Ernst Waldfried Josef Wenzel Mach ( ; ; 18 February 1838 – 19 February 1916) was an
Austrian physicist
A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate cau ...
and
philosopher
Philosophy ('love of wisdom' in Ancient Greek) is a systematic study of general and fundamental questions concerning topics like existence, reason, knowledge, Value (ethics and social sciences), value, mind, and language. It is a rational an ...
, who contributed to the understanding of the physics of
shock wave
In physics, a shock wave (also spelled shockwave), or shock, is a type of propagating disturbance that moves faster than the local speed of sound in the medium. Like an ordinary wave, a shock wave carries energy and can propagate through a me ...
s. The ratio of the speed of a flow or object to
that of sound is named the
Mach number in his honour. As a
philosopher of science
Philosophy ('love of wisdom' in Ancient Greek) is a systematic study of general and fundamental questions concerning topics like existence, reason, knowledge, Value (ethics and social sciences), value, mind, and language. It is a rational an ...
, he was a major influence on
logical positivism
Logical positivism, also known as logical empiricism or neo-positivism, was a philosophical movement, in the empiricist tradition, that sought to formulate a scientific philosophy in which philosophical discourse would be, in the perception of ...
and
American pragmatism. Through his criticism of
Isaac Newton
Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed ...
's theories of space and time, he foreshadowed
Albert Einstein
Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...
's
theory of relativity
The theory of relativity usually encompasses two interrelated physics theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical ph ...
.
Biography
Early life
Mach was born in Chrlice (),
Moravia
Moravia ( ; ) is a historical region in the eastern Czech Republic, roughly encompassing its territory within the Danube River's drainage basin. It is one of three historical Czech lands, with Bohemia and Czech Silesia.
The medieval and early ...
,
Austrian Empire
The Austrian Empire, officially known as the Empire of Austria, was a Multinational state, multinational European Great Powers, great power from 1804 to 1867, created by proclamation out of the Habsburg monarchy, realms of the Habsburgs. Duri ...
(now part of
Brno
Brno ( , ; ) is a Statutory city (Czech Republic), city in the South Moravian Region of the Czech Republic. Located at the confluence of the Svitava (river), Svitava and Svratka (river), Svratka rivers, Brno has about 403,000 inhabitants, making ...
in the
Czech Republic
The Czech Republic, also known as Czechia, and historically known as Bohemia, is a landlocked country in Central Europe. The country is bordered by Austria to the south, Germany to the west, Poland to the northeast, and Slovakia to the south ...
). His father Jan Nepomuk Mach, who had graduated from
Charles-Ferdinand University in
Prague
Prague ( ; ) is the capital and List of cities and towns in the Czech Republic, largest city of the Czech Republic and the historical capital of Bohemia. Prague, located on the Vltava River, has a population of about 1.4 million, while its P ...
, acted as tutor to the noble Brethon family in
Zlín
Zlín (in 1949–1989 Gottwaldov; ; ) is a city in the Czech Republic. It has about 75,000 inhabitants. It is the seat of the Zlín Region and it lies on the Dřevnice River. It is known as an industrial centre. The development of the modern city ...
in eastern Moravia. His grandfather, Wenzl Lanhaus, an administrator of the Chirlitz estate, was also master builder of the streets there. His activities in that field later influenced Ernst Mach's theoretical work. Some sources give Mach's birthplace as Tuřany (, also part of Brno), the site of the Chirlitz registry office. It was there that Mach was baptised by Peregrin Weiss. Mach later became a
socialist
Socialism is an economic ideology, economic and political philosophy encompassing diverse Economic system, economic and social systems characterised by social ownership of the means of production, as opposed to private ownership. It describes ...
and an
atheist, but his theory and life was sometimes compared to
Buddhism
Buddhism, also known as Buddhadharma and Dharmavinaya, is an Indian religion and List of philosophies, philosophical tradition based on Pre-sectarian Buddhism, teachings attributed to the Buddha, a wandering teacher who lived in the 6th or ...
.
Heinrich Gomperz called Mach the "Buddha of Science" because of his
phenomenalist approach to the "Ego" in his ''Analysis of Sensations''.

