CHAP. I. Of certain terms of Geometry, necessary to be known of the unlearned, before the proceeding in this Art of Dialling.

BEing intended in this Treatise of Dialling, to proceed by Geometricall proportion: I have thought fit, first, to declare unto you the meaning of some terms of Geo­metry which are necessary for the unlearn­ed to know before they enter into this Art of Dialling.

CHAP. II. To a line given, to draw a parallel line, at any distance required.

SUppose the line given to be A B, unto which line it is required to draw a parallel line. First, open your Com­passes to the distance required, then set one foot in the end A, and with the other strike an arch line, on that side the given line

whereunto the parallel line is to be drawn, as the arch line C, this being done, draw the like arch line upon the end B, as the arch line D, and by the convexity of those two arch lines C and D, draw the line C D, which shall be parallel to the given line, as was required.

CHAP. III. To perform the former proposition at a distance required, and by a point limited.

SUppose the line given to be D E, unto which line it is required to draw a parallel line, at the distance, and by the point F.

First there­fore, place one foot of the compasses in the point F, from whence take the shortest extention to the [Page 10] line D E, as F E, at which distance, place one foot of the compasses in the end D, and with the other, strike the arch line G, by the convexity of which arch line, and the limited point F, draw the line F G, which is parallel to the given line D E, as was required.

CHAP. IIII. The manner how to raise a perpendicular line, from the middle of a line given.

LEt the line given be A B; and let C be a point therein, whereon it is required to raise a perpendicular. First therefore, open the compasses to any convenient distance,

& setting one foot in the point C, and with the other foot mark on either side thereof, the equall distances C A, and C B: then opening your compasses to any convenient wi­der distance, with one foote in the point A, with the other, strike the arch line E over the point C, then with the same distance of your compasses, set one foot in B, and with the other draw the arch line F, crossing the arch E in the point D, from which point D, draw the line D C, which line is perpendicular unto the given line A B, from the point C, as was required.

CHAP. V. To let a Perpendicular fall from a point assigned, unto the middle of a line given.

LEt the line given whereupon you would have a perpen­dicular let fall, be the line D E F, and the point assign­ed to be the point C, from whence you would have a per­pendicular let

fall upon the given line D E F. First, set one foote of your compas­ses in the point C, and opening your compasses to any conveni­ent distance, so that it be more then the distance C E, make an arch of a circle with the other foot, so that it may cut the line D E F twice, that is, at I and G: then finde the middle between those two intetsections, which will be in the point E, from which point E, draw the line C E, which is the perpendicular which was desired to be let fall from the given point C, unto the middle of the given line D E F.

CHAP. VI. To raise a Perpendicular upon the end of a line given.

SUppose the line whereupon you would have a perpen­dicular to be raised, be the line B C, & from the point B a [Page 12] perpendicular is to be raised. First, open your Compasses unto any convenient distance, which here we suppose to be the distance B E, and set one foot of your compasses in

B, with the other draw the arch ED, then this distance being kept, set one foot of your com­passes in the point E, and with the o­ther make a mark in the former arch E D, as at D, stil kee­ping the same di­stance, set one foot in the point D, and with the other draw the arch line F o­ver the given point B: now laying a ruler upon the two points E and D, see where it crosseth the arch line F, which will be at F, from which point F, draw the line F B, which shall be a perpendicular line unto the given line B C, raised from the end B, as was required.

CHAP. VII. To let a perpendicular fall from a point assigned, unto the end of a line given.

LEt the line D E be given, unto which it is required to let a perpendicular fall from the assigned point A, unto the end D. First, from the assigned point A, draw a line unto any part of the given line D E, which may be the line A B C, then finde the middle of the line A C, which will [Page 13] be at B, place therefore one foot of your compasses in the point B, and extend the other unto A or C, with which distance draw the

Semicircle A D C, so shall it cut the given line D E in the point D, from which point D, draw the line A D, which shall be the perpendicular let fall from the assign­ed point A unto the end D of the given line D E, as was required.

CHAP. VIII. Certain Definitions Astronomicall, meet to be understood of the unlearned, before the proceeding in this Art of Dialling.

IN the former Chapter I have shewed the meaning of some terms of Geometry, which be most helpfull unto this art of Dialling, with the drawing of a parallel line at any distance, or by a point assigned; so likewise have I shew­ed the manner either how to raise or let fall a perpendicular either from or unto any part of a line given. So likewise now I think it will not be unnecessary for to shew unto the unlearned, the meaning of some of the most usefullest terms in Astronomie, and most fitting this art of Dialling.

CHAP. IX. Of the six greater Circles.

CHAP. X. Of the foure lesser Circles.

CHAP. I. The description of the Scale, whereby this work may be performed

THis Scale for this work, needs to be divi­ded but into two parts, the first whereof may be a Scale of equall divisions of 16 in an inch, and may serve for ordinary mea­sure. The second part of the Scale may be a single Chord of a Circle, or a Chord of 90, and is divided into 90 unequall divisions, representing the 90 degrees of the Quadrant, and are numbered with 10, 20, 30, 40, &c. unto 90. This Chord is in use to measure any part or arch of a Circle, not surmounting 90 [Page 24] degrees, the number of these degrees from 1 unto 60, is called the Radius of the Scale, upon which distance all

Circles are to be drawn, whereupon 60 of these degrees are the semidiameter of any Circle that is drawn upon that Radius.

CHAP. II. How speedily with Rule and Compasse, to make an angle containing any degrees assigned, or to get the degrees of any angle made.

FIrst, therefore to make an angle of any quantity, open your Compasses to the Radius of our Scale, and [Page 26] setting one foot thereof in the point A, with the other foot describe the arch B C, then draw the line A B, then ope­ning your Compasses to so many degrees upon your line of Chords, as you would lay down, which here we will suppose to be 40 degrees, and setting one foot in B, with the other make a mark in the arch B C, as at C, from which point C, draw the line C A, which shall make the angle B A C, containing 40 degrees, as was required.

And if you desire to finde the quantity of an angle, open the Compasses to the Radius of your Scale, & set one foot thereof in the point A, and with the other describe the arch B C, then taking the distance betwixt B and C (that is, where the two legs and the arch line crosseth) and ap­ply it unto the line of Chords, and there it will shew you the number of degrees contained in that angle, which here will be found to be 40 degrees.

CHAP. III. To finde the altitude of the Sun by the shadow of a Gnomen set perpendicular to the Horizon.

FFirst, draw the line A B, then opening your Compasses to the Radius of your Scale, set one foot in the end A, [Page 27] and with the other describe the arch B C D, then ope­ning your Compasses unto the whole 90 degrees, with one foot in B, with the other marke the arch B C D in the point D, from which point D, draw the line D A, which shall be perpendicular unto the line A B, and make the Quadrant A B

C D, then suppose the height of your Gnomon or sub­stance yeilding sha­dow, to be the line A E, which here we will suppose to be 12 foot, therefore take 12 of your e­quall divisions from your Scale, as here I have taken 12 quar­ters for this our pur­pose, and set them down from A to E, and draw the line E F parallel to A B, then suppose the length of the shadow to be 9 foot, for this 9 foot must I take 9 of the same divisions as I did be­fore, and place them from E to G, by which point G, draw the line A G C, from the center A through the point G, until it cutteth the arch B F C D in the point C, so shall the arch B C be the height of the Sun desired, which in this example will be found to be 53 degrees 8 minutes, the thing desired.

CHAP. IIII. To finde the altitude of the Sun by the shadow of a Gnomon standing at right angles with any perpendicular wall, in such manner that it may lie parallel unto the Horizon.

FIrst, draw your Quadrant A B C D, as is taught in the last Chapter, & place the length of your Gnomon from A to E, which here we will suppose to be 12 foot as before in the last Chapter; then draw the line E F parallel to A B, then suppose the length of the shadow to be 9 foot as before, this I place from E to G, by which point G, draw the line A G C, as was formerly done in the last Chapter, by which we have proceeded thus far, but as in the last Chapter the arch B C was the height of the Sun desired; so by this Chapter the arch C D shall be the height of the Sun, which being applyed unto your Scale, will give you 36 deg. 52 min. for the height of the Sun desired.

CHAP. V. The Almicanter, or height of the Sun being given, to finde the length of the right shadow.

BY right shadow is meant the shadow of any staffe, post, steeple, or any Gnomon whatsoever, that stand­eth at right angles with the Horizon, the one end thereof respecting the Zenith of the place, and the other the Nadir. First, therefore according unto the third Chapter, describe the Quadrant A B D, then suppose the height of [Page 29] your Gnomon, or substance yeilding shadow, to be 12 foot, as in the former Chapter: this doe I set down from A to E, and

from the point E draw the line E F parallel to A B, then set the Almicanter (which here we will suppose to be 53 de. and 8 min. as it was found by the third Chapter) from B unto C, from which point C, draw the line C A, cutting the line E F in the point G; so shall E G be the length of the right shadow desired, which being taken betwixt your Compasses, and applyed unto your Scale, will give you 9 of those divisions, whereof A E was 12, which here doth signifie 9 foot.

CHAP. VI. The Almicanter, or height of the Sun being given, to finde the length of the contrary shadow.

BY the contrary shadow is understood the length of any shadow that is made by a staffe or Gnomon standing at right angles against any perpendicular wall, in such a man­ner that it may lie parallel unto the Horizon; the length of the contrary shadow doth increase as the Sun riseth in [Page 30] height: whereas, contrariwise, the right shadow doth de­crease in length as the Sun doth increase in height. There­fore the way to finde out the length of the Versed shadow is as followeth. First draw your Quadrant, as is taught in the third Chapter, now supposing the length of your Gno­mon to be 12 foot, place it from A to E, likewise from E draw the line E F parallel to A B, as before: now sup­posing the height of the Sun to be 36 deg. 52 min. take it from your Scale, and place it from D to C, from which point C, draw the line C A, cutting the line E F in the point G, so shall G E be the length of the contrary sha­dow, which here will be found to be 9 foot, the thing desired.

CHAP. VII. Having the distance of the Sun from the next Equinoctiall point, to finde his Declination.

FIrst, draw the line A B, then upon the end A, raise the perpendicular A D; then opening your Compas­ses to the Radius of the Scale, place one foot in the center A, and with the other draw the Quadrant B C D: then supposing the Sun to be either in the 29 degree of Taurus, or in the first degree of Leo, both which points are 59 degrees distant from the next Equinoctiall point Aries. Or if the Sun shall be in the 29 degree of Scorpio, or in the first degree of Aquarius, both which are also 59 de­grees distant from the Equinoctiall point Libra, therefore [Page 31] take 59 degrees from your Scale, and place it from B to C, and draw the line C A, then place the greatest decli­nation of the Sun from B unto F, which is 23 degrees 30 minutes, then

fixing one foot of your Compasses in the point F, with the other take the neerest distance un­to the line A B, which you may doe by opening or shut­ing of your Com­passes, still turning them to and fro, till the moving point thereof doe only touch the line A B: this distance being kept, set one foot of your Compasses in the point A, and with the other make a mark in the line A C, as at E, from which point E, take the neerest extent unto the line A B, this distance be­twixt your Compasses being kept, fix one foot in the arch B C D, moving it either upwards or downwards, still keeping it directly in the arch line, untill by moving the other foot to and fro, you finde it to touch the line A B and no more, so shall the fixed foot rest in the point G, which shall be the Declination of the Sun accounted from B, which in this example will be found to be about 20. degrees, the thing desired.

CHAP. VIII. The Declination of the Sun, and the quarter of the Ecliptique which he possesseth being given, to finde his true place.

LEt the declination given be 20 degrees, and the quar­ter that he possesseth, be betwixt the head of Aries and Cancer, first draw the Quadrant A D E F, then set the greatest declination of the Sun upon the Chord from D unto B, which is 23 degrees and 30 minutes, then from the point B take the shortest extent unto the line A D, this distance being kept, set one foot in the point A, and with the other describe the small Quadrant G H I, then set the

declination of the Sun (which in this example is 20 de­grees) from D un­to C, from which point C, take the shortest extent un­to the line A D, this distance being kept, place one foot in the arch line G H I, after such a manner, that the other foot being turned about, may but onely touch the line A D, so shall the fixed foot stay upon the point H, through which point H, draw the line A H E, cutting the arch D F in the point E: so shall the arch D E be the distance of the Sun from the head [Page 33] of Aries, which here will be found to be 59 degrees, so that the Sun doth hereby appear to be in 29 degrees of Taurus, at such time as he doth possesse that quarter of the Ecliptique, betwixt the head of Aries and Cancer.

CHAP. IX. Having the Latitude of the place, and the distance of the Sun from the next Equinoctiall point, to finde his Amplitude.

