If Newton actually did perform his Moon test calculation in 1666, he might have been inspired by an idea that only appears in print in A Treatise of the System of the World. This attempt to popularize Book III of the Principia, System of the
World, came out in 1728, a year after Newton's death.
Imagine a cannon on a mountain top V shooting cannon balls horizontally. As larger charges of powder are employed, the cannon balls go farther. The first lands at D, the second at E, the third at F. Increasingly larger charges of powder propel the next three cannon balls to B, to A, and finally to V, in effect orbiting the Earth.
Now think of the Moon's orbit as the result of the same force (gravity) causing cannon balls to fall to the Earth or remain in orbit around the Earth.
The Moon would move in a straight line (as if shot from a cannon) from M toward M' were it not pulled toward the Earth by the force of gravity. The Moon's resulting curved path bends below the horizontal line; the Moon falls from X to X'.
If the time from M to M' is 1 second, the distance the Moon is observed to fall from X to X' is 1.37 millimeters. A cannon ball near the Earth's surface falls about 5 meters (3,600 times more than 1.37 millimeters) in one second. Thus the force on the cannon ball is calculated to be 3,600 times as great as the force on the Moon.
The radius of the Earth is approximately 4,000 miles. The radius of the Moon's orbit is approximately 240,000 miles, 60 times the Earth's radius. Sixty times 60 is 3,600, which is the ratio of the forces on a cannon ball and on the Moon. Thus the force of gravitational attraction from the Earth pulling down a cannon ball and holding the Moon in its orbit decreases in proportion to the square of the distance.
Or it would have been so calcu-
Figure 17.3: The Moon lated, had Newton had at hand the cor-
in orbit around the rect value for the radius of the Earth.
Earth. Gravity pulls the But he was in the country, away from
Moon down the distance the university library. Supposedly he between X and X' in the used an incorrect value in his calcula-
time the Moon otherwise tion and thus concluded, temporarily would have moved hori- and erroneously, that some other force zontally from M to M'. must also be involved. Years later he would plug the correct value into the calculation and all would be right.
out the phenomena of the celestial motions by the supposition of a gravitation towards the center of the Sun decreasing as the squares of the distances therefrom reciprocally" (Westfall, Never at Rest, 444-45). Halley undertook to see the book published at his own expense when the Royal Society shirked its duty.
The title page reflects Newton's intent to refute Descartes' philosophical principles, which had appeared under the title Principia philosophiae. Descartes' title is incorporated in Newton's, and lest readers miss the allusion, set in boldface and larger type. In the third edition, the words Philosophiae and Principia were in scarlet color. Newton's philosophy was not the pretentiously new metaphysical philosophy of Descartes, but a natural philosophy founded securely on mathematics; in place of philosophical principles was a natural philosophy consisting of mathematical principles.
In his Principia, Newton "laid down the principles ofphilosophy; principles not philosophical but mathematical: such, namely, as we may build our reasonings upon in philosophical inquiries. These principles are the laws and conditions of certain motions, and powers or forces" (Principia III, Introduction). His four rules of reasoning were the following:
I. We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances.
II. Therefore to the same natural effects we must, as far as possible, assign the same causes.
III. The qualities of bodies, which admit neither intensification nor remission of degrees, and which are found to belong to all bodies within the reach of our experiments, are to be esteemed the universal qualities of all bodies whatsoever. (Principia III, Rules of Reasoning)
From this rule it followed that:
Lastly, if it universally appears, by experiments and astronomical observations, that all bodies about the Earth gravitate towards the Earth, and that in proportion to the quantity of matter which they severally contain; that the Moon likewise, according to the quantity of its matter, gravitates towards the Earth; that on the other hand, our sea gravitates towards the Moon; and all the planets one
Figure 17.4: Title page from the Principia, 1687. Latin was still the scholarly language of Europe in Newton's time. The c in Principia is pronounced as if it were a k. The Principia was first published in 1687 in London in an edition of perhaps as many as 500 copies. Another 750 copies were printed at Cambridge in 1713, and 800 more (plus 25 on large paper and 10 on extra large paper) at London in 1726. Also during Newton's lifetime unauthorized editions appeared in Amsterdam in 1714 and 1723. Image copyright History of Science Collections, University of Oklahoma Libraries.
towards another; and the comets in like manner towards the Sun; we must, in consequence of this rule, universally allow that all bodies whatsoever are endowed with a principle of mutual gravitation. (Principia III, Rules of Reasoning)
Newton's grand generalization "that all bodies whatsoever are endowed with a principle of mutual gravitation" was based on a narrow range of evidence. He didn't know anything about distant stars, about tiny entities visible only in microscopes, about particles of light, or even very much about common bodies on the Earth that might be manipulated in ordinary laboratory settings to yield evidence of mutual gravitation. For example, in 1798 the British scientist Henry Cavendish would measure the attraction between two lead spheres with a torsion balance, and then from that value calculate the mass of the Earth (to within 1% of the currently accepted value). Still, what scientist has ever had all the possible facts in hand before leaping to a generalization? Uniformity is a necessary assumption of science. From a limited body of data scientists draw overly broad generalizations, which they then test and often subsequently refine by dividing a class of objects previously presumed universal into different types with different laws.
The fourth and final rule of reasoning followed:
IV. In experimental philosophy we are to look upon propositions inferred by general induction from phenomena as accurately or very nearly true, notwithstanding any contrary hypotheses that may be imagined, till such time as other phenomena occur, by which they may either be made more accurate, or liable to exceptions. (Principia III, Rules of Reasoning)
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