Dialling diagram HOROMETRIA: Or the Complete DIALLIST: Wherein the whole mystery of the Art of DIALLING is plainly taught three several ways; two of which are performed Geometrically by Rule and Compass only: And the third Instrumentally, by a QUADRANT fitted for that purpose. With the working of such Propositions of the Sphere, as are most useful in ASTRONOMY and NAVIGATION, both Geometrically and Instrumentally. By Thomas Stirrup, Philomath. Whereunto is added an APPENDIX, showing how the parallels of Declination; the Jewish, Babylonish, and Italian hours; the Azimuths, Almicanters, etc. may be easily inscribed on any Dial whatsoever, by Rule and Compass only. Also how to draw a Dial on the ceiling of a room, By W. L. LONDON, Printed by R. & W. Leybourn, for Thomas Pierrepont, at the Sun in Paul's Churchyard. 1652. TO THE READER: Courteous Reader, HEre is presented unto thee a short Treatise of the Art of Dialling. Concerning the antiquity, excellency, and necessity thereof, I shall in this place say little, the Antiquity thereof being well known to all who have diligently read the Sacred Scriptures, wherein mention is made of that of K. Ahaz, upon which the Almighty was pleased to express a Miracle for the recovery of King Hezekiah, by causing the Sun to go back ten degrees upon the said Dial, and this is the first that was ever recorded, being above 2400 years ago, since which time, Learning spreading itself over the whole Universe, hath made this Art more common. For the excellency of it, the skill which is required in the Mathematics, especially in Geometry, Astronomy, and Optics, for the making a man complete and excellent, is an evident proof, for without good knowledge in the Elements of Geometry, with a competency of knowledge in the circles of the Sphere, and some insight in the Optics, 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 business with another, and not prefix a time, without the loss of that which cannot be regained, and aught 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 fetched 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 itself so excellent and necessary, may induce any industrious person to the practice thereof, the perfect knowledge whereof in this ensuing Treatise is sufficiently taught, and that by such brief, easy, and familiar ways, that not any Treatise hitherto published, can for convenience, ease, and quick dispatch, be compared thereto. The whole Treatise consisteth of five Books, in which the whole mystery of the Art of Dialling is plainly taught three several ways, 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 showeth how to perform most propositions in Astronomy and Navigation, Geometrically, with Scale and Compass only. III. The third showeth how to find 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 situate, by Rule and Compass only, two several ways, in both which, the two grand inconveniencies of the common ways (viz. of outrunning the limits of the plane; and drawing of many unnecessary lines) are totally avoided, you having no lines to draw, but such as will be comprised within the bounds of your plane, and those so few, that you need not fear confusion. IU. & V The fourth, and fifth, sheweth the construction and use of a Quadrant, by which all the most usual Propositions in Astronomy may be wrought with great facility, and by which the Inclination and Declination of a plane may be speedily attained, and also the houre-lines drawn upon all kind of planes in any Latitude. Unto these five Books is added an Appendix, showing how to furnish any kind of Dial with Astronomical Variety, as to draw thereon the parallels of Declination, by which the place of the Sun may be known: the parallels of the length of the day, by which the day of the Month, the Suns Rising and Setting, the length of the day and night may be known: the Babylonish and Italian hours, by which you may know how many hours are passed since Sun rising, and how many remain to Sun setting: The old unequal hours, by which the day is divided into twelve equal 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 Dial on the Ceiling of a Room by reflection: All which are performed Geometrically by Rule and Compass 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 Astronomical 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 find the sun's altitude page 26 To find the length of Right and Contrary shadow page 28 To find the sun's declination page 30 How to find the sun's place page 32 To find the sun's Amplitude page 33 How to find the height of the Pole page 34 How to find the sun's Amplitude page 35 How to find the sun's Declination page 37 To know at what time the sun shall be East or West page 39 How to find the height of the sun at the hour of six page 40 To find the Azimuth at the hour of six page 41 To find the Azimuth page 42 How to find the hour of the day page 44 How to find the ascensional difference page 49 How to find the Right or Obliqne Ascension page 50 How to find the sun's altitude without Instrument page 52 How to find the Latitude of a place page 54 How to find the declination, and inclination of any plane page 59 How to draw a Meridian line upon an Horizontal plane. page 64 To make an Equinoctial Dial page 65 How to draw a Dial upon a Polar plane page 66 How to make an East or West Dial page 68 How to draw a Dial upon an Horizontal plane page 71 How to draw a Dial upon a full North or South plane page 74 To draw a Dial upon a Vertical inclining plane page 79 To draw a Dial upon a North or South declining plane page 81 Another way to draw an horizontal Dial page 87 Another way to draw a full North or South Dial page 89 In an upright declining plane, to find the deflexion, the height of the stile, and the inclination of Meridian's page 92 To draw a Dial upon an upright declining plane page 94 To draw a Dial up a Meridian inclining plane page 101 In declining inclining planes, to find the height of the stile, the deflexion, etc. and to draw the Dial 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 Astronomy by the Quadrant page 124 To find the inclination and declination of a plane page 139 How to draw Dial's on all kind of planes by the Quadrant. page 144 If the Cock of a Dial be lost, to find the height thereof page 156 How to describe the Equinoctial, Tropics, and other parallels of the sun's course and declination in all kind 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 unequal hours may be drawn upon any plane page 185 How to draw the Azimuths and Almicanters on all kind of planes page 188 How to draw a dial on the ceiling of a Room. page 197 Courteous Reader, These Books are to be sold by Thomas Pierrepont at the Sun in Paul's Church yard. Dr. Twisse learned Treatise in defence of the Sabath day. Dr. Stoughton 17 choice 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 unboweled, by Henry Church. Dr. Ames Exposition of both the Epistles of Peter. The Works of Mr. Nicholas Lokyer in three Treatises. The spiritual man's directory by Mr. Fennor. Useful instructions for evil times, by Mr. Nicholas Lokyer. Natural Philosophy reform, by I. A Comenius. Artificers plain Scale, by Mr. Thomas Stirrup. Dr. Sibbs Christian Charter. Certain select cases resolved by Mr. Hen. Shepard. A vindication of Ordination and laying on of hands, by Mr. Lazarus Seaman. ALL the work of this Book is performed either Geometrically or Instrumentally: For the Geometrical performance there is required only a line of Chords (or rather a Sector,) and the Instrumental part is performed by a Quadrant fitted for that purpose: Therefore, if any be desirous to have either Scale, Sector, Quadrant, or any other Mathematical Instrument whatsoever, they may be furnished by Master Anthony Thompson in Hosier lane near Smithfield. NOte, that the line of Chords which is drawn on the edge of the Quadrant A C should issue from the centre, but in the figure it is drawn short thereof, which defect the Instrument maker will easily supply. THE FIRST BOOK. Showing the meaning of some of the usefullest terms 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 intent to practice either in the Art of NAVIGATION, ASTRONOMY, OR DIALLING. CHAP. I. Of certain terms of Geometry, necessary to be known of the unlearned, before the proceeding in this Art of Dialling. BEing intended in this Treatise of Dialling, to proceed by Geometrical proportion: I have thought fit, first, to declare unto you the meaning of some terms of Geometry which are necessary for the unlearned to know before they enter into this Art of Dialling. Definition 1. Point A 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 itself in length, without breadth or thickness, 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. Line B Definition 3. An Angle is the meeting of two lines in any sort, so as they both make not one line. Angle CDE Angle CDE 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 always the angle intended. Definition 4. If a right line fall on a right line, making the angles on either side equal, each of those angles are called right angles, and the line Perpendicular lines AB and CBD illustrating right angles 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 equal, as here you see, and therefore are called Right angles. Definition 5. A Perpendicular is a line raised from, or let fall upon another line, making equal angles on both sides, as you may see declared in the former figure, where the line A B is perpendicular unto the line C B D, making equal angles in the point B. Definition 6. A Circle is a plain figure, and contained under one line which is called the Circumference thereof, as in the figure following, the very Ring C B D E is called the Circumference of that Circle. Definition 7. The Centre of a Circle is that point which is in the midst thereof, from which point, all right lines drawn to the Circumference are equal, as you may see in the following figure, where the point by the letter A represents the Centre, 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 centre of any Circle, in such sort that it may divide the whole Circle into two equal parts, as you may see the line C A D, or B A E, either of which is the Diameter of the circle B C E D, because either of them passeth through the centre A, & divideth the whole circle into two equal parts. Definition 9 The Semidiameter of a Circle, is half of the Diameter, and is contained betwixt the centre, 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 half 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 half 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 centre thereof, dividing the Semicircle into two equal parts, of the which parts, the one is the Quadrant or fourth part of the same Circle. As for example, the Diameter of the Circle B D E C, is the line C A D, dividing the Circle into two equal parts: then from the centre A, raise the perpendicular A B, dividing the Semicircle likewise into two equal parts, so is A B D, or A B C, the Quadrant of the Circle C B D E. Definition 12. A Segment or portion of a Circle, is a figure contained under a right Circle CBDE with segment FBGH line, and a part of a circumference, either greater or lesser than the semicircle, as in the figure you may see that F B G H is a segment or part of the Circle C B D E, and is contained under the right line FHG (which is less than 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 (less than the Diameter) drawn through, and applied 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. Parallel lines A and B Definition 14. Circular parallel: circle FGHI within circle BCDE having the same center A A Circular parallel is a Circle drawn either within or without another Circle upon the same centre, as you may plainly see by the two Circles B C D E, and F G H I, these Circles are both drawn upon the same centre A, and therefore are parallel the one to the other. Definition 16. A Degree is the 360th part of the circumference of any Circle, so that divide the circumference of any circle into 360 parts, and each of those parts is called a degree; so shall the Semicircumference contain 180 of those Degrees; and 90 of those degrees make a Quadrant, or a quarter of the circumference of any Circle. Definition 16. A Minute is the 60th part of a degree, being understood of measure: but in time a Minute is the 60th part of an hour, or the fourth part of a degree, 15 degrees answering to an hour, and 4 minutes to a degree. Definition 17. The quantity or 60-degree angle ABC and 60-degree arc (arch) AC 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 between 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 Compliment of an arch less than 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 compliment thereof, which is 30 degrees. depiction of complementary arcs (arches): semicircle with E as center and A, B, C, and D as points on the circumference, showing 60-degree angle AEB (arc AB), 30-degree angle BEC (arc BC), and 90-degree angle CED (arc CD) Definition 19 The compliment of an arch less than 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 compliment thereof, which will be found to be 60 degrees. And here note, that what is said of the compliments of arches, the same is meant by the compliments 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 Dialling diagram: how to draw a parallel line by first drawing arch (arc) lines with a compass 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. Dialling diagram: how to draw a parallel line at a specific distance and opposite a specific point First therefore, place one foot of the compasses in the point F, from whence take the shortest extension to the 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. FOUR 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, Dialling diagram: how to draw a perpendicular line CD from the middle of a given line ACB & setting one foot in the point C, and with the other foot mark on either side thereof, the equal distances C A, and C B: then opening your compasses to any convenient wider distance, with one foot in the point A, with the other, strike the arch line E over the point C, 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 Dialling diagram: how to draw a perpendicular line down from a given point C to the middle of an existing line DEF fall upon the given line D E F. First, set one foot of your compasses in the point C, and opening your compasses to any convenient distance, so that it be more than the distance C E, make an arch of a circle with the other foot, so that it may cut the line D E F twice, that is, at I and G: then find the middle between those two 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 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 Dialling diagram: how to draw a perpendicular line from the end of a given line using compass and ruler B, with the other draw the arch ED, than 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, still keeping the same distance, set one foot in the point D, and with the other draw the arch line F over the given point B: now laying a ruler upon the two points E and D, see where it crosseth the arch line F, which will be at F, from which point F, draw the line F B, which shall be a perpendicular line unto the given line B C, raised from the end B, as was required. CHAP. VII. To let a perpendicular fall from a point assigned, unto the end of a line given. LEt the line D E be given, unto which it is required to let a perpendicular fall from the assigned point A, unto the end D. First, from the assigned point A, draw a line unto any part of the given line D E, which may be the line A B C, then find the middle of the line A C, which will be at B, place therefore one foot of your compasses in the point B, and extend the other unto A or C, with which distance draw the Dialling diagram: how to draw a perpendicular line down from a given point to the end of a line Semicircle A D C, so shall it cut the given line D E in the point D, from which point D, draw the line A D, which shall be the perpendicular let fall from the assigned point A unto the end D of the given line D E, as was required. CHAP. VIII. Certain Definitions Astronomical, meet to be understood of the unlearned, before the proceeding in this Art of Dialling. IN the former Chapter I have showed the meaning of some terms of Geometry, which be most helpful unto this art of Dialling, with the drawing of a parallel line at any distance, or by a point assigned; so likewise have I shown 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 show unto the unlearned, the meaning of some of the most usefullest terms in Astronomy, and most fitting this art of Dialling. Definition 1. A Sphere is a certain solid superficies, in whose middle is a point, from which all lines drawn unto the circumference are equal, which point is the centre of the Sphere. Definition 2. The Pole is a prick or point imagined in the heavens, whereof are two, the North pole being the centre 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 pass through the centre of the Sphere, and passing directly to the opposite part of the heavens, showeth there to be the South Pole, and this line so imagined to pass from one Pole to the other, through the centre of the Sphere, is called the Axletree of the World, because it hath been formerly supposed, that the Sun, Moon, and Stars, together with the whole heavens hath been turned about from East to West, once round in 24 hours, by a true equal course, like much in like time; which diurnal revolution is performed about this Axletree of the World, and this Axletree is set out unto you in the following figure by the line P A D, the Poles whereof are P and D. Definition 3. A Sphere accidentally is divided into two parts, that is to say, into a right Sphere, and an obliqne Sphere, a right Sphere is only unto those that dwell under the Equinoctial, to whom neither of the Poles of the World are seen, but lie hid in the Horizon. An obliqne Sphere is unto those that inhabit on either side of the Equinoctial, 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 equal parts, and they are in number six, viz. the Equinoctial, the Ecliptic line, the two Colours, the Meridian, and the Horizon. The lesser Circles are such as divide the Sphere into two parts unequally; and they are four in number, as, the Tropic of Cancer, the Tropic of Capricorn, the Circle Arctic, and the Circle Antartique. CHAP. IX. Of the six greater Circles. Definition 5. THe Equinoctial is a Circle that crosseth the Poles of the world at right angles, and divideth the Sphere into two equal parts, and is called the Equinoctial, because when the Sun cometh unto it (which is twice in the year, viz. at the Sun's entrance into Aries and Libra) it maketh the days & nights of equal length throughout the whole world, and in the figure following, is described by the line S A N. Definition 6. The Meridian is a great Circle, passing through the Poles of the world, and the poles of the Horizon, or Zenith point right over our heads, and is so called, because that in any time of the year, or in any place of the world, when the Sun (by the motion of the heavens) cometh unto that Circle, it is then noon, or 12 of the clock: and it is to be understood, that all Towns and Places that lie East and West one of another, have every one a several Meridian; but all places that lie North and South one of another, have one and the same Meridian: this circle is declared in the figure following by the Circle E B W C. Definition 7. The Horizon is a Circle dividing the superior Hemisphere from the 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. Colours are two great Circles, passing through both the Poles of the World, crossing one the other in the said Poles at right angles, and dividing the Equinoctial and the Zodiaque into four equal parts, making thereby the four Seasons of the year, the one Colour passing through the two Tropical points of Cancer and Capricorn, showing the beginning of Summer, & also of Winter, at which times the days and nights are longest and shortest. The other Colour passing through the Equinoctial points Aries and Libra, showing the beginning of the Spring time and Autumn, at which two times the days and nights are of equal length throughout the whole World. Definition 9 The Ecliptic is a great Circle also, dividing the Equinoctial into two equal parts by the head of Aries and Libra, the one half thereof