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.

THE SECOND PART OF THIS BOOKE. (Book 2)

Shewing the vse of the Second side of the Instrument, for the working of most questi∣ons, which may be performed by the Globe: And the declination of Dyals, vpon any kinde of Plaine. (Book 2)

VPon the second side of the Instrument, is deli∣neated the proiection of the vpper Hemisphaere vpon the plaine of the Horizon: The Horizon it selfe is vnderstood to bee the innermost circle of the limbe: and is diuided on both sides, from the points of East, and West into degrees, no∣ted with 10, 20, 30, &c. vnto 90. And the center of the Instrument is the Zenith, or Ʋertical point.

Within the Horizon, the middle straight line, or Dia∣meter pointing North and South, is the Meridian, or 12 a clock line: and the other shortarching lines, on both sides of it are the houre lines, distinguished accordingly by their

figures. These houre lines should indeede bee drawne through the whole plaine, crossing one another in the Pole of the world: but that the Instrument may be more faire, they are onely drawne short.

And because diuers excellent vses, doe require the to∣tall delineation of the houre circles, I haue in a seuerall pa∣per, inscribed intirely, both the houre lines, and also two other circles betweene them, containing euery one fiue degrees. (But if the Instrument were large enough to re∣ceiue them, it were best if euery degree had his circle: and so euery 15 circle should bee an houre line.) And of the parallels, there needes no more, but the Aequinoctial, and both the Tropics.

For as much as there will be great vse of this paper In∣strument; I haue in the 24 Vse shewed the manner of making it: so that any that 1•• ingenuous, and ready handed may himselfe delineate one sufficient enough to serue his turne, for any eleuation.

The two arches which crosse the houre lines meeting on both sides in the points of intersection of the sixe a clocke lines with the Horizon, are the two Semicircles of the Ecliptick, or Annuall circle of the Sunne: the vpper of which arches serueth for the Summer halfe yeare, and the lower for the Winter halfe yeare: and are therefore diuided in 365 dayes: which are also distinguished into 12 moneths with longer lines, hauing their names set downe: and into tenthes, and fifthes with shorter lines: and the rest of the dayes with pricks: as may plainly bee seene in the Instrument.

And this is for the ready finding out of the place of the Sunne euery day: and also for shewing of the Sunnes yearely motion: because by this motion the Sunne goeth round about the heauens in the compasse of a yeare, make∣ing

the foure parts, or seasons thereof. Namely the Spring, in that quarter of the Ecliptick which beginneth at the intersection on the West side of the Instrument, and is therefore called the Vernall intersection. Then the Sum∣mer in that quarter of the Ecliptick which beginneth with the intersection of the Meridian in the highest point next the Zenith. And after that Autumne in that quar∣ter of the Ecliptick, which beginneth at the intersection on the East side of the Instrument, and is therefore cal∣led the Autumnall intersection. And lastly, the Winter in that quarter of the Ecliptick, which beginneth at the in∣tersection with the Meridian, in the lowest point next the Horizon.

But besides this yearely motion, the Sunne hath a Di∣urnall or dayly motion, whereby it maketh day and night with all the diuersities, and inequalities thereof: which is expressed by those other circles drawne crosse the houre lines: the middlemost whereof being grosser then the rest, meeting with the Ecliptick in the points of the Vernall, and Autumnall intersections, is the Aequinoctiall: and the rest on both sides of it, are called the Parallels, or Diurnall arches of the Sunne: the two outermost where∣of are the Tropics, because in them the Sunne hath his furthest digression, or Declination from the Aequinocti∣all, which is degrees 23½: and thence beginneth againe to returne to the Aequinoctiall. The vpper of the two Tropics next the center (in this our Northerne Hemi∣sphaere) is the Tropick of Cancer: and the Sunne being in it, is highest into the North, making the longest day of Summer. And the lower next the Horizon, is called the Tropick of Capricorne; and the Sunne being in it, is lowest into the South, making the shortest day of Winter.

