hauing the greater latitude is And say As the tang of the compl of the lesser distance from the pole is to the Radius So is the Sine of the compl or excesse of the difference of longitudes to the tang of the first base Take this first base out of the greater distance from the pole and the remaines shal be the second base Then say As the Sine of the compl of the first base is to the Sine of the compl of the second base So is the Sine of the compl of the lesser distance from the pole to the Sine of the compl of the distance of the two Starres If any man will take paines to calculate by this last Rule the distances of some noted starrs of the first second and third magnitudes round about the heauens which are not aboue 5 or 6 degrees at the most one from the other and shall keepe them written in his booke they may serue as a Rule or Instrument whereby he may reasonably estimate with his eye the distance of any Planet or Comet or other apparition from a knowen fixed starre not very farre remote by comparing the distance which hee would know with some of those knowen distances which he shall find either to be equall or else to haue some proportion thereto 42. The longitude and latitude of any two Cities being giuen to find their distance The manner of the operation is the very same with the former vnto which therefore I referre the Reader onely will note that in the heauens the longitude and latitude is taken in respect of the Ecliptic which being the way of the Sunne all the starrs in their proper motion haue reference vnto it as vnto their measure and rule But in the Earth the principall Circle is the Equinoctiall diuiding it into the Northerne and Southerne he misphaeres And therefore in the earth the longitude and latitude is reckoned by the Aequinoctiall The distance of two places vpon the Earth being found in degrees may bee conuerted into English miles by taking 60 miles for euery degree and one mile for every minute 43. To find at what hower a fixed starre commeth into the Meridian any day Seeke the Right ascension of the Sunne for that day by Sect 〈◊〉 and subduct it out of the Right ascension of the Starre And reduct the degrees remaining into howers by Sect 1. The same shall shew how long time from the Noone before the same starre shall come into the Meridian Wherefore if at any time of the night a Starr whose Right ascension is knowne be in the Meridian the hower of the night is easily found 44. The height of any knowne Starre aboue the Horizon being by any means given to find the hower of the night First seeke out the hower of that starrs comming into the Meridian the same day by Sect. 43. Againe seeke out the horary distance of that starr from the Meridian according to Sect 19. And then if the starr bee on the East side not yet come to the Meridian take the difference of those two numbers but if the starre bee past the Meridian take the Summe of them for the houre of the night 45. The height of the Pole being given to find the comming of any fixed Starre in the due East or West Say As the Radius is to the tang of the starres declination So is the tang of the compl of the Pole to the Sine of the compl of the Starres horary distance from the Meridian 46. The height of the Pole being giuen to find the Altitude of any fixed starr aboue the Horizon being due East or West Say As the Sine of the height of the Pole is to the Radius so is the Sine of the Starrs declination to the Sine of the Altitude at due East or West 47. By the Altitudes of any two knowne fixed starrs taken when they are both in the same Azumith to find the height of the Pole First say As the Sine of the difference of the starrs altitudes is to the Sine of the difference of their Right ascensions so is the sine of the neerer starrs distance from the apparent Pole to the Sine of an angle to be kept Againe compare the furthest starrs distance from the Pole with the distance from the Zenith and say As the Radius is to the Sine of the compl of the Angle kept so is the tang of the lesser of the compared arches to the tang of the first base Subduct the first base out of the greater of the two compared arches and the remaines shall bee the second base Then lastly say As the Sine of the complement of the first base is to the Sine of the compl of the second base so is the Sine of the compl of the lesser of the two compared arches to the Sine of the height of the Pole 48. To find out the horizontall Parallax of the Moone First the distance of the Moone from the Center of the earth must be knowne in Semidiameters of the earth which vnto them that are acquainted with the Theorie of the Planets is not very difficult And whereof peraduenture I may hereafter teach the practise by most easie and exact instruments which I haue long since framed Say As the distance of the Moone from the center of the earth is to the Semidiameter of the earth So is the Radius to the Sine of the Moones horizontall parallax in that distance 49. The horizontall Parallax of the Moone being given to find her Parallax in any apparent altitude As the Radius is to the Sine of the altitude of the Moone so is the Sine of the horizontall Parallax to the Sine of the Parallax in that altitude 50. The place of the Moone in the Ecliptic hauing little or no latitude as in the Eclips of the Sunne together with her Parallax of altitude being giuen to find the Parallaxes of her longitude and latitude If the Moone bee in the 90th degree of the Ecliptic shee hath no Parallax of longitude and the Parallax of latitude is the very Parallax in that altitude But if the Moone be not in the 90th deg say As the Radius is to the tang of the angle of the Ecliptick with the horizon So is the Sine of the compl of the distance of the Moone from the Ascendent or descendent degree of the Ecliptick to the tang of the compl of the angle of the Ecliptick with the Azumith of the Moone Againe say As the Radius is to the Sine of that angle So is the Parallax of the Moones altitude to the Parallax of her latitude Lastly say As the Radius is to the Sine of the compl of that former angle So is the Parallax of the Moones altitude to the Parallax of her longitude which is to bee added to the true motion of the Moone if she be on the East part of the 90th degree of the Ecliptick or to be subducted out of it if she be on the West
