Royal Society. 133 
A c the time of the remoteft oulervation A from the 7'ropic. But 
the other cafe, when the middle oblervation is not exactly in the 
middle between the other two times, as at F, is lomething more 
operoie, and the whole time trom A to C, being put = r, and 
from A to F :zi; S, <; ?, zz; t, and ^ c m: Z? 3 the theorem will Hand 
thus l.^' -'^J ^ -^z hcy the time fought. 
1 T C zJS 
To illuftrate this method of calculation, it may perhaps be re- 
quifite to give an example or two for the lake of thole aftrono- 
mers, that are lels inflrufted in the geometrical part of their art. 
Anno 1 500 Bernard Walt her, in the month of jfune, at Nu- 
remberg^ oblerved the chord of the diftance of the lun from the 
zenith, by a large paralladic inflrument of *iPro/^»y\ as follows 5 
June 2, 454<^7 7 ^June 8, 44975 
jfune 9, 44934 > and «^y//w 12, 44883 
jfune Id, 44990S Cjfune 15, 44990 
In both thefe cales, the middle term is exactly in the middle 
between the extremes, and therefore in the former three, ac 
r= 593, be ^z /^-ji, and r, the time between being 14 days, by 
the fir It rule, the time of the Iblftice will be found by this pro- 
portion; as 589 to 827 T ; lb is Y r or 7 days, to 9 days, 20 h. 
2' 5 whence the folftice An. 1500 is concluded to have fallen 
^une II, 20 h. 2'. In the latter ihree, ac is equal to 107, and 
^^ = 15, and the whole interval of time is 8 days = z^ j whence 
as 199 is to 106 T -J lo is 4 d. to 4 d. 3 h. 97', which, taken 
from the i6th day at noon,- leaves 11 d. 20 h. z^' for the time 
of the folllice, agreeing with the former to the third part of an 
hour. 
Again, An. 1 6 '^6, Gaffendus obferved at Marfeilles the fum- 
mer folftice by a gnomon of 55 foot high, in order to determine 
the projDortion of the gnomon to the folftitial fhadow, and he 
hath lefr us the following obfcrvations, which may lerve as an 
example for the fecond rule. 
June 20 C|i,adow) ^^ ' ^ ^' ^C P^'*^^' ^^''^'^/ '\ 
jfune zi C V5 17 5 iT gnomon was 89428 
y//;7^22 J O^^iyj^ 
Thefe being divided into two fets of 3 oblervations each ; viz, 
the i9tb, 20th and 22d3 and the loth, 21ft and 22d3 we fhall 
have 