doth decline unto the Northward, and the other towards the South, the greatest declination thereof (according to the observation of that late famous Mathematician Master Edward Wright) is 23 degrees, 31 minutes, 30 seconds. Note also that this Circle is divided into 12 equal parts, which parts are attributed unto the 12 Signs, 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 sometimes on the one side, and sometimes on the other side thereof: this line may be represented in the following figure, by that line whereon the characters of the 12 Signs standeth. CHAP. X. Of the four lesser Circles. Definition 10. THe Sun having ascended unto his highest Solsticiall point, doth describe a Circle, which is the nearest that he can approach unto the North Pole, whereupon it is called the Circle of the Summer Solstice, or the Tropic 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 departed from the Equinoctial line towards the Antartique Pole, whereupon it is called the Circle of the Winter Solstice, or the Tropic of Capricorn, and is described in the figure following by the line H I ♑. Definition 12. So much as the Ecliptic declineth from the Equinoctial, so much doth the Poles of the Ecliptic decline from the Poles of the World, whereupon the pole of the Ecliptic, 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 Tropic of Cancer is from the Equator, and it is called the Circle Arctic, 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 Ecliptic, 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 Ecliptic 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 Equinoctial maketh with the Pole. Definition 15. The Nadir is a point in the heavens under our feet, making right angles with the Horizon under the earth, as the Zenith doth above, and therefore is opposite unto the Zenith: both these may be represented in the figure by the line E W, where the letter E standeth for the Zenith, and W for the Nadir. Definition 16. The Declination of the Sun is the arch of a Circle contained betwixt the Ecliptic and the Equinoctial, making right angles with the Equinoctial, and may be set forth unto you by the arch S ♋. But the Declination of a Star, is the arch of a Circle let fall from the centre of a Star, perpendicular unto the Equinoctial. This Declination may Dialling diagram: Circle EBWC (where E stands for Zenith and W for Nadir) illustrating astronomical definitions in chapters 9 and 10 be counted either Northward or Southward, according to the situation of the Sun or Star, whether it be nearer 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 centre of any star and the Ecliptic line, making right angles with the Ecliptic, and counted either Northward or Southward according to the situation of the Star, whether it be nearer unto the North or South Poles of the Ecliptic. And here note, that the Sun hath no Latitude, but always keepeth in the Ecliptic line. Definition 18. The Latitude of a Town or Country, is the height of the Pole above the Horizon, or the distance betwixt the Zenith and the Equinoctial, and may be represented in this figure by the arch of the Meridian B P, where the North Pole P is elevated above the horizontal line C A B according to the angle 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 Ecliptic which is contained betwixt the Stars place in the Ecliptic and the beginning of Aries, counting them from Aries according to the order or succession of the Signs. Definition 20. The Longitude of a Town or Country, are the number of degrees which are contained in the Equinoctial, betwixt the meridian that passeth over the Isles of Azores, (from whence the beginning of Longitude is accounted) Eastwards, and the Meridian that passeth over the Town or Country desired. Definition 21. The Altitude of the Sun or Star, is the arch of a circle contained betwixt the centre of the Sun or any Star, and the Horizon. As for example, in the former figure, suppose the Sun to be in the Meridian at S, than 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 Equinoctial Circle S A N. Definition 22. Azimuths are Circles which meet together in the Zenith, and cross the horizon at right angles, and serve to find the point of the Compass which the Sun is upon at any hour 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 centre 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 Ecliptic above the Horizon. Definition 24. Right Ascension, is the number of degrees and minutes of the Equinoctial (counted from the beginning of Aries) which cometh to the Meridian with the Sun, Moon, Star, or any portion of the Ecliptic. Definition 25. Obliqne Ascension, is a part of the Equinoctial contained betwixt the beginning of Aries, and that part of the Equinoctial that riseth with the centre of a Star, or any portion of the Ecliptic in an Obliqne Sphere. Definition 26. The ascensional Difference, is the difference betwixt the right and obliqne ascension, or it is the number of degrees contained betwixt that place of the Equinoctial that riseth with the centre of a Star, and that place of the Equinoctial that cometh to the Meridian with the centre of the same Star. Definition 27. Almicanters are circles drawn parallel unto the Horizon one over another, until they come unto the Zenith, these are Circles that do 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 centre 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 intent to practise either in the Art of Navigation, Astronomy, or Dialling. Therefore now I intent to proceed with Scale and Compass to perform some questions Astronomical, before we enter upon the Art of Dialling, seeing they are both delightful, and also helpful unto all such as shall be practitioners in this Art of Dialling. The end of the First Book. THE SECOND BOOK. Showing Geometrically how to resolve all such Astronomical Propositions as are of ordinary use, as well in the Art of Navigation, as in this Art of Dialling. CHAP. I. The description of the Scale, whereby this work may be performed THis Scale for this work, needs to be divided but into two parts, the first whereof may be a Scale of equal divisions of 16 in an inch, and may serve for ordinary measure. The second part of the Scale may be a single Chord of a Circle, or a Chord of 90, and is divided into 90 unequal divisions, representing the 90 degrees of the Quadrant, and are numbered with 10, 20, 30, 40, etc. unto 90. This Chord is in use to measure any part or arch of a Circle, not surmounting 90 degrees, the number of these degrees from 1 unto 60, is called the Radius of the Scale, upon which distance all Dialling diagram: ruler or scale 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 Mathematical Instruments, yet I would not have them that shall make use of this Book, be ignorant of the dividing of this line: Therefore, first, draw the Diameter A D C, which being done, upon the centre D describe the semicircle A B C, which semicircle divide into two equal parts or Quadrants by the point B, then dividing one of these Quadrants into 90 equal parts, or degrees, you are prepared, as here you see in the Quadrant A B. Now this being done, set one foot of your Compasses in the point A, and let the other be extended unto each degree 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 unequal divisions from the point A, (and numbered by 10, 20, 30, 40, etc. 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 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 centre D, whereby it doth plainly appear that 60 of those degrees are equal to the semidiameter of the same Circle, and therefore is the Radius upon which all circles are drawn, as was showed before in this Chapter. Dialling diagram: illustration of how to divide a line of chords into degrees of a circle CHAP. II. How speedily with Rule and Compass, 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 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. arch (arc) BC and 40-degree angle BAC And if you desire to find 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 show you the number of degrees contained in that angle, which here will be found to be 40 degrees. CHAP. III. To find the altitude of the Sun by the shadow of a 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, 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 mark 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 Dialling diagram: Quadrant ABCD (where A is the center of a circle and B, C, and D points along the circumference) C D, then suppose the height of your Gnomon or substance yielding shadow, to be the line A E, which here we will suppose to be 12 foot, therefore take 12 of your equal divisions from your Scale, as here I have taken 12 quarters for this our purpose, and set them down from A to E, and draw the line E F parallel to A B, then suppose the length of the shadow to be 9 foot, for this 9 foot must I take 9 of the same divisions as I did before, and place them from E to G, by which point G, draw the line A G C, from the centre 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. FOUR To find the altitude of the Sun by the shadow of a Gnomon standing at right angles with any perpendicular wall, in such manner that it may lie parallel unto the Horizon. FIrst, draw your Quadrant A B C D, as is taught in the last Chapter, & 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 applied 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 find the length of the right shadow. BY right shadow is meant the shadow of any staff, 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 your Gnomon, or substance yielding shadow, to be 12 foot, as in the former Chapter: this do I set down from A to E, and Dialling diagram 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 applied unto your Scale, will give you 9 of those divisions, whereof A E was 12, which here doth signify 9 foot. CHAP. VI The Almicanter, or height of the Sun being given, to find the length of the contrary shadow. BY the contrary shadow is understood the length of any shadow that is made by a staff or Gnomon standing at right angles against any perpendicular wall, in such a manner that it may lie parallel unto the Horizon; the length of the contrary shadow doth increase as the Sun riseth in height: whereas, contrariwise, the right shadow doth decrease in length as the Sun doth increase in height. Therefore the way to find out the length of the Versed shadow is as followeth. First draw your Quadrant, as is taught in the third Chapter, now supposing the length of your Gnomon to be 12 foot, place it from A to E, likewise from E draw the line E F parallel to A B, as before: now supposing the height of the Sun to be 36 deg. 52 min. take it from your Scale, and place it from D to C, from which point C, draw the line C A, cutting the line E F in the point G, so shall G E be the length of the contrary shadow, which here will be found to be 9 foot, the thing desired. CHAP. VII. Having the distance of the Sun from the next Equinoctial point, to find 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 centre 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 Equinoctial 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 Equinoctial point Libra, therefore take 59 degrees from your Scale, and place it from B to C, and draw the line C A, then place the greatest declination of the Sun from B unto F, which is 23 degrees 30 minutes, then Dialling diagram fixing one foot of your Compasses in the point F, with the other take the nearest distance unto the line A B, which you may do by opening or shuting of your Compasses, still turning them to and fro, till the moving point thereof do 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 nearest extent unto the line A B, this distance betwixt your Compasses being kept, fix one foot in the arch B C D, moving it either upwards or downwards, still keeping it directly in the arch line, until by moving the other foot to and fro, you find 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 Ecliptic which he possesseth being given, to find 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 Dialling diagram 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 only 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 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 possess that quarter of the Ecliptic, betwixt the head of Aries and Cancer. CHAP. IX. Having the Latitude of the place, and the distance of the Sun from the next Equinoctial point, to find 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 compliment of the Latitude, and place it from B unto E, then taking the nearest 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 Sun's greatest declination from B unto I, from which point I, take the nearest extent unto the Dialling diagram line A B, which distance being kept, place one foot of your compasses in the arch line F G H, so that the moving foot may but only touch the line A B at the shortest extent, so shall the fixed foot rest in the arch line F G H at G, through which point G, draw the line A G C, then supposing the Sun to be in the 29 degree of Taurus, that is 59 degrees distant from the next Equinoctial point, take 59 degrees Dialling diagram from your Scale and place them from B to L, from which point L, take the nearest distance unto the line A B, with this distance, setting one foot in the point A, with the other make a mark in the line A C, as at O, from which point O, take the shortest extent unto the line A B, this distance being kept, fix one foot of your Compasses in the arch B C D, in such a manner, that the moving foot thereof may but only touch the line A B, so shall the fixed foot rest in the point R, which is the Amplitude counted from B, and will be found in this example to be 34 deg. 9 min. CHAP. X. Having the Declination and Amplitude of the Sun, to find the height of the Pole. FIrst, make the Quadrant A B C D, then supposing the Amplitude to be 34 deg. 9 min. (as it was found by the last Chapter) take it from your Scale, and place it from B to E, then taking the nearest extent from the point E unto the line A B, set one foot of your Compasses in the centre 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 nearest extent, so shall the fixed foot rest at the point H, through which point H, draw the line A H C, cutting Dialling diagram the arch B C D in the point C, so shall the arch B C be the height of the Equinoctial, and the compliment thereof which is the arch C D, shall be the elevation of the Pole above the Horizon, or the distance of the Equinoctial from the Zenith, which in this example will be found to be 52 degrees 30 minutes, the thing desired. CHAP. XI. Having the Latitude of the place, and the Declination of the Sun, to find his Amplitude. FIrst, make the Quadrant A B C D, then supposing the Latitude to be 52 degrees 30 minutes, take it from your Scale, and place it from D to E, or (which is all one) if you place the compliment thereof from B to E, from which point E take the nearest extent unto the line A B, with this distance, setting one foot of your Compasses in the centre 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, fix one foot in the arch F G H, so that the other Dialling diagram may but only touch the line A B at the nearest distance, so shall the fixed foot rest at the point G, through which point G draw the line A G C, cutting the arch B C D in the point C, so shall the arch 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 find his Declination. FIrst, draw the Quadrant A B C D, then supposing the Amplitude to be 34 deg. 9 min. place it from B to E, and from the point E take the nearest extent unto the line A B, with which distance, setting one foot of your Compasses in the centre A, describe the arch G H I: then supposing the Latitude Dialling diagram 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 nearest 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 nearest extent, so shall the fixed foot rest at the point F, which shall be 20 degrees distance from the point B, the declination of the Sun desired. CHAP. XIII. Having the Latitude of the place, and the Declination of the Sun, to find his height in the Vertical Circle, or when he shall come to be due East or West. FIrst, draw the Quadrant A B C D, then supposing the Latitude to be 52 deg. 30 min, take it from your Scale and place it from B to C, then taking the nearest extent from the point C unto the line A B, with one foot set in the centre A, with the other describe the arch G H I: Dialling diagram 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 nearest 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 cometh to be due East or West, which being taken between your Compasses, and applied unto your Scale, will give you 25 deg. 32 min. the thing desired in this example. CHAP. XIV. Having the Latitude of the place, and the Declination of the Sun, to find the time when the Sun cometh to be due East or West. FIrst, draw the Quadrant A B C D, then placing the Latitude of the place (which here we will suppose to be 52 deg. 30 min.) from B to C, and draw the line C E, then with the nearest distance from the point C unto the line A B, which is the line C E, setting one foot of your Compasses in the Dialling diagram centre A, with the other draw the arch G H I: then supposing the declination of the Sun to be 20 degrees, place it from B to F, from which point F, lay a rule unto the centre 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 nearest extent: 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 cometh 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 hour, and four minutes for a degree) will make four hours and 54 minutes of an hour, 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 find what altitude the Sun shall have at the hour 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 Dialling diagram 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 centre A, with the other draw the arch G H I, then supposing the declination of the Sun to be 20 deg. place it from B to E, and draw the line 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 hour of six, which in this example will be found to be 15 degrees 44 minutes the thing desired. CHAP. XVI. Having the Latitude of the place, and the declination of the Sun, to find what Azimuth the Sun shall have at the hour 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 Dialling diagram it from B to E, from which point E draw the line E G, parallel to A B, until 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 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 find 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 Dialling diagram Sun to be 11 deg. 30 min. Northward, set it from B to E, from which point E, take the nearest 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 point G, take the nearest 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 bade fallen betwixt the centre A and the point F, then should the arch D L have showed 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 the convexity of which arch and the point A, draw the line A L, cutting the arch B C D in the point L, so shall the arch B L be the Azimuth from the East or West, and the arch L D the Azimuth from the South, which in this example will be found to be 66 deg. 43 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 nearest 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 the nearest extent unto the line A D, and place it from A unto N in the line A B, then take the nearest 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 Dialling diagram touch the line A D, and where the fixed foot so resteth, there make a mark, as at O, from which point O take the shortest extent unto the line A B, and place it from F to H, then with the distance 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 centre point A, draw the line A L, cutting the arch B C D in the point L, so shall the arch D L be the Azimuth of the Sun from the South, which here will be found to be 49 deg. 49 min. the thing desired. CHAP. XVIII. The Latitude of the place, the Declination of the Sun, and the Altitude of the Sun being given, to find the hour of the day. Dialling diagram But if the declination had been North, than the distance from the point H to the line A B, should have been placed from the centre A towards the point I, and the distance I P taken instead of A I, as by the next figure I will make more plain. Dialling diagram And here note, that if