Horologiographia, or, The art of dyalling being the second book of the use of the trianguler-quadrant : shewing the natural, artificial, and instrumental way, of making of sun-dials, on any flat superficies, with plain and easie directions, to discover their nature and affections, by the horizontal projection : with the way of drawing the usual ornaments on any plain : also, a familiar easie way to draw those lines on the ceiling of a room, by the trianguler quadrant : also, the use of the same instrument in navigation, both for observation, and operation : performing the use of several sea-instruments still in use / by John Brown, philomath.

Horologiographia, or, The art of dyalling being the second book of the use of the trianguler-quadrant : shewing the natural, artificial, and instrumental way, of making of sun-dials, on any flat superficies, with plain and easie directions, to discover their nature and affections, by the horizontal projection : with the way of drawing the usual ornaments on any plain : also, a familiar easie way to draw those lines on the ceiling of a room, by the trianguler quadrant : also, the use of the same instrument in navigation, both for observation, and operation : performing the use of several sea-instruments still in use / by John Brown, philomath.

Horologiographia, or, The art of dyalling being the second book of the use of the trianguler-quadrant : shewing the natural, artificial, and instrumental way, of making of sun-dials, on any flat superficies, with plain and easie directions, to discover their nature and affections, by the horizontal projection : with the way of drawing the usual ornaments on any plain : also, a familiar easie way to draw those lines on the ceiling of a room, by the trianguler quadrant : also, the use of the same instrument in navigation, both for observation, and operation : performing the use of several sea-instruments still in use / by John Brown, philomath.

Table of contents

[1]

Title page

To the Courteous Reader.

§

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

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

1. And first for the first Equinoctial-DIAL.

2. To draw a Direct Polar Dial.

3. To draw a Direct East or West-DIAL.

4. To draw the Horizontal-Plain.

5. To draw a North or South Plain.

6. To draw the Hours on a Direct Recliner.

7. To draw a Direct East or West-Recliner.

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

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

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.

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

CHAP. VII. Of Declining and Inclining-PLAINS.

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.

1. To draw the Tropicks, or Parallels of the Suns Declination, or the length of the Day Artificial, on any Dial.

II. To find the Horizontal-line in all manner of Plains.

III. To draw the old unequal Hours.

IV. To draw the Italian or Babylonish-Hours.

V. To draw the Azimuth-Lines.

VI. To describe the Almicanters, or the Parallels of the Suns Altitude above the Horizon.

VII. To draw the Circles of Position, or Houses.

The Description and some Uses of the Sphear for Dialling, and for the better understanding of the general and particular Scheams.

Use I. The Day of the Month being given, to find the Suns true Place.

Use II. To find his Declination any day.

Use III. The day given, to find the Suns Right-Ascention.

Use IV. The Suns Declination and Latitude being given, to find the Suns Meridian Altitude.

Use V. The Suns Declination and Latitude being given, to find the Suns Rising or Setting, and Amplitude, East or West.

Use VI. The Declination & Latitude, & Suns Altitude given, to find both Hour & Azimuth.

Use VII. The Hour, or Azimuth, and the Suns Declination given, to find the Altitude.

Use VIII. To find the Suns Height in the Vertical-Circle.

Use IX. To find the Suns Altitude at 6.

Use X. To find the Hour of the Day, when the Sun shineth.

The uses of the Sphear IN DYALLING.

The Poor-Mans Dial-Sphear; Or another way to demonstrate the Mystery of Dyalling, both for Declining and Inclining Plains, in a very plain, easie way, for one 6th part of the cost of the other Brass-Sphear.

Use I. To find the Declination of a Plain by the Sun-shining.

Use II. The Declination of any Erect Decliner given, to find the Substile, and Stile, Inclination of Meridians, and every hour and quarters distance from 12, being the Perpendiculer Line on the Plain.

Use III. Any Declining North-east, or North-west-Dial being given, to find the former Requisites for those Dials.

Use IV. To find the Requisites, and to draw the Hours on a far Declining Erect-Dial, S. W. 80.

Use V. To find the Requisites, and Hour-Arks, from the Perpendiculer of a Declining Inclining Plain, with its Affections.

Use VI. To find the Requisites in a North-east Reclining-Dial, and the Hour-Lines.

Use VII. To find what are the most Hours, that the Sun can shine on any Plain, whatsoever.

Use VIII. The Declination of any Plain given, to find what Reclination will make it a Polar-Dial, and the contrary.

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

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Title page

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

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

Use I. To find the Suns, or a Stars Altitude, by a forward Observation, as by a Fore-staff.

Use II. To observe the Sun or a Stars Altitude, by a forward Observation, using the Thred and Plummet.

Use III. To find the Suns Altitude by a backward Observation, as with a Back-staff, or Davis-Quadrant.

Use IV. To find the Suns Distance from the Zenith, by the Trianguler-Quadrant.

Use V. To find the Suns Altitude when near the Zenith, or above 90 degrees above some part of the Horizon.

Use VI. To find the Latitude at Sea, by a forward Meridian Observation of the Altitude, according to Mr Gunter's Bow.

Use VII. To find the Latitude at Sea, by a backward Meridian Observation, according to the way of Mr. Gunter's Bow.

Use VIII. To find the Latitude by a Meridian Observation, by the Thred and Plummet, by the Sun or Stars.

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

CHAP. IV. The use of the Trianguler-Quadrant in the Operative part of Navigation.

Use I. To find how many Leagues, or Miles, answer to one Degree of Longitude, in any Latitude between the Equinoctial and Pole.

Use II. To find how many Leagues, or Miles, answer to Raise, or to Depress the Pole one degree on any Rumb from the Meridian.

Use III. To find how many Miles or Leagues answer to any number of degrees in any parallel of Latitude.

Use IV. To work the six Problems of Plain Sayling by Gunter's Lines on the edge, or the Trianguler-Quadrant.

Problem I. The Course, and Distance run on that Course, being given, to find the difference in Latitude, and Departure, or difference in Longitude.

Problem II. The Course and Difference of Latitude given, to find the Distance run, and Departure.

Problem III. The Course and Departure given, to find the distance run, and difference of Latitude.

Problem IV. The Distance run, and difference in Latitude given, to find the Course and Departure.

Problem V. The Distance run, and Departure given, to find the Course and Difference in Latitude.

Problem VI. The Difference of Latitude, and the Departure given, to find the Course and Distance run.

Use V. The use of the Meridian Line, and his Scale.

Problem I. Two places being propounded, one under the Equinoctial, the other in any Latitude, to find their Meridional difference in degrees and minuts, or 100 parts.

Problem II. Any two places having both Southerly or Northerly Latitude, to find the Meridional difference between them.

Problem III. When one place hath South Latitude, and the other North Latitude, to find the Meridional difference.

Problem IV· The Latitudes of two places, together with their difference in Longitude being given, to find the Rumb directing from one to the other.

Problem V. By the two Latitudes and the Rumb, to find the Distance on the Rumb.

Problem VI. By the two Latitudes, and distance between two places given, to find the Rumb.

Problem VII. Both Latitudes and the Rumb given, to find the difference of Longitude.

Problem VIII. By one Latitude, Distance and Rumb, to find the other Latitude.

Use VI. To find the distance of places on the Globe of Earth and Sea; Or, Geography by the Trianguler-Quadrant.

Problem I.

Problem II.

Problem III.

Problem IV.

Problem V.

I Way. When one place hath no Latitude, and the other North or South, with difference in Longitude also; then,

II Way. When both the places have either North or South Latitude, that is, both toward one Pole; then thus,

III Way. But when one Latitude is on one side the Equinoctial, and the other on the otherside, viz. one having North-latitude, and the other South. Then,

Use VII. To find the distance of places by the Natural Versed Sines in the way of a Sector on the Trianguler-Quadrant, being much more easie than the two former wayes.

Use VIII. Having the Latitudes and Distance of two places, to find their Difference in Longitude.

CHAP. V. Of Sayling by the Arch of a great Circle.

The Table of the Things contained in this Second Part.

Errata for the Second Part.

Advertisements.

Horologiographia, or, The art of dyalling being the second book of the use of the trianguler-quadrant : shewing the natural, artificial, and instrumental way, of making of sun-dials, on any flat superficies, with plain and easie directions, to discover their nature and affections, by the horizontal projection : with the way of drawing the usual ornaments on any plain : also, a familiar easie way to draw those lines on the ceiling of a room, by the trianguler quadrant : also, the use of the same instrument in navigation, both for observation, and operation : performing the use of several sea-instruments still in use / by John Brown, philomath.[1]

§

§

§

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Horologiographia: OR, The Art of Dyalling, BEING The Second Book of the Use of the Trianguler-Quadrant. Shewing the Natural, Artificial, and Instrumental way, of making of Sun-Dials, on any flat Superficies: With plain and easie Directions, to discover their Nature and Affections, by the Horizontal Projection. With the way of Drawing the usual Ornaments on any Plain: Also, a familiar easie way to draw those Lines on the Ceiling of a Room, by the Trianguler Quadrant. Also, the Use of the same Instrument in NAVIGATION; Both for Observation, and Operation. Performing the use of several Sea-Instruments still in use.

By Iohn Brown, Philomath.

London, Printed by Iohn Darby, for Iohn Wingfield, and are to be sold at his house in Crutched-Fryers; and by Iohn Brown at the Sphear and Sun-Dial in the Minories; and by Iohn Seller at the Hermitage-stairs in Wapping. 1671.

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To the Courteous Reader.

THou hast here presented to thy view (Courteous Reader) in this second Part, a plain discourse of Dialling, both Natural, Artificial, & Instrumental. Natural I call it, from the plain illustration thereof, by the Armilary Sphear of Brass herein described, or by the Poorman's Dial-Sphear, as I fancy to call it, being only a moving Horizontal-Dial, and a moving Plain, according to the Figure thereof in the Book annexed, whereby all the Arks, Angles, Scituations and Affections, are very plainly represented to an ordinary capacity. Artificial I call it, from the lively delineation of the Horizontal-projection, the fittest in my opinion for the making plain the mystery of Dialling. Instrumental I call it, from the applying of the Trianguler-Quadrant, to the ready resolving all the Arithmetical and Astronomical work, [Page] needful thereunto; and that to competent exactness as in the first Part, and also in this second Part is sufficiently seen, in finding the requisites and delineating the hour-lines to small parts, exactly & speedily by the natural Sines, Tangents and Secants on the Sector and Quadrant.

Also, the ready way of finding the Suns Altitude, Hour, Azimuth, Angle of the Plain, and any such business relating to Dyalling, as in the first Part is largely treated on.

Further, in this second Part you have Tables of the Suns Declination to every day of the years, 1, 2, & 3, after the Bissextile, as near as any extent. Also, a short, but plain direction, how to use the Trianguler-Quadrant, at any manner of way of Observation used at Sea; as backward or forward, as the Davis-Quadrant, and the Cross-staff is used; also, as Gunter's Bow is used both for the Sun or Stars.

Also, how it is used, as a Cinical, or Traverse Quadrant, with the manner of [Page] [Page]

A figure of the Stereographick Projection Pag [...]

[Page] working the six usual Problems of plain Sailing by the Natural and Artificial Line of Sines, Tangents, and equal Parts and Numbers; and plainly and conveniently to apply it, to the resolving that hard Question, of Sayling by the Arch of a great Circle, to shew plainly the Longitudes and Latitudes alteration the whole Course or Voyage.

The Prints of the Lines of Numbers, as you see here inserted, are in part according to Mr. Windgates, as to a single and broken line of Numbers: But the addition of the line of the Fractional parts of a pound, and the several Gage-points, were never before used as I know of; but do much ease & expedite the Operations by the Line of Numbers, Sines and Tangents. Also, these Scales of Reduction are convenient for the finding the Decimal-fraction, equal to the other Sexagenary-fraction, and are agreeable to those Tables in Mr. Windgates Book of Arithmetick, pag. 82.

Also note, that the figure of the Rule [Page] at the beginning of the Book, pasted on a Board, is the very same with that spoken of Chap. XV. Use 28. pag. 397, of the first Part, and will work all Questions wrought by the Trianguler-Quadrant, to exercise them that are out of the way to have them made, and may serve as good directions to the young Instrument-Maker, though these are made too too small a Radius to arrive at exactness.

The like may I say of the Gunters-Lines in the Figures annexed, yet as large as the Book will bear.

Thus I have given you a brief account of my present Thoughts about this matter, and somewhat more particularly in the First Part, disclaiming all boasting or vain ostentation, knowing that at the next Impression it may be amended in many places; I shall rest and remain, ready to make amends in the making of these, or any other Mathematical Instruments, at my House at the Sphear and Sun-Dial in the Great Minories,

John Browne.

February 16. 1670.

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[figure]

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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] 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]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.

[Page]

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] 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] 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]

[figure]

[Page] 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.

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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] 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] 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] 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]

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

1. And first for the first Equinoctial-DIAL.

AN Equinoctial Plain, as before is shewed, is that whose plain or flat Superficies lieth parallel to the Equinoctial, and is represented by the Line WAEE in the general Scheam, and therefore needs no other Scheam to represent it; In which Dial all the Hour-lines are equal one to the other, being just 15 degrees assunder; so that to draw the Hour-lines here, describe a Circle as the Circle 12.6.12.6. and fit the Radius in the Sine of 30 degrees (or the Chord of 60) and take out the parallel Sine of 7 degrees 30 minuts, the half of 15 degrees, [Page] and number it from 12 round about, and that shall divide the whole Circle into 24 equal parts, for the 24 hours, for the true Hour-lines on the Equinoctial-plain, and is the same in all latitudes; only in the setting of it, the Poles of it are to be set due North and South; the Horizontal-line on the Plain, lying Parallel to the East and West-points of the Horizon, and the Stile thereof, only a Wyre or sharp Edge standing perpendiculerly on the Center; which being so set, must point directly to the North (and South) Poles of the World. The reclining Dials-Stile pointing to the North-Pole, and the inclining Dials-Stile pointing to the South Pole; then is the Dial truly placed.

To set a Plain (or to try whether a Plain be set) Polar, or Parallel to the Equinoctial, do thus;

Set the Trianguler-Quadrant together, (by putting the Tennons of the Loose-piece into the Mortise holes) and on the Center hang a Thred and Plummet, and apply the Moveable-leg to the Meridian-line on the Plain (which on all direct Plains is the same with the Perpendiculer as here) with the Head-end uppermost; then if the Thred [Page] [Page]

[figure]

[Page] falls right on the complement of the latitude, the Plain lies parallel to the Equinoctial, or else not.

But to try the Inclining Plain, apply the Loose-piece to the Plain with the Head-end downwards; or else apply the Head-leg to the Plain with the Head-end downwards, and the Thred shall cut on 38-30 in London latitude, if the Plain be set parallel to the Equinoctial.

2. To draw a Direct Polar Dial.

The next Dial, shall be a Direct Polar-Dial, which is represented in the general Scheam by the Hour-line of 6, viz. the Line EPW; And here also the Horizontal-line on the Plain, is parallel to the East & West-points of the Horizon; and the Pole (or Point opposite to the Plain) is in the Equinoctial-point.

The Hour-lines, in this Plain, are all parallels, because the Axis, or Stile-line, in all Plains, is parallel to the Poles of the World; and this Plain it self, being so parallel, the Stile or Axis therein makes no Angle; therefore the Hour-lines must needs be parallels also.

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And the way of drawing those Hour-Lines, is thus;

First draw the Perpendiculer-line on the Plain, which is done thus by the Trianguler-Quadrant; Hang a Plummet and Thred on the Center, and apply the Moveable-leg to the Plain, to and fro, till the Thred falls neatly on 60|0, and draw that Line along by the Moveable-leg, which shall be a true Horizontal-line on any reclining Plain; and a Perpendiculer-line thereunto, is the perpendiculer Line on the Plain.

Or else;

When the Sun shineth (the Sun begins in the Pole of the Plain) hold up a Thred and Plummet, till the shadow of the Thred fall on the Plain, making two Points in that shadow at the remotest distance asunder; then a Line drawn through those two points shall be a true Perpendiculer-line, (this shall need no more Repetition).

This is general for Upright Plains, but for Recliners, the Sun must be in the Pole of the Plain; then the shadow of the Plumb-line is the Perpendiculer. This Perpendiculer-line in these Plains, ought to be just in the midst thereof, being the Hour-line for 12; draw also, at any convenient distance, two Horizontal-lines, [Page] [Page]

[figure]

[Page] Perpendiculer to the hour 12; then consider what Hours shall be the first and last hour, which in no latitude can well be more than 10 hours, viz. from 5 in the morning till 7 at night, viz. 5 hours on a side; then take the measure on the Plain, from 12 in the Horizontal-line, to the place that you intend for 7 or 5, and make it a = Tangent of 75▪ then is the Sector set to a fit Radius, to fit and fill the Plain with the hours you resolve [...]o put on.

Then,

The several = Tangents of 60-45, 30 & 15, laid both wayes from 12 on both the Horizontal-lines, shall give you Po [...]n [...]s whereby to draw all the Hour-lines in their true places.

Also,

The = Tangent of 45, shall be the true breadth of the Plate that must be a S [...]e to this Dial; or the length of an upright Wyre set any where in the Line 12.

Note,

That for the hours under 45, you may take = 45 from the small Tangents, and make it a = Tangent of 45 in the great Tangents; and then take o [...] = Tangent of 30 & 30, for 2 & 10; an [...] the = Tangent of 15 for 11 & 1; and if you want them above 45, then take the = Tangent of [Page] 60 & 60 from the small Tangents, and turn that Extent 4 times from 12 both wayes, on both the Horizontal-lines, and those shall be the Points for 8 in the forenoon, and 4 afternoon.

And lastly,

The = Tangent of 75, taken and turned 4 times from 12 to 7 in the morning, and to 5 in the afternoon, will fit and fill a Plain of 4 foot in breadth, with a Sector of one foot, shut.

3. To draw a Direct East or West-DIAL.

The next Dial, in the third place, is the Direct East or West-Dial, which is represented in the general Scheam by the Line NZS, whose Poles are in the Line EZW, whose Plain also is = to the Pole, & drawn in the same manner as the Polar Dial was; yet with this difference, the Equinoctial-line, whereon to prick the hours, is not the Horizontal-line, but is thus found.

First, by the 3d of the 5th Chapter, or 2d of the 18th Chapter, Draw a Horizontal-line on the upper part of the Plain, as AB; and divide that Line into 3 parts, counting one third part from A to C; then on C, as a Center, draw a Semi-circle as large as you [Page] [Page]

[figure]

[Page] can, as AEFD, which distance of the Compasses, make a = Sine of 30 degrees; and take out the Sine of half the latitude, and lay it from the South-corner of the semi-Circle at A, to E; and draw the Line CE for 6 a Clock hour-line: Then lay the = Sine of half the complement of the latitude from the North-end of the Semi-circle at D to F, and draw the Line CF for the Equinoctial-line, and another = to it. (Or, you might draw these Lines by the Trian [...]uler-Quadrant, by applying the flat-side to [...]he Wall, and making the Thred to play on [...]8-30, and 51-30, counting from 60|0 on [...]e Loose-piece and Moveable-leg, as by aplying it, you will see better than many [...]ords can dictate). Then having the Equi [...]octial-lines, and the Hour-line of 6, ap [...]oint also the place where 1 or 11 shall be, [...]en make that distance, viz. C 1, in the [...]est-dial, or C 11, in the East-dial, a = [...]angent of 75 degrees; and the Sector so [...]t, lay off the = Tangents of 75, 60, 45▪ [...]d if you will not move the Sector, take = Tangent out of the great Line that is [...]ght against 30 & 15 in the little Line of [...]angents, or see, by the = Line of Lines, [...]hat the great Tangent of 30 & 15 is, and [...]ke one 4th part.

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Thus;

You may take the Tangents under 45, when the Sector is set to the small Tangent, and by turning 4 times, you have the remainder of the great Tangent above 45; when the Sector is set to the great Line of 45, as in the Polar Dial.

Or else;

Alter the Sector to the Radius of 45 in the great Tangent that goes but to 45, and take out the = Tangents of 30, and lay it from 6 both wayes, for 4 & 8; and the = Tangent of 15, and lay it from 6, both wayes, for 7 & 5.

And lastly,

By all those Points, draw Lines = to 6, for the Hour-lines required; and number the East-dial with the morning-hours, and the West with the afternoon-hours, the Stile is to be a Plate, or an upright Point; the top of whose edge, or point, is to be equal to the Tangent of 45, as the Sector stood to prick down the Hour-lines.

4. To draw the Horizontal-Plain.

The fourth Plain next, in a natural order of easiness to apprehend as I judge, is the Horizontal Dial, that lies with its plain = [Page] to the Horizon; and the Zenith is the Pole thereof, represented by the primitive Circle S.N.E.W. in the general Scheam, wherein only the Hour, Arks, and Stile is required.

The Stiles Elevation is alwayes equal to the Latitude, and therefore given; the Substile is alwayes in the Hour 12, being the Meridian-line.

The Hour-lines are found by this general Canon;

As the Sine of 90, the right Angle PN 1,See the General-Scheam. to the Sine of PN, a side alwayes equal to the Latitude or Stiles elevation 51-30;

So is the Tangent of the Angle NP 1, 15; or NP 2 30, &c. the Angles at the Pole, to the Tangent of N 1 a side, or N 2 a second side, the several Hour-arks on the Plain required; found by the Artificial Sines and Tangents, as fast as one can write them down.

Thus;

The Extent of the Compasses, from the Sine of 90, the Sine of the latitude 51-30, being laid the same way from the Tangent of 15, shall reach to the Tangent of 11-50; and if you turn the Compasses the other way [Page] from the Tangent of 15, it shall give the Tangent of 71-6, for the hour of 5 as well as 11; which Numbers being gathered into a Table, and laid off by Chords or Sines in a Semi-circle, shall be the true Hour-points to draw the Lines by.

But I shall not insist further thereon, but shew how to draw it more readily, and as truly by the Sector, thus;

First, draw a streight Line (in the Meridian, if the Plain be fixed) for 12, as the Line AB; then design a Point in that Line to serve for a Center, as at C; then on the Center C, erect a Perpendiculer-line to AB, and draw it through the Center C, for the two 6 a clock Hour-lines, as the Line DE; then draw two Lines equally distant from, and = to the first Line AB, on either side, as large as the Plain will give leave, as DF and EG; (which may commonly serve for margents to put the figure in).

Then,

Take the distance CD, and make it a = Secant of 00, and take out the = Secant of the complement of the latitude, and lay it from D to F, and from E to G, on the two Parallel-Lines, and draw the Line FG.

Then lastly,

Fit DF, or EG, as a = Tangent of 45, and take out the = Tangents of every 3 deg. [Page] [Page]

[figure]

[Page] and 45 minuts, counting from 45, and closing the Compassing, and lay them both wayes, from both the sixes, on both the = Lines, for all the morning hours from 3 to 9; and for all the afternoon hours and quarters from 3 to 9; then take out BE, or BG, and make this also a = Tangent of 45, and take out the former = Tangents, and lay them both wayes from B, on the Line FG, for the mid-day-hours and quarters, from 9 to 3 afternoon; and by those Points draw for the hours and quarters required.

For pricking down the Stile, Note, That the = Tangent of 38-30, the complement of the latitude, as the Sector stands for the Noon hours, laid from D to H, gives a Point to draw it truly by; or the Sine of 51-30 the latitude, laid from B at nearest distance about H, as the Sector stood for the morning-hours will do as well. The Stile is to be a Plate, or a bended Wyre, cut or bended according to the Angle HCB, and erected Perpendicularly on the Line 12, so long, as the Sun being 62 degrees high may cause the shadow thereof to reach the hour of 12; and then set duly North & South, and Horizontal, the shadow will shew the true hour of the day. Note the Figure.

[Page] Note also, That a Horizontal Dial drawn for any one latitude, may serve for any other latitude North or South, elevating or depressing the Stile, till it look to the Polepoint; that is, by making it to recline Northward, or Southward, as much as the difference of the latitudes, viz. that the Dial was made for, and that wherein it is to be used, shall be.

5. To draw a North or South Plain.

The next Plain to this, and most like it, is the Direct North and South Dial; whose Plain lies = to the prime Virtical, or Circle of East and West, and its Poles in the South and North part of the Horizon, and represented by the Line EZW, in the general Scheam, whose Stile is alwayes equal to the complement of the Latitude, as the Horizontals was equal to the Latitude, and consequently given.

The Hour Arks on the Plain, are found by the former Canon, viz.

As the Sine of 90, viz. the Angle PZE, is to the Sine of the Side PZ, the Colatitude or Stiles Elevation;

So is the Tangent of the Angles at the Pole AEP 1, AEP 2, &c. to the Tangents [Page] [Page]

[figure]

[Page] of the Sides Z 1, Z 2, Z 3, &c.

The Arks on the Plain, found as before by Artificial Sines and Tangents; and being drawn into a Table, to be laid off by Chords, or Sines, or by the Sector,

Thus;

Draw a Perpendiculer-line for the Substile, or 12 a clock Line; and in that Line design a Point for a Center, as the Point A in the Line AB; through which Point A, draw another Line, crossing the former at Right Angles, for a Horizontal-line, and the two sixes, as you did in the Horizontal; then, on each side, and equi-distant from 12, make two Lines = to AB, as marginal-lines, as CF and DE; The distance AD, of the Parallel make a = Secant of 00, and take out the = Secant of 51-30, the latitude of the place, and lay it from C to F, and from D to E, and draw the Line FE; then make DF a Tangent of 45, and lay off the hours and quarters as you did in the Horizontal in all respects.

Also,

Make BF a = Tangent of 45, and lay off the = Tangents of every hour and quarters (if you please) from B, both wayes, toward E and F; and by those Points draw Lines for the hours required.

[Page]The Angle of the Stile may be laid off by Sines, Tangents, or Chords, as before is shewed, to the quantity of the complement of the Latitude, and may be a Plate or Wyre, as you please, as the Angle GAB.

The North Dial is the same with the South, for manner of making, only the Noon-hours are neglected, and the Morning and Evening-hours, both before and after 6 on each side only inserted; and the Center of the Dial for that cause appointed in the middle of the Plain, and not on the upper-part, as in the South, and the Stile-points upwards (as in the South it points downwards). Note the Figures.

6. To draw the Hours on a Direct Recliner.

The next Plain to be considered, being also Direct, but not Erect, or Upright, but Leaning from you; and may be either a North or South Recliner;

That is thus;

As the Poles, of a Direct South Plain, are in the meeting-point of the Meridian and Horizon, viz. the Point S. in the general Scheam; and the Point N, in the same Scheam, is the Pole of the North Plain; and the Point Z, is the Pole of the Horizontal Plain.

[Page]So the Pole of these Plains is a Point in the Meridian, elevated as many degrees above the Horizon, as the Plain shall recline from the Zenith, or upright toward the Horizon.

As thus;

Suppose the Hour-circle of 6 in the general Scheam, to represent a Reclining Plain, the Point AE, in the Meridian, is the Pole of it, being as many degrees above the Horizon S, as P is below the Zenith Z.

So also is P the Pole to the North Reclining Plain WAEE; for the Point P, is as much above the Horizon N, as AE is below the Zenith Z.

Thus you see what the Pole of a Plain means, viz. a Point 90 degrees every way from it.

Now therefore North Direct Recliners have their Poles any where between Z & N, and South Direct Recliners have their Poles any where between Z & S, according to the degrees and minuts of their Reclination.

This being premised, for drawing the Hour-lines, observe, That for South Recliners Direct, the difference between the Reclination and the complement of your Latitude, is alwayes the Stiles height for that reclining Plain.

[Page] But note, That when the Reclination is more than the complement of the Latitude, that then the contrary Pole is elevated, viz. the North Pole on South Recliners.

But for North Recliners, the sum of the Co-latitude, and the Plains reclination is alwayes the Stiles elevation; but note, when the Sum is above 90, then the complement to 180 is the Stiles elevation; but it must be turned the other way, viz. contrary to the nature of a North Recliner, for the Stile will look downwards in the North Recliner, and upwards in his opposite South Incliner.

