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Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
end_n	divide_v	equal_a	line_n	1,364	5	9.1106	5	false

Text snippets containing the quad

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

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

you farther directions in a matter of so great perspicuity as to lay down the severall wayes for projecting the Sphere on every severall plane but proceed to shew the making of a general Dial for the whole World which we will use as our Declinatorie to finde the scituation of any wall or plane as shall be required to make a Diall thereon as followeth in the next Chapter CHAP XII Shewing the making of a Diall on a Crosse form as also a Universall Quadrant drawn from the same projection as also to describe the Tropicks on Meridian or Polar planes THis Universall Diall is described by Clavius in his eighth Book de Gnomonicis But because the Artists of these times have found out a more commodious contrivance of it in the fabrique I shall describe it according to this Figure Now to know the houre of the day you shall turn the plane by the helpe of the needle so as the end A shall be toward the North and E toward the South and elevate the end E to the complement of the elevation then bringing the Box to stand in the Meridian the shoulder of the Crosse shall shew you the houre Upon this also is grounded the Universall Quadrant hereafter described which Instrument is made in Brasse by Mr. Walter Hayes as it is here described Prepare a Quadrant of Brasse divide it in the limbe into 90 degrees and at the end of 45 degrees from the center draw the line A B which shall represent the Equator divide the limbe into 90 degrees as other Quadrants are usually divided then number both wayes from the line AB the greatest declination of the Sun from the North and South at the termination whereof draw the arch CD which shall be the Tropicks then out of the Table of declination pag. 45 from B both wayes let there be numbered the declinatiō of the Signes according to this Table   G M   ♈ 00 00 ♎ ♉ ♍ 11 30 ♏ ♓ ♊ ♌ 20 30 ♐ ♒ ♋ 23 30 ♑ Now the plane it selfe is no other then an East or West Diall numbred on one side with the morning houres and on the other with the evening houres the middle line AB representing the Equator And to set it for the houre you shall project the Tropicks and other intermediate parallels of the Signes upon them as is hereafter shewed but that the plane may not run out of the Quadrant you shal work thus opening the Compasses to 15 degrees of the Quadrant prick that down both wayes at which distance draw parallels to the line AB and with the same distance as if it were the semidiameter of the Equator describe the semidiameter of the Equator on the top of the line AB which divide into 12 parts and laying a ruler through the center and each of those divisions in the semicircle to those parallel lines on each side of AB marke where they cut and from side to side draw the parallel houre lines as is taught in the making of an East and West Diall make those parallel lines also divided as a tangent line on each side AB so if this Quadrant were held on an East or West wall and a plummet let fall from the center of the Equator where the style stands which may be a pin fitted to take out and in fitted to the height of the distance between the line A B and the other parallels which is all one with the Radius of the small Circle it shall I say be in its right scituation on the East or West wall if you let the plummet and threed fall on the elevation of the Pole in that place But because we desire to make it generall we must describe the Tropicks and other parallels of declination upon it as is usuall to be done on your Polar and East and West Diall which how to doe is thus Having drawn the houre lines and Equator as is taught from E the height of the style take all the distances between it and the houre lines where they doe crosse the line AB and prick them down on the line representing the Equator in this figure from the center B. Then describe an occult arch of a Circle whereon describe a Chorde of 23 degrees 30 minutes with such other declinations as you intend on your plane Then on the line representing the Equator noted here with the figures of the houres they were taken from 6 7 8 9 10 11 at the marks formerly made that was taken from E the height of the style and every of the houres from these distances I say raise perpendiculars to cut the other lines of declination so those perpendiculars shall represent those houre lines from whence they were taken and the distances between the Equator and the severall lines of declination shall be the same distances from the Equator and the other parallels of declination upon your plane through which marks being pricked down upon the severall hourelines from the Equinoctiall If you draw those Hyperbolicall lines you shall have described the parallels of declination required But if you will performe the same work a second and easie way worke by this Table following which is universall and is composed out of the Table of Right Versed shadow Put this Table before thee for the point of each houre line whereby the severall parallels of the Signes shall pass worke thus The style being divided into known parts if into 12 take the parts of shadow out of the