HOROMETRIA: OR, THE COMPLEAT DIALLIST.

Wherein the whole mystery of the Art of DIALLING is plainly taught three several wayes; two of which are performed Geometrically by Rule and Compass onely: And the third Instrumentally, by a QUADRANT fitted for that purpose.

With the working of such Propositions of the Sphere, as are most usefull in Astronomy and Navigation, both Geometri­cally and Instrumentally.

By THOMAS STIRRUP, Philomath.

Whereunto is added an APPENDIX, shewing how the parallels of Declination; the Jewish, Babylonish, & Italian hours; the Azimuths, Almicanters, &c. may be easily in­scribed on any Dial whatsoever, by Rule and Compasse onely.

And to draw a Dial on the seeling of a Room, By W. LEYBOURN.

Also, Dialling Ʋniversal, performed by an easie and most speedy way, shewing how to describe the hour-lines on all sorts of Planes whatsoever, and in any Latitude: Performed by certain Scales set on a small portable Ruler, By G. S. Practitioner in the Mathematicks.

The Second Edition with Additions.

London, Printed by R. & W. Leybourn, for Thomas Pirrepont, at the Sun in Paul's Church-yard. 1659.

[...]

TO THE READER.

Courtous Reader,

HEre is presented unto thee a short Treatise of the Art of Dialling. Concerning the antiquity, excel­lencie, and necessity thereof, I shall in this place say little, the antiquity thereof being well known to all who have diligently read the Sacred Scriptures, where­in mention is made of that of King Ahaz, upon which the Almighty was pleased to express a Miracle for the recovery of K. Hezekiah, by causing the Sun to go back ten deg. upon the said Dial, and this is the first that was ever recorded, be­ing above 2400 years agoe, since which time, Learning spreading it selfe over the whole Universe, hath made this Art more common.

For the excellencie of it, the skill which is required in the Mathematicks, especially in Geometry, Astronomy and Optiques, for the making a man compleat and excellent, is an evident proof, for without good knowledge in the Elements of Geometry, with a competencie of knowledge in the Circles of the Sphere, and some in-sight in the Optiques, in vain doth a man bestow his time in the study of this Art of Dialling.

Now for the necessity of it, what is more necessary in a well ordered Common-wealth? what action can be performed in due season without it? or what man can appoint any bu­siness with another, and not prefix a time, without the losse of that which cannot be re-gained, and ought therefore to most be prized.

Now because that all the light which we receive is from the great and glorious light of the World, (the Sun) we have fetcht the beams thereof from Heaven, to enlighten the un­derstandings of men upon earth, and from whose Light we receive and retein the benefit of all our knowledge.

Therefore, this Art of Dialling being in it self so excellent and necessary, may induce any industrious person to the practise thereof, the perfect knowledge whereof in this in­suing Treatise is sufficiently taught, and that by such briefe, easie, and familiar wayes, that not any Treatise hitherto published, can for convenience, ease, and quick dispatch, be compared thereto.

The whole Treatise consisteth of five Books, in which the whole mystery of the Art of Dialling is plainly taught three several wayes, namely, two Geometrically, and the third Instrumentally.

  • I The first Book containeth certain Elements of Geometry and Astronomy, as also how to perform divers Propositions in Geometry.
  • II The Second sheweth how to perform most Propositions in Astronomy and Navigation, Geometrically, with Scale and Compasse only.
  • III The Third sheweth how to find the Inclination and Declination of any Plane without Instrument, as also how to draw the hour-lines upon any Plane howsoever, and in what Latitude soever scituate, by Rule and Com­passe only, two several wayes, in both which, the two grand inconveniencies of the common wayes (viz. of our-running the limits of the Plane; and drawing of many un-necessary lines) are totally avoided, you having no lines to draw, but such as will be comprised within the bounds of your Plane and those so few, that you need not fear confusion.
  • IV & V The Fourth, and Fifth, sheweth the constinction and use of a Quadrant, by which all the most usual Propositions in Astronomy may be wrought with great facility, and by which the Inclination and Declination of a Plane may be speedily attained, and also the hour-lines drawn upon all kind of Planes in any Latitude.

Unto these five Books is added an Appendix, shewing how to furnish any kind of Dial with Astronomical variety, as to draw thereon the parallels of declination, by which the place of the Sun may be known: the parallel of the lenght of the day, by which the day of the Moneth, the Sun rising and setting, the length of the day and night may be known: how many hours are past since Sun rising, and how many remain to Sun setting: The old un-equal hours, by which the day is divided into twelve equal parts according to the Jewish account: The azimuths, by which you may know in what quarter of the heavens the Sun is at any time of the day: The Almicanters or Circles of altitude, by which the height of the Sun, the proportion of shadows to their bodies may be easily discovered. And lastly, How to draw a Dial on the seeling of a Room by reflection: all which are per­formed Geometrically by Rule and Compasse only, afford­ing great delight and pleasure in the practice of this most excellent Art. All which is here presented to thee as freely as it was given from God, who is the Author and giver of all good things.

THE CONTENTS.

  • TErms of Geometry. pag. 1
  • How to draw parallel lines pag. 7
  • How to raise and let fall perpendiculars. pag. 8, 9
  • stronomical Definitions. pag. 11
  • Of the Circles of the Sphere both great and smal. pag. 13
  • A Table of the Suns Declination for four years. pag. 21
  • The Description of the Scale for Dialling. pag. 29
  • How to make a Line of Chords. pag. 31
  • How to make an Angle of any quantity of degrees and minutes. pag. 31
  • How to find the Suns Altitude. pag. 32
  • To find the length of Right and contrary shadow. pag. 33
  • To find the Suns Declination. pag. 35
  • How to find the Suns place. pag. 37
  • To find the Suns Amplitude. pag. 38
  • How to find the height of the Pole. pag. 38
  • How to find the Suns Amplitude. pag. 39
  • How to find the Suns Declination. pag. 40
  • To know at what time the Sun shall be East or West. pag. 41
  • How to find the height of the Sun at the hour of six. pag. 43
  • To find the Azimuth at the hour of six. pag. 44
  • To find the Azimuth. pag. 44
  • How to find the hour of the day. pag. 47
  • How to find the Ascensional difference. pag. 51
  • How to find the Right or Oblique Ascension. pag. 52
  • How to find the Suns Altitude without Instrument. pag. 53
  • How to find the Latitude of a place. pag. 54
  • How to find the Declination, and Inclination of any plane. pag. 57
  • How to draw a Meridian line upon an Horizontal plane. pag. 63
  • To make an Equinoctial Dial. pag. 63
  • How to draw a Dial upon a Polar plane. pag. 65
  • How to make an East or West Dial. pag. 66
  • How to draw a Dial upon an Horizontal plane. pag. 69
  • How to draw a Dial upon a ful North or South plane. pag. 72
  • To draw a Dial upon a Vertical inclining plane. pag. 75
  • To draw a Dial upon a North and South declining plane. pag. 78
  • Another way to draw an Horizontal Dial. pag. 82
  • Another way to draw a full North or South Dial. pag. 84
  • In an upright declining plane, to find the deflexion, the height of the stile, and the inclination of Meridians. pag. 86
  • [Page]To draw a Dial upon an upright declining plane. pag. 81
  • To draw a Dial upon a Meridian inclining plane. pag. 91
  • In declining inclining planes, to find the height of the stile, the deflexion, &c. and to draw the Dial. pag. 96
  • The Description of a Quadrant. pag. 105
  • The working of divers propositions in Geometry by the Quadrant. pag. 107
  • To work propositions in Astronomie by the Quadrant. pag. 115
  • To find the Inclination and Declination of a plane. pag. 128
  • How to draw Dials on all kind of planes by the Quadrant. pag. 131
  • If the Cock of a Dial be lost, to find the height thereof. pag. 141
  • How to describe the Equinoctial, Tropicks, and other parallels of the Suns course and declination in all kind of planes. pag. 142
  • To draw the parallels of the length of the day on all planes. pag. 157
  • How the Babylonish & Italian hours may be drawn on all kind of planes. pag. 161
  • How the Jewish hours may be drawn upon any plane. pag. 165
  • How to draw the Azimuths, or Vertical Circles on all kind of planes. pag. 168
  • Of the Almicanters or Circles of Altitude. pag. 175
  • How to draw Dial on the seeling of a Room. pag. 176

Note, that the line of Chords which is drawn on the edge of the quadrant A C should issue from the center, but in the figure it is drawn short thereof, which defect the Instrument maker will easily supply.

THE FIRST BOOK.
Shewing the meaning of some of the usefullest termes of GEOMETRY, which be most attendant unto this Art of DIALLING: With a description of some of the chief Points, Lines, and Circles imagined in the Sphere: Being very fit to be understood of all those that intend to practise either in the Art of NAVIGATION, ASTRONOMIE, Or DIALLING.

CHAP. I. Of certain terms of Geometry, necessary to be known of the unlearned, before the proceeding in this Art of Dialling.

BEing intended in this Treatise of Dialling, to proceed by Geometrical Proportion: I have thought fit, first, to declare unto you the meaning of some terms of Geo­metry which are necessary for the un­learned to know before they enter into this Art of Dialling.

Definition 1.

First,A. therefore a point or prick is that which is the least of all materials, having neither part nor quantity, and therefore void of length, breadth, and depth: as is set forth unto you by the point or prick noted with the letter A.

Definition 2.

A line is a supposed length, or a thing extending it selfe in length, without breadth or thicknesse, whether it be right lined or crooked, and may be divided into parts in respect of his length, but admitteth no other division, as is set forth unto you by the line B.

[diagram]

Definition 3.

An Angle is the meeting of two lines in any sort, so as they both make not one line.

[diagram]

As for example, suppose the lines C D and E D to be drawn in such sort so as they may both meet in the point D, so shall the point D be the angle included between the two lines, as C D E: and here note, that an Angle is usually descri­bed by three letters, of which, the second, or middle letter, representeth alwayes the angle intended.

Definition 4.

[diagram]

If a right line fall on a right line, ma­king the angles on either side equal, each of those angles are called right an­gles, and the line e­rected is called a Perpendicular line unto the other. As for example, the line AB here in this [Page 3]figure, falling upon the line C B D, in such sort, that the angles on both sides are thereby made equal, as here you see, and therefore are called right angles.

Definition 5.

A Perpendicular is a line raised from, or let fall upon an­other line, making equal angles on both sides, as you may see declared in the former figure, where the line A B is per­pendicular unto the line C B D, making equal angles in the point B.

Definition 6.

A Circle is a plain figure, and contained under one line which is called the Circumference thereof, as in the figure following, the very Ring C B D E is called the circumfe­rence of that circle.

Definition 7.

The center of a Circle is that point which is in the midst thereof, from which point, all right lines drawn to the Circumference are equal, as you may see in the fol­lowing figure, where the point by the letter A represents the Center, and is the very middle point upon which the circumference was drawn.

Definition 8

The Diameter of a Circle is a right line drawn through the center of any Circle, in such sort that it may divide the whole Circle into two equal parts, as you may see the line C A D, or B A E, either of which is the Diameter of the circle B C E D, because either of them passeth through the center A, and divideth the whole circle into two equal parts.

Definition 9.

The Semidiameter of a Circle, is halfe of the Diameter and is contained betwixt the Center, and the one side of the circle, as the line A D, or A B, or A C, or A E, are either of them the Semidiameters of the circle B C E D.

Definition 10.

A Semicircle is the one halfe of a Circle drawn upon his Diameter, and is contained upon the superficies or surface of the Diameter, as the Semicircle C B D, which is halfe of the Circle C B D E, and is contained above the Diameter C A D.

Definition 11.

A Quadrant is the fourth part of a Circle, and is contain­ed betwixt the Semidiameter of the Circle, and a line drawn perpendicular unto the Diameter of the same Circle, from the Center thereof, dividing the Semicircle into two equal parts, of the which parts the one is the quadrant or fourth part of the same Circle. As for Example, the Diameter of the Circle B D E C, is the line C A D, dividing the Circle into two equal parts: then from the Center A, raise the perpendicular A B, dividing the Semicircle likewise into two equal parts, so is A B D, or A B C, the quadrant of the Circle C B D E.

Definition 12.

[diagram]

A Segment or portion of a Circle, is a fi­gure contained under a right line, and a part of a circumfe­rence, either greater or les­ser then the se­micircle, as in the figure you may see that F B G H is a Segment or part of the cir­cle C B D E, & [Page 5]is contained under the right line F H G (which is less then the Diameter C A D) and a part of the whole circumference as F B G.

And here note, that these parts, and such like of the cir­cumference so divided, are commonly called arches or arch lines, and all lines (lesse then the Diameter) drawn through, and applyed to any part of the circumference, are called chords, or chord lines, of those arches which they subtend.

Definition 13.

A Parallel line is a line drawn by the side of another line, in such sort that they may be equidistant in all places, & of such parallels, two only belong unto this work of Dialling, that is to say the right lined parallel, & the circular parallel,

Right lined parallels, are two right lines equidistant in all places one from the other, which being drawn forth in­finitely, would never meet or concur; as may be seen by these two lines A and B.

[diagram]

Definition 14.

A circular pa­rallel is a circle drawn either within or with­out another cir­cle upon the same center, as you may plain­ly see by the two Circles B C D E, & F G H I these circles are both drawn up­on the same cen­ter A, and there­fore are parallel the one to the other.

Definition 15.

A Degree is the 360th part of the circumference of any circle, so that divide the circumference of any circle into 360 parts, and each of those parts is called a degree; so shall the semi-circumference contain 180 of those Degrees; and 90 of those degrees make a quadrant, or a quarter of the circumference of any circle.

Definition 16.

A minute is the 60th part of a degree, being understood of measure: but in time a Minute is the 60th part of an houre, or the fourth part of a degree, 15 degrees answering to an houre, and 4 minutes to a degree.

Definition 17

[diagram]

The quantity or measure of an Angle, is the number of de­grees contained in the arch of a circle, described from the point of the same an­gle, and intercepted betweene the two sides of that angle. As for example, the measure of the angle A B C is the number of degrees contained in the arch A C, which subtendeth the angle B, being found to be 60

Definition 18.

The Complement of an arch lesse then a quadrant, is so much as that arch wanteth of 90 degrees.

As for example, the arch A B being 60 degrees, which being taken from 90 degrees, leaveth B C for the comple­ment thereof, which is 30 degrees.

[diagram]

Definition 19.

The complement of an arch lesse then a Semicircle, is so much as that arch wanteth of a Semicircle, or of 180 deg. As for Example, the arch D C B being 120 degrees, this be­ing taken from 180 deg. the whole Semicircle, leaveth A B for the complement thereof, which will be found to be 60 degrees.

And here note, that what is said of the complements of arches, the same is meant by the complements of angles.

CHAP. II. To a line given, to draw a parallel line, at any distance required.

SUppose the line given to be A B, unto which line it is required to draw a parallel line. First, open your Com­passes to the distance required,

[diagram]

then set one foot in the end A, and with the other strike an arch line, on that side the given line [Page 8]whereunto the parallel line is to be drawn, as the arch line C, this being done, draw the like arch line upon the end B, as the arch line D, and by the convexity of those two arch lines C and D, draw the line C D, which shall be pa­rallel to the given line, as was required.

CHAP. III. To perform the former proposition at a distance required, and by a point limited.

SUppose the line given to be D E, unto which line it is required to draw a parallel line, at the distance, and by the point F. First therefore, place one foot of the compasses in the point F, from whence take the shortest extention to the line DE as F E, at

[diagram]

which di­stance, place one foot of the Com­passes in the end D, and with the other, strike the arch line G by the convexity of which arch line, and the limited point F, draw the line F G, which is parallel to the given line D E, as was required.

CHAP. IV. The manner how to raise a perpendicular line, from the middle of aline given.

LEt the line given be A B; and let C be a point therein, whereon it is required to raise a perpendicular. First therefore, open the compasses to any convenient distance, and setting one foot in the point C, with the other foot mark on either side thereof, the equal distances C A, and [Page 9]C B: then opening your compasses to any convenient wider distance, with one foot in the point A, with the other strike the arch line E over the point C,

[diagram]

then with the same distance of your compasses, set one foot in B, and with the other draw the arch line F, crossing the arch E in the point D, from which point D, draw the line DC, which line is perpendicular un­to the given line A B, from the point C, as was required.

CHAP. V. To let a Perpendicular fall from a point assigned, unto the middle of a line given.

LEt the line given whereupon you would have a perpen­dicular let fall, be the line D E F, and the point assign­ed to be the point C, from whence you would have a perpendicu­lar let fall up­on the given

[diagram]

line D E F. First, set one foot of your compasses in the point C, and opening your compasses to any convenient distance, so [Page 10]that it be more then the distance C E, make an arch of a circle with the other foot, so that it may cut the line D E F twice, that is, at I and G: then find the middle between those two intersections, which will be in the point E, from which point E, draw the line C E, which is the perpendicular which was desired to be let fall from the given point C, unto the middle of the given line D E F.

CHAP. VI. To raise a Perpendicular upon the end of a line given.

SUppose the line whereupon you would have a perpen­dicular to be raised, be the line B C, and from the point B a perpendicular is to be raised. First, open your Compas­ses unto any convenient distance, which here we suppose to be the distance B E, and set one foot of your compasses in B, with the other draw the arch E D, then this distance being kept, set one

[diagram]

foot of your com­passes in the point E, & with the other make a mark in the former arch E D, as at D, still keep­ing the same di­stance, set one foot in the point D, and with the other draw the arch line F over the given point B: now laying a ruler upon the two points E and D, see where it crosseth the arch line F, which will be at F, from which point F, draw the line F B, which shall be a perpendicular line unto the given line B C, raised from the end B, as was required.

CHAP. VII. To let a Perpendicular fall from a point assigned, unto the end of a line given.

LEt the line D E be given, unto which it is required to let a perpendicular fall from the assigned point A, unto the end D. First, from the assigned point A, draw a line unto any part of the given line D E, which may be the line ABC, then find the middle of the line A C, which will be at B, place therefore one foot of your compasses in the point B, and extend the o­ther unto A or C,

[diagram]

with which distance draw the Semicir­cle A D C, so shall it cut the given line D E in the point D, from which point D, draw the line A D, which shall be the perpendicular let fall from the assigned point A unto the end D of the given line D E, as was required.

CHAP. VIII. Certain Definitions Astronomical, meet to be understood of the unlearned, before the proceeding in this Art of Dialling.

IN the former Chapter I have shewed the meaning of some terms of Geometry, which be most helpfull unto this Art of Dialling, with the drawing of a Parallel line at [Page 12]any distance, or by a point assigned; so likewise have I shew­ed the manner either how to raise or let fall a perpendicular either from or unto any part of a line given. So likewise now I think it will not be un-necessary for to shew unto the unlearned, the meaning of some of the most usefullest terms in Astronomie, and most fitting this art of Dialling.

Definition 1.

A Sphere is a certain solid superficies, in whose middle is a point, from which all lines drawn unto the circumfe­rence are equal, which point is the Center of the Sphere.

Definition 2.

The Pole is a prick or point imagined in the Heavens, whereof are two, the North pole being the center to a circ l described by the motion of the North Star, or the taile of the little Bear, from which point aforesaid is a line imagi­ned to passe through the center of the Sphere, and passing directly to the opposite part of the heavens, sheweth there to be the South Pole, and this line so imagined to passe from one Pole to the other, through the Center of the Sphere, is called the Axletree of the World, because it hath been formerly supposed, that the Sun, Moon, and Stars, together with the whole Heavens hath been turned about from East to West, once round in 24 houres, by a true equal course, like much in like time; which diurnal revolution is performed about this Axletree of the World, and this Axletree is set out unto you in the following figure by the line P A D, the Poles whereof are P and D,

Definition 3.

A Sphere accidentally is divided into two parts; that is to say, into a right Sphere, and an oblique Sphere, a right Sphere is only unto those that dwell under the Equinoctial, to whom neither of the Poles of the World are seen, but lie hid in the Horizon. An oblique Sphere is unto those hat in habit on either side of the Equinoctial, unto whom [Page 13]one of the Poles is ever seen, and the other hid under the Horizon.

Definition 4

The Circles whereof the Sphere is composed are divided into two sorts; that is to say, into greater Circles, and lesser: The greater Circles are those that divide the Sphere into two equal parts, and they are in number six, vix. the Equi­noctial, the Ecliptique line, the two Colures, the Meridian, and the Horizon. The lesser Circles are such as divide the Sphere into two parts unequally; and they are foure in number, as, the Tropick of Cancer, the Tropick of Capricorn, the Circle Artique, and the Circle Antartique.

CHAP. IX. Of the six greater Circles.

Definition 5.

THe Equinoctial is a circle that crosseth the Poles of the World at right Angles, and divideth the Sphere into two equal parts, and is called the Equinoctial, because when the Sun commeth unto it (which is twice in the year, viz. at the Suns entrance into Aries and Libra) it maketh the dayes and nights of equal length throughout the whole World, and in the figure following, is described by the line S A N.

Definition 6.

The Meridian is a great Circle, passing through the Poles of the World, and the Poles of the Horizon, or Ze­nith point right over our heads, and is so called, because that in any time of the year, or in any place of the World, when the Sun (by the motion of the Heavens) commeth unto that circle, it is then Noon, or 12 of the clock: and it [Page 14]is to be understood, that all Towns and Places that lie East and West one of another, have every one a several Meri­dian; but all places that lie North and South one of ano­ther, have one and the same Meridian: this circle is decla­red in the figure following by the circle E B W C.

Definition 7.

The Horizon is a Circle, dividing the superior Hemis­phere from the inferiour, whereupon it is called Horizon, that is to say, the bounds of sight, or the farthest distance that the eye can see, and is set forth unto you by the line C A B in the following figure.

Definition 8.

Colures are two great circles, passing through both the Poles of the World, crossing one the other in the said Poles at right Angles, and dividing the Equinoctial and the Zodiaque into four equal parts, making thereby the four Seasons of the year, the one Colure passing through the two Tropical points of Cancer and Capricorn, shewing the beginning of Summer, and also of Winter, at which times the dayes and nights are longest and shortest. The other Colure passing through the Equinoctial points Aries and Libra, shewing the beginning of the Spring time and Autumne, at which two times the dayes and nights are of equal length throughout the whole World.

Definition 9.

The Ecliptique is a great Circle also, dividing the Equi­noctial into two equal parts by the head of Aries and Li­bra, the one halfe thereof doth decline unto the North­ward, and the other towards the South, the greatest decli­nation thereof (according to the observation of that late famous Mathematician Master Edward Wright) is 23 de­grees, 31 minutes, 30 seconds. Note also that the Circle is divided into 12 equal parts, which parts are attributed unto the 12 Signes, Aries, Taurus, Gemini, Cancer, Leo, Virgo, Li­bra, Scorpio, Sagittarius, Capricornus, Aquarius, and Pisces. [Page 15]Out of this line doth the Sun never move, but the Moon and other Planets are sometimes on the one side, and som­times on the other side thereof: this line may be represen­ted in the following figure, by that line whereon the cha­racters of the 12 Signes standeth.

CHAP. X. Of the four lesser Circles.

Definition 10.

THe Sun having ascended unto his highest Solsticial point, doth describe a Circle, which is the neerest that he can approach unto the North Pole, whereupon it is cal­led the Circle of the Summer Solstice, or the Tropick of Cancer, and is noted in the figure following by the line ♋ F G.

Definition 11.

The Sun also approaching unto the first scuple of Capri­corn, or the Winter Solstice, describeth another Circle, which is the utmost bounds that the Sun can depart from the Equinoctial line towards the Antartique Pole, where­upon it is called the Circle of the Winter Solstice, or the Tropick of Capricorn, and is described in the figure follow­ing by the line H I ♑.

Definition 12.

So much as the Ecliptique declineth from the Equino­ctial, so much doth the Poles of the Ecliptique decline from the Poles of the World, whereupon the Pole of the Eclip­tique, which is by the North Pole of the World describeth a Circle as it passeth about the Pole of the World, being just so far from the Pole, as the Tropick of Cancer is from the Equator, and it is called the Circle Artick, or the Circle of the North Pole, it is described in the following Diagram by [Page 16]the line T O, where the letter O doth stand for the Pole of the Ecliptique, and the line T O for the Circle which the point O doth describe about P the Pole of the World.

Definition 13.

The fourth and last of the lesser Circles is described in like manner, by the other Pole of the Ecliptique about the South Pole of the World, and therefore called the Antar­tique Circle, or the Circle of the South Pole, and is de­monstrated in the following figure by the line L R.

Definition 14.

The Zenith is an imaginary point in the Heavens over our heads, making right angles with the Horizon, as the Equinoctial maketh with the Pole.

Definition 15.

The Nadir is a point in the Heavens under our feet, ma­king right Angles with the Horizon under the earth, as the Zenith doth above, and therefore is opposite unto the Ze­nith: both these may be represented in the figure by the line E W, where the letter E standeth for the Zenith, and W for the Nadir.

Definition 16.

The Declination of the Sun is the arch of a Circle con­tained betwixt the Ecliptique and the Equinoctial, making right Angles with the Equinoctial, and may be set forth unto you by the arch S ♋. But the Declination of a Star, is the arch of a Circle let fall from the Center of a Star, perpendicular unto the Equinoctial. This Declination may be counted either Northward or Southward, according to the scituation of the Sun or Star, whether it be neerer unto the North or South Pole of the World.

Definition 17.

The Latitude of a Star is the arch of a circle contained betwixt the center of any Star and the Ecliptique line, ma­king right angles with the Ecliptique, and counted either Northward or Southward according to the situation of the [Page 17]Star, whether it be neerer unto the North or South Poles of the Ecliptique. And here note, that the Sun hath no Lati­tude, but alwayes keepeth in the Ecliptique line.

[diagram]

Definition 18.

The Latitude of a Town or Countrey, is the height of the Pole above the Horizon, or the distance betwixt the Zenith and the Equinoctial, and may be represented in this figure by the arch of the Meridian B P, where the North Pole P is elevated above the Horizontal line C A B according to the Angle BAP, which here is 52 degr. 25 min. the Latitude of Thurning.

Definition 19.

The Longitude of a Star is that part of the Ecliptique which is contained betwixt the Stars place in the Ecliptique and the beginning of Aries, counting them from Aries ac­cording to the order or succession of the Signes.

Definition 20.

The Longitude of a Town or Countrey, are the number of degrees which are contained in the Equinoctial, betwixt the Meridian that passeth over the Isles of Azores, (from whence the beginning of Longitude is accounted) East­wards, and the Meridian that passeth over the Town or Countrey desired.

Definition 21.

The Altitude of the Sun or Star, is the arch of a circle contained betwixt the center of the Sun or any Star, and the Horizon. As for example, in the former figure, suppose the Sun to be in the Meridian at S, then the angle of altitude will be the angle SAC, the measure whereof will be the arch C S, contained betwixt the Sun at S, & the Horizon C, which here will be found to be 37 deg. and 35 min. the height of the Sun at noon when it is in the Equinoctial circle S A N.

Definition 22.

Azimuths are Circles which meet together in the Zenith, and crosse the horizon at right angles, and serve to finde the point of the Compasse which the Sun is upon at any houre of the day: or the Azimuth of the Sun or Star is a part of the Horizon, contained betwixt the true East or West point, and that Azimuth which passeth by the center of the same Star to the Horizon, and may be represented in the former figure by the arch line E V W.

