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 ...

SECT. VII. 1. How to draw and divide the Ellipsis upon any Plain, to an Index that stands upright (not to the Plain, but) to the Zenith line of the place, or perpendicular to the plain of the Horizon. (Book 7)

THis may be done artificially by calculation, or by a lineary way, but not without too much trouble and incumberance, which would deter any man quite from putting it in practise. I shall therefore let that way alone, and fall upon another more feasible and delightfull.

The way that I intend, is partly by projection of such lines as are usefull, and partly by inscription of such points as the Ellipticall line is to passe through, as shall be seen hereafter. And therefere we shall not need to looke after any position of the plain in respect of declination or leaning.

But first in generall note thus much; that the Ellipticall Diall cannot possibly be drawn upon any plain that lies pa∣rallel to the Index. And consequently to these upright In∣dexes there can no such Diall be made upon any upright plain. And therefore again to such an Index, it is requisite that the plain make some kinde of angle in respect of

[illustration]

Reclination or Inclination from the Zenith line towards the Horizon, which position (notwithstanding) we shall have nothing to do withall in this way of working.

2. The manner of the work.

1. LEt the plain be A B C, and the upright Index A D. First then, assume any point in the Index as at E, and from thence (by the projecting Quadrant, or some Levell) cast an Horizontall levell line as F G upon the plain; or if it cannot be cast upon the plain it selfe, set any board (for a time) that may receive it; lying in the same Levell with the assigned point at E. And further, the same Horizontall line must be continued and returned about the point E, by help of threeds stretched out from some points or other of the Horizontall line before drawn, as from F or G, unto two supporters set up (for a while) for this purpose (either upon the plain or otherwise neer to it) as is to be seen at H and I: so that now the points H F G I are all in one and the same Horizontall levell to one another, and likewise to the first point of the Index assumed at E.

2. At some convenient time of the day, when the Azi∣muth may be found distinctly, observe where the shadow of the Index cutteth the Horizontall line, suppose at K, and there make a mark, and immediately take the Suns Al∣titude.

3. By the Suns Altitude observed, compute what Azi∣muth the Sun was then in, which will tell you what Azi∣muth the observed point K (in the Horizontall line) doth represent.

4. Take a Pastboard as E L, and fit it to the Horizontall

line of the plain F G, and to the assumed centerat ••: and applying it to its proper place (as in the figure is represented) draw a line from the point K to the center E, quite over the pastboard. And then knowing what Azim••••h it is, you may (from it) set off the Meridian in the true Co••st of it, such as will be answerable to the heavens, which Meridian line suppose to be P E upon the Pastboard.

5. You are next of all, from this Meridian line P E, to set of all the other houres of an Horizontall Diall (not the common Horizontall Diall, but) according to su••h num∣bers or arks as are expressed in the Table of Angles, Pag. 29.) And when this is done upon the Pastboard, as you see done at o P q κ s t u: Then,

6. Apply the Pastboard to its former position upon the plain, and from the center E project the houres from the Pastboard to the Horizontall line and threeds. And then taking away your Pastboard, you may draw lines from A to such points as are in the Horizontall line i•• selfe, as A o, A p, A q, A κ, A s, A t, A u. And for the others that stand not upon the Horizontall line of the plain, but upon the threeds which are separate from the plain (such as are ε ρ π) you may project them by the Index A D, reposing the same upon the said points (ε ρ π) of the threed; and marking where the shadow or appearance of the threed traceth the plain, there draw the lines A m, A n, A u. And so all the lines (that the plain is capable of) may be drawn. And it is to be noted, that these lines so drawn are not such houre lines as usually are upon other Dials, but they are those Azimuth lines in which the Sun is at every houre of the Equinoctiall Circle. So the houres and quarters may be put in by the Table, Pag. 29, and the Column of Angles in the same Table.

[illustration]

7. Looke again into the Table Pag. 29, for the Column of Altitudes; for by that must the Ellipsis it selfe be desc••ibed, for the former Azimuthall-ho••re l••••es will give it the true divisions into its re∣quisite parts. The manner of the division is this. Make a Scale of 90 right Sines, but number them versedly, as is here expressed. And let the Scale be of a fit ••ignes••e for your plain, which your own judgement must direct you to do. Out of this Scale (setting alwayes one foot of your Compasses in the point of 90) take such Altitudes as are (in the Table) appropriated to every houre, and put them into those severall houres in this wise. Suppose the altitude for 12 a clock were to be put in, which altitude is 38 gr. 30 min. with my Compasses I take (in this Scale of Sines) from 90 to 38 gr. 30 min. and one foot of th••t length I put into the line of 12, namely P A upon the plain, and (alwayes keeping it upon some part of that line) I remove it thereon, neerer to, or further from the center A, till the other foot of the Compasses being turned about will justly touch the edge (or fiduciall line) of the Index. Then I diligently observe where, or in what point of the line P A the first foot of my Compasses stayeth, for that is the point through which the Ellipsis must passe upon the houre-line of 12. So for any other houre (and the intermediate quarters) taking their altitudes, (set down in the Table) out of the same Scale of Sines, and inserting them in the same manner as was shewed before, you shall finde all the points (for eve∣ry houre and quarter) through which the Ellipsis is to be drawn. Through these points therefore you must draw it carefully, as one continued line, without any breaches or an∣gles in it.

3. Concerning the motion that is to be made either by the Index, or by the Ellipsis it selfe.

THe motion must be made according to the 12 a clock line, which is alwayes the proper Meridian to the In∣dex or Zenith line. That is, the removall or sliding must be either in the Meridian it selfe, or else parallel to it.

Now if the Diall it selfe be made to slide upon the plain, that is, if the Ellipsis be to be described upon another plate that s••all slip over the former plain, then must this plate be first of all laid fast upon the plain, and the houres and Ellip∣ticall line must be described thereon in the same manner that was shewed before. And this plate must move in the Meri∣dian line, that is, the 12 a clock line of the plate (being ima∣gined to be continued forth right) must move through the fiduciall edge of the Index.

And so again, If the Diall be drawn upon the plain it self, and the Index be made to move (the Diall it selfe standing still) then must care be had that the fiduciall edge of the In∣dex do precisely move in the Meridian or 12 a clock line, and that the same Index move alwayes precisely upright in the Zenith line.

4. Concerning the Place of the Suns Annuall course or Zodiac.

THe place of it may be best upon the plain. It must stand (if it be inscribed into a streight line) parallel to the line of 12, and needs not to be in or upon the said line. The mover (whether plate or Index) must have another peculiar

Index in it, called he••e the Zodiacall Index, by which it is to be rectified according to the time of the year, If the Zodiac be put into a Circle (as is mentioned Pag. 15.) then that Diame••er (of the Annuall Circle) which passeth from Tropick to Tropick, must be either in or parallel to the line of 12.

