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 Compasse 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 ASTRONOMIE and NAVIGATION, both Geometrically and Instrumentally.

By Thomas Stirrup, Philomath.

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

Also how to draw a Diall on the seeling of a room, By W. L.

LONDON, Printed by R. & W. Leybourn, for Thomas Pierrepont, at the Sun in Paul's Church-yard. 1652.

# TO THE 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 wel known to all who have diligently read the Sacred Scriptures, wherein mention is made of that of K. Ahaz, upon which the Almighty was pleased to expresse a Miracle for the recovery of King Hezekiah, by causing the Sun to go back ten degrees upon the said Diall, 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 [Page] 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 insight 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 Commonwealth? What action can be performed in due season without it? Or what man can appoint any businesse with another, and not prefix a time, without the losse of that which cannot be re-gained, and ought therefore to be most 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 retain the benefit of all our knowledge.

Therefore this Art of Dialling being in it selfe so excellent and necessary, may induce any industrious person to the practise thereof, the perfect knowledge whereof in this ensuing 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 severall 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 Astronomie and Navigation, Geometrically, with Scale and Compasse only.

## III.

The third sheweth how to finde the Inclination and declination of any plane without Instrument, as also how to draw the houre-lines upon any plane howsoever, and in what Latitude soever scituate, by Rule and Compasse only, two severall wayes, in both which, the two grand inconveniencies of the common wayes (viz. of outrunning the limits of the plane; and drawing of many unnecessary lines) are totally avoyded, 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 construction and use of a Quadrant, by which all the most usuall Propositions in Astronomie may be wrought with great facility, and by which the Inclination and Declination of a plane may be speedily attained, and also the houre-lines drawn upon all kinde of planes in any Latitude.

Unto these five Books is added an Appendix, shewing how to furnish any kinde of Diall with Astronomicall Variety, as to draw thereon the parallels of Declination, by which the place of the Sun may be known: the parallels of the length of the day, by which the day of the Moneth, the Suns Rising and Setting, the length of the day and night may be known: the Babylonish and Italian houres, by which you may know how many houres are past since Sun rising, and how many remain to Sun setting: The old unequall houres, by which the day is divided into twelve equall paats 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 Diall 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.

Vale.

# THE CONTENTS.

- TErms of Geometry. page 1
- How to draw parallel lines page 9
- How to raise and let fall perpendiculars page 10
- Astronomicall definitions page 13
- Of the Circles of the Sphere both great and small. page 15
- The description of the Scale for Dialling. page 23
- How to make a line of Chords. page 24
- How to make an angle of any quantity of degrees and min. page 25
- How to finde the suns altitude page 26
- To finde the length of Right and Contrary shadow page 28
- To finde the suns declination page 30
- How to finde the suns place page 32
- To finde the suns Amplitude page 33
- How to finde the height of the Pole page 34
- How to finde the suns Amplitude page 35
- How to finde the suns Declination page 37
- To know at what time the sun shall be East or West page 39
- [Page]How to finde the height of the sun at the houre of six page 40
- To finde the Azimuth at the houre of six page 41
- To finde the Azimuth page 42
- How to finde the houre of the day page 44
- How to finde the Ascensionall difference page 49
- How to finde the Right or Oblique Ascension page 50
- How to finde the suns altitude without Instrument page 52
- How to finde the Latitude of a place page 54
- How to finde the declination, and inclination of any plane page 59
- How to draw a Meridian line upon an Horizontal plane. page 64
- To make an Equinoctiall Diall page 65
- How to draw a Diall upon a Polar plane page 66
- How to make an East or West Diall page 68
- How to draw a Diall upon an Horizontal plane page 71
- How to draw a Diall upon a full North or South plane page 74
- To draw a Diall upon a Verticall inclining plane page 79
- To draw a Diall upon a North or South declining plane page 81
- Another way to draw an horizontall Diall page 87
- Another way to draw a full North or South Diall page 89
- In an upright declining plane, to finde the deflexion, the height of the stile, and the inclination of Meridians page 92
- To draw a Diall upon an upright declining plane page 94
- To draw a Diall up a Meridian inclining plane page 101
- In declining inclining planes, to finde the height of the stile, the deflexion, &c. and to draw the Diall page 104
- The description of a Quadrant page 111
- The working of divers propositions in Geometry by the Quadrant page 117
- To work propositions in Astronomie by the Quadrant page 124
- To finde the inclination and declination of a plane page 139
- [Page]How to draw Dials on all kinde of planes by the Quadrant. page 144
- If the Cock of a Diall be lost, to finde the height thereof page 156
- How to describe the Equinoctiall, Tropicks, and other parallels of the suns course and declination in all kinde of planes page 159
- To draw the parallels of the length of the day on all planes page 175
- How the Babylonish and Italian hours may be drawn on all kind of planes page 181
- How the unequall houres may be drawn upon any plane page 185
- How to draw the Azimuths and Almicanters on all kinde of planes page 188
- How to draw a diall on the seeling of a Room. page 197

# Courteous Reader, These Books are to be sold by Thomas Pierrepont at the Sun in Pauls Church yard.

- Dr. Twisse learned Treatise in defence of the Sabath day.
- Dr. Stoughton 17 choyce Sermons.
- Mr. Fenners Works in five Treatises.
- Puerilis Confabula, in English.
- The whole Treatise of the cases of conscience by Mr. William Perkins.
- Miscellanea Philo Theologica, or God and Man, wherein many secrets of Scripture and Nature are un-bowelled, by Henry Church.
- Dr. Ames Exposition of both the Epistles of Peter.
- The Works of Mr. Nicholas Lokyer in three Treatises.
- The spirituall mans directory by Mr. Fennor.
- Usefull instructions for evill times, by Mr. Nicholas Lokyer.
- Naturall Philosophie reformed, by I. A Comenius.
- Artificers plain Scale, by Mr. Thomas Stirrup.
- Dr. Sibbs Christian Charter.
- Certain select cases resolv'd by Mr. Hen. Shepard.
- A vindication of Ordination and laying on of hands, by Mr. Lazarus Seaman.

ALL the worke of this Book is performed either Geometrically or Instrumentally: For the Geometricall performance there is required only a line of Chords (or rather a Sector,) and the Instrumentall part is performed by a Quadrant fitted for that purpose: Therefore, if any be desirous to have either Scale, Sector, Quadrant, or any other Mathematicall Instrument whatsoever, they may be furnished by Master Anthony Thompson in Hosier lane neer Smithfield.

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 practice 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 Geometricall 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, 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.

### Definition 3.

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

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.

If a right line fall on a right line, making the angles on either side equall, 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 A B here in this figure, falling upon the line C B D, in such sort, that the angles on both sides are thereby made equall, 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 equall 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 equall 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 [Page 4] the Circumference are equall, 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 equall 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, & 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, & a line drawn perpendicular unto the Diameter of the same Circle, from the center thereof, dividing the Semicircle into two equall 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 equall parts: then from the center A, raise [Page 5] the perpendicular A B, dividing the Semicircle likewise into two equall parts, so is A B D, or A B C, the Quadrant of the Circle C B D E.

### Definition 12.

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 segmēt or part of the Circle C B D E, and is contained under the right line FHG (w^{ch} 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, and of such parallels, two only belong unto this work of Dialling, that is to say, the right lined Parallel, and 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.

### Definition 14.

A Circular parallel is a Circle drawn either within or without another Circle upō the same center, as you may plainly see by the two Circles B C D E, and F G H I, these Circles are [Page 7] both drawn upon the same center A, and therefore are parallel the one to the other.

### Definition 16.

A Degree is the 360^{th} 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 60^{th} part of a degree, being understood of measure: but in time a Minute is the 60^{th} 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.

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.

### Definition 19.

The complement of an arch lesse then a Semicircle, is so much as that arch wanteth of a Semicircle, or of 180 degrees. As for example, the arch D C B being 120 degrees, this being taken from 180 degrees, 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, then set one foot in the end A, and with the other strike an arch line, on that side the given line

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 [Page 10] line D E, as F E, at 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. IIII. The manner how to raise a perpendicular line, from the middle of a line 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,

& setting one foot in the point C, and with the other foot mark on either side thereof, the equall distances C A, and C B: then opening your compasses to any convenient wider distance, with one foote in the point A, with the other, strike the arch line E over the point C, 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 D C, 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 line D E F. First, set one foote of your compasses in the point C, and opening your compasses to any convenient distance, so 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 finde the middle between those two intetsections, 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, & from the point B a [Page 12] 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 ED, then this distance being kept, set one foot of your compasses in the point E, and with the other make a mark in the former arch E D, as at D, stil 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 A B C, then finde the middle of the line A C, which will [Page 13] be at B, place therefore one foot of your compasses in the point B, and extend the other unto A or C, 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 Astronomicall, 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 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 unnecessary 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 equall, 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 circle described by the motion of the North star, or the tale 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 beene 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 equall course, like much in like time; which diurnall 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 Equinoctiall, to whom neither of the Poles of the World are seen, but lie hid in the Horizon. An oblique Sphere is unto those that inhabit on either side of the Equinoctiall, unto whom one of the Poles is ever seen, and the orher 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 equall parts, and they are in number six, viz. the Equinoctiall, 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 Equinoctiall is a Circle that crosseth the Poles of the world at right angles, and divideth the Sphere into two equall parts, and is called the Equinoctiall, 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 & nights of equall 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 16] is to be understood, that all Towns and Places that lie East and West one of another, have every one a severall 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 inferior, 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 Equinoctiall and the Zodiaque into four equall parts, making thereby the four Seasons of the year, the one Colure passing through the two Tropicall points of Cancer and Capricorn, shewing the beginning of Summer, & also of Winter, at which times the dayes and nights are longest and shortest. The other Colure passing through the Equinoctiall points Aries and Libra, shewing the beginning of the Spring time and Autumne, at which two times the dayes and nights are of equall length throughout the whole World.

