think may fit the Plain set off the Point G then making GA Radius of 45 Tang. set off on AB from A the Tang. of the Stiles Elevation to F and draw the Line FG as an obscure Line Then come to the Dial Plain and measure from the Center to the place on the Substile-line where you would have your remotest Line of the sign ♋ or ♑ to pass and take this distance between your Compasses and carry it in above or below the Line FG first drawn and produced to ♋ or ♑ till you find one Point to stay in A ♋ and the other in AG so as to draw a Line = to FG first drawn if that doth not fit then dele FG and draw this = to it in its stead to fit and fill the Plain with the Tropicks to your mind to make them large and yet convenient Then note The point G represents the Center of the Dial AG is the length of the Stile from the Center to the Nodus a Perpendiculer let fall from A to FG shews the point H GH is the measure on the Substile-line on the Plain from the Center to the Horizontal-line HA is the Perpendiculer height of the Stile A the Apex or top of the Stile or Nodus to give the shadow Then Draw a Line from G = to AB as LK and any where between AB draw LM = to AG and wheresoever FG cuts LM make a mark as at M then make LM a = sine of 90 degrees and the Sector so set take out the sine complements of the Arks at the Pole for every hour and lay them from L towards M on the Line LM and to all those Points draw Lines from G and mark them with 12 1 2 3 4 c. as in the Table Or else Take the measure from G to F and lay it on the Dial from the Center on the Substile and draw that Line precisely Perpendiculer to the Substile for the true Equinoctial-line on the Plain Then The measure from the Center of the Dial to the crossing of every Hour-line and the Equinoctial-line taken and laid from G to the Line AB gives Points to draw the Hour-lines on the Trygon As in the Figure Wherein you may note That if the Substile happens to fall on an even whole or half hour then one Line will serve on both sides of the Substile but if not you must draw as many more and set Figures to them to avoid confusion Then I say that the several distances from G to the crossings of those Hour-lines last drawn on the Trygon and the Signs being laid on their correspondent Hour-lines from the Center of the Dial shall give Points in those Hour-lines to draw the signs of the Zodiack with a thin Rule that will bend to those Hyperbolick Sections The same way serves to draw the Parallels of the length of the Day if you lay the distunce from G the crossings of the pricked Lines and Hours on the Trygon and is as true as any other way by Calculation which must afterward be performed by protraction in this manner Thus you have the way to proportion the Height of the Stile to fit the Plain and the place of the Horizontal-line in all Erect-Dials which is alwayes Perpendiculer to 12 and drawn through that point a-cross the Plain And this way of drawing the Signs is general in all Plains whatsoever that will admit them II. To find the Horizontal-line in all manner of Plains First The Horizontal-plain can have none nor many other both Reclining and Inclining whose Reclination or Inclination is above the complement of the Suns Meridian Altitude in ♑ if the Stile have any considerable Altitude In all other Plains the best Mechanick way is thus The Dial being set in or as in his place apply the Moveing-leg to the top of the Stile one corner of it to the Plain and at the same time let the Thred play evenly on 600 and the corner at the Plain will make as many marks as you please to draw it by Otherwise note That wheresoever the Hour-line of 6 and the Equinoctial-line do meet there is one Point Then find at what hour and minute the Sun doth Rise or Set at in the beginning of any other whole Sign most remote from the first Point and that shall be another and so as many as you please to draw that Line by This is general for all Plains To find that Point by the Trianguler-Quadrant Lay the Thred to the Sign given and in the Hour-line is the hour and minut required Thus the Sun being in ♈ riseth and setteth at 6 or 1 quarter of a minut before or after and in ♉ at just 5 and sets at 7 in ♊ at 9 minuts after 4 or sets 9 minuts before 8 The like for Winter signs III. To