A Horizontal Dyal for the Latitude of London 51 degrees 32 minutes Cross the same with a perpendicular to wit the Line of Six and draw an Oblong or square figure as here is done wherein to write the hours from M to C set off such a Radius as you intend to draw a Circle withall which for convenience may be as big as the Plain will admit and therewith upon C as a Center describe a Circle and set off the Radius of the said Circle from F to d from M to k and draw the right Line Fd then prick the Poles height or Latitude of the place from f to L and from L to S and through the point S draw the Line MS and it shall represent the Cock or Stiles height from the point L take the nearest distance to FM and prick that extent from F to N on the Line Fd then a Ruler laid from N to k where it intersects the Meridian is the Regulating point or point ☉ then from M divide the Circle into twelve equal parts for the whole hours setting the Letters of the Alphabet thereto and lay a ruler from the Regulating point to each of those Divisions and it will intersect the Circle on the opposite side from which Intersections Lines drawn into the Center at M shall be the hours Lines required to be produced beyond the Center as many as are needful which shall be the hours before or after 6 in the Summer half year the halfs and quarters are after the same manner to be inscribed by dividing each equal division of the Circle into halfs and quarters This Circular Dyalling was in effect published and invented by Mr. Foster in his book of a Quadrant in Anno 1638. for the demonstration of this Work will demonstrate the truth of those Circular performances which he Operates on the back of that Quadrant but is more expresly hinted in his Posthuma the demonstration whereof shall follow Those that have plain Tables with a Frame may have Tangent Lines put on the sides of their Frame and then if a Center be found upon the paper under the Frame by the Intersection of a Ruler laid over those Tangents the requisite Divisions of a Circle to any Radius that can be described upon the paper will be most readily given without dividing any Circle or setting of Marks or Letters thereto the Frame keeping fast that paper on which the Draught of the Dyal is made which may also be supplyed from a Circle divided on Pastboard cut out which is to be Laced upon a Board over the paper whereon the Draught is to be made An upright full South Dyal This is no other then a Horizontal Dyal in that Latitude which is equal to the complement of the Latitude of the place you are in onely the hours must not be continued beyond the Center and the Delineation requires no other directions then the former On a direct North Dyal in this Latitude there will not be above two hours above and two hours beneath each end of the Horizontal Line to be expressed and the Stile will have the same Elevation and point upward Horizontall Diall An Vpright South Diall In all upright North and South Plains the Meridian or hour line of 12 is perpendicular to the Plains Horizontal line if the Plain be direct the height of the Stile above the substile is equal to the complement of the Latitude If the Plain be South Reclining or North Inclining the height of the Stile above the substile is equal to the Ark of difference between the complement of the Latitude and the Ark of Re-Inclination and if this latter Ark be greater then the former the contrary Pole is elevated If the Plain be South Inclining or North Reclining the height of the Stile above the substile is equal to the sum of the Colatitude and of the Re-Inclination and when this latter Ark is greater then the Latitude the Stiles height will be greater then a Quadrant Such Horizontal or direct South Dyals both upright and leaning whereon the stile hath but small Elevation are to be drawn with a double Tangent Line without a Center wherein the following directions for direct Polar plains and those for other Oblique plains whereon the stile hath but small elevation will fully direct you All Plains that cut the Axis of the World have a Center but if they be parallel thereto the stile hath no Elevation Such direct South Reclining or North Inclining Plains whose Arch of Reclination is equal to the complement of the Latitude are parallel to the Axis and are called Polar plains in these the Hour Lines will be Tangent Lines of any assumed Radius and the parallel height of the stile above the plain must be made equal to the Radius of the Tangent by which the hours were set off First draw the Tangent line CG and perpendicular thereto ACB upon C as a Center with any Radius describe the Quadrant of a Circle HB and prick off the Radius from C to A from H to E from B to D from D to F from F twice to G. A Ruler laid from A to E Findes the point 1 on the Tangent CG for the hour line of one and laid from A to D findes the point 2 for the hour line of two the hour line of three is at the point H of four at the point F of five at the point G thus are the whole hours easily inserted Now to insert the halfs and quarters Divide the Arches BE ED DH into halfs and a Ruler laid from A to those divisions wil finde points upon CH where all the half hours under 45d are to be graduated and if a Ruler be laid from B to those respective divisions of the Quadrant BH it will finde points on the Tangent CG where the whole hours and halfs are to be graduated above 45d after the same manner are the quarters to be inserted But in regard the halfs and quarters above 45d will by this direction be found with much uncertainty I have added the help following First Divide the halfs and quarters under 45d as now directed then for those above make use of this Table In a direct South Polar Dyal The distances of these hour lines Are equal to the distances of these hour lines 3 and 3 ¼ 3 ½ and 4 2 ¼ and 3 ¾ 4 and 4 ¼ 4 ¼ and 4 ½ 4 ¼ and 4 ¼ ¾ and 1 ¼ 1 ¼ and 2 ½ 1 ½ and doubled ¾ and 1 ¾ 3 ½ and 3 ¾ doubled otherwise 1 ¾ and 2 ¾ 2 and 3 ¾ This I do not assert to be absolutely true but so neer the truth that there will not arive above one thousandth part of the Radius difference in the greatest Dyal that is made and will be more certain then any Sector though of a Vast Radius or then they can with convenience be prickt down the common way by a Contingent Line the meaning of the Table will
half the Declination To the Tangent of a fourth Arch. Again As the Cosine of half the sum of the former Complements Is to the Cosine of half their difference So is the Contangent of half the Declination To the tangent of a seventh Arch. Get the sum and difference of the fourth and seventh Arch then if the Colatitude be greater then the complement of the Reclination the sum is the Substiles distance from the plains perpendicular and the difference the Inclination of Meridians But if it be less the difference is the Substiles distance from the Plains perpendicular and the sum the Inclinations of Meridians To place the Substile For Plains leaning Southwards when the Angle of the Substile from the Plains perpendicul is less then a Quadrant it will on the upperface lye above that end of the Horizontal line that is opposite to the Coast of Declination and on the under face lye beneath it but when it is greater it will lye below the said end on the upper face and above it on the under face but this will not be till the Reclination be more then the complement of the Reclination of a Polar plain that hath the same Declination for plains leaning Northwards the Directions of the first Method suffice To place the Meridian Either Calculate it and place it according to the directions of the first and second Method or else Calculate it by this Proportion As the Radius Is to the Sine of the Stiles height So is the tangent of the Inclination of Meridians when it is Obtuse take its complement to a Semicircle To the tangent of the Meridian line from the substilar For Plains leaning Northward the first directions must serve but for Southern Plains the second because the distance of the Meridian is Calculated from the Substile supposed to be placed and here the work is converse to that for in that we supposed the Meridian placed and not the