2 Archdeacon Pratt on Chinese Astronomical Epochs. 



and the shadows were 1 foot 5 inches (pouccs) and 13 feet long, 

 there being 10 pouces in 1 pied. These data give at once, by a 

 table of tangents, 79° 7' 11" and 31° 18' 42" for the altitudes of 

 the sun. To these Laplace applies corrections for refraction, 

 parallax, and the sun's semidiameter, and makes them finally 

 79° 6' 52" and 31° 18' 48". Half the sum and half the differ- 

 ence of these should give the colatitude of the place of observa- 

 tion and the obliquity of the ecliptic. They give the latitude 

 = 34° 47' 10", and the obliquity =23° 54' 2". This latitude is 

 equal to the mean of the three observations mentioned above, 

 but is greater by 4' than the best of the three. Laplace 

 shows, by a formula in Mecanique Celeste, that 23° 51' 58 ;/ was 

 the obliquity in 1100 B.C. This differs by 2' 4" from that 

 obtained from the observations, which, at the rate of 48" a cen- 

 tury (the mean decrease of obliquity, see HerschePs 'Astronomy/ 

 art. 640), would throw the date back to 1358 b.c. Laplace 

 thinks the obliquity deduced from the observations as perfect an 

 accordance as could be desired, " seeing the uncertainty which 

 this kind of observations presents, especially because of the ill- 

 defined limit of the shadow" [Con. des Terns, 1809, pp. 433,434). 

 3. It is the extent of uncertainty arising from this cause which 

 I wish now to determine. Let h be the height of the style, s and 

 w the lengths of the shadows at the summer and winter solstices, 

 I and (p the latitude and obliquity, a and /3 the altitudes of the 

 sun. Then 



7 1 , J 1 li , h , 1, Ji 1 . h 



90— /=-rtan- ] - + p: tan" 1 -, 6= -tan- 1 -tan" 1 -: 



2 s 2 w * T 2 s 2 w 



~ 1 h 2 8s 1 A 2 Bw _ 1 — cos 2a, Bs 1— cos 2/3 Sw 

 ''• bl= 2 ~W+? J + 2 FT^ 3 T ~ 4 J + 4~~ " T' 



Similarly, 



~. I— cos 2a hs 1 — cos 2/3 Bw 



*</>= 4~- A + 4 ~T 



Put 



a = 79°6'52", 18=31° 18' 48", cos 2«= -0-92867, cos 2/3= 0459781 



.-. S/=0-482|.+0-365^=27 o -6^ + 20 o -9^, 

 h h h h 



and 



The recorded lengths of the shadows contain no fractions of 

 an inch. It may therefore be supposed that fractions equal to, 

 or less than half an inch, were thought too trifling to observe ; 

 or the undefined appearance of the shadow made greater nicety 

 impracticable. Put, therefore, Bs and Bw each equal to half an 

 inch, in excess or in defect, as errors to which the measure of the 