5. Of what limitation or length the Zodiac must be, and how to be described, and where to be set.

IT must be regulated by the former Scale of Sines mentio∣ned Pag. 73. For you may take from 90 (in that Scale) to the number of the Latitude of your place (not comple∣ment as that Scale is numbered) and count that length as a new Radius. Unto th••s new Radius finde the Secant of the Meridians inclination (which what it is shall be presently shewed in the 6 Proposition following) and then make this Secant to be a Tangent of 45 gr. and out of that Scale of Tangents take 23½ gr. for that length being set both wayes from the Equinoctiall, will give the length of the Zodiac (in a streight line, or the Diameter of the Zodiac put into a Circle:) this work is to be understood for the describing of the Zodiacall Scale upon the plain it selfe. But first, you are to pitch the Equinoctials place, which is that place only upon which the Zodiacall Index lyeth, when the mover (whether plate or Index) is placed in the same position that it had when the Ellipticall line was described. Now when this Tangent of 45 gr. is thus found, you may take the De∣clinations (in the Tables of the 4 and 5 Pages:) out of it, and so prick on the Suns yearly course. Or else make the

forementioned Secant a Decimall Scale, and out of it take the Tangents noted in the Tables Pag. 6, 7. So shall you make the same Annuall course of the Sun that was produced the former way.

Note, that if the Index be made to move, it is not neces∣sary that it should move upon or parallel to the superficies of the plain, but may be made to move either horizontall, or at any other inclination: only the fiduciall edge must necessarily be made to move in the very Meridian line, and must also be (in all motions) alwayes upright, or in the Ze∣nith line of the place; and therefore must slide just under the Meridian line.

Note also, That the Zodiac though it be not necessarily confined to be set upon the superficies of the plain, yet it will be most conveniently limited thereunto.

6. What the Meridians inclination meaneth, and how to finde the quantity of it.

BY the Meridians inclination is meant what angle the same line maketh with the plain of the Horizon, and in the former figure Pag. 70 it is expressed by the complement of the angle D A P, or the excesse of it. To finde it, you may either apply the edge of an Inclinatorie square to the said Meridian line, and then the threed (being drawn per∣pendicularly down by the plummet) will shew the said ele∣vation (or inclination) of the Meridian line above the Ho∣rizon, in the degrees of the limbe, if the same degrees be taken to the threed from that side of the Square that stands perpendicularly upon the Meridian line. Or else another way may be by protracting or measuring the angle D A P,

which is the complement of the former required inclination. It may be done thus. Set one foot of your Compasses at A, and let the other foot be extended to any point of the Index, suppose to 8, then measure the same length A 8 upon the Meridian line, which let be A 7, and to this length fit your line of Chords (upon the Sector, or some like opening Scale of Chords.) Then take the length from 7 to 8, and measure the same upon your Scale of Chords, so shall you finde the quantity of the angle D A P, whose complement (or excesse) is the inclination, or elevation, or depression of the Meridian, which is here required.

7. Other things to be noted concerning the Zodiacall Scale.

1. IF the Diall be described upon an irregular superficies, such as is not flat but writhen (as by this course it may very well be) then it is most convenient that the former Scale be set in some other place, and not upon the superfici∣es whereon the Ellipsis is described. It may be contrived two wayes.

First, If the Diall superficies (how irregular soever it be on that face whereon the houres are inserted) be flat at the bottome, and be made to move (the Index standing still) upon some other plain below it) then the best way will be to make the Zodiacall Index upon the moving bottome of it, and to describe the Zodiac upon the other plain upon which the motion is made. And to do this aright, you must project the Meridian line upon that nether plain, and finde out the inclination of it, and so finde the Scale by the Se∣cant of that inclination, just as you were before directed. And so this work will be compleat.

The motion of dials unequall Superficies must be in or pa∣rallel to this last projected Meridian line, alwayes so as that the Meridian of the Diall must passe through the fiduc••all edge of the Index.

Secondly, If the Index be made to move (the Diall standing still) and that the Diall be upon some unequall su∣per••icies, such as is unfit to receive the Zodiac, then the foot of the Index may have it inscribed upon it. Now in this case of unequall Superficies, it is supposed that this foot of the Index cannot move parallel to that uneven Superfici∣es, but must move streight forward in some right line, just along with the Meridian line. In this case you must finde the inclination of the foot of the Index to the Horizon, that is, what inclination a line drawn upon the foot of the Index, either in or parallel to the Meridian line, hath to the Horizon, which must be done by some Inclinatorie Instru∣ment, or some such way as is used in taking the Reclination or Inclination of Plains. And when this is done, you are to finish the Zodiac Scale by the ••ecant of that inclination, in the same manner as was before shewed. So much for this.

2. For the other Scales of the Suns Declination, Am∣plitude, Ascensionall difference, and graduall motion in the 12 Signes, they are to be done in the selfe same man∣ner, and by the same Tables as before (without any diffe∣rence) after that you have found your Tangent Scale or De∣cimall Scale out of which to describe them.

8. Another Observation.

ACcording to that generall observation Pag. 36. Note here; That if to an Index standing in the Zenith line, a Plain be set in the Equinoctiall, the same plain shall have a Circular Diall upon it, equally divided. Then whether the said Equinoctiall plain or the Index move, if they be made to move upon the Horizontall plain, the Zodiac for the Horizontall plain must serve. But if the upright (or Zenith line) Index be made to move upon the Equinoctiall plain, the former Horizontall Zodiac must be set upon the Equinoctiall plain, and must there be enlarged above the Horizontall Zodiac, that is, every part of the Horizontall Zodiac must be made greater upon the Equinoctiall plain in proportion as the Radius to the Co-secant of the Latitude. That is, the Scales that made the Horizontall, being taken as Radius, must now here be enlarged to be as Co-secant of your Latitude; and from them (so enlarged) must the parts of the Zodiac be inscribed upon the Equinoctiall plain.

Note further. That for this Equinoctiall plain which is to descend downwards from South towards North, you may insteed of it set another plain quite contrarily, that is, des∣cending from North towards South; which will be the most convenient of the two, because the upper face of this will give the houre all the Seasons of the year, whereas the other will be only for Sommer upon the upper face of it, and will require an under face for Winter.

SECT. VIII. Hitherto of Ellipticall Dials to all Superfi∣cies whether plain or curved, whose In∣dexes stand upright in the Zenith line of the place: There now followeth some other directions how the same thing may be done to any Superficies, and to an Index set casu∣ally in any position whatsoever. But first are premised some usefull Propositions tend∣ing to the same purpose.

1. An Index or streight line being set casually, how to finde the Re / In-clination and declination thereof.

IF it stand upright, it is free from both those accidents, and falls to be the same case with the upright Index before treated of.

But if it lie leaning, then is it to be dealt withall in this place.

And first for the re / in-clination, it is best to be taken by some inclinatory (square or other) Instrument. The manner of the work is the very same that is performed

in finding the re / in-clination of a plain: because the fiduciall edge of the Index is (here) like the Verticall line of a re / in-clining plain, and the application of the Square must here be the same to this edge that was there to the verticall line of the plain, and the degrees of re / in-clination are to be numbered in both these cases alike. So that it will not be needfull here to make repetition of that which is so often elswhere decla∣red. You are here also to observe whether the re / in-clination be towards the North or towards the South.

Then for the declination do thus.

[illustration]

Suppose the Index to be A B. From the end of it that is furthest from the plain, as B, let fall or raise the per∣pendicular B C, and let B C be some materiall object unto which the side of a Quadrant may be applyed. Or if they should be threeds (and so too

[illustration]

weake for such a purpose) you may extend another threed, as is D E, which may touch the two former threeds A B and C D, and so lie in the same plain with them, and then

to this threed D E apply some upright plain or materiall ob∣ject which may serve the intended work.

