Hour-Angles at the Pole by the Directions Chap. 2. which being made as in the Table draw the Dial in this manner 12 33-28 Â 1 18-28 Â 2 3-28 Â Â Â Sub. 3 11-32 Â 4 26-32 Â 5 41-32 Â 6 56-32 Â 7 71-32 Â 8 86-32 Â 9 78-28 Â 10 63-28 Â 11 48-28 Â 12 33-28 Â Upon AB the Horizontal-line of your Plain describe the semi-circle AEB and from the Perpendiculer-line CE of the Plain lay off 13-28 Eastward for the 12 a clock Line on the Plain or the complement thereof 76-32 from the east-East-end at B to + draw the Line C + Again Set further Eastward from 12 7-58 the distance of the Substile from 12 to F and draw the Line CF for the Substile and beyond that set off from F 12-13 the Stiles height above the Substile to G and draw CG also Then Draw a contingent Line perpendiculer to the Substile CF as far from the Center as you can as the Line HI then take the nearest distance from the point F to the Line CG and make it a = Tangent of 45 then the Sector being so set take out the = Tangents of all the Hour-Arks in the Table and lay them both wayes from F toward H and I as they proceed then Lines drawn from the Center C and those Points shall be the Hours required Or Having in that manner pricked down 12 6 3 or any other Hours 3 hours distant draw two Lines on each side 12 = to 12 and measure the distance from 6 to 3 in the = and lay it from C the Center on the Line 12 and by those two Points draw a third Line = to the 6 a clock-line then 6-3 and 12-3 made a = Tangent of 45 shall be the two Radiusses to lay off the Hour-lines from 6 12 as before in the former Dials And the = Tangent of Inclination of Meridians doth prove the truth of your Work here also as well as in the Decliners Erect But note That this Dial is better to be augmented by the losing the Hours of 8 and 9 in the morning which makes the Hours more apparent as you see Also the Requisites formerly sound may Geometrically be found by the Scheam being large and truly drawn as before is shewed in the other Dials Thus 1. A Rule laid from Q the Pole-point of the Plain to G the Point of 12 on the Plain gives in the Limb the point 12 D 12 13-28 is the distance of 12 a clock-clock-line on the Plain from the Plains perpendiculer-perpendiculer-line ZD and to be laid from the perpendiculer-perpendiculer-line on the Plain Eastwards in the Dial and the distance on the Limb from A to 12 is the Meridians distance from the east-East-end of the horizontal-Horizontal-line on the Plain namely 76-32 2. A Rule laid from Q to F on the Limb gives the Point Sub for the Substile and the Ark Sub. 12 7-58 is the distance from 12 or the Ark Sub. D 21-26 the distance from the Perpendiculer 3. A Rule laid from Q to 6 the place where the 6 a clock hour-line on the Scheam cuts the Plain gives on the Limb the Point 6 the Ark 6 12 25-38 or 6 D 38-56 is the distance of the Hour-line of 6 on the Plain from the Hour-line 12 or the Perpendiculer 4. A Rule laid from Y the Pole-point of the Circle QFP to P F on the limb gives two points IK and the Ark IK is the Stiles Elevation 12-13 5. A Rule laid from P to Y on the limb gives the Point M EM is the Inclination of Meridians or a Rule laid from P to the intersection of the Circle PFQ and the Equinoctial-line gives a Point in the Limb near C which Ark CS is more naturally the Angle between the two Meridians 33-28 Or If you like the way of referring this Plain to a new Latitude and to a new Declination in that new Latitude Then thus by the Scheam 6. A Rule laid from E to P and G in the Limb gives L and O the Ark LO is the complement of the new Latitude being the Ark PG the second requisite in the former Calculation being 14-33 the distance on the Meridian from the Pole to the Plain But note That this Dial is better to be augmented by the losing the Hours of 8 and 9 in the morning which makes the Hours more apparent as you see Also the Requisites formerly found may Geometrically be found by the Scheam being large and truly drawn as before is shewed in the other Dials Thus 1. A Rule laid from Q the Pole-point of the Plain to G the Point of 12 on the Plain gives in the Limb the point 12 D 12 13-28 is the distance of 12 a clock-clock-line on the Plain from the Plains perpendiculer-perpendiculer-line ZD and to be laid from the perpendiculer-perpendiculer-line on the Plain Eastwards in the Dial and the distance on the Limb from A to 12 is the Meridians distance from the east-East-end of the horizontal-Horizontal-line on the Plain namely 76-32 2. A Rule laid from Q to F on the Limb gives the Point Sub for the Substile and the Ark Sub. 12 7-58 is the distance from 12 or the Ark Sub. D 21-26 the distance from the Perpendiculer 3. A Rule laid from Q to 6 the place where the 6 a clock hour-line on the Scheam cuts the Plain gives on the Limb the Point 6 the Ark 6 12 25-38 or 6 D 38-56 is the distance of the Hour-line of 6 on the Plain from the Hour-line 12 or the Perpendiculer 4. A Rule laid from Y the Pole-point of the Circle QFP to P F on the limb gives two points IK and the Ark IK is the Stiles Elevation 12-13 5. A Rule laid from P to Y on the limb gives the Point M EM is the Inclination of Meridians or a Rule laid from P to the intersection of the Circle PFQ and the Equinoctial-line gives a Point in the Limb near C which Ark CS is more naturally the Angle between the two Meridians 33-28 Or If you like the way of referring this Plain to a new Latitude and to a new Declination in that new Latitude Then thus by the Scheam 6. A Rule laid from E to P and G in the Limb gives L and O the Ark LO is the complement of the new Latitude being the Ark PG the second requisite in the former Calculation being 14-33 the distance on the Meridian from the Pole to the Plain 7. A Rule laid from G to Q on the limb gives R the Ark SR is the new declination in that new Latitude 32-37 Or else find it by this Rule As sine of 90 to the Co-sine of the Reclination or Inclination So is the sine of the old Declination to the sine of the new in this Example being 32-37 and generally the same way as the old Declination is Only observe That when the North-pole is Elevated on South Recliners you must draw them as North-decliners and North-west
and North-east incliners that have the South-pole Elevated you must draw them as South-east and West-decliners which will direct as to the right way of placing the Substile and Hour of 6 from 12. In this place I shall also insert the general way by Calculation to find the new Latitude as well as new Declination Which is thus As Radius or Sine of 90 to the Co-sine of the Plains old Declination So is the Co-tangent of the Reclination or Inclin to the Tang. of a 4th Ark. Then In South Recliners and in North Incliners get the difference between this 4th Ark and the Latitude of your place and the complement of that difference is the new Latitude if the 4th Ark be less then the old Latitude then the contrary Pole is Elevated but if it be equal to the old Latitude it is a Polar-plain But in South Incliners and in North Recliners the difference between the 4th Ark and the complement of the Latitude of the place or old Latitude shall be the new Latitude when the 4th Ark and old Latitude is equal it is an Equinoctial-plain Thus in this Example As sine 90 to Co-sine of 35 the old Declination So is Co-tangent of 20 the Reclination to 66-03 for a 4th Ark from which taking 51-32 the old Latitude rests 14-31 the complement of the new Latitude which will be found to be 75-29 the new Latitude By which new Latitude and new Declination if you work as for an Erect Dial you shall find the same Requisites as by the former Operations you have done and the distance of the Perpendiculer and Meridian will set all right The Second Variety of South Recliners reclining just to the Pole 1. The Scheam is drawn as before to the same Declination and the same way viz. 35 degrees Westward and reclines 33-3′ Now to try whether such a Plain be just a Polar-plain or no use this Proportion By the Sector As the sine of 90 DA 90-0 To Co-sine of Declin NA 55-0 So Co-tang of Reclin DE 56-57 To Tang. of Latitude NP 51-32 As Co-sine Declination NA 55-00 To = sine of AD 90-00 So is = Co-tang of Reclin DE 56-57 being taken from the small Tangents To Tangent of NP 51-32 being measured from the Center on the same small Tangents Which 4th Ark if it hit to be right the Latitude then it is a declining Polar-plain or else not 2. If you have a Declination given to which you would find a Reclination to make it Polar then reason thus By the Sector As the Co-sine of the Declin AN 55-0 To the Radius or Sine of AD 90-0 So is the Tang. of the Lat. PN 51-32 To the Co-tang of the Reclin DE 56-57 As Tangent of NP 51-30 To = Sine of AN 55-00 So = Sine of AD 90-00 To Tangent of DE 56-57 3. If the Reclination were given and the Declination required to make it a Polar then the Canon may be thus By the Sector As the Co-tang of the Reclin DE 56-57 To the Radius or Sine of AD 90-00 So is the Tang. of the Lat. NP 51-32 To the Co-sine of the Declin NA 55-00 As Co-tang Reclination DE 56-57 To = sine of AD 90-00 So Tang. of Latitude ND 51-32 To = Co-sine of Declination NA 55-00 But by the Scheam these three Operations are found by drawing the Scheam 1. For if the Line or Circle representing the Plain cut the Pole P it is a Polar-Dial 2. If AB the Co-declination be given then draw the Circle APB and it gives E then ZE is the Reclination measured by half Tangents or a Rule laid from A to E on the Limb gives an Ark from B which measured on fit Chords is the Reclination 3. If P the Pole-point and ZE the Reclinatin be given then with the distance ZE on Z as a Center draw an Ark of a Circle in that Quadrant which is contrary to the Coast of Declination observing the letters in the Scheam then by the Convexity of that Ark and the Pole-point P draw the Circle PE cutting the Limb into two equal parts which are the points A B the declination required This being premised there are two things requisite to be found before you can draw the Dial. viz. the Substile from the Perpendiculer or Horizon and the Inclination of Meridians 1. And first for the Substile by the Sector As the sine of PEZ 90-0 To the Co-sine of the Lat. PZ 38-28 So the sine of the Declination PZE 35-00 To the sine of Substile from Perp. PE 20-54 As sine of Declination PZE 35-0 To = sine of PEZ 90-0 So = sine of Co-latitude PZ 38-28 To sine of Substile from Perp. FE 20-54 The distance of the Substile from the Perpendiculer whose complement 69-06 is the Elevation above the Horizon Or A Rule laid from Q to P gives I DI is 20-54 2. For the Inclination of Meridians say By the Sector As the Co-sine of the Latitude PZ 38-28 To the sine of PEZ 90-00 So the sine of the Reclin ZE 33-03 To the Co-sine of Incl. Mer. ZPE 61-15 Whose complement ZPQ 28-45 is the Inclination of Meridians required As sine of Reclination ZE 33-3 To = Co-sine of Latitude PZ 38-28 So = sine of 90 PEZ 90-00 To Co-sine of Incl. Mer. ZPE 61-15 Whose complement QPZ 28-45 is the Inclin of Meridians required Or A Rule laid from P to Y gives M EM is 28-45 the Inclination of Meridians Again 8 88 45  81 15 9 73 45  56 15 10 58 45  51 15 11 43 45  36 15 12 28 45  21 15 1 13 45  6 15 2 1 15  8 45 3 16 15  23 45 4 31 15  38 45 5 46 15  53 45 6 61 15  68 45 7 76 15 If I take 15 the quantity in degrees of one Hour out of 28-15 the Inclination of Meridians there remains 13-45 for the first Hour on the other-side of the Substile Then again by continual addition of 15 degrees to 13-45 and the increase thereof I make up the other half Or else Against 12 set 28-45 and add 15 successively to it its increase till it come to 90 Then to 13-45 the residue of 15 taken from 28-45 add 15 as often as you can to 90 and thus is the Table made To draw the Dial. First Draw a perpendiculer Line on your Plain as CB by crossing the Horizontal-line at Right Angles then from the perpendiculer-Line lay off from the upper-upper-end toward the left-hand as the Scheam directs ZD being the Perpendiculer and ZN the Meridian and EP on the Plain the distance between being toward the left hand 20-54 for the substile-Substile-line as CD then on that Line any where draw two perpendiculer Lines quite through the Plain crossing the Substile at right Angles for two Equinoctial-lines as EF GH Then consider what hours shall be put o◠your Plain as here is convenient from
