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 Dials 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 Dials that have Centers; where the Altitude of the Stile above the Plain is not lesse then 10 degrees; when that happens there are other Scales to performe it with speed and exactnesse 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 shew 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 [Page 2]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 Dialls, also all Polar Dialls that decline.

Both these Scales together serve for the speedy draw­ing the hour-lines on all sorts of Planes, where the height of the Stile is but small, and the Center of the Diall 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.

WIth your line of chords describe the Circle ABCD, then holding up a thred and plummet, so as the shadow of the thred may pass through the Center 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 Suns [Page 3]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,

and crosse 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 thred and plummet, so as that the plummet may by means of the Ruler, play under the Center of the Qua­drant, the thred passing through the Center, 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 Complement 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 alwayes 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 Suns 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 Threed and Plummet, so as the shadow of the Threed may pass through the Center of the Quadrant, then observe the degrees cut off by the Threeds shadow on the limbe, 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.

2 Secondly, take the Suns altitude, and finde his Azimuth, these compared together will help you to the declination, as may be plainly demōstrated 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 Suns altitude, I [Page 5]find the Azimuth from East towards South to be 50 de­grees, these I place from F the Suns place, to G North­ward, and draw the line G H through the Center, so shal 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

G both wayes, 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 [Page 6]from G both wayes, 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 wayes, 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 wayes, 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 houre 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.

Draw first the Horizontal line A B, then with the Ra­dius of your line of Chords, one foot fixed in B, with the other draw the arch C D, and in the arch C D, set off [Page 7]the height of the Equinoctial 38 d. 30 min. from C to D, and draw the line B D E, then in some convenient place of the line B E, as at K, draw the line G K F at right an­gles to the line E D B, so shall G F be the Substile and hour of 6, then to E D B draw a parallel as I H, having so done work as in the last Chapter was shewn, by taking the distance from the beginning of either of the Polar Scales to 1 hour, setting that extent from G to 5, and from G to 7, and on its parallel E B, from K, both wayes, and draw the hours of 5 & 7 parallel to the substile and hour of 6, then take in your Compasses the distance from the beginning of the Scale to 2 hours, and set it off from G to 4, and from G to 8, and also from K both wayes, and draw the hours of 8 and 4, parallel to the other, so you have no more hours to set off towards I, being 4 is the hour of Sun rising, but proceed to set off the rest of the hours towards H,

as you did the o­ther, taking in your Compasses, the di­stances from the beginning of the Scale to 3, 4 and 5, setting each di­stance off respe­ctively from G to­wards H at 9, 10, and 11, and also from K towards B & draw the, hours, of 9, 10, 11: and so you have all the hours proper for this Dial.

The distance from the beginning of the Scale to the hour of 3, gives the height of the Stile which must [Page 8]stand directly over the Substile, making right angles therewith, and may be made of a thin plate of Iron or Brasse, or a pin of either sharpned; whose extremity must give the shadow to the hour lines on the Diall; Note that the making of the West Diall differeth from this only in its scituation and changing the hours, this Diall looking to the East and the other to the West, this serving from Sun rising to past 11, the other from be­fore 1 to Sun-set.

Note if you turne the East Dyall drawn in paper from you, and look on the backside, you shall there see the perfect form of the West Diall, onely 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 Diall 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.

Draw a line at length as B A C, then repaire to your Scale of Latitudes and with one foot fixed at the begin­ning of the same Scale, extend the other to 51 deg. 30 min. and set off that extent from A to B and from A to C, and draw the lines A B, A C, then open your Com­passes to the length of the whole Scale of hours, or first Scale, and with that extent, one foot fixed in B, with the other foot make an arch at D, then with the same extent, [Page 9]

one foot fixed in C with the othercross that arch at D, and draw the lines B D and C D also draw the line A D for the substile & houre of 12, thus done re­turn to your first Scale on your Ruler extending your Compasses from the beginning of that Scale to 1 hour, and set off that extent from D to 1, and from D to 11, also the same extent must be set from B to 5, and from C to 7. Then extend your Compasses from the beginning of the said scale to 2, and set it off from D to 10, and from D to 2: also from B to 4, and from C to 8. Lastly, extend your Compasses from the beginning of the said scale to 3, and set that off from D to 3, and from D to 9, so have you points for 12 hours and lines drawn from the center A to each of these points shall be the hours requi­red. Note that the hours of 4 and 5 in the morning, and 7 and 8, in the evening are to be drawn through the cen­ter, if you desire the halves and quarters, they are to be drawn in all respects as the hours.

