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	north_n	south_n	5,687	5	9.7162	5	true

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A08582	The circles of proportion and the horizontal instrument The former shewing the maner how to work proportions both simple and compound: and the ready and easy resolving of quæstions both in arithmetic, geometrie, & astronomie: and is newly increased with an additament for navigation. All which rules may also be wrought with the penne by arithmetic, and the canon of triangles. The later teaching how to work most quæstions, which may be performed by the globe: and to delineat dialls upon any kind of plaine. Invented, and written in latine by W.O. Translated into English, and set out for the public benefit, by William Forster.	Oughtred, William, 1575-1660.; Forster, William, mathematician.; Oughtred, William, 1575-1660. Addition unto the use of the circles of proportion. aut	1633 (1633)	STC 18899B; ESTC S120820	74,822	159

at that point and the pinne vpright in the Center hold or set your instrument parallel to the plaine of the Horizon with the pinne toward the Sunne and moue it gently till the shadow of the pinne shall fall exactly vpon the fiduciall edge of the Labell For then the Meridian line of the instrument shall be in the true Meridian of the place and the foure quarters of the instrument shall looke into the foure cardinall points of East West North and South Wherefore if with a bodkin you make a prick at each end of the Meridian of your instrument where it standeth and with a Ruler draw a line through them the same shall bee the Meridian of that place This is a most excellent practise for finding out the Meridian in any place and is in an instant performed and that easily And hereby you may examine the Variation of the Compasse And also exactly place any Sunne Dyall XX. Vse Consid●rations for the vse of the instrument in the night In such questions as concerne the night or the time before Sun rising and after Sunne setting the instrument representeth the lower Hemisphaere wherin the Southerne Pole is eleuated And therefore the Parallels which are aboue the Aequinoctiall shall bee for the Southerne or Winter Parallels and those beneath the Aequinoctiall for the Northerne or Summer parallels And the East shall be accounted for West and the West for East and the North shall bee accounted for South and the South for North contrary to that which was before when the Instrument represented the vpper Hemisphaere XXI Vse To find how many degrees the Sunne is vnder the Horizon at any time of the night Seeke the declination of the Sunne for the day proposed and at the same declination on the contrary side imagine a Parallel for the Sunne that night and marke what point of it is in the very hower and minute proposed then set the Index or Labell to that point of the Parallel and it will shew you thereon the degree of the Sunnes depression vnder the Horizon XXII Vse To find out the length of the Crepusculum or Twilight euery day Because the question concerneth the night time you must seeke out the Sunnes Parallel for the night on the other side of the Aequinoctiall hauing the same declination with that which the day of the moneth sheweth then moue about the Labell vntill the said Parallel cutteth the edge thereof in the 18th deg on the West side for the Morning Twilight and on the East side for the Evening Twilight of the same day And note that in the height of Summer the Twilight in our Horizon continueth all night long because the same goeth not vnder the Horizon full 18 degrees XXIII Vse To find the Declination of any Wall or Plaine Take a board hauing one streight edge and a line drawne perpendicular vnto that edge apply the streight edge vnto the Wall at what time the Sunne shineth theron holding the board parallel to the plaine of the Horizon and hang vp a thread with a plummet so that the shadow of the thread may fall on the board crossing that perpendicular line Then take with your Instrument the height of the Sunne and instantly make two pricks in the shadow of the thread on the board a good way distant one from the other and laying a Ruler to those two pricks draw a line which line shall be the Azumith of the Sunne on the board againe with the height of the Sunne lastly taken find out on your instrument the Azumith of the Sunne or the Angle which the Sunnes Azumith maketh with the Meridian by the XV. Vse And on the board taking the intersection of the shadow line with the perpendicular for the Center describe a Circle equall to the innermost Circle of the Limbe which you may easily doe if you set one foot of your compasses vpon the East or West point and extend the other foot vnto 60 degrees on the same innermost Circle for this distance is equall to the Radius thereof Againe with your compasses take of the Arch betweene the Azumith of your Instrument and the Meridian and set that on the Circle of the board that way that the true South is and through the end of that Arch measured on the board draw a streight line for the Meridian Lastly take with your compasses the Arch intercepted between the Meridian on the board and the perpendicular line and by applying it to the in most Circle of the limbe from the East or West points see how many degrees it containeth for that is the