CHAP. I. The use of the Trianguler Quadrant, IN Making of DIALS.

SVn-Dials may be made on any Plain, and all kind of Plains are either Flat, as Horizontal; or Vpright, or Lean­ing.

The Horizontal hath two faces, the one beholding the Zenith, called the Horizontal-Plain; the other, beholding the Nadir, as the Ceiling of a Room is.

The Upright Plains, are those that make right Angles with the Horizon, and do be­hold neither the Zenith or Nadir, but are parallel to them.

The Leaning Plains are of two sorts ge­nerally; the one called Recliners, behold­ing the Zenith; the other sort called In­cliners, [Page 8] beholding the Nadir, as the out­side, and in-side of a Roof of a House, may represent.

The two last sorts, viz. Upright and Leaning, may be Direct, or Declining, viz. beholding the South, or North, or East, or West Point of the Horizon; or Declining therefrom, viz. Declining from South, or North, toward the East or West.

All which Plains, are lively represented by a Sphear, made for that purpose, in Brass or Pasteboard, or by the Projection of the Sphear in Plano, Thus;

Equal to the Radius of the smaller Tan­gents, describe the Circle ESWN, repre­senting the Horizon, crossing it precisely in the the Center Z, with the Lines SN and EW, denoting the Points of South and North, East and West.

Then counting the smaller Tangent on the Sector-side doubly, as thus, calling 5, 10; & 10, 20; & 20, 40; & 30, 60; & 40, 80; & 45, 90; &c. Lay off from Z, to­wards S, the complement of the Suns Me­ridian Altitude, in ♋, in ♈, and ♑; for those Points on the Meridian-line, between Z and S; and consequently the half Tan­gent of the complement of the Suns Meri­dian Altitude in every degree of Declinati­on, (if you proceed so far).

[Page 9]Then for the Intersections of all those Lines and Parallels of Declination on the North-side of the Meridian, Observe, That the same number of degrees and minuts, that any Point is above the Horizon on the South part of the Meridian in Summer, just so many degrees and minuts is his opposite Parallel in Winter below the Horizon.

As thus for Example.

The Sun being in ♋, or 23 deg. 31 min. of Declination North, hath for his Meridian Altitude 62 degrees, and so many degrees is his opposite Parallel of 23-31, or ♑, be­low the North part of the Horizon, at mid­night.

As thus;

Let the Center, at the beginning of the Line of Tangents, represent the Center Z; and let the Tangent of 45, represent the Ho­rizon in the Scheam, viz. S. and N.

Then,

As the distance from S. to ♑, is 15 deg. taken from 45 toward 0, and laid from S. to ♑ inwards toward the Center Z, as the distance was taken from the Tangent of 45, toward the beginning of the Line of Tangents, that repre­sents the Center;

So the Point Cancer from N. is 15 deg. counted beyond 45, toward the end, below or beyond the Horizon.

Again,

As S. ♋ is 62 degrees from 45 towards 00;

So is the other Point 62 degrees below N, taken from 45, viz. at 76 degrees; which being laid from N, doth over­reach this little page.

So that to draw the Tropick of ♑, the Point ♋ being his opposite, is 28 degrees from Z, or 62 deg. from S; and the other Point of ♑, on the North part of the Meri­dian, is 62 degrees, counting from 45 doubly also; or 28 degrees from 90, the supposed end of the Tangent, which is naturally in­finite, being the Tangent of 76 degrees, or the Semi-tangent of 152, reading the Tan­gents doubly from the Center; which di­stance from the Center, to the Tangent of 76; or as half-tangents, 152, laid from Z, gives the Point ♑ on the North-part of the Meridian, below the Horizon; the midst between which two Points of ♑ on the South and North part of the Meridian, is the Center to draw the Tropick of Capri­corn.

Again, to illustrate this difficulty, to draw the Tropick of Cancer, the Suns Meridian-Altitude in ♑, his opposite sign is 15 degrees above the Horizon on the South part of the [Page 11] Meridian, and 15 degrees below the Hori­zon, on the North-part of the Meridian, viz. the Extent from the Center to the Tan­gent of 52 deg. 30 min. or the Semi-tan­gent of 105, reading it doubly; being laid from Z, gives the Point ♋ below the Hori­zon; the middle between which two Points is the Center to draw the Tropick of Can­cer.

Again, for the Equinoctial or Parallel of ♈; the Meridian Altitude in ♈, is 38-28; and the Meridian Altitude likewise in ♎, his opposite Parallel is 38-28 also; so that if you count 38-28 doubly beyond 45, which will be at the Tangent of 64 degrees and 14 minuts, and take from thence to the Center; this distance laid from Z, shall give the Point AE below the Horizon, and the the middle between the two Points AE, is the Center to draw the Aequinoctial.

Then for the Hour-Lines; first, set off the Semi-tangent of 38-28 from Z to P; and the Secant of 38-28 to the same Radi­us from Z to L, and draw the Line L 45 parallel to EW; then make PL a Tangent of 45 degrees, and lay off the Tangents of 15-30, and 45, from L both-wayes, as you see in the Figure.

Also, As the Sector stands, take out the = Tangents of 60 and 75 severally, and [Page 12] turn them four times from L both-wayes, and note those Points with 6, 7, 8, 9, 10, 11.

Lastly, Set one Point of the Compasses in L, and open the other to P, and draw the Line WPE for the hour of 6.

Again,

Set one Point in 7-15 degrees from L, and open the other to P, and draw the Hour­line 5 P 5; Set the same Extent also in 7, or 5, on the other side of L, and draw the Hour-line 7 P 7, as the Figure sheweth.

Then,

Set one Point of the Compasses in 8, 30 degrees from L, and open the other Point to P, and draw the Hour-line 8 P 8, and remove it to the other side of L, and draw the Hour-line 4 P 4: And so for all the rest in order.

Thus having drawn the Figures; to draw Lines therein, which shall truly repre­sent any Plain whatsoever, observe the following Rules.

1. The Horizontal-Plain, is represented by the Circle E.S.W.N.

2. A direct South or North-Diall, is re­presented by the Line E.Z.W.

3. A South or a North declining-Plain, is [Page] [Page]

[Page 13] represented by the Right-line 60 Z. 60, whose Poles are at C and D, declining 30 degrees from S. or N. toward E. or W.

4. An East or West Plain, is represented by the Meridian-line of 12, viz. S. & N.

5. A Polar Plain, is represented by the hour of 6, viz. the Line E.P.W.

6. An Equinoctial Plain, is represented by the Equinoctial-line E.AE.W.

7. Any Direct Reclining, or Inclining-Plain, between the two last, is called, A di­rect Recliner, whose Poles are alwayes in the Meridian, and are represented by any Re­clining Circle, as the two Circles W. ♋. E. and E. ☉. W. do shew.

8. An East or West Recliner or Incliner, represented by the Circle N.F.S.

9. A Declining and Reclining, or Incli­ning Polar-Plain; that is, it so Declines and Reclines, or Inclines, as to lie parallel to the Pole, as the Circle 8 P 8 doth represent.

10. A Declining Reclining-Plain, that so Declines and Reclines, as not to fall in the Pole or Equinoctial, as generally they will do, as the Circle 60 G 60 doth represent, which Declines from the South-eastwards, and Reclines 62 deg. which kind of Plains are various and infinite, yet confined to six varieties, as afterward.

Now, the way of Drawing these Scheams, to represent these varieties, is briefly thus, by the Sector.

First, to the Radius of the small Tangents, draw the Circle N. E. S. W. observing this Method, if it be a South Recliner, to set the letter N above, and E on the right hand; and contrarily, in North Recliners; for we meddle not with Incliners till afterwards, (and alwayes observe, that a South Incliner is the same with a North Recliner, and the contrary) then cross that Circle with two Diameters, precisely in the Center, as the Letters shew; then according to your Plains Declination from North or South, toward either East or West, set off the Declinati­on with a Line of Chords or Sines, as before is shewed; and draw that Line for the Per­pendiculer Line of the Plain, and laying the same distance as much from E. and W. draw another Line Perpendiculer to the former, representing the Plain; then, on the first Line, viz. the Plains Perpendicu­ler, lay off from Z, the half Tangent of the Plains Reclination from Z to E, and the half Tangent of the complement thereof from Z to Q the contrary way; and the whole Tangent of the complement thereof from Z, contrary to E, on the same Line, extended [Page 15] for a Center, to draw the Reclining Circle that represents the Plain.

Lastly, You must draw a Circle through Q and P, (P being alwayes the Semi­tangent of the complement of the Latitude laid alwayes from Z toward N for the North Pole) so as to cut the Primitive Circle N.E. S.W. into two equal parts, as is shewed in the 10th Proposition of the third Chapter; part of which Line, doth represent the Stile-Line of the Dial; which last work shall be again shewed in the Example.

Example. To draw the Scheam for a Plain, Decli­ning from the South to the West 35 de­grees; and Reclining 20 degrees, for the Latitude of 51-30.

First, to the Radius of your small Line of Tangents, being the Latteral distance from the Center to 45, (or larger if you please) draw the Circle N.E.S.W. representing the Horizon, crossing it in the Center with the Lines N.S. & W.E. for the North and South, and East and West Lines.

Then,

Take out the latteral Tangent of half the latitude, viz. 19-15, for 38-30, calling the Tangent of 10, the half Tangent of 20; [Page 16] and lay it from Z at the Center, to P for the Pole-point.

Then consider the Declination of your Plain, and which way, as here 35 deg. −0′ from the South towards the West; take out the Chord of 35 deg. and lay if from S to C, and from W to A, and from N to D, and from E to B, for the more exact drawing of the Lines AB, CD; the Lines CD repre­senting the Poles of the Plain, and the Line AB the Declining Plain it self; then from Z towards D, lay off the Tangent of 10 deg. (being the half Tangent of 20 degrees, the given Declination) to E.

Also, Take out the Secant of 70 degrees, the complement of 20, to the same Radius; and that laid from the Point E, on the Line DC produced, shall be the Center to draw the Circle AFEB, that represents the De­clining, Reclining Plain, that declines 35 degrees, and reclines 20 degrees. Also, Lay off half the Tangent of the complement of the Reclination, viz. 35 degrees (for the Reclination is 20, the complement whereof is 70, and the half of 70 is 35) from Z to Q.

Then to draw the Line QP, do thus;

Observe how many degrees you count from Z to the Point E, counting from the Center, count so many in the manner of half Tan­gents [Page 17] from 45; and the latteral distance from thence to the Center, laid from the Center Z, on the Line CD, gives a third Point, viz. the Point I; which three Points, QPI, brought into a Circle, will cut the Circle N.E.S.W. into two equal parts.

Or thus;

The Semi-tangent of the complement of the Reclination to 180 degrees, laid from Z on the Line CD, will find the Point I.

As thus;

The Reclination is 20, the complement 70, being taken from 180 rests 110, whose half is 55, the Tangent of ZI.

Or more briefly thus;

Set one Point of the Compasses in the small Tangent of 45, and count the Recli­nation from thence in the way of Semi-tan­gents, both wayes, both above and under 45; and lay one, viz. that under 45, from C to Q; and the other, viz. that above 45 from D to I; then on the middle Point, between Q and I, last found, raise a Per­pendiculer to CD, and in that Line will be the Center to draw IPQ.

