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. VI. AN ADVERTISEMENT Concerning s••me ot••er uses of the Instrument that was last described. (Book 6)

SInce the writing of that which is gone be∣fore, other things there are which came into my thoughts concerning the further use of this Instrument.

It were more expedient therefore that the Ellipsis were divided into houres and degrees, rather then into quarters and half quarters of houres. Which division (as also the description of it) may by protraction be performed by those rules that are given before. But if it be thought better to do it by Tables; then by the former rules of calculation given for this work, you may frame two Tables, the one of Angles, the other of Altitudes to your own Latitude, such as in the following Table are computed to ••he Latitude of London. B•• these Tables you may put in every houre and third de∣gree,

the intermediate degrees may be equally divided. The numera∣tion will be either by 15, 30, 45, &c. accor∣ding as the signing of it by hours wil requir••; Or ••lse besides the houres you may set on 10, 20, 30, &c. in small figures, which will stand without hindring the numbers (or nume∣rall letters) that are set for the houres.

Ad Latitudinem 51 gr. 30 min.
Hor	gr.	Angles	Altit.	gr.	Hor
XII	00	00	00	38	30	00	XII
	12	3	51	38	26	3
	9	7	39	38	16	6
	6	11	26	37	57	9
	3	15	12	37	31	12
XI.	00	18	54	36	58	00	I.
	12	22	33	36	18	3
	9	26	08	35	32	6
	6	29	38	34	40	9
	3	33	04	33	41	12
X.	00	36	25	32	37	00	II.
	12	39	41	31	27	3
	9	42	52	30	14	6
	6	45	59	28	56	9
	3	49	00	27	33	12
IX.	00	51	57	26	07	00	III.
	12	54	50	24	37	3
	9	57	38	23	04	6
	6	60	23	21	28	9
	3	63	04	19	49	12
VIII	00	65	41	18	08	00	IV.
	12	68	16	16	25	3
	9	70	47	14	40	6
	6	73	17	12	53	9
	3	75	44	11	05	12
VII	00	78	09	09	17	00	V.
	12	80	33	07	26	3
	9	82	56	05	35	6
	6	85	18	03	44	9
	3	87	39	01	52	12

This Ellipsis thus divided, may represent either the Horizon, or else any of the (Almi∣cantars or) parallel cir∣cles to the Horizon; & the divisions of it must then signifie the Azi∣muths. And if it be too big (especially when it comes to represent any of the higher Almican∣tars) you may adde an∣other sliding plate of the same breadth that the former was, na••••ly

M O here equall to M O in the former, and M N must in both be justly equall, that it may both slip in the former cavity, so as just to fill it, and that the same limbs, centers, and threeds, may in both agree.

But then a lesser Ellipsi•• must be here described and so the length G E or F H will be diminished.

[illustration]

A lesser Ellipsis (I say) must be described, which though it be lesse, yet it must be like and propor∣tionall to the former.

Therefore assuming any length as M P, for the longer Radius, you must (to that Radius) make M R and M S equall•• to the Sine of your Latitude, and so describe and divide it as formerly was ordered.

Then again, to this new Ellipsis there must a new Scale of Tangents be made, whose length must be limited as the former was, in this manner. To M P as Radius, finde the Co-sine of your Latitude, and make that Co-sine a Radius or Tangent of 45 gr. and according thereunto, continue the Scale of Tangents as far as it will go. It must begin upon the same line that the other Scale of Tangents began, and must go the same way with it. And in this new way of ordering the Scales and the Ellipses, it must be conside∣red what the other parts of the Instrument do signifie

The end of the lower plate, upon which the Scales be∣gin, must be accounted for the North, the other end C D is to be taken for South. The side lines are the Meridian. The degrees of the limbs are to be used as the degrees of

the Equinoctiall. The threed applyed to those degrees are the Meridians comming from the Centers, which are the North Pole. The Tangents are the degrees of Altitude. The Ellipsis notes out that Almicantar which the threed Index stands unto upon the Tangent Scale.

