1824: Carnot Cycle

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The Carnot cycle is the ideal cycle of operations that provides the maximum theoretical efficiency for any engine that utilizes heat.  It was first proposed by the French physicist Sadi Carnot in 1824 prior to its expansion by other scientists.

Reflections on the Motive Power of Fire

Nicolas Leonard Sadi Carnot is often referred to as the “father of thermodynamics”.  In 1824, at the age of 27, he published his only book titled Reflections on the Motive Power of Fire in which he introduced the concept of the Carnot cycle while laying down the foundations for a new discipline – thermodynamics.  By this time the steam engines industrial and economic importance had been established, although little scientific work had been done by studying them. Carnot’s book was one of the first scientific studies of steam engines.

Carnot explore some key ideas in his book. He recognized heat as a form of energy that can be converted from one state to another. This concept eventually led to the first law of thermodynamics, which states that energy is conserved in any thermodynamic process. He introduced the notion of an idealized heat engine, known as the Carnot engine, as a theoretical construct to study the maximum efficiency of heat conversion. This engine could be served as a reference point against which all other real work engines could be compared. Additionally, he noted that the efficiency of a heat engine depends of the temperature difference between the source and the sink, and that maximum efficiency is only achieved when it operates in a recoverable manner.

To understand the principles of an ideal heat engine, Carnot devised the Carnot cycle which is a series of four reversible processes. It is a theoretical cycle that provides an upper limit on the efficiency that any classical thermodynamic engine can achieve during the conversion of heat into work.  The Carnot cycle heat engine is not a practical engine that can be made because its processes are reversible, which violate the second law of thermodynamics.

The Four Stages of the Carnot Cycle

The Carnot cycle consists of several operations.  First, the engine absorbs energy in the form of heat from a reservoir.  Work is done by the heat which causes an expansion.  Next is compression where the heat is given out.  The net result is that heat is taken from a hot source to a cooler one, while some of the heat does work.  The Carnot cycle establishes the highest possible efficiency – measured by dividing the work that is done by the energy absorbed from the reservoir – for any heat engine.  In the real work efficiency of an engine is never as high as that predicted by the Carnot cycle due to factors such as friction.

Here are the four stages of the Carnot cycle.

The Four Stages of the Carnot Cycle
The Four Stages of the Carnot Cycle
  1. Isothermal expansion – gas expands on its surroundings as heat is transferred from hot reservoir at a constant temperature; work being done = heat supplied.
  2. Adiabatic expansion – gas continues to expand on its surroundings with no heat exchange, causing temperature to cool; work being done > heat supplied.
  3. Isothermal compression – surroundings compress gas as heat is transferred to a low temperature reservoir at a constant temperature.
  4. Adiabatic compression – surroundings compress gas with no heat exchange, resetting the system to its original state (before stage 1).

The Importance of the Carnot Cycle

Sadi Carnot’s work brought about a revolution in our understanding of the underlying principles of heat transfer and energy conversion. Most directly, it set the stage for subsequent advancements in the field of thermodynamics. Work on the Carnot cycle heat engine led to the fundamental thermodynamic principle of entropy, a fundamental concept related the the second law of thermodynamics.  Rudolf Clausius introduced the term entropy in the mid-19th century to explain the irreversible nature of energy transfer and transformations in physical systems. Clausius observed that heat naturally flows from a higher temperature region to a lower temperature region and that the process is irreversible. He recognized the importance of Carnot’s work with heat and sought to put in on a more rigorous, mathematical footing.
Another Rudolf was to later be inspired by the Carnot cycle when he was developing a more efficient engine. In 1982 Rudolf Diesel submitted design patents for an internal combustion engine.  He knew gasoline engines were very inefficient, wasting around 90% of its heat in the process. He sought to make an engine closer to the theoretical framework of the Carnot cycle engine.

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Theodor Schwann

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Theodor Schwann portrait
Theodor Schwann

Theodor Schwann (1810 – 1882) was a German physician and physiologist who proposed the cellular nature of all living things.  Along with Matthias Schleiden, Schwann laid down the foundation to Cell Theory.

Schwann was born in Neuss, Germany, attended a Jesuit school in Cologne, enrolled in the University of Bonn, transferred the the University of Wurzburg for clinical training in medicine before he finally moved the the University of Berlin where he obtained his M.D. degree.  Much of his moving involved him following physiologist Johannes Muller, a leading physiologist of his time.