### Definition 9.

The Ecliptique is a great Circle also, dividing the Equinoctiall into two equall 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 [Page 17] famous Mathematician Master Edward Wright) is 23 degrees, 31 minutes, 30 seconds. Note also that this Circle is divided into 12 equal parts, which parts are attributed unto the 12 Signes, Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagitarius, Capricornus, Aquarius, and Pisces. Out of this line doth the Sun never move, but the Moon and other Planets are somtimes 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 foure lesser Circles.

### Definition 10.

THe Sun having ascended unto his highest Solsticiall 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, & is noted in the figure following by the line ♋ FG.

### Definition 11.

The Sun also approaching unto the first scruple of Capricorn, or the Winter Solstice, describeth another Circle, which is the utmost bounds that the Sun can depart from the Equinoctiall 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 Equinoctiall, [Page 18] 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 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, & 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 Equinoctiall 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 [Page 19] right angles with the Equinoctiall, 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 Equinoctiall. 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 [Page 20] right angles with the Ecliptique, and counted either Northward or Southward according to the scituation of the 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.

### Definition 18.

The Latitude of a Town or Countrey, is the height of the Pole above rhe Horizon, or the distance betwixt the Zenith and the Equinoctiall, and may be represented in this figure by the arch of the Meridian B P, where the North Pole P is elevated above the horizontall line C A B according to the angle B A P, which here is 52 degrees 25 minutes, 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 Equinoctiall, 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 [Page 21] the Sun to be in the Meridian at S, then the angle of altitude will be the angle S A C, the measure whereof will be the arch C S, contained betwixt the Sun at S, and the Horizon C, which here will be found to be 37 degr. and 35 min. the height of the Sun at noon when it is in the Equinoctiall 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 Equinoctiall (counted from the beginning of Aries) which commeth to the Meridian with the Sun, Moon, Star, or any portion of the Ecliptique.

### Definition 25.

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

### Definition 26.

The Ascensionall Difference, is the difference betwixt [Page 22] the right and 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 Equinoctiall 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 Star: the arch of the Sun or Stars Almicanter, is a portion of an Azimuth contained 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 attendent 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, or Dialling. Therefore now I intend to proceed with Scale and Compasse to perform some questions Astronomicall, before we enter upon the Art of Dialling, seeing they are both delightfull, and also helpfull unto all such as shall be practitioners in this Art of Dialling.

# 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 equall 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 unequall divisions, representing the 90 degrees of the Quadrant, and are numbered 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 [Page 24] degrees, the number of these degrees from 1 unto 60, is called the Radius of the Scale, upon which distance all

Circles are to 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 Mathematicall 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 equall parts or Quadrants by the point B, then dividing one of these Quadrants into 90 equall 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 or the Quadrant A B, and these extents transfer into the line A D C, as here you see is done. This line so divided into 90 unequall 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 degree of the Quadrant [Page 25] 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 equall 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.

## 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 our Scale, and [Page 26] setting one foot thereof in the point A, with the other foot describe the arch B C, then draw the line A B, then opening your Compasses to so many degrees upon your line of Chords, as you 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 B C, as at C, from which point C, draw the line C A, which shall make the angle B A C, containing 40 degrees, as was required.

And if you desire to finde the quantity of an angle, open the Compasses to the Radius of your Scale, & 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 finde the altitude of the Sun by the shadow of a Gnomen set perpendicular to the Horizon.

FFirst, draw the line A B, then opening your Compasses to the Radius of your Scale, set one foot in the end A, [Page 27] and with the other describe the arch B C D, then opening your Compasses unto the whole 90 degrees, with one foot in B, with the other marke the arch B C D in the point D, from which point D, draw the line D A, which shall be perpendicular unto the line A B, and make the Quadrant A B

C D, then suppose the height of your Gnomon or substance yeilding shadow, to be the line A E, which here we will suppose to be 12 foot, therefore take 12 of your equall 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 C, so shall the arch B C be the height of the Sun desired, which in this example will be found to be 53 degrees 8 minutes, the thing desired.

## CHAP. IIII. To finde 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, & 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 E F 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 line 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 min. for the height of the Sun desired.

## CHAP. V. The Almicanter, or height of the Sun being given, to finde 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 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 [Page 29] 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 line E F parallel to A B, then set the Almicanter (which here we will suppose to be 53 de. and 8 min. as it was found by the third Chapter) 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 finde 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 [Page 30] height: whereas, contrariwise, the right shadow doth decrease in length as the Sun doth increase in height. Therefore the way to finde 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 Equinoctiall point, to finde 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 the other draw the Quadrant B C D: 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 degrees distant from the next Equinoctiall point Aries. Or if the Sun shall be in the 29 degree of Scorpio, or in the first degree of Aquarius, both which are also 59 degrees distant from the Equinoctiall point Libra, therefore [Page 31] take 59 degrees from your Scale, and place it from B to C, and draw the line C A, then place the greatest declination of the Sun from B unto F, which is 23 degrees 30 minutes, then

fixing one foot of your Compasses in the point F, with the other take the neerest distance unto the line A B, which you may doe by opening or shuting of your Compasses, still turning them to and 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, untill 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 finde his true place.

LEt the declination given be 20 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 a manner, that the other foot 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 from the head [Page 33] of 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 Latitude of the place, and the distance of the Sun from the next Equinoctiall point, to finde his Amplitude.

FIrst, make the Quadrant A B C D, 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 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 [Page 34] 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 Equinoctiall point, take 59 degrees

from your Scale and place them from B to L, frō which point L, take the neerest distance unto the line A B, with this distance, seting one foot in the point A, with the other make a mark in the line A C, as at O, frō 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 finde 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 [Page 35] 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 point F, take the shortest extent unto the line A B: this distance being kept, fix one foot of your Compasses in the arch line G H I, so that the other foot may but touch the line A B at the neerest extent, so shall the fixed foot rest at the point H, through which point H, draw the line A H C, cuting

the arch B C D in the point C, so shall the arch B C be the height of the Equinoctiall, 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 Equinoctiall from the Zenith, which in this example will be found to be 52 degrees 30 minutes, the thing desired.

## CHAP. XI. Having the Latitude of the place, and the Declination of the Sunne, to finde his Amplitude.

FIrst, make the Quadrant A B C D, then supposing the Latitude to be 52 degrees 30 minutes, take it [Page 36] 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 F G H, so that 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 w^{ch} point G draw the line A G C, cutting the arch B C D in the point C, so shall the arch B C be the Amplitude desired, which in this example will be found to be 34 degrees 9 minutes, as before in the 9 Chapter.

## CHAP. XII. The elevation of the Pole, and the Amplitude of the Sun being given, to finde 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 finde his height in the Verticall 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 which point O take the shortest extent unto the line A B, with this distance, fixing one foot in the arch G H I, so that the other may but only touch the line A B at the neerest extent, so shall the fixed foot rest in the point H, through which point H draw the line A H E, 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 [Page 39] 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 finde the time when the Sunne 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 A, with the other draw the arch G H I: then supposing the declination of the Sun to be 20 degrees, place it frō B to F, from which point F, lay a rule unto the center A, and 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: [Page 40] So shall the fixed foot stay in the point H, through which point H draw the line A H N, 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 for a degree) will make foure houres and 54 minutes of an houre, that is, either at 4 a clock and 55 minutes in the afternoon, or at 7 a clock and 5 min. in the morning.

## CHAP. XV. Having the Latitude of the place, and the Declination of the Sun, to finde 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 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 [Page 41] line A H E, 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 sort 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 houre 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 finde what Azimuth the Sun shall have at the houre of six.

FIrst, draw the Quadrant A B C D, then supposing the Latitude to be 52 deg, 30 min. 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, frō which point E draw the line E G, parallel to A B, untill it cutteth the line A D in the point G, and with the distance A G, describe the arch G H I, cutting the line A C in the point H, through [Page 42] 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 P unto 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 finde the Azimuth.

FIrst, draw the Quadrant A B C D, then supposing the Latitude to be 52 deg. 30 min. setting it from B to C, draw the line A C, then supposing the declination of the

Sun 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 F in 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 [Page 43] point G, take the neerest extent unto the line A D, and 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, then take the distance F H betwixt your Compasses,Here note, that if the Sun had been so low that the point H bad 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. 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 A B, with which distance, setting one foot of your Compasses in the point N, with the other foot describe the arch I, by rhe 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 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 min. 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 A D, 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 [Page 44] the neerest 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, frō which point O take the shortest extent unto the line A B, and place it from F to H, then with the distance HA, 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 finde the houre 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 [Page 45] 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 which point F draw 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 from A to R in the line A B: then supposing the altitude of the Sun to be 15 deg. 24 min.

place it from B unto H, from which point H take the shortest extent unto the line A B, & 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 A G, 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 deg. 20 min. the Sun having 11 deg. 30 min. of South declination, & 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 distance 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 line A B, and place it from A to F in the line A C, and through the point F, draw the line O P parallel to A B, then from the aforesaid point E, take the shortest extent unto the line A D, 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 degrees 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 [Page 47] 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 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 10 a clock, if it had been in the afternoon 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 finde the houre of the day.

FIrst, draw the Quadrant A B C D, then supposing the Sun to have 11 deg. 30 min. declination, place it from B to E, and from the point E take the shortest extent 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. 43 min. as it was found by the former part of the 17 Chapter, which place from B to F, and from the point F take the neerest extent unto the line A B, which distance place from G to H, in [Page 48] 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 min. 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 NR, then with the distance NR, 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 somthing better then 34 min. from the Meridian.

