HOROLOGIOGRAPHIA OPTICA dialling universal and particular. speculative and practical together with the description of the court of Arts by a new Method By Sylvanus Morgan JJ. sculp Horologiographia Optica. DIALLING universal and Particular: Speculative and practical. In a threefold PRAECOGNITA, viz. geometrical, philosophical, and astronomical: and a threefold practice, viz. arithmetical, geometrical, and instrumental. With diverse Propositions of the use and benefit of shadows, serving to prick down the signs, Declination, and Azimuths, on Sun-Dials, and diverse other benefits. Illustrated by diverse optical 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 astronomy. Together with many useful Instruments and Dials in brass, made by Walter Hayes, at the cross Daggers inMore Fields Written by Silvanus Morgan. LONDON, Printed by R & W. Leybourn, for Andrew Kemb, and Robert Boydell▪ ● and are to be sold at St Margaret's Hill in Southwark, and at the Bulwark near the Tower. 1657. TO WILLIAM BATEMAN, Esqrs. TO ANTHONY BATEMAN, Esqrs. TO THOMAS BATEMAN, Esqrs. Sons to the late Honourable THOMAS BATEMAN, Esq Chamberlain of LONDON, Deceased. GENTLEMEN, YOur late Father being a Patron of this Honourable City, doth not a little invite me to you, though young, yet to patronise no less than the aspiring of Coelum, which, as the Poets feign, was the ancientest of the gods, and where you may see Sol only of the Titans, favouring Jupiter's▪ sign, and by their power and operation hath established Arts or Learning, the fable rather according to that establishment which God hath given them, they are, I say, sought out of those that take their pleasure therein: Pardon my boldness, I beseech you, if like Prometheus I have made a man of clay; and now come to light my bundle of twigs at the Chariot of the Sun, desiring that you would infuse vigour in that which cannot at all move of itself, & if your benevolence shall but shine upon it, the angles of incidence & reflection shall be all one: your love invites me to be so bold as to think you worthy of my labour, wherein, if faults shall arise in the Cuspis of the Ascendent, they shall also have their fall upon my self. And if any shall be offended at this work, my device shall be a dial with this Mottoe, Aspicio ut aspiciar, only to all favourers of Art I am direct erect plain, as I am, Gentlemen, to you, and desire to be Yours in the best of my services, S. M. TO THE READER. REader, I here present thee with some celestial operations drawn from the Macrocosmall World, if I should tell you of plurality, it may seem absurd, but I'll distinguish the word. Mundus the World is sometimes taken Archtypically, and so is God, only in whose divine mind is an example of all things. Mundus the World is sometimes taken angelical, and this is the hierarchical government of Angels in Ceruphins, Cherubins, and Thrones. Mundus the World is sometimes taken elementary, and this is the Philosophers common place: the Salamander in Fire, the Birds in Air, the Fish in Water, and Men and Beasts on Earth. Sometimes Macrocosmally, considering the whole Universe, as well etherial as Subterene, yea, and every Orb, and this is imaginarily set down in the Praecognita astronomical. Sometimes Microcosmally, as in the little World man, and this is described in the last Chapter of the Praecognita philosophical. Sometimes Typically, and that either geographical or Gnomonicall, or mentally in the mind of the workman. Geographically in Maps or Globes, or spheres in plano. Gnomonicall in this present Art of Dialling, of which it may be said that Umbra horas Phoebi designat climate nostro Nodus, quod signum Sol tenet arte docet. And by which they must necessarily trace out our times by the orbiculation of the Rady of the circle of the body of the sun. Again, the World is mentally considered in the mind of an Artist, as in Painting, Graving, Carving, &c. But having thus defined the word, you may think from hence that I am with Democrates Platonissans, acquainting thee with infinity of Worlds, and in his words, Stanza 20. —— and To speak out though I detest the Sect Of Epicurus for their manner vile. Yet what is true I may not well neglect Of truths incorruptible, ne can the stile Of vicious pen her sacred worth defile. If we no more of truth should deign to speak Then what unworthy mouths did never soil, No truths at all 'mongst men would find a place But make them speedy wings, & back to heaven apace. Howsoever thou hast here a field large enough to walk in, which if thou affect the light, thou mayst trace out the truth, and I presume I have done that for thee who art a learner, the most plain ways that were ever published, and have studied not to make it the Art of shadows, so much as the shadow of that art whose Gnomon may be said to touch the Poles, and whose planes may be several Planispheres, a Scale to the Geometrician, a Pole to the Navigator, a Chart to the Geographer, a Zodiaque to the Astronomer, a Table of Houses to the Astrologian, the Meridian and Needle to the Surveyor, a dial to us all, to put us in mind of that precious time which saith to us Fugio, Fuge, and which time shall be swallowed up of Eternity, when there shall be but one day without tropical distinctions, where thou shalt not need helps from any other, nor from me who am thine, S. M. In Solarium. HIc tibi cum numero spectantur Nodus & umbra, Quae tria quid doceant, commemorare libet Umbra notat dextrè quota cursitet hora dici, Hincque monet vitam sic properare tuam Ast in quo signo magni lux publica mundi Versetur mira nodulus arte docet Si vis scire, dies quot quilibet occupet horas, Id numerus media sede locatus habet. On my Friend Mr. Silvanus Morgan, his Book of Dialling. THe use of Dials all men understand; To make them few: & I am one of those. I am not of the mathematic Band: Nor know I more of verse, than verse from Prose. But though nor Diallist I am, nor Poet: I honour those in either do excel; Our Author's skilled in both alike, I know it, Shadows, and Substance, here run parallel. Consider then the pains the Author took, And thank him, as thou benefitest by's Book. Edward Barwick. On the Author and his Book. DAres Zoil or Momus for to carp at thee, And let such idiots as some Authors be Boldly to prosecute or take in hand Such noble subjects they not understand, Only for ostentation, pride, or fame, Or else because they'd get themselves a name, Like that lewd fellow, who with hateful ire, Flinched not, but set Diana's Court on fire: His name will last and be in memory From age to age▪ although for infamy. What more abiding tomb can man invent Then Books, which (if they're good) are permanent And monuments of fame, the which shall last Till the late evening of the World be past: But if erroneous, soothed with virtue's face, Their author's 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'll forbear, lest that my Verses do Belie that praise that's only due to you. Good Wine requires no Bush, and Books will speak Their author's credit, whether strong or weak. W. Leybourn. ERRATA. REader, I having writ this some years since, while I was a child 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 error in Art, as in Chap. 17, which is only true at the time of the equinoctial, 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 sign, and know that Optima prima cadunt, pessimas aeve manent. pag. line Correct. ● 10 Equal lines 18 16 Galaxia 21 1 Galaxia 21 8 Mars▪ 24 12 Scheme 35 1 Hath 38 8 of the Tropics & polar Circles 40 22 AB is 44 31 Artificial 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 centre 126 4 happen 126 6 Toward 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 dial 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 centre 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 end of the line where 9 is 116 I & L omitted on the little Epicicle. 122 THE argument OF THE Praecognita geometrical, and of the Work in general. WHat shall I do? I stand in doubt To show thee to the light; For Momus still will have a flout, And like a satire bite: His Serpentarian tongue will sting, His tongue can be no slander, He's one to wards all that hath a fling His finger's ends hath scanned 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 through, and if then This Momus needs must bite At shadows which dependent 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 find fault with me. How e'er 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 useful to this art. THE PRAECOGNITA GEOMETRICAL. THe Arts, saith Arnobius, are not together with our minds, sent out of the heavenly places, but all are found out on earth, and are in process of time, soft and fair, forged by a continual 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 practical 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 philosopher's atom, such a Nothing, as that it is the very energy of all things, In God it carrieth its extremes from eternity to eternity: in the World it is the same which Moses calls the beginning, and is his Genesis: 'tis 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 'tis the Alpha, and is in the cusp of the Ascendant in every Science, and the house of Life in every operation. Again, a Point is either centrical or excentrical, both which are considered Geometrically or Optically, that is, a point, or a seeming point: a point Geometrically considered is indivisible, and being central 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. 'Tis the birth of any thing, and indeed is to be considered as our principal significator, which being increased doth produce quantity which is the required to Magnitude; for Magnitude is no other than 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 thickness, the form either right or curved. The quantity of a Superficies consisteth in length and breadth, without thickness, the form is divers, either regular or irregular; Regular are Triangles, Squares, Circles, Pentagons, Hexagons, &c. An equilateral Triangle consisteth of three right lines & as many angles, his inscribed side in a Circle contains 120 degrees. A Square of four equal right lines, and as many right angles, and his inscribed side is 90 degrees. A Pentagon consisteth of five equal lines and angles, and his inscribed side is 72 degrees of a Circle. A Hexagon is of six equal lines and angles, and his side within a Circle is 60 degrees, which is equal to the Radius or Semidiameter. An Angle is the meeting of two lines not in a straight concurring, but which being extended will cross each other; but if they will never cross, than 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 itself is the centre 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 Platonic Bodies are, the Tetrahedron, Hexahedron, Octohedron, Dodecahedron, Icosahedron. Tetrahedron is a Solid Body consisting of four equal equilateral 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 equilateral Triangles. A Dodecahedron is a Solid Body consisting of 12 equal Pentagons. An Icosahedron is a Solid Body consisting of 20 equal equilateral 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 mark, from whence draw the line DA, which is perpendicular to AB. And so much shall suffice for the Praecognita geometrical, the philosophical followeth. The end of the Praecognita geometrical. THE argument OF THE Praecognita philosophical. NOt to maintain with nice philosophy, What unto reason seems to be obscure, Or show you things hid in obscurity, Whose grounds are nothing sure. 'Tis not the drift of this my BOOK, The world in two to part, Nor show you things whereon to look But what hath ground by Art. If Art confirm what here you read, Sure you'll confirmed be, If reason wont demonstrate it, Learn somewhere else for me. There's showed to you what shadow is, And the earth's proper place, How it the middle doth possess, And how heavens run their race. Resolving many a Proposition, Which are of use, and needful 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 rendered as hath been, it is not a reality, but a confused imitation of a Body, arising from the objecting of light, So than there can be no other definition than 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 ceiling 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 do condescend to the glory of God, and especially good and lawful Arts. And that shadow may appear to be but dependent on light, it is thus proved, Quod est & existit in se, id non existit in alio: that which is, and subsisteth in itself, that subsisteth not in another: but shadow subsisteth not in itself, 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 annual revolution, or his proper motion that limits our Year? Is it not his tropical distinctions that limits our Seasons? Is it not his diurnal motion that limits out our days and hours? And man truly, that arch type of perfection, hath limited these motions even in the small type of a dial 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? 'tis rendered, Quia omnia reliqua mundi corpora in se includit. I'll 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 pass that it is a Gnomon also to cast a shadow on the body of the Moon far less than itself, and so by consequence a greater light cannot seem to be darkened on a lesser or duller light, and if not darkened, no shadow can appear? But from this common place the World with all its parts, shall we descend to a second grave 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 dispossess 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 heavy things tend, in quo motus terminantur, in which their motion is ended. CHAP III. Showing how the Earth is to be understood to be the centre. A centre 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 centre of magnitude, or a centre of ponderosity, or a centre of rotundity: if it be a seeming point, that is increased or diminished according to the ocular aspect, as being sometime nearer, and sometime farther from the thing in the visual line, the thing is made more or less apparent. A centre 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 centre. A centre of ponderosity is such a point in which an unequal thing hangs in equi libra, in an equal distribution of the weight, though one end be longer or bigger than the other of the quantity of the ponderosity. A centre of rotundity is such a centre as is the centre of a Globe or Circle, being equally distant from all places. Now the earth is to be understood to be such a centre as the centre of a Globe or sphere, being equally distant from the concave superficies of the Firmament, neither is it to be understood to be a centre 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 seem, by the weakness of the sense, to be conceived as a centre 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 celestial 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, do as truly show the hour as if they stood in the centre. CHAP IV. Declaring what reason might move the Philosophers and others to think the Earth to be the centre, 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 instrumental observation, in respect that by the eye it is observed that one place of the sky is semper apparens, neither making cosmical, Haeliacall or Achronicall rising or setting, but still remaining as a point, as it were, immovable, 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 showing why the World may be imagined to be turned on an Axis, the demonstration proving that the earth is the centre, is thus, not in maintaining unlikely arguments, but verity of observation; for all Gnomon 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 all the Gnomon. 