132 M E M O I R S o//;&« 
cuius of thefe famous authors , lo that it is an eafier matter to be 
aflurcd of the moments of the lolilices or r>f the times of the fun's 
entrance into Cancer and Capikorn^ than it is to obierve the true 
times of the equinoxes, or his ingrefs into Aries and Libra, 
"Mr. HciUev. before he proceeds, thinks it neceffary to premife 
the following Lemmata^ which lerve to demonftrate this method, 
ijiz. I. That the motion of the fun in the eel :ptic, ab«^ut the 
time of the folftices, is {o nearly equable, that the difference 
from equality is not ienfible, from 5 days before the Iblflice to 5 
days after it, by reaion of the nearnels of the apogason of the fun 
to the '^I'rofic of Cancer. 2. That f )r five degrees before and 
after the folftices, the differences, whereby the fun falls fhort of 
the 'Tropes^ arc as the verled fines of the fun's diilance in longi- 
tude from the Tropics:, which veried fines in arches under 5 de- 
grees, are beyond the utmoft nicety of lenle, as the fquares of 
thofe arches^ from thele two there follows a third. 5. That 
for 5 days before and after the folftices, the declination of the 
fun falls fliort of the utmoft tropical declination, by ipnces which 
are as the fquares of the times, the fun is fhort of, or paft, the 
folfticicd moment. Hence it is evident, that if the ihadows of 
the fun, either in the meridian or any other azimuth, be carefully 
obferved about the time of the folftices, the fpaces, whereby the 
tropical Ihadow falls ihort of, or exceeds, thole at other times, 
are always proportional to the iquares of the intervals of time 
between thofe oblervations, and the true time of the Iblftice ^ and 
confequcntly, if the line, on which the limits of the fhadow are 
taken, be m.ade the axis, and the correfpondent times fi-om the 
Iblftice, expounded by lines, be erected on their relpe61:ive points 
in the axis as ordinates, the extremities of thole lines will touch 
the curve of a parabola: Thus, a^ by c, Fig. 8. Plate V. being 
fuppofed to be the points obferved, the lines ^B, ^C, <: A, ^F, 
are refoe^livcly proportional to the times of each oblervation be- 
fore or after the Iblftitial moment in Cancer. 
This being prcmiicd, we (hall be able to bring the problem of 
finding the true time of the folfticc by 5 oblervations to this geo- 
metrical one; having 5 points in a parabola, A, B, C, or A, F, 
C, given, together with the die^tion of the axis, to find the dil- 
tance of thole points from the axis. Of this there are two cales, 
the one, when the time of the fecond obfervation B is precifely in 
the middle between A and C; in this cale, putting t f^r the 
whole time between A and C, we fliall have kc, the interval of 
the remoteft obfervation A, from the "Tropic^^ by the following 
analogy : As a ^ f — be is to zac — f be, lb is r iT or A £ to 
A.- 