Also,

If you count 51-32, the latitude on the Line of Tangents from 45 forwards (as Semi-tangents) and lay it on the Meridian-Line [Page 18]

CHAP. II. To Draw the Hour-Lines on all Ordinary Dials; the easiest in the first place.

CHAP. III. To find the Declination of any PLAIN.

FOr finding the Declination of a Plain, the most easie way is by a Magnetical-Needle, fitted according to Mr. Failes way, in the Index of a Declinatory (as he calls it) being 180 degrees of a Semi-circle, divided on an Oblong-Board, or Quadrant, or a longer Needle in a square Box, (or) fitted with Hinges and a Cover; after all which wayes, you may have them made at the sign of the Sun-dial in the Minories, by Iohn Brown; or of any other manner you shall think fit.

But, to our Trianguler Quadrant, is a Box and Needle also to be fitted of another form, in some things more convenient.

Whose form is thus;

First, in a piece of Box 5 inches long, [Page 49] 2 ½ broad, and 6 tenths of one inch thick, is a hole made near 4 inches long, 1 inch ¾ broad, and 4 tenths deep for a Needle to play in; about 50 degrees at each end; with brass-hinges, and a cover, and a brace to keep the lid upright, & an Axis of Th [...]ed, and a Plummet playing in the lid, and a Horizontal and a South-dial, drawn on the Box and Cover; also a hasp and glass to keep the Needle close covered, and on the bottom a Grove one tenth of an inch dee [...], made just as broad as one leg of the Sector is.

The use whereof is thus;

Put your Box and Needle, on that leg of the Sector, as will be most convenient for your purpose; the North or cross-end of the Needle toward the Wall, when it is a South decliner; and the contrary when it is applyed to a North decliner, as the play­ing of the Needle will tell you better than many words; then open or close the Rule, till the Needle play right over the Line in the bottom of the Box, (unless there be va­riation, then you must allow for it East­wards or Westwards what it is). Then, I say, the quantity of the Angle in degrees and minuts the Sector stands at, above or under 90, is the degrees and minuts of Declinati­on; [Page 50] being counted from 00 in the little Semi-circle, as complements to the Angle of opening; as in the 4th Use of the 5th Chapter is largely and plainly shewed.

Thus you have the quantity of degrees and minuts of Declination: but to deter­mine which way, consider thus;

If the Needle will stand still in the middle, when the North-end is toward the Wall, then the first denomination is South, if not North.

Again,

When you know where North and South is, you may resolve which way the East and West is; For, observe alwayes, if the North be before you, then the East is on the right-hand, and the West on the left; and contrarily, If the South be before you, the West is on the right-hand, and the East on the left.

Then,

If the Sun, being in the East-point of the Horizon, can look on the Plain, it is a South-east Plain; but if it beholds it when in the West-point, it is a South-west Plain.

Likewise,

If the Cross-end of the Needle will not stand toward the Wall (the Needle playing well) and the Sun being due East, beholds [Page 51] the Plain, then it is so many degrees North-east; but if it cannot look on the Plain, being due East, then it is a North-west Plain, declining so many deg. as the Sector stands at, under or above 90, being alwayes the complement of the Angle the legs of the Sector stand at, and found by taking the Angle the legs stand at, from 90, when the Angle is less than 90.

Or,

Taking 90 out of the Angle, when it stands at an Angle above 90 degrees as a look at the little Semi-circle on the Head sheweth.

Example.

Suppose I come to a Wall, and putting the Box and Needle on the Leg of the Se­ctor, and applying the other Leg to the Wall (or on a streight piece of Wood, ap­plied to the Wall, because of the Walls un­evenness); and open or close the Legs, till the Needle playes right over the Meridian-line, drawn on the bottom of the Box; then, I say, the complement of the Angle the Legs of the Sector stands at, being al­wayes what it wants of, or is above 90 degrees, is the degrees of Declination; and the Coast which way, the Needle and Suns being East and West, tells you.

For,

If the North or Cross-end of the Needle be toward the Wall, it is a South Plain; and if the Sun, being in the East, can be­hold it, then it is South-east; if not, a South-west Plain.

A ready way of counting the Angle found, may be thus;

Take the = distance between Center and Center, in the middle of the innermost-lines, and lay it latterally from the Center, and co [...]nt two degrees more than the Point sheweth, after the manner of Chords from 90 (at the sine of 45) toward the Compass-point, and that shall be the degrees and minuts required.

Example.

Suppose the Legs are so opened, that the = distance between the two Centers, makes the — sine of 25; then, I say, the Lines do stand at an Angle of 50 degrees, and the Legs at 48, two degrees less, the com­plement whereof is 42; as if you count thus from 45, you will find, 40 from 45 is 10, 35 is 20, 30 is 30, 25 is 40, and 2 de­grees more makes 42, the thing desired.

But,

If you like not the abating of two de­grees, then the = distance taken just be­ [...]ween the two legs right against the Cen­ [...]ers, shall be just the — sine of 24 degrees, [...]r 42, counting after the manner of Chords, viz. every 5 degrees on the Sines, for 10 on [...]he Chords backwards from 45 of the Sines, which is 90 in Chords.

Or,

If you use the first Rule, of the 4th Use [...]f the 5th Chapter, viz. by taking the — [...]ine of 30, and put one Point of the Com­ [...]asses in the middle Center in the Tangent- [...]ne, and apply the other to the Line of [...]ines, you shall find it reach to the sine com­ [...]lement of the Angle the Lines stand at, [...]iz. 40 degrees and 2 degrees more, viz. [...]2, is the Angle or thing desired; as pra­ [...]tice with consideration will make easie.

Thus, by the Needle, you may find the [...]eclination of a Wall, which in cloudy [...]eather may stand you in good stead; or [...] prove a declination taken by the Sun, to [...]revent mistakes. And if nothing draw the [...]eedle from its right position, but that it [...]ay well, and you find the Angle truly, [...]ou may come to less than half a degree: [Page 54] And this convenience it hath, that it carries the Needle a competent distance from the Wall, to prevent that attraction; but if it happen to be so near a Meridian, or East and West-plain, that the Angle, by the Sector, cannot well be taken; then you may only apply the side of the Box and Needle to the Wall, and the Needle it self will shew the Declination, on the degrees on the bottom of the Box.

Yet for exactness, the way by the Sun is alwayes the best, where you may come to make a good Observation, and then the Needle only is not to be trusted to; a bet­ter way with opportunity offering it self,

To f [...]nd a Declination of a Wall by the Sun.

For this purpose you (must or) ought to have another Thred and Plummet, which Thred may be a fine even small Pack-thred, and it is convenient to have it ready hanged up near the Wall, so far off, as the Triangu­ler-Quadrant may pass along between it and the Wall, that you may not be troubled to hold it up, and lay it down, and be annoyed with the inconveniencies of your hand shaking, and time wasting, to more uncertainty than needs be.

Also,

You must needs take notice of the two Meridians, viz. one of the place which is the Meridian, or 12 a clock; to which place, when the Sun or a Star comes, it is said to be in the Meridian.

And the other is the Meridian of the Plain, in which Line the Pole-point of e­very Plain is, being 90 degrees distant from the Plain every way, and in all upright-Dials their Pole is in the Horizon; and that degree of Azimuth in which the Pole-point lies, counted from South or North toward East or West, is alwayes the declination thereof; so that by finding the Suns Azi­muth at any time, and the distance of the Sun at the same time from the Meridian of the Plain, is gotten the declination.

The Azimuth of the Sun from the Meri­dian of the place, is found by the 26, 27, 28, 30, 32, 34, 39 Uses of the 15th Chap­ter.

But the Azimuth of the Sun from the Meridian of the Plain, is found by apply­ing the Head-leg against the Plain Horizon­tally, slipping it to and fro, till the shadow of the Thred, hung (or held) up, play right over the Center of the Trianguler-Quadrant [Page 56] on the Head-leg; then what deg. soever the thred cuts, counted from 60 [...]0 on the Loose-piece (being the Perpendiculer or Pole-point of the Plain) shall be the Azimuth of the Sun from the Meridian of the Plain.

This is the Operation; the Application or Use is worded several wayes by several men; I hope I shall do it as fully, and as briefly as some others.

The Sun, to our appearance, passeth from East, by the Meridian, to the West every day; therefore in the morning it wants of coming to the Meridian; at noon it is for a moment just in the Meridian, and in the afternoon it is past the Meridian of the place.

Even so it begins to shine on, and is di­rectly against, and leaveth to shine upon most Plains, when it begins to shine upon, or is not directly against; I say, it wants of coming to the Pole or Meridian of the Plain. When it is directly against the Plain, then it is in the Meridian or Pole of the Plain; when it [...]s past, it is past, or begins to leave the Plain.

Which Th [...]ee Varieties I intend thus briefly to e [...]press;

Azimuth Want, or W in the morning only; Azimuth Direct at noon; Azimuth [Page 57] past, or P being in the afternoon.

The other Three Varieties let be Sha­dow Want, Shadow Direct, Shadow Past; all which may be in several Plains at several times; that is to say, at morning, noon, and night.

These Observations, and Cautions pre­mised, the Rule is thus;

1. If the Azimuth and Shadow are both wanting, or both past; substract the losser out of the greater, and the residue is the Declination. But if one want, and the o­ther be past, then the sum of them is the Declination.

2. If the Sun come to the Meridian of the Plain, before it come to the Meridian of the place, it is an East Plain. But if it come to the Meridian of the place, before it come to the Meridian of the Plain, it is a West Plain.

3. If the sum or remainder, after Addi­tion or Substraction, be under 90, it is a South-east, or South-west Plain, declining so many degrees, as the sum or residue is. But if the sum or remainder be above 90, it is a North-east or a North-west Plain, and the complement of the sum or remainder to [Page 58] 180, is the quantity of Declination North-east, or North-west.

4. If the sum or remainder be 00, it is just South; If 90, just East or West. But, If it be 180, it is a direct North Plain.

It shall be further Explained by two or three Examples.

Suppose that on the first of May, in the forenoon, I come and apply the Head-leg of the Trianguler-Quadrant to the Wall, and holding of it level, the shadow of the Thred, held up steady, cuts the Center and 60 degrees on the Moving-leg; that is, 60 deg. want; which I presently set down in a Paper ready prepared, thus;

Then, as soon as possible, or rather by some body else, at the same moment, find the Suns Altitude, which suppose to be 20 degrees; (but if you are alone, and have a [Page 59] Thred ready hanged up; then take the Al­titude first, and the shadow will be had presently after, the Thred hanging steadily) and set that down also, as here you see.

Then by the 26th Use of the 15th Chap­ter, you shall find the Suns Azimuth at that time and Altitude to be 94 degrees, and after Substraction remains 34-0, for the Walls declination Eastward, becau [...]e the re­mainder is under 90, and the Sun comes to the Meridian of the Plain, before it comes to the Meridian of the place, or South.

Again,

In a morning, Iune 13, I observe the Al­titude, and find it 15 degrees, and instant­ly the shadow, and find it to be 10 degrees past the Plain, viz. on the Loose-piece, to­ward the Head-leg, I set both Altitude and Shadow, with the day and time down thus;

[Page 60] And then find the Azimuth at that time and Altitude to be 109 degrees; here the terms being unlike, I add them together, and the sum being above 90, I know it must be a North Plain; and because the Sun comes to the Plain before it comes to the Meridian of the place, it is North-east; and the complement of 119 to 180, is 61-0 North-east.

