Johann Heinrich Lambert (1728 - 1777)

As the son of a tailor, Lambert’s family life was not particularly affluent. His father hoped that Lambert would follow into the family trade, but his son had other ideas!

Lambert decided instead to teach himself and work his way up through life, studying Latin and French, mathematics and science. At this time he also began his lifelong interest in astronomy while observing the night sky.

His career began as a clerk, and after a move into newspapers he was appointed as a private tutor to the family of Count Peter von Salis, in Switzerland, where he stayed for 8 years.

This opportunity allowed him access to the family’s extensive library where he spent considerable time reading and learning.

During this time he observed the Klinkenberg-De Cheseaux comet and attempted to calculate its orbit. He also started a number of investigations which became the foundations of his later scientific work, including the idea of a disc-shaped Milky Way.

Between 1756 and 1758, Lambert had the good fortune to be able to travel around Europe, visiting the centres of academic excellence.

This was part of his duties as a tutor - he took two members of the Count’s family on a grand tour of Europe to complete their formal education.

During the tour Lambert visited Paris, Utrecht and Italy, broadening his knowledge and meeting the achievers of the day. While in Germany, Lambert was elected a member of the (University of) Göttingen Society. It was while he was in Germany that Lambert achieved some of his most important work, becoming acknowledged (amongst physicists) as the founder of the theory of light measurement (Photometria), which was completed just prior to his death in 1777.

At the Berlin Academy, which he joined in 1763, Lambert continued his work in astronomy, mathematics and philosophy publishing many papers on all these subjects. His lasting contribution to astronomy concerned a startling idea about the nature of our galaxy.

Lambert proposed that our galaxy - The Milky Way - was a flat disc, composed of thousands of stars and each such star as a sun with its own planetary system.

Also, he argued that there were other Milky Way systems, and along with others at the time (Thomas Wright and Immanuel Kant) he assumed that all these celestial bodies, even stars and comets, were inhabited!

His work in mathematics is significant but mostly underrated and unacknowledged. He worked in geometry and in the theory of functions.

In geometry he was a pioneer of non-Euclidean geometry along with Bolyai, Lobachevsky, Riemann and Gauss.

Lambert mathematics

The background to Non-Euclidean geometry is easy to describe. Euclidean geometry is the geometry of flat surfaces, unlike say, the geometry on the surface of a sphere, which not unaturally is called Spherical geometry.

Euclidean geometry starts off with a set of assumptions and basic ideas - called axioms and postulates. These are taken as given. Using them, Theorems are progressivley discovered and proved. Each Theorem when proved, may be used in the proof of a new Theorem. Here is an example of this at work.

Theorem already proved: alternate angles of a pair of parallel lines are equal; corresponding angles of a pair of parallel lines are equal;

Postulates - angles on a straight line sum to 180^\circ; lines may be extended indefinitely; through a given point may be drawn a line parallel to any other line;

THEOREM: The angles of a triangle add up to 180^\circ

PROOF: Take any triangle. Extend a side of the triangle, and draw a line parallel to another side as shown. Mark equal alternate and corresponding angles as shown:

Because of the properties of parallel lines (alternate angles, the angles marked \angle B; corresponding angles, the angles marked ) there are two sets of equal angles as marked. The angles A, B and C now make a straight line and so

\angle A+\angle B+\angle C=180^\circ \mbox{as required. The proof is complete.}

Bolyai, like many others, didn’t consider the fifth postulate obvious - at least not as obvious as the other postulates. The fifth postulate is one of the ones we have just made use of: though a given point may be drawn a line paralell to a given line.

Notice that it says ‘a’ line, not lines - there is just one such parallel. On this simple point much would emerge.

In attempting to prove Euclid’s parallel postulate, Lambert continued the work of another mathematician Saccheri. Lambert quickly realised that taking different forms of the parallel postulate meant that the angle sum of a triangle could take different forms - less than 180^\circ if there was no parallel; exactly 180^\circ if there was one such parallel and greater than 180^\circ if there were more than one such parallel.

There are equivalent forms when applied to quadrilaterals. Using his quadrilaterals he sought to prove that the first and last forms of this postulate led to contradictions, and therefore that the middle form had to be true.

In this he failed, and he compounded his failure by not recognizing the significance of what he had achieved. Had he done so, he would be credited with the discovery of non-Euclidean geometry.

Lambert continued the work of Saccheri but his work was published only after his death. Lambert employed a quadrilateral ABCD (now called a Lambert quadrilateral) with three right angles.

He then considered the possible angles at the remaining vertex. He made the astonishing discovery that each form of the postulate gave rise to a different geometry, each of which was valid. In the process, Lambert made an extensive study of area and utilized the concept of defect of a triangle - the defect of a triangle is just the difference between its angle sum and 180^\circ. Rather surprisingly, in non-Euclidean geometry, the area of a triangle determines the defect.

He also contributed much to the study of trigonometric functions and even gave us their familiar names sine and cosine - abbreviated sin and cos respectively.

These trigonometric functions enable us to calculate the sides and angles of triangles according to Euclidean geometry - that is flat geometry.

He had the clever idea of exploiting Demoivre’s work to create a set of new functions that operated like their trigonometric counterparts but provided the means to calculate angles and sides of triangles in non-Euclidean geometries. He called his new functions hyperbolic functions and denoted them by sinh and cosh.

In 1765 he found a proof for the irrationality of the number \pi. In other words, no fraction can ever represent \pi, so that statements such as:

\pi={22\over7} should in fact be written as \pi\approx{22\over7}

He also produced a proof of the irrationality of the exponential number e, but this had already been given before.