deg − 0′ from the South towards the West take out the Chord of 35 deg and lay if from S to C and from W to A and from N to D and from E to B for the more exact drawing of the Lines AB CD the Lines CD representing the Poles of the Plain and the Line AB the Declining Plain it self then from Z towards D lay off the Tangent of 10 deg being the half Tangent of 20 degrees the given Declination to E. Also Take out the Secant of 70 degrees the complement of 20 to the same Radius and that laid from the Point E on the Line DC produced shall be the Center to draw the Circle AFEB that represents the Declining Reclining Plain that declines 35 degrees and reclines 20 degrees Also Lay off half the Tangent of the complement of the Reclination viz. 35 degrees for the Reclination is 20 the complement whereof is 70 and the half of 70 is 35 from Z to Q. Then to draw the Line QP do thus Observe how many degrees you count from Z to the Point E counting from the Center count so many in the manner of half Tangents from 45 and the latteral distance from thence to the Center laid from the Center Z on the Line CD gives a third Point viz. the Point I which three Points QPI brought into a Circle will cut the Circle N.E.S.W. into two equal parts Or thus The Semi-tangent of the complement of the Reclination to 180 degrees laid from Z on the Line CD will find the Point I. As thus The Reclination is 20 the complement 70 being taken from 180 rests 110 whose half is 55 the Tangent of ZI Or more briefly thus Set one Point of the Compasses in the small Tangent of 45 and count the Reclination from thence in the way of Semi-tangents both wayes both above and under 45 and lay one viz. that under 45 from C to Q and the other viz. that above 45 from D to I then on the middle Point between Q and I last found raise a Perpendiculer to CD and in that Line will be the Center to draw IPQ. Also CHAP. II. To Draw the Hour-Lines on all Ordinary Dials the easiest in the first place 1. And first for the first Equinoctial DIAL AN Equinoctial Plain as before is shewed is that whose plain or flat Superficies lieth parallel to the Equinoctial and is represented by the Line WAEE in the general Scheam and therefore needs no other Scheam to represent it In which Dial all the Hour-lines are equal one to the other being just 15 degrees assunder so that to draw the Hour-lines here describe a Circle as the Circle 12.6.12.6 and fit the Radius in the Sine of 30 degrees or the Chord of 60 and take out the parallel Sine of 7 degrees 30 minuts the half of 15 degrees and number it from 12 round about and that shall divide the whole Circle into 24 equal parts for the 24 hours for the true Hour-lines on the Equinoctial-plain and is the same in all latitudes only in the setting of it the Poles of it are to be set due North and South the Horizontal-line on the Plain lying Parallel to the East and West-points of the Horizon and the Stile thereof only a Wyre or sharp Edge standing perpendiculerly on the Center which being so set must point directly to the North and South Poles of the World The reclining Dials-Stile pointing to the North-Pole and the inclining Dials-Stile pointing to the South Pole then is the Dial truly placed To set a Plain or to try whether a Plain be set Polar or Parallel to the Equinoctial do thus But to try the Inclining Plain apply the Loose-piece to the Plain with the Head-end downwards or else apply the Head-leg to the Plain with the head-Head-end downwards and the Thred shall cut on 38-30 in London latitude if the Plain be set parallel to the Equinoctial 2. To draw a Direct Polar Dial. The next Dial shall be a Direct Polar-Dial which is represented in the general Scheam by the hour-Hour-line of 6 viz. the Line EPW And here also the Horizontal-line on the Plain is parallel to the East West-points of the Horizon and the Pole or Point opposite to the Plain is in the Equinoctial-point The Hour-lines in this Plain are all parallels because the Axis or Stile-line in all Plains is parallel to the Poles of the World and this Plain it self being so parallel the Stile or Axis therein makes no Angle therefore the Hour-lines must needs be parallels also And the way of drawing those Hour-Lines is thus First draw the Perpendiculer-line on the Plain which is done thus by the Trianguler-Quadrant Hang a Plummet and Thred on the Center and apply the Moveable-leg to the Plain to and fro till the Thred falls neatly on 600 and draw that Line along by the Moveable-leg which shall be a true Horizontal-line on any reclining Plain and a Perpendiculer-line thereunto is the perpendiculer Line on the Plain Or else When the Sun shineth the Sun begins in the Pole of the Plain hold up a Thred and Plummet till the shadow of the Thred fall on the Plain making two Points in that shadow at the remotest distance asunder then a Line drawn through those two points shall be a true Perpendiculer-line this shall need no more Repetition Then The several = Tangents of 60-45 30 15 laid both wayes from 12 on both the Horizontal-lines shall give you Poâ—nâ—s whereby to draw all the Hour-lines in their true places Also The = Tangent of 45 shall be the true breadth of the Plate that must be a Sâ—â—â—e to this Dial or the length of an upright Wyre set any where in the Line 12. Note That for the hours under 45 you may take = 45 from the small Tangents and make it a = Tangent of 45 in the great Tangents and then take oâ— = Tangent of 30 30 for 2 10 anâ— the = Tangent of 15 for 11 1 and if you want them above 45 then take the = Tangent of 60 60 from the small Tangents and turn that Extent 4 times from 12 both wayes on both the Horizontal-lines and those shall be the Points for 8 in the forenoon and 4 afternoon And lastly The = Tangent of 75 taken and turned 4 times from 12 to 7 in the morning and to 5 in the afternoon will fit and fill a Plain of 4 foot in breadth with a Sector of one foot shut 3. To draw a Direct East or West-DIAL The next Dial in the third place is the Direct East or West-Dial which is represented in the general Scheam by the Line NZS whose Poles are in the Line EZW whose Plain also is = to the Pole drawn in the same manner as the Polar Dial was yet with this difference the Equinoctial-line whereon to prick the hours is not the Horizontal-line but is thus found Thus You may take the Tangents under 45 when the
10 in the morning to 6 afternoon though the Sun may shine on it from 8 to 7 bu◠then the Lines will be too close together and the Radius too small And also when you would have those two utmost hours 〈◊〉 be as at E and F on the upper Equinoctial-line or at G H on the lower contingent-line Then Then Take the whole distance EF or GH and make it a = Tangent of 73-55 then the Sector to set take out the = Tangent of 58-45 and lay it from the point E to I on the Equinoctial-line Also take out the = Tangent 61-15 and lay it from the point F and if your work be true it must needs meet in the point I then draw the Line IK for the true Substile and from thence lay the = Tangent of 45 to draw a Line near 5 for the Stiles Elevation parallel to IK the Substile for being a Polar-plain it hath no Elevation but what you please to augment it to as here from I to L. Then As the Sector stands prick on all the whole hours halfs and quarters according to the Numbers in your Table at least those that be above 45 and for those under 45 make = Tangent of 45 in small Tangents a = Tangent of 45 in the great Tangents and then the Sector shall be set to that Radius which is most convenient for your use Note That this way of Augmenting the Stile is general in all Dials 3. The third Variety of South-Recliners The next and last kind of South Recliners are such as recline or fall from you below the Pole viz have their Plains lying between the Pole and the Horizon as by the Scheam is more apparent In which work the drawing the Scheam and the things required are the same as in the first Example as the Figure and following words do make make manifest The Example here is of a Plain that declines from the South toward the West 35 degrees and reclines upon its proper Azimuth ZE 60 degrees from the Zenith 1. Having drawn the Scheam then first for the distance of the Meridian from the Perpendiculer or Horizon By the Sector or Quadrant As the sine of ZD 90-00 To the Tangent of Declination ND 35-00 So the sine of Reclination ZE 60-00 To the Tang. of Perp. Merid. EG 31-12 As Tangent of Declination ND 35-00 To = sine of 90 ZD 90-00 So = sine of Reclination ZE 60-00 To Tang. of Perp. Merid. EG 31-12 Whose complement is 58-48 AG the distance between the west-West-end of the horizontal-Horizontal-line and the Meridian Or by the Scheam A Rule laid from Q to G cuts the limb at L the DL and AL are the Arks required DL from Perpendiculer and AL from the Horizon 2. To find PG the Ark on the Meridian from the Pole to the Plain By the Sector As sine of AD 90-0 To Co-tang of the Reclin DE 30-0 So Co-sine of the Declination AN 55-0 To Tang. of dist Plain Horiz NG 25-19 As Co-tangent Reclin ED 30-0 To = sine of 90 AD 90-0 So = sine of Reclination AN 55-0 To Tang. dist on Mer. P. Hor. NG 25-19 Which being taken from NP 51-32 leaveth GP 26-13 the distance on the Meridian from the Pole to the Plain or the complement of the new Latitude Or A Rule laid from E to P and G gives on the limb 2 Points whose distance between is ab 26-13 the Ark required 3. To find the Stiles Elevation above the Plain By the Sector As sine dist Merid. Horizon GA 58-48 To Co-sine Declination AN 55-00 So sine dist Pole to Plain GP 26-13 To sine Stiles Elevation PF 25-02 As sine of GP 26-13 To = sine of GA 58-48 So = sine of AN 55-00 To sine of PF 25-02 Being found by the Scheam by laying a Rule from Y to P and F on the limb gives the distance between being 25-02 the Stiles Elevation 4. To find the Substile from 12. By the Sector As Co-tang of the Declin AN 55-00 To S. dist on Mer. fr. Pl. to Hor. NG 25-19 So Tang. of the Stiles height PF 25-02 To S. of the Substile from 12 FG 8-05 As Co-tang of Declin Plain AN 55-00 To = S. dist on Mer. fr. Pl. to Hor. NG 25-19 So Tang. of the Stiles height PF 25-02 To = S. of the Substile from 12 FG 08-03 By the Scheam a Rule laid from Q to G and F on the limb gives L and M 8-3 Or else the Ark MD is the distance of the Substile from the Perpendiculer 23-19 5. To find the Inclination of Meridians By the Sector As the sine of the distance on Mer. from Pole to Plain PG 25-19 To the sine of the Angle GFP 90-00 So the sine of dist of Sub. fr 12 GF 08-03 To the sine of the Incl. of Mer. GPF 18-27 As sine GF 08-03 To = sine PG 25-19 So = sine GFP 90-00 To sine GPF 18-27 By the Scheam a Rule laid from P to Y on the limb gives O the Ark EO is 18-27 the Inclination of Meridians by help of which to make the Table of Hour-Arks at the Pole as before is shewed and as in the Table following 12 18 27 8-3  10 57  1 3 27 1-27   Subst  4 03  2 11 33 4-58  18 03  3 26 33 11-55  34 03  4 41 33 20-35  48 03  5 56 33 32-45  64 03  6 71 33 51-45  78 03     7 86 33 81-52  85-57  8 78 27 64-10  70 57  9 63 27 40-20  55 57  10 48 27 25-36  40 57  11 33 27 15-33  25 57  12 18 27 8-3 To draw the Dial. First for the Affections consult the Scheam wherein laying the Perpendiculer-line CD right before you you see that the Substile and the Meridian are to be laid from the Perpendiculer toward the left-hand the Substile lying between the Perpendiculer and the Meridian and the Stile or Cock of the Dial must look upwards the North-Pole being Elevated above this Plain which will guide all the rest Then First draw the Horizontal-line AB and on C as a Center raise a Perpendiculer and set off by Chords Sines or Tangents the Meridian or 12 a clock Line the Substile and Stile as exactly as you may and draw the Lines 12 C Substile C and Stile C. Then As far from the Center C as you conveniently may draw a long Line perpendiculer to the Substile as the Line EHF then setting one Point of a pair of Compasses in H open the other till it touch the Stile-line at the nearest distance Then Make this distance a = Tangent of 45 and take out the = Tangents of every whole Hour as in the Table as far as the Tangent of 76. will give leave and then from the Center C to those Points draw Lines for the even whole Hours
