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. I. Of the Elliptical Dial, where the Index stands perpendicular to the Plain: How to draw it for an Horizontal Plain, or any other Plain that declineth not. (Book 1)

BEcause here is no declination supposed, there∣fore the draught will be the more easie.

1. Make B C for the longer Diameter of your Ellipsis, and count one halfe of it, that is A B or A C as the Radius, and through the point A, draw an infinite line (as D E) at right angles to B C. Then for the shorter Diameter, you must consider the eleva∣tion of the Pole above your plain. Looke therefore to make A D or A E equall to the Sine of the Poles elevation, which Sine must be estimated to the Radius A B. Thus you have the two extream diameters.

2. Through the four points D B E C you may describe the Ellipsis, either by Elliptical Compasses, or by finding many points through which it is to passe: for effecting which there are multitudes of wayes prescribed.

One may be this. Describe two Circles upon the two extream Diameters B C and D E. Divide them both into any like parts, as B f h n p q C, and V i k E o t X; and from each couple of those like divisions (as f and i) draw f g i g, (the first parallel to A n, the second parallel to A B) cut∣ting each other at right angles in g, so the point g shall be one of the points through which the Ellipsis is to be de∣scribed. In the same manner, from h and k must be drawn

two lines concurring at right angles in m, which will there∣fore be another point through which the Ellipsis must passe. So again from p and o, t and q, lines are to be drawn, con∣curring at right angles in s and r, which are two other points of passage for the Ellipsis. After this manner may the other halfe B D C be described.

[illustration]

How to divide the Ellipsis into its requisite parts.

1. If the parts B f h n p q C were all equall, and 12 of them in number, then would the points g m, &c. be the houre points. And accordingly, if those houres were sub∣divided into equall halves and quarters, &c. there would be found points for the halves and quarters, &c. of the houres.

And if the Ellipsis be first drawn, then the exteriour Circle it selfe by lines issuing out of f h p q, drawn parallel to A n) will divide the Ellipsis: or, where these lines cut it more obliquely (as at g and r, &c.) there the lesser Circles equall parts (at i and t, &c.) will cut the same Ellipsis at more right angles, so that one of them may help the other in this division.

2. Or, If the quadrants B n C, or V E X (or of any other Circle described upon the center A) were divided into Horizontall Spaces, such as are proper for this Diall (as you see them done upon the quadrants B y C) and a Ruler laid from the center A did transfer those divisions into the El∣lipsis (as is done upon that halfe of it which is noted with B D C) this division would be the same with the former.

The proportion whereby to make the Table of Horizon∣tall Spaces for this Ellipsis, is inverse to that which is for common Horizontall Dials: thus,

As the Sine of the Poles elevation above the plain, Is to the Radius;

So is the Tangent of 15 gr. 30, 45, &c.

To the Tangent of the spaces of every houre from 12, upon the plain whereon the Ellipsis is described.

And according to these spaces must every houre be set on from n or y towards B or C.

How to make the Zodiac, or dayes of the yeare, whereby the Ellipsis and Index are to be set in a right position, that they may daily stand true to shew the houre.

Whether the 12 Signes or the 12 Moneths be fittest for

use, is left to every mans choise. Bu•• in both these it is re∣quired that the declinations due to the Signes or Dayes be known, because these are to be inscribed by them. So that now it only remains to be shewed what kinde of Scale that must be, out of which the foresaid Declinations are to be taken.

In generall, the Scale must be a Scale of Naturall Tan∣gents.

In particular, there are these two Rules, one of which will with due convenience serve for all kindes of Latitudes, or elevations. You may use which is most sutable to your purpose.

[illustration]

The first for great Latitudes. Make the lesser semidia∣meter A D or A E, to be a Tangent of the Poles elevation

above the plain; and that Scale is the Scale of Tangents out of which the Declinations (before mentioned) must be taken.

The second for lesser Latitudes. To the Radius B A, finde the Co-sine of the plains latitude or elevation; and make that Co-sine a Radius, or Tangent of 45 gr. and this will be the same Scale with the former▪

By these wayes the Declinations of these two Tables (one for the Signes, the other for the Dayes of the Moneths) may be inserted from the Equinoctiall line, or middle of the Scale, by the graduations of such a Tangent line as is before mentioned.

