these I put in their places as in the Figure The Gnomon must stand square upon the Substyle at an angle of 34 degrees Note that the Reclination must alwayes be reckoned from the Limb inwards upon the Finitor because where the Finitor touches the Limb there is our Zenith for this turn Inclination is reckoned from the Zenith line which here is both the Diameter of the Horizon and Horizon itself For the Opposite or Inclining face of this Dyal-plain use the general way I shewed you Chapter 12. that is strike the Substyle and all the hour lines through the Center and set the same figures to every hour line beyond the Center which he had on this side and set the Gnomon upon the Substyle downward to behold the South Pole and it is done And so by the Inclining Dyal if you had him first drawn you might presently make the Recliner CHAP. XVI How to find the Arches and Angles that are requisite for the making of the Reclining Declining Dyal BEfore you can Intelligently make a Reclining-Declining Dyal which is the most Irregular of all having two Anomalies viz. Declination and Reclination you must be acquainted with 3 Triangles in the Sphear wherein certain arches and angles lie which are neefull to be known I advise you therefore first to draw though it be but by aime an Horizontal Projection of the Sphear such as here I have drawn for a South Dyal Declining West 50 degrees and Reclining 60 degrees in the Latitude 52. 10. minutes which shall be our Example The Circle E S W N is our Horizon N S our Meridian D T d the plain Z T the Reclination thereof D d the Base or Horizontal line of the plain V u the Vertical of the plain cutting it right in T and cutting the Pole thereof at H for u is the Pole of a plain erected upon D d but the Pole of the Reclined plain D T d is H. S u or N V the Declination of the plain M P H m the Meridian of the plain cutting the North Pole at P the plain in Right angles at R and the Pole thereof at H. Now see your three Triangles all adjoyning in this Scheam viz. D N O and O R P Rectangled at N and R and P Z H. Obtuse-angled at Z. It is true that the last Triangle alone may do your work or the two first may do it without the last but you shall do well to be acquainted with them all In the first Triangle D N O you have given D N 40 degrees the Complement of the plains Declination N the Right angle of our Meridian with our Horizon D the Complement of Reclination whereby you may find D O the Oblique Ascension of our Meridian that is how many degrees of the plain the noon-line shall lie above the Horizontal line also you may find N O the Perpendicular Altitude of the noon-line or the Inclination of the noon-line of the Dyal to the Horizon where you shall note that when this Altitude of the noon-noon-line N O is equal to N P the Elevation of the Pole then is the second Triangle P R O quite lost in the point P and the plain then becometh a Declining Equinoctial plain also you may find the angle O called the Position-Reclination for a reason hereafter to be shewed Wittekindus calls it Complementum repetendum because he means to have a Bout with it again to find other arches by it In the second Triangle O R P you have given O as before for this angle was in the former Triangle or his equal for Anguli precrucem oppositi sunt aequales R the Right angle of the plain with his Meridian O P the position Latitude that is the Latitude of that Place wherein the Reclining plain O R T d Q shall be a Circle of Position this is given if you subtract N O the Perpendicular Altitude of the noon-line out of N P your Latitude this O P is Wittekindus his Differentia Retenta And hence may be found O R the Declination of the Gnomon or distance of the Meridian of the plain from the Meridian of the Place R P the Elevation of the Pole above the plain in the Plains own Meridian P the angle between the Meridian of the plain and the Meridian of the Place this angle is called the difference of Longitude because it shewes how far the places are distant from us in Longitude wherein this Dyal shall be a Direct Dyal without Declination having his Gnomon in the noon-line of the Place and and shews also how many degrees of the Equinoctial or how many hours and minutes there are between our Meridian and the Meridian of the plain as the arch O R shews how many degrees of the plain come between the said Meridians Let this be well observed by Learners In the Third Triangle P Z H you have given P Z the Complement of your Latitude Z H the Complement of the plains Reclination and Z the Supplement of the plains Declination Hence may be found H P whose Complement is P R the Elevation of the Pole above the plain P the difference of Longitude H whose measure is R T the arch of the plain between the Meridian of the plain or Substyle and the Vertical line of the plain the Complement whereof is R D the Substyle's distance from the Horizontal line of the plain Every arch and angle therefore in these Triangles is given or may presently be found by the Problemes of Spherical Triangles Book 3. But I shall shew you a short and pleasant way to find them by setting the whole Scheam at once on your Planisphear where you shall have them almost all at one view For this purpose I use my Planisphear in the Horizontal Projection But note that I make use onely of the Reet and Label and one of the Meridians of the Mater Thus I set D d on the Axis of the Reet D T d on the 60th Azimuth from the Center V u on the Finitor Z N on the Label fixing it in the Limb of the Reet 40 degrees from the Zenith according to the arch D N. Then in the Label I make a prick with ink at P for the North Pole 52. 15. minutes within the Limb and another prick in the Finitor at H for the Pole of the plain 30 degrees from the Center which done and keeping my Label fixed to my Reet I turn the Reet till I see some one Meridian cutting both the pricks P and H as the 15 Meridian from the Axis shall do in this Example and that Meridian shall serve for the Meridian of the plain for this time And by this time I see my three Triangles on my Planisphear their sides divided into degrees as a Carpenters Rule into Inches and I find for the Latitude 52. 10. minutes that D O the Oblique Ascension of the Meridian is 44.06 N O the Altitude of the Meridian 20. 22 minutes O P the Position Latitude 31. 48. minutes
