Born: about 160 BC in Bithynia, Anatolia (now Turkey)
Died: about 90 BC

Theodosius of Bithynia was for a long time thought to have been born in Tripolis. The reason for this comes from an error in the Suda Lexicon (a work of a 10^th century Greek lexicographer) which states that Theodosius was a (see for example [1]):-

... philosopher [who] wrote "Sphaerics" in three books, a commentary on the chapter of Theudas, two books "On Days and Nights", a commentary on the "Method of Archimedes", Descriptions of Houses in three books, "Skeptical Chapters", "Astrological works", "On Habitations". Theodosius wrote verses on the spring and other types of works. He was from Tripolis.

We are indeed interested in this article in the Theodosius who wrote Sphaerics in three books. He must have lived close to 100 BC. Yet Theudas who is next referred to in the Suda was a sceptic philosopher of the second century AD so we see immediately that there is an error. It would now appear that, apart from the work on Theudas and Skeptical Chapters which was almost certainly written by the same person (one assumes another author by the name of Theodosius), the rest of the entry is correct except for the final two sentences "Theodosius wrote verses on the spring and other types of works. He was from Tripolis." which must refer to what one has to assume is yet a third author called Theodosius.

So Theodosius was the author of Sphaerics, a book on the geometry of the sphere, written to provide a mathematical background for astronomy. It is thought that Sphaerics is based on some pre-Euclidean textbook which is now lost. It is conjectured, on rather little evidence one would have to say, that Eudoxus wrote this earlier text. There seems to be no way in which the speculation on this point can ever be settled.

Sphaerics contains no trigonometry although it is likely that Hipparchus introduced spherical trigonometry before Sphaerics was written (although, one has to assume, after the book on which Sphaerics is based, which would certainly be the case if this earlier book was written by Eudoxus). Sphaerics was written to supplement Euclid's Elements in particular to make up for the lack of results on the geometry of the sphere in Euclid's work.

Theodosius defines a sphere to be a solid figure with the property that any point on its surface is at a constant distance from a fixed point (the centre of the sphere). He gives theorems which generalise those given by Euclid in Book III of the Elements for the circle. The second book of Theodosius's work considers touching circles on a sphere. It then goes on to consider geometry results which are relevant to astronomy and these continue to be studied through Book III. Heath writes [2]:-

It is evident that Theodosius was simply a laborious compiler, and that there was practically nothing original in his work.

Perhaps it is worth remarking that despite our comment above that the work contains no trigonometry, there are some results which we could easily interpret in trigonometrical terms. For example Theodosius proves that for a spherical triangle with angles A, B, C (C a right angle) and sides a, b, c where side a is opposite angle A, etc. then

tan a = sin b tan A.

Neugebauer, in [3], is highly critical of the Sphaerics calling it dull and pedantic only surviving because it was used as a textbook. More specifically Neugebauer writes:-

Theodosius comes nowhere near recognising the fundamental importance of the great-circle triangle and his theorems rarely go beyond the geometrically obvious in the relations between a few special great circles and their parallels, without ever mentioning that one is dealing with configurations of interest to astronomy.

Two other works by Theodosius have survived in the original Greek. These are On habitations containing 12 theorems and On days and nights. The first of these explains the views of the universe due to the rotation of the Earth and, in particular, it considers how the view is affected by the different places on the Earth in which people live.

Theodosius considers the length of the night and day at various points on the earth and claims that the day lasts for seven months at the north pole and the night for five months. On moving south one reaches the circle where at the summer solstice the day is 30 days long. Of course this is very strange and partly explained by Theodosius's definition of night as period of darkness and day as a period of light. Theodosius considered that it was 'day' if the sun was less than 15° below the horizon for then no stars were visible and he seemed to fail to understand that in the polar regions the sun can move almost parallel to the horizon.

The other work On days and nights is in two books, the first of which has 13 propositions, the second 19 propositions, which give conditions on the lengths of the night and day depending on the location of the observer. Theodosius also considers the two possibilities, that the length of the year is a rational multiple of the length of the day and that it is an irrational multiple.

Neugebauer [3] makes some interesting comments on the diagrams in ancient texts and how they may have been totally changed by both early editors and even by modern editors. Referring to Theodosius's On days and nights he says that:-

... errors occur in diagrams. Letters are easily misplaced or sometimes an arc may be missing but by and large figures are well drawn. In many cases the extant diagrams show an axial symmetry which is not wrong but which is not required by the theorem or proof in question. Such symmetries detract ... from the general validity of the proposition. It is impossible to tell if such symmetrisations, caused either by the greater simplicity of construction or its aesthetic appeal, belong to the archetype or are copyist or editorial "improvements".

Of the other works mentioned in the Suda we have no reason to doubt that Theodosius wrote a commentary on the Method by Archimedes but there is no other evidence to prove whether this is correct or not.

1. T L Heath, A History of Greek Mathematics (2 Vols.) (Oxford, 1921).
2. O Neugebauer, A history of ancient mathematical astronomy (New York, 1975).
3. J Tropike, Geschichte der Elementar-Mathematik V (Berlin, 1923).

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