Substile For the Stiles height As the Sine of the fourth Arch Is to the Sine of the seventh Arch So is the tangent of half the difference of the complements both of the Latitude and Reclination To the tangent of an Arch sought How much the said Ark being doubled wants or exceeds 90d is the Stiles height In South Recliners if the said Ark being doubled is less then 90d its complement is the elevation of the North Pole and the Plain falls below the Pole But if the said Arch exceed 90d the Plain passeth above the Pole and the excess is the elevation of the North Pole on the under face of a South Recliner called a North Incliner and the affections were determined in the first method where the Declination hath its Denomination from that Coast of the Meridian to which the Plain looketh These methods of Calculation may not precisely agree one with another though all true unless the parts Proportional be exactly Calculated from large Tables in every Operation which to do as to the Examples in this Book my leisure would not permit This last method is derived also from the former Oblique Triangle the Proportions here applyed being demonstrated in Trigonometria Brittanica by Mr. Newton The Demonstration of the former Proportions In projecting the Sphere it is frequently required to draw an Arch through any two different Points within a Circle that shall divide the said Circle into two equal Semicircles Construction 1. Draw a line from one of the given points through the Center for conveniency through that point which is most remote 2. From the Center raise a Radius perpendicular to that line 3. And from the said point draw a line to the end of the Radius 4. From the end of the Radius raise a line perpendicular to the line last drawn and where it intersects the former line drawn through the first point and Center is a third point given describe a Circle through these three Points and the Proposition will be effected Example Let it be required to draw the Arch of a Circle through the two points E and F that shall divide the Circle BD into two equal parts Operation From E draw EG through the Center A make AD perpendicular thereto joyn ED and make DG perpendicular to ED cutting EG in G through E F and G draw the Arch of a Circle which will divide the Circumference BDC into two equal parts in B and C that is if CA be drawn it will pass through B if not let it pass above or below as let it pass below and cut BFE in H. Demonstration By construction EDG and DAG are right Angles therefore □ AD ■EAG by 13 Prop. 6 Euclid because EDG being a right Angle AD is a mean Proportional between EA and AG but ■EAG should be ■CAH by 35. Prop. of 3 Euclid therefore ■CAH □ AD But □ AD □ AI that is ■CAI therefore ■CAH ■CAI which is absurd therefore CI cannot pass below B the same absurdity will follow if it be thought to pass above it therefore CA produced will fall in the point B wherefore BDC is a Semicircle which was to be proved And hereof I acknowledge I have seen a Demonstration by the Learned teacher of the Mathematicks Mr. John Leak to this effect To project the Sphere and measure off the Arks of an upright Decliner Upon Z as a Center describe the Arch of a Circle and cross it with two Diameters at right Angles in the Center whereto set NESW to represent the North East South and West Prick off the Latitude from N to L and lay a Ruler to it from E and where it cuts NZ set P to represent the pole Prick off the Declination of the plain from E to A and from S to D and draw the Diameter AZB which represents the plain and DZC which represents the Poles thereof Through the three points CPD draw the Arch of a Circle and there will be framed aright Angled Triangle ZHP right Angled at H in which there will be given the side ZP the complement of the Latitude with the Angle PZH the complement of the Declination Whereby may be found the Stiles height represented by the side PH the Substiles distance from the plains perpendicular represented by ZH and the Angle between that Meridian which makes right Angles with the plain and the Meridian of the place represented by the Angle ZPH shewing the Arch of Time between the Substile and meridian called the Inclination of Meridians from which Triangle are educed those proportions delivered for upright Decliners To measure off these Arks 1. The Substile A Ruler laid from D to H findes the point F in the Limbe and the Arch CF is the measure of the Substiles distance from the meridian to wit 21d 41′ 2. The Stiles height Set off a Quadrant from F to G lay a Ruler from G to D and where it intersects BZ set ☉ which is the Pole of the Circle CPD lay a Ruler from ☉ to
measure off all the Arks that can be found by Calculation With the Demonstration of all the former Proportions from p. 52 to 59 To determine what Hours are proper to all Plains p. 60 to 61 Another Method of inscribing the Hour-lines in all Plains by a Parallelogram p. 62 To draw the Tangent Scheme suited thereto p. 63. The Hour-lines so inscribed in a Horizontal and South Dyal p. 63 64 As also in an upright Decliner p. 65 With another Tangent Scheme suited thereto for pricking them down without the use of Compasses p. 66 67 A general Method without proportional work for fitting the Parallelogram into Oblique Plains that have the Requisites first placed p. 69 70 By help of three shadows to finde a Meridian-line p. 70 71 Another Scheme suited to that purpose p. 73 A Method of Calculation for finding the Azimuth Latitude Amplitude c. by three shadows p. 75 From three shadows to inscribe the Requisites and Hour-lines in any Plain p. 77 Which is to be performed by Calculation also p. 97 Characters used in this Book Plus for more or Addition Minus less or Substraction Equal q For square □ Square ■The Rectangle or Product of two terms ∷ for Proportion If the Brass Prints in this Book be thought troublesome to binde up they may be placed at the end thereof for the Pages to which they relate are graved upon them A Quadrant of a Circle being divided into 90d all that is required in this Treatise is to prick off any number of the said degrees with their Sub-divisions which may be easily done anywhere from a Quadrant drawn and divided into nine equal Parts and one of those parts into ten sub-divisions called degrees but for readiness the equal divisions of a whole Quadrant are transferred into a right line as in the Frontispiece called a Line of Chords serving more expeditly to prick off any number of degrees or minutes in the Arch of a Circle An Advertisement The Reader may possibly desire to be furnished with such Scales to any Radius these and all manner of other Mathematical Instruments either for Sea or Land are exactly made in Brass or Wood by Heury Sutton in Threed-needle-street behinde the Exchange or by William Sutton in Upper Shadwell a little beyond the Church Mathematical Instrument-Makers A direct South Diall Reclining 60d Lat 51d 32′ An erect direct North Diall Lat 51d 32′ A South Diall Declin 40d East Reclining 60d Lat 51d 32′ A North Diall Declin 40d West Reclining 75d Latitude 52d 32′ THE DISTINCTION OF DYALS THough much of the subject of Dyalling hath been wrote already by divers in diverse Languages notwithstanding the Reader will meet with that in this following Treatise which will abundantly satisfie his expectation as to the particulars in the Preface not yet divulged in any Treatise of this nature In this Treatise that we might not be too large divers definitions are passed over supposing that the Reader understands what a Dyal is what hour Lines are that that part of the Stile that shews the hour ought to be in the Axis of the World that the hour Lines being projected on any Regular Flat will become straight Lines Dyals are by Clavius in the second Chapter of his Gnomonicks distinguished into Seventeen Kindes 1. The Horizontal being parallel to the Horizon 2. South and North direct by some called a Vertical Dyal because parallel to the prime Vertical or Circle of East and West 3. 4. South direct Reclining or North Inclining less more then the Pole 5. 