5 N 30 Dolphins head 3 307 53 20 32 15 0 N. 31 Pegassus mouth 1 321 50 21 27 8 19 N 32 Pomahant 3 339 30 22 38 31 17 S. 33 Pegassus lower wing 2 358 50 23 55 13 22 N. As for Example To find the Suns Declination for the year 1670 on the 12th day of May First if you divide 70 being the tens only of the year of our Lord by 4 rejecting the 100s s you shall find 2 as a remainder which notes it to be the second after Leap-year and if 0 remain then it is Leap-year Then Look in the Table of Declination for 1666 the second after Leap-year as the year 1670 is and find the Month in the head of the Table and the day on one side and in the meeting-point you shall find 20 deg 31 min. for the Declination on that day at noon required Or If you use the Trianguler Quadrant extend the Thred from the Center over the 12th of May and you shall find it to cut in the degrees just 20 deg 31 min. the true Declination for that year and day Note That if you have occasion to use the Declination before noon then observe that the difference between stroke and stroke is the difference of Declination for one day and by consequence one half of that space for half a day and a quarter for a quarter of a day c. As thus for Example Suppose I would have the Suns Declination the 18th of August 1666 at 6 in the morning here you must note that the 18th stroke from the beginning of August represents the 18th day at noon just Now the time required being 6 hours before noon Lay the Thred one fourth part of the distance for one day toward the 17th day and then in the degrees the Thred shall cut on 9-43′ whereas at noon just it will be but 9-38 and the next or 19th day at noon it is 9 degrees 16 min. and 3 quarters of a min. as the three pricks thus ... in the Table doth plainly shew but by the Rule a minut is as much as can be seen and so near with care may you come Note also farther That if you shall use it in places that be 4 hours 6 or 8 10 or 12 hours more Eastward or Westward in Longitude the same Rule will tell you the minuts to be added in Western-Longitudes or to be substracted in Eastern-Longitudes as Reason and Experience will dictate unto you with due consideration For if being Eastwards the Sun comes to the Meridian of that place before it comes to the Meridian of London then lay the Thred as in morning hours But if the place be to the Westwards where it comes later then lay the Thred so many hours beyond the Noon-stroke for London as the place hath hours of Western-longitude more than London counting 15 degrees for an hour and 4 minuts for every degree and then shall you have the Declination to one minut of the very truth But if it happens to be the Leap-year or the first or third year after the Leap-year then thus Suppose for the 5th of October 1671 being the third after Leap-year I would have the Declination First if you lay the Thred over the 5th of October in the degrees it gives 08 deg 42 minuts for the Declination in the second year after Leap-year then because this is the third year look in the Rectifying-Table for the 5th of October and there you find s. 4 for substract 4 minuts and a half from 8-42 rests 8-38 the true declination required for the 5th of October 1671. The like work serves for any other day or year but for every 5th and 10th day you have the Declination set down in a Table for all 4 years to prove and try the truth of your Operations and by that and the Line of Numbers or the Rule of Three you may continue it to every day by this proportion As 5 dayes or 120 hours to the difference of Declination in the Table between one 5th day and another So is any part of 5 dayes or 120 hours to the difference in Declination to be added or substracted to the 5 dayes Declination immediately fore-going the day required Example Suppose for the 18th of February 1669 the first after Leap-year I would know the Declination by the Table made to every 5th day only On the 20th of February I find 6-53 ½ On the 15th day 8-47 the difference between them is 1-53 ½ then the Extent of the Compasses from 5 the Number of dayes to 1-53 the minuts difference counted properly every 10th for 6 minuts shall reach from 3 the dayes from 15 toward 18 to 1 degree 7 minuts and a half which taken from 8-47′ the Declination for the 15th day leaves 7 degrees 38 minuts and a half the true Declination for the 18th day of February in the first after Leap-year Or by the Line of Numbers thus The Extent from 5 the difference in dayes to 113 ½ the difference in min. for 5 dayes shall reach from 3 the difference in dayes to 68 the difference in minuts for 3 dayes to