Betweene the two Tropics, and the Aequinocti∣all infinite such parallel circles are vnderstoode to bee

contained: for the Sunne is what point soeuer of the Ecliptick it is caried, describeth by his lation, a circle pa∣rallel to the Aequinoctial. Yet those parallels which are in the Instrument, though drawne but to euery second de∣gree of Declination, may be sufficient to direct the eye, in imagining and tracing out, through euery day of the whole yeare in the Ecliptick, a proper circle, which may be the Diurnall arch of the Sunne for that day. For vpon the right estimation of that imaginary parallel, doth the manifold vse of this Instrument especially rely: because the true place of the Sunne, all that day, is in some part, or point of that circle. Wherefore for the better concei∣uing, and bearing in minde thereof, euery fift parallel, is herein made a little grosser then the rest.

I Vse. And thus by the eye, and view only, to behold and comprehend the course of the Sunne, both for his Annuall and Diurnall motion, may be the first vse of this Instrument.

II Vse. To take the height of the Sun aboue the Horizon.

Set vp the pinne, (which is therefore made fit for the hole at the center) perpendicular in the center: and put the Indices on both sides, downe vpon the Meridian, that they with their waight, may not sway the Instrument any way as it hangeth: then with a threed put into the hole aboue in the handle, hang it perpendicularly, bearing the edge toward the Sunne, that the pinne may cast a sha∣dow, vpon the degrees in the limbe: for that degree which the shadow of the pinne cutteth in the limbe, is the height of the Sunne aboue the Horizon, at that present.

III. Vse. To find the Declination of the Sunne euery day.

Looke the day of the moneth proposed in the Eclip∣tic, and marke how many degrees the prick shewing that day, is distant from the Equinoctiall, either on the Sum∣mer, or Winter side, viz North, or South.

Example. I. What will the Declination of the Sunne be, vpon the 11^th day of August? Looke the 11^th day of August: and you shall find it in the sixt Circle a∣boue the Aequinoctiall: now because each Parallel, stan∣deth (as hath beene said before) for 2 degrees, the Sunne shall that day decline North-wards 12 degrees.

Example. II. What Declination hath the Sunne, vp∣on the 24^th day of March? Looke the 24^th day of March, and you shall find it, betweene the second, and third Northerne parallels, as it were an halfe and one fift part more of that distance from the second: reckon therefore 4 degrees for the two Circles, and one degree for the halfe space: so shall the Sunnes declination bee 5 degrees, and about one fift part of a degree Northward, that same day.

Example. III. What Declination hath the Sunne vpon the 13^th day of Nouember? Looke the 13^th day of Nouember, and you shall find it below the Equinoctiall, tenne parallels and about one quarter, which is 20 de∣grees, and an halfe South-wards. So much is the Decli∣nation. And according to these examples iudge of all the rest.

IIII. Vse. To find the Right as∣cension of the Sunne euery day.

Imagine an hower line through the day of the moneth giuen, and marke in what point it will cr••sse the

Aequinoctiall: then lay a Ruler, or a streight Scroule of paper, to the Pole of the world (noted in the Instru∣ment with P W) and that same point. For the Ruler shall in the innermost Circle of the limbe, of the South side, cut the Right ascension of the Sunne for that day, to be reckoned from the West, to the point of intersecti∣on, for the first, or vpper Semicircle of the Ecliptic: or from the East together with 180, for the second, or low∣er Semicircle of the Ecliptic.

V. Vse. To find the longitude of the Sunne, or in what degree of the Signe he is euery day.

The Pole of the first Semicircle of the Ecliptic is no∣ted P I. and the Pole of the second Semicircle is noted P I I. Lay a Ruler, or a streight Scroule of paper, to the day of the moneth, and the proper Pole of the Semicir∣cle of the Ecliptic, in which it is: for the Ruler shall in the innermost Circle of the limbe, on the South side, cut the degree of the Sunnes place in the Ecliptic, reckoning it in the same manner as you did in finding the Sunnes Right ascension: and the Arch thus found is called the longitude of the Sunne. which may bee expansed into signes, by reckoning on the limbe, from the West to South ♈, ♉, ♊, and from South to East ♋, ♌, ♍: then backe againe from East to South ♒, ♏, ♐; and lastly from South to West ♑ ♒, ♓, allowing 30 degrees, for each of those twelue signes.

