PAR 



PAR 



circle, and at that instant take the alti- 

 tudes of both horns : the difference of 

 these two altitudes being halved and add- 

 ed to the leas,, or subtracted from the 

 greatest, gives nearly the visible or appa- 

 rent altitude of the moon's centre ; and 

 the true altitude is nearly equal to the al- 

 titude of the centre of the shadow at that 

 time. Now we know the altitude of the 

 shadow, because we know the place of 

 the sun in the ecliptic, and its depression 

 under the horizon, which is equal to the 

 altitude of the opposite point of the eclip- 

 tic in which is the centre of the sha- 

 dow. And therefore, having both the 

 true altitude of the moon and the ap- 

 parent altitude, the difference of these 

 is the parallax required. But as the pa- 

 rallax of the moon increases as she ap- 

 proaches towards the earth, or the peri- 

 gxum of her orbit, therefore astrono- 

 mers have made tables, winch shew the 

 horizontal parallax for every degree of 

 its anomaly. 



The parallax always diminishes the al- 

 titude of a phenomenon, or makes it ap- 

 pear lower than it would do, if viewed 

 from the centre of the earth ; and this 

 change of the altitude may, according to 

 the different situation of the ecliptic and 

 equator in respect of the horizon of the 

 spectator, cause a change of the latitude, 

 longitude, declination, and right ascen- 

 sion of any phenomenon, which is called 

 their parallax. The parallax, therefore, 

 increases the right and oblique ascension ; 

 diminishes the descension ; diminishes the 

 northern declination and latitude in the 

 eastern part, and increases them in the 

 western ; but increases the southern both 

 in the eastern and western part ; dimi- 

 nishes the longitude in the western part, 

 and increases it in the eastern. Hence it 

 appears, that the parallax has just op- 

 posite effects to refraction. See REFRAC- 

 TION . 



PARALLAX, annual, the change of the 

 apparent place of a heavenly body, which 

 is caused by being viewed from the earth 

 in different parts of its orbit round the 

 sun. The annual parallax of all the 

 planets is found very considerable, but 

 that of the fixed stars is imperceptible. 



PARALLAX, in levelling, denotes the 

 angle contained between the line of the 

 true level, and that of the apparent level. 

 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- 



haps, there is none which can be said to 

 have given entire satisfaction. The diffi- 

 culty consists in converting 1 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 

 rcnifdy this defect, three sorts of me- 

 thods have been adopted a new defini- 

 tion of parahel 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, 

 viz. 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 the\definition in T. Simpson's first 

 edition, must equally hold against ours ; 

 but we think that if his objection real- 

 ly hold 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.) If one side of a 

 triangle be produced, the outward angle 

 is greater than either of the inward oppo- 



