Tag Archives: mathematics

Samos Island

By Kathleen Mcilvenna

Samos Coast
Samos Coast

Lying in the eastern Aegean Sea and off the coast of Turkey is an unassuming Greek island called Samos. It could be easily dismissed as just another beautiful Greek island, abundant with lush olive groves and secluded bays, but Samos has a surprising history and a legacy that every British GCSE student can appreciate.

With such a strategically important position Samos’ history has been somewhat turbulent, but loyally Greek. The island has been ruled by a number of different empires, and after the fall of the Byzantine Empire and a number of earthquakes the island was left virtually empty by the start of the sixteenth century. Samos was slowly re-inhabited which resulted in many towns adopting names related to its populations’ place of origin. However the island’s national identity developed into aligning itself with Greece. This meant that the condition of the London Treaty of 1830, that brought an end to the Greek War of Independence but gave Samos over to the Ottoman Empire, was strongly opposed in Samos. A revolutionary movement reached its peak in 1912 and Samos was officially united with Greece in 1913.

With such an unsettled history it’s surprising that so many features from Samos’ ancient past still survive. These include the ruins of the temple of Hera at Ireon and the tunnel of Eupalinos near the town of Pythagoreion, both declared UNESCO world heritage sites in 1992. Samos is said to be the birthplace of the Greek goddess Hera, and the site of her ancient temple dates back to the eighth century BC and was later also the site of a Christian basilica. Most of the sites artefacts are in the Archaeological Museum in Samos’ modern capital, Vathy, though other artefacts are also housed in Pythagorio’s Archaeological Museum.

The Eupalinos tunnel was rediscovered in the nineteenth century and is a popular tourist attraction today. It dates back to the sixth century BC and is an astounding piece of Ancient engineering. Acting as an underground aqueduct, it was designed to transport fresh water from an inland spring to the ancient coastal capital, today called Pythagorio. What is notable about this tunnel, apart from its ancient origins, is that is one of only two that are known to have been dug out from both ends to meet in the middle, and was the first to do so with a geometric approach. Commissioned by Samos’ tyrant leader, Polycrates, the tunnel was designed by Eupalino from Megara, and took ten years to build. Its achievement was even acknowledged by the ancient historian Herotodus.

Pythagoras Monument in Pythagorio
Pythagoras Monument in Pythagorio

Though these ancient echoes invite interested modern visitors, Samos is most famous for one of its Ancient Greek sons, Pythagoras. Pythagoras was born in Samos in 580 BC and was a philosopher and Mathematician. His triangle theorem, regarding right-angled triangles, is still an important part of GCSE maths courses in Britain. Pythagoras didn’t stay in Samos but his link is greatly celebrated, the town of Pythagorio is named after him and has a commemorative monument. Furthermore the ‘Just Cup’, reported to be a design by Pythagoras as he got fed up with the inaccuracy of measurements, are on sale throughout the island.

Like most of the Greek islands, tourism is important to their economy, and Samos’ legacy to the history of science is at the heart of its offer to tourists today. I’ve only covered a small part, and among the numerous small towns there are a number of museums dedicated to all sorts of topics. These include the Natural History Museum, the Folklore Museum and the Museum of Wine. With such a glorious climate and such friendly and generous inhabitants I would heartily recommend a visit.

View from Pythagorio with Greek flag
View from Pythagorio with Greek flag

Torun, Poland

By Charles Tanford & Jacqueline Reynolds

Copernicus Statue, Torun

The birthplace of Copernicus is a picturesque town on the Vistula River and provides some of the best examples of Gothic urban architecture in Central Europe. (And it’s famous for its gingerbread, too.) The tower on the Town Hall dates from 1274 and is the oldest in Poland. The house in which Copernicus was born is now a museum, devoted to his life and work.

Further information on Nicolaus Copernicus House Museum: http://www.visittorun.pl/301,l2.html

The Stadt Friedhof, Gottingen

By Charles Tanford & Jacqueline Reynolds

Otto Hahn's gravemarker, Stadt Friedhof

The most conspicuous memorial site in Gottingen is a cemetery, the Stadt Friedhof, located on the road to Kassel. There is a scientists’ corner here, where many famous scientists who worked or studied in Gottingen are buried close together. They include Max Planck, the original discoverer of the need for energy quantization; Otto Hahn, one of the authors of the famous paper on the splitting of the atom; Walther Nernst and his entire family: and several more. Hahn’s tombstone bears an enigmatic, perhaps ominous inscription:

92U + on


The top line is standard chemical language for the reaction of an atom of uranium (isotope of mass 92) with a neutron. But how are we to interpret the down-pointing arrow? The end of the world or maybe descent into hell?