To this new plain therefore, or to the two lines A B, C D, apply one of the streight edges of a Quadrant, the limbe of it being turned alwayes towards the Sun, and observe the Horizontall distance of the Sun, as is usually done in finding the declinations of plains. Only here take this cau∣tion, that you count not this Horizontall distance from that side of the Quadrant which is perpendicular to the plain, but from that side which lies in the line E D, or upon the two lines A B, B C. And further, you are alwayes to account this Horizontall distance from that end of the Quadrants side which lookes the same way that the point B (of the In∣dex, which is furthest remote from the plain A C) doth look: as in the second figure is exprest, where the Horizontall di∣stance is more than a Quadrant, it being there to be accounted from F to H. Or for more easie conceit, you may alwayes suppose your Quadrant in that posture to be continued to a Semicircle, as is done in the second figure: and then count your Horizontall distance from that term of the Semicircles Diameter which respecteth the same coast of the heavens with that point of the Index which is most remote from the plain, which is B; that is, you must count it from F, and so the Horizontall distance will be F G H, more than a Qua∣drant, in this particular case.

The Horizontall distance being thus accounted and ob∣served, you are immediately to take the Suns Altitude and to finde the Azimuth. Then for the declination of the Index, that is to know into what coast of the world, that is into what Azimuth of the Heavens (it being continued from the plain at A infinitely forwards toward B) it would point into: I say to finde this, you have only the same work to do

which is usually done in finding the declinations of plains, the same work without any difference at all. And therefore I shall not here give any further directions in this particular, because I have done it often enough in other places, to which the Reader may have recourse.

And so I suppose the position of the Index in respect of re / in-clination and declination to be fully found out, both for coast and quantity; by which two things known, we are further to enquire what Longitude from our Meridian, and what Latitude the said Index pointeth unto, which will be the next Proposition.

¶ Note, that though I call A C a plain, yet it may be any curved Superficies as well as a plain, for the Di∣all will be described upon one as well as the other indifferently.

2. By having the Re / In-clination and Declination of any right line, to finde the Longitude and Latitude thereof.

THis shall not here be much spoken of: for we may suppose the In∣dex A B in the first Fi∣gure, to point up to O in this Projectiō (where∣in the Horizon is N E W S, the Zenith Z, the North Pole P) and that the reclination Z O, with

[illustration]

the Declination O Z S, or O Z P be known, together with Z P the complement of the Latitude of your place, so it shall be easie thereby to finde the side O P, whose Complement is the North Latitude, or whose excesse is the South Lati∣tude required: and the angle at P will be found also, which is the difference of Longitude sought for.

Now in the second case before given where the Index points downwards, or under the Horizon, where the plain is inclining, and so looking downward, we may shape the Projection a little otherwise, and the work then will be as easie as the former. Let n E S W be the Hori∣zon, n the North, S the South, P the South Pole N the Nadir point op∣posite to the Zenith, O the point or place to which the Index respect∣eth. In the Triangle N P O, are given, N P the complement of the Lati∣tude of the place, N O the inclination of the In∣dex; and P N O the de∣clination of the same In∣dex, by which we may finde P O, whose complement is the South Latitude, or whose excesse is the North Latitude required: and the angle at P will be found, which is the difference of Longitude here sought after.

[illustration]

The work is easie to them that understand the like work in re / in-clining plains, and may be performed either by calcula∣tion, or else Instrumentally, as every one shall like best

of. And this is all that needs to be said of this parti∣cular.

3. How to finde a Meridian line, and to erect a true Axis of the World from the foot of the Index.

THe foot of the Index I call that point where the Index enters into the Diall ••••perficies, whatsoever that super∣ficies be, whether plain or curved. As in the two former figures may be understo•••• by the point A.

From the foot or po••••••at A, set some upright threed which may represent th•• ••••••enith-line of the place, as is to be seen in that Figure Pag. ••••. and is there represented by the line A D. Then one way will be to make use of some other Meridian line, observing when the Sun comes to it, and at the same moment to note out the shadow cast by the line A D, for that is the Meridian line. But this is for such su∣perficies upon which the Sun comes at noon: and it ties a man to the Meridian moment. But without this we may (according to the four first Precepts) set off the Meridian line P E upon the Pastboard, and by the Zenith-line A D you may project it upon all objects that shall stand or be set in the way. For which purpose you must place some object upon that Coast of A D upon which any one of the Poles of the World may be projected, and also elevated above your Diall superficies. And upon this obiect project the Meridi∣an neer that place whereabout you conjecture the Axis of the World will passe through. Then with your Semicircle (or projecting Quadrant rather) project the Axis of the World, as is usually done in my wayes of Dialling, and by that means you shall finde a point in the formerly projected Meridian,

which shall represent the Pole of the World. And then further, if from the foot of the Index at A, to this Pole, you fasten a threed, the same threed will represent the true Axis of the world comming from the foot of the Index.

4. Having an Axis raised from the foot of the Index, how to finde in what Longitude and Latitude the Index it selfe lyeth, by a way easier and differing from the former, without looking after any re / in-clination or declination of the said Index.

THis way will not prove (perhaps) so good as I expected, especially in the Longitude, and therefore the former may be used.

The former way that I used was troublesome enough for effecting of what was by it intended, which is the cause that I shall here endeavour to give another more easie.

Let the Diall superficies be supposed to have upon it an Index, an Axis truly placed, issuing from the foot of the Index, with a Meridian line projected also to, or directly under the same Axis, according as was done in the work of the preceding Proposition.

First then, for the Latitude of the Index, it is but enqui∣ring what angle is contained between the Axis and the In∣dex, and the thing is resolved. Now to do this, for the manner of it, we may suppose A B to be the Index, and A X to be the Axis. Opening therefore your Compasses to any extent, set one foot of it at A, and suppose the other to reach up to m upon the Axis, at which point m fasten some knot of threed, which may first be put on, and then slipped to the point of the Compasses at m. So again, set

[illustration]

on the same extent from A to n upon the Index. and at n fasten another knot of threed. Then also to the same ex∣tent open the Radius of a line of Chords. When this is done, take the distance between the two knots m and n, and measuring it upon the same line of Chords, you shall finde what angle is contained between X A and A B. If A X therefore point from A towards the North Pole, and that X A B be lesse than a Quadrant, then doth the complement of that angle X A B give the North Latitude whereinto the Index pointeth: but if X A B be more than a Quadrant,

then doth the excesse give that South Latitude whereinto the Index pointeth. And so contrarily, if A X had pointed from A towards the South Pole, then the angle X A B being lesse than a Quadrant would give the quantity of South La∣titude: but if X A B had been greater than a Quadrant, then the excesse would give the North Latitude into which the Index tendeth.

All the difficulty in this work is, that there is no stable place whereon to fix the Compasses for measuring of the distance between the two knots m and n: but you may hold somthing underneath close to one of the points (with∣out disturbing the threeds place) on which setting one foot of your Compasses, you may measure the distance of the other point. Or else, because it is supposed that the threed Axis A X is fixed upon some solid thing at X, you may open your Compasses to the Radius A X (to which also you are to have a Radius of Chords equall) and set the same ex∣tent from A to R upon the Index (prolonging the Index if need be by a threed till it come to be of a competent length) and there fasten a slipping knot. Then because X is a stable point, you may take the extent from X (setting one foot of your Compasses in that point) to R, and measure the same upon your Scale of Chords, where it will give the former angle X A R, which is to be used as is shewed before.

