ELLIPTICAL, OR AZIMUTHAL Horologiography. Comprehending several ways of describing DIAL'S upon all kinds of Superficies either plain or curved: And unto upright Styles in whatsoever position they shall be placed.  
Invented and Demonstrated By SAMVEL FOSTER, Late Professor of Astronomy in Gresham-Colledge.  
LONDON, 〈…〉 for Nicholas T 〈…〉   

To the READER.  
Courteous Reader,  
AMongst other Treatises of this deceased Author, of which, in their due time, we intent to make thee partaker, we have, in the first place, made choice of this, as well in some measure to keep our word with thee, as also to stay thy expectation till other things can be made ready. We intent not to spend time either in the commendation of It or the Author, being w●ll assured that our words will not, where the matter itself is 〈…〉 capable to, satisfy therein the judicious Reader. Only let 〈◊〉 say thus much, that though the general Subject of this Book 〈◊〉 dialing, yet 'tis handled in a way which no man whosoever, that we know of, hath hitherto fully traced. For by it thou wilt see, that the representing the true Hour by the shadow made by the Axis of the World is but one of those infinite ways which may be invented, and that it is as possible to do the same thing by the shadow of the Axis of one of the vertical Circles, and by the projection of one Ellipsis upon a plain, supply the place of all the Parallels comprehended within the Tropics. 'Tis true, that Mr. Va●lezard, a learned Mathematician, we think yet living in France, hath some years since published a short Treatise in that Language, in which he showeth by the projection of an Ellipsis upon the plain of the Horizon, and by the help of an upright movable stile to find the He●re and Azimuth, with some other uses of the same. But this Treatise of our Author is very different from that, and most of the things here handled, such as are not appliable to his, and in themselves wholly new.  
In the next place, if it shall seem strange to any that amongst other things, as well of this, as of different natures, which we intent shortly to Publish; we begin first with this of Dialling (a Subject upon which something too hath already been published by our Author, and from whence some might perhaps take occasion to carp at him, as either unable for other things, or too much busied in this) we desire them first to con●●der how difficult this Subject hath been thought by the 〈◊〉, and withal what large Volumes have been writ by som● of the best rank of later Mathematicians, (such were Clavius, Maigran, and others) and then compare their ways with what they shall here and hereafter (God willing) find in our Author, and we doubt not, but they will be forced to yield him it this honour, to have made that Art, in all the Cases of it, and all Circumstances thereunto belonging, more easy and expedite both in the Understanding and Practice, and with much more brevity than any that have gone before him either of our own or other Nations.  
Lastly, we advertise thee (Reader) that our Author, in regard of his great and long infirmities, could not fit either this, or any other of his Treatises for the Press, as he desired and intended; and therefore they must needs want much of that accomplishment which otherwise they would have had. But we hope, notwithstanding, thou wilt find so much of worth in them as they now are, whereby thou wilt judge them (as we do) fitter to be made public, then wholly suppressed.  


John Twysden,  
Edmund Wingate.     

ERRATA.  
Page 85. line 11. for Pag. 81. read Pag. 70. There are divers other literal faults, but this is most material.   
ELLIPTICAL OR AZIMUTHAL Horologiography. Of the Elliptical Dial, whose Index stands perpendicular to the plain: How to draw and divide the Ellipsis upon an Horizontal plain, or any other plain that declineth not.  

1. LEt the two right lines B C and A D, cut each other at right angles in A. Then making A C or A B as a Radius, let A D be the Right Sine of the Poles elevation above the plain; and make up the Rectangled Parallelogram B E F C, and continue it further to G and H.  
2. Divide the Radius A C into the Sins of 15, 30, 45, 60, 75 gr. at a b c d e, which stand for the six equal hours: ●o that A a may be the Sine of one hour, A b of two hours, A c of three hours, A d, A e of four and five hours. And in the same sort divide the other lines which are equal to A C, namely D F, A B, D E, into the same parts, at a b c d e, and at l m n o p: and draw the parallel lines a a, b b, etc. l l, m m, etc.  
Then taking A D, or C F, or B E, for a Radius, set on again the Sins of the six equal hours from B and C, at f g h i k, and as many as shall be needful on the other side, at q r s: and draw the parallel lines f f, g g, etc. and q q, r r, etc.  
3. First, If a curved line be begun at D or 12, and carried both ways diagonally through the succeeding Parallellograms, the same line shall be the Ellipsis required. Thus you see it carried from D to 11, from 11 to 10, from 10 to 9, and so through the successive opposite angles of the following Parallellograms, at 8, 7, 6, 5, 4, 3. And on the other side in the same manner, from D, diagonally through 1, 2, 3, 4, 5, 6, 7, 8, 9 This line thus regularly described without angles, and extravagant inordinate flexures, is the Ellipsis here intended.  
Secondly, If from the centre A, through each of the angular points in the small Parallellograms, certain short lines be drawn, (as in the figure you see done at 11, 10, 9, etc. and at 1, 2, 3, etc.) the same lines shall be the requisite horary divisions of the Elliptical line.  
4. If these divisions into single hours shall not be enough, but that more exactness is required, into halves and quarters; then must you (besides the Sins of whole hours at a, b, etc. l, m, etc.) put in the right Sins of the intermediate halves and quarters of the equal hours; which will be easy to do by the former directions.  

Declinations of the Sun from the Equinoctial.  
 Ianu. Febru. March April May june july Augu. Septen. Octob. Noven. Decemb. 1 21 49 13 57 3 34 8 30 18 01 23 11 22 09 15 15 4 29 7 10 17 36 23 07 2 21 39 13 37 3 11 8 52 18 16 23 14 22 00 14 57 4 06 7 33 17 52 23 12 3 21 29 13 17 2 47 9 13 18 31 23 18 21 51 14 39 3 43 7 56 18 08 23 16 4 21 18 12 57 2 23 9 34 18 46 23 21 21 42 14 21 3 20 8 18 18 24 23 19 5 21 07 12 36 2 00 9 56 19 00 23 24 21 33 14 02 2 57 8 40 18 40 23 22 6 20 56 12 15 1 36 10 17 19 14 23 26 21 23 13 43 2 34 9 02 18 55 23 25 7 20 44 11 54 1 12 10 38 19 27 23 28 21 13 13 24 2 10 9 24 19 10 23 27 8 20 32 11 33 0 48 10 59 19 40 23 29 21 02 13 05 1 47 9 46 19 24 23 29 9 20 19 11 12 0 24 11 20 19 53 23 30 20 51 12 45 1 24 10 08 19 38 23 30 10 20 06 10 50 Aequin.  11 40 20 06 23 31 20 40 12 25 1 00 10 30 19 52 23 31 11 19 53 10 28 0 24 12 01 20 18 23 31 20 29 12 05 0 37 10 51 20 05 23 31 12 19 39 10 06 0 48 12 22 20 30 23 31 20 17 11 45 Aequinoct. 14 11 12 20 18 23 31 13 19 25 9 44 1 12 12 42 20 42 23 31 20 05 11 25  10 11 33 20 31 23 31 14 19 10 9 22 1 36 13 02 20 53 23 30 19 53 11 04 0 34 11 54 20 43 23 30 15 18 55 9 00 2 00 13 21 21 04 23 28 19 40 10 43 0 58 12 15 20 55 23 28 16 18 40 8 38 2 23 13 40 21 14 23 26 19 27 10 22 1 22 12 36 21 06 23 25 17 18 25 8 15 2 47 14 00 21 24 23 24 19 13 10 01 1 46 12 57 21 17 23 22 18 18 10 7 52 3 11 14 19 21 34 23 20 18 59 9 40 2 10 13 17 21 27 23 19 19 17 54 7 29 3 34 14 38 21 43 23 17 18 45 9 19 2 33 13 37 21 37 23 16 20 17 37 7 06 3 57 14 57 21 52 23 14 18 30 8 58 2 56 13 57 21 47 23 12 21 17 20 6 43 4 20 15 15 22 01 23 11 18 15 8 36 3 19 14 16 21 57 23 08 22 17 03 6 20 4 43 15 32 22 10 23 06 18 00 8 14 3 42 14 35 22 06 23 03 23 16 46 5 57 5 06 15 49 22 18 23 01 17 45 7 52 4 06 14 54 22 15 22 58 24 16 28 5 34 5 29 16 06 22 25 22 56 17 30 7 30 4 29 15 13 22 23 22 52 25 16 10 5 11 5 52 16 23 22 32 22 51 17 14 7 08 4 52 15 32 22 32 22 45 26 15 52 4 47 6 15 16 40 22 39 22 45 16 58 6 46 5 15 15 51 22 38 22 38 27 15 34 4 24 6 38 16 57 22 45 22 39 16 42 6 24 5 38 16 09 22 45 22 31 28 15 15 4 01 7 00 17 13 22 51 22 32 16 25 6 01 6 01 16 27 22 51 22 24 29 14 56   7 22 17 29 22 57 22 25 16 08 5 38 6 24 16 45 22 57 22 16 30 14 37   7 45 17 45 23 02 22 17 15 51 5 15 6 47 17 02 23 02 22 08 31 14 17   8 08   23 07   15 33 4 52   17 19   21 59   

Tangents of the Sun's declination from the Equinoctial.  
 Ianu. Febru. March April May june july Augu. Septen. Octob. Noven. Decemb. 1 4005 2484 0623 1498 3251 4281 4071 2727 0784 1257 3170 4272 2 3971 2422 0554 1562 3300 4295 4042 2671 0717 1325 3222 4287 3 3936 2360 0485 1626. 3348 4307 4012 2615 0650 1392 3275 4300 4 3900 2297 0416 1690 3396 4317 3981 2558 0584 1458 3326 4311 5 3863 2234 0347 1754 3443 4326 3949 3500 0517 1524 3377 4321 6 3825 2170 0278 1818 3489 4334 3916 2442 0450 1590. 3427 4330 7 3785 2106 0209 1881 3534 4341 3882 2383 0383 1656 3476 4338 8 3744 2042 0140 1944 3576 4347 3847 2323 0316 1722 35●3 4345 9 3702 1978 0070 2007 3618 4350 3811 2263 0249 1788 3568 4350 10 3659 1913 Aequin. 2069 3659 4351 3776 2202 0181 1853 3613 4352 11 3615 1847 0070 2131 3700 4352 3739 2141 0112 1917 3657 4352 12 3570 1781 0140 2193 3740 4352 3700 2079 Aequin. 42 1981 3700 4351 13 3524 1715 0209 2254 3779 4351 3660 2017 29 2045 3742 4350 14 2477 1649 0278 2315 3817 4348 3618 1955 0099 2109 378● 4347 15 3429 1582 0347 2375 3854 4343 3575 1893 0169 2172 3821 4341 16 3380 1515 0416 2434 3889 4336 3532 1831 0239 2235 3859 4333 17 3330 1448 0485 2493 3922 4327 3487 1768 0309 2298 3896 4324 18 3280 1381 0554 2552 3952 4317 3442 1705 0378 2360 39●2 4314 19 3230 1314 0623 2610 3982 4306 3396 1642 0446 2422 3907 4302 20 3177 1246 0692 2668 4012 4294 3349 1578 0514 2483 4000 4288 21 3123 1178 0761 2725 4042 4281 3301 1513 0582 2543 4031 4273 22 3068 1110 0829 2780 4071 4266 3253 1448 0650 2603 4061 4256 23 3013 1042 0897 2834 4099 4250 3204 1382 0718 2662 4090 4238 24 2957 0974 0965 2888 4125 4232 3154 1316 0786 2721 4118 4217 25 2900 0906 1033 ●941 4149 4213 3103 1250 0854 2779 4146 4195 26 2842 0837 1101 2994 4171 4193 3052 1184 0921 2837 4172 4172 27 2784 0768 1168 3047 4192 4172 3000 1118 0988 2895 4196 4148 28 2726 0699 1●35 3099 4212 4149 2947 1052 1055 2952 4218 4123 29 2667  1301 3150 4231 4124 2893 0985 1122 3008 4238 4096 30 2607  1367 3201 4249 4098 2838 0918 1189 3063 4256 4068 31 2546  1433  4266  2783 0851  3117  4038    

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.  
BEcause here is no declination supposed, therefore the draught will be the more easy.  
1. Make B C for the longer Diameter of your Ellipsis, and count one half 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 elevation of the Pole above your plain. Look therefore to make A D or A E equal to the Sine of the Poles elevation, which Sine must be estimated to the Radius A B. Thus you have the two extreme 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 pass: for effecting which there are multitudes of ways prescribed.  
One may be this. Describe two Circles upon the two extreme 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 oh 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) cutting each other at right angles in g, so the point g shall be one of the points through which the Ellipsis is to be described. In the same manner, from h and k must be drawn two lines concurring at right angles in m, which will therefore be another point through which the Ellipsis must pass. So again from p and o, t and q, lines are to be drawn, concurring at right angles in s and r, which are two other points of passage for the Ellipsis. After this manner may the other half B D C be described.  

How to divide the Ellipsis into its requisite parts.  
1. If the parts B f h n p q C were all equal, and 12 of them in number, than would the points g m, etc. be the hour points. And accordingly, if those hours were subdivided into equal halves and quarters, etc. there would be found points for the halves and quarters, etc. of the hours. And if the Ellipsis be first drawn, than the exterior Circle itself 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, etc.) there the lesser Circles equal parts (at i and t, etc.) 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 centre A) were divided into horizontal Spaces, such as are proper for this Dial (as you see them done upon the quadrants B y C) and a Ruler laid from the centre A did transfer those divisions into the Ellipsis (as is done upon that half 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 horizontal Spaces for this Ellipsis, is inverse to that which is for common horizontal Dial's: thus,  
As the Sine of the Poles elevation above the plain, Is to the Radius;  
So is the Tangent of 15 gr. 30, 45, etc.  
To the Tangent of the spaces of every hour from 12, upon the plain whereon the Ellipsis is described.  
And according to these spaces must every hour be set on from n or y towards B or C.   

How to make the Zodiac, or days of the year, whereby the Ellipsis and Index are to be set in a right position, that they may daily stand true to show the hour.  
Whether the 12 Signs or the 12 Months be fittest for use, is left to every man's choice. Bu● in both these it is required that the declinations due to the Signs or Days be known, because these are to be inscribed by them. So that now it only remains to be showed what kind of Scale that must be, out of which the foresaid Declinations are to be taken.  
In general, the Scale must be a Scale of Natural Tangents.  
In particular, there are these two Rules, one of which will with due convenience serve for all kinds of Latitudes, or elevations. You may use which is most suitable to your purpose.  
The first for great Latitudes. Make the lesser semidiameter 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, find 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 ways the Declinations of these two Tables (one for the Signs, the other for the Days of the Months) may be inserted from the Equinoctial line, or middle of the Scale, by the graduations of such a Tangent line as is before mentioned.  
Signs. Declin. Signs. ♑ ♋ 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 ♈ ♎  
Month Declin. Month Declin. Month Declin. Month Declin. Ianu. 5 21 7 April 5 9 56 july 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 june 5 23 24 Septe. 5 2 56 Dece. 5 23 22 10 A 1 10 23 31 10 1 0 10 23 31 15 2 0 15 23 28 15 A 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 Declinations (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 express the said Tangents.  
Signs. Tang Signs. Month Tangle Month Tangle Month Tangle Month Tangle ♑ ♋ 0 4348 30  Ianu. 5 2862 April 5 1751 july 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 june 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 Months.  
The former descriptions do suppose that the yearly course of the Sun is to be set in a straight Scale, in which the parts nearest to the two Tropics will be exceeding close together; and those at the Equinoctial 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 showed.   

How to divide the Annual Circle into its requisite parts.  
You are first to divide the Circle (it self, or one equal to it) into 360 equal degrees, and by them to divide the said Circle into such unequal parts as the numbers of degrees 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 Sun's place or Declination, in any of the Signs, or of any days of the 12 months. The Tables are here placed. The manner of dividing the Circle is the same with other like things of this kind.  
 Mon R. Ascen. Mon R. Ascen Mon R. Ascen Mon R. Ascen. ja 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. Ascen 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 equal 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 preceding figure, especially if the fiducial Circle be cut through with small divisions, that so the intersections may be the more discernible, and the Circle more distinguishable from the rest; and more clear from mistakes.   

Concerning the Index.  
The Index must stand perpendicular to the plain, and must, according to the time of the year, be nearer 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 man's judgement and best liking. But the mover with its peculiar Index (called here the zodiacal Index) must move according to the length of the plains Meridian line, either in it, or parallel to it, always so that the fiducial edge of the Dial's Index may ever stand in the Meridian line.  
But if to this Ellipticall Dial it be thought fit to join an ordinary Horizontal Dial (fitted to the Elevation of the Pole above the plain) that so it may set itself true North and South, which by this double kind of Dial it will do; for the hours in both Dial's 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 horizontal fixed plate, according to the graduations of the Zodiac.   

Of the place for the Sun's Annual course or Zodiac.  
Whichsoever of the two (the Ellipsis or Index) is movable or fixed, it matters not. In both ways it is necessary to set the Zodiac right, which must thus be done.  
Let the fiducial edge of the Index be placed exactly in the centre of the Ellipsis. And being there set, let the place of the line or thread (or whatever it be that serves for the zodiacal Index) be marked upon the standing plate. Then through that mark or point draw a straight line parallel to the line of the two six, or perpendicular to the line of twelve. The same line is to be esteemed for the Equinoctial: and from thence are all the parts of the Scale (or Circle) to be insctibed by help of the former Tables.   

The Uses 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 itself with its upright Index, will show the true hour 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 adjoined a common Horizontal Dial with an Axis, than there needs no finding of a Meridian line before hand, because they two betwixt them will find 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 adjoined to the Zodiac.  
1. If a Tangent line of 23½ gr. 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 show (upon this Tangent Scale) the Sun's Declination.  
2. In the 11 page there are two Rules given whereby to find that Scale of Tangents out of which the Zodiacal Scale is to be divided. If then out of that Scale of Tangents so found and limited, you take the Co-tangent of your Latitude, and divide that length as a Radius or whole line of Sins into hours and parts of hours, and put in so many of them as will reach the Sun's greatest Declination, or the length of the Zodiac both ways from the Equinoctial; then the former rectification of the Zodiacal Index to the day of the month or Sun's place, will perfectly show (in this Scale, and for that Latitude to which all the work is made) the Ascensional difference, with what depends thereon: 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 rectified, and the time of the Sun rising or setting is known (as before) then with your eye project the fiducial edge of the upright Index upon that time of Sun rising or setting counted in the Ellipsis, and the same edge will show in the heavens whereabout the Sun will ascend or descend upon the plain. That is, it shows the Amplitude of Sun rising upon the plain.  

1. Note, That if the plain be horizontal, than the two last Uses serve for the place where a man lives. But if the plain be not Horizontal, than it serves not for the place, but is proper to the plain itself, or to that Horizon or Latitude which the plain represents, according as they are set down in the former Rules: because 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 show the second of these last preceding conclusions, namely,) to show 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 horizontal plain, and then the Zodiacal Index being rectified will effect the thing required 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 Amplitude proper to those plains, or else for the Amplitude 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, etc. or 5, 10, 15, etc. 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 half Zodiac) the forenamed Amplitudes be inscribed. Or having the Declinations, you may look out their Tangents in the Canon, by which Tangents the said Amplitudes may be inferred with help of the decimal Scale made to the Co-sine of the plains Latitude, as was before mentioned.  
The Amplitudes proper to the plain are of no great use, unless the plain do justly represent the Horizon of some known place, whose Amplitude you desire to be acquainted withal. I have therefore here added a particular Table belonging to the Latitude of London 51 gr. 30 min. that by it the said Amplitudes may be inserted into such direct Recliners 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.     

SECT. II. How to frame the Elliptical Dial to other Plains which are not direct but declining: To an Index that standeth perpendicular to the plain.  
IT is not here to be enquired whether the plain declining, be erect or leaning; for one rule serves both these kinds. But than it must first be supposed, that the plains situation is in all respects known, how much and which way it declines and reclines or inclines.  
Secondly, These three things must further be found (either by calculation or otherwise) namely, 

1. The Poles elevation above the plain.  
2. The plains difference of Longitude.  
3. The departure of the substylar (which is the plains proper Meridian) from the Vertical line of the plain.    
These are pre-requisites to that which comes after, in which you must proceed by the following directions.  

How to limit and draw the Ellipsis.  
Having the substylar or proper Meridian drawn upon the plain, in its true position, and also knowing the elevation of the Pole above the plain, you may, upon the substylar line, set out your shorter diameter, and on a line drawn perpendicular thereto, you may set off the longer diameter, in this proportion: Let the longer Diameter be as the Radius, the shorter as the Sine of the plains Latitude taken to the same Radius. Or, the longer semidiameter may be the Radius, and the shorter semidiameter must then be the Sine of the elevation of the Pole above the plain.  
Upon these two extreme diameters thus limited, may the Ellipsis be described with Elliptical Compasses. Or otherwise it may be done by two Circles, the one circumscribed, the other inscribed, both divided into like parts, and so points found through which the Ellipsis is to pass, as in the former direct plains was prescribed, and through those points the Ellipsis may, with a steadfast hand be described. In right Orisons or Polar plains there will be no Ellipsis at all, but it vanisheth into a straight line only.   