〈…〉 〈…〉 Planispherium Catholicum Quod vulgo dicitur The Mathematical Jewel Per Joh. Palmer M.A. LONDINI Sumptibus Josephi Moxon The Catholique PLANISPHAER Which Mr Blagrave calleth The Mathematical Jewel Briefly and plainly discribed in Five Books The first shewing The making of the Instrument The rest shewing the manifold Vse of it 1. For representing several Projections of the Sphere 2. For resolving all Spherical Triangles 3. For resolving all Problemes of the Sphere Astronomical Astrological and Geographical 4. For making all sorts of Dials both without Doors and Within upon any Walls Cielings or Floores be they never so Irregular where-so-ever the Direct or Reflected Beams of the Sun may come All which are to be done by this Instrument with wondrous Ease and Delight A Treatise very usefull for Marriners and for all Ingenious Men who love the Arts Mathematical By JOHN PALMER M. A. Hereunto is added a brief Description of the CROS-STAF And a Catalogue of Eclipses Observed by the same I. P. The Heavens declare the Glory of God and the Firmament sheweth his Handi-work Psal 19. London Printed by Joseph Moxon and sold at his Shop on Corn-hill at the Signe of Atlas 1658. To my Honoured Friend JOHN TWYSDEN Doctour of Physick Sir MAny Learned Men have complained that Mr. Blagrave's Mathematical Jewel as he calls it both the Instrument and the Book are rarely to be found That the Book also by reason of the Interpolation of Gemma Frisius his precepts is longer then needed and by reason of the Authors frequent interruptions by vexatious Suits in Law is somewhat confused whereof himself complains in his Preface to the Reader and in the Conclusion of the Fourth Book and elswhere wishing an amendment At your request especially I undertook the reformation of that Treatise which now at length I have simshed Gemma Frisius was the first that brought this Instrument to some good perfection calling it Astrolabium Catholicum but he did that by a Cursor and Brachialum which Mr Blagrave happily devised to perform by a Reet and Label with more ease and delight I have no designe to deprave the labours or to obscure the Names or Fame of those ingenious Men by whom this Instrument was contrived and advanced to so great perfection but as Mr Blagrave said when he took upon him to reform Gemma Frisius his Treatise so I say of my last Edition of the Instrument after both Facile est Inventis addore Surely if men deceased have any knowledge or regard to what is done after them in this World and could have communication with those that remain here I suppose Mr Blagrave's Ghost would give me thanks for doing that which he heartily wished to be done but for want of leasure left unfinished and I should like wise thank him by whose means I became acquainted with this excellent Instrument which next to the Sphear or Globe it self is the best Instrument in my judgement that ever was devised for Astronomy and for the easier making and portableness is to be preferred before the Sphear it self My aime hath been throughout this Treatise to Write Plainly Methodically and with as much brevity as might consist with perspicuity remembring that of the Poet Brevis esse laboro Obscurus fio How far I have attained my Intention the Reader will judge If this work shall be found usefull to the World the thanks is due to you who first ingaged me in it and for the furtherance thereof took the pains to delineate the Instrument for me with your own hands in Brass Plates of fifteen inches Diameter which I esteem very highly both for the exactness of the work and for the work-man's sake to whom for more then twenty Years past I am also many other wayes obliged I confess I have been somewhat slow in performing my promise to you because this Treatise hath taken up onely my spare hours which by reason of infirm health and more necessary emploiments are not many Sir I am Your Servant John Palmer Ecton Apr. 1. 1658. The first book of the Catholique Planisphear Wherin The whole Fabrick or making thereof is plainly Described CHAP. I. Of the parts of the Planisphear And of the Mater his matter and Lineaments THis Planisphere is made up of five parts 1 The lying plate called Mater 2 The moving plate called the Rete Reet or Net 3 The Ring or Limb. 4 The Label 5 The Sights The Mater is a round plate of metal or past-board flat smooth and stiff the larger the better And if you will have the circles actually drawn for every degree it had need be ten Inches at least in Diameter If it be made of metal as Silver Brass or Tin and Tin-glass in equal quantity melted together it must be well pollished but it may very well be made on a fair pastboard pasted on a Messie board for thereon the Lineaments may be distinguished with inkes of several colours which cannot be if it be made in metall The Lineaments of the Mater though they be fitted to represent other circles of the Sphear also as shall be shewed yet most aptly represent the Meridians and Parallels and therefore so we call them here while wee speak of the Fabrique Among the Parallels the two Tropiques and the two Poler circles are to be inserted And lastly to these you shall add the Ecliptique and so you have all the Lineaments of the Mater For the Declination whereof 1 Upon the center of the Mater plate describe the fundamental circle about an inch within the edge if your plate be 12 inches Diameter that so a convenient space may be left for the Limb. This circle shall be the great Meridian passing through the Poles of the world and also through the Zenith and Nadir of your Countrey and is the bound by which all the Lineaments of the Mater are inclosed Draw two Diameters crossing one another in the center at right angles and dividing this Meridian into his quarters let one of these Diameters be A B the Axtree-line the other C D the Equator or his Diameter divide also every quarter of this Meridian into 90 equal parts or degrees This Meridian onely of all the circles of the Mater falleth out to be a full circle in this projection because the bisection of the Sphear is supposed here to be made in the plain thereof and because the eye is supposed to be the Pole thereof and so equidistant from every part thereof The rest of the Meridians and Parallels their Semicircles are in this Planisphere fore-shortned according to Optique reason as shall be further explained Ch 2 because they all either are great circles passing through the eye-point and cutting the Meridian at Right angles as the axtree-Axtree-line or East Meridian the Equator and the Ecliptique which are therefore projected upon their Diameters and become straight lines or else lie Oblique to the eye as do all the rest of the Meridians and Parallels which are all of them projected
