PAR 



PAR 



equals H G D. Lastly, B N C and H G D 

 are together equal to A H G and B H G 

 together; and therefore (5) are equal to- 

 gether to the sum of two right angles. 



15 Theorem V. If a straight line fall- 

 ing upon two other straight lines makes 

 the alternate angles equal to one another, 

 those two straight lines will be parallel. 



Let the straight line EF, (fig. 6) which 

 falls upon the two straight line's A B. C D, 

 make the alternate angles A E F, E F I* 

 equal to one another, then A B is parallel 

 to G D If not, through E draw G H parallel 

 to C D. Then the alternate angle G E F 

 equals the alternate angle EFD. But 

 A E F equals li F D ; therefore A E F is 

 equal to G E F, the le.-s to the greater. 

 Hence G H is not parallel to C D; and in 

 like manner it may be shown that no 

 other line passing through the point E, 

 and not coinciding with A B, is parallel 

 to C D. Therefore A B is parallel to 

 CD. 



16. Cor. If a straight line, falling upon 

 two other straight lines, makes the exte- 

 rior angle equal to the interior and oppo- 

 site one on the same nide of the line ; 

 or makes the interior angles on the same 

 side equal to two right angles; the two 

 straight lines shall be parallel to one 

 another. 



PABALLEL planes, are such planes as 

 have all the perpendiculars drawn be- 

 twixt them equal to each oilier. 



PARALLEL rays, in optics, are those 

 which keep at an equal distance from the 

 visible object to the eye, which is sup- 

 posed to be infinitely remote from the ob- 

 ject. 



PARALLEL ruler, an instrument con- 

 sisting of two wooden, brass, &c. rulers, 

 equally broad every where; and so join- 

 ed together by the cross blades as to open 

 to different intervals, accede and recede, 

 and yet still retain their parallelism. See 

 FEXT AGRAPH. 



PARALLELS, or parallel circles, in geo- 

 graphy, called also parallels, or circles of 

 latitude, are lesser circles of the sphere 

 conceived to be drawn from west to east, 

 through all the points of the meridian, 

 commencing from the equator to which 

 they are parallel, and terminating with 

 the poles. They are called parallels of 

 latitude, because all places lying under 

 the same parallel, have the same lati- 

 tude. 



PARALLELS of latitude, in astronomy, 

 are lesser circles of the sphere parallel 

 to the ecliptic, imagined to pass through 

 every degree and minute of the colures. 

 They are represented on the globe by the 



divisions on the quadrant of altitude, in 

 its motion round the globe, when screwed 

 over the pole of the ecliptic. See GLOBE. 

 PARALLELS of altitude, or ALMUCAN- 

 TARS, are circles parallel 10 the horizon, 

 imagined to pass through every degree 

 and minute of the meridian between the 

 horizon and zenith, having iheir poles in 

 the lenith. They are represented on the 

 globe by the divisions on the quadrant of 

 altitude, in its motion about the body of 

 the globe, when screwed to the zenith. 



PARALLELS of declination, in astrono- 

 my, are the same with parallels of lati- 

 tude in geography. 



PARALLEL sphere, that situation of the 

 sphere, wherein the equator coincides 

 with the horizon, and the poles with the 

 zenith and nadir. In this sphere all the 

 parallels of the equator become parallels 

 of the horizon, consequently, no stars 

 ever rise or set, but all turn round in cir- 

 cles parallel to the horizon ; and the sun 

 when in the equinoctial, wheels round the 

 horizon the whole day. Atter his rising 

 to the elevated pole, he never sets for six 

 months ; and after his entering again on 

 the other side of the line, never rises for 

 six months longer. This is the position 

 of the sphere to such as live under the 

 poles, and to whom the sun is never above 

 23 3tf high. 



PARALLEL sailing, in navigation, is 

 the sailing under a parallel of latitude. 

 See NAVIGATION. 



PARALLELEPIPED, or PARALLELO- 

 PIPED, in geometry, a regu;ar solid com- 

 prehended under six parallelograms, the 

 opposite ones whereof are similar, paral- 

 lel, and equal. All parallelepipeds, 

 prisms, cylinders, Stc. whose bases and 

 heights are equal, are themselves equal. 

 A diagonal plane divides a parallelepiped 

 into two equal prisms ; so that a triangu- 

 lar prism is half a parallelepiped, upon 

 the same base, and of the same altitude. 

 All parallelepipeds, prisms, cylinders, 

 &c. are in a ratio compounded of their 

 bases and altitudes ; wherefore, if their 

 bases be equal, they are in proportion to 

 their altitudes, and conversely. Ail pa- 

 rallelepipeds, cylinders, cones, &c. are in 

 a triplicate ratio of their homologous 

 sides, and also of their altitudes. 



Equal parallelepipeds, prisms, cones, 

 cylinders, &c. reciprocate their bases and 

 altitudes. 



PARALLELISM, the situation or qua- 

 lity whereby any thing is denominated 

 parallel. See PARALLEL. 



PARALLELISM of the earth's axis, in 

 astronomy, that situation of the earth's 



