The English globe being a stabil and immobil one, performing what the ordinary globes do, and much more / invented and described by the Right Honorable, the Earl of Castlemaine ; and now publish't by Joseph Moxon ...

SECT. VI. Of the STARS. (Book 6)

AS for the Stars, it is not my set business to meddle with them, nor is there any Instrument that so naturally re∣solves all the usual Questions concerning them, as the Coele∣stial Globe; for there things appear as they do in the Hea∣vens themselves; But because such a Globe is not always at hand, I will shew you how its most necessary Operations may (in case of necessity) be perform'd, even by our present Globe, and then I shall treat of the Pedestal, on which all the appear∣ing Stars are truly projected.

The usual and most necessary Operations of the Coelestial Globe in relation to the Stars are these.

• 1. To find the Declination of any Star.
• 2. To find the Right Ascension of any Star.
• 3. To find the difference between the Suns Right Ascension, and that of any Star, or the difference between the Right As∣censions of any two Stars.
• 4. To find the true place of any Star on the Globe, i. e. the Point that corresponds there, with its then Place in the Hea∣vens.
• 5. To find the Bearing of any Star according to the Points of the Compass.
• 6. To take the height of any Star you see.
• 7. To find the height of any Star by the Hour tho' unseen.
• 8. To find the Azimuth of any Star.
• 9. To find how many Hours any Star stays above or below the Horizon.
• 10. To find when any Star rises or sets.
• 11. To find what a Clock 'tis by any Star.
• 12. To know the name of any Star you see.

OPERATION I. To find the Declination of any Star.

AS for the Declension of the Stars, since the Coelestial Globe and other Instruments, that shew their motions, perform this Operation by their make (for on them the Stars are always plac'd, according to their respective Declensions) we shall require here a proportionable Concession, viz. That in some of the vacant parts of our Globe there may be a little Table, containing the Declension and Magnitude of the most noted Stars, as also the Degree of the Ecliptick, which agrees with their respective Right Ascensions: that is to say, a Table containing their Declension and Magnitude, with the Suns Place in the Ecliptic, when his and their Right Ascension are the same; and the said Table may be made in the following manner.

The Table for the Stars.

Mag.	Name.	Declension.	Deg.	Eclip.
1.	Bulls Eye.	15	48.	♊	6.
1.	Lions Heart.	13	33.	♌	26.
1.	Arcturus.	51	4.	♏	2.
2.	Little Dog.	6	3.	♋	19.
2.	Medusa, or Algol.	39	40.	♉	13.

And so for as many as the Globe maker thinks fit to express.

OPERATION II. To find the Right Ascension of any Star, v. g. of the Lion's Heart.

HAving found by the Table, that the Lions Heart has the same Right Ascension with the Sun, when he is in the 26 of ♌, draw your String over that Degree of the Ecliptic, and it will cut the Aequator at almost 148 Degrees, for the Stars Right Ascension.

OPERATION III. To find the difference between the Suns Right Ascension, and that of any Star, as also the Difference of the Right Ascensions of any two Stars.

HAving found by the former Operation, that the Right As∣cension of the Lion's Heart is towards 148 Degrees, and that the * 1.1 Suns (on v. g. the 10 of April) is near 28, the difference (by Substraction) will appear to be about 120 Degrees, or (by the intermediate Hour Circles) 8 hours. In like manner ha∣ving found (for example sake) the Little Dogs Right Ascen∣sion, to be about 110 Degrees, the difference between it and that of the Lions Heart is 38, or 2 hours and a half.

OPERATION IV. To find the Place of any Star on the Globe, i. e. the Point that corresponds with its then Place in the Heavens.

HAving found by the foregoing Operation, that the difference between the Suns Right Ascension and the Lions Heart, is about 120 Degrees, or 8 hours, substract the said hours (for so much the Sun Rises and Sets now before the Star) from the time given (suppose from 10 at Night) and the remaining 2 hours shows you, that the Star is at that moment in some part of the Hour-circle of 2 in the afternoon, or thereabouts; so that the Bead rectify'd to the Stars Declension, and moved on its Noose from the Pole to the said Hour-Circle, determins the very Point or Place requir'd, The like may also be per∣formed by your Compasses open'd from the Pole, at the Com∣plement of the Stars Declension.

But here you must remember that the readiest way still to* 1.2 find the present place of a Star is by its Almucantar and Azi∣muth; for where these Circles intersect, there the Stars then Place will be; and as for the Almucantar and Azimuth of a∣ny Star, they are found by the 6th, and 8th, Operation of this Section.

OPERATION V. To find the Bearing of a Star at all times.

HAving found the true Place of the Lions Heart by the for∣mer Operation, if you draw your String over it from the Zenith, 'twill cut the Horizon at or about S. W. for its then Bearing.

OPERATION VI. To take the Almucantar or height of any Star you see.

BEcause Stars cast no shade, you must take their Height as you do the Suns when he is overcast, and therefore con∣sult the first Operation of the first * 1.3 Section, or the 5th, of the * 1.4 4th. Section.

OPERATION VII. To find the height of a Star at any time, by the hour tho' un∣seen.

