Johann Heinrich Lambert

How to use this course

Johann Heinrich Lambert left school at twelve and never attended a university. By his death at 49 he had written more than 150 works on mathematics, physics, astronomy, cartography and philosophy. Several of his results are now standard in their fields under his name: Lambert’s cosine law in photometry, the Lambert projections in cartography, Lambert’s theorem in orbital mechanics, and the Lambert W function in analysis. He gave the first proof that π is irrational, and his unpublished study of parallel lines came closer to hyperbolic geometry than anyone before Gauss.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proofs or derivations, and exercises with folded solutions. The lessons can be read in any order. They follow, roughly, the order of the questions Lambert took up: series (1758), light (1760), orbits and the cosmos (1761), π (1761–68), parallels (1766) and maps (1772).

Timeline

Mulhouse and Basel, 1728–1748

Johann Heinrich Lambert was born on 26 August 1728 in Mulhouse, in southern Alsace. Some sources give 28 August. Mulhouse was then a small Protestant republic allied to the Swiss Confederacy; it joined France only in 1798. The Lamberts were Calvinists whose ancestors had left Lorraine for Mulhouse as religious refugees in 1635. His father, Lukas Lambert, was a tailor. Johann Heinrich was one of five sons, and the family had little money.

He attended the town school until about the age of twelve, then left to help in his father’s workshop. He kept studying on his own in the evenings, from whatever books he could obtain, and he never had any further formal education. At about fifteen he became a clerk at the ironworks at Seppois, south of the town, and took private pupils. At seventeen he moved to Basel as secretary to Johann Rudolf Iselin, a scholar who edited the Basler Zeitung. The post gave him time to read and access to a learned city.

Chur and the grand tour, 1748–1759

In 1748, at twenty, Lambert became tutor in the household of Count Peter von Salis at Chur, in the Grisons. He stayed about eight years. The family had a large library, and Lambert used it to study mathematics, astronomy, physics and philosophy systematically. He built his own astronomical instruments and kept regular meteorological records. He was elected to the Literary Society of Chur and to the Swiss Scientific Society in Basel, whose Acta Helvetica printed his first paper, on heat, in 1755.

The grand tour

In 1756 he set out with two of his pupils on a tour of the learned cities of Europe. At Göttingen he met the mathematician Abraham Gotthelf Kästner and the astronomer Tobias Mayer, and was elected to the Göttingen Society of Sciences. When French troops occupied Göttingen in 1757, he took the pupils to Utrecht for about two years. His first book, on the passage of light through various media, was printed at The Hague in 1758. In Paris he met Jean le Rond d’Alembert; the party returned through Marseille, Nice, Turin and Milan.

Also in 1758 he published the paper on series that contains his trinomial equation (Lesson 05). He was thirty, and had a growing European reputation but no position.

Augsburg, Munich and the road to Berlin, 1759–1764

After leaving the Salis household, Lambert settled for a time in Augsburg, where he found publishers. In quick succession there appeared Die freye Perspektive (1759) on perspective drawing, Photometria (1760) on the measurement of light, and in 1761 both Cosmologische Briefe, letters on the structure of the universe, and Insigniores orbitae cometarum proprietates on the orbits of comets. These are the subjects of Lessons 07 and 10.

In 1760 Leonhard Euler recommended him for a professorship of astronomy at the St Petersburg Academy, but the post was not filled. In the same period Lambert was drawn into the founding of the Bavarian Academy of Sciences in Munich, which offered him a salary; he quarrelled with colleagues there and left in 1762. He then helped survey the border between the Duchy of Milan and the Grisons.

Berlin

The Berlin Academy, on Euler’s proposal, had elected him in 1761, but King Frederick II did not confirm the election. In 1764 Lambert went to Berlin to press his case. The King found his dress and manner strange and at first refused him. In the same year Lambert’s major philosophical work, the Neues Organon, appeared in Leipzig. On 10 January 1765 he was admitted as a member of the Academy’s physical class, with a salary. Frederick came to value him highly.