How to divide the Ellipsis into such requisite parts as the plain shall require.  
If you should put in the hours proper to the plain there will be no difficulty, for then the substilar being taken for the line of 12, the hours must be drawn as before was showed in direct plains. For in this case the Dial is Horizontal, showing the hour of the day to all places that lie under that Meridian under which the plain itself lieth▪ But  
If you would put in the hours of the place (as most frequently is desired) than you are to work (for the division of the Ellipsis) either by calculation or by protraction. If you calculate for the Ellipsis, you must first form all the angles at the Pole as the manner is in these kinds of Dial's: Than you are to invert the terms of proportion from the common way of calculating ordinary Dial's, and say thus:  
As the Sine of the Poles elevation above the plain, Is to the Radius;  
So is the Tangent of each angle at the Pole, To the Tangent of the angular distances of all the hours from the substylar line of the plain.  
Supposing therefore that you had your Ellipsis before described, and the substilar line set off from the plains Vertical line in its true position, you may from that substylar line (and by the horary spaces found by calculation) set in every hour point into the Ellipsis, by help of any Circle described upon the centre A. As you see here are two Circles, the circumscribed and the inscribed, either of which, or any other circle else, will serve to prick on the hours by. The hour points are here set upon the circumscribed Circle, and transferred into the Ellipsis by a Ruler laid to A, the common centre of the Ellipsis and of the Circle. This is one way, and is best done by calculation: here you may also put in the halves, and quarters, and half quarters of hours, or what other parts you shall best like of.   

How to limit the Ellipsis, and divide it, without Calculation.  
But if you are desirous to do it rather by protraction, then must you work somewhat as formerly was done, the manner I will in brief show, wherein you may see the way both how to describe the Ellipsis, and also how to divide it, all under one.  
1. Having prepared all necessary requisites before hand, you must first set off the substilar line in its true position from the Vertical line, which to do, I here suppose already known. So in the following figure you see A C B drawn for the substilar line, or the proper meridian of the plain.  
2. Upon this plain, and upon some point of the Substylar line, as at A, as upon a centre, describe the Circle B M E, and quarter it, and let the Semidiameter of that Circle be counted as the Radius  
3. Upon one of the quarters, set off the Poles elevation as from E to F, and draw A F, and from E to A F take the least distance, as E G, and with that distance, upon the former centre A, describe another lesser Circle as C H.  
4. From B (where the greater Circle cuts the substylar) set B M equal to the plains difference of Longitude: and from M divide that exterior Circle into 24 equal parts, which signify the 24 hours; and by a Ruler laid to the common centre A, transfer them into the lesser Circle, as M H, 10 ay, 6 K, etc.  
5. From every point in the great Circle, draw lines parallel to the substylar, and from every point of the lesser Circle draw lines perpendicular to the substylar, or parallel to the longest diameter P A R, and so each couple of these lines shall cut each other at right angles.  
6. Note where every line thus drawn from any point of the greatest Circle, meets with the other line which is drawn from the like point in the lesser Circle. For the concourses of these answerable lines are the points here required. That is, Those are the points which show both where the Ellipsis is to be drawn, and also where it is to be divided into its requisite parts.  
Thus in the precedent figure: M O and H O meet in the point O, 10 L and I L meet in the point L, 6 N and K N meet in the point N, and so the rest will meet in their due places. These points show the way where through the Ellipsis is to be drawn, and the same points show where the hours are to be marked out. The like may be done for quarters and half quarters of hours, or any other division that shall be best liked of.  
There are many other ways to do the same things, but I suppose this to be most expedient.   

Note: that  
1. The Index in these Elliptical Dial's must stand perpendicularly upright upon the plain, making right angles with it every way, as it was ordered to do upon the former sort of direct plains.  
2. The annual course of the Sun must be limited as formerly in the other plains, that is, making the greater semidiameter as a Badius, you must find the Co-sine of the Poles elevation above the plain. This Co-sine is to be made either as a Tangent of 45 gr. whereof you are only to use 23½: Or else it is to be made a Decimal Scale. By both these you have Tables and Rules how to complete the Sun's annual course, either in degrees of the Signs in the Zodiac, or by the days of the 12 months.  
3. Either the Index must move and the Ellipsis lie still, or contrarily. Every man in this must do as his invention shall best suggest. And that motion must be made either upon the substylar line, or else parallel to it, which way soever it be, it must be precisely and punctually ordered.  
4. All other things concerning the time of Sun rising and setting, the Sun's Amplitudes and Declinations, may here (in the same manner as before) be inserted either proper to the plain, or proper to the place, as shall be desired.  
5. In right Orisons or Polar plains, the Ellipsis closeth quite up into a straight line: and so the division of it is only by the exterior Circles parts projected upon it by those lines that are drawn parallel to the substylar.  
Thus far of the Elliptical Dial, as it is to be described upon declining plains whose Indices stand perpendicular to the plains, and which do not lie under the Meridian of the place but swerve from it.    

SECT. III. Another way to prick down the Ellipsis upon an Horizontal Plain.  
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 IU.  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 joined together, both of them made for the Latitude of London, 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 centre 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 Treatise. Namely,  
As the Sine of your Poles elevation (which is here 51 gr. 30 m.) is to the Radius;  
So is the Tangent of each hour and their quarters (counted from 12 a clock,)  
To the Tangent of the angle required.  
The second Table is of the Altitudes of each hour 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 hour 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 hours 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 centre describe the Circle 6 D 6 of any convenient bigness; and upon the same Circle from D (on both sides of it) set on the hours and their quarters, as 11 and 1, 10 and 2, etc. by help of the degrees and minutes of the angles set to every hour in the former Table. And through every of these points inscribed into the Circle, draw lines from the centre A, as A 11, A 10, A 1, A 2, with the rest of the hours and their quarters if you will.  
Fourthly, look in the second Table for the altitudes of the Equinoctial at every hour. Then count those Altitudes in the line of Sines, and take with your Compasses the several distances of them from A, & transfer the said distances from the centre A to every hour 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 Sins from A towards S, is inserted into 4 hours 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 hours at n, do give four points more. The rest of the altitudes give (each of them) two hours only, as at o, i, and r, is done. And the last of all at 12, giveth 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 applied 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 upright) 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 got 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 applied 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, unless it be required to put in the hours that are proper to the plain; for in that case the work is the same which was in them. But if the hours of the place are to be inscribed, as most usually they are, than there will be some trouble, by reason that the difference of the plains Longitude seldom falls upon any just hour. I purpose not here to show the way, it being such as will prove un-pleasant to the unskilful, and such as have knowledge will find it out quickly. Perhaps the pleasure of the thing done, will recompense their labour.    

SECT. IV. Here follow some Uses and Varieties of this Ellipticall Dial.  

USES.  
FIrst, If the Index stand still, and the Ellipsis be made movable; you may upon that plate which is immovable, and on which the Index standeth, describe a Circle, whose centre must be the foot of the Index. And drawing a Meridian line upon the said immovable plate (just under the Meridian of the Ellipsis, and) from the foot of the Index you may divide that Circle (beginning from the Meridian line) into 360 gr. or so many of them as shall be needful. By these degrees you shall find the Sun's Azimuth by the shadow of the upright Index, and by the divisions of the Ellipsis you may know the hour of the day. So both Azimuth and Hour are showed upon the two plates by one upright Index.  
This kind of Dial must be set in its true position (for it will not set itself) and must there be fixed. If it be upon an horizontal, or any direct plain, you may draw a Meridian line thereon, and either draw the Azimuths true upon it, and so make the Ellipsis to move to and fro according to that Meridian: Or else, if both your plates be to be fixed upon any Base prepared for them, you may, first, upon that Base draw a true Meridian line, and to that line make the Meridian of the Dial to agree.  
The like must be done in all declining flats, only you must be careful to make the shortest Diameter of the Ellipsis (which is the proper Meridian of the plain) to agree with the same proper Meridian, as it shall first be drawn upon the Base in its true position.  
2. A second excellent use may be for the setting of movable Dial's, such as are made to stand out of the weather, and are only to be set in the Sun at pleasure, when the hour shall be required. Of this kind are Polyhedrall Bodies made of some light matter, and Concaves, etc. But it is supposed that some of the superficies of such Bodies, (etc.) have the hours of the p●ace described upon them the ordinary way, by the Axis of the Equinoctial, and that this Ellipsis is fitted to some one plain (which principally is the Horizontal plain (because that alone will show all the hours of the day) or in some Horizontal position to the concave, or any such Dial) in its just position and always with an upright Index, (that is, an Index perpendicular to the plain whereon it stands) which is not always upright in respect of us, but only in respect of the plain) and with the hours also of the place. These things being so fitted, you may place your Body or Concave in the Sun, and turn it about till you discern that both kinds of Dial's do agree to show one hour, which when they do, than all stand right in their true situation, which will never else happen but only when the hours do in both agree.— To this place is referred the Double Horizontal Dial, the one of them giving the hour by the Axis, the other by the upright Index.— And here it must be noted, that the worst time of the day for this setting the Dial's in a true position, is at the nearest to noon, and the best time is about the Sun's b●ing in the East and West Azimuth.  
3. This kind of Dial may be placed (in a movable posture) before a casement in any window that hath a flat board upon which it may move. And the Index to it may be an upright thread (precisely upright it must be) that is fixed at the top and bottom of the window, to and from which the Ellipsis must move in a Meridian line that is drawn upon the board of the window, & from or justly even upon the thread.— And where I say the window must be flat, it is to be understood that there is no extraordinary exactness required in that. For if it should rise or incline 5 gr. to the Horizon, it would not err one minute of time about our Latitude; provided that the Meridian be truly drawn, and the Index perfectly upright, in which all possible exactness is required.  
4. If any choice plain reclining, or however situated plain that shall represent the Horizon of some notable place, be fixed as a Base whereon to place such a Dial thus fitted, and drawn to the proper hours of that same place: this double Dial being placed upon that Base, and turned about thereon, will show the true hour at that place when both of them do agree, and the Meridian of it will show how the proper Meridian of that place is to be placed, and which way drawn.  
5. If such a Recliner (or Incliner, or Erect) shall be furnished with such a double Dial to some intended Declination, though they stand and be framed properly to the plain in respect (chiefly) of the upright Index, and shall (notwithstanding) have the proper hours of the place upon both Dial's: and if further such a Recliner as was mentioned, have an horizontal base, upon which it may be turned round: I say, though this Body be horizontally movable, and may be turned any way, yet the two Dial's will never agree to show the same hour, until the Base stand to the true Declination to which the double Dial was drawn.    

SECT. V. Some varieties of the structure of it do here follow.  
FIrst, Instead of the Ellipsis drawn to the proper Meridian and Latitude of the plain, there may be used a true Circle equally divided: but then let these things be observed.  
First, That the Index must be perpendicular every way to the plain on which the Circle is elevated, standing upright upon it. Secondly, That the motion (either of Index or Circle) must be according to the proper Meridian of the plain. Thirdly, That the Zodiac must also be proper to the plain, and must lie upon the plain or Base itself, and must be the same with that which should serve for the Ellipsis. But then, Fourthly, this equally divided Circle must be elevated over the proper Meridian of the plain (that is, the Base of it must rest upon the proper East and West line of the plain) so much it must be elevated as the elevation of the Equinoctial above that plain comes unto. And the elevation of it may be either way; that is, either upon the South Coast, and then it will be answerable to the Equinoctial in the heavens; or else it may be towards the North, but still so much as the Equinoctials (not the Poles) elevation comes unto, and in some cases it must be done both ways that the Sun may come at it, and it must then have two Indices. Both ways it is alike true. It must be divided into equal hours and parts, and so much of it must stand above the Base or plain on which it moves, as the longest day comes unto. The beginning of the division must be from the line of 12, and if the plain be not direct, then must the difference of Longitude be counted in the circular degrees, from the highest point, or from the Vertical line of this Equinoctial Circle.  
2. This Dial is not so tied to a plain, but that it may also be brought to an un-even Superficies, yet so, that the motion of it, or the Zodiacal Scale must be upon some plain. But to show the manner how this should be performed, is more proper for those Projecting ways which I have in another place had some assays of. Only remember that the Horizontal Dial must be projected, not by the Axis, but by the Zenith.  
3. A double Dial depending upon this kind, may be made to set itself, and to show the hour in a Craticular form, which is also more properly to be handled in my other ways.  

There remains yet one structure more, by which it is fitted for some uses that follow after, and is made a portable Instrument, not at all to be used in observation.  
1. MAke a right angled Parallelogram, as A B C D, whose length may be about three times the breadth: and let the limb (upon which the graduations stand) be higher than the Plain or Area of it.  
2. Describe a Semiellipsis to your own Latitude, whose longest Radius let be M O, a little longer than A B, that when the centre M is left out and cut off, so much as M N comes to (which may be about half a quarter of an hour from 12 in the Ellipsis) the remaining breadth N O, or E F, or G H, may be just fit to A B the breadth of the hollow Area, in which this lesser rectangled Parallelogram E F G H may be moved to and fro as there shall be occasion. And let the thickness or depth of this moving plate be the same with the thickness of the limb of the lower plate. The division of the Ellipsis may be made according to the rules before given.  
3. In the midst of this lesser plate, let a hole be pierced quite through it; and in that hole let a thread be extended, as I K, which is to serve for an Index, by which the said Semiellipsis is to be rectified. And therefore it is best that the thread I K be laid just in the longer diameter, which is the 6 a clock line.  
4. Take the lesser Parallelogram and put it into its place, as close to the end A B as it will go, and when it is there laid, observe where the thread lies upon the lower Area, and draw the line upon which it lies, which must be the line 90, 90; upon which the Scales (that are inscribed) must take their beginning. And likewise you are to take the longer Radius O M, and as the plate there lies, set one foot of that Radius upon O, and let the other reach as far as it will, which must be upon the rising limb beyond the inner edge of it, so much as M N comes to: in that point you are to make one centre: so again, turning your lesser plate about, you must lay it in the contrary way, close to A B, as you did before, and from the point O you may find the second centre: both these centres (in each of which must a thread be fastened) will happen to fall in the former line 90, and 90, near the points T and V.   

Being thus far prepared, you may take away the lesser plate, and divide the Scales upon the greater plate, in this manner.  
5. First, For the two limbs of it, they are nothing else but the degrees of a Circle projected thereon from the two centres: which may therefore without difficulty be done either from the degrees of a Circle, or else by Tangent lines, as any man will easily perceive.  
6. Secondly, For the other Scales you must make them parallel to the edge of the limbs. There are three of them in number: The longest is a Scale of Tangents: The middlemost is a Scale of hours, made in the form of a line of Sines: The third is a Zodiac, or Scale of the year.  
The Tangents are made thus. Taking M O as Radius, find the Co-sine of your Latitude agreeable thereto: and making that Co-sine as a Radius, or Tangent of 45 degrees, all the other Tangents are to be fitted accordingly. Then to know how far of this line will be useful, put the lesser plate again into its place, as close to the end C D as it will go, so shall the thread I K show the furthest point of it that will be useful.  
The middle Scale of Sines is made thus. Count the compliment of your Latitude upon the new made Tangent line, and from the beginning of the Scale to that point, make a line of Sins for six hours and their quarters, so shall this Scale be fitted.  
The Zodiac or Annual Scale is equal in length to the Tangent of 23 gr. 30 min. and the manner of inscription with Tables whereby to do it, are set down before, and need not be here repeated.   

The reason why this Instrument is thus contrived.  
The intent of the contrivance is, that it should show both Azimuth and hour, and that therefore there should be a Ci●●● and an Ellipsis sliding over it, both of them made to one centre, so that the centre should be in the midst, and the Circle should be a whole entire Circle, and the Ellipsis also whole. But because the Ellipsis sliding upon the nether plain, doth sometimes of necessity cover the centre, so as that there can be no thread fixed for use, I was therefore compelled to order it in this manner that is before expressed, that the motion of the upper plate and the centres might stand clear without hindering each other.  
Then further it must be noted, that the Instrument is (not general to all, but) particular to one La●●tude.   

The uses of it do here follow, which by reason of the Semiellipsis will not be so easy, because many changes are required. In general note thus much.  

1. In using this Instrument you must always suppose that you look upon the North.  
2. If the Declination of the Sun or Star be North, hold A B towards you. If South, then hold C D towards you.  
3. If the Sun or Star be Eastward of the Meridian, let the morning hours be towards you. If they be Westward, let the evening hours be towards you.  
4. Always use that thread which is on the same side with the centre of the Ellipsis. And that side lin● of the larger plate is ever to be taken for the Meridian, and consequently the other side must be esteemed for one of the two Coasts of East or West.  
5. Because North is furthest from you, therefore the furthest quarter of the limb must be either North-east or Northwest: the nearest Southeast or South-west.   
These cautions being observed, the conclusions that are to be wrought will be easy.   

1. By having the Azimuth to find the Hour.  
First, Order your Instrument by the former cautions, according to the present case of your observation. And rectify the upper ●late (by help of the thread Index) either to the Declination of the Star, counted in the Tangent line, or to the Sun's place in the Zodiac. Then count the Azimuth given upon the right quarter of the limb, and apply the thread thereto: So shall the thread show (upon the Ellipsis) the hour in which the Sun or Star is.  
If the work be for the Sun, than the hour thus showed is the Sun's hour, or hour of the day.  
If the work be for a Star, it than shows the Stars hour only; which must afterwards be reduced into the Sun's hour, that it may be the true hour of the night. This change may most easily be done by a common Nocturnal.   

2. By having the Hour in which the Sun or any Star is, the Azimuth thereof may be found.  
The Instrument being rectified; lay the thread upon the hour (counted in the Ellipsis) the same thread will show the Azimuth in the graduations upon the limb.   

3. To find the ascensional difference, the semidiurnal and seminocturnal arkes, with the time of their risings and settings.  
In general note this: If the Sun or Star decline Northward, than the semidiurnal ark is greater than six hours; if Southward, then less.  
The upper plate therefore being rectified, either to the Sun's place, or to the Stars Declination, the thread will show upon the middle Scale, the ascensional difference, if you count the hours how many they are from 6. It shows the seminocturnal arks, by the hours of setting and rising being rightly taken according to the condition of North or South declination. It shows the time of Sun rising and setting, according to the hours upon which it lies. But for the Stars, it shows only upon what houre-circles they rise and set; but the true hour or time thereof must be found by turning the Stars hour into the Sun's hour, which (as I said before) will be done with most ease by an ordinary Nocturnal, in which the Stars (that you deal with) are placed.   

4. To find the Amplitude of their Risings and Setting.  
When your upper plate lies so rectified, lay the thre●● upon the Ellipsis, to the ascensional difference, or S●midiurnall ark before found. The thread so laid, will show the Amplitude in the limb of the Instrument.   

Note here.  
In the first of these Propositions I suppose the Azimuth, in the second I suppose the Hour to be given by observation. For the observing of these two, you may make in any convenient window, both an horizontal Dial, with a thread Axis lying in the Axis of the world, and an Azimuth Circle with a thread Axis, standing upright in the Zenith line. The divisions suitable to both these, may be inscribed upon the board of the window; and for the Sun, the shadow of the thread will give the hour of it upon the horizontal Dial; or the Azimuth upon the Azimuthall Circle. But for the Stars that cast no shadows, you may make sliding sights, that may move upon the edge of the window board, or may be upon the bo●rd itself, (which must in these respects be level, or at least near to it.) And those moving sights must be furnished with short wires or pins which may best stand upon the edge of the sight, and go down to the window board, and to the divisions that are drawn upon it, which may be both for sights by which to look at the standing Index and Star both together, and must serve as Indices to show the hour or Azimuth on which a Star is at any time. Only this general caution must be given, that th●se pins be set parallel to the thread Axis that is set in the window (that is) for the Azimuths they must stand upright, but for the hour they must lie aslope, parallel to the Axis of the world. This I have sometimes done, and this any man may at his pleasure do, whereby either the Hour or Azimuth may be observed, and when any one of them is known; this Instrument (now described) will find the other.  
In such like cases this Instrument will be delightful and of good use.   

5. To find the Declination of the Sun.  
Set the thread of the upper pla●e upon the Sun's place in the Zodiac, or upon the day of the month in the Annual Scale, the same thread will give the Declination thereto belonging, upon the Tangent Scale.   

6. How to make an horizontal Dial by this Instrument.  
The meaning is to know how many degrees are intercepted between the Meridian line, and any other hour line. There are two ways to do it. The first is the converse of the making of the Instrument. For if the thread Index be laid to the line where the Scales of the lower plate begin, so that the line of six do lie evenly between the two ●●nters where the threads are fastened, t●en the centre thread applied to the hours severally will show how much they are gradually distant from the line of 12. But you must here remember to take 6 for 12, and 7 or 5 for 11 or 1, and so the rest of the hours in that order. And the gradual space● or distances from the Meridian or line of 12, must be counted from 90 contrary to the numeration of the limb. Bu● this is not so artificial as this that now follows.  
Set the moving plate to any hour upon the middle Scale, and apply the centre thread to the like hour in the Ellipsis taken on that side of it that is furthest from ●he beginning of the three Scales, and it will s●ew you (upon the limb) the angle or horizontal space that is du● to that hour. The like is to be done in every other of the hours.    