Signes.	Declin.	Signes.
♑ ♋	0	23	31	30
	5	23	26	25
	10	23	9	20
	15	22	41	15
	20	22	2	10
	25	21	12	5	♊ ••
♒ ♌	0	20	13	30
	5	19	5	25
	10	17	48	20
	15	16	24	15
	20	14	52	10
	25	13	14	5	♉ ♏
♓ ♍	0	11	31	30
	5	9	43	25
	10	7	51	20
	15	5	56	15
	20	3	58	10
	25	2	00	5	♈ ♎

Moneth	Declin.	Moneth	Declin.	Moneth	Declin.	Moneth	Declin.
Ianu. 5	21	7	Aprill 5	9	56	Iuly 5	21	33	Octob. 5	8	40
10	20	6	10	11	41	10	20	41	10	10	29
15	18	55	15	13	20	15	19	39	15	12	15
20	17	37	20	14	55	20	18	30	20	13	56
25	16	10	25	16	23	25	17	14	25	15	32
31	14	17	30	17	45	31	15	33	31	17	18
Febr. 5	12	36	May 5	19	0	Aug. 5	14	1	Nove. 5	18	40
10	10	50	10	20	6	10	12	24	10	19	51
15	9	00	15	21	4	15	10	42	15	20	54
20	7	6	20	21	53	20	8	56	20	21	47
25	5	11	25	22	32	25	7	7	25	22	30
28	4	1	31	23	6	31	4	52	30	23	2
Mar. 5	2	0	Iune 5	23	24	Septe. 5	2	56	Dece. 5	23	22
10	Ae	1	10	23	31	10	1	0	10	23	31
15	2	0	15	23	28	15	Ae	57	15	23	28
20	3	57	20	23	14	20	2	55	20	23	11
25	5	52	25	22	50	25	4	51	25	22	45
31	8	7	30	22	17	30	6	46	31	21	58

Or if it be required rather to set them on by a Decimal Scale equally divided, then must the Tangents of these De∣clinations (here specified) be used; and the Decimal Scale must be equal to the Co-sine of the plains Latitude, which Co-sine must be limited to B A, the greater Radius of the Ellipsis. And for this purpose here are two Tables more which do expresse the said Tangents.

Signes.	Tang	Signes.	Moneth	Tang	Moneth	Tang	Moneth	Tang	Moneth	Tang
♑ ♋	0	4348	30		Ianu. 5	2862	Aprill 5	1751	Iuly 5	3949	Octob. 5	1524
	5	4334	25		10	3659	10	2068	10	3775	10	1850
	10	4276	20		15	3427	15	2370	15	3571	15	2171
	15	4180	15		20	3175	20	2664	20	3346	20	2481
	20	4047	10		25	2899	25	2940	25	3102	25	2780
	25	3879	5	♊ ••	31	2546	30	3201	31	2783	31	3115
♒ ♌	0	3683	30		Febr. 5	2235	May 5	3443	Aug. 5	2496	Nove. 5	3378
	5	3460	25		10	1914	10	3659	10	2199	10	3610
	10	3211	20		15	1584	15	3852	15	1890	15	3819
	15	2943	15		20	1246	20	4017	20	1572	20	3996
	20	2655	10		25	0907	25	4149	25	1249	25	4142
	25	2352	5	♉ ♏	28	0702	31	4265	31	0851	30	4252
♓ ♍	0	2038	30		Mar. 5	0349	Iune 5	4327	Septe. 5	0512	Dece. 5	4320
	5	1712	25		10	Aeq 3	10	4348	10	0175	10	4348
	10	1379	20		15	0349	15	4341	15	ae166	15	4341
	15	1039	15		20	0690	20	4293	20	0509	20	4283
	20	0693	10		25	1028	25	4210	25	0849	25	4193
	25	0349	5	♈ ♎	31	1426	30	4098	30	1187	31	4033

Another way to describe and divide the same Zodiac, or Scale of Moneths.

The former descriptions do suppose that the yearly course of the Sun is to be set in a streight Scale, in which the parts neerest to the two Tropicks will be exceeding close toge∣ther; and those at the Aequinoctiall or middle part of the Scale will be wide. But if it shall better be liked that the parts should stand distinctly one from the other; it will then be most expedient, First, to limit out the whole length of the Scale: then upon that length (as a diameter) to describe a Circle, and to divide it as is here under shewed.

How to divide the Annuall Circle into its requisite parts.

You are first to divide the Circle (it selfe, or one equall to it) into 360 equall degrees, and by them to divide the said Circle into such unequall parts as the numbers of de∣grees and minutes expressed in the Tables will require. The two Tables are the Right Ascensions of such parts of the Ecliptic as are due to the Suns place or Declination, in any of the Signes, or of any dayes of the 12 moneths. The Tables are here placed. The manner of dividing the Circle is the same with other like things of this kinde.