And further. It is to be here noted, that if the Meridians (or threeds comming from the center) were rightly divided, those divisions or parts of it should signifie the degrees of Declination from the Equinoctiall. But the inconvenience of it is, that because one Ellipsis is to represent every Almi∣cantar, therefore the Radius cannot possibly be at all times the same in length, but must varie according to the removing or severall positions of the said Ellipsis. That is, when the Ellipsis signifies the Horizon (or stands at the beginning of the Scales, without any altitude at all) then the Radius M O in the first moving plate, or M P in the second moving plate, is the Scale of Declinations, but at other times, when the Ellipsis signifies any Almicantar, then must the Secant of that Almicantar (of Altitude) be taken as the Radius for that particular work. But alwayes the Radius (of what length soever it be) must be divided as a whole line of Sines, the greatest parts beginning towards the center, the least parts ending at the remotest end from the center, and yet then a∣gain, the numeration of the parts or declinations of it must begin at the remotest end, and must end in the center at the Pole.

It might be applyed to use in the motions of the Stars, but that would be troublesome; therefore it shall suffice to make it usefull only for the Sun.

The best way that (for the present) I know whereby to divide this Radius (thus of severall lengths) is this. Divide the length from the center T to C into 90 Sines (the

greatest parts beginning at the center, but numbered the contrary way.) This line may be drawn from the center T into some part of the limbe, where it may have room to re∣ceive divisions. Then with your Compasses take the Ra∣dius T V, equall to M O, and from the end of the former Scale, open the threed to the least distance, and where the threed stayes, there make a mark upon the limbe V D, for there is a new Scale to begin, for the Secants that were men∣tioned before.

Then lay the threed from T to 80 in the opposite limb, and move the Ellipsis till the point O fall just under the threed, and take T O in your Compasses, and put in that distance as before, so shall you put in 10 gr. into this new Scale. In like manner lay the threed to 70, 60, &c. and bring the Ellipsis till the point O lies under it, so shall T O (in these severall positions) give the lengths to be inscribed in this new Scale, for 20 gr. and 30 gr. &c. all the rest must be done in like manner, till you have put in as many as will come within the reach of the limb, which will be upon 60 gr. or thereabout.

The like should also be done for the lesser Ellipsis, work∣ing in the same manner by the threed and the point P. These are called the new inscribed Scales.

The former line of Sines serves to them both indiffe∣rently.

Now when things are thus fitted, the uses of them will be such as in particular Astrolabes is vulgarly known.

Some of the Uses are here mentioned.

1. Having the Suns Declination and Alti∣tude to finde the Houre and Azimuth.

First, For North Declinations. Set the Ellipsis to the Altitude counted in the Tangent Scale. Then count the same Altitude in the new inscribed Scale of Secants, and thereto lay the threed. Afterwards, take the least distance from the Suns declination, (counted in the fore-mentioned inscribed Scale of Sines) to the threed. Set one foot of that distance in the center, and extend the other till it crosse the Ellipsis, and where it crosseth, thither apply the threed, which (in the limb) will shew the Houre and scruple (in de∣grees) required. The same point of the Compasses doth also immediately shew (upon the Ellipsis) the Azimuth sought for. If the Compasses do crosse the Ellipsis twice (as somtimes it will) take that crossing that is furthest from the Meridian.

Secondly, for South declinations. The work is in a man∣ner the same: only you must note that the threed will crosse the Ellipsis twice, and you are in this case to take that cros∣sing which is neerest to the Meridian.

By supposing the Altitude to be 00, that is, by laying the Ellipsis upon the line T V, and using the Declination as before, you may finde both the Amplitude and Ascen∣sionall difference as before, and what else doth thereon depend.