FIrst, make the Quadrant A B C D, then take from your Scale 37 deg. 30 min. which here we will suppose to be the complement of the Latitude, and place it from B unto E, then taking the neerest distance betwixt the point E, and the line A B, with one foot set in A, with the other draw the arch F G H, then place the Suns greatest declina­tion from B unto I, from which point I, take the neerest extent unto the

line A B, which distance being kept, place one foot of your com­passes in the arch line F G H, so that the moving foot may but on­ly touch the line A B at the shortest extent, so shall the fixed foot rest in the arch line F G H at G, through [Page 34] which point G, draw the line A G C, then supposing the Sun to be in the 29 degree of Taurus, that is 59 degrees distant from the next Equinoctiall point, take 59 degrees

from your Scale and place them from B to L, frō which point L, take the neerest distance unto the line A B, with this distance, set­ing one foot in the point A, with the other make a mark in the line A C, as at O, frō which point O, take the shortest extent unto the line A B, this distance being kept, fixe one foot of your Compasses in the arch B C D, in such a man­ner, that the moving foot thereof may but only touch the line A B, so shall the fixed foot rest in the point R, which is the Amplitude counted from B, and will be found in this example to be 34 deg. 9 min.

CHAP. X. Having the Declination and Amplitude of the Sun, to finde the height of the Pole.

FIrst, make the Quadrant A B C D, then supposing the Amplitude to be 34 deg. 9 min. (as it was found by the last Chapter) take it from your Scale, and place it from B [Page 35] to E, then taking the neerest extent from the point E un­to the line A B, set one foot of your Compasses in the center A, and with the other draw the arch G H I, then supposing the Sun to have 20 degrees of Declination, take them from your scale, and place them from B unto F, from which point F, take the shortest extent unto the line A B: this distance being kept, fix one foot of your Compasses in the arch line G H I, so that the other foot may but touch the line A B at the neerest extent, so shall the fixed foot rest at the point H, through which point H, draw the line A H C, cut­ing

the arch B C D in the point C, so shall the arch B C be the height of the E­quinoctiall, and the complement thereof which is the arch C D, shall be the ele­vation of the Pole above the Horizon, or the distance of the Equinoctiall from the Zenith, which in this example will be found to be 52 degrees 30 minutes, the thing desired.

CHAP. XI. Having the Latitude of the place, and the Declination of the Sunne, to finde his Amplitude.

FIrst, make the Quadrant A B C D, then supposing the Latitude to be 52 degrees 30 minutes, take it [Page 36] from your Scale, and place it from D to E, or (which is all one) if you place the complement thereof from B to E, from which point E take the neerest extent unto the line A B, with this distance, setting one foot of your Compas­ses in the center A, with the other describe the arch F G H, then supposing the declination of the Sun to be 20 de­grees, place them from B to O, from which point O, take the shortest extent unto the line A B, which distance be­ing kept, fixe one foot in the arch F G H, so that the other

may but only touch the line A B at the nee­rest distance, so shall the fixed foot rest at the point G, thro­ugh w^ch point G draw the line A G C, cutting the arch B C D in the point C, so shall the arch B C be the Amplitude desired, which in this example will be found to be 34 de­grees 9 minutes, as before in the 9 Chapter.

CHAP. XII. The elevation of the Pole, and the Amplitude of the Sun being given, to finde his Declination.

FIrst, draw the Quadrant A B C D, then supposing the Amplitude to be 34 deg. 9 min. place it from B to E, and from the point E take the neerest extent unto the line A B, with which distance, setting one foot of your Compasses in the center A, describe the arch G H I: then supposing the Lati­tude

to be 52 deg. 30 min. place it from D to C, from which point C, draw the line C H A, cutting the arch G H I in the point H, from which point H, take the nee­rest extent unto the line A B, with this distance, fixing one foot of your Com­passes in the arch B C D, as the other may but only touch the line A B at the neerest extent, so shall the fixed foot rest at the point F, which shall be 20 degrees distance from the point B, the declination of the Sun desired.

CHAP. XIII. Having the Latitude of the place, and the Declination of the Sun, to finde his height in the Verticall Circle, or when he shall come to be due East or West.

FIrst, draw the Quadrant A B C D, then supposing the Latitude to be 52 deg. 30 min, take it from your Scale and place it from B to C, then taking the neerest extent from the point C unto the line A B, with one foot set in the center A, with the other describe the arch G H I:

then supposing the Sun to have 20 deg. of decli­nation, place it from B to O, from which point O take the shortest extent unto the line A B, with this distance, fixing one foot in the arch G H I, so that the other may but only touch the line A B at the nee­rest extent, so shall the fixed foot rest in the point H, through which point H draw the line A H E, cutting the arch B C D in the point E, so shall the arch B E be the height of the Sun when he commeth to be due East or West, which being taken between your Compasses, and [Page 39] applyed unto your Scale, will give you 25 deg. 32 min. the thing desired in this example.

CHAP. XIV. Having the Latitude of the place, and the Declination of the Sun, to finde the time when the Sunne cometh to be due East or West.

FIrst, draw the Quadrant A B C D, then placing the Latitude of the place (which here we will suppose to be 52 deg. 30 min.) from B to C, and draw the line C E, then with the neerest distance from the point C unto the line A B, which is the line C E, setting one foot of your Compasses in the

center A, with the other draw the arch G H I: then supposing the de­clination of the Sun to be 20 de­grees, place it frō B to F, from which point F, lay a rule unto the center A, and where it cros­seth the line C E, there make a mark as at O, then with the distance O E, fix one foot of your Compasses in the arch G H I, after such manner, that the other foot may but only touch the line A B at the neerest extent: [Page 40] So shall the fixed foot stay in the point H, through which point H draw the line A H N, so shall D N be the quan­tity of time from the Meridian, when the Sun commeth to be due East or West, which in degrees will here be found to be 73 deg. 30 min. and these converted into time (by allowing 15 degrees to an houre, and four minutes for a degree) will make foure houres and 54 minutes of an houre, that is, either at 4 a clock and 55 minutes in the af­ternoon, or at 7 a clock and 5 min. in the morning.

CHAP. XV. Having the Latitude of the place, and the Declination of the Sun, to finde what altitude the Sun shall have at the houre of six.

FIrst, draw the Quadrant A B C D, then supposing the Latitude of the place to be 52 deg. 30 min. place it from

B to C, and from the point C, take the shortest extent unto the line A B, with this distance setting one foot in the center A, with the other draw the arch G H I, then supposing the de­clination of the Sun to be 20 deg. place it from B to E, and draw the [Page 41] line A H E, cutting the arch G H E in the point H, from which point H take the shortest extent unto the line A B, this distance being kept, fix one foot of your Compasses in the arch B C D, in such sort that the other may but only touch the line A B, so shall the fixed foot rest in the point O, whose distance from the point B shall be the al­titude of the Sun at the houre of six, which in this exam­ple will be found to be 15 degrees 44 minutes the thing desired.

CHAP. XVI. Having the Latitude of the place, and the declination of the Sun, to finde what Azimuth the Sun shall have at the houre of six.

FIrst, draw the Quadrant A B C D, then supposing the Latitude to be 52 deg, 30 min. place it from D to C, and draw the line A C, then supposing the Declination to be 20 deg, place

it from B to E, frō which point E draw the line E G, paral­lel to A B, untill it cutteth the line A D in the point G, and with the di­stance A G, de­scribe the arch G H I, cutting the line A C in the point H, through [Page 42] which point H draw the line O H P, parallel to A B, then taking the length of the line G E betwixt your Compas­ses, place it upon the line P O from P unto R, through which point R draw the line A R M, cutting the arch B C D in M, so shall the arch B M be the Azimuth from the East or West, which is here found to be 12 degrees 30 minutes.

CHAP. XVII. The Latitude of the place, the Almicanter, and Declination of the Sun being given, to finde the Azimuth.

FIrst, draw the Quadrant A B C D, then supposing the Latitude to be 52 deg. 30 min. setting it from B to C, draw the line A C, then supposing the declination of the

Sun to be 11 deg. 30 min. North­ward, set it from B to E, from which point E, take the neerest extent un­to the line A B, and with this di­stance, fixing one foot in the line A C, so as the other may but only touch the line A B, make the mark F in the line A C: then supposing the height of the Sun to be 30 deg. 45 min. place it from B to G, from which [Page 43] point G, take the neerest extent unto the line A D, and lay it down from A to N, then from the aforesaid point G, take the shortest extent unto the line A B, and place it from A to H, in the line A C, then take the distance F H betwixt your Compasses,Here note, that if the Sun had been so low that the point H bad fallen betwixt the cen­ter A and the point F, then should the arch D L have shewed the A­zimuth from the North part of the Meridian. and fix one foot in the line A C, so as the other may but touch the line A D, so shall the Compasses stay in the point O, from whence take the shortest ex­tent unto the line A B, with which distance, setting one foot of your Compasses in the point N, with the other foot describe the arch I, by rhe convexity of which arch and the point A, draw the line A L, cutting the arch B C D in the point L, so shall the arch B L be the Azimuth from the East or West, and the arch L D the Azimuth from the South, which in this example will be found to be 66 deg. 43 min. the true Azimuth from the South: this is to be understood when the Sun hath North Declination.

But if the Sun hath South Declination, then draw the following Quadrant A B C D, and set 52 deg. 30 min. from B to C, for the elevation of the Pole, and draw the line C A, then supposing the Sun to have 11 deg. 30 min. of South declination, place it from B to E, and from the point E, take the neerest extent unto the line A B, with which distance, fix one foot in the line A C, so as the o­ther foot may but only touch the line A D, and where the fixed foot so resteth, there make a mark, as at F: then place the height of the Sun, which here we will suppose to be 13 deg. 20 min. from B to G, from which point G, take [Page 44] the neerest extent unto the line A D, and place it from A unto N in the line A B, then take the neerest extent from the former point G unto the line A B, with this distance fix one foot in the line A C, so as the other may but only

touch the line A D, and where the fixed foot so resteth, there make a mark, as at O, frō which point O take the shortest extent unto the line A B, and place it from F to H, then with the distance HA, setting one foot in the point N, with the other describe the arch line I, by the convexity of which arch line, and the center point A, draw the line A L, cutting the arch B C D in the point L, so shall the arch D L be the Azimuth of the Sun from the South, which here will be found to be 49 deg. 49 min. the thing desired.

CHAP. XVIII. The Latitude of the place, the Declination of the Sun, and the Altitude of the Sun being given, to finde the houre of the day.

FIrst, draw the Quadrant A B C D, then supposing the Latitude to be 52 deg. 20 min. place it from B unto C, and draw the line A C, then place the declination of the [Page 45] Sun (which here let be 11 deg. 30 min. South) from B un­to E, from which point E, take the neerest extent unto the line A B, this distance place from A to F in the line A C, through which point F draw the line P F O parallel to A B, then from the aforesaid point E, take the shortest extent unto the line A D, which lay down from A to R in the line A B: then supposing the altitude of the Sun to be 15 deg. 24 min.

place it from B un­to H, from which point H take the shortest extent un­to the line A B, & place it in the line A D from D unto I, and take the di­stance A I, with which distance fix one foot in the line A C so as the other may but just touch the line A D, and where the fixed foot shall rest, make a mark as at N, then with the distance A N, place one foot in R, with the other draw the arch L, by the convexity of which arch line and the center A, draw the line A G, so shall the arch B G give the time from the houre of six, and D G the time from noon, which in this example will be found to be 45 deg. which in time makes 3 houres; so that in the Latitude of 52 deg. 20 min. the Sun having 11 deg. 30 min. of South declination, & the altitude in the morning to be 15 deg. 24 min. it will appear to be 9 a clock; but in the afternoon it would have been 3 a clock.

But if the declination had been North, then the distance from the point H to the line A B, should have been placed from the center A towards the point I, and the distance I P taken instead of A I, as by the next figure I will make more plaine.

First, draw the Quadrant A B C D as before, then place the Latitude 52 deg. 20 min. from B to C, and draw the line A C, then let the declination be 11 deg. 30 min. as before, place it from B to E, from which point E take the neerest distance unto the line A B, and place it from A to F in the line A C, and through the point F, draw the line O P parallel to A B, then from the aforesaid point E, take the shortest extent unto the line A D, and place it in the line A B from A unto R: then the altitude of the Sun being observed in the morning to be 42 degrees 33 min.

I place it from B to H, from which point H take the neerest extent un­to the line A B. Now seeing the declination of the Sun is supposed to be Northward, therefore place this last distance from the center A unto I, and take the distance P I, with which di­stance fix one foot in the line A C, so as the other may but only touch the line A D, and where the fixed foot shall [Page 47] so rest, there make a mark as at N, then with the distance A N, place one foot in the point R, and with the other draw the arch line L, by the convexity of which arch line and the center A, draw the line A G, cutting the arch B C D in the point G, so shall the arch B C give the quan­tity of time from the houre of six, and the arch D G the quantity of time from the Meridian, which in this exam­ple will be found to be about 30 degrees, that is in time two houres; so that the observation being before noon as here, it will be 10 a clock, if it had been in the afternoon it would have been two a clock.

And here note, that if the altitude of the Sun had been so small that the point I had falne betwixt the center A and the point P (which is the altitude of the Sun at the houre of six) then should that part of the arch B C D towards B give the quantity of time, either before six in the morning, or after six in the evening.

CHAP. XIX. Having the Azimuth, the Suns Altitude, and the Declination, to finde the houre of the day.