6. South direct Inclining or North Reclining less more then the Equinoctial 7. South direct Reclining or North Inclining to the Pole called a direct Polar Plain because Parallel thereto 8. South direct Inclining or North Reclining to the Equator called an Equinoctial Plain because parallel thereto 9. South and North Declining East or West 10. East and West direct 11. East and West direct Reclining or Inclining 12. A South North Plain declining East or West Reclining Inclining to the Pole called a Polar Decliner 13. A South North Plain Declining East or West Inclining Reclining to the Equinoctial called an Equinoctial Decliner 14. 15. A South North Plain Declining East or West Reclining Inclining Whereof are two sorts the one passing above the other beneath the Pole 16. 17. A. South North Plain Declining East or West Inclining Reclining whereof are 2. sorts the one passing above the Equinoctial the other beneath it Of each of these we shall say something but before we proceed it will be necessary in the first place to shew how to finde the Scituation of any Plain 1. To finde whether a Plain be Level or Horizontal The performance hereof is already shewed by Mr. Stirrup in his Compleat Dyallist Page 58. where he hath a Scheme to this purpose I shall onely mention it Get a smooth Board let it have one right edge and near to that edge let a hole be cut in it for a Plummet to play in draw a Line cross the Board Perpendicular to the streight edge thereof passing through the former hole if then setting the Board on its smooth edge and holding it Perpendicularly so that the Plummet may play in the hole if which way soever the Board be turned the threed will fall being held upright with the Plummet at the end of it playing in the former hole directly on the Perpendicular Line drawn cross the Board the said Line no wayes Reclining from it the Plain is Horizontal otherwise not 3. To take the Altitude of the Sun without Instrument Upon any Smooth Board draw two Lines at right Angles as AB and AC and upon A as a Center with 60d of a Line of Chords describe the Arch BC and into the Center thereof at A drive in a Pin or Steel Needle upon which hang a threed and Plummet then when you would observe an Altitude hold the Board so to the Sun that the shaddow of the Pin or Needle may fall on the Line AC and where the threed intersects the Arch BC set a mark suppose at E then measure the Ark BE on your Line of Chords and it shews the Altitude sought for want of a Line of Chords you may first divide the Arch BC into 9 parts and the first of those Divisions into 10 smaller parts 3. To draw a Horizontal and Vertical Line upon a Plain The readyest and most certainest way to do this especially if the Plain lean downwards from the Zenith will be by help of a threed and Plummet held steadily to make two pricks at a competent distance in the shadow of the threed on the Plain projected by the eye and a Line drawn through those or parallel to those two points shall be the Plains perpendicular or Vertical Line and a Line drawn perpendicular to the said Line shall be the Plains Horizontal Line To finde the Reclination of a Plain The Reclination of a Plain is the Angle comprehended between the Plains perpendicular and the
be illustrated by one or two examples The distance between the hour lines of two and a quarter and three and three quarters will be equal to twice the distance between the hour lines of twelve and one and a half Also the distance between four and a quarter and four and a half is equal to twice the distance between three and a half and three and three quarters or it is equal to the distance between one and three quarters and two and three quarters but the former is nearer the truth It will be inconvenient in such plains as also in direct East or West Dyals to express any hour line from the substile beyond 75d or 5 hours To these I may add some other observations which were communicated by Doctor Richard Sterne to Mr. Sutton which may be of use to try the truth of these kinde of Plains 1. The distance between the hours of 3 and 4 is equal to the distance between the hours of 1 and 3. 2. The distance between the hours of 2 and 4 is double to the distance between the hours of 12 and 2. 3. The distance between the hours of 11 and 4 is doubte to the distance between 12 and 3 or to that between 9 and 12 and so equal to the distance between 9 and 3 also equal to the distance between 4 and 5. 4. The distance between the hours of 11 and 5 is double to the distance between the hours of 11 and 4 as also to the distance between the hours of 4 and 5 and between 9 and 3 and quadruple to the distance between 12 and 9 or between 12 and 3. These are absolutely true as may be found by comparing the differences of the respective Tangents from the natural Tables To draw a Polar direct South Dyal Having drawn the Plains perpendicular in the middle of the Plain let that be the hour line of 12 then assuming the stile to be of any convenient parallel height that will suit the plain making that Radius divide a Tangent line into hours and quarters by the former directions and prick them down on the Plain upon a line drawn perpendicular to the Meridian or hour line of 12 on each side thereof and through the points so prickt off draw lines parallel to the Meridian line and they shall be the hour lines required as in the Example To describe an Equinoctial Dyal Such direct North Recliners or South Incliners whose Re-Inclination is equal to the Latitude are parallel to the Equinoctial Circle and are therefore called Equinoctial Dyals there is no difficulty in describing of these Divide a circle into 24 equal parts for the whole hours afterwards into halfs quarters and place the Meridian line in the Plains perpendicular assuming as many of the former hours as the Sun can shine upon for either face and then placing a round Wyre in the Center for the Stile perpendicular to the Plain and the Dyal is finished A South plaine Declining 30d East Latitude 51-32′ page 13 page 17 A South Diall Declin 30d East Latitude 51-32′ To prick off the Substile and Stiles height on upright Decliners in their true Coast and quantity On such Plains draw a Horizontal Line and cross the same with a perpendicular or Vertical at the intersection set V at the upper end of the Vertical Line set S at the lower end N at the East end of the Horizontal Line set E at the West end W prick off the Declination of the Plain in its proper Coast from S or N to D and draw DV through the Center the same way count off the Latitude to L and from it draw a Line into the Center in the same quarter make a Geometrical square of any proportion at pleasure so that two sides thereof may be parallel to the Horizontal and Vertical Line at the intersection of one of the sides thereof with the Vertical set A and of the other side with the Horizontal Line set B and where the Latitude Line intersects the side of the square set F. To prick off the Substilar For Latitudes above 45d take BF and prick it on the Line of Declination beyond the Center from V to O and from the point O draw the Line OC parallel to the Vertical Line and produced beyond O if need require and thereon from C to I prick the side of the Square and a line drawn into the Center shall be the Substilar line but for Latitudes under 45d prick the side of the square from V to O and draw OC as before and make CI equal to AF and a line from I drawn into the Center shall be the Substilar line Stiles Height From the Point I erect the extent OC perpendicularly to the substilar OI at the extremity thereof set K from whence draw a line into the Center and the Angle IVK will shew how much the stile is to be elevated above the substilar line In order to the Demonstration hereof let it be observed that an Angle may be prickt off by Sines or Tangents in stead of Chords To prick off an Angle by Sines or Tangents As in the Scheme annexed let BC be Radius and let there be an arch prickt off with Chords as BE I say if the Tangent of the said Ark BA be taken out of a line of Tangents to the same Radius and be erected perpendicular to the end of the Radius as BA a line drawn from A into the center shall include the same Angle as was prickt off by Chords as is evident