that imaginary parallel doth the manifold vse of this Instrument especially rely because the true place of the Sunne all that day is in some part or point of that circle Wherefore for the better conceiuing and bearing in minde thereof euery fift parallel is herein made a little grosser then the rest I Vse And thus by the eye and view only to behold and comprehend the course of the Sunne both for his Annuall and Diurnall motion may be the first vse of this Instrument II Vse To take the height of the Sun aboue the Horizon Set vp the pinne which is therefore made fit for the hole at the center perpendicular in the center and put the Indices on both sides downe vpon the Meridian that they with their waight may not sway the Instrument any way as it hangeth then with a threed put into the hole aboue in the handle hang it perpendicularly bearing the edge toward the Sunne that the pinne may cast a shadow vpon the degrees in the limbe for that degree which the shadow of the pinne cutteth in the limbe is the height of the Sunne aboue the Horizon at that present III. Vse To find the Declination of the Sunne euery day Looke the day of the moneth proposed in the Ecliptic and marke how many degrees the prick shewing that day is distant from the Equinoctiall either on the Summer or Winter side viz North or South Example I. What will the Declination of the Sunne be vpon the 11th day of August Looke the 11th day of August and you shall find it in the sixt Circle aboue the Aequinoctiall now because each Parallel standeth as hath beene said before for 2 degrees the Sunne shall that day decline North-wards 12 degrees Example II. What Declination hath the Sunne vpon the 24th day of March Looke the 24th day of March and you shall find it betweene the second and third Northerne parallels as it were an halfe and one fift part more of that distance from the second reckon therefore 4 degrees for the two Circles and one degree for the halfe space so shall the Sunnes declination bee 5 degrees and about one fift part of a degree Northward that same day Example III. What Declination hath the Sunne vpon the 13th day of Nouember Looke the 13th day of Nouember and you shall find it below the Equinoctiall tenne parallels and about one quarter which is 20 degrees and an halfe South-wards So much is the Declination And according to these examples iudge of all the rest IIII. Vse To find the Right ascension of the Sunne euery day Imagine an hower line through the day of the moneth giuen and marke in what point it will crâ—sse the Aequinoctiall then lay a Ruler or a streight Scroule of paper to the Pole of the world noted in the Instrument with P W and that same point For the Ruler shall in the innermost Circle of the limbe of the South side cut the Right ascension of the Sunne for that day to be reckoned from the West to the point of intersection for the first or vpper Semicircle of the Ecliptic or from the East together with 180 for the second or lower Semicircle of the Ecliptic V. Vse To find the longitude of the Sunne or in what degree of the Signe he is euery day The Pole of the first Semicircle of the Ecliptic is noted P I. and the Pole of the second Semicircle is noted P I I. Lay a Ruler or a streight Scroule of paper to the day of the moneth and the proper Pole of the Semicircle of the Ecliptic in which it is for the Ruler shall in the innermost Circle of the limbe on the South side cut the degree of the Sunnes place in the Ecliptic reckoning it in the same manner as you did in finding the Sunnes Right ascension and the Arch thus found is called the longitude of the Sunne which may bee expansed into signes by reckoning on the limbe from the West to South ♈ ♉ ♊ and from South to East ♋ ♌ â™ then backe againe from East to South â™’ â™ â™ and lastly from South to West ♑ â™’ ♓ allowing 30 degrees for each of those twelue signes VI. Vse To find the Diurnall Arch or Circle of the Sunnes course euery day The Sunne euery day by his motion as hath beene said describeth a Circle parallell to the Aequinoctiall which is either one of the Circles in the Instrument or some-where betweene two of them First then seeke 〈◊〉 of the moneth and if it fall vpon one of those 〈◊〉 that is the Circle of the Sunnes course that same day But if it fall betweene any two of those Parallelâ— imagine in your minde and estimate with your eye another Parallel through that point betweene those two Parallels keeping still the same distance from each of them As in the first of the three former Examples The circle of the Sunnes course vpon the 11th day of August shall be the very sixt Parallel aboue the Aequinoctiall towards the Center In Example II. The Circle of the Sunnes course vpon the 24th day of March shall bee an imaginary Circle betweene the second and third Parallels still keeping an halfe of that space and one fift part more of the rest from the second In Example III. The Circle of the Sunnes course vpon the 13th day of Nouember shall be an imaginary Circle betweene the tenth and eleauenth Parallels below the Aequinoctiall still keeping one quarter of that space from the tenth VII Vse