the altitude of the Sun had been so small that the point I had fall'n betwixt the centre A and the point P (which is the altitude of the Sun at the hour 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 Sun's Altitude, and the Declination, to find the hour 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 centre 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 nearest extent unto the line A B, which distance place from G to H, in the arch G H I, through which point H draw the line A H C, then the altitude of the Sun being 30 deg. 45 min. place it from B to L, and from the point K take the nearest Dialling diagram extent unto the line A D, with which distance, setting one foot in the centre 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 show the hour from the Meridian, which will be found in this example to be 53 deg. 40 min. that is three hours, and something better than 34 min. from the Meridian. CHAP. XX. Having the hour of the day, the Sun's altitude, and the declination, to find the Azimuth. FIrst, make the Quadrant A B C D, then supposing the declination of the Sun to be 11 deg. 30 min. North as before, place it from B to E, 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 A D, with this distance, setting one foot in the centre A, describe the arch N R, now let the arch for the hour 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 Dialling diagram 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 nearest extent unto the line A D, with which distance upon the centre 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 will show the Azimuth from the South, which in this example will be found to be 66 deg. 43 min. the thing desired. CHAP. XXI. Having the Latitude of the place, and the Declination of the Sun, to find the ascensional difference. FIrst, draw the Quadrant A B C D, then place the Latitude (which here let be 52 deg. 30 min.) from D to C, & draw the line COP parallel to AD, then with the distance C P, upon the centre A, describe the arch G H I, then place the declination, being 20 degrees, from B to F, then lay your rule from the centre A upon the point F, Dialling diagram 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 ascensional, and will be found in this example to be 28 deg. 19 min. which resolved into time doth give one hour, and something better than 53 min. for the difference betwixt the Sun's rising or setting, and the hour of six, according to the time of the year. CHAP. XXII. Having the Declination of the Sun to find the Right Ascension. FIrst, describe the Quadrant A B C D, then place the greatest declination of the Sun from B to E, and draw the line 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 Declination of the Sun given 20 degrees, from B unto F, and laying your Rule upon the centre A and the point F, draw the line F O Dialling diagram 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 shall B C be 56 deg. 50 min. the right Ascension desired. CHAP. XXIII. Having the Right Ascension of the Sun or Star, together with the difference of their Ascensions, to find the Obliqne 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 situated in the Northern or Southern Signs. As for example: If the Sun be in any of these six Northern Signs Aries, Taurus, Gemini, Cancer, Leo, or Virgo, than the difference of the Ascensions is to be substracted from the right Ascension, and the remainder is the Obliqne Ascension; therefore let the fourth degree of Gemini be given, the Right Ascension whereof is found to be 62 degrees, or 4 hours, and 8 min. and the difference of Ascension (where the Pole is elevated 52 deg. 30 min.) is 30 deg. 3 min. or in time, 2 hours and something better, which being taken from the Right Ascension, leaves 2 hours and 8 min. or 32 deg. 5 min. which is the Obliqne Ascension of the Sun in the fourth degree of Gemini. But if the Sun be upon the South side of the Equinoctial, either in Libra, Scorpio, Sagitarius, Capricornus, Aquarius, or Pisces, than the difference of the Ascensions is to be added unto the right Ascension, and the sum of them both will be the Obliqne Ascension. As suppose the fourth degree of Sagitarius to be given, the Right Ascension whereof is found to be 242 degrees, or 16 hours 8 min. and the difference of Ascensions is 30 deg. 3 min. or 2 hours, which being added unto the Right Ascension, doth make 18 hours 8 min. or in degrees, 272 deg. 3 min. which is the Obliqne Ascension of the Sun in the fourth degree of Sagitarius. But if you would find the Obliqne Descension, you must work directly contrary to these Rules given. CHAP. XXIV. How to find the Altitude of the Sun without Instrument. IN the third Chapter of this Book it is showed how to find 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 may have occasion to find the altitude of the Sun, and thereby the Azimuth or hour 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 unneedfull to show the finding of the same without the Gnomon or other Instrument. Take therefore a Trencher, or any simple board's end, of what fashion soever, such as you can get, make thereon two pricks, as A and B, then prick in a pin, nail, or short wire in one of Dialling diagram the points, as at A, whereupon hang a thread with a plummet, 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 thread than falleth, there make a mark as at E, under the thread, then with your Rule and Compass draw the lines A B and A E, and find the angle B A E (by the second Chapter) for that is the compliment 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 find out the Latitude of a place, or the Poles elevation above the Horizon, by the Sun. SEeing that throughout this Book, the Latitude of the place is supposed to be known, when as every one perhaps cannot tell which way to find it out, therefore it will not be unneedfull to show how it may be readily attained, sufficiently for our purpose. First, therefore you must get the Meridian Altitude, which you may do by observing diligently about noon a little before, and a little after, still observing until you perceive the Sun to begin to fall again, then marking what was his greatest altitude, will serve for this our present purpose. Having gotten the Meridian altitude by this, and the Declination by the seventh Chapter, you may find the Latitude of the place, or the elevation of the Pole above the Horizon after this manner. If the Sun hath North Declination, then subtract the Declination out of the Meridian altitude, and the remainder shall be the height of the Equinoctial. But if the Sun hath South Declination, then add the Declination to the Meridian altitude, so shall the sum of them give the altitude of the Equinoctial, which being taken out of the Quadrant or 90 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 Meridian altitude of the Sun being observed to be 55 deg. 35 min. upon which day I find the Sun's place to be in 20 deg. 48 min. of Taurus, and the declination 18 deg. 00 min. and because the Declination is North, I subtract 18 deg. 00 min. out of the Meridian altitude 55 deg. 35 min. and there remains 37 deg. 35 min. the height of the Equinoctial, and this taken out of 90 degrees, leaveth 52 deg. 25 min. for the Latitude of Thuring. But it may be required sometimes for you to make a Dial for a Town or Country whose Latitude you know not, neither can come thither conveniently to observe it. Here is therefore added a Table showing the Latitude of the most principal Cities and Towns in England, so that being required to make a Dial for any of those places, you need but look in your Table, and there you have the Latitude thereof. But if the Town you seek be not in the Table, look what Town in the Table lies near unto it, and make your Dial to that Latitude, which will occasion little difference. A TABLE showing the Latitude of the most principal Cities and Towns in ENGLAND. Names of the places. Latitude Names of the places. Latitude d m d m St. Albon 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 Lincoln 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 Derby 53 6 Winchester 51 10 Grantham 52 58 Worcester 52 20 Halifax 53 49 Yarmouth 52 45 Hereford 52 14 York. 54 0 The end of the second Book. THE THIRD BOOK. Showing Geometrically how to describe the Houre-lines upon all sorts of Planes, howsoever, (or in what Latitude soever) situated, two manner of ways, without exceeding the limits of the Plane. CHAP. I How to examine a Plane for a horizontal Dial. FOrasmuch as it is necessary before the drawing of any Dial, to know how your plane is already placed, or how it ought afterwards to be placed; it is therefore expedient to show how it may be attained unto without the help of a Quadrant, (or any such like Instrument) which for this purpose is very useful. Dialling diagram As for example, let the figure A B C D be a plane supposed to stand level with the Horizon, and for to try the same, I take the simple board GOH, having one straight side, as G H, then drawing a parallel thereto, I cross 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 thread with a plummet to play in the hole E, and finding the thread to fall directly on the perpendicular E O, which way soever I turn the board, I therefore conclude it to be a horizontal plane. CHAP. II. Of the trying of Planes, whether they be Erect or Inclining, and to find the quantity of Inclination. FOr the distinguishing of planes, because their inclination and declination may be divers, we will consider three lines belonging to every plane: the first is the horizontal line; the second the perpendicular line, crossing the horizontal at right angles; the third is the Axis of the plane, crossing both the horizontal line, and his perpendicular, and the plane itself, at right angles; the extremity of which Axis may be called, the pole of the planes horizontal line. The perpendicular line doth help to find the inclination, the horizontal line with his Axis to find the Declination, and the pole of the planes horizontal line, to give denomination unto the plane. When the plane standeth upright, pointing directly into the Zenith, it maketh right angles with the Horizon, and is therefore called an erect plane, and a plumb line drawn thereon is called a Vertical line; as in this figure, the plane G H L I is erect, & the line H I is the Vertical line. Now for the trying of this plane, if you apply the straight side of your board vertically thereto, as here you see done in the figure C, and either hanging a thread and plummet in the point M, or holding up a thread and plummet with your hand, you find the thread to fall directly on the parallel line M N it is an erect plane, but if the thread will cross the line M N, it is no erect plane, but inclineth to the Horizon. And if you find your plane to be erect, you may by applying your board thereto, with the thread and plummet falling on the parallel line M N, draw the vertical line H I by the edge of your board, the vertical line being drawn, you may cross it at right angles with the line G L, which shall be level with your Horizon, and therefore called the horizontal line of the plane. If the plane shall be found to incline to the horizon, you may find out the inclination after this manner. Apply your board to the plane, as you see here by the figure B, in the plane F K E H, then holding up a thread and plummet that it may fall upon the perpendicular line O E, and turning about your board, till the straight side thereof lie close with the plane, and the thread fall on the perpendicular line O E, so the line drawn by the straight side of the board, shall be an horizontal line, which here in this figure will be the line F E. This being done, cross the horizontal line at right angles with the perpendicular H K, than set the straight side of your board upon the line H K, which is perpendicular to the horizontal line, with holding the board up right, and holding up a thread and plummet, so that it may have free course to play by the side of the board, which it may have by letting the plumber fall within the hole, a little below X in the figure A: the thread thus hanging, mark two points directly under it, the one at X where the thread crosseth the parallel line on the board, the other at T, at the upper side thereof, and so by drawing the line T X, you have the angle T X V, which is called the angle of Reclination, and it is the angle contained betwixt the plane and the vertical line passing from the Zenith to the Nadir, the compliment 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. Dialling diagram 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 straight side of your board to the perpendicular line of the under face, and hang the thread and plummet in any part of the parallel line, the angle that is made by that parallel line and the thread, shall be the compliment of the angle of the inclination of the plane to the Horizon. CHAP. III. To find out the Declination of a Plane. THe Declination of a plane is always reckoned in the Horizon, and it is the angle contained between the line of East & West, & the horizontal line upon the plane. For the finding out of this Declination, first, take any board that hath but one straight side, and draw a line parallel thereto, as was done in the first Chapter, and having drawn an horizontal line upon your plane, apply the straight side of your board thereunto, holding it parallel to the Horizon, as in the figure of the last Chapter, where the board D is applied to the horizontal line G L, then the Sun shining upon the board, hold out a thread and plummet, so as the thread being vertical, the shadow of the thread may cross 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 horizontal line of the plane, and the Azimuth wherein the Sun is at the time of observation: at this same instant, or as near as may be, must you take the Altitude of the Sun; these two being done diligently, will help you to the planes Declination as followeth. Dialling diagram Now the angle of declination here required, is the angle B A H, or E A S, for look how much the horizontal line of the plane declineth from the line of East and West, so much doth the poles of the planes horizontal line decline from the North and South towards either East or West, either of which angel's in this example will be found to be about 16 deg. the declination of the plane from the South point E, Eastward. CHAP. IU. How to draw the Meridian line upon an horizontal plane, the Sun shining thereon. IF your plane be level with the Horizon, describe thereon a Circle as B C D E, then holding up a thread and plummet, so as the thread being vertical, the shadow thereof may fall upon the centre A, and draw the line of shadow Dialling diagram C E, then take the altitude of the Sun at the same instant (or as near 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 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 Equinoctial Dial. Dialling diagram This Dial is no other than a Circle divided into 24 equal parts, by which divisions and the centre A, you may draw so many hour lines as shall be necessary. As you may see here done in the Circle B D C E, which is divided equally into 24 equal parts, and hour lines drawn from the centre A to so many of them as is needful, the line D E, which is the line of inclination, is the Meridian or 12 of clock line: his Style is no more but a straight pin or wire plumb erected in the centre. This Dial, though, of all other, he be the simplest, yet is he mother to all the rest, for out of him, as from a root, is derived the projectment of those 24 hour lines on any other great Circle or plane whatsoever. CAAP. VI The drawing of a Dial upon the direct Polar Plane. A Direct Polar plane, is that which is parallel to the Circle of the hour of six, therefore, having drawn the horizontal line A B, rnd crossed it at right angles about the middle of the line at C, with the perpendicular C E, if you shall find the planes inclination towards the North to be equal to the Latitude of the place, and the horizontal line directly in the line of East and West, and so to have no declination, you may be sure this plane lieth parallel to the hour of six, and is therefore called a Polar plane. The horizontal line being drawn at the length of the plane, divide it into seven equal parts, and set down one of them in the line of inclination from C unto D, and upon the centre D describe the Equinoctial Circle, which you may divide into 24 equal parts if you will, but one quarter thereof into six will serve as well: then at the Dialling diagram distance C E draw the line F G parallel to A B: Then having divided the Equator either into 24 equal parts, or one quarter thereof into six, you may by a rule laid upon the centre D, and each of those six parts, make marks in the horizontal line A B (which here is instead of the contingent line) as you may see by the pricked lines, these distances from the Meridian, being applied upon the same line on the other side of the Meridian, and also on both sides the Meridian in the upper line, the lines drawn from point to point, parallel to the Meridian C E, shall be the hour lines, the line C E shall be the Meridian line, the hour of 12, and must also be the substilar line, whereon the stile must stand, which may be a plate of iron or some other metal, being so broad as the semidiameter of the Circle is, as is showed in the figure. This style must be placed along upon the line of 12, making right angles therewith, the upper edge whereof must be parallel to the plane; so shall it cast a true shadow upon the hour lines. The under face of this Polar plane, and also of the former Equinoctial plane, is made altogether like unto the upper faces here described, without any difference at all. CHAP. VII. The making of an erect Meridian Dial. A Meridian plane is that which is parallel to the Meridian Circle of the Sphere; therefore having drawn the horizontal line A B, and finding it to decline 90 deg. from the South, the plane being erect, I conclude it to lie parallel to the Meridian Circle of the Sphere, and is therefore called a Meridian plane. Dialling diagram For the style of this Dial it may be either a plate of some metal, being so broad as the semidiameter of the circle is, and so placed perpendicularly along over the line of the hour of six, the upper edge thereof being parallel to the plane, or it may be a straight pin fixed in the centre of the circle, making right angles with the plane, being just so long as the Semidiameter of the Circle is, only showing the hour with the very top or end thereof. Dialling diagram This plane hath two faces, one to the East, the other to the West, the making whereof are both alike, only in naming the hours, for the one containeth the hours for the forenoon, and the other for the afternoon, as you may perceive by the figures. CHAP. VIII. To draw a Dial upon an horizontal plane. AN horizontal plane is that which is parallel to the horizontal Circle of the Sphere, which being found by the first Chapter to be level with the Horizon, you may by the fourth Chapter draw the Meridian line A B serving for the Meridian, the hour of 12, and the substilar: in this Meridian make choice of a centre 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 equal to the Latitude of the place, which here we will suppose to be 52 deg. 30 min. this line shall represent the cock of the Dial, and the Axletree of the World, then at the North end of the Meridian line, draw another line as F G, crossing the Meridian in the point A at right angles, this line is called the Touch-line, or line of contingence. Then set one foot of your Compasses in the point A, and with the other take the nearest extent unto the line S C or the Style, 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 centre of the Equinoctial, upon which describe the Equinoctial Circle A D B E, with this same distance, setting one foot in the point A, make a mark●● F on the one side of the Meridian, and another at G on the other side thereof, both which must be in the line of Contingence, by which two points, and the centre C, you may draw the hour lines of 3 and 9 Dialling diagram This same distance of your Compasses being kept, with one foot still in the centre A, with the other make the marks T and V in the Equinoctial Circle, each of which distances is an arch of 60 degrees, or four hours of time, the half of which arch is 30 degrees, or two hours from the Meridian, this divided