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THis Diagram of the Gnomical Scale I have described Book 2. Chap. 3. to which I refer you.
I shall only describe the Lines on the Scale, viz.
There are six Scales or Lines on this Instrument.
1. A Scale or Line of Chords, AD, in Degrees.
2. A Gnomon Scale or Line, as BD, the use thereof you may see in Chap. 7. and so forward in Degrees.
3. A Scale or Line of six Hours, for drawing the Hour-lines in any Dial, divided into every 10 Minutes; the use you may read from Chap. 4. forward, as BEA.
4. A Scale of Inclination of Meridians, divided in Degrees as the Diam. BA; the use thereof is in Chap. 12. and so forward.
5. Upon the other side are two Lines or Scales, for the inlarging the Hour-Lines on any Plain: The greater Pole is marked with +; the lesser is marked with −, called the lesser Pole for distinction sake; and these Lines are divided into every 10 Minutes, but may be by the Table into every 5 Minutes.
6. These Scales or Lines are on the Mathematical Scale, with the rest that are descri¦bed Book 2. called The Scale of Scales.
ALthough Gnomoniques pertain to Astronomy, yet I think it not amiss for the ease of the Reader, to place these in a distinct Book by themselves.
Sun-Dials may be reduced to two sorts. Some shew the Hour by the Altitude of the Sun, as Quadrants, Rings, Cylinders; and for the making thereof, you must know the Suns Altitude for every day, or at least every tenth day of the year, and for every hour of those days.
The other sort shew the hour by the shadow of a Gnomon or Stile parallel to the Axis of the World; and of that I treat chiefly in this Book. These be all Projecti∣ons of the Sphere, upon a Plane which lies parallel to some Horizon or other in the World. And if upon such a Plane the Meridians only be projected, they shall suf∣fice to shew the Hour, without projecting the other Circles, as the Ecliptique, the Aequator with his Parallels of Declination, the Horizon with his Almicanters and Azimuths, which are sometimes drawn upon Dials more for Ornament than for Necessity.
FOr the better understanding of the Reasons of Dials, these Theorems would be known.
I. That every Plane whereupon any Dial is drawn, is part of the Plane of a Great Circle of the Heaven, which Circle is an Horizon to some Country or other; That the Center of the Dial, representeth the Center of the Earth and World; and the Gnomon which casteth the Shade, representeth the Axis, and ought to point directly to the two Poles.
II. That these Dial Planes are not Mathematically in the very Planes of Great Cir∣cles; for then they should have their Centers in the Center of the Earth, from which they are removed almost 4000 miles; and yet we may say they lye in the Planes of Circles parallel to the first Horizon, because the Semidiameter of the Earth beareth so small proportion to the Suns Distance, that the whole Earth may be taken for one Point or Center, without any perceivable Error.
III. That as all Great Circles of the Sphere, so every Dial Plane hath his Axis, which is a straight Line passing through the Center of the Plane, and making Right An∣gles with it; and at the end of the Axis be the two Poles of the Plane, whereof that above our Horizon is called the Pole Zenich, and the other the Pole Nadir of the Dial.
IV. That every Plane hath two Faces or Sides: and look what respect or situation the North Pole of the World hath to the one side, the same hath the South Pole to the other; and these two Sides will receive 24 Hours always: so that what one Side wanteth, the other Side shall have; and the one is described in all things as the other.
V. That as Horizons, so Dial Planes are with respect to the Aequator divided into first, Parallel or Aequinoctial; secondly, Right; thirdly, Oblique Planes.
VI. A Parallel or Polar Plane maketh no Angles with the Aequator, but lies in the Plane of it, or parallel to it; that is, hath the Gnomon erected on the Plane at Right Angles, as the Axis of the World is upon the Plane of the Aequator: be∣cause the Axis and Poles of the Dial are here all one with the Axis and Poles of the World, and the Hour-lines here meet all at the Center, making equal Angles, and dividing the Dial Circle into 24 equal parts, as the Meridians do the Aequator, in whose Plane the Dial lies.
VII. A Right Horizon or Dial Plane cutteth the Aequator at Right Angles, and so cutteth through the Poles of the World, that it hath the Gnomon parallel to the Plane, and so the Hour-lines parallel one to another; because their Planes, though infinitely extended, will never cut the Axis of the World: yet have those Dials a Center, though not for the meeting of the Hour-lines, viz. through which the Axis of the Dial Circle passeth, cutting the Plane at Right Angles, and cutting also (neer enough for the projecting of a Dial) the Circle of the World.
VIII. An Oblique Horizon or Dial Plane cutteth the Aequator at Oblique Angles; that is, hath for their Gnomon the side of a Triangle, whose Angles vary according to the more or less Obliquity of the said Horizon: and the Gnomon shall always make an Angle with the Plane, of so many Degrees as the Axis of the World maketh with the Plane, or as either of the Poles of the World is elevated above the Plane.
IX. Every Oblique Horizon is divided by the Meridians or Hour-circles of the Sphere into 24 unequal parts; which parts are always lesser, as they are neerer to the Meridian of that Horizon or Plane; and greater, as they are farther off: and on both sides of the Meridian of the Plane, the Hour-circles which are equally distant in time, are also equally distant in space. Whence it is, that the divisions of one Quadrant of your Dial Plane being known, the divisions of the whole Circle are likewise known.
X. The Hour-lines in an Oblique Dial, are the Sections of the Planes of the Hour-circles of the Sphere, with the Dial Plane: and because the Planes of Great Circles do always cut one another in Halves by Diameters, which are straight Lines pas∣sing
XI. Every Dial Plane being an Horizon to some place in the Earth (as was said Theo∣rem I.) hath his proper Meridian, which is the Meridian cutting through the Poles of the Plane, and making Right Angles with the Plane. If the Poles of the Dial Plane lie in the Meridian of the Place, then is the Meridian of the Plane all one with the Meridian of the Place, and the Gnomon or Style shall stand erected upon the Noon-line, or Line of 12 a Clock, as in all direct Dials. But if the Plane decline, then shall the Substyle Line, or Line which the Gnomon standeth upon, which is the Meridian of the Plane, vary from the Line which is the Meri∣dian of the Place; and this Variation shall be East, if the Declination be West of the Plane: And contrarily, because the Visual Lines, by which the Sphere is pro∣jected on Dial Planes, do, like the Beams of a Burning-glass, intersect or cross one another in a certain Point of the Gnomon (to be assigned at pleasure, and cal∣led Nodus) and so do all place and depaint themselves on the Dial Plane, beyond the Nodus, the contrary way.
XII. Dials are most aptly denominated from that part of the Sphere where their Poles lie, though some Authors have chosen to denominate them from the Circles in which their Planes lie; as the Dial Plane which lieth in the Aequinoctial, or Parallel to it, is called by many an Aequinoctial Plane; but I concur with those who would ra∣ther call it a Polar Plane, because the Poles thereof are in the Poles of the World.
THe Plane of the Polar Dial lieth in the Aequinoctial, where the 12 chief Meridians or Hour-circles divide both the Aequinoctial and this Plane into 24 Hours or equal parts; the Gnomon stands upon the Center at Right An∣gles with the Plane.
First draw the Horizontal Line AB, and cross the same at Right Angles with the Line CD; now on the Center at G, with the Chord of 60 Degrees, or with the Tangent of three hours, you may describe the Circle ACBD, and about it make the Square EFHI; then take out of the Hour-line one Hour, and lay it from each corner, as EFHI both ways: also do the like with two hours as you see done, and from the Center at G draw Lines to those Hour-points: so shall you have the Hour-lines in the Aequinoctial Dial; CD being the Meridian or 12 a clock Line, and AB the East and West Line, serving for 6 in the morning at B, and 6 in the afternoon at A, and so number the rest of the Hours in order: You need draw no more hours than from 4 in the morning unto 8 at night, for this Latitude of Bristol, being neer 51 d. 30 min.
For the Gnomon or Stile, you must have a straight Pin or Wyre set upright in the Center, of such length as you see convenient; but if you will have it of such a length as may neither be too short nor too long, then take this Rule.
IF it were required to proportion the Stile to the Plane, suppose the Semidiameter of the greatest Parallel upon the Plane were but 6 Inches, and the Parallels should be the 5 d. of Declination, the Rule is general.
| As the Tangent of 45 deg. | 1000000 |
| Is to the Tangent-parallel of Declination 5 deg. | 894195 |
| So is the Semidiameter of the Plane 6 inches OA | 277813 |
| To the length of the Stile 53 parts | 172010 |
SUppose the length of the Stile above the Plane to be 10 inches, and you were to find the Semidiameter of the Tropick, whose Declination is known to be 23 deg. 30 min. the Rule is for this and any other Declination,
| As the Tangent of 45 deg. | 1000000 |
| Is to the length of the Stile 10 inches | 100000 |
| So is the Co-tangent of Declination 23 deg. 30 min. | 1036169 |
| To the Semidiameter of his Circle 23 inches | 136169 |
which shews the Semidiameter of the Tropick to be 23 inches: So if the Declination be 20 d. the Semidiameter will be 27 inches 47/100; if 15 d. then 3732/100; if 10 d. then 56 71/100; if 5 d. then 114 305/1000; and so of any other height of the Stile: as admit it were 53/100 parts of an inch high, then the Semidiameter of 23 deg. 30 min. would be 1 21/100, and for 20 deg. it will be 1 57/100, and for 15 deg. 1 96/100; if 10, then 2 98/100; if 5 deg. then 6 inches; if Stile be 13/100 and 75/100, the Semidiameter 23 deg. 30 min. is 37/100 parts, as you may see the Figure makes all plain; and so of any other.
Of all Dials this is the plainest; for it is no more but divide a whole Circle into 24 equal parts: and this is the very ground to all the rest.
With this Dial, the Hour-lines being equally divided into 24 equal parts, on the inner circle you may make a Mariners Compass, with the 32 Points drawn upon it, to know in all Latitudes whether the Moon being upon such a Point maketh High-wa∣ter; or upon what Point the Moon must be, when at those Places set together it ma∣keth High-water or Full-Sea.
For to know upon what Point the Moon is, may be done two manner of ways; by setting it by the Compass, or by reckoning according to the age of the Moon, and the Hour of the day. The setting according to any Point, may not be done with a common flat Compass as the Mariners steer by (as many, wanting better reason, think they may, to their great mistake) by reason it doth only divide the Horizon into equal Points, and sheweth in what Vertical Circle or Azimuth the Sun or Moon stands: But this must be done with a Compass, which being elevated according to the Superficies of the Aequinoctial, divideth the Aequinoctial so likewise into equal parts, as the common flat Mariners Compass doth divide the Horizon. Such an Aequinoctial Compass, with a Dial in, as abovesaid, is of fashion as hereafter fol∣loweth pourtrayed. Whereof the Wheel ABC sheweth the Superficies of the Aequinoctial, the Wyre ED the Axle-tree of the World. The foresaid Wheel must be all alike marked on both sides, as well under as above, with the 32 Points of the Compass, and with twice 12 Hours: and right against the East and West at Land M, must so hang upon two Pins, as upon an Axletree, that it may be turned
IN the height of 50 deg. or thereabouts, the Sun being in the beginning of Cancer, at his greatest Declination to the North, by a common Compass cometh not before half an hour after seven of the Clock to the East, and at half an hour after four to the West; that is, he goeth from the East to the South and round to the West in nine hours; but from the West through the North, until again in the East, in 15 hours.
IN the height of 30 deg. he cometh a little before half an hour past nine of the clock to the East, and a little after half an hour past two of the Clock in the West, and so goeth in less than five hours and a half from the East through the South to the West; but from the West through the North, until again in the East, he go∣eth more than 18 hours. Thirdly, Being under the Line, and the Sun having no Declination, he ariseth in the morning right in the East, and so rising higher and higher, continueth East until that he goeth over our heads through the Zenith into the West; and so continueth West, still going down West, until he cometh again to the Horizon: and so according to a flat Compass he is the one half of the day East, and the other West, without coming upon any other Point. It is not so with this Aequi∣noctial Compass. The Sun and Moon go always a like time on every one Point of the Compass, to wit, from the East to the South 6 hours, from the South to the West 6 hours, from the West through the North to the East in twice 6 hours.
This Dial will serve for all Latitudes, if you put the end of the Wyre at D, to the height of the Pole or Latitude of the place as beforesaid; so the shadow of the other end at E will fall upon the Hours and true Points of the Compass, all the time the Sun is to the North of the Aequinoctial; but when the Sun is to South of the Aequi∣noctial, you must look for the Hours and Points of the Compass upon the under side of the Dial.
THe Aequinoctial Dial we call that which hath his Poles in the Aequinoctial Circle, of which there be three kinds.
1. The Direct or South Aequinoctial Dial, which faceth the Meridian directly, not looking from him to the one side more than to the other, having his Poles in the Intersections of the Aequinoctial and Meridian.
2. The East or West Aequin••••tial Dials, which may also be called Aequinoctial Horizontal Dials, for an Horizontal Di••l declaring just 90 Degrees from the South or North, becomes an Aequ••noc••ial Dial, as well as Horizontal, because there is his Polar height, upon the Intersection o•• the Horizon with the Aequinoctial: and though this Dial be of kin to both, yet his Gnomon shews that he should be sorted rather with the Aequinoctial Dials, than with the Horizontal. These two sorts are regular, having the Poles in the four notablest Points of the Aequator. The third is somewhat irregular, but may be brought to Rule.
How to make the first of these, draw the Horizontal Line AB, and about the midst at C let fall the Perpendicular CD, which is the Meridian or 12 a clock Line. Let CD be equal to a Chord of 60 Degrees, or the Tangent of three hours, and through D draw the Line FE, parallel to AB; make also DE and CB equal to D, so have you a true Square CDEB. Now take one hour with your Compasses off your Scale, and lay the same both ways from E towards B and D, as E 1. Do the like with two hours, and draw the pricked Tangent-lines from C to these Marks.