To find the Rising and Setting of the Sunne euery day Seeke out as was last shewed the imaginary Circle or Parallel of the Sunnes course for that day and marke the point where it meeteth with the Horizon both on the East and West sides thereof for that is the very point of the Sunnes rising and setting that same day and the hower lines which are on both sides of it by proportioning the distance reasonably according to 15 minuts for the quarter of the hower will shew the hower of the Sunnes rising on the East side and the Sunnes setting on the West side VIII Vse To know the reason and manner of the Increasing and Decreasing of the dayes and nights throughout the whole yeare When the Sunne is in the Aequinoctiall it riseth and setteth at 6 a Clock for in the instrument the intersection of the Aequinoctiall and the Ecliptic with the Horizon is in the 6 a clock Circle on both sides But if the Sunne bee out of the Aequinoctiall declining toward the North the intersections of the Parallel of the Sunne with the Horizon is before 6 in the Morning and after 6 in the Euening and the diurnall Arch of the Sunne greater then 12 howers and so much more great the greater the Northerne Declination is Againe if the Sun be declining toward the South the intersections of the Parallel of the Sunne with the Horizon is after 6 in the Morning and before 6 in the Euening and the diurnall Arch lesser then 12
at that point and the pinne vpright in the Center hold or set your instrument parallel to the plaine of the Horizon with the pinne toward the Sunne and moue it gently till the shadow of the pinne shall fall exactly vpon the fiduciall edge of the Labell For then the Meridian line of the instrument shall be in the true Meridian of the place and the foure quarters of the instrument shall looke into the foure cardinall points of East West North and South Wherefore if with a bodkin you make a prick at each end of the Meridian of your instrument where it standeth and with a Ruler draw a line through them the same shall bee the Meridian of that place This is a most excellent practise for finding out the Meridian in any place and is in an instant performed and that easily And hereby you may examine the Variation of the Compasse And also exactly place any Sunne Dyall XX. Vse Considâ—rations for the vse of the instrument in the night In such questions as concerne the night or the time before Sun rising and after Sunne setting the instrument representeth the lower Hemisphaere wherin the Southerne Pole is eleuated And therefore the Parallels which are aboue the Aequinoctiall shall bee for the Southerne or Winter Parallels and those beneath the Aequinoctiall for the Northerne or Summer parallels And the East shall be accounted for West and the West for East and the North shall bee accounted for South and the South for North contrary to that which was before when the Instrument represented the vpper Hemisphaere XXI Vse To find how many degrees the Sunne is vnder the Horizon at any time of the night Seeke the declination of the Sunne for the day proposed and at the same declination on the contrary side imagine a Parallel for the Sunne that night and marke what point of it is in the very hower and minute proposed then set the Index or Labell to that point of the Parallel and it will shew you thereon the degree of the Sunnes depression vnder the Horizon XXII Vse To find out the length of the Crepusculum or Twilight euery day Because the question concerneth the night time you must seeke out the Sunnes Parallel for the night on the other side of the Aequinoctiall hauing the same declination with that which the day of the moneth sheweth then moue about the Labell vntill the said Parallel cutteth the edge thereof in the 18th deg on the West side for the Morning Twilight and on the East side for the Evening Twilight of the same day And note that in the height of Summer the Twilight in our Horizon continueth all night long because the same goeth not vnder the Horizon full 18 degrees XXIII Vse To find the Declination of any Wall or Plaine Take a board hauing one streight edge and a line drawne perpendicular vnto that edge apply the streight edge vnto the Wall at what time the Sunne shineth theron holding the board parallel to the plaine of the Horizon and hang vp a thread with a plummet so that the shadow of the thread may fall on the board crossing that perpendicular line Then take with your Instrument the height of the Sunne and instantly make two pricks in the shadow of the thread on the board a good way distant one from the other and laying a Ruler to those two pricks draw a line which line shall be the Azumith of the Sunne on the board againe with the height of the Sunne lastly taken find out on your instrument the Azumith of the Sunne or the Angle which the Sunnes Azumith maketh with the Meridian by the XV. Vse And on the board taking the intersection of the shadow line with the perpendicular for the Center describe a Circle equall to the innermost Circle of the Limbe which you may easily doe if you set one foot of your compasses vpon the East or West point and extend the other foot vnto 60 degrees on the same innermost Circle for this distance is equall to the Radius thereof Againe with your compasses take of the Arch betweene the Azumith of your Instrument and the Meridian and set that on the Circle of the board that way that the true South is and through the end of that Arch measured on the board draw a streight line for the Meridian Lastly take with your compasses the Arch intercepted between the Meridian on the board and the perpendicular line and by applying it to the in most Circle of the limbe from the East or West points see how many degrees it containeth