Proclus Diadochus: Poet,
Philosopher, and Scientist.
"Wherever
there is number, there is beauty."
Born: 8 Feb 411 in Constantinople
(now Istanbul), Byzantium (now Turkey)
Died: 17 April 485 in Athens, Greece
Proclus's father, Particius, and his mother, Marcella, were citizens of high social position in Lycia. Particius was a senior law official in the courts at Byzantium. Proclus was brought up at Xanthus, on the south coast of Lycia, where he attended school.
It was intended that Proclus should follow his father and enter the legal profession. With this aim in mind he was sent to Alexandria but, while in the middle of his studies, he visited Byzantium and he became convinced that his calling in life was the study of philosophy. He returned to Alexandria where now he studied philosophy under Olympiodorus the Elder, in particular making a deep study of the works of Aristotle. He also learnt mathematics in Alexandria .
Not entirely satisfied with the education he was receiving in philosophy in Alexandria, while still a teenager, he moved from Alexandria to Athens where he studied at Plato's Academy under the philosophers Plutarch and Syrianus (a pupil of Plutarch). He progressed from being a student at the Academy to teaching there. On the death of Syrianus, Proclus became head of the Academy. The title Diadochus was given to him at this time, the meaning of the word being successor.
He never married, did not eat meat and tried to live a religious life, composing hymns to the gods. His hymns were clearly highly thought of since seven of them have been preserved and are seen today as having considerable literary merit. Proclus was to remain as head of the Academy until his death.
Proclus was a capable mathematician and wrote Commentary on Euclid which is our principal source about the early history of Greek geometry. At the Academy he had access to books which are now lost and others( already lost during his time) were described based on extracts in other books available to him. In particular. he used the History of Geometry by Eudemus, and the works of Geminus both of which are now lost. Heath, describing Proclus's Commentary on Euclid writes, ”Proclus deals historically and critically with all the definitions, postulates and axioms in order. The notes on the postulates and axioms are preceded by a general discussion of the principles of geometry, hypotheses, postulates and axioms, and their relation to one another”.
Proclus was not a creative mathematician; but he was an acute expositor and critic, with a thorough grasp of mathematical method and a detailed knowledge of the thousand years of Greek mathematics from Thales to his own time. The earliest source of information on attempts to prove the fifth postulate is the commentary of Proclus on Euclid's Elements. Proclus lived more than 700 years after Euclid. He mentions Ptolemy's (2nd century) attempts to prove the postulate and demonstrates that Ptolemy had unwittingly assumed what in later years became known as Playfair's axiom. Proclus left a proof of his own, but the latter rests on the assumption that parallel lines are always a bounded distance apart, and this assumption can be shown to be equivalent to the fifth postulate. There is one instance in which he attempts to alter a "difficulty " he finds in Euclid's Elements. This difficulty is what we commonly refer to as the "parallel postulate”.
The statement Proclus proves instead of the parallel postulate is, "Given a + b < 2d , prove that the straight lines g' and g'' meet at a certain point C."
In his proof of this, Proclus draws a straight line, g''' through a given point a parallel to g'. Then taking a point B on g '' he drops a perpendicular to g ''' from it. From this he reasons that since the distance from g ''' increases without limit as the distance between A and B grows and the distance between g' and g''' is constant then there must be a point C on g'' belonging to g'. And it is this point where g' and g'' meet, thus completing his proof. However, as with most of the other alternatives to the parallel postulate, this one had faults. It is observed by Pogorelov that the parallel straight lines this proof relies on are not explicitly contained in the other postulates or axioms and therefore cannot be deduced from them.
Proclus’ axiom: If a line intersects one of two parallel lines, both of which are coplanar, with the original line, then it must intersect the other also. This axiom is equivalent to the parallel postulate.
Mathematics played a central role in his Elements of Theology. The central principles of limit and unlimited have a universal provenance in Proclus’ whole system of reality. He compared Geometrical forms to Soul. He said the diagrams used (products of the imagination) are really projections by the higher intellect onto a lower level so as to facilitate the study of geometrical objects. He opined that in the mathematical order of being there are ratios proceeding to infinity but controlled by the principle of limit. For example, a number can be increased indefinitely, yet any number by itself is finite.
Some other works of Proclus: Although more
highly regarded as a systematizer and commentator than as an original thinker, Proclus was the author of numerous
philosophical and non -philosophical writings, including astronomical,
mathematical and grammatical works. He wrote seven hymns and two epigrams, one
of which is on his tomb.
1. Proclus wrote Hypotyposis, an introduction to the astronomical theories of Hipparchus and Ptolemy in which he described the mathematical theory of the planets based on epicycles and on eccentrics. He combined his geometrical skills and his knowledge of astronomy to give a geometrical proof that the epicycle theory for the planets is equivalent to the eccentric theory. In the epicycle theory the Earth is in the centre of a circle which has smaller circles rotating round its circumference. In the eccentric theory the planets move round in circles whose centres do not coincide with the Earth.
2. In his astronomical writings, Proclus described how the water clock invented by Heron could be used to measure the apparent diameter of the Sun. Proclus's method can be used at the equinox. Water is collected from the clock in a container while the sun rises. As soon as the Sun has risen the water is collected in another container and this measurement continues until sunrise the following day. Then the ratio of the weights of water in the two containers gives the apparent diameter of the Sun.
Among Proclus's many works are: Liber de causis (Book of Causes), Institutio theologica (Elements of Theology), a concise exposition of metaphysics, Elements of Physics, largely giving Aristotle's views, and In Platonis theologiam (Platonic Theology) giving Plato's metaphysics.
Proclus deserves to be
remembered ... for the qualities he possessed that are exceedingly rare in any
age and were almost unique in his: the logical clarity and firmness of his
thought, the acuteness of his analyses, his eagerness to understand and
readiness to present the views of his predecessors on controversial issues, the
sustained coherence of his lengthy expositions, and the large horizon, as broad
as the whole of being, within which his thinking moved.
Some of Proclus’s quotations
This, therefore, is mathematics:
she reminds you of the invisible form of the soul;
she gives light to her own discoveries;
she awakens the mind and purifies the intellect;
she brings light to our intrinsic ideas;
she abolishes oblivion and ignorance which are ours by birth
References:
http://www-history.mcs.st-andrews.ac.uk/Biographies/Proclus.html
http://www.math.rutgers.edu/~cherlin/History/Papers2000/eder.html
http://www.mathworld.wolfram.com/ProclusAxiom.html
http://www.bu.edu/wcp/Papers/Anci/AnciClea.htm
Sites on Proclus
http://www.esotericarchives.com/proclus/metaelem.htm#p30
http://www.hiw.kuleuven.be/dwmc/plato/proclus/probiblio.htm
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Proclus.html
Interesting and informative page, dedicated to Proclus as a mathematician and
astronomer.
Proclus
Page at the Shrine of
the Goddess Athena - includes, biography, list of works, introduction
to Elements of Theology. An introduction The Theology of Plato is in the
works. Excellent
Proclus' Elements -
some interesting observations on Plotinus' argument for the existence of an
immortal soul in Elements
of Theology, from Metaphysics by
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