October 20: Christopher Wren

Sir Christopher Wren

October 20, 1632 – February 25, 1723

Christopher Wren was one of the best known and highest acclaimed English architects in history, responsible for rebuilding 55 churches in the City of London after the Great Fire in 1666, including his masterpiece St Paul's Cathedral, completed in 1710.

Educated in Latin and Aristotelian physics at the University of Oxford, Wren was a notable astronomer, geometer, mathematician-physicist as well as an architect. He was a founder of the Royal Society (president 1680–82), and his scientific work was highly regarded by Sir Isaac Newton and Blaise Pascal.

One of Wren's friends, another great scientist and architect in his time, Robert Hooke said of him "Since the time of Archimedes there scarce ever met in one man in so great perfection such a mechanical hand and so philosophical mind."

When a fellow of All Souls, Wren constructed a transparent beehive for scientific observation; he began observing the moon, which was to lead to the invention of micrometers for the telescope. He experimented on terrestrial magnetism and had taken part in medical experiments, performing the first successful injection of a substance into the bloodstream (of a dog).

In Gresham College, he did experiments involving determining longitude through magnetic variation and through lunar observation to help with navigation, and helped construct a 35-foot (11 m) telescope with Sir Paul Neile. Wren also studied and improved the microscope and telescope at this time. He had also been making observations of the planet Saturn from around 1652 with the aim of explaining its appearance. His hypothesis was written up in De corpore saturni but before the work was published, Huygens presented his theory of the rings of Saturn. Immediately Wren recognized this as a better hypothesis than his own and De corpore saturni was never published. In addition, he constructed an exquisitely detailed lunar model and presented it to the king.

A year into Wren's appointment as a Savilian Professor in Oxford, the Royal Society was created and Wren became an active member. As a Savilian Professor, Wren studied thoroughly in mechanics, especially in elastic collisions and pendulum motions, which he studied extensively. He also directed his far-ranging intelligence to the study of meteorology, and fabricated a "weather-clock" that recorded temperature, humidity, rainfall and barometric pressure, which could be used to predict the weather.

Another topic to which Wren contributed was optics. He published a description of an engine to create perspective drawings and he discussed the grinding of conical lenses and mirrors. Out of this work came another of Wren's important mathematical results, namely that the hyperboloid of revolution is a ruled surface. These results were published in 1669. In subsequent years, Wren continued with his work with the Royal Society, although after the 1680s his scientific interests seem to have waned: no doubt his architectural and official duties absorbed all his time.

Mentioned above are only a few of Wren’s scientific works. He also studied in other areas not mentioned, ranging from agriculture, ballistics, water and freezing, to investigating light and refraction only to name a few. Thomas Birch's History of the Royal Society is one of the most important sources of our knowledge not only of the origins of the Society, but also the day to day running of the Society. It is in these records that the majority of Wren’s scientific works are recorded.

October 18: Pascual Jordan

Pascual Jordan

October 18, 1902 - July 31, 1980

Pascual Jordan was a theoretical and mathematical physicist who made significant contributions to quantum mechanics and quantum field theory. He contributed much to the mathematical form of matrix mechanics, and developed canonical anticommutation relations for fermions. While the Jordan algebra he invented is no longer employed in quantum mechanics, it has found other mathematical applications.

Jordan joined the Nazi party and became a brownshirt, a political affiliation which isolated him within the physics community.

Jordan enrolled in the Hanover Technical University in 1921 where he studied an eclectic mix of zoology, mathematics, and physics. As was typical for a German university student of the time, he shifted his studies to another university before obtaining a degree. Göttingen University, his destination in 1923, was then at the very zenith of its prowess and fame in mathematics and the physical sciences. At Göttingen Jordan became an assistant first to mathematician Richard Courant and then to physicist Max Born.

Together with Max Born and Werner Heisenberg he was co-author of an important series of papers on quantum mechanics. He went on to pioneer early quantum field theory before largely switching his focus to cosmology before World War II.

Jordan devised a type of non-associative algebras, now named Jordan algebras in his honor, in an attempt to create an algebra of observables for quantum mechanics and quantum field theory. Today, von Neumann algebras are employed for this purpose. Jordan algebras have since been applied in projective geometry and number theory.

Had Jordan not joined the Nazi party, it is conceivable that he could have shared the 1954 Nobel Prize in Physics awarded to Max Born.

