Radius of your Scale set one soot in the end A and with the other describe the arch BCD then opening your Compasses unto the whole 90 deg with one foot in B with the other mark the arch B C D in the point D from which point D draw the line DA which shall be perpendicular unto the line AB and make the quadrant ABCD then suppose the height of your Gnomon or substance yeilding shadow to be the line A E which here we will suppose to be 12 foot therefore take 12 of your equal divisions from your Scale as here I have taken 12 quarters for this our purpose and set them down from A to E and draw the line E F parallel to A B then suppose the length of the shadow to be 9 foot for this 9 foot must I take 9 of the same divisions as I did before and place them from E to G by which point G draw the line A G C from the center A through the point G until it cutteth the arch B F C D in the point G so shall the arch B C be the height of the Sun desired which in this example will be found to be 53 deg 8 min. the thing desired CHAP. IV. To find the altitude of the Sun by the shadow of a Gnomon standing at right angles with any perpendicular wall in such manner that it may lie parallel unto the Horizon FIrst draw your quadrant A B C D as is taught in the last Chapter and place the length of your Gnomon from A to E which here we will suppose to be 12 foot as before in the last Chapter then draw the line EF parallel to A B then suppose the length of the shadow to be 9 foot as before this I place from E to G by which point G draw the sine A G C as was formerly done in the last Chapter by which we have proceeded thus far but as in the last Chapter the arch B C was the height of the Sun desired so by this Chapter the arch C D shall be the height of the Sun which being applyed unto your Scale will give you 36 deg 52 m. for the height of the Sun desired CHAP. V. The Almicaâ—ter or height of the Sun being given to find the length of the right shadow BY right shadow is meant the shadow of any staffe post steeple or any Gnomon whatsoever that standeth at right angles with the Horizon the one end thereof respecting the Zenith of the place and the other the Nadir First therefore according unto the third Chapter describe the quadrant A B D then suppose the height of your Gnomon or substance yeilding shadow to be 12 foot as in the former Chapter this doe I set down from A to E and from the point E draw the line EF parallel to A B then set the Almicanter which here we will suppose to be 53 deg 8 m. as it was found by the third chap. from B unto C from which point C draw the line C A cutting the line E F in the point G so shall E G be the length of the right shadow desired which being taken betwixt your compasses and applyed unto your Scale will give you 9 of those divisions whereof A E was 12 which here doth signifie 9 foot CHAP. VI. The Almicanter or height of the Sun being given to find the length of the contrary shadow BY the contrary shadow is understood the length of any shadow that is made by a Staffe or Gnomon standing at right angles against any perpendicular wall in such a manner that it may lie parallel unto the Horizon the length of the contrary shadow doth increase as the Sun riseth in height whereas contrariwise the right shadow doth decrease in length as the Sun doth increase in height Therefore the way to find out the length of the Versed shadow is as followeth First draw your quadrant as is taught in the third Chapter now supposing the length of your Gnomon to be 12 foot place it from A to E likewise from E draw the line E F parallel to A B as before now supposing the height of the Sun to be 36 deg 52 min. take it from your Scale and place it from D to C from which point C draw the line C A cutting the line E F in the point G so shall G E be the length of the contrary shadow which here will be found to be 9 foot the thing desired CHAP. VII Having the distance of the Sun from the next Equinoctial point to his Declination FIrst draw the line A B then upon the end A raise the perpendicular A D then opening your compasses to the Radius of the Scale place one foot in the center A and with the other draw the quadrant BCD then supposing the Sun to be either in the 29 degree of Taurus or in the first degree of Leo both which points are 59 deg distant from the next Equinoctial point Aries Or if the Sun shall be in the 29 degree of Scorpio or or in the first degree of Aquarius both which are also 59 degrees distant from the Equinoctial point Libra therefore take 59 degrees from your Scale and place it from B to C and draw the line C A then place the greatest declination of the Sun from B unto F which is 23 deg 30 min. then fixing one foot or your compasses in the point F with the other take the nearest distance unto the line A B which you may do by opening or shuting of your compasses still turning them to fro till the moving point thereof doe only touch the line A B this distance being kept set one foot of your compasses in the point A and with the other make a mark in the line A C as at E from which point E take the neerest extent unto the line A B this distance betwixt your compasses being kept fix one foot in the arch B C D moving it either upwards or downwards still keeping it directly in the arch line until by moving the other foot to and fro you finde it to touch the line A B and no more so shall the fixed foot rest in the point G which shall be the Declination of the Sun accounted from B which in this example will be found to be about 20 degrees the thing desired CHAP. VIII The Declination of the Sun and the quarter of the Ecliptique which he possesseth being given to find his true place LEt the Declination given be â—0 degrees and the quarter that he possesseth be betwixt the head of Aries and Cancer first draw the quadrant A D E F then set the greatest Declination of the Sun upon the Chord from D unto B which is 23 degrees and 30 minutes then from the point B take the shortest extent unto the line A D this distance being kept set one foot in the point A and with the other describe the small quadrant G H I then set the declination of
C is the line C A D dividing the Circle into two equal parts then from the Center A raise the perpendicular A B dividing the Semicircle likewise into two equal parts so is A B D or A B C the quadrant of the Circle C B D E. Definition 12. A Segment or portion of a Circle is a figure contained under a right line and a part of a circumference either greater or lesser then the semicircle as in the figure you may see that F B G H is a Segment or part of the circle C B D E is contained under the right line F H G which is less then the Diameter C A D and a part of the whole circumference as F B G. And here note that these parts and such like of the circumference so divided are commonly called arches or arch lines and all lines lesse then the Diameter drawn through and applyed to any part of the circumference are called chords or chord lines of those arches which they subtend Definition 13. A Parallel line is a line drawn by the side of another line in such sort that they may be equidistant in all places of such parallels two only belong unto this work of Dialling that is to say the right lined parallel the circular parallel Right lined parallels are two right lines equidistant in all places one from the other which being drawn forth infinitely would never meet or concur as may be seen by these two lines A and B. Definition 14. A circular parallel is a circle drawn either within or without another circle upon the same center as you may plainly see by the two Circles B C D E F G H I these circles are both drawn upon the same center A and therefore are parallel the one to the other Definition 15. A Degree is the 360th part of the circumference of any circle so that divide the circumference of any circle into 360 parts and each of those parts is called a degree so shall the semi-circumference contain 180 of those Degrees and 90 of those degrees make a quadrant or a quarter of the circumference of any circle Definition 16. A minute is the 60th part of a degree being understood of measure but in time a Minute is the 60th part of an houre or the fourth part of a degree 15 degrees answering to an houre and 4 minutes to a degree Definition 17 The quantity or measure of an Angle is the number of degrees contained in the arch of a circle described from the point of the same angle and intercepted betweene the two sides of that angle As for example the measure of the angle A B C is the number of degrees contained in the arch A C which subtendeth the angle B being found to be 60 Definition 18. The Complement of an arch lesse then a quadrant is so much as that arch wanteth of 90 degrees As for example the arch A B being 60 degrees which being taken from 90 degrees leaveth B C for the complement thereof which is 30 degrees Definition 19. The complement of an arch lesse then a Semicircle is so much as that arch wanteth of a Semicircle or of 180 deg As for Example the arch D C B being 120 degrees this being taken from 180 deg the whole Semicircle leaveth A B for the complement thereof which will be found to be 60 degrees And here note that what is said of the complements of arches the same is meant by the complements of angles CHAP. II. To a line given to draw a parallel line at any distance required SUppose the line given to be A B unto which line it is required to draw a parallel line First open your Compasses to the distance required then set one foot in the end A and with the other strike an arch line on that side the given line whereunto the parallel line is to be drawn as the arch line C this being done draw the like arch line upon the end B as the arch line D and by the convexity of those two arch lines C and D draw the line C D which shall be parallel to the given line as was required CHAP. III. To perform the former proposition at a distance required and by a point limited SUppose the line given to be D E unto which line it is required to draw a parallel line at the distance and by the point F. First therefore place one foot of the compasses in the point F from whence take the shortest extention to the line DE 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. IV. The manner how to raise a perpendicular line from the middle of aline given LEt the line given be A B and let C be a point therein whereon it is required to raise a perpendicular First therefore open the compasses to any convenient distance and setting one foot in the point C with the other foot mark on either side thereof the equal distances C A and 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 DC which line is perpendicular unto the given line A B from the point C as was required CHAP. V. To let a Perpendicular fall from a point assigned unto the middle of a line given LEt the line given whereupon you would have a perpendicular let fall be the line D E F and the point assigned to be the point C from whence you would have a perpendicular let fall upon the given 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 then the distance C E make an arch of a circle with the other foot so that it may cut the line D E F twice that is at I and G then find the middle between those two intersections which will be in the point E from which point E draw the line C E which is the perpendicular which was desired to be let fall from the given point C unto the middle of the given line D E F. CHAP. VI. To raise a Perpendicular upon the end of a line given SUppose the line whereupon you would have a perpendicular to be raised be the line B C and from the point B a perpendicular is to be raised First open your Compasses unto any convenient distance which here we suppose to be the distance B E and set one foot of your compasses in B with the other draw the
