Selected quad for the lemma: end_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
end_n	cut_v	draw_v	line_n	1,514	5	9.0981	5	false

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A51553	A tutor to astronomie and geographie, or, An easie and speedy way to know the use of both the globes, coelestial and terrestrial in six books : the first teaching the rudiments of astronomy and geography, the 2. shewing by the globes the solution of astronomical & geographical probl., the 3. shewing by the globes the solution of problems in navigation, the 4. shewing by the globes the solution of astrological problemes, the 5. shewing by the globes the solution of gnomonical problemes, the 6. shewing by the globes the solution of of [sic] spherical triangles : more fully and amply then hath ever been set forth either by Gemma Frisius, Metius, Hues, Wright, Blaew, or any others that have taught the use of the globes : and that so plainly and methodically that the meanest capacity may at first reading apprehend it, and with a little practise grow expert in these divine sciences / by Joseph Moxon ; whereunto is added Antient poetical stories of the stars, shewing reasons why the several shapes and forms are pictured on the coelestial globe, collected from Dr. Hood ; as also a Discourse of the antiquity, progress and augmentation of astronomie.	Moxon, Joseph, 1627-1691.; Hood, Thomas, fl. 1582-1598.	1659 (1659)	Wing M3021; ESTC R23159	189,557	267

that shadow shall be a Meridian li●e Secondly on the backside the Clinatory discribe a Circle and draw a line through the Center to both sides the Circumference cross this line with an other line at R●ght Angles in the Center so shall the Circle be divided into four equal parts These four parts you must ma●k with East West North South and divide each of them into 90. degrees In the Center of this Plain erect a straight wyer prependicularly when you would find a Meridian line examine by the tenth Prob. of the second Book the Amplitude of the Suns Rising or Setting from the East or West points and waiting the just Rising or Setting that Day turn the Instrument about till the shadow of the wyer falls upon the same degree from the East or West the Amplitude is of for then the North and South line in the Instrument will be the same with the North and South line in Heaven Thirdly by the Suns Azimuth Find the Azimuth of the Sun by Prob. 22. of the second Book and at the same instant turn the Instrument till the shadow of the wyer fall upon the degree on the Instrument opposite to the degree of the Suns Azimuth so shall the Meridional line of the Instrument agree with the Meridional line in Heaven You may the same way work by the Azimuth of any Star Only whereas the shadow of the wyer should fall upon the opposite degree aforesaid Now you must place a Sight or Perpendicular upon that opposite degree and turn the Instrument about till the wyer at the Center the Sight in the opposite degree of the Stars Azimuth and the Star in Heaven come into one straight line so shall the Meridian line of the Instrument agree with the Meridional line in Heaven Fourthly It may be found by any Star observed in the Meridian if two Perpendiculars be erected in the Meridian line of your Instrument for then by turning the Instrument till the two Perpendiculars and the Star come into a straight line the Meridian line of your Instrument will be the same with the Meridian line in Heaven See more waies in Mr. Palmer on the Planisphear Book 4. Chap. 9 If your Plain either Recline or Incline apply one of the sides of your Clinatory Parallel to one of the Semi-diameters of the Quadrant to the Plain in such sort that the Plumb-line hanging at liberty may fall upon the Circumference of the Quadrant for then the number of degrees of the Quadrant comprehended between the side of the Quadrant Parallel to the Plain and the Plumb-line shall be the number of degrees of Reclination if th● Center of the Quadrant points upwards or Inclination if th● Center points downwards If your Reclining or Inclining Plain Decline draw upon it a line Parallel to the Horizon which you may do by applying the back-side of the Clinatory and raising or depressing the Center of the Quadrant till the Plumb-line hang just upon one of the Semi-diameters for then you may by the upper side of the Clinatory draw an Horizontal line if the Plain Incline or by the under side if it Recline If it neither Incline or Recline you may draw● an Horizontal line both by the upper and under sides of the Clinatory Having drawn the Horizontal line apply the North 〈◊〉 ● of the Clinatory to it and if the North end of the Needle 〈◊〉 directly towards the Plain it is then a South Plain If the 〈◊〉 point of the Needle points directly from the Plain it is a Nor●● plain but if it points towards the East it is an East Plain if towards the West a West Plain If it do not point directly 〈◊〉 East West North or South then so many degrees as the 〈◊〉 declines from any of these four points to any of the other of 〈◊〉 four points so many degrees is the Declination of the Plain 〈◊〉 respect as aforesaid had to the Variation of the Compass Or if you find the Azimuth of the Sun by its Altitude observed just when its beams are coming on or going off you● Plain that Azimuth shall be the Azimuth of your Plain Or you may erect a wyer Perpendicularly on your Plain and wait till the shadow of that wyer comes to be Perpendicular with the Horizon which you may examine by applying a Plumb-line to it for then the shadow of the Plumb-line and the shadow of the Perpendicular will be in one then taking the Altitude of the Sun you may by Prob. 22. of the second Book find its Azimuth and thereby know in what Azimuth the Plain of your Dyal lies for the Azimuth your Plain lies in is distant from the Azimuth of the Sun just 90. degrees PROB. I. How by