FIrst, draw the Quadrant A B C D, then supposing the Sun to have 11 deg. 30 min. declination, place it from B to E, and from the point E take the shortest ex­tent unto the line A D, with which distance, place one foot in the center A, and with the other describe the arch G H I, then let the Azimuth be 66 deg. 43 min. as it was found by the former part of the 17 Chapter, which place from B to F, and from the point F take the neerest extent unto the line A B, which distance place from G to H, in [Page 48] the arch G H I, through which point H draw the line A H C, then the altitude of the Sun being 30 deg. 45 min. place it from B to L, and from the point K take the nee­rest

extent unto the line A D, with which distance, set­ting one foot in the center A, with the other describe the arch NR, then with the distance NR, fix one foot in the arch line B C D, so as the other may but only touch the line A B, so shall the fixed foot rest in the point M, and the arch B M shall shew the houre from the Meridian, which will be found in this example to be 53 deg. 40 min. that is three houres, and somthing better then 34 min. from the Meridian.

CHAP. XX. Having the houre of the day, the Suns altitude, and the declination, to finde the Azimuth.

FIrst, make the Quadrant A B C D, then supposing the declination of the Sun to be 11 deg. 30 min. North as before, place it from B to E, then suppose the altitude of the Sun to be 30 deg. 45 min. place it from B to L, from which point L, take the shortest extent unto the line [Page 49] A D, with this distance, setting one foot in the center A, describe the arch N R, now let the arch for the houre be 53 deg. 40 min. as it was found by the last Chapter, this set from B unto M, and from the point M take the shortest extent unto the line

A B, and place it from N to R, in the arch N R, and by the point R, draw the line A R C, then from the point E take the neerest ex­tent unto the line A D, with which di­stance upon the center A, draw the arch G H, then with the distance G H, fix one foot in the arch B C D, so that the other may but touch the line A B, then will the fixed foot rest in the point F, and the arch BF wil shew the Azimuth from the South, which in this example will be found to be 66 deg. 43 min. the thing desired.

CHAP. XXI. Having the Latitude of the place, and the Declination of the Sun, to finde the Ascensionall difference.

FIrst, draw the Quadrant A B C D, then place the Latitude (which here let be 52 deg. 30 min.) from D to C, & draw the line COP parallel to AD, then with the [Page 50] distance C P, upon the center A, describe the arch G H I, then place the declination, being 20 degrees, from B to F, then lay your rule from the center A upon the point F,

and draw F O, cut­ting C P in the point O, through which point O draw the line O H R, cutting the arch G H I in the point H, through which point H draw the line A H E, cutting the arch B C D in E, so shall the arch B E be the diffe­rence ascensionall, and will be found in this example to be 28 deg. 19 min. which resolved into time doth give one houre, and somthing better then 53 min. for the difference betwixt the Suns rising or setting, and the houre of six, according to the time of the yeare.

CHAP. XXII. Having the Declination of the Sun to finde the Right Ascension.

FIrst, describe the Quadrant A B C D, then place the greatest declination of the Sun from B to E, and draw the line E P parallel to A D, and with the distance E P, with one foot in A, describe the arch G H I, then set the [Page 51] Declination of the Sun given 20 degrees, from B unto F, and laying your Rule upon the center A and the point F, draw the line F O

cutting E P in O, through which point O draw the line O H R parallel to A B, cutting the arch G H I in H, and through the point H draw the line A H C, cutting the arch B C D in the point C, so shal B C be 56 deg. 50 min. the right As­cension desired.

CHAP. XXIII. Having the Right Ascension of the Sun or Star, together with the difference of their Ascensions, to finde the Oblique Ascension.

THe right Ascension of any point of the heavens being known, the difference of the Ascension is either to be added thereunto, or else to be substracted from it, accor­ding as the Sun or Star is scituated in the Northern or Southern Signes. As for example: If the Sun be in any of these six Northern Signes Aries, Taurus, Gemini, Cancer, Leo, or Virgo, then the difference of the Ascensions is to be substracted from the right Ascension, and the remainder [Page 52] is the Oblique Ascension; therefore let the fourth degree of Gemini be given, the Right Ascension whereof is found to be 62 degrees, or 4 houres, and 8 min. and the diffe­rence of Ascension (where the Pole is elevated 52 deg. 30 min.) is 30 deg. 3 min. or in time, 2 houres and som­thing better, which being taken from the Right Ascension, leaves 2 houres and 8 min. or 32 deg. 5 min. which is the Oblique Ascension of the Sun in the fourth degree of Ge­mini. But if the Sun be upon the South side of the E­quinoctiall, either in Libra, Scorpio, Sagitarius, Capricornus, Aquarius, or Pisces, then the difference of the Ascensions is to be added unto the right Ascension, and the sum of them both will be the Oblique Ascension. As suppose the fourth degree of Sagitarius to be given, the Right As­cension whereof is found to be 242 degrees, or 16 houres 8 min. and the difference of Ascensions is 30 deg. 3 min. or 2 houres, which being added unto the Right Ascen­sion, doth make 18 houres 8 min. or in degrees, 272 deg. 3 min. which is the Oblique Ascension of the Sun in the fourth degree of Sagitarius. But if you would finde the Oblique Descension, you must work directly contrary to these Rules given.

CHAP. XXIV. How to finde the Altitude of the Sun without Instrument.

IN the third Chapter of this Book it is shewed how to finde the Altitude of the Sun by a Gnomon set perpen­dicular to the Horizon, but seeing the ground is so unlevel it is not so ready for this our purpose, and perhaps some [Page 53] may have occasion to finde the altitude of the Sun, and thereby the Azimuth or houre of the day, according to the 17 or 18 Chapters, and yet may be unprovided of In­struments to perform the same, or at least may be absent from them, therefore it will not be un-needfull to shew the finding of the same without the Gnomon or other Instru­ment.

Take therefore a Trencher, or any simple boards end, of what fashion soever, such as you can get, make thereon two pricks, as A and B, then prick in a pin, naile, or short wire in one of

the points, as at A, where­upon hang a threed with a plūmet, then lift up this board toward the Sun, till the shadow of the pin at A, come directly on the point B, and direct­ly where the threed then falleth, there make a mark as at E, under the threed, then with your Rule and Compasse draw the lines A B and A E, and finde the angle B A E (by the second Chapter) for that is the complement of the altitude of the Sun: or, when you have drawn A B and A E, you may make the Quadrant B A F by the third Chap­ter, and then the angle E A F shall be your altitude de­sired.

CHAP. XXV. How to finde out the Latitude of a place, or the Poles elevation above the Horizon, by the Sunne.

SEeing that throughout this Book, the Latitude of the place is supposed to be known, when as every one per­haps cannot tell which way to finde it out, therefore it will not be un-needfull to shew how it may be readily at­tained, sufficiently for our purpose.

First, therefore you must get the Meridian Altitude, which you may doe by observing diligently about noon a little before, and a little after, still observing untill you perceive the Sun to begin to fall again, then marking what was his greatest altitude, will serve for this our present purpose.

Having gotten the Meridian altitude by this, and the Declination by the seventh Chapter, you may finde the Latitude of the place, or the elevation of the Pole above the Horizon after this manner. If the Sun hath North Declination, then substract the Declination out of the Meridian altitude, and the remainder shall be the height of the Equinoctiall. But if the Sun hath South Decli­nation, then adde the Declination to the Meridian altitude, so shall the sum of them give the altitude of the Equinocti­all, which being taken out of the Quadrant or 90 degrees, leaveth the Latitude of your place, or the elevation of the Pole above your Horizon.

As for example, upon the first day of May 1650, the [Page 55] Meridian altitude of the Sun being observed to be 55 deg. 35 min. upon which day I finde the Suns place to be in 20 deg. 48 min. of Taurus, and the declination 18 deg. 00 min. and because the Declination is North, I sub­stract 18 deg. 00 min. out of the Meridian altitude 55 deg. 35 min. and there remains 37 deg. 35 min. the height of the Equinoctiall, and this taken out of 90 degrees, leaveth 52 deg. 25 min. for the Latitude of Thuring.

But it may be required somtimes for you to make a Di­all for a Town or Countrey whose Latitude you know not, neither can come thither conveniently to observe it. Here is therefore added a Table shewing the Latitude of the most principall Cities and Towns in England, so that being required to make a Diall for any of those places, you need but look in your Table, and there you have the La­titude thereof. But if the Town you seeke be not in the Table, looke what Town in the Table lies neere unto it, and make your Diall to that Latitude, which will occasi­on little difference.

CHAP. I How to examine a Plane for a Horizontall Diall.

FOrasmuch as it is necessary be­fore the drawing of any Diall, to know how your plane is al­ready placed, or how it ought afterwards to be placed; it is therefore expedient to shew how it may be attained unto without the help of a Quadrant, (or any such like Instrument) which for this purpose is very usefull.

First, take any board that hath one streight side, and an inch or more from the streight side, draw a line parallel thereto, about the middle of which line erect a perpendi­cular line, and at the center, where these two lines meet, cut out a hollow piece from the edge of the parallel line for a plummet to hang in: then if your plane seem to be levell with the horizon, you may try it by applying the streight side of your board thereunto, and holding the perpendicular line upright, and holding a threed and

plummet in your hand, so as the plummet may have free play in the hole; for then if the threed shall fall on the perpendicular line, which way soever you turn the board, it is an horizontall plane.

As for example, let the figure A B C D be a plane supposed to stand levell with the Horizon, and for to try the same, I take the simple boarde GOH, ha­ving one streight side, as G H, then drawing a pa­rallel thereto, I crosse it at right angles with the per­pendicular O E, and at the point of intersection I cut out a little bit as at E, for the plummet to play in, then applying the side G H to the plane, with holding the perpendicular O E upright, and holding a threed with a plummet to play in the hole E, and [Page 59] finding the threed to fall directly on the perpendicular E O, which way soever I turn the board, I therefore con­clude it to be a horizontall plane.

CHAP. II. Of the trying of Planes, whether they be Erect or Inclining, and to finde the quantity of Inclination.

FOr the distinguishing of planes, because their inclinati­on and declination may be diverse, we will consider three lines belonging to every plane: the first is the Ho­rizontall line; the second the perpendicular line, crossing the horizontall at right angles; the third is the Axis of the plane, crossing both the horizontall line, and his per­pendicular, and the plane it selfe, at right angles; the ex­tremity of which Axis may be called, the pole of the planes horizontall line.

The perpendicular line doth help to finde the inclinati­on, the horizontall line with his Axis to finde the Decli­nation, and the pole of the planes horizontall line, to give denomination unto the plane.

When the plane standeth upright, pointing directly into the Zenith, it maketh right angles with the Horizon, and is therefore called an erect plane, and a plumb line drawn thereon is called a Verticall line; as in this figure, the plane G H L I is erect, & the line H I is the Verticall line.

Now for the trying of this plane, if you apply the streight side of your board vertically thereto, as here you see done in the figure C, and either hanging a threed and plummet in the point M, or holding up a threed and plummet with your hand, you finde the threed [Page 60] to fall directly on the parallel line M N it is an erect plane, but if the threed will crosse the line M N, it is no erect plane, but inclineth to the Horizon.

And if you finde your plane to be erect, you may by applying your board thereto, with the threed and plum­met falling on the parallel line M N, draw the verticall line H I by the edge of your board, the verticall line being drawn, you may crosse it at right angles with the line G L, which shall be levell with your Horizon, and therefore called the horizontall line of the plane.

If the plane shall be found to incline to the horizon, you may finde out the inclination after this manner.

Apply your board to the plane, as you see here by the figure B, in the plane F K E H, then holding up a threed and plummet that it may fall upon the perpendicular line O E, and turning about your board, till the streight side thereof lie close with the plane, and the threed fall on the perpendicular line O E, so the line drawn by the streight side of the board, shall be an horizontall line, which here in this figure will be the line F E.

This being done, crosse the horizontall line at right an­gles with the perpendicular H K, then set the streight side of your board upon the line H K, which is perpendicular to the horizontall line, with holding the board up right, and holding up a threed and plummet, so that it may have free course to play by the side of the board, which it may have by letting the plummer fall within the hole, a little below X in the figure A: the threed thus hanging, mark two points directly under it, the one at X where the threed crosseth the parallel line on the board, the other at T, at the upper side thereof, and so by drawing the line T X, you have the angle T X V, which is called the angle of Recli­nation, [Page 61] and it is the angle contained betwixt the plane and the verticall line passing from the Zenith to the Nadir, the complement of which angle is the inclination of the plane, and it is the angle that the plane maketh with the Horizon, the thing here desired.

By what is said here of finding the inclination of the upper face, the inclination of the under face may soon be had, seeing they are both of one quantity in themselves, therefore, if you apply the streight side of your board to [Page 62] the perpendicular line of the under face, and hang the threed and plummet in any part of the parallel line, the angle that is made by that parallel line and the threed, shall be the complement of the angle of the inclination of the plane to the Horizon.

CHAP. III. To finde out the Declination of a Plane.

THe Declination of a plane is alwayes reckoned in the Horizon, and it is the angle contained between the line of East & West, & the horizontall line upon the plane.

For the finding out of this Declination, first, take any board that hath but one streight side, and draw a line pa­rallel thereto, as was done in the first Chapter, and having drawn an horizontall line upon your plane, apply the streight side of your board thereunto, holding it parallel to the Horizon, as in the figure of the last Chapter, where the board D is applyed to the horizontall line G L, then the Sun shining upon the board, hold out a threed and plummet, so as the threed being verticall, the shadow of the threed may crosse the parallel line S P upon the board, in which shadow make two points, the one where the sha­dow crosseth the parallel, as at P, the other about R, so have you the angle S P R, which is made between the horizontall line of the plane, and the Azimuth wherein the Sun is at the time of observation: at this same instant, or as neere as may be, must you take the Altitude of the Sun; these two being done diligently, wil help you to the planes Declination as followeth.