Table in the same known parts by which the style is divided prick them down on each houre line as you finde it marked in the Table answering the houre both before and after noon As suppose that a Polar plane I finde when the Sun is in Aries or Libra at 12 a clock the shadow hath no latitude but at 1 and 11 it hath 3 parts 13 min. of the parts of the style which I prick from the foot of the style on the houres of 1 and 11 both above and beneath the Equator and for 2 and 10 I finde 6 parts 56 min. which I prick down also from the center to the houre lines of 10 and 2 and so of the other houre lines and parallels through which if I draw those lines they shall represent the parallels of the Declination A Table of the Latitude of shadows   Cancer Gemini Leo Virgo Taurus Libra Aries   p m p m p m p m p m a m 12 5 13 4 25 2 26 0 0 12 1 6 17 5 35 4 5 3 13 11 2 8 11 8 35 7 27 6 56 10 3 14 5 13 31 12 39 12 0 9 4 23 15 22 45 21 21 20 27 8 5 49 6 47 57 45 45 44 47 7 6 Vmbra infinita 6 Having promised in the description of the use of this Instrument to shew how to finde the inclination and reclination of a plane I shal proceed to give you some cautions First then the quadrant is divided in the limbe as other

name will last and be in memory From age to age although for infamie What more abiding Tombe can man invent Then Books which if they 'r good are permanent And monuments of fame the which shall last Till the late evening of the World be past But if erroneous sooth'd with vertues face Their Authors cridit's nothing but disgrace If I should praise thy Book it might be thought Friends will commend although the work be nought But I 'le forbeare lest that my Verses doe Belie that praise that 's only due to you Good Wiue requires no Bush and Books will speak Their Authors credit whether strong or weak W. Leybourn ERRATA REader I having writ this some years since while I was a childe in Art and by this appear to be little more for want of a review hath these faults which I desire thee to mend with thy pen and if there be any errour in Art as in Chap. 17 which is only true at the time of the Equinoctiall take that for an oversight and where thou findest equilibra read equilibrio and in the dedication in some Copies read Robert Bateman for Thomas and side for signe and know that Optima prima cadunt pessimas aeve manent pag. line Correct ● 10 equall lines 18 16 Galaxia 21 1 Galaxia 21 8 Mars 24 12 Scheame 35 1 Hath 38 8 of the Tropicks polar Circles 40 22 AB is 44 31 Artificiall 46 ult heri 49 4 forenoon 63 29 AB 65 11 6 80 16 BD 92 17 Arch CD 9 ult in some copies omit center 126 4 happen 126 6 tovvard B 127 26 before 126 prop. 10 for sine read tang elev   Figure of the Dodicahedron false cut pag. 4 LF omitted at end of Axis 25 For A read D 26 In the East and West Diall A omitted on the top of the middle line C on the left hand B on the right 55 Small arch at B omitted in the first polar plane 58 For E read P on the side of the shadowed line toward the left hand I omitted next to M and L in the center omitted 81 K omitted in figure 85 On the line FC for 01 read 6 for 2 read 12 line MO for 15 read 11 96 A small arch omitted at E F G H omitted at the ende of the line where 9 is 116 I L omitted on the little Epicicle 122 THE ARGVMENT OF THE Praecognita Geometricall and of the Work in generall WHat shall I doe I stand in doubt To shew thee to the light For Momus still will have a flout And like a Satyre bite His Serpentarian tongue will sting His tongue can be no slander He 's one to wards all that hath a fling His fingers ends hath scan'd her But seeing then his tongue can't hurt Fear not my little Book His slanders all last but a spurt And give him leave to look And scan thee thorough and if then This Momus needs must bite At shadows which dependant is Only upon the light Withdraw thy light and be obscure And if he yet can see Faults in the best that ever writ He must finde fault with me How ere proceed in private and deline The time of th' day as oft as sun shall shine And first define a Praecognitiall part Of magnitude as usefull to this art THE PRAECOGNITA GEOMETRICAL THe Arts saith Arnobius are not together with our mindes sent out of the heavenly places but all are found out on earth and are in processe of time soft and fair forged by a continuall meditation our poor and needy life perceiving some casual things to happen prosperously while it doth imitate attempt and try while it doth slip reform and change hath out of these same assiduous apprehensions made up small Sciences of Art the which afterwards by study are brought to some perfection By which we see that Arts are found out by daily practice yet the practice of Art is not manifest but by speculative illustration because by speculation Scimus ut sciamus we know that we may the better know And for this cause I first chose a speculative part that you might the better know the practice and therefore have first chose this speculative part of practicall Geometry which is a Science declaring the nature quantity and quality of Magnitude which proceeds from the least imaginable thing To begin then A Point is an indivisible yet is the first of all dimension it is the Philosophers Atome such a Nothing as that it is the very Energie of all things In God it carryeth its extreams from eternity to eternity in the World it is the same which Moses