[132] This sentence within the brackets, is wholly omitted in the printed Greek.
[133] In i. De Cælo.
[134] This sentence within the brackets, which is very imperfect in the Greek, I have supplied from the excellent translation of Barocius. In the Greek there is nothing more than λὲγω δὲ ἑνὸν τῂν γραμμὴν δυαδός πρὸς τὸ στερεόν.
[135] In the Greek, γὰρ ἡ μονὰς ἐκεῖ πρῶτον, ὅπου πατρικὴ μονάς ἐστι φησὶ τὸ λόγιον. The latter part only of this oracle, is to be found in all the printed editions of the Zoroastrian oracles; though it is wonderful how this omission could escape the notice of so may able critics, and learned men. It seems probable, from hence, that it is only to be found perfect in the present work.
[136] The word τανάη, is omitted in the Greek.
[137] This and the following problems, are the 1st, 22d, and 12th propositions of the first book. But in the two last, instead of the word ἄπειρος or infinite, which is the term employed by Euclid, Mr. Simson, in his edition of the Elements, uses the word unlimited. But it is no unusual thing with this great geometrician, to alter the words of Euclid, when they convey a philosophical meaning; as we shall plainly evince in the course of these Commentaries. He certainly deserves the greatest praise for his zealous attachment to the ancient geometry: but he would (in my opinion) have deserved still more, had he been acquainted with the Greek philosophy; and fathomed the depth of Proclus; for then he would never have attempted to restore Euclid’s Elements, by depriving them of some very considerable beauties.
[138] This is doubtless the reason why the proportion between a right and circular line, cannot be exactly obtained in numbers; for on this hypothesis, they must be incommensurable quantities; because the one contains property essentially different from the other.
The cornicular angle is that which is made from the periphery of a circle and its tangent; that is, the angle comprehended by the arch L A, and the right line F A, which Euclid in (16. 3.) proves to be less than any right-lined angle. And from this admirable proposition it follows, by a legitimate consequence, that any quantity may be continually and infinitely increased, but another infinitely diminished; and yet the augment of the first, how great soever it may be, shall always be less than the decrement of the second: which Cardan demonstrates as follows. Let there be proposed an angle of contact B A E, and an acute angle H G I. Now if there be other lesser circles described A C, A D, the angle of contact will be evidently increased. And if between the right lines G H, G I, there fall other right lines G K, G L, the acute angle shall be continually diminished: yet the angle of contact, however increased, is always less than the acute angle, however diminished. Sir Isaac Newton likewise observes, in his Treatise on Fluxions, that there are angles of contact made by other curve lines, and their tangents infinitely less than those made by a circle and right line; all which is demonstrably certain: yet, such is the force of prejudice, that Mr. Simson is of opinion, with Vieta, that this part of the 16th proposition is adulterated; and that the space made by a circular line and its tangent, is no angle. At least his words, in the note upon this proposition, will bear such a construction. Peletarius was likewise of the same opinion; but is elaborately confuted by the excellent Clavius, as may be seen in his comment on this proposition. But all the difficulties and paradoxes in this affair, may be easily solved and admitted, if we consider, with our philosopher, that the essence of an angle does not subsist in ether 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.