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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

ponderosity or a center of rotundity if it be a seeming point that is increased or diminished according to the ocular aspect as being somtime neerer and somtime farther from the thing in the visuall line the thing is made more or lesse apparent A center of magnitude is an equal distribution from that point an equality of distribution of the parts giving to each end alike and to each a like vicinity to that point or center A center of ponderosity is such a point in which an unequall thing hangs in equi libra in an equall distribution of the weight though one end be longer or bigger than the other of the quantity of the ponderosity A center of rotundity is such a center as is the center of a Globe or Circle being equally distant from all places Now the earth is to be understood to be such a center as the center of a Globe or Sphear being equally distant from the concave superficies of the Firmament neither is it to be understood to be a center as a point indivisible but either comparatively or optically comparatively in respect of the superior Orbs Optically by reason of the far distance of the one from the earth as that the fixed Stars being far distant seeme by the weaknesse of the sense to be conceived as a center indivisible when by the force and vigour of reason and demonstration they are found to exceed this Globe of earth much in magnitude so that what our sense cannot apprehend must be comprehended by reason As in the Circles of the Coelestiall Orbs because they cannot be perceived by sense yet must necessarily be imagined to be so Whence it is observable that all Sun Dials though they stand on the surface of the earth doe as truly shew the houre as if they stood in the center CHAP IV. Declaring what reason might move the Philosophers and others to think the Earth to be the center and that the World moves on an axis circa quem convertitur OCular observations are affirmative demonstrations so that what is made plain by sense is apparent to reason hence it so happeneth that we imagine the Earth to move as it were on an axis because both by ocular and Instrumentall observation in respect that by the eye it is observed that one place of the Skie is semper apparens neither making Cosmicall Haeliacall or Achronicall rising or setting but still remaining as a point as it were immoveable about which the whole heavens are turned These yet are necessary to be imagined for the better demonstration of the ground of art for all men know the heavens to be supported only by the providence of God Thus much for the reason shewing why the World may be imagined to be turned on an Axis the demonstration proving that the earth is the center is thus not in maintaining unlikely arguments but verity of observation for all Gnomons casting shadow on the face of the earth cast the like length or equality of shadow they making one the same angle with the earth the Sun being at one and the same angle of height to al the Gnomons As in example let the earth be represented by the small circle within the great circle marked ABCD and let a Gnomon stand at E of the lesser Circle whose horizon is the line AC and let an other gnomon of the same length be set at I whose horizon is represented by the line BD now if the Sun be at equall angles of height above these two Horizons namely at 60 degrees from C to G and 60 from B to F the Gnomons shall give a like equality of shadows as in example is manifest Now from the former appears that the earth is of no other form then round else could it not give equality of shadows neither could it be the center to all the other inferior Orbs For if you grant not the earth to be the middle this must necessarily follow that there is not equality of shadow For example let the great Circle represent the heavens and the lesse the earth out of the center of the greater now the Sunne being above the Horizon AC 60 d. and a gnomon at E casts his shadow from E to F and if the same gnomon of the same length doth stand till the Sun come to the opposite side of the Horizon AC and the Sun being 60 degrees above that Horizon casts the shadow from E to H which are unequall in length the reason of which inequality proves that then it did not stand in the center and the equality of the other proves that it is in the center Hence is also most forceably proved that the earth is compleatly round in the respect of the heavens as is shewed by the equality of shadows for if it were not round one and the same gnomon could not give one and the same shadow the earth being not compleatly round as in the ensuing discourse and demonstration is more plainly handled and made manifest And that the earth is round may appeare first by the Eclipses when the shadow of the earth appeareth on the body of the Moon darkning it in whole or in part and such is the body such is the shadow Again it appears to be round by the orderly appearing of the Stars for as men travell farther North or South they discover new Stars which they saw not before and lose the sight of them they did see As also by the rising or setting of the Sun or Stars which appear not at the same time to all Countries but by difference of Meridians and by the different observations of Eclipses appearing sooner to the Easterly Nations then those that are farther West Neither doe the tops of the highest hils nor the sinking of the lowest valleys though they may seeme to make the earth un-even yet compared with the whole greatnesse doe not at all hinder the roundnesse of it and is no bigger then a point or pins head in comparison of the highest heavens Thus having run over the Systeme of the greater WORLD now let us say somthing of the Compendium thereof that is MAN CHAP V. Of Man or the little World MAn is the perfection of the Creation the glory of the Creator the compendium of the World the Lord of the Creatures He is truly a Cosmus of beauty whose eye is the Sunne of his body by which he beholds the never resting motions of the heavens contemplatively to behold the place of motion the place of his eternall rest Lord what is man that thou shouldest be so mindefull of him or the son of man that thou so regardest him thou hast made a World of wonder in his face Thou hast made him to be a rationall creature endowed him with reason so that his intellect becomes his Primum mobile to set his action at work nevertheles man neither moves nor reigns in himselfe and therefore not for himselfe but is born not to himselfe but for his Countrey therefore he ought to employ

