to the constellations and of the vernal equinoxes from the constellation Aries in the time of Hipparchus to the present location in the constellation of Pisces. This can be interpreted as a slow rotation of the earth's rotational axis, similar to the rotation of the axis of a spinning top about a vertical line. Newton showed that it was the effect of the gravitational attraction of the moon and sun on the earth's equatorial bulge. Since the orbit of the moon lies very nearly in the plane of the ecliptic, the moon constantly exerts an oblique force on the earth's equatorial bulge which steadily changes the direction of the earth's axis, but not its angle of inclination. Newton was able to account quantitatively for this secondary, or precessional, rotation of the earth's axis.

    7. Determination of the masses of celestial bodies.
      Once the value of the gravitational constant, G, was determined, then it can be used to "weigh the earth," as it is often called, that is, to calculate the mass of the earth. Cavendish is said to have succeeded in "weighing the earth" (although the proper meaning of the term "weight" is the gravitational force exerted on a body by the earth). The acceleration a imparted to a body by the gravitational attraction of M is found by
      a = GM / r2. Solving for M,
      M = ar2 / G.
      To find the mass of the earth, substitute for r the radius of the earth, for a the acceleration due to gravity at the surface of the earth, and for G the gravitational constant in this formula, converting to CGS units of measurement:

      M = 980 cm/sec2 × (4000 mi × 5280 ft/mi × 12 in/ft × 2.54 cm/in)2 / 6.67 × 10-8 dynes · cm2/gm2.

      When the arithmetic is performed, the result is very nearly 6 × 1027 gm, which is equivalent to 6 × 1024 kg, or 6.6 × 1021 tons. The number is so large it hardly has any real meaning. A more meaningful number is the mass of the earth divided by its total volume which is called the average density of the earth, and turns out to be 5.52 times the density of water, which is 1 gm per cm3.

      Once the mass of the earth is known, the masses of the sun, the moon, and the other planets can be determined by further application of the law of gravitation. Most of the mass of the solar system is concentrated in the mass of the sun, as would expected from the fact that it appears to stand still at the center of the solar system. It has a mass that is 333,000 times the mass of the earth. The volume of the sun is so great that its mass per unit of volume, density, is only 1.4 times that of water.

      Masses and Density of Solar System Bodies
      Body Mass relative to earth (1.00 = 5.974 × 1024 kg) Radius (km) Radius (earth radii) Density (kg/m³ ) Surface Gravity (earth = 1) Rotation Period (equatorial)
      Sun 333,000 696,000 109.12 1400 28.0 25.4 days D (31 d)
      Mercury 0.0562 2,439 0.38 5430 0.38 58.65 days
      Venus 0.815 6,052 0.95 5240 0.91 243.01 days R
      Earth 1.000 6,378.14 1.00 5520 1.00 23 h 56m 4.1s
      Moon 0.012 1,738 0.27 3340 0.16
      Mars 0.1074 3,393 0.53 3940 0.39 24h 37m 22.6s
      Jupiter 317.9 71,398 11.19 1330 2.54 9h 50.5m
      Saturn 95.1 60,000 9.41 700 1.07 10h 14m
      Uranus 14.56 25,559 4.01 1240 0.90 17h 14m R
      Neptune 17.24 24,800 3.89 1610 1.14 16h 3m
      Pluto 0.0018 1,140 0.18 2100 0.06 6.39 days R

      R indicates retrograde axial rotation.
      D differential rotation equator rotates different from the poles.

    8. Stellar Parallax.
      The lack of a measured stellar parallax was the major obstacle to the acceptance of the heliocentric system. Tycho rejected the heliocentric system of Copernicus mainly because of the failure of his attempts to measure it by naked-eye observations. Galileo with his telescope was not able to observe any change in the angular separation of a pair of stars close together. Neither were the Herschels able to observe in spite of improvement in technique and larger telescopes. It was not unit 1838 that German astronomer, F. W. Bessel, by telescopic observations actually measured a parallax of a third of a second of arc for the faint star 61 Cygni. Because the stars are so far away and the angles are so small that the shift (actually an elliptical loop) against the stellar background is hard to observe. The maximum parallactic displacement occurs in observation made six months apart, and defines the angle of the displacement. The largest parallactic angle observed is 0.756 seconds of arc, for the star Alpha Centauri, our sun's closest neighbor in space, by T. Henderson in 1839. Knowing the base line of the observation (the diameter of the earth's orbit of 186 million miles) it has been possible to calculate that Alpha Centauri is 4.3 light-years away (light-year is the distance that light travels in one year at 186,000 miles per second), or 25 thousand billion miles. In 1840 F. G. Struve reported a parallactic angle of a quarter of a second of arc for Vega. Today the parallactic angle of thousand of stars have determined by photographic methods using large telescopes. The majority of the stars are so far away that their parallactic displacement cannot be detected, even with the best telescopes. This long sought observational proof the earth's orbital motion about the sun was only observed 150 years ago.

