line C E which is the perpendicular which was desired to be let fall from the given point C unto the middle of the given line D E F. CHAP. VI. To raise a Perpendicular upon the end of a line given SUppose the line whereupon you would have a perpendicular to be raised be the line B C from the point B a perpendicular is to be raised First open your Compasses unto any convenient distance which here we suppose to be the distance B E and set one foot of your compasses in B with the other draw the arch ED then this distance being kept set one foot of your compasses in the point E and with the other make a mark in the former arch E D as at D stil keeping the same distance set one foot in the point D and with the other draw the arch line F over the given point B now laying a ruler upon the two points E and D see where it crosseth the arch line F which will be at F from which point F draw the line F B which shall be a perpendicular line unto the given line B C raised from the end B as was required CHAP. VII To let a perpendicular fall from a point assigned unto the end of a line given LEt the line D E be given unto which it is required to let a perpendicular fall from the assigned point A unto the end D. First from the assigned point A draw a line unto any part of the given line D E which may be the line A B C then finde the middle of the line A C which will be at B place therefore one foot of your compasses in the point B and extend the other unto A or C with which distance draw the Semicircle A D C so shall it cut the given line D E in the point D from which point D draw the line A D which shall be the perpendicular let fall from the assigned point A unto the end D of the given line D E as was required CHAP. VIII Certain Definitions Astronomicall meet to be understood of the unlearned before the proceeding in this Art of Dialling IN the former Chapter I have shewed the meaning of some terms of Geometry which be most helpfull unto this art of Dialling with the drawing of a parallel line at any distance or by a point assigned so likewise have I shewed the manner either how to raise or let fall a perpendicular either from or unto any part of a line given So likewise now I think it will not be unnecessary for to shew unto the unlearned the meaning of some of the most usefullest terms in Astronomie and most fitting this art of Dialling Definition 1. A Sphere is a certain solid superficies in whose middle is a point from which all lines drawn unto the circumference are equall which point is the center of the Sphere Definition 2. The Pole is a prick or point imagined in the heavens whereof are two the North pole being the center to a circle described by the motion of the North star or the tale of the little Bear from which point aforesaid is a line imagined to passe through the center of the Sphere and passing directly to the opposite part of the heavens sheweth there to be the South Pole and this line so imagined to passe from one Pole to the other through the center of the Sphere is called the Axletree of the World because it hath beene formerly supposed that the Sun Moon and Stars together with the whole heavens hath been turned about from East to West once round in 24 houres by a true equall course like much in like time which diurnall revolution is performed about this Axletree of the World and this Axletree is set out unto you in the following figure by the line P A D the Poles whereof are P and D. Definition 3. A Sphere accidentally is divided into two parts that is to say into a right Sphere and an oblique Sphere a right Sphere is only unto those that dwell under the Equinoctiall to whom neither of the Poles of the World are seen but lie hid in the Horizon An oblique Sphere is unto those that inhabit on either side of the Equinoctiall unto whom one of the Poles is ever seen and the orher hid under the Horizon Definition 4. The Circles whereof the Sphere is composed are divided into two sorts that is to say into greater Circles and lesser The greater Circles are those that divide the Sphere into two equall parts and they are in number six viz. the Equinoctiall the Ecliptique line the two Colures the Meridian and the Horizon The lesser Circles are such as divide the Sphere into two parts unequally and they are foure in number as the Tropick of Cancer the Tropick of Capricorn the Circle Artique and the Circle Antartique CHAP. IX Of the six greater Circles Definition 5. THe Equinoctiall is a Circle that crosseth the Poles of the world at right angles and divideth the Sphere into two equall parts and is called the Equinoctiall because when the Sun commeth unto it which is twice in the year viz. at the Suns entrance into Aries and Libra it maketh the dayes nights of equall length throughout the whole world and in the figure following is described by the line S A N. Definition 6. The Meridian is a great Circle passing