DIALLING UNIVERSAL: Performed by an easy and most speedy way. SHOWING How to describe the hour lines on all sorts of Planes whatsoever, and in any Latitude. Performed by certain Scales set on a small Portable Ruler. By G. S. practitioner in the Mathematics. LONDON Printed by R. and W. Leybourn for Thomas Pierrepont at the Sign of the Sun in Paul's Churchyard. 1657. TO THE READER. COurteous Reader, I here present to thy view a short Tract containing the description and uses of certain Scales to be put on a small Ruler, serving for the easy and speedy drawing of the Hour lines on all sorts of Planes, in what Latitude soever, and howsoever situated. The first Scale of Hours and the Scale of Latitudes, I acknowledge to be the Invention of that famous Mathematician Mr. Samuel Foster, laid down in his book of the uses of a Quadrant published Anno 1638. His Scale of Hours on the Quadrant, being really divided only into Hours and half hours. But by applying of the Thread and Bead, (by which means the hours are there found) the Scale is divided into any other parts requisite. Now having perused the labours of divers Authors that have written on this Subject and finding none (in my judgement) so easy, pleasant, and of so quick dispatch, for all dials with Centres; I caused that Scale to be put on a Ruler for my own use, subdivided into as many parts as the length of the Ruler would afford, which being exactly done. I found that the several hours and parts might be taken off with the Compasses from the Scale on the Ruler, with more ease and exactness than could be done from the Quadrant, by the application of the Thread and Bead; which whosoever shall carefully try both, will find to be true. The other Scales on the Ruler, (by which are made all sorts of Polar dials direct or declining, as also direct Meridian dials, together with fare Decliners, and generally all such as fall near the Pole, to be drawn without a Centre) are of my own framing; which although I esteem not worth publishing, yet I am confident will yield some contentment, and give some satisfaction to the ingenious; who will accept of my good will. Note that whereas in divers of the Chapters, I have set down short Tables, it was done only to show the manner of the work, not that there is any need at all thereof, for having the Inclination of the Meridian's converted into time, you have sufficient directions for setting off all the hours of any Dial; Note also that if your Ruler shall be a foot in length, the first hour Scale may have every several minute expressed on it, and the Polar Scales every fifth minute, and any Dial may be drawn in half a sheet of ordinary paper. But if it be of the length in the Book, any Dial may then he drawn in a quarter of a sheet of paper. I conclude referring thee to the Book itself, which if it shall find acceptance with thee in yielding thee in any kind the satisfctation expected I have my ends and desires. G. S. The Contents. Chap. 1 The Description of the Scales on the Ruler. Pag. 1 Chap. 2 How to draw a Meridian line. Pag. 2 Chap. 3 How to find the Inclination of a Plane. Pag. 3 Chap. 4 How to find the Declination of a Plane. Pag. 4 Chap. 5 How to draw a direct Polar Dial. Pag. 5 Chap. 6 How to draw an Erect Meridian Dial. Pag. 6 Chap. 7 How to draw an Horizontal Dial. Pag. 8 Chap. 8 How to draw an Erect direct vertical. Pag. 10 Chap. 9 How to make a vertical inclining Dial. Pag. 11 Chap. 10 How to make an Equinoctial Dial. Pag. 12 Chap. 11 How to find the height of the Pole above the Plane, etc. for Decliners and Declining Inclining Planes, Geometrically, only by a line of Chords. Pag. 13 Chap. 12 How to make a vertical declining Dial. Pag. 17 Chap. 13 How to make a vertical declining Dial, where the Evation is small. Pag. 19 Chap. 14 How to make a Meridian Inclining Dial. Pag. 22 Chap. 15 How to make a Declining Inclining Dial. Pag. 24 Chap. 16 How to make a Polar Declining Dial. Pag. 27 diagram An Advertisement to the Reader. NOte that the Diagrams are all contracted, otherwise they would have taken up a great deal of room to little purpose, for as they are, they give as much light to the ingenious practitioner as if they had been drawn by the Scale annexed, and every Diagram have filled a whole page or near upon, whereas now it takes up but a small portion of a page. And note that these Scales and all other Mathematical Instruments whatsoever, are made and sold by Mr. Anthony Thompson, dwelling in Hosier lane near Smithfield. DIALLING: CHAP. I. The Description of the Scales on the Ruler. THe first Scale is a Scale of six hours, each hour divided into halves and quarters