THE Hour being (v. g.) 10 at Night, on the 10 of April, the Suns place is where his Parallel cuts the 10 a Clock Hour Circle, so that knowing by the difference of their Right Ascensions that the Lions Heart is (v. g.) 8 hours behind the Sun, you may conclude the Star to be somewhere in the Hour-Cir∣cle of 2 in the afternoon, to wit, in that Point, which answers to the said Stars Declension; having therefore its Place, draw but your String from the Zenith over it, and mounting your Bead to it, if you move your said Bead to the Meridian or Quadrant of Altitude, 'twill lye on or about the 45th. Degree for the required Height.

OPERATION VIII. To find the Azimuth of any Star.

FInd but the Lions Heart's Bearing, or his then true Place in the Heavens as before, and the String will cut the Hori∣zon on its true Azimuth. Now if you see the Star, you may perform this Operation without any of the former Postulats; for placing your Globe on a Meridian Line, and holding your String streight from the Zenith, do but move it in that posture by the direction of your Eye (as we show'd you in the first * 1.5 Section) till it be in the same Plane with the Star, and the Degrees of the Horizon under your String, give you the re∣quired Azimuth, which will now be about 45 Degrees West∣ward.

OPERATION IX. To know how many hours any Star stays above or under the Horizon.

REctify the Bead to the Declension of the Lions Heart, and moving your String on the Noose from the Pole, till the said Bead touches the Horizon on the West side, see what Hour-Circle cuts with it there, and you will find it to be that of 7 and a quarter or thereabouts; and this doubled (making in all some 14 hours and a half) gives the true time of its stay above the Horizon; so 4 and three quarters doubled (I mean the Hour-Circle which Intersects with it on the East side) gives you 9 hours and a half for its stay below the Horizon.

OPERATION X. To find when any Star Rises or Sets.

HAving found by the third Operation, the difference between the Suns Right Ascension and that of the Lyons Heart to be 8 hours on the 10th of April; and having also found by the

foregoing Operation, that it Rises where the Hour-Circle of 4 and 3 quarters cuts the Horizon, and Sets where that of 7 and a quarter do's the like, add the 8 hours difference (because the Star is now so much behind, or too slow for the Sun) to 4 and 3 quarters, which making 12 and 3 quarters in all, shows that the Star rises at 12 a Clock, and 3 quarters in the after∣noon; and by adding it to the aforesaid 7 and a quarter, that it sets at 15 and a quarter, to wit, at 3 and a quarter in the mor∣ning.

OPERATION. XI. To find what a Clock 'tis by any Star.

HAving (v. g.) on the 10th. of April found the true place of the Lyons Heart, on the Globe, by some of the former ways, as (for Example) by its Height and Azimuth: I say; ha∣ving thus found the Stars true place on your Globe (which hap∣ning (v. g.) to be in the 2 a Clock Circle) find by the third Ope∣ration the difference between its and the Suns right Ascension, which being 8 hours, add it to the said 2 (for the Star is as we said 8 hours now behind or too slow for the Sun) and the then true hour will be 10 at night.

OPERATION XII. To know the Name of any remarkable Star which you see.

BY a Stars Height and Azimuth you may (as we have show'd you) quickly find its present true place on the Globe, and consequently its Declension, as being the nearest distance between its said Place and the Aequator; so that your Tables of Declen∣sion gives you its Name; and if there should be two of the same Declension then their right Ascensions (being different) will resolve the Doubt.

To conclude, all the former Operations may be yet more readi∣ly performed, and that without any Table, if the Globe-maker* 1.6 place 10 or 20 of the most noted Stars (which will be enough to

[illustration]

These Schemes are properly Sch 3^d & 4 belonging to Page 141 as showing y^e way to describe y^e Projectiō but they may serve for Sch 1 & 2 Pag. 135. if on y^e higher y^e Constellation are supposd to be Engravd & on y^e lower a line was draw̄ frō y^e Center to lack hour in y^e limb & no Almucantar or Azi••••th exprest but by Pricks

satisfy any ordinary curiosity) on the Globe it self, according to their true Longitude and Latitude; for then their Declensions, Parallels, and right Ascensions appear in a manner at first view, which must needs therefore facilitate the other Operations.

Of the PEDESTAL.

THUS you see that our Globe (tho' it be a Terrestial one) may (in case of necessity) be serviceable in relation to the very Stars; but because all Operations that have the least Reflection in them, seem intricate and troublesome to some, I have here adjoyned (for them that will be at the Expence of the best sort of these Globes) a most Facile way, that shall re∣solve in an instant, all the former Questions and more; for there is not only a Steriographical Projection on the Pedestal of the appearing Stars in our Horizon, but one also so ordered, that it obviates the inconveniences which make Stofflers admirable Astrolabe so much neglected of late; for some say, there is no finding a Star on it without much poring, tho' we should know near what Constellation it lyes; others, that when we see a Star there, we are still ignorant to what Constellation it belongs; ma∣ny quarrel at the great confusion which the Azimuths, Almu∣cuntars, and other Circles exprest on it make; and some again object, that the numeral Figures belonging to the said Circles are oftentimes so hid by the solid part of the Rete, that we cannot without a new trouble and motion perform the intended Opera∣tion. I say, this Projection on the Pedestal (besides several other things) obviates these inconveniences, as you will presently see.