square to the line A B but if any of these have failed you shall never bring the three points into an arch while the foot of your compass â—andeth in the line C D. Therefore in such case set one foot in A and draw with the other foot a short arch crossing the line C D then set the standing foot in B and with the running foot cross the short arch last drawn where these arches cross will be the center by which you may draw an arch cutting A and B and if it cut 2 also you have your desire But if this arch over reach 2 widen your Compass if it come short of 2 the middle point narrow your Compass and try again in like manner till you can compass all the three points in the same arch 3. A third way I suppose you may know that every Meridian cuts the Equator twice viz. in two opposite points distant 180 degr one from another as for example the Meridian which cutteth the Equator in 60 degr of Right Ascension cuts it again in the opposite degr viz. 240. Now if you can find these two points in the Equator line C D the center will be in the just middle betwixt them One of these points is already given within the fundamental Circle the other without is thus found Prolong your Equator infinitely beyond your plate both wayes and divide the extension thereof by like reason as you divided his Diameter viz. as by a Ruler laid from A to the several deg of the Quadrant B C you devided the Semidiameter E C into 90 degr so keeping still one end of your Ruler fixed at A and carrying about the other end thereof to the severall degrees of the Quadrant C A you may divide the excurrence of the line E C into 90 degr more and so E C and his exccurrence or continuation will be half the Equator divided into his degr and E D with his excurrence on the other side will be the other half divided by like reason And thus the whole Equator is projected in one straight line and divided into degr also Then having a point given within your fundamental circle through which the sixtieth Meridian must pass viz the 60 deg of the Semidimeter E C or E D number thence over the center to 180 deg and there is the point where the other semicircle must cross and the middle between those points is the center But because the two points taken in the quadrant A C are very near together especially towards A and the Ruler also will cross the prolongation of the line E C very obliquely you may therefore do better to divide this line into his degr by a Scale of Tangents for if upon the Equinoctial line D E C you prick down from E both wayes the Tangents of the half degr in order from 0 to 90 those pricks shall be the whole degr of the Equinoctial line in this Projection to be numbred from E both wayes to 180 deg where the Tangent becomes infinite Thus taking A E for Radius E D is the Tangent of 45 deg by the structures yet the arch or Diameter E D is a Quadrant or 90 deg of the Equinoctial in this Projection because the Tangents of the half deg of the Quadrant E A I measure out the whole degr of the line E D as was above-said 4 If you consider well what hath been said you will find or you may take it here upon trust that for the 90 Meridians to be drawn between C and E half the centers will be found in the opposite Semidiameter E D and the other half without D in the said Semidiameter prolonged And that every second division of the line E D from E toward D and forwards shall be the centers of the Meridians which cut the Semidiameter C E. As for example to draw every fifth Meridian from C to E you take every tenth deg from E toward D for the centers And further if you would not be at more trouble then needs to divide the extension of the Diameter beyond the fundamental circle you shall but do thus Begin with the crookedest Meridians first whose centers are within the fundamental Circle and first pitching one foot of your Compass in the point 1 near E extend the other foot beyond the center to 2 there is the center from which you shall draw the first Meridian A 1 B and also turning about your Compass you shall make a marke in the extension beyond D at 1 where the other Semicircle of this Meridian would cross the Equator So for the next Meridian in the line C E marked with 2 your center is beyond E at 4 and after you have drawn his arch A 2 B marke with your compass his other crossing at 2 beyond D and so with one labour you shall both draw the 45 crookeder Meridians and also make the out lying division of the line E D prolonged of which division every second or even number will be a center to some of the straighter Meridians This is a very good and easie way and this way Mr Blagrave alwaies used 5 Or lastly You may frame a decimal Scale of 1000 or 10000 parts equal to the Semidiameter of the Mater by which Scale with the help of the Cannon of Triangles you may presently find the length of any S ne Tangent or Secant you shall desire Now look what inclination any Meridian hath to your fundamental Circle that is what angle they make between them the Secant of that inclination is the Semidiameter of that Meridian and the Tangent of the same inclination is the distance of his center from E the center of the Mater For example the Meridian A 2 B his inclination is 20 deg for the angle C A 2 and likewise the arch C 2 which measures it is 20 degr the S cant of 20 degr is 10641. by the Cannon of Triangles which every Mathematician ought to have at hand Take with your Compasses from your decimal Scale 10641. and setting one foot in A with the other foot cross the Semidiameter E D in that crossing is the center or take with your Compasses 3639. the Tangent of 20 degr and set it from E toward D and it shall give you the same center at 4. For A E being Radius E 4 is the Tangent and A 4 the Secant of 20 deg by the structure And if you like to work this way it will help you much to have a short Cannon of Tangents and Secants of whole degr of the Quadrant gathered into one page which Cannon for your ease is here annexed A Table of Tangents and Secants to every degree of the Quadrant Degr Tan. Secan 1 00174 010001 2 00349 010006 3 00524 010013 4 00699 010024 5 00374 010038 6 01051 010055 7 01227 010075 8 01405 010098 9 01583 010124 10 01763 010154 11 01943 010187 12 02125 01022â— 13 02308 010263 14 02493 010306 15 02679
