Why did Leibniz's notation win out over Newton's?
Newton had calculus first, yet nearly every textbook uses the symbols of his rival. A good way of writing an idea can be as powerful as the idea.
▶ Start the storyLeibniz's notation caught on because it was designed to be used. Although Newton first developed his theory in 1666, Leibniz's notation has been favoured as the conventional and more exact expression of calculus. Its original form is used throughout mathematics today, while Newton's version survives in a narrower corner.
The two men wrote the same idea in different ways. Newton's notation for differentiation, also called the dot notation or , places a dot over the , and it is generally used when the independent variable denotes time. Leibniz wrote the derivative as a ratio of two tiny changes, dy over dx, and the integral with an elongated S chosen to suggest summation. His notation allows one to specify the variable for differentiation, in the denominator, which is especially helpful when considering . And the dx in an integral is meant to suggest dividing the area under the curve into an infinite number of rectangles whose width becomes the infinitesimally small dx.
Newton's dots
- A dot over the
- Mostly used when the variable is time
Leibniz's ratio
- dy over dx, naming the variable changed
- Used throughout mathematics
Part of the reason lies in the man. Leibniz thought symbols were important for human understanding. He attached so much importance to good notation that he attributed all his discoveries in mathematics to it. In the manuscripts of late 1675 he recorded his discoveries and experiments with various forms of notation.
The choice also had a political tail. The priority dispute separated English-speaking mathematicians from those in continental Europe for many years, and only in the 1820s, thanks to the , did Leibnizian calculus become accepted in England. Even today there is no single standard notation for differentiation: Newton's dot and Lagrange's prime still have their uses.
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Recap
Leibniz designed symbols that show what is being changed and what it is changing against, and that design helped them win.
💡 A trick to remember it · Newton's dots are a pencil flick over a letter; Leibniz's dy over dx is a label that tells you who changes with whom.
Surprising fact · English mathematicians accepted Leibniz's notation only in the 1820s.
Sources (5)
No source, no claim. Every fact in this lesson (15 claims) cites at least one of these.