VI. Vse. To find the Diurnall Arch, or Circle of the Sunnes course euery day.

The Sunne euery day by his motion (as hath beene said) describeth a Circle parallell to the Aequinoctiall, which is either one of the Circles in the Instrument, or

some-where betweene two of them. First then seeke 〈◊〉〈◊〉 of the moneth; and if it fall vpon one of those 〈◊〉〈◊〉, that is the Circle of the Sunnes course that same day: But if it fall betweene any two of those Pa∣rallel••, imagine in your minde, and estimate with your eye, another Parallel through that point, betweene those two Parallels, keeping still the same distance from each of them.

As in the first of the three former Examples, The cir∣cle of the Sunnes course, vpon the 11^th day of August, shall be the very sixt Parallel aboue the Aequinoctiall to∣wards the Center.

In Example II. The Circle of the Sunnes course vpon the 24^th day of March, shall bee an imaginary Circle be∣tweene the second, and third Parallels, still keeping an halfe of that space, and one fift part more of the rest from the second.

In Example III. The Circle of the Sunnes course vp∣on the 13^th day of Nouember, shall be an imaginary Cir∣cle, betweene the tenth, and eleauenth Parallels, below the Aequinoctiall, still keeping one quarter of that space from the tenth.

VII. Vse. To find the Rising and Setting of the Sunne euery day.

Seeke out (as was last shewed) the imaginary Circle or Parallel of the Sunnes course, for that day, and marke the point where it meeteth with the Horizon, both on the East and West sides thereof, for that is the very point of the Sunnes rising, and setting that same day, and the hower lines which are on both sides of it, by proportio∣ning the distance reasonably, according to 15 minuts, for the quarter of the hower, will shew the hower of the Sunnes rising, on the East side, and the Sunnes setting on the West side.

VIII. Vse. To know the reason, and manner, of the Increasing, and De∣creasing of the dayes and nights throughout the whole yeare.

When the Sunne is in the Aequinoctiall, it riseth, and setteth at 6 a Clock, for in the instrument, the intersecti∣on of the Aequinoctiall, and the Ecliptic with the Hori∣zon, is in the 6 a clock Circle on both sides. But if the Sunne bee out of the Aequinoctiall, declining toward the North, the intersections of the Parallel of the Sunne with the Horizon, is before 6 in the Morning, and after 6 in the Euening: and the diurnall Arch of the Sunne, greater then 12 howers; and so much more great, the greater the Northerne Declination is. Againe if the Sun be declining toward the South, the intersections of the Parallel of the Sunne with the Horizon, is after 6 in the Morning, and before 6 in the Euening; and the diurnall Arch lesser then 12 howers; and by so much lesser, the greater the Southerne Declination is.

And in those places of the Ecliptic in which the Sun most speedily changeth his Declination, the length also of the day is most altered, and where the Ecliptic goeth most parallel to the Aequinoctiall changing the Declina∣tion but little, the length of the day also is but little altered.

As for example, when the Sunne is neere vnto the Aequinoctiall, on both sides, the dayes increase, and also decrease suddenly and apace: because in those places, the Ecliptic inclineth to the Aequinoctiall in a manner like a streight line, making sensible declination. Againe when the Sunne is neere his greatest Declination, as in the height of the Summer, and the depth of Winter, the

dayes keepe for a good time, as it were at one stay, be∣cause in those places the Ecliptic is in a manner parallel to the Aequinoctiall, scarce altering the declination: and because in those two times of the yeare, the Sunne stan∣deth as it were still, at one declination; they are called the Summer Solstice, and the Winter Solstice. And in the meane spaces, the neerer every place is to the Aequinocti∣all, the greater is the diversitie of dayes.

