·
Born:
15-Nov-1794 in Bad König,
Odenwald, Germany
o
Father was a
court official of the counts of Erbach-Schöneberg
o
Mother’s
maiden name was Luise Juliane Schweikart
o
Mother’s
younger brother, F K Schweikart, professor of law at University of
Königsberg
o
Taurinus
followed in uncle’s footsteps, studying the law at Heidelberg, Giessen and
Gottingen
o
Starting in
1820 Taurinus and his uncle corresponded about mathematical topics and Taurinus
was influenced by his uncle to look into the problem of parallel lines and
Euclid’s fifth postulate.
o His uncle is known for his investigation into the new geometry (he called astral geometry) where the angle sum of a triangle is less that 180o and this angle sum increased as the area of the triangle decreased which was noticed by Lambert. He had read Lambert’s work of 1807 and Schweikart come up with new results that he corresponded to Gauss in his 1818 letter. Gauss’ response was “…I shall only note that I can solve completely all the problems in the Astral Geometry – so far it has been developed – as soon as the constant C is given”.
o In his first publication, “Die Theorie der Parallellinien” in 1825, Taurinus was convinced that Euclidean geometry was the geometry.
o In his second book, “Geometriae Prima Elementa” in 1826, he had to admit that the “third system of geometry”, where the angle sum of a triangle is less than two right angles, is not contradicted and he called this the “logarithmic-spherical geometry”. But he felt that this would not be a possible geometry on the plane.
o Taurinus and his uncle were the first two investigators of non-Euclidean geometry to move their work into the area of analysis.
o Taurinus uses trigonometry to derive his geometry by starting with the formulas of spherical trigonometry
§ first fundamental formula connecting the sides and angles of a spherical triangle is
cos(a/K) = cos(b/K)cos(g/K) + sin(b/K)sin(g/K)cosA (Spherical triangle)
where a,b and g are the center angles with the center as the vertex and the sides incident
respectively, with the surface triangle vertices of B and C, A and C, and A and B and K is the radius
of the sphere.
§ Taurinus now said replace K with the imaginary number, iK. One can doubt a geometry based upon an imaginary sphere. This yields the following hyperbolic trigonometric equation,
cosh(a/K) = cosh(b/K)cosh(g/K) - sinh(b/K)sinh(g/K)cosA
§ Angle sum is less than 180o; look at equilateral triangle: a = b = g
· cosh(a/K) = cosh2(a/K) – sinh2(a/K) cosA
cosA = [cosh (a/K)] / [cosh (a/K) + 1]
cosh (a/K) > 1 since a ¹ 0 Þ cosA > ½ Þ A < 60o
Therefore, the angle sum is less than 180o
As a approaches 0 or as K approaches ¥ these triangles on the sphere approach Euclidean
§ Second fundamental formula in spherical geometry,
cosA = cosBcosC + sinBsinCcos(a/K) which, under hyperbolic functions becomes
cosA = -cosBcosC + sinBsinCcosh(a/K)
§ As a special case, let A = 0 and C = 90o
· 1 = sinBcosh(a/K) Þ cosh(a/K) = 1 / sinB
· ÐBCA is a right angle and side BA is asysmptotic to side CA
· This yields a new, precise description of Saccheri and Gauss’ figure for the HAA (hypothesis – acute angle). The angle at B is called the angle of parallelism for line through BC
§ From this Taurinus calculates Schweikart’s constant in terms of the sphere’s radius,
C = Klog(1 + Ö2)
Taurinus also determined results for
areas of triangles
circumferences of circles
area and volume of spheres
o In his book of 1826, Taurinus commented
“This [the book] was already written and it seemed to me to remain to state my views on the true essence of this Geometry. I am led at last to the certainty that my answer is really proved. From the very beginning I have conjectured that a geometry so to speak inverse spherical (the logarithmic) with all formulae derivable from the spherical can exist. And I have further wondered that this fact, which is so clear and lies so readily to hand, was not spotted, and such a vast an extent not explored until I recalled that every self-evident matter is often hidden even for a long time even from the most perceptive of men. Moreover, I thought that in everything that was earlier deduced concerning the analytic formulae there was nothing that refers to the geometry and that pure substitution the formulae remain valid”.
o Taurinus could not fully accept this non-Euclidean geometry based upon an imaginary sphere. He wanted a “real” surface. The big issue seemed to be with HOA (hypothesis – obtuse angle)
o This is the first method that was not pictorial but merely formal.
o Taurinus also stated that elliptic geometry can be “realized” on the sphere. This concept was first taken up again by Bernhard Riemann.
· Died: 13-Feb-1874 in Cologne, Germany
Taurinus seems to get lost in the time period between [Saccheri and Lambert] and [Lobachevsky and Bolyai].
References:
Franz
Adolf Taurinus, article by J.J.O'Connor &
E.F.Robertson
Dictionary of Scientific Biography (New York 1970-1990) page 264.
Ideas of space : Euclidean, non-Euclidean, and Relativistic (New York, 1989) by Jeremy Gray