The post Eudoxus and the Method of Exhaustion appeared first on SciHi Blog.

]]>**Eudoxus of Cnidus** was a Greek astronomer, mathematician, scholar and student of Plato. All of his works are lost or have survived as fragments in the texts of other classical writers. He is best known for having developed the method of exhaustion, a precursor to the integral calculus.

Eudoxus of Cnidus was born around 408 BC as the son of Aischines of Cnidus. His name Eudoxus means “honored” or “of good repute”. It is analogous to the Latin name *Benedictus*. As to his teachers, we know according to the 3rd-century CE historian Diogenes Laërtius that Eudoxus travelled to Tarentum, Italy, where he studied with Archytas who was a follower of Pythagoras,[4] from whom he learned mathematics. Eudoxus also visited Sicily, where he studied medicine with Philiston, before making his first visit to Athens in the company of the physician Theomedon in about 387 BC. Eudoxus spent two months in Athens on this visit and he certainly attended lectures on philosophy by Plato and other philosophers at the Academy which had only been established a short time before.[1] Eudoxus was quite poor and could only afford an apartment at the Piraeus. To attend Plato’s lectures, he walked the seven miles each direction, each day.

Due to his poverty, his friends raised funds sufficient to send him to Heliopolis, Egypt to pursue his study of astronomy and mathematics. He lived there for 16 months. From Egypt, he then traveled north to Cyzicus, located on the south shore of the Sea of Marmara, the Propontis. He traveled south to the court of Mausolus. During his travels he gathered many students of his own. After a brief interlude in Athens, he eventually returned to his native Cnidus, where he served in the city assembly. However he continued his scholarly work, writing books and lecturing on theology, astronomy and meteorology. He had built an observatory on Cnidus and we know that from there he observed the star Canopus. The observations made at his observatory in Cnidus, as well as those made at the observatory near Heliopolis, formed the basis of two books referred to by Hipparchus. These works were the *Mirror* and the *Phaenomena* which are thought by some scholars to be revisions of the same work. Hipparchus tells us that the works concerned the rising and setting of the constellations but unfortunately these books, as all the works of Eudoxus, have been lost.

In mathematical astronomy, his fame is due to the introduction of the astronomical globe, and his early contributions to understanding the movement of the planets. According to Eudoxus‘ model, the spherical earth is at rest at the center. Around this center, 27 concentric spheres rotate. The exterior sphere caries the fixed stars, the others account for the sun, moon, and five planets. Each planet requires four spheres, the sun and moon, three each. Eudoxus is considered by some to be the greatest of classical Greek mathematicians, and in all antiquity, second only to Archimedes.[5] His work on proportions shows tremendous insight into numbers Richard Dedekind,[6] who himself emphasised that his work was inspired by the ideas of Eudoxus.

Another remarkable contribution to mathematics made by Eudoxus was his early work on integration using his method of exhaustion. This work developed directly out of his work on the theory of proportion since he was now able to compare irrational numbers. It was also based on earlier ideas of approximating the area of a circle by Antiphon where Antiphon took inscribed regular polygons with increasing numbers of sides. According to Eratosthenes of Cyrene, Eudoxus also contributed a solution to the problem of doubling the cube—that is, the construction of a cube with twice the volume of a given cube. Aristotle preserved Eudoxus’s views on metaphysics and ethics. Unlike Plato, Eudoxus held that forms are in perceptible things. He also defined the good as what all things aim for, which he identified with pleasure.

Although Eudoxus made his most significant achievements in the field of geometry, not a single title of a relevant work has survived. His mathematical discoveries are therefore known only from writings of other authors. Eudoxus’ students included the physician Chrysippus, who accompanied him to Egypt, the mathematicians Menaichmus and Deinostratus, and the astronomer Polemarchus of Cyzicus. There is an account by Aristotle of two statements by Eudoxus on philosophical questions. One concerns the doctrine of ideas, the other the doctrine of the good. In both questions, Eudoxus holds a view that fundamentally contradicts that of Plato. The problem was the question how the participation of the single things in the ideas comes about. Eudoxus thought he could solve it with a doctrine of mixture; the ideas were mixed with the perceptible objects. Aristotle compares this with the admixture of a color to the colored by it. It is uncertain whether this comparison goes back to Eudoxus. How Eudoxus conceived of mixture is unclear; apparently he started from a natural philosophical notion of mixture common among the Presocratics and, unlike Plato, assumed a local presence of ideas in things. In contrast to Aristotle, he at the same time wanted to adhere to Plato’s doctrine of an existence of ideas separate from things. This earned him the reproach of contradictoriness. The traditional counter-argumentation is that he materialized the ideas and included them in the transitoriness of the material world and that they thereby lost their simplicity and immutability. Thereby they would lose their specific ontological status, thus they would no longer be ideas in the sense of the Platonic doctrine of ideas.

NJ Wildenberger, *Infinity in Greek mathematics | Math History* [7]

**References and Further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F., “Eudoxus of Cnidus”,
*MacTutor History of Mathematics archive*, University of St Andrews - [2] Eudoxus of Cnidus at Encyclopedia Britannica
- [3] Eudoxus of Cnidus Donald Allen, Professor, Texas AM University
- [4] Pythagoras and his Eponymous Theorem, SciHi Blog
- [5] Archimedes lifted the world off their Hinges, SciHi Blog
- [6] Richard Dedekind and the Real Numbers, SciHi Blog
- [7] NJ Wildenberger,
*Infinity in Greek mathematics | Math History, Insights into Mathematics @ youtube* - [8] Laërtius, Diogenes (1925). .
*Lives of the Eminent Philosophers*.**2:8**. Translated by Hicks, Robert Drew (Two volume ed.). Loeb Classical Library. - [9] Eudoxos of Knidos (Eudoxus of Cnidus): astronomy and homocentric spheres Henry Mendell, Cal State U, LA
- [10] Eudoxus of Cnidos at Wikidata
- [11] Timeline of Ancient Greek Astronomers, via Wikidata and DBpedia

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]]>The post Richard Dedekind and the Real Numbers appeared first on SciHi Blog.

]]>On October 6, 1831, German mathematician **Julius Wilhelm Richard Dedekind** was born. He is known for making important contributions to abstract algebra (particularly ring theory), algebraic number theory and the foundations of the real numbers.

“Numbers are the free creation of the human mind.” Richard Dedekind

Richard Dedekind, the son of the Braunschweig lawyer and university teacher Julius Dedekind, attended the Martino-Katharineum Braunschweig and studied mathematics at the Collegium Carolinum there from 1848. He continued his studies from 1850 in Göttingen, where he received his doctorate in 1852 after only four semesters with Carl Friedrich Gauss as his last student on the theory of Euler integrals.[3] However, he mainly studied mathematics with Moritz Abraham Stern and Georg Ulrich at the mathematical-physical seminar that Stern had just established, and physics with Wilhelm Weber and Johann Benedict Listing. With Gauss he heard in the winter semester 1850/51 about the method of least squares, which Dedekind remembered as one of the most beautiful lectures he ever heard, and in the following semester about higher geodesy.

