What Is Calculus? The Mathematics of Change and Accumulation
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Arithmetic and algebra deal with quantities that stay put. Calculus deals with quantities that are changing while you look at them: the speed of a falling stone at this instant, the slope of a curve at this point, the area under a line that never straightens out. It was invented twice, independently, in the 1660s and 1670s, by two men who then spent decades accusing each other of theft, and it turned out to be the language in which physics, engineering, economics and most of modern science are written.
The problem of the instant
Speed is distance divided by time. A car that goes sixty miles in an hour averaged sixty miles an hour, but its speed at any one moment is a different question, since in one moment it goes no distance in no time, and zero divided by zero is nothing. Calculus answers by asking what the average speed becomes over shorter and shorter intervals, a minute, a second, a thousandth of a second, and noticing that the averages settle towards a limit, and calling that limit the speed at the instant. The same move gives the slope of a curve at a point: take two points on the curve, find the slope of the line between them, and slide the second towards the first. The limiting slope is called the derivative, and finding it is differentiation.
The derivative turns out to obey simple rules. The derivative of x squared is 2x; of x cubed, 3x squared; of the sine is the cosine; and there are rules for sums, products and functions inside other functions, so that the slope of almost any formula can be written down without drawing anything. Where the derivative is zero the curve is momentarily flat, at a peak or a trough, which is how calculus finds the best of anything: the cheapest, the strongest, the fastest.
The problem of the area
The second question is older. How much area lies under a curve, or how much water flows through a pipe whose rate keeps changing, or how far the car went if only its varying speed is known? The Greeks had answers for a few shapes by cutting them into thin slices and adding up, and calculus makes that general: divide the region into strips so thin that each is nearly a rectangle, add their areas, and take the limit as the strips shrink to nothing. The result is the integral, and finding it is integration. Some things calculus computes:
- •The speed and acceleration of anything whose position is known as a formula, and the position of anything whose speed is
- •The maximum or minimum of a quantity, from the shape of a soap bubble to the price that maximises profit
- •Areas, volumes, lengths of curves and centres of gravity of any shape that can be described
- •The total effect of a rate that varies: rainfall from a varying downpour, drug in the blood from a varying dose
- •The behaviour of anything governed by a differential equation, which is to say planets, pendulums, populations, circuits, heat and waves
The theorem that joins them
Differentiation and integration look like different subjects, one about slopes and one about areas, and the discovery that made calculus a single thing is that they are opposites. The fundamental theorem of calculus says that integrating a function and then differentiating the result gives the function back, and that the area under a curve between two points can be found by finding a function whose derivative is the curve and subtracting its values at the two ends. The slicing that the Greeks did by hand is replaced by running differentiation backwards. Newton and Leibniz both saw this, and it is why the two of them are credited with the invention rather than the many mathematicians before them who had found slopes and areas one at a time.
Two inventors
Isaac Newton worked it out in 1665 and 1666, at home in Lincolnshire while Cambridge was closed by plague, as a tool for the physics he was building, and told almost no one. Gottfried Leibniz, a German diplomat and philosopher, found it independently in Paris around 1675, invented the notation that is still used, the elongated S for the integral and the d for the differential, and published in 1684. Newton's followers accused Leibniz of having seen Newton's unpublished papers, the Royal Society, with Newton as its president, ruled in Newton's favour in a report Newton wrote himself, and the quarrel cut British mathematics off from the Continent for a century, to Britain's cost, since Leibniz's notation was better. The modern verdict is that both invented it, Newton first and Leibniz better.
Neither could say what a limit was. The idea of a quantity smaller than any number but not zero, which both of them used, was attacked by the philosopher Berkeley as the ghost of a departed quantity, and it was not until the 1820s that Cauchy, and then Weierstrass, defined the limit precisely enough to make the subject rigorous. The calculations had been right for a hundred and fifty years before anyone could prove why.
Where it went
Newton's laws of motion are differential equations, statements about rates of change, and solving them is calculus; so are Maxwell's equations for electricity and light, the equations of fluids, heat, quantum mechanics and general relativity, and the models economists use for growth and finance. Every bridge, aircraft, engine and circuit is designed with it, and the algorithms that train machine-learning systems are a form of differentiation applied to billions of parameters at once. A subject invented to find the speed of a falling stone became the general method for reasoning about anything that changes smoothly, which is most of the physical world.
A worked example with a falling stone
Suppose you drop a stone from a cliff and ignore air resistance. Near the Earth's surface it falls roughly 5t squared metres in t seconds, using ten metres per second squared as a round figure for gravity. After three seconds it has fallen 45 metres, so its average speed over that time was 15 metres per second. To find its speed at exactly three seconds, differentiate: the derivative of 5t squared is 10t, which gives 30 metres per second. The stone is moving twice as fast at the end as it averaged over the whole fall.
Now run it the other way. If all you know is that the speed is 10t, a straight line rising steadily on a graph, the distance fallen is the area under that line. Between nought and three seconds the area is a triangle with base 3 and height 30, which comes to 45, the same answer as before. Integrating 10t gives 5t squared, so the fundamental theorem and the geometry agree. Engineers use this same pair of moves, from position to speed and from speed back to distance, for rockets, cars and satellites.
Calculus before calculus
Newton and Leibniz built on ideas that had been taking shape for nearly two thousand years. Archimedes, in the third century BC, proved that the area of a segment cut from a parabola by a straight line is exactly four thirds of a particular triangle drawn inside it, by filling the region with ever smaller triangles and summing them. A lost work of his, The Method, turned up in 1906 hidden beneath the text of a medieval prayer book written on reused parchment. It showed him balancing imaginary thin slices of shapes against each other, a way of thinking strikingly close to integration.
In southern India, mathematicians of the Kerala school, beginning with Madhava of Sangamagrama around 1400, found infinite series for the sine, cosine and arctangent, including the series that gives a quarter of pi as 1 minus a third plus a fifth minus a seventh and so on, later rediscovered in Europe. Their proofs survive in a sixteenth-century text written in Malayalam. In France in the 1630s, Pierre de Fermat found maxima and tangents by a method that looks very like taking a derivative. What none of them had was a general set of rules and the link between slopes and areas.
Calculus running quietly inside everyday machines
You use the two halves of calculus every time you drive. A speedometer shows a rate of change, how fast distance is changing at this instant, which is a derivative. The distance counter adds up speed over time to give the total travelled, which is an integral. The two dials are linked in exactly the way the fundamental theorem describes.
Computers rarely solve these problems with neat formulas. A video game working out how a ball flies, or a satellite navigation system predicting where you will be, usually advances the situation in tiny time steps, updating position from velocity and velocity from acceleration again and again. This numerical approach is calculus in practice, and it lets machines handle situations far too messy for any equation in a textbook to solve exactly.
The takeaway
Calculus has two halves: differentiation finds the rate at which something is changing at an instant, the slope of a curve at a point, as the limit of averages over shrinking intervals; integration finds the total that a changing rate accumulates, the area under a curve, as the limit of sums of shrinking slices. The fundamental theorem shows they are inverses. Newton and Leibniz invented it independently in the seventeenth century, and it became the language of physics, engineering and every science that deals with continuous change.