The irrationality of π

For the general reader. The ratio of a circle’s circumference to its diameter, π, is about 3.14159. For two thousand years nobody knew whether it could be written exactly as a fraction. In a memoir presented to the Berlin Academy, Lambert proved that it cannot. His proof expands the tangent function as an infinite continued fraction and shows that such a fraction cannot have a rational value.

The memoir, Mémoire sur quelques propriétés remarquables des quantités transcendantes circulaires et logarithmiques, is dated 1761 and was printed in the Berlin Academy’s volume of 1768. Euler had believed π irrational but had not proved it.

Where the fraction comes from

For \(k\ge-1\) define the entire functions

Since \(2^n n!\,(2n-1)!!=(2n)!\) and \(2^n n!\,(2n+1)!!=(2n+1)!\), we have \(t_{-1}(x)=\cos x\) and \(t_0(x)=\frac{\sin x}{x}\). Comparing coefficients of \(x^{2n}\) gives the three-term recurrence

Dividing by \(t_k\) gives \(\frac{t_{k-1}}{t_k}=(2k+1)-\frac{x^2}{t_k/t_{k+1}}\). Iterating from \(k=0\) gives \(\frac{\cos x}{\sin x/x}=1-\cfrac{x^2}{3-\cfrac{x^2}{5-\cdots}}\), which is the theorem. Convergence follows because \(t_{k+1}/t_k\to0\) as \(k\to\infty\) (Exercise 4.3). Lambert reached the expansion by a long direct calculation, dividing series by series; the recurrence is the modern way to organize it.

The irrationality criterion

Proof. Convergence under \(|b_k|\ge|a_k|+1\) is Pringsheim’s criterion, which also gives \(|\xi_k|\le1\) for every tail \(\xi_k=\cfrac{a_k}{b_k+\cfrac{a_{k+1}}{b_{k+1}+\cdots}}\). Then \(|\xi_k|=\frac{|a_k|}{|b_k+\xi_{k+1}|}\le\frac{|a_k|}{|b_k|-1}\le\frac{|a_k|}{|a_k|+1}<1\), and \(\xi_k\ne0\) since \(a_k\ne0\). Suppose \(\xi_1=u_1/u_0\) with integers \(u_0,u_1\). From \(\xi_k=a_k/(b_k+\xi_{k+1})\) we get \(\xi_{k+1}=a_k/\xi_k-b_k\). Define integers by \(u_{k+1}=a_k u_{k-1}-b_k u_k\). Induction shows \(\xi_k=u_k/u_{k-1}\) for all \(k\). Since \(0<|\xi_k|<1\), we get \(|u_0|>|u_1|>|u_2|>\cdots>0\), an infinite strictly decreasing sequence of positive integers. Contradiction. ∎

Proof. Let \(x=p/q\). Multiplying numerators and denominators through by \(q\),

Choose \(K\) with \((2K+1)q\ge p^2+2\). The tail from level \(K\) on has \(a_k=-p^2\) and \(b_k=(2k+1)q\), so by the lemma it is irrational. The value \(\tan(p/q)\) is obtained from that tail by finitely many maps \(t\mapsto a/(b+t)\) with rational \(a\ne0\) and \(b\), each of which sends irrationals to irrationals. So \(\tan(p/q)\) is irrational. If \(\pi\) were rational, so would be \(\pi/4\), but \(\tan(\pi/4)=1\). ∎

The same memoir treats the hyperbolic tangent, whose continued fraction has plus signs, and shows in the same way that \(e^x\) is irrational for rational \(x\ne0\). Lambert conjectured that \(e\) and \(\pi\) are transcendental. Adrien-Marie Legendre proved in 1794 that \(\pi^2\) is irrational, Charles Hermite proved \(e\) transcendental in 1873, and Ferdinand von Lindemann proved \(\pi\) transcendental in 1882.