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 equal angles of height above these two orisons, namely, at 60 degrees from C to G, and 60 from B to F, the Gnomon 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 centre 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 less the earth out of the centre of the greater, now the sun 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 unequal in length; the reason of which inequality proves that then it did not stand in the centre, and the equality of the other proves that it is in the centre. Hence is also most forceably proved that the earth is completely round in the respect of the heavens, as is showed 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 completely round, as in the ensuing discourse and demonstration is more plainly handled and made manifest. And that the earth is round may appear, first, by the Eclipses, when the shadow of the earth appeareth on the body of the Moon, darkening 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 travel 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 than those that are farther West: Neither do the tops of the highest hills, nor the sinking of the lowest valleys, though they may seem to make the earth uneven, yet compared with the whole greatness, do not at all hinder the roundness of it, and is no bigger than a point or pin's head in comparison of the highest heavens. Thus having run over the system of the greater WORLD, now let us say something 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 sun 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 eternal rest. Lord, what is man that thou shouldest be so mindful 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 rational creature, endowed▪ him with reason, so that his intellect becomes his Primum mobile, to set his action at work, nevertheless, man neither moves nor reigns in himself, and therefore not for himself, but is born not to himself, but for his country; therefore he ought to employ himself in such Arts as may be, and prove to be profitable for his country. Man is the Atlas that supports the Earth, A perfect World, though in a second birth: I know not which the complete World to call, The senseless World, or man the rational: One claims complete in bigness and in birth, Saith she's complete, for man was last brought forth. Man speaks again, and stands in his defence Because he's rational, hath complete sense. Nature now seeing them to disagree, Sought for a means that they united be: Concluded man, that he should guide the spheres, Limit their motion in days, and months, and Years: He thinking now his Office not in vain, Limits the Sun unto a dial plain: Girdles the World in Circles, Zones, and Climes, To show his Art unto the after times. Nature that made him thus complete in all, To please him more, him Microcosmus call, A little world, only in this respect Of quantity, and not for his defect: Pray, Gentle Reader, view but well their feature, Which being done, pray tell me who's the greater? For he hath given me certain knowledge of the things that are, namely to know how the World was made, and the operations of the Elements, the beginning, ending, and midst of times, the alteration of the turning of the Sun, and the change of Seasons, the circuit of years and position of Stars, Wisd. 7. 17. The end of the Praecognita philosophical. THE argument OF THE Praecognita astronomical. YOu're come to see a sight, the World's the stage, Perhaps you'll sayt's but a stargazing age, What come you out to see? one use an Instrument? Can speculation yield you such content? That you can rest in learning but the name Of Pegasus, or of swift Charles's wain? And would you learn to know how he doth move About his axis, set at work by Jove? If you would learn the practice, read and then I need not thus entreat you by my pen To tread in Arts fair steps, or to attain the way, Go on, make haste, Relinquent do not stay: Or will you scale Olympic hills so high? Be sure you take fast hold, astronomy: Then in that fair spread canopy no way From thee is hid, no not Galezia. They that descend the waters deep do see God's wonders in the deep, and what they be: They that contemplate on the starry sky Do see the works that he hath framed so high. Learn first division of the World, and how 'Tis seated, I do come to show you now. THE PRAECOGNITA ASTRONOMICAL. CHAP I. Of the division of the World, by accidental situation of the Circles. COSMUS, the World, is divided by Microcosmus the little World, into substantial and imaginary parts: Now the substantial are those material parts or substance of which the World is compacted and made a Body, by the inter-folding of one sphere within another, as is the sphere of Saturn, Jupiter, Mars, Sol, &c. And these of themselves have a gentle and proper motion, but by violence of the first mover, have a racked motion contrary to their own proper motion: whence it appears, that the motion of the heavens are two, one proper to the spheres as they are different in themselves, the other common to all. By Phoebus' motion plainly doth appear, How many days do constitute one year. Will you know how many days do constitute a year, he telleth you who saith, Ter centum ter viginti cum quinque diebus Sex horas, neque plus integer Annus habet. Three hundred sixty five days, as appear, With six hours added, make a complete Year. The just period of the sun's proper revolution. Perpetuus Solis distinguit tempora motus. The Imaginary part traced out by man's imagination, are Circles, such is the Horizon, the Equator, the Meridian, these Circles have of themselves no proper motion, but by alteration of place have an accidental division, dividing the World into a right sphere, cutting the parallels of the Sun equally or oblique, making unequal days and nights: whence two observations arise: First, Where the parallels of the Sun are cut equally, there is also the days and nights equal. Secondly, Where they are cut oblique, there also the days and nights are unequal. The variety of the heavens are diversely divided into spheres, or several Orbs, and as the Poets have found out a Galazia, the milky way of Juno her breasts, or the way by which the gods go to their Palaces, so they will assign to each sphere his several god. Goddess of heralds. Calliope in the highest spheres doth dwell, astrology. Amongst the Stars Urania doth excel, philosophy. Polimnia, the sphere of Saturn guides, gladness, Sterpsicore with Jupiter abides. history, And Clio reigneth in man's fixed sphere. Tragedic. Melpomene guides him that guides the year Solace. Yea, and Erata doth fair Venus' sway. Loud Instruments. Mercury his orb Euturpe doth obey. Ditty. And horned Cynthia is become the Court Of Thalia to sing and laugh at sport. Where they take their places as they come in order. The sphere is said to be right where the Poles have no elevation, but lie in the Horizon, so that to them the equinoctial is in the Zenith, that is, the point just over their heads. The sphere is oblique in regard of its accidental division, accidentally divided in regard of its orbicular form; orbicular in regard of its accidental, equal variation orbicular, it appears before in the Praecognita philosophical, his equal variation is seen by the equal proportion of the earth answering to a celestial degree, for Circles are in proportion one to another, and parallel one to another are cut equally, so is the earth to the heavens; having considered them as before, we will now consider another sort of sphere, which is called parallel. This parallel sphere is so that the parallels of the Sun are parallel to the Horizon, having the Poles in their Zenith, being the extreme intemperate, cold, and frozen Zone: Ovid in his banishment complains thus thereof. Hard is the fright in Scythia I sustain, Over my head heaven's Axis doth remain. CHAP II. Of the Circles of the Horizon, the Equator, and the Meridian. THe greatest Circle of a sphere is that which divides it in two equal parts, and that because it crosseth diametrically, and the diameter is the longest line as can be struck in a Circle, and therefore the greatest, which great Circles are represented in the following figure, representing the Circles of a sphere in an oblique Latitude, according to the Latitude or elevation of the Pole here at London, which is 51 deg. 32 min. being North Latitude, because the North Pole is elevated. The Horizon is a great Circle dividing the part of heaven seen, from where we imagine an Antipodes, the inhabitants being to us an Antipheristasin, our direct opposites, so that while the Sun continues visible to us, it is above our Horizon, and so continues day with us, while it is night with our opposites; and when the Sun goes down with us it appears to 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, than 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 sphere. The Meridian is a great Circle situated 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 steadfastly fixed, it is represented by the great Circle marked NDSC, and is accidentally divided, if we travel East or West, but in travailing North or South altereth not, & when the Sun touches this Circle, it is then midday or Noon: Now if you imagine a Circle to pass 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 hours. The equinoctial likewise divides the World in two equal 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 days and nights are then equal, and to those that live under this Citcle the days and nights hang in equilibra continually, and the Sun doth move every hour 15 degrees of this Circle, making the hour lines equal, passing 15 degrees in one hour, 30 degrees in two hours, 45 degrees in three hours, 60 degrees for four, and so increasing 15 degrees as you increase in hours. This I note to the intent you may know my meaning at such time as I shall have occasion ro mention the equinoctial distances. The Axis of the World is that which the style in every dial represents, being a line imaginary, supposed to pass through the centre 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 sphere having no elevation, in an oblique to be elevated above the Horizon and the angle at the centre, numbered on the arch of the Meridian between the apparent Pole and the Horizon, is the elevation thereof, represented by the straight 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 do attend the Arctic or North Pole, are the greater and lesser bear, the last star in the lesser Bears tale is called the Pole Star, by reason of its nearness to it: this is the guide of Mariners, as appeareth by Ovid in his exile, thus You great and lesser Bear whose Stars do guide Sydonian and Grecian ships that glide Even you whose Poles do view this lesser Ball, Under the Western Sea near 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 several sorts of Planes, and how they are known. Dial's are the days limiters, and the bounders of time, whereof there are three sorts: horizontal, Erect, Inclining: horizontal are always 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 compass shall explain the demonstration: Now if you imagine Circles to pass through the Zenith A, crossing the Horizon in his opposite points, as from SW through the vertical point A, passing to the opposite point of South-west to North-East, those, or the like circles, are called azimuths, parallel to which azimuths all erect Sciothericals' do stand. Those Planes that lie parallel to the horizontal Circle are called horizontal planes, and his Style makes an angle with the Pole equal 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 vertical 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 Cardinal 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 cross the Meridian due East and West, and the Poles are their Styles, equally elevated according to the equinoctial altitude, being the compliment 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 less than a right angle, and a right angle is 90 degrees: these declining Planes lying in some accidental azimuth. For supposing a dial to turn from the South or North towards the East or West, the Meridian line of the South declines Eastward, happening in these azimuths 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 Southeast 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, do happen in these azimuths 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 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 compass 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 dial 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 equinoctial planes, as infinite more according to the inclination or reclination of the plane, or they are as erect planes do become declining recliners, which look oblique to the Cardinal 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 equinoctial 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 equinoctial, 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 accidental Azimuth, and so not beholding one of the Cardinal Quarters is said to decline: Again, the said plain may happen not to stand vertical, 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 vertical 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. Showing the finding out of a Meridian Line after many ways, 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 coming 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 situation to their points respective, as in the definition of this Circle I showed there was accidental 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 shows why noon is sooner to some then others. The Meridian may be found divers ways, as most commonly by the mariner's compass, but by reason the needle hath a point attractive subject to error, and so overthroweth the labour, I cease to speak any further. It may be found in the night, for when the star called Aliot, seems to be over the polestar, 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 book 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 centre 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; do 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 half, then from that middle point to the centre draw a line which is a true Meridian. Or thus, Draw a great many Circles concentrical one within another, then observe by the Circles about noon 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 will cast its shadow from a line, and so my eye may deceive me, when as by Circles the Sun casting shadow round about, still meets 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 nearest the vertical 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 find 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 horizontal line, and then draw a perpendicular to that, and in the perpendicular line strike a Style or small wire 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 azimuth reckoned from the South, for that is the true declination of the wall from the South. The distance of the azimuths from the South, or other points, are mentioned in degrees and minutes in the third Chapter, in the definition of the several sorts of planes: or by holding the straight 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 third and plummet in your hand at E, the Sun shining, and it cast shadow the line of, and at the same instant take the altitude of the Sun, thereby getting the azimuth as is taught following, then from the point F, as the centre of the Horizon., and from the line FE, reckon the distance of the South, which suppose I find the azimuth to be 60 degrees from the East or West, by the propositions that are delivered in the end of this book, and because there is a Quadrant of a Circle