There will be found other wayes to do this last work. So much therefore for the Latitude.

Secondly, For the Longitude of the Index, it will not be so easily had as the other was. The way that best likes me (amongst many others) for the present, is this. By the Precepts, Pag. 70. you must make some observation of the Sun's Azimuth (and so must you also do by the last prece∣dent Proposition) and you finde what Azimuth that is too.

You may also (further) project it upon your Diall superfici∣es, and then you have two Azimuths upon the said superfi∣cies, namely, the Meridian, and the observed Azimuth. You are to know yet further, what Altitudes the Equinocti∣all hath upon these two Azimuths. And for the Meridian it is certain enough that the Altitude there, is equall to the complement of your Latitude. For the other Azimuth say thus.

As the Radius, Is to the Co-tangent of your Latitude;

So is the Co-sine of your Azimuth, To the Tangent of the Equinoctials altitude upon that Azimuth Circle.

After this you must make some mark or knot upon your Axis A X, which suppose to be at m, and then with your projecting Quadrant (upon your Meridian and Azimuth) from the point m, set on the respective Equinoctiall Alti∣tudes, as the manner of projecting by that Instrument useth to be.

Next, you must project this Equinoctiall Circle all over the Diall-superficies (and upon some other objects where need shall be,) suppose it here to be the line S V.

And furthermore, you are to proiect the Axis A X, and the Index A B, one upon the other, and to observe the line that they both (in that position) do make upon the Diall-superficies, and other objects if need be. I say you must, first, principally observe this line, for it is the proper Meri∣dian line to the Index or Zenith line A B, because it passeth (quoad superficiem) through both the Axis of the world and the Zenith line. Suppose it in the former plain to be A ☉. And secondly you must note the intersection that this line makes with the Equinoctiall line which was now pro∣jected,

which we may imagine to be at V, upon some object laid in the way of purpose. And so also you must (thirdly) take notice of the intersection of the Equinoctiall line with the Meridian line, which suppose at S. For these two E∣quinoctiall intersections and the point m, all three together, must do what we now intend.

The thing here now intended is, to know what angle is contained between the two superficies X A B, and X A T, which are two Meridian circles, or (which is the same) the angle V m S in the plain of the Equinoctiall, which angle measures the said inclination of the two forenamed Meridi∣ans, as it doth of all Meridians.

Now to measure this angle V m S, the best way will be first to extend two threeds, one from V to m, and the other from S to m, crossing each other at m, and they may be con∣tinued further till they meet with some object (in the way standing, or else set for that purpose) where they may be both fixed as at F and G. Then measure from S to m, and from V to m, which is the shortest, suppose S m to be the shortest, and V m the longest. First, take V m, and lay it down in a line, as there is done. Then take S m, and measure it on the same line from M to S or D. Thirdly, take the distance from V to D out of this line, and set it from V to D, upon the line V m, so shall m S and m D be equall one to the other. To this distance therefore m S or m D as a Radius, open some line of Chords, then take the Chord or subtense from S to D (S is a stable point upon which you may firmly set your Compasses) and measure the same length upon your line of Chords, where you shall see how many degrees the angle S m D or S m V containeth.

Now all this businesse is to finde how great an angle is contained between the South part of the Meridian of the

place, and the Meridian belonging properly to the Zenith line. But here (if it be observed well) the angle now measu∣red is made between the proper Meridian and the North part of the Meridian of the place, so that this angle must be the supplement (to a Semicircle) of what is required. The supplement therefore of the angle now found, is (in this case) the Longitude or Meridian into which the Zenith line A B pointeth. I say the supplement of that angle is the proper Meridian of the Zenith line; or the difference of Longitude proper to that line, from the Meridian of the place. But for∣asmuch as the angle V m S doth measure the inclination or angle of this proper Meridian from the North part of the Meridian of the place, and that the shadow of the Index A B doth first come to the proper Meridian A O before it comes to A T the Meridian of the place: we may therefore in such case say, that the angle V m S gives the difference of Lon∣gitude; and that the said difference of Longitude lies East∣ward from the South part of the Meridian of the place, as there in the Figure it appears to lie Westward from the North part. Suppose that the difference of Longitude were 39 gr. 25 min. and the proper Latitude 40 gr. And so much for this also.

If this way be thought not feasible enough, the former in the second Proposition may be used; yet variety in all kindes is delightfull, and not to be rejected in this. It is done, you see, without having respect to any declination or re / in-clination, which the former was founded upon.

5. How to forme the Angles at the Pole.

THis Proposition makes way for computing Tables of Horizontall spaces, and Equinoctiall Altitudes to any Longitude or Meridian differing from the Longitude of the

East hour	generall Table	West hours	east hou	generall Table	west hour
13	0	00	12	6	90	00	6
	3	45—	a		93	45
	7	30—b		97	30
	11	15—c		101	15
11	15	00	1-d	5	105	00	7
	18	45			108	45
	22	30			112	30
	26	15			116	15
10	30	00	2	4	120	00	8
	33	45			123	45
	37	30			127	30
A-	1	55			131	15
A-	39	25		3	135	00	9
A-	1	50			138	45
	41	15			142	30
9	45	00	3		146	15
	48	45		2	150	00	10
	52	30			153	45
	56	15			157	30
8	60	00	4		161	15
	63	45		1	165	00	11
	67	30			168	45
	71	15			172	30
7	75	00	5		176	15
	78	45		12	180	00	12
	82	30
	86	15

Angles at the Pole.	Angles at the Pole.
	s	88	10	n	Proper Merid.	00	00-
3	t	84	25	n	p	01	50	r
	v	80	40	n	9q	05	35	e
		76	55	n	r	09	20	f
		73	10	n		13	05	g
2		69	25	n		16	50	h
		65	40	n	8	20	35	i
		61	55	m		24	20	k
		58	10	m		28	05	l
1		54	25	m		31	50	m
		50	40	m	7	35	35	m
		46	55	m		39	20	m
		43	10	m		43	05	m
12		39	25	m		46	50	m
		35	40	m	6	50	35	m
		31	55	m		54	20	m
		28	10	l		58	05	m
11		24	25	k		61	50	n
		20	40	i	5	65	35	n
		16	55	h		69	20	n
		13	10	g		73	05	n
10	y	09	25	f		76	50	n
	x	05	40	e	4st	80	35	n
	w	01	55	r	u	84	20	n
					z	88	05	n

place, and to any Latitude differing from the Latitude of your place. After that the difference of Longitude is known, you are first to frame a Table of angles at the Pole, which will not be hard to perform, and the manner of it will be seen best in an Example, being altogether like the computa∣tion of the said angles at the Pole for the houres of any place to the same declining plain.

I have here first of all in a generall Table set down the graduall distances of houres and quarters from 12 at noon or mid-day, which will be some help in performing the work. The houres that are on the left side of the degrees are the East or forenoon houres, and those on the right hand are the West or afternoon houres. Suppose (as before Prop. 4.) a difference of Longitude were given 39 gr. 25 min. towards the East, and it were required to know what angles our hours (namely those 12 that stand neerest to it, six upon one side, and six upon the other) with their intermediate quarters do make with this Longitude or Proper Meridian, for so I in∣tend to call it.

