Euler's Eyes
2026-10-28
The Instituzioni analitiche
In the middle of the eighteenth century, anyone who wanted to learn calculus had a problem. Newton and Leibniz’s invention was only seventy years old, and its results were scattered across papers in scientific journals, letters and books written in Latin, French and English, in different notations. The only famous textbook, the Marquis de L’Hôpital’s Analysis of the Infinitely Small, of 1696, dealt only with differential calculus. There was no book that led the student, step by step, from algebra to differential equations.
Agnesi’s Instituzioni analitiche were that book.
Four books in two volumes
The work runs to about a thousand pages and is divided into four books:
- The analysis of finite quantities (volume I, 1748): algebra, solving equations, analytic geometry, conics and other curves, and the “construction of equations”, that is, solving equations by means of geometric figures. This is where the versiera appears.
- Differential calculus (volume II, 1749): derivatives, tangents, maxima, minima and points of inflection.
- Integral calculus: integrals, areas, volumes, lengths of curves and series. See the area under the curve.
- The inverse method of tangents: what we now call differential equations.
Agnesi did not set out to present discoveries of her own. The book’s merit lies in its organization: each subject comes after the ones it depends on, each method is illustrated by many worked examples, and all of the calculus is written in Leibniz’s notation, with $dx$ and $\int$, which would prove the most practical.
Solving equations with ruler and compass
The oldest part of the book shows a custom of the time that has disappeared: besides solving an equation by calculation, mathematicians liked to construct the solution, that is, draw it as a segment.
The simplest example is the equation $x^2 = ab$, where $a$ and $b$ are given segments. Place $a$ and $b$ in a straight line, one after the other, and draw the semicircle that has the whole segment $a + b$ as its diameter. At the point where $a$ meets $b$, raise a perpendicular up to the semicircle. The length $h$ of that perpendicular solves the equation: $h^2 = ab$.
Why? The triangle formed by the ends of the diameter and the point on the semicircle is right-angled (it is inscribed in a semicircle), and the altitude of a right triangle divides the hypotenuse into two pieces whose product is the square of the altitude. Agnesi treats much more complicated cases, such as equations of the third and fourth degree, constructed by intersecting two conics.
An extraordinary reception
In 1749 a commission of the Paris Academy of Sciences examined the book and judged it the most complete and best organized treatise ever written on the subject. The second volume was translated into French in 1775, with additions by Charles Bossut. The English translation by John Colson, a Cambridge professor, appeared in 1801.
Exercise: the geometric mean
Prove that, in the construction above, $h^2 = ab$.
Call $C$ the point on the semicircle and $D$ the foot of the perpendicular. The centre is at the middle of the diameter, and the radius is $r = \frac{a + b}{2}$. The distance from $D$ to the centre is $\left|\frac{a+b}{2} - a\right| = \frac{|b - a|}{2}$. By Pythagoras’s theorem in the triangle formed by the centre, $D$ and $C$:
$$h^2 = r^2 - \left(\frac{b-a}{2}\right)^2 = \frac{(a+b)^2 - (b-a)^2}{4} = \frac{4ab}{4} = ab.$$The segment $h = \sqrt{ab}$ is the geometric mean of $a$ and $b$.
Exercise: constructing a square root
Use the construction to draw a segment of length $\sqrt{7}$, having only a ruler, a compass and a segment of length 1.
Take $a = 1$ and $b = 7$: a segment of length 8, divided at the point 1 from one end. Draw the semicircle of diameter 8 and raise the perpendicular at the dividing point. The altitude is $h = \sqrt{1 \cdot 7} = \sqrt{7}$. With the same trick, any square root of a constructible number can be drawn.
Sources: Maria Gaetana Agnesi, Instituzioni analitiche ad uso della gioventù italiana, 2 vols. (1748–1749) · Guillaume de L’Hôpital, Analyse des infiniment petits pour l’intelligence des lignes courbes (1696) · Maria Gaetana Agnesi, Traités élémentaires de calcul différentiel et de calcul intégral, trans. P. T. d’Antelmy, with notes by C. Bossut (1775) · Massimo Mazzotti, The World of Maria Gaetana Agnesi, Mathematician of God (2007).
All topics on Maria Gaetana Agnesi · ← The curve of Agnesi · Maxima, minima and points of inflection →