Again,

Suppose the same day, in the Afternoon, I find the Suns Altitude 15-0, and the sha­dow 20 degrees want; the Azimuth at the same Altitude, and the same day, will be near the same number of degrees; but in the Afternoon it is past the South, or Meri­dian of the place; here also it is a North-plain, because the sum is above 90; and a [Page] [Page]

[Page 61] North-west, because it is against the South, before it comes to be right against the Plain.

But if you happen to come when the Sun is in the Meridian of the Plain, then the Suns Azimuth is the Declination, East or West, as the Azimuth is.

Also,

If you take the shadow, when the Sun is just in the South, or Meridian of the place, the shadow is the Declination; if it is past the Plain, it is Eastward; if it wants, it is Westwards.

Thus I have (I hope) shewed the true manner of finding the Declination of a Wall by the Sun shining on the Plain, as plainly and as briefly as the matter will bear, speak­ing to young Tiroes therein.

It may be done also, by observing when the Sun just begins to shine on a Chimny, or Wall, or high place you cannot for the present come near, conceiving the Sun to be then just 90 degrees from the Meridian of the place wanting, or just when it leaves it being then 90 degrees past the Plain, then take the Altitude and Azimuth, and work accordingly to the former Rules.

CHAP. IV. To Draw a South, or North Erect Declining-Dial▪

FOr better illustrations sake, I will draw a particular Scheam for this Dial also, as I did for the East Recliner; whose De­clination let it be 20 degrees declining from the South toward the West, in the latitude of 51-32 for London.

The Scheam is drawn by the former di­rections; the Pole of the Plain being at D, declining 20 degrees from S toward W, and the Plain it self is represented by the Line AB; the Circuler pricked Line DHPC is a certain Meridian drawn through the three given Points DPC, whose Center will be in the intersection of the Plain AB, and the Tangent Line for the hours, which being drawn, whatsoever ZH is in the half Tangents, ZQ is the complement thereof, in the same half Tangents.

[Page 63]The Scheam thus drawn, ZH is the Sub­stile, PH is the Stile, QZ the distance of 6 from 12, HPZ the inclination of Meridians, or Angle between the two Me­ridians, viz. of the Place PZ, and of the Plain PH, found by the following Canons.

By Artificial Sines and Tangents.

1. First for the Substile from 12.

As the sine of 90 ZN, to the sine of the Declination NC 20 degrees;

So is the Co-tangent of the Latitude PZ 38-28, to the Tangent of ZH 15-12.

2. For the Stiles Elevation.

As the sine of 90 ZN, to the Co-sine of the Plains Declination NA 70-0;

So is the Co-sine of the Latitude ZP 38-28, to the sine of PH 35-46, the Stiles Elevation.

3. For the Distance between 6 & 12.

As the sine of 90 ZW, to the sine of the Plains Declination WA 20-0;

So is the Tangent of the Latitude NP 51-32, to the Tangent of AQ, the Co-tangent of 6 from 12, 23-18.

Or thus;

As Co-tangent Latitude ZP 38-28, to the sine 90 ZPQ;

So is S. declination WZA 20-0, to the Tangent of QA 23-18.

4. For the Inclination of Meridians.

As the sine of the Latitude ZAE 51-32, to the sine of 90 ZS;

So is the Tangent of the declination SD 20-00, to the Tangent of AEK 24-56, the Inclination of Meridians.

Or,

As Co-sine Latitude 38-28 PZ, to the sine 90 PHZ;

So is the sine of the Substile ZH 15-12, to the sine of ZPH 24-56, IM.

5. Then having made a Table of Arks at the Pole, by this Canon you may find the Hour-Arks on the Plain.

Thus;

As the sine of 90 PK, to the sine of the Stiles height PH 35 46;

So is the Tangent of the Hour from 12, 19-56 for I, AEI, to the Tangent of the Hour from the Substile on the Plain, H 1, 12-14.

[Page 65]But I prefer the way by Tangents before it, as followeth.

All these requisites may be found by the ge­neral Scale and Sector, the Canons where­of in brief are thus;

By the Trianguler-Quadrant and Sector.

These Requisites are also found by the particular Quadrant, very really and truly, for that Latitude the Rule is made for, in this manner.

1. First, for the Substile.

Lay the Thred to the complement of the Plains declination, counted on the Azimuth Line, and on the degrees it giveth the Sub­stile from 12, counting from 60|0 on the Moveable-leg.

Example.

The Thred laid to 70, the complement of 20 on the degrees, gives 15-12 for the Substile.

2. For the Stiles height.

Take the distance between 90, and the Plains declination on the Azimuth-line, and measure it on the particular Scale from the begin [...]ing, and it shall give the Angle of the Stiles-Elevation above the Substile, 35-46.

[Page]

3. For the Inclination of Meridians.

Take the Substile from the particular Scale of Altitudes, and measure on the Azimuth-line from 90, and it shall give the complement of the Inclination of Meridians, or the Angle counting from 90. viz. here 24-56.

4. To find the Angle between 12 & 6.

Take the Plains Declination from the particular Scale of Altitudes (less by the sine of the Declination, to a Radius equal to 45 minuts of the first degree on the par­ticular Scale of Altitudes), and lay it from 90 on the Azimuth Scale, and to the Com­pass-point lay the Thred, then on the Line of degrees, the Thred gives the complement of 6 from 12, counting from 60 toward the end, as here it is in this Dial 23-18.

Also, the requisites may be found Geo­metrically by the Scheam, thus;

As,

1. A Ruler laid from D to H in the Limb, gives F; the Ark CF is the Sub­stile.

2. A Ruler laid from Q to P in the Limb gives I, the Ark AI is the Stiles height.

[Page 68]3. A Rule laid from P to Q, cuts the Limb at T; the Ark TE is the Inclination of Meridians.

Or,

A Rule laid from P to K, cuts the Limb at L; then SL is the Inclination of Meri­dians.

4. A Rule laid from D to Q, cuts the Limb at 6, the Ark C 6, is the Angle be­tween 12 and 6.

5. A Rule laid from D, to the intersecti­on of any other Hour-line, with the Plain AB on the Limb, gives Points, whose di­stances from C, are their Angles from 12, or their distances from F, or their Angles from the Substile.

To Delineate the Dial by the Sector.

Thus by any of these wayes, having gotten the Requisites, proceed to draw the Dial thus;

First, Draw a Perpendiculer-line on the Plain CB, by a Thred and Plummet; then if it be a South Decliner, at the upper-end make a Center, as C; and on that Center describe the Arch of a Quadrant, as the Arch DE on the Center C; then in that Arch by [Page] [Page]

[Page 69] Sines or Chords lay off from D the Substile, and upon the Substile, the Stiles height; and the Hour-line of 6 by the Angle be­tween 12 and 6; and draw those Lines as you see in the Figure, contrary to the Coast of Declination; then draw two Lines paral­lel to 12, as 6 G, and [...] H; then fit the distance of the Parallels from 12, in the Se­cant of the Plains declination 20; and take out the = Secant of the Latitude 51-32, and lay it from 6 to G, and from C on the Line of 12 to F, & draw the Line GF to H, which Line is a = to the Hour-line of 6; then make FG Radius, or the = Tangent of 45; and before you prick off the Hours, take out the = Tangent of the Inclination of Meridians; and if it reach from F to the Substile, on the Line FG, your work done is true, else not.

Then,

Take out the = Tangent of 30, & 15, and the respective quarters, and &c. as before; then make 6 G a = Tangent of 45, and do likewise as before, in the Hori­zontal and South Dials, and to those Points draw the Hour-lines required.

2. To Draw the Hour-lines on a North-Declining Dial.

The Requisites, as Substile, and Stile, In­clination of Meridians, 6 & 12, are found the same way, and by the same Rules, as the South Decliners are done. But when you come to delineate the Dial, there is some alteration; which I conceive is best seen by an Example, as Northeast declining 35 deg. Lat. 51-32, at London.

First, as be [...]ore, draw a Perpendiculer-line for 12 a Clock, as AB; then about the middle, or toward the lower-part of that Line, as at C, make a Point for a Center, as C; then on the Center C describe the Arch of a Circle, that way, from the Line AB, as is contrary to the Coast of declina­tion, as if the Plain declines Eastward, as here, draw the Arch Westward from AB, as BD; and the contrary way in North-west plains; and on that Ark lay down the Substile from 12, and the Stiles height above the Substile, and the Hour of 6, by the Angle of 6 & 12; and then, by those Points and the Center, draw these Lines.

Then, at any distance, draw a Line = to 12▪ (or AB) as the Line EF, and make [Page 71] that distance a = Secant of 35, the decli­nation; the Sector so set, take out the = Secant of the Latitude 51-32, and lay it on the Parallel-line from 6 to 9, then make 6-9, the measure the Compass stands at, a = Tangent of 45; and take out the = Tangent of 15-30, &c. and lay them both wayes from 6, upwards and downwards; also, for the hour of 10, as the Sector stands, take out the = Tangent of 60, and turn it 4 times from 6 on the Line EF; and (when you want it) the = Tangent of 75, and turn that also 4 times from 6, for 11 a Clock-line; and then by those Points, draw Lines for the Hours required.

CHAP. V. To Draw the Hour-Lines on a Dial falling near the Meridi­an, whose Stile hath but a small Elevation, and therefore no Center.

THe former Examples may be sufficient to the considerate, to draw any Erect Declining-Dial having a Center; but when the Stile happens to be less than 15 deg. of Elevation; then, if it be not augmented by casting away the Center, the usefulness, and handsomness of the Dial is l [...]st; now if you draw the Dial by the former Rules on a Table, and cut off so much, and as many Hours as you care for, the work is per­formed.

But then you shall find, that long Rulers and Lines will be wanting for a small Dial; [Page] [Page]

[Page 73] [...]erefore I prefer this way following by the [...]ctor, general in all Dials.

First, on or near the North Edge of the Plain, in far South de­cliners, (but near to the South-edge of the Plain, in far North-decliners) draw a per­pendicular-Line, re­presenting the Hour-line of 12, as the Line AB in our Example, being a Southwest de­clining 80 deg. 25 min. then, in the upper-part of that Line, in South-decliners; or about the middle, or lower-part in North-decliners, ap­point a Center, as here at A; then upon A, as a Center, as large as you may, draw an Arch as BD; and in that Arch, or rather by a Tangent-line, lay [...] the Substile from B to D, and draw the [...]ine AD, as an obscure Line, for the present [Page 74] only to be seen; and upon that, the Stile-line, as before: Then at any convenient places, as far from the Center as you can, draw two Lines Perpendiculer to the Sub­stile, as the Lines CE, FG, for two contin­gent Lines, (antiently and properly so cal­led); then by the Inclination of Meridians, by the directions in the East and West Re­cliner, being the 7th Dial in the 2d Chapter, make the Table of Hour-Arks at the Pole, by setting down against 12, 82-30; and taking out 7-30 for every half hour, till you come to 00 at the Substile; and then by adding 7-30 for every half hour, and 15 for every hour, to 8 ½, as long as the Sun shines; which in regard it falls on an even half hour, is the most easie, and fits the Points in the Tangent ready made for hours and quarters.