of hours, being proper for Dial's that have no declination: as the Horizontal, the direct North or South, whether upright or inclining. Moreover each hour is subdivided into as many unequal parts, as the length of the Scale will permit, this Scale is of special use in describing all other Dial's that have Centres; where the Altitude of the Style above the Plain is not less than 10 degrees; when that happens there are other Scales to perform it with speed and exactness enough. This Scale hath the letters, (Hour,) at the beginning. The second is a Scale of 90 degrees, answerable to the Scale of Hours, every degree being also subdivided into as many parts as quantity will give leave. This Scale is known by the letters (Incl.) at the beginning; Note that the first and second Scales together show how many degrees and minutes of time are contained in any number of degrees and minutes of the Equinoctial under 90 & contra. Example. Against 20 degrees on the second Scale; you shall find on the first Scale 1 hour 20 minutes, and against 50 degrees on the second Scale, you shall find 3 hours 20 minutes on the first Scale, and so of any other number what ever, if carefully computed. The third Scale is a Scale of Latitudes, known by the letters (Lat.) at the beginning. The fourth and fifth Scales are 2 Scales of hours of different lengths, either of which serve to make all direct Meridian or Polar dials, also all Polar dials that decline. Both these Scales together serve for the speedy drawing the hour-lines on all sorts of Planes, where the height of the Style is but small, and the Centre of the Dial left out, these Scales are known by the letters Pol. at the beginning. These are the chief Scales, there is also a Scale of Chords, whose use is common. CHAP. II. To draw a Meridian line on an Horizontal Plane. diagram WIth your line of chords describe the Circle ABCD, then holding up a thread and plummet, so as the shadow of the thread may pass through the Centre E, draw the line of shadow DEB: Take then the Altitude of the Sun (or rather let another take it at the same instant,) which done get the Sun's Azimuth, which in this Example shall be 50 degrees from South towards East, which 50 deg. I take out of the line of Chords, and set it from the point D, Southward to A, drawing the line AEC for the Meridian line. CHAP. III. To find the Inclination of a Plane. THe Inclination of a Plane is the angle which it maketh with the Horizon: Draw first an Horizontal line as AB, diagram and cross it with the perpendicular CD, then apply a Quadrant to the vertical line CD, on a Ruler as the Figure directeth you, and holding up a thread and plummet, so as that the plummet may by means of the Ruler, play under the Centre of the Quadrant, the thread passing through the Centre, observe the degrees cut in the limb, and count them from that side of the Quadrant which is perpendicular to the plane, that shall be the Inclination of the Plane, as the angle HFE: the Compliment whereof EFG, is the angle of Reclination of the Plane from the Zenith. CHAP. iv To find the Declination of a Plane. THe Declination is always reckoned in the Horizon, and is the angle contained between the line of East and West and the Horizontal line upon the Plane: Now to get the Declination two observations are required, First, the Horizontal distance of the Sun from the Pole of the Plane; Secondly, the Sun's Altitude. Draw first an Horizontal line, as the line I K in the figure of the last Chapter, then apply thereto the side of a Quadran, holding it parallel to the Horizon, then hold up a Thread and Plummet, so as the shadow of the Thread may pass through the Centre of the Quadrant, then observe the degrees cut off by the Threads shadow on the limb, and reckon them from that side of the Quadrant, that is perpendicular to the Plane, as the angle L M N, that is the Horizontal distance. diagram 2 Secondly, take the Sun's altitude, and find his Azimuth, these compared together will help you to the declination, as may be plainly demonstrated by this figure: describe the Circle A B C D: representing the Horizontal circle, draw A E C for the Horizontal line of the Plane, then set off the Horizontal distance L M N, which in this Example shall be 35 degrees: these I place from A to F, then having taken the Sun's altitude, I find the Azimuth from East towards South to be 50 degrees, these I place from F the Sun's place, to G Northward, and draw the line G H through the Centre, so shall G H be the true points of East and West, and I K drawn at right angles to G H shall be the points of North and South: Now the angle of declination is A