Switzerland/Germany, 1760. The intensity emitted in any direction from a region of a diffuse surface is proportional to the cosine of the angle between the direction of radiation and the normal to the surface.
Cross Reference: Beer’s Law of Absorption, the Lambert-Beer Law, and the Bouguer-Beer Law.
In 1760, during the French and Indian War, Cherokee natives allied with French forces and attacked a North Carolina militia stationed at Fort Dobbs. The Russians occupied and burned Berlin. German cabinetmaker Kaspar Faber made preparations for the first commercial production of pencils.
Lambert’s Law of Emission, also known as Lambert’s Cosine Law or the Cosine Law of Emission, states that the intensity (flux per unit solid angle) emitted in any direction from a region of a perfect diffuse radiating surface is proportional to the cosine of the angle between the direction of radiation and the normal to the surface. (An ideal diffuse surface is usually a rough surface, like chalk, such that the small variations in the surface cause an incoming light ray to be reflected in all directions equally.) Thus, the region, or element, of the surface that obeys Lambert’s Cosine Law will appear equally bright when observed from any direction.
Here’s another way of stating the law: The total radiant power observed from a perfect radiating surface is proportional to the cosine of the angle θbetween the observer’s line of sight and a line drawn perpendicular to the surface. The radiating surface appears equally bright regardless of the viewing angle because, purely from geometrical considerations, the apparent size of a portion of the surface is proportional to the cosine of the angle.
Lambert’s Cosine Law may be stated as Ie∝cosθ,
where Ie is the intensity of emitted light, θ is the angle between the observed emitted intensity and the normal to the surface, and∝ means
“is proportional to.”
An untreated piece of lumber from the lumber yard exhibits nearly Lambertian reflectance (i.e., it obeys Lambert’s Cosine Law), but the same piece of wood with a glossy coat of varnish is probably not a Lambertian reflector because the observer will see specular highlights (bright spots
of light that appear on shiny objects when illuminated) when viewing the wood from specific angles.
Textbooks frequently give additional examples of Lambertian reflec- tors that include “sand-blasted opal glass” and “scraped plaster of Paris.”
Such surfaces are said to have a “matte” finish. When we have a matte surface, luminance is sometimes expressed in terms of total luminous flux, in units of lumens, emitted by a unit area of surface. A surface emitting one lumen per square centimeter has a luminance of one lambert, named in honor of Johann Lambert.
An example of Lambert’s law in action can be found in our observa- tions of visible light from the Sun. Because the Sun is nearly a Lambertian radiator, its brightness is almost the same everywhere on an image of the solar disk.
Johann Heinrich Lambert (1728–1777), Swiss-German mathematician and physicist famous for his work on the mathematical constantπand for his laws of reflection and absorption of light.
CURIOSITYFILE: Philosopher Immanuel Kant called Lambert “the greatest genius of Germany.” • Lambert developed theorems regarding conic sections that made it possible to simplify calculation of the orbits of comets.
• Lambert invented the first practical hygrometer and photometer. • He introduced the word “albedo” when studying the brightness of planets.
• Lambert’s accomplishments are particularly impressive, considering he was almost entirely self-taught.
The first object of my endeavors was the means to become perfect and happy. I understood that the will could not be improved before the mind had been enlight- ened.
—Johann H. Lambert, quoted inDictionary of Scientific Biography
In geometry, Lambert goes beyond the previously assumed concept of space, by establishing the properties of incidence. Lambert’s physical erudition indicates yet another clear way in which it would be possible to elimi- nate the traditional myth of three-dimensional geometry through the parallels with the physical dependence of functions. A number of questions that were formulated
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by Lambert in his metatheory in the second half of the 18th century have not ceased to remain of interest today.
—J. Folta, “Remarks on the Axiomatic Development of Mathematics in the Second Half of the Eighteenth Century”
Johann Lambert is another example of an extreme polymath, having pub- lished more than 150 works on topics ranging from geometry and proba- bility to optics, cartography, philosophy, meteorology, astronomy, and per- spective in art. He is also considered to be the greatest eighteenth-century logician. When Lambert was asked by the King of Prussia, Frederick the Great, in which science he was most proficient, Lambert modestly replied,
“All.”
