VOL. LIII.j PHILOSOPHICAL TRANSACTIONS. Jl 



inculcated, in an age in which it has been so little known, and almost totally 

 neglected, Dr. P. closes this paper with an example in each species of problems. 

 Thus, were it proposed to draw a triangle given in species, that two of its angles 

 might touch each a right line given in position, and the 3d angle a given point. 

 It is obvious how difficult it would be to adapt a commodious algebraic calcula- 

 tion to this problem; yet it admits of more than one very concise solution, as 

 follows. Let the lines given in position be ab, ac, and the given point d, the 

 triangle given in species being edf, fig. 19, 20, 21. 



In the first place suppose a circle to pass through the 3 points a, e, d, which 

 shall intersect AC in g, fig. IQ. Then kg, dg being joined, the angle deg will 

 be equal to the given angle dac, both insisting on the same arch dg: also the 

 angle edg is the complement to two right ones of the given angle bag: these 

 angles therefore are given, and the whole figure efgd given in species. Conse- 

 quently the angle egf, and its equal ade, will be given, together with the side 

 DE of the triangle in position. 



Again, suppose a circle to pass through the 3 points a, e, f, cutting ad in h, 

 and EH, FH joined, fig. 20. Here the angle eph will be equal to the given angle 

 eah, and the angle feh equal to the given angle fah. Therefore the whole 

 figure EHFD is given in species, and consequently the angle ade, as before. 



In the last place, suppose a circle to circumscribe the triangle, and intersect 

 one of the lines, as ac, in i, fig. 21. Here di being drawn, the angle dip will 

 be equal to the given angle dep in the triangle ; consequently di is inclined to 

 AC in a given angle, and is given in position, as also the point i given ; whence, 

 IE being drawn, the angle pie will be the complement of the angle edf in the 

 triangle to two right ones. Therefore ie is given in position, and by its intersection 

 with the line ab gives the point E, with the position of de, and thence the 

 whole triangle, as before. Here it may be observed, that the angle d of the 

 triangle edf given in species touching a given point d, and another of its angles 

 touching AC, the line ie here found is the locus of the 3d angle e. 



Again, in the astronomical lectures of Dr. Keil, it is proposed to find the 

 place of the earth in the ecliptic, whence a planet in any given point of its orbit 

 shall appear stationary in longitude, and a solution is given from the late eminent 

 astronomer Dr. Halley, on the assumption, that the orbit of the earth be con- 

 sidered as a circle concentric to the sun. But for a complete solution of this 

 problem let the following lemma be premised. The velocity of a planet in lon- 

 gitude bears to the velocity of the earth, the ratio which is compounded of the 

 subduplicate ratio of the latus rectum of the greater axis of the planet's orbit, to 

 the latus rectum of the greater axis of the earth's orbit, of the ratio of the co- 

 sine of the angle which the orbit of the planet makes with the plane of the eclip- 

 tic, to the radius, and of the ratio of a line drawn in any angle from the centre 