one position of the Globe to find the distances of the Hour-lines on all manner of Plains YOu may have Meridian lines drawn from Pole to Pole through every 15. degrees of the Equinoctial to represent the Horary motion of the Sun both Day and Night and when the Pole of the Globe is Elevated to the height of the Pole in any Place and one of these Meridian lines be brought to the Brazen Meridian all the rest of the Meridian lines shall cut any Circle which you intend shall represent the Plain of a Dyal in the number of degrees on the same Circle that each respective Hour-line is distant from the Noon-line point in the same Circle Thus if you should enquire the distance of the Hour-lines upon an Horizontal Plain in Londons Latitude The Pole of the Globe as aforesaid must be Elevated 51½ degrees and one of the Meridian lines you may chuse the Vernal Colure be brought to the Brazen Meridian which being done you are only to examine in the Horizon Because it is an Horizontal Plain at what distance from the Meridian which in Horizontals is the Noon-line the several Meridians drawn on the Globe intersect the Horizon for that distance in degrees shall be the distance on a Circle divided into 360. degrees that each respective Hour-line must have from the Meridian or a Noon line chosen in the same Circle and lines drawn from the Center of that Circle through those degrees shall be the Hour lines of an Horizontal Plain If your Plain be not Direct but declines East or West 〈◊〉 must number the Declination Eastwards or Westwards re●pectively in the degrees of the Horizon and the Quadrant 〈◊〉 Altitude screwed to the Zenith as aforesaid bring the lower end of the Quadrant of Altitude to the said degrees of Declination and the number of degrees cut by the Meridians in the Quadrant of Altitude numbred downwards is the number of degrees that the hour-Hour-lines are distant from the Noon line in a Circle of 360 degrees And lines drawn from the Center of that Circle through those degrees be the Hour lines of half the Day And if you turn about the Quadrant of Altitude upon the Zenith point till the lower end of it come to the degree of the Horizon

Thus The 1 a clock Hour-line 11.40 whose Complement 90. is 78.20 2 24.15 65.45 3 38.4 51.56 4 53.36 36.24 I measure in a Quadrant of the same Radius with those arches already drawn from the Equinoctial line for the 1 a clock Hour 78.20 2 65.45 3 51.56 4 36.24 and transfer these distances to the Arches drawn on the Ceeling For then straight lines drawn through the mark in the Arch and through the mark in the Equator and prolonged both waies to a convenient length shall be the several Hour-lines a foresaid And when the Sun shines upon the Glass at Nodus its Beams shall reflect upon the Hour of the Day PROB. XVI To make a Dyal upon a solid Ball or Globe that shall shew the Hour of the Day without a Gnomon THe Equinoctial of this Globe or which is all one the middle line must be divided into 24 equal parts and marked with 1 2 3 4 c to 12. and then beginning again with 1 2 3 c. to 12. Then if you Elevate one of the Poles so many degrees above an Horizontal line as the Pole of the World is Elevated above the Horizon in your Habitation and place one of the 12 s directly to behold the North and the other to behold the South when the Sun shines on it the Globe will be divided into two halfs the one enlightened with the Sun-shine and the other shadowed and where the enlightned half is parted from the shadowed half there you shall find in the Equinoctial the Hour of the Day and that on two places on the Ball because the Equinoctial is cut in two opposite points by the light of the Sun A Dyal of this fort was made by M r John L●●k and set up on a Composite Columne at Leaden Hall Corner in London in the Majoralty of S r John Dethick Knight The Figure whereof I have inserted because it is a pretty peece of Ingenuity and may perhaps stand some Lover of Ar● in stead either for Imitation or help of Invention PROB. XVII To make a Dyal upon a Glass Globe whose Axis shall cast a shadow upon the Hour of the Day FIrst divide the Equinoctial of your Globe into 24 equal parts and having a Semi-Circle cut out of some Brass plate or thin Wood to the same Diameter your Globe is of or a very little wider Apply this Semi-Circle to the Globe so as the upper edge of each end of the Semi Circle may touch the Poles of the Globe and the middle of the Semi Circle may at the same edge cut through some division made in the Equinoctial for then a line drawn by the edge of the Semi Circle thus posited shall be a Meridian line The same way you must draw Meridian lines through every division of the Equinoctial and set figures to them beginning with 1 2 3 4 c. to 12 and then beginning again with 1 2 3 4 c. to 12 again This Globe being made of Glass and having an Axis of Wyer passing through ●t from Pole to Pole will be an Horizontal Dyal all the World over if its Axis be set Parallel to the Axis of the World in the same Place and one of the Meridians marked 12 be set so as it may directly behold the North point in Heaven and the other the South point in Heaven for then the Axis of the Globe shall cast a shadow upon the Hour of the Day And if you divide the upper half of the Glass Globe from the under half when the Axis stands Parallel to the Axis of the World by a Circle drawn round about the Globe that Circle shall represent the Horizon and the Meridian lines drawn on the Globe shall be the Hour lines and have in the Horizontal Circle the same distance from the 12 a clock line that the same respective Hour line was found to have as by Prob. 3. of this Book But because the shadow of this Axis will not be discerned through the Glass Body therefore you may with Water and white Lead ground together lay a Ground on the Inside of the under half of the Glass to the Horizontal Circle