from the definition of a Tangent In like manner an Angle may be prickt off by Sines the nearest distance from E to BC is the Sine of the Arch BE so in like manner the nearest distance from B to EC is the Sine of the same Arch wherefore if with the Sine of an Arch from the end of the Radius be described another Ark as F and from the Center or other extremitie of the Radius a line drawn just touching the same the Angle included between the said line and the Radius shall be an Ark equal to the Ark belonging to the said Sine and what is here done by a line of natural Sines or Tangents by help of a Decimal line of equal parts equal to the Radius ' may be done by help of the natural Tables without them Any Proportion relating to the sixteen cases of Sphoerical Triangles amongst which the Radius is always ingredient may be so varyed that the Radius may be in the third place and a Tangent or a Sine in the fourth place and then if the Ark belonging to the fourth Proportional be known an Angle equal thereto may be prickt off by Tangents or Sines according to the nature of the fourth term as beforth if it be unknown notwithstanding an Angle equal thereto may be prickt off with Sines or Tangents according to the nature of the fourth term from the two first terms of the Proportion because they
the Plains perpendicular by some termed the Plains Vertical line or line of Reclination at the intersection of these two lines set V and upon it as a Center describe a Circle the Radius whereof may be equal to 60d of a line of Chords at the upper end of the Vertical line set S and at the lower end N. As the Declination is set it off with Chords from S or N towards the true Coast and at it set D from whence draw a line through the Center also set off the Latitude of the place the same way and in the same quarter and at it set L from which draw a line to the Center From that end of the Horizontal line towards which the declination was counted set the Inclination which as well as the Reclinais reckoned from the Zenith the former being the Denomination of the under the latter of the upper face upwards towards S and the Reclination downwards towards N and at it set R from whence draw a line through the Center to the other side of the Circle In the same quarter of Declination draw HA parallel to the Horizontal line and FG parallel to the Vertical line in a Geometrical square of like and of any convenient distance from the Center at pleasure and where the Latitude line intersects the side of the square let the letter F be placed On the line of Declination beyond the Center make VO equal to FG and draw OB parallel to the Horizontal line continued till it meet with the side of the square FG produced at the point of concurrence set B and where it intersects the Plains perpendicular set P and draw OC parallel to the Vertical line cutting WE at C and make HA equal to OC or BG now by help of the three points A B C thus found the requisites will be easily prickt off 1. The Substilar line The nearest distance from A to RV set on the line CO produced if need be from C to I the same way the distance was taken from A that is if downward or upward the other must be so too will shew where VI the Substilar Line is to be drawn 2. The Stiles height The least distance from B to RV set on a perpendicular raised upon the Substile from the point I will finde the point K from whence draw a Line into the center at V and the Angle IVK will shew how much the Stile is elevated above the Substile and if the work be true VK and VF will be equal whence it follows that the trouble of raising the mentioned perpendicular may be shunned 3. The Meridian line The least distance from C to RV set upon the Line OPB from P on that side which is farthest from the Line RV will finde the Point M from whence a Line drawn into the Center shall be the meridian Line And I adde that on all North Recliners in the Northern Hemisphere the meridian Line must be drawn through the center on the other side and then the construction of the Scheam will place it below the Plains Horizontal Line which is its proper Scituation for the said upper face and for the under face the Scheam placeth it true without caution 4. A Polar Plain how known If the line RV fall just into B the Plain is a Polar Plain in such a Plain the Stile hath no height but is parallel to the Axis in this case the Inclination of Meridians must be known directions for such Plains must afterwards follow But if the line RV fall between the Points B and P then must the Substile Stile and Meridian be all drawn through the Center and stand beyond on the other side Annotations on the former Scheam 1. That for Latitudes under 45d this construction of the Scheam supposeth the sides of the square produced which will therefore be lyable to large excursions or other inconveniences wherefore for such Latitudes I shall somewhat vary from the construction prescribed 2. In finding the Substilar line in stead of erecting CI upon VC you may prick the same on the Vertical line VN and thereto erect VC and get the point I possibly with more certainty by finding the intersection of two Arks where the said Point is to pass 3. In pricking off the Meridian line the distance of C from the Center may be doubled or tripled but so must likewise VP and the nearest distance from C to RV erected on a line drawn parallel to WE passing through the Point P so found and in stead of drawing such a parallel the Point M may be found by the intersection of two Arks 4. That this Scheam placeth the requisites of all Dyals in their true coast and quantity yet notwithstanding if this Scheam be held before a Looking-Glass the Effigies thereof in the Glass shews how the Scheam would happen and place the Requisites namely the Stile Substile and Meridian for a Plain of the same Denomination but declining to the contrary Coast And if the face of the said Scheam be laid upon a Window and the Substile Stile and Meridian be continued through the Center on the backside thereof it shews you how these requisites are to be placed on the opposite side of the Plain which being done may be held before a Looking-Glass as before and will be represented for the contrary Declination of that opposite face the truth of all which will be confirmed from the Scheam it self This Scheam for Declining Reclining or Inclining Plains useth a new method of Calculation derived from an Oblique Triangle in the Sphere wherein there is two sides with the Angle comprehended given to finde both the other Angles which is reduced by a perpendicular to two right Angled Triangles from which the following proportions are derived I shall therefore first deliver the said method then demonstrate that the said proportions are carried on in the Scheam and lastly from the Sphere shew how those proportions do arise 1. To finde a Polar Plains Reclination or Inclination As the Radius Is to the Cosine of the Plains Declination So is the Cotangent of the Latitude To the Tangent of the Reclination or Inclination sought 2. To finde the distance of the Substile or Meridian line from the plains perpendicular for a Polar plain As the Radius Is to the Sine of a Polar Plains Reclination So is the Tangent of the Declination To the Tangent of the Substilar line from the Plains perpendicular 3. The Inclination of Meridians As the Radius Is to the Sine of the Latitude So is the Tangent of the Declination To the Tangent of the Inclination of Meridians Affections of a Polar Plain The Substilar on the upper face lies above that end of the Horizontal line towards the Coast of Declination and the Meridian lyes parallel to the Substile beyond it towards that end of the Horizontal line that is towards the Coast of Declination For Declining Reclining or Inclining Plains First finde a Polar Plains Reclination for the same