Note also by the way, That when the South Recliner reclines equal to the complement of the Latitude, it is called a Direct Polar Dial, or rather an Equinoctial in respect of his Poles, (but I mind not to be singular).

And when the Reclination of a North Recliner, is equal to the Latitude, then the Stiles height is just 90 degrees; and the Plain, called an Equinoctial-plain, or rather Polar, in respect of his Poles, (being first in order treated on).

Thus understanding, and right conceiving what the Plains are, the drawing of any of them is the same with the North and South; for the Stiles height is alwayes to be [Page] counted the complement of the Latitude, and by consequence you have the Latitude.

As thus for Example. Comparing the reading, and Figure VII.

Suppose a Plain recline from the Zenith toward the North part of the Horizon 10 degrees, his Pole is 10 degrees above the Horizon; and then 10 taken from 38-30, there remains 28-30, for the Stiles Elevation; or the Latitude to draw it as a Horizontal Dial; and 61-30 for the Latitude for which place you are to draw a Direct Erect South Dial.

Again,

Suppose a South Recliner, recline 50 degrees, being more than 38-30, the Co-latitude; then take the Co-latitude from thence, and there remains 11-30 for the Latitude or Stiles height, to draw a Horizontal Dial by; and 78-30 for a Latitude to draw a South Dial by; but the Cock must look up to the North, therefore must be turned the other way.

Again,

For a North Recliner, reclining 60 degrees; 60 & 38-30, added, makes 98-30, whose complement to 180, is 81-30, the [Page] Stiles height; but the contrary way, as you may well perceive by the Horizontal.

As once more thus;

Suppose a North Plain recline 85 degrees; that, and 38-30, added, makes 123-30, whose complement to 180, is 56-30, the Stiles height: but put the con [...]ary way, as a South Incliner, being almost a Horizontal Dial; so that to draw this Dial, let 56-30 be the Stiles height, or Colatitude▪ then, 33-30, is the Latitude to draw a Direct South Dial by.

Or,

You may count the Stiles height the Latitude, and then draw it as a Horizontal-Dial, by taking out the Secant of the Colatitude, and the work will be the same; As in the Figure North Reclining 60 degrees.

7. To draw a Direct East or West-Recliner.

This Plain being a Direct Plain also, and taking no cognizance of Declination, (otherwise should have come after); is only an East or a West Plain, reclining or falling from you; or inclining, or falling to [Page] [Page]

[figure]

[Page] you; and the Poles of these Plains are in the prime Vertical, or Circle of East and West, as the other were in the Meridian; and this Plain is represented by the pricked Circle NFS, reclining 45 degrees, whose Pole is at Q 45 degrees above W; for the drawing of which, it will not be amiss, but very convenient to draw a particular Scheam by the last Rule in the 17 Chapter.

As thus;

With 45 degrees of the small Tangents, draw the Circle N.E.S.W. crossing it in the Center with the Lines WE, SN; then lay off the half Tangent of 38-30, from Z to P; and the half Tangent of the Reclination 45, from Z to E, and from Z to Q, and draw the Circles NES, and FPQ; in which Scheam, PF represents the Stile, FG the distance of the Substile from the Meridian, and GPF the Angle between the two Meridians, viz. ZPN of the place, QPF of the Plain. All which requisites are thus found out by the Artificial, or Natural Sines and Tangents.

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1. And first for the Stiles Elevation.

As the Sine of 90 NZ, to the Sine of ZE, the Reclination 45;

So is the Sine of NP the Latitude 51-32, to the Sine of PF, the Stiles height, 33-37.

To work this by the Trianguler Quadrant, or Sector, do thus;

As — sine of NP, the Latitude 51-32, to = sine 90 ZE;

So is = sine of ZE, the Reclination 45, to — sine of PF 33-17, the Stiles height.

2. For the Distance of the Substile from 12, thus, by Artificial Sines and Tangents.

As the Tangent of the Reclination, ZE 45, to sine of EN 90;

So is the Tangent of the Stiles elevation, PF 33-37, to the sine of FG 41-40, the distance of the Substile from 12.

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Or, by Natural Sines and Tangents, thus;

As — sine 90, to = Tangent of the Reclination 45;

So is = Tangent of the Stiles height 33-37, to — sine of the Substile from 12, 41-40. For,

If you only take the Tangent of 33-37, from the Moveable-leg, and measure it on the Sines from the Center, it shall reach to the sine of 41-40, the Substiles distance from 12.

3. For the Inclination of Meridians, thus;

As the sine of the Latitude NP 51-32, to the sine of PFG 90;

So is the sine of the Substile from 12, GF 41-40, to the sine of GPF 58-7, the Angle between the two Meridians.

By Natural Lines thus;

As — sine of the Substile 41-40 GF, to = sine of the Latitude 51-32 NP;

So is = sine of 90 PFG, to — sine of GPF 58-7, the Incliner.

[Page]If this Rule fails, use a less Radius, or a = Answer, as is largely shewed before.

Thus having found the Requisites, proceed to draw the Dial thus;

First consider the Scheam, where you shall find the Reclining Plain to be represented by the Line SEN, as the upper-edge thereof; SZN is the Horizontal-line and Meridian-line also; N or G is the place where the Meridian cuts the Plain, being in the Horizon; Therefore here the Hour-line of 12 is a Horizontal-line, and the Sun being in the South part of the Meridian, doth cast his shadow Northwards; and being in the East, casts his shadow Westwards: Therefore laying the Scheam before you, as the Plain reclines from you, you shall see that the Meridian must lie to the right-hand from Z, toward N; and the Substile upwards from N, towards E, at F. And the the Stile must look upwards, as the Angle GPF doth plainly shew; and the morning-hours are chiefly fit for the Plain, because the Sun rising Eastward, is opposite to the Plain. Thus the Affections and Scituations of the Cardinal-lines are naturally and demonstratively shewed, the Delineation followes.

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[figure]

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First,

Draw the Horizontal-line SN; and on Z, [...]s a Center, describe a Semi-cirle as SEN, [...]nd from N toward E lay off 41-40, the distance of the Substile from 12, and draw [...]he Line ZF for the Substile; also beyond [...]hat, from F to E, prick off 33-17, and [...]raw that Line for the Stile-line.

Then for drawing the Hour-Iines, you [...]ust first make the Table of Equinoctial- [...]istances, or Angles at the Pole, thus;

First, in all Direct Plains, it is orderly [...]us; 3-45, for the first quarter of an hour [...]om 12; 7-30, for half an hour; 11-15, [...]or three quarters; and 15 degrees for an [...]our; and so successively to 90: So also [...]ill it be in all Plains, whose inclination of Meridians is just 15, 30, 45, 60, 75, or [...]0 degrees, being even whole hours; and [...]ear as well, when it falls on an even quar [...]er of an hour also. But when it doth not [...] here, then the best Rule or Method I [...]now is thus;

First, set down 12, 11, 10, 9, 8, 7, 6, [...], 4, 3, 2, 1, & 12, as in the Table follow [...]g.

Then right against 12, set down the In [...]ination of Meridians; then substract 15 [Page] degrees for every hour; and 3-45 for every quarter, as often as you can, setting down the remainder; then draw a Line a-cross, and what the last number remaining wants of 15, or 3-45 (for hours, or hours and quarters) set down on the other side below the Line, as you see in the Table following; and so proceed, adding of 3-45 to that sum, for every quarter; or, 15 degrees for every whole hour, till you come to 90 both wayes; so is the Table of Hour-Arks at the Pole, compleated for all Hours that can come on this Dial, or on any other.

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The Table.

| 91 | 52 | 8 | 01 | 53
2 | 88 | 07 |  | 05 | 38
| 84 | 22 |  | 09 | 23
| 80 | 37 |  | 13 | 08
| 76 | 52 | 7 | 16 | 53
1 | 73 | 07 |  | 20 | 38
| 69 | 22 |  | 24 | 23
| 65 | 37 |  | 28 | 08
| 61 | 52 | 6 | 31 | 53
12 | 58 | 07 |  | 35 | 38
| 54 | 22 |  | 39 | 23
| 50 | 37 |  | 43 | 08
| 46 | 52 | 5 | 46 | 53
11 | 43 | 07 |  | 50 | 38
| 39 | 22 |  | 54 | 23
| 35 | 37 |  | 58 | 08
| 31 | 52 | 4 | 61 | 53
10 | 28 | 07 |  | 65 | 38
| 24 | 22 |  | 69 | 23
| 20 | 37 |  | 73 | 08
| 16 | 52 | 3 | 76 | 53
9 | 13 | 07 |  | 80 | 38
| 09 | 22 |  | 84 | 23
| 05 | 37 |  | 88 | 08
| 01 | 52 | 2 | 91 | 53

[Page]Thus you see that the Substile falls on near a quarter past 8, or 3 hours 3 quarters and better from 12; then if you will, by the former Canon, you may find all the Hour-Arks on the Plain.

Thus;

As the sine 90, to the sine of the Stiles height 33-37;

So is the Tangent of the Hours, in the Table last made, called Arks, at the Pole 31-53 for 6, to the Tangent of the respective Hour-Arks at the Plain from the Substile 90-0 for 6.

More br [...]f thus;

As the sine of 90, to the sine of the Stiles Elevation;

So is the Tangent of the Hour from 12, to the Tangent of the Hour from the Substile.

Which being brought into a Table, may be pricked down in a Semi-circle by Sines, or a Line of Chords, from the Substile on the Plain.

But I prefer this Geometrical way before it thus;

Having drawn 12, the Substile and Stile, [Page] [...]aw also a Line at any convenient distance [...]rallel to 12, as DI; then at any conve [...]nt distance from the Center Z, draw a [...] Perpendiculer to the Substile quite [...]ough the Plain, as the Line KL.

Then,

Take the nearest distance from that meet [...] Point at F, to the Stile-line, and make [...] = Tangent of 45; then the = Tangent [...] every hour and quarter, as in the Table [...]ken from the Sector, and laid from F the [...] way, as the hours go, shall be the true [...] whereby to draw the Hour-lines re [...]red.

But in regard that this way will some [...]e be troubled with Excursions in some of [...] hours, you may help it thus; Having [...] some hours, as suppose 6 & 3; or [...] 9; or indeed any 2 hours, 3 hours di [...]ce assunder, as here 6 & 9; take the di [...]nce between 6 and 9, and lay it from the [...] to N on the Meridian, and draw the [...] 9 N = to 6 at length, beyond 12; [...], as before, make 6-9 a = Tangent of [...], and lay off every hour and quarter, as [...] the South Erect Dial, both wayes from

Also, make N 9 a = Tangent of 45, [...]nd do likewise laying the hours both wayes [...]rom N, and you shall have Points enough to draw the Dial by.

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Otherwise, make these Dials thus.

Count the complement of the Latitude where the Dial is to stand, for the Latitude; And the complement of the Reclination for a new Declination; and then draw them as Upright Decliners by the following Rules, and you shall do as well and speedily as any way. But note, That all East Recliners, are North-east Decliners; and West Recliners, are North-west Decliners; And East and West Incliners (being the under faces) are South-east, and South-west Decliners.

Also note, That if you draw your Scheam true, and large, you may from thence Geometrically find the Substile, Stile, Inclination of Meridians, and every hours distance on the Plain, by Scale and Compass, thus; As Captain Lankford hath shewed.

First, For the Stiles Elevation set off a Quadrant or quarter, as W.N. from B to I; then a Rule laid from Z to I, cuts the Plain at A; then a Rule laid from A, to P & F, cuts the Circle at O and L; The Ark OL, is the Stiles Elevation, and measured by fit Chords, gives 33-17.

Secondly, For the distance of the Substile from 12, a Rule laid from Q to F, cuts the [Page] Limb or Circle at M, the Ark MN measured on fit Chords, gives 41-40, the Substile from 12.

Thirdly, For the Inclination of Meridians, a Rule laid from P to A, in the Limb gives C; the Ark WC 58-7, is the Angle between the two Meridians.

Fourthly, To find the Hour-Arks on the Plain; a Rule laid from Q to the intersecting of every Hours Ark (in the Scheam) and the Plain as here, a Rule laid from Q to 6, cuts the Limb at R, the Ark MR 19-0 is the distance of 6 from the Substile on the Plain; and so for all others, as 12 & 6 is NR 60-40, both hours and quarters if you have them truly drawn on a large general Scheam, as Mr. Lankford hath done.

Thus much for Direct Plains, both Erect and Reclining, before I come to speak of Decliners; It will not be amiss to shew how to find the declination of a Plain, both by the Sun-shine, or without, by a Magnetical-Needle, as followeth.

As — sine of the Substile 41-40 GF, to = sine of the Latitude 51-32 NP;

So is = sine of 90 PFG, to — sine of GPF 58-7, the Incliner.

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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] 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] 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] 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.

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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.

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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] 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.

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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] 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] 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] 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;

May 1, 1669. Forenoon.

Shadow | 60 | 00 | want.
Altitude | 20 | 00 |
Azimuth | 94 | 00 | want.
Substract. | 34 | 00 | South-east.

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] 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;

Iune 13, Forenoon.

Altitude | 15 | 0 |
Shadow | 10 | 0 | Past.
Azimuth | 109 | 0 | Want.
Sum is | 119 | 0 |
| 180 | 0 |
| 061 | 0 | North-east.

[Page] 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,

Iune 13, Afternoon ▪

Altitude | 15 | 0 |
Shadow | 20 | 0 | Want.
Azimuth | 109 | 0 | Past.
Sum is | 129 | 0 |
| 180 | 0 |
| 051 | 0 | North-west.

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]

[figure]

[Page] 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.

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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]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.

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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]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.

Substile.

As — Co-tang. Lat. | 38-28 ZP,
To = sine 90 | 90-00 ZN;
So = sine Declination | 20-00 NC,
To — I Substiler | 15-12 ZH.

Stile.

As — Co-sine Lat. | 38-28 ZP,
To = sine 90 | 90-00 ZN;
So = Co-sine declin. | 70-00 NA,
To — sine Stile | 35-46 PH.

Distance between 6 & 12.

As — sine Declin. | 20-0 WA,
To = sine of 90 | 90-0 ZW;
So is = C.T. of Lat. | 51-32 NP,
To — C.T. 6 & 12 | 23-18 AQ.
Tangent 66-42. |

Inclinations of Meridians.

As — I declination | 20-0 SD,
To = sine Latitude | 51-32 ZAE;
So = sine 90 | 90-00 ZS,
To — T. Inclin. Merid. | 24-56 AEK.

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.

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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]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]

[figure]

[Page] 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.

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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] 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.

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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]

[figure]

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

Lat. | 51-32 d. mi. |
S.W. | 80-25 |
Sub. | 38-4 |
Stile | 5-56 |
[...] & 12 | 38-51 |
I. M. | 82-30 | 12
| 75 [...]00 |
| 67-30 | 1
| 60-00 |
| 52-30 | 2
| 45-00 |
| 37-30 | 3
| 30-00 |
| 22-30 | 4
| 15-00 |
| 10-30 | 5
| 00-00 | —
| 07-30 | 6
| 15-00 |
| 22-30 | 7
| 30-00 |
| 37-30 | 8

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] 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] 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] 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] 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] 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.

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By the Sector and Quadrant.

As sine 90 Radius, | ZD | 90—00
To Tang. of Declin. Plain | ND | 35—00
So sine of Reclin. Plain | ZE | 20—00
To Tang. of Perp. & 12 | EG | 13—28
Whose complement AC 76—32 is the distance from the East-end of the Horizon to 12.

As — Tangent of | ND | 35— 0
To = Sine of | ZD | 90— 0
So = Sine of | ZE | 20— 0
To — Tangent of EG 13—28, the distance of 12, from the Perpendiculer.

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.

By the Quadrant and Sector.

As the sine of the Perpendiculer from 12 | GE | 13-28
To the sine of declination | GZE | 35-00
So is the sine of 90 | GEZ | 90-0
To the sine of the distance on the Meridian, from the Plain to the Zenith — | GZ | 23-55
[Page]As — sine | 90-0 | GEP,
To = sine | 35-0 | GZE;
So = sine | 13-28 | GE,
To — sine 23-55 GZ, which taken from 38-28 gives PG 14-33.

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.

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Then thus; by the Quadrant and Sector.

As sine of the Arch of the Merid. from the Zenith to the Plain, | ZG | 23-57
To sine of the Reclination | ZE | 20 00
So is the sine of the Arch on the Meridian, from the Pole to the Plain | PG | 14-33
To sine of the Stiles height | PF | 12-13

As — sine of PG | 14-33
To = sine of ZG | 23-58
So = sine of ZE | 20-00
To — sine of PF | 12-13

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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Therefore, by the Quadrant and Sector.

As the Tang. of the Reclin. | ZE | 20-0
To the sine of the Distance on the Plain, from the Perpend. to the Merid. | GE | 13-28
So is the T. of the Stils height | PF | 12-13
To the S. of the Subst. fr. 12 | FG | 7-58

As — sine of GE | 13-28
To = Tang. of ZE | 20-0
So = Tang. of PF | 12-13
To — sine of FG | 7-58

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.

Therefore, by the Quadrant and Sector.

As the sine of the Dist. on the Merid. from the Pole to Plain | PG | 14-33
To the S. of 90, the opp. Angle | PFG | 90-00
So is the S. of the Subst. fr. 12, | FG | 07-58
To the S. of the Inclin. Merid. | FPG | 33-28

As = sine of the side | 07-58 | FG
To = sine of the side | 14-33 | PG
So = sine of the Angle | 90-00 | PFG
To — sine of the Angle | 33-28 | FPG

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;

12 | 33-28 |
1 | 18-28 |
2 | 3-28 |
|  | Sub.
3 | 11-32 |
4 | 26-32 |
5 | 41-32 |
6 | 56-32 |
7 | 71-32 |
8 | 86-32 |
9 | 78-28 |
10 | 63-28 |
11 | 48-28 |
12 | 33-28 |

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] 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] 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] 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] 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] 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]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] 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:

By the Sector;

As the sine of 90 | DA | 90-0
To Co-sine of Declin. | NA | 55-0
So Co-tang. of Reclin. | DE | 56-57
To Tang. of Latitude | NP | 51-32
[Page]As — Co-sine Declination | NA | 55-00
To = sine of | AD | 90-00
So is = Co-tang. of Reclin. DE 56-57 being taken from the small Tangents,
To — Tangent of NP 51-32 being measured from the Center on the same small Tangents.

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▪

By the Sector;

As the Co-sine of the Declin. | AN | 55-0
To the Radius or Sine of | AD | 90-0
So is the Tang. of the Lat. | PN | 51-32
To the Co-tang. of the Reclin. | DE | 56-57

As — Tangent of | NP | 51-30
To = Sine of | AN | 55-00
So = Sine of | AD | 90-00
To — Tangent of | DE | 56-57

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

By the Sector;

As the Co-tang. of the Reclin. | DE | 56-57
To the Radius, or Sine of | AD | 90-00
So is the Tang. of the Lat. | NP | 51-32
To the Co-sine of the Declin. | NA | 55-00

As — Co-tang. Reclination. | DE | 56-57
To = sine of | AD | 90-00
So — Tang. of Latitude | ND | 51-32
To = Co-sine of Declination | NA | 55-00

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]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.

As the sine of | PEZ | 90-0
To the Co-sine of the Lat. | PZ | 38-28
So the sine of the Declination | PZE | 35-00
To the sine of Substile from Perp. | PE | 20-54

As — sine of Declination | PZE | 35-0
To = sine of | PEZ | 90-0
So = sine of Co-latitude | PZ | 38-28
To — sine of Substile from Perp. | FE | 20-54

[Page]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.

As the Co-sine of the Latitude | PZ | 38-28
To the sine of | PEZ | 90-00
So the sine of the Reclin. | ZE | 33-03
To the Co-sine of Incl. Mer. | ZPE | 61-15
Whose complement ZPQ 28-45 is the Inclination of Meridians required.

As — sine of Reclination | ZE | 33-3
To = Co-sine of Latitude | PZ | 38-28
So = sine of 90 | PEZ | 90-00
To — Co-sine of Incl. Mer. | ZPE | 61-15
Whose complement QPZ 28-45 is the Inclin. of Meridians required.

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]

[figure]

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

Again,

8 | 88—45
| 81—15
9 | 73—45
| 56—15
10 | 58—45
| 51—15
11 | 43—45
| 36—15
12 | 28—45
| 21—15
1 | 13—45
| 6—15
2 | 1—15
| 8—45
3 | 16—15
| 23—45
4 | 31—15
| 38—45
5 | 46—15
| 53—45
6 | 61—15
| 68—45
7 | 76—15

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] 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]

[figure]

[Page] 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.

[Page]

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.

As the sine of | ZD | 90-00
To the Tangent of Declination | ND | 35-00
So the sine of Reclination | ZE | 60-00
To the Tang. of Perp. & Merid. | EG | 31-12
[Page]As — Tangent of Declination | ND | 35-00
To = sine of 90 | ZD | 90-00
So = sine of Reclination | ZE | 60-00
To — Tang. of Perp. & Merid. | EG | 31-12

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.

As sine of | AD | 90-0
To Co-tang. of the Reclin. | DE | 30-0
So Co-sine of the Declination | AN | 55-0
To Tang. of dist. Plain & Horiz. | NG | 25-19

As — Co-tangent Reclin. | ED | 30-0
To = sine of 90 | AD | 90-0
So = sine of Reclination | AN | 55-0
To — Tang. dist. on Mer. P. Hor. | NG | 25-19

[Page]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.

As sine dist. Merid. Horizon. | GA | 58-48
To Co-sine Declination | AN | 55-00
So sine dist. Pole to Plain | GP | 26-13
To sine Stiles Elevation | PF | 25-02

As — sine of | GP | 26-13
To = sine of | GA | 58-48
So = sine of | AN | 55-00
To — sine of | PF | 25-02

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,

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4. To find the Substile from 12.

By the Sector.

As Co-tang. of the Declin. | AN | 55-00
To S. dist. on Mer. fr. Pl. to Hor. | NG | 25-19
So Tang. of the Stiles height | PF | 25-02
To S. of the Substile from 12 | FG | 8-05

As — Co-tang. of Declin. Plain | AN | 55-00
To =S. dist. on Mer. fr. Pl. to Hor. | NG | 25-19
So — Tang. of the Stiles height | PF | 25-02
To = S. of the Substile from 12 | FG | 08-03

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.

As the sine of the distance. on Mer. from Pole to Plain | PG | 25-19
To the sine of the Angle | GFP | 90-00
So the sine of dist. of Sub. fr▪ 12 | GF | 08-03
To the sine of the Incl. of Mer. | GPF | 18-27

As — sine | GF | 08-03
To = sine | PG | 25-19
So = sine | GFP | 90-00
To — sine | GPF | 18-27

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.

12 | 18—27 | 8-3
| 10—57 |
1 | 3—27 | 1-27
|  | Subst.
| 4—03 |
2 | 11—33 | 4-58
| 18—03 |
3 | 26—33 | 11-55
| 34—03 |
4 | 41—33 | 20-35
| 48—03 |
5 | 56—33 | 32-45
| 64—03 |
6 | 71—33 | 51-45
| 78—03 |
7 | 86—33 | 81-52
| 85-57 |
8 | 78—27 | 64-10
| 70—57 |
9 | 63—27 | 40-20
| 55—57 |
10 | 48—27 | 25-36
| 40—57 |
11 | 33—27 | 15-33
| 25—57 |
12 | 18—27 | 8-3

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[figure]

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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.

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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.

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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] 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;

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

By the Secctor.

As sine 90 Radius | ZD | 90-0
To Tangent Declination Plain | SD | 55-0
So sine Reclination Plain | ZE | 20-0
To Tangent Merid. & Perpend. | GE | 26-2

As — Tangent of Declin. | SD | 55-0
To = sine of Radius | ZD | 90-0
So = sine of Reclination | ZE | 20-0
To — Tang. of 12 from Perp. | GE | 26-02

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.

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By the Sector.

As sine declin. of the Plain | GZE | 55-0
To sine dist. of Mer. & Perp. | GE | 26 [...]02
So sine of the Radius | GEZ | 90 [...]00
To sine of dist. on Merid. from Pole to Plain | GZ | 32-03

As — sine of | GEZ | 90-0
To = sine of | GZE | 55-0
So = sine of | GE | 26-2
To — sine of | GZ | 32-03

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.

As sine of distance on Mer. from Zenith to the Plain | GZ | 32-03
To sine of the Plains Reclin. | ZE | 20-00
So sine of dist. on Mer. from Pole to the Plain | GP | 70-31
To sine of the Stiles Elevat. above the Plain | PF | 37-01

As the — sine | GP | 70-31
To the = sine | GZ | 32-03
So the = sine | ZE | 20-00
To the — sine | PF | 37-01

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.

As Tangent of the Reclin. | ZE | 20-00
To sine of dist. of 12 from Perp. | GE | 26-02
So Tang. of the Stiles Elevat. | PF | 37-01
To sine of the Substile from 12 | GF | 65-24

As — sine | EG | 26-02
To = Tangent | ZE | 20-0
So = Tangent | PF | 37-01
To — sine | GE | 65-24

By the Scheam.

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

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5. To find the Inclination of Meridians FPG.

By the Sector.

As sine dist. on Merid. from Pole to Plain | GP | 70-31
To sine Radius opposite Angle | GFP | 90-00
So sine dist. on Plain from 12 to Substile | GF | 65-24
To sine of the Inclin. of Mer. | GPF | 74-38

As — sine | GF | 65-24
To = sine | GP | 70-31
So = sine | GFP | 90-00
To — sine | GPF | 74-38

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.

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To draw the Dial.

3 | 29—38
2 | 44—38
1 | 59—38
12 | 74—38
11 | 89—38
10 | 75—22
9 | 60—22
8 | 45—22
7 | 30—22
6 | 15—22
5 | 0—22
4 | 14—38

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] 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.

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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.

As the sine of 90 | AD | 90-0
To Tang. of the Reclination | DE | 54-10
So Co-sine of Declin. Plain | AS | 35-0
To Co-tang. of the Latitude | SG | 38-28

As — Tangent Reclination | DF | 54-10
To = sine 90 | AD | 90-0
So = Co-sine of Declination | AS | 35-0
To — Co-tang. of the Lat. | SG | 38-28

[Page]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;

As the Co-sine of the Declin. | AS | 35-0
To the Co-tang. of the Lat. | SG | 38-28
So is the sine of 90 | AD | 90-00
To the Co-tang. of the Reclin. | DE | 54-10

As the — Co-tang. Lat. | SG | 38 28
To the = Co-sine Declin. | AS | 35-00

So the = sine Radius | AD | 90-00
To the — Co-tang. Reclin. | DE | 54-10

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.

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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.

As Co-tang. of the Reclin. | ED | 54-10
To sine of 90 | AD | 90-00
So Co-tang. of the Latitude | SG | 38-28
To Co-sine of the Declin. | SA | 35-00

As — Co-tang. Reclin. | ED | 54-10
To = sine | AD | 90-0
So — Co-tang. Latitude | SG | 38-28
To = Co-sine Declination | SA | 35-00

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.

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By the Sector.

As sine of 90 | ZEG | 90-0
To sine of dist. on the Mer. from Z, to the Plain | GZ | 51-30
So sine of Declin. of the Plain | GZE | 55-0
To sine of dist. on the Plain from Perpend. to Merid. | GE | 39-54

As — sine | GZE | 55-0
To = sine | ZEG | 90-0
So = sine | GZ | 51-32
To — sine | GE | 39-54

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.

As sine of the Latitude | GZ | 51-32
To sine of the Reclination | ZE | 35-50
So sine of dist. Mer. Pole to Plain | GP | 90-00
To sine of the Stiles Elevation | PF | 48-24

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As — sine 90 | GP | 90-0
To = sine Latitude | GZ | 51-32
So = sine Reclination | ZE | 35-50
To — sine Stiles height | PF | 48-24

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.

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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.

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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.

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

By the Sector;

As sine 90 | ZD | 90-00
To Tang. Declin. Plain | SD | 55-00
So sine Reclin. Plain | ZE | 65-00
To Co-tang. Elevation Merid. | GE | 52-18

As — Tangent Declin. | SD | 55-0
To = sine 90 | ZD | 90-0
So = sine Reclination | ZE | 65-0
To — Co-tang. Merid. Elev. | GE | 52-18

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.

