Elliptical or azimuthal horologiography comprehending severall wayes of describing dials upon all kindes of superficies, either plain or curved, and unto upright stiles in whatsoever position they shall be placed / invented and demonstrated by Samuel Foster ...

CIRCULAR Horologiography.

How to make an Horizontall Diall in a Circle equally divided, to shew the Houre of the day, and Azi∣muth of the Sun.

HEre before we have had ELLIPTICAL HOROLOGIOGRAPHY, now shall fol∣low CIRCULAR HOROLOGIO∣GRAPHY, which sheweth how to make a Diall in a perfect Circle equally divided into houres, (whereby to finde the houre) upon any plain what∣soever.

Divide a Circle into 24 equall parts, and take so many of

them as your Horizon hath houres in the longest day, or ra∣ther so many as the degrees of your greatest Amplitude (East and West) from the South do arise unto; which here at London will be neer 18 of them. Then divide each of these parts into 15, which for the Azimuth will signifie degrees, for the houre will stand for four minutes of time apiece.

The altitude of the Horary Index may thus be found.

Adde your Latitude to 90 gr. halfe that summe is the elevation of the Horary Index above the Horizon. Thus at London, 〈 math 〉〈 math 〉 45 min. which is the altitude or elevation required. Or, adde halfe the complement of your Latitude (which is 19¼) to your Latitude (51½) the sum (70¾) is the elevation of the Index.

The standing and looking of it.

It must stand right over the line of 12, elevated above the said line 70¼ gr. and must looke toward (but not into) the North Pole.

The motion of the Horary Index.

It must move to and fro, directly over the line of 12. Or else the houres must move to and from it, according to the line of 12, so as that the same line may alwayes lie under the foot of the said Index. The motion of the one or the other is necessary, because else the Circle of equall parts can never shew the true houre all the year long.

How the Zodiac is to be limited, and laid, and charactred.

The motion of the Index must be regulated by the Zodi∣ac. The Zodiac therefore must lie either in, or else parallel to the Meridian line. The length of it is thus to be limi∣ted. Count the Semidiameter of your Circle (viz. A B) for the Radius: to that Radius, Either

1. Make the Sine of 70¾ (taken to the Radius of your Circle) a Tangent of 70¾; and to the Radius of that Tan∣gent finde the Secant of 19¼ (the complement of 70¾) that length shall be the Radius of the degrees, or the Decimall of the Tangents of the Zodiac, to be inserted by the Tables Pag. 4, 5, 6, and 7.

2. Or else, Make the Sine of 19¼ (estimated to the Semi∣diameter of the Circle A B as Radius) a Radius, and to that Radius finde the Secant of 19¹, this last length or Secant shall be the quantity of the Tangent of 45 gr. or of the Decimall Scale by which the numbers Pag. 4, 5, &c. are to be inserted. And in these Northern Horizons ♋ and ♈ must be placed so that 12 may be neerest to the Index in Summer, and furthest off in Winter.

The manner how to fashion the Cock which holdeth the Index.

The fashion may be seen by the figure A C D. At A is the place of the fiduciall point of the foot of the Index to be assigned: and in that point a hole must be pierced, and a threed fixed. Then the Cock must have two holes more pierced, one at C, the other at D, both to stand perpendi∣cularly over the line of 12. That at C must be so placed, that the angle C A B may be 70¾ gr. That at D must be placed so, that D A B may be an angle of 90 gr.

[illustration]

How to place this Diall for use.

YOu must either fix your Diall plate in the Meridian line, and truly Horizontall▪ or else upon an Horizontall or levell flat you must draw a Meridan line, whereby to place it upon any occasion. Then,

To finde the Houre.

Make use of the Index A C, and rectifie the foot of the Index to the requisite place in the Zodiac, (either to the day of the moneth, or the degree of the Signe.) When it is thus rectified and set, the shadow of the threed A C will shew the houre of the day.

To finde the Azimuth.

Put the threed from A to D, and let A D be your Index. Then (alwayes) place the foot of the threed in the center of the Circle at A, So shall the threed A D give the Azimuth.

¶ Note, That the Scales of Declination, Amplitude, Ascensionall difference, may be placed by the Zodiac, and used as is before shewed in the Ellipticall Dials.

Another way to make the same Horizontall Diall equally divided, to finde the Houre and Azimuth.

THe former way maketh the Index to point up neer to the elevated Pole. This other will re∣quire the Index to point towards the contrary Coast, namely, that of Noon or Mid day, or toward (but not into) that Coast of the hea∣vens where is the depressed Pole under the Horizon, but still the Index is to be above the Horizon.

