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-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-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
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-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-Horizontal-line at Right Angles then from the perpendiculer-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
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
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-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-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
But when the Sun is in the Equinoctial it beholds the South-plain at the Rising being at 6 a clock in the morning and shines on it all day till Sun set being at 6 at night and then the North Dial is useless 2. For a Declining-Plain Suppose 30 degrees South-east first set the Scheam in his right scituation for a South-east Plain then if you count 30 degrees from S toward E for the Pole of the Plain and 30 degrees from W toward S or from E toward N and draw that Line that shall represent the Plain then you shall find that the Sun being in Cancer will begin to shine on this Plain just a quarter before 5 in the morning and continue till near half an hour after 2. But about the middle of Ianuary it will shine on it till a quarter after 4 viz. till Sun set and all the hours after 2 belong to the North-west Plain that declines 30 degrees and one hour in the morning also viz. from a quarter before till three quarters after 4. The like work serves for any Decliner whatsoever in any Latitude 3. But for Decliners and Recliners Draw a long Line as AB and cross it with a Perpendiculer in the Center C and lay off from C toward A and B the Tangent of 45 or the Semi-tangent of 90 equal to the largeness of your Scheam then lay off the Semi-tangent of the Reclination from C to D up and down both wayes then take out the Secant of the complement of the Reclination which will be a Radius to draw the Arks ADB which Paper you must cut out and apply the two Points of the Paper ADBD to the two Points of Declination of the Plain noted in the Scheam with A and B that is put A to A and B to B then the round or convex-edge of the Paper represents the reclining Plain and the same edge on the other part next the Horizon Southwards represents the South-west Incliner Example Suppose I make the Paper ADB to recline 35-50 the Reclination of the Equinoctial-plain then first set the Scheam right before you in its right scituation and putting the Points A in the Paper on A on the Scheam and B in the Paper to B on the Scheam I shall find it to be even with the reclining Circle AEB then following the Tropick of Cancer I find that it shines on the North Recliner from the Rising till near 2 at which time it leaves the North-recliner declining Eastward and begins to shine upon the opposite Plain viz. the South-west Incliner declining 55-0 and reclining 35-50 and so continues till Sun-set But note That if the Line that represents the Plain cuts the Tropick twice as the Line EW for a North-plain then though the Sun leave the Plain in the morning it will shine on it again in the afternoon Note also That a North-east Recliner is represented by the other Convex-edge of the Paper as here a North-east Decliner 55 and Inclining 35-50 the Sun will shine but till 3 quarters after 8 in Cancer but in Capricorn it shines till half an hour after 9 and comes no more on it that day And note alwayes That when it leaves any Plain that then it begins to shine on his opposite as here the opposite to this North-east Incliner is the South-west Recliner being represented by the same Line or Circle ADB that the North Recliner was Only you must count that side of the Line next to the Horizon the Inclining-plain and that side next the Zenith the Reclining-plain For the Line that represents it having no bredth can be no otherwise distinguished unless you will make a material Armilary Sphear of Pastboard or Brass as the following Discourse doth plainly demonstrate in these several Operations for the better conceiving of these Mathematical Excercitations Thus you have the way of making all manner of Sun Dials upon any plain Superficies the Axis of the World being the supposed Stile to all these Plains As for those curiosities of Upright Stiles and Eliptical Dials and drawing of Dials by the Horizontal or Equinoctial Dials you have them in the Works of Mr. Samuel Foster and others and in Kerkers Ars