be added or substracted according to the increasing or decreasing of the Declination at that time of the year Proved thus If you substract 5′ ½ from 7 deg 44 ... the declination in the second year there remains 7 deg 38′ ½ the Declination for the 18th of February 1669. These Tables may serve very well for 30 years and not differ 6 minuts in Declination about the Equinoctial where the difference is most and in Iune and December not at all to be perceived Thus you may by the Rule and Rectifying Table find the Suns Declination to a minut at any time without the trouble of Calculation CHAP. IV. The use of the Trianguler-Quadrant in the Operative part of Navigation Use I. To find how many Leagues or Miles answer to one Degree of Longitude in any Latitude between the Equinoctial and Pole FIrst it is convenient to be resolved how many Leagues or Miles are in one Degree in the Meridian or Equinoctial which Mr. Norwood and Mr. Collins hath stated about 24 leagues or 72 miles Or. If you keep the old number making the miles greater viz. 60 miles or 20 leagues then the proportion by the Numbers Sines and Tangents runs thus As Sine 90 to 20 on the Numbers for leagues So is Co-sine of the Latitude to the leagues on the Numbers contained in one degree of Longitude in that Latitude But in Miles to have the Answer work thus As Sine 90 to 60 on Numbers So Co-sine Latitude to the number of miles Example Latitude 51° 32′ As Sine 90 to 60 So Sine 38-28 to 37 miles 30 100. But by the Trianguler-Quadrant or Sector work thus Take the latteral 20 for leagues or 60 for miles from the Line of Lines from the Center downwards and make it a parallel in the sine of 90 laying the Thred to the nearest distance Then The nearest distance
Sector is set to the small Tangent and by turning 4 times you have the remainder of the great Tangent above 45 when the Sector is set to the great Line of 45 as in the Polar Dial. Or else Alter the Sector to the Radius of 45 in the great Tangent that goes but to 45 and take out the = Tangents of 30 and lay it from 6 both wayes for 4 8 and the = Tangent of 15 and lay it from 6 both wayes for 7 5. And lastly By all those Points draw Lines = to 6 for the Hour-lines required and number the East-dial with the morning-hours and the West with the afternoon-hours the Stile is to be a Plate or an upright Point the top of whose edge or point is to be equal to the Tangent of 45 as the Sector stood to prick down the Hour-lines 4. To draw the Horizontal-Plain The fourth Plain next in a natural order of easiness to apprehend as I judge is the Horizontal Dial that lies with its plain = to the Horizon and the Zenith is the Pole thereof represented by the primitive Circle S.N.E.W. in the general Scheam wherein only the Hour Arks and Stile is required The Stiles Elevation is alwayes equal to the Latitude and therefore given the Substile is alwayes in the Hour 12 being the Meridian-line The Hour-lines are found by this general Canon As the Sine of 90 the right Angle PN 1 to the Sine of PN a side alwayes equal to the Latitude or Stiles elevation 51-30 So is the Tangent of the Angle NP 1 15 or NP 2 30 c. the Angles at the Pole to the Tangent of N 1 a side or N 2 a second side the several Hour-arks on the Plain required found by the Artificial Sines and Tangents as fast as one can write them down Thus The Extent of the Compasses from the Sine of 90 the Sine of the latitude 51-30 being laid the same way from the Tangent of 15 shall reach to the Tangent of 11-50 and if you turn the Compasses the other way from the Tangent of 15 it shall give the Tangent of 71-6 for the hour of 5 as well as 11 which Numbers being gathered into a Table and laid off by Chords or Sines in a Semi-circle shall be the true Hour-points to draw the Lines by But I shall not insist further thereon but shew how to draw it more readily and as truly by the Sector thus First draw a streight Line in the Meridian if the Plain be fixed for 12 as the Line AB then design a Point in that Line to serve for a Center as at C then on the Center C erect a Perpendiculer-line to AB and draw it through the Center C for the two 6 a clock Hour-lines as the Line DE then draw two Lines equally distant from and = to the first Line AB on either side as large as the Plain will give leave as DF and EG which may commonly serve for margents to put the figure in Then Take the distance CD and make it a = Secant of 00 and take out the = Secant of the complement of the latitude and lay it from D to F and from E to G on the two Parallel-Lines and draw the Line FG. Then lastly For pricking