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
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
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
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
the same line on the other side of the Meridian and also on both sides the Meridian in the upper line the lines drawn from point to point parallel to the Meridian C E shall be the houre lines the line C E shall be the Meridian line the houre of 12 and must also be the substilar line whereon the stile must stand which may be a plate of iron or some other metall being so broad as the semidiameter of the Circle is as is shewed in the figure This style must be placed along upon the line of 12 making right angles therewith the upper edge whereof must be parallel to the plane so shall it cast a true shadow upon the houre lines The under face of this Polar plane and also of the former Equinoctiall plane is made altogether like unto the upper faces here described without any difference at all CHAP. VII The making of an erect Meridian Diall A Meridian plane is that which is parallel to the Meridian Circle of the Sphere therefore having drawn the horizontall line A B and finding it to decline 90 deg from the South the plane being erect I conclude it to lie parallel to the Meridian Circle of the Sphere and is therefore called a Meridian plane For the style of this Diall it may be either a plate of some metall being so broad as the semidiameter of the circle is and so placed perpendicularly along over the line of the houre of six the upper edge thereof being parallel to the plane or it may be a streight pin fixed in the center of the circle making right angles with the plane being just so long as the Semidiameter of the Circle is only shewing the houre with the very top or end thereof This plane hath two faces one to the East the other to the West the making whereof are both alike only in naming the houres for the one containeth the houres for the forenoon and the other for the afternoon as you may perceive by the figures CHAP. VIII To draw a Diall upon an Horizontall plane AN Horizontall plane is that which is parallel to the Horizontall Circle of the Sphere which being found by the first Chapter to be levell with the Horizon you may by the fourth Chapter draw the Meridian line A B serving for the Meridian the houre of 12 and the substilar in this Meridian make choice of a center as at C through which point C draw the line D E crossing the Meridian at right angles this line shall be the line of East and West and is the six a clock line both for morning and evening Then by the second Chapter of the second Book draw the line S C making the angle S C A equall to the Latitude of the place which here we will suppose to be 52 deg 30 min. this line shall represent the cock of the Diall and the Axletree of the World then at the North end of the Meridian line draw another line as F G crossing the Meridian in the point A at right angles this line is called the Touch-line or line of contingence Then set one foot of your Compasses in the point A and with the other take the neerest extent unto the line S C or the Stile with this distance turning your Compasses about with one foot still in the point A with the other make a mark in the Meridian as at I which shall be the center of the Equinoctiall upon which describe the Equinoctiall Circle A D B E with this same distance setting one foot in the point A make a markâ—â— 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 houre lines of 3 and 9. This same distance of your Compasses being kept with one foot still in the center A with the other make the marks T and V in the Equinoctiall Circle each of which distances is an arch of 60 degrees or four houres of time the halfe of which arch is 30 degrees or two houres from the Meridian this divided in the halfe will be 15 deg or one houre from the Meridian then laying your rule upon the center I of the Equinoctiall and upon these two last divisions in the circle thereof where the rule shall touch the line of contingence there mark it as at H and O by which points and the center C you may draw the hour-lines of 10 and 11 the like may you do on the other side of the meridian so have you six of your hour-lines drawn and now because the contingent will out-run our plane we may from the intersection of the houres of 9 and 3 with the touch line draw the lines F D and G E parallel to the meridian A B untill they cut the line of East and West in the points D and E then draw the lines A D and A E this being done set one foot of your compasses in the point H and with the other take the neerest extent unto the line A E this distance being kept fix one foot in the line G E so as the other may but touch the line A E so shall the fixed foot rest in the point N by which and the center C you may draw the 7 a clock hour-line in like manner may you place one foot in the point O and with the other take the shortest extent unto the line A E with this distance fixing one foot in the line G E so as the other may but onely touch the line A E so shall the fixed foot rest in the point R by which and the center C you may draw the 8 a clock hour-line the like may be done on the other side of the meridian or you may by these distances thus found prick out the like on the other side of the meridian Thus by dividing but half a quarter of the Equinoctiall Circle into three equall parts you may describe your whole Diall And whereas in Summer the 4 and 5 in the morning and also 7 and 8 at evening shall be necessary in this kind of Diall prolong or draw the lines of 4 and 5 at evening beyond the center C and they shall shew the hour of 4 and 5 in the morning and likewise the 7 and 8 in the morning for 7 and 8 at evening What is here spoken concerning the hours the like is to be done in drawing the half hours as well in this kinde as in all them which follow The style must be fixed in the center C hanging directly over the meridian line A C with so great an angle as the lines S C A maketh which is the true pattern of the cock This and all other kinds of Dials may be drawn upon a clean paper and then with the help of your compasses placed on your plane CAAP. IX To draw a Diall upon an erect direct verticall Plane commonly called a South or
