XXXIIX. which I shall not repeat here, but leave the Reader the pleasure of examining it himself; nor will it be amiss to consult Dr. Barrow's Learned Lectures on that Book, Publish'd at London, Anno 1684, viz. Probl. IX. and the Corollaries thereof.
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 (Tab. 4. Fig. 4.) which is the Analemma projected on the Plain of the Meridian, Z the Zenith, P the Pole, HH the Horizon, ææ the Æquinoctial, ♋♋, ♑♑ the two Tropicks, ♋1 the Sine of the Meridian Altitude in ♋; and equal thereto, but perpendicular to the Tropick, erect ♋I, and draw the Line TI intersecting the Horizon in T, and the Hour Circle of 6, in the Point 4, and 64 shall be equal to 6R, or to the Sine of the Altitude at 6: And the like for any other Point in the Tropick, erecting a Perpendicular thereat, terminated by the Line T I: Through the Point 4 draw the Line 4, 5, 7 parallel to the Tropick, and representing a Circle equal thereto; then shall the Tropick ♋♋ in Fig. 4. answer to the Circle NOPQ, in Fig. 3. the Circle 457 shall answer the Circle IKLM, T4I shall answer to the Elliptick Segment QIBKP, 6R or 64 shall answer to
SP, and 5I to BL, and the Arch ♋T, to the Arch LS, being the semidiurnal Arch in that Latitude and Declination; the Sine whereof, tho' not expressible in Fig. 4. must be conceived as Analogous to the Line TS or UP in Fig. 3.
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 Tropicks, (and the like for any two opposite Parallels) being multiplied by the Sine of the semidiurnal Arch, will give an Area*