draw the old unequal Hours The unequal Jewish or Planetary hours divide the Day be it long or short into 12 equal Hours for the drawing of which in the Equinoctial the common hours gives Points For the Tropicks do thus Divide the number of minuts in the longest and shortest dayes by 12 viz. divide 986 the minuts in one day in ♋ at London by 12 the Quotient is 82 ½ and divide 454 the number of minuts in one day at London in ♑ and the Quotient shall be 37′ ¾ then if you fasten an Index or lay a Rule to the Center and to every 1 hour and 22′ ½ in ♋ from 12 and to every 37′ ¾ in ♑ it shall give Points to draw the Jewish or Planetary hours required according to this Table thus made for London by the Line of Numbers against 12 set 6 and the rest in order as the day proceeds for our 12 is the 6th hour according to the Jewes To make this Table readily by the Line of Numbers A Table to divide the Planetary hours in ♋ and in ♑ for London 51-32 Latitude H ♑ M hou H ♋ M 8 43 1 5 10 9 28 2 6 31 10 16 3 7 52 10 44 4 9 15 11 22 5 10 37 12 00 6 12 00 12 37 7 1 22 1 15 8 2 44 1 53 9 4 06 2 31 10 5 29 3 8 11 6 51 3 47 12 8 13 Extend the Compasses from 16-26 the length of the longest day in hours and minuts to 1 the same Extent shall reach the contrary way from 60 to 986 the Number of minuts in one day Or rather As 1 hour to 60 minuts So is 16 hours 26′ to 986 minuts Then As 1 to 82 minuts ¼ So is 2 to 164 minuts ½ So is 3 to 246 minuts ¾ Or you may say As 12 to 1 So is 986 to 82-2 the minuts in 1 hour Which properly is one hour 22 minuts the length of one hour in Cancer then the second hour is 2 hours 44 ½ the third hour is 4 hours and 6′ ¾ from 12 and so for the rest as in the foregoing Table for London But if you draw the Parallels of
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
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
Horologiographia OR The Art of Dyalling BEING The Second Book of the Use of the Trianguler-Quadrant Shewing the Natural Artificial and Instrumental way of making of Sun-Dials on any flat Superficies With plain and easie Directions to discover their Nature and Affections by the Horizontal Projection With the way of Drawing the usual Ornaments on any Plain Also a familiar easie way to draw those Lines on the Ceiling of a Room by the Trianguler Quadrant Also the Use of the same Instrument in NAVIGATION Both for Observation and Operation Performing the use of several Sea-Instruments still in use By Iohn Brown Philomath London Printed by Iohn Darby for Iohn Wingfield and are to be sold at his house in Crutched-Fryers and by Iohn Brown at the Sphear and Sun-Dial in the Minories and by Iohn Seller at the Hermitage-stairs in Wapping 1671. To the Courteous Reader THou hast here presented to thy view Courteous Reader in this second Part a plain discourse of Dialling both Natural Artificial Instrumental Natural I call it from the plain illustration thereof by the Armilary Sphear of Brass herein described or by the Poor-man's Dial-Sphear as I fancy to call it being only a moving Horizontal-Dial and a moving Plain according to the Figure thereof in the Book annexed whereby all the Arks Angles Scituations and Affections are very plainly represented to an ordinary capacity Artificial I call it from the lively delineation of the Horizontal-projection the fittest in my opinion for the making plain the mystery of Dialling Instrumental I call it from the applying of the Trianguler-Quadrant to the ready resolving all the Arithmetical and Astronomical work needful thereunto and that to competent exactness as in the first Part and also in this second Part is sufficiently seen in finding the requisites and delineating the hour-lines to small parts exactly speedily by the natural Sines Tangents and Secants on the Sector and Quadrant Also the ready way of finding the Suns Altitude Hour Azimuth Angle of the Plain and any such business relating to Dyalling as in the first Part is largely treated on Further in this second Part you have Tables of the Suns Declination to every day of the years 1 2 3 after the Bissextile as near as any extent Also a short but plain direction how to use the Trianguler-Quadrant at any manner of way of Observation used at Sea as backward or forward as the