must, and no more: He goeth obliquum iter, a crooked way, which goeth more then he needeth, as Proclus saith.
6. A right line is the shortest betweene the same bounds.
Linea recta, a straight or right line is that, as Plato defineth it, whose middle points do hinder us from seeing both the extremes at once; As in the eclipse of the Sunne, if a right line should be drawne from the Sunne, by the Moone, unto our eye, the body of the Moone beeing in the midst, would hinder our sight, and would take away the sight of the Sunne from us: which is taken from the Opticks, in which we are taught, that we see by straight beames or rayes. Therfore to lye equally betweene the boundes, that is by an equall distance: to bee the shortest betweene the same bounds; And that the middest doth hinder the sight of the extremes, is all one.
7. A crooked line is touch'd of a right or crooked line, when they both doe so meete, that being continued or drawne out farther they doe not cut one another.
Tactus, Touching is propper to a crooked line, compared either with a right line or crooked, as is manifest out of the 2. and 3. d 3. A right line is said to touch a circle, which touching the circle and drawne out farther, doth not cut the circle, 2 d 3. as here ae, the right line toucheth the periphery iou. And ae. doth touch the helix or spirall. Circles are said to touch one another, when touching they doe not cutte one another, 3. d 3. as here the periphery doth aej. doth touch the periphery ouy.
Therefore
8. Touching is but in one point onely. è 13. p 3.
This Consectary is immediatly conceived out of the definition; for otherwise it were a cutting, not touching. So Aristotle in his Mechanickes saith; That a round is easiliest mou'd and most swift; Because it is least touch't of the plaine underneath it.
9. A crooked line is either a Periphery or an Helix. This also is such a division, as our Authour could then hitte on.
10. A Periphery is a crooked line, which is equally distant from the middest of the space comprehended.
Peripheria, a Periphery, or Circumference, as eio. doth stand equally distant from a, the middest of the space enclosed or conteined within it.
Therefore
11. A Periphery is made by the turning about of a line, the one end thereof standing still, and the other drawing the line.
As in eio. let the point a stand still: And let the line ao, be turned about, so that the point o doe make a race, and it shall make the periphery eoi. Out of this fabricke doth Euclide, at the 15. d. j. frame the definition of a Periphery: And so doth hee afterwarde define a Cone, a Spheare, and a Cylinder.
Now the line that is turned about, may in a plaine, bee either a right line or a crooked line: In a sphericall it is onely a crooked line; But in a conicall or Cylindraceall it may bee a right line, as is the side of a Cone and Cylinder. Therefore in the conversion or turning about of a line making a periphery, there is considered onely the distance; yea two points, one in the center, the other in the toppe, which therefore Aristotle nameth Rotundi principia, the principles or beginnings of a round.
12. An Helix is a crooked line which is unequally distant from the middest of the space, howsoever inclosed.
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,
14. If a right line be perpendicular unto a right line, it is from the same bound, and on the same side, one onely. ê 13. p. xj.
Or, there can no more fall from the same point, and on the same side but that one. This consectary followeth immediately upon the former: For if there should any more fall unto the same point and on the same side, one must needes reele, and would not ly indifferently betweene the parts cut: as here thou seest in the right line ae. io. eu.
15. Parallell lines they are, which are everywhere equally distant. è 35. d j.
Parallelismus, Parallell-equality doth now follow: And this also is common to crooked lines and right lines: As heere thou seest in these examples following.
Parallell-equality is derived from perpendicularity, and is of neere affinity to it. Therefore Posidonius did define it by a common perpendicle or plum-line: yea and in deed our definition intimateth asmuch. Parallell-equality of bodies is no where mentioned in Euclides Elements: and yet they may also bee parallells, and are often used in the Optickes, Mechanickes, Painting and Architecture.