## CHAP. XX. Having the houre of the day, the Suns altitude, and the declination, to finde 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, then suppose the altitude of the Sun to be 30 deg. 45 min. place it from B to L, from which point L, take the shortest extent unto the line [Page 49] 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 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 BF wil 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 finde the Ascensionall difference.

FIrst, draw the Quadrant A B C D, then place the Latitude (which here let be 52 deg. 30 min.) from D to C, & draw the line COP parallel to AD, then with the [Page 50] distance C P, upon the center A, describe the arch G H I, then place the declination, being 20 degrees, from B to F, then lay your rule from the center A upon the point F,

and draw F O, cutting C P in the point O, through which point O draw the line O H R, cutting the arch G H I in the point H, through which 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 ascensionall, and will be found in this example to be 28 deg. 19 min. which resolved into time doth give one houre, and somthing better then 53 min. for the difference betwixt the Suns rising or setting, and the houre of six, according to the time of the yeare.

## CHAP. XXII. Having the Declination of the Sun to finde 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 E P parallel to A D, and with the distance E P, with one foot in A, describe the arch G H I, then set the [Page 51] Declination of the Sun given 20 degrees, 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 O H R 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 shal 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 finde the Oblique Ascension.

THe right Ascension of any point of the heavens being known, the difference of the Ascension is either to be added thereunto, or else to be substracted from it, according as the Sun or Star is scituated 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 substracted from the right Ascension, and the remainder [Page 52] is the Oblique Ascension; therefore let the fourth degree of Gemini be given, the Right Ascension whereof is found to be 62 degrees, or 4 houres, 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 min. 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 Equinoctiall, either in Libra, Scorpio, Sagitarius, 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 Sagitarius to be given, the Right Ascension whereof is found to be 242 degrees, 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 Sagitarius. But if you would finde the Oblique Descension, you must work directly contrary to these Rules given.

## CHAP. XXIV. How to finde the Altitude of the Sun without Instrument.

IN the third Chapter of this Book it is shewed how to finde the Altitude of the Sun by a Gnomon set perpendicular to the Horizon, but seeing the ground is so unlevel it is not so ready for this our purpose, and perhaps some [Page 53] may have occasion to finde 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 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 wire in one of

the points, as at A, whereupon hang a threed with a plūmet, then 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 finde 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 altitude desired.

## CHAP. XXV. How to finde out the Latitude of a place, or the Poles elevation above the Horizon, by the Sunne.

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 finde it out, therefore it will not be un-needfull 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 untill 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 seventh Chapter, you may finde the Latitude of the place, or the elevation of the Pole above the Horizon after this manner. If the Sun hath North Declination, then substract the Declination out of the Meridian altitude, and the remainder shall be the height of the Equinoctiall. 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 Equinoctiall, which being taken out of the Quadrant or 90 degrees, 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 [Page 55] Meridian altitude of the Sun being observed to be 55 deg. 35 min. upon which day I finde the Suns place to be in 20 deg. 48 min. 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 Equinoctiall, and this taken out of 90 degrees, leaveth 52 deg. 25 min. for the Latitude of Thuring.

But it may be required somtimes 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 principall Cities and Towns in England, so that being required to make a Diall for any of those places, you need but look in your Table, and there you have the Latitude thereof. But if the Town you seeke be not in the Table, looke what Town in the Table lies neere unto it, and make your Diall to that Latitude, which will occasion little difference.

## A TABLE shewing the Latitude of the most principall Cities and Towns in ENGLAND.

Names of the places. | Latitude | Names of the places. | Latitude | ||

d | m | d | m | ||

St. Albons | 51 | 55 | Hull | 53 | 50 |

Barwick | 55 | 49 | London | 51 | 32 |

Bedford | 52 | 18 | Lancaster | 54 | 8 |

Bristol | 51 | 32 | Leicester | 52 | 40 |

Boston | 53 | 2 | Lincolne | 53 | 15 |

Cambridge | 52 | 17 | Newcastle | 54 | 58 |

Chester | 53 | 20 | Northampton | 52 | 18 |

Coventry | 52 | 30 | Oxford | 51 | 54 |

Chichester | 50 | 56 | Shrewsbury | 52 | 48 |

Colchester | 52 | 4 | Warwick | 52 | 25 |

Darby | 53 | 6 | Winchester | 51 | 10 |

Grantham | 52 | 58 | Worcester | 52 | 20 |

Halifax | 53 | 49 | Yarmouth | 52 | 45 |

Hereford | 52 | 14 | York. | 54 | 0 |

# THE THIRD BOOK. Shewing Geometrically how to describe the Houre-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 a Horizontall Diall.

FOrasmuch as it is necessary before the drawing of any Diall, 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 usefull.

First, take any board that hath one streight side, and an inch or more from the streight 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 levell with the horizon, you may try it by applying the streight 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 soever you turn the board, it is an horizontall plane.

As for example, let the figure A B C D be a plane supposed to stand levell with the Horizon, and for to try the same, I take the simple boarde GOH, 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 [Page 59] finding the threed to fall directly on the perpendicular E O, which way soever I turn the board, I therefore conclude it to be a horizontall plane.

## CHAP. II. Of the trying of Planes, whether they be Erect or Inclining, and to finde the quantity of Inclination.

FOr the distinguishing of planes, because their inclination and declination may be diverse, we will consider three lines belonging to every plane: the first is the Horizontall line; the second the perpendicular line, crossing the horizontall at right angles; the third is the Axis of the plane, crossing both the horizontall line, and his perpendicular, and the plane it selfe, at right angles; the extremity of which Axis may be called, the pole of the planes horizontall line.

The perpendicular line doth help to finde the inclination, the horizontall line with his Axis to finde the Declination, and the pole of the planes horizontall 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 Verticall line; as in this figure, the plane G H L I is erect, & the line H I is the Verticall line.

Now for the trying of this plane, if you apply the streight 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 finde the threed [Page 60] to fall directly on the parallel line M N it is an erect plane, but if the threed will crosse 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 by applying your board thereto, with the threed and plummet falling on the parallel line M N, draw the verticall line H I by the edge of your board, the verticall line being drawn, you may crosse it at right angles with the line G L, which shall be levell with your Horizon, and therefore called the horizontall line of the plane.

If the plane shall be found to incline to the horizon, you may finde 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 streight side thereof lie close with the plane, and the threed fall on the perpendicular line O E, so the line drawn by the streight side of the board, shall be an horizontall line, which here in this figure will be the line F E.

This being done, crosse the horizontall line at right angles with the perpendicular H K, then set the streight side of your board upon the line H K, which is perpendicular to the horizontall line, with holding the board up right, 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 plummer 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, [Page 61] and it is the angle contained betwixt the plane and the verticall line passing from the Zenith to the Nadir, the complement of which angle is the inclination of the plane, and it is the angle 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 [Page 62] 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 and the threed, shall be the complement of the angle of the inclination of the plane to the Horizon.

## CHAP. III. To finde out the Declination of a Plane.

THe Declination of a plane is alwayes reckoned in the Horizon, and it is the angle contained between the line of East & West, & the horizontall line upon the plane.

For the finding out of this Declination, first, take any board that hath but one streight side, and draw a line parallel thereto, as was done in the first Chapter, and having drawn an horizontall 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 applyed to the horizontall line G L, then the Sun shining upon the board, hold out a threed and plummet, so as the threed being verticall, 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 S P R, which is made between the horizontall line of the plane, and the Azimuth wherein the Sun is at the time of observation: at this same instant, or as neere as may be, must you take the Altitude of the Sun; these two being done diligently, wil help you to the planes Declination as followeth.

First, describe the Circle B C D E, which shall represent the Horizontall Circle, and draw the diameter B A C, representing the horizontall line of the plane, in the last chapter set out by the line G L, then having found S P R [Page 63] in the last Chapter (which is the angle made by the horizontall 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 horizontall line to G

Southwards, because the angle was taken at that end of the line, at which instant the altitude of the Sun (by the 24 Chapter of the second Book) being found to be 13 deg. 20 m, having 11 deg. 30 min. South Declination, it will, by the 17 Chapter 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, & crossing the line H R at right angles, in the center A, you shall have the line D E for the North and South; & if you crosse the horizontall 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 S and N, shall be the poles of the planes horizontall line.

Now the angle of declination here required, is the angle B A H, or E A S, for look how much the horizontall line of the plane declineth from the line of East and West, so much doth the poles of the planes horizontall 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 Horizontall plane, the Sun shining thereon.

IF your plane be levell with the Horizon, describe thereon a Circle as B C D E, then holding up a threed and plummet, so as the threed being verticall, the shadow thereof may fall upon the center A, and draw the line of shadow

C E, then take the altitude of the Sun at the same instant (or as neere as may be) and 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 deg. 43 min. from the [Page 65] 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 B A D, which is the meridian line desired.

## CHAP. V. Of making the Equinoctiall Diall.

AN Equinoctiall plane is that which is parallel to the Equinoctiall Circle of the Sphere, and therefore having drawn the horizontall line B C, and crossed it with the perpendicular D E, at right angles in the point A, if by

[Page 66] the second Chapter you shall finde the inclination of the plane towards the South to be equall to the complement of your Latitude, and by the third Chapter you finde the horizontall 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 Equinoctiall Circle, and is therefore called an Equinoctiall plane.

This Diall is no other then a Circle divided into 24 equall parts, by which divisions and the center A, you may draw so many houre lines as shall be necessary. As you may see here done in the Circle B D C E, which is divided equally into 24 equall parts, and houre lines drawn from the center A to so many of them as is needfull, the line D E, which is the line of inclination, is the Meridian or 12 of clock line: his Stile is no more but a streight pin or wire plumb erected in the center.

This Diall, 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 houre lines on any other great Circle or plane whatsoever.