arch E D then this distance being kept set one foot of your compasses in the point E with the other make a mark in the former arch E D as at D still keeping the same distance set one foot in the point D and with the other draw the arch line F over the given point B now laying a ruler upon the two points E and D see where it crosseth the arch line F which will be at F from which point F draw the line F B which shall be a perpendicular line unto the given line B C raised from the end B as was required CHAP. VII To let a Perpendicular fall from a point assigned unto the end of a line given LEt the line D E be given unto which it is required to let a perpendicular fall from the assigned point A unto the end D. First from the assigned point A draw a line unto any part of the given line D E which may be the line ABC then find the middle of the line A C which will be at B place therefore one foot of your compasses in the point B and extend the other unto A or C with which distance draw the Semicircle A D C so shall it cut the given line D E in the point D from which point D draw the line A D which shall be the perpendicular let fall from the assigned point A unto the end D of the given line D E as was required CHAP. VIII Certain Definitions Astronomical meet to be understood of the unlearned before the proceeding in this Art of Dialling IN the former Chapter I have shewed the meaning of some terms of Geometry which be most helpfull unto this Art of Dialling with the drawing of a Parallel line at any distance or by a point assigned so likewise have I shewed the manner either how to raise or let fall a perpendicular either from or unto any part of a line given So likewise now I think it will not be un-necessary for to shew unto the unlearned the meaning of some of the most usefullest terms in Astronomie and most fitting this art of Dialling Definition 1. A Sphere is a certain solid superficies in whose middle is a point from which all lines drawn unto the circumference are equal which point is the Center of the Sphere Definition 2. The Pole is a prick or point imagined in the Heavens whereof are two the North pole being the center to a circ l described by the motion of the North Star or the taile of the little Bear from which point aforesaid is a line imagined to passe through the center of the Sphere and passing directly to the opposite part of the heavens sheweth there to be the South Pole and this line so imagined to passe from one Pole to the other through the Center of the Sphere is called the Axletree of the World because it hath been formerly supposed that the Sun Moon and Stars together with the whole Heavens hath been turned about from East to West once round in 24 houres by a true equal course like much in like time which diurnal revolution is performed about this Axletree of the World and this Axletree is set out unto you in the following figure by the line P A D the Poles whereof are P and D Definition 3. A Sphere accidentally is divided into two parts that is to say into a right Sphere and an oblique Sphere a right Sphere is only unto those that dwell under the Equinoctial to whom neither of the Poles of the World are seen but lie hid in the Horizon An oblique Sphere is unto those hat in habit on either side of the Equinoctial unto whom one of the Poles is ever seen and the other hid under the Horizon Definition 4 The Circles whereof the Sphere is composed are divided into two sorts that is to say into greater Circles and lesser The greater Circles are those that divide the Sphere into two equal parts and they are in number six vix the Equinoctial the Ecliptique line the two Colures the Meridian and the Horizon The lesser Circles are such as divide the Sphere into two parts unequally and they are foure in number as the Tropick of Cancer the Tropick of Capricorn the Circle Artique and the Circle Antartique CHAP. IX Of the six greater Circles Definition 5. THe Equinoctial is a circle that crosseth the Poles of the World at right Angles and divideth the Sphere into two equal parts and is called the Equinoctial because when the Sun commeth unto it which is twice in the year viz. at the Suns entrance into Aries and Libra it maketh the dayes and nights of equal length throughout the whole World and in the figure following is described by the line S A N. Definition 6. The Meridian is a great Circle passing through the Poles of the World and the Poles of the Horizon or Zenith point right over our heads and is so called because that in any time of the year or in any place of the World when the Sun by the motion of the Heavens commeth unto that circle it is then Noon or 12 of the clock and it is to be understood that all Towns and Places that lie East and West one of another have every one a several Meridian but all places that lie North and South one of another have one and the same Meridian this circle is declared in the figure following by the circle E B W C. Definition 7. The Horizon is a Circle dividing the superior Hemisphere from the inferiour whereupon it is called Horizon that is to say the bounds of sight or the farthest distance that the eye can see and is set forth unto you by the line C A B in the following figure Definition 8. Colures are two great circles passing through both the Poles of the World crossing one the other in the said Poles at right Angles and dividing the Equinoctial and the Zodiaque into four equal parts making thereby the four Seasons of the year the one Colure passing through the two Tropical points of Cancer and Capricorn shewing the beginning of Summer and also of Winter at which times the dayes and nights are longest and shortest The other Colure passing through the Equinoctial points Aries and Libra shewing the beginning of the Spring time and Autumne at which two times the dayes and nights are of equal length throughout the whole World Definition 9. The Ecliptique is a great Circle also dividing the Equinoctial into two equal parts by the head of Aries and Libra the one halfe thereof doth decline unto the Northward and the other towards the South the greatest declination thereof according to the observation of that late famous Mathematician Master Edward Wright is 23 degrees 31 minutes 30 seconds Note also that the Circle is divided into 12 equal parts which parts are attributed unto the 12 Signes Aries Taurus Gemini Cancer Leo Virgo Libra Scorpio Sagittarius Capricornus Aquarius and Pisces Out of this line
other great circle or plane whatsoever CHAP. VI. The drawing of a Dial upon the direct Polar Plane A Direct Polar Plane is that which is parallel to the circle of the hour of six therefore having drawn the Horizontal line A B and crossed it at right angles about the middle of the line at C with the perpendicular C E if you shall find the Planes inclination towards the North to be equal to the Latitude of the place and the horizontal line directly in the line of East and VVest and so to have no declination you may be sure this plane lieth parallel to the hour of six and is therefore called a Polar plane The horizontal line being drawn at the length of the plane divide it into seven equal parts and set down one of them in the line of inclination from C unto D upon the center D describe the Equinoctial circle which you may divide into 24 equal parts if you will but one quarter thereof into 6 will serve as well then at the distance C E draw the line F G parallel to AB Then having divided the Aequator either into 24 equal parts or one quarter thereof into 6 you may be a rule laid the center D and each of those six parts make marks in the horizontal line A B which here is instead of the contingent-line as you may see by the pricked lines these distances from the Meridian being applied upon the same line on the other side of the Meridian and also on both sides the Meridian in the upper line the lines drawn from point to point parallel to the Meridian C E shall be the hour-lines the line C E shall be the Meridian-line the hour of 12 and must also be the substilar-line whereon the stile must stand which may be a plate of iron or some other metal being so broad as the semidiameter of the Circle is as is shewed in the figure This stile must be placed along upon the line of 12 making right angles therewith the upper edge whereof must be parallel to the plane so shall it cast a true shadow upon the hour-lines The under face of this Polar plane and also of the former Equinoctial plane is made altogether like unto the upper faces here described without any difference at all CHAP. VII The making of an erect Meridian Diall A Meridian plane is that which is parallel to the Meridian Circle of the Sphere therefore having drawn the horizontal line A B and finding it to decline 90 deg from the South the plane being erect I conclude it to lie parallel to the Meridian Circle of the Sphere and is therefore called a Meridian plane For the style of this Dial it may be either a plate of some metal being so broad as the semidiameter of the Circle is and so placed perpendicularly along over the line of the hour of six the upper edge thereof being parallel to the plane or it may be a straight pin fixed in the center of the Circle making right angles with the plane being just so long as the Semidiameter of the circle is only shewing the hour with the very top or end thereof This Plane hath two faces one to the East the other to the west the making whereof are both alike onely in naming the hours for the one containeth the hours for the forenoon the other for the afternoon as you may perceive by the figures CHAP. VIII To draw a Dial upon an horizontal plane AN Horizontal plane is that which is parallel to the Horizontal circle of the Sphere which being found by the first Chapter to be level with the Horizon you may by the fourth Chapter draw the Meridian line A B serving for the Meridian the hour of 12 and the substilar in this Meridian make choice of a center as at C through which point C draw the line D E crossing the Meridian at right angles this line shall be the line of East and VVest and is the six a clock line both for morning and evening Then by the second Chapter of the second Book draw the line S C making the angle S C A equal to the latitude of the place which here we will suppose to be 52 deg 30 min. this line shall represent the cock of the Dial and the Axletree of the world then at the North end of the Meridian line draw another line as F G crossing the Meridian in the point A at right angles this line is called the touch-Touch-line or line of contingence Then set one foot of your Compasses in the point A and with the other take the necrest extent unto the line S C or the stile with this distance turning your compasses about with one foot still in the point A with the other make a mark in the Meridian as at I which shall be the center of the Equinoctial upon which describe the Equinoctial Circle A D B E with this same distance setting one foot in the point A make a mark at F on the one side of the Meridian and another at G on the other side thereof both which must be in the line of contingence by which two points and the center C you may draw the hour-lines of 3 and 9. Thus by dividing but half a quarter of the Equinoctial Circle into three equal parts you may describe your whole Dial. And whereas in Summer the 4 and 5 in the morning and also 7 and 8 at evening shall be necessary in this kind of Dial prolong or draw the lines of 4 and 5 at evening beyond the center C and they shall shew the hour of 4 and 5 in the morning and likewise the 7 and 8 in the morning for 7 and 8 at evening What is here spoken concerning the hours the like is to be done in drawing the half hours as well in this kind as in all them which follow The stile must be fixed in the center C hanging directly over the Meridian line A C with so great an angle as the lines S C A maketh which is the true pattern of the cock This and all other kinds of Dials may be drawn upon a clean paper and then with the help of your compasses placed on your plane CHAP. IX To draw a Dial upon an erect direct vertical plane commonly called a South or North Dial. A Vertical plane is that which is parallel to the prime vertical circle it hath two faces one to the South the other to the North therefore having drawn the horizontal line A B and from the middle thereof let fall the perpendicular C D which if you find by the second Chapter to be erect and the Horizontal line A B to lie in the line of East west and so to have no declination you may be sure this plane is parallel to the prime vertical circle of the Sphere and therefore is called a vertical plane This perpendicular C D shall serve for the meridian the hour of 12 and the substilar line which
