THE ART OF DIALLING; BY A NEW, EASIE, AND MOST SPEEDY WAY. SHOWING, HOW TO DESCRIBE THE Houre-lines upon all sorts of Plains, Howsoever, or in what Latitude soever Situated: As also, To find the Sun's Azimuth, whereby the sight of any Plain is examined. Performed by a Quadrant, fitted with lines necessary to the purpose. Invented and Published by SAMVEL FOSTER, Professor of Astronomy in Grosham College. LONDON, Printed by john Dawson for Francis Eglesfield, and are to be sold at the sign of the Marigold in Paul's Churchyard. 1638. To the Reader. READER, HEre is presented to thy view a short and plain Treatise; it was written for mine own use, it may become thine if thou like it; The subject indeed is old; but the manner of the Work is all new. If any be delighted with recreation of this nature, and yet have not much time to spend, they are here fitted, the instrument will dispatch presently. If they fear to lose themselves in a wilderness of lines, or to outrun the limits of a Plain, by infinite excursions (two inconveniences unto which the common ways are subject) they are here acquitted of both, having nothing to draw but the Dial itself, contracted within a limited equicrural triangle. If want of skill in the Mathematics should deter any from this subject, let them know that here is little or none at all required, but what the most ignorant may attain. If others shall think the Canons more exact; so do I, but not so easy to be understood, not so ready for use, not so speedy in performance, nor so well fitting all sorts of men: and withal an instrument in part must be used, this will do all, and is accurate enough. If it must needs be disliked, let a better be showed and I will dislike it too; It is new, plain, brief, exact, of quick dispatch. Accept it, and use it, till I present thee with some other thing, which will be shortly. Imprimatur. Decemb. 1. 1637. SA. BAKER. THE DESCRIPTION OF THE QVADRANT, and the manner how the lines are inscribed and divided. CHAP. I. 1. The description of the fore-side. THe limb is divided into 90 degrees, and subdivided into as many parts as quantity will give leave. The manner of division, and distinction of the subdivided parts is such as is usual in all other Quadrants. To describe the other Work in the superficies; Take from the upper edge of the limb about 3 degrees, and set off that space from the centre R to A. Then divide A into seven parts, whereof let EBB contain two. Or in greater instruments, if A be 1000 let EBB contain 285. Make SC equal to EBB, and draw the line BC. From C, draw CD parallel, and of equal length to AB. Upon AB and CD, and BE also (as far as it is capable) insert the 90 sins, from B towards A and E, and from C towards D, but let them be numbered from A unto B to 90, and so to E 113 degrees 30 minutes, from D to C unto 90 degrees. Again: Draw ES cutting CD at F; so shall BCFEB contain a parallelogram, whose opposite sides, being parallel, are divided alike, and in this manner. BE and CF as whole sins, do contain the 90 sins, or as many of them as can distinctly be put in: and from the divisions are drawn parallel lines, having every tenth, or fifth, distinguished from the rest. These serve for the 12 Signs and their degrees, and therefore you see upon every 30th degree, the characters of the 12 Signs inserted, in such manner as the figure showeth. And these lines may be called, The Parallels of the Sun's place. In like manner, The lines BC, OF, being first bisected at X and Z, shall make 4 lines of equal length. These 4 lines XB, XC, ZE, and ZF, are each of them divided as a scale of Sines, beginning at X and Z, and from each others like parts are parallel lines protracted, having every tenth and fifth distinguished from the rest. They are numbered; upon BC, from B to X 90, to C 180; upon FE, from F to Z 90, to E 180. These lines are called, The lines of the Sun's Azimuth. This done; Upon the centre R describe the two quadrants VT, and BC, let their distance VC be one sixth part of Rc, or more if you will. Divide them each into 6 equal parts, at e, o, y, n, s; and a, i, u, m, r, drawing slope-lines from each others parts, as Va, ei, on, y, nr, sb: and these lines so drawn are to be accounted as Hours. Then dividing each space into two equal parts, draw other slope-lines standing for half hours, which may be distinguished from the other, as they are in the figure. Then from the points V and T draw the right line VT. Lastly, Having a decimal scale equal to TR, you must divide the same TR into such parts as this Table here following alloweth, the numbers beginning at T, and rising upto 90 at R. Upon your instrument (for memory and directions sake) near to the line AB, write, The sum of the latitude and Sun's altitude in Summer; The difference in Winter. Over VT, write, The line of Hours. near to CD write, The sum of the latitude and Sun's altitude in Winter; The difference in Summer. By TR, write, The line of latitudes for the delineation of dials. A Table to divide the line of Latitudes. 