quantity, quality, or inclination, taken singly, but in the aggregate of them all. For if we regard the inclination of a circular line to its tangent, we shall find it possess the property, by which Euclid defines an angle: if we respect its participation of quantity, we shall find it capable of being augmented and diminished; and if we regard it as possessing a peculiar quality, we shall account for its being incommensurable with every right-lined angle. See the Comment on the 8th Definition. [140] In i. De Cælo. [141] It is from this cylindric spiral that the screw is formed. [142] [Illustration] The present very obscure passage, may be explained by the following figure. Let A B C, be a right angle, and D E the line to be moved, which is bisected in G. Now, conceive it to be moved along the lines A B, B C, in such a manner, that the point D may always remain in A B, and the point E in B C. Then, when the line D E, is in the situations _d e_, _δ ε_, the point G, shall be in _g, γ_, and these points G, _g, γ_, shall be in a circle. And any other point F in the line D E, will, at the same time, describe an ellipsis; the greater axis being in the line A B, when the point F is between D and G; and in the line B C, when the point F is between G and E. [143] That is, the soul of the world. [144] In Timæo. [145] The ellipsis. [146] The cissoid. For the properties of this curve, see Dr. Wallis’s treatise on the cycloid, p. 81. [147] The conchoid. [148] Thus, a right line, when considered as the side of a parallelogram, moving circularly, generates a cylindrical superficies: when moving circularly, as the side of a triangle, a conical surface; and so in other lines, the produced superficies varying according to the different positions of their generative lines. [149] Inv ii. De Rep. [150] In multis locis. [151] This definition is the same with that which Mr. Simson has adopted instead of Euclid’s, expressed in different words: for he says, “a plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies.” But he does not mention to whom he was indebted for the definition; and this, doubtless, because he considered _it was not worth while to relate the trifles of Proclus at full length_: for these are his own words, in his note to proposition 7, book i. Nor has he informed us in what respect Euclid’s definition is _indistinct_. [152] In the Greek ἐννοιὰς, but it should doubtless be read εἰκόνας, _images_, as in the translation of Barocius.