However, Dedekind preferred to continue his studies at the University of Berlin, which was known to have the best reputation in the field of mathematics. There, he was awarded the habilitation and returned to Göttingen shortly after, where he was appointed Privatdozent of geometry and probability. It is assumed that Dedekind was the first to teach Galois’ theory and to understand the notion of groups in algebra and arithmetic. After a short period of teaching in Zurich, Switzerland, Dedekind returned to his native Braunschweig where he spent the rest of his working career, because the Collegium Carolinum was upgraded to an Institute of Technology. [1]

“What is provable is not to be believed without proof in science.”,

Richard Dedekind, Was sind und was sollen die Zahlen? (1888)

One of Dedekind’s best known contributions to mathematics is the ‘Dedekind cut‘. The idea behind a cut is that an irrational number divides the rational numbers into two classes, with all the members of one class being strictly greater than all the members of the other class. His thought on irrational numbers and Dedekind cuts was published in his pamphlet “Continuity and irrational numbers”, in modern terminology better known as completeness. In the later 1870s, Dedekind began his first works fundamental to ring theory, even though the term ‘ring‘ never appeared in his writings. Dedekind defined an ideal as a subset of a set of numbers, composed of algebraic integers that satisfy polynomial equations with integer coefficients. The concept underwent further development in the hands of Hilbert and, especially, of Emmy Noether. [2,4]

In 1888 Richard Dedekind gave the first exact introduction of the natural numbers by axioms in the paper *Was sind und was sollen die Zahlen*? *(What are and what should be the numbers). *In the appendix of the number theory of his teacher Dirichlet, he presented his construction of the ideal theory, which at that time was in competition with that of Leopold Kronecker.[5] The Dedekind rings and further the Dedekind η-function in the theory of modular forms, the Dedekind ζ-function of an algebraic number field, the Dedekind complementary modulus, Dedekind number, Dedekind sums as well as the terms “Dedekind-infinity” and “Dedekind-finity” are named after him. Dedekind played an essential role in the elaboration of abstract algebra. The algebraic notion of ring was introduced by Dedekind, as well as unity and the notion of “field”. Dedekind was furthermore a pioneer of group theory: In his lectures of 1855/56 he gave the first modern exposition of Galois theory (which, along with transformation groups in geometry and along with number theory as the third root, was important for the formation of the notion of group in the 19th century) with introduction of the abstract group notion as automorphism group of field extensions. In 1897 he introduced commutators and commutator groups independently of George Abram Miller. The notion of a lattice also goes back to Dedekind and Ernst Schröder at the end of the 19th century, but initially remained unnoticed.

After the death of Gauss, Peter Gustav Dirichlet succeeded him in 1855 and became friends with Dedekind. Dedekind became a full professor at the Polytechnikum in Zurich in 1858 and was professor of mathematics at the Technische Hochschule in Braunschweig from 1862 until his retirement in 1894. From 1872 to 1875 he was its director. Although he received several calls to prestigious universities, he preferred to remain in his hometown of Braunschweig. One of the main reasons was the close ties with his family (he had a brother and a sister, but was not married). Even after his retirement in 1894, he still lectured occasionally. In 1859 he visited Berlin with Riemann,[6] where he also met Leopold Kronecker, Ernst Eduard Kummer,[7] and Karl Weierstrass. In 1878 he visited Paris on the occasion of the World’s Fair.

Dedekind was a corresponding member of the Göttingen Academy of Sciences from 1862, a corresponding member of the Berlin Academy of Sciences from 1880, a corresponding member of the Académie des sciences in Paris from 1900, and a foreign member from 1910. Dedekind died on February 12, 1916, and was buried in Braunschweig’s main cemetery.

Dedekind Domains, [11]

**References and Further Reading:**

- [1] Short Dedekind Biography
- [2] Dedekind’s Contributions to the Foundations of Mathematics, Stanford Encyclopedia of Philosophy
- [3] Carl Friedrich Gauss – The Prince of Mathematicians
- [4] Emmy Noether and the Love for Mathematics
- [5] God made the integers, all the rest is the work of man – Leopold Kronecker, SciHi Blog
- [6] Bernhard Riemann’s innovative approaches to Geometry, SciHi Blog
- [7] Ernst Kummer and the Introduction of Ideal Numbers, SciHi Blog
- [8] Richard Dedekind at zbMATH
- [9] Richard Dedekind at Mathematics Geneology Project
- [10] Richard Dedekind at Wikidata, Timeline for Richard Dedekind via Wikidata
- [11] Yuli Billig, Dedekind Domains, Yuli Billig @ youtube

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]]>The post Bernard Bolzano and the Theory of Knowledge appeared first on SciHi Blog.

]]>On October 5, 1781, Bohemian mathematician, logician, philosopher, theologian and Catholic priest of Italian extraction **Bernard Bolzano** was born. Bolzano made significant contributions to both mathematics and the theory of knowledge. He provided a more detailed proof for the binomial theorem and suggested the means of distinguishing between finite and infinite classes. His major work, *Wissenschaftslehre* (1837), contains various contributions to logic and semantics concerning the relations of compatibility, derivability, and consequence, the deduction theorem, and the logic of classes, entailment, and probability.

Bolzano was born as the fourth of twelve children into a pious catholic family to Bernard Pompeius Bolzano, an Italian art dealer who had moved to Prague, Bohemia, Austrian Habsburg domain (now Czech Republic). When he was ten years old, Bolzano entered the Gymnasium of the Piarists in Prague, which he attended from 1791 to 1796. He subsequently entered the University of Prague in 1796 and studied mathematics, philosophy and physics. Starting in 1800, he also began studying theology. Bolzano got ordained as Roman Catholic priest on 7 April 1805. A few days later, on 17 April 1805, he received his doctorate of philosophy at the University of Prague. Just two days later, on 19 April 1805, he took up the newly established chair for religious doctrine in the Philosophical Faculty at the University of Prague.[2]

The appointment of Bolzano was viewed with suspicion by the Austrian rulers in Vienna. He criticised discrimination wherever he saw it, principally by the German speaking Bohemians against their Czech fellow citizens, and also he criticised the anti-Semitism displayed by both the German and Czech Bohemians. Some members of the Roman Catholic Church were also unhappy because Bolzano’s lectures contains elements of rationalism.[1]

Bolzano should hold his position at the university until 1819, and he was even elected dean of the philosophy department in 1818. During this time, he published his first book, *Beyträge zu einer begründeteren Darstellung der Mathematik* (*Contributions to a More Well-founded Presentation of Mathematics*), in which he opposes German philosopher Immanuel Kant’s views on mathematics.[6] His sermons and lectures on philosophy and religion were highly popular with the students but disturbing to Church and government officials, because he voiced his own liberal opinions, advocating pacifism and socialism instead of reinforcing the Catholic doctrine. He openly criticized the government for discrimination and pleaded the cause of minority groups within the empire, such as the Jews and the Czechs. He was forced to resign from his position when he refused to recant his political beliefs. After a lengthy trial held by the Catholic Church, he was forbidden from preaching in public or publishing any of his writing.[3]

Bolzano was exiled to the countryside and at that point devoted his energies to his writings on social, religious, philosophical, and mathematical matters. Although forbidden to publish in mainstream journals, Bolzano continued to develop his ideas and publish them either on his own or in obscure Eastern European journals. In 1842 he moved back to Prague to spent his later days. Bolzano also died in Prague on December 18, 1848, leaving behind an extensive handwritten estate.

In his 1837 *Wissenschaftslehre (Philosophy of Science)* Bolzano attempted to provide logical foundations for all sciences, building on abstractions like part-relation, abstract objects, attributes, sentence-shapes, ideas and propositions in themselves, sums and sets, collections, substances, adherences, subjective ideas, judgments, and sentence-occurrences. In the *Wissenschaftslehre*, Bolzano is mainly concerned with three realms:

- The realm of language, consisting in words and sentences.
- The realm of thought, consisting in subjective ideas and judgements.
- The realm of logic, consisting in objective ideas (or ideas in themselves) and propositions in themselves.