Davis-Quadrant and the Cross-staff is used also as Gunter's Bow is used both for the Sun or Stars A figure of the Stereographick Projection Pag ◠The Prints of the Lines of Numbers as you see here inserted are in part according to Mr. Windgates as to a single and broken line of Numbers But the addition of the line of the Fractional parts of a pound and the several Gage-points were never before used as I know of but do much ease expedite the Operations by the Line of Numbers Sines and Tangents Also these Scales of Reduction are convenient for the finding the Decimal-fraction equal to the other Sexagenary-fraction and are agreeable to those Tables in Mr. Windgates Book of Arithmetick pag. 82. Also note that the figure of the Rule at the beginning of the Book pasted on a Board is the very same with that spoken of Chap. XV. Use 28. pag. 397 of the first Part and will work all Questions wrought by the Trianguler-Quadrant to exercise them that are out of the way to have them made and may serve as good directions to the young Instrument-Maker though these are made too too small a Radius to arrive at exactness The like may I say of the Gunters-Lines in the Figures annexed yet as large as the Book will bear Thus I have given you a brief account of my present Thoughts about this matter and somewhat more particularly in the First Part disclaiming all boasting or vain ostentation knowing that at the next Impression it may be amended in many places I shall rest and remain ready to make amends in the making of these or any other Mathematical Instruments at my House at the Sphear and Sun-Dial in the Great Minories John Browne February 16. 1670. CHAP. I. The use of the Trianguler Quadrant IN Making of DIALS SVn-Dials may be made on any Plain and all kind of Plains are either Flat as Horizontal or Vpright or Leaning The Horizontal hath two faces the one beholding the Zenith called the Horizontal-Plain the other beholding the Nadir as the Ceiling of a Room is The Upright Plains are those that make right Angles with the Horizon and do behold neither the Zenith or Nadir but are parallel to them The Leaning Plains are of two sorts generally the one called Recliners beholding the Zenith the other sort called Incliners beholding the Nadir as the outside and in-side of a Roof of a House may represent The two last sorts viz. Upright and Leaning may be Direct or Declining viz. beholding the South or North or East or West Point of the Horizon or Declining therefrom viz. Declining from South or North toward the East or West All which Plains are lively represented by a Sphear made for that purpose in Brass or Pasteboard or by the Projection of the Sphear in Plano Thus Equal to the Radius of the smaller Tangents describe the Circle ESWN representing the Horizon crossing it precisely in the the Center Z with the Lines SN and EW denoting the Points of South and North East and West Then counting the smaller Tangent on the Sector-side doubly as thus calling 5 10 10 20 20 40 30 60 40 80 45 90 c. Lay off from Z towards S the complement of the Suns Meridian Altitude in ♋ in ♈ and ♑ for those Points on the Meridian-line between Z and S and consequently the half Tangent of the complement of the Suns Meridian Altitude in every degree of Declination if you proceed so far Then for the Intersections of all those Lines and Parallels of Declination on the North-side of the Meridian Observe That the same number of degrees and minuts that any Point is above the Horizon on the South part of the Meridian in Summer just so many degrees and minuts is his opposite Parallel in Winter below the Horizon As thus for Example The Sun being in ♋ or 23 deg 31 min. of Declination North hath for his Meridian Altitude 62 degrees and so many degrees is his opposite Parallel of 23-31 or ♑ below the North part of the Horizon at midnight As thus Let the Center at the beginning of the Line of Tangents represent the Center Z and let the Tangent of 45 represent the Horizon in the Scheam viz. S. and N. Then As the distance from S. to ♑ is 15 deg taken from 45 toward 0 and laid from S. to ♑ inwards toward the Center Z as the distance was taken from the Tangent of 45 toward the beginning of the Line of Tangents that represents the Center So the Point Cancer from N.
