**Aristaeus the Elder** was probably older than, but still a contemporary of, Euclid. We know practically nothing of his life except that Pappus refers to him as Aristaeus the Elder which presumably means that Pappus was aware of another later mathematician also named Aristaeus. We have no record of such a person but we do point out below a possible confusion which may result from there being two mathematicians called Aristaeus.

Pappus gave Aristaeus great credit for a work entitled *Five Books concerning Solid Loci* which was used by Pappus but has now been lost. 'Solid loci' is the Greek name for conic sections so it is rather confusing that there is another reference by a later writer to a work by Aristaeus called *Five Books concerning Conic Sections*. However these two works are now thought to be the same.

Pappus describes the work as:-

... five books of Solid Loci connected with the conics.

and also claims (if this is not a latter addition to the text) that Euclid compiled elementary results on conics in his treatise *Conics* while Aristaeus's results, much deeper, original and specialised, were not included by Euclid who preferred to leave them in their original presentation due to Aristaeus.

Heath makes a guess at the possible contents of the *Solid Loci* and writes [3]:-

A very large portion of the standard properties of conics admit of being stated in the form of locus theorems ... But it may be assumed that Aristaeus's work was not merely a collection of the ordinary propositions transformed in this way; it would deal with new locus theorems not implied in the fundamental definitions and properties of the conics, such as ... the theorems of the three- and four-line locus. But one(to us)ordinary property, the focus directrix property, was, as it seems to me, in all probability included.

Heath refers to theorems of the three- and four-line locus in the above quote and we should explain what these are. For the three line locus we are given a point *P* and three directed lines *a*, *b*, and *c* drawn to meet at given angles, three fixed straight lines. Then the locus of *P* such that *ac* : *b*^{2} is a given constant is a conic. The four-line locus is similar. We are given a point *P* and four directed lines *a*, *b*, *c*, and *d* drawn to meet at given angles, four fixed straight lines. Then the locus of *P* such that *ac* : *bd* is a given constant is a conic.

There is a reference to Aristaeus in the works of Hypsicles where he refers to Aristaeus as the author of a book *Concerning the Comparison of Five Regular Solids.* Heath believes that, although it is not certain whether this is Aristaeus the Elder, the results described make it quite probable that it is. Hypsicles tells us that, in this work, Aristaeus proved that [3]:-

... the same circle circumscribes both the pentagon of the dodecahedron and the triangle of the icosahedron inscribed in the same sphere.

Heath's opinion that the Aristaeus referred to by Hypsicles this is Aristaeus the Elder has been disputed by some historians, and there is a possibility that Hypsicles refers to Aristaeus the Younger thus making sense of Pappus's comments which we referred to in the first paragraph.

The work of both Aristaeus and Euclid on conics was, almost 200 years later, further developed by Apollonius. This work by Apollonius made the theory of conics as developed by Aristaeus and Euclid obsolete.

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

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