258 Similarly But BOOK X t is incommensurable with u, v. t, u, v, w are all rational and ~ σ. Therefore (t+u+v+w) is a quadrinomial and therefore irrational. whence (x + y + z) is irrational. (4) The major made up of three straight lines. The commentator describes this as "the line composed of three straight lines incommensurable in square and such that one of them gives with each of the other two a sum of squares (which is) rational, while the rectangle contained by the two lines is medial." If x, y, z are the three straight lines, this would indicate (x2+y2) rational, (x2 + z2) rational, 2yz medial. Woepcke points out (pp. 696-8, note) the difficulties connected with this supposition or the supposition of (x2 + y2) rational, (x2+z2) rational, 2xy (or 2xz) medial, and concludes that what is meant is the supposition (x+2) rational xy medial xz medial (though the text is against this). The assumption of (x+y) and (x2+z2) being concurrently rational is certainly further removed from Euclid, for x. 33 only enables us to find one pair of lines having the property, as x, y. But we will not pursue these speculations further. As regards further irrationals formed by subtraction the commentator writes as follows. "Again, it is not necessary that, in the irrational straight lines formed by means of subtraction, we should confine ourselves to making one subtraction only, so as to obtain the apotome, or the first apotome of the medial, or the second apotome of the medial, or the minor, or the straight line which produces with a rational area a medial whole, or that which produces with a medial area a medial whole; but we shall be able here to make two or three or four subtractions. "When we do that, we show in manner analogous to the foregoing that the lines which remain are irrational and that each of them is one of the lines formed by subtraction. That is to say that, if from a rational line we cut off another rational line commensurable with the whole line in square, we obtain, for remainder, an apotome; and, if we subtract from this line (which is) cut off and rational-that which Euclid calls the annex (pоσapμóČovσα)— another rational line which is commensurable with it in square, we obtain, as the remainder, an apotome; likewise, if we cut off from the rational line cut I. BOOK XI. DEFINITIONS. A solid is that which has length, breadth, and depth. 2. An extremity of a solid is a surface. 3. A straight line is at right angles to a plane, when it makes right angles with all the straight lines which meet it and are in the plane. 4. A plane is at right angles to a plane when the straight lines drawn, in one of the planes, at right angles to the common section of the planes are at right angles to the remaining plane. 5. The inclination of a straight line to a plane is, assuming a perpendicular drawn from the extremity of the straight line which is elevated above the plane to the plane, and a straight line joined from the point thus arising to the extremity of the straight line which is in the plane, the angle contained by the straight line so drawn and the straight line standing up. 6. The inclination of a plane to a plane is the acute angle contained by the straight lines drawn at right angles to the common section at the same point, one in each of the planes. 7. A plane is said to be similarly inclined to a plane as another is to another when the said angles of the inclinations are equal to one another. 8. Parallel planes are those which do not meet. than two lines which meet one another and are not in t same surface, towards all the lines. Otherwise: A solid angle is that which is contained more than two plane angles which are not in the same pla and are constructed to one point. 12. A pyramid is a solid figure, contained by plane which is constructed from one plane to one point. 13. A prism is a solid figure contained by planes tv of which, namely those which are opposite, are equal, simil and parallel, while the rest are parallelograms. 14. When, the diameter of a semicircle remaining fixe the semicircle is carried round and restored again to the sar position from which it began to be moved, the figure comprehended is a sphere. 15. The axis of the sphere is the straight line whi remains fixed and about which the semicircle is turned. 16. The centre of the sphere is the same as th of the semicircle. 17. A diameter of the sphere is any straight li drawn through the centre and terminated in both directio by the surface of the sphere. 18. When, one side of those about the right angle in right-angled triangle remaining fixed, the triangle is carri round and restored again to the same position from which began to be moved, the figure so comprehended is a cone. And, if the straight line which remains fixed be equal the remaining side about the right angle which is carri round, the cone will be right-angled; if less, obtuse-angle and if greater, acute-angled. 19. The axis of the cone is the straight line wh remains fixed and about which the triangle is turned. 262 20. And the base is the circle described by the straight line which is carried round. 21. When, one side of those about the right angle in a rectangular parallelogram remaining fixed, the parallelogram is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a cylinder. 22. The axis of the cylinder is the straight line which remains fixed and about which the parallelogram is turned. 23. And the bases are the circles described by the two sides opposite to one another which are carried round. 24. Similar cones and cylinders are those in which the axes and the diameters of the bases are proportional. 25. A cube is a solid figure contained by six equal squares. 26. An octahedron is a solid figure contained by eight equal and equilateral triangles. 27. An icosahedron is a solid figure contained by twenty equal and equilateral triangles. 28. A dodecahedron is a solid figure contained by twelve equal, equilateral, and equiangular pentagons. DEFINITION 1. Στερεόν ἐστι τὸ μῆκος καὶ πλάτος καὶ βάθος ἔχον. This definition was evidently traditional, as may be inferred from a number of passages in Plato and Aristotle. Thus Plato speaks (Sophist, 235 D) of making an imitation of a model (mapádeуμa) “in length and breadth and depth" and (Laws, 817 E) of "the art of measuring length, surface and depth" as one of three μalnμara. Depth, the third dimension, is used alone as a description of "body" by Aristotle, the term being regarded as connoting the other two dimensions; thus (Metaph. 1020 a 13, 11) "length is a line, breadth a surface, and depth body"; "that which is continuous in one direction is length, in two directions breadth, and in three depth." Similarly Plato (Rep. 528 B, D), when reconsidering his classification of astronomy as next to (plane) geometry: "although the science dealing with the additional dimension of depth is next in order, yet, owing to the fact that it is studied absurdly, I passed it over and put next to geometry astronomy, the motion of (bodies having) depth." In Aristotle (Topics VI. 5, 142 b 24) we find “the definition of body, that which has three dimensions (diaσráσes)"; elsewhere he speaks of it as "that which has all the dimensions" (De caelo 1. 1, 268 b 6), "that which has dimension every way” (Tò #árty diáotaσiv čxov, Metaph. 1066 b 32) etc. In the Physics |