First, describe the Circle B C D E, which shall repre­sent the Horizontall Circle, and draw the diameter B A C, representing the horizontall line of the plane, in the last chapter set out by the line G L, then having found S P R [Page 63] in the last Chapter (which is the angle made by the hori­zontall line of the plane, and the Azimuth wherein the Sun was at that time of observation) to be about 24 deg. 10 min. I place it here from B the West end of the hori­zontall line to G

Southwards, be­cause the angle was taken at that end of the line, at which instant the altitude of the Sun (by the 24 Chapter of the second Book) being found to be 13 deg. 20 m, having 11 deg. 30 min. South Declination, it will, by the 17 Chapter of the said second Book, be made to appear, that the Azimuth of the Sun is 40 deg. 11 min. from the West toward the South, and therefore the West point is 40 deg. 11 min. from the Sun towards the North; Now taking this 40 deg. 11 min. and placing them from G the place of the Sun at the time of observation, to H Northwards, you shall have the true West point, and if you draw the line H R, it shall represent both East and West, & crossing the line H R at right angles, in the center A, you shall have the line D E for the North and South; & if you crosse the horizontall line B A C in the point A, at right angles with the line S N, it shall be the Axis of the plane, the two poles whereof S and N, shall be the poles of the planes horizontall line.

Now the angle of declination here required, is the an­gle B A H, or E A S, for look how much the horizontall line of the plane declineth from the line of East and West, so much doth the poles of the planes horizontall line de­cline from the North and South towards either East or West, either of which angles in this example will be found to be about 16 deg. the declination of the plane from the South point E, Eastward.

CHAP. IV. How to draw the Meridian line upon an Horizontall plane, the Sun shining thereon.

IF your plane be levell with the Horizon, describe there­on a Circle as B C D E, then holding up a threed and plummet, so as the threed being verticall, the shadow there­of may fall upon the center A, and draw the line of shadow

C E, then take the altitude of the Sun at the same instant (or as neere as may be) and by the 17 Chapter of the former Book get the Azimuth of the sun, which let be (as in that example it was found) 66 deg. 43 min. from the [Page 65] South towards the East, this 66 deg. 43 min. I place from E the point of the shadow Southwards to B, and draw the line B A D, which is the meridian line desired.

CHAP. V. Of making the Equinoctiall Diall.

AN Equinoctiall plane is that which is parallel to the Equinoctiall Circle of the Sphere, and therefore ha­ving drawn the horizontall line B C, and crossed it with the perpendicular D E, at right angles in the point A, if by

[Page 66] the second Chapter you shall finde the inclination of the plane towards the South to be equall to the complement of your Latitude, and by the third Chapter you finde the horizontall line directly in the line of East and West, and so to have no declination, you may be sure this plane is parallel to the Equinoctiall Circle, and is therefore called an Equinoctiall plane.

This Diall is no other then a Circle divided into 24 e­quall parts, by which divisions and the center A, you may draw so many houre lines as shall be necessary. As you may see here done in the Circle B D C E, which is divided equally into 24 equall parts, and houre lines drawn from the center A to so many of them as is needfull, the line D E, which is the line of inclination, is the Meridian or 12 of clock line: his Stile is no more but a streight pin or wire plumb erected in the center.

This Diall, though, of all other, he be the simplest, yet is he mother to all the rest, for out of him, as from a root, is derived the projectment of those 24 houre lines on any other great Circle or plane whatsoever.

CAAP. VI. The drawing of a Diall upon the direct Polar Plane.

A Direct Polar plane, is that which is parallel to the Circle of the houre of six, therefore, having drawn the horizontall line A B, rnd crossed it at right angles a­bout the middle of the line at C, with the perpendicular C E, if you shall finde the planes inclination towards the North to be equall to the Latitude of the place, and the horizontall line directly in the line of East and West, and [Page 67] so to have no declination, you may be sure this plane lyeth parallel to the houre of six, and is therefore called a Polar plane. The horizontall line being drawn at the length of the plane, divide it into seven equall parts, and set down one of them in the line of inclination from C unto D, and upon the center D describe the Equinoctiall Circle, which you may divide into 24 equall parts if you will, but one quarter thereof into six will serve as well: then at the

distance C E draw the line F G parallel to A B: Then having divided the Equator either into 24 equall parts, or one quarter thereof into six, you may by a rule laid upon the center D, and each of those six parts, make marks in the horizontall line A B (which here is instead of the contin­gent line) as you may see by the pricked lines, these di­stances from the Meridian, being applyed upon the same line on the other side of the Meridian, and also on both sides the Meridian in the upper line, the lines drawn from point to point, parallel to the Meridian C E, shall be the houre lines, the line C E shall be the Meridian line, the houre of 12, and must also be the substilar line, whereon the stile must stand, which may be a plate of iron or some other metall, being so broad as the semidiameter of the Circle is, as is shewed in the figure. This style must be [Page 68] placed along upon the line of 12, making right angles therewith, the upper edge whereof must be parallel to the plane; so shall it cast a true shadow upon the houre lines. The under face of this Polar plane, and also of the former Equinoctiall plane, is made altogether like unto the upper faces here described, without any difference at all.

CHAP. VII. The making of an erect Meridian Diall.

A Meridian plane is that which is parallel to the Meridi­an Circle of the Sphere; therefore having drawn the horizontall line A B, and finding it to decline 90 deg. from the South, the plane being erect, I conclude it to lie paral­lel to the Meridian Circle of the Sphere, and is therefore called a Meridian plane.

The horizontall line being drawn, at the North end thereof as at A, make an angle equall to the elevation of the Equinoctiall, which is in this example 37 deg. 30 min. and draw the line A C D so long as your plane will give you leave, making an angle in the point A equall to the Equinoctials height, so shall the South end of this line behold precisely the Equinoctiall Circle, this line divide into five equall parts, and with the same widenesse of the Compasses, with one foot in E (which is the fourth divi­sion from A, or the first from D) describe the Circle D H, through the center whereof draw a Diameter, cutting the former Diameter D H at right angles in the center E, this Diameter shall lie parallel to the Axl [...]tree of the world, and be the line for the houre of six, then at the out-sides of the Circle draw two touch lines, one beneath, the other above the Circle, so that they may be both parallel to the [Page 69] middle line, then divide one quarter of the Circle into six equall parts, and place the Rule upon the center E, and each of those parts, mark where it toucheth the line of con­tingence; as here you may see it doth in the points 7, 8, 9, 10, and 11, from which points, if you draw lines parallel to the line of six, they shall be the houre lines here desi­red, and shall be parallel one to the other: the distance betwixt 6 and 7 is the same with 6 and 5, and the di­stance

between 6 and 8 is the same with the distance betwixt 6 and 4, and so you have all the houre points in the upper touch line, and if you transfer these distances from the houre of six into the other touch line likewise from the line of six, you may the better by the opposite points draw the lines parallel to the line of six.

For the style of this Diall it may be either a plate of some metall, being so broad as the semidiameter of the cir­cle is, and so placed perpendicularly along over the line of the houre of six, the upper edge thereof being parallel to the plane, or it may be a streight pin fixed in the center of the circle, making right angles with the plane, being just so long as the Semidiameter of the Circle is, only shewing the houre with the very top or end thereof.

This plane hath two faces, one to the East, the other to the West, the making whereof are both alike, only in na­ming the houres, for the one containeth the houres for the forenoon, and the other for the afternoon, as you may perceive by the figures.

CHAP. VIII. To draw a Diall upon an Horizontall plane.

AN Horizontall plane is that which is parallel to the Horizontall Circle of the Sphere, which being found by the first Chapter to be levell with the Horizon, you may by the fourth Chapter draw the Meridian line A B serving for the Meridian, the houre of 12, and the substi­lar: in this Meridian make choice of a center as at C, through which point C draw the line D E, crossing the Meridian at right angles, this line shall be the line of East and West, and is the six a clock line both for morning and evening.

Then by the second Chapter of the second Book draw the line S C; making the angle S C A equall to the Lati­tude of the place, which here we will suppose to be 52 deg. 30 min. this line shall represent the cock of the Di­all, and the Axletree of the World, then at the North end of the Meridian line, draw another line as F G, crossing the Meridian in the point A at right angles, this line is called the Touch-line, or line of contingence. Then set one foot of your Compasses in the point A, and with the other take the neerest extent unto the line S C or the Stile, with this distance turning your Compasses about, with one foot still in the point A, with the other make a mark in the Meridian as at I, which shall be the center of the Equinoctiall, upon which describe the Equinoctiall Circle A D B E, with this same distance, setting one foot in the point A, make a mark [...] F on the one side of the [Page 72] Meridian, and another at G on the other side thereof, both which must be in the line of Contingence, by which two points, and the center C, you may draw the houre lines of 3 and 9.

This same distance of your Compasses being kept, with one foot still in the center A, with the other make the marks T and V in the Equinoctiall Circle, each of which distances is an arch of 60 degrees, or four houres of time, the halfe of which arch is 30 degrees, or two houres from the Meridian, this divided in the halfe will be 15 deg. or one houre from the Meridian; then laying your rule upon [Page 73] the center I of the Equinoctiall, and upon these two last divisions in the circle thereof, where the rule shall touch the line of contingence, there mark it as at H and O, by which points and the center C, you may draw the hour­lines of 10 and 11; the like may you do on the other side of the meridian, so have you six of your hour-lines drawn: and now because the contingent will out-run our plane, we may from the intersection of the houres of 9 and 3 with the touch line draw the lines F D and G E parallel to the meridian A B, untill they cut the line of East and West in the points D and E, then draw the lines A D and A E, this being done, set one foot of your compasses in the point H, and with the other take the neerest extent unto the line A E; this distance being kept, fix one foot in the line G E, so as the other may but touch the line A E; so shall the fixed foot rest in the point N, by which and the center C, you may draw the 7 a clock hour-line: in like manner may you place one foot in the point O, and with the other take the shortest extent unto the line A E, with this distance fixing one foot in the line G E, so as the other may but onely touch the line A E; so shall the fixed foot rest in the point R, by which and the center C you may draw the 8 a clock hour-line, the like may be done on the other side of the meridian, or you may by these di­stances thus found prick out the like on the other side of the meridian.

Thus by dividing but half a quarter of the Equinoctiall Circle into three equall parts, you may describe your whole Diall.

And whereas in Summer, the 4 and 5 in the morning, and also 7 and 8 at evening, shall be necessary in this kind of Diall: prolong or draw the lines of 4 and 5 at evening [Page 74] beyond the center C, and they shall shew the hour of 4 and 5 in the morning, and likewise the 7 and 8 in the mor­ning for 7 and 8 at evening.

What is here spoken concerning the hours, the like is to be done in drawing the half hours, as well in this kinde as in all them which follow.

The style must be fixed in the center C, hanging direct­ly over the meridian line A C, with so great an angle as the lines S C A maketh, which is the true pattern of the cock.

This and all other kinds of Dials may be drawn upon a clean paper, and then with the help of your compasses pla­ced on your plane.

CAAP. IX. To draw a Diall upon an erect direct verticall Plane, com­monly called a South or North Diall.

A Verticall plane is that which is parallel to the prime verticall Circle, it hath two faces, one to the South the other to the North; therefore having drawn the Ho­rizontall line A B, and from the middle thereof let fall the perpendicular C D, which if you finde by the second Chapter to be erect, and the Horizontall line A B to lie in the line of East and West, and so to have no declination, you may be sure this plane is parallel to the prime verti­call Circle of the Sphere, and therefore is called a verti­call plane.

This perpendicular C D shall serve for the meridian, the hour of 12, and the substilar line, which is the line over which the Stile or Gnomon in your Diall directly hang­eth.

The Horizontall line A B shall serve for the hour-lines of six, both for morning and evening.