in the half will be 15 deg. or one hour from the Meridian; then laying your rule upon the centre I of the Equinoctial, and upon these two last divisions in the circle thereof, where the rule shall touch the line of contingence, there mark it as at H and O, by which points and the centre C, you may draw the hour-lines 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 outrun our plane, we may from the intersection of the hours of 9 and 3 with the touch line draw the lines F D and G E parallel to the meridian A B, until 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 nearest 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 centre 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 only touch the line A E; so shall the fixed foot rest in the point R, by which and the centre 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 Equinoctial Circle into three equal parts, you may describe your whole Dial. And whereas in Summer, the 4 and 5 in the morning, and also 7 and 8 at evening, shall be necessary in this kind of Dial: prolong or draw the lines of 4 and 5 at evening beyond the centre C, and they shall show the hour of 4 and 5 in the morning, and likewise the 7 and 8 in the morning for 7 and 8 at evening. What is here spoken concerning the hours, the like is to be done in drawing the half hours, as well in this kind as in all them which follow. The style must be fixed in the centre 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 Dial's may be drawn upon a clean paper, and then with the help of your compasses placed on your plane. CAAP. IX. To draw a Dial upon an erect direct vertical Plane, commonly called a South or North Dial. A Vertical plane is that which is parallel to the prime vertical Circle, it hath two faces, one to the South the other to the North; therefore having drawn the horizontal line A B, and from the middle thereof let fall the perpendicular C D, which if you find by the second Chapter to be erect, and the horizontal line A B to lie in the line of East and West, and so to have no declination, you may be sure this plane is parallel to the prime vertical Circle of the Sphere, and therefore is called a vertical plane. This perpendicular C D shall serve for the meridian, the hour of 12, and the substilar line, which is the line over which the Style or Gnomon in your Dial directly hangeth. The horizontal line A B shall serve for the hour-lines of six, both for morning and evening. Dialling diagram Thus you may see, that by dividing the Radius of the Equinoctial Circle into four equal parts, you may describe your whole Dial, if it hath no declination; for having with this Radius pricked out the lines of 3 & 9, & placed the lines for 10 and 11, these two distances from the hour of 12 shall 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 Dial from 6 to 5 & 4: so have you all your hour-lines drawn, and yet we have not outrun 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 Dial's cock, and that this is the South face of this plane, therefore the centre 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 only in placing the Dial, and naming the hours; for if it be the North face, the centre 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. Dialling diagram CHAP. X. To draw a Dial upon a direct vertical plane, inclining to the Horizon. ALL those planes that have their horizontal line lying East and West, are in that respect said to be direct vertical planes; if they be also upright passing thorough the Zenith, they are erect direct vertical planes; if they incline to the pole, they are direct polar planes; if to the Equinoctial, they are called Equinoctial planes, and are described before: if to none of these three points, they are then called direct verticals inclining. In all Dial's that decline not, two things must be had before you can make the Dial, the first is the meridian or 12 a clock line, wherein the cock must stand, and the second is the elevation of the pole above the same line. For the horizontal plane, the meridian line is drawn by the 4 Chapter, and the elevation of the pole above the plane is always equal to the latitude of the place; and in erect direct verticals, the perpendicular or vertical line is the meridian or 12 a clock line, and the elevation of the pole above the plane is always equal to the compliment of the latitude, the North pole being elevated above the North face, and the South pole above the South face thereof. And in these verticals inclining being direct, the line of inclination is their meridian, and line wherein the cock must stand, but for the elevation of the cock, we must first consider, whether the plane inclines towards the South or towards the North; if it inclineth towards the South, add the inclination to your latitude, the sum of both shall be the elevation of the pole above the plane, and if the sum shall be just 90 degrees, it is an Equinoctial plane, and is described before in the 5 Chapter, but if the sum shall exceed 90 degrees, take it out of 180, and that which remains shall be the elevation of the pole above the plane. As for example in the latitude of 52 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 sum will be 72 degr. 30 min. the elevation of the pole above the plane, with which you may proceed to draw a Dial by the eighth or ninth Chapters as if it were a horizontal 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 sixth 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 Dial, as if you were to make an horizontal Dial in that country. 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 only 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 less than the latitude of the place, on the under faces of all North incliners, whose inclination is greater than 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 less than the latitude of the place, on the upper faces of all North incliners, whose inclination is greater than 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 styles of all dials must point directly. CAAP. XI. To draw a Dial upon an erect, or vertical plane declining, commonly called a South or North erect declining Dial. ALL upright planes whereon a man may draw a vertical line; are in that respect said to be erect or vertical; if their horizontal line shall lie directly East and West, they are direct vertical planes; if directly North and South, they are properly called Meridian planes, and are described before. If they behold none of these four principal parts of the world, but shall stand between the prime vertical circle, and the Meridian, they are then called by the general name of declining verticals, or by the name of South or North erect declining planes. In all such declining planes, because the Meridian of the place (which in all upright planes is the vertical line, and serveth for the hour of 12) and the Meridian of the plane deflecteth one from the other; therefore we must find out and place the Meridian of the plane, (which is the line over which the stile directly hangeth, and is here called the Substile) and likewise the elevation of the pole above the plane: both which may be easily performed in this manner. First, draw a blind 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 centre 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 compliment 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 compliment 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, 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. Dialling diagram Now having drawn the line NT through the centre C, and parallel to the touch-line I K, which will be square to the Substile C O, I take the distance O y, which is the Radius upon which the Equinoctial circle was drawn, and place it on both sides the substile, in the line of contingence from O to I, and from O to K, and in his parallel line from C to G, and from C to R, and draw G I and R K, then laying your rule upon the point y, the centre of the Equinoctial, and a the point of intersection of the touch-line with the meridian, and where it cutteth the circle's circumference, there must you begin to divide it into 24 equal parts, but those six shall be only 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 centre y, and upon each of these six points of the Equator, and where it toucheth the line of contingence, there make marks, by which and the centre 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 cross the line G I, shall be the point through which you may draw the six a clock hour-line; in like manner, take the distance from O to the line for 11 a clock, and set it from O to c, and from G to Z, then lay the Rule upon C and Z, and where it shall cut the line, G I shall be the point through which you shall draw the 5 a clock hour-line; and so placing the distance O 10 from O to g, and from G to V, and laying the Rule upon V and g, you shall find the point in the line G I, through which you may draw the 4 a clock hour-line: in like manner may you proceed with the other side. For taking the distance from the point O to the line of 9, and setting it from O to h, and from R to x, and laying the Rule upon the points h and x, where you shall see it cut the line R K, there shall be the point, through which you shall draw the 3 a clock hour-line. And so you may take the distances from O, to the line of 8, and 7, and place them from O to d and m, and from R to W and T, and so by laying your Rule upon the points W d and T m, where it shall cross the line R K, there shall be the points through which the houre-lines of 2, and 1, shall be drawn. The Dial being thus drawn upon the Southeast face of this plane, let the stile be fixed in the centre C, so that it may hang directly over the Substile C O, making an angle therewith equal to the angle P C O. The stile with the substile must here point downwards, because in all upright planes declining from the South, the South pole is elevated; and in all upright planes declining from the North, the North pole is elevated. Therefore if you were to make a Dial to the North face of this plane, you must make choice of your centre C in the lower part of the Meridian C D, that the stile with the substile may have room to point upwards. This Dial being made on paper for the Southeast face of this plane, will also serve for the Northwest face thereof, if you turn it upside down; so that the stile with the substile may point upwards; and the paper being oiled or pricked through, so that you may take the backside 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; only altering the numbers set to the hours. This paper being oiled, if you do but change the backside for the foreside; and the numbers set to the hours, still keeping the centre 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 Dial's, which being well understood, will be a great help to the Artist. CHAP. XII. How to draw a Dial upon an horizontal plane, otherwise then in the eighth Chapter was showed. ALthough I have plainly and perfectly shown the making of the horizontal, the direct South or North, as well erect as inclining; and the South or North erect declining Dial's, in the four former Chapters: yet to satisfy 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 easy to be understood) nor to any one Instrument (which may be absent from me, when I should need it) although in this Treatise I do perform the whole by a plain Quadrant. Therefore by the first Chapter, having found the plane to be horizontal; by the fourth Chapter draw the meridian line A B, and cross 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. Dialling diagram 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 show better than 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 centre, as was showed in the 8 Chapter. CHAP. XIII. To draw a Dial upon a direct vertical plane, as well erect as inclining, otherwise then in the ninth Chapter was showed. THe work of this, is almost like unto the other before; the difference is only in the elevation of the pole above the plane: for in the horizontal plane, the elevation is equal to the latitude of the place; and in all direct verticals being erect, the elevation of the pole above the plane is equal to the compliment of the latitude, but if they shall incline towards the horizon, then shall you find 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 Dial's are all alike; therefore by the second Chapter draw the line EWE 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. Dialling diagram This being done, divide each Quadrant of your Semicircle into six equal 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 centre 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 nearest 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 centre of your Dial, as the shortest extent from the point A unto the meridian, set in the Chord N from the meridian both ways, 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 centre 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 centre Z in the upper part of the meridian line Z N, over which the style must directly hang, making therewith an angle equal to the angle N Z S. But if it had been the North face, then must the centre 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 find the elevation of the pole above the same, with the angle of Deflexion, or the distance of the substile from the Meridian: and also the angle of inclination betwixt both Meridian's. IN all erect declining planes, when the declination is found, there is three things more to be considered before we can come to the drawing of the Dial. I. The elevation of the pole above the plane. II. The distance of the substile from the Meridian. III. The angle contained betwixt the Meridian of the plane, and the Meridian of the place; which here we call the inclination of Meridian's: This angle is made at the Pole, and serveth to show us where we shall begin to divide our Diall-circle into 24 equal parts. These three may be both artificially, easily, and speedily, performed after this manner following. First, describe a Quadrant, as A B C, then supposing your Latitude to be 52 deg. 30 min. take it from your Scale, and set it from B to E in the arch of the Quadrant, and draw the line E D parallel to A B, cutting the line A C in the point D, then take the distance D E, and setting one foot in the centre 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. Dialling diagram 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 Meridian's, and is found by this example to be 38 deg. 13 min. so that by this example the Meridian of the plane will fall betwixt the hours of 2 and 3 if the plane shall decline Westward; but if it shall decline Eastward, then shall it fall betwixt the hours of 9 and 10 before noon. CHAP. XV. To draw a Dial upon an erect or Vertical plane declining; otherwise then in the 11 Chapter was showed. 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 Meridian's to be 38 deg. 13 min. we may proceed to make the Dial after this manner. First, draw the horizontal W E, and the perpendicular line Z N, crossing the horizontal line at right angles, which is the Meridian of the place and the line of 12. Then in the meridian make choice of some point with most convenience, as the centre C, whereupon describe your Dial 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 Dial circle from the Meridian at N unto M, I draw the line C M for the substile: then through the centre 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 Style, then from M the end of the Style draw the crooked line M S, cutting the line of the Style in the point S, so shall the triangle S C M be the true pattern for the Cock of the Dial. Dialling diagram This being done, take 38 deg. 13 min. and set them always on that side the substile whereon the line of 12 lieth, as here from M to H, so shall the point H be the point where you shall begin to divide your dial circle into 24 equal parts, but those points shall be only in use which do fall below the Diameter A C B. And if the line of the substile falleth not directly upon one of the hour lines, then shall you have six points on each side thereof, from which you may let perpendiculars fall unto the line of the substile as here you see done. Dialling diagram When you have drawn and described your Dial upon paper, for any plane whatsoever, you may cut off the hour-lines Cock and all, with a lesser Circle than the Dial Circle, either with a concentrique or an excentrique Circle, and so make a Dial less than the Circle by which you framed it. Or if you extend the hour-lines beyond the Dial Circle, you may cut them off either with a greater concentrique Circle, and so make a bigger Dial, or else you may cut them off with a Square, as here you see in the following figure; or any other form what you shall think most convenient. Of a Plane falling near the Meridian. When as the declination of a plane shall cause it to lie near the Meridian, as that the Declination and inclination shall cause it to lie near the Pole, then doth the elevation of the Pole above the plane grow so small, & the hour-lines so exceeding near together, that except the plane be very large they will hardly serve to good purpose; as here in this figure, being a plane which is erect, and declining from the South 80 deg. towards the East. Therefore first, draw your Dial very true (as before hath been taught) upon a large paper, making your circle as big as you can: then extend the houre-lines, with the substile, and the line of the stile, a great way beyond the Dial's circle, until they do spread, so that they will fill the plane indifferent well, and then cut them off with a long square as O N in the following figure, so will it show almost Dialling diagram An erect Dial declining from the South 80 deg. towards the East: the distance of the substile from the Meridian 37 deg. 4 min. the elevation of the Pole above the plane 6 deg. 4 min. and the inclination of both Meridian's 82 deg. 5 min. like the Meridian Dial of the 7 Chapter, for the hours will be almost parallel the one to the other, and the stile almost parallel to the substile, as you may see by the figure. CHAP. XVI. The inclination of a Meridian plane being given, how thereby to find the elevation of the pole above the plane, the distance of the substile from the Meridian; and the angle of the inclination of the Meridian of the plane to the Meridian of the place. ALL those planes wherein the horizontal line is the same with the Meridian line, are therefore called Meridian planes, if they make right angles with the Horizon, they are called erect Meridian planes, and are described before. But if they lean to the Horizon, they are then called Incliners. These may incline either to the East part of the Horizon, or to the West, and each of them hath two faces, the upper towards the Zenith, the lower towards the Nadir, wherein knowing the Latitude of the place, and the inclination of the plane to the Horizon, we are to consider three things more before we can come to the drawing of the Dial. I. The elevation of the pole above the plane. II. The distance of the substile from the Meridian. III. The angle of Inclination betwixt both Meridian's. 