After he graduated he began to make a series of discoveries.  In 1835 he discovered the enzyme Pepsin.  Next he performed experiments with yeast and fermentation.  He successfully demonstrated that fermentation was an organic process; that living yeast was necessary to produce more yeast.

Schwann’s most important work was in the development of Cell Theory.  He began by taking the idea that all plants are made from cells and extended them to animals.  In 1838 Schleiden published Contributions to our Knowledge of Phytogenesis, outlining his theories of the roles of cells in plant development.  This influenced Schwann and the next year he published Microscopical Researches into the Accordance in the Structure and Growth of Animals and Plants, a landmark work where he proposed his own Cell Theory.  In it he exclaimed ” “All living things are composed of cells and cell products”, extending Schleiden’s idea that new plant cells are formed from old plant cells to the domain of animals.   He also coined the term “metabolism” to describe chemical reactions taking place within the cell.

Cell Theory and his work on yeast and fermentation provided strong evidence against the idea of spontaneous generation – the idea that living organisms could develop from nonliving matter.  This won him tremendous respect from his peers.  In 1879 Schwann was elected to both the Royal Society and to the French Academy of Science.  He died three years later.

Matthew Maury

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Matthew Maury portrait
Matthew Maury

Matthew Maury (1806 – 1873) was an American Naval officer and oceanographer.  He is credited with the moniker “Father of Modern Oceanography” thanks to the comprehensive book on oceanography he published in 1855, The Physical Geography of the Sea.

Maury was born into a Huguenot family in Virginia but had moved to Tennessee by the time he turned five.  His brother John was a Navy officer and Matthew was determined to follow suit.  He obtained a naval appointment at age 19 from Tennessee Representative Sam Houston.  He immediately began studying the sea on a four-year voyage aboard the Vincennes that began in 1826.  It was the first US Naval warship to circumnavigate the globe.  Sadly, his seafaring days came to an abrupt end at the age of 33 after his leg was maimed in a stagecoach accident.  Henceforth he would devote his time to studying the ocean.

In 1842 Maury was placed as head of the Depot of Charts and Instruments in Washington DC, which offered him a tremendous amount of maritime data in terms of log books and various other records.  He would eventually turn this institution into the United States Naval Observatory and become its first superintendent.

In 1855 he published the first modern oceanography textbook, The Physical Geography of the Sea, describing the winds, currents, climate, and physical geography over the worlds oceans.  That same year Maury proposed sea lanes in his book Sailing Directions.  This idea was taken up by the major shipping companies to the benefit of lives and dollars saved.  He also sent out survey ships to take depth readings on the Atlantic Ocean’s floor, which revealed the Mid-Atlantic ridge.  His books and his surveys helped to prove the feasibility of laying a first transatlantic cable, which occurred in July 1866.

The American Civil War interrupted his career, sending him to Europe and then to Mexico before he finally returned to Virginia where he took the post of professor of meteorology at the Virginia Military Institute.  He stayed there until his death in 1873.

820s: Algebra

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Algebra is the study of mathematics by using a combination of symbols and values and the rules for manipulating those symbols and values. In its most basic form, it involves using equations to find the unknown. Linear equations, the quadratic formula, functions, and much more are all familiar examples of algebra. Algebra became recognized as a separate branch of mathematics thanks to work of the Persian mathematician Muhammad ibn Musa al-Khwarizmi. 

The Roots of Algebra

The Quadratic Formula with Examples
The Quadratic Formula with Examples
(Credit: www.onlinemathlearning.com)

This history of any field of mathematics rides on a curvy road. This is no-less true for algebra. The roots of algebra can be traced back to Babylonian and Greek mathematics, at least 2000 BCE.  We have evidence of stone tablets from Babylonian mathematicians who were hitting on the same ideas of algebra. The representation was not the same and the symbols they used were unique to their culture, but the fundamental spirit of algebra is evident. The Babylonians used complex arithmetic methods to solve modern algebraic problems. The Egyptians also worked with algebraic ideas but they were much less advanced that the Babylonians and did not advance much past solving linear equations.

The next big reservoir of algebraic thought came courteous of the Greek mathematicians, in particular a person named Diophantus of Alexandria. Diophantus lived in Alexandria, Egpyt in the 3rd century. Little is known of his life except his works and his age. He authored a thirteen book series titled Arithmetica that unfortunately has not survived in its full form. The portions that have survived show algebraic equations being solved. Diophantus and the Greeks devised a system of geometric algebra, using squares to solve for equations. 

