Irrationality of π
Johann Heinrich Lambert (August 26, 1728 – September 25, 1777) was a Swiss mathematician, physicist and astronomer.
Lambert was born in 1728 in the city of Mulhouse (now in Alsace, France), at that time an exclave of Switzerland. Leaving school he continued to study in his free time whilst undertaking a series of jobs. These included assistant to his father (a tailor), a clerk at a nearby iron works, a private tutor, secretary to the editor of Basler Zeitung and, at the age of 20, private tutor to the sons of Count Salis in Chur. Travelling Europe with his charges (1756–1758) allowed him to meet established mathematicians in the German states, The Netherlands, France and the Italian states. On his return to Chur he published his first books (on optics and cosmology) and began to seek an academic post. After a few short posts he was rewarded (1764) by an invitation from Euler to a position at the Prussian Academy of Sciences in Berlin, where he gained the sponsorship of Frederick II of Prussia. In this stimulating, and financially stable, environment he worked prodigiously until his death in 1777.
Lambert was the first to introduce hyperbolic functions into trigonometry. Also, he made conjectures regarding non-Euclidean space. Lambert is credited with the first proof that π is irrational in 1768. Lambert also devised theorems regarding conic sections that made the calculation of the orbits of comets simpler.
Lambert devised a formula for the relationship between the angles and the area of hyperbolic triangles. These are triangles drawn on a concave surface, as on a saddle, instead of the usual flat Euclidean surface. Lambert showed that the angles cannot add up to π (radians), or 180°. The amount of shortfall, called defect, is proportional to the area. The larger the triangle's area, the smaller the sum of the angles and hence the larger the defect CΔ = π — (α + β + γ). That is, the area of a hyperbolic triangle (multiplied by a constant C) is equal to π (in radians), or 180°, minus the sum of the angles α, β, and γ. Here C denotes, in the present sense, the negative of the curvature of the surface (taking the negative is necessary as the curvature of a saddle surface is defined to be negative in the first place). As the triangle gets larger or smaller, the angles change in a way that forbids the existence of similar hyperbolic triangles, as only triangles that have the same angles will have the same area. Hence, instead of expressing the area of the triangle in terms of the lengths of its sides, as in Euclid's geometry, the area of Lambert's hyperbolic triangle can be expressed in terms of its angles.
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