I first take the given Longitude or Proper Meridian, 39 gr. 25 min and enter it among the East houres of the generall Table, and I since it to fall in at A; that is, between a quarter and halfe an houre past 9 before noon, and I finde also the circumstant numbers to be 37.30 the lesser, and 41.15 the greater. The differences between these two num∣bers and the Longitude given, are 1 gr. 55 min. and 1 gr. 50 min. as you see them set down in the said generall Table right against the letter A.

These two differences are to be first placed in this second Table of Angles at the Pole, as you see in the Example, where, first of all is written Proper Meridian 00.00 separa∣ted from the rest as signifying only the place where it is to

stand, and above it is placed (the difference between 37.30 and the proper Meridian 39.25, namely) 1 gr. 55 min. So again, below it is placed 1 gr. 50 min. (which is the diffe∣rence of 39 gr. 25 min. the proper Meridian from the next greater number 41 gr. 15 min.) Now to these two radicall numbers, I adde ¼ ½ ¾ and one whole houre, namely a, b, c, and thereby produce e, f, g, on both sides the pro∣per Meridian. Then to r, e, f, g, I adde 15 gr. d, name∣ly, the number of one houre, and thereby do further produce h, i, k, l, on both sides the Proper Meridian, as before. And afterwards for more ease, to the numbers r, e, f, g, h, i, k, l, I adde 30 gr. or two houres, and do thereby pro∣duce on both sides the eight numbers noted with the letter m. And so still adding 30 gr. to the last eight numbers no∣ted with m, or else adding 60 gr. to the first eight numbers noted with r, e, f, g, h, i, k, l, I shall make the eight re∣maining numbers noted with the letter n, both above and below, on both sides the Proper Meridian. And so this Table of Angles at the Pole is compleated, for if it should (in the same manner) be further continued, the next num∣bers (above) would be 91 gr. 55 min. and (below) 91 gr. 50 min. both greater than 90 gr. beyond which there will be neither need nor expedience to go.

This way of forming the angles I thought best to take, because it is more plain and easie than any other, which else might have been in this work used.

You are afterward to place the houres (in this last Table) about the Proper Meridian, just as you see their order to be about the letter A in the general Table, and as you see they are in this particular Table of Angles at the Pole.

The like work serves for West Longitudes.

Thus you have the angles for 12 houres and their quarters,

the same angles serve for the other 12 houres: and whatsoe∣ver is hereafter computed for any of these 12 houres, must be understood to serve also for the 12 opposites, and so the work will be done for the whole Circle, or 24 houres of the naturall day.

6. By knowing the angles at the Pole, and the Latitude of the place or Horizon, how to finde the Horizontall Spaces thereto belonging.

WE must suppose that this Longitude or Meridian which we have before mentioned, is proper to some Horizon or other, above which also the elevation of the Pole must be imagined known, which we may suppose to be 40 gr. as before, Prop. 4. In respect of this plain or Ho∣rizon it is, that the Spaces now mentioned are called Hori∣zontall Spaces.

The way to finde them for this Ellipticall work is con∣verse to that which is usuall in declining plains. Namely thus,

As the Sine of the Poles elevation above this Horizon or Plain, Is to the Radius;

So is the Tangent of each of these angles at the Pole, To the Tangent of the Horizontall space belonging to each of those foresaid angles.

So having found all these Horizontall spaces severally by this form of calculation, you may set them into a Table, as here you see it done in the first of the two main Columns. They signifie the distances of every of the houres and

quarters from the proper Meridian, and are to be accounted as numbered in the degrees of the said plain or Horizon.

7. By knowing the angles at the Pole, and the Latitude of the plain or Horizon, how to finde the Equinoctiall Altitudes or depressions (above or under the same Horizon) due to the said Angles at the Pole or points of the Equinoctiall.

THe proportion by which this is to be effected is this.

As the Radius, Is to the Co-sine of the Poles elevation above the plain or Horizon;

So is the Co-sine of any of the Angles specified in the former Table of Angles at the Pole; To the Sine of the Altitude required, and due to that angle, or houre (rather) in the Equinoctiall Circle.

So in the former example. The Latitude or elevation of the Pole above the Horizon was 40 gr. the complement of that is 50 gr. which is the Meridian altitude and depression of the Equinoctiall Circle above and under the said Hori∣zon or plain, and is therefore to be set for the altitude of the proper Meridian. Now,

As the Radius, Is to the altitude of 50 gr.

So is the Co-sine of every angle at the Pole, or arke of the Equinoctiall in the former Table, To the Altitude due thereunto.

ho∣urs	Horizon Spaces	altitudes or profu.	ho∣urs
	88	49	1	24
3	86	24	4	16	3
	83	58	7	08
	81	30	9	59
	79	00	12	49
3	76	26	15	37	2
	73	48	18	24
	71	04	21	08
	68	15	23	50
1	65	18	26	28	1
	62	13	29	03
	59	00	31	33
	55	35	33	58
12	51	58	36	17	12
	48	09	38	29
	44	06	40	33
	39	48	42	29
11	35	14	44	14	11
	30	24	45	47
	25	19	47	8
	20	00	48	14
10	14	28	49	05	10
	8	47	49	40
	2	59	49	58

ho∣urs	Horizon Spaces.	altitudes or profun	ho∣urs
Proper Merid.	00	00	50	00
	2	51	49	58
9	8	39	49	40	9
	14	21	49	06
	19	53	48	16
	25	13	47	09
8	30	18	45	49	8
	35	08	44	16
	39	42	42	31
	44	00	40	36
7	48	04	38	32	7
	51	53	36	20
	55	30	34	01
	58	55	31	36
6	62	09	29	06	6
	65	14	26	32
	68	11	23	53
	71	00	21	12
5	73	44	18	28	5
	76	22	15	41
	78	56	12	53
	81	27	10	03
4	83	55	7	12	4
	86	21	4	20
	88	46	1	28

And note that r e f g h, &c. below the propet Meridian in the Table of Angles, are complements to n n n n n, &c. at the top of the Table of Angles: and again, those r e f g h, &c. above the proper Meridian, are complements to the lowest n n n n n, &c. successively rising in order one after the other. So that, As the Radius, Is to the Sine of 50 gr. So is the Sine of p q r, &c. To the Sines of the Altitudes

belonging to s t v, &c. And so again, the Sines of w x y, &c. to the Sines of the Altitudes belonging to z u st, &c. And accordingly this Table of Altitudes and depressions is calculated.

If the Pole be elevated upon the plain or Horizon, above 50 gr. it will be sufficient to compute these two Tables to halfe houres only, and so to save halfe your labour. But in lesser elevations it is best to do it to quarters, as here is done n this Example.

8. How to finde the proper Meridian line duly belonging to any Zenith line casually placed, and to draw it upon the plain.

THough the 4. Proposition be not made use of for finding the Longitude and Latitude (which of due pertaineth to any Zenith line) but instead of it the second precedent be thought most fit to be used; yet so much of the fourth will be best to be used, as shall concern the finding out of the proper Meridian line. That is, you must raise an Axis from the foot of the Index, as is A X, and then projecting this Axis A X and the Index A B one upon the other, as if they were both one, you shall thereby also project (by them both together) the Proper Meridian belonging to the Zenith line A B, such as (in the figure of the fourth) is A O, which must be drawn upon the plain accordingly. The manner, of all the particular workings that do hereunto tend, is set down in the fourth precedent Proposition, and therefore will not here again need to be repeated.