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 [Page 10]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.

A Vertical Plane is that which is Parallel to the Prime Vertical Circle of the Sphere, it hath two fa­ces one to the South, the other to the North: our exam­ple shall be of a South Plane in the latitude of 51 deg. 30 min. The making of this Dial differeth little from the Horizontal only, you must take notice that this Plane re­specteth the South Pole, which Pole is elevated thereon 38 deg. 30 min. being the complement of the latitude, and so much you must take out of your Scale of Lati­tudes, and set off from A to B, and from A to C, then take in your Compasses the whole scale of hours or first scale,

and with that extent one foot fixed in B with the other make an arch at D, againe with the same extent, one foot fixed in Cwith the other cross the arch at D, and draw the lines B D and C D, also A D [Page 11]for the Substile and hour of 12. Finish the Diall in all respects as you did the Horizontal; onely observe that whereas in that Diall the hours of 1, 2, 3, &c. were set off on the line B D, towards the left hand, the same houres must in this Diall be set off on the line C D, to­wards the right hand.

The Stile E A D must containe an angle of 38 deg. 30 min. and must stand directly over the Substile A D.

For the North face, the Center must be below and the Stile 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 com­plement of the elevation, and the remainer is the new Latitude.

Example. Of a South Diall 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 Stile must contain an Angle of 13.30 and must be set directly over the Substile A D, and the Diall is fi­nished.

But if the Inclination be towards the South part of the Horizon, then adde the Inclination to the Latitude and the summe is the elevation of the Pole above the Plane, if the summe exceed 90, take it out of 180, and the re­mainer 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 summe is 66.30 that is the elevation of the Pole above the Plane, so observing the former directions you may proceed to make the Diall as is before taught.

CHAP. X. How to draw an Equinoctial Diall.

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 Diall, and set up a sharpned point in the Center of any convenient length.

But the best way to draw this Diall, being the houres 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 Diall containe as many houres as the Horizontal and so numbred.

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 Meridi­ans, for upright declining and Meridian in­clining Dialls; as also what ever else is ne­cessary to be found for all other Dialls, 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 onely 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.

Describe a Quadrant as ABC: Set off the Latitude 51.30 from B to E, draw ED parallel to AB: then take the distance DE with your Compasses, setting one foot in A, with the other draw the Arch GHOR: then set off the declination 28, from B to F, draw FA, cutting [Page 14]the Arch GR in the point H,

through which point H draw the line SHN, so shall CN be the height of the Pole a­bove the Plane 33 d. 20′, then take the di­stance HS, and set it off from D to K, and draw the line AKL, so shall CL be the di­stance of the Substile from the Meridian.

Then from the point L, draw LT cutting the arch GR in O; draw AOI: so shall CI be the Inclinati­on of Meridians 34 deg. 12 minutes.

The second Example is of a Meridian Inclining Plane in the Latitude of 51 deg. 30 min. inclining 40 deg.

This worke differs little from the former, describe a Quadrant as ABC, set off the Latitude 51 deg. 30 min. from C to E draw ER pa­rallel to AB, then with the distance ER raw the arch GH OD: then set off the Inclination 40 from B to F, and ppraw AF, cutting the arch GOD in H, then draw SHN: so shall CN be [Page 15]the elevation of the Pole above the Plane 36 deg. 49 min. Then take the distance SH, and set it from R to K, draw AKL, so shall CI, be the distance of the sub­stile from the Meridian 38 deg. 57 min. Lastly, draw LT cutting the arch GD, in O draw AOI, so shall CI be the Inclination of Meridians 53 deg. 26 min.

The third Example is of a Declining Inclining Plane, in the Latitude of 51 deg. 30 min. declining from South towards East 34, and Inclining to the Horizon 24 deg. First draw the Quadrant ABC, set off the declination from C to E: draw AE,

