ABSTRACT

BOOK I I ol'ens with a description of the different classes of curves. The Greeks classified geometrical problems according to the means by which they were solved, and this led to the classification of such problems

into plane, solid or linear problems, according as they required for their construction, straight lines and circles, aconic section, or curves more complicated. The distinction is clearly enunciated in Pappus, and as references to it in La Geometrie are of frequent occurrence, we quote it at length:

"Tria genera problematum geometricorum statuimus, quorum alia plana, alia solida, alia linearia vocamus. Quae igitur per rectas lineas et circuli circumferentias solvi possunt, ea merito plana dicantur, quoniam lineae, per quas ejusmodi problemata solvuntur, in plano originem habent. Quorum autem problematum resolutio adsumptis una pluribusve coni sectionibus invenitur, haec solida appellata sunt; nam ad eorum constructionem solid arum figurarum superficiebus, nimirum conicis, uti necesse est. Tertium autem relinquitur problematum genus quod line are vocatur; nam praeter eas quas diximus lineas aliae variam et contortiorem originem habentes, quae ex superficiebus minus ordinatis et motibus implicatis gignuntur, ad constructionem adhibentur ... Atque ejusdem generis aliae sunt, velut helices sive spirales, tetragonizusae sive quadratrices, conchoides sive conchiformes, cissoides, sive hederae similes."*

Nevertheless, observes Descartes, the ancients did not proceed further and distinguish between the different degrees of these more complex curves, which for some obscure reason, they styled mechanical rather than geometrical. The fact that some sort of instrument is required in their construction hardly justifies the term mechanical, because even the straight line and the circle cannot be constructed without recourse to an appliance of some sort. N or can it be argued that the instruments employed in their construction are less accurate than the' ruler and compasses because they are more complicated; if that were so we should have to exclude them from mechanics where accuracy of construction is even more important than it is in geometry, which demands only accuracy in reasoning. Still less can it be on account of the reluctance of the ancient geometers to add to the two existing postulates, namely, a straight line can be drawn between any two points, and secondly, a circle can be drawn with any given point as centre to pass through another given point. At a very early

I02 Scient~fic Work of Descartes stage the Greeks must have recognised the inadequacy of the means sanctioned by Euclid, and so they sought to discover others. Therefore they did not hesitate to add a third postulate, namely, that a given cone could be cut by a given plane, and this opened up the way to a new series of curves, the conic sections.