The altitude of the horary Index above the Horizon, must be halfe the complement of the Latitude, or halfe the height of the Equinoctiall. Here at London 19¼ gr. which is the complement of the former wayes elevation 70¾

T••e standing and looking of it, must be right over the line of 12, drawn out at competent length, and must justly point into the Coast of 12 mid day: Or towards (but not into) the South (or depressed) Pole.

It must move as the former did.

The Zodiac is to be laid as before, and to be limited and charactered thus. The Semidiameter of your Circle being taken for Radius, you must use the two former rules in this manner.

1. Either make the Sine of 19¼ a Tangent of 19¼, and to the Radius of that Tangent finde the Secant of (the com∣plement of 19¼, namely) 70¾, that length shall be the Ra∣dius of the degrees, or else the Decimall Scale of the Tan∣gents of the Zodiac, and they must be inserted by the Tables Pag. 4, 5, &c.

2. Or else, make the Sine of 70¾ (estimated to the Se∣midiameter of the Circle A B as to a Radius) to be a Radius; and to that Radius finde the Secant of 70¾, this last length or Secant, shall be the length of the Tangent of 45 gr. or of the Decimall Scale, by which the numbers in the Tables Pag. 4, 5, &c. are to be inserted.

By this instance for the Horizon of London, where the numbers 19¼ and 70¾ are proper, may the like work be per∣formed in other places of different Latitudes from London, only by altering the numbers, according as the proper La∣titudes of such places shall require. And in all Northern Horizons ♋ (or the longest dayes) must be so placed that the point of 12 in the Circle may then be neerest to the Index, and in ♑ (or the shortest dayes) it may be furthest of. But in Southern Latitudes, the places of ♋ and ♑ are quite con∣trary, because there the Sun being in ♑ makes the longest day, and in ♋ the shortest.

The manner of fashioning the Index, of placing the Di∣all, and using it, is the same as was shewed before.

How to make the like Houres and Azimuths by an equally divided Circle, upon all Plains whatsoever.

YOU are first to finde these three things. 1. How much, and which Pole is elevated above your plain, which is equall to the La∣titude of your plain. 2. The position of the (usuall substylar or) proper Meridian of the plain. 3. The difference of the Plains Lon∣gitude from your own, or the angle at the Pole made be∣tween yours and the plains Meridian. And when these things are had, the work will be like that which was before. For the way will be twofold▪ as there it also was.

The first way.

Take halfe the complement of the plains Latitude, and adde it to the plains Latitude it selfe, the summe will give the elevation of the Index above the plain. This Index is to point towards (but not into) that Pole which is elevated above the plain.

The other circumstantiall things about the Index are as those before, if insteed of the line of 12 there mentioned, we here take the proper Meridian (or Substilar drawn in its right position) of the plain.

The limitation of the Zodiac must be somewhat as before.

That is, you must count the Semidiameter of your Diall Circle to be as the Radius. And to that Radius, 1. Either finde the Sine of the elevation of the Index, and make that Sine a Tangent of the same elevation of the Index; then to the Radius of this Tangent finde the Co-secant of the former elevation, the same Co-secant shall be the Radius or Deci∣mall Scale whereby to insert those dayes or Ellipticall de∣grees set down in the Tables Pag. 4, 5, &c.

2. Or else to that same Radius of the Circle, finde the Co-sine of the elevation of the Index, and to this being made a Radius, finde the Co-secant of the Indexes elevati∣on, and so shall this Co-secant be the Radius or Decimall Scale as before.

The second way.

The altitude of the Index above the plain is the halfe com∣plement of the plains Latitude. The Index must (in this way) point towards (but not into) that Pole which is depres∣sed under the plain (I mean toward that Coast of the world, but must still remain above the plain.)

The standing and motion of the Index, with the chara∣ctering of the Zodiac, and other circumstantiall matters, as of the Cock for one Index, and the usage for the houre, must be agreeable to the former rules given before.

The limitation of the Zodiac must be done by the two former rules given (in generall terms) in this Page.

How to divide the Circle of houres.

It must be divided into 24 equall parts, and those of the 24 taken into use as will be serviceable, the rest left out.

Every of these 24 parts may be divided into 15 gr. or into quarters and halfe quarters (for here in such plains there will be no Azimuth shewed for the place, nor well for the plain it selfe, and therefore it is best to omit it in these altogether.) All the doubt will be where you must begin your division.

From the intersection of the proper Meridian (of the plain) with the Circle, set off (upon the same Circle, and to the true Coast) an arke equall to the plains difference of Longitude. And from that point so inserted, (which you must suppose to be the point of 12) you are to begin the di∣vision of your Circle. The numbring of the houres on both sides 12 will not be hard to finde, in respect of their course, for the course of the shadow of the Index will give that.