    9. Foucault Pendulum.

      The first popular demonstration of the earth's rotation on its axis was given by the French physicist, J. B. I. Foucault, at the Paris Exhibition of 1851. Foucault suspended a heavy iron ball by a long wire from the dome of the Pantheon at the Exhibition. When it was set swinging back and forth in a north-south line, it was observed that the plane of the oscillations of the pendulum would change direction slowing rotating in a clockwise direction when observed from above. Its rate of rotation was such that it would take about 32 hours to complete one rotation. In reality the plane of the oscillating pendulum is fixed in space as the earth rotates beneath it. If a Foucault pendulum was moved to the north pole, the plane of the oscillating pendulum would complete a 360 degree rotation in 24 hours. At the equator, a Foucault pendulum would show no rotation; that is, if the pendulum were set to swing in a north-south line (a meridian), this direction in space is maintained throughout the rotation of the earth. The apparent rotation of the Foucaut pendulum depends upon the latitude of the location of the Foucault pendulum.

  2. Philosophy of Science in 17th century.
    During the early 17th century, two schools of philosophy of science developed about the nature, purpose and methods of science: empiricism and rationalism. Although geographically oriented, the influence of these schools was widely felt and contributed significantly to the general awareness of science. Empiricism was championed by the English philosopher, Sir Francis Bacon (1561-1626 A.D.) and rationalism was lead by the French philosopher, Rene Descartes (1596-1650 A.D.).

    Bacon held that the goal and purpose of science was to give man control over nature and he worked out a rigid set of rules for gaining this control. Bacon believed that this control was obtained by knowledge: "knowledge is power." Truth and utility are two sides of the same coin; so gaining the one the other is found. He was impressed by inadequacy of the knowledge inherited from the past; ancient writers such as Aristotle, Galen, etc., were taken as the only sources of knowledge. Bacon urged that science be reorganized for efficient and systematic discovery of new knowledge, and not rely on the old knowledge. After warning about the major sources of error ("the idols") of the past, he championed a new empiricism in science and glorified a new method, the Novum Organum (New Organon), which would replace the old Organon of Aristotle, with its reliance on the deductive method of the syllogism. Bacon misunderstood the method of Aristotle, accepting the medieval distortions of him. Bacon's new method was actually a revision of Aristotle's inductive method, which the medieval scholastics ignored, emphasizing the deductive method of the syllogism. Bacon's new method began with the collection and organization of all available facts on the relevant subject and checking them with meticulous care. After all possible knowledge related to the subject was collected and organized, it is examined to find those features common to all facts. These form the basis of grand generalization. Generalizations obtained in this manner, in turn, might suggest new avenues of observation and experimentation, leading to new generalizations. Bacon warned of the danger of making inferences that go beyond the evidence that has been gathered. He dismissed the Copernican theory, criticizing Copernicus for inventing "fictions," which are not based on sound philosophical [scientific] foundations, but are introduced to make his calculations come out right. Bacon writes,

    "In the system of Copernicus there are many and grave difficulties; for the threefold motion [rotation, revolution, and changes in the tilt of the axis] with which he encumbered the earth is a serious inconvenience, and the separation of the sun from the planets, with which it has so many affections in common, is likewise a harsh step; and the introduction of so many immovable bodies into nature, as when he makes sun and stars immovable, the bodies which are peculiarly lucid and radiant, and his making the moon adhere to the earth in a sort of epicycle, and some other things which he assumes, are proceedings which marks a man who thinks nothing of introducing fictions of any kind into nature, provided his calculations turn out well."

    He heartily disapproved of Galileo and his use of "thought experiments." Bacon's methodology made no place for mathematics or the use of deductive reasoning.

    Descartes, on the other hand, was a mathematician and believed profoundly in the deductive system of reasoning. He believed that it was possible, on the basis of a limited number of affirmed premises or "primary truths," to deduce the grand generalizations of nature correctly, and thus to explain individual facts. He ignored the inductive method in establishing the individual facts and relied on the deductive method in explaining facts by a deductive system. He developed the geometrical interpretation of algebraic relationships, called analytical geometry, by which it is possible to make graphical representation of them by means of rectangular coordinate system named for him, the Cartesian coordinates. Using his analytic deductive method, he also developed a mechanistic view of the world based on natural law. He reasoned that God rules his creation by natural law. Thus the universe for Descartes was a machine set in motion at the Creation and maintained, not by the active intervention of the Deity, but by the operation of the laws of nature. His mechanism set the pattern of scientific explanation for the next two hundred years.