calls the beginning and is his Genesis 't is the Clotho that gives Clio the matter to work upon and spins it forth from terminus à quo to terminus ad quem in the Alphabet 't is the Alpha and is in the Cuspe of the Ascendant in every Science and the house of Life in every operation Again a Point is either centricall or excentricall both which are considered Geometrically or Optically that is a point or a seeming point a point Geometrically considered is indivisible and being centrall is of magnitude without consideration of form or of rotundity with reference to Figure as a Circle or a Globe c. or of ponderosity with reference to weight and such a point is in those Balances which hang in equilibra yet have one beam longer than the other If it be a seeming point it is increased or diminished Optically that is according to the distance of the object and subject 'T is the birth of any thing and indeed is to be considered as our principall significator which being increased doth produce quantity which is the required to Magnitude for Magnitude is no other then a continuation of Quantity which is either from a Line to a plain Superficies or from a plain Superficies to a Solid Body every of which are considered according to the quantity or form The quantity of a Line is length without breadth or thicknesse the forme either right or curved The quantity of a Superficies consisteth in length and breadth without thicknesse the form is divers either regular or irregular Regular are Triangles Squares Circles Pentagons Hexagons c. An equilaterall Triangle consisteth of three right lines as many angles his inscribed side in a Circle contains 120 degrees A Square of four equall right lines and as many right angles and his inscribed side is 90 degrees A Pentagon consisteth of five equall lines and angles and his inscribed side is 72 degrees of a Circle A Hexagon is of six equall lines and angles and his side within a Circle is 60 degrees which is equall to the Radius or Semidiameter An Angle is the meeting of two lines not in a streight concurring but which being extended will crosse each other but if they will never crosse then they are parallel The quantity of an angle is the measure of the part of a Circle

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

N W 67 30 E by N 78 45 W by N 78 45 East 90 00 West 90 00 By which it appeareth that every point of the Compasse is distant from the Meridian 11 degrees 15 minutes The third sort of planes are inclining or rather reclining whose upper face beholds the Zenith and in that respect is called Reclining but if a Diall be made on the nether side and thereby respect the Horizon it is then called an incliner so that the one is the opposite to the other These planes are likewise accidentally divided for they are either direct recliners reclining from the direct points of East West North and South and in this sort happens the direct Polar and Aequinoctiall planes as infinite more according to the inclination or reclination of the plane or they are as erect planes doe become declining recliners which looke oblique to the Cardinall parts of the World and obtusely to the parts they respect Suppose a plane to fall backward from the Zenith and by consequence it falls towards the Horizon then that represents a Reclining plane such you shall you suppose the Aequinoctiall Circle in the figure to represent reclining from the North Southwards 51 degrees from the Zenith or suppose the Axis to represent a plane lying parallel to it which falls from the Zenith Northward reclining 38 degrees one being Aequinoctiall the other a Polar plane But for the inclining decliners you shall know them thus forasmuch as the Horizon is the limiter of our sight and being cut at right angles representeth the East West North and South points it may happen so that a plane may lie between two of these quarters in an accidentall Azimuth and so not beholding one of the Cardinall Quarters is said to decline Again the said plain may happen not to stand Verticall which is either Inclining or Reclining and so are said to be Inclining Decliners First because they make no right angle with the Cardinal Quarters Secondly because they are not Verticall or upright There are other Polar planes which lie parallel to the Poles under the Meridian which may justly be called Meridian plains and these are erect direct East and West Dials where the poles of the plane remain which planes if they recline are called Position planes cutting the Horizon in the North and South points for Circles of position are nothing but Circles crossing the Horizon in those points CHAP IV. Shewing the finding out of a Meridian Line after many wayes and the Declination of a Plane A Meridian Line is nothing else but a line whose outmost ends point due North and South and consequently lying under the Meridian Circle and the Sun comming to the Meridian doth then cast the shadow of all things Northward in our Latitude so that a line drawn through the shadow of any thing perpendicularly eraised the Sun being in the Meridian that line so drawn is a Meridian line the use whereof is to place planes in a due scituation to their points respective as in the definition of this Circle I shewed there was accidentall Meridians as many as can be imagined between place and place which difference of Meridians is the Longitude or rather