Two distinctions play a prominent role in his system. Firstly, the distinction between parts and wholes. For instance, words are parts of sentences, subjective ideas are parts of judgments, objective ideas are parts of propositions in themselves. Secondly, all objects divide into those that exist, which means that they are causally connected and located in time and/or space, and those that do not exist.

As a mathematician he conducted basic research in analysis. He was probably the first to construct a function that is everywhere continuous but nowhere differentiable. He also worked with large and infinitely small numbers. In an essay of 1817 he proved the intermediate value theorem and introduced Cauchy sequences, four years before Augustin-Louis Cauchy.[7] Bolzano’s work on a stricter foundation of analysis was hardly noticed by his contemporaries, in contrast to that of Cauchy, and was only appreciated in the second half of the 19th century (for example by Hermann Hankel, Hermann Amandus Schwarz, Otto Stolz). The mathematical theorem of Bolzano-Weierstraß is named after him. In his 1851 posthumous work “Paradoxien des Unendlichen” (Paradoxes of the Infinite), he collected a great deal of knowledge about (finite and infinite) sets and introduced the concept of set as a technical term into mathematics.

Most of Bolzano’s works remained in manuscript and did not become noticed and therefore did not influence the development of the subject. Many of his works were not published until 1862 or later. Bolzano’s theories of mathematical infinity anticipated Georg Cantor’s theory of infinite sets.[1,8] Bolzano’s posthumously published work *Paradoxien des Unendlichen* *(The Paradoxes of the Infinite*) (1851) was greatly admired by many of the eminent logicians who came after him, including Charles Sanders Peirce, Georg Cantor, and Richard Dedekind.

Marc Walker, *Lecture 23(A): Compact Sets and Metric Spaces; Bolzano-Weierstrass Theorem* [12]

**References and Further Reading: **

- [1] O’Connor, John J.; Robertson, Edmund F. (2005), “Bernard Bolzano“, MacTutor History of Mathematics archive.
- [2] Bernard Bolzano at Stanford Encyclopedia of Philosophy
- [3] Bernard Bolzano at NewWorldBiography
- [4] Bernardo Bolzano at zbMATH
- [5] Bernard Bolzano at Wikidata
- [6] Immanuel Kant – Philosopher of the Enlightenment, SciHi Blog
- [7] Augustin-Louis Cauchy and the Rigor of Analysis, SciHi Blog
- [8] Georg Cantor and the Beauty of Infinity, SciHi Blog
- [9] Works by or about Bernard Bolzano at German Digital Library
- [10] The Philosophy of Bernard Bolzano: Logic and Ontology
- [11] Works by or about Bernard Bolzano at Internet Archive
- [12] Marc Walker,
*Lecture 23(A): Compact Sets and Metric Spaces; Bolzano-Weierstrass Theorem, Arizona Math Camp @ youtube* - [13] Timeline of 19th Century Mathematicians, via Wikidata and DBpedia

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]]>The post Pierre Louis Maupertuis – The Man who flattened the Earth appeared first on SciHi Blog.

]]>On September 28, 1698, French mathematician, philosopher and man of letters **Pierre Louis Maupertuis** was born. Maupertuis made an expedition to Lapland to determine the shape of the Earth. He is also credited with having invented the principle of least action, an integral equation that determines the path followed by a physical system.

“Nature always uses the simplest means to accomplish its effects.”

– Formulation of the principle of least action, as stated in Mémoires de l’académie royale des sciences (1748)

Pierre Louis Maupertuis was born at Saint-Malo, France. He was the son of the naval officer, capitaine malouin and deputy to the Conseil royal de commerce René Moreau de Maupertuis (1664-1746), his mother was Jeanne-Eugénie Baudran (1672-1724). Maupertuis attended the Collège de la Marche in Paris from 1714, where he studied philosophy, returning to Saint-Malo in 1716 at the request of his mother, where he began to study music in 1717 and his interest in mathematics awakened. Maupertuis was educated in mathematics by a private tutor, Nicolas Guisnée and after completing his formal education at the age of 20, Maupertuis joined the army in Lille as a musketeer lieutenant and later became a cavalry captain, serving there for five years. In his free time, he studied mathematics. In 1722, he went to Paris and joined intellectual circles, including being friends with Marivaux, where he began building his reputation as a mathematician and literary wits.

In 1723, at the age of 25, he was admitted to the French Académie des sciences and later became a full member of this Parisian academy of sciences in 1725. During these years, he received his doctorate there with the thesis *Sur la forme des instruments de musique* (*On the shape of musical instruments, 1724*), which was accepted by the academy in 1723. His first work dealt with the relationship between musical acoustics and the form of instruments. He published on mathematics and biology (for example, the salamander). In 1728 he traveled to London, where he was elected a member of the Royal Society. In 1729/30 he was in Basel, where he studied under one of the leading continental European mathematicians of the time, Johann I Bernoulli.[6] In particular, he studied Isaac Newton‘s *Principia* and his theory of gravitation. He became a fierce proponent of this theory, which at the time, as a seemingly inexplicable theory of action at a distance, was mostly met with skepticism on the Continent (including by the Bernoullis) in favor of Descartes’ theory.

“It is interesting to note that Newton was not impressed by Descartes’ great argument for God’s existence derived from the idea of a perfect Being, nor by other metaphysical arguments that we have mentioned; yet Newton’s own arguments for God’s existence from the uniformity and suitability of different parts of the universe would not have seemed like proofs to Descartes.”

– Pierre Louis Maupertuis, Les Loix du Mouvement et du Repos, déduites d’un Principe Métaphysique (1746)

Among other things, people in France believed they could deduce from the survey of the land a shape of the earth contrary to Newton’s theory – an elongation toward the poles rather than, as Newton predicted, a flattening. To verify this, geodetic measurements were needed in more distant areas. Thus, the shape of the Earth became a major issue of science in the 1730s. Pierre Louis Maupertuis predicted the Earth to be more oblate shaped, which he based on his studies of Newton’s works. However, his rival Cassini measured Earth to be prolate. In 1736 Maupertuis acted as chief of the French Geodesic Mission sent by King Louis XV to Lapland to measure the length of a degree of arc of the meridian. Simultaneously with a second group in today’s Ecuador (Pierre Bouguer,[8] Charles Marie de La Condamine, Louis Godin) an exact degree measurement of a long meridian arc should be made, in order to determine from the differences in the radius of curvature of the earth its size and form. The measurements of both groups confirmed the Newtonian theory of polar oblateness. Maupertuis was so proud of his scientific achievement under the difficult conditions that he subsequently often wore the costume of the lobes. His results, which he published in a book detailing his procedures, essentially settled the controversy in his favor. The book also included an adventure narrative of the expedition, and an account of the Käymäjärvi Inscriptions.

After returning from the Lapland expedition, Pierre Louis Maupertuis increased his reputation greatly and then indended to continue his earler scientific work. In 1740, Frederick the Great, on Voltaire‘s recommendation, invited him to Berlin to take charge of the Prussian Academy of Sciences. However, since Frederick was preoccupied with military matters, Maupertuis accompanied him and fell into Austrian captivity during the Battle of Mollwitz, was taken to Vienna, but was treated kindly there and soon released on Maria Theresa’s orders. He returned first to Berlin, but already in June 1741 to Paris to await the end of the Second Silesian War.