then to any one whole Hour as suppose the Hour-line of 3 draw two = Lines equally distant on both sides the Line of 3 as IK LM Then Count any way 3 hours and 6 hours from 3 as here 12 and 9 so as the = line may cross the 3 remotest hours as here you see 9 and 12 a clock Hour-lines do cross the = line at I and K then take the distance IK and lay on the Hour-line of 3 from C to N and draw INL = to 9 C Which Work doth constitute the Parallellogram KILM Then lastly Make KI and NI = Tangents of 45 and pâ—ick off every hour half and quarter and minut if you please on the two Lines IK and IL from K and N both wayes as before is already shewed in the Erect Decliners Note also That to supply the defect on the other side when the point M falls out of the Plain the distance from I to the Hour-point from 11 will reach from L to 7 and from I to 10 from L to 8. This is general in all Dials Also note If you like not to lay off the â—irst Hours by the Tangents having made the Table as before you may soon find the Hour-Arks on the Plain for 3 Hours as â—ere 3 12 and 9 Or 4 1 and 8 which â—ould have made the Parallellogram more â—â—uare and consequently more better and â—â—en to draw the rest by the Sector Thus â—ou may see how your Work accords The â—ay by the Table and Contingent-line and 〈◊〉 way by the Sector on the Parallellogram 〈◊〉 by Calculation at last use the Mystery 〈◊〉 Dialling made plain and ready to an â—â—dinary capacity Of North Declining Recliners The other kind viz. North Declining Recliners have also three Varieties as those â— That fall back or recline between the Zenith and Equinoctial 2d Those that recline to the Equinoctial And 3d. Those that recline below the Equinoctial And first of the first Variety reclining less then to the Equinoctial The drawing the Scheam is the same as in the former except in the placing of the Points and Letters For first these Plains behold the North-part of the Horizon and then when you look on the Plain the South is before you and the West on your right-hand and the East on the left then the South and North are alwayes opposite and the point P representing the Elevated Pole of the place which with us being North must be placed towards N downwards as before in South Recliners it was upwards Also It is necessary in the Scheam to draw the Equinoctial-line by laying the half Tangent of 51-32 from Z to AE then the Secant of 38-28 the complement of ZE laid from AE on the Line SN shall be the Center to draw EAEW for the Equinoctial-Circle Thus the Scheam being drawn to find the Requisites thus 1. For the Meridians Elevation or distance from the Perpendiculer AG or GE. By the Secctor As sine 90 Radius ZD 90-0 To Tangent Declination Plain SD 55-0 So sine Reclination Plain ZE 20-0 To Tangent Merid. Perpend GE 26-2 As Tangent of Declin SD 55-0 To = sine of Radius ZD 90-0 So = sine of Reclination ZE 20-0 To Tang. of 12 from Perp. GE 26-02 Whose complement AG 63-58 is the Meridians Elevation above the East-end of the Horizon By the Scheam A Rule laid from Q to G on the Limb gives L then DL and AL are the Arks required 2. To find the Distance on the Meridian from the Pole to the Plain GP By the Sector As sine declin of the Plain GZE 55-0 To sine dist of Mer. Perp. GE 26â—02 So sine of the Radius GEZ 90â—00 To sine of dist on Merid. from Pole to Plain GZ 32-03 As sine of GEZ 90-0 To = sine of GZE 55-0 So = sine of GE 26-2 To sine of GZ 32-03 Which added to 38-28 ZP makes up GP to be 70-31 Or By the Scheam A Rule laid from E to P and G gives on the limb ab the Ark ab is 70-31 3. To find the Stiles height above the Plain PF By the Sector As sine of distance on Mer. from Zenith to the Plain GZ 32-03 To sine of the Plains Reclin ZE 20-00 So sine of dist on Mer. from Pole to the Plain GP 70-31 To sine of the Stiles Elevat above the Plain PF 37-01 As the sine GP 70-31 To the = sine GZ 32-03 So the = sine ZE 20-00 To the sine PF 37-01 By the Scheam A Rule laid from Y to P and F on the limb gives c and d the Stiles height 4. To find the distance of the Substile from the Meridian GF when it is above 90 deg take the comp to 108 deg By the Sector As Tangent of the Reclin ZE 20-00 To sine of dist of 12 from Perp. GE 26-02 So Tang. of the Stiles Elevat PF 37-01 To sine of the Substile from 12 GF 65-24 As sine EG 26-02 To = Tangent ZE 20-0 So = Tangent PF 37-01 To sine GE 65-24 By the Scheam A Rule laid from Q to G and F gives on the limb LF the Ark required 5. To find the Inclination of Meridians FPG By the Sector As sine dist on Merid. from Pole to Plain GP 70-31 To sine Radius opposite Angle GFP 90-00 So sine dist on Plain from 12 to Substile GF 65-24 To sine of the Inclin of Mer. GPF 74-38 As sine GF 65-24 To = sine GP 70-31 So = sine GFP 90-00 To sine GPF 74-38 By the Scheam A Rule laid from P to Y on the Limb gives g the Ark Eg is 74-38 the Inclination of Meridians Or A Rule laid from P to K gives h Sh is the Inclination of Meridians by which to make the Table as before is shewed and as followeth To draw the Dial. 3 29 38 2 44 38 1 59 38 12 74 38 11 89 38 10 75 22 9 60 22 8 45 22 7 30 22 6 15 22 5 0 22 4 14 38 For drawing the Dial consult with the Scheam laying the Plain AEB and his Perpendiculer CD right before you then note SN is the Meridian-line ZE the Plains perpendiculer with the Meridian G on the left-hand and the Subtile F on the right-hand Also note That the Sun being in the South as S casts â—is beams and consequently the shadow of â—he Stile into the North So that though G be the true Meridian found yet it is the North-part that is drawn as an Hour-line â—ut the Substile and other Hours are counâ—ed from the south-South-end thereof as the Tableâ—nd â—nd the Figure of the Dial do plainly make â—anifest being drawn in this manner First draw the horizontal-Horizontal-line AB then 〈◊〉 C as a Center draw a semi-circle equal 〈◊〉 60 of the Chords and lay off the Meriâ—ian Substile and Stile in their right Sciâ—â—ations as last was declared then draw â—â—ose lines and to the Substile erect a Perâ—endiculer as DE then take
the Extent or nearest distance from the place where the Perpendiculer or Contingent-Line last drawn cuts 12 and the Stile-line and make it a = Tangent of 45 Then is the Sector set to lay off all the Hours by the = Tangents of the Arks in the Table except 11 and 10 which do excur For If you prick the Nocturnal-Hours 12 1 2 3 and draw them through the Center on the other side they shall be the Hours of 12 1 2 3 4 c. on the North-part of the Plain where they are only used As for the Hours of 10 and 11 do thus Draw a Line = to any one Hour which = line may conveniently cut those Hour-lines As Suppose the Line 6 12 which is = to the Hour-line of 3 then make the distance from 9 to 12 or from 6 to 9 in that Line last drawn a = Tangent of 45 and lay off hours and quarters or else the whole Hours by the distances from 9 to 7 and 8 for 10 and 11 turning the Compasses the other way from 9 then to all those Points Lines drawn shall be the Hour-lines required Or Having only the hours of 3 6 9 12 in a Parallellogram design the rest by Sector The Second Variety of North-Recliners Reclining to the Equinoctial By the bare drawing of the Scheam you see that the Circle AEB representing the reclining Plain doth cut the Meridian just in the Equinoctial Now to try by Arithmetick whether it be a just Equinoctial-plain or no say 1. By the Sector As the sine of 90 AD 90-0 To Tang. of the Reclination DE 54-10 So Co-sine of Declin Plain AS 35-0 To Co-tang of the Latitude SG 38-28 As Tangent Reclination DF 54-10 To = sine 90 AD 90-0 So = Co-sine of Declination AS 35-0 To Co-tang of the Lat. SG 38-28 Which happening so to be it is a declining Equinoctial or Polar in respect of its Poles which are in the Poles of the World 2. If the Declination were given and to it you would have a Reclination to make it Equinoctial By the Sector As the Co-sine of the Declin AS 35-0 To the Co-tang of the Lat. SG 38-28 So is the sine of 90 AD 90-00 To the Co-tang of the Reclin DE 54-10 As the Co-tang Lat. SG 38 28 To the = Co-sine Declin AS 35-00 So the = sine Radius AD 90-00 To the Co-tang Reclin DE 54-10 By the Scheam The Points AB of Declination being given and the Point G on the Meridian if you draw the Reclining Circle AGB it will intersect the Perpendiculer at E then the measure of ZE is the Reclination measured by half-Tangents or by Chords by laying a Rule from A to E on the limb gives a the Chord B a is the Reclination 35-50 3. But on the contrary if the Reclination be given and a Declination required to make an Equinoctial Plain Then contrarily say thus By the Sector As Co-tang of the Reclin ED 54-10 To sine of 90 AD 90-00 So Co-tang of the Latitude SG 38-28 To Co-sine of the Declin SA 35-00 As Co-tang Reclin ED 54-10 To = sine AD 90-0 So Co-tang Latitude SG 38-28 To = Co-sine Declination SA 35-00 But by the Scheam By the Point G and the touch of an Arch about E draw the Circle GE to cut the limb into two equal parts and you have the Points AB 4. The Plain thus made or proved to be Equinoctial to find the Meridians Elevation above the Horizon AG Or his Distance from the Perpendiculer EG By the Sector As sine of 90 ZEG 90-0 To sine of dist on the Mer. from Z to the Plain GZ 51-30 So sine of Declin of the Plain GZE 55-0 To sine of dist on the Plain from Perpend to Merid. GE 39-54 As sine GZE 55-0 To = sine ZEG 90-0 So = sine GZ 51-32 To sine GE 39-54 Whose complement is AG 50-06 the Elevation above the Horizon By the Scheam A Rule laid from Q to G gives b on the limb DB is 39-54 as before 5. To find the Stiles Elevation above the Substile on the Plain By the Sector As sine of the Latitude GZ 51-32 To sine of the Reclination ZE 35-50 So sine of dist Mer. Pole to Plain GP 90-00 To sine of the Stiles Elevation PF 48-24 As sine 90 GP 90-0 To = sine Latitude GZ 51-32 So = sine Reclination ZE 35-50 To sine Stiles height PF 48-24 By the Scheam A Rule laid from Y to F on the limb gives C NC is 48-24 the Stiles height The distance of the Substile from 12 in these Equinoctial Dials is alwayes 90 degrees for a Rule laid from Q the Pole of the Plain to G on the limb gives b a Rule also laid from Q to F the Substile on the limb gives d the Ark bd is 90 degrees both for the distance of the Substile from 12 and also for the Inclination of Meridians for the Substile stands on the hour of 6 being part of the Circle EPW which is the hour of 6 90 degrees distant from the hour of 12. Or A Rule laid as before from Y to P on the limb gives N the Ark EN or WN is 90 for the Inclination of Meridians Which being just 90 the Table is easily made viz. 15 30 45 60 75 90 twice repeated from 12 to 6 both way s. To draw the Dial. On the Horizontal-line AB draw an obscure Semi-circle and set off the Meridian as the Scheam sheweth viz. 50 degrees 6 min. above the east-East-end of the horizontal-Horizontal-line but make visible only the north-North-end thereof as the line C 12 Then 90 degrees from thence toward the right-hand as the Scheam sheweth when the perpendiculer-Perpendiculer-line is right before you draw a Line that serves both for 6 and the Substile as C 6. Also lay off the Chord of 36-47 from 6 to 9 and draw the Line C 9 also which is found by Calculation as before is shewed Or thus Draw a Line = to 12 or Perpendiculer to 6 being in this Dial all one as the Line FEG then setting one Point in E the Substile take the nearest distance to the Stile-line and it shall reach from E to G the Point for 9. The same distance EG lay also on the line 12 from C to H and draw the line GHI then make EG a = Tangent of 45 and lay off the = Tangents of 15-30-45 both wayes from E as hath been often shewed Also Make the distance of HG a = Tangent of 45 and lay the same = Tangents both wayes from H and to those Points draw the Hour-lines required The third Variety of North-Recliners This third and last sort of North-Recliners are those that recline beyond the Equinoctial that is lie between the Equinoctial and the Horizon and it differs somewhat from the other five before in the Scheam and Operation also For first the Ark of the Plain is extended below the Horizon till it meet with the