Next, Let the length or height of the Gnomon or Stile be GH, equal to C••, or 3 hours; so drawing a Line through GH, parallel to the Horizon, you shall find it cut the former Lines drawn to the Center C, in the Points l, m, n, o, p: through which Points, if you draw Parallel-lines to the 12 a clock Line CD, you shall have all the afternoon hours as far as V: and the morning hours must be drawn in like manner and distance, to the left hand or West side, beginning from 7 in the morn∣ing unto 12, as in the Figure following.
Note, that the height or length of the Stile is always 3 hours from the Meridian, as you see HG, which you may make with Copper or Brass Plate, or Iron, in form as you see shadowed, whose breadth on the top is here HR, which may be made more or less as you please.
This Dial will serve in any Latitude, if the Plane be placed parallel to the hour of 6, so that the Plane be even with the Pole of the World.
SUppose the length of the Horizontal Line AB or FE be 12 Inches, and that it were required to put on all the Hours from 7 in the morning to 5 in the even∣ing; here we have 5 hours and 6 inches on either side of the Meridian, ••herefore I allow 15 Degrees for an hour. The Rule to find the height of the Stile is,
| As the Tangent-compl. of the given Hour 15 deg. | 1057194 |
| Is to half the Horizon or Distance from the Meridian 6 inches | 277815 |
| So is the Tangent of 45 Degrees | 1000000 |
| To the ••eight of the 〈…〉〈…〉 inches and parts | 220621 |
And likewise the distance of the Hour points of 9 and 3 from the Meridian will be 1 61/100, or 1 inch and 61 parts of 100.
HAving found the length of the Stile in our Example to be 1 inch 61 parts of 100, then in this Example, as we find the first Hour, so find the rest.
| As the Tangent of 45 deg. | 1000000 |
| Is to the Tangent of the Hour from the Meridian 15 deg. | 942805 |
| So is the height of the Stile 1 61/100 inches | 220621 |
| To the length of the Tangent-line between the Meridian or Sub∣stiler 43/100 inch | 163426 |
| Hours | An. Po. | Tang. | |||
| deg. | mi. | In. | par. | ||
| 12 | 0 | 0 | 0 | 0 | |
| 11 | 1 | 15 | 0 | 0 | 43 |
| 10 | 2 | 30 | 0 | 0 | 93 |
| 9 | 3 | 45 | 0 | 1 | 61 |
| 8 | 4 | 60 | 0 | 2 | 79 |
| 7 | 5 | 75 | 0 | 6 | 0 |
| 6 | 6 | 90 | 0 | Infinit. |
THis Plane is a right Horizon of those People who dwell under the Aequator, distant from us 90 deg. of Longitude; as the South Aequinoctial Plane of the last Chapter was the Horizon of those who dwell under the Aequator in the same Longitude with us: Therefore these Dials are in all Points alike, only the Substiler Line, which in the South Aequinoctial Dial is at 12, is but 6 in the morning for our Country, because of the difference of Longitude.
To pourtraict this on a Wall or Plane, first draw the Horizontal Line AB; then upon the Center C describe the Semicircle ADB, whereon lay the Latitude of the place 51 d. 30 m. from A unto D; so drawing GD continued, you shall have the Hour of 6: then with your Compasses take off your Scale 15 deg. of the Line of Chords, and turning them off 6 times, divide the Arch DF into 6 equal parts, and
draw prick'd or blind Lines to those Divisions, which would be all one as if you had done it thus. CD being equal to the Chord of 60, or Tangent of 3 hours, you shall make the Quadrant or true Square equal to the side thereof CDEF, and from the corner at E, you shall lay down both ways towards D and F the hours of 1 and 2, from whence draw Lines to the Center C: Next make choice of the length or height of your Pin or Stile, which you must lay down from C to G on the 6 hour Line, drawing from the Point G a Line perpendicular to the Line of 6, or parallel to the side CF, as GH, which cuts the former Lines in the Points IKLMH; through which Points drawing Lines parallel to the hour of 6, you shall have the morning hours from 6 to 11, and the hours before 6, from 4 in the morning, are equal as from 6 to 7 and 8.
THe West Aequinoctial Dial erect, serving for the afternoon, is drawn by the same Rules contrariwise like the East in all points, only it shews but the after∣noon hours, as the East shews the forenoon hours: When you have drawn on paper the East Dial, and set it by guess in its scituation, go on the West side of it, and you may see through the paper the picture by reflection of the West Dial; and so will the picture of the backside of the West shew you the true picture of the East Dial.
The way to calculate the height of the
| Hours. | Ang. Po. | Tang. | |||
| deg. | min. | In. | par. | ||
| 5 | 7 | 15 | 0 | 2 | 68 |
| 4 | 8 | 30 | 0 | 5 | 77 |
| 3 | 9 | 45 | 0 | 10 | 0 |
| 10 | 2 | 60 | 0 | 17 | 32 |
| 11 | 1 | 75 | 0 | 37 | 32 |
| 12 | 90 | 0 | Infinit. |
WHat an Oblique Dial is, and why it hath been so called, hath been shewed Chap. 2.
They be
The Regular lie in some notable Circle of the Sphere; as first the Vertical Dial, whose Plane lieth in the Horizon, for which cause many call it the Horizontal Dial. Secondly, the South and North Horizontal Dials, whose Plane lieth in the East Azi∣muth, and is commonly called the South or North erect direct Dial. As for the East and West Dials, they belong to another place, as was said Chap. 5.
The Irregular are such as lie Oblique to the Horizon, as Reclining or Inclining Di∣als; or else lie Oblique to the Meridians, as Decliners; or else Oblique to both, as Recliners or Incliners declining, which are esteemed the hardest of all, because of their double irregularity.
The Declination of a Plane is the Azimuthal Distance of his Poles from the Meri∣dian of the place East or West.
The Reclination is the distance of his Poles from the Zenith and Nadir of your place.
Inclination is the neerest distance of the Poles of the Plane from your Horizon; and whatsoever the reclination of the upper face of a Plane is, the inclination of the lower face is the Complement thereof.
DRaw first the Horizontal-line AB, which is the Hour-line of 6;* 1.1 then take the Latitude of 51 d. 30 m. from the Gnomon-line, and lay it down both ways from the Center C, as to A and B: Next take the whole Hour-line of 6, fixing one leg of your Compasses on A, describe a little Arch towards D; do the like from the Point B, crossing the Arch at D; so draw the Line AD and BD. Now upon these Lines you must transport the 6 hours from D unto A, and also from D unto B, as you see by the Figures 1, 2, 3, 4, 5, from whence drawing Lines ••rom the Center C, you shall have the Hours as you see numbred from 4 in the mo••ning until 8 in the afternoon, which sufficeth for this Latitude 51 d. 30 m.
For the height of the Stile, take off your Line of Chords with your Compasses the Latitude of the Place 51 d. 30 m. and lay from K to E, from 12 to neer 4, and so drawing CE, you have the height of the Stile, which may be made in Brass or Copper Plate, as you see shadowed in the Dial following.
Thus by Calculation,
| As the Sine of 90 d. Is to the Sine of the Latitude 51 d. 30 m. | 989354 |
| So is the Tangent of the Ho. 15 d. | 942805 |
| To the Tangent of the Hour-line from the Meridian 11 d. 50 m. | 932159 |
As in this Table, which take from the
| Hours. | Gr. mi. | Gr. mi. Tangents. |
| 12 | 00 00 | 00 00 |
| 1 | 15 00 | 11 50 |
| 2 | 30 00 | 24 20 |
| 3 | 45 00 | 28 3 |
| 4 | 60 00 | 53 35 |
| 5 | 75 00 | 71 06 |
| 6 | 90 00 | 90 00 |
Line of Chords, and prick from the Meridian 12 from K on each side, the Degree or Tangent of each hour; And by the same Rule you may find the Quarters, and that you may prick off in like manner, which is a way how to make an Horizontal Dial, as before, Lati∣tude 51 deg. 30 min.
THis belongs to an upright Wall looking full North or South, and the Plane of it lies in the East Azimuth.
First draw the Horizontal-line AB, which serveth for 6 in the morning at A, and 6 in the afternoon at B; then from the Center lay down from C the Gno∣mon of the Latitudes Complement 33. 30 both ways, as to A and B: Now with the whole Line of 6 hours from A describe an Arch towards D, and with the same di∣stance from B cross the same Arch, and draw the two Lines AD and BD, whereon from D you must transport the hours, as you see by the Figures 1, 2, 3, 4, 5. So drawing Lines through those parts from the Center C, you shall have the hours from 6 a clock in the morning to six a clock in the afternoon.
* 1.2With the Chord of 60 on the Center C describe the Semicircle ADB from which Line of Chords take the Complement of the Latitude 38, 30, and lay down from the Meridian at E unto F; so drawing CF, you shall have the height of the Stile
above the Plane. This, if it be for a large Dial, as against a Wall, is best to be made of a Rod of Iron; for small Dials a Brass Plate is best, and your Dial is done.
This Dial shews the Hours from 6 in the morning to 6 at night: The other hours before and after 6, as far as four and eight, belong to the North face of this Dial. Because the Almicantars may oft obscure the Intersections of the Hour-circles, you may avoid that if you reduce this Dial to a Vertical Dial, for the South Horizontal Dial, being the Vertical Dial of those People who live 90 degrees Southward from us, that is, in 38 d. 30 m. of South Latitude.
Secondly, For the North face, imagine you had for the Gnomon a Wire thrust aslope through the center of the Plane from the Southside Northward, and you will presently conceive, that in the North Dial the Horizontal or 6 a clock Line will be lowest, and that the Stile or Gnomon will turn upwards towards the North Pole, as much as it turned downwards on the other side; and that all the Hours save 6 in the morning, and 6, 7, 8 at night, may be left out in our Latitude, because the Sun shi∣neth no longer upon it; and those Hour-distances you may find and set off from 6 a clock Line, as you did the Hours of like distance in the South face. Note in a South erect direct, or a South erect declining Dial, the Stile always points downwards; but if it be a North erect declining Dial, the Stile points upwards.
SUppose that the Plane be so inclining, that the face thereof be towards the South, and the North part be elevated 23 deg. above the Horizon, and that the South part be dipped as much under the Horizon; then to find the height of the Stile above the Plane,* 1.3 you must substract the Inclination 23 deg. from the Latitude of the Place, which is here 51 deg. 30 min. so the Remainer being 28 deg. 30 min. shall be the height of the Stile. Now for drawing the Hour-lines, you shall do no otherwise than you have done before in making the Horizontal Dial according to the Stiles height 28. 30, as you may perceive in the Dial following.
Note, That if the Inclination of the Plane be more than the Latitude, then you must substract the Latitude from it, so there shall remain the height of the Stile above rhe Plane.
But if the inclination be South, that so the upward face of the Plane looks North∣ward, then you are to add the Inclination to the Latitude of the Place; and if it exceed 90 degrees, you must then substract it from 180 degrees, so shall you have the Poles height above the Plane towards the South part of the Dial. The Figure of this Dial followeth.
ALL perpendicular Planes, as Walls, lie in the Planes of one of the Azimuths; which Planes cut always both Zenith and Nadir, and the Center of the Earth, as in the Figure Z is Zenith and Nadir ESWN. Horizon EW is the Base or Ground-line, or any Horizontal Line, drawn upon a Wall or Plane, look∣ing full South or North: his Poles are at S and N in the Meridian; wherefore he declineth not, but lieth in the East Azimuth EW.
AB is a Wall or Plane declining East by the Arch SP, to which AB or WE are equal: for so much as the Wall bendeth from the East Azimuth, so much doth his Pole at P decline or bend from the Meridian.
Now to find how much any Plane declineth, and so in what Azimuth he lies, one good way is this. When the Sun begins to enlighten the Wall, or when he leaves it, then is the Sun in the same Azimuth with the Wall: therefore take at that instant his Altitude, and thereby get his Azimuth, according to Chap. 14. of the Sixth Book, so you shall have the Declination of the Wall.
Another way, if you have not time until the Sun cometh unto the Azimuth of the Wall, or the Vertical of it, which cutteth the Pole thereof, then get the Suns Azi∣muth as before when you can, and at the same time observe by the sight of your Cir∣cumferenter the Suns Horizontal Distance from the Pole of the Plane; and by compa∣ring of those together, you may easily gather the Declination of the Wall: As in Example.
I observed the Sun to be gone West from the Pole of the Plane 72 deg. and by the Altitude of the Sun then taken, I found his Azimuth 62 deg. Here I reason thus: The Sun is gone from the Pole Vertical of the Wall 72 deg. and from the Meridian 62 deg. therefore the Meridian lies between the Pole of the Plane, and the Sun: And because ☉ P is 72, and ☉ S 62, therefort SP the Declination of the Plane is 10 deg. the Difference of 72 and 62; and the Declination is East. for the Sun is neerer to the Meridian, than to the Vertical of the Plane.
And thus if you draw a rude Scheme of your Case, you may soon reason out the Declination, better than do it blindfold, by the Rules commonly given.
And by those two last ways you may take the Declination not only of upright Planes, but of Recliners also.
FOr Example. Suppose SNDE represent a Wall, or the face of the Plane where∣on I am to make a Dial, and I desire to know the Declination thereof from the Meridian Eastward or Westward.
If you have no Instrument, take a plain Board, having one planed or straight side or edge, which Board let be repre∣sented by DEVQ: apply the straight edge of the Board ED to the side of the Wall or Plane, as in this Figure, and in the middle of the Board at C, I set one foot of the Compasses, and the other opened to 60 deg. of my Line of Chords, I describe the Circle ZBHA. In the Center C, I erect or place a Stile or Wire, as CO perpendicular to the Horizon, pla∣cing the Board as neer Horizontal as I can. I find by observation, that the shadow of the top of the Pin or Wire toucheth the Circle in the forenoon at the Point B, where I make a little mark; and like∣wise I observe in the afternoon that it toucheth the said Circle in the Point A: Then I measure the half thereof from B or A to X, and drawing a Line through the Center to X, as KCX, you shall have the Meridian Line exactly described KCX. Lastly, I take the Distance ZX, which I apply to my Scale of Chords, and find the Arch thereof 18 deg. 10 min. and so much is the Declination of th•• Plane EDNS, which you may see by the Meridian Line XK to be towards the East; therefore it is a South Plane declining West 18 d. 10 m. This way is the most easie way, and requires time for the making the two Ob∣servations; therefore I will lay down some other ways, that may resolve at one mo∣ment, or at one observation.