the elevation of the Pole above the plane 6 deg 4 min. and the inclination of both Meridians 82 deg 5 min. But if they lean to the Horizon they are then called Incliners These may incline either to the East part of the Horizon or to the West and each of them hath two faces the upper towards the Zenith the lower towards the Nadir wherein knowing the latitude of the place and the inclination of the plane to the Horizon we are to consider three things more before we can come to the drawing of the Dial. I. The elevation of the pole above the plane II. The distance of the substile from the Meridian III. The angle of inclination betwixt both Meridians These three may be found after this manner little differing from the 14. Chap. Then take the distance S H and set it in the line E R from R to K through which point K draw the line A K L cutting the arch of the quadrant in the point L so shall the arch CL be the distance of the Substile from the Meridian and is in this example 33 deg 5 min. This being done from the point S draw the line L T parallel to the line A C cutting the arch G D in the point O through which point O draw the line A O I cutting the arch of the quadrant B C in the point I so shall the arch CI be the inclination of the Meridian of the plane to the Meridian of the place and in this example is found to be 43 deg 28 min. which being resolved into time doth give about two hours and 54 min from the Meridian for the place of the substile amongst the hour-lines CHAP. XVII To draw a Dial upon the Meridian inclining plane HAving by the second Chapter found the inclination of this plane to be 30 degrees and so by the last Chapter found the elevation of the pole above the plane to be 43 degr 23 min. and the distance of the substile from the Meridian to be 33 deg 5 min. and likewise the angle of inclination to be 43 deg 38 min. we may proceed to make the Dial after this manner First draw the horizontal line A B serving for the Meridian and hour of 12 about the middle of this line make choice of a center at C upon which describe a Circle for your Dial as A D B E. Then seeing this is the upper face of the plane set 33 degrees 5 minutes the distance of the substile from the Meridian in the Dials Circle from the North end of the Horizontal line upwards as from B to H and draw the line C H for the Substile But if this had been the under face the substile must have fallen below the horizontal line now through the center C draw the Diameter E F making right angles with the substile C H. Then set 43 degrees 23 minutes from H unto D for the stile and draw the line C D unto S and from the end of the 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. Now take each perpendicular betwixt your Compasses and with one foot in the center C with the other make marks in the line of the stile from which take the neerest extents unto the substile and lay them down upon their own proper perpendiculars from the substile so may you make marks through which and from the center you may draw the hour-lines This Dial being thus drawn for the upper face of a Meridian plane inclining towards the West you must fix the Cock in the center C hanging over the substile C H with an angle equal to the angle S C H so that it may point to the North Pole because upon the upper faces of all Meridian incliners the North Pole is elevated and therefore contrarily the South Pole must needs be elevated above their under faces This Dial being drawn in paper for the the upper face of this plane will also serve for the under face thereof if you turn the pattern about so that the Horizontal line A B may lie still parallel to the Horizon and the stile with the substile lying under the Horizontal line may point downwards to the South Pole the paper being oiled or pricked through so that you may take the back side thereof for the fore-side without altering the numbers set to the hours CHAP. XVIII The inclination and declination of any plane being given in a known Latitude to find the angle of intersection botween the plane and the Meridian the aseension and elevation of the Meridian with the arch thereof between the Pole and the plane and also the elevation of the Pole above the plane the distance of the substile from the Meridian with the inclination between both Meridians IF a plane shall decline from the South and also incline to to the Horizon it is then called by the name of a declining inclining plane Of these there are several sorts for the inclination being Northward the plane may fall betwixt the Horizon and the Pole or betwixt the Zenith and the Pole or else they may lie in the Poles of the World or the inclination may be southward and so fall below the intersection of the Meridian and the Equator or above it or the plane may fall directly in the intersection of the Meridian with the Equator each of these planes have two faces the upper towards the Zenith and the lower towards the Nadir Now having the Latitude of the place with the declination and inclination of the plane we have seven things more to consider before we can come to the drawing of the Dial. I. The angle of intersection betwixt the plane and the Meridian II. The arch of the plane betwixt the Horizon and the Meridian III. The arch of the Meridian betwixt the Horizon and the plane IV. The arch of the Meridian between the Pole and the plane V. The elevation of the Pole above the plane VI. The distance of the substile from the meridian VII The angle of inclination betwixt the Meridian of the plane and the Meridian of the place All these seven may be found out after this manner First Describe the Quadrant A B C then suppose the plane to decline from the South towards the East 35 deg and to incline towards the Horizon 25 deg set 35 deg the declination of the plane from C to E in the Quadrants arch C B and draw the line A E then set 25 deg the inclination of the plane in the same arch from B to F and draw the line E Z parallel to A C cutting the line A B in the point Z and with the distance F Z and one foot placed in the center A with the other describe the arch G H I. cutting the line A E in the point H through which point H draw the line K L parallel to A C
Sun be upon the South side of the Equinoctial either in Libra Scorpio Sagittarius Capricornus Aquarius or Pisces then the difference of the ascensions is to be added unto the right ascension and the sum of them both will be the Oblique ascension As suppose the fourth degree of Sagittariou to be given the right ascension whereof is found to be 242 deg or 16 houres 8 min. and the difference of ascensions is 30 deg 3 min. or 2 houres which being added unto the right ascension doth make 18 houres 8 min. or in degrees 272 deg 3 min. which is the Oblique ascension of the Sun in the fourth degree of Sagittarius But if you would finde the Oblique descension you must work directly contrary to these Rules given CHAP. XXIV How to find the altitude of the Sun without Instrument IN the third Chapter of this Book it is shewed how to find the altitude of the Sun by a Gnomon set perpendicular to the Horixon but seeing the ground is 〈◊〉 unlevel it is not so ready for this our purpose and perhaps some may have occasion to find the altitude of the Sun and thereby the azimuth or houre of the day according to the 17 or 18 chapters and yet may be unprovided of Instruments to perform the same or at least may be absent from them therefore it will not be un-needfull to shew the finding of the same without the Gnomon or other Instrument Take therefore a Trencher or any simple boards end of what fashion soever such as you can get make thereon two pricks as A and B then prick in a pin naile or short wire in one of the points as at A whereupon hang a threed with a plummet the lift up this board toward the Sun till the shadow of the pin at A come directly on the point B and directly where the threed then falleth there make a mark as at E under the threed then with your Rule and Compasse draw the lines A B and A E and find the angle B A E by the second chapter for that is the complement of the altitude of the Sun or when you have drawn A B and A E you may make the quadrant B A F by the third chapter and then the angle E A F shall be your altitudo desired CHAP. XXV How to find out the latitude of a place or the Poles elevation above the Horizon by the Sun SEeing that throughout this Book the latitude of the place is supposed to be known when as every one perhaps cannot tell which way to find it out therefore it will not be un-needful to shew how it may be readily attained sufficiently for our purpose First therefore you must get the Meridian altitude which you may doe by observing diligently about noon a little before and a little after still observing until you perceive the Sun to begin to fall again then marking what was his greatest altitude will serve for this our present purpose Having gotten the Meridian altitude by this and the Declination by the 7 chapter you may find the latitude of the place or the elevation of the Pole above the Horizon after this manner If the Sun hath North Declination then subtract the Declination out of the Meridian altitude and the remainder shall be the height of the Equinoctial But if the Sun hath South Declination then adde the Declination to the Meridian altitude so shall the sum of them give the altitude of the Equinoctial which being taken out of the quadrant or 90 deg leaveth the latitude of your place or the elevation of the Pole above your Horizon As for example upon the first day of May 1650 the Meridian altitude of the Sun being observed to be 55 deg 35 m. upon which day I find the Suns place to be in 20 deg 48 m. of Taurus and the declination 18 deg 00 min. and because the declination is North I substract 18 deg 00 min. out of the Meridian altitude 55 deg 35 min. and there remains 37 deg 35 min. the height of the Equinoctial and this taken out of 90 deg leaveth 52 deg 25 m. for the latitude of Thuring But it may be required sometimes for you to make a Diall for a Town or Countrey whose Latitude you know not neither can come thither conveniently to observe it Here is therefore added a Table shewing the latitude of the most principal Cities and Towns in England so that being required to make a Dial for any of those places you need but look in your Table and there you have the Latitude thereof But if the Town you seek be not in the Table look what Town in the Table lies neer unto it and make your Dial to that Latitude which will occasion little difference A TABLE shewing the Latitude of the most principal Cities and Towns in ENGLAND Names of the Places Latitude  D M St. Albons 51 55 Barwick 55 49 Bedford 52 18 Bristol 51 32 Boston 53 2 Cambridge 52 17 Chester 53 20 Coventry 52 30 Chichester 50 56 Colchester 52 4 Darby 53 6 Grantham 52 58 Halifax 53 49 Horeford 52 14 Hull 53 50 London 51 32 Lancaster 54 8 Leicester 52 40 Lincolne 53 15 Newcastle 54 58 Northampton 52 18 Oxford 51 54 Shrewsbury 52 48 Warwick 52 25 Winchester 51 10 Worcester 52 20 Yarmouth 52 45 York 54 0 The end of the second Book THE THIRD BOOK Shewing Geometrically how to describe the hour-Hour-lines upon all sorts of Planes howsoever or in what Latitude soever scituated two manner of wayes without exceeding the limits of the Plane CHAP. I. How to examine a Plane for an Horizontal Dial. FOrasmuch as it is necessary before the drawing of any Dial to know how your plane is already placed or how it ought afterwards to be placed it is therefore expedient to shew how it may be attained unto without the help of a Quadrant or any such like Instrument which for this purpose is very useful First take any board that hath one straight side and an inch or more from the straight side draw a line parallel thereto about the middle of which line erect a perpendicular line and at the center where these two lines meet cut out a hollow piece from the edge of the parallel line for a plummet to hang in then if your plane seem to be level with the horizon you may try it by applying the straight side of your board thereunto and holding the perpendicular line upright and holding a threed and plummet in your hand so as the plummet may have free play in the hole for then if the threed shall fall on the perpendicular line which way soevreyou turn the board it is an horizontal plane As for example let the figure ABCD be a Plane supposed to stand level with the Horizon for to try the same I take the simple board G O H having one streight side as G H then drawing a parallel thereto I crosse it at right angles