between the South, and the East or West points, I subtract the distance of the azimuth from 90 degrees, and it shall leave 30, which is the declination of the wall, equal to the angle EFG: but to find the inclination or reclination, I shall show when I come to the use of the universal Quadrant, or having first found the Meridian line, you may prick down the azimuth. CHAP V. Showing what hourlines may be drawn upon any Plane. LIght being the cause primary of shadows, shadows being but the imitation of the secondary cause, that is substance, doth delineate unto us the passing away of time, by receiving light on the substance casting shadow. The Sun, though he never moves from the line ecliptic wherein he hath his annual or yearly motion, yet have a declination from the equinoctial North or South, making his diurnal or daily motion, altering the days and nights according to all the diversities thereof: for the Sun being in the equinoctial hath no declination, but in his diurnal motion still declining from the equinoctial makes his progress towards the North or South, describeth many parallel Circles, being parallel to the equinoctial, whose farthest distance from either side is 23 deg. 30 minutes, so that so many degrees that the Sun is distant from the equinoctial, so much is its declination. Now if you imagine the Circle before described to represent the Meridian Circle which crossed diametrically, which diameter shall represent the equinoctial, then laying down the greatest declination, on either side of it, drawing two lines at that distance, on either side of the equinoctial, parallel to it, represent the Tropics, the upper representing the tropic of Cancer, marked with GE, the other the tropic of Capricorn, marked with HI: and if from each several degree you draw parallels too, they do represent the parallels of the Sun, which shall show the diurnal motion of the Sun: now if you cross these parallels with a line from E to H, that then represents the ecliptic; now if you cross the equinoctial at right angles with another line, that line represents the Axis of the World: then if you lay down from the Poles the elevation thereof, to wit, the North and South Poles, according to the elevation of the North Pole downward, where the number of degrees end make a mark; then account the same elevation from the South Pole upward, and there also make a mark, from which two marks draw a right line, which shall represent your Horizon, and cuts the parallels of the Sun according to the time of his abiding above the Horizon. First, An East and West dial lies parallel to the Meridian, therefore the Sun in the Meridian cannot shine on them; nevertheless, though an East and West dial cannot have the hour of 12 on it, yet an East or West position may, because it crosseth the Horizon in the North and South. Secondly, A direct North dial can have but morning and evening hours on it, and then of no use but when the Sun hath North declination, for then his Amplitude or distance from the East and West is Northward, and so at morning or night shines on the face thereof. Thirdly, A North reclining may show all the hours all the year, if it recline from the North Southward, the quantity of the compliment of the least Meridian altitude, but if but the compliment of the elevation of the equinoctial, and so become a Polar Plane, it can then but show while the Sun is in the North signs, for the dial lying parallel to the equinoctial while the Sun is in South declination cannot shine on the plane because it lies under. All upright planes declining from the South may have the hour line of 12, so also may all North decliners, but not in the Temperate Zone, which is contained between the degrees. South incliners also may have the line of 12, whose upper face is not below the least Meridian altitude, as also if greater than the greatest Meridian altitude, than doth the upper face want it. Fifthly, all North recliners reclining more than the greatest meridian altitudes compliment, may have all the hours but will show but one part of the year. Sixthly, All South declinets or recliners may have the line of 12 on them. And now having proceeded thus far in some theorical demonstration or grounds of Dials for the geometrical projection, we will in the next Chapter lay down the theorical demonstration for the arithmetical Calculation, and so proceed to our practical way of operation as ensueth. CHAP VI. Being the definition of the several lines of Sines, Tangents, and Secants, to be understood before we can come to arithmetical Calculation. A Tangent is a right line without the periphery to the extremity of the Secant to the Radius being perpendicular eraised, such is represented by the line BC. A Secant is a right line drawn from the centre through the circumference to the Tangent, such is represented by the line AB, the Semidiameter of the same Circle is called the Radius. You may furthermore for very convenient uses have those lines placed on a Ruler, for if from one degree of one Quadrant of a Semicircle you draw lines to the same degree of the other Quadrant, cutting the line GA, that line so cut shall be a line of Sines, and if from the centre you draw lines to the Tangent line through every degree of the Quadrant, that line so cut is a Tangent line, whose use is most exquisite and infinite for the solution of many excellent propositions. CHAP VII. Being the fundamental Diagram for the geometrical projection of Dials. THe Style being the representation of the Axis of the World, doth become the Gnomon or substance casting shadow on all Planes lying parallel to some Circle or other, as to circles of azimuths in all vertical Dials. So that the figure following is a representation of divers semidiameters, doth plainly show the theorical ground of the practic part hereof. Where the line in the demonstration, noted the semidiameter of the Horizon, signifies the Horizon, for so supposing it to represent an horizontal dial, the style or Axis must be elevated above it, according to the Poles elevation above the Horizon, and then the semidiameter or Axis of the World represents the style or Axis casting shadow being the line AC. The geometrical projection of Dials. Where note by the way, that if you set one foot of the Compasses in B, and with the Semidiameter of the Equator, fix the other foot in the line BC, keeping that last foot fast, and at that centre draw a Quadrant divided into six parts, & a ruler from the centre of the Equator through each division, shall divide the line AB as a contingent line, and if from C to these marks on the line AB you draw lines, it shall be the hour lines of a vertical dial. But supposing a dial to stand vertical, or upright to the Horizon AB, as the line BC, then that is represented by the semidiameter of the vertical, and his style again represented by the semidiameter or Axis AC, being distant from the vertical equal to the compliment of the Poles elevation, and again, the equinoctial crossing the Axis at right angles, the semidiameter thereof is represented by the line BD, the reason why the angle at A hath to his opposite angle at C, the compliment of the angle at A, is grounded on this, the three angles of any right lined triangle are equal to two right angles, and a right angle consists of 90 degrees: now the angle at B is 90 degrees, being one right angle, and the angle at A being an angle of 51 degrees, which wants of 90 39 degrees, which is the angle at C, all which being added together do make 180 degrees, being two right angles: here you see that having two angles, the third is the compliment of 180 degrees. CHAP VIII. Of the proportion of shadows to their Bodies. SEeing the Zenith makes right angles with the Horizon, and a right angle consisteth of 90 degrees, the middle point betwixt both is 45 degrees, the Sun being at that height, the shadow of all things perpendicularly raised, are equal to their bodies, so also is the Radius of a Circle equal to the Tangent of 45 degrees: and if the sun be lower than 45 degrees it must necessary follow the shadow must exceed the substance, because the Sun is nigh the Horizon, and this is called the adverse or contrary shadow. Contrarily, if the Sun exceed this middle point, the substance than exceeds the shadow, because the Sun is nearer the vertical point. Mr. Diggs in his Pantometria laying down the manifold uses of his Quadrant geometrical, doth there show, that having received the Sun beams through the Pinacides or Sights, that when the sun's altitude cuts the parts of right shadow, than the shadow exceeds the substance erected casting shadow as 12 exceeds the parts cut: But in contrary shadow contrary effects. CHAP ix.. To find the Declination of the Sun. TO give you Orontius his words, it is convenient to take the beginning from the greatest obliquation of the Sun, because on that almost the whole harmony of all astronomical matters seem to depend, as shall be manifest from the discourse of the succeeding Canons. Wherefore prepare of commodious and elect substance, a Quadrant of a Circle parted into 90 equal parts, on whose right angled Radius let be placed two pinacides or sights to receive the beams of the Sun. Then erect it toward the South in the time of the Solsticials, either in Cancer the highest annual Almicanther, or in Capricorn the lowest annual▪ meridian altitude, also observe the equilibra, or equality of day and night in the time of the equinoctialss, from the Meridian altitude thereof subtract the least Meridian altitude, which is, when the Sun enters in the first minute of Capricorn, the remainder is the Declination, or subtract the equinoctial altitude from the greatest Meridian altitude, the remainder is the Declination of the greatest obliquity of the Sun in the Zodiaque. The height of the Sun is observed by the Quadrant when the beams are received through the sights by a plummet proceeding from the centre, noting the degree of altitude by the third falling thereon. You may also take notice that for the continual variation of the Suns greatest declination it ought to be observed by faithful Instruments: for as Orontius notes that Claudius, Ptolemy found it to be 23 degrees 51 minutes and 20 seconds, but in the time of Albatigine the same number of degrees yet but 35 minutes, Alcmeon found it of little less, to wit 33 minutes, Purbachis and some of his Disciples do affirm the same to be 23 degrees only 28 minutes, yet Johanes Regiomontan. in the tables of Directions, hath allotted the minutes to be 30, but since Dominick Maria an Italian, and Johannes Varner of Norimburg testify to have found it to be 29 minutes, to which observation our works do exactly agree. Albeit all did observe the same well near by like Instruments, nevertheless, not justly by exact construction, or by insufficient dexterity of observation some small difference might happen, but not so much as from Ptolemy to our time. Having this greatest Declination, to find the present Declination is thus, by calculation: As the Radius, is to the Sine of the greatest Declination; so is the Sine of the sun's distance from the next equinoctial point, that is Aries or Libra, to the declination required: wherefore in the natural Sines, as in the Rule of Proportion, multiply the second by the third, divide by the first, the Quotient is the Sine of the Declination. Or by the natural Sines, add the second and third, and subtract the first, the remainder is the Sine of the present Declination. Degree. ♈ ♎ ♉ ♏ ♊ ♐ Degree. D m D m D m 0 0 0 11 29 20 10 30 1 0 24 11 50 20 23 29 2 0 47 12 11 20 35 28 3 1 11 12 31 20 47 27 4 1 35 12 52 20 58 26 5 1 59 13 12 21 9 25 6 2 23 13 32 21 20 24 7 2 47 13 52 21 30 23 8 3 10 14 11 21 40 22 9 3 34 14 30 21 49 21 10 3 58 14 50 21 58 20 11 4 21 15 8 22 7 19 12 4 45 15 27 22 15 18 13 5 8 15 45 22 23 17 14 5 31 16 3 22 30 16 15 5 55 16 21 22 37 15 16 6 18 16 38 22 43 14 17 6 41 16 56 22 50 13 18 7 4 17 12 22 55 12 19 7 27 17 29 23 0 11 20 7 49 17 45 23 5 10 21 8 12 18 1 23 9 9 22 8 34 18 17 23 13 8 23 8 57 18 32 23 17 7 24 9 19 18 47 23 20 6 25 9 41 19 2 23 22 5 26 10 3 19 16 23 24 4 27 10 25 19 30 23 26 3 28 10 46 19 44 23 27 2 29 11 8 19 57 23 27 1 30 11 29 20 10 23 28 0 De ♓ ♍ ♒ ♌ ♑ ♋ De But I have here added a Table of Declination of the part of the ecliptic from the equinoctial, the use whereof you may discern is very plain, for if you find the sign on the top, and the degrees downward, the common angle shall be the Declination of the Sun that day. As if the Sun being in the 10 degree of Taurus or Scorpio, the declination shall be 14 degrees 50 minutes, and if you find the sign in the bottom, you shall seek the degrees on the right hand upward, so the 20 degree of Leo or Aquarius hath the same declination with the former. The end of the Praecognita astronomical. THE argument OF practical Sciothericy Optical. REader, read this, for I dare this defend, Thy posting life on Dials doth depend, Consider thou how quick the hour's gone, Alive to day, to morrow life is done: Then use thy time, and always bear in mind, Times hary forehead, yet he's balled behind, Here's that that will deline to thee and show How quick time runs, how fast thy life doth go: Yet (festina lente) learn the praecognit part, And so attain to practice of this art, Whereby you shall be able for to trace Out such a path, where Sol shall run his race, And make the greater Cosmus to appear, Delineating day and time of year. Horologium Vitae. Latus ad occasum, nunquam rediturus ad ortum Vivo hodie, moriar cras, here natus eram. HOROLOGIOGRAPHIA OPTICA. CHAP I. Showing the making of an horizontal plane to an Oblique sphere. FRom the theorical Demonstration before, take the semidiameter of the Horizon with your Compasses, then draw the line AB, representing the Meridian or line of 12, and setting one foot in A, describe the Quadrant CAB, and CA must be at right angles to AB, to which Quadrant draw the tangent line FA, which is the line of contingence, then take from the Theorical demonstration the semidiameter of the Aequator, and placing that on the line AB desctibe a quadrant touching the line of contingence also within the other, represented by the Quadrant H e I which divide into 6 parts, and a Ruler laid to the centre e, make marks where the Ruler toucheth the line of contingence, which must be continued beyond F, that so the hour lines may meet with the line BF, where it crosseth that line make marks: then removing the Ruler to the centre A of the horizontal Semicircle, draw lines through each mark of the line of contingence which shall be the hours, number the morning hours from the Meridian towards your left hand, and evening or afternoon hours towards the right. The Style must be an angle equal to the elevation of the Pole, the 12 hour must lie under the Meridian Circle. The arithmetical Calculation. As the Radius, Is to the Tangent of the equinoctial distance of the hour from the Meridian; So is the sign of the elevation of the Pole, To the Tangent of the hour's distance from the Meridian. The definition of the equinoctial distance is in the definition of the equinoctial Circle, Chap. 1. Praecognita astronomical. The figure of an horizontal dial, for the Latitude of London 51d. 30m. South The hours of the afternoon must be the same distance from the Meridian, 1 and 11, 2 and 10, 3 and 9, and so of the rest, this is very plain, neither wants any expositor, only you may on the Horizontal plane, prick down beyond the hour of 6 a clock, the morning hours of 4 and 5, and the evening hours of 7 and 8, by reason that the Sun will shine on the horizontal plane as soon as it is above the Horizon. The figure of a South vertical plane, for the Latitude of