Maupertuis was not only a mathematician and a good connoisseur of Newton’s and Leibniz’s theories, he also realized that Newton’s theories were not sufficient to explain biological phenomena. In this sense, he was one of the most advanced thinkers of his time, speaking out not only against preformism and Newtonian determinism, but also against creationism. Maupertuis proposed the principle of least action as a metaphysical principle that underlies all the laws of mechanics. However, the scientist also expanded into the biological realm, anonymously publishing a book that was part popular science, part philosophy, and part erotica: *Vénus physique*. In his work, Maupertuis proposed a theory of generation in which organic matter possessed a self-organizing intelligence that was analogous to the contemporary chemical concept of affinities. The work was widely read and commented upon favorably by Georges-Louis Leclerc, Comte de Buffon.[7] Maupertuis was an essentialist in the sense that he put any species that differed from its neighbors on the taxonomic map. However, even though he assumed the creation of new traits, he could not conceive of a gradual adaptation of the species by selecting the best adapted members of that species.

Frederick II of Prussia invited Maupertuis a few times and appointed him as president of the Royal Prussian Academy of Sciences in 1746. He lead the organisation with the help of Leonhard Euler until his death. In 1756 Maupertuis moved to Basel. There he died in 1759 in the house of Johann II Bernoulli.

Kenneth Young on “*A Special Lecture: Principle of Least Action*” [12]

**References and Further Reading:**

- [1] The man who flattened the Earth – Maupertuis and the sciences in the Enlightenment
- [2] O’Connor, John J.; Robertson, Edmund F., “Pierre Louis Maupertuis”,
*MacTutor History of Mathematics archive*, University of St Andrews - [3] Chisholm, Hugh, ed. (1911). “Maupertuis, Pierre Louis Moreau de“.
*Encyclopædia Britannica*(11th ed.). Cambridge University Press. - [4] Works by or about Pierre Louis Maupertuis at Wikisource
- [5] Maupertuis (1751).
*Essai de cosmologie* - [6] What is a Mathematical Function – according to Johann Bernoulli, SciHi Blog
- [7] Sir Isaac Newton and the famous Principia, SciHi Blog
- [8] Comte de Buffon and his Histoire Naturelle, SciHi Blog
- [9] Pierre Bouguer – Child Prodigy and ‘Father of Photometry’, SciHi Blog
- [10] Pierre Louis Maupertuis at Wikidata
- [11] Timeline for Pierre Louis Maupertuis, via Wikidata
- [12] Kenneth Young “
*A Special Lecture: Principle of Least Action*“, hNeg @ youtube

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]]>The post Read Euler, he is the Master of us all… appeared first on SciHi Blog.

]]>On September 18, 1783, Swiss mathematician and physicist **Leonhard Euler** passed away. Euler is considered to be the pre-eminent mathematician of the 18th century and one of the greatest mathematicians to have ever lived. He is also one of the most prolific mathematicians. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. A statement attributed to Pierre-Simon Laplace expresses Euler‘s influence on mathematics: “*Read Euler, read Euler, he is the master of us all.*“

“Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.”

– Leonhard Euler

Leonhard Euler was born on April 15, 1707, in Basel, Switzerland, to Paul Euler, who had studied theology at the University of Basel and had attended the lectures of famous mathematician Jacob Bernoulli there. Thus, Paul Euler was able to teach his son elementary mathematics along with other subjects. Euler’s formal education started in Basel, where he was sent to live with his maternal grandmother. In 1720, at age thirteen, Euler enrolled at the University of Basel, and in 1723, he received a Master of Philosophy with a dissertation that compared the philosophies of Descartes and Newton. At that time, he was receiving Saturday afternoon lessons from Johann Bernoulli, then regarded as Europe’s foremost mathematician, who quickly discovered his new pupil’s incredible talent for mathematics. At that time Euler’s main studies included theology, Greek, and Hebrew at his father’s urging, in order to become a pastor, but Bernoulli convinced his father that Leonhard was destined to become a great mathematician.

In 1726, Euler completed a dissertation on the propagation of sound with the title *De Sono*. In 1727 he published another article on reciprocal trajectories and submitted an entry for the 1727 Grand Prize of the Paris Academy on the best arrangement of masts on a ship. But, the went to Pierre Bouguer,[7] an expert on mathematics relating to ships, who became known as “the father of naval architecture”. Nevertheless Euler’s essay won him second place which was a fine achievement for the young graduate. However, Euler later won this annual prize twelve times.

Euler now had to find himself an academic appointment and when Nicolaus Bernoulli died in St Petersburg in July 1726 creating a vacancy there, Euler was offered the post which would involve him in teaching applications of mathematics and mechanics to physiology.[1] Euler eagerly accepted the offer, but delayed making the trip to St Petersburg while he unsuccessfully applied for a physics professorship at the University of Basel. Euler arrived in St Petersburg on 17 May 1727. Daniel Bernoulli held the senior chair in mathematics at the Academy in St. Petersburg, which was originally founded by Peter the Great,[9] intended to improve education in Russia and to close the scientific gap with Western Europe. When Daniel Bernoulli left St Petersburg to return to Basel in 1733 it was Euler who was appointed to this senior chair of mathematics. In 1734, Euler married Katharina Gsell, the daughter of a painter from the St Petersburg Gymnasium. Katharina, like Euler, was from a Swiss family. They had 13 children altogether although only five survived their infancy. Euler claimed that he made some of his greatest mathematical discoveries while holding a baby in his arms with other children playing round his feet.[1]

Euler’s health problems began in 1735 when he had a severe fever and almost lost his life. However, he kept this news from his family and friends in Basel until he had recovered. In his autobiographical writings Euler says that his eyesight problems began in 1738 with overstrain due to his cartographic work and that by 1740 he had alread lost the sight of one eye completely. In 1741 Euler moved to Berlin at the request, or rather command, of Frederick the Great. He lived for twenty-five years in Berlin, where he wrote over 380 articles. In Berlin, he published the two works for which he would become most renowned:* The Introductio in analysin infinitorum* (1748), a text on functions, and the *Institutiones calculi differentialis* (1755) on differential calculus.

Euler undertook an unbelievable amount of work for the Academy: he supervised the observatory and the botanical gardens; selected the personnel; oversaw various financial matters; and, in particular, managed the publication of various calendars and geographical maps, the sale of which was a source of income for the Academy. Frederick the Great also charged Euler with practical problems, such as the project in 1749 of correcting the level of the Finow Canal or the supervision of the work on pumps and pipes of the hydraulic system at Sans Souci, the royal summer residence.[9] Despite Euler’s immense contribution to the Academy’s prestige, he eventually incurred the ire of Frederick and ended up having to leave Berlin. The Prussian king had a large circle of intellectuals in his court and he found the mathematician unsophisticated and ill-informed on matters beyond numbers and figures. Euler stayed in Berlin until 1766, when he returned to Russia, and was succeeded at Berlin by Lagrange.

During his years in Berlin, Euler wrote books on the calculus of variations, on the calculation of planetary orbits, on artillery and ballistics, on analysis, on shipbuilding and navigation, on the motion of the moon, lectures on the differential calculus, and a popular scientific publication *Letters to a Princess of Germany*, which became rather famous. Euler developed a cataract in his left eye, which was discovered in 1766. Just a few weeks after its discovery, he was rendered almost totally blind. However, his condition appeared to have little effect on his productivity, as he compensated for it with his mental calculation skills and exceptional memory. For example, Euler could repeat the Aeneid of Virgil from beginning to end without hesitation, and for every page in the edition he could indicate which line was the first and which the last. With the aid of his scribes, Euler’s productivity on many areas of study actually increased. He produced on average, one mathematical paper every week in the year 1775. In 1771 his home was destroyed by fire and he was able to save only himself and his mathematical manuscripts.