Up to the age of 14, Mach was educated at home by his parents. He then entered a
gymnasium in
Kroměříž (), where he studied for three years. In 1855 he became a student at the
University of Vienna
The University of Vienna (, ) is a public university, public research university in Vienna, Austria. Founded by Rudolf IV, Duke of Austria, Duke Rudolph IV in 1365, it is the oldest university in the German-speaking world and among the largest ...
, where he studied
physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
and for one semester medical physiology, receiving his doctorate in physics in 1860 under
Andreas von Ettingshausen with the thesis ''Über elektrische Ladungen und Induktion'', and his
habilitation
Habilitation is the highest university degree, or the procedure by which it is achieved, in Germany, France, Italy, Poland and some other European and non-English-speaking countries. The candidate fulfills a university's set criteria of excelle ...
the following year. His early work focused on the
Doppler effect
The Doppler effect (also Doppler shift) is the change in the frequency of a wave in relation to an observer who is moving relative to the source of the wave. The ''Doppler effect'' is named after the physicist Christian Doppler, who described ...
in
optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of optical instruments, instruments that use or Photodetector, detect it. Optics usually describes t ...
and
acoustics
Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician ...
.
Professional research
In 1864, Mach became professor of mathematics at the
University of Graz after having declined a chair in surgery at the
University of Salzburg
The University of Salzburg (, ), also known as the Paris Lodron University of Salzburg (''Paris-Lodron-Universität Salzburg'', PLUS), is an Austrian public university in Salzburg, Salzburg municipality, Salzburg (federal state), Salzburg State, ...
. In 1866 he was appointed professor of physics. During this period, Mach continued his work in
psycho-physics and in sensory perception. In 1867, he took the chair of experimental physics at the
Charles-Ferdinand University, where he stayed for 28 years before returning to Vienna.
In 1871 he was elected a member of the
Royal Bohemian Society of Sciences.
Mach's main contribution to physics involved his description and photographs of spark shock-waves and then ballistic shock-waves. He described how when a bullet or shell moved faster than the speed of sound, it created a compression of air in front of it. Using
schlieren photography
Schlieren photography is a process for photographing fluid flow. Invented by the Germans, German physicist August Toepler in 1864 to study supersonic motion, it is widely used in aeronautical engineering to photograph the airflow, flow of air ar ...
, he and his son Ludwig photographed the shadows of the invisible shock waves. During the early 1890s Ludwig invented a modification of the
Jamin interferometer that allowed for much clearer photographs.
But Mach also made many contributions to psychology and physiology, including his anticipation of
gestalt phenomena, his discovery of the
oblique effect and of
Mach bands, an inhibition-influenced type of visual illusion, and especially his discovery of a
non-acoustic function of the inner ear that helps control human balance.
One of the best-known of Mach's ideas is the so-called
Mach principle, the physical origin of inertia. This was never written down by Mach, but was given a graphic verbal form, attributed by
Philipp Frank
Philipp Frank (; March 20, 1884 – July 21, 1966) was an Austrian-American physicist, mathematician and philosopher of the early-to-mid 20th century. He was a logical positivism, logical positivist, and a member of the Vienna Circle. He was infl ...
to Mach: "When the subway jerks, it's the fixed stars that throw you down."
In 1900 Mach became the
godfather of the physicist
Wolfgang Ernst Pauli, who was also named after him.
Mach was also well known for his philosophy, developed in close interplay with his science. He defended a type of
phenomenalism, recognizing only
sensations as real. That position seemed incompatible with the view of atoms and molecules as external, mind-independent things. After an 1897 lecture by
Ludwig Boltzmann
Ludwig Eduard Boltzmann ( ; ; 20 February 1844 – 5 September 1906) was an Austrian mathematician and Theoretical physics, theoretical physicist. His greatest achievements were the development of statistical mechanics and the statistical ex ...
at the
Imperial Academy of Science in
Vienna
Vienna ( ; ; ) is the capital city, capital, List of largest cities in Austria, most populous city, and one of Federal states of Austria, nine federal states of Austria. It is Austria's primate city, with just over two million inhabitants. ...
, Mach said, "I don't believe that atoms exist!"