The Explanation of the Circles and Lines of the whole Projection or Pedestal.

THE uttermost Circle (in Sch. 1) or Limb SENW of the* 1.7 lower or first Plane, represents Circulum maximum sem∣per latentium, or (if you think that too large) what Parallel you please. It may be conveniently nine Inches or a little more in Diameter, if the Globes be a Foot, and being of fine Pastboard or the like substance, it is to be let into the Pedestal,

which is purposely made Cradle or Frame wise, that it may (by your hand underneath) be easily turn'd round, and be also ta∣ken quite out, if any particular or extraordinary occasion should require it; Nay, the whole Pedestal may be pulled off, (if you think fit) from the handle or Fulerum, and us'd apart as a distinct Instrument.

2. The great Circles described on it are only two, viz. the* 1.8 Aequator (♈ AE ♎ ae) and the Ecliptic (♈ ♋ ♎ ♑) divided into the 12 Signs, with their gradual subdivisions. Now (since it will be no incumbrance to your Plane) you may express on it also (if you please) the two Tropics, by two fine Circles, that of Cancer touching the Ecliptic at ♋, and that of Capricorn at ♑. And as for the Limb, it is divided in∣to 360 Degrees, for being in Projection greater than the Aequa∣tor, 'twill prove more useful in all the Operations, that con∣cern such Divisions. Nor are the Circles or Stars placed here as on the Globe (I mean according to the Degrees of a Qua∣drant equally divided) but Steriographically projected by half Tangents, i. e. as they would appear and fall on an Aequinoctial Plane, or a Plane parallel to it, were our Eye in the Pole, of which more hereafter, as also the Centers and Radius's of each Circle, when we come to the Description and Demonstration of the whole Projection; and in this manner also (to wit, by half Tangents) the Line P. E. is divided, which shows the Declension of any Star.

Thirdly. The Stars being all plac'd on this Plane accord∣ing to their respective Right Ascensions and Declensions; and by the way, when you once know how to find by this Proje∣ction the Right Ascension and Declension of a Star (as you will presently do by the following Instructions, that concern operation) you will then also know by the help of Astrono∣mical Tables (which give each Star's Right Ascension and Declension) how to place them here: I say, the Stars being all plac't on this Plane, according to their respective Right Ascensions and Declensions, they are to be Marshall'd and re∣duc'd into Constellations; and therefore you must suppose either fit Pictures drawn about them to express what they are, or that the uttermost Stars of each be join'd by a fine Prick't Line, which will give you perchance, the most clear and just representation of them, and consequently prove the

easiest way for the finding them out in the Heavens; But since Pictures have conveniences and great ones also; for thus without consulting the written names, we cannot only find presently (even a far off) the Constellation we seek after, but know at the same time the Place of each Star in it, which Place for the most part gives the Star its ordinary Name: I say, since Pictures have great Conveniences, let them be us'd; but then they must be as faintly and simply express'd as can be; for deep shadows, and unnecessary Flourishes both distract the Fancy, and cause even the Stars that are express'd to be less conspicuous and observ'd.

Fourthly, When the first Plane is thus garnished and plac'd* 1.9 in its Frame, there is another of the same bigness, either of Glass, or Talk (represented by Scheme the second) to be put over it, and fixt or fastned in the uttermost Molding or Ledge of the Pedestal. And here be pleas'd (for distinction sake) to remember, that by the Terms First, and Second, these two Planes are distinguish'd, and that by Projection is meant the whole Pedestal, or Astronomical Machin, which (as I said) may be taken off, and used apart, as a particular In∣strument.

Lastly, the second Plane (represented, as I said by Scheme the second) has its Limb S. E. N. W: divided (besides the subdivisions or Quarters) into 24 equal parts, by so many streight Lines, drawn from the. Center P, and figur'd (I. II. III, &c.) according to the hours of a natural Day. As for the Circle HRST, it represents the Horizon; and the Cir∣cular Pricks within it give the Almucantars and Azimuths of every 10 Degrees; for (on the one side) if you consider the said Pricks as so many Circles ascending from the Ho∣rizon towards the Zenith, the Figures along the Lines, PS and PN give you from the Horizon upwards the height of that Star which touches any of them. On the other side, if you consider them in File, (I mean as so many Arches pas∣sing thro' the Zenith, and terminating in the Horizon) their distance from PS (the Southern part of the Meridian) shows the Azimuth of the Star next any of them, by the Figures round the Horizon; and least you might not readily distin∣guish Arch from Arch, if the Pricks were all of the same kind or Species, there are two sorts here, viz. one of plain

and simple Pricks the other of small Astricks alternatively plac'd; so that 'tis but observing of what Species the Prick next a Star is, (as suppose an Astrisk,) and then following with your Eye a File or Arch of Astrisks 'till you come to the Horizon; for the Figures at their termination there give you the requir'd Azimuth. Thus then the confusion which the several Almucantars and Azimuths would make (were they all describ'd on the Plane) is avoided, seeing the Plane is now less fill'd than if the Almucantars were only exprest on it; for disjoyn'd Pricks circularly plac'd occupy not the room of a continued Circle, and yet each Row or Circle of the said Pricks perform both the forementioned Offices.

How to operate by the Projection or Pedestal.

