Newton to Euler

The law of gravitation is enunciated in the first book. Its discovery envelops the name of Newton in a halo of perpetual glory. The current version of the discovery is as follows: it was conjectured by Hooke, Huygens, Halley, Wren, Newton,

and others, that, if Kepler's third law was true (its absolute

accuracy was doubted at that time), then the attraction between the earth and other members of the solar system varied inversely as the square of the distance. But the proof of the truth or falsity of the guess was wanting. In 1666 Newton reasoned, in substance, that if $g$ represent the acceleration of gravity on the surface of the earth, $r$ be the earth's radius, $R$ the distance of the moon from the earth, $T$ the time of lunar revolution, and $a$ a degree at the equator, then, if the law is true, $g\frac{r^2}{R^2}=4{\pi }^2\frac{R}{T^2},\text{ or }g=\frac{4\pi }{T^2}{\left(\frac{R}{r}\right)}^3·180a.$

The data at Newton's command gave $R=60.4r$, $T=2,360,628$ seconds, but $a$ only $60$ instead of $69\frac{1}{2}$ English miles. This wrong value of $a$ rendered the calculated value of $g$ smaller than its true value, as known from actual measurement. It looked as though the law of inverse squares were not the true law, and Newton laid the calculation aside. In 1684 he casually ascertained at a meeting of the Royal Society that Jean Picard had measured an arc of the meridian, and obtained a

more accurate value for the earth's radius. Taking the corrected

value for $a$, he found a figure for $g$ which corresponded to the known value. Thus the law of inverse squares was verified. In a scholium in the Principia, Newton acknowledged his indebtedness to Huygens for the laws on centrifugal

force employed in his calculation.

The perusal by the astronomer Adams of a great mass of

unpublished letters and manuscripts of Newton forming the Portsmouth collection (which remained private property until 1872, when its owner placed it in the hands of the University of Cambridge) seems to indicate that the difficulties encountered by Newton in the above calculation were of a different nature. According to Adams, Newton's numerical verification was fairly complete in 1666, but Newton had not been able to determine what the attraction of a spherical shell upon an external point would be. His letters to Halley show

that he did not suppose the earth to attract as though all its mass were concentrated into a point at the centre. He could not have asserted, therefore, that the assumed law of gravity was verified by the figures, though for long distances he might have claimed that it yielded close approximations. When Halley visited Newton in 1684, he requested Newton to determine what the orbit of a planet would be if the law of attraction were that of inverse squares. Newton had solved a similar problem for Hooke in 1679, and replied at once that it

was an ellipse. After Halley's visit, Newton, with Picard's

new value for the earth's radius, reviewed his early calculation,