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 centre 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. Dialling diagram 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 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 Dial 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 Dial after this manner. First, draw the horizontal line A B, serving for the Meridian and the hour of 12, about the middle of this line make choice of a centre at C, upon which describe a Circle for your Dial, as A D B E. Then seeing this is the upper face of the plane, set 33 deg. 5 min. the distance of the substile from the Meridian, in the Dial's Circle from the North end of the horizontal line upwards, as from B to H, and draw the line C H for the substile: But if this had been the under face, the substile must have fallen below the horizontal line: now through the centre 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 Dial. Then set 43 deg. 28 min. from H unto M, for the difference betwixt the Meridian of the plane and the Meridian of the place. Now here at M must you begin to divide your Circle into 24 equal parts, from which points let down so many perpendiculars to the substile, as there shall be points on that side the diameter E F next the substile. Now take each perpendicular betwixt your Compasses, and with one foot in the centre C, with the other make marks in the line of the stile, from which take the nearest 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 centre, you may draw the houre-lines. This dial being thus drawn, for the upper face of a Meridian plane inclining towards the West, you must fix the Cock in the centre C, hanging over the substile C H, with an angle equal to the angle S C H, so that it may point to the North Pole, because upon the upper faces of all Meridian incliners the North Pole is elevated; and therefore contrarily, the South Pole must needs be elevated above their under faces. This Dial being drawn in paper, for the upper face of this plane, will also serve for the under face thereof, if you turn the pattern about so that the horizontal line A B may lie still parallel to the Horizon, and the stile with the substile (lying under the horizontal line) may point downwards to the South Pole, the paper being oiled or pricked through, so that you may take the back side thereof for the fore side, without altering the numbers set to the hours. Dialling diagram CHAP. XVIII. The inclination and declination of any plane being given in a known Latitude, to find 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 Meridian's. 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 several sorts, for the inclination being Northward, the plane may fall betwixt the Horizon and the Pole, or betwixt the Zenith and the Pole, or else they may lie in the Poles of the World: or the inclination may be southward, and so fall below the intersection of the Meridian and the Equator or above it, or the plane may fall directly in the intersection of the Meridian with the Equator, each of these planes have two faces, the upper towards the Zenith, and the lower towards the Nadir: Now having the Latitude of the place, with the declination and inclination of the plane, we have seven things more to consider before we can come to the drawing of the Dial. I. The angle of intersection betwixt the plane and the Meridian. II. The arch of the plane betwixt the Horizon and the Meridian. III. The arch of the Meridian betwixt the Horizon and the plane. iv The arch of the Meridian between the Pole and the plane. V The elevation of the Pole above the Plane. VI The distance of the substile from the Meridian. VII. The angle of inclination betwixt the Meridian of the plane, and the Meridian of the place. All these seven may be found out after this manner. First, Describe the Quadrant A B C, then suppose the plane to decline from the South towards the East 35 deg. and to incline towards the Horizon 25 deg. set 35 deg. the declination of the plane from C to E in the Quadrants arch C B, and draw the line A E, then set 25 deg. the inclination of the plane in the same arch from B to F, and draw the line 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 centre 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 Meridian's ascension, or the arch of the plane, betwixt the Horizon and the Meridian, and the arch B M shall be 20 deg. 54 min. for the elevation of the Meridian, or the arch of the Meridian betwixt the Horizon and the plane. Now if the plane shall incline toward the South, add 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. Dialling diagram 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 centre A, describe the arch Y M W, cutting the line A T in the point M, through which point M draw the line R S parallel to A B, cutting the arch B C in the point S, so shall the arch B S be 30 degrees 33 minutes, the height of the Pole above the plane. Then lay your rule upon the point S and the centre 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 Dial upon a declining inclining Plane. HAving by the second Chapter found the inclination to be 25 deg. towards the North, and by the third Chapter the declination from the South towards the East to be 35 deg. and so by the last Chapter the Meridian's 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 Meridian's 17 deg. 30 min. we may proceed to make the Dial after this manner. First, Draw the line A B parallel to the Horizon, in which line make choice of a centre as at C, whereon describe your Dial circle A D B E A, then take 57 deg. 36 min. the Meridian's ascension, and set it from B that end of the horizontal line with the declination of the plane, as from B to N, and draw the line C N for the hour of 12. Then set 8 deg. 35 min. the distance of the substile from the Meridian from N to M (on that side the Meridian which is contrary to the declination of the plane) and draw the line C M for the substile. And set 30 deg. 33 min. from M to D, and draw the line C D unto S, and from the end of the substile draw the crooked line M S, cutting the line of the Style in S, so shall the Triangle M C S be the true pattern of this Dial's Cock. Then set 17 deg. 30 min. the inclination of Meridian's from M unto O, which is the point where you must begin to divide your Dial circle into 24 equal parts; from which points let down so many perpendiculars to the substile, as there shall be points on that side the Diameter F E next the substile, and so by working as before hath been showed, 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 kind of Dial's, 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 Dial's 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. Dialling diagram And upon the under faces of all North incliners whose meridians elevation is less than the Latitude of the place: On the upper faces of all North incliners, whose meridians elevation is greater than 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 horizontal line; upon the upper faces of all South incliners, whose meridians elevation is greater than the Latitudes compliment: on the under faces of all South incliners whose meridians elevation is less than the Latitudes compliment: on the under faces of all North incliners, whose meridians elevation is greater than the Latitude of the place: and on the upper faces of all North incliners whose Meridian's elevation is less than the Latitude of the place; the Meridian must be placed above the horizontal line, as here in this example. And therefore by the contrary; Upon the upper faces of all South incliners, whose meridians elevation is less than the Latitudes compliment: On the under faces of all South incliners, whose meridians elevation is greater than the Latitudes compliment: On the under faces of all North incliners, whose meridians elevation is less than the Latitude of the place: And on the upper faces of all North incliners, whose meridians elevation is greater than the Latitude of the place; the Meridian must be placed below the horizontal line. But here you must observe, that if it be either the upper or under faces of a South inclining plane, whose meridians elevation is greater than the Latitudes compliment: or either the upper or under faces of a North inclining plane, whose meridians elevation is less than the Latitude of the place; that then the Meridian must be placed from that end of the horizontal line with the declination of the plane: But on all the other faces of these kind of planes, the Meridian must be placed from that end of the horizontal line which is contrary to the declination of the plane. And here note, that if the inclination shall be Southward, and the elevation of the Meridirn equal to the compliment of your Latitude; then shall the substile lie square to the Meridian. And if the inclination shall be Northward, and the elevation of the meridian equal to the Latitude of the place, then shall neither Pole be elevated above this plane, and therefore shall be a Polar declining plane. Wherein the Meridian being placed according to his ascension from the horizontal line, shall be in place of the substile, unto which if you draw a line square, it shall serve for the Equator. Then set one foot of your Compasses in the intersection of the substile with the Equator, and open the other to any convenient distance upon the substile, and describe the Equinoctial Circle (as in the sixth Chapter of this Book was showed:) upon the centre whereof, make an angle with the line of the substile, equal to the inclination of both meridians, namely the meridian of the plane and the meridian of the place, which shall show you where to begin to divide your Equinoctial Circle into twenty four equal parts. These things being known, you may proceed to make your Dial 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 Meridian's ascension by the 18 Chapter will be found to be 69 deg. 33 min. and his elevation 52 deg. 30 min. equal to the Latitude of the place, and therefore neither pole is elevated above this plane, and so no distance between the Substile and the Meridian: for the Meridian and the stile with the substile, will be as it were all one line, which is the Axletree of the World: so that here the stile must be parallel to the plane, and the hour lines parallel one to the other, as in the Meridian and direct Polar planes. Therefore, first draw the horizontal line A B, wherein make choice of a centre as at C, whereon describe an occult arch of a Circle as B E: then into this arch inscribe the Meridian's 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 wideness, 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 Meridian's 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. Dialling diagram Now from the point O, divide your Equinoctial circle into 24 equal parts, with which you may proceed to make your Dial, and set up the cock according to the 6 Chapter. The end of the third Book. THE FOURTH BOOK. Showing how to resolve all such Astronomical Propositions (as are of ordinary use in this Art of Dialling) by help of a Quadrant fitted for the same purpose. CHAP. I. The description of the Quadrant. HAving in the second and third Books shown Geometrically the working of most of the ordinary Propositions Astronomical, with the delineation of all kind of plain wall Dial's howsoever, or in what latiitude soever situated, still keepng within the limits of our plane, and yet not tied to the use of any Instrument. I will now show how you may perform the former work exactly, easily and speedily, by a plain Quadrant fitted for that purpose; the description whereof is after this manner. Having prepared a piece of Box or Brass 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 centre A, upon which centre A, with the Semidiameter A B or A C, describe the arch B C, this arch is called the limb, and is divided into 90 equal parts or degrees; and subdivided into as many parts as quantity will give leave, being numbered from the left hand towards the right after the usual manner. Then let the Semidiameter A B be divided into 90 unequal parts, (called right Sines) either from the Table of natural Sins by help of a decimal Scale, equal to the Semidiameter A B, or else by taking the nearest 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 centre A towards B, you shall exactly divide the Semidiameter A B into 90 unequal divisions called right Sins. This being done, draw the line D E from the Sine of 45 degrees counted in the line of Sins 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 diagonal line A E, which you may call the line of Latitudes. Then upon the centre A, with the distance A D or A F describe the arch D F, which you may divide into six equal parts, by laying your Rule upon each 15th. degree in the Quadrant, and the centre A as at g h I k l F, from which points draw slope lines to each 15th. degree in the Quadrant, numbered backward, as F P, l O, k E, I n, h m, g B; these lines so drawn are to be accounted as hours, then dividing each space into two equal parts, draw other slope lines standing for half hours, which may be distinguished from the other, as they are in the figure. Dialling diagram Now because in the latter part of this Book there is often required to use a line of Chords to several Radiuses, 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 less than 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 thread, bead, and plummet according to the usual manner. CAAP. II. Of the use of the line of Sines. Any Radius not exceeding the line of Sines being known, to find the right Sine of any arch or angle thereunto belonging. IF the Radius of the Circle given be equal to the line of Sines, there needs no farther work, but to take the other Sins 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 thread to the nearest distance, which you may do by turning the compasses about, till the moving point thereof do only touch the thread and no more: the threrd lying still in this position, take the nearest extent thereunto, from any Sine you think good, and it shall be the like Sine agreeable to the Radius given. As for example, let the circle B C D E in the following chapter represent the meridian circle, let B D be the Horizon, and C E the vertical circle; and let F G be the diameter of an almicanter, and so F H the Semidiameter thereof; which being given it is required to find the Sins 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 thread to the nearest distance, the thread lying still in this position, take the nearest extents thereunto, from the Sine of 30 and likewise of 50, these distances place upon the Radius F H from H to N, and from H to R, so shall H N be the Sine of 30 degrees, and H R the Sine of 50 degrees, agreeable to the Radius F H the thing desired. CHAP. III. The Right Sine of any arch being given to find the Radius. TAke the Sine given betwixt your compasses, and setting one foot in the like Sine in the line of Sines, with the other lay the thread to the nearest distance, the thread 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 thread to the nearest 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. Dialling diagram CHAP. IU. The right Sine, or the Radius of any Circle being given, and a straight line resembling a Sine, to find the quantity of that unknown Sine. FIrst, take the Radius, or the right sine given, and setting one foot of your Compasses either in the like sine or in the Radius of the line of Sines, and with the other, lay the thread to the nearest distance, then take the right line given, and six one foot in the line of sins, moving it till the movable foot touch the thread at the nearest 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 straight line given resembling a Sine, first with the distance F H from the Sine of 90 lay the thread to the nearest distance; the thread lying still in this position, take the line H N and fixing one foot of your compasses in the line of Sines, still moving it to and fro till the movable foot thereof, doth only touch the thread, so shall the fixed foot rest at the Sine of 30 degrees in the line of Sines; this 30 degrees is the arch, of which H N is the Sine, F H in the last chapter being the Radius. CHAP. V The Radius of a circle not exceeding the line of Sines being given, to find the chords of every arch. IF the Radius given, shall be equal to the line of Sines, then double the Sine of half 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 equal to the Semiradius (the sine of 30 deg.) of the line of sins; than you need not to double the lines of sins as before, but only 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 less than the semiradius of your line of sins, then take it betwixt your compasses and setting one foot in the sine of 30 deg. with the other lay the thread to the nearest distance, the thread lying still in this position, take it over at the nearest extent in what Sine you think good, only 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 chord 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 thread to the nearest 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 than the Sine of 30. and yet less than the Radius of the line of Sines; then with the Radius given, and from the Sine of the compliment of half the arch required, lay the thread to the nearest distance; then taking it over at the nearest extent from the sine of the whole arch, you shall have your desire. As for example, let the Radius A C of the circle in the third Chapter be given; and a Chord of 30 deg. required: the half of 30 deg. is 15 deg. the compliment 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 thread to the nearest distance: the thread 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 extreme and mean proportion. A Right line is said to be divided by extreme and mean proportion, when the lesser Segment thereof, is to the greater, as the greater is to the whole line. Let A B be the line to be so divided, this line I take with my compasses, and setting one foot in the sine of 54 deg. and with the other I lay the thread to the nearest 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 thread 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 extreme 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 Dialling diagram CHAP. VII. To find a mean proportional line between two right lines given. A Mean proportional line is that, whose square is equal to the long square, contained under his two extremes. First, join the two given lines together, so as they may make both one right line; the which divide into two equal parts; and with the one half thereof, setting one foot in the Sine of 90 deg. with the other lay the thread to the nearest 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 sins, so as the other may but only touch the thread; now from the compliment of the sine where the fixed foot so resteth, take the shortest extent unto the thread, which shall be the mean proprtional line required. Dialling diagram As for example, let A and B be two lines given, between which it is required to find a mean proportional line, first join the two lines together in F, so as they both make the right line C D, which divide into two equal parts in the point E, then with either half of which, setting one foot in the sine of 90 deg. with the other lay the thread to the nearest distance: then keeping the thread in this position, take the distance between the middle point E and F, the place of meeting of the two given lines, and fixing one foot in the line of sins, so as the other may but only touch the thread, and the fixed foot will stay about 22 deg. 30 min. the compliment whereof is 67 deg. 30 min. from which take the shottest extent unto the thread lying as before, which shall be the line H, the mean proportional line betwixt the two extremes A and B, which was required. CHAP. VIII. Having the distance of the Sun from the next equinoctial point, to find his declination. FIrst, lay the thread upon 23 deg. 30 min. the sun's greatest declination, counted on the limb of the quadrant, the thread lying still open at this angle, take the shortest extent thereunto from the sine of the distance of the Sun from the next equinoctial point, this distance being applied to the line of sins from the centre A, shall give you the sine of the sun's 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 equinoctial 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 ecliptic which he posseseth being given, to find his place. TAke the sine of the sun's declination from the line of sins, & setting one foot in the sine of the sun's greatest declination, with the other lay the thread to the nearest distance so shall it show upon the limb, the distance of the sun from the next equinoctial 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 sun's greatest declination, the line C K will be found to be 59 deg. for the distance of the sun from the next equinoctial point which was required. CHAP. X. Having the latitude of the place, and the distance of the sun from the next equinoctial point, to find his amplitude. TAke the sine of the sun's greatest declination betwixt your compasses, and setting one foot in the co-sine of the latitude, with the other lay the thread to the nearest distance; which lying still in this position, set one foot in the sine of the sun's distance from the next equinoctial point, and with the other take the nearest extent unto the thread, 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 compliment of the latitude, and K the angle of the Sun's greatest declination, and C K 59 deg. the distance of the Sun from the equinoctial point Aries, the line C N will be found to be the Sine of 34 deg. 9 min. the amplitude required CHAP. XI. Having the declination and amplitude to find the height of the pole. FIrst, take the sine of the Sun's declination, and set one foot in the sine of the Amplitude, and with the other lay the thread to the nearest distance, so shall the thread upon the limb, show the compliment 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 to be 37 deg. 30 min. the compliment 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 cousin of the latitude, and with the other lay the thread to the nearest distance: so shall it show upon the limb the amplitude required: so in the figure of the next Chapter, the angle CNM being 37 deg. 30 min. the cousin of the latitude, and C M the declination here 20 deg. and the angle at M being right, we shall find the base C N to be the Sine of 34. which was required for the Sun's amplitude. CHAP. XIII. Having the elevation of the pole, and amplitude of the Sun, to find his declination. FIrst, lay the thread to the amplitude counted in the limb, then take it over at the shortest extent, from the cousin of the latitude, so shall you have the Sine of the sun's declination betwixt your compasses. So in this figure, the amplitude C N being. 