Johann Heinrich Lambert
Johann Heinrich Lambert (1728–1777)
|Died||25 September 1777 49) (aged|
|Known for||First Proof that π is irrational |
Lambert's cosine law
Transverse Mercator projection
Lambert W function
|Fields||Mathematician, physicist, astronomer, and philosopher|
|Influences||Aristotle, Bacon, Euler, Wolff|
Johann Heinrich Lambert (German: [ˈlambɛʁt] , Jean-Henri Lambert in French; 26 August 1728 – 25 September 1777) was a Swiss polymath who made important contributions to the subjects of mathematics, physics (particularly optics), philosophy, astronomy and map projections. Edward Tufte calls him and William Playfair "The two great inventors of modern graphical designs" (Visual Display of Quantitative Information, p. 32).
Lambert was born in 1728 into a Huguenot family in the city of Mulhouse (now in Alsace, France), at that time an exclave of Switzerland. Leaving school at 12, 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 (1763) by an invitation to a position at the Prussian Academy of Sciences in Berlin, where he gained the sponsorship of Frederick II of Prussia, and became a friend of Euler. 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 by using a generalized continued fraction for the function tan x.Euler believed the conjecture but could not prove that π was irrational, and it is speculated that Aryabhata also believed this, in 500 CE. 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 added up to less than π (radians), or 180°. The amount of shortfall, called the defect, increases with the 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 (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.
Lambert was the first mathematician to address the general properties of map projections (of a spherical earth).In particular he was the first to discuss the properties of conformality and equal area preservation and to point out that they were mutually exclusive. (Snyder 1993 p77). In 1772, Lambert published seven new map projections under the title Anmerkungen und Zusätze zur Entwerfung der Land- und Himmelscharten, (translated as Notes and Comments on the Composition of Terrestrial and Celestial Maps by Waldo Tobler (1972) ). Lambert did not give names to any of his projections but they are now known as:
The first three of these are of great importance.Further details may be found at map projections and in several texts.
Lambert invented the first practical hygrometer. In 1760, he published a book on photometry, the Photometria . From the assumption that light travels in straight lines, he showed that illumination was proportional to the strength of the source, inversely proportional to the square of the distance of the illuminated surface and the sine of the angle of inclination of the light's direction to that of the surface. These results were supported by experiments involving the visual comparison of illuminations and used for the calculation of illumination. In Photometria Lambert also formulated the law of light absorption (the Beer–Lambert law) and introduced the term albedo .Lambertian reflectance is named after Johann Heinrich Lambert, who introduced the concept of perfect diffusion in his 1760 book Photometria. He wrote a classic work on perspective and contributed to geometrical optics.
The non-SI unit of luminance, Lambert, is named in recognition of his work in establishing the study of photometry. Lambert was also a pioneer in the development of three-dimensional colour models. Late in life, he published a description of a triangular colour pyramid (Farbenpyramide), which shows a total of 107 colours on six different levels, variously combining red, yellow and blue pigments, and with an increasing amount of white to provide the vertical component.His investigations were built on the earlier theoretical proposals of Tobias Mayer, greatly extending these early ideas. Lambert was assisted in this project by the court painter Benjamin Calau.
In his main philosophical work, Neues Organon (New Organon, 1764), Lambert studied the rules for distinguishing subjective from objective appearances. This connects with his work in the science of optics. In 1765 he began corresponding with Immanuel Kant who intended to dedicate to him the Critique of Pure Reason but the work was delayed, appearing after his death.
Lambert also developed a theory of the generation of the universe that was similar to the nebular hypothesis that Thomas Wright and Immanuel Kant had (independently) developed. Wright published his account in An Original Theory or New Hypothesis of the Universe (1750), Kant in Allgemeine Naturgeschichte und Theorie des Himmels, published anonymously in 1755. Shortly afterward, Lambert published his own version of the nebular hypothesis of the origin of the solar system in Cosmologische Briefe über die Einrichtung des Weltbaues (1761). Lambert hypothesized that the stars near the sun were part of a group which travelled together through the Milky Way, and that there were many such groupings (star systems) throughout the galaxy. The former was later confirmed by Sir William Herschel. In astrodynamics he also solved the problem of determination of time of flight along a section of orbit, known now as Lambert's problem. His work in this area is commemorated by the Asteroid 187 Lamberta named in his honour.