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-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
the length of the day in the Dial you shall find these hours to cross the even Hour-lines and quarters in the Parallels for 15 and 9 hours as well as in the Equinoctial IV. To draw the Italian or babylonish-Babylonish-Hours First draw the common Hours and the Parallels of the Signs or rather the length of the day Then note that these Hour-lines meet with the common hours in the Equinoctial only the Italians who account from Sun-setting call our 12 in the Equinoctial 18 And the Babylonians who reckon from the Sun-rising call our 12 in the Equinoctial 6 hours Then to mark these in the Tropicks do thus But if you draw the Parallels of the length of the Day then you shall find the 18th hour after Sun Setting to cut the Hour-line of 10 in the Parallel of the Day being 8 hours long and 12 in the Parallel of 12 hours long and the common Hour-line of 2 in the Parallel of 16 hours long and so successively for the rest for so many hours from the last Sun-setting For from 6 the last night in the Equinoctial to 12 this noon is 18 hours but in ♑ from 47′ after 3 at Sun-set to the next noon is 20 hours and 13′ as in the Figure foregoing But for the Babylonish-hours who reckon by equal hours from the Sun Rising as before count 2 hours and 13 minuts after 6 in ♑ and 2 hours and 13′ before 6 in ♋ and just 6 in ♈ and that shall draw the Line of the Suns rising then count 3 hours and 13′ after 6 in ♑ and 7 in ♈ and 1 hour 13′ before 6 in ♋ and that shall be the first hour after Sun rising and so successively till night But if you use the Parallel of the length of the day the work is easier for then 5 7 and 09 in the Parallels of 16 12 and 8 hours shall be Points for the first from Sun-rising and 6 8 and 10 shall shew the second hour from Sun-rising and so forwards as in the Table following V. To draw the Azimuth-Lines For all Erect-Dials both Direct or Decliners deal with the Declination of the Plain as you did with the Inclination of Meridians and at the Meridian or 12 set the Plains declination and then for Rumbs take 11 deg 15′ as often as you can and what the last number wants of 11-15 set on the other side of the Substile and to that add 11-15 till you have enough as in the Table annexed for a Dial whose declination was 35 degrees Westwards Then make the Perpendiculer height of the Stile Radius or Tangent of 45 and on the Horizontal-line lay off the = Tangents of the Rumbs last made in the Table from the foot of the Stile their right way and draw Lines through those Points all Parallel to 12 for the Rumbs or Virtical Circles required on the Meridian write South and the rest in their due order To draw the Azimuth or Virtical-Circles on Reclining or Inclining-Plains Points D. M S. E. 80 00 S. E. by S. 68 45 S. S. E. 57 30 S. by E. 46 15 South 35 00 S. by W. 23 45 S. S. W. 12 30 S. W. by S. 1 15 Substile  S. W. 10 00 S.W. b. W. 21 15 W. S. W. 32 30 W. by S. 43 45 West 55 00 W. by N. 66 15 W. N. W. 77 30 N. W. b. W. 88 45 N. W.  In all Reclining or Inclining Plains these Azimuths virtical Circles or Rumbs do meet in a Point called the Vertical Point found in the Meridian or 12 a clock Line right over in Incliners or under in Recliners the Apex or top of the Stile that is to give the shadow when set in its right place right over the Substile-line And as far off the foot of the Stile being a Point in the Substile Square or Perpendiculer to the Apex or top of the Stile in a Vertical Line drawn through the foot of the Stile = to the Perpendiculer Line of the Plain as the Co-tangent of the Reclination making the Perpendiculer height of the Stile to be Radius or Tangent of 45 degrees Also The Co-tangent of the Reclination of the Plain to the same Radius laid from the foot of the Stile in the same Virtical-Line shall give the Point in the Vertical-line to draw the Horizontal-line by for a Rule laid to this Point and the crossing the Equinoctial-line and hour of 6 shall draw the true Horizontal-line Then make the distance between this Point and the meeting of the Equinoctial and 6 a = Tangent of the West or East Azimuth in the Table and then the Sector is set to lay off all the rest by taking the = Tangents of the Numbers in the Table and laying them from the Vertical-point in