Definition 23.

Ascension, is the rising of any Star, or of any part or por­tion of the Ecliptique above the Horizon.

Definition 24.

Right Ascension, is the number of degrees and minutes of the Equinoctial (counted from the beginning of Aries) which commeth to the Meridian with the Sun, Moon, Stars, or any portion of the Ecliptique.

Definition 25.

Oblique Ascension, is a part of the Equinoctial contain­ed betwixt the beginning of Aries, and that part of the Equi­noctial that riseth with the center of a Star, or any portion of the Ecliptique in an Oblique Sphere.

Definition 26.

The Ascensional difference, is the difference betwixt the right & oblique ascension, or it is the number of degrees con­tained betwixt that place of the Equinoctial that riseth with the center of a Star, and that place of the Equinoctional that commeth to the Meridian with the center of the same Star.

Definition 27.

Almicanters are circles drawn parallel unto the Horizon one over another, untill they come unto the Zenith, these are circles that doe measure the elevation of the Pole, or height of the Sun, Moon, or Stars, above the Horizon, which is called the Almicanter of the Sun, Moon, or Stars, the arch of the Sun or Stars Almicanter, is a portion of an Azimuth coutained betwixt that Almicanter which passeth through the center of the Star and the Horizon.

Thus having set forth unto the view of the unlearned (for whose sake this Treatise was intended) the meaning of some of the usefullest terms of Geometry, which be most attendant unto this Art of Dialling, and also a description of some pe­culiar things concerning the Points, Lines, and Circles imagin­ed in the Sphere, being very fit to be understood of all such as intend to practise either in the Art of Navigation, Astronomie, [Page 20]or Dialling. Therefore now I intend to proceed with Scale and Compass to perform some questions Astronomical, before we enter upon the Art of Dialling, seeing they are both de­lightful, and also helpful unto all such as shall be practitio­ners in this Art of Dialling.

But first of all I will adde a Table of the Suns Declination for four years to come, commencing this present year of our Lord 1658, and continuing till 1662, and may for this age serve for many years farther, exact enough for the Practice of the Art of Dialling, and the resolution of other Problemes of the Sphere, which follow in the next Book.

The Table followeth.

For the years 1657, 1661, 1665, 1669.
DayesJanna.Febru.Marc.April.MayJune
SouthSouthSouthNorth.NorthNorth
121 4413 463 2408 3618 0523 12
221 3413 263 0008 5818 2033 16
321 2313 052 3709 2018 3523 19
421 1312 452 1309 4218 5023 22
521 0212 251 4910 0319 0423 25
620 5012 041 2510 2419 1823 27
720 3811 431 0110 4519 3123 29
820 2611 210 3811 0619 4423 30
920 1311 000 1411 2719 5723 31
1020 0010 380N1011 4720 1023 31
1119 4610 160 3312 0720 2223 32
1219 3209 540 5712 2820 3423 31
1319 1809 321 2112 4820 4523 30
1419 0309 101 4413 0720 5623 29
1518 4808 482 0813 2721 0723 28
1618 3308 252 3113 4621 1723 26
1718 1708 032 5414 0521 2723 23
1818 0207 403 1814 2421 3723 20
1917 4507 173 4114 4221 4623 17
2017 2806 544 0515 0121 5523 14
2117 1106 314 2815 1922 0423 10
2216 5406 084 5115 3722 1223 06
2316 3605 455 1415 5422 2023 01
2416 1805 215 3716 1222 2722 55
2516 0004 586 0016 2922 3422 50
2615 4204 346 2216 4622 4122 44
2715 2304 116 4517 0222 4722 37
2815 0403 477 0717 1822 5322 31
2914 45 7 3017 3422 5822 23
3014 26 7 5217 5023 0322 16
3114 06 8 14 23 08 
For the years 1657, 1661, 1665, 1669.
DayesJulyAug.Septē.Octob.Novē.Decem
NorthNorthNorthSouthSouthSouth
122 0815 124 2407 1517 4023 90
222 0014 544 0107 3817 5623 13
321 5114 363 3808 0018 1223 17
421 4214 173 1508 2218 2823 20
521 3213 582 5208 4518 4323 23
621 2213 392 2909 0718 5823 26
721 1213 202 0509 2919 1323 28
821 0213 011 4209 5119 2723 30
920 5112 411 1910 1319 4123 31
1020 4012 210 5510 3519 5523 31
1120 2812 010 3210 5620 0823 32
1220 1611 410 0811 1820 2123 31
1320 0411 210S1611 3920 3423 30
1419 5111 090 3912 9020 4623 29
1519 3810 391 0312 2120 5823 27
1619 2510 181 2612 4120 0923 25
1719 1209 571 5013 1221 2023 22
1818 5809 362 1313 2221 3123 19
1918 4309 152 3713 4221 4123 16
2018 2908 533 0014 0221 5023 12
2118 1408 313 2314 2122 0023 07
2217 5908 093 4714 4122 0923 02
2317 1407 474 1015 0022 1722 57
2417 2807 254 3315 1922 2522 51
2517 1207 034 5715 3722 3322 44
2616 5606 415 2015 5522 4022 37
2716 3906 185 4316 1322 4622 30
2816 2205 566 0616 3122 5222 22
2916 0505 336 2916 4922 5822 14
3015 4805 106 5217 0623 0422 05
3115 3004 47 17 23 21 56
For the years 1658, 1662, 1666, 1670.
DayesJanua.Febru.Mar.April.May.June
SouthSouthSouthNorthNorthNorth
121 4713 513 2908 3118 0223 11
221 3713 313 0608 5318 1723 15
321 2713 102 4209 1518 3223 18
421 1612 502 1809 3618 4623 21
521 0512 301 5509 5819 0023 24
620 5312 091 3110 1919 1423 26
720 4111 481 0710 4019 2823 28
820 3911 260 4311 0119 4123 30
920 1611 050 2011 2219 5423 31
10200 310 430N0411 4220 0723 31
1119 4910 230 2812 0320 1923 32
1219 3509 500 5112 2320 3123 31
1319 2109 381 1512 4320 4223 31
1419 0709 151 3913 0320 5323 30
1518 5208 532 0213 3221 0423 28
1618 3708 312 2513 4121 1523 26
1718 2108 082 4914 0021 2323 24
1818 0507 453 1214 1921 3523 21
1917 4907 223 3614 3821 4423 18
2017 3206 593 5914 5621 5323 15
2117 1506 364 2215 1422 0223 11
2216 5806 134 4515 3222 1023 06
2316 4105 505 0815 5022 1823 02
2416 2305 275 3116 0822 2522 57
2516 0505 045 5416 2522 3222 51
2615 4704 406 1716 4222 3922 45
2715 2804 176 4016 5822 4522 39
2815 0903 537 0217 1422 5122 32
2914 50 7 2517 3022 5722 25
3014 31 7 4717 4623 0222 18
3114 11 8 09 23 07 
For the years 1658, 1662, 1666, 1670.
DayesJuly.Augu.Septē.Octob.Novē.Decem
NorthNorthNorthSouthSouth.South
122 1015 174 3007 0917 3623 08
222 0214 594 0707 3217 5223 12
321 5314 403 4407 6518 0823 16
421 4414 223 2108 1718 2423 20
521 3514 032 5808 3918 4023 23
621 2513 442 3409 0218 5523 26
721 1513 252 1109 2419 0923 28
821 0413 051 4809 4619 2423 29
920 5412 461 2410 0819 3823 30
1020 4312 261 0110 2919 5223 31
1120 3112 060 3710 5120 0523 32
1220 1911 460 1411 1220 1823 31
1320 0711 260S1011 3420 3123 31
1419 5411 050 3311 5520 4323 29
1519 4110 440 5712 1520 5523 28
1619 2810 231 2012 3621 0623 26
1719 1510 021 4412 5721 1723 23
1819 0109 412 0713 1721 2823 20
1918 4709 202 3113 3721 3823 17
2018 3308 582 5413 5721 4823 13
2118 1808 363 1814 1621 5723 08
2218 0308 153 4114 3622 0623 03
2317 4707 534 0414 5522 1522 58
2417 3207 314 2815 1422 2322 52
2517 1607 094 5115 3322 3122 46
2617 0006 465 1415 5122 3822 39
2716 4306 245 3816 0922 4522 32
2816 2606 016 0016 2722 5122 24
2916 0905 386 2316 4522 5722 16
3015 5205 166 4617 0223 0322 07
3115 3404 53 17 19 21 58
For the years 1659, 1663, 1667, 1671.
DayesJanua.Febru.Mar.April.May.June
SouthSouthSouthNorthNorthNorth
121 4913 563 3508 2617 5823 10
221 3913 363 1108 4818 1323 14
321 2913 162 4809 0918 2823 18
421 1812 552 2409 3118 4323 21
521 0712 352 0009 5318 5723 24
620 5612 141 3710 1419 1123 26
720 4411 531 1310 3519 2523 28
820 3211 320 4910 5619 3823 29
920 1911 100 2611 1719 5123 30
1020 0610 490 0211 3720 0423 31
1119 5310 270N2211 5820 1623 32
1219 3910 050 4612 1820 2823 31
1319 2509 431 0912 3820 4023 31
1419 1009 211 3312 5820 5123 30
1518 5508 581 5613 1721 0223 29
1618 4008 362 2013 3721 1223 27
1718 2508 142 4313 5621 2323 25
1818 0907 513 0714 1521 3323 22
1917 5307 283 3014 3421 4223 19
2017 3607 053 5314 5221 5123 16
2117 1906 424 1715 1022 0023 12
2217 0206 194 4015 2822 0823 08
2316 4505 565 0315 4622 1623 03
2416 2705 325 2616 0322 2422 58
2516 0905 095 4916 2122 3122 53
2615 5104 466 1216 3822 3822 47
2715 3204 226 3416 5422 4422 41
2815 1403 596 5717 1122 5022 34
2914 55 7 1917 2722 5622 27
3014 35 7 4217 4323 0122 30
3114 16 8 04 23 06 
For the years 1659, 1663, 1667, 1671.
DayesJuly.Augu.Septē.Octob.Novē.Decem
NorthNorthNorthSouthSouth.South
122 1215 214 3507 0417 3223 06
222 0415 034 1207 2617 4823 11
321 5514 454 4907 4918 0423 15
421 4614 263 2608 1218 2023 39
521 3714 083 0308 3418 3623 22
621 2713 492 4008 5618 5123 25
721 1713 302 1709 1819 0623 27
821 0713 101 5309 4019 2023 29
920 5612 511 0310 0219 3423 30
1020 4512 311 0610 2419 4823 31
1120 3412 110 4310 4620 0223 32
1220 2211 510 2011 0720 1523 31
1320 1011 310S0411 2820 2823 31
1419 5711 100 2811 4920 4023 30
1519 4510 490 5112 1020 5223 28
1619 3210 391 1512 3121 0423 26
1719 1810 081 3812 5221 1523 24
1819 0409 462 0213 1221 2523 21
1918 5009 252 2513 3221 3623 17
2018 3609 042 4813 5221 4623 13
2118 2108 423 1214 1221 5523 09
2218 0608 203 3614 3122 0423 04
2317 5107 583 5914 5022 1322 59
2417 3507 364 2215 0922 2122 53
2517 1907 144 4615 2822 2922 47
2617 0306 515 0915 4622 3622 41
2716 4706 295 3216 0522 4322 34
2816 3006 075 5516 2322 5022 26
2916 1305 446 1816 4122 5622 18
3015 5605 226 4116 5823 0122 10
3115 3904 59 17 15 22 01
For the years 1660, 1664, 1668, 1672.
DayesJanna.Febru.Marc.April.MayJune
SouthSouthSouthNorth.NorthNorth
121 5114 013 1708 4218 1023 13
221 4213 412 5409 0418 2523 17
321 3213 212 3009 2618 3923 20
421 2113 002 0609 4718 5423 23
521 1012 401 4210 0919 0823 25
620 5912 191 1910 3019 2223 27
720 4711 580 5510 5119 3523 29
820 3511 370 3211 1219 4823 30
920 2211 160 0811 3220 1023 31
1020 0911 540N1611 5320 1323 31
1119 1610 320 4012 1320 2523 32
1219 4210 101 0312 3320 3723 31
1319 2809 481 2712 5320 4823 03
1419 1409 261 5013 1220 5923 29
1518 5909 042 1413 3221 1023 27
1618 4408 422 3813 5121 2023 25
1718 2908 193 0214 1021 3023 23
1818 1307 563 2514 2921 4023 20
1917 5707 333 4814 4821 4923 16
2017 4007 104 1115 0621 5823 13
2117 2406 474 3415 2422 0623 09
2217 0706 244 5715 4222 1423 04
2316 4906 015 2015 5922 2223 59
2416 3205 385 4316 1722 2922 54
2516 1405 156 0616 3422 3622 48
2615 5505 526 2916 5022 3422 42
2715 3704 296 5117 0722 4922 36
2815 1804 047 1417 2322 5422 29
2914 5903 417 3617 3922 5922 21
3014 40 7 5817 5423 0422 13
3114 21 8 20 23 09 
For the years 1660, 1664, 1668, 1672.
DayesJulyAug.Septē.Octob.Novē.Decem.
NorthNorthNorthSouthSouthSouth
122 0515 074 1807 2117 4423 10
221 5714 494 5507 4418 0023 14
321 4814 313 3208 0618 1623 18
421 3914 123 0908 2918 3223 21
521 3013 532 4508 5118 4723 24
621 2013 342 2209 1319 0223 27
721 1013 151 5909 3519 1723 29
820 5912 551 3509 5719 3123 30
920 4812 361 1210 1919 4523 31
1020 3712 160 4910 4119 5923 31
1120 2511 560 2511 0220 1223 32
1220 1311 360 0211 2320 2523 31
1320 0011 150S2211 4420 3723 30
1419 4810 540 4512 0520 4923 29
1519 3510 341 0912 2621 0123 27
1619 2110 131 3312 4721 1223 24
1719 0809 511 5613 0721 2323 21
1818 5409 302 1913 2721 3323 18
1918 3909 092 4313 4721 4323 15
2018 2508 473 0614 0721 5323 11
2118 1008 253 3014 2622 0223 06
2217 5508 033 5314 4622 1123 01
2317 4007 414 1715 0522 1922 55
2417 2407 194 4015 2422 2722 49
2517 0706 575 0315 4222 3522 42
2616 5106 345 2616 0022 4222 35
2716 3406 125 4916 1822 4822 28
2816 1705 506 1216 3622 5422 20
2916 0005 276 3516 5423 0022 12
3015 4405 046 5817 1123 0522 03
3115 2504 41 17 28 21 51
The end of the first Book.

THE SECOND BOOK.
Shewing Geometrically how to resolve all such Astronomical Propositions as are of ordinary use, as well in the Art of Navigation, as in this Art of Dialling.

CHAP. I. The description of the Scale, whereby this work may be performed.

THis Scale for this work, needs to be divided but into two parts, the first whereof may be a Scale of equal divisions of 16 in an inch, and may serve for ordinary measure. The second part of the Scale may be a single Chord of a Circle, or a Chord of 90, and is divided into 90 unequal divisions, repre­senting the 90 deg. of the quadrant, and are numbred with 10, 20, 30, 40, &c. unto 90. This Chord is in use to measure any part or arch of a Circle, not surmounting 90 degrees, the number of these degrees from 1 unto 60, is called the Radius of the Scale, upon which distance all circles are to

[diagram]

be drawn, whereupon 60 of these degrees are the semidia­meter of any circle that is drawn upon that Radius.

The manner how to divide the line of Chords.

Although the making or dividing of this line of Chords be well known unto all those that do make Mathematical Instruments, yet I would not have them that shall make use of this Book: be ignorant of the dividing of this line: Therefore, first, draw the diameter A D C, which being done, upon the center D describe the semicircle A B C, which semicircle divide into two equal parts or quadrants by the Point B, then dividing one of these quadrants into 90 equal parts, or degrees, you are prepared, as here you see in the quadrant A B.

Now this being done, set one foot of your Compasses in the point A, and let the other be extended unto each degree of the quadrant A B, and these extents transfer into the line ADC, as here you see is done. This line so divided into 90 unequal divisions from the point A, (and numbred by 10, 20, 30, 40, &c. unto 90) is called a line of Chords, and may be set on your Rule, as here you see is done. And this may be as well performed within the quadrant D A B, by transferring the degrees of the quadrant A B into the line A E B, or into any other line: and here you may see that when you open your compasses unto 60 degrees in the quadrant, and transfer it into the line A D, that it will light upon the center D, whereby it doth plainly appear that 60 of those degrees are equal to the semidiameter of the same circle, and therefore is the Radius upon which all circles are drawn, as was shewed before in this Chapter.

[diagram]

CHAP. II. How speedily with Rule and Compasse, to make an angle containing any degrees assigned; or to get the degrees of any angle made.

FIrst, therefore to make an angle of any quantity, open your compasses to the Radius of your Scale, and setting

[diagram]

one foot there­of in the point A, with the o­ther foot de­scribe the arch BC, then draw the line A B, then opening your compas­ses to so many degrees upon your line of Chords, as you [Page 32]would lay down, which here we will suppose to be 40 de­grees, and setting one foot in B, with the other make a mark in the arch BC, as at C, from which point C, draw the line CA, which shall make the angle B A C, containing 40 deg. as was required.

And if you desire to finde the quantity of an angle, open the compasses to the Radius of your Scale, and set one foot thereof in the point A, and with the other describe the arch B C, then taking the distance betwixt B and C (that is, where the two legs and the arch line crosseth) and apply it unto the line of Chords, and there it will shew you the number of degrees contained in that angle, which here will be found to be 40 degrees.

CHAP. III. To find the Altitude of the Sun by the shadow of a Gnomon set perpendicular to the Horizon.

FIrst, draw the line A B, then opening your compasses to the Radius of your Scale, set one soot in the end A, and

[diagram]

with the other de­scribe the arch BCD, then opening your Compasses unto the whole 90 deg. with one foot in B, with the other mark the arch B C D, in the point D, from which point D, draw the line DA, which shall be perpendicular un­to the line AB, and make the quadrant ABCD, then suppose the height of your Gnomon or sub­stance [Page 33]yeilding shadow, to be the line A E, which here we will suppose to be 12 foot, therefore take 12 of your equal divisions from your Scale, as here I have taken 12 quarters for this our purpose, and set them down from A to E, and draw the line E F parallel to A B, then suppose the length of the shadow to be 9 foot, for this 9 foot must I take 9 of the same divisions as I did before, and place them from E to G, by which point G, draw the line A G C, from the center A through the point G, until it cutteth the arch B F C D in the point G, so shall the arch B C be the height of the Sun desired, which in this example will be found to be 53 deg. 8 min. the thing desired.

CHAP. IV. To find the altitude of the Sun by the shadow of a Gnomon stan­ding at right angles with any perpendicular wall, in such manner that it may lie parallel unto the Horizon.

FIrst, draw your quadrant A B C D, as is taught in the last Chapter, and place the length of your Gnomon from A to E, which here we will suppose to be 12 foot, as before, in the last Chapter; then draw the line EF parallel to A B then suppose the length of the shadow to be 9 foot as before, this I place from E to G, by which point G, draw the sine A G C, as was formerly done in the last Chapter, by which we have proceeded thus far, but as in the last Chapter the arch B C was the height of the Sun desired; so by this Chapter the arch C D shall be the height of the Sun, which being applyed unto your Scale, will give you 36 deg. 52 m. for the height of the Sun desired.

CHAP. V. The Almica [...]ter, or height of the Sun being given, to find the length of the right shadow.

BY right shadow is meant the shadow of any staffe, post, steeple, or any Gnomon whatsoever, that standeth at [Page 34]right angles with the Horizon, the one end thereof respect­ing the Zenith of the place, and the other the Nadir. First, therefore according unto the third Chapter, describe the quadrant A B D, then suppose the height of your Gnomon or substance yeilding shadow, to be 12 foot, as in the former Chapter: this doe I set down from A to E, and from the point E draw the

[diagram]

line EF parallel to A B, then set the Almicanter (which here we will sup­pose to be 53 deg & 8 m. as it was found by the third chap.) from B unto C, from which point C, draw the line C A, cutting the line E F in the point G, so shall E G be the length of the right shadow desired which being taken betwixt your compasses, and applyed unto your Scale, will give you 9 of those divisions, whereof A E was 12, which here doth signifie 9 foot.

CHAP. VI. The Almicanter, or height of the Sun being given, to find the length of the contrary shadow.

BY the contrary shadow is understood the length of any shadow that is made by a Staffe or Gnomon standing at right angles against any perpendicular wall, in such a manner that it may lie parallel unto the Horizon; the length of the contrary shadow doth increase as the Sun riseth in height: whereas, contrariwise, the right shadow doth de­crease in length as the Sun doth increase in height. There­fore [Page 35]the way to find out the length of the Versed shadow is as followeth. First, draw your quadrant, as is taught in the third Chapter, now supposing the length of your Gnomon to be 12 foot, place it from A to E, likewise from E draw the line E F parallel to A B, as before: now supposing the height of the Sun to be 36 deg. 52 min. take it from your Scale, and place it from D to C, from which point C, draw the line C A, cutting the line E F in the point G, so shall G E be the length of the contrary shadow, which here will be found to be 9 foot, the thing desired.

CHAP. VII. Having the distance of the Sun from the next Equino­ctial point, to his Declination.

FIrst, draw the line A B, then upon the end A, raise the perpendicular A D, then opening your compasses to the Radius of the Scale, place one foot in the center A and with

[diagram]

the other draw the quadrant BCD: then supposing the Sun to be either in the 29 degree of Taurus, or in the first degree of Leo, both which points are 59 deg. distant from the next Equinoctial point A­ries. Or if the Sun shall be in the 29 degree of Scorpio, or or in the first degree of Aquarius, both which are also 59 de­grees distant from the Equinoctial point Libra, therefore, take 59 degrees from your Scale, and place it from B to C and draw the line C A, then place the greatest declination [Page]of the Sun from B unto F, which is 23 deg. 30 min., then fix­ing one foot or your compasses in the point F, with the other take the nearest distance unto the line A B, which you may do by opening or shuting of your compasses, still turning them to & fro, till the moving point thereof doe only touch the line A B: this distance being kept, set one foot of your compasses in the point A, and with the other make a mark in the line A C, as at E, from which point E, take the neerest extent unto the line A B, this distance betwixt your com­passes being kept, fix one foot in the arch B C D, moving it either upwards or downwards, still keeping it directly in the arch line, until by moving the other foot to and fro, you finde it to touch the line A B and no more, so shall the fixed foot rest in the point G, which shall be the Declination of the Sun accounted from B, which in this example will be found to be about 20 degrees, the thing desired.

CHAP. VIII. The Declination of the Sun, and the quarter of the Ecliptique which he possesseth being given, to find his true place.

LEt the Declination given be [...]0 degrees, and the quar­ter that he possesseth, be betwixt the head of Aries and Cancer, first draw the quadrant A D E F, then set the greatest Declination of the Sun upon the Chord from D unto B, which is 23 degrees and 30 minutes, then from the point B take the shortest extent unto the line A D, this di­stance being kept, set one foot in the point A, and with the other describe the small quadrant G H I, then set the decli­nation of the Sun (which in this example is 20 degrees) from D unto C, from which point C, take the shortest extent unto the line A D, this distance being kept, place one foot in the arch line G H I. after such manner, that the other foce being turned about, may but onely touch the line A D, so shall the fixed foot stay upon the point H, through which point H, draw the line A H E, cutting the arch D F in the point E: so shall the arch D E be the distance of the Sun [Page 37]from the head of

[diagram]

Aries, which here will be found to be 59 degrees, so that the Sun doth here­by appear to be in 29 degrees of Tau­rus, at such time as he doth possesse that quarter of the Ecliptique, betwixt the head of Aries and Cancer.

CHAP. IX. Having the Latiude of the place, and the distance of the Sun from the next Equinoctial point, to find his Amplitude.

FIrst, make the the quadrant

[diagram]

ABCD, then take from your Scale 37 deg. 30 min. which here we will suppose to be the comple­ment of the Lati­tude, and place it from B unto E, then taking the neerest distance betwixt the point E, and the line A B, with one [Page 38]foot set in A, with the other draw the arch F G H, then place the Suns greatest declination from B unto I, from which point I, take the neerest extent unto the line A B, which distance being kept, place one foot of your compas­ses in the arch line F G H, so that the moving foot may but only touch the line A B at the shortest extent, so shall the fixed foot rest in the arch line F G H at G, through which point G, draw the line A G C, then supposing the Sun to be in the 29 degree of Taurus, that is, 59 degrees distant from the next Equinoctial point, take 59 degrees from your Scale and place themfrom B to L, from which point L, take the neerest distance unto the line A B, with this distance, setting one foot in the point A, with the other make a mark in the line A C, as at O, from which point O, take the shor­test extent unto the line A B, this distance being kept, fixe one foot of your compasses in the arch B C D, in such a manner, that the moving foot thereof may but only touch the line A B, so shall the fixed foot rest in the point R, which is the amplitude counted from B, and will be found in this example to be 34 deg. 9 min.

CHAP. X. Having the Declination and Amplitude of the Sun, to find the height of the Pole.

FIrst, make the quadrant A B C D, then supposing the amplitude to be 34 deg. 9 min. (as it was found by the last chapter) take it from your Scale, and place it from B to E, then taking the neerest extent from the point E unto the line A B, set one foot of your compasses in the center A, and with the other draw the arch G H I, then supposing the Sun to have 20 degrees of declination, take them from your Scale, and place them from B unto F, from which [Page 39]point F, take the shor­test extent unto the

[diagram]

line A B: this distance being kept, fix one foot of your cōpasles in the arch line GHI, so that the other foot may but touch the line AB at the nearest extent, so shall the fix­ed foot rest at the point H, through which point H, draw the line A H C, cut­ting the arch B C D in the point C, so shall the arch B C be the height of the Equinoctial, and the complement thereof which is the arch C D, shall be the elevation of the Pole above the Horizon, or the distance of the Equinoctial from the Zenith, which in this example will be found to be 52 deg. 30. min. the thing desired.

CHAP. XI. Having the Latitude of the place, and the Declination of the Sun, to find his Amplitude.

FIrst, make the quadrant A B C D, then supposing the Latitude to be 52 deg. 36 min. take it from your Scale, and place it from D to E, or (which is all one) if you place the complement thereof from B to E, from which point E take the neerest extent unto the line A B, with this distance setting one foot of your compasses in the center A, with the other describe the arch F G H, then supposing the de­clination of the Sun to be 20 degrees, place them from B to O, from which point O, take the shortest extent unto the line A B, which distance being kept, fixe one foot in the arch [Page 40]F G H, so that

[diagram]

the other may but only touch the line A B at the neerest di­stance, so shall the fixed foot rest at the point G, through which point G, draw the line A G C, cutting the arch B C D in the point C, so shall the arch BC be the amplitude de­sired, which in this example will be found to be 34 degrees 9 min. as before in the 9 chapter.

CHAP. XII. The elevation of the Pole, and the Amplitude of the Sun being given, to find his Declination.