the arch of the Ecliptick from the Ascendant to the Midheaven and his match taken so many degrees on the other side the Center gives the other arch of the Ecliptick from the Midheaven to the Descendant The rest of the Meridians and the Parallels are in this Mode of no use The Almicanters and Azimuth of the Reet here shew you the Altitude and Azimuth of every degree of the Ecliptick at one view CHAP. II Of the Equinoctial Projection shewing the Northern or Southern Hemisphears THe Equinoctial Projection representeth the Northern or Southern Hemisphear projected upon the plain of the Equator Here the Limb or outmost Circles of the Mater and Reet are Equator The eye-point is the North or South Pole which you will by turns Which Poles are here expressed on the Center of the Equator because the Sphear is pictured on a plain or flat The Axtree line of the Mater A B is Colurus Equinoctiorum the Diameter C D crossing him is Colurus Solstitiorum But contrary on the Reet the Axletree is Colurus Solstitiorum and the Finiter Colurus Equinoctiorum The Colurus Solstitiorum on the Mater is also the Meridian of your place and therefore is marked with Septentrio and Meridies and the ends of the Axtree with Oriens and Occidens The rest of the Meridians being all straight lines meeting in the Poles or Center are casily supplyed by the Label and so may the Parallels also being Concentrick with the Equator For if you lay the Label on the 15. degree in the Limb from Meridies toward Occidens the fiduciall edge of the Label there designeth the 15 Meridian or the One a clock line the North Quadrant of the said Meridian proceeding from the Center now the North Pole outward to the Limb or Equinoctial and the South Quadrant returning in the same line from the Equinoctial to the Center now the South Pole and if you remove the Label 180 degrees from One a clock of the day there it shall designe One a clock at night made by the other Semicircle of the same Meridian which joyneth with his match in the Center without any angle that is into the same straight line and so of the rest And for the Parallels if you set the point of your Compass or a needles point in the 23. degree ½ of the Label and turn about the Label with the point it shall describe a Circle which will serve for both the Tropicks and so may you make any other of the parallels I do not advise you to draw the Meridians and Parallels in this form least you cumber your Instrument but I shew you how you may represent any of them in a moment when ocasion requireth The Meridians of the Mater that were so called in the Meridional Projection are here turned into the severall Horizons of the World And the Parallels here serve only to graduate those Horizons Out of these Horizons choose your own Horizon and distinguish him if you will that you may readily find him when you shall looke for him Your Horizon is thus inquired Because the Elevation of the Pole at Northampton is 52. degrees 15. minutes therefore from the Center now North Pole number in the Meridian line Northward 52. degrees 15. minutes and there cutteth the the North Semicircle of our Horizon or there you may Imagine him between the 52 and 53 Horizons and the Southern Semicircle thereof lies 52 degrees 15 minutes on the other side the Center towards Meridies This may seeme strange that the North and South points of the Horizon which in the Sphear are unequally distant from the North Pole viz. the one but 51. degrees 15. minutes and the other 127. degrees 45. minutes the supplement thereof should be equally distant in this Projection But the reason is because the Center is both North and South Pole here at pleasure and the Northern and Southern Hemisphears are both here represented by turns Carry this in your head and then lay the Eabel upon the South part of the Meridian and number thereon from the Center now North Pole outward to the Equator at the Limb 90. degrees thence number backward toward the Center now the South Pole the Elevation of the Equator which is always complement of the Elevation of the Pole and is here 37. degrees 45 minutes there is the Southern point of the Horizon and is distant from the Center now South Pole onely 52 degrees 15 minutes but from the Center being North Pole 127. degrees 45. minutes and from the Northern point of the Horizon before found just 180. degrees as it is in the Sphear Having found the North arch of your Horizon 52. degr 15. min. behind the Center count as many degrees and minutes forward in the Meridian before the Center toward Meridies and the arch crossing there shall be his match to make up the whole Circle and so may you find your whole Horizon upon the Mater whatsoever your Latitude be Here you must remember that Stars which have Northern Declination rise and set upon the Northern arch of the Horizon and those which have Southern declination upon the Southern arch Remember also that many Stars between the Tropicks which have Northern Latitude have nevertheless Southern Declination and contrary many which have Southern Latitude have Northern Declination The lineaments of the Reet serving you in this Projection are onely the Ecliptick and the fixed Stars the Almicanters and Azimuths here are of no use The Meridians and Parallels are supplyed by the Label for the Reet as well as for the Mater And whereas the Ecliptick here seemes to be irregular seeing the Solstitial points of Cancer and Caprcorn are not distant 180 degrees as they should be you must imagine that the Southern arch of the Ecliptick is Projected by the eye placed in the North Pole and for the Northern arch the eyes place in the South Pole and the Center serveth for both the Poles alike as hath been shewed number therefore as you were taught for the Horizon in this Projection For the reason of the draught of the Horizon and of the Ecliptick in this Projection is the same CHAP. III. Of the Nonagesimal Projection shewing the Eastern and Western parts of the Sphear being divided by the Azimuth of the Nonagesimus gradus NUmber in the Limb from the Equinoctial line toward the Pole the Altitude of the Nonagesimus gradus which is the highest degree of the Ecliptick and thereto set the Finitor turning the Almicanters either to the North or to the South as your work proposed shall require Now is the Finiter Ecliptick his point at the Limb-is Nonagesimus gradus The Center of the Planisphear is Ascendant and Descendant the East and west points of the Horizon are here distant from the Center as much as the Amplitude of the Ascendant cometh to to be counted from the Center upon the Eqinoctial line of the Mater which here stands for Horizon the Meridians and Parallels of the Mater are here