as Looking-glass makers do their Looking Glasses with Tinfoil for then the shadow will appear Such a Glass Globe Dyal hath the Lord Robert Titchborn standing in his Garden supported by Atlas The End of the Fifth Book The Sixth BOOK Shewing the Practical Use of the GLOBES Applying them to the Solution of Spherical Triangles PRAEFACE THe Solution of Spherical Triangles is to know the length of its Sides and the width of its Angles These have already by many learned Men been taught to be performed by a Canon of Sines and Tangents and also by many Instruments some serving as Tables of Sines and Tangents such as are the Sectors Scales the Spiral line c. and others serving to represent the Globe such as be the Mathematical Jewel Astrolabium Catholicum and several other Projections of the Sphear But none hath as yet taught the Solution of Spherical Triangles by the Globe it self though it be the most natural and most demonstrative way of all and indeed ought first to be learnt before the Learner enters upon any other way To this Authors of Trigonometry agree for the most of them in their Books give Caution that the Learner be already sufficiently grounded in the Principles of the Globe For those Lines or Circles which either in Tables or other Instuments your force your Imagination to conceive represents your Line or Circle in question those Lines and Circles I say you have Actually and Naturally discribed on the Globe and therefore may at a single Operation or perhaps only by a sudden inspection have an Answer annexed according as the nature of your Question shall require and that more Copiously then by Tables of Sines and Tangents For therein you find but one Question at once resolved but by the Globe you have alwaies two resolved together Of the Parts and Kindes of Spherical Triangles THEOREMS 1. ALL Spherical Triangles are made of six parts Three Sides and three Angles The Sides are joyned together at the Angles and measured by degrees of a Great Circle from one end to the other The Angles are the distance of the two joyned sides and they are also measured by an Arch of a Circle discribed on the Angular point If any three of these parts be known the rest may be found 2. All Spherical Triangles are either Right Angled or Oblique Angled A Right Angle contains 90. degrees An Oblique Angle either more or less 3. If a Spherical Triangle have one or more Right Angles it is called a Right Angled Spherical Triangle But if it have no Right Angle it is called an Oblique Angled Spherical Triangle 4. If an Oblique Spherical Triangle have one Angle greater then a Right Angle it is called an Obtuse Angled Spherical Triangle But if it have no Angle greater it is called an Accute Angled Spherical Triangle 5. In Right

opposite to the degree of Declination found before the Meridian lines on the Globe as before shall cut the Quadrant of Altitude in the number of degrees counted downward that each hour-Hour-line is distant from the other side the noon-Noon-line And lines drawn from the Center of that Circle through those degrees shall be the Hour-lines of the other half of the Day If your Plane Decline and also Recline or Incline you must use the Gnomonical Semi-Circle discribed in Prob. 12 which must be Elevated on the Quadrant of Altitude when it is set to the Declination as by the former Rule according to the complement of Reclination or Inclination But if your Plane be Direct and Recline or Incline it must be set to the Meridian and the Meridians on the Globe shall cut that Semi Circle in the number of degrees counted from the Quadrant of Altitude if the Plane Declines or from the Brasen Meridian if it be Direct that the several Hour lines are distant from a line Perpendicular to an Horizontal line in a Circle divided into 360 degrees And lines drawn from the Center through those degrees shall be the Hour-lines of such Reclining or Inclining Planes The moving this thred from wyer to wyer represents the motion of the Sun which as it passes over all the Meridians causes the shadow of that Meridional Semi Circle which it is directly over and the Axis and the Meridional Semi-Circle directly opposite to the upper Meridional Semi-Circle to fall all into one straight line And upon what point in the East and West line mentioned before that shadow-line shall fall is marked ●ut by the application of the thred as aforesaid and is an Hour-line on any of the foresaid Planes If you understand this Probleme rightly you do already know how to draw the Hour lines upon all manner of Planes and need no further Instructions yet partly fearing a raw Student should not clearly understand these Rules and partly doubting because other Authors have been more Copious upon this Subject that I should be censured to be too sparing of my pains if I should lightly touch so eminent a Doctrine as Dyalling is Therefore I shall more distinctly handle Dyalling by the Globe according to the way or Method that other Authors have used and that after so plain a manner as possibly my Genius can devise PROB. II. To make an Equinoctial Dyal DIscribe a Circle on a square board or Plane as B C E D and through A the Center thereof draw a straight line Parallel to one of the sides as B E Cross that straight line with another straight line as C D at Right Angles so shall the Circle be divided into 4 equal parts Divide each of these four equal parts into 90. degrees as in the Figure This Circle shall represent the Horizon Erect a wyer exactly perpendicular to the Center of the Plane and that wyer shall be the Gnomon or Style of the Dyal Then Elevate one of the Poles of your Globe into the Zenith and bring the Equinoctial Colure to the Meridian And because in every hours Time 15 degrees of the Equator passes through the Meridian in Heaven therefore turn the Globe till 15 degrees of the Equator pass through the Meridian