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calculating the area under a hyperbola to 55 places in an infinite series
“In the beginning of the year 1665 I found the Method of approximating
series & the Rule for reducing any dignity of any Binomial into such a series.
The same year in May I found the method of Tangents of Gregory & Slusius,
& in November had the direct method of fluxions & the next year in January
had the Theory of Colors & in May following I had entrance into the inverse
method of fluxions. And the same year I began to think of gravity extending
to the orb of the Moon & (having found out how to estimate the force with
which a globe revolving within a sphere presses the surface of the sphere)
from Kepler’s rule of the periodical times of the Planets being in sesquialterate
proportion of their distances from the centers of their Orbs, I deduced that the
forces which keep the Planets in their Orbs must be reciprocally as the squares
of their distances from the centers about which they revolve: and thereby
compared the force requisite to keep the Moon in her Orb with the force
of gravity at the surface of the earth, and found them answer pretty nearly.
All this was in the two plague years of 1665 and 1666. For in those days
I was in the prime of my age of invention & minded Mathematics &
Philosophy more than at any time since.”
In the 18 months away from Cambridge, during the Plague, Newton developed, as he described,
– mathematics of infinite series, binomial theorem, and expansions of series
– solution for the area under hyperbola
– inventing the derivative
–inventing the integral
– optics research and the spectrum theory of color
– a possible solution balancing falling and inertial tendency, to yield the moon’s orbit
The later, standard account of Newton’s “discovery of gravitation” is by Henry Pemberton, friend of Newton and editor of the 3rd edition of Principia. He spent much time with the older Newton, and tells us
“In his age of celebrity, Newton was asked how he discovered the law of universal gravitation. ‘By thinking on it continually’ was his reply. Sitting in the garden, a falling apple set his mind “into a speculation on the power of gravity: that as this power in not found sensibly diminished at the remotest distance from the center of the earth, to which we can rise, neither at the tops of the loftiest buildings, nor even on the summits of the highest mountains; it appear to him reasonable to conclude, that this power must extend much farther than is usually thought; why not as high as the moon, said he to himself? and if so, her motion must be influenced by it; perhaps she is retained in her orbit thereby.”
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