As we saw in the previous section, the year 1666 marked
the peak of Newton's achievements. Here we will explore the details
and significance of these astonishing accomplishments, which included
the invention of calculus, groundbreaking work in optics, and the invention
of the notion of gravity as a universal force.

The branch of mathematics known as calculus is difficult
to define. Very roughly, it can be defined as the calculation
of variable quantities, such as weight, distance, or time, using
forms of algebraic notation. For example, as water is poured at
a uniform rate into an inverted cone, its level rises less and
less rapidly; calculus can be used to determine how much the level
will rise in any given interval. In more complicated forms, calculus
can be used to find the slope of curves, and to determine the area
under, and within, curves. It has proved an indispensable tool
for engineers and architects, and yet it effectively did not exist
before the 17th century. Isaac Newton cannot claim to be its sole
inventor--credit must go to any number of mathematicians, particularly
the German Gottfried von Leibniz--however, Newton unquestionably
made significant contributions to the field. In 1666 he formulated
the binomial theorem, which enabled one to calculate any power of
a binomial (an algebraic expression involving two variables being
added or subtracted, such as [x + y] or [4y - 7z]) without multiplying
the entire expression out. Also in 1666, he discovered how find
the slope of a curve at any point on a curve, by a process he called
"fluxions." However, while he mentioned the "fluxions" in a letter
to Isaac Barrow in 1669, he did not publish the system until 1704,
and so must share credit for the innovation with Leibniz, who developed
his own method in the 1670s.

Newton's work in optics, the study of light, was equally
pioneering. For decades scholars had debated the nature of light,
its composition and its properties, without reaching conclusions.
One puzzling characteristic of light was its ability to break
down into various shards of color when shining through a prism.
Now Newton, using a prism that he had purchased at a local fair,
made a groundbreaking discovery: in his own words, Newton "procured me
a Triangular glass-Prisme, to try therewith the celebrated Phaenomena
of colours. And in order thereto having darkened my chamber, and
made a small hole in my window shuts, to let in a convenient quantity
of the Sun light, I placed my Prisme at its entrance, that it might
be thereby refracted to the opposite wall." What appeared was
a row of bands, a spectrum of color, with red at one end and violet
at the other, each refracted at a slightly greater angle. He hypothesized
that white light was composed of "a Heterogeneous mixture of differently
refrangible rays," each a different color, and each refracted at
a different angle by the prism. Using a lens, he was able to prove
his hypothesis by bending the colored rays back together, into
a single beam of white light.

This idea--that white light is a combination of differently
colored rays--was a completely new notion in the 17th century:
most people assumed that red light, green light, and so on, were
all just slight modifications of white light, not components of
it. Yet while Newton's insight caused changes in the way people
of his lifetime thought about light, its vast consequences for
science would not be realized until the 20th century. Newton's
discoveries in optics have allowed modern scientists to make much
progress in astronomy, for example: because different substances
radiate different colors in the spectrum when they burn, astronomers
have been able to determine the chemical composition of distant
stars by observing which colors they produce. Close examination
of the spectrum of colors produced by stars has also enabled scientists
to calculate these stars' rate of motion toward or away from Earth;
these calculations have in turn enabled us to estimate its distance
from us, and, more generally, the size of galaxies and the universe
itself.

The innovations of calculus and optics alone would have
made the year 1666 famous in the annals of science. But it was
also in this year that the twenty- four-year old first began to
conceive his greatest idea: the concept of gravity. Years later,
Voltaire would recount the legend that arose around the discovery:
"One day, in the year 1666, Newton, then retired to the country,
seeing some fruit fall from the tree... fell into a profound meditation
upon the cause which draws all bodies in a line which, if prolonged,
would pass very nearly through the center of the earth." This
story of how Newton came to his revelation by watching a falling
apple amounts, alas, to no more than popular fiction; however,
the event of the revelation itself is quite true. Gravitation,
the invisible force exerted between objects, was by no means original
to the English scientist--minds as eminent as Johannes Kepler, the
German astronomer, had speculated on the attraction of interstellar
bodies, and contemporaries like Robert Hooke and Edmund Halley
discussed the idea in the 1660s and '70s. But Descartes's idea
of particles and planet-propelling vortices now seemed to make
a gravitational force unnecessary. Newton, however, was suspicious
of Descartes's theory, and continued to calculate the interaction
of the heavenly bodies: in 1666 he calculated the force of attraction
that held planets in their orbits, and the Moon in its orbit around
Earth, as varying inversely with the square of their distance from
the sun. This was *the* fundamental law of gravity,
and Newton knew it as he sat alone in his mother's house in Woolsthorpe.

But Newton did not publish the idea immediately. Indeed,
Newton did not publish any of his great discoveries directly after
making them--as we have seen, his "fluxions" would not come into
print for nearly four decades; his work in optics waited six years
to be published. But in the case of his work on gravitational
attraction, publication was stalled for a particularly ironic reason:
he delayed because he could not get the sizes of the Earth and
the Moon to agree with his inverse square equation. In fact, the
existing data on these two sizes was faulty, but he would not realize
that until the 1670s, when new research proved that he had been
right all along.