of your Globe so shall the Colure pass by 15 degrees of the Horizon also Therefore from the Center of your Plane draw straight lines through 15 degrees from one of the Semidiameters both waies and those straight lines shall be two Hour-lines Then turn the Globe till 15 degrees more of the Equator pass through the Meridian and you will find as before the Colure pass by 15 degrees more of the Horizon therefore on your Plane number 15. degrees further beyond both the former lines and from the Center draw straight lines through both those 15. degrees and they shall be two Hour lines more Fór all the other Hour lines turn the Globe till 15. degrees of the Equator at a time pass through the Meridian as before and you will find that for every 15. degrees of the Equator that passes through the Meridian the Colure will pass through 15. degrees of the Horizon therefore those Hour lines must be drawn from the Center according to the succession of every 15 degrees on your Plane Having drawn the Hour lines you may set figures to them beginning to number your Hour lines from one of the Diameters marking it with XII and the next Hour line to the left hand with I and the next II the next III c. to XII and begin again with I II III c. till you come to the other XII where you began and then your Dyal is finished See the Figure This is an Universal Dyal and serves in all Latitudes therefore when you place it you must set one of the XII s downwards and the Axis Parallel to the Axis of the World But note Both faces of this Dyal ought to be divided and the Gnomon must appear on both sides like the stick in a Whirligig which childeren use or else you must turn it upside down so oft as the Sun passes the Equinoctial PROB. III. To make an Horizontal Dyal DIscribe a Circle on your Plane as C B D E and through the Center A of that Circle draw a Meridian line as B E cross that line at Right angles with another line as C D so shall your Circle be divided into four equal parts Divide each of these four parts into 90. degrees so shall the whole be divided into 360. These 360 degrees represent the 360 degrees of the Horizon which a Meridian line drawn through the place of the Sun runs through in every 24. Hours The motion of which Meridian line through the degrees of the Horizon is Regular in a Parallel Sphear for in equal Time it moves an equal Space throughout the whole Circle viz. it will pass through 15. degrees of the Horizon in one Hours Time or which is all one whiles 15. degrees of the Equator passes through the Meridian as was shewed in the last Probleme But in an Oblique Sphear its motion through the Horizon is Irregular and that more or less according to the more or less Obliquity of the Sphear For far Northwards or Southwards you may see this Meridian line pass through 40 50 yea 60. degrees of the Horizon in one Hours time viz whiles 15. degrees of the Equator passes through the Meridian but in an other Hours time you will scarce have 4 or 5 degrees pass through the Horizon whiles 15 degrees of the Equator passes through the Meridian But that you may know the motion of the Sun represented by this Meridian line through the Horizon in all Latitudes Elevate the Pole to the Elevation of your Place and chuse instead of a Meridian line drawn through the Place of the Sun the Vernal Colure to be your Meridian line both because it is most visible and because from thence the degrees of the Equator are begun to

on the Meridian till the graduated edge cut the degree of the Ecliptick the Sun is in Then I examine on the Meridian what degree the upper end of the Quadrant of Altitude touches which in this example I find is 38½ degrees Therefore I substract 38½ from 51½ Londons Latitude and there remains 13. Then counting on the Meridian 13. degrees backwards from the Place where the Quadrant of Altitude touched the Meridian I come to 25½ on the Meridian Northwards Therefore I say In the North Latitude of 25½ degrees and in the Longitude of London which is in Africa in the Kingdom of Numidia the Sun May 10. at 53. minutes past 8. a clock in the Morning hath the same Altitude above the Horizon it hath here at London The Quadrant of Altitude thus applyed to the East point of the Horizon makes right angles with all points on the Meridian even as all the Meridians proceeding from the Pole do with the Equator therefore the Quadrant being applyed both to the East point and the Suns Place projects a line to intersect the Meridian Perpendicularly in equal degrees from which intersection the Sun hath at the same time equal Heighth be the degrees few or many for those 5. degrees to the Northwards of this intersection have the Sun in the same heighth that they 5 degrees to the Southwards have it and those 10 20 30. degrees more or less to the Northwards have the Sun in the same heighth that they have that are 10 20. 30. degrees more or less to the Southwards So that this Prob. may be performed another way more easily with your Compasses Thus Having first rectified the Globe and Hour Index Turn about the Globe till the Hour Index point to the Hour of the Day Then pitch one foot of your Compasses in the Suns Place and extend the other to the degree of Latitude on the Meridian which in this example is 51½ degrees North then keeping the first foot of your Compasses on the degree of the Sun turn about the other foot to the Meridian and it will fall upon 25½ as before Blaew commenting upon this Probleme takes notice how grosly they ere that think they can find the heighth of the Pole at any Hour of the Day by the Suns height because they do not consider that it is impossible to find the Hour of the Day unless they first know the height of the Pole PROB. XLVIII To find the length of the Longest and Shortest Artificial Day or Night THe Artificial Day is that space of Time which the Sun is above the Horizon