NOTE

1. In all Polar plains (such as our upright East and West, &c.) the Index in both these cases or wayes (here before given) will be elevated 45 gr. and the Radius of the Diall Circle will be the Radius or Decimall Scale by which the Zodiac Scale is to be inserted out of the Tables, Pag. 4, 5, &c.

2. If at any time the Zodiac prove longer than the Dia∣meter of the Horologicall Circle, and so the Index do go without the Circle; at such time as it is without, the sha∣dow of the Index will go one way till it touch the Circle, and then back again the same way it formerly came.

A briefe DEMONSTRATION of these Circular wayes of making DIALS.

THe Circular wayes have dependance, and are de∣duced out of the precedent Ellipticall wayes; and the Cases are but two, wherein the same may be done upon any Horizon or plain, as may be per∣ceived by the former Precepts, wherein the altitudes of the Indexes above the plain are ever made to be either halfe the complement of the Plains Latitude, or else (the summe of the Latitude and that halfe complement, which summe is equall to) the complement of that forenamed halfe com∣plement.

According to these two cases here are two Schemes fit∣ted. The projections are made upon the plain of the Meri∣dian Circle P Z H X. P is the Pole, Z the Zenith of the place, H A the Horizon of the place, Ae A the Equinoctiall Circle, D A and R A are drawn in the middle of Z P and Ae H or Ae O. D B C X represents one Azimuth or Ver∣ticall Circle proper to the plain R A, and by that one all the rest of them may be understood. D A represents the Index in both the former Cases, and R A the Horizontall plain (not of the place but that Horizontall plai•• which is) pro∣perly

belonging to the In∣dex or Zenith line D A. The first Scheme shews the first of the two former Cases, the second shews the second of them. Their Demonstrations will be both one. For the questi∣on in them both is, How to these inclinations of the Index, the Diall, or hora∣ry line, that falls upon the Horizontall plain of the place, comes to be a per∣fect Circle.

[illustration]

First, therefore consi∣der, that in all the Ellipti∣call Dials, the horary line hath relation to the Equi∣noctiall Circle, and to the Index or line that is to give the shadow (whatsoever the Superficies be upon which the projecture is made, whether plain or o∣therwise it matters not, as appears before, but here we deale only with plains, because Ellipses and Circles are plain Figures.

[illustration]

Secondly, consider, that through the severall houre points of the Equinoctiall Circle, right lines are to be sup∣posed to passe infinitely extended, till they meet with the plain whereon the projecture is to be designed. Which

issuing out of these infinite lines (if they be regularly cloath∣ed about with an inflected Superficies must comprehend a Cylindricall concave, either round or compressed according as the forenamed infinite lines are either perpendicular to the plain of the Equinoctiall, or not perpendicular to it.

Thirdly, These parallel lines so drawn through the houre points of the Equinoctiall, parallel to the Index, must all fall perpendicular to the Horizontall plain which is pro∣perly belonging to the Index as a Zenith line. And the same lines upon that proper Horizon do alwayes make an Ellipsis, except only these two Cases. First, If the Horizon be the same with the Equinoctiall Circle, and the Zenith line or Index the same with the Axis of the world, for then those lines do there make a perfect Circle. Secondly, If the Ho∣rizon be a Polar or Meridian Horizon, and the Index or Ze∣nith line fall into the Equinoctiall (having no Latitude from it) for then those lines do all fall into (or coincide with) the plain of the Equinoctiall, and consequently do make (upon their proper Horizon) a direct right line: ••ea more, because the said lines do lie all upon the plain of the Equi∣noctiall, and are all drawn out of the equall divisions of the Equinoctiall Circle, and besides are all parallel one to the other; therefore they make (in this case) a double line of Sines, as is said before in the fifth note Pag. 37. Now to come to my intended purpose, which is, to prove 〈◊〉〈◊〉 In∣dexes so laid (as is mentioned before in the two precedent Cases) will require (upon the Horizontall plain of the place) a true Circle and not an Ellipsis.

Suppose therefore D A to be the Index (according to one of the two precedent conditions) then must R A be the pro∣per Horizon to that Index. A••d according to the former Doctrine, if Ae, and B, and A, be taken for three of the

