PARALLEL. The subject of parallel lines, says Playfair, is one of the most dif ficult in the Elements of Geometry. It has accordingly been treated in a great variety of different ways, of which, per there is none which can be said to have given entire satisfaction. The diffi. culty consists in converting the twenty seventh and twenty-eighth of Euclid, or in demonstrating, that parallel straight lines (or such as do not meet one another) when they meet a third line, make the al ternate angles with it equal, or which comes to the same, are equally inclined to it, and make the exterior angle equal to the interior and opposite. In order to demonstrate this proposition, Euclid as sumed it as an axiom, that if a straight line meet two straight lines, so as to make the interior angles on the same side of it less than two right angles, these straight lines being continually produced, will at length meet on the side on which the an gles are that are less than two right an gles. This proposition, however, is not self-evident; and ought the less to be re ceived, without proof that the converse of it is a proposition that confessedly re quires to be demonstrated. In order to remedy this defect, three sorts of me thods have been adopted—a new defini tion of parallel lines ; a new manner of reasoning on the properties of straight lines without any new axiom ; and the introduction of a new axiom less excep tionable than Euclid's. Playfair adopts the latter plan ; but we do not perceive that his axiom is by any means self-evident upon Euclid's definition which he retains, vie. Parallel straight lines are such as are in the same plane, and which being pro duced ever so far both ways do not meet. A more intelligible, and we think an equally rigid, demonstration of the pro perty of parallels, may be obtained with out any axiom, by means of a new defi nition. It may at first sight be thought, that the objection urged by Playfair against thetdefinition in T. Simpson's first edition, must equally hold against ours ; but we think that if his objection real ly bold good against that definition, (though we confess we cannot feel the force of it,) it is obviated by distinguish ing, as ought to be done, between the dis tance and the measure of that distance.
We must of course suppose our read ers acquainted with the propositions in Euclid preceding the twenty-seventh ; but to save the necessity of reference, we shall give an enunciation of those which we shall have to employ in our demonstra tion, in the form in which we employ them. 1.' (Prop. 16.) lf one side of a triangle be produced, the outward angle is greater than either of the inward oppo.
site angles. 2. (Prop. 19.) The greater angle of every triangle has the greater side opposite to it. 3. (Prop. 4.) If two triangles have two sides of the one respec tively equal to two sides of the other, and have the included angles equal, the other angles will be respectively equal, viz. those to which the equal sides are oppo. site. 4. (Prop. 15.) If two straight hues cut each other, the vertical or opposite angles will be equal. 5. (Prop. 13.) If a straight line meet another, the sum of the adjacent angles is equal to the sum of two right angles.
6. Definition. Parallel straight lines are those whose least distances from each other are every where equal.
7. Theorem 1. The perpendicular drawn to a straight line from any point, is the least line that can be drawn from that point to the given line.
Let C 13, (Plate X11 Miscell. fig. 2) be a straight line drawn from C perpendicu lar to A B ; and let C B be any other straight line from C to A B ; then is C D less than C E. For the angle C D 11 equals angle C D A by construction ; and C DA is greater than C E D (1) ; therefore C DE is greater than C E D. Hence (2) C D is less than C E.
8. Cur. 1. Hence the perpendicular from any point to a straight line is the true measure of the least distance of that point from that line.