Editing Johann Heinrich Lambert (section)

== Work ==

===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.

===Map projection===
Lambert was the first mathematician to address the general properties of [[map projection]]s (of a spherical Earth).<ref>{{Cite book|url=http://atena.beic.it/webclient/DeliveryManager?pid=13452701&search_terms=DTL47|title=Acta Eruditorum|year=1763|location=Leipzig|pages=143}}</ref> 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<ref name=flattening>{{cite book | author=Snyder, John P. | title=Flattening the Earth: Two Thousand Years of Map Projections | publisher =[[University of Chicago Press]]| date=1993|isbn=0-226-76747-7}}.</ref>  p77). In 1772, Lambert published<ref name=lambert>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).</ref><ref name="lambert1772republished">{{cite book |last=Lambert |first=Johann Heinrich |title=Anmerkungen und Zusätze zur Entwerfung der Land- und Himmelscharten (1772) |editor = A. Wangerin |publisher=W. Engelmann |location=Leipzig |date=1894 |url=https://archive.org/details/anmerkungenundz00lambgoog | access-date=2018-10-14 }}
</ref>
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)<ref name="tobler">Tobler, Waldo R, ''Notes and Comments on the Composition of Terrestrial and Celestial Maps'', 1972. (University of Michigan Press), reprinted (2010) by Esri: [http://store.esri.com/esri/showdetl.cfm?SID=2&Product_ID=1284&Category_ID=38].</ref>).
Lambert did not give names to any of his projections but they are now known as:
# [[Lambert conformal conic projection|Lambert conformal conic]]
# [[Transverse Mercator projection|Transverse Mercator]]
# [[Lambert azimuthal equal-area projection|Lambert azimuthal equal area]]
# Lagrange projection
# [[Lambert cylindrical equal-area projection|Lambert cylindrical equal area]]
# Transverse cylindrical equal area
# [[Lambert equal-area conic projection|Lambert conical equal area]]
The first three of these are of great importance.<ref name=flattening/><ref>Corresponding to the Lambert azimuthal equal-area projection, there is a Lambert [[zenith]]al equal-area projection.  ''The Times Atlas of the World'' (1967), Boston:  Houghton Mifflin, Plate 3 et passim.</ref> Further details may be found at [[map projections]] and in several texts.<ref name=flattening/><ref name=snyder>{{cite book | author=Snyder, John P. | title=Map Projections - A Working Manual. U.S. Geological Survey Professional Paper 1395 | publisher =United States Government Printing Office, Washington, D.C. | date=1987}}This paper can be downloaded from
[https://pubs.er.usgs.gov/pubs/pp/pp1395 USGS pages.] {{Webarchive|url=https://web.archive.org/web/20080516070706/http://pubs.er.usgs.gov/pubs/pp/pp1395 |date=2008-05-16 }}</ref><ref name="mulcahy_lambert">{{cite web
  | last =Mulcahy
  | first =Karen
  | title =Cylindrical Projections
  | publisher = [[City University of New York]]
  | url =http://www.geo.hunter.cuny.edu/mp/cylind.html
  | access-date = 2007-03-30 }}</ref>

===Physics===
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 [[Lambert's cosine law|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 cited a law of light absorption, formulated earlier by [[Pierre Bouguer]] he is mistakenly credited for<ref>{{Cite web|url=https://www.britannica.com/biography/Pierre-Bouguer#ref149726|title = Pierre Bouguer &#124; French scientist}}</ref> (the [[Beer–Lambert law]]) and introduced the term ''[[albedo]]''.<ref>{{cite book|last=Mach|first=Ernst|title=The Principles of Physical Optics|publisher=Dover|date=2003|isbn=0-486-49559-0|pages=14–20}}</ref> [[Lambertian reflectance]] is named after him. He wrote a classic work on [[Perspective (visual)|perspective]] and contributed to [[geometrical optics]].