horizontal plane otherwise then in the eighth Chapter was shewed ALthough I have plainly and perfectly shewed the making of the horizontal the direct South or North as well erect as inclining and the South or North crect declining Dials in the four former Chapters yet to satisfie them that delight in variety I have here declared another way whereby you may make them most artificially and geometrically not being tied to the use of the Canons which indeed of all others is most exact but not so easie to be understood nor to any one Instrument which may be absent from me when I should need it although in this Treatise I do perform the whole by a plain Quadrant Therefore by the first Chapter having found the plane to be horizontal by the fourth Chapter draw the Meridian line A B and crosse it at right angles in the middle with the line D E which is the line of East and West and serveth for the hour of six at morning and six at evening Then upon the center C which is the point of intersection describe a circle for your Dial as large as your plane will give leave which let be the circle A D B E then take the latiude of the place which is here 52 deg 30 min. and set it from A to N in the quadrant A D and draw the line C N S then from A raise the perpendicular A S to cut the line C S at S so shall the Triangle A C S be the true pattern for your cock this being done divide the two quarters of your circle A E and A D each into six equal parts so shall you have in each Quadrant five ponits by which you may draw the five Chord lines I F G H and A as here you see then take one half of the Chord-line A and set it in the line of the stile from C to O from which point O take the neerest extent unto the meridian with this distance setting one foot in the point A with the other make a mark on each side of the Meridian in the same Chord-line A through which points you shall draw the hour-lines of 1 and 11. So likewise you may take one half of the chord H and place it in the line of the stile from C to K from which point K take the shortest extent unto the meridian with this distance set one foot in H and with the other make on each side the meridian a mark in the same chord-line through which you shall draw the hour-lines of 2 and 10. And thus you may proceed with the rest of the lines as the Figure will shew better then many words for this is sooner wrought then spoken And if you would have the hours before and after six you may extend them through the center as was shewed in the eighth Chapter CHAP. XIII To draw a Dial upon a direct vertical plane as well erect as inclining otherwise then in the ninth Chapter was shewed THe work of this is almost like unto the other before the difference is onely in the elevation of the Pole above the plane for in the horizontal plane the elevation is equal to the latitude of the place and in all direct verticals being erect the elevation of the pole above the plane is equal to the complement of the latitude but if they shall incline towards the horizon then shall you finde the elevation of the pole above the plane by the 10 Chapter The elevation of the pole above the plane being known the making of these Dials are all alike therefore by the second Chapter draw the line E W parallel to the horizon and from the middle thereof let fall the perpendicular Z N which shall be the meridian of the plane and also the meridian of the place serving for the line of 12 and also for the substile over which the stile must hang both in erect and inclining planes being direct Then upon the center Z describe your Dials circle or rather the Semicircle E N W and seeing this plane is erect and also direct therefore the elevation of the pole above the plane is 37 d. 30 m. equal to the complement of our latitude which take from your Scale place it from N to H in your Dials semicicircle and draw the line Z H S for the line of the stile then from the end of the meridian as at N draw the crooked line N S cutting the line of the stile in the point S so shall the triangle S Z N be the true pattern for your cock This being done divide each quadrant of your Semicircle into six equal parts so shall you have five points by which you may draw five chord-lines cutting the Meridian at right angles in the points I K L M N. This being done take the half of each chord and place it from the center Z along upon the line of the stile as here you see the half of the Chord N from Z to A and one half the chord M from Z to B and half the chord L from Z to C and one half the chord-lines I and K set from Z unto D and G now from each of these points take the neerest extent unto the Meridian Z N place them upon their proper chord-lines from the meridian on both sides thereof so shall you have two points on each Chord through which you shall draw the hour-lines from the center of your Dial as the shortest extent from the point A unto the meridian set in the Chord N from the Meridian both wayes shall give you the points for 1 and 11 so shall the shortest extent from the point B being placed from the Meridian both wayes in the Chord M give you the two points for 10 and 2 and so you may proceed with the rest thus doing you shall have in each chord two points on each side the meridian one through which and from the center Z you may draw your hour-lines at pleasure without exceeding the limits of your plane And seeing this is the South face of this plane therefore the stile must point downwards being fixed in the center Z in the upper part of the meridian line Z N over which the stile must directly hang making therewith an angle equal to the angle N Z S. But if it had been the North face then must the center be placed in the lower part of the Meridian and the stile with the substile and also the hour-lines must point upwards CHAP. XIV The declination of an upright plane being given how thereby to find the elevation of the Pole above the same with the angle of Deflexion or the distance of the substile from the Meridian and also the angle of inclination betwixt both Meridians IN all erect declining planes when the declination is found there is three things more to be considered before we can come to the drawing of the Dial. I. The elevation of the pole above the plane II. The distance of the substile
from the Meridian III. The angle contained betwixt the Meridian of the plane and the Meridian of the place which here we call the inclination of Meridians this angle is made at the Pole and serveth to shew us where we shall begin to divide our Dial-circle into 24 equal parts These three may be both artificially easily and speedily performed after this manner following First describe a Quadrant as A B C then supposing your Latitude to be 52 deg 30 min. take it from your Scale and set it from B to E in the arch of the Quadrant and draw the line E D parallel to A B cutting the line A C in the point D then take the distance D E and setting one foot in the center A with the other describe the arch G H O R. Then suppose your declination to be 32 deg this set from B to F in the arch B E C and draw the line F A cutting the arch G R in the point H. through which point draw the line S H N cutting the arch B E C in N so shall the arch C N be the elevation of the pole above the plane which in this example is found to be 31 deg 5 min. Now from the point L draw the line L T parallel to the line A C cutting the arch G R in the point O through which point O draw the line A O I cutting the arch B C in the point I so shall the arch C I be the inclination of both Meridians and is found by this example to be 38 deg 13 min. so that by this example the Meridian of the plane will fall betwixt the hours of 2 and 3 if the plane shall decline Westward but if it shall decline Eastward then shall it fall betwixt the hours of 9 and 10 before noon CHAP. XV. To draw a Dial upon an erect or vertical plane declining otherwise then in the 11 Chapter was shewed HAving by the third Chapter found the declination of this plane to be 32 degrees and so by the last Chapter found the elevation of the pole above the plane to be 31 deg 5 min. and the distance of the substile from the meridian to be 22 degr 8 minutes and likewise the angle of inclination between both meridians to be 38 degreees 13 minutes we may proceed to make the Diall after this manner First draw the horizontal W E and the perpendicular line Z N crossing the horizontal line at right angles which is the meridian of the place and the line of 12. Then in the meridian make choice of some point with most convenience as the center C whereupon describe your Dial-circle E N W. Then take a chord of 22 degrees 8 minutes from your Scale for the distance of the substile from the Meridian and inscribe it into this circle from the Meridian upon these conditions that if the plane declineth west then must the substile be placed East of the plumb line but if the declination shall be East then must the substile be placed west from the Meridian as here it is This 22 degrees 8 minutes being set in the Dial-circle from the Meridian at N unto M I draw the line C M for the substile then through the center C draw the diameter A B making right angles with the substile C M above this Diameter there needs no hour lines to be drawn if the plane be erect Then take 31 deg 5 min. and set them from M to D and draw the line C D S for the stile then from M the end of the stile draw the crooked line M S cutting the line of the stile in the point S so shall the triangle S C M be the true pattern for the cock of the Dial. This being done take 38 deg 13 min. and set them alwayes on that side the substile whereon the line of 12 lieth ' as here from M to A so shall the point H be the point where you shall begin to divide your Dial-circle into 24 equal parts but those points shall be onely in use which do fall below the Diameter A C B. And if the line of the substile falleth not directly upon one of the hour-lines then shall you have six points on each side thereof from which you may let perpendiculars fall unto the line of the substile as here you see done Now take each perpendicular betwixt your Compasses and with one foot in the center C with the other make marks in the line of the stile from which take the neerest extents unto the line of the substile and lay them upon their own proper perpendiculars from the Substile so may you make points through which you may draw hour-lines and by thus doing with each perpendicular on both sides the substile you may describe your whole Dial as here you see which may serve for four faces by observing what was spoken in the 11 Chapter When you have drawn and described your Dial upon paper for any plane whasoever you may cut off the hour-lines cock and all with a lesser Circle then the Dial circle either with a concentrique or an excentrique circle and so make a Dial lesse then the Circle by which you framed it Of a Plane falling neer the Meridian When as the declination of a plane shall cause it to lie neer the Meridian as that the declination and inclination shall cause it to lie neer the Pole then doth the elevation of the Pole above the plane grow so small and the hour-lines so exceeding neer together that except the plane be very large they will hardly serve to good purpose as here in this figure being a plane which is erect and declining from the South 80 deg towards the East Therefore first draw your Dial very true as before hath been taught upon a large paper making your circle as big as you can then extend the hour-lines with the substile and the line of the stile a great way beyond the Dials circle until they do spread so that they will fill the plane indifferent well and then cut them off with a long square as O N in the following figure so will it shew almost like the Meridian Dial of the 7 Chapter for the hours will be almost parallel the one to the other and the stile almost parallel to the substile as you may see by the figure CHAP. XVI The inclination of a Meridian plane being given how thereby to find the elevation of the pole above the plane the distance of the Substile from the meridian and the angle of the inclination of the meridian of the plane to the meridian of the place ALL those planes wherein the horizontal line is the same with the Meridian line are therefore called Meridian planes if they may make right angles with the Horizon they are called erect Meridian planes and are described before An erect Dial declining from the South 80 deg towards the East the distance of the Substile from the meridian 37 d. 4 min.