9. How to draw and divide the Elli••sis into houres and quarters, to an Index casually set, whose Latitude and difference of Longitude is discovered by the former works.

WHen you know the position of your Index in respect of Longitude and Latitude, you may then compute two Tables to the same Meridian or difference of Longi∣tude, and to the Latitude of your Index, as is done before in the 7 Proposition, one of which Tables is of Horizontall Spaces, the other of Equinoctiall Altitudes or Depressions above and under that Horizon which is proper to the In∣dex or Zenith line casually placed.

By these two Tables the work will be done in such man∣ner as was shewed before, Prop. 2. The manner of the work is this.

1. You are to assume some point in your Index A B, let the point be C, where you may fasten some knot of threed that it may not be lost again.

2. From this point you must draw an Horizontall line, not in the levell of your own Horizon, but in that Horizon which is proper to that Index or Zenith line A B: that is, it must lie perpendicular to A B, making right angles (in e∣very part of it) to that line, and must have respect (in this perpendicularity) to the point C. The meaning is, you must imagine a plain to passe through the point C, and the same plain to be perpendicular to the line A B; or that the line A B is a perpendicular surgent line to the said plain passing through the point at C.

Now this work will be somewhat hard to perform if A B

[illustration]

be a threed only, and not of some more stur∣dy substance. Yet the best of it is, that there is no very great pre∣cisenesse here required, for the work to be done will be good though this Horizon∣tall plain be not placed so exactly perpendicu∣lar.

[And so it may be noted, that if other Dials be described by the Equinoctiall Circle and not by the Horizon∣tall, the work of drawing the Equinoctiall perpen∣dicular to the Axis will be difficult, but though no great accuratenesse be used, yet the work will be per∣fect enough, and no way defective for the losse of a degree or two in the perpendicularity required, which I thought good here also to note, because I have omitted it in all my other Precepts of Projecting Dials.]

Wherefore you may do it by some Pastboard, applying one edge of it to any line projected from A B the Index, as to A D, and keeping the edge there, you may turn the flat of it to the Index A B, and draw a line by it, or make two pricks through it into the pastboard, whereby a line may be drawn, but above all note the point C upon it. Then to this line of the Index thus drawn, and from the noted point of it at C, erect a perpendicular: So applying your Pastboard

to its former place again (the edge of it lying upon A D, and the flat of it applyed to the Index A B, and the point in it, noted for C, being again fitted to C in the Index, I say thus doing) you may note where the last drawn perpendicular doth cut the line A D (which must be extended by help of some threed if need be) suppose at E: at the point E (then) you must say that one point of the proper Horizontall line is to be taken. Then in like manner you must seeke another Horizontall point: first by projecting a line from the Index (any where) such as is A F, and by applying one edge of a pastboard to that line, and the plain of that pastboard to the Index A B, and so noting the point C, and drawing or mark∣ing the line A B upon it, &c. as was done before. So you shall finde another point of the same Horizontal line or plain (rather) which suppose to fall at F.

Now having three points of the Horizontall plain at C, E, and F, (which I suppose not to lie in one and the same right line, for that must with carefulnesse be avoided here) you may project some part of that line upon the plain, as E F, the rest of it (so much as shall be found usefull) may be made up with returns of threed, and regulated or kept in the same plain by projection, as the man∣ner of working that way useth to be, and as here you see exprest, by the line H E F G, lying in the same plain with the point C, and that whole plain lying perpendi∣cular to the Index A B.

[illustration]

3. The next thing to be done is the drawing of the houre lines upon the plain. And the first

thing h••re presupposed to be done, is the drawing of the proper Meridian, performed by the 8 Prop. That (I say) is supposed as already done before any of this work is begun. Let the proper Meridian be A V. Having then made Tables (by the 7 Prop.) for the Horizontall Spaces of your hours from this proper Meridian A B; you must first apply a past∣board to the Horizontall line E F, and fit the center of it to C: and upon the Pastboard, project the proper Meridian from C to A V, or from C to P. And then by the Table of Horizontall Spaces in the 7 Prop. (for we now here suppose that this is the Diall for which those Tables were computed) you may (upon that pastboard) set off all the houres and quarters from the proper Meridian upon this pastboard: and applying the pastboard into its proper place, namely to the Horizontall line E P F, and to the point C, you may project the houres and parts of houres from the Pastboard to the ho∣rizontall line H E P F G, as the manner in this way of Di∣alling is well known.

4. Having transferred the houre-points into the Hori∣zontall line, you may (by help of your Index A B) project and draw the houre lines upon the plain, which we will sup∣pose done, because the manner of doing it is the same with that which was done before for upright Indexes.

5. To know where the Ellipticall line must come, or to finde the points in those houre lines, through which it must passe, we must work in the same manner as before (in the 7 § of Pag. 7••.) is exprest, namely thus. We must make a Scale of right Sines of a fit length, and number them Ver∣sedly, and out of that Scale we must take such Altitudes or Profundities (which you will, one or both) as the Table Pag 97. giveth, which Table we here suppose to be compu∣ted for this Example whereabout we now are. And taking

these Altitudes from that Scale (that is, from the end at 90, to the altitude numbered upon the parts of the Scale) we must insert them into their respective houres to which they belong. They must be inserted in this manner. Having ta∣ken any altitude out of the Scale, and found the houre upon which it must be placed, you must set one foot of that extent upon the houre line (keeping it alwayes thereon, but) remo∣ving it untill the other foot being turned about, may only touch the fiduciall edge of the Index: and when the feet of the Compasses are thus fitted, you must note upon what point of the houre line the foot that is thereon doth stand, for through that point of that houre must the Ellipticall line passe.

The same manner of work you must perform upon every houre, untill you have gone through 12 of them, which do make up halfe the houres of the whole Diall. And if you strike the lines through the center, you shall have all the 24. And looke what is done upon any one houre line, the same is to be done upon the opposite. That is, looke what distance (upon a plain) the Ellipsis hath from the center upon any one line, the selfe same distance from the center must the oppo∣site line have. But if the description be made upon an uneven superficies, then this rule may not, perhaps, hold: yet this will; namely, Looke what distance from the Index (the least or perpendicular Index I mean) any point of the Ellip∣sis hath, the same perpendicular distance is to be given for the Ellipsis upon the opposite houre line. And by this means you may put in as many houres and Ellipticall points as you please. And through these points you are to draw the El∣lipticall line.

10. Concerning the motion that is to be made, either by the Index, or by the Ellipsis it selfe.

THe motion must be either in, or else parallel to, the pro∣per Meridian, and not elsewhere. Now if the Diall Superficies be made to slide, and the Index stand still; then (though the Diall be upon an unequall superficies, yet) the motion or sliding must be upon a plain. And it must be up∣on the proper Meridian, projected by the Index from the Diall plate, unto or upon that said plain, and the motion must be in or parallel to it. And it must be noted, that this proper Meridian thus projected never falls directly under any Index that is (not direct but) declining, but it falls aside from it, according as the Index stands aside from the proper Meridian upon the Diall superficies, and as it shall be laid by projecting it.