SECT. VI AN ADVERTISEMENT Concerning s●me ot●er uses of the Instrument that was last described.  
SInce the writing of that which is gone before, 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 hours and degrees, rather than into quarters and half quarters of hours. 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 hour and third degree, the intermediate degrees may be equally divided. The numeration will be either by 15, 30, 45, etc. according as the signing of it by hours will requir●; Or ●lse besides the hours you may set on 10, 20, 30, etc. in small figures, which will stand without hindering the numbers (or numeral letters) that are set for the hours.  
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 IU.  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 (Almicantars' or) parallel circles to the Horizon; & the divisions of it must then signify the Azimuths. And if it be too big (especially when it comes to represent any of the higher Almicantars') you may add another sliding plate of the same breadth that the former was, namely M O here equal to M O in the former, and M N must in both be justly equal, that it may both slip in the former cavity, so as just to fill it, and that the same limbs, centres, and threads, may in both agree.  
A lesser Ellipsis (I say) must be described, which though it be less, yet it must be like and proportional 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 equally 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, find 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 considered what the other parts of the Instrument do signify  
The end of the lower plate, upon which the Scales begin, 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 Equinoctial. The thread applied to those degrees are the Meridian's coming from the Centres, which are the North Pole. The Tangents are the degrees of Altitude. The Ellipsis notes out that Almicantar which the thread Index stands unto upon the Tangent Scale.  
And further. It is to be here noted, that if the Meridian's (or threads coming from the centre) were rightly divided, those divisions or parts of it should signify the degrees of Declination from the Equinoctial. But the inconvenience of it is, that because one Ellipsis is to represent every Almicantar, therefore the Radius cannot possibly be at all times the same in length, but must vary according to the removing or several 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) than 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 always the Radius (of what length soever it be) must be divided as a whole line of Sines, the greatest parts beginning towards the centre, the least parts ending at the remotest end from the centre, and yet then again, the numeration of the parts or declinations of it must begin at the remotest end, and must end in the centre at the Pole.  
It might be applied to use in the motions of the Stars, but that would be troublesome; therefore it shall suffice to make it useful only for the Sun.  
The best way that (for the present) I know whereby to divide this Radius (thus of several lengths) is this. Divide the length from the centre T to C into 90 Sins (the greatest parts beginning at the centre, but numbered the contrary way.) This line may be drawn from the centre T into some part of the limb, where it may have room to receive divisions. Then with your Compasses take the Radius T V, equal to M O, and from the end of the former Scale, open the thread to the least distance, and where the thread stays, there make a mark upon the limb V D, for there is a new Scale to begin, for the Secants that were mentioned before.  
Then lay the thread from T to 80 in the opposite limb, and move the Ellipsis till the point O fall just under the thread, 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 thread to 70, 60, etc. and bring the Ellipsis till the point O lies under it, so shall T O (in these several positions) give the lengths to be inscribed in this new Scale, for 20 gr. and 30 gr. etc. 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, working in the same manner by the thread and the point P. These are called the new inscribed Scales.  
The former line of Sines serves to them both indifferently.  
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 Sun's Declination and Altitude to find the Hour 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 thread. Afterwards, take the least distance from the Sun's declination, (counted in the forementioned inscribed Scale of Sines) to the thread. Set one foot of that distance in the centre, and extend the other till it cross the Ellipsis, and where it crosseth, thither apply the thread, which (in the limb) will show the Hour and scruple (in degrees) required. The same point of the Compasses doth also immediately show (upon the Ellipsis) the Azimuth sought for. If the Compasses do cross the Ellipsis twice (as sometimes it will) take that crossing that is furthest from the Meridian.  
Secondly, for South declinations. The work is in a manner the same: only you must note that the thread will cross the Ellipsis twice, and you are in this case to take that crossing which is nearest 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 find both the Amplitude and ascensional difference as before, and what else doth thereon depend.  
But because these things being done near the Horizon, will not prove good, especially when the Declination is little from the Equinoctial: 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 Dial upon an horizontal Plain: showing the reason why the same Dial, by an upright Index should show the true Hour.  
THE reason is principally deduced from the Sphere itself: and secondly, from the Orthographical Projection of the Sphere upon the plain of the Horizon; which as it doth represent the Sphere itself, so it doth perform the same conclusions with the like certainty that the Sphere doth; and that upon this general ground, that look 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 itself, which is abundantly made good by those that have treated upon this subject. Then from th●s Projection I shall show how the Ellipticall Dial upon the Horizon 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.  
Note, that in no form of Projection besides the Orthographical, this kind of Dial 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 half or 180 gr. on one side, are greater than the other half or 180 gr. on the other side: these will not do here, but only such as whose extreme Diameters are the six and twelve, that is, such as are produced by infinite Orthographical Projection.  

LEMMA I.  
First, The Equinoctial 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 Meridian's) alike or proportionally one to the other.  
The reason of this may be conceived upon this ground. Since the Equinoctial 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 horizontal 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 Axis, 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 proportional one to the other (not all the parts of one and the same Ellipsis are like, that is equal, 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, etc. with another.) And the reason briefly is, because 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 proportional Projection of every of those parallels with their like situated parts. But if the eye should stand at a finite distance, than 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 position. But in the Orthographical 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 Equinoctial Circle, or Ellipsis rather, may represent both itself, and likewise any other of the Parallels, if it be so contrived to stand from the Zenith point, as that it may have a proportional 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 Dial 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 proportional parts from those two like figures.  
This is general to all like figures, whether they be two or more; plain or solid: but for our purpose it will be enough to show it to be true in like Ellipses. 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 will, as a b, a c, you must divide them into proportional 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 greater▪ and alike situated to the point a, because all those lines are proportional, and the subtenses b c, d e, will likewise be parallel and proportional. And if the lines a b, a c, had been augmented proportionally (as here they were diminished) there would have been a greater Ellipsis described, like to the other.  

COROLLARY.  
Hence will follow▪ That if the centres 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 centre 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 individual position.    

LEMMA III.  
Thirdly, All the Azimuths of the Horizon are (in this Orthographical Projection) cast into straight lines meeting all in the Zenith point.  
The reason is, because all these Circles do cross 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 cross through the eye, and consequently must all be projected into straight lines. And their concourse being upon the Zenith line, and the same Zenith line coming 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 itself we are to consider these things.  
If it be furnished with parallels and hours, as the manner is, and laid horizontally with the Meridian line of it just North and South: and if further, there be an upright Index 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 show (among the hours of that parallel) the time of the day.  
This conclusion is true upon any Horizontal Projection (the Stereographicall as well as the Orthographical) for the straight lines or Azimuths whether shadow or Index coming from the Zenith, and cutting through any hour in any parallel, doth show the Azimuth in which the Sun must be at that hour 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 fundamental 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 centre of it, and so answering to the Zenith line of the Sphere, and the degrees of the horizontal Circle answering to the degrees of the Horizon in the Sphere. I say the shadow of that Index doth really show 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 Equinoctial, by the Meridian Circles issuing out of the Poles) it notes out also the hour of the day.  
So that the Projection made in this manner with an upright Index and divided Parallels, will (by the shadow of that Index) show the true hour 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 showed, is how the Ellipticall Equinoctial may supply the use of all the Parallels.  
That is to say, How the Equinoctial, being made movaable, and the Index standing always still, may be fitted to represent any of the Sun's 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 Equinoctial may be at any time placed in a like position 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 Equinoctial that it doth in the Parallel.  
Let the Parallel be H E, I would remove the Equinoctial A 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 easy. For first, the two Ellipses H E and A A, have their centres upon the Meridian line A 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 equal in radiall degrees to H A (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 A, must be equal.) So that E A is the Declination of that Parallel from the Equinoctial. 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 Parallel) counted in the Circle of six; and then the Equinoctial (still keeping its centre upon the common Diameter A P) be slipped up till the point A thereof do concur with the infinite line (drawn before) at G; in thus doing, I say that the Equinoctial is in a like situation to the point Z, that the Parallel 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 Corollary 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 angle 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 Parallels Declination, to the Tangent of A O, that is, to the line A G, which is the Tangent of A O, because the ark A O is expressed in its full quantity▪ without any enlarging 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 Parellels declination, estimated to the same Radius. And because Z P is in the self same proportion to A C, therefore: If Z P (the Co-sine of the Latitude) be estimated for a Radius, 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 Equinoctial] 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 Equinoctial to such a position in respect of the Zenith 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 pass through any hour point of the Parallel, will also pass through the same point of the Equinoctial, and so this one may serve for them all.  
This gives the reason of making the Zodiac or Annual Scale, mentioned in the former Treatise, I say it gives the reason of the second way mentioned for lesser Latitudes. For if either the Signs of the Zodiac, or the months 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 Sign or Month, 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 according to the former proportion.  
What is before spoken of North Parallels of Declination, is the same in South also, and one Scale (for the kind of it) serves both; only in North Declinations the Equinoctial goes nearer to the Zenith point, in South it goes further of.  
All other Scales annexed to the Ellipsis have their dependence 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 Zodiac stands, so that it give the just quantity of removal for the Equinoctial Ellipsis.  
And so also, it is no matter whether the Index move, and the Equinoctial stand, or contrarily: It is only required that their situation be such as may make the foot of the Index stand in a proportional position to the station of the Parallel, as was formerly showed.  
Further. For the circular year or Zodiac, the ground of it is thus. The Circle is supposed to be divided into equal parts or degrees: and so every Sine (as is D F) is a proper Sine to any ark (as D C) to which it is annexed. Again, A B in another consideration is to be esteemed (as formerly for the Semidiameter of the Circle, so now again) for the Tangent of 23½ degrees, whence this proportion will stand:  
As the Radius A B, Is to the Sine F D; So the same A B, tangent of 23½▪ To the Tangent A E. Which is the Tangent belonging to the Right Ascension C D.  
[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 equal degrees, as C D here is, the Sine of that Right Ascension, viz. F D, will be equal to A E the Tangent belonging to that Right Ascension, if B A be taken for the Tangent of 23½ gr. So that this way of putting in Months and Signs is the same in effect with the former Zodiac or Annual course of the Sun.  
Thus far for demonstration of what was hard in the first Section, Pag. 8, etc. concerning horizontal 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, etc. Of framing it to Declining Plains.  
DEclining plains, if they had their proper hours upon them, would al●o (in this Ellipticall respect) be the same with the former direct, or horizontal plains. The distinction is, because they are made to another Meridian than is their own, that is, to the Meridian of the place, and all the account of hours is deduced from it. The difference therefore of these from the former, is only this, that the hours 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 Horizonta●l plains, you do but suppose the Equinoctial and Parallels to be divided from any casual Meridian as P M (not P A) into hours and parts, then will follow still the same things in substance that was before. Namely, That if the Equinoctial 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 Sun's parallel of declination) with those divisions now spoken of, then still, when the Equinoctial 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 hour will be showed by the Equinoctial 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, etc.  
THat pricking down of the horizontal Dial mentioned in this Section, is the pricking down of the Equinoctial with its hours or parts, just as the Orthographical Projection itself hath them. The first Table shows what Angles or upon what Azimuths from the Meridian every hour point lies. The second gives the Altitudes, or rather the distances of the same hour points from the Zenith, And therefore no more will be required of this.  
The fourth Section requires no explication nor Demonstration.   

Sect. 5. Pag. 37, etc.  
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 (instead of the horizontal Ellipsis) elevated to the height of the Equinoctial. The reason of that will best be seen out of the Sphere itself, For there we know, that the Equinoctial and all the Parallels to it are both equally divided, and equally elevated, and so being all alike, the Equinoctial may supply the room of each and all of them. Only it must be required (whether the Index move toward the Circle; or the Circle towards it) that this motion (if the Zodiac be upon the horizontal plain) be limited by an horizontal Zodiac, such as was used for the Ellipsis upon any plain, (which plain, what ever it be, I now count as an horizontal plain.) But if the Index be made to move upon the Equinoctial (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 equal to the Latitude of the place, and not the Latitude of the plain.) If, I say, the Index be made to move upon the Equinoctial 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 horizontal Scale, to the parts and whole of this Equinoctial 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 horizontal plain (for the Ellipsis thereon described) you look 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 decimal Scale; and by it you may put on the Zodiac or months as was prescribed before. This Co▪ sine mentioned 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 Equinoctial Circle, whose Radius must be conceived to be the same with the greater Radius of the Ellipsis. For from this Circle projected Orthographically (that is, by perpendiculars let fall from the Periphery to the subjacent plain) is the Ellipsis deduced, and from that Ellipsis (again) is this Circle raised or restored. And so the reason of this Circular Dial 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 centre of it lie in a just distance from the Index or Zenith line, the same Index must show the same point or hour, because in both ways they are situated from it upon the same Coast.   

Sect. 6. Pag. 47, etc.  
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 repeat 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.    

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.  
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 practice. I shall therefore let that way alone, and fall upon another more feasible and delightful.  
The way that I intent, is partly by projection of such lines as are useful, and partly by inscription of such points as the Ellipticall line is to pass through, as shall be seen hereafter. And therefore we shall not need to look after any position of the plain in respect of declination or leaning.  

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 Level) cast an horizontal level line as F G upon the plain; or if it cannot be cast upon the plain itself, set any board (for a time) that may receive it; lying in the same Level with the assigned point at E. And further, the same horizontal line must be continued and returned about the point E, by help of threads stretched out from some points or other of the horizontal 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 near 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 horizontal level 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 Azimuth may be found distinctly, observe where the shadow of the Index cutteth the horizontal line, suppose at K, and there make a mark, and immediately take the Sun's Altitude.  
3. By the Sun's Altitude observed, compute what Azimuth the Sun was then in, which will tell you what Azimuth the observed point K (in the horizontal line) doth represent.  
4. Take a Pasteboard as E L, and fit it to the horizontal 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 centre E, quite over the pasteboard. And then knowing what Azimuth 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 Pasteboard.  
5. You are next of all, from this Meridian line P E, to set of all the other hours of an horizontal Dial (not the common horizontal Dial, but) according to su●h numbers or arks as are expressed in the Table of Angles, Pag. 29.) And when this is done upon the Pasteboard, as you see done at o P q κ s t u: Then,  
6. Apply the Pasteboard to its former position upon the plain, and from the centre E project the hours from the Pasteboard to the horizontal line and threads. And then taking away your Pasteboard, you may draw lines from A to such points as are in the horizontal line i● self, as A o, A p, A q, A κ, A s, A t, A u. And for the others that stand not upon the horizontal line of the plain, but upon the threads 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 thread; and marking where the shadow or appearance of the thread 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 hour lines as usually are upon other Dial's, but they are those Azimuth lines in which the Sun is at every hour of the Equinoctial Circle. So the hours and quarters may be put in by the Table, Pag. 29, and the Column of Angles in the same Table.  
7. Look again into the Table Pag. 29, for the Column of Altitudes; for by that must the Ellipsis itself be described, for the former Azimuthall-ho●re l●●es will give it the true divisions into its requisite parts. The manner of the division is this. Make a Scale of 90 right Sins, but number them versedly, as is here expressed. And let the Scale be of a fit dignesse for your plain, which your own judgement must direct you to do. Out of this Scale (setting always one foot of your Compasses in the point of 90) take such Altitudes as are (in the Table) appropriated to every hour, and put them into those several hours 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 (always keeping it upon some part of that line) I remove it thereon, nearer to, or further from the centre A, till the other foot of the Compasses being turned about will justly touch the edge (or fiducial 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 pass upon the houre-line of 12. So for any other hour (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 showed before, you shall find all the points (for every hour 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 angles in it.   

3. Concerning the motion that is to be made either by the Index, or by the Ellipsis itself.  
THe motion must be made according to the 12 a clock line, which is always the proper Meridian to the Index or Zenith line. That is, the removal or sliding must be either in the Meridian itself, or else parallel to it.  
Now if the Dial itself 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 hours and Ellipticall line must be described thereon in the same manner that was showed before. And this plate must move in the Meridian line, that is, the 12 a clock line of the plate (being imagined to be continued forth right) must move through the fiducial edge of the Index.  
And so again, If the Dial be drawn upon the plain itself, and the Index be made to move (the Dial itself standing still) then must care be had that the fiducial edge of the Index do precisely move in the Meridian or 12 a clock line, and that the same Index move always precisely upright in the Zenith line.   

4. Concerning the Place of the Sun's Annual course or Zodiac.  
THe place of it may be best upon the plain. It must stand (if it be inscribed into a straight 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 zodiacal 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 Diameter (of the Annual Circle) which passeth from Tropic to Tropic, 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 mentioned Pag. 73. For you may take from 90 (in that Scale) to the number of the Latitude of your place (not compliment as that Scale is numbered) and count that length as a new Radius. Unto th●s new Radius find the Secant of the Meridian's inclination (which what it is shall be presently showed 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 ways from the Equinoctial, will give the length of the Zodiac (in a straight line, or the Diameter of the Zodiac put into a Circle:) this work is to be understood for the describing of the zodiacal Scale upon the plain itself. But first, you are to pitch the Equinoctials place, which is that place only upon which the zodiacal Index lieth, 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 Declinations (in the Tables of the 4 and 5 Pages:) out of it, and so prick on the Sun's yearly course. Or else make the forementioned Secant a decimal Scale, and out of it take the Tangents noted in the Tables Pag. 6, 7. So shall you make the same Annual course of the Sun that was produced the former way.  
Note, that if the Index be made to move, it is not necessary that it should move upon or parallel to the superficies of the plain, but may be made to move either horizontal, or at any other inclination: only the fiducial edge must necessarily be made to move in the very Meridian line, and must also be (in all motions) always upright, or in the Zenith 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 Meridian's inclination meaneth, and how to find the quantity of it.  
BY the Meridian's 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 compliment of the angle D A P, or the excess of it. To find it, you may either apply the edge of an Inclinatorie square to the said Meridian line, and then the thread (being drawn perpendicularly down by the plummet) will show the said elevation (or inclination) of the Meridian line above the Horizon, in the degrees of the limb, if the same degrees be taken to the thread 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 compliment 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 find the quantity of the angle D A P, whose compliment (or excess) is the inclination, or elevation, or depression of the Meridian, which is here required.   

7. Other things to be noted concerning the zodiacal Scale.  
1. IF the Dial be described upon an irregular superficies, such as is not flat but writhe (as by this course it may very well be) than it is most convenient that the former Scale be set in some other place, and not upon the superficies whereon the Ellipsis is described. It may be contrived two ways.  
First, If the Dial superficies (how irregular soever it be on that face whereon the hours are inserted) be flat at the bottom, and be made to move (the Index standing still) upon some other plain below it) than the best way will be to make the zodiacal Index upon the moving bottom 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 find out the inclination of it, and so find the Scale by the Secant of that inclination, just as you were before directed. And so this work will be complete. The motion of dials unequal Superficies must be in or parallel to this last projected Meridian line, always so as that the Meridian of the Dial must pass through the fiducial edge of the Index.  
Secondly, If the Index be made to move (the Dial standing still) and that the Dial be upon some unequal superficies, such as is unfit to receive the Zodiac, than the foot of the Index may have it inscribed upon it. Now in this case of unequal Superficies, it is supposed that this foot of the Index cannot move parallel to that uneven Superficies, but must move straight forward in some right line, just along with the Meridian line. In this case you must find 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 Instrument, 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 showed. So much for this.  
2. For the other Scales of the Sun's Declination, Amplitude, ascensional difference, and gradual motion in the 12 Signs, they are to be done in the self same manner, and by the same Tables as before (without any difference) after that you have found your Tangent Scale or decimal Scale out of which to describe them.   

8. Another Observation.  
ACcording to that general observation Pag. 36. Note here; That if to an Index standing in the Zenith line, a Plain be set in the Equinoctial, the same plain shall have a Circular Dial upon it, equally divided. Then whether the said Equinoctial plain or the Index move, if they be made to move upon the horizontal plain, the Zodiac for the horizontal plain must serve. But if the upright (or Zenith line) Index be made to move upon the Equinoctial plain, the former horizontal Zodiac must be set upon the Equinoctial plain, and must there be enlarged above the horizontal Zodiac, that is, every part of the horizontal Zodiac must be made greater upon the Equinoctial plain in proportion as the Radius to the Co-secant of the Latitude. That is, the Scales that made the horizontal, 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 Equinoctial plain.  
Note further. That for this Equinoctial plain which is to descend downwards from South towards North, you may instead of it set another plain quite contrarily, that is, descending from North towards South; which will be the most convenient of the two, because the upper face of this will give the hour all the Seasons of the year, whereas the other will be only for Summer upon the upper face of it, and will require an under face for Winter.   

SECT. VIII. Hitherto of Ellipticall Dial's to all Superficies whether plain or curved, whose Indices 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 casually in any position whatsoever. But first are premised some useful Propositions tending to the same purpose.  