APply the North side of the Instrument wherein the Needle is placed unto the Wall, and hold it Horizontally as neer as you can, that the Needle may have li∣berty to play to and fro; and when it stands, observe upon the Limb of the Chard over which it moves, upon what Degree the Needle stands; for that is the Declinati∣on of the Plane, reckoned from the South Point of the Needle: And if you would know the Coast, observe, That if the Needle stand upon the East side of the Meri∣dian Line, then is the Declination West; but if it stand on the West side of the Meridian Line, the Declination is East. By the Sea-Compass described Book V. as it hangs in the Box, you may also find the Declination. Set the slit of the Brass Di∣ameter North and South, as before directed; then set the square side of the Compass-box next the Plane, reckon outwards 180 deg. and set the Index to it; so reckon the number of Degrees betwixt 180 the Index, and the Meridian, and that number of Degrees is the Declination of the Plane required; and by the Chard you may see what Coast it is, that is, whether he declines from the North or South Eastward or Westward. And note, That all Lines parallel to any Horizontal Line be Horizontal, and all Lines parallel to Vertical Lines be also Vertical.
HEre three things are required; for besides the Distance of the several hours from 12, and the Elevation of the Gnomon, which are requisite to the ma∣king of all direct and regular Dials, we must here also know the Declinati∣on of the Gnomon, which some call the Distance of the Substile from the Meridian, or the distance of the Meridian of the Plane from the Meridian of the Place. For in all Dials the Noon-line in the Meridian of the Place, projected on the Dial, and in all Horizontal or Mural Dials, not reclining or inclining, the Noon-line is a Perpen∣dicular cutting the Center of the Dial, how much soever they decline.
But declining Dials which look awry from our Meridian, have a Meridian of their own, which is called the Meridian of the Plane and the Substile (because the Stile or Gnomon stands upon it) and is indeed the Meridian of that Place where this Decli∣ning Dial would be a Vertical Dial, and where the Substile would be Noon-line; and to this Substile, the Hours of the Plane are always so conformed, that the neerer they be to the Substile, the narrower are the Hour-spaces; and contrarily, because the Meridians do cut every Oblique Horizon, that is thickest neer the Meridian of the place; and this Declining Dial being a Stranger with us, followeth the fashion of his own Country, and so hath his narrowest Hour-spaces neer his own Meridian, rather than ours: And now, as that is the Meridian of our place, which cutteth our Horizon at Right Angles, passing through his Poles, Zenith, and Nadir; so the Meridian of any Plane is that which cutteth the Plane at Right Angles, and passeth through his Poles.
Before we draw the Hour-lines in these sort of Dials, it will be very convenient to shew a general way for all Latitudes in a Diagram by it self, and how to find the Substiler Distance from the Meridian or 12 a clock Line, and the height of the Gno∣mon or Stile above the Plane. First, Draw the Horizontal-line AB, and upon the Center at C, take off your Scale with your Compasses a Chord of 60 Degrees, de∣scribe the Semicircle ADB, and with a Chord of 90 you may lay from A to D, and from B to D, so shall you draw CD from the Meridian-line of 12 a Clock; Then take the Complement of the Latitude 38 deg. 30 min. and lay from D to E, and so draw EF parallel to the Horizon AC; next take the Declination of the Dial 32 d. 30 m. and lay from D to G, drawing the Radius OG thereon, you must lay the Di∣stance EF from the Center at C, as CH. Now with the neerest distance from H to the Meridian CD, as HI, make FL; and drawing a Line from C through L, it will cut the Limb in the Point M; so measuring DM on the Line of Chords, you shall have the Substiler Distance 23 deg. 8 min. all which you may see in this Scheme following.
By Calculation,
| As the Radius 90 deg. | 10 |
| Is to the Sine of the Declination SE 32 deg. 30 m. | 973021 |
| So is the Co-tangent of the Latitude 51 d. 30 m. | 990060 |
| To the Tang. of the Substiler Dist. from the Meridian 23 d. 8 m. | 963081 |
For the height of the Stile, take the neerest Distance from H to the Horizon K, and lay the same from L to cut the Arch in N: So measure MN, you shall have on the Line of Chords the Height of the Stile neerest 31 deg. 40 min.
By Calculation, viz.
| As the Radius 90 deg. | 10 |
| Is to the Co-sine of the Latitude 51 deg. 30 min. | 979414 |
| So is the Co-sine of the Declination 32 deg. 30 min. | 992602 |
| To the Height of the Stile 31 deg. 40 min. | 972016 |
TAke the neerest Distance from M to FE, and lay it on the Meridian from F to O: Then take the Distance from O unto G, and lay it from O unto P on the Meridian; so the Distance from P to M, measured on a Line of Chords, will be found to be 39 deg. 9 min. or thereabouts; which in time, allowing 15 deg. for an hour, and four Minutes to a Degree, you shall have 2 ho. 36 min. 36 sec. which is the di∣stance of the Substile Line from the 12 a Clock Line, which in this Dial is between 9 and 10 of the Clock in the morning. And by Calculation,
| As the Radius 90 deg. | 10 |
| Is to the Sine of the Latitude 51 deg. 30 min. | 989354 |
| So is the Co-tangent of the Declination 32 deg. 30 min. | 1019581 |
| To the Co-tangent of the Inclination 39 deg. 9 min. | 1008935 |
Thus is shadowed a Geometrical way, and by Calculation, for any Latitude: But for one particular Latitude, Mr. Philip Staynred, which first composed the Scale and Gnomon Line, and Inclination of Meridians, and the greater and lesser Pole on the Dialling Scale, for 37 years since, as I have seen by him calculated, and the Projecti∣on Geometrical in his Study: he hath for the more ease set two Lines upon the Dial∣ling Scale, as he usually makes, to find the Substile for the Latitude of 51 deg. 30 m. against the Lines stands the Letters Sub or Stile joyned with it; so if you take from off the Substile-line the Declination of the Dial, and lay it from D unto M, which in the last Example was 32 deg. 30 min. you shall find it to reach in the Diagram from D unto M, as in the Line of Chords 23 deg for the Substile, as before. Also, the other Line noted with the word Stile, you shall likewise take from thence the Declination 32 deg. 30 min. which you shall find to reach in the Diagram from M unto N, or in the Line of Chords 31 deg. 40 min.
FIrst draw the Horizontal Line AB, and on the Center at C describe the Semi∣circle AEB, with the Chord of 60 deg. and from A and B lay down 90 deg. unto E; so shall you draw CE the Meridian Line or Hour of 12; then in
the former Diagram take the Substile distance DM 23 deg. and lay the same in the Dial following from E unto F, and from the Center C through F you shall draw CFK the Substiler Line. Next take the Chord of 90 deg. and lay it from F both ways upon the Arch, so shall you draw the Gnomon Line GH, whereon from C, with the Stiles Altitude before found, 31 deg. 40 min. taken from the Gnomon Line, you shall make CG and CH. Then take the whole Line of 6 hours, and with the same distance from G describe an Arch at K, and with the like from •• cross the same Arch, and draw the Lines GK and HK, which last Line cuts the Meridian at N. Now if you measure KN on the Hour-lines, you shall find it neer 2 ho. 36 m. ½ as you found in the last Diagram. Then take one Hour more, which is 3 ho. 36 min. and lay the same from K unto M; and so increasing one Hour more, you shall have the Hour points l and i; also diminishing one Hour less than 2 ho. 36 min. which is 1 ho. 36 min. the same will reach from K to O, and so 38 min. from K to P. Now as you have divided KH, the very same distance as is from K towards H, must be from G towards K; so drawing the Lines from the Center C through those Points, you shall have the Hour-lines, as you see in the Dial following.
By Calculation,
| As the Radius 90 deg. | 10 |
| Is to the Sine of the Stiles or Gnomons Height. 31 d. 40 m. | 972013 |
| So is the Tangent of the Dist. of an Hour from the Subst. 9 d. 9 m. | 920701 |
| To the Tangent of the Hour-Arch from the Substile 4 d. 50 m. betwixt the 10 a clock Line, and the Subst. Line on the Arch | 892714 |
By this Rule was this Table made;
| Hours. | Equal. Dist. | Hour Arches. |
| D. M. | D. M. | |
| 4 | 80 51 | 72 57 |
| 5 | 65 51 | 49 30 |
| 6 | 50 51 | 32 from 49 |
| 7 | 35 51 | 20 46 |
| 8 | 20 51 | 11 18 |
| 9 | 5 51 | 3 5 |
| Meridian. | Substile. | |
| 10 | 9 9 | 04 50 |
| 11 | 24 9 | 13 15 |
| 12 | 39 9 | 23 8 |
| 1 | 54 9 | 36 0 |
| 2 | 69 9 | 54 2 |
| 3 | 84 9 | 78 57 |
and by the same you may make one for any Latitude, and for any Declining Dial; and you may by it prove your former Work: for if you prick from the Substiler Line F the Chord of 4 deg. 50 m. and draw a Line from the Center, it will be the Hour-line of 10; and prick the Chord of 3 deg. 5 min. from the Substile, and draw a Line through that Point to the Center, and it will be the Hour-line of 9 a clock; and so of the rest, as you find them in the last Column.
Note, That the Height of the Stile FS being equal unto MN in the former Dia∣gram, which is the Chord of 31 d. 40 m. now because the Plane declines East, there∣fore the Gnomon shall decline West: for the Dial being such a Projection of the Sphere, wherein all the usual Lines cross in the Nodus of the Gnomon, and thence disperse themselves again towards the Plane; therefore that which is East in the Sphere, will be expressed West on the Plane, and contrarily, as was said Chap. 2. Theorem 2. Also I consider, that howsoever the Plane be turned East or West, the Gnomon place is fixed, because it is a part of the Axis of the World, or a Line Parallel to it. Now therefore I turn a South Dial, and make him decline East, and hold the Gnomon unmovable, the West side of the Dial will approach neerer to the Gnomon, as reason and sense will require. Likewise the Hours which are found on the same side of the Meridian or Noon-line with the Substile, must be set the same way with it from the Noon-line in the Dial.
And if you would draw the North Dial of this Plane, do but prolong those Hour lines, and the Substile upwards beyond the Center, and you have the North Dial beyond C, or above the Horizontal Line AB, as the South Dial below it. And note, Because the Sun sets after 8 a Clock in Summer, therefore the three hours next before and after midnight, may be left out in this Dial, and all others which must serve in our Latitude.
This is the most ready way to delineate the opposite face of any Dial. Note, That if a Wall decline from the South Eastwards 32 deg. 30 min. therefore the Plane which lieth 90 deg. from his Pole, is in the 32½ Azimuth from the East Northward.
Note this well: Extend the Compasses as before from K to N, the Intersection of the Meridian with the Line KH at N, before found to be 2 ho. 36 min. which converted into Degrees, by allowing 15 Degrees to an Hour, and 4 Minutes to a Degree, it makes 39 deg. 9 min. which 39 deg. 9 min. shew me the Difference of Longitude between our Country and the Country of this Dial.
You may apply this Distance to the Line of Inclination of Meridians, and it will give you the Distance before 39 deg. 9 min.
Note, I allow this Countries Longitude to be 27 deg. 44 min. at Bristol, to the Eastward of the Grand Meridian Flowers and Calfs one of the Isles of Azores, which added to 39 deg. 9 min. shews the Longitude of the Country of the Dial to be 66 d. 53 min. Eastward, and Latitude 31 deg. 40 min. which I find by my Globe is in the Desarts of Arabia at Asichia neer Soar.
WHat Reclination and Inclination are, hath been shewed Chap. 8. and you will have it following in a Diagram by it self.
All Reclining and Inclining Planes have their Bases or Horizontal Diameters lying in the Horizontal Diameter of some Azimuth; but the top of the Plane leaneth back from the Zenith of your place in the Vertical of the Plane (which is the Azimuth cutting the Plane at Right Angles) so much as the Reclination hap∣neth to be: and the Pole of the Plane, on that side the Plane inclines to, is sunk as much below the Horizon, as the top of the Plane is sunk below the Zenith; and the opposite Pole is mounted as much.
Let ESWN be Horizon, Z the Zenith, EW the Horizontal Diameter of the Plane and of the East Azimuth, EOW a Plane not declining but reclining South∣wards from the Zenith by the Arch ZO 45 deg. and his opposite Face inclining to the Horizon according to the Arch OS 45 deg. the Pole of the reclining Face is at P in the Meridian CP, which here is also Vertical of the Plane, and is elevated 45 deg. in the Arch NP, equal to the Arch of Reclination ZO, the Pole of the inclining Face is depressed as much on the other side under the Horizon.
To find the Quantity of the Reclination, you shall draw a Vertical Line on the Plane by Chap. 3. and thereto apply a long Ruler, which may overshoot the Plane either above or below: to that Ruler apply any Semidiameter of a Quadrant, and the Degrees, between that Semidiameter and the Plumb-line, shall be the Degrees of Reclination. Or stick up in the Vertical Line two Pins of equal height, and perpen∣dicular, and placing your self either above or below the Plane, as you find most easie, direct the Sights of your Quadrant to the Heads of the two Pins, being in a right Line with your eye; and the Plummet shall shew the Reclination on the Side of the Quadrant, and the Inclination, which is always the Complement thereof, on the other.