houre points in the Equinoctiall, and Azimuths drawn through them from D the Verticall point of the Index, then I say (in these two cases and not otherwise) looke how the Equinoctiall is divided by those Circles, in the same manner is H O the Horizon of the place divided. But because the Equinoctiall is ever equally divided in respect of houres, therefore the Circles from D to X (from the Zenith of the proper Horizon R A to the Nadir, which are Azimu∣thall Circles) shall cut the Horizon of the place (H O) into equall parts. This, I say, is true, Because D Ae B and X H C are equall Sphericall Triangles in three of their quantities [For D Ae is equall to X H, and the angles at D and X are equall, as also the right angles at Ae and H] and therefore equall in all the rest, if like be compared to like: that is, H C is equall to Ae B; and consequently A C equal to A B: so also C X equall to B D. And if so, then the Chord B C must be parallel to the Index or Zenith line D X, and so Ae H is parallel to D X, and so contrarily (in these two Ca∣ses) right lines drawn from the houre points in the Equino∣ctiall Circle, parallel to the Index, must cut (upon the Ho∣rizon of the place) equall parts, equall to one another, and equall to the houres upon the Equinoctiall Circle. That is to say, they cut the horizontall Circle it selfe into 24 equall parts or hours. And consequently (in these two Cases) those forementioned right lines must designe (the horizontall Circle it selfe, that is) a Circle upon the horizontall plain of the place, though upon the proper Horizon it selfe (R A) they must designe an Ellipsis (falling perpendicularly there∣on) according to the third observation in the 128 Pag. And from that Ellipsis these lines B C, H Ae, may be conceived as surgent lines (such as are mentioned before Pag. 113) con∣taining a compressed Cylinder, which Cylinder being cut

two wayes, (that is subcontrarily) will again reveile the ori∣ginall Circle (at least in one of the two Sections) from whence it selfe and the surgent lines tooke their forme and places.

NOTE.

1. The same reason holds in all plains, that in them also there may be found such a Circular Horologiography. For they are Horizons to some place of the World or other, and are therefore capable of the same accidents that the forementioned Horizon of a particular place is.

2. That these Cases are the same (in a manner) with those §. 5. For here a new Zenith line (as D A) is found to H O the Horizon of the place, such a Zenith line (or Index) as may lie equidistant from the Horizon of the place H O, and from the Aequinoctiall Circle Ae A: And there there is a plain set up which may stand so, as that the Zenith line of the place may stand equally distant from the Aequi∣noctiall Circle and this plain. In both, the Index shands equally distant from the Equinoctiall, and from the plain whereon the houre points are to be inserted.

For the limitation of the Zodiac.

The first work in the limitation of it is for the Ellipsis that should be described upon the proper Horizon, and the ground of that is the same with what was before delivered Pag. 75. The reason of both will be plain enough out of the forego∣ing Diagrams. For the Zenith line D A, if it were to move upon the proper Horizon R A, it must move in the proper horizontall Zodiac, as formerly it did, Pag. 17, 18. Which Zodiac is to be made as before, and here in the first work is shewed: the reason of the division of it is given in Pag. 61, 62. Here is only to be shewed the reason of the enlarge∣ment.

If the Index were set upon the proper Horizon R A, then it must move thereon (perpendicularly) from A to S suppose. But it moves not upon that but upon H O the ho∣rizontall plain of the place, and yet to keepe the same di∣stances in motion from the first position of it in D A X, that is, it must move parallel alwayes to D A, but not upon A S but upon A H, which to do, it must not have the parts of the Zodiac in A S (to guide upon H O) but proportionall ones to them, that is, instead of A S must be taken A m; which is all one as to say, As A S the Radius, to A m the Co∣secant of S m A, or D A O, the Indexes elevation (above the Horizon H O;) So are the parts of A S the prime or fundamentall Zodiac, to the parts of A m, the secondary or inlarged Zodiac.

Thus much of Circular Dialling.

APPENDIX.

SEE the Figure in page 120. To that Zodiacs length you may finde such a Radius as shall be thereto justly competent so as to make the Ellipticall Horizontall Diall R T S; and ha∣ving made that Diall, the upright Index D A will give the hour upon it, as the slope Index A C will give the houre upon the Circle. And these two being severall In∣dexes and Dialls must set themselves as the other do, and wil not stand right till they agree. Onely this cannot be so punctuall, because the two Indexes A C and A D are no•• so far distan••••s in the other horizontall Di••ll. But the A∣zimuth

will here be still shewed, as was mentioned before.

Having the Zodiac limited, how to finde the two extream Diameters of the Ellipsis.

The length of halfe the Zodiac is alwayes to be esteemed the Tangent of 23½ gr. Now because we here suppose this Zodiac already made, and to be intended for an horizontall Diall; therefore, Either thus,

1 As the Tangent of 23½, is to the Tangent of your La∣titude; So the length of the Semizodiac upon your Diall, to the Sine of your latitude, which is the shorter Semidia∣meter A T.

Or else thus,

2 As the Tangent of 23½, is to the Radius or Tangent of 45 gr. So is your known and limited Zodiac, to the Cosine of your Latitude. The Radius, to which Cosine is the lar∣ger Semidiameter A W.

Thus having fitted your Ellipsis (to the Zodiac) you may describe and divide it in such manner, as was formerly shew∣ed, pag. 1.

FINIS.