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

By the Sector.

As sine of AD Radius, or sine of | AD | 90-0
To Co-tang. of Reclin. Plain | DE | 25-0
So Co-sine of Declin. Plain | AS | 35-0
To Co-tang. dist. on Merid. from Plain to Zenith | SG | 14-58

As — Tang. of | ED | 25-0
To = sine of | AD | 90-0
So = sine of | AS | 35-0
To — Tang. of | GS | 14-58

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] 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.

As the sine dist. Mer. from Zenith and Plain | GZ | 75-02
To the sine of the Reclin. Plain | ZE | 65-00
So the sine dist. Mer. from the Pole to the Plain | PH | 66-30
To the sine of the Stiles Elev. | PF | 59-21

As — | GZ | 75-0
To = | ZE | 65-0
So — | PH | 68-0
To = | PF | 59-21

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.

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

By the Sector.

As Tang. Reclin. of the Plain | ZE | 65-00
To Co-sine dist. Mer. & Horiz. | EG | 52-18
So Tang. of the Stiles Elevat. | PF | 59 21
To sine of the Substile from North part Merid. | FH | 38-30

As sine dist. Mer. from Perp. | EG | 52-18
To Tang. of the Reclin. | ZE | 65-00
So Tang. of the Stiles height | PE | 59-21
To sine of the Substile from 12 | FH | 38-30

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.

As sine dist. Mer. fr. Pole to Plain | PH | 65-00
To sine of 90 Radius | PFH | 90-00
So sine of dist. from Subst. & 12 | FN | 38-30
To sine of Inclin. Merid. | FPN | 42-45
[Page]As — sine of | PFN | 90-00
To = sine of | PN | 65-00
So = sine of | FN | 38-30
To — sine | FPH | 42-45

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.

12 | 42-45 |
1 | 57-45 |
2 | 72-45 |
3 | 87-45 |
4 | 77-15 |
5 | 62-15 |
6 | 47-15 | 42-55
7 | 32-15 |
8 | 17-15 |
9 | 02-15 | 1-56
10 | 17-15 |
11 | 2-15 |
12 | 42-45 |

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]

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[Page] [...]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.

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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]

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[Page] 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] 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]

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[Page] 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] 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] 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] 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] 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.

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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.

1. To draw the Tropicks, or Parallels of the Suns Declination, or the length of the Day Artificial, on any Dial.

THe speediest and easiest way for this, is by the Trygon, or by the Trianguler-Quadrant, (which is a proper Trigon) the [Page] [Page]

[figure]

[Page] Axis or Stile being first set into the Dial; For, if you apply the Moveable-leg to the Axis, and extend a Thred of a competent length, fastened in the Center of the Trianguler-Quadrant, and cause it to cut the Suns place in the Line of the Suns place, or the Line of the Suns Rising, at an even, whole, or half hour, in the Hour-line for the length of the Day; or the degree of Declination in the degrees; and at the same time the Notch, or place, where you intend the Nodus or Apex shall be: Then, I say, if the Thred shall be extended to touch the Plain, and the Superficies of the Trianguler-Quadrant, and any one of the fore-mentioned Points on the Rule, and the place of the Nodus, you may find Points as many as you please on both sides of the Substile, by which Points, by the help of a thin Rule that will bend to them, you may draw these Parallels of the 12 Signs, length of the Day, or Suns Declination required; one of which Lines, viz. the Equinoctial-line, or ♈ and ♎ on the Signs; or the Line for 12 hours, the length of the Day, will be a streight Line.

But note, That if it be a Perpendiculer Stile, whose upper Point, or Apex, is to be the Nodus to give the Shadow; then you [Page] must strain a Thred very hard, or apply a Rule for the present whereon to rest the Moving-leg on, instead of the Axis; or, else you may do it thus, as Mr. Gunter sheweth.

First, to make the Trygon, if the Rule or Quadrant prove too large for your small Dial.

On a sheet of Pastboard, or Slate, draw a long streight Line, as AB; to which Line erect two Perpendiculers, one at the upper, and the other at the lower end, as CD, and EF; then make AB a Tangent of 45 degrees, (then having first made these little Tables that follow, by the Trianguler-Quadrant, which is only the Suns Declination, at his entrance into the whole Signs, or at an even half-hour of Rising); lay of both wayes from B, the Tangents of the Suns declination at ♈ ♉ ♊ ♋, as in the Table following; and draw Lines to these Points from the Center A, as in the Figure annexed; and then set the marks to them, and this is the Trigon. Figure I.

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Suns declinations for the Parallels of the length of the Day.

Hours | Declin
16-26 | 23-31
16-0 | 21-41
15-0 | 16-55
14 | 11-37
13 | 5-53
12 | 0-00
11 | 5-53
10 | 11-37
9 | 16-55
8 | 21-41
7-34 | 23-31

For the Signs of the Zodiack.

Signs. | Declin
♋ | 23-31
♌ ♊ | 20-14
♉ ♍ | 11-31
♈ ♎ | 0-00
♓ ♏ | 11-31
♒ ♐ | 20-14
♑ | 23-31

Declinations.

5-0
10-0
15-0
20-0
23-31 both ways

Then from the Center A, any way on the Line CD, at such a convenient distance as you think may fit the Plain, set off the Point G; then making GA Radius of 45 Tang. set off on AB from A, the Tang. of the Stiles Elevation to F, and draw the Line FG, as an obscure Line.

[Page]Then come to the Dial Plain, and measure from the Center to the place on the Substile-line, where you would have your remotest Line of the sign ♋ or ♑ to pass; and take this distance between your Compasses, and carry it in, above, or below the Line FG, first drawn and produced to ♋, or ♑, till you find one Point to stay in A ♋, and the other in AG, so as to draw a Line = to FG first drawn; if that doth not fit, then dele FG, and draw this = to it in its stead, to fit and fill the Plain with the Tropicks to your mind, to make them large and yet convenient.

Then note, The point G represents the Center of the Dial; AG is the length of the Stile from the Center to the Nodus; a Perpendiculer let fall from A to FG, shews the point H; GH is the measure on the Substile-line on the Plain from the Center to the Horizontal-line, HA is the Perpendiculer height of the Stile; A the Apex or top of the Stile or Nodus to give the shadow.

Then,

Draw a Line from G, = to AB, as LK; and any where between AB, draw LM = to AG; and wheresoever FG cuts LM, make a mark as at M; then make LM a = sine of 90 degrees, and the Sector so [Page] set, take out the sine complements of the Arks at the Pole for every hour, and lay them from L towards M, on the Line LM, and to all those Points, draw Lines from G, and mark them with 12, 1, 2, 3, 4, &c. as in the Table.

Or else,

Take the measure from G to F, and lay it on the Dial from the Center on the Substile, and draw that Line precisely Perpendiculer to the Substile, for the true Equinoctial-line on the Plain.

Then,

The measure from the Center of the Dial, to the crossing of every Hour-line, and the Equinoctial-line, taken and laid from G, to the Line AB, gives Points to draw the Hour-lines on the Trygon; As in the Figure.

Wherein you may note,

That if the Substile happens to fall on an even, whole, or half hour, then one Line will serve on both sides of the Substile; but if not, you must draw as many more, and set Figures to them, to avoid confusion.

Then, I say, that the several distances from G, to the crossings of those Hour-lines last drawn on the Trygon; and the Signs being laid on their correspondent Hour-lines from the Center of the Dial, shall give [Page] Points in those Hour-lines, to draw the signs of the Zodiack, with a thin Rule that will bend to those Hyperbolick Sections.

The same way serves to draw the Parallels of the length of the Day, if you lay the distunce from G, the crossings of the pricked Lines and Hours on the Trygon, and is as true as any other way by Calculation, which must afterward be performed by protraction in this manner.

Thus you have the way to proportion the Height of the Stile, to fit the Plain, and the place of the Horizontal-line in all Erect-Dials, which is alwayes Perpendiculer to 12, and drawn through that point a-cross the Plain▪ And this way of drawing the Signs, is general in all Plains whatsoever, that will admit them.

II. To find the Horizontal-line in all manner of Plains.

First, The Horizontal-plain can have none, nor many other both Reclining and Inclining, whose Reclination or Inclination is above the complement of the Suns Meridian Altitude in ♑, if the Stile have any considerable Altitude.

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[figure]

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In all other Plains, the best Mechanick way is thus;

The Dial being set in (or, as in) his place, apply the Moveing-leg to the top of the Stile & one corner of it to the Plain; and at the same time let the Thred play evenly on 60|0, and the corner at the Plain will make as many marks as you please to draw it by.

Otherwise note,

That wheresoever the Hour-line of 6, and the Equinoctial-line do meet, there is one Point: Then find at what hour and minute the Sun doth Rise or Set at, in the beginning of any other whole Sign, most remote from the first Point, and that shall be another; and so as many as you please to draw that Line by: This is general for all Plains:

To find that Point by the Trianguler-Quadrant.

Lay the Thred to the Sign given, and in the Hour-line is the hour and minut required; Thus the Sun being in ♈, riseth and setteth at 6, or 1 quarter of a minut before, or after; and in ♉ at just 5, and sets at 7; in ♊ at 9 minuts after 4, or sets 9 minuts before 8: The like for Winter signs.

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III. To draw the old unequal Hours.

The unequal, Jewish, or Planetary hours, divide the Day, be it long or short, into 12 equal Hours; for the drawing of which, in the Equinoctial the common hours gives Points. For the Tropicks do thus;

Divide the number of minuts in the longest and shortest dayes by 12; viz. divide 986, the minuts in one day in ♋ at London, by 12, the Quotient is 82 ½; and divide 454, the number of minuts in one day at London in ♑, and the Quotient shall be 37′ ¾; then if you fasten an Index, or lay a Rule to the Center, and to every 1 hour and 22′ ½ in ♋ from 12; and to every 37′ ¾ in ♑, it shall give Points to draw the Jewish or Planetary hours required, according to this Table, thus made for London, by the Line of Numbers; against 12 set 6, and the rest in order as the day proceeds, for our 12 is the 6th hour, according to the Jewes.

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To make this Table readily by the Line of Numbers.

A Table to divide the Planetary hours in ♋ and in ♑, for London, 51-32 Latitude.

H ♑ M | hou | H ♋ M
8 43 | 1 | 5 10
9 28 | 2 | 6 31
10 16 | 3 | 7 52
10 44 | 4 | 9 15
11 22 | 5 | 10 37
12 00 | 6 | 12 00
12 37 | 7 | 1 22
1 15 | 8 | 2 44
1 53 | 9 | 4 06
2 31 | 10 | 5 29
3 8 | 11 | 6 51
3 47 | 12 | 8 13

Extend the Compasses from 16-26, the length of the longest day in hours and minuts to 1, the same Extent shall reach the contrary way from 60, to 986, the Number of minuts in one day.

Or rather,

As 1 hour, to 60 minuts;

So is 16 hours 26′, to 986 minuts.

Then,

As 1, to 82 minuts ¼;

So is 2, to 164 minuts ½;

So is 3, to 246 minuts ¾.

Or you may say,

As 12, to 1; So is 986 to 82-2, the minuts in 1 hour.

[Page]Which properly is one hour 22 minuts, the length of one hour in Cancer; then the second hour, is 2 hours 44 ½; the third hour is 4 hours and 6′ ¾ from 12; and so for the rest, as in the foregoing Table for London.

But if you draw the Parallels, of the length of the day in the Dial, you shall find these hours to cross the even Hour-lines and quarters in the Parallels for 15 and 9 hours, as well as in the Equinoctial.

IV. To draw the Italian or Babylonish-Hours.

First, draw the common Hours, and the Parallels of the Signs, or rather the length of the day; Then note, that these Hour-lines meet with the common hours in the Equinoctial; only the Italians who account from Sun-setting, call our 12 in the Equinoctial 18; And the Babylonians, who reckon from the Sun-rising, call our 12 in the Equinoctial 6 hours.

Then to mark these in the Tropicks, do thus;

The Sun being in either of the Tropicks, doth rise in ♋, two hours 13 minuts before 6; and in ♑, so many hours and minuts [Page] [Page]

[figure]

[Page] after 6; therefore if you count 2 hours and 13 minuts before 12 in ♑, and 2 hours and 13 minutes after 12 in ♋, and there make

A Table to draw the Italian hours from Sun-setting in the Parallels [...]f 16, 12, & 8 Hours.

H. | 8 | 12 | 16
7 | 11 | 1 | 3
8 | 12 | 2 | 4
9 | 1 | 3 | 5
10 | 2 | 4 | 6
11 | 3 | 5 | 7
12 | 4 | 6 | 8
13 | 5 | 7 | 9
14 | 6 | 8 | 10
15 | 7 | 9 | 11
16 | 8 | 10 | 12
17 | 9 | 11 | 1
18 | 10 | 12 | 2
19 | 11 | 1 | 3
20 | 12 | 2 | 4
21 | 1 | 3 | 5
22 | 2 | 4 | 6
23 | 3 | 5 | 7
24 | 4 | 6 | 8

2 pricks, and a Rule laid to those 2 Points, shall cut 12 in the Equinoctial for the 6th hour from the Suns Rising, or the 18th from his last Setting, and so successively 13 minuts after every hour.

But, if you draw the Parallels of the length of the Day, then you shall find the 18th hour after Sun Setting, to cut the Hour-line of 10 in the Parallel of the Day, being 8 hours long, and 12 in the Parallel of 12 hours long; and the common Hour-line of 2 in the Parallel of 16 hours long, and so successively for the rest, for so many hours from the last Sun-setting: For, from 6 the last night in the Equinoctial, to 12 this noon, is 18 hours; [Page] but in ♑, from 47′ after 3 at Sun-set, to the next noon, is 20 hours and 13′, as in the Figure foregoing.

But for the Babylonish-hours, who reckon by equal hours from the Sun Rising, as before, count 2 hours and 13 minuts after 6 in ♑; and 2 hours and 13′ before 6 in ♋; and just 6 in ♈, and that shall draw the Line of the Suns rising; then count 3 hours and 13′ after 6 in ♑, and 7 in ♈, and 1 hour 13′ before 6 in ♋; and that shall be the first hour after Sun rising, and so successively till night.

But if you use the Parallel of the length of the day, the work is easier; for then 5, 7, and 09, in the Parallels of 16, 12, and 8 hours, shall be Points for the first from Sun-rising; and 6, 8, and 10, shall shew the second hour from Sun-rising, and so forwards, as in the Table following.

V. To draw the Azimuth-Lines.

For the Horizontal-Dial, on the foot of the Perpendiculer Stile, as on a Center, describe a Circle; and then divide that Circle into 32 Points of the Compass, or Rumbs, or 360 degrees, and Lines drawn from the foot of the Stile to those Points, [Page] [Page]

[figure]

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Hour from rising | Paral of 16 hours | Paral of 12 hours | Paral of 8 hours
1 | 5 | 7 | 9
2 | 6 | 8 | 10
3 | 7 | 9 | 11
4 | 8 | 10 | 12
5 | 9 | 11 | 1
6 | 10 | 12 | 2
7 | 11 | 1 | 3
8 | 12 | 2 | 4
9 | 1 | 3 | 5
10 | 2 | 4 | 6
11 | 3 | 5 | 7
12 | 4 | 6 | 8
13 | 5 | 7 | 9
14 | 6 | 8 | 10
15 | 7 | 9 | 11
16 | 8 | 10 | 12

shall be the Azimuth or Virtical Circles required, counting the Meridian, or 12 a Clock Line, as South, and the rest in order.

For all Erect-Dials, both Direct or Decliners, deal with the Declination of the Plain, as you did with the Inclination of Meridians; and at the Meridian, or 12, set the Plains declination; and then for Rumbs, take 11 deg. 15′ as often as you can; and what the last number wants of 11-15, set on the other side of the Substile, and to that add 11-15 till you have enough, as in the Table annexed for a Dial, whose declination was 35 degrees Westwards.

[Page]Then make the Perpendiculer height of the Stile Radius, or Tangent of 45, and on the Horizontal-line lay off the = Tangents of the Rumbs last made▪ in the Table, from the foot of the Stile their right way, and draw Lines through those Points, all Parallel to 12, for the Rumbs, or Virtical Circles required; on the Meridian write South, and the rest in their due order.

To draw the Azimuth or Virtical-Circles on Reclining, or Inclining-Plains.

Points. | D. M
S. E. | 80 00
S. E. by S. | 68 45
S. S. E. | 57 30
S. by E. | 46 15
South. | 35 00
S. by W. | 23 45
S. S. W. | 12 30
S. W. by S. | 1 15
Substile |
S. W. | 10 00
S.W. b. W. | 21 15
W. S. W. | 32 30
W. by S. | 43 45
West. | 55 00
W. by N. | 66 15
W. N. W. | 77 30
N. W. b. W. | 88 45
N. W. |

In all Reclining, or Inclining Plains, these Azimuths, virtical Circles, or Rumbs, do meet in a Point (called the Vertical Point) found in the Meridian, or 12 a clock Line, right over (in Incliners) or under (in Recliners) the Apex or top of the [Page] Stile, that is to give the shadow, when set in its right place, right over the Substile-line; And as far off the foot of the Stile (being a Point in the Substile, Square, or Perpendiculer to the Apex or top of the Stile) in a Vertical Line drawn through the foot of the Stile, = to the Perpendiculer Line of the Plain), as the Co-tangent of the Reclination, making the Perpendiculer height of the Stile to be Radius or Tangent of 45 degrees.

Also, The Co-tangent of the Reclination of the Plain, to the same Radius, laid from the foot of the Stile, in the same Virtical-Line, shall give the Point in the Vertical-line, to draw the Horizontal-line by; for a Rule laid to this Point, and the crossing the Equinoctial-line and hour of 6, shall draw the true Horizontal-line.

Then make the distance between this Point, and the meeting of the Equinoctial and 6, a = Tangent of the West or East Azimuth in the Table, and then the Sector is set, to lay off all the rest, by taking the = Tangents of the Numbers in the Table, and laying them from the Vertical-point in the Horizontal-line, both wayes on the Horizontal-line.

For, from hence you may note, That the Sun, being in the Equinoctial, doth rise and [Page] set near 6; and also doth rise near the East-point, and set near the West; therefore the same Point in the Dial, must be for the hour 6 in the morning; and the East Aximuth, or the hour 6 at night, and the West Azimuth, according as the Plain declines Eastwards or Westwards.

Then Right Lines drawn from the Vertical-point in the Meridian, and to all these Points in the Horizontal-line, shall be the Azimuth-lines required.

A thus for Example in the Figure annexed, being the Third sort of South-Recliners before-going; Declining 35 degrees Southwest, and Reclining 60 degrees, CH is the Substile, CG the Stile, H the Foot of the Stile, IK the Vertical-line drawn through the foot of the Stile HI the Vertical-point in the crossing of 12 (and the Vertical-line) and yet right under G the Apex (considering the Reclination) and the raising of G the Apex, Square, or Perpendiculer to H the foot of the Stile; Then, I say, a Plumb-line let fall from G, will rest in I, the Vertical-point; The Dial being set in its due place.

Then GH, the Perpendiculer height, made a = Tangent of 45; HI is the Co-tangent of the Reclination, viz. 30; and HK the Tangent o [...] the Reclination 60, being [Page] the Vertical-point in the Horizontal-line, from whence to lay the = Tangents, of the Rumbs in the Table last made, into the Horizontal-line.

Then Lines drawn from the Vertical-point I, to those Points in the Horizontal-line, shall be the Rumbs or Points of the Compass, Vertical Circles, or Azimuths required.

Otherwise,

When you have made the Tables of the Angles at the Zenith, as before, you may by this Canon make Tables of Angles at the Vertical-point, between the Vertical-line and the Rumb, to be drawn on the Plain.

As the sine of 90,

To the Co-sine of Reclination, or Inclination;

So the Tangent of the Angle at Zenith,

To the Tangent at the Vertical.

This Table being made, you may set one Point in the Vertical-point, and describe a Circle to any Radius, and therein prick off from the Vertical-line, the several Chords of the Rumbs, as in the Table you shall make by the last Canon.

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A Table, shewing at what Hour and Minute the Sun is in, in an even Azimuth, or Point of the Compass in ♑, ♈, ♋, for 51-32.

Degr. | Rumbs. | Alt ♋ | H. M. | Alt. ♈ | H. M. | Alt ♑ | H. M.
00-00 | South [...] | 62-00 | 12-00 | 38-28 | 12-00 | 15-00 | 12-0
11 15 | S. by E. | 61 39 | 11 38 | 37 58 | 11 24 | 14 17 | 11 16
22 30 | S. S. E. | 60 33 | 11 15 | 36 19 | 10 48 | 12 05 | 10 27
33 45 | S.E.b.S. | 58 31 | 10 46 | 33 29 | 10 10 | 8 17 | 9 33
45 00 | S. E. | 55 40 | 10 17 | 29 22 | 9 28 | 3 00 | 8 39
56 15 | S.E.b.E | 51 35 | 9 42 | 23 51 | 8 42 |  |
67 30 | E. S. E. | 46 06 | 9 02 | 16 56 | 7 51 |  |
78 45 | E. by S. | 39 03 | 8 15 | 8 49 | 6 57 |  |
90 00 | East. | 30 38 | 7 21 | 0 0 | 6 00 |  |
78 45 | E. by N. | 21 21 | 6 24 |  |  |  |
67 30 | E. N. E. | 11 10 | 5 19 |  |  |  |
56 15 | N.E.b E | 3 48 | 4 18 |  |  |  |
45 00 | N.E. |  |  |  |  |  |
33 45 | N.E.b.N |  |  |  |  |  |
22 30 | N. N. E |  |  |  |  |  |
11 15 | N. by E |  |  |  |  |  |
00 00 | North. |  |  |  |  |  |

Lastly, by help of this Table, being general for all Dials in the Latitude 51-32, it is done thus;

First note, The Table doth shew the hour and minut, together with the Suns Altitude in ♑, ♈, and ♋, when the Sun is just on [Page] [Page]

[figure]

[Page] an even Rumb, or Point of the Compass; so that having drawn the hours, and quarters of hours (by the Sector easie to do) through the Plain in light Lines, make pricks in ♑, ♈, and ♋, as the Table sheweth.

Then you shall see that a Rule laid to the Vertical-point, and any one of those three Points shall cut the other two, if the former Lines be true, and you estimate the minuts well.

Note, That this last Table in the Equinoctial, is thus readily made, by Sines and Tangents.

As the sine of 90, to the sine of the Latitude;

So is the Tangents of the Azimuths from the Meridian, being the first column in the Table,

To the Tangent of the Angle between the Meridian, and Azimuth Line on the Equator, which are the numbers in the 6th column, reduced into hours and minuts.

So that you see the Azimuth of 45, or Rumb of S.E. will cross the Equinoctial at [...]8 minuts past 9, as in the Table; which Table is easily made by the Trianguler-Qua [...]rant, by the Rules in Chap. XV.

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VI. To describe the Almicanters, or the Parallels of the Suns Altitude above the Horizon.

First, on the Equinoctial, these Lines shewing the Suns Altitude, cannot be expressed.

On the Horizontal Dial they are Circles, making the Perpendiculer height of the Stile Radius, or Tangent of 45; prick off on the Hour-line 12, from the foot of the Stile, the = Tangent of 10, 20, 30, 40, 50, 60, &c. Then one Point of a pair of Compasses set in the foot of the Stile, and the other opened to 10, 20, 30, &c. draw those Circles for the Parallels of the Altitude required.

For all Erect Dials, whether Direct or Declining, they are best done thus;

If the Stile be in, and right set, then the distance from the Nodus, to the crossing of the Horizontal-line, and Azimuth line, on which you would prick down the Altitudes shall be the = Tangent of 45; then the Sector so set, the = Tangents of 10, 20, 30, &c. laid from the Horizontal-line on the respective Azimuths, shall be Points to draw the Parallels of Altitude by (or by applying [Page] the Rule to the Nodus and Plain, and the Thred to the Almicanter) as afterward is plainly shewed.

But if the Stile is not in, then the Secants to the same Numbers and Radius, that pricked down the Azimuth Lines, shall be the several Radiusses to use as before; where you may note, That the Suns Meridian Altitude in the whole even Signs, will help to prove the truth of your work.

The East and West Erect Dials, are fitted with Parallels of Altitude in the same manner; for the Perpendiculer height of the Stile, is a Tangent of 45, and the = Tangents of 11-15, 22-30, 33-45, &c. laid from the foot of the Stile in the Horizontal-line, draws down-right Lines for the Azimuths; and the Secant of 11-15, 22-30, 33-45, &c. shall be the several Radiusses to prick off the = Tangents of 10, 20, 30, 40, 50, 60, (or what you will) on those Perpendiculer Azimuth Lines, for the Almicanters, or Parallels of Altitude required.

But for Declining Reclining Plains, you must first draw the Azimuth Lines, as before is shewed, and then find also the length of the Axis of the Horizon, as Mr. Gunter calls it, which is thus done;

[Page]Make the length of the Perpendiculer-Stile a = Tangent of 45, viz. GH Fig. II. then HI is the Co-tangent of the Reclination, and HK the Tangent of the Reclination; and then, as the Sector stands, the Secant of the complement of the Reclination, shall be the length of the Axis of the Horizon required, viz. GI, or by the Sines and Tangents Artificial.

As the sine of the Reclination, to the sine of 90;

So is the length of the Stile on the Line of Numbers, being taken in inches and 100 parts, to the length of the Axis in the same parts.

Which is an imaginary Diagonal Line, reaching from the Apex to the Vertical-point.

This being found, you must find the Angles between this Axis and the Horizontal-line, on every particular Azimuth; and lastly, the distance between the Vertical-point, and the Parallels of Altitude, on every particuler Azimuth last drawn.

For the doing whereof, you must work as you did before, to lay off the Signs, or the Parallels of the length of the Day, for these [Page] Almicanters, bear the same respect or proportion to the Horizon, as the Parallels of the length of the day have to the Equator, and are described in the same manner, as followeth. See Figure III.

First, draw the Line AB, and make AB a Chord of 60, and sweep the Arch of a Circle, and lay off 10, 20, 30, &c. and draw the Lines from A the Center, and mark them with 10, 20, 30, 40, 50, 60, the even 10th degrees; or, 45 for equal, 26-34 for double, 11-19 for 5 times, the length of the shadow and object, or what you please.

Then,

Draw AC Perpendiculer to AB▪ and lay off the length of the Axis of the Horizon from A to C; then make AC the Cosine of the Reclination, and as the Sector stands, take out the sine of the Reclination, and lay it from A to D, and this will be the distance from the Apex to the Horizon; Also, the sine of 90 shall reach from C to D, the distance between the Vertical-point and the Horizon; also the nearest distance from A to CD, is the Perpendiculer height of the Stile AH.

Then,

Take the distance from I, the Vertical-point [Page] point on the Plain, to the Horizon on every particular Azimuth Line, and lay them in the Trygon, or III Figure, from C to the Horizontal-line AD, produced if need be; and draw those obscure Lines, as in the Figure, and mark them with the Names of the Rumbs, to avoid confusion; then is your Trygon made ready for use.

Then,

Take the distance from C in the Trigon, to every crossing of the Azimuth-line and Almicanter, and lay it on the Plain from the Vertical Point I, on its proper Azimuth, finishing one Almicanter before you meddle with another, and the work with patience and diligence will be performed; the line [...] are to be drawn from Point to Point, with a steady hand, or a bending thin Ruler, being Conical Sections.

Note,

That when the Vertical-line of the Plain falls on an even Azimuth, then half the number of Rumbs will serve, being laid each way on both sides at once.

Or,

Having a Table of the Angles at the Zenith, the same as you made to draw the Azimuth-lines, draw a Line at any convenient distance, Parallel to AC; the further [Page] from AC, the larger and better, as DEF in the Figure; and note, where CD crosses the last Line EF, as at D; make DE a Parallel sine of 90, and lay off the sine complements of the Angles at the Zenith in the Table, from E towards D, and draw and mark the Lines, as in the Figure.

