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 Analogous to the Curve Surface RIMSQDP; and thereto adding in Summer, or substracting 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 the Meridian Altitude: The sum in one case, and difference in another, shall be as the Aggregate of all the Sines of the Sun's Altitude, during his appearance above the Horizon; and consequently of all his Heat and Action on the Plain of the Horizon in the proposed Day. And this may also be extended to the parts of the same Day; for if the aforesaid Sum of the Sines of the Meridian Altitudes, be multiplied by half the Sum of the Sines of the Sun's Horary distance from Noon, when the Times are before and after Noon; or by half their difference, when both are on the same side of the Meridian; and thereto in Summer, or therefrom in Winter, be added or substracted the Product of half the Arch answerable to the proposed interval of Time, into the difference of the Sines of Meridian Altitudes, the Sum in one case and Difference in the other, shall be proportional to all the Action of the Sun during that space of time.
I fore-see it will be Objected, that I take the Radius of my Circle on which I erect my Perpendiculars always the same, whereas the Parallels of Declination are unequal; but to this I answer, That our said Circular Bases ought not to be Analogous to the Parallels, but to the Times of Revolution, which are equal in all of them.
It may perhaps be useful to give an Example of the Computation of this Rule, which may seem difficult to some. Let the Solstitical Heat in ♋ and ♑ be required at London, Lat. 51° 32'.
| 380- 2'28 | Co-Lat | |
| 23-30 | Decl. ⨀ | |
| 61-58 | Sinus = | ,8826 4 |
| 14-58 | Sinus = | ,258257 |
| Summa | 1,140931 | |
| Diff. | ,624417 |
| Diff. Ascen. | 3300-1'1. |
| Arch. Semid. æstiv. | 123-11. |
| Ar. Sem. hyb. 56-49. S. | 638923 |
| Arch. æstiv. mensura | 2,149955 |
| Arc. hyb. mensura | 991683 |
Then 1,140931 in ,836923 + 624417 in 2,149955 = 2,29734. And 1,140931 in 836929 - ,624417 in ,991638 = 33895.
So that 2,29734 will be as the Tropical Summers Day Heat, and 0,33895 as the Action of the Sun in the Day of the Winter Solstice.
After this manner I computed the following Table for every tenth Degree of Latitude, to the Æquinoctial and Tropical Sun, by which an Estimate may be made of the intermediate Degrees.