cutting the arch C B in the point K then take the distance H L and set it in the line F Z from Zunto O through which point O draw the line AOM cutting the arch B C in the point M from which point M draw the line M P N parallel to A B cutting the arch G I in the point P through which point P draw the line A P Q cutting the arch B C in the point G so shall the arch B K be 75 deg 58 min. the inclination of the plane to the Meridian and the arch B Q will be 57 deg 36 min. for the Meridians ascension or the arch of the plane betwixt the Horizon and the Meridian and the arch B M shall be 20 deg 54 min. for the elevation of the Meridian or the arch of the Meridian betwixt the Horizon and the plane Now if the plane shall incline toward the South adde this elevation of the Meridian to your Latitude and the sum of both shall be the position Latitude or the arch of the Meridian betwixt the Pole and the plane and if the sum shall exceed 90 deg take it out of 180 deg and that which remaines shall be the position latitude or the arch of the Meridian between the Pole and the plane But if the inclination shall be northward then compare the elevation of the meridian with your Latitude and take the lesser out of the greater and so shall the difference be the position Latitude As here in this example supposing the inclination to be Northward we take 20 deg 54 min. the elevation of the meridian out of 52 deg 30 min. the Latitude proposed and there will remain 31 deg 36 min for the position Latitude or the arch of the meridian between the Pole and the plane This being done set 31 deg 36 min. the position Latitude from B to T in the arch B C and draw the line A T then with the distance K L upon the center A describe the arch Y M W cutting the line A T in the point M through which point M draw the line R S parallel to A B cutting the arch B C in the point S so shall the arch B S be 30 degrees 33 minutes the height of the Pole above the plane Then lay your rule upon the point S and the center A and where it shall cut the line K L there make a mark as at V through which point V draw the line D V N W parallel to A B cutting the arch Y W in the point N and the arch B C in W so shall the arch B W be 8 deg 35 min the distance of the substile from the meridian Lastly through the point N draw the line Y X parallel to AC cutting the arch BC in the point X so shall the arch BX be 16 deg 20 min. the inclination of the meridian of the plane to the meridian of the place CHAP. XIX To draw a Dial upon a declining inclining Plane HAving by the second Chapter found the inclination to be 25 deg towards the North and by the third Chapter the declination from the South towards the East to be 35 deg and so by the last Chapter the meridians ascension to be 57 deg 36 min. The elevation of the Pole above the plane 30 deg 33 min. The distance of the substile from the meridian 8 degrees 35 min. And the inclination of both meridians 17 deg 30 min. we may proceed to make the Dial after this manner First Draw the line A B parallel to the Horizon in which line make choice of a center as at C whereon describe your Dial circle A D B E A then take 57 deg 36 min. the meridians ascension and set it from B that end of the Horizontal line with the declination of the plane as from B to N and draw the line C N for the hour of 12. Then set 8 deg 35 min. the distance of the substile from the Meridian from N to M on that side the meridian which is contrary to the declination of the plane and draw the line C M for the substile And set 30 deg 33 min. from M to D and draw the line C D unto S and from the end of the substile draw the crooked line M S cutting the line of the Stile in S so shall the Triangle M C S be the true pattern of this Dials Cock. Then set 17 deg 30 min. the inclination of meridians from M unto O which is the point where you must begin to divide your Dial circle into 24 equal parts from which points let down so many perpendiculars to the substile as there shall be points on that side the Diameter F E next the substile and so by working as before hath been shewed you may draw the hour-lines and set up the stile as in the former planes Now here I would have you well to consider what hath been here spoken concerning these kind of Dials and also what followeth the same for if you mark the diversity which doth arise by reason of the elevation of the meridian you may perceive thereby three sundry kinds of Dials to arise out of a North inclining plane declining and also in a South inclining declining plane yet in effect they are but one if you consider what followeth here concerning them in all which the stile with the substile and such like materials are found out according to the last Chapter Therefore having drawn your horizontal line you must consider which pole is elevated above your plane and how to place the meridian from the Horizontal line For upon the upper faces of all North incliners whose meridians elevation is lesse then the Latitude of the place on the under faces of all North incliners whose meridians elevation is greater then the Latitude of the place and on the upper faces of all South incliners the North Pole is elevated Now for placing the Meridian from the horizontal line upon the upper faces of all South incliners whose meridians elevation is greater then the Latitudes complement on the under faces of all South incliners whose meridians elevation is lesse then the Latitudes complement on the under faces of all North incliners whose meridians elevation is greater then the Latitude of the place and on the upper faces of all North incliners whose Meridians elevation is less then the Latitude of the place the Meridian must be placed above the Horizontal line as here in this example And therefore by the contrary Upon the upper faces of all South incliners whose meridians elevation is lesse then the Latitudes complement On the under faces of all South incliners whose meridians elevation is greater then the Latitudes complement On the under faces of all North incliners whose meridians elevation is lesse then the Latitude of the place And on the upper faces of all North incliners whose meridians elevation is greater then the Latitude of the place the
Meridian must be placed below the Horizontal line But here you must observe that if it be either the upper or under faces of a South inclining plane whose meridians elevation is greater then the Latitudes complement or either the upper or under faces of a North inclining plane whose meridians elevation is lesse then the Latitude of the place that then the Meridian must be placed from that end of the horizontal line with the declination of the plane But on all the other faces of these kinds of planes the meridian must be placed from that end of the horizontal line which is contrary to the declination of the plane And here note that if the inclination shall be Southward and the elevation of the meridian equal to the complement of your Latitude then shall the substile lie square to the Meridian And if the inclination shall be Northward and the elevation of the Meridian equal to the Latitude of the place then shall neither Pole be elevated above this plane and therefore shall be a Polar declining plane Wherein the meridian being placed according to his ascension from the horizontal line shall be in place of the substile unto which if you draw a line square it shall serve for the Equator Then set one foot of your compasses in the intersection of the substile with the Equator and open the other to any convenient distance upon the substile and describe the Equinoctial circle as in the sixth Chapter of this Book was shewed upon the center whereof make an angle with the line of the substile equal to the inclination of both meridians namely the meridian of the plane and the meridian of the place which shall shew you where to begin to divide your Equinoctial circle into 24 equal parts These things being known you may proceed to make your Dial and set up the cock according to the 6 chapter As for example in our Latitude of 52 deg 30 min. a plane is proposed to decline from the South towards the East 35 deg as before but inclining Northward 57 deg 50 min. the Meridians ascension by the 18 chapter will be found to be 69 deg 33 min. and his elevation 52 deg 30 min. equal to the latitude of the place and therefore neither pole is elevated above this Plane and so no distance between the Substile and the Meridian for the Meridian and the stile with the substile will be as it were all one line which is the Axletree of the world so that here the stile must be parallel to the plane and the hour-lines parallel one to the other as in the Meridian and direct Polar Planes Therefore first draw the Horizontal line A B wherein make choice of a center as at C whereon describe an occult arch of a circle as B E then into this arch inscribe the meridians ascension 69 deg 33 min. from B to E and draw the line C E for the meridian of the plane and for the substilar and if you draw a line square to this substilar it shall be the Equator Then set one foot of your compasses in the point of intersection D and with the other opened to a convenient widenesse draw a circle for the Equator unto which you may draw two touch-lines square to the substile as in the direct Polar plane The end of the Third Book THE FOURTH BOOK Shewing how to resolve all such Astronomical Propositions as are of ordinary use in this Art of Dialling by help of a Quadrant fitted for the same purpose CHAP. I. The description of the Quadrant HAving in the second and third Books shewed Geometrically the working of most of the ordinary Propositions Astronomical with the delineation of all kind of plain wall Dials howsoever or in what latitude soever scituated 〈◊〉 I keeping within the limits of our plane and yet not tyed to the use of any Instrument I will now shew how you may performe the former work exactly easily and speedily by a plain Quadrant fitted for that purpose the description whereof is after this manner Having prepared a piece of Box or Brasse in manner of a Quadrant draw thereon the two Semidiameters A B and A C equally distant or parallel to the edges cutting one the other at right angles in the center A upon which center A with the Semidiameter A B or A C describe the arch B C this arch is called the limb and is divided into 90 equal parts or degrees and sub-divided into as many parts as quantity will give leave being numbered from the left hand towards the right after the usual manner Then let the Semidiameter A B be divided into 90 unequal parts called right Sines either from the Table of natural Sines by help of a decimal Scale equal to the Semidiameter A B or else by taking the neerest extents from each degree of your Quadrant unto the side A B and placing them upon the side A B each after other from the center A towards B you shall exactly divide the Semidiameter A B into 90 unequal divisions called right Sines This being done draw the line D E from the Sine of 45 degrees counted in the line of Sines unto 45 degrees counted in the Quadrant then from the point E draw the line E F parallel to A B making the square A D E F the side D E whereof for distinction may be called a Tangent line and the side F F a Co-tangent line then draw the Diagonal line A E which you may call the line of Latitudes Then upon the center A with the distance A D or A F describe the arch D F which you may divide into six equal parts by laying your Rule upon each 15th degree in the Quadrant and the center A as at g h I k l F from which points draw slope lines to each 15th degree in the Quadrant numbered backward as F P l O k E I n h m g B these lines so drawn are to be accounted as hours then dividing each space into two equal parts draw other slope lines standing for half hours which may be distinguished from the other as they are in the figure Now because in the latter part of this Book there is often required to use a line of Chords to several Radiusses therefore upon the edge of the Quadrant A C you may have a line of Chords divided as in the figure and so the Quadrant being at hand will supply the uses of the Scale mentioned in the preceding Book and also a Chord of any Circle whose Radius is lesse then the line A C may be taken off and in that case supply the use of a Sector To this Quadrant as to all others of this kind in their use is added Sights with a threed bead and plummet according to the usual manner CHAP. II. Of the use of the line of Sines Any Radius not exceeding the line of Sines being known to find the right Sine of any arch or angle thereunto belonging