The trinomial equation and the W function

For the general reader. Many equations, like \(x=q+x^3\), have no solution in radicals that is easy to use. Lambert found an infinite series that solves every equation of this shape at once. Euler turned his series into the function now called Lambert’s W.

The trinomial equation (1758)

In Observationes variae in mathesin puram (1758) Lambert considered the equation \(x=q+x^m\) and found the solution as a power series in \(q\). In modern notation, for the root that tends to 0 with \(q\):

The modern proof is Joseph-Louis Lagrange’s inversion formula (1770): if \(x=q+\phi(x)\) then \(H(x)=H(q)+\sum_{k\ge1}\frac{1}{k!}\frac{d^{k-1}}{dq^{k-1}}\big[\phi(q)^kH'(q)\big]\). With \(\phi(x)=x^m\) and \(H(x)=x^n\) the coefficients reduce to those above. Lagrange’s formula contains Lambert’s series as a special case.

From Lambert to W

In 1783 Euler rewrote Lambert’s equation in the symmetric form \(x^\alpha-x^\beta=(\alpha-\beta)v\,x^{\alpha+\beta}\) and, letting \(\beta\) tend to \(\alpha\), arrived at the equation \(\ln x=v\,x^{\alpha}\). Its solution is the function now written \(W\), defined by

The name “Lambert W function” and its standard notation were fixed by Robert Corless, Gaston Gonnet, David Hare, David Jeffrey and Donald Knuth in 1996. It appears in combinatorics (counting rooted trees), delay differential equations, and quantum mechanics.

Hyperbolic functions

For the general reader. Sine and cosine describe a point moving round a circle. Hyperbolic sine and cosine do the same for a point moving along a hyperbola, and they turn up in hanging chains, in relativity and in non-Euclidean geometry. Lambert gave them their first systematic treatment, with formulas and tables, and linked them to ordinary trigonometry by a single angle.

Vincenzo Riccati had introduced hyperbolic functions in the 1750s. Lambert developed them systematically, with their addition formulas and tables, in his Observations trigonométriques (Berlin, 1770), giving credit to Riccati and to Daviet de Foncenex. With \(\cosh u=\frac{e^u+e^{-u}}2\), \(\sinh u=\frac{e^u-e^{-u}}2\):

Lambert related the circular and hyperbolic functions through the angle \(\varphi\) defined by \(\tan\varphi=\sinh u\), now called the Gudermannian \(\varphi=\operatorname{gd}u\). It satisfies \(\sin\varphi=\tanh u\), \(\cos\varphi=1/\cosh u\), and \(u=\ln\tan(\tfrac\pi4+\tfrac\varphi2)\). The last formula is the vertical coordinate of the Mercator projection (Lesson 08).

Photometria: the measurement of light

For the general reader. How bright is a surface lit at a slant, and how bright does a matte wall look from different angles? Lambert’s Photometria (1760) answered these questions with a few simple laws and so made the measurement of light a part of mathematical physics. Computer graphics, astronomy and the design of lighting still use his laws.

Lambert worked in a period with no light detectors except the eye. His method was to compare two surfaces side by side and adjust one source until they looked equally bright. From such comparisons and from the assumption that light travels in straight lines, he obtained the laws below. He also introduced the word albedo for the fraction of incident light that a surface reflects diffusely.

Derivation of \(M=\pi L\)

The flux per unit area into a solid angle \(d\Omega\) at angle \(\theta\) is \(L\cos\theta\,d\Omega\). In spherical coordinates \(d\Omega=\sin\theta\,d\theta\,d\phi\), so

A Lambertian surface looks equally bright from every direction, because the factor \(\cos\theta\) in its emitted intensity is exactly compensated by the foreshortening of its apparent area. The full Moon is close to a uniformly bright disc partly for this kind of reason, though lunar soil is not perfectly Lambertian.

The law of absorption

The law follows from the assumption that each thin layer \(dx\) removes the same fraction \(\kappa\,dx\) of the light reaching it: \(dI=-\kappa I\,dx\). Pierre Bouguer had stated it in 1729, and Lambert cited him. August Beer added in 1852 the proportionality of \(\kappa\) to concentration, and the combined statement is now called the Beer–Lambert law.

