Selected quad for the lemma: end_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
end_n	east_n	lie_v	west_n	3,601	5	9.4636	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

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

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

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