down the Stile Note That the = Tangent of 38-30 the complement of the latitude as the Sector stands for the Noon hours laid from D to H gives a Point to draw it truly by or the Sine of 51-30 the latitude laid from B at nearest distance about H as the Sector stood for the morning-hours will do as well The Stile is to be a Plate or a bended Wyre cut or bended according to the Angle HCB and erected Perpendicularly on the Line 12 so long as the Sun being 62 degrees high may cause the shadow thereof to reach the hour of 12 and then set duly North South and Horizontal the shadow will shew the true hour of the day Note the Figure Note also That a Horizontal Dial drawn for any one latitude may serve for any other latitude North or South elevating or depressing the Stile till it look to the Pole-point that is by making it to recline Northward or Southward as much as the difference of the latitudes viz. that the Dial was made for and that wherein it is to be used shall be 5. To draw a North or South Plain The next Plain to this and most like it is the Direct North and South Dial whose Plain lies = to the prime Virtical or Circle of East and West and its Poles in the South and North part of the Horizon and represented by the Line EZW in the general Scheam whose Stile is alwayes equal to the complement of the Latitude as the Horizontals was equal to the Latitude and consequently given The Hour Arks on the Plain are found by the former Canon viz. As the Sine of 90 viz. the Angle PZE is to the Sine of the Side PZ the Co-latitude or Stiles Elevation The Arks on the Plain found as before by Artificial Sines and Tangents and being drawn into a Table to be laid off by Chords or Sines or by the Sector Thus Draw a Perpendiculer-line for the Substile or 12 a clock Line and in that Line design a Point for a Center as the Point A in the Line AB through which Point A draw another Line crossing the former at Right Angles for a Horizontal-line and the two sixes as you did in the Horizontal then on each side and equi-distant from 12 make two Lines = to AB as marginal-lines as CF and DE The distance AD of the Parallel make a = Secant of 00 and take out the = Secant of 51-30 the latitude of the place and lay it from C to F and from D to E and draw the Line FE then make DF a Tangent of 45 and lay off the hours and quarters as you did in the Horizontal in all respects Also Make BF a = Tangent of 45 and lay off the = Tangents of every hour and quarters if you please from B both wayes toward E and F and by those Points draw Lines for the hours required The Angle of the Stile may be laid off by Sines Tangents or Chords as before is shewed to the quantity of the complement of the Latitude and may be a Plate or Wyre as you please as the Angle GAB. The North Dial is the same with the South for manner of making only the Noon-hours are neglected and the Morning and Evening-hours both before and after 6 on each side only inserted and the Center of the Dial for that cause appointed in the middle of the Plain and not on the upper-part as in the South and the Stile-points upwards as in the South it points downwards Note the Figures 6. To draw the Hours on a Direct Recliner The next Plain to be considered being also Direct but not Erect or Upright but Leaning from
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-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
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
prick after notes a quarter of a minut and two pricks half a minut and three pricks three quarters of a minut more Now by the Rule you may count to a minut and the Rectifying Table tells you how many minuts more you must add to or substract from the degrees and minuts the Table or Rule shall shew it is in the second year A Table of the Suns Declination every day at Noon for London in the year 1666 the second year after the Leap-year according to Mr. Street's Tables of Longitude Calculated by Iohn Brown 1668. Month Dayes Ianu. Febr. March April May. Iune D.M. D.M. D.M. D.M. D.M. D.M. 1 21 45 ... 13 50. 03 29. 08 31. 18 02 23 11. 2 21 36 13 30. 03 05. 08 53. 18 17. 23 14. 3 21 25. 13 10. 02 42 09 15. 18 32 23 18. 4 21 14 ... 12 49 ... 02 18. 09 36 ... 19 46 ... 23 21 5 21 03. 12 29 01 54 ... 09 58 19 01 23 23 ... 6 20 51 12 08. 01 31 10 19. 19 14 ... 23 26 7 20 40 11 47. 01 07 10 40 ... 19 28. 23 27 ... 8 20 27. 11 25 ... 00 43. 11 01 19 41. 23 29. 9 20 15 11 04. S. 19 ... 11 22 19 54. 23 30. 10 20 01 ... 10 43 N. 04. 11 42 ... 20 07 23 30 ... 11 19 48. 10 21 00 27 ... 12 03 20 19 23 31 12 19 34. 09 59. 00 51. 12 23 20 31 23 30. 