E G, or K E M, which in this Example is 16 degrees. CHAP. V To draw a direct Polar Dial. A Direct Polar Plane is that which is parallel to the hour circle of 6. First, draw the Horizontal line A B, with its parallel C D, then draw the perpendicular E F, which shall serve for the substile and hour of 12, then repair to either of your Scales known by the letters Pol. and fixing one foot of your Compasses at the beginning of the said Scale, extend the other foot to the hour of 1, set off that extent from E to 11, and from E to 1, as also from diagram G both ways, on the line A B, and draw the hours of 11 and 1, parallel to the substile E G F: then again extend your Compasses from the beginning of the Scale to two hours, set off that extent from E to 10, and from E to 2, also from G both ways, and draw the hours of 10 and 2: Then extend the Compasses from the beginning of the Scale to 3, and set off that from E to 9, and from E to 3, and draw the hours 9 and 3 parallel to the former: then extend the Compasses from the beginning of the Scale to 4, and set off that from E to 8, and from E to 4, also from G both ways, and draw the hours of 8 and 4. Lastly, extend the Compasses from the beginning of the Scale to 5, and set it off from E to 7, and from E to 5, also from G both ways, and draw the hours of 7 and 5. So have you all the hours proper for this Plane. The stile may be a thin Plate of brass or Iron that must stand directly perpendicular to the Plane, and in the substilar line or hour line of 12. The length whereof must be the distance from the beginning of the Scale to the hour of 3. This done, the Dial is finished. CHAP. VI To draw an erect Meridian Dial. A Meridian Plane is that which is parallel to the circle of the hour of six, having one face to the East, the other to the West, in each of them the stile and substile will be parallel to the Plane, and the hour-lines parallel one to the other, as in the direct Polar of the last Chapter, our example shall be of a Meridian Dial, with the face to the East, in the Latitude of 51 d. 30 min. and is thus described. diagram The distance from the beginning of the Scale to the hour of 3, giveth the height of the Style which must stand directly over the Substile, making right angles therewith, and may be made of a thin plate of Iron or Brass, or a pin of either sharpened; whose extremity must give the shadow to the hour lines on the Dial; Note that the making of the West Dial differeth from this only in its situation and changing the hours, this Dial looking to the East and the other to the West, this serving from Sun rising to past 11, the other from before 1 to Sunset. Note if you turn the East Dial drawn in paper from you, and look on the backside, you shall there see the perfect form of the West Dial, only instead of the hours 11, 10, 9, 8, 7, 6, 5, 4, you must write 1, 2, 3, 4, 5, 6, 7, 8. CHAP. VII. To draw an Horizontal Dial for the Latitude of 51 deg. 30 min. AN Horizontal Plane is that which is parallel to the Horizontal Circle of the Sphere, to draw the hours proceed thus. diagram With your line of Chords make an angle of 51 deg. 30 min. for the stile or cock of your dial, and set it over the substile A D, at right angles as the angle E A D, in the Diagram, and the Dial is finished. CHAP. VIII. To draw a Dial on an erect direct vertical Plane. diagram The Style E A D must contain an angle of 38 deg. 30 min. and must stand directly over the Substile A D. For the North face, the Centre must be below and the Style point upward. The hours fit for that Plane are the hours of 4, 5, 6, 7, in the morning, and 5, 6, 7, 8, in the evening. CHAP. IX. How to draw a vertical Inclining Dial. IF the Inclination be towards the North part of the Horizon, you are to subtract the Inclination out of the compliment of the elevation, and the remainder is the new Latitude. Example. Of a South Dial in the Latitude of 51 deg. 30 min. inclining Northward 25 deg. I deduct 25 out of 51.30, and there remaineth 13.30. the elevation of the Pole above the Plane, having found the elevation of the Pole above the Plane I proceed to make the Dial, as if it were an upright South Dial, for the Latitude of 13 deg. 30 min making A B and A C equal to 13.30 taken out of the line of Latitudes, following the rules of the last Chapter for the rest of the work. The Style must contain an Angle of 13.30 and must be set directly over the Substile A D, and the Dial is finished. But if the Inclination be towards the South part of the Horizon, then add the Inclination to the Latitude and the sum is the elevation of the Pole above the Plane, if the sum exceed 90, take it out of 180, and the remainder is the elevation of the Pole above the Plane. Example. In the Latitude of 51 d. 30. a Plane found to incline Southwards 15 degrees, I add 15 d. to 51. 