But because these things being done neer the Horizon, will not prove good, especially when the Declina∣tion is little from the Equinoctiall: and because the work at the best is but troublesome, I shall here break off, supposing that I have already written too much.

A DEMONSTRATION of the Ellipticall Diall upon an Hori∣zontall Plain: shewing the reason why the same Diall, by an upright Index should shew the true Houre.

THE reason is principally deduced from the Sphere it self: and secondly, from the Or∣thographicall Projection of the Sphere up∣on the plain of the Horizon; which as it doth represent the Sphere it selfe, so it doth perform the same conclusions with the like certainty that the Sphere doth; and that upon this generall ground, that looke what Circles, and in what parts they cut each other upon the former, the same Circles and the same parts of those Circles do upon this latter cut each other in the same manner. I shall here take for granted, that the Projecture is (for truth) in all respects answerable to the Sphere it selfe, which is abundantly made good by those that have treated upon this subject. Then from th••s Pro∣jection I shall shew how the Ellipticall Diall upon the Hori∣zon must be of the same truth with it. And therefore these things following must be considered, as so many Lemmaes to prove what is here required.

[illustration]

Note, that in no form of Projection besides the Or∣thographicall, this kinde of Diall can be true; that is, that no figure but an Ellipsis will do it, nor yet all Ellipses, or Ellipses divided into all 〈◊〉〈◊〉, as in some Projections the••

are, when one halfe or 180 gr. on one side, are greater than the other halfe or 180 gr. on the other side: these will not do here, but only such as whose extream Diameters are the sixes and twelves, that is, such as are produced by infinite Orthographicall Projection.

LEMMA I.

First, The Equinoctiall Circle, and all the Parallels of Declination, are projected upon the plain of the Horizon into like Ellipses. And that they are all of them divided (by the Meridians) alike or proportionally one to the other.

The reason of this may be conceived upon this ground. Since the Equinoctiall and all the Parallels are Circles of a just parallel situation one to the other (as their name imports:) therefore on what point soever of the horizontall Axis the eye is placed, the said Circles must be Ellipses (excepting only if the eye be placed upon the superficies of the Sphere, for then, and only then, they will be all perfect Circles,) as all that write upon this subject do sufficiently make good: and consequently, when the eye is placed upon the same Ax∣is, in an infinite distance, as this Projection supposeth. But the second thing is more to our purpose, and that is, that all their degrees and quarters, and other like parts, art like and proportionall one to the other (not all the parts of one and the same Ellipsis are like, that is equall, one to the other, as in the Circles themselves upon the Sphere they are, but each one of these Ellipses is justly like any other of them, both in the whole, and likewise when any of the like parts of one be compared with the like parts of the other, as one quarter, &c. with another.) And the reason briefly is, be∣cause the eye is supposed (in this Orthographical Projection)

to stand in an infinite distance from the plain on which the projection is made; and because it doeh stand in an infinite distance, therefore all these Circles are alike situated to the eye, and consequently must make a like or proportionall Projection of every of those parallels with their like situa∣ted parts. But if the eye should stand at a finite distance, then these parallels would not have a like situation to the eye, neither in respect of distance, nor in respect of position, and so the like parts of the parallels could not be like one to the other in the Projection, because of their different positi∣on. But in the Orthographicall Projection at an infinite distance all things will be alike, being alike projected from a like position. I say in this it will be so, and in none besides this.

COROLLARY.

Whence will follow: That the Equinoctiall Circle, or Ellipsis rather, may represent both it selfe, and likewise any other of the Parallels, if it be so contrived to stand from the Zenith point, as that it may have a propor∣tionall situation from it, suiting with the position of any other Parallel which it is to represent. And this is the main ground upon which this Diall will be proved to be true.

LEMMA II.

Secondly, Any two like plain Figures may be alike situated. to any point assigned; so as that from that point any two right lines being drawn infinitely, shall cut proportionall parts from those two like figures.

