## ABSTRACT

PROBLEMS OF GEOMETRICAL CONSTRUCTION

3.1. THE REGULAR HEPTAGON

3.1.1. Introduction

One of the geometrical construction problems that most interested mathematicians between the ninth and eleventh centuries was that of constructing the regular heptagon. If we are to believe what the Arabic mathematicians themselves say, this problem belongs to a whole group of three-dimensional (solid) problems inherited from Greek mathematicians; but historians have placed too little emphasis upon what makes it different from other such problems. Unlike, for example, investigations of the two means or the trisection of an angle, work on the regular heptagon was resumed rather late. Whereas the first two problems were subjects of intense research from the mid ninth century onwards, from such early scholars as the Banº Mºsæ, Thæbit ibn Qurra and others, it was not until the second half of the tenth century that study of the regular heptagon was resumed. But once it started, such research became something of a craze among eminent mathematicians: it is as if each of them wanted to leave his mark on it. This is how, a little later on, Ibn al-Haytham presented such research:

Why was there such enthusiasm, and why did work start relatively late? The fashion for studying this topic may be explained by the high standing of Archimedes. The mathematicians of the time, including Ibn al-Haytham, refer to the lemma that Archimedes proposed for this construction. It is clear that they are concerned with a text attributed to Archimedes, and one of which no trace has come down to us. If we are to believe the mathematicians, Archimedes had not, however, proved his lemma, but had

simply assumed it was accepted as true. So the regular heptagon basked in the reflected glory of Archimedes, but did so without having been constructed by him: a state of affairs peculiarly intriguing and exciting. But that does not explain why it was not until the second half of the tenth century that mathematicians revived a problem already known, they say, to Thæbit ibn Qurra and his contemporaries.2 The time that elapsed might be explained by reference to the new interests that emerged during this period. These interests, which have various origins – algebraic and geometrical – encouraged mathematicians to investigate this problem, along with many others previously neglected to which we shall return. Interest in matters of algebra, as we have seen elsewhere,3 is directed to the development of a theory of algebraic equations of degree less than or equal to three, and to research in algebraic geometry. As we shall see, the interest in geometry is bound up with the changes brought about in work on geometrical construction by the introduction of the use of conic sections.