for that is the declination of the Wall Or else you may find the Meridian vpon the board by XIX Vse If the Angle of the Meridian with the perpendicular on the board be a right Angle the Wall is direct East or West But if the Meridian fall vpon the perpendicular or be parallel there to making no Angle with it the Wall is direct North or South XXIIII Vse The Art of Dyalling And first how to make the Instrument in paper promised in the beginning of this second part For the Delincation of this instrument in paper it will bee necessary first to shew the manner how the Semidiameter is to bee graduated or diuided into degrees and how the Centers and Semidiameters of the seuerall kinds of Arches are to be found Vpon halfe a sheet of strong large Dutch paper the larger the better draw two streight lines making a right Angle neere one of the corners the one through the length and the other through the breadth of the paper which two lines I therefore call the longer and the shorter perpendicular Vpon the right Angle point being the Center with a Semidiameter equall to that by which you intend to delineate your instrument describe a quadrant of a Circle and on the point where it meeteth with the shorter perpendicular draw a long tangent line parallel to the longer pependicular Divide the Quadrant into 90 degrees among which from the beginning at the shorter perpendicular reckon the eleuation of the Pole for which you will make your instrument and applying a Ruler to the end thereof and to the Center where the Ruler cutteth the tangent line make a prick And taking with your compasses the distance from the Center to that prick measure it vpon the shorter perpendicular this shall be the Semidiameter of the sixt hower Circle At the end thereof draw another long line parallel also to the longer perpendicular Then out of the Center vnto the second parallel through every degree of the quadrant draw fine streight lines cutting also the first Parallel The intersection of those lines with the first Parallel shall be The scale of centers of Arches And their intersection with the second Parallel shall be The scale of centers of hower Circles And the segments of those lines intercepted betweene the Center and the first Parallel shall be the Semidiameters of Arches and the whole lines betweene the Center and the second Parallel
howers and by so much lesser the greater the Southerne Declination is And in those places of the Ecliptic in which the Sun most speedily changeth his Declination the length also of the day is most altered and where the Ecliptic goeth most parallel to the Aequinoctiall changing the Declination but little the length of the day also is but little altered As for example when the Sunne is neere vnto the Aequinoctiall on both sides the dayes increase and also decrease suddenly and apace because in those places the Ecliptic inclineth to the Aequinoctiall in a manner like a streight line making sensible declination Againe when the Sunne is neere his greatest Declination as in the height of the Summer and the depth of Winter the dayes keepe for a good time as it were at one stay because in those places the Ecliptic is in a manner parallel to the Aequinoctiall scarce altering the declination and because in those two times of the yeare the Sunne standeth as it were still at one declination they are called the Summer Solstice and the Winter Solstice And in the meane spaces the neerer every place is to the Aequinoctiall the greater is the diversitie of dayes Wherefore we may hereby plainely see that the common receiued opinion that in every moneth the dayes doe equally increase is erroneous Also wee may see that in Parallels equally distant from the Aequinoctiall the day on the one side is equall to the night on the other side IX Vse To find the Ascensionall difference of the Sunne every day Seeke out the time of the Suns Rising or Setting that same day by the VII Vse and see how much it differeth from sixe a clocke then conuert the same difference into degrees as was taught in 1 Part. Chap. 12. Sect. 1. by multiplying the howers with their decimall parts by 15. And so haue you the Ascensionall difference for that day X. Vse To find out the Oblique ascension of the Sunne every day Seeke out the Sunnes Right ascension by the IIII Vse and the Ascensionall difference by the IX Vse And if the Sunne be in the first Semicircle of the Ecliptic Subduct the Ascensionall difference out of the Right ascension But if the Sunne be in the second Semicircle of the Ecliptic adde the Ascensionall difference to the Right ascension and you shall haue the oblique ascension XI Vse To find how farre the Sunne riseth and setteth from the true East and West points which is called the Sunnes Amplitude ortiue and occasiue Seeke out as was shewed in the VI. Vse the imaginary Circle or Parallel of the Sunnes course and the points of that Circle in the Horizon on the East and West sides cutteth the degree of the Amplitude Ortiue and occasiue XII Vse To find the length of every day and night Double the hower of the Sunnes setting and you shall haue the length of the day and double the hower of the Sunnes rising and you shall haue the length of the night XIII Vse To find the true place of the Sunne vpon the Instrument which answereth to the point wherein the Sunne is in the heauens and is the ground of all the questions following Take with your Instrument the height of the Sunne and reckon it on the moueable Index or Labell and then moue the said Labell till you find the height of the Sunne exactly to fall vpon the Parallel of the Sunne for that day on the East side if it bee in the Fore-noone and on the West side if it bee in the After-noone the point of intersection