London, which is parallel to the Prime vertical. The semidiameter of the vertical is but the Tangent of the elevation of the Pole to the Radius of the Horizon. And the semidiameter of the Horizon, the Tangent of the elevation of the Equator to the Radius of the vertical. CHAP II. Showing the making of a direct vertical dial for an Oblique sphere, that is, a direct North or South dial plane. EVery plane hath a vertical point, and for the making of a vertical dial for the Latitude of London, out of the theorical demonstration Chap. 7. praecog.. Astron▪ take the semidiameter of the vertical, and with that, as with the semidiameter of the Horizon, describe a Quadrant, & draw the tangent line FG, and with the semidiameter of the Aequator finish all as in the horizontal: the Style must proceed from the centre A, and be elevated from the Meridian line of, so much as is the compliment of the Elevation of the Pole, and must point toward the invisible Pole, viz. the South Pole, and hath but 12 hours on it. The arithmetical Calculation. As the Radius, Is to the tangent of the equinoctial distance of the hour from the Meridian; So is the cousin, that is, the compliment sine of the elevation, to the tangent of the hour distance from the Meridian required. CHAP III. Showing the making of a direct North vertical dial for an Oblique sphere, as also a more easy way of drawing the South or horizontal Planes. THe North dial is but the back side of the South dial; and differeth little from it, but in naming of the hours, for accounting the sixth hour from the Meridian in the direct South vertical, to be the same in the direct North vertical, and accounting the first hours on the East side of the South, on the West side of the North plane, and so vice versa, the first hours on the West side of the South, on the East side of the North plane, as by the figure appeareth. And because the North Pole is elevated, the Style must point up toward it the visible Pole. It must have but the first and last hours of the South plane, because the Sun never shines but at evening or morning on a North wall in an oblique sphere, and but in summer, because then the Sun hath North Declination, but in a right sphere, it may show all the hours as a South dial, but for a season of the year. But if you will make the vertical plane or horizontal in a long angled Parallelogram, you shall take the Secant of the elevation of the Pole, which is the same with AC in the fundamental Diagram, and make that your Meridian line, and shall take the Sine of the elevation of the Pole above the Meridian, which in a direct South or North is equal to the elevation of the equinoctial, and in the fundamental Diagram is the line DE, and prick it down from A and C at right angles with the line AC, and so enclose the long square BADBCD, it shall be the bounds of a direct North or South dial; lastly, if from the fundamental diagram you prick down the several tangents of 15, 30 45, from Band D on the lines BB and DD, & the same distances from C toward B and D, & lastly if from the centre A, you draw lines to every one of those marks, they shall be the hourlines of an erect direct South dial. To make an horizontal dial by the same projection you shall take the Secant of 38 deg. 30 min. the elevation of the Equator, which in the fundamental Scheme is the line of, for the Meridian, and the Sine of the elevation of the Pole, which in the fundamental diagram is the same with DA, and prick that down from the Meridian at right angles both ways, as in the former planes, and so proceed as before from the six of clock hour and the Meridian, with the several Tangents of 15, 30, 45, you shall have constituted a horizontal plane. I have caused the pricked line that goes cross, and the other pricked lines which are above the hour line of six, to be drawn only to save the making of a figure for the North direct dial, which is presented to you if you turn the Book upside down, by this figure, contained between the figures of 4, 5, 6, the morning hours, 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 centre 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 arithmetical calculation is the same with the former, also a North plane may show all the hours of the South by consideration of reflection: For by optical 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 find it. The Figure of a direct East and West dial for the Latitude of London,▪ 51 deg. 30 min. East dial. West dial. CHAP IV. Showing the making of the Prime vertical planes, that is, a direct East or West dial. FOr the effecting of this dial, 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 hours are marked: divide the Equator in the lower semicircle in 12 equal parts, then apply a ruler to the centre, 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 hours, and at the end of those lines mark the Easterly hours from 6 to 11, and of the West from 1 to 6. These planes, as I told you, want the Meridian hour, because it is parallel to the Meridian. Now for the placing of the East dial, 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 dial number the elevation to B; fasten a plummet and third in the centre A, and hold it so that the plummet may fall on the line AC for the East dial, and AB for the West dial, 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 hour's distance is ●●t the Tangent of the equinoctial distance. The arithmetical Calculation. 1 Having drawn a line for the hour of 6, whether East or West, As the tangent of the hour distance, is to the Radius, so is the distance of the hour 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 hour distance from 6, to the distance of the same hour from the substyle. The style must be equal in height to the semidiameter of the Equator, and fixed on the line of 6. CHAP V. Showing the making a direct parallel Polar plane, or opposite equinoctial. I Call this a direct parallel Polar plane for this cause, because all planes may be called by their situation of their Poles, and so an equinoctial 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 equal to the semidiameter of the Equator, as in the East and West Dials, so likewise these hours are Tangents to the Equator. Arithmetical calculation. Draw first a line representing the Meridian, or 12 a clock line, and another parallel to the said line for some hour which may have place on the line, say, As the tangent of that hour is to the Radius, so is the distance of that hour 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 hour, to the distance of that hour from the Meridian. CHAP VI. Showing the making of a direct opposite polar plane, or parallel equinoctial dial. AN equinoctial plane lieth parallel to the equinoctial 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 centre at right angles to the plane. As the Radins, is to the sine of declination, so is the cotangent 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 hour lines of a parallel polar plane, now if you draw to each opposite from the pricked lines, those lines shall be the hour lines of the former plane. CHAP VII. Showing the making of an erect vertical declining dial. IF you will work by the fundamental Diagram, you shall first draw a line, such is the line AB, representing the Meridian, then shall you take out of the fundamental diagram the Secant of the Latitude, viz. AC, and prick it down from A to B, and at B you shall draw a horizontal line at right angles, such is the line CD, than you shall continue the line AB toward i, and from that line, and where the line AB crosseth in CD, describe an arch equal 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 compliment of the Latitude, viz. AG in the fundamental Diagram, and the Sine of 51 degrees, viz. DA, which is all one with the semidiameter of the Equator, and therewithal 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 several 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 horizontal line CD, and where they end on that line to each point draw the hour lines from the point A, which plane in our example is a vertical 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 geometrical example to prick down a declining dial in a right angled parallelogram. Now for the arithmetical calculation, the first operation shall be thus: As the Radius, to the cotangent of the elevation, so is the sine of the declination, to the tangent of the substile's distance from the meridian of the place. then, II Operation. Having the compliment of the declination and elevation, find the styles height above the sub-stile, thus, As the Radius, to the cousin of the declination, so the cousin of the elevation, to the sine of the styles height above the substyle. III Operation. As the sine of elevation, is to the Radius, so the tangent of declination, to the tangent of the inclination of the Meridian of the plane to the Meridian of the place. IV Operation. Having the styles height above the substyle, and the angle at the pole comprehended between the hour given and the meridian of the plane say. As the Radius, to the sine of the styles height above the substyle; so is the tangent of the angle at the pole, comprehended between the hour given and the meridian of the plane, to the tangent of the hour distance from the substyle. Thus the arithmetical way being laid down, another geometrical follows. YOu shall first on the semidiameter of the Horizon, viz. AB, describe the arch BC the declination of the plane, and BD the compliment of the elevation of the pole, then shall you draw the lines AC and AD, and at B you shall raise the perpendicular DCB. The Figure of an upright plane declining from the South Eastward 30 degrees. Now good Reader, labour to understand my plain meaning in this, labouring only not to confound thy memory or capacity, & therefore give you also to understand that such are the hour distances of a Westerly declining plane, as are those of an Easterly, only changing the side of the plane, and naming it by the complemental hours, the complemental hours I call those that added together make 12, as followeth. Forenoon hours of the declining East plane. 6 Complimental hours are 6 are afternoon hours of a declining West plane. 7 5 8 4 9 3 10 2 11 1 So that if the hours of the Easterly declining plane be 6, 7, 8, 9, 10, 11, 12, 1, 2, 3, the hours of the Westerly declining dial is 6, 5, 4, 3, 2, 1, 12, 11, 10, 9, still keeping the same distances of the hour lines in one as the other, so that if an Easterly declining be but turned the back side, it represents a Westerly declining Dial as much, and the style must stand over his substyle, and whereabouts the hour lines are closest or nearest together, thereabout is the substyle. Now having showed you the making of all horizontal and vertical, whether direct or declining, Polar or equinoctial, I shall proceed to show the projecting of those which are oblique, whether declining reclining, or inclining, reclining, &c. whereto, for the more ease, I have calculated to every degree of a Quadrant the hour arches of the horizontal planes, from one degree of elevation till the Pole is in the Zenith. The Table and use followeth in several Chapters. Here followeth the Table of the arches of the hour lines distance from the Meridian in all orisons, from one degree of elevation, till the Pole is elevated 30 degrees, by which is made all direct mural, whether upright, or reclining Dials. 1 11 2 10 3 9 4 8 5 7 6 6 1 0 16 0 35 1 00 1 44 3 43 2 0 32 1 9 2 00 3 27 7 25 3 0 48 1 44 3 00 5 11 11 3 4 1 5 2 19 4 00 6 54 14 36 5 1 20 2 53 4 59 8 65 18 1 6 1 36 3 27 5 58 10 16 21 19 7 1 52 4 1 6 57 11 55 24 27 8 2 8 4 35 7 54 15 9 30 3 9 2 24 5 9 8 54 15 9 30 3 10 2 40 5 44 9 51 16 45 32 57 11 2 56 6 18 10 48 18 18 35 27 12 3 11 6 51 11 44 19 48 37 49 13 2 27 7 24 12 41 21 17 40 1 14 3 46 7 57 13 36 22 44 42 5 15 3 59 8 30 14 31 24 9 44 0 16 4 14 9 2 15 25 25 31 45 49 17 4 28 9 35 16 17 26 51 47 30 18 4 44 10 7 17 10 28 9 49 4 19 5 15 11 10 18 53 30 39 51 55 20 5 15 11 10 18 53 30 39 51 55 21 5 29 11 41 19 43 31 50 53 13 22 5 44 12 13 21 20 34 5 55 34 23 5 58 12 43 21 20 34 5 55 34 24 6 13 13 13 22 8 35 10 56 37 25 6 28 13 43 22 54 36 12 57 37 26 6 42 14 12 23 40 37 13 58 34 27 6 57 14 41 24 25 38 11 59 27 28 7 10 15 00 25 9 39 11 60 37 29 7 24 15 39 25 52 40 2 61 4 30 7 38 16 6 26 36 40 54 61 49 The continuation of the arches of the horizontal planes, from 30 to 60 deg. of elevation of the Pole 1 11 2 10 3 9 4 8 5 7 6 6 31 7 51 16 34 27 15 41 44 62 30 32 8 5 17 1 27 55 42 32 63 11 33 8 19 17 27 28 37 43 20 63 49 34 8 31 17 54 29 13 44 5 64 24 35 8 44 18 20 29 50 44 49 64 58 36 8 57 18 45 30 27 45 31 65 30 37 9 10 19 9 31 2 46 12 66 ●0 38 9 22 19 34 31 37 46 50 66 29 39 9 24 19 58 32 11 47 28 66 56 40 9 47 20 22 32 44 48 4 67 23 41 9 58 20 45 33 16 48 39 67 47 42 10 10 21 7 33 47 49 13 68 10 43 10 21 21 30 34 18 49 45 68 33 44 10 32 21 51 34 47 50 16 68 55 45 10 44 21 45 35 16 50 46 69 15 46 10 54 22 33 35 53 51 15 69 34 47 11 6 22 54 36 11 51 43 69 53 48 11 16 23 14 36 37 52 9 70 11 49 11 26 23 33 37 2 52 35 70 27 50 11 36 23 51 37 27 53 0 70 43 51 11 46 24 10 37 52 53 24 71 13 52 11 55 24 57 38 14 53 46 71 24 53 12 5 24 45 38 37 54 8 71 27 54 12 14 25 2 38 58 54 30 71 40 55 12 23 25 19 39 19 54 49 71 53 56 12 32 25 35 39 39 55 9 72 5 57 12 40 25 51 39 59 55 28 72 17 58 12 48 26 5 40 18 55 45 72 28 59 12 56 26 20 40 36 56 2 72 39 60 13 4 26 33 40 54 56 19 72 49 The continuation of the arches of the horizontal planes, from 60 deg. of elevation, till the Pole is in the Zenith. 