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Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
end_n	draw_v	line_n	perpendicular_a	3,095	5	14.0786	5	true

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ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A89305	Horlogiographia optica. Dialling universall and particular: speculative and practicall. In a threefold præcognita, viz. geometricall, philosophicall, and astronomicall: and a threefold practise, viz. arithmeticall, geometricall, and instrumentall. With diverse propositions of the use and benefit of shadows, serving to prick down the signes, declination, and azimuths, on sun-dials, and diverse other benefits. Illustrated by diverse opticall conceits, taken out of Augilonius, Kercherius, Clavius, and others. Lastly, topothesia, or, a feigned description of the court of art. Full of benefit for the making of dials, use of the globes, difference of meridians, and most propositions of astronomie. Together with many usefull instruments and dials in brasse, made by Walter Hayes, at the Crosse Daggers in More Fields. / Written by Silvanus Morgan.	Morgan, Sylvanus, 1620-1693.; Goddard, John, fl. 1645-1671, engraver.	1652 (1652)	Wing M2741; Thomason E652_16; ESTC R202919	57,946	133

divided into 360 degrees between the open ends and the angle it self is the Center of the Circle The quantity of a Solid consists of length breadth and thickness the form is various regular or irregular The five regular or Platonick Bodies are the Tetrahedron Hexahedron Octohedron Dodecahedron Icosahedron Tetrahedron is a Solid Body consisting of four equall equilaterall Triangles A Hexahedron is a Solid Body consisting of six equal Squares and is right angled every way An Octahedron is a Solid Body consisting of eight equal Equilaterall Triangles A Dodecahedron is a Solid Body consisting of 12 equall Pentagons An Icosahedron is a Solid Body consisting of 20 equal Equilaterall Triangles All which are here described in plano by which they are made in pasteboard Or if you would cut them in Solid it is performed by Mr. Wells in his Art of Shadows where also he hath fitted planes for the same Bodies A Parallel line is a line equidistant in all places from another line which two lines can never meet A Perpendicular is a line rightly elevated to another at right angles and is thus erected Suppose AB be a line and in the point A you would erect a perpendicular set one foot of your Compasses in A extend the other upwards anywhere as at C then keeping the foot fixed in C remove that foot as was in A towards B till it fall again in the line AB then if you lay a Ruler by the feet of your Compasses keep the foot fixed in C and turn the other foot toward D by the side of the Ruler and where that falls make a marke from whence draw the line DA which is perpendicular to AB And so much shall suffice for the Praecognita Geometricall the Philosophicall followeth The end of the Praecognita Geometricall THE ARGVMENT OF THE Praecognita Philosophicall NOt to maintain with nice Philosophie What unto reason seems to be obscure Or shew you things hid in obscurity Whose grounds are nothing sure 'T is not the drift of this my BOOK The world in two to part Nor shew you things whereon to looke But what hath ground by Art If Art confirm what here you read Sure you 'l confirmed be If reason wonte demonstrate it Learn somwhere else for me There 's shew'd to you what shadow is And the Earths proper place How it the middle doth possesse And how heavens run their race Resolving many a Proposition Which are of use and needfull to be known THE PRAECOGNITA PHILOSOPHICAL CHAP I. Of Light and Shadows HE that seeketh Shadow in its predicaments seeketh a reality in an imitation he is rightly answered umbram per se in nullo praedicamento esse the reason is thus rendred as hath been it is not a reality but a confused imitation of a Body arising from the objecting of light So then there can be no other definition then this Shadow is but the imitation of substance not incident to parts caused by the interposition of a substance for Umbra non potest agere sine lumine And And it is twofold caused by a twofold motion of light that is either from a direct beam of light which is primary or from a secondary which is reflective hence it is that Sun Dials are made where the direct beams can never fall as on the seeling of a Chamber or the like But in vain man seeketh after a shadow what then shall we proceed no farther surely not so for qui semper est in suo officio is semper orat for there are no good and lawful actions but doe condescend to the glory of God and especially good and lawfull Arts And that shadow may appear to be but dependant on light it is thus proved Quod est existit in se id non existit in alio that which is and subsisteth in it selfe that subsisteth not in another but shadow subsisteth not in it selfe for take away the cause that is light and you take away the effect that is shadow Hence we also observe the Sun to be the fountain of light whose daily and occurrent motions doth cause an admirable lustre to the glory of God seeing that by him we measure out our Times Seasons and Years Is it not his annuall revolution or his proper motion that limits our Year Is it not his Tropicall distinctions that limits our Seasons Is it not his Diurnall motion that limits out our Dayes and Houres And man truly that arch type of perfection