90 10000 62 9360 46 8259 30 6325 14 3325 85 9982 61 9311 45 8165 29 6169 13 3104 80 9924 60 9258 44 8068 28 6010 12 2879 78 9888 59 9203 43 7968 27 5846 11 2650 76 9849 58 9147 42 7865 26 5678 10 2419 75 9825 57 9088 41 7738 25 5505 9 2186 74 9801 56 9026 40 7647 24 5328 8 1949 72 9745 55 8962 39 7532 23 5146 7 1711 70 9685 54 8895 38 7414 22 4961 6 1470 69 9651 53 8825 37 7292 21 4772 5 1228 68 9615 52 8753 36 7166 20 4577 4 984 67 9378 51 8678 35 7036 19 4378 3 739 66 9519 50 86●● 34 6902 18 4176 2 493 65 9496 49 8519 33 6764 17 3969 1 247 64 9454 48 8436 32 6622 16 3758 0 ● 63 9408 47 8348 31 6475 15 3543 S●… 2. The description of the backside. Upon the backside is a circle only described, of as large extent as the Quadrant will give leave, noted with ABCD, divided into two equal parts by the Diameter AC. The semicircle ABC is divided into 90 equal parts or degrees, every fifth and tenth being distinguished from the rest by the longer line; They are numbered by 10, 20, 30, etc. unto 90. The same parts are also projected upon the diameter AC, by a ruler applied to them from the point D. These are numbered also from A to C by 10, 20, etc. unto 90. The other semicircle ADC, is first divided into two Quadrants at D. And then upon these two quadrants are inscribed 90 such parts as this Table ensuing doth allow. The inscription is made by help of a Quadrant of a circle equal to AD or CD, being divided into 45 equal degrees, out of which you may take such parts as the Table giveth, and so prick them down, as the figure showeth. Every fifth and tenth of these parts is distinguished from the rest by a longer line; they are numbered from A and C, by 10, 20, etc. unto 90 ending in D. A Table to divide the upper and nether Quadrants of the Circle. 1 1.00 14 13.36 27 24.25 40 32.44 53 38.37 66 42.25 2 2.00 15 14.31 28 25.09 41 33.16 54 38.59 67 42.38 3 3.00 16 15.25 29 25.52 42 33.47 55 39.19 68 42.50 4 3.59 17 16 18 30 26.34 43 34.18 56 39.40 69 43.02 5 4.59 18 17.10 31 27.15 44 34.47 57 39.59 70 43.13 6 5.58 19 18.02 32 27.55 45 35.16 58 40.18 72 43.34 7 6.57 20 18.53 33 28.35 46 35.44 59 40.36 74 43.52 8 7.55 21 19.43 34 29.13 47 36.11 60 4054 75 44.00 9 8.53 22 20.32 35 29.50 48 36.37 61 41.10 76 44.08 10 9.51 23 21.21 36 30.27 49 37.03 62 41.27 78 44.22 11 10.48 24 22.08 37 31.02 50 37.27 63 41.43 80 44.34 12 11.45 25 22.55 38 31.37 51 37.51 64 41.57 85 44.53 13 12.41 26 22 40 39 32. 1● 52 38.15 65 42.11 90 45.00 Thus have you both sides decribed. Besides all this, there are two sights added, with a thread and plummet like as in other instruments. The thread hath a moovable bead upon it for special use. The same thread passeth through the centre R. quite behind the Quadrant, and is hung upon a pin at the bottom of the Quadrant, noted with W. The reason of the threads length will be seen when we come to the uses of the instrument. CHAP. II. The use of the Quadrant in general. FIrst upon the fore-side. The limb serveth especially for observation of all necessary angles. The lines A, CD, with the Parallelogram BCEF, are to find out the Sun's Azimuth in any latitude whatsoever. The slope-lines within the arkes VT, cb, by help of the thread and bead, do serve artificially to divide the line of Hours TV, into its requisite parts; which together with TR the line of latitudes, do serve to protract all plain dials howsoever situated. Secondly upon the backside. Note that ABC is called the Semicircle: AC is called the Diameter: AD the Upper quadrant: CD the Nether quadrant. The uses of these are to find out the necessary arkes and angles, either for preparation to the dials description, or serving after for the dials situation upon the Plain. In all these uses the thread bearing part, and therefore having asufficient extent of length, that being loosed it may with facility reach over either side of the Quadrant. CHAP. III. To find the Azimuth of the Sun in any Latitude whatsoever. BEfore you can make any draught of your Dial, you must know the situation of your plain, both for declination and inclination. The best way to come to the plains declination is by help of the Sun's Azimuth. By having the Latitude of the place; The place of the Sun in the Ecliptic, and the altitude of the Sun above the Horizon, you may find out the Azimuth thereof in this manner. Add the Sun's altitude, and your latitude together, and subtract the lesser of them from the greater; So shall you have the sum of them, and the Difference of them. With this sum and difference, come to your Quadrant, and according to the time of the year (as the lines will direct you) Count the said Sum and Difference respectively, and applying the thread unto them, find out the Sun's place in the Parallels serving thereto, and where the thread cuts this Parallel, observe the Azimuth there intersecting, for that is the Azimuth from the South, if you number it from the line whereon the sum was numbered. Example 1. In the North latitude of 52 gr. 30 min. in the Summertime the Sun entering into 8, and the altitude being observed 30 gr: 45 min. I add the latitude 52 gr. 30 min. and the Sun's altitude 30 gr. 45 min So I find the sum of them 83 gr. 15 min. and substracting the lesser of them from the greater, I find the difference of them 21 gr. 45 min. The sum I number in the line A, and the difference in DC (because it is in Summer) and to the terms I apply the thread, and where it crosseth the parallel of the beginning of 8, there I meet with 66 gr. 43 min. which is the Azimuth from the South, being reckoned from the line A whereon the Sum was counted. Example 2. The latitude and Sun's place being the same if the altitude had been 9 gr. 15 min. The sum of the latitude and altitude would be 61 gr. 45 min. The difference 43 gr. 15 min. and so the thread applied to these terms would have showed 96 gr. 52 min. for the Azimuth from the South. A third Example. In the same Latitude of 52 gr. 30 min. in the Wintertime, the Sun entering the tenth degree of ♏, and the altitude being 9 gr. 30 min. I would know the Azimuth of the Sun from the South. I add the Altitude 9 gr. 30 min. to the Latitude 52 gr. 30 min. and so find the sum of them 62 gr. 0 min. And substracting the Altitude out of the Latitude, I find the Difference of them 43 gr. 0 min. The sum (because it is in Winter) I count upon the line DC in the Quadrant, and the Difference upon AE. So the thread applied to these terms cutteth the tenth of ♏, at 49 gr. 50 min. which is the Azimuth numbered from DC the South. The Amplitude. Note here by the way, That the thread applied to the Latitude of your place numbered upon both