FIrst, draw the quadrant A B C D, then supposing the am­plitude to be 34 deg. 9 min. place it from B to E, and from the point E take the neerest extent unto the line A B, with which distance, setting one foot of your compasses in the center A, describe the arch G H I: then supposing the Latitude to be 52 deg. 30 min. place it from D to C, from which point C, draw the line C H A, cutting the arch G H I, in the point H, from which point H, take the neerest extent unto the line A B; with this distance, fixing one foot of your compasses in the arch B C D, as the other may but only touch the line A B at the neerest extent, so shall the fixed foot rest at the point F, which shall be 20 degrees di­stance from the point B, the declination of the Sun desired.

CHAP. XIII. Having the Latitude of the place, and the declination of the Sun, to find his height in the Vertical Circle, or when he shall come to be due East or West.

FIrst, draw the quadrant A B C D, then supposing the Latitude to be 52 deg. 30 min. take it from your Scale and place it from B to C, then taking the neerest extent from the point C unto the line A B, with one foot set in the center A, with the other describe the arch G H I: then supposing the Sun to have 20 deg. of declination, place it from B to O, from

[diagram]

which point O take the shortest extent unto the line A B, with this distance, fixing one foot in the arch GHI, so that the other may but only touch the line AB, at the nee­rest extent, so shall the fixed foot rest in the point H, through which point H draw the line AHE, cutting the arch B C D in the point E, so shall the arch B E be the height of the Sun when he commeth to be due East or West, which being taken between your compasses, and applyed unto your Scale, will give you 25 deg. 32 min. the thing desired in this example.

CHAP. XIV. Having the Latitude of the place, and the declination of the Sun, to find the time when the Sun cometh to be due East or West.

FIrst, draw the quadrant A B C D, then placing the Latitude of the place (which here we will suppose to be 52 deg. 30 min.) from B to C, and draw the line C E, then with the neerest distance from the point C unto the line A B, which is the line C E, setting one foot of your com­passes in the cen­ter

[diagram]

A, with the o­ther draw the arch GHI, then suppo­sing the declina­tion of the Sun to be 20 deg. place it from B to F, from which point F, lay a Rule unto the center A, & where it crosseth the line C E, there make a mark as at O, then with the distance O E, fix one foot of your compasses in the arch G H I, after such manner, that the other foot may but only touch the line A B at the nee­rest extent: So shall the fixed foot stay in the point H, through which point H draw the line AHN, so shall D N be the quantity of time from the meridian, when the Sun com­meth to be due East or West, which in degrees will here be found to be 73 deg. 30 min. and these converted into time (by allowing 15 degrees to an houre, and four minutes [Page 43]for a degree) will make four houres and 54 minutes of an houre, that is, either at four a clock and 55 minutes in the afternoon, or at 7 a clock and 5 minutes in the morn­ing.

CHAP. XV. Having the latitude of the place, and the declination of the Sun, to find what altitude the Sun shall have at the houre of six.

FIrst, draw the quadrant A B C D, then supposing the latitude of the place to be 52 deg. 30 min. place it from B to C, and from the point C, take the shortest extent unto the line A B, with this distance setting one foot in

[diagram]

the center A, with the other draw the arch G H I, then supposing the De­clination of the Sun to be 20 deg. place it from B to E, and draw the line AHE, cutting the arch G H E in the point H, from which point H take the shortest extent unto the line A B, this distance being kept, fix one foot of your compasses in the arch B C D, in such fort that the other may but only touch the line A B, so shall the fixed foot rest in the point O, whose distance from the point B shall be the altitude of the Sun at the hour of six, which in this example will be found to be 15 degrees 44 minutes the thing desired.

CHAP. XVI. Having the Latitude of the place, and the Declination of the Sun, to find what Azimuth the Sun shall have at the houre of fix.

FIrst, draw the quadrant A B C D, then supposing the la­titude to be 52 deg. 30 m. place it from D to C, and draw the line A C, then supposing the Declination to be 20 deg. place it from B to E, from which point E draw the line E G, parallel to

[diagram]

A B, until it cut­teth the line A D in the point G, & with the distance AG, describe the arch GHI, cutting the line AC in the point H, through which point H draw the line O H P, parallel to A B, then taking the length of the line G E, betwixt your compasses, place it upon the line P O from Punto R, through which point R draw the line A R M, cutting the arch B C D in M, so shall the Arch B M be the Azimuth from the East or West which is here found to be 12 degrees 30 minutes.

CHAP. XVII. The Latitude of the place; the Almicanter, and Declination of the Sun being given, to find the Azimuth.

FIrst, draw the quadrant A B C D, then supposing the la­ritude to be 52 deg. 30 min. setting it from B to C, draw [Page 45]the line A C, then supposing the Declination of the Sun

[diagram]

to be 11 deg. 30 min. Northward, set it from B to E, from which point E, take the neerest extent unto the line A B, and with this distance, fix­ing one foot in the line A C, so as the other may but on­ly touch the line A B, make the mark Fin the line A C: then suppo­sing the height of the Sun to be 30 deg. 45 min. place it from B to G, from which point G, take the neerest extent unto the line A D, & lay it down from A to N, then from the aforesaid point G, take the shortest extent unto the line A B, and place it from A to H, in the line A C,Here note, that if the Sun had been so low that the point H had fallen betwixt the cen­ter A and the point F, then should the arch D L have shewed the A­zimuth from the North part of the Meridian. then take the distance F H betwixt your compasses, and fix one foot in the line A C, so as the other may but touch the line A D, so shall the compasses stay in the point O, from whence take the shortest ex­tent unto the line AB, with which distance, setting one foot of your compasses in the point N, with the other foot describe the arch I, by the convexity of which arch and the point A, draw the line A L, cutting the arch B C D in the point L, so shall the arch B L be the azimuth from the East or West, and the arch L D the azimuth from the South, which in this example will be found to be 66 deg. 43 [Page 46]min. the true Azimuth from the South: this is to be under­stood when the Sun hath North Declination.

But if the Sun hath South Declination, then draw the following quadrant A B C D, and set 52 deg. 30 minutes, from B to C, for the elevation of the Pole, and draw the line C A, then supposing the Sun to have 11 deg. 30 min. of South declination, place it from B to E, and from the point E, take the neerest extent unto the line A B, with which distance, fix one foot in the line A C, so as the other foot may but only touch the line AD, and where the fixed foot so resteth, there make a mark, as at F: then place the height of the Sun, which here we will suppose to be 13 deg. 20 min. from B to G, from which point G, take the neerest

[diagram]

extent unto the line A D, and place it from A unto N in the line A B, then take the neerest ex­tent from the for­mer point G unto the line A B, with this distance fix one foot in the line A C, so as the other may but only touch the line A D, and where the fixed foot so re­steth, there make a mark, as at O, from which point O take the shortest extent unto the line A B, and place it from F to H, then with the distance H A, setting one foot in the point N, with the other describe the arch line I, by the convexity of which arch line and the center point A, draw the line A L, cutting the arch B C D in the point L, so shall the arch D L be the Azimuth of the Sun from the South, which here, will be found to be 49 deg. 49 min. the thing desired.

CHAP. XVIII. The latitude of the place, the Declination of the Sun, and the altitude of the Sun being given, to find the hour of the day.

FIrst, draw the quadrant A B C D, then supposing the La­titude to be 52 deg. 20 min. place it from B unto C, and draw the line A C, then place the declination of the Sun (which here let be 11 deg. 30 min. South) from B unto E, from which point E, take the neerest extent unto the line A B, this distance place from A to F in the line A C, through wch point F draw

[diagram]

the line P F O pa­rallel to A B, then from the aforesaid point E, take the shortest extent un­to the line A D, which lay down frō A to R in the line A B: then suppo­sing the altitude of the Sun to be 15 d. 24 m. place it from B unto H, from which point H, take the shortest extent unto the line A B, and place it in the line A D from D unto I, and take the distance A I, with which distance fix one foot in the line A C so as the other may but just touch the line A D, and where the fixed foot shall rest, make a mark as at N, then with the distance A N, place one foot in R, with the other draw the arch L, by the convexity of which arch line and the center A, draw the line AG, so shall the arch B G give the time from the houre of six, and D G the time from noon, which in this example will be found to be 45 deg. which in time makes 3 houres: so that in the La­titude of 52 d. 20 m. the Sun having 11 d. 30 min. of South [Page 48]declination, and the altitude in the morning to be 15 deg. 24 min. it will appear to be 9 a clock; but in the afternoon it would have been 3 a clock.

But if the declination had been North, then the distrnce from the point H to the line A B, should have been placed from the center A towards the point I, and the distance I P taken instead of A I, as by the next figure I will make more plaine.

First, draw the quadrant A B C D as before, then place the latitude 52 deg. 20 min. from B to C, and draw the line A C, then let the declination be 11 deg. 30 min. as before, place it from B to E, from which point E take the neerest distance unto the

[diagram]

line AB, and place it from A to F in the line AC, and through the point F, draw the line O P parallel to AB, then from the aforesaid point E, take the shortest extent unto the line AD, and place it in the line A B from A unto R: then the altitude of the Sun being ob­served in the morning to be 42 deg. 33 min. I place it from B to H from which point H take the neerest extent unto the line A B. Now seeing the declination of the Sun is supposed to be Northward, therefore place this last distance from the center A unto I, and take the distance P I, with which di­stance fix one foot in the line A C, so as the other may but only touch the line A D, and where the fixed foot shall so rest, there make a mark, as at N, then with the distance A N, place one foot in the point R, and with the other draw the [Page 49]arch line L, by the convexity of which arch line and the center A, draw the line A G, cutting the arch B C D in the point G, so shall the arch B C give the quantity of time from the houre of six, and the arch D G the quantity of time from the Meridian, which in this example will be found to be about 30 degrees, that is in time two houres, so that the ob­servation being before noon, as here, it will be ten a clock, if it had been in the afteruoon it would have been two a clock.

And here note, that if the altitude of the Sun had been so small that the point I had falne betwixt the center A and the point P (which is the altitude of the Sun at the houre of six) then should that part of the arch B C D towards B give the quantity of time, either before six in the morning, or after six in the evening.

CHAP. XIX. Having the Azimuth, the Suns altitude, and the Declination, to find the houre of the day.

FIrst, draw the quadrant A B C D, then supposing the Sun to have 11 deg. 30 m. declination, place it from B to E, & from the point E take the shortest ex­tent

[diagram]

unto the line A D, with which distance, place one foot in the center A, and with the other describe the arch G H I, then let the A­zimuth be 66 deg. 34 min. as it was found by the for­mer part of the 17 chapter, which place from B to F, & from the point F take the [Page 50]neerest extent unto the line A B, which distance place from G to H, in the arch G H I, through which point H draw the line A H C, then the altitude of the Sun being 30 deg. 45. m. place it from B to L, and from the point K take the neerest extent unto the line A D, with which distance, setting one foot in the center A, with the other describe the arch N R, then with the distance N R, fix one foot in the arch line B C D, so as the other may but only touch the line A B, so shall the fixed foot rest in the point M, and the arch B M shall shew the houre from the Meridian, which will be found in this Example to be 53 deg. 40 min. that is three houres, and something better then 34 min. from the Meridian.

CHAP. XX. Having the hour of the day, the Suns altitude, and the declination, to find the Azimuth.

FIrst, make the quadrant A B C D, then supposing the de­clination of the Sun to be 11 deg. 30 min. North as before place it from B to E,

[diagram]

then suppose the alti­tude of the Sun to be 30 deg. 45 min. place it from B to L, from wch point L, take the shortest extent unto the line A D, with this distance, setting one foot in the center A, describe the arch N R, now let the arch for the houre be 53 deg. 40 min. as it was found by the last chapter, this set from B unto M, and from the point M take the shortest extent unto the line A B, and place it from N to R, in the arch N R, and [Page 51]by the point R, draw the line A R C, then from the point E take the neerest extent unto the line A D, with which di­stance upon the center A, draw the arch G H, then with the distance G H, fix one foot in the arch B C D, so that the o­ther may but touch the line A B, then will the fixed foot rest in the point F, and the arch B F will shew the azimuth from the South, which in this example will be found to be 66 deg. 43 min. the thing desired.

CHAP. XXI. Having the Latitude of the place, and the Declination of the Sun, to find the Ascensional difference.

FIrst, draw the quadrant A B C D, then place the Lati­titude (which here let be 52 deg. 30 min.) from D to C, and draw the line C O P parallel to A D, then with the di­stance C P, upon the center A, describe the arch G H I, then

[diagram]

place the declina­tion, being 20 deg. from B to F, then lay your rule from the center A upon the point F, and draw F O, cutting CP in the point O, through which point O draw the line OHR, cutting the arch G H I in the point H, through wch point H draw the line A H E, cutting the arch B C D in E, so shall the arch B E be the difference ascensional, and will be found in this example to be 28 deg. 19 min. which resolved into time doth give one houre, and something better then [Page 52]53 min. for the difference betwixt the Suns rising or setting, and the houre of six, according to the time of the year.

CHAP. XXII. Having the Declination of the Sun to find the right ascension.

FIrst, describe the quadrant A B C D, then place the grea­test declination of the Sun from B to E, and draw the line

[diagram]

EP parallel to AD, and with the di­stance EP, with one foot in A, describe the arch GHI, then set the Declination of the Sun given 20 deg. from B un­to F, and laying your rule upon the center A and the point F, draw the line F O cutting E P in O, through which point O draw the line OHR parallel to A B, cutting the arch G H I, in H, and through the point H draw the line A H C, cutting the arch B C D in the point C, so shall B C be 56 deg. 50 min. the right ascen­sion desired.

CHAP. XXIII. Having the right Ascension of the Sun or Star, together with the difference of their ascensions, to find the Oblique Ascension.

THe right ascension of any point of the heavens being know, the difference of the Ascension is either to be ad­ded thereunto, or else to be substracted from it, according [Page 53]as the Sun or Star is situated in the Northern or Southern signes. As for example: if the Sun be in any of these six Northern signes Aries, Taurus Gemini, Cancer, Leo, or Virgo, then the difference of the ascensions is to be subtracted from the right ascension, and the remainder is the oblique ascen­sion; therefore let the fourth degree of Gemini be given, the right ascension whereof is found to be 62 deg. or 4 hours, and 8 min. and the difference of ascension (where the Pole is elevated 52 deg. 30 min.) is 30 deg. 3 min. or in time, 2 houres and somthing better, which being taken from the right ascension, leaves 2 houres and 8 min. or 32 deg. 5 m. which is the Oblique ascension of the Sun, in the fourth degree of Gemini. But if the Sun be upon the South side of the Equinoctial, either in Libra, Scorpio, Sagittarius, Capri­cornus, Aquarius, or Pisces, then the difference of the ascen­sions is to be added unto the right ascension, and the sum of them both will be the Oblique ascension. As suppose the fourth degree of Sagittariou to be given, the right ascension whereof is found to be 242 deg. or 16 houres 8 min. and the difference of ascensions is 30 deg. 3 min. or 2 houres, which being added unto the right ascension, doth make 18 houres 8 min. or in degrees, 272 deg. 3 min. which is the Oblique ascension of the Sun in the fourth degree of Sagittarius. But if you would finde the Oblique descension, you must work directly contrary to these Rules given.

CHAP. XXIV. How to find the altitude of the Sun without Instrument.

IN the third Chapter of this Book it is shewed how to find the altitude of the Sun by a Gnomon set perpendicular to the Horixon, but seeing the ground is [...] unlevel it is not so ready for this our purpose, and perhaps some may have occasion to find the altitude of the Sun, and thereby the azimuth or houre of the day, according to the 17 or 18 chapters, and yet may be unprovided of Instruments to per­form the same, or at least may be absent from them; there­fore [Page 54]it will not be un-needfull to shew the finding of the same without the Gnomon or other Instrument.

Take therefore a Trencher, or any simple boards end, of what fashion soever, such as you can get, make thereon two pricks, as A and B, then prick in a pin, naile, or short

[diagram]

wire in one of the points, as at A, where­upon hang a threed with a plummet, the lift up this board toward the Sun, till the shadow of the pin at A, come directly on the point B, and direct­ly where the threed then falleth, there make a mark as at E, under the threed, then with your Rule and Compasse draw the lines A B and A E, and find the angle B A E (by the second chapter) for that is the complement of the altitude of the Sun: or, when you have drawn A B and A E, you may make the quadrant B A F, by the third chapter, and then the angle E A F shall be your altitudo desired.

CHAP. XXV. How to find out the latitude of a place, or the Poles elevation above the Horizon, by the Sun.

SEeing that throughout this Book, the latitude of the place is supposed to be known, when as every one perhaps cannot tell which way to find it out, therefore it will not be un-needful to shew how it may be readily attained, suffi­ciently for our purpose.

First, therefore you must get the Meridian altitude, which you may doe by observing diligently about noon a little before, and a little after, still observing until you perceive the Sun to begin to fall again, then marking what was his greatest altitude, will serve for this our present purpose.

Having gotten the Meridian altitude by this, and the De­clination by the 7 chapter, you may find the latitude of the place, or the elevation of the Pole above the Horizon after this manner. If the Sun hath North Declination, then sub­tract the Declination out of the Meridian altitude, and the remainder shall be the height of the Equinoctial. But if the Sun hath South Declination, then adde the Declination to the Meridian altitude, so shall the sum of them give the alti­tude of the Equinoctial, which being taken out of the qua­drant or 90 deg. leaveth the latitude of your place, or the ele­vation of the Pole above your Horizon.

As for example upon the first day of May 1650, the Meri­dian altitude of the Sun being observed to be 55 deg. 35 m. upon which day I find the Suns place to be in 20 deg. 48 m. of Taurus, and the declination 18 deg. 00 min. and because the declination is North, I substract 18 deg. 00 min. out of the Meridian altitude 55 deg. 35 min. and there remains 37 deg. 35 min. the height of the Equinoctial, and this taken out of 90 deg. leaveth 52 deg. 25 m. for the latitude of Thuring.

But it may be required sometimes for you to make a Di­all for a Town or Countrey whose Latitude you know not, neither can come thither conveniently to observe it. Here is therefore added a Table shewing the latitude of the most principal Cities and Towns in England, so that being requi­red to make a Dial for any of those places, you need but look in your Table, and there you have the Latitude thereof. But if the Town you seek be not in the Table, look what Town in the Table lies neer unto it, and make your Dial to that Latitude, which will occasion little difference.

A TABLE shewing the Latitude of the most principal Cities and Towns in ENGLAND.
Names of the Places.Latitude
 DM
St. Albons5155
Barwick5549
Bedford5218
Bristol5132
Boston532
Cambridge5217
Chester5320
Coventry5230
Chichester5056
Colchester524
Darby536
Grantham5258
Halifax5349
Horeford5214
Hull5350
London5132
Lancaster548
Leicester5240
Lincolne5315
Newcastle5458
Northampton5218
Oxford5154
Shrewsbury5248
Warwick5225
Winchester5110
Worcester5220
Yarmouth5245
York540
The end of the second Book.

THE THIRD BOOK.
Shewing Geometrically how to describe the Hour-lines upon all sorts of Planes, howsoever, (or in what Latitude soever) scituated, two manner of wayes, without exceeding the limits of the Plane.

CHAP. I. How to examine a Plane for an Horizontal Dial.

FOrasmuch as it is necessary before the drawing of any Dial to know how your plane is alrea­dy placed, or how it ought afterwards to be placed; it is therefore expedient to shew how it may be attained unto without the help of a Quadrant, (or any such like Instrument) which for this purpose is very useful.

First, take any board that hath one straight side, and an inch or more from the straight side draw a line parallel thereto, about the middle of which line erect a perpendicular line, and at the center where these two lines meet, cut out a hollow piece from the edge of the parallel line for a plum­met to hang in: then if your plane seem to be level with the horizon, you may try it by applying the straight side of your board thereunto, and holding the perpendicular line up­right, and holding a threed and plummet in your hand, so as the plummet may have free play in the hole; for then if the threed shall fall on the perpendicular line, which way soevreyou turn the board, it is an horizontal plane.

[diagram]

As for example, let the figure ABCD be a Plane supposed to stand level with the Horizon, & for to try the same, I take the simple board G O H, ha­ving one streight side, as G H, then drawing a pa­rallel thereto I crosse it at right angles with the per­pendicular O E, and at the point of intersection I cut out a little bit, as at E, for the plummet to play in, then applying the side G H to the plane, with hol­ding the perpendicular O E upright, and holding a threed with a plummet to play in the hole E, and finding the threed to fall directly on the perpendi­cular E O which way soever I turn the board, I therefore conclude it to be an horizontal plane.

CHAP. II. Of the trying of Planes, whether they be erect or inclining, and to find the quantity of Inclination.

FOr the distinguishing of Planes, because their inclinati­on and declination may be divers, we will consider three lines belonging to every plane; the first is the Horizon­tal line, the second the perpendicular line, crossing the ho­rizontal at right angles, the third is the axis of the plane, crossing both the horizontal line and his perpendicular, and the plane itself, at right angles; the extremity of which axis [Page 59]may be called the pole of the Planes horizontall line.

The perpendicular line doth help to find the inclination, the horizontal line with his Axis to finde the Declination, and the pole of the Planes horizontal line, to give denomi­nation unto the plane.

When the plane standeth upright, pointing directly into the Zenith, it maketh right angles with the Horizon, and is therefore called an erect Plane, and a plumb-line drawn thereon is called a Vertical line, as in this figure, the plane G H L I is erect, and the line H I is the vertical line.

Now for the trying of this Plane, if you apply the straight side of your board vertically thereto, as here you see done in the figure C, and either hanging a threed and plummet in the point M, or holding up a threed and plummet with your hand, you find the threed to fall directly on the parallel line M N, it is an erect plane, but if the threed will cross the line M N, it is no erect plane, but inclineth to the Horizon.

And if you finde your plane to be erect, you may be ap­plying your board thereto, with the threed and plummet fal­ling on the parallel line M N, draw the vertical line H I by the edge of your board, the vertical line being drawn, you may crosse it at right angles with the line G L, which shall be level with your Horizon, and therefore called the hori­zontal line of the plane.

If the plane shall be found to incline to the Horizon, you may find out the inclination after this manner,

Apply your board to the plane, as you see here by the fi­gure B, in the plane F K E H, then holding up a threed and plummet, that it may fall upon the perpendicular line O E, and turning about your board, till the straight side thereof lie close with the plane, and the threed fall on the perpendi­cular line O E, so the line drawn by the straight side of the board, shall be an horizontal line, which here in this figure will be the line F E.

This being done, crosse the horizontal line at right an­gles with the perpendicular H K, then set the straight side of your board upon the line H K, which is perpendicular [Page 60]to the horizontal line, with holding the board upright, and holding up a threed and plummet, so that it may have free course to play by the side of the board, which it may have, by letting the plummet fall within the hole, a little below X in the figure A: the threed thus hanging, mark two points directly under it, the one at X where the threed crosseth the parallel line on the board, the other at T, at the upper side thereof, and so by drawing the line T X, you have the an­gle T X V, which is called the angle of reclination, and it is the angle contained betwixt the plane and the vertical line passing from the Zenith to the Nadir, the complement of which angle is the inclination of the plane, and it is the an­gle

[diagram]

[Page 61]that the plane maketh with the Horizon, the thing here desired.

By what is said here of finding the inclination of the up­per face, the inclination of the under face may soon be had, seeing they are both of one quantity in themselves, there­fore if you apply the streight side of your board to the per­pendicular line of the under face, and hang the threed and plummet in any part of the parallel line, the angle that is made by that parallel line & the threed shall be the comple­ment of the angle of the inclination of the plane to the Ho­rizon.

CHAP. III. To find the Declination of a Plane.

THe Declination of a plane is always reckoned in the ho­rizon, and it is the angle contained between the line of East & West, and the horizontal line upon the plane.

For the finding out of this Declination, first, take any board that hath but one straight side, & draw a line parallel there­to, as was done in the first Chapter, & having drawn an ho­rizontal line upon your plane, apply the streight side of your board thereunto, holding it parallel to the Horizon as in the figure of the last Chapter, where the board D is applied to the horizontal line GL, then the Sun shining upon the board, hold out a threed and plummet, so as the threed being verti­cal, the shadow of the threed may crosse the parallel line S P upon the board, in which shadow make two points, the one where the shadow crosseth the parallel, as at P, the other about R, so have you the angle SPR, which is made between the horizontal line of the plane, and the Azimuth wherein the Sun is at the time of Observation: at this same instant or as neer as may be must you take the Altitude of the Sun these two being done diligently, will help you to the planes' Declination, as followeth.

First, describe the Circle B C D E, which shall represent the Horizontal Circle, and draw the diameter B A C, repre­senting [Page 62]the horizontal line of the plane, in the last Chapter set out by the line G L, then having found S P R in the last Chapter (which is the angle made by the horizontal line of the plane, and the Azimuth wherein the Sun was at that time of observation) to be about 24 deg. 10 min. I place it here from B the West end of the horizontal line to G South­wards,

[diagram]

because the angle was ta­ken at that end of the line, at wch instant the alti­tude of the Sun (by the 24 Chap­ter of the second Booke) being found to be 13d. 20 m. having 11 deg. 30 m. South Delination, it will, by the 17 chap. of the said second Book, be made to appear that the Azimuth of the Sun is 40 deg. 11 min. from the West toward the South, and therefore the West point is 40 deg. 11 min. from the Sun towards the North; Now taking this 40 deg. 11 min. and placing them from G, the place of the Sun at the time of observation, to H Northwards, you shall have the true West point, and if you draw the line H R, it shall represent both East and West, and crossing the line H R at right angles, in the center A, you shall have the line D E for the North and South; and if you crosse the ho­rizontal line B A C in the point A, at right angles with the line S N, it shall be the Axis of the plane, the two Poles whereof Sand N shall be the poles of the planes horizontal line. Now the angle of Declination here required, is the angle B A H, or E A S, for look how much the horizontal [Page 63]line of the plane declineth from the line of East & West, so much doth the poles of the Planes horizontal line decline from the North and South towards either East or West, ei­ther of which angles in this example will be found to be a­bout 16 deg. the declination of the plane from the South point E, Eastward.

CHAP. IV. How to draw the Meridian line upon an Horizontal Plane, the Sun shining thereon.

IF your Plane be level with the Horizon, describe thereon a Circle, as B C D E, then holding up a threed and plum­met, so as the threed being vertical, the shadow thereof may fall upon

[diagram]

the center A, & draw the line of shadow CE, then take the altitude of the Sun at the same instant (or as neer as may be) & by the 17 chapter of the former book get the Azimuth of the Sun, which let be (as in that example it was found) 66 d. 43 min. from the South towards the East, this 66 deg. 43 min. I place from E the point of the shadow Southwards to B, and draw the line BAD, which is the meridian line desired.

CHAP. V. Of making the Equinoctial Dial.

AN Equinoctial plane is that which is parallel to the E­quinoctial circle of the Sphere, and therefore having draw the horizontal line B C, and crossed it with the perpendicular D E at right angles in the point A, if by [Page 64]the second Chapter, you shall finde the inclination of the plane towards the South to be equal to the complement of your Latitude, and by the the third Chapter you find the horizontal line directly in the line of East and West, and so to have no declination, you may be sure this plane is paral­lel to the Equinoctial Circle, and is therefore called an Equi­noctial plane.