Otherwise,

The Stile being fixed, and the Dial set in its place where it must be, or at least set to the same Reclination, and Declination that it must be; then if you apply the side of the Trianguler Quadrant to the Nodus, and the corner at the end of the same edge that toucheth the middle of the Nodus to the Plain; and at the same time, the Thred and Plummet playing neatly on the Almicanter you would draw, you may find as many Points, and mark them as you please, without all the former trouble, and it may be every whit as true; if the under-side be in [...]onvenient, you may use the upper; only be sure, that the side you apply, and the Thred and Plummet play at the Angle of the Almicanter required.

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VII. To draw the Circles of Position, or Houses.

The Circles of Position, or 12 Houses, meet and cross one another in the crossing of the Meridian and Horizon; therefore the Horizon is the begining of the 1st and 7th Houses, beginning at the East, and reckoning under the Earth, by Imum Coeli, to the Descendant, or 7th House, at the West-part of the Horizon; and so to Medium Coeli, the beginning of the 10th House, to the Ascendant, or Horoscope, the beginning of the 1st House.

To draw these on the Horizontal-Dial, where they are Parallel Lines to the Hour 12, do thus;

Take the distance from the Apex to the Equinoctial-line, and make it a = Tangent of 45; then the = Tangent of 30 degrees laid both wayes on the Equinoctial, shall give Points to draw Lines by, = to 12, for the Houses required.

For all Vpright Dials and Recliners with Centers, streight Lines drawn from the meeting of the Meridian-line and Horizon to every two hours, counting from 12 on [Page] [Page]

[figure]

[Page] the Equinoctial, shall be the Houses or Circles of Position required.

For East and West Dials, take the Radius as before, viz. from the Apex to the Equinoctial-line on the Plain, which here is the Meridian (and but the length of the Stile) a Tangent of 45; then the = Tangents of 30, 60, and laid from 6 on the Equinoctial-line, gives Points to draw Lines Parallel to the Horizon, for the Houses required.

For East and West Recliners, the Perpendiculer height of the Stile made a Secant of 0; then the Secant of the Stiles Elevation, shall be Radius to prick off the = Tangents of 30, 60, on the Equinoctial-line from the foot of the Stile, whereby to draw Lines Parallel to the Horizon for the Circles of Position required.

All these Lines may most elegantly and [...]asily be drawn and expressed, on a large Ceiling, with competent exactness in this manner following.

First provide a Quadrant of Brass, or thin Wood, of about a foot Radius, or 14, 15, or 16 inches; also, a Semi-circle of Brass, of about half an inch broad, and about an inch less Radius than the Quadrant: the Semi-circle must have at each end, somewhat more than to make up 180 degrees, [Page] to nail to the Transum, or stroke of the Window, where your Glass is to lie.

Also, to one Ray of the Quadrant must be fastened two strong Wyres, to fasten the Quadrant to play after the manner of a Casement, one Point in the Ray of the Quadrant next the Center, sticking in the hole where you intend the Glass shall lie; and the other end fastened to a piece of Wood nailed on the two upright Posts of the Window, so that howsoever you turn the Quadrant, fixed on those two Points, it may be precisely Perpendiculer, the Semi-circle playing all the while through a hole in the other Ray of the Quadrant, that lies Horizontally; having a Skrew to stay the Quadrant at any Azimuth, as in Figure IV, is plainly expressed to your view.

Then having degrees on the Semi-circle, and also on the Quadrant, and having fitted the Quadrant on his Points to play precisely Perpendiculer, which the Plummet in the Quadrant will shew, by turning it round about, and put in the Semi-circle through the hole in the Horizontal Ray of the Quadrant, and nailed it so to the Stoole or Transum of the Window, by putting two little bits of Wood under the ends, that the Quadrant may play evenly and smoothly on [Page] the Semi-circle-to almost the half-round, for quite the half-round will not be necessary, or useful.

Then is the Instrument set fit for its Operation.

Then first, to find the Declination, or rather the true Meridian-line.

Turn the Quadrant till the edge be just against the Sun, and at the same instant get the Suns Azimuth; then if you count so much as the Suns Azimuth is, on the Brass Semi-circle, from the place the Quadrant stands at, the right way, a Line drawn from the Center of the Semi-circle, or Quadrant, to that place, is the true Meridian Line; which place you must carefully find by two or three tryals, and then mark it with Ink or otherwise, on the Brass Semi-circle to count from thence, in setting the Quadrant to the Suns Azimuth, at every hour and quarter in those Points you intend to draw on the Ceiling; which a crooked Rule set to 00, on the Semi-circle▪ to pass to and fro with the Quadrant, will make easie.

Then, having a Table of the Suns Altitude, and Azimuth, at every hour in that Latitude you draw the Dial for;

First, set the Quadrant to the Azimuth at [Page] the hour, counted the right way from the marked Meridian-line on the Semi-circle, and there skrew it fast; Then extend the Thred fastened in the Center of the Quadrant, till it cut the Altitude of the Sun at the same hour and Azimuth, on the degrees of the Quadrant, and extending the Thred to the Ceiling, make a mark for that Hour and Altitude; that Point at that time, gives the true place where the reflected spot will fall, at that Hour, Azimuth, and Altitude on the Ceiling of the Room.

This work repeated as many times as there be hours and quarters in the Summer, and Winter Tropicks, for about 5 hours, (and in the Equinoctial, and any where between, if you please) shall give Points enough to draw the Dial, and also the Tropicks, and Azimuths, and Altitudes also, if it were convenient to mark it; Or, to any other Altitude you mind to have at that Azimuth, all at once, or at most with two slips of the Thred; the Italian, Babylonish, or Iewish-Hours, as easily drawn by Points found in the other Lines.

Also, On the Meridian-line, you may add the day of the month, or any thing that depends on the Suns Meridian Altitude; which work being well done, and drawn [Page] with smoth Lines, and well ornamented, would be a comely & pleasant Ornament to a Ceiling, and far cheaper then some fret Ceilings are done, and more useful.

Lastly, When all is done, to put the Glass in right, the Foile being first rubbed off, to to cause it to give but one spot, let the Superficies of the Glass lie just so high as the Center of the Quadrant was, in the drawing the Lines, and put some Putty under it, and the Sun shining, make it to play right on the true Hour, Altitude, and Azimuth; or, if it be just at noon, then bend it on the Putty with your finger, till it fall just on the Meridian, and day of the Month also in the Meridian-line.

Also note, That look what Altitude the Sun hath at any time, the same will the reflected Altitude be, at the same time, if the Glass lie true, which two Observators at the same time may carfully prove.

The making of the Tables of the Suns Altitude and Azimuth, is very largely shewed in the 15th Chapter, Vse the 37th and 38th, where you have wayes both general and particular, for any one or more Latitudes.

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The Figure Explained.

A, the place on the Transum for the Glass to lie on, and in the middest thereof one Point, in the Ray of the Quadrant, is to play: IH, a piece of Wood to be nailed fast at H and I, for the other Point to play in at G: L, the hole for the Plummet to play in, being cut through the thin Quadrant: B and C, the ends of the Semi-circle, nailed on the Transum or Stoole: K, the hole in the Quadrant for the Semi-circular-Ring to pass through: FE, the Posts of the Window: D, the beginning of the degrees on the Semi-circle: AM, a Thred extended from the Center of the Quadrant to the Ceiling.

Thus you have the usual wayes of Dialing in a competent measure, plainly, and practically handled, which may be useful to many a Learner; and I hope will be as well accepted, as with free-will (though with little ability, and less leisure) readily imparted.

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A Table of the Suns Azimuth from the South, at every Hour and Quarter, 51-32.

| ♋ | ♊ ♌ | ♉ ♍ | ♈ ♎ | ♓ ♏ | ♒ ♐ | ♑
| D. M. | D. M. | D. M | D. M | D. M | D. M | D. M
12 | 00 00 | 00 00 | 00 00 | 00 00 | 00 00 | 00 00 | 00 00
| 07 10 | 05 53 | 5 37 | 4 55 | 03 54 | 03 53 | 3 52
| 14 22 | 13 24 | 11 10 | 9 40 | 07 57 | 07 40 | 7 10
| 21 27 | 19 53 | 16 44 | 14 13 | 12 11 | 11 10 | 10 37
11 1 | 27 54 | 26 00 | 22 13 | 18 52 | 16 22 | 14 45 | 14 13
| 34 14 | 32 01 | 27 32 | 23 22 | 20 15 | 18 17 | 17 38
| 40 12 | 37 40 | 32 41 | 27 41 | 24 06 | 21 53 | 20 59
| 45 39 | 43 04 | 37 32 | 32 08 | 29 44 | 26 30 | 24 25
10 2 | 50 51 | 47 57 | 42 05 | 36 24 | 31 50 | 28 52 | 27 49
| 55 31 | 52 41 | 47 04 | 40 30 | 35 35 | 32 18 | 31 00
| 61 43 | 57 11 | 50 51 | 44 26 | 39 10 | 35 39 | 34 17
| 64 11 | 61 18 | 54 33 | 48 13 | 42 36 | 38 50 | 37 34
9 3 | 68 10 | 65 16 | 58 47 | 51 56 | 46 02 | 42 04 | 40 36
| 71 52 | 69 56 | 62 33 | 55 33 | 49 25 | 45 16 | 43 40
| 75 26 | 72 37 | 66 10 | 58 59 | 52 43 | 48 21 | 46 42
| 78 48 | 76 05 | 69 38 | 62 22 | 55 55 | 51 22 | 49 40
8 4 | 82 00 | 79 20 | 72 57 | 65 40 | 59 00 | 54 22 |
| 85 08 | 82 42 | 76 12 | 68 53 | 62 08 |  |
| 88 10 | 85 38 | 79 23 | 72 06 | 65 12 |  |
| 91 09 | 87 50 | 82 28 | 75 03 | 68 12 |  |
7 5 | 94 05 | 91 34 | 85 30 | 78 06 | 71 10 |  |
| 96 51 | 94 25 | 88 27 | 81 09 |  |  |
| 99 38 | 97 16 | 91 25 | 84 06 |  |  |
| 102 25 | 100 07 | 94 22 | 87 04 |  |  |
6 6 | 105 08 | 102 54 | 97 12 | 90 00 |  |  |
| 107 52 | 104 58 | 100 02 |  |  |  |
| 110 36 | 108 26 | 103 00 |  |  |  |
| 113 18 | 111 14 | 105 54 |  |  |  |
5 7 | 116 03 | 114 03 |  |  |  |  |
| 118 50 | 116 54 |  |  |  |  |
| 121 41 | 119 47 |  |  |  |  |
| 124 29 | 122 37 |  |  |  |  |
4 8 | 127 24 |  |  |  |  |  |
| 130 28 |  |  |  |  |  |

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[figure]

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A Table of the Suns Altitude, at every Hour and Quarter in each Sign, for 51-32.

| ♋ | ♊ ♌ | ♉ ♍ | ♈ ♎ | ♓ ♏ | ♒ ♐ | ♑
| D. M | D. M | D. M | D. M | D. M | D. M | D. M
12 | 62 90 | 58 42 | 50 00 | 38 28 | 27 00 | 18 18 | 14 59
| 61 50 | 58 34 | 49 52 | 38 24 | 26 54 | 18 13 | 14 55
| 61 25 | 58 09 | 49 31 | 38 07 | 26 40 | 18 00 | 14 38
| 60 41 | 57 29 | 49 00 | 37 38 | 26 15 | 17 37 | 14 20
11 | 59 43 | 56 34 | 48 12 | 36 58 | 25 40 | 17 06 | 13 52
| 58 32 | 55 26 | 47 10 | 36 07 | 24 02 | 16 26 | 13 10
| 57 04 | 54 06 | 46 02 | 35 06 | 23 59 | 15 38 | 12 35
| 55 29 | 52 36 | 44 38 | 33 56 | 22 58 | 14 49 | 11 30
10 | 53 45 | 50 55 | 43 12 | 32 37 | 21 51 | 13 38 | 10 36
| 51 52 | 49 07 | 41 33 | 31 10 | 20 33 | 12 27 | 9 20
| 49 54 | 47 12 | 39 46 | 29 36 | 19 10 | 11 09 | 8 9
| 47 50 | 45 12 | 37 56 | 27 54 | 17 35 | 9 44 | 6 41
9 | 45 42 | 43 06 | 36 00 | 26 07 | 15 58 | 8 13 | 5 16
| 43 30 | 40 57 | 33 55 | 24 14 | 14 14 | 9 49 | 3 41
| 41 16 | 38 45 | 31 50 | 22 16 | 12 27 | 4 54 | 2 04
| 39 00 | 36 30 | 29 40 | 20 14 | 10 32 | 3 07 | 0 17
8 | 36 41 | 34 13 | 27 31 | 18 08 | 8 35 | 1 15 |
| 34 22 | 31 55 | 25 09 | 15 59 | 6 30 |  |
| 32 02 | 29 13 | 22 56 | 13 46 | 4 27 |  |
| 29 42 | 27 16 | 20 37 | 11 33 | 2 17 |  |
7 | 27 23 | 24 56 | 18 18 | 09 17 | 0 06 |  |
| 25 03 | 22 36 | 15 58 | 06 58 |  |  |
| 22 45 | 20 16 | 13 38 | 04 40 |  |  |
| 20 28 | 17 58 | 11 18 | 02 20 |  |  |
6 | 18 11 | 15 41 | 09 00 | 00 00 |  |  |
| 15 56 | 13 25 | 06 41 |  |  |  |
| 13 44 | 11 11 | 04 23 |  |  |  |
| 11 35 | 8 59 | 02 08 |  |  |  |
5 | 9 32 | 6 50 | 00 06 |  |  |  |
| 7 23 | 4 44 |  |  |  |  |
| 5 26 | 2 41 |  |  |  |  |
| 3 36 | 0 41 |  |  |  |  |
4 | 1 32 |  |  |  |  |  |

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The Description and some Uses of the Sphear for Dialling, and for the better understanding of the general and particular Scheams.

NExt the Foot and Semi-circle Frame for supporting of it, you may consider;

1. The fixed Horizon, to which the Foot is fastened with 4 skrews, numbred and divided into 360 degrees, or four 90 deg. whose count begins at the Dividees side of the Meridian-Circle.

2. The Meridian Circle, whose fore-side at the Nadir-point stands in the Center of the Foot; this is also divided into 4 90s, and begins to be numbered at the South and North part of the Horizon, upwards toward the Zenith, and downwards toward the Nadir; which Circle is alwayes fixed as the Horizon is.

3. The Equinoctial Circle, made fast at the East and West Points of the Horizon, moving up and down upon the Meridian-Circle, according to the Elevation of the [Page] Equinoctial in any Latitude; this is divided [...]kewise into four 90s, & numbred from the Meridian each wayes to the East and West Points of the Horizon.

4. On the Meridian Circle, is set 2 moveable Poles, to be elevated or depressed fit to the Latitude of any place; on the Fiducialedge of which, is fastened the Thred, representing the Axis of the World, at any Elevation of the Pole.

5. On the 2 Pole Points, is fastened the Hour Circle, which delineates or represents the motion of the Sun, or any fixed Star, moving in its supposed Diurnal motion about the Poles of the World, and may not improperly be called the moveable Meridian Circle, or Hour Circle, divided as before.

6. The Moveable Horizon, that moveth about to any Azimuth, and slideth or moveth in the fixed Horizon.

7. The Plain, fixed in 2 opposite Points to the moving Horizon; being set, either Horizontal, when it lies Parallel to the fixed Horizon; or Erect, when Perpendiculer thereunto; or set to any Reclination or Inclination, by help of the Semi-circle of Reclination, fastened to the backside of the Plain in the 2 Poles thereof.

[Page]8. You have the upper moving Semi-circle, in turning about of which, whateve [...] degree the fore-side of the Semi-circle cuts the Perpendiculer-point cuts the comple [...]ment thereof, and to be called the upper Semi-circle, or Circle alwayes Perpendicu [...]ler to the Plain.

9. There ought to be a Thred fastened in the Center of the Plain, to be extended to any Altitude or Azimuth required.

Thus much for Description, repeated again in short thus;

The Horizon; The Meridian; The Equinoctial Circles; The 2 Pole Points, and Axis; The Hour Circle, or Moveable Meridian; The Moveable Horizon; The Plain; The Semi-circle of Reclination; The upper Semi-circle, and, The Thred.

Note also, Every Circle is divided into 4 times 90, and numbred the most useful way.

Also, on the Plain is set the 12 Months, and every single Day; on which every respective day, if you extend the Thred, then in the degrees, is the Suns Right Ascention in degrees (on the innermost Circle, the same in hours and quarters) from the next Equinoctial-point, on the Line of Declination, his mean Declination; on the Line of [Page] [...]he Suns place, his mean true place, sufficiently true for any illustration in Mathematical practice.

The Uses whereof in some part follow.

1. To rectifie the Sphear to any Latitude, count the Elevation of the Pole on the Meridian Circle, from the Horizon upwards, and downwards from the North and South parts of the Horizon; and there make fast, with the help of the small skrew, the Fiducial-edge of the Poles Points, carrying the Hour Circle fixed upon them, then the Pole is rightly elevated.

2. Count the complement of the Poles Elevation on the Meridian, from the South part of the Horizon, and to it set the divided side of the Equinoctial Circle, then is that rectified also (in the Northern Hemisphere, or in the Southern, if you call the North Pole the South Pole).

3. Extend the Thred or Axis passing through the Center to the South Pole, and there make it fast, and then the Sphear is rectified for many Uses in that Latitude.

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Use I. The Day of the Month being given, to find the Suns true Place.

Lay the Thred in the Center of the Plain on the day of the Month, and in the Line of the Suns place, you have his place.

Example.

On the 5th of November, it is 23 degrees in ♐; or if the Suns place be given, look for that, and just against it, in the Months, is the day required.

Example.

The Suns place being 15 degrees ♌, I look for it in the Line of his place, and just against it I find Iuly 28 day.

Use II. To find his Declination any day.

Look for the day given, and right against it in the Line of Declination, is his due Declination required.

Example.

August the 5th; The Declination is 14 degrees 5 minuts from the next Equinoctial-point, viz. ♎.

Note, In the Northern Sines, or Summertime, [Page] the Sun hath North declination; or in Southern Sines, or Winter-months, the Sun hath South declination. Or if you have the Suns declination, find that in the Line Declination, and right against it in the Months is the day required.

Example.

21 degrees South declination, beginning from the Equinoctial towards the Winter Solstice, I find Novemb. 15. The like work had been, if the Suns place had been given, to find his declination.

Use III. The day given, to find the Suns Right-Ascention.

This is usually reckoned from ♈ to ♈, round, in 24 hours; but twice 12 is as useful, and then it is thus;

Find the day amongst the Months and Dayes, and just against it, in the time of Hours, is the Suns Right Ascention; (but note, it is not right figured for this use) counting onwards from ♈, or the 10th of March, to the 13th of Septemb. and from thence to Aries again; Likewise the degrees are to be reckoned from ♈ onwards, as the Months proceed.

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Example.

On the 12 of May, what is the Suns Right Ascention? Lay the Thred on the 12th of May, and in the Line of Hours it cuts 9-57′ counting from Aries onwards; or in degrees 59-15, counting as before. Thus, if any one of these 4 general things be given, the other may be found.

Use IV. The Suns Declination and Latitude being given, to find the Suns Meridian Altitude.

The Sphear being rectified, count the declination on the Meridian, from the Equinoctial, that way the declination is, either North or South; and where the count ends; there is the Meridian Altitude required for that day, or Declination.

Example.

Iune 11. Declination 23-30′; Count 23-30, from 38-30, the place where the Equinoctial stands, for 51-30 Latitude, and the account will end at 62 degrees, the Suns Meridian Altitude at that Declination Northwards: But, if it had been 23-30 South Declination; then count as much from the Equinoctial downwards, and the [Page] count will end at 15 degrees, for the Suns Meridian Altitude, at 23-30 South Declination.

Use V. The Suns Declination and Latitude being given, to find the Suns Rising or Setting, and Amplitude, East or West.

Count the Suns declination on the Hour-Circle towards his proper Pole, that is South-declination toward the South-Pole, and North-declination towards the North-Pole; and thereunto lay the Thred that is fastned in the Center; then bring the Hour-circle and Thred both together, till the Thred touch the Horizon; then the Thred on the Horizon shews the Amplitude, and the divided-side of the Hour-Circle, shews the Suns Rising and Setting on the Equinoctial, counting the Meridian alwayes 12, and the 2 East and West-points 6, and 15 degr. for an Hour, and every deg. 4 min.

Example.

Iune 11. Declination 23-30, the Sun Riseth at 13′ before 4, and the Amplitude is near 40 deg.

Again,

April 10. Declination 11-30, the Amplitude is 18-30 from the East to the North, [Page] and Riseth at 5, the Hour-circle cutting 15 degrees on the Equinoctial.

Use VI. The Declination & Latitude, & Suns Altitude given, to find both Hour & Azimuth.

Rectifie the Sphear, and set the Plain horizontal; that is, Level or Parallel to the Horizon; then apply the Thred to the Declination, counted the right way on the Hour-circle; then turn the Hour-circle and upper Semi-circle about, till the Thred cuts the degrees of Altitude in the upper Semi-circle, and the Hour-circle, shews the hour in the Equinoctial, and the Semi-circle cuts the Suns Azimuth in the deg. on the Horizon or Plain.

Example.

Declination 10, Latitude 51-30, and the Suns Altitude 30; the hour will be 8-27, and the Azimuth 66, from South Eastwards if in the morning, or the contrary if in the afternoon.

Use VII. The Hour, or Azimuth, and the Suns Declination given, to find the Altitude.

The Sphear rectified, as before, and the [Page] Hour being given, set the Hour-Circle to the hour on the Equinoctial; then bring the Thred to the Declination, counted on the Hour-circle; then bring the upper Semi-circle, till the fore-side do just touch the Thred, and the Thred on the Semi-circle, shall shew the Altitude required; and on the Horizon, the Azimuth at that Hour, and Altitude.

But if the Azimuth be first given, then set the upper Semi-circle thereunto, counted on the fixed Horizon; then the Thred laid to the declination, on the Hour-circle, and turned about till it touch the upper Semi-circle, there it shews the Altitude; and the Hour-circle on the Equinoctial, shews the hour.

Use VIII. To find the Suns Height in the Vertical-Circle.

Set the upper Semi-circle to the East or West-Point, or 90 degrees of Azimuth; then lay the Thred to the declination on the Hour-circle, and then bring it and the Thred together, till it just touch the upper Semi-circle, and it shall there shew the Altitude at East or West required.

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Example.

At 10 degrees Declination North, it will be East at 16 degrees of Altitude.

Use IX. To find the Suns Altitude at 6.

Set the Hour-circle to 6 on the Equinoctial, and the Thred to the Declination; then bring the Semi-circle to the Thred, and it shewes the Altitude at 6 required.

Example.

At 23-30 Declination, the Altitude 18-15 above the Horizon in North-declination; and as much under in South-declination; for, you must observe that the surest working is from the upper or divided-sides of the Rings, on every occasion to use it.

Use X. To find the Hour of the Day, when the Sun shineth.

Rectifie the Sphear, and set the Plain Parallel to the Equinoctial-circle; then set the Meridian-circle due North and South, and the shadow of the Axis shall on the Plain, shew the true hour.

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[figure]

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Or, otherwise thus;

At the true place of the Suns Declination, on the Hour-circle make a mark, or stick the point of a Pin, then turn the Sphear about, till the shadow of that mark, fall on the Center; (the Sphear standing Horizontal, as near as may be) then the Hour-circle shall, on the Equinoctial, shew the hour of the day required.

Note, A small Bead, or knot on the Thred, will do the business as well as may be.

Thus any the like Questions may be wrought for the Stars; or the manner of raising the Canon for any Spherical Triangle whatever, to work the same exactly by the Logarithms.

As thus;

Suppose I would make the Canon, or Proposition, to find the Suns height in the Vertical Circle at any declination. First, The Sphear being rectified, and the Plate set Horizontal, bring the upper Semi-circle to the East-point, and laying the Thred to the declination on the Hour-circle, bring it and the Thred together, till it just touch the upper Semi-circle.

The Rings or Circles so standing, and [Page] being great Circles of the Sphear, there is constituted a Spherical-Triangle in this form;

Wherein you have,

ZAE, 51-30 the Latitude, the Angle at the Equinoctial; and ZE 90, the upper Semi-circle; and AB 23-30, the Declination, part of the Hour-circle; to find BE, part of the upper Semi-circle: Now this being a right-angled Spherical-Triangle, and the parts which are given, being one right Angle, viz. The Angle at A, and the Side AB, the Suns Declination; and the Angle at E 51-30, to find the Side BE; Now the Sines of the sides of Spherical-Triangles are proportional to the Sines of their opposite Angles, and the contrary.

Therefore,

As the sine of the Triangle | BEA | 51-30
Is to the sine of the Side | AB | 23-30
So is the sine of the Angle | BAE | 90-00
To the sine of | BE | 30-39

And the like for any other, as by comparing the Rules in Mr. Norwood's Trigonometry, and the Circles of the Sphear together, the use and convenience thereof will evidently appear unto you▪

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Only note this plain Observation.

That the side of a right-Angled Triangle, which subtends the Right Angle, is most properly called the Hypothenusa; the other which you make or suppose Radius, the Base. The other, the Perpendiculer. Or more short, The Hypothenusa and Leggs: Therefore if the Hypothenusa and one Leg be given, the proportion is wrought by Sines alone; but if the two Legs be given, and first and second in the Question, then the Proportion is wrought by Sines and Tangents together.

As for Example.

As the sine of | ♈ ♋ | 90-00
To the sine of | ♋ AE | 23-31
So is the sine of | ♈ ♉ | 30-00
To the sine of | ♉ R | 11-31
The Suns Declination in ♉.

Again secondly,

As the sine of | ♈ AE | 90-00
To the Tangent of | AE ♋ | 23-31
So is the sine of | ♈ R | 27-54
To the Tangent of | R ♉ | 11-31
The Declination as before. |

But if the one acute Angle, and his opposite [Page] Leg or Side be given, then the Proportion is made by Sines only, as in the foregoing Example.

Again,

In Vertical Triangles that have the same acute Angle at the Base, as the Triangle P ♉ ♋, and ♈ ♉ R, being equal Angled at ♉; the sines of the Bases are proportional to the Tangents of the Perpendiculer, and the contrary.

Likewise,

The Sines of the Perpendiculers, as proportional to the Sines of the Hypothenusaes, and the contrary.

As for Example.

Thus for Perpendiculers and Bases.

As the sine of the Base | ♉ ♋ | 60-00
To the Tangent of Perpend. | ♋ P | 66-29
So the sine of the Base | ♉ R | 11-31
To the Tangent of Perpend. | ♈ R | 27-54

Or,

As the Tang. | P ♋ | the Perpend. | 66-29
To the sine | ♉ ♋ | the Base | 60-00
So the Tang. | ♈ P | Perpend. | 27-54
To the sine | R ♉ | the Base | 11-31

[Page]

Also for the Second, viz. Hypothenusaes and Perpendiculers.

As the sine of Hypothen. | P ♉ | 78-29
To the sine of Perpend. | P ♋ | 66-29
So the sine of Hypothen. | ♈ ♉ | 30-00
To the sine of Perpend. | ♈ R | 27-54

Or the contrary thus;

As sine of Perpend. | P ♋ | 66-29
To sine of Hypothen. | P ♉ | 78-29
So sine of Perpend. | ♈ R | 27-54
To sine of Hypothen. | ♈ ♉ | 30-00

This being premised, when to use Sines alone, and when to use Sines and Tangents together, you may rectifie the Scheam to your present purpose, and see there how the Triangle lies in its Natural parts, very plain and demonstratively to be apprehended.

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The uses of the Sphear IN DYALLING.

TO this purpose, you must take notice, That the Sphear is very excellent to demonstrate that Art; especially all those Dials whose Stiles have any competent Elevation.

Therefore, first to explain the terms.

The Sphear being rectified to the Latitude; Then, first the Plain, or Broad-plate, is to represent any Plain howsoever scituate, either Horizontal, or Erect Direct, or Direct Reclining or Inclining, or East and West Erect, or Reclining, or Inclining, or Erect and Declining, or South Declining, or Reclining or Inclining, less or more than to the Pole or North Declining; or Re-inclining less, to, or beyond the Equinoctial.