The Indian mathematician Brahmagupta was another person who influenced the development of algebra. Brahmagupta lived during the 7th century in northwest India. He wrote many influential works with a focus on mathematics and astronomy. His most famous work, Brahmasphutasiddhanta, provided solutions to linear and quadratic equations and is one of the earliest known texts to treat zero as a number. Much of his work moved from India the the Middle East, and was not known by Western Europe until many centuries later.

The Compendious Book on Calculation by Completion and Balancing

The Compendious Book on Calculation by Completion and Balancing
The Compendious Book on Calculation by Completion and Balancing

These earlier systems, especially the Greek and Indian, provided the inspiration for Persian mathematician al-Khwarizmi. Al-Khwarizmi was born in 780 and fortunate enough to have studied and worked in the House of Wisdom in Baghdad. The House of Wisdom was an enormous library and a major intellectual center of the time. In the 820s he wrote The Compendious Book on Calculation by Completion and Balancing, forming the foundation of algebra and establishing it as an independent discipline from arithmetic and geometry. 

The Arabic title of his work is Al-jabr wa’l muqabalah, and it is from “al-jabr” that we get the term algebra. As the title indicates the text stresses the completion and balancing of equations. Here is a simple example of each type of operation:

  1. Completion – Take the equation x+6=36. To complete this equation, we subtract 6 from each side to get x=36-6, or x=30.
  2. Balancing – Take the equation x+y=y+30. To balance this equation we cancel y from both sides and get x=30.

His treatise is important because it presented the first systematic solution of linear and quadratic equations.

Algebra Today

Today algebra is used in a variety of mathematical fields, practical applications, and everyday life situations. Numbers and equations are used in everyday life whether we realize it or not. We use it in finance when we calculate loan interest, our return on investment, or a currency exchange rate. We use algebra when calculating rations. Ratios are relationships between different quantities. Twice as many guests are now showing up to your party? We need to balance the equation and add twice as many ingredients to that soup we are cooking. Are you a United States resident traveling outside the country? Most of the world uses the metric system for measurements and we’d use algebra to convert these measurements. We use algebra in statistics, graphing, computer coding, measuring calculations such as area, volume, and mass, and more.

Intuitively, we use algebra all the time when we solve for unknown variables. Abstractly, algebra helps us with our critical thinking and problem solving skills. Lastly many other branches of mathematics are dependent on algebra. Finding the area under a curve requires the use of calculus, and calculus would not be possible without algebra.

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1848: Absolute Zero

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Everyone is familiar with the concept of temperature.  Temperature is a way to describe how hot or cold something is.  But what is it that determines how hot or cold something is?

All of matter is made of atoms, and those atoms are always moving.  Temperature then, is a measure of the kinetic energy (the energy of motion) of the particles in a substance or system.  The faster the atoms move, the higher the temperature.  Temperature also determines the direction of heat transfer, which is always from objects of a higher temperature to objects of a lower temperature.

Absolute zero is the lowest temperature theoretically possible.  It corresponds to a bone-chilling -459.67 degrees on the Fahrenheit scale and -273.15 on the Celsius scale.  At this temperature there is the complete absence of thermal energy, as the particles of a substance have no kinetic energy.

The History of Absolute Zero

The roots of the idea of absolute zero can be traced back to the early 17th century when scientists began to explore the behavior of gases.  In 1665, Robert Boyle formulated Boyle’s Law, which stated that the volume of a gas is inversely proportional to its pressure at a constant temperature.  This law laid the foundation for the study of gases and eventually lead to the concept of absolute zero.

Absolute Zero Temperature Scale
Absolute Zero Temperature Scale

Over the next 200 years additional discoveries were made that brought scientists closer and closer to the concept of an absolute zero temperature point.  Then in 1848 the distinguished British scientist, William Thomson (later Lord Kelvin), published a paper titled On an Absolute Thermometric Scale where he made the case for a new temperature scale with the lower limit to be absolute zero. At this time temperature was measured on various scales, such as Celsius and Fahrenheit scales.  However these scales had certain limitations and were based on arbitrary reference points.  Thomson recognized the need for a temperature scale that would provide a universal standard and be based on fundamental physical principles.  Scientists could now rely on a scale for temperature measurements without the need for using negative numbers.

Thomson’s key insight was to base his new scale on the behavior of an ideal gas.  According to Boyle’s Law, the pressure of an ideal gas is directly proportional to its temperature when the volume is held constant.  He realized that if a gas were to be cooled to a temperature at which its volume reached zero, then this temperature would represent the absolute zero of temperature.  Thomson correctly calculated its value and used Celsius as the scale’s unit increment.