[diagram]

This Dial is no other then a Circle divided into 24 equal parts, by which divisions and the center A you may draw so many hour-lines as shall be necessary. As you may see here done in the Circle B C D E, which is divided equally into 24 equal parts, and hour-lines drawn from the center A to so many of them as is needful, the line D E, which is [Page 65]the line of inclination, is the Meridian or 12 of clock line, his stile is no more but a straight pin or wire plumb erected in the center.

This Dial, though of all other he be the simplest, yet is he mother to all the rest; for out of him, as from a root, is derived the projectment of those 24 hour-lines on any other great circle or plane whatsoever.

CHAP. VI. The drawing of a Dial upon the direct Polar Plane.

A Direct Polar Plane is that which is parallel to the circle of the hour of six, therefore having drawn the Hori­zontal line A B, and crossed it at right angles about the middle of the line at C, with the perpendicular C E, if you shall find the Planes inclination towards the North to be equal to the Latitude of the place, and the horizontal line directly in the line of East and VVest, and so to have no declination you may be sure this plane lieth parallel to the hour of six, and is therefore called a Polar plane. The hori­zontal line being drawn at the length of the plane, divide it into seven equal parts, and set down one of them in the line of inclination from C unto D, & upon the center D describe the Equinoctial circle, which you may divide into 24 equal parts if you will, but one quarter thereof into 6 will serve as well: then at the distance C E draw the line F G parallel to AB: Then having divided the Aequator either into 24 equal parts, or one quarter thereof into 6, you may be a rule laid

[diagram]

[Page 66]the center D, and each of those six parts, make marks in the horizontal line A B, (which here is instead of the contin­gent-line) as you may see by the pricked lines, these distan­ces from the Meridian being applied upon the same line on the other side of the Meridian, and also on both sides the Meridian in the upper line, the lines drawn from point to point, parallel to the Meridian C E, shall be the hour-lines, the line C E shall be the Meridian-line, the hour of 12, and must also be the substilar-line, whereon the stile must stand, which may be a plate of iron or some other metal, being so broad as the semidiameter of the Circle is, as is shewed in the figure. This stile must be placed along upon the line of 12, making right angles therewith, the upper edge whereof must be pa­rallel to the plane, so shall it cast a true shadow upon the hour-lines. The under face of this Polar plane, and also of the former Equinoctial plane, is made altogether like unto the upper faces here described, without any difference at all.

CHAP. VII. The making of an erect Meridian Diall.

A Meridian plane is that which is parallel to the Meridian Circle of the Sphere; therefore having drawn the hori­zontal line A B, and finding it to decline 90 deg. from the South, the plane being erect, I conclude it to lie parallel to the Meridian Circle of the Sphere, and is therefore called a Meridian plane.

The horizontal line being drawn, at the North end there­of as at A, make an angle equal to the elevation of the Equi­noctial, which is in this example 37 deg. 30 min. and draw the line A C D so long as your plane will give you leave, making an angle in the point A, equal to the Equinoctials height, so shall the South end of this line behold precisely the Equinoctial Circle, this line divide into five equal parts, and with the same widenesse of the Compasses, with one foot in E, (which is the fourth division from A, or the first from D) describe the Circle DH, through the center where­of draw a Diameter, cutting the former Diameter D H at [Page 67]right angles in the center E, this Diameter shall lie parallel to the Axletree of the world, and be the line for the hour of six, then at the out-sides of the Circle draw two touch-lines, one beneath, the other above the Circle, so that they may be both parallel to the middle line; then divide one quarter of the Circle into six equal parts, and place the Rule upon the center E, and each of those parts, mark where it touch­eth the line of contingence; as here you may see it doth in the points 7, 8, 9, 10, and 11, from which points, if you draw lines parallel to the line of six, they shall be the hour-lines here desired, and shall be parallel the one to the other: the

[diagram]

distance betwixt 6 and 7 is the same with 6 and 5, and the distance between 6 and 8 is the same with the distance be­twixt 6 and 4, and so you have all the hour-points in the up­per [Page 68]touch-line, and if you transfer these distances from the hour of six into the other rouch-line; likewise from the line of six you may the better by the opposite points draw the lines parallel to the line of six.

For the style of this Dial, it may be either a plate of some metal, being so broad as the semidiameter of the Circle is, and so placed perpendicularly along over the line of the hour of six, the upper edge thereof being parallel to the plane, or it may be a straight pin fixed in the center of the Circle, making right angles with the plane, being just so long as the Semidiameter of the circle is, only shewing the hour with the very top or end thereof.

[diagram]

This Plane hath two faces, one to the East, the other to the west, the making whereof are both alike, onely in naming [Page 69]the hours, for the one containeth the hours for the forenoon the other for the afternoon, as you may perceive by the fi­gures.

CHAP. VIII. To draw a Dial upon an horizontal plane.

AN Horizontal plane is that which is parallel to the Ho­rizontal circle of the Sphere, which being found by the first Chapter to be level with the Horizon, you may by the fourth Chapter draw the Meridian line A B, serving for the Meridian, the hour of 12, and the substilar: in this Meridi­an make choice of a center, as at C, through which point C draw the line D E, crossing the Meridian at right angles, this line shall be the line of East and VVest, and is the six a clock line both for morning and evening.

Then by the second Chapter of the second Book draw the line S C, making the angle S C A equal to the latitude of the place, which here we will suppose to be 52 deg. 30 min. this line shall represent the cock of the Dial, and the Axle­tree of the world; then at the North end of the Meridian line draw another line as F G, crossing the Meridian in the point A at right angles, this line is called the Touch-line, or line of contingence. Then set one foot of your Compasses in the point A, and with the other take the necrest extent unto the line S C or the stile, with this distance turning your compasses about, with one foot still in the point A, with the other make a mark in the Meridian, as at I, which shall be the center of the Equinoctial, upon which describe the E­quinoctial Circle A D B E, with this same distance setting one foot in the point A, make a mark at F on the one side of the Meridian, and another at G on the other side thereof, both which must be in the line of contingence, by which two points and the center C you may draw the hour-lines of 3 and 9.

This same distance of your Compasses being kept, with one foot still in the center A, with the other make the marks T and V in the Equinoctial Circle, each of which [Page 70]distances is an arch of 60 degrees, or four hours of time, the half of which arch is 30 degrees, or two hours from the Me­ridian this divided in the half will be 15 deg. or one hour from the Meridian; then laying your rule upon the center I of the Equinoctial, and upon these two last divisions in the circle thereof, where the rule shall touch the line of con­tingence, there mark it as at Hand O, by which points and the center C, you may draw the hour-lines of 10 and 11;

[diagram]

the like may you do on the other side of the Meridian, so have you six of your hour-lines drawn: and now because the contingent will out-run our plane, we may from the intersection of the hours of 9 and 3 with the touch-line draw the line F D and GE parallel to the Meridian A B, until they [Page 71]cut the line of East and West in the points D and E, then draw the lines A D and A E, this being done, set one foot of your compasses in the point H, and with the other take the neerest extent unto the line A E; this distance being kept, fix one foot in the line GE, so as the other may but touch the line A E, so shall the fixed foot rest in the point N, by which and the center C you may draw the 7 a clock hour-line: in like manner may you place one foot in the point O, and with the other take the shortest extent unto the line A E; with this distance fixing one foot in the line G E, so as the o­ther may but onely touch the line A E, so shall the fixed foot rest in the point R, by which & the center C you may draw the 8 a clock hour-lline, the like may be done on the other side of the Meridian, or you may by these distances thus found prick out the like on the other side of the Meridian.

Thus by dividing but half a quarter of the Equinoctial Circle into three equal parts, you may describe your whole Dial.

And whereas in Summer the 4 and 5 in the morning, and also 7 and 8 at evening shall be necessary in this kind of Di­al: prolong or draw the lines of 4 and 5 at evening beyond the center C, and they shall shew the hour of 4 and 5 in the morning, and likewise the 7 and 8 in the morning for 7 and 8 at evening.

What is here spoken concerning the hours, the like is to be done in drawing the half hours, as well in this kind as in all them which follow.

The stile must be fixed in the center C, hanging directly over the Meridian line A C, with so great an angle as the lines S C A maketh, which is the true pattern of the cock.

This and all other kinds of Dials may be drawn upon a clean paper, and then with the help of your compasses pla­ced on your plane.

CHAP. IX. To draw a Dial upon an erect direct vertical plane, commonly called a South or North Dial.

A Vertical plane is that which is parallel to the prime ver­tical circle, it hath two faces, one to the South, the other to the North; therefore having drawn the horizontal line A B, and from the middle thereof let fall the perpendicular C D, which if you find by the second Chapter to be erect, and the Horizontal line A B to lie in the line of East & west, and so to have no declination, you may be sure this plane is parallel to the prime vertical circle of the Sphere, and there­fore is called a vertical plane.

This perpendicular C D shall serve for the meridian, the hour of 12, and the substilar line, which is the line over which the Stile or Gnomon in your Dial directly hangeth.

The Horizontal line A B shall serve for the hour-lines of six, both for morning and evening.

Then (by the second Chapter of the second Book) draw the line S C, making the angle S C D equal to the Latitudes complement, which in this example is 37 deg. 30 min. this line S D shall represent the axletree of the world, and if you draw the line S D square to the Meridian, you shall have the triangle S C D for the true pattern of your Dials cock: then crosse the meridian (in some point thereof, as at E) at right angles, with the line F G, serving for the touch-line, then taking the shortest extent from the point E, unto the line S C, place it in the meridian from E to I, which point shall be the center of the Equinoctial circle, therefore upon the center I describe the circle A E B for the Aequator; this di­stance of your compasses being kept, set one foot in the point E, and with the other set out the equal distances E G and E F in the contingent line, which shall be the points thorow which you must draw the hour-lines of 9 and 3 from the point C, the center of the Dial; the same distance of your compasses being still kept, with one foot in the point E, with the other make a mark in the Equinoctial Circle, as at H, So shall the arch E H contain 60 deg. or four hours of time, [Page 73]which arch you may divide into four equal parts, and by laying your Rule upon the center I, and those two divisi­ons next the meridian C D you may make two marks in the line of contingence, as at N and O, thorow which two points, and from the center C may you draw the hour-lines of 10 and 11, and now because the touch-line will out-run the plane, you may from the intersections of the hour-points of 9 and 3 in the line of contingence, draw the lines A F and B G, parallel to the meridian C D, until they cut the

[diagram]

Horizontal line A B, then draw the lines A E and B E, this being done, set one foot in the point O, and with the other take the shortest extent unto the line A E: with this distance set one foot in the line A F, so as the other may but touch [Page 74]the line A E at the neerest extent, and the fixed foot shall rest in the point T, by which & the center C you may draw the 7 a clock hour-line: in like manner may you place one foot in the point N, and with the other take the shortest ex­tent unto the line A E, with which distance fix one foot in the line A F, as that the other being turned about, may but onely touch the line A B, so shall the fixed foot rest in the point V, by which and the center C you may draw the 8 a clock hour-line, or you may extend the line A B a little be­yond the points A and B, and take the distance E O, and set it from A to R, or from B to R, then lay a Rule upon the points R and O, and where you shall see it crosse either the line A F or B G, there make a mark as at T, which shall be the point thorow which the hour of 7 or 5 must be drawn from the center, as before; so likewise may you take the distance E N, and set it from A to P, and from B to P, and laying your Rule upon the two points P and N, where you shall see it crosse the lines A F or B G, there make a mark, as at V, which shall be the point for the hour of 8 or 4; this is in effect no more but to draw a line parallel to the line A E or B E by the point O, and this line shall cut the line A F or B G in the point T, which shall be the points for the hours of 7 or 5, as before: and so you may by the point N draw a line parallel to the line A E or B E, and where it shall crosse the line A F or B G, there make a mark as at V, which shall be the points for the hours of 8 or 4.

Thus you may see, that by dividing the Radius of the E­quinoctial circle into four equal parts, you may describe your whole Dial, if it hath no Declination; for having with this Radius pricked out the lines of 3 and 9, and placed the lines for 10 and 11, these two distances from the hour of 12 shall give the like for the hour of 1 and 2 on the other side of the meridian, and having drawn the lines for 7 and 8, by the former rules, you may take their distances from the hour of 6, and place them on the other side of the Dial from 6 to 5, and 4: so have you all your hour-lines drawn, and yet we have not out-run the limits of our plane, which is an [Page 75]inconvenience, unto which the most are subject. Now see­ing the triangle S C D is the true pattern of this Dials cock, and that this is the South face of this plane, therefore the center will be upward, and the stile point downward, hang­ing directly over the meridian-line.

But if it had been the North face of this plane, you must have proceeded in all things, as before, but onely in placing the Dial, and naming the honrs; for if it be the North face, the center must be in the lower part of the meridian line, and the stile and hour-lines point upwards, as you may see in this figure following.

[diagram]

CHAP. X. To draw a Dial upon a direct vertical plane, inclining to the Horizon.

ALL those planes that have their Horizontal line lying [Page 76]East and West, are in that respect said to be direct vertical planes; if they be also upright, passing thorow the Zenith, they are erect direct vertical planes; if they incline to the pole, they are direct Polar planes; if to the Equinoctial, they are called Equinoctial planes, and are described before: if to none of these three points, they are then called direct verticals inclining.

In all Dials that decline not, two things must be had be­fore you can make the Dial, the first is, the meridian or 12 a clock line, wherein the cock must stand, and the second is the elevation of the pole above the same line.

For the Horizontal plane, the meridian line is drawn by the fourth Chapter, and the elevation of the pole above the plane is alwayes equal to the latitude of the place; and in e­rect direct verticals, the perpendicular or vertical line is the meridian or 12 a clock line, and the elevation of the pole above the plane is alwayes equal to the complement of the Latitude, the North Pole being elevated above the North face, and the South pole above the South face thereof.

And in these verticals inclining being direct, the line of inclination is their meridian, and line wherein the cock must stand, but for the elevation of the cock, we must first consider, whether the plane inclines towards the South, or towards the North, if it inclineth towards the South, adde the inclination to your latitude, the summe of both shall be the elevation of the pole above the plane, and if the summe shall be just 90 degrees, it is an Equinoctial plane, and is de­scribed before in the fifth Chapter; but if the summe shall exceed 90 degrees, take it out of 180, and that which re­mains shall be the elevation of the pole above the plane.

As for example, in the latitude of 52 degr. 30 min let a plane be found to incline Southwards 20 degrees, this 20 degrees added to 52 degrees 30 min. the latitude of the place, the summe will be 72 deg. 30 min. the elevation of the pole above the plane, with which you may proceed to draw a Dial by the eighth or ninth Chapters as if it were a hori­zontal [Page 77]plane, for their difference is nothing, but in the height of the stile, which is the elevation of the Pole above the plane. For this plane shall be parallel to that Horizon, whose latitude is 72 deg. 30 min. lying both under one and the same meridian.

But if the inclination be Northward, compare the incli­nation with your latitude, and take lesser out of the greater; so shall the difference be the elevation of the Pole above the plane, but if there be no difference, it is a direct polar plane, and is described before in the sixth chapter.

As in the latitude of 52 deg. 30 min. a plane being pro­posed to incline towards the North 25 deg. this 25 deg. be­ing taken out of 52 deg. 30 min. leaveth 27 deg. and 30 m. for the elevation of the Pole above the plane. Now this plane being parallel to that Horizon, whose latitude is 27 d. 30 min. lying both under one and the same Meridian, there­fore you may proceed to make this Dial, as if you were to make an horizontal Dial in that Countrey.

Each of these planes have two faces; one towards the Zenith, the other towards the Nadir: but what is said of the one is common to the other; they onely differ in this, the one hath the South, the other hath the North Pole ele­vated above their faces.

For upon the upper faces of all North incliners, whose in­clination is lesse then the latitude of the place, on the under faces of all North incliners, whose inclintion is greater then the latitude of the place: and on the upper faces of all South incliners, the North Pole is elevated; and therefore con­trarily, on the under faces of all North incliners, whose in­clination is lesse then the latitude of the place, on the upper faces of all North incliners, whose inclination is greater then the latitude of the place; and on the under faces of all South incliners, the South Pole is elevated: unto one of which Poles, the stiles of all Dials must point directly.

CHAP. XI. To draw a Dial upon an erect, or vertical plane declining, com­monly called a South or North erect declining Dial.

ALL upright planes whereon a man may draw a verti­cal line; are in that respect said to be erect or vertical, if their horizontal line shall lie directly East and West, they are direct vertical planes; if directly North and South, they are properly called Meridian planes, and are described before.

If they behold none of these four principal parts of the world, but shall stand between the prime vertical circle, and the Meridian, they are then called by the general name of declining verticals, or by the name of South or North erect declining planes.

In all such declining planes, because the Meridian of the place (which in all upright planes is the vertical line, and serveth for the hour of 12) and the Meridian of the plane deflecteth one from the other, therefore we must find out and place the Meridian of the plane, (which is the line over which the stile directly hangeth, and is here called the Sub­stile) and likewise the elevation of the Pole above the plane: both which may be easily performed in this manner.

First, draw a blinde line parallel to the Horizon, which may be the line A B and from a point therein as at C, let fall the perpendicular C D, serving for the Meridian of the place, and the houre of 12, and through some place of this Meri­dian as at E, draw the line F G at right angles.

Then having by the third chapter examined this plane, and finding it to decline 30 deg. from the South towards the East, I draw an arch of a circle upon the center C, with my compasses opened to the Radius of the Scale; in which arch I place the declination of the plane from E to H, on the same side of the Meridian with the declination of the plane, as here you see, then set the complement of the latitude, which is 37 degrees 30 minutes in the same arch from E to M, & draw the line C H for the declination of the plane, and the line C M G cutting the line F G in the point G for the complement of your latitude.

This being done, take the distance E G, and set it in the line of declination from C to S, from which point S draw the line S L square to the Meridian C D,Here note, that in all decliners, the Substile go­eth from the meridian, to­wards that coast which is contrary to the coast of the planes declina­tion. then take S L and set it from E to O in the line E F, and draw the line C O, which is the Meridi­an of this plane, or the line of the Substile, wherein the stile must stand directly up from the plane, then through the point O draw the line P K, square to the Substile C O, which shall be the touch-line, or line of contingence.

Then take the distance C L, and set it in the line of contingence from O to P, and draw the line C P, for the stile.

[diagram]

This done, set one foot in the point O, and with the other take the shortest: extent unto the stile C P, with this distance, one foot remaining still in the point O, the other turned towards C, make a mark at y in the line of the Substile, which shall be the center whereon you must de­scribe the Aequinoctial circle.

Now having drawn the line N T through the center C, and parallel to the touch-line I K, which will be square to the Substile C O, I take the distance O y, which is the Radius upon which the Equinoctial circle was drawn, and place it on both sides the Substile, in the line of contingence from O to I, and from O to K, and in his parallel line from C to G, and from C to R, and draw G I and R K, then laying your rule upon the point y, the center of the Equinoctial, and a the point of intersection of the touch-line with the Meri­dian, and where it cutteth the circles circumference, there must you begin to divide it into 24 equal parts, but those six shall be onely in use which are next the line of contin­gence, that is, three of each side of the substile next thereunto.

Then place your Ruler upon the center y, and upon each of these six points of the Equator, and where it toucheth the line of contingence, there make markes, by which and the center C, draw those six hour-lines next the Substilar; which shall all fall between the points I and K, in the touch-line, three whereof shall fall betwixt O and K, and three betwixt O and I, thus have you 6 of your 12 hours, viz. 12, 11, 10, 9, 8 and 7, then take the distance from the point O to the intersection of the hour of 12 with the touch-line, and place it from O to b, and from G to N, then laying your Rule upon these two points N and b, where it shall crosse the line G I, shall be the point through which you may draw the six a clock hour-line, in like manner, take the distance from O to the line for 11 a clock, and set it from O to c, and from G to Z, then lay the Rule upon C and Z, and where it shall cut the line, G I shall be the point through which you shall draw the 5 a clock hour-line, and so placing the distance [Page 81]O 10 from O to g, and from G to V, and laying the Rule upon V and g, you shall find the point in the line G I, through which you may draw the 4 a clock hour-line: in like manner may you proceed with the other side.

For taking the distance from the point O to the line of 9, and setting it from O to h, and from R to x, and laying the Rule upon the points h and x, where you shall see it cut the line R K, there shall be the point, through which you shall draw the 3 a clock hour-line.

And so you may take the distances from O to the line of 8 and 7. and place them from O to d and m, and from R to W and T, and so by laying your Rule upon the points W d and T m, where it shall crosse the line R K, there shall be the points through which the hour-lines of 2 & 1 shall be drawn.

The Dial being thus drawn upon the South-east face of this plane let the stile be fixed in the center C, so that it may hang directly over the Substile C O, making an angle there­with equal to the angle P C O.

The stile with the substile must here point downwards, because in all upright planes declining from the South, the South pole is elevated; and in all upright planes declining from the North, the North pole is elevated.

Therefore if you were to make a Dial to the North face of this plane, you must make choice of your center C in the lower part of the meridian C D, that the stile with the sub­stile may have room to point upwards.

This Dial being made on paper for the South-east face of this plane, will also serve for the North-west face thereof, if you turn it upside down, so that the stile with the Substile may point upwards, and the paper being oiled or pricked through, so that you may take the back-side thereof for the foreside, without altering the numbers set to the hours.

And the fore-side of this pattern, turned upside down, so that the cock may point upwards, shall serve for the North­east face of a plane having the same declination; onely al­tering the numbers set to the hours.

This paper being oiled, if you do but change the back­side [Page 82]for the foreside; and the numbers set to the hours, still keeping the center upwards, and the stile pointing down­wards, this pattern will serve for the South-west face of a plane, whose declination is the same as before.

And thus you see by diligent observation, this pattern may be made to serve for four Dials, which being well un­derstood, will be a great help to the Artist.

CHAP. XII. How to draw a Dial upon an horizontal plane, otherwise then in the eighth Chapter was shewed.

ALthough I have plainly and perfectly shewed the ma­king of the horizontal, the direct South or North, as well erect as inclining; and the South or North crect declining Dials, in the four former Chapters: yet to satisfie them that delight in variety, I have here declared another way, whereby you may make them most artificially and geome­trically, not being tied to the use of the Canons; (which in­deed of all others is most exact, but not so easie to be under­stood) nor to any one Instrument, (which may be absent from me, when I should need it) although in this Treatise I do perform the whole by a plain Quadrant.

Therefore by the first Chapter, having found the plane to be horizontal, by the fourth Chapter draw the Meridian line A B, and crosse it at right angles in the middle with the line D E, which is the line of East and West, and serveth for the hour of six at morning, and six at evening.

Then upon the center C (which is the point of intersecti­on) describe a circle for your Dial as large as your plane will give leave, which let be the circle A D B E; then take the latiude of the place, which is here 52 deg. 30 min. and set it from A to N, in the quadrant A D, and draw the line C N S, then from A raise the perpendicular A S, to cut the line C S at S, so shall the Triangle A C S be the true pattern for your cock; this being done, divide the two quarters of your circle A E and A D, each into six equal parts, so shall you have in each Quadrant five ponits, by which you may [Page 83]draw the five Chord lines I F G H and A, as here you see; then take one half of the Chord-line A, and set it in the line of the stile from C to O, from which point O take the nee­rest extent unto the meridian; with this distance setting one foot in the point A, with the other make a mark on each side of the Meridian, in the same Chord-line A, through which points you shall draw the hour-lines of 1 and 11.

[diagram]

So likewise you may take one half of the chord H, and place it in the line of the stile from C to K, from which point K take the shortest extent unto the meridian, with this di­stance set one foot in H, and with the other make on each side the meridian a mark in the same chord-line, through which you shall draw the hour-lines of 2 and 10.

And thus you may proceed with the rest of the lines, as the Figure will shew better then many words; for this is sooner wrought then spoken.

And if you would have the hours before and after six, you may extend them through the center, as was shewed in the eighth Chapter.

CHAP. XIII. To draw a Dial upon a direct vertical plane, as well erect as incli­ning, otherwise then in the ninth Chapter was shewed.

THe work of this is almost like unto the other before; the difference is onely in the elevation of the Pole above the plane: for in the horizontal plane, the elevation is equal to the latitude of the place; and in all direct verticals being erect, the elevation of the pole above the plane is e­qual to the complement of the latitude, but if they shall in­cline towards the horizon, then shall you finde the elevation of the pole above the plane, by the 10 Chapter.

The elevation of the pole above the plane being known, the making of these Dials are all alike: therefore by the second Chapter draw the line E W parallel to the horizon, and from the middle thereof let fall the perpendicular Z N which shall be the meridian of the plane, and also the meri­dian of the place, serving for the line of 12, and also for the substile, over which the stile must hang, both in erect and inclining planes being direct.

Then upon the center Z describe your Dials circle, or ra­ther the Semicircle E N W, and seeing this plane is erect, and also direct, therefore the elevation of the pole above the plane is 37 d. 30 m. equal to the complement of our latitude, [Page 85]which take from your Scale, & place it from N to H in your Dials semicicircle, and draw the line Z H S for the line of the stile; then from the end of the meridian, as at N, draw the crooked line N S, cutting the line of the stile in the point S, so shall the triangle S Z N be the true pattern for your cock.

[diagram]

This being done, divide each quadrant of your Semi­circle into six equal parts, so shall you have five points, by which you may draw five chord-lines, cutting the Meridi­an at right angles in the points I K L M N.

This being done, take the half of each chord, and place it from the center Z, along upon the line of the stile, as here you see; the half of the Chord N from Z to A, and one half the chord M, from Z to B, and half the chord L from Z to C, and one half the chord-lines I and K, set from Z unto D and G: now from each of these points take the neerest ex­tent unto the Meridian Z N, & place them upon their proper chord-lines from the meridian on both sides thereof, so shall [Page 86]you have two points on each Chord, through which you shall draw the hour-lines from the center of your Dial, as the shortest extent from the point A unto the meridian, set in the Chord N from the Meridian both wayes, shall give you the points for 1 and 11; so shall the shortest extent from the point B (being placed from the Meridian both wayes in the Chord M) give you the two points for 10 and 2, and so you may proceed with the rest; thus doing, you shall have in each chord two points, on each side the meridian one; through which, and from the center Z, you may draw your hour-lines at pleasure, without exceeding the limits of your plane. And seeing this is the South face of this plane, there­fore the stile must point downwards, being fixed in the cen­ter Z in the upper part of the meridian line Z N, over which the stile must directly hang, making therewith an angle e­qual to the angle N Z S.

But if it had been the North face, then must the center be placed in the lower part of the Meridian, and the stile with the substile, and also the hour-lines must point upwards.

CHAP. XIV. The declination of an upright plane being given, how thereby to find the elevation of the Pole above the same, with the angle of Deflexion, or the distance of the substile from the Me­ridian: and also the angle of inclination betwixt both Meridians.

IN all erect declining planes, when the declination is found, there is three things more to be considered before we can come to the drawing of the Dial.

  • I. The elevation of the pole above the plane.
  • II. The distance of the substile from the Meridian.
  • III. The angle contained betwixt the Meridian of the plane and the Meridian of the place, which here we call the inclination of Meridians: this angle is made at the Pole, and serveth to shew us where we shall begin to divide our Dial-circle into 24 equal parts.

These three may be both artificially, easily, and speedily performed after this manner following.

