VOL. XIX.] PHILOSOPHICAL TRANSACTIONS. 7 



is -L z or A E : to A c, the time of the remotest observation A froin the tropic. 

 But the other case, when the middle observation is not exactly in the middle 

 between the other two times, as at f, is something more operose, and the 

 whole time from a to c being put = t, and from a to p = *, c e z=z c, and 



ftp I I, ft s s 



b c ■= b, the theorem will stand thus — —r- z=z a c, the time sought. 



' 2 t c — 2 u s 



To illustrate this method of calculation, it may perhaps be requisite to give 

 an example or two, for the sake of those astronomers that are less instructed in 

 the geometrical part of their art. Anno 1500 Bernard Walther, in the month 

 of June, at Nuremburg, observed the chord of the distance of the sun from 

 the zenith by a large parallactic instrument of Ptolemy, as follows : 



June 2, 45467 June 8, 44975 



June 9, 44934 and June 12, 44883 



June 16, 44990 June 16, 4499O. 



In both which cases the middle lime is exactly in the middle between the 

 extremes, and therefore in the former three, a c ^ 533, b c = 477, and t the 

 time between, being 14 days; by the first rule, the time of the tropic will be 

 found by this proportion, as 589 to 827-i- :: so -|- t or 7 days, to 9 days 20h. 

 2m. whence the tropic. Anno 1500, is concluded to have fallen June lid. 

 20h. 2m. In the latter three, a c is = 107, and b c = 15, and the whole in- 

 terval of time is 8 days = t ; whence as 199 : to 206^^ :: so is 4 days to 4d. 

 3h. 37m., which taken from the l6th day at noon, leaves lid. 20h. 23m. for 

 the time of the tropic, agreeing with the former to the third part of an hour. 



Again, Anno l63d, Gassendus, at Marseilles, observed the summer solstice 

 by a gnomon of 55 feet high, in order to determine the proportion of the 

 gnomon to the solstitial shade ; and he has left us these observations, which may 

 serve as an example for the second rule. 



June 19, N.S. shadow 31 766 parts, whereof the gnomon was 89428. 



June 20, 31733 



June 21, 31751 



June 22, 31759- 



These being divided into two sets, of three observations each, viz. the 19th, 

 20th, and 22d, and the igth, 2ist, and 22d ; we shall have in the first three, 

 c = 1 3, and b = 7, t = 3 days, * = 1 ; and in the second, c = 15, and b = 7, 

 t = 3, and s = 2 ; whence, according to the rule, the 19th day at noon the 

 sun wanted of the tropic a time proportionate to one day, as tlc — ssb to 

 Itc — lbs, that is, as 1 10 to 64 in the first set, or 107 to 62 in the second 

 set; that is, id, 17h. 15m. in the first, or id. 17h. 25m. in the second set; 