Of which in their Order.

1. By Horizontal I mean, when the [Page] Plain is set even with the fixed Horizon, and the Notch which the Semi-circle of Reclination passeth in just against the Meridian; then if you stretch the Axis streight, and bring the upper Semi-circle just to touch the Axis; then the Axis, on the Semi-circle, sheweth the Stiles height; and the Edge of the Semi-circle on the Plain, shews the Substile to be in the Meridian.

For all the Hour-Arks on the Plain, do thus;

Set the Hour-circle to every hour and quarter on the Equinoctial; and then if you bring the loose Thred, fastened in the Center of the Plain, along the Plain till it just touch the Hour-circle, then on the Plain it shall shew the Angle from 12, for that respective hour and quarter the Hour-circle stands at on the Equinoctial, accounting 3-45 for a quarter, and 7-30 for half an hour, and 15 deg. for every whole hour, as was hinted before.

2. For an Erect Direct South or North-Dial.

Just as the Plain stood before, that is to say, the Notches of the Moving Horizon against the Meridian; turn the fixed Semi-circle, [Page] till the divided side of the Horizon cuts no deg. on the fixed Semi-circle, then the upper-edge of the Plain respects the Zenith, and the lower the Nadir; and the two Notches in the Moveing Horizon (being alwayes the Poles of every Plain) are just in the Meridian; therefore it is a Direct Plain, and Erect, because Upright without any Reclination, as the fixed Semi-circle sheweth. Then being so fixed, and made fast there, pull the Axis streight, and bring the upper Moving Semi-circle just to touch the Thred or Axis; then on the upper and lower Semi-circles, the Axis sheweth the Stiles Elevation; and on the Plain the Semi-circle cuts the Substiles distance from 12, viz. 00, because a Direct Plain.

And for all the Hour-Arks on the Plain, set the Hour-circle to every hour, quarter, and half hour on the Equinoctial; and bring the Thred easily along the Plain, till it just touch the Hour-circle; then on the Plain it sheweth the Ark from 12 required.

Also note, The several Triangles made on the Meridian, Equinoctial, and Hour-circle, at every hour it is set unto.

As thus;

Suppose at the Pole, I set P; at the cutting of the Equinoctial, and Meridian, AE; [Page] at the upper-end, or Zenith, set Z; on the Meridian, and where the Hour-circle cuts the Equinoctial, at 1 & 11, set 15; at 2 & 10, 30; at 3 & 9, 45; at 4 & 8, 60; at 5 & 7, 75; and at 6 & 6, 90.

Then the Triangle runs thus;

As the whole sine PAE 90-00

To the Tang. of AE 15 15-00 one Hour on the Equinoctial.

So the sine of PZ 38-30

To the Tang. of Z 11 11-28 the measure on the Plain for 11-1.

The like work serves for all the rest.

But note, Because the Hour-circle cannot pass by 12, you must turn the other-side, or half, for the afternoon hours.

Also note, That if the back-side of the Plain do not well represent the South-side, being the more useful Dial; then if you hold the Sphear with the foot upward, the Zenith becomes the Nadir, and the North Plain a South Plain, to appear more Plain to the apprehension.

3. For a Direct Reclining Dial.

For these Dials, set the Plain Direct, as [Page] before, and let the upper part of the Horizon cut the Semi-circle of Reclination, according to the Plains Reclination, and there make it fast; then the Axis drawn streight, and the upper Semi-circle brought to it, sheweth the Stile and Substile; and the Thred and Hour-circle, laid as before, giveth the Hour-Arks on the Plain, and sheweth also how the Proportion runs.

To find any Requisite also you may observe for all North-Recliners and South-Incliners, that the complement of Latitude and Reclination put together, doth give the Poles Elevation, or Stiles height, for all those Plains, which sometime will be above 90 from the South part of the Meridian; and then the complement to 180, is to be set from the North part of the Meridian: But if it be a South-Recliner, then substract the Reclination out of the Comp. Lat. and the remainder is the Stiles Elevation: But if the Reclination be more than the complement Latitude, then substract the complement Latitude out of the Reclination or Inclination, and the remainder is the Stiles Elevation.

Note also, That the upper-face of the Plain, that beholds the Zenith, is the Recliner; and the under-face that beholds the [Page] Nadir, is the Inclining-plain. And note, That both Plains, viz. both Incliners and Recliners have the same Requisites in each of them. But, the hours proper to the Recliner, are not to be put on the Incliner; for when the Sun shines on the one, it can't shine on the other. Therefore to know what hours are fit for these or any Plains whatever, do thus;

The Sphear rectified, and the Plain set to his true scituation, lay the Thred on the Suns declination, on the Hour-circle (according to what time of year you would know when the Sun begins and ceases to shine on any Plain) and turn the Hour-circle, with the Thred so laid, till the Thred do but just touch the Plain, and the Hour-circle doth on the Equinoctial, cut the Hour and Minuit required; when the Sun comes on the East-side, and when it goes off from the West-side of the Plain.

Example.

Suppose you have a Direct North-plain that Reclines from the Zenith towards the Equinoctial 25 degrees, you shall find the Stiles Elevation to be 63-30, the Substile from 12. The North-Pole to be elevated on the Recliner, and the South-Pole on the Incliner; and that the Sun shines on the [Page] North-recliner in the longest dayes, viz. 23-31, declination, from the Rising 13′ before 4, till 10; and then it begins to shine on the South-incliner, and shines till 2 afternoon; then it comes on the North-recliner again, and continues till it sets.

But in the shortest dayes, when the Declination is 23-30 towards South, then on the North-recliner it shines not at all, but only on the South-incliner, from Rising to Setting; and so doth it all the time the Sun hath South-declination.

This Rule serves for all sorts of Dials whatsoever.

Note, That the Circles of the Sphear shews the Canon to work this Question exactly, whereof you have a large Discourse in Wells his Art of Shadows, from pag. 391, to 408, in 35 Chap.

4. For a Direct East or West Erect-Dial.

The Sphear being rectified to the Latitude, bring the Notch in the Moveing Horizon, to the East or West-points on the fixed Horizon, viz. to 90 degrees; then set the Plain Erect, and make it fast there; then you shall perceive the Axis lie close to the Plain, it shews the Stile to have no Elevation, [Page] but must be set Parallel to the Plain, at any quantity you please, which is to be the Radius of a Tangent-line, whereby to pr [...]k down the Hours; and that the S [...]bstile or place where the Cock or Stile must stand is in 6, being the Hour-circle, till it be [...]ust against the upper Semi-circle, touch [...]ng the Thred, and in the Equinoctial it cuts 6, the true place where the Stile must stand.

Also, By the fore-going Rule you shall find the Sun shine all the year from the Rising, till 12 on the East-side; and on the West-side from 12, till his Setting.

5. For an East or West-Recliner.

Turn the moving Horizon to 90 degrees in the fixed, as before; then set the Plain to his due Reclination, and make it fast there, and pull the Axis streight, and bring the upper Semi-circle just to touch it, and straitway you have the Stile, and Substile, and 12, the Inclination, Meridian, and Hour-Arks on the Plain.

As for Example.

An East-plain reclining from the Zenith towards the Horizon 45 degrees, hath his Meridian, or 12 a clock Line in the Horizon; for if you extend the Thred from the Center [Page] to the fore-side of the Meridian, just there the 12 a clock Line must alwayes be, which in this Plain lies in the Horizon. The Substile doth lie 41-40 from thence upward, as the upper Semi-circle doth shew; the Inclination Meridian is thus found; Bring the Hour-circle, till it stand even and parallel to the upper Semi-circle; then on the Equinoctial it cuts 58-7′, the Inclination of the Meridian, with which you must make a Table of Hours, or Arks at the Pole, to calculate the Arks on the Plain, if you work Arithmetically. But by the Sphear, Set the Hour-circle to the hours on the Equinoctial, and the Thred being brought along the Plain till it touch the Hour-circle, shall shew on the Plain the Angle from the Horizon or Perpendiculer; or with some more trouble, from Substile or 12.

Also, It shews, that the North-Pole is Elevated on the West-reclining; and the South, on the East-inclining opposite thereunto; and that the Recliner in ♋, shews from 9 in the forenoon, till 8 at night; and the East Incliner from the Rising, till 9 forenoon in Summer; and in Winter, till a 11 in the forenoon.

Now to make these Plains, as Erect Decliners, let the complement Latitude become [Page] a new Latitude; and the complement Declination a new Declination; then they may become Erect Decliners, as in the next sort following.

6. Of Erect Decliners East or West.

By Declination, I mean the quantity of the Angle that the Meridian or Pole of place makes between the Meridian, or Pole of the Plain; therefore to set the Sphear to any Declination, do thus;

The Sphear being set to the Latitude, turn the Sphear as well as you can guess, to the scituation of the place; that is, put the North part of the Meridian towards the North; and the South part towards the South; then turn the Notch of the movable Horizon, alwayes to the degrees of the Plains Declination, from North or South, towards either East or West, and Upright also as in Erect Dials: Then is the Plain set to his Declination, viz. the distance of the Horizon between the Meridian, or Pole-place, which is alwayes 12 a clock, and the Meridian, and Pole of the Plain, being alwayes just where the Notch is in the Moving Horizon.

[Page] Now according to these Rules, A Plain that declines 30 degrees from South to West, The Stiles Elevation is 32-35. The Substile from 12, 21-40. The Inclination of the Meridian 36-24. The South Pole is elevated on the South-side, and the North Pole on the North-side: And the Sun shines on the North-side from Rising, to 8; and on the South-side, from 8 to 7 at night; and on the North again, till Sun-setting, by working as in the former Directions is expressed.

Note, In those Erect Decliners, whose declinations is above 60 degrees, you shall find the Stiles Elevation to be very small; therefore to make it exact, you must use Arithmetical Calculation; for the doing of which, the Sphear, with due consideration, gives the best directions, with these Proportions or Canons.

As sine 90 | ZN | 90-00
To sine Declination | NC | 30-00
So Co-tangent Latitude | PZ | 38-30
To Tang. Subst. from 12. | ZH | 21-40
As sine 90 | ZN | 90-00
To Co-sine Declination | NA | 60-00
So Co-sine Latitude | ZP | 38-30
To sine Stiles Elevation | PH | 32-25
[Page]As sine Latitude | PN | 51-30
To sine 90 | PAE | 90-00
So Tangent Declination | NC | 30-00
To Tangent Elevation Merid. | AEI | 36-24
As Co-tangent Latitude | ZP | 38-30
To sine of 90 | ZPQ | 90-00
So sine Declination | ZIA | 30-00
To Co-tang. 6 from 12 | AQ | 57-50

Note, If you set P, at the Pole. Z, at the Zenith. N, at the North-end of the Horizon, at the Declination, or Pole-plain. H, on the Plain, just against the moving Semi-circle, or Substile. A, at the Plain on the Horizon. AE, on the Equinoctial. I, at the Hour-circle, cutting the Equinoctial, set just against the upper Semi-circle. Note, Q is to be set on the Plain, right against the Hour-circle, being set to the Hour.

Having, I say, by these Rules, and the like, made and found the Requisites, then proceed to draw the Dial thus; by help of a Sector with Sines and Tangents, to 7-5; such as are usually made▪

But for very far Decliners, use that help as directed in Chap. 4.

The like work serves to help all sorts of [Page] Dials with low Stiles, Polar, and Meridian. Dials also.

The other 6 sorts, yet behind, I shall demonstrate only in two of them, which do properly enough comprehend them all; and the work of one, is as easie as the work of the other, especially by the help of the Sphear, where the hardest is as plain as the Horizontal. Therefore,

7. Of Declining, Reclining-Dials.

1. For South Recliners, they may recline short of, to, or beyond the Pole, at any Declination, as the putting up and down the Plain, doth plainly demonstrate. Therefore, first,

Of one that Declines South-west 35, and Reclines 20 from the Zenith.

Set the Notch, or Pole of the Plain to the Declination, and the Reclining Circle to its Reclination, and there make it fast; then extend the Axis streight, and bring the upper Semi-circle just to touch it, and the Hour-circle exactly even with the moving Semi-circle.

[Page]

Then, First, The Axis shews the Stiles height on the Semi-circle to be 12-13.

The Thred brought along the Plain while it touches the Meridian, and that shews the Meridians Elevation above the Horizon, on the North Recliner to be 76-32; or its Depression below the Horizon in South-Recliners, and that from the East-end, as the Sphear sheweth.

Then,

3. The Substile from the Perpendiculer Line of the Plain, is 21-6, as the upper Semi-circle sheweth; but from the hour 12, or Meridian 7-58, and stands on the East-side of the Meridian.

The Inclination of the Meridian is 33-29, as the degrees on the Equinoctial, between the Meridian and Hour-circle, shew.

All the Hour-Arks are easily found from the Plains Perpendiculer Eastwards and Westwards, by applying the Thred to the Hour-circle and Plain, being set to the Hours on the Equinoctial.

The South Pole is elevated in the South-Recliner, and the North, on the North Incliner.

[Page]If you set Letters to the Sides and Angles, according to the former discourse, you will see how all the Canons in the Arithmetical Calculation lie, as I shewed you before in the Declining Dials. And as again thus;

On the Pole set P. On the Zenith Z. At the West-end of the Plain, set A. At the East-end B. At the South Pole of the Plain C. At the North Pole D. At the East-end of the Horizon E. At the West-end W. At the North-end of the Meridian, set N. At the South-end S. Where the Hour-circle cuts the Plain F. Where the Meridian cuts the Plain G. Where the fixed Semi-circle cuts the Plain, set E. As in the Figure before. Then these Canons in short run thus;

As sine Base | ZD | 90-00
To Tang. Perpend. | ND | 35-00
So sine of Base | ZE | 20-00
To Tang. Perpend. | GE | 13-28
Whose complement AG 70-32, is the Meridians elevation.
As sine of the Side | GE | 13-28
To sine of the Angle | CZE | 35-00
So sine of the Angle | GFZ | 90-00
To sine of the Side | GZ | 23-57

Which taken from ZP 38-28, leaves 14-33, the distance of the Meridians place from the Pole to the Plain, viz. GF.

[Page]

As sine of Hypothen. | GZ | 23-57
To sine of Perpend. | ZE | 20-00
So sine of Hypothen. | PG | 14-33
To sine of Perpend. | PF | 12-13
| the Stile.

As Tangent of Perpend. | ZF | 20-00
To sine of Base | GE | 13-28
So Tangent of Perpend. | PF | 12-13
To sine of Base | FG | 7-58
| the Substile to 12.

As the sine of the Side | ZE | 20-00
To the sine of the Side | GE | 13-28
So is the sine of the Angle | PFG | 90-00
To the sine of the Angle | FPG | 33-28
| Inclin. Merid.

For the Hours in all Dials, say thus;

As sine of 90,

To sine of Stiles height;

So Tangent of the Angle at the Pole,

To Tangent of the Angle on the Plain.

8. For North Declining Reclining-Dials.

For these Plains also, you must rectifie the Sphear to the Latitude, and set the Plain to his Declination, and Inclination, which is [Page] given, and for which you are to make a North Declining Reclining Dial. As you did in the South-Recliner, so work in all respects, as you shall bring forth the Quesita's, either by the Sphear or Arithmetical-Calculation, as is largely shewn. And for a Plain that declines 55 degrees from the North towards the East, and relines 20 from the Zenith, you shall find the Requisites to be as followeth.

1. The Meridians Elevation above the Horizon, is found to be 63 deg. 58 min. But yet observe, You must make use of that part of it which is below the Horizon, because the Sun being Elevated high on the South-part of the Meridian, must needs cast a shadow on the North-part thereof; therefore in drawing the Dial-part, part is only to be made use of for the Sun to shine on.

2. The Stiles Elevation is 37 degrees 00 minuts.

3. The Substile from 12, 65-24; or from the Plains perpendiculer 39-22.

4. The distance of the Meridian of the place from the Pole of the Plain, is 70-33. The Angle between the 2 Meridians 74-38. The Hour-Arks are found by applying of a Thred to the Hour-circle and Plain; the [Page] [Page]

[figure]

[Page] Hour-circle being first set to the hour on the Equinoctial.

The North Pole is Elevated; and in regard the Plain declines to the East, the Stile must be set towards the West, and it shines on the Plain in Summer-time, from the Rising unto 12: But in the Winter-time, but a few hours.

Note also, That these Declining Reclining-plains, may be referred to a new Latitude and Declination, wherein they shall become Upright Decliners, as before is hinted.

The Poor-Mans Dial-Sphear; Or another way to demonstrate the Mystery of Dyalling, both for Declining and Inclining Plains, in a very plain, easie way, for one 6th part of the cost of the other Brass-Sphear.

First, as to the Description, and afterward for the Vse.

AS to the Description, the Figure annexed, and a few words shall suffice; wherein consider,

First, The plain flat-Board, representing [Page] the Horizon, as ABCD.

Secondly, The two upright pieces, as East and West-points, as AE, and BF, to support the moving Plain.

Thirdly, The Moving-plain, moving to any Inclination, on the two Points E and F, with 180 degrees upon the Plain, and noted by ABEF.

Fourthly, Also a Brass-circle as G, fastened to the Plain, to set it to any degree of Inclination; and a skrew, as at H, that may stay it steady, when set to any Reclination.

Fiftly. On the middle of the Horizontal-board, is fastened at the Point M, a true Horizontal-Dial, drawn fit for your Latitude, and to turn round on the Point M, as IMKL.

Sixtly, A Thred fastened in L, the Center of the Horizontal-Dial; and in N, the Center of the Plain; to be both a Stile for the Horizontal-Dial, and to represent the Axis of the World; also a small Woodden-Quadrant will be useful, such a one as half the Plain is, to draw Perpendiculers, and measure Angles, as afterwards in the Uses. The Uses follow.

[Page]

Use I. To find the Declination of a Plain by the Sun-shining.

Apply the side AB to the Wall, and hold the Instrument level, as by help of a Point Plummer, fastened at N, and the Point playing right on M, it is easie to do; then by the Trianguler-Quadrant, having first observed the true hour, turn the Horizontal-Dial about on the Point M, till the shadow of the Thred (or Axis) shew the same Hour; then the Point on the North-end of the Horizontal-Dial, shall shew the true Declination of the Plain. For any South Decliner, the use is obvious.

But for North-Decliners, you must turn the Plain out of the way of the Thred, still keeping the same side, AB, to the Wall; and if the Horizontal-Dial hinder, put a Parallel-piece between, as your Rule, or any other thing, and you shall have the Point give the Declination on the Southern Semi-circle on the fixed Horizon.

[Page]

Use II. The Declination of any Erect Decliner given, to find the Substile, and Stile, Inclination of Meridians, and every hour and quarters distance from 12, being the Perpendiculer Line on the Plain.

First, Set the Point at 12 on the Horizontal-Dial, to the Declination of the Plain, toward the East or West, and set the Plain Upright.

Then first for the Substile.

Apply the side of the Quadrant to the Plain, and cause the shadow of the Thred to play Parallel to the perpendicular Ray of the Quadrant, and at the same time it shall shew on the degrees on the Plain, the true Substiles distance from 12.

Example.

Suppose the Plain decline 20 degrees South-west, you shall find the Substile to be 15 deg. and 12′ from 12, and to stand on the East-side of 12, in a South declining West 20 degrees, Latitude 51-30.

Again, for the Stiles Elevation.

Apply the Quadrant to the flat of the Plain, on the Substile Line, so as the Thred may cut the Center of the Quadrant; and then [Page] the Thred shall cut on the Quadrant 35-46 for the Stiles height.

Again, for the Inclination of Meridians.

The shadow of the Thred when it cuts the Substile 15 deg. 12′ on the Plain, shall on the Horizontal-Dial cut 1 hour 36 min. which reduced to degrees, is 24 deg. 50 min. the Inclination of Meridians.

Again, for every Hours distance, in degrees and minuts from 12.

Turn the whole Instrument about, (as it is then first set) till the shadow of the Thred shall fall on every hour and quarter, and then the shadow shall cut on the degrees on the Plain, the distance of every hour and quarter from 12, for that declination, in degrees and minuts; which you may draw into a Table, for your use and purpose; or hereby examine your more exact Calculation, and prevent all gross mistakes in your former work.

Use III. Any Declining North-east, or North-west-Dial being given, to find the former Requisites for those Dials.

In the true proper using the Sphear for [Page] North-Dials, the Stile should look upwards, which will appear so to do, if you turn the Instrument the bottom upwards, for the further help to your fancy; but observe that the Hour-Arks, and Angles, are the same for the North, as for the South, only the difference is in the Scituation, as to the contrary-side, and looking upward instead of the South Decliner, looking downward, as by turning the Instrument appears; so that if you draw the Dial as a South-west, when you would make a North-west; and set right figures, and the right way, and then your work is effected to your mind, to the right intent and purpose.

Example of a North-East, 30 degrees, Latitude 51-30.

Set the Point at 12, to 30 degrees Westward, and apply the Square to the Plain, till it just touch the Thred; and on the degrees on the Plain, it cuts 21-40 for the Substile; and at the same time almost half an hour past 2 for Inclination of the Meridians; and applying the Quadrant to the Substile-Line on the Plain, and to the Thred; it cuts 32-35 for the Stiles height, being the same, and the same way found as for the South Decliner East.

[Page] But observe, That for the Hour-Arks, you must note, That the North-Dial cannot shew 12 at Noon, nor any Hours very near Noon, which will be seen on the South Decliner East; Therefore 4 in the morning, is here called 8; and 5 is called 7; and 6 is 6: 7 in the morning, is called 5; and 8 is to be named 4: And if you turn the Instrument, that the shadow of the Thred may fall on those hours, it will also cut on the degrees on the Plain, the true Hour-Arks required.

As thus;

For 8, it sheweth it not; at 7, it sheweth 77-00; at 6, it sheweth 58 deg. 52 minuts; at 5, it cuts on the degrees on the Plain 45-38; at 4 in the morning, it cuts on the Plain 35-27; but the shadow falls then on Hour of 8, on the Horizontal-Dial.

Also note, That these numbers are not laid from the Substile, but from the Plains Perpendiculer, which in all Upright Plains is a Perpendiculer Line; and in all other Plains, a Perpendiculer to the Horizontal-Line, drawn on the Plain. And thus proceed with any other; the affections are best seen when you turn the Instrument the upper-part downwards.

[Page]

Use IV. To find the Requisites, and to draw the Hours on a far Declining Erect-Dial, S. W. 80.

Set the Point to 80, as before S. W. then the Thred and the Quadrant shall shew 38-2 for the Substile; and 82-8 on the Horizontal, for the Inclination of Meridians; and 6-12, for the Stiles Elevation; and the shadow of the Thred on the Horizontal-Dial, will shew you how close and inconvenient the Hours will be, if not helped by the former directions; and in like manner will the North-East or West be, and likewise helped.

Use V. To find the Requisites, and Hour-Arks, from the Perpendiculer of a Declining Inclining Plain, with its Affections.

Set the Point at 12 to the Declination, and move the Plain by help of the Arch, or Circle of Brass, to the Inclination, and with the skrew make it fast and steady in that place.

Then for the Substile, Apply the Quadrant to the Plain, and also Perpendiculer [Page] to the Axis, as the edge of the Quadrant being thick, will neatly shew; then the Thred will shew on the degrees on the Plain, the distance of the Substile from the Perpendiculer, or the complement thereof from the Horizon; which Point note with a spot of Ink; for, when the shadow of the Thred falls on that spot, on the Horizontal Dial, it sheweth the Inclination of Meridians; that is to say, on what hour and minut, the Cock of the Dial should stand right over.

Also, The Quadrant, applied to the Plain and Thred, on the Substile-Line, sheweth the true Stiles Elevation above the Plain.

And lastly, making the shadow of the Thred to fall on every Hour on the Horizontal-Dial, it shall at the same time shew how many degrees and minuts on the Plain, that Hour-line ought to be from the Perpendiculer, or from the Horizon; and also which way, either to the Right or Left, East or West; or from the Substile, or 12; if you will trouble your self to count it, from the place found out for the Substile, or 12.

[Page]

Example of a Plain Declining 30 S.E. and Inclining 20.

The Substile, by applying the Square, you shall find to be 30 degrees on the left-hand of the Perpendiculer Westward, and the Inclination of Meridians 48-20, the Stiles height 51-36, and the Meridian on the right-hand of the Perpendiculer-line 11-30 Eastward; and the shadow of the Thred playing on every hour and quarter, on the Horizontal-Dial, will shew on the Plain the quantity in degrees from the Perpendiculer-Line.

Use VI. To find the Requisites in a North-east Reclining-Dial, and the Hour-Lines.

Set the Instrument as before, and find the Substile, Stile, and Inclination of Meridians as before; But note, as to the Affections, which way do thus;

Turn the Instrument the bottom upward, and as near as you can guess, turn the Plain to its scituation; then you shall first see the Stile to look upward in the North-east Recliner, which before was downward in the South-east Incliner.

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Also,

The Substile stands on the right-hand of the Perpendiculer, 30 degrees Westward (for observe this alwayes, If a Plain declines Eastward, the Substile will stand Westward, and the contrary).

Also note,

That the Meridian-Line is to be drawn quite through the Center on the other-side; because, when the Sun is in the Meridian above, it must needs cast the shadow of the Axis, or Stile, the contrary way downwards.

Use VII. To find what are the most Hours, that the Sun can shine on any Plain, whatsoever.

First, on all South Direct, or Declining Inclining-Dials, the mid-day-Meridian is proper to it, unless it incline above 75 degrees, and then it becomes useless in London Latitude; then what hour soever you can make the Sun to shine on the Plain, and Horizontal-Dial both together, (the Sun being at that hour above the Horizon) by bending or turning the Instrument any way, [Page] (when the Point at 12 is first set to the Declination) that, and all those Hours are proper to that Plain, at one time of the year or other.

Also note,

That several Hours that serve for the South-plain, do, at some time of the year, belong to the North-plain also; as by turning the Instrument about, you may plainly see, either by the Sun-shine, or by the Thred, and your Eye cutting the Hour-Lines and the Plain.

Also observe,

That if you would delineate a South Reclining Plain, you may bring the Plain toward the Thred, till it becomes a Polar-Plain.

But if it Reclines below the Pole, then conceive it to become a North Reclining-Dial, and work as is before directed, and you shall obtain your desire; for the Dials will be the same, the one as the other, as before was hinted at, in the Inclining-Plains.

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Use VIII. The Declination of any Plain given, to find what Reclination will make it a Polar-Dial, and the contrary.

Set the North-point to the Declination, and bring the Plain to touch the Thred; then on the Brass Circle is cut the Reclination required.

Or contrary;

Set the Plain to the Reclination given, and then bring the Thred to the Plain, by turning the Horizontal-Dial, and the Point at 12 shall shew the Declination required, to make it Polar.

In like manner you may discover a declining Equinoctial, but not so easily, when the Substile and Meridian are 90 degrees assunder; the Substile being then alwayes in the hour of 6, as by moving the Plain, if the Declination be given; or by moving the Thred, if the Inclination be given, till the Square, touching the Thred, it shall shadow or bourn, just upon 6 on the Horizontal-Dial.

Note also,

That East and West Recliners, and Incliners, are discovered after the same manner; [Page] So also Direct Recliners, and Incliners, as by moving the Plain to and fro, you shall see the plain and true reason, how the Stile is Elevated or Depressed, and how the Hour-lines are inlarged or contracted, according to the Elevation of the Stile.

Also,

In East and West-Dials, that the Stile hath no Elevation, but is parallel to the Plain; and how the Meridian lieth in the Horizon, in East and West Recliners, and Incliners.

Many more Uses might be insisted on, which I shall leave to the scruteny of the industrious Practitioner, in the Art of Shadows.

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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] 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]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] 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] 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;

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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]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] 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.

The end of the Book of Dyalling.