October 13: Peter Barlow

Peter Barlow

October 13, 1776 – March 1, 1862

Peter Barlow was an English mathematician and physicist.

In 1801, Barlow was appointed assistant mathematics master at the Royal Military Academy, Woolwich, and retained this post until 1847. He contributed articles on mathematics to The Ladies' Diary as well as publishing books such as: An Elementary Investigation of the Theory of Numbers (1811); A New Mathematical and Philosophical Dictionary (1814); and New Mathematical Tables (1814).

The latter became known as Barlow's Tables and gives squares, cubes, square roots, cube roots, and reciprocals of all numbers from 1 to 10,000. These tables were regularly reprinted until 1965, when computers rendered them obsolete. Barlow also contributed largely to the Encyclopaedia Metropolitana.

Barlow made an important contribution to astronomy by inventing the Barlow lens, an achromatic telescope lens in which two glass layers enclosed a colourless liquid (later replaced by a different kind of glass).

In 1823 he was made a fellow of the Royal Society and two years later received its Copley Medal for his work on correcting the deviation in ship compasses caused by the presence of iron in the hull. Some of his magnetic research was done in collaboration with Samuel Hunter Christie. Barlow's Wheel is an early electric motor he invented, while Barlow's law is an (incorrect) law describing electrical conductance.

September 30: Michael Maestlin

Michael Maestlin

September 30, 1550 - October 20, 1631

Michael Maestlin was a German astronomer and mathematician, known for being the mentor of Johannes Kepler.

Maestlin studied theology, mathematics, and astronomy at the Tübinger Stift in Tübingen, a town in Württemberg. In 1580 he became a Professor of mathematics, first at the University of Heidelberg, then at the University of Tübingen were he taught for 47 years from 1583. In 1582 Maestlin wrote a popular introduction to astronomy.

Among his students was Johannes Kepler (1571-1630). Although he primarily taught the traditional geocentric Ptolemaic view of the solar system, Maestlin was also one of the first to accept and teach the heliocentric Copernican view. Maestlin corresponded with Kepler frequently and played a sizable part in his adoption of the Copernican system. Galileo Galilei's adoption of heliocentrism was also attributed to Maestlin .

The first known calculation of the (inverse) golden ratio as a decimal of "about 0.6180340" was written in 1597 by Maestlin in a letter to Kepler.

Maestlin catalogued the Pleiades cluster on December 24, 1579. Eleven stars in the cluster were recorded by Maestlin, and possibly as many as fourteen were observed. He also observed the occultation of Mars by Venus on October 3, 1590, at Heidelberg.

In Jules Verne's Cinq semaines en ballon (Five Weeks in a Balloon) the character of Joe, the manservant, is described as enjoying, "in common with Moestlin, Kepler's professor, the rare faculty of distinguishing the satellites of Jupiter with the naked eye, and of counting fourteen of the stars in the group of Pleiades, the remotest of them being only of the ninth magnitude."

The Lunar crater Maestlin, the Lunar rille Rimae Maestlin and the asteroid 11771 Maestlin, discovered in 1973, are named in his honor.

September 26: Christopher Hansteen

Christopher Hansteen

September 26, 1784 – April 11, 1873

Christopher Hansteen was a Norwegian geophysicist, astronomer and physicist.

In 1807 Hansteen began the inquiries in terrestrial magnetism with which his name is especially associated. His first scientific publication was printed in Journal de Physique, following a contest on magnetic axes created in 1811 by the Royal Danish Academy of Sciences and Letters. In 1813 he was given a research scholarship by the recently established (in 1811) Royal Frederick University in Christiania, with a promise of a future academic position.

Working as a lecturer from 1814, in 1816 Hansteen was promoted to professor of astronomy and applied mathematics. He was the editor of the official Norwegian almanac from 1815, manager of the city astronomical observatory from the same year and co-director of the Norwegian Mapping and Cadastre Authority from 1817. In 1819 he published a volume of researches on terrestrial magnetism, which was translated into German under the title of Untersuchungen über den Magnetismus der Erde, with a supplement containing Beobachtungen der Abweichung und Neigung der Magnetnadel and an atlas. By the rules there framed for the observation of magnetical phenomena Hansteen hoped to accumulate analyses for determining the number and position of the magnetic poles of the Earth. In 1822 he co-founded Norway's first journal on natural sciences, Magazin for Naturvidenskaberne. He sat as editor-in-chief for eight years.