This is generall to all like figures, whether they be two or more; plain or solid: but for our purpose it will be enough to shew it to be true in like Ellip∣ses. One way for the making of one Ellipsis like to another, is by assuming of any point, as a in the greater Ellipsis, and from thence drawing as many lines as you wil, as a b, a c, you must divide them into proportio∣nall parts at d & e, that as a b to a d; so a c to a e: & then through those points the lesser Ellipsis may be drawn, which must be like to the great∣er▪ and alike situated to the point a, because all those lines are proportionall, and the subtenses b c, d e, will likewise be parallel and proportionall. And if the lines a b, a c, had been augmented proportionally (as here they were dimi∣nished) there would have been a greater Ellipsis described, like to the other.

[illustration]

COROLLARY.

Hence will follow▪ That if the centers of two like Ellipses be laid upon one line, as upon one of their common Diameters, and from any point assigned (unto which they are to be alike situated) there be an infinite line drawn to any point of one of the Ellipses, and the like point of the other Ellipsis be brought to that ••••finite line (the center of it still keeping upon the common Diameter,) those Ellipses in that situation shall stand alike posited to the assigned point.

Because there are two lines (the common diameter and the infinite line) that from the point assigned do cut like parts of both Ellipses. The common diameter makes them alike situate in respect of one dimension (suppose the length) of the locus planus whereon they lie, and the infinite line limits them in respect of the other dimension (suppose the breadth) of the same locus planus. So that they are quite limited in respect of their like situation to the point, and can (in this respect) lie in no other place for this individuall position.

LEMMA III.

Thirdly, All the Azimuths of the Horizon are (in this Orthographicall Projection) cast into streight lines meeting all in the Zenith point.

The reason is, because all these Circles do crosse each other upon the Axis of the Horizon (or upon the Zenith line,) upon which line the eye is imagined to be placed at an infinite distance. Therefore they all (that is, the plains of

them infinitely continued) do crosse through the eye, and consequently must all be projected into streight lines. And their concourse being upon the Zenith line, and the same Zenith line comming into the eye, the said whole line, and their whole concourse will be cast or resolved into one single point, answering to the Zenith point of the Sphere.

The application of these things to the purpose intended.

First, In the Projection it selfe we are to consider these things.

If it be furnished with parallels and houres, as the man∣ner is, and laid horizontally with the Meridian line of it just North and South: and if further, there be an upright In∣dex set in the Zenith point at Z, then shall the shadow of that erect Index represent the Azimuth in which the Sun is, and if you note where it cuts the Parallel wherein the Sun (for that day) is, the same shadow will shew (among the houres of that parallel) the time of the day.

This conclusion is true upon any Horizontal Projection (the Stereographicall as well as the Orthographicall) for the streight lines or Azimuths whether shadow or Index com∣ming from the Zenith, and cutting through any houre in any parallel, doth shew the Azimuth in which the Sun must be at that houre for the day of that parallel.

And further, because all the Projection is made upon the plain of the Horizon (which therefore must be the funda∣mentall Circle of the Projection, and) which for that cause is, and must be (as all fundamental Circles of any Projection are) equally divided; Therefore an upright Index standing

in Z the center of it, and so answering to the Zenith line of the Sphere, and the degrees of the Horizontall Circle an∣swering to the degrees of the Horizon in the Sphere. I say the shadow of that Index doth really shew the true Azimuths of the Sun, or the true angles of position that the Sun at any time maketh with the Meridian line. And further, the same shadow or Azimuth where it crosseth the Parallel of the day (which Parallel is divided into its proper parts, like to the parts of the Equinoctiall, by the Meridian Circles issu∣ing out of the Poles) it notes out also the houre of the day.

So that the Projection made in this manner with an up∣right Index and divided Parallels, will (by the shadow of that Index) shew the true houre of the day upon that parallel that is proper for the day, if the Meridian of the Projection lie in the true Meridian of the Horizon.