the Horizontal-line both wayes on the Horizontal-line For from hence you may note That the Sun being in the Equinoctial doth rise and set near 6 and also doth rise near the East-point and set near the West therefore the same Point in the Dial must be for the hour 6 in the morning and the East Aximuth or the hour 6 at night and the West Azimuth according as the Plain declines Eastwards or Westwards Then Right Lines drawn from the Vertical-point in the Meridian and to all these Points in the Horizontal-line shall be the Azimuth-lines required A thus for Example in the Figure annexed being the Third sort of South-Recliners before-going Declining 35 degrees Southwest and Reclining 60 degrees CH is the Substile CG the Stile H the Foot of the Stile IK the Vertical-line drawn through the foot of the Stile HI the Vertical-point in the crossing of 12 and the Vertical-line and yet right under G the Apex considering the Reclination and the raising of G the Apex Square or Perpendiculer to H the foot of the Stile Then I say a Plumb-line let fall from G will rest in I the Vertical-point The Dial being set in its due place Then GH the Perpendiculer height made a = Tangent of 45 HI is the Co-tangent of the Reclination viz. 30 and HK the Tangent o◠the Reclination 60 being the Vertical-point in the Horizontal-line from whence to lay the = Tangents of the Rumbs in the Table last made into the Horizontal-line Then Lines drawn from the Vertical-point I to those Points in the Horizontal-line shall be the Rumbs or Points of the Compass Vertical Circles or Azimuths required Otherwise When you have made the Tables of the Angles at the Zenith as before you may by this Canon make Tables of Angles at the Vertical-point between the Vertical-line and the Rumb to be drawn on the Plain As the sine of 90 To the Co-sine of Reclination or Inclination So the Tangent of the Angle at Zenith To the Tangent at the Vertical This Table being made you may set one Point in the Vertical-point and describe a Circle to any Radius and therein prick off from the Vertical-line the several Chords of the Rumbs as in the
if you please shall give Points enough to draw the Dial and also the Tropicks and Azimuths and Altitudes also if it were convenient to mark it Or to any other Altitude you mind to have at that Azimuth all at once or at most with two slips of the Thred the Italian Babylonish or jewish-Iewish-Hours as easily drawn by Points found in the other Lines Also On the Meridian-line you may add the day of the month or any thing that depends on the Suns Meridian Altitude which work being well done and drawn with smoth Lines and well ornamented would be a comely pleasant Ornament to a Ceiling and far cheaper then some fret Ceilings are done and more useful Lastly When all is done to put the Glass in right the Foile being first rubbed off to to cause it to give but one spot let the Superficies of the Glass lie just so high as the Center of the Quadrant was in the drawing the Lines and put some Putty under it and the Sun shining make it to play right on the true Hour Altitude and Azimuth or if it be just at noon then bend it on the Putty with your finger till it fall just on the Meridian and day of the Month also in the Meridian-line Also note That look what Altitude the Sun hath at any time the same will the reflected Altitude be at the same time if the Glass lie true which two Observators at the same time may carfully prove The making of the Tables of the Suns Altitude and Azimuth is very largely shewed in the 15th Chapter Vse the 37th and 38th where you have wayes both general and particular for any one or more Latitudes The Figure Explained A the place on the Transum for the Glass to lie on and in the middest thereof one Point in the Ray of the Quadrant is to play IH a piece of Wood to be nailed fast at H and I for the other Point to play in at G L the hole for the Plummet to play in being cut through the thin Quadrant B and C the ends of the Semi-circle nailed on the Transum or Stoole K the hole in the Quadrant for the Semi-circular-Ring to pass through FE the Posts of the Window D the beginning of the degrees on the Semi-circle AM a Thred extended from the Center of the Quadrant to the Ceiling Thus you have the usual wayes of Dialing in a competent measure plainly and practically handled which may be