IF a Plane shall decline from the Prime Vertical, and incline to the Horizon, and yet not lie even with the Poles of the World, it is then called a Declining, In∣clining Plane. Of these there be several sorts, you may see 19 Planes in the following Diagram, and directed how to know them in the 25th Chapt r: but to shew the Recliners in order as they come, viz. the North Recliner 45 deg. and South Incliner falls between the Aequator and the Tropick of ♋, as the Circle EQW; the South Recliner 45 deg. and North Incliner, falls between the Horizon and the Pole, and is represented by the Circle EAW. The East and West Recliner and Incliner 45 deg. may be seen by the Circle NVS, the Inclination may be Northward 45 deg. and declining 45, as the Circle FKC, or the Plane may decline S 45 Westward, and recline 45 deg. from the Zenith, as the Circle CLF. If your Inclination or Reclina∣tion fall more or less, you may see the way. Each of these Planes have two Faces, the upper toward the Zenith, the lower towards the Nadir, wherein having the Latitude of the Place, and the Declination, with the Inclination of the Plane, you are farther to consider what must be found before you can draw the Dial, which will follow in order, and is represented in this Fundamental Diagram; only I will mention the Arches and Angles in the hardest, which is a South declining West 45 deg. and recli∣ning from the Zenith 45 deg. In such you must consider.
1. The Arch of the Plane between the Horizon and Meridian CO or FO.
2. The Arch of the Meridian between the Horizon and the Plane ON or OS.
3. The Angle of Inclination between the Meridian and the Plane CON or FOS.
4. The Substile-distance from the Meridian OR or OR.
5. The Height of the Pole above the Plane or Stiles Height PR.
6. The Inclination of both Meridians or Angles at P.
7. The Difference of the Hour from the Substile. But first we will describe the Diagram.
THe Description of this Diagram is set down at large by that worthy Mathematici∣an Mr. Edmond Gunter, in the Use of his Sector, Chap. 3. But for this purpose it may suffice, if it have the Vertical Circle, the Hour Circles, the Aequator, and the Tro∣picks first drawn in it; other Circles may be supplied afterward, as we shall have use of them; and those may be readily drawn, as I have borrowed of him, in this manner.
Let the outward Circle, representing the Horizon, be drawn and divided into four equal parts, with SN the Meridian, EW the Vertical, and each fourth part into 90 deg. That done, lay a Ruler to the Point S, and each Degree in the Qua∣drant EN, and note the Intersections where the Ruler crosseth the Vertical; so shall the Semidiameter EZ be divided into other 90 deg. and from thence the other Semi∣diameters may be divided in the same sort; those may be numbred with 10, 20, 30, from E towards C; and for variety, with 10, 20, 30, from C towards W: But for the Meridian, the South part would be best numbered according to the Declination from the Aequator, and the North part according to the Distance from the Pole.
Then with respect unto the Latitude, which here we suppose to be 51 deg. 30 min.
open the Compasses unto 38 deg. 30 min. from C toward W, and prick them down in the Meridian from C unto P; so this Point P shall represent the Pole of the World, and through it must be drawn all the Hour-circles.
Having three Points EPW, find their Centers, which will fall in the Meridian, a little without the Point S, and draw them in a Circle EPW, which will be the Circle of the Hour of 6.
Through this Center of the Hour of 6 draw an occult Line at length parallel to E W, so this Line shall contin••e the Centers of all the other Hour-circles: where the Circles of the Hour of 6 crosseth this occult Line, there will be the Centers of 9 and 3 their Hour-circles.
The distance between these Centers of 9 and 3, will be equal to the Semidiameters of the Hour-circles of 10 and 2: where these two Circles of 10 and 2 shall cross this occult Line, there will be the Center of 7 and 5. And again, take with your Com∣passes off the Diameter EW 75 deg. under W, and turn the Compasses three times over on the occult Line, from the Center of the Hour of 6, and you have the Center of the Hours of 1 and 11. Again, take 78 deg. from E towards C, and lay it both ways from the Center of the first Hour-circle of 6, on the occult Line, and you have the Center of the Hour-circle of 4. So practice for any other Latitude.
The Hour-circles being thus drawn, take 51 deg. 30 min. from C toward W, and prick them down in the South part of the Meridian from C unto A, and bring the third Point EAW into a Circle; this Circle so drawn shall represent the Aequator.
The Tropick of Cancer is 23 deg. 30 min. above the Aequator, and 66 deg. 30 min. distant from the Pole; and so in this Latitude it will cross the South part of the Me∣ridian at 28 deg. from the Zenith, and the North part of the Meridian 15 deg. be∣low the Horizon. Take therefore 28 deg. from C towards W, and prick them down in the Meridian from C unto D, so have you the South Intersection: Then lay the Ruler to the Point W and 15 deg. in the Quadrant NW, numbred from N toward W, and note where it crosseth the Meridian; so shall you have the North Intersecti∣on. The half way between these two Intersections in the Meridian Line is the Center of the Tropick of Cancer: Which being truly drawn, will cross the Horizon before 4 in the Morning, and after 8 in the Evening, about 40 deg. Northward from E and W, according to the rising and setting of the Sun at his entrance into Cancer.
The Tropick of Capricorn is 23 deg. 30 min. below the Aequator, and 113 deg. 30 min. distant from the North Pole; so that in this Latitude it crosseth the South part of the Meridian at 75 deg. from the Zenith, and the North part of the Meridian at 62 deg. below the Horizon. Take therefore 75 deg. towards W, and prick them down in the Meridian from Z unto ♑; so have you the South Intersection: Then lay the Ruler to the Point W, and 62 deg. in the Quadrant NW, numbred from N to∣wards W, and note where it crosseth the Meridian, so shall you have the North In∣tersection: the half way shall be the Center, whereon you may describe the Tropick of Capricorn ♑. This Tropick will cross the Horizon after 8 in the morning, and be∣fore 4 in the Evening, about 40 deg. Southward from E and W, according to the rising and setting of the Sun at his entrance into Capricorn. Now we will proceed to draw the Hour-lines in a North Recliner and a South Incliner, and shew the Height of the Stile above the Plane on the Meridian, and a South Recliner, and a North In∣cliner; and so in order to the rest.
THe Base or Horizontal-line of such a Dial lieth in the East Azimuth, and his Pole in the Meridian, as you may see Chap. 14.
The Plane of the 14 Chapter was a North Plane reclining Southward 45 deg. the Zenith is distant from the North Pole 38 deg. 30 min. the Complement
of the Latitude 51 deg. 30 min. toward the South, and I see the Reclination is 45 deg. more Southward, because I see my Plane reclines so much that way. I add the Com∣plement of the Latitude 38 deg. 30 min. and the Reclination 45 deg. together, and I see then by the same the North Pole is elevated 83 deg. 30 min. which is the Height of the Stile above the Plane on the Meridian; which 83 deg. 30 min. taken off the Gnomon Line of the Scale, you may proceed and draw the Dial, and lay that on the Line WE, and work as you did in the other Dials.
The opposite face to this is the South Incliner; and if you would draw it, do but prolong those Hour-Lines, as was said in Chap. 13. and you have the South Recliner below the Horizontal Line WE.
Note, Had this Reclination been 51 deg. 30 min. and the Complement of the La∣titude added to it would have made 90 deg. then it would have fallen into the Plane of the Aequinoctial, and so the Dial would have been a Polar Dial, and all the Hours would have had equal spaces, and the Gnomon would have stood perpendicular, which are the properties of a Polar Dial, as hath been shewed Chap. 5.
* 1.4As for the South reclination, which is 45 deg. which substract from the Latitude 51 deg. 30 min. and you have the height of the Gnomon or Stile above the Plane, which by reason the Hour-lines will be so neer together, continue them, and cut them off as they may fit your Plane, by leaving out one of the Hours, or more, as you will; so will you have the Stile a handsom height above the Plane, and the Hour-circles a good distance asunder: But for the North Incliner, his Lines and Stile must be drawn from the Center of the Plane,* 1.5 although they do come neer. If the Recli∣nation had been 31 deg. then you should have substracted the Reclination from the Latitude 51 deg. 30 min. and the Remainder would have been 20 deg. 30 min. the Height of the Stile or Gnomon above the Plane.
AS it hath been shewn Chap. 15. That the Base or Horizontal-line of a South Recliner lieth always in the East Azimuth; so the Base of an East Recliner lieth always in the Meridian of the Place: And as all Declining Planes lie in some Azimuth, and cross one another in the Zenith and Nadir, by Chap. 13. So these Reclining Planes lie in some Circle of Position, and cross one another in the North and South Points of the Horizon; which being considered; these East Recli∣ners, West Incliners, and West Recliners, and East Incliners, shall be made as easily as the former.
For these East Recliners be in very deed South Decliners to those that live 90 deg. from us Northward or Southward, and have one of those Poles elevated as much as the Complement of our Latitude; for the Perpendicular or Plumb-line of those Peo∣ple is parallel to the Horizontal Diameter of our Meridian.
I Have an East Plane reclining 45 deg. which I would make a Dial.
In the former Diagram I number 45 deg. from E to F, and then lay a Ruler from N to F, and it will cut the Semidiameter ZW in 45 deg. in V. And then draw the Arch SVN, which Circle shall represent the Plane proposed.
Then the Arch of the Plane between the Horizon and the Substiler Distance is re∣presented in the Diagram by NQ, and may be found by resolving the Triangle QN P, wherein the Angle at Q is known to be Radius, and the Angle at N to be Recli∣nation, and the Angle at P the Latitude. Then work thus.
| As the Radius or Sine of 90 deg. Q | 1000000 |
| Is to the Sine of Reclination 45 deg. N | 984948 |
| So is the Tangent of the Latitude 51 deg. 30 min. PN | 1009939 |
| To the Tangent of the Substile QN 41 deg. 38 min. | 994887 |
Or upon Gunter's Ruler, Extend the Compasses from the Sine of 90 deg. to the Sine of 45 deg. the same will reach from the Tangent of the Latitude 51 deg. 30 min. to neer 41 deg. 38 min. as before, in the Line of Sines; and such is the Substiler distance.
Secondly, The Height of the Pole above the Plane may be represented by the Arch PQ, and may be found, by which we have given in the Triangle QNP: For,
| As the Sine of 90 Q | 1000000 |
| To the Sine of 51 deg. 30 min. PN | 989354 |
| So is the Sine of Reclination 45 deg. N | 984948 |
| To the Sine of the Stiles Height 33 deg. 36 min. or Pole above the Plane PQ | 974302 |
Extend the Compasses from the Sine of 90 deg. to the Sine of 51 deg. 30 min. the same Extent will reach from the Sine of Reclination 45 deg. to 33 deg. 36 min. as be∣fore, which is the Height of the Stile.
Thirdly, The Inclination of Meridians (or indeed you may call it Longitude) is here represented by the Angle PQN; for having drawn the Arch of the Meridian of the Plane SQN, or let fall a Perpendicular PQ, and that from the Pole unto the Plane, this Perpendicular shall be the Meridian of the Plane; so that from Q to N is the Distance of Inclination of both Meridians, which will be found as before: For,
| As the Sine of 51 deg. 30 min. PN | 989354 |
| To the Sine of 90 deg. Q | 1000000 |
| So is the Sine of the Substiler Distance 41 deg. 38 min. | |
| To the Sine of Inclination of both Meridians | 971594 |
Extend the Compasses from the Sine 51 deg. 30 min. to the Sine of 90; the same Extent will reach from 41 deg. 38 min. the Substiler Distance, to 58 deg. 40 min. and such is the Angle PQN of the Inclination, between the Meridian of the Place and the proper Meridian of the Plane: which resolved into time, doth make about 3 ho. 54 min. and so the Substiler must be placed neer the Hour of 8 in the morning.
For to draw the Hour-lines on the Plane, first draw the Horizontal-line SN: Then take off your Line of Chords with your Compasses the Chord of 60 deg. from your Scale, and sweep the Semicircle: Then take off your Line of Chords with your Com∣passes the Substiler Distance, and lay it from N on the Arch to A; Then draw through the Center and A in the Arch, the Substiler Line, crossing it in the Center at Right Angles with the Line KF: Then take off with your Compasses the Height of the Pole above the Plane, or the Stiles Height 33 deg. 36 min. from the Gnomon Line of the Scale, and lay it from the Center of the Dial both ways from K to F. Then take the whole Line of 6 Hours, and sweep the two small Arches from K towards G, and the like from F toward G: Then draw the Lines KG and FG: Then extend the Compasses from G to N, and apply it to the Hour-line, and you shall find the Incli∣nation of Meridians to be as before 3 ho. 54 min. Then take 4 ho. 54 min. and lay it to O, and the same Distance from K unto O toward G; and the like do with 2 ho. 54 min. and make such marks on the Line GF and KG as you see in the Figure to
draw the Hour-lines by; and then take off the Line of Chords 33 deg. 36 min. the Stiles Height, and lay from A to B, so drawing the Hour-lines, and you have done. And then you may see as in a Glass the West Recliner, the opposite Face, as you were shew'd before Chap. 13. that is, strike the Substiler Line, and all the Hour Lines through the Center, and the same Figures to every Hour beyond the Center, which you had on the first side, and set the Gnomon upon the Substile downwards, to behold the South Pole, and you have done both: So have you on the back side, looking through the Paper, the West Recliner and East Incliner, if you draw in the like manner, or prick on the back side, for 11 in the East 1 in the West Recliner, and so contrarily of the rest.
BEfore you can intelligently make a Reclining Declining Dial, which is the most irregular of al, having two Anomalies, viz. Declination and Reclination, you must be acquainted with those three Triangles in the Sphere, wherein cer∣tain Arches and Angles lie, which are needful to be known. I advise you first to draw, though it be but by aim, an Horizontal Projection of the Sphere, such as here I have drawn for a South declining West 45 deg. and reclining from the Zenith 45 deg. in the Latitude of 51 deg. 30 min. which shall be our Example. The same also is shewed in the Fundamental Diagram; only I shew you this, to let you know, there is seve∣ral ways to the Wood besides one.