shall the arch C I be the inclination of both Meridians and is found by this example to be 38 deg 13 min. so that by this example the Meridian of the plane will fall betwixt the houres of 2 and 3 if the plane shall decline Westward but if it shall decline Eastward then shall it fall betwixt the houres of 9 and 10 before noon CHAP. XV. To draw a Diall upon an erect or Verticall plane declining otherwise then in the 11 Chapter was shewed HAving by the third Chapter found the declination of this plane to be 32 degrees and so by the last Chapter found the elevation of the pole above the plane to be 31 deg 5 min. and the distance of the substile from the Meridian to be 22 deg 8 min. and likewise the angle of inclination between both Meridians to be 38 deg 13 min. we may proceed to make the Diall after this manner First draw the horizontall W E and the perpendicular line Z N crossing the horizontall line at right angles which is the Meridian of the place and the line of 12. Then in the meridian make choice of some point with most convenience as the center C whereupon describe your Diall circle E N W. Then take a Chord of 22 deg 8 min. from your Scale for the distance of the substile from the Meridian and inscribe it into this Circle from the Meridian upon these conditions that if the plane declineth West then must the substile be placed East of the plumb line but if the declination shall be East then must the substile be placed West from the Meridian as here it is This 22 deg 8 min. being set in the Diall circle from the Meridian at N unto M I draw the line C M for the substile then through the center C draw the diameter A B making right angles with the substile C M above this Diameter there needs no houre-houre-lines to be drawn if the plane be erect Then take 31 deg 5 min. and set them from M to D and draw the line C D S for the Stile then from M the end of the Stile draw the crooked line M S cutting the line of the Stile in the point S so shall the triangle S C M be the true pattern for the Cock of the Diall This being done take 38 deg 13 min. and set them alwayes on that side the substile whereon the line of 12 lyeth as here from M to H so shall the point H be the point where you shall begin to divide your diall circle into 24 equall parts but those points shall be only in use which doe fall below the Diameter A C B. And if the line of the substile falleth not directly upon one of the houre lines then shall you have six points on each side thereof from which you may let perpendiculars fall unto the line of the substile as here you see done When you have drawn and described your Diall upon paper for any plane whatsoever you may cut off the hour-lines Cock and all with a lesser Circle then the Diall Circle either with a concentrique or an excentrique Circle and so make a Diall lesse then the Circle by which you framed it Or if you extend the hour-lines beyond the Diall Circle you may cut them off either with a greater concentrique Circle and so make a bigger Diall or else you may cut them off with a Square as here you see in the following figure or any other form what you shall think most convenient Of a Plane falling neer the Meridian When as the declination of a plane shall cause it to lie neere the Meridian as that the Declination and inclination shall cause it to lie neer the Pole then doth the elevation of the Pole above the plane grow so small the hour-lines so exceeding neer together that except the plane be very large they will hardly serve to good purpose as here in this figure being a plane which is erect and declining from the South 80 deg towards the East Therefore first draw your Diall very true as before hath been taught upon a large paper making your circle as big as you can then extend the houre-lines with the substile and the line of the stile a great way beyond the Dials circle untill they doe spread so that they will fill the plane indifferent well and then cut them off with a long square as O N in the following figure so wil it shew almost An erect Diall declining from the South 80 deg towards the East the distance of the substile from the Meridian 37 deg 4 min. the elevation of the Pole above the plane 6 deg 4 min. and the inclination of both Meridians 82 deg 5 min. like the Meridian Diall of the 7 Chapter for the hours wil be almost parallel the one to the other and the stile almost parallel to the substile as you may see by the figure CHAP. XVI The inclination of a Meridian plane being given how thereby to finde the elevation of the pole above the plane the distance of the substile from the Meridian and the angle of the inclination of the Meridian of the plane to the Meridian of the place ALL those planes wherein the horizontall line is the same with the Meridian line are therefore called Meridian planes if they make right angles with the Horizon they are called erect Meridian planes and are described before But if they leane to the Horizon they are then called Incliners These may incline either to the East part of the Horizon or to the West and each of them hath two faces the upper towards the Zenith the lower towards the Nadir wherein knowing the Latitude of the place and the inclination of the plane to the Horizon we are to consider three things more before we can come to the drawing of the Diall I. The elevation of the pole above the plane II. The distance of the substile from the Meridian III. The angle of Inclination betwixt both Meridians These three may be found after this manner little differing from the 14 Chap. First describe a Quadrant as A B C then set 52 deg 30 min. your Latitude from C to E in the arch of your Quadrant C B and draw the line E R parallel to A B cutting the line A C in the point R and with the distance E R with one foot in the center A with the other draw the arch G H O D then let your inclination be 30 deg which set in the arch of the Quadrant from B to F and draw the line A F cutting the arch G O D in the point H through which point H draw the line S H N cutting the arch of the Quadrant in the point N so shall the arch C N be the elevation of the pole above the plane which in this example is 43 deg 23 min. This being done from the point L draw the line L T parallel to the line A C cutting the arch
G D in the point O through which point O draw the line A O I cutting the arch of the Quadrant B C in the point I so shall the arch C I be the inclination of the Meridian of the plane to the Meridian of the place and in this example is found to be 43 deg 28 min. which being resolved into time doth give about two hours and 54 min. from the Meridian for the place of the substile amongst the houre-lines CHAP. XVII To draw a Diall upon the Meridian inclining plane HAving by the second Chapter found the inclination of this plane to be 30 deg and so by the last chapter found the elevation of the pole above the plane to be 43 deg 23 min. and the distance of the substile from the meridian to be 33 deg 5 min. and likewise the angle of inclination to be 43 deg 38 min. we may proceed to make the Diall after this manner First draw the horizontall line A B serving for the Meridian and the houre of 12 about the middle of this line make choice of a center at C upon which describe a Circle for your Diall as A D B E. Then seeing this is the upper face of the plane set 33 deg 5 min. the distance of the substile from the Meridian in the Dials Circle from the North end of the Horizontall line upwards as from B to H and draw the line C H for the substile But if this had been the under face the substile must have fallen below the horizontall line now through the center