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Humanity has pondered infinities since time immemorial. It has even tried to define them: What happens, for example, if you continually increase a number by one? Is it possible to eventually reach a value that exceeds all natural numbers? For that matter, what happens if you make the absolute value of a number ever smaller?
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The oldest known records of such considerations come from Archimedes, who lived in the third century B.C.E. But infinities and their opposites, infinitely tiny values called infinitesimals, became truly significant in the 17th century, spurring one of the most intense scientific debates the world had ever seen.
In the 1660s and 1670s, both Isaac Newton and Gottfried Wilhelm Leibniz laid the foundations for what we now call calculus. This branch of mathematics deals with measuring geometric figures and investigating variable quantities, among other things.
At that time, Newton was conducting research at Trinity College Cambridge. Then, in 1665, the bubonic plague swept across Europe, forcing the university to close its doors. Newton dedicated himself even more intensely to his research during this period, making impressive strides in optics and kinematics and even developing a theory of gravitation.
The young physicist was particularly fascinated by variable systems. To describe these, however, he needed a new kind of mathematics—which Newton, then in his 20s, invented for this purpose.
Calculation and Competition
Imagine you drop a ball to the ground. Its velocity will increase over time as the Earth’s gravitational pull accelerates it. The distance, s, traveled by the ball can be described by the equation of motion s = 1/2 gt², where grepresents the acceleration due to gravity, which is 9.81 meters per second squared near Earth’s surface. To keep the formulas as concise as possible, we will omit units and approximate g as 10. This gives us an approximate result of s ≈ 5t² in meters for the distance traveled.
But what if we want to know the ball’s speed after it has traveled for one second? We know that the average velocity, v, of an object over a distance traveled is given by the time it takes to travel that distance: v = s/t . But what if the velocity changes continuously and we are interested in v(t) at a specific time, t?
Newton realized that the result for the instantaneous velocity becomes more accurate when the time interval from which the value is calculated is smaller. To this end, he eventually introduced an “infinitesimal” quantity (dt, in modern notation), which corresponds to a tiny time interval—smaller than any ordinary number.
The velocity at time t = 1 could therefore be calculated by dividing the distance traveled between t = 1 and t = 1 + dt, written as s(1 + dt) – s(1), by the time interval dt. Substituting the values for t into the distance formula, we obtain: s(1 + dt) = 5(1 + dt)2 = 5 + 10dt + 5(dt)2 and s(t = 1) = 5.
Thus, for the velocity, v, at time t = 1, it follows that: v(t = 1) = (s(1+dt)−s(1)) / dt = (10dt + 5(dt)2) / dt = 10 + 5dt.
What does this result, which still depends on dt, mean? According to Newton, it’s 10 meters per second: he simply omitted the dt term.
About a decade later, philosopher Leibniz considered similar ideas—albeit for entirely different reasons. He was looking for a way to calculate the area under a curve. To do this, he divided the area into increasingly smaller intervals to approximate it by summing the areas of numerous rectangular shapes. The narrower the intervals, the more accurate the calculation becomes. And so Leibniz, too, introduced an infinitesimal: dx.
Claims and Credit
More detailed investigations of such infinitesimal calculations were then sidelined by a dispute between Leibniz and Newton. Although Leibniz had published his theory before his English colleague, a supporter of Newton—and later Newton himself—claimed that Leibniz had seen his unpublished work on infinitesimal calculus beforehand and had stolen his ideas.
The British Royal Society subsequently launched an investigation that said it had confirmed Newton’s accusations. But Newton himself was president of the society and hand-picked the investigating committee. (Newton, for all his intelligence, is also remembered for ruthlessness; he attacked other brilliant contemporaries, including scientist Robert Hooke.)
Leibniz’s reputation was ruined. He died in 1716, and his grave was unmarked. Only about 75 years later did people add the inscription “OSSA LEIBNITII” (meaning “the bones of Leibniz”).
Today historians of science agree that there were significant differences between Leibniz’s and Newton’s work on calculus, concluding that both scientists had probably arrived at their findings independently of each other.
Next time, I’ll dig into the controversy within the mathematical community surrounding the concept of infinitesimals.
This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission. It was translated from the original German version with the assistance of artificial intelligence and reviewed by our editors.
