Eudoxus of Cnidus (410 or 408 BC – 355 or 347 BC) was a Greek astronomer, mathematician, scholar and student of Plato. Since all his own works are lost, knowledge of him is obtained from secondary sources, such as Aratus's poem on astronomy. Theodosius of Bithynia's Sphaerics may be based on a work of Eudoxus.

Life[link]

His name Eudoxus means "honored" or "of good repute" (in Greek Εὔδοξος, from eu "good" and doxa "opinion, belief, fame"). It is analogous to the Latin name Benedictus.

Eudoxus's father Aeschines of Cnidus loved to watch stars at night. Eudoxus first travelled to Tarentum to study with Archytas, from whom he learned mathematics. While in Italy, Eudoxus visited Sicily, where he studied medicine with Philiston.

Around 387 BC, at the age of 23, he traveled with the physician Theomedon, who according to Diogenes Laërtius some believed was his lover,^[1] to Athens to study with the followers of Socrates. He eventually became the pupil of Plato, with whom he studied for several months, but due to a disagreement they had a falling out. Eudoxus was quite poor and could only afford an apartment at the Piraeus. To attend Plato's lectures, he walked the seven miles (11 km) 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.

Around 368 BC, Eudoxus returned to Athens with his students. According to some sources, around 367 he assumed headship of the Academy during Plato's period in Syracuse, and taught Aristotle^{[citation needed]}. He eventually returned to his native Cnidus, where he served in the city assembly. While in Cnidus, he built an observatory and continued writing and lecturing on theology, astronomy and meteorology. He had one son, Aristagoras, and three daughters, Actis, Philtis and Delphis.

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.

His work on proportions shows tremendous insight into numbers; it allows rigorous treatment of continuous quantities and not just whole numbers or even rational numbers. When it was revived by Tartaglia and others in the 16th century, it became the basis for quantitative work in science for a century, until it was replaced by the algebraic methods of Descartes.

Eudoxus rigorously developed Antiphon's method of exhaustion, which was used in a masterly way by Archimedes. This method is a precursor to the integral calculus.

An algebraic curve (the Kampyle of Eudoxus) is named after him

a²x⁴ = b⁴(x² + y²).

Also, craters on Mars and the Moon are named in his honor.

Mathematics[link]

The Pythagoreans had discovered that the diagonal of a square does not have a common unit of measurement with the sides of the square; this is the famous discovery that the square root of 2 cannot be expressed as the ratio of two integers. This discovery had heralded the existence of incommensurable quantities beyond the integers and rational fractions, but at the same time it threw into question the idea of measurement and calculations in geometry as a whole. For example, Euclid provides an elaborate proof of the Pythagorean theorem, by using addition of areas instead of the much simpler proof from similar triangles, which relies on ratios of line segments.

Ancient Greek mathematicians calculated not with quantities and equations as we do today, but instead they used proportionalities to express the relationship between quantities. Thus the ratio of two similar quantities was not just a numerical value, as we think of it today; the ratio of two similar quantities was a primitive relationship between them.

Eudoxus was able to restore confidence in the use of proportionalities by providing an astounding definition for the meaning of the equality between two ratios. This definition of proportion forms the subject of Euclid's Book V.

In Definition 5 of Euclid's Book V we read:

Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.

Let us clarify it by using modern-day notation. If we take four quantities: a, b, c, and d, then the first and second have a ratio Failed to parse (Missing texvc executable; please see math/README to configure.): a/b

similarly the third and fourth have a ratio Failed to parse (Missing texvc executable; please see math/README to configure.): c/d

.

Now to say that Failed to parse (Missing texvc executable; please see math/README to configure.): a/b = c/d

we do the following:

For any two arbitrary integers, m and n, form the equimultiples m·a and m·c of the first and third; likewise form the equimultiples n·b and n·d of the second and fourth.

If it happens that m·a > n·b, then we must also have m·c > n·d. If it happens that m·a = n·b, then we must also have m·c = n·d. Finally, if it happens that m·a < n·b, then we must also have m·c < n·d.

Notice that the definition depends on comparing the similar quantities m·a and n·b, and the similar quantities m·c and n·d, and does not depend on the existence of a common unit of measuring these quantities.