13 19 20. 09 37 01 15 12 43. 20 42. 23 30 14 19 06 01 15 01 38. 13 03 20 54 23 29 15 18 51 08 52 02 02. 13 22. 21 04. 23 37. 16 18 35 ... 08 29 ... 02 26 13 42 21 15. 23 15. 17 18 20. 08 07. 04 49. 14 01 21 25. 23 23 18 18 04 07 44 ... 03 13 14 20 21 35 23 20. 19 17 48 07 22 03 36 14 38. 21 44. 23 17. 20 17 31. 06 59 03 59. 14 57 21 53. 23 14 21 17 14. 06 36 04 22. 15 15 22 01 ... 23 10 22 16 57. 06 13 04 45 ... 15 33 22 10. 23 05. 23 16 40 05 50 05 0â— ... 15 50 ... 22 17 ... 23 01 24 16 22. 05 26. 05 32 16 08 22 25. 22 55 ... 25 16 03 ... 05 03. 05 34. 16 25. 22 32. 22 50. 26 15 45 04 39 ... 06 17. 16 42 22 39 22 44. 27 15 27 04 16. 06 40 16 58. 22 45. 22 37 ... 28 15 08 03 52 ... 07 02. 17 14 ... 22 51. 22 31 29 14 49 Â 07 25 17 30 ... 22 57 22 23 ... 30 14 29. Â 07 47 17 47 23 02 22 16. 31 14 10. Â 08 â—9 Â 23 06. Â A Table of the Suns Declination every day at Noon c. Month Dayes Iuly Augu. Septem Octob. Novem Decem D.M. D.M. D.M. D.M. D.M. D.M. 1 22 09 15 14. 04 26. 07 12. 17 37 ... 23 07. 2 22 00. 14 56 04 03 ... 07 35. 17 35 ... 23 12 3 21 51. 14 37. 03 40 ... 07 58 ... 18 10. 23 16 4 21 42. 14 19. 03 17. 08 20. 18 25. 23 19. 5 21 33 14 00 ... 02 54. 08 42. 18 41. 23 22. 6 21 23. 13 41. 02 31 09 04 ... 18 56. 23 25 7 21 13. 13 22. 02 08. 09 27 19 10 ... 23 27. 8 21 02. 13 03 01 44. 09 49 19 25. 23 28 ... 9 20 52 12 43. 01 20 ... 10 11 19 39. 23 30 10 20 41 12 23 00 57. 10 32. 19 53 23 30 ... 11 20 29 12 02 ... N 34 10 53 ... 20 06. 23 31 12 20 17. 11 43 N 10. 11 15. 20 19 23 30. 13 20 05 11 21. S 13. 11 36. 20 31 ... 23 30 14 19 52 ... 11 02 00 36 ... 11 57 20 44 23 29 15 19 39. 10 41. 01 00 ... 12 18. 20 55 ... 23 27 16 19 26 ... 10 20. 01 24 12 38 ... 21 07 23 25. 17 19 13. 09 59. 01 47. 12 59. 21 18. 23 22. 18 18 59. 09 38 02 10 ... 13 20 21 28 ... 23 19. 19 18 44 ... 09 16 ... 02 34. 13 39 ... 21 38 ... 23 16 20 18 30. 08 55. 02 58 13 59. 21 48. 23 11 ... 21 18 15 ... 08 33. 03 21. 14 19 21 58 23 07. 22 18 00. 08 11. 03 44. 14 38 22 07 23 02. 23 17 45. 07 49 ... 04 08 14 57. 22 15. 22 57 24 17 29. 07 27. 04 31. 15 16 22 23. 22 51. 25 17 31. 07 05. 04 54. 15 35 22 31 22 44 ... 26 16 57 06 43 05 18. 15 53. 22 38. 22 38 27 16 40. 06 20. 05 41 16 11. 22 45 22 30. 28 16 24 05 57. 06 04 16 29. 22 51. 22 23 29 16 06. 05 35. 06 27 16 46 ... 22 57. 22 15 30 15 49. 05 12. 06 49. 17 03 ... 23 02 ... 22 06. 31 15 32 Â Â 17 20. Â 21 57. A Table of the Suns Declination for every 5th and 10th Day of every Month of the Four years Calculated from Mr. Street's Tables of the Suns place made for the years 1665 1666 1667 and 1668 the nearest extant M. D 1. 1665 2. 1666 3. 1667 L.Y. 68 Ianuary 05 21 01 21 03. 21 06. 21 09. 10 19 58. 20 01 ... 20 05 20 08. 15 18 47. 18 51 18 54. 18 58. 20 17 27. 17 31. 17 35 17 39. 25 15 59 ... 16 03 ... 16 08 16 12 ... 30 14 25 14 29 ... 14 34. 14 39 February 05 12 24 12 29 12 34 12 39 10 10 39. 10 43 10 48 10 53. 15 08 47 08 52. 08 57 ... 09 03. 20 06 53. 06 59 07 04 ... 07 10. 25 04 57. 05 03. 05 08 ... 05 14. 28 03 47. S 03 52 ... 03 58. 04 04. March 05 01 48 ... 01 54 ... 02 00. 01 42. S 10 00 10 N 00 04. N 00 01 ... S 00 16. N 15 02 08 02 02. 01 56. N 02 14. 20 04 05 03 59. 03 54 04 11. 25 06 00. 05 54. 05 49. 06 06. 30 07 52 ... 07 47 07 42. 07 58. April 05 10 03. 09 58 09 53 10 09 10 11 47 ... 11 42 ... 11 37 ... 11 35. 15 13 27. 13 22. 13 17. 13 32. 20 15 01. 14 57 15 52. 15 06. 25 16 29. 16 29. 16 21. 16 34 30 17 50. 17 47 17 42 ... 17 54. May. 05 19 04. 19 01 18 57. 19 08 10 20 10 20 07 20 04 20 13. 15 21 07 21 04. 21 02 21 10 20 21 55. 21 53. 21 51 ... 21 57 ... 25 22 34 22 32. 22 30 ... 22 36 30 23 03 23 02 23 01 23 04. Iune 05 23 24 23 23 ... 23 23 23 25. 10 23 30 ... 23 30 ... 23 30. 23 31 15 23 27 23 27. 23 28 23 26 ... 20 23 13 23 14 23 15 ... 23 12 25 22 48 ... 22 50. 22 51 ... 22
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
that it carries the Needle a competent distance from the Wall to prevent that attraction but if it happen to be so near a Meridian or East and West-plain that the Angle by the Sector cannot well be taken then you may only apply the side of the Box and Needle to the Wall and the Needle it self will shew the Declination on the degrees on the bottom of the Box. Yet for exactness the way by the Sun is alwayes the best where you may come to make a good Observation and then the Needle only is not to be trusted to a better way with opportunity offering it self To fâ—nd a Declination of a Wall by the Sun For this purpose you must or ought to have another Thred and Plummet which Thred may be a fine even small Pack-thred and it is convenient to have