In St. Petersburg on 18 September 1783, after a lunch with his family, Euler was discussing the newly discovered planet Uranus and its orbit with a fellow academician Anders Johan Lexell, when he collapsed from a brain hemorrhage and died shortly after. After his death the St Petersburg Academy continued to publish Euler’s unpublished work for nearly 50 more years.

Euler’s work in mathematics is so vast that here only a very superficial account of it can be given. He was the most prolific writer of mathematics of all time. He made large bounds forward in the study of modern analytic geometry and trigonometry where he was the first to consider sin, cos etc. as functions rather than as chords as Ptolemy had done. He made decisive and formative contributions to geometry, calculus and number theory. He integrated Leibniz’s differential calculus and Newton’s method of fluxions into mathematical analysis. He introduced beta and gamma functions, and integrating factors for differential equations. He studied continuum mechanics, lunar theory with Clairaut, the three body problem, elasticity, acoustics, the wave theory of light, hydraulics, and music. We owe to Euler the notation *f (x)* for a function (1734), *e* for the base of natural logarithms (1727), *i* for the square root of -1 (1777), π for pi, ∑ for summation (1755), the notation for finite differences Δy and Δ2y and many others.[1]

William Dunham, *A Tribute to Euler*, [8]

**References and Further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F., “Leonhard Euler“, MacTutor History of Mathematics archive, University of St Andrews.
- [2] Leonhard Euler in A Short Account of the History of Mathematics by W. W. Rouse Ball (4th Edition, 1908)
- [3] How to calculate Fortune – Jacob Bernoulli, SciHi blog, Aug 16, 2012.
- [4] Frederick the Great’s Cunning Plan to Introduce the Potato, SciHi blog, March 24, 2013.
- [5] Leonard Euler at zbMATH
- [6] Leonard Euler at Wikidata, Timeline of Leonard Euler via Wikidata
- [7] Pierre Bouguer – Child Prodigy and ‘Father of Photometry’, SciHi blog

- [8] William Dunham,
*A Tribute to Euler*, (2008), Poincare Duality @ youtube - [9] Peter the Great and the Grand Embassy, SciHi Blog
- [10] Timeline for Leonhard Euler, via Wikidata

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]]>The post Bernhard Riemann’s innovative approaches to Geometry appeared first on SciHi Blog.

]]>On September 17, 1826, influential German mathematician **Bernhard Riemann** was born. Riemann‘s profound and novel approaches to the study of geometry laid the mathematical foundation for Albert Einstein’s theory of relativity. He also made important contributions to the theory of functions, complex analysis, and number theory.

“Nevertheless, it remains conceivable that the measure relations of space in the infinitely small are not in accordance with the assumptions of our geometry [Euclidean geometry], and, in fact, we should have to assume that they are not if, by doing so, we should ever be enabled to explain phenomena in a more simple way.”

– Bernhard Riemann, Memoir (1854)

Bernhard Riemann was born in Breselenz, a village near Dannenberg in the Kingdom of Hannover, today in Germany, as the second of six children to Friedrich Bernhard Riemann, a poor Lutheran pastor who had fought in the Napoleonic Wars. Friedrich Riemann acted as teacher to his children and he taught Bernhard until he was ten years old. Bernhard Riemann exhibited exceptional mathematical skills, such as calculation abilities, from an early age but suffered from timidity and a fear of speaking in public.

In 1840 Bernhard entered directly into the third class at the Lyceum in Hannover, living with his grandmother but, in 1842, his grandmother died and Bernhard moved to the Johanneum Gymnasium in Lüneburg. He showed a particular interest in mathematics and the school director allowed Bernhard to study mathematics texts from his own library. On one occasion he lent Bernhard Legendre’s book on the theory of numbers and Bernhard read the 900 page book in six days.[1] In 1846, at the age of 19, he started studying philology and theology at the University of Göttingen in order to become a pastor and help with his family’s finances. However, once there, he began studying mathematics under Carl Friedrich Gauss, who recommended that Riemann give up his theological work and enter the mathematical field.[3]

Riemann moved from Göttingen to Berlin University in the spring of 1847 to study under Jakob Steiner, Carl Gustav Jakob Jacobi, Peter Gustav Lejeune Dirichlet and Gotthold Eisenstein.[4] It was during his time at the University of Berlin that Riemann worked out his general theory of complex variables that formed the basis of some of his most important work.

In 1849 he returned to Göttingen and his Ph.D. thesis, supervised by Gauss, was submitted in 1851. Riemann’s thesis studied the theory of complex variables and, in particular, what we now call Riemann surfaces. It therefore introduced topological methods into complex function theory. In 1851 and in his more widely available paper of 1857, Riemann showed how such surfaces can be classified by a number, later called the genus, that is determined by the maximal number of closed curves that can be drawn on the surface without splitting it into separate pieces. This is one of the first significant uses of topology in mathematics.[2] The work builds on Cauchy’s foundations of the theory of complex variables built up over many years. In proving some of the results in his thesis Riemann used a variational principle which he was later to call the Dirichlet Principle since he had learnt it from Dirichlet’s lectures in Berlin.[1]

On Gauss’s recommendation Riemann was appointed to a post in Göttingen and he worked for his Habilitation on the foundations of geometry. Over many months, Riemann developed his theory of higher dimensions and delivered his lecture at Göttingen in 1854 entitled Über die *Hypothesen welche der Geometrie zu Grunde liegen* (“*On the hypotheses which underlie geometry*“), published posthumously by Dirichlet in 1868. This should be the foundation of the field of Riemannian geometry and thereby set the stage for Einstein’s general theory of relativity.[5] Riemann found the correct way to extend into n dimensions the differential geometry of surfaces. The fundamental object is called the Riemann curvature tensor. For the surface case, this can be reduced to a number (scalar), positive, negative or zero; the non-zero and constant cases being models of the known non-Euclidean geometries. Riemann’s idea was to introduce a collection of numbers at every point in space (i.e., a tensor) which would describe how much it was bent or curved. Riemann found that in four spatial dimensions, one needs a collection of ten numbers at each point to describe the properties of a manifold, no matter how distorted it is. This is the famous construction central to his geometry, known now as a Riemannian metric.

In 1857, there was an attempt to promote Riemann to extraordinary professor status at the University of Göttingen. Although this attempt failed, it did result in Riemann finally being granted a regular salary. In 1859, following Lejeune Dirichlet’s death, he was promoted to head the mathematics department at Göttingen. In the same year he was elected a corresponding member of the Berlin Academy of Sciences. Also in 1859 Riemann introduced complex function theory into number theory. He took the zeta function, which had been studied by many previous mathematicians because of its connection to the prime numbers, and showed how to think of it as a complex function.[2] About his eponymous zeta function, which is important for understanding the distribution of prime numbers, he stated the famous (and still not completely proven) Riemann hypothesis, a conjecture that the non-trivial zeros of the Riemann zeta function all have real part 1/2.

Riemann’s influence was initially less than it might have been. Göttingen was a small university, Riemann was a poor lecturer, and, to make matters worse, several of his best students died young. His few papers are also difficult to read, but his work won the respect of some of the best mathematicians in Germany.[2]

Physicists were still far removed from such a way of thinking: space was still, for them, a rigid, homogeneous something, susceptible of no change or conditions. Only the genius of Riemann, solitary and uncomprehended, had already won its way by the middle of the last century to a new conception of space, in which space was deprived of its rigidity, and in which its power to take part in physical events was recognized as possible. (Albert Einstein)

However, in 1862, shortly after his marriage to Elise Koch, Riemann fell seriously ill with tuberculosis. Repeated trips to Italy failed to stem the progress of the disease. Riemann fled Göttingen when the armies of Hanover and Prussia clashed there in 1866. He died during his journey to Italy in the same year.