lines A, DC, will show you, for any place of the Sun, the due Amplitude of his Rising or Setting, or the Azimuth whereon he riseth or setteth, if you number the same from the middle line noted with XZ which here representeth the East and West Azimuths. CHAP. four To find out the Declination of a Plain. THe declination of a Plain is numbered from the South or North points towards either East or West. And it is the ark of the Horizon comprehended between the South-North, and a line infinitely extended upon the Horizon perpendicular to the horizontal line of the Plain; which line may be called the Axis, and the extremity of it, the Pole of the Plains horizontal line. To find out this declination you must make two observations by the Sun: The first is of the Distance or angle made between the Axis of the horizontal line of the Plain, and the Azimuth wherein the Sun is at the time of observation. The second is of the Sun's Altitude. Both these observations should be made at one instant, which may be done by two observers, but if they be made by one, the less distance of time between them, will make the work to agree together the better. 1. For the Distance. Upon your Plain draw a line parallel to the horizon, to this line apply the side of your Quadrant, holding it parallel to the horizon. Then holding up a thread and plumber, which must hang at full liberty, so as the shadow of the thread may pass through the centre of the Quadrant, observe the Angle made upon the Quadrant by the shadow of the thread, and that side that lieth perpendicular to the horizontal line, for that angle is the Distance required. 2. At the same instant as near as may be, take the Sun's Altitude; These two being heedfully done, will help you to the plains Declination by these rules following. When you have taken the Altitude, you may find the Sun's Azimuth by the former Chapter. Then observe, whether the Sun be between the Pole of the horizontal line and the South North point or not. If the Sun be between them, add the Azimuth and Distance together, and the sum of them is the Declination of the plain. If the Sun be not between them, subduct the lesser of them from the greater, and the difference shall be the Declination of the plain. ¶ By your observation you may know to what coast a Plain declineth. For if the South North point be in the midst between the Sun's Azimuth and the pole of the Plains horizontal line, then doth the Plain decline to the coast contrary to that wherein the ☉ is: If otherwise, the declination is upon the same coast with the Sun. CHAP. V. To find the Inclination of a Plain. THe Inclination of a Plain is the angle that it maketh with the Horizon. When you have described your horizontal line upon a Plain, as in this figure OF, cross it with a perpendicular GH, for the Vertical line. And because the inclinations of the Upper and Under faces of the Plain, are both of one quantity in themselves, if therefore you apply the side of the Quadrant noted with AB unto the vertical line of the under face, or to the under side of a Ruler applied to the vertical line of the upper face, as is here showed in this figure; Then shall the degrees of the Quadrant give you GOD the angle of inclination required. CHAP. VI Of upright declining Plains. THose Plains are upright, which point up directly into the Zenith or vertical point of the Horizon, and may be tried by a perpendicular or plumb-line. In these, as in the rest that follow, before the Hours can be drawn, two things must be found; 1. The Rectifying ark; 2. The Elevation of the Pole above the Plain. 1. To find the Rectifying ark. Extend the thread from your Latitude counted in the upper Quadrant of the circle on the backside, to the compliment of the Plains declination numbered in the Semicircle; so shall the thread show you on the Diameter the Ark required. 2. To find the Elevation of the Pole above the Plain. Extend the thread from the Rectifying ark numbered in he upper quadrant, to your Latitudes compliment taken in the Semicircle; so shall the thread show upon the Diameter, the Elevation of the Pole above the Plain. According to these rules, in the latitude of 52 gr. 30 min. supposing an upright Plain to decline 55. gr. 30 min. I find the Rectifying ark to be 28 gr. 36 min. And the elevation of the Pole above the Plain to be 20 gr. 10 minutes. CHAP. VII. In East and West incliners. THose plains are called East and West incliners, whose horizontal line lieth full North and South, and their inclination is directly towards either East or West. 1. To find the Rectifying Ark. Extend the thread from your Latitudes compliment taken in the upper quadrant of the Circle on the backside, to the compliment of the Plains inclination counted in the semicircle; so shall the thread show upon the Diameter the Ark required. 2. To find the Elevation of the Pole above the Plain. Extend the thread from the Rectifying-arke counted in the upper quadrant, to your latitude taken in the Semicircle; so the thread upon the Diameter gives the elevation of the Pole above the Plain. Thus in the latitude of 52 gr. 30 min. If a Plain incline Eastward 40 gr. to the horizon, the Rectifying-arke will be 35 gr. 58 min. And the elevation of the Pole 37 gr. 26 min. above the plain. CHAP. VIII. In North and South incliners. SUch Plains are called North and South incliners, whose horizontal line lieth full East and West, and their inclination is directly upon either North or South. 1. For the Rectifying-Arke. There is no use of it in these plains, because they all lie directly under the Meridian of the place. 