find in the next rule 2. Another way Mark what is the Right Ascension of the point proposed being counted from the next Equinoctial point as of ♉ 9 degr the Right Ascension is 36.36 min. count so many degrees in the Arctick circle from the Axeltree at the end of this number is the Pole of the Ecliptick Lay the Label to him and you shall make a Quadrantal Triangle whose Sides shall be equal to the Angles of the former Triangle which was made of the Longitude Declination and Right Ascension of the point proposed for the Right Angle you have a Radius or Quadrant of the Axis for the Angle of the greatest Declination between the Equator and Ecliptick 23 ½ you have the arch of a Meridian between the Pole of the Equator and the Pole of the Ecliptick for the angle sought you have the arch of the Label between the Pole of the Ecliptick and the Center 71.20 minutes as before the least angle of this Quadrantal Triangle is at the Center and you shall find his measure in the Limb 14.32 minutes that is the measure of the least Side of the former Triangle viz. the Declination of the point proposed Here you see If the Declination had been given you should have set it in the Limb between the Pole and the Label and so had you made the same Quadrantal Triangle and might have found on the Label between the Arctick Circle and the Center the measure of the angle sought and likewise in the Arctick Circle between the Label and the Axtree-line the Right Aseension though it be more then was required The reason hereof you may learn from Book 3.7 CHAP. X. To find the point of the Ecliptick in which the Longitude and Right Ascension have greatest difference Move the Label on the Polar circle till you find the degrees of the Label between the Polar circle and the Limb to be equal to the degr of the Limb between the Label and the Pole so have you a Rectangled aeqaicrurall Triangle made by the Limb Label and the Meridian 46 ¼ like to that in the second Variety Book 3.10 Here the angle B at the Pole between the 46 ¼ Meridian and the Limb is equal to the Longitude of the point sought 46¼ and either Leg is equal to the Declination thereof 16 ¼ Therefore I conclude that when the Sun is 46 ¼ in Longitude that is in ♉ 16 ¼ then his Longitude hath furthest out run the Right Ascension Subtract now the Right Ascension of ♉ 16 ¼ which is 43 ¾ out of the Longitude 46 ¼ there remains 2 deg ½ which being converted into Time is 10 min. the greatest inequality of Ascension in a Right Sphear CHAP. II. To find the Latitude of your Place or the Elevation of the Pole above your Horizon by the Meridional Altitude and Declination of the Sun Meridional Projection GEographers call the distance of a place from the nearest point of the Equator upon Earth the Latitude of that Place as the Latitude of London is 51 deg 32 min. from the Equator Northward the Latitude of St Thomas Island upon the coast of Africk is 0 deg 0 min. because the middle of that Island lyeth under the Equator And because the Latitude of your Place and the Elevation of the Pole above your Horizon are alwaies equal therefore the Elevation of the Pole is oft called Latitude of the Place or Latitude simply and so for brevity sake we shall often call it But when we speak of the Latitude of the Moon or Stars you must understand Astronomers thereby mean their distance from the neerest point of the Ecliptick To find the Latitude of your Place get the Suns Declination by the 6 or 7th and his Meridian Altitude by the second of this Book Then find the parallel of the Suns Declination North or South as the Declination is and where it toucheth the Limb here Meridian there is the point where you observed the Sun at Noon set the South end of the Finiter so many degr below this point as the Meridian Altitude had then is your Finiter set to your Latitude and you shall find the measure of it between the Equator and the Zenith which is properly the Latitude and the same measure shall you find between the North point of the Finiter and the North Pole where it is more properly called the Elevation of the Pole Example June 20 1651. I observed the Meridian Altitude of the Sun here at Ecton four miles Eastward from Northampton 60 degr 59 min. the Longitude of the Sun was then ♋ 8 degr 19 min. ½ his Declination 23 degr 14 min. Northward Therefore having found in the Limb the point where the Parallel 23 degr 14 min. toucheth above the Equator I put the South end of the Finiter 60 degr 59 min. below that point toward the South Pole which done I see the North Pole Elevated above the Finiter 52 degr 15 min. and the Zenith of my Horizon likewise to be removed from the Equator Northward 52 degr 15 min. which is the Latitude of Ecton Note that you may best observe the Latitude when the Sun is near the Summer Tropick for then you shall not be troubled with Refraction and then the Declination varyeth slowly which varyeth almost one minute every hour near the Equinoctial CHAP. XII To do the same by the Meridian Altitudes of the Stars about the Poles MAny of the Stars near the Northern Pole may be seen with us twice in the Meridian in one Winters Night that is one while above the Pole and 12 hours after again below the Pole As for Example the Pole-star called Alrucabe about December 18 will be in the Meridian above the Pole at 6 of the clock at Night and at 6 next morning he will be in the Meridian below the Pole Observe both the Meridian Altitudes and add them together half that sum is the Elevation of the Pole Example I observed at Ecton the greatest Altitude of the Pole-star to be 54 deg 45 min. and his least Altitude 49 degr 45 min. the sum is 104 deg 30 min. the half 52 degr 15 min. the Latitude of Ecton and here I have gotten also the Pole-stars distance from the Pole and consequently his Declination which is the complement thereof for the Latitude being subducted from the greater Altitude leaves the Stars distance from the Pole 2 degr 30 min. and consequently shewes his Declination to be 87 degr 30 min. which is 39 min. more then Gemma Frisius observed it Anno Dom. 1547. for in our age the Pole-star approcheth about 1 min. nearer the Pole in every 3 years Note that these Stars which are distant from the Pole less then the Latitude and more then the complement thereof have their less Meridian Altitude in the North part of the Meridian and their greater Meridian Altitude in the Southern part of the Meridian beyond the Zenith Wherefore for them you shall take the complement of