In this Figure the Arch FLC is the Plane, ZL the Reclination thereof, FE the Base or Horizontal Line of the Plane, and AE n the Vertical of the Plane, cutting it right at L, and cutting the Pole thereof at H: for n is the Pole of a Plane erected upon FE; but the Pole of the Reclined Plane FLE is H n AE; or S••n is the Declination of the Plane, PH m the Metidian of the Plane, cutting the North Pole at P, the Plane at Right Angles at R, and the Pole thereof at H.
In the first Triangle FNO you have given FN 45 deg. the Complement of the Planes Reclination N, the Right Angle of our Meridian with our Horizon F, the Complement of Reclination 45 deg. whereby you may find FO the Oblique Ascen∣sion, or the Arch of the Plane between the Horizon and our Meridian, that is, how many Degrees the Noon-line shall lie above the Horizontal-line. Also you may find NO the Perpendicular Altitude of the Noon-line, or the Inclination of the Noon-line of the Dial to the Horizon, which taken out of 51 deg. 30 min. remains the Arch of the Meridian between the Pole and the Plane. But note, That when this Altitude of the Noon-line NO is equal to NP the Elevation of the Pole, then is the second Triangle PRO quite lost in the Point P, and the Plane becometh then a De∣clining Aequinoctial Plane: Also you may find the Angle at O, called the Angle of Inclination between the Meridian and the Plane. In the second Triangle ORP you have given O as before, R the Right Angle of the Plane with his Meridian, OP the Position Latitude, that is, the Latitude of the Place wherein the Reclining Plane ORLEQ shall be a Circle of Position; this is given if you substract NO, the Al∣titude of the Noon-line, as before, out of the Latitude, and hence may be found OR the Declination of the Gnomon or Substiler Distance, or Distance of the Meri∣dian of the Plane from the Meridian of the Place; RP the Elevation of the Pole above the Plane, in the Planes own Meridian or Stiles Height; P the Angle be∣tween the Meridian of the Plane, and the Meridian of the Place. This Angle is called the Difference of Longitude, because it shews how far the Places are distant from us in Longitude; wherein this Dial shall be a Direct Dial, without Declina∣tion, having his Gnomon in the Noon-line of the Place, and shews also how many Degrees of the Plane comes between the said Meridians. Let this be well observed by Learners.
Hence may be found, if you will, the third Triangle PZH: You have given PZ the Complement of our Latitude, ZH the Complement of the Planes Reclination, Z the Supplement of the Planes Declination.
Also hence may be found HP, whose Complement is PR the Elevation of the Pole above the Plane, the Difference of Longitude H, whose measure is RL, the Arch of the Plane between the Meridian of the Plane or Substile, and the Vertical Line of the Plane; the Complement thereof is RF, the Substiles Distance from the Horizontal Line of the Plane.
Every Arch and Angle is given and may be found by the Problems of Spherical Triangles, as before; but we will make short our business.
First find the Arch of the Plane between the Horizon and Meridian FO.
| As the Sine of 90 N | 1000000 |
| To the Sine of Reclination for the Zenith F 45 | 984948 |
| So is the Co-tangent of Declination 45 NF | 1000000 |
| To the Co-tangent of FO 36. 16 | 984948 |
Extend the Compasses from 90 to 45; the same Extent will reach from the Co-tangent 45, to 35. 16, as before, by the Ruler.
Secondly, The next to find is the Arch of the Meridian between the Pole and the Plane; but we will by opposite Work find it thus.
| As the Sine of the Angle at N 90 | 1000000 |
| To the Sine of his opposite Side FO 54. 44 | 991194 |
| So is the Sine of the Angle at F 45 | 984948 |
| To the Sine of the Side NO 35 deg. 16 min. | 976142 |
Or, Extend the Compasses from the Sine of 90 deg. to the Sine of 54 and 44; the same Extent will reach from the Sine of 45 deg. to 35. 16 NO, which Substracted from the Latitude 51 deg. 30 min. remains the Arch of the Meridian between the Pole and the Plane OP 16 deg. 14 min.
Thirdly, Now you must find the Angle NOF, which will be ROP, which is called the Angle of Inclination between the Meridian and the Plane, thus:
| As the Sine of FO 54. 44 | 008805 |
| To the Radius or Sine of 90 | 1000000 |
| So is the Sine of the Side FN 45 | 984948 |
| To the Sine of the Angle ROP 60. 0 | 993753 |
Extend the Compasses from the Sine of 90, to the Sine of 54. 44; the same Ex∣tent will reach from the Sine of 45, unto 60, as before-found, the Angle of Inclina∣tion between the Meridians and the Plane.
Fourthly, The next to be found is the Substile Distance, or the Meridian of the Plane from the Meridian of the Place.
| As the Sine of the Angle at R 90 | 1000000 |
| To the Co-sine of the Angle at O 30. 0 | 969897 |
| So is the Side of the Tangent PO 16. 14 | 946412 |
| To the Tangent of the Substile OR 8 deg. 17 min. | 916309 |
And thus extend the Compasses from the Sine of 90 deg. to the Co-sine of 30 deg. the same Distance will reach from the Tangent of 16. 14, to the Tangent of the Sub∣stile from the Meridian 8 deg. 12 min.
Fifthly, Next find the Elevation of the Pole above the Plane or Stiles Height: For,
| As the Sine of 90 deg. at R | |
| To the Sine of his Opposite PO 16. 14 | 944645 |
| So is the Sine of the Angle at O 60 | 993753 |
| To the Sine of the Stiles Height above the Plane 14. 1 | 438398 |
Or, Extend the Compasses from the Sine of 90 deg. to the Sine of 16 deg. 14 min the same Distance will reach from the Sine of 60 the Angle at O, to the Height of the Pole above the Plane 14 deg. 1 min. or Stiles Height, as before.
Sixthly, Now for the Difference of Longitude or Angle at P, or Inclination of the Meridian of the Plane to the Meridian of the Place, it is thus.
| As the Sine of the Arch PO 16. 14 | 944645 |
| Is to the Sine of 90 deg. R | 1000000 |
| So is the Sine of the Substile Distance 8 deg. 17 | 915856 |
| To the Angle at P or Difference of Longitude 31 deg. 1 min. | 97••211 |
Extend the Compasses from the Sine of 16 deg. 14 min. to the Sine of 90 deg. the same Extent will reach from the Substile Distance 8 deg. 17 min. to the Inclination of both Meridians 2 ho. 4 min. as before.
Seventhly, The Distance of the Hours from the Substiler are here also represented by those Arches of the Plane which are intersected between the proper Meridian and the Hour-circles. The Angle at R, between the Pole and the proper Meridian, is a Right Angle; the Side RP is the Height of the Pole above the Plane; and then the Angles at the Pole between the proper Meridian and the Hour-circles may be gathered into a Table: For,
Extend the Compasses from the Sine of 90, to the Sine of 14. 1; the same Extent will reach from 16. 14, to 3 deg. 50 min.
The Arch of the Plane between the Horizon and the Meridian FO 54 deg. 44.
The Arch of the Meridian between the Horizon and the Plane ON 35. 16.
The Angle of Inclination between the Meridian and the Plane FON 60.
The Substile Distance from the Meridian OR 8 deg. 17 min.
The Height of the Stile above the Plane PR 14 deg. 1 min.
The Inclination of both Meridians or Angle at P 31 deg. 1 min.
The Difference of the Hour from the Substile 3 deg. 50 min.
NOte, You have got the three principal Arches for the drawing the Hours.
How to draw the South declining West 45 deg. and reclining from the Zenith 45 d. is thus. First draw the Horizontal Line Fe, with your Compasses take off the Scale and Line of Chords the Radius 60 deg. and sweep the Circle FORPe; then take
of Chords 8 deg. 17 min. the Substile Distance, and lay it from O to R; then from the Center draw the Substile Line CG; then take of the Quadrant 90 deg. of the Line of Chords, and lay it both ways from R to M and N, and draw that obscure Line; then take off the Poles Height above the Plane or Stiles Height 14 deg. 1 min. from the Gnomon Line of the Scale, and lay it both ways from C to K; and then with your Compasses take off your Scale the whole Line of 6 Hours, and from K to∣ward G, and from I toward G, strike the two small Arches; then draw the Lines KG and IG; then extend the Compasses from G to the Meridian Line at L, and apply that Distance to your Scale and Hour-line, and you will find it to be 2 ho. 4 m. which is the Inclination of Meridians or Angle at P, as before: Then lay that Distance from L toward G, and take (as before directed in Chap. 17.) one Hour less, and lay it off in the same manner from G toward K, and from I toward G; and the like do with 3 ho. 4 min. until you have put all the 6 Hours on the Line from G to K, and from I to G. Then draw the Hour Lines according to your Plane, whether it be a Triangle, or Circle, or Square, or what shape soever; then lay off the Height of your Stile 14 deg. 1 min. from R to P, taken off the Line of Chords, and draw it from the Center, and cut it fit to your Plane in what shape you will, as you may see in the Figure, and your Dial is done. The opposite Face or Incliner to the Horizon is but to continue the Stile and Substile and Hour-lines through the Center, as you may see, and you have it. In like manner you may draw the South East Decliner 45 deg. and Reclining from the Zenith 45 deg. if you once draw on Paper this before, and the Hour-lines over the same, and the like the Substile, and the Stile to stand upon the Substile upright, as of the rest: And let the highest part of the Stile be towards the North Pole, pointing upwards; and where the Hour of 10 is in the South West Re∣cliner, on the back side put 2 a clock, and for 11 put 1, and for 1 put 11, and for 2 put 10, and so contrary all the rest. And if you observe your Work, you have the South West and the South East Recliners, and the North West and North East Incliners; or you may draw them by what was given in the first, in the same manner.
YOu have seen Chap. 17. how easily East and West Reclining Dials are to be made; and by the Figure in Chap. 18. how they fall out to be Circles of Position, as you may see by PORCQ.
I will shew you how all reclining Dials may be reduced to East or West Recliners, for some Latitude or other; and so the Hour-distance found by the Method of Chap. 17.
The Circles of Position, as have been shewed, do all cross one another in the North and South Points of the Meridian: Now therefore by the Point O, where the Plane cuts our Meridian, draw a new Horizon, as OBQC, and then shall you see your Plane in that Horizon to be a very Circle of Position.
But now we are gotten into a new Latitude OP, called before Chap. 18. the Posi∣tion Latitude; and we have here a new Reclination: for whereas this Plane reclineth in our Latitude ZFL 45 deg. his Position Reclination is O, viz. ZOL or POR 60 deg. In the making of this Dial therefore you shall forget your own Latitude, and the Planes Reclination in your Horizon; and with this new Latitude and Reclina∣tion make the Dial after the manner of the East Recliner, Chap. 17. not regarding the Declination at all: for the Base of this Plane is now fallen into the Horizontal Line of the Meridian; and his Declination being a Quadrant, he is become a Regular Plane, and neither his Declination nor Reclination shall much trouble you.
How to place your Noon-line from the Horizontal or Vertical Line of the Plane, you have found already.
Note, Your new Latitude is PO 16 d. 14 m. then you know your Plane is the 60 d. Azimuth from the Axis, because POR is 60 deg. as before 90 deg. farther from the said Azimuth you have the Pole of the Plane, and therefore is the Meridian of my Plane, and shall make the Substile of my Dial OPR 3 deg. 1 min. his Distance from the Meridian of the Place in that Aequinoctial, and is therefore the Difference of Lon∣gitude, as before.
Then have you the Side PR the Height of the Stile 14 deg. 1 min. or Elevation of the Gnomon, as before: Likewise have you OR the Declination or Substile Distance from the Meridian 8 deg. 17 min. as before, and you may proceed to draw the Dial in like manner as you have been directed in Chap. 16.
EVery declining Plane, whether it recline or not, hath two great Meridians much spoken of.
1 The Meridian of the Plane, which is the proper Meridian of that Country to whose Horizon the Plane lieth Parallel.
2 The Meridian of the Place, which is the Meridian of our Country in which you set up this Declining Plane, to shew the Hours; and so either of these Meridian Dials may be conformed.
How to draw the Hours of our Country on such a Plane, is the harder work, be∣cause
the Plane is irregular to our Horizon: yet I suppose I have made the way very easie in the former Chapters. But to draw the Hours of the Country to which the Plane belongs, is most easie; for if you take the Substiler for the Noon-line, and the Elevation of the Pole above the Plane for the Latitude, you may make this Dial in all points like the Vertical Dial, after the Precept of Chap. 8.
IF the Dial decline East, add the Difference of Longitude found in Chap. 13. & 18. to the Longitude of the Place, and the sum or the excess above 360 is the number of the Longitude sought. If the Dial decline West, substract the said Difference of Longitude out of the Longitude of your Place and the Difference is the Longitude inquired: but when the Longitude of your place happens to be less than the Difference of Longitude, you must add to it 360 deg. before you substract the Difference of Longitude. Note, The Elevation of the Pole above the Plane, or Stiles Height, is the Latitude of the Place inquired.
Example. The Declining Plane of Chap. 13. will be a Vertical Plane in the Lon∣gitude of 66 deg. 46 min. in the Desarts of Arabia neer Zoar: and the declining re∣clining Plane of Chap. 18. & 19. is parallel to the Horizon of those that sail in Lon∣gitude 357 deg 41 min. and North Latitude 14 deg. that is, as the Terrestrial Globes and Maps shew me, between Bonavista, one of the Cape Verd Islands, and Barbadoes.
I Could not pass by this Example of the North Recliner Decliner, and South In∣cliner Decliner, although it is shewed in the Fundamental Diagram; but it may be too obscure, and harder to be apprehended by the Industrious Practitio∣ner there; therefore I would advise him to draw a Scheme of the Dial, as was shewed Chap. 15. & 18. or draw the Fundamental Diagram in Paper, and with a small Needle prick the Hour-lines, Horizon, and Meridians; Aequinoctial, and the Tro∣picks; and then you have a Figure ready to be stamped with a little Charcoal-dust as often as you have occasion: Or if you apprehend your Work in any manner, the Figure following, or the like, may serve your turn, to shew you the Angles you are to find, and Arches for the making of your Dial. I shall be short in this, and refer you to Chap. 15. & 18.