2. To find the Elevation of the Pole above the Plain. If the inclination be toward the South, add the inclination to your latitude; for the sum is the Elevation of the pole above the Plain. If the sum exceed 90 degrees, take it out of 180, and the supplement gives you the Poles elevation. If the inclination be Northward, compare the inclination with your latitude, and subduct the lesser out of the greater: the Difference is the elevation of the Pole above the Plain, If there be no difference, it is a Direct polar Plaine. CHAP. IX. In declining Incliners. THose Plains are called Declining incliners, whose horizontal line declineth from the East or West, towards either North or South, and their inclination also deflecteth from the coasts of North and South towards either East or West. The best way to find the Rectifying-arke, and the poles elevation for these Plains, will be First, to refer them to a New latitude, wherein they may lie as East or West incliners. For which purpose you are first to find out an Ark, which in respect of its use may fitly be called, The Prosthaphaereticall ark, it is found by this rule: ¶ Extend the thread from the compliment of the Plains declination counted in the upper quadrant, to the inclination numbered in the Semicircle; so the thread shall give you upon the Diameter the Prosthaphaereticall-arke required. This Prosthaphaereticall-arke is to be used as the Inclination was in the precedent Chapter. For, If the Plain do incline towards the South, it must be added to your Latitude: and so the sum (if less than 90 degrees) gives you the New Latitude: but if the sum be greater than 90, than the residue, or supplement of it to 180 degrees will be the New Latitude required. If the Plain incline toward the North, compare this Prosthaphaereticall-arke with your Latitude, and subduct the lesser of them out of the greater; So the Difference shall give you the New Latitude. If there be no difference, it is a declining Polar plain. Secondly, it will be required to know what Inclination these Plains shall have in this their New latitude; and that is done by this rule: ¶ Extend the thread from the Prosthaphaereticall-arke taken in the upper quadrant to the Plains declination counted in the Semicircle: so the thread shows on the Diameter, the New-inclination in their New latitude. Being thus prepared, you may now proceed as in East and West incliners you did before. 1. To find the Rectifying-Arke. Extend the thread from the New latitudes compliment taken in the upper quadrant, to the New-inclinations compliment numbered in the Semicircle; so the thread upon the Diameter shows the Ark required. 2. To find the Elevation of the Pole above the Plain. Extend the thread from the Rectifying-arke in the Vpper-quadrant to the New latitude in the Semicircle; so the thread upon the Diameter gives the Elevation of the pole above the plain. According to these rules, supposing a Plain to incline towards the North 30 degrees, and to decline from the South towards the West 60 degrees in the latitude of 52 gr. 30 min. First I find the Prosthaphi-arke 60 gr. 6 min. and because the Plain inclineth toward the North; I compare this ark with the Latitude of the place, and taking it out of the Latitude there remaineth 36 gr. 24 min. for the New Latitude. Then I find the New inclination to be 25 gr. 40 min. and so the Rectifying-arke 59 gr. 8 min. and the Elevation of the Pole above the Plain to be 32 gr. 20 minutes. CHAP. X. To draw the Houre-lines upon the horizontal, the full North or South plains, whether standing upright or inclining. IN the four last Chapters we have seen the uses of the Circle on the backside of the Quadrant: in this and the next Chapter we shall show the use of TR the line of latitudes, and of TV the line of Hours; which two lines with the help of the limb VCTB, and of the thread and Bead, will serve to prick down any Dial, by the Precepts hereafter delivered. And first we begin with those Plains which have no declination, whose Poles lie directly under the Meridian of the place; of which sort are the horizontal, the Erect South and North plains, with all Incliners looking directly North or South. Having then by the former Precepts found the Elevation of the pole above your Plain, you may begin your draught in this manner. First, Draw the line RAT of sufficient length, and out of the line of Latitudes in your Quadrant, take off the Elevation of the pole above the plain, and prick it down from the point A, unto R and T both ways. 2. Take the line of Hours TV also out of the Quadrant, and with that extent of your Compasses upon R and T as upon two centres, draw the arkes BV and CV, crossing each other in V; and draw the lines RV and TV: then coming to your Quadrant again; 3. Apply the thread to every hour point in the limb VT or CB, as first to s, or r, so shall it cut the Line of hours TV in 1; Then take off with your Compasses T1, and prick it down here from V to 1, and from T to 7. Again, Apply your thread to the next hour in the limb at n or m, it will cut the Line of hours TV in 2 take off T2, and prick it down here from V to 2, and from T to 8. So again, the thread applied to the third noure at y, or u, cuts the line TV, in 3; take off T3, and prick it down here from V to 3, and from T to 9 In like manner, the thread applied to the fourth hour at o, or i, will cut the line TV in 4 take off T4, and prick it down here from V to 4, and from T to 10. So also the thread laid upon the fifth hour at e, or a, cutteth TV in 5; take off T5, and prick it down here from V to 5, and from T to 11. Thus are all the Hours pricked down. An horizontull Dial to 52 gr: 30 m: lat: Lastly then, laying your Ruler to the centre A, through each of these points, you shall draw the houre-lines A7, A8, A9, A10, A11, AV which is 12, A1, A2, A3, A4, A5, RAT is the line of the two six. So having 12 hours, which is half the Dial, drawn, you may extend the necessary lines, as many as you will, beyond this centre, as 5A5, 4A4, 7A7, 8A8, etc. In the same manner