Map projections

For the general reader. A globe cannot be flattened without distortion. A map can keep the shapes of small regions or their areas, but not both. In a single essay of 1772 Lambert posed the problem of mapping in general terms, proved that no map has both properties, and introduced seven new projections. Three of them are still among the most widely used in cartography.

The essay Anmerkungen und Zusätze zur Entwerfung der Land- und Himmelscharten appeared in the third volume of Lambert’s Beiträge zum Gebrauche der Mathematik und deren Anwendung (1772). He did not name his projections. Today they are known as the Lambert conformal conic, the transverse Mercator, the Lambert azimuthal equal-area, the Lagrange projection, the Lambert cylindrical equal-area, the transverse cylindrical equal-area, and the Lambert conic equal-area.

Local distortion

Write a point of the sphere of radius \(R\) by latitude \(\varphi\) and longitude \(\lambda\). The metric is \(ds^2=R^2(d\varphi^2+\cos^2\varphi\,d\lambda^2)\). For a map whose image curves of latitude and longitude meet at right angles, let \(h\) be the scale factor along meridians and \(k\) the scale factor along parallels. The map is conformal when \(h=k\) everywhere and equal-area when \(hk=1\) everywhere.

For the cylindrical map, \(h=\frac{1}{R}\frac{dy}{d\varphi}=\cos\varphi\) and \(k=\frac{R\,d\lambda}{R\cos\varphi\,d\lambda}=\frac1{\cos\varphi}\), so \(hk=1\). This is Archimedes’ theorem that the sphere and its circumscribed cylinder have equal lateral areas between any two parallel planes. For the azimuthal map, the area of the cap of angular radius \(c\) is \(2\pi R^2(1-\cos c)=4\pi R^2\sin^2\frac c2=\pi\rho^2\), and rotational symmetry then gives equal area for every region. The whole sphere maps onto a disc of radius \(2R\), as in the figure at the top of this page.

Proof. If \(h=k\) and \(hk=1\), then \(h=k=1\): the map preserves all lengths and is a local isometry. By Gauss’s Theorema Egregium (1827), isometries preserve Gaussian curvature. The sphere has curvature \(1/R^2\) and the plane has curvature 0. ∎ Lambert reached the incompatibility by analysing the differential conditions directly. Gauss’s theorem explains it, and also shows that no map at all can preserve lengths.

The theory of parallels

For the general reader. Like Saccheri thirty years before, Lambert tried to prove Euclid’s parallel postulate by assuming it false and looking for a contradiction. He found none. Instead he found that if the postulate fails, the angles of a triangle determine its area, and he observed that the formulas look like those of a sphere with an imaginary radius. This is exactly the geometry Lobachevsky and Bolyai published sixty years later.

Lambert wrote Theorie der Parallellinien in 1766 and did not publish it. Johann Bernoulli and Carl Friedrich Hindenburg printed it in the Leipziger Magazin für reine und angewandte Mathematik in 1786, nine years after his death. Its three parts review earlier attempts (§§1–11), develop propositions that do not use the postulate (§§12–26), and examine three hypotheses (§§27–88). Lambert knew Georg Simon Klügel’s 1763 dissertation, which examined about thirty attempted proofs, Saccheri’s among them.

The quadrilateral

Lambert’s figure has three right angles. Such a quadrilateral is half of a Saccheri quadrilateral, cut along the line joining the midpoints of base and summit. The three hypotheses concern its fourth angle: right, obtuse or acute.

• Right angle (§§40–51): implies the parallel postulate.
• Obtuse angle (§§52–64): Lambert shows that the perpendiculars from one side of the quadrilateral to the opposite line shrink steadily and would reach zero, so two lines would meet twice. He rejects the hypothesis, as Saccheri did, because Euclid’s lines meet at most once.
• Acute angle (§§65–88): no contradiction appears.