In 1898, Mach survived a paralytic stroke, and in 1901, he retired from the University of Vienna and was appointed to the upper chamber of the Austrian Parliament. On leaving Vienna in 1913, he moved to his son's home in
Vaterstetten, near
Munich
Munich is the capital and most populous city of Bavaria, Germany. As of 30 November 2024, its population was 1,604,384, making it the third-largest city in Germany after Berlin and Hamburg. Munich is the largest city in Germany that is no ...
, where he continued writing and corresponding until his death in 1916, one day after his 78th birthday.
Politics
Born to a liberal family, Mach lamented that a "very reactionary-clerical" period followed the
1848 revolutions, prompting him to plan to emigrate to America, although he never did.
In 1901, Mach accepted an appointment to the Austrian
House of Lords
The House of Lords is the upper house of the Parliament of the United Kingdom. Like the lower house, the House of Commons of the United Kingdom, House of Commons, it meets in the Palace of Westminster in London, England. One of the oldest ext ...
but declined a nobility because he thought it inappropriate for a scientist to accept such a thing. He was on good personal terms with the Social Democrat politician
Viktor Adler and left money in his will to the Social Democrat newspaper
Arbeiter-Zeitung.
Mach was critical of the European powers' colonial conquests, saying that they "will constitute...the most distasteful chapter of history for coming generations".
Physics
Most of Mach's initial studies in experimental physics concentrated on the
interference
Interference is the act of interfering, invading, or poaching. Interference may also refer to:
Communications
* Interference (communication), anything which alters, modifies, or disrupts a message
* Adjacent-channel interference, caused by extra ...
,
diffraction
Diffraction is the deviation of waves from straight-line propagation without any change in their energy due to an obstacle or through an aperture. The diffracting object or aperture effectively becomes a secondary source of the Wave propagation ...
,
polarization and
refraction
In physics, refraction is the redirection of a wave as it passes from one transmission medium, medium to another. The redirection can be caused by the wave's change in speed or by a change in the medium. Refraction of light is the most commo ...
of light in different media under external influences. From there followed explorations in
supersonic
Supersonic speed is the speed of an object that exceeds the speed of sound (Mach 1). For objects traveling in dry air of a temperature of 20 °C (68 °F) at sea level, this speed is approximately . Speeds greater than five times ...
fluid mechanics. Mach and physicist-photographer
Peter Salcher presented their paper on this subject in 1887; it correctly describes the sound effects observed during the supersonic motion of a
projectile
A projectile is an object that is propelled by the application of an external force and then moves freely under the influence of gravity and air resistance. Although any objects in motion through space are projectiles, they are commonly found ...
. They deduced and experimentally confirmed the existence of a
shock wave
In physics, a shock wave (also spelled shockwave), or shock, is a type of propagating disturbance that moves faster than the local speed of sound in the medium. Like an ordinary wave, a shock wave carries energy and can propagate through a me ...
of conical shape, with the projectile at the apex. The ratio of the speed of a fluid to the local
speed of sound
The speed of sound is the distance travelled per unit of time by a sound wave as it propagates through an elasticity (solid mechanics), elastic medium. More simply, the speed of sound is how fast vibrations travel. At , the speed of sound in a ...
''v''
p/''v''
s is called the
Mach number after him. It is a critical parameter in the description of high-speed fluid movement in
aerodynamics
Aerodynamics () is the study of the motion of atmosphere of Earth, air, particularly when affected by a solid object, such as an airplane wing. It involves topics covered in the field of fluid dynamics and its subfield of gas dynamics, and is an ...
and
hydrodynamics
In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids – liquids and gases. It has several subdisciplines, including (the study of air and other gases in ...
. Mach also contributed to
cosmology
Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe, the cosmos. The term ''cosmology'' was first used in English in 1656 in Thomas Blount's ''Glossographia'', with the meaning of "a speaking of the wo ...
the hypothesis known as
Mach's principle
In theoretical physics, particularly in discussions of gravitation theories, Mach's principle (or Mach's conjecture) is the name given by Albert Einstein to an imprecise hypothesis often credited to the physicist and philosopher Ernst Mach. The ...
.