30 S.E. and Inclining 20. The Substile by applying the Square you shall find to be 30 degrees on the left-hand of the Perpendiculer Westward and the Inclination of Meridians 48-20 the Stiles height 51-36 and the Meridian on the right-hand of the Perpendiculer-line 11-30 Eastward and the shadow of the Thred playing on every hour and quarter on the Horizontal-Dial will shew on the Plain the quantity in degrees from the Perpendiculer-Line Use VI. To find the Requisites in a North-east Reclining-Dial and the Hour-Lines Set the Instrument as before and find the Substile Stile and Inclination of Meridians as before But note as to the Affections which way do thus Turn the Instrument the bottom upward and as near as you can guess turn the Plain to its scituation then you shall first see the Stile to look upward in the North-east Recliner which before was downward in the South-east Incliner Also The Substile stands on the right-hand of the Perpendiculer 30 degrees Westward for observe this alwayes If a Plain declines Eastward the Substile will stand Westward and the contrary Also note That the Meridian-Line is to be drawn quite through the Center on the other-side because when the Sun is in the Meridian above it must needs cast the shadow of the Axis or Stile the contrary way downwards Use VII To find what are the most Hours that the Sun can shine on any Plain whatsoever First on all South Direct or Declining Inclining-Dials the mid-day-Meridian is proper to it unless it incline above 75 degrees and then it becomes useless in London Latitude then what hour soever you can make the Sun to shine on the Plain and Horizontal-Dial both together the Sun being at that hour above the Horizon by bending or turning the Instrument any way when the Point at 12 is first set to the Declination that and all those Hours are proper to that Plain at one time of the year or other Also note That several Hours that serve for the South-plain do at some time of the year belong to the North-plain also as by turning the Instrument about you may plainly see either by the Sun-shine or by the Thred and your Eye cutting the Hour-Lines and the Plain Also observe That if you would delineate a South Reclining Plain you may bring the Plain toward the Thred till it becomes a Polar-Plain But if it Reclines below the Pole then conceive it to become a North Reclining-Dial and work as is before directed and you shall obtain your desire for the Dials will be the same the one as the other as before was hinted at in the Inclining-Plains Use VIII The Declination of any Plain given to find what Reclination will make it a Polar-Dial and the contrary Set the North-point to the Declination and bring the Plain to touch the Thred then on the Brass Circle is cut the Reclination required Or contrary Set the Plain to the Reclination given and then bring the Thred to the Plain by turning the Horizontal-Dial and the Point at 12 shall shew the Declination required to make it Polar In like manner you may discover a declining Equinoctial but not so easily when the Substile and Meridian are 90 degrees assunder the Substile being then alwayes in the hour of 6 as by moving the Plain if the Declination be given or by moving the Thred if the Inclination be given till the Square touching the Thred it shall shadow or bourn just upon 6 on the Horizontal-Dial Note also That East and West Recliners and Incliners are discovered after the same manner So also Direct Recliners and Incliners as by moving the Plain to and fro you shall see the plain and true reason how the Stile is Elevated or Depressed and how the Hour-lines are inlarged or contracted according to the Elevation of the Stile Also In East and West-Dials that the Stile hath no Elevation but is parallel to the Plain and how the Meridian lieth in the Horizon in East and West Recliners and Incliners Many more Uses might be insisted on which I shall leave to the scruteny of the industrious Practitioner in the Art of Shadows CHAP. IX How to remedy several Inconveniencies in the using of the Artificial Lines of Numbers Sines and Tangents as they are usually made 1. IF the term required happen to be under one degree of Sines and Tangents then the Line of Numbers will supply it having due respect to the increase of the Radius or Caracteristick As thus As the sine of 90 to the sine of 23-31 the greatest Declination So is the sine of 1 deg 10′ the Suns distance from the Equinoctial to 0-28 the Declination which falls beyond the end of the Rule Now to remedy this the 1 deg 10′ is 70 minuts therefore by the Numbers say So is 70 minuts the Suns distance from the Equinoctial to 28 the Suns Declination on the Line of Numbers observing to extend the same way as from the first to the second term 2. When you have occasion to use a sine above 90 degrees then you must count the sine of 80 for the sine of 100 and 70 for 110 and 60 for 120. So also the distance from 90 to 60 in the Sines is the Secant of 30 degrees and the distance from 90 to 50 is the Secant of 40 or the Point beyond 90 that represents the Secant of 40. 3. If the Extent be too large for your Compasses as from 45 or 90 to 3 or 4 degrees then instead of 90 or 45 make use of a Point in the Sines or Tangents right against the middle 1 in the Line of Numbers where you may have two Brass Center-pins viz. in the Tangent of 5-43 and the sine of 5-45 and the extent from thence backward or forward shall reach in the Numbers to the 4th proportional Number required Example As Tang. 45 to 1-61 in the Numbers So is Tang. of 15-0 to 0-43 in the Numbers Instead of which you may say As the Tang. of 5-43 to 1-61 on the Numbers so is the Tang. of 15 to 0-43 on the Numbers diminishing a Radius for as Tang. 45 to 1-15 a greater than that so is the Tang. of 15 to a greater than 15 also viz. 0-43 Secondly in Sines Tangents or Sines only where there is another Caution to be observed As sine 90 to sine 10 so is sine 20 to sine of 3-24 ⅓ To work this with small Compasses on a large Line do thus Note that at 10 on the Line of Numbers or Sine of 90 or Tang. of 45 is one compleat Radius but at the middle 1 on the Line of Numbers is a place or Radius less wherein the Logarithm Sines the Characteristick is 8. Again at the sine of 0-34 ½ the Characteristick is 7 and at 3 minuts it is 6 which do note the several decreasings of the Radiusses Therefore set the distance from one Number given to the next nearest place against 1 or next Radius as far
count the fiducial Line in which the Leg-Center-pin stands alwayes for the Meridian of one place and some where in that Line according to the latitude thereof counting the Leg-center the Pole of the World and the Index being hung thereon by the Tangents prick down the Latitude there I say knock in a Pin to stay a Thred for one place then on the degrees count the difference of Longitude from the Head-leg and lay the Index to it and bring the Thred fastened as before till on the Index it cuts the degree and part of the other Latitude and there make the Thred fast with another Pin in the Loose-piece Then If you move the Index to any degree of Longitude between those places the Thred shall cut on the Index the degree of Latitude that answers unto it or if you make the Thred cut any degree of Latitude the Index gives the Longitude required for that Latitude Note If the Latitude be small as between 10 and 30 the small Tangents are most convenient but if it be between 40 and 80 the greater Tangent Line is best Note That two Threds and a pair of Compasses may serve but the Index is much better and quicker in Operation Example Let the two places be the Summer-Islands and the Lizard-point the same Example that you find in Mr. Norwood pag. 126 and in Mr. Phillip's Geometrical-Seaman pag. 55. that you may the more readily compare the truth thereof by their Operations The Latitude of the Lizard Point is 50 degrees the Longitude is 10. The Latitude of the Summer Islands is 32-25 the Longitude is 300. The Difference of Longitudes is 70 as is computed by their Observation Then Hanging or putting the Center-hole of the Index over the Leg-center-pin and counting the fiducial-line on the Head-leg for the Meridian of one place count on the Tangent Line on the Index the Co-tangent of one Latitude as suppose the Latitude of the Lizard-point the Center alwayes counted as 90 and there knock in a Pin in a small hole to hang a Thred on Then count 70 degrees the difference in Longitude on the degrees from the Head-leg and there stay it then draw the Thred put over the first Pin till it cut the complement of the other Latitude and by help of another Pin stay it there which you may conveniently do by one of the sliding-sights then the Thred being so laid slide the Index to every single degree or fifth degree of Longitude and then the Thred shall shew on the Index the Co-tangent of the Latitude answerable to that degree of Longitude as in the Table annexed Also If you would have equal degrees of Latitude and would find the Longitude according to it then slide the Index to and fro till the Thred cuts on the Index an even degree of Latitude then on the degrees you have the difference of Longitude from either place Also note That the drawing of one Line only on the Trianguler Instrument in the beginning according to the directions of laying of the Thred with the Thred and Compasses will perform this work also The Table Long D.L. Latitude 300 09 32 25 305 05 35 52 310 10 38 51 315 15 41 24 320 20 43 34 325 25 45 24 330 30 46 54 335 35 48 07 340 40 49 04 345 45 49 47 350 50 50 15 355 55 50 31 360 60 50 33 05 65 50 23 10 70 50 00 If this work fit not any case that may happen there is another way mentioned in Page 75 of the Geometrical Seaman by the Steriographick Projection and that Scheam is drawn the same way as the Horizontal-Projection for Dyalling was and somewhat easier and any two Points given in a Circle you may draw a great Circle to cut them and the first Circle into two equal-parts by the directions in Page 15 And the Application thereof you have very plainly in Mr. Phillips his Book to which I refer you having said more than at first I intended which was chiefly the use thereof in Observation only So for the present I conclude this Discourse and shall endeavour a further Advantage in the next Impression according as Time and Opportunity shall offer Farewel The End of the Second Part. The Table of the Things contained in this Second Part. THe difinition and kind of Dials Page 7 Directions to draw the Scheam 9 To draw Lines to represent the several sorts of Plains in the Scheam 13 To draw a Scheam particularly for one Dial 14 To draw the Equinoctial Dial 19 To try when a Plain lies Equinoctial 20 To draw a Polar-Dial 21 To draw an Erect East or West-Dial 24 To draw a Horizontal-Dial 27 The dâ—monstration of the Canon for Hours ib. To draw a Direct Erect South or North-Dial 30 To draw a Direct Recliner 33 The use of the Figure 35 To draw a Direct East or West Recliner 37 To make the Table of Arks at the Pole 42 To refer those Dials to a new Latitude and a new Declination wherein they may become Erect Decliners 46 To find the Requisites by the Scheam ibid. To find the Declination of a Plain by the Needle or by the Sun 49 To take off an Angle or set the Sector to any Angle required 53 Precepts to find the Declination by the Sun and Examples also of the same 58 To draw an Erect Declining-Dial 62 The Proportions for the Requisites of Erect Decliners 64 To find the Requisites Three wayes 66 To draw the Erect South Decliner 67 To draw the Lines on a North Decliner 70 To draw the Hour-Lines on a Plain that declines above 60 degrees 73 Of Declining Reclining Plains 77 The first sort of South Recliners 79 The second sort of South Recliners being Polars 90 The third sort of South-Recliners 98 The first sort of North Recliners 106 The second sort of North Recliners being Equinoctial 114 The third sort of North Recliners 119 Of Inclining Diâ—ls 126 To find the useful Hours in all Plains 130 To draw the Mathematical Ornaments on all sorts of Dials 134 To draw the Tropicks or length of the Day 136 To make the Trygon 138 To draw the Planetary or Iewish Hours 142 To draw the Italian Hours 144 To draw the Babylonish Hours 145 To draw the Azimuth Lines 146 To draw the Almicanters 154 To draw the Circles of Position or Houses 160 To draw the Hours and all the rest on the Ceiling of a Room 165 The Figure of the Instrument Explained 166 A Table of the Suns Azimuth at every Hour and Quarter in the whole Signs 168 A Table of the Suns Altitude the same time 169 The Description and Use of the Armilary-Sphear for Dyalling several wayes 172 The Description and Use of the Poor-man's Dial-Sphear for Dyalling and several Uses thereof 203 How to remedy several Inconveniences in the use of the Gunter's Rule 220 The Use and a further Description of the Trianguler-Quadrant for Navigation or Observation at Sea 227 For a fore-Observation with