[illustration]

	Mon	R. Ascen.	Mon	R. Ascē	Mon	R. Ascen	Mon	R. Ascen.
Ia 10	302	43	A 10	28	36	I. 10	119	47	O. 10	205	06
20	313	16	20	37	56	20	129	40	20	214	53
31	324	21	30	47	33	31	140	15	31	225	50
Fe 10	334	03	M 10	57	23	A 10	149	37	N 10	236	11
20	343	30	20	67	27	20	158	48	20	246	51
28	351	07	31	78	45	31	168	45	30	257	57
M 10	0	14	I. 10	89	05	S. 10	177	44	D 10	269	02
20	9	18	20	99	27	20	186	47	20	280	11
31	19	19	30	109	43	30	195	53	31	292	08

	Signs	R. Ascē	Signs	R. Ascen.	Signs	R. Ascen	Signs	R. Ascen.
♈	10	9	11	♋	10	100	53	♎	10	189	11	♑	10	280	53
	20	18	27		20	111	39		20	198	27		20	291	39
	30	27	54		30	122	12		30	207	54		30	302	12
♉	10	37	34	♌	10	132	28	♏	10	217	34	♒	10	312	28
	20	47	32		20	142	26		20	227	32		20	322	26
	30	57	48		30	152	06		30	237	48		30	332	06
♊	10	68	21	♍	10	16••	33	♐	10	248	21	♓	10	341	33
	20	79	07		20	170	49	20	259	07		20	350	49
	30	90	00		30	180	00		30	270	00		30	360	00

The beginning of the equall parts or degrees, must be upon that middle Diameter of the Circle, which lies parallel to the two six a clock lines, or perpendicular to the twelve a clock line.

The best manner of dividing the Circle is as in the pre∣ceding figure, especially if the fiduciall Circle be cut through with small divisions, that so the intersections may be the more discernable, and the Circle more distinguishable from the rest; and more cleere from mistakes.

Concerning the Index.

The Index must stand perpendicular to the plain, and must, according to the time of the year, be neerer or further from the Ellipsis.

Now, Whether the Index should move upon the plate lying still, or the elliptical plate move towards the Index fixed upon another plate, must be left to every mans judge∣ment and best liking. But the mover with its peculiar Index

(called here the Zodiacall Index) must move according to the length of the plains Meridian line, either in it, or paral∣lel to it, alwayes so that the fiduciall edge of the Dials In∣dex may ever stand in the Meridian line.

But if to this Ellipticall Diall it be thought fit to joyn an ordinary Horizontal Dial (fitted to the Elevation of the Pole above the plain) that so it may set it selfe true North and South, which by this double kinde of Diall it will do; for the houres in both Dials will never agree to be the same till it stand right, and the best time for setting it, is when the Sun is in the East or West Azimuth, the worst time is at Noon: then the common Horizontal Dial may be the standing plate, the Index of the Ellipsis being fixed to it, the elliptical plate may move too and fro upon the horizon∣tal fixed plate, according to the graduations of the Zodiac.

Of the place for the Suns Annual course or Zodiac.

Whichsoever of the two (the Ellipsis or Index) is move∣able or fixed, it matters not. In both wayes it is necessary to set the Zodiac right, which must thus be done.

Let the fiduciall edge of the Index be placed exactly in the center of the Ellipsis. And being there set, let the place of the line or threed (or whatever it be that serves for the Zodiacall Index) be marked upon the standing plate. Then through that mark or point draw a streight line parallel to the line of the two sixes, or perpendicular to the line of twelve. The same line is to be esteemed for the Aequino∣ctiall: and from thence are all the parts of the Scale (or Cir∣cle) to be insctibed by help of the former Tables.

The Vses of the Elliptical Dial.

The Index and Ellipsis being used alone, and set to their true distance, by the help of the Zodiac, the Ellipsis it selfe with its upright Index, will shew the true houre of the day.

But it is supposed hitherto, that the plain be either direct, looking full North or South, or else Horizontal: and in every such plain, there must be drawn the Meridian line of the place or plain, which in direct flats are one and the same. And according to the Coast of this Meridian line must the moving or sliding be, as also the line of 12 upon the Ellipsis must lie in or parallel to it.

This way therefore requires, first, a Meridian line to be drawn, whereby to place the Ellipsis. And in all direct plains the Meridian line is the same with the Vertical line of the plain.

But if to this Elliptical Dial, be adjoyned a common Ho∣rizontal Dial with an Axis, then there needs no finding of a Meridian line before hand, because they two betwixt them will finde one, and consequently will place themselves in a true position. Only with this proviso, that if the plain be not Horizontal, it must yet be such as looketh directly North or South. This was intimated before.

There may other uses be made of it, if it have other Scales adjoyned to the Zodiac.

1. If a Tangent line of 23½gr. be inscribed according to the length of the Zodiac, then when the situation of the Ellipsis to the Index is fitted, the Zodiacal Index will shew

(upon this Tangent Scale) the Suns Declination.