through the Poles of the world and the poles of the Horizon or Zenith point right over our heads and is so called because that in any time of the year or in any place of the world when the Sun by the motion of the heavens commeth unto that Circle it is then noon or 12 of the clock and it is to be understood that all Towns and Places that lie East and West one of another have every one a severall Meridian but all places that lie North and South one of another have one and the same Meridian this circle is declared in the figure following by the Circle E B W C. Definition 7. The Horizon is a Circle dividing the superior Hemisphere from the inferior whereupon it is called Horizon that is to say the bounds of sight or the farthest distance that the eye can see and is set forth unto you by the line C A B in the following figure Definition 8. Colures are two great Circles passing through both the Poles of the World crossing one the other in the said Poles at right angles and dividing the Equinoctiall and the Zodiaque into four equall parts making thereby the four Seasons of the year the one Colure passing through the two Tropicall points of Cancer and Capricorn shewing the beginning of Summer also of Winter at which times the dayes and nights are longest and shortest The other Colure passing through the Equinoctiall points Aries and Libra shewing the beginning of the Spring time and Autumne at which two times the dayes and nights are of equall length throughout the whole World Definition 9. The Ecliptique
surface of the Diameter as the Semicircle C B D which is halfe of the Circle C B D E and is contained above the Diameter C A D. Definition 11. A Quadrant is the fourth part of a Circle and is contained betwixt the Semidiameter of the Circle a line drawn perpendicular unto the Diameter of the same Circle from the center thereof dividing the Semicircle into two equall parts of the which parts the one is the Quadrant or fourth part of the same Circle As for example the Diameter of the Circle B D E C is the line C A D dividing the Circle into two equall parts then from the center A raise the perpendicular A B dividing the Semicircle likewise into two equall parts so is A B D or A B C the Quadrant of the Circle C B D E. Definition 12. A Segment or portion of a Circle is a figure contained under a right line and a part of a circumference either greater or lesser then the semicircle as in the figure you may see that F B G H is a segmēt or part of the Circle C B D E and is contained under the right line FHG 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 and of such parallels two only belong unto this work of Dialling that is to say the right lined Parallel and the circular Parallel Right lined Parallels are two right lines equidistant in all places one from the other which being drawn forth infinitely would never meet or concur as may be seen by these two lines A and B. Definition 14. A Circular parallel is a Circle drawn either within or without another Circle upō the same center as you may plainly see by the two Circles B C D E and F G H I these Circles are both drawn upon the same center A and therefore are parallel the one to the other Definition 16. A Degree is the 360th part of the circumference of any Circle so that divide the circumference of any circle into 360 parts and each of those parts is called a degree so shall the 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 degrees As for example the arch D C B being 120 degrees this being taken from 180 degrees the whole Semicircle leaveth A B for the complement thereof which will be found to be 60 degrees And here note that what is said of the complements of arches the same is meant by the complements of angles CHAP. II. To a line given to draw a parallel line at any distance required SUppose the line given to be A B unto which line it is required to draw a parallel line First open your Compasses to the distance required then set one foot in the end A and with the other strike an arch line on that side the given line whereunto the parallel line is to be drawn as the arch line C this being done draw the like arch line upon the end B as the arch line D and by the convexity of those two arch lines C and D draw the line C D which shall be parallel to the given line as was required CHAP. III. To perform the former proposition at a distance required and by a point limited SUppose the line given to be D E unto which line it is required to draw a parallel line at the distance and by the point F. First therefore place one foot of the compasses in the point F from whence take the shortest extention to the line D E as F E at which distance place one foot of the compasses in the end D and with the other strike the arch line G by the convexity of which arch line and the limited point F draw the line F G which is parallel to the given line D E as was required