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

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

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

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

them making day with them while it remaineth night with us and according to the demonstration is expressed by the greot Circle marked NSEW signifying the East West North and South parts of the Horizon So now if you imagine a Circle to be drawn from the Suns leaving our sight through those Azimuth points of heaven then that Circle there imagined is the Horizon and is accidentally divided as a man changes his place and divides the World in a right or oblique Sphear The Meridian is a great Circle scituated at right angles to the Horizon equally passing between the East and West points and consequently running due North and South and passeth through the Poles of the World being stedfastly fixed it is represented by the great Circle marked NDSC and is accidentally divided if we travell East or West but in travailing North or South altereth not when the Sun touches this Circle it is then mid-day or Noon Now if you imagine a Circle to passe from the North to the South parts of the Horizon through your Zenith that Circle so imagined is your Meridian from which Meridian we account the distance of houres The Aequinoctiall likewise divides the World in two equall parts crossing at right angles between the two Poles and is therefore distant from each Pole 90 degrees and is elevated from the Horizon on the contrary side of the Poles elevation so much as the Pole wants of 90 deg. elevation demonstrated in the Scene by the Circle passing from A to B and is accidentally elevated with the Poles as we change our Horizon and when the Sun touches this Circle the dayes and nights are then equall and to those that live under this Citcle the dayes and nights hang in equilibra continually and the Sun doth move every houre 15 degrees of this Circle making the houre lines equall passing 15 degrees in one houre 30 degrees in two houres 45 degrees in three houres 60 degrees for four and so increasing 15 degrees as you increase in houres This I note to the intent you may know my meaning at such time as I shall have occasion ro mention the Aequinoctiall distances The Axis of the World is that which the Stile in every Diall represents being a line imaginary supposed to passe through the center of the World from the South to the North part of the Meridian whose outmost ends are the Poles of the World this becomes the Diameter about which the World is imagined to be turned in a right Sphear having no elevation in an oblique to be elevated above the Horizon and the angle at the center numbred on the arch of the Meridian between the apparent Pole and the Horizon is the elevation thereof represented by the streight line passing from E to F the arch EN being accounted the elevation thereof which according to our demonstration is the Latitude of London The Stars that doe attend the Artick or North Pole are the greater and lesser Beare the last star in the lesser Bears tale is called the Pole Star by reason of its neerness to it this is the guide of Mariners as appeareth by Ovid in his exile thus You great and lesser Bear whose Stars doe guide Sydonian and Graecian ships that glide Even you whose Poles doe view this lesser Ball Under the Western Sea neere set at all The stars that attend the Southern Pole is the Cross as is seen in the Globes Lord be my Pole make me thy Style Lord then Thy name shall be my terminus ad quem Video Coelos opera manuum tuarum lunam stellas que tu fundasti CHAP III. Of the severall sorts of Planes and how they are known DYals are the dayes limiters and the bounders of time whereof there are three sorts Horizontall Erect Inclining Horizontall are alwayes parallel to the Horizon Erect some are erect direct others erect declining Inclining also are direct or declining for more explanation the figure following shall give you better satisfaction where the Horizon marked with diverse points of the Compasse shall explain the demonstration Now if you imagine Circles to passe through the Zenith A crossing the Horizon in his opposite points as from SW through the verticall point A passing to the opposite point of South-west to North-East those or the like circles are called Azimuthes parallel to which Azimuthes all erect Sciothericals doe stand Those Planes that lie parallel to the Horizontall Circle are called Horizontall planes and his Style makes an angle with the Pole equall to the elevation thereof then the elevation of the Pole is the elevation of the Style Erect Verticals are such which make right angles with the Horizon and lie parallel to the Verticall point and these as I told you before were either direct or declining Direct are those that stand in a direct Azimuth beholding one of the four Cardinall Quarters of the World as either direct East West North or South marked with these letters NEWS or declining from them to some other indirect Azimuth or side-lying points Erect North and South are such as behold those Quarters and cuts the Meridian at right angles so that the planes crosse the Meridian due East and West and the Poles are their Styles equally elevated according to the aequinoctiall altitude being the complement of the Poles elevation For in all North Faces Planes or Dials the Style beholds the North Pole and in all South faces the Style beholds the South Pole therefore where the North Pole is elevated there the North Pole must be pointed out by the Style and where the South Pole is elevated vice versa The second sort of Verticals are declining which ate such that make an acute angle with the Quarter from which they decline for an acute angle is lesse then a right angle and a right angle is 90 degrees these declining Planes lying in some accidentall Azimuthe For supposing a Diall to turn from the South or North towards the East or West the Meridian line of the South declines Eastward happening in these Azimuthes or between them South declining East South declining West S by E 11 15 Or to these points of the West decliners or between them S by W 11 15 S S E 22 30 S S W 22 30 S E by S 33 45 S W by S 33 45 South-East 45 00 South West 45 00 S. E by E 56 15 S W by W 56 15 E S E 67 30 W S W 67 30 E by S 78 45 W by S 78 45 East 90 00 West 90 00 Again North decliners declining toward the East and West doe happen in these Azimuthes or between them North declining East North declining West N by E 11 15 Or to these points of the West decliners or between them N by W 11 15 N N E 22 30 N N W 22 30 N E by N 33 45 N W by N 33 45 North-East 45 00 North West 45 00 N E by E 56 15 N W by W 56 15 E N E 67 30 W