Hæc tortuosa linea, This crankled line is of Proclus called Helicoides. But it may also be called Helix, a twist or wreath: The Greekes by this word do commonly either understand one of the kindes of Ivie which windeth it selfe about trees & other plants; or the strings of the vine, whereby it catcheth hold and twisteth it selfe about such things as are set for it to clime or run upon. Therfore it should properly signifie the spirall line. But as it is here taken it hath divers kindes; As is the Arithmetica which is Archimede'es Helix, as the Conchois, Cockleshell-like: as is the Cittois, Iuylike: The Tetragonisousa, the Circle squaring line, to witt that by whose meanes a circle may be brought into a square: The Admirable line, found out by Menelaus: The Conicall Ellipsis, the Hyperbole, the Parabole, such as these are, they attribute to
Menechmus: All these Apollonius hath comprised in eight Bookes; but being mingled lines, and so not easie to bee all reckoned up and expressed, Euclide hath wholly omitted them, saith Proclus, at the 9. p. j.
13. Lines are right one unto another, whereof the one falling upon the other, lyeth equally: Contrariwise they are oblique. è 10. d j.
Hitherto straightnesse and crookednesse have beene the affections of one sole line onely: The affections of two lines compared one with another are Perpendiculum, Perpendicularity and Parallelismus, Parallell equality; Which affections are common both to right and crooked lines. Perpendicularity is first generally defined thus:
Lines are right betweene themselves, that is, perpendicular one unto another, when the one of them lighting upon the other, standeth upright and inclineth or leaneth neither way. So two right lines in a plaine may bee perpendicular; as are ae. and io. so two peripheries upon a sphearicall may be perpendiculars, when the one of them falling upon the other, standeth indifferently betweene, and doth not incline or leane either way. So a right line may be
perpendicular unto a periphery, if falling upon it, it doe reele neither way, but doe ly indifferently betweene either side. And in deede in all respects lines right betweene themselves, and perpendicular lines are one and the same. And from the perpendicularity of lines, the perpendicularity of surfaces is taken, as hereafter shall appeare. Of the perpendicularity of bodies, Euclide speaketh not one word in his Elements, & yet a body is judged to be right, that is, plumme or perpendicular unto another body, by a perpendicular line.
Therefore,