Horologiographia, or, The art of dyalling being the second book of the use of the trianguler-quadrant : shewing the natural, artificial, and instrumental way, of making of sun-dials, on any flat superficies, with plain and easie directions, to discover their nature and affections, by the horizontal projection : with the way of drawing the usual ornaments on any plain : also, a familiar easie way to draw those lines on the ceiling of a room, by the trianguler quadrant : also, the use of the same instrument in navigation, both for observation, and operation : performing the use of several sea-instruments still in use / by John Brown, philomath.[2]

§

§

§

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AN APPENDIX To the Use of the Trianguler-Quadrant IN NAVIGATION.

Where it performs the Uses of the Davis-Quadrant, the Cross-Staff, Bow, Sinical-Quadrant, and Sector, with as much ease and exactness as any, or all of them, will do in Observation or Operation, Naturally or Artificially.

Being first thus Contrived, and made by Iohn Brown, dwelling at the sign of the Sphear and Sun-Dial in the Minories, near to Aldgate, London.

London, Printed by Iohn Darby, for Iohn Wingfield, and are to be sold at his house in Crutched Fryers; and by Iohn Brown at the Sphear and Sun-Dial in the Minories; and by Iohn Selle at the Hermitage-stairs in Wapping. 1671.

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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]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] 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.

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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]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]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.

Use I. To find the Suns, or a Stars Altitude, by a forward Observation, as by a Fore-staff.

Skrew the turning-Sight to the Head-Center, [Page] and put the object-Sight into the hole at the end of the Head-leg, and put the sliding Horizon-sight on the in-side of the Loose-piece; Then setting the turning-sight to your eye, and holding the Loose-piece in your right-hand, and the moveable-Leg toward your body, then with your Thumb on the right-hand, thrust upwards, or pull downwards the Horizon-sight, till you see the Sun through the Object-sight, and the Horizon through the Horizon-sight; then the degrees cut by the Line on the middle of the Horizon-sight, shall shew the true Altitude required.

Also observe, That if you like to use the upper or lower-edge of the Horizon-sight, instead of the small bar a-cross the open-hole, after the manner of the ends of a Fore-staff, that then the degrees and minuts cut by the edge of the Brass, is the Altitude required, to be counted as it is figured from the Object-sight, toward the Horizon-sight; the degrees between them being the Angle required.

Note also, That if the Altitude of the Sun, or Star, be above 30 degrees, you will find it a hard matter, to behold the Horizon and Sun with a bare roling the ball of the eye only, and a stirring of the head, will [Page] easily cause a stirring of the hand, which will spoil the exactness of Observation, unless the Instrument shall be fixed to a Ball-socket and Three-legged-staff, which is not usual at Sea.

Therefore to remedy this, you may observe with the open oval-hole in the turning-sight set to the eye, or taking the turning-sight quite away; Observe just as you do with a Fore-staffe, setting the round part of the head, to the hollow-part beside your eye, so as the Head-Center-pin may be as near the very sight of your eye as possibly as you can; which Center is the Center to the degrees now used in a forward way of Observation. Or, rather use this way when the Weather will suffer, by a Thred and Plummet, which I shall add as a second Use.

Use II. To observe the Sun or a Stars Altitude, by a forward Observation, using the Thred and Plummet.

Skrew the turning-sight to the Head-Center, as before, and put the two Object-sights into the two holes at the two ends of the Rule; and on the Leg-Center-pin hang the Thred with a weighty Plummet of two [Page] pound, or above a pound at least.

Then hold up the Trianguler-Quadrant, setting the small-hole on the turning-sight close to your eye; and if the Sun, or Star, be under 25 degrees high, then look to the Sun or Star through the turning-sight, and that object-sight, which stands in the end of the moveable-Leg, letting the Thred and Plummet play between your Thumb and Fore-finger, as a Brick-layers Plummet in his Plum-Rule doth in a bendid hole, that you may keep it in order whilst you look at the Sun or Star, and the weighty Plummet will pull the Thred streight, and let you know by feeling which way it is playing, till it playeth evenly and truly, whilst you have the Object precisely in the midst thereof, whether it be Sun, Moon, or any Star, or other Object, whose Altitude you would observe; Then, I say, when the Plummet playes well, and you behold the Object right, bend back the Quadrant, and see what the Thred cuts on the degrees on the moveable-Leg, which shall be the true Altitude required; And in my opinion, must needs be more exact than any other way of a forward Observation, because you are not troubled to mind the Horizon and Sun both at at once.

[Page]An Objection may be, The boisterous Winds, and the rouling of the Ship, will hinder such an Observation.

Answ. So it will any other way, though happily not so much.

Again, I answer, One Object is better and more certainly seen, than two at any time together; and though the Wind blow hard, if you can stand to observe at all, the heavy Plummet will be sure to draw the Thred Perpendiculer; and for ought I know, you may come as near this way as any other; however this, at most times, may confirm and prove the other, and may be useful in Rivers, and Harbours, and misty-Dayes, when you may see the Sun well enough, but not the Horizon at all.

Use III. To find the Suns Altitude by a backward Observation, as with a Back-staff, or Davis-Quadrant.

Skrew the turning-sight to the Leg-Center (or Center to the degrees on the moveable-Leg); and set the object-sight to the long stroke by 00-60 on the out-side of the Loose-piece, and put the sliding Horizon-sight on the out-side of the moveable-Leg; [Page] then hold the Object-sight upwards, and the small-hole in the piece turning on the [...]dge (or to the small-hole in the middle) of the Horizon-sight (which you please) close to your eye; and looking through that hole, and the middle-hole of the turning-sight, to the true Horizon, turning your self about, and lifting up, or pressing down the Horizon-sight, close to the moveable-Leg, till the shadow of the upper-edge of the shadow-sight, being next to the Sun, fall at the same time just on the middle of the turning-sight; Then, I say, the edge or middle of the Horizon-sight, that you looked through, shall cut the true Altitude of the Sun required.

Being the same way as you do observe with a Davis-Quadrant, or Back-staff.

Use IV. To find the Suns Distance from the Zenith, by the Trianguler-Quadrant.

Skrew the turning-sight to the Leg-Center, and put the Object-sight, whose oval-hole is remotest from the Quadrant, in the hole in the end of the Head-Leg, or rather in a hole on the general Scale, between the turning-sight, and the Sun; and put the [Page] Horizon-sight on the out-side of the moveable-Leg; then hold the turning-sight toward the Sun, and the small-hole in the edge of the Horizon-sight to your eye; then look through that hole and the turning-sight, till you see the shadow, the Object-sight, to fall just on the turning-sight, or the shadow of the turning-sight to fall just on the object-sight, which is all one, though the first be more easie, because you shall see the Horizon through the turning-sight, and that, both at once; Then, I say, the degrees cut by the Horizon-sight, shall be the Suns distance from the Zenith required; Being the very same work, and done in the same manner, and producing the same Answer, viz. the Suns distance from the Zenith, that the Davis-Quadrant doth.

Note, That this way you may observe very conveniently, till the Sun be 20 degrees distance from the Zenith; and by the adding of a 60 Arch, as in Davis Quadrant, or to 45 will be enough, it will do as well as any Davis Quadrant, being then the same thing.

But I conceive, the complement of the Altitude being the same, will do as well; which Altitude is better found by this Instrument, than the distance from the Zenith [Page] by a Davis Quadrant is, as in the next Use will be seen.

Use V. To find the Suns Altitude when near the Zenith, or above 90 degrees above some part of the Horizon.

In small Latitudes, or in places near the Equinoctial, or under it; the Sun will be found to be in, or near the Zenith: and if you count from some part of the Horizon, above 90 degrees distant from it; then instead of setting the sliding Object-sight, to the long stroke at 00 on the Loose-piece, you must set it 30 degrees more towards the Head-leg; then observe, as you did before, and whatsoever the Horizon-sight cuts, you must add 30 degrees more to it, and the sum shall be the true Altitude required.

Example.

Suppose that in the Latitude of 10 deg. North, on the 10th of Iune, when the Suns Declination is 23 degrees and 31 min. Northward; Suppose that at noon, I observe the Suns Meridian Altitude, skrewing the Turning-sight to the Leg-Center, and setting the Object-sight to the 30 degrees on the Loose-piece, near the end of the Head-leg, and the Horizon-sight on the movable-Leg; [Page] then hold up the Quadrant, with the shadow-sight toward the Sun, and the small-hole in the Horizon-sight toward your eye, and look to the Horizon through that, and the turning-sight, the shadow of the right-edge of the shadow-sight, that cuts the degree of 30, at the same time falling on the middle of the turning-sight, you shall find the Horizon-sight to cut on 46-29 minuts; to which if you add 30, the degrees, the shadow-sight is set forwards, it makes up 76-29, the Suns true Altitude on that day in that Latitude; 76-29 the Meridian Altitude, and 23-31 the Declination, added together, make 100 deg. 00; from which taking 90, there remains 10, the Latitude of the place.

1. In this Observation, first you may note this, That if you had stood with your back toward the South, you would have had 103 degrees and 31 minuts, for the sliding Horizon-sight would have stayed at 73 degrees 30; to which if you add 30, it makes 103-31; which a Davis Quadrant will not do.

2. In the holding it, you may lean the head of the Rule to your breast, and command it the better, as to steady holding.

[Page]3. You may turn the Turning-sight about, to any convenient Angle, to make it fit to look through to the Horizon, and also to receive the shadow of the shadow-sight.

If the brightness of the Sun offend the eyes, you may easily apply a red or a blue Glass, to darken the Sun beams, and the Sights may be painted white, to make a shadow be seen better.

Use VI. To find the Latitude at Sea, by a forward Meridian Observation of the Altitude, according to Mr Gunter's Bow.

Skrew the Turning-sight to the Leg-Center, and set the shadow-sight to the Suns-Declination, and the Horizon-sight to the moving Leg (or Loose-piece), and the Turning-sight to your eye; then let the shadow-sight cut the Horizon, and the Horizon-sight the Sun, moving it higher or lower till it fits; then whatsoever the sight sheweth, adding 30 degrees to it, is the Latitude of that place required.

Example.

Suppose on the 10th of March, when the Declination is only 10′ to the Northward, [Page] as in the first after Leap-year it is; then set the edge, or stroke on the middle of the shadow-sight to 10′ of Declination toward the Head, and the Horizon-sight, on the same Leg toward the end, and slide only the Horizon-sight till it cuts the Sun, and the other the Horizon; then suppose [...]t shall stay at 21-30: then if you count the degrees between the two Sights, it will amount to the Suns Meridian Altitude; but if you add 30 degrees to what the Sight cuts, it shall give the Latitude of the place where the Observation was made for 21 and 30, to which if you add 30, it makes 51-30, the Latitude of London, the place where the Observation was made.

Note here, That in small Latitudes the Sun will be very high, in Summer time especially, and then the sliding-sight, must be set on the loose peice.

As thus for Example.

Suppose on the 10th of May 1670, when the Declination is 20-7 in the Latitude of 30 Degrees, I observe at a Meridian Altitude, I shall find the sliding-sight to stay at 00. on the loose peice; then it is apparent that 30 added to 0, makes but 30 degrees for the Latitude [...]equired.

But if the sliding-sight shall happen to [Page] pass beyond 00 on the loose peice, then whatsoever it is you must take it out of 30, and the remainder is the Latitude required.

Example.

Suppose on the 11th of Iune 1670 you were in the Latitude of 10 degres to the Northward, and standing with your back to the North, as you must needs do in all forward Observations in more Northern Latitudes, you shall find the sight to pass just 20 deg. beyond 00 on the Loose peice; therefore 20 taken from 30 the residue is 10, the Latitude required.

Again.

Suppose that in the same place you had obsereved on the 11th of December, when the Sun is most Southwards, if you set the one sight to 23-31 Southwards against the 11th of December; then if you observe forwards with your face toward the South, as before, you shall find the moving-Sight to stay at 20 degrees beyond 00 on the Loose-piece; then, I say, 20 taken from 30, rests 10, the Latitude required, because the sight passed beyond 30 on the Loose-piece.

Lastly, if the Moving-sight shall happen to pass above 30 degrees beyond 00 on the Loose-piece, when the other Sight is set to the Suns Declination, and you observe with [Page] your face toward the South (part of the Meridian); then, I say, the Latitude is Southwards as many degrees as the Moving-sight stands beyond 30, on the Loose-piece toward the Head-leg.

So that the general Rule is alwayes, in North Latitudes (observing the Suns Meridian Altitude, to find the Latitude, by a forward Observation, according to Mr. Gunters Bow), your face must be toward the South; Although that thereby in some Latitudes, the Altitude may seem to be (as indeed it is above the South-part of the Horizon) above 90 degrees.

Then,

If the sliding-sight stay any where on the Moving-leg, or Loose-piece, short of 00, add it alwayes to 30, and the sum shall be the true Latitude North; if it pass beyond 00, then so much as it doth, take out of 30, and the remainder is the Latitude North; but if it shall stay just at 30 on the Loose-piece, then the Latitude is 00; but if it pass beyond 30, then so much as it is, the Latitude is Southward.

The same Rule serves, if you were in South Latitude, then you must in forward Observations, to find the Latitude, as with a Gunters Bow, stand with your face to the [Page] North, and in setting the fixed-sight to the Declination, you must count South Declination toward the Head; because those that have Southern-Latitude, have their longest dayes, when those that live in Northern-Latitude have their shortest dayes.

The same Rule serves for the Stars also, for being in North-Latitudes; and observing a Latitude forwards, have your face alwayes toward the South, and set one Sight to the Declination, counting the Stars Northern or Southern Declination▪ the same way as the Suns, (and the contrary in South-Latitude); then holding your eye close to the great-hole of the Turning-sight, slide the Moving-sight till its middle-bar or edge (as is most convenient) cuts the Star, and the other the Horizon; then whatsoever the edge of the Moving-sight cuts short of 0, added to 30, or beyond 00, taken out of 30, shall be the Latitude required.

Example.

Suppose the middle-Star of Orions-Girdle, whose Declination is 1-28 South, being in the Meridian, I set one Sight to 1-28 of South-declination; and slipping the other-sight till it cuts the Star, the fixed-sight being set to the Horizon, you shall find it stay in the Latitude of 51-30, at 21-30 on the [Page] Moving-leg, which added to 30, makes 51-30.

Note, That if the corner of the Instrument be inconvenient for the sight to slide on, as for about 5 degrees it will, then you may remedy it by slipping the Sight set to the Declination 10 degrees more, any way that is convenient, increasing or decreasing; But then note, That instead of adding even 30, to what the Moving-sight stayeth at, you must add 10 degrees more, viz. 40, when you slip it towards the end of the Moving-leg; or 10 degrees less, viz. 20 degrees, when you slip it 10 degrees more toward the Head, as is easie to conceive of.

Use VII. To find the Latitude at Sea, by a backward Meridian Observation, according to the way of Mr. Gunter's Bow.

Skrew the Turning-sight to the Leg-Center, and set the sliding-Horizon-sight to the Suns Declination (the middle or edge of it, as you can best like of) and the Shadow-sight on the Loose-piece, or Moving-leg, with your face alwayes to the North, in North Latitudes; or supposing your self to be so, though it may be you are not.

[Page]Then looking through the hole in the Horizon-sight (standing at the Declination) and the Turning-sight to the Horizon, with your hand gently slide the shadow-sight till the shadow fall just on the middle of the Turning-sight, as you do in observing the Altitude with a Davis Quadrant; then, I say, whatsoever the shadow-sight shall stay at under 0, add to 30; or over 0, take out of 30, and the sum or remainder, shall be the Latitude North: but if it happen to stay at just 30, the Latitude is 00; if beyond, it is so much to the Southwards: This is only the converse of the former, and needs no Example, but a few words to demonstrate it; which may be thus,

In the way of an Example.

Suppose that on the 11th of December, in sayling toward the East-Indies, about the Isle of St. Matthews, supposing our Ship to be in North-Latitude, I set the Horizon-sight to 23-31, South Declination; and the Shadow-sight on the Loose-piece, then standing with my face to the North, as another then would do, as at other times, and looking through the Horizon, and Turning-sight to the North-part of the Horizon, I find the shadow-sight when it playes well over [Page] the Turning-sight, to stay at 33 degrees on the Loose-piece.

Then,

Consider that the distance between the wo Sights, is the Altitude of the Sun above the South-part of the Horizon; which if you do count on the Trianguler-Quadrant, you will find to be 36-29, and 33, which put together, make 69-29, for the Suns Meridian Altitude; to which if you add 23-31, his declination, it makes 93-00 the distance of the North-Pole and Zenith, or 3 degrees of South Latitude; for had you been just under the Equinoctial, the Altitude would be 66-30; or had you been more Northward, it would have been less; therefore by considering, you may soon see the reason of the Operation.

Also,

If the Shadow-sight be too near the corner, or too far from the Turning-sight to cast a clear shadow; then, set the Horizon-sight that stands at the Declination 10, 20, or 30 degrees more toward the end of the Moving-leg, and you shall see the inconvenience removed; but then you must take 10, 20, or 30 degrees less than the shadow-sight sheweth, for the reason abovesaid.

The reason why even 30 is added, is because [Page] that 0 degrees of Declination, stands at 60 on the Moving-leg, instead of 90, or 00.

Note, If you had rather move the lower-sight than the upper, then count like Latitudes and Declinations from 00 on the Loose-piece toward the Head-leg, and unlike the contrary, and then set the shadow-sight fixed there; then observing, as in a back-Observation, the Horizon-sight shall shew the complement of the Latitude required, without any adding of 30.

Thus you see, That the Trianguler-Quadrant, containing 180 degrees in a Triangle, brings the shadow-sight near the Center, and with one manner of figuring, gives the Suns Altitude above the Horizon, backwards or forwards, and his distance form the Zenith, and the Latitude of the place South or North, or North or South, backwards or forwards, by the Sun or Stars, by one side only, as conveniently and with fewer Cautions, and as exactly, if well used, as any other Instrument whatsoever; So that by this time you see it is a Fore-staff, Quadrant, and Bow. The other Uses follow.

[Page]

Use VIII. To find the Latitude by a Meridian Observation, by the Thred and Plummet, by the Sun or Stars.

This way of Observing without a Horizon, must be done by an Astrolabe, which is a Plummet it self, or else with a Plummet fitted to another Instrument, and at some times may do better service than the Horizon, and for an Altitude barely, is shewed already.

For the Latitude thus;

Count the Declination, which is the same with the Latitude, from 00 on the Loo [...]epiece toward the Moveable-leg; and contrary Declinations, both of Sun or Stars, count the other-way toward the Head-leg, and thereunto set the edge of the Horizon-sight, that hath the small-hole on it.

Then let the Sun-beams shine through the small-hole on the Turning-sight, to the small-hole on the Horizon-sight, the Thred and Plummet duly playing, shall shew the Latitude of the place required.

But if you look at a Star, having the same Declination, then set your eye to the Horizon-sight, and behold the Star through the Turning-sight, and the Thred shall fall [Page] on the Latitude required, when you look toward the South, being in Northern Latitudes.

So also, When you turn your face toward the North, in observing those Stars, it is best done when they come to the Meridian below the Pole; but for their coming to the Meridian above the Pole, then their Declination is increased by the quantity of their distance from the Pole, or the complement of their Declination.

As thus;

The Declination of the Pole-Star, when in the Meridian below the Pole, is 87-20 from the Equinoctial; but when the same Star is on the Meridian above the Pole, then it is 92 deg. 40′ distant from the same Northern part of the Equinoctial.

So that if you make 60 on the moveable-leg, to represent the North-pole; then you may count or observe any Star that is 25 degrees distant from the Pole, both above or below the Pole; then adding 30 degrees to what the Thred falls on, shall be the true Latitudes complement required; because you have removed the Pole from 90 to 60, 30 degrees backward.

[Page]

Example.

The declination of the uppermost Star in the great Bears back, is 63-45; that is, 26-15 below the Pole; or, 25 degrees 17 minuts above the Horizon, when on the Meridian below the Pole; but the same Star, when on the South-part of the Meridian, is 77-47 above the Horizon, or 26-15 above the Pole.

Therefore,

The Star being below the Pole, you may set the hole in the middle of the Horizon-sight, to the Declination, counting 90 the Pole, and looking up to the Star, as usually, the Plummet will fall on 38-28, the Latitudes complement required.

Again,

The same Star being on the South-part of the Meridian, above the Pole, I count 60 on the Moving-leg for the Pole, and 26-15 beyond that Pole further, viz. to 86-15, which is as far as you can well go, counting 60 the Pole; then observing, as you did before, you shall find the Thred to play on 08-28, the Latitudes complement required, for the distance between 08-28, and 86-15 is 77-47, adding 30 degrees, because of 60 instead of 90, for the Pole-point.

[Page] Note, That the Thred playing near the corner, may prove somewhat troublesome to observe, without help of another person; but if you will be exact in this or any other Observation, a Staff and a Ball-socket, should be applied to this, as well as to other Instruments, to stand steady and sure in the time of Observation.

These wayes are ready and easie, without taking notice of those Regulations and Cautions, which are to be observed in finding the Altitude, barely▪ as in the Seamans Kalender, and Mr. Wrights Errors in Navigation, is plainly seen.

But if you know them all, and had rather use those Rules in those Books; then, I say, a Thred and Plummet by this Instrument, will do as conveniently as any other, or the three Sights and Horizon, as before is shewed, to find the Altitude.

[Page]

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]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.

[Page]

A Table of the Suns Declination every day at Noon for London in the year 1666, the second year after the Leap-year, according to Mr. Street's Tables of Longitude. Calculated by Iohn Brown, 1668.

Month Dayes. | Ianu. | Febr. | March | April. | May. | Iune.
D.M. | D.M. | D.M. | D.M. | D.M. | D.M.
1 | 21 45... | 13 50.. | 03 29. | 08 31. | 18 02 | 23 11.
2 | 21 36 | 13 30.. | 03 05.. | 08 53. | 18 17. | 23 14..
3 | 21 25.. | 13 10. | 02 42 | 09 15. | 18 32 | 23 18.
4 | 21 14... | 12 49... | 02 18. | 09 36... | 19 46... | 23 21
5 | 21 03.. | 12 29 | 01 54... | 09 58 | 19 01 | 23 23...
6 | 20 51 | 12 08. | 01 31 | 10 19. | 19 14... | 23 26
7 | 20 40 | 11 47. | 01 07 | 10 40... | 19 28. | 23 27...
8 | 20 27.. | 11 25... | 00 43. | 11 01 | 19 41. | 23 29.
9 | 20 15 | 11 04. | S. 19... | 11 22 | 19 54.. | 23 30.
10 | 20 01... | 10 43 | N. 04.. | 11 42... | 20 07 | 23 30...
11 | 19 48. | 10 21 | 00 27... | 12 03 | 20 19 | 23 31
12 | 19 34.. | 09 59. | 00 51.. | 12 23 | 20 31 | 23 30..
13 | 19 20. | 09 37 | 01 15 | 12 43. | 20 42.. | 23 30
14 | 19 06 | 01 15 | 01 38.. | 13 03 | 20 54 | 23 29
15 | 18 51 | 08 52 | 02 02.. | 13 22. | 21 04. | 23 37..
16 | 18 35... | 08 29... | 02 26 | 13 42 | 21 15. | 23 15..
17 | 18 20. | 08 07.. | 04 49. | 14 01 | 21 25. | 23 23
18 | 18 04 | 07 44... | 03 13 | 14 20 | 21 35 | 23 20..
19 | 17 48 | 07 22 | 03 36 | 14 38. | 21 44. | 23 17.
20 | 17 31. | 06 59 | 03 59.. | 14 57 | 21 53. | 23 14
21 | 17 14.. | 06 36 | 04 22.. | 15 15 | 22 01... | 23 10
22 | 16 57. | 06 13 | 04 45... | 15 33 | 22 10. | 23 05..
23 | 16 40 | 05 50 | 05 0 [...]... | 15 50... | 22 17... | 23 01
24 | 16 22. | 05 26.. | 05 32 | 16 08 | 22 25. | 22 55...
25 | 16 03... | 05 03. | 05 34.. | 16 25. | 22 32.. | 22 50.
26 | 15 45 | 04 39... | 06 17. | 16 42 | 22 39 | 22 44.
27 | 15 27 | 04 16. | 06 40 | 16 58.. | 22 45.. | 22 37...
28 | 15 08 | 03 52... | 07 02.. | 17 14... | 22 51.. | 22 31
29 | 14 49 |  | 07 25 | 17 30... | 22 57 | 22 23...
30 | 14 29.. |  | 07 47 | 17 47 | 23 02 | 22 16.
31 | 14 10. |  | 08 [...]9. |  | 23 06.. |

[Page]

A Table of the Suns Declination every day at Noon, &c.

Month Dayes. | Iuly. | Augu. | Septem | Octob. | Novem | Decem
D.M. | D.M. | D.M. | D.M. | D.M. | D.M.
1 | 22 09 | 15 14. | 04 26.. | 07 12.. | 17 37... | 23 07.
2 | 22 00. | 14 56 | 04 03... | 07 35. | 17 35... | 23 12
3 | 21 51. | 14 37.. | 03 40... | 07 58... | 18 10. | 23 16
4 | 21 42.. | 14 19. | 03 17. | 08 20. | 18 25.. | 23 19..
5 | 21 33 | 14 00... | 02 54. | 08 42.. | 18 41. | 23 22.
6 | 21 23. | 13 41.. | 02 31 | 09 04... | 18 56. | 23 25
7 | 21 13. | 13 22. | 02 08.. | 09 27 | 19 10... | 23 27.
8 | 21 02.. | 13 03 | 01 44. | 09 49 | 19 25. | 23 28...
9 | 20 52 | 12 43. | 01 20... | 10 11 | 19 39. | 23 30
10 | 20 41 | 12 23 | 00 57. | 10 32.. | 19 53 | 23 30...
11 | 20 29 | 12 02... | N 34 | 10 53... | 20 06. | 23 31
12 | 20 17. | 11 43 | N 10.. | 11 15. | 20 19 | 23 30..
13 | 20 05 | 11 21.. | S 13. | 11 36. | 20 31... | 23 30
14 | 19 52... | 11 02 | 00 36... | 11 57 | 20 44 | 23 29
15 | 19 39.. | 10 41.. | 01 00... | 12 18. | 20 55... | 23 27
16 | 19 26... | 10 20.. | 01 24 | 12 38... | 21 07 | 23 25.
17 | 19 13. | 09 59. | 01 47.. | 12 59. | 21 18. | 23 22.
18 | 18 59. | 09 38 | 02 10... | 13 20 | 21 28... | 23 19..
19 | 18 44... | 09 16... | 02 34.. | 13 39... | 21 38... | 23 16
20 | 18 30. | 08 55. | 02 58 | 13 59. | 21 48.▪ | 23 11...
21 | 18 15... | 08 33.. | 03 21. | 14 19 | 21 58 | 23 07..
22 | 18 00.. | 08 11.. | 03 44.. | 14 38 | 22 07 | 23 02..
23 | 17 45. | 07 49... | 04 08 | 14 57.. | 22 15.. | 22 57
24 | 17 29. | 07 27.. | 04 31.. | 15 16 | 22 23.. | 22 51..
25 | 17 31.. | 07 05.. | 04 54.. | 15 35 | 22 31 | 22 44...
26 | 16 57 | 06 43 | 05 18. | 15 53. | 22 38. | 22 38
27 | 16 40.. | 06 20.. | 05 41 | 16 11. | 22 45 | 22 30..
28 | 16 24 | 05 57.. | 06 04 | 16 29. | 22 51.. | 22 23
29 | 16 06.. | 05 35. | 06 27 | 16 46... | 22 57. | 22 15
30 | 15 49. | 05 12.. | 06 49.. | 17 03... | 23 02... | 22 06.
31 | 15 32 |  |  | 17 20. |  | 21 57.

[Page]

A Table of the Suns Declination for every 5th and 10th Day of every Month of the Four years; Calculated from Mr. Street's Tables of the Suns place, made for the years 1665, 1666, 1667, and 1668; the nearest extant.