C draw the Diameter E F making right angles with the substile C H. Then set 43 deg 23 min. from H unto D for the stile and draw the line C D unto S and from the end of the substile draw the crooked line H S cutting the line of the stile in the point S so shall the Triangle S C H be the true pattern of your Cock for this Diall Then set 43 deg 28 min. from H unto M for the difference betwixt the Meridian of the plane and the Meridian of the place Now here at M must you begin to divide your Circle into 24 equall parts from which points let down so many perpendiculars to the substile as there shall be points on that side the diameter E F next the substile Now take each perpendicular betwixt your Compasses and with one foot in the center C with the other make marks in the line of the stile from which take the neerest extents unto the substile and lay them down upon their own proper perpendiculars from the substile so may you make marks through which and from the center you may draw the houre-lines This diall being thus drawn for the upper face of a Meridian plane inclining towards the West you must fixe the Cock in the center C hanging over the substile C H with an angle equall to the angle S C H so that it may point to the North Pole because upon the upper faces of all Meridian incliners the North Pole is elevated and therefore contrarily the South Pole must needs be elevated above their under faces This Diall being drawn in paper for the upper face of this plane will also serve for the under face thereof if you turn the pattern about so that the horizontall line A B may lie still parallel to the Horizon and the stile with the substile lying under the Horizontall line may point downwards to the South Pole the paper being oyled or pricked through so that you may take the back side thereof for the fore side without altering the numbers set to the hours CHAP. XVIII The inclination and declination of any plane being given in a known Latitude to finde the angle of intersection between the plane and the Meridian the ascension and elevation of the Meridian with the arch thereof between the Pole and the plane and also the elevation of the Pole above the plane the distance of the substile from the Meridian with the inclination between both Meridians IF a plane shall decline from the South and also incline to the Horizon it is then called by the name of a declining inclining plane Of these there are severall sorts for the inclination being Northward the plane may fall betwixt the Horizon and the Pole or betwixt the Zenith and the Pole or else they may lie in the Poles of the World or the inclination may be southward and so fall below the intersection of the Meridian and the Equator or above it or the plane may fall directly in the intersection of the Meridian with the Equator each of these planes have two faces the upper towards the Zenith and the lower towards the Nadir Now having the Latitude of the place with the declination and inclination of the plane we have seven things more to consider before we can come to the drawing of the Diall I. The angle of intersection betwixt the plane and the Meridian II. The arch of the plane betwixt the Horizon and the Meridian III. The arch of the Meridian betwixt the Horizon and the plane IV. The arch of the Meridian between the Pole and the plane V. The elevation of the Pole above the Plane VI. The distance of the substile from the Meridian VII The angle of inclination betwixt the Meridian of the plane and the Meridian of the place All these seven may be found out after this manner First Describe the Quadrant A B C then suppose the plane to decline from the South towards the East 35 deg and to incline towards the Horizon 25 deg set 35 deg the declination of the plane from C to E in the Quadrants arch C B and draw the line A E then set 25 deg the inclination of the plane in the same arch from B to F and draw the line F Z parallel to A C cutting the line A B in the point Z and with the distance F Z and one foot placed in the center A with the other describe the arch G H I cutting the line A E in the point H through which point H draw the line K L parallel to A C cutting the arch C B in the point K then take the distance H L and set it in the line F Z from Z unto O through which point O draw the line A O M cutting the arch B C in the point M from which point M draw the line M P N parallel to A B cutting the arch G I in the point P through which point P draw the line A P Q cutting the arch B C in the point G so shall the arch B K be 75 deg 58 min. the inclination of the plane to the Meridian and the arch B Q will be 57 deg 36 min. for the Meridians ascension or the arch of the plane betwixt the Horizon and the Meridian and the arch B M shall be 20 deg 54 min. for the elevation of the Meridian or the arch of the Meridian betwixt the
Horizon and the plane Now if the plane shall incline toward the South adde this elevation of the Meridian to your Latitude and the sum of both shall be the position Latitude or the arch of the Meridian betwixt the Pole and the plane and if the sum shall exceed 90 deg take it out of 180 deg and that which remains shall be the position Latitude or the arch of the Meridian between the Pole and the plane 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 A C cutting the arch B C in the point X so shall the arch B X be 16 deg 20 min. the inclination of the meridian of the plane to the meridian of the place CHAP. XIX To draw a Diall upon a declining inclining Plane HAving by the second Chapter found the inclination to be 25 deg towards the North and by the third Chapter the declination from the South towards the East to be 35 deg and so by the last Chapter the Meridians ascension to be 57 deg 36 min. The elevation of the Pole above the plane 30 deg 33 min. The distance of the substile from the Meridian 8 degrees 35 min. And the inclination of both Meridians 17 deg 30 min. we may proceed to make the Diall after this manner First Draw the line A B parallel to the Horizon in which line make choice of a center as at C whereon describe your Diall circle A D B E A then take 57 deg 36 min. the Meridians ascension and set it from B that end of the horizontall line with the declination of the plane as from B to N and draw the line C N for the houre of 12. Then set 8 deg 35 min. the distance of the substile from the Meridian from N to M on that side the Meridian which is contrary to the declination of the plane and draw the line C M for the substile And set 30 deg 33 min. from M to D and draw the line C D unto S and from the end of the substile draw the crooked line M S cutting the line of the Stile in S so shall the Triangle M C S be the true pattern of this Dials Cock. Then set 17 deg 30 min. the inclination of Meridians from M unto O which is the point where you must begin to divide your Diall circle into 24 equall parts from which points let down so many perpendiculars to the substile as there