declination of the Wall Or else you may find the Meridian vpon the board by XIX Vse If the Angle of the Meridian with the perpendicular on the board be a right Angle the Wall is direct East or West But if the Meridian fall vpon the perpendicular or be parallel there to making no Angle with it the Wall is direct North or South XXIIII Vse The Art of Dyalling And first how to make the Instrument in paper promised in the beginning of this second part For the Delincation of this instrument in paper it will bee necessary first to shew the manner how the Semidiameter is to bee graduated or diuided into degrees and how the Centers and Semidiameters of the seuerall kinds of Arches are to be found Vpon halfe a sheet of strong large Dutch paper the larger the better draw two streight lines making a right Angle neere one of the corners the one through the length and the other through the breadth of the paper which two lines I therefore call the longer and the shorter perpendicular Vpon the right Angle point being the Center with a Semidiameter equall to that by which you intend to delineate your instrument describe a quadrant of a Circle and on the point where it meeteth with the shorter perpendicular draw a long tangent line parallel to the longer pependicular Divide the Quadrant into 90 degrees among which from the beginning at the shorter perpendicular reckon the eleuation of the Pole for which you will make your instrument and applying a Ruler to the end thereof and to the Center where the Ruler cutteth the tangent line make a prick And taking with your compasses the distance from the Center to that prick measure it vpon the shorter perpendicular this shall be the Semidiameter of the sixt hower Circle At the end thereof draw another long line parallel also to the longer perpendicular Then out of the Center vnto the second parallel through every degree of the quadrant draw fine streight lines cutting also the first Parallel The intersection of those lines with the first Parallel shall be The scale of centers of Arches And their intersection with the second Parallel shall be The scale of centers of hower Circles And the segments of those lines intercepted betweene the Center and the first Parallel shall be the Semidiameters of Arches and the whole lines betweene the Center and the second Parallel

plaine perpendicular to the Meridian is that which standeth directly North or South which if it be also perpendicular to the Horizon is called North or South direct vpright But if it stoope from the Zenith forward it is called North or South inclining if backeward it is called North or South reclining And note that in a stooping Plaine that side which is toward the Horizon is inclining and that which is toward the Zenith is reclining The Plaine oblique to the Meridian is that which standeth not directly North or South but declineth one side into the East and the other into the West and is therefore called Declining Eastward or Westward according as either side of the Plaine looketh As if an vpright Wall being Southerne declineth from the South into the East it is called South declining Eastwards vpright But if it be not vpright it is called South declining Eastward and inclining or reclining The Plaine parallel to the Meridian is that which looketh directly East or West and accordingly hath his denomination whether it bee Vpright Inclining or Reclining The Plaine Parallel to the Horizon is called Horizontall and is represented by the instrument it selfe or at least by the inner most Circle of the limbe thereof And note that the Arch of Declination is reckoned from the next East or West point And that the Arch of Inclination or Reclination is reckoned from the Zenith or the complement of it from the Horizon So that euery vpright Plaine is vnderstood to passe through the Zenith which in the instrument is the Center And thus having shewed the seuerall affections of Plaines wee will now proceed to shew the manner how to set them vpon the Instrument A Direct North or South vpright Plaine is represented in the instrument by a line drawne through the Center from the East point to the West which is also the Horizontall intersection of the Pla●ne And by it you shall see that the Southerne side or face of the plaine is open to all the houres betweene sixe in the morning and sixe in the evening And that about London the Northerne side onely in the Summer enioyeth the Sunne from his rising till after seven in the morning and from before 5 a clocke in the afternoone till his setting A direct East or West upright plaine is represented in the Instrument by the Meridian which is also the Horizontall intersection of the plaine And in it you shall see that all the forenoone houres are open to the East side and all the afternoone houres to the West side A Declining Plaine is thus set upon the Instrument reckon on the Horizon the arch of Declination from the East or West point and at the end draw a line through the Center vnto the opposite point of the Horizon So that each side thereof may be open to that point either East or West into which the Declination is supposed That line so drawne throught the center is the Horizontall intersection of the plaine and representeth the plaine it selfe if it bee vpright For example there is about London an vpright Wall declining