Lambert was born in Mülhausen, which is now Mulhouse, Alsace, France. Encyclopedias list Lambert variously as a German scientist, a Swiss-German scientist, or a German-French scientist. (In my country count in the introduction for this book, I counted him as German.) The town of Mülhausen was a member of an association of ten free towns in Alsace that were allied to the Swiss Confederation, which was a free republic until it was absorbed into France in 1798.
Lambert was one of five sons and two daughters. His father was a tailor.
During Lambert’s preteen years, he had a diverse education with studies that included Latin and French. At age 12, he left school to help his father in the tailor shop but studied science on his own whenever time permitted.
At age 17, Lambert worked as a secretary to the editor of a conservative newspaper, and he studied science, philosophy, and mathematics after work. His keen interest in these subjects is obvious in a letter that he wrote while still in his teens:
I bought some books in order to learn the first principles of philos- ophy. The first object of my endeavors was the means to become perfect and happy. . . . I studied Christian Wolff’s “On the power of the human mind,” Nicolas Malebranche’s “On the investigation of truth,” and John Locke’s “Essay concerning human understanding.”
The mathematical sciences, in particular algebra and mechanics, provided me with clear and profound examples to confirm the rules I had learned. Thereby, I was able to penetrate into other sciences more easily and more profoundly, and to explain them to others, too.
While in the town of Chur, which was at the time part of the Swiss Confederation, he was elected to the Literary Society of Chur and to the
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Swiss Scientific Society based in Basel. He made regular meteorological observations as part of his society responsibilities. In 1755, he published his first paper, which was on his theories of caloric heat. In 1758, he published a book dealing with the passage of light through various substances. The book was followed by his 1760 bookPhotometria—one of his most famous works because it contained the emission law that often bears his name.
Christoph Scriba in theDictionary of Scientific Biographywrites:
Lambert carried out his experiments with few and primitive instru- ments, but his conclusions resulted in laws that bear his name. The exponential decrease of the light in a beam passing through an absorbing medium of uniform transparency is often called “Lam- bert’s law of absorption,” although Bouguer discovered it earlier.
“Lambert’s cosine law” states that the brightness of a diffusely radiating plane surface is proportional to the cosine of the angle formed by the line of sight and the normal to the surface.
As alluded to by Scriba, Lambert’s name is also associated with an absorption law of light. Under “Beer’s Law of Absorption” in part III, I discuss how, in 1729, French mathematician Pierre Bouguer (1698–1758) formulated an absorption law for light—namely, that the fraction of light absorbed by a particular material is directly proportional to the thickness of the material. In Bouguer’s 1729 paper Essai d’optique sur la grada- tion de la lumière(“Optical Experiment on the Gradation of Light”), he defined the quantity of light lost by passing through a given extent of the atmosphere, and perhaps Bouguer should be considered as the first known discoverer of “Beer’s Law.” Lambert—a scientist more prominent than Bouguer—rediscovered and published Bouguer’s law. When additional careful experiments were made, scientists noticed that the amount of light absorbed by solutions also probably depended on additional factors. In 1852, August Beer (1825–1863) announced a more complete law of absorp- tion that is known variously as Beer’s Law, the Lambert-Beer Law, and the Bouguer-Beer Law. See the entry on “Beer’s Law” for further discussion.
In 1761, Lambert published his cosmological theories inCosmologische Briefe über die Einrichtung des Weltbaues(Cosmological Letters on the Arrangement of the World Structure). Here, he proposed that we live in a finite universe composed of galaxies of stars. Lambert came to believe that all planets, comets, and moons in the universe were likely to contain life.
In hisCosmologische Briefe, Lambert asserted,
The Creator is much too efficient not to imprint life, forces and activity on each speck of dust. . . . [I]f one is to form a correct notion of the world, one should set as a basis God’s intention in its true 140 | a r c h i m e d e s t o h a w k i n g
extent to make the whole world inhabited. . . . All possible varieties which are permitted by general laws ought to be realized. . . .
According to Lambert, an omnipotent God would populate all parts of the universe with diverse beings, and humanoids were likely to be everywhere. In order to protect such life forms, God would rarely allow collisions between bodies such as planets and comets. Both Immanuel Kant (1724–1804) and Lambert conceived of a fractal, or hierarchical, universe of stars clustered into larger systems, which today we call galaxies.