where the Index or Labell crosseth the Parallel in that point of the Sunnes altitude shall bee the true place of the Sunne on the Instrument XIIII Vse To find the Hower of the day The true place of the Sunne on the Instrument found out as was last shewed sheweth among the hower lines the true hower of the day XV. Vse To find out the Azumith or verticall Circle in which the Sunne is or the Horizontall distance of the Sunne from the Meridian The Index or Labell fastned at the Center is a moueable Azumith apply therefore the edge thereof vnto the true place of the Sunne on the instrument found out as was shewed by XIII Vse And marke what point of the Horizon or Limbe the same edge of the Labell cutteth reckon how many degrees of the Hâ—rizon are intercepted betweene that point and the Meridian line or South point either on the East or West side and that Arch shall be the Horizontall distance sought for whereby is shewed the Azumith of the Sunne at that instant and consequently the Angle which the verticall Circle or Azumith of the Sunne maketh with the Meridian XVI Vse The Azumith of the Sunne being knowne to find out the Altitude of the Sunne and the Hower of the day Set the edge of the Labell to the Azumith given and marke in what point the same edge crosseth the Parallel of the Sunne for that day that point of intersection sheweth the height of the Sunne aboue the Horizon vpon the Labell and also it sheweth the hower of the day among the hower lines XVII Vse To find at what hower the Sunne commeth to be full East or West every day in Summer Apply the edge of the Labell vnto the East or West points of the Limbe and marke in what point the said edge cutteth the Parallel of the Sunne for that day for that same point among the hower lines shall shew the time of the Sunnes comming to be full East or West in that day and likewise of what altitude the Sunne will be aboue the Horizon at that time of his being full East or West XVIII Vse To find the height of the Sunne at high Noone every day and likewise at every other hower Marke in what point the Parallel of the Sunne for that day cutteth the line of that hower for which you would know the Sunnes altitude And vnto that point of intersection apply the edge of the moueable Labell or Index and thereon shall you find the very degree of the Sunnes altitude at that hower By this XVIII Vse and by the XVI are made the Quadrants described by Gemma Frizius Munster Clauins Mr. Gunter and others and also all manner of Rings Cylinders inuumerable other topicall Instruments for the finding out of the hower and other like conclusions And likewise the reason of finding the hower of the day by a mans shadow or by the shadow of any Gnomon set vp perpendicular to the Horizon or else parallel to it XIX Vse To find out the Meridian line and the points of the compasse without a Magneticall needle yea more exactly then with a needle Take the height of the Sunne by the shadow of the pinne and apply the same height reckoned on the Index or Labell to the parallel of the Sunne for that day whereby you haue the true place of the Sunne in the instrument as hath beene shewed in the XIII Vse Then keeping both the Labell
plaine perpendicular to the Meridian is that which standeth directly North or South which if it be also perpendicular to the Horizon is called North or South direct vpright But if it stoope from the Zenith forward it is called North or South inclining if backeward it is called North or South reclining And note that in a stooping Plaine that side which is toward the Horizon is inclining and that which is toward the Zenith is reclining The Plaine oblique to the Meridian is that which standeth not directly North or South but declineth one side into the East and the other into the West and is therefore called Declining Eastward or Westward according as either side of the Plaine looketh As if an vpright Wall being Southerne declineth from the South into the East it is called South declining Eastwards vpright But if it be not vpright it is called South declining Eastward and inclining or reclining The Plaine parallel to the Meridian is that which looketh directly East or West and accordingly hath his denomination whether it bee Vpright Inclining or Reclining The Plaine Parallel to the Horizon is called Horizontall and is represented by the instrument it selfe or at least by the inner most Circle of the limbe thereof And note that the Arch of Declination is reckoned from the next East or West point And that the Arch of Inclination or Reclination is reckoned from the Zenith or the complement of it from the Horizon So that euery vpright Plaine is vnderstood to passe through the Zenith which in the instrument is the Center And thus having shewed the seuerall affections of Plaines wee will now proceed to shew the manner how to set them vpon the Instrument A Direct North or South vpright Plaine is represented in the instrument by a line drawne through the Center from the East point to the West which is also the Horizontall intersection of the Plaâ—ne And by it you shall see that the Southerne side or face of the plaine is open to all the houres betweene sixe in the morning and sixe in the evening And that about London the Northerne side onely in the Summer enioyeth the Sunne from his rising till after seven in the morning and from before 5 a clocke in the afternoone till his setting A direct East or West upright plaine is represented in the Instrument by the Meridian which is also the Horizontall intersection of the plaine And in it you shall see that all the forenoone houres are open to the East side and all the afternoone houres to the West side A Declining Plaine is thus set upon the Instrument