hath limited these motions even in the small type of a Dyall plane as shall be made manifest in things of the second notion that is Demonstration by which all things shall be made plain CHAP II. Of the World proving that the Earth possesseth its own proper place WE have now with the Philosopher found out that common place or place of being that is the World will you know his reason 't is rendred Quia omnia reliqua mundi corpora in se includit I 'le tell you of no plurality not of planetary inhabitants such as the Lunaries lest you grabble in darkness in expecting a shadow from the light without interposition for can the light really without a substance be its own Gnomon surely no neither can we imagine our earth to be a changing Cynthia or a Moon to give light to the Lunary inhabitants For if our Earth be a light as some would have it how comes it to passe that it is a Gnomon also to cast a shadow on the body of the Moon far lesse then it selfe and so by consequence a greater light cannot seem to be darkned on a lesser or duller light and if not darkned no shadow can appear But from this common place the World with all its parts shall we descend to a second grade of distinction and come now to another which is a proprius locus and divide it into proper places considering it as it is divided into Coelum Solum Salum Heaven Earth Sea we need not so far a distinction but to prove that the earth is in its own proper place I thus reason Proprius locus est qui proxime nullo alio interveniente continet locatum but it is certain that nothing can come so between the earth as to dispossesse it of its place therefore it possesseth its proper place furthermore ad quod aliquid movetur id est ejus locus to what any thing moves that is its place but the earth moves not to any other place as being stable in its own proper place And this proper place is the terminus ad quem to which as the place of their rest all heavie things tend in quo motus terminantur in which their motion is ended CHAP III. Shewing how the Earth is to be understood to be the Center A Center is either to be understood Geometrically or Optically either as it is a point or seeming a point If it be a point it is conceived to be either a center of magnitude or a center of

elevation or greater or lesser If the arke be equal to the complement of the poles elevation by it is a token the plane is oblique under the Meridian to be inclined unto the Pole in that case the meridian of the place and of the plane and also the Axis doe concur in the same line G L if the plane be supposed to fall in the same great circle KN but if the plane be not supposed but in some parallel of the same and the Axis be somwhat carryed away as necessarily it is done if the Sciotericall be absolved the Meridian of the plane and place are two lines parallel between themselves and are mutually joyned together according to the difference of longitude of the place and of the plane which difference is according to the angle HGC which is the complement of the angle BNK late found because the angle KGH is right by 57. p. 1. yea forasmuch as the meridian of the plane may goe by the poles of the plane but concurring at G or N are equall to two right by 20 p. 1. Example Let the plane meridionall declined to the right hand 29 de 59 m. inclining toward the pole artick 23 de 3 m. the elevation of the pole 49 de 35 m. and there are to be sought in the same the meridian of the place the plane and the elevation of the pole or Axis above the plane The calculation shall be thus To 67874 the tangent of the arke KN the distance of the meridian of the place from the Verticall of the plane 34 de 10 m. per ax 2 The sine of the arke NC 49de 35 m. whose complement is the arke BN 40de 25 m per axi. 4. To 60388 the sine of the angle BNK 37d 9m whose complement is the angle HNC or HGC 52 de 51. m. the difference of the longitude of the plane from the longitude of the place or the distance of the meridians of the place and plane Therefore let the horizon of the place be LC the verticall of the plane KL the circle of the plane of the horizon KNC in which there is numbred from K towards C 34 de 10m and at the terme of the numeration N draw the right line L N E which shall be the meridian of the plane and place if the center of the Sciotericie L or F is taken for the center of the World and the right line L N F for the Axis but because in the perfection of the Diall IG remaineth the Axis with E the center of the world not in the right line L N F but above the same with props at pleasure but notwithstanding it is raised equall in height with EI and OG and moreover the plane is somwhat withdrawn frō the axis of the world therefore the line L N F is now not altogether the meridian of the place but only the meridian of the plane or as vulgarly they speake the substilar But you may finde the meridian of the place thus draw IH at right angles to the meridian of the plane which they vulgarly call the Contingence to the common section of the Equator which in the plane let E the center of the world be set from the axis IG in the meridian of the plane L N F. Then to the center E consisting in the line L N E le the circle of the Equator FK be described and in the same toward the East because the horizon of the plane is more easterly then the horizon of the place and moreover the beame is cast sooner or later upon the meridian of the plane then the place