magna c. But I intended not a Volumn of Shadows but only a further improvment of the Trianguler-Quadrant as you will see in the next Chapter of drawing the Furniture or Ornament of Dials which being but seldom used I shall here crave an Apology for the brevity therein fearing lest that to the young Practitioner it may seem somewhat hard to conceive though to the exercised in these matters it may be plain enough Then for a Conclusion you shall have an easie Mechanick way to draw a Dial on the Ceiling of a Room that lieth Flat or Horizontal which will be very good for Painters or Plaisterers to Ornament a Room withal and is not yet treated on that way as ever I read of CHAP. VIII To furnish any Dial with the usual Mathematical Ornaments by the Trianguler-Quadrant as Parallels of the Suns Declination or the Suns place or length of the Day to find the Horizontal and Virtical Lines and Points to draw the Azimuths and Almicanters the Iewish Italian Babylonish Hours and 12 Houses on any Plain before mentioned 1. To draw the Tropicks or Parallels of the Suns Declination or the length of the Day Artificial on any Dial. But note That if it be a Perpendiculer Stile whose upper Point or Apex is to be the Nodus to give the Shadow then you must strain a Thred very hard or apply a Rule for the present whereon to rest the Moving-leg on instead of the Axis or else you may do it thus as Mr. Gunter sheweth First to make the Trygon if the Rule or Quadrant prove too large for your small Dial. On a sheet of Pastboard or Slate draw a long streight Line as AB to which Line erect two Perpendiculers one at the upper and the other at the lower end as CD and EF then make AB a Tangent of 45 degrees then having first made these little Tables that follow by the Trianguler-Quadrant which is only the Suns Declination at his entrance into the whole Signs or at an even half-hour of Rising lay of both wayes from B the Tangents of the Suns declination at ♈ ♉ ♊ ♋ as in the Table following and draw Lines to these Points from the Center A as in the Figure annexed and then set the marks to them and this is the Trigon Figure I. Suns declinations for the Parallels of the length of the Day Hours Declin 16-26 23-31 16-0 21-41 15-0 16-55 14 11-37 13 5-53 12 0-00 11 5-53 10 11-37 9 16-55 8 21-41 7-34 23-31 For the Signs of the Zodiack Signs Declin ♋ 23-31 ♌ ♊ 20-14 ♉ ♠11-31 ♈ ♎ 0-00 ♓ ♠11-31 ♒ ♠20-14 ♑ 23-31 Declinations 5-0 10-0 15-0 20-0 23-31 both ways Then from the Center A any way on the Line CD at such a convenient distance as you
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
00 00     15 56 13 25 06 41      13 44 11 11 04 23      11 35 8 59 02 08     5 9 32 6 50 00 06      7 23 4 44       5 26 2 41       3 36 0 41      4 1 32       The Description and some Uses of the Sphear for Dialling and for the better understanding of the general and particular Scheams NExt the Foot and Semi-circle Frame for supporting of it you may consider 1. The fixed Horizon to which the Foot is fastened with 4 skrews numbred and divided into 360 degrees or four 90 deg whose count begins at the Dividees side of the Meridian-Circle 2. The Meridian Circle whose fore-side at the Nadir-point stands in the Center of the Foot this is also divided into 4 90s s and begins to be numbered at the South and North part of the Horizon upwards toward the Zenith and downwards toward the Nadir which Circle is alwayes fixed as the Horizon is 3. The Equinoctial Circle made fast at the East and West Points of the Horizon moving up and down upon the Meridian-Circle according to the Elevation of the Equinoctial in any Latitude this is divided â—â—kewise into four 90s s numbred from the Meridian each wayes to the East and West Points of the Horizon 4. On the Meridian Circle is set 2 moveable Poles to be elevated or depressed fit to the Latitude of any place on the Fiducial-edge of which is fastened the Thred representing the Axis of the World at any Elevation of the Pole 5. On the 2 Pole Points is fastened the Hour Circle which delineates or represents the motion of the Sun or any fixed Star moving in its supposed Diurnal motion about the Poles of the World and may not improperly be called the moveable Meridian Circle or Hour Circle divided as before 6. The Moveable Horizon that moveth about to any Azimuth and slideth or moveth in the fixed Horizon 7. The Plain fixed in 2 opposite Points to the moving Horizon being set either Horizontal when it lies Parallel to the fixed Horizon or Erect when