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Stile thereon Erected: Make the Perpendicular Stile the Radius (or Equal to the Tangent of 45 degrees) and make that part of the Sub-file which lies beyond the foot of the Stile and towards. the Centre, equal to the Tangent Complement of the heighth of the Pole (or Stile) above the Plain and the other part of the Subfile, below the Foot of the Stile, Equal to the Tangent Complement of the Meridian Altitude of the Sm, when he is in that Tropick which is to be moft remote from the Centre of the Dial
CHA P. XIV.
Of the Inscription of the Signs or Parallels of the Sun's
Sign is the Twelfth part of the the Ecliptick, and therefore contains 30 degrees.
A Parallel, is the Sun's Diurnal Motion Day by Day; and be-caufe there are 47 deg. between the two Tropicks, there may be fo many Parallels, that is, Circles, which the Sun defcribeth every 24 Hours: and although there be 47 of thefe, yet in the Lati tude of $1 deg. 30 min. we account but nine, viz. thofe, which are the Day from Sam to Sun, when it is 8, 9, 10, 11, 12, 13, 14, 15, or 16 juft hours long. The Defcription of thefe Parallels and Signs is made the fame way; only due refpect must be had to the quantity of the Sun's Declination: For (in all Direct Horizontals) the Perpendicular Stile being made Radius, the Tangent Complement of the Sun's height, in any Sign or Parallel at any hour of the Day, fet off from the Foot of the Stile, and extended to that Hour-line, gives a Mark upon the Hour-line, by which the Parallel of that Day fhall pafs: So that this Work, repeated fo often as the number of Parallels to be infcribed, and the Hour-lines require; fhall give refpective Points enough, in each Hour, to draw each Parallel by..
Example. In the Latitude of 51 de. 32 m. the Sun being in the beginning (or entring) of Pifces; The Sim's height above the Horizon at every Hour may be found (by CASE IX of O. AS. T) to be as followeth, viz..
10 21. 49
4 8 8. 32
Now, the Perpendicular Stile being Radius, the Tangents of the Complements of the refpective Altitudes, as 62 deg. 59 m. the Complement of 27 de. oi m. fet from the foot of the Per. Stile, on the Hour-line of 12 (or Sub-file) fhall give a point thereon, by which the Parallel of Pifces muft pafs: And fo, the Tangent of 64 de. 23 m. fet from the foot of the Per. Stile, upon the Hour-lines of 11 and 1 a Clock, fhall give you two other points by which the faid Parallel fhall pafs: And fo for all the reft of the Hourlines, through which points found upon all the Hour-lines, a Line drawn by an even hard, fhall be the Parallel required; for along that Line will the Shadow of the Top of the Per. Stile (as it creepeth along) pafs, when the Sun is in the beginning of Pifces, viz. about the 9th of February.
And therefore, generally in Verticals, as alfo in all Recliners; that is to fay, upon all Plains whatsoever: Draw an Horizontal Dial proper to the Plain, and inferibe the Signs or Parallels upon it, by fetting off from the Foot of the per. Stile, the Tangents Complements of the Sun's height at every hour in the beginning of every Sign above that Plain (taken as an Horizontal, the Foot of the per. Stile being ever, Radius) and at the end of thefe Tangents fo fet off upon every refpective Hour-line, will be a Point: By which Points, Lines drawn with an even Hand, fhall trace out upon the Dial Plain, the Parallels required.
Example. Suppofe a Plain Decline 30 deg. and Recline 55 deg. the height of the Pole above the Plain 19 deg. 25 min. And the Sun's height at the beginning of Taurus to be at the feveral Hours, as in this Table.
Then, The Tangents of the Complements of thefe Hour-diftan -ces (as 7 deg. 55 m. for 12: 16 deg. 30 m. for the Hours of 11 and 1) fet off from the Foot of the per. Stile (the faid Stile being the Radius to thofe Tangents) to the obfcure Horizontal Hours of 12, 11, 10: and 1, 2, 3, &c. give the true diftances between the Foot of the Stile, and thofe auxiliary Hours, for the Parallel of "Taurus; and fo points for the defcribing of other Parallels of Declination: Having firft (by Trigonometrical Calculation) found the Horizontal Diflances, and the Sun's Altitude at his entrance into thofe Parallels of Signs or Declination, in fuch Latitude as you have need of. All which are taught how to do in the foregoing parts of this Book.
CHA P. XV.
Of the Infcription of the Vertical Circles (commonly called Azimuths) upon all Dial Plains.
HESE are great Circles of the Sphere, whofe Poles lie in the Horizon, and interfect one another in the Zenith and Nadir Points of the Place wherein the Dial is to ftand.
The whole Horizon being divided into 32 equal parts; thefe Circles paffing through thofe Divifions, are called Points of the Compafs, and denominated accordingly, as South, S by E, SSE, &c. But the better way of accounting them is by 10, 20, 30, &'c. Degrees from the Meridian on either fide thereof.
