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The fundamental theorem of calculus is the reason you can find the area under y = x² by "anti-differentiating". Most textbooks credit Newton and Leibniz, but one of the clearest early statements came from Newton's teacher, Isaac Barrow, written entirely in the language of geometry. This post tells Barrow's story from reliable sources, translates his theorem into school notation, and checks it with numbers.
Who was Isaac Barrow?
The MacTutor biography records that Barrow was born in London in October 1630 and died there on 4 May 1677. His school life started badly; at Charterhouse he had a reputation as a bully. At Felstead he made rapid progress in Greek, Latin, Hebrew and logic, and he entered Trinity College, Cambridge in 1646, at first doing menial work in return for teaching.
He taught himself geometry and in 1655 published a simplified edition of Euclid's Elements that stayed the standard textbook for half a century. He then spent four years travelling through France, Italy and Turkey; pirates attacked his ship, and in Venice a fire destroyed everything he had. Back in Cambridge he became Professor of Greek, then in 1662 Professor of Geometry at Gresham College in London.
How did Barrow come to teach Newton?
In 1663 an endowment left by Henry Lucas created a new chair of mathematics at Cambridge, and Barrow became the first Lucasian Professor. MacTutor records that Isaac Newton attended his lectures and had many private discussions with him, which influenced his work greatly. In 1669 Barrow resigned the chair and did no further mathematical work, and Newton took it over. Barrow later became Master of Trinity, helped start the building of the Wren Library, and was buried in Westminster Abbey.
What did Barrow's theorem say?
Barrow's Lectiones Geometricae was published in 1670. J M Child's 1916 English translation, The Geometrical Lectures of Isaac Barrow, titles Lecture X "Differentiation as the inverse of integration" and Lecture XI "Integration the inverse of differentiation". MacTutor calls Barrow the first to recognise that the two operations are inverse.
In Lecture X, Barrow takes a rising curve ZGE above an axis AD. He builds a second curve AIF so that, at every point D, a fixed length R times the height DF equals the area ADEZ under the first curve. He then proves that the tangent to AIF at F has a slope fixed by the height DE of the first curve.
In modern notation, with f(x) the first curve and R = 1, his second curve is the area function A(x) = ∫₀ˣ f(t) dt, and his result is A′(x) = f(x).
Worked example: checking the theorem with numbers
Take f(x) = x². Its area function from 0 is A(x) = x³/3.
Area up to x = 3. A(3) = 27/3 = 9.
Area up to x = 3.01. 3.01³ = 27.270901, so A(3.01) = 9.090300.
Slope of the area function. (A(3.01) − A(3)) / 0.01 = 0.090300 / 0.01 = 9.03.
Compare with the curve. f(3) = 3² = 9. The slope of the area function is very close to the height of the curve, and gets closer as the step shrinks.
Why it works: adding a thin strip of width h under the curve at x adds an area of about f(x) × h. Divide by h and you get f(x). That is Barrow's theorem in one line.
Common mistakes:
Confusing the area function with the curve. A(x) is a new function. Its slope, not its height, equals f(x).
Dropping the lower limit. ∫₀ˣ and ∫₁ˣ differ by a constant, which is why indefinite integrals carry + C.
Using the theorem when the curve is not continuous. Textbooks state the theorem for continuous f for a reason.
These ideas are central to JEE Main mathematics and physics; see our guides to the calculus you need for JEE physics and to properties of definite integrals.
What can a student take from Barrow's way of working?
Teach what you are learning. Barrow built Cambridge's mathematics teaching while still working things out himself.
Notice when two problems are one problem. Tangents and areas looked unrelated until he linked them.
Make room for others. He stepped aside from the Lucasian chair, and his student used it well.
If the step from "area" to "integral" still feels like magic, the Fermi Concept Library has every concept in mathematics with its own lesson, so you can rebuild the link one idea at a time. The next part of the story, how Leibniz gave calculus its notation, is in our post on Gottfried Leibniz.
Repeat the numerical check for f(x) = 2x + 1 at x = 2, using steps of 0.1 and 0.01.