1. An Index or straight line being set casually, how to find 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 withal 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 fiducial edge of the Index is (here) like the Vertical 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 vertical 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 needful here to make repetition of that which is so often elsewhere declared. You are here also to observe whether the re/ in-clination be towards the North or towards the South.  
To this new plain therefore, or to the two lines A B, C D, apply one of the straight edges of a Quadrant, the limb of it being turned always towards the Sun, and observe the horizontal distance of the Sun, as is usually done in finding the declinations of plains. Only here take this caution, that you count not this horizontal 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 always to account this horizontal distance from that end of the Quadrants side which looks the same way that the point B (of the Index, which is furthest remote from the plain A C) doth look: as in the second figure is expressed, where the horizontal distance is more than a Quadrant, it being there to be accounted from F to H. Or for more easy conceit, you may always suppose your Quadrant in that posture to be continued to a Semicircle, as is done in the second figure: and then count your horizontal 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 horizontal distance will be F G H, more than a Quadrant, in this particular case.  
The horizontal distance being thus accounted and observed, you are immediately to take the Sun's Altitude and to find 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 find 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 inquire 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 Dial 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 find the Longitude and Latitude thereof.  
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 easy as the former. Let n E S W be the Horizon, n the North, S the South, P the South Pole N the Nadir point opposite to the Zenith, O the point or place to which the Index respecteth. In the Triangle N P O, are given, N P the compliment of the Latitude of the place, N O the inclination of the Index; and P N O the declination of the same Index, by which we may find P O, whose compliment is the South Latitude, or whose excess is the North Latitude required: and the angle at P will be found, which is the difference of Longitude here sought after.  
The work is easy to them that understand the like work in re/ in-clining plains, and may be performed either by calculation, or else Instrumentally, as every one shall like best of. And this is all that needs to be said of this particular.   

3. How to find 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 Dial superficies, whatsoever that superficies be, whether plain or curved. As in the two former figures may be understo●● by the point A.  
From the foot or polecat A, set some upright thread 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 superficies 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 Pasteboard, 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 Dial superficies. And upon this object project the Meridian near that place whereabout you conjecture the Axis of the World will pass through. Then with your Semicircle (or projecting Quadrant rather) project the Axis of the World, as is usually done in my ways of Dialling, and by that means you shall find 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 thread, the same thread will represent the true Axis of the world coming from the foot of the Index.   

4. Having an Axis raised from the foot of the Index, how to find in what Longitude and Latitude the Index itself lieth, 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 easy.  
Let the Dial 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.  
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 something underneath close to one of the points (without disturbing the threads 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 thread 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 equal) and set the same extent from A to R upon the Index (prolonging the Index if need be by a thread 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 showed before.  
There will be found other ways 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 precedent Proposition) and you find what Azimuth that is too. You may also (further) project it upon your Dial superficies, and then you have two Azimuths upon the said superficies, namely, the Meridian, and the observed Azimuth. You are to know yet further, what Altitudes the Equinoctial hath upon these two Azimuths. And for the Meridian it is certain enough that the Altitude there, is equal to the compliment 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 Equinoctial Altitudes, as the manner of projecting by that Instrument useth to be.  
Next, you must project this Equinoctial 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 project 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 Meridian 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 Equinoctial line which was now projected, 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 Equinoctial line with the Meridian line, which suppose at S. For these two Equinoctial 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 Equinoctial, which angle measures the said inclination of the two forenamed Meridian's, as it doth of all Meridian's.  
Now to measure this angle V m S, the best way will be first to extend two threads, one from V to m, and the other from S to m, crossing each other at m, and they may be continued 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 equal 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 business is to find 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 measured 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 forasmuch 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 Longitude; and that the said difference of Longitude lies Eastward 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 kinds is delightful, 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 form the Angles at the Pole.  
THis Proposition makes way for computing Tables of horizontal spaces, and Equinoctial Altitudes to any Longitude or Meridian differing from the Longitude of the East hour general Table West hours east how general 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 4 saint 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 computation of the said angles at the Pole for the hours of any place to the same declining plain.  
I have here first of all in a general Table set down the gradual distances of hours and quarters from 12 at noon or midday, which will be some help in performing the work. The hours that are on the left side of the degrees are the East or forenoon hours, and those on the right hand are the West or afternoon hours. 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 angel's our hours (namely those 12 that stand nearest 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 intent to call it.  
I first take the given Longitude or Proper Meridian, 39 gr. 25 min and enter it among the East hours of the general Table, and I since it to fall in at A; that is, between a quarter and half an hour past 9 before noon, and I find also the circumstant numbers to be 37.30 the lesser, and 41.15 the greater. The differences between these two numbers and the Longitude given, are 1 gr. 55 min. and 1 gr. 50 min. as you see them set down in the said general 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 separated 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 difference of 39 gr. 25 min. the proper Meridian from the next greater number 41 gr. 15 min.) Now to these two radical numbers, I add ¼ ½ ¾ and one whole hour, namely a, b, c, and thereby produce e, f, g, on both sides the proper Meridian. Then to r, e, f, g, I add 15 gr. d, namely, the number of one hour, 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 add 30 gr. or two hours, and do thereby produce on both sides the eight numbers noted with the letter m. And so still adding 30 gr. to the last eight numbers noted 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 remaining 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 completed, for if it should (in the same manner) be further continued, the next numbers (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 easy than any other, which else might have been in this work used.  
You are afterward to place the hours (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 hours and their quarters, the same angles serve for the other 12 hours: and whatsoever is hereafter computed for any of these 12 hours, must be understood to serve also for the 12 opposites, and so the work will be done for the whole Circle, or 24 hours of the natural day.   

6. By knowing the angles at the Pole, and the Latitude of the place or Horizon, how to find the horizontal 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 Horizon it is, that the Spaces now mentioned are called horizontal Spaces.  
The way to find them for this Ellipticall work is converse to that which is usual 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 horizontal space belonging to each of those foresaid angles.  
So having found all these horizontal 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 signify the distances of every of the hours 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 find the Equinoctial Altitudes or depressions (above or under the same Horizon) due to the said Angles at the Pole or points of the Equinoctial.  
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 hour (rather) in the Equinoctial Circle.  
So in the former example. The Latitude or elevation of the Pole above the Horizon was 40 gr. the compliment of that is 50 gr. which is the Meridian altitude and depression of the Equinoctial Circle above and under the said Horizon 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 ark of the Equinoctial in the former Table, To the Altitude due thereunto.  
hours Horizon Spaces altitudes or profu. hours'  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  hours Horizon Spaces. altitudes or profun hours 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, etc. below the proper Meridian in the Table of Angles, are compliments to n n n n n, etc. at the top of the Table of Angles: and again, those r e f g h, etc. above the proper Meridian, are compliments to the lowest n n n n n, etc. 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, etc. To the Sins of the Altitudes belonging to s t v, etc. And so again, the Sins of w x y, etc. to the Sins of the Altitudes belonging to z u st, etc. 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 half hours only, and so to save half your labour. But in lesser elevations it is best to do it to quarters, as here is done n this Example.   

8. How to find 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 Ellipsis into hours 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 Longitude, and to the Latitude of your Index, as is done before in the 7 Proposition, one of which Tables is of horizontal Spaces, the other of Equinoctial Altitudes or Depressions above and under that Horizon which is proper to the Index or Zenith line casually placed.  
By these two Tables the work will be done in such manner as was showed 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 thread that it may not be lost again.  
2. From this point you must draw an horizontal line, not in the level 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 every 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 pass 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.  
[And so it may be noted, that if other Dial's be described by the Equinoctial Circle and not by the horizontal, the work of drawing the Equinoctial perpendicular to the Axis will be difficult, but though no great accurateness be used, yet the work will be perfect enough, and no way defective for the loss 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 Dial's.]  
Wherefore you may do it by some Pasteboard, 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 pasteboard, 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 Pasteboard to its former place again (the edge of it lying upon A D, and the flat of it applied 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 thread if need be) suppose at E: at the point E (then) you must say that one point of the proper horizontal line is to be taken. Then in like manner you must seek another horizontal point: first by projecting a line from the Index (any where) such as is A F, and by applying one edge of a pasteboard to that line, and the plain of that pasteboard to the Index A B, and so noting the point C, and drawing or marking the line A B upon it, etc. as was done before. So you shall find another point of the same Horizontal line or plain (rather) which suppose to fall at F.  
Now having three points of the horizontal plain at C, E, and F, (which I suppose not to lie in one and the same right line, for that must with carefulness 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 useful) may be made up with returns of thread, and regulated or kept in the same plain by projection, as the manner of working that way useth to be, and as here you see expressed, by the line H E F G, lying in the same plain with the point C, and that whole plain lying perpendicular to the Index A B.  
3. The next thing to be done is the drawing of the hour 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 horizontal Spaces of your hours from this proper Meridian A B; you must first apply a pasteboard to the horizontal line E F, and fit the centre of it to C: and upon the Pasteboard, project the proper Meridian from C to A V, or from C to P. And then by the Table of horizontal Spaces in the 7 Prop. (for we now here suppose that this is the Dial for which those Tables were computed) you may (upon that pasteboard) set off all the hours and quarters from the proper Meridian upon this pasteboard: and applying the pasteboard into its proper place, namely to the horizontal line E P F, and to the point C, you may project the hours and parts of hours from the Pasteboard to the horizontal line H E P F G, as the manner in this way of Dialling is well known.  
4. Having transferred the houre-points into the horizontal line, you may (by help of your Index A B) project and draw the hour lines upon the plain, which we will suppose done, because the manner of doing it is the same with that which was done before for upright Indices.  
5. To know where the Ellipticall line must come, or to find the points in those hour lines, through which it must pass, we must work in the same manner as before (in the 7 § of Pag. 7●.) is expressed, namely thus. We must make a Scale of right Sins of a fit length, and number them Versedly, 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 computed 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 hours to which they belong. They must be inserted in this manner. Having taken any altitude out of the Scale, and found the hour upon which it must be placed, you must set one foot of that extent upon the hour line (keeping it always thereon, but) removing it until the other foot being turned about, may only touch the fiducial edge of the Index: and when the feet of the Compasses are thus fitted, you must note upon what point of the hour line the foot that is thereon doth stand, for through that point of that hour must the Ellipticall line pass.  
The same manner of work you must perform upon every hour, until you have gone through 12 of them, which do make up half the hours of the whole Dial. And if you strike the lines through the centre, you shall have all the 24. And look what is done upon any one hour line, the same is to be done upon the opposite. That is, look what distance (upon a plain) the Ellipsis hath from the centre upon any one line, the self same distance from the centre must the opposite line have. But if the description be made upon an uneven superficies, than this rule may not, perhaps, hold: yet this will; namely, Look what distance from the Index (the least or perpendicular Index I mean) any point of the Ellipsis hath, the same perpendicular distance is to be given for the Ellipsis upon the opposite hour line. And by this means you may put in as many hours and Ellipticall points as you please. And through these points you are to draw the Ellipticall line.   

10. Concerning the motion that is to be made, either by the Index, or by the Ellipsis itself.  
THe motion must be either in, or else parallel to, the proper Meridian, and not elsewhere. Now if the Dial Superficies be made to slide, and the Index stand still; then (though the Dial be upon an unequal superficies, yet) the motion or sliding must be upon a plain. And it must be upon the proper Meridian, projected by the Index from the Dial 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 Dial superficies, and as it shall be laid by projecting it.  
But if the Index move (the Dial standing still) then the fiducial edge of it must be made to move always according to, and in the very proper Meridian line itself, and not any where else, which may be contrived sundry ways, as every man shall invent to his own liking. Great care must be had that the fiducial edge of the Index (in the motion of it) keep always one justly parallel situation.   

11. Of the place of the Sun's Annual course or Zodiac.  
THe best place for it is to be considered of according to that which moveth.  
If the Dial 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 thread Index (serving only for the Zodiac) be set upon the Diall-moving-plate, whereby 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 zodiacal Index may be set somewhere where it may stand to point at the Zodiac. Or else the Zodiac itself may have its place upon some part of the standing plate or body, and the zodiacal Index may be placed upon the foot of the Dial Index; from whence it may be made to show the zodiacal parts, as occasion 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 Dial plate (that is the mover) must move according to the proper Meridian of the Dial Superficies, and precisely so, as that fiducial edge of the Index must move (not elsewhere, but) in the very proper Meridian itself. This is, if the Index move. But if the Dial Superficies move, than the proper Meridian thereof (or the Superficies of the proper Meridian, namely the Superficies made by the proper Meridian cutting through the Dial plate, which is of any thickness) must move directly upon, or through, the fiducial edge of the Index. These things are intimated before. And that the Zodiac is to be described upon a plain, though the Dial itself be not so. And further, that if the Dial plate move, (though it be not a plain itself, 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 inculcated, because they are hard to be conceived, and had need of the better consideration for that reason. Now further.  
1. If the Dial plate be supposed thus to move upon a plain, and on it the proper Meridian be drawn, than first of all, The angle is to be inquired that is made between the Index 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, than 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 keep it. Then when you have found the forenamed angle (of finding which more is said Prop. 13. following) to this new Radius find the Secant of the compliment or excess of that angle: this length or Secant will be the Tangent of 45 gr. or the decimal Scale by which you are to describe the Zodiac on both sides from the Equinoctial point or line, according to the numbers in the general Tables made for this purpose, Pag. 4, 5, 6, and 7.  
[For placing the Equinoctial 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 Dial was described: and when you have made fit place for the Zodiac, and set on some peculiar zodiacal Index, you must note the place upon which the said zodiacal Index pointeth, for that place is the peculiar place for the Equinoctial, from whence all the other parts of the Sun's annual course must be set on.]  
2. If the Index be made to move in any depth under the Dial superficies, than a slit must be made in the same Superficies that the proper Meridian would cut, and the Index continued therein, down into one straight line till it meet with the foot whereon the Index is fixed, and which being moved, carrieth the Index along with it. And the Zodiac must be described either upon this foot (the zodiacal Index being made to stand still while that said foot with its divisions moveth) and that also parallel to (or in) the proper Meridian projected upon it: or else the foot must carry the zodiacal Index, and the Zodiac must be described upon some other convenient place, as the sides (etc.) of that forementioned flat. And here, for limitation of the Zodiacs length, you must find the Inclination of this moving foot to the Index 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 Radius, and the Secant of the compliment (or excess) of the angle. By which you shall find the Tangent of 45 gr. or the length of your decimal Scale, out of which the Zodiac may be described by the Tables▪ Pag. 4, 5, etc. as is mentioned 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 described.  
For Occurrences of other sorts of Cases than are here mentioned (whereof there will not be many) he that can understand 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 find 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 think of is this. Apply a straight edge of pasteboard to the projected line, and the plain of the same pasteboard to the fiducial edge of the Index, and make two points upon the pasteboard, by or through which the same fiducial edge passeth. Then taking away the Pasteboard, draw a straight line through those two points or marks, and so measure the angle made between the forenamed edge of the pasteboard and this right line. If it be so that the line crosseth not the edge of the Pasteboard, then draw such a parallel line to the edge as may cross 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 Dial and Index, for setting them true.  
THe best way is to make the Dial plate move, and the Index to stand still (in these obliquely situated Indices) for the Zodiac will (in such cases) be most easily described and made useful. And in this case the Index may be also set fast first, and quite finished before the Dial be drawn at all. Then also the Dial will be drawn more easily, and the motion of the Dial plate may sooner (this way) be contrived, then can the motion of the Index be contrived when the said Index is to move and the Dial stand still. The motion (as hath been often said) must be according (that is parallel) to the proper Meridian: and the slit (for the fiducial edge of the Index) may be so contrived that the fiducial edge itself (which is best to be a fine thread) may be also the proper Index, and the Zodiac may be described upon the Diall-moving-plate, closely contrived to the thread. Or thus at least, The length of the whole Zodiac may be so limited as is before mentioned, and then if you desire to have it drawn upon some other place of the plain upon which the Dial superficies moveth, and the same Dial 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 fiducial 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 Dial plate moveth, for the new zodiacal Index set fast into the moving plate, will (upon the standing plate) show the terms of the same Zodiac suiting 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 kind would be too too many.  
2. If the Index be thought best to move, then must a pattern of it (that is some thread) be first set up. And to this vice-Index must the Dial 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 Index is to move (for a plain it must move upon, whatever be the Superficies whereon the Dial is described) and that same foot to carry the zodiacal Index upon it; or else upon the foot of the Index, and the zodiacal or peculiar Index to stand fixed upon the plain (or Dial superficies) And then when all this is done, the vice-Index must be taken away and a true substantial Index put in the place of it. And for that purpose, you were best, before you take it away, to find 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 substantial 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 ways care must be had what inclination the zodiacal 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 kinds of Ellipticall hours, 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 Dial's as are upon fixed walls or Superficies that cannot be removed. But if it be to be done for movable Bodies, if they be regularly cut, than the declinations of every of the plains are known by their regularity; so that such plains of them as are capable of any Dial before mentioned (for the Index and the plain must not be parallel in any case, they may be perpendicular) may have such Dial's upon them as they are capable of, either to Indices perpendicular to their plains, or to Indices lying in the Zenith line of the place. But for these casual Indices upon them, you will be put to it, to know what declinations 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 horizontal or upright position, that is, let the foot of it stand upon some just horizontal plain, though in respect of declination it stand at all adventures. Then from the fiducial edge of your casual 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 Indices concourse with the plain, draw a right line; this line shall represent the Azimuth wherein the said Index lieth. Then again, with your eye repose a perpendicular hanging thread upon the two points (namely, the centre or concourse of the Index with the plain, and the late found perpendicular point) both together, and project the umbrage of the thread so hanging upon the horizontal 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 also the perpendicular thread must be projected upon them both joined before into one, so that Thread, Style, and Meridian must now be all as one line; and the thread so hanging project the shadow of it upon the same horizontal 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 look what angle they make at their concurrence, the same being counted the right way is the angle of the casual Indices declination from the Meridian line. Thus the Declination is found.  
Then the Reclination may be found as before is declared, and by help of them the Longitude and Latitude of the Index may be had, and so the Dial made by the precedent Precepts.   

A brief DEMONSTRATION of the 7th. and 8th. Sections.  
IN the 7 § all are made to the Zenith line of the place. If therefore you imagine an horizontal Ellipsis to be described to that Zenith line, and upon that Ellipsis a kind of compressed Cylinder to rise upright, parallel to the said Zenith line, with hour lines raised from the several points of the horizontal Ellipsis, then will the Zenith line show the hour upon those surgent lines: and the same Index (or Zenith line) must show the hour among tho●e upright lines in the compressed Cylinder, by the same reason that it doth upon the Ellipsis itself, upon which the said Cylinder is raised, And if so, than 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 hours (by help of the Table in 29 Pag.) as it will serve to find the Ellipticall points upon the horizontal plain itself, so must it serve to find some one point of these surgent hour lines upon any other Superficies, because they keep the same distances always from the Zenith line (from whence the projection of them is made, and whereon they depend altogether) and therefore it finds (upon any Superficies) the points of those lines which pass through, or do intersect the Dial Superficies. Therefore it is that these Ellipticall points (upon all Superficies) are points of those surgent lines rising from the horizontal plain, and that the Zenith line must have the same relation and situation to them, that it had to the points upon the horizontal plain itself, And consequently, that this way must be of the same truth upon any Superficies that it is upon the horizontal. So much for any upright Index, set upon all sorts of Superficies, with the hours thereon depending.  
In the 8 §. are handled casual Indices; 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 seem to be stuffed with many impertinencies, and some need less difficulties, which the Author acknowledgeth willingly, and excuseth, by reason that they were his first med●tat●ons in th●s kind; and so much the more undigested by how much the less practice hath been by him used therein. The truth is, he never described any thing suitable 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 Mathematical Subject wherein hath preceded no real representation, he will not only excused difficulties and impert●nencies in the tradition, 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 Dial plate to move (let the Coast of the motion be casual) how to fit a steadfast Index to it, and to describe an Ellipticall Dial upon the said moving plain.  
THis Case was forgotten before, but now here supplied. You must observe the line (upon the immovable plate) which the midst of the moving plate doth describe, and suppose 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 thread (or Index) any where, only so as that both this Thread, the Axis, and the Proper Meridian may all three appear in one line. There six the thread, and then find the Longitude and Latitude of it, and afterwards describe the Dial to it, according to the rules given before.   

16. If an Index should be set up and made movable upon a standing plain, there can no Dial be described thereto,  
Unless the Index be made of wire or some such bending substance, but to such there may. For if you observe what straight 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 fiducial edge of the Index, and the Proper Meridian, may all three appear in one line. And then finish your work as is before directed.     
FINIS.  

CIRCULAR Horologiography.  
SHOWING, How to make an horizontal Dial in a Circle equally divided, to show the Hour of the day, and Azimuth of the Sun.  
Invented and written by Mr. SAMVEL FOSTER, Late professor of Astronomy in Gresham-Colledge.  
London, Printed for Nicholas Bourn. 1654.   