The next work, is to resolve what hours shall come on the Plain, as will be best de­termined by the discreet Orderer, or Sur­veyor, or experimental Dialist, as here 8 and 1; and for those two hours, mark the upper contingent Line in two places where you would have them to be, as at E and C; then take the — Tangent of 37-30 for 8, from the small Tangents, and add it to the = Tangent of 67-30, the Tangent for 1; [Page 75] and behold! it makes the — Tang. of 72-33.

Then,

Take the whole space CE, and make it a = Tangent of 72-33; then take out the = Tangent of 67-30, and lay it from C to H; and take also in the same common-line, right against the small Tangent of 37-30, which is in the large Tangent 10-50, the = Tangent of 10-50, taken for 37-30, be­ing laid from E, the place for 8, will meet just at H; which Point H, is the true place for the Substile, to fit and fill the Plain, with the hours determined.

Then,

The Sector so set, Take out all the = Tangents above 45, as in the Table, and lay them the right way from H, toward C, and E; then take out = Tangent of small 45; and setting one Point in H, strike the touch of an Arch, as at I; then make HI a = Tangent in great 45, and take out the = Tangents of the rest of the hours under 45, as in the Table, and lay them both wayes from H, because the Substile falls on an even half hour.

Then,

Draw the Line HK, = to the first Line AD, for the true Substile; then make HK Radius, or the Tangent of 45, and take out the Parallel Tangent of 5-56, the Stiles [Page 76] height, and lay it from K to L; then take HI, the first Radius, and setting one Point in L, draw the touch of an Ark as by M; then draw a Line by the Convexity of the Arches by I and M, for the true Stile-line.

Then,

Take the nearest distance from the Point K, to the Line IM, and make it a = Tan­gent of 45, the greater Radius, and take out the = Tangents, as in the Table, and lay them from K both wayes; and then lastly, by those Points draw Lines from the hours required.

Note, That if in striving to put too many hours, the sum of the two extream hours come to above 76, it will make the hours too close together, and put you to much more trouble.

Also note, If your Rule prove too small, then take the half of the sum of both the Tangents, and turn the Compasses twice.

Also, If you be curious, you may use the Natural Logarithm Tangents, instead of the Line of Tangents, but this will serve very well.

This is a general way of augmenting all manner of Dials, when the Stiles height is low, as under 15 degrees; and as ready a way, as you meet with in any Author what­soever.

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CHAP. VI. To Draw the Hour-Lines on De­clining Reclining Dials.

FOR the compleat and true drawing of these Dials, that you may plainly see their Affections and Properties, it will be ne­cessary to have a Scheam for every variety; in doing whereof, I shall follow the Method that Mr. Wells, of Deptford, used in his Art of Shadows; which will comprehend any sort of Reclining and Declining Dial, under 6 varieties, viz. 3 South Recliners, and 3 North Recliners (the Inclining being their opposites, and no other, as afterwards is shewed).

Wherein I shall be very brief, yet suffici­ently plain, to a Mathematical Genius, and render the Canons, by Artificial and Na­tural Sines and Tangents, and draw the Dial by the Sector, the fittest Instrument for that use; With other occurrent Obser­vations, [Page 78] as they come in place, and the way by the Scheam Geometrically also.

1. And first for a South-Declining, Re­clining Dial, declining from the South to­ward the West 35 degrees, and reclining from the Zenith 20 degrees, being less than to the Pole, viz. falling from you, between the Zenith and the Pole: As the Circle AEB, representing the Reclining Plain, plainly sheweth P being the Pole, and Z the Zenith.

The manner of drawing this Scheam, is plainly shewed before, (Chap. 1.) both ge­nerally and particularly for the drawing of Dials, and the Example there, is the very Scheam for this Dial; wherein you may further consider, That the Perpendiculer-line is right before you, and when you look right on this Plain that declines Southwest, the North is before you on the left hand, the South behind you on the right hand 35 degrees, the East on the right hand, the West on the left; the Line CD the perpendicu­ler-line right before you, representing the Perpendiculer-line on the Plain, AB the Horizontal-line, ZE the quantity of Recli­nation, PF the Stiles Elevation above the Plain, having the South Pole elevated above the lower-part of the Plain, because the [Page 79] North-Pole is behind the Plain, EG the di­stance on the Plain, between the Plains per­pendiculer, and the Meridian, (being to be laid Eastwards, as the Dial-draught shew­eth, besides that general Rule before hinted, that whensoever a Plain declines Eastward, the Substile Line must stand Westward, and the contrary; for the Arch whereon to prick the Substile, and Stile is alwayes to be drawn on that side of the Plain, which is contrary to the coast of declination) EF the distance from the Substile and Perpendiculer, to be laid the same way; GF the distance on the Plain, from the Substile to the Meridian, to be laid the same way also; the Angle FPG, is the Inclination of Meridians; All which Requisites are found by these Canons Arith­metically, or by the Artificial and Natural Sines and Tangents.

1. To find the Distance of 12, from the Perpendiculer EG, or Horizon AG, by the second Axiome of Mr. Gellibrand, viz. that the Sines of the Base and Tan­gent of the Perpendiculer are proportio­nal.

[Page 80]

2. To find the Distance on the Meridian; from the Pole to the Plain, PG, by the 3 Propositions of Mr. Gellibrand, the Sines of the Sides are proportional to the Sines of their opposite Angles.

Which being taken from 38-28, the di­stance on the Meridian from the Pole to the Zenith, leaveth the distance on the Meri­dian of the place, from the Pole to the Plain, viz. 14-33, as a help to get the next.

3. To find the Height of the Stile a­bove the Plain PF.

In the two Triangles ZGE, and PGF, which are vertical, by the second Consecta­ry of Mr. Gellibrand; If two Perpendicu­ler Arks subtend equal Angles, on each side of the meeting, then the Sines of their Hy­pothenusaes, and Perpendiculers are pro­portional, (and the contrary); for the Angles ZGE, and PGF are equal Angled at G, and ZE, and PF, are both two per­pendiculer Arks on the Plain AB.

Therefore,

As the sine of the Hypothenusa GZ, to the sine of the Perpendiculer ZE;

So is the sine of the Hypothenusa PG, to the sine of the Perpendiculer PF, and the contrary.

[Page 82]

4. To find the Distance of the Substile from the Meridian, GF.

In the same Vertical Triangle, having the same acute Angle at the Base, the Tangents of the Perpendiculers, are proportional to the Sines of the Base, by the second Axiome of Mr. Gellibrand.

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[Page 83]

5. To find the Angle between the two Me­ridians, of the Place and Plain, viz. the Angle, PFG.

By the third Proposition of Mr. Gellibrand, it is proved, That the Sines of the Sides are proportional to the Sines of their opposite Angles, and the contrary.

The Angle between the 2 Meridians.

By Angle of Inclinations of Meridians, make the Table of the Hour-Angles at the Pole, by the Directions, Chap. 2. which be­ing made as in the Table, draw the Dial in this manner;

Upon AB, the Horizon­tal-line of your Plain, de­scribe the semi-circle AEB, and from the Perpendicu­ler-line CE of the Plain, lay off 13-28 Eastward for the 12 a clock Line, on the Plain, or the comple­ment thereof 76-32, from the East-end at B to +, & draw the Line C +.

Again, Set further East­ward from 12, 7-58, the distance of the Substile from 12, to F, and draw the Line CF for the Sub­stile; and beyond that, [Page 85] set off from F 12-13, the Stiles height above the Substile to G, and draw CG also.

Then,

Draw a contingent Line perpendiculer to the Substile CF, as far from the Center as you can, as the Line HI; then take the nearest distance from the point F, to the Line CG, and make it a = Tangent of 45; then the Sector being so set, take out the = Tan­gents of all the Hour-Arks in the Table, and lay them both wayes from F toward H and I, as they proceed; then Lines drawn from the Center C, and those Points shall be the Hours required.

Or,

Having in that manner pricked down 12, 6 & 3 (or any other Hours 3 hours distant) draw two Lines on each side 12 = to 12, and measure the distance from 6 to 3 in the =, and lay it from C the Center on the Line 12; and by those two Points draw a third Line, = to the 6 a clock-line; then 6-3, and 12-3, made a = Tangent of 45, shall be the two Radiusses to lay off the Hour-lines from 6 & 12, as before in the former Dials. And the = Tangent of In­clination of Meridians, doth prove the truth of your Work here also, as well as in the Decliners Erect.

[Page 86] But note, That this Dial is better to be augmented by the losing the Hours of 8 and 9 in the morning, which makes the Hours more apparent, as you see.

Also, the Requisites formerly sound, may Geometrically be found by the Scheam, be­ing large and truly drawn, as before is shewed in the other Dials. Thus,

1. A Rule laid from Q, the Pole-point of the Plain, to G the Point of 12 on the Plain, gives in the Limb the point 12; D 12, 13-28, is the distance of 12 a clock-line on the Plain from the Plains perpendiculer-line ZD, (and to be laid from the perpen­diculer-line on the Plain Eastwards in the Dial); and the distance on the Limb from A to 12, is the Meridians distance from the East-end of the Horizontal-line on the Plain, namely 76-32.

2. A Rule laid from Q to F, on the Limb, gives the Point Sub, for the Substile; and the Ark Sub. 12, 7-58, is the distance from 12, or the Ark Sub. D 21-26, the distance from the Perpendiculer.

3. A Rule laid from Q to 6, the place where the 6 a clock hour-line on the Scheam cuts the Plain, gives on the Limb the Point 6, the Ark 6 12, 25-38, or 6 D, 38-56, is the distance of the Hour-line of 6 on the [Page 87] Plain, from the Hour-line 12, or the Per­pendiculer.

4. A Rule laid from Y, the Pole-point of the Circle QFP, to P & F, on the limb, gives two points IK, and the Ark IK is the Stiles Elevation 12-13.

5. A Rule laid from P to Y on the limb, gives the Point M; EM is the Inclination of Meridians: or, a Rule laid from P, to the intersection of the Circle PFQ, and the Equinoctial-line, gives a Point in the Limb near C, which Ark CS, is more naturally the Angle between the two Meridians, 33-28.

Or,

If you like the way of referring this Plain to a new Latitude, and to a new De­clination in that new Latitude,

Then thus by the Scheam;

6. A Rule laid from E, to P and G, in the Limb gives L and O; the Ark LO is the complement of the new Latitude, being the Ark PG, the second requisite, in the former Calculation being 14-33, the di­stance on the Meridian from the Pole to the Plain.

[Page 86] But note, That this Dial is better to be augmented by the losing the Hours of 8 and 9 in the morning, which makes the Hours more apparent, as you see.

Also, the Requisites formerly found, may Geometrically be found by the Scheam, be­ing large and truly drawn, as before is shewed in the other Dials. Thus,

1. A Rule laid from Q, the Pole-point of the Plain, to G the Point of 12 on the Plain, gives in the Limb the point 12; D 12, 13-28, is the distance of 12 a clock-line on the Plain from the Plains perpendiculer-line ZD, (and to be laid from the perpen­diculer-line on the Plain Eastwards in the Dial); and the distance on the Limb from A to 12, is the Meridians distance from the East-end of the Horizontal-line on the Plain, namely 76-32.