it ready hanged up near the Wall so far off as the Trianguler-Quadrant may pass along between it and the Wall that you may not be troubled to hold it up and lay it down and be annoyed with the inconveniencies of your hand shaking and time wasting to more uncertainty than needs be Also You must needs take notice of the two Meridians viz. one of the place which is the Meridian or 12 a clock to which place when the Sun or a Star comes it is said to be in the Meridian And the other is the Meridian of the Plain in which Line the Pole-point of every Plain is being 90 degrees distant from the Plain every way and in all upright-Dials their Pole is in the Horizon and that degree of Azimuth in which the Pole-point lies counted from South or North toward East or West is alwayes the declination thereof so that by finding the Suns Azimuth at any time and the distance of the Sun at the same time from the Meridian of the Plain is gotten the declination The Azimuth of the Sun from the Meridian of the place is found by the 26 27 28 30 32 34 39 Uses of the 15th Chapter But the Azimuth of the Sun from the Meridian of the Plain is found by applying the Head-leg against the Plain Horizontally slipping it to and fro till the shadow of the Thred hung or held up play right over the Center of the Trianguler-Quadrant on the Head-leg then what deg soever the thred cuts counted from 60â—0 on the Loose-piece being the Perpendiculer or Pole-point of the Plain shall be the Azimuth of the Sun from the Meridian of the Plain This is the Operation the Application or Use is worded several wayes by several men I hope I shall do it as fully and as briefly as some others The Sun to our appearance passeth from East by the Meridian to the West every day therefore in the morning it wants of coming to the Meridian at noon it is for a moment just in the Meridian and in the afternoon it is past the Meridian of the place Even so it begins to shine on and is directly against and leaveth to shine upon most Plains when it begins to shine upon or is not directly against I say it wants of coming to the Pole or Meridian of the Plain When it is directly against the Plain then it is in the Meridian or Pole of the Plain when it â—s past it is past or begins to leave the Plain Which Thâ—ee Varieties I intend thus briefly to eâ—press Azimuth Want or W in the morning only Azimuth Direct at noon Azimuth past or P being in the afternoon The other Three Varieties let be Shadow Want Shadow Direct Shadow Past all which may be in several Plains at several times that is to say at morning noon and night These Observations and Cautions premised the Rule is thus 1. If the Azimuth and Shadow are both wanting or both past substract the losser out of the greater and the residue is the Declination But if one want and the other be past then the sum of them is the Declination 2. If the Sun come to the Meridian of the Plain before it come to the Meridian of the place it is an East Plain But if it come to the Meridian of the place before it come to the Meridian of the Plain it is a West Plain 3. If the sum or remainder after Addition or Substraction be under 90 it is a South-east or South-west Plain declining so many degrees as the sum or residue is But if the sum or remainder be above 90 it is a North-east or a North-west Plain and the complement of the sum or remainder to 180 is the quantity of Declination North-east or North-west 4. If the sum or remainder be 00 it is just South If 90 just East or West But If it be 180 it is a direct North Plain It shall be further Explained by two or three Examples Suppose that on the first of May in the forenoon I come and apply the Head-leg of the Trianguler-Quadrant to the Wall and holding of it level the shadow of the Thred held up steady cuts the Center and 60 degrees on the Moving-leg that is 60 deg want which I presently set down in a Paper ready prepared thus May 1 1669. Forenoon Shadow 60 00 want Altitude 20 00 Â Azimuth 94 00 want Substract 34 00 South-east Then as soon as possible or rather by some body else at the same moment find the Suns Altitude which suppose to be 20 degrees but if you are alone and have a Thred ready hanged up then take the Altitude first and the shadow will be had presently after the Thred hanging steadily and set that down also as here you see Then by the 26th Use of the 15th Chapter you shall find the Suns Azimuth at that time and Altitude to be 94 degrees and after Substraction remains 34-0 for the Walls declination Eastward becauâ—e the remainder is under 90 and the Sun comes to the Meridian of the Plain