34 deg. 9 min. and the angle at N being cousin to the latitude, the angle at M being a right angle, we shall find CM to be 20 deg. for the declination of the sun which was required. Dialling diagram 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 thread to the nearest distance; so shall it show upon the limb the height of the Sun in the Vertical circle. So in the figure of the last Chapter, the angle C I O being 52 deg. 30 min. the latitude of the place, and C O the Sun's declination 20 degrees, and the angle C O I being a right angle we may find C I to be a sine of 25 deg. 32 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 Equinoctial Point, to find his height in the vertical circle. FIrst, take the sine of the sun's greatest declination, & setting one foot in the sine of the latitude, with the other lay the thread to the nearest distance: the thread lying still in this position; from the sine of the sun's place take the nearest extent thereunto, which shall be the sine of the sun's 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 sun's greatest declination, and K C being 59 deg. the sun's distance from the next equinoctial point, we shall find C I to be 25 deg. 32 min. for the height of the sun in the Vertical circle. CHAP. XVI. Having the latitude of the place and the declination of the Sun, to find the time when the Sun cometh to be due east or west. WIth the sine of the declination, set one foot in the sine of the latitude, and with the other lay the thread to the nearest distance: then take it over at the nearest extent from the co-sine of the latitude; which distance keep; and setting one foot in the co-sine of the declination, with the other lay the thread to the nearest distance: so shall it show upon the limb, the quantity of degrees betwixt the hour 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 compliment 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 cousin of the declination, therefore set one foot therein, and with the other distance kept, lay the thread to the nearest distance: so shall it show upon the limb 16 deg. 30 min. which converted into time maketh 1 hour, and 6 min. for the quantity of time between the hour of six and the suns being in the East or West points. CHAP. XVII. Having the latitude of the Place, and the declination of the sun, to find his altitude at the hour of six. FIrst, take the thread, and lay it upon the declination counted in the limb; then from the sine of the latitude, take it over at the shortest extent; which distance shall be the sine of the height of the sun at the hour of six. So in the figure of the 13 Chapter, the angle at L being a right angle, and L O C being 52 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 hour 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 hour of six. FIrst, with the sine of the sun's height at the hour of six, set one foot in the sine of the latitude, and with the other lay the thread to the nearest distance: then take the least distance thereunto from the cousin of the latitude, now with this distance setting one foot in the cousin of the altitude, with the other lay the thread to the nearest distance as before: so shall it show upon the limb, 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 cousin of the latitude, the angle L O C must be 52 deg. 30 min. the latitude of the place being the compliment of the angle L C O, and C L being 15 deg 44 min. (as by the last chapter it did appear) we shall find L O to be the sine of 12 deg. 30 min. for the azimuth of the sun from the East or West, at the hour of six as was required. CHAP. XIX. Having the declination of the sun, to find his Right Ascension. FIrst, with your compasses take the sine of the sun's declination given, and setting one foot in the sine of the sun's greatest declination, with the other lay the thread to the nearest distance: then at the least distance from the cousin of the sun's greatest declination take it over: now again, with this distance lay the thread to the nearest distance from the cousin of the declination given, so shall it show upon the limb the right ascension of the sun. So in the figure of the 13 chapter, C O the sun's declination being 20 deg. and the angle O K C being 23 deg. 30 min. the sun's greatest declination, and the angle K C O being the compliment 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 find the ascensional difference. FIrst, take the sine of the Sun's declination, and setting one foot in the co-sine of the Latitude, with the other lay the thread to the nearest 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 thread again to the nearest distance, so shall it show upon the limb the Sun's ascensional difference. So in the figure of the 13 Chapter, the angle M C N being 52 deg. 30 min. and the angle C N M being the compliment thereof, the one being the latitude, and the other the co-latitude, and C M being 20 deg. the sine of the Sun's declination, we shall find M N 28 deg. 19 min. for the difference of Ascensions, which being converted into time, maketh 1 hour, and something better than 53 min. Now when the Sun hath North declination, if you take this difference of Ascension (which is 1 hour 53 min.) out of 6 hours, there will be left 4 hours 7 min. for the time of Sun rising, and if you add it unto 6 hours, the same will be 7 hours 53 min. for the time of Sun setting. And so contrarily, when the Sun hath South declination, if you add this ascensional 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 find the Azimuth. IF the Sun's declination be Northward, then by the 14 or 15 Chapters get his height in the Vertical Circle for the day proposed: from the sine of which take the distance unto the sine of the Sun's altitude observed: with this distance, setting one foot in the co-sine of the Latitude, with the other lay the thread to the nearest 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 Sun's altitude, with the other lay the thread again to the nearest distance, so shall it show upon the limb the Sun's Azimuth from the East or West, either Northward or Southward. Dialling diagram And here note, when the declination is Northward, that as when the latitude of the Sun given, and his height in the Vertical circle is equal, he is directly in the East or West, 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, find the Amplitude for the day proposed. Now first, take the sine of the Sun's altitude, and setting one foot in the co-sine of the Latitude, with the other lay the thread to the nearest distance, which thread lying still in this position, take it over at the shortest extent from the sine of the Latitude, this distance add to the sine of the Amplitude, by setting one foot in the sine of the Amplitude, and extending the other upon the line of sins, these two being thus joined, take them betwixt your Compasses, setting one foot in the co-sine of the Sun's altitude, and with the other lay the thread to the nearest distance: so shall it show upon the limb the Sun's 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 Sun's 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 compliment thereof; we shall find X N to be the sine of 40 deg. 11 min. the Azimuth of the Sun from the East or West points Southward, which was required. CHAP. XXII. The latitude of the place, the declination and altitude of the Sun being given, to find the hour of the day. IF the declination of the Sun be Northward, find the height of the Sun at the hour of six by the 17 Chapter, betwixt which sine, and the sine of the Sun's altitude given, take the distance upon the line of sins, with which distance, setting one foot in the co-sine of the latitude, with the other lay the thread to the nearest distance, the thread 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 thread again to the nearest distance: so shall it show upon the limb the quantity of time from the hour of six. So in this figure, having M N the distance betwixt the sine of 9 deg. 5 min (the height of the Sun at the hour 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 compliment M N O, we shall find N O to be the sine of 60 deg. the quantity of time from the hour of six, which 60 deg. is four hours of time. And here also note, that if the altitude given be greater than the altitude of the Sun at the hour of six, then is the time found to the Southward of the hour of six; but if it be lesser than is it to the Northward. But if the declination of the Sun be Southward, find his depression at the hour of six, by the 17 Chapter, for the day proposed, which will be equal to his height at six, if the quantity of declination be alike. Dialling diagram Now take the sine of this depression, and add 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 sins: These two being thus joined 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 thread to the nearest distance: which lying still in this position, take it over at the shortest extent from the sine of 90 deg. with this distance, setting one foot in the co-sine of the declination as before, with the other lay the thread again to the nearest distance: so shall it show upon the limb the quantity of time from the hour 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 hour of six, joined both together in one straight 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 hour of six, which converted into time will make three hours. The end of the fourth Book. THE FIFTH BOOK. Showing how to describe the houre-lines upon all sorts of planes howsoever, or in what Latitude soever situated, by a Quadrant fitted for the purpose. CHAP. I. How to examine a plane for an horizontal Dial. IF your plane seem to be level 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 thread with the plummet doth fall directly upon his level line A C; which way soever you turn it, it is a horizontal 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 thread and plummet, so that it may play upon the centre, if it shall fall directly upon his level line A C, making no angle therewith, it is an horizontal plane, as here you may see by this figure. Dialling diagram CHAP. II. Of the trying of planes, whether they be erect or inclining, and to find the quantity of their inclination. IF the plane seem to be erect, you may try it by holding the Quadrant, so that the thread may fall on the plumb line A C, for than if that side of the Quadrant shall lie close to the plane, it is erect, and a line drawn by that side of the Quadrant shall be a Vertical line; and the line which crosseth this vertical line at right angles, will be the horizontal line, as here you may see in this figure, the plane D E F G being erect, and the line D E being vertical, the line F G must be horizontal. But if the plane shall incline, the quantity of inclination may be found out after this manner. First, you must draw thereon the horizontal line, which you may do upon the under face, by applying the side A B of your Quadrant thereunto, so as the thread 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 horizontal 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, until the thread and plummet doth fall directly upon the plumb line A C, the Rule lying thus close to the plane, you may thereby draw a line parallel to the Horizon. Having drawn this horizontal line M N, cross 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 thread 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 showed in this figure, so shall the degree of the Quadrant give you C A H, the angle of inclination required. Dialling diagram Here you must be needful that both edges of your Ruler be straight, and one parallel to the other. CHAP. III. To find the Declination of a plane. TO find out this declination you must make two observations by the Sun: the first is of the angle made between the horizontal line of the plane, and the Azimuth wherein the Sun is at the time of observation: the second is of the Sun's altitude: both these observations should be made at one instant. First, for the horizontal distance, having drawn upon your plane a line parallel to the Horizon, apply the side of your Quadrant thereunto; holding it parallel to the Horizon, then holding up a thread and plummet at full liberty, so as the shadow thereof may pass through the centre of the Quadrant, observe the angle made upon the Quadrant by the shadow of the thread, and that side with the horizontal line, for that is the distance here required. Then at the same instant, as near as may be, take the Sun's altitude, that so you may find the Sun's Azimuth from the East or West points; by the 21 Chapter of the fourth Book. Having thus gotten the horizontal distance, with the Azimuth of the Sun for the same time, describe a Circle as A B C D, representing the horizontal circle, and draw the diameter A C, which shall represent the horizontal line F G of the last Chapter. Now supposing the horizontal distance to be 38 deg. 30 min. the angle O A B of the last Chapter, place it from C Southward to E (that is from the same end of the horizontal line with which the angle was made upon the plane) and draw the line E Z: Then supposing the altitude of the Sun at the same time to be 30 deg. 45 min. with 11 deg. 30 min. North declination, and so by the 21 Chapter of the fourth Book, the Azimuth will be found to be 23 deg. 17 min. from the East Southward, being the observation was made in the 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 Vertical Circle, so shall the angle made between the horizontal line of the plane, and the line of East and West, be the declination of the plane, which in this example is found to be 15 deg. 13 min. the angle C Z R. Dialling diagram Or you may observe the angle made between the shadow of the thread, and that side of the Quadrant which lieth perpendicular unto the horizontal line of the plane, which in this example is 51 deg. 30 min. the compliment of the former angle, and it is the angle O A C in the former Chapter upon the Quadrant. Now having drawn your horizontal Circle as before, and the diameter A C for the horizontal line of the plane, you may cross it at right angles with the diameter B D, for the Axis of the planes horizontal line, from which as from D, you may set your horizontal distance on the same side thereof, as before you found it by your observation, as here from D to E, and draw the line E Z for the line of the shadow, and having found the Azimuth of the Sun 23 deg. 17 min. from the East Southward, you may set it from E (the place of the Sun) Northward to R, and draw the line R Z H for the line of East and West, as before. Or if you take the Sun's Azimuth from the South, which in this example will be 66 deg. 43 min. the compliment 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 horizontal, the full North or South planes, whether erect or inclining. SEeing the making of these Dial's are all after one manner, we will here proceed to make an horizontal Dial, 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 nearest extent unto A E the line of latitudes, and set it from A to D, and from A to F both ways, 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 kind of planes that declines not from the North or South. Dialling diagram For the hour-lines before six and after, you may extend their opposite hour lines beyond the centre as was showed in the 8 chapter of the third book. What is here showed concerning the hours, the like may be understood for the half hours, by applying the thread thereunto in the limb. CHAP. V To draw a Dial upon a South or North erect declining plane. IN the drawing of all these kind of Dial's by help of this Quadrant, when the Latitude of the place, and the declination of the plane is known, two things more is to be considered: First, the elevation of the Pole above the plane,: Secondly, the inclination of the Meridian of the plane, to the Meridian of the place; both which will speedily be found when you are ready for them. First therefore, draw the line D A F as before, from the middle whereof let fall the perpendicular A B for the substilar, and at the distance of the tangent line D E, draw the line C B E parallel to the line D A F. Now to find the elevation of the pole above the plane, lay the thread upon the co-sine of the Latitude counted on the limb, and take it over at the nearest 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 nearest extent unto the line of Latitudes, this distance set from A to D and F both ways, 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 Meridian's, take the sine of the declination of the plane, and setting one foot in the co-sine of the styles elevation, with the other lay the thread to the nearest distance, so shall it show upon the limb the inclination of Meridian's to be 38 deg. 13 min. Dialling diagram The thread lying still in this position, observe which of the hours 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 Dial: the thread 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 thread to the least distance, and taking 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 thread, 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 thread 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 applied to the line k E; the thread 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 thread 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 applied to the lines l O and F P, the thread 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 thread 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 hours 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 supplied. The Dial being thus drawn on paper, you must place it so upon the plane, that the twelve a clock houre-line may be perpendicular unto the Horizon, according to the 11 or 15 Chapters of the third Book. Note, that if the inclination of Meridian's shall be more than 45 deg. so that the thread doth cut the co-tangent line F E, than you must take the distance from F to the thread, and prick it down either from D or from F upon the line D C or F E for the twelve a clock point, according as the plane shall decline either Eastward or Westward, and his parallel taken from the sine of the height of the stile, shall give the distance from B to 6, and so of the rest. CHAP. VI To draw a Dial upon an East or West Inclining plane. IN these planes, as in the former, when we have the latitude of the place and the inclination of the plane, we have two things more to consider before we can draw the houre-lines upon the plane. First, the elevation of the Pole above the plane: Secondly, the inclination of both Meridian's. For the elevation of the Pole above the plane, lay the thread upon the Latitude counted in the limb, and take it over at the nearest 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 Meridian's, 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 thread to the nearest distance, and it will show upon the limb the inclination required. Dialling diagram And you may also see where the thread cutteth the co-tangent line E F, that so you may take the distance from the point F upon the Quadrant, unto the point of intersection of the thread with the co-tangent line; this distance you must set here from D to 12, and with the same, open the thread 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 showed. 