Page:The Works of Archimedes.djvu/195

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ON THE SPHERE AND THE CYLINDER I.
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3. Similarly also there are certain terminated surfaces, not themselves being in a plane but having their extremities in a plane, and such that they will either be wholly on the same side of the plane containing their extremities, or have no part of them on the other side.

4. I apply the term concave in the same direction to surfaces such that, if any two points on them are taken, the straight lines connecting the points either all fall on the same side of the surface, or some fall on one and the same side of it while some fall upon it, but none on the other side.

5. I use the term solid sector, when a cone cuts a sphere, and has its apex at the centre of the sphere, to denote the figure comprehended by the surface of the cone and the surface of the sphere included within the cone.

6. I apply the term solid rhombus, when two cones with the same base have their apices on opposite sides of the plane of the base in such a position that their axes lie in a straight line, to denote the solid figure made up of both the cones.

Assumptions.

1. Of all lines which have the same extremities the straight line is the least[1].

  1. ↑ This well-known Archimedean assumption is scarcely, as it stands, a definition of a straight line, though Proclus says [p. 110 ed. Friedlein] "Archimedes defined (ωρίσατο) the straight line as the least of those [lines] which have the same extremities. For because, as Euclid's definition says, ἐξ ἴσου κεῖται τοῖς ἐφ' ἑαυτῆς σημείοις, it is in consequence the least of those which have the same extremities." Proclus had just before [p. 109] explained Euclid's definition, which, as will be seen, is different from the ordinary version given in our textbooks; a straight line is not "that which lies evenly between its extreme points," but "that which ἐξ ἴσου τοῖς ἐφ' ἑαυτῆς σημείοις κεῖται." The words of Proclus are, "He [Euclid] shows by means of this that the straight line alone [of all lines] occupies a distance (κατέχειν διάστημα) equal to that between the points on it. For, as far as one of its points is removed from another, so great is the length (μέγεθος) of the straight line of which the points are the extremities; and this is the meaning of τὸ ὲξ ἴσου κεῖσθαι τοῖς ἐφ' ἑαυτῆς σημείοις. But, if you take two points on a circumference or any other line, the distance cut off between them along the line is greater than the interval separating them; and this is the case with every line except the straight line." It appears then from this that Euclid's definition should be understood in a sense very like that of
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