The complexity of the definition reflects the deep conceptual and methodological innovation involved. It brings to mind the famous fifth postulate of Euclid concerning parallels, which is more extensive and complicated in its wording than the other postulates.

The Eudoxian definition of proportionality uses the quantifier, "for every ..." to harness the infinite and the infinitesimal, just as do the modern epsilon-delta definitions of limit and continuity.

Additionally, the Archimedean property stated as definition 4 of Euclid's book V is originally due not to Archimedes but to Eudoxus.^[2]

Astronomy[link]

In ancient Greece, astronomy was a branch of mathematics; astronomers sought to create geometrical models that could imitate the appearances of celestial motions. Identifying the astronomical work of Eudoxus as a separate category is therefore a modern convenience. Some of Eudoxus' astronomical texts whose names have survived include:

• Disappearances of the Sun, possibly on eclipses
• Oktaeteris (Ὀκταετηρίς), on an eight-year lunisolar cycle of the calendar
• Phaenomena (Φαινόμενα) and Entropon (Ἔντροπον), on spherical astronomy, probably based on observations made by Eudoxus in Egypt and Cnidus
• On Speeds, on planetary motions

We are fairly well informed about the contents of Phaenomena, for Eudoxus' prose text was the basis for a poem of the same name by Aratus. Hipparchus quoted from the text of Eudoxus in his commentary on Aratus.

Eudoxan planetary models[link]

A general idea of the content of On Speeds can be gleaned from Aristotle's Metaphysics XII, 8, and a commentary by Simplicius of Cilicia (6th century CE) on De caelo, another work by Aristotle. According to a story reported by Simplicius, Plato posed a question for Greek astronomers: "By the assumption of what uniform and orderly motions can the apparent motions of the planets be accounted for?" (quoted in Lloyd 1970, p. 84). Plato proposed that the seemingly chaotic wandering motions of the planets could be explained by combinations of uniform circular motions centered on a spherical Earth, apparently a novel idea in the 4th century.

In most modern reconstructions of the Eudoxan model, the Moon is assigned three spheres:

• The outermost rotates westward once in 24 hours, explaining rising and setting.
• The second rotates eastward once in a month, explaining the monthly motion of the Moon through the zodiac.
• The third also completes its revolution in a month, but its axis is tilted at a slightly different angle, explaining motion in latitude (deviation from the ecliptic), and the motion of the lunar nodes.

The Sun is also assigned three spheres. The second completes its motion in a year instead of a month. The inclusion of a third sphere implies that Eudoxus mistakenly believed that the Sun had motion in latitude.

The five visible planets (Venus, Mercury, Mars, Jupiter, and Saturn) are assigned four spheres each:

• The outermost explains the daily motion.
• The second explains the planet's motion through the zodiac.
• The third and fourth together explain retrogradation, when a planet appears to slow down, then briefly reverse its motion through the zodiac. By inclining the axes of the two spheres with respect to each other, and rotating them in opposite directions but with equal periods, Eudoxus could make a point on the inner sphere trace out a figure-eight shape, or hippopede.

Importance of Eudoxan system[link]

Callippus, a Greek astronomer of the 4th century, added seven spheres to Eudoxus' original 27 (in addition to the planetary spheres, Eudoxus included a sphere for the fixed stars). Aristotle described both systems, but insisted on adding "unrolling" spheres between each set of spheres to cancel the motions of the outer set. Aristotle was concerned about the physical nature of the system; without unrollers, the outer motions would be transferred to the inner planets.

A major flaw in the Eudoxan system is its inability to explain changes in the brightness of planets as seen from Earth. Because the spheres are concentric, planets will always remain at the same distance from Earth. This problem was pointed out in Antiquity by Autolycus of Pitane. Astronomers responded by introducing the deferent and epicycle, which caused a planet to vary its distance. However, Eudoxus' importance to Greek astronomy is considerable, as he was the first to attempt a mathematical explanation of the planets.