FIRST the Reader must remember, that I call Re∣ctifying the first Plane; the placing and adjusting it so that all the Stars may appear above and below the Horizon, as they then really do in the Heavens themselves; which O∣peration being a main and principal matter (for all the other are in Truth but so many Deductions or Corollaries) I will now begin with it; nor is there any thing here requir'd but the height of some Star in view (as the Lion's Heart, or the like) which you may find by the Globe as you do the * 1.10 Sun's or † 1.11 Moons height as I mentioned * 1.12 before. Now for cleerness sake, let us suppose this Star to be about 45 Degrees high Westwardly, and then if you move your Plane till the said Star, lyes thus under a Prick of this height, you have (without ever moving more the Plane) the several following Opera∣tions at a time.

First, You see all the Stars that are then above the Ho∣rizon and below it; for all the painted ones within the Circle HRST, on the second Plane represent the real ones then in sight, and the rest those that are below the Horizon. Se∣condly, You see what Stars are Rising, what are Setting, what are Culminating, and what are in their Lowest Depression. Thirdly, If you look after any particular Star (suppose the Lion's Heart) by seeing him on the West-side of PS (the Meridian of the said second Plane) you are sure he is not on∣ly

in a Declining state but also (by following the Prick next him to the Horizon, according to its Species) that his Azi∣muth is 45 Degrees. Fourthly, You will see his Bearing, to be about S. W. if you follow the Azimuthal Arch to the Nautical Characters there. Fifthly, You see that the Hour of the Night is 10, by observing under what Hour-Line the 10th. of April (i. e. the day of the Month, the Suns place in the Ecliptick) lyes. Sixthly, By any real or imaginary Hour Line that runs over the said Star, you find his Right Ascension to be near 148 Degrees; for thus the said Hour Line cuts the Limb. Se∣venthly, By his being behind the Sun about 8 hours (as appears by the Hour Lines that pass over the Star and the Suns place) you have the difference of their Right Ascensions, which amounts to about 120 Degrees. Eighthly, Which is the most surpri∣sing (and not performable even by a Coelestial Globe) you no sooner see these things in relation to this or any other parti∣cular Star, but at the same time also (even without touching your Projection) you have them in relation to all the Stars in general; for when the First Plane is rectify'd, we have (be∣sides the Hour) the Heighths, Azimuths, Bearings, Right A∣scensions, &c. of all the other Stars above the Horizon.

Concerning the other Operations, they are more restrain'd, as being peculiar to the Star you enquire after; for if you would know when the Lions Heart Sets, (which for continuation's sake we will call the ninth Operation) do but move your first Plane till the said Star touches the Horizon, and the imaginary Hour Line that passeth over the Sun's place in the Ecliptic, show's you, that 'twill be then about 3 and a quarter in the morning.

10ly. By the Figures about the Horizon, you will see at the same time, that his Occasive Amplitude is near 23 Degrees▪ Northward, and his then Bearing (by the Nautecal Caracters) to be WNW, or thereabouts.

11. By the imaginary Hour-line that then passes over the said Star (viz. that of about 7 and a quarter) you have half the time of his constant aboad above the Horizon, and conse∣quently know, that from his Rising to his Setting there are a∣bout 14 hours and an half.

12. By reason that the imaginary Hour-line of about 7 and a quarter passes over the Star (as we said) at his Setting, it fol∣lows that it's Ascensional difference (i. e. the difference between its

Right and Oblique Ascension) is about an Hour and a quarter, or 18 Degrees.

13. By the Degree of the Ecliptic that Sets with the Star (which is the 26 of ♌) and by the opposite Degree which then Rises (viz. the 26. of ♒) you see that on the 8th. of August he Sets Achronically, and on the 2. of February Cosmi∣cally.

14. Remove the said Plane, till the said Star brushes the Horizon on the East-side, and by the precedent method (muta∣tis mutandis) you will find when he Rises, what his Ortive Am∣plitude is, how he then Bears, how long he is under the Horizon, when he Rises Cosmically, and when Achronically.

15. By placing the point of a Pin or Needle, on the Class over the Lions Heart, and then moving the first Plane, till the divided 6 a Clock Hour-line PE, lyes just under the said point the Divisions there will show its Declination to be about 13 Degrees and 33 Minutes. The like you may do with your Compasses; for if you take the Distance between the Pole and Star, and measure it on PE, you have what you seek for.

Many other Operations are performable by the Projection, touching the Stars; but since these are the most material ones, and since I have not time to treat more fusely, I leave the rest to be found out by my Reader himself, who may easily do it, if he understands either the Caelestial Globe, or any Instrument belonging to the Stars. And here he is to remember, that knowing but the Hour at any time, let him put the Suns place,* 1.13 or day of the Month under the Hour-line, that corresponds with it, and the Projection will be rectified, and consequently (having a true view of the then posture of the Heavens) he may opperate as before. In the next place, if he knows but the Suns place in the Ecliptic of the first Plane, and opperates with the said place as if it were a Star, he may find out the former Operations in relation to the Sun it self; that is to say, he may at that moment know his Height, Azimuth, Bearing, Amplitude, &c.