First, describe a Quadrant, as A B C, then supposing your Latitude to be 52 deg. 30 min. take it from your Scale, and set it from B to E in the arch of the Quadrant, and draw the line E D parallel to A B, cutting the line A C in the point D, then take the distance D E, and setting one foot in the center A, with the other describe the arch G H O R.

Then suppose your declination to be 32 deg. this set from B to F in the arch B E C, and draw the line F A, cutting the arch G R in the point H. through which point draw the line S H N, cutting the arch B E C in N, so shall the arch C N be the elevation of the pole above the plane, which in this example is found to be 31 deg. 5 min.

Then take the di­stance

[diagram]

H S, and set it in the line D E from D to K, tho­row which point K draw the line A K L cutting the arch B C in the point L, so shall the arch C L be the distance of the substile from the Meridian, and in this example is 22 deg. 8 min.

Now from the point L, draw the line L T parallel to the line A C, cutting the arch G R in the point O, through which point O draw the line A O I, cutting the arch B C in the point I, so shall the arch C I be the inclination of both Meridians, and is found by this example to be 38 deg. 13 min. so that by this example the Meridian of the plane will fall betwixt the hours of 2 and 3, if the plane shall decline Westward; but if it shall decline Eastward, then shall it fall betwixt the hours of 9 and 10 before noon.

CHAP. XV. To draw a Dial upon an erect, or vertical plane declining, other­wise then in the 11 Chapter was shewed.

HAving by the third Chapter found the declination of this plane to be 32 degrees, and so by the last Chapter found the elevation of the pole above the plane to be 31 deg. 5 min. and the distance of the substile from the me­ridian to be 22 degr. 8 minutes, and likewise the angle of inclination between both meridians to be 38 degreees 13 minutes, we may proceed to make the Diall after this manner.

First, draw the horizontal W E, and the perpendicular line Z N, crossing the horizontal line at right angles, which is the meridian of the place, and the line of 12.

Then in the meridian make choice of some point with most convenience, as the center C, whereupon describe your Dial-circle E N W.

Then take a chord of 22 degrees 8 minutes from your Scale, for the distance of the substile from the Meridian, and inscribe it into this circle from the Meridian; upon these conditions, that if the plane declineth west, then must the substile be placed East of the plumb line; but if the decli­nation shall be East, then must the substile be placed west from the Meridian, as here it is.

This 22 degrees 8 minutes being set in the Dial-circle from the Meridian at N unto M, I draw the line C M for the substile: then through the center C, draw the diameter A B, making right angles with the substile C M, above this Diameter there needs no hour lines to be drawn, if the plane be erect.

Then take 31 deg. 5 min. and set them from M to D, and draw the line C D S for the stile, then from M the end of the stile draw the crooked line M S, cutting the line of the stile in the point S, so shall the triangle S C M be the true pattern for the cock of the Dial.

This being done, take 38 deg. 13 min. and set them al­wayes on that side the substile whereon the line of 12 lieth' as here from M to A, so shall the point H be the point where you shall begin to divide your Dial-circle into 24 equal parts but those points shall be onely in use which do fall below the Diameter A C B.

[diagram]

And if the line of the substile falleth not directly upon one of the hour-lines, then shall you have six points on each side thereof, from which you may let perpendiculars fall un­to the line of the substile, as here you see done.

Now take each perpendicular betwixt your Compasses, and with one foot in the center C, with the other make marks in the line of the stile, from which take the neerest extents unto the line of the substile, and lay them upon their [Page 90]own proper perpendiculars from the Substile, so may you make points, through which you may draw hour-lines, and by thus doing with each perpendicular on both sides the substile, you may describe your whole Dial, as here you see, which may serve for four faces, by observing what was spoken in the 11 Chapter.

When you have drawn, and described your Dial upon paper for any plane whasoever, you may cut off the hour­lines, cock and all, with a lesser Circle then the Dial circle, either with a concentrique or an excentrique circle, and so make a Dial lesse then the Circle by which you framed it.

Or if you extend the hour-lines beyond the Dial-circle,

[diagram]

[Page 91]you may cut them off either with a greater concentrique circle, and so make a bigger Dial, or else you may cut them off with a Square, as here you see in the following fi­gure, or any other form what you shall think most conve­nient,

Of a Plane falling neer the Meridian.

When as the declination of a plane shall cause it to lie neer the Meridian, as that the declination and inclination shall cause it to lie neer the Pole, then doth the elevation of the Pole above the plane grow so small, and the hour-lines so exceeding neer together, that except the plane be very large, they will hardly serve to good purpose; as here in this figure, being a plane which is erect, and declining from the South 80 deg. towards the East.

Therefore first, draw your Dial very true (as before hath been taught) upon a large paper, making your circle as big as you can: then extend the hour-lines, with the substile, and the line of the stile, a great way beyond the Dials cir­cle, until they do spread, so that they will fill the plane in­different well, and then cut them off with a long square, as O N in the following figure, so will it shew almost like the Meridian Dial of the 7 Chapter, for the hours will be almost parallel the one to the other, and the stile almost parallel to the substile, as you may see by the figure.

CHAP. XVI. The inclination of a Meridian plane being given, how thereby to find the elevation of the pole above the plane, the distance of the Substile from the meridian, and the angle of the inclination of the meridian of the plane to the meridian of the place.

ALL those planes wherein the horizontal line is the same with the Meridian line, are therefore called Meridian planes, if they may make right angles with the Horizon, they are called erect Meridian planes, and are described before.

An erect Dial decli­ning from the South 80 deg. towards the East: the distance of the Substile from the meridian 37 d. 4 min. the elevation of the Pole above the plane 6 deg. 4 min. and the in­clination of both Meri­dians 82 deg. 5 min.

But if they lean to the Horizon, they are then called In­cliners.

These may incline either to the East part of the Horizon, or to the West, and each of them hath two faces, the upper towards the Zenith, the lower towards the Nadir, wherein knowing the latitude of the place, and the inclination of the plane to the Horizon, we are to consider three things more before we can come to the drawing of the Dial.

  • I. The elevation of the pole above the plane.
  • II. The distance of the substile from the Meridian.
  • III. The angle of inclination betwixt both Meridians.

These three may be found after this manner, little dif­fering from the 14. Chap.

First, describe a quadrant, as A B C, then set 52 deg. 30 min. (your Latitude) from C to E in the arch of your qua­drant C B, and draw the line E R parallel to A B, cutting the line A C in the point R, and with the distance E R, with one foot in the center A, with the other draw the arch G H O D; then let your inclination be 30 deg. which set in the arch of the quadrant from B to F, and draw the line A F, cutting the arch C O D in the point H, through which point

[diagram]

H draw the line S H N, cutting the arch of the quadrant in the point N, so shall the arch C N be the elevation of the pole above the plane, wch in this example is 43 deg. 23 min.

Then take the di­stance S H, and set it in the line E R from R to K, through which point K, draw the line A K L, cut­ting the arch of the [Page 94]quadrant in the point L, so shall the arch CL be the distance of the Substile from the Meridian, and is in this example 33 deg. 5 min.

This being done, from the point S, draw the line L T pa­rallel to the line A C, cutting the arch G D in the point O, through which point O draw the line A O I, cutting the arch of the quadrant B C in the point I, so shall the arch CI be the inclination of the Meridian of the plane to the Me­ridian of the place, and in this example is found to be 43 deg. 28 min. which being resolved into time, doth give a­bout two hours and 54 min from the Meridian, for the place of the substile amongst the hour-lines.

CHAP. XVII. To draw a Dial upon the Meridian inclining plane.

HAving by the second Chapter found the inclination of this plane to be 30 degrees, and so by the last Chapter found the elevation of the pole above the plane to be 43 degr. 23 min. and the distance of the substile from the Meridian to be 33 deg. 5 min. and likewise the angle of in­clination to be 43 deg. 38 min. we may proceed to make the Dial after this manner,

First, draw the horizontal line A B, serving for the Me­ridian and hour of 12, about the middle of this line make choice of a center at C, upon which describe a Circle for your Dial, as A D B E.

Then seeing this is the upper face of the plane, set 33 de­grees 5 minutes the distance of the substile from the Meri­dian, in the Dials Circle from the North end of the Hori­zontal line upwards, as from B to H, and draw the line C H for the Substile. But if this had been the under face, the substile must have fallen below the horizontal line: now through the center C draw the Diameter E F, making right angles with the substile C H.

Then set 43 degrees 23 minutes from H unto D for the stile, and draw the line C D unto S, and from the end of the [Page 95]substile draw the crooked line H S, cutting the line of the stile in the point S, so shall the Triangle S C H be the true pattern of your Cock for this Dial.

Then set 43 deg. 28 min. from H unto M, for the diffe­rence betwixt the Meridian of the plane and the Meridian of the place. Now here at M must you begin to divide your.

[diagram]

[Page 96]Circle into 24 equal parts, from which points let down so many perpendiculars to the substile, as there shall be points on that side the diameter E F next the substile.

Now take each perpendicular betwixt your Compasses, and with one foot in the center C, with the other make marks in the line of the stile, from which take the neerest ex­tents unto the substile, and lay them down upon their own proper perpendiculars from the substile, so may you make marks, through which and from the center, you may draw the hour-lines.

This Dial being thus drawn, for the upper face of a Meri­dian plane inclining towards the West, you must fix the Cock in the center C, hanging over the substile C H, with an angle equal to the angle S C H, so that it may point to the North Pole, because upon the upper faces of all Meridian incliners the North Pole is elevated, and therefore con­trarily, the South Pole must needs be elevated above their under faces.

This Dial being drawn in paper, for the the upper face of this plane, will also serve for the under face thereof, if you turn the pattern about, so that the Horizontal line A B may lie still parallel to the Horizon, and the stile with the sub­stile (lying under the Horizontal line) may point down­wards to the South Pole, the paper being oiled or pricked through, so that you may take the back side thereof for the fore-side, without altering the numbers set to the hours.

CHAP. XVIII. The inclination and declination of any plane being given, in a known Latitude, to find the angle of intersection botween the plane and the Meridian, the aseension and elevation of the Meridian, with the arch thereof between the Pole and the plane, and also the elevation of the Pole above the plane, the distance of the substile from the Meridian, with the inclination between both Meridians.

IF a plane shall decline from the South, and also incline to [Page 97]to the Horizon, it is then called by the name of a declining inclining plane.

Of these there are several sorts, for the inclination being Northward, the plane may fall betwixt the Horizon and the Pole, or betwixt the Zenith and the Pole, or else they may lie in the Poles of the World: or the inclination may be southward, and so fall below the intersection of the Me­ridian and the Equator, or above it, or the plane may fall directly in the intersection of the Meridian with the Equa­tor, each of these planes have two faces, the upper towards the Zenith, and the lower towards the Nadir: Now having the Latitude of the place, with the declination and inclina­tion of the plane, we have seven things more to consider before we can come to the drawing of the Dial.

  • I. The angle of intersection betwixt the plane and the Meridian.
  • II. The arch of the plane betwixt the Horizon and the Meridian.
  • III. The arch of the Meridian betwixt the Horizon and the plane.
  • IV. The arch of the Meridian between the Pole and the plane.
  • V. The elevation of the Pole above the plane.
  • VI. The distance of the substile from the meridian.
  • VII. The angle of inclination betwixt the Meridian of the plane, and the Meridian of the place. All these seven may be found out after this manner.

First, Describe the Quadrant A B C, then suppose the plane to decline from the South towards the East 35 deg. and to incline towards the Horizon 25 deg. set 35 deg. the declination of the plane from C to E in the Quadrants arch C B, and draw the line A E, then set 25 deg. the inclinati­on of the plane in the same arch from B to F, and draw the line E Z parallel to A C, cutting the line A B in the point Z, and with the distance F Z, and one foot placed in the center A, with the other describe the arch G H I. cutting the line A E in the point H, through which point H draw [Page 98]the line K L parallel to A C cutting the arch C B in the point K, then take the distance H L, and set it in the line F Z from Zunto O, through which point O draw the line AOM, cutting the arch B C in the point M, from which point M draw the line M P N parallel to A B, cutting the arch G I in the point P, through which point P draw the line A P Q cutting the arch B C in the point G, so shall the arch B K be 75 deg. 58 min. the inclination of the plane to the Meridi­an, and the arch B Q will be 57 deg. 36 min. for the Meridi­ans ascension, or the arch of the plane, betwixt the Hori­zon and the Meridian, and the arch B M shall be 20 deg. 54 min. for the elevation of the Meridian, or the arch of the Meridian betwixt the Horizon and the plane. Now if the plane shall incline toward the South, adde this elevation of the Meridian to your Latitude, and the sum of both shall be the position Latitude, or the arch of the Meridian betwixt the Pole and the plane, and if the sum shall exceed 90 deg. take it out of 180 deg. and that which remaines shall be the position latitude, or the arch of the Meridian between the Pole and the plane.

[diagram]

But if the incli­nation shall be northward, then compare the ele­vation of the me­ridian with your Latitude, and take the lesser out of the greater, and so shall the difference be the position La­titude: As here in this example, sup­posing the inclina­tion to be North­ward, we take 20 deg. 54 min. the elevation of the meridian, out of 52 deg. [Page 99]30 min. the Latitude proposed, and there will remain 31 deg. 36 min, for the position Latitude, or the arch of the meri­dian between the Pole and the plane.

This being done, set 31 deg. 36 min. the position Lati­tude, from B to T, in the arch B C, and draw the line A T, then with the distance K L upon the center A, describe the arch Y M W, cutting the line A T in the point M, through which point M draw the line R S parallel to A B, cutting the arch B C in the point S, so shall the arch B S be 30 degrees 33 minutes, the height of the Pole above the plane.

Then lay your rule upon the point S, and the center A, and where it shall cut the line K L, there make a mark as at V, through which point V, draw the line D V N W paral­lel to A B, cutting the arch Y W in the point N, and the arch B C in W, so shall the arch B W be 8 deg. 35 min, the distance of the substile from the meridian.

Lastly, through the point N, draw the line Y X parallel to AC, cutting the arch BC in the point X, so shall the arch BX be 16 deg. 20 min. the inclination of the meridian of the plane to the meridian of the place.

CHAP. XIX. To draw a Dial upon a declining inclining Plane.

HAving by the second Chapter found the inclination to be 25 deg. towards the North, and by the third Chapter the declination from the South towards the East to be 35 deg. and so by the last Chapter the meridians ascension to be 57 deg. 36 min. The elevation of the Pole above the plane 30 deg. 33 min. The distance of the substile from the meridian 8 degrees 35 min. And the inclination of both meridians 17 deg. 30 min. we may proceed to make the Di­al after this manner.

First, Draw the line A B parallel to the Horizon, in which line make choice of a center as at C, whereon describe your Dial circle A D B E A, then take 57 deg. 36 min. the me­ridians [Page 100]ascension, and set it from B that end of the Horizon­tal line with the declination of the plane, as from B to N, and draw the line C N for the hour of 12.

Then set 8 deg. 35 min. the distance of the substile from the Meridian from N to M (on that side the meridian which is contrary to the declination of the plane) and draw the line C M for the substile. And set 30 deg. 33 min. from M to D, and draw the line C D unto S, and from the end of the substile draw the crooked line M S, cutting the line of the Stile in S, so shall the Triangle M C S be the true pat­tern of this Dials Cock.

Then set 17 deg. 30 min. the inclination of meridians from M unto O, which is the point where you must begin to divide your Dial circle into 24 equal parts: from which points let down so many perpendiculars to the substile, as there shall be points on that side the Diameter F E next the substile, and so by working as before hath been shewed, you may draw the hour-lines, and set up the stile as in the for­mer planes.

Now here I would have you well to consider what hath been here spoken concerning these kind of Dials, and also what followeth the same, for if you mark the diversity which doth arise by reason of the elevation of the meridian, you may perceive thereby three sundry kinds of Dials to a­rise out of a North inclining plane declining, and also in a South inclining declining plane, yet in effect they are but one if you consider what followeth here concerning them, in all which, the stile with the substile, and such like mate­rials, are found out according to the last Chapter.

Therefore having drawn your horizontal line, you must consider which pole is elevated above your plane, and how to place the meridian from the Horizontal line. For upon the upper faces of all North incliners, whose meridians ele­vation is lesse then the Latitude of the place: on the under faces of all North incliners, whose meridians elevation is greater then the Latitude of the place: and on the upper fa­ces of all South incliners, the North Pole is elevated.

And upon the under faces of all North incliners, whose meridians elevation is lesse then the Latitude of the place: On the upper faces of all North incliners, whose meridians

[diagram]

elevation is greater then the Latitude of the place: and on the under faces of all South incliners the South pole is cle­vated.

Now for placing the Meridian from the horizontal line; upon the upper faces of all South incliners, whose meridians elevation is greater then the Latitudes complement: on the under faces of all South incliners, whose meridians elevati­on is lesse then the Latitudes complement; on the under fa­ces of all North incliners, whose meridians elevation is greater then the Latitude of the place: and on the upper fa­ces of all North incliners, whose Meridians elevation is less then the Latitude of the place: the Meridian must be pla­ced above the Horizontal line as here in this example.

And therefore by the contrary; Upon the upper faces of all South incliners, whose meridians elevation is lesse then the Latitudes complement: On the under faces of all South incliners, whose meridians elevation is greater then the La­titudes complement: On the under faces of all North in­cliners, whose meridians elevation is lesse then the Latitude of the place: And on the upper faces of all North incliners, whose meridians elevation is greater then the Latitude of the place; the Meridian must be placed below the Horizon­tal line.

But here you must observe, that if it be either the upper or under faces of a South inclining plane, whose meridians elevation is greater then the Latitudes complement: or ei­ther the upper or under faces of a North inclining plane, whose meridians elevation is lesse then the Latitude of the place; that then the Meridian must be placed from that end of the horizontal line with the declination of the plane: But on all the other faces of these kinds of planes the meridian must be placed from that end of the horizontal line, which is contrary to the declination of the plane.

And here note, that if the inclination shall be Southward and the elevation of the meridian equal to the complement of your Latitude, then shall the substile lie square to the Meridian.

And if the inclination shall be Northward, and the ele­vation of the Meridian equal to the Latitude of the place, then shall neither Pole be elevated above this plane, and therefore shall be a Polar declining plane. Wherein the me­ridian [Page 103]being placed according to his ascension from the ho­rizontal line, shall be in place of the substile, unto which if you draw a line square, it shall serve for the Equator. Then set one foot of your compasses in the intersection of the sub­stile with the Equator, and open the other to any convenient distance upon the substile, and describe the Equinoctial cir­cle, as in the sixth Chapter of this Book was shewed:) upon the center whereof make an angle with the line of the sub­stile, equal to the inclination of both meridians, namely, the meridian of the plane, and the meridian of the place, which shall shew you where to begin to divide your Equi­noctial circle into 24 equal parts.

These things being known, you may proceed to make your Dial, and set up the cock according to the 6 chapter.

As for example, in our Latitude of 52 deg. 30 min. a plane is proposed to decline from the South towards the East 35 deg. as before, but inclining Northward 57 deg. 50 min. the Meridians ascension, by the 18 chapter will be found to be 69 deg. 33 min. and his elevation 52 deg. 30 min. equal to the latitude of the place, and therefore neither pole is eleva­ted above this Plane, and so no distance between the Sub­stile and the Meridian: for the Meridian, and the stile with the substile will be as it were all one line, which is the Axle­tree of the world: so that here the stile must be parallel to the plane, and the hour-lines parallel one to the other, as in the Meridian and direct Polar Planes. Therefore first draw the Horizontal line A B, wherein make choice of a center, as at C, whereon describe an occult arch of a circle, as B E: then into this arch inscribe the meridians ascension 69 deg. 33 min. from B to E, and draw the line C E for the meridi­an of the plane, and for the substilar: and if you draw a line square to this substilar, it shall be the Equator.

Then set one foot of your compasses in the point of inter­section D, and with the other opened to a convenient wide­nesse, draw a circle for the Equator, unto which you may draw two touch-lines square to the substile, as in the direct Polar plane.

This being done, and the inclination of both Meridians [Page 104]being found by the last Chapter to be 29 deg. 3 min. set it in this circle from H unto O, and draw the line D O F, cutting the contingent in F, from which point F you shall draw the 12 a clock hour-line parallel to the substile.

[diagram]

Now from the point O divide your Eqninoctial circle into 24 equal parts, with which you may proceed to make your Dial, and set up the cock, according to the 6 chapter.

The end of the Third Book.

THE FOURTH BOOK.
Shewing how to resolve all such Astro­nomical Propositions (as are of ordinary use in this Art of Dialling) by help of a Quadrant fitted for the same purpose.

CHAP. I. The description of the Quadrant.

HAving in the second and third Books shewed Geometrically the working of most of the ordinary Propositions Astronomical, with the delineation of all kind of plain wall Di­als howsoever, or in what latitude soever sci­tuated, [...] I keeping within the limits of our plane, and yet not tyed to the use of any Instrument.

I will now shew how you may performe the former work exactly, easily and speedily, by a plain Quadrant fitted for that purpose; the description whereof is after this manner.

Having prepared a piece of Box or Brasse in manner of a Quadrant, draw thereon the two Semidiameters A B and A C, equally distant or parallel to the edges, cutting one the other at right angles in the center A, upon which center A, with the Semidiameter A B or A C, describe the arch B C, this arch is called the limb; and is divided into 90 equal parts or degrees; and sub-divided into as many parts as quantity will give leave, being numbered from the left hand towards the right after the usual manner.

Then let the Semidiameter A B be divided into 90 une­qual parts, (called right Sines,) either from the Table of na­tural Sines by help of a decimal Scale, equal to the Semidi­ameter A B, or else by taking the neerest extents from each degree of your Quadrant, unto the side A B, and placing them upon the side A B each after other, from the center A towards B, you shall exactly divide the Semidiameter A B into 90 unequal divisions called right Sines.

[diagram]

This being done, draw the line D E from the Sine of 45 degrees counted in the line of Sines unto 45 degrees counted in the Quadrant, then from the point E draw the line E F [Page 107]parallel to A B, making the square A D E F, the side D E whereof (for distinction) may be called a Tangent line, and the side F F a Co-tangent line, then draw the Diagonal line A E, which you may call the line of Latitudes.

Then upon the center A, with the distance A D or A F describe the arch D F, which you may divide into six equal parts, by laying your Rule upon each 15th. degree in the Quadrant, and the center A as at g h I k l F, from which points draw slope lines to each 15th. degree in the Qua­drant, numbered backward, as F P, l O, k E, I n, h m, g B; these lines so drawn are to be accounted as hours, then divi­ding each space into two equal parts, draw other slope lines standing for half hours, which may be distinguished from the other, as they are in the figure.

Now because in the latter part of this Book there is often required to use a line of Chords to several Radiusses, there­fore upon the edge of the Quadrant A C, you may have a line of Chords, divided as in the figure, and so the Quadrant being at hand will supply the uses of the Scale mentioned in the preceding Book, and also a Chord of any Circle, whose Radius is lesse then the line A C may be taken off, and in that case supply the use of a Sector.

To this Quadrant, as to all others of this kind in their use is added Sights, with a threed, bead, and plummet accor­ding to the usual manner.

CHAP. II. Of the use of the line of Sines.
Any Radius not exceeding the line of Sines being known, to find the right Sine of any arch or angle thereunto belonging.

IF the Radius of the Circle given be equal to the line of Sines, there needs no farther work, but to take the other Sines also out of the line of Sines.

But if it be lesser, then take it betwixt your compasses, and [Page 108]set one foot in the Sine of 90 degrees, and with the other lay the threed to the neerest distance, which you may doe by turning the Compasses about till the moving point thereof doe onely touch the threed, and no more: the threed lying still in this position, take the neerest extent thereunto; from any Sine you think good, and it shall be the like Sine agree­able to the Radius given.

As for example let the circle B C D E in the following chapter represent the meridian circle, let B D be the Hori­zon, and C E the vertical circle; and let F G be the diame­ter of an almicanter, and so F H the Semidiameter thereof; which being given it is required to find the Sines both of 30 and 50 degrees, agreeable to that Radius, first therefore, take the given Radius betwixt your compasses, and with one foot set in the Sine of 90 degrees, with the other lay threed to the neerest distance, the threed lying still in this position, take the neerest extents thereunto, from the Sine of 30 and like­wise of 50, these distances place upon the Radius F H from H to N, and from H to R, so shall H N be the Sine of 30 degrees, and H R the Sine of 50 degrees, agreeable to the Radius F H the thing desired.

CHAP. III. The Right Sine of any arch being given to find the Radius.

TAke the Sine given betwixt your compasses, and setting one foot in the like Sine in the line of Sines, with the other lay the threed to the neerest distance, the threed lying still in this position, take the shortest extent thereunto from the Sine of 90 degrees, which distance shall be the Radius required.

As for example, let H R be the given Sine of 50 degrees, and it is required to find the Radius answering thereunto, take H R with your compasses, and set one foot in the Sine of 50 deg. and with the other lay the threed to the neerest distance, which being kept in this position; if you take [Page 109]the shortest extent thereunto, From the Sine of 90 you shall have the line H F for the Radius required.

[diagram]

CHAP. IV. The right Sine, or the Radius of any Circle being given, and a streight line resembling a Sine, to find the quantity of that unknown Sine.

FIrst, take the Radius, or the right sine given, and setting one foot of your Compasses either in the like sine or in the Radius of the line of Sines, and with the other, lay the threed to the neerest distance, then take the right line given, and fix one foot in the line of sines, moving it till the movea­ble foot touch the threed at the neerest extent, so shalll the fixed foot stay at the degree of the sine required.

As for example, let F H be the Radius given, and H N the straight line given resembling a Sine, first with the di­stance F H from the sine of 90 lay the threed to the neerest distance; the threed lying still in this position, take the line H N and fixing one foot of your compasses in the line of Sines, still moving it to and fro, till the moveable foot there­of doth onely touch the threed, so shall the fixed foot rest at the Sine of 30 degrees in the line of Sines; this 30 degrees is the arch, of which H N is the Sine, F H in the last chap­ter being the Radius.

CHAP. V. The Radius of a circle not exceeding the line of Sines being given, to find the chords of every arch.

IF the Radius given, shall be equal to the line of Sines, then double the Sine of halfe the arch, and you shall have the chord of the whole arch, that is, a Sine of 10 deg. doubled giveth a chord of 20 deg. and a Sine of 15 deg. doubled gi­veth a chord of 30 deg. and so of the rest, as in the third chapter, the line I O the Sine of I C an arch of 30 deg. be­ing doubled giveth I L the chord of I C L, which is an arch of 60 deg.

And if the Radius of the circle given, be equal to the Semi-radius (the sine of 30 deg. of the line of sines; then you need not to double the lines of sines as before, but onely dou­ble the numbers: so shall a sine of 10 deg. be a chord of 20 deg. and a sine of 15 deg. be a chord of 30 deg. and so of the rest, but if the Radius of the circle given, be lesse then the semi-radius of your line of sines, then take it betwixt your compasses, and setting one foot in the sine of 30 deg. with the other lay the threed to the neerest distance, the threed lying still in this position, take it over at the neerest extent in what Sine you think good, onely doubling the number, and you shall have the Chord desired.