as 12 6 is NR 60-40 both hours and quarters if you have them truly drawn on a large general Scheam as Mr. Lankford hath done Thus much for Direct Plains both Erect and Reclining before I come to speak of Decliners It will not be amiss to shew how to find the declination of a Plain both by the Sun-shine or without by a Magnetical-Needle as followeth As sine of the Substile 41-40 GF to = sine of the Latitude 51-32 NP So is = sine of 90 PFG to sine of GPF 58-7 the Incliner CHAP. III. To find the Declination of any PLAIN FOr finding the Declination of a Plain the most easie way is by a Magnetical-Needle fitted according to Mr. Failes way in the Index of a Declinatory as he calls it being 180 degrees of a Semi-circle divided on an Oblong-Board or Quadrant or a longer Needle in a square Box or fitted with Hinges and a Cover after all which wayes you may have them made at the sign of the Sun-dial in the Minories by Iohn Brown or of any other manner you shall think fit But to our Trianguler Quadrant is a Box and Needle also to be fitted of another form in some things more convenient Whose form is thus First in a piece of Box 5 inches long 2 ½ broad and 6 tenths of one inch thick is a hole made near 4 inches long 1 inch ¾ broad and 4 tenths deep for a Needle to play in about 50 degrees at each end with brass-hinges and a cover and a brace to keep the lid upright an Axis of Thâ—ed and a Plummet playing in the lid and a Horizontal and a South-dial drawn on the Box and Cover also a hasp and glass to keep the Needle close covered and on the bottom a Grove one tenth of an inch deeâ— made just as broad as one leg of the Sector is The use whereof is thus Put your Box and Needle on that leg of the Sector as will be most convenient for your purpose the North or cross-cross-end of the Needle toward the Wall when it is a South decliner and the contrary when it is applyed to a North decliner as the playing of the Needle will tell you better than many words then open or close the Rule till the Needle play right over the Line in the bottom of the Box unless there be variation then you must allow for it Eastwards or Westwards what it is Then I say the quantity of the Angle in degrees and minuts the Sector stands at above or under 90 is the degrees and minuts of Declination being counted from 00 in the little Semi-circle as complements to the Angle of opening as in the 4th Use of the 5th Chapter is largely and plainly shewed Thus you have the quantity of degrees and minuts of Declination but to determine which way consider thus If the Needle will stand still in the middle when the North-end is toward the Wall then the first denomination is South if not North. Again When you know where North and South is you may resolve which way the East and West is For observe alwayes if the North be before you then the East is on the right-hand and the West on the left and contrarily If the South be before you the West is on the right-hand and the East on the left Then If the Sun being in the East-point of the Horizon can look on the Plain it is a South-east Plain but if it beholds it when in the West-point it is a South-west Plain Likewise If the Cross-end of the Needle will not stand toward the Wall the Needle playing well and the Sun being due East beholds the Plain then it is so many degrees North-east but if it cannot look on the Plain being due East then it is a North-west Plain declining so many deg as the Sector stands at under or above 90 being alwayes the complement of the Angle the legs of the Sector stand at and found by taking the Angle the legs stand at from 90 when the Angle is less than 90. Or Taking 90 out of the Angle when it stands at an Angle above 90 degrees as a look at the little Semi-circle on the Head sheweth Example Suppose I come to a Wall and putting the Box and Needle on the Leg of the Sector and applying the other Leg to the Wall or on a streight piece of Wood applied to the Wall because of the Walls unevenness and open or close the Legs till the Needle playes right over the Meridian-line drawn on the bottom of the Box then I say the complement of the Angle the Legs of the Sector stands at being alwayes what it wants of or is above 90 degrees is the degrees of Declination and the Coast which way the Needle and Suns being East and West tells you For If the North or Cross-end of the Needle be toward the Wall it is a South Plain and if the Sun being in the East can behold it then it is South-east if not a South-west Plain A ready way of counting the Angle found may be thus Take the = distance between Center and Center in the middle of the innermost-lines and lay it latterally from the Center and coâ—nt two degrees more than the Point sheweth after the manner of Chords from 90 at the sine of 45 toward the Compass-point and that shall be the degrees and minuts required Example Suppose the Legs are so opened that the = distance between the two Centers makes the sine of 25 then I say the Lines do stand at an Angle of 50 degrees and the Legs at 48 two degrees less the complement whereof is 42 as if you count thus from 45 you will find 40 from 45 is 10 35 is 20 30 is 30 25 is 40 and 2 degrees more makes 42 the thing desired But If you like not the abating of two degrees then the = distance taken just beâ—ween the two legs right against the Cenâ—ers shall be just the sine of 24 degrees â—r 42 counting after the manner of Chords viz. every 5 degrees on the Sines for 10 on â—he Chords backwards from 45 of the Sines which is 90 in Chords Or If you use the first Rule of the 4th Useâ—f â—f the 5th Chapter viz. by taking the â—ine of 30 and put one Point of the Comâ—asses in the middle Center in the Tangent-â—â—ne and apply the other to the Line of â—ines you shall find it reach to the sine comâ—lement of the Angle the Lines stand at â—iz 40 degrees and 2 degrees more viz. â—2 is the Angle or thing desired as praâ—tice with consideration will make easie Thus by the Needle you may find the â—eclination of a Wall which in cloudy â—eather may stand you in good stead or 〈◊〉 prove a declination taken by the Sun to â—revent mistakes And if nothing draw the â—eedle from its right position but that it â—ay well and you find the Angle truly â—ou may come to less than half a degree And this convenience it hath
will serve very well This is a general way of augmenting all manner of Dials when the Stiles height is low as under 15 degrees and as ready a way as you meet with in any Author whatsoever CHAP. VI. To Draw the Hour-Lines on Declining Reclining Dials FOR the compleat and true drawing of these Dials that you may plainly see their Affections and Properties it will be necessary to have a Scheam for every variety in doing whereof I shall follow the Method that Mr. Wells of Deptford used in his Art of Shadows which will comprehend any sort of Reclining and Declining Dial under 6 varieties viz. 3 South Recliners and 3 North Recliners the Inclining being their opposites and no other as afterwards is shewed Wherein I shall be very brief yet sufficiently plain to a Mathematical Genius and render the Canons by Artificial and Natural Sines and Tangents and draw the Dial by the Sector the fittest Instrument for that use With other occurrent Observations as they come in place and the way by the Scheam Geometrically also 1. And first for a South-Declining Reclining Dial declining from the South toward the West 35 degrees and reclining from the Zenith 20 degrees being less than to the Pole viz. falling from you between the Zenith and the Pole As the Circle AEB representing the Reclining Plain plainly sheweth P being the Pole and Z the Zenith The manner of drawing this Scheam is plainly shewed before Chap. 1. both generally and particularly for the drawing of Dials and the Example there is the very Scheam for this Dial wherein you may further consider That the Perpendiculer-line is right before you and when you look right on this Plain that declines Southwest the North is before you on the left hand the South behind you on the right hand 35 degrees the East on the right hand the West on the left the Line CD the perpendiculer-line right before you representing the Perpendiculer-line on the Plain AB the Horizontal-line ZE the quantity of Reclination PF the Stiles Elevation above the Plain having the South Pole elevated above the lower-part of the Plain because the North-Pole is behind the Plain EG the distance on the Plain between the Plains perpendiculer and the Meridian being to be laid Eastwards as the Dial-draught sheweth besides that general Rule before hinted that whensoever a Plain declines Eastward the Substile Line must stand Westward and the contrary for the Arch whereon to prick the Substile and Stile is alwayes to be drawn on that side of the Plain which is contrary to the coast of declination EF the distance from the Substile and Perpendiculer to be laid the same way GF the distance on the Plain from the Substile to the Meridian to be laid the same way also the Angle FPG is the Inclination of Meridians All which Requisites are found by these Canons Arithmetically or by the Artificial and Natural Sines and Tangents 1. To find the Distance of 12 from the Perpendiculer EG or Horizon AG by the second Axiome of Mr. Gellibrand viz. that the Sines of the Base and Tangent of the Perpendiculer are proportional By the Sector and Quadrant As sine 90 Radius ZD 90 00 To Tang. of Declin Plain ND 35 00 So sine of Reclin Plain ZE 20 00 To Tang. of Perp. 12 EG 13 28 Whos 's complement AC 76 32 is the distance from the east-East-end of the Horizon to 12. As Tangent of ND 35 0 To = Sine of ZD 90 0 So = Sine of ZE 20 0 To Tangent of EG 13 28 the distance of 12 from the Perpendiculer 2. To find the Distance on the Meridian from the Pole to the Plain PG by the 3 Propositions of Mr. Gellibrand the Sines of the Sides are proportional to the Sines of their opposite Angles By the Quadrant and Sector As the sine of the Perpendiculer from 12 GE 13-28 To the sine of declination GZE 35-00 So is the sine of 90 GEZ 90-0 To the sine of the distance on the Meridian from the Plain to the Zenith GZ 23-55 As sine 90-0 GEP To = sine 35-0 GZE So = sine 13-28 GE To sine 23-55 GZ which taken from 38-28 gives PG 14-33 Which being taken from 38-28 the distance on the Meridian from the Pole to the Zenith leaveth the distance on the Meridian of the place from the Pole to the Plain viz. 14-33 as a help to get the next 3. To find the Height of the Stile above the Plain PF In the two Triangles ZGE and PGF which are vertical by the second Consectary of Mr. Gellibrand If two Perpendiculer Arks subtend equal Angles on each side of the meeting then the Sines of their Hypothenusaes and Perpendiculers are proportional and the contrary for the Angles ZGE and PGF are equal Angled at G and ZE and PF are both two perpendiculer Arks on the Plain AB Therefore As the sine of the Hypothenusa GZ to the sine of the Perpendiculer ZE So is the sine of the Hypothenusa PG to the sine of the Perpendiculer PF and the contrary Then thus by the Quadrant and Sector As sine of the Arch of the Merid. from the Zenith to the Plain ZG 23-57 To sine of the Reclination ZE 20 00 So is the sine of the Arch on the Meridian from the Pole to the Plain PG 14-33 To sine of the Stiles height PF 12-13 As sine of PG 14-33 To = sine of ZG 23-58 So = sine of ZE 20-00 To sine of PF 12-13 4. To find the Distance of the Substile from the Meridian GF In the same Vertical Triangle having the same acute Angle at the Base the Tangents of the Perpendiculers are proportional to the Sines of the Base by the second Axiome of Mr. Gellibrand Therefore by the Quadrant and Sector As the Tang. of the Reclin ZE 20-0 To the sine of the Distance on the Plain from the Perpend to the Merid. GE 13-28 So is the T. of the Stils height PF 12-13 To the S. of the Subst fr. 12 FG 7-58 As sine of GE 13-28 To = Tang. of ZE 20-0 So = Tang. of PF 12-13 To sine of FG 7-58 5. To find the Angle between the two Meridians of the Place and Plain viz. the Angle PFG By the third Proposition of Mr. Gellibrand it is proved That the Sines of the Sides are proportional to the Sines of their opposite Angles and the contrary Therefore by the Quadrant and Sector As the sine of the Dist. on the Merid. from the Pole to Plain PG 14-33 To the S. of 90 the opp Angle PFG 90-00 So is the S. of the Subst fr. 12 FG 07-58 To the S. of the Inclin Merid. FPG 33-28 As = sine of the side 07-58 FG To = sine of the side 14-33 PG So = sine of the Angle 90-00 PFG To sine of the Angle 33-28 FPG The Angle between the 2 Meridians By Angle of Inclinations of Meridians make the Table of the
