The witch of Agnesi.

Pierre de Fermat (1601-1663), who must ne conslcterect one ot the inventors of analytic geometry, at one time interested himself in the cubic curve, which in present-day notation would be indicated by the Cartesian equationy(x² + a²) = a³.
The curve is pictured in Figure 33. Fermat did not name the curve, but it was later studied by Guido Grandi (1672-1742), who named it versoria. This is a Latin word for a rope that guides a sail. It is not clear why Grandi assigned this name to the cubic curve. There is a similar obsolete Italian word, versorio, which means "free to move in every direction," and the doubly-asymptotic nature of the cubic curve suggests

that perhaps Grandi meant to associate this word with the curve. At any rate, when Maria Gaetana Agnesi wrote her widely read analytic geometry, she confused
Grandi's versoria or versorio with versiera, which in Latin means "devil's randmother" or "female goblin." Later, in 1801, when John Colson translated Agnesi's text into English, he rendered versiera as "witch." The curve has ever since in English been called the "witch of Agnesi," though in other languages it is generally more simply referred to as the " curve of Agnesi. "

The witch of Agnesi possesses a number of pretty properties. First of all, the curve can be neatly described as the locus of a point P in the following manner. Let a variable secant OF (see Figure 33) through a given point O on a fixed circle cut the circle again in Q and cut the tangent to the circle at the diametrically opposite point R to O in A. The curve is then the locus of the point P of intersection of the lines QP and UP, parallel and perpendicular, respectively, to the aforementioned tangent. If we take the tangent through O as the x-axis and OR as the y-axis of a Cartesian coordinate system,
and denote the diameter of the fixed circle by a,
the equation of the witch is found to be y(x² + a²) = a³.
The curve is symmetrical in the y-axis and is asymptotic to the x-axis in both directions. The area between the witch and its asymptote is eras, exactly four times the area of the fixed circle. The centroid of this area lies at the point (0, a/4), one fourth the way from O to R.
The volume generated by rotating the curve about its asymptote is p²a³/2.
Points of inflection on the curve occur where OQ makes angles of 60° with the asymptote.

An associated curve called the pseudo-witch is obtained by doubling the ordinates (the y-coordinates) of the witch. This curve was studied byJames Gregory in 1658 and was used by Leibuiz in 1674 in deriving his famous expression

pi/4 = 1 – 1/3 + 1/5 – 1/7 + …..