James Gregory and the Pappus-Guldin Theorem - Historical Background: Guldin

The contribution of Paul Guldin (1577-1643) to the Pappus-Guldin Theorem occurs toward the end of a long road of re-discovery and invention related to centers of gravity. Archimedes had initiated the classical study of centers of gravity in the two books On the Equilibrium of Planes [2]. Given its rather marginalized status in today's mathematics curriculum, it might be surprising to learn that finding centers of gravity was an important research topic in mathematics in the 16th and 17th centuries which was completely on par with computation of areas, volumes, and tangents. The translation of Archimedes' work on centers of gravity by Frederico Commandino (1506-1575) in 1565 had established the computation of centers of gravity as a problem very much of interest in mathematics, and it is perhaps noteworthy that Commandino himself, who was familiar with the length and breadth of Greek mathematics, wrote only one major original mathematical work, Liber De centro gravitatis solidorum, the focus of which was to determine the center of gravity of a parabolic conoid [3].

Later authors would add tremendously to this body of research on centers of gravity. Among others, Luca Valerio (1552-1618) computed centers of gravity for the "Archimedean Solids" in his work De centro gravitatis solidorum libri tres in 1604 [4]. In 1632, Jean Charles della Faille (1597-1652) devoted an entire work to determining the center of gravity of the sector of a circle [5]. Guldin's 1641 book, usually referred to as the Centrobaryca, was perhaps the most extensive of all of these works, amounting to more than 700 pages devoted solely to the study of centers of gravity. Indeed, the Pappus-Guldin Theorem occupies a small part of the overall work, appearing as Proposition 3 of Chapter Eight of Book II of the Centrobaryca. Unfortunately, it is dressed in technical language which makes it almost incomprehensible to the casual reader.

In fact, neither Pappus nor Guldin has given us a legitimate proof of the Pappus-Guldin Theorem. Pappus' proof is lost and, as H. Bussard notes in Guldin's DSB entry, in the Centrobaryca Guldin "attempted to prove his theorem by metaphysical reasoning" [6]. Indeed, for quite a while, there were serious questions about whether Guldin had actually plagiarized his result from Pappus. (See [7, p. 139].) However, Ivor Bulmer-Thomas has successfully argued that this is not the case [8]. Ironically, given the result's attribution to Pappus and Guldin, the first acceptable published proof was presented by the obscure Italian mathematician John Antonio Roccha in Evangelista Torricelli's Opera Geometrica in 1644. (See also [9, p. 157].)