CIRCULAR Horologiography.  

How to make an horizontal Dial in a Circle equally divided, to show the Hour of the day, and Azimuth of the Sun.  
HEre before we have had ELLIPTICAL HOROLOGIOGRAPHY, now shall follow CIRCULAR HOROLOGIOGRAPHY, which showeth how to make a Dial in a perfect Circle equally divided into hours, (whereby to find the hour) upon any plain whatsoever.  
Divide a Circle into 24 equal parts, and take so many of them as your Horizon hath hours in the longest day, or rather so many as the degrees of your greatest Amplitude (East and West) from the South do arise unto; which here at London will be near 18 of them. Then divide each of these parts into 15, which for the Azimuth will signify degrees, for the hour will stand for four minutes of time apiece.  

The altitude of the Horary Index may thus be found.  
Add your Latitude to 90 gr. half that sum is the elevation of the Horary Index above the Horizon. Thus at London, 45 min. which is the altitude or elevation required. Or, add half the compliment of your Latitude (which is 19¼) to your Latitude (51½) the sum (70¾) is the elevation of the Index.   

The standing and looking of it.  
It must stand right over the line of 12, elevated above the said line 70¼ gr. and must look toward (but not into) the North Pole.   

The motion of the Horary Index.  
It must move to and fro, directly over the line of 12. Or else the hours must move to and from it, according to the line of 12, so as that the same line may always lie under the foot of the said Index. The motion of the one or the other is necessary, because else the Circle of equal parts can never show the true hour all the year long.   

How the Zodiac is to be limited, and laid, and charactered.  
The motion of the Index must be regulated by the Zodiac. The Zodiac therefore must lie either in, or else parallel to the Meridian line. The length of it is thus to be limited. Count the Semidiameter of your Circle (viz. A B) for the Radius: to that Radius, Either  
1. Make the Sine of 70¾ (taken to the Radius of your Circle) a Tangent of 70¾; and to the Radius of that Tangent find the Secant of 19¼ (the compliment of 70¾) that length shall be the Radius of the degrees, or the decimal of the Tangents of the Zodiac, to be inserted by the Tables Pag. 4, 5, 6, and 7.  
2. Or else, Make the Sine of 19¼ (estimated to the Semidiameter of the Circle A B as Radius) a Radius, and to that Radius find the Secant of 19 ¹, this last length or Secant shall be the quantity of the Tangent of 45 gr. or of the decimal Scale by which the numbers Pag. 4, 5, etc. are to be inserted. And in these Northern Orisons ♋ and ♈ must be placed so that 12 may be nearest to the Index in Summer, and furthest off in Winter.   

The manner how to fashion the Cock which holdeth the Index.  
The fashion may be seen by the figure A C D. At A is the place of the fiducial point of the foot of the Index to be assigned: and in that point a hole must be pierced, and a thread fixed. Then the Cock must have two holes more pierced, one at C, the other at D, both to stand perpendicularly over the line of 12. That at C must be so placed, that the angle C A B may be 70¾ gr. That at D must be placed so, that D A B may be an angle of 90 gr.  
  

How to place this Dial for use.  
YOu must either fix your Dial plate in the Meridian line, and truly horizontal▪ or else upon an horizontal or level flat you must draw a Meridan line, whereby to place it upon any occasion. Then,   

To find the Hour.  
Make use of the Index A C, and rectify the foot of the Index to the requisite place in the Zodiac, (either to the day of the month, or the degree of the Sign.) When it is thus rectified and set, the shadow of the thread A C will show the hour of the day.   

To find the Azimuth.  
Put the thread from A to D, and let A D be your Index. Then (always) place the foot of the thread in the centre of the Circle at A, So shall the thread A D give the Azimuth.  
¶ Note, That the Scales of Declination, Amplitude, ascensional difference, may be placed by the Zodiac, and used as is before showed in the Ellipticall Dial's.    

Another way to make the same horizontal Dial equally divided, to find the Hour and Azimuth.  
THe former way maketh the Index to point up near to the elevated Pole. This other will require the Index to point towards the contrary Coast, namely, that of Noon or Mid day, or toward (but not into) that Coast of the heavens where is the depressed Pole under the Horizon, but still the Index is to be above the Horizon.  
The altitude of the horary Index above the Horizon, must be half the compliment of the Latitude, or half the height of the Equinoctial. Here at London 19¼ gr. which is the compliment of the former ways elevation 70¾  
T●e standing and looking of it, must be right over the line of 12, drawn out at competent length, and must justly point into the Coast of 12 mid day: Or towards (but not into) the South (or depressed) Pole.  
It must move as the former did.  
The Zodiac is to be laid as before, and to be limited and charactered thus. The Semidiameter of your Circle being taken for Radius, you must use the two former rules in this manner.  
1. Either make the Sine of 19¼ a Tangent of 19¼, and to the Radius of that Tangent find the Secant of (the compliment of 19¼, namely) 70¾, that length shall be the Radius of the degrees, or else the decimal Scale of the Tangents of the Zodiac, and they must be inserted by the Tables Pag. 4, 5, etc.  
2. Or else, make the Sine of 70¾ (estimated to the Semidiameter of the Circle A B as to a Radius) to be a Radius; and to that Radius find the Secant of 70¾, this last length or Secant, shall be the length of the Tangent of 45 gr. or of the decimal Scale, by which the numbers in the Tables Pag. 4, 5, etc. are to be inserted.  
By this instance for the Horizon of London, where the numbers 19¼ and 70¾ are proper, may the like work be performed in other places of different Latitudes from London, only by altering the numbers, according as the proper Latitudes of such places shall require. And in all Northern Orisons ♋ (or the longest days) must be so placed that the point of 12 in the Circle may then be nearest to the Index, and in ♑ (or the shortest days) it may be furthest of. But in Southern Latitudes, the places of ♋ and ♑ are quite contrary, because there the Sun being in ♑ makes the longest day, and in ♋ the shortest.  
The manner of fashioning the Index, of placing the Dial, and using it, is the same as was showed before.   

How to make the like Hours and Azimuths by an equally divided Circle, upon all Plains whatsoever.  
YOU are first to find these three things. 1. How much, and which Pole is elevated above your plain, which is equal to the Latitude of your plain. 2. The position of the (usual substylar or) proper Meridian of the plain. 3. The difference of the Plains Longitude from your own, or the angle at the Pole made between yours and the plains Meridian. And when these things are had, the work will be like that which was before. For the way will be twofold▪ as there it also was.  

The first way.  
Take half the compliment of the plains Latitude, and add it to the plains Latitude itself, the sum will give the elevation of the Index above the plain. This Index is to point towards (but not into) that Pole which is elevated above the plain.  
The other circumstantial things about the Index are as those before, if instead of the line of 12 there mentioned, we here take the proper Meridian (or Substilar drawn in its right position) of the plain.  
The limitation of the Zodiac must be somewhat as before. That is, you must count the Semidiameter of your Dial Circle to be as the Radius. And to that Radius, 1. Either find the Sine of the elevation of the Index, and make that Sine a Tangent of the same elevation of the Index; then to the Radius of this Tangent find the Co-secant of the former elevation, the same Co-secant shall be the Radius or decimal Scale whereby to insert those days or Ellipticall degrees set down in the Tables Pag. 4, 5, etc.  
2. Or else to that same Radius of the Circle, find the Co-sine of the elevation of the Index, and to this being made a Radius, find the Co-secant of the Indices elevation, and so shall this Co-secant be the Radius or decimal Scale as before.   

The second way.  
The altitude of the Index above the plain is the half compliment of the plains Latitude. The Index must (in this way) point towards (but not into) that Pole which is depressed under the plain (I mean toward that Coast of the world, but must still remain above the plain.)  
The standing and motion of the Index, with the charactering of the Zodiac, and other circumstantial matters, as of the Cock for one Index, and the usage for the hour, must be agreeable to the former rules given before.  
The limitation of the Zodiac must be done by the two former rules given (in general terms) in this Page.   

How to divide the Circle of hours.  
It must be divided into 24 equal parts, and those of the 24 taken into use as will be serviceable, the rest left out. Every of these 24 parts may be divided into 15 gr. or into quarters and half quarters (for here in such plains there will be no Azimuth showed for the place, nor well for the plain itself, and therefore it is best to omit it in these altogether.) All the doubt will be where you must begin your division.  
From the intersection of the proper Meridian (of the plain) with the Circle, set off (upon the same Circle, and to the true Coast) an ark equal to the plains difference of Longitude. And from that point so inserted, (which you must suppose to be the point of 12) you are to begin the division of your Circle. The numbering of the hours on both sides 12 will not be hard to find, in respect of their course, for the course of the shadow of the Index will give that.   

NOTE  
1. In all Polar plains (such as our upright East and West, etc.) the Index in both these cases or ways (here before given) will be elevated 45 gr. and the Radius of the Dial Circle will be the Radius or decimal Scale by which the Zodiac Scale is to be inserted out of the Tables, Pag. 4, 5, etc.  
2. If at any time the Zodiac prove longer than the Diameter of the horological Circle, and so the Index do go without the Circle; at such time as it is without, the shadow of the Index will go one way till it touch the Circle, and then back again the same way it formerly came.    

A brief DEMONSTRATION of these Circular ways of making DIAL'S.  
THe Circular ways have dependence, and are deduced out of the precedent Ellipticall ways; and the Cases are but two, wherein the same may be done upon any Horizon or plain, as may be perceived by the former Precepts, wherein the altitudes of the Indices above the plain are ever made to be either half the compliment of the Plains Latitude, or else (the sum of the Latitude and that half compliment, which sum is equal to) the compliment of that forenamed half compliment.  
According to these two cases here are two Schemes fitted. The projections are made upon the plain of the Meridian Circle P Z H X. P is the Pole, Z the Zenith of the place, H A the Horizon of the place, A A the Equinoctial Circle, D A and R A are drawn in the middle of Z P and A H or A O. D B C X represents one Azimuth or Vertical Circle proper to the plain R A, and by that one all the rest of them may be understood. D A represents the Index in both the former Cases, and R A the horizontal plain (not of the place but that horizontal plai● which is) properly belonging to the Index or Zenith line D A. The first Scheme shows the first of the two former Cases, the second shows the second of them. Their Demonstrations will be both one. For the question in them both is, How to these inclinations of the Index, the Dial, or horary line, that falls upon the horizontal plain of the place, comes to be a perfect Circle.  
First, therefore consider, that in all the Ellipticall Dial's, the horary line hath relation to the Equinoctial Circle, and to the Index or line that is to give the shadow (whatsoever the Superficies be upon which the projecture is made, whether plain or otherwise it matters not, as appears before, but here we deal only with plains, because Ellipses and Circles are plain Figures.  
Secondly, consider, that through the several hour points of the Equinoctial Circle, right lines are to be supposed to pass infinitely extended, till they meet with the plain whereon the projecture is to be designed. Which issuing out of these infinite lines (if they be regularly clothed about with an inflected Superficies must comprehend a cylindrical concave, either round or compressed according as the forenamed infinite lines are either perpendicular to the plain of the Equinoctial, or not perpendicular to it.  
Thirdly, These parallel lines so drawn through the hour points of the Equinoctial, parallel to the Index, must all fall perpendicular to the horizontal plain which is properly belonging to the Index as a Zenith line. And the same lines upon that proper Horizon do always make an Ellipsis, except only these two Cases. First, If the Horizon be the same with the Equinoctial Circle, and the Zenith line or Index the same with the Axis of the world, for then those lines do there make a perfect Circle. Secondly, If the Horizon be a Polar or Meridian Horizon, and the Index or Zenith line fall into the Equinoctial (having no Latitude from it) for then those lines do all fall into (or coincide with) the plain of the Equinoctial, and consequently do make (upon their proper Horizon) a direct right line: ●ea more, because the said lines do lie all upon the plain of the Equinoctial, and are all drawn out of the equal divisions of the Equinoctial Circle, and besides are all parallel one to the other; therefore they make (in this case) a double line of Sines, as is said before in the fifth note Pag. 37. Now to come to my intended purpose, which is, to prove 〈◊〉 Indices so laid (as is mentioned before in the two precedent Cases) will require (upon the horizontal plain of the place) a true Circle and not an Ellipsis.  
Suppose therefore D A to be the Index (according to one of the two precedent conditions) then must R A be the proper Horizon to that Index. A●d according to the former Doctrine, if A, and B, and A, be taken for three of the hour points in the Equinoctial, and Azimuths drawn through them from D the Vertical point of the Index, than I say (in these two cases and not otherwise) look how the Equinoctial is divided by those Circles, in the same manner is H O the Horizon of the place divided. But because the Equinoctial is ever equally divided in respect of hours, therefore the Circles from D to X (from the Zenith of the proper Horizon R A to the Nadir, which are Azimuthall Circles) shall cut the Horizon of the place (H O) into equal parts. This, I say, is true, Because D A B and X H C are equal Spherical Triangles in three of their quantities [For D A is equal to X H, and the angles at D and X are equal, as also the right angles at A and H] and therefore equal in all the rest, if like be compared to like: that is, H C is equal to A B; and consequently A C equal to A B: so also C X equal to B D. And if so, than the Chord B C must be parallel to the Index or Zenith line D X, and so A H is parallel to D X, and so contrarily (in these two Cases) right lines drawn from the hour points in the Equinoctial Circle, parallel to the Index, must cut (upon the Horizon of the place) equal parts, equal to one another, and equal to the hours upon the Equinoctial Circle. That is to say, they cut the horizontal Circle itself into 24 equal parts or hours. And consequently (in these two Cases) those forementioned right lines must design (the horizontal Circle itself, that is) a Circle upon the horizontal plain of the place, though upon the proper Horizon itself (R A) they must design an Ellipsis (falling perpendicularly thereon) according to the third observation in the 128 Pag. And from that Ellipsis these lines B C, H A, may be conceived as surgent lines (such as are mentioned before Pag. 113) containing a compressed Cylinder, which Cylinder being cut two ways, (that is subcontrarily) will again reveile the original Circle (at least in one of the two Sections) from whence itself and the surgent lines took their form and places.  

NOTE.  
1. The same reason holds in all plains, that in them also there may be found such a Circular Horologiography. For they are Orisons to some place of the World or other, and are therefore capable of the same accidents that the forementioned Horizon of a particular place is.  
2. That these Cases are the same (in a manner) with those §. 5. For here a new Zenith line (as D A) is found to H O the Horizon of the place, such a Zenith line (or Index) as may lie equidistant from the Horizon of the place H O, and from the Equinoctial Circle A A: And there there is a plain set up which may stand so, as that the Zenith line of the place may stand equally distant from the Equinoctial Circle and this plain. In both, the Index shands equally distant from the Equinoctial, and from the plain whereon the hour points are to be inserted.   

For the limitation of the Zodiac.  
The first work in the limitation of it is for the Ellipsis that should be described upon the proper Horizon, and the ground of that is the same with what was before delivered Pag. 75. The reason of both will be plain enough out of the foregoing Diagrams. For the Zenith line D A, if it were to move upon the proper Horizon R A, it must move in the proper horizontal Zodiac, as formerly it did, Pag. 17, 18. Which Zodiac is to be made as before, and here in the first work is showed: the reason of the division of it is given in Pag. 61, 62. Here is only to be showed the reason of the enlargement. If the Index were set upon the proper Horizon R A, than it must move thereon (perpendicularly) from A to S suppose. But it moves not upon that but upon H O the horizontal plain of the place, and yet to keep the same distances in motion from the first position of it in D A X, that is, it must move parallel always to D A, but not upon A S but upon A H, which to do, it must not have the parts of the Zodiac in A S (to guide upon H O) but proportional ones to them, that is, instead of A S must be taken A m; which is all one as to say, As A S the Radius, to A m the Cosecant of S m A, or D A O, the Indices elevation (above the Horizon H O;) So are the parts of A S the prime or fundamental Zodiac, to the parts of A m, the secondary or enlarged Zodiac.  
Thus much of Circular Dialling.    

APPENDIX.  
SEE the Figure in page 120. To that Zodiacs length you may find such a Radius as shall be thereto justly competent so as to make the Ellipticall horizontal Dial R T S; and having made that Dial, the upright Index D A will give the hour upon it, as the slope Index A C will give the hour upon the Circle. And these two being several Indices and dials must set themselves as the other do, and will not stand right till they agree. Only this cannot be so punctual, because the two Indices A C and A D be no● so far distances in the other horizontal Di●ll. But the Azimuth will here be still showed, as was mentioned before.  

Having the Zodiac limited, how to find the two extreme Diameters of the Ellipsis.  
The length of half the Zodiac is always to be esteemed the Tangent of 23½ gr. Now because we here suppose this Zodiac already made, and to be intended for an horizontal Dial; therefore, Either thus,  
1 As the Tangent of 23½, is to the Tangent of your Latitude; So the length of the Semizodiac upon your Dial, to the Sine of your latitude, which is the shorter Semidiameter A T.  
Or else thus,  
2 As the Tangent of 23½, is to the Radius or Tangent of 45 gr. So is your known and limited Zodiac, to the Cousin of your Latitude. The Radius, to which Cousin is the larger Semidiameter A W.  
Thus having fitted your Ellipsis (to the Zodiac) you may describe and divide it in such manner, as was formerly showed, pag. 1.  
FINIS.     

RECTILINEAL OR DIAMETRAL Horologiography.  
SHOWING The manner how to describe the Hours upon a finite right line, or the Diameter of a Circle, and to fit a movable Index unto the Dial so described.  
Invented and written by Mr. SAMVEL FOSTER, Late professor of Astronomy in Gresham-Colledge.   

RECTILINEAL OR DIAMETRAL Horologiography.  

Concerning Dial's made upon a finite straight line, with a movable Index.  
THis may seem no difficult nor new thing, since the most vulgar known way of protracting Dial's is either by one contingent line (which is a right line;) or else by two Tangent lines, as by the Sector is usually done. But it must be considered, that the contingent line is infinite, and this line (here mentioned) is finite. Again, that other way by the Sector, is not by one finite line, but by two at the least; whereas this is by one finite line▪ Some difference there is. And besides; that other by Tangents may be as well done upon a single crooked line, this cannot.  
This straight line is mentioned before, that in right Orisons or Polar plains the Ellipsis shutreth up into a straight line. And it is true, that this falls out to be so, only in right Orisons (if the Index stand perpendicular to the plain, as is before intended) which cases happen but seldom, and to some plains only. But here my purpose is to show how it may be done to all plains whatsoever: observing only this one condition; always to set the Index so, as to point up into the Equinoctial Circle: for so it must do, and will not be done any other way, according to this form of casting the hours.  
There are two Cases which will offer themselves.  
The first is, when the Index pointeth up both into the Equinoctial Circle, and likewise into the plains proper Meridian Circle.  
The second is, when it points up into the Equinoctial Circle of the heavens (for so it must ever do) and into some other Meridian Circle, such as is not proper to the plain.  

¶ 1. How to make a Dial upon a finite straight line, drawn upon any plain, the Index being so ordered that it point up into the Equinoctial Circle, and into the plains proper Meridian.  
IN this Case, the Index is to lie just over the proper Meridian of the plain, and is to be elevated so much above that line as the height of the Equinoctial above that plain comes unto, and must point up into the Equinoctial Circle itself.  
It must therefore here be supposed that the situation of the plain, with such things as are thereon depending, are known. Such are these that follow. 

1 The plains reclination or inclination and declination.  
2 The position of the Substylar, or plains proper meridian  
3. The elevation of the Pole above the plain.  
4. The plains difference of Longitude from your Meridian.    
These things are to be found by the common ways of making of Dial's upon plains, which need not here be repeated, but shall be supposed to be already known, and placed in their true positions upon the plain. Let A B be the proper Meridian of the plain truly placed, and A D the Index standing over it, and pointing up into the Equinoctial. Let the degrees of the difference of the plains longitude be C L, and let the point L stand from C upon the right Coast according to the exigence of the plain. Then for drawing and dividing the line that is to give the hours, do thus.  
1. Supposing A to be the point of the Equinoctial (in the zodiacal motion of the Index) through that point draw R A S perpendicular to B A C the proper Meridian; and upon A as the centre, and to any convenient extent, describe the Circle R C S, in which, C L is to be set according to the difference of Longitude, as was before mentioned. So that L is the point from whence you must begin to divide your Circle, for from L is the point of the hour 12 deduced. Therefore,  
2. Divide your Circle from L into 24 hours at a, b, c, d, e, f, etc. l, m, n, o, etc. and from each of these points let perpendiculars fall upon the diameter R S, as L m, a, bp, etc. the same Perpendiculars shall divide the Diameter into the hours required: m shall be the point of 12, because it falls from L, the point of the difference of Longitude: n and o shall be the next hours to 12 on both sides: p and q shall be the second hour points on both sides 12, and so the rest in their orders.  
But then in these Dial's, when the difference of Longitude falls not just upon an hour (as very seldom (and then but casually) it doth) the line R S will be cross divided as you see it in this example, for e R u is one hour, and unequally broken and set upon the Diameter R S: so t S o is the space of 5 hours; and the points t and o stand cross or distant one from the other, and are not coincident; which I say must always happen when the difference of Longitude is not equal to one or more just full hours.  
3. In these cross hours (for so I may call them) you may divide each side of the line severally, as thus:  
Or else you may make a double line in this manner, supposing the fiducial line to be the white that lies entire and undivided betwixt them.  
This kind of Dial will be somewhat difficult in the use, because of this cross division of the hours. Yet I thought best not to omit it, because it is the ground of the next Case which will be more useful.   