    Newton combined both the inductive and the deductive method in his methodology, and made extensive use of mathematics. In the preface to his Principia he wrote,

    "Since the ancients (as we are told by Pappus) esteemed the science of mechanics of greatest importance in the investigation of natural things, and the moderns, rejecting substantial forms and occults qualities, have endeavored to subject the phenomena of nature to the laws of mathematics, I have in this treatise cultivated mathematics as far as it relates to philosophy [physical sciences] .... for the whole burden of philosophy seems to consist in this from the phenomena of motions to investigate [by induction] the forces of nature, and then from these forces to demonstrate [by deduction] the other phenomena, and to this end the general propositions of the first and second Books are directed. In the third Book I give an example of this in the explanation of the System of the World; for by propositions mathematically demonstrated in the former Books, in the third I derive from the celestial phenomena the forces of gravity with which bodies tend to the sun and the several planets. Then from these forces, by other propositions which are also mathematical, I deduce the motion of the planets, the comets, the moon, and the sea [tides]...."

    This discussion would seem to indicate that he used mostly a deductive method, but this would be a misunderstanding of his methodology. That he also used the inductive method is made clear in his remarks about hypotheses at the end of his Principia;

    "But to hitherto I have not been able to discover the cause of the properties of gravity from phenomena, and I frame no hypothesis; for whatever is not deduced from the phenomena is to be called an hypothesis; and hypotheses, whether metaphysical or physical, whether of occult qualities or mechanical, have no place in experimental philosophy. In this philosophy particular propositions are inferred from the phenomena, and afterward rendered general by induction. Thus it was that the impenetrability, the mobility, and the impulsive force of bodies, and the laws of motion and of gravitation, were discovered. And to us it is enough that gravity does really exist, and act according to the laws which we have explained, and abundantly serves to account for all the motions of the celestial bodies, and of our sea [tides]."

    Newton's statement, "I frame no hypothesis," has been misunderstood to mean that Newton rejected the use all hypotheses. From the context of the statement it is clear that he was refusing to form a specific hypotheses, about the nature or "the cause of the properties of gravity." He was rejecting "metaphysical or physical" hypotheses about the nature of gravity, like Descartes' explanation that vortices or whirlpools of ether was the cause of the motion of the planets. Newton rejected all attempted mechanical explanations of gravity, whether imputing it to the action or pressure of a subtle matter pervading the universe, or considering it as a form of magnetism. Newton showed that mathematically a whirlpool could not behave in the way that Descartes assumed; a planet caught in a whirlpool would not act as observed and described in Kepler's laws of planetary motion. Newton was troubled that he could not give an explanation of gravity. In a letter to the theologian William Bentley , he wrote,

    "That gravity should be innate, inherent and essential to matter, so that one body may act upon another at a distance, through a vacuum, without the mediation of anything else by and through which their action may be conveyed from one to another, is to me so great an absurdity that I believe no man, who has in philosophical matters a competent faculty of thinking, can ever fall into it."

    To account for the action of gravity, scientists began to use Newton's phrase, "action at a distance," as an explanation, even though Newton never used these words as an explanation.

    Newton's contemporary, the Dutch physicist, Christian Huygens Huygens (1629-1695 A.D.), criticized both Bacon and Descartes, the former for lack of emphasis on mathematical theory, and the latter for lack of sufficient confirmation of his theories by experiment. It has often been said that Huygens was in many respects Galileo's and Newton's peer. Huygens was not only an outstanding mathematician (Newton referred to him as one of the three "greatest geometers of our time."), he made an improved telescope, resolving the rings of Saturn, and, building on the work of Galileo, invented the first practical pendulum clock. Among his many accomplishments in physics are his theorem on centripetal force, the conservation principle of elastic collision, theory of oscillating motion, and a treatise that laid the foundations for the wave theory of light. Huygens and the German philosopher and scientist, Gottfried Wilhelm Leibniz (1646-1716 A.D.), who independently developed the mathematical methods of the calculus, severely criticized the Newtonian system, defending a mechanical explanation of gravity as an effect of the whirlpools of matter that filled the universe, so that the philosophy of Descartes dominated Europe for many years. The English in general supported Newton, and the French clung to Descartes; the result was a controversy that continued well into the eighteenth century. Ultimately the Newtonian view of gravity acting-at-a-distance through empty space carried the day against a Cartesian universe filled with matter agitated with whirlpools, for the existence of which there was no scientific evidence. This vindicated the Newtonian method of both mathematics and experiment over against the deductive method of Descartes which built an elaborate deductive mechanical system on slim or no evidence.