Eastwards 35 degrees which I would set vpon the Instrument Hold the Southerne part of the Instrument to you and reckon from the East backward into the North vpon the Horizon 35 degrees there draw a line through the Center this line shall not onely vpon the South side represent a Southerne Plaine declining Eastward 35 degrees But also vpon the North side shall represent a Northerne Plaine declining Westward 35 deg And moreover it will appeare that on the Southerne side shall bee drawne the houres from almost 4 a clocke in the morning till 3 in the afternoone And that in the Northerne side shall bee drawne vpon one side 4 a clocke in the morning onely and vpon the other side all the houres from 3 in the afternoone till Sunne set And ●o consequently the declination of an vpright wall or Window being given it may be found at what houre the Sunne vpon any day in the yeare will come to that Wall or Window and when it will goe from it As in the former example There is about London a Northerne wall declining Westward 35 deg I would know at what time of the day the Sunne will begin to shine vpon it on the 24th day of March. Set the Index at 35 deg from West toward South and because that day the Sunnes Declination is 6 degrees Northward Looke at what houre the sixt Parallel aboue the Aequinoctiall toward the Center meeteth with the Index so placed and you shall find it at 3 ● clock in the Afternoone Wherefore at that time the Sunne will begin to shine vpon the Wall that same day The Poles of every vpright Wall are in the Horizon 90 deg that is a quarter of a Circle distant from the line representing the Plaine Wherefore if vpon that line in the Center you erect a perpendicular the ends therof in the Horizon shall be the poles of that Plaine and are so farre distant from the North and South points as the Plaine it selfe is from the East and West XXVI Vse To set an Inclining and Reclining Wall or Plaine vpon the Instrument and to find how many houres the Sunne shall shine thereon at some time of the yeare When you haue an Inclining or Reclining Plaine to be described on the Instrument First the Horizontall intersection is to be set thereon as if it were vpright together with the line perpendicular thereto in which are the Poles of the Plaine according as was taught in the XXV Vse Then vpon the scale of degrees in your paper reckon the arch of Inclination or Reclination and with your compasses take set it in your Instrument vpon the line perpendicular to the Horizontall intersection of your Plaine from the Center that way into which the Inclination or Reclination tendeth the same shall bee the vppermost point of your Plaine Againe with your Compasses take the Complement of inclination or reclination both upon the scale of degrees and also upon the scale of centers of arches in your paper and set both spaces upon the same perpendicular line but on the other side of the center extended if need be At the shorter of those spaces shal be the pole of your plaine and at the longer of them shal be the Center of it Lastly setting one foot of your Compasses in the center of your Plaine and extending the other foot to the vppermost point describe in your Instrument an Arch of a Circle which if you haue done well will exactly fall vpon the ends of the Horizontal intersection of your Plaine That Arch shall represent your Plaine inclining vpon the lower side which is toward the Horizon or Limbe but reclining vpon the vpper-side which is toward the Zenith or Center And so either side shall shew in what hower lines the Sunne at some time of the yeare will shine vpon it that in delineating a Dyall thereon it may not be combered

substile The due situation of those distances vpon the Dyal plaine dependeth on the true placing of the Meridian or 12 a clocke line for that being truely described all the rest will be easie enough First therefore I will shew the manner how the Meridian or 12 a clocke line is to be described Take in your Dyal some point for the Center where you shall thinke fit and through it draw a line parallel to the plaine of the Horizon Crosse it in the Center with a perpendicular line And hauing opened your compasses to the length of the Semidiameter of your paper Instrument describe on the Center a Circle equall to the innermost Limbe thereof In which Circle the line parallel to the Horizon is for the Horizontal intersection and the other for the line perpendicular to it and the Circle it selfe representeth the plaine Marke therein the East and West sides of the Plaine with E and W. In the Horizontal and in al North and South direct Plaines both vpright and stooping and in all vpright declining plaines the Meridian is perpendicular to the Line parallel to the Horizon In North ●●clining and South reclining plaines the Meridian is to bee drawne on that side of the Dyal plaine either East or West into which the declination is But in North inclining and South reclining on the contrary side And if the plaine bee Northerne the Meridian shall be aboue the Line parallel to the Horizon and if the plaine be Southerne it shal be vnder it And if the contrary Pole be eleuated it shall be drawne through the Center into the opposite Quadrant of the Circle in your Dyal plaine Lastly in a direct East and West plaine both inclining and reclining the Meridian is