of any Place and the Artificial Night is that space of Time which the Sun is under the Horizon of any Place They are measured in the Hour Circle by Hours and Minutes There is a constant unequallity of proportion in the Length of these Daies and Nights which is caused both by the alteration of the Suns Declination and the difference of the Poles Elevation Those that inhabite on the North side the Equator have their longest Day when the Sun enters ♋ and those that inhabite on the South side the Equator have their longest Day when the Sun enters ♑ But to know how long the longest Day is in any North or South Elevation Raise the North or South Pole according to the Elevation of the Place and bring ♋ for North Elevation or ♑ for South Elevation to the Meridian and the Index of the Hour Circle to 12. Then turn the Globe about till ♋ for North Elevation or ♑ for South Elevation come to the West side the Horizon and the number of Hours and minutes pointed at on the Hour Circle doubled is the number of Hours and minutes of the Longest Day The length of the Night to that Day is found by substracting the length of the day from 24. for the remainder is the length of the Night The shortest Day in that Latitude is the length of the shortest Night found as before And the longest Night is of the same length with the longest Day Example I would know the length of the longest Day at London Therefore I Elevate the North Pole 51½ degrees and bring ♋ to the Meridian and the Index of the Hour Circle to 12. Then I turn ♋ to the Western side the Horizon and find the Index point at 8. hours 18. minutes which being doubled makes 16. hours 36. minutes for the length of the longest Day here at London PROB. XLIX To find how much the Pole is Raised or Depressed where the longest Day is an Hour longer or shorter then it is in your Habitation REctifie the Globe to the Latitude of your Place and make a prick at that point of the Tropick which is at the Meridian I mean at the Tropick of ♋ if your Habitation be on the North side the Equator or ♑ if your Habitation be on the South side the Equator And if you would know where the longest Day is just an hour longer then it is at your Habitation turn the Globe to the Westward till 7½ degrees of the Equato● pass through the Meridian and make there another prick on the Tropick Then turn about the Globe till the first prick come to the Horizon and move the Meridian through the notches of the Horizon till the second prick on the Tropick come to the Horizon so shall the arch of the Meridian contained between the Elevation of your Place and the Degree of the Meridian at the Horizon be the number of Degrees that the Pole is Elevated higher then it is in your Latitude Example I would know in what Latitude the longest Day is an Hour longer then it is at London Therefore I Rectifie the Globe to 51½ deg and where the Meridian cuts the Tropick of ♋ I make a prick then I note what degree of the Equator is at the Meridian and from that degree on the Equator count 7½ degrees to the Eastwards and bring those 7½ degrees to the Meridian also and again where the Meridian cuts the Tropick of ♋ I make another prick so shall 7½ degrees of the Tropick be contained between those 〈◊〉 pricks Then I turn the Globe about till the first prick comes to the Horizon and with a Quill thrust between the Meridian and the Ball I fasten the Globe in this position Afterwards I move the Meridian through the 〈◊〉 of the Horizon till the second prick rises up to the Horizon and then I find 56½ degrees of the Meridian cut by the Superficies of the Horizon Therefore I say In the Latitude of 56½ degrees the longest Day is an Hour longer then it is here at London But if you would know in what Latitude the Dayes are an Hour shorter you must make your second prick 7½ degrees to the Westwards of the first and after you have brought the first prick to the Horizon you must depress the Pole till the second prick descends to the Horizon so shall the degree of the Meridian at the Horizon shew in

Order or Later Authors gave the Plains their Names upon the same grounds you may also learn to know them I confess both waies admit of some just exception against for in the Older Rule a Plain about the Pole is called an Equinoctial Plain when as to a sudden apprehension it would sound more significant to call it a Polar Plain as Later Authors do Again Later Authors call an Horizontall Plain a Vertical Plain when as it sounds more significant to call it an Horizontal Plain as Older Authors do because it lie flat upon the Horizon But I shall give you the names according to both Rules and leave you to your liberty to accept of which you please First therefore you have an Equinoctial Plain otherwise called a Polar Plain This Plain hath two Faces upper and under These two Faces ly in the Plain of the Equinoctial the upper Face beholding the Elevated Pole the under Face the depressed Pole 2. An Horizontal Plain otherwise called a Vertical Plain it lies in the Plain of the Horizon directly beholding the Zenith Erect Plains otherwise called Horizontal Plains are the sides of Walls and these are of seven sorts viz 1. Erect Direct Vertical North or South 2. Erect Direct East or West 3. Erect Vertical Declining 4. Erect Inclining Direct 5. Erect Inclining Declining 6. Erect Reclining Direct 7. Erect Reclining Declining 3. Erect Vertical North or South Direct otherwise called Direct North or South Horizontals behold the North or South Directly and ly in the East or West Azimuth 4. Erect Direct East or West otherwise called Direct East or West Equinoctials behold the East or West Directly and lies in the Plain of the Meridian having its Poles in the Equinoctial 5. Erect Vertical Declining Plains otherwise called Declining Horizontals do not behold the North or South Directly but swerves from them so much as the