Why area is proportional to defect

For a polygon \(P\) with \(n\) sides define \(\delta(P)=(n-2)\pi-\Sigma(P)\), where \(\Sigma\) is the angle sum. When a polygon is cut into two along a segment joining two points of its boundary, the defects add (Exercise 9.1). So \(\delta\) is a positive additive function on polygons, unchanged by congruence, exactly like area. Under mild continuity assumptions, any two such functions are proportional: \(\text{area}=k^2\,\delta\) for a constant \(k\). Lambert argued along these lines. Since the defect of a triangle is less than \(\pi\), it follows that no triangle has area as large as \(\pi k^2\), however long its sides.

The imaginary sphere

On a sphere of radius \(R\), a triangle has area \(R^2(A+B+C-\pi)\), the formula of Albert Girard. Replace \(R\) by \(ik\): the area becomes \(-k^2(A+B+C-\pi)=k^2(\pi-A-B-C)\), Lambert’s formula. He remarked (§§81–82) that one could almost conclude that the third hypothesis holds on an imaginary sphere. Lobachevsky made the same substitution systematic in his Imaginary Geometry of 1835, and Eugenio Beltrami’s pseudosphere of 1868 gave the observation a real meaning.

Comets, orbits and the Milky Way

For the general reader. To find the path of a newly seen comet, astronomers had to work back from a few observed positions to an orbit around the Sun. Lambert discovered a surprising fact: the time a body takes to travel between two points of its orbit depends on only three lengths. In the same year he published a picture of the universe as a vast system of star systems, each like the Milky Way.

Lambert’s interest in comets began with the great comet of 1744. In Insigniores orbitae cometarum proprietates (Augsburg, 1761) he developed geometric methods for comet orbits, including the theorem below. Euler had found the case of a parabolic orbit earlier, and Lagrange later gave an analytic proof of the general statement.

Lambert established the first statement by geometric arguments. One modern way to see it: the time is unchanged when the orbit is deformed keeping \(a\), \(r_1+r_2\) and \(c\) fixed, and the orbit can be deformed into a degenerate ellipse, a straight segment through the centre, on which the time is easy to compute. Today the inverse problem, finding the orbit that joins two given positions in a given time, is called Lambert’s problem. Its solution is used daily to plan spacecraft transfers and rendezvous.

The Cosmologische Briefe (1761)

In a series of letters Lambert described the Milky Way as a flattened system of stars around the Sun, with its plane serving as an ecliptic for the stars. Each star was a sun, presumably with planets. Stars move in groups, and the Milky Way itself might be one member of a system of higher order, and so on, though he held the whole to be finite. He developed these ideas independently of Thomas Wright (1750) and Immanuel Kant (1755), whose work he learned of only later. Unlike Kant he did not consistently regard the nebulae as other Milky Ways.

The Berlin Academy, 1765–1777

Lambert spent the last twelve years of his life in Berlin as a member of the Academy’s physical class. He was the only member who regularly presented papers to other classes as well, and his output was enormous: more than 150 works in those twelve years. The four volumes of his Beiträge zum Gebrauche der Mathematik und deren Anwendung (1765–1772) collected work on applied mathematics, including the map projections of Lesson 08 and, in 1772, a mathematical treatment of mortality statistics. His Observations trigonométriques (1770) systematized hyperbolic functions.

His relations with Euler, who had brought him to Berlin, became strained over the Academy’s finances, which depended on its monopoly on selling calendars. Euler left for St Petersburg in 1766. When Lagrange arrived later that year as director of the mathematics class, Lambert became one of his colleagues. In 1770 Frederick II appointed him Oberbaurat, a senior councillor for public works, which raised his income substantially.