setting one foot in the sine of 54 deg and with the other I lay the threed to the neerest distance which lying still in this position I take it over from the sine of 30 d. which distance shall be the greater segment A C dividing the whole line in the point C or the threed lying in the former position if you shall take the shortest extent thereunto from 18 deg you shall have B C for the lesser segment which will divide the whole line by extream and mean proportion in the point C from the end B so that as B C the lesser segment is to A C the greater segment so is A C the greater segment to A B the whole line as was required CHAP. VII To find a mean proportional line between two right lines given A Mean proportional line is that whose square is equal to the long square contained under his two extreams First joyn the two given lines together so as they may make both one right line the which divide into two equal parts and with the one half thereof setting one foot in the sine of 90 deg with the other lay the threed to the neerest extent which lying still in this position take the distance betwixt the middle point and the point of meeting of the two given lines and fixing one foot in the line of sines so as the other may but onely touch the threed now from the complement of the sine where the fixed foot so resteth take the shortest extent unto the threed which shall be the mean proportional line required As for example let A and B be two lines given between which it is required to find a mean proportional line first joyne the two lines together in F so as they both make the right line C D which divide into two equal parts in the point E then with either halfe of which setting one foot in the sine of 90 deg with the other lay the threed to the neerest distance then keeping the threed in this position take the distance between the middle point E and F the place of meeting of the two given lines and fixing one foot in the line of sines so as the other may but only touch the threed and the fixed foot will stay about 22 deg 30 min. the complement whereof is 67 deg 30 min. from which take the shortest extent unto the threed lying as before which shall be the line H the meane proportional line betwixt the two extreames A and B which was required CHAP. VIII Having the distance of the Sun from the next equinoctial point to find his declination FIrst lay the threed upon 23 d. â—0 m. the suns greatest declination counted on the limb of the quadrant the threed lying still open at this angle take the shortest extent thereunto from the sine of the distance of the Sun from the next Equinoctial point this distance being applyed to the line of sines from the center A shall give you the sine of the Suns declination So in the figure of the 13 chapter the Sun being in the 29 d. of Taurus at K which is 59 d. from C the Equinoctial point Aries the declination of the Sun will be found about 20 d. the line C M which was required CHAP. IX The declination of the Sun and the quarter of the ecliptick which he possesseth being given to find his place TAke the sine of the Suns declination from the line of sines and setting one foot in the sine of the Suns greatest declination with the other lay the threed to the neerest distance so shall it shew upon the limb the distance of the Sun from the next Equinoctial point So in the figure of the 13 chap. C M the declination of the Sun being 20 d. and K the angle of the suns greatest declination the line C K will be found to be 59 d. for the distance of the Sun from the next equinoctial point which was required CHAP. X. Having the latitude of the place and the distance of the Sun from the next equinoctial point to find his amplitude TAke the sine of the suns greatest declination betwixt your compasses and setting one foot in the co-sine of the latitude with the other lay the threed to the neerest distance which lying still in this position set one foot in the sine of the Suns distance from the next equinoctial point and with the other take the neerest extent unto the threed so shall you have betwixt your compasses the Sine of the Amplitude As in the figure of the 13 chapter the angle at N being 37 deg 30 min. the complement of the latitude and K the angle of the Suns greatest declination and C K 59 deg the distance of the Sun from the equinoctial point Aries the line C N will be found to be the sine of 34 deg 9 min. the amplitude required CHAP. XI Having the declination and amplitude to find the height of the pole FIrst take the sine of the suns declination and set one foot in the sine of the Amplitude and with the other lay the threed to the neerest distance so shall the threed upon the limb shew the complement of the latitude So in the figure of the 13 chapter the declination C M being 20 deg and the amplitude C N being 34 d. 9 m. and the angle at M being right we shall find the angle at N to be 37 deg 30 m. the complement whereof is 52 deg 30 m. which was required for the latitude of the place CHAP. XII Having the latitude of the place and the declination of the Sun to find his amplitude WIth the sine of the declination set one foot in the co-sine of the latitude and with the other lay the threed to the neerest distance so shall it shew upon the limb the amplitude required so in the figure of the next chapter the angle C N M being 37 deg 30 min. the co-sine of the latitude and C M the declination here 20 deg and the angle at M being right we shall find the base C N to be the sine of 34 which was required for the Suns amplitude CHAP. XIII Having the elevation of the pole and amplitude of the Sun to find his declination FIrst lay the threed to the amplitude counted in the limb then take it over at the shortest extent from the co-sine of the latitude so shall you have the sine of the Suns declination betwixt your compasses So in this figure the Amplitude C N being 34 deg 9 m. and the angle at N being co-sine to the latitude the angle at M being a right angle we shall find C M to be 20 deg for the declination of the sun which was required CHAP. XIV Having the latitude of the place and the declination of the Sun to find his height in the Vertical Circle FIrst take the sine of the declination of the Sun and setting one foot in the sine of the
for the time of Sun setting And so contrarily when the Sun hath South declination if you adde this ascentional difference to 6 hours you shall have the time of his rising and if you take it away from 6 houres that which is left shall be the time of Sun setting CHAP. XXI The Latitude of the place the Almicanter and declination of the Sun being given to find the Azimuth IF the suns declination be Northward then by the 14 or 15 chapters get his height in the Vertical circle for the day proposed from the sine of which take the distance unto the sine of the Suns altitude observed with this distance setting one foot in the co-sine of the latitude with the other lay the threed to the neerest distance unto which being kept still in this position take the least distance from the sine of the latitude with this distance setting one foot in the co-sine of the Suns altitude with the other lay the threed again to the neerest distance so shall it shew upon the limb the Suns Azimuth from the East or West either Northward or Southward So in this figure having N M the distance betwixt the sine of 14 deg 33 min. the height of the Sun in the Vertical circle and the sine of 30 deg 45 min. the height of the Sun at the time of observation and 52 deg 30 m. the angle N O M the latitude of the place the complement whereof is 37 deg 30 min. the angle M N O we shall find M O to be the sine of 23 deg 17 min. the Azimuth from the East or West points Southward And here note when the declination is Northward that as when the latitude of the Sun given and his height in the Vertical circle is equal he is directly in the East or West so when his altitude given is greatest then is the Azimuth towards the South and when his altitude given is least then is the Azimuth towards the North. But if the declination of the Sun be Southward then by the 10 or 12 chapters find the Amplitude for the day proposed Now first take the sine of the Suns altitude and setting one foot in the co-sine of the Latitude with the other lay the threed to the neerest distance which threed lying still in this position take it over at the shortest extent from the sine of the Latitude this distance adde to the sine of the Amplitude by setting one foot in the sine of the Amplitude and extending the other upon the line of sines these two being thus joyned take them betwixt your compasses setting one foot in the co-sine of the Suns altitude and with the other lay the threed to the neerest distance so shall it shew upon the limb the Suns Azimuth from the East or West towards the South So in this figure having V C or T N 19 deg 7. min. the amplitude for the day proposed and T V the sine of the Suns altitude being 13 deg 20 min. and 52 deg 30 min. the angle V X T the latitude of the place and the angle T V X the complement thereof we shall find X N to be the sine of 40 deg 11 min. the Azimuth of the Sun from the East or West points Southward which was required CHAP. XXII The latitude of the place the declination and altitude of the Sun being given to find the hour of the day IF the declination of the Sun be Northward find the height of the Sun at the hour of six by the 17 chapter betwixt which sine and the sine of the Suns altitude given take the distance upon the line of sines with which distance setting one foot in the co-sine of the latitude with the other lay the threed to the neerest distance the threed lying still in this position take it over at the shortest extent from the sine of 90 deg with this distance setting one foot in the co-sine of the declination with the other lay the threed again to the neerest distance so shall it shew upon the limb the quantity of time from the hour of six So in this figure having M N the distance betwixt the sine of 9 deg 5 min. the height of the Sun at the houre of six and the sine of 42 deg 33 min. the height of the Sun given and the angle M O N 52 deg 20 min. the Latitude of the place and his complement M N O we shall find N O to be the sine of 60 deg the quantity of time from the houre of six which 60 deg is four hours of time And here also note that if the altitude given be greater then the altitude of the Sun at the houre of six then is the time found to the Southward of the houre of six but if it be lesser then is it to the Northward But if the declination of the Sun be Southward find his depression at the hour of six by the 17 chapter for the day proposed which will be equal to his height at six if the quantity of declination be alike Now take the sine of this depression and adde it to the sine of this altitude observed by setting one foot in the sine of his altitude and extending the other upon the line of sines These two being thus joyned together in one take them betwixt your compasses and setting one foot in the co-sine of the latitude as before and with the other lay the threed to the neerest distance which lying still in this position take it over at the shortest extent from the sine of 90 deg with this distance setting one foot in the co-sine of the declination as before with the other lay the threed again to the neerest distance so shall it shew upon the limb the quantity of time from the houre of six So in this figure having the sine of 15 deg â—4 min. the altitude of the Sun given and the sine of 9 deg 5 min. his depression at the houre of six joyned both together in one straight line as T V and having the angle T X V 52 deg 30 min. the latitude given and the angle T V X the co-latitude we shall find T X to be the sine of 45 deg the quantity of time from the hour of six which converted into time will make three hours The end of the fourth Book THE FIFTH BOOK Shewing how to describe the hour-hour-lines upon all sorts of Planes howsoever or in what Latitude soever scituated by a Quadrant fitted for the purpose CHAP. I. How to examine a plane for an Horizontal Dial. IF your Plane seem to be levell with the Horizon you may try it by laying a Ruler thereupon and applying the side A B of your quadrant to the under side thereof and if the threed with the plummet doth fall directly upon his level line A C which way soever you turn it it is an Horizontal Plane Or if you set the side A B of your