Azimuth Parallel to their Plains swerves or Declines from them 6. Erect Inclining Direct Plains have the upper side of their Plains Inclining or coming towards you and their Plains do exactly behold either the East West North or South 7. Erect Reclining Direct Plains have the upper side of their Plains Reclining or falling from you and their Plains exactly beholding either the East West North or South 8. E●●ct Reclining Declining or Erect Inclining Declining Plains are those Plains which are either Inclining or Reclining but 〈◊〉 behold the East West North or South Directly but 〈◊〉 or Decline more or less from them 9. Polar Plains are Parallel to the Axis of the World and to the M●ridians that cuts the East and West or North and South points of the Horizon All these kinds of Plains have two Faces the one beholding the North Pole with the same respect that the other beholds the South Pole except the Equinoctial Plain which because neither Pole is Elevated hath but one Face yet that one contains as many Hour lines as two other Faces These two Faces or Plains will receive just 24. hour lines fo● the 24 Hour-lines of Day and Night for so much as the one side or Face wanteth or exceedeth 12. the other side shall either exceed or want of 12. Every Dyal Plain is Parallel to the Horizon of some Country or other in the World therefore a Dyal made for any Horizon in the World may be set to such a Position that it will shew you the Hour of the Day in your own Habitation At least for so long as the Sun continues upon that Plan● All Plains may be aptly demonstrated by the Globe by setting it correspondent to all the Circles in Heaven as by Prob. 2. of the second B ok for if you imagine the Globe in that Position were prest flat into the Plain of any Circle that Flat shall represent a Dyal plain which shall be called after the name of that Circle it is prest into Thus if the Quadrant of Altitude be applyed to any degree of Azimuth and you imagine the Globe were prest flat to the edge of the Quadrant of Altitude so much as that Azimuth Declines from the East West North or South in the Horizon so much shall that flat on the Globe be said to Decline either from the East West North or South Or if you imagine the Globe were prest flat down even with the Plain of the Horizon that flat shall represent an Horizontal Plain because as was said before the Plain lies in that Circle cal'd the Horizon The Style or Gnomon is that straight wyre that casts the shadow upon the Hour of the Day it is alwaies placed Parallel to the Axis of the World There are several waies to find the scituation of all Plains but the readiest and speediest is by a Clinatory The Clinatory is made of a square board as A B C D of a good thickness and the larger the better between two of the sides is discribed on the Center A a Quadrant as E F divided into 90 equal parts or degrees which are figured with 10 20 30 to 90 and then back again with the Complements of the same numbers to 90 between the Limb and the two Semidiameters is made a Round Box into which a Magnetical Needle is fitted and a Card of the Sea Compass divided into 4 Nineties beginning their numbers at the East West North and South points of the Compass from which points the opposite sides of the Clinatory receives their Names of East West North or South Upon the Center A whereon the Quadrant was discribed is fastned a Plumb-line having a Plumbet of Lead or Brass fastned to the end of it which plumb-Plumb-line is of such length that the Plumbet may fall just into the Grove G H below the Quadrant which is for that purpose made of such a depth that the Plumbet may ride freely within it without stopping at the sides of it See the Figure annexed But admit there be Variation Having by Prob. 19. of the third Book found the number of degrees of this Variation towards the East or West count the same number of degrees from the North point in the Card either to the Eastwards or Westwards and note the degree in the Card terminating at that number for that degree shall be the North point and its opposite degree the South point 90. degrees from it either way shall be the East and West points Therefore whereas before you were directed to turn the Clinatory till the North point of the Needle point to the Flower-de-luce on the ●aid you m●st now turn or move the Clinatory till the North point of the Needle ●arg just over the degree of Variation thus sound and then a line drawn as aforesaid by the side of the Clinatory Paral●el to the Needle shall be a North and South line or to speak more properly a Meridional line You may fi●d a M●ridian li●e several other waies as first If the Sun shine just at Noon hold up a Plumb-line so as the shadow of it may fall upon your Plain and

Contingence elevated to the Height of the Equinoctial draw line from the Center through every 15 degrees of the Circle of Position and by continuing them intersect the line of Conti●gence in the points from whence the Hour lines of an East or West Dyal is to be drawn Example But because in our Latitude the Sun Rises before 4. in the Morning therefore two Hour-lines are yet wanting viz. 5 and 4 which I may find either by applying the thred first to 15 and next to 30 degrees from 0 towards g in the Semi-Circle and so marking where it cuts the Contingent line as before Or else by transfering the distance of the same number of Hour lines from the 6 a clock line already drawn on the side e 〈◊〉 to the side e g as in Prob. 2. of this Book is more fully shewed Having thus marked out on the Contingent line the distances of each Hour I draw a line Parallel to the Contingent line and draw lines from every Hour markt on the Contingent to cross the Contingent line at Right Angles and continue each line to the line Parallel to the Contingent and these lines shall be the Hour lines of an East Plane To these