CHAP. IIII. The manner how to raise a perpendicular line from the middle of a line given LEt the line given be A B and let C be a point therein whereon it is required to raise a perpendicular First therefore open the compasses to any convenient distance setting one foot in the point C and with the other foot mark on either side thereof the equall distances C A and C B then opening your compasses to any convenient wider distance with one foote in the point A with the other strike the arch line E over the point C then with the same distance of your compasses set one foot in B and with the other draw the arch line F crossing the arch E in the point D from which point D draw the line D C which line is perpendicular unto the given line A B from the point C as was required CHAP. V. To let a Perpendicular fall from a point assigned unto the middle of a line given LEt the line given whereupon you would have a perpendicular let fall be the line D E F and the point assigned to be the point C from whence you would have a perpendicular let fall upon the given line D E F. First set one foote of your compasses in the point C and opening your compasses to any convenient distance so that it be more then the distance C E make an arch of a circle with the other foot so that it may cut the line D E F twice that is at I and G then finde the middle between those two intetsections which will be in the point E from which point E draw the
come unto the Zenith these are Circles that doe measure the elevation of the Pole or height of the Sun Moon or Stars above the Horizon which is called the Almicanter of the Sun Moon or Star the arch of the Sun or Stars Almicanter is a portion of an Azimuth contained betwixt that Almicanter which passeth through the center of the Star and the Horizon Thus having set forth unto the view of the unlearned for whose sake this Treatise was intended the meaning of some of the usefullest terms of Geometry which be most attendent unto this Art of Dialling and also a description of some peculiar things concerning the Points Lines and Circles imagined in the Sphere being very fit to be understood of all such as intend to practise either in the Art of Navigation Astronomie or Dialling Therefore now I intend to proceed with Scale and Compasse to perform some questions Astronomicall before we enter upon the Art of Dialling seeing they are both delightfull and also helpfull unto all such as shall be practitioners in this Art of Dialling The end of the First Book THE SECOND BOOK Shewing Geometrically how to resolve all such Astronomical Propositions as are of ordinary use as well in the Art of Navigation as in this Art of Dialling CHAP. I. The description of the Scale whereby this work may be performed THis Scale for this work needs to be divided but into two parts the first whereof may be a Scale of equall divisions of 16 in an inch and may serve for ordinary measure The second part of the Scale may be a single Chord of a Circle or a Chord of 90 and is divided into 90 unequall divisions representing the 90 degrees of the Quadrant and are numbered with 10 20 30 40 c. unto 90. This Chord is in use to measure any part or arch of a Circle not surmounting 90 degrees the number of these degrees from 1 unto 60 is called the Radius of the Scale upon which distance all Circles are to be drawn whereupon 60 of these degrees are the semidiameter of any Circle that is drawn upon that Radius The manner how to divide the line of Chords Although the making or dividing of this line of Chords be well known unto all those that do make Mathematicall Instruments yet I would not have them that shall make use of this Book be ignorant of the dividing of this line Therefore first draw the Diameter A D C which being done upon the center D describe the semicircle A B C which semicircle divide into two equall parts or Quadrants by the point B then dividing one of these Quadrants into 90 equall parts or degrees you are prepared as here you see in the Quadrant A B. Now this being done set one foot of your Compasses in the point A and let the other be extended unto each degree or the Quadrant A B and these extents transfer into the line A D C as here you see is done This line so divided into 90 unequall divisions from the point A and numbred by 10 20 30 40 c. unto 90 is called a line of Chords and may be set on your Rule as here you see is done And this may be as well performed within the Quadrant D A B by transferring the degree of the Quadrant A B into the line A E B or into any other line and here you may see that when you open your compasses unto 60 degrees in the Quadrant and transfer it into the line A D that it will light upon the center D whereby it doth plainly appear that 60 of those degrees are equall to the semidiameter of the same Circle and therefore is the Radius upon which all circles are drawn as was shewed before in