Philosophy of science
Empirio-criticism
From 1895 to 1901, Mach held a newly created chair for "the history and philosophy of the inductive sciences" at the University of Vienna. In his historico-philosophical studies, Mach developed a phenomenalistic
philosophy of science
Philosophy of science is the branch of philosophy concerned with the foundations, methods, and implications of science. Amongst its central questions are the difference between science and non-science, the reliability of scientific theories, ...
that became influential in the 19th and 20th centuries, empirio-criticism, a rigorously
positivist and radically
empiricist philosophy established by the German philosopher
Richard Avenarius and further developed by Mach,
Joseph Petzoldt, and others, according to which all we can know is our sensations.
Mach originally saw scientific laws as summaries of experimental events, constructed for the purpose of making complex data comprehensible, but later emphasized mathematical functions as a more useful way to describe sensory appearances. Thus, scientific laws, while somewhat idealized, have more to do with describing sensations than with reality as it exists beyond sensations.
Mach's positivism influenced many Russian
Marxist
Marxism is a political philosophy and method of socioeconomic analysis. It uses a dialectical and materialist interpretation of historical development, better known as historical materialism, to analyse class relations, social conflic ...
s, such as
Alexander Bogdanov. In 1908,
Lenin
Vladimir Ilyich Ulyanov ( 187021 January 1924), better known as Vladimir Lenin, was a Russian revolutionary, politician and political theorist. He was the first head of government of Soviet Russia from 1917 until Death and state funeral of ...
wrote a philosophical work, ''
Materialism and Empirio-criticism'', in which he criticized Machism and the views of "
Russian Machists". His main criticisms were that Mach's philosophy led to
solipsism and to the absurd conclusion that nature did not exist before humans:
In accordance with empirio-critical philosophy, Mach opposed
Ludwig Boltzmann
Ludwig Eduard Boltzmann ( ; ; 20 February 1844 – 5 September 1906) was an Austrian mathematician and Theoretical physics, theoretical physicist. His greatest achievements were the development of statistical mechanics and the statistical ex ...
and others who proposed an atomic theory of physics. Since one cannot observe things as small as atoms directly, and since no atomic model at the time was consistent, the atomic hypothesis seemed unwarranted to Mach, and perhaps not sufficiently "economical". Mach had a direct influence on the
Vienna Circle philosophers and
logical positivism
Logical positivism, also known as logical empiricism or neo-positivism, was a philosophical movement, in the empiricist tradition, that sought to formulate a scientific philosophy in which philosophical discourse would be, in the perception of ...
in general.
Several principles are attributed to Mach that distill his ideal of physical theorization, called "Machian physics":
# It should be based entirely on directly observable phenomena (in line with his positivistic leanings)
# It should completely eschew
absolute space and time
Absolute space and time is a concept in physics and philosophy about the properties of the universe. In physics, absolute space and time may be a preferred frame.
Early concept
A version of the concept of absolute space (in the sense of a prefe ...
in favour of
relative motion
In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents a point ''P'' in space. Its length represents the distance in relation to an arbitrary reference origin ''O'', and ...
# Any phenomena that seem attributable to absolute space and time (e.g.,
inertia
Inertia is the natural tendency of objects in motion to stay in motion and objects at rest to stay at rest, unless a force causes the velocity to change. It is one of the fundamental principles in classical physics, and described by Isaac Newto ...
and
centrifugal force
Centrifugal force is a fictitious force in Newtonian mechanics (also called an "inertial" or "pseudo" force) that appears to act on all objects when viewed in a rotating frame of reference. It appears to be directed radially away from the axi ...
) should instead be seen as emerging from the distribution of matter in the universe.
The last is singled out, particularly by Einstein, as "the"
Mach's principle
In theoretical physics, particularly in discussions of gravitation theories, Mach's principle (or Mach's conjecture) is the name given by Albert Einstein to an imprecise hypothesis often credited to the physicist and philosopher Ernst Mach. The ...