As for example, let A C be the diameter of the circle in [Page 111]the third Chapter, and it is required to find a Chord of 30 degrees, therefore first, I take A G betwixt my compasses, and setting one foot in the Sine of 30 deg with the other I lay the threed to the neerest distance: which being kept at this angle, I take it over from the sine of 15 deg which doth give me I C the Chord of 30 deg. which was desired.

And if the Radius given, be greater then the Sine of 30, and yet lesse then the Radius of the line of Sines; then with the Radius given, and from the Sine of the complement of half the arch required, lay the threed to the neerest distance, then taking it over at the neerest extent from the sine of the whole arch, you shall have your desire.

As for example, let the Radius A C of the circle in the third Chapter be given; and a Chord of 30 deg. required: the half of 30 deg. is 15 deg. the complement whereof is 75 deg. therefore I take the Radius with my compasses, and setting one foot in the sine of 75 deg. with the other I lay the threed to the neerest distance: the threed lying still in this po­sition, I take the shortest extent thereunto from the Sine of 30 deg. which giveth I C the Chord of 30 deg. which was desired.

Now by the converse of this Chapter, if you have the Chord of any arch given, you may thereby find out the Ra­dius.

CHAP. VI. To divide a line by extream and mean proportion.

A Right line is said to be divided by extream and mean proportion, when the lesser Segment thereof, is to the greater, as the greater is to the whole line.

Let A B be the line to be so divided, this line I take with my Compasses, and setting one foot in the sine of 54 deg. and with the other I lay the threed to the neerest distance: which lying still in this position, I take it over from the sine of 30 d. which distance shall be the greater segment A C dividing the whole line in the point C; or the threed lying in the former position, if you shall take the shortest extent thereunto from 18 deg. you shall have B C for the lesser segment, which [Page 112]will divide the whole line by extream and mean proporti­on in the point C from the end B, so that as B C the lesser segment, is to A C the greater segment; so is A C the grea­ter segment, to A B the whole line, as was required.

[diagram]

CHAP. VII. To find a mean proportional line between two right lines given.

A Mean proportional line is that, whose square is equal to the long square, contained under his two extreams.

First, joyn the two given lines together, so as they may make both one right line; the which divide into two equal parts; and with the one half thereof, setting one foot in the sine of 90 deg. with the other lay the threed to the neerest extent, which lying still in this position, take the distance betwixt the middle point, and the point of meeting of the two given lines, and fixing one foot in the line of sines, so as the other may but onely touch the threed; now from the complement of the sine where the fixed foot so resteth take the shortest extent unto the threed, which shall be the mean proportional line required.

[diagram]

As for example, let A and B be two lines given, between which it is required to find a mean proportional line, first joyne the two lines together in F, so as they both make the right line C D, which divide into two equal parts in the [Page 113]point E, then with either halfe of which, setting one foot in the sine of 90 deg. with the other lay the threed to the neerest distance: then keeping the threed in this position, take the distance between the middle point E and F, the place of meeting of the two given lines, and fixing one foot in the line of sines, so as the other may but only touch the threed, and the fixed foot will stay about 22 deg. 30 min. the com­plement whereof is 67 deg. 30 min. from which take the shortest extent unto the threed lying as before, which shall be the line H, the meane proportional line betwixt the two extreames A and B, which was required.

CHAP. VIII. Having the distance of the Sun from the next equinoctial point, to find his declination.

FIrst, lay the threed upon 23 d. [...]0 m. the suns greatest decli­nation, counted on the limb of the quadrant the threed ly­ing still open at this angle, take the shortest extent thereunto from the sine of the distance of the Sun from the next Equi­noctial point, this distance being applyed to the line of sines from the center A, shall give you the sine of the Suns decli­nation.

So in the figure of the 13 chapter, the Sun being in the 29 d. of Taurus at K, which is 59 d. from C the Equinoctial point Aries; the declination of the Sun will be found about 20 d. the line C M which was required.

CHAP. IX. The declination of the Sun, and the quarter of the ecliptick which he possesseth being given, to find his place.

TAke the sine of the Suns declination from the line of sines, and setting one foot in the sine of the Suns greatest declination, with the other lay the threed to the neerest distance so shall it shew upon the limb, the distance of the Sun from the next Equinoctial point.

So in the figure of the 13 chap. C M the declination of the Sun being 20 d. and K the angle of the suns greatest declina­tion, the line C K will be found to be 59 d. for the distance of the Sun from the next equinoctial point which was required.

CHAP. X. Having the latitude of the place, and the distance of the Sun from the next equinoctial point, to find his amplitude.

TAke the sine of the suns greatest declination betwixt your compasses, and setting one foot in the co-sine of the latitude, with the other lay the threed to the neerest distance, which lying still in this position, set one foot in the sine of the Suns distance from the next equinoctial point and with the other take the neerest extent unto the threed, so shall you have betwixt your compasses the Sine of the Amplitude.

As in the figure of the 13 chapter, the angle at N being 37 deg. 30 min. the complement of the latitude, and K the angle of the Suns greatest declination, and C K 59 deg. the distance of the Sun from the equinoctial point Aries, the line C N will be found to be the sine of 34 deg. 9 min. the amplitude required.

CHAP. XI. Having the declination and amplitude to find the height of the pole.

FIrst, take the sine of the suns declination, and set one foot in the sine of the Amplitude, and with the other lay the threed to the neerest distance, so shall the threed upon the limb, shew the complement of the latitude.

So in the figure of the 13 chapter, the declination C M be­ing 20 deg. and the amplitude C N being 34 d. 9 m. and the angle at M being right, we shall find the angle at N to be 37 deg. 30 m. the complement whereof is 52 deg. 30 m. which was required for the latitude of the place.

CHAP. XII. Having the latitude of the place and the declination of the Sun, to find his amplitude.

WIth the sine of the declination set one foot in the co-sine of the latitude, and with the other lay the threed [Page 115]to the neerest distance: so shall it shew upon the limb the amplitude required: so in the figure of the next chapter, the angle C N M being 37 deg. 30 min. the co-sine of the la­titude, and C M the declination here 20 deg. and the angle at M being right, we shall find the base C N to be the sine of 34 which was required for the Suns amplitude.

CHAP. XIII. Having the elevation of the pole, and amplitude of the Sun, to find his declination.

FIrst, lay the threed to the amplitude counted in the limb, then take it over at the shortest extent, from the co-sine of the latitude, so shall you have the sine of the Suns decli­nation betwixt your compasses.

[diagram]

So in this figure, the Amplitude C N being 34 deg. 9 m. and the angle at N being co-sine to the latitude, the angle at M being a right angle, we shall find C M to be 20 deg. for the declination of the sun which was required.

CHAP. XIV. Having the latitude of the place, and the declination of the Sun, to find his height in the Vertical Circle.

FIrst, take the sine of the declination of the Sun, and set­ting one foot in the sine of the latitude, with the other lay the threed to the neerest distance; so shall it shew upon the limb the height of the Sun in the Vertical Circle.

So in the figure of the last chapter, the angle C I O be­ing 52 deg. 30 min. the latitude of the place, and C O the Suns declination 20 degrees, and the angle C O I being a right angle we may find C I to be a sine of 25 deg. 32 minutes, the height of the Sun in the Vertical Circle which was required.

CHAP. XV. Having the Latitude of the place, and the distance of the Sun from the next Equinoctial Point, to find his height in the Vertical Circle.

FIrst, take the sine of the Suns greatest declination, and setting one foot in the sine of the latitude, with the other lay the threed to the neerest distance: the threed lying still in this position; from the sine of the Suns place take the neerest extent thereunto, which shall be the sine of the Suns height in the Vertical Circle.

So in the figure of the 13 chapter, the angle at I being 52 deg. 30 min. which is the latitude of the place, and the angle at K the Suns greatest declination, and K C being 59 deg. the Suns distance from the next Equinoctial point, we shall find C I to be 25 deg. 32 min. for the height of the Sun in the Vertical circle.

CHAP. XVI. Having the latitude of the place and the declination of the Sun to find the time when the Sun cometh to be due East, or West.

WIth the sine of the declination, set one foot in the sine of the latitude, and with the other lay the threed, to the neerest distance: then take it over at the nec­rest extent from the co-sine of the latitude; which distance keep; and setting one foot in the co-sine of the declination, with the other lay the threed to the neerest distance: so shall it shew upon the limbe, the quantitie of degrees betwixt the houre of six and the East or West points.

So in the figure in the 13 chapter, the declination C O being 20 deg. and the angle O I C being 52 deg. 30 min. the complement whereof is the angle O C I, we may find the sine O I which distance keep; now seeing O I is a sine of the Radius O F and not of AE C, therefore by the 4 chapt. you may find the quantity of that unknown sine; for see­ing the Radius O F is the co-sine of the declination, there­fore set one foot therein, and with the other distance kept, lay the threed to the neerest distance: so shall it shew upon the limb 16 deg. 30 min. which converted into time maketh 1 houre, and 6 min. for the quantity of time between the hour of six and the Suns being in the East or West points.

CHAP. XVII. Having the latitude of the Place, and the declination of the Sun, to find his altitude at the hour of six.

FIrst, take the threed, and lay it upon the declination counted in the limb; then from the sine of the latitude, take it over at the shortest extent; which distance shall be the sine of the height of the Sun at the hour of six.

So in the figure of the 13 Chapter, the angle at L being a right angle, and L O C being 52 degr. 30 min. the latitude of the place, and C O the declination of the Sun being 20 deg. we shall find C L to be the sine of 15 deg. 44 min. for the height of the Sun at the houre of six, which was enquired.

CHAP. XVIII. Having the latitude of the place, and the height of the Sun at the hour of six, to find what Azimuth he shall have at the houre of six.

FIrst, with the sine of the Suns height at the houre of six set one foot in the sine of the Latitude, and with the o­ther lay the threed to the neerest distance: then take the least distance thereunto from the co-sine of the latitude; now with this distance setting one foot in the co-sine of the alti­tude, with the other lay the threed to the neerest distance as before: so shall it shew upon the limb, the Azimuth of the Sun from the East or West points.

So in the figure of the 13 chapter, the angle C L O being a right angle, and the angle L C O being 37 deg. 30 min. the co-sine of the latitude, the angle L O C must be 52 deg. 30 min. the latitude of the place being the complement of the angle L C O, and C L being 15 deg. 44. min. (as by the last chapter it did appeare) we shall find L O to be the sine of 12 deg. 30 min. for the Azimuth of the Sun from the East▪ or West, at the hour of six as was required.

CHAP. XIX. Having the declination of the Sun, to find his Right Ascension.

FIrst, with your compasses take the sine of the Suns de­clination given, and setting one foot in the sine of the Suns greatest declination, with the other lay the threed to the neerest distance: then at the least distance from the co-sine of the Suns greatest declination take it over: now again, with this distance lay the threed to the neerest di­stance from the co-sine of the declination given, so shall it shew upon the limb the right ascension of the Sun.

So in the figure of the 13 chapter, C O the Suns declina­tion being 20 deg. and the angle O K C being 23 deg. [...]0 m. the Suns greatest declination, and the angle K C O being the complement of the angle O K C, we shall find K O to be the sine of 56 deg. 50 m. for the right ascension of the Sun required.

CHAP. XX. Having the latitude of the place, and the declination of the Sun, to find the ascensional difference.

FIrst, take the sine of the Suns declination, and setting one foot in the co-sine of the latitude, with the other lay the threed to the neerest distance: then at the least distance take it over from the sine of the latitude: with which, setting one foot in the co-sine of the declination, with the other lay the threed again to the neerest distance, so shall it shew upon the limb the Suns ascensional difference.

So in the figure of the 13 chapter, the angle M C N being 52 deg. 30 min. and the angle C N M being the comple­ment thereof, the one being the latitude, and the other the co-latitude, and C M being 20 deg. the sine of the Suns de­clination, we shall find M N 28 deg. 19 min. for the diffe­rence of ascensions, which being converted into time, maketh 1 houre, and something better then 53 min.

Now when the Sun hath North declination, if you take this difference of ascension (which is 1 hour 53 min.) out of 6 hours, there will be left 4 hours 7 min. for the time of Sun rising, and if you adde it unto 6 hours, the same will be 7 houres 53 min. for the time of Sun setting.

And so contrarily, when the Sun hath South declination if you adde this ascentional difference to 6 hours, you shall have the time of his rising, and if you take it away from 6 houres, that which is left shall be the time of Sun setting.

CHAP. XXI. The Latitude of the place, the Almicanter, and declination of the Sun being given, to find the Azimuth.

IF the suns declination be Northward, then by the 14 or 15 chapters get his height in the Vertical circle for the day proposed: from the sine of which take the distance unto the sine of the Suns altitude observed; with this di­stance, setting one foot in the co-sine of the latitude, with the other lay the threed to the neerest distance; unto which (being kept still in this position) take the least distance from the sine of the latitude, with this distance, setting one foot in the co-sine of the Suns altitude, with the other lay the threed again to the neerest distance so shall it shew upon the limb the Suns Azimuth from the East or West, either North­ward or Southward.

So in this figure, having N M the distance betwixt the sine of 14 deg. 33 min. (the height of the Sun in the Verti­cal circle) and the sine of 30 deg. 45 min. the height of the Sun at the time of observation, and 52 deg. 30 m. the angle N O M the latitude of the place, the complement whereof is 37 deg. 30 min. the angle M N O, we shall find M O to be the sine of 23 deg. 17 min. the Azimuth from the East or West points Southward.

And here note, when the declination is Northward, that as when the latitude of the Sun given, and his height in the Vertical circle is equal, he is directly in the East or West, [Page 121]so when his altitude given is greatest, then is the Azimuth towards the South, and when his altitude given is least, then is the Azimuth towards the North.

[diagram]

But if the declination of the Sun be Southward, then by the 10 or 12 chapters, find the Amplitude for the day pro­posed.

Now first, take the sine of the Suns altitude, and setting one foot in the co-sine of the Latitude, with the other lay the threed to the neerest distance, which threed lying still in this position, take it over at the shortest extent from the sine of the Latitude, this distance adde to the sine of the Amplitude, [Page 122]by setting one foot in the sine of the Amplitude, and extend­ing the other upon the line of sines, these two being thus joyned, take them betwixt your compasses, setting one foot in the co-sine of the Suns altitude, and with the other lay the threed to the neerest distance: so shall it shew upon the limb the Suns Azimuth from the East or West, towards the South.

So in this figure, having V C or T N, 19 deg. 7. min. the amplitude for the day proposed, and T V the sine of the Suns altitude being 13 deg. 20 min. and 52 deg. 30 min. the angle V X T, the latitude of the place; and the angle T V X, the complement thereof; we shall find X N to be the sine of 40 deg. 11 min. the Azimuth of the Sun from the East or West points Southward, which was required.

CHAP. XXII. The latitude of the place, the declination and altitude of the Sun being given, to find the hour of the day.

IF the declination of the Sun be Northward, find the height of the Sun at the hour of six by the 17 chapter, betwixt which sine, and the sine of the Suns altitude given, take the distance upon the line of sines, with which distance, setting one foot in the co-sine of the latitude, with the other lay the threed to the neerest distance, the threed lying still in this position, take it over at the shortest extent from the sine of 90 deg. with this distance, setting one foot in the co-sine of the declination, with the other lay the threed again to the neerest distance: so shall it shew upon the limb the quantity of time from the hour of six.

So in this figure, having M N the distance betwixt the sine of 9 deg. 5 min. (the height of the Sun at the houre of six) and the sine of 42 deg. 33 min. the height of the Sun given, and the angle M O N 52 deg. 20 min. the Latitude of the place, and his complement M N O, we shall find N O to be the sine of 60 deg the quantity of time from the houre [Page 123]of six, which 60 deg. is four hours of time. And here also note, that if the altitude given be greater then the altitude of the Sun at the houre of six, then is the time found to the Southward of the houre of six; but if it be lesser, then is it to the Northward.

[diagram]

But if the declination of the Sun be Southward, find his depression at the hour of six, by the 17 chapter, for the day proposed, which will be equal to his height at six, if the quantity of declination be alike.

Now take the sine of this depression, and adde it to the sine of this altitude observed, by setting one foot in the sine of his altitude, and extending the other upon the line of sines; These two being thus joyned together in one, take them betwixt your compasses, and setting one foot in the co-sine of the latitude as before, and with the other, lay the threed to the neerest distance, which lying still in this po­sition, take it over at the shortest extent from the sine of 90 deg. with this distance, setting one foot in the co-sine of the declination, as before, with the other lay the threed again to the neerest distance: so shall it shew upon the limb the quantity of time from the houre of six.

So in this figure, having the sine of 15 deg. [...]4 min. the altitude of the Sun given, and the sine of 9 deg. 5 min. his depression at the houre of six, joyned both together in one straight line, as T V, and having the angle T X V 52 deg. 30 min. the latitude given, and the angle T V X the co-la­titude, we shall find T X to be the sine of 45 deg. the quan­tity of time from the hour of six, which converted into time will make three hours.

The end of the fourth Book.

THE FIFTH BOOK.
Shewing how to describe the hour-lines upon all sorts of Planes howsoever, or in what Latitude soever scituated; by a Quadrant fitted for the purpose.

CHAP. I. How to examine a plane for an Horizontal Dial..

IF your Plane seem to be levell with the Horizon, you may try it by laying a Ruler thereupon, and applying the side A B of your quadrant to the under side thereof, and if the threed with the plummet doth fall directly upon his level line A C, which way soever you turn it, it is an Horizontal Plane. Or if you set the side A B of your quadrant upon the upper side of your Ruler, so that the Center may hang a little over the end of your ruler, and holding up a threed and plum­met so that it may play upon the Center, if it shall fall di­rectly upon his level line A C, making no angle therewith, it is an Horizontal plane, as here you may see by this figure.

[diagram]

CHAP. II. Of the trying of planes, whether they be erect or inclining, and to find the quantity of their inclination.

IF the plane seeme to be erect, you may try it by holding the quadrant, so that the threed may fall on the plumb line A C, for then if that side of the quadrant shall lie close to the plane, it is erect, and a line drawn by that side of the quadrant shall be a Vertical line: and the line which cros­seth this Vertical line at right angles, will be the Horizon­tal line, as here you may see in this figure, the plane D E F G being erect, and the line D E being vertical, the line F G must be horizontal. But if the plane shall incline, the quantity of inclination may be found out after this manner.

First, you must draw thereon the Horizontal line, which you may doe upon the under face, by applying the side A B of your quadrant thereunto, so as the threed and plum­met may fall upon the plumb line A C, the side A B lying close with the plane, by which if you draw a line, it shall be parallel to the Horizon.

Or you may draw an horizontal line upon the uppe face, by laying a Ruler thereupon, and applying the side A B of your quadrant to the under side thereof, still moving your Rule, untill the threed and plummet doth fall dire­ctly upon the plumb line A C, the Rule lying thus close to the plane, you may thereby draw a line parallel to the Horizon.

Having drawn this Horizontal line M N, crosse it at right angles with the perpendicular K D, unto which, if it be the under face, apply the side A B of your quadrant, so shall the threed upon the limb give you the angle of incli­nation required.

But if it be the upper face of the plane, then lay a Ruler to the perpendicular K D, unto the under side whereof, apply the side A B of your quadrant, as is here shewed in this figure, so shall the degree of the quadrant give you C A H, the angle of inclination required.

But if it be so, that you cannot apply the side of your quadrant to the under side of your Ruler, then set it upon the upper side thereof, so that the Center thereof may hang a little over the end of the Ruler, and holding up a threed and plummet, so that it may fall upon the center A, and it shall shew upon the limb, the inclination of the plane; which is the angle C A H, equal to the former angle.

Here you must be heedful that both edges of your Ruler be straight, and one parallel to the other.

[diagram]

CHAP. III. To find the Declination of a plane.

TO find out this Declination you must make two ob­servations by the Sun: the first is of the angle made between the Horizontal line of the plane, and the Azimuth wherein the Sun is at the time of observation: the second is of the Suns altitude: both these observations should be made at one instant.

First, for the Horizontal distance, having drawn upon your plane a line parallel to the Horizon, apply the side of [Page 129]your quadrant thereunto, holding it parallel to the Hori­zon, then holding up a threed and plummet at full liberty, so as the shadow thereof may passe through the center of the quadrant, observe the angle made upon the quadrant by the shadow of the threed, and that side with the Horizontal line, for that is the distance here required.

Then at the same instant, as neer as may be, take the Suns altitude, that so you may find the Suns Azimuth from the East or West points, by the 21 chap. of the fourth Book.

Having thus gotten the horizontal distance, with the A­zimuth of the Sun for the same time, describe a circle as A B C D, representing the horizontal circle, and draw the diameter A C, which shall represent the horizontal line F G of of the last chapter. Now supposing the horizontal di­stance to be 38 deg. 30 min. the angle O A B of the last chapter, place it from C South ward to E (that is from the same end of the horizontal line, with which the angle was made upon the plane) and draw the line E Z: Then sup­posing the altitude of the Sun at the same time to be 30 deg. 45 min. with 11 deg. 30 min. North declination, and so by the 21 chapter of the fourth Book, the Azimuth will be found to be 23 deg. 17 min. from the East Southward, being the observation was made in the fore noon: this 23 deg. 17 min. I place from E (the place of the Sun at the time of observation) unto R, (which is the true point of the East) and draw the line H R representing the Vertical circle, so shall the angle made between the horizontal line of the plane and the line of East and West, be the declination of the plane, which in this example is found to be 15 deg. 13 m. the angle C Z R.

Or you may observe the angle made between the shadow of the threed, and that side of the quadrant which lyeth perpendicular unto the horizontal line of the plane, which in this example is 51 deg. 30 min. the complement of the former angle, and it is the angle O A C in the former chap upon the quadrant.

Now having drawn your Horizontal Circle, as before, and the diameter A C for the horizontal line of the plane, you may crosse it at right angles with the diameter B D, for the axis of the planes horizontal line, from which as from D, you may set your horizontal distance on the same side thereof, as before you found it by your observation, as here from D to E, and draw the line E Z for the line of the sha­dow, and having found the Azimuth of the Sun 23 deg. 17 min. from the East Southward, you may set it from E (the place of the Sun) Northward to R, and draw the line RZH for the line of East and West, as before.

[diagram]

Or if you take the Suns Azimuth from the South, which [Page 131]in this example will be 66 deg. 43 min. the complement of the former, 23 deg. 17 min. you may set it from E (the place of the Sun) unto S Southward, and draw the line S Z N for the meridian, so shall the arch S D or R C be 15 deg. 13 min. for the declination, as before.

CHAP. IV. To draw the hour-lines upon the Horizontal, the full North or South planes, whether erect or inclining.

SEeing the making of these Dials are all after one man­ner, we will here proceed to make an Horizontal Diall, by help of the lines upon the Quadrant fitted for that pur­pose.

Therefore having, by the 10 Chapter of the third book, found the elevation of the pole above the plane, we may proceed after this manner.

First, draw the line D A F of sufficient length, out of the middle whereof let fall the perpendicular A B for the Meri­dian and Substile, then take the line D E or E F out of your Quadrant, and set it from A to B in the meridian, through which point B draw the line D B C parallel to D A F, now supposing the elevation of the pole above the plane to be 52 deg. 30 min. the latitude of the place, from the sine thereof take the neerest extent unto A E the line of latitudes, and set it from A to D, and from A to F both wayes, and from B to C, and from B to E, and draw the lines D C and E F, making the long square C D F E: the two angles whereof C and E shall be the points for the hours of 3 and 9 in all these kinde of planes that declines not from the North or South.

Then applying the threed to the first hour-point in the limb B C or D F, as to m or g, it will cut the tangent line D E in 5, then take the distance D 5, and set it down here from D to 5, and from F to 7, with this distance setting one foot in the sine of 90 deg. with the other lay the threed to the neerest distance, unto which take the shortest extent from the sine of the elevation of the pole above the plane: [Page 132]this distance set from B to 1, and from B to 11, then again apply your threed to the next hour in the limb as at n or h, and it will cut the tangent line D E at 4, therefore take D 4 from your quadrant, and set it from D to 4, and from F to 8, with this distance from the sine of 90 degr. open the threed as before, and take it over from the sine of the height of the stile, this distance prick down from B to 2, and from B to 10, so have you all the hour-points pricked down; by which and the center A you may draw all the hour-lines, as here you see done, the line A B for 12, and the line D A F for the two sixes.

[diagram]

For the hour-lines before six and after, you may extend their opposite hour-lines beyond the center as was shewed in the 8 chapter of the third Book.

What is here shewed concerning the hours, the like may be understood for the half hours; by applying the threed thereunto in the limb.

CHAP. V. To draw a Dial upon a South or North erect declining plane.

IN the drawing of all these kinde of Dials, by help of this quadrant, when the latitude of the place and the declina­tion of the plane is known, two things more is to be consi­dered; First, the elevation of the Pole above the plane: Secondly, the inclination of the Meridian of the plane, to the Meridian of the place, both which will speedily be found when you are ready for them.

First therefore, draw the line D A F as before, from the middle whereof let fall the perpendicular A B for the sub­stilar, and at the distance of the tangent line D E, draw the line C B E parallel to the line D A F.

Now to find the elevation of the pole above the plane, lay the threed upon the co-sine of the latitude counted on the limb, and take it over at the neerest extent from the co-sine of the declination, which distance shall be the sine of the elevation of the pole above the plane.

So the declination of the plane being 32 deg in the lati­tude of 52 deg. 30 min. the elevation of the pole above the plane will be 31 deg. 5 min. from the sine of which take the neerest extent unto the line of latitudes, this distance set from A to D and F both wayes, and from B to C and E, and draw the lines D C and E F, making the long square C D F E, as in the former chapter.

For the inclination of Meridians, take the sine of the de­clination of the plane, and setting one foot in the co-sine of the stiles elevation, with the other lay the threed to the nee­rest distance, so shall it shew upon the limb the inclination of Meridians to be 38 deg. 13 min.

The threed lying still in this position, observe which of the hours and where it cutteth, which will be the slope [Page 134]line n I in the point y, to this point y set the bead, which by this means is fitted to the description of this Dial: the threed lying still in the former position, you shall see it cut the tangent D E in the point S upon the quadrant, therefore take D S with your Compasses, and prick it down here from D to 6, with this same distance from the sine of 90 deg. open the threed to the least distance, and taking it over

[diagram]

from the sine of the height of the stile; you shall have the distance from B to 12. And here note, that as in the 13 chap. of the third Book, the substile is placed on that side the Me­ridian which is contrary to the planes declination: so here the Meridian is placed on that side the substile whereon the declination of the plane is.

The bead being thus fitted, apply it to every hour-line, by removing the threed, as first, I remove it to the lines m h and B g, and it will cut the line D E upon the quadrant in r and q, therefore I take D r and D q, and prick them down from D to 5 and 4, with these same distances open the threed as before from the sine of 90; and by taking it over from the sine of the height of the stile, you shall have the distances B 11 and B 10: again, the bead being applyed to the line k E: the threed will cut the co-tangent line in T, therefore take F T from your quadrant, and prick it down here from F to 1, with this same distance, open the threed from the sine of 90 deg. as before, and by taking it over from the sine of the height of the pole above the plane, you shall have the distance B 7: then again, the bead being ap­plyed to the lines l O and E P, the threed will cut the co­tangent line F E in V and W, therefore take F V and F W, and prick them down here from F to 2 and 3, with these distances open the threed from the sine of 90 deg. and take them over from the sine of the height of the stile, so shall you have the distances B 8 and B 9, thus have you the twelve houres pricked down, by which points and the center A, you may draw the houre-lines, as here you see.