Ethics[link]

Aristotle, in The Nicomachean Ethics^[3] attributes to Eudoxus an argument in favor of hedonism, that is, that pleasure is the ultimate good that activity strives for. According to Aristotle, Eudoxus put forward the following arguments for this position:

1. All things, rational and irrational, aim at pleasure; things aim at what they believe to be good; a good indication of what the chief good is would be the thing that most things aim at.
2. Similarly, pleasure's opposite − pain − is universally avoided, which provides additional support for the idea that pleasure is universally considered good.
3. People don't seek pleasure as a means to something else, but as an end in its own right.
4. Any other good that you can think of would be better if pleasure were added to it, and it is only by good that good can be increased.
5. Of all of the things that are good, happiness is peculiar for not being praised, which may show that it is the crowning good.^[4]

See also[link]

• Incommensurable magnitudes

References[link]

• Evans, James (1998). The History and Practice of Ancient Astronomy. Oxford University Press. ISBN 0-19-509539-1. OCLC 185509676. 

• Huxley, GL (1980). Eudoxus of Cnidus p. 465-7 in: the Dictionary of Scientific Biography, volume 4. 

• Lloyd, GER (1970). Early Greek Science: Thales to Aristotle. W.W. Norton. 

Notes[link]

Further reading[link]

• De Santillana, G. (1968). "Eudoxus and Plato: A Study in Chronology". Reflections on Men and Ideas. Cambridge, MA: MIT Press. 
• Huxley, G. L. (1963). "Eudoxian Topics". Greek, Roman, and Byzantine Studies 4: 83–96. 
• Knorr, Wilbur Richard (1986). The Ancient tradition of geometric problems. Boston: Birkhäuser. ISBN 0-8176-3148-8. 
• Knorr, Wilbur Richard (1978). "Archimedes and the Pre-Euclidean Proportion Theory". Archives Intemationales d'histoire des Sciences 28: 183–244. 
• Neugebauer, O. (1975). A history of ancient mathematical astronomy. Berlin: Springer-Verlag. ISBN 0-387-06995-X. 
• Van der Waerden, B. L. (1988). Science Awakening (5th ed.). Leiden: Noordhoff. 

External links[link]

• Models of Planetary Motion—Eudoxus
• The Universe According to Eudoxus (Java applet)
• Eudoxus of Cnidus
• O'Connor, John J.; Robertson, Edmund F., "Eudoxus of Cnidus", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Eudoxus.html .
• Diogenes Laërtius, Life of Eudoxus, translated by Robert Drew Hicks (1925).
• Application of mathematical principles associated with Eudoxus
• Herodotus Project: Extensive B+W photo essay of Cnidus
• Dennis Duke, "Statistical dating of the Phaenomena of Eudoxus", DIO, volume 15, pages 7 to 23.

Sir Charles Hubert Hastings Parry, 1st Baronet (27 February 1848 – 7 October 1918) was an English composer, teacher and historian of music.

Parry's first major works appeared in 1880. As a composer he is best known for the choral song "Jerusalem", the coronation anthem "I was glad" and the hymn tune "Repton", which sets the words "Dear Lord and Father of Mankind". He was director of the Royal College of Music from 1895 until his death and was also professor of music at the University of Oxford from 1900 to 1908. He also wrote several books about music and music history. Some contemporaries rated him as the finest English composer since Henry Purcell, but his academic duties prevented him from devoting all his energies to composition.

Biography[link]

Early years[link]

Parry was born in Bournemouth, the youngest of six children of Thomas Gambier Parry (1816–1888) of Highnam Court, Gloucestershire, a painter, art collector and inventor of the "spirit fresco" process,^[1] and his first wife, Isabella née Fynes-Clinton (1816–1848). Three of their children died in infancy, and Isabella Parry died twelve days after the birth of her sixth child. Parry grew up at Highnam Court with his surviving siblings, (Charles) Clinton and Lucy. Thomas Parry remarried in 1851, and had a further six children.^[2]

Parry was sent to Twyford Preparatory School in Hampshire ^[3] and Eton, where his interest in music was encouraged and developed. At Eton he distinguished himself at sports as well as music, despite early signs of the heart trouble that was to dog him for the rest of his life. He took music lessons with Sir George Job Elvey, the organist of St George's Chapel, Windsor, and composed many prentice works.^[4]