this Chapter CHAP. II. How speedily with Rule and Compasse to make an angle containing any degrees assigned or to get the degrees of any angle made FIrst therefore to make an angle of any quantity open your Compasses to the Radius of our Scale and setting one foot thereof in the point A with the other foot describe the arch B C then draw the line A B then opening your Compasses to so many degrees upon your line of Chords as you would lay down which here we will suppose to be 40 degrees and setting one foot in B with the other make a mark in the arch B C as at C from which point C draw the line C A which shall make the angle B A C containing 40 degrees as was required And if you desire to finde the quantity of an angle open the Compasses to the Radius of your Scale set one foot thereof in the point A and with the other describe the arch B C then taking the distance betwixt B and C that is where the two legs and the arch line crosseth and apply it unto the line of Chords and there it will shew you the number of degrees contained in that angle which here will be found to be 40 degrees CHAP. III. To finde the altitude of the Sun by the shadow of a Gnomen set perpendicular to the Horizon FFirst draw the line A B then opening your Compasses to the Radius of your Scale set one foot in the end A and with the other describe the arch B C D then opening your Compasses unto the whole 90 degrees with one foot in B with the other marke the arch B C D in the point D from which point D draw the line D A which shall be perpendicular unto the line A B and make the Quadrant A B C D then suppose the height of your Gnomon or substance yeilding shadow to be the line A E which here we will suppose to be 12 foot therefore take 12 of your equall divisions from your Scale as here I have taken 12 quarters for this our purpose and set them down from A to E and draw the line E F parallel to A B then suppose the length of the shadow to be 9 foot for this 9 foot must I take 9 of the same divisions as I did before and place them from E to G by which point G draw the line A G C from the center A through the point G until it cutteth the arch B F C D in the point C so shall the arch B C be the height of the Sun desired which in this example will be found to be 53 degrees 8 minutes the thing desired CHAP. IIII. To finde the altitude of the Sun by the shadow of a Gnomon standing at right angles with any perpendicular wall in such manner that it may lie parallel unto the Horizon FIrst draw your Quadrant A B C D as is taught in the last Chapter place the length of your Gnomon from A to E which here we will suppose to be 12 foot as before in the last Chapter then draw the line E F parallel to A B then suppose the length of the shadow to be 9 foot as before this I place from E to G
the distance C e and set in your Trigon from O unto e and draw the line O e 4 representing the houres of 4 and 8. And thus must you doe with the rest of the houres in your plane if occasion require These lines O a O b O c O d and O e in your Trigon being extended doe cut the Tropick of Cancer P A in the points 12 1 2 and 3 therefore out of your Trigon take the distances O 12 O 1 O 2 O 3 O 4 and set them upon their correspondent houre lines of your plane from the center C unto g h i k and l so shall the points g h i k and l be the points upon the houre-lines through which the Tropick of Cancer must passe and is therefore noted with the character of Cancer ♋ at both ends ¶ Now before you draw the Tropick of Capricorn it is necessary to draw the horizontall line of your plane A B which line in all upright planes must be drawn through the point L the foot of the perpendicular stile and perpendicular to the Meridian or line of 12 And in all planes whatsoever this line must be drawn through the intersection of the Equinoctiall with the houre of six This line ought first to be drawn because it is very improper to extend the Tropicks or other parallels of declination above the Horizontall line because at what houre any parallel of declination cutteth this line on either side of the Meridian at that time doth the Sun rise or set as was instanced in the last Now the Tropick of Capricorn must be put upon your plane in the same manner as that of Cancer by taking out of your Trigon the distances from O where the severall houre-lines a b c d e doe cut the Tropick of Capricorn P C and place them on your plane from the center C upon the respective houre-lines and through those points so found draw the line ♑ ♑ representing the Tropick of Capricorn ¶ And in the same manner may the parallels of the other Signes be drawn upon your plane by placing them into your Trigon according to their declinations and afterwards transfer them into your plane as you see in the former figure The rules that have been here given for the describing of the parallels of the Signes in this erect direct plane is universall in all planes observing this one exception that whereas in all erect direct planes the Equinoctiall is drawn perpendicular to the Meridian or line of 12 so in all other planes whatsoever the Equinoctioll must be drawn perpendicular to the substile and then the work will be the same in all respects as may appear more largely in the next Section §. 