The Circle ESWN is our Horizon, as before; NS our Meridian, FLC the Plane, ZL the Reclination thereof, FC the Base or Horizontal Line of the Plane, AEN the Vertical of the Plane, cutting it right at L, and cutting the Pole thereof at H: for N is the Pole of a Plane erect upon FC; but the Pole of the Reclining Plane FLC is H; SE or nN the Declination of the Plane.
Now you see your three Triangles all adjoyning in this Scheme, viz. FSO and ORP rectangled at S and R, and PZH obtuse angled at Z.
It is true, That the two first may do the Work, and so we will be brief. Observe, you are to find as followeth.
1. To find the Arch of the Meridian between the Horizon and the Plane, it is thus.
| As the Sine of 90 at S | 1000000 |
| To the Sine of Reclination at F 45 | 984948 |
| So is the Co-tangent of Declination FS 45 | 1000000 |
| To the Complement Tangent of FO 35 deg. 16 min. | 984948 |
Or thus: Extend the Compasses from the Sine of 90, to the Sine of 45; the same Extent will reach from the Tangent of 45 deg. to the Sine of 35 deg. 16 min. as be∣fore; which substracted from 90 deg. there remains 54 deg. 44 min. the Arch of the Plane between the Horizon and Meridian FO.
2. To find the Arch of the Meridian between the Pole and the Plane, first find the Arch of the Meridian between the Horizon and the Plane SO thus.
| As the Sine of 90 at S | 1000000 |
| To the Sine of the Arch of the Plane between the Horizon and Plane FO 54. 44 | 991194 |
| So is the Tangent of Reclination at F 45 | 1000000 |
| To the Tangent of the Arch of the Meridian between the Horizon and Plane SO 39 deg. 14 min. | 991194 |
Or thus: Extend the Compasses from the Sine of 90, to the Sine of 54 deg. 44 m. the same Extent will reach from the Tangent of 45 deg. to the Tangent of SO 39 d. 14 min. by the Tables found before: which 39 deg. 14 min. taken out of 90 deg.
there remains 50 deg. 46 min. the Arch of the Meridian between the Zenith and the Plane OZ; which being added to the Complement of the Latitude ZP, there will be 89 deg. 16 min. for the Arch of the Meridian between the Pole and the Plane PO.
3. For the Angle between the Meridian and the Plane FOS or ZOL, it is
| As the Sine of the Arch FO 54 deg. 44 min. | 991194 |
| To the Radius or Sine of 90 deg. S | 1000000 |
| So is the Sine of the Declination FS 45 deg. | 98494•• |
| To the Sine of the Angle at O 60. 0 | 993754 |
Or, by Gunter's Rule, Extend the Compasses from the Sine of 54 deg. 44 min. to the Sine of 90 deg. the same Extent will reach from the Sine of 45 deg. O, to the Sine of 60 deg. 0, the Angle between the Meridian and the Plane.
4. Before we find the Inclinations of both Meridians, or the Difference of Longi∣tude or Angle at P, we will find the Substiler Distance OR.
| As the Sine of the Angle at R 90 deg. | 1000000 |
| To the Co-sine of the Angle at O 60 deg. | 969897 |
| So is the Tangent of the Side PO 89 deg. 16 min. | 1189279 |
| To the Tangent of the Substile OR 88 deg. 32 min. | 1159176 |
Extend the Compasses from the Sine of 90 deg. to the Sine of 30 deg. the same Extent will reach from the Tangent of 89 deg. 16 min. to the Tangent of the Substile 88 deg. 32 min.
5. Now for the Inclination of both Meridians, or Difference of Longitude, or Angle at P, it is thus. For,
| As the Sine of the Arch PO 89 deg. 16 m. | 999996 |
| To the Radius or Sine of 90 at R | 1000000 |
| So is the Substiler Distance OR 88 deg. 32 min. | 999985 |
| To the Sine of the Angle of Inclinations between both Meridians 88 deg. 45 min. Difference of Longitude | 999989 |
6. For the Height of the Pole above the Plane, or Stiles Height,
| As the Radius or Sine of 90 deg. | 1000000 |
| To the Sine of the Arch PO 89 deg. 16 min. | 999996 |
| So is the Sine of the Angle at O 60 deg. | 993753 |
| To the Sine of the Stiles Height PR 60 deg. | 993749 |
AS was said Chap. 18. the three principal Arches for drawing the Hour-lines in a South declining West and inclining to the Horizon, and a North Recli∣ner Decliner, as before, or South East Incliner, or North West Recliner, are these three.
1. The Arch of the Plane between the Horizon and Meridian 54 deg. 44 FO.
2. The Substile Distance from the Meridian of the Place 88 deg. 32 OR.
3. The Height of the Pole above the Plane, or Stiles Height 60 deg. PR.
And so follow the Directions of Chap. 19. and draw the Dial as you see. In what shape soever the Plane is, proportion the Hours and Substile to your Plane, and let the Gnomon or Stile stand upright on the Substile Line; and so have you the lower Face the South West Incliner to the Horizon 45 deg. Recliner the like North Face, and de∣clining 45 deg. as before, and your Dial is done.
Note, This South West Incliner Recliner is parallel to the Horizon of those that live on the South Land in the South Sea at Terra vista Decleros, in Longitude 297, and South Latitude 60, neer the Straights of Magellanicum.
And the North declining East and Recliner will be parallel to the Cossacks, neer Tartaria, in Longitude 114 deg. and Latitude North 60 deg. as the Globe shews me, and I have shewed you how to know the like in Chap. 21.
And observe, The 5 ho. 55 min. or 88 deg. 45 Difference of Longitude, as before, shews that the Sun rises and comes to the Meridian, or 12 a Clock with us at Bristol, 5 ho. 55 min. before it doth with those Inhabitants in the South Sea, as before: And on the contrary, the Sun is risen or on the Meridian with the Cossacks to the Eastward of us, the like time as the Sun rises with us before it doth with those of Terra Vista Decleros in the South Sea: So that you see of what use the Line of Inclination of Meridians on your Scale is, as likewise of all Declining Dials.
THese several sorts of Planes take their denomination from those Great Circles to which they are Parallels, and may be known by their Horizontal and Perpendicular Lines, of such as know the Latitude of the Place, and the Circles of the Sphere.
1. An Aequinoctial Plane, parallel to the Aequinoctial, which passeth through the Points of East and West, being right to the Meridian, but inclining to the Horizon, with an Angle equal to the Complement of the Latitude; this here is represented by EOW.
2. A Polar Plane, parallel to the Hour of 6, which passeth through the Pole and Points of East and West, being right to the Aequinoctial and Meridian, but incli∣ning to the Horizon, with an Angle equal to the Latitude; this is here represented by EPW.
3. A Meridian Plane, parallel to the Meridian the Circle of the Hour of 12, which passeth through the Zenith, the Pole, and the Points of South and North, being right to the Horizon, and the Prime Vertical; this is here represented by SZN.
4. An Horizontal Plane, parallel to the Horizon, here represented by the outward Circle ESWN.
5. A South and North erect direct Dial, parallel to the Prime Vertical Circle, which passeth through the Zenith, and the Points of East and West in the Horizon, and
is right to the Horizon and Meridian; that is, makes Right Angles with them both: this is represented by EZW.
6. A South Declining Plane Eastward is represented by BD.
7. A South Incliner and North Recliner is represented by EQW.
8. The South Recliner and North Incliner is represented by EAW.
9. A Meridian Plane, which is the East and West Incliners and Recliners, and from the Zenith parallel to any Great Circle which passeth through the Points of South and North, being right to the Prime Vertical, but inclining to the Horizon; this is represented by SVN.
10. A declining, reclining, inclining Plane, which is parallel to any Great Circle which is right to none of the former Circles, but declining from the Prime Vertical, reclining from the Zenith, inclining to the Horizon and Meridian, and all the Hour Circles; this may here be represented either by FLC or FKC, or any such Great Circles which pass neither through the South and North, nor East and West Points, nor through the Zenith nor the Pole.
Each of those Planes, except the Horizontal, and South inclining 23 deg. hath two F••ces whereon Hour-lines may be drawn; and so there are 19 Planes in all. The Meridian Plane you see hath one Face to the East, and the other to the West: Re∣member, that it is an East and West Dial. The other Vertical Planes have one to the South, another to the North; and the rest, one to the Zenith, and another to the Nadir. What is said of the one, may be understood of the other.
THe Projection of some other Circles of the Sphere besides the Meridians (though it be not necessary for finding the Hours, yet) may be both an Or∣nament to Dials, and useful also for finding the Meridian, and placing the Dial in its due situation, if it be made upon a movable Body, as shall hereafter be shewed.
The Circles fittest to be projected in all Dials for those purposes, are, the Aequator with his Tropicks, and other his Parallels, which may be accounted Parallels of Declination, as they pass through equal Degrees, as every 5 or 10 of Declination: Or Parallels of the Signs, as they pass through such Degrees of Declination as the Sun declineth when he entreth into any Sign, or any notable Degree thereof; or Paral∣lels of the Length of the Day, as they pass through such Degrees of Declination wherein the Sun increaseth or decreaseth the Length of the Day by Hours or half Hours.
Also the Horizon, with his Azimuths and Almicantars, are as an Ornament to Horizontal and Vertical Dials, and are likewise useful for projecting the Aequator and his Parallels in all Dials. My intent is to be brief in this Treatise of the Furni∣ture here following, because I will have a President of some other Country Dials: I shall therefore think it sufficient if I shew you one way to furnish any Dial with the Circles of the Sphere, leaving you to devise others which I could have shewn.
FRom any Point of the Gnomon taken at pleasure let fall a Perpendicular upon the Substile; that Perpendicular shall be part of the Axis of the Plane, and shall be reputed Radius to the Horizon of your Plane. The top of this Radius in the Gnomon is called Nodus, because there you must set a Knot, Bead, or Button, or else cut there a Notch in the Gnomon, to give shade; or cut off the Gno∣mon in the place of the Nodus, that the end may give the shadow for those Linea∣ments. Let not your Nodus stand too high above the Plane, for too great a part of the Planes day; nor let it stand too low, for then the Lineaments will run too close together: a mean must be chosen.
At the foot of this Radius take your Center and describe a Circle of the Plane,* 1.6 and divide it into equal Degrees, and from the Center draw Lines through those Degrees infinitely, that is, so far as your Dial Plane will bear; these Lines shall be the Azimuths of the Horizon of the Plane, and shall be numbred from his Meridian or Substile.
Divide any of these Azimuth Lines into Degrees, by Tangents agreeable to the said Radius; and having made a prick at every Degree, through every of these pricks you shall draw parallel Circles, which shall be Almicantars or Parallels of Altitude, to be numbred inwards; so that at the Center be 90 for the Zenith, and from the Center outwards you shall number 80, 70, 60, until you come within 10 or 5 deg. of the Horizon; for the Plane is too narrow to receive its own Horizon, or the Pa∣rallel neer, if the Nodus have any competent Altitude.
SUch Planes whose Stiles or Gnomons lie low, cannot have their Hour-lines di∣stinctly severed, unless the Center of the Dial be placed out of the Plane, as you may see in Chapter 18. Now to inlarge the Stile in such Dials, there are two Lines or Scales in the large Mathematical Scale, which I call Polar or Tangent Lines of two several Radius's, the larger of them marked thus ✚, and the lesser of them marked -, as you may see in the Second Book, Chap. 3. The Use whereof I shall here shew you.
In Chapter 18. of this Book there is descibed a Plane that declines from the South, Westward 45 deg. and reclines from the Zenith 45 deg. Now suppose it were re∣quired to inlarge that Stile and Dial.
First you shall find that there is given the Arch of the Plane between the Horizon and the Meridian, 54 deg. 44 min. Secondly, The Substiler Distance from the Meri∣dian to the Substile is 8 deg. 17 min. Thirdly, The Height of the Pole above the Plane is 14 deg. 1 min. Fourthly, The Difference of Longitude or Inclination of Me∣ridians is 31 deg. 1 min. as you may see in Chap. 18. These being found, you must thus proceed in the delineation of your Dial.
First with your Compasses take from off your Scale a Chord of 60 deg. and on the Center C sweep the Arch HLRP, and draw CH the Horizontal Line blindly. Then from H set off the several Distances before found, upon this Arch of a Circle, viz. First, HL 54 deg. 44 min. for the Arch of the Plane between the Horizon and the Meridian. Then set off LR 8 deg. 17 min. for the Distance of the Substile from
the Meridian, and there draw the Line of the Substile CRB. Thirdly, From this Point R set off RP, which is 14 deg. 1 min. for the Height of the Stile above the Substile, and draw the prick'd Line CPK for the Stile or Axis.
Now this Stile being but somewhat low, for the enlarging hereof first chuse some convenient place in your Substiler Line, as in this Example at B, and there draw the Line FBA squire wise to the Substiler Line. Then take with your Compasses from off your larger Polar Line marked ✚ the Distance of three Hours, and prick it down in this Line from B to I; then from this Point I take with your Compasses ••••e neerest distance to the Line of the Stile, and with that distance draw the Line I▪ pa∣rallel to the Stile, and this will be the Line of the Stile inlarged.
Now to set on the lesser Polar Line, that so you may draw the Hour-lines, first take from off your lesser Polar Line marked -, the distance of 3 Hours, and with that distance draw a little white Line parallel to the Line of the Stile, and note where it cuts the Line of the Stile inlarged, which is in the Point R. Mark this Point R well, and through this Point R draw the Line OG parallel to the former Polar Line FBA. And now if you take the distance ER in this Line, and measure it on your Scale, if you have done your work right, you shall find it just equal to 3 Hours upon the lesser Polar Line. And now by these two Polar Lines you must draw the Hour Lines after this manner.
