Deg. M.

4

9 15.57
8 8. 32

74. 03

81. 28

Now, the Perpendicular Stile being Radius, the Tangents of the Complements of the respective Altitudes, as 62 deg. 59 m. the Complement of 27 de. of m. fet from the foot of the Per. Stile, on the Hour-line of 12 (or Sub-ftile) shall give a point thereon, by which the Parallel of Pisces must pass: And so, the Tangent of 64 de. 23 m. set from the foot of the Per. Stile, upon the Hour-lines of 11 and 1 a Clock, shall give you two other points by which the faid Parallel shall pass: And so for all the rest of the Hourlines, through which points found upon all the Hour-lines, a Line drawn by an even hard, shall be the Parallel required; for along that Line will the Shadow of the Top of the Per. Stile (as it creepeth along) pass, when the Sun is in the beginning of Pisces, viz. about the 9th of February.

And therefore, generally in Verticals, as also in all Recliners; that is to say, upon all Plains whatsoever: Draw an Horizontal Dial proper to the Plain, and infcribe 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 these Tangents fo set off upon every respective Hour-line, will be a Point: By which Points, Lines drawn with an even Hand, shall trace out upon the Dial Plain, the Parallels required.

Example. Suppose 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 several Hours, as in this Table.

:

De. M. 507. 55 16.30, 29. 57

Complements

:

Then,

Then, The Tangents of the Complements of these Hour-diftan-ces (as 7 deg. 55 m. for 12: 16 deg. 30 m. for the Hours of II and 1) fet off from the Foot of the per. Stile (the faid Stile being the Radius to these Tangents) to the obfcure Horizontal Hours of 12, 11, 10: and 1, 2, 3, &c. give the true distances between the Foot of the Stile, and those auxiliary Hours, for the Parallel of Taurus; and so points for the defcribing of other Parallels of Declination: Having first (by Trigonometrical Calculation) found the Horizontal Distances, and the Sun's Altitude at his entrance into those 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. '

CHAP. XV.

Of the Inscription of the Vertical Circles (commonly called Azimuths) upon all Dial Plains.

T

HESE are great Circles of the Sphere, whose Poles lie in the Horizon, and interfect one another in the Zenith and

Nadir Points of the Place wherein the Dial is to stand.

The whole Horizon being divided into 32 equal parts; these Circles paffing through those Divifions, are called Points of the Compass, 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.

First, in all Horizontal Dials; the Perpendicular Stile being chofen, making the Foot thereof the Centre; at any convenient distance, defcribe a Circle; and account from the Meridian both ways, Arches equal to 10, 20, 30, &c. Degrees: From which Divisions, right Lines drawn to the Foot of the Stile aforesaid, shall represent those 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, 6. Degrees; through which Divifions, Right-lines drawn, all at Right Angles with the Parallel Line, shall be the Azimuths.

300

Thirdly, Upon any Declining Vertical, the fame being done, shall give the Azimuths of 10, 20, 30, &c. degrees from the Meridian of the Plain; or from the Meridian of the Place, just allowance being made for the Difference of Meridians.

Fourthly, In South Declining Reclining Plains, the Per. Stile being chosen, and made the Radius, the Tangent Complement of the Reclination, applyed from the Foot of the Per. Stile to the Meridian of the Place, shall 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, shall be a Perpendicular to the Horizontal Line, and fhall concur with the Æquator 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 set off the Tangent of the Reclination, a Line drawn from the end thereof, at Right Angles with it, shall 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, shall 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 ease. For Example, let that distance be the Tangent of 20 deg. Then that Azimuth which is 10, from the one; is 10 from the other also: And that which is 30 on the same side of the Sub-stile, is 10 on the other fide of the Meridian of the Place: And the like method serves for any distance.

CHAP. XVI.

Of the Inscription of Almicanters or Circles of the Sun's Al
titude upon Dial Plains.

T
THESE 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 these be counted by 15 deg. and those usually by 10.

And

And therefore, as in the description of the Signs and Parallels; fo in these,

Let an Horizontal Dial, proper to the Plain, be first (obfcurely) described; and then, as it was there shewed, that the points through which the Signs or Parallels must pass, upon every Hourline, might be had by applying the Tangents of the Complements of the Sun's height of those Hours in those Parallels, from the Foot of the Per. Stile, to the respective Hour-lines: - So here, making use of that Azimuth which is perpendicular to the Plain, (which in all Plains is that which passeth through the Foot of the Per. Stile) the rest of the Azimuths being also 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 must pass.

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 Astronomical Problems; Or, by the Resolving of an Oblique Angled Spherical Triangle, where in is always Given, Two Sides, and the Angle opposite to one of them to find the third Side, (By CASE V. of O. A. S. T.) Which third Side so found, is the Complement of the Altitude which is in this case required; and must accordingly be set from the Foot of the Per. Stile unto the Azimuths, &c.

CHAP XVII.

How to inscribe the Jewish, Babylonish and Italian Hours, upon all Dial-Plains.

FOR OR the Inscription of these Hours upon Dial-plains, there needs no Trigonometrical Calculation : For the two Tropicks, the Æquator, and other Parallels of Declination being already described (or fuch of them as shall be needful) together with the common Hour-lines proper for the Plain, Points through which these Hour-lines inay pass, may be found by these following Di

rections.

The Babylonish Hours are accounted Equal Hours from Sun-rifing, and may be infcribed upon any Plain, by help of those two Pa

rallels of Declination which shew the Lorgest and Shortest Day, confifting of whole (or entire) Hours: As with us 16 and 8; and. the Equator. For,

Σ

7

rallel of

of Declination shall 16 of Declinations

Equinoctial

be the Hour of One from the Sun's rising. And likewife in the fame order; through 6, 8 and 10, shall give the Second Hour from Sun-rifing; and in the like Order, all the reft.

In Winter, when the Parallel of 8 Hours shall fail, the other. two Points will ferve to draw it by, because those Hours are Streight Lines. But after the first Six Hours are infcribed, the Equinoctial also failing, some other Diurnal Arch, (as of 9 or 10 Hours) must be described to supply that want.

The Italian Hours are accounted by 1, 2, 3, &c. from Sun-fetting: And for the Inscription of these, the fame Parallels of 8 and 16 Hours, with the Æquator, will ferve: For a Line drawn through them, in the Hours of 9, 7 and 5, Afternoon (observing the fame as before) shall be the hour of One; the like through 7, 5 and 3; shall be the hour 23.: The Night hours of 9, 10, &c. are the Morning hours produced.

The Jewish hours are reckoned like the Babylonish, from Sun-rifing, but unequally, their Sixth hour being (always) Noon; and every hour, one Twelfth part of the Artificial Day, of what length foever that be.

For the Infcription of them: The Vulgar Hours proper for the Plain being first drawn, and the Diurnal Arches of 15, 12, and 9 hours, divide the degrees in each by 12, and that Quotient by 15; or elfe (which is all one) divide the faid. Arches by 180, the three Quotients shall give the just Times, in hours, and usual parts of hours from 12 a Clock, upon the two Parallels and the Aquator: through which, Lines drawn by a Ruler, shall be the Jewish hours required.

Example. In Latitude 51 deg. 32 min. the Diturnal Arch of 15 hours is in Degrees 225, which divided by 180, the Quotient is 1h. and fo much the Jewish hours of 5 and 7 are diftant from Noon; one hour and a quarter being a twelfth part of the Diurnal Arch of 15 hours: And this hour and quarter being doubled, gives the place for 4 and 8: Tripled, the place of 3 and 9, &c. from Noon, upon that Parallel of 15 hours...

In