Johann Heinrich Lambert (German:[ˈlambɛɐ̯t];French:Jean-Henri Lambert; 26 or 28 August 1728 – 25 September 1777) was apolymath from theRepublic of Mulhouse, generally identified as eitherSwiss orFrench, who made important contributions to the subjects ofmathematics,physics (particularlyoptics),philosophy,astronomy andmap projections.

Johann Heinrich Lambert
Johann Heinrich Lambert (1728–1777)
Born	26 or 28 August 1728 Republic of Mulhouse (city-state allied to theSwiss Confederacy; currentlyAlsace, France)
Died	25 September 1777(1777-09-25) (aged 49) Berlin,Prussia
Known for	FirstProof that π is irrational Beer–Lambert law Lambert's cosine law Transverse Mercator projection Lambert W function
Scientific career
Fields	Mathematician,physicist,astronomer, andphilosopher

Lambert was born in 1728 into aHuguenot family in the city ofMulhouse^[1] (now inAlsace,France), at that time a city-state allied to theSwiss Confederacy.^[2] Some sources give 26 August as his birth date and others 28 August.^[3][4][1] Leaving school at 12, he continued to study in his free time while 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 ofBasler Zeitung and, at the age of 20, private tutor to the sons of Count Salis inChur. 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 thePrussian Academy of Sciences in Berlin, where he gained the sponsorship ofFrederick II of Prussia, and became a friend ofLeonhard Euler. In this stimulating and financially stable environment, he worked prodigiously until his death in 1777.^[1]

Lambert was the first to systematize and popularize the usehyperbolic functions intotrigonometry. He credits the previous works ofVincenzo Riccati andDaviet de Foncenex. Lambert developed exponential expressions and identities and introduced the modern notation.^[5] Lamber also made conjectures aboutnon-Euclidean space.

Lambert is credited with the firstproof that π is irrational using ageneralized continued fraction for the function tan x.^[6]Euler believed the conjecture but could not prove that π was irrational, and it is speculated thatAryabhata also believed this, in 500 CE.^[7] Lambert also devised theorems aboutconic sections that made the calculation of theorbits ofcomets simpler.

Lambert devised a formula for the relationship between the angles and the area ofhyperbolic triangles. These are triangles drawn on a concave surface, as on asaddle, instead of the usual flat Euclidean surface. Lambert showed that the angles added up to less thanπ (radians), 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 thecurvature 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 ofsimilar 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.

Lambert was the first mathematician to address the general properties ofmap projections (of a spherical Earth).^[8] In particular he was the first to discuss the properties of conformality and equal areapreservation and to point out that they were mutually exclusive.(Snyder 1993^[9] p77). In 1772, Lambert published^[10][11]seven new map projections under the titleAnmerkungen und Zusätze zur Entwerfung der Land- und Himmelscharten, (translated asNotes and Comments on the Composition of Terrestrial and Celestial Maps by Waldo Tobler (1972)^[12]).Lambert did not give names to any of his projections but they are now known as:

1. Lambert conformal conic
2. Transverse Mercator
3. Lambert azimuthal equal area
4. Lagrange projection
5. Lambert cylindrical equal area
6. Transverse cylindrical equal area
7. Lambert conical equal area

The first three of these are of great importance.^[9][13] Further details may be found atmap projections and in several texts.^[9][14][15]

Lambert invented the first practicalhygrometer. In 1760, he published a book on photometry, thePhotometria. 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 thesine 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. InPhotometria Lambert also cited a law of light absorption, formulated earlier byPierre Bouguer he is mistakenly credited for^[16] (theBeer–Lambert law) and introduced the termalbedo.^[17]Lambertian reflectance is named after him. He wrote a classic work onperspective and contributed togeometrical optics.

The non-SI unit of luminance,lambert, is named in recognition of his work in establishing the study ofphotometry. Lambert was also a pioneer in the development of three-dimensionalcolour 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.^[18] His investigations were built on the earlier theoretical proposals ofTobias Mayer, greatly extending these early ideas.^[19] Lambert was assisted in this project by the court painterBenjamin Calau.^[20]

In his main philosophical work,Neues Organon (New Organon, 1764, named afterAristotle'sOrganon), Lambert studied the rules for distinguishingsubjective fromobjective appearances, connecting with his work inoptics. TheNeues Organon contains one of the first appearances of the termphenomenology,^[21] and it includes a presentation of the variouskinds of syllogism. According toJohn Stuart Mill,

The German philosopher Lambert, whoseNeues Organon (published in the year 1764) contains among other things one of the most elaborate and complete expositions of thesyllogistic 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.^[22]

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

In 1765 Lambert began corresponding withImmanuel Kant. Kant intended to dedicate theCritique of Pure Reason to Lambert, but the work was delayed, appearing after Lambert's death.^[23]

Lambert also developed a theory of the generation of theuniverse that was similar to thenebular hypothesis thatThomas Wright andImmanuel Kant had (independently) developed. Wright published his account inAn Original Theory or New Hypothesis of the Universe (1750), Kant inAllgemeine Naturgeschichte und Theorie des Himmels, published anonymously in 1755. Shortly afterward, Lambert published his own version of the nebular hypothesis of the origin of theSolar System inCosmologische Briefe über die Einrichtung des Weltbaues (1761). Lambert hypothesized that the stars near theSun were part of a group which travelled together through theMilky Way, and that there were many such groupings (star systems) throughout thegalaxy. The former was later confirmed by SirWilliam Herschel. Inastrodynamics he also solved the problem of determination of time of flight along a section of orbit, known now asLambert's problem. His work in this area is commemorated by theAsteroid187 Lamberta named in his honour.

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:

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.

— Johann Heinrich Lambert^[24]

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.^[24]

• Lambert, Johann Heinrich. "Pyrometrie; oder, Vom maasse des feuers und der wȧrme. Mit acht kupfertafeln." Berlin, Bey Haude und Spener, 1779.

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  1779 copy of "Pyrometrie oder vom Maasse des Feuers und der Wärme"
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  Title page to "Pyrometrie oder vom Maasse des Feuers und der Wärme"
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  First page of "Pyrometrie oder vom Maasse des Feuers und der Wärme"

• List of things named after Johann Lambert
• Asteroid187 Lamberta
• Lambert (Martian crater)

• 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.
• Caddeo, R.; Papadopoulos, A. (2022).Mathematical geography in the eighteenth century: Euler, Lagrange and Lambert. Cham: Springer.ISBN 978-3-031-09569-6.
• Eisenring, Max E. (Nov 1941).Johann Heinrich Lambert und die wissenschaftliche Philosophie der Gegenwart(PDF) (Ph.D. dissertation) (in German). ETH Zürich.Archived(PDF) from the original on 2016-03-04.

• 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 worksArchived 2014-03-29 at theWayback Machine atUniversité de Strasbourg
• "Mémoire sur quelques propriétés remarquables..." (1761), demonstration of irrationality of π, online and analyzedBibNumArchived 2015-04-02 at theWayback Machine (PDF).