The non-[[SI]] unit of luminance, [[lambert (luminance)|lambert]], is named in recognition of his work in establishing the study of [[photometry (optics)|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.<ref>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.</ref> His investigations were built on the earlier theoretical proposals of [[Tobias Mayer]], greatly extending these early ideas.<ref>Sarah Lowengard (2006) [http://www.gutenberg-e.org/lowengard/A_Chap03.html "Number, Order, Form: Color Systems and Systematization"] and [http://www.gutenberg-e.org/lowengard/glossShell.html?l#l03 Johann Heinrich Lambert] in ''The Creation of Color in Eighteenth-Century Europe'', [[Columbia University Press]]</ref> Lambert was assisted in this project by the court painter [[Benjamin Calau]].<ref>Introduction to {{cite book|date=2011|title=Johann Heinrich Lambert's ''Farbenpyramide''|type=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|url=http://www.iscc.org/pdf/LambertFarbenpyramide.pdf|url-status=dead|archive-url=https://web.archive.org/web/20160304034654/http://www.iscc.org/pdf/LambertFarbenpyramide.pdf|archive-date=2016-03-04}}</ref>

=== Logic and philosophy ===
In his main philosophical work, ''Neues Organon'' (''New Organon'', 1764, named after [[Aristotle]]'s ''[[Organon]]''), Lambert studied the rules for distinguishing [[Subjectivity|subjective]] from [[Objectivity (science)|objective]] appearances, connecting with his work in [[optics]]. The ''Neues Organon'' contains one of the first appearances of the term ''phenomenology'',<ref>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.</ref> and it includes a presentation of the various [[syllogism#Types|kinds of syllogism]]. According to [[John Stuart Mill]], {{blockquote| 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 [[syllogism|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.<ref>[[J. S. Mill]] (1843) [https://archive.org/details/systemofratiocin00milluoft/page/130 A System of Logic], page 130 via [[Internet Archive]]</ref>}}

A modern edition of the ''Neues Organon'' was published in 1990 by the Akademie-Verlag of Berlin.

In 1765 Lambert began corresponding with [[Immanuel Kant]].  Kant intended to dedicate the ''[[Critique of Pure Reason]]'' to Lambert, but the work was delayed, appearing after Lambert's death.<ref>O'Leary M., ''Revolutions of Geometry'', London:Wiley, 2010, p.385</ref>

=== Astronomy ===
Lambert also developed a theory of the generation of the [[universe]] that was similar to the [[nebular hypothesis]] that [[Thomas Wright (astronomer)|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 system]]s) 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.

=== Meteorology ===
Lambert propounded the ideology of observing periodic phenomena first, try to derive their rules and then gradually expand the theory. He expressed his purpose in meteorology as follows:
{{Blockquote
|text=It seems to me that if one wants to make meteorology more scientific than it currently is, one should imitate the astronomers who began with establishing general laws and middle movements without bothering too much with details first. [...] Should one not do the same in meteorology? It is a sure fact that meteorology has general laws and that it contains a great number of periodic phenomena. But we can but scarcely guess these latter. Only few observations have been made so far, and between these one cannot find connections.
|author=Johann Heinrich Lambert<ref name="Bullynck 2010" />
}}

To obtain more and better data of meteorology, Lambert proposed to establish a network of weather stations around the world, in which the various weather configurations (rain, clouds, dry ...) would be recorded – the methods that are still used nowadays. He also devoted himself to the improvement of the measuring instruments and accurate concepts for the advancement of meteorology. This results in his published works in 1769 and 1771 on hygrometry and hygrometers.<ref name="Bullynck 2010">{{cite journal | last=Bullynck | first=Maarten | title=Johann Heinrich Lambert's Scientific Tool Kit, Exemplified by His Measurement of Humidity, 1769–1772 | journal=Science in Context | volume=23 | issue=1 | date=2010-01-26 | url= https://halshs.archives-ouvertes.fr/halshs-00663305/document | archive-url=https://web.archive.org/web/20181103164408/https://halshs.archives-ouvertes.fr/halshs-00663305/document | archive-date=2018-11-03 | issn=1474-0664 | doi=10.1017/S026988970999024X | pages=65–89 | s2cid=170241574 }}</ref>