Wherefore we may hereby plainely see, that the com∣mon receiued opinion, that in every moneth, the dayes doe equally increase, is erroneous.

Also wee may see that in Parallels equally distant from the Aequinoctiall, the day on the one side, is equall to the night on the other side.

IX. Vse. To find the Ascensionall difference of the Sunne every day.

Seeke out the time of the Suns Rising, or Setting that same day (by the VII Vse) and see how much it differeth from sixe a clocke, then conuert the same difference into degrees (as was taught in 1 Part. Chap. 12. Sect. 1.) by multiplying the howers with their decimall parts, by 15. And so haue you the Ascensionall difference for that day.

X. Vse. To find out the Oblique ascension of the Sunne every day.

Seeke out the Sunnes Right ascension (by the IIII Vse) and the Ascensionall difference (by the IX Vse:) And if the Sunne be in the first Semicircle of the Ecliptic, Sub∣duct the Ascensionall difference, out of the Right ascen∣sion: But if the Sunne be in the second Semicircle of the Ecliptic, adde the Ascensionall difference to the Right as∣cension: and you shall haue the oblique ascension.

XI. Vse. To find how farre the Sunne riseth, and setteth, from the true East and West points, which is called the Sunnes Amplitude or∣tiue, and occasiue.

Seeke out (as was shewed in the VI. Vse) the imagina∣ry Circle, or Parallel of the Sunnes course, and the points of that Circle in the Horizon, on the East, and West sides, cutteth the degree of the Amplitude Ortiue, and occasiue.

XII. Vse. To find the length of every day and night.

Double the hower of the Sunnes setting, and you shall haue the length of the day; and double the hower of the Sunnes rising, and you shall haue the length of the night.

XIII. Vse To find the true place of the Sunne, vpon the Instrument, which answereth to the point, wherein the Sunne is in the heauens: and is the ground of all the questions follow∣ing.

Take with your Instrument the height of the Sunne, and reckon it on the moueable Index, or Labell: and then moue the said Labell, till you find the height of the Sunne, exactly to fall vpon the Parallel of the Sunne for that day, on the East side if it bee in the Fore-noone, and on the West side, if it bee in the After-noone; the point of in∣tersection, where the Index, or Labell crosseth the Pa∣rallel,

in that point of the Sunnes altitude, shall bee the true place of the Sunne on the Instrument.

XIIII. Vse. To find the Hower of the day.

The true place of the Sunne on the Instrument (found out as was last shewed) sheweth among the hower lines the true hower of the day.

XV. Vse. To find out the Azumith or verticall Circle in which the Sunne is, or the Horizontall distance of the Sunne from the Meridian.

The Index or Labell fastned at the Center, is a mouea∣ble Azumith: apply therefore the edge thereof, vnto the true place of the Sunne on the instrument (found out as was shewed by XIII. Vse.) And marke what point of the Horizon, or Limbe, the same edge of the Labell cutteth; reckon how many degrees of the H••rizon, are intercep∣ted betweene that point, and the Meridian line, or South point, either on the East, or West side: and that Arch shall be the Horizontall distance sought for, whereby is shewed the Azumith of the Sunne at that instant: and consequently the Angle which the verticall Circle, or A∣zumith of the Sunne maketh with the Meridian.

XVI. Vse. The Azumith of the Sunne being knowne, to find out the Altitude of the Sunne, and the Hower of the day.

Set the edge of the Labell to the Azumith given, and marke in what point the same edge crosseth the Parallel of the Sunne for that day: that point of intersection

sheweth the height of the Sunne aboue the Horizon, vp∣on the Labell: and also it sheweth the hower of the day among the hower lines.

XVII. Vse. To find at what hower the Sunne commeth to be full East, or West every day in Summer.

Apply the edge of the Labell, vnto the East, or West points of the Limbe, and marke in what point, the said edge cutteth the Parallel of the Sunne for that day, for that same point among the hower lines, shall shew the time of the Sunnes comming to be full East, or West in that day, and likewise of what altitude the Sunne will be aboue the Horizon, at that time of his being full East, or West.

