Selected quad for the lemma: end_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
end_n	draw_v	line_n	perpendicular_a	3,095	5	14.0786	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

shall bee The Semidiameters of hower Circles And that you may know for what Circle every Center and Semidiameter serueth you shall note every fift line from the beginning with the figures 5 10 15 20 c. Set vnder the second Parallel vnto 90 which will fall vpon the longer perpendicular that so you may readily find the Center and Semidiameter of any Circle required Againe divide the first 45 degrees of the Quadrant in the middest and applying your Ruler to the Center and to every one of those halfe divisions where in each place the Ruler cutteth the first Parallel or tangent line make a prick So shall you haue vpon the tangent line betweene the shorter perpendicular and the midlemost line 45 a third scale which is The scale of 90 degrees for the graduating of the Semidiameter of your instrument on the paper In which you shall also distinguish every fift degree with figures set vnder the tangent line Having thus prepared your paper of scales with lines neatly and exactly drawne keepe it by you to haue it still in a readinesse for the making and vsing of the Instrument in paper The making whereof is thus Take with your compasses the Semidiameter of the Quadrant in your paper of scales and therewith vpon a peice of strong Dutch paper Describe the Horizontall Circle which you shall cut into two Semicircles with a Meri●ian line drawne through the Center divide them into Quadrants in the points of East and West and each Q●adr●nt into 90 degrees to be marked with figures iust as is done in the Instrument Then with your compasses take the eleuation of the pole vpon the scale of degrees in your paper set it vpon the Meridian line from the Center which way you please that shall bee the intersection of the Aequinoctiall with the Meridian Also reckon the complement of the height of the Pole vpon the scale of Centers of Arches and with your compasses take the distance from the end therof to the Center the same shall bee the Semidiameter of the Aequinoctiall to bee drawne from the East point of the Horizon through the point of intersection with the Meridian vnto the West point Againe take with your compasses vpon the scale of degrees in your paper the complement of the height of the Pole and set it vpon the Meridian on the other side of the Center from the Aequinoctiall there shall bee the Pole of the Aequinoctiall or of the World in which all the houre lines shall crosse one another Moreover with your compasses take the distance betweene the Center and the second Parallel in your paper which is the Semidiameter of the sixt houre Circle and set it on the Meridian from the Pole beyond the Aequinoctiall that shall be the Center of the sixt houre Circle vpon which you may draw the same Circle from the East point of the Horizon through the Pole to the West point Then through the center of the sixt houre Circle erect a line perpendicular to the Meridian extending it infinit●y on both sides of the Meridian and in that line both wayes pricke downe the Centers of the horary Circles out of the scale in paper And lastly opening your compasses from every one of those Centers vnto the Pole severally describe all the horary Circles or at least every fift of them and so is your paper instrument perfectly finished The vse of this instrument on paper is that lines and arches may bee designed vpon it with a fine pennicell of blacke lead and afterward be wiped out againe Wherefore it will bee needfull for him that will vse this instrument to all the purposes thereof to get a good paire of large compasses with three points ●ne sharpe another for inke a third for blacke Lead And I suppose it would doe well to fast ●n over your instrument a peice of thinne oyled paper through which the lineaments may be conspicuous and vpon it to trace such lines and arches as you haue occasion to vse that so your instrument may be kept cleane and last longer For as much as in delineating the horary Circles which are within 30 degrees of the Meridian the Semidiameters will be too long for your compasses you may in that streight thus helpe your selfe First say As the Radius is to the Sine of the Eleuation of the Pole So is the tang of the distance of any Horary Circle from the Meridian suppose 25 or 20 or 15 or 10 or 5 degrees to the arch of the Horizon betweene the Meridian and that horary Circle Reckon this distance on the limbe of your Instrument from both ends of the Meridian and marke it Thus doe for the 25th 20th 15th 10th and 5th horary Circle on both sides of each end of the Meridian Then in any peice of cleane paper through the middest of the longer way draw aline and toward one end which I call the upper ende crosse the same with a perpendicular line exactly equall to the Diameter of your Instrument the point of Intersection being the center Take with your Compasses