useful to many a Learner and I hope will be as well accepted as with free-will though with little ability and less leisure readily imparted A Table of the Suns Azimuth from the South at every Hour and Quarter 51-32  ♋ ♊ ♌ ♉ ♠♈ ♎ ♓ ♠♒ ♠♑  D. M. D. M. D. M D. M D. M D. M D. M 12 00 00 00 00 00 00 00 00 00 00 00 00 00 00  07 10 05 53 5 37 4 55 03 54 03 53 3 52  14 22 13 24 11 10 9 40 07 57 07 40 7 10  21 27 19 53 16 44 14 13 12 11 11 10 10 37 11 1 27 54 26 00 22 13 18 52 16 22 14 45 14 13  34 14 32 01 27 32 23 22 20 15 18 17 17 38  40 12 37 40 32 41 27 41 24 06 21 53 20 59  45 39 43 04 37 32 32 08 29 44 26 30 24 25 10 2 50 51 47 57 42 05 36 24 31 50 28 52 27 49  55 31 52 41 47 04 40 30 35 35 32 18 31 00  61 43 57 11 50 51 44 26 39 10 35 39 34 17  64 11 61 18 54 33 48 13 42 36 38 50 37 34 9 3 68 10 65 16 58 47 51 56 46 02 42 04 40 36  71 52 69 56 62 33 55 33 49 25 45 16 43 40  75 26 72 37 66 10 58 59 52 43 48 21 46 42  78 48 76 05 69 38 62 22 55 55 51 22 49 40 8 4 82 00 79 20 72 57 65 40 59 00 54 22   85 08 82 42 76 12 68 53 62 08    88 10 85 38 79 23 72 06 65 12    91 09 87 50 82 28 75 03 68 12   7 5 94 05 91 34 85 30 78 06 71 10    96 51 94 25 88 27 81 09     99 38 97 16 91 25 84 06     102 25 100 07 94 22 87 04    6 6 105 08 102 54 97 12 90 00     107 52 104 58 100 02      110 36 108 26 103 00      113 18 111 14 105 54     5 7 116 03 114 03       118 50 116 54       121 41 119 47       124 29 122 37      4 8 127 24        130 28       A Table of the Suns Altitude at every Hour and Quarter in each Sign for 51-32  ♋ ♊ ♌ ♉ ♠♈ ♎ ♓ ♠♒ ♠♑  D. M D. M D. M D. M D. M D. M D. M 12 62 90 58 42 50 00 38 28 27 00 18 18 14 59  61 50 58 34 49 52 38 24 26 54 18 13 14 55  61 25 58 09 49 31 38 07 26 40 18 00 14 38  60 41 57 29 49 00 37 38 26 15 17 37 14 20 11 59 43 56 34 48 12 36 58 25 40 17 06 13 52  58 32 55 26 47 10 36 07 24 02 16 26 13 10  57 04 54 06 46 02 35 06 23 59 15 38 12 35  55 29 52 36 44 38 33 56 22 58 14 49 11 30 10 53 45 50 55 43 12 32 37 21 51 13 38 10 36  51 52 49 07 41 33 31 10 20 33 12 27 9 20  49 54 47 12 39 46 29 36 19 10 11 09 8 9  47 50 45 12 37 56 27 54 17 35 9 44 6 41 9 45 42 43 06 36 00 26 07 15 58 8 13 5 16  43 30 40 57 33 55 24 14 14 14 9 49 3 41  41 16 38 45 31 50 22 16 12 27 4 54 2 04  39 00 36 30 29 40 20 14 10 32 3 07 0 17 8 36 41 34 13 27 31 18 08 8 35 1 15   34 22 31 55 25 09 15 59 6 30    32 02 29 13 22 56 13 46 4 27    29 42 27 16 20 37 11 33 2 17   7 27 23 24 56 18 18 09 17 0 06    25 03 22 36 15 58 06 58     22 45 20 16 13 38 04 40     20 28 17 58 11 18 02 20    6 18 11 15 41 09 00
the Equinoctial stands for 51-30 Latitude and the account will end at 62 degrees the Suns Meridian Altitude at that Declination Northwards But if it had been 23-30 South Declination then count as much from the Equinoctial downwards and the count will end at 15 degrees for the Suns Meridian Altitude at 23-30 South Declination Use V. The Suns Declination and Latitude being given to find the Suns Rising or Setting and Amplitude East or West Count the Suns declination on the Hour-Circle towards his proper Pole that is South-declination toward the South-Pole and North-declination towards the North-Pole and thereunto lay the Thred that is fastned in the Center then bring the Hour-circle and Thred both together till the Thred touch the Horizon then the Thred on the Horizon shews the Amplitude and the divided-side of the Hour-Circle shews the Suns Rising and Setting on the Equinoctial counting the Meridian alwayes 12 and the 2 East and West-points 6 and 15 degr for an Hour and every deg 4 min. Example Iune 11. Declination 23-30 the Sun Riseth at 13′ before 4 and the Amplitude is near 40 deg Again April 10. Declination 11-30 the Amplitude is 18-30 from the East to the North and Riseth at 5 the Hour-circle cutting 15 degrees on the Equinoctial Use VI. The Declination Latitude Suns Altitude given to find both Hour Azimuth Rectifie the Sphear and set the Plain horizontal that is Level or Parallel to the Horizon then apply the Thred to the Declination counted the right way on the Hour-circle then turn the Hour-circle and upper Semi-circle about till the Thred cuts the degrees of Altitude in the upper Semi-circle and the Hour-circle shews the hour in the Equinoctial and the Semi-circle cuts the Suns Azimuth in the deg on the Horizon or Plain Example Declination 10 Latitude 51-30 