2. A Rule laid from Q to F, on the Limb, gives the Point Sub, for the Substile; and the Ark Sub. 12, 7-58, is the distance from 12, or the Ark Sub. D 21-26, the distance from the Perpendiculer.

3. A Rule laid from Q to 6, the place where the 6 a clock hour-line on the Scheam cuts the Plain, gives on the Limb the Point 6, the Ark 6 12, 25-38, or 6 D, 38-56, is the distance of the Hour-line of 6 on the [Page 87] Plain, from the Hour-line 12, or the Per­pendiculer.

4. A Rule laid from Y, the Pole-point of the Circle QFP, to P & F, on the limb, gives two points IK, and the Ark IK is the Stiles Elevation 12-13.

5. A Rule laid from P to Y on the limb, gives the Point M; EM is the Inclination of Meridians: or, a Rule laid from P, to the intersection of the Circle PFQ, and the Equinoctial-line, gives a Point in the Limb near C, which Ark CS, is more naturally the Angle between the two Meridians, 33-28.

Or,

If you like the way of referring this Plain to a new Latitude, and to a new De­clination in that new Latitude,

Then thus by the Scheam;

6. A Rule laid from E, to P and G, in the Limb gives L and O; the Ark LO is the complement of the new Latitude, being the Ark PG, the second requisite, in the former Calculation being 14-33, the di­stance on the Meridian from the Pole to the Plain.

[Page 88]7. A Rule laid from G to Q on the limb, gives R, the Ark SR is the new declination in that new Latitude, 32-37.

Or else find it by this Rule;

As sine of 90, to the Co-sine of the Recli­nation, or Inclination;

So is the sine of the old Declination, to the sine of the new, in this Example, being 32-37, and generally the same way as the old Declination is.

Only observe,

That when the North-pole is Elevated on South Recliners, you must draw them as North-decliners; and North-west and North-east incliners, that have the South-pole Elevated, you must draw them as South-east and West-decliners, which will direct as to the right way of placing the Substile, and Hour of 6 from 12.

In this place I shall also insert the general way, by Calculation, to find the new Lati­tude, as well as new Declination:

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Which is thus;

As Radius, or Sine of 90, to the Co-sine of the Plains old Declination;

So is the Co-tangent of the Reclination, or Inclin. to the Tang. of a 4th Ark.

Then,

In South Recliners, and in North Incli­ners, get the difference between this 4th Ark, and the Latitude of your place, and the complement of that difference is the new Latitude: if the 4th Ark be less then the old Latitude, then the contrary Pole is Elevated; but if it be equal to the old Lati­tude, it is a Polar-plain.

But in South Incliners, and in North Recliners, the difference between the 4th Ark, and the complement of the Latitude of the place (or old Latitude) shall be the new Latitude, when the 4th Ark and old Latitude is equal, it is an Equinoctial-plain.

Thus in this Example;

As sine 90, to Co-sine of 35, the old De­clination;

So is Co-tangent of 20, the Reclination to 66-03, for a 4th Ark; from which taking 51-32, the old Latitude, rests [Page 90] 14-31, the complement of the new Lati­tude, which will be found to be 75-29, the new Latitude.

By which new Latitude, and new Decli­nation, if you work as for an Erect Dial, you shall find the same Requisites, as by the for­mer Operations you have done; and the distance of the Perpendiculer and Meridian will set all right.

The Second Variety of South Recliners, reclining just to the Pole.

1. The Scheam is drawn, as before, to the same Declination, and the same way, viz. 35 degrees Westward, and reclines 33-3′, Now, to try whether such a Plain be just a Polar-plain or no, use this Proportion:

Which 4th Ark, if it hit to be right the Latitude, then it is a declining Polar-plain, or else not.

2. If you have a Declination given, to which you would find a Reclination to make it Polar, then reason thus▪

[Page 92]3. If the Reclination were given, and the Declination required to make it a Polar, then the Canon may be thus;

But by the Scheam, these three Operations are found by drawing the Scheam.

1. For if the Line or Circle, represent­ing the Plain, cut the Pole P, it is a Polar-Dial.

2. If AB, the Co-declination, be given, then draw the Circle APB, and it gives E; then ZE is the Reclination, measured by half Tangents; or a Rule laid from A to E on the Limb, gives an Ark from B; which measured on fit Chords, is the Reclinati­on.

[Page 93]3. If P, the Pole-point, and ZE the Reclinatin, be given; then, with the di­stance ZE, on Z as a Center, draw an Ark of a Circle in that Quadrant which is con­trary to the Coast of Declination, observing the letters in the Scheam; then by the Convexity of that Ark, and the Pole-point P, draw the Circle PE, cutting the Limb into two equal parts, which are the points A & B, the declination required.

This being premised, there are two things requisite to be found, before you can draw the Dial. viz. the Substile from the Perpendiculer or Horizon, and the In­clination of Meridians.

1. And first for the Substile, by the Sector.

[Page 94]The distance of the Substile from the Perpendiculer, whose complement 69-06, is the Elevation above the Horizon.

Or,

A Rule laid from Q to P, gives I; DI is 20-54.

2. For the Inclination of Meridians, say, By the Sector.

Or,

A Rule laid from P to Y, gives M; EM is 28-45, the Inclination of Meridians.

The Scheam being true drawn, which be­ing one degree and 15 minuts less than 30, the quantity of two hours, I set 1-15 a­gainst 2; and by continual addition of [Page] [Page]

[Page 95] [...]5 deg. to 1-15, and the increase thereof, [...] make up the one half of my Table, as fol­ [...]oweth.

Again,

If I take 15, the quantity in degrees of one Hour, out of 28-15, the Inclination of Meridians; there remains 13-45, for the first Hour on the other-side of the Substile.

Then again, by continual addition of 15 degrees to 13-45, and the increase thereof, I make up the other half.

Or else, Against 12, set 28-45, and add 15 suc­cessively to it, & its increase, till it come to 90, Then, to 13-45, the residue of 15, taken from 28-45; add 15 as often as you can to 90, and thus is the Table made.

To draw the Dial.

First, Draw a perpendicu­ler Line on your Plain, as CB, by crossing the Horizon­tal-line at Right Angles; [Page 96] then from the perpendiculer-Line lay off from the upper-end, toward the left-hand (as the Scheam directs, ZD being the Per­pendiculer, and ZN the Meridian, and EP on the Plain, the distance between, be­ing toward the left hand) 20-54, for the Substile-line, as CD; then on that Line (any where) draw two perpendiculer Lines quite through the Plain, crossing the Sub­stile at right Angles, for two Equinoctial-lines, as EF, & GH.

Then consider what hours shall be put o [...] your Plain, as here is convenient, from 10 in the morning, to 6 afternoon; (though the Sun may shine on it from 8 to 7, bu [...] then the Lines will be too close together, and the Radius too small). And also when you would have those two utmost hours [...] be, as at E and F on the upper Equinoctial-line; or, at G & H on the lower contin­gent-line.

Then,

Take the — Tangent of 58-45, in the Table for 10, and add it to the — Tan­gent of 61-15 for 6, on the Line of smal [...] Tangents, and you shall see the Compasse point to reach to the Tangent of 73-55, the sum of both the two extream hours, which happening under 76, will be convenien [...] [Page] [Page]

[Page 97] and handsome, in their distances assunder.

Then,

Take the whole distance EF, or GH, and make it a = Tangent of 73-55; then the Sector to set, take out the = Tangent of 58-45, and lay it from the point E to I, on the Equinoctial-line; Also, take out the = Tangent 61-15, and lay it from the point F; and if your work be true, it must needs meet in the point I; then draw the Line IK for the true Substile, and from thence lay the = Tangent of 45, to draw a Line near 5, for the Stiles Elevation, paral­lel to IK the Substile; for being a Polar-plain it hath no Elevation, but what you please to augment it to▪ as here from I to L.

Then,

As the Sector stands, prick on all the whole hours, halfs, and quarters, according to the Numbers in your Table, at least those that be above 45; and for those under 45, make = Tangent of 45 in small Tangents, a = Tangent of 45 in the great Tangents, and then the Sector shall be set to that Ra­dius, which is most convenient for your use.

Note,

That this way of Augmenting the Stile, is general in all Dials.

3. The third Variety of South-Recliners.

The next and last kind of South Recliners, are such as recline, or fall from you below the Pole, viz have their Plains lying between the Pole and the Horizon, as by the Scheam is more apparent.

In which work, the drawing the Scheam, and the things required, are the same as in the first Example, as the Figure, and fol­lowing words, do make make manifest.

The Example here, is of a Plain that de­clines from the South toward the West 35 degrees, and reclines upon its proper Azi­muth ZE, 60 degrees from the Zenith.

1. Having drawn the Scheam, then first for the distance of the Meridian from the Perpendiculer, or Horizon.

By the Sector, or Quadrant.

Whose complement is 58-48 AG, the distance between the West-end of the Hori­zontal-line, and the Meridian.

Or by the Scheam;

A Rule laid from Q to G, cuts the limb at L; the DL, and AL, are the Arks re­quired; DL from Perpendiculer, and AL from the Horizon.

2. To find PG, the Ark on the Meridi­an from the Pole to the Plain.

By the Sector.

[Page 100]Which being taken from NP 51-32, leaveth GP 26-13, the distance on the Me­ridian from the Pole to the Plain, or the complement of the new Latitude.

Or,

A Rule laid from E, to P and G, gives on the limb 2 Points, whose distance be­tween, is ab 26-13, the Ark required.

3. To find the Stiles Elevation above the Plain.

By the Sector.

Being found by the Scheam, by laying a Rule from Y, to P and F, on the limb, gives the distance between being 25-02, the Stiles Elevation,

4. To find the Substile from 12.

By the Sector.

By the Scheam, a Rule laid from Q, to G and F on the limb, gives L and M 8-3; Or else, the Ark MD, is the distance of the Substile from the Perpendiculer 23-19.

5. To find the Inclination of Meridians.

By the Sector.

By the Scheam, a Rule laid from P to Y, on the limb, gives O, the Ark EO is 18-27; the Inclination of Meridians, by help of which, to make the Table of Hour-Arks at the Pole, as before is shewed, and as in the Table following.

[Page]

To draw the Dial.

First, for the Affections, consult the Scheam, wherein, laying the Perpendiculer-line CD right before you, you see that the Substile, and the Meridian, are to be laid from the Perpendiculer toward the left-hand, the Substile lying between the Perpendiculer and the Meridian, and the Stile or Cock of the Dial must look upwards, the North-Pole being Elevated above this Plain, which will guide all the rest.

Then,

First, draw the Horizontal-line AB, and on C as a Center raise a Perpendiculer, and set off by Chords, Sines, or Tangents, the Meridian or 12 a clock Line, the Sub­stile, and Stile, as exactly as you may; and draw the Lines 12 C, Substile C, and Stile C.

Then,

As far from the Center C, as you conve­niently may, draw a long Line perpendicu­ler to the Substile, as the Line EHF; then setting one Point of a pair of Compasses in H, open the other till it touch the Stile-line at the nearest distance.

Then,

Make this distance a = Tangent of 45, and take out the = Tangents of every whole Hour, as in the Table, as far as the Tangent of 76. will give leave; and then from the Center C, to those Points draw Lines for the even whole Hours; then to any one whole Hour, as suppose the Hour-line of 3; draw two = Lines equally distant on both sides the Line of 3, as IK, LM.