In the course of his research he travelled over Finland and the greater part of his own country; and from 1828 to 1830 he undertook, in company with Georg Adolf Erman and with the cooperation of Russia, a government-funded mission to Western Siberia. Shortly after the return of the mission, in 1833 Hansteen moved with his family into the observatory, which was created from drawings by the architect Christian Heinrich Grosch. A magnetic observatory was added in 1839.

From 1835 to 1838 he published textbooks on geometry and mechanics, largely a reaction to his former research assistant Bernt Michael Holmboe's textbooks. In 1842 Hansteen wrote his Disquisitiones de mutationibus, quas patitur momentum acus magneticae. He also contributed various papers to different scientific journals, especially Magazin for Naturvidenskaberne.

Hansteen was a member of the Royal Norwegian Society of Sciences and Letters from 1818 and of the Norwegian Academy of Science and Letters from 1857, as well as several learned societies in other countries, including the Royal Swedish Academy of Sciences (1822). He was a member of the board of the Royal Norwegian Society for Development for many years, and also chaired the board of the Norwegian National Academy of Craft and Art Industry.

The crater Hansteen and the mountain Mons Hansteen on the Moon are named in his honor.

September 19: Jean-Baptiste Delambre

Jean Baptiste Joseph, chevalier Delambre

September 19, 1749 - August 19, 1822

Jean-Baptiste Delambre was a French mathematician and astronomer. He was also director of the Paris Observatory, and author of well-known books on the history of astronomy from ancient times to the 18th century.

After a childhood fever, he suffered from very sensitive eyes, and believed that he would soon go blind. For fear of losing his ability to read, he devoured any book available to him and practised his ability to memorise. He thus immersed himself in Greek and Latin literature, acquired the ability to recall verbatim entire pages of books he may have read weeks beforehand, became fluent in Italian, English and German and even published Règles et méthodes faciles pour apprendre la langue anglaise (Easy rules and methods for learning English). In 1788, he was elected a foreign member of the Royal Swedish Academy of Sciences.

In 1801, First Consul Napoléon Bonaparte took the presidency of the Academy of Sciences and appointed Delambre its Permanent Secretary for the Mathematical Sciences, a post he held until his death.

After Méchain's death in 1804, he was appointed director of the Paris Observatory. He was also professor of Astronomy at the Collège de France. The same year he married Elisabeth-Aglaée Leblanc de Pommard, a widow with whom he had lived already for a long time. Her son, Achille-César-Charles de Pommard (1781-1807) assisted Delambre on several occasions in his astronomical and geodetical surveys, notably the measuring of the baselines for the meridian survey, and the latitude definition for Paris in December 1799 which was presented to the Conference of Savants.

Delambre was one of the first astronomers to derive astronomical equations from analytical formulas, was the author of Delambre's Analogies and, after the age of 70, also the author of works on the history of astronomy like the Histoire de l'astronomie.

He was a knight (chevalier) of the Order of Saint Michael and of the Légion d'honneur.

The crater Delambre on the Moon is named in his honor.

September 16: Francesco Maurolico

Francesco Maurolico

September 16, 1494 - July 21, 1575

Francesco Maurolico (in Latin, Franciscus Maurolycus) was an Italian mathematician and astronomer. Throughout his lifetime, he made contributions to the fields of geometry, optics, conics, mechanics, music, and astronomy.

Between 1548 and 1550, Maurolico stayed at the castle of Pollina in Sicily as a guest of the marquis Giovanni II Ventimiglia, and utilized the castle tower in order to carry out astronomical observations.

Maurolico's astronomical observations include a sighting of the supernova that appeared in Cassiopeia in 1572. Tycho Brahe published details of his observations in 1574; the supernova is now known as Tycho's Supernova.

In 1569, he was appointed professor at the University of Messina.

Maurolico's Photismi de lumine et umbra and Diaphana concern the refraction of light and attempted to explain the natural phenomenon of the rainbow. He also studied the camera obscura. His Arithmeticorum libri duo (1575) includes the first known proof by mathematical induction.