Max Born is buried with his wife in a totally different part of the cemetery, the family plot of his wife and her forebears. His epitaph, too, is in the form of an equation, a mathematical formula in this case: pq – qp = hI27ri, and what will strike the layman about it is the fact that pq – qp is not zero, as he would expect. It turns out that p and q stand respectively for the momentum and the position of a particle in space and the significance of the inequality of their forward and reverse products is the underlying basis for Heisenberg’s uncertainty principle. This may be Born’s claim to posterity for at least an equal share of the credit.

There is an amusing anecdote about the interment of Walther Nernst, a none too popular physical chemist (but sufficiently proficient to have won a Nobel Prize in 1920). He died in 1945 on his estate in East Prussia and was buried there, with two colleagues, Karl Bonhoeffer and Max Bodenstein, serving as pallbearers. When the Russians annexed East Prussia, the remains were removed to German soil (to Berlin) and there WaS another ceremony with Bonhoeffer and Bodenstein again in attendance. Some years later the family thought he should really lie in Gottingen, where he had been professor for most of his career, and so the body was moved once more, still with the same honorary escort. “I’m getting tired of this,” Bodenstein is reported to have remarked to his partner, who, however, responded more cheerfully: “You can’t bury Nernst too often” was his reported reply.

Emmy Noether’s Grave, Philadelphia

By Paul Halpern

Emmy Noether’s grave, the Cloisters, Bryn Mawr
Emmy Noether’s grave, the Cloisters, Bryn Mawr

One of the great intellectual heroes of the 20th century was German-Jewish mathematician Emmy Noether. Born in 1882, she was remarkable in that her accomplishments in the field of abstract algebra emerged despite considerable prejudice against her, first because of being a woman, and second because of her ethnic background. She taught in the 1920s and early 1930s at the University of Goettingen. Then in 1933, with the rise of the Nazi regime, an act was passed “The Law of the Restoration of the Civil Service” forbidding those of Jewish background to teach in Germany, unless they had been World War I veterans (a concession made to placate Hindenburg). Noether fled Germany and obtained a position at Bryn Mawr College in the US. She died two years later.

Einstein wrote a beautiful obituary about her in the New York Times:

“In the judgment of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began. In the realm of algebra, in which the most gifted mathematicians have been busy for centuries, she discovered methods which have proved of enormous importance in the development of the present-day younger generation of mathematicians. Pure mathematics is, in its way, the poetry of logical ideas.”
-Albert Einstein, New York Times, May 1, 1935.

Emmy Noether’s grave is in a quiet, monastery-like part of Bryn Mawr campus, known as The Cloisters.

Emmy Noether’s simple grave marker
Emmy Noether’s simple grave marker

The grave marker, with her initials and years of birth and death only, is very plain and right in the pavement.

Map location (Emmy Noether’s grave, Bryn Mawr College): 101 N Merion Ave, Bryn Mawr, PA, USA

This article is adopted from a piece posted on the Philadelphia Area Center for the History of Science (PACHS) blog. Photos by Aden Halpern.

Related link: Emmy’s Noether’s birthplace in Germany by Thony Christie

David Rittenhouse in Philadelphia

Ask Philadelphians what is the fanciest address in the city and they are likely to say Rittenhouse Square. Located at the intersection of Walnut Street and 19th Street, Rittenhouse Square houses many posh hotels and restaurants. Few locals know, however, that the prime location is named after astronomer and mathematician David Rittenhouse.

David Rittenhouse was born in 1732 in an early industrial community, set on a stream, that was then outside the boundaries of Philadelphia. Later incorporated into the city, the enclave is now called Historic RittenhouseTown.

Circa 1702 Mill, photographed in 1890
Circa 1702 Mill, photographed in 1890, courtesy of Historic RittenhouseTown.

The story of RittenhouseTown dates back to the 17th century, when German-born papermaker William Rittenhouse emigrated from Holland to Philadelphia and established the first paper mill in the Colonies. Because of its success, a second mill and bakehouse were constructed, along with the Rittenhouse Homestead. Generations of papermakers, weavers, and other industrial workers lived in the community supplying important products for the colonies.

Historic RittenhouseTown is open for tours. Surrounded by parkland, it is located at 206 Lincoln Drive in the northwest part of Philadelphia. It is open summer weekends from 12:00 pm to 4:00 pm. It is also open on weekdays, if arranged in advance, for groups of 10 or more.

Rittenhouse Homestead, photographed in 2006
Rittenhouse Homestead, photographed in 2006, courtesy of Historic RittenhouseTown.

The great-grandson of William, young David demonstrated great mathematical and scientific prowess, studying Newton’s Principia on his own. He became adept at building mechanical devices and established his own clock-making and instrument-making business.