M. | D | 1. 1665 | 2. 1666 | 3. 1667 | L.Y. 68
Ianuary. | 05 | 21 01 | 21 03.. | 21 06. | 21 09.
10 | 19 58.. | 20 01... | 20 05 | 20 08..
15 | 18 47. | 18 51 | 18 54.. | 18 58.
20 | 17 27. | 17 31. | 17 35 | 17 39..
25 | 15 59... | 16 03... | 16 08 | 16 12...
30 | 14 25 | 14 29... | 14 34.. | 14 39
February. | 05 | 12 24 | 12 29 | 12 34 | 12 39
10 | 10 39.. | 10 43 | 10 48 | 10 53..
15 | 08 47 | 08 52.. | 08 57... | 09 03.
20 | 06 53.. | 06 59 | 07 04... | 07 10.
25 | 04 57.. | 05 03. | 05 08... | 05 14..
28 | 03 47. S | 03 52... | 03 58.. | 04 04.
March. | 05 | 01 48... | 01 54... | 02 00. | 01 42. S
10 | 00 10 N | 00 04. N | 00 01...S | 00 16. N
15 | 02 08 | 02 02.. | 01 56..N | 02 14..
20 | 04 05 | 03 59.. | 03 54 .. | 04 11..
25 | 06 00. | 05 54.. | 05 49. | 06 06..
30 | 07 52... | 07 47 | 07 42. | 07 58.
April. | 05 | 10 03. | 09 58 | 09 53 | 10 09
10 | 11 47... | 11 42... | 11 37... | 11 35.
15 | 13 27. | 13 22. | 13 17.. | 13 32.
20 | 15 01.. | 14 57 | 15 52.. | 15 06..
25 | 16 29. | 16 29. | 16 21. | 16 34
30 | 17 50.. | 17 47 | 17 42... | 17 54..
May. | 05 | 19 04. | 19 01 | 18 57.. | 19 08
10 | 20 10 | 20 07 | 20 04 | 20 13..
15 | 21 07 | 21 04. | 21 02 | 21 10
20 | 21 55.. | 21 53. | 21 51... | 21 57...
25 | 22 34 | 22 32.. | 22 30... | 22 36
30 | 23 03 | 23 02 | 23 01 | 23 04..
Iune. | 05 | 23 24 | 23 23... | 23 23 | 23 25..
10 | 23 30... | 23 30... | 23 30.. | 23 31
15 | 23 27 | 23 27.. | 23 28 | 23 26...
20 | 23 13 | 23 14 | 23 15... | 23 12
25 | 22 48... | 22 50. | 22 51... | 22 47
30 | 22 19... | 22 16. | 22 18 | 22 12.

[Page]

A Table of the Suns Declination, &c.

M. | D | 1. 1665 | 2. 1666 | 3. 1667 | L.Y. 68.
Iuly. | 05 | 21 31 | 21 33 | 21 35. | 21 28
10 | 20 38 | 20 41 | 20 43. | 20 35
15 | 19 36 | 19 39.. | 19 42.. | 19 32...
20 | 18 25.. | 18 30. | 18 34 | 18 23
25 | 17 09... | 17 13.. | 17 17.. | 17 05
30 | 15 45.. | 15 49. | 15 53.. | 15 40..
August. | 05 | 13 55.. | 14 00... | 14 05 | 13 50
10 | 12 18... | 12 23 | 12 28 | 12 13
15 | 10 36.. | 10 41.. | 10 46.. | 10 31
20 | 08 50 | 08 55. | 09 00.. | 08 44
25 | 07 00 | 07 05.. | 07 10.. | 06 53...
30 | 05 07 N | 05 12.. | 05 18 | 05 00...
September. | 05 | 02 48.. | 02 54. | 02 59... | 02 42.
10 | 00 51..N | 00 57. N | 01 03 N | 00 45. N
15 | 01 07 S | 01 00...S | 00 54 S | 01 12. S
20 | 03 05.. | 02 58 | 02 52. | 03 09...
25 | 05 00. | 04 54.. | 04 49 | 05 06..
30 | 06 55. | 06 49 | 06 44. | 07 01.
October. | 05 | 08 48 | 08 42.. | 08 38.. | 08 54
10 | 10 37... | 10 32.. | 10 27. | 10 43..
15 | 12 23 | 12 18. | 12 13. | 12 28...
20 | 14 04 | 13 59. | 13 54.. | 14 09.
25 | 15 39. | 15 35 | 15 30.. | 15 44
30 | 16 08. | 17 03... | 16 59... | 17 12...
Novemb. | 05 | 18 45 | 18 41. | 18 37. | 18 49.
10 | 19 56 | 19 53 | 19 49.. | 19 59...
15 | 20 58. | 20 55... | 20 53 | 21 01..
20 | 21 51... | 21 48.. | 21 46.. | 21 53.
25 | 22 32... | 22 31 | 22 29.. | 22 35
30 | 23 04 | 23 02... | 22 01.. | 23 05...
December. | 05 | 23 20. | 23 22. | 23 21 | 23 23...
10 | 23 30... | 23 30... | 23 30... | 23 31
15 | 23 27 | 23 27 | 23 27.. | 23 26
20 | 23 11 | 23 11... | 23 13 | 23 09
25 | 22 43 | 22 44... | 22 46. | 22 41
30 | 22 05... | 22 06... | 22 08... | 22 01...

[Page]

A Rectifying Table of the minuts and quarters that are to be added or substracted from the fore-going Table of the Suns Declination, made for the second year after Leap-year, for every day at noon in the Meridian of London.

M. | D | 1 year | 3 year | L. year.
Ianuary. | 05 | s. 2.. | a. 2... | a. 5...
10 | s. 3 | a. 3. | a. 6..
15 | s. 3.. | a. 3.. | a. 7
20 | s. 4 | a. 4 | a. 8.
25 | s. 4.. | a. 4. | a. 9..
30 | s. 4... | a. 5. | a. 9...
February. | 05 | s. 5 | a. 5.. | a. 10
10 | s. 5. | a. 5.. | a. 10.
15 | s. 5.. | a. 5... | a. 10..
20 | s. 5... | a. 5... | a. 10...
25 | s. 5... | a. 5... | a. 11
28 | s. 6 | a. 5... | a. 11..
March. | 05 | s. 6. | a. 5.. | s. 12
10 | a. 6.. | s 5.. | a. 12..
15 | a. 6. | s 5. | a. 12
20 | a. 5... | s. 5. | a. 11...
25 | a. 5... | s. 5 | a. 11..
30 | a. 5... | s. 4... | a. 11.
April. | 05 | a. 5.. | s. 4... | a. 11
10 | a. 5.. | s. 4.. | a. 10..
15 | a. 5... | s. 4.. | a. 10
20 | a. 5 | s. 4.. | a. 09
25 | a. 4... | s. 4... | a. 08
30 | a. 4.. | s. 4 | a. 07...
May. | 05 | a. 3.. | s. 3.. | a. 7
10 | a. 3 | s. 3 | a. 6
15 | a. 2... | s. 2 | a. 5..
20 | a. 2 | s. 1... | a. 4
25 | a. 1.. | s. 1.. | a. 3
30 | a. 1. | s. 1 | a. 2
Iune. | 05 | a. 0.. | s. 0... | a. 1
10 | a. 0 | s. 0.. | a. 0
15 | s. 0.. | a. 0 | s. 0..
20 | s. 1 | a. 1 | s. 2
25 | s. 1.. | a. 1. | s. 3
30 | s. 1... | a. 1.. | s. 4
Iuly. | 05 | s. 2.. | a. 2.. | s. 05
10 | s. 3 | a. 2... | s. 06
15 | s. 3. | a. 3 | s. 06...
20 | s. 3.. | a. 3.. | s. 07.
25 | s. 3... | a. 4 | s. 08..
30 | s. 4 | a. 4. | s. 09.
August. | 05 | s. 4. | a. 4.. | s. 09...
10 | s. 4.. | a. 4... | s. 10
15 | s. 5 | a. 5 | s. 10..
20 | s. 5. | a. 5. | s. 11
25 | s. 5.. | a. 5. | s. 11.
30 | s. 5... | a. 5.. | s. 11..
September. | 05 | s. 5... | a. 5... | s. 11...
10 | s. 5... | a. 5... | s. 12
15 | a. 6.. | s. 6.. | a. 12.
20 | a. 6. | s. 5... | a. 12..
25 | a. 5... | s. 5 | a. 11
30 | a. 5.. | s. 4... | a. 11..
October. | 05 | a. 5.. | s. 4.. | a. 11..
10 | a. 5. | s· 4... | a. 11
15 | a. 4... | s. 5 | a. 10..
20 | a. 4... | s. 5... | a. 10
25 | a. 4... | s. 5. | a. 09..
30 | a. 4.. | s. 4.. | a. 09
November. | 05 | a. 4 | s. 4 | a. 08
10 | a. 3 | s. 3.. | a. 07
15 | a. 2.. | s. 3. | a. 06
20 | a. 2 | s. 2.. | a. 05
25 | a. 1... | s. 2 | a. 03..
30 | a. 1.. | s. 1.. | a. 02
December. | 05 | a. 1 | s. 0... | a. 01..
10 | a. 0 | s. 0 | a. 00.
15 | a. 0. | [...]. 0. | s. 01
20 | s. 0.. | [...]. 1. | s. 02
25 | s. 0... | a. 1.. | s. 03
30 | s. 1 | a. 2 | s. 04...

1665 | 1667 | 1668 |  | 1665 | 1667 | 1668
1669 | 1671 | 1672 | For these years. | 1669 | 1671 | 1672
1673 | 1675 | 1676 | 1673 | 1675 | 1676

[Page]

A Table of the Magnitudes, Right Ascention in Hours and Minuts, and Degrees and Minuts, and the Declination North or South of 33 fixed Stars.

N. | Names of the Stars. | M. | R. Asc. | R. Asc | Decli. | N.
| D. M. | H. M. | D. M. | S.
01 | Pole-Star, or last in little Bear. | 2 | 7 53 | 0 32 | 87 33 | N.
02 | Andromedas Girdles | 2 | 12 31 | 0 50 | 33 50 | N.
03 | Medusaes head | 3 | 41 27 | 2 46 | 39 35 | N.
04 | Perseus right side | 2 | 44 30 | 2 58 | 48 33 | N.
05 | Middle of the Pleides | 5 | 51 22 | 3 26 | 23 06 | N.
06 | Bulls eye | 1 | 64 0 | 4 16 | 15 48 | N.
07 | Hircus or Goat | 1 | 72 44 | 4 51 | 45 36 | N.
08 | Orions left foot | 1 | 74 30 | 4 58 | 8 38 | S.
09 | Mid-star in Orions Girdle | 2 | 79 45 | 5 19 | 1 28 | S.
10 | Orions right shoulder | 2 | 84 5 | 5 36 | 7 18 | N.
11 | Auriga, or Waggoner | 2 | 84 45 | 5 39 | 44 56 | N.
12 | Great Dog | 1 | 97 24 | 6 30 | 16 13 | N.
13 | Castor, or Apollo | 2 | 108 00 | 7 12 | 32 30 | N.
14 | Little Dog | 1 | 110 20 | 7 21 | 6 6 | N.
15 | Pollux, or Hercules | 2 | 110 25 | 7 22 | 28 48 | N.
16 | Hydraes heart | 1 | 137 36 | 9 10 | 7 10 | S.
17 | Lyons heart | 1 | 147 30 | 9 50 | 13 39 | N·
18 | Great Bears fore-guard | 2 | 160 48 | 10 43 | 63 32 | N.
19 | Lyons tayl | 1 | 172 45 | 11 31 | 16 32 | N·
20 | Virgins Spike | 1 | 196 43 | 13 07 | 9 11 | N·
21 | Last in great Bears tayl | 2 | 203 36 | 13 34 | 51 5 | N·
22 | Arcturius | 1 | 209 56 | 14 00 | 21 4 | N.
23 | Little Bears fore-guard | 2 | 222 46 | 14 52 | 75 36 | N.
24 | Brightest in the Crown | 3 | 231 00 | 15 24 | 27 43 | N.
25 | Scorpions heart | 1 | 242 23 | 16 09 | 25 37 | S.
26 | Hercules head | 3 | 254 40 | 16 59 | 14 51 | N.
27 | Lyra, or Harp | 1 | 276 17 | 18 25 | 38 30 | N.
28 | Eagle, or Vulture | 1 | 293 28 | 19 35 | 8 1 | N.
29 | Swans tayl | 2 | 307 30 | 20 30 | 44 5 | N
30 | Dolphins head | 3 | 307 53 | 20 32 | 15 0 | N.
31 | Pegassus mouth | 1 | 321 50 | 21 27 | 8 19 | N
32 | Pomahant | 3 | 339 30 | 22 38 | 31 17 | S.
33 | Pegassus lower wing | 2 | 358 50 | 23 55 | 13 22 | N.

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As for Example.

To find the Suns Declination for the year 1670, on the 12th day of May: First, if you divide 70 (being the tens only of the year of our Lord by 4, rejecting the 100s) you shall find 2, as a remainder, which notes it to be the second after Leap-year; and if 0 remain, then it is Leap-year.

Then,

Look in the Table of Declination for 1666, the second after Leap-year, as the year 1670 is, and find the Month in the head of the Table, and the day on one side, and in the meeting-point you shall find 20 deg. 31 min. for the Declination on that day at noon required.

Or,

If you use the Trianguler Quadrant, extend the Thred from the Center over the 12th of May, and you shall find it to cut in the degrees just 20 deg. 31 min. the true Declination for that year and day.

Note, That if you have occasion to use the Declination before noon, then observe that the difference between stroke and stroke, is the difference of Declination for one day; and by consequence, one half of [Page] that space for half a day; and a quarter for a quarter of a day, &c.

As thus for Example.

Suppose I would have the Suns Declination the 18th of August 1666, at 6 in the morning; here you must note, that the 18th stroke from the beginning of August, represents the 18th day at noon just.

Now the time required being 6 hours before noon, Lay the Thred one fourth part of the distance for one day, toward the 17th day, and then in the degrees, the Thred shall cut on 9-43′, whereas at noon just, it will be but 9-38; and the next, or 19th day at noon, it is 9 degrees 16 min. and 3 quarters of a min. as the three pricks thus ... in the Table doth plainly shew; but by the Rule, a minut is as much as can be seen, and so near with care may you come.

Note also farther,

That if you shall use it in places that be 4 hours, 6 or 8, 10 or 12 hours more Eastward, or Westward in Longitude, the same Rule will tell you, the minuts to be added in Western-Longitudes, or to be substracted in Eastern-Longitudes, as Reason and Experience will dictate unto you with due consideration.

[Page]For if being Eastwards, the Sun comes to the Meridian of that place before it comes to the Meridian of London; then lay the Thred as in morning hours: But if the place be to the Westwards where it comes later, then lay the Thred so many hours beyond the Noon-stroke for London, as the place hath hours of Western-longitude more than London, counting 15 degrees for an hour, and 4 minuts for every degree; and then shall you have the Declination to one minut of the very truth.

But if it happens to be the Leap-year, or the first or third year after the Leap-year, then thus;

Suppose for the 5th of October 1671, being the third after Leap-year, I would have the Declination.

First, if you lay the Thred over the 5th of October, in the degrees, it gives 08 deg. 42 minuts .., for the Declination in the second year after Leap-year; then, because this is the third year, look in the Rectifying-Table for the 5th of October, and there you find s. 4 .., for substract 4 minuts and a half from 8-42 .. rests 8-38, the true declination required for the 5th of October 1671. The like work serves for any other day or year; but for every 5th and 10th [Page] day, you have the Declination set down in a Table for all 4 years, to prove and try the truth of your Operations; and by that, and the Line of Numbers, or the Rule of Three, you may continue it to every day by this proportion.

As 5 dayes, or 120 hours, to the difference of Declination in the Table, between one 5th day and another;

So is any part of 5 dayes, or 120 hours, to the difference in Declination to be added or substracted to the 5 dayes Declination immediately fore-going the day required.

Example.

Suppose for the 18th of February 1669, the first after Leap-year, I would know the Declination by the Table made to every 5th day only; On the 20th of February, I find 6-53 ½; On the 15th day, 8-47; the difference between them is 1-53 ½; then the Extent of the Compasses from 5, the Number of dayes, to 1-53▪ the minuts difference (counted properly every 10th for 6 minuts) shall reach from 3, the dayes from 15 toward 18, to 1 degree 7 minuts and a half, which taken from 8-47′, the Declination for the 15th day, leaves 7 degrees 38 minuts and a half, the true Declination [Page] for the 18th day of February, in the first after Leap-year.

Or, by the Line of Numbers thus;

The Extent from 5, the difference in dayes, to 113 ½, the difference in min. for 5 dayes, shall reach from 3, the difference in dayes, to 68, the difference in minuts for 3 dayes, to be added or substracted, according to the increasing or decreasing of the Declination at that time of the year.

Proved thus;

If you substract 5′ ½ from 7 deg. 44... the declination in the second year, there remains 7 deg. 38′ ½, the Declination for the 18th of February, 1669.

These Tables may serve very well for 30 years, and not differ 6 minuts in Declination about the Equinoctial, where the difference is most; and in Iune and December not at all to be perceived.

Thus you may by the Rule and Rectifying Table, find the Suns Declination to a minut at any time, without the trouble of Calculation.

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CHAP. IV. The use of the Trianguler-Quadrant in the Operative part of Navigation.

Use I. To find how many Leagues, or Miles, answer to one Degree of Longitude, in any Latitude between the Equinoctial and Pole.

FIrst, it is convenient to be resolved how many Leagues or Miles are in one Degree in the Meridian or Equinoctial, which Mr. Norwood and Mr. Collins hath stated about 24 leagues, or 72 miles.

Or.

If you keep the old number, making the miles greater, viz. 60 miles, or 20 leagues; then the proportion, by the Numbers, Sines and Tangents, runs thus;

[Page]As Sine 90, to 20 on the Numbers for leagues;

So is Co-sine of the Latitude, to the leagues, on the Numbers, contained in one degree of Longitude in that Latitude.

But in Miles, to have the Answer, work thus;

As Sine 90, to 60 on Numbers;

So Co-sine Latitude, to the number of miles.

Example, Latitude 51° 32′.

As Sine 90, to 60;

So Sine 38-28, to 37 miles 30/100.

But by the Trianguler-Quadrant, or Sector, work thus;

Take the latteral 20 for leagues (or 60 for miles) from the Line of Lines from the Center downwards; and make it a parallel in the sine of 90, laying the Thred to the nearest distance.

Then,

The nearest distance from the Co-sine of the Latitude, to the Thred, measured latterally from the Center, shall shew the true number of Leagues required.

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Example, Latitude 51° 32′.

As — 60, to = sine of 90;

So = sine of 38-28, to — 37-30, on the Lines.

As — 20, to = sine of 90;

So is = sine of 38-28, to — 12-40 for Leagues.

Or,

As — 24, to = sine of 90;

So is = sine of Co-lat. to — 15, the number of Leagues, after the experiment made by Mr. Norwood, of which true measure you may read more in the Second Part of the Plain Scale, by Mr. Collins.

Or,

If you multiply the Natural Sine of the Co-lat. by 2, it gives the Leagues; or by 6, it gives the Miles in one degree, cutting off the Radius from the Product.

Note also, That if you take the Natural-Number of the Secant of the Course or Rumb, and multiply it by 2, cutting off the Radius from the Product, it shall give the Leagues required, to raise one degree, at the rate of 20 Leagues to one Degree of a great Circle.

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Use II. To find how many Leagues, or Miles, answer to Raise, or to Depress the Pole one degree on any Rumb from the Meridian.

First, by the Artificial Sines, Tangents, and Numbers.

As the Co-sine of the Rumb from the Meridian, to 20 Leagues (or 24 Leagues) on the Numbers;

So is the sine of 90, to the number of Leagues required.

Which, when you have sayled on that Rumb, you shall raise or depress the Pole one degree.

But by the Trianguler-Quadrant, thus;

As — 20, taken from the Line of Lines, or any equal parts, to the = Co-sine of the Rumb, laying the Thred to the nearest distance.

So is the = sine of 90, or nearest distance from sine 90, to the Thred, to — Number of Leagues required, to sayl on that Rumb, and to raise the Pole one degree.

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Use III. To find how many Miles or Leagues answer to any number of degrees in any parallel of Latitude.

Suppose you sayling in the Latitude of 48 degrees, have altered your Longitude 30 degrees, and would then thereby know how many leagues you had sailed.

First, bring (or reduce) the 30 degrees to Leagues, by multiplying them by 20, or 24, (the leagues resolved to be in one degree) which makes 600, (or 720).

Then by the Numbers and Sines.

The Extent from the sine of 90, to 42 the Co-sine of the Latitude, shall reach the same way, from 600 on the Numbers, to 400 the leagues required; or from 720, to 480, according to Mr. Norwood.

By the Trianguler-Quadrant.

Take — 600 from the Line of Lines, or any equal parts, and make it a = in the sine of 90, laying the Thred to the nearest distance.

Then,

The nearest distance from the sine of 42, [Page] (the Co-sine of the Latitude to the Thred) and it shall give 400 on the Lines, or equal parts, the leagues required.

Which is thus more briefly;

As — 600, to = 90;

So is = sine 42, to — 400, as frequently before.

Use IV. To work the six Problems of Plain Sayling by Gunter's Lines on the edge, or the Trianguler-Quadrant.

Note, That in this Art of Navigation, or Plain Sayling, that the Angle that any degree of the Quadrant, or Point of the Compass makes with the Meridian, or North and South-line, that is called the Rumb or Course.

But the Angle that it maketh with the East and West-line, or parallel, is called the complement of the Rumb or Course.

Note, That in plain Triangles, the Sines and Tangents give Angles, and the Numbers give Sides.

Note also, That in Plain Sayling, the distance run, or Course, is the same with the Hypothenusa in plain Triangles.

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[Page]

[figure]

[Page] Also note, That the difference of Latitude is counted on the Meridian, and the difference of Longitude or Departure from the Meridian, is counted on the Equinoctial, or on a Parallel of Latitudes.

One of which Lines, in plain Triangles, is called the Base; and the other, the Perpendiculer. The Base being a sine, and the Perpendiculer a sine complement.

Note also, That in North Latitude, Sailing Southerly, the Latitude doth decrease; therefore you must substract the difference in Latitude, from the Latitude you parted from; but if you sayl Northerly, then you must add it to the Latitude you parted from: The like in South Latitudes.

But when one Latitude is South, and the other North, then you must add them both together.

Note also, That the difference in Latitude and Longitude, (and Departure) when given in degrees, are to be reduced to Leagues, by multiplying by 20, and counted alwayes on the Line of Numbers, or equal-parts, when you use the Trianguler-Quadrant.

So then in using the Index and Square in Plain Sailing, the distance sayled, is alwayes counted on the Index from the Center.

[Page]The Course is counted on the degrees from the Head toward the Loose-piece.

The Difference of Latitude on the Head-Leg, from the Leg-center to the Head.

The Departure or Longitude, is counted on the Square.

The complement of the Course or Rumb is counted on the degrees, beginning at 00 on the Loose-piece.

When your number of Leagues exceed 100, you must double the Numbers on the Index, the Square, and Head-leg, or count 10 for a 100, &c.

Problem I. The Course, and Distance run on that Course, being given, to find the difference in Latitude, and Departure, or difference in Longitude.

As sine of 90, to the distance run (or Leagues sayled) on the Line of Numbers;

So is Co-sine of the Course or Rumb, to difference in Latitude on the Numbers.

Again, for the Longitude or Departure.

So is the sine of the Course, to the Departure, or difference in Longitude.

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By the Trianguler-Quadrant.

As — Leagues sailed, to = sine 90, laying the Thred to the nearest distance;

So = Co-sine of the Rumb or Course, to — difference in Latitude. Or,

So is = sine of the Rumb or Course, to = Departure, or difference in Longitude.

By the Index and Square, after the manner of a Synical Quadrant, thus;

Set the Index (being put over the Leg-Center-pin) to the Course counted on the degrees from the Head, toward the Loose-piece.

Then slide the Square perpendiculer to th [...] Head-leg, till the divided edge thereof cuts the distance run on the Index; then shall the Index, on the Square, give the Departure or Difference in Longitude; and the Square on the Head-leg, shall shew the Difference in Latitude.

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Problem II. The Course and Difference of Latitude given, to find the Distance run, and Departure.

As Co-sine of Course, to the Difference in Latitude;

So is sine 90, to the Distance run.

Then,

As sine of 90, to the Distance run;

So is sine of Course, to the Departure.

By the Trianguler-Quadrant, without the Square.

As — difference of the Latitude, to = Co-sine of the Course;

So = sine 90, to — distance run.

So is = sine of the Course, to the Departure.

With the Index and Square, thus;

Set the Index to the Course, and the Square to the difference in Latitude; then on the Index, is cut the Distance; and on the Square, the Departure.

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Problem III. The Course and Departure given, to find the distance run, and difference of Latitude.

As sine Course, to the departure on Numbers;

So is sine 90, to the distance.

Again,

As the sine 90, to distance run;

So is Co-sine Course, to difference in Latitude.

By the Trianguler-Quadrant.

As — Departure taken from any fit Scale, to = Co-sine of the Course;

So is = sine 90, to — distance run on the same Scale.

So is = sine of the Course, to the difference in Latitude.

With the Index and Square.

Set the Index to the Course, and slide the Square perpendiculer to the head-leg, till the Index cuts the departure on the Square; then the Index sheweth the Distance, and the Square the Latitude on the Head-leg, counting from the Center.

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Problem IV. The Distance run, and difference in Latitude given, to find the Course and Departure.

As the Leagues run, to sine 90;

So the difference in Latitude, to Co-sine Course.

Again,

As sine 90, to the distance run;

So is sine of the Course, to the Departure.

By the Trianguler-Quadrant.

As — Radius, or a small sine of 90, to = distance run on the Line of Lines;

So is = difference in Latitude, to Co-sine of the Course, measured on the small Sine.

So is — sine of the Course, to the = departure, carried = in the Lines.

By the Index and Square.

Set the Square to the difference in Latitude, and move the Index till the Square cuts the distance run on the Index; then shall the Index shew on the Square, the Departure; and on the Degrees, the Course required.

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Problem V. The Distance run, and Departure given, to find the Course and Difference in Latitude.

As the Distance run, to sine 90;

So is the Departure, to sine of the Course.

Then,

As sine 90, to the Distance run;

So is Co-sine Course, to the Difference in Latitude.

By the Trianguler-Quadrant.

As — distance run, to = sine 90;

So — departure, to = sine of the Course.

So is = Co-sine Course, to — difference in Latitude.

By the Square and Index.

Slide the Square and Index, till the Index cuts the Departure on the Square, and the Square cuts the Distance run on the Index;

Then,

On the Degrees, the Index shall shew the Course; and on the Head-leg, the Square shall shew the difference in Latitude.

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Problem VI. The Difference of Latitude, and the Departure given, to find the Course and Distance run.

As the Difference in Latitude, to 45 degrees;

So the Departure, to the Tangent of the Course.

Again,

As sine Course, to the Departure;

So is sine 90, to the Distance run.

By the Trianguler-Quadrant.

As — Radius, or Tangent of 45, to = Difference in Latitude;

So is = Departure, to — Tangent of the Course on the Loose-piece from whence you took 45.

Then,

As — Departure, to = sine of the Course;

So is = 90, to — Distance run.

By the Index and Square.

Set the Square to the Difference in Latitude, on the Head-leg, counted from the Center; and bring the Index to cut the [Page] Departure on the Square; then the Square on the Index shews the distance; and the Index, on the degrees, gives the Course required.

In all these 6 Problems, which Mr. Gunter makes 12 Problems, of Plain Sayling, I have set no Example, nor drawn no figure, because the way by the Index and Square is so plain; and of it self makes a figure of the work: For the Index is alwayes the Distance run, the Hypothenusa▪ [...] Secant: The Square sheweth the Departure; and the Line of Lines on the Head-leg, the difference of Latitude: And you may not only perform the work, but also see the reason thereof, being a help to the fancy of young Learners in these Nautical Operations: And if your Square playes true, you may be more exact than you can by Scale and Compass, and much more quick and ready; not only in this, but any thing else in right-Angled plain Triangles, as in Heights and Distances, and the like.

Use V. The use of the Meridian Line, and his Scale.

