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. III. Another way to prick down the Ellipsis upon an Horizontal Plain. (Book 3)

Latitude 51 gr. 30 min.
Hor	Ang.	Altit.	Hor
XII	00	••0	38	30	XII
	04	47	38	24
	09	33	38	7
	14	16	37	38
XI.	18	54	36	58	I.
	23	27	36	7
	27	53	35	7
	32	13	33	56
X.	36	25	32	37	II.
	40	29	31	10
	44	26	29	36
	48	15	27	54
IX.	51	57	26	7	III.
	55	32	24	14
	59	1	22	16
	62	24	20	14
VIII	65	41	18	8	IV.
	68	54	15	59
	72	2	13	47
	75	7	11	33
VII	78	9	9	17	V.
	81	9	6	59
	84	7	4	39
	87	4	2	20
VI.	90	0	0	0	VI.

FOr this purpose here are two Tables joyned together, both of them made for the Latitude of Lon∣don, 51 gr. 30 min. the like to which every man may calculate for his own place.

The first of them is a Table of such angles as are made by the hour lines (coming through the center of the Ellipsis) with the Meridian line or line of 12. And it was made by that rule which was given (in this case) at the beginning of this Trea∣tise. Namely,

As the Sine of your Poles elevation (w^ch is here 51 gr. 30 m.) is to the Radius;

So is the Tangent of each houre and their quarters (counted from 12 a clock,)

To the Tangent of the angle required.

The second Table is of the Altitudes of each houre and quarter, in the Equinoctial Circle, above the Horizon; and it is calculated by this Rule.

As the Radius,

To the Co-sine of your Latitude (which is here the Sine of 38 gr. 30 min.)

So is the Sine of each houre and their quarters, (counted from 6 a clock,)

To the Sine of the altitude which is here sought.

Then having computed these two Tables, you may (by help of them) both draw and divide the Ellipsis into its true houres and quarters by these following directions.

First, draw the two lines 6 6 and D E, crossing each other at right angles in A, and let A D be the Meridian, and 6 A 6 the two six a clock lines.

Then upon A as a center describe the Circle 6 D 6 of any convenient bignesse; and upon the same Circle from D (on both sides of it) set on the houres and their quarters, as 11 and 1, 10 and 2, &c. by help of the degrees and minutes of the angles set to every houre in the former Table. And through every of these points inscribed into the Circle, draw lines from the center A, as A 11, A 10, A 1, A 2, with the rest of the houres and their quarters if you will.

Thirdly, supposing A 6 to be the Radius, you may divide

[illustration]

it into 90 Sines; the beginning or largest parts whereof must be at A, but the numeration must be the contrary way, be∣ginning at 6 and ending at A in 90. Or you may take the complements of the former Altitudes in the Table, and take them out of a line of Sines numbered the right way, which will be best for the Sector, or such Instruments which will best perform this work without any great trouble.

Fourthly, looke in the second Table for the altitudes of the Equinoctiall at every houre. Then count those Altitudes in the line of Sines, and take with your Compasses the se∣veral distances of them from A, & transfer the said distances

from the center A to every houre respectively, so shall those intersections give you the points through which the Ellipsis must be drawn.

Thus the altitude of 7 and 5 a clock (which is 9 gr. 17 m.) being taken upon the line of Sines from A towards S, is in∣serted into 4 houres at the note m. And the altitude at 8 and 4 (which in the Table is 18 gr. 8 min.) being taken and transferred to the four houres at n, do give four points more. The rest of the altitudes give (each of them) two houres only, as at o, i, and r, is done. And the last of all at 12, gives one only point at t. The like may be done for the quarters. And so through these points thus found the Ellipsis may easily be drawn, and the lines formerly drawn give the divisions that are due unto it.

Note.

1. This is propounded only for Horizontal plains, but it may without difficulty be applyed to any other direct plain. The trouble that is, comes by reason that the foresaid plains have a different elevation of the Pole, from that elevation that belongs to the Horizon or place where they are to stand: and consequently there will be required two new Tables for that elevation which is proper to the plain; the calculation of which will easily be done by the two former proportions set down for that end. Note further that by these two Tables you may prick down an Ellipsis upon any leaning (not up∣right) plain whatsoever. But the Index must then lie in the Zenith line of the place, (not of the plain) and the Ellipsis (or Index) must move in the Meridian of the place (not of the plain) and the Zodiac must be like or proportional to the Zodiac of the Horizon, but augmented, for the most part, in this proportion.

As the Radius, To the Secant of that angle that the Meridian line (upon the plain) makes with the Horizon; (which angle must be gott••n by a Clinatorie;)

So the Radius of the Horizontal Zodiac, (which is the Co-sine of the Latitude,) To the Radius of the Zodiac proper to the plain: which Zodiac must be set according to the line of 12 upon the plain.

2. This way of delineation may likewise be applyed to all other plains which decline, and are not direct. But in these will be found more difficulty then in the former plains which declined not, unlesse it be required to put in the houres that are proper to the plain; for in that case the work is the same which was in them. But if the houres of the place are to be inscribed, as most usually they are, then there will be some trouble, by reason that the difference of the plains Lon∣gitude seldome falls upon any just houre. I purpose not here to shew the way, it being such as will prove un-pleasant to the unskilfull, and such as have knowledge will finde it out quickly. Perhaps the pleasure of the thing done, will recompense their labour.