Agnesi’s Problem III … Could the Textbook Be Wrong?

Summary: In the pre-calculus portion of her book Instituzioni Analitiche (1748), Maria Gaetana Agnesi presents the following problem and its solution: “Given two circles EGF, BNO, and given their centers C, A; if from any point G on the periphery of the circle EGF one draws the tangent GNO that meets the other circle BNO in the points N, O, and from these two points draw two tangents NM, OM, find the locus of all the points M in which said tangents meet.” The problem’s solution is complicated by the need to consider several cases, and Agnesi’s work contained errors. We discuss her work and present an illustrated and complete solution. In doing so, we hope not only to introduce a problem of geometric interest, but also to contribute to the historical understanding of Agnesi’s work.