then set off the Inclination 24 from B to F: draw FZ parallel to AC, then draw the arch GHI, cutting AE in H, through the point H, draw KL cutting the arch CB in K, take the distance HL, and set it in the line FZ unto O, and draw the line AOM, cutting the arch BC in M, from which point M draw MPN parallel to AB, cutting the arch GI in P, through which point P draw APQ, cutting the arch BC in Q, so shall the arch BK, be the Inclination of the Plane to the Meridian 76 deg. 51 min. and the arch BQ, the Meri­dians ascension 58 deg. 22 min, or the arch of the Plane betwixt the Horizon and Meridian, and the arch BM shall be the Elevation of the Meridian, or the arch of the Meridian between the Horizon and the Plane 20 deg. [Page 16]16 min. Now if the Plane incline towards the South add the elevation of the Meridian, to your Latitude, and the Summe shall be the Position Latitude, or the arch of the Meridian betwixt the Pole and the Plane: if the sum exceed 90, take it out of 180, and that which remains shall be the Position Latitude: But if the Inclination shall be Northward, then compare the Elevation of the Meridian with your Latitude, and take the lesser out of the greater, so the difference shall be the Position Lati­tude as in this example supposing the Inclination to be Northward, I take 20 deg. 16 min. the elevation of the Meridian out of 51 deg. 30 min. the Latitude, and there remains 31 degrees 14 minutes for the Position latitude, or the arch of the Meridian between the Pole and the Plane: This being done set 31 degees 14 minutes from B to T in the arch BC, and draw the line AT, then with the distance KL upon the Center A describe the arch YMW cutting the line AT in M through M draw RS. Parallel to AB cutting the arch BC in S, so shall BS be 30 d. 20′. the height of the Pole above the Plane. Then lay your Rule upon the point S, & the Center A, & where it shall cut the line KL there make a mark as at V, through V draw the line Dunw cutting YW in the point n, and the arch BC in w, so shall the arch B w be 7 deg. 51 min. the distance of the Substile from the Meridian. Lastly, through the point n draw YX pa­rallel to AC, so shall BX be 15 deg. 17 min. the Incli­nation of Meridians.

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 Dials, 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, chusing A for the Center 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 Meridians 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′ [Page 18]and set that from D to 10,

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 houres, 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, then last­ly, open your Compasses to 5 houres 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 Columne 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

respective hours on the Plane, contained in the se­cond and third Co­lumns of the Table: consider also that the Inclination of Meri­dians; [Page 19]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 Diall is finished. I have been more large in this Chapter then I shall be in what shall follow, touching all other Dials with Centers, 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 farre declining vertical Plane, by the 2 Polor Scales on the Ruler.

THose Planes whose declination, or declination and Inclination shal cause them to fall neer the Pole, so as that the hours can hardly be distinguished they falling so neer together, must have the Center left out and the Stile increased, and then the hours may be easily and speedily drawn by the directions following.

The Example shall be of a vertical Diall declining East from South 80 d. By the directions of the 11th. Chapter, I find the Inclination of Meridians to be 82 d. 8′ the height of the Pole above the Plane 6 d. 12′, and the distance of the Substile from the Meridian 38 d. 5 °: Draw first the perpendicular GH, then set off the Stile [Page 20]and Substile according to their angles, and draw the prick'd lines GBA, for the Substile, and GIK for the Stile: Then chuse any point in the Substile as A, through which point A, draw a line at length, making right an­gles therewith as the line EAC, that done fix one foot

of your Compasses at the beginning of your longest Po­lar Scale, and extend the other to the houre of 3, set off this extent from A to C, and through the point C draw a line Parallel to GIK, the stile as the line CD. So shall CD be the stile increased then open your Com­passes from the beginning of your second or lesser Polar Seale to 3 hours, place one foot of the Compasses in the [Page 21]substile ABG, carrying it along the line until the other point of the Compasses, may just touch the line CD, and and there make a mark as at B, through which point B draw a line at length perpendicular thereunto and pa­rallel to the line EAC: Then considering the Inclina­tion of Meridians to be 82 deg. 8 min. I find that on the second Scale, & against it on the first Scale I find 5 hours 29 min. by which it appears that the substile falleth be­twen the hours of 6 and 7 in the morning. According to the Inclination of Meridians 5 hours 29. I proceed to make a Table, and to set offt he hours accordingly as fol­loweth. I open my Compasses to 0 hours, 29 min. of the greater Polar Scale, and set it off from A to the hour of 7 in the line CAE, then I open the compasses to 1 houre 29 minutes of the same scale, and place it from A to 8, then I open the compasses to two hours 29 min. and set it off from A to 9. then I take in my compasses 3 houres 29 min. and set it from A to 10. Lastly, I take in my Com­passes, 4 d. 29′ & set it from A to 11. Now having found the points for the hours on one side of the substile from A towards E, I must do the like for the other side of the substile: and considering that the first hour from the Sub­stile

to­wards E, namely A 7, contains the dist­ance of 0 hours 29′, as by the Table appears, the complement thereof to 60 or 1 hour: is 0 ho. 31′, as by the said Table appears, so I proceed to set off 0 ho. 31′, taken from the same Polar Scale; [Page 22]from A to the hour of 6, and the extent of the compas­ses from 1 ho. 31′; I set from A to 5, also 2 ho. 31′, I set from A to 4. and 3 hours 31 min. from A to 3, and lastly, the extent of the Compasses from the beginning of the Scale to 4 hours 31 min. I set off from A to 2. And so I have found points for all the hours of the Plane on the perpendicular CAE.