1 11 2 10 3 9 4 8 5 7 6 6 61 13 11 26 48 41 10 56 34 72 58 62 13 18 27 1 41 26 56 49 73 7 63 13 25 27 13 41 42 57 3 73 16 64 13 32 27 26 41 57 57 17 73 24 65 13 39 27 37 42 11 57 30 73 32 66 13 45 27 49 42 25 57 42 73 39 67 13 52 27 59 42 38 57 54 73 46 68 13 56 28 9 42 50 58 5 73 53 69 14 3 28 19 43 2 58 16 73 59 70 14 8 28 29 43 13 58 26 74 5 71 14 13 28 38 43 24 58 36 74 11 72 14 18 28 46 43 34 58 44 74 16 73 14 22 28 55 43 43 58 53 74 21 74 14 26 29 2 43 52 59 1 74 25 75 14 30 29 9 44 0 59 8 74 29 76 14 34 29 5 44 8 59 15 74 34 77 14 38 29 22 44 15 59 21 74 37 78 14 41 29 27 44 22 59 26 74 41 79 14 44 29 32 44 28 59 32 74 44 80 14 47 29 37 44 34 59 37 74 47 81 14 49 29 42 44 39 59 41 74 49 82 14 51 29 44 44 43 59 43 74 50 83 14 53 29 49 44 47 59 49 74 53 84 14 55 29 52 44 51 59 52 74 55 85 14 57 29 54 44 53 59 54 74 57 86 14 58 29 56 44 56 59 57 74 58 87 14 59 29 58 44 58 59 58 74 59 88 14 59 29 59 44 59 59 58 74 59 89 14 59 30 44 59 59 59 75 90 15 30 45 60 75 CHAP VIII. Showing the use of this Table both in vertical and horizontal planes. FOr an horizontal dial enter the Table with the elevation of the Pole on the left hand, and the arches noted against the hours and the elevation found, are the distance of the hours from the Meridian. For a vertical or direct South or North, enter the Table with the compliment of the elevation on the right side, and the common meeting of the hours at top, and the compliment of elevation, is the distance of the hours from the Meridian in the said plane. For every horizontal plane is a direct vertical in that place whose Latitude or distance of their Zenith from the Aequator, is equal to the compliment of the elevation of the horizontal planes Axis or style. As to make an horizontal dial for the Latitude of 51 degrees, I enter the Table and find these Arches for 1 and 11, for 2 and 10, &c. Now the same distances are the distances of the hour lines of a direct South plane, where the Pole is elevated the compliment of 51 degrees, that is 39 degrees, for 51 and 39 together do make 90. So to make a vertical dial, I enter the Table with 39, the compliment of the elevation of the pole, and find the arches answering to 1 and 11, to 2 and 10, &c. Thus much in general of the use of the Table, now followeth the use in special. CHAP ix.. Showing the use of the Tables in making any Declining or Inclining direct Dials. LEt the great Circle ABCD represent the Meridian, A the North, and C the South, than the line of represents a South reclining plane, while it falls back from the South Northward, and represents an inclining plane while it respects the Horizon. This is sufficiently discussed before. So much as the plane reclines northward beyond the compliment of the elevation of the Pole, so much is the North pole elevated above the plane, as here the plane is represented by of, the elevation of the style or Axis the arch EG, therefore in this case subtract the compliment of the reclination of the plane from the elevation of the elevated pole, and the remainder is the arch of the poles elevation above the plane, with which elevation enter the Table in the left margin, and there are the hour arches from the meridian. If the reclination of the plane be less than the compliment, as is IK, subtract the arch of reclination from the compliment of the elevation, there is left the elevation of the South pole above the plane, and with the compliment of the elevation of the pole above the plane enter the table on the right margin, and there shall you find the distance of the hours: and herein Mr. Faile failed, for instead of subtracting one from the other, he addeth one to another, causing a great error. The distance of every hour of the North incliner on the back side of the South incliner as much are equal, saving that the hours on the North side must be named by the compliment hours to 12, and as the North pole is above one plane, so is the South pole above the other, you may also conceive the like in making of all South incliners and recliners, by framing the position of the plane on the South side as the figure is on the North: and in North recliners less than the elevation of the pole, add the reclination of the flat, which is the elevation of the North pole above the plane: herein Mr. Fail failed also, as depending on the former, following the doctrine of contraries, which foremost well examined would have saved the opening of a gap to this second error: With the said elevation found enter the Table for the Horizontal arches, and thereby make a horizontal▪ plane as is showed, so is the dial also prepared. If it recline that it lie between the Horizon and the Equator, then to the elevation of the Pole add the compliment of the reclination, which is the height of the style above the plane, and finish it as a horizontal plane for that latitude, and not as a vertical, as Mr. Faile would have it, because every reclining plane is a horizontal plane where the pole is elevated according to the style. In a given plane oblique to the Meridian, and to the Horizon, and to the prime vertical, that is, a given plane Inclining declining, to find as well the Meridian of the place as of the plane, and the elevation of the pole above the plane: Prob. 3, Petici, Liber Gnomonicorum. TO give you the parallel of Pitiscus his example, we will prosecute it according to the natural Tangents in his example, and give you his words. Let the Meridian of the place be ABCD▪ the Horizon AEC, the prime vertical BED, the Oriental point E, the vertical declined BKD, and right angled at K, the poles of the World G and I: the poles of the planes H, the Meridian GHI, the angle of declination EBF, the arch of inclination BK. But before all things the arch K, or the distance of the meridian of the place NL is from the Vertical plane KL should be sought by the second axiom, than the ark BN by the third or fourth axiom, after these the angle BKN, that is, in one word, the Triangle BKN is found, by which discharged, the ark BN is found either equal to the poles elevation, or greater or lesser. If the ark be equal to the compliment 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 do 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 somewhat carried 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 joined 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 compliment 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 go by the poles of the plane, but concurring at G or N are equal to two right, by 20 p. 1. Example, Let the plane meridional declined to the right hand 29 de. 59 m. inclining toward the pole Arctic 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 ark KN the distance of the meridian of the place from the vertical of the plane, 34 de. 10 m. per axe. 2▪ The sine of the ark NC 49de. 35 m. whose compliment is the ark BN 40de. 25 m per axi. 4. To 60388 the sine of the angle BNK 37d 9m. whose compliment 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 vertical of the plane KL, the circle of the plane of the horizon KNC, in which there is numbered from K towards C 34 de. 10m. and at the term of the numeration N, draw the right line L N E, which shall be the meridian of the plane and place, if the centre of the scioteric L or F is taken for the centre of the World, and the right line L N F for the Axis, but because in the perfection of the dial, IG remaineth the Axis, with E the centre of the world, not in the right line L N F, but above the same, with props at pleasure, but notwithstanding it is raised equal in height with EI and OG, and moreover the plane is somewhat withdrawn from 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 speak, the substilar. But you may find 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 centre of the world be set from the axis IG in the meridian of the plane L N F. Then to the centre 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 than the horizon of the place, and moreover the beam is cast sooner or later upon the meridian of the plane than 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 problem of Pitiscus his Liber Gnomonicorum. Sivero arcus BN, repertus fuerit, &c. But if the ark BN shall be found less than the compliment of the poles elevation, it is a sign the plane doth consist on this side the pole Arctic, and moreover above such a plane not the pole Arctic, but the pole Antarctic shall be extolled to such an angle as ILM is, whose measure is the ark IM, to which, out of the doctrine of opposites, the ark GO is equal, which you may certainly find together with the ark 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 meridional declined to the right hand 34 de. 30 m. inclined toward the pole Arctic 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 country 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 compliment 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 vertical of the plane, per axi. 2. The sine of the ark N C 62 de, 532/3 m. whose compliment is B N 27 de. 61/3 m. by which subtracted from BG the compliment of the poles elevation 40 de. 25 m. there is remaining the ark 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 axe. 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 vertical 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 numbered the distance of the meridian of the place from the vertical of the plane 22 de. 3 m. and by the end 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 term of the numeration M, let the meridian of the plane LM be drawn. Finally, from the point M, into whatsoever part, let the proper elevation of the pole be numbered, or the distance of the axis from the meridian of the plane 8 de▪ 51/3m. and by the term of the numeration I, let the axis▪ LI be drawn, to be extolled or lifted up on the meridian of the plane LM, to the angle MLN. The third case of the third problem of Pitiscus his liber Gnomonicorum. Si denique arcus BN repertus fuerit major, &c. Lastly, if the ark BN be found greater than the compliment of the poles elevation BG, it is a token the plane to be inclined beyond the pole arctic, and moreover the pole arctic should be extolled above such a plane to so great an angle as the angle glow, which the ark GO measureth, which ark, together with the ark ON in the end you may find in such sort as in the precedent case. Example, Let there be a meridian plane declining to the right hand 35 de. 54 m. inclining towards the pole arctic 75 de. 43 m. and let the elevation of the pole be 49 de. 35½ m. but there is sought the meridian of the plane and place, together with the elevation of the pole above the plane, the calculation shall be thus. to 133874 tangent of the ark KN, the distance of the meridian of the place from the vertical of the plane, 53 de. 14½ m, by axi. 2. The sine of the ark NC 8 de. 29● m. whose compliment is BN 81 de. 30½ m. from whence if you subtract BG 40 de. 25 m. there remaineth the ark GN 41 de. 5½ m. to 97982, the sine of the angle BNK, or ONG, by axi. 3. to 64399 the sine of the arch OG, the distance of the axis from the meridian of the plane 40 de. 51/3 m. by axi. 3. to 17483 the sine of the ark O N the distance of the meridian of the plane from the meridian of the place, 10 de. 4 m. by axi. & comp. 2. The calculation being finished, let the horizon of the place be AC, the vertical of the plane KD, the horizon of the plane AKCD, in which let be numbered from the vertical point K toward C the distance of the meridian of the place from the vertical of the plane 53 de. 14½ m. and by the end of the numeration let be drawn the meridian of the place LN, then from the meridian of the place, to wit, from the point N backward, let the distance of the meridian of the plane 10 de. 4m. be numbered, and by O the end of the numeration, let LO the meridian of the plane be drawn, from which afterwards let the proper elevation of the pole be numbered, or the distance of the axis from the meridian of the plane 48d. 5½m. and by the term of the numeration G, let the axis LG be drawn, being extolled above the plane boy, to the angle glow. CHAP X. In which is showed the drawing of the hourlines in these last planes not there mentioned, being also part of Pitiscus his example in the fourth problem of his liber Gnom. SO then, saith he, Si axis, &c. If the axis be oblique to the plane, as the foregoing are, as in any plane oblique to the Equator many of the hourlines do concur at the axis with equal angles, but they are easily found thus. But because Pitiscus is mute in defining which part he takes for the right hand and which the left, we must search his meaning. Pitiscus was a Divine is evident by his own words in his dedication, Celsitudini tuae tota vita mea prolix me excusarem quod ego homo Theologus▪ &c. If we take him as he was a Divine, we imagine his face to be towards the East, than the South is his right hand, and the North is his left hand. That he was an Astronomer too, appeareth by his Books both of proper and common motion, than we must imagine his face representing the South, the East on his left hand, which cannot be, as shall appear. Neither must we take him according to the Poets, whose face must be imagined toward the West. In short, take him according to geography, representing the Pole, and this shows the right hand was the East, and left the West, as is evident by the dial before going, for it is a plane declining from the South to the right hand 30 degrees, that is, the East, because it hath the morning hours not the evening, because the Sun shines but part of the afternoon on the plane. Thus in brief I have run throng all planes, and proceed to show you farther conclusions: But I desire the Reader to take notice that in these examples of Pitiscus. I have followed his own steps, and made use of the natural Sines and Tangents. CHAP XI. Showing how by the help of a horizontal dial, or other, to make any dial in any position how ever. HAving prepared a horizontal dial as is taught before: on the 12 hour, as far distant as you please from the foot of the style, draw a line perpendicular to the line of 12, on that describe a Semicircle, plasing the foot of the Compasses in the crossing of the lines, this Semicircle divide into 180 parts, each Quadrant into 90, to number the declination thereon, let the arch of the Semicircle be toward the North part of the dial. Then prepare a plane slate, such as will blot out what hath been formerly made thereon, and make it to move perpendicularly on the horizontal plane on the centre of the semicircle, which will represent any declining plane by moving it on the semicircle. Now knowing the declination of the plane turn this slate towards the easterly part, if it decline towards the East, if contrary to the West, if toward the West, and set it on the semicircle to the degree of declination, then taking a candle and moving the dial till the shadow fall on all the hours of the horizontal plane, mark also where the shadow falls on the declining plane, that also is the same hour on the plane so situated, drawn from the joining of the style with the plane. It is so plain it needs no figure. So may you do in all manner of declining reclining, or reclining and inclining Dials, by framing your instrument to represent the position of the plane. Note also that the same angle the axis of the Horizontal Dial makes with the plane, the same elevation must the axis of that plane have, and where it shadows on the representing plane when the shadow of the horizontal axis is on 12, that is the meridian of the place. By the same also may you describe all the conclusions astronomical, the Almicanthers, circles of height: the parallels of the Sun, showing the declination: the azimuths, showing the point of the compass the Sun is in: and all the propositions of the Sphere. Seeing this is so plain and evident, nay a delightful conclusion, I will not give you farther directions in a matter of so great perspicuity, as to lay down the several ways for projecting the Sphere on every several plane, but proceed to show the making of a general Dial for the whole World, which we will use as our declinatory to find the situation of any wall or plane, as shall be required to make a dial thereon, as followeth in the next Chapter. CHAP XII. Showing the making of a dial on a cross form, as also a universal Quadrant drawn from the same projection, as also to describe the Tropics on Meridian or Polar planes. THis universal dial 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 fabric, I shall describe it according to this Figure. Now to know the hour of the day, you shall turn the plane by the help 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 compliment of the elevation, then bringing the Box to stand in the Meridian, the shoulder of the cross shall show you the hour. Upon this also is grounded the universal Quadrant hereafter described, which Instrument is made in brass by Mr. Walter Hayes as it is here described. Prepare a Quadrant of brass, divide it in the limb into 90 degrees, and at the end of 45 degrees from the centre draw the line A B, which shall represent the Equator, divide the limb into 90 degrees, as other Quadrants are usually divided, than number both ways 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 Tropics, than out of the Table of declination, pag. 45, from B both ways let there be numbered the declination of the signs according to this Table. G M ♈ 00 00 ♎ ♉ ♍ 11 30 ♏ ♓ ♊ ♌ 20 30 ♐ ♒ ♋ 23 30 ♑ Now the plane itself is no other than an East or West dial, numbered on one side with the morning hours, and on the other with the evening hours, the middle line AB representing the Equator. And to set it for the hour, you shall project the Tropics and other intermediate parallels of the signs upon them as is hereafter showed, but that the plane may not run out of the Quadrant you shall work thus, opening the Compasses