Then,

Count any way 3 hours, and 6 hours from 3, as here 12, and 9, so as the = line may cross the 3 remotest hours, as here you see 9 and 12 a clock Hour-lines do cross the = line at I and K; then take the distance IK, and lay on the Hour-line of 3 from C to N, and draw INL = to 9 C; Which Work doth constitute the Parallellogram KILM.

Then lastly,

Make KI, and NI, = Tangents of 45, and p [...]ick off every hour, half, and quarter (and minut if you please) on the two Lines IK, and IL, from K and N both wayes, as before is already shewed in the Erect De­cliners.

[Page]

Note also,

That to supply the defect on the other side, when the point M falls out of the Plain, the distance from I, to the Hour-point from 11, will reach from L to 7, and from I to 10, from L to 8. This is general in all Dials.

Also note, If you like not to lay off the [...]irst Hours by the Tangents, having made the Table, as before, you may soon find the Hour-Arks on the Plain for 3 Hours, as [...]ere 3, 12, and 9; Or, 4, 1, and 8, which [...]ould have made the Parallellogram more [...]uare, and consequently more better, and [...]en to draw the rest by the Sector. Thus [...]ou may see how your Work accords; The [...]ay by the Table and Contingent-line, and [...] way by the Sector on the Parallellogram, [...] by Calculation, & at last use the Mystery [...] Dialling made plain and ready, to an [...]dinary capacity.

Of North Declining Recliners.

The other kind, viz. North Declining Recliners, have also three Varieties; as those, [...]. That fall back or recline between the Ze­nith [Page 106] and Equinoctial: 2d. Those that re­cline to the Equinoctial: And 3d. Those that recline below the Equinoctial. And first of the first Variety, reclining less then to the Equinoctial.

The drawing the Scheam, is the same as in the former, except in the placing of the Points and Letters; For first, these Plains behold the North-part of the Horizon, and then when you look on the Plain, the South is before you, and the West on your right-hand, and the East on the left; then the South and North are alwayes opposite, and the point P, representing the Elevated Pole of the place, which with us being North, must be placed towards N downwards, as before in South Recliners it was upwards.

Also,

It is necessary in the Scheam, to draw the Equinoctial-line, by laying the half Tan­gent of 51-32 from Z to AE; then the Se­cant of 38-28, the complement of ZE, laid from AE on the Line SN, shall be the Cen­ter to draw EAEW for the Equinoctial-Circle.

Thus the Scheam being drawn, to find the Requisites, thus;

1. For the Meridians Elevation, or di­stance from the Perpendiculer, AG, or GE.

By the Secctor.

Whose complement AG, 63-58, is the Meridians Elevation above the East-end of the Horizon.

By the Scheam, A Rule laid from Q to G, on the Limb gives L; then DL and AL are the Arks required.

2. To find the Distance on the Meridian, from the Pole to the Plain GP.

[Page 108]

Which added to 38-28 ZP, makes up GP to be 70-31. Or, By the Scheam, A Rule laid from E, to P and G, gives on the limb ab; the Ark ab is 70-31.

3. To find the Stiles height above the Plain PF.

By the Sector.

By the Scheam.

A Rule laid from Y, to P and F, on the limb gives c and d, the Stiles height.

4. To find the distance of the Substile from the Meridian GF; when it is above 90 deg. take the comp. to 108 deg.

By the Sector.

By the Scheam.

A Rule laid from Q to G and F, gives on the limb LF, the Ark required.

5. To find the Inclination of Meri­dians FPG.

By the Sector.

By the Scheam.

A Rule laid from P to Y, on the Limb gives g, the Ark Eg is 74-38, the Inclinati­on of Meridians.

Or,

A Rule laid from P to K, gives h, Sh is the Inclination of Meridians, by which to make the Table as before is shewed, and as followeth.

[Page]

To draw the Dial.

For drawing the Dial, con­sult with the Scheam, laying the Plain AEB, and his Perpendiculer CD right be­fore you; then note, SN is the Meridian-line, ZE the Plains perpendiculer, with the Meridian G on the left-hand, and the Subtile F on the right-hand.

Also note, That the Sun being in the South as S, casts [...]is beams, and consequently the shadow of [...]he Stile into the North; So that though G be the true Meridian found, yet it is the North-part that is drawn as an Hour-line; [...]ut the Substile, and other Hours, are coun­ [...]ed from the South-end thereof, as the Table [...]nd the Figure of the Dial, do plainly make [...]anifest; being drawn in this manner.

First, draw the Horizontal-line AB, then [...] C, as a Center, draw a semi-circle equal [...] 60 of the Chords, and lay off the Meri­ [...]ian, Substile, and Stile, in their right Sci­ [...]ations, as last was declared; then draw [...]ose lines, and to the Substile erect a Per­ [...]endiculer, as DE; then take the Extent, [Page 112] or nearest distance from the place where the Perpendiculer or Contingent-Line last drawn, cuts 12 and the Stile-line, and make it a = Tangent of 45; Then is the Sector set, to lay off all the Hours by the = Tan­gents of the Arks in the Table, except 11 and 10, which do excur.

For▪

If you prick the Nocturnal-Hours 12, 1, 2, 3; and draw them through the Cen­ter, on the other side, they shall be the Hours of 12, 1, 2, 3, 4, &c. on the North-part of the Plain, where they are only used. As for the Hours of 10 and 11, do thus;

Draw a Line = to any one Hour, which = line may conveniently cut those Hour-lines.

As,

Suppose the Line 6 12, which is = to the Hour-line of 3; then make the distance from 9 to 12, or from 6 to 9, in that Line last drawn, a = Tangent of 45, and lay off hours and quarters, or else the whole Hours, by the distances from 9 to 7, and 8 for 10 and 11, turning the Compasses the other way from 9; then to all those Points Lines drawn, shall be the Hour-lines re­quired.

[Page]

Or,

Having only the hours of 3, 6, & 9, & 12 in a Parallellogram, design the rest by Sector.

The Second Variety of North-Recli­ners, Reclining to the Equi­noctial.

By the bare drawing of the Scheam, you see, that the Circle AEB, representing the reclining Plain, doth cut the Meridian just in the Equinoctial; Now to try by Arith­metick, whether it be a just Equinoctial-plain, or no, say:

1. By the Sector.

[Page 114]Which happening so to be, it is a decli­ning Equinoctial, or Polar in respect of its Poles, which are in the Poles of the World.

2. If the Declination were given, and to it you would have a Reclination, to make it Equinoctial.

By the Sector;

By the Scheam.

The Points AB of Declination, being given, and the Point G on the Meridian, if you draw the Reclining Circle AGB, it will intersect the Perpendiculer at E; then the measure of ZE is the Reclination, mea­sured by half-Tangents, or by Chords, by laying a Rule from A, to E on the limb, gives a; the Chord B a, is the Reclinati­on 35-50.

3. But on the contrary, if the Reclination be given, and a Declination required, to make an Equinoctial Plain; Then con­trarily say thus,

By the Sector.

But by the Scheam.

By the Point G, and the touch of an Arch about E, draw the Circle GE, to cut the limb into two equal parts, and you have the Points AB.

4. The Plain thus made, or proved to be Equinoctial; to find the Meridians Elevation above the Horizon, AG; Or, his Distance from the Perpendicu­ler EG.

By the Sector.

Whose complement is AG 50-06, the E­levation above the Horizon.

By the Scheam.

A Rule laid from Q to G, gives b on the limb, DB is 39-54, as before.

5. To find the Stiles Elevation above the Substile on the Plain.

By the Sector.

[Page]

[Page 117]

By the Scheam.

A Rule laid from Y to F on the limb, gives C, NC is 48-24, the Stiles height.

The distance of the Substile from 12, in these Equinoctial Dials, is alwayes 90 de­grees; for a Rule laid from Q, the Pole of the Plain, to G, on the limb gives b; a Rule also laid from Q to F, the Substile, on the limb gives d; the Ark bd, is 90 degrees, both for the distance of the Substile from 12, and also for the Inclination of Meridi­ans, for the Substile stands on the hour of 6, being part of the Circle EPW, which is the hour of 6, 90 degrees distant from the hour of 12.

Or,

A Rule laid, as before, from Y to P, on the limb, gives N; the Ark EN, or WN, is 90, for the Inclination of Meridians.

Which being just 90, the Table is easily made, viz. 15, 30; 45, 60; 75, 90; twice repeated, from 12 to 6 both way s.

To draw the Dial.

On the Horizontal-line AB, draw an ob­scure Semi-circle, and set off the Meridian, as the Scheam sheweth, viz. 50 degrees 6 min. above the East-end of the Horizontal-line; but make visible only the North-end thereof, as the line C 12; Then 90 degrees from thence, toward the right-hand, as the Scheam sheweth, when the Perpendiculer-line is right before you, draw a Line that serves both for 6 and the Substile, as C 6. Also, lay off the Chord of 36-47 from 6 to 9, and draw the Line C 9 also, which is found by Calculation, as before is shewed.

Or thus;

Draw a Line = to 12, or Perpendiculer to 6, being in this Dial all one, as the Line FEG; then setting one Point in E the Sub­stile, take the nearest distance to the Stile-line, and it shall reach from E to G, the Point for 9.

The same distance EG lay also on the line 12, from C to H, and draw the line GHI; then make EG a = Tangent of 45, and lay off the = Tangents of 15-30-45, both wayes from E, as hath been often shewed.

[Page]

Also,

Make the distance of HG a = Tangent of 45, and lay the same = Tangents both wayes from H, and to those Points draw the Hour-lines required.

The third Variety of North-Recliners.

This third and last sort of North-Recli­ners, are those that recline beyond the E­quinoctial, that is, lie between the Equi­noctial and the Horizon; and it differs somewhat from the other five before, in the Scheam and Operation also.

For first, the Ark of the Plain is extend­ed below the Horizon, till it meet with the North-part of the Meridian below the Ho­rizon at H; and the Center of the Ark AQB, is in the Line ZD, as much di­stance from Q, as the Secant of 65 deg. to the Radius of the Scheam, being the complement of ZQ 25-0; Here also the same requisites are to be found as in the other Dials.

1. First, for the Meridians Elevation above the Horizon, AG.

By the Sector;

Whose complement GA 37-42, is the Meridians Elevation above the Horizon.

By the Scheam;

A Rule laid from Q to G, gives on the limb a; then D a is the distance from the Perpendiculer 52-18; and A a the distance from the Horizon 37-42.

2. To find the Distance on the Meridian from the Pole to the Plain GP.

By the Sector.

Whose complement 75-02 GZ, added to ZP, the complement of the Latitude, makes 113-30, for the distance of the North-pole P, on the Meridian of the place, from (the North-pole P, to) the Plain below the Equa­tor at G; which being more than 90, find the complement thereof to 180, viz. 66-30, being the distance on the Meridian from P the Pole, to the Plain on the North-part of the Meridian, viz. PH, found on the Scheam, by laying a Rule from E or W, to [Page 122] P and H, on the limb gives b and c; the Ark bc is 66-30, the distance on the Meridian from the Pole to the Plain.

3. To find th [...] Stiles height above the Plain PE.

By the Sector.

By the Scheam.

A Rule laid from Y, to P and F on the limb, gives d and e; the Ark de is 59-21, the Stiles Elevation.