Johann-Heinrich Lambert is the author of a treatise on logic, which he called Neues Organon (1764), that is to say, the New Organon. The most recent edition of this work named after Aristotle's Organon was issued in 1990 by the Akademie-Verlag of Berlin. This contains one of the first appearances of the term phenomenology,and one can find therein a very pedagogical presentation of the various kinds of syllogism. According to John Stewart Mill,
In optics, Lambert's cosine law says that the radiant intensity or luminous intensity observed from an ideal diffusely reflecting surface or ideal diffuse radiator is directly proportional to the cosine of the angle θ between the direction of the incident light and the surface normal. The law is also known as the cosine emission law or Lambert's emission law. It is named after Johann Heinrich Lambert, from his Photometria, published in 1760.
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises when either the metric requirement is relaxed, or the parallel postulate is replaced with an alternative one. In the latter case one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras which give rise to kinematic geometries that have also been called non-Euclidean geometry.
The Gauss–Bonnet theorem, or Gauss–Bonnet formula, is an important statement about surfaces in differential geometry, connecting their geometry to their topology. It is named after Carl Friedrich Gauss, who was aware of a version of the theorem but never published it, and Pierre Ossian Bonnet, who published a special case in 1848.
In mathematics, hyperbolic geometry is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
August Beer was a German physicist, chemist, and mathematician of Jewish descent.
The year 1772 in science and technology involved some significant events.
The transverse Mercator map projection is an adaptation of the standard Mercator projection. The transverse version is widely used in national and international mapping systems around the world, including the UTM. When paired with a suitable geodetic datum, the transverse Mercator delivers high accuracy in zones less than a few degrees in east-west extent.
Lambertian reflectance is the property that defines an ideal "matte" or diffusely reflecting surface. The apparent brightness of a Lambertian surface to an observer is the same regardless of the observer's angle of view. More technically, the surface's luminance is isotropic, and the luminous intensity obeys Lambert's cosine law. Lambertian reflectance is named after Johann Heinrich Lambert, who introduced the concept of perfect diffusion in his 1760 book Photometria.
In hyperbolic geometry, a hyperbolic triangle is a triangle in the hyperbolic plane. It consists of three line segments called sides or edges and three points called angles or vertices.
In mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle. The triangle can be an ordinary Euclidean triangle, a triangle on the sphere, or a hyperbolic triangle. Each triangle group is the symmetry group of a tiling of the Euclidean plane, the sphere, or the hyperbolic plane by congruent triangles called Möbius triangles, each one a fundamental domain for the action.
A Lambert conformal conic projection (LCC) is a conic map projection used for aeronautical charts, portions of the State Plane Coordinate System, and many national and regional mapping systems. It is one of seven projections introduced by Johann Heinrich Lambert in his 1772 publication Anmerkungen und Zusätze zur Entwerfung der Land- und Himmelscharten.
The Lambert azimuthal equal-area projection is a particular mapping from a sphere to a disk. It accurately represents area in all regions of the sphere, but it does not accurately represent angles. It is named for the Swiss mathematician Johann Heinrich Lambert, who announced it in 1772. "Zenithal" being synonymous with "azimuthal", the projection is also known as the Lambert zenithal equal-area projection.
In cartography, the Lambert cylindrical equal-area projection, or Lambert cylindrical projection, is a cylindrical equal-area projection. This projection is undistorted along the equator, which is its standard parallel, but distortion increases rapidly towards the poles. Like any cylindrical projection, it stretches parallels increasingly away from the equator. The poles accrue infinite distortion, becoming lines instead of points.
In a Euclidean space, the sum of angles of a triangle equals the straight angle . A triangle has three angles, one at each vertex, bounded by a pair of adjacent sides.
In hyperbolic geometry an ideal triangle is a hyperbolic triangle whose three vertices all are ideal points. Ideal triangles are also sometimes called triply asymptotic triangles or trebly asymptotic triangles. The vertices are sometimes called ideal vertices. All ideal triangles are congruent.
There are several projections used in maps carrying the name of Johann Heinrich Lambert:
In cartography, the cylindrical equal-area projection is a family of cylindrical, equal-area map projections.
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry:
If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.
Photometria is a book on the measurement of light by Johann Heinrich Lambert published in 1760. It established a complete system of photometric quantities and principles; using them to measure the optical properties of materials, quantify aspects of vision, and calculate illumination.
In complex analysis, the Schwarz triangle function or Schwarz s-function is a function that conformally maps the upper half plane to a triangle in the upper half plane having lines or circular arcs for edges. Let πα, πβ, and πγ be the interior angles at the vertices of the triangle. If any of α, β, and γ are greater than zero, then the Schwarz triangle function can be given in terms of hypergeometric functions as:
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