Combining his scientific interests, particularly in astronomy, with his mechanical skills and craftsmanship, Rittenhouse set out in 1767 to build an orrery: a machine replicating the motions of the planets and moons in the Solar System using Kepler’s laws as a guide. He also constructed a modified refracting telescope to record the transit of Venus.

In 1786, politician Francis Hopkinson, a friend of Rittenhouse, sent him an intriguing question, “why… when he looked through a fine silk handkerchief at a light source, did he see a grid of dark lines which did not move at all, even though he moved the handkerchief back and forth?” (Hindle, p. 276)

After repeating Hopkinson’s experiment, Rittenhouse decided to resolve the issue by developing what became the first diffraction grating. By placing fine hairs parallel to each other, he constructed a grating with about 250 lines per inch. He then turned to the subject of precision timekeeping and astronomical measurement, constructing the first collimating telescope.

Rittenhouse was honoured much in his life. In the 1780s he was appointed the University of Pennsylvania’s first Professor of Astronomy and Vice-Provost. In 1791, one year after the death of Franklin, Rittenhouse was elected to be the second president of the American Philosophical Society. Rittenhouse held that position five years, until his own death. In 1825, Philadelphia renamed what was previously called Southwest Square after him, and Rittenhouse Square soon became known as one of the fanciest locations in the city. The physics and mathematics building at the University of Pennsylvania is named David Rittenhouse Laboratory in his honour.

Map location (Historic RittenhouseTown): 206 Lincoln Drive, Philadelphia, PA 19144, USA


I thank Chris Owens, Director of Historic RittenhouseTown, for his help and for supplying the photos for this article.

Parts of this article are adopted from “Philadelphia: Life, Liberty, and the Pursuit of Physics,” by Paul Halpern, published in Physics in Perspective.


  1. Brooke Hindle, David Rittenhouse, (Princeton,: Princeton University Press, 1964)
  2. Paul Halpern, “Philadelphia: Life, Liberty, and the Pursuit of Physics,” Physics in Perspective 11, No. 2, (2009), pp. 209-227.
  3. Historical RittenhouseTown website: http://www.rittenhousetown.org

Island of Samos, Northern Aegean

By Charles Tanford & Jacqueline Reynolds

This article has been superseded by a more up-to-date article by Kathleen McIlvenna at http://www.bshs.org.uk/travel-guide/samos-island.

Island of Samos

This is the island of philosopher/mathematician Pythagoras and of Aristarchus, who lived 300 years later, when Samos was under control of first the Egyptians and later the Syrians; he is famous for being the first proponent of a heliocentric system of planets and stars. The island is physically attractive, with a backbone of mountains reaching to nearly 5,000 feet (1,500 m) above sea level. There are wooded pine forests on the mountain slopes, vineyards below, and sandy coves for bathing at the seashore. The island has been less spoiled by tourism than many others. We can still see islanders tending herds of goats or carting their produce to market on the backs of donkeys.

The ancient capital of Samos, formerly Tigani, was renamed Pythagorio in 1955, in honor of the island’s famous native son. There is a tiny museum in the town hall, space shared with the mayor’s offices, but nothing within is about Pythagoras-all we have is his bust on a pedestal outside. An adjacent street is named for Aristarchus, but there are no plaques to proclaim his espousal of heliocentricity or to explain how he came to the idea. All we can do is to wander around the island, imagining the astronomer doing likewise, dreaming up new geometrical methods for measuring distances and sizes of the objects he saw in skies. Strangely enough, our imagination gets some help, for on a hill above Pythagorio is the Tunnel of Eupalinus, 3,385 feet (1,026 m) in length and tall enough for a man to stand within it. It was built during the reign of the island’s most ambitious ruler, polycrates, not long after the time of Pythagoras. It used to have pipes on the floor to carry water to the town from springs on the other side of the hill. The digging was done by two teams of workers, one from each side, and they met properly in the middle. Pretty good geometry for 525 B.C.!

Also relevant to our story is a short drive or taxi ride east from the town of Samos, to the Strait of Mykali, for here we see the Turkish mainland not much over a mile (2 km) away, almost within swimming distance. And at this point we are but 33 miles (50 km) from Miletus, the earliest of all sources of Greek scientific philosophy. Samos may be formally an island in the geographic sense, but at the time of Pythagoras it was far from insular in its intellectual life, being always in close communication with the mainland coastal cities. Today we have Greeks on one side of the Strait and Turks on the other, and not much love between them, but back then it was all Greek-the center of Greek civilization, in fact, for the age of Pericles in Athens was still some decades away.