The next thing to be shewed, is how the Ellipticall Equino∣ctiall may supply the use of all the Parallels.

That is to say, How the Equinoctiall, being made mova∣able, and the Index standing alwayes still, may be fitted to represent any of the Suns parallels. And because (as was said before) it is every way like to each of the Parallels, it will therefore be only required to give some rule how the said Equinoctiall may be at any time placed in a like positi∣on to the Zenith point at Z, that any parallel hath to the same point, so as that any right line (or Azimuth) being drawn from thence may cut a like part or point in the Equi∣noctiall that it doth in the Parallel.

Let the Parallel be H E, I would remove the Equinocti∣all Ae A so, as that it might have a like position to the point Z, that the Parallel H E hath.

By the Corollary of the first Lemma, it may represent it. By the Corollary of the second Lemma, the way of it will be

easie. For first, the two Ellipses H E and Ae A, have their centers upon the Meridian line Ae H Z P, as upon one of their common Diameters. Secondly, the Parallel H E cuts the six a clock Circle, or Meridian P E A, in the point E, making E A equall in radiall degrees to H Ae (for the Parallel is every where equidistant to the Equator, and the Meridional Arks, therefore, intercepted between them, such as are E A and H Ae, must be equall.) So that E A is the Declination of that Parallel from the Equinoctiall. If therefore the line (or Azimuth) Z E be drawn infinitely through the point of 6 in the Parallel, or through the Declination (of that Pa∣rallel) counted in the Circle of six; and then the Equinocti∣all (still keeping its center upon the common Diameter Ae P) be slipped up till the point A thereof do concur with the in∣finite line (drawn before) at G; in thus doing, I say that the Equinoctiall is in a like situation to the point Z, that the Pa∣rallel H E is in, to the same point. And consequently, that any right line (or Azimuth) from Z, will cut the same (rather like) parts in one that it doth in the other. As by the Corol∣lary of the second Lemma will appear.

The next thing to be enquired is, what the line A G is, or how it must be found and estimated.

For this purpose, consider, that Z P E and Z C O are two Triangles rectangled at P and C, and having a common an∣gle at Z; therefore, As Z P Co-sine of the Latitude, to Z C the Radius; So P E, the Co-tangent of the Declination, to the Tangent of C O. Or, As the Radius Z C, to the Co-sine of the Latitude Z P; so is the Tangent of E A, the Pa∣rallels Declination, to the Tangent of A O, that is, to the line A G, which is the Tangent of A O, because the arke A O is expressed in its full quantity▪ without any en∣larging or fore-shortning, and the right line A G stands

to it as a Tangent thereof, standing at the end of the Radius Z A. Now consider, that if Z C be considered as Radius, then must the Tangent of E A be the Tangent of the Pa∣rellels declination, estimated to the same Radius. And be∣cause Z P is in the selfe same proportion to A C, therefore: If Z P (the Co-sine of the Latitude) be estimated for a Ra∣dius, then must A G be esteemed as the Tangent (of this Parallels declination) to the said Radius Z P. And hence it will follow that

If Z P the Co-sine of the Latitude [being taken (as that Co-sine) to the Radius Z C or Z A, which is the longer Radius of the Ellipticall Equinoctiall] be counted as a Radius or Tangent of 45 gr. and so be divided and continued as occasion shall be, It is to be noted I say, That this Scale thus limited will be a right Scale for the requisite motion of the Ellipticall Equinoctiall to such a position in respect of the Ze∣nith point, as that it may represent any parallel, whose Declination is known, if it be removed to the same Declination counted upon that Scale. And being set in that like posture, any Azimuth or shadow of the upright Index that would passe through any houre point of the Parallel, will also passe through the same point of the Equinoctiall, and so this one may serve for them all.