While still at Eton Parry successfully sat the Oxford Bachelor of Music examination, the youngest person who had ever done so.^[2] His examination exercise, a cantata, O Lord, Thou hast cast us out, "astonished" the Oxford Professor of Music, Sir Frederick Ouseley, and was triumphantly performed and published in 1867.^[5] On going to Oxford after leaving Eton, Parry did not study music, being intended by his father for a commercial career, and instead read law and modern history. From 1870 to 1877 he was an underwriter at Lloyd's of London.^[4] In 1872 he married Lady Elizabeth Maude Herbert (1851–1933), second daughter of the politician Sidney Herbert and his wife Elizabeth. His in-laws, like his father, preferred a conventional career for him, although Parry proved as unsuccessful in insurance as he was successful in music.^[1] Parry and his wife had two daughters, their elder, Dorothea (d. 11 July 1963)^[6] married the politician Arthur Ponsonby in 1898^[7] and had a son and a daughter. Their younger daughter Gwendolen (b. 1877) married the baritone Harry Plunket Greene (1865–1936) and had two sons and a daughter.^[8]

Double harness[link]

Parry continued his musical studies alongside his work in insurance. He had already studied with the English-born composer Henry Hugo Pierson in Stuttgart, and on moving to London he took lessons from William Sterndale Bennett, but finding them insufficiently demanding^[2] he sought lessons from Johannes Brahms.^[2] Brahms was not available, and Parry was recommended to the pianist Edward Dannreuther, "wisest and most sympathetic of teachers".^[5] Dannreuther started by giving Parry piano lessons, but soon extended their studies to analysis and composition. At this stage in his musical development, Parry moved away from the classical conventions inspired by Mendelssohn. Dannreuther introduced him to the music of Wagner, which influenced his compositions of these years.

At the same time as his compositions were coming to public notice, Parry was taken up as a musical scholar by George Grove, first as his assistant editor for his new Dictionary of Music and Musicians, to which post Parry was appointed in 1875 and contributed 123 articles, and then, in 1883, Grove, as the first director of the new Royal College of Music, appointed him as the college's professor of composition and musical history.^[4]

Parry's first major works appeared in 1880: a piano concerto, which Dannreuther premiered, and a choral setting of scenes from Shelley's Prometheus Unbound. The first performance of the latter has often been held to mark the start of a "renaissance" in English music, but was regarded by many critics as too avant garde.^[5] Parry scored a greater contemporary success with the ode Blest Pair of Sirens (1887), commissioned by and dedicated to Charles Villiers Stanford, one of the first British musicians to recognise Parry's talent. Stanford described Parry as the greatest English composer since Purcell.^[5] Blest Pair of Sirens, a setting of Milton's "At a Solemn Musick", suggested as a text by Grove, established Parry as the leading English choral composer of his day; this had the drawback of bringing him a series of commissions for conventional oratorios, a genre with which he was not in sympathy.^[4]

Peak years[link]

Now well established as a composer and scholar, Parry received many commissions. Among them were choral works such as the Ode on Saint Cecilia's Day (1889), the oratorios Judith (1888) and Job (1892), the psalm-setting De Profundis (1891) and a lighter work, The Pied Piper of Hamelin (1905), described later as "a bubbling well of humour."^[5] The biblical oratorios were well received by the public, but Parry's lack of sympathy with the form was mocked by George Bernard Shaw, then writing musical criticism in London. He denounced Job as "the most utter failure ever achieved by a thoroughly respectworthy musician. There is not one bar in it that comes within fifty thousand miles of the tamest line in the poem."^[9]

Contemporary critics generally regarded Parry's orchestral music as of secondary importance in his output,^[10] but in the late twentieth and early twenty-first centuries many of Parry's orchestral pieces have been revived. These include five symphonies, a set of Symphonic Variations in E minor, the Overture to an Unwritten Tragedy (1893) and the Elegy for Brahms (1897). In 1883 Parry wrote music to accompany the Cambridge Greek Play The Birds by Aristophanes, a production which starred the mediaevalist and ghost-story writer, M. R. James. Parry received an honorary degree from Cambridge University in the same year.^[11] Subsequently, he wrote music for Oxford productions of Aristophanes: The Frogs (1892), The Clouds (1905) and The Acharnians (1914). He had also provided elaborate incidental music for a West End production by Beerbohm Tree Hypatia (1893). ^[12] In contrast to this involvement with music for the theatre, his only attempt at opera, Guenever was turned down by the Carl Rosa Opera Company.^[2]

