23
cotton, which are now well known throughout Europe , andare so common in England, is that of the equality ojparallels comprised between parallels. Besides having thegreat advantage of spinning 40, 50, 60, or even many morethreads by the movement of one frame or chariot, all thethreads are spun of an equal thickness; which could neverhave been effected in spinning each one sepaiatefy, andwithout the geometrical means here brought into notice.
Hitherto we have only compared parallels with per-pendiculars, let us now compare them with oolique lines.Draw AB, CD, fig. 8, pi. 2, oblique with respect to EACF;if the two angles, EAB, ECD, called corresponding angles,are equal, the two right lines, AB, CD, are parallel.
If they are not parallel, by producing them they will meet atsome point or other, either above or below EACF: let us see ifthis he possible.
Produce BA and DC to J and d, and take the figure BACD,which turn upside down, so that A is placed where C, and C whereA now is.
But the angle BAF, which equals EAJ, is equal to DCF, whichequals EC d; the side AB, therefore, when the figure is reversed, willplace itself on C d, and the side CD, will be placed on 4J. If,therefore, the two lines JAB, dCD, were to meet at any point onone side of AC, it would be necessary that they should meet at asecond point, on the other side of AC ; but this is impossible, forthere would then be two right lines, which would meet each other intwo points.
Thus it is an invariable rule, when two right lines, JAB, dCD,forming acute equal angles, a, a', a", a'”, with the oblique line EACF,and consequently the obtuse equal angles, o, o', o", O', these linesare parallel.
The converse is equally true; that is to say, when thelines are parallel every oblique line intersects them, so asto form with them four acute angles, equal to one another,and four obtuse angles, also equal to one another.
To convince the student of this, it is only necessary to observethat the right line <2CD, fig. 8, pi. 2, drawn from the point C, sothat the angles a", and U y are equal to a, and «', is parallel toJAB. More than one line paraUei to JAB, cannot be drawn fromthe point C, it is therefore the right line by which a o', a", a"are equal, as well as o, o', o", o'".