out of the paper of scales the semidiameter of 60. degr which you may well doe for an ordinary instrument and setting one foot on either end of the Diameter that point wherein the other foot shall cutt the first long line make your Center and thereon draw an Arch through both ends of the Diameter and cutting the vpper part of the first long line this Arch is equall to that horary Circle which is distant from the Meridian 30 degrees the complement of 60. Divide each halfe of the Diameter into 3 equall parts with 4 points and from every of those points vnto the Arch draw lines parallel to the first long line And having divided euery one of those fiue parallel lines intercepted betweene the Diameter and the Arch into 6 equall parts for the 6 times 5 degrees which remaine to the Meridian draw through those divisions from the ends of the Diameter with a smooth and even hand the Arches 25 20 15 10 and 5. Those Arches you may transferre from the paper to your instrument in this manner Rubb the backe side of the paper against the Arches with fine powder of blacke lead then applying the paper with Arches to your instrument that the ends of the Diameter may exactly fall vpon the two opposite markes in the limbe of your instrument which serue for the horary Circle that you would draw either 25 20 15 10 or 5 trace over that Arch with the point of any hard peice of wood sharpened and the blacke lead on the backe side will vpon the instrument leaue the print of that Arch. XXV Vse To set an vpright Wall or plaine vpon the instrument and to find how many houres the Sunne shall shine thereon at some time of the yeare The situation of Wals or Plaines is considered either in respect of the Meridian or of the Horizon And vnto both it is either perpendicular or oblique or parallel The

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

reclining and also declining in which As the tangent of the Eleuation of the Pole is to the Radius So is the Sine of the compl of Declination to the tang of the compl of Inclination or Reclination The Plaine being set vpon the Instrument by the Arches of Declination and stooping thereof as hath beene taught in XXV ● Vse shall cut the pole of the Aequinoctiall Apply therefore a Ruler to the Pole of the plaine and to the Pole of the Aequinoctial and the point in which it cutteth the Limbe marke for the substile which is to bee transferred vnto the Circle of the Dyal plaine by taking the distance betweene that point and the next Horizontal intersection and setting it on that Circle from the line parallel to the Horizon vpward if the plaine be North or downe-ward if the plaine bee South and on that side which is contrary to the Declination The substile being thus sound draw a long line perpendicular to it for the Aequinoctial intersection And taking a conuenient distance for the stile from the substile measure it vpon the substile from the Aequinoctiall intersection and on the end of that space describe the Semicircle of the Aequinoctial Then looke in your Instrument how many degrees of the Aequinoctial are intercepted betweene the Meridian and the Arch representing your Plaine and reckoning the same number of degrees vpon the Aequinoctial of the Dyal plaine from the substile towards the side of Declination there make a marke for the Meridian point thereof in which you must begin to diuide the Aequinoctial semicircle into houres both wayes And that being diuided apply a ruler to the center and to euery one of the diuisions and at the points of the seuerall intersections of the ruler with the Aequinoctial intersection draw the houre lines parallel to the substile Set 12 at that houre line which was drawne at the intersection through the Meridian point of the Equinoctial and 11 10 9 8 c. on the West side and 1 2 3 4 c on the East side Lastly hang the stile directly ouer the substile and parallel to it at the distance formerly taken XXX Vse How by Sines and tangents to calculate the places of the Meridian and Substile and the height of the Stile aboue it and the distance of the Meridian of the Aequinoctiall from the Substile together with the places of houre lines both by calculation and also Geometrically I haue already taught the making of all manner of plaine Dyals most easily by the Instrument for the same height of the Pole But if any man either want an Instrument or else desireth greater exactnesse I will also here shew how to performe the same by calculation on the other side of the Instrument In a plaine erect Dyall declining As the Radius is to the Sine of doclination So is the tang of the compl of the Poles height to the tang of the distance of the substile from the Meridian In plaine Dyalls both declining and also inclining and reclining As the Radius is to the Sine of inclination reclination So is the tang of Declination to the tang of the compl of the distance of the Meridian from the line parallel to the horizon Againe As the Radius is to the Sine of the compl of Declination So is the