Maurolico published a Cosmographia in which he described a methodology for measuring the earth, which was later employed by Jean Picard in measuring the meridian in 1670.

Maurolico published an edition of Aristotle's Mechanics, and a work on music. He summarized Ortelius's Theatrum orbis terrarum and also wrote Grammatica rudimenta (1528) and De lineis horariis. He made a map of Sicily, which was published in 1575.

Maurolico worked on ancient mathematical texts: Theodosius of Bithynia, Menelaus of Alexandria, Autolycus of Pitane, Euclid, Apollonius of Perga and Archimedes. His didn't make new translations, but working on the existing ones, he provided new and sound interpretations of Greek mathematics.

The lunar crater Maurolycus is named after him.

September 14: Johann Kies

Johann Kies

September 14, 1713—July 29, 1781

Johann Kies was a German astronomer and mathematician. Born in Tübingen, Kies worked in Berlin in 1751 alongside Jérôme Lalande in order to make observations on the lunar parallax in concert with those of Nicolas Louis de Lacaille at the Cape of Good Hope.

From 1742 to 1754, at the recommendation of the mathematician Leonhard Euler, he was made professor of mathematics at Berlin's Academy of Sciences and astronomer at its observatory. His reports from this time include De la Situation la plus avantageuse des planètes pour découvrir les irrégularités de leurs mouvemens, Sur les Éclipses des étoiles fixes par la lune, and Description d'un instrument qui se trouve à l'observatoire de Berlin. In his Rapport de quelques observations célestes faites à l'observatoire Royal, Kies wrote, "

Observations on eclipses are extremely useful, especially the ones that are exact. They serve primarily to understand, if the theories on the Sun and the Moon are well or badly ascertained in the astronomical tables, and can either confirm them, or give us a need to reform them."

He subsequently taught also at the Collegium of Tübingen. From 1754 to 1755, Kies served as director of the Astronomisches Rechen-Institut in Heidelberg.

Kies was one of the first to propagate Newton's discoveries in Germany, and dedicated two of his works to the Englishman: De viribus centralibus (Tübingen, 1758) and De lege gravitatis (Tübingen, 1773). Kies is also the author of a work on lunar influences: De influxu lunae in partes terrae mobiles (Tübingen, 1769). He wrote many other works, both in French and in Latin, on astronomy.

Kies corresponded with Euler from 1747 to 1767. Their correspondence consists of 8 letters, all of which were written by Kies.

The crater Kies on the Moon is named in his honor.

July 22: Friedrich Bessel

Friedrich Wilhelm Bessel

July 22, 1784 – March 17, 1846

Friedrich Bessel was a German mathematician, astronomer, and systematizer of the Bessel functions (which were discovered by Daniel Bernoulli). He was a contemporary of Carl Gauss, also a mathematician and astronomer.

Bessel was born in Minden in Minden-Ravensberg, the son of a civil servant. At the age of 14 Bessel was apprenticed to the import-export concern Kulenkamp. He soon became the company's accountant. The business's reliance on cargo ships led him to turn his mathematical skills to problems in navigation. This in turn led to an interest in astronomy as a way of determining longitude.

Bessel came to the attention of a major figure of German astronomy at the time, Heinrich Wilhelm Olbers, by producing a refinement on the orbital calculations for Halley's Comet. Within two years Bessel had left Kulenkamp and become an assistant at Lilienthal Observatory near Bremen. There he worked on James Bradley's stellar observations to produce precise positions for some 3,222 stars.

This work attracted considerable attention, and at the age of 26 Bessel was appointed director of the Königsberg Observatory by King Frederick William III of Prussia. There he published tables of atmospheric refraction based on Bradley's observations, which won him the Lalande Prize from the Institut de France. Bessel was able to pin down the position of over 50,000 stars during his time at Königsberg.

With this work under his belt, Bessel was able to achieve the feat for which he is best remembered today: he is credited with being the first to use parallax in calculating the distance to a star. Astronomers had believed for some time that parallax would provide the first accurate measurement of interstellar distances—in fact, in the 1830s there was a fierce competition between astronomers to be the first to measure a stellar parallax accurately. In 1838 Bessel won the race, announcing that 61 Cygni had a parallax of 0.314 arcseconds; which, given the diameter of the Earth's orbit, indicated that the star was about 3 parsecs (9.8 light years) away. He narrowly beat Friedrich Georg Wilhelm Struve and Thomas Henderson, who measured the parallaxes of Vega and Alpha Centauri in the same year.

