578 PHILOSOPHICAL TRANSACTIONS. [aNNO I693. 



meet with the said circle IKLM in the points R, S, and draw the lines RTS, 

 QVP, bisected in T and V ; then the curve surface RMSQDP is equal to the 

 rectangle of BL or MD and RS, or of 2BL or AD and STor V P; and the 

 curve surface QNPD is equal to RS X MD— the arch RMS X S P, or the 

 archMS X 'ZSP; or it is equal tothesurface RMSQDP, subtractingthe surface 

 RMSQNP. So likewise the curve surface QBPO, is equal to the sum of the 

 surface RMSQDP, or RS X MD, and of the surface RLSQOP, or the 

 archLSx2SP. 



This is most easily demonstrated from the consideration, that the cylindric 

 surface IKLB is to the inscribed spherical surface IKLE, either in the whole 

 or in its analogous parts, as the tangent BL is to the arch EL, and from the 

 demonstrations of Archimedes de Sphaera et Cylindro, lib. 1, prop, 30, 37, 39. 

 Now to reduce our case, of the sum of all the sines of the sun's altitude, in a 

 given declination and latitude, to the aforesaid problem, let us consider fig. 5, 

 which is the analemma projected on the plane of the meridian ; Z is the zenith, 

 P the pole, HHthe horizon, EE the equator, 25 25, Vy VJ the two tropics, 

 05 1 the sine of the meridian altitude in 25 ; and equal thereto, but perpendi- 

 cular to the tropic, erect 25 1, and draw the line T 1 intersecting the horizon 

 in T, and the hour circle of 6 : in the point 4, and 6 4 will be equal to 6R, or 

 to the sine of the altitude at 6 : and the like for any other point in the tropic, 

 erecting a perpendicular to it, terminated by the line T 1 : through the point 4 

 draw the line 4 5 7 parallel to the tropic, and representing a circle equal to it ; 

 then will the tropic 25 25 in fig. 5, answer to the circle N OP Q, in fig. 4 ; the 

 circle 4 5 7 will answer to the circle IKLM ; and T 4 1 will answer to the elliptic 

 segment QIBKP; also 6 R or 6 4 will answer to SP, and 5 1 to BL, and the 

 arch 25 T, to the arch LS, being the semidiurnal arch in that latitude and de_ 

 clination ; the sine of which, though not expressed in fig. 5, must be con- 

 ceived as analogous to the line TS or VP in fig. 4. 



The relation between these two figures being well understood, it will follow 

 from what precedes, that the sum of the sines of the meridian altitudes of the 

 sun in the two tropics, (and the like for any two opposite parallels) being mul- 

 tiplied by the sine of the semidiurnal arch, will give an area analogous to the 

 curve surface RMSQDP ; and thereto adding in summer, or subtracting in 

 winter, the product of the length of the semidiurnal arch, (taken according to 

 Van Ceulen's numbers) into the difference of the above-said sines of meridian 

 altitude ; the sun in one case, and difference in the other, will be as the aggre- 

 gate of all the sines of the sun's altitude, during his appearance above the 

 horizon ; and consequently of all his heat or action on the plane of the horizon, 

 in the proposed day. And this may also be extended to the parts of the same. 