16. If you would know the Stars in the Heavens, you may also do it by the help of this Projection; for your first Plane being rectified, it gives you (as I said) the true posture of all the Stars; so that if those you seek after be near the Horizon, Meridian, or any other noted Quarter, those on your Plane

near its Horizon, Meridian, or corresponding Quarter will re∣solve the Question. Or, if you take the height of a Star, and its Azimuth (according to any of the former Directions) then whatever Star on your Plane has the same, it will be that you seek after, and consequently you have its Name. Now know∣ing once a Star, your said first Plane shows you what they are that lye about it, and so by degrees you may run from one to another round the Heavens. Nor need you, as to the knowing of the Stars, be so exact (either in rectifying your Projection, or in having the hour of the night, or in taking the Heights, and the like) as in other Operations: for, by the bigness of the Star, by its nearness to some remarkable one, and by twenty other particular properties, you will be so regulated and confined, that you may safely conclude, when you examine your Projecti∣on, that the real Star you see, can be no other than such and such a one.

How to Describe the PROJECTION.

HAving thus show'd you the use of the Pedestal or Projecti∣on, I shall fall on the way of Describing it, and (accor∣ding to my manner all along) on the Demonstration of it also, especially since it conduces to a more easy comprehension of all Steriographical Projections; and if I be a little longer than or∣dinary, it is now no great matter, for I have ended all the Ope∣rations I intend at present, so that what is here further said may be omitted without inconvenience, if the Reader be disgusted at Speculation.

As for the nature of the Projection, tis Optical, representing* 1.14 all things in the Heavens, as they appear to the Eye, from such and such a Station, and not according to their true and real distances. 'Tis chiefly founded on the 20th. Proposition of the third Book of Euclid, which proves, that the Angle at the Peri∣phery is but ½ that at the Center; for from thence 'tis infer'd, that if placeing our Eye on the superficies of the Sphere (v. g. at the South Pole) we look into its Cavity, the Angle made at our Eye, by the two Rayes that issue from it (the one along or throu' the Axis to the opposite Pole, and the other to a deter∣mined Point) will be the Angle only of half the value of the Arch, or real distance between the two Objects, i. e. between the

said Opposite Pole and Point; now since any Diameter on the Plane of the Aequator (for that, or some Parallel Circle to it, we now suppose to be the Plane of our present Projection) meeting with those Rays, will be the Tangent of the Angle they make, which being in value (as we said) but half the real di∣stance between the said Objects, it must need follow, if any Star or Point in the Heavens be distant from this opposite Pole, suppose 20. Degrees, That the Tangent of 10 Degrees from the Center of the Projection (which represents the said Pole) gives its true apparrent place there, and the like is to be said of any other distance.

I shall not trouble the Reader with any Scheme to demon∣strate this further, because (being fusely treated of by Aguilo∣nius and others) 'tis obvious enough to all Mathematicians; and as for new Beginners (if they desire a fuller conception of it) let them but apply themselves to any man vers't in Projections, and in the space of ten Minutes he will shew it them more clear∣ly and naturally, by Strings fitly placed on an Armillary Sphere, than I can here in many hours; therefore supposing (if to such, what I have already said be not evident) that the Heavens may be thus projected by half Tangents, let us proceed to the way of doing it, that is to say, to the finding of the Cen∣ters and Radius's of all the Circles which conduce to the before mentioned Operations.

As for the Concentric Circles of the first Plane, to wit, the* 1.15 Aequator, the Tropics, and the Limb, which is (as I said) Circulus maximus semper latentium, or some Parallel▪ to it, there is no difficulty in describing them; for having drawn at right Angles the Lines NS and EW (representing the four Cardinal Points) throu'P, the Center, or projected Pole, if you open your Compasses at the Tangent of 45 Degrees, and place one foot on the said P, you must needs project the Aequator; because being distant from either Pole 90 De∣grees, the Ray that touches it, and that which runs along the Axis to the opposite or North Pole, makes an Angle at your Eye (as we said before) of only half so much. In like man∣ner, the Tropic of Cancer being 66 g. 30 m. from this Pole, the Tangent of 33 g. 15 m. gives his Radius, as the Tangent of 56. g. 45 m. does Capricorn, whose real distance from the said Pole is 113 g. 30 m. for it lies 47 Degrees beyond the former

Tropick. And lastly, the Tangent of 64 g. 15 m. projects the Limb or uttermost Circle, if it be Circulus maximus super la∣tentium, as being yet 15 Degrees further; for the true di∣stance of that Circle from the said Pole 128 Degrees and 30 Minutes.

Now for the main matter, to wit, the great Circles which fall* 1.16 obliquely on the Plane, take this easy general Rule for them all, viz. That their Centers are distant from the Center of the Pro∣jection the Tangent of as many Degrees as their Poles are di∣stant from the Pole of the Plane, on which the Projection is made (that is to say, in our present Case, from the North-Pole of the World) and the Secant of the said Degrees is their Radius.