Declination Then for South Recliners and North Incliners get the difference but for North Recliners and South Incliners the sum of a Polar plains Reclination and of the Re Inclination of the plain proposed and then it holds 1. For the Substile As the Cosine of the said Ark of difference or sum according as the Plain leans Northward or Southwards Is to the Sine of the Polar Plains Reclination So is the Tangent of the Declination To the Tangent of the Substilar from the Plains perpendicular 2. For the Stiles height As the Radius Is to the Cosine of the Substiles distance from the Plains perpendicular So is the Tangent of the Sum or difference of Reclinations as before limited To the Tangent of the Stiles height 3. Meridians distance from the plains perpendicular As the Radius Is to the Sine of the Re Inclination So is the Tangent of the Declination To the Tangent of the Meridian from the Plains perpendicular 4. Inclination of Meridians As the Sine of the Stiles height Is to the tangent of the distance between the Meridian and Substile So is the Radius to the tangent of the Inclination of Meridians For South Recliners or North Incliners the difference between the substiles distance from the plains perpendicular and the Meridians distance therefrom is equal to the distance between the Meridian and Substile the like for such North Recliners or South Incliners as Recline or Incline more then an Equinoctial Plain having the same declination but if they lean above it or have a lesser Reclination the sum is the distance between the Meridian and the Substile The three first Proportions besides the finding of a Polar Reclination are used in the Scheam for the placing of the Requisites and the latter proportion in the Circular Scheam for drawing the hours Another proportion for finding the Inclination of Meridians by Calculation is As the Cosine of the Latitude Is to the sine of the substiles distance from the Plains perpendicular So is the Cosine of the Re Inclination of the Plain To the sine of the Inclination of Meridians The Reclination of an Equinoctial Plain to any assigned Declination is necessary for the determining of divers affections The Proportion to finde it is As the Radius Is to the Cosine of the Plains Declination So is the Tangent of the Latitude To the Tangent of the Reclination sought The upper face of an Equinoctial Plain is called a North Recliner the Meridian descends from the end of the Horizontal line opposite to the Coast of Declination the Substilar line is the hour-line of six and maketh right Angles with the Meridian line Directions for the true Scituating of the Meridian and Substile suited to the former method of Calculation 1. For Plains leaning Northwards If a South Plain recline more then a Polar Plain having the same Declination the Plain passeth beneath the Pole of the World the North Pole is elevated upon the upper face the Substile and Meridian line lye above that end of the Plains Horizontal line towards the Coast of Declination the Substilar line being next the Plains perpendicular For the under face being a North Incliner the South Pole is elevated the lines lye in the same position below the plains Horizontal line and on the contrary side of the plains perpendicular If a South Plains Reclination be less then the Polar Plains Reclination the Plain passeth above the Pole and the North Pole is elevated on the under face being the inclining side The Substile and meridian lye above that end of the Plains Horizontal line that is opposite to the Coast of Declination the meridian being nearest the Plains perpendicular for the upper face being a South Recliner the South Pole is elevated and the lines lye in the same Position below the Plains horizontal line but on the contrary side of the plains perpendicular descending below that end of the Horizontal line opposite to the Coast of Declination 2. For Plains leaning Southwards generally on the upper face the North Pole is elevated on the under face the South Pole To place the Substile Such North Recliners whose Reclination is less then the complement of a Polar plains Reclination the Substile is elevated above the end of the Horizontal line contrary to the Coast of Declination and on the under face being a South Incliner the Substilar is depressed below the end of the Horizontal line opposite to the Coast of Declination But when the Reclination is more then the complement of the Reclination of a Polar plain the Substile is to lye below the plains Horizontal Line from that end opposite to the Coast of Declination But for South Incliners being the under face the Substile is elevated above the end of the Horizontal line opposite to the Coast of Declination To place the Meridian On all North Recliners the Meridian lies below the Horizontal Line from that end thereof opposite to the Coast of Declination because at noon the Sun being South casts the shadow of the Stile to the Northwards On the under face being a South Incliner it must always be placed below the Horizontal Line below that end of it toward the Coast of Declination These Directions suppose the Declination to be denominated from the Scituation of that face of the plain on which the Dyal is to be made and the Horizontal line for all Dyals that have Centers is supposed to pass through the same Now to the Demonstration of the former Scheam 1. 'T is asserted that if RV fall into the point B the plain is a Polar plain in which case the Stile is parallel to the Axis of the world Demonstration Every Declining plain may have such a Reclination found thereto as shall make the said plain become a Polar plain and the proportion to finde it may be thus As the Tangent of the Latitude Is to the Cosine of the Declination So is the Radius To the Tangent of the Reclination sought In the former Scheam if we make FG the Cotangent of the Latitude Radius the side of the square will be the Tangent of the Latitude now VO equal to FG being Radius OC equal to GB is the Cosine of the Declination wherefore a Line drawn into the Center from B shall include the Angle of a Polar plains Reclination agreeable to the two first terms of the proportion and to the directions for pricking off an Angle by Tangents 2. That the substile is true prickt off Upon V as a Center with the Radius VB imagine or describe a Circle then is BG equal to VP the Sine of a polar plains Reclination which is equal to HA and the Ark comprehended between A and B will be a Quadrant But in a Quadrant any line being drawn from the Limbe passing through the Center the nearest distance from the end of one of the Radij will be the Sine of the Ark thence counted and the nearest distance from the other Radius thence counted
Cancer under the Horizon is equal to the least Meridian Altitude and the depression of the Tropick of Capricorn to the greatest Example 38d 28′ Colatitude 23d 31′ 61d 59′ greatest 14d 57′ least Meridian Altitude having drawn the Primitive Circle c. as before Prick 14d 15′ from S to C and 61d 59′ from S towards W a Ruler laid from the points found will intersect the meridian ZS at the point L for the Winter Tropick and K for the Summer Tropick through which the Circles that represent them are to pass to finde the Semidiameters whereof set off their depression from N towards E thus 14d 57′ the depression of the Summer Tropick terminates at O a Ruler laid from E to ☉ findes the point X in the meridian SZN produced so is XK the Diameter of the Summer Tropick which being divided into halfs will finde the Center thereof whereon to describe it In like manner is the Diameter of the Winter Tropick to be found or if the Amplitude be given or found as elsewhere is shewed which at London is 39d 54′ and set off both ways from G and E we shall have three points given through which to draw each Tropick and the Centers falling in the Meridian Line will be found with half the trouble as to finde a Center to three Points Also Project the Pole Point P as before being thus prepared FKG will represent the Summer and HLI the Winter Tropick Let it be required to know what hours are proper for a South plain Declining 30d Eastwards through the three points BPA describe the the Arch of a Circle BQP then laying a Ruler from B to Q finde the point R in the Limbe and from it set off a Quadrant to M then a Ruler laid from B to M findes the point ☉ the Pole of the hour Circle BQP then laying a Ruler from P to ☉ it findes the point T in the Limbe and the Arch ET being 65d 40′ is the measure