quadrant upon the upper side of your Ruler so that the Center may hang a little over the end of your ruler and holding up a threed and plummet so that it may play upon the Center if it shall fall directly upon his level line A C making no angle therewith it is an Horizontal plane as here you may see by this figure CHAP. II. Of the trying of planes whether they be erect or inclining and to find the quantity of their inclination IF the plane seeme to be erect you may try it by holding the quadrant so that the threed may fall on the plumb line A C for then if that side of the quadrant shall lie close to the plane it is erect and a line drawn by that side of the quadrant shall be a Vertical line and the line which crosseth this Vertical line at right angles will be the Horizontal line as here you may see in this figure the plane D E F G being erect and the line D E being vertical the line F G must be horizontal But if the plane shall incline the quantity of inclination may be found out after this manner First you must draw thereon the Horizontal line which you may doe upon the under face by applying the side A B of your quadrant thereunto so as the threed and plummet may fall upon the plumb line A C the side A B lying close with the plane by which if you draw a line it shall be parallel to the Horizon Or you may draw an horizontal line upon the uppe face by laying a Ruler thereupon and applying the side A B of your quadrant to the under side thereof still moving your Rule untill the threed and plummet doth fall directly upon the plumb line A C the Rule lying thus close to the plane you may thereby draw a line parallel to the Horizon Having drawn this Horizontal line M N crosse it at right angles with the perpendicular K D unto which if it be the under face apply the side A B of your quadrant so shall the threed upon the limb give you the angle of inclination required But if it be the upper face of the plane then lay a Ruler to the perpendicular K D unto the under side whereof apply the side A B of your quadrant as is here shewed in this figure so shall the degree of the quadrant give you C A H the angle of inclination required But if it be so that you cannot apply the side of your quadrant to the under side of your Ruler then set it upon the upper side thereof so that the Center thereof may hang a little over the end of the Ruler and holding up a threed and plummet so that it may fall upon the center A and it shall shew upon the limb the inclination of the plane which is the angle C A H equal to the former angle Here you must be heedful that both edges of your Ruler be straight and one parallel to the other CHAP. III. To find the Declination of a plane TO find out this Declination you must make two observations by the Sun the first is of the angle made between the Horizontal line of the plane and the Azimuth wherein the Sun is at the time of observation the second is of the Suns altitude both these observations should be made at one instant First for the Horizontal distance having drawn upon your plane a line parallel to the Horizon apply the side of your quadrant thereunto holding it parallel to the Horizon then holding up a threed and plummet at full liberty so as the shadow thereof may passe through the center of the quadrant observe the angle made upon the quadrant by the shadow of the threed and that side with the Horizontal line for that is the distance here required Then at the same instant as neer as may be take the Suns altitude that so you may find the Suns Azimuth from the East or West points by the 21 chap. of the fourth Book Having thus gotten the horizontal distance with the Azimuth of the Sun for the same time describe a circle as A B C D representing the horizontal circle and draw the diameter A C which shall represent the horizontal line F G of of the last chapter Now supposing the horizontal distance to be 38 deg 30 min. the angle O A B of the last chapter place it from C South ward to E that is from the same end of the horizontal line with which the angle was made upon the plane and draw the line E Z Then supposing the altitude of the Sun at the same time to be 30 deg 45 min. with 11 deg 30 min. North declination and so by the 21 chapter of the fourth Book the Azimuth will be found to be 23 deg 17 min. from the East Southward being the observation was made in the fore noon this 23 deg 17 min. I place from E the place of the Sun at the time of observation unto R which is the true point of the East and draw the line H R representing the Vertical circle so shall the angle made between the horizontal line of the plane and the line of East and West be the declination of the plane which in this example is found to be 15 deg 13 m. the angle C Z R. Or you may observe the angle made between the shadow of the threed and that side of the quadrant which lyeth perpendicular unto the horizontal line of the plane which in this example is 51 deg 30 min. the complement of the former angle and it is the angle O A C in the former chap upon the quadrant Now having drawn your Horizontal Circle as before and the diameter A C for the horizontal line of the plane you may crosse it at right angles with the diameter B D for the axis of the planes horizontal line from which as from D you may set your horizontal distance on the same side thereof as before you found it by your observation as here from D to E and draw the line E Z for the line of the shadow and having found the Azimuth of the Sun 23 deg 17 min. from the East Southward you may set it from E the place of the Sun Northward to R and draw the line RZH for the line of East and West as before Or if you take the Suns Azimuth from the South which in this example will be 66 deg 43 min. the complement of the former 23 deg 17 min. you may set it from E the place of the Sun unto S Southward and draw the line S Z N for the meridian so shall the arch S D or R C be 15 deg 13 min. for the declination as before CHAP. IV. To draw the hour-lines upon the Horizontal the full North or South planes whether erect or inclining SEeing the making of these Dials are all after one manner we will here proceed to make an Horizontal Diall by help of the lines
latitudes which are equal to the height of the stile above such reclining plains they are Horizontal planes One example therefore in one plane will be sufficient for the rest Therefore in Latitude of 52 deg 30 min. Let it be required to describe the Equinoctial and the two Tropicks in a full South erect plane Having drawn your Diall with the houres halves and quarters as also the line C Q for the stile you must make choice of some convenient point in the stile as at S for the Nodus or knot which must give the shadow to the Tropicks and other parallels of declination for all these Astronomical conclusions are not shewed by the shadow of the whole length of the stile or Axis as the houre is but by some point therein which representeth the center of the earth which in the Dial following is the points and the triangular stile in that Dial is represented by the triangle C S L whereof C L is called the substilar C S the Axis of the stile and S L the perpendicular stile the top of which viz. S is the point we are in this place to respect The Dial being drawn and the Triangle C S L made equal to the Cock of the Dial you must upon a piece of pastboard draw the Triangle O P R equal to the stile in your Dial C S L making R O equall to C L the substilar P O equal to C S the Axis of the stile and P R equal to S L the length of the perpendicular stile Then from the point P raise a perpendicular as P B representing the Equinoctial and on Pas a center describe the arch A B C now because the Tropicks of Cancer and Capricorn doe decline 23 deg 31 min. from the Equinoctial therefore take 23 deg 31 min. from your Scale of Chords and set it off upon the arch A B C from B t A and from B to C and draw the lines P A and P C representing the two Tropicks of Cancer and Capricorn This done extend the line of the substilar R O which 〈◊〉 North or South erect direct planes I told you was alwayes the same with the twelve a clock line from O to 12 cutting the Equinoctial line P B in the point a then with your compasses take the distance O a out of your Trigon and place it in your plane from the center C unto a and draw the line ♈ a ♎ perpendicular to the substile or line of 12. The Equinoctial being drawn First take out of your plane the distance C b and place that distance in your Trigon from O unto b and draw the line O b 1 representing the hour of 1 or 11 in your Dial. Secondly take out of your plane the distance C c and place that in your Trigon from O unto c and draw the line O c 2 representing the hour-lines of 2 or 10. Thirdly take out of your plane the distance C d and place it in your Trigon from O unto d and draw the line O d 3 for the hours of 9 and 3. Fourthly from our plane take the distance C e and set in your Trigon from O unto e and draw the line O e 4 representing the hours of 4 and 8. And thus must you doe with the rest of the hours in your plane if occasion require These lines O a O b O c O d and O e in your Trigon being extended doe cut the Tropick of Cancer P A in the points 12 1 2 and 3 therefore out of your Trigon take the distances O 12 O 1 O 2 O 3 O 4 and set them upon their correspondent hour-lines of your plane from the center C unto g h i k and l so shall the points g h i k and l be the points upon the hour-lines through which the Tropick of Cancer must passe and is therefore noted with the character of Cancer ♋ at both ends ¶ Now before you draw the Tropick of Capricorn it is necessary to draw the Horizontal line of your plane A B which line in all upright planes must be drawn through the point L the foot of the perpendicular stile and perpendicular to the Meridian or line of 12 And in all planes whatsoever this line must be drawn through the intersection of the Equinoctial with the houre of six This line ought first to be drawn because it is very improper to extend the Tropicks or other parallels of Declination above the Horizontal line because at what hour any parallel of Declination cutteth this line on either side of the Meridian at that time doth the Sun rise or set as was instanced in the last Now the Tropick of Capricorn must be put upon your plane in the same manner as that of Cancer by taking out of your Trigon the distance from O where the several hour-lines a b c d e doe cut the Tropick of Capricorn P C and place them on your plane from the center C upon the respective hour-lines and through those points so found draw the line ♑ ♑ representing the Tropick of Capricern ¶ And in the same manner may the parallels of the other Signes be drawn upon your plane by placing them into your Trigon according to their Declinations and afterwards transfer them into your plane as you see in the former figure The rules that have been here given for the describing of the parallels of the Signes in this erect direct plane is universal in all planes observing this one exception that whereas in all erect direct planes the Eqninoctial is drawn perpendicular to the Meridian or line of 12 so in all other planes whatsoever the Equinoctial must be drawn perpendicular to the substile and then the work will be the same in all respects as may appear more largely in the next Section § 3. In Declining or Declining Reclining Dials THe last caution preceding is sufficient for the performing of this work and therefore needeth no example However suppose an upright plane to decline 32 deg from the South Eastwards in the Latitude of 52 deg 30 min. and let it be required to describe the two Tropicks and the Equinoctial upon such a plane And here note that whatsoever is said of upright decliners the same is also to be understood of those planes which both decline and recline and for the horizontal line in all reclining or inclining planes it must passe through the foot of the perpendicular stile and the intersection of the Equinoctial with the houre of six CHAP. II. Shewing how to inscribe tho parallels of the length of the day on any plane THe parallels of the length of the day and those of the Signes are inscribed upon all kinde of planes by one and the same Rules they being in the Sphere the same Circles so that as when you put on the parallels of the Suns entrance into the 12 Signes you seek what declination he hath and accordingly proceed as before so now for the parallels of the length of the
