Selected quad for the lemma: end_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
end_n	north_n	side_n	south_n	2,400	5	9.1749	5	true

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A33999	Geometrical 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 ... : whereto is added four new methods of calculation, for finding the requisites in all leaning plains ... : also how by projecting the sphere, to measure off all the arks found by calculation ... : lastly, the making of dyals from three shadows of a gnomon ... / written by John Collins ...	Collins, John, 1625-1683.	1659 (1659)	Wing C5373; ESTC R17003	56,110	109

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

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

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

the Stiles height the Inclination of Meridians is found in the Oblique Spherical Triangle by the proportion of Opposite sides and Angles Lastly in the right Angled Triangle ZRC there is given RC and RCZ to finde RZ the distance of the Meridian from the plains perpendicular To measure the respective Arks abovesaid 1. The New Latitude A Ruler laid from W to Z and P will give you the Arch MO in the Limbe the complement of the new Latitude to wit 27d 40′ 2. The New Declination The Ruler laid from the Plains Zenith at Z to its Pole at ☉ findes the point Q in the Limbe and the Arch SQ is the new Declination to wit 18d 56′ 3. The Substiles distance from the Plains perpendicular A Ruler laid from ☉ to H findes the point T in the Limbe and the Arch CT being 26d 26′ is the Substiles distance from the Plains perpendicular 4. The Meridians distance from the Plains perpendicular A Ruler laid from ☉ to Z findes the point V in the Limbe and the Arch DV being 35d 56′ is the Meridians distance from the Plains perpendicular 5. The Stiles height Set off a Quadrant from G to X and draw XC where it intersects the plain as at Y is the Pole of the Arch FG then laying a Ruler from Y to H and P you shall finde the Stiles height in the Limbe to be the Arch 2 3 namely 26d 6′ A South plaine Declin 40d East Reclining 60d Lat 51-32′ page 55 A South plaine Declin 30d East Reclining 25d Lat 51 32′ 6. The Inclination of Meridians A Ruler laid from P to Y intersects the Limbe at the Arch W is 20d 58′ and so much is the Inclination of Meridians The Polar Reclination CS is 31d 19′ The Scheam determineth all the affections of the plain 1. It shews that H the point of the Substilar lies on that side the plains perpendicular that is towards the Coast of Declination 2. That Z the point for the place of the Meridian lyes towards the same Coast as before but below the Substilar Line 3. The Arch PH shews you that the North Pole is elevated above the upper or Reclining face After the same manner may all the Requisite Arks be measured and affections determined for all plains whatsoever A South Plain Declining 30 degrees Eastwards Reclining 25 degrees or a North Plain Declining 30 degrees Westwards Inclining ●● degrees In this Scheam we have the same Oblique Triangle PC ☉ reduced to two right Angled Triangles PSC and PS ☉ SC is the Inclination of a Polar plain and RC the Inclination of the plain proposed the difference is SR and the complement of it is the complement of S ☉ to a Semicircle because S ☉ is greater then a Quadrant and the proportions are wholly the same though the Triangle have sides greater then a Quadrant The North Pole is elevated on the Inclining face the Meridian Z lyes from the plains perpendicular towards that end of the Horizontal Line opposite to the Coast of Declination the same way and beneath it lyeth the Substilar The complement of the new Latitude ZP is 10d 10′ The new Declination viz. the complement of NZA is 26d 57′ The meridians distance from the Plains perpendicular RZ 13d 43′ The Substiles distance therefrom RH 18d 22′ The Stiles height PH is 9d 3′ The Inclination of Meridians to wit the Angle CP ☉ 27d 18′ The Polar Reclination CS 34d 31′ A South Plain Declining 40 degrees East Inclining 15 degrees or rather a North Plain Declining 40 degrees West Reclining 15 degrees Here again the Oblique Triangle CP ☉ is reduced to two right Angled Triangles PSC and PS ☉ and SR is the sum of the Polar Reclination SC and the Re Inclination of the Plain proposed CR and S ☉ is the complement hereof because ☉ R is a Quadrant finde the Equinoctial point AE The Polar Reclination CS 31d 19′ The new Latitude ZAE is 32d 15′ The new Declination being the complement of NZA is 38d 23′ The Meridians distance from the plains perpendicular ZR 12d 33′ The Substiles distance from the plains perpendicular RH 32d 16′ The Stiles height PH is 41d 30′ The Inclination of Meridians ☉ PC rather the Acute Angle ☉ PN is 55d 58′ The Meridian Z lies from the plains perpendicular towards the Coast of Declination on the Reclining side but must be drawn through the Center because the Sun at noon casts his shadow Northwards unless in the Torrid or Frozen Zone and the Substile Hlyes on the other side the plains perpendicular A North Plain Declining 40 degrees Eastwards Reclining 75 degrees In this plain likewise the Oblique Triangle C ☉ P is reduced to two right Angled Triangles PS ☉ and PSC by the perpendicular PS which is part of the Arch of a polar plain here CR more CS is equal to the sum of the Plains Reclination proposed and of the Polar plains Reclination which is greater then a Quadrant for the Arch R ☉ is a Quadrant now the Cosine of an Ark greater then a Quadrant is the Sine of that Arks excess above a Quadrant wherefore the Sine of S ☉ is the Cosine of the sum of both the Reclinations and the Case the same as before A South plaine Declin 40d East Inclining 15d Lat 51d 72′ page 56 A North plain Declin 40d East Reclin 75d lat 51d 32′ Prick off a Quadrant from G to X and draw XC it cuts the plain at Y a Ruler laid from Y to H and P findes the points 2 3 in the Limbe and the Arch 2 3 being 61d 31′ is the Stiles height orther complement thereof to a Semicircle might be found by measuring the Arch PF A Ruler laid from ☉ to Z findes the point V in the Limbe for the Meridian Line from which draw a Line through the Center on the other side and it will be placed in its true Coast and quantity from the Plains perpendicular at A to wit 39d 2′ The Inclination of Meridians to wit the Angle CP ☉ is 20d 30′ The new Latitude ZAE 26d 38′ The new Declination is 9d 35′ to wit the complement of KRB. The Polar Reclination CS is 31d 19′ The truth of this Stereographick Projection is fully handled by Aguilonius in his Opticks and how to determine the affection of any Angle of an Oblique Spherical Triangle I have fully shewed in a Treatise called the Sector on a Quadrant For the Resolution of Spherical propositions Delineations from proportions or the Analemma will be more speedy and certain though they may also be thus resolved which I have handled at large in the Mariners Plain Scale new plain'd To determine what hours are proper to all kinde of Plains To do this it will be necessary to project upon the Plain of the Horizon the Summer and Winter Tropicks Get the Sum and difference of the Colatitude and of the Suns greatest Declination so we shall obtain his greatest and least Meridian Altitudes The depression of the Tropick of

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