Editing Johann Heinrich Lambert (section)

===Mathematics===
[[File:Acta Eruditorum - I geometria, 1763 – BEIC 13452701.jpg|thumb|Illustration from ''De ichnographica campi'' published in [[Acta Eruditorum]], 1763]]
[[Image:Lambert - Perspective affranchie de l'embarras du plan géometral, 1759 - 1445566.jpg|thumb|''La perspective affranchie de l'embarras du plan géometral'', French edition, 1759]]
Lambert was the first to systematize and popularize the use of [[hyperbolic functions]] into [[trigonometry]]. He credits the previous works of [[Vincenzo Riccati]] and [[François Daviet de Foncenex|Daviet de Foncenex]]. Lambert developed exponential expressions and identities and introduced the modern notation.<ref>{{Cite book |url=https://archive.org/details/eulerat300apprec0000unse/mode/2up?q=foncenex&view=theater |title=Euler at 300 : an appreciation |date=2007 |publisher=[Washington, D.C.] : Mathematical Association of America |others=Internet Archive |isbn=978-0-88385-565-2}}</ref> Lamber also made conjectures about [[non-Euclidean]] space. 

Lambert is credited with the first [[proof that π is irrational]] using a [[generalized continued fraction]] for the function tan x.<ref>{{cite journal|last=Lambert|first=Johann Heinrich|title=Mémoire sur quelques propriétés remarquables des quantités transcendentes circulaires et logarithmiques |trans-title=Memoir on some remarkable properties of circular and logarithmic transcendental quantities |publication-date=1768|date=1761|journal=Histoire de l'Académie Royale des Sciences et des Belles-Lettres de Berlin |volume=17|pages=265–322 |url=https://babel.hathitrust.org/cgi/pt?id=nyp.33433009864251;view=1up;seq=303 |language=fr }}</ref> [[Euler]] believed the conjecture but could not prove that π was irrational, and it is speculated that [[Aryabhata]] also believed this, in 500 CE.<ref>{{cite book |first= S. Balachandra |last=Rao |title = Indian Mathematics and Astronomy: Some Landmarks | publisher = Jnana Deep Publications |date = 1994 |location = Bangalore | isbn = 81-7371-205-0}}</ref> Lambert also devised theorems about [[conic section]]s that made the calculation of the [[orbit]]s of [[comet]]s simpler.

Lambert devised a formula for the relationship between the angles and the area of [[hyperbolic triangle]]s. These are triangles drawn on a concave surface, as on a [[saddle]], instead of the usual flat Euclidean surface. Lambert showed that the angles added up to less than [[Pi|π]] ([[radian]]s), or 180°. The defect (amount of shortfall) increases with area. The larger the triangle's area, the smaller the sum of the angles and hence the larger the defect C△ = π — (α + β + γ). That is, the area of a hyperbolic triangle (multiplied by a constant C) is equal to π (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 by definition negative). As the triangle gets larger or smaller, the angles change in a way that forbids the existence of [[similar triangle|similar]] hyperbolic triangles, as only triangles that have the same angles will have the same area. Hence, instead of the area of the triangle's being expressed in terms of the lengths of its sides, as in Euclidean geometry, the area of Lambert's hyperbolic triangle can be expressed in terms of its angles.