Philosophy and other work

The Neues Organon (1764) treats logic, the theory of truth, signs, and appearances; it contains one of the first uses of the word “phenomenology”. From 1765 Lambert corresponded with Immanuel Kant, who valued his judgement and intended to dedicate the Critique of Pure Reason to him; Lambert died before it appeared. The Anlage zur Architectonic (1771) tried to build philosophy deductively in the manner of Euclid. Lambert also invented a practical hygrometer, wrote on hygrometry and on heat (his Pyrometrie appeared posthumously in 1779), argued for a worldwide network of weather stations, and in 1772, with the court painter Benjamin Calau, published a colour pyramid arranging mixtures of three pigments with white.

Death and legacy

Lambert fell ill in 1775 and declined medical treatment. He died in Berlin on 25 September 1777, aged 49, of a lung disease; the standard German biography attributes it to tuberculosis. He never married. His friend Johann Bernoulli, the astronomer and grandson of Johann I Bernoulli, edited his correspondence and some unpublished papers, including, with Hindenburg, the Theorie der Parallellinien in 1786.

Legacy

• Number theory: the first proof that \(\pi\) is irrational, the starting point for the theory of irrationality and transcendence developed by Legendre, Hermite and Lindemann.
• Geometry: the Lambert quadrilateral and the area–defect relation; his remark on the imaginary sphere anticipated the hyperbolic geometry of Lobachevsky and Bolyai.
• Cartography: the Lambert conformal conic is standard for aeronautical charts and for mapping regions of large east–west extent; the azimuthal equal-area projection is used for continental and polar maps; the transverse Mercator underlies the Universal Transverse Mercator grid.
• Physics and astronomy: Lambert’s cosine law, Lambertian reflectance, albedo, and the Beer–Lambert law; Lambert’s theorem and Lambert’s problem in orbital mechanics. The lambert was a unit of luminance. The asteroid 187 Lamberta and a lunar crater bear his name.
• Analysis: the trinomial equation, the Lambert W function, and the systematic use of hyperbolic functions.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Johann Heinrich Lambert, MacTutor History of Mathematics.
2. Johann Heinrich Lambert, Wikipedia.
3. Lambert, Johann Heinrich, Neue Deutsche Biographie and Allgemeine Deutsche Biographie, Deutsche Biographie.
4. H. K. Strick, Johann Heinrich Lambert, MacTutor (translated from Spektrum der Wissenschaft).
5. H. Frommert, Johann Heinrich Lambert, SEDS (on his astronomy and cosmology).

Mathematics and science

1. J. H. Lambert, Mémoire sur quelques propriétés remarquables des quantités transcendantes circulaires et logarithmiques, Histoire de l’Académie Royale des Sciences et des Belles-Lettres de Berlin (1761), printed 1768. Reprinted in L. Berggren, J. Borwein and P. Borwein, Pi: A Source Book, Springer.
2. Proof that π is irrational, Wikipedia (Lambert’s, Niven’s and Laczkovich’s proofs).
3. M. Laczkovich, On Lambert’s proof of the irrationality of π, Amer. Math. Monthly 104 (1997) 439–443.
4. J. H. Lambert, Photometria, sive de mensura et gradibus luminis, colorum et umbrae, Augsburg, 1760. English translation by D. L. DiLaura, Illuminating Engineering Society, 2001.
5. J. H. Lambert, Notes and Comments on the Composition of Terrestrial and Celestial Maps (1772), translated by W. R. Tobler, University of Michigan, 1972.
6. J. P. Snyder, Flattening the Earth: Two Thousand Years of Map Projections, University of Chicago Press, 1993.
7. A. Papadopoulos and G. Théret, Hyperbolic geometry in the work of Johann Heinrich Lambert, arXiv:1503.01865.
8. A. Papadopoulos and G. Théret, La théorie des lignes parallèles de Johann Heinrich Lambert, Blanchard, Paris, 2014 (critical edition with French translation).
9. P. Stäckel and F. Engel, Die Theorie der Parallellinien von Euklid bis auf Gauss, Teubner, Leipzig, 1895 (includes Lambert’s memoir).
10. R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth, On the Lambert W function, Adv. Comput. Math. 5 (1996) 329–359.
11. Lambert’s problem, Wikipedia.