and the Suns Altitude 30 the hour will be 8-27 and the Azimuth 66 from South Eastwards if in the morning or the contrary if in the afternoon Use VII The Hour or Azimuth and the Suns Declination given to find the Altitude The Sphear rectified as before and the Hour being given set the Hour-Circle to the hour on the Equinoctial then bring the Thred to the Declination counted on the Hour-circle then bring the upper Semi-circle till the fore-side do just touch the Thred and the Thred on the Semi-circle shall shew the Altitude required and on the Horizon the Azimuth at that Hour and Altitude But if the Azimuth be first given then set the upper Semi-circle thereunto counted on the fixed Horizon then the Thred laid to the declination on the Hour-circle and turned about till it touch the upper Semi-circle there it shews the Altitude and the Hour-circle on the Equinoctial shews the hour Use VIII To find the Suns Height in the Vertical-Circle Set the upper Semi-circle to the East or West-Point or 90 degrees of Azimuth then lay the Thred to the declination on the Hour-circle and then bring it and the Thred together till it just touch the upper Semi-circle and it shall there shew the Altitude at East or West required Example At 10 degrees Declination North it will be East at 16 degrees of Altitude Use IX To find the Suns Altitude at 6. Set the Hour-circle to 6 on the Equinoctial and the Thred to the Declination then bring the Semi-circle to the Thred and it shewes the Altitude at 6 required Example At 23-30 Declination the Altitude 18-15 above the Horizon in North-declination and as much under in South-declination for you must observe that the surest working is from the upper or divided-sides of the Rings on every occasion to use it Use X. To find the Hour of the Day when the Sun shineth Rectifie the Sphear and set the Plain Parallel to the Equinoctial-circle then set the Meridian-circle due North and South and the shadow of the Axis shall on the Plain shew the true hour Or otherwise thus At the true place of the Suns Declination on the Hour-circle make a mark or stick the point of a Pin then turn the Sphear about till the shadow of that mark fall on the Center the Sphear standing Horizontal as near as may be then the Hour-circle shall on the Equinoctial shew the hour of the day required Note A small Bead or knot on the Thred will do the business as well as may be Thus any the like Questions may be wrought for the Stars or the manner of raising the Canon for any Spherical Triangle whatever to work the same exactly by the Logarithms As thus Suppose I would make the Canon or Proposition to find the Suns height in the Vertical Circle at any declination First The Sphear being rectified and the Plate set Horizontal bring the upper Semi-circle to the East-point and laying the Thred to the declination on the Hour-circle bring it and the Thred together till it just touch the upper Semi-circle The Rings or Circles so standing and being great Circles of the Sphear there is constituted a Spherical-Triangle in this form Wherein you have ZAE 51-30 the Latitude the Angle at the Equinoctial and ZE 90 the upper Semi-circle and AB 23-30 the Declination part of the Hour-circle to find BE part of the upper Semi-circle Now this being a right-angled Spherical-Triangle and the parts which are given being one right Angle viz. The Angle at A and the Side AB the Suns Declination and the Angle at E 51-30 to find the Side BE Now the Sines of the sides of Spherical-Triangles are proportional to the Sines of their opposite Angles and the contrary Therefore As the sine of the Triangle BEA 51-30 Is to the sine of the Side AB 23-30 So is the sine of the Angle BAE 90-00 To the sine of BE 30-39 And the like for any other as by comparing the Rules in Mr. Norwood's Trigonometry and the Circles of the Sphear together the use and convenience thereof will evidently appear unto you Only note this plain Observation That the side of a right-Angled Triangle which subtends the Right Angle is most properly called the Hypothenusa the other which you make or suppose Radius the Base The other the Perpendiculer Or more short The Hypothenusa and Leggs Therefore if the Hypothenusa and one Leg be given the proportion is wrought by Sines alone but if the two Legs be given and first and second in the Question then the Proportion is wrought by Sines and Tangents together As for Example As the sine of ♈ ♋ 90-00 To the sine of ♋ AE 23-31 So is the sine of ♈ ♉ 30-00 To the sine of ♉ R 11-31 The Suns Declination in ♉ Again secondly As the sine of ♈ AE 90-00 To the Tangent of AE ♋ 23-31 So is the sine of ♈ R 27-54 To the Tangent of R ♉ 11-31 The Declination as before  But if the one acute Angle and his opposite Leg or Side be given then the Proportion is made by Sines only as in the foregoing Example Again In Vertical