These clusters are clustered into superclusters, and so on, at different size scales.
Lambert’s philosophical works focus on the nature of human knowl- edge and thought, mathematical logic, and methods for scientific proof.
In the field of mathematics, Lambert is most famous for being the first to prove thatπis irrational, that is, cannot be written as the ratio of two integers.
Lambert developed a means of organizing colors by using a triangular pyramid representation. The triangular base is black at its center with vertices colored cinnabar (red), yellow, and azurite (blue). As one gazes upward along the pyramid, the colors increase in brightness until reaching the white tip at the top. Lambert suggested that his system could help tex- tile merchants decide which colors they had available. He also hoped that printers would get ideas for aesthetic combinations of colors by studying his color pyramid.
Around the year 1772, Lambert developed a map projection that is now called the Lambert conformal conic projection. The shapes of the countries on a globe are well preserved when represented in this flat map.
Cartographers still use this projection today and consider it one of the more useful projections for regions of Earth near the middle latitudes with an east–west orientation.
Lambert died of tuberculosis in Berlin at age 49, having never married.
A lunar crater with a diameter of 30 kilometers was named after Lambert and approved in 1935 by the International Astronomical Union General Assembly. A Martian crater is also named in his honor.
Throughout his life, Lambert had been awed by the power of science and scientists—and he was concerned about potential dangers from outer space. Sara Schechner, inComets, Popular Culture, and the Birth of Mod- ern Cosmology, comments on Lambert and other scientists of his era with similar interests:
In affirming a role for comets in the beginning and end of the world, natural philosophers derived a new sense of power. If comets were indeed divine tools for reforming the world, it became conceivably
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possible for astronomers to predict when certain scriptural prophe- cies might be fulfilled.
Lambert had written on these thoughts in 1761 inCosmologische Briefe:
I was not far from looking at astronomers as authorized prophets and . . . from seeing in the invention of the telescope and in the rapid growth of astronomy the herald of an impending disaster. How could, I thought, a genie suggest to Copernicus the structure of the world, to Kepler its laws, and to Newton that terrible attraction and the doctrine about the course and impacts of comets, so that everything might be available for prediction of the calamity, and the inhabitants of the Earth might see to it that instead of having all come to an end, a seed for propagation might remain alive on the changed Earth.
FURTHER READING
Folta, J., “Remarks on the Axiomatic Development of Mathematics in the Second Half of the Eighteenth Century,”DVT-Dejiny Ved a Techniky, 6: 189–205, 1973.
Lambert, Johann,Cosmologische Briefe über die Einrichtung des Weltbaues (Augsburg, 1761).
Schechner, Sara,Comets, Popular Culture, and the Birth of Modern Cosmology (Princeton, N.J.: Princeton University Press, 1999).
Scriba, Christoph, “Johann Lambert,” in Dictionary of Scientific Biography, Charles Gillispie, editor-in-chief (New York: Charles Scribner’s Sons, 1970).
INTERLUDE: CONVERSATION STARTERS Nature is showing us only the tail of the lion, but I have no doubt that the lion belongs to it even though, because of its large size, it cannot totally reveal itself all at once. We can see it only the way a louse that is sitting on it would.
—Albert Einstein to Heinrich Zangger, March 10, 1914 Our attempts at modeling physical reality normally con- sist of two parts: (1) A set of local laws that are obeyed by the various physical quantities. These are usually formu- lated in terms of differential equations. (2) Sets of bound- ary conditions that tell us the state of some regions of the universe at a certain time. . . . Many people would claim that the role of science is confined to the first of these and 142 | a r c h i m e d e s t o h a w k i n g
that theoretical physics will have achieved its goal when we have obtained a complete set of local physical laws.
—Stephen Hawking,Black Holes and Baby Universes Science, like life, feeds on its own decay. New facts burst old rules; then newly divined conceptions bind old and new together into a reconciling law.
—William James,The Will to Believe and Other Essays in Popular Philosophy
We materialists don’t deny the force of ideas; we merely say that the minds precipitating them are wholly situated within brains that . . . seem to have emerged without any need for miracles.
—Frederick Crews, “Follies of the Wise,” Skeptical Inquirer, March/April, 2007
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