Thus let a part A E B cut off by the diameter A B (fig. I.) of the circle A E B D be placed on the other part A D B, as in fig. II.. Then, if it is not equal to the other part, either A E B will fall within A D B, or A D B within A E B: but in either case, C E will be equal to C D, which is absurd. [172] This objection is urged by Philoponus, in his book against Proclus on the eternity of the world; but not, in my opinion, with any success. See also Simplicius, in his third digression against Philoponus, in his commentary on the 8th book of Aristotle’s Physics. [173] This definition is no where extant but in the commentaries of Proclus. Instead of it, in almost all the printed editions of Euclid, the following is substituted. _A segment of a circle is the figure contained by a diameter, and the part of the circumference cut off by the diameter._ This Mr. Simson has marked with commas, as a symbol of its being interpolated: but he has taken no notice of the different reading in the commentaries of Proclus. And what is still more remarkable, this variation is not noticed by any editor of Euclid’s Elements, either ancient or modern. [174] As in every hyperbola. [175] The Platonic reader must doubtless be pleased to find that Euclid was deeply skilled in the philosophy of Plato, as Proclus every where evinces. Indeed, the great accuracy, and elegant distribution of these Elements, sufficiently prove the truth of this assertion. And it is no inconsiderable testimony in favour of the Platonic philosophy, that its assistance enabled Euclid to produce such an admirable work. [176] Concerning these crowns, or annular spaces, consult the great work of that very subtle and elegant mathematician Tacquet, entitled _Cylindrica et Annularia_. [177] In the preceding tenth commentary. [178] This in consequence of every triangle possessing angles alone equal to two right. [179] This too, follows from the same cause as above. [180] Thus the following figure A B D C has four sides, and but three angles. [Illustration] [181] The Greek in this place is very erroneous, which I have restored from the version of Barocius. [182] For the Greek word ῥόμβος is derived from the verb ῥέμβω, which signifies to have a circumvolute motion. [183] See the Orphic Hymns of Onomacritus to these deities; my translation of which I must recommend to the English reader, because there is no other. [184] These twelve divinities, of which Jupiter is the head, are, _Jupiter_, _Neptune_, _Vulcan_, _Vesta_, _Minerva_, _Mars_, _Ceres_, _Juno_, _Diana_, _Mercury_, _Venus_, and _Apollo_. The first triad of these is demiurgic, the second comprehends guardian deities, the third is vivific, or zoogonic, and the fourth contains elevating gods. But, for a particular theological account of these divinities, study Proclus on Plato’s Theology, and you will find their nature unfolded, in page 403, of that admirable work.