difference of Longitude which is the space of two Meridians which shews why noon is sooner to some then others The Meridian may be found divers wayes as most commonly by the Mariners compasse but by reason the needle hath a point attractive subject to errour and so overthroweth the labour I cease to speake any further It may be found in the night for when the starre called Aliot seems to be over the Pole-starre they are then true North the manner of finding it Mr. Foster hath plainly laid down in his book of Dyalling performed by a Quadrant which is the fourth part of a circle being parted into 90 degrees It may also be fouhd as Master Blundevile in his Booke for the Sea teacheth being indeed a thing very necessary for the Sea which way is thus Strike a Circle on a plain Superficies and raise a wire or such like in the center to cast a shadow then observe in the forenoon when the shadow is so that it just touches the circumference or edge of the Circle and there make a mark doe so again in the afternoon and at the edge where the shadow goes out make another mark between which two marks draw a line which part in halfe then from that middle point to the center draw a line which is a true Meridian Or thus Draw a great many Circles concentricall one within another then observe by the Circles about noone when the Sun casts the shortest shadow and that then shall represent a true Meridian the reason why you must observe the length of the shadow by circles not by lines is because if the Sun have not attained to the true Meridian it wil cast its shadow from a line and so my eye may deceive me when as by Circles the Sun casting shadow round about still meetes with one circumference or other and so we may observe diligently Secondly it is proved that the shadow in the Meridian is the shortest because the Sun is neerest the Verticall point Thirdly it is proved that it is a true Meridian for this cause the Sun as all other Luminous bodies casts his shadow diametrically and so being in the South part casts his shadow northward and is therefore a true Meridian But now to finde the declination of a wall if it be an erect wall draw a perpendicular line but if it be a declining reclining plane draw first an horizontall line and then draw a perpendicular to that and in the perpendicular line strike a Style or small Wyre to make right angles with the plane then note when the shadow of the Style falleth in one line with the perpendicular and at that instant take the altitude of the Sun and so get the Azimuthe reckoned from the South for that is the true declination of the wall from the South The distance of the Azimuthes from the South or other points are mentioned in degrees and minutes in the third Chapter in the definition of the severall sorts of planes or by holding the streight side of any thing against the wall as is the long Square ABCD whose edge AB suppose to be held to a wall and suppose again that you hold a thrid and plummet in your hand at E the Sun shining and it cast shadow the line EF and at the same instant take the altitude of the Sun thereby getting the Azimuthe as is taught following then from the point F as the center of the Horizon and from the line FE reckon the distance of the South which suppose I finde the Azimuthe to be 60 degrees from the East or West by the propositions that are delivered in the end of this Booke and because there is a Quadrant of a Circle between the South and the East or West points I substract the distance of the Azimuthe from 90

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

Meridian the one neere the window the other farther in through severall points whereof we must draw the houre-lines Let AB be the Meridian line found on the seeling now suppose the Sun being in the highest degree of Cancer should shine into the Glasse that is fixed in C it shall again reflect unto D where I make a mark then letting a plummet fall from the top of the seeling till it fall just on C the glasse from the point E from which draw the line A B through D and E which shall be the Meridian required if you do this just at noon Now if you would finde out the places where the hour-lines shall crosse the Meridian the Center lying without the window EC you may work thus CHAP X. Shewing the making and use of the Cylinder Dial whose hour-lines are straight as also a Diall drawn from the same form having no Style THis may be used on a Staff or other round made like a Cylinder being drawn as is here described where the right side represent the Tropicks and the left side the Equinoctial or it may be used flat as it is in the Book the Instrument as you see is divided into months and the bottom into signs and the line on the right side is a tangent to the radius of the breadth of the Parallelogram serving to take the height of the Sun the several Parallels downward running through the pricked line in the midle are the lines of Altitude and the Parallels to the Equator are the Parallels of Declination numbred on the bottom on a Sine of 23 de and a half For the Altitude of the Sun The use of it is first if it be described on the head of a staff to have a gnomon on the top equal to the radius and just over the tangent of Altitudes to turn it till you bring the shadow of it at right angles to it self which shal denote the height required For the Houre of the Day Seek the Altitude