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Since thePclarPlane POQS (Fig-u.) paffcs through the Pall an< l ^ , ' e " : P°' nts and S, a Quadrant of the Equa- tor is intercepted between it, and the Meridian : Confe- quently the Upper Surface is illumined by the Sun from fix in the Morning to fix at Night ; and the Lower from the gun's Rife to fix in the Morning, and from fix in theEvcn- ino to Sun-fet. A Lower "Polar Dial, therefore, fhews the Hour of the Morning from Sun-Rife to fix a-Cloclc, and that of the Evening from fix to Sun-fet ; And an Upper, the Hours from fix in the Morning to fix in the Evening.
"to draw an Upper "Polar Diki..
Draw i right Line AB (Fig.i 3.) parallel to the Horizon, jtnd if <hc Plane be immoveable, find the Meridian Line CE. Divide CE into two equal Parts, and through C draw a right Line FG parallel to AB. Then from the Centre D with the Interval DE, defcribe a Quadrant, which divide into fix equal Parts. And from the fame Centre D through the feveral Points ofDivifion draw right Lines Dl, Di, B;, D4, D 5 ; and the Intervals E 1, E 2, E 3, E 4, E 5, frt off the contrary Way from E 11, 10, 0, 8, and 7. From the Points 5, 4, 5, 2, t, &c. raife Perpendiculars meeting the Line FG in the correfpondent Points. Laftly, in D erect a perpendicular Style equal to DE ; or on two equal Pieces E and C, fix a crofs Iron Rod.
Then will 12,12, 1,1, 2,2, 3,3; tSc. be Hour-Lines, to be pointed out at the proper times by the Shadow of the Indices.
An Upper "Polar Dial only differs in Situation, and the Manner of writing in the Hours, from Eaft, and Weft SJials joyn'd together in the Line of fix a-Clock,
A Lower "Polar "Dial is had by putting out the Hours of the Fore-noon 0, 10, and ir, and thofe of the After-noon 1, 2. and 3, with the Noon-hour 12 it {elf : and only leaving the Hours 7 and 8 of the Morning, and 4 and 5 in the Evening.
"to draw all the Primary Dials on the fame "Block, or Poft.
1. Let thePlane ABCD (Fig.14.) in the proper Pofition of the Block be fuppofed Horizontal 3 And thereon de- fcribe a Horizontal 'DM. See Horizontal Dial.
2. Draw the right Lines EM, and FL parallel to DC, which accordingly, in the proper Pofition of the Block, will be parallel to the Horizon. Then let the Plane BNMC make an Angle with EM, equal to the Elevation of the Pole, CME : and thereon defcribe an Uj-per 'Polar Dial. See "Polar Dial.
3. Let the Oppofite Plane ADE make with EM an An- gle DEM, equal to the Elevation of the Equator : And on this draw an Upper Equinoctial Dial. See Equinoctial Dial.
4. Let the Plane KLHI make with FL an Angle HLF : And on this inferibe a Lower Equinoctial Dial. See Equinoctial Dial.
5. Let the oppofite Plane FG make with FL an Angle FGL equal to the Elevation of the Pole : And here draw a Lower Polar Dial. See "Polar Dial.
6. Let the Plane MNKL, and the oppofite one EE be perpendicular to FL : And on that draw a South Dial, and on this a North Dial. See South North and Dial.
7. In the Plane EMLF defcribe a Weft Dial; and on the oppofite Plane an Eaft Dial. See Eaft and Weft Dials.
If then the Block be fo placed, as that the Plane MNKL looks to the South, and the Plane of the Meridian biftect it by the Line of 12 a- Clock in the Horizontal Dial ABCD, and South Dial MNKL ; They will all tell the Hour oftheDay.
Secnndary Dials.
Seamdary Dials are all thofe drawn on the Planes of other Circles, befide the Horizons, prime Vertical, Equi- noctial, and Polar Circles : Or thofe, which either De- cline, Incline, Recline, or Deincline.
Decliningt)iAi.s are EreS, or Vertical Dials, which decline from any of the CardinalPoints 5 Or, more ftrictly, which cut either the Plane of the Prime Vertical, or of the Horizon at oblique Angles;
The Ufe of Declining Dials is very frequent ; as the Walls ofHoufes, whereon Dials are ufually drawn, com- monly deviate from the Cardinal Points. See Declines..
Of Declining Dials there are feveral Kinds, denomi- nated from the Cardinal Points, which they feem moll to refpeft, but from which they have a real Declination : Zlecliners from the South, and from the North, and even from the Zenith.
Vertical Declining Dials, "Trigonometrically-.
"Prob.i. The Declination of the plane, and the Eleva- tion of the Pole of the Plane, being given : To find the Angle form'd in the Centre of the Dial, by the Meridian, and Subftyle.