before it comes to the Meridian of the place or South Again In a morning Iune 13 I observe the Altitude and find it 15 degrees and instantly the shadow and find it to be 10 degrees past the Plain viz. on the Loose-piece toward the Head-leg I set both Altitude and Shadow with the day and time down thus Iune 13 Forenoon Altitude 15 0 Â Shadow 10 0 Past. Azimuth 109 0 Want Sum is 119 0 Â Â 180 0 Â Â 061 0 North-east And then find the Azimuth at that time and Altitude to be 109 degrees here the terms being unlike I add them together and the sum being above 90 I know it must be a North Plain and because the Sun comes to the Plain before it comes to the Meridian of the place it is North-east and the complement of 119 to 180 is 61-0 North-east Again Iune 13 Afternoon Altitude 15 0 Â Shadow 20 0 Want Azimuth 109 0 Past. Sum is 129 0 Â Â 180 0 Â Â 051 0 North-west But if you happen to come when the Sun is in the Meridian of the Plain then
according to what time of year you would know when the Sun begins and ceases to shine on any Plain and turn the Hour-circle with the Thred so laid till the Thred do but just touch the Plain and the Hour-circle doth on the Equinoctial cut the Hour and Minuit required when the Sun comes on the East-side and when it goes off from the West-side of the Plain Example Suppose you have a Direct North-plain that Reclines from the Zenith towards the Equinoctial 25 degrees you shall find the Stiles Elevation to be 63-30 the Substile from 12. The North-Pole to be elevated on the Recliner and the South-Pole on the Incliner and that the Sun shines on the North-recliner in the longest dayes viz. 23-31 declination from the Rising 13′ before 4 till 10 and then it begins to shine on the South-incliner and shines till 2 afternoon then it comes on the North-recliner again and continues till it sets But in the shortest dayes when the Declination is 23-30 towards South then on the North-recliner it shines not at all but only on the South-incliner from Rising to Setting and so doth it all the time the Sun hath South-declination This Rule serves for all sorts of Dials whatsoever Note That the Circles of the Sphear shews the Canon to work this Question exactly whereof you have a large Discourse in Wells his Art of Shadows from pag. 391 to 408 in 35 Chap. 4. For a Direct East or West Erect-Dial The Sphear being rectified to the Latitude bring the Notch in the Moveing Horizon to the East or West-points on the fixed Horizon viz. to 90 degrees then set the Plain Erect and make it fast there then you shall perceive the Axis lie close to the Plain it shews the Stile to have no Elevation but must be set Parallel to the Plain at any quantity you please which is to be the Radius of a Tangent-line whereby to prâ—â—k down the Hours and that the Sâ—bstile or place where the Cock or Stile must stand is in 6 being the Hour-circle till it be â—ust against the upper Semi-circle touchâ—ng the Thred and in the Equinoctial it cuts 6 the true place where the Stile must stand Also By the fore-going Rule you shall find the Sun shine all the year from the Rising till 12 on the East-side and on the West-side from 12 till his Setting 5. For an East or West-Recliner Turn the moving Horizon to 90 degrees in the fixed as before then set the Plain to his due Reclination and make it fast there and pull the Axis streight and bring the upper Semi-circle just to touch it and straitway you have the Stile and Substile and 12 the Inclination Meridian and Hour-Arks on the Plain As for Example An East-plain reclining from the Zenith towards the Horizon 45 degrees hath his Meridian or 12 a clock Line in the Horizon for if you extend the Thred from the Center to the fore-side of the Meridian just there the 12 a clock Line must alwayes be which in this Plain lies in the Horizon The Substile doth lie 41-40 from thence upward as the upper Semi-circle doth shew the Inclination Meridian is thus found Bring the Hour-circle till it stand even and parallel to the upper Semi-circle then on the Equinoctial it cuts 58-7′ the Inclination of the Meridian with which you must make a Table of Hours or Arks at the Pole to calculate the Arks on the Plain if you work Arithmetically But by the Sphear Set the Hour-circle to the hours on the Equinoctial and the Thred being brought along the Plain till it touch the Hour-circle shall shew on the Plain the Angle from the Horizon or Perpendiculer or with some more trouble from Substile or 12. Also It shews that the North-Pole is Elevated on the West-reclining and the South on the East-inclining opposite thereunto and that