may the half hours be pricked down and drawn, by applying the thread to the half hours in the limb, etc. And note also that in these Plains before mentioned; As the extent from V to 1, is the same with that from T to 7, so likewise is it the same with V11, R5; And as V2 is the same with T8, so likewise is it the same with V10, R4: So likewise V9 and T9 are all one, and both equal to R3 and V3. So that the three first hours taken from the Quadrant, that is to say, T1, T2, T3, will give all the hours for these dials. T1, gives V1, V11, R5, T7. T2, gives V2, V10, R4, T8. T3, gives V3 or R3, V9 or T9. But in other Plains it is not so, for which cause I have rather set down this way before at length, as a direction for what comes after, for that is general. Here note again, that if you desire to make your draught greater, you may in your description either double or triple every length which you take in your Compasses. And so I proceed to all declining Plains. CHAP. XI. To draw the Hours upon all sorts of declining Plains, whether erect or inclining. BY the former precepts you must first get the Rectifying-arke, with the Elevation of the pole above the Plain. After they are had, you may prick down the Hour points in this manner following, little differing from the former. A Plain▪ inclining Eastward 40 gr: The horizointall line, parallel to the line of 12. 1. Asbefore; Upon the line RAT, set off the Elevation of the pole above the plain, being taken out of the line of latitudes in the Quadrant, from A both ways, to R and T. 2. Take the line of Hours TV out of the Quadrant, and with that extent upon R and T as upon two centres, describe the two arkes BV and CV crossing at V, and draw the lines RV, TV, and AU. Thus far we go along with the last Chapter. 3. If we take the example in the seventh chapter, that plain is the upper face of an East incliner, whose Elevation is 37 gr. 26 min. and so much doth this line TA reach unto in the line of Latitudes: the Rectifying ark is 35 gr. 58 min. This ark I number below in the limb of the Quadrant ES, and thereto applying the thread I observe in the upper limb Vcb T which of the Hours and where it cutteth, I find it to cut the slope line o u in the point P; too this point P I set the Bead, which by this means is rectified and fitted to the description of the Dial. Here you see the use of the Bead, and the reason why this ark counted upon the limb is called the Rectifying ark: and here be careful that you stretch not the thread. 4. The thread and Bead being thus placed and rectified, you shall see the thread to cut the line TV at a upon the Quadrant; take T a in your Compasses, and prick it down here from V to 12, and from R to 6. Here by the way observe, that because this plain is an Eeast-incliner, the face of it looketh toward the West, and then if you imagine the true situation of this Dial upon the plain whereon it must stand, you will easily conceive that the line of 12 is to stand on the right hand from the line AU. and so the line of 6 on the left hand, whereas if this plain had faced toward the East, the line of 12 must have stood on the left hand, and 6 on the right hand. Your own conceit, together with the precepts of the chapter following, must help in this, and in other things concerning the right scituating of the lineaments of your Dial. To proceed then, In the same manner must you apply the Bead to every hour line, as in the next place I remove it to the line y m in the Quadrant, and then I see it to cut the line TV in b; I take 1 b in my Compasses, and with it do prick down from V to 1, and from R to 7. Again, the Bead being applied to the lines nr, sb, the thread will cut the line TV upon the Quadrant in c and d; I take the points TC, Td, in my Compasses, and prick them down from V to 2 and 3, and from R to 8 and 9 Then again, the Bead applied to the lines ei, Va, the thread will cut the line TV in the points e and o; I take then Te and Tf, and prick them down from U ●o 11 and 10, and from R to 5 and 4. 5. Lastly, lay your rule to A, and draw A10, A11, A12, A1, A2, A3, A4, A5, A6, A7, A8, A9. Thus have you twelve hours, and if you extend these beyond the Centre, you shall have the whole 24 hours, of which number you may take those that shall be fit for the Plain in this situation. The half hours may thus be pricked on and drawn also, by applying the Bead to the half hours pricked down in Vcb T the upper limb of the Quadrant, for so the thread will give you the half hour points upon the line TV, which may be taken off, and set down upon the Dial as the hours themselves were. CHAP. XII. How to place the Dial in a right Situation upon the Plain. AFter the houre-lines are drawn by the last Chapter, they are to be placed in a right situation upon their Plain. Which to do, upon some Plains is more difficult than the Description of the Dial itself. To give some directions herein, I have added this Chapter, where you have 9▪ Questions with their Answers, giving light sufficient to what is here intended and required: but first be admonished of three things. 1. That the inclination mentioned Chap. 8. is the very same in Use with the Prosthaphaereticall ark mentioned Chapter 9 And therefore when I mention the Prosthaphaereticall ark, because it is of most frequent use, you must remember I mean both the Prosthaph: ark, Chap. 9, and the Inclination, Chap. 8. 2. That these rules, though given primarily for places of North-latitude, lying within the Temperate, Torrid, and Frigid Zones, yet are also as true, and may be applied to all places of South-latitude, if we exchange the names of North and South, for South and North. Here by the way note, that the North part of the Torrid Zone extendeth from 0 degrees of latitude to 23 gr. 30 min. the Temperate Zone reacheth from 23 gr. 30 min. to 66 gr. 30 min. the Frigid Zone extendeth from 66 gr. 30 min. to 90 gr. of latitude. And so I come to the 9 Questions. 