of the Sun in the midle prick't line and the Declination in the Parallels from the Equator and mark where the traverse lines crosse through the crossing of the two former lines and at the end you shal finde the figures of 2 or 10 3 or 9 c. only the summer Houres are sought in the right side where the Sun is highest and the traverse lines longest and in the winter the Hour is sought on the left side where the traverse lines are shorter For the Declination and degree of the Signe Seek the day of the moneth on the top marked with J. for January F for February c. and by the help of a horse hair or threed extended from that all along of Parallel of Declination till it cut on the bottom where the signes are numbred the down right lines that are parallel to the Equator counted toward the right hand is the degree of the Declination of that part of the Ecliptick which is in the bottom right against the day of the moneth sought on the top The pricked line passing through the 18 degree of the Parallel of Altitude is the line of Twy-light this projection I had of my very good friend John Hulet Master of Arts and Teacher of the Mathematicks You may also make a Dyal by preparing of a hollow Cylinder and if you doe number on both ends of the Circle on top and bottom 15 de from line to line or divide it into 24 parts and if from top to bottom you draw streight lines first by dividing the Cylinder through the middle and only making use of one half it shal have 12 houres upon it Lastly if you cut off a piece from the bottom at an angle according to the Elevation and turn the half Cylinder horizontal on that bottom til the shadow of one of the sides fal parallel with any one of those lines from top to bottom which numbred as they ought shal shew the hour without the use of a Style So also may you project a Dyal on a Globe having a round brim on the top whose projection will seem strange to those that look upon it who are ignorant of these Arts CHAP XVI Shewing the making of a universall Dyall on a Globe and how to cover it if it be required If you desire to cover the Globes and make other inventions thereon first learn here to cover it exactly with a pair of compasses bowed toward the points measure the Diameter of the Globe you intend to cover which had finde the Circumference thus Multiply the Diameter by 22 and divide that product by 7 and you have your desire That Circumference let be the line A B which divide into 12 equal parts and at the distance of three of those parts draw the Parallel C D and E F A Parallel is thus drawn take the distance you would have it asunder as here it is three of those 12 divisions set one foot in A and make the Arch at E another at B and make the Arch with the other foot at F the compasses at the wideness taken then by the outward bulks of those Arches draw the line E F so also draw the line C D. And to divide the Circumference into parts as our example is into 12 work thus set your compasses in A make the Ark B F the compasses so opened set again in B and make the Ark A C then draw the line from A to F then measure the distance from F to B on the Ark and place it on the other Arch from A to C thence draw the line C B then your compasses open at any distance prick down one part less on both those slanting lines then you intend to divide thereon which is here 11 because we would divide the line A B into 12 then draw lines from each division to the opposite that cuts the line A B in the parts of division But to proceed continue the Circumference at length to G and H numbring from A toward G9 of those equal parts and from B toward H as many which shal be the Centers for each Arch. So those quarters so cut out shall exactly cover the Globe whose Circumference is equal to the line A B. Thus have you a glance of the Mathematicks striking at one thing through the side of an other for I here made one figure serve for three several operations because I would not charge the Press with multiplicity of figures CHAP XVII Shewing the finding of the Elevation of the Pole and therewithall a Meridian without the Declination of Sun or Starre THis is done by erecting a gnomon horizontal and at 3 times of the day to give a mark at the end of the Shadows now it is certain that represents the Parallel of the Sunne for that day then take three thin sticks or the like and lay them from the top of the gnomon to the places where the shadows fell and on these three so standing lay a board to ly on all

three flat and a gnomon in the midle of that board points to the Pole because every Parallel the Sun moves in is parallel to the Equinoctial and that is at right Angles with the pole of the World Now the Meridian passeth through the most elevated place of that board or circle so laid neither can the Sun's Declination make any sensible difference in the so small proportion of 3 or 4 houres time CHAP XVIII Shewing how to finde the Altitude of the Sun only by Scale and Compasses WIth your Compasses describe the Circle A B C D place it horizontal with a gnomon in the Center crosse it with two Diameters then turn the board till the shadow be on one of the Diameters at the end of the shadow mark as here at E lay down also the length of the gonmon from the Center on the other Diameter to F from E to F drawe a right line then take your Compasses and on the chord of 90 take out the Radius the Ark of 