When Grove retired as director of the Royal College of Music, Parry succeeded him from January 1895, and held the post until his death. In 1900 he succeeded John Stainer as professor of music at Oxford. An obituarist in 1918 lamented these calls on Parry's time: "A composer who counts is rare enough anywhere, any time. Do not try to use him as a mixture of university don, cabinet minister, city magnate, useful hack, or a dozen things besides. A great blow was delivered against English music when Parry was appointed to succeed Sir George Grove as director of the RCM."^[13] Despite the demands of these posts his personal beliefs, which were Darwinian and humanist, led him to compose a series of six "ethical cantatas", experimental works in which he hoped to supersede the traditional oratorio and cantata forms. They were generally unsuccessful with the public, though Elgar admired The Vision of Life (1907), and The Soul's Ransom (1906) has had several modern performances.

Influenced as a composer principally by Bach and Brahms, Parry evolved a powerful diatonic style which itself greatly influenced future English composers such as Elgar and Vaughan Williams. His own full development as a composer was almost certainly hampered by the immense amount of work he took on; but his energy and charisma, not to mention his abilities as a teacher and administrator, helped establish art music at the centre of English cultural life. As head of the Royal College of Music, he numbered among his leading pupils Ralph Vaughan Williams, Gustav Holst, Frank Bridge and John Ireland.^[1]

Parry was created a Knight Bachelor in 1898. He was made a baronet in 1902. The baronetcy became extinct on his death.^[2]

Books on music[link]

Parry wrote about music throughout his adult life. As well as his 123 articles in Grove's Dictionary, his publications include Studies of Great Composers (1886); The Art of Music (1893) enlarged as The Evolution of the Art of Music (1896); The Music of the Seventeenth Century, (Volume III of the Oxford History of Music (1902); Johann Sebastian Bach: the Story of the Development of a Great Personality (1909); and Style in Musical Art, collected Oxford lectures (1911).^[4]

Last years[link]

Parry resigned his Oxford appointment on doctor's advice in 1908 and, in the last decade of his life, produced some of his best-known works, including the Symphonic Fantasia '1912' (also called Symphony No. 5), the Ode on the Nativity (1912), Jerusalem (1916) and the Songs of Farewell (1916–1918). The piece by which he is best known, the setting of William Blake's poem "Jerusalem" mentioned above, was immediately taken up by the suffragette movement, with which both Parry and his wife were strongly in sympathy.^[2]

Parry held German music and its traditions to be the pinnacle of music, and was a friend of German culture in general. He was, accordingly, certain that Britain and Germany would never go to war against each other, and was in despair when World War I broke out. In the words of the Oxford Dictionary of National Biography: "During the war he watched a life's work of progress and education being wiped away as the male population, particularly the new fertile generation of composing talent—of the Royal College, dwindled."

In the autumn of 1918, Parry contracted Spanish flu during the infamous global pandemic and died at Knightscroft, Rustington, West Sussex, aged 70. At the suggestion of Stanford, he was buried in St Paul's Cathedral. The site of his birthplace, in Richmond Hill, Bournemouth, next door to The Square, is marked with a blue plaque, and there is a memorial tablet, with an inscription by the Poet Laureate, Robert Bridges, in Gloucester Cathedral, unveiled during the Three Choirs Festival of 1922.^[2]

Media[link]

	There rolls the deep (1.9 Mb)
Problems listening to this file? See media help.

Notes[link]

References[link]

• Boden, Anthony (1998). The Parrys of the Golden Vale. London: Thames Publishing. ISBN 0-905210-72-7. 
• Dibble, Jeremy (1992). C. Hubert H. Parry: His Life and Music. Oxford: Oxford University Press. ISBN 0-19-315330-0. 

External links[link]

Wikimedia Commons has media related to: Hubert Parry

Wikisource has original works written by or about: Charles Hubert Hastings Parry

• Free scores by Hubert Parry in the Choral Public Domain Library (ChoralWiki)
• Free scores by Hubert Parry at the International Music Score Library Project