These six Problems of Plain Sayling for [Page]short Distances, may come very near the matter; as in making a Traverse of the Ships way from place to place Coasting, as in the Streights, and the Channel, and the like: But for great Distances, it is not so certain as the Sayling by Mercators Chart; therefore to that purpose the Meridian-line was invented, to reduce degrees on the Globe, to degrees in Planò, as Mr. Wright hath largely shewed.

On the innermost-edge of the Rule, or Trianguler-Quadrant, you may have a Meridian-line so large, as to have half an inch for one degree of the Equinoctial; and the inches for measure, to go along by it; or rather you may have it lie near to the Line of Lines on the Head-leg, as you shall think most convenient, for then it will be the same as Mr. Gunter's is, and perform his very Operations, as near as may be, after his way, by the Thred and Compasses, or Index and Compasses.

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Problem I. Two places being propounded, one under the Equinoctial, the other in any Latitude, to find their Meridional difference in degrees and minuts, or 100 parts.

Look for the Latitude of the place, scituate out of the Equinoctial in the Meridian-line, and right against in the equal-parts is the Meridional difference of those two places.

Example.

Let the River of Amazones, under the Equinoctial, be one place; and the Lizard, in the Latitude of 50 degrees North, another place; look for 50 on the Meridian-line, and right against it, on the equal-parts, is 57-54, for 57 degrees 54 minuts; or in Decimal parts of a degree 57-90.

Problem II. Any two places having both Southerly or Northerly Latitude, to find the Meridional difference between them.

Extend the Compasses on the Meridian-line, from one of the Latitudes to the other; the same Extent laid from the beginning of [Page] the Scale of equal-parts, by the Meridian-line, shall reach to the Meridional difference required.

Or,

The measure from the least Latitude, to the beginning on the Meridional-line, shall reach the same way from the greater, to the difference on the equal-parts.

Example.

If the Latitude of one place be 30 degrees, and the other 50 degrees; Extend the Compasses from 30 to 50 on the Meridian-line, and that Extent shall reach on the equal-parts, from the beginning of the Line, to 26 degrees 26 minuts.

Problem III. When one place hath South Latitude, and the other North Latitude, to find the Meridional difference.

Extend the Compasses from the beginning of the Line of Meridians, to the lesser Latitude; then that Extent applied the same way on the Meridian-line from the greater Latitude, shall shew on the Scale of equal-parts the Meridional difference required.

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Example.

Suppose one Latitude be 10 deg. South, and the other 30 deg. North; The Extent from 0 to 10 degrees, shall reach from 30, to 41° 31′, the Meridional degrees required.

Problem IV· The Latitudes of two places, together with their difference in Longitude being given, to find the Rumb directing from one to the other.

As the Meridional difference in Latitude, to the difference in Longitude;

So is the Tangent of 45, to the Tangent of the Rumb or Course.

Example.

Let one place be in the Latitude of 50 North, the other in 15 deg. and 30 min. North, as the Lizard-point, and St. Christophers; and let the difference in Longitude be 68 degrees, 30 minuts; and let the Rumb, leading from the Lizard to St. Christophers, be required·

First, by the Meridian-line, and the Scale of Equal-parts, by Problem II. find the Meridional difference in Latitudes, which in our Example will be 42 degrees, and 12 parts of a 100.

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Then,

The Extent on the Line of Numbers, from 42 degrees and 12 minuts, the Meridional difference in Latitude, to 68 degrees and 30 minuts, the difference in Longitude shall reach the same way from the Tangent of 45, to the Tangent of 58 degrees and 26 minuts, the Rumb from the Meridian of the Lizard Westwards, being two degrees, and better, beyond the 5th Rumb from the Meridian.

By the Trianguler-Quadrant thus;

As the — Tangent of 45, taken from the Loose-piece, is to the = Meridional-difference in Latitudes on the Line of Lines;

So is the = difference in Longitudes, to the — Tangent of the Course 58 degrees 25 minuts.

But by the Index and Square, this is wrought very easily and demonstratively thus;

Count the Meridional difference of Latitudes on the Head-leg down-wards from the Center, as 42 and 12 on the Line of Lines, and set the Square to it.

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Then,

Count the difference of Longitudes on the Square, viz. 68-50, and to that Point lay the Index; and then the Index on the degrees shall cut the complement of the Course, viz. 31-35, or 58-25, if you count from the Head.

Having been so large in this, I shall contract the rest.

Problem V. By the two Latitudes and the Rumb, to find the Distance on the Rumb.

As the Co-sine of the Rumb, to the true difference of the Latitudes, (on the Numbers);

So is the sine of 90, to the distance on the Rumb required, (on the Numbers).

Being given in degrees and Decimal parts, and brought to Leagues by multiplying by 20, or 24, according to Mr. Norwood, as before.

Note also, That the true difference of Latitudes, is found by Substraction, of the less Latitude out of the greater.

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By the Quadrant.

As — true difference of Latitudes, to = Co-sine of the Course or Rumb;

So is = sine of 90, to — distance on the Rumb (in the same Line of Lines).

The Index and Square is used as in the second Problem of Plain Sayling.

Problem VI. By the two Latitudes, and distance between two places given, to find the Rumb.

As the distance sayled, in the degrees and 100 parts, counted on the Lins of Numbers, is to the true difference of Latitudes, found as before, by Substraction;

So is the sine of 90, to the Co-sine of the Rumb required.

As — sine of 90, to = distance sailed;

So is = difference of Latitudes, to — Co-sine of the Course.

By the Index and Square, work as in Problem IV. of Plain Sayling.

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Problem VII. Both Latitudes and the Rumb given, to find the difference of Longitude.

As the Tangent of 45, to the Tangent of the Rumb;

So is the Meridional difference of Latitudes, to the difference of Longitude required.

As — Tangent of 45, to = Tangent of the Rumb, (first laid on the Lines from the Loose-piece);

So is the — Meridional difference of Latitudes, to the difference of Longitudes.

By the Index and Square, work as in the 4th Problem last past.

Problem VIII. By one Latitude, Distance and Rumb, to find the other Latitude.

As sine 90, to the Co-sine of the Rumb;

So is the distance, to the true difference of Latitude.

As — Co-sine of the Course, to = sine 90;

So is = distance, in degrees and parts, on the Lines, to the — true difference in Latitudes, to be added or substracted from the Latitude you are in, according as you [Page] have increased, or depressed the Latitude in the Voyage.

By the Index and Square, work as in the 5th Problem last past, or 2d or Plain Sayling.

Use VI. To find the distance of places on the Globe of Earth and Sea; Or, Geography by the Trianguler-Quadrant.

Problem I.

When two places are scituated under the same Meridian (or Longitude) and on the same side of the Equinoctial; then substract the lesser Latitude out of the greater, and the remainder shall be the distance in degrees required, counting 20 (or 24) Leagues to a degree on every great Circle of the Sphear.

Problem II.

When one place is on one side of the Equinoctial, and the other on the other side; and yet both on one Meridian, as was the former; then the two Latitudes (viz. the North-latitude, and the South-latitude) added together, shall give the distance in degrees required.

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Problem III.

When the two places differ only in Longitude, and are both under the Equinoctial, then substract the lesser Longitude from the greater, and the residue is the distance in degrees.

Problem IV.

When the two places have both one Latitude, or near it, North or South, and differ only in Longitude.

Then work thus;

As sine 90, to Co-sine of the (middle) Latitude;

So is the sine of half the difference in Longitude, to the sine of half the distance.

By the Trianguler-Quadrant, or Sector.

As — Co-sine of the mean Latitude, to the = sine of 90, laying the Thred to the nearest distance;

So is = sine of half the difference in Longitude, to — sine of half the distance.

Problem V.

When both places have different Longitudes and Latitudes, as these Three Wayes following▪

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I Way. When one place hath no Latitude, and the other North or South, with difference in Longitude also; then,

As sine 90, to Co-sine of difference in Longitude;

So the Co-sine of the Latitude, to the Co-sine of the distance required.

By the Trianguler-Quadrant, thus;

As — Co-sine of difference in Longitudes, to the = sine of 90;

So the = Co-sine of the Latitude, to the = Co-sine of the distance.

II Way. When both the places have either North or South Latitude, that is, both toward one Pole; then thus,

As the sine of 90, to the Co-sine of the difference in Longitude;

So the Co-tangent of the lesser Latitude, to Tang. of a 4th Ark.

Which 4th Ark, must be taken out of the complement of the greater Latitude, when the difference of Longitudes is less than a Quadrant, or added to it when more, then [Page] the sum or difference shall be a 5th Ark.

Then,

As the Co-sine of the 4th Ark, to Co-sine of the 5th Ark;

So is the sine of the lesser Latitude, to the sine of the distance.

By the Trianguler-Quadrant.

As — Co-sine of difference in Longitudes, to = sine of 90;

So is the = Co-tangent of the lesser Latitude, taken from the Loose-piece, and laid from the Center, and from thence taken parallely to the — Tangent of a 4th Ark, which do with, as before is shewed, to find a 5th Ark.

And then,

As the — Co-sine of the 4th Ark, to the = Co-sine of the 5th Ark;

So is the — sine of the lesser Latitude, to = Co-sine of the distance.

III Way. But when one Latitude is on one side the Equinoctial, and the other on the otherside, viz. one having North-latitude, and the other South. Then,

As the sine of 90, to the Co-sine of the difference in Longitude;

[Page]So is the Co-tangent of one Latitude, to the Tangent of a 4th Ark.

Which taken out of the other Latitude, and 90 deg. added, when the difference of Longitude is less than a Quadrant, but added to it if more than a Quadrant, and that sum or difference shall be the 5th Ark.

Then,

As the Co-sine of the 4th Ark, to the Co-sine of the 5th Ark;

So is the sine of the Latitude, first taken, to the Co-sine of the distance in degrees.

By the Trianguler-Quadrant, or Sector;

As the — Co-sine of the difference of Longitudes, to = sine of 90;

So the = Co-tangent of one Latitude (being first taken from the Loose-piece, or Moveable-leg, and laid from the Center downwards, and from thence taken parallely) to the — Tangent of a 4th Ark.

Which 4th Ark you must do with, as before, to obtain a 5th Ark.

Then,

As — Co-sine of the 4th Ark, to = Co-sine of the 5th Ark;

So — sine of the Latitude, first taken, to = Co-sine of the distance.

[Page]That is, when the 4th Ark is substracted; or, to the Co-sine of the comp. distance when added.

Example.

Suppose I would know how far it is from the Lizard, to the Cape of Good Hope; the Lizard having 50 degrees of North-latitude, and the Cape of good Hope 35 degrees of South-latitude, and the difference in Longitude 30 degrees.

As the sine of 90, to the Co-sine of the difference in Longitude 30, being best counted from 90 backwards;

So is the Co-tangent of 50, (viz. at 40) to 36 degrees 01 minut, a 4th Ark.

Then 90 degrees, and 35 degrees, the other Latitude added, makes 125; from which sum, taking the 4th Ark, remains 88-59, for a 5th Ark.

Then say,

As the Co-sine of the 4th Ark 36-1, to the Co-sine of the 5th Ark 88-59;

So is the sine of 50, the Latitude first taken, to the Co-sine of the distance 89 deg. 3 min. the nearest distance in the Arch of a great Circle.

[Page] Note, That here you will have occasion to make use of that help mentioned p. 218, Sect. 3. As thus for instance.

The Proportion being as the Co-sine of 36-1, to the Co-sine of 88-59; which is all one, as the sine of 54 and 59, to the sine of 1 degree and 1 minut, which is too large for ordinary Compasses, on ordinary Gunters Rules; therefore first lay the distance from the sine of 90, to the sine of 54 and 59, the same way from the sine of 5 degrees and 45 minuts, and note the place.

Also, Lay the distance from the sine of 90, to the sine of 50, the same way from the sine of 5 degrees and 45 minuts; and note that place also.

Then,

As the Extent first noted for 54-59, is to 1 degree and 1 minut, the Co-sine of 88-59;

So is the second mark noted for 50, to 89-3, the distance in degrees required.

Which multiplyed by 72, gives the distance in the Arch of a great Circle, viz. 6412 miles Statute-measure; Or, 5340 miles, whereof 60 make one degree, on a great Circle on the superficies of the Sea.

[Page]

Use VII. To find the distance of places by the Natural Versed Sines in the way of a Sector on the Trianguler-Quadrant, being much more easie than the two former wayes.

First, by the Pen, find the sum and difference of the complements of the two Latitudes, and count that sum and difference on the versed Sines latterally, and take the distance between your Compasses, and make it a parallel versed Sine of 180 degrees.

Or, by the Trianguler-Quadrant.

If you have not the Line set on from the Leg-center, then the small Line of Sines beyond the Leg-center, being doubled, will do the work, by taking the distance between the sum and difference, and setting one Point in the Center-prick at two times the Radius of the Sines from the Leg-center; and then laying the Thred to the nearest distance, or the Line of Right Asceneions under the Months, is a fit Line.

Then,

Take out the = difference of Longitude, [Page] and that shall reach latterally from the difference to the distance required.

Example.

London and Ierusalem, two places in North Latitude; London 51-32, Ierusalem 32-0, whose two complements 38-28 and 58 added, make 96-28 for a sum, and one taken from the other, leave 19-32 for a difference.

Now the — distance between the versed Sines of 96-28, and 19-32, make a = versed Sine in 180, keeping the Sector so, or laying the Thred to the nearest distance, (and noting where it cuts in degrees).

Then,

The = distance between 47, the difference of Longitude between the two places, shall reach on the versed Sines from 19-32, the difference to 39-14, the distance required; which, at 72 miles to a degree, makes 2805 miles.

Note, This one Rule comprehends all the Three last Wayes, and is not troubled with half so many Cautions as the former.

[Page]

Use VIII. Having the Latitudes and Distance of two places, to find their Difference in Longitude.

Find the sum and difference of the two Co-latitudes, as before, by Addition and Substraction; count them on the versed Sines, and take the — distance between, and make it a = versed Sine of 180.

Then,

The — distance, between the difference and distance on versed Sines, shall stay at the = difference in Longitudes required.

Example.

Let one place be Burmudas Isle, and the Latitude thereof 32-25; let the other place be the Lizard-point, and the Latitude thereof 50 degrees; the Co-latitudes are 57-35 and 40-0; the sum of them is 97-35; the difference between them is 17-35. The distance in the great Circle, according to Mr. Norwood, is 44-30, or 886 Leagues, counting 20 leagues to one degree.

Then,

The — distance between the versed Sines of 17-35, and 97-35 made a = versed Sine of 180, the Sector is set.

[Page]

Then,

The — distance taken between 44-30, and 17-37 on the versed Sines, and carried parallelly, shall stay at 55, the difference in Longitude required between those two places.

CHAP. V. Of Sayling by the Arch of a great Circle.

IN the Book called, The Geometrical Seaman, by Mr. Phillips, is a very ready Figure to shew in a Quadrant, or more, by what Longitudes and Latitudes a Ship is to pass in any long-run, which is contained under 90 degrees, or 120 difference of Longitude, and the two places having both North Latitude.

Which Figure, or Quadrant, is neatly and readily performed by the Trianguler-Quadrant, thus;

[Page]Upon the back-side of the Index, before spoken of, may be graduated from the Center, two Tangent-Lines, one equal to the Radius on the Loose-piece, the other to the Radius on the Moving-leg; then in the use, count the fiducial Line in which the Leg-Center-pin stands, alwayes for the Meridian of one place; and some where in that Line, according to the latitude thereof, counting the Leg-center the Pole of the World; and the Index being hung thereon, by the Tangents prick down the Latitude; there, I say, knock in a Pin to stay a Thred for one place; then, on the degrees, count the difference of Longitude from the Head-leg, and lay the Index to it, and bring the Thred fastened, as before, till on the Index it cuts the degree and part of the other Latitude, and there make the Thred fast with another Pin in the Loose-piece.

Then,

If you move the Index to any degree of Longitude between those places, the Thred shall cut on the Index the degree of Latitude that answers unto it; or if you make the Thred cut any degree of Latitude, the Index gives the Longitude required for that Latitude.

[Page] Note, If the Latitude be small, as between 10 and 30, the small Tangents are most convenient; but if it be between 40 and 80, the greater Tangent Line is best.

Note, That two Threds and a pair of Compasses may serve; but the Index is much better and quicker in Operation.

Example▪

Let the two places be the Summer-Islands and the Lizard-point; the same Example that you find in Mr. Norwood, pag. 126; and in Mr. Phillip's Geometrical-Seaman, pag. 55. that you may the more readily compare the truth thereof by their Operations.

The Latitude of the Lizard Point is 50 degrees, the Longitude is 10.

The Latitude of the Summer Islands is 32-25, the Longitude is 300.

The Difference of Longitudes is 70, as is computed by their Observation.

Then,

Hanging or putting the Center-hole of the Index over the Leg-center-pin, and counting the fiducial-line on the Head-leg for the Meridian of one place, count on the Tangent Line on the Index the Co-tangent of one Latitude, as suppose the Latitude of the Lizard-point (the Center alwayes counted [Page] as 90) and there knock in a Pin in a small hole to hang a Thred on.

Then count 70 degrees, the difference in Longitude, on the degrees from the Head-leg, and there stay it; then draw the Thred put over the first Pin, till it cut the complement of the other Latitude, and by help of another Pin stay it there, which you may conveniently do by one of the sliding-sights; then the Thred being so laid, slide the Index to every single degree, or fifth degree of Longitude, and then the Thred shall shew on the Index, the Co-tangent of the Latitude answerable to that degree of Longitude, as in the Table annexed.

Also, If you would have equal degrees of Latitude, and would find the Longitude according to it; then slide the Index to and fro till the Thred cuts on the Index an even degree of Latitude; then on the degrees you have the difference of Longitude from either place.

Also note, That the drawing of one Line only on the Trianguler Instrument in the beginning, according to the directions of laying of the Thred; with the Thred and Compasses, will perform this work also.

[Page]

The Table.

Long | D.L. | Latitude
300 | 09 | 32—25
305 | 05 | 35—52
310 | 10 | 38—51
315 | 15 | 41—24
320 | 20 | 43—34
325 | 25 | 45—24
330 | 30 | 46—54
335 | 35 | 48—07
340 | 40 | 49—04
345 | 45 | 49—47
350 | 50 | 50—15
355 | 55 | 50—31
360 | 60 | 50—33
05 | 65 | 50—23
10 | 70 | 50—00

If this work fit not any case that may happen, there is another way mentioned in Page 75 of the Geometrical Seaman, by the Steriographick Projection; and that Scheam is drawn the same way, as the Horizontal-Projection for Dyalling was, and somewhat easier; and any two Points given, in a [Page] Circle, you may draw a great Circle to cut them, and the first Circle into two equal-parts, by the directions in Page 15; And the Application thereof you have very plainly in Mr. Phillips his Book, to which I refer you, having said more than at first I intended, which was chiefly the use thereof in Observation only.

So for the present I conclude this Discourse, and shall endeavour a further Advantage in the next Impression, according as Time and Opportunity shall offer. Farewel.

The End of the Second Part.

The Table of the Things contained in this Second Part.

THe difinition and kind of Dials,

Page. 7

Directions to draw the Scheam,

9

To draw Lines to represent the several sorts of Plains in the Scheam,

13

To draw a Scheam particularly for one Dial,

14

To draw the Equinoctial Dial,

19

To try when a Plain lies Equinoctial,

20

[Page]To draw a Polar-Dial,

21

To draw an Erect East or West-Dial,

24

To draw a Horizontal-Dial,

27

The d [...]monstration of the Canon for Hours,

ib.

To draw a Direct Erect South or North-Dial,

30

To draw a Direct Recliner,

33

The use of the Figure,

35

To draw a Direct East or West Recliner,

37

To make the Table of Arks at the Pole,

42

To refer those Dials to a new Latitude, and a new Declination, wherein they may become Erect Decliners,

46

To find the Requisites by the Scheam,

ibid.

To find the Declination of a Plain by the Needle, or by the Sun,

49

To take off an Angle, or set the Sector to any Angle required,

53

Precepts to find the Declination by the Sun, and Examples also of the same,

58

To draw an Erect Declining-Dial,

62

The Proportions for the Requisites of Erect Decliners,

64

To find the Requisites Three wayes,

66

To draw the Erect South Decliner,

67

To draw the Lines on a North Decliner,

70

To draw the Hour-Lines on a Plain, that declines above 60 degrees▪

73

[Page]Of Declining Reclining Plains,

77

The first sort of South Recliners,

79

The second sort of South Recliners, being Polars,

90

The third sort of South-Recliners,

98

The first sort of North Recliners,

106

The second sort of North Recliners, being Equinoctial,

114

The third sort of North Recliners,

119

Of Inclining Di [...]ls,

126

To find the useful Hours in all Plains,

130

To draw the Mathematical Ornaments on all sorts of Dials,

134

To draw the Tropicks, or length of the Day,

136

To make the Trygon,

138

To draw the Planetary or Iewish Hours,

142

To draw the Italian Hours,

144

To draw the Babylonish Hours,

145

To draw the Azimuth Lines,

146

To draw the Almicanters,

154

To draw the Circles of Position, or Houses,

160

To draw the Hours, and all the rest, on the Ceiling of a Room,

165

The Figure of the Instrument, Explained,

166

[Page]A Table of the Suns Azimuth, at every Hour and Quarter, in the whole Signs,

168

A Table of the Suns Altitude the same time,

169

The Description and Use of the Armilary-Sphear for Dyalling, several wayes,

172

The Description and Use of the Poor-man's Dial-Sphear for Dyalling, and several Uses thereof,

203

How to remedy several Inconveniences in the use of the Gunter's Rule.

220

The Use, and a further Description of the Trianguler-Quadrant, for Navigation, or Observation at Sea,

227

For a fore-Observation with Sights,

233

For a fore-Observation with Thred and Plummet,

235

For a back-Observation, as a Davis Quadrant,

237

To find the Suns distance from the Zenith, or the Co-altitude,

238

To find the Altitude, when near the Zenith,

239

To find a Latitude by a forward Observation, as with a Gunter's Bow,

241

To find the Latitude by a back Observation,

247

[Page]To find the Latitude by a Meridian Observation, with Thred and Plummet,

252

To find the Suns Declination,

254

A Table of the Suns Declination for the second after Leap-year,

256, 257

A Table of the Suns Declination for every 5th Day the intermediate years,

258, 259

A Rectifying Table for the intermediate years,

260

A Table of the Magnitudes Declinations and Right Ascentions of 33 fixed Stars, in Degrees, and Hours, and Minuts,

261

The use of the Trianguler-Quadrant, in the Operative part of Navigation,

267

Of Sayling by the Arch of a great Circle,

300

FINIS.

Errata for the Second Part.

[Page]

PAge 6. line 10. for too, read to. P. 18. l. 10. f. H, r. the ends of the Arch QP. p. 22. l. 14. f. begins, r. being. p. 15. l. 27. f. Latitude, r. Co-latitude p. 34. l. 8. f. Sun, r. sum. p. 39. l. 24. f. incliner, r. inclination of Meridians. p. 61. l. 23. f. place, r. plain. p. 62. l. 20. f. ☉ r. Q. p. 66. l. 2. f. I, r. T. p. 69. l. 7. f. 12. r. 7. p. 92 l. 22. r. gives a mark near E, whose measure on the Limb from, B. p. 87. l. 8. f. gi [...]es, r. gives. l. 11. add at R near C. p. 93. l. 25. f. FE. r. PE. p. 100. l. 21. add cd next gives. p. 10 [...]. l. 6. f. 8-5, r. 8-3. p. 105. l. 19. f. use, r. have. p. 108. l. 6. f. Pole, r. Zenith. p. 112. l. 3. f. cuts 12, r. cuts the substile. p▪ 113. l. 19. f. DF, r. DE. l▪ 19. f. T. r. CT. Also in l. [...]8. r. CT. p. 122. l. 6. f. E, r. F. p. 122. l. 13, 14, 15, 16, add Sine. p. 128. l. 26. f. I, r. L. p. 139. l. 21. add, as in this Example. p. 140. l. 6. add to. p 170. l. 10. f. divides, r. divided. p. 181. l. 20. f. popsition, r. proportion. p. 119. l. 27. f. from, r. on. p. 193. l. 6. f. being, r. bring. p. 197 l. 4. f. elevation, r. inclination. p. 200 l. 24. f. C, r. G. p. 204. l. 3. f. F, r. E. p 209. add in the last line, or by the upper part of the Plain. p. 124. l. 18. add, or remove the Thred to turn it further when it reclines beyond the Pole. p. 238. l. 7. add of. l. 20 add but. p. 247. l. 13. r. and much better in small Latitudes. p. 248. l. 5. f. wo, r. two. p. 251. l. 14. f. 20, r. 33. Also, l. 17. f. 40, r. 27.

Advertisements.

[Page]

The use of these, or any other Instruments concerning the Mathematical Practice, or further Instructions in any part thereof, is taught by Iohn Colson near the Hermitage-stairs.

Also by Euclide Speidwel, dwelling near to White-Chappel Church in Capt. Canes Rents, or at the Custom-house.

Also, by William Northhall Mariner, dwelling at the Crooked-Billet in Meeting-house Alley on Green-Bank near Wapping.

Also the Instruments may be had at the house of Iohn Brown, Iohn Seller, or Iohn Wingfield, as in the Title-page is expressed.

There is now extant a large Treatise of Navigation in Folio, describing the Sea-Coasts, Capes, Head-lands, the Bayes, Roads, Rivers, Harbours and Sea-marks in the whole Northern-Navigation; shewing the Courses and Distances from one Place to another, the ebbing and flowing of the Sea, with many other things belonging to the practick part of Navigation.

A Book (not heretofore printed in England) Collected from the practice and experience of divers able and experienced Navigators of our English Nation. Published by Iohn Seller, Hydrographer to the Kings most excellent Majesty, and to be sold by him at the sign of the Marine [...]s Compass at the Hermitage-stairs in Wapping, and by Iohn Wingfield right against St. Olaves Church in Crouched-Fryers.

Practical Navigation; or, an Introduction to that whole Art. Sold by Iohn Seller and Iohn Wingfield aforesaid.

Information about this book

Title statement

Horologiographia, or, The art of dyalling being the second book of the use of the trianguler-quadrant : shewing the natural, artificial, and instrumental way, of making of sun-dials, on any flat superficies, with plain and easie directions, to discover their nature and affections, by the horizontal projection : with the way of drawing the usual ornaments on any plain : also, a familiar easie way to draw those lines on the ceiling of a room, by the trianguler quadrant : also, the use of the same instrument in navigation, both for observation, and operation : performing the use of several sea-instruments still in use / by John Brown, philomath.

Brown, John, philomath.

Edition statement

1671

Publication

Publisher

Text Creation Partnership,

Place of publication

Ann Arbor, MI ; Oxford (UK) :

Date

2004-05 (EEBO-TCP Phase 1).

ID [DLPS]

A29762

ID [STC]

Wing B5042

ID [STC]

ESTC R17803

ID [EEBO-CITATION]

12547379

ID [OCLC]

ocm 12547379

ID [VID]

63104

Availability

This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal. The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission.

Series

Early English books online.

Notes

(EEBO-TCP ; phase 1, no. A29762)

Transcribed from: (Early English Books Online ; image set 63104)

Images scanned from microfilm: (Early English books, 1641-1700 ; 955:8)

Source

— Horologiographia, or, The art of dyalling being the second book of the use of the trianguler-quadrant : shewing the natural, artificial, and instrumental way, of making of sun-dials, on any flat superficies, with plain and easie directions, to discover their nature and affections, by the horizontal projection : with the way of drawing the usual ornaments on any plain : also, a familiar easie way to draw those lines on the ceiling of a room, by the trianguler quadrant : also, the use of the same instrument in navigation, both for observation, and operation : performing the use of several sea-instruments still in use / by John Brown, philomath., Brown, John, philomath..

Extent

305 [i.e. 304], [6] p. : 40 ill.

Printed by John Darby, for John Wingfield ... and by John Brown ... and by John Seller ...,. London :: 1671.. (Reproduction of original in Cambridge University Library.) ("An appendix to the use of the trianguler-quadrant in navigation ..." (p.[225]-305 [i.e. 304]) has special t.p.) (Errata: p. [5])

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