Hour-lines I set Figures as in the Scheam may be seen The Style D K of this Dyal as well as of others must stand Parallel to the Axis of the World it must be also Parallel to all the Hour lines and stand directly over the 6 a clock line and that so high as is the distance between the Center of the Semi-Circle of Position and the point where the 6 a clock line cuts the Contingent line Or which is all one at such a height as when it is laid flat down upon the Plane it may just reach the 3 a clock line PROB. VII To make an Erect Direct West Dyal AN Erect Direct West Dyal is the same in all respects with an Erect Direct East Dyal Only as the East shews the Fore-noon Hours the West shews the After-noon Hours Thus if you should draw the East Dyal on any transparent Plane as on Glass Horn or an Oyled Paper on the one side will appear an East Dyal and on the other a West Only the Figures as was said before must be changed for that which in the East Dyal is 11 in the West must be 1 that which in the East Dyal is 10 in the West must be 2 that which in the East Dyal is 9 in the West must be 3. c. PROB. VIII To make a Polar Dyal POlar Dyals are Horizontal Dyals under the Equinoctial They are of the same kind with East and West Dyals Only whereas East and West Dyals have but the Hour lines of half the longest Day discribed on them these have all the Hour lines of the whole Day and are marked on both sides the Noon line as in the following Figure The Style of this Dyal must stand over the Noon line Parallel to the Plane for then it will also be Parallel to the Axis of the World and its height above the Plane must be the distance between the Center i of the Semi-Circle and the point in the Contingent line cut by the Noon-line But I have inserted the Figure which alone is sufficient Instructions PROB. IX To make Erect South Dyals Declining Eastwards or Westwards DRaw on your Plane an Horizontal line and on it discribe a Semi-Circle as you were taught in Prob 4. Then Rectifie the Globe Quadrant of Altitude Colure and Hour Index as by the same Probleme and bring the lower end of the Quadrant of Altitude to the degree of Declination from the East or West point according is your Declination is Eastwards or Westwards for then the Quadrant of Altitude shall represent a Plane declining from the South E●stwards or Westwards accordingly Then tu●n the Globe Eastwards till the Index of the Hour-Circle points to all the Hours before Noon and examine in what number of degrees from the Zenith the Colure cuts the Q●●drant of Altitude when the Index points to each Hour For a line drawn from the Center A through the same number of degrees reckoned from the Perpendicular A B which is the 12 a clock line towards Con the Plane shall be the same Hour-lines the Index points at Example I would make an Erect Dyal declining from the South towards the East 27. degrees The Globe Quadrant of Altitude Vernal Colure and Hour Index Rectified as before I bring the lower end of the Quadrant of Altitude to 27. degrees counted from the East point of the Horizon towards the North Then I turn the Globe East-wards till the Index points to 11 a clock or till 15. deg of the Equator pass through the Meridian and find the Colure cut the Quadrant of Altitude in 9.43 counted from the Zenith 10 19.0 9 25.57 8 35.10 7 45.56 6 60.15 5 79.45 And these are the distances of the Fore-noon Hour-lines which I seek in the West side of the Plane viz. from B towards C and through these distances I draw lines from the Center and these lines shall be the Fore-noon Hour-lines Now herein is a difference between Declining Dyals and Direct Dyals For having found the distances of the Hour lines for one half of the Day be it either for Before Noon or After Noon in a Direct Dyal you have also found the distances for the other half Day because as was said Prob. 3. Equal number of Hours have equal distance from the Noon line But in Declining Dyals it is not so Because the Sun remaining longer upon that side of the Plane which it declines to then it doth upon the contrary side there will be a greater number of Hour lines upon it and by consequence the distance of the Hour lines less then on the contrary side of the Plane Therefore for finding the After Noon Hour lines I turn about the Quadrant of Altitude upon the Zenith point till the lower end of it come to the degree of the Horizon opposite to that degree of Declination that the Quadrant of Altitude was placed at when I sought the Fore Noon Hour lines viz to 27. degrees counted ●om the West towards the South and bring the Ver●al Colure again to the Meridian and the Index as before to 12. Then turning the Globe Westwards till the Index poin●s to 1 a clock or till 15 degr of the Equator pass through the Meridian I find the Colure cut the Quadrant of Altitude in 11.20 counted from the Zenith 2 26.47 3 49.20 4 75.52 And these are the distances of the After Noon Hour lines which dista●●●● I seek in the East side of the Plane viz. from B towards D as before and so drawing lines from the Center A through these distances I have all the Afternoon Hour lines also drawn on my Plane You may note that this Plane is capable to receive no more Hour lines After Noon then 4. for when the Colure goes off the Quadrant of Altitude the Sun goes