But if the Index move (the Diall standing still) then the fiduciall edge of it must be made to move alwayes according to, and in the very proper Meridian line it selfe, and not any where else, which may be contrived sundry wayes, as eve∣ry man shall invent to his own liking. Great care must be had that the fiduciall edge of the Index (in the motion of it) keepe alwayes one justly parallel situation.

11. Of the place of the Suns Annuall course or Zodiac.

THe best place for it is to be considered of according to that which moveth.

If the Diall Superficies move, it must (though it be rough (as I said before) yet it must) move upon a plain, this plain

therefore in this case is fittest to receive the Zodiac. And then there may a peculiar point or threed Index (serving only for the Zodiac) be set upon the Diall-moving-plate, where∣by it may be rectified according to the time of the year.

If the Index be made to move, then either the Zodiac may be set upon the foot of the Index which guides the motion of it, and the peculiar or Zodiacall Index may be set somewhere where it may stand to point at the Zodiac. Or else the Zodiac it selfe may have its place upon some part of the standing plate or body, and the Zodiacall Index may be placed upon the foot of the Diall Index; from whence it may be made to shew the Zodiacall parts, as oc∣casion shall be.

12. How the Zodiac is to be limited in regard of length, and how to be described and set in its true place.

THe limitation of it must be according to the Scale of Sines by which the Ellipsis is described, which Scale is mentioned Pag 102 § 5. and Pag. 73 § 7▪

The Index or Diall plate (that is the mover) must move according to the proper Meridian of the Diall Superficies, and precisely so, as that fiducial edge of the Index must move (not elsewhere, but) in the very proper Meridian it self. This is, if the Index move. But if the Diall Superficies move, then the proper Meridian thereof (or the Superficies of the proper Meridian, namely the Superficies made by the pro∣per Meridian cutting through the Diall plate, which is of any thicknesse) must move directly upon, or through, the fiduciall edge of the Index. These things are intimated be∣fore. And that the Zodiac is to be described upon a plain,

though the Diall it selfe be not so. And further, that if the Diall plate move, (though it be not a plain it selfe, yet) it must move upon a plain, and that the proper Meridian is to be described upon the same plain, and the motion to be directed according thereunto. These things are often incul∣cated, because they are hard to be conceived, and had need of the better consideration for that reason. Now further.

1. If the Diall plate be supposed thus to move upon a plain, and on it the proper Meridian be drawn, then first of all, The angle is to be inquired that is made between the In∣dex and that part of the proper Meridian which is projected upon the plain whereon the motion is made, which how to measure will be a hard matter to give rules for, because the variety of cases and positions of one to the other will be so various. It is first to be supposed, that it makes a just right angle with it, and consequently that the Zodiac is described upon the proper horizon of the Index. And if upon this supposition the Zodiac be to be limited, then the rule will be the same with the former given in the like case, namely thus. Upon your Scale of Sines (by which you described your Ellipticall line) take from 90, to the Latitude of the Index, and count that length for a new Radius, and keepe it. Then when you have found the forenamed angle (of finding which more is said Prop. 13. following) to this new Radius finde the Secant of the complement or excesse of that angle: this length or Secant will be the Tangent of 45 gr. or the Decimall Scale by which you are to describe the Zo∣diac on both sides from the Equinoctiall point or line, ac∣cording to the numbers in the generall Tables made for this purpose, Pag. 4, 5, 6, and 7.

[For placing the Equinoctiall point in the Zodiac (upon which all the other parts of that Scale do depend)

you must set all in the same posture that they had when the Diall was described: and when you have made fit place for the Zodiac, and set on some pecu∣liar Zodiacall Index, you must note the place upon which the said Zodiacall Index pointeth, for that place is the peculiar place for the Equinoctiall, from whence all the other parts of the Suns annuall course must be set on.]

2. If the Index be made to move in any depth under the Diall superficies, then a slit must be made in the same Super∣ficies that the proper Meridian would cut, and the Index continued therein, down into one streight line till it meet with the foot whereon the Index is fixed, and which being moved, carryeth the Index along with it. And the Zodiac must be described either upon this foot (the Zodiacall Index being made to stand still while that said foot with its divisi∣ons moveth) and that also parallel to (or in) the proper Me∣ridian projected upon it: or else the foot must carry the Zo∣diacall Index, and the Zodiac must be described upon some other convenient place, as the sides (&c.) of that foremen∣tioned flat. And here, for limitation of the Zodiacs length, you must finde the Inclination of this moving foot to the In∣dex or Zenith line (which angle of inclination is mentioned before, and spoken of afterwards in the 13 Prop. following) and work directly as you did before by finding the new Ra∣dius, and the Secant of the complement (or excesse) of the angle. By which you shall finde the Tangent of 45 gr. or the length of your Decimall Scale, out of which the Zodi∣ac may be described by the Tables▪ Pag. 4, 5, &c. as is men∣tioned before in this proposition. Thus (in these cases) is the Zodiac to be set in its true place, thus it is to be limited and d••scribed.

For Occurrences of other sorts of Cases than are here mentioned (whereof there will not be many) he that can un∣derstand to do these things aright, will be able to grapple with them; and for such as do not understand what is here said, their best course is to let these difficulties alone.

13. How to finde what angle is made between the Index and that part of the proper Meridian which is projected upon the plain whereon the motion is made, or which drawn upon the foot of the Index which maketh the same motion.

THe way that for the present I thinke of is this. Apply a streight edge of pastboard to the projected line, and the plain of the same pastboard to the fiduciall edge of the Index, and make two points upon the pastboard, by or through which the same fiduciall edge passeth. Then taking away the Pastboard, draw a streight line through those two points or marks, and so measure the angle made between the forenamed edge of the pastboard and this right line. If it be so that the line crosseth not the edge of the Pastboar••, then draw such a parallel line to the edge as may crosse the former line, and then with a Scale of Chords you may measure the angle.

14. Further observations concerning the motion and daily fitting of the Diall and Index, for setting them true.

THe best way is to make the Diall plate move, and the Index to stand still (in these obliquely situated Index∣es)

for the Zodiac will (in such cases) be most easily descri∣bed and made usefull. And in this case the Index may be also set fast first, and quite finished before the Diall be drawn at all. Then also the Diall will be drawn more easily, and the motion of the Diall plate may sooner (this way) be con∣trived, then can the motion of the Index be contrived when the said Index is to move and the Diall stand still. The mo∣tion (as hath been often said) must be according (that is pa∣rallel) to the proper Meridian: and the slit (for the fiduciall edge of the Index) may be so contrived that the fiduciall edge it selfe (which is best to be a fine threed) may be also the proper Index, and the Zodiac may be described upon the Diall-moving-plate, closely contrived to the threed. Or thus at least, The lenght of the whole Zodiac may be so li∣mited as is before mentioned, and then if you desire to have it drawn upon some other place of the plain upon which the Diall superficies moveth, and the same Diall plate to carry the peculiar Index for the Zodiac; then (I say) by these two terms prefixed to the length of the Zodiac (by the fiduciall edge of the Index) having reference to the Diall-moving-plate (as before) you may determine the length of the same Zodiac upon the standing plate whereon the Diall plate mo∣veth, for the new Zodiacall Index set fast into the moving plate, will (upon the standing plate) shew the terms of the same Zodiac suting with the terms upon the moving plate fitted to the standing Index. This will be direction enough because one way may serve as well as many. Yet I doubt not to set down many more which would be without end, if they were all put together, for inventions in this kinde would be too too many.