XVIII. Vse. To find the height of the Sunne at high Noone every day, and likewise at every other hower.

Marke in what point the Parallel of the Sunne for that day, cutteth the line of that hower, for which you would know the Sunnes altitude: And vnto that point of inter∣section, apply the edge of the moueable Labell, or Index: and thereon shall you find, the very degree of the Sunnes altitude, at that hower.

By this XVIII Vse, and by the XVI, are made the Quadrants, described by Gemma Frizius, Munster, Clauins, Mr. Gunter, and others: and also all manner of Rings, Cylinders, & inuumerable other topicall Instruments, for the finding out of the hower, and other like conclusi∣ons. And likewise the reason, of finding the hower of the day, by a mans shadow, or by the shadow of any Gnomon, set vp perpendicular to the Horizon, or else parallel to it.

XIX. Vse. To find out the Meridian line, and the points of the compasse without a Magneticall needle, yea more exactly then with a needle.

Take the height of the Sunne, by the shadow of the pinne: and apply the same height, reckoned on the Index, or Labell to the parallel of the Sunne for that day, where∣by you haue the true place of the Sunne, in the instru∣ment, as hath beene shewed in the XIII. Vse. Then kee∣ping both the Labell 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 in∣strument, 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 Me∣ridian 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 be∣fore Sun rising, and after Sunne setting, the instrument re∣presenteth 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 de∣grees the Sunne is vnder the Ho∣rizon, at any time of the night.

Seeke the declination of the Sunne for the day propo∣sed: 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 propo∣sed: then set the Index, or Labell to that point of the Pa∣rallel, 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 declina∣tion with that which the day of the moneth sheweth: then moue about the Labell, vntill the said Parallel cut∣teth the edge thereof in the 18^th 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 Declina∣tion 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 ther∣on, holding the board parallel to the plaine of the Hori∣zon: 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 di∣stant 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 Cir∣cle, for this distance is equall to the Radius thereof.) A∣gaine 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 Semidia∣meter 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 pa∣per; 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 de∣lineate your instrument, describe a quadrant of a Circle: and on the point where it meeteth with the shorter per∣pendicular, 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 di∣stance 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 cen∣ters 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 instru∣ment 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 Instru∣ment 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 ther∣of 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 de∣grees 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.

Nextly vnto the height of the Pole both adde, and al∣so subduct 23½ degrees: and with your compasses take both those Arches in the scale of degrees: and set them in the Meridian from the Center, one falling beyond the Aequinoctiall, and the other short of it: those shall be the intersections of the two Tropics with the Meridian. Seek

[illustration] diagram

also by Sect: 8, Chapt: 12 of the first part, the Ampli∣tude ortiue of the Sun, having 23½ degrees declination: and reckon it being found, in the Horizontall Circle from the East and West points both wayes: those are the points of intersection of the Tropics with the Horizon on both sides: and so having three points in each Tropic, you may easily through them drawe the Circles.

Moreover with your compasses, take the distance be∣tweene 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 Aequi∣noctiall: 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. Where∣fore it will bee needfull for him that will vse this instru∣ment, 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 con∣spicuous: 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 Semi∣diameters 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 Instru∣ment from both ends of the Meridian, and marke it. Thus doe for the 25^th, 20^th, 15^th, 10^th, and 5^th horary Circle on both sides of each end of the Meridian.

Then in any peice of cleane paper, through the mid∣dest of the longer way, draw aline: and toward one end which (I call the upper ende) crosse the same with a per∣pendicular line exactly equall to the Diameter of your In∣strument, 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 ei∣ther 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 Diame∣ter, 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.

[illustration] diagram

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 Meridi∣an, 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 in∣strument, that the ends of the Diameter may exactly fall vpon the two opposite markes, in the limbe of your in∣strument, 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 sharpe∣ned: 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 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 backe∣ward,

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 Ze∣nith is reclining.