## CAAP. VI. The drawing of a Diall upon the direct Polar Plane.

A Direct Polar plane, is that which is parallel to the Circle of the houre of six, therefore, having drawn the horizontall line A B, rnd crossed it at right angles about the middle of the line at C, with the perpendicular C E, if you shall finde the planes inclination towards the North to be equall to the Latitude of the place, and the horizontall line directly in the line of East and West, and [Page 67] so to have no declination, you may be sure this plane lyeth parallel to the houre of six, and is therefore called a Polar plane. The horizontall line being drawn at the length of the plane, divide it into seven equall parts, and set down one of them in the line of inclination from C unto D, and upon the center D describe the Equinoctiall Circle, which you may divide into 24 equall parts if you will, but one quarter thereof into six will serve as well: then at the

distance C E draw the line F G parallel to A B: Then having divided the Equator either into 24 equall parts, or one quarter thereof into six, you may by a rule laid upon the center D, and each of those six parts, make marks in the horizontall 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 applyed 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 houre lines, the line C E shall be the Meridian line, the houre of 12, and must also be the substilar line, whereon the stile must stand, which may be a plate of iron or some other metall, being so broad as the semidiameter of the Circle is, as is shewed in the figure. This style must be [Page 68] 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 houre lines. The under face of this Polar plane, and also of the former Equinoctiall 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 horizontall 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 horizontall line being drawn, at the North end thereof as at A, make an angle equall to the elevation of the Equinoctiall, 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 equall to the Equinoctials height, so shall the South end of this line behold precisely the Equinoctiall Circle, this line divide into five equall 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 D H, through the center whereof draw a Diameter, cutting the former Diameter D H at right angles in the center E, this Diameter shall lie parallel to the Axl [...]tree of the world, and be the line for the houre 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 [Page 69] middle line, then divide one quarter of the Circle into six equall 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 houre lines here desired, and shall be parallel one to the other: the 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 houre points in the upper touch line, and if you transfer these distances from the houre of six into the other touch 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 Diall it may be either a plate of some metall, being so broad as the semidiameter of the circle is, and so placed perpendicularly along over the line of the houre of six, the upper edge thereof being parallel to the plane, or it may be a streight 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 houre with the very top or end thereof.

This plane hath two faces, one to the East, the other to the West, the making whereof are both alike, only in naming the houres, for the one containeth the houres for the forenoon, and the other for the afternoon, as you may perceive by the figures.

## CHAP. VIII. To draw a Diall upon an Horizontall plane.

AN Horizontall plane is that which is parallel to the Horizontall Circle of the Sphere, which being found by the first Chapter to be levell with the Horizon, you may by the fourth Chapter draw the Meridian line A B serving for the Meridian, the houre 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 West, 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 equall 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 Diall, 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 neerest 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 Equinoctiall, upon which describe the Equinoctiall Circle A D B E, with this same distance, setting one foot in the point A, make a mark [...] F on the one side of the [Page 72] 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 houre 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 Equinoctiall Circle, each of which distances is an arch of 60 degrees, or four houres of time, the halfe of which arch is 30 degrees, or two houres from the Meridian, this divided in the halfe will be 15 deg. or one houre from the Meridian; then laying your rule upon [Page 73] the center I of the Equinoctiall, and upon these two last divisions in the circle thereof, where the rule shall touch the line of contingence, there mark it as at H and O, by which points and the center C, you may draw the hourlines of 10 and 11; 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 houres of 9 and 3 with the touch line draw the lines F D and G E parallel to the meridian A B, untill they 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 G E, 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 and the center C you may draw the 8 a clock hour-line, 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 Equinoctiall Circle into three equall parts, you may describe your whole Diall.

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 Diall: prolong or draw the lines of 4 and 5 at evening [Page 74] 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 kinde as in all them which follow.

The style 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.

## CAAP. IX. To draw a Diall upon an erect direct verticall Plane, commonly called a South or North Diall.

A Verticall plane is that which is parallel to the prime verticall Circle, it hath two faces, one to the South the other to the North; therefore having drawn the Horizontall line A B, and from the middle thereof let fall the perpendicular C D, which if you finde by the second Chapter to be erect, and the Horizontall line A B to lie in the line of East and West, and so to have no declination, you may be sure this plane is parallel to the prime verticall Circle of the Sphere, and therefore is called a verticall 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 Diall directly hangeth.

The Horizontall 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 equall to the Latitudes complement, which in this example is 37 degr. 30 minutes, 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 Equinoctiall 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 equall 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 centre of the Diall; the same distance of your compasses being still kept, with one foot in the point E, with the other make a mark in the Equinoctiall Circle as at H. So shall the arch E H contain 60 degr. or four hours of time, which arch you may divide into four equall 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, untill [Page 76] they cut the Horizontall 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 the line A E at the neerest extent, and the fixed foot shall rest in the point T, 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 N, and with the other take the shortest extent unto [Page 77] 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 hour 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 shal be the points for the hour of 8 or 4.

Thus you may see, that by dividing the Radius of the Equinoctiall Circle into four equall parts, you may describe your whole Diall, if it hath no declination; for having with this Radius pricked out the lines of 3 & 9, & placed the lines for 10 and 11, these two distances from the hour of 12 shal give the like for the hour of 1 & 2 on the other side of the meridian, and having drawn the lines for 7 & 8 by the former rules, you may take their distances from the hour of 6, and place them on the other side of the Diall from 6 to 5 & 4: so have you all your hour-lines drawn, and [Page 78] yet we have not out-run the limits of our plane, which is an 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 Diall, and naming the houres; 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.

## CHAP. X. To draw a Diall upon a direct verticall plane, inclining to the Horizon.

ALL those planes that have their Horizontall line lying East and West, are in that respect said to be direct verticall planes; if they be also upright passing thorow the Zenith, they are erect direct verticall planes; if they incline to the pole, they are direct polar planes; if to the Equinoctiall, they are called Equinoctiall 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 Diall, 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 Horizontall plane, the meridian line is drawn by the 4 Chapter, and the elevation of the pole above the plane is always equall to the latitude of the place; and in erect direct verticals, the perpendicular or verticall line is the meridian or 12 a clock line, and the elevation of the pole above the plane is always equall 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 [Page 80] 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 Equinoctiall plane, and is described before in the 5 Chapter, but if the summe shall exceed 90 degrees, take it out of 180, and that which remaines shall be the elevation of the pole above the plane.

As for example in the latitude of 52 deg. 30 min. let a plane be found to incline Southwards 20 degrees, this 20 deg. added to 52 deg. 30 min. the latitude of the place, the summe will be 72 degr. 30 min. the elevation of the pole above the plane, with which you may proceed to draw a Diall by the eighth or ninth Chapters as if it were a horizontall 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 the 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 sixt Chapter.

As in the latitude of 52 deg. 30 min. a plane being proposed to incline towards the North 25 deg. this 25 degr. being taken out of 52 degrees 30 minutes leaveth 27 deg. and 30 minutes for the elevation of the pole above the plane. Now this plane being parallel to that Horizon, whose latitude is 27 deg. 30 min. lying both under one & the same meridian, therefore you may proceed to make this Diall, as if you were to make an Horizontall Diall 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 inclination 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 Dialls must point directly.

## CAAP. XI. To draw a Diall upon an erect, or verticall plane declining, commonly called a South or North erect declining Diall.

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

If they behold none of these four principall parts of the world, but shall stand between the prime verticall circle, [Page 82] and the Meridian, they are then called by the generall 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 verticall line, and serveth for the hour of 12) and the Meridian of the plane deflecteth one from the other; therefore we must finde 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 hour 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 degr. 30 min. in the same arch from E to M, and 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, then take S L and set it from E to O in the line E F, and draw the line C O, [Page 83] which is the Meridian of this plane, or the line of the Substile,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. 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.

This done, set one foot in the point O,

and with the other take the shortest extent unto the stile [Page 84] 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 Equinoctiall circle.

Now having drawn the line NT 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 Equinoctiall 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 Equinoctiall, 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 equall 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 [Page 85] 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 O 10 from O to g, and from G to V, and laying the Rule upon V and g, you shall finde 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 houre-lines of 2, and 1, shall be drawn.

The Diall 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 equall 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 [Page 86] planes declining from the North, the North pole is elevated.

Therefore if you were to make a Diall 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 Diall 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 oyled or pricked through, so that you may take the back-side thereof for the fore-side, without altering the numbers, set to the hours.

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

This paper being oyled, if you do but change the backside 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 Diall upon an horizontall plane, otherwise then in the eighth Chapter was shewed.

ALthough I have plainly and perfectly shewed the making of the horizontall, the direct South or North, as well erect as inclining; and the South or North erect 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 horizontall; 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 Diall as large as your plane will give leave; which let be the circle A D B E, then take the latitude of the place which is here 52 degr. 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 [Page 88] the two quarters of your circle A E and A D, each into six equall parts, so shall you have in each Quadrant five points by which you may 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 [Page 89] 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.

So likewise you may take one half 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 8 Chapter.

## CHAP. XIII. To draw a Diall upon a direct verticall 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 horizontall plane, the elevation is equall to the latitude of the place; and in all direct verticals being erect, the elevation of the pole above the plane is equall 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 EW parallel to the horizon, and from the middle thereof set fall the perpendicular ZN 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 degr. 30 min. equall to the complement of our latitude, which take from your scale, and place it from N to H in your Dialls semicircle, 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 [Page 91] line of the stile in the point S, so shall the Triangle S Z N be the true pattern for your cock.

This being done, divide each Quadrant of your Semicircle into six equall parts, so shall you have 5 points, by which you may draw 5 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, and place them upon their proper Chord lines from the meridian on both sides thereof; so shall you have two points on each Chord, through which you shall draw the hour-lines from the center of your Diall, 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 ways 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 houre-lines at pleasure, without exceeding the limits of your plane. And seeing this is the South face of this plane, therefore the style must point downwards being fixed in the center Z in the upper part of the meridian line Z N, over which the style must directly hang, making therewith an angle equall to the angle N Z S.