- Analogy: As the whole Sine is to the Tanrent of the
Complement of the Height of the Pole of ,L ll "c C GF (Fig.!,.) /o is theSmeoftheffl.c&^f^^^l GFD to the Tangent GD, of the Angle required GAD
"Prob. 2. The Declination of the Plane S d^T ™4 the Elevation of the Pole of the Place; » fiMJMc form'd in the Centre of a Vertical Declining Dial bv the Subftyle and Axis. ' '
Analogy : As the whole Sine is to the Sine of the Com plement of the Elevation of the Pole GFj fo is flu Sine of the Complement of the Declination of the Plane DGF to the Side DF, the Sine of the Angle DAB required. '
"Prob. 3. The Declination of the Plane, and [he Eleva- tion of the Pole, given; to find the Difference of Longi- tudes, that is, the Arch of rhe Equator comprehended be- tween the Meridian of the Place, and the Meridian of the Plane.
Analogy : As the whole Sine is to the Sine of the Height of the Pole ; fo is the Tangent of the Complement o the Declination of thePlane, to the Tangent of the Com- plement of the Difference ofLongitudes.
"Prob*. The Angle of the Difference ofLongitudes, and that of the Axis, with the Subftyle, being given ; to find the Angles form'd in the Centre of a Vertical Declining Dial between the Subftyle and Hour-Lines.
This "Problem admits of three Cafes. For the Hour- Lines, whofe Angles are fought, may be, either, 1. Be- tween the Meridian, and Subftyle; or, 2. Beyond the Sub- ftyle ; or, 3. On that Side the Meridian the Subftyle is not. In the two firft Cafes, the Difference is to be taken between the Suns Diftance from rhe Meridian that Hour, and the Angle of the Difference of Longitudes found by the laft Problem: And in the third Cafe, the Sum of thofe two Angles is to be taken ; and the following Analogy ufed.
Analogy : As the whole Sine is to the Sine of the An- gle between the Axis and Subftyle, CM ; fo is the Tan- gent of the Difference of the Sun'sDiftance from the Meri- dian, and the Difference of Longitudes, or of the Sum of thofe two Angles ; to the Tangent of the Angle required.
Prob. 5. The Angle form'd by rhe Subftyle, with the! Hour-Lines, and that of the Subftyle, with the Meridian, given ; to find the Angles form'd between the Meridian) and Hour-Lines in the Centre of Vertical Declining Dials±
1. The Angles of the Hour-Lines between the Meridian, and Subftyle, are found by fubftrafting the Angle form'd by the Subftyle, with the Hour-Line, from the Angle form'd by the Subftyle, with the Meridian.
2. The Angles beyond the Subftyle, and on the Side oppofite to that of the Meridian, are found by addino thofe two Angles.
3. Thofe on the other Side the Meridian, are found by taking their Difference.
To defcribe a Vertical Dim, declining from the South to the Eaft, or Weft, Geometrically.
Find the Declination of the Plane, as already taught un- der the Article Declination, andDECLjNEK.
Then draw upon Paper a Horizontal Dial, fuppo- fing the Line of Contingency of the Horizontal, with the Equinoaial Plane to be GH, (Fig. 16.) Through the Point E, wherein the Meridian Line AE cuts the fame, draw a right Line IK, making with GH an Angle HEK equal to the Declination of the given Plane. Thus, as GH repre- fents the Interfefiion of the Prime Vertical, with the Ho- rizon ; IK will be the Interieflion of the Inclining Plane, and the Horizon; Whence we alfo conceive, that the Part IE muft be railed above GE, in Cafe the given Plane de- cline to the Weft: Or be deprefs'd below the fame GE in Cafe it decline to the Eaft. Draw a right Line parallel to the Horizon on the given Plane, or Wall, to anfwer to IK, and affirming a Point therein, anfwering to E fet off from the right Line IK on the Paper, the feveral Horary Diftances E 1, E 2, E 3, SSc. Then from rhe Point E erect a Perpendicular EC, equal to the Diftance of the Centre of the Horizontal Dial, fron its Line of Contingency. Draw Lines thence to the feveral Points of the Hour-Lines E I E 2, E 3, e?c. let fall a Perpendicular AD from the Cen- tre of the Horizontal Dial A to the Line of Contingency IK, and transfer the Diftance ED from the Point i ? up- on the Wall ; then will CD be the Subftylar Line. See
SUBSTYLE.
Wherefore, joyning AD and DC at right Angles ; the Hypothenufe AC will be an oblique Index to be faften'd on the Wall in the Point C, according to the Angle DCA.
"Fo draw a Vertical Dial declining from the
North towards the Eaft, and Weft.
Take the Declination of the Plane, as already taught s
Then, as North Dials are only South Dials inverted, draw
a Vertical Dial declining from the South, and invert it
in fuch manner, as that the Centre C look to the Horizon;
and the Point E to the Zenith; and the Hours on the ripht
L* Ffand