let there be numbered the difference of longitude of the place and plane 52 de 51 m. and by K the end of the numeration let a right line be drawn as it were the certain beams of the Equator EKH which where it toucheth the common section of the Equator with the plane to wit the right line FH by that point let C the meridian of the place be drawn perpendicular The second case of the third Probleme of Pitiscus his Liber Gnomonicorum Sivero arcus BN repertus fuerit c. But if the arke BN shall be found lesse then the complement of the poles elevation it is a signe the plane doth consist on this side the pole artick and moreover above such a plane not the pole Artick but the pole Antartick shall be extolled to such an angle as ILM is whose measure is the arke IM to which out of the doctrine of opposites the arke GO is equall which you may certainly finde together with the arke NO thus As MOG the right angle to NG the difference between BN and BG so ONG the angle before found to OG per axi. 3. As the tangent ONG to Radius so the tangent OG to the sine O N by axi. 2. Example Let the plane be meridionall declined to the right hand 34 de 30 m. inclined toward the pole artick 16 de 10 m. and again let the elevation of the pole be 49 de 35 m. and there are sought The meridian of the place the longitude of the countrey The meridian of the plane the longitude of the plane The elevation of the pole above the plane The Calculation 1. As BF Radius 100000 to FC tangent complement of declination 55 de 30 m. 14550 so 27843 the sine of the inclination 16 de 10 m. to 40511 the tangent of K N 22 de 31 3 m. the distance of the meridian of the place from the Verticall of the plane per axi. 2. The sine of the arke N C 62 de 532 3 m. whose complement is B N 27 de 61 3 m. by which substracted from BG the complement of the poles elevation 40 de 25 m. there is remaining the arke N G 13 de 182 3 m. by axi. 4. To 61108 the sine of the angle B N K or O N G 37d 40 m. per axi. 3. comp. 1. To 14069 the sine of the arch OG the distance of the axis GL from the meridian of the plane OL 8de 51 3m by ax 3. To 18410 the sine of the arch N O the distance of the meridian of the plane OL from the meridian of the place N L 30 deg. 36½ m by axi. 2. The calculation being absolved let there be drawn the horizon of the place AC secondly the verticall of the plane BQ thirdly the horizon of the plane ABCQ in whose Quadrant AQ to wit according to the pole antartique which alone appeareth above such a plane First let be numbred the distance of the meridian of the place from the verticall of the plane 22 de 3 m. and by the ende of the numeration at P let the meridian of the plane LP be drawn then from the point P let the distance of the meridian of the plane from the meridian of the place be numbered by the terme of the numeration M let the meridian of the plane LM be drawn Finally

0 11 1 59 43 56 34 48 12 36 58 25 40 17 6 13 52 10 2 53 45 50 55 43 12 32 37 21 51 13 38 10 30 9 3 45 42 43 6 36 0 26 7 15 58 8 12 5 15 8 4 36 41 34 13 27 31 18 8 8 33 1 15     7 5 27 17 24 56 18 18 9 17 0 6         6 6 18 11 15 40 9 0                 5 7 9 32 6 50                 11 37 4 8 1 32                     21 40 This Table is in Mr. Gunters Book page 240 which if you desire to have the point of the Equinoctiall for a Horizontall plane on the houre of 12 enter the Table of shadows with 38 de 30 m. and you shall finde the length of the shadow to be 15 parts 5 m. of the length of the style divided into 12 which prick down on the line of 12 for the Equinoctiall point from the foot of the style So if I desire the points of the Tropick of Cancer I finde by this Table that at 12 of the clock the Sun is 62 de high with which I enter the Table of shadows finding the length of the shadow which I prick down on the 12 a clock line for the point of the Tropick of Cancer at the houre of 12. If for the houre of 1 I desire the point through which the parallel must pass looke for the houre of 1 and 11 in this last table under Cancer and I finde the Sun to have the height of 59 de 43 m. with which I enter the table of shadows and prick down the length thereof from the bottome of the style reaching till the other foot of the Compasses fall on the houre for which it was intended Doe so in all the other houres till you have pricked down the points of the parallels of declination through which points they must be drawn Hyperbolically Proceed thus in the making of a Horizontall Diall but if it be a direct verticall Diall you shall then take the length of the verticall shadow out of the said Table or work it as an Horizontal plane only accounting the complement of the elevation in stead of the whole elevation For a declining plane you may consider it as a verticall direct in some other place and having found out the Equator of the plane and the substyle you may proceed in the same manner from the foot of the style accounting where the style stands to be no other wayes then the meridian line or line of 12 in a Horizon whose pole is elevated according to the complement height of the style above the substyle and so prick down the length of the shadows from the foot of the style on every one of the Houre lines as if it were a horizontal or Verticall plane But in this you must be wary remembring that you have the height of the sun calculated for every houre of that Latitude in the entrance of the 12 signes in that Place where your