Firft, in all Horizontal Dials; the Perpendicular Stile being chofen, making the Foot thereof the Centre; at any convenient diftance, defcribe a Circle; and account from the Meridian both ways, Arches equal to 10, 20, 30, &c. Degrees: From which Divifions, right Lines drawn to the Foot of the Stile aforefaid, fhall reprefent thofe Azimuths upon that Dial.
Secondly, Upon a Prime Vertical (or South) Dial: Through the Foot of the Per-ftile, draw a Right-Line Parallel to the Horizon; and making the faid Stile Radius; upon the Parallel Line, fet off, both ways from the Meridian Tangents of 10, 20, 30, &c. Degrees; through which Divifions, Right-lines drawn, all at Right Angles with the Parallel Line, fhall be the Azimuths.
Thirdly, Upon any Declining Vertical, the fame being done, fhall give the Azimuths of 10, 20, 30, &c. degrees from the Meridian of the Plain; or from the Meridian of the Place, juft allowance being made for the Difference of Meridians.
Fourthly, In South Declining Reclining Plains, the Per. Stile being chofen, and made the Radius, the Tangent Complement of the Reclination, applyed from the Foot of the Per. Stile to the Meridian of the Place, hall determine the Zenith of the Place: through. which, and the Foot of the Stile, (that is the Zenith of the Plain) a right Line drawn, fhall be a Perpendicular to the Horizontal Line, and fhall concur with the Equator in the Hour-line of 6 and therefore, if from the Foot of the Stile upon the faid Perpen-.. dicular, towards the North (for the former application was made towards the South) be fet off the Tangent of the Reclination, a Line drawn from the end thereof, at Right Angles with it, fhall be the Horizontal Line: Upon which, the Tangents of 10, 20, 30,&c. . (the Secant of the Reclination being now made Radius) fet from the faid Right Angle; Lines drawn from them to the Zenith of the Place, fhall be the Azimuths.
Fifthly, The Distance between the Meridians being known, upon the Horizontal Line; the Azimuths which were accounted from the Meridian of the Plain, may be fitted for the account from the Meridian of the Place, with eafe.. -For Example, let that diftance be the Tangent of 20 deg. Then that Azimuth which is 10, from the one; is 10 from the other alfo: And that which is 30 on the fame fide of the Sub-ftile, is 10 on the other fide of the Meridian of the Place: And the like method ferves for any diftance.
Of the Infcription of Almicanters or Circles of the Sun's Al
HESE are leffer Circles of the Sphere; and may be called the Parallels of Declination from the Horizon; they having. in all refpects, the fame relation and habitude to the Azimuths, as the Signs and Parallels of Declination have to the Meridians; although thefe be counted by 15 deg, and thofe ufually by 10.
And therefore, as in the defcription of the Signs and Parallels; fo in thefe,
Let an Horizontal Dial, rroper to the Plain, be firft (obfcurely) defcribed; and then, as it was there fhewed, that the points through which the Signs or Parallels muft pafs, upon every Hourline, might be had by applying the Tangents of the Complements of the Sun's height of thofe Hours in thofe Parallels, from the Foot of the Per. Stile, to the refpective Hour-lines: So here, making ufe of that Azimuth which is perpendicular to the Plain, (which in all Plains is that which pafieth through the Foot of the Per. Stile) the reft of the Azimuths being alfo infcribed, the Tangents Complements of the Sun's height above the Plain, when he is in any Azimuth, applyed from the Foot of the Per. Stile to the faid Azimuth, gives a Point, through which that Circle or Almicanter, upon that Azimuth muft pafs.
Now to know what Altitude the Sun will have, when he will be upon any Azimuth, in any Parallel of Declination (or degree of the Ecliptick) is taught in the Section of Aftronomical Problems; Or, by the Refolving of an Oblique Angled Spherical Triangle, where in is always Given, Two Sides, and the Angle oppofite to one of them to find the third Side, (By CASE V. of O. A. S. T.) Which third Side fo found, is the Complement of the Altitude which is in this cafe required; and muft accordingly be fet from the Foot of the Per. Stile unto the Azimuths, &c.
How to infcribe the Jewish, Babylonifh and Italian Hours, upon all Dial-Plains.
OR the Infcription of thefe Hours upon Dial-plains, there needs no Trigonometrical Calculation: For the two Tropicks, the Equator, and other Parallels of Declination being already de fcribed (or fuch of them as fhall be needful) together with the common Hour-lines proper for the Plain, Points through which thefe Hour-lines may pafs, may be found by thefe following Di
The Babylonish Hours are accounted Equal Hours from Sun-rifing, and may be infcribed upon any Plain, by help of thofe two PaEee 2