1. What Pole is elevated above the Plain. Upon all Upright plains declining from the North: Upon the upper faces of all East or West incliners: Upon the upper faces of all North-incliners, whose Prosthaph: ark is less than the latitude of the place: On the under faces of all North-incliners, whose Prosthaph: ark is greater than the Latitude of the place; and on the upper faces of all South-incliners, The North pole is elevated. And therefore contrarily, Upon all upright Plains declining from the South: On the under faces of all East and West, and South incliners: On the under faces of all North-incliners, whose Prosthaphaereticall ark is less than the Latitude of the place: On the upper faces of all North-incliners, whose Prosthaph: ark is greater than the Latitude of the place, The South pole is elevated. 2. What part of the Meridian ascendeth or descendeth from the horizontal line of the Plain? In all Upright plains the Meridian lieth in the Vertical line, and if they decline from the South it descendeth, if from the North it ascendeth. Upon both faces of East and West Incliners the Meridian lieth in the horizontal line. In all North-incliners, the North part of the Meridian ascendeth, the South part descendeth: in all South incliners the South part of the Meridian ascendeth, the North part descendeth: upon both upper and under faces. And if these North and South incliners be direct, than the Meridian lieth in the Vertical line, and so maketh a right angle with the horizontal line: but if they decline, than the Meridian on the one side maketh an acute angle with the horizontal line. 3. To which part of the Meridian is the style with the substyle to be referred, as making with it an acute angle? The style is the cock of the Dial; the substyle is the line whereon it standeth, signed out in the former descriptions by the letters AU. In all Plains whereon the North pole is elevated, it is referred to the North part of the Meridian, and maketh an acute angle therewith. In all Plains whereon the South pole is elevated, it is referred to the South part of the Meridian, and is to make an acute Angle therewith. Except here only those South-incliners, whose Prosthaph: ark is more than the compliment of your Latitude: for on these plains the substyle standeth on that part of the Meridian, whose denomination is contrary to the Pole elevated above the Plain. For on the upper faces the North pole is elevated, but the substyle standeth toward the South end of the Meridian: and on the under 〈◊〉 the South pole is elevated, but the substyle lieth toward the North end of the Meridian. Note here, that in South-incliners whose Prosthaphaereticall ark is equal to the compliment of your Latitude, the substyle lieth square to the Meridian upon the line of 6 a clock; which line in such plains always lieth perpendicular to the Meridian line. Amongst these falleth the Equinoctial plain. 4. On which side of the Meridian lieth the substyle? In all direct plains it lieth in the Meridian. In all Decliners it goeth from the Meridian toward that coast which is contrary to the coast of the plains declination. And so do all hours also go upon the Plain to that coast which is contrary to the coast whereon they are; As all the morning or Eastern hours go to the Western coast of the Plain, and all the Evening or Western hours go to the Eastern coast of the Plain. Which being observed will be a great help to place them aright. 5. What plains have the line of 12 upon them, and which not? All upright Plains, in all latitudes whatsoever, declining from the South have the line of 12; and decliners from the North in the temperate Zone have it not, but in the other Zones they also have it. The upper faces of East and West incliners in all Latitudes have it, the underfaces have it not. The upper faces of all North incliners whatsoever have it; their under faces in the Temperate Zone want it, in the Frigid Zone have it, and in the Torrid Zone likewise if the Prosthaph: ark be greater than the Suns least North Meridian altitude, but if it be less they want it also. For South incliners, consider the Sun's greatest and least Meridian altitude upon the South coast. For if the Prosthaphaereticall ark be such as falleth between them, that is, if it be greater than the least, or less than the greatest, then have bothsides the line of 12 upon them; but if it be less than the least, than doth the Underface want it universally, and the upper face alone hath it▪ if greater than the greatest, then doth the Upper face want it, and the under face alone hath it: Except in the Frigid Zone where the upper face hath it also, by reason of the Suns not setting there for a time. 6. Whether the North or South part of the Meridian serveth for the line of 12? In those Plains that have the line of 12, where the North pole is elevated, there the North part of the Meridian serveth for 12. and where the South pole is elevated, there the South part of the Meridian serveth for the line of 12 or midday. Except, in all Latitudes, the under faces of those South incliners, whose Prosthaphaereticall ark falleth between the Sun's greatest and least Meridian altitudes, for in them the South pole is elevated, but the North part of the Meridian serveth for the line of 12. Except in special those Upright Plains in the Torrid-zone which look toward the North, and the Under faces of North-incliners also, whose Prosthaphaereticall ark is greater than the least North-meridian-altitude; for these have the South or lower part of the Meridian serving for 12, though the North pole be elevated. 