mark them with the Names of the Rumbs to avoid confusion then is your Trygon made ready for use Then Take the distance from C in the Trigon to every crossing of the Azimuth-line and Almicanter and lay it on the Plain from the Vertical Point I on its proper Azimuth finishing one Almicanter before you meddle with another and the work with patience and diligence will be performed the lineâ— are to be drawn from Point to Point with a steady hand or a bending thin Ruler being Conical Sections Note That when the Vertical-line of the Plain falls on an even Azimuth then half the number of Rumbs will serve being laid each way on both sides at once Or Having a Table of the Angles at the Zenith the same as you made to draw the Azimuth-lines draw a Line at any convenient distance Parallel to AC the further from AC the larger and better as DEF in the Figure and note where CD crosses the last Line EF as at D make DE a Parallel sine of 90 and lay off the sine complements of the Angles at the Zenith in the Table from E towards D and draw and mark the Lines as in the Figure Otherwise The Stile being fixed and the Dial set in its place where it must be or at least set to the same Reclination and Declination that it must be then if you apply the side of the Trianguler Quadrant to the Nodus and the corner at the end of the same edge that toucheth the middle of the Nodus to the Plain and at the same time the Thred and Plummet playing neatly on the Almicanter you would draw you may find as many Points and mark them as you please without all the former trouble and it may be every whit as true if the under-side be inâ—onvenient you may use the upper only be sure that the side you apply and the Thred and Plummet play at the Angle of the Almicanter required VII To draw the Circles of Position or Houses The Circles of Position or 12 Houses meet and cross one another in the crossing of the Meridian and Horizon therefore the Horizon is the begining of the 1st and 7th Houses beginning at the East and reckoning under the Earth by Imum Coeli to the Descendant or 7th House at the West-part of the Horizon and so to Medium Coeli the beginning of the 10th House to the Ascendant or Horoscope the beginning of the 1st House To draw these on the Horizontal-Dial where they are Parallel Lines to the Hour 12 do thus Take the distance from the Apex to the Equinoctial-line and make it a = Tangent of 45 then the = Tangent of 30 degrees laid both wayes on the Equinoctial shall give Points to draw Lines by = to 12 for the Houses required For East and West Dials take the Radius as before viz. from the Apex to the Equinoctial-line on the Plain which here is the Meridian and but the length of the Stile a Tangent of 45 then the = Tangents of 30 60 and laid from 6 on the Equinoctial-line gives Points to draw Lines Parallel to the Horizon for the Houses required For East and West Recliners the Perpendiculer height of the Stile made a Secant of 0 then the Secant of the Stiles Elevation shall be Radius to prick off the = Tangents of 30 60 on the Equinoctial-line from the foot of the Stile whereby to draw Lines Parallel to the Horizon for the Circles of Position required All these Lines may most elegantly and â—asily be drawn and expressed on a large Ceiling with competent exactness in this manner following First provide a Quadrant of Brass or thin Wood of about a foot Radius or 14 15 or 16 inches also a Semi-circle of Brass of about half an inch broad and about an inch less Radius than the Quadrant the Semi-circle must have at each end somewhat more than to make up 180 degrees to nail to the Transum or stroke of the Window where your Glass is to lie Also to one Ray of the Quadrant must be fastened two strong Wyres to fasten the Quadrant to play after the manner of a Casement one Point in the Ray of the Quadrant next the Center sticking in the hole where you intend the Glass shall lie and the other end fastened to a piece of Wood nailed on the two upright Posts of the Window so that howsoever you turn the Quadrant fixed on those two Points it may be precisely Perpendiculer the Semi-circle playing all the while through a hole in the other Ray of the Quadrant that lies Horizontally having a Skrew to stay the Quadrant at any Azimuth as in Figure IV is plainly expressed to your view Then having degrees on the Semi-circle and also on the Quadrant and having fitted the Quadrant on his Points to play precisely Perpendiculer which the Plummet in the Quadrant will shew by turning it round about and put in the Semi-circle through the hole in the Horizontal Ray of the Quadrant and nailed it so to the Stoole or Transum of the Window by putting two little bits of Wood under the ends that the Quadrant may play evenly and smoothly on the Semi-circle-to almost the half-round for quite the half-round will not be necessary or useful Then is the Instrument set fit for its Operation Then first to find the Declination or rather the true meridian-Meridian-line Turn the Quadrant till the edge be just against the Sun and at the same instant get the Suns Azimuth then if you count so much as the Suns Azimuth is on the Brass Semi-circle from the place the Quadrant stands at the right way a Line drawn from the Center of the Semi-circle or Quadrant to that place is the true Meridian Line which place you must carefully find by two or three tryals and then mark it with Ink or otherwise on the Brass Semi-circle to count from thence in setting the Quadrant to the Suns Azimuth at every hour and quarter in those Points you intend to draw on the Ceiling which a crooked Rule set to 00 on the Semi-circle to pass to and fro with the Quadrant will make easie Then having a Table of the Suns Altitude and Azimuth at every hour in that Latitude you draw the Dial for First set the Quadrant to the Azimuth at the hour counted the right way from the marked Meridian-line on the Semi-circle and there skrew it fast Then extend the Thred fastened in the Center of the Quadrant till it cut the Altitude of the Sun at the same hour and Azimuth on the degrees of the Quadrant and extending the Thred to the Ceiling make a mark for that Hour and Altitude that Point at that time gives the true place where the reflected spot will fall at that Hour Azimuth and Altitude on the Ceiling of the Room This work repeated as many times as there be hours and quarters in the Summer and Winter Tropicks for about 5 hours and in the Equinoctial and any where between
help of a Sector with Sines and Tangents to 7-5 such as are usually made But for very far Decliners use that help as directed in Chap. 4. The like work serves to help all sorts of Dials with low Stiles Polar and Meridian Dials also The other 6 sorts yet behind I shall demonstrate only in two of them which do properly enough comprehend them all and the work of one is as easie as the work of the other especially by the help of the Sphear where the hardest is as plain as the Horizontal Therefore 7. Of Declining Reclining-Dials 1. For South Recliners they may recline short of to or beyond the Pole at any Declination as the putting up and down the Plain doth plainly demonstrate Therefore first Of one that Declines South-west 35 and Reclines 20 from the Zenith Set the Notch or Pole of the Plain to the Declination and the Reclining Circle to its Reclination and there make it fast then extend the Axis streight and bring the upper Semi-circle just to touch it and the Hour-circle exactly even with the moving Semi-circle Then First The Axis shews the Stiles height on the Semi-circle to be 12-13 The Thred brought along the Plain while it touches the Meridian and that shews the Meridians Elevation above the Horizon on the North Recliner to be 76-32 or its Depression below the Horizon in South-Recliners and that from the east-East-end as the Sphear sheweth Then 3. The Substile from the Perpendiculer Line of the Plain is 21-6 as the upper Semi-circle sheweth but from the hour 12 or Meridian 7-58 and stands on the East-side of the Meridian The Inclination of the Meridian is 33-29 as the degrees on the Equinoctial between the Meridian and Hour-circle shew All the Hour-Arks are easily found from the Plains Perpendiculer Eastwards and Westwards by applying the Thred to the Hour-circle and Plain being set to the Hours on the Equinoctial The South Pole is elevated in the South-Recliner and the North on the North Incliner If you set Letters to the Sides and Angles according to the former discourse you will see how all the Canons in the Arithmetical Calculation lie as I shewed you before in the Declining Dials And as again thus On the Pole set P. On the Zenith Z. At the West-end of the Plain set A. At the East-end B. At the South Pole of the Plain C. At the North Pole D. At the East-end of the Horizon E. At the West-end W. At the North-end of the Meridian set N. At the South-end S. Where the Hour-circle cuts the Plain F. Where the Meridian cuts the Plain G. Where the fixed Semi-circle cuts the Plain set E. As in the Figure before Then these Canons in short run thus As sine Base ZD 90-00 To Tang. Perpend ND 35-00 So sine of Base ZE 20-00 To Tang. Perpend GE 13-28 Whose complement AG 70-32 is the Meridians elevation As sine of the Side GE 13-28 To sine of the Angle CZE 35-00 So sine of the Angle GFZ 90-00 To sine of the Side GZ 23-57 Which taken from ZP 38-28 leaves 14-33 the distance of the Meridians place from the Pole to the Plain viz. GF As sine of Hypothen GZ 23-57 To sine of Perpend ZE 20-00 So sine of Hypothen PG 14-33 To sine of Perpend PF 12-13 Â the Stile As Tangent of Perpend ZF 20-00 To sine of Base GE 13-28 So Tangent of Perpend PF 12-13 To sine of Base FG 7-58 Â the Substile to 12. As the sine of the Side ZE 20-00 To the sine of the Side GE 13-28 So is the sine of the Angle PFG 90-00 To the sine of the Angle FPG 33-28 Â Inclin Merid. For the Hours in all Dials say thus As sine of 90 To sine of Stiles height So Tangent of the Angle at the Pole To Tangent of the Angle on the Plain 8. For North Declining Reclining-Dials For these Plains also you must rectifie the Sphear to the Latitude and set the Plain to his Declination and Inclination which is given and for which you are to make a North Declining Reclining Dial. As you did in the South-Recliner so work in all respects as you shall bring forth the Quesita's either by the Sphear or Arithmetical-Calculation as is largely shewn And for a Plain that declines 55 degrees from the North towards the East and relines 20 from the Zenith you shall find the Requisites to be as followeth 1. The Meridians Elevation above the Horizon is found to be 63 deg 58 min. But yet observe You must make use of that part of it which is below the Horizon because the Sun being Elevated high on the South-part of the Meridian must needs cast a shadow on the North-part thereof therefore in drawing the Dial-part part is only to be made use of for the Sun to shine on 2. The Stiles Elevation is 37 degrees 00 minuts 3. The Substile from 12 65-24 or from the Plains perpendiculer 39-22 The North Pole is Elevated and in regard the Plain declines to the East the Stile must be set towards the West and it shines on the Plain in Summer-time from the Rising unto 12 But in the Winter-time but a few hours Note also That these Declining Reclining-plains may be referred to a new Latitude and Declination wherein they shall become Upright Decliners as before is hinted The Poor-Mans Dial-Sphear Or another way to demonstrate the Mystery of Dyalling both for Declining and Inclining Plains in a very plain easie way for one 6th part of the cost of the other Brass-Sphear First as to the Description and afterward for the Vse AS to the Description the Figure annexed and a few words shall suffice wherein consider First The plain flat-Board representing the Horizon as ABCD. Secondly The two upright pieces as East and West-points as AE and BF to support the moving Plain Thirdly The Moving-plain moving to any Inclination on the two Points E and F with 180 degrees upon the Plain and noted by ABEF Fourthly Also a Brass-circle as G fastened to the Plain to set it to any degree of Inclination and a skrew as at H that may stay it steady when set to any Reclination Fiftly On the middle of the Horizontal-board is fastened at the Point M a true Horizontal-Dial drawn fit for your Latitude and to turn round on the Point M as IMKL Sixtly A Thred fastened in L the Center of the Horizontal-Dial and in N the Center of the Plain to be both a Stile for the Horizontal-Dial and to represent the Axis of the World also a small Woodden-Quadrant will be useful such a one as half the Plain is to draw Perpendiculers and measure Angles as afterwards in the Uses The Uses follow Use I. To find the Declination of a Plain by the Sun-shining Apply the side AB to the Wall and hold the Instrument level as by help of a Point Plummer fastened at N and the Point playing right on M it is easie to
