MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

158 Biographical Vignettes
give the correct scientific explanation of the cause of the wind (differences of
air temperature and density) and he discovered the principle of the mercury
barometer. He was a skilled lens grinder and made excellent telescopes and
microscopes. The Fermat-Torricelli Problem/Point has already been discussed
at length in Property 12 of Chapter 2. (This problem was solved by Torricelli
and Cavalieri for triangles of less than 120 ◦ and the general case was solved by
Viviani.) Torricelli’s Trumpet (Gabriel’s Horn) is a figure with infinite surface
area yet finite volume. Torricelli’s Law/Theorem relates the speed of fluid
flowing out of an opening to the height of the fluid above the opening (v =
√ 2gh). Torricelli’s Equation provides the final velocity of an object moving
with constant acceleration without having a known time interval (v 2 f = v2 i +
2a∆d). He died in Florence, aged 39, shortly after having contracted typhoid
fever. Viviani agreed to prepare his unpublished materials for posthumous
publication but he failed to accomplish this task, which was not completed until
1944 nearly 300 years after Torricelli’s death. Source material for Torricelli is
available in [297].
Vignette 14 (Vincenzo Viviani: 1622-1703).
Vincenzo Viviani, the last pupil of Galileo, was born in Florence, Italy
[144]. His exceptional mathematical abilities brought him to the attention
of Grand Duke Ferdinando II de’ Medici of Tuscany in 1638 who introduced
him to Galileo. In 1639, at the age of 17, he became Galileo’s assistant at
Arcetri until the latter’s death in 1642. During this period, he met Torricelli
and they later became collaborators on the development of the barometer.
(He was Torricelli’s junior colleague not his student although he collected and
arranged his works after the latter’s death.) The Grand Duke then appointed
him Mathematics teacher at the ducal court and engaged him as an engineer
with the Uffiziali dei Fiumi, a position he held for the rest of his life. From
1655 to 1656, he edited the first edition of Galileo’s collected works and he also
wrote the essay Life of Galileo which was not published in his lifetime. In 1660,
he and Giovanni Alfonso Borelli conducted an experiment involving timing the
difference between seeing the flash and hearing the sound of a cannon shot at
a distance which provided an accurate determination of the speed of sound. In
1661, he experimented with rotation of pendula, 190 years before the famous
demonstration by Foucault. In 1666, the Grand Duke appointed him Court
Mathematician. Throughout his life, one of his main interests was ancient
Greek Mathematics and he published reconstructions of lost works of Euclid
and Apollonius and also translated a work of Archimedes into Italian. He
calculated the tangent to the cycloid and also contributed to constructions
involving angle trisection and duplication of the cube. Viviani’s Theorem has
been previously described in Property 8 of Chapter 2. Viviani’s Curve is a
space curve obtained by intersecting a sphere with a circular cylinder tangent