This gives the reason of making the Zodiac or Annuall Scale, mentioned in the former Treatise, I say it gives the reason of the second way mentioned for lesser Latitudes. For if either the Signes of the Zodiac, or the moneths of the year be put in according to their declinations taken in this

Scale, it will be all one to set your Ellipsis to the Signe or Moneth, that it is to set it to the Declination, since they are made one to answer to the other.

Then for the first way for greater Latitudes: because,

As the Co-sine of the Latitude, Is to the Sine of the Latitude;

So the Radius, (or Tangent of 45 gr.) To the Tangent of the Latitude.

You see that if you divide the Co-sine of your Latitude into 45 Tangents; or the Sine of your Latitude into the Tangent of your Latitude, the Scale will be all one accor∣ding to the former proportion.

What is before spoken of North Parallels of Declinati∣on, is the same in South also, and one Scale (for the kinde of it) serves both; only in North Declinations the Equi∣noctiall goes neerer to the Zenith point, in South it goes further of.

All other Scales annexed to the Ellipsis have their de∣pendance wholly upon this Scale of Declinations, and will not need any further explication in this place.

Then again, it matters not where the little Scale or Zo∣diac stands, so that it give the just quantity of removall for the Equinoctiall Ellipsis.

And so also, it is no matter whether the Index move, and the Equinoctiall stand, or contrarily: It is only required that their situation be such as may make the foot of the In∣dex stand in a proportionall position to the station of the Parallel, as was formerly shewed.

Further. For the circular year or Zodiac, the ground of it is thus. The Circle is supposed to be divided into equall

parts or degrees: and so every Sine (as is D F) is a proper Sine to any arke (as D C) to which it is annexed. Again, A B in another con∣sideration is to be e∣steemed (as formerly for the Semidiameter of the Circle, so now again) for the Tan∣gent of 23½ degrees, whence this propor∣tion will stand:

As the Radius A B, Is to the Sine F D; So the same A B, tan∣gent of 23½▪ To the Tangent A E. Which is the Tangēt belonging to the Right Ascension C D.

[illustration]

[Because in the Sphere, As the Radius,

To the Sine of any Right Ascension, So the Tangent of 23½

To the Tangent of the Declination answering to that Right Ascension.]

If therefore you have the Right Ascension belonging to any day, and put it into the Circle of equall degrees, as C D here is, the Sine of that Right Ascension, viz. F D, will be equall to A E the Tangent belonging to that Right Ascen∣sion, if B A be taken for the Tangent of 23½ gr. So that this way of putting in Moneths and Signes is the same in

effect with the former Zodiac or Annuall course of the Sun.

Thus far for demonstration of what was hard in the first Section, Pag. 8, &c. concerning Horizontall plains.

What is added more in the first Section of framing it to other plains that are direct, is the same with this. For there is no difference but only in the latitude of the plain, which is no re••ll discrepance from the former, they both going upon one ground, and therefore no more to be said here of such plains.

Sect. 2. Pag. 22, &c. Of framing it to Declining Pl••ins.

DEclining plains, if they had their proper houres up∣on them, would al••o (in this Ellipticall respect) be the same with the former direct, or Horizontall plains. The distinction is, because they are made to ano∣ther Meridian than is their own, that is, to the Meridian of the place, and all the account of houres is deduced from it. The difference therefore of these from the former, is only this, that the houres in the Ellipsis are not evenly fitted to the quarters or diameters of the Ellipsis, but fall to stand as casually they may. Now if in the former case of Horizon∣ta••l plains, you do but suppose the Equinoctiall and Paral∣lels to be divided from any casuall Meridian as P M (not P Ae) into houres and parts, then will follow still the same things in substance that was before. Namely, That if the Equinoctiall Circle be removed according to A G (which

must be supposed to be suited to the Latitude of the place as well as to the Suns parallel of declination) with those divisions now spoken of, then still, when the Equinoctiall by that motion hath got a like situation to the Parallel, the parts of one will be answerable to the parts of the other, in respect of the shadows or Azimuths that are cast from the Zenith point and upright Index standing in it at Z. So that even in these plains also, the same houre will be shewed by the Equinoctiall that would be by the Parallel. And so the ground is (in these) like (rather the same) in substance with the former direct plains. So much of these.

