78 ME M O I R S of the 
R, S, and dmw the lines RTS, Q^VP biffeaed in T and V? 
then is the curve furface R/;;SQJ)P equal to the rectangle of 
BL or;;;DandRS, or of ^BLor AD and STor UP5 and 
the curve furface QJN P D is equal to R S x mT) — the arch 
R;;?S X S P, or the arch MS x 2 S P ; or it is equal to the 
furface R;;?SQ^DP, fubftraaing the furface RwSQ^NP^ fa 
3ikewife the curve furface QJ3 P O is equal to the fum of the 
furface R^;?SQpP, or the arch LSx.SP: This is very 
eafily demonilrated from the confideration, that the cylindrical 
iurface IK LB is to the infcribed fpherical iurface IKLE, 
cither in the whole, or in its analogous parts, as the tangent B L 
is to the arch E L, and from the dcmonftrations of Archimedes^ 
ds Sphfjera ^ Cylindro I. i. \P)'op. 30, and 37, 39. 
Now to reduce our cale of the fum of all the fines of the fun's 
altitude in a given declination and latitude to the afbrefaid pro- 
blem, let us confider Fig. 6. Plate II. which is the Aialemma 
projeaed on the plane of the meridian ^ Z the Zenith 3 P the 
'^Fok'^ HH the horizon 5 dS ee the Equator^ 05 ©, ^'y Yf the two 
tropics y Sl the fine of the meridian altitude in S, and erea 
S I equal thereto perpendicular to the Tropic ^ and draw ths 
line T i interleaing the horizon in T, and the hour-circle of r) in 
the point 4, and 54 will be equal to 5R, or to the fine of the 
altitude at (^ ; and the lame holds for any other point in the 
tropic, ereaing a perpendicular thereat, terminated by the line 
T 1 5 thro' the point 4 draw the line 457, parallel to the T'ropic, 
and reprcfenting a circle equal thereto; then the ^Tropic ^ ^ \x\ 
Fig. d>. will anfwer to the circle N O P Q^ in Fig. 5 • the circle 
457 will anfwer the circle I KLM; T4 i will anfwer the elliptic 
fegment Q^I BKPj (jRor64 will anfwer to S P, and 5 i B L, 
and the arch vj" T to the arch L S, being the femidiurnal arch 
in that latitude and declination 3 the fine whereof, tho' not ex- 
preiled in Fig. 6. muft be conceived as analogous to the line T S 
or V P Fig. 5. 
The relation between thefe two figures being well underllood, 
it will follow from what preceeds, that the fum of the fines of the 
meridian altitudes of the fun in the two y5'-c;/)/Vi (and the like 
for any two oppofite parallels) being multiplied by the fine of the 
femidiurnal arch, will give an area analogous to the curve furface 
R ;;; S QJD P 3 and thereto adding in fummer, or fullraaing in 
winter, the produa of the length of the lemidiurnal arch, (taken 
axording to Van Qv///:;z's numbers ) into the difference of the 
aSovcfaid fines of meridian altitude 3 the fum in one cale, and 
ditfirrcncc in the other, will be as the a^^rei^ate of all the fines of 
the 
