MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

154 Biographical Vignettes
He discovered that if a triangle is moved so that one vertex moves along a line
while another vertex moves along a second line then the third vertex describes
an ellipse [322, p. 65] and this observation is the basis of a commercial instrument
for drawing an ellipse using trammels [322, p. 66]. He observed that the
angle between an emerging leaf and its predecessor, known as the divergence, is
a constant and thereby explained the resulting logarithmic spiral arrangement
[139, p. 4]. He was also the first to direct attention to “curves of pursuit” [41,
p. 273]. His architectural studies led him to Leonardo’s Symmetry Theorem
which states that all planar isometries are either rotations or reflections [327,
pp. 66, 99]. The remainder of his mathematical discoveries concerned the areas
of lunes, solids of equal volume, reflection in a sphere, inscription of regular
polygons and centers of gravity [56, pp. 43-60]. His greatest contribution to
geometry came in the latter area where he discovered that the lines joining
the vertices of a tetrahedron with the center of gravity of the opposite faces
all pass through a point, the centroid, which divides each of these medians in
the ratio 3 : 1. These diverse and potent mathematical results have certainly
earned Leonardo the title of Mathematician par excellence! He died at the
castle of Cloux in Amboise, France, aged 67.
Vignette 9 (Niccolò Fontana (Tartaglia): 1499-1557).
Niccolò Fontana was a Mathematician, engineer, surveyor and bookkeeper
who was born in Brescia in the Republic of Venice (now Italy) [144]. Brought
up in dire poverty, he became known as Tartaglia (“The Stammerer”) as a
result of horrific facial injuries which impeded his speech that he suffered in
his youth at the hands of French soldiers. He was widely known during his
lifetime for his participation in many public mathematical contests. He became
a teacher of Mathematics at Verona in 1521 and moved to Venice in 1534 where
he stayed for the rest of his life, except for an 18 month hiatus as Professor
at Brescia beginning in 1548. He is best known for his solution to the cubic
equation sans quadratic term (which first appeared in Cardano’s Ars Magna)
but also is known for Tartaglia’s Formula for the volume of a tetrahedron.
His first book, Nuova scienzia (1551), dealt with the theory and practice of
gunnery. His largest work, Trattato generale di numeri e misure (1556), is
a comprehensive mathematical treatise on arithmetic, geometry, mensuration
and algebra as far as quadratic equations. It is here that he treated the “three
jugs problem” described in Recreation 21 of Chapter 4. He also published the
first Italian translation of Euclid (1543) and the earliest Latin version from the
Greek of some of the principal works of Archimedes (1543). He died at Venice,
aged 58.
Vignette 10 (Johannes Kepler: 1571-1630).
Johannes Kepler was born in the Free Imperial City of Weil der Stadt which
is now part of the Stuttgart Region in the German state of Baden-Württemberg