shall be points on that side the Diameter F E next the substile and so by working as before hath been shewed you may draw the houre-lines and set up the stile as in the former planes Now here I would have you well to consider what hath been here spoken concerning these kinde of Dials and also what followeth the same for if you mark the diversity which doth arise by reason of the elevation of the Meridian you may perceive thereby three sundry kinds of Dials to arise out of a North inclining plane declining and also in a South inclining declining plane yet in effect they are but one if you consider what followeth here concerning them in all which the stile with the substile and such like materials are found out according to the last Chapter And upon the under faces of all North incliners whose meridians elevation is lesse then the Latitude of the place On the upper faces of all North incliners whose meridians elevation is greater then the Latitude of the place and on the under faces of all South incliners the South Pole is elevated Now for placing the Meridian from the horizontall line upon the upper faces of all South incliners whose meridians elevation is greater then the Latitudes complement on the under faces of all South incliners whose meridians elevation is lesse then the Latitudes complement on the under faces of all North incliners whose meridians elevation is greater then the Latitude of the place and on the upper faces of all North incliners whose Meridians elevation is lesse then the Latitude of the place the Meridian must be placed above the Horizontall line as here in this example And therefore by the contrary Upon the upper faces of all South incliners whose meridians elevation is lesse then the Latitudes complement On the under faces of all South incliners whose meridians elevation is greater then the Latitudes complement On the under faces of all North incliners whose meridians elevation is lesse then the Latitude of the place And on the upper faces of all North incliners whose meridians elevation is greater then the Latitude of the place the Meridian must be placed below the Horizontall line But here you must observe that if it be either the upper or under faces of a South inclining plane whose meridians elevation is greater then the Latitudes complement or either the upper or under faces of a North inclining plane whose meridians elevation is lesse then the Latitude of the place that then the Meridian must be placed from that end of the horizontall line with the declination of the plane But on all the other faces of these kinde of planes the Meridian must be placed from that end of the horizontall line which is contrary to the declination of the plane And here note that if the inclination shall be Southward and the elevation of the Meridirn equall to the complement of your Latitude then shall the substile lie square to the Meridian And if the inclination shall be Northward and the elevation of the meridian equall to the Latitude of the place then shall neither Pole be elevated above this plane and therefore shall be a Polar declining plane Wherein the Meridian being placed according to his ascension from the horizontall line shall be in place of the substile unto which if you draw a line square it shall serve for the Equator Then set one foot of your Compasses in the intersection of the substile with the Equator and open the other to any convenient distance upon the substile and describe the Equinoctiall Circle as in the sixt Chapter of this Book was shewed upon
the center whereof make an angle with the line of the substile equall to the inclination of both meridians namely the meridian of the plane and the meridian of the place which shall shew you where to begin to divide your Equinoctiall Circle into twenty four equall parts These things being known you may proceed to make your Diall and set up the Cock according to the sixth Chapter As for example in our Latitude of 52 deg 30 min. a plane is proposed to decline from the South towards the East 35 deg as before but inclining Northward 57 deg 50 min. the Meridians ascension by the 18 Chapter will be found to be 69 deg 33 min. and his elevation 52 deg 30 min. equall to the Latitude of the place and therefore neither pole is elevated above this plane and so no distance between the Substile and the Meridian for the Meridian and the stile with the substile will be as it were all one line which is the Axletree of the World so that here the stile must be parallel to the plane and the houre lines parallel one to the other as in the Meridian and direct Polar planes Therefore first draw the Horizontall line A B wherein make choice of a center as at C whereon describe an occult arch of a Circle as B E then into this arch inscribe the Meridians ascension 69 deg 33 min. from B to E and draw the line CE for the meridian of the plane and for the substilar and if you draw a line square to this substilar it shall be the Equator Then set one foot of your Compasses in the point of intersection D and with the other opened to a convenient widenesse draw a Circle for the Equator unto which you may draw two touch lines square to the substile as in the direct polar plane This being done and the inclination of both Meridians being found by the last Chapter to be 29 deg 3 min. set it in this Circle from H unto O and draw the line D O F cutting the contingent in F from which point F you shall draw the 12 a clock hour-hour-line parallel to the substile Now from the point O divide your Equinoctiall circle into 24 equall parts with which you may proceed to make your Diall and set up the cock according to the 6 Chapter The end of the third Book THE FOURTH BOOK Shewing how to resolve all such Astronomicall Propositions as are of ordinary use in this Art of Dialling by help of a Quadrant fitted for the same purpose CHAP. I. The description of the Quadrant HAving in the second and third Books shewed Geometrically the working of most of the ordinary Propositions Astronomicall with the delineation of all kinde of plain wall Dials howsoever or in what latiitude soever scituated still keepng within the limits of our plane and yet not tyed to the use of any Instrument I will now shew how you may performe the former work exactly easily and speedily by a plain Quadrant fitted for that purpose the description whereof is after this manner Having prepared a piece of Box or Brasse in manner of a Quadrant draw thereon the two Semidiameters A B and A C equally distant or parallel to the edges cutting one the other at right angles in the center A upon which center A with the Semidiameter A B or A C describe the arch B C this arch is called the limbe and is divided into 90 equall parts or degrees and subdivided into as many parts as quantity will give leave being numbered from the left hand towards the right after the usuall manner Then let the Semidiameter A B be divided into 90 unequall parts called right Sines