Mathematics , 1763 Lambert was the first to systematize and popularize the use of
hyperbolic functions into
trigonometry. He credits the previous works of
Vincenzo Riccati and
Daviet de Foncenex. Lambert developed exponential expressions and identities and introduced the modern notation. Lambert 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.
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 about
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 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 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 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: •
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. Further details may be found at
map projections and in several texts.
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
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 (the
Beer–Lambert law) and introduced the term
albedo.
Lambertian reflectance is named after him. 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.
Logic and philosophy In his main philosophical work,
Neues Organon (
New Organon, 1764, named after
Aristotle's
Organon), Lambert studied the rules for distinguishing
subjective from
objective appearances, connecting with his work in
optics. The
Neues Organon contains one of the first appearances of the term
phenomenology, and it includes a presentation of the various
kinds of syllogism. According to
John Stuart Mill, 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.
Astronomy 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.
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: 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. == Published works ==