Absolute zero is the temperature to which you all atoms would stop moving and kinetic energy equals zero. This temperature has never been achieved in the laboratory, but it’s been close. Sophisticated technology involving laser beams to trap clouds of atoms held together by magnetic fields generated by coils have cooled elements such as helium to within fractions of a degree of absolute zero.  The current world record for the coldest temperature is held by a team of researchers at Standford University in 2015. They used sophisticated laser beams to slow rubidium atoms, cooling them to an incredible 50 trillionth of a degree, or 0.00000000005 degrees Celsius, above absolute zero! This is extremely impressive since according to theory, it is suggested that we will never be able to achieve absolute zero.

Thomson’s temperature scale was later named the Kelvin Scale in his honor, and kelvin is the International System of Units (SI) base unit of temperature.

Practical Uses of Absolute Zero

The concept of absolute zero is relevant to many modern technologies, such as cryogenics and quantum computing.  Below is a summary of its applications:

  1. Cryogenics – cryogenics is the study of very low temperatures. cryogenics is the study of very low temperatures. Its technologies are used in various industries such as medical science, where they assist in the preservation and storage of biological materials.
  2. Superconductivity – superconductivity is the phenomenon where certain materials can conduct electric current with zero electrical resistance. Superconductivity is needed in several fields including medical imaging (MRI) and particle accelerator technologies.
  3. Quantum Computing – at very low temperatures quantum mechanical effects become more pronounced.at very low temperatures quantum mechanical effects become more pronounced. In order to create and manipulate qubits, the basic unit of quantum information, quantum computing systems require extremely low temperatures.
  4. Space Exploration – extremely low temperatures are encountered in deep space. Understanding the properties of materials at these temperatures is crucial for designing spacecraft components.

As you can see, absolute zero holds profound implications for various fields of study and cutting-edge technology.

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Henry Cavendish

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Henry Cavendish portrait
Henry Cavendish

Henry Cavendish (1731 – 1810) was one of the great experimental and theoretical chemist and physicist of the 18th century.  So meaningful were his contributions to science that James Clerk Maxwell named the University of Cambridge’s physics laboratory in his honor after he founded it in 1874.

Henry was born in Nice, France, due to his family traveling at the time of his birth.  He was educated in a private school in London then attended the University of Cambridge in 1748 where he stayed for three more years.  His father, Lord Charles Cavendish, was involved with members of the Royal Society of London and took Henry to meetings in the last 1750s.  By 1760 Henry was became an elected member of the Royal Society and from there on lived a life dedicated to science.

His interest and achievements in science were vast and wide ranging.  He began his work at the Royal Society by heading a committee to review the society’s meteorological instruments.  This initiated his research in chemistry and in particular gas chemistry.  His is credited with being the first person to isolate hydrogen (which he termed “inflammable air”), to correctly calculate its density, and determine that it was contained in water in a two to one proportion.  As with most scientists of his time, Henry also experimented with electricity.  He wrote many papers on electricity for the Royal Society but most of his experiments did not become known until many years after his death.

He was known for his extremely careful and accurate measurements.  This quality came in handy when it came time for him to measure the composition of the atmosphere and the density of the Earth.  Both measurements he obtained compare very nicely with the values accepted today.

Henry Cavendish amassed incredible wealthy throughout his life.  He used his wealthy mainly in the pursuit of science as he was not very sociable.  It is thought that he had Asperger syndrome, a form of autism.  He died in 1810 as one of the wealthiest men in Britain.

Heinrich Hertz

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Heinrich Hertz
Heinrich Hertz

Heinrich Hertz (1857 – 1894) was a German who lived a short yet impactful life.  He was interested in meteorology and assisted in making advances in weather forecasting.  He also conducted groundbreaking research in electromagnetic waves, making him the first person to conclusively prove James Clerk Maxwell’s theory of electromagnetism.

Hertz was born in Hamburg into a wealthy and affluent family.  He showed early aptitude in the sciences and went on the receive his PhD from the University of Berlin in 1880.  He was able to study under the physicist and physician Hermann von Helmholtz, whom he became an assistant to in his post-doctorate studies.  In 1885, Hertz obtained a full time professorship at the University of Karlsruhe.

During his time studying at the University of Berlin Helmholtz encouraged the university’s Philosophy Department to offer a prize to anyone who could solve the problem of whether electricity moves with inertia.  Hertz showed that it did through a series of clever experiments and won the prize.  Impressed by his work and capabilities, Helmholtz then asked Hertz to compete for a different prize offered by the Berlin Academy: verifying Maxwell’s theory of electromagnetism.  He declined to work on this problem after he decided it would be too difficult and time consuming, instead electing to establish his reputation by doing work less tedious.