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
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
Sights 233 For a fore-Observation with Thred and Plummet 235 For a back-Observation as a Davis Quadrant 237 To find the Suns distance from the Zenith or the Co-altitude 238 To find the Altitude when near the Zenith 239 To find a Latitude by a forward Observation as with a Gunter's Bow 241 To find the Latitude by a back Observation 247 To find the Latitude by a Meridian Observation with Thred and Plummet 252 To find the Suns Declination 254 A Table of the Suns Declination for the second after Leap-year 256 257 A Table of the Suns Declination for every 5th Day the intermediate years 258 259 A Rectifying Table for the intermediate years 260 A Table of the Magnitudes Declinations and Right Ascentions of 33 fixed Stars in Degrees and Hours and Minuts 261 The use of the Trianguler-Quadrant in the Operative part of Navigation 267 Of Sayling by the Arch of a great Circle 300 FINIS Errata for the Second Part. PAge 6. line 10. for too read to P. 18. l. 10. f. H r. the ends of the Arch QP p. 22. l. 14. f. begins r. being p. 15. l. 27. f. Latitude r. Co-latitude p. 34. l. 8. f. Sun r. sum p. 39. l. 24. f. incliner r. inclination of Meridians p. 61. l. 23. f. place r. plain p. 62. l. 20. f. ☉ r. Q. p. 66. l. 2. f. I r. T. p. 69. l. 7. f. 12. r. 7. p. 92 l. 22. r. gives a mark near E whose measure on the Limb from B. p. 87. l. 8. f. giâ—es r. gives l. 11. add at R near C. p. 93. l. 25. f. FE r. PE. p. 100. l. 21. add cd next gives p. 10â— l. 6. f. 8-5 r. 8-3 p. 105. l. 19. f. use r. have p. 108. l. 6. f. Pole r. Zenith p. 112. l. 3. f. cuts 12 r. cuts the substile p 113. l. 19. f. DF r. DE. l 19. f. T. r. CT Also in l. â—8 r. CT p. 122. l. 6. f. E r. F. p. 122. l. 13 14 15 16 add Sine p. 128. l. 26. f. I r. L. p. 139. l. 21. add as in this Example p. 140. l. 6. add to p 170. l. 10. f. divides r. divided p. 181. l. 20. f. popsition r. proportion p. 119. l. 27. f. from r. on p. 193. l. 6. f. being r. bring p. 197 l. 4. f. elevation r. inclination p. 200 l. 24. f. C r. G. p. 204. l. 3. f. F r. E. p 209. add in the last line or by the upper part of the Plain p. 124. l. 18. add or remove the Thred to turn it further when it reclines beyond the Pole p. 238. l. 7. add of l. 20 add but. p. 247. l. 13. r. and much better in small Latitudes p. 248. l. 5. f. wo r. two p. 251. l. 14. f. 20 r. 33. Also l. 17. f. 40 r. 27. Advertisements The use of these or any other Instruments concerning the Mathematical Practice or further Instructions in any part thereof is taught by Iohn Colson near the Hermitage-stairs Also by Euclide Speidwel dwelling near to White-Chappel Church in Capt. Canes Rents or at the Custom-house Also by William Northhall Mariner dwelling at the Crooked-Billet in Meeting-house Alley on Green-Bank near Wapping Also the Instruments may be had at the house of Iohn Brown Iohn Seller or Iohn Wingfield as in the Title-page is expressed There is now extant a large Treatise of Navigation in Folio describing the Sea-Coasts Capes Head-lands the Bayes Roads Rivers Harbours and Sea-marks in the whole Northern-Navigation shewing the Courses and Distances from one Place to another the ebbing and flowing of the Sea with many other things belonging to the practick part of Navigation A Book not heretofore printed in England Collected from the practice and experience of divers able and experienced Navigators of our English Nation Published by Iohn Seller Hydrographer to the Kings most excellent Majesty and to be sold by him at the sign of the Marineâ—s Compass at the Hermitage-stairs in Wapping and by Iohn Wingfield right against St. Olaves Church in Crouched-Fryers Practical Navigation or an Introduction to that whole Art Sold by Iohn Seller and Iohn Wingfield aforesaid See the General-Scheam