Suppose then you were to project (v. g.) the Ecliptic, which* 1.17 is the only oblique Circle of your first Plane; you know that its Northern Pole, (being in your Meridian) is distant from the North Pole of the World 23 g. 30 m. Open therefore your Compasses at the Tangent of those Degrees, and place one Foot in P, and the other will give you in the Line PN (the Northern half of the Meridian of your Plane) or in the Line PS, (the Southern half of the said Meridian) the point D, for the re∣quir'd Center. D then being the Center, open but your Com∣passes at the Secant of the said Degrees, and you have the Ra∣dius; Nay, the Distance from D to e, or from D to w, the East and West Points of the Aequator (or points where the Ecliptic intersects with the Aequator on the Sphere) gives this Secant; for if PD be the Tangent of 23 g. 30 m. then D e and D w are (you see) the Secants. But before we demonstrate the aforesaid Rule, let us make an end with the great Oblique Cir∣cles of the Transparent or second Plane, which are only the Ho∣rizon HRST, and the Azimuths of every 10 Degrees, exprest (as I said) by plain Pricks and Astrisks.

As for the Pole of the Horizon, it is (you know) the Ze∣nith,* 1.18 which being distant in your Meridian 38 g. 30 m. South∣wards from the North Pole of the World, it must follow by the former Rule, that the Tangent of 38 g. 30 m. (or Comple∣ment of the Elevation) from P (the Center of the Projection) giving you (Southwards in the Meridian of your Plane) h, the requir'd Cent••••, the Secant of these Degrees will be the re∣quir'd Radius; Nay the distance from h to e, or from h to w

the East and West points of the Aequator, (or Points where the Horizon cuts the Aequinoctial Colure) gives this Secant; for if P h be the Tangent of 38 g. 30 m. h e and h w are the Se∣cants.* 1.19

The Poles of all the Azimuths, are (as every body knows) in the Horizon; now that of the Primary Vertical, being in the Meridian also, 'tis distant in the Heavens (on the North side of your Meridian) the value of the Elevation, or 51 g. 30 m. so that by the foregoing Rule (PV) the Tangent of those De∣grees will, from the Center P (Northward,) give you in the Me∣ridian of the Plane the Center of this Circle, and the Secant the Radius. Nay, the distance from V to e, or from V to w, the East and West Points of the Aequator, (or points where the said Primary Vertical cuts the Aequinoctial Colure) gives this Secant; for if PV be the Tangent of 51 g. 30 m. V e and V w are the Secants. Besides, where the moving foot of your Com∣passes (thus extended) touches the Meridian of the Plane, there will be the Zenith in projection, and consequently distant from P (Southward,) the Tangent of 19 d. 15 m. or half the Complement of the Elevation; for our Zenith lyes in the Me∣ridian 38 g. 30 m. beyond the Pole on the South-side of the Sphere or Heavens.

As for the Centers of the other Azimuths, tho' there be no* 1.20 Tables calculated to shew how their repective Poles are di∣stant from that of the Plane or Projection, and consequent∣ly the aforesaid Rule may seem useless, yet by resolving a Tri∣angle, these Distances may be found, as also the value of the Angle, made by your Meridian (or 12 a Clock hour Circle) with the Meridian that passes throu' the proposed Degree of the Horizon, so that the Rule serves as before; for if you draw a blind Line thro' P, that makes an Angle with PN, answe∣rable to the value of the Angle of those two Meridians in the said Triangle, the Tangent of the distance found between the Pole of the Plane and that of the propos'd Azimuth will still give you its Center from P in the said blind Line, and the Se∣cant its Radius.

But you may avoid the Resolution of a Triangle, by the usual* 1.21 expedite way, viz. by drawing thro' V (the Center of the Pri∣mary Vertical, found as before) the blind Line K. M. at Right Angles with P. N. (the Northern part of the Meridian of your

Plane) and then pricking on both sides of the said V (ZV be∣ing Radius) the Tangents of all the Azimuths you would ex∣press, as (for example) those of 10, 20, 30 Degrees, &c. for the said Pricks give their Centers, and the Secant of those Deg. their Radius. This Way also agrees not a little with the above menti∣oned Rule; for if the distance from V (the Center of the prima∣ry Vertical) to 10 (the Center of the Azimuth of 10 De∣grees) be the Tangent of those Degrees, 'tis evident, that the Radius Z 10 is the Secant; and if this be the Secant, the di∣stance from V to 10 is the Tangent. Thus then in short may be drawn (mutatis mutandis) all other great oblique Circles in any Steriographical Projection, when their Poles lye in one and the same Circle; and now since the aforesaid Rule agrees even with this usual way of describing these Circles, I will here Demonstrate it, having done with the great Circles on both our Planes; for as to the Hour Circles, they are all seen in Cul∣tro, (that is to say, they lye directly under your Eye, and con∣sequently are in projection streight Lines, and distant (as on the Sphere) 15 Deg. asunder; I say, since the Rule agrees not a little with this way, and that I have done with the great Circles both Planes, I will now demonstrate it by the two Lemmas that follow.

The Demonstration.

[illustration]

Sch. 5.

* 1.22I. THe Secant of any Arch is equal to the Tangent of the same Arch more by the Tangent of half its Comple∣ment. That is to say, CE the Secant (for example sake) of 60 Degrees (in Scheme 5) is equal to EB (the Tan∣gent of 60) and to BA the Tangent of 15, or half the Complement of 60: For the Angle ECA being equal by Hyp. to the Angle ACH, becomes equal to * 1.23 the Angle EAC. therefore EA is equal to † 1.24 EC, and consequently EB plus BA is equal to EC. QED.