of the Angle BPS which turned into Time is 4 ho 23′ prope and sheweth that at no time of the year the Sun will shine longer on the South side of this plain then 23 minutes past 4 in the afternoon In like manner if the Arch of a Circle be drawn through the two points PV we may finde the time when the Sun will soonest in the morning begin to shine on the South side of this plain A South plaine Declin 30d East lat 51d 32′ page 61 A South plaine Declin 60d East Reclining 40d Lat 51-32′ So if there were a South plain Declining 60d Eastwards Reclining 40 degrees here represented by BRA if it were required to know what hours are proper for the upper and what for the under face then where the plain intersects the Tropicks as at I and K draw two Meridians into the Pole at P to wit IP and KP and first finde the Angle IPZ as was before shewed to wit 53d 14′ which in time is 3 hours 33 minuutes shewing that the Sun never shines longer on the upper face of the Plain then 33 minutes past 3 in the afternoon which is capable of receiving all hours from Sun rising to that period of time and the Angle KPZ to wit 39d 50′ in time 2 hours 39 minutes shews that the Sun never begins to shine sooner on the under face then 39 minutes past 2 in the afternoon after which all the hours to Sun-set may be expressed To finde these Arks of time by Calculation there must be given the Stiles height above the Plain 15d 22′ PH and the complement of the Inclination of Meridians to a Semicircle 136d 32′ to wit HPS then in the right Angled Triangle PHI there is given PH the Stiles height PI the complement of the Declination besides the right Angle at H to finde the Angle IPH 83d 18′ which taken from the complement of the Inclination of Meridians HPS there rests the Angle IPS the Arch of time sought to wit 53d 14′ The ascensional difference may be found by drawing the Arch of a Circle through the three points TPF and thereby the length of the longest day determined that no hours be expressed on which the Sun can never shine Another manner of Inscribing the hour-lines in all Plains having Centers The method here intended is to do it in a parallelogram from the Meridian line whence the hour-lines may be prickt down by a Tangent of three hours with their halfs and quarters from a Sector without collecting Angles at the Pole or by help of a Scheam which I call the Tangent Scheam the foundation of this Dyalling supposeth the Axis of the world to be inscribed in a Parallelipiped on continued about the Axis the sides whereof are by the plains of the respective hour Circles in the Sphere divided into tangent-Tangent-lines that is to say each side is divided into a double tangent of 45d set together in the middle and the said parallelipiped on being cut by any Plain the end thereof supposed to be intersected shal be either a right or Oblique Angled parallelogram and then if from the opposite tangent hour points on the sides of the intersected parallelipipedon lines be drawn on the Plain they shall cross one another in a Center and be the hour-lines proper to the said plain but of the Demonstration hereof I shall say no more at present the inquisitive Reader will finde it in the Works of Clavius To draw the Tangent Scheam I have before in Page 10 shewed how to divide a Tangent Line into hours and quarters which in part must be here repeated draw any right Line as MABH from any point therein as at B raise a perpendicular and upon B as a Center describe the Quadrant CH and prick the Radius from B to A from C to G from H to F and laying a Ruler from A to F and G you will finde the points D and E upon the perpendicular CB I say the said perpendicular is divided into a Tangent line of three hours and the halfs and quarters may be also divided thereon by dividing the Arches CF FG and GH into halfs and quarters then from those subdivisions laying a Ruler to A the halfs and quarters may be divided on as were the whole hours Being thus prepared draw the Lines MB LE KD and IC all parallel one to another passing through the points B 1 2 3. In this Scheam they are perpendicular to BC but that is not material provided they pass through the same points and are parallel one to another yet notwithstanding the points A and H must be in a right Line perpendicular to CB. This Scheam thus prepared I call the Tangent Scheam because a Line ruled any way over it shall be divided also into a Tangent of the like hours and quarters whence it follows that one of these Scheams may serve to inscribe the hour-lines into many Dyals which I shall next handle To inscribe the Hour lines in a Horizontal Dyal Having
Geometricall DYALLING or Dyalling by a line of CHORDS onely by IOHN COLLINS Accomptant Philomath London Printed for Francis Cossinet in Tower Street and for H Sutton behind Exchang Generall Scheme Projection of ye Sphere Anno 1659 In a Circle In a Parralellogram Chords fitted to ye Schemes in ye Booke Chords in time fitted to the Booke Geometricall Dyalling OR DYALLING Performed by a Line of CHORDS onely Or by the PLAIN SCALE Wherein is contained two several Methods of Inscribing the Hour-Lines in all Plains with the Substile Stile and Meridian in their proper Coasts and Quantities Being a full Explication and Demonstration of divers difficulties in the Works of Learned Mr. Samuel Foster deceased late Professor of Astronomy in Gresham Colledge Also a Collection of divers things from the Works of Clavius and others Whereto is added four new Methods of Calculation for finding the Requisites in all Leaning Plains with full directions suited to each Method for placing them in their proper Coasts without the help of any Dclinations Also how by Projecting the Sphere to measure off all the Arks found by Calculation and to determine what hours are proper to all kinde of Plains omitting superfluity Lastly the making of Dyals from three Shadows of a Gnomon placed in a Wall at random with a Method of Calculation suited thereto and divers ways from three Shadows to finde a meridian-Meridian-line Written by John Collins of London Accomptant Philomath London Printed by Thomas Johnson for Francis Cossinet and are to be sold at his Shop at the Anchor and Mariner in Tower-street at the end of Mincing-lane with other Mathematical Books Also to be sold by Henry Sutton Mathematical Instrument-Maker living in Threed-needle Street behinde the Exchange 1659. To the Reader BEing in conference with my loving friend Mr. Thomas Rice one of the Gunners of the Tower much exercised in making of Dyals in many eminent places of the City he was pleased to communicate unto me the knowledge of a General Scheme for Inscribing the Requisites in all Oblique Leaning Plains which he added was the useful Invention of the Learned Master Samuel Foster Professor of Astronomy in Gresham Colledge London deceased from whom he received Instructions concerning the same in the year 1640. And the said Mr. Rice not having his Papers about him did dictate to me from his Memory the Construction and Practice of the said Scheme which I afterwards Methodized just as it is delivered in page 25 and 26. At the same time also I received directions for Inscribing the Substile and Stile in upright Decliners and East or West Reclining or Inclining Plains but they were of another mans Invention and did not seem to be derived from the former general Scheme and therefore I have not used them in this Treatise but derived the performance thereof from the said general Scheme Moreover Mr. Rice added that in regard of the death of the Author and since of his Executor who had the care and inspection of his Papers I should do well to Study out the Demonstration of the former Scheme and make it publick the rather because it hath been neglected The manner of Inscribing the hour-lines being already published in a Treatise of the Authors Intituled Posthumi Posteri this desire of his which was also furthered by Mr. Sutton and others I am confident is fully effected in the following Treatise and much more then was in his Request and here let me adde that I have had no other light from the Endeavors of the Learned Author then what was as above communicated unto me