Sect. 3. Pag. 29, &c.

THat pricking down of the Horizontall Diall men∣tioned in this Section, is the pricking down of the Equinoctiall with its houres or parts, just as the Or∣thographicall Projection it selfe hath them. The first Ta∣ble shews what Angles or upon what Azimuths from the Meridian every houre point lies. The second gives the Al∣titudes, or rather the distances of the same houre points from the Zenith, And therefore no more will be required of this.

The fourth Section requires no explication nor Demon∣stration.

Sect. 5. Pag. 37, &c.

FOr the varieties it must be known, that the substance is the same with that which went before, which being well understood will give light enough to this.

But in this Section there is mention made of a Circle (in∣steed of the Horizontall Ellipsis) elevated to the height of the Equinoctiall. The reason of that will best be seen out of the Sphere it selfe, For there we know, that the Equino∣ctiall and all the Parallels to it are both equally divided, and equally elevated, and so being all alike, the Equinoctiall may supply the room of each and all of them. Only it must be required (whether the Index move toward the Cir∣cle; or the Circle towards it) that this motion (if the Zodi∣ac be upon the Horizontall plain) be limited by an Hori∣zontall Zodiac, such as was used for the Ellipsis upon any plain, (which plain, what ever it be, I now count as an Ho∣rizontall plain.) But if the Index be made to move upon the Equinoctiall (which though it do, yet it must still lie in the Zenith line, not perpendicular to the circular plain, but making an angle with it equall to the Latitude of the place, and not the Latitude of the plain.) If, I say, the Index be made to move upon the Equinoctiall plain, then must a new Scale be made like to that upon the Horizontal plain, but somewhat larger; that is, it must be augmented so as that the parts and whole of the Horizontall Scale, to the parts and whole of this Equinoctiall Scale; must be as the Radius, to the Co-secant of the Latitude. Or thus. Because, As the Radius, to the Co-secant of the Latitude; so the Sine of the Latitude to the Radius: and so again, the Co-sine of the Latitude, to the Co-tangent of the Latitude. Therefore, whereas on the Horizontall plain (for the Ellipsis thereon described) you looke the Co-sine of your Latitude, and make it a Radius or Tangent of 45 gr. you must in this take the Co-tangent of your Latitude, and make it a Radius or Decimall Scale; and by it you may put on the Zodiac or moneths as was prescribed before. This Co▪sine mentioned

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before was taken to the greater Radius, of the Ellipsis, and so this Co-tangent here mentioned must be taken to the (same greater Radius of the Ellipsis, if the Ellipsis were here used, or to the) Radius of the Equinoctiall Circle, whose Radius must be conceived to be the same with the greater Radius of the Ellipsis. For from this Circle proje∣cted Orthographically (that is, by perpendiculars let fall from the Periphery to the subjacent plain) is the Ellipsis de∣duced, and from that Ellipsis (again) is this Circle raised or restored. And so the reason of this Circular Diall with an upright Index will be understood well enough.

Then whereas it is said that it makes no matter which way this Circle be raised, that is, whether toward North or South, there is no difficulty in this, for which way soever it is turned, if the plain and center of it lie in a just distance from the Index or Zenith line, the same Index must shew the same point or houre, because in both wayes they are situ∣ated from it upon the same Coast.

Sect. 6. Pag. 47, &c.

THat which is here done will not be difficult, if it be considered that what is aimed at, hath relation to Stofflerius Astrolabe: and that here one Ellipsis must serve to represent every Almicantar. I purpose not to re∣peat any thing that is before said: and the rather because this Structure will not be so expedient and ready in some uses as could be wished. They that desire the reason of it, may fetch it out of this that hath been already said: which may be done without any great labour.