Barry Mazur, *A Lecture on Primes and the Riemann Hypothesis*, [8]

**References and Further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F., “Bernhard Riemann“, MacTutor History of Mathematics archive, University of St Andrews.
- [2] Bernhard Riemann at Britannica Online
- [3] Carl Friedrich Gauss – The Prince of Mathematicians, SciHi blog, April 30, 2013.
- [4] Lejeune Dirichlet and the Mathematical Function, SciHi blog, Feb 13, 2015.
- [5] Albert Einstein revolutionized Physics, SciHi blog, Mar 14, 2013.
- [6] Bernhard Riemann at zbMATH
- [7] Bernhard Riemann at Wikidata
- [8] Barry Mazur,
*A Lecture on Primes and the Riemann Hypothesis, Harvard University,*Gratuate Mathematics @ youtube - [9] The Mathematical Papers of Georg Friedrich Bernhard Riemann
- [10] Timeline of Bernhard Riemann via Wikidata

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]]>The post Giovanni Saccheri and his Problems with Euclidian Geometry appeared first on SciHi Blog.

]]>On September 5, 1667, Italian Jesuit priest, scholastic philosopher, and mathematician **Giovanni Girolamo Saccheri** was born. He is primarily known today for his last publication, *Euclide Ab Omni Naevo Vindicatus (Euclid Freed of Every Flaw, 1733)*, now considered the second work in non-Euclidean geometry.

Saccheri was born in Sanremo to his father Giovanni Felice Saccheri, a lawyer and notary.[6] As a child Saccheri ‘was notably precocious’.[1] He entered the Jesuit order in 1685, and from 1687 onwards he began teaching at the College as well as studying philosophy and theology. In 1690 the Superiors of the Company of Jesus sent Saccheri to the Collegio di Brera in Milan, where he was encouraged to take up mathematics by one of his teachers Tommaso Ceva who suggested that he read Christopher Clavius‘s edition of Euclid’s *Elements*.[4] It was through the influence of Giovanni Ceva that Saccheri published his first mathematical work *Quaesita geometrica* (1693), in which he solved many problems in elementary geometry. It was not a particularly significant work but showed that Saccheri was becoming deeply involved in thinking about Euclidean geometry.

Saccheri was ordained a priest in March 1694 at Como and then later in the year he was sent by the Superiors of the Jesuit Order to teach philosophy at the Jesuit College in Turin, which led to the publication of *Logica demonstrativa* (1697), which covered the course on logic that he had been teaching in Turin. In 1697, Saccheri was sent to the Jesuit College of Pavia to teach philosophy and theology, where he stayed until his death, but he also held the chair of mathematics at the University of Pavia from 1699 until his death.

However, Saccheri today is primarily remembered for his last publication E*uclides ab Omni Naevo Vindicatus* (*Euclid Freed of Every Flaw,* 1733), which was published shortly before his death and is now considered the second work in non-Euclidean geometry, although he did not see it as such, rather an attempt to prove the parallel postulate of Euclid. Interestingly, it remained in obscurity until its rediscovery by Eugenio Beltrami 150 years later.[2] Many of Saccheri’s ideas have a precedent in the 11th Century Persian polymath Omar Khayyám‘s *Discussion of Difficulties in Euclid*, a fact ignored in most Western sources until recently.[7] It is unclear whether Saccheri had access to this work in translation, or developed his ideas independently.

The intent of Saccheri’s work was ostensibly to establish the validity of Euclid by means of a reductio ad absurdum proof of any alternative to Euclid’s parallel postulate. To do this he assumed that the parallel postulate was false, and attempted to derive a contradiction. Since Euclid’s postulate is equivalent to the statement that the sum of the internal angles of a triangle is 180°, he considered both the hypothesis that the angles add up to more or less than 180°. The first led to the conclusion that straight lines are finite, contradicting Euclid’s second postulate. So, Saccheri correctly rejected it. However, today this principle is accepted as the basis of elliptic geometry, where both the second and fifth postulates are rejected.

The second possibility turned out to be harder to refute. In fact he was unable to derive a logical contradiction and instead derived many non-intuitive results. He finally concluded that: “*the hypothesis of the acute angle is absolutely false; because it is repugnant to the nature of straight lines*“. Today, his results are theorems of hyperbolic geometry. There is some minor argument on whether Saccheri really meant that, as he published his work in the final year of his life, came extremely close to discovering non-Euclidean geometry and was a logician. Some believe Saccheri concluded as he did only to avoid the criticism that might come from seemingly-illogical aspects of hyperbolic geometry.

Gierolamo Saccheri died in Milan on 25 October 1733. It should take 170 years until the significance of the work was realized.[1]

*Non-Euclidean Geometry [Topics in the History of Mathematics]* [10]

**References and Further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F., “Giovanni Gierolamo Saccheri“, MacTutor History of Mathematics archive, University of St Andrews.
- [2] Eugenio Beltrami and Non-Euclidian Geometry, SciHi blog, Nov 16, 2014.
- [3] Nikolai Lobachevsky – The Copernicus of Geometry, SciHi blog, Feb, 24, 2015.
- [4] Euclid – the Father of Geometry, SciHi blog, Jan 30, 2015.
- [5] Works by or about Girolamo Saccheria at German National Library
- [6] Saccheri, Giovanni Gierolamo, at The Galileo Project, Rice University
- [7] Omar Khayyam – Mathematics and Poetry, SciHi Blog
- [8] Girolamo Saccheri,
*Euclides Vindicatus*(1733), edited and translated by G. B. Halsted, 1st ed. (1920) - [9] Gierolamo Saccheri at Wikidata
- [10]
*Non-Euclidean Geometry [Topics in the History of Mathematics], via marshare @ youtube* - [11] Timeline of Italian 17th century mathematicians, via Wikidata and DBpedia

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]]>The post Al-Biruni – Mathematician, Astronomer and Founder of Indology appeared first on SciHi Blog.

]]>On September 4, 973, Muslim scholar **Al-Biruni** was born. He is regarded as one of the greatest scholars of the medieval Islamic era and was well versed in physics, mathematics, astronomy, and natural sciences, and also distinguished himself as a historian, chronologist and linguist. He is referred to as the founder of Indology for his remarkable description of early 11th-century India.

“You well know … for which reason I began searching for a number of demonstrations proving a statement due to the ancient Greeks … and which passion I felt for the subject … so that you reproached me my preoccupation with these chapters of geometry, not knowing the true essence of these subjects, which consists precisely in going in each matter beyond what is necessary. … Whatever way he [the geometer] may go, through exercise will he be lifted from the physical to the divine teachings, which are little accessible because of the difficulty to understand their meaning … and because the circumstance that not everybody is able to have a conception of them, especially not the one who turns away from the art of demonstration.”

– Al Biruni, Book on the Finding of Chords.

Abu Rayhan al-Biruni was born in Khwarazm now better known as Karakalpakstan. It is believed that Biruni started his studies at early age under the famous astronomer and mathematician Abu Nasr Mansur and he was probably engaged in his own scientific work starting from the age of 17. By 995, al-Biruni had written several short works including his Cartography, a work on map projections. Also, al-Biruni managed to describe his own projection of a hemisphere onto a plane. However, Abu Rayhan al-Biruni’s quiet life came to an end with the unrest in the Islamic world during the end of the 10th century and beginning of the 11th century. Due to several civil wars, al-Biruni fled, but his exact destination is not clear to this day. He may have gone to Rayy, near today’s city of Tehran, but most likely lived in poverty at that time. Clear is however, that the astronomer al-Khujandi discussed his observations and methods with al-Biruni, who managed to point out several of al-Khujandi’s errors.

