Torricelli's Trumpet or Gabriel's Horn

The paradox of  y  =  1/x .

Evangelista Torricelli ( 1608 - 1647 ) was a student of Galileo.  As a young man he studied in Galileo's home at Arcetri near Florence.  Upon Galileo's death, Torricelli succeeded his teacher as mathematician and philosopher for their good friend and patron, the Grand Duke of Tuscany.

Torricelli's own words fully describe his amazement at discovering an infinitely long solid with a surface that calculates to have an infinite area, but a finite volume.  "It may seem incredible that although this solid has an infinite length, nevertheless none of the cylindrical surfaces we considered has an infinite length but all of them are finite."  This "incredible" paradox prompted Torricelli to try several alternate proofs.

Torricelli was born a bit too soon.  The study of infinitesimal was too new.  Recall that Newton was born the year Galileo had died (1642) and Leibniz was yet four years younger.  Unfortunately, Torricelli did not live to see the methods of calculus fully emerge to confirm his painstaking calculations largely based on his friend Cavalieri's "summation of plane slices" method.
 

Torricelli is better known for his work in physics.  Along with his mentor, Galileo, he pioneered telescopic astronomy and the construction of barometers. The objective lens in the upper left-hand corner was ground by Torricelli in 1646 and is now on display in the Istituto e Museo di Storia della Scienza (Florence, Italy).  The Museo also displays other fascinating curiosities such as Galileo's preserved finger!      http://www.treasure-troves.com/bios/Torricelli.html A Brief Listing of references that should be in most university libraries: Boyer, Carl B.  A History of Mathematics, various editions and publishers. Katz, Victor J.  A History of Mathematics, 2nd ed., Addison-Wesley, 1998. Fleron, Julian.  Gabriel's Wedding Cake, The College Mathematics Journal,  Vol. 30, No. 1, January, 1999, 35 - 38.