Search

Whether or not it was deserved—and it may well have been deserved—Halley gives Aldrich the credit for pointing the way on how to reconstruct Book VIII. In the preface to the reconstruction, Halley recounts Aldrich’s observation that “…in Pappus’ Mathematical Collection, [Pappus] himself passed on lemmas serving [what] was to be demonstrated in the seventh and the eighth book of the Conics at the same time; whereas in the other books, different [lemmas] to the different [books] are provided. Hence, to you [Aldrich] we owe the discovery that the [two] books are conjoined; that the problems in [Book] eight are allotted their determination from the theorems on diorisms () in [Book] seven.” Surely, by the time Halley wrote this, he would have known the connection between the contents of Books VII and VIII which Apollonius himself points out in the preface to Book VII, as described above. Still, Aldrich’s observation regarding Pappus may truly have been the trigger that set the project going.

When I had set out to publish the [Conics] of Apollonius, it disturbed us not to a small degree that in the Arabic Codex the last book of the Conics was missing. But you, as it is by [your] happy nature, felt at once that there is no need to deplore [the loss], but that surely to some degree it could be restored, given the fact that in Pappus’ Mathematical Collection, [Pappus] himself passed on lemmas serving [what] was to be demonstrated in the seventh and the eighth book of the Conics at the same time; whereas in the other books, different [lemmas] to the different [books] are provided.

1.1

IN every problem in which it is asked to investigate a locus, be this on a straight line or on a curve, one arrives at an equation determining an arbitrarily chosen point on the required locus [1.1] if one considers as known and determinate two unknown and indeterminate line segments enclosing a given or chosen angle [1.2].

From Apollonius to Attalus. Peace be on you. I have sent to you with this letter of mine the seventh book of the treatise on Conics. In this book are many wonderful and beautiful things on the topic of diameters and the figures constructed on them, set out in detail. All of this is of great use in many types of problems,1 and there is much need for it in the kind of problems which occur in conic sections which we mentioned,2 among those which will be discussed and proven in the eighth book of this treatise (which is the last book in its. I shall strive to send that to you speedily. Farewell.

From Apollonius to Attalus, peace be on you. In this fifth book I have composed propositions1 on the maximum and minimum lines. You should realize that our predecessors and contemporaries paid (a little) attention only to the minimum [lines]: they proved thereby which straight lines are tangent to the sections and also the reverse, i.e. what properties are possessed by the tangents to the sections such that when those properties are possessed by lines they are tangents. But as for us, we have proven those things in Book 1 without making use, in our proof of that, of the topic of minimum lines; for we wanted to make the place where those [things] were put near to our discussion of the derivation of the three sections, in order to show in this way that in each of the sections there may occur an infinite number of properties and necessities of these things, as is the case with the original diameters.2

In this chapter I explain the concepts and terminology in the extant Books of the Conics of Apollonius (fl. 200 B.C.; DSB 1,179–193) that are needed to understand the text of Ibn al-Haytham’s reconstruction of Book VIII. In §3.3 I shall discuss Book VII in some detail, since this book seems to be related to the lost Book VIII. A more detailed summary of Conics I–VII may be found in Heath, HGM 11,126–174.