The Plaine oblique to the Meridian is that which stan∣deth not directly North, or South, but declineth one side into the East, and the other into the West: and is there∣fore 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 loo∣keth directly East, or West; and accordingly, hath his denomination, whether it bee Vpright, Inclining, or Re∣clining.

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 eue∣ry vpright Plaine is vnderstood to passe through the Ze∣nith, 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 represen∣ted in the instrument by a line drawne through the Cen∣ter 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 o∣pen to all the houres betweene sixe in the morning and sixe in the evening. And that about London, the Nor∣therne side, onely in the Summer enioyeth the Sunne from his rising till after seven in the morning: and from be∣fore 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 Hori∣zontall 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 Horizon∣tall 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 Sou∣therne 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 Sou∣therne Plaine declining Eastward 35 degrees. But also vp∣on the North side shall represent a Northerne Plaine de∣clining 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 after∣noone. 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 24^th 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 ther∣of 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 In∣strument: 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 in∣tersection is to be set thereon, as if it were vpright; to∣gether 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 Inclina∣tion, or Reclination tendeth: the same shall bee the vp∣permost 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 pa∣per: 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 cen∣ter 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 to∣ward 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 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 de∣cline 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 ele∣uated: and if it Incline, the South Pole.

XXVII. Vse. The Plaine being set vpon the Instrument, to find the distan∣ces 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 be∣ing 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 Horizon∣tall 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 Hori∣zon, 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 perpen∣dicular 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 sub∣stile, 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 Cir∣cle 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 ob∣scure 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 man∣ner of plaine Dyals with Centers.

I haue already shewed how to find out in our Instru∣ment 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, toge∣ther with 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 Meri∣dian, 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 Instru∣ment, describe on the Center a Circle equall to the inner∣most 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 con∣trary Pole be eleuated, it shall be drawne through the Cen∣ter into the opposite Quadrant of the Circle in your Dy∣al plaine.

Lastly in a direct East and West plaine, both inclining and reclining, the Meridian is the same with the line pa∣rallel 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 or∣derly 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 fi∣gures 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 Cen∣ter 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 be∣tweene 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 Hori∣zontall intersection whereof is the 12 a clocke Line drawne from North to South: and the Dyall made there∣on, is a direct East, or West vpright Dyal: In which the substile is distant from the Line, in the Circle of the Dy∣all 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 conueni∣ent 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 inter∣section: 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 in∣tersection, 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 Horizon∣tal 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 Hori∣zon is the Aequinoctial intersection: and the line perpen∣dicular 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, measu∣ring it vpon the Substile from the Aequinoctial inter∣section: and on the end of that space describe the Semi∣circle 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 Aequi∣noctial 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 recli∣ning, 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 Ar∣ches of Declination and stooping thereof (as hath beene taught in XXV •• Vse) shall cut the pole of the Aequinocti∣all. 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 Declina∣tion. The substile being thus sound, draw a long line per∣pendicular to it, for the Aequinoctial intersection. And ta∣king 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 num∣ber 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 ru∣ler 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 paral∣lel 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 Meri∣dian of the Aequinoctiall from the Sub∣stile; together with the places of houre lines, both by calculation, and also Geome∣trically.

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 Instru∣ment, 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 Me∣ridian, 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 in∣clining, 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 Aequi∣noctiall.

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 sub∣stile 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 diffe∣rence of the two distances, namely of 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 Me∣ridian,

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 shew∣ed 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 in∣tersection. Againe from the same point let fall a perpen∣dicular 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 ha∣uing thus the center of the Aequinoctiall, describe there∣vpon toward the contingent line one halfe of the Aequi∣noctiall circle: which if the substile be the Meridian, or 12 a clock line of your Dyall, you must begin to diuide in∣to 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 inter∣section 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 pa∣rallel, 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 de∣liuered, to one that is any whit pregnant and ingenious, will neede no other exemplification, then the inspection of the instru∣ment it selfe, and of these seuerall Dyalls following

FINIS.