it because the Sun is then in the Horizon of this Dyal where he projects the shadows of all upright things infinite And as you found the points for the hours afternoon so may you by like reason find the points for the 5. morning hours and their quarters also if you please which had if by these points found you draw lines Parallel to the Vertical line of the plain which is here the Meridian of the plain and of the place they shall be the true hour-lines And the Gnomons edge must stand over the Meridian and Parallel to it at the same distance that the Axtree line of your Planisphear was situate in projecting the hour points If you cannot fix your Planisphear on a board as abovesaid or if your plain require a Gnomon of a greater or lesser height you may upon any board presently draw so much of your Planisphear as serves for this purpose Or to say more breifly Do but make the height of your Gnomon Radius and the Tangents of 15 30 45 60 75. shall give the distances of the hour points in the Horizontal line on both sides from the Center of the Dyal as may appear in the Figure CHAP. VI. How to make the East Equinoctial Dyal or the West THis plain is a Right Horizon of those People who dwell under the Equator distant from us 90 degrees of Longitude as the South Equinoctial-plain of the last Chapter was the Horizon of those who dwel under the Equator in the same Longitude with us Therefore these Dyals are in all points alike onely the Substylar-line which in the South Equinoctial Dyal is 12. in this East Equinoctial Dyal is but 6. in the Morning for our Country because of the difference of Longitude Lastly draw a line Parallel to the Contingent line at such distance as the plain will afford as the line E I and to this you shall protract your hour lines drawing them from every point of the Contingent to this so that they make Right angles with the Contingent and with this Parallel even as the rounds of a Ladder do with the sides but that the distance of the rounds of a Ladder are equal and these distances be unequal The Gnomon must be set like a Bridge Perpendicularly over the 6. a clock hour-line the edge that casteth shadow being Parallel to it and of such height as the line K G of your Planisphear or so that if the Gnomon fall his edge may lie in the line of 3. or of 9. of the clock This also may be made speedily by help of the Tangents as the South Equinoctial Dyal For the West Equinoctial Dyal it is made like the East in all points onely it shews but the after Noon hours as the East shews but those of the fore noon When you have drawn on paper the East Dyal and set it by guess in its Situation go on the West side of it and you may see through the paper the picture of the West Dyal and so will the back side of the West Dyal shew you the true picture of the East CHAP. VII How to make the Declining Equinoctial Dyal ANy Declining plain may be so Reclined that he shall become a Right Horizon or Equinoctial plain and at what Reclination this shall happen you may easily find by Chapter 19. Latitude 52. 10 minutes Declination 50. 00. West Reclination 26.32 D. L. 43.16 Ascension of Noon-liee 61.59 Set the line M E N for the Axis of the World or Gnomon and prop him up over the line P Z with two props of equal height and Perpendicular to the plain and make the point E which standeth Perpendicularly over Z the Center of the World then from this Axis or Gnomon mounted in the Air shall the hour-lines be projected distinctly and all of them shall be Parallel to the Axis and one to another as it hapneth in all sorts of Equinoctial Dyals The line Z P shall remain now onely the Meridian of the plain or Substylar And to find the hour-lines you shall do thus Draw through Z an Equinoctial or Contingent line E Z making Right angles with the Axis M E N or Z P then setting Z O equal to Z E draw upon the Center O with an extension of the Compasses the arch of the Equinoctial b Z d or the Parallel arch passing by D. Then number in this arch from the Substyle Westward if your plain decline West or Eastward if it decline East the difference of Longitude and where it ends there is the point of Noon in this arch from that point begin to divide the said arch by fifteens of degrees or 24th parts of the whole Circle And remember that when you come to 90 degrees from the Substyle you need divide no further for the Sun is no longer upon this plain Also you may leave out those hours at which the Sun is alwayes under our Horizon as the hours from 8. at Night to 4 in the Morning then lay a Ruler from the Center O to every one of these divisions of the Circle and where the Ruler cuts the Contingent there make points for the hours respectively and through these points you may draw the hour-lines Parallel to the Substyle of what length you please and mark them from the Noon line Eastward 1 2 3. c. because the Suns Diurnal course is Westward and the course of the shadow is contrary He that will may make use of his Planisphear for dividing the hours as was taught Chapter 4 and 5. or use a Quadrant or a Scale of Chords or the Tables of Tangents with a Sector or a Scale of equal parts But it needs not Note that this Dyal may compare with the hardest however Mr Blagrave and other Dyalists have omitted it as seeming easy and here Wittekindus to whom all later Dyalists are much beholden and after him Fale were mistaken using the Declination of the plain where they should have used the difference of Longitude in the making of this Dyal CHAP. VIII Of the kinds of Oblique Dyals WHat an Oblique Dyal is and why it is so called hath been shewed Chapter 2. They be Regular Irregular The Regular lie in some notable Circle of the Sphear as 1. The Vertical Dyal whose plain lieth in the Horizon for which cause many call it the Horizontal Dyal 2. The South and North Horizontal Dyal whose plain lies in the East Azimuth and it is commonly called the South or North Erect Direct Dyal As for the East and West Dyals they belong to another place as was said Chapter 5. The Irregular are such as lie Oblique to the Horizon as Reclining or Inclining Dyals or lie Oblique to the Meridian as Decliners or else Oblique to both as Recliners or Incliners Declining which are esteemed the hardest of all because of their double Irregularity though by the Planisphear they are made almost as easily as the rest The Declination of a plain is the Azimuthal distance of his Poles from the Meridian