the Recliner in ♋ shews from 9 in the forenoon till 8 at night and the East Incliner from the Rising till 9 forenoon in Summer and in Winter till a 11 in the forenoon Now to make these Plains as Erect Decliners let the complement Latitude become a new Latitude and the complement Declination a new Declination then they may become Erect Decliners as in the next sort following 6. Of Erect Decliners East or West By Declination I mean the quantity of the Angle that the Meridian or Pole of place makes between the Meridian or Pole of the Plain therefore to set the Sphear to any Declination do thus The Sphear being set to the Latitude turn the Sphear as well as you can guess to the scituation of the place that is put the North part of the Meridian towards the North and the South part towards the South then turn the Notch of the movable Horizon alwayes to the degrees of the Plains Declination from North or South towards either East or West and Upright also as in Erect Dials Then is the Plain set to his Declination viz. the distance of the Horizon between the Meridian or Pole-place which is alwayes 12 a clock and the Meridian and Pole of the Plain being alwayes just where the Notch is in the Moving Horizon Now according to these Rules A Plain that declines 30 degrees from South to West The Stiles Elevation is 32-35 The Substile from 12 21-40 The Inclination of the Meridian 36-24 The South Pole is elevated on the South-side and the North Pole on the North-side And the Sun shines on the North-side from Rising to 8 and on the South-side from 8 to 7 at night and on the North again till Sun-setting by working as in the former Directions is expressed Note In those Erect Decliners whose declinations is above 60 degrees you shall find the Stiles Elevation to be very small therefore to make it exact you must use Arithmetical Calculation for the doing of which the Sphear with due consideration gives the best directions with these Proportions or Canons As sine 90 ZN 90-00 To sine Declination NC 30-00 So Co-tangent Latitude PZ 38-30 To Tang. Subst from 12. ZH 21-40 As sine 90 ZN 90-00 To Co-sine Declination NA 60-00 So Co-sine Latitude ZP 38-30 To sine Stiles Elevation PH 32-25 As sine Latitude PN 51-30 To sine 90 PAE 90-00 So Tangent Declination NC 30-00 To Tangent Elevation Merid. AEI 36-24 As Co-tangent Latitude ZP 38-30 To sine of 90 ZPQ 90-00 So sine Declination ZIA 30-00 To Co-tang 6 from 12 AQ 57-50 Note If you set P at the Pole Z at the Zenith N at the North-end of the Horizon at the Declination or Pole-plain H on the Plain just against the moving Semi-circle or Substile A at the Plain on the Horizon AE on the Equinoctial I at the Hour-circle cutting the Equinoctial set just against the upper Semi-circle Note Q is to be set on the Plain right against the Hour-circle being set to the Hour Having I say by these Rules and the like made and found the Requisites then proceed to draw the Dial thus by
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
from a greater or a less Radius as your occasion serves and note the place As thus for Example In this Operation the extent from the Point at 5-45 on the Sines to the sine of 10 degrees I set the same way from the Point at 0-34 ½ and note the place which will be at near 1 degree then the work is thus As the place against the middle 1 instead of 90 is to the place last found for 10 so is the sine of 20 to sine of 3 deg 24′ ⅓ the 4th term required But in those Lines of Numbers Sines and Tangents where the Number is double this is performed by working a-cross only 4. When the last term in Tangents happens to be above 45 then the remedy is two wayes As thus As sine of 30 to sine of 90 So is the Tang. of 30 to Tang. 49-07 which here happens beyond 45. Apply the end of the Rule next 90 close and even with any thing on which the Point of the Compasses may stay till you take from thence to 45 for that distance laid from 45 shall reach to 49-07 reading the Tangents as numbred beyond 45. Or more neatly thus The Compasses being set from the sine of 30 to the sine of 90 set one Point in the Tangent of 45 and turn the other on the Tangents and keep it there fixed then remove the other from 45 and close it to the third term being here the Tangent of 30 then this last Extent laid from 45 shall reach to 49-07 the Tangent required 5. When the first term is a Tangent above 45 and the second under 45. Take the excess of the first Number above 45 and set it the same way from the second Number then the Extent from the second Number to 45 shall be the true distance between the first and second