Concerning the Zodiac and the Index.  
FOr the limitation of the Zodiac, let A R or A S be as Radius. To that Radius let A F be the Tangent of 23½. This Tangent is to be augmented according to the proportion that the Radius bears to the Secant of the compliment of the angle D A B, which is the elevation of the Equinoctial above the plain: That is,  
As the Radius, To the Secant of the Poles elevation above the plain (which is the same with the said compliment;) So is the Tangent of A F, to the line A B suppose. Therefore now A B is the length of half your Zodiac, and must be taken as a Tangent of 23½ gr. (or as a Scale of 4348/10000 equal parts) by which you may insert the Months or Signs as is before showed, Pag. 4, 5, etc. These must be set on and charactered as the nature of the plain shall require them to be ordered.  
The Index though it move, yet must always lie just over the proper Meridian A B, and must also keep the angle D A B unchanged, and must be moved from B to A, and so to B, as the time of the year shall require. Or the Index may stand and the Dial move upon the line B B, as the manner of these kinds of Dial's may be, and which have been too often mentioned. The second case follows.   

¶ 2. How to make a Dial upon a finite straight line, drawn upon any plain, the Index being so ordered, that it point up into the Equinoctial Circle, and into a just hour point in the same Circle.  
AS before, so here, it must be supposed that the situation of the plain with the other things (beforementioned) which depend thereupon are known and placed upon the plain. And the matter must be so ordered, that the Index must both look into the Equinoctial Circle, and also it must point to some just hour therein, that so one division of the Dial line may very well serve for accounting the hours both ways, the whole hours and parts upon one side justly corresponding to the just hours and parts upon the other side. This will be for more expedite use than the former way could be, and no such trouble nor confusion in the numeration of hours and parts of hours, as will appear in the next Example.  
In this Case that we are now going upon, we must consider these things.  

1. How the Index is to be ordered, so as that it may look up into some just hour point of the Equinoctial?  
2. How the line of the Dial or hours (the horological line) is to be placed, and limited and divided?  
3. How the Zodiac is to be placed and limited?   
¶ First, for the Index: The best way will be to find what Declination and Reclination (in respect of the Meridian and Zenith of the plain) the hour point (into which the Index must point) hath. And that is done by these two Rules.  
1. As the Radius, to the Sine of the Equinoctials altitude above the plain:  
So is the Co-sine of the distance of that hour point from the Meridian of the plain, to the Co-sine of the Reclination; or to the Sine of the elevation of the Index above the plain.  
2. As the Radius, to the Sine of the Poles altitude above the plain:  
So is the Co-tangent of the former distance (of the intended hour point from the Meridian of the plain,) to the Co-tangent of the declination of the Index from the Meridian of the plain. Or thus, 2.  
As the Sine of the Poles altitude above the plain, to the Radius, So is the Tangent of the former distance, to the Tangent of the declination of the Index from the plains Meridian.  
Thus,] suppose A B were the proper Meridian of the plain, make the angle B A C on the right Coast, and equal to the declination of the Index (now found) and elevate A D just perpendicularly over A C, to the angle D A C equal to the elevation of the Index above the plain, and so you shall place the Index in a just position, that it shall look up into a just hour point of the Equinoctial. That is the first thing.  
¶ Secondly. The horological line is to be set for position as it was before in the precedent Case, namely, perpendicular to the proper Meridian, as in the former and this Scheme you see R A S.  
Then again, this line which in the former figure was equal to R S the Diameter of the Circle (or two assumed Radiusses) must now in this case be enlarged, according as you have ordered your Index to wry from the proper Meridian of the plain to any other just hour; as in this figure it is made to be distant from that proper Meridian unto the hour of 12: that is, it is supposed to be turned so, as to look into our 12 a clock point in the Equinoctial: which point of 12 in the former figure was noted at L, and so in this figure let it be supposed again. I say, let it be supposed that (because the former horological line was crossly divided, and therefore untoward for use) this way that is now spoken of shall be to take away that inconvenience in use which was in the former. And therefore suppose the. Style or Index to be wried from the proper Meridian of the plain to the Meridian of the place, or to the point of 12. It may as well be put to any other just hour, as to 12, but be it so in this Example, and let the point be L selected from all the rest.  
From the selected point L, draw the diameter L A M, and from the selected point L divide the Circle into 24 equal parts or hours. Then from each two hour points correspondently distant on both sides the Diameter L M draw right lines, as G h and n O, and G O, which will be all parallel to the Diameter L M. Draw these lines (I say) and continue them until they meet with the horological line R A S, sufficiently prolonged both ways. By this draught thus made, the horological line R A S shall be both prolonged to h O, and also divided just as a line of Sines, so the parts of it are in the same proportion that the parts of the Diameter G A G are, which are Right Sines both ways from the centre, or Versed Sins from G through A to G.  
¶ Note that here (as formerly) you may assume the Circle of what extent you list, and shall find most fitting your purpose.  
It is plain then by this Example, First, that the Radius of the Circle A R, is to be enlarged, according to the secant of the Equinoctial distance between the proper Meridian of the plain, and that just hour (whatever it be) inro which the Index is made to point. As you see that ark to be here L C, or G R, or G S, whose secant is A h, or A o, in respect of the assumed Radius A R. And secondly, that h o thus enlarged is divided like a double line of Sines, as it ought to be.  
And for numbering the hours, if A be 12, and the Index look toward the South, then must o be 2, and the other o must be 6; and the first o, 10 at night. And h must be 9 in the morning; the other h, 6 in the morning; and the first h 3 in the morning.  
But if the Index had not pointed up into 12 at L, but into some other hour as P, then must a Diameter have been drawn at first from P to the centre A quite through, and all the hours parallel to P A, and then the horological line would have been thus numbered with 7 at the two extremes, and 1 in the midst at A.  
And here is one special thing to be noted (as formerly Pag. 125) but this here is always so, and in all Oblique Orisons. That in our Climates where the days are in Summer longer than 12 hours, the shadow of the Index doth really go forward and backward. As to make it plain. If the Dial were described upon an horizontal plain (for instance) and it were our longest day, than the shadow of the Index would begin at 4 a clock (figured below the line) and would go thence towards the left hand to 5, 6, and 7. At 7 it would return back towards the right hand through 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6, until 7, and then being arrived at 7, the utmost point on the right hand, it will come back again towards 8 and 9 to the left hand. This for the placing, limiting, and dividing of the horological line.  
¶ Thirdly, Concerning the Zodiac how it must be placed and limited, consider of these things. The Zodiac (in this case of wrying the Index from the plains proper Meridian) must be augmented in length, and changed in situation from what it was in the former case where the Index looked up into the plains proper Meridian. Yet I will here first begin with that way so far as it goes, and then proceed to what is further needful. Let the Figure Pag. 137 be here repeated. To the Radius A R, the Tangent of 23 and a half is A F, and the same is there augmented thus.  
As the Radius, To the Secant of the Poles elevation above the plain;  
So let OF be supposed to be, To AB.  
So that if the Index were pointing directly to the Meridian point (in the Equinoctial) which is proper to the plain, than the business was at an end; for A B must be the length of the Zodiac, and standing square to R A S. But now we here suppose the Index to be wried from that point, as much as the ark C L comes too; which turning aside will both lengthen the Zodiac again, and turn it out of its perpendicular standing to R A S. Which thing is thus to be effected. To A F as a Radius, find the Tangent of the Poles elevation above the plain; suppose that Tangent were G H. Then say, As the Radius, to the Tangent of C L the quantity of wrying, so G H to X Z. Make B K equal to X Z, and parallel to A R, and draw A K, so shall A K be the length, and shall also give (in this position) the situation of the Zodiac. The manner how to divide it need not be repeated, because it hath been often set down heretofore, You must only remember to place this line upon that Coast of A B which is contrary to the Coast of the Indices deviation or wrying from the proper Meridian point of the plain: this work may either be protracted, or calculated, or wrought Instrumentally, as shall best be liked of.  
Another way I will here add to do the same work, which may perhaps be not inferior to the former. Let all things preparatory in the former Figure, be here again repeated, R A the Radius, and A F the Tangent of 23 degrees and a half to that Radius: then to A F as Radius, find the Sine and Co-sine of the Poles elevation above the plain: let F P be the Sine, and A P the Co-sine. Make A m equal to A P; and from m draw the line m n, making the angle n m C equal to the angle B A C Pag. 142, and standing on the contrary Coast, which is the angle of the declination of the Index from A C in this Figure, the proper Meridian of the plain.  
Then say, As the Radius, is to the Tangent of the Reclination of the Index (which is the Compliment of D A C (Pag. 142) the elevation of the Index above the plain,) So is the Sine F P to a fourth quantity, suppose the same to be D G: make m n equal to D G, and draw the straight line A ●, so shall A n in this Figure be equal to A K in the precedent, and the angle n A C be (here) equal to the angle K A B in the former. So that A n must be the length of the half Zodiac, and will lie in its true position by this work. This may be done either by calculation, protraction, or Instrument, as all other works may also be. Care must always be had to place these Zodiacs on the right Coast from F A C, and that the Zodiac be rightly divided and charactered according to the Seasons of the year, as they shall have respect to the nature of the plain.  
Or thirdly, it may be done thus. First,  
1. As the Radius, To the been of the Poles elevation above the plain;  
So the Tangent of the Equinoctial discession or wrying, To the Tangent of the Declination of the Zodiac; which is the angle n A C in this Figure, o● B A K in the former.  
2. As the Co-sine of the discession, or Equinoctial wrying,  
Is to the Radius; So is the Tangent of the Poles elevation above the plain,  
To the Tangent of another Ark.  
3. And, As the Radius,  
Is to the Secant of that ark (or as the Co-sine of that ark to the Radius;)  
So is the Tangent of 23½ gr. To a number.  
If now you make the Radius of the Circle first assumed, namely R A or A S to be a decimal scale, and out of that decimal scale do take this number, the same shall give the length of the Zodiac A n in this figure, or A K in the former. The division of this Zodiac now it is thus placed and limited, must be done by the Tables and directions in the 4 and 5 Pages. For A n must be taken as a Tangent of 23½ gr. or as the number 4348 of a decimal scale of 10000 parts.  
Thus much for this kind of Dial upon a straight line, both crossly and evenly divided.    

A DEMONSTRATION of the former way.  
IN Pag. 28, Note 5, there is mention made of Polar plains or right Orisons, that in them, the Ellipsis closeth up into a right line. That right line is here the line that we have to do withal. For we here suppose always that the Index pointeth up into the Equinoctial Circle; and the Index we always suppose to be a Zenith line, and therefore the Horizon proper to that Index as a Zenith line, must needs make right angles w●th the Equinoctial Circle, and must therefore be a right Horizon or Polar plain.  
When the Index is pitched in a true posture, all the hour points of the Equinoctial Circle must be supposed to have lines issuing from them all parallel to the Index. This is a general truth in the whole course of the Ellipticall Dialling, Wherefore,  
First of all, These lines will (in this case) fall all upon the plain of the Equinoctial; and because the common section of any two plains must be a right line, therefore it is that all these lines so issuing, will fall upon one and the same straight line, namely, whensoever the Equinoctial meets with or cuts any other plain. Hence comes it to be Rectilineal Horologiography, being performed upon a plain.  
Secondly, Because they are all parallel to the Index, therefore they must all be parallel to one another; and because they come originally from a Circle, and are parallel, they will all also keep distance one from another as the Sins in a Circle do.  
Thirdly, Because the Index moves always in a parallel position to itself, and would show the hour upon the circular divisions of the Equinoctial Circle (at all times of the year, that is, when this Equinoctial is made to represent all other parallels of Declination) it must follow that the Index when (the shadow of) it falleth upon any one such point, it must also fall upon the whole line that issueth through that point. The reason is, because the line is always parallel to the Index, therefore is it touch one point it must shadow the whole.  
Fourthly, Hence will follow, that any line drawn upon the plain of the Equinoctial, crossing those forementioned parallel lines, shall be a line of Sines (or a double line of Sines, if you will rather express it properly) and if it pass through the centre o● the Equinoctial Circle (where the Index also doth always pass) then is such a line one of the Dial's that we here are to deal with. Because every plain is taken for a great Circle cutting the Equinoctial in the centre. And the divisions of this line (the Cynical divisions) must be the hour points, as is said before.  
Now all that we have here to say is the making good of the former rules of limiting out and placing the hour line, the zodiacal line, and the Index. If we therefore (in all cases) suppose the Index to be a Zenith line, and then an Horizontal plain to be placed perpendicular to it, this plain must be the proper Horizon to that Index (or Zenith line) and must (in these Cases here handled) go through the Axis of the world (a part of which I must call the primary Zodiac) and will cut the Equinoctial plain in a right line placed perpendicularly to the Index, which I call the primary horizontal line.  
And again, all that is done in other plains (not perpendicular to the Index) must be imagined primarily (or originally) to be deduced from this proper horizontal plain.  
First therefore, for the hour line. If the Index lie in the proper Meridian of the plain (as is supposed in the first of the former Cases) than the Diameter of the Circle that is at first assumed, serves for the horological line without any alteration, because the proper Horizon, and the plain upon which you work, do (both of them) cut the Equinoctial in one and the same line, which is to be the horological line. And therefore in plain reason, so much of the Equinoctial as the Index is wried from this proper Meridian of the plain, so much of the same Equinoctial must the proper Horizon be wried with it, and consequently, so much will be between the proper horizontal line and the horological line of the plain, but still both these lines will continue upon the same plain of the Equinoctial, and consequently they must in both cases be one and the same line, namely that which is made by the projection of the Equinoctial Circle upon the plain, on●y it must b● augmented according to the proper horizontal lines departure from the section of the plain with the Equinoctial, that is, according to the wrying of the Index, as is plain enough by what is said in the 142 Pag▪ And so for the division of that line in the same Page. For G G and h o, do both lie in one and the same plain of the Equinoctial: and all the projecture must be made parallel to the Index, whence h A is the secant (to G A Radius) of R G or C L, the wrying of the Index.  
Then for the Zodiac, you are to deduce it from the Axis which is perpendicular to the Zenith line or Index: and from so much of it as containeth the Tangent of 23½ gr. to the Radius G A. Now though the Index do keep the proper Meridian of the plain, yet the Axis is elevated above the plain. And because the projecture of that Tangent or part of the Axis must be parallel to the Index, therefore look what elevation the Axis or Pole of the World hath, to such a Secant of the Radius equal to the Tangent of 23½ before mentioned must the Zodiac be extended. By which means the Index keeps the same situation and distance from the Equinoctial plain that it would do if it moved according to the proper and primary Tangents upon the Axis, and no otherwise. And so doth the former enlarging of the horological line produce the like effect, namely, that the Index doth upon that enlarged line show the same hours that it would do upon the proper horizontal line.  
Then for the Zodiac to a wried Index, it is both displaced and enlarged too, according as right lines drawn through the divisions of the Axis, and parallel to the Index, would make a projection of it. According to which, there are three ways given.  
The first makes A F equal to the Tangent of 23½ to the Radius R A, to be Radius, and to that Radius finds F H, and to that again as Radius, find H Z the Tangent of wrying. And so A Z is the Semizodiac placed and enlarged.  
As F P Radius, To P Z the Tangent of the angle P F Z;  
So is F P in the former known length (found to A F Tangent of 23½ gr.) To P Z.  
Therefore Z is the point terminating the extremity of the projected Zodiac, and so A Z must be the Semizodiac, according to what was delivered in the second way Pag. 147.  
The third way resolves a Spherical rectangled Triangle, made between the proper Meridian of the plain P A, the plain itself A B, and the Meridian in which the Index lieth, or into which it pointeth, P B. In which three quantities are known. P A the altitude of the Pole above the plain. The angle at A a right angle. And the angle at P equal to the ark of the Equinoctial or wrying of the Index from the proper Meridian of the plain. And there is first required A B.  
As the Radius P n, To the Tangent of (n m =) P;  
So the Sine of P A, To the Tangent of A B, = n A C, Pag. 147,  
Than secondly, the ark P B is required, and that 〈◊〉 wrought by the second work, Pag. 148, which in the second Figure Pag. 153, may be understood to be the angle F A Z, Then because A F Z is a plain Triangle rectangled at F, and the angle F A Z is now known, and the side A F (equal to the Tangent of 23½) it will be required to find A Z in the same parts. Therefore in the third work (Pag. 148) it is said,  
As A F, Radius, To A Z, Secant of F A Z;  
So is A F in parts, (as a Tangent of 23½) To A Z in the same parts.  
And so the Zodiac is both placed and limited.   

APPENDIX.  
IT hath been forwerly showed (in the VIII. Section) how to make the Ellipticall Dial to any Index however placed, and to any Superficies flat or curved. That Doctrine will show how to project this kind of Rectilineall Dial (and the former Circular Dial too) only I shall add a word or two to show that that way will produce a single line in this case, and such a line as must answer to a double line of Sines.  
1. See the 6 and 7 Propositions Pag. 95, 96. The first of them is to find the horizontal spaces. And in this case (because the Index looks always up into the Equinoctial Circle) the Latitude of the Horizon proper to this Index line, is nothing. So that,  
As the Sine of the Poles elevation, which is nothing, Is to the Radius;  
So is the Tangent of each hour from the proper Meridian of the Index,  
To the Tangent of an infinite length: [For so it must be because the first term is 00.]  
Which infinite Tangent is belonging to 90 gr. therefore it will follow that all the hours and each of them are 90 gr. from the proper Meridian of the Index: that is, that they are all coincident into one line, which is perpendicular to the proper Meridian of the Index. This shows that this Dial will be a single line, and that by the former general Doctrine, Section VIII.  
2. Then for the second thing, that it will be a line of Sines, or (though a curved line, yet) to be thence deduced or divided, it will appear by the seventh Proposition Pag. 96. Because,  
As the Radius, To the Co-sine of the Poles elevation; which here is the Radius;  
So the Co-sine, etc. To the same term again.  
For if the two first terms be the same, than the two latter must be equal too. And since the third terms are a line of Sines, the fourth terms (which are to divide the Dial line) being the same with the third, will be also such as will make a line of Sines. So this is made good likewise.  
And there may other ways be found to effect this same thing upon a curved Superficies. As first to project the Equinoctial line or section with the uneven supersicies, upon the said Superficies: and when that is done, to lay the edge of some board or ruler, in the Superficies of the Equinoctial, with the Index lying in the plain of the Equinoctial, will (by the former Equinoctial line drawn) direct you in, and afterwards to proceed answerably to the nature of the Dial, etc. But what need I to mention any such thing thus obscurely. They that understand the former ways will be able to invent many more, and so I ce●se from further troubling myself or the Reader. One Proposition more must here be added, which conduceth to this Rectilineal Horologiography, namely:  

How to lay a line that shall point up to any assigned place or point in the heavens.  
THis is useful for the laying of an Index to any condition that shall be required: namely, so as to cut through any just hour, either in the Equinoctial Circle, or any other Circle in the heavens.  
The first work must be to find what Reclination and Declination the said point shall have in respect of our own Zenith and Meridian. This is an easy Spherical work, and therefore I shall omit it. Only note, that the Reclination here mentioned is the distance of it from our Zenith; and the declination is, what Azimuth from the Meridian the said point falls into.  
The next work is, how to lay or stretch a line to any given Declination and Reclination. The way that for the present I think on shall be this.   

For the hour points.  
You are here left to your choice to take which (of three ways) you like best. Either, 

1. These points in the Circle may serve for the hour points without any more draughts, and then this will be another Circular Dial, differing from those that were handled before in Circular Dialling.  
2. The perpendicular Diameter D E may have the hours inserted by drawing lines from each pare of hours equally distant from D and E hour points of the Circle, parallel to the Index C A, and so you shall have the Diameter D E divided like a double line of Sines (unequally) at 12, 11, 1, 2, 3, and 10, 9, 8, etc.  
3. You may likewise (if you allow not the two other ways) draw any line as F G, and by lines drawn through the hour points of the Circle, all parallel to the Index C A, the same line F G shall be divided as a double line of Sines, at 12, 1, 2, etc. and 11, 10, 9, etc. and will show the hour as the other do, and the two extreme parallels 4 F, and 4 G, shall limit the length of the line F G.     