two parallels justly upon the houres halves and quarters of the common hour-lines and so be the easier drawn Now the points through which every one of the Jewish hours must passe is exactly shewed by this little Table wherein you may see that the first Jewish hour must be drawn through 5 hours 45 mi. or 5 hours three quarters in the parallel of 15 through 7 hours in the Equinoctial and through 8 hours and a quarter in the parallel of 9 hours In like manner the second Jewish hour must be drawn in your plane through 7 of the clock in the parallel of 15 through 8 a clock in the Equinoctial and through 9 of the clock in the parallel of 9 hours and so of all the rest according as you see in thsi Table and as you may perceive them drawn in the South plane the numbers belonging to these hours being set at both ends of each hour-hour-line CHAP. V. Shewing how to draw the Azimuths or Vertical Circles in all kinde of planes THe Azimuths are great Circles of the Sphere meeting together in the Zenith of the place and are variously inscribed on all planes according to their scituation In the Horizontal plane they meet in a center with equal angles In all upright planes whether direct or declining they are parallel to the Meridian or line of 12. And in all reclining planes they meet together in a point which is the Zenith of the place These Azimuths being great circles in the Sphere are therefore straight lines in all planes and may be drawn as followeth §. 1 In the Horizontal plane IN the Horizontal plane these Azimuths are most easily inserted for your Dial being drawn with the Tropicks thereon you have no more to doe but upon the foot of the perpendicular stile to describe a Circle which you may divide into 2 equall parts beginning at the Meridian answering to the 3 points of the Mariners Compasse Or else you may divide the same Circle into 90 equal parts according to the Astronomical division and through each of those points draw straight lines from the foot of the stile and set numbers or letters to them either by 10 20 30 40 c if you divide it into 90 or else by South S by W S S W S W by S c if you divide the Circle according the Mariners Compasse This is so plain that it needeth no example § 2. In the East or West erect planes YOur Dial being finished you may draw upon a piece of pastboard the line M E N representing the Horizontal line MEN in your Dial then on the point E raise the perpendicular E Q equal to the line E G in your Dial and on Q as a center describe the semicircle K E L and divide one halfe thereof namely the quadrant into eight equal parts representing one quarter of the Mariners compasse and from the center Q draw lines through each of those divisions extending them till they cut the line M E N in the points ☉ ☉ ☉ ☉ ☉ ☉ then with your compasses take the distances from E to every one of these points ☉ ☉ c. and prick them down in the Horizontal line of your plane from E to ☉ ☉ ☉ ☉ ☉ ☉ from which points draw lines perpendicular to the horizontal line M E N which shall be the Azimuths or points of the compasse between the East and the South Divide likewise the other quadrant of the Circle EK into eight equal parts and draw lines from the center Q through three of them till they cut the horizontal line as you see in the figure and there also draw lines perpendicular to the Horizon and these lines shall be the azimuths between the East and the North viz. so many of them as your plane is capable to receive which the following figure doth most plainly shew ¶ Here note that as the East Dial sheweth all the morning hours from Sun rising to the Meridian and the West Dial sheweth all the afternoon hours from the Meridian to his setting so doth the East Dial shew all the azimuths from the Suns rising till Noon and the West Dial all the azimuths from noon till his setting § 3. In the full North and South erect planes THe drawing of the azimuths upon the full North or South erect planes is very little different from the drawing of the same circles upon the East or West planes But for example let it be required to draw the Azimuths upon the full South Dial the Tropicks and the Equinoctial being drawn together with the Horizontal line you must upon a piece of Pastboard draw the line A L B representing the Horizontal line A L B in the South Dial next following then on the point L raise the perpendicular L S making L S equal to L S the perpendicular stile of the Dial and on S as a center describe the semicircle E L F and divide each quadrant thereof namely E L and L F into 8 equal parts each quadrant representing one quarter of the Mariners compasse and through each of those divisions draw lines from the center S till they cut the line A L B in the points m n o p and q then with your compasses take the distance L q and set that distance upon the Horizontal line of your plane from L unto q both wayes Likewise take the distance L p and set that distance in your plane upon the horizontal line thereof from L unto p both wayes Also take the distances L o L n and L m and set them upon the horizontal line in your Dial from L to o and n and m on each side of the Meridian Lastly if from the points m n o p and q you draw lines parallel to the Meridian or line of 12 they shall be the true azimuths upon your plane and these Azimuths may be put on either according to the Astronomical account by 10 20 30 40 c. or else by the points of the Compasse as in this figure according as you shall divide the Semicircle E L F And thus much concerning erect direct planes § 4. In erect declining planes IN upright declining planes the azimuths are easily in scribed little differing from the former Draw therefore your Dial which we will suppose to be the South declining plane before used in the third Section of the first chapter of this Appendix which declineth from the South Eastward 32 deg Lastly with your Compasses take the distances O a O b O c O d c. out of your pastboard and prick the same distances down in your plane from O to a b c d e f g h i k l and m and from those points draw lines parallel to the Meridian which lines shall be the azimuths required which you must number according to the scituation of the plane viz. the Western azimuths on the East side of the Meridian and the Eastern azimuths on the West side of your Dial as you see them here numbred
in this figure § 5. In East and West incliners and also in North and South incliners declining IN all these planes because the Zenith of the place cutteth the plane obliquely making oblique angles ther with there is in all these planes two points to be found in each plane before the azimuths can be drawn the one is the Zenith of the plane the other the Zenith of the place in which all the Azimuths must meet with unequal angles I. Therefore suppose a direct South plane to recline 25 deg from the Zenith the complement thereof is 65 deg the inclination of the under face of the same plane to the horizon therefore make the perpendicular side of the stile Radius then the Meridian will be a Tangent line thereunto upon which Meridian from the foot of the perpendicular stile prick down 65 deg for the Zenith point where all the azimuths must meet and 25 deg for the horizontal point through which the horizontal line must passe Then describing a Semicircle divide it into 16 parts and lay a ruler from the center and each of those divisions till it cut the horizontal line and thereon make marks then lay a ruler upon the Zenith point and each of these marks in the horizontal line and they shall be the true Azimuths belonging to your plane which you must number according to the scituation thereof II. In the East and West Incliners and in the North and South decliners inclining because the 12 a clock line and the substilar are several lines you must therefore draw a line perpendicular to the base of the plane which must passe directly through the foot of the perpendicular stile then make the perpendicular stile the Radius and the other line last drawn shall be a Tangent line thereto upon which line set off the inclination of the plane to the Horizon and that shall give you the Zenith point and the horizontal point shall be found by setting off the reclination of the plane from the Zenith and here note that the Zenith point will alwayes fall upon the Meridian CHAP. VI. Of the Almicanters or Circles of Altitude THe circles of altitude have the same relation to Azimuths as the Tropicks and parallels of declination have to the hour-lines and therefore as the parallels of declination in the Equinoctial plane are perfect circles so are the circles of altitude in an horizontal plane The inscription of these into all kinde of planes is in a manner the same with the parallels of declination but whereas in the drawing the parallels of declination you take the hour-lines out of your plane and put them in a Trigon so in this you must take the Azimuths out of your plane and put them into a Trigon for that purpose and so tranfer them to the plane again as you did the other and because these are small circles therefore they become conick sections except on such planes as lie parallel to the Zenith which is only the horizontal CHAP. VII How to draw a Dial on the seeling of a Room BEcause the direct beams of the Sun can never shine upon the seeling of a Room they must therefore be reflected thither by help of a small piece of Looking-glasse conveniently fixed in some Transam of the window so that it may lie exactly parallel to the Horizon The place being chosen and the glasse therein fixed you must draw upon the seeling of the Room a Meridian line as you are taught in the former Books which Meridian line must be so drawn that it may passe directly over the glasse before placed which you may perceive how to doe by holding a threed and plummet from the top of the seeling till it fall directly upon the superficies of the glasse The foundation being thus laid we will now proceed to the work which among so many wayes as these are to perform it I shall make choice of that which I suppose to be most familiar and easie Draw therefore upon paper or otherwise an horizontal Dial for the Latitude in which you are as is the horizontal Dial fore-going for the Latitude of 52 deg 30 min. Then upon the center thereof at A with the Radius of your line of Chords describe the Semicircle B C D cutting the hour-lines in the points a b c d e then with your compasses you may measure the quantity of each hours distance from the Meridian by taking the distance from C to a b c d and e so shall you find the distance between the Meridian and 11 or 1 to be 12 degrees Likewise the distance between the Meridian and 10 or 2 to be 24 degr 37 min. and the distance between the Meridian and 9 or 3 to be 38 deg 25 min. and so of the rest as by the figure and the second column of the Table doth appear This done take the complement of every of these angles so shall the complement of 12 deg be 78 deg and the complement of 24 deg 37 min. be 65 deg 23 min. and so of all the rest as by the third column of the Table may appear Having these things prepared Let the line L R in the following figure represent a Meridian line drawn upon the seeling of a Room and let K be the glasse fixed directly under the said Meridian upon some transom of the window The Hours The angle that each hour-line makes with the Meridian The complement of each hourlines angle with the Merid.   D M D M 12 00 00 90 00 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 then laying one end of a string upon the glasse at K extend the other up to the Meridian at L which point L you may finde by moving the string to and fro upon the Meridian line till another holding the side of a Quadrant to the moveable string he shall find the threed and plummet to fall directly upon the complement of the Latitude which in this example is 37 deg 30 min. The point L being thus found upon the Meridian draw the line L AE perpendicular to the Meridian L R which line shall be the Equinoctial Having thus done upon a Table or such like draw a line which shall be of equal length with L K the distance from the glasse to the point L on the seeling which line divide into 10 equal parts and each of those or at least one of them into 10 other parts so shall you have in all 100 parts each of which you must suppose to be divided into 10 other smaller parts so shall the whole line contain 1000 parts as in the figure is expressed by the line S. Your line thus supposed to be divided into 1000 parts you must take with your compasses out of the said line 268 of them which is the natural Tangent of 15 degrees and place them upon the Equinoctial line