First you must consider the Inclination of the Meridians, which in this Example is 31 deg. 1 min. which reduced into time, is 2 ho. 4 min. or 2 of the Clock and 4 min. in the afternoon, from whence you may frame a Table for your direction in placing the Hours, after this manner.
| The Hour-lines of the Plane. | Their dist. from the Substile. |
| Ho. Mi. | |
| X | 4 F 4 |
| XI | 3 4 |
| XII | 2 4 |
| I | 1 4 |
| II | 0 4 |
| Substile | 0 B 0 |
| III | 0 56 |
| IIII | 1 56 |
| V | 2 56 |
| VI | 3 A 56 |
Now according to this Table, take the distance of these Hour-lines first out of your larger Polar Line ✚, and prick them down in the Line FBA, from the point B towards F and A. Thus the Line of II must be set 4 min. from the Substile B towards F; the Line of I must be set 1 ho. 4 min. from the Substile B toward F; the Line of XII must be set 2 ho. 4 min. from B towards F. And so on the other side of the Substile, the Line of III must be set 56 min. from the Point B toward A; the Line of IIII must be set 1 ho. 56 min. from B toward A. And so you must do till you have set down all the Hour-lines upon the Line FBA, taking them out of the larger Polar Line upon your Scale.
Having thus set down the Hour-lines in the Line FBA, you must set them also in the Line OEG, taking their Distances off from your lesser Polar Line -, as be∣fore you did from the larger Polar Line, making use of the Table to direct you, as before.
And thus having made marks for the Hour-lines in both these Tangent-lines, you must draw the Hour-lines through these Marks; and so shaping your Dial into a Triangle, Square, or Circle, as your Plane will best allow, number the Hour-lines with their proper Figures, and so finish your Dial, as the Scheme will direct you better than many words.
THE Stile or Gnomon in this Example is very low, lying very neer to the Pole of the World, as you may see before in the Fundamental Diagram. This Dial only reclines from the Zenith; and therefore to know the Stiles Height,
you need only substract the Reclination from the Latitude of the Pole; which being 51 deg. 30 min. the Reclination 45 deg. substracted out of it, there remains 6 deg. 30 min. for the Height of the Stile.
This Dial hath no Declination, and therefore the Substile must be the Meridian-line; draw that Line therefore first about the middle of the Plane, and then with a Chord of 60 deg. describe a short Arch RP, and from R prick off 6 deg. 30 min. according to the Height of the Stile before-found, and thereby draw the prickt Line CP from the Center C, representing the lesser Stile of the Plane.
Now make choice of some fit place upon the Substiler Line, as B, and there cross the Meridian at Right Angles with the Line FBA: Then take from off your larger Polar Scale ✚ the distance of 3 Hours, and prick it from B to q. Then take the neerest distance from this Point q, to the Line of the Stile or Axis, and therewith draw a Line parallel to the Stile, and this shall be the Line of the Stile inlarged. Then take the distance of 3 Hours off from your lesser Scale marked -, and with that distance draw a blind Line parallel to the Meridian or Substile, and mark where it crosseth the Line of the Stile Inlarged, which is in the Point ♈: Then take the neerest distance from this Point ♈, to the former Polar Line FBA, and so draw the Line OEG parallel thereunto, through the Point ♈: This shall be your lesser Polar Line.
Now to set off the Hour-lines upon these two Lines, first take with your Compasses the distance of one Hour off from your ✚ Polar Scale, and prick it both ways from the Point B toward A and F: Likewise do the same for 2, 3, and 4 Hours, and prick them down in the Line FBA.
Then take the distances of each Hour also off from the lesser Polar Scale -, and prick them down from E on both sides the Meridian Line, in the Line OEG. Then draw through these Points the Hour-lines by their several marks, as you may see in the Figure, and put the numbers of the Hours thereunto; so your Dial will be finished, as in the Figure. And if you well understand this, you may do the like in any other Dial which shall need inlarging.
THe greatest part, and as much as you shall use of the Vertical or Horizontal Dial, described Chap. 8. may by reflection be turned upside down, and placed upon a Cieling; but the Center will be in the Air without doors.
The first thing to do is to fasten a piece of Looking-glass, as brood as a Groat or Sixpence, set level; or a Gally-pot of Fair-water, which will set it self level being placed upon the Sole of the Window, shall supply the use of the Nodus in the Gno∣mon; and the Beams of the Sun being reflected by the Glass or Water, shall shew the Hours upon the Cieling.
Before you can draw a Figure for this Dial well, I would advise you first to find the Meridian of the Room, which may be done thus.
Hang a Plumb line in the Window, directly over the Nodus or place of the Glass; for the shadow which the Plumb-line gives upon the Floor at Noon, is the Meridian-line sought; and by a Ruler, or a Line stretched upon it, you may prolong it as far as you shall need.
Then take the perpendicular Height thereof from the Glass to the Cieling of the Room, which suppose it be 40 Inches, as DB, the Glass being fixed at D: Now from B draw the Meridian-line upon the Cieling, which shall be represenred both ways continued, as ABCK, and from D erect a Perpendicular to DC: Or let a stander by stop one end of a Thred on the Glass at D; extend the same to the Meri∣dian-line, moving the end of the s••ring shorter and longer upon the Meridian, till
another holding the Side of a Quadrant, shall find the Thred and Plummet to fall directly upon the Complement of the Latitude, which in this Example is 38 deg. 30 min. and that is the Intersection of the Aequinoctial. Then raise the Perpendicu∣lar, as DA; take a Chord of 60 deg. and from A sweep the Arch at P, and from Play down the Latitude 51 deg. 30 min. to q, and draw the Line AD.
Then let fall a Perpendicular from C, as CEFGHI, which is the Aequinoctial Line; and so likewise draw a Parallel Line to the Meridian of 40 Inches at D. Now note, That the Hours must be drawn all one as the Horizontal Dials are.
Then draw a Line Parallel to the Aequinoctial, as KO, at what distance you think convenient, on the Cieling of the Room, which let be here 50 Inches, as CK, as you may measure by the Scale. Now for the placing of the Hours on the Cieling of the Room, you must measure how much by the Scale of Inches each Distance on the Aequinoctial Line is from C to E, and from C to F, and from C to G, H, and I, and likewise on the Parallel from K, collecting them into a Table; so will it be ready to transport on the Cieling.
| Inches. par. | |
| C E | 17 0 |
| C F | 38 0 |
| C G | 63 3 |
| C H | 111 4 |
| C I | 222 5 |
| K L | 27 7 |
| K M | 59 7 |
| K N | 101 0 |
| K O | 173 0 |
| K P | 385 8 |
So by this Table you shall find the Distance CE, which is from 12 to 1 in the afternoon, or 11 in the morning, to be 17 Inches; which you may prick upon the Cieling. Likewise KL on the Parallel, between 12, 11, and 1, will be found to be 27 Inches 7/10 parts of an Inch in 10 parts; which Hour-line mark out upon the Ciel∣ing. Then draw a straight Line through those two Points L and E; this Line conti∣nued shall be the first Hour from the Meridian, which is 11 in the morning, or 1 in the afternoon: So do for all the rest of the Hours.
Now by this you may know how far the Center is without the Window; measure it, and you will find it 31 82/100 Inches from A to B, and from B to C 52 Inches, and from C to K 50 Inches, as before. I hope now I have given the Practicioner content, in making this so easie to be understood, although I may be condemned by others.
I will give you one Example more, to find the Meridian Line on the Cieling, which is this. Fit a plain smooth Board, about a Foot square, to lie level from the Sole of the Window inwards; then neer the outward edge thereof make a Center in the Board, in the very place of Nodus, or a little under it: Then by Chap. 3. get the Meridian Line from the Glass on the Board; after you have drawn the Line on the Board upon the Center, describe as much of a Circle as you may with the Semidia∣meter of your Quadrant, which Circle shall be Horizon; then from the Meridian you may with your Degrees on the Quadrant graduate your Horizon into Degrees of Azi∣muths both ways as far as you can.
Next you may devise to make your Quadrant stand firm and upright upon one of his straight Sides, which I will call his Foot for this time; and that you may thus do, take a short space of a Ruler or Transom, and saw in one side of it a Notch perpendi∣cularly, in which Notch you may stick fast or wedge the heel of the toe of your Qua∣drant, in such sort as his Foot may come close to the Board, and the other Triangular Side or Leg may stand perpendicular upon it. Let the Foot be round, and with your Compasses strike a Circle round it: when you have fitted the Diameter of the Foot on the Meridian Line on the Board, draw a Circle round the Glass, that so you may set the edge of the Circle according as you may have need, for to lay off the Suns Altitude at every Hour. Now to find the Meridian on the Cieling, you may make a Table for the Suns Altitude every Hour of the Day, in this manner as here is for the Latitude 51 deg. 30 min. and place the Foots Diameter directly on the Meridian of the Board, and elevate the Quadrant to the Tropick of Capricorn, which in this Lati∣tude is 15 deg.
| Hours. | Cancer. | Gemini. Leo. | Taurus. Virgo. | Aries. Libra. | Pisces. Scorpio. | Aquari. Sagitta. | Capric. |
| 12 | 62 0 | 58 45 | 50 0 | 38 30 | 27 0 | 18 18 | 15 0 |
| 11 1 | 59 43 | 56 34 | 48 12 | 36 58 | 25 40 | 17 6 | 13 52 |
| 10 2 | 53 45 | 50 55 | 43 12 | 32 37 | 21 51 | 13 38 | 10 30 |
| 9 3 | 45 42 | 43 6 | 36 0 | 26 7 | 15 58 | 8 12 | 5 15 |
| 8 4 | 36 41 | 34 13 | 27 31 | 18 8 | 8 33 | 1 15 | |
| 7 5 | 27 17 | 24 56 | 18 18 | 9 17 | 0 6 | ||
| 6 6 | 18 11 | 15 40 | 9 0 | ||||
| 5 7 | 9 32 | 6 50 | 11 37 | ||||
| 4 8 | 1 32 | 21 40 |
Let a stander by stop on the Glass a Thred, and extend the other part straight on to the Cieling, the Thred touching only the Plane of the Quadrant, and making no Angle with it, but held parallel; and where the Thred thus extended touches the Cieling, make a Point; then the Quadrant unmov'd, elevated to 62 deg. of Altitude, and extend the Line, and make another Point as before; and between those two Points draw a straight Line, and that shall be your Meridian, and shall be long
enough for your use. Then elevate the Quadrant to 38 deg. 30 min. and hold the Thred to the Meridian on the Cieling, and where he touches mark; and cross the Meridian at Right Angles with an Infinite Line, which shall be the Aequator: So you may do as you did before. but if the Plane of the Cieling of the Wall is inter∣rupted, and made irregular by Beams, Wall-plates, Cornishes, Wainscot, or Chim∣ney-piece, and such like Bodies, I will shew you the Remedy to carry on your Hour-lines over all.
Extend the Thred from any Hour-line to the Tropick of Cancer in the Cieling, as you were taught before, and fix it there; and extend another Thred in like manner to the Tropick of Capricorn, where-ever it shall happen beyond the middle Beam, or quite beyond the Cieling upon the Wall, and fix the Threds also. Then place your eye so behind these Threds, that one of them may cover another; and at the same instant where the upper Line to your sight or Imagination cuts the Cieling, Beam, Wall, or any irregular Body, about the end of the lower Line, there shall the Hour-line pass from Tropick to Tropick: Direct any By-stander to make Marks, as many as you shall need, and by those Marks draw the Hour-lines according to your desire. This is in Mr. Palmer, pag. 202.
If the Arch of the Horizon, between the Tropicks, be within view of your Win∣dow, you shall draw the same on the Wall to bound the Parallels. The Horizon Altitude is nothing, and therefore it will be a level Line: and the Suns Azimuth when he riseth, commonly called Ortine Latitude, is in Cancer 40 deg. East North∣ward, and in Capricorn as much Southward; and these will be reflected to the contra∣ry Coasts on the Dial.
A Globe, saith Euclid, is made by the turning about of a Semicircle, keeping the Diameter fixed. This Dial, if Universal, will want the aid of a Magneti∣cal Needle to set it, and it must move on an Axis in an Horizon, as the usual Globes do; whose Aequator let be divided into 24 Hours, the proportion of the Day Natural.
You may see the Figure on the top of the Dial in the Title, but that you cannot see the two Poles, and the Semicircle, and the Horizontal Circle.
You may imagine this Globe set to the Elevation of the Pole, as that is, with two Gnomons of the length of the Suns greatest Declination, proportion∣ed to the Poles Circle, with the 24 Hours, according to the 24 Meridians, and serves for a North and South Polar Dial.
But in the Meridian let be placed the 12 a Clock Line; then turn the Semi∣circle till it cast no shadow: then doth it cross the Hours, which Hours are drawn from the Pole to each of the 24 Divisions, as before.
If you desire to cover the Globe, and make other Inventions thereon, first learn here to cover it exactly. With a Pair of Compasses bowed towards the Points (like a Pair of Calapers the Gunners use) measure the Diameter of the Globe you intend to cover; which being known, find the Circumference thus.
Multiply the Diameter by 22, and divide the Product by 7, and you have your desire.
Let the Circumference found be the Line EF, which divide into 12 Equal parts; draw the Parallel AB and CD, at the distance of three of those Parts from E to A and from F to C; then by the outward Bulks of those Arches draw the Line AB and CD.
And to divide the Circumference in∣to 12 parts, as our Example is, work thus.
Set your Compasses in E, and make the Arch FC: The Compasses so open∣ed set again in F, and make Arch E A; then draw the Line from A to F, and from E to C. Then your Com∣passes opened at any distance, prick down one part less on both those slant∣ing Lines, than you intend to divide thereon; which is here 11, because we would divide the Line EF into 12: Then draw Lines from each Division to his opposite, that cuts the Line EF in the parts or Division. —But to proceed, It is Mr. Morgan's Conceit, page 116. Continue the Circumference at length to G and H, numbring from E towards G 12 of those Equal Parts, and from F towards H as many, which shall be the Center for each Arch; so those Quarters so cut out, shall exact∣ly cover the Globe, whose Circumfe∣rence is equal to EF.