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 showed the true situation of this Dial upon the plane. CHAP. VII. To draw a Dial upon a declining inclining plane. IN the making of these kind of Dial's by this Quadrant, when the latitude of the place, and the declination, and inclination of the plane is known, there is six things more to be considered before we can come to the drawing of this Dial upon the plane. 1 The inclination of the plane to the Meridian. 2 The Meridian's ascension. 3 The elevation of the Meridian. 4 The position latitude. 5 The elevation of the pole above the plane. 6 The inclination of Meridian's. All these six may speedily be found out upon the Quadrant after this manner. 1 To find the inclination of the plane to the Meridian. Lay the thread 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 compliment of the inclination of the plane to the Meridian. 2 To find the Meridian's 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 thread to the nearest distance; so shall it show upon the limb, the Meridian's ascension required. The thread 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, add the elevation of the Meridian to your Latitude; so shall the sum (if less than 90 deg.) be the position Latitude: but if the sum shall exceed 90 deg. take the compliment thereof to 180 deg. for the position Latitude here required. And if the plane shall incline toward the North, compare the Meridian's elevation with your Latitude, and subduct the lesser out of the greater, so shall the difference give you the position Latitude; if there be no difference, it is a declining polar plane, and may be described as in the latter part of the last Chapter of the third Book. 3 To find the elevation of the Pole above the plane. Lay the thread to the position Latitude counted in the limb, and take it over at the nearest extent from the sine of the inclination of the plane to the Meridian; and you shall have the sine of the elevation of the Pole above the plane. 4 To find the inclination of Meridian's. 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 thread to the nearest extent, so shall it show 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 find the inclination of the plane to the Meridian to be 64 deg. 20 min. Then I find the Meridian's ascension to be 33 deg. 41 min. In like manner I find the elevation of the Meridian to be 16 deg. 6 min. and because the plane inclineth towards the North, I compare this arch with the Latitude of the place, and finding it lest 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 Meridian's 30 deg. 52 min. The elevation of the Pole above the plane, with the inclination of Meridian's being thus found out, you may proceed to draw the Dial as in the former planes. The Dial being thus drawn on paper, you may place it in a right situation upon the plane, by help of the Meridian's ascension here found out, with the directions given in the last Chapter of the third Book. Dialling diagram In all kind of plane Dial's, the stile must be placed over the substile, making an angle therewith equal to the elevation of the Pole above the plane, as hath been fully showed in the third Book. CHAP. VIII. In any erect declining Dial, having the distance of the substile from the Meridian, in a known Latitude; how thereby to get the Cock's elevation, and the declination. IN any of these Dial's, 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 deflexion by the second Chapter of the second Book, which is the angle included between the line of 12 and the substilar, which admit to be 22 deg. 8 min. as in the Dial of the 15 Chap. of the third Book. This being found, take the sine of the Latitude betwixt your Compasses, and setting one foot in the co-sine of the angle of deflexion found, with the other lay the thread to the nearest distance: so shall it show upon the limb the elevation of the Pole above the plane. So in the Latitude of 52 deg. 30 min. the distance of the substile from the Meridian, being 22 deg. 8 min. as in the said 15 Chap. the elevation of the Pole above the plane will be 31 deg. 3 min. by which you may fashion a new Cock to the Dial at your pleasure. To find the Declination. Take the sine of the height of the stile, and setting one foot of your Compasses in the sine of the compliment of the Latitude, with the other lay the thread to the nearest distance: so shall it show upon the limb the compliment of the declination of the plane. So shall you find the declination of the former plane to be 32 deg. The end of the fifth Book. AN APPENDIX. SHOWING How the Parallels of Declination, the Parallels of the length of the day, the Jewish, Babylonish and Italian Hours; the Azimuths, Almicanters, and the like, may be easily inscribed in any Dial whatsoever, by Rule and Compass only. Whereby the Sun's place, the day of the Month, the Rising and setting of the Sun, the length of the Day and Night, the point of the Compass, and other necessaries, may be discovered at first sight, only by looking upon the Dial. Also how to draw a Dial on the ceiling of a Room. By W. L. LONDON, Printed by R. & W. Leybourn, for Thomas Pierrepont, at the Sun in Paul's Churchyard: 1652. THE APPENDIX. CHAP. I. How to describe the Equinoctial, Tropics; and other parallels of the Sun's course or Declination, in all kind 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 situate; but notwithstanding this, all great Circles, viz▪ such as divide the Sphere into two equal parts, as all Hours, Azimuths, and Orisons, are straight lines, though variously projected, according as the planes on which they are drawn do lie in respect of them. And all small Circles, viz. such as divide the Sphere unequally, are Conic 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 horizontal plane, are perfect Circles. For the Equinoctial Dial lying in the very plain of the Equinoctial Circle is parallel to all the parallels of declination; as the horizontal Dial lying in the very plain of the Horizon, is parallel to all the Almicanters or Circles of Altitude. Now because the Sun in his course moveth continually between the two Tropics; and never exceedeth those bounds; so likewise, all Astronomical conclusions that are to be drawn upon any Dial plane, are limited either by the Equinoctial, or by one or both of the Tropics: therefore it is requisite; first, to show you how to describe the Equinoctial and the Tropics upon all kind of planes, because it is them that limit and confine all other intermediate parallels, whether they be of the Sun's entrance into the Signs, or the Diurnal arches for the length of the day. And therefore I shall first show you how to perform this work upon such planes as lie parallel to the Axis of the World, as do the East and West Dial's, and the Polar, whether direct or declining. §. 1. In the East, West, and Polar Dial's. HOw to make an East or West Dial you are taught before, therefore let the Square A B C D be a plane, on which there is an East Dial drawn, the height of the stile being equal to the distance between the hours of 9 and 6, noted there with the letters E G, and let it be required to draw upon the same plane the Equinoctial and the two Tropics. Now the Equinoctial being a great circle of the Sphere, it is therefore a straight line, and is represented in the Dial following by the line H F. The houre-lines and the Equinoctial being thus drawn, we may proceed to the rest of the work in this manner. Dialling diagram Having fitted your Trigon, you must have recours to your Dial, and from thence with your Compasses you must first take out the distance E G (equal to the height of the stile of the same Dial) and prick it down in the Trigon from O to P, and draw the line P 6 perpendicular to O R. Secondly, going to your plane again, take the distance from G the top of the stile, to 7 in the Equinoctial of your plane; and place that distance in the Trigon from O to q, and draw the line q 7 perpendicular to O R. Thirdly, take out of your plane the distance G 8, and prick it down in your Trigon from O to r, and draw the line r 8 perpendicular 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 show you how they must be again transferred from the Trigon to the plane. Therefore, to find upon the houre-lines of your plane, the points through which the Tropic of Cancer must pass 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 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. Dialling diagram These points a b c d e f g h, being found upon the several and respective hour-lines, shall be the points through which the Tropic of Cancer, shall pass, therefore draw the line a b c d e f g h, and that shall be the Tropic of Cancer, so that when the Sun is in Cancer, (which is about the 11 of June) the top of the shadow of the stile of your Dial will run directly along that line a b c d e f g h, and when the Sun is in the Equinoctial, that is, in the beginning of Aries or Libra, (which is on the 10 of March, or the 12 of September) the top of the shadow of the stile will run along the Equinoctial line E F. The Tropic of Cancer being drawn, I will now show you how to draw the Tropic of Capricorn, which differeth nothing from that of Cancer, because they have both of them like declination from the Equinoctial: therefore the distance 8 k being made equal to the distance 8 e, and the distance 9 l equal to 9 f; and the distance 10 m equal to 10 g, you shall have the points k l m upon the hours of 8, 9 and 10, through which points k l m draw the line k l m, etc. which line shall represent the Tropic 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 shown you how to insert the Equinoctial and Tropics into your plane, I will now give you one rule by which you may put on any other intermediate parallels of the Sun's course, they differing nothing at all from the directions formerly given you to insert the Tropics. Consider therefore what parallels you would put on your plane, and find what declination the sun hath when he is in such a parallel, and accordingly insert those degrees of declination into your Trigon, as before you did for the Tropics. Example, Let it be required to put upon your plane, the parallels of the Sun's entrance into the 12 signs of the Zodiaque: You must, first, find what declination the Sun hath when he enters any of those Signs, which this little Table doth plainly show, 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. A TABLE showing what Declination the Sun hath at his entrance into the twelve SIGNS. 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 Equinoctial, they are to be transferred into your plane in all respects as the Tropics were, by taking out of your Trigon the distances from the line O R, to the several points where the hours cross the parallel, and place the same distances upon your plane from the Equinoctial 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 Sun's course at his entrance into all the 12 Signs of the Zodiaque, to which you may set the Characters of the Signs, as you see done in the figure. ¶ And here note, that if you draw upon your plane the halves and quarters of hours, and put them into your Trigon, and transfer them to your plane again, you shall then have more points, through which your parallels must pass, which will much help you in the drawing thereof, (especially in large planes) for there is no better way to draw these kind of lines, but by finding a great number of points, and so draw them by hand. ¶ Note also, that whatsoever is here spoken of the East and West Dial's, the same in all respects is to be observed in putting on the parallels of the Sun's course in all planes that lie parallel to the Axis of the world as the Polar, whether direct or declining. In all these kinds of planes, as the East, West, and Polar, the stile were best to be made of a straight piece of wire, equal in length to the line E G, fixed in the point E, standing perpendicular unto the plane, the end thereof at G being filled very fine and sharp, proportionable to the greatness of the plane, for all these Astronomical conclusions are showed (not by the shadow of the whole length of the stile, but) by the very Apex or top thereof, and therefore the more care ought to be had in the forming and making of it. ¶ The line MEN in the former East Dial is called the horizontal line, because it lieth parallel to the Horizon, and by the meeting of the parallels of the Sun's course with this line, the rising of the Sun may be nearly estimated, for there you see that the Tropic of Cancer cutteth this line near the point M, which is a little before the four a clock hour-line, which showeth, that when the Sun is in the Tropic of Cancer, he riseth somewhat before four in the morning, in like manner the Tropic of Capricorn cutteth the horizontal line something after 8, at which time the Sun riseth being in Capricorn, but this by the way the farther use of this line shall be showed hereafter. I have been the larger in the work of this plane, because I intent 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 show you how to do the like in the Horizontal, full South or Norrh planes, which are the next in order. §. 2. In the North, South, and horizontal Dial's. IN all these planes the substiler and the Meridian are all one, and the height of the stile, in the horizontal Dial is always equal to the latitude of the place, and you are to take notice, that whatsoever is here said of the full North & South upright planes, the same is to be understood of the full North & South reclining or inclining, all which in those latitudes whose compliment is equal to the height of the stile they are erect direct planes, & in those latitudes which are equal 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 Equinoctial, and the two Tropics in a full South erect plane. Having drawn your Dial, with the hours, 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 Tropics and other parallels of declination, for all these Astronomical conclusions are not showed by the shadow of the whole length of the stile or Axis, as the hour is, but by some point therein which representeth the centre of the earth, which in the Dial following is the point S, and the triangular stile in that Dial is represented by the triangle C S L, whereof C L is called the substilar, C S the Axis of the stile, and S L the perpendicular stile, the top of which, viz. S, is the point we are in this place to respect. The Dial being drawn, and the Triangle C S L made equal to the Cock of the Dial, you must upon a piece of pasteboard draw the Triangle O P R equal to the stile in your Dial C S L, making R O equal to C L the substilar, P O equal to C S the axis of the stile, and P R equal to S L the length of the perpendicular stile. Then from the point P, raise a perpendicular as P B, representing the Equinoctial, and on P as a centre, describe the arch A B C, now because the Tropics of Cancer and Capricorn do 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 Tropics 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 always the same with the twelve aclock line) from O to 12, cutting the Equinoctial line P B in the point a, then with your compasses take the distance O a out of your Trigon, and place it in your plane from the centre C unto a, and draw the line ♈ a ♎ perpendicular to the substile or line of 12. The Equinoctial being drawn: First, take out of your plane the distance C b, and place that distance in your Trigon from O unto b, and draw the line O b 1, representing the hour of 1 or 11 in your Dial. Secondly, take out of your plane the distance C c, and place that in your Trigon from O unto c, and draw the line O c 2, representing the 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 hours of 9 and 3. Fourthly, from our plane take the distance C e, and set in your Trigon from O unto e, and draw the line O e 4 representing the hours of 4 and 8. And thus must you do with the rest of the hours in your plane if occasion require. These lines O a; O b, O c, O d, and O e, in your Trigon, being extended, do cut the Tropic of Cancer P A in the points 12, 1, 2, and 3: therefore out of your Trigon take the distances O 12, O 1, O 2, O 3, O 4, and set them upon their correspondent hour lines of your plane, from the centre 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 Tropic of Cancer must pass, and is therefore noted with the character of Cancer ♋ at both ends. Dialling diagram ¶ Now before you draw the Tropic of Capricorn, it is necessary to draw the horizontal line of your plane A B, which line in all upright planes must be drawn through the point L, the foot of the perpendicular stile, and perpendicular to the Meridian or line of 12: And in all planes whatsoever, this line must be drawn through the intersection of the Equinoctial with the hour of six. This line ought first to be drawn, because it is very improper to extend the Tropics or other parallels of declination, above the horizontal line, because at what hour any parallel of declination cutteth this line, on either side of the Meridian, at that time doth the Sun rise or set, as was instanced in the last. Dialling diagram Now the Tropic 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 several houre-lines a b c d e do cut the Tropic of Capricorn P C, and place them on your plane from the centre C upon the respective houre-lines, and through those points so found, draw the line ♑ ♑, representing the Tropic of Capricorn. ¶ And in the same manner may the parallels of the other Signs 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 Signs in this erect direct plane, is universal in all planes, observing this one exception; that whereas in all erect direct planes the Equinoctial 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 Dial's. 