Westwards Having thus found out where this Plane becomes Horizontal make your Dyal to this Plane as by the second Rule in this Probleme Find also the Style as is there directed 5. If your Plane be a Declining Incliner The Globe and Quadrant of Altitude Rectified Bring the Colure to the Meridian and the Quadrant of Altitude to the degree of the Horizon opposite to the degree of the Planes Declination and count upwards on the Quadrant of Altitude the degrees of Inclination and make a 〈◊〉 there For in the 〈◊〉 of that prick found as by 〈◊〉 〈◊〉 of the Second Book that Declining In 〈◊〉 shall become an Horizontal Plane Then find the Latitude and difference of Longitude of this 〈◊〉 by the 〈◊〉 〈◊〉 and make a ●yal to that 〈◊〉 by the second 〈◊〉 in this Probleme Find also the Style as therein is directed PROB. XV. To make a Dyal on the Ceeling of a Room where the Direct Beams of the Sun never come FInd some convenient place in the Transum of a Window to place a smal round peece of Looking-Glass about the bigness of a Groat or less so as it may ly exactly Horizontal The point in the middle of this Glass we will marke A and for distinctions sake with Mr Palmer call it Nodus Through this Nodus you must draw a Meridian line on the Floor Thus Hang a Plumb line in the Window exactly over Nodus and the shadow that that Plumb line casts on the Floor just at Noon will be a Meridian line Or you may find a Meridian line otherwise as by the Preface Having drawn the Meridian line on the Floor find a Meridian line on the Ceeling thus Ho●d a Plumb line to the Ceeling over that end of the Meridian line next the Window If the Plumbet hang not exactly on the Meridian line on the Floor remove your hand on the Ceeling one way or other as you see cause till it do hang quietly just over it and at the point where the Plumb line touches the Ceeling make a mark as at B that mark B shall be directly over the Meridian line on the Floor then remove your Plumb line to the other end of the Meridian line on the Floor and find a point on the Ceeling directly over it as you did the former point as at C and through those two points B and C on the Ceeling strain and strike a line blackt with Smal Cole or any other Culler as Carpenters do and that line B C on the Ceeling shall be a Meridian line as well as that on the Floor Then examine the Altitude of the Equinoctial as by Prob. 6. of the Second Book you did the Meridian Altitude of the Sun and fasten a string just on the Nodus and remove that string in the Meridian line on the Ceeling till it have the same Elevation in a Quadrant that the Equinoctial hath in your Habitation and through the point where the string touches the Meridian line in the Ceeling shall a line be drawn at right Angles with the Meridian to represent the Equinoctial line Thus in our Latitude the Elevation of the Equator being 38½ degrees I remove the string fastned to the Nodus forwards or backwards in the Meridian line of the Ceeling till the Plumb line of a Quadrant when one of the sides are applyed to the string falls upon 38½ degrees and then I find it touch the Meridian line at D in the Ceeling therefore at D I make a mark and through this mark strike the line D E as before I did the Meridian line to cut the Meridian line at Right Angles This line shall be the Equinoctial line Then I place the Center of the Semi-Circle of Position upon Nodus and under-prop it so that the flat side of it may ly Parallel to the string when it is strained between the Nodus and the Equinoctial and also so as the string may ly on the division of the Semi-Circle marked o when it is help up to the Meridian line in the Ceeling Then removing the string the space of 15. degrees in the Circle of Position to the Eastwards and extending it to the Equator on the Ceeling where the string touches the Equator there shall be a point through which the 1 a clock Hour-line shall be drawn and Removing the string yet 15. degrees further to the Eastwards in the Semi-Circle of Position and extending it also to the Equator where it touches the Equator there shall be a point through which the 2 a clock Hour-line shall be drawn Removing the string yet 15. degrees further to the Eastwards in the Semi-Circle of Position and extending it to the Equator there shall be a point through which the 3 a clock Hour-line shall be drawn The like for all the other After-Noon Hour lines so oft as the string is removed through 15. degrees on the Semi-Circle of Position so oft shall it point out the After-Noon distances in the Meridian line on the Ceeling The scituation of the Semi-Circle of Position cannot conveniently be shewn in this Figure unless it be drawn by the Rules of Perspective Neither if it were would it suit with the other demonstrations expect they were drawn by the same Rules also which to do would be hard for young Learners to understand Therefore I have left out the Semi-Circle of Position in this Figure and refer you for a demonstration thereof to the sixth Probleme For even as the lines drawn through every 15 degrees of the Semi-Circle there denote in a Contingent line the distance of any Hour line from the Meridian line even so a line drawn through every 15. degrees of the Semi-Circle of Position posited as aforesaid point out in the Equinoctial line on the Ceeling the distance of each respective Hour line from the Meridian line Having thus found out the points in the Equator through which the After-Noon Hour-lines are to be drawn I may find the Fore-Noon Hour distances also the same way viz. by bringing the string to the several 15. degrees on the West side the Semi-Circle of Position or else I need only measure the distances of each Hour distance found in the Equator from the Meridian line on the Ceeling for the same number of Hours from 12 have the same distance in the Equinoctial line on the other side the Meridian both Before and Afternoon The 11 a clock Hour distance is the same from the Meridian line with the 1 a clock distance on the other side the Meridian the 10 a clock distance the same with the 2 a clock distance the 9 with the 3 c. And thus the distances of all the Hour lines are found out on the Equator Thus upon the point markt for each Hour distance in the Equinoctial line on the Ceeling I discribe the Arches I II III IIII as in the Figure and finding the distance from the Meridian of the Hour-lines of an Horizontal Dyal to be according to the third Probleme