[Page]

4. To find the Substile from 12, viz. FG from the South part, or HF from the North part.

By the Sector.

By the Scheam.

A Rule laid from Q, to H and F, on the [...]imb, gives f and e; the Ark f and e, is the Substiles distance on the Plain from 12.

5. To find the Angle between the two Me­ridians, viz. PF, and PH.

By the Sector.

By the Scheam.

A Rule laid from P to Y, on the limb gives g, then W g is the Angle of the In­clination of Meridians, viz. 42-45; by which make the Table, as is several times before shewed, and as followeth.

To draw the Dial.

On the Horizontal-line AB, describe a Semi-circle, and lay off the Meridians Ele­vation, in its proper place, as the Scheam directs; and then the Substile, and Stile, be­yond the Perpendicu­ler, as by laying the Perpendiculer-Line of the Scheam right be­fore you; then the Line AZB, represents the Horizon; the Line ZG, the Meridian on the South-part; and ZH, on the North-part; [Page] [Page]

[Page 125] [...]art; the Line ZF, represents the Substile [...]ear the Horizon: which things being ob­ [...]erved, and the method thereof understood, [...]t will prove better than dry Rules, or Pre­cepts, without such representations, and far more easie to apprehend and conceive in the imagination.

Then prick down the Hours of 9 & 6, by the Table on the Contingent-line, as be­fore, or on the Semi-circle, having Calcu­lated only those two Hour-lines, by the ge­neral Canon.

Then,

Draw a Line = to 12, at any convenient distance from it, as GH; Then, take the distance between 6 and 9 in that = Line, and lay it from the Center to I, on the 12 a clock Hour-line, and draw the Line HI; then make the distances GH, and IH, se­verally one after another, = Tangents of 45; and take out the = Tangents of 45, 30, 15; and lay them both wayes from 12 and 6, on those two Lines, as hath been of­ten shewed, in the former Dials; then lines drawn to those Points, shall be the Hours required.

CHAP. VII. Of Declining and Inclining-PLAINS.

INclining Plains are but the under faces of Recliners, beholding the Nadir, at the same Angle that the Recliners behold the Zenith.

And the making of them differs nothing from the Recliners already mentioned; and all the Requisites, as Meridians Elevation, Substiles distance from 12, and the Stiles E­levation, & the Hour distances, are the same both in the Incliner, as they were in the Re­cliner, and have the same Numbers also set to the Hour-lines; So that in drawing of these Dials that have Centers, if you draw all the 24 Hours, you then draw 4 Dials at once; as thus, in the Example of the declining E­quinoctial, being a North declining East 55 degrees, and reclining 35-50. If you draw the 24 hours, being done at the same time [Page] [Page]

[Page 127] and stroke, by drawing the Hour-lines through the Center on the other-side, and the Substile and Stile also, as here you see in the Dial annexed, being the Equinoctial-Dial, belonging to the second Variety of North Recliners declining Eastward 55, and reclining 35-50; the Lines drawn through the Center, and complemented to 12, is a South-east declining, and Inclining 35-50.

Also,

If you turn the Paper, and look against the light, and then the North-east becomes a North-west Decliner 55, and reclining 35-50; and the South-east becomes a South-west, Declining and Inclining as much.

Thus you see, that every Draught of a Dial will serve for 4 Plains, that is for the place you draw it, and his opposite; and for another Plain, declining as many degrees the contrary way, and reclining as much also, and for the opposite thereunto, as by the two Draughts of the two sides, may plainly be seen to appear. And the like holds in all sorts, as Upright Decliners also.

As a North-east and a North-west, a South-east and a South-west, declining 30; one Dial drawn round about, serves all 4 Dials; But note, that no South Erect or In­clining Dial, can have the Sun to shine on [Page 128] any Hour-line that falls above the Horizon­tal-line; and those hours on the North-re­cliners, that fall below the Horizontal-line, belong also to the South Dials.

But for a plain general Rule, to know what hours belong to any Plain whatso­ever in any Latitude, do thus.

To know what Hours belong to any Plain.

First, draw a general Scheam to your Latitude, as this is done for 51-32; and mark the 4 Cardinal-points with E.W.N. & S. as is usual for setting the Scheam right be­fore you.

Then,

For all Declining Upright Dials, draw on­ly a streight Line for the Plain, Perpendicu­ler to the Line that doth represent the Pole of the Plain, counting so many degrees as the Declination of the Plain shall happen to be from S. or N. toward E. or W. then all the Hour-lines of the Scheam that that Line of the Plain shall intersect, are the Hour-lines proper to that Plain.

1. Example.

The Line E and W, being Perpendicul [...]r to S and N, the Poles of a South and North-plain; [Page] [Page]

[Page 129] doth therefore represent a South-plain on one side, and a North-plain on the other.

Therefore,

If you conceive the Sun to be in Cancer, and going of his Diurnal Motion, at his Rising about a quarter before 4, beholds the North-side of the Line EW, and continu­eth so to do till 25 minuts after 7; and then it shines on the South-plain till 35 minuts after 4, and then begins again to shine on the North-plain, and so continues till Sun setting.

But when the Sun is in the Equinoctial, it beholds the South-plain at the Rising, be­ing at 6 a clock in the morning; and shines on it all day, till Sun set, being at 6 at night; and then the North Dial is useless.

2. For a Declining-Plain.

Suppose 30 degrees South-east; first set the Scheam in his right scituation for a South-east Plain; then if you count 30 de­grees from S toward E, for the Pole of the Plain; and 30 degrees from W toward S, or from E toward N, and draw that Line that shall represent the Plain; then you shall find that the Sun being in Cancer will be­gin [Page 130] to shine on this Plain, just a quarter be­fore 5 in the morning, and continue till near half an hour after 2.

But about the middle of Ianuary, it will shine on it till a quarter after 4, viz. till Sun set; and all the hours after 2, belong to the North-west Plain that declines 30 de­grees, and one hour in the morning also, viz. from a quarter before, till three quarters after 4.

The like work serves for any Decliner whatsoever, in any Latitude.

3. But for Decliners and Recliners.

Draw a long Line, as AB, and cross it with a Perpendiculer in the Center C, and lay off from C, toward A and B, the Tan­gent of 45; or the Semi-tangent of 90, e­qual to the largeness of your Scheam; then lay off the Semi-tangent of the Reclination from C to D, up and down, both wayes; then take out the Secant of the complement of the Reclination, which will be a Radius to draw the Arks ADB, which Paper you must cut out, and apply the two Points of the Paper ADBD, to the two Points of Declination of the Plain, noted in the Scheam with A and B; that is, put A to A, [Page 131] and B to B; then the round or convex-edge of the Paper, represents the reclining Plain; and the same edge, on the other part next the Horizon Southwards, represents the South-west Incliner.

Example.

Suppose I make the Paper ADB, to re­cline 35-50, the Reclination of the Equi­noctial-plain; then, first set the Scheam right before you in its right scituation, and putting the Points A, in the Paper, on A on the Scheam; and B in the Paper▪ to B on the Scheam; I shall find it to be even with the reclining Circle AEB; then following the Tropick of Cancer, I find that it shines on the North Recliner from the Rising till near 2, at which time it leaves the North-recliner declining Eastward, and begins to shine upon the opposite Plain, viz. the South-west Incliner, declining 55-0, and reclining 35-50, and so continues till Sun­set.

But note, That if the Line that represents the Plain, cuts the Tropick twice, as the Line EW for a North-plain; then, though the Sun leave the Plain in the morning, it will shine on it again in the afternoon.

Note also, That a North-east Recliner, is represented by the other Convex-edge of the [Page 132] Paper, as here a North-east Decliner 55, and Inclining 35-50, the Sun will shine but till 3 quarters after 8 in Cancer; but in Capricorn it shines till half an hour after 9, and comes no more on it that day: And note alwayes, That when it leaves any Plain, that then it begins to shine on his opposite, as here the opposite to this North-east In­cliner, is the South-west Recliner, being re­presented by the same Line or Circle ADB, that the North Recliner was: Only, you must count that side of the Line next to the Horizon, the Inclining-plain; and that side next the Zenith, the Reclining-plain; For, the Line that represents it, having no bredth, can be no otherwise distinguished, unless you will make a material, Armilary Sphear, of Pastboard or Brass, as the following Discourse doth plainly demonstrate, in these several Operations, for the better conceiving of these Mathematical Excercitations.

Thus you have the way of making all manner of Sun Dials, upon any plain Su­perficies, the Axis of the World being the supposed Stile to all these Plains; As for those curiosities of Upright Stiles, and Elip­tical Dials, and drawing of Dials by the Horizontal, or Equinoctial Dials, you have them in the Works of Mr. Samuel Foster, [Page 133] and others; and in Kerkers Ars magna, &c. But I intended not a Volumn of Sha­dows, but only a further improvment of the Trianguler-Quadrant, as you will see in the next Chapter, of drawing the Furniture or Ornament of Dials; which being but seldom used, I shall here crave an Apology for the brevity therein, fearing, lest that to the young Practitioner it may seem some­what hard to conceive, though to the exer­cised in these matters it may be plain e­nough.

Then for a Conclusion, you shall have an easie Mechanick way, to draw a Dial on the Ceiling of a Room, that lieth Flat or Hori­zontal, which will be very good for Painters or Plaisterers, to Ornament a Room withal, and is not yet treated on that way, as ever I read of.

CHAP. VIII. To furnish any Dial, with the usual Mathematical Ornaments by the Trianguler-Quadrant, as Paral­lels of the Suns Declination, or the Suns place, or length of the Day, to find the Horizontal and Virtical Lines, and Points, to draw the Azimuths, and Almicanters; the Iewish, Italian & Babylonish Hours, and 12 Houses on any Plain before mentioned.

CHAP. IX. How to remedy several Inconve­niencies in the using of the Artificial Lines of Numbers, Sines and Tangents, as they are usually made.

1. IF the term required happen to be un­der one degree of Sines and Tangents, then the Line of Numbers will supply it, having due respect to the increase of the Radius, or Caracteristick.

As thus;

As the sine of 90, to the sine of 23-31, the greatest Declination;

So is the sine of 1 deg. 10′, the Suns di­stance from the Equinoctial, to 0-28, the Declination which falls beyond the end of the Rule.

Now to remedy this, the 1 deg. & 10′, [Page 218] is 70 minuts; therefore by the Numbers say, So is 70 minuts, the Suns distance from the Equinoctial, to 28 the Suns Declination on the Line of Numbers, observing to ex­tend the same way, as from the first to the second term.

2. When you have occasion to use a sine above 90 degrees, then you must count the sine of 80, for the sine of 100; and 70, for 110; and 60, for 120.

So also, the distance from 90 to 60 in the Sines, is the Secant of 30 degrees; and the distance from 90 to 50, is the Secant of 40; or the Point beyond 90, that represents the Secant of 40.

3. If the Extent be too large for your Compasses, as from 45 or 90, to 3 or 4 de­grees; then instead of 90 or 45, make use of a Point in the Sines or Tangents right a­gainst the middle 1 in the Line of Numbers, where you may have two Brass Center-pins, viz. in the Tangent of 5-43, and the sine of 5-45; and the extent from thence back­ward or forward, shall reach in the Num­bers, to the 4th proportional Number re­quired.

Example.