shewed Also 60 degrees on the innermost-edge of the Loose-piece The Kalendar of Months and Dayes and degrees of the Suns Place and Right Ascention on the moveable-Leg For the speedy and ready finding the Suns place and declination which you may do to a minut at all times by help of the Rectifying Table and Astronomical Cautions of Time and Longitude Also on the Head-leg is the general Scale of Sines and Lines to the great and lesser Radius as in the Figure And thus much will serve both for Observation and Operation as in the following Discourse will fully appear 4. To this Instrument doth chiefly belong the Sights for the Observations at Sea where the Horizon is made use of in the taking the Sun or Stars Altitude And to this Instrument belongs the Index and Square that makes it a most compleat Sinical-Quadrant for the plain and easie resolving of all plain Triangles Also a weighty Plummet and Thred and a pair of large Wood or Brass Compasses for Operation Thus much for Description being all put on one side only unless you shall be pleased to add the Artificial Numbers Sines and Tangents on the outer-edge and a Meridian-line and his Scale on the inner-edge and Natural Sines and Natural Versed-Sines on the Sector-side But these as you please CHAP. II. The use of the Trianguler-Quadrant in Observation THat the Discourse may be plain and brief and general there are 10 terms to be named and described before I come to the Vses and Examples which are as followeth 1. First the Head-leg of the Instrument in which the Brass-Rivit is fixed and about which the other Leg turns as AB in the Figure on which Leg the general Scale of Sines and Lines are usually set 2. The moveable-Leg on which the Months and Dayes be as in the Figure noted by BD which Leg turns about the Head-Leg 3. The Loose-piece that is joyned to the Head and moving-Leg by two Tennons at each end thereof noted by DA in the Figure 4. The Head-Center or Center-pin on the round-part of the Head-leg being Center to the 60 degrees on the in-side of the Loose-piece which Point is known by B in the Figure 5. The Leg-Center being near the end of the Head-leg which is the Center to the degrees on the moving-Leg and out-side of the Loose-piece being in all 180 degrees and noted in the Figure by the Letter C. 6. The great Radius or greater Line of Sines issuing from the Leg-Center toward the Head having the Tangents on the moveable-Leg to the same Radius and the measure from the Leg-Center to the Tangent on the moving-Leg a Secant to the same Radius as CE in the Figure 7. The little Radius that issues from the Leg-Center toward the end having the Tangents on the out-side of the Loose-piece to the same Radius and the measure from the Center to those Tangents for Secants to the same Radius as CF. 8. The Turning Sight alwayes to be skrewed to the Head or Leg-Center known by his shape and skrew-hole as 9. The sliding Horizon-sight to slide on the moving-Leg and Loose-piece noted with its bigness and hole to look through as 10. The shadow Sight and 2 others to pin the Instrument together which you may call the Object-Sights alwayes fixed in the two holes at the ends of the moving-Leg and the Head-leg and the shadow-Sight is to set to and fro to any place required noted in the Figure with 〈◊〉 and the other two with 〈◊〉 And Thus you have their Name and Description at large which in brief take thus for easie remembring 1. The Head-Leg 2. The Moveable-Leg 3. The Loose-Piece 4. The Head-Center 5. The Leg-Center 6. The great Radius 7. The less Radius 8. The turning-Sight 9. The Horizon sliding-Sight 10. The shadow-Sight and the two Objest-Sights the open-part in one is next to and the other remoter from the Rule to answer to the upper or lower-hole in the turning-Sight according as you please to use them in Observation Thus much for the Terms the Vses follow Use I. To find the Suns or a Stars Altitude by a forward Observation as by a Fore-staff Skrew the turning-Sight to the Head-Center and put the object-Sight into the hole at the end of the Head-leg and put the sliding Horizon-sight on the in-side of the Loose-piece Then setting the turning-sight to your eye and holding the Loose-piece in your right-hand and the moveable-Leg toward your body then with your Thumb on the right-hand thrust upwards or pull downwards the Horizon-sight till you see the Sun through the Object-sight and the Horizon through the Horizon-sight then the degrees cut by the Line on the middle of the Horizon-sight shall shew the true Altitude required Also observe That if you like to use the upper or lower-edge of the Horizon-sight instead of the small bar a-cross the open-hole after the manner of the ends of a Fore-staff that then the degrees and minuts cut by the edge of the Brass is the Altitude required to be counted as it is figured from the Object-sight toward the Horizon-sight the degrees between them being the Angle required Note also That if the Altitude of the Sun or Star be above 30 degrees you will find it a hard matter to behold the Horizon and Sun with a bare roling the ball of the eye only and a stirring of the head will easily cause a stirring of the hand which will spoil the exactness of Observation unless the Instrument shall be fixed to a Ball-socket and Three-legged-staff which is not usual at Sea Therefore to remedy this you may observe with the open oval-hole in the turning-sight set to the eye or taking the turning-sight quite away Observe just as you do with a Fore-staffe setting the round part of the head to the hollow-part beside your eye so as the Head-Center-pin may be as near the very sight of your eye as possibly as you can which Center is the Center to the degrees now used in a forward way of Observation Or rather use this way when the Weather will suffer by a Thred and Plummet which I shall add as a second Use. Use II. To observe the Sun or a Stars Altitude by a forward Observation using the Thred and Plummet Skrew the turning-sight to the Head-Center as before and put the two Object-sights into the two holes at the two ends of the Rule and on the Leg-Center-pin hang the Thred with a weighty Plummet of two pound or above a pound at least Then hold up the Trianguler-Quadrant setting the small-hole on the turning-sight close to your eye and if the Sun or Star be under 25 degrees high then look to the Sun or Star through the turning-sight and that object-sight which stands in the end of the moveable-Leg letting the Thred and Plummet play between your Thumb and Fore-finger as a Brick-layers Plummet in his Plum-Rule doth in a bendid hole that you may keep it in order whilst you look
at the Sun or Star and the weighty Plummet will pull the Thred streight and let you know by feeling which way it is playing till it playeth evenly and truly whilst you have the Object precisely in the midst thereof whether it be Sun Moon or any Star or other Object whose Altitude you would observe Then I say when the Plummet playes well and you behold the Object right bend back the Quadrant and see what the Thred cuts on the degrees on the moveable-Leg which shall be the true Altitude required And in my opinion must needs be more exact than any other way of a forward Observation because you are not troubled to mind the Horizon and Sun both at at once An Objection may be The boisterous Winds and the rouling of the Ship will hinder such an Observation Answ. So it will any other way though happily not so much Again I answer One Object is better and more certainly seen than two at any time together and though the Wind blow hard if you can stand to observe at all the heavy Plummet will be sure to draw the Thred Perpendiculer and for ought I know you may come as near this way as any other however this at most times may confirm and prove the other and may be useful in Rivers and Harbours and misty-Dayes when you may see the Sun well enough but not the Horizon at all Use III. To find the Suns Altitude by a backward Observation as with a Back-staff or Davis-Quadrant Skrew the turning-sight to the Leg-Center or Center to the degrees on the moveable-Leg and set the object-sight to the long stroke by 00-60 on the out-side of the Loose-piece and put the sliding Horizon-sight on the out-side of the moveable-Leg then hold the Object-sight upwards and the small-hole in the piece turning on the â—dge or to the small-hole in the middle of the Horizon-sight which you please close to your eye and looking through that hole and the middle-hole of the turning-sight to the true Horizon turning your self about and lifting up or pressing down the Horizon-sight close to the moveable-Leg till the shadow of the upper-edge of the shadow-sight being next to the Sun fall at the same time just on the middle of the turning-sight Then I say the edge or middle of the Horizon-sight that you looked through shall cut the true Altitude of the Sun required Being the same way as you do observe with a Davis-Quadrant or Back-staff Use IV. To find the Suns Distance from the Zenith by the Trianguler-Quadrant Skrew the turning-sight to the Leg-Center and put the Object-sight whose oval-hole is remotest from the Quadrant in the hole in the end of the Head-Leg or rather in a hole on the general Scale between the turning-sight and the Sun and put the Horizon-sight on the out-side of the moveable-Leg then hold the turning-sight toward the Sun and the small-hole in the edge of the Horizon-sight to your eye then look through that hole and the turning-sight till you see the shadow the Object-sight to fall just on the turning-sight or the shadow of the turning-sight to fall just on the object-sight which is all one though the first be more easie because you shall see the Horizon through the turning-sight and that both at once Then I say the degrees cut by the Horizon-sight shall be the Suns distance from the Zenith required Being the very same work and done in the same manner and producing the same Answer viz. the Suns distance from the Zenith that the Davis-Quadrant doth Note That this way you may observe very conveniently till the Sun be 20 degrees distance from the Zenith and by the adding of a 60 Arch as in Davis Quadrant or to 45 will be enough it will do as well as any Davis Quadrant being then the same thing But I conceive the complement of the Altitude being the same will do as well which Altitude is better found by this Instrument than the distance from the Zenith by a Davis Quadrant is as in the next Use will be seen Use V. To find the Suns Altitude when near the Zenith or above 90 degrees above some part of the Horizon In small Latitudes or in places near the Equinoctial or under it the Sun will be found to be in or near the Zenith and if you count from some part of the Horizon above 90 degrees distant from it then instead of setting the sliding Object-sight to the long stroke at 00 on the Loose-piece you must set it 30 degrees more towards the Head-leg then observe as you did before and whatsoever the Horizon-sight cuts you must add 30 degrees more to it and the sum shall be the true Altitude required Example Suppose that in the Latitude of 10 deg North on the 10th of Iune when the Suns Declination is 23 degrees and 31 min. Northward Suppose that at noon I observe the Suns Meridian Altitude skrewing the Turning-sight to the Leg-Center and setting the Object-sight to the 30 degrees on the Loose-piece near the end of the Head-leg and the Horizon-sight on the movable-Leg then hold up the Quadrant with the shadow-sight toward the Sun and the small-hole in the Horizon-sight toward your eye and look to the Horizon through that and the turning-sight the shadow of the right-edge of the shadow-sight that cuts the degree of 30 at the same time falling on the middle of the turning-sight you shall find the Horizon-sight to cut on 46-29 minuts to which if you add 30 the degrees the shadow-sight is set forwards it makes up 76-29 the Suns true Altitude on that day in that Latitude 76-29 the Meridian Altitude and 23-31 the Declination added together make 100 deg 00 from which taking 90 there remains 10 the Latitude of the place 1. In this Observation first you may note this That if you had stood with your back toward the South you would have had 103 degrees and 31 minuts for the sliding Horizon-sight would have stayed at 73 degrees 30 to which if you add 30 it makes 103-31 which a Davis Quadrant will not do 2. In the holding it you may lean the head of the Rule to your breast and command it the better as to steady holding 3. You may turn the Turning-sight about to any convenient Angle to make it fit to look through to the Horizon and also to receive the shadow of the shadow-sight If the brightness of the Sun offend the eyes you may easily apply a red or a blue Glass to darken the Sun beams and the Sights may be painted white to make a shadow be seen better Use VI. To find the Latitude at Sea by a forward Meridian Observation of the Altitude according to Mr Gunter's Bow Skrew the Turning-sight to the Leg-Center and set the shadow-sight to the Suns-Declination and the Horizon-sight to the moving Leg or Loose-piece and the Turning-sight to your eye then let the shadow-sight cut the Horizon and the Horizon-sight the Sun moving it higher or lower
