Johann Heinrich Lambert | |
---|---|

Johann Heinrich Lambert (1728–1777) | |

Born | |

Died | 25 September 1777 49) | (aged

Nationality | Swiss |

Known for | First Proof that π is irrational Beer–Lambert law Lambert's cosine law Transverse Mercator projection Lambert W function |

Scientific career | |

Fields | Mathematician, physicist, astronomer, and philosopher |

Influences | Aristotle, Bacon, Euler, Wolff |

Influenced | Kant, Mendelssohn |

**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.^{ [1] }

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.^{ [2] } Euler believed the conjecture but could not prove that π was irrational, and it is speculated that Aryabhata also believed this, in 500 CE.^{ [3] } 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).^{ [4] } 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^{ [5] } p77). In 1772, Lambert published^{ [6] }^{ [7] } 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)^{ [8] }). Lambert did not give names to any of his projections but they are now known as:

- Lambert conformal conic
- Transverse Mercator
- Lambert azimuthal equal area
- Lagrange projection
- Lambert cylindrical equal area
- Transverse cylindrical equal area
- Lambert conical equal area

The first three of these are of great importance.^{ [5] }^{ [9] } Further details may be found at map projections and in several texts.^{ [5] }^{ [10] }^{ [11] }

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 *.^{ [12] } 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.^{ [13] } His investigations were built on the earlier theoretical proposals of Tobias Mayer, greatly extending these early ideas.^{ [14] } Lambert was assisted in this project by the court painter Benjamin Calau.^{ [15] }

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.^{ [16] }

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*,^{ [17] } and one can find therein a very pedagogical presentation of the various kinds of syllogism. According to John Stewart Mill,

- The German philosopher Lambert, whose
*Neues Organon*(published in the year 1764) contains among other things one of the most elaborate and complete expositions of the syllogistic doctrine, has expressly examined which sort of arguments fall most suitably and naturally into each of the four figures; and his investigation is characterized by great ingenuity and clearness of thought.^{ [18] }

- ↑ W. W. Rouse Ball (1908) Johann Heinrich Lambert (1728 — 1777) via Trinity College, Dublin
- ↑ Lambert, Johann Heinrich (1761). "Mémoire sur quelques propriétés remarquables des quantités transcendentes circulaires et logarithmiques" [Memoir on some remarkable properties of circular transcendental and logarithmic quantities].
*Histoire de l'Académie Royale des Sciences et des Belles-Lettres de Berlin*(in French) (published 1768).**17**: 265–322. - ↑ Rao, S. Balachandra (1994).
*Indian Mathematics and Astronomy: Some Landmarks*. Bangalore: Jnana Deep Publications. ISBN 81-7371-205-0. - ↑
*Acta Eruditorum*. Leipzig. 1763. p. 143. - 1 2 3 Snyder, John P. (1993).
*Flattening the Earth: Two Thousand Years of Map Projections*. University of Chicago Press. ISBN 0-226-76747-7.. - ↑ Lambert, Johann Heinrich. 1772.
*Ammerkungen und Zusatze zurder Land und Himmelscharten Entwerfung*. In Beitrage zum Gebrauche der Mathematik in deren Anwendung, part 3, section 6). - ↑ Lambert, Johann Heinrich (1894). A. Wangerin (ed.).
*Anmerkungen und Zusätze zur Entwerfung der Land- und Himmelscharten (1772)*. Leipzig: W. Engelmann. Retrieved 2018-10-14. - ↑ Tobler, Waldo R,
*Notes and Comments on the Composition of Terrestrial and Celestial Maps*, 1972. (University of Michigan Press), reprinted (2010) by Esri: . - ↑ Corresponding to the Lambert azimuthal equal-area projection, there is a Lambert zenithal equal-area projection.
*The Times Atlas of the World*(1967), Boston: Houghton Mifflin, Plate 3 et passim. - ↑ Snyder, John P. (1987).
*Map Projections - A Working Manual. U.S. Geological Survey Professional Paper 1395*. United States Government Printing Office, Washington, D.C.This paper can be downloaded from USGS pages. - ↑ Mulcahy, Karen. "Cylindrical Projections". City University of New York . Retrieved 2007-03-30.
- ↑ Mach, Ernst (2003).
*The Principles of Physical Optics*. Dover. pp. 14–20. ISBN 0-486-49559-0. - ↑ Lambert,
*Beschreibung einer mit dem Calauschen Wachse ausgemalten Farbenpyramide wo die Mischung jeder Farben aus Weiß und drey Grundfarben angeordnet, dargelegt und derselben Berechnung und vielfacher Gebrauch gewiesen wird*(Berlin, 1772). On this model, see, for example, Werner Spillmann ed. (2009).*Farb-Systeme 1611-2007. Farb-Dokumente in der Sammlung Werner Spillmann*. Schwabe, Basel. ISBN 978-3-7965-2517-9. pp. 24 and 26; William Jervis Jones (2013).*German Colour Terms: A study in their historical evolution from earliest times to the present*. John Benjamins, Amsterdam & Philadelphia. ISBN 978-90-272-4610-3. pp. 218–222. - ↑ Sarah Lowengard (2006) "Number, Order, Form: Color Systems and Systematization" and Johann Heinrich Lambert in
*The Creation of Color in Eighteenth-Century Europe*, Columbia University Press - ↑ Introduction to
*Johann Heinrich Lambert's*Farbenpyramide*(PDF) (Translation of "Beschreibung einer mit dem Calauischen Wachse ausgemalten Farbenpyramide" ("Description of a colour pyramid painted with Calau's wax"), 1772, with an introduction by Rolf Kuehni). 2011. Archived from the original (PDF) on 2016-03-04.* - ↑ O'Leary M.,
*Revolutions of Geometry*, London:Wiley, 2010, p.385 - ↑ In his Preface, p. 4, of vol. I, Lambert called phenomenology "the doctrine of appearance." In vol. ii, he discussed sense appearance, psychological appearance, moral appearance, probability, and perspective.
- ↑ J. S. Mill (1843) A System of Logic, page 130 via Internet Archive

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:

- Asimov, Isaac (1972).
*Asimov's Biographical Encyclopedia of Science and Technology*. Doubleday & Co., Inc. ISBN 0-385-17771-2. - Papadopoulos, A.; Théret, G. (2014).
*La théorie des parallèles de Johann Heinrich Lambert: French translation, with historical and mathematical commentaries*. Paris: Collection Sciences dans l'histoire, Librairie Albert Blanchard. ISBN 978-2-85367-266-5. - Eisenring, Max E. (Nov 1941).
*Johann Heinrich Lambert und die wissenschaftliche Philosophie der Gegenwart*(PDF) (Ph.D. dissertation) (in German). ETH Zürich.

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- Johann Heinrich Lambert (1728-1777): Collected Works - Sämtliche Werke Online
- O'Connor, John J.; Robertson, Edmund F., "Johann Heinrich Lambert",
*MacTutor History of Mathematics archive*, University of St Andrews . - Britannica
- Digitized works at Université de Strasbourg
- "Mémoire sur quelques propriétés remarquables..." (1761), demonstration of irrationality of π, online and analyzed
*BibNum*(PDF).

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