Meridian then to the Vertical of the plain and thus if you draw a rude Scheam of your Case you may soon reason out the Declination better then do it blind-fold by the rules commonly given And by these two last wayes you may take the Declination not only of upright plains but of Recliners also for which the first way will not serve CHAP. XII How to make a Declining Horizontal Dyal But Declining Dyals which look awry from our Meridian have a Meridian of their own which is called the Meridian of the plain and the Substyle because the Style or Gnomor stands upon it and is indeed the Meridian of that Place where this Declining Dyal would be a Vertical Dyal and where the Substyle would be the Noon line and to this Substyle the hours of the plain are alwayes so conformed that the nearer they be to the Substyle the narrower are the hour spaces and contrarily because the Meridians do so cut every oblique Horizon that is thickest near the Meridian of the place and this Declining Dyal being a stranger with us followeth the fashion of his own Country and so hath his narrowest hour spaces near his own Meridian rather then ours Now as that is the Meridian of our place which cutteth our Horizon at Right angles passing through his Poles Zenith and Nadir so the Meridian of any plain is that which cutteth the plain at Right angles and passeth through his Poles You may find all these requisites in the Meridional Projection not only for one but for all Declinations lying as in a Table before you with admirable ease and delight for there is no Declining Wall or Horizontal plain but we have an Azimuth in the Reet which shall picture him and look how the Meridians divide these Azimuths so do they divide the Horizons or Circles of the Declining plains The Pole of any Azimuth is found in the Finitor 90 degrees distant from him the Meridian that cuts the Pole of the Azimuth cuts also the Azimuth and the plain thereby represented at Right angles and is the Meridian of the plain or Substylar Chapter 2. Theorem 11. and the degrees of that Meridian between the plain and the next Pole of the World are the Elevation of the Pole above the plain and so the Elevation of the Gnomon or Style and the arch of the plain comprehended between this Meridian of the plain and the Limb is the Declination of the Gnomon or distance of the Substyle from the Meridian or distance of the Meridian of the plain from the Meridian of the Place What would you more Example If a Wall Decline East 30 degrees I say because the face of the Wall looketh 30 degrees from the South Eastward therefore the plain which lieth 90 degrees from his Pole is in the 30th Azimuth from the East Northward therefore I go to the 30th Azimuth from the East line of Axis counted cither way and take that Azimuth and his Match which is equally distant from the Axis for the very picture of my Declining plain Then seeking the Substyle or Meridian of the plain I say the Pole of the plain is in the Finitor at the 30th Azimuth from Meridies in the Limb because the plain it self is the 30th Azimuth beyond the Axis the Meridian that cuts this Pole is the 36 ¼ exactly 36. 8 minutes the number whereof shewes me the difference of Longitude between our Country and the Country of this Dyal This 36 ¼ Meridian being the Meridian of the plain I follow toward the Pole and find him cutting both the arches of my plain on both sides the Axis but I regard the cuttng only in that arch which is nearest to the Pole because there the angle looks more like a Right angle and there is the nearest distance of the Pole from the plain and there I see the hour spaces least from that Intersection therefore I reckon in the same Meridian to the Pole 32 degrees and perhaps a minute more you may find it by Calculation this is the Elevation of the Pole above the plain and of the Gnomon likewise also from the same Intersection I reckon in the plain to the Limb or Meridian 21. degrees 10. minutes the distance of the Meridian of the plain from the Meridian of the Place the same is the Declination of the Gnomon or of his Substyle Then for the hours I begin at the Zenith of the Reet where is our Meridian and numbring first toward the Substyle I seek at what number of degrees from the Zenith the hour Circles cut my plain and I find as followeth deg min. 11. 9. 35 10. 17. 54 9. 25. 54 8. 34. 22 7. 44. 17 6. 57. 10 5. 75. 4 Then in the other arch of the plain I have the afternoon hours thus deg min. 1 12. 10 2 28. 59 3 52. 26 4 80. 17 And further I cannot go because I see the next hour is above 90 from the Substyle therefore my Dyal receives him not on this side but on the North side there is use of him Now to draw the Dyal I consider that because the plain declines East therefore the Gnomon shall decline West for the Dyal being such a projection of the Sphear wherein all the Viâ—ual lines cross in the Nodus of the Gnomon and thence disperse themselves again toward the plain therefore that which is East in the Sphear will be expressed West on the plain and contrarily as was shewed Chapter 2. Theorem 11. Also I consider that howsoever the plain be turned East or West the Gnomon's place is fixed because it is a part of the Axis of the World or a line Parallel to it Now therefore if I turn a South Dyal and make him Decline East and hold the Gnomon unmoveable the West side of the Dyal will approach nearer to the Gnomon as reason and sence will tell me likewise the hours which are found on the same side of the Meridian or noon-Noon-line with the Substile must be set the same way with it from the Noon-line in the Dyal Therefore having drawn an Horizontal line E W on the Wall from the Center taken at A I let fall the Perpendicular A B for the Noon-line then upon the Center A. I draw a blind Semi-circle with the Semi-diameter of my Planisphear or of some Quadrant as E B W and therein I prick down the Substyle and the hours after the manner used the 10th Chapter This is the most ready way to delineate the opposite face of any Dyal See another way to make this Declining Horizontal Dyal Chapter 21. CHAP. XIII How to Observe the Reclination or Inclination or any Plain WHat Reclination and Inclination are hath been shewed Chapter 8. All Reclining and Inclining plains have their Bases or Horizontal Diameters lying in the Horiz ontal Diameter of some Azimuth but the top or Nonagesimus gradus of the plain from the Horizon leaneth back from the Zenith of your Place in the Vertical of the plain which is the