7. Which way the style pointeth, and how it is to be placed? In Plains where the North pole is elevated, it pointeth up towards it; and where the South pole is elevated, it pointeth down towards it. The style lieth perpendicularly over the substyle, noted in the former figures with AV, and is to be elevated above it to such an angle as the Elevation of the pole above the Plain shall be found to be by the 6, 7, 8, and 9 Chapters. 8. When is it that that part of the Meridian next the substyle, and the line of twelve do go contrary ways? In all Latitudes, Upon the upper faces of South-incliners, whose Prosthaphaereticall ark is greater than the compliment of the Latitude, but less than the Sun's greatest South Meridian altitude: And on the Under faces of those South-incliners also, whose Prosthaph: is less than the compliment of the Latitude, but greater than the Suns least South meridian altitude: In the Torrid-Zone alone you must add hither also, North upright Plains, and those North-incliners on the Underface, whose Prosthaphaereticall-arke is greater than the least North-meridian altitude of the Sun; for these have the line of midday standing on that coast which is contrary to the coast of that part of the Meridian next the substyle, and none else. The line of 12. I call heareth line of midday because in the Frigid-zone, where the Sun setteth not in many days together, there are two twelves, the one answering to our midday, and the other to our midnight: and so all Upper faces of South-incliners, whose Prosthaphaereticall ark falleth between the least and greatest South meridian altitudes, have there two 12 a clockelines upon them. 9 How much the Meridian line ascendeth or descendeth from the horizontal line? The quantity of the Angle is to be found upon the circle on the backside of your Quadrant, in this manner; Extend the thread from the compliment of the Plains inclination taken in the lower Quadrant, to the compliment of the Plains declination counted in the Semicircle, and the thread will show you upon the Diameter, the degrees and minutes of the Meridian's Ascension or Descension. In the example of the 9 Chapt. taking the Upper face of that Plain, I find the Meridian to ascend above the horizontal line 33 gr. 41 minutes. ¶ These directions are sufficient for the bestowing of every line into its proper place and coast. As may be seen in the Example of the ninth Chapter. For, First, upon the upper face of that North incliner, because his Prosthaph: ark 16 gr. 6 min. is less than 52 gr. 30 min. the Latitude of the place, therefore the North pole is elevated above it: by the Answer to the first Quest. 2. Because it is a North-incliner, therefore the North part of the Meridian ascendeth above the horizontal line, by the answer to the second Question. 3. Because the North pole is elevated, therefore the Style with the substyle maketh an acute angle with the North end of the Meridian, by the Answer to the third Question. 4. Because this Plain declineth toward the West, therefore the substyle lieth on the East-side of the Meridian, and so do the hours of the afternoon: by the Answer to the fourth Question. 5. This Plain, being the Upper face of the North-incliner, will have the line of 12 to be drawn upon it, by the Answer to the fifth Question. 6. Because the North Pole is elevated, therefore the North part of the Meridian serveth for the line of 12: by the Answer to the sixth Question. 7. Because the North pole is elevated, therefore the style pointeth upward toward the North pole; by the Answer to the seventh Question. 8. That part of the Meridian next the Substyle, and the line of 12 are both one, and so therefore go both one way: by the Answer to the eight Question. 9 By the second the Meridian line ascendeth, and the quantity of the ascent is 33 gr. 41 min. above the horizontal line: by the Answer of the ninth Question. Thus you see every doubt cleared in this example: the like may be done in all others. CHAP. XIII. The making and placing of Polar Plains. Place this Diagram between folio 32. and 33. The horizontal line of the Plain. These Plains may have dials described upon them by this Quadrant, but the better way is the common way, to protract them by an equinoctial circle, for otherwise the style will be always of one distance from the Plain, be the Dial greater or lesser. The Polar plains that decline, before they can be described, must have their New-inclination known, and then their delineation will be easy, the manner of it may be seen in this Example. Suppose the upper face of a North-inclining Plain, lying in the Latitude of 52 gr. 30 min. to decline from the South toward the East 68 gr. and to incline towards the North 73 gr. 57 min. you shall find by the ninth Chapter, the Prosthaph: ark to be 52 gr. 30 min. the same with the Latitude of the place, and therefore you may conclude this plain to be Polar. By the same Chapter you shall find the New inclination to be 63 degrees. When you have these you may draw your Semicircle AB4, and divide it into 12 equal parts for the hours: so signing the new-inclination 63 degrees from A to B, draw CB: and supposing the altitude of your style to be CD, through D draw the perpendicular D 12; and where the lines drawn from C through the divisions of the semicircle do cut the line D 12, there raise perpendiculars for the hours, and so finish it up as the manner is. The style lieth directly over and parallel to the substyle CB, & the distance of it from the plain is CD, and in this Example the substyle CB standeth from the line of 12 Westward, because the plain declineth Eastward, according to the rules in the former Chapter, and so do the morning hours also. For the placing of the Dial in a true site upon the Plain, you shall find by the answer to the 9 Quest. in the former Chapter, that the Meridian ascendeth 55 gr. 38 min. for other necessaries, the precepts of the former Chapter will direct you. Only observe, that in Upright East and West plain, the line of 6 is always the substyle, and it ascendeth above the North end of the horizontal line, as much as the Latitude of the place cometh to. FINIS. AN APPENDIX Showing a ready way to find out the Latitude of any place by the Sun. BEcause in the third Chapter, and quite through this Treatise, the Latitude of the place is supposed to be known, when as every one perhaps cannot tell which way to find it out; I thought good therefore to add this Appendix as a ready help to show how it may be attained sufficiently for our purpose. Know then that for the finding out of the Latitude of a place by the Sun, these things are required. 