VOL. XIX.] PHILOSOPHICAL TRANSACTIONS. 5 



more than the advantage to be gained by it. And, for the most part, it will be 

 safe enough, and least trouble, to divide by 5 a*, which gives a quotient some- 

 what too great ; which we may either rectify at discretion, by taking a number 

 somewhat less, or proceed to another b, affirmative or negative, as the case 

 shall require, and so onward to what exactness we please. Which is, for sub- 

 stance, in a manner coincident with Mr. Raphson's method, even for affected 

 equations. How far this method may be coincident with some of those before- 

 mentioned, I do not trouble myself to inquire, nor whether, or for what causes, 

 all or any of those may be more eligible. My design being only to show the 

 true natural ground from whence such rules of approach are, or might have 

 been derived, and by which they may be examined. 



In affected equations, especially where the coefficients are great, and some 

 affirmative, others negative, the cases will be more perplexed. And to multi- 

 ply rules for each case would increase the trouble, with no great advantage. 

 Which therefore I leave to the prudence of each to take some intermediate, 

 between a greater and a less. Or, if they please, to accommodate that in my 

 Commerc. Epistol. to the present case, which is there applied to a case not less 

 intricate. Or to make use of some of the methods delivered by others to this 

 purpose. Where this is to be considered, that such affected equations are 

 capable of more roots than one, according to the number of dimensions to 

 which they arise. 



A Method of discovering the true Moment of the Suns Ingress into the Tropical 

 Signs. By E. Halley. N° 215, p. 12. 



It may perhaps pass for a paradox, if not seem extravagant, if I should assert 

 that it is an easier matter to be assured of the moments of the tropics, or of 

 the times of the sun's entrance into Cancer and Capricorn, than it is to observe 

 the true times of the equinoctials, or ingress into Aries and Libra. I know the 

 opinion both of ancient and modern astronomers to the contrary, as Plolemy, 

 Riccioli, &c. and this because of the exceeding slowness of the change of the 

 sun's declination on the day of the tropic, being not a quarter of a minute in 

 24 hours. This indeed would make it very difficult, nor would any instru- 

 ments suffice to do it, were the moment of the tropic to be determined from 

 one single observation. But by three subsequent observations made near the 

 tropic, at proper intervals of time, I hereby design to show a method to find 

 the moment of the tropics, capable of all the exactness the most accurate can 

 desire ; and that without any consideration of the parallax of the sun, of the 

 refractions of the air, of the greatest obliquity of the ecliptic, or latitude of 

 the place ; all which are required to ascertain the times of the equinoctials from 