terms Example As the Tangent of 51-30 to the Tangent of 30 So is the Tangent of 40 to Tangent 21-04′ For the Extent from 45 to 51-30 on the Tangents set the same way from 30 does reach to about 24-30 then the Extent from thence to 45 shall reach from 40 to 21-04 on the Tangents the 4th Number required Or If it had been from a Tangent above 45 to a sine the same way would have remedied the defect 6. When the third term exceeds 45 of Tangents then thus Example As sine 90 to sine 30 So is the Tang. of 50 to Tang. of 30-48 The Compasses set from the first term sine 90 to sine of 30 the second a less then set one Point in the Tangent of 45 and extend the other backwards in the Tangents and note the place keeping one Point there close the other to 50 the third term being above 45 by counting backwards Then I say that Extent laid from Tangent 45 shall reach to Tangent 30-48 the 4th proportional Tangent required If the Proportion had been increasing then there had been no trouble at all Also note That working a-cross or changing the terms is a good remedy also As thus As sine 90 to Tang. 50 which is properly increasing for the Tang. of 50 being more than the sine of 90 yet taken on the Rule from 90 to 40 the complement thereof as if it were decreasing So is sine 30 to Tang. 30-48 the contrary way Therefore As from the first term properly counting to the second 7. Lastly When one or two Radiusses or Alterations of the Characteristick falls between the first and second term As thus for Example First By the Line of Numbers only As 8000 is to 10 So is 5000 to 6 ¼ or 25. To work this properly and naturally the unite on the Numbers should be four times repeated which is seldom more done than twice as here But this and any other by the Line of Numbers is not interrupted having a due respect to the Number of Places For to work this the best way is changing of terms thus As 8000 to 5000 in the same Radius so is 10 to 6-25 in the same Radius also Or without changing As 8000 to the next 1 so is 5000 turning the Compasses the same way to 6-25 But to call it so and not 625 your reason must guide you more than precepts But in using Sines and Tangents the way in the third remedy will fit you Example As sine 90 to 1 degree or under so is sine 30 degrees to sine 30 minuts This being too wide an Extent for the Compasses the third Rule is a remedy for it which on a large Radius several times repeated as in Mr. Oughtred's Circles of Proportion is as easie as may be being sure to remember the number of Radiusses between the first and second term that you may have so many between the third and fourth term also Much more might have been said as to this but this Observation being alwayes kept That as the Extent from the first term to the second is either increasing or decreasing So alwayes must the Extent be from the third to the fourth increasing or decreasing in like manner when you use Sines and Tangents And Numbers also except as before in a few particuler Rules then you will be truly resolved The end of the Book of Dyalling AN APPENDIX To the Use of the Trianguler-Quadrant IN NAVIGATION Where it performs the Uses of the Davis-Quadrant the Cross-Staff Bow Sinical-Quadrant and Sector with as much ease and exactness as any or all of them will do in Observation or Operation Naturally or Artificially Being first thus Contrived and made by Iohn Brown dwelling at the sign of the Sphear and Sun-Dial in the Minories near to Aldgate London 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 Selle at the Hermitage-stairs in Wapping 1671. CHAP. I. The Description thereof FOR SEA-USES THe Description of the Instrument is largely and plainly set down in the First Part and First Chapter But in regard that is the general Description of all the Lines that can conveniently be put on and those necessary for this use being far less I shall repeat the Description again as far as concerns the use thereof for Sea-Observations 1. First for length it ought to be two foot-long at least when shut together and not above 3 foot at any time for Sea-uses but for Land-uses it may be 6 8 10 or 12 foot in length to find Altitudes or distances to Seconds of a degree certainly 2. The Form of it is the same as before viz. an opening Joynt of about an inch and quarter or half quarter broad each Leg and 6 tenth parts of an inch in thickness with a Loose-piece of the same length breadth and thickness to make it an Equilateral-Triangle As the Figure sheweth 3. The Lines necessary for Sea-uses are first the 180 degrees upon the moving-Leg and Loose-piece numbred as before is