either from the Table of naturall Sines by help of a decimall Scale equall to the Semidiameter A B or else by taking the neerest extents from each degree of your Quadrant unto the side A B and placing them upon the side A B each after other from the center A towards B you shall exactly divide the Semidiameter A B into 90 unequall divisions called right Sines This being done draw the line D E from the Sine of 45 degrees counted in the line of Sines unto 45 degrees counted in the Quadrant then from the point E draw the line E F parallel to A B making the square A D E F the side D E whereof for distinction may be called a Tangent line and the side E F a Co-tangent line then draw the Diagonall line A E which you may call the line of Latitudes Then upon the center A with the distance A D or A F describe the arch D F which you may divide into six equall parts by laying your Rule upon each 15th degree in the Quadrant and the center A as at g h I k l F from which points draw slope lines to each 15th degree in the Quadrant numbered backward as F P l O k E I n h m g B these lines so drawn are to be accounted as hours then dividing each space into two equall parts draw other slope lines standing for half hours which may be distinguished from the other as they are in the figure Now because in the latter part of this Book there is often required to use a line of Chords to severall Radiuses therefore upon the edge of the Quadrant A C you may have a line of Chords divided as in the figure and so the Quadrant being at hand will supply the uses of the Scale mentioned in the preceding Book and also a Chord of any Circle whose Radius is lesse then the line A C may be taken off and in that case supply the use of a Sector To this Quadrant as to all others of this kind in their use is added Sights with a threed bead and plummet according to the usuall manner CAAP. II. Of the use of the line of Sines Any Radius not exceeding the line of Sines being known to finde the right Sine of any arch or angle thereunto belonging IF the Radius of the Circle given be equall to the line of Sines there needs no farther work but to take the other Sines also out of the line of Sines But if it be lesser then take it betwixt your compasses and set one foot in the Sine of 90 degrees and with the other lay the threed to the neerest distance which you may doe by turning the compasses about till the moving point thereof doe onely touch the threed and no more the threrd lying still in this position take the neerest extent thereunto from any Sine you think good and it shall be the like Sine agreeable to the Radius given As for example let the circle B C D E in the following chapter represent the meridian circle let B D be the Horizon and C E the verticall circle and let F G be the diameter of an almicanter and so F H the Semidiameter thereof which being given it is required to finde the Sines both of 30 and 50
degrees agreeable to that Radius first therefore take the given Radius betwixt your compasses and with one foot set in the Sine of 90 degrees with the other lay the threed to the neerest distance the threed lying still in this position take the neerest extents thereunto from the Sine of 30 and likewise of 50 these distances place upon the Radius F H from H to N and from H to R so shall H N be the Sine of 30 degrees and H R the Sine of 50 degrees agreeable to the Radius F H the thing desired CHAP. III. The Right Sine of any arch being given to finde the Radius TAke the Sine given betwixt your compasses and setting one foot in the like Sine in the line of Sines with the other lay the threed to the neerest distance the threed lying still in this position take the shortest extent thereunto from the Sine of 90 degrees which distance shall be the Radius required As for example let H R be the given Sine of 50 degrees it is required to find the Radius answering thereunto take H R with your compasses and set one foot in the Sine of 50 deg and with the other lay the threed to the neerest distance which being kept in this position if you take the shortest extent thereunto from the Sine of 90 you shall have the line H F for the Radius required CHAP. IV. The right Sine or the Radius of any Circle being given and a streight line resembling a Sine to finde the quantity of that unknown Sine FIrst take the Radius or the right sine given and setting one foot of your Compasses either in the like sine or in the Radius of the line of Sines and with the other lay the threed to the neerest distance then take the right line given and six one foot in the line of sines moving it till the moveable foot touch the threed at the neerest extent so shall the fixed foot stay at the degree of the sine required As for example let F H be the Radius given and H N the streight line given resembling a Sine first with the distance F H from the Sine of 90 lay the threed to the neerest distance the threed lying still in this position take the line H N and fixing one foote of your compasses in the line of Sines still moving it to and fro till the moveable foote thereof doth onely touch the threed so shall the fixed foote rest at the Sine of 30 degrees in the line of Sines this 30 degrees is the arch of which H N is the Sine F H in the last chapter being the Radius CHAP. V. The Radius of a circle not exceeding the line of Sines being given to finde the chords of every arch IF the Radius given shall be equall to the line of Sines then double the Sine of halfe the arch and you shall have the chord of the whole arch that is a Sine of 10 deg doubled giveth a chord of 20 deg a Sine of 15 deg doubled giveth a chord of 30 deg and so of the rest as in the third chapter the line I O the Sine of I C an arch of 30 deg being doubled giveth I L the chord of ICL which is an arch of 60 deg And if the Radius of the circle given be equall to the Semi-radius the sine of 30 deg of the line of sines then you neede not to double the lines of sines as before but onely double the numbers so shall a sine of 10 deg be a chord of 20 deg and a sine of 15 deg be a chord of 30 deg and so of the rest but if the Radius of the circle given be lesse then the semi-radius of your line of sines then take it betwixt your compasses and setting one foot in the sine of 30 deg with the other lay the threed to the neerest distance the threed lying still in this position take it over at the neerest extent in what Sine you think good onely doubling the number and you shall have the Chord desired As for example let A C be the diameter of the circle in the third Chapter and it is required to find a Chorde of 30 degrees therefore first I take A C betwixt my compasses and setting one foot in the Sine of 30 deg with the other I lay the threed to the neerest distance which being kept at this angle I take it over from the sine of 15 deg which doth give me I C the Chord of 30 deg which was desired And if the Radius given be greater then the Sine of 30. and yet lesse then