all which might be expressed in half a page of paper or little more to which I have made this large Access and Collection nam facile est inventis adore not that I would hereby any thing endeavor to ecclipse the Authors Work the excellency of whose Inventions in this and other kindes will speak forth his Renown to all posterity And though in probability I have not performed so much nor so well as was obvious to the knowledge of the Learned Author yet I am confident when the Reader understands what is written he would be as loath to be without the knowledge thereof as my self and I am induced to believe that the Author left nothing written about many particulars in this Treatise throughout which we suppose the Reader furnished with the common Rudiments of Geometry that he can raise Perpendiculars draw Parallels describe a Parralellogram bring three points into a Circle understands Definitions and Tearms of Art knows what a Line of Chords is can prick off an Arch thereby all which with Delineations for all the usual Cases of Triangles both from Projection and Proportions the Reader will meet with in my Treatise called The Mariners Plain Scale new plain'd now in the Press Vale fruere I remain thy friend and a Well-willer to the Publique Advancement of Knowledge JOHN COLLINS The Contents DYals distinguished Page 1 2 To take the Suns Altitude without Instrument p. 3 To finde the Reclination of a Plain p. 4 Also the Declination thereof p. 5 A general proportion and Scheam for finding the Suns Azimuth or true Coast p. 6 To draw a Horizontal Dyal p. 7 Also a South Dyal p. 8 A new way to divide a Tangent liue into five hours and their quarters p. 10 11 A direct South Polar Dyal p. 12 To prick off the Requisites of upright Decliners p. 13 To prick off an Arch or Angle by Sines or Tangents p. 14 The Scheam for placing the Requisites of upright Decliners demonstrated p. 15 16 To inscribe the hour-lines in an upright Decliner p. 17 The Demonstration thereof p. 18 to 21 An East Dyal p. 22 Requisites placed in East or West leaning Plains p. 23 The Demonstration thereof p. 25 The Construction of the general Scheam for placing the Requisites in Declining Re-Inclining Plains p. 25 to 27 The first Method of Calculation for Oblique Plains p. 28 to 30 And directions for the true placing the Requisites suited thereto p. 30 31 The general Scheam demonstrated p. 32 to 34 The hour-lines inscribed in an Oblique Plain ibid. The general Scheam fitted for Latitudes under forty five degrees p. 35 To draw the hour-lines in a Declining Polar Plain p. 36 Also how to delineate the hour-lines in Plains having small height of Stile p. 37 38 Another way to perform the same p. 39 to 41 A second Method of Calculation for Oblique Plains p. 42 Proportions for upright Decliners p. 43 A third Method of Calculation for Oblique Plains p. 44 Directions for placing the Requisites suited thereto p. 46 A fourth Method of Calculation for Oblique Plains p. 47 Through any two points assigned within a Circle to draw an Arch of a Circle that shall divide the primitive Circle into two Semicircles p. 49 To measure the Arks of upright Decliners by Projection p. 50 Also the Arks of leaning East or West Plains thereby p. 51 To project the Sphere for Oblique Plains To
and NK is GO because KL is LG now in the equiangled Triangles ABH AOG as BH HA ∷ OG GA and in the Triangles AHE AGK as HA HE ∷ GA to GK therefore ex equo it will be as BH HE ∷ OG GK but as BH HE ∷ BI IM for IH is parallel to EM therefore as BI IM ∷ OG GK but NK is OG therefore BI IM ∷ NK KG but IC is half of IM and KL half of KG therefore BI IC ∷ NK KL therefore by Composition of Proportion as BC IC ∷ NL KL alternately as BC NL ∷ IC KL In the second figure BC is NL therefore IC is KL wherefore because IC is also parallel to KL IK is parallel to CL by like reason GM is also parallel to CL but in the third and fourth figures the sides PC PL of the Triangle PCL are cut Proportionally by the parallel BN and as before as BC NL ∷ IC KL and as PC PL ∷ BC NL therefore PC PL ∷ IC KL wherefore IK is parallel to CL Again as before BC NL ∷ IC KL But CM is IC and IG is KL therefore as BC NL ∷ CM LG but as hath been said as BC NL ∷ PC PL therefore as PC PL ∷ CM LG wherefore because the sides PM PG of the Triangle PMG are cut proportionally in C and L CL and MG are parallel one to another Now in all the three figures it hath been proved that IK and MG are either of them parallel to CL therefore they are parallel one to another wherefore the Angle GMI is equal to the Angle KIB which before was proved to be equal to AMF therefore the Angles GMI and AMF are equal one to another wherefore FM and MG is one and the same right line which was to be proved To draw an East or West Dyal Let the hour line of six which is also the Substilar line make an Angle equal to the Latitude of the place with the Plains Horizontal line above that end of it that points to the Coast of the elevated Pole then draw a line perpendicular to the Substilar line which some call a Contingent line and with such a Radius as you determine the Stile shall have parallel height above the Substile divide a Tangent line of hours and quarters according to the direction for direct Polar Recliners then through those divisions draw lines parallel to the Substile and they shall be the hour lines required thus the hours of 5 and 7 are each of them Tangents of 15d from six and so for the rest let them be numbred on each side of six being the Substile for an East dyal with the morning hours for a West dyal with the afternoon hours the one being the complement of the other to 12 hours and therefore we have but one example namely an East dyal for the Latitude of London How to fill this or any other Plain with any determined number of hours shall afterwards be handled A West Diall Reclining 50d Latitude 51-32′ An East Diall Inclining 40d Latitude 50d To prick off the requisites of an East or West Reclining or Inclining Dyal in their true Scituation and quantity In these Plains first draw the Plains perpendicular and cross it with a Horizontal line which is also the Meridian line in any one of the quarters make a Geometrical square as before directed for upright decliners and from the Plains Vertical in the said quarter count off the Latitude to L and from the Horizontal line in the said quarter count off the Reclination or Inclination to R and from L and R draw lines into the Center and where the Latitude line intersects the side of the square set F. 1. To prick off the Substile For Latitudes above 45d place BF from V the Center in the Horizontal line for Recliners Northwards for Incliners Southwards and thereto set O and through the said point draw a line parallel to the Vertical and place the nearest distance from A to RV on the former parallel from O for Recliners upwards to I but for Incliners downwards and a line thence drawn into the Center shall be the Substile 2. The Stiles height Place the nearest distance from B to RV on a perdendicular raised from the point I in the Substile and it findes the Point K whence a line drawn into the Center shall represent the Stile In the other Hemisphere the words Northwards and Southwards must be mutually changed For Latitudes under 45′ The side of the square must be placed from V to O and AF must be placed on the line of Reclination or Inclination from V to C the nearest distance from C to VA placed on the perpendicular passing through O for Recliners upwards but for Incliners downwards findes the Point I through which the Substile is to pass and the nearest distance from C to VB raised perpendicularly on the point I in the Substile findes the point K for the Stile as before Demonstration 1. For the Substile In these Plains VB or VA being Radius FB is the Contangent of the Latitude and the nearest distance from A to RV is the Cosine of Reclination or Inclination to the same Radius and the nearest distance from B to RV the Sine thereof this for Latitudes above 45 but for lesser Latitudes AF the Tangent of the Latitude was made Radius and thereupon the Radius or side of the square be-became the Cotangent of the Latitude and the nearest