German physicist Gustav Theodor Fechner discovers that an increase in the intensity of a stimulus does not produce a one-to-one increase in the intensity of sensation. Instead, the sensation seems to increase as the logarithm of the excitation.

Mathematician Bernard Bolzano's Paradoxien des Unendlichen is published, two years after the Bolzano's death. It is the first work to use the word "set" in a modern mathematical sense, and also treats infinite sets.

In Über die bewegende Kraft der Wärme, mathematical physicist Rudolf Julius Emanuel Clausius articulates what comes to be known as the Second Law of Thermodynamics. The law states that heat can only be transferred from a warmer body to a colder body. In 1865, he'll restate the Law, saying that in the closed system entropy always increases.

Mathematician Joseph Liouville demonstrates the existence of transcendental numbers (transcendentals are numbers that are not the solution to any algebraic equation).

Léon Foucault demonstrates that the Earth rotates using a pendulum, suspended from the ceiling of a church.

Physicist William Thomson proposes a concept of "absolute zero", at which the energy of molecules is zero. He draws on Charles' Law to show that such a condition would hold at -273 degrees Celsius.

Albrecht von Kölliker publishes the first textbook of histology, Handbuch der Gewebelehre.

Hermann Ludwig von Helmholtz becomes the first to discover how fast a message travels along a nerve, measuring the speed of transmission in a frog nerve cell.

James Prescott Joule and William Thompson, later Lord Kelvin, establish that an expanding gas becomes cooler. This is now known as the Joule-Thompson effect.

Rudolph Albert von Kölliker discovers the cellular origin of spermatozoa, which were believed by some to originate from a fermentation process.

In Britain, vaccination against small pox is made compulsory.

Léon Foucault demonstrates that the velocity of light is lesson water than in air, tending to confirm the wave theory of light.

William John Macquorn Rankine introduces the concept of potential energy, or energy of position.

Mathematician George Boole, whose "Calculus of Logic," appeared in 1848, publishes An Investigation of the Laws of Thought, on Which are Founded the Mathematical Theories of Logic and Probabilities, which articulates a system of "symbolic logic".

Alfred Russel Wallace publishes "On the Law which has Regulated the Introduction of New Species," anticipating Darwin's theory of evolution by natural selection.

Rudolf Virchow states the principle that new cells come into being only by division of previously existing cells: Omnis cellula e cellula.

Recherches sur la putréfaction, by Louis Pasteur, is published. The book describes how fermentation is caused by microorganisms.

The remains of the first known example of what come to be known as the "Neanderthals" is found in a cave near Düsseldorf, in the Neander Valley. The discovery was made by limestone quarry miners and consists of a skullcap, two femora, the three right arm bones, two of the left arm bones, ilium, and fragments of a scapula and ribs. The fossils were given to Johann Carl Fuhlrott, a local teacher and amateur naturalist. The first description of the remains was made by anatomist Hermann Schaaffhausen and the find was announced jointly in 1857.In 1997, the specimen was the first to yield Neanderthal mitochondrial DNA fragments. The description of this discovery represents the beginning of paleoanthropology as a scientific discipline.

Louis Agassiz publishes Essay on Classification advocating a theory of multiple creations and contradicting both evolution and Noah's ark.

Gregor Mendel, a monk at the Augustinian monastery of St. Thomas in Brünn, Austria (now Brno, Czechoslovakia), begins breeding experiments with the garden pea, Pisum sativum.

Heinz Christian Pander describes conodonts — toothy microfossils that will puzzle paleontologists for many years.

Über die Art der Bewegung, welche wir Wäe nennen (On the type of motion turned heat) by Rudolf Clausius is establishes his kinetic theory of heat of the mathematical basis. It also explains how evaporation occurs.

Rudolf Virchow finalizes the cell theory originally announced by Schleiden and Schwann 11 years earlier by declaring that cells are the basic units of all living things, and all cells are formed by the division of existing cells.

Alfred Russel Wallace sends to Darwin a manuscript — "On the Tendency of Varieties to Depart Indefinitely from the Original Type" — that shows clearly that Wallace has independently formulated a model of evolution by natural selection.

Darwin's and Wallace's ideas are jointly presented to the Linnaean Society of London.

The first relatively complete dinosaur skeleton, of Hadrosaurus foulkii, is found in New Jersey.

Physician Henry Gray publishes Anatomy of the Human Body, Descriptive and Surgical. Gray's Anatomy will be a standard textbook of anatomy for more than a century.

The concept of "bonds" is introduced by chemist Archibald Scott Couper, who believes that carbon atoms are the fundamental building blocks of organic compounds.

Julius Plücker shows that cathode rays bend under the influence of a magnet, suggesting they are connected in some way with charge. This is an early step along the path that will lead in 1897 to the discovery that cathode rays are composed of electrons.

Charles Darwin publishes On the Origin of Species.

An exceptionally well-preserved skeleton is discovered in Bavaria. Two years later, this bird-like, bipedal dinosaur will be named Compsognathus, meaning "dainty jaw."

Catholic priest Jean-Jacques Pouech describes fossil eggshell fragments. They will eventually prove to be the first described dinosaur eggs.

Matrix algebra begins, as mathematician Arthur Cayley discusses the matrix in the context of a theory of invariant transformations.

Gustav Kirchoff recognizes that sodium is found on the sun and discovers his black-body radiation law.

Physicist Gustav Robert Kirchhoff and chemist R. W. Bunsen explain that when light passes through a gas, or heated material, only certain wavelengths of the light are absorbed. Therefore, an analysis of the spectrum of the light can reveal the chemical makeup of the gas or material.