the Radius of the line of Sines then with the Radius given and from the Sine of the complement of halfe the arch required lay the threed to the neerest distance then taking it over at the neerest extent from the sine of the whole arch you shall have your desire As for example let the Radius A C of the circle in the third Chapter be given and a Chord of 30 deg required the halfe of 30 deg is 15 deg the complement whereof is 75 deg therefore I take the Radius with my compasses and setting one foot in the Sine of 75 deg with the other I lay the threed to the neerest distance the threed lying still in this position I take the shortest extent thereunto from the Sine of 30 deg which giveth I C the Chord of 30 deg which was desired Now by the converse of this Chapter if you have the Chord of any arch given you may thereby find out the Radius CHAP. VI. To divide a line by extream and mean proportion A Right line is said to be divided by extream and meane proportion when the lesser Segment thereof is to the greater as the greater is to the whole line Let A B be the line to be so divided this line I take with my compasses and setting one foot in the sine of 54 deg and with the other I lay the threed to the neerest distance which lying still in this position I take it over from the sine of 30 deg which distance shall be the greater segment A C dividing the whole line in the point C or the threed lying in the former position if you shall take the shortest extent thereunto from 18 deg you shall have B C for the lesser segment which will divide the whole line by extream and mean proportion in the point C from the end B so that as B C the lesser segment is to A C the greater segment so is A C the greater segment to A B the whole line as was required CHAP. VII To find a mean proportionall line between two right lines given A Mean proportionall line is that whose square is equall to the long square contained under his two extreams First ioyn the two given lines together so as they may make both one
a horizontall line upon the upper face by laying a Ruler thereupon and applying the side A B of your Quadrant to the under side thereof still moving your Rule untill the threed and plummet doth fall directly upon the plumb line A C the Rule lying thus close to the plane you may thereby draw a line parallel to the Horizon Having drawn this horizontall line M N crosse it at right angles with the perpendicular K D unto which if it be the under face apply the side A B of your Quadrant so shall the threed upon the limb give you the angle of inclination required But if it be the upper face of the plane then lay a Ruler to the perpendicular K D unto the under side whereof apply the side A B of your Quadrant as is here shewed in this figure so shall the degree of the Quadrant give you C A H the angle of inclination required Here you must be needfull that both edges of your Ruler be streight and one parallel to the other CHAP. III. To finde the Declination of a plane TO finde out this declination you must make two observations by the Sun the first is of the angle made between the horizontall line of the plane and the Azimuth wherein the Sun is at the time of observation the second is of the Suns altitude both these observations should be made at one instant First for the horizontall distance having drawn upon your plane a line parallel to the Horizon apply the side of your Quadrant thereunto holding it parallel to the Horizon then holding up a threed and plummet at full liberty so as the shadow thereof may passe through the centre of the Quadrant observe the angle made upon the Quadrant by the shadow of the threed and that side with the horizontall line for that is the distance here required Then at the same instant as neer as may be take the Suns altitude that so you may finde the Suns Azimuth from the East or West points by the 21 Chapter of the fourth Book Having thus gotten the horizontall distance with the Azimuth of the Sun for the same time describe a Circle as A B C D representing the horizontall circle and draw the diameter A C which shall represent the horizontall line F G of the last Chapter Now supposing the horizontall distance to be 38 deg 30 min. the angle O A B of the last Chapter place it from C Southward to E that is from the same end of the horizontall line with which the angle was made upon the plane and draw the line E Z Then supposing the altitude of the Sun at the same time to be 30 deg 45 min. with 11 deg 30 min. North declination and so by the 21 Chapter of the fourth Book the Azimuth will be found to be 23 deg 17 min. from the East Southward being the observation was made in the forenoon this 23 deg 17 min. I place from E the place of the Sun at the time of observation unto R which is the true point of the East and draw the line H R representing the Verticall Circle so shall the angle made between the horizontall line of the plane and the line of East and West be the declination of the plane which in this example is found to be 15 deg 13 min. the angle C Z R. Or you may observe the angle made between the shadow of the threed and that side of the Quadrant which lyeth perpendicular unto the horizontall line of the plane which in this example is 51 deg 30 min. the complement of the former angle and it is the angle O A C in the former Chapter upon the Quadrant Now having drawn your Horizontall Circle as before and the diameter A C for the horizontall line of the plane you may crosse it at right angles with the diameter B D for the Axis of the planes horizontall line from which as from D you may set your horizontall distance on the same side thereof as before you found it by your observation as here from D to E and draw the line E Z for the line of the shadow and having found the Azimuth of the Sun 23 deg 17 min. from the East Southward you may set it from E the place of the Sun Northward to R and draw the line R Z H for the line of East and West as before Or if you take the Suns Azimuth from the South which in this example will be 66 deg 43 min. the complement of the former 23 deg 17 min. you may set it from E the place of the Sun unto S Southward and draw the line S Z N for the Meridian so shall the arch S D or R C be 15 deg 13 min. for the declination as before CHAP. IV. To draw the houre-lines upon the Horizontall the full North or South planes whether erect or inclining SEeing the making of these Dials are all after one manner we will here proceed to make an Horizontall Diall by help of the lines upon the Quadrant fitted for that purpose Therefore having by the 10 Chapter of the third book found the elevation of the pole above the plane we may proceed after this manner First draw the line D A F of sufficient length out of the middle whereof let fall the perpendicular A B for the Meridian and Substile then take the line D E or E F out of your Quadrant and set it from A to B in the Meridian through which point B draw the line E B C parallel to D