the same with the line parallel to the Horizon Wherefore with your compasses take the distance in the limbe of your Instrument from the next Horizontall point vnto the marke of the Meridian and measure it vpon the Circle of the Dyal plaine in that part and on that side according as in consideration of the eleuated Pole and of the qualitie of the Plaine was shewed to be agreeable And at the end of that arch through the Center draw a line for the Meridian Againe with your compasses take the distances in the limbe of your Instrument betweene the marke of the Meridian and the markes of all the houre Lines seuerally and setting them vpon the Circle of the Dyal plaine orderly from the Meridian the Forenoone houres on the West side of it and the Afternoone houres on the East side at the end of euery one of those arches draw the houre Lines and distinguish them with their proper figures accordingly Lastly fasten the stile in the Center so that it may hang perpendicular vnto the plaine in the Substile at the iust height And because the stile in every Dyal is vnderstood to be a segment of the Axis of the world which is a line imagined to passe from the North to the South Pole through the Center of the earth the stile being rightly placed shal still with the end point towards the eleuated Pole that is vpward from the Center if the North Pole be eleuated or downeward from the Center if the South Pole be eleuated XXIX Vse The making of all manner of plaine Dyals not hauing Centers If the plaine represented on the Instrument as was taught before in the XXV and XXVI Vses cut the Pole of the Aequinoctiall it is an horary Circle either one of them which are drawne in the Instrument or falling betweene some two of them and the Dyall plaine it selfe shall not crosse the axis of the world but lye parallel to it without any Angle of eleuation And therefore such a Dyal can haue no Center But the stile the substile and all the houre lines shall be parallel one to another Every such Plaine represented on the Instrument Either First it is the Meridian of the place the Horizontall intersection whereof is the 12 a clocke Line drawne from North to South and the Dyall made thereon is a direct East or West vpright Dyal In which the substile is distant from the Line in the Circle of the Dyall plaine parallel to the Horizon with an Arch equall to the eleuation of the Pole and vpward toward the Pole And is also the 6 a clocke line in your Dyal The rest of the houre lines are thus described Draw through the substile in any point a long Line at right Angles that line shall bee the Aequinoctial intersection vsually called the Contingent line And taking a conuenient distance for the stile to hang parallel over the substile according to the greatnesse of your Dyall plaine measure it vpon the substile from the Aequinoctiall intersection and vpon the end of that measure describe halfe a Circle for the Aequinoctiall it selfe Diuide each Quadrant thereof from the substile into 6 equall parts or houres Then applying a Ruler to the Center and to euery one of those diuisions seuerally where in euery place the Ruler shall cut the long line of Aequinoctiall intersection make pricks and through those pricks draw the houre lines parallel to the Substile or 6 a clocke line distinguishing so many of them as bee needfull with their figures that is all the Forenoone houres on the East plaine and all the Afternoone houres on the West plaine But in these Dyals there is no 12 a clock line it being infinitly distant from the Substile Lastly hang the stile directly over the Substile and parallel to it at the distance formerly taken And thus are your East and West Dyals finished Or Secondly it is the sixt houre Circle the Horizontal intersection whereof is the line of East and West and the Dyal made thereon is direct North inclining or South reclining with an Arch equal to the complement of the height of the Pole And the parallel to the Horizon is the Aequinoctial intersection and the line perpendicular to it is the 12 a clocke line and also the Substile The rest of the houre Lines from 7 a clocke in the morning to 5 in the euening are thus described Take a conuenient distance for the Stile from the Substile measuring it vpon the Substile from the Aequinoctial intersection and on the end of that space describe the Semicircle of the Aequinoctial to bee diuided on both sides of the Substile into 6 houres through euery one of which out of the Center a Ruler being applyed at the points of the seuerall intersections of the Ruler with the Aequinoctial intersection draw the houre Lines parallel to the Substile or 12 a clocke Line distinguishing them with their figures namely 11 10 9 8 7 on the West side and 1 2 3 4 5 on the East side but in these Dyals there is no sixe a clocke Line it being infinitly distant from the Substile Lastly hang the Stile directly over the Substile and parallel to it at the distance formerly taken Or Thirdly it is North inclining or South