3. In Declining or Declining Reclining Dials THe last caution preceding is sufficient for the performing of this work and therefore needeth no example However suppose an upright plane to decline 32 deg from the South Eastwards in the Latitude of 52 deg 30 min. and let it be required to describe the two Tropicks and the Equinoctiall upon such a plane And here note that whatsoever is said of upright decliners the same is also to be understood of those planes which both decline and recline and for the horizontal line in all reclining or inclining planes it must passe through the foot of the perpendicular stile and the intersection of the Equinoctiall with the houre of fix CHAP. II. Shewing how to inscribe the parallels of the length of the day on any plane THe parallels of the length of the day and those of the Signes are inscribed upon all kinde of planes by one and the same rules they being in the Sphere the same Circles so that as when you put on the parallels of the Suns entrance into the 12 Signes you seek what declination he hath and accordingly proceed as before so now for the parallels of the length of the day you must seeke what declination the Sun hath at such a length of the day as you would put into your plane which that you may do I have here added the rule following ¶ Consider how much longer or shorter your day proposed is then 12 houres and take the difference then the proportion will be As the Sine of 90 degrees Is to the Sine of halfe the difference So is the Tangent complement of the Latitude of the place To the Tangent of the declination that the Sun shall have when the day is at such a length as you require As for example Let it be required to know what declination the Sun shall have when the day is 16 houres long in the Latitude of 52 deg 30 min. The difference betwixt 16 houres and 12 houres is 4 houres or 60 deg the halfe of which is 30 deg Therefore say As the Sine of 90 deg 10,000000 Is to the Sine of 30 deg which is halfe the difference 9,698970 So the Tangent complement of the Latitude 37 deg 30 min. 9,884980 To the Tangent of the Declination of the sun 20 deg 59 m. 9,583950 And such declination shall the Sun have when the day is either 16 houres or 8 houres long in the Latitude of 52 deg 30 min. Now if the day be above 12 houres long the Sun hath North declination but if lesse then 12 houres long he hath South declination For those who are ignorant of these kinde of prportions they had best to read Mr. Norwoods Doctrine of Triangles But that nothing might be wanting and not much to trouble the learner I have here added a Table shewing what declination the Sun hath at such time that the day is either 8 9 10 11 12 13 14 15 or 16 houres long in the Latitude of 52 deg 30 min. which Table was made by the preceding rules By which table you may see Length of the day The Suns Declinatiō D M 8 20 59 9 16 22 10 11 14 11 5 43 12 0 0 13 5 43 14 11 14 15 16 22 16 20 59 that when the day is 12 houres long the Sun hath then no declination but is in the Equinoctiall but when the day is either 11 or 13 houres long the declination is then 5 deg 43 min. and when the day is either 9 or 15 houres long the Sun hath 16 deg 22 min. of declination and so for the rest as in the Table For the placing of these parallels of the length of the day upon any of the forementioned planes you must insert these angles of declination into your Trigon between the Tropicks and proceed in all respects as before I will therefore give you but one example which shall be in a full South plane upon which and the Horizontall these arches doe appear most uniform Now let it be required to draw the parallels of the Suns course when the day is 8 9 10 11 12 13 14 15 and 16 houres long upon a full South plane in the Latitude of 52 deg 30 min. Having drawn your Diall with Houres