60 set the compasses so in E describe an Arch then take the distance between the line E F and the Diameter D B which measure on the chord of 90 and so many degrees as the compasses extend over such a quantity is the height of the Sun in like manner any Angles being given you must measure it by the parts of a circle Here followeth the problematical Propositions of the Office of shadow and the benefit we receive thereof Prop. 1 By shadow we have a plain demonstration that the Sphere of Sol is higher than the Sphere of Luna to confirm such as think they move in one Orbe Let the Sun be at A in the great Circle and the Moon at B in the lesser let the Horizon be C D now they make one Angle of height in respect of the Center of the Earth notwithstanding though they so equally respect the Earth as one may hinder the sight of the other yet the shadow of the Sun shall passe by the head of the gnomon E and cast it to F and the beames of the Moon shall passe by E to G much longer which shewes shee is much lower for the higher the light is the shorter is the shadow I call the Moon a feminine if you ask my reason shee is cold and moist participating of the nature of Women and we call her the Mother of moisture but that 's not all for I have a rule for it Nomen non crescens Prop. 2. By shadow we are taught the Earth is bigger then the Moon seeing in time of a total Obscurity the Moone is quite overshadowed for the shadow is cast in this manner By the same we learn also that seeing the shadow comes to a point the Earth is less then the Sun for if the opacous body be equal to the luminous body then like two parallels they will never meet but concurre in infinitum as these following figures shew Or if the luminous body were less then the opacous body then the shadow would be so great in so long a way as from the Earth to the Starry Firmament that most of the Starres as were in opposition to the Sun would not appear seeing they borrow their light of the Sun It is also sufficiently proved by shadow in the Praecognita Philosophical that the Earth is round and that it possesseth the middle as proprius locus from which it cannot passe and to which all heavie things tend in a right line as their terminus ad quem From which the semidiameter of the Sun 15 min. substracted doth remain the Altitude of the center of the Sun 50 de 3 m. the Altitude required or From this or the former Proposition we may take notice that there is no Dial can shew the exact time without the allowance of the Suns semidiameter which in a strict acception is true but hereto Mr. Wells hath answered in the 85 page of his Art of shadows where saith he because the shadow of the center is hindered by the style the shadow of the hour-line proceeds from the limbe which alwayes precedeth the center one min. of time answerable to 15 min. the semidiameter of the Sun which to allow in the height of the Style were erroneous wherefore let the al●owance be made in the hour-lines detracting from the true Equinoctial distances of every 15 deg. 15 primes and so the Arches of the horizontall plane from the Meridian shall stand thus Prop. 4. By shadow we may finde the natural Tangent of every degree of a quadrant as appeares by the former example Houres Equinoctial distances True hour distances 12 0 de m. de m. se 11 1 14 45 11 38 51 10 2 29 45 24 6 31 9 3 44 45 37 4 2 8 4 59 45 53 19 12 7 5 74 45 70 48 6 6 6 89 45 89 40 51 For the Sun being 46 deg 13 min. of Altitude makes a shadow of 95. parts of such as the gnomon is 100 so then multiply the length of the gnomon 100 by the Radius and divide by 95 and it yeelds 105263 the natural Tangent of that Ark Prop. 5. By shadow we may take the height of any Building by the Rule of Proportion if a gnomon of 6 foot high give a shadow of 10 foot how high is that house whose shadow is 25 foot resolved by the Rule of Three Prop. 6. By shadow also we learn the magnitude of the Earth according to Eratosthenes his proposition Prop. 7. By shadow we learne the true Equinoctial line running from East to West which crossed at right Angles is a true Meridian where note that in the times of the Equinoctiall that the shadows of one gnomon is all in one right line Prop. 8. By shadow we know the Earth to be but as a point as may appear by the shadow of the Earth on the body of the Moon Prop. 9. By shadow we may learn the distance of places by the quantity of the obscurity of an Eclipse Prop. 10. By gnomonicals we make distinctions of Climates and People some Hetorezii some Perezii some Amphitii Prop. 11. By shadow the Climates are known in the cold intemperate Zones the shadow goes round In the hot intemperate Zones the shadow is toward the West at the rising Sun and toward the East at the setting Sun and no shadow at Noones to them as dwel under the Parallels And to them in the temperate Zones always one way toward the North or toward the South Prop. 12. By shadow we are taught the Rule of delineating painting according to the perspective way how much is to be light or dark accordingly drawn as the center is disposed to the eye so the Office of shadow is manifold as in the Optical conclusions are more amply declared therefore I referre you to other more learned works and desist to speak But for matter of Information I will here insert certain definitions taken out of Optica Agulion ii lib. 5. First saith he we call that a