houre line from noone and of the Meridian of the Aequinoctiall from the substile But if the substile bee vpon one side of the Meridian and the houre on the other side it shall be equall to the summe thereof Then say As the Radius is to the Sine of the height of the stile aboue the substile So is the tang of the arch of the aequinoctiall betweene the substile and the houre line to the tang of the arch of the circle of your Dyall plaine betweene the substile and that lower line Or else you may without calculation Geometrically inscribe the houre lines in Dyals hauing centers for how to doe it in Dyals not hauing centers I haue already shewed in the XIX Vse thus Describe in your Dyall plaine a line for the stile at the same height or distance from the substile that the true stile ought to haue Take also in the substile as in reason you shall see fit a point and through it draw at right angles a long line for the contingent or Aequinoctiall intersection Againe from the same point let fall a perpendicular vnto the stile the length of this perpendicular is the nearest distance betweene that point and the stile and it is also the distance of the center of the Aequinoctiall from that point measure it therefore vpon the substile the contrary way from the center of the Dyall and hauing thus the center of the Aequinoctiall describe therevpon toward the contingent line one halfe of the Aequinoctiall circle which if the substile be the Meridian or 12 a clock line of your Dyall you must begin to diuide into houres at the substile But if the substile and Meridian of your Dyall be seuerall lines apply a ruler to the center of the Aequinoctial and to the intersection of the 12 a clock line with the contingent and there draw a line this line shal bee the Meridian of the Aequinoctial at which you must begin to diuide the Aequinoctiall circle into houres both wayes Then applying a ruler vnto the center of the Aequinoctial euery one of those diuisions where the ruler in euery place shal cut the contingent line there make a marke and lastly through euery one of those marks from the center of the Dyall draw the houre lines themselues And if in any houre line it shall happen that the ruler so applied will not reach to intersecat the contingent line you may thus help your selfe Which rule also may serue you to find the Meridian of the Aequinoctiall as often as the intersection of the Meridian of the Dyall with the contingent falleth without your paper or plaine Draw the houre line as farre as it will goe And take with your Compasses the distance of the intersection point of the contingent with the substile both from the center of the Dyal frō the center of the Aequinoctial And taking at all aduenture a point in the contingent line on that side in which the houre line is measure from that point on the contingent both those distances and at the ends of them both draw two lines parallel to the substile crossing the contingent Then applying a ruler to the point which you tooke at all aduenture and to the intersection of the parallel which hath the distance of that center whence the houre line giuen proceedeth with that houre line where the ruler shall cut the other parallel make a prick and measure the distance betweene that prick and the contingent vpon the former parallel on the other side of the contingent Lastly out of the proper center through the end of that measure draw a line which shall be that you desire An example of this Geometricall way of delineating th● houre lines you shall finde in the description of a South vpright Dyall declining 35 degr and reclining degr 41 min. 30. by considering whereof these rules will be found exceeding plainely set downe As also all the other rules and obseruations here deliuered to one that is any whit pregnant and ingenious will neede no other exemplification then the inspection of the instrument it selfe and of these seuerall Dyalls following FINIS A North Dyall declining Eastwards 35 deg reclining 41 deg 30 min. Latitude 51 deg 30 min. A Horizontall Dyall South direct vpright South direct inclining 24 deg North direct reclining 24 deg South direct reclining aequall to the complement of the poles height West direct vpright East direct reclining 32 deg South vpright declining Eastward 35 deg South declining Westward 76 degr reclining 48 deg South declining Westward 61 degr Reclining 21 deg ●½ min. The Translator to the Reader Gentle Reader by reason of my absence whilest this Booke was in the Presse it is no meruaile though some faults haue escaped which you will be pleased to amend thus Pag. 14 lin 14 2.0413927 Pag. 15. lin 1 the first terme of a progression Pag. 17 lin 17 the Antecedent arme Pag. 18 lin 19 terme giuen from Pag. 19 lin 11 in the fift circle Pag. 20 lin 19 lye hid As in this lin 20 D. Rat. mul ta 1 in R. lin 28 and Rat mul ta in R in α R in α Also in the aequations pag. 21 24 26 which haue a magnitude equall to a fraction the same magnitude together with the note of equality ought to be set right against the line that is betweene the Numerator and Denominator of the fraction as in these And And so of the rest Pag. 25 lin 20 arme at 71 ⌊ 382 Pag. 26 lin 16 Ratiocination Pag. 29 lin 29. number of figures Pag. 35 lin 5 61 49 144 Pag. 36 lin 11 17 ⌊ 48 lin 14 17 ⌊ 48 · 3 ⌊ 26 + Pag. 44 lin 8 is a roofe line 26 therof Pag. 94 lin 21 a circle or 90 degrees Pag. 100 lin 3 and then the side DC Pag. 113 lin 7 delineation Pag. 127 lin 1 the Sunne goeth not Pag. 131 lin 19 in the paper Pag. 132 lin 10 to the tangent of the arch of Pag. 143 lin 3 in North reclining and South inclining Pag. 152 lin 30 North Dyall declining Eastward 35 deg