distance from C to VA was the Cosine and from C to VB the Sine of the Reclination or Inclination so that the prescribed Construction in both cases erects the Cosine of the Re-Inclination on the Cotangent of the Latitude which is made good from this proportion As the Cotangent of the Latitude Is to the Cosine of the Re-Inclination from the Zenith So is the Radius To the tangent of the Substilar from the Meridian 2. For the Stile A proportion that will serve to prick it off is As the Cosecant of the Latitude is to the Sine of the Re-Inclination So is the Radius To the Sine of the Stiles height page 25 A West Diall Reclining 50d Latitude 51 32′ A South Plaine Declining 40d East Inclining 15 deg A West Plain Reclining 50 degrees Latitude 51 degrees 32 minutes The Point Sol Substile Stile and hour-lines are all found after the same manner as in an upright Decliner the Substile being set off from the Meridian line here after the same manner as it was from the Meridian there A South Plain Declining East 40 degrees Inclining 20 degrees Latitude 51 degrees 32 minutes To prick off the Requisites in all Declining Reclining and Incliing Plains in their true Coast and quantity Now followeth those Directions inlarged which as I said in the Epistle I received from Mr. Thomas Rice UPon any Plain first draw a true Horizontal line at the East end thereof set E and at the West end W cross the said line with a perpendicular which may be called
Calculate all the Requisites In all upright Plains the Meridian lyeth in the plains perpendicular and if they Decline from the South in this Hemisphere it is to descend or run downward if from the North it ascends and the Substile lyeth on that side thereof opposite to the Coast of Declination In East or West Re-Incliners it lyeth in the plains horizontal line on the Inclining side the South Pole is elevated but on the upper side the North Pole and the Substile lyeth above or below that end of the Meridian line which points to the Pole elevated above the Plain On all plains whatsoever to Calculate the hour distances As the Radius Is to the Sine of the stiles height above the substile So is the tangent of the Angle at the Pole To the tangent of the hour-lines distance from the substilar line By the Angle at the Pole is meant the Ark of difference between the Ark called the Inclination of Meridians and the distance of any hour from the Meridian for all hours on the same side of the Meridian the Substile falls and the sum of these two Arks for all hours on the other side the Meridian All hours on any Plain go to the contrary Coast of their Scituation in the Sphere thus all the morning or Eastern hours go to the Western Coast of the plain and all the evening or Western hours go to the Eastern Coast of the Plain A third Method of Calculation for leaning Plains that is for all sorts of Plains that do both Decline and also Incline or Recline They may be referred to a new Latitude in which they shall stand as upright Plains and then they will have a new Declination in that new Latitude which two things being found the former Proportions for upright Decliners will serve to Calculate all the Arks required How this may be done on a Globe is not difficult to apprehend having set the Globe to your Latitude let one of the Meridians of the Ecliptick or Longitude in the heavens represent a Declining Reclining Plain this Circle intersects the Meridian of the place in two Points the one above the other beneath the Horizon Imagine the Globe to be so fixed that it cannot move upon its Poles then elevate or depress the Globe so in the Meridian that the point of Intersection above the Horizon may come under the Zenith then will the Pole of the world be elevated above the Horizon to the new Latitude sought and where the Meridian of Longitude that represents the Plain intersects the Horizon it shews the new Declination Or it may be thus apprehended The distance between the Pole of the world and that point of Intersection that represents the Zenith of the new Latitude is the complement of the said new Latitude and the distance between that point and the Equinoctial is the new Latitude it self the new Declination is the complement of the Angle between the plain and the meridian of the place an Ark usually found in Calculation under this denomination To finde these Arks by Calculation As the Radius Is to the Cosine of the Plains Declination So is the Cotangent of the Re-Inclination from the Zenith To the tangent of the Meridional Ark namely the Ark of the Meridian between the Plain and the Horizon And this is the first thing Master Gunter and others finde for South Recliners North Incliners the one being the upper the other the under face get the difference between this Ark and the Latitude of the place the complement of the said residue remainder or difference is the new Latitude sought but for North Recliners or South Incliners the difference between this fourth Arch and the complement of the old Latitude is the new Latitude To finde the new Declination As the Radius Is to the Cosine of the Re-Inclination So is the sine of the old Declination To the sine of the new This method is hinted to us in Mr. Fosters Posthuma also in his Book of Dyalling in Anno 1638 where he refers leaning plains to such a Latitude wherein they may become East or West Recliners but that method is to be deserted as multiplying more proportions then this and doth not afford that instrumental ease for pricking down the hours that this doth Affections determined Such South Recliners whose meridional Arch is less then the Latitude pass beneath the Pole and have the North Pole elevated above them but if the meridional Ark be greater then the Latitude they pass above the Pole the North Pole is elevated on the under face all other affections are before determined If the meridional Arch be equal to the Latitude the plain is a Polar plain for plains leaning Southwards if the meridional Arch be equal to the complement of the Latitude the plain is an Equinoctial plain if it be more the plain hath less Reclination then an Equinoctial plain if it be less it hath more and all affections necessary for placing and Calculating the meridian line were before determined This method of Calculation findes the Substiles distance from the meridian not from the plains perpetdicular wherefore it must be shewed how to place it in Plains leaning Southwards for plains leaning Northwards use the former directions To place the Substile in North Recliners In these plains the Meridian and Substilar are to meet at the Center and not being drawn through will make sometimes an Acute sometimes an obtuse Angle When the Plains Meridional Ark is greater then the Colatitude they make an Obtuse Angle in this Case having first placed the Meridian line above it prick off the complement of the distance of the Substile from Meridian to a Semicircle But when the Meridional Ark is less then the Colatitude prick off the said distance it self above the Meridian line In South Incliners When the Plains Meridional Ark is greater then the Colatitude the Substile and Meridian make an Acute Angle when it is equal to the Colatitude they make a right Angle when it is less then the Colatitude they make an Obtuse Angle and must be prickt off by the complement of their distance to a Semicircle the Substile always lying on that side of the Meridian opposite to the Coast of Declination A fourth Method of Calculation for leaning Plains An Advertisement In this method of Calculation for all Plains leaning Northward both upper and under side their Declination is the Arch of the Horizon between the North and the Azimuth of the plains South Pole so that their Declination is always greater then a Quadrant But for all Plains leaning Southwards both upper and under face their Declination is the Arch of the Horizon between the North and the Plains North Pole wherefore it is always less then a Quadrant in this sense Declination is used in the following Proportions As the Sine of half the sum of the complements both of the Latitude and of the Reclination Is to the Sine of half their difference So is the Contagent of