Then (by the second Chapter of the second Book) draw the line S C, making the angle S C D equall to the Lati­tudes complement, which in this example is 37 degr. 30 minutes, this line S D shall represent the axletree of the world, and if you draw the line S D square to the meridian, you shall have the Triangle S C D for the true pattern of your Dials cock: then crosse the meridian (in some point thereof as at E) at right angles, with the line F G, serving for the touch-line, then taking the shortest extent from the point E unto the line S C, place it in the meridian from E to I, which point shall be the center of the Equi­noctiall Circle, therefore upon the center I, describe the circle A E B for the Aequator: this distance of your com­passes being kept, set one foot in the point E, and with the other set out the equall distances E G and E F in the con­tingent line, which shall be the points thorow which you must draw the hour-lines of 9 and 3 from the point C, the centre of the Diall; the same distance of your compasses being still kept, with one foot in the point E, with the other make a mark in the Equinoctiall Circle as at H. So shall the arch E H contain 60 degr. or four hours of time, which arch you may divide into four equall parts, and by laying your rule upon the center I, and those two divisions next the meridian C D you may make two marks in the line of contingence, as at N and O; thorow which two points, and from the center C may you draw the hour-lines of 10 and 11, and now because the touch-line will out-run the plane, you may from the intersections of the hour-points of 9 and 3 in the line of contingence, draw the lines A F and B G parallel to the meridian C D, untill [Page 76] they cut the Horizontall line A B, then draw the lines A E and B E, this being done, set one foot in the point O, and with the other take the shortest extent unto the

line A E: with this distance set one foot in the line A F, so as the other may but touch the line A E at the neerest extent, and the fixed foot shall rest in the point T, by which and the center C, you may draw the 7 a clock hour-line: in like manner may you place one foot in the point N, and with the other take the shortest extent unto [Page 77] the line A E, with which distance fix one foot in the line A F, as that the other being turned about, may but onely touch the line A B, so shall the fixed foot rest in the point V, by which and the center C, you may draw the 8 a clock hour-line, or you may extend the line A B a little beyond the points A and B, and take the distance E O, and set it from A to R, or from B to R, then lay a Rule upon the points R and O, and where you shall see it crosse either the line A F or B G, there make a mark as at T, which shall be the point, thorow which the hour of 7 or 5 must be drawn from the center as before, so likewise may you take the distance E N, and set it from A to P and from B to P, and laying your Rule upon the two points P and N, where you shall see it crosse the lines A F or B G, there make a mark as at V, which shall be the point for the hour of 8 or 4; this is in effect no more but to draw a line paral­lel to the line A E or B E by the point O, and this line shall cut the line A F or B G in the point T, which shall be the points for the hour of 7 or 5 as before: and so you may by the point N draw a line parallel to the line A E or B E, and where it shall crosse the line A F or B G, there make a mark as at V, which shal be the points for the hour of 8 or 4.

Thus you may see, that by dividing the Radius of the Equinoctiall Circle into four equall parts, you may de­scribe your whole Diall, if it hath no declination; for ha­ving with this Radius pricked out the lines of 3 & 9, & pla­ced the lines for 10 and 11, these two distances from the hour of 12 shal give the like for the hour of 1 & 2 on the o­ther side of the meridian, and having drawn the lines for 7 & 8 by the former rules, you may take their distances from the hour of 6, and place them on the other side of the Diall from 6 to 5 & 4: so have you all your hour-lines drawn, and [Page 78] yet we have not out-run the limits of our plane, which is an inconvenience, unto which the most are subject. Now seeing the Triangle S C D is the true pattern of this Di­als cock, and that this is the South face of this plane, therefore the center will be upward, and the stile-point downward, hanging directly over the meridian line.

But if it had been the North face of this plane, you must have proceeded in all things as before, but onely in placing the Diall, and naming the houres; for if it be the North face, the center must be in the lower part of the meridian line, and the stile and hour-lines point upwards, as you may see in this figure following.

CHAP. X. To draw a Diall upon a direct verticall plane, inclining to the Horizon.

ALL those planes that have their Horizontall line lying East and West, are in that respect said to be direct verticall planes; if they be also upright passing thorow the Zenith, they are erect direct verticall planes; if they incline to the pole, they are direct polar planes; if to the Equinoctiall, they are called Equinoctiall planes, and are described before: if to none of these three points, they are then called direct verticals inclining.

In all Dials that decline not, two things must be had before you can make the Diall, the first is the meridian or 12 a clock line, wherein the cock must stand, and the se­cond is the elevation of the pole above the same line.

For the Horizontall plane, the meridian line is drawn by the 4 Chapter, and the elevation of the pole above the plane is always equall to the latitude of the place; and in erect direct verticals, the perpendicular or verticall line is the meridian or 12 a clock line, and the elevation of the pole above the plane is always equall to the complement of the latitude, the North pole being elevated above the North face, and the South pole above the South face thereof.

And in these verticals inclining being direct, the line of inclination is their meridian, and line wherein the cock must stand, but for the elevation of the cock, we must first consider, whether the plane inclines towards the South or towards the North; if it inclineth towards the [Page 80] South, adde the inclination to your latitude, the summe of both shall be the elevation of the pole above the plane, and if the summe shall be just 90 degrees, it is an Equinoctiall plane, and is described before in the 5 Chapter, but if the summe shall exceed 90 degrees, take it out of 180, and that which remaines shall be the elevation of the pole above the plane.

As for example in the latitude of 52 deg. 30 min. let a plane be found to incline Southwards 20 degrees, this 20 deg. added to 52 deg. 30 min. the latitude of the place, the summe will be 72 degr. 30 min. the elevation of the pole above the plane, with which you may proceed to draw a Diall by the eighth or ninth Chapters as if it were a horizontall plane, for their difference is nothing, but in the height of the stile, which is the elevation of the pole above the plane. For this plane shall be parallel to that Horizon, whose latitude is 72 deg. 30 min. lying both under one and the same meridian.

But if the inclination be Northward, compare the in­clination with your latitude, and take the lesser out of the greater; so shall the difference be the elevation of the pole above the plane, but if there be no difference, it is a di­rect polar plane, and is described before in the sixt Chapter.

As in the latitude of 52 deg. 30 min. a plane being pro­posed to incline towards the North 25 deg. this 25 degr. being taken out of 52 degrees 30 minutes leaveth 27 deg. and 30 minutes for the elevation of the pole above the plane. Now this plane being parallel to that Horizon, whose latitude is 27 deg. 30 min. lying both under one & the same meridian, therefore you may proceed to make this Diall, as if you were to make an Horizontall Diall in that countrey.

Each of these planes have two faces; one towards the Zenith, the other towards the Nadir; but what is said of the one is common to the other; they onely differ in this, the one hath the South, the other hath the North pole e­levated above their faces.

For upon the upper faces of all North incliners, whose inclination is lesse then the latitude of the place, on the un­der faces of all North incliners, whose inclination is great­er then the latitude of the place; and on the upper faces of all South incliners, the North pole is elevated; and therefore contrarily, on the under faces of all North in­cliners, whose inclination is lesse then the latitude of the place, on the upper faces of all North incliners, whose inclination is greater then the latitude of the place; and on the under faces of all South incliners, the South pole is e­levated: unto one of which poles, the stiles of all Dialls must point directly.

CAAP. XI. To draw a Diall upon an erect, or verticall plane decli­ning, commonly called a South or North erect declining Diall.

ALL upright planes whereon a man may draw a verti­call line; are in that respect said to be erect or verti­call; if their horizontall line shall lie directly East and West, they are direct verticall planes; if directly North and South, they are properly called Meridian planes, and are described before.

If they behold none of these four principall parts of the world, but shall stand between the prime verticall cir­cle, [Page 82] and the Meridian, they are then called by the generall name of declining verticals, or by the name of South or North erect declining planes.

In all such declining planes, because the Meridian of the place (which in all upright planes is the verticall line, and serveth for the hour of 12) and the Meridian of the plane deflecteth one from the other; therefore we must finde out and place the Meridian of the plane, (which is the line over which the stile directly hangeth, and is here called the Substile) and likewise the elevation of the pole above the plane: both which may be easily performed in this manner.

First, draw a blinde line parallel to the Horizon, which may be the line A B, and from a point therein as at C, let fall the perpendicular C D, serving for the Meridian of the place, and the hour of 12, and through some place of this Meridian as at E, draw the line F G at right angles.

Then having by the third chapter examined this plane, and finding it to decline 30 deg. from the South towards the East, I draw an arch of a circle upon the center C, with my compasses opened to the Radius of the Scale; in which arch I place the declination of the plane from E to H, on the same side of the meridian with the declination of the plane, as here you see; then set the complement of the latitude, which is 37 degr. 30 min. in the same arch from E to M, and draw the line C H for the declination of the plane, and the line C M G cutting the line F G in the point G for the complement of your latitude.

This being done, take the distance E G; and set it in the line of declination from C to S, from which point S draw the line S L square to the meridian C D, then take S L and set it from E to O in the line E F, and draw the line C O, [Page 83] which is the Meridian of this plane, or the line of the Substile,Here note, that in all decliners, the Substile go­eth from the Meridian, to­wards that coast which is contrary to the coast of the planes declina­tion. wherein the stile must stand direct­ly up from the plane, then through the point O draw the line P K, square to the Substile C O, which shall be the touch-line or line of contingence.

Then take the distance C L, and set it in the line of contingence from O to P, and draw the line C P for the stile.

This done, set one foot in the point O,

and with the other take the shortest extent unto the stile [Page 84] C P with this distance, one foot remaining still in the point O, the other turned towards C, make a mark at y in the line of the Substile, which shall be the center where­on you must describe the Equinoctiall circle.

Now having drawn the line NT through the center C, and parallel to the touch-line I K, which will be square to the Substile C O, I take the distance O y, which is the Radius upon which the Equinoctiall circle was drawn, and place it on both sides the substile, in the line of contin­gence from O to I, and from O to K, and in his parallel line from C to G, and from C to R, and draw G I and R K, then laying your rule upon the point y, the center of the Equinoctiall, and a the point of intersection of the touch-line with the meridian, and where it cutteth the cir­cles circumference, there must you begin to divide it into 24 equall parts, but those six shall be onely in use which are next the line of contingence, that is, three of each side of the substile next thereunto.

Then place your Ruler upon the center y, and upon each of these six points of the Equator, and where it toucheth the line of contingence, there make markes, by which and the center C, draw those six hour lines next the Substilar; which shall all fall between the points I and K in the touch-line, three whereof shall fall betwixt O and K, and three betwixt O and I, thus have you 6 of your 12 hours, viz. 12. 11. 10. 9. 8. and 7. then take the distance from the point O to the intersection of the hour of 12 with the touch-line, and place it from O to b, and from G to N, then laying your Rule upon these two points N and b, where it shall crosse the line G I, shall be the point through which you may draw the six a clock [Page 85] hour-line; in like manner, take the distance from O to the line for 11 a clock, and set it from O to c, and from G to Z, then lay the Rule upon C and Z, and where it shall cut the line, G I shall be the point through which you shall draw the 5 a clock hour-line; and so placing the distance O 10 from O to g, and from G to V, and laying the Rule upon V and g, you shall finde the point in the line G I, through which you may draw the 4 a clock hour-line: in like manner may you proceed with the other side.

For taking the distance from the point O to the line of 9, and setting it from O to h, and from R to x, and laying the Rule upon the points h and x, where you shall see it cut the line R K, there shall be the point, through which you shall draw the 3 a clock hour-line.

And so you may take the distances from O, to the line of 8, and 7, and place them from O to d and m, and from R to W and T, and so by lay­ing your Rule upon the points W d and T m, where it shall crosse the line R K, there shall be the points through which the houre-lines of 2, and 1, shall be drawn.

The Diall being thus drawn upon the South-east face of this plane, let the stile be fixed in the center C, so that it may hang directly over the Substile C O, making an angle therewith equall to the angle P C O.

The stile with the substile must here point down­wards, because in all upright planes declining from the South, the South pole is elevated; and in all upright [Page 86] planes declining from the North, the North pole is ele­vated.

Therefore if you were to make a Diall to the North face of this plane, you must make choice of your center C in the lower part of the Meridian C D, that the stile with the substile may have room to point upwards.

This Diall being made on paper for the South-east face of this plane, will also serve for the North-west face thereof, if you turn it upside down; so that the stile with the substile may point upwards; and the paper being oyl­ed or pricked through, so that you may take the back-side thereof for the fore-side, without altering the numbers, set to the hours.

And the foreside of this pattern, turned upside down, so that the cock may point upwards, shall serve for the North-east face of a plane having the same declination; onely altering the numbers set to the hours.

This paper being oyled, if you do but change the back­side for the foreside; and the numbers set to the hours, still keeping the center upwards, and the stile pointing downwards, this pattern will serve for the South-west face of a plane, whose declination is the same as before.

And thus you see by diligent observation, this pattern may be made to serve for four Dials, which being well understood, will be a great help to the Artist.

CHAP. XII. How to draw a Diall upon an horizontall plane, otherwise then in the eighth Chapter was shewed.

ALthough I have plainly and perfectly shewed the ma­king of the horizontall, the direct South or North, as well erect as inclining; and the South or North erect declining Dials, in the four former Chapters: yet to sa­tisfie them that delight in variety, I have here declared another way, whereby you may make them most artifici­ally and geometrically; not being tied to the use of the Canons; (which indeed of all others is most exact, but not so easie to be understood) nor to any one Instrument (which may be absent from me, when I should need it) although in this Treatise I do perform the whole by a plain Quadrant.

Therefore by the first Chapter, having found the plane to be horizontall; by the fourth Chapter draw the me­ridian line A B, and crosse it at right angles in the middle with the line D E, which is the line of East and West, and serveth for the hour of six at morning, and six at e­vening.

Then upon the center C (which is the point of inter­section) describe a circle for your Diall as large as your plane will give leave; which let be the circle A D B E, then take the latitude of the place which is here 52 degr. 30 min. and set it from A to N, in the quadrant A D, and draw the line C N S, then from A raise the perpendicular A S, to cut the line C S at S, so shall the Triangle A C S be the true pattern for your cock; this being done, divide [Page 88] the two quarters of your circle A E and A D, each into six equall parts, so shall you have in each Quadrant five points by which you may draw the five Chord lines I F

G H and A, as here you see; then take one half of the Chord line A, and set it in the line of the stile from C [Page 89] to O, from which point O take the neerest extent unto the meridian; with this distance setting one foot in the point A, with the other make a mark on each side of the meridi­an, in the same Chord line A, through which points you shall draw the hour-lines of 1 and 11.