A F now supposing the elevation of the pole above the plane to be 52 deg 30 min. the latitude of the place from the Sine thereof take the neerest extent unto A E the line of latitudes and set it from A to D and from A to F both wayes and from B to C and from B to E and draw the lines D C and E F making the long square C D F E the two angles whereof C and E shall be the points for the hours of 3 and 9 in all these kinde of planes that declines not from the North or South For the hour-lines before six and after you may extend their opposite houre lines beyond the centre as was shewed in the 8 chapter of the third book What is here shewed concerning the hours the like may be understood for the half hours by applying the threed thereunto in the limbe CHAP. V. To draw a Diall upon a South or North erect declining plane IN the drawing of all these kinde of Dials by help of this Quadrant when the Latitude of the place and the declination of the plane is known two things more is to be considered First the elevation of the Pole above the plane Secondly the inclination of the Meridian of the plane to the Meridian of the place both which will speedily be found when you are ready for them First therefore draw the line D A F as before from the middle whereof let fall
halves and quarters and also made choise of some convenient point in the stile to give the shadow and draw the horizontall line C D then make the triangle S A R in this Trigon equall to the triangle S A R in the following South Diall as S A equall to the Axis of the stile A R equall to the substilar and R S equall to the perpendicular stile then draw the perpendiculars S G for the Equinoctiall and describe the arch O G P making G O and G P each of them 23 deg 31 min. for the two Tropicks which you must transfer into your plane as before Now for the drawing of the parallels of the length of the day you must have recourse to the little Table before going and therein see what declination the Sun hath at such a day as you would put into your plane as when the day is either 8 or 16 houres long the declination is 20 deg 59 min. therefore place in your Trigon 20 deg 59 min. from G unto a both wayes and draw the lines S a and S a marking them at the ends with 8 and 16 the length of the day for which they serve Likewise when the day is either 9 or 15 houres long then the Suns declination is 16 deg 22 min. therefore set 16 deg 22 min. from G unto b both wayes and draw S b and S b. Also when the day is either 10 or 14 houres long then the declination is 11 deg 14 min. which set from G to c both wayes and draw S c and S c. Lastly when the day is 11 or 13 houres long the declination is 5 deg 43 min. which set from G unto d both wayes and draw S d and S d noting them with numbers answering to the length of the day as you see in the Trigon when the day is just 12 houres long it is Equinoctiall and hath no declination and is signified in the Trigon by the line S G. For the manner how to transfer these parallels of the length of the day into the plane it is to be performed in all respects as in the former Chapter for the inserting of the Signes not at all differing therefrom and therefore I shall forbeare to give you any farther instructions for the performance thereof but give you the figure of a South plane with these parallels drawn thereon which will instruct more then a whole Chapter of information And thus much for the drawing of the parallels of the Signes and Diurnall arches in all kinde of planes I will now proceed to shew you how some other Astronomicall conclusions which are very pleasing and delightfull may be inscribed upon all sorts of Dials CHAP. III. Shewing how the Italian and Babylonish houres may be drawn upon all kinde of planes THe Italians account their houres from the Suns setting and the Babylonians from his rising so that these kinde of houre-lines being drawn upon any plane you may know by inspection only how many houres are past since the last setting or rising of the Sun The inscription of these houre-lines into any of the former planes is very easie the work of the last Chapter being well understood Because that upon a full South or an Horizontall plane these houre-lines shew themselves most uniform I have therefore for example sake made choice of a full South Diall upon which it shall be shewn how to draw both the Italian and Babylonish houres Your Diall being drawn and the two Tropicks and the Equinoctial thereon inscribed and also the Horizontall line you must draw in your Diall two obscure parallels of the length of the day one when the day is 8 houres and the other when the day is 16 houres long expressed in the following Dial by the two pricked arches neer the two Tropicks the uppermost of which is the parallel of the Suns course when the day is 8 houres long and the undermost is the parallel of his course when the day is 16 houres long the Aequinoctiall is the parallel of the Suns course when the day is 12 houres long Your Diall being thus prepared and these parallels thus inserted the inscription of these houre-lines is very easie and plain to be understood To begin then with the inscription of the Babylonish houres which are the houres from the Suns rising First It is apparent that when the day is 8 houres long that the Sun riseth at 8 in the morning so that at that time the first houre after the suns rising is 9 in the morning Secondly when the day is 12 hours long the Sun riseth at 6 in the morning so that at that time the first houre after the suns rising is 7 in the morning Thirdly when the day is 16 houres long the Sun riseth at 4 in the morning so that the first houre after his rising is 5 in the morning as plainly appeareth by this Table  Length of the Day 8 12 16 Hours from Sun rising 1 9 7 5 2 10 8 6 3 11 9 7 4 12 10 8 5 1 11 9 6 2 12 10 7 3 1 11 8 4 2 12 9 5 3 1 10 6 4 2 11 7 5 3 By which you may perceive that when the day is 8 houres long the seventh houre from sun rising is 3 in the afternoon When the day is 12 houres long the seventh houre from sun rising is 1 in the afternoon And when the day is 16 houres long the seventh houre from the suns rising is 11 before noon as by this Table doth evidently appear And therefore a streight line drawn in your Diall through those points where the common houre-lines of your Diall crosse the respective parallels of the dayes length shall shew the true quantity of houres since the suns rising at all times of the yeare which is the Babylonish houre For example let it be required to draw the seventh hour from the suns rising in your Diall First by the Table you see that in the parallel of 8 houres for the length of the day the seventh houre from the suns rising is 3 in the afternoon therefore observe where the houre-line of three crosseth the parallel of 8 houres which is at a. Secondly by the Table you see that in the parallel of 12 hours for the length of the day the seventh houre from sun rising is then 1 in the afternoon wherefore observe where the houre-line of 1 crosseth the Equinoctiall which is at b. Thirdly by the Table you see that in the parallel of 16 houres for the length of the day the seveth houre from the suns rising is 11 before noon therefore observe where the houre-line of 11 crosseth the parallel of 16 hours which is at c then draw the streight line a b c which shall be the seventh Babylonish houre or the seventh hour from the suns rising all the year long And by this rule and the help of the Table you may draw all the other houres from sun rising as you see them drawn in the figure