Plane is a Horizontall plane or otherwayes by considering of it as a horizontall or Verricallplane in another latitude For the Azimuths or verticall circles shewing one what point of the compasse the sun is in every houre of the day it is performed with a great deale of facility if first when the sun is in the Equator we doe know by the last Table of the height of the sun for every houre of the day and by his meridian altitude with the help of the table of shadows find out the Equinoctiall line whether it be a Horizontall or upright direct plane for having drawn that line at right angles with the meridian and having the place of the Style and length thereof in parts and the parts of shadow to all altitudes of the sun being pricked down from the foot of the Style on the Equinoctiall line through each of those points draw parallel lines to the meridian or 12 a clock line on each side which shall be the Azimuths which you must have a care how you denominate according to the quarter of heaven in which the sun is in for if the Sun be in the easterly points the Azimuths must be on the Western side of the plane so also the morning houres must be on the opposite side There are many other Astronomical conclusions that are used to be put upon planes as the diurnall arches shewing the length of the day and night as also the Jewish or old unequal houres together with the circles of position which with the meridian and horizon distinguisheth the upper hemispheare into 6 parts commonly called the houses of Heaven which if this I have writ beget any desire of the reader I shall endeavour to inlarge my self much more in shewing a demonstrative way in these particulars I have last insisted upon I might heare also shew you the exceeding use of the table of Right and versed shadow in the taking of heights of buildings as it may very wel appear in the severall uses of the quadrant in Diggs his Pantometria in Mr. Gunters quadrant having the parts of right and versed shadow graduated on them to which Books I refer you CHAP XIV Shewing the drawing of the Seeling Diall IT is an Axiom pronounced long since by those who have writ of Opticall conceipts of Light and Shadow that Omnis reflectio Luminis est secundum lineas sensibiles latitudinem habentes And it hath with as great reason bin pronounced by Geometricians that the Angles of Incidence and Reflection is all one as appeareth to us by Euclides Catoptriques and on this foundation is this conceipt of which we are now speaking Wherefore because the direct beams cannot fall on the face of this plane we must by help of a piece of glasse apt to receive and reflect the light placed somwhere horizontally in a window proceed to the work which indeed is no other then a Horizontall Diall reversed to which required a Meridian line which you must endeavour to draw and finde according as you are before taught or by the helpe of the Meridian altitude of the Sun your glasse being fixed marke the spot that reflects upon the seeling just at 12 a clock make that one point and for the other point through which you must draw your meridian line you may finde by holding up a threed and plummet till the plummet fall perpendicular on the glasse and at the other end of the line held on the seeling make another mark through both which draw the Meridian line Now for so much as the center of the Diall is a point without and the distance between the glasse and the seeling is to be considered as the height of the style the glasse it selfe representing the center of the world or the very apex of the style wee must finde out those two Tangents at right angles with the

the morning houres and 6 7 8 the evening And because the North pole is elevated above this plane 38 deg. 30 min. the Axis must be from the center according to that elevation pointing upward as the South doth downward so as A is the Zenith of the South C must be in the North The Arithmeticall calculation is the same with the former also a North plane may shew all the houres of the South by consideration of reflection For by Opticall demonstration it is proved that the angles of incidence is all one to that of reflection if any be ignorant thereof I purposely remit to teach it to whet the ingenious Reader in labouring to finde it The Figure of a direct East and West Diall for the Latitude of London 51 deg. 30 min. East Diall West Diall CHAP IV. Shewing the making of the Prime Verticall planes that is a direct East or West Diall FOr the effecting of this Diall first draw the line AD on one end thereof draw the circle in the figure representing the Equator then draw two touch lines to the Equator parallel to the line AD these are they on which the houres are marked divide the Equator in the lower semicircle in 12 equall parts then apply a ruler to the center through each part and where it touches the lines of contingence make marks from each touch point draw lines to the opposite touch point which are the parallels of the houres and at the end of those lines mark the Easterly houres from 6 to 11 and of the West from 1 to 6. These planes as I told you want the Meridian houre because it is parallel to the Meridian Now for the placing of the East Diall number the elevation of the Axis to wit the arch DC from the line of the Equator to wit the line AD and in the West Diall number the elevation to B fasten a plummet and thrid in the center A and hold it so that the plummet may