30 the sum is 66.30 that is the elevation of the Pole above the Plane, so observing the former directions you may proceed to make the Dial as is before taught. CHAP. X. How to draw an Equinoctial Dial. AN Equinoctial Plane is that which is parallel to the Equinoctial Circle of the Sphere: Make AB and AC of the former Diagrams, equal to the whole line of Latitudes, and proceed as if you were to make an Horizontal Dial, and set up a sharpened point in the Centre of any convenient length. But the best way to draw this Dial, being the hours are equidistant, is to divide 360 by 24, the quotient is 15, so having with the Radius of your line of Chords described a Circle and drawn the Diameter for the 2 hours of six, and the perpendicular for the hour of 12, take 15 degrees out of your line of Chords, and set it off from hour to hour, let this Dial contain as many hours as the Horizontal and so numbered. CHAP. XI. How to find the height of the Pole above the Plane, the distance of the Substile from the Meridian, and the Inclination of Meridian's, for upright declining and Meridian inclining dials; as also what ever else is necessary to be found for all other dials, hereafter treated of before the hour lines can be drawn. THe best and exactest way to find these, is by the Tables, or Canons of Logarithms; but of that I will not touch in this small Treatise, but make use of some Geometrical way whereof there are divers, but the best and easiest in my judgement (it requiring only a line of Chords,) is that of Mr. Stirrup, which I here make use of (I hope without offence. The first Example shall be of a Plane declining from South 28 degrees towards the East: as by the first Diagram of this Chapter is Demonstrated as followeth. diagram Then from the point L, draw LT cutting the arch GRACCUS in O; draw AOI: so shall CI be the Inclination of Meridian's 34 deg. 12 minutes. diagram diagram CHAP. XII. How to draw the hours on an erect vertical declining Plane. HAving by the last Chapter obtained those necessary requisites there mentioned I now proceed to draw the hour-lines on all thoses Planes treated of in that Chapter, by our first Scale of Hours, as also all Polar declining Dial's, and all far declining vertical, by the Polar Scales. And first for the vertical Decliner mentioned in the last Chapter declining from South towards East 28 degrees in the Latitude of 51 deg. 30 min. Proceed thus: first, draw a line at length as BAC, then considering that the height of the Pole above the Plane was 33 deg. 20 min. I take 33 deg. 20 min. out of the Scale of Latitudes, and set it off from A to B, and from A to C, choosing A for the Centre of the Dial. Then take in your Compasses the whole first Scale, or Scale of Hours, and with one foot fixed in B with the other make an arch at D, and with the same extent, one foot fixed in C, with the other cross the arch at D, and draw the lines BD and CD, as also the line AD for the Substile: Then having found the Inclination of the Meridian's to be 34 deg. 12 min. I seek that in the second Scale known by the Letters Ind. and just against it on the first Scale, I find two hours 17 min. then take off two hours 17 min. from that Scale, and set it from D to 12, and from B to 6, then extend your Compasses from the beginning of the same Scale too hours 17′ and set that from D to 10, diagram and from B to 4, then open your Compasses to 1 hour 17′ of the same Scale and set it from D to 11, and from B to 5, then open the Compasses to 3 hours, 17 min. and set it from D to 1, and from B to 7, then open the Compasses to 4 ho. 17′, and set it from D to 2, and from B to 8, than lastly, open your Compasses to 5 hours 17 min. and set it from D to 3, and from B to 9 So lines drawn from A to those points shall be the hours proper for this Plane: if you please you may set off the largest extents first, as 5 hours 17 min. and so go on closing of the Compasses till you come to o hours 17 min. for D 10, and B 4. Note that the first Column of the Table contains the hours and minutes that are to be taken in your Compasses off of the first Scale, and set from D or B to their Ho. M. from D toward C, from B towards D. 5. 17 D— 3 And B— 9 4. 17 D— 2 And B— 8 3. 17 D— 1 And B— 7 2. 17 D— 12 And B— 6 1. 17 D— 11 And B— 5 0. 