Very little is known of **Serenus**'s life. In fact the article [4] claims that Serenus was born in Antissa but this has been shown by more recent historians of mathematics to be based on an error. That he was born in Antinoupolis is confirmed from two sources. The information has been added to one of the manuscripts of his works by at a later stage but we have no reason to doubt the authority of the addition. It can also be deduced from a copy of the second treatise of Serenus which has survived.

Serenus was a commentator on the texts of others but, unlike some commentators, he was a fine mathematician in his own right. He wrote two original mathematical works which show that he was indeed a mathematician of considerable ability.

The two treatises by Serenus are *On the Section of a Cylinder* and *On the Section of a Cone* both of which have survived. In the preface to the first of these Serenus gives his reasons for writing the work [2]:-

... many persons who were students of geometry were under the erroneous impression that the oblique section of a cylinder was different from the oblique section of a cone known as an ellipse, whereas it is of course the same curve.

The work consists of 33 propositions. Typical of these are the following two problems.

(i) Suppose we are given a cone and an ellipse *E* on the cone. Serenus shows how to find the cylinder which is cut in the ellipse *E*.

(ii) Given a cone, find a cylinder so that when both are cut by the same plane the sections of the cuts form similar ellipses.

The final five propositions, involving rays of light, are designed to support his friend Peithon who wrote a tract giving what he considered a better definition of parallels to that given by Euclid. It appears that Peithon's work treated as a bit of a joke and Serenus tries to in these propositions to show that Peithon's ideas are mathematically sound. Peithon [1]:-

... had defined parallels to be such lines as are cast on a wall or a roof by a pillar with a light behind it.

As Heiberg comments in [3], even though Greek geometry was in decline by this time mathematicians were sufficiently knowledgeable to find this definition funny.

In the first 57 propositions in *On the Section of a Cone* Serenus examined triangular sections of right and scalene cones made by planes passing through the vertex. He also considered some problems relating to maximising areas. The remaining 12 propositions deal with the volumes of right cones given its height, its base, and the area of a triangular section through its axis.

Serenus wrote a commentary on Apollonius's *Conics* unfortunately is lost, except for a fragment preserved by Theon of Smyrna. That he wrote such a work is confirmed by Serenus in his own writings. The result described by Theon of Smyrna is introduced with the words (see for example [1]):-

From Serenus the philosopher out of the lemmas.

The result is that if a number of angles are subtended at points on a diameter of a circle so that the arcs of the circle subtended by the angles are all equal, then the closer to the centre of the circle is the point on the diameter, the greater is the angle.

**Article by:** *J J O'Connor* and *E F Robertson*

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