to 15 degrees of the Quadrant, prick that down both ways, 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 centre and each of those divisions in the semicircle to those parallel lines on each side of AB, mark where they cut, and from side to side draw the parallel hour lines as is taught in the making of an East and West dial, 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 centre 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 situation on the East or West wall if you let the plummet and thread fall on the elevation of the Pole in that place. But because we desire to make it general, we must describe the Tropics and other parallels of declination upon it, as is usual to be done on your Polar and East and West dial, which how to do is thus. Having drawn the hour lines and Equator as is taught from E the height of the style, take all the distances between it and the hour lines where they do cross the line AB, and prick them down on the line representing the Equator in this figure from the centre B. Then describe an occult arch of a Circle, whereon describe a chord 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 hours 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 hours, from these distances I say raise perpendiculars to cut the other lines of declination, so those perpendiculars shall represent those hour lines from whence they were taken, and the distances between the Equator and the several 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 several hourelines from the equinoctial. If you draw those hyperbolical lines, you shall have described the parallels of declination required. But if you will perform the same work a second and easy way, work by this Table following, which is universal, and is composed out of the Table of Right & Versed shadow. Put this Table before thee, & for the point of each hour line whereby the several parallels of the signs shall pass work 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 hour line as you find it marked in the Table answering the hour both before and after noon. As suppose that a Polar plane I find 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 hours of 1 and 11 both above and beneath the Equator: and for 2 and 10 I find 6 parts 56 min. which I prick down also from the centre to the hour lines of 10 and 2, and so of the other hour 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 show how to find the inclination and reclination of a plane, I shall proceed to give you some cautions; First then, the quadrant is divided in the limb, as other quadrants are into 90 degrees, by which is measured the angles of inclination or reclination, for if it be a declining plane only, the declination is accounted from the North or South toward the East or West, if it decline from the North, the North Pole is elevated above it, and the meridian-line ascendeth, if it decline from the South, the south pole is elevated above that plane, if it decline from the South Eastward, then is the style and sub-style refered toward the west side of the plane, if to the contrary the contrary, and may have the line of 12 except north decliners in the temperate Zone, you may make use of the side of the quadrant to find the declination, as is taught before page 33, observing the angle as is cut by the shadow of the thread held by the limb, & through the centre, and that side that lieth perpendicular to the Horizontal line which shall be the angle, as is before taught: And if the south point is between the poles of the plane and the Azimuth, than doth the plane decline Eastward, if it be the afternoon you take the Azimuth in, if it be the forenoon you take the Azimuth in, and the south point be between it and the Poles of the planes horizontal line, it doth decline Westward, if contrary it is in the same quarter where the sun is: For an inclining plane, which is the angle that it maketh with the Horizon▪ draw a horizontal line and cross it again with a square, or vertical line, then apply the side of the quadrant to the vertical line at the beginning of the numeration of the deg. on the quadrant, and the angle contained between the thread & plummet, and the applied side is the inclination; in all north incliners the north part of the meridian ascendeth, in south incliners the south part, and in east and west incliners, the meridian lieth parallel with the Horizon. And for the reclination it being all one with the inclination, considered as an upper and under face of the same plane, if you cannot apply the side of the quadrant, you may set a square or ruler at right angles with the vertical line drawn on the upper face and apply the side of the quadrant to the edge of the ruler, and measure the quantity of the angle by the thread and plummet: but this is of direct, howsoever these are subject to another passion of declining and inclining together, which must be sought severally, and such are those whose Horizontal line declineth toward the north or south and inclination from north or south, toward the east or west, which must be sought severally. Here followeth the Tables of Right and Contrary shadows. A Table of Right and Contrary shadow, to every Degree and tenth minute of the Quadrant. ☉ Alt 0 1 2 3 4 5 6 7 8 9 ☉ Alti● S S S S S S S S S S p m p m p m p m p m p m p m p m p m p m Horizontal shadow 0 41378, 54 687 34 143 44 229 0 171 37 137 10 114 11 97 44 85 23 75 46 60 Vertical shadow 10 4137, 53 589 16 317 14 216 54 164 44 132 43 111 4 95 26 83 37 74 22 50 20 2065, 23 515 46 294 31 206 3 158 23 128 33 108 7 93 15 81 55 73 1 40 30 1376, 6 458 22 274 54 196 13 152 29 124 38 105 19 91 9 80 18 71 43 30 40 1031, 45 412 29 257 40 187 16 147 1 120 56 102 40 89 9 78 44 70 27 20 50 825, 13 374 55 242 28 179 6 141 56 117 28 100 8 87 14 77 13 69 14 10 60 687, 34 343 54 229 0 171 37 137 10 114 11 97 44 85 23 75 46 68 3 0 10 11 12 13 14 15 16 17 18 19 S S S S S S S S S S p m p m p m p m p m p m p m p m p m p m Horizontal shadow 0 68 3 61 44 56 27 51 59 48 8 44 47 41 51 39 15 36 57 34 51 60 Vertical shadow. 10 66 55 60 47 55 40 51 18 47 32 44 16 41 24 38 51 36 34 34 31 50 20 65 49 95 52 54 53 50 38 46 58 43 46 40 57 38 27 36 13 34 12 40 30 64 45 85 59 54 8 49 59 46 24 43 16 40 31 38 4 35 52 33 53 30 40 63 43 85 7 53 24 49 21 45 51 42 47 40 5 37 41 35 31 33 35 20 50 62 43 57 16 52 41 48 44 45 19 42 19 39 40 37 18 35 11 33 16 10 60 61 44 56 27 51 59 48 8 44 47 41 51 39 15 36 56 34 51 32 58 0 20 21 22 23 24 25 26 27 28 29 S S S S S S S S S S p m p m p m p m p m p m p m p m p m p m Horizontal shadow 0 32 58 31 16 29 42 28 16 26 57 25 44 24 36 23 33 22 34 21 39 60 Vertical shadow. 10 32 40 31 0 29 27 28 3 26 45 25 52 24 25 23 23 22 25 21 ●0 50 20 32 23 30 44 29 13 27 49 26 32 25 21 24 15 23 13 22 15 21 21 40 30 32 6 30 28 28 58 27 36 26 20 25 10 24 4 23 3 22 6 21 13 30 40 31 49 30 12 28 44 27 23 26 8 24 58 23 54 22 53 21 57 21 4 20 50 31 32 29 57 28 30 27 10 25 56 24 47 23 43 22 44 21 48 20 56 10 60 31 16 29 42 28 16 26 57 25 44 24 36 23 33 22 34 21 39 20 47 0 30 31 32 33 34 35 36 37 38 39 S S S S S S S S S S p m p m p m p m p m p m p m p m p m p m Horizontal shadow 0 20 47 19 58 19 12 18 29 17 47 17 8 16 31 15 55 15 22 14 49 60 Vertical shadow. 10 20 ●9 19 50 19 5 18 21 17 41 17 2 16 25 15 50 15 16 14 44 50 20 20 31 19 43 18 57 18 15 17 34 16 56 16 19 15 44 15 11 14 39 40 30 20 22 19 35 18 50 18 8 17 28 16 49 16 13 15 38 15 5 14 33 30 40 20 14 19 27 18 43 18 1 17 21 16 43 16 7 15 33 15 0 14 28 20 50 20 6 19 20 18 36 17 54 17 15 16 37 16 1 15 27 14 54 14 23 10 60 19 58 19 12 18 29 17 47 17 8 16 31 15 55 15 22 14 49 14 18 0 40 41 42 43 44 45 46 47 48 49 S S S S S S S S S S p m p m p m p m p m p m p m p m p m p m Horizontal shadow 0 14 18 13 48 13 20 12 52 12 26 12 0 11 35 11 11 10 48 10 26 60 Vertical shadow. 10 14 13 13 43 13 15 12 48 12 21 11 56 11 31 11 8 10 45 10 22 50 20 14 8 13 39 13 10 12 42 12 17 11 52 11 27 11 4 10 41 10 19 40 30 14 3 13 34 13 6 12 39 12 13 11 48 11 23 11 0 10 37 10 15 30 40 13 58 13 29 13 1 12 34 12 8 11 43 11 19 10 56 10 33 10 11 20 50 13 53 13 24 12 57 12 30 12 4 11 39 11 15 10 52 10 30 10 8 10 60 13 48 13 20 12 52 12 26 12 0 11 35 11 11 10 48 10 26 10 4 0 50 51 52 53 54 55 56 57 58 59 S S S S S S S S S S p m p m p m p m p m p m p m p m p m p m Horizontal shadow 0 10 4 9 43 9 23 9 3 8 43 8 24 8 6 7 48 7 30 7 13 60 Vertical shadow. 10 10 1 9 40 9 19 8 59 8 40 8 21 8 3 7 45 7 27 7 10 50 20 9 57 9 36 9 16 8 56 8 37 8 18 8 0 7 42 7 24 7 7 40 30 9 54 9 33 9 12 8 53 8 34 8 15 7 57 7 39 7 21 7 4 30 40 9 50 9 29 9 9 8 50 8 30 8 12 7 54 7 36 7 18 7 1 20 50 9 47 9 26 9 6 8 46 8 27 8 9 7 51 7 33 7 15 6 59 10 60 9 43 9 23 9 3 8 43 8 24 8 6 7 48 7 30 7 13 6 56 0 60 61 62 63 64 65 66 67 68 69 S S S S S S S S S S p m p m p m p m p m p m p m p m p m p m Horizontal shadow 0 6 56 6 39 6 23 6 7 5 51 5 36 5 21 5 6 4 51 4 36 60 Vertical shadow. 10 6 53 6 36 6 20 6 4 5 49 5 33 5 18 5 3 4 48 4 34 50 20 6 50 6 34 6 17 6 2 5 46 5 31 5 16 5 1 4 46 4 32 40 30 6 47 6 31 6 15 5 59 5 43 5 28 5 13 4 58 4 44 4 29 30 40 6 45 6 28 6 12 5 56 5 41 5 26 5 11 4 56 4 41 4 27 20 50 6 42 6 26 6 10 5 54 5 38 5 23 5 8 4 53 4 39 4 24 10 60 6 39 6 23 6 7 5 51 5 36 5 21 5 6 4 51 4 36 4 22 0 70 71 72 73 74 75 76 77 78 79 S S S S S S S S S S p m p m p m p m p m p m p m p m p m p m Horizontal shadow 0 4 22 4 8 3 54 3 40 3 26 3 13 3 0 2 46 2 33 2 20 60 Vertical shadow. 10 4 20 4 6 3 52 3 38 3 24 3 11 2 56 2 44 2 31 2 18 50 20 4 17 4 3 3 49 3 36 3 22 3 8 2 55 2 42 2 29 2 16 40 30 4 15 4 1 3 47 3 33 3 20 3 6 2 53 2 40 2 26 2 13 30 40 4 13 3 59 3 45 3 31 3 17 3 4 2 51 2 37 2 24 2 11 20 50 4 10 3 56 3 42 3 29 3 15 3 2 2 48 2 35 2 22 2 9 10 60 4 8 3 54 3 40 3 22 3 13 3 0 2 46 2 33 2 20 2 7 0 80 81 82 83 84 85 86 87 88 89 S S S S S S S S S S p m p m p m p m p m p m p m p m p m p m Horizontal shadow 0 2 7 1 54 1 41 1 28 1 16 1 3 0 50 0 38 0 25 0 13 60 Vertical shadow. 10 2 5 1 52 1 39 1 26 1 14 1 1 0 48 0 36 0 23 0 10 50 20 2 3 1 50 1 37 1 24 1 11 0 50 0 46 2 34 0 21 0 8 40 30 2 0 1 48 1 35 1 22 1 9 0 57 0 44 0 31 0 19 0 6 30 40 1 58 1 45 1 33 1 20 1 7 0 55 0 42 0 29 0 17 0 4 20 50 1 56 1 43 1 31 1 18 1 5 0 32 0 40 0 27 0 15 0 2 10 60 1 54 1 41 1 28 1 16 1 3 0 50 0 38 0 25 0 13 0 0 0 CHAP XIII. Of the general description and use of the preceding Tablein, the pricking down and drawing the circles of declination and Aximuths in any planes. THe Table you see consisteth of 11 columns, the first being the minutes of the sun's altitude, and the greater figures on the top are the degrees of altitude, all the other columns consist of the parts of shadow, and minutes of shadow, noted above with S for shadow, and p m for parts and minutes of shadow, answerable to a gnomon divided into 12 equal parts, and it is, As the sine of a known altitude of the sun, is to the fine compliment of the same altitude; so the length of the Gnomon in 10 or 12 parts, to the parts of right shadow: or for the versed shadow, as the sine compliment of the given altitude of the sun, to the right sine of the same altitude; so the style in parts, to the length of the versed shadow So if we enter the Table with the given altitude of the Sun in the great figures, and if we seek the minutes in the sides, either noted with horizontal or vertical shadow, according as your plane is, it shall give you the length of the shadow in parts and minutes in the common angle of meeting together. As if we look for 50 de. 40 m. the meeting of both in the Table shall be 9 parts 50 min. for the length of the right shadow on a horizontal plane: But for the versed shadow, take the compliment of the altitude of the Sun, and the minutes in the right side of the Table, titled vertical shadow, and the common area of both shall give your desire. By this Table it appeareth first, that the circles of altitude either on the horizontal or vertical planes are easily drawn, consicering they are nothing else but circles of altitude, which by knowing the altitude you will know the length of the shadow, which in the horizontal dial are perfect circles, and have the same respect unto the Horizon, as the parallels of declination have to the Equator, but in all upright planes they will be conical Sections, and by having the length of the style, the altitude of the Sun may be computed by the foregoing Table with much facility, but for the more expediating of the work in pricking down the parallels of declination with the Tropics, I have here added a Table of the altitude of the Sun for every hour of the day when the Sun enters into any of the 12 signs. A Table for the altitude of the Sun in the beginning of each sign, for all the hours of the day for the Latitude of London. Hours. Cancer. Gemini Leo Taurus Virgo Aries Libra Pisces Scorpio Aquar Sagitta. Capric. 12 62 0 58 43 50 0 38 30 27 0 18 18 15 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. gunter's Book, page 240 which if you desire to have the point of the equinoctial for a horizontal plane on the hour of 12, enter the Table of shadows with 38 de. 30 m. and you shall find 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 equinoctial point, from the foot of the style. So if I desire the points of the tropic of Cancer, I find 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 tropic of Cancer at the hour of 12. If for the hour of 1, I desire the point through which the parallel must pass, look for the hour of 1 and 11, in this last table under Cancer, and I find 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 bottom of the style reaching till the other foot of the Compasses fall on the hour for which it was intended. Do so in all the other hours, 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 horizontal dial, but if it be a direct vertical dial, you shall then take the length of the vertical shadow out of the said Table, or work it as an Horizontal plane, only accounting the compliment of the elevation in stead of the whole elevation. For a declining plane you may consider it as a vertical 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 ways than the meridian line or line of 12 in a Horizon whose pole is elevated according to the compliment 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 hour lines, as if it were a horizontal or vertical plane. But in this you must be wary, remembering that you have the height of the sun calculated for every hour of that Latitude in the entrance of the 12 signs, in that Place where your Plane is a horizontal plane, or otherways, by considering of it as a horizontal or Verricallplane in another latitude For the Azimuths, or vertical circles, showing one what point of the compass the sun is in every hour of the day it is performed with a great deal of facility, if first, when the sun is in the Equator, we do know by the last Table of the height of the sun for every hour of the day and by his meridian altitude with the help of the table of shadows, find out the equinoctial line, whether it be a horizontal 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 equinoctial 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 hours must be on the opposite side. There are many other Astronomical conclusions that are used to be put upon planes, as the diurnal arches, showing the length of the day and night, as also the Jewish or old unequal hours together with the circles of position, which with the meridian and horizon distinguisheth the upper hemisphere 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 enlarge myself much more, in showing a demonstrative way, in these particulars I have last insisted upon. I might hear also show you the exceeding use of the table of Right and versed shadow in the taking of heights of buildings as it may very well appear in the several uses of the quadrant in Diggs