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
and put numbers to them as you see there done ¶ 1 Note That if any of the points you are to make use of for the drawing of any of these houres fall without your plane you must in this case extend your houre-line parallel and Equinoctiall beyond the limits of your Diall-plane and there make use of the points but you need extend the line you draw no farther then the bounds of the plane as here in the figure you see the first houre from Sun rising crosseth not the Equinoctiall and the houre-line of 7 within the plane but if the Equinoctiall and the houre-line of 7 were extended it would crosse ¶ 2 Note That if any of the three points you are to make use of doe so far exceed the limits of your plane that it will be either impossible or at least very troublesome to extend the houre-lines so far that then in that case any two of the three points will sufficiently serve the turn ¶ 3 Note That as the houres from sun rising were put into the plane by the same rule may the hours from sun setting or Italian houres be inserted the difference being only in the numbring of them the houres from the sun rising being numbered from the West side of the Horizontall line by 1 2 3 4 5 6 7 8 9 10 and 11 and the houres from the suns setting are denominated from the East side of the Horizon and numbered backwards by 23 22 21 20 19 18 17 16 15 14 and 13 as in the figure doth evidently appear ¶ 4 Note That these Italian and Babylonish hours are inscribed on all planes by help of this little Table and the rules and cautions delivered in this Chapter and therefore more examples were superfluous CHAP. IV. Shewing how the Jewish hours may be drawn upon any plane IT was the custome of the Ancients to divide their day and also their night whether long or short into 12 equal parts beginning their day at the suns rising and their night at the suns setting so that 12 of the clock at noon was alwayes the sixth houre of their day and 12 at night was alwayes the sixth hour of their night and according to this division were their Dials drawn so that all the Somer the houres of their day were longer then the houres of their night and all the Winter the houres of their night were longer then those of their day and when the Sun is in the Equinoctiall then the houres of their day and night were equall and the same with those of our account but at all other times of the year different The inscribing of these hours into all kinde of planes is very easie being much like the drawing of the Babylonish and Italian houres before taught Having therefore drawn your Diall which in this example for the avoyding of many figures we will have to be the full South plane used before in Chap. 2. of this Appendix with the houres halves and quarters and also drawn the two Tropicks and the parallels of the length of the day thereupon as you see here done in this figure Then make choice of two parallels of the length of the day which must be both of them equidistant from the Equinoctiall which let be the parallels of 9 houres and 15 houres both which are three houres different from the Equinoctiall on either side thereof and these two parallels are the most convenient for this our purpose because the Jewish hours will fall in these two parallels justly upon the houres halves and quarters of the common houre-lines and so be the easier drawn Now the points through which every one of the Jewish hours must passe is exactly shewed by this little Table wherein you may see that the first Jewish houre must be drawn through 5 houres 45 min. or 5 houres three quarters in the parallel of 15 through 7 hours in the Equinoctial Jewish hours The parallel of 15 hours Equinoctiall The parallel of 9 hours H. M. H. H. M. 1 5 45 7 8 15 2 7 0 8 9 0 3 8 15 9 9 45 4 9 30 10 10 30 5 10 45 11 11 15 6 12 0 12 12 0 7 1 15 1 0 45 8 2 30 2 1 30 9 3 45 3 2 15 10 5 0 4 3 0 11 6 15 5 3 45 12 7 30 6 4 30 through 8 houres a quarter in the parallel of 9 hours In like manner the second Jewish houre must bee drawn in your plane through 7 of the clock in the parallel of 15 through 8 a clock in the Equinoctiall and through 9 of the clock in the parallel of 9 houres and so of all the rest according as you see in this Table and as you may perceive them drawn in the South plane the numbers belonging to these hours being set at both ends of each houre-line CHAP. V. Shewing how to draw the Azimuths or Verticall Circles in all kinde of planes THe Azimuths are great Circles of the Sphere meeting together in the Zenith of the place and are variously inscribed on all planes according to their scituation In the Horizontall plane they meet in a center with equall angles In all upright planes whether direct or declining they are parallel to the Meridian or line of 12. And in all reclining planes they meet together in a point which is the Zenith of the place These Azimuths being great Circles in the Sphere are therefore streight lines in all planes and may be drawn as followeth §. 1. In the Horizontall plane IN the Horizontall plane these Azimuths are most easily inserted for your Diall being drawn with the Tropicks thereon you have no more to doe but upon the foot of the perpendicular stile to describe a Circle which you may divide into 32 equall parts beginning at the Meridian answering to the 32 points of the Mariners Compasse Or else you may divide the same Circle into 90 equall parts according to the Astronomicall division and through each of those points draw streight lines from the foot of the stile and set numbers or letters to them either by 10 20 30 40 c. if you divide it into 90 or else by South S by W S S W S W by S c. if you divide the Circle according to the Mariners Compasse This is so plain that it needeth no example §. 2. In the East or West erect planes YOur Diall being finished you may draw upon a piece of Pastboard the line M E N representing the Horizontall line M E N in your Dial then on the point E raise the perpendicular E Q equall to the line EG in your Diall and on Q as a center describe the semicircle K E L and divide one halfe thereof namely the Quadrant E L into eight equall parts representing one quarter of the Mariners Compasse and from the center Q draw lines through each of those divisions extending them till they cut the line M E N in the points ☉ ☉ ☉ ☉ ☉ ☉ then with your compasses