In the like manner may the half hours be supplyed.

The Dial being thus drawn on paper, you must place it so upon the plane, that the twelve a clock houre-line may be perpendicular unto the horizon, according to the 11 or 15 chapters of the third Book.

Note, that if the inclination of Meridians shall be more then 45 deg. so that the threed doth cut the co-taugent line F E, then you must take the distance from F to the threed, and prick it down either from D or from F, upon the line D C or F E for the twelve a clock point, according as the plane shall decline either Eastward or Westward, and his parallel taken from the sine of the height of the stile, shall give the distance from B to 6, and so of the rest.

CHAP. VI. To draw a Dial upon an East or West inclining plane.

IN these planes, as in the former, when we have the lati­tude of the place and the inclination of the plane, we have two things more to consider, before we can draw the hour-lines upon the plane.

[diagram]

First, the elevation of the Pole above the plane: Second­ly, the inclination of both Meridians.

For the elevation of the Pole above the plane, lay the threed upon the Latitude counted in the limb, and take it over at the neerest extent from the co-sine of the inclinati­on, which distance shall be the sine of the elevation of the Pole above the plane.

So in the Latitude of 52 deg. 30 minutes, if a plane shall incline 40 degrees to the Horizon, the height of the stile will be 37 deg. 26 min. with which you may proceed to make your parallelogram or long square, as in the former chapters.

Then for the inclination of Meridians, take the sine of the inclination of the plane with your Compasses, and setting one foot in the co-sine of the Poles elevation above the plane, with the other lay the threed to the nee­rest distance, and it will shew upon the limb the inclina­tion required.

Thus in the latitude of 52 deg. 30 min. if a plane shall in­cline 40 deg. to the horizon, the inclination of both Meri­dians, will be 54 deg. 2 min. The threed lying still upon this inclination of Meridians, you may see both which and where it cutteth the hour-lines, and so accordingly rectifie the bead as before was shewed.

And you may also see where the threed cutteth the co-tangent line E F, that so you may take the distance from the point F upon the quadrant, unto the point of intersection of the threed with the co-tangent line; this distance you must set here from D to 12, and with the same, open the threed from the sine of 90 deg. as before, and take it over from the sine of the height of the stile, which shall be the distance from B to 6, and so you may proceed to prick down the rest of the hours as in the last chapter was shewed.

In all these planes you must place the line of twelve pa­rallel to the Horizon, according to the 17 chapter of the third Book, in which chapter is fully shewed the true scitua­tion of this Dial upon the plane.

CHAP. VII. To draw a Dial upon a declining inclining plane.

IN the making of these kinde of Dials by this quadrant, when the latitude of the place, and the declination, and inclination of the plane is known, there is six things more to be considered before we can come to the drawing of this Dial upon the plane.

  • 1 The inclination of the plane to the Meridian.
  • 2 The Meridians ascension.
  • 3 The elevation of the Meridian.
  • 4 The position latitude.
  • 5 The elevation of the pole above the plane.
  • 6 The inclination of Meridians.

All these six may speedily be found out upon the quadrant after this manner.

1 To find the inclination of the plane to the Meridian.

Lay the threed upon the inclination of the plane count­ed in the limb, and taking it over at the shortest extent from the sine of the planes declination, you shall have the sine of the complement of the inclination of the plane to the Meridian.

2 To find the Meridians ascension and elevation.

Take with your Compasses the co-sine of the declination and setting one foot of your Compasses in the sine of the inclination of the plane to the Meridian, with the other lay the threed to the neerest distance: so shall it shew upon the limb, the Meridians ascension required.

The threed lying still in this position, take it over from the sine of the inclination given, and you shall have the ele­vation of the Meridian his sine, which was required.

Now if the planes inclination shall be Southward, adde the elevation of the Meridian to your Latitude, so shall the sum (if lesse then 90 deg.) be the position Latitude, but [Page 139]if the sum shall exceed 90 deg. take the complement thereof to 180 deg. for the position Latitude, here required.

And if the plane shall incline toward the North: com­pare the Meridians elevation with your Latitude, and sub­duct the lesser out of the greater, so shall the difference give you the position Latitude, if there be no difference, it is a de­clining polar plane, and may be described as in the latter part of the last chapter of the third Book.

3 To find the elevation of the Pole above the plane.

Lay the threed to the position Latitude counted in the limb, and take it over at the neerest extent from the sine of the inclination of the plane to the Meridian, and you shall have the sine of the elevation of the pole above the plane.

4 To find the inclination of Meridians.

Take the co-sine of the inclination of the plane to the Meridian, and setting one foot in the co-sine of the height of the stile, with the other lay the threed to the neerest ex­tent, so shall it shew upon the limb, the inclination of the Meridian of the plane, to the Meridian of the place, as was required.

According to these Rules, Suppose a plane to incline towards the North 30 deg. and to decline from the South towards the East 60 deg. in the Latitude of 52 deg. 20 min. First, I find the inclination of the plane to the Meridian to be 64 deg. 20 min Then I find the Meridians ascension to be 33 deg. 41 min. In like manner I find the elevation of the Meridian to be 16 deg. 6 min. and because the plane in­clineth towards the North, I compare this arch with the Latitude of the place, and finding it least I take it there­from, and there remaineth 36 deg. 24 min. for the position Latitude: and so the elevation of the pole above the plane is 32 d. 20 m. and the inclination of Meridians 30 d. 52 m.

The elevation of the pole above the plane, with the incli­nation of Meridians being thus found out, you may pro­ceed to draw the Dial as in the former planes.

The Dial being thus drawn on paper, you may place it in a right scituation upon the plane, by help of the Meridians ascension here found out, with the directions given in the last chapter of the third Book.

[diagram]

In all kind of plain Dials, the stile must be placed over the substile, making an angle therewith equal to the eleva­tion of the pole above the plane, as hath been fully shewed in the third Book.

CHAP. VIII. In any erect declining Dial, having the distance of the substile from the Meridian, in a known Latitude, how thereby to get the Cocks elevation, and the declination.

IN any of these Dials, if the Cock be lost, you may hereby get the height thereof again, & make it a new, for though it be gone, the substile where it stood will remain.

First, therefore get the quantity of the angle of deflexion, by the second chapter of the second Book, which is the angle included between the line of 12 and the substilar, which ad­mit to be 22 deg. 8 min. as in the Dial of the 15 chap. of the third Book.

This being found, take the sine of the latitude betwixt your compasses; and setting one foot in the co-sine of the angle of deflexion found, with the other lay the threed to the neerest distance; so shall it shew upon the limb the ele­vation of the Pole above the plane.

So in the latitude of 52 deg. 30 min. the distance of the substile from the Meridian, being 22 deg. 8 min. as in the said 15 chap. the elevation of the Pole above the plane will be 31 deg. 3 min. by which you may fashion a new Cock to the Dial at your pleasure.

To find the declination.

Take the sine of the height of the stile, and setting one foot of your compasses in the sine of the complement of the Latitude, with the other lay the threed to the neerest di­stance; so shall it shew upon the limb the complement of the declination of the plane. So shall you find the declina­tion of the former plane to be 32 deg.

The end of the fifth Book.

AN APPENDIX.
SHEWING How the Parallels of Declination, the Parallels of the length of the day, the Jewish, Ba­bylonish and Italian houres: the Azimuths, Almicanters, and the like, may be easily inscribed in any Dial whatso­ever, by Rule and Compasse onely. Whereby the Suns place, the day of the moneth, the Rising and setting of the Sun, the length of the day and night, the point of the Compasse, and other necessaries, may be discovered at first sight, only by looking upon the Dial. Also how to draw a Dial on the seeling of a Room.

CHAP. I. How to describe the Equinoctial, Tropicks, and other parallels of the Suns course or declination, in all kind of planes.

ALL Circles of the Sphere whether great or small, that may be projected upon any Dial­plane, become various, according as the planes on which they are to be drawn are scituate; but notwithstanding this, all great Circles, viz. such as divide the Sphere into two equal parts, as all Houres, Azimuths, and Horizons, are streight lines, though variously projected, according as the planes on which they are drawn doe lie in respect of them. And [Page 143]all smal Circles, viz. such as divide the Sphere unequally, are Conick Sections, namely, either Eclypses, Hyper­bola's, or Parabola's, except they be drawn upon such planes as lie parallel to those smaller Circles, and therefore the parallels of declination in the Equinoctial plane, and the circles of Altitude in an Horizontal plane, are perfect circles. For the Equinoctial Dial lying in the very plain of the Equinoctial circle is parallel to all the parallels of de­clination: as the Horizontal Dial lying in the very plain of the Horizon, is parallel to all the Almicanters or circles of Altitude.

Now because the Sun in his course moveth continually between the two Tropicks, and never exceedeth those bounds: so likewise, all Astronomical conclusions that are to be drawn upon any Dial plane, are limited either by the Equinoctial: or by one or both of the Tropicks: therefore it is requisite: first, to shew you how to describe the Equi­noctial and the Tropicks upon all kinde of planes, because it is them that limit and consine all other intermediate pa­rallels, whether they be of the Suns entrance into the Signes or the Diurnal arches for the length of the day. And there­fore I shall first shew you how to perform this work upon such planes as lie parallel to the Axis of the world, as doe the East and West Dials, and the Polar, whether direct or declining.

§ 1. In the East, West, and Polar Dials.

HOw to make an East or West Dial you are taught be­fore, therefore let the square A B C D be a plane, on which there is an East Dial drawn, the height of the stile being equal to the distance between the houre of 9 and 6, noted there with the letters E G, and let it be re­quired to draw upon the same plane the Equinoctial and the two Tropicks. Now the Equinoctial being a great cir­cle of the Sphere, it is therefore a straight line, and is re­presented in the Dial following by the line H F. The hour­lines and the Equinoctial being thus drawn, we may pro­ceed to the rest of the work in this manner.

[diagram]

Upon a piece of fine pastboard, or other conve­nient matter, draw a line as O R, and upon O, as a center, describe the arch of a circle R S, and because the declination of the Tropick of Cancer or Ca­pricorn is 23 degr. 31 min. distant from the Equino­ctial, on either side there­of, therefore on the arch R S set 23 deg. 31 minutes from R to S, and draw the line O S, then shall the line O R represent the Equinoctial, and the other line O S either of the Tropicks, and this trian­gular figure O R S, we shall hereafter call the Trigon.

Having fitted your Tri­gon, you must have re­course to your Dial, and from thence with your Compasses you must first take out the distance E G (equal to the height of the stile of the same Dial) and prick it down in the Trigon from O to P, and draw the line P 6 perpendicular to O R.

Secondly, going to your plane again, take the distance from G the top of the stile, to 7 in the Equinoctial of your plane; and place that distance in the Trigon from O to q, and draw the line q 7 perpendicular to O R. Thirdly, take out of your plane the distance G 8, and prick it down in your Trigon from O to r, and draw the line r 8 perpendi­cular [Page 145]to O R. Fourthly, take out of your plane the distance G 9, and prick it down in your Trigon from O to s, and draw s 9 perpendicular to O R. Fifthly, take out of your plane the distance G 10; and prick it down in your Trigon from O to t, and draw t 10 perpendicular to O R. Lastly, take the distance G 11, and prick it down in your Trigon from O to v, and draw v 11 perpendicular to O R, as be­fore.

These distances being, all of them, taken out of your plane, and placed on your Trigon, it resteth now to shew you how they must be again transferred from the Trigon to the Plane. Therefore, to find upon the hour-lines of your plane, the points through which the Tropick of Cancer must passe you have no more to do but thus. First, out of your Trigon, take the distance P 6, and set that same distance upon your plane from 6 to c upon the hour­line of six. Secondly, take out of your Trigon the di­stance q 7, and place that distance upon the plane from 5 to b, and from 7 to d, upon the hour-lines of 5 and 7. Thirdly, take out of your Trigon the distance r 8, and set that distance on your plane from 4 to a, and from 8 to c. Fourthly, take out of your Trigon the distance s 9, and set it on your plane from 9 to f. Fifthly, take from your Trigon the distance t 10, and set it on your plane from 10 to g. Lastly, take out of your Trigon the distance v 11, and set it on your plane from 11 to h.

These points a b c d e f g h, being found upon the several and respective hour-lines, shall be the points through which the Tropick of Cancer, shall passe, therefore draw the line a b c d e f g h, and that shall be the Tropick of Cancer, so that when the Sun is in Cancer, (which is about the 11 of June) the top of the shadow of the stile of your Dial will run di­rectly along that line a b c d e f g h, and when the Sun is in the Equinoctial, that is, in the beginning of Aries or Libra, (which is on the 10 of March, or the 12 of September) the top of the shadow of the stile will run along the Equinocti­al line E F.

[diagram]

The Tropick of Cancer being drawn, I will now shew you how to draw the Tropick of Capricorn, which diffe­reth nothing from that of Cancer, because they have both of them like declination from the Equinoctial, therefore the distance 8 k being made equal to the distance 8 e, and the distance 9 l equal to 9 f; and the distance 10 m equal to 10 g, you shall have the points k l m upon the hours of 8, 9 and 10, through which points k l m draw the line k l m, &c. which line shall represent the Tropick of Capricorn, along which line the top of the shadow of the stile shall run about the 11 of December, when the Sun is in Capricorn.

Having thus plainly shewn you how to insert the E­quinoctial and Tropicks into your plane, I will now give you one Rule by which you may put on any other interme­diate parallels of the Suns course, they differing nothing at all from the directions formerly given you to insert the Tropicks.

Consider therefore what parallels you would put on your plane, and find what declination the Sun hath when he is in such a parallel, and accordingly insert those degrees of declination into your Trigon, as before you did for the Tropicks.

Example, Let it be required to put upon your plane, the paralls of the Suns entrance into the 12 Signes of the Zodiaque: You must, first, find what declination the Sun hath when he enters any of those Signes, which this little Table doth plainly shew, by which you may see, that when the Sun enters into Taurus, Virgo, Scorpio, or Pisces, his declination is 11 deg. 30 min. and when he is in the be­ginning of Gemini, Leo, Sagitarlus or Aquartus, his declination is 20 deg. 12 min.

A TABLE shewing what declination the Sun hath at his entrance into the twelve SIGNES.
North decli.  DM  South decli.
 Aries0000Libra 
 TaurusVirgo1130ScorpioPisces 
 GeminiLeo2012SagittariusAquarius 
 Cancer2331Capricorn 

Therefore take 11 deg. 30 min. in your Compasses, and place it in your Trigon from R unto V, and draw the line [Page 148]O V, which shall represent the Parallel of Taurus, Virgo, Scorpio and Pisces. Also take 20 deg. 12 min. in your com­passes and place it in your Trigon from R unto X, and draw O X, which shall represent the parallel of Gemini, Leo, Sagittarius and Aquarius.

These parallels being placed in your Trigon according to their true declination from the Equinoctial, they are to be trans erred into your plane in all respects as the Tro­picks were, by taking out of your Trigon the distances from the line O R, to the several points where the hours crosse the parallel, and place the same distances upon your plane from the Equinoctial upon the respective hour-lines, from which they were taken our of the Trigon, and through these points draw the lines in your plane, which shall be the true parallels of the Suns course at his entrance into all the 12 Signes of the Zodiaque, to which you may set the characters of the Signes, as you see done in the figure.

¶ And here note, that if you draw upon your plane the halves and quarters of houres, and put them into your Trigon and transfer them to your plane again, you shall then have more points, through which your parallels must passe, which will much help you in the drawing thereof, (especially in large planes) for there is no better way to draw these kind of lines, but by finding a great number of points, and so draw them by hand.

¶ Note also; that whatsoever is here spoken of the East and West Dials, the same in all respects is to be observed in putting on the parallels of the Suns course in all planes that lie parallel to the Axis of the world, as the Polar, whether direct or declining.

In all these kindes of planes, as the East, West, and Polar, the stile were best to be made of a straight piece of wyer, equal in length to the line E G, fixed in the point E, standing [Page 149]perpendicular unto the plane, the end thereof at G being filed very fine and sharpe, proportionable to the greatnesse of the plane, for all these Astronomical conclusions are shewed (not by the shadow of the whole length of the stile, but) by the very Apex or top thereof, and therefore the more care ought to be had in the forming and making of it.

¶ The line M E N in the former East Dial is called the Horizontal line, because it lyeth parallel to the Hori­zon, and by the meeting of the parallels of the Suns course with this line, the rising of the Sun may be neerly estimated. for there you see that the Tropick of Cancer cutteth this line neer the point M, which is a little before the four a clock hour-line, which shewed, that when the Sun is in the Tropick of Cancer, he riseth somwhat before four in the morning, in like manner the Tropick of Ca­pricorn cutteth the Horizontal line something after 8, at which time the Sun riseth being in Capricorn, but this by the way, the farther use of this line shall be shewed here­after.

I have been the larger in the work of this plane, because I intend to be more brief in those which follow, and this being well and truly understood, the others will need very few precepts or examples; yet I shall not omit any thing, but make it apparent to the meanest capacity. Having thus finished the East or West planes, I will now shew you how to doe the like in the Horizoatal, full South or North planes, which are the next in order.

§ 2. In the North, South, and Horizontal Dial.

IN all these planes the substiler and the Meridian are all one, and the height of the stile, in the Horizontal Diall is always equal to the latitude of the place, and you are to take notice, that whatsoever is here said of the full North and South upright planes, the same is to be understood of [Page 150]the full North and South reclining or inclining, all which in those latitudes, whose complement is equal to the height of the stile they are erect direct planes, and in those lati­tudes which are equal to the height of the stile above such reclining plains, they are Horizontal planes. One example therefore in one plane will be sufficient for the rest. There­fore, in Latitude of 52 deg. 30 min. Let it be required to describe the Equinoctial, and the two Tropicks in a full South erect plane.

Having drawn your Diall with the houres, halves and quarters, as also the line C Q for the stile, you must make choice of some convenient point in the stile, as at S, for the Nodus or knot which must give the shadow to the Tro­picks and other parallels of declination, for all these Astro­nomical conclusions are not shewed by the shadow of the whole length of the stile or Axis, as the houre is, but by some point therein which representeth the center of the earth, which in the Dial following is the points, and the triangular stile in that Dial is represented by the triangle C S L, whereof C L is called the substilar, C S the Axis of the stile and S L the perpendicular stile, the top of which viz. S, is the point we are in this place to respect.

The Dial being drawn, and the Triangle C S L made equal to the Cock of the Dial, you must upon a piece of pastboard draw the Triangle O P R equal to the stile in your Dial C S L, making R O equall to C L the substi­lar, P O equal to C S the Axis of the stile, and P R equal to S L the length of the perpendicular stile.

Then from the point P, raise a perpendicular as P B, re­presenting the Equinoctial, and on Pas a center, describe the arch A B C, now because the Tropicks of Cancer and Capricorn doe decline 23 deg. 31 min. from the Equinoctial, therefore take 23 deg. 31 min. from your Scale of Chords, and set it off upon the arch A B C from B t, A, and from B to C, and draw the lines P A and P C representing the two Tropicks of Cancer and Capricorn. This done, extend the line of the substilar R O (which [...] North or South [Page 151]erect direct planes, I told you was alwayes the same with the twelve a clock line) from O to 12, cutting the Equino­ctial line P B in the point a, then with your compasses take the distance O a out of your Trigon, and place it in your plane from the center C unto a, and draw the line ♈ a ♎ perpendicular to the substile or line of 12. The Equinoctial being drawn: First, take out of your plane the distance C b, and place that distance in your Trigon from O unto b, and draw the line O b 1; representing the hour of 1 or 11 in your Dial. Secondly, take out of your plane the distance C c, and place that in your Trigon from O unto c and draw the line O c 2, representing the hour-lines of 2 or 10. Thirdly, take out of your plane the distance C d, and place it in your Trigon from O unto d, and draw the line O d 3, for the hours of 9 and 3. Fourthly, from our plane take the distance C e, and set in your Trigon from O unto e, and draw the line O e 4 representing the hours of 4 and 8. And thus must you doe with the rest of the hours in your plane if occasion require.

These lines O a, O b, O c, O d, and O e, in your Trigon, being extended, doe cut the Tropick of Cancer P A in the points 12, 1, 2, and 3, therefore out of your Trigon take the distances O 12, O 1, O 2, O 3, O 4, and set them upon their correspondent hour-lines of your plane, from the center C unto g h i k and l, so shall the points g h i k and l be the points upon the hour-lines, through which the Tropick of Cancer must passe, and is therefore noted with the character of Cancer ♋ at both ends.

¶ Now before you draw the Tropick of Capricorn, it is necessary to draw the Horizontal line of your plane A B, which line in all upright planes must be drawn through the point L, the foot of the perpendicular stile, and per­pendicular to the Meridian or line of 12: And in all planes whatsoever, this line must be drawn through the interse­ction of the Equinoctial with the houre of six. This line ought first to be drawn, because it is very improper to ex­tend [Page 152]the Tropicks or other parallels of Declination, above the Horizontal line, because at what hour any parallel of Declination cutteth this line, on either side of the Meri­dian, at that time doth the Sun rise or set, as was instanced in the last.

[diagram]

Now the Tropick of Capricorn must be put upon your plane in the same manner as that of Cancer, by taking out of your Trigon the distance from O, where the several [Page 153]hour-lines a b c d e doe cut the Tropick of Capricorn P C, and place them on your plane from the center C upon the respective hour-lines, and through those points so found, draw the line ♑ ♑, representing the Tropick of Capricern.

[diagram]

¶ And in the same manner may the parallels of the o­ther Signes be drawn upon your plane, by placing them into your Trigon, according to their Declinations, and afterwards transfer them into your plane, as you see in the former figure.

The rules that have been here given for the describing of the parallels of the Signes in this erect direct plane; is universal in all planes, observing this one exception; that whereas in all erect direct planes the Eqninoctial is drawn perpendicular to the Meridian or line of 12, so in all other planes whatsoever, the Equinoctial must be drawn per­pendicular to the substile, and then the work will be the same in all respects, as may appear more largely in the next Section.

§ 3. In Declining, or Declining Reclining Dials.

THe last caution preceding is sufficient for the performing of this work, and therefore needeth no example: How­ever, suppose an upright plane to decline 32 deg. from the South Eastwards, in the Latitude of 52 deg. 30 min. and let it be required to describe the two Tropicks and the Equino­ctial upon such a plane.

The Dial being drawn, with the stile and substile, make choice of some convenient point in the stile or Axis, as at C; for the knot that must give the shadow, and from that point C, let fal a perpendicular to the substile, as C B, and through the point B, draw the Horizontal line D E perpendicular to the line of 12 a clock, then shall the Tri­angle A B C represent the stile of the Dial. Then pro­vide a Trigon, as this figure sheweth, making the Triangle F G H equal to the cock of your Dial, viz. FH equal to the Axis of the stile, G H equal to the substilar, and F G equal to the perpendicular of the stile, extending the line O H to O; then from the point F, raise the perpendicular F K; and on F as a center describe the arch of a circle L K M, setting 23 deg. 31 min. (the declination of the Tropicks) from K to L and M. Then with your compasses take out of your Trigon the distance H c, and set that distance on the substile of your plane from the center A unto P, and draw the line ♈ P ♎ perpendicular to the substilar. This done, the manner of inserting the Tropicks will be directly the same as before, for you take with your compasses the [Page 155]distance from A the center of your Dial, to the several points where the hour-lines crosse the Equinoctial, and put them into your Trigon from H upon the line F K, and draw lines from H through those points and both the Tro­picks F L and F M, setting the number of the hour from whence the distances were taken in the plane at the end of of each line as is done in the Trigon, then you have no more to doe, but to take the distance from H to the intersections of these hour-lines with the Tropicks, and transfer those

[diagram]

[Page 156]distances to your plain again upon the correspondent hour­lines, in all respects as in the work of the former Sections So shall you have described the two Tropicks and the Equi­noctial. And by the same rules and reason any other inter­mediate parallels of declination.

And here note, that whatsoever is said of upright decli­ners, the same is also to be understood of those planes which both decline and recline, and for the horizontal line in all reclining or inclining planes, it must passe through the foot of the perpendicular stile, and the intersection of the Equinoctial with the houre of six.

[diagram]

CHAP. II. Shewing how to inscribe tho parallels of the length of the day on any plane.

THe parallels of the length of the day, and those of the Signes are inscribed upon all kinde of planes by one and the same Rules, they being in the Sphere the same Circles, so that as when you put on the parallels of the Suns entrance into the 12 Signes, you seek what declination he hath, and accordingly proceed as before; so now for the parallels of the length of the day you must seeke what d [...] ­clination the Sun hath at such a length of the day as you would put into your plane, which that you may do, I have here added the Rule following.

¶ Consider how much longer or shorter your day pro­posed is then 12 hours, and take the difference, then the proportion will be,

As the Sine of 90 deg.

Is to the Sine of halfe the difference.

So is the Tangent complement of the latitude of the place,

To the Tangent of the declination that the Sun shall have when the day is at such a length as you require.

As for example, Let it be required to know what de­clination the Sun shall have when the day is 16 hours long in the latitude of 52 deg. 30 min. The difference betwixt 16 hours and 12 hours is 4 hours, (or 60 deg.) the half of which is 30 deg. Therefore say,

As the Sine of 90 deg.10,000000
Is to the sine of 30 d. which is half the difference9,698970
So the Tangent complement of the latitude 37 d. 30. m.9,884980
To the Tangent of the declination of the Sun. 20 d. 59 m.9,583950

And such declination shall the Sun have when the day is either 16 hours or 8 hours long in the Latitude of 52 deg. 30 min.

Now if the day be above 12 hours long, the Sun hath North declination, but if lesse then 12 hours long he hath South declination. For those who are ignorant of these kinde of proportions, they had best to read Mr. Norwoods Dctorine of Triangles. But that nothing might be want­ing, and not much to trouble the learuer, I have here added a Table shewing what declination the Sun hath at such time that the day is either 8, 9, 10, 11, 12, 13, 14, 15, or 16 hours long, in the latitude of 52 d. 30 m. which Table was made by the preceding Rules.

Length of the day.The Suns Declina­tion.
 DM
82059
91622
101114
11543
1200
13543
141114
151622
162059

By which table you may see that when the day is 12 hours long the Sun hath then no declination, but is in the Equinoctial: but when the day is either 11 or 13 hours long, the declination is then 5 deg. 43 min. and when one day is either 9 or 15 hours long, the Sun hath 16 deg. 22 min. of declination, and so for the rest, as in the Table.

For the placing of these parallels of the length of the day upon any of the fore-mentioned planes: you must insert these angles and declination unto your Trigon between the Tro­picks; and proceed in all respects as before. I will therefore give you but one example, which shall be in a full South plane, upon which and the Horizontal these arches doe appear most uniform.

Now let it be required to draw the parallels of the Suns course, when the day is 8, 9, 10, 11, 12, 13, 14, 15, and 16 hours long: upon a full South plane in the Latitude of 52 deg. 30 min.