So likewise you may take one half the Chord H, and place it in the line of the stile from C to K, from which point K, take the shortest extent unto the meridian, with this distance set one foot in H, and with the other make on each side the meridian a mark in the same Chord line, through which you shall draw the hour-lines of 2 and 10.

And thus you may proceed with the rest of the lines, as the Figure will shew better then many words; for this is sooner wrought then spoken.

And if you would have the hours before and after six, you may extend them through the center, as was shewed in the 8 Chapter.

CHAP. XIII. To draw a Diall upon a direct verticall plane, as well erect as inclining, otherwise then in the ninth Chapter was shewed.

THe work of this, is almost like unto the other before; the difference is onely in the elevation of the pole a­bove the plane: for in the horizontall plane, the elevation is equall to the latitude of the place; and in all direct ver­ticals being erect, the elevation of the pole above the plane is equall to the complement of the latitude, but if they shall incline towards the horizon, then shall you finde the elevation of the pole above the plane by the 10 Chapter.

The elevation of the pole above the plane being known, the making of these Dials are all alike; therefore by the second Chapter draw the line EW parallel to the horizon, and from the middle thereof set fall the perpendicular ZN which shall be the meridian of the plane, and also the me­ridian of the place, serving for the line of 12, and also for the Substile, over which the stile must hang, both in erect and inclining planes being direct.

Then upon the center Z describe your Dials circle, or rather the Semicircle E N W, and seeing this plane is e­rect, and also direct, therefore the elevation of the pole

above the plane is 37 degr. 30 min. equall to the comple­ment of our latitude, which take from your scale, and place it from N to H in your Dialls semicircle, and draw the line Z H S for the line of the stile; then from the end of the meridian as at N, draw the crooked line N S, cutting the [Page 91] line of the stile in the point S, so shall the Triangle S Z N be the true pattern for your cock.

This being done, divide each Quadrant of your Semi­circle into six equall parts, so shall you have 5 points, by which you may draw 5 Chord lines, cutting the meridian at right angles in the points I K L M N.

This being done, take the half of each Chord, and place it from the center Z along upon the line of the stile, as here you see; the half of the Chord N from Z to A, and one half the Chord M from Z to B, and half the Chord L from Z to C, and one half the Chord-lines I and K, set from Z unto D and G; now from each of these points, take the neerest extent unto the meridian Z N, and place them upon their proper Chord lines from the meridian on both sides thereof; so shall you have two points on each Chord, through which you shall draw the hour-lines from the center of your Diall, as the shortest extent from the point A unto the meridian, set in the Chord N from the meridian both wayes, shall give you the points for 1 and 11; so shall the shortest extent from the point B (being placed from the meridian both ways in the Chord M) give you the two points for 10 and 2, and so you may pro­ceed with the rest; thus doing, you shall have in each Chord two points, on each side the meridian one; through which, and from the center Z, you may draw your houre-lines at pleasure, without exceeding the limits of your plane. And seeing this is the South face of this plane, therefore the style must point downwards being fixed in the center Z in the upper part of the meridian line Z N, over which the style must directly hang, making there­with an angle equall to the angle N Z S.

But if it had been the North face, then must the center [Page 92] be placed in the lower part of the Meridian, and the style, with the substyle, and also the houre-lines, must point up­wards.

CHAP. XIV. The declination of an upright plane being given, how thereby to finde the elevation of the pole above the same, with the angle of Deflexion, or the distance of the substile from the Meridian: and also the angle of inclination betwixt both Meridians.

IN all erect declining planes, when the declination is found, there is three things more to be considered before we can come to the drawing of the Diall.

I. The elevation of the pole above the plane.

II. The distance of the substile from the Meridian.

III. The angle contained betwixt the Meridian of the plane, and the Meridian of the place; which here we call the inclination of Meridians: This angle is made at the Pole, and serveth to shew us where we shall begin to divide our Diall-circle into 24 equall parts.

These three may be both artificially, easily, and speedily, performed after this manner following.

First, describe a Quadrant, as A B C, then supposing your Latitude to be 52 deg. 30 min. take it from your Scale, and set it from B to E in the arch of the Quadrant, and draw the line E D parallel to A B, cutting the line A C in the point D, then take the distance D E, and setting one foot in the center A, with the other describe the arch G H O R.

Then suppose your declination to be 32 deg. this set from B to F in the arch B E C, and draw the line F A, cutting the arch G R in the point H, through which point draw the line S H N cutting the arch B E C in N, so shall the arch C N be the elevation of the pole above the plane, which in this example is found to be 31 deg. 5 min.

Then take the

distance H S, and set it in the line D E, from D to K, through which point K draw the line A K L, cut­ting the arch B C in the point L, so shall the arch C L be the distance of the substile from the Meridian, and in this example is 22 deg. 8 min.

Now from the point L, draw the line L T parallel to the line A C, cutting the arch G R in the point O, through which point O draw the line A O I, cutting the arch B C in the point I, so shall the arch C I be the inclination of both Meridians, and is found by this example to be 38 deg. 13 min. so that by this example the Meridian of the plane will fall betwixt the houres of 2 and 3 if the plane shall decline Westward; but if it shall decline Eastward, then shall it fall betwixt the houres of 9 and 10 before noon.

CHAP. XV. To draw a Diall upon an erect or Verticall plane declining; otherwise then in the 11 Chapter was shewed.

HAving by the third Chapter found the declination of this plane to be 32 degrees, and so by the last Chap­ter found the elevation of the pole above the plane to be 31 deg. 5 min. and the distance of the substile from the Meridian to be 22 deg. 8 min. and likewise the angle of in­clination between both Meridians to be 38 deg. 13 min. we may proceed to make the Diall after this manner.

First, draw the horizontall W E, and the perpendicular line Z N, crossing the horizontall line at right angles, which is the Meridian of the place and the line of 12.

Then in the meridian make choice of some point with most convenience, as the center C, whereupon describe your Diall circle E N W.

Then take a Chord of 22 deg. 8 min. from your Scale, for the distance of the substile from the Meridian, and in­scribe it into this Circle from the Meridian, upon these con­ditions, that if the plane declineth West, then must the substile be placed East of the plumb line; but if the de­clination shall be East, then must the substile be placed West from the Meridian, as here it is.

This 22 deg. 8 min. being set in the Diall circle from the Meridian at N unto M, I draw the line C M for the substile: then through the center C draw the diameter A B, making right angles with the substile C M, above this Diameter there needs no houre-lines to be drawn if the plane be erect.

Then take 31 deg. 5 min. and set them from M to D, and draw the line C D S for the Stile, then from M the end of the Stile draw the crooked line M S, cutting the line of the Stile in the point S, so shall the triangle S C M be the true pattern for the Cock of the Diall.

This being done, take 38 deg. 13 min. and set them al­wayes on that side the substile whereon the line of 12 ly­eth, as here from M to H, so shall the point H be the point where you shall begin to divide your diall circle into 24 e­quall parts, but those points shall be only in use which doe fall below the Diameter A C B.

And if the line of the substile falleth not directly upon [Page 98] one of the houre lines, then shall you have six points on each side thereof, from which you may let perpendiculars fall unto the line of the substile as here you see done.

Now take each perpendicular betwixt your compasses, and with one foot in the center C, with the other make marks in the line of the Stile, from which take the neerest extents unto the line of the substile, and lay them upon their own proper perpendiculars from the substile, so may you make points through which you may draw houre-lines, and by thus doing with each perpendicular on both

[Page 97] sides the substile, you may describe your whole Diall as here you see, which may serve for four faces, by observing what was spoken in the 11 Chapter.

When you have drawn and described your Diall upon paper, for any plane whatsoever, you may cut off the hour-lines Cock and all, with a lesser Circle then the Diall Circle, either with a concentrique or an excentrique Cir­cle, and so make a Diall lesse then the Circle by which you framed it.

Or if you extend the hour-lines beyond the Diall Cir­cle, you may cut them off either with a greater concen­trique Circle, and so make a bigger Diall, or else you may cut them off with a Square, as here you see in the follow­ing figure; or any other form what you shall think most convenient.

CHAP. XVI. The inclination of a Meridian plane being given, how thereby to finde the elevation of the pole above the plane, the distance of the substile from the Meridian; and the angle of the inclination of the Meridian of the plane to the Meridian of the place.

ALL those planes wherein the horizontall line is the same with the Meridian line, are therefore called Meridian planes, if they make right angles with the Hori­zon, they are called erect Meridian planes, and are descri­bed before.

But if they leane to the Horizon, they are then called Incliners.

These may incline either to the East part of the Hori­zon, or to the West, and each of them hath two faces, the upper towards the Zenith, the lower towards the Nadir, wherein knowing the Latitude of the place, and the incli­nation of the plane to the Horizon, we are to consider three things more before we can come to the drawing of the Diall.

I. The elevation of the pole above the plane.

II. The distance of the substile from the Meridian.

III. The angle of Inclination betwixt both Meridi­ans.

These three may be found after this manner, little dif­fering from the 14 Chap.

First, describe a Quadrant, as A B C, then set 52 deg. 30 min. (your Latitude) from C to E, in the arch of your Quadrant C B, and draw the line E R parallel to A B, cut­ting the line A C in the point R, and with the distance E R, with one foot in the center A, with the other draw the arch G H O D, then let your inclination be 30 deg. which set in the arch of the Quadrant from B to F, and draw the line A F, cutting the arch G O D in the point H, through which point H, draw the line S H N, cutting the arch of the Quadrant in the point N, so shall the arch C N be the elevation of the pole above the plane, which in this exam­ple is 43 deg. 23 min.

Then take the di­stance S H, and set it in the line E R from R to K, through which point K, draw the line A K L, cut­ting the arch of the quadrant in the point L, so shall the arch C L be the distance of the substile from the Meridian, and is in this example 33 deg. 5 min.

This being done, from the point L, draw the line L T parallel to the line A C, cutting the arch G D in the point O, through which point O draw the line A O I, cutting the arch of the Qua­drant B C in the point I, so shall the arch C I be the incli­nation of the Meridian of the plane to the Meridian of the [Page 101] place, and in this example is found to be 43 deg. 28 min. which being resolved into time, doth give about two hours and 54 min. from the Meridian, for the place of the sub­stile amongst the houre-lines.

CHAP. XVII. To draw a Diall upon the Meridian inclining plane.

HAving by the second Chapter found the inclination of this plane to be 30 deg. and so by the last chapter found the elevation of the pole above the plane to be 43 deg. 23 min. and the distance of the substile from the me­ridian to be 33 deg. 5 min. and likewise, the angle of in­clination to be 43 deg. 38 min. we may proceed to make the Diall after this manner.

First, draw the horizontall line A B, serving for the Meridian and the houre of 12, about the middle of this line make choice of a center at C, upon which describe a Circle for your Diall, as A D B E.

Then seeing this is the upper face of the plane, set 33 deg. 5 min. the distance of the substile from the Meridian, in the Dials Circle from the North end of the Horizon­tall line upwards, as from B to H, and draw the line C H for the substile: But if this had been the under face, the substile must have fallen below the horizontall line: now through the center C draw the Diameter E F, making right angles with the substile C H.

Then set 43 deg. 23 min. from H unto D for the stile, and draw the line C D unto S, and from the end of the substile draw the crooked line H S, cutting the line of the stile in the point S, so shall the Triangle S C H be the true pattern of your Cock for this Diall.

Then set 43 deg. 28 min. from H unto M, for the dif­ference betwixt the Meridian of the plane and the Meridian of the place. Now here at M must you begin to divide your Circle into 24 equall parts, from which points let down so many perpendiculars to the substile, as there shall be points on that side the diameter E F next the substile.

Now take each perpendicular betwixt your Compas­ses, and with one foot in the center C, with the other make marks in the line of the stile, from which take the neerest extents unto the substile, and lay them down upon their own proper perpendiculars from the substile, so may you make marks, through which and from the center, you may draw the houre-lines.

This diall being thus drawn, for the upper face of a Me­ridian plane inclining towards the West, you must fixe the Cock in the center C, hanging over the substile C H, with an angle equall to the angle S C H, so that it may point to the North Pole, because upon the upper faces of all Meridian incliners the North Pole is elevated; and therefore contrarily, the South Pole must needs be eleva­ted above their under faces.

This Diall being drawn in paper, for the upper face of this plane, will also serve for the under face thereof, if you turn the pattern about so that the horizontall line A B may lie still parallel to the Horizon, and the stile with the sub­stile (lying under the Horizontall line) may point down­wards to the South Pole, the paper being oyled or pricked [Page 103] through, so that you may take the back side thereof for the fore side, without altering the numbers set to the hours.

CHAP. XVIII. The inclination and declination of any plane being given in a known Latitude, to finde the angle of intersection between the plane and the Meridian, the ascension and ele­vation of the Meridian, with the arch thereof between the Pole and the plane, and also the elevation of the Pole above the plane, the distance of the substile from the Meridian, with the inclination be­tween both Me­ridians.

IF a plane shall decline from the South, and also incline to the Horizon, it is then called by the name of a decli­ning inclining plane.