As well as helping determine the parallax of 61 Cygni, Bessel's precise measurements allowed him to notice deviations in the motions of Sirius and Procyon, which he deduced must be caused by the gravitational attraction of unseen companions. His announcement of Sirius's "dark companion" in 1844 was the first correct claim of a previously unobserved companion by positional measurement, and eventually led to the discovery of Sirius B.

Despite lacking a university education, Bessel was a major figure in astronomy during his lifetime. He was elected a fellow of the Royal Society, won the Gold Medal of the Royal Astronomical Society in 1841 and the largest crater in the Moon's Mare Serenitatis is named Bessel after him. The asteroid 1552 Bessel was also named in his honor.

July 21: Jean-Felix Picard

Jean-Felix Picard

July 21, 1620 – July 12, 1682

Jean Picard was a French astronomer and priest born in La Flèche, where he studied at the Jesuit Collège Royal Henry-Le-Grand. He was the first person to measure the size of the Earth to a reasonable degree of accuracy in a survey conducted 1669-70, for which he is honored with a pyramid at Juvisy-sur-Orge.

Guided by Maurolycus's methodology and Snellius's mathematics for doing so, Picard achieved this by measuring one degree of latitude along the Paris Meridian using triangulation along thirteen triangles stretching from Paris to the clocktower of Sourdon, near Amiens. His measurements produced a result of 110.46 km for one degree of latitude, which gives a corresponding terrestrial radius of 6328.9 km. The polar radius has now been measured at just over 6357 km. This was an error only 0.44% less than the modern value. This was another example of advances in astronomy and its tools making possible advances in cartography.

Picard was the first to attach a telescope with crosswires (developed by William Gascoigne) to a quadrant, and one of the first to use a micrometer screw on his instruments. The quadrant he used to determine the size of the Earth had a radius of 38 inches and was graduated to quarter-minutes. The sextant he used to find the meridian had a radius of six feet, and was equipped with a micrometer to enable minute adjustments. These equipment improvements made the margin of error only ten seconds, as opposed to Tycho Brahe's four minutes of error. This made his measurements 36 times more accurate. Isaac Newton was to use this value in his theory of universal gravitation.

Picard also travelled to Tycho Brahe's Swedish observatory, Uraniborg, in order to assess its position accurately to compare Tycho's readings to others'.

Picard collaborated and corresponded with many scientists, including Newton, Christian Huygens, Römer, Bartholin, Hudde, and even his main competitor, Giovanni Cassini, although Cassini was often less than willing to return the gesture. These correspondences led to Picard's contributions to areas of science outside the field of geodesy, such as the aberration of light he observed while in Uraniborg, or his discovery of mercurial phosphorescence upon his observance of the faint glowing of a barometer. This discovery led to Newton's studies of spectrometry.

Picard also developed the system of right ascension for measuring positions of celestial objects in relation to the Earth.

His book "Mesure de la Terre" was published in 1671.

The Lunar crater Picard, on the northwest quadrant of Mare Crisium and the PICARD mission, an orbiting solar observatory, are named in his honor.

July 17: Barnaba Oriani

Barnaba Oriani

July 17 1752 - November 12 1832

Barnaba Oriani was an Italian geodesist, astronomer and scientist.

After getting his elementary education in Carignano, he went on to study at the College of San Alessandro in Milan, under the tutelage and with the support of the Order of Barnabus, which he later joined. After completing his studies in the humanities, physical and mathematical sciences, philosophy, and theology, he was ordained as a priest in 1775.

When Napoleon established the Republic of Lombardy, Oriani refused to swear an oath against the monarchy, and the new republican government modified the oath of allegiance on his behalf. He was retained in his position at the observatory and was made president of the commission appointed to regulate the new system of weights and measures.

When the republic became a Napoleonic kingdom, Oriani was awarded the Iron Crown and the Legion of Honour, was made a count and senator, and was appointed to measure the arc of the meridian between the zeniths of Rimini and Rome.

Oriani was a devoted friend of the Theatine monk, Giuseppe Piazzi, the discoverer of Ceres. Oriani and Piazzi worked together for thirty-seven years, cooperating on many astronomical observations.

