As the object of the present treatise is not to teach geometry, we shall describe, in very general terms, the manner in which Newton, who was the first who systematically extended the laws of motion to the heavenly bodies, identified their results with the two remaining laws of Kepler. His "Principles of Natural Philosophy" contain general propositions with regard to any law of centripetal force, but that which he supposed to be the true one in our system, is expressed in mathematical language, by saying that the centripetal force varies inversely as the square of the distance, which means, that if the force at any distance be taken for the unit of force, at half that distance, it is two times twice, or four times as strong; at one-third the distance, three times thrice, or nine times as strong, and so for other distances. He shewed the probability of this law in the first instance by comparing the motion of the moon with that of heavy bodies at the surface of the earth.
Taking LP to represent part of the moon's orbit described in one minute, the line PM between the orbit and the tangent at L would shew the space through which the central force at the earth (assuming the above principles of motion to be correct) would draw the moon. From the known distance and motion of the moon, this line PM is found to be about sixteen feet. The distance of the moon is about sixty times the radius of the earth, and therefore if the law of the central force in this instance were such as has been supposed, the force at the earth's surface would be 60 times 60, or 3600 times stronger, and at the earth's surface, the central force would make a body fall through 3600 times 16 feet in one minute. Galileo had already taught that the spaces through which a body would be made to fall, by the constant action of the same unvarying force, would be proportional to the squares of the times during which the force was exerted, and therefore according to these laws, a body at the earth's surface ought (since there are sixty seconds in a minute) to fall through 16 feet in one second, which was precisely the space previously established by numerous experiments.
With this confirmation of the supposition, Newton proceeded to the purely geometrical calculation of the law of centripetal[198] force necessary to make a moving body describe an ellipse round its focus, which Kepler's observations had established to be the form of the orbits of the planets round the sun. The result of the inquiry shewed that this curve required the same law of the force, varying inversely as the square of the distance, which therefore of course received additional confirmation. His method of doing this may, perhaps, be understood by referring to the last figure but one, in which Cd, for instance, representing the space fallen from any point C towards S, in a given time, and the area CSD being proportional to the corresponding time, the space through which the body would have fallen at C in any other time (which would be greater, by Galileo's law, in proportion to the squares of the times), might be represented by a quantity varying directly as Cd, and inversely in the duplicate proportion of the triangular area CSD, that is to say, proportional to Cd/ (SC × Dk)², if Dk be drawn from D perpendicular on SC. If this polygon represent an ellipse, so that CD represents a small arc of the curve, of which S is the focus, it is found by the nature of that curve, that Cd/ (Dk)² is the same at all points of the curve, so that the law of variation of the force in the same ellipse is represented solely by 1/ (SC)². If Cd, &c. are drawn so that Cd/ (Dk)² is not the same at every point, the curve ceases to be an ellipse whose focus is at S, as Newton has shewn in the same work. The line to which (Dk)²/ Cd is found to be equal, is one drawn through the focus at right angles to the longest axis of the ellipse till it meets the curve;—this line is called the latus rectum, and is a third proportional to the two principal axes.
Kepler's third law follows as an immediate consequence of this determination; for, according to what has been already shown, the time of revolution round the whole ellipse, or, as it is commonly called, the periodic time, bears the same ratio to the unit of time as the whole area of the ellipse does to the area described in that unit. The area of the whole ellipse is proportional in different ellipses to the rectangle contained by the two principal axes, and the area described in an unit of time is proportional to SC × Dk, that is to say, is in the subduplicate ratio of SC² × Dk², or Dk²/ Cd, when the force varies inversely as the square of the distance SC; and in the ellipse, as we have said already, this is equal to a third proportional to the principal axes; consequently the periodic times in different ellipses, which are proportional to the whole areas of the ellipses directly, and the areas described in the unit of time inversely, are in the compound ratio of the rectangle of the axes directly, and subduplicately as a third proportional to the axes inversely; that is to say, the squares of these times are proportional to the cubes of the longest axes, which is Kepler's law.
FOOTNOTES:
[192] This mode of verifying configurations, though something of the boldest, was by no means unusual. On a former occasion Kepler, wishing to cast the nativity of his friend Zehentmaier, and being unable to procure more accurate information than that he was born about three o'clock in the afternoon of the 21st of October, 1751, supplied the deficiency by a record of fevers and accidents at known periods of his life, from which he deduced a more exact horoscope.
[193] Kepler probably meant his own mother, whose horoscope he in many places declared to be nearly the same as his own.
[194] See Preliminary Treatise, p. 13.
[195] In allusion to the Harmonics of Ptolemy.