But if it had been the North face, then must the center [Page 92] be placed in the lower part of the Meridian, and the style, with the substyle, and also the houre-lines, must point upwards.

## CHAP. XIV. The declination of an upright plane being given, how thereby to finde 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 Diall.

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 Diall-circle into 24 equall 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 H S, and set it in the line D E, from D to K, through 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 houres of 2 and 3 if the plane shall decline Westward; but if it shall decline Eastward, then shall it fall betwixt the houres of 9 and 10 before noon.

## CHAP. XV. To draw a Diall upon an erect or Verticall 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 deg. 8 min. and likewise the angle of inclination between both Meridians to be 38 deg. 13 min. we may proceed to make the Diall after this manner.

First, draw the horizontall W E, and the perpendicular line Z N, crossing the horizontall 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 Diall circle E N W.

Then take a Chord of 22 deg. 8 min. 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 deg. 8 min. being set in the Diall 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 houre-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 Diall.

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

And if the line of the substile falleth not directly upon [Page 98] one of the houre 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 own proper perpendiculars from the substile, so may you make points through which you may draw houre-lines, and by thus doing with each perpendicular on both

[Page 97] sides the substile, you may describe your whole Diall 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 Diall upon paper, for any plane whatsoever, you may cut off the hour-lines Cock and all, with a lesser Circle then the Diall Circle, either with a concentrique or an excentrique Circle, and so make a Diall lesse then the Circle by which you framed it.

Or if you extend the hour-lines beyond the Diall Circle, you may cut them off either with a greater concentrique Circle, and so make a bigger Diall, 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 neere 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, & 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 Diall very true (as before hath been taught) upon a large paper, making your circle as big as you can: then extend the houre-lines, with the substile, and the line of the stile, a great way beyond the Dials circle, untill they doe 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 wil it shew almost [Page 98]

[Page 99] like the Meridian Diall of the 7 Chapter, for the hours wil 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 finde 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 horizontall line is the same with the Meridian line, are therefore called Meridian planes, if they make right angles with the Horizon, they are called erect Meridian planes, and are described before.

But if they leane 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 Diall.

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 G O D in the point H, through which point 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, which 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 quadrant in the point L, so shall the arch C L be the distance of the substile from the Meridian, and is in this example 33 deg. 5 min.

This being done, from the point L, 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 C I be the inclination of the Meridian of the plane to the Meridian of the [Page 101] 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 houre-lines.

## CHAP. XVII. To draw a Diall upon the Meridian inclining plane.

HAving by the second Chapter found the inclination of this plane to be 30 deg. and so by the last chapter found the elevation of the pole above the plane to be 43 deg. 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 Diall after this manner.

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

Then seeing this is the upper face of the plane, set 33 deg. 5 min. the distance of the substile from the Meridian, in the Dials Circle from the North end of the Horizontall 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 horizontall line: now through the center C draw the Diameter E F, making right angles with the substile C H.

Then set 43 deg. 23 min. from H unto D for the stile, and draw the line C D unto S, and from the end of the 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 Diall.

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 Circle into 24 equall 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 houre-lines.

This diall being thus drawn, for the upper face of a Meridian plane inclining towards the West, you must fixe the Cock in the center C, hanging over the substile C H, with an angle equall 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 Diall being drawn in paper, for the upper face of this plane, will also serve for the under face thereof, if you turn the pattern about so that the horizontall line A B may lie still parallel to the Horizon, and the stile with the substile (lying under the Horizontall line) may point downwards to the South Pole, the paper being oyled or pricked [Page 103] 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 finde the angle of intersection between the plane and the Meridian, the ascension 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 the Horizon, it is then called by the name of a declining inclining plane.

Of these there are severall 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 Diall.

I. The angle of intersection betwixt the plane and the Meridian.

[Page 105]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 F 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 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 Z unto O, through which point O draw the line A O M, 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 [Page 106] 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 remains shall be the position Latitude, or the arch of the Meridian between the Pole and the plane.

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. 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 [Page 107] 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 A C, cutting the arch B C in the point X, so shall the arch B X be 16 deg. 20 min. the inclination of the meridian of the plane to the meridian of the place.

## CHAP. XIX. To draw a Diall 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 Diall 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 Diall circle A D B E A, then take 57 deg. 36 min. the Meridians ascension, and set it from B that end of the horizontall line with the declination of the plane, as from B to N, and draw the line C N for the houre 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 Diall circle into 24 equall 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 houre-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 kinde 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 horizontall line, you must consider which pole is elevated above your plane, and how to place the Meridian from the Horizontall line. For

upon the upper faces of all North incliners, whose [Page 110] 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 elevation is greater then the Latitude of the place: and on the under faces of all South incliners the South Pole is elevated.

Now for placing the Meridian from the horizontall 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 lesse then the Latitude of the place; the Meridian must be placed above the Horizontall 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 Horizontall 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 horizontall line with the declination of the plane: But on all the other faces of these kinde of planes, the Meridian must be placed from that end of the horizontall 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 Meridirn equall 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 equall 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 being placed according to his ascension from the horizontall 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 Equinoctiall Circle (as in the sixt Chapter of this Book was shewed:) upon the center whereof, make an angle with the line of the substile, equall 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 Equinoctiall Circle into twenty four equall parts.

These things being known, you may proceed to make your Diall and set up the Cock according to the sixth 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. equall 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 houre lines parallel one to the other, as in the Meridian and direct Polar planes. Therefore, first draw the Horizontall 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 CE 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 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.

Now from the point O, divide your Equinoctiall circle into 24 equall parts, with which you may proceed to make your Diall, and set up the cock according to the 6 Chapter.

# THE FOURTH BOOK. Shewing how to resolve all such Astronomicall 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 Astronomicall, with the delineation of all kinde of plain wall Dials howsoever, or in what latiitude soever scituated, still keepng 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 [Page 115] 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 limbe, and is divided into 90 equall parts or degrees; and subdivided into as many parts as quantity will give leave, being numbered from the left hand towards the right after the usuall manner.

Then let the Semidiameter A B be divided into 90 unequall parts, (called right Sines) either from the Table of naturall Sines by help of a decimall Scale, equall 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 unequall divisions called right Sines.

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 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 E F a Co-tangent line, then draw the Diagonall 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 equall parts, by laying your Rule upon each 15th. degree in the [Page 116] 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 equall 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 severall Radiuses, [Page 117] 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 usuall manner.

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

IF the Radius of the Circle given be equall 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 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 threrd 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 verticall circle; and let F G be the diameter [Page 118] of an almicanter, and so F H the Semidiameter thereof; which being given it is required to finde 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 the 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 finde 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, & 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 the shortest extent thereunto, from the Sine of 90 you shall have the line H F for the Radius required.

## CHAP. IV. The right Sine, or the Radius of any Circle being given, and a streight line resembling a Sine, to finde 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 six one foot in the line of sines, moving it till the moveable foot touch the threed at the neerest extent, so shall 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 streight 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 foote of your compasses in the line of Sines, still moving it to and fro till the moveable foote thereof, doth onely touch the threed, so shall the fixed foote 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 finde the chords of every arch.

IF the Radius given, shall be equall 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. & 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 ICL which is an arch of 60 deg.

And if the Radius of the circle given, be equall to the Semi-radius (the sine of 30 deg.) of the line of sines; then you neede 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 [Page 121] 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 the third Chapter, and it is required to find a Chorde of 30 degrees, therefore first, I take A C 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 halfe 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 halfe 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 meane 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 deg. 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 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

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

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

First, ioyn the two given lines together, so as they may make both one right line; the which divide into two equall parts; and with the one halfe 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, & 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 proprtional line required.

As for example, let A and B be two lines given, between which it is required to find a mean proportionall line, first joyne the two lines together in F, so as they both make the right line C D, which divide into two equall parts in the 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 positiō, take the distance betweene 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 onely 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 shottest 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 equinoctiall point, to find his declination.

FIrst, lay the threed upon 23 deg. 30 min. the suns greatest declination, counted on the limbe of the quadrant, the threed lying still open at this angle, take the shortest extent thereunto from the sine of the distance of the Sunne from the next equinoctiall point, this distance being applied to the line of sines from the centre A, shall give you the sine of the suns declination.

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

## CHAP. IX. The declination of the sun, and the quarter of the ecliptick which he posseseth being given, to find his place.

TAke the sine of the suns declination from the line of sines, & 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 equinoctiall point.

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

## CHAP. X. Having the latitude of the place, and the distance of the sun from the next equinoctiall 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 equinoctiall 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 equinoctiall 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 finde 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 limbe, 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 deg. 9 min. & the angle at M being Right, we shall find the angle at N [Page 126] to be 37 deg. 30 min. the complement whereof is 52 deg. 30 min. 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 cosine of the latitude, and with the other lay the threed to the neerest distance: so shall it shew upon the limb the amplitude required: so in the figure of the next Chapter, the angle CNM being 37 deg. 30 min. the cosine of the latitude, and C M the declination here 20 deg. and the angle at M being right, we shall finde 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 cosine of the latitude, so shall you have the Sine of the suns declination betwixt your compasses.

So in this figure, the amplitude C N being. 34 deg. 9 min. and the angle at N being cosine to the latitude, the angle at M being a right angle, we shal find CM to be 20 deg. for the declination of the sun which was required.

## CHAP. XIIII. 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 Verticall 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 [Page 128] being a right angle we may find C I to be a sine of 25 deg. 32 min. 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 Equinoctiall Point, to find his height in the verticall circle.