[illustration]

Sch. 6.

* 1.25II. The Tangent of any Arch greater than 45 Degrees is equal to the Tangent and Secant of double its Excess above 45 Degrees; that is to say, AB Tangent (for example) of 46 Deg. (in Sch. 6th) is equal to CD Secant of 2 Degrees plus DB Tangent of the said Degrees; for the Angle DCA be∣ing by Hyp. † 1.26 equal to the Angle ACH becomes equal to the

Angle DAC; therefore CD is equal to * 1.27 AD, and conse∣quently AD plus DB is equal to CD plus DB.

These two Lemmas being premis'd, let us consider the Pro∣jection* 1.28 (for example sake) of the Ecliptic, and see how it a∣grees with our said Rule, to wit, That the Centers of all the projected great oblique Circles are distant from the Center of the Projection, the Tangent of as many Degrees as their Poles are distant from the Pole of the Plane on which the Projection is made and that the Secant of those Degrees gives their Radius's. The Ecliptic is to touch both Tropics on the Solstitial Colure or Me∣ridian of the Plane, because it touches that Colure thus in the Heavens, and on the Sphere; so that by Construction P ♋ (the distance in Projection between the Center of the Plane and the Point where the Ecliptic touches the Tropic of Cancer) is the Tangent 33. 15′. or half 66. 30′, (its real distance on the Sphere from the North Pole) and on the other side P ♑ (the distance in Projection between the Center and the Point where the Ecliptic touches Capricorn) is the Tangent of 56. 45. or half 113. 30′. its real distance as before. Now D by Construction being distant (on the Meridian or Diameter of the Plane) from the Center P the Tangent of 23 d. 30 m. (or real distance be∣tween the Pole of the Plane of the Projection and that of the Ecliptic) must needs be, according to our Rule, the Center of this Circle in Projection, and the Secant of those Degrees its Radius, if we prove the said D to be the middle of the Line ♋ ♑ (or Diameter of the Ecliptic) and D ♋ and D ♑ to be Secants of 23 d. 30 m.

'Tis manifest that D ♋ is Secant of 23 d. 30 m. because 'tis equal (by Lem. 1.) to PD, Tangent of 23 d. 30 m. plus P 69 Tan∣gent of 33 d. 15 m.

Again D ♑ is Secant of 22 d. 30 m. because P ♑ (Tangent of 56 d. 45 m.) is equal by Lemma the second to the Tangent and Secant of 23 d. 30 m. Now PD being Tangent of those Degrees, D ♑ must be Secant; therefore D ♋ and D ♑ being equal, D is the middle of the Line ♋, ♑, and consequently PD (the Tangent of 23 d. 30 m. from the Center of your Plane) gives in its Meridian the Center of the Ecliptic, and the Secant of those Degrees the Radius, Q. E. D. and in this maner the way of projecting the other great oblique Circles is to be de∣monstrated.

Nor do's this Rule solely serve for the Description of the* 1.29 great Oblique Circles on the present Planes, but for all that are expressed on Stofflers Astrolabe, or Mr. Oughtreds Horizontal; Nay it shews not only how to draw the Meridians in Gemma Frisius his Projection, but, by the bare conversion of the Terms, the Parallels themselves, tho little Circles.

For first as to the Meridians, whose Poles, (as every body* 1.30 knows) lye all in the Aequator, suppose you would describe the 10th. from the Limb or grand Meridian, which is to be the Solstitial Colure, since, in this Projection, your Eye lies in the East or West points of the Aequator, to wit in the Pole of the said Colure; I say, suppose you were to describe the 10th. from the Limb, it follows by our Rule, because their Poles are 10 Degrees asunder on the Sphere and in the Heavens, that the Tangent of those Degrees gives from A (the Center of the Pro∣jection in Sch. 7th.) the requir'd Center B, and the Secant the Radius; For this Circle on the Sphere cutting the Aequator at the 80th. Deg. from the Pole of your Plane, (or point opposite to your Eye) its extremity C must in Projection be distant from A the Tangent of 40 Deg. only; Now since BC (to wit BA plus AC the Tangents of 10 and 40 Deg) is equal by Lemma the first) to the Secant of 10 Degrees, and since BN. and BS (or distance from B to the two Poles of the World) are visibly the Secants of those Degreees, it necessarily follows, that the Meridian to be describ'd (which pass we know throu' the said three points C. N. and S.) can have no other Center but B, nor Radius but the said Secant. Besides if if we make this Arch an entire Circle (by the prickt Arch NAES) then AB the Tangent of 10 Degrees plus BAE = BC (the Secant of 10) is equal (by Lemma the 2d. to the Tangent of 50 Degrees, but the other part of the said Meridian lyes (we know on the Sphere) 100 Degrees from the forementioned Pole of the Plane, and in projection the Tan∣gent of 50 from the Center A; ergo B is the true Center of the requir'd Meridian, and the Secant of 10 Degrees the Ra∣dius.

[illustration]

Sch. 7.