“For it is the same whether you take it that the Earth is in motion or the Sky. For, in both the cases, it does not affect the Astronomical Science. It is just for the Physicist to see if it is possible to refute it.”

– Abu Rayhan al-Biruni

In the following years, al-Biruni probably traveled around often, and historians managed to determine some dates and places through the astronomical events he described. He went back to his homeland probably around 1004 and its ruler Abu’s’l Abbas Ma’mun provided great support for al-Biruni’s scientific work. For instance, the scientist was able to build an instrument at Jurjaniyya to observe solar meridian transits and he made 15 such observations with the instrument between 7 June 1016 and 7 December 1016. Unfortunately for al-Biruni, the political events took its toll once more. Al-Biruni had to leave the region, probably as prisoners, after their ruler had been executed. It is assumed that he was supported to do some scientific work even though he suffered great hardships. In 1018, he was probably in Kabul for some time and even though he had no access to any accurate instruments, the scientist managed to observe an eclipse of the sun. While being a prisoner of Mahmud, al-Biruni made an excursion to India and published his famous work ‘*India*, which covered many aspects of the country including its religion and cast system, and science. Apparently, al-Biruni even studied the original studied Indian literature and translated a few texts into Arabic. With Mahmud’s death, his son Mas’ud turned out to treat al-Biruni better and he was now free to travel and to his research as he pleased to.

Another major work by al-Biruni is known as ‘*Shadows*‘ which he is believed to have written in 1021 and e.g. contains work on the Arabic nomenclature of shade and shadows. The book turned out to be an important source for the history of mathematics, astronomy, and physics. The work also gives a decent overview of al-Biruni’s abilities and contributions to mathematics and astronomy, as he wrote about theoretical and practical arithmetic as well as the summation of series, combinatorial analysis, and much more. He was the first Islamic scholar to study Brahmanic science and reported extensively on it in the *Kitab al-Hind*. Al-Biruni, whose mother tongue was chorusmic, translated numerous Arabic and Greek works into Sanskrit, including the elements of Euclid.[3]

Bīrūnī devised a novel method of determining the earth’s radius by means of the observation of the height of a mountain. He carried it out at Nandana in Pind Dadan Khan (present-day Pakistan). He used trigonometry to calculate the radius of the Earth using measurements of the height of a hill and measurement of the dip in the horizon from the top of that hill. He determined the radius of the earth’s globe to be 6339.6 km, which is very close to the real value today at the equator of 6378.1 km. In his *Codex Masudicus* (1037), Al-Biruni theorized the existence of a landmass along the vast ocean between Asia and Europe, or what is today known as the Americas. He argued for its existence on the basis of his accurate estimations of the Earth’s circumference and Afro-Eurasia’s size, which he found spanned only two-fifths of the Earth’s circumference, reasoning that the geological processes that gave rise to Eurasia must surely have given rise to lands in the vast ocean between Asia and Europe. Abu ‘r-Raihan Muhammad al-Biruni constructed the first pycnometer. He used it to determine the density (the specific weight) of different materials. Al-Biruni wrote some 146 books, estimated to be 13,000 pages long, and exchanged correspondence with colleagues such as Avicenna (Ibn Sina).[4] About one fifth of his work has been preserved.

How Abū al-Rayhān Ahmad al-Bīrūnī Measured the Size of the Earth [8]

**References and Further Reading:**

- [1] John J. O’Connor, Edmund F. Robertson:
*Al-Biruni.*In:*MacTutor History of Mathematics archive.* - [2] Al-Biruni: A Master of Scholarship at Lost Islamic History
- [3] Euclid of Alexandria – the Father of Geometry, SciHi Blog
- [4] Avicenna – The Most Significant Polymath of the Islamic Golden Age, SciHi Blog
- [5]“Al-Biruni (973–1048).” Encyclopedia of Occultism and Parapsychology. 2001. Encyclopedia.com. 5 Feb. 2015.
- [6] Hogendijk, Jan: The works of al-Bīrūnī – manuscripts, critical editions, translations and online links
- [7] Works by or about Al-Biruni at Wikisource
- [8] How Abū al-Rayhān Ahmad al-Bīrūnī Measured the Size of the Earth, via Sajid Ali Mir at youtube
- [9] Al-Biruni at Wikidata
- [10] Timeline of Medieval Persian Astronomers, via Wikidata and DBpedia

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]]>The post James Joseph Sylvester – Lawyer and Mathematician appeared first on SciHi Blog.

]]>On September 3, 1815, English mathematician **James Joseph Sylvester** was born. He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory and combinatorics. He also was the founder of the *American Journal of Mathematics*.

“It seems to be expected of every pilgrim up the slopes of the mathematical Parnassus, that he will at some point or other of his journey sit down and invent a definite integral or two towards the increase of the common stock.”

– James Joseph Sylvester, Collected Mathematical Papers, Vol. 2 (1908), p. 214.

Sylvester was born James Joseph in London, England, to Abraham Joseph, a Jewish merchant. James adopted the surname Sylvester when his older brother did so upon emigration to the United States — a country which at that time required all immigrants to have a given name, a middle name, and a surname. In 1828, at the age of 14, Sylvester was a student of Augustus De Morgan at the University of London. His family withdrew him from the University after he was accused of stabbing a fellow student with a knife. Subsequently he attended the Liverpool Royal Institution. Sylvester began his study of mathematics at St John‘s College, Cambridge in 1831. Although his studies were interrupted for almost two years due to a prolonged illness, he nevertheless ranked second in Cambridge‘s famous mathematical examination, the tripos, for which he sat in 1837. However, Sylvester was not issued a degree, because graduates at that time were required to state their acceptance of the Thirty-Nine Articles of the Church of England, and Sylvester, refused to do so because of his Jewish religion.

For the same reason he was not eligible for a Smith’s prize nor was he eligible to compete for a Fellowship.[1] In 1838 Sylvester became professor of natural philosophy at University College London. In 1841, he was awarded a BA and an MA by Trinity College, Dublin. In the same year he moved to the United States to be appointed to the chair of mathematics in the University of Virginia in Charlottesville for about six months, and returned to England in November 1843, following an altercation with a student for which the school’s administration did not take his side.[4]

On his return to England he studied law, alongside fellow British lawyer/mathematician Arthur Cayley,[2] with whom he made significant contributions to matrix theory while working as an actuary. One of his private pupils was Florence Nightingale.[3] In 1851 he discovered the discriminant of a cubic equation and first used the name ‘discriminant’ for such expressions of quadratic equations and those of higher order. In particular he used matrix theory to study higher dimensional geometry. He also contributed to the creation of the theory of elementary divisors of lambda matrices.[1] He did not obtain a position teaching university mathematics until 1855, when he was appointed professor of mathematics at the Royal Military Academy, Woolwich, from which he retired in 1869, because the compulsory retirement age was 55. The Woolwich academy initially refused to pay Sylvester his full pension, and only relented after a prolonged public controversy, during which Sylvester took his case to the letters page of *The Times*.

“Number, place, and combination . . . the three intersecting but distinct spheres of thought to which all mathematical ideas admit of being referred.”

– James Joseph Sylvester, Collected Mathematical Papers, Vol. 1 (1904), p. 91.