Of these there are severall sorts, for the inclination be­ing Northward, the plane may fall betwixt the Horizon and the Pole, or betwixt the Zenith and the Pole, or else they may lie in the Poles of the World: or the inclinati­on may be southward, and so fall below the intersection of the Meridian and the Equator or above it, or the plane may fall directly in the intersection of the Meridian with the Equator, each of these planes have two faces, the up­per towards the Zenith, and the lower towards the Nadir: Now having the Latitude of the place, with the declinati­on and inclination of the plane, we have seven things more to consider before we can come to the drawing of the Di­all.

I. The angle of intersection betwixt the plane and the Meridian.

[Page 105]II. The arch of the plane betwixt the Horizon and the Meridian.

III. The arch of the Meridian betwixt the Horizon and the plane.

IV. The arch of the Meridian between the Pole and the plane.

V. The elevation of the Pole above the Plane.

VI. The distance of the substile from the Meridian.

VII. The angle of inclination betwixt the Meridian of the plane, and the Meridian of the place. All these seven may be found out after this manner.

First, Describe the Quadrant A B C, then suppose the plane to decline from the South towards the East 35 deg. and to incline towards the Horizon 25 deg. set 35 deg. the declination of the plane from C to E in the Quadrants arch C B, and draw the line A E, then set 25 deg. the in­clination of the plane in the same arch from B to F, and draw the line F Z parallel to A C, cutting the line A B in the point Z, and with the distance F Z, and one foot pla­ced in the center A, with the other describe the arch G H I, cutting the line A E in the point H, through which point H draw the line K L parallel to A C cutting the arch C B in the point K, then take the distance H L, and set it in the line F Z from Z unto O, through which point O draw the line A O M, cutting the arch B C in the point M, from which point M draw the line M P N parallel to A B, cut­ting the arch G I in the point P, through which point P draw the line A P Q cutting the arch B C in the point G, so shall the arch B K be 75 deg. 58 min. the inclination of the plane to the Meridian, and the arch B Q will be 57 deg. 36 min. for the Meridians ascension, or the arch of the plane, betwixt the Horizon and the Meridian, and the [Page 106] arch B M shall be 20 deg. 54 min. for the elevation of the Meridian, or the arch of the Meridian betwixt the Hori­zon and the plane. Now if the plane shall incline toward the South, adde this elevation of the Meridian to your La­titude, and the sum of both shall be the position Latitude, or the arch of the Meridian betwixt the Pole and the plane, and if the sum shall exceed 90 deg. take it out of 180 deg. and that which remains shall be the position La­titude, or the arch of the Meridian between the Pole and the plane.

But if the in­clination shall be northward, then compare the ele­vation of the Meridian with your Latitude, and take the les­ser out of the greater, and so shall the diffe­rence be the po­sition Latitude: As here in this example, supposing the inclination to be Northward, we take 20 deg. 54 min. the elevation of the Meridian, out of 52 deg. 30 min. the Latitude proposed, and there will remain 31 deg. 36 min. for the position Latitude, or the arch of the meridian between the Pole and the plane.

This being done, set 31 deg. 36 min. the position La­titude, from B to T in the arch B C, and draw the line A T, then with the distance K L, upon the center A, describe [Page 107] the arch Y M W, cutting the line A T in the point M, through which point M draw the line R S parallel to A B, cutting the arch B C in the point S, so shall the arch B S be 30 degrees 33 minutes, the height of the Pole above the plane.

Then lay your rule upon the point S and the center A, and where it shall cut the line K L, there make a mark as at V, through which point V, draw the line D V N W pa­rallel to A B, cutting the arch Y W in the point N, and the arch B C in W, so shall the arch B W be 8 deg. 35 min. the distance of the substile from the Meridian.

Lastly, through the point N, draw the line Y X pa­rallel to A C, cutting the arch B C in the point X, so shall the arch B X be 16 deg. 20 min. the inclination of the meridian of the plane to the meridian of the place.

CHAP. XIX. To draw a Diall upon a declining inclining Plane.

HAving by the second Chapter found the inclination to be 25 deg. towards the North, and by the third Chapter the declination from the South towards the East to be 35 deg. and so by the last Chapter the Meridians as­cension to be 57 deg. 36 min. The elevation of the Pole above the plane 30 deg. 33 min. The distance of the substile from the Meridian 8 degrees 35 min. And the in­clination of both Meridians 17 deg. 30 min. we may pro­ceed to make the Diall after this manner.

First, Draw the line A B parallel to the Horizon, in which line make choice of a center as at C, whereon de­scribe your Diall circle A D B E A, then take 57 deg. 36 min. the Meridians ascension, and set it from B that end of the horizontall line with the declination of the plane, as from B to N, and draw the line C N for the houre of 12.

Then set 8 deg. 35 min. the distance of the substile from the Meridian from N to M (on that side the Meri­dian which is contrary to the declination of the plane) and draw the line C M for the substile. And set 30 deg. 33 min. from M to D, and draw the line C D unto S, and from the end of the substile draw the crooked line M S, cutting the line of the Stile in S, so shall the Triangle M C S be the true pattern of this Dials Cock.

Then set 17 deg. 30 min. the inclination of Meridians from M unto O, which is the point where you must be­gin to divide your Diall circle into 24 equall parts; from which points let down so many perpendiculars to the sub­stile, as there shall be points on that side the Diameter F E next the substile, and so by working as before hath been shewed, you may draw the houre-lines, and set up the stile as in the former planes.

Now here I would have you well to consider what hath been here spoken concerning these kinde of Dials, and also what followeth the same, for if you mark the di­versity which doth arise by reason of the elevation of the Meridian, you may perceive thereby three sundry kinds of Dials to arise out of a North inclining plane declining, and also in a South inclining declining plane, yet in effect they are but one if you consider what followeth here concerning them, in all which, the stile with the substile, and such like materials, are found out according to the last Chapter.

Therefore having drawn your horizontall line, you must consider which pole is elevated above your plane, and how to place the Meridian from the Horizontall line. For

upon the upper faces of all North incliners, whose [Page 110] Meridians elevation is lesse then the Latitude of the place: on the under faces of all North incliners whose meridians elevation is greater then the Latitude of the place: and on the upper faces of all South incliners, the North Pole is elevated.

And upon the under faces of all North incliners whose meridians elevation is lesse then the Latitude of the place: On the upper faces of all North incliners, whose meridi­ans elevation is greater then the Latitude of the place: and on the under faces of all South incliners the South Pole is elevated.

Now for placing the Meridian from the horizontall line; upon the upper faces of all South incliners, whose meridians elevation is greater then the Latitudes comple­ment: on the under faces of all South incliners whose me­ridians elevation is lesse then the Latitudes complement: on the under faces of all North incliners, whose meridians elevation is greater then the Latitude of the place: and on the upper faces of all North incliners whose Meridians ele­vation is lesse then the Latitude of the place; the Meridi­an must be placed above the Horizontall line, as here in this example.

And therefore by the contrary; Upon the upper faces of all South incliners, whose meridians elevation is lesse then the Latitudes complement: On the under faces of all South incliners, whose meridians elevation is greater then the Latitudes complement: On the under faces of all North incliners, whose meridians elevation is lesse then the Latitude of the place: And on the upper faces of all North incliners, whose meridians elevation is greater then the Latitude of the place; the Meridian must be placed below the Horizontall line.

But here you must observe, that if it be either the up­per or under faces of a South inclining plane, whose meri­dians elevation is greater then the Latitudes complement: or either the upper or under faces of a North inclining plane, whose meridians elevation is lesse then the Latitude of the place; that then the Meridian must be placed from that end of the horizontall line with the declination of the plane: But on all the other faces of these kinde of planes, the Meridian must be placed from that end of the horizontall line which is contrary to the declination of the plane.

And here note, that if the inclination shall be South­ward, and the elevation of the Meridirn equall to the com­plement of your Latitude; then shall the substile lie square to the Meridian.

And if the inclination shall be Northward, and the ele­vation of the meridian equall to the Latitude of the place, then shall neither Pole be elevated above this plane, and therefore shall be a Polar declining plane. Wherein the Meridian being placed according to his ascension from the horizontall line, shall be in place of the substile, unto which if you draw a line square, it shall serve for the Equator. Then set one foot of your Compasses in the intersection of the substile with the Equator, and open the other to any convenient distance upon the substile, and describe the E­quinoctiall Circle (as in the sixt Chapter of this Book was shewed:) upon the center whereof, make an angle with the line of the substile, equall to the inclination of both meridians, namely the meridian of the plane and the meridian of the place, which shall shew you where to begin to divide your Equinoctiall Circle into twenty four equall parts.

These things being known, you may proceed to make your Diall and set up the Cock according to the sixth Chapter.

As for example, in our Latitude of 52 deg. 30 min. a plane is proposed to decline from the South towards the East 35 deg. as before, but inclining Northward 57 deg. 50 min. the Meridians ascension by the 18 Chapter will be found to be 69 deg. 33 min. and his elevation 52 deg. 30 min. equall to the Latitude of the place, and therefore neither pole is elevated above this plane, and so no distance between the Substile and the Meridian: for the Meridian and the stile with the substile, will be as it were all one line, which is the Axletree of the World: so that here the stile must be parallel to the plane, and the houre lines pa­rallel one to the other, as in the Meridian and direct Polar planes. Therefore, first draw the Horizontall line A B, wherein make choice of a center as at C, whereon describe an occult arch of a Circle as B E: then into this arch in­scribe the Meridians ascension 69 deg. 33 min. from B to E, and draw the line CE for the meridian of the plane, and for the substilar, and if you draw a line square to this sub­stilar, it shall be the Equator.

Then set one foot of your Compasses in the point of intersection D, and with the other opened to a convenient widenesse, draw a Circle for the Equator, unto which you may draw two touch lines square to the substile as in the direct polar plane.

This being done, and the inclination of both Meridians being found by the last Chapter to be 29 deg. 3 min. set it in this Circle from H unto O, and draw the line D O F, cutting the contingent in F, from which point F you shall draw the 12 a clock hour-line parallel to the substile.

Now from the point O, divide your Equinoctiall circle into 24 equall parts, with which you may proceed to make your Diall, and set up the cock according to the 6 Chap­ter.

CHAP. I. The description of the Quadrant.

HAving in the second and third Books shewed Geometrically the working of most of the ordinary Propositions Astro­nomicall, with the delineation of all kinde of plain wall Di­als howsoever, or in what lati­itude soever scituated, still keep­ng within the limits of our plane, and yet not tyed to the use of any Instrument.

I will now shew how you may performe the former [Page 115] work exactly, easily and speedily, by a plain Quadrant fit­ted for that purpose; the description whereof is after this manner.

Having prepared a piece of Box or Brasse in manner of a Quadrant, draw thereon the two Semidiameters A B and A C, equally distant or parallel to the edges, cutting one the other at right angles in the center A, upon which cen­ter A, with the Semidiameter A B or A C, describe the arch B C, this arch is called the limbe, and is divided into 90 equall parts or degrees; and subdivided into as many parts as quantity will give leave, being numbered from the left hand towards the right after the usuall manner.

Then let the Semidiameter A B be divided into 90 un­equall parts, (called right Sines) either from the Table of naturall Sines by help of a decimall Scale, equall to the Semidiameter A B, or else by taking the neerest extents from each degree of your Quadrant, unto the side A B, and placing them upon the side A B each after other, from the center A towards B, you shall exactly divide the Se­midiameter A B into 90 unequall divisions called right Sines.

This being done, draw the line D E from the Sine of 45 degrees counted in the line of Sines unto 45 degrees counted in the Quadrant, then from the point E draw the line E F parallel to A B, making the square A D E F, the side D E whereof (for distinction) may be called a Tangent line, and the side E F a Co-tangent line, then draw the Diagonall line A E, which you may call the line of Latitudes.

Then upon the center A, with the distance A D or A F describe the arch D F, which you may divide into six equall parts, by laying your Rule upon each 15th. degree in the [Page 116] Quadrant, and the center A as at g h I k l F, from which points draw slope lines to each 15th. degree in the Qua­drant, numbered backward, as F P, l O, k E, I n, h m, g B; these lines so drawn are to be accounted as hours, then dividing each space into two equall parts, draw other slope lines standing for half hours, which may be distin­guished from the other, as they are in the figure.

Now because in the latter part of this Book there is often required to use a line of Chords to severall Radiuses, [Page 117] therefore upon the edge of the Quadrant A C, you may have a line of Chords, divided as in the figure, and so the Quadrant being at hand will supply the uses of the Scale mentioned in the preceding Book, and also a Chord of any Circle, whose Radius is lesse then the line A C may be taken off and in that case supply the use of a Sector.

To this Quadrant, as to all others of this kind in their use is added Sights, with a threed, bead, and plummet ac­cording to the usuall manner.

CAAP. II. Of the use of the line of Sines. Any Radius not exceeding the line of Sines being known, to finde the right Sine of any arch or angle thereunto belonging.

IF the Radius of the Circle given be equall to the line of Sines, there needs no farther work, but to take the other Sines also out of the line of Sines.