Of the Almicanters or circles of Altitude THe circles of altitude have the same relation to the Azimuths as the Tropicks and parallels of declination have to the houre-lines and therefore as the parallels of declination in the Equinoctiall plane are perfect circles so are the circles of Altitude in an Horizontall plane The inscription of these into all kinde of planes is in a manner the same with the parallels of declination but whereas in the drawing the parallels of declination you take the houre-lines out of your plane and put them in a Trigon so in this you must take the Azimuths out of your plane and put them into a Trigon for that purpose and so transfer them to the plane again as you did the other and because these are small circles therefore they become Conick sections except on such planes as lie parallel to the Zenith which is only the Horizontall CHAP. VII How to draw a Diall on the Seeling of a Room BEcause the direct beams of the Sun can never shine upon the seeling of a Room they must therefore be reflected thither by help of a small piece of Looking glasse conveniently fixed in some Transam of the window so that it may lie exactly parallel to the Horizon The place being chosen and the glasse therein fixed you must draw upon the seeling of the Room a Meridian line as you are taught in the former Books which Meridian line must be 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 Hours The angle that each houre-line makes with the Merid. The complement of each houre-lines angle with the Meridiā  12  00d 00m 90d 00m 1  11 12 00 78 00 2  10 24 37 65 23 3  9 38 25 51 35 4  8 53 58 36 2 5  7 71 20 18 40  6  90 00 00 00 Having these things prepared Let the line L R in the following figure represent a Meridian line drawn upon the seeling of a room and let K be the glasse fixed directly under the said Meridian upon some transam of the window then laying one end of a string upon the glasse at K extend the other up to the Meridian at L which point L you may finde by moving the string to and fro upon the Meridian line till another holding the side of a Quadrant to the moveable string he shall finde the threed and plummet to fall directly upon the complement of the Latitude which in this example is 37 deg 30 min. The point L being thus found upon the Meridian draw the line L AE perpendicular to the Meridian L R which line shall be the Equinoctiall Having thus done upon a table or such like draw a line which shall be of equall length with LK the distance from the glasse to the point L on the seeling which line divide into 10 equall parts and each of those or at least one of them into 10 other parts so shall you have in all 100 parts each of which you must suppose to be divided into 10 other smaller parts so shall the whole line contain 1000 parts as in the figure is expressed by the line S. Now because the center of the Diall is without the Roome so that you cannot make use of that to draw the hours by you must therefore place one foot of your compasses in the points L M N P and Q with the other draw obscure arches of Circles as ***** and out of the last column of the former Table take the complement of every hours arch from the Meridian and place them upon the respective houre arches from the Equinoctiall to the points ***** as you see in the figure Lastly if you draw the lines * M * N * P * Q they shall be the true houres upon the seeling In the inscription of the Azimuths in declining reclining planes and in drawing the circles of Altitude in all kinde of planes I confesse I should have been somwhat larger in giving you an example in each plane as I did with the other varieties before but pre-supposing the ingenious practitioner sufficiently to understand that which goes before he cannot but with small pains overcome the rest But I should not have been so briefe could I possibly have procured more time which by no means would be granted Also my intent in this place was to have shewn you the inscription of the Circles of position and other varieties Also the framing of divers Geometricall Bodies and to furnish them with variety of Dials and the making of divers Instrumentall Dials But these with many other HOROLOGICAL conceits and inventions I reserve till a more convenient opportunity and therefore in the mean time Farewell FINIS
by which point G draw the line A G C as was formerly done in the last Chapter by which we have proceeded thus far but as in the last Chapter the arch B C was the height of the Sun desired so by this Chapter the arch C D shall be the height of the Sun which being applyed unto your Scale will give you 36 deg 52 min. for the height of the Sun desired CHAP. V. The Almicanter or height of the Sun being given to finde the length of the right shadow BY right shadow is meant the shadow of any staffe post steeple or any Gnomon whatsoever that standeth at right angles with the Horizon the one end thereof respecting the Zenith of the place and the other the Nadir First therefore according unto the third Chapter describe the Quadrant A B D then suppose the height of your Gnomon or substance yeilding shadow to be 12 foot as in the former Chapter this doe I set down from A to E and from the point E draw the line E F parallel to A B then set the Almicanter which here we will suppose to be 53 de and 8 min. as it was found by the third Chapter from B unto C from which point C draw the line C A cutting the line E F in the point G so shall E G be the length of the right shadow desired which being taken betwixt your Compasses and applyed unto your Scale will give you 9 of those divisions