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 Tropics and the Equinoctial upon such a plane. Dialling diagram 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 pass through the foot of the perpendicular stile, and the intersection of the Equinoctial with the hour of fix. Dialling diagram CHAP. II. Showing 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 Signs are inscribed upon all kind of planes by one and the same rules, they being in the Sphere the same Circles; so that as when you put on the parallels of the Sun's entrance into the 12 Signs, you seek what declination he hath, and accordingly proceed as before; so now for the parallels of the length of the day you must seek 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 hours, and take the difference, than the proportion will be; As the Sine of 90 degrees, Is to the Sine of half the difference; So is the Tangent compliment of the Latitude of the place, To the Tangent of the declination that the Sun shall have when the day is at such a length as you require. As for example, Let it be required to know what declination the Sun shall have when the day is 16 hours long in the Latitude of 52 deg. 30 min. The difference betwixt 16 hours and 12 hours is 4 hours, (or 60 deg.) the half of which is 30 deg. Therefore say, As the Sine of 90 deg. 10,000000 Is to the Sine of 30 deg. which is half the difference 9,698970 So the Tangent compliment 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 hours or 8 hours long in the Latitude of 52 deg. 30 min. Now if the day be above 12 hours long, the Sun hath North declination, but if less than 12 hours long he hath South declination. For those who are ignorant of these kind 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 showing what declination the Sun hath at such time that the day is either 8, 9, 10, 11, 12, 13, 14, 15, or 16 hours long, in the Latitude of 52 deg. 30 min. which Table was made by the preceding rules. By which table you may see Length of the day. The Sun's Declination 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 hours long the Sun hath then no declination, but is in the Equinoctial; but when the day is either 11 or 13 hours long, the declination is then 5 deg. 43 min. and when the day is either 9 or 15 hours long, the Sun hath 16 deg. 22 min. of declination, and so for the rest, as in the Table. For the placing of these parallels of the length of the day upon any of the forementioned planes; you must insert these angles of declination into your Trigon between the Tropics, and proceed in all respects as before. I will therefore give you but one example, which shall be in a full South plane, upon which and the horizontal these arches do appear most uniform. Now let it be required to draw the parallels of the Sun's course, when the day is 8, 9, 10, 11, 12, 13, 14, 15, and 16 hours long; upon a full South plane in the Latitude of 52 deg. 30 min. Having drawn your Dial with Hours halves and quarters, and also made choice of some convenient point in the stile to give the shadow, and draw the horizontal line C D, then make the triangle S A R in this Trigon equal to the triangle S A R in the following South Dial; as S A equal to the Axis of the stile, A R equal to the substilar, and R S equal to the perpendicular stile: then draw the perpendiculars S G for the Equinoctial, and describe the arch O G P, making G O and G P each of them 23 deg. 31 min. for the two Tropics, 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 hours long, the declination is 20 deg. 59 min. therefore place in your Trigon 20 deg. 59 min. from G unto a both ways, and draw the lines S a and S a, marking them at the ends with 8 and 16 the length of the day for which they serve. Likewise; when the day is either 9 or 15 hours long, than the Sun's declination is 16 deg. 22 min. therefore set 16 deg. 22 min. from G unto b both ways; and draw S b and S b. Also when the day is either 10 or 14 hours long, than the declination is 11 deg. 14 min. which set from G to c both ways, and draw S c and S c. Lastly, when the day is 11 or 13 hours long the declination is 5 deg. 43 min. which set from G unto d both ways, and draw S d and S d, noting them with numbers answering to the length of the day, as you see in the Trigon, when the day is just 12 hours long it is Equinoctial and hath no declination, and is signified in the Trigon by the line S G. Dialling diagram 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 Signs, not at all differing therefrom, and therefore I shall forbear to give you any farther instructions for the performance thereof but give you the figure of a South plane with these parallels drawn thereon, which will instruct more than a whole Chapter of information. Dialling diagram And thus much for the drawing of the parallels of the Signs and Diurnal arches in all kind of planes. I will now proceed to show you how some other Astronomical conclusions (which are very pleasing and delightful) may be inscribed upon all sorts of Dial's. CHAP. III. Showing how the Italian and Babylonish hours may be drawn upon all kind of planes. THe Italians account their hours from the Suns setting, and the Babylonians from his rising, so that these kind of houre-lines being drawn upon any plane, you may know (by inspection only) how many hours are passed since the last setting or rising of the Sun. The inscription of these houre-lines into any of the former planes is very easy, the work of the last Chapter being well understood. Because that upon a full South or an horizontal plane, these houre-lines show themselves most uniform, I have therefore for example sake, made choice of a full South Dial, upon which it shall be shown how to draw both the Italian and Babylonish hours. Your Dial being drawn, and the two Tropics and the Equinoctial thereon inscribed, and also the horizontal line, you must draw in your Dial two obscure parallels of the length of the day, one when the day is 8 hours, and the other when the day is 16 hours long, expressed in the following Dial by the two pricked arches near the two Tropics, the uppermost of which is the parallel of the Sun's course when the day is 8 hours long, and the undermost is the parallel of his course when the day is 16 hours long, & the Equinoctial is the parallel of the Sun's course when the day is 12 hours long. Your Dial being thus prepared, and these parallels thus inserted, the inscription of these houre-lines is very easy and plain to be understood. To begin then with the inscription of the Babylonish hours (which are the hours from the Suns rising.) First, It is apparent that when the day is 8 hours long, that the Sun riseth at 8 in the morning, so that at that time, the first hour after the suns rising is 9 in the morning. Secondly, when the day is 12 hours long, the Sun riseth at 6 in the morning, so that at that time the first hour after the suns rising is 7 in the morning. Thirdly, when the day is 16 hours long, the Sun riseth at 4 in the morning, so that the first hour 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 hours long, the seventh hour from sun rising is 3 in the afternoon. When the day is 12 hours long, the seventh hour from sun rising is 1 in the afternoon. And when the day is 16 hours long, the seventh hour from the suns rising is 11 before noon, as by this Table doth evidently appear. And therefore a straight line drawn in your Dial through those points where the common houre-lines of your Dial cross the respective parallels of the day's length, shall show the true quantity of hours since the suns rising at all times of the year, which is the Babylonish hour. Dialling diagram For example, let it be required to draw the seventh hour from the suns rising in your Dial. First, by the Table you see, that in the parallel of 8 hours for the length of the day, the seventh hour from the suns rising is 3 in the afternoon, therefore observe where the houre-line of three crosseth the parallel of 8 hours, which is at a. Secondly, by the Table you see that in the parallel of 12 hours for the length of the day, the seventh hour from sun rising is then 1 in the afternoon, wherefore observe where the houre-line of 1 crosseth the Equinoctial, which is at b. Thirdly, by the Table you see that in the parallel of 16 hours for the length of the day, the seveth hour 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 straight line a b c, which shall be the seventh Babylonish hour, or the seventh hour from the suns rising all the year long. And by this rule, and the help of the Table, you may draw all the other hours from sun rising, as you see them drawn in the figure, and put numbers to them as you see there done. ¶ 1 Note, That if any of the points you are to make use of for the drawing of any of these hours fall without your plane, you must in this case extend your houre-line; parallel and Equinoctial, beyond the limits of your Diall-plane, and there make use of the points, but you need extend the line you draw no farther than the bounds of the plane, as here in the figure you see the first hour from Sun rising crosseth not the Equinoctial and the houre-line of 7 within the plane, but if the Equinoctial and the houre-line of 7 were extended, it would cross. ¶ 2 Note, That if any of the three points you are to make use of do 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 hours from sun rising were put into the plane, by the same rule may the hours from sun setting (or Italian hours) be inserted, the difference being only in the numbering of them; the hours from the sun rising being numbered from the West side of the horizontal line by 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11; and the hours from the suns setting are denominated from the East side of the Horizon, and 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. IU. Showing how the Jewish hours may be drawn upon any plane. IT was the custom of the Ancients to divide their day and also their night (whether long or short) into 12 equal parts, beginning their day at the suns rising, and their night at the suns setting; so that 12 of the clock at noon was always the sixth hour of their day, and 12 at night was always the sixth hour of their night, and according to this division were their Dial's drawn; so that all the Summer the hours of their day were longer than the hours of their night; and all the Winter, the hours of their night were longer than those of their day; and when the Sun is in the Equinoctial, than the hours of their day and night were equal, and the same with those of our account, but at all other times of the year different. The inscribing of these hours into all kind of planes is very easy, being much like the drawing of the Babylonish and Italian hours before taught. Having therefore drawn your Dial (which in this example (for the avoiding of many figures) we will have to be the full South plane used before in Chap. 2. of this Appendix) with the hours, halves, and quarters, and also drawn the two Tropics and the parallels of the length of the day thereupon, as you see here done in this figure. Then make choice of two parallels of the length of the day, which must be both of them equidistant from the Equinoctial, which let be the parallels of 9 hours and 15 hours, both which are three hours different from the Equinoctial on either side thereof, and these two parallels are the most convenient for this our purpose, because the Jewish hours will fall (in these two parallels) justly upon the hours, 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 pass is exactly showed by this little Table, wherein you may see that the first Jewish hour must be drawn through 5 hours 45 min. (or 5 hours three quarters) in the parallel of 15: through 7 hours in the Equinoctial: Jewish hours. The parallel of 15 hours. Equinoctial. 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 hours & a quarter in the parallel of 9 hours. In like manner, the second Jewish hour must be drawn in your plane through 7 of the clock in the parallel of 15: through 8 a clock in the Equinoctial: and through 9 of the clock in the parallel of 9 hours, and so of all the rest, according as you see in 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. Dialling diagram CHAP. V Showing how to draw the Azimuths, or Vertical Circles in all kind 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 situation. In the horizontal plane they meet in a centre with equal angles. In all upright planes, whether direct or declining, they are parallel to the Meridian or line of 12. And in all reclining planes they meet together in a point which is the Zenith of the place. These Azimuths being great Circles in the Sphere, are therefore straight lines in all planes, and may be drawn as followeth. §. 1. In the horizontal plane. IN the horizontal plane these Azimuths are most easily inserted, for your Dial being drawn, with the Tropics thereon, you have no more to do, but upon the foot of the perpendicular stile to describe a Circle, which you may divide into 32 equal parts (beginning at the Meridian) answering to the 32 points of the Mariner's Compass: Or else you may divide the same Circle into 90 equal parts, according to the Astronomical division, and through each of those points draw straight lines from the foot of the stile, and set numbers or letters to them, either by 10, 20, 30, 40, etc. if you divide it into 90, or else by South, S by W, S S W, S W by S, etc. if you divide the Circle according to the Mariner's Compass. This is so plain that it needeth no example. §. 2. In the East or West erect planes. YOur Dial being finished, you may draw upon a piece of Pasteboard the line M E N, representing the horizontal line M E N in your Dial then on the point E, raise the perpendicular E Q equal to the line EGLANTINE in your Dial, and on Q as a centre describe the semicircle K E L, and divide one half thereof, namely, the Quadrant E L into eight equal parts, representing one quarter of the Mariner's Compass, and from the centre 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 ☉ ☉, etc. and prick them down in the horizontal line of your plane from E to ☉ ☉ ☉ ☉ ☉ ☉, from which points draw lines perpendicular to the horizontal line M E N, which shall be the Azimuths or points of the Compass between the East and the South. Divide likewise the other Dialling diagram Quadrant of the Circle E K into eight equal parts, and draw lines from the centre Q through three of them, till they cut the horizontal line as you see in the figure, and there also draw lines perpendicular to the Horizon, and these lines shall be the Azimuths between the East and the North, viz. so many of them as your plane is capable to receive, which the following figure doth most plainly show. ¶ Here note, that as the East Dial showeth all the morning hours from sun rising to the Meridian; and the West Dial showeth all the afternoon hours from the Meridian to his setting; so doth the East Dial show all the Azimuths from the Suns rising till Noon, and the West Dial all the Azimuths from noon till his setting. Dialling diagram §. 3. In the full North and South erect planes. THe drawing of the Azimuths upon the full North or South erect planes is very little different from the drawing of the same Circles upon the East or West planes. But for example, let it be required to draw the Azimuths upon the full South Dial: the Tropics and the Equinoctial being drawn, together with the horizontal line, you must upon a piece of Pasteboard draw the line A L B, representing the horizontal line A L B in the South Dial next following; then on the point L raise the perpendicular L S, making L S equal to L S the perpendicular stile of the Dial, and on S as a centre describe the semicircle E L F, and divide each Quadrant thereof, namely E L and L F into 8 equal parts (each Quadrant representing one quarter of the Mariner's Compass) and through each of those divisions draw lines from the centre S, till they cut the line A L B in the points m n oh p and q, then Dialling diagram with your Compasses take the distance L q, and set that distance upon the horizontal line of your plane from L unto q both ways. Likewise, take the distance L p, and set that distance in your plane upon the horizontal line thereof from L unto p both ways. Also take the distances L o, L n, and L m, and set them upon the horizontal line in your Dial from L to o, and n, and m, on each side of the Meridian. Lastly, if from the points m, n, o, p, & q, you draw lines parallel to the Meridian or line of 12, they shall be the true Azimuths upon your plane, and these Azimuths may be put on either according to the Astronomical account by 10, 20, 30, 40, etc. or else by the points Dialling diagram of the Compass, 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 Dial, which we will suppose to be the South declining plane before used in the third Section of the first Chapter of this Appendix, which declineth from the South Eastward 32 deg. Your Dial being drawn and the Equinoctial, and Tropics, and also the horizontal line, thereon inscribed, upon a piece of pasteboard draw the line D B E, representing the horizontal line D B E in your Dial, then on the point B raise the perpendicular B C, making B C equal to B C the perpendicular stile in your Dial; then on the point C as a centre describe the Semicircle R B S, then out Dialling diagram of your plane take the distance between B the foot of the stile, and the point O, where the Meridian crosseth the horizontal line, and set that distance on your pasteboard from B unto O, and draw the line C O. cutting the Semicircle in the point K. at which point K you must begin to divide your Semicircle into 16 equal parts, then from the centre 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, etc. out of your pasteboard, 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 situation of the Dialling diagram plane, viz. the Western Azimuths on the East side of the Meridian, and the Eastern Azimuths on the West side of your Dial, as you see them here 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, 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 compliment 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, than the Meridian will be a tangent line thereunto, upon which Meridian, from the foot of the perpendicular stile, prick down 65 deg. for the Zenith point where all the Azimuths must meet. and 25 deg. for the horizontal point, through which the horizontal line must pass, Then describing a Semicircle, divide it into 16 parts, and lay a ruler from the centre and each of those divisions till it cut the horizontal line, and thereon make marks, then lay a ruler upon the Zenith point and each of these marks in the horizontal line, and they shall be the true Azimuths belonging to your plane, which you must number according to the situation 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 several lines, you must therefore draw a line perpendicular to the base of the plane, which must pass directly through the foot of the perpendicular stile, then make the perpendicular stile the Radius, and the other line last drawn shall be a Tangent line thereto, upon which line set off the inclination of the plane to the Horizon, and that shall give you the Zenith point, and the horizontal point shall be found by setting off the reclination of the plane from the Zenith, and here note that the Zenith point will always 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 Tropics and parallels of declination have to the houre-lines, and therefore, as the parallels of declination in the Equinoctial plane are perfect circles, so are the circles of Altitude in an horizontal plane. The inscription of these into all kind 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 Conic sections, except on such planes as lie parallel to the Zenith, which is only the horizontal. CHAP. VII. How to draw a Dial on the Ceiling of a Room. BEcause the direct beams of the Sun can never shine upon the ceiling of a Room, they must therefore be reflected thither by help of a small piece of Looking glass conveniently fixed in some Transam of the window, so that it may lie exactly parallel to the Horizon. The place being chosen, and the glass therein fixed, you must draw upon the ceiling of the Room a Meridian line, as you are taught in the former Books, which Meridian line must be so drawn that it may pass directly over the glass before placed, which you may perceive how to do by holding a thread and plummet from the top of the ceiling till it fall directly upon the superficies of the glass. Dialling diagram The Hours. The angle that each houre-line makes with the Merid. The compliment of each houre-lines angle with the Meridian 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 ceiling of a room, and let K be the glass fixed directly under the said Meridian upon some transam of the window, then laying one end of a string upon the glass at K, extend the other up to the Meridian at L, which point L you may find by moving the string to and fro upon the Meridian line, till another holding the side of a Quadrant to the movable string, he shall find the thread and plummet to fall directly upon the compliment 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 A perpendicular to the Meridian L R, which line shall be the Equinoctial. Having thus done, upon a table or such like draw a line which shall be of equal length with LK, the distance from the glass to the point L on the ceiling, which line divide into 10 equal parts, and each of those (or at least one of them) into 10 other parts, so shall you have in all 100 parts, each of which you must suppose to be divided into 10 other smaller parts, so shall the whole line contain 1000 parts, as in the figure is expressed by the line S. Dialling diagram Now because the centre of the Dial is without the Room, 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 compliment of every hours arch from the Meridian, and place them upon the respective hour arches from the Equinoctial to the points ***** as you see in the figure. Lastly, if you draw the lines * M, * N, * P, * Q, they shall be the true hours upon the ceiling. In the inscription of the Azimuths in declining reclining planes, and in drawing the circles of Altitude in all kind of planes, I confess I should have been somewhat larger in giving you an example in each plane, as I did with the other varieties before, but pre-supposing the ingenious practitioner sufficiently to understand that which goes before, he cannot but with small pains overcome the rest; But I should not have been so brief, could I possibly have procured more time, which by no means would be granted. Also my intent in this place was to have shown you the inscription of the Circles of position and other varieties: Also the framing of divers Geometrical Bodies, and to furnish them with variety of Dial's, and the making of divers Instrumental Dial's: But these, with many other HOROLOGICAL conceits and inventions, I reserve till a more convenient opportunity, and therefore in the mean time, Farewell. FINIS.