Analytical Institutions in Four Books

about #MariaGaetanaAgnesi

According to Dirk Jan Struik, Agnesi is "the first important woman mathematician since Hypatia (fifth century A.D.)". The most valuable result of her labours was the Instituzioni analitiche ad uso della gioventù italiana, (Analytical Institutions for the Use of Italian Youth) which was published in Milan in 1748 and "was regarded as the best introduction extant to the works of Euler." In the work, she worked on integrating mathematical analysis with algebra. The first volume treats of the analysis of finite quantities and the second of the analysis of infinitesimals. A French translation of the second volume by P. T. d'Antelmy, with additions by Charles Bossut (1730–1814), was published in Paris in 1775; and Analytical Institutions, an English translation of the whole work by John Colson (1680–1760), the Lucasian Professor of Mathematics at Cambridge, "inspected" by John Hellins, was published in 1801 at the expense of Baron Maseres. The work was dedicated to Empress Maria Theresa, who thanked Agnesi with the gift of a diamond ring, a personal letter, and a diamond and crystal case. Many others praised her work, including Pope Benedict XIV, who wrote her a complimentary letter and sent her a gold wreath and a gold medal.

The witch of Agnesi

The first woman known as mathematichian was the italian Maria Gaetana Agnesi (1718-1799) and hermost famous work was Instituzioni analitiche ad uso della gioventù italiana (Analytical institutions for italian young people), in which she studied an old famous curve callde versiera, in anglosaxon worlds the witch of Agnesi. The name derives from a mistranslation by John Colson who intend the original italian therm averisera, that means versed sine curve, like avversiera, that in english became witch or wife of devil.
The curve was studied for the first time by Pierre de Fermat in 1630, and after by Guido Grandi in 1703 and in 1748 by Maria in her treatise. The curve was defined by:

(commons)

Starting with a fixed circle, a point $O$ on the circle is chosen. For any other point $A$ on the circle, the secant line $OA$ is drawn. The point $M$ is diametrically opposite $O$. The line $OA$ intersects the tangent of $M$ at the point $N$. The line parallel to $OM$ through $N$, and the line perpendicular to $OM$ through $A$ intersect at $P$. As the point $A$ is varied, the path of $P$ is the witch.

Its cartesian equation is \[y = \frac{8a^3}{x^2 + 4a^2}\] In 1918 Frederick H. Hodge proofed that the witch is generated from the following curve