Perpendiculer thereunto or set to any Reclination or Inclination by help of the Semi-circle of Reclination fastened to the backside of the Plain in the 2 Poles thereof 8. You have the upper moving Semi-circle in turning about of which whateveâ— degree the fore-side of the Semi-circle cuts the Perpendiculer-point cuts the compleâ—ment thereof and to be called the upper Semi-circle or Circle alwayes Perpendicuâ—ler to the Plain 9. There ought to be a Thred fastened in the Center of the Plain to be extended to any Altitude or Azimuth required Thus much for Description repeated again in short thus The Horizon The Meridian The Equinoctial Circles The 2 Pole Points and Axis The Hour Circle or Moveable Meridian The Moveable Horizon The Plain The Semi-circle of Reclination The upper Semi-circle and The Thred Note also Every Circle is divided into 4 times 90 and numbred the most useful way Also on the Plain is set the 12 Months and every single Day on which every respective day if you extend the Thred then in the degrees is the Suns Right Ascention in degrees on the innermost Circle the same in hours and quarters from the next Equinoctial-point on the Line of Declination his mean Declination on the Line of â—he Suns place his mean true place sufficiently true for any illustration in Mathematical practice The Uses whereof in some part follow 1. To rectifie the Sphear to any Latitude count the Elevation of the Pole on the Meridian Circle from the Horizon upwards and downwards from the North and South parts of the Horizon and there make fast with the help of the small skrew the Fiducial-edge of the Poles Points carrying the Hour Circle fixed upon them then the Pole is rightly elevated 2. Count the complement of the Poles Elevation on the Meridian from the South part of the Horizon and to it set the divided side of the Equinoctial Circle then is that rectified also in the Northern Hemisphere or in the Southern if you call the North Pole the South Pole 3. Extend the Thred or Axis passing through the Center to the South Pole and there make it fast and then the Sphear is rectified for many Uses in that Latitude Use I. The Day of the Month being given to find the Suns true Place Lay the Thred in the Center of the Plain on the day of the Month and in the Line of the Suns place you have his place Example On the 5th of November it is 23 degrees in â™ or if the Suns place be given look for that and just against it in the Months is the day required Example The Suns place being 15 degrees ♌ I look for it in the Line of his place and just against it I find Iuly 28 day Use II. To find his Declination any day Look for the day given and right against it in the Line of Declination is his due Declination required Example August the 5th The Declination is 14 degrees 5 minuts from the next Equinoctial-point viz. ♎ Note In the Northern Sines or Summer-time the Sun hath North declination or in Southern Sines or Winter-months the Sun hath South declination Or if you have the Suns declination find that in the Line Declination and right against it in the Months is the day required Example 21 degrees South declination beginning from the Equinoctial towards the Winter Solstice I find Novemb. 15. The like work had been if the Suns place had been given to find his declination Use III. The day given to find the Suns Right-Ascention This is usually reckoned from ♈ to ♈ round in 24 hours but twice 12 is as useful and then it is thus Find the day amongst the Months and Dayes and just against it in the time of Hours is the Suns Right Ascention but note it is not right figured for this use counting onwards from ♈ or the 10th of March to the 13th of Septemb. and from thence to Aries again Likewise the degrees are to be reckoned from ♈ onwards as the Months proceed Example On the 12 of May what is the Suns Right Ascention Lay the Thred on the 12th of May and in the Line of Hours it cuts 9-57′ counting from Aries onwards or in degrees 59-15 counting as before Thus if any one of these 4 general things be given the other may be found Use IV. The Suns Declination and Latitude being given to find the Suns Meridian Altitude The Sphear being rectified count the declination on the Meridian from the Equinoctial that way the declination is either North or South and where the count ends there is the Meridian Altitude required for that day or Declination Example Iune 11. Declination 23-30′ Count 23-30 from 38-30 the place where
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