Six years later Hertz was working at the University of Karlsruhe and decided it was time to return to experimental physics. After several months of experiments some breakthroughs began to emerge.  In November 1886 Hertz devised an experiment in the effects of electromagnetic waves were observed.  They were originally called Hertzian waves, but were later renamed radio waves.  These experiments also allowed Hertz to report on the photoelectric effect which would soon be explained by Albert Einstein.

Hertz successful scientific career was cut short becoming very ill and eventually passed away at the age of 36. The SI unit for frequency – hertz – was named in his honor in 1960, replacing the term “cycles per second.”

1866: Laws of Inheritance

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The laws of inheritance are a set of fundamental principles that govern the transmission of genetic traits from one generation to the next. Its discovery and understanding have changed our view of life while having a profound impact on a diverse range of topics such as medicine, agriculture and biotechnology. The ideas behind the laws of inheritance, the theory of evolution by natural selection, and population genetics has formed what scientist’s call the modern synthesis, a cornerstone of modern biology.

Gregor Mendel and the Pea Plant Experiments

For most of history peoples understanding about inheritance came from anecdotal evidence and observations of certain traits being passed down from parents to offspring. It wasn’t until the mid 19th century that the Augustinian monk Gregor Mendel conducted his now famous experiments with pea plants that established the principles of heredity. Prior to Mendel’s experiments the prevailing theory of inheritance suggested a blending of traits and characteristics from both parents to their offspring.

Gergor Mendel's pea plant experiment (Credit: Encycleopedia Britannica)
Gergor Mendel’s pea plant experiment
(Credit: Encycleopedia Britannica)

In 1866 the Augustinian monk Gregor Mendel published Experiments in Plant Hybridization that explained his pea plant experiments and the resulting laws of inheritance.  His work was first read to the Natural History Society of Brünn then published in the Proceedings of the Natural History Society of Brünn. 

During the years 1856 to 1863 Mendel cultivated over 28,000 plants and tested for seven specific traits   The traits he tested for were:

  • Pea shape (round or wrinkled)
  • Pea color (green or yellow)
  • Pod shape (constricted or inflated)
  • Pod color (green or yellow)
  • Flower color (purple or white)
  • Plant size (tall or dwarf)
  • Position of flowers (axial or terminal)

The results of his careful experimentation allowed Mendel to formulate some general laws of inheritance. His three laws of inheritance are:

  • Law of Segregation – allele pairs (one form of a gene) segregate during gamete (sex cells: sperm or egg) formation. Stated differently: each organism inherits at least two alleles for each trait but only one of these alleles are randomly inherited when the gametes are produced.
  • Law of Independent Assortment – allele pairs separate independently during the formation of gametes.
  • Law of Dominance – when two alleles of a pair are different, one will be dominate while the other will be recessive.

Mendel’s laws of inheritance suggested a particulate inheritance of traits in which traits are passed from one generation to the next in discrete packets.  As already noted, this differed from the most popular theory at the time which suggested a blending of characteristics in which traits are blended from one generation to the next.

Unfortunately for the progress of science, Mendel’s work was largely unnoticed and forgotten during his lifetime.  This was for a few reasons.  First, he lived in relative isolation at the Augustinian St. Thomas’s Abbey, now the modern day Czech Republic, and did not have a network of scientific colleagues.  He published his work in relatively obscure scientific journal and did not have the means to promote his findings.  His work, in a sense, was also ahead of his time.  The scientific community was simply focused on other areas of study during his lifetime and the concept of discrete hereditary units (now called genes) did not fit in with the prevailing scientific paradigm.  Lastly Mendel did little follow up to his work and soon shifted his attention to administrative and educational duties within the abbey.  It wasn’t until the turn of the 20th century that his work was rediscovered and popularized independently by three scientists – Hugo de Vries, Carl Correns, and Erich von Tschermak.  

A Journey into Genetics

Mendel’s laws of inheritance laid the groundwork for the 20th century field of genetics.  The field of genetics is the study of heredity that incorporates the structure and function of genes as the mechanism of biological inheritance.  

The emergence of molecular genetics began to take shape after it was discovered that the mechanism of hereditary transfer was contained in nucleic acids.  The race was on to discover the mechanism by which nucleic acids transferred the hereditary material.  The final breakthrough culminated with the discovery of the double-helical structure of DNA by James Watson and Francis Crick in 1953, as it provided the definitive explanation for how genetic information is encoded and transmitted within living organisms.  