plain The top of this Radius in the Gnomon is called Nodus because you must there set a Knot Bead or Button to give shade or else cut there a notch in the Gnomon or cut off the Gnomon in the Place of Nodus that the end may give the shadow for those lineaments Let not your Nodus stand too high above the plain for then the shadow will fall beside your plain for too great a part of the plains day nor let it stand too low for then the lineaments will run too close together A mean must be chosen At the foot of this Radius take your Center and describe a Circle on the plain and divide it into equal degrees and from the Center draw lines through those degrees infinitely that is so far as your Dyal-plain will bear these lines shall be the Azimuths of the Horizon of the plain and shall be numbred from his Meridian or Sub style And to divide the said Azimuth lines you use the Tables of Tangents with a Scale of equal parts or else plot the Tangents thus on paper set A B equal to the Radius of your Horizon and with that Radius draw the Quadrant A B C or A b c and divide the Quadrant numbring the degrees from C to B and having drawn the Tangent B D or B d Parallel to A C draw lines from the Center through the several degrees to the said Tangent-line so shall this Tangent-line be divided for your purpose and from it you may transfer the divisions to your plain Now if your plain lie in the Horizon of your place as the Vertical plain doth these Azimuths and Almicantars may be of some use to shew you the Altitude and Azimuth of the Sun for any time See them in the Scheam Chapter 30. But if your plain lie not in the Horizon of your place then you shall draw the said Almicantars or Azimuths or so many of them as you shall need in deleble lines because here they serve only the Horizon of the plain yet shall they help you to describe the Equator and his Parallels with the Horizon of your Place in any Dyal and when they have done this unless your Dyal be Vertical they may be gone CHAP. XXVIII How by help of the proper Azimuths and Almicantars of the Plain to describe the Equator and his Parallels on the Polar or Orthognomonical Dyal IT shall suffice here to shew how the Parallels of the Signes may be described because the Parallels of Declination and of the length of the day are described by like reason And know that in the Polar plain because the Gnomon is Perpendicular to the plain the same Gnomon shall serve both Hours and Azimuths for the hour-Hour-lines be Azimuths in this plain Note also that the Sun is never Elevated above this plain more then he Declineth from the Equator which at the most is 23 ½ degrees and that if the height of Nodus be above a sixth part of the Semidiameter of the plain the ten first Almicantars will fall beside the plain A sixth part therefore must serve and that will give you all the Altitudes above to degrees and the Parallels of the Signes whose Declination is more then 10. Describe therefore the Almicantars here as you were taught Chapter 27. for in the Hour lines you have already every 15th Azimuth and may draw more if you please CHAP. XXIX How to inscribe the Equator and his Parallels in the Equinoctial or Parallelognomonical Dyal IF this plain Decline not the Hour lines of your Country will serve you for they be also the Hour lines of the plain and the Noon-line is Sub style if it do Decline you shall draw in deleble lines the proper Dyal of the plain by Chapter 23. which Declineth not And having here the Azimuths or Almicantars of the plain drawn by Chapter 27. you shall observe upon your Planisphear at what Altitude or at what Azimuth the Parallels cut the several Hour lines and where the like Altitude or Azimuth cuts the same Hour lines upon the plain you shall make marks and through those marks draw the Parallels which shall be all Conical sections except the Equator which because he is a great Circle shall be a straight line on those plains and in all other plains except the Polar where he is a Circle In the Equator the Azimuth is always the same full East or West and so upon your plain he must needs be a straight line The Altitudes in the Equator are 15 30 45 60 75 90. The Hours alike distant from the Meridian on both sides are alike and so are the Parallels alike distant from the Equator alike also When you have therefore gathered a Table out of your Planisphear for the Morning Hours of the North Parallels and of the Equator as I have done here in hast for the Equator and Tropique of Cancer you may by that Table prick down the Parallels upon one quarter of your Dyal and by that also draw the rest for as you may see upon your Planisphear all the 4. quarters are alike Note that the Azimuths cut the Hour lines too Obliquely it is best therefore to trust to the Almicantars and so shall you have easier and surer work though you meddle not with the Azimuths at all CHAP. XXX How to inscribe the Equator and his Parallels in an Oblique or Scalenognomonical Dyal IF the plain neither Decline nor Recline and so be a Vertical plain the Hour lines of your Place will serve you for they be also the Hour lines of the plain and the Noon line is the Sub-style If it either Decline or Recline or both Decline and Recline you shall draw in deleble lines the proper Dyal of the plain by Chapter 23. and so this Dyal shall be reduced to a Vertical Dyal and be as casily furnished with the Parallels as the Vertical and when you have by the deleble Hour lines of the plain inscribed the Parallels you may wipe out those Hour lines of the plain and let the Hour lines of the place and the Parallels stand Having therefore drawn the Azimuths or Almicantars of your plain by Chapter 27 take your Planisphear in the Meridional Projection setting the Finitor to the Latitude of your plain Then find your Equator and Parallels on the Mater and where the several Hour lines intersect them above the Finitor mark what Azimuth or rather what Almicantar passeth through these intersections for in the same Azimuth and Almicantar shall the Parallels cut the Hour lines of the plain upon the plain Example In the Vertical Dyal for our Latitude 52.15 minutes I set the Finitor to this Latitude and going first to the Tropique of Cancer I begin at the Limb that is at Noon there I see the Azimuth full South the Altitude 61.15 minutes at 1. a clock Azimuth 27. Altitude 59. at 2. Azimuth 49 ½ Altitude 53 ⅓ at 3. Azimuth 67. Altitude 45 ½ c. Therefore where I find