2. If the Index be thought best to move, then must a pattern of it (that is some threed) be first set up. And to this

vice-Index must the Diall be described upon the plain (or whatever curved Superficies it be.) And the Zodiac is best to be fitted either upon the plain whereon the foot of the In∣dex is to move (for a plain it must move upon, whatever be the Superficies whereon the Diall is described) and that same foot to carry the Zodiacall Index upon it; or else upon the foot of the Index, and the Zodiacall or peculiar Index to stand fixt upon the plain (or Diall superficies) And then when all this is done, the vice-Index must be taken away and a true substantiall Index put in the place of it. And for that purpose, you were best, before you take it away, to finde at what elevation it stood from the plain, and likewise to draw the perpendicular just under it. For by these two helps you may set up a true substantiall Index, regulating it thereby into the same position; which to do, must be left to the judgement of every man to contrive as he shall see requisite.

But in both these wayes care must be had what inclinati∣on the Zodiacall Scale hath to the Index, according to what is said in the 13 Prop.

3. It must here again be taken for a Rule, thot a plain, if it lie parallel to the index is not possibly capable of these kindes of Ellipticall houres, but the Index must have some inclination to the plain, and make some angle with the same.

4. All these Precepts do serve to make such Dials as are upon fixed walls or Superficies that cannot be removed. But if it be to be done for moveable Bodies, if they be regularly cut, then the declinations of every of the plains are known by their regularity; so that such plains of them as are capable of any Diall before mentioned (for the Index and the plain must not be parallel in any case, they may be perpendicular) may have such Dials upon them as they are capable of, either to Indexes perpendicular to their plains, or to Indexes lying

in the Zenith line of the place. But for these casuall Index∣es upon them, you will be put to it, to know what declinati∣ons they have from the Meridian upon the regular body (for I suppose that there is a Meridian drawn upon some one of the plains or other.) The way that for the present I think upon is this. Set the Body upon the foot true in respect of horizontall or upright position, that is, let the foot of it stand upon some just horizontall plain, though in respect of declination it stand at all adventures. Then from the fidu∣ciall edge of your casuall Index, let fall a perpendicular (I mean perpendicular to the Horizon of the place, not to the plain) and to that perpendicular point, from the point of the Indexes concourse with the plain, draw a right line; this line shall represent the Azimuth wherein the said Index lyeth. Then again, with your eye repose a perpendicular hanging threed upon the two points (namely, the center or concourse of the Index with the plain, and the late found perpendicu∣lar point) both together, and project the umbrage of the threed so hanging upon the Horizontall plain. So also you must repose the shadow of some one of the Cocks or Axes with your eye, upon the Meridian or line of 12, and then al∣so the perpendicular threed must be projected upon them both joyned before into one, so that Threed, Stile, and Me∣ridian must now be all as one line; and the threed so hang∣ing project the shadow of it upon the same Horizontall plain. Now these two lines thus projected either do concur and make an angle, or else (by drawing a parallel to one of them through the other) they must be made to concur, and looke what angle they make at their concurrence, the same being counted the right way is the angle of the casuall Indexes declination from the Meridian line. Thus the Declinati∣on is found.

Then the Reclination may be found as before is declared, and by help of them the Longitude and Latitude of the In∣dex may be had, and so the Diall made by the precedent Precepts.

A briefe DEMONSTRATION of the 7^th. and 8^th. Sections.

IN the 7 § all are made to the Zenith line of the place. If therefore you imagine an Horizontall Ellipsis to be described to that Zenith line, and upon that Ellipsis a kinde of compressed Cylinder to rise up∣right, parallel to the said Zenith line, with houre lines raised from the severall points of the Horizontall Ellipsis, then will the Zenith line shew the houre upon those surgent lines: and the same Index (or Zenith line) must shew the houre among tho••e upright lines in the compressed Cylinder, by the same reason that it doth upon the Ellipsis it selfe, upon which the said Cylinder is raised, And if so, then it matters not what part of these lines is taken in for this use, since any one point of them will serve to do the work. The way therefore that is used for the projecting of these houres (by help of the Table in 29 Pag.) as it will serve to finde the Ellipticall points upon the Horizontall plain it selfe, so must it serve to finde some

one point of these surgent houre lines upon any other Super∣ficies, because they keepe the same distances alwayes from the Zenith line (from whence the projection of them is made, and whereon they depend altogether) and therefore it findes (upon any Superficies) the points of those lines which passe through, or do intersect the Diall Superficies. Therefore it is that these Ellipticall points (upon all Su∣perficies) are points of those surgent lines rising from the Horizontall plain, and that the Zenith line must have the same relation and situation to them, that it had to the points upon the Horizontall plain it selfe, And consequently, that this way must be of the same truth upon any Superficies that it is upon the Horizontall. So much for any upright Index, set upon all sorts of Superficies, with the houres thereon depending.

In the 8 §. are handled casuall Indexes; which if well considered will fall to be the same with them in the 7 §. For the Table of Angles and Altitudes by which they are made, are calculated to that Horizon which is proper to the Index, considered as a Zenith line. And so from that Horizon we may imagine the like surgent lines to rise all parallel to the Index or Zenith line, and the very same reason to hold in these, that held in the former.

To the Reader that shall have the view of this first draught of Precepts.

THese Rules here given may seeme to be stuffed with many impertinencies, and some need lesse difficulties, which the Author acknowledgeth wil••ingly, and excuseth, by reason that they were his first med••tat••ons in th••s kinde; and so much the more undigested by how much the lesse pra••••••se hath been by him used therein. The truth is, he never described any thing sutable to the Cases of these two last Sections. And if the Reader be any way able to discern what it is to write upon a Mathematicall Subject wherein hath preceded no reall ••epre∣sentation, he will not only excused difficulties and impert••nencies in the trad••tion, but will wonder if there be not some miscarriages in point of truth: of which (notwithstanding) the Author is confident this Treatise is clear.

S. Foster.

15. Having placed a Diall plate to move (let the Coast of the motion be casual••) how to fit a stedfast Index to it, and to describe an Ellipticall Diall upon the said moving plain.

THis Case was forgotten before, but now here supplyed. You must observe the line (upon the immoveable plate) which the midst of the moving plate doth describe, and sup∣pose that line to be the proper Meridian. Then from some convenient point of that proper Meridian, raise a true Axis. Project the Axis upon the proper Meridian, and from any point of the said Meridian raise a threed (or Index) any where, only so as that both this Threed, the Axis, and the Proper Meridian may all three appear in one line. There six the threed, and then finde the Longitude and Latitude of it, and afterwards describe the Diall to it, according to the rules given before.

16. If an Index should be set up and made moveable upon a standing plain, there can no Diall be described thereto,

UNlesse the Index be made of wire or some such bend∣ing substance, but to such there may. For if you ob∣serve what streight line the foot makes in its motion, you must count that as the proper Meridian, and so setting up an Axis to some point of it, you may put in an Index into that foot, so as the Axis, and fiduciall edge of the Index, and the Proper Meridian, may all three appear in one line. And then finish your work as is before directed.