reckon on the Horizon the arch of Declination from the East or West point and at the end draw a line through the Center vnto the opposite point of the Horizon So that each side thereof may be open to that point either East or West into which the Declination is supposed That line so drawne throught the center is the Horizontall intersection of the plaine and representeth the plaine it selfe if it bee vpright For example there is about London an vpright Wall declining Eastwards 35 degrees which I would set vpon the Instrument Hold the Southerne part of the Instrument to you and reckon from the East backward into the North vpon the Horizon 35 degrees there draw a line through the Center this line shall not onely vpon the South side represent a Southerne Plaine declining Eastward 35 degrees But also vpon the North side shall represent a Northerne Plaine declining Westward 35 deg And moreover it will appeare that on the Southerne side shall bee drawne the houres from almost 4 a clocke in the morning till 3 in the afternoone And that in the Northerne side shall bee drawne vpon one side 4 a clocke in the morning onely and vpon the other side all the houres from 3 in the afternoone till Sunne set And â—o consequently the declination of an vpright wall or Window being given it may be found at what houre the Sunne vpon any day in the yeare will come to that Wall or Window and when it will goe from it As in the former example There is about London a Northerne wall declining Westward 35 deg I would know at what time of the day the Sunne will begin to shine vpon it on the 24th day of March. Set the Index at 35 deg from West toward South and because that day the Sunnes Declination is 6 degrees Northward Looke at what houre the sixt Parallel aboue the Aequinoctiall toward the Center meeteth with the Index so placed and you shall find it at 3 â— clock in the Afternoone Wherefore at that time the Sunne will begin to shine vpon the Wall that same day The Poles of every vpright Wall are in the Horizon 90 deg that is a quarter of a Circle distant from the line representing the Plaine Wherefore if vpon that line in the Center you erect a perpendicular the ends therof in the Horizon shall be the poles of that Plaine and are so farre distant from the North and South points as the Plaine it selfe is from the East and West XXVI Vse To set an Inclining and Reclining Wall or Plaine vpon the Instrument and to find how many houres the Sunne shall shine thereon at some time of the yeare When you haue an Inclining or Reclining Plaine to be described on the Instrument First the Horizontall intersection is to be set thereon as if it were vpright together with the line perpendicular thereto in which are the Poles of the Plaine according as was taught in the XXV Vse Then vpon the scale of degrees in your paper reckon the arch of Inclination or Reclination and with your compasses take set it in your Instrument vpon the line perpendicular to the Horizontall intersection of your Plaine from the Center that way into which the Inclination or Reclination tendeth the same shall bee the vppermost point of your Plaine Againe with your Compasses take the Complement of inclination or reclination both upon the scale of degrees and also upon the scale of centers of arches in your paper and set both spaces upon the same perpendicular line but on the other side of the center extended if need be At the shorter of those spaces shal be the pole of your plaine and at the longer of them shal be the Center of it Lastly setting one foot of your Compasses in the center of your Plaine and extending the other foot to the vppermost point describe in your Instrument an Arch of a Circle which if you haue done well will exactly fall vpon the ends of the Horizontal intersection of your Plaine That Arch shall represent your Plaine inclining vpon the lower side which is toward the Horizon or Limbe but reclining vpon the vpper-side which is toward the Zenith or Center And so either side shall shew in what hower lines the Sunne at some time of the yeare will shine vpon it that in delineating a Dyall thereon it may not be combered
and then adde together the products In the first Multiplication 1 · 47 ∷ 240 · 11280 · For set the Armes of the Index at 1 and 47 in the fourth circle and then bring the Antecedent arme which stood at 1 vnto 240 and the Consequent arme will shew 11280. Againe in the second Multiplication 1 · 9 ∷ 12 · 108 · For set the two Armes of the Index at 1 and 9 in the fourth circle and bring the Antecedent arme unto 12 and the Consequent arme will shew 108. Lastly adde together 11280 and 108 and the summe 11388 will be the number of pence contained in the said summe of 47 li 9sh. 9 An example of Division How many pounds and shillings are in 11388 pence Divide 11388 by 240 the division is thus 240 · 1 ∷ 11388 · 47 ⌊ 5 For set the two Armes of the Index at 240 and 1 in the fourth circle and then bring the Antecedent arme which stood at 240 unto 11388 and the Consequent arme will shew 47 and almost an halfe But how many shillings that excesse doth containe wil appeare if first you finde by Multiplication that 11280 pence are contained in 47 li which subducted from 11388 there will remaine for the excesse 108 pence Afterwards by division you may seeke how many shillings are in 108 pence the diuision is thus 12 · 1 ∷ 108 · 9. For set the Armes of the Index at 12 and 1 then bring the Antecedent arme which stood at 12 unto 108 and the Consequent arme will shew 9. 