Following his early retirement, Sylvester (1870) published a book entitled *The Laws of Verse* in which he attempted to codify a set of laws for prosody in poetry. For three years Sylvester appears to have done no mathematical research but then Chebyshev visited London and the two discussed mechanical linkages which can be used to draw straight lines.[9] After working on this topic Sylvester lectured on it at an evening lecture entitled* On recent discoveries in mechanical conversion of motion* which he gave at the Royal Institution. One mathematician in the audience at this lecture was Kempe and he became absorbed by this topic. Kempe and Sylvester worked jointly on linkages and made important discoveries.[1] In 1872, he finally received his B.A. and M.A. from Cambridge, having been denied the degrees due to him being a Jew.

In 1877 Sylvester accepted a chair at Johns Hopkins University in Baltimore, Maryland, USA, and he founded in 1878 the *American Journal of Mathematics*, the first mathematical journal in the United States, whose first editor he was. In 1883, he returned to England to take up the Savilian Professor of Geometry at Oxford University. He held this chair until his death, although in 1892 the University appointed a deputy professor to the same chair.

Sylvester was primarily an algebraist. He did brilliant work in the theory of numbers, particularly in partitions (the possible ways a number can be expressed as a sum of positive integers) and Diophantine analysis (a means for finding whole-number solutions to certain algebraic equations). Sylvester invented a great number of mathematical terms such as “matrix”, “graph” and “discriminant”. He coined the term “totient” for Euler’s totient function φ(n). His collected scientific work fills four volumes. In 1880, the Royal Society of London awarded Sylvester the Copley Medal, its highest award for scientific achievement.

J. J. Sylvester died in London on 15 March 1897, at age 82. The Sylvester Medal of the Royal Society, of which he had been a member (Fellow) since 1839, is named in his honor.

Christof Paar, *Lecture 11: Number Theory for PKC: Euclidean Algorithm, Euler’s Phi Function & Euler’s Theorem* [10]

**References and Further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F., “James Joseph Sylvester“, MacTutor History of Mathematics archive, University of St Andrews.
- [2] Arthur Cayley and the Love for Pure Mathematics, SciHi blog Jan 26, 2013.
- [3] Florence Nightingale – The Lady with the Lamp, SciHi blog, May 12, 2014.
- [4] James Joseph Sylvester at Britannica Online
- [5] Collected Works of and secondary literature about J. J. Sylvester at the University of Edinburgh</li>
- [6] Augustus De Morgan and Formal Logic, SciHi blog June 27, 2013.
- [7] James Joseph Sylvester at zbMATH
- [8] James Joseph Sylvester at Mathematics Genealogy Project
- [9] Pafnuty Chebyshev and the Chebyshev Inequality, SciHi Blog
- [10] Christof Paar,
*Lecture 11: Number Theory for PKC: Euclidean Algorithm, Euler’s Phi Function & Euler’s Theorem*, Introduction to Cryptography by Christof Paar @ youtube - [11] Grattan-Guinness, I. (2001), “The contributions of J. J. Sylvester, F.R.S., to mechanics and mathematical physics”,
*Notes and Records of the Royal Society of London*,**55**(2): 253–265 - [12] Karen Hunger Parshall:
*James Joseph Sylvester. Jewish mathematician in a Victorian world.*Johns Hopkins University Press, Baltimore MD 2006 - [13] James Joseph Sylvester at Wikidata
- [14] Timeline for James Joseph Sylvester, via Wikidata

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]]>The post John Venn and the Venn Diagram appeared first on SciHi Blog.

]]>On August 4, 1834, English logician and philosopher **John Venn** was born. He is best known for his contribution of the eponymous Venn diagram, used in the fields of set theory, probability, logic, statistics, and computer science.

“I began at once somewhat more steady work on the subjects and books which I should have to lecture on. I now first hit upon the diagrammatical device of representing propositions by inclusive and exclusive circles. Of course the device was not new then, but it was so obviously representative of the way in which any one, who approached the subject from the mathematical side, would attempt to visualise propositions, that it was forced upon me almost at once.”

— John Venn, as quoted in [4]

John Venn was born in Kingston upon Hull, Yorkshire, UK, as the son of Martha Venn (née Sykes) and the pastor and social reformer Henry Venn. He grew up with two sisters, Henrietta and Susan. His mother died when he was three years old. Venn was descended from a long line of Lutheran clergy, including his grandfather, John Venn Senior, who worked to abolish slavery. John Venn Junior followed his family tradition and became an Anglican minister. He worked first in Cheshunt, Hertfordshire and later in Mortlake, Surrey. Venn attended Highgate School in London. From 1853 to 1857 he studied at Gonville and Caius College in Cambridge. In 1858 he was ordained a deacon in Ely and in 1859 a priest of the Church of England. In 1862 he returned to Cambridge as a university lecturer and under the influence of the work of Augustus De Morgan,[9] George Boole [3] and John Stuart Mill [8] he devoted himself to logic and probability theory.

In 1862, Venn returned to Cambridge as a lecturer in Moral Science and studying and teaching logic and probability theory. It is believed that Venn became more enthusiastic about the field of logic after reading the works of De Morgan, Boole, John Austin, and John Stuart Mill. He also began extending Boole’s mathematical logic and created what he is best known for on this day, his diagrammatic way of representing sets, and their unions and intersections. Venn considered three discs R, S, and T as typical subsets of a set U. The intersections of these discs and their complements divide U into 8 non-overlapping regions, the unions of which give 256 different Boolean combinations of the original sets R, S, T. Keynes later described his new method as very original and a great contribution to the theory of statistics.

In his later career, John Venn was elected a member of the Royal Society and wrote his book ‘*The Biographical History of Gonville and Cauis College*‘, which was published in 1897. His son, John Archibald Venn became president of Queen’s College, Cambridge in 1932. Together with his son John Archibald Venn he was the editor of the reference book *Alumni Cantabrigienses*. As professor of logic and natural philosophy Venn taught at Cambridge for over 30 years. From about 1890 onwards, he was primarily concerned with the history of his university. Venn was the first to work out the frequency concept of probability in the form of a mathematical theory. His ideas on probability theory were later taken up and further developed by Hans Reichenbach. Venn also was a prominent supporter of votes for women. He co-signed with his wife Susanna, a letter to the Cambridge Independent Press published 16 October 1908, encouraging women to put themselves forward as candidates for the up-and-coming Cambridge town council elections.

John Venn died on 4th April 1923.

At yovisto academic video search, you can learn more about ‘*The beauty of data visualization*‘ in a lecture by David McCandless.

**References and Further Reading:**

- [1] John J. O’Connor, Edmund F. Robertson:
*John Venn.*In:*MacTutor History of Mathematics archive.* - [2] John Venn at Famous Mathematicians
- [3] George Boole – The Founder of Modern Logics, SciHi Blog
- [4] Edwards, Anthony William Fairbank (2004).
*Cogwheels of the Mind: The Story of Venn Diagrams*. Baltimore, Maryland, USA: Johns Hopkins University Press. p. 3 - [5] Venn, John (January 1876). “Consistency and Real Inference”.
*Mind*.**1**(1). *[6]*Venn, John (1881).*Symbolic Logic*. London: Macmillan and Company.- [7] Obituary of John Venn (New York Times)
- [8] Liberty vs. Authority according to John Stuart Mill, SciHi blog
- [9] Augustus de Morgan and Formal Logic, SciHi Blog
- [10] John Venn at Wikidata
- [11] John Venn at zbMATH
- [12]
**Timeline of English Logicians,**via Wikidata and DBpedia

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