. Einstein cited it as one of the three principles underlying
general relativity
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
. In 1930, he wrote, "it is justified to consider Mach as the precursor of the general theory of relativity" and "the whole direction of thought of this theory conforms with Mach's". Einstein further reported that he had read
David Hume
David Hume (; born David Home; – 25 August 1776) was a Scottish philosopher, historian, economist, and essayist who was best known for his highly influential system of empiricism, philosophical scepticism and metaphysical naturalism. Beg ...
and Mach's work "with eagerness and admiration shortly before finding relativity theory" and that "very possibly, I wouldn't have come to the solution without those philosophical studies". Before his death, Mach apparently rejected Einstein's theory. Einstein knew his theories did not fulfill all Mach's principles, and no subsequent theory has either.
Phenomenological constructivism
According to Alexander Riegler, Mach's work was a precursor to the influential perspective known as
constructivism. Constructivism holds that all knowledge is constructed rather than received by the learner. He took an exceptionally non-dualist, phenomenological position. The founder of radical constructivism,
Ernst von Glasersfeld, gave a nod to Mach as an ally.
On the other hand, there is also a reasonable case for viewing Mach simply as an empiricist and a precursor of the logical empiricists and the Vienna Circle. On this view, the purpose of science is to detail functional relationships between observations: "The goal which it (physical science) has set itself is the simplest and most economical abstract expression of facts."
Influence
Friedrich Hayek
Friedrich August von Hayek (8 May 1899 – 23 March 1992) was an Austrian-born British academic and philosopher. He is known for his contributions to political economy, political philosophy and intellectual history. Hayek shared the 1974 Nobe ...
wrote that, when he attended the
University of Vienna
The University of Vienna (, ) is a public university, public research university in Vienna, Austria. Founded by Rudolf IV, Duke of Austria, Duke Rudolph IV in 1365, it is the oldest university in the German-speaking world and among the largest ...
from 1918 to 1921, "as far as philosophical discussion went it essentially revolved around Mach's ideas". Mach's work has also been cited as an influence on the
Vienna Circle, being described as a "major precursor of logical positivism". Members of the Circle organized the "Ernst Mach Society" as a vehicle for discussion of their ideas.
Mach's work was a "forerunner" of
Gestalt psychology
Gestalt psychology, gestaltism, or configurationism is a school of psychology and a theory of perception that emphasises the processing of entire patterns and configurations, and not merely individual components. It emerged in the early twent ...
.
[Pojman, Paul, "Ernst Mach", The Stanford Encyclopedia of Philosophy (Winter 2023 Edition), Edward N. Zalta & Uri Nodelman (eds.)]
Physiology
In 1873, independently of each other, Mach and the physiologist and physician
Josef Breuer discovered how the
sense of balance (i.e., the perception of the head's imbalance) functions, tracing its management by information the brain receives from the movement of
a fluid in the
semicircular canals of the
inner ear
The inner ear (internal ear, auris interna) is the innermost part of the vertebrate ear. In vertebrates, the inner ear is mainly responsible for sound detection and balance. In mammals, it consists of the bony labyrinth, a hollow cavity in the ...
. That the sense of balance depends on the three semicircular canals was discovered in 1870 by the physiologist
Friedrich Goltz, but Goltz did not discover how the balance-sensing apparatus functions. Mach devised a swivel chair to test his theories, and Floyd Ratliff has suggested that this experiment may have paved the way to Mach's critique of a physical conception of absolute space and motion.
Psychology

In the area of sensory perception, psychologists remember Mach for the
optical illusion called
Mach bands. The effect exaggerates the contrast between edges of the slightly differing shades of gray as soon as they make contact, by triggering edge-detection in the human visual system.
More clearly than anyone before or since, Mach made the distinction between what he called ''physiological'' (specifically
visual) and ''geometrical'' spaces.
Mach's views on mediating structures inspired
B. F. Skinner's strongly
inductive position, which paralleled Mach's in the field of psychology.
Eponyms
In homage his name was given to:
*
3949 Mach, an asteroid
*
Mach
The Mach number (M or Ma), often only Mach, (; ) is a dimensionless quantity in fluid dynamics representing the ratio of flow velocity past a Boundary (thermodynamic), boundary to the local speed of sound.
It is named after the Austrian physi ...
