6 PHILOSOPHICAL TRANSACTIONS. [aNNO 1694-5. 



observation, and which being faultily assumed, have occasioned an error of 

 near 3 hours in the times of the equinoctials deduced from the tables of Tycho 

 Brahe and Kepler ; the vernal beinfj so much later, and the autumnal so much 

 earlier, than by the calculus of those famous authors. 



Now before we proceed, it will be necessary to premise the following lem- 

 mata, serving to demonstrate this method, viz. 1. That the inotion of the 

 sun in the ecliptic, about the time of the tropics, is so nearly equable, tiiat the 

 difference from equality is not sensible, from 5 days before the tropic, to 5 days 

 after ; and the difference arising from the little inequality that there is, never 

 amounts to above 4- of a single second in the declination, and this by reason 

 of the nearness of the apogoeon of the sun to the tropic of Cancer. 2. That 

 for 5 degrees before and after the tropics, the differences by which the sun falls 

 short of the tropics are as the versed sines of the sun's distance in longitude 

 from the tropics ; which versed sines, in arciies under 5 degrees, are be)ond 

 the utmost nicety of sense, as the squares of tliose arches. From these two 

 follow a third ; 3. That for 5 days before and after the tropics, the declination 

 of the sun falls short of the utmost tropical declination, by spaces which are 

 in duplicate proportion, or as the squares of the times, by which the sun is 

 wanting of or past the moment of the tropic. 



Hence it is evident, that if the shadows of the sun, either in the meridian 

 or any other azimuth, be carefully observed about the time of the tropics, the 

 spaces by which the tropical shade falls short of, or exceeds, those at other 

 times, are always proportionable to the squares of the intervals of time between 

 those observations and the true time of the tropic; and consequently if the 

 line, on which the limits of the shade is taken, be made the axis, and the cor- 

 respondent times from the tropic expounded by lines be erected on their res- 

 pective points in the axis, as ordinates, the extremities of those lines shall touch 

 the curve of a parabola, as may be seen in fig. 1, pi. l. Where a, b, c, e, 

 being supposed points observed, the lines ah, b c, ca, ep, are. respectively pro- 

 portional to the times of each observation before or after the tropical moment 

 in Cancer. 



This premised, we siiall be able to bring the problem, of finding the true 

 time of the tropic by three observations, to this geometrical one, having three 

 points in a parabola a, b, c, or a, f, c, given, together with the direction of 

 the axis, to find the distance of those points from the axis. Of this thtre are 

 two cases, the one when the time of the second observation b is precisclv in 

 the middle time between a and c ; in this case putting i for the whole time be- 

 tween a and c, we shall have At, the interval of the remotest observation a 

 from the troj ic, by the following analogy : as 2 a i: ~ b c to 2 a c — -^ b c :: so 



