VOL. XI.] PHILOSOPHICAL TRANSACTIONS. 331 



let the sine of the angle AB C = S, and its cosine = s. Then as c : Z> :: 2 a — 

 Z,:^iizL^and,i^^^^^=BDby36.3. Eucl. and as / : rf:: 2a + d: 



^° "*" — ; and • • ~ ^ "'~ ^ ' = B G by the same proposition ; and to lessen 



the trouble of calculation, let^^^^^ — = g, and - = h, likewise let - ^-^ ~ — 



= kf and -2. =z I; then BD :=z g -\- ha, and BG =z h •— la ; and because in 



obtuse angled trianerle S the square of the base 1 the sum /.^, 



every , , . i "1 • , r , ,m ofthesum 



•^ acute angled triangle I is equal to J the diirerence 



of the squares of the sides and of double the rectangle of the sides multiplied 

 into the cosine of the comprehended angle, it will he gg -\- Igha + hhaa -\- 

 kk — q,kla + llaa + 2^-^^ — Iglsa + ihhsa — ihlsaa equal to the square 

 of DG ; but DG is equal to the sine of the angle DFG or DBG multiplied 

 into a, that is, into FB ; for the quadrilateral FBDG is inscribed in a circle 

 whose diameter is FB: therefore SS aa = gg -|- Igha + hhaa -\- kk — 

 Qjkla. ■\- llaa + 2gks — Iglsa -\- ikhsa — ihlsaa, which equation is easily 

 resolved, since it does not exceed an adfected quadratic, and always consists of 

 these squares and rectangles; but the signs -\- and — are to be very cautiously 

 used to the rectangles, on account of the different nature of the three lines. 

 We have fitted our equation to tig. 8; but in any other case it will not be dif- 

 ficult from what is said above to constitute a similar one; and thus it is shown 

 how from three heliocentric places of a planet, and the observed distances from 

 the sun, its orbit may be described ; which formerly required five such observa- 

 tions. 



An Extract of an account given by Mr. Flamsteedj of his own and Mr. Edmund 

 Halleys Observations concerning the Spots in the Sun, appearing in July and 

 August, 1676. N"^ 128, p. 687. 



The following ephemeris of their daily places, was reduced from careful ob- 

 servations, made with the micrometer, of the distances of the spots from the 

 limb of the sun, and the differences of altitudes and azimuths from the upper and 

 under parts and sides of him. The comparing of the observations made in two dis- 

 tant places, Greenwich and Oxford, evince the diligence of the observers, and the 

 goodness of their instruments ; the differences between them being easily excu- 

 sable ; for that the spot had a diameter more considerable than any of the dif- 

 ferences, and was broken into several pieces. 



Mr. Halley says, that he saw the spot again on the fifth day at 8h. 30 m. mane, 

 very near the limb of the sun, so that it appeared only as a fine line; but by 



u u 2 



