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In the mid-1700s, calculus was barely a lifetime old and scattered across papers written in different styles and notations. A student who wanted to learn it had no single, orderly book in their own language. Maria Gaetana Agnesi wrote one, in Italian, and printed it in her family home in Milan. This post covers who she was, what her book did, how a curve in it came to be called a witch, and a worked example on that curve that uses JEE calculus.
Who was Maria Gaetana Agnesi?
Her MacTutor biography gives her birth in Milan on 16 May 1718 and her death there on 9 January 1799. She was the eldest of 21 children of Pietro Agnesi, whose family wealth came from silk, and she learnt Latin, Greek and Hebrew young. In 1738 she published Propositiones Philosophicae, 191 theses on philosophy and natural science that she defended in Latin before guests her father invited.
She disliked these public displays and asked to become a nun. Her father refused, and she stayed at home on condition that she could live simply and avoid balls and theatres. She turned to mathematics, guided by the monk Ramiro Rampinelli, who had taught at Rome and Bologna.
What did her textbook do?
Instituzioni analitiche ad uso della gioventù italiana, or analytical methods for the use of Italian youth, came out in two volumes, in 1748 and 1749. The Dictionary of Scientific Biography entry describes two huge quarto volumes of more than a thousand pages, running from elementary algebra through coordinate geometry to differential and integral calculus, infinite series and simple differential equations. It calls her "the first woman in the Western world who can accurately be called a mathematician".
MacTutor adds the working details. She paid for private printing at home so she could supervise it, and sent drafts to the mathematician Riccati, who passed them to his sons for comments. A committee of the Paris Académie des Sciences praised the book for reducing scattered discoveries to nearly uniform methods, with order and clarity throughout. Pope Benedict XIV had her appointed to the chair of mathematics at Bologna in 1750; her name stayed on its rolls for 45 years, but she never went. After her father died in 1752 she gave herself to caring for the sick and poor, and she died in poverty in the poorhouse she directed.
The two sources disagree on one point. MacTutor says the book contains no original mathematics and teaches by examples, while the DSB says she admitted some methods were her own. Reading both is a good reminder to check claims against more than one source.
Why is a curve in her book called a witch?
Her book includes the curve now called the witch of Agnesi. MacTutor's page on the curve notes that Fermat had studied it and Guido Grandi had constructed it in 1703. Grandi called it versoria, a rope that turns a sail, and in Italian versiera, the name Agnesi correctly used. John Colson, who translated her book into English before 1760, mistook versiera for aversiera, meaning witch.
You can see the slip in Colson's translation, published in 1801. Problem III in section 238 takes a semicircle on diameter AC = a and asks for points M with AB : BD = AC : BM, where the perpendicular to AC at B meets the circle at D. With AB = x and BM = y, the circle gives BD = √(ax − x²), and the curve is y = a√((a − x)/x), "which is vulgarly called the Witch".
Worked example: the witch with a = 2
Take y = 8/(x² + 4).
Highest point. The denominator is smallest at x = 0, so y = 8/4 = 2 = a.
Slope. y′ = −16x/(x² + 4)². This is zero only at x = 0, positive for x < 0 and negative for x > 0, so the curve rises then falls.
Inflection points. Differentiating again gives y″ = −16(4 − 3x²)/(x² + 4)³, which is zero when x² = 4/3, so x = ±2/√3 ≈ ±1.155. There y = 8/(4/3 + 4) = 8/(16/3) = 1.5, which matches MacTutor's 3a/4.
Area under the whole curve. ∫ 8/(x² + 4) dx = 4 arctan(x/2). As x runs from −∞ to ∞, arctan(x/2) runs from −π/2 to π/2, so the area is 4 × π = 4π. The generating circle has diameter 2 and area π, so the area under the witch is four times the circle.
What mistakes do students make?
Quotient-rule slips in y″. Differentiate −16x/(x² + 4)² carefully, then factor out (x² + 4)³.
Thinking the curve meets the x-axis. y is always positive; the x-axis is an asymptote.
Mixing up Agnesi's axes. Her x runs along the diameter. Convert before comparing with modern books.
Treating ∞ as a number. Write the area as a limit of the integral from −t to t as t grows.
Where does this sit in your syllabus?
The JEE (Main) 2026 syllabus covers derivatives up to second order, increasing and decreasing functions, maxima and minima, and definite integrals. Integrals over an infinite range are not listed, so treat step 4 as enrichment and check your year's syllabus. Our post on increasing and decreasing functions practises step 2.
When a curve-sketching question stalls, the Fermi Problem Library has a step-by-step solver that explains the reasoning, so you can see why each derivative step follows.
What can a student take from Agnesi's way of working?
Write for the learner. She chose Italian and an orderly path from algebra to calculus.
Ask experts to check your work. She sent every part to Riccati before printing.
Check the story. Even her curve's name comes from a mistranslation.
Next step: redo the worked example with a = 3 and confirm the inflection points at y = 2.25 and the total area 9π.
