ASTRONOMY. 339 



requisite to say, having carried one of the chords 200 times over upon 

 the arc, he found it exhausted ; and that the other chords could only 

 be applied 164 times upon the quadrant. 



We see, also, that Archimedes had not the means of computing the 

 angle at the vertex of an isosceles triangle, of which he knew the base 

 and the two equal sides. He was obliged to recur to a graphical 

 operation as uncertain as the observation itself. Thus he was entirely 

 ignorant even of rectilinear trigonometry, and he had not any notion 

 of computing the chords of circular arcs. 



We now come to the great father of true astronomy, Hipparchus ; 

 but our limits will not admit of our entering very deeply into his 

 discoveries and improvements. One of his first cares was to rectify Finds the 

 the length of the year, which before his time we have seen had been ^J ' 

 made to consist of 365 days and 6 hours. By comparing one of his 

 own observations at the summer solstice with a similar observation 

 made 145 years before by Aristarchus, he shortened the year about 

 7 minutes; making it to consist of 365 days, 5 hours, 53 minutes; 

 which, however, was not sufficient: but the cause of the mistake is said 

 to have rested principally with Aristarchus and not with Hipparchus ; 

 for the observations of the latter, compared with those of modern times, 

 give 365 days, 5 hours, 48 min. 49^- sec. for the duration of the year ; a 

 result which exceeds the truth very little more than a second. It is to 

 be observed, however, that this is 110 very exact criterion, unless the 

 same be compared with the observation of the more ancient observer ; 

 for supposing all the error on the side of Hipparchus, it is more divided 

 by comparing it with others at the distance of 19 or 20 centuries, than 

 in comparing it with one, where the distance of time is only 145 years. 



One of the greatest benefits which astronomy derived from this Introduction 

 philosopher was his enunciation and demonstration of the method of ^eSyTy" 

 computing triangles, whether plane or spherical. He constructed a chords, 

 table of chords, which he applied nearly in the same manner as we 

 now do our tables of sines. As an observer, however, he rendered 

 great service to the doctrine of astronomy, having made much more 

 numerous observations than any of his predecessors, and upon far 

 more accurate principles. He established the theory of the sun's Establishes 

 motion in such a manner, that Ptolemy, 130 years afterwards, found o^elu'n's 

 no essential alteration requisite ; he determined also the first lunar motion, 

 inequality, and gave the motions of the moon's apogee and of its fij st lunar 

 nodes, which Ptolemy afterwards very slightly modified. Hipparchus inequality, 

 also prepared the way for the discovery of the second lunar inequality, 

 and from his observation it was, that the fact of the precession of the 

 equinoxes was first inferred. He employed the transit of the stars Hour of the 

 over the meridian to find the hour of the night, and invented the S^the stars, 

 planisphere, or the means of representing the concave sphere of the 

 stars, on a plane, and thence deduced the solution of problems in 

 spherical astronomy, with considerable exactness and facility. To him 

 also we owe the happy idea of marking the position of towns and 



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