tang of the compl of the Poles height to the tang of Base I. If the Dyall be South reclining or North inclining the summe of Base I and of the complement of Inclination or reclination shall be Base II. But if the Plaine be South inclining or North reclining the difference of Base I and of the complement of inclination or reclination shall bee Base II. Then say thirdly As the Sine of the compl of Base I is to the Sine of the compl or excesse of Base II So is the Sine of the height of the Poles to the Sine of the height of the stile aboue the substile Fourthly As the Sine of the compl of the height of the stile aboue the substile is to the Sine of Declination So is the Sine of the compl of the height of the Pole to the Sine of the compl of the distance of the substile from the line parallel to the horizon Againe As the Radius is to the Sine of the compl of Declination So is the Sine of the compl of the Poles height to the Sine of the height of the stile aboue the substile Thirdly As the Sine of the Poles height is to the Radius So is the tang of Declination to the tang of the Distance of the Meridian of the Aequinoctiall from the substile And this distance is euer lesse then 90 degrees In a plaine East and West Inclining and Reclining Dyall As the Radius is to the Sine of the compl of inclination reclination So is the tang of the height of the Pole to the tang of the distance of the substile from the Meridian Againe As the Radius is to the Sine of inclination reclination So is the Sine of the height of the Pole to the Sine of the height of the stile aboue the substile Thirdly As the Sine of the compl of the Poles height is to the Radius So is the tang of the compl of inclination reclination to the tang of the distance of the Meridian of the Aequinoctiall from the substile And this distance is euer greater then 90 degrees Fiftly As the Sine of the compl of the height of the stile aboue the substile is to the Sine of Declination So is the Sine of the compl of inclination reclination to the Sine of the distance of the Meridian of the Aequinoctiall from the substile And note that in South reclining and North inclining Plaines if Base II be lesse then a quadrant the contrary pole is eleuated aboue the Plaine And if Base II be equall to a quadrant the Plaine doth cut the Pole of the Aequinoctiall Now concerning the placing of the substile vpon the Dyall plaine as I haue already in the XXVIII Vse shewed for the Meridian Wee are to know First that the substile is to be drawne vpward from the line parallel to the horizon if the Plaine be Northerne or downeward from it if it bee Southerne Except in North reclining and South inclining Dyalls in which the Base I exceedeth the complement of inclination and reclination for in them it is quite contrary And secondly that the substile is to be drawne in the contrary side from the Declination But in North inclining and South reclining Dyalls in which the contrary Pole is eleuated the substile must be drawne through the center into the opposite quadrant of your Dyall circle Lastly the houre lines in all manner of plaine Dyalls are thus to be found If the substile and houre bee both on the same side of the Meridian the arch of the Aequinoctiall betweene the substile and the houre line shall bee equall to the difference of the two distances namely of the

with vnnecessary houre lines For Example suppose that the former Plaine which with the South declined Eastward 35 deg doe also incline 41 deg 30 min. Wherefore also with the North side it shall decline Westward 35 deg and Recline 41 deg 30 min. Describe this plaine vpon your Instrument with an Arch of a Circle found out as was taught last before And it will appeare that vpon the Inclining side shall be drawne all the houre lines from almost 4 in the morning to 4 in the afternoone And vpon the Reclining side shall bee drawne first 4 and 5 in the morning and then beginning at 0 a clocke all houres to Sunne set And note that in all Northerne plaines the North Pole is eleuated and in all Southerne plaines the South Pole is eleuated Except such North inclining and South reclining Plaines that in the Instrument fall below the North Pole betweene it and the Horizon For in them the contrary Pole is eleuated And also that a direct East and West plaine if it Recline hath the North Pole eleuated and if it Incline the South Pole XXVII Vse The Plaine being set vpon the Instrument to find the distances of the houre lines and Substile from the Horizontall Intersection And also the height of the Stile aboue the Substile Euery Dyall either hath a Center in which all the houre lines together with the Substile and Stile doe meet or else it hath no center so they are al parallel one to another If the Plaine being set vpon the Instrument cutteth the Pole of the Aequinoctiall that is the point in which all the horary Circles crosse the Meridian the Dyal to be drawne vpon that Plaine shall haue no Center But if it cutteth not the Pole the Dyal shall haue