As Tang. 45, to 1-61 in the Numbers;

So is Tang. of 15-0, to 0-43 in the Num­bers.

[Page 219]Instead of which, you may say, As the Tang. of 5-43, to 1-61 on the Numbers; so is the Tang. of 15, to 0-43 on the Num­bers diminishing a Radius; for as Tang. 45 to 1-15, a greater than that; so is the Tang. of 15, to a greater than 15 also, viz. 0-43.

Secondly, in Sines & Tangents, or Sines only, where there is another Caution to be observed, As sine 90, to sine 10; so is sine 20, to sine of 3-24 ⅓.

To work this with small Compasses on a large Line, do thus; Note, that at 10 on the Line of Numbers, or Sine of 90, or Tang. of 45, is one compleat Radius; but at the middle 1, on the Line of Numbers, is a place, or Radius, less; wherein the Loga­rithm Sines, the Characteristick is 8. A­gain, at the sine of 0-34 ½, the Characteri­stick is 7, (and at 3 minuts it is 6,) which do note the several decreasings of the Ra­diusses; Therefore set the distance from one Number given, to the next nearest place a­gainst 1, or next Radius, as far from a greater or a less Radius, as your occasion serves, and note the place.

As thus for Example.

In this Operation, the extent from the Point at 5-45 on the Sines, to the sine of 10 degrees, I set the same way from the [Page 220] Point at 0-34 ½ and note the place, which will be at near 1 degree; then the work is thus; As the place against the middle 1, instead of 90, is to the place last found for 10; so is the sine of 20, to sine of 3 deg. 24′ ⅓, the 4th term required.

But in those Lines of Numbers, Sines, and Tangents, where the Number is double, this is performed by working a-cross only.

4. When the last term in Tangents hap­pens to be above 45, then the remedy is two wayes,

As thus;

As sine of 30, to sine of 90;

So is the Tang. of 30, to Tang. 49-07. which here happens beyond 45.

Apply the end of the Rule, next 90, close and even with any thing on which the Point of the Compasses may stay, till you take from thence to 45, for that distance laid from 45, shall reach to 49-07, reading the Tangents as numbred beyond 45.

Or more neatly thus;

The Compasses being set from the sine of 30, to the sine of 90; set one Point in the Tangent of 45, and turn the other on the Tangents, and keep it there fixed; then re­move [Page 221] the other from 45, and close it to the third term, being here the Tangent of 30; then this last Extent laid from 45, shall reach to 49-07, the Tangent required.

5. When the first term is a Tangent a­bove 45, and the second under 45.

Take the excess of the first Number a­bove 45, and set it the same way from the second Number; then the Extent from the second Number to 45, shall be the true di­stance between the first and second terms.

Example.

As the Tangent of 51-30, to the Tan­gent of 30;

So is the Tangent of 40, to Tangent 21-04′.

For the Extent from 45, to 51-30 on the Tangents, set the same way from 30, does reach to about 24-30; then the Ex­tent from thence to 45, shall reach from 40 to 21-04 on the Tangents, the 4th Number required.

Or,

If it had been from a Tangent above 45, to a sine, the same way would have re­medied the defect.

6. When the third term exceeds 45 of Tangents, then thus;

Example.

As sine 90, to sine 30;

So is the Tang. of 50, to Tang. of 30-48.

The Compasses set from the first term sine 90, to sine of 30 the second, a less; then set one Point in the Tangent of 45, and extend the other backwards in the Tangents, and note the place, keeping one Point there close, the other to 50 the third term (being above 45, by counting backwards) Then, I say, that Extent laid from Tangent 45, shall reach to Tangent 30-48, the 4th proportio­nal Tangent required.

If the Proportion had been increasing, then there had been no trouble at all.

Also note,

That working a-cross, or changing the terms, is a good remedy also.

As thus;

As sine 90, to Tang. 50; which is proper­ly increasing, for the Tang. of 50 being more than the sine of 90, yet taken on the Rule from 90 to 40, the complement there­of, as if it were decreasing;

So is sine 30, to Tang. 30-48, the contrary way: Therefore,

As from the first term, properly counting to the second.

[Page 223]7. Lastly, When one or two Radiusses (or Alterations of the Characteristick) falls between the first and second term.

As thus for Example.

First, By the Line of Numbers only; As 8000 is to 10, So is 5000 to 6 ¼, or 25.

To work this properly, and naturally, the unite on the Numbers should be four times repeated, which is seldom more done than twice, as here: But this, and any other, by the Line of Numbers is not interrupted, ha­ving a due respect to the Number of Places. For to work this, the best way, is changing of terms thus; As 8000, to 5000 in the same Radius; so is 10, to 6-25 in the same Radius also. Or, without changing; As 8000, to the next 1; so is 5000 turning the Compasses the same way, to 6-25.

But to call it so, and not 625, your rea­son must guide you more than precepts.

But in using Sines and Tangents, the way in the third remedy will fit you.

Example. As sine 90, to 1 degree (or under); so is sine 30 degrees, to sine 30 minuts. This being too wide an Extent for the Compasses, the third Rule is a remedy for it; which on a large Radius several times [Page 224] repeated, as in Mr. Oughtred's Circles of Proportion, is as easie as may be; being sure to remember the number of Radiusses be­tween the first and second term, that you may have so many between the third and fourth term also.

Much more might have been said as to this; but this Observation being alwayes kept, That as the Extent from the first term to the second, is either increasing or de­creasing; So alwayes must the Extent be from the third to the fourth, increasing or decreasing, in like manner, when you use Sines and Tangents; And Numbers also, except, as before, in a few particuler Rules; then you will be truly resolved.

CHAP. I. The Description thereof FOR SEA-USES.

THe Description of the Instrument, is largely and plainly set down in the First Part, and First Chapter.

But, in regard that is the general Descrip­tion of all the Lines that can conveniently be put on, and those necessary for this use being far less, I shall repeat the Description again, as far as concerns the use thereof for Sea-Observations.

1. First for length, it ought to be two foot-long at least, when shut together, and not above 3 foot at any time for Sea-uses; (but for Land-uses it may be 6, 8, 10, or 12 foot in length, to find Altitudes or di­stances to Seconds of a degree certainly).

[Page 228]2. The Form of it is the same, as before, viz. an opening Joynt of about an inch and quarter, or half quarter broad each Leg; and 6 tenth parts of an inch in thickness, with a Loose-piece of the same length, breadth, and thickness, to make it an Equi­lateral-Triangle. As the Figure sheweth.

3. The Lines necessary for Sea-uses are, first, the 180 degrees upon the moving-Leg and Loose-piece, numbred as before is shew­ed. Also, 60 degrees on the innermost-edge of the Loose-piece. The Kalendar of Months and Dayes, and degrees of the Suns Place, and Right Ascention, on the move­able-Leg.

For the speedy and ready finding the Suns place, and declination, which you may do to a minut at all times, by help of the Recti­fying Table, and Astronomical Cautions of Time and Longitude.

Also, on the Head-leg, is the general Scale of Sines and Lines, to the great and lesser Radius, as in the Figure. And thus much will serve both for Observation and Operation, as in the following Discourse will fully appear.

4. To this Instrument doth chiefly be­long the Sights for the Observations at Sea, [Page 229] where the Horizon is made use of in the ta­king the Sun or Stars Altitude.

And to this Instrument belongs the In­dex and Square, that makes it a most com­pleat Sinical-Quadrant, for the plain and easie resolving of all plain Triangles.

Also, a weighty Plummet and Thred, and a pair of large Wood or Brass Com­passes for Operation.

Thus much for Description, being all put on one side only, unless you shall be pleased to add the Artificial Numbers, Sines, and Tangents on the outer-edge, and a Meri­dian-line, and his Scale on the inner-edge; and Natural Sines, and Natural Versed-Sines on the Sector-side: But these as you please.

CHAP. II. The use of the Trianguler-Quadrant in Observation.

THat the Discourse may be plain, and brief, and general; there are 10 terms to be named and described, before I come to the Vses and Examples, which are as fol­loweth.

1. First, the Head-leg of the Instrument in which the Brass-Rivit is fixed, and about which the other Leg turns, as AB, in the Figure; on which Leg, the general Scale of Sines and Lines are usually set.

2. The moveable-Leg, on which the Months and Dayes be, as in the Figure, no­ted by BD; which Leg turns about the Head-Leg.

3. The Loose-piece that is joyned to the Head, and moving-Leg, by two Tennons at each end thereof, noted by DA in the Fi­gure.

[Page 231]4. The Head-Center, or Center-pin on the round-part of the Head-leg, being Cen­ter to the 60 degrees on the in-side of the Loose-piece; which Point is known by B, in the Figure.

5. The Leg-Center, being near the end of the Head-leg, which is the Center to the degrees on the moving-Leg, and out-side of the Loose-piece, being in all 180 degrees; and noted in the Figure by the Letter C.

6. The great Radius, or greater Line of Sines, issuing from the Leg-Center toward the Head, having the Tangents on the move­able-Leg to the same Radius; and the mea­sure from the Leg-Center to the Tangent on the moving-Leg, a Secant to the same Radius; as CE in the Figure.

7. The little Radius that issues from the Leg-Center toward the end, having the Tangents, on the out-side of the Loose-piece to the same Radius, and the measure from the Center to those Tangents for Secants to the same Radius; as CF.

8. The Turning Sight alwayes to be skrewed to the Head, or Leg-Center, known by his shape and skrew-hole, as

9. The sliding Horizon-sight to slide on the moving-Leg and Loose-piece, noted with its bigness and hole to look through, as

[Page 232]10. The shadow Sight, and 2 others, to pin the Instrument together, which you may call the Object-Sights, alwayes fixed in the two holes at the ends of the moving-Leg, and the Head-leg; and the shadow-Sight is to set to and fro to any place required; noted in the Figure with [...] and the other two with [...] And

Thus you have their Name and Descrip­tion at large, which in brief take thus for easie remembring.

1. The Head-Leg. 2. The Moveable-Leg. 3. The Loose-Piece. 4. The Head-Center. 5. The Leg-Center. 6. The great Radius. 7. The less Radius. 8. The turn­ing-Sight. 9. The Horizon sliding-Sight. 10. The shadow-Sight, and the two Ob­jest-Sights; the open-part in one is next to, and the other remoter from the Rule, to an­swer to the upper or lower-hole in the turn­ing-Sight, according as you please to use them in Observation.

Thus much for the Terms, the Vses fol­low.

CHAP. III. To Rectifie the Table of the Suns Declination.

THus much as for the way of Observa­tion; now, that your Operation may be true also, it is necessary that you have a Table of the Suns Declination, for the first, second, and third year, after the Leap-year.

But in consideration, that the second after the Leap-year, is a mean between the other three; I have made a Table for that, and the Months on the Trianguler-Quadrant are agreeable thereunto; and for the first, third, and Leap-year, have added a Recti­fying Table to bring it to a minut at least to the real truth, wherein I have followed the Suns place, according to Mr. Streets Table of the Suns place, for 1666.

[Page 255]In which Table, you have degrees and minuts; and a prick after, notes a quarter of a minut; and two pricks, half a minut; and three pricks, three quarters of a minut more.

Now, by the Rule, you may count to a minut, and the Rectifying Table tells you how many minuts more you must add to, or substract from the degrees and minuts the Table or Rule shall shew it is, in the second year.

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