ferred it either from fome fuch experiment made or imagined; or from knowing, that the angles of a plane triangle are equal to two right angles. The axiom is; "if a 66 ftraight meets two straight lines, so as to "make the two interior angles on the fame "fide of it taken together less than two 66 66 right angles, these straight lines being continually produced, fhall at length "meet upon that fide on which are the angles which are less than two right angles." B D Let a ftraight line fo fall upon two other ftraight lines, that the angles ABC and CBD, are equal to one another, and alfo the the angles AEF and FED, (which may be done either by experiment or by former propofitions) then the lines CB and FE ftand upright with refpect to AD, or the angles, being adjacent and equal, are right angles according to the definition. Now in a model conftructed according to this diagram, let CB, or EF, or both, be moveable, or have a joint at в and E, then by turning either of them inwards, or in the direction of the dotted lines EC or BF, they will meet and form a triangle, if produced. So that if these angles, which by the construction are equal to two right angles, become at all less, the ftraight lines, continually produced, fhall at length meet upon that fide, on which are the angles which are lefs than two right angles; which was to bę fhewn. The following experiments would furnish a fatisfactory demonstration of the lead 津 ing properties of parallel lines; let parallel lines be fhewn to be fuch as always to preserve the fame distance; hence all the straight lines meeting them internally at equal angles fhall be equal. Thus let AB be a straight line; and let DC make with it the angle DCB. At the point F make the angle G F B equal to DC B, and the line FG equal to DC, and through G and D draw the line HI. Now the angles DCB and GFB being equal, and GF equal to DC, the lines AB and HI are parallel. Now let us suppose a model made according to this diagram, fo that the line HI might move along GF and DC, upon each of which certain small equal distances distances are marked off, as n, 0, p. It is evident, that if I were brought down through these equal fmall diftances fucceffively, it would be parallel, for Fn is equal to cn, and fo on; and at laft it would coincide with AB, and not cut it. Therefore, the internal angle DCB is equal to the external, and oppofite KDI, (fuppofing DC to be prolonged to K) because they would be the fame angle; and the alternate angles GDC, DCB, would be equal, as being now the vertical angles DCB and FCL; and the interior angles, IDC coinciding with BCL and BCD with IDK, would, by propofition the 13th, be equal to two right angles; and any line parallel to AB would be parallel to HI, prop. 30th. On feparating the straight strips again, it would appear, that if the alternate angles be equal, the exterior equal to the interior and opposite, and both interior equal to two right angles, the lines would keep parallel, and if any of these circumftances change, the angles would not be parallel, for the lines forming the equal angles, would become unequal. E B H A courfe of experiments on parallel lines may begin in a manner different still. Mr. Ludlam proposes to demonstrate the equality of the alternate angles, for which alone the 12th axiom is needed, by the following experiment, which he mifcalls an axiom. Two ftraight lines, meeting in a point, are not BOTH parallel to a third line. For, fays he, if AGH be not equal to GHD (the alternate angle) one must be greater; lẹt AGH |