a Center And of these Dyals with Centers wee will first intreate as being most proper for the vse of the Instrument Behold therefore the Pole of your plaine heedily what horary Circle it falleth vpon Or if it fall betweene any two the distance of each being reasonably apportioned imagine a horary Circle passing through it Marke in what point that horary Circle either real or imagined doth cut the Plaine that same point shall bee the place of the Substile in your Plaine and the height of the Stile aboue it is the Arch of that horary Circle intercepted betwixt the Pole of the Aequinoctial and the point of the Substile noted in the Plaine Therefore applying a Ruler to the Pole of your Plaine carry it about vnto all the intersections of the Plaine with the houre Circles and the substile seuerally and where in euery place the Ruler shall cut the innermost Circle of the Limbe there make a visible marke For the arches of the Limbe intercepted betweene the Horizontall points of your Plaine and euery one of those marks shall be the distance intended to be sought But for the Horizontall plaine the ends of the houre Circles in the Limbe of your instrument doe giue the distance without any more adoe Concerning the height of the Stile aboue the Substile It is apparent by the instrument that in a Horizontal dyal the 6 a clocke line lyeth directly East and West and the Meridian perpendicular to it directly North and South And that the Meridian is the Substile And that the height of the Stile aboue the Substile is equall to the height of the Pole in that place It is also apparent that in all direct North and South Dyals the 6 a clocke Line is drawne parallel to the Horizon and the Meridian perpendicular to it And that the Meridian is the substile And that the height of the Stile aboue the substile if the Plaine bee vpright is equall to the complement of the height of the Pole But if it be North inclining or South Reclining it is equall to the difference of the height of the Pole and the Arch of Inclination or Reclination And if the Plaine bee North Reclining or South Inclining it is equall to the Summe of the complement of the height of the Pole and of the Arch of Inclination or Reclination And if the Plaine fall vpon the Aequinoctiall the stile shall stand vp perpendicular vpon it in the Center and the hower lines shall be drawne all at equall Angles viz. 15 degrees one from another In such Dyals as haue not the Meridian for the substile the height of the Stile aboue the Substile is thus sound by the Instrument It was shewed before that the height of the stile aboue the substile is the Arch of the horary Circle through the Pole of the Plaine intercepted betweene the pole of the Aequinoctiall and the point noted in the Plaine for the substile Therefore from the Horizontall points or intersections of that horary Circle reckon 90 degrees both wayes and thereto through the Center draw an obscure line in which line shall be both the Inclination of that horary Circle which is the distance of the intersection or vppermost point thereof from the Center and also the Pole Then with your compasses take that distance or Inclination and setting it vpon the scale of degrees in your paper see how many degrees it containeth vpon that scal● Againe vpon the same scale take with your compasses the complement of that inclination or distance which being set vpon the obscure line on the other side of the Center shall shew the Pole of that horary Circle Lastly applying a Ruler to the Pole of that horary Circle and both to the Pole of the Aequinoctiall and to the point of the substile in the Plaine seuerally marke where in both places the Ruler cutteth the innermost limbe of your instrument For the degrees of the limbe intercepted betwixt those markes shall be the height of the stile aboue the substile which was sought for And by this which hath beene taught you shall find that in an vpright Dyall declining as before 35 deg from the South into the East or from the North into the West the substile shall fall vpon that horary Circle which is about 3 deg after 9 a clock in the morning and the stile eleuated aboue the substile about 31 deg And also that in a South Dyall declining Eastward 35 deg and inclining 41 deg 30 min Or in a North declining Westward 35 deg and reclining 41 deg 30 min the substile shall fall vpon that horary Circle which is about 8 degrees after 6 a clock and the stile eleuated aboue the substile about 63 deg 30 minutes XXVIII Vse The making of all manner of plaine Dyals with Centers I haue already shewed how to find out in our Instrument the distances of all the houre lines and the substile from the Horizontal intersection Now the delineating of a Dyal is nothing else but to transferre those distances out of your Instrument into the Dyal plaine euery one in his due situation and then through them out of the Center to draw such houre lines as shall be of vse together with the

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