9 Any Fraction given may bee reduced into Decimal parts thus Set the Antecedent arme of the Index at the Denominator of the Fraction given in the fourth circle and the Consequent arme at the Numerator then keeping the same distance bring the Antecedent arme unto 1 and the consequent arme will shew the decimal parts So â—40 â—â—â— is 0 ⌊ 75. And 19 48 is 0 ⌊ 396 CHAP. III. Now follow certaine examples of Proportion Example I. IF 54 Elnes of Holland bee solde for 96 shillings for how many shillings was 9 elnes sold The termes given are 54 · 96 ∷ 9 · According to the 2 Chap. 3 Sect. Set one of the armes of the Index at the Antecedent terme 54 in the fourth circle and the other arme at the consequent terme 96 then keeping that distance bring the Antecedent arme vnto 9 and the consequent arme beyond the line of the Radius will shew 16 for the fourth proportionall according to the considerations in the 2 Chap. Sect. 3. Therefore 54 · 96 ∷ 9 · 16 · are proportionals And 16 shillings is the price of 9 Elnes Example II. If 108 bushels of corne be sufficient for a company of Souldiers keeping a Fort for 36 dayes How many dayes will 12 bushels suffice that same number of Souldiers The termes giuen are 108 · 36 ∷ 12 · Set one Arme of the Index at the Antecedent terme 108 in the fourth circle and the other Arme at the consequent terme 12 being mindfull of the considerations in the 2 Chap. 3 Sect. then keeping that same distance bring the Antecedent arme vnto 36 and the consequent arme will shew 4. Therefore 108 · 36 ∷ 12 · 4 · shall bee proportionals And 4 is the number of dayes sought for Example III. There is layd vp in a Fort so much corne as will suffice for 756 Souldiers which keepe that Fort for 196 dayes how many Souldiers will that same corne suffice for 364 dayes The Proportion is reciprocall therefore the termes given are 364 ∶ 756 ∷ 196 · Set one Arme of the Index at the Antecedent terme 364 and the other Arme at the consequent 196 and keeping the same distance bring the Antecedent arme vnto 756 and the consequent arme will shew 407+ And therefore for so many Souldiers will the corne laid vp suffice for 364 dayes or 13 moneths Example IIII. There is a Tower whose height I would measure with a Quadrant I take two Stations in the same right line from the Tower and at either Station having observed the height through the sights I finde that the perpendicular cutteth in the nearer Station 28 degrees 7 minutes almost and in the further Station 21 degr 58 min. almost and betweene both the Stations the distance was 76 feet The Rule of measuring heights by two Stations is contained in these Theoremes Theor. As the difference of the Tangents of the arches cut in either station is to the distance betweene the stations so is the Tangent of the lesser arch to the nearer distance from the Tower Againe Theor. As the Radius is to the Tangent of the greater arch so is the nearer distance found to the height And therefore because by the 1 Chap. 6 Sect. the Tangents of the arches 28° 7′ and 21° 58′ are 5342 and 4032 whose difference is 1310 the proportions will be First 1310 · 76 ∷ tang 21° 58′ · 234 · Wherefore 234 feet is the nearer distance Second Radius · tang 28° 7′ ∷ 234 · 125 · Wherefore 125 feet is the height sought for Example V. To finde the Declination of the Sunne the 9th day of May. The place of the Sunne for every day may be had nere inough out of this Table by Adding vnto the place of the Sun in the beginning of that moneth so many degrees as there are dayes past in that moneth But if the number of degrees exceed 30 the excesse is to be accounted in the Signe next following Wherefore the 9th of May the place of the Sun is ♉ 20+9 that is ♉ 29 which is 59 degr distant from the next Aequinoctiall point The place of the Sunne in the beginning of every Moneth Ianuary ♑ 21 February â™’ 22 March ♓ 20 April ♈ 21 May ♉ 20 Iune ♊ 19 Iuly ♋ 18 August ♌ 13 Septemb. â™ 18 October ♎ 17 Novemb â™ 18 Decemb â™ 19 These things being knowne the Rule is delivered in this Theoreme Theor. As the Radius is to the sine of the sunnes distance from the next Aequinoctiall point so is the sine of the sunnes greatest declination to the sine of the declination sought for The proportion will be Radius · sine 59° ∷ sine 2â—° 30′ · sine 19° â—9′· And so much is the Declination sought for Example VI. To finde the Right ascension of the Sunne the 9th day of May. Seeke the place of the Sunne for the day proposed in the former Table and the Sunnes distance from the next Aequinoctial point as in the former example These things being knowne the Rule is by one of these two Theoremes Theor. As the Radius is to the sine of the complement of the sunnes greatest declination so is the the tangent of the sunnes distance from the next Aequinoctiall point to the tangent of the distance of the right ascension of the sunne from the same Aequinoctiall point Or Theor. As the tangent of the greatest declination of the Sunne is to the Radius so is the tangent of
then going backe againe to your first pole looke what knowne starre is directly ouer your last pole for that starre is in the Meridian You may therefore instantly goe to some conuenient place and take a marke whereby you may at all times know the Meridian as is afore taught When therefore at any time of the night you would know what a clocke it is goe to that place where you stood and looking directly ouer your marke see if any of the 12 fixed starres bee in the Meridian or if none of them be therein obserue which two of them are on either side thereof and what part of that space is in the Meridian Then goe into the light and take your instrument and set the Index to that starre or point which you saw in the Meridian marke what houre it cutteth for that same houre being added to the houre which the day of the moneth sheweth shall giue you the true houre of the night so that you cast out 12 houres from the said summe if it shall chance to be