From 0 to Infinity in 26 Centuries (13 page)

Scientifically, Newton’s insight was to recognize gravity as a force that is caused by the earth’s mass, and his ability to understand intuitively how objects would behave when the
earth’s gravity was not present. Newton’s critics saw gravity, which acts invisibly and at a distance, as some kind of demonic force and that Newton, an alchemist, was obviously in
league with such forces. However, Newton’s Law of Universal Gravitation and his Equations of Motion were perfectly sufficient to allow us to send men to the moon three hundred years
later.

Change afoot

Mathematically, Newton’s greatest achievement is
calculus
, which was also developed independently by Gottfried Leibniz at approximately the same time. Calculus is a
tool used today in a range of different disciplines to describe and predict change. Building on the work of Descartes (see
here
), calculus can be split into two main branches.

We can see that for every square the line moves to the right (the positive direction) the line goes up two squares. The gradient of the line, therefore, is 2.

However, not all equations give straight lines. Any equation with x
²
, x
³
or higher powers produces a curved line:

The gradient of this line constantly changes. However, the gradient of the
tangent
– the line that meets the curve at a point – is the same as the gradient of
the line at that point:

Differentiation lets us find a formula for the gradient of the line at
any
point so we no longer need to draw the tangent, which eliminates an area prone to error.

If we have the formula for the gradient at any point, we can find the places where the gradient is zero. These are called the
turning points
of the equation and finding them can be very
helpful. Many problems in mathematics, banking and business involve finding the maximum or minimum values of an equation – differentiation lets us find these points. Scientists have also
found that many phenomena are governed by
differential equations
. For example, Newton’s Second Law, force = mass x acceleration, is derived by differentiating momentum.

Again, drawing the graph might enable us to estimate the area under the curve, and there are various numerical methods that allow us to calculate an approximation of the area.
One method would be to divide the area into thin rectangular strips and add each area together:

As you increase the number of rectangles you get closer to the actual value. Newton and Leibniz took this one stage further.
They imagined the rectangles
became infinitely thin, in which case you get the true value of the area.

You can use this method if you want to calculate the precise area of the shape under a curve. As with differentiation, there are many scientific laws that rely on integration.

G
OTTFRIED
L
EIBNIZ
(1646–1716)

A mathematician and philosopher from Saxony, in the present-day state of Germany, Gottfried Leibniz was the son of a philosophy professor who died when Leibniz was just six
years old. Leibniz inherited his father’s extensive library, through which he gained much knowledge, after first having taught himself Latin so he could read the books. Leibniz began his
working life as a lawyer and diplomat, and, while on secondment
in Paris, he met a Dutch astronomer called Christiaan Huygens, who assisted him in his learning of science and
mathematics.

Things turn ugly

Leibniz is important for several reasons, although, perhaps unfortunately, he is mainly remembered for the bitter dispute he had with Isaac Newton over the invention of
calculus. Newton was based in Cambridge and Leibniz in Paris, where both men devised the concept of calculus independently of each other. Newton began work on calculus as early as 1664 but he
failed to publish his findings. That responsibility was left to Leibniz, who published his first paper on the subject in 1684.

The argument centred on whether or not Leibniz had been exposed to Newton’s prior work. No evidence exists to prove whether or not Leibniz did have access to Newton’s work, and there
is no reason to assume that Leibniz could not have come up with calculus independently. However, he died with the matter still unsettled.

Leibniz and Newton developed different notations for calculus and both are used in different areas of mathematics. Leibniz’s is perhaps more commonly used.

To evaluate the area shown on the graph you would use Leibniz’s notation to write:

Which is shorthand for: ‘integrate x
²
between x=1 and x=2 with respect to x’.

If you wanted to find the gradient of the line at a point you would need to use differentiation, for which the notation is:

A new dawn

Leibniz was also instrumental in a new field of mathematics that was emerging at the time: computing. In our Hindu-Arabic numeral system, each column in a number is 10 times
larger than the one on its right – a decimal system. Leibniz was interested in a way of writing numbers in which each column is twice the value of the column on its right – a binary
system. The binary system required only the digits 0 and 1, and the columns have values of 1, 2, 4, 8, 16, etc., doubling each time. So the decimal number 13 would be written as 1101:

Column: 8 4 2 1

Digit: 1 1 0 1 because 8 + 4 + 1 = 13

This system seems quite peculiar but it has the advantage of using only two digits, which makes calculations easier. The binary system would later become very important for
electronics and computers.

J
OHANN
B
ERNOULLI
(1667–1748)
AND
J
ACOB
B
ERNOULLI
(1654–1705)

The Bernoulli brothers were Swiss mathematicians. Although they both pursued alternative professions – Johann was trained
in medicine and Jacob as a
minister – both siblings loved mathematics, especially the calculus of Leibniz. The Bernoulli brothers were able to push on with the fledgling field of calculus and became proficient in its
use, turning it from an intellectual and political curiosity into a useful mathematical tool. They were also fiercely competitive with each other, which spurred on their discoveries even more.