2. In the 11 page there are two Rules given whereby to finde that Scale of Tangents out of which the Zodiacal Scale is to be divided. If then out of that Scale of Tan∣gents so found and limited, you take the Co-tangent of your Latitude, and divide that length as a Radius or whole line of Sines into houres and parts of houres, and put in so many of them as will reach the Suns greatest Declination, or the length of the Zodiac both wayes from the Equino∣ctiall; then the former rectification of the Zodiacal Index to the day of the moneth or Suns place, will perfectly shew (in this Scale, and for that Latitude to which all the work is made) the Ascensional difference, with what depends there∣on: namely, the length of the Day and Night, and the time of Sun Rising and Setting.

3. The Dial being fixed in a true position, the Place or Coast of the Sun rising and setting upon the plain, may be discerned in the heavens. For when the Ellipsis is rectifi∣ed, and the time of the Sun rising or setting is known (as be∣fore) then with your eye project the fiduciall edge of the upright Index upon that time of Sun rising or setting counted in the Ellipsis, and the same edge will shew in the heavens whereabout the Sun will ascend or descend upon the plain. That is, it shews the Amplitude of Sun rising upon the plain.

• 1. Note, That if the plain be Horizontall, then the two last Uses serve for the place where a man lives. But if the plain be not Horizontal, then it serves not for the place, but is proper to the plain it selfe, or to that Horizon or Latitude which the plain represents, ac∣cording as they are set down in the former Rules: be∣cause things are done to the plains Latitude, and not to the places.
• ...

• 2. Note again, that if upon plain•• that are not Horizontal, you would yet have them (to shew the second of these last preceding conclusions, namely,) to shew the time of Sun rising or setting for the place in which you are, and not for the plains, then must you take (not the Co-tangent of the plains Latitude, but) the Co-tangent of the Latitude of the place, just as you did in the Horizontall plain, and then the Zodi∣acal Index being rectified will effect the thing requi∣red for the place, and not for the plain.
• 3. Note, that both in the Horizontal and the other plains, there may be a peculiar Scale put, either for the Am∣plitude proper to those plains, or else for the Ampli∣tude proper to the place, if a Table of Amplitudes be computed for either of them, by this proportion.

As the Radius, to the Sine of 1, 2, 3, &c. or 5, 10, 15, &c. degr. of Amplitude;

So the Co-sine of the poles elevation, To the Sine of the competent declination:

And then by the said competent declination (taken in the Scale of Tangents of 23½ equal to the halfe Zodiac) the forenamed Amplitudes be inscribed. Or having the Declinations, you may look out their Tan∣gents in the Canon, by which Tangents the said Amplitudes may be inferred with help of the Deci∣mall Scale made to the Co-sine of the plains Lati∣tude, as was before mentioned.

The Amplitudes proper to the plain are of no great use,

unlesse the plain do justly represent the Horizon of some known place, whose Amplitude you desire to be acquainted withall. I have therefore here added a particular Table be∣longing to the Latitude of London 51 gr. 30 min. that by it the said Amplitudes may be inserted into such direct Recli∣ners and Incliners as shall any way stand in the said Latitude.

A Table of Amplitudes, with their answerable Declinations, for the Latitude of 51 gr. 30 min.

A.	Declin	A.	Declin.	A.	Declin.	A.	Decl••n▪
1	0	37	11	6	43	21	12	53	31	18	42
2	1	14	12	7	26	22	13	29	32	19	16
3	1	52	13	8	03	23	14	04	33	19	49
4	2	29	14	8	40	24	14	40	34	20	22
5	3	07	15	9	16	25	15	15	35	20	55
6	3	44	16	9	53	26	15	50	36	21	28
7	4	21	17	10	29	27	16	25	37	22	0••
8	4	58	18	11	05	28	17	00	38	22	32
9	5	36	19	11	41	29	17	34	39	23	04
10	6	12	20	12	18	30	18	08	40	••3	35

Another Table of Amplitudes, with the Tangents of their answerable Declinations, for the Latitude of 51 gr. 30 mi.

A.	Tan.	A.	Tan.	A.	Tan.	A.	Tan.
1	0109	11	1196	21	2289	31	3385
2	0217	12	1305	22	2398	32	3495
3	0326	13	1414	23	2507	33	3••04
4	0434	14	1524	24	2617	34	3712
5	0543	15	1632	25	2726	35	3822
6	0652	16	1741	26	2836	36	3932
7	0761	17	1851	27	2946	37	4040
8	0869	18	1960	28	3056	38	4149
9	0981	19	2069	29	3166	39	4258
10	1087	20	2180	30	3275	40	4365

So far of this Elliptical Dial, as it is to be described upon any direct plain which lies under the same Meridian with the Meridian of the place, and which declineth not from it.