more Example Suppose the fifth of December that the middle point of the space betweene the bright starre in the head of Aries and the Bulls eye bee in the Meridian Set the Index to the middle point of the space betweene those two starres in the Instrument and it will cut in the houre circle 2 and an halfe then againe set the Index to the fifth of December and in the houre circle it wil cut 7 which added vnto 2 and an halfe giueth 9 and an halfe for the true houre of the night Another example Suppose the 19th of December that one third part of the space betweene the Bulls eye and the right shoulder of Orion be in the Meridian Set the Index to one third part of that space in the Instrument and it will cut in the houre circle 4 and halfe a quarter almost againe set the Index to the 19th of December and in the houre circle it will cut 6 which being added vnto 4 and halfe a quarter almost giueth 10 and almost halfe a quarter for the houre of the night THE SECOND PART OF THIS BOOKE Shewing the vse of the Second side of the Instrument for the working of most questions which may be performed by the Globe And the declination of Dyals vpon any kinde of Plaine VPon the second side of the Instrument is delineated the proiection of the vpper Hemisphaere vpon the plaine of the Horizon The Horizon it selfe is vnderstood to bee the innermost circle of the limbe and is diuided on both sides from the points of East and West into degrees noted with 10 20 30 c. vnto 90. And the center of the Instrument is the Zenith or Vertical point Within the Horizon the middle straight line or Diameter pointing North and South is the Meridian or 12 a clock line and the other shortarching lines on both sides of it are the houre lines distinguished accordingly by their figures These houre lines should indeede bee drawne through the whole plaine crossing one another in the Pole of the world but that the Instrument may be more faire they are onely drawne short And because diuers excellent vses doe require the totall delineation of the houre circles I haue in a seuerall paper inscribed intirely both the houre lines and also two other circles betweene them containing euery one fiue degrees But if the Instrument were large enough to receiue them it were best if euery degree had his circle and so euery 15 circle should bee an houre line And of the parallels there needes no more but the Aequinoctial and both the Tropics For as much as there will be great vse of this paper Instrument I haue in the 24 Vse shewed the manner of making it so that any that 1◠ingenuous and ready handed may himselfe delineate one sufficient enough to serue his turne for any eleuation The two arches which crosse the houre lines meeting on both sides in the points of intersection of the sixe a clocke lines with the Horizon are the two Semicircles of the Ecliptick or Annuall circle of the Sunne the vpper of which arches serueth for the Summer halfe yeare and the lower for the Winter halfe yeare and are therefore diuided in 365 dayes which are also distinguished into 12 moneths with longer lines hauing their names set downe and into tenthes and fifthes with shorter lines and the rest of the dayes with pricks as may plainly bee seene in the Instrument And this is for the ready finding out of the place of the Sunne euery day and also for shewing of the Sunnes yearely motion because by this motion the Sunne goeth round about the heauens in the compasse of a yeare makeing the foure parts or seasons thereof Namely the Spring in that quarter of the Ecliptick which beginneth at the intersection on the West side of the Instrument and is therefore called the Vernall intersection Then the Summer in that quarter of the Ecliptick which beginneth with the intersection of the Meridian in the highest point next the Zenith And after that Autumne in that quarter of the Ecliptick which beginneth at the intersection on the East side of the Instrument and is therefore called the Autumnall intersection And lastly the Winter in that quarter of the Ecliptick which beginneth at the intersection with the Meridian in the lowest point next the Horizon But besides this yearely motion the Sunne hath a Diurnall or dayly motion whereby it maketh day and night with all the diuersities and inequalities thereof which is expressed by those other circles drawne crosse the houre lines the middlemost whereof being grosser then the rest meeting with the Ecliptick in the points of the Vernall and Autumnall intersections is the Aequinoctiall and the rest on both sides of it are called the Parallels or Diurnall arches of the Sunne the two outermost whereof are the Tropics because in them the Sunne hath his furthest digression or Declination from the Aequinoctiall which is degrees 23½ and thence beginneth againe to returne to the Aequinoctiall The vpper of the two Tropics next the center in this our Northerne Hemisphaere is the Tropick of Cancer and the Sunne being in it is highest into the North making the longest day of Summer And the lower next the Horizon is called the Tropick of Capricorne and the Sunne being in it is lowest into the South making the shortest day of Winter Betweene the two Tropics and the Aequinoctiall infinite such parallel circles are vnderstoode to bee contained for the Sunne is what point soeuer of the Ecliptick it is caried describeth by his lation a circle parallel to the Aequinoctial Yet those parallels which are in the Instrument though drawne but to euery second degree of Declination may be sufficient to direct the eye in imagining and tracing out through euery day of the whole yeare in the Ecliptick a proper circle which may be the Diurnall arch of the Sunne for that day For vpon the right estimation of