Horologiographia OR The Art of Dyalling BEING The Second Book of the Use of the Trianguler-Quadrant Shewing the Natural Artificial and Instrumental way of making of Sun-Dials on any flat Superficies With plain and easie Directions to discover their Nature and Affections by the Horizontal Projection With the way of Drawing the usual Ornaments on any Plain Also a familiar easie way to draw those Lines on the Ceiling of a Room by the Trianguler Quadrant Also the Use of the same Instrument in NAVIGATION Both for Observation and Operation Performing the use of several Sea-Instruments still in use By Iohn Brown Philomath London Printed by Iohn Darby for Iohn Wingfield and are to be sold at his house in Crutched-Fryers and by Iohn Brown at the Sphear and Sun-Dial in the Minories and by Iohn Seller at the Hermitage-stairs in Wapping 1671. To the Courteous Reader THou hast here presented to thy view Courteous Reader in this second Part a plain discourse of Dialling both Natural Artificial Instrumental Natural I call it from the plain illustration thereof by the Armilary Sphear of Brass herein described or by the Poor-man's Dial-Sphear as I fancy to call it being only a moving Horizontal-Dial and a moving Plain according to the Figure thereof in the Book annexed whereby all the Arks Angles Scituations and Affections are very plainly represented to an ordinary capacity Artificial I call it from the lively delineation of the Horizontal-projection the fittest in my opinion for the making plain the mystery of Dialling Instrumental I call it from the applying of the Trianguler-Quadrant to the ready resolving all the Arithmetical and Astronomical work needful thereunto and that to competent exactness as in the first Part and also in this second Part is sufficiently seen in finding the requisites and delineating the hour-lines to small parts exactly speedily by the natural Sines Tangents and Secants on the Sector and Quadrant Also the ready way of finding the Suns Altitude Hour Azimuth Angle of the Plain and any such business relating to Dyalling as in the first Part is largely treated on Further in this second Part you have Tables of the Suns Declination to every day of the years 1 2 3 after the Bissextile as near as any extent Also a short but plain direction how to use the Trianguler-Quadrant at any manner of way of Observation used at Sea as backward or forward as the Davis-Quadrant and the Cross-staff is used also as Gunter's Bow is used both for the Sun or Stars A figure of the Stereographick Projection Pag ◠The Prints of the Lines of Numbers as you see here inserted are in part according to Mr. Windgates as to a single and broken line of Numbers But the addition of the line of the Fractional parts of a pound and the several Gage-points were never before used as I know of but do much ease expedite the Operations by the Line of Numbers Sines and Tangents Also these Scales of Reduction are convenient for the finding the Decimal-fraction equal to the other Sexagenary-fraction and are agreeable to those Tables in Mr. Windgates Book of Arithmetick pag. 82. Also note that the figure of the Rule at the beginning of the Book pasted on a Board is the very same with that spoken of Chap. XV. Use 28. pag. 397 of the first Part and will work all Questions wrought by the Trianguler-Quadrant to exercise them that are out of the way to have them made and may serve as good directions to the young Instrument-Maker though these are made too too small a Radius to arrive at exactness The like may I say of the Gunters-Lines in the Figures annexed yet as large as the Book will bear Thus I have given you a brief account of my present Thoughts about this matter and somewhat more particularly in the First Part disclaiming all boasting or vain ostentation knowing that at the next Impression it may be amended in many places I shall rest and remain ready to make amends in the making of these or any other Mathematical Instruments at my House at the Sphear and Sun-Dial in the Great Minories John Browne February 16. 1670. CHAP. I. The use of the Trianguler Quadrant IN Making of DIALS SVn-Dials may be made on any Plain and all kind of Plains are either Flat as Horizontal or Vpright or Leaning The Horizontal hath two faces the one beholding the Zenith called the Horizontal-Plain the other beholding the Nadir as the Ceiling of a Room is The Upright Plains are those that make right Angles with the Horizon and do behold neither the Zenith or Nadir but are parallel to them The Leaning Plains are of two sorts generally the one called Recliners beholding the Zenith the other sort called Incliners beholding the Nadir as the outside and in-side of a Roof of a House may represent The two last sorts viz. Upright and Leaning may be Direct or Declining viz. beholding the South or North or East or West Point of the Horizon or Declining therefrom viz. Declining from South or North toward the East or West All which Plains are lively represented by a Sphear made for that purpose in Brass or Pasteboard or by the Projection of the Sphear in Plano Thus Equal to the Radius of the smaller Tangents describe the Circle ESWN representing the Horizon crossing it precisely in the the Center Z with the Lines SN and EW denoting the Points of South and North East and West Then counting the smaller Tangent on the Sector-side doubly as thus calling 5 10 10 20 20 40 30 60 40 80 45 90 c. Lay off from Z towards S the complement of the Suns Meridian Altitude in ♋ in ♈ and ♑ for those Points on the Meridian-line between Z and S and consequently the half Tangent of the complement of the Suns Meridian Altitude in every degree of Declination if you proceed so far Then for the Intersections of all those Lines and Parallels of Declination on the North-side of the Meridian Observe That the same number of degrees and minuts that any Point is above the Horizon on the South part of the Meridian in Summer just so many degrees and minuts is his opposite Parallel in Winter below the Horizon As thus for Example The Sun being in ♋ or 23 deg 31 min. of Declination North hath for his Meridian Altitude 62 degrees and so many degrees is his opposite Parallel of 23-31 or ♑ below the North part of the Horizon at midnight As thus Let the Center at the beginning of the Line of Tangents represent the Center Z and let the Tangent of 45 represent the Horizon in the Scheam viz. S. and N. Then As the distance from S. to ♑ is 15 deg taken from 45 toward 0 and laid from S. to ♑ inwards toward the Center Z as the distance was taken from the Tangent of 45 toward the beginning of the Line of Tangents that represents the Center So the Point Cancer from N.
is 15 deg counted beyond 45 toward the end below or beyond the Horizon Again As S. ♋ is 62 degrees from 45 towards 00 So is the other Point 62 degrees below N taken from 45 viz. at 76 degrees which being laid from N doth over-reach this little page So that to draw the Tropick of ♑ the Point ♋ being his opposite is 28 degrees from Z or 62 deg from S and the other Point of ♑ on the North part of the Meridian is 62 degrees counting from 45 doubly also or 28 degrees from 90 the supposed end of the Tangent which is naturally infinite being the Tangent of 76 degrees or the Semi-tangent of 152 reading the Tangents doubly from the Center which distance from the Center to the Tangent of 76 or as half-tangents 152 laid from Z gives the Point ♑ on the North-part of the Meridian below the Horizon the midst between which two Points of ♑ on the South and North part of the Meridian is the Center to draw the Tropick of Capricorn Again to illustrate this difficulty to draw the Tropick of Cancer the Suns Meridian-Altitude in ♑ his opposite sign is 15 degrees above the Horizon on the South part of the Meridian and 15 degrees below the Horizon on the North-part of the Meridian viz. the Extent from the Center to the Tangent of 52 deg 30 min. or the Semi-tangent of 105 reading it doubly being laid from Z gives the Point ♋ below the Horizon the middle between which two Points is the Center to draw the Tropick of Cancer Again for the Equinoctial or Parallel of ♈ the Meridian Altitude in ♈ is 38-28 and the Meridian Altitude likewise in ♎ his opposite Parallel is 38-28 also so that if you count 38-28 doubly beyond 45 which will be at the Tangent of 64 degrees and 14 minuts and take from thence to the Center this distance laid from Z shall give the Point AE below the Horizon and the the middle between the two Points AE is the Center to draw the Aequinoctial Then for the Hour-Lines first set off the Semi-tangent of 38-28 from Z to P and the Secant of 38-28 to the same Radius from Z to L and draw the Line L 45 parallel to EW then make PL a Tangent of 45 degrees and lay off the Tangents of 15-30 and 45 from L both-wayes as you see in the Figure Also As the Sector stands take out the = Tangents of 60 and 75 severally and turn them four times from L both-wayes and note those Points with 6 7 8 9 10 11. Lastly Set one Point of the Compasses in L and open the other to P and draw the Line WPE for the hour of 6. Again Set one Point in 7-15 degrees from L and open the other to P and draw the Hour-line 5 P 5 Set the same Extent also in 7 or 5 on the other side of L and draw the Hour-line 7 P 7 as the Figure sheweth Then Set one Point of the Compasses in 8 30 degrees from L and open the other Point to P and draw the Hour-line 8 P 8 and remove it to the other side of L and draw the Hour-line 4 P 4 And so for all the rest in order Thus having drawn the Figures to draw Lines therein which shall truly represent any Plain whatsoever observe the following Rules 1. The Horizontal-Plain is represented by the Circle E.S.W.N. 2. A direct South or North-Diall is represented by the Line E.Z.W. 4. An East or West Plain is represented by the Meridian-line of 12 viz. S. N. 5. A Polar Plain is represented by the hour of 6 viz. the Line E.P.W. 6. An Equinoctial Plain is represented by the Equinoctial-line E.AE.W. 7. Any Direct Reclining or Inclining-Plain between the two last is called A direct Recliner whose Poles are alwayes in the Meridian and are represented by any Reclining Circle as the two Circles W. ♋ E. and E. ☉ W. do shew 8. An East or West Recliner or Incliner represented by the Circle N.F.S. 9. A Declining and Reclining or Inclining Polar-Plain that is it so Declines and Reclines or Inclines as to lie parallel to the Pole as the Circle 8 P 8 doth represent 10. A Declining Reclining-Plain that so Declines and Reclines as not to fall in the Pole or Equinoctial as generally they will do as the Circle 60 G 60 doth represent which Declines from the South-eastwards and Reclines 62 deg which kind of Plains are various and infinite yet confined to six varieties as afterward Now the way of Drawing these Scheams to represent these varieties is briefly thus by the Sector First to the Radius of the small Tangents draw the Circle N. E. S. W. observing this Method if it be a South Recliner to set the letter N above and E on the right hand and contrarily in North Recliners for we meddle not with Incliners till afterwards and alwayes observe that a South Incliner is the same with a North Recliner and the contrary then cross that Circle with two Diameters precisely in the Center as the Letters shew then according to your Plains Declination from North or South toward either East or West set off the Declination with a Line of Chords or Sines as before is shewed and draw that Line for the Perpendiculer Line of the Plain and laying the same distance as much from E. and W. draw another Line Perpendiculer to the former representing the Plain then on the first Line viz. the Plains Perpendiculer lay off from Z the half Tangent of the Plains Reclination from Z to E and the half Tangent of the complement thereof from Z to Q the contrary way and the whole Tangent of the complement thereof from Z contrary to E on the same Line extended for a Center to draw the Reclining Circle that represents the Plain Lastly You must draw a Circle through Q and P P being alwayes the Semi-tangent of the complement of the Latitude laid alwayes from Z toward N for the North Pole so as to cut the Primitive Circle N.E. S.W. into two equal parts as is shewed in the 10th Proposition of the third Chapter part of which Line doth represent the Stile-Line of the Dial which last work shall be again shewed in the Example Example To draw the Scheam for a Plain Declining from the South to the West 35 degrees and Reclining 20 degrees for the Latitude of 51-30 First to the Radius of your small Line of Tangents being the Latteral distance from the Center to 45 or larger if you please draw the Circle N.E.S.W. representing the Horizon crossing it in the Center with the Lines N.S. W.E. for the North and South and East and West Lines Then Take out the latteral Tangent of half the latitude viz. 19-15 for 38-30 calling the Tangent of 10 the half Tangent of 20 and lay it from Z at the Center to P for the Pole-point Then consider the Declination of your Plain and which way as here 35