fall on the line AC for the East Diall and AB for the West Diall and then the line AD is parallel to the Equator and the Dial in its right position And thus the West as well as East for according to the saying Contrariorum eadem est doctrina contraries have one manner of doctrine Here you may perceive the use of Tangent line for it is evident that every houres distance is ●●t the Tangent of the Aequinoctiall distance The Arithmeticall Calculation 1 Having drawn a line for the houre of 6 whether East or West As the tangent of the houre distance is to the Radius so is the distance of the houre from 6 to the height of the Style 2 As the Radius is to the height of the Style so is the tangent of the houre distance from 6 to the distance of the same houre from the substyle The style must be equall in height to the semidiameter of the Equator and fixed on the line of 6. CHAP V. Shewing the making a direct parallel Polar plane or opposite Aequinoctiall I Call this a direct parallel Polar plane for this cause because all planes may be called by their scituation of their Poles and so an Aequinoctiall parallel plane may be called a Polar plane because the Poles thereof lie in the poles of the World The Gnomon must be a quadrangled Parallelogram whose height is equall to the semidiameter of the Equator as in the East and West Dials so likewise these houres are Tangents to the Equator Arithmeticall calculation Draw first a line representing the Meridian or 12 a clock line and another parallel to the said line for some houre which may have place on the line say As the tangent of that houre is to the Radius so is the distance of that houre from the Meridian to the height of the Style 2 As the Radius is to the height of the style so the tangent of any houre to the distance of that houre from the Meridian CHAP VI Shewing the making of a direct opposite polar plane or parallel Aequinoctiall Diall AN Aequinoctiall plane lyeth parallel to the Aequinoctiall Circle making an angle at the Horizon equal to the elevation of the said Circle the poles of which plane lie in the poles of the world The making of this plane requires little instruction for by drawing a Circle and divide it into 24 parts the plane is prepared all fixing a style in the center at right angles to the plane As the Radins is to the sine of declination so is the co-tangent of the Poles height to the tangent of the distance of the sub-stile from the Meridian If you draw lines from 7 to 5 on each side those lines so cut shall be the places of the houre lines of a parallel polar plane now if you draw to each opposite from the pricked lines those lines shall be the houre lines of the former plane CHAP VII Shewing the making of an erect Verticall declining Diall IF you will work by the fundamentall Diagram you shall first draw a line such is the line AB representing the Meridian then shall you take out of the fundamentall diagram the Secant of the Latitude viz. AC and prick it down from A to B and at B you shall draw a horizontall line at right angles such is the line CD then you shall continue the line AB toward i and from that line and where the line AB crosseth in CD describe an arch equall to the angle of Declination toward F if it decline Eastward and toward G if the plane decline Westward Then shall you prick down on the line BF if it bean Easterly declining plane or from B to G if contrary the Secant complement of the Latitude viz. AG in the fundamentall Diagram and the Sine of 51 degrees viz. DA which is all one with the semidiameter of the Equator and therewithall prick it down at right angles to the line of declination viz. BF from B to H and G and from F towards K and L then draw the long square KIKL and from B toward H and G prick down the severall tangents of 15 30 45 and prick the same distance from K and L towards H and G lastly draw lines through each of those points from F to the horizontall line CD and where they end on that line to each point draw the houre lines from the point A which plane in our example is a Verticall declining eastward 45 degrees and it is finished But because the contingent line will run out so far before it be intersected I shall give you one following Geometricall example to prick down a declining Diall in a right angled parallelogram Now for the Arithmeticall calculation the first operation shall be thus As the Radius to the co-tangent of the elevation so is the sine of the declination to the tangent of the substiles distance from the meridian of the place then II Operation Having the complement of the declination and elevation finde the

from the point M into whatsoever part let the proper elevation of the pole be numbered or the distance of the axis from the meridian of the plane 8 de 51 3m and by the term of the numeration I let the axis LI be drawn to be extolled or lifted up on the meridian of the plane LM to the angle MLN The third case of the third probleme of Pitiscus his liber Gnomonicorum Si denique arcus BN repertus fuerit major c. Lastly if the arke BN be found greater then the complement of the poles elevation BG it is a token the plane to be inclined beyond the pole artique and moreover the pole artique should be extolled above such a plane to so great an angle as the angle GLO which the arke GO measureth which arke together with the arke ON in the end you may find in such sort as in the precedent case Example Let there be a meridian plane