☞ Upon this Circular way Note thus much.  
That though the Index A C should not lie upon a just hour, but otherwise accidentally, yet because the hour points are upon a Circle, the hours will be distant from each other, and not breed that distraction in computing that was before noted necessarily to fall upon the straight horological line. And so you are left at liberty to set your Index how you will, only so as ever to look into one and the same Meridian Circle. But you must then be careful to place some one hour line true, so as to answer to some just hour circle in the heavens, which upon this Equinoctial plain (when the Axis is once truly placed) will be easy. For if the Axis give a shadow, and you at the same instant time (noting the shadow) do observe the hour, that shadow will represent the same hour. If therefore upon that shadow line (as a Diameter) you describe a Circle, you may (from the intersections of the said line, with the Circle) set off any one full and just hour, and then from that just hour you may divide the Circle into the 24 hours. This equal setting of the Index is allowable upon the Circle, but not upon the right lines D E or F G, for in them there will be confusion of parts, which in the Circle are distant.  
Note further, that you need not make the Centre of the horological Circle just in the point A, where the extremity of the Index which came down upon the plain, but you may make it in any point where the Index A C will touch the plain, as at H, or in other like places.   

Concerning the Zodiac.  
You must account the Semidiameter D A of the horological Circle to be as Radius. And to that length (and none other) you must make the length of the Semizodiac A B upon the Axis to be the Tangent of 23 and a half, and by that Scale of Tangents you must put in the Sins or months as hath been often showed before. And whereas I have hitherto said that the Index must move upon the Axis, and agree to a Zodiac thereon inscribed, it must be understood that in some sense this is necessary, but in some sense not so. As thus, The Index D A or D E, must always move perpendicular 
A X the Axis. A C the Zodiac. A D the Index, A B another supporter differing from the Axis.  to the Axis A X. But it is not tied to move upon the Axis itself. It may move according to any line that lies in the same plain with the Index D E and the Axis A X, that is, it may move according to the line A B, which we here suppose to be in the same plain or Meridian with A D and A X. This condition is always requisite, and it must ever keep one & the same angle with the ●ight line A B, and such an angle that may keep it always perpendicular to the true Axis, that is, parallel to the Equinoctial plain. And this being observed, you may insert the Zodiac upon the line A B, but it must be proportioned from the Axis A X, thus; As A C Radius, to A E the Secant of the angle C A E, so is A C the Tangent of 23 and a half (in respect of the Radius of the horological Circle,) to a fourth quantity; the length whereof gives A E. And now this line A E must be esteemed as a Tangent of 23 degrees and a half, and by it must the Semizodiac or half year be put in, as hath been showed formerly. So this is done also, For the Index will move upon A E, according to the parts upon A C, as it is bound to do. Only note that these Dial's upon the Equinoctial plains can serve but half a year upon one side▪ & therefore for the other half year you must do the like work upon the under side of the Equinoctial plain. The work I say is like without any real difference, and therefore there will need no more words concerning it.    

For DEMONSTRATION of these things.  
THere needs nothing to be added more than what hath formerly been done for Rectilineal Dial's. For what is done there in the proper Horizon (proper to the Index I mean) by a line of Sines, is here expressed by the line of Sines D A E, Pag. 162: only there, all the lines issuing from the Equinoctial hour points did meet with the proper Horizontal plain, and did thereon create a single line of ●80 Sines, and so the Equinoctial Circle standing towards the eye (being infinitely removed from it) edgewise is projected upon his own diameter, and appears nothing else but as a Diameter (or straight line) to the eye. But here the same forementioned parallel lines issuing from the hour points of the Equinoctial Circle, and flowing out upon the plain of that Circle, do so appear in full view to the eye, because the eye stands so as fully to view the said plain and not looking edgewise upon it. When the plain stood edgewise to the eye, the eye did project it into the Diameter, but now we conceive the eye to stand only to look what was effected upon the plain by the former Projection.  
And further. Look what projection was formerly made (in this case of the Indices pointing into the Equinoctial Circle) upon the Indices proper Horizon, and was there received by a straight line, the same is here done by drawing the Diameter D A E, Pag. 162, which line must be supposed to be the common section of the Indices proper Horizon with the Equinoctial plain. And so in effect both these Cases are nothing different.  
Then for the line F G, the parts of it, and the points of the Circle, and the points of D E must necessarily all fall into one hour line, and all show the true hour (whichsoever of them you take for your horological line) because as the lines issuing from the hour points of the Equinoctial are all parallel one to the other, and all again to the Index, therefore if any part of the shadow of the Index do fall upon any part of one of those lines, the whole shadow must fall upon the whole line, and consequently the correspondent points upon D E and F G and the Circle, will all jointly, or each of them severally, give the hour of the day. Other things are plain enough.  

Of the parallelisme of the Index and Plain.  
It must further be noted, that an Index may be made to point up into the Equinoctial Circle, and yet also be parallel to any plain whatsoever. For every plain cuts the Equinoctial Circle somewhere: Now if an Index be made to lie in that common section of these two Circles or plains, then will it lie parallel both to the Equinoctial plain and to the other plain, and it may be made to move always parallel to them both, if it be moved at right angles to any line that shall stand perpendicular to the said common Section.  
But for all this, there can no such Dial (as we have now spoken of) be made upon any other plain besides the Equinoctial. I say there can no parallelisme of the Index to the plain be (in this kind of Dialling) allowed, more than what is said may be done upon the Equinoctial plain.  
For though the Index may be parallel to the Equinoctial and plain both, yet the Equinoctial itself must make angles with all such plains. And because the Index is parallel to the common Section, therefore all the hour points projected by parallel lines to the Index, must be upon the Equinoctial plain itself, and must run parallel to the line of common section: that is, they will all run so, as never to meet so much as with the line of common section (which line is upon the Equinoctial plain) much less ever to meet with any part of the plain itself, which is every where distant more or less from the Equinoctial, except only in the line of common section. So that there is no more to be said of this subject.  
FINIS.     

ELLIPTICAL Horologiography.  
SHOWING How upon any plain to draw an Elliptical Dial, to an Index set any way, by Spherical (and not Projective) work.  
Invented and written by Mr. SAMVEL FOSTER, Late professor of Astronomy in Gresham-Colledge.  
London, Printed for Nicholas Bourn. 1654.   

ELLIPTICAL Horologiography.  
HItherto we have had the whole business of Elliptical Horologiography, so far as that more cannot seem to be thought of, or required. Yet, Because to any Index set either on purpose or casually, the hours have been (formerly) found out by Projection, I thought fit here to add another way (but for plains only) that shall not need that projective manner of working.  
And so the Problem will be this:  

How upon any plain to draw an Ellipticall Dial, to an Index set any 〈◊〉, by Spherical (and not Projective) work.  
The substance of what I shall here say is most of it delivered before, only I must now show what Propositions a●e to be referred hither, and how they must be used.   

Things that are pre-requisite to this work.  
1. You must know the situation of your plain, that is, what Declination and Reclination it hath in respect of your own Horizon. This is to be done the ordinary way, as in all plains is usual.  
2. You must know the Declination and Reclination of your Index. That is, if the Index be set casually you must then find out the Declination and Reclination thereof, which is showed, Pag. 80. how it must be done, namely, by observation from the Sun, and cannot be otherwise performed. Or if you choose a Longitude and Latitude wherein to lay your Index, then must you (by that Longitude and Latitude given) find what Declination and Reclination is due thereunto, in respect of your own Horizon: and according to this Declination and Reclination you are to lay your Index. See the Schemes Pag. 83, 84, for by those or the like you are to conceive of your Spherical work in this kind.  
3. Having thus found the situation both of plain and Index, in respect of your own Horizon, you must then find what situation the Index hath in respect of the plain; that is, what Declination it hath from the proper Meridian of the plain, and what Reclination it hath from the Plains Zenith. This Proposition hath not been handled before, but must be put here in this place.  
The next work is to find what situation an Index (set up upon the plain) shall have to the plain, and the fundamental lines thereon described, viz. the Vertical line, and proper Meridian.  
I suppose first, that the Declination and Reclination of the Index (in respect of your own Horizon) is known by the former things already spoken of, and from thence you may gather what is here required. Thus  
Let the Index point up to X: and let Z X (the reclination from your Zenith) be known, and S Z X (the declination from your Meridian) be also known. Now therefore in the triangle T Z X, there are three things known.  
1 T Z the Compliment of the Plains reclination in respect of your Horizon.  
2 Z X the compliment of the Indices reclination in respect of your Horizon.  
3 Thirdly, the angle T Z X, which is the difference of the two declinations, namely of the plain and of the Index. By these all the rest will be found; as namely,  
1 T X, the reclination of the Index X from T the pole of the plain, which is called the proper reclination of the Index.  
2 Z T X, or V Y, or rather V M, which shows how much the 
* Subindicall line I call that line upon the plain which lies perpendicularly under the Index, as the substilar doth under the Style. I mean perpendicularly in respect of the plain, not of the Horizon.  Subindicall line lies from the Vertical line of the plain (in degrees of a Circle described upon the plain) and consequently, because P T Z was known before, therefore P T X is now also become known, which shows the declination of the * Subindicall line from the plains proper Meridian, which may therefore be called the proper declination of the Index.  
3 The angle Z X T (if it be of any use here) shows how much (in Azimuthal position) your Zenith Z lies from T, the Zenith of the plain upon that Hotizon whose Zenith line is the same with the Index, which is therefore the proper Horizon to the Index.   

Or if you would set an Index so as to fit it to any assigned difference of Longitude, and to any assigned Latitude also, you must work thus.  
LEt X P S be the assigned difference of Longitude, and P X the compliment of the Latitude, into both which the Index shall point up in X. Because the plains difference of Longitude Z P T. is known before, therefore the angle T P X is known. And P T the compliment of the plains Latitude is also before known. Therefore in the triangle T P X these three things are known, viz. P X, P T, and the interjacent angles T P X. By which three you may find all the other quantities: namely, P T X, or B Y, the departure of the Subindicall line from the plains proper Meridian: T X the proper reclination of the Index X from T the pole of the plain, or the compliment of it (X Y) which is the elevation of the Index above the plain just over (perpendicularly over in respect of the plain) the Subindicall line: with the angle P X T (if there were any use of it) which shows how much the pole of the plain declines from the proper Meridian of the Index. I say, it shows this declination upon that Horizon (whatever it should be) which shall be properly belonging or perpendicularly placed to the said Index as the proper Zenith-line.  
Thus the situation of the Index (whether casually, or by election set) is found to the true coasts of the plain itself.   

What remains to be done.  
THe rest of the work for making the Elliptical Dial itself to the Index assigned, and for limiting and dividing the zodiacal Scale depends upon this one Rule. Namely,  
To project the hour points of the Equinoctial, and the zodiacal Scale upon the Axis (which zodiacal Scale upon the Axis itself is made by the Tangents of 23 gr. and an half, to the Radius of that Sphere wherein you shall imagine your Circles to be) I say. To project (all such points) upon the plain, by lines passing through each one of them, and all and each of the same lines to be drawn parallel to the Index given. This is the sum of what is here now to be done, and of what hath been done formerly quite through this Book. For all that is before effected upon any plain or curved Superficies, is nothing more than what I have now said; namely, a determining of the true places of every such (hour and zodiacal) point, as they shall be projected upon the dials Superficies, by lines going through them, and going parallel also to the Index that is to give the hour. For so, the Equinoctial Circle will very well supply every parallel of the Sun, the Index and horological line being rectified by the Zodiac in a true site one to the other, as hath been often mentioned before; And then the Index, casting its shadow upon the hour point, either in the Equinoctial or in the parallel Circle, must also cast it upon the whole projecting line, which passeth through the hour point, (because that line is made always parallel to the Index) and consequently upon that point of the hour, which (by the said line) is carried to the Diall-Superficies. And so for the points of the Zodiac, they are also projected in such manner, as that the Index moving in them (however they may seem to stand awry and out of order) shall always cut the Axis of the World (if it were there placed) and cut it according to the Tangents of declination thereon expressed. And this will serve for a brief demonstration of that work which is ●ow to come afterwards.  
In which we are first to bring down the hour points and Zodiac to the plain considered as an Horizon, and the projection of those points to be made into an Ellipsis, by lines all perpendicular to the said plain, that is, parallel to the plains proper Zenith line. Secondly, we are to project or transpose these first (both Hour and zodiacal) points from the primary Ellipsis into a secondary Ellipsis, such as the position of the true intended Index will require: And so also the points of the Zodiac must be dealt withal.   

How to make an Ellipticall Dial upon any Plain, to an Index perpendicular to the Plain.  
THe first work is done before, partly by Calculation, and partly by protraction, Pag. 25, 26, but more fully and better, Proposition 5, 6, 7, Pag. 92, 95, 96, by which you make two Tables, one of Horizontal spaces, the other of Altitudes. So that I shall not need to say any more of it. By that work (I say) you may make two Tables which may serve to prick down the hour points, which points you shall see to wind about the centre in form of an Ellipsis.  
1. It is first requisite that the proper Meridian of the plain be rightly situated upon the plain (as is there also expressly required. And it is again requisite that the proper line of the two six (proper I mean to the plain, considered as an Horizon by itself) be also drawn: which line (as in all Horizontal Dial's is usual) must be perpendicular (upon the plain) to the proper Meridian. And the points also of the two six must be carefully set on, whose distance from the centre is the longer Radius of the Ellipsis. The reason why these points of the proper Sins must be had, is, because the ●ew Ellipsis (which we now look for) must pass through them both. So also the just point of the proper 12 (or the section of the Ellipsis with the plains proper Meridian) must be with the same care designed upon the plain. And the Subindicall line must be drawn in its true position upon the plain, according to the proper declination of the Index.  
2. After this is done, you must call to mind what the Declination and Reclination of your Index is (I speak now of the proper Declination and Reclination before mentioned) as Pag. 172 is mentioned.  
3. Then from every hour point lately made upon the plain, you are to draw right lines, parallel to the subindicall line, and draw them both ways or on both sides of each hour point, because you know not which of them (as yet) you are to make use of for the pricking down of your New or Distorted Ellipsis.  
4. Take the larger Radius of your primary Ellipsis (that is the length from the centre of it, to the point of the proper six a clock) and (upon the Sector, for that Instrument is most fit for this use or practice) entering it upon the Tangent of 45 degrees, open your Sector to tha● length: then take over from the reclination of the Index (the proper reclination I now speak of) counted on the same Scale of Tangents, to the other leg, and reserve this length. That is, As the Radius, to the Tangent of the proper reclination of the Index; So is the larger Semidiameter of the primary Ellipsis, to a fourth quantity, which is equal to the ●●rmer reserved length, and to that large Semidiameter or Radius, is the Tangent of the Indices proper Reclination.  
5. This reserved length must again be entered from 90 in the line of Sines, and the Sector set according to it, and being so set it is fit for the rest of the work that is to ensue. Because this proportion we are to go upon: As the Radius, to the Tangent of the Indices proper reclination (which two terms are they that now do set the Sector,) so is the Sine of every altitude of the horary points in the Equinoctial, (which altitudes your Table of altitudes computed by the forementioned Proposition, Pag. 96, 97, will help you unto,) so is the Sine of every (horary-equinoctiall-points) altitude, to the Tangent of the reclination of every one of those several hour points of the Equinoctial Circle. Note that the reclinations here mentioned are, in respect, and from the proper Zenith line of the plain. And note also that these Tangents thus found, are to be estimated to the larger semidiameter of the primary Ellipsis as to their proper Radius.  
6. By these Tangents so found, you are to set on the new hour points, by which both the new Ellipsis is to be described and also divided. I say the new hour points are to be set on upon the lines drawn through each of the old houre-points parallel to the Subindicall line, and they are to be s●t on from each of the said old hour points in those lines. But which way, or on which side of them, will be still doubted▪ as was before mentioned § 3. And to take away this doubt, it will be best to consider the point of the proper 12 upon the primary Ellipsis, when the Sun is (any day) in that proper 12 point of the heavens, then must the upright (or proper Index of the plain) cast its shadow into the proper 12 point of the primary Ellipsis. Then further, supposing the Sun to stand still there, imagine how your slope Index is to lie from the proper Index, and by that you shall easily discern which way the inclination of that Index will wry the proper point of 12 from the primary Ellipsis, and consequently you shall be able (thereby) to judge upon which part of that line which is drawn through the point of proper 12 (parallel to the subindicall line) the hour (of proper 12) shall be cast into the secondary Ellipsis from the primary. This point being set right, you may place all the hour points from that proper 12 to the two proper six, the same way with this first point: but all the other points on the remoter side of the proper 12, must go the contrary way, as you will easily perceive, the very figure of the Ellipsis will require it should so be.  
As here. R S is the Subindical line, over which the Index leaneth.  
O the point of proper 12 upon P O Q the primary Ellipsis.  
O A, 9 B, ● C, etc. parallel lines through the hour points in the primary Ellipsis, parallel I say to the Subindicall line R S.  
P Q the two proper fixes, through which always the distorted Ellipsis must pass.  
Then because the Index leaneth from the upright Index (or Zenith line of the plain) towards S, therefore the proper 12 at O must go towards S, viz. to A: and therefore also 9 B, 8 C, etc. 11 G, etc. must go the same way till you come to the two proper six at P and Q, and then the hour points D E F must be set from 3 2 1, the contrary way to that wherein the first hours were placed; as you may plainly see the form of the new Ellipsis will require. For it cannot possibly be made to turn again from P towards M, if it must make up an Ellipsis, as in this case it is necessary it should do.  
Another way followeth.   

A second way to make the Ellipsis and hours to a slope Index upon a plain: and without trouble of making that Table before mentioned.  
THis way will be more expedite than the former. I shall here suppose these particulars to be again done by the former precepts.  
1. That the plains proper Meridian R O, is rightly placed.  
2. That P R Q is drawn perpendicular thereunto.  
3. That the Subindicall line R S is also rightly placed from R O the proper Meridian.  
4. That R P or R Q being determined for the Radius, R O and R K are each of them the Sins of the plains proper Latitude.  
5. From the points O and K (which are the two proper points of 12 to an Index standing perpendicularly to the plain) let O A and K L be drawn parallel to R S the Subindicall line.  
6. Then to the Radius R P, find the Sine of the plains difference of Longitude from your Meridian [because I purpose here to show how to work all from your 12 a clock line, though there is no obligation to that hour, but all may be done just in the same manner from any other hour point, if you account the plains difference of Longitude from some one of them, which is easy to do: as if in this Example the difference of Longitude from 12 be 36 gr. then from 11 it must be 21 gr. from 10 it must be 6 gr. from 1 a clock it must be 51 gr. etc. to all or any of the other hours] which here for example sake we will say to be 36 gr. With that Sine set off R T, and draw T I parallel to R O the proper Meridian. Then again to the Radius R O, find out the Co-sine of the difference of Longitude, and set it upon the line T I, from T to I. All this work is only to find the point of 12 in the primary Ellipsis, as you see it expressed in the former Figure at 12. The like work you must do to find the point of one of the hours of six in the primary Ellipsis, because six is 90 gr. from 12. [If you had chosen the point of 11 to work by, you must have looked for the point of 5 a clock, because that point is the 90th gr. from 11, if 10, then 4, etc.] That is, to the Radius R O find the Sine of the difference of Longitude, and set it from R to V, and then draw V X parallel to P R the line of the plains proper six. Then to the Radius R P find the Co-sine of the said difference of Longitude. and set it upon V X from V to X: So shall X be the point of your own six a clock upon the proper Ellipsis; as you may see in the Figure is expressed.  
Now through these two points I and X draw right lines parallel to the Subindical line R S. 
If this way of finding the two points I and X, seem not so expedite, you must remember that all the work here is done without the two forementioned calculated Tables.   
For if you divide R 12 (both ways) as a line of Sines (R e and R f being Sines of 15 gr. R d and R g Sines of 30 gr. etc. R 12 and R 12 Sins of 90 gr.) at a, b, c, d, e, R, f, g, h, i, k; you must then draw right lines through each of those points (each one parallel to the conjugate diameter 6 R 6.) And when that is done, make R 6 a Radius, and to that Radius make e 7, e 5, f 5, f 7, to be the Sine of 75 gr. or one hour under 90. Make d 8, d 4, g 4, g 8, the Sine of 60 gr. or two hours less than 90. So c 9, c 3, is the Sine of three hours under 90: b 10, b 2, of four hours under 90: a 11, a 1, is the Sine of five hours under 90, or of one hour. The like you may do for halves and quarters, or what other parts you desire.  
Thus much likewise for this second way.  
Always you must remember to put all things upon their right Coasts.   