2. For the Parallels or Circles of Latitude, the same* 1.31 Rule (the Terms as I said being converted) finds both their Centers and Radius's; for if you would project (suppose) the 80th. Parallel from the Aequator, that is to say, the 10th. from the Pole of the World, 'tis but saying, That the Secant of

10 Degrees from the Center of the Plane gives you the Center of the Parallel requir'd, and the Tangent of the same Degrees the Radius. To prove this, let AF (in Sch. 7.) be by Construction the Secant of 10 Degrees, and opening your Compasses at the Tangent of those Degrees place one foot on the said F, and de∣scribe the Circle KLPO; Now because AF the Secant of 10 Degrees is equal (by Lemma the 1st.) to the Tangent of 10 and Tangent of 40 Degrees, therefore AK is the Tangent of 40 De∣grees. Again because AF (Secant of 10) plus FP = FK (Tangent of 10 Degrees) is (by Lemma the 2d) equal to the Tan∣gent of 50, ergo AP is Tangent of 50; but the Parallel requir'd is a Circle which on one side cuts (in the Sphere) the Aequinoctial Colure 80 Degrees from the Pole of your Plane (or point opposite to your Eye) and on the other side at 100, or supplement of the said 80 Degrees, therefore seeing K and P the two extreme points of the projected Circle OPLK are distant from the Center A on the produc'd Axis (or in∣tersection of the Aequinoctial Colure with the Plane) the Tan∣gents of 40 and 50 Degrees (to wit, the Tangents of half the real value of these Arches) it must follow that the said OPLK truly represents the requir'd Parallel, and consequently that the Arch OKL is that part of it, which is farthest from your Eye, to wit, so much of the whole Circle as falls on the Plane. Thus much then for these Parallels, since all are to be describ'd after the same manner, and now having mention'd little Circles, 'tis fit the Reader should know how the Circles of Altitude are to be describ'd on the second Plane of the Pedestal or Projecti∣on which are little Circles also.

The way is easy for if you would have the Almucantar, (suppose) of 10 Degrees (viz. abcd in Scheme 4th) you* 1.32 must proceed thus. Because the Horizon in projection (as we show'd you before) is distant from the Center, (on the North side of the Meridian) the Tangent of 25, 45m. or half the E∣levation, to wit from P to H, and (on the South side) from P to S, the Tangent of 64. 15, or half 128. 30′. the supplement of the said Elevation, therefore the Almucantar of 10 Degrees (being on the Sphere 10 Degrees neerer the Pole than the Ho∣rizon,) will in projection be nearer the Center 5 Degrees. So that the Tangent of 20, 45′ from the Center P giving (a) its ex∣tremity on the North side of the Meridian, and the Tangent of 59. 15. giving (c) its extremity on the South side, it follows that (g) half the distance between the said (a) and c becomes the Center to describe it by; For since all the Circles of the Sphere

are still Circles in projection (except those that are seen in Cultro (as we said) if you have the Diameter (or streight Line that joyns the extreme points of any of them) half of it must needs give you the Center; and in this manner then are the o∣ther Circles of Altitude, to be describ'd. But here take notice that whereas in Sch. 2. (representing the second or trasparent Plane) the Azimuths and Almucantars are found (as I * 1.33 show'd you) by the consideration of the Pricks or Asterisks there ex∣prest: Now, that the Reader may know how to Place them, the very Circles and Arches are describ'd on it, Sch. 4. as it represents for the said Pricks and Asterisks are ever to be in their inter∣sections. And by way the Instrument maker may (if he pleases) make use of Pricks, and no Asterisks on the real Tran∣sparent Plane of the Pedestal; for they will upon second thoughts perform better the Operation.

Notes

• * 1.1

  Op. 17. Sect. 1. pag. 19.
• * 1.2

  A Memoran∣dum.
• * 1.3

  p. 6.
• * 1.4

  p. 68.
• * 1.5

  Op. 5. way 2. p. 10.
• * 1.6

  A Memoran∣dum.
• * 1.7

  Of the first Plane and its bigness.
• * 1.8

  Of the Circles and Stars on it, and how they are pla∣ced.
• * 1.9

  Of the second Plane, and it's bigness.
• * 1.10

  p. 6.
• † 1.11

  p. 68.
• * 1.12

  p. 132.
• * 1.13

  Two Memo∣randums.
• * 1.14

  The Principle on which this Projection de∣pends.
• * 1.15

  Of the Concen∣tric Circles.
• * 1.16

  The general Rule for pro∣jecting the great Oblique Circles.
• * 1.17

  Of projecting the Ecliptic.
• * 1.18

  Of projecting▪ the Horizon.
• * 1.19

  Of projecting the primary Vertical.
• * 1.20

  Of projecting the rest of the Azimuths.
• * 1.21

  An expedite way of finding the said Cen∣ters and Ra∣dius.
• * 1.22

  Lemma 1.
• * 1.23

  Eucl. 27. 1▪
• † 1.24

  6. 1.
• * 1.25

  Lemma 2.
• † 1.26

  Eucl. 27. 1.
• * 1.27

  6. 1.
• * 1.28

  The Ecliptic truly proje∣cted.
• * 1.29

  A Memoran∣dum.
• * 1.30

  The way of de∣scribing G. Frisius's Me∣ridians.
• * 1.31

  The way of describing G. F. hia Pa∣rallels.
• * 1.32

  How to des∣cribe the Cir∣cles of Alti∣tude on the 2 Plane.
• * 1.33

  p. 138 and 139.