Given his strong interest in astronomy, Oriani was appointed on the staff of the Observatory of Brera in Milan in 1776, becoming assistant astronomer in 1778 and director in 1802. In 1778 he began publishing various in-depth dissertations on astronomical objects, the Effemeridi di Milano (Ephemerides of Milan).

A very capable astronomer, Oriani's work began to attract considerable attention. His research in the areas of astronomic refraction, the obliquity of the ecliptic and orbital theory were of considerable noteworthiness in themselves; but his greatest achievement was his detailed research of the planet Uranus, which had been discovered by Sir William Herschel in 1781. Oriani devoted significant time to observations of Uranus, calculating its orbital properties, which he published as a booklet of tables in 1793.

After others had shown that Uranus was not on a parabolic orbit but rather in a roughly circular orbit, he calculated the orbit in 1783. In 1789, Oriani improved his calculations by accounting for the gravitational effects of Jupiter and Saturn.

In addition to his continual contributions to the Effemeridi, he published a series of memoirs on spherical trigonometry: the Memorie dell' Istituto Italiano, 1806-10, and the Istruzione suelle misure e sui pesi, 1831.

For his work in astronomy, Oriani was honored by naming asteroid (4540) "Oriani". This asteroid had been discovered at the Osservatorio San Vittore in Bologna, Italy on November 6, 1988.

June 23: Alan Turing

Alan Mathison Turing

June 23, 1912 – June 7, 1954

Alan Turing, OBE, FRS was a British mathematician, logician, cryptanalyst and computer scientist.

Turing is often considered to be the father of modern computer science. He provided an influential formalisation of the concept of the algorithm and computation with the Turing machine. Of his role in the modern computer, Time Magazine in naming Turing one of the 100 most influential people of the 20th century, states:

"The fact remains that everyone who taps at a keyboard, opening a spreadsheet or a word-processing program, is working on an incarnation of a Turing machine."

With the Turing test, meanwhile, he made a significant and characteristically provocative contribution to the debate regarding artificial intelligence: whether it will ever be possible to say that a machine is conscious and can think. He later worked at the National Physical Laboratory, creating one of the first designs for a stored-program computer, the ACE, although it was never actually built in its full form. In 1948, he moved to the University of Manchester to work on the Manchester Mark 1, then emerging as one of the world's earliest true computers.

During the Second World War, Turing worked at Bletchley Park, Britain's codebreaking centre, and was for a time head of Hut 8, the section responsible for German naval cryptanalysis. He devised a number of techniques for breaking German ciphers, including the method of the bombe, an electromechanical machine that could find settings for the Enigma machine.

Near the end of his life Turing became interested in chemistry. He wrote a paper on the chemical basis of morphogenesis and he predicted oscillating chemical reactions such as the Belousov–Zhabotinsky reaction, which were first observed in the 1960s.

June 22: Hermann Minkowski

Hermann Minkowski

June 22, 1864 – January 12, 1909

Hermann Minkowski was a German mathematician of Jewish and Polish descent, who created and developed the geometry of numbers and who used geometrical methods to solve difficult problems in number theory, mathematical physics, and the theory of relativity.

Minkowski taught at the universities of Bonn, Göttingen, Königsberg and Zürich. At the Eidgenössische Polytechnikum, today the ETH Zurich, he was one of Einstein's teachers.

Minkowski explored the arithmetic of quadratic forms, especially concerning n variables, and his research into that topic led him to consider certain geometric properties in a space of n dimensions. In 1896, he presented his geometry of numbers, a geometrical method that solved problems in number theory.

In 1902, he joined the Mathematics Department of Göttingen and became one of the close colleagues of David Hilbert, whom he first met in Königsberg. Constantin Carathéodory was one of his students there.

By 1907 Minkowski realized that the special theory of relativity, introduced by Einstein in 1905 and based on previous work of Lorentz and Poincaré, could be best understood in a four dimensional space, since known as "Minkowski spacetime", in which the time and space are not separated entities but intermingled in a four dimensional space-time, and in which the Lorentz geometry of special relativity can be nicely represented. The beginning part of his address delivered at the 80th Assembly of German Natural Scientists and Physicians (September 21, 1908) is now famous:

The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.

The asteroid 12493 Minkowski, M-matrices and the Lunar crater Minkowski are named in his honor.