Having drawn your Dial with hours, halves and quarters and also made choise of some convenient point in the stile to give the shadow, and draw the horizontal line C D, [Page 159]then make the triangle S A R in this Trigon equal to the triangle S A R in the following South Dial: as S A equal to the Axis of the stile, A R equal to the substilar, and R S, equal to the perpendicular stile: then draw the perpendi­cular S G for the Equinoctial, and describe the arch O G P, making G O and G P each of them 23 deg. 31 min. for the two Tropicks, which you must transfer into your plane as before.

[diagram]

Now for the drawing of the parallels of the length of the day, you must have recourse to the little table before go­ing, and therein see what declination the Sun hath at such a day as you would put into your plane, as when the day is either 8 or 16 hours long, the declination is 20 deg. 59 min. therefore place in your Trigon 20 deg. 59 m. from G unto a both wayes, and draw the lines S a and S a, marking them at the ends with 8 and 16 the length of the day for which they serve. Likewise, when the day is either 9 or 15 hours long, then the Suns declination is 16 deg. 22 min. therefore set 16 deg. 22 min. from G unto b both wayes, and draw S b and 8 b. Also when the day is either 10 or 14 hours long, then the declination is 11 deg. 14 min which set from G to c both wayes: and draw S c and S c. Lastly, when the day is 11 or 13 hours long the declination is 5 deg. 43 min. which set from G unto d both wayes, and draw S d and S d, noting them with numbers answering to the length of the day, as you see in the Trigon, when the day is just 12 hours long it is Equinoctial and hath no declination, and is signified in the Trigon by the line S G.

For the manner how to transfer these parallels of the length of the day into the plane, it is to be performed in all respects as in the former chapter for the inserting of the Signes, not at all differing therefrom, and therefore I shall forbear to give you any farther instructions for the perfor­mance thereof, but give you the figure of a South plane with these parallels drawn thereon, which will instruct more then a whole chapter of information.

And thus much for the drawing of the parallels of the Signes and Diurnal arches in all kinde of planes. I will now proceed to shew you how some other Astronomical conclusions (which are very pleasing and delightfull) may be inscribed upon all sorts of Dials.

[diagram]

CHAP. III. Shewing how the Italian and Babylonish hours, may be drawn upon all kind of planes.

THe Italians account their hours from the Suns setting; and the Babylonians from his rising, so that these kinde of hour-lines being drawn upon any plane, you know (by inspection only) how many hours are past since the last setting or rising of the Sun. The inscription of [Page 162]these hour-lines into any of the former planes is very easie, the work of the last chapter being well understood.

Because that upon a full South or an Horizontal plane, these hour-lines shew themselves most uniform, I have therefore for example sake, made choice of a full South Dial, upon which it shall be shewn how to draw both the Italian and Babylonish hours.

Your Dial being drawn and the two Tropicks and the Equinoctial thereon inscribed, and also the Horizontal line, you must draw in your Dial two obscure parallels of the length of the day, one when the day is 8 hours, and the other when the day is 16 hours long, expressed in the following Dial by the two pricked arches neer the two Tro­picks, the uppermost of which is the parallel of the Suns course when the day is 8 hours long, and the undermost is the parallel of his course when the day is 16 hours long, and the Aequinoctial is the parallel of the Suns course when the day is 12 hours long.

  Length of the day.
  81216
Hours from Sun rising.1975
21086
31197
412188
51119
621210
73111
84212
9531
10642
11753

Your Dial being thus pre­pared, & these parallels thus inserted, the inscription of these hour-lines is very easie and plain to be understood. To begin then with the in­scription of the Babylonish hours (which are the hours from the Suns rising.) First, It is apparent that when the day is 8 hours long, that the Sun riseth at 8 in the morn­ing, so that at that time, the first hour after the Suns ri­sing is 9 in the morning. Secondly, when the day is 12 hours long, the Sun riseth at 6 in the morning, so that at that time the first houre after the Suns rising is 7 in the morning. Thirdly, when the day is 16 hours long, the Sun riseth at 4 in the morning, so that [Page 163]the first hour after his rising is 5 in the morning, as plainly appeareth by this Table: By which you may perceive that when the day is 8 hours long, the seventh hour from Sun rising is 3 in the afternoon. When the day is 12 hours long, the seventh hour from the Sun rising is 1 in the afternoon. And when the day is 16 hours long, the seventh hour from the Suns rising is 11 before noon as by this Table doth evidently appear. And therefore a straight line drawn in your Dial through those points where the common hour­lines of your Dial crosse the respective parallels of the dayes length, shall shew the true quantity of hours since the Suns rising at all times of the yeare, which is the Baby lo­nish hour.

For example, let it be required to draw the seventh hour from the Suns rising in your Dial. First, by the Table you see, that in the parallel of 8 hours for the length of the day, the seventh hour from the Suns rising is 3 in the afternoon, therefore observe where the hour-line of three crosseth the parallel of 8 hours, which is at a. Secondly, by the Table you see that in the parallel of 12 hours for the length of the day, the seventh hour from Sun rising is then 1 in the afternoon, wherefore observe where the hour-line of 1 crosseth the Equinoctial, which is at b. Thirdly, by the Table you see that in the parallel of 16 hours for the length of the day, the seventh hour from the Suns rising is 11 before noon, therefore observe where the hour-line of 11 crosseth the parallel of 16 hours, which is at c: then draw the straight line a b c, which shall be the seventh Ba­bylonish hour, or the seventh hour from the Suns rising all the year long.

And by this Rule, and the help of the Table, you may draw all the other hours from Sun rising, as you see them drawn in the figure, and put numbers to them as you see there done.

[diagram]

¶ 1 Note, That if any of the points you are to make use of for the drawing of any of these hours fall without your plane, you must in this case extend your hour-line, parallel and Equinoctial, beyond the limits of your Dial­plane, and there make use of the points, but you need ex­tend the line you draw no farther then the bounds of the plane as here in the figure you see the first hour from Sun rising crosseth not the Equinoctial and the hour-line of 7 within the plane, but if the Equinoctial and the hour­line of 7 were extended, it would crosse.

¶ 2 Note, That if any of the three points you are to make use of doe so far exceed the limits of your plane, that it will be either impossible (or at least very troublesome) to extend the hour-lines so far that then in that case any two of the three points will sufficiently serve the turn.

¶ 3 Note, that as the hours from Sun rising were put into the plane, by the same Rule may the hours from Sun setting (or Italian hours) be inserted, the difference being only in the numbring of them; the hours from the Sun rising being numbred from the West side to the Hori­zontal line by 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11, and the hours from the Suns setting are denominated from the East side of the Horizon, and numbred backwards by 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, and 13, as in the figure doth evidently appear.

¶ 4 Note, That these Italian and Babylonish hours are in­scribed on all planes by help of this little Table, and the Rules and cautions delivered in this chapter, and there­fore more examples were superfluous.

CHAP. IV. Shewing how the Jewish hours may be drawn upon any plane.

IT was the custome of the Ancients to divide their day and also their night (whether long or short) into 12 equal parts, beginning their day at the Suns rising, and their night at the Suns setting: so that 14 of the clock at noon was alwayes the sixth hour of their day, and 12 at night was alwayes the sixth hour of their night, and according to this division were their Dials drawn; so that all the So­mer the hours of their day were longer then the hours of their night; and all the Winter, the hours of their night were longer then those of their day and when the Sun is in the Equinoctial, then the hours of their day and night were equal, and the same with those of our account, but at all other times of the year different.

The inscribing of these hours into all kinde of planes is very easie, being much like the drawing of the Babylonish and Italian hours before taught.

Having therefore drawn your Dial (which in this exam­ple (for the avoiding of many figures) we will have to be the full South plane used before in chap. 2. of this Appen­dix) with the hours, halves, and quarters, and also drawn the two Tropicks and the parallels of the length of the day thereupon, as you see here done in this figure. Then make choice of two parallels of the length of the day, which must be both of them equidistant from the Equinoctial, which let be the parallels of 9 hours and 15 hours, both which are three hours different from the Equinoctial on either side thereof, and these two parallels are the most convenient for

Jewish hours.The paral­lel of 15 hours.Equinoctial.The paral­lel of 9. hours.
HMHM
15457815
270890
38159945
4930101030
51045111115
612012120
71151045
82302130
93453215
1050430
116155345
127306430

this our purpose, because the Jewish hours, will fall (in these two parallels) justly upon the houres, halves and quarters of the common hour-lines: and so be the easier drawn. Now the points through which every one of the Je­wish hours must passe is ex­actly shewed by this little Table, wherein you may see that the first Jewish hour must be drawn through 5 hours 45 mi. (or 5 hours three quarters) in the parallel of 15, through 7 hours in the Equinoctial, and through 8 hours and a quarter in the parallel of 9 hours.

In like manner, the second Jewish hour must be drawn in your plane through 7 of the clock in the parallel of 15: [Page 167]through 8 a clock in the Equinoctial: and through 9 of the clock in the parallel of 9 hours, and so of all the rest, accord­ing as you see in thsi Table, and as you may perceive them drawn in the South plane, the numbers belonging to these hours being set at both ends of each hour-line.

[diagram]

CHAP. V. Shewing how to draw the Azimuths, or Vertical Circles in all kinde of planes.

THe Azimuths are great Circles of the Sphere, meeting together in the Zenith of the place, and are variously inscribed on all planes according to their scituation. In the Horizontal plane they meet in a center with equal an­gles. In all upright planes, whether direct or declining, they are parallel to the Meridian or line of 12. And in all reclining planes they meet together in a point which is the Zenith of the place. These Azimuths being great circles in the Sphere; are therefore straight lines in all planes, and may be drawn as followeth.

§. 1 In the Horizontal plane.

IN the Horizontal plane these Azimuths are most easily inserted, for your Dial being drawn, with the Tropicks thereon, you have no more to doe, but upon the foot of the perpendicular stile to describe a Circle, which you may divide into 2 equall parts (beginning at the Meridian) answering to the 3 points of the Mariners Compasse; Or else you may divide the same Circle into 90 equal parts, according to the Astronomical division, and through each of those points draw straight lines from the foot of the stile, and set numbers or letters to them, either by 10, 20, 30, 40, &c if you divide it into 90, or else by South, S by W, S S W, S W by S, &c if you divide the Circle according the Mariners Compasse. This is so plain that it needeth no example.

§ 2. In the East or West erect planes.

YOur Dial being finished, you may draw upon a piece of pastboard the line M E N, representing the Hori­zontal line MEN in your Dial then on the point E, raise the perpendicular E Q equal to the line E G in your Dial, and on Q as a center describe the semicircle K E L, and [Page 169]divide one halfe thereof, namely, the quadrant into eight equal parts, representing one quarter of the Mariners com­passe, and from the center Q draw lines through each of those divisions extending them till they cut the line M E N in the points ☉ ☉ ☉ ☉ ☉ ☉, then with your compasses take the distances from E to every one of these points ☉ ☉, &c. and prick them down in the Horizontal line of your plane from E to ☉ ☉ ☉ ☉ ☉ ☉, from which points draw lines perpendicular to the horizontal line M E N, which shall be the Azimuths or points of the compasse between the East and the South.

[diagram]

Divide likewise the other quadrant of the Circle EK into eight equal parts, and draw lines from the center Q through three of them, till they cut the horizontal line as you see in the figure, and there also draw lines perpendicular to the Horizon, and these lines shall be the azimuths be­tween the East and the North, viz. so many of them as your plane is capable to receive, which the following figure doth most plainly shew.

¶ Here note, that as the East Dial sheweth all the morning hours from Sun rising to the Meridian: and the West Dial sheweth all the afternoon hours from the Meridian to his setting: so doth the East Dial shew all the azimuths from the Suns rising till Noon, and the West Dial all the azimuths from noon till his setting.

[diagram]

§ 3. In the full North and South erect planes.

THe drawing of the azimuths upon the full North or South erect planes is very little different from the draw­ing of the same circles upon the East or West planes. But for example, let it be required to draw the Azimuths upon the full South Dial: the Tropicks and the Equino­ctial being drawn, together with the Horizontal line, you must upon a piece of Pastboard draw the line A L B, re­presenting the Horizontal line A L B in the South Dial next following: then on the point L raise the perpendicular L S, making L S equal to L S the perpendicular stile of the [Page 171]Dial, and on S as a center describe the semicircle E L F, and divide each quadrant thereof, namely E L and L F into 8 equal parts (each quadrant representing one quarter of the Mariners compasse) and through each of those divi­sions draw lines from the center S, till they cut the line A L B in the points m n o p and q, then with your compasses take the distance L q, and set that distance upon the Hori­zontal

[diagram]

line of your plane from L unto q both wayes. Like­wise, take the distance L p, and set that distance in your plane upon the horizontal line thereof from L unto p both wayes. Also take the distances L o, L n, and L m, and set them upon the horizontal line in your Dial from L to o, and n, and m, on each side of the Meridian. Lastly, if from the points m, n, o, p, and q, you draw lines parallel to the Meri­dian or line of 12, they shall be the true azimuths upon your plane, and these Azimuths may be put on, either ac­cording to the Astronomical account by 10, 20, 30, 40, &c. or else by the points of the Compasse, as in this figure, ac­cording as you shall divide the Semicircle E L F: And thus much concerning erect direct planes.

[diagram]

§ 4. In erect declining planes.

IN upright declining planes the azimuths are easily in scribed, little differing from the former. Draw there­fore your Dial, which we will suppose to be the South declining plane before used in the third Section of the first chapter of this Appendix, which declineth from the South Eastward 32 deg.

Your Dial being drawn and the Equinoctial, and Tro­picks, and also the Horizontal line, thereon inscribed: upon a piece of pastoard draw the line D B E, represent­ing the horizontal line D B E in your Dial, then on the point B raise the perpendicular B C, making B C equal to B C the perpendicular stile in your Dial: then on the point Cas a Center describe the Semicircle R B S, then out of

[diagram]

your plane take the distance between B the foot of the stile, and the point O, where the Meridian crosseth the hori­zontal line, and set that distance on your pastboard from B unto O, and draw the line C O, cutting the Semicircle in the point K, at which point K you must begin to divide your Semicircle into 16 equal parts, then from the Center C draw lines through each of those divisions till they cut the line D E in the points a b c d e f g h i k l and m.

Lastly, with your Compasses take the distances O a, O b, O c, O d, &c. out of your pastboard, and prick the same distances down in your plane from O to a b c d e f g h i k l and m, and from those points draw lines parallel to the Meridian, which lines shall be the azimuths required which you must number according to the scituation of the plane, viz. the Western azimuths on the East side of the Meridian, and the Eastern azimuths on the West side of your Dial, as you see them here numbred in this figure.

[diagram]

§ 5. In East and West incliners, and also in North and South incliners declining.

IN all these planes, because the Zenith of the place cutteth the plane obliquely, making oblique angles ther with, there is in all these planes two points to be found in each plane be­fore the azimuths can be drawn, the one is the Zenith of the plane, the other the Zenith of the place, in which all the Azimuths must meet with unequal angles.

  • I. Therefore suppose a direct South plane to recline 25 deg. from the Zenith, the complement thereof is 65 deg. [Page 175]the inclination of the under face of the same plane to the horizon, therefore make the perpendicular side of the stile Radius, then the Meridian will be a Tangent line thereun­to, upon which Meridian, from the foot of the perpendi­cular stile, prick down 65 deg. for the Zenith point where all the azimuths must meet, and 25 deg. for the horizon­tal point, through which the horizontal line must passe, Then describing a Semicircle, divide it into 16 parts, and lay a ruler from the center and each of those divisions till it cut the horizontal line, and thereon make marks, then lay a ruler upon the Zenith point and each of these marks in the horizontal line, and they shall be the true Azimuths belonging to your plane, which you must number accor­ding to the scituation thereof.
  • II. In the East and West Incliners, and in the North and South decliners inclining, because the 12 a clock line and the substilar are several lines, you must therefore draw a line perpendicular to the base of the plane, which must passe directly through the foot of the perpendicular stile, then make the perpendicular stile the Radius, and the other line last drawn shall be a Tangent line thereto, upon which line set off the inclination of the plane to the Horizon, and that shall give you the Zenith point, and the horizontal point shall be found by setting off the reclination of the plane from the Zenith, and here note that the Zenith point will alwayes fall upon the Meridian.

CHAP. VI. Of the Almicanters or Circles of Altitude.

THe circles of altitude have the same relation to Azi­muths, as the Tropicks and parallels of declination have to the hour-lines, and therefore, as the parallels of declination in the Equinoctial plane are perfect circles, so are the circles of altitude in an horizontal plane.

The inscription of these into all kinde of planes is (in a manner) the same with the parallels of declination, but whereas in the drawing the parallels of declination, you [Page 176]take the hour-lines out of your plane and put them in a Trigon: so in this you must take the Azimuths out of your plane, and put them into a Trigon for that purpose, and so tranfer them to the plane again as you did the other: and because these are small circles, therefore they become co­nick sections, except on such planes as lie parallel to the Zenith, which is only the horizontal.

CHAP. VII. How to draw a Dial on the seeling of a Room.

BEcause the direct beams of the Sun can never shine upon the seeling of a Room, they must therefore be reflected thither by help of a small piece of Looking-glasse conve­niently fixed in some Transam of the window, so that it may lie exactly parallel to the Horizon. The place being chosen, and the glasse therein fixed, you must draw upon the seeling of the Room a Meridian line, as you are taught in the former Books, which Meridian line must be so drawn that it may passe directly over the glasse before placed, which you may perceive how to doe by holding a threed and plummet from the top of the seeling till it fall directly upon the superficies of the glasse.

The foundation being thus laid, we will now proceed to the work, which among so many wayes as these are to perform it, I shall make choice of that which I suppose to be most familiar and easie. Draw therefore upon paper or otherwise an horizontal Dial for the Latitude in which you are, as is the horizontal Dial fore-going for the Lati­tude of 52 deg. 30 min. Then upon the center thereof at A, with the Radius of your line of Chords describe the Semi­circle B C D, cutting the hour-lines in the points a b c d & e then with your compasses you may measure the quantity of each hours distance from the Meridian, by taking the di­stance from C to a, b, c, d, and e, so shall you find the di­stance between the Meridian and 11 or 1 to be 12 degrees. Likewise the distance between the Meridian and 10 or 2 to be 24 degr. 37 min. and the distance between the Meridian [Page 177]and 9 or 3, to be 38 deg. 25 min. and so of the rest as by the figure, and the second column of the Table doth appear, This done, take the complement of every of these angles. so shall the complement of 12 deg. be 78 deg. and the com­plement of 24 deg. 37 min. be 65 deg. 23 min. and so of all the rest, as by the third column of the Table may appear.

[diagram]

Having these things prepared, Let the line L R in the following figure represent a Meridian line drawn upon the seeling of a Room, and let K be the glasse fixed directly un­der the said Meridian upon some transom of the window, [Page 178]

The Hours.The angle that each hour-line makes with the Meridi­anThe com­plement of each hourlines angle with the Merid.
  DMDM
1200009000
11112007800
21024376523
3938255135
485358362
5771201840
690000000

then laying one end of a string upon the glasse at K, extend the other up to the Meridian at L, which point L you may finde by moving the string to and fro upon the Meridian line, till another holding the side of a Quadrant to the moveable string, he shall find the threed and plummet to fall directly upon the complement of the Latitude, which in this exam­ple is 37 deg. 30 min.

The point L being thus found upon the Meridian, draw the line L AE perpendicular to the Meridian L R, which line shall be the Equinoctial.

Having thus done, upon a Table or such like, draw a line which shall be of equal length with L K, the distance from the glasse to the point L on the seeling, which line divide into 10 equal parts, and each of those (or at least one of them) into 10 other parts, so shall you have in all 100 parts, each of which you must suppose to be divided into 10 other smaller parts, so shall the whole line contain 1000 parts, as in the figure is expressed by the line S.

Your line thus supposed to be divided into 1000 parts, you must take with your compasses out of the said line 268 of them, (which is the natural Tangent of 15 degrees,) and place them upon the Equinoctial line from L to M. Then take 577 the natural Tangent of 30 deg. and place it from L to N. Then take the whole line, and set it from L to P. Lastly, take 732 parts and set them from P to Q; so shall the points M N P & Q be the points through which the hours of 1, 2, 3 and 4, must passe, and the same work being done on the other side of the Meridian, you shall find points through which the hours of 11, 10, 9, and 8 shall passe: the hours of 5 in the morning and 7 at night will seldome fall [Page 179]upon the plane except they be supplyed from East and West windows.

[diagram]

Now because the center of the Dial is without the Room [Page 180]so that you cannot make use of that to draw the hours by, you must therefore place one foot of your compasses in the point L M N P and Q with the other draw obscure arches of Circles as *****, and out of the last column of the former Table take the complement of every hours arch from the Meridian, and place them upon the respective hour arches from the Equinoctial to the points ***** as you see in the figure. Lastly, if you draw the lines * M, * N, * P, * Q, they shall be the true hours upon the seeling.

In the inscription of the Azimuths in declining reclining planes, and in drawing the circles of altitude in all kind of planes, I confesse, I should have been somewhat larger in gi­ving you an example in each plane, as I did with the other varieties before, but pre-supposing the ingenious practitioner sufficiently to understand the precedent, he cannot but with small pains overcome the rest: but I should not have been so briefe, could I possibly have procured more time.

CHAP. VIII. The manner how to put on the parallels of the Signes, Azimuths, Al­micanters, and other varieties upon a seeling Dial.

A Seeling Dial is no other then an Horizontal Dial in­verted, and therefore the variety to be thereon inser­ted is to be performed by the same Rules as have been before delivered in this Appendix. But in respect that the seelings of Rooms are for the most part large those rules will be deficient, and some other way must be found for the effecting thereof: which way in brief may be this, which is general, you must by the Tables of Sine, and Tan­gents, or by some planisphere, compute the Suns Altitude at every hour, half, and quarter, upon these dayes on which the Sun enters the 12 Signes, or the diurnal arches, and having thus made a Table for your purpose you may fix a string in the place where the Glasse is to lie, and let it be ex­tended up to the seeling upon any hour-line, and there mo­ved to and fro till a quadrant being applyed to the string, the line and plummet hangs directly upon the deg. of alti­tude [Page 181]proper to that hour, and there make a mark for that is one point for the parallel of that day, you must doe thus for every houre, half, and quarter, so shall you have points enough to draw your parallel by, for the Azimuths they may be drawn upon the Floor, and tranferred to the seeling by perpendiculars. The Almicanters are perfect Circles whose center is that point in the seeling which is directly over the Glasse.

FINIS.

COVRTEOVS READER
These Books following are to be sold by Tho. Pierrepont at the Sun in Pauls Church-yard.

Books of Divinity.
  • A Learned Commentary upon the whole Book of the Revelation of St. John by David Pa [...]us D. D. fol.
  • Select Sermons preached upon several occasions, with two positions for explication and confirmation of these two questions, first, [...]ota Christi justitia eredentibus imputatur, 2 Fides justificat sub ratione instrumen [...], by John Froft, B. D. late fellow of St. Johns Colledge in Cambridge, and since Pastor of the Church of St. Olaves-Hart-street, London. fol.
  • Annotations upon the five Poëtical Books of the old Te­stament, viz. Job, Psalmes, Proverbs, Ecclesiastes, and Canticles, by Edward Leigh M. A. of Magdalen-ha [...] in Oxford. fol.
  • [Page]Practical exposition upon both Epistles of Peter, by W. Ames D. D. fol.
  • A brief Exposition on the Lords Prayer, by Tho. Hooker.
  • Isaiahs Prophecy of Christ Passion, for mans Redemp­tion: being a practical exposition on the 53 chap. of Isaiah, by Tho. Calvert, Minister of Sods-word at Munster in the City of York.
  • Return of Mercies, or the Saints advantage by losses, by John Goodwin.
  • God a Good Master, and Protector to his People, by John Goodwin. 12
Mathematical Books.
  • TRigonometria Britannica, or the Doctrine of Triangles, in two Books, 1 shewing the Construction of Natu­ral and Artificial Sines, Tangents, and Secants: a Table of Logarithmes, with their use in the ordinary questions of Arithmetick, extraction of Roots, in finding the increase and rebate of money, and annuities at any rate or time pro­pounded, the other the use of the Canons of Artificial sines, and Tangents, and Logarithmes, in the most easie way of Resolution of all Triangles. The one composed the other translated, from the Latine Copy, written by Hen. Gillebrand. A Table of Logarithmes to 100000 therto annexed with the Artificial Sines, and Tangents to the hundred part of a deg. and the first 3 deg. to a 100 part. by John Newton, M. A.
  • Astrononomia Britannica exhibiting the Doctrine of the Sphere, and Theory of the Planets Decimally, by Trigono­metry and by Tables, by John Newton. M. A
  • The Sector on a quadrrant a Treatise containing the de­scription and use of three several quadrants, fitted for Dial­ling, and resolving of all Proportions Instrumentally, & for the ready finding the hour and azimuth, in the equal limb, of great use to Seamen, and practitioners in the Mathema­ticks, by John Collins Accountant, and Student in the Mathe­maticks: also an Appendix touching Reflecting Dialling from [Page]a Glass however posited, with large Cuts and quadrants.
  • The Compleat Diallist, shewing the whole Art thereof three several wayes, two of which performed Geometrically by Rule and Compass only, and the third Instrumentally by a quadrant fitted for that purpose with an Appendix, and Additions to this second Edition, by Tho. Stirrup, Philomath.
  • Universal Dialling, by a most easie way, shewing how to describe the hour-lines on all sorts of Planes, and in any Altitude performed by certain Scales set on a smal Ruler by G. S. Philomath.
  • Description and use of the Universal quadrant by which is performed the whole Doctrine of Triangles two several wayes: Resolution of such propositions, as are most use­ful in Astronomy, Navigation, and Dialling; by which is also performed the proportioning of lines, and superficies, the measuring of all manner of Land, Board, Glass, Timber, Stone, &c by Tho. Stirrup Philomat.
  • Carpenters new Rule in two parts, 1 shewing how to mea­sure all superficies, as Timber, Stone, Glass, &c. Geometrical­ly without Aritnmetick, 2 the measuring of Timber & Stone Instrumentally, upon the Scale without Arithm or Geo­metry, also to take heights and distances, and to draw the Plat of a Town or City, with an Appendix, &c. T. S. Phil.
  • Learners help for the understanding of the Hebrew tongue, whereby may presently bee found out the root of any He­brew word in the Bible.
  • Natural Phylosophy reformed: by J. A. Commenies.
  • Schollers Companion, containing all the interpretation of the Hebrew, and Greek Bible in Latine and English, very useful for all that desire the knowledge of both Testaments in the Original tongues, by A. R.
  • The Purchasers Patern in two parts: 1 shewing the true value of the purchase of any land and house by lease or otherwise: new Tables of Interest, and rebate at 6 per centum, 2 shewing the measuring of Land, Board, and Timber, and gauging of Vessels, with Rules about weights and measures, and Tables of accounts with other Rules, and Tables of [Page]daily use to most men, third Edit. and much enlarged: by H. Phillippes.
  • Geometrical Trigonometry, or an explication of such Geome­trical Problems, as are most usefull in the making of the Canon of Triangles or in the solution of them whether plain or Spherical.
  • Tabulae Mathematicae, or Table of natural Sines, and Tan­gents, and Secants, and the Logarithmes of sines and Tan­gents to every degree, and hundred part of a degree in the quadrant: with a Table of Loga rithmes. By John Newton, M. A.

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