whereof A E was 12 which here doth signifie 9 foot CHAP. VI. The Almicanter or height of the Sun being given to finde the length of the contrary shadow BY the contrary shadow is understood the length of any shadow that is made by a staffe or Gnomon standing at right angles against any perpendicular wall in such a manner that it may lie parallel unto the Horizon the length of the contrary shadow doth increase as the Sun riseth in height whereas contrariwise the right shadow doth decrease in length as the Sun doth increase in height Therefore the way to finde out the length of the Versed shadow is as followeth First draw your Quadrant as is taught in the third Chapter now supposing the length of your Gnomon to be 12 foot place it from A to E likewise from E draw the line E F parallel to A B as before now supposing the height of the Sun to be 36 deg 52 min. take it from your Scale and place it from D to C from which point C draw the line C A cutting the line E F in the point G so shall G E be the length of the contrary shadow which here will be found to be 9 foot the thing desired CHAP. VII Having the distance of the Sun from the next Equinoctiall point to finde his Declination FIrst draw the line A B then upon the end A raise the perpendicular A D then opening your Compasses to the Radius of the Scale place one foot in the center A and with the other draw the Quadrant B C D then supposing the Sun to be either in the 29 degree of Taurus or in the first degree of Leo both which points are 59 degrees distant from the next Equinoctiall point Aries Or if the Sun shall be in the 29 degree of Scorpio or in the first degree of Aquarius both which are also 59 degrees distant from the Equinoctiall point Libra therefore take 59 degrees from your Scale and place it from B to C and draw the line C A then place the greatest declination of the Sun from B unto F which is 23 degrees 30 minutes then fixing one foot of your Compasses in the point F with the other take the neerest distance unto the line A B which you may doe by opening or shuting of your Compasses still turning them to and fro till the moving point thereof doe only touch the line A B this distance being kept set one foot of your Compasses in the point A and with the other make a mark in the line A C as at E from which point E take the neerest extent unto the line A B this distance betwixt your Compasses being kept fix one foot in the arch B C D moving it either upwards or downwards still keeping it directly in the arch line untill by moving the other foot to and fro you finde it to touch the line A B and no more so shall the fixed foot rest in the point G which shall be the Declination of the Sun accounted from B which in this example will be found to be about 20. degrees the thing desired CHAP. VIII The Declination of the Sun and the quarter of the Ecliptique which he possesseth being given to finde his true place LEt the declination given be 20 degrees and the quarter that he possesseth be betwixt the head of Aries and Cancer first draw the Quadrant A D E F then set the greatest declination of the Sun upon the Chord from D unto B which is 23 degrees and 30 minutes then from the point B take the shortest extent unto the line A D this distance being kept set one foot in the point A and with the other describe the small Quadrant G H I then set the declination of the Sun which in this example is 20 degrees from D unto C from which point C take the shortest extent unto the line A D this distance being kept place one foot in the arch line G H I after such a manner that the other foot being turned about may but onely touch the line A D so shall the fixed foot stay upon the point H through which point H draw the line A H E cutting the arch D F in the point E so shall the arch D E be the distance of the Sun from the head of Aries which here will be found to be 59 degrees so that the Sun doth hereby appear to be in 29 degrees of Taurus at such time as he doth possesse that quarter of the Ecliptique betwixt the head of Aries and Cancer CHAP. IX Having the Latitude of the place and the distance of the Sun from the next Equinoctiall point to finde his Amplitude FIrst make the Quadrant A B C D then take from your Scale 37 deg 30 min. which here we will suppose to be the complement of the Latitude and place it from B unto E then taking the neerest distance betwixt the point E and the line A B with one foot set in A with the other draw the arch F G H then place the Suns greatest declination from B unto I from which point I take the neerest extent unto the line A B which distance being kept place one foot of your compasses in the arch line F G H so that the moving foot may but only touch the line A B at the shortest extent so shall the fixed foot rest in the arch line F G H at G through which point G draw the line A G C then supposing the Sun to be in the 29 degree of