Newton was the greatest genius who ever lived, and the most fortunate, for there is only one universe to discover, and it can be discovered only once.

— Joseph-Louis Lagrange, as reported by his contemporaries

The calculus war

Newton created the method of fluxions in 1665–1666 but did not publish it. Leibniz reached calculus independently in 1675 and published in 1684. For some years nobody quarrelled. In 1676 the two even exchanged cordial letters through Henry Oldenburg, and Newton hid his ideas in an anagram.

The mood soured around 1700, when Newton’s followers began to suggest that Leibniz had copied ideas from manuscripts he had seen in London. Leibniz complained to the Royal Society and asked for an impartial inquiry. The Royal Society appointed a committee, and in 1712 it published a report that found for Newton.

What few people knew at the time: the President of the Royal Society was Newton himself, and he was the one who wrote the report. Afterwards he anonymously published a review praising it.

Historians today agree that the two invented calculus independently. Newton got there first; Leibniz published first and created the notation that won. The quarrel came at a high price: out of national pride, English mathematicians stayed loyal to Newton’s dots for over a century, while the Continent advanced with Leibniz’s $dx$, with the Bernoullis and with Euler. Only in 1812 did a group of Cambridge students found the Analytical Society to bring Leibniz’s notation back to England.

See the other side of the story in the Leibniz trail.

Where Newton’s mechanics stops

For more than two centuries, Newton’s laws seemed to hold everywhere. At the start of the 20th century, two theories revealed their limits:

  • Relativity. At speeds close to that of light, time and space are no longer absolute, as Newton assumed. In relativity, a time interval $t'$ measured on a moving clock corresponds to a longer interval for someone at rest:

    $$t=\frac{t'}{\sqrt{1-\dfrac{v^2}{c^2}}}.$$

    When the speed $v$ is small compared with the speed of light $c$, the square root is practically 1 and we are back in Newton’s world: $\displaystyle\lim_{v\to0}t=t'$. When $v\to c$, the difference grows without bound.

  • Quantum mechanics. For atoms and particles, well-defined trajectories no longer make sense.

That does not make Newton wrong in everyday life. For bridges, cars, aircraft and rockets, his mechanics is so accurate that it is still the tool in use. Relativity did not erase Newton: it showed how far he holds.

Tributes

Newton was buried in Westminster Abbey, among kings and queens. His monument shows cherubs holding a prism, a reflecting telescope and newly minted coins, together with a celestial globe marked with the path of the comet of 1680.

Key points

  • Newton was born in 1642 by the English calendar of the time, and in 1643 by ours.
  • At 22, isolated by the plague, he created binomial series, fluxions and the theory of colours.
  • The binomial $(1+x)^\alpha$ works for any exponent, provided $|x|<1$.
  • The fluxion $\dot x$ measures the rate of change; area is the inverse operation.
  • Newton’s method approximates roots by following tangents: $x_{n+1}=x_n-f(x_n)/f'(x_n)$.
  • The Principia (1687) contains the three laws of motion and universal gravitation.
  • The Moon test showed that the Earth’s gravity falls off with the square of the distance.
  • The prism separates the colours already present in white light; it does not create them.
  • Newton and Leibniz invented calculus independently.

Sources: introduction to the manuscripts, Newton Project · MacTutor biography.

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