, a lunar crater
*
Mach bands, an optical illusion
*
Mach diamonds, seen in supersonic exhausts
*
Mach Five, the car used by
Speed Racer
*
Mach number, the unit for speed relative to the speed of sound
Bibliography
* (Later editions were published under the title ''Analyse der Empfindungen und das Verhältnis des Physischen zum Psychischen'')
*
*
*
*
Mach's principal works in English:
*
* with Peter Slacher
*
* ''Popular Scientific Lectures'' (1895)
Revised & enlarged 3rd edition (1898)* with S.J.B. Sugden
''History and Root of the Principle of the Conservation of Energy''(1911)
* ''The Principles of Physical Optic''s (1926)
* ''Knowledge and Error'' (1976)
* ''Principles of the Theory of Heat'' (1986)
* ''Fundamentals of the Theory of Movement Perception'' (2001)
See also
*
Energeticism
*
Mach (kernel)
*
Mach bands
*
Mach disk
*
Mach reflection
*
Mach's principle
In theoretical physics, particularly in discussions of gravitation theories, Mach's principle (or Mach's conjecture) is the name given by Albert Einstein to an imprecise hypothesis often credited to the physicist and philosopher Ernst Mach. The ...
*
Mach–Zehnder interferometer
*
Stereokinetic stimulus
*
Visual space
Visual space is the experience of space by an aware observer. It is the subjective counterpart of the space of physical objects. There is a long history in philosophy, and later psychology of writings describing visual space, and its relationsh ...
References
Notes
Citations
Sources
*
*
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*
*
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*
*
*
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*
Further reading
* Erik C. Banks: ''Ernst Mach's World Elements. A Study in Natural Philosophy''. Dordrecht: Kluwer (now Springer), 2013.
* John Blackmore and
Klaus Hentschel (eds.): ''Ernst Mach als Außenseiter''. Vienna: Braumüller, 1985 (with select correspondence).
*
* John T. Blackmore, Ryoichi Itagaki and Setsuko Tanaka (eds.): ''Ernst Mach's Science''. Kanagawa: Tokai University Press, 2006.
* John T. Blackmore, Ryoichi Itagaki and Setsuko Tanaka: ''Ernst Mach's Influence Spreads''. Bethesda: Sentinel Open Press, 2009.
* John T. Blackmore, Ryoichi Itagaki and Setsuko Tanaka: ''Ernst Mach's Graz (1864–1867), where much science and philosophy were developed''. Bethesda: Sentinel Open Press, 2010.
* John T. Blackmore: ''Ernst Mach's Prague 1867–1895 as a human adventure'', Bethesda: Sentinel Open Press, 2010.
*
*
*
*
*
* {{citation, editor1-first=V. , editor1-last=Prosser , editor2-first=J., editor2-last= Folta , title=Ernst Mach and the development of Physics – Conference Papers, location= Prague, publisher= Universitas Carolina Pragensis, date= 1991
* {{citation, first=Joachim , last=Thiele, title=Wissenschaftliche Kommunikation – Die Korrespondenz Ernst Machs, language=de, location=Kastellaun, publisher= Hain, date= 1978 (with select correspondence).
{{refend
External links
{{Commons
{{Wikiquote
{{EB1911 poster, Mach, Ernst
{{wikiquote, The Science of Mechanics
Ernst Mach bibliography of all of his papers and books from 1860 to 1916 compiled by Vienna lecturer Dr. Peter Mahr in 2016
Various Ernst Mach links compiled by Greg C Elvers
*
Klaus Hentschel: Mach, Ernst, in
''Neue Deutsche Biographie'' 15 (1987), pp. 605–609.* {{Gutenberg author , id=40055, name=Ernst Mach
* {{Internet Archive author , sname=Ernst Mach
* {{Librivox author , id=10401
* {{cite SEP , url-id=ernst-mach , title=Ernst Mach , last=Pojman , first=Paul , author-link=Paul Pojman
Short biography and bibliographyin the Virtual Laboratory of the
Max Planck Institute for the History of Science
Ernst Mach: ''The Analysis of Sensations'' (1897) ranslation of ''Beiträge zur Analyse der Empfindungen'' (1886)* {{MathGenealogy , id=113010
"The critical positivism of Mach and Avenarius" entry in the ''
Britannica Online Encyclopedia''
{{Positivism
{{Authority control
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