his Pantometria, & in Mr. gunter's quadrant, having the parts of right and versed shadow graduated on them, to which Books I refer you. CHAP XIV. Showing the drawing of the ceiling dial. IT is an Axiom pronounced long since, by those who have writ of optical conceits of Light and Shadow, that Omnis reflectio Luminis est secundum lineas sensibiles, latitudinem habentes. And it hath with as great reason been 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 conceit 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 glass apt to receive and reflect the light, placed somewhere horizontally in a window, proceed to the work, which indeed is no other than a horizontal dial reversed, to which required a Meridian line, which you must endeavour to draw and find according as you are before taught, or by the help of the Meridian altitude of the Sun, your glass being fixed mark the spot that reflects upon the ceiling just at 12 a clock, make that one point, and for the other point through which you must draw your meridian line, you may find by holding up a thread and plummet till the plummet fall perpendicular on the glass, and at the other end of the line held on the ceiling make another mark, through both which draw the Meridian line. Now for so much as the centre of the dial is a point without, and the distance between the glass and the ceiling is to be considered as the height of the style, the glass itself representing the centre of the world, or the very apex of the style, we must find out those two Tangents at right angles with the Meridian, the one near the window, the other farther in, through several points whereof we must draw the hourlines. Let AB be the Meridian line found on the ceiling, now suppose the Sun being in the highest degree of Cancer should shine into the glass 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 ceiling till it fall just on C the glass, 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 find out the places where the hour-lines shall cross the Meridian, the centre lying without the window EC, you may work thus CHAP X. Showing the making and use of the Cylinder Dial, whose hour-lines are straight, as also a dial 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 Tropics, 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 middle, are the lines of Altitude, and the Parallels to the Equator are the Parallels of Declination, numbered 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 itself, which shall denote the height required. For the hour of the Day. Seek the Altitude of the Sun in the middle pricked line, and the Declination in the Parallels from the Equator, and mark where the traverse lines cross; through the crossing of the two former lines, and at the end, you shall find the figures of 2 or 10, 3 or 9, &c. only the summer hours 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 sign. Seek the day of the month on the top marked with J. for January, F for February, &c. and by the help of a horse hair or thread extended from that all along of Parallel of Declination, till it cut on the bottom where the signs are numbered: 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 ecliptic which is in the bottom, right against the day of the month sought on the top. The pricked line passing through the 18 degree of the Parallel of Altitude, is the line of twilight; this projection I had of my very good friend John Hulet, Master of Arts▪ and Teacher of the mathematics. You may also make a dial, by preparing of a hollow Cylinder, and if you do 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 straight lines, first, by dividing the Cylinder through the middle, and only making use of one half, it shall have 12 hours 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, till the shadow of one of the sides fall parallel with any one of those lines from top to bottom: which numbered as they ought, shall show the hour without the use of a Style; So also may you project a dial 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. Showing the making of a universal dial 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, find 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, numbering from A toward G9 of those equal parts, and from B toward H as many, which shall be the centres 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 mathematics, 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. Showing the finding of the Elevation of the Pole, and therewithal a Meridian without the Declination of Sun or star. 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 sun 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 lie on all three flat, and a gnomon in the middle of that board points to the Pole: because every Parallel the snn 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 hours' time. CHAP XVIII. Showing how to find 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 centre, cross 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 centre on the other Diameter to F, from E to F draw 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 orb. 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 centre 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 pass by the head of the gnomon E, and cast it to F, and the beams of the Moon shall pass by E to G much longer, which shows she 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, she 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 than the Moon; seeing in time of a total Obscurity, the moon 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 than the Sun: for if the opacous body be equal to the luminous body, then like two parallels they will never meet, but concur in infinitum, as these following figures show. Or if the luminous body were less than 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 stars 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 pass, and to which all heavy things tend in a right line, as their terminus ad quem. From which the semidiameter of the Sun 15 min. subtracted doth remain the Altitude of the centre 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 show the exact time without the allowance of the sun's 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 centre is hindered by the style, the shadow of the hour-line proceeds from the limb which always precedeth the centre 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 horizontal plane from the Meridian shall stand thus. Prop. 4. By shadow we may find the natural Tangent of every degree of a quadrant, as appears by the former example. Hours. 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 yields 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 learn 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 equinoctial 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 noons to them as dwell 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 centre is disposed to the eye: so the Office of shadow is manifold, as in the Optical conclusions are more amply declared; therefore I refer 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 light body from whence light doth proceed; truly saith he, the definition is plain, and wants not an Expositor, so say I, it matters not whether you understand the luminous body: only that which doth glister by proper brightness as doth the Sun, or that which doth not shine but by an external overflowing light, as doth the Moon. 2. That we call a diaphon body, through which light may pass, and is the same that Aristotle calls perspicuous. 3. It is called Adiopton, or Opacous; through which the light cannot pass, so saith he, you may easily collect from a diaphon body the definition of shadow: for as that is transparent through which the light may pass: so also is that opacous, or of a dense nature wherein the light cannot pass. 4. That is generated from a shining body, is called the first light, that hath his immediate beginning from the luminous body, it is called the second light, which hath his beginning from the first, the third which hath his beginning from the second, and so the rest in the same order. Whence we make this distinction of day and light, day is but the second light, receiving from the Sun the first, so that day is light, but the Sun is the light. 5. Splendour is light repercussed from a pure polished body; and as light is called so from the luminous body: so this is called splendent from the splendour. Theor. Light doth not only proceed from the centre, but from every part of the superficies. Theor. Light also is dispersed in right lines. Theor▪ Light dispersed about everywhere, doth collect into a Spherical body. 6. The beams of light, some are equi distant parallels, some intersect each other, and some diversely shaped. Let A be the Light, a beam from A to B, and another from C to D are parallel, A D and C B intersect; and the other two do diversely happen, one ascending, the other descending: its plain. 7. That is called a full and perfect shadow, to which no beam of light doth come. 8. That is called a full and perfect light; which doth proceed from all parts of that which gives light; but that which giveth light but in part, is imperfect: this he exemplified by an Eclipse, the Moon interposing herself between the Sun and Earth, doth eclipse the perfect light of the Sun: whereby there appears but a certain obscure, dim, glimmering light, and is so made imperfect. Hence we may learn to distinguish day from night; for day is but the presence of the Sun by a perfect light received, which we count from Sun rising to Sun setting. Twilight is but an imperfect light from the partial shining or neighbourhood with Sun: whereas Night is a total deprivation or perfect shadow, to which no beam of light doth appettain. Yet from the overflowing light of the Sun, the stars are illuminated; yet because shadow is always in the opposite, those Stars that are in direct opposition to the Sun, are obscure for that season, and hence proceeds the Eclipse of the Moon. Hence it is with the Sciothericalls as it is with the Dutch Emblamist, comparing Love to a dial, and the Sun with the Motto, Nil sine te, and his comparison to Coelestis cum me Sol aspicit ore sereno, Protinùs ad numeros mens reddit apta suos. Implying that as soon as the Sun shines it returns to the number, so a Lover seeing his Love on a high Tower, and a Sea between, yet (protinùs ad numeros) he will swim the Sea and scale tbe Castle to return to her: So here lies the gradation, first, from the sun's light, from the light by the opoacus body, interposition, shadow, and from the shadow of the Axis is demonstrated the hour. Add also, the beam and shadow of a gnomon, have one and the same termination or ending, toward which I now draw my pen; desiring you to take notice that the whole Method of Dialling, as may appear by the former discourse, doth seem to be fourfold, viz. Geometrical, Arithmetical, or by Tables mechanically, or by Observation. So that the Art of shadows is no other than a certain and demonstrative motion of the Heavens in any plain or Superficies, and a Gnomonical hour is no other than a direct projecting of the hour-lines of any Plain; so as that it shall limit a Style so to cast its shadow from one line to another, as that it shall be just the twenty-fourth part of the natural day, which consisteth of 24 hours; and this I have laid down after a most plain manner following: A gnomonical day is the same that the Artificial day is; which the shadow of a gnomon maketh from the rising of the Sun, till the setting of the same in a concave superficies: which length of the day is also projected from the motion of the shadow of the Style, a gnomonical month is also described on Planes, which is the space that the shadow of a Gnomon maketh from one Parallel of the sign, to an other succeeding Parallel of a sign, again, a gnomonical year is limited by the shadow of a Gnomon, from a point in the Meridian of the tropic of Cancer, till it shall revolve to the same Meridian Altitude and point of the tropic, and is the same as is a tropical year: wherefore, above all things we ought first tobe acquainted with the knowledge of the Circles of the sphere▪ Secondly, to have a judicious and exact discerning of those Planes in which we ought to project Dials. Thirdly, to consider the Style, Quality, and Position of the axis or Style, with consideration of the cause, nature and effects in such or such Planes as also an artificial projecting of the same, either on a Superficies by a geometrical Knowledge, and reducing them to Tables by arithmetic, which we have afore demonstrated, and come now to the conclusion: So that as I began with the dial of Life, So we shall Dye-all, For, Mors ultima linea. TO ABRAHAM CHAMBRELAN Esq.. S M. consecrateth his Court of Arts. SIR. IF the original Light be manifestatiu, by it I have made a double discovery, your genius did so discover itself according to the quality of the Sun, that I am umbrated and passive like the eclipsed Moon, yet cannot but reflect a beam which I have received from the fountain of Light; 'tis you which I make the Patron to my fancy (which perhaps you may wonder at the Idleness of my head, to tell you a dream, or a praeludium of the several Arts: howsoever knowing you are a Lover of them, I did easily believe you could not but delight in the scene; though in most I have written, I have in some sort imitated Nature itself, which dispenseth not her Light without Shadows, which will truly follow them from whom they proceed, and shall Sir, in time to come render me like Pentheus whose curiosity in prying into Secrets makes me uncertain. Et Solem Geminum duplices se ostendere Thebas, & while I know neither Copernicus, nor Ptolemy's system of the World, dare affirmatively reject neither, but run after both; and submitting my wisdom to the wisest of men, must conclude, that Cuncta fecit tempestatibus suis pulchra, and hath also set the World in their meditation: Yet can not Man find out the Work that God hath wrought. Sir, pardon my boldness, in fastening this on your Patronage, who indeed are called to this Court of Arts, as being Nobly descended, whom only it concerns; and only whose virtue hath arrived them to the Temple of Honour, who are all invited as appeareth in the conclusion of this imaginary description, wherein, whilst I seem to be in a dream; yet Sir, I am certain, I know myself to be Yours in all that I am able to serve you, S. M. TOPOTHESIA. OR An IMAGINARY DESCRIPTION of the court of ART. Coming into a library of Learning, where there was more Languages than I had Tongues, that if I had been asked to bring brick I should have brought mortar, and going gradually along, as then but passus Geometricus, there I met Minerva, which said unto me (Vade mecum) & had not the expression of her gesture bespoke my company, I should have shunned her; she then taking me by the hand, led me to the end, where sat one which was called as I did inquire, Clemency, the name indeed I understood, but the Office I did not, whose Inscription was Custos Artis, I being touched now with a desire to understand this Inscription; began with Desire, & craving leave, used diligence to peruse the Library, and found then a book entitled the Gate of Languages, by that I had perused it, I understood the forenamed Inscription, and craving leave of Clemency in what respect she might be called the Keeper of Arts, who answered with Claudanus thus; Principio magni Custos Clementia Mundi, Quae Jovis incoluit Zonam quae temper at Aethrum, Frigoris & flammae mediam quae maxima natu, Coelicolum: nam prima Chaos Clementia solvit, Congeriem miserata rudem, vultuque sereno, Discussus tenebris in lucem saecula fundit. And