1. To find the Meridian line. The readiest way to find the Meridian line is by the North-star. This star is within 2 degr. 37 min. of the North-pole. The North-pole lies very near between Allioth, or the root of the great Bear's tail, and this star; You may therefore imagine where the Pole is, if you conceive a right line drawn from the Polestar to Allioth, and by your imagination suppose ⅔ parts of the distance of the next star of the little Bear's tail from the Polestar towards Allioth, for there is the very Pole-point. Now than if you set up two poles aslope, and from the tops of them hang two cords with weights at the ends of them, and turn them till you standing on the Southside of them may see them both together with the Pole-point, as it were all in one line, then be sure these two cords do hang in the Meridian line, or very near it, yea so near it, that though you should err 3 degrees herein (wherein you need not to err one degree) yet will not the Meridian altitude in these Climates (especially more Northward) fail you above 3 minutes, which is near enough to our purpose. I have here given you the chief stars of the great and little Bears, that by them you may come to know the stars used in this observation, and so find the very Pole-point itself. 2. To find the Sun's Meridian altitude. Observe diligently about noon when the shadow of the South cord shall fall upon the North cord, for than is the Sun in the Meridian. At that instant observe the Sun's altitude steadyly and carefully, for that is the Meridian and greatest altitude of the Sun for that day. 3. To find the Sun's declination. For this purpose the limb hath the characters of the 12 Signs fixed to each 30 degree, and a scale of declinations under the limb noted with MN. The Scale is divided by this table; for look what degr. and min. of the Yclept. do answer to the degr. of declination in the table, the same are to be numbered in the limb, and by a ruler applied to them, the degrees of declination are drawn upon the Scale. A Table to make the Scale for the declination of every part of the Ecliptic. Degr. of decls. Deg. of the ecls. Degr. of decls Deg. of the ecls. Degr. declin. Degr. yclept. Degr. declin. Degr. Yclept Degr. declin. Degr. Yclept. Degr. declin. Degr. Yclept. 0.0 0.00 4.0 10.04 8.0 20.26 12.0 31.26 16.0 43.44 2.0 59.04 0.15 0.38 4.15 10.43 8.15 21.06 12.15 31.09 16.15 44.34 20.15 60.14 0.30 1.15 4.30 11.21 8.30 21.46 12.30 32.52 16.30 45.25 20.30 61.26 0.45 1.53 4.45 11.59 8.45 22.26 12.45 33.36 16.45 46.17 20.45 62.41 1.0 2.31 5.0 12.37 9.0 23.06 13.0 34.21 17.0 47.09 20.0 46.00 1.15 3.08 5.15 13.16 9.15 23.46 13.15 35.05 17.15 48.03 21.15 65.22 1.30 3.46 5.30 13.54 9.30 24.27 13.30 35.50 17.30 48.57 21.30 66.48 1.45 4.24 5.45 14.33 9.45 25.08 13.45 36.35 17.45 49.52 21.45 68 ●● 2.0 5.01 6.0 15.12 10.0 25.49 14.0 37.21 18.0 50.48 22.0 69.58 2.15 5.39 6.15 15.51 10.15 26.30 14.15 38.07 18.15 51.45 22.15 71.44 2.30 6.16 6.30 16.30 10.30 27.12 14.30 38.54 18.30 52.43 22.30 73.41 2.45 6.55 6.45 17.08 10.45 27.53 14.45 39.41 18.45 53.43 22.45 75.53 3.0 7.33 7.0 17.48 11.0 28.36 15.0 40.28 19.0 54.44 23.0 78.30 3.15 8.10 7.15 18.27 11.15 29.17 15.15 41.16 19.15 55.47 23.15 81.52 3.30 8.48 7.30 19.6 11.30 30.00 15.30 42.05 19.30 56.50 23.30 90.00 3.45 9.26 7.45 19.46 11.45 30.43 15.45 42.54 19.45 57.56 Finis. Before you can find the Declination, you must know the Sun's place, and for such as know not the use of the Astronomical tables, an Almanac will serve, where for every day at noon, you shall find the Sun's place in signs, degrees and minutes. The degr. and min. must be numbered in their Signs upon the limb, and the thread applied thereto will show the declination answerable. As for example. September 21. 1637 in the Almanac for this year, the Sun is found to be in 8 gr. 23 min. of ♎. In the Quadrants limb I look for the Sign ♎ and number there, 8 gr. 23 min. whereto apply the thread, I find it to cut in the scale of Declinations 3 gr. 20 min. 4. By the Meridian Altitude, and declination of the Sun had; how to find the Latitude of the place, or the Elevation of the Pole above the Horizon. Compare the Sun's Meridian altitude and declination together, and if the Sun be in a North Sign as ♈ ♉ ♊ ♋ ♌ ♍, then subtract the declination out of the Meridian altitude, so shall the difference give you the height of the Equinoctial. But if the Sun be in the South Signs, as ♎ ♏ ♐ ♑ ♒ ♓, then add the declination to the Meridian altitude, so shall the sum give you the height of the Equinoctial, which being taken out of the Quadrant or 90 degrees, leaveth the Latitude of your place, or the Elevation of the Pole above your Horizen. For Example. Upon the 21 of September 1637. I observed the Sun's altitude in the Meridian to be 34 gr. 10 min. Upon which day I find the Sun's place to be (as before) 8 gr. 23 min. of ♎, and the declination 3 gr. 20 min. And because the Sun is in a South sign, I add this declination and Meridian altitude together; the sum 37 gr. 30 min. is the altitude of the Aequator, and this taken out of 90 degrees leaveth 52 gr. 30 min. for the Latitude of Coventrie.