FIrst, take the sine of the suns greatest declination, & 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 equinoctiall 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 finde the time when the Sun commeth 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 neerest extent [Page 129] 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 & not of AEC, therefore by the 4 Chapter, you may find the quantity of that unknown sine; for seeing the Radius O F is the cosine 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 limbe 16 deg. 30 min. which converted into time maketh 1 houre, and 6 min. for the quantitie of time betweene 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 houre of six.

FIrst, take the threed, and lay it upon the declination counted in the limbe; 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 houre of six.

So in the figure of the 13 Chapter, the angle at L being a right angle, and L O C being 52 deg. 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 equired.

## 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 cosine of the latitude, now with this distance setting one foot in the cosine of the altitude, with the other lay the threed to the neerest distance as before: so shall it shew upon the limbe, the azimuth of the sun from the East or West points.

So in the figure of the 13 chapter, the angle CLO being a right angle, and the angle L C O being 37 deg. 30 min. the cosine 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 finde 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 [Page 131] the neerest distance: then at the least distance from the cosine of the suns greatest declination take it over: now again, with this distance lay the threed to the neerest distance from the cosine of the declination given, so shall it shew upon the limbe 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. 30 min. 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 min. for the right ascension of the sun required.

## CHAP. XX. Having the latitude of the place, and the declination of the Sun, to finde the Ascensionall 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 limbe the Suns Ascensionall 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 finde M N 28 deg. 19 min. for the difference of Ascensions, which being converted [Page 132] into time, maketh 1 houre, and somthing better then 53 min.

Now when the Sun hath North declination, if you take this difference of Ascension (which is 1 houre 53 min.) out of 6 houres, 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 ascensionall difference to 6 hours, you shall have the time of his rising, and if you take it away from 6 hours, 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 finde the Azimuth.

IF the Suns declination be Northward, then by the 14 or 15 Chapters get his height in the Verticall 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 [Page 133] sine of 14 deg. 33 min. (the height of the Sun in the Verticall circle) and the sine of 30 deg. 45 min. the height of the Sun at the time of observation, and 52 deg. 30 min.

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 finde 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 [Page 134] Verticall circle is equall, he is directly in the East or West, so when his altitude given is greatest, then is the Azimuth toward the South, and when his altitude given is least, then is the Azimuth towards the North.

But if the declination of the Sun be Southward, then by the 10 or 12 Chapters, finde 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, 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 finde 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 finde the houre of the day.

IF the declination of the Sun be Northward, finde the height of the Sun at the houre 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 houre 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. 30 min. the Latitude of the place, and his complement M N O, we shall finde N O to be the sine of 60 deg. the quantity of time from the houre 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.

But if the declination of the Sun be Southward, finde his depression at the houre of six, by the 17 Chapter, for [Page 136] the day proposed, which will be equall 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 his 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 [Page 137] 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. 24 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 streight line, as T V, and having the angle T X V 52 deg. 20 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 houre of six, which converted into time will make three houres.

# THE FIFTH BOOK. Shewing how to describe the houre-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 Horizontall Diall.

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 levell line A C; which way soever you turn it, it is a horizontall plane. Or if you set the side A B of your Quadrant upon the upper side of your Ruler, so that the centre may hang a little over the end of your ruler, and holding up a threed and plummet, so that it may play upon [Page 139] the centre, if it shall fall directly upon his levell line A C, making no angle therewith, it is an horizontall plane, as here you may see by this figure.

## CHAP. II. Of the trying of planes, whether they be erect or inclining, and to finde 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 [Page 140] that side of the Quadrant shall be a Verticall line; and the line which crosseth this verticall line at right angles, will be the Horizontall line, as here you may see in this figure, the plane D E F G being erect, and the line D E being verticall, the line F G must be horizontall. But if the plane shall incline, the quantity of inclination may be found out after this manner.

First, you must draw thereon the horizontall 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 a horizontall line upon the upper 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 horizontall 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 [Page 141]

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 limbe, the inclination of the plane, which is the angle C A H, equall to the former angle.

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

## CHAP. III. To finde the Declination of a plane.

TO finde out this declination you must make two observations by the Sun: the first is of the angle made between the horizontall 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 horizontall distance, having drawn upon your plane a line parallel to the Horizon, apply the side of 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 centre of the Quadrant, observe the angle made upon the Quadrant by the shadow of the threed, and that side with the horizontall 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 finde the Suns Azimuth from the East or West points; by the 21 Chapter of the fourth Book.

Having thus gotten the horizontall distance, with the Azimuth of the Sun for the same time, describe a Circle as A B C D, representing the horizontall circle, and draw the diameter A C, which shall represent the horizontall line F G of the last Chapter. Now supposing the horizontall distance to be 38 deg. 30 min. the angle O A B of the last Chapter, place it from C Southward to E (that is from the same end of the horizontall line with which the angle was made upon the plane) and draw the line E Z: Then [Page 143] 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 forenoon: 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 Verticall Circle, so shall the angle made between the horizontall 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 min. 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 horizontall 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 Chapter upon the Quadrant.

Now having drawn your Horizontall Circle as before, and the diameter A C for the horizontall line of the plane, you may crosse it at right angles with the diameter B D, for the Axis of the planes horizontall line, from which as from D, you may set your horizontall 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 R Z H for the line of East and West, as before.

Or if you take the Suns Azimuth from the South, which 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 houre-lines upon the Horizontall, 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 Horizontall Diall, [Page 145] 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 E 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 houre-point in the limbe 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: this distance set from B to 1, and from B to 11, then again apply your threed to the next hour in the limbe 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 deg. open the threed as before, and take it over from the [Page 146] 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 houre-points pricked down; by which and the centre 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.

For the hour-lines before six and after, you may extend their opposite houre lines beyond the centre 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 limbe.

## CHAP. V. To draw a Diall 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 finde 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 [Page 148] 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 houres and where it cutteth; which will be the slope 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 Diall: 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 [Page 149] it over 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 Chapter 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 houre-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 fame 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 F P, the threed will cut the co-tangent 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 centre A, you may draw the houre-lines, as here you see.

In the like manner may the half hours be supplyed.

The Diall being thus drawn on paper, you must place it so upon the plane, that the twelve a clock houre-line may [Page 150] 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-tangent 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 Diall 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 houre-lines upon the plane.

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 min. if a plane shall incline 40 deg. 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 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,

[Page 152] 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 houre-lines, and so accordingly rectifie the bead as before was shewed.

And you may also see where the threed cutteth the cotangent 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 Chap 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 Diall upon the plane.

## CHAP. VII. To draw a Diall 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 Diall 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.
- [Page 153]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 finde 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 finde 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 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 finde 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 finde 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. 30 min. First, I finde the inclination of the plane to the Meridian to be 64 deg. 20 min. Then I finde the Meridians ascension to be 33 deg. 41 min. In like manner I finde 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 deg. 20 min. and the inclination of Meridians 30 deg. 52 min.

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

The Diall 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.

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

## CHAP. VIII. In any erect declining Diall, 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, and make it anew, for though it be gone, the substile where it stood will remain.

First, therefore get the quantity of the angle of deflexiō 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 Diall 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 Diall at your pleasure.

To finde 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 finde the declination of the former plane to be 32 deg.

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 Diall whatsoever, by Rule and Compasse only.

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 Diall.

Also how to draw a Diall on the seeling of a Room.

By W. L.

LONDON, Printed by R. & W. Leybourn, for Thomas Pierrepont, at the Sun in Paul's Church-yard: 1652.

# THE APPENDIX.

## CHAP. I. How to describe the Equinoctiall, Tropicks; and other parallels of the Suns course or Declination, in all kinde of planes.

ALL Circles of the Sphere whether great or small, that may be projected upon any Diall-plane, 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 160] all small Circles, viz. such as divide the Sphere unequally, are Conick Sections, namely, either Elypses, 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 Horizontall plane, are perfect Circles. For the Equinoctiall Diall lying in the very plain of the Equinoctiall Circle is parallel to all the parallels of declination; as the Horizontall Diall 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 Astronomicall conclusions that are to be drawn upon any Diall plane, are limited either by the Equinoctiall, or by one or both of the Tropicks: therefore it is requisite; first, to shew you how to describe the Equinoctiall and the Tropicks upon all kinde of planes, because it is them that limit and confine all other intermediate parallels, whether they be of the Suns entrance into the Signes, or the Diurnall 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 Diall you are taught before, therefore let the Square A B C D be a plane, on which there is an East Diall drawn, the height of the stile being equall to the distance between the houres of 9 and 6, [Page 161] noted there with the letters E G, and let it be required to draw upon the same plane the Equinoctiall and the two Tropicks. Now the Equinoctiall being a great circle of the Sphere, it is therefore a streight line, and is represented in the Diall following by the line H F. The houre-lines and the Equinoctiall being thus drawn, we may proceed to the rest of the work in this manner.

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 deg. 31 min. distant frō the Equinoctiall, on either side thereof, therefore on the arch R S set 23 deg. 31 min. from R to S, and draw the line O S, then shall the line O R represent the Equinoctiall, 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 recours to your Diall, and from thence with your Compasses you must first take [Page 162] out the distance E G (equall to the height of the stile of the same Diall) 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 Equinoctiall 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 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 finde upon the houre-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 hour-line of six. Secondly, take out of your Trigon the distance q 7, & 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 e. Fourthly, take out of your Trigon the distane s 9, and set it on your plane from 9 to f. Fifthly, take from your Trigon [Page 163] 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 severall 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 [Page 164] Diall will run directly along that line a b c d e f g h, and when the Sun is in the Equinoctiall, 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 wil run along the Equinoctiall line E F.