arising from a Globe which was then her seat, she began to discourse of the Nature and Magnitude of the terrestrial body, and propounded to me questions: as first, If one degree answerable to a celestial degree yield 60 miles, what shall 360 degrees yield, the proportion was so plainly propounded, that I resolved it by the ordinary rule of Proportion, she seeing the resolution, propounded again, and said, if this solid Body were cut from the centre how many solid obtuse angles might be cut from thence, at this I stumbled, and desired, considering my small practice, that she would reduce this Chaos also, and turn darkness into light: seeing then my desire and diligence bid me make observation for those three were the ways to bring me to peace, and resolved, that as from the centre of a Circle but three obtuse angles could be struck, so from the centre of a Globe, but three such angles could be struck and from thence fell to another question, & asked what I thought of the motion of that body: I answered, Motion I thought it had none, seeing I had such Secretaries of Nature on my side, and was loath to join my forces with the Copernicans. She answered, it was part of folly to condemn without knowing the reasons, I said it should still remain a Hypothesis to me, but not a firm axiom: for the resolution of which I will only sing as sometimes other Poets sang concerning the beginning of the world, and invert the sense only, as that in another case, so this for our purppse. If Tellus winged be The Earth a motion round, Then much deceived are they That it before ne'er found. Solomon was the wisest, His wit ne'er this attained; Cease then Copernicus, Thy Hypothesis vain. And began to discourse of the longitude of the earth, and then I demanded what benefit might incur from thence to a young Diallist, she answered above all one most necessary problem, which we may find in Petiscus his example, and propounded it thus; The difference of meridians given, to find the difference of hours. If the place be easterly, add the difference of longitude converted into time to the hours given: if it be westerly, subtract the easterly places, whose longitude is greater & contra, as in Petiscus his example, the meridian of Cracovia is 45 deg. 30 min. the longitude of the meridian of Heidelberge is 30 degrees, 45 minutes, therefore Heidelberg is the more westerly. One subtracted from 45 30 30 45 the other showeth the difference of longitude, to which degrees and minutes doth answer o ho. 59 m. for as Therefore when it is 2 hours post merid. at Cracovia; at Neidelberg, it is but 1 hour, 1 minute past noon. For, There is left 1 hour 1 minute. Thus out of the difference of meridians, the divers situation of the heavens is known, and from the line of appearances of the heavens, the divers hours of divers places is known, and this is the foundation of observing the longitude: if it be observed what hour an Eclipse appears in one place, and what in another, the difference of time would show the longitude, and hereby you may make a dial that together with the proper place of elevation, shall show for any other country; for this Proposition I did heartily gratify Geographia, and turning, said Astronomy, why stand you so sad? she answered, Art is grown contemptible, and every one was ready to say (Astrologus est Gastrologus) then I said, what though virtue was despised, yet let them take this answer: Thou that contemnest Art And makes it not regarded, In Court of Art shall have no part None there but Arts rewarded. Gnashing the teeth as if ye strive to blame it, Yet know I'll spare no cost for to obtain it. Perceiving your willingness said Astronomy, I will yet extend my charity and lay down the numbers, so that if you add the second and third and subtract the first, it shall give the fourth; the question demanded, and then I being careful of the tuition of what she should say, took a Table-book and writ them as follows. 1 The fine comp. elevation pole 38½, sine 90; sine of the decl. of the sun yields the sine of the amplitude ortive: which is the distance of the suns rising from due East. 2 The sine 90, the sine eel. pole 51d½; the sine of decli. yields the sine of the sun's height at six a clock. 3 Sine comp. of altitude of the sun, sine comp. declina. sine 90; the sine of the angle of the vertical circle, and the meridian for the Azimuth of the sun at the hour of 6: The Azimuth is that point of the compass the sun is on. 4 Sine comp. decli. of the sun: sine compl. eleva. pole 38d½, sine Altitude of the sun; the hour distance from six. 5 Sine compl. of decli. sine 90; compl, of fine sun's amplitude to sine compl. of the Assentional difference. 6 The sine of the difference of ascension, Tang. decli. sun; sine 90: Tangent compliment of the elivation. 7 Sine altitude of the sun, sine declina. of the sun; sine 90: Elevation of the pole. 8 Sine 90, sine come. of distance from 6; fine come. declination of the sun: sine comp. of the altitude sun. 9 Sine 90, sine eleva. pole; sine alti. of a star: sine decli. of that star. 10 The sine of a stars altitude in an east Azimuth, sine amplitude ortive; sine 90: sine of the elevation. 11 The greatest meridian altitude, the less subtracted sines; the distance of the Tropics, whose half distance is the greatest declination of the sun; which added to the least meridian altitude, or subtracted from the greater, leaves the altitude of the Equator: the compliment whereof, is the elevation of the pole. 12 Tang. eleva. pole, sine 90; Tang. decli. of the sun, to the cousin of the hour from the meridian, when the sun will be due east or west. By these Propositions said Astronomy, you may much benefit yourself; but let us now go see the Court of Art: I liked the motion, and we went and behold the sight had like to made me a Delinquent, for I saw nought but a poor Anatomy sitting on the earth naked exposed to the open air, which made me think on the hardness of a Child of Art, that it had neither house nor bed, and now being at a pitch high enough resolve never to follow it: this Anatomy also it seems was ruled by many, both Rams, and bulls and Lions, for he was descanted thus on. Anatomy why dost not make thy moan, ☞ So many limbs, and yet canest govern none; Thy head although it have a manly sign, Yet art thou placed on watery feminine. 'Tis true, yet strong, but prithee let me tell ye, Let not the Virgin always rule your belly: For what, although the Lion rule your heart; The weakest vessel will get the strongest part. Then be content set not your foot upon A slippery fish, that's in an instant gone; A slippery woman, who at Cupid's call Will slip away, and so give you a fall: And if Rams horns she do on your head place; It is a dangerous slip, may spoil your face. Here at I smiled, than said Astronomy, what is your thought? then said I, do men or Artists so depend on women, as that their strength consists in them? she said, I misunderstand him, for the Ram that rules the head is a sign masculine, because it is hot and dry, the Fish that rules the feet is cold and moist is therefore called feminine. Pisces the Fish you knows a watery creature, ☞ 'Tis slippery, and shows a woman's nature; So women in their best performance fail, There's no more hold than in a fish's tail. But the more to affect the beholder, I will typigraphe this Court of Art. Under was written these lines, to show man's misery by the fall, which I will deliver you, as follows: When Chaos became Cosmos, oh Lord! than How excellent was Microcosmus, Man When he was subject to the maker's will, Stars influence could no way work him ill: But since his fall his stage did open lie, And Constellations work his destiny. Thus man no sooner in the World did enter, But of the Circumference is the centre. And then came in virtue, making a speech, and said; Honour to him, that honour doth belong: You stripling Artist, coming through this throng, Have found out virtue that doth stand to take You by the hand, and Gentleman you make. For Geometry, I care not who doth hear it, May bear in shield Coat armour by his merit: We respect merit, our love is not so cold, We love men's worth (not in love with men's gold) Not Herald-like to sell, an arms we give; Honour to them that honourably live. The noble professors of the Sciences, may bear as is here blazoned, (viz.) the field is Jupiter, Sun and Moon in conjunction proper, in a chief of the second, Saturn, Venus, Mercury in trine or perfect amity; and Mars in the centre of them; Mantled of the Light, doubled of the night, and on a wreath of its colours a Helitropian or Marigold of the colour of Helion with this Motto, Quod est superius, est sicut inferius; then did I desire to know, what did each Planet signify in colour, she then told me as followeth. ☉ Or Gold ☽ Argent Silver ♂ Gules Red ♃ Azure Blue ♄ Sable Black ♀ Vert Green ☿ Purpre Purple And by mantled of Light, she meant Argent and of the night she meant an Azure mantle, powdered with Estoiles, or Stars Silver. I indeed liked the Blazon, and went in, where also I found a fair genealogy of the Arts proceeding from the Conjunction of arithmetic and Geometry collected by the famous Beda Dee in his mathematical preface. Both number and magnitude saith he have a certain original seed of an incredible property; of number a unit, of magnitude a point. Number, is the union and unity of Unites, and is called arithmetic. ☽ Magnitude is a thing mathematical▪ and is divisible for ever, and is called Geometry. Geodesie, or Land measuring Geographia, showing ways either in spheric, plane, or other the situation of Cities, Towns, Villages, etc Chorographia, teaching how to describe a small proportion of ground, not regarding what it hath to the whole &c. Hydrographia, showing on a Globe or Plane the analogical description of the Ocean, seacoasts, through the world, &c. N●vigation, demonstrating how by the shortest way, and in the shortest time a sufficient Ship, between any two places in passages navigable assigned, may be conducted, &c. Perspective, is an art mathematical which demonstrateth the properties of Radiations, direct, broken, & reflected astronomy demonstrates the distance of magnitudes and natural motions, appearances and passions proper to the Planets and fixed stars. cosmography; the whole & perfect description of the heavenly, & also elemental part of the world & their homologal & mutual collation necessary. stratarithmetry. is the ki●● appertaining to the War●●, to set in figure any number of men appointed: differing from tacticy which is the wisdom & foresight. music, saith Plato, is sister to astronomy, & is a Science mathematical, which teacheth by sense & reason perfectly, to judge & order the diversity of sounds high & low. astrology, several from, but an offspring of astronomy, which demonstrated reasonably the operation and effects of the natural beams of light, and secret influence of the Stars. Statick; is an Art mathematical, demonstrating the causes of heaviness and lightness of things. ●●thropographie, being the description of the number, weight, figure, s●●uation and colour of every diverse thing contained in the body of man. trochilic, descended of number and measure, demonstrating the properties of wheel or circular motions, whether simple or compound, near Sister to whom is holicosophy, which is seen in the describing of the several conical Sections and Hyperbolicalline in plants of dials or other by spiral lines, Cylinder, Cone, &c. Pneumatithmie, demonstrating by close hollow figures geometrical, the strange properties of motion, or stay of water, air, smoke, fire in their continuity. Menadrie, which demonstrateth how above nature's virtue and force, power may be multiplied▪ hypogeody, being also a child of mathematical Arts, showing how under the spherical superficies of the earth at any depth to any perpendicular assigned, to know both the distance and Azimuth from the entrance. hydragogy, demonstrating the possible leading of water by nature's law, and by artificial help. H●rometrie, or this present work of Horologiographia, of which it is said, the commodity thereof no man would want that could know how to bestow his time. ●ographie, demonstrating how the intersection of all visual Pyramids made by any plane assigned, the centre, distance and lights, may be by lines and proper colours represented. Then followed Architecture, as chief Master, with whom remained the demonstrative reason and cause of the mechanic work in line, plane and solid, by the help of all the forementioned Sciences. Thaumaturgike, giving certain order to make strange works, of the sense to be perceived, and greatly to be wondered at. Arthemeastire, teaching to bring to actual experience, all worthy conclusions by the Arts mathematical. While I was busied in this employment which indeed is my calling, I questioned Caliopy, why she put the note of Illegitimacy upon astrology; she said, it indeed made Astronomy her father, but it was never owned to participate of the inheritance of the Arts, and therefore the pedigree doth very fitly say, doth reasonably not, quasi intellectiuè▪ but imperfectiuè; then did I ask again, why arithmetic had the distinction of an elder brother the label, she told me, because it was the unity of units, and hath three files united in one Lambeaux, and did therefore signify a mystery, then said I, why do you represent magnitude by the distinction of a second Brother, to which she said, because as the Moon, so magnitude in increasing or decreasing is the same in reason, than did she being the principal of the nine Muses, and goddess of Heralds summon to Urania, and so to all the other to be silent, at which silence was heard Harmonicon Coeleste by the various Motions of the Heavens, and Fame her Trumpeter sounded forth the praise of men, famous in their generation; and concluded with the Dedication and Consecration of the Court of Arts in these words of the learned Vencelaus Clemens. Templum hoc sacrum est, Pietati, Virtuti, Honori, Amori, Fidei, semi Deûm ergò, & Coelo Ductum genus, vos magni minoresque Dei, Vos turba ministra Deorum vos inquam. Sancti Davides, magnanimi Hercules, generosi Megistanes Bellicosi Alexandri, gloriosi Augusti, docti Platones, Facundi Nestores, imici Jonathanes, fidi Achatae. Uno verbo boni Huc adeste, praeiste, prodeste Vos verò orbis propudia Impii Holophernes, dolosi Achitopheles, superbi Amanes, Truculenti Herodes, proditorus Judae, impuri Nerones, Falsi Sinones, seditiosi Catilinae, apostatae Juliani. Adeoque, quicunque, quacunque, quodcunque es Malus, mala, malum, exeste Procul hinc procul ite prophani. Templi hujus Pietas excubat antefores, Virtute & Honore vigilantibus Amore & fide assistentibus Reliqua providente aedituo Memoria, Apud quam nomin● profiteri Fas & Jura sin●nt. Quantum hoc est? tantum Vos caetera, quos demisse compellamus, praestabitis, Vivite, vincite, valete, Favete. Et vos ô viri omnium ordinum, Dignitatum, Honorum, spectatissimi amplissimi, christianissimi, &c. Which being done, the Muses left me, and I found myself like Memnon, or a youth too forward, who being as the learned Sir Francis Bacon saith, animated with popular applause, did in a rash boldness come to encounter in single combat with Achilles the valiantest of the Grecians, which if like him I am overcome by greater Artists, yet I doubt not but this work shall have the same obsequies of pity shed upon it, as upon the son of Aurora's Bright Armour, upon whose statue the sun reflecting with its morning beams, did usually send forth a mourning sound. And if you say, I had better have followed my Heraldry (being it is my calling) henceforth you shall find me in my own sphere. FINIS.