17 D— 10 And B— 4 respective hours on the Plane, contained in the second and third Columns of the Table: consider also that the Inclination of Meridian's; being 34 d. 12′ or 2 ho. 17′ of time and the declination towards East, the Substile falls between the hours of 9 and 10 in the morning, whereas if the declination had been West the Substile would have fain betwixt the hours of 2 and 3 in the afternoon, and the hour of 12 would have been towards the left hand of the Substile, whereas the declination being East, it stands on the right hand of the Substile. The stile must have elevation 33.20. as the angle EAD, and must stand directly over the Substile AD, and the Dial is finished. I have been more large in this Chapter than I shall be in what shall follow, touching all other Dial's with Centres, by reason the drawing and setting off the hours is the same with the work of this Chapter. CHAP. XIII. To draw the hours on a fare declining vertical Plane, by the 2 Polor Scales on the Ruler. THose Planes whose declination, or declination and Inclination shall cause them to fall near the Pole, so as that the hours can hardly be distinguished they falling so near together, must have the Centre left out and the Style increased, and then the hours may be easily and speedily drawn by the directions following. diagram Now for the other line DIBF, you must set off the hours thereon taken out of the lesser Polar Scale from B, according to the Table in all respects changing only A for B: this done draw lines through the respective points in the lines CAESAR and DBF, which shall be the hour lines of the Dial. Note that a Dial of this kind is near as socn made as spoken of, The Style must be a thin plate to stand directly over the Substile, as in the figure is demonstrated by AB and CD. CHAP. XIV. To draw the Hours on a Meridian Inclining Plane. THose Planes whose Horizontal line is the same with the Meridian line, are called Meridian Planes, as the direct East and West. But if they lean to the Horizon they are called Incliners. Those Planes may incline either to the East or West part of the Horizon, and each of them hath two faces, the upper towards the Zenith, the lower towards the Nadir. diagram Take notice that when you are to draw any of the dials with Centres, let the line BAC stand towards you, as if it were the Horizontal line, and the line AD a Plumb line, and so you will set off the hours with more ease as the direct vertical. Note that this Example is of a Meridian Plane inclining East, and therefore the Substile must stand to the left hand of the Meridian or hour of 12. CHAP. XV. To draw the Hour lines on a Declining, Inclining Plane. A Plane that declines from the prime Vertical and Inclines to the Horizon, and yet lieth not even with the Poles of the World, is called a Declining Inclining Plane. Of these there are several sorts, for the Inclination being Northward, the Plane may fall between the Horizon and the Pole, or between the Zenith and the Pole. Or the Inclination may be Southward, and may fall either below the intersection of the Meridian, and the Aequator, or above it: and each of these have two faces, the upper towards the Zenith, and the lower towards the Naidir. diagram Then considering the Inclination of Meridian's was 15 deg. 17 min. I find it on the second Scale, and against it on the first Scale, I find 1 hour 1 min. so I set off the hours as hath been formerly shown and this Table demonstrateth. from D towards B from C towards D. Ho. Min. Hours on the Plane. 0. 1 D— 11 And C— 5 1. 1 D— 12 And C— 6 2. 1 D— 1 And C— 7 3. 1 D— 2 And C— 8 4. 1 D— 3 And C— 9 5. 1 D— 4 And C— 10 The Style must contain an angle of 30 deg. 20 min. as JAK in the Diagram to stand directly over the Substile AD, and so the Dial is finished. Now when you meet with Declining Inclining Planes you must consider which Pole is elevated above your Plane and how to place the Meridian from the Horizontal line, for upon the upper faces of all North Incliners, whose Meridian's Elevation is less than the Latitude of the Place, on the under faces of all North Incliners, whose Meridian's Elevation is greater than the Latitude of the place, and on the upper faces of all South Incliners the North Pole is elevated. And upon the under faces of all North Incliners, whose Meridian's Elevation is less than the Latitude of the place, on the upper faces of all North Incliners, whose Meridian's Elevation is greater than the Latitude of the place, and on the under faces of all South Incliners the South Pole is elevated. Now for placing the Meridian from the Horizontal line, upon the upper faces of all South Incliners whose Meridian's Elevation is greater than the compliment of Latitude, on the under faces of all South Incliners, whose Meridian's Elevation is less than the