The field of genetics continues to advance into the 21st century at a blistering pace.  In addition to unraveling the fundamental principles of life, scientists are now able to exploit the mechanics of genes and are learning novel ways to edit them to cure disease.  As of late 2023, the United States Food and Drug Administration (FDA) and medical regulators in the United Kingdom have approved the world’s first gene-editing treatment for Sickle Cell Disease using a gene-editing tool called Crispr.  Crispr technology has the potential to revolutionize the field of genetics and various related fields through its precise genome editing capabilities, potentially leading to another exciting development in the exciting history of science!

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Gregor Mendel

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Gregor Mendel portrait
Gregor Mendel

Gregor Mendel (1822 – 1884) was a German-speaking, Augustinian monk who did pioneering work on genetics.  His claim to fame however, was posthumous.  Mendel’s initial work was more or less unknown while he was alive and went unnoticed until it was rediscovered 50 years after death.

Mendel was born in the Austrian Empire in what is now present day Czech Republic.  During his childhood he worked on the family farm until he was sent to school at age 11.  However money was tight for him and his family and financial struggles weighted on his decision to join the monastery.  He joined the Augustinian Saint Thomas’ Abbey in Brünn and began his theological studies.  Mendel was always interested in more than just is theological studies and under a sponsorship he sent to study for two years at the University of Vienna to receive a broader education in the sciences.

When Mendel returned from the University of Vienna he began to carry out experiments on plants in the monastery’s experimental gardens.  He chose the common pea and began his experiments in 1856.  He identified seven traits that seemed to be inherited independently of other traits.  Mendel tested over 28,000 pants in the eight years of experimentation and was able to generalize a few laws of inheritance from his results.

His first law, the Law of Segregation, states that allele pairs (one form of a gene) segregate during gamete (sex cells: sperm or egg) formation.  Stated differently: each organism inherits at least two alleles for each trait but only one of these alleles are randomly inherited when the gametes are produced.  His second law, the Law of Independent Assortment, states the allele pairs separate independently during the formation of gametes.  His third law, the Law of Dominance, states that when two alleles of a pair are different, one will be dominate while the other will be recessive.

Mendel did various other experiments in biology and in other areas of science but the burdens of his administrative duties became too great and he stopped going his scientific studies.  He presented his work a handful of other people but nobody at the time realized the significance of his work.  The conventional wisdom was that there was a general blending of heritable traits rather than Mendel’s particulate inheritance of traits, where traits are passed in discrete packets.  Mendel’s work was rediscovered by three botanists each working independently in 1900 – Hugo DeVries, Carl Correns and Erich von Tschermak.  They gave priority to his work as well as confirmation to their own research.

Pierre de Fermat

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Pierre de Fermat
Pierre de Fermat

Pierre de Fermat (1607 – 1665) was a French mathematician whose mathematical work lead to the development of probability and statistics, the infinitesimal calculus, and of analytic geometry. He may be best know for his famous Fermat’s Last Theorem, a mathematical problem that went unsolved for centuries before it was finally solved by British mathematician Andrew Wiles in 1994.

Fermat was born in Beaumont-de-Lomagne, France, attended the University of Orleans in 1623 where he studied law despite showing an early interest in mathematics.  He received the title of councillor at the High Court of Judicature in Toulouse in 1631 and held this position for the rest of his life.

In the 1630s Fermat began some of his pioneering work in analytic geometry.  His work was circulated around in manuscript form and he showed to how find maximum point, minimum point, and tangents to curves.  He found techniques that were equivalent to differentiation and integral calculus.  These techniques were helpful to Isaac Newton and Gottfried Wilhelm Leibniz when they formulated their theories of calculus.

Along with his pioneering work in analytic geometry and laying the groundwork for the invention of calculus, Fermat had a tremendous contribution to number theory – a branch of mathematics devoted to the study of integers.   His study of Pell’s equation, perfect numbers, amicable numbers, and prime numbers ultimately led to the discovery of a new set of numbers that would be named after him: Fermat numbers.

The work he is best known for is called Fermat’s Last Theorem.  While working on number theory he had scribbled in the margin on a text that he had discovered a proof for an equation, but that the proof was too large for him to fit in the margin.  For over 350 years mathematicians were unable to obtain the proof until his theorem was finally proven in 1994 by Andrew Wiles.