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 CAE and DBF, which shall be the hour lines of the Diall.

Note that a Diall of this kind is neer as socn made as spoken of, The Stile 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.

Having by the 11th. Chapter & second Diagram found the heighth of the Pole above the Plane to be 36 d. 49′, the Inclination of Meridians 53 deg. 26 min. and the

distance of the substile from the Meridian 38 d. 57. I proceed to make the Diall thus. Draw the Horizontal line F AE, make A the center of the Di­al, then with the Radius of your line of Chords, draw an arch & set off the dist­ance of the Sub­stile from the Meridian from F to G, and from the center draw AGD for the Substile, again through the Center A, draw B C at right angles to the Substile A D. Then make A B and A C equal to the height of the Stile ta­ken ont of the line of Latitudes 36 d. 49′, find then the

[Page 24]Inclination of Meridians on the second Scale, which is 53.26. and against it on the first Scale I find 3 ho. 34′, which being taken out of the same Scale, and set from D and B gives points for the hours of 12 and 6, the rest of the worke differs nothing from the directions of the twelfth Chapter: Draw lines from the Center A to each of those points, and they shall be the houres for this Plane. The Stile must stand directly over the Snbstile AD and containe an angle of 36 d. 49″ as the angle EAD.

Take notice that when you are to draw any of the Di­alls with Centers, 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 lyeth 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.

I take the Example of the 11th Chapter, according to the third Diagram of that Chapter, being a Plane that declineth from the South towards the East 34 degrees, and Inclining to the Horizon 24 degrees, in the Latitude of 51 deg. 30 min. I found the height of the Pole above the Plane to be 30 degrees 20 min. The arch of the Plane from the Horizon to the Meridian 58 deg. 22 min. the Inclination of Meridians 15 deg. 17 min. and distance substile from the Meridian 7 d. 51′. this is all we need to know before we begin to make the Dial; which is ef­fected after this manner, draw first the horizontal line of EF, then w^th the radius of your line of Chords draw the arch FH,

and set off the Altitude of the Meridian 58 deg. 22 min. from F to H. and the distance of the substile from the Meri­dian 7 d. 51 min. from H to K: Let A be the center of the Dial then draw the line AKD for the substile then draw BAC at right angles thereto, & [Page 26]make AB and AC equal to the height of the Pole a­bove the Plane 30 deg. 20 min. taken out of the Scale of Latitudes, then set off the whole first Scale or hour line from B and C to D. as hath been before shewn.

Then considering the Inclination of Meridians 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 shewn and this Table demonstrateth.

The Stile must containe an angle of 30 deg. 20 min. as IAK in the Diagram to stand directly over the Sub­stile AD, and so the Diall 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 Meridians Elevation is lesse then the Latitude of the Place, on the under faces of all North Incliners, whose Meridians Elevation is greater then the Latitude of the place, and on the upper faces of all South Incli­ners the North Pole is elevated. And upon the under [Page 27]faces of all North Incliners, whose Meridians Elevati­on is lesse then the Latitude of the place, on the upper faces of all North Incliners, whose Meridians Elevati­on is greater then 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 Meridians Elevation is greater than the complement of Latitude, on the under faces of all South Incliners, whose Meridians Elevation is lesse than the Latitudes, complement, on the under faces of all North Incliners, whose Meridian Elevation is greater then the Latitudes of the place, and on the upper faces of all North Incli­ners, whose Meridians Elevation is lesse 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 Meridians Elevation is lesse than the Latitudes comple­ment. On the under faces of all South Incliners, whose Meridans Elevation is greater than the complement of Latitudes, on the under faces of all North Incliners, whose Meridians Elevation is lesse than the Latitude of the place, and on the upper faces of all North In­cliners, whose Meridians Elevation is greater then 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 Meridians Elevation is greater than the Latitudes com­plement, or either the upper or under faces of a North Inclining Plane, whose Meridians Elevation is lesse [Page 28]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 comple­ment 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 its a declining Polar.

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: Then considering the Inclination of Me­ridians 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 farre 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 sheweth.

The extent of the Compasses from the beginning of the Scale to the hour of 3 gives the height of the Stile, which must be a plate of Iron or Brasse, set up just over the Substile HK, and the Diall is finished.