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 Equinoctiall: therefore the distance 8 k being made equall to the distance 8 e, and the distance 9 l equall to 9 f; and the distance 10 m equall 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 Equinoctiall 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 finde 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 parallels of the Suns entrance into the 12 signes of the Zodiaque: You must, first, finde what declination the Sun hath when he enters any of those Signes, which this [Page 165] 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 degrees 30 minutes, and when he is in the beginning of Gemini, Leo, Sagitarius or Aquarius, his declination is 20 degrees 12 minutes.

North Decli. | D | M | South Decli. | ||||

Aries | 00 | 00 | Libra | ||||

Taurus | Virgo | 11 | 30 | Scorpio | Pisces | ||

Gemini | Leo | 20 | 12 | Sagitarius | Aquarius | ||

Cancer | 23 | 31 | Capricorn |

Therefore take 11 deg. 30 min. in your Compasses, and place it in your Trigon from R unto V, and draw the line 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, Sagitarius and Aquarius.

These parallels being placed in your Trigon according to their true declination from the Equinoctiall, they are to be transferred 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 severall points where the hours [Page 166] crosse the parallel, and place the same distances upon your plane from the Aequinoctiall upon the respective hour-lines, from which they were taken out 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 kinde 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 streight piece of wyer, equall in length to the line E G, fixed in the point E, standing 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 Astronomicall conclusions are shewed (not by the shadow of the whole [Page 167] 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 MEN in the former East Diall is called the Horizontall 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 sheweth, that when the Sun is in the Tropick of Cancer, he riseth somewhat before four in the morning, in like manner the Tropick of Capricorn cutteth the Horizontall 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 Horizontal, full South or Norrh planes, which are the next in order.

### §. 2. In the North, South, and Horizontall Dials.

IN all these planes the substiler and the Meridian are all one, and the height of the stile, in the Horizontall Diall is always equall to the latitude of the place, and you are to [Page 168] take notice, that whatsoever is here said of the full North & South upright planes, the same is to be understood of the ful North & South reclining or inclining, all which in those latitudes whose complement is equall to the height of the stile they are erect direct planes, & in those latitudes which are equall to the height of the stile above such reclining planes, they are horizontal planes. One example therefore in one plane will be sufficient for all the rest. Therefore, in the latitude of 52 deg. 30 min. Let it be required to describe the Equinoctiall, 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 Gnodus or knot which must give the shadow to the Tropicks and other parallels of declination, for all these Astronomicall 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 Diall following is the point S, and the triangular stile in that Diall 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 Diall being drawn, and the Triangle C S L made equall to the Cock of the Diall, you must upon a piece of pastboard draw the Triangle O P R equall to the stile in your Diall C S L, making R O equall to C L the substilar, P O equall to C S the axis of the stile, and P R equall to S L the length of the perpendicular stile.

Then from the point P, raise a perpendicular as P B, representing the Equinoctiall, and on P as a center, describe [Page 169] 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 to 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 in North or South erect direct planes, I told you was alwayes the same with the twelve aclock line) from O to 12, cutting the Equinoctiall 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 Equinoctiall 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 houre of 1 or 11 in your Diall. 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 houre-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 houres 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 houres of 4 and 8. And thus must you doe with the rest of the houres 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 houre lines of your plane, from [Page 170] 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 houre-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 horizontall 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 [Page 171] to the Meridian or line of 12: And in all planes whatsoever, this line must be drawn through the intersection of the Equinoctiall with the houre of six. This line ought first to be drawn, because it is very improper to extend the Tropicks or other parallels of declination, above the Horizontall line, because at what houre 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.

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 distances from O, where the severall houre-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 houre-lines, and through those points so found, draw the line ♑ ♑, representing the Tropick of Capricorn.

¶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 universall in all planes, observing this one exception; that whereas in all erect direct planes the Equinoctiall is drawn perpendicular to the Meridian or line of 12, so in all other planes whatsoever, the Equinoctioll 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 Equinoctiall upon such a plane.

The Diall being drawn, with the stile and substile, make [Page 173] 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 fall a perpendicular to the substile, as C B, and through the point B, draw the horizontall line D E perpendicular to the line of 12 a clock, then shall the Triangle A B C represent the stile of the Diall. Then provide a Trigon, as this figure sheweth, making the Triangle F G H equall to the cock of your Diall, viz. F H equall

[Page 174] to the Axis of the stile, G H equall to the substilar, and F G equall to the perpendicular of the stile, extending the line G 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 if you take with your compasses the distance from A the center of your Diall, to the severall points where the houre-lines crosse the Equinoctiall, 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 houre from whence the distances were taken in the plane at the end 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 houre-lines with the Tropicks, and transfer those distances to your plane again upon the correspondent houre-lines, 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 Equinoctiall with the houre of fix.

## CHAP. II. Shewing how to inscribe the 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 [Page 176] 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 declination 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 houres, and take the difference, then the proportion will be;

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

- As the Sine of 90 deg. 10,000000
- Is to the Sine of 30 deg. which is halfe the difference 9,698970
- So the Tangent complement of the Latitude 37 deg. 30 min. 9,884980
- To the Tangent of the Declination of the sun. 20 deg. 59 m. 9,583950

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

Now if the day be above 12 houres long, the Sun hath [Page 177] North declination, but if lesse then 12 houres long he hath South declination. For those who are ignorant of these kinde of prportions, they had best to read Mr. Norwoods Doctrine of Triangles. But that nothing might be wanting, and not much to trouble the learner, 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 houres long, in the Latitude of 52 deg. 30 min. which Table was made by the preceding rules.

By which table you may see

Length of the day. | The Suns Declinatiō | |

D | M | |

8 | 20 | 59 |

9 | 16 | 22 |

10 | 11 | 14 |

11 | 5 | 43 |

12 | 0 | 0 |

13 | 5 | 43 |

14 | 11 | 14 |

15 | 16 | 22 |

16 | 20 | 59 |

that when the day is 12 houres long the Sun hath then no declination, but is in the Equinoctiall; but when the day is either 11 or 13 houres long, the declination is then 5 deg. 43 min. and when the day is either 9 or 15 houres 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 forementioned planes; you must insert these angles of declination into 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 Horizontall 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 [Page 178] houres long; upon a full South plane in the Latitude of 52 deg. 30 min.

Having drawn your Diall with Houres halves and quarters, and also made choise of some convenient point in the stile to give the shadow, and draw the horizontall line C D, then make the triangle S A R in this Trigon equall to the triangle S A R in the following South Diall; as S A equall to the Axis of the stile, A R equall to the substilar, and R S equall to the perpendicular stile: then draw the perpendiculars S G for the Equinoctiall, 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.

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 houres long, the declination is 20 deg. 59 min. therefore place in your Trigon 20 deg. 59 min. 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 houres 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 S b. Also when the day is either 10 or 14 houres 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 houres 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 houres long it is Equinoctiall [Page 179] 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 [Page 180] forbeare 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 Diurnall arches in all kinde of planes. I will now proceed to shew you how some other Astronomicall conclusions (which are very pleasing and delightfull) may be inscribed upon all sorts of Dials.

## CHAP. III. Shewing how the Italian and Babylonish houres may be drawn upon all kinde of planes.

THe Italians account their houres from the Suns setting, and the Babylonians from his rising, so that these kinde of houre-lines being drawn upon any plane, you may know (by inspection only) how many houres are past since the last setting or rising of the Sun. The inscription of these houre-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 Horizontall plane, these houre-lines shew themselves most uniform, I have therefore for example sake, made choice of a full South Diall, upon which it shall be shewn how to draw both the Italian and Babylonish houres.

Your Diall being drawn, and the two Tropicks and the Equinoctial thereon inscribed, and also the Horizontall line, you must draw in your Diall two obscure parallels of the length of the day, one when the day is 8 houres, and the other when the day is 16 houres 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 houres long, and the undermost is the parallel of his course when the day is 16 houres [Page 182] long, & the Aequinoctiall is the parallel of the Suns course when the day is 12 houres long.

Your Diall being thus prepared, and these parallels thus inserted, the inscription of these houre-lines is very easie and plain to be understood. To begin then with the inscription of the Babylonish houres (which are the houres from the Suns rising.) First, It is apparent that when the day is 8 houres long, that the Sun riseth at 8 in the morning, so that at that time, the first houre 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 houres long, the Sun riseth at 4 in the morning, so that the first houre after his rising is 5 in the morning, as plainly appeareth by this Table:

Length of the Day. | ||||

8 | 12 | 16 | ||

Hours from Sun rising. | 1 | 9 | 7 | 5 |

2 | 10 | 8 | 6 | |

3 | 11 | 9 | 7 | |

4 | 12 | 10 | 8 | |

5 | 1 | 11 | 9 | |

6 | 2 | 12 | 10 | |

7 | 3 | 1 | 11 | |

8 | 4 | 2 | 12 | |

9 | 5 | 3 | 1 | |

10 | 6 | 4 | 2 | |

11 | 7 | 5 | 3 |

By which you may perceive that when the day is 8 houres long, the seventh houre from sun rising is 3 in the afternoon. When the day is 12 houres long, the seventh houre from sun rising is 1 in the afternoon. And when the day is 16 houres long, the seventh houre from the suns rising is 11 before noon, as by this Table doth evidently appear. And therefore a streight line drawn in your Diall through those points where the common houre-lines [Page 183] of your Diall crosse the respective parallels of the dayes length, shall shew the true quantity of houres since the suns rising at all times of the yeare, which is the Babylonish houre.

For example, let it be required to draw the seventh hour from the suns rising in your Diall. First, by the Table you see, that in the parallel of 8 houres for the length of [Page 184] the day, the seventh houre from the suns rising is 3 in the afternoon, therefore observe where the houre-line of three crosseth the parallel of 8 houres, 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 houre from sun rising is then 1 in the afternoon, wherefore observe where the houre-line of 1 crosseth the Equinoctiall, which is at b. Thirdly, by the Table you see that in the parallel of 16 houres for the length of the day, the seveth houre from the suns rising is 11 before noon, therefore observe where the houre-line of 11 crosseth the parallel of 16 hours, which is at c: then draw the streight line a b c, which shall be the seventh Babylonish houre, 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 houres from sun rising, as you see them drawn in the figure, and put numbers to them as you see there done.

