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 ...

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.