Concerning the Zodiac, how it is to be limited and placed.  
TO omit other ways, I shall here pitch upon that which is most suitable to this manner of working upon which we now are. To the Radius P R (which is the larger Semidiameter of the primary Ellipsis) first, find the Tangent of 23 degrees and a half; which suppose to be R ξ: and then to this line, being taken as a new Radius, find the Co-sine of the plains Latitude, or the Sine of the Equinoctials altitude, which suppose to be the length R W, which is also to be set from the centre R upon the proper Meridian of the plain, as here it is from R to W. Then through the point W, draw a line, as W Z, parallel to the subindical line R S. When this is done you are to find the Sine of the Poles elevation above the plain, to the same Radius R ξ, which Sine you may suppose to be the length of the line φ ψ. Then say again; As the Radius, to the Tangent of the Indices reclination; so φ ψ (the Sine of the Poles elevation above the plain) to a fourth line; which line let be represented by λ ω. Take this line λ ω therefore, and set it upon W Z, the line formerly drawn, from W to Z. And lastly, draw the line R Z, and make R y equal to it, so shall Z y be the Zodiac in its true length and position: and R Z or R y shall give the half of it, which half is to be estimated as a Tangent of 23 degrees and a half, and so the Zodiac to be set on as hath formerly been often showed.  
You are here to take care that all things be set in their right position. In this example, Z turns from W towards the left hand, because it is here supposed that the North Pole is elevated upon the plain, and so the Axis riseth out of R above the plain: and therefore that 23½ tangent degree upon that Axis a●e above the plain. From which extreme point of 23½ above the plain, you may suppose a line to issue out parallel to the Index, which inclines from R towards S, and therefore that line inclining so too, must project that extreme point from W towards Z. All the reason of the work depends upon that which hath been formerly said: Namely, that both hour points of the Equinoctial, and declination points in the Axis, are all projected from their true positions in the Sphere (upon or unto a plain) by right lines passing through each of them, parallel to the assigned Index. I shall need to say no more, because I think enough (if not too much) hath been said already.  
This shall serve for the Effection and Demonstration also of this Proposition or Problem.   

CORONIDIS LOCO: I shall here further add; How upon an Horizontal Plain, to describe an Ellipticall Dial to an Index lying aslope, and pointing into some assigned Longitude and Latitude.  
THis I shall add as a Corollary to that which hath last of all been delivered in general for all Plains and Indices. Out of which Doctrine this (as a particular branch) is deduced. It containeth not any new thing, but because it may be more frequently made use of (in practice) than other Cases, therefore I add it, that it may also be the more plain and easy in operation. It is (for more ease) intended that the Index should point up into some just hour Circle, and to have some just degrees of Reclination from the Zenith of the Horizon.  
I purpose not to give variety of ways, for they are showed in the last precepts now given which are general, and may easily be brought and applied to this particular Case, as a branch of that general before.  
Suppose in our Horizon 51 gr. 30 min. in Latitude, and to two Indices set upon an horizontal plain, the one pointing Southeast into the morning Circle of 7 a clock, the other pointing South-west into the evening Circle of 5 a clock, and each of them reclining 45 gr. it were required to make two Ellipticall Dial's: How must they be done?  
It must be noted that the making of one will make the other, because they are to be set upon one plain, and their differences of the Indices Longitudes, is in each 5 hours, and the reclination of the Index in each 45 degrees. Only the Coasts in respect of situation must be changed, because the positions of the Indices are not parallel, but contrary one to the other, and so will the Ellipses be, as you may see in the next page. The first thing to be done, is,  
By having the difference of the Indices Longitude (75 gr.) and the reclination of the Index (45 gr.) to find what declination the Index must have: Or to find the declination of the subindicall line from our Meridi●n.  
When this work is done;  
Draw A R B for the Meridian line of North and South, and 6 R 6 for the line of East and West, crossing each other at R, which must be the centre of the Dial, and draw the Subindicall line R S 60 gr. 5 min. from A R upon the right Coast, that is, North-east in the first plain, and Northwest in the other plain.  
Then making 6 R for Radius, find thereto the Sine of the Latitude of the place 51 gr. 30 min. which let be the line R C, and set it upon the North part of the Meridian line A B (I call it the Meridian line because it is the said line upon the Horizontal plain whereupon these descriptions are now made) from the centre R to C. Then from the point C, draw C D parallel, and the same way with R S the Subindicall line. So shall C be the point of 12 upon the primary Ellipsis which should be drawn, if an Ellipticall Dial were to be made upon the horizontal plain to the upright Index, which is the proper Zenith line of the horizontal plain.  
After this, to the same Radius R 6, find the Co-sine of your Latitude, or the Sine of the Equinoctials height, or rather (in this particular work) the depth of it under the Horizon, which Sine let be the line E F. Then say,  
As the Radius, is to the Tangent of 45 gr. the Indices reclination; so is the Sine E F, to a fourth; which fourth will be (in this Example) the same line with E F, because the Tangent of 45 (the Reclination) is the same length with the Radius, yet I will call it the fourth quantity.  
Take this fourth quantity and set it off upon the line C D, from C to D, so shall D give the point of 12 upon the distorted Ellipsis, which is the 12 that fitteth with the reclining Index with which we have now to deal. Then draw (from R to D) the line R D, both ways, as D R G is done.  
By these means we have now got two conjugate Diameters. And now the rest of the work will be easy. For you are only to  
Divide the Semidiameters R D and R G (from R towards the extremities) each of them as a line of Sines. That is, R m, R o, make the Sine of 1 hour or 15 gr. So R l, R p, the Sine of 30, or 2 hours: R k, R q, the Sine of 45 gr. or 3 hours: R i the Sine of 60 gr. or 4 hours: R h the Sine of 5 hours: R D or R 12, the whole Sine or Sine of 90 gr.  
Then through each of these points at h, i, k, l, m, R, oh, p, q, draw a line parallel to the other conjugate Diameter, namely, to 6 R 6.  
This done, make R 6 a Radius, and to that Radius make m7, m5, 07, 05 (each of them) a Sine of 5 hours or 75 degrees. Make l 8, l 4, p 4, p 8, a Sine of 4 hours or 60 gr. Make k9, k3; q3; q 9, (each of them) a Sine of 3 hours or 45 gr.: So i 10, i●, must be the Sine of 2 hours: and h 11, h 1, the Sine of 1 hour or 15 gr. By this work you shall (upon the former parallel lines drawn through h, i, k, l, m, oh, p, q,) find the hour points of 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9; In which you are to sign out the hour points, and through which you are to draw the Ellipsis, as you see the figure for a pattern.  
You might also have divided each of the Radiusses, noted with R 6 into parts correspondent to a line of Sines, as you see partly done at the letters s t u, in the last of the two former Schemes) as at s t u is expressed, and through those points you must draw parallels to the other Diameter D R G. Then to the Radius D R make s 7, s 5, the Sine of 15 gr. or 1 hour: t 8, t 4, the Sine of 30 gr. or too hours: u 9, u 3, the Sine of 45 gr. or 3 hours; and so forward: by which work you shall find the same hour points (3, 4, 5, 6, 7, 8, 9, etc.) that you did before, and so you may find out and draw the Ellipsis by them.  
The same manner of work is to be performed for halves and quarters of the several hours.   

How the Zodiac is to be placed and limited.  
IT is the same work with that which went before, Pag. 185. yet I will here insert it again.  
First, work thus. As the Radius, is to the Tangent of 23½ gr. So let E F (which was formerly found to be the Sine of the Equinoctials altitude above the Horizon 38 gr. 30 mn. in respect of the Radius R 6, and must now again be here used) be to a fourth quantity, which suppose to be F W. Take therefore that length F W, and set it from R to X on the North side (in this example:) and through X draw X γ parallel to the Subindicall line R S.  
Then say again. As the Radius, is to the Tangent of 23 gr. and an half; So let R C (which was before found to be the Sine of the Latitude of the place 51 gr. and an half in respect of the Radius R 6, and comes now to be here used again) be to a fourth quantity, which suppose to be F z. Take therefore F z, and set it upon the line formerly drawn through X, from X, contrary to the going of the Subindical line, namely Southward (in this example) I say from X to γ Then from γ through R, draw γ R π, and make R π equal to R γ, So shall γ π be the Zodiac both placed rightly and justly limited.  
For division of it, you must conceive that R γ or R π is a Tangent of 23 and an half gr. the Sun's greatest declination, and consequently you may by it set on either 12 months, or 12 signs, as you will; the manner whereof hath been often showed in this Book, and especially Pag. 10, 11. and Pag. 14, 15, etc. so that now all this work is come to an end. The Index must slide to and fro upon the line γ π (or the Index standing, the Ellipsis must move after the same rate) γ π is the proper Meridian of the Index. And if your work (when you make this Dial, according to this case here before put) be exactly true, you shall find the zodiacal line π γ to fall just into the hour lines of 5 and 7, as they ought to do, because the difference of the Indices Longitude was assumed to be 75 gr. from the South Meridian of the place, which is the same with those hours.  
The sign ♋ must be placed at γ, and ♈ at π in this Dial.  
The Index must lie (when it is in the Centre R) just over the line R S, and must always move in a parallel position to itself. And it must in this example recline 45 gr. or rise above the Horizon (or above R S making an angle above that line 45 gr. also. But though it lie thus, yet it must move in the line γ π, and it must be so set on, that when you project the fiducial edge of it into a point (as workmen use to do, when with their eye they tr●e the straightness of a line) all that edge may at once justly appear to run into the line γ π, wherever the Index be set in the Zodiac. This is to be done carefully.  
I have here set two of them, whose proper Meridian's, (I mean proper to the Index, which are the lines of their Zodiacs, viz. γ π) do not lie in one and the same right line, or parallelism (which is the sameness of position) but make angles one to the other. This is done, because they should not lie in one Meridian: for by that means they will always set one another, which the Horizontal Dial with the single Ellipsis to an upright Index will not do at all times. These two, I say, will set themselves: and if you add a common horizontal Dial more, to stand between them, as you see done there▪ they will be the more specious and useful for setting each other.  
I have set that whose Index looks North-east upon the left hand, you may set it on the right hand if you will, and the other on the left. For this makes no material change.  
The Index is best to be a thread. Other circumstances I remember none but what the Workman will be well able to go through.   

Another way for the description of an Ellipticall Dial upon the horizontal plain of 51 gr. 30 min. Latitude, whose Zodiac, and the motion of the Index is performed upon the ●oures of 7 in the morning, and of 5 in the afternoon, as the same hours are drawn upon the common horizontal Dial.  
THis way was formerly written, and though it need not to have been here placed, in respect the way last given is the best, yet because it doth somewhat differ from what was set down before, I thought best not to omit it: because it is delightful to see different way ●● meet both in one effect.  
1. First, you are to get (this way) the altitudes of each hour point in the Equino●tiall, above the Horizon, by this R●le.  
As the Radius, to the Sine of 38 gr. 30 min.  
So ●he Sines of 75, 60, 45, 30, 15 gr. to so many 4th Sins. The Co-sines of these will limit out the points (upon the hours of the horizontal Dial formerly drawn) through which the Ellipsis upon the horizontal plain (made for an upright Index) is to pass: as is showed before in the 28, 29, 30, and 31, Pages, which points in the figure of the next Page may be represented by 8, 9, 10, 11, 12, 1, 2, 3, etc.  
2. As the Radius, to the Tangent of the reclination of the Index;  
So are the 4th Sins before found, to so many 7th terms.  
3. Describe the regular horizontal Ellipsis, as you see P O Q which you must know to be for the Latitude of 51 gr. 30 min. and that P Q are the points of the two six, and O K the points of the two twelves, viz. at Noon and Midnight: and also that all the hours are regularly set down to O P Q K, etc.  
Suppose here an Index to  gr. m. Recline Northwards 45 00 Differ in Longit. S E/ W 75 00 Consequently it must  Decline North East/ West 60 05 Have North Latitude 50 37  
4. From each point there set down, draw lines parallel to the Subindicall line R S, as O A, 8 C, 9 B, etc. the angle O R S must be 60 gr. 5 min.  
6. Through these new points B F D A E G, etc. you are to draw a new Ellipsis, which will be distorted from the former Ellipsis first drawn.  
Thus the Dial itself for the hour points is limited, you may add halves and quarters.   

For the Zodiac.  
IN this Example remember, that the Index declineth North East/ West 60 gr. 5 min. Reclineth 45 gr. Differeth in Longitude South East/ West 75 gr. Hath Latitude 50 gr. 37 min.  
1. Because the difference of Longitude is 75 gr. from the South, therefore our two hours of 7 in the morning, and 5 at evening, are the proper Meridian's upon which the Indices must slide.  
2. For that cause you must consider how much of those two Meridian's is intercepted between the Equinoctial and our Horizon. You may find it by this Rule.  
As the Radius, is to the Sine of 15 gr. (the Conpl. of 75,)  
So is the Tangent of 38 gr. 30 min. to the Tangent of 11 gr. 38 min. which is the ark required.  
3. Because the Latitude of the Index is 50 gr. 37 min. North, the compliment of it must be 39 gr. 23 min. and so much under the Equinoctial points of 7 and 5 (and upon the hour circles of 7 and 5, which are the proper Meridian's to the two Indices) doth the proper Horizon of each of those Indices cut the said hours of 7 and 5 upon the Southeast and South-west; and so much do they cut the same Meridian's above the Equinoctial upon the South-west and Southeast. So that upon the Southerly Coast, that portion of these Meridian's which is comprehended between the Equinoctial and our Horizon, namely 11 gr. 38 min. must be taken from 39 gr. 23 min. and the remainder will be 27 gr. 45 min. which is the ark comprehended between the proper Horizon of the Index and our own Horizon, which upon the South part (in this particular Example) is below our Horizon, on the North part above.  
4. First, therefore the limitation of the Semi-Zodiacs length must be to that Latitude of the Index or proper Horizon 50 gr. 37 min. And then secondly, the said Zodiac first found and limited must be augmented (in this Example) in proportion of the Radius to the Secant of 27 gr. 45 min. In this manner, by the second rule given at the beginning of this Book Pag. 11. To the Radius P R find the Cofine of the Latitude of the Index, namely the Sine of 39 gr. 23 min. which suppose to be N T.  
To this length N T as a Radius, find out the Secant of the forementioned ark 27 gr. 45 min. which suppose to be N V.  
This length N V shall give the Radius or Tangent of 45 gr. which is the Tangent or decimal Scale, out of which (either of them) you are to make your Zodiac by the numbers of those Tables in the 6 and 7 Pages.  
To this Radius or decimal Scale, or Tangent of 45 gr. N W is the Tangent of 23 gr. and a half, wherefore the Semizodiac, R L or R M is to be made equal in length to N W. Thus the Zodiacs length is limited.  
For the position of the Zodiac, it was before said, that it must lie in the line of 7 in the morning, or 5 in the afternoon. So in this last figure you see it points to Z, which is 7 in the morning, (and consequently also to X, which is 7 in the evening) in the distorted Ellipsis: because this is that Dial whose Index looks North-east: you see it do so likewise in the first figure Pag. 189, as in the second figure there it lies to 5 in the evening, and consequently also to 5 in the morning.  
Observe in the last figure (which is as true and just in the two other Pag. 189.) that the angle P R Z, and so Q R X, is equal to the angle made (upon our common horizontal Dial) between 6 and 7 a clock, namely, 18 gr. 54 min. and consequently that 1 R 1 (a line drawn from 1 to 1 in the primary Ellipsis) is perpendicular to the Zodiac L M: but P R C is not equal to the angle thereon between 6 and 8 a clock; nor is it so in any other of the hours.  
Observe again, that if a line be drawn parallel to O K. through A the point of 12 in the distorted Ellipsis, till it meet with the Zodiac (extended) at y; observed (I say) that if R M be taken as a Tangent of 23½ gr. (as it is taken in this work) then will R y be the Tangent of (the compliment of your Latitude, namely) 38 gr. 30 min. And this must be so, because if the Sun should decline 38 gr. 30 min. Southward, than the Sun must only peep up upon the Horizon at 12, and consequently the shadow of the Index (namely the point of it that intersects with your Horizon, for the Sun hath no altitude at that place and time) must run full from South to North to show the point of 12 at A.  
Other observations might be made, but I go no further.  
The reason of the precedent work will be demonstrated out of the former things delivered in this Book: especially out of this last Treatise, which begins Pag. 171, and ends Pag. 186. And therefore I wholly omit to add any thing more to this particular subject.  
They that understand the things before, will not be to seek in the necessary Demonstrations of these particulars.   

A Note concerning the framing of Dial's to find the Azimuth.  
AS the Axis of the world is to the Equinoctial Circle, so is the Zenith line to the Horizon of any place.  
Whence it will follow: That whatsoever is in this Book declared concerning the projection of the hour points of the Equinoctial Circle upon any plain, etc. the same may be applied to the projection of Azimuths or points of the Horizon upon any plain, etc. Only mutatis mutandis, (i.e.) instead of the Equinoctial and Poles of the World, you are to use the Horizon and the Poles of it, (i.e.) Zenith and Nadir.  
Then for the Zodiac (which depends upon the Scale or Tangents of Declination, which exceed not 23 and a half gr. you are to use (in the Azimuthal work) a Scale of Tangents (rightly limited) going up from oo to so much altitude as you mean to use (if it be for the Sun's course, to 62 gr. here at London.)  
As there you rectify the Index to the Declination of the Sun: So here you must (answerably) rectify it to the Sun's altitude taken by observation, and then the Index being set to the altitude gives the Azimuth for that moment and no more.  

COROLLARY.  
Therefore, to find the Azimuth by this way, you must take the Sun's Altitude, and so find the Azimuth for one moment only.  
Quest. But why is there so little said here of this?  
Answ. Because it is a thing of no use, but of a great deal of trouble: for the altitude is to be observed, and the Index set thereto every moment,  
Quest. Is it not so too in the Dial's before described?  
Answ. No; for there the Index set once, may serve in that place one whole day very well, because the Sun's declination doth not alter much in one day.  
Quest. Why then is this Azimuthal business mentioned at all?  
Answ. Because the Reader might see, that the Author of this Treatise was not ignorant of it: yet it is mentioned also that if any delight in such a curiosity he may to his liking effect it.  
¶ And for a full conclusion note, that, All the former Dial's may be made in a Craticular way.     
FINIS.  

AN INDEX Of the Chief PARTICULARS:  
OF the Elliptical Dial, with an Index perpendicular to the plain.  
Upon an horizontal, or any other direct plain, see from 
Page 8 to page 22  Upon a declining plain, from 
22 to 29  Another way to prick down and divide the Ellipsis, 
29 to 34  Some uses and varieties of this Elliptical Dial, 
34 to 37  Some varieties of the structure of it, 
37 to 47  An Advertisement concerning some other uses of the last description, 
47 to 53  A demonstration of what soever went before, 
53 to 69   

Of the Elliptical Dial with an Index standing in the Zenith-line.  
Upon any plain, or other curved Superficies: done by projection, from 
Page 69 to 80  To any Superficies, and to an Index casually set: done also by projection, 
80 to 112  A demonstrations of all, 
112 to 115  Circular Horologiographie, performed and demonstrated, 
115 to 133  Rectilineal Horologiography, how done, and demonstrated, 
133 to 169  Elliptical Horologiography, 
169  Upon all plain (not curved) Superficies to an Index (casually, or by election) placed any way: performed by Spherical operations, (not by projection, and demonstrated, 
169 to 201  A Note concerning Azimuths how they may be found by such kind of Dial's made in Analogy to the former, 
201  All the former works may be craticularly performed, 
202    
FINIS.   

Courteous Reader, be pleased to take notice of these Books following, which are very useful and necessary for all Merchants, Tradesmen, Accomptants, they are sold by Nicholas Bourn at the South entrance of the Royal Exchange.  
INtroduction to Merchant's Accounts, compiled by john Gollines, Student in the Mathematics, and Professor of Merchant's Accounts.  
Tabula Foeneratoria, or Tables for the forbearance and discompts of money; likewise Tables for the forbearance, discompt and purchase of Annuities to 31 years, at the rate of 6 per Centum, per Annum, Calculated by Roger Clavell, Gent. Student in the Mathematics.  
Manuel of Millions or Accounts ready cast up, whereby you may both suddenly and truly know the value of any Commodity at any price whatsoever: the second Edition: to which is aded a second Part of very great use, by Richard Hodgetts.  
Posthuma Fosteri, the description of a Ruler, upon which is inscribed divers Scales, and the uses thereof; with divers Propositions in Astronomy, Navigation and dialing, with the delineating of Horizontal dials, by Samuel Foster, late Professor of Astronomy in Gresham-Colle●ge.  

These following are Books of other sorts:  
Bread for the poor, an Advancement of the English Nation; promised by the Enclosure of the Wastes and Common grounds of England, by Adam Moor, Gent.  
Bills of Lading in all Languages, for shipping forth of Merchant's goods to all Countries of Trading.  
Indentures for Virginia, Bermoodas, New-England, and all other new Plantations.  
A Confutation of Atheism, by Dr. Dove.  
Bishop Downame, on the Lords prayer.  
Popular Errors in Physic, by Dr. Primrose.  
Sweedish Wars complete in four Volumes, 4ᵒ.  
Youth's Guide, or the Father's Legacy to his Son.  
Dr. Preston, of Faith and Love.  
Merchant's Mirror, or Directions for the perfect Booking or Surveying of his Estate; Framed by way of Debtor and Creditor, after the so termed Italian manner; and Compiled by Richard Dafforne, of London Accountant.  
Certain Sermons, preached by Dr. Preston at Cambridge and London upon these Texts following, 2d to the Chronicles 7, 14.5 of the Ephesians, 32. and the 5 to the Ephesians, 22, 23, 24. called the Golden Sceptre, 4ᵒ.  
Dr. Featly, Handmaid to Devotion.  
Surgeon's Mate, by Mr. Woodall, Folio.