Latitudes, compliment, on the under faces of all North Incliners, whose Meridian Elevation is greater than the Latitudes of the place, and on the upper faces of all North Incliners, whose Meridian's Elevation is less than the Latitude of the place the Meridian must be placed above the Horizontal line as in our Example. Again, contrariwise for the upper faces of all South Incliners, whose Meridian's Elevation is less than the Latitudes compliment. On the under faces of all South Incliners, whose Meridans Elevation is greater than the compliment of Latitudes, on the under faces of all North Incliners, whose Meridian's Elevation is less than the Latitude of the place, and on the upper faces of all North Incliners, whose Meridian's Elevation is greater than the Latitude of the place, the Meridian must be placed below the Horizontal line. But if it be either the upper or under faces of a South Inclining Plane, whose Meridian's Elevation is greater than the Latitudes compliment, or either the upper or under faces of a North Inclining Plane, whose Meridian's Elevation is less than the Latitude of the place, that then the Meridian must be placed from that end of the Horizontal line with the Declination of the Plane but on all the other faces of these kinds of Planes, the Meridian must be placed from that end of the Horizontal line, which is contrary to the Declination of the Plane. Note also that if the Inclination be Southward and the Elevation of the Meridian, equal to the compliment of your Latitude, then shall the Substile lie square to the Meridian. CHAP. XVI. To make a Dial on a Polar declining Plane. THese Planes if the Inclination be Northward and the Elevation of the Meridian equal to the Latitude of the Place then neither Pole is elevated above the Plane, and therefore it's a declining Polar. diagram Example. A Plane declining East from South 30 deg. and inclining North 34 deg. 30 min by the rules of the eleventh Chap. I find the distance of the Horizon and Meridian to be 71 deg. 53 min. which I set off from A to B in the arch AB, and draw BD for the Substile. Then at right angles to the Substile, I draw the lines FG and CE: Than considering the Inclination of Meridian's to be 24 deg. 19 min. I find it on the second Scale, and against it on the first Scale I find 1 ho. 37 min. Then having recourse to the Table, I set off the hours from K and H, according to the directions of the 13 Chapter, for the fare Decliner, taking the several distances with my Compasses out of either of the Polar Scales, and setting them off from K towards C or E, as the Table plainly showeth. from K toward E, from K towards C. Ho. M. Ho. Plane Ho. M. Hours Plane. 0. 37 K 11 0. 23 K 10 1. 37 K 12 1. 23 K 9 2. 37 K 1 2. 23 K 8 3. 37 K 2 3. 23 K 7 4. 37 K 3 4. 23 K 6 The extent of the Compasses from the beginning of the Scale to the hour of 3 gives the height of the Style, which must be a plate of Iron or Brass, set up just over the Substile HK, and the Dial is finished. Some observations relating chief to the Dial of the 12 Chap. being a vertical Decliner. WHereas in the Diagram of the 12th Chapter the Substile stands square to the Horizon, you see here how it ought to stand, that is to say the 12 of clock hour must in this and all other vertical Decliners be the Plumb line as the Diagram here showeth. diagram Note that no hour lines are to be drawn beyond the line BC. Note also that if you oil the Pattern of this Dial drawn in paper, it will serve for 3 other dials that have the same Declination. First, a Dial for the South-west face of the Plane if you change the side, and the numbers set to the hours, the Centre of the Dial upwards and the Style and Substile pointing downwards. Secondly, a Dial for the Northwest face, if you turn the pattern upside down, and changing the side taking the backside for the foreside, not altering the hours the Style and Substile pointing upward. Lastly, a Dial for the North-east face, if you take the foreside only turning it upside down, and altering the numbers set to the hours, the Style and Substile pointing upward. A Table showing the Latitude of the most principal Cities and Towns in England. Name of the places. Latitude d. m. St. Alban 51 55 Barwick 55 49 Bedford 52 18 Bristol 51 32 Boston 53 2 Cambridge 52 17 Chester 53 20 Coventry 52 30 Chichester 50 56 Colchester 52 4 Derby 53 6 Exon 50 40 Grantham 52 58 Halifax 53 49 Hereford 52 14 Hull 53 50 Launston 50 41 London 51 32 Lancaster 54 8 Leicester 52 40 Lincoln 53 15 Newcastle 54 58 Northampton 52 18 Oxford 51 54 Shrewsbury 52 48 Warwick 52 25 Winchester 51 10 Worcester 52 20 Yarmouth 52 45 York 54 0 FINIS.