ASTRONOMY of the BRAHMINS, 171 



and ingenious. At the place for which they calculate, they 

 obferve the fhadow of a gnomon on the day of the equinox, 

 at noon, when the fun, as they exprefs it, is in the middle of 

 the world. The height of the gnomon is divided into 72c 

 equal parts, in which parts the length of the fhadow is alfo 

 meafured. One third of this meafure is the number of mi- 

 nutes by which the day, at the end of the firft month after the 

 equinox, exceeds twelve hours ; four-fifths of this excels is the 

 increafe of the day during the fecond month ; and one third 

 of it is the increafe of the day, during the third month *. 



46. It is plain, that this rule involves the fuppoiition, that, 

 when the fun's declination is given, the fame ratio every where 

 exifts between the arch which meafures the increafe of the day 

 at any place, and the tangent of the latitude ; for that tangent 

 is the quotient which arifes from dividing the length of the 

 fhadow by the height of the gnomon. Now, this is not flriclly 

 true ; for fuch a ratio only fubfifts between the chord of the 

 arch, and the tangent above mentioned. The rule is, therefore, 

 but an approximation to the truth, as it neceffarily fuppofes 

 the arch in queftion to be fo fmall as to coincide nearly with 

 its chord. This fuppoiition holds only of places in low lati- 

 tudes ; and the rule which is founded on it, though it may 

 fafely be applied in countries between the tropics, in thofe that 

 are more remote from the equator, would lead into errors too 

 confiderable to efcape obfervation f. 



y 2 As 



* Mem. Acad, des Scien. II. P. 175. 



f To judge of the accuracy of this approximation, fuppofe to be the obliquity of 

 the ecliptic, and x the excefs of the femidiurnal arch, on the longeft day, above an arch 

 of 90 , then^Jn. a? = tan. X tan. /at. But if G be the height of a gnomon, and S 



S S 



the length of its fhadow on the equinoctial day, — - — tan. /at. and Jin. x — tan. x . 



G G 



_, . _ ,S tan.O'xSi tan.O'xS* ''"'.'.". 



Therefore x = tan.O x -pr H 7 %. + — — + &c. or in minutes of 



G ou } 24u* 



time, 