¶1 Note, That if any of the points you are to make use of for the drawing of any of these houres fall without your plane, you must in this case extend your houre-line; parallel and Equinoctiall, beyond the limits of your Diall-plane, 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 houre from Sun rising crosseth not the Equinoctiall and the houre-line of 7 within the plane, but if the Equinoctiall and the houre-line of 7 were extended, it would crosse.

[Page 185]¶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 houre-lines so far that then in that case any two of the three points will sufficiently serve the turn.

¶3 Note, That as the houres from sun rising were put into the plane, by the same rule may the hours from sun setting (or Italian houres) be inserted, the difference being only in the numbring of them; the houres from the sun rising being numbered from the West side of the Horizontall line by 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11; and the houres from the suns setting are denominated from the East side of the Horizon, and numbered 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 [Page 186] night at the suns setting; so that 12 of the clock at noon was alwayes the sixth houre 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 houres of their day were longer then the houres of their night; and all the Winter, the houres of their night were longer then those of their day; and when the Sun is in the Equinoctiall, then the houres of their day and night were equall, 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 houres before taught.

Having therefore drawn your Diall (which in this example (for the avoyding of many figures) we will have to be the full South plane used before in Chap. 2. of this Appendix) with the houres, 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 Equinoctiall, which let be the parallels of 9 houres and 15 houres, both which are three houres different from the Equinoctiall on either side thereof, and these two parallels are the most convenient for this our purpose, because the Jewish hours will fall (in these two parallels) justly upon the houres, halves and quarters of the common houre-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 houre must be drawn through 5 houres 45 min. (or 5 houres three quarters) in the parallel of 15: through 7 [Page 187] hours in the Equinoctial:

Jewish hours. | The parallel of 15 hours. | Equinoctiall. | The parallel of 9 hours. | ||

H. | M. | H. | H. | M. | |

1 | 5 | 45 | 7 | 8 | 15 |

2 | 7 | 0 | 8 | 9 | 0 |

3 | 8 | 15 | 9 | 9 | 45 |

4 | 9 | 30 | 10 | 10 | 30 |

5 | 10 | 45 | 11 | 11 | 15 |

6 | 12 | 0 | 12 | 12 | 0 |

7 | 1 | 15 | 1 | 0 | 45 |

8 | 2 | 30 | 2 | 1 | 30 |

9 | 3 | 45 | 3 | 2 | 15 |

10 | 5 | 0 | 4 | 3 | 0 |

11 | 6 | 15 | 5 | 3 | 45 |

12 | 7 | 30 | 6 | 4 | 30 |

& through 8 houres & a quarter in the parallel of 9 hours.

In like manner, the second Jewish houre must bee drawn in your plane through 7 of the clock in the parallel of 15: through 8 a clock in the Equinoctiall: and through 9 of the clock in the parallel of 9 houres, and so of all the rest, according as you see in this 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 houre-line.

## CHAP. V. Shewing how to draw the Azimuths, or Verticall 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. [Page 189] In the Horizontall plane they meet in a center with equall 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 streight lines in all planes, and may be drawn as followeth.

### §. 1. In the Horizontall plane.

IN the Horizontall plane these Azimuths are most easily inserted, for your Diall 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 32 equall parts (beginning at the Meridian) answering to the 32 points of the Mariners Compasse: Or else you may divide the same Circle into 90 equall parts, according to the Astronomicall division, and through each of those points draw streight 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 to the Mariners Compasse. This is so plain that it needeth no example.

### §. 2. In the East or West erect planes.

YOur Diall being finished, you may draw upon a piece of Pastboard the line M E N, representing the Horizontall line M E N in your Dial then on the point E, raise the perpendicular E Q equall to the line EG in your Diall, and on Q as a center describe the semicircle K E L, and divide one halfe thereof, namely, the Quadrant E L into [Page 190] eight equall 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 Horizontall line of your plane from E to ☉ ☉ ☉ ☉ ☉ ☉, from which points draw lines perpendicular to the horizontall line M E N, which shall be the Azimuths or points of the Compass between the East and the South. Divide likewise the other

Quadrant of the Circle E K into eight equall parts, and draw lines from the center Q through three of them, till they cut the horizontall 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 Diall sheweth all the morning houres from sun rising to the Meridian; and the West Diall sheweth all the afternoon hours from the Meridian to his setting; so doth the East Diall shew all the Azimuths from the Suns rising [Page 191] till Noon, and the West Diall all the Azimuths from noon till his setting.

### §. 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 [Page 192] upon the full South Diall: the Tropicks and the Equinoctiall being drawn, together with the Horizontall line, you must upon a piece of Pastboard draw the line A L B, representing the Horizontall line A L B in the South Diall next following; then on the point L raise the perpendicular L S, making L S equall to L S the perpendicular stile of the Diall, 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 equall 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 horizontall 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 horizontall line thereof from L unto p both wayes. Also take the distances L o, L n, and L m, and set them upon the horizontall line in your Diall from L to o, and n, and m, on each side of the Meridian. Lastly, if from the points m, n, o, p, & 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 Astronomicall account by 10, 20, 30, 40, &c. or else by the points [Page 193]

of the Compasse, as in this figure, according as you shall divide the Semicircle E L F: And thus much concerning erect direct planes.

### §. 4. In erect declining planes.

IN upright declining planes the Azimuths are easily inscribed, little differing from the former. Draw therefore your Diall, which we will suppose to be the South [Page 194] declining plane before used in the third Section of the first Chapter of this Appendix, which declineth from the South Eastward 32 deg.

Your Diall being drawn and the Equinoctial, and Tropicks, and also the Horizontall line, thereon inscribed, upon a piece of pastboard draw the line D B E, representing the horizontall line D B E in your Diall, then on the point B raise the perpendicular B C, making B C equall to B C the perpendicular stile in your Diall; then on the point C as a center describe the Semicircle R B S, then out

of your plane take the distance between B the foot of the stile, and the point O, where the Meridian crosseth the horizontall 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 equall 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 on 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 [Page 195]

plane, viz. the Western Azimuths on the East side of the Meridian, and the Eastern Azimuths on the West side of your Diall, as you see them here numbered in this figure.

### §. 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 oblikely, making oblike angles therewith, [Page 196] 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. 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 horizontall point, through which the horizontall 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 horizontall line, and thereon make marks, then lay a ruler upon the Zenith point and each of these marks in the horizontall 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 of clock line and the substilar are severall 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 horizontall 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 the Azimuths, as the Tropicks and parallels of declination have to the houre-lines, and therefore, as the parallels of declination in the Equinoctiall plane are perfect circles, so are the circles of Altitude in an Horizontall 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 take the houre-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 transfer 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 Horizontall.

## CHAP. VII. How to draw a Diall 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 [Page 198] 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 amongst so many wayes as there are to perform it, I shall make choice of that which I suppose to

[Page 199] be most familiar and easie. Draw therefore upon paper or otherwise an horizontall Diall for the Latitude in which you are, as is the Horizontall Diall foregoing 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 houre-lines in the points a b c d and e, then with your Compasses you may measure the quantity of each houres distance from the Meridian, by taking the distance from C to a b c d and e, so shall you finde the distance between the Meridian and 11 or 1 to be 12 deg. Likewise the distance between the Meridian and 10 or 2 to be 24 deg. 37 min. and the distance between the Meridian 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.

The Hours. | The angle that each houre-line makes with the Merid. | The complement of each houre-lines angle with the Meridiā | ||||

12 | 00d. | 00m. | 90d. | 00m. | ||

1 | 11 | 12 | 00 | 78 | 00 | |

2 | 10 | 24 | 37 | 65 | 23 | |

3 | 9 | 38 | 25 | 51 | 35 | |

4 | 8 | 53 | 58 | 36 | 2 | |

5 | 7 | 71 | 20 | 18 | 40 | |

6 | 90 | 00 | 00 | 00 |

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 transam of the window, 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 finde 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 Equinoctiall.

Having thus done, upon a table or such like draw a line which shall be of equall length with LK, the distance from the glasse to the point L on the seeling, which line divide into 10 equall 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 naturall tangent of 15 deg.) and place them upon the Equinoctiall line from L to M. Then take 577 the naturall 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 houres of 1 2 3 and 4 must passe, and the same work being done on the other side of the Meridian, you [Page 201]

shall finde points through which the houes of 11 10 9 and 8 shall passe: the houres of 5 in the morning and 7 [Page 202] at night wil seldome fall upon the plane except they be supplyed from East and West windows.

Now because the center of the Diall is without the Roome, so that you cannot make use of that to draw the hours by, you must therefore place one foot of your compasses in the points 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 houre arches from the Equinoctiall to the points * * * * * as you see in the figure. Lastly, if you draw the lines *M, *N, *P, *Q, they shall be the true houres upon the seeling.

In the inscription of the Azimuths in declining reclining planes, and in drawing the circles of Altitude in all kinde of planes, I confesse I should have been somwhat 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 that which goes before, he cannot but with small pains overcome the rest; But I should not have been so briefe, could I possibly have procured more time, which by no means would be granted.

Also my intent in this place was to have shewn you the inscription of the Circles of position and other varieties: Also the framing of divers Geometricall Bodies, and to furnish them with variety of Dials, and the making [Page 203] of divers Instrumentall Dials: But these, with many other HOROLOGICAL conceits and inventions, I reserve till a more convenient opportunity, and therefore in the mean time, Farewell.