declining to the right hand 35 de 54 m. inclining towards the pole artique 75 de 43 m. and let the elevation of the pole be 49 de 35½ m. but there is sought the meridian of the plane and place together with the elevation of the pole above the plane the calculation shall be thus to 133874 tangent of the arke KN the distance of the meridian of the place from the verticall of the plane 53 de 14½ m by axi. 2. The sine of the arke NC 8 de 29● m. whose complement is BN 81 de 30½ m. from whence if you substract BG 40 de 25 m. there remaineth the arke GN 41 de 5½ m. to 97982 the sine of the angle BNK or ONG by axi. 3. to 64399 the sine of the arch OG the distance of the axis from the meridian of the plane 40 de 51 3 m. by axi. 3. to 17483 the sine of the arke O N the distance of the meridian of the plane from the meridian of the place 10 de 4 m. by axi. comp. 2. The calculation being finished let the horizon of the place be AC the verticall of the plane KD the horizon of the plane AKCD in which let be numbered from the vertical point K toward C the distance of the meridian of the place from the vertical of the plane 53 de 14½ m. and by the end of the numeration let be drawn the meridian of the place LN then from the meridian of the place to wit from the point N backward let the distance of the meridian of the plane 10 de 4m be numbred and by O the end of the numeration let LO the meridian of the plane be drawn from which afterwards let the proper elevation of the pole be numbred or the distance of the axis from the meridian of the plane 48d 5½m and by the term of the numeration G let the axis LG be drawn being extolled above the plane BO to the angle GLO CHAP X. In which is shewed the drawing of the houre-lines in these last planes not there mentioned being also part of Pitiscus his example in the fourth Probleme of his liber Gnom SO then saith he Si axis c. If the axis be oblique to the plane as the foregoing are as in any plane oblique to the Equator many of the houre-lines doe concur at the axis with equal angles but they are easily found thus But because Pitiscus is mute in defining which part he takes for the right hand and which the left we must search his meaning Pitiscus was a Divine is evident by his own words in his dedication Celsitudini tuae tota vita mea prolixe me excusarem quod ego homo Theologus c. If we take him as hee was a Divine we imagine his face to be towards the East then the South is his right hand and the North is his left hand That he was an Astronomer too appeareth by his Books both of proper and common motion then we must imagine his face representing the South the East on his left hand which cannot be as shall appear Neither must we take him according to the Poets whose face must be imagined toward the West In short take him according to Geographie representing the Pole and this shews the right hand was the East and left the West as is evident by the Diall before going for it is a plane declining from the South to the right hand 30 degrees that is the East because it hath the morning houres not the evening because the Sun shines but part of the afternoon on the plane Thus in briefe I have run throngh all planes and proceed to shew you farther conclusions But I desire the Reader to take notice that in these examples of Pitiscus I have followed his own steps and made use of the Naturall Sines and Tangents CHAP XI Shewing how by the helpe of a Horizontall Diall or other to make any Diall in any position how ever HAving prepared a Horizontall Diall as is taught before on the 12 houre as far distant as you please from the foot of the style draw a line perpendicular to the line of 12 on that describe a Semicircle plasing the foot of the Compasses in the crossing of the lines this Semicircle divide into 180 parts each Quadrant into 90 to number the declination thereon let the arch of the Semicircle be toward the North part of the Diall Then prepare a plane slate such as will blot out what hath been formerly made thereon and make it to move perpendicularly on the horizontal plane on the center of the semicircle which wil represent any declining plane by moving it on the semicircle Now knowing the declination of the plane turn this slate towards the easterly part if it decline towards the East if contrary to the West if toward the West and set it on the semicircle to the degree of declination then taking a candle and moving the Diall till the shadow fall on all the houres of the horizontall plane mark also where the shadow falls on the declining plane that also is the same houre on the plane so scituated drawn from the joyning of the style with the plane It is so plain it needs no figure So may you doe in all manner of declining reclining or reclining and inclining Dials by framing your instrument to represent the position of the plane Note also that the same angle the axis of the Horizontal Dial makes with the plane the same elevation must the axis of that plane have and where it shadows on the representing plane when the shadow of the horizontal axis is on 12 that is the meridian of the place By the same also may you describe all the conclusions Astronomicall the Almicanthers circles of height the parallels of the Sun shewing the declination the Azimuthes shewing the point of the Compasse the Sun is in and all the propositions of the Sphere Seeing this is so plain and evident nay a delightful conclusion I will not give