Thus have you a glance of the Ma∣thematicks, striking at one thing through the side of another: For here one Fi∣gure is made for several Operations, to save the Press the charge of Fi∣gures.
THis Dial is made all one as the South Dial you may see Chap. 9. Only observe this, That you are 35 deg. to the Southward of the Aequinoctial, and that the Sun, when he is on the Tropick of Capricorn, wants 11 ½ Degrees of the Zenith of that Place Northward: As the Sun goes always to the Southward of us in England, so it goes to the Northward of them; therefore must the Stile or Gnomon point downward in the North Face, and upward in the South Face. So likewise as in our South Dial the afternoon Hours are put on the East side the Dial Plane, and
the morning Hours on the West side; so in their North Dials they will stand con∣trarily, by reason the Sun casts a Shadow (as the Plane must stand there) in the morning to the West side, and in the afternoon to the East: So you see the Plane is only turned to face the Sun. If you do but conceive in your mind how the Sun
HAving the Age of the Moon by the Epact; as for Example, the 9th. day of August 1665. the Epact 23, to which add 9 the day of the Month, and 6 the Months from March, makes 38; from it substract 29 a whole Moon, with 12 ho. 44 min. which multiply the remaining 9 by 4, makes 36; that divide by 5, and you have 7 ho. and 12 min. for the odd Unite, and 24 min. for the half day, or 12 hours added together, makes 7 ho. 36 min. for the Moon being South.
Now having the Moons coming to the South by the former way, add this Southing of the Moon and the Shadow of the Moon upon the Dial together, and that is the time of the Night. If the Sum exceed 12 Hours, take only the overplus.
* 1.7Or thus you may do: If the Moon be 9 days old at Noon, she will be 9 days and an half at night; therefore you may add about a quarter or half an hour thereunto, as it is more early or late in the night, and add the Southing of the Moon, which makes 7 ho. 30 min. added to the Shadow of the Moon 3 ho. upon the Sun-dial, it makes 10 ho. 30 min. for the time of the night: So you see there is 6 min. difference betwixt these two ways, which cannot well be estimated; but either way will give neer enough satisfaction for the time of the Night.
HAving a Gold Ring and a Silver Drinking Bowl, take a small Thred or Silk and measure the compass of the top of the Silver Bowl, Glass, or other Ves∣sel, which will be a convenient length for your use: Then put this Thred through the Ring, and tie the ends thereof together, taking up as little as you can with the knots. Put this Thred over your Thumb, where you feel the Pulse beat, upon the lower Joynt it may be; then stretch our your hand, and hold it so that the in∣side of your Thumb may be upward; and hold your Hand so over the Bowl, that the Ring may hang as neer the midst of the Bowl as you can guess: and you shall see that the beating of your Pulse (holding your Hand a while as still as you can) will give a motion to the Ring, causing it to swing cross the Bowl by Degrees more and more, till at last it will beat against the Sides thereof.
Now mark when it begins to strike, and tell the strokes as you would do a Clock; for it will strike what Hour of the Day or Night it is, and then leave off striking, and swinging also by degrees: Which hath been approved of by the experience and judg∣ment of many.
WE may take notice, That there is no Dial can shew the exact time, without the allowance of the Suns Semidiameter, which in a strict acceptation is true. But hereto Mr. Wells hath answered in page 85. of his Art of Shadows, where saith he, Because the Shadow of the Center is hindered by the Stile, the Shadow of the Hour-line proceeds from the Limb, which always precedeth the Center one minute of time, an∣swerable to 15 minutes the Semidiameter of the Sun: which to allow in the Height of the Stile were erroneous; but there may be allowance in the Hour-line, detracting from the true Aequinoctial Distance of every Hour or 15 degrees, 15 minutes. But I will go no further with this Subject, to put the Learners in doubt of the true Hour; for this is as neer a way which I have shewed you, as any projected upon Dial Planes. You may see a Geometrical Figure of it in my Problems of the Sphere.
ALthough I never saw any man Make a Dial, nor paint one, but what I made, painted, and guilded my self; as you may see the Piece in the Title, on which I made 26 Dials, and put in the Brass Gnomons or Stiles into the Freestone with Lead, and guilded them with all the Figures, and on the Globe drew the Aequi∣noctial Circle, the Ecliptick, the Tropicks, and Polar Circles, and the 24 Meridians, with such Constellations in the North and South Hemisphere, and Stars, in such Co∣lours as was fit to set out the Dial, with Pole Dials, and Globe Dials, Chap. 32.
FOr to fasten the Gnomon to the Plane, be it of Wood or Freestone, you must have a small thin Chisel, or Googe, or Gimblet, as is fit for the Stile, be it round as a Rod of Iron, or a piece of Brass, let in with a Foot an inch and half, or more or less, as you will; and in the Wood make such little Mortises as just the breadth and length of the Foot of the Stile; and if it comes thorow to clinch it on the other side, then it is fast.
If it is in Freestone, your Dial drawn first in Paper, lay it upon the Plane ••s it should be; then cut out the Substile-line as neer its breadth as you can, an•• ••••ly leave so much as will just hold it together. The Paper laid as before on the Plane, with a Black-lead Pensil, or such like, draw the Substile Line where it stood in the Paper, and with a small Chisel make such Mortises in that Line as are answerable to the Foot of the Stile; and crook his Foot, and put it into its place, with a small Ladle and some Lead melted, put the Stile perpendicular with the Plane, and pour in the Lead into the Mortise until it is full; and when it is cold, then with a blunt Chisel har∣den the Lead in one Inch side of the Stile or Gnomon: And if the Mortise should be too wide, or broken, and not even enough with the Plane, then wet some Flower of Alabaster, as you may have it fit for that purpose at any Masons, and as soon as 'tis wet make a Plaster, and so smooth it, and spread it even and plain with the Plane; it is presently dry. Now have you the Stile or Gnomon as fast as if it grew there.
To Paint them, you must first Prime them: The Prime is made thus. Take an equal quantity of Bole Armoniack and Red Lead, well ground together with Lin∣seed Oyl, and well rubb'd in with a Brush or Pensil into the Plane; that being dry, for the outside Colour, it is White Lead or Ceruse well ground together with Linseed Oyl. How to know the best. Buy the White Lead, and grind it to a Powder, and put it into Water until it become as thick as Pap, and let it dry; then it is for your use.
For the Hour Lines a Vermillion, and a part Red Lead, well ground together with Linseed Oyl, with a small quantity of Oyl of Spike, or Turpentine that will dure, and make the Lines shine.
For a Gold Border, Rub the Border well with the white Ceruse Paint; be sure it be very thick in the Border: Then with Blew Smalts strew very thick the Border while it is wet; and when it is dry, wing that which is loose off, and save it in a Pa∣per; and for the rest that clings, it is fast enough.
Take Red Lead and White Lead, and as much Red Lead again as White, or Yel∣low Oker, well ground with Oyle of Spike or Turpentine; this is the Sise: Then draw with that the Figure you would have in Gold, and when it is so dry that it will not come off on your Fingers by a slight touch, lay on the Gold; and when it is thorowly dry, wing it off.
How to make a good Black, to shadow or make Figures. Grind well with Linseed Oyl Lam-black, with some Verdigrease, and that is a firm Black. The like you may do with all other Colours, as you fancy for such Work.
FIrst, Steep one penny-worth of Brazeel Wood all night in Piss or Urin; then boil it well and strain it; then bruise two penny-worth of Cochineel, and boil it, and put in it the bigness of a Hens Egg of Roch Allom, that brings it to a colour, and then it is for your turn.
To Paint Freestone, wash the Stone with Oil, and it will last; and then all the Colours before may be used, as directed.
TAke blew Smalts, temper it in Water, and rub the Picture with it, and after wipe it with a Linnen Cloth, which Cloth should be dipp'd in Beer, or other∣wise with a dry Cloth, and it is clean.
WAsh it with Beer, and dry it, and then cleanse it with Linseed Oyl.
Masticous is a fine Yellow, ground with some Oyl of Spike or Turpentine.
Bice is a good Blew Colouring, to be ground with Linseed Oyl and Red Lead.
And Spanish Brown will make a lasting Colour for Course Work.
TAke as many Leaves of Gold as you please, Honey three or four drops; mix and grind these, and keep it in some Bone Vessel. If you will write with it, add some Gum-water, and it will be Excellent.
AMongst the many admirable ways that have been from time to time invented for propagating the Arts Mathematical, and especially that of Trigonometry, Logarithmes, invented by the Right Honourable the Lord Napier, Baron of Marcheston, may challenge the priority, and the Tables of Artificial Sines and Tan∣gents, composed by Mr. Edmund Gunter Professor of Astronomy in Gresham-College London; for that they expedite the Arithmetical Work in most Questions; Multipli∣cation being performed by Addition, and Division by Substraction, the Square Root extracted by Bipartition, and the Cubique Root by Tripartition: So that by help of these Numbers, and the aforesaid Sines and Tangents, more may be performed in the space of an Hour, than by Natural Numbers or by Vulgar Arithmetick can be in six. Now of what frequent use the Doctrine of Triangles, both Plain and Spherical, is in Astronomy (for the Resolution of which the Tables following chiefly serve) let the precedent Work testifie. And as Mr. Newton in his Mathematical Institutions, or in the same form as Mr. Vincent Wing in his Harmonicon Coeleste, are these Tables follow∣ing: And therefore I think it not amiss here in this place to insert some few Propositi∣ons, to shew the Use of the Canon and Tables of Sines and Tangents following.
EVery Page in the Table of Logarithmes is divided into 11 Columns; the first of which Columns, having the Letter N at the Head thereof, are all Numbers suc∣cessively continued from 1 to 1000: So that to find the Logarithmes of any Num∣ber,
is no more but to find the Number in the first Column, and in the second Co∣lumn you shall have the Logerithme answering thereunto.
Example. Let the Number given be 415, and if it is required to find the Loga∣rithme thereof, in the Table of Logorithmes, in the first Column thereof, under the Letter N, I find the Number 415, and right against it in the next Column I find 618048, which is the Logarithme of 415. In the same manner you may find the Logarithme under 1000; as the Logarithme of 506 is 704151, and the Logarithme of 900 is 954243, &c.
But here is to be noted, That before every Logarithme must be placed his proper Characteristick; viz. If the Number consist but of one Figure, as all Numbers un∣der 10, then the Characteristick is 0; if the Number consist of two Figures, as all Numbers between 10 and 100, then the Characteristick is 1; if the Number consist of three Figures, as all Numbers betwixt 100 and 1000, then the Characteristick is 2; and if the Number consist of four Figures, as all between 1000 and 10000, the Characteristick must be 3. In brief, the Characteristick of any Logarithme must con∣sist of an Unit less than the given Number consisteth of Di••its or Places: And by ob∣serving this Rule, the Logarithme of 415 will be 2.618048, and the Logarithme of 506 is 2.704151, and the Logarithme of 900 is 2.954243, &c.
Example. Let the Logarithme given be 2.164353; now because the Characteri∣stick is 2, I know by it the Absolute Number consisteth of three places, and therefore may be found in the second Column of the Logarithme Tables, having 0 at the top thereof, against which I find 146, which is the Absolute Number answering to the Logarithme of 2.164353.
You must find the three first Figures of the given Number in the first Column, as before, and seek the last Figure thereof amongst the great Figures in the head of the Page; and in the common Area or meeting of these two Lines is the Logarithme you desire, if before it you add or prefix its proper Characteristick.
Example. Let it be required to find the Logarithme of 5745; I find 574, the three first Figures, in the first Column, and 5, the last Figure, in the head of the Table; then going down from 5 in the head of the Table, until I come against 574 in the first Column, there I find 759290, before which I place 3 for the Characte∣ristick, which is 3.759290, and that is the Logarithme sought for.
Admit it were required to find the Sine of 21 deg. 24 min. I turn to the Sines in the Table, and in the head thereof I find Degrees 21; then in the first Column (under M) I find 24, and right against it is 9.562146 for the Sine, and 9.593170 for the Tangent of 21 deg. 24 min. But suppose it were required to find the Sine or Tan∣gent of 56 deg. 35 min. look for all else under 45 deg. are found in the head, and the odd min. in the left hand; and all above 45 deg. are found in the foot of the Table, and the min. in the last Column toward the right hand; as in this Example the Sine of 56 deg. 35 min. is 9.921524, and the Tangent is 16.180590.
Suppose 9.584663 were a Sine given, I look for the Number in the Table of Sines, and I find it stand against 22 d. 36 m. and therefore is the Sine thereof. As admit 9.624330 were a Tangent given, look for the Number in the Column of Tangents, and I find stand against it 22 d. 50 m. The same must be done for Sines and Tangents in the foot of the Tables.
* 1.8Cuncta fluunt omnisque vagans formatur Imago. Ipsa quoque assiduo labuntur tempora motu.All things pass on: Those Creatures which are made Fail, and by Time's assiduate motion fade. Much like the Running Stream, which cannot stay, No more can the light Hours that poste away. But as one Billow, hast'ning to the Shore, Impels another, and still that before Is by the following driv'n; so we conclude Of Time, it so flies, and is so pursu'd. The Hours are always new; and what hath been Is never more to be perceiv'd or seen: That daily grows, which had before no ground; And Minutes, past once, never more are found.
* 1.9Labitur occulte fallitque volubilis aetas.The fretting Age deceives, and stealing glides; And the swift Year on loose-rein'd Horses rides.Quid non longa dies? quid non consumitis Anni?
Take off the Scale of Hours the hasts and quarters, and prick from D as you did the Hours, and draw the quar∣ters in the Dial as you were di∣rected for the Hours.
Tropicks and other Circles of Declination is shewed in Chap. 3. in the Aequi∣noctial Dial, where they be calculated and drawn by help of a Scale of Equal Parts.
Ovid. lib. Met.
Lib. Eleg. 1.