But if it be lesser, then take it betwixt your compasses, and set one foot in the Sine of 90 degrees, and with the o­ther lay the threed to the neerest distance, which you may doe by turning the compasses about, till the moving point thereof doe onely touch the threed and no more: the threrd lying still in this position, take the neerest extent thereunto, from any Sine you think good, and it shall be the like Sine agreeable to the Radius given.

As for example, let the circle B C D E in the following chapter represent the meridian circle, let B D be the Ho­rizon, and C E the verticall circle; and let F G be the dia­meter [Page 118] of an almicanter, and so F H the Semidiameter there­of; which being given it is required to finde the Sines both of 30 and 50 degrees, agreeable to that Radius, first there­fore, take the given Radius betwixt your compasses, and with one foot set in the Sine of 90 degrees, with the o­ther lay the threed to the neerest distance, the threed lying still in this position, take the neerest extents thereunto, from the Sine of 30 and likewise of 50, these distances place upon the Radius F H from H to N, and from H to R, so shall H N be the Sine of 30 degrees, and H R the Sine of 50 degrees, agreeable to the Radius F H the thing desired.

CHAP. III. The Right Sine of any arch being given to finde the Radius.

TAke the Sine given betwixt your compasses, and set­ting one foot in the like Sine in the line of Sines, with the other lay the threed to the neerest distance, the threed lying still in this position, take the shortest extent thereunto from the Sine of 90 degrees, which distance shall be the Radius required.

As for example, let H R be the given Sine of 50 de­grees, & it is required to find the Radius answering there­unto, take H R with your compasses and set one foot in the Sine of 50 deg. and with the other lay the threed to the neerest distance, which being kept in this position, if you take the shortest extent thereunto, from the Sine of 90 you shall have the line H F for the Radius required.

CHAP. IV. The right Sine, or the Radius of any Circle being given, and a streight line resembling a Sine, to finde the quantity of that unknown Sine.

FIrst, take the Radius, or the right sine given, and set­ting one foot of your Compasses either in the like sine or in the Radius of the line of Sines, and with the other, lay the threed to the neerest distance, then take the right line given, and six one foot in the line of sines, moving it till the moveable foot touch the threed at the neerest ex­tent, so shall the fixed foot stay at the degree of the sine required.

As for example, let F H be the Radius given, and H N the streight line given resembling a Sine, first with the di­stance F H from the Sine of 90 lay the threed to the neerest distance; the threed lying still in this position, take the line H N and fixing one foote of your compasses in the line of Sines, still moving it to and fro till the moveable foote thereof, doth onely touch the threed, so shall the fixed foote rest at the Sine of 30 degrees in the line of Sines; this 30 degrees is the arch, of which H N is the Sine, F H in the last chapter being the Radius.

CHAP. V. The Radius of a circle not exceeding the line of Sines being given, to finde the chords of every arch.

IF the Radius given, shall be equall to the line of Sines, then double the Sine of halfe the arch, and you shall have the chord of the whole arch, that is, a Sine of 10 deg. dou­bled giveth a chord of 20 deg. & a Sine of 15 deg. doubled giveth a chord of 30 deg. and so of the rest, as in the third chapter, the line I O the Sine of I C an arch of 30 deg. be­ing doubled giveth I L the chord of ICL which is an arch of 60 deg.

And if the Radius of the circle given, be equall to the Semi-radius (the sine of 30 deg.) of the line of sines; then you neede not to double the lines of sines as before, but onely double the numbers: so shall a sine of 10 deg. be a chord of 20 deg. and a sine of 15 deg. be a chord of 30 deg. and so of the rest, but if the Radius of the circle given, be lesse then the semi-radius of your line of sines, then take it [Page 121] betwixt your compasses and setting one foot in the sine of 30 deg. with the other lay the threed to the neerest distance, the threed lying still in this position, take it over at the neerest extent in what Sine you think good, onely doubling the number, and you shall have the Chord desired.

As for example, let A C be the diameter of the circle in the third Chapter, and it is required to find a Chorde of 30 degrees, therefore first, I take A C betwixt my compas­ses and setting one foot in the Sine of 30 deg. with the other I lay the threed to the neerest distance: which being kept at this angle, I take it over from the sine of 15 deg. which doth give me I C the Chord of 30 deg. which was desired.

And if the Radius given, be greater then the Sine of 30. and yet lesse then the Radius of the line of Sines; then with the Radius given, and from the Sine of the comple­ment of halfe the arch required, lay the threed to the nee­rest distance; then taking it over at the neerest extent from the sine of the whole arch, you shall have your desire.

As for example, let the Radius A C of the circle in the third Chapter be given; and a Chord of 30 deg. required: the halfe of 30 deg. is 15 deg. the complement whereof is 75 deg. therefore I take the Radius with my compasses, and setting one foot in the Sine of 75 deg. with the other I lay the threed to the neerest distance: the threed lying still in this position, I take the shortest extent thereunto from the Sine of 30 deg. which giveth I C the Chord of 30 deg. which was desired.

Now by the converse of this Chapter, if you have the Chord of any arch given, you may thereby find out the Radius.

CHAP. VI. To divide a line by extream and mean proportion.

A Right line is said to be divided by extream and meane proportion, when the lesser Segment thereof, is to the greater, as the greater is to the whole line.

Let A B be the line to be so divided, this line I take with my compasses, and setting one foot in the sine of 54 deg. and with the other I lay the threed to the neerest distance: which lying still in this position, I take it over from the sine of 30 deg. which distance shall be the greater segment A C dividing the whole line in the point C; or the threed lying in the former position, if you shall take the shortest extent thereunto from 18 deg. you shall have B C for the lesser segment, which will divide the whole line by extream and mean proportion in the point C from the end B, so that as B C the lesser segment, is to A C the greater segment; so is A C the greater segment, to A B the whole line, as was required

CHAP. VII. To find a mean proportionall line between two right lines given.

A Mean proportionall line is that, whose square is equall to the long square, contained under his two extreams.

First, ioyn the two given lines together, so as they may make both one right line; the which divide into two equall parts; and with the one halfe thereof, setting one foot in the Sine of 90 deg. with the other lay the threed to the nee­rest extent, which lying still in this position, take the di­stance betwixt the middle point, & the point of meeting of the two given lines, and fixing one foot in the line of sines, so as the other may but onely touch the threed; now from the complement of the sine where the fixed foot so resteth, take the shortest extent unto the threed, which shall be the mean proprtional line required.

As for example, let A and B be two lines given, between which it is required to find a mean proportionall line, first joyne the two lines together in F, so as they both make the right line C D, which divide into two equall parts in the point E, then with either halfe of which, setting one foot in the sine of 90 deg. with the other lay the threed to the neerest distance: then keeping the threed in this positiō, take the distance betweene the middle point E and F, the place of meeting of the two given lines, and fixing one foot in the line of sines, so as the other may but onely touch the threed, and the fixed foot will stay about 22 deg. 30 min. the complement whereof is 67 deg. 30 min. from which take the shottest extent unto the threed lying as before, which shall be the line H, the meane proportional line be­twixt the two extreames A and B, which was required.

CHAP. VIII. Having the distance of the Sun from the next equinoctiall point, to find his declination.

FIrst, lay the threed upon 23 deg. 30 min. the suns great­est declination, counted on the limbe of the quadrant, the threed lying still open at this angle, take the shortest ex­tent thereunto from the sine of the distance of the Sunne from the next equinoctiall point, this distance being ap­plied to the line of sines from the centre A, shall give you the sine of the suns declination.

So in the figure of the 13 chapter, the sun being in the 29 deg. of Taurus at K, which is 59 deg. from C the equi­noctiall point Aries; the declination of the sun will be found about 20 deg. the line C M which was required.

CHAP. IX. The declination of the sun, and the quarter of the ecliptick which he posseseth being given, to find his place.

TAke the sine of the suns declination from the line of sines, & setting one foot in the sine of the suns greatest declination, with the other lay the threed to the neerest di­stance so shall it shew upon the limb, the distance of the sun from the next equinoctiall point.

So in the figure of the 13 chapter C M the declination of the sun being 20 deg. and K the angle of the suns great­est declination, the line C K will be found to be 59 deg. for the distance of the sun from the next equinoctiall point which was required.

CHAP. X. Having the latitude of the place, and the distance of the sun from the next equinoctiall point, to find his amplitude.

TAke the sine of the suns greatest declination betwixt your compasses, and setting one foot in the co-sine of the latitude, with the other lay the threed to the neerest di­stance; which lying still in this position, set one foot in the sine of the suns distance from the next equinoctiall point, and with the other take the neerest extent unto the threed, so shall you have betwixt your compasses the Sine of the Amplitude.

As in the figure of the 13 chapter, the angle at N being 37 deg. 30 min. the complement of the latitude, and K the angle of the Suns greatest declination, and C K 59 deg. the distance of the Sun from the equinoctiall point Aries, the line C N will be found to be the Sine of 34 deg. 9 min. the amplitude required

CHAP. XI. Having the declination and amplitude to finde the height of the pole.

FIrst, take the sine of the Suns declination, and set one foot in the sine of the Amplitude, and with the other lay the threed to the neerest distance, so shall the threed upon the limbe, shew the complement of the latitude.

So in the figure of the 13 Chapter, the declination C M being 20 deg. and the amplitude C N being 34 deg. 9 min. & the angle at M being Right, we shall find the angle at N [Page 126] to be 37 deg. 30 min. the complement whereof is 52 deg. 30 min. which was required for the latitude of the place.

CHAP. XII. Having the latitude of the place, and the declination of the Sun, to find his amplitude.

WIth the Sine of the declination set one foot in the co­sine of the latitude, and with the other lay the threed to the neerest distance: so shall it shew upon the limb the amplitude required: so in the figure of the next Chapter, the angle CNM being 37 deg. 30 min. the cosine of the la­titude, and C M the declination here 20 deg. and the angle at M being right, we shall finde the base C N to be the Sine of 34. which was required for the Suns amplitude.

CHAP. XIII. Having the elevation of the pole, and amplitude of the Sun, to find his declination.

FIrst, lay the threed to the amplitude counted in the limb, then take it over at the shortest extent, from the cosine of the latitude, so shall you have the Sine of the suns decli­nation betwixt your compasses.

So in this figure, the amplitude C N being. 34 deg. 9 min. and the angle at N being cosine to the latitude, the an­gle at M being a right angle, we shal find CM to be 20 deg. for the declination of the sun which was required.

CHAP. XIIII. Having the latitude of the place, and the declination of the Sun, to find his height in the Vertical circle.

FIrst, take the sine of the declination of the Sun, and setting one foot in the Sine of the latitude, with the o­ther lay the threed to the neerest distance; so shall it shew upon the limb the height of the Sun in the Verticall circle.

So in the figure of the last Chapter, the angle C I O being 52 deg. 30 min. the latitude of the place, and C O the Suns declination 20 degrees, and the angle C O I [Page 128] being a right angle we may find C I to be a sine of 25 deg. 32 min. the height of the sun in the Vertical circle which was required.

CHAP. XV. Having the latitude of the place, and the distance of the Sun from the next Equinoctiall Point, to find his height in the verticall circle.

FIrst, take the sine of the suns greatest declination, & set­ting one foot in the sine of the latitude, with the other lay the threed to the neerest distance: the threed lying still in this position; from the sine of the suns place take the nee­rest extent thereunto, which shall be the sine of the suns height in the Vertical circle.

So in the figure of the 13 chapter, the angle at I being 52 deg. 30 min. which is the latitude of the place, and the angle at K the suns greatest declination, and K C being 59 deg. the suns distance from the next equinoctiall point, we shall find C I to be 25 deg. 32 min. for the height of the sun in the Vertical circle.

CHAP. XVI. Having the latitude of the place and the declination of the Sun, to finde the time when the Sun commeth to be due east or west.

WIth the sine of the declination, set one foot in the sine of the latitude, and with the other lay the threed to the neerest distance: then take it over at the neerest extent [Page 129] from the co-sine of the latitude; which distance keep; and setting one foot in the co-sine of the declination, with the other lay the threed to the neerest distance: so shall it shew upon the limbe, the quantitie of degrees betwixt the houre of six and the East or West points.

So in the figure in the 13 Chapter, the declination C O being 20 deg. and the angle O I C being 52 deg 30 min. the complement whereof is the angle O C I, we may find the sine O I which distance keep; now seeing O I is a sine of the Radius O F & not of AEC, therefore by the 4 Chapter, you may find the quantity of that unknown sine; for seeing the Radius O F is the cosine of the declination, therefore set one foot therein, and with the other distance kept, lay the threed to the neerest distance: so shall it shew upon the limbe 16 deg. 30 min. which converted into time maketh 1 houre, and 6 min. for the quantitie of time betweene the hour of six and the suns being in the East or West points.

CHAP. XVII. Having the latitude of the Place, and the declination of the sun, to find his altitude at the houre of six.

FIrst, take the threed, and lay it upon the declination counted in the limbe; then from the sine of the latitude, take it over at the shortest extent; which distance shall be the sine of the height of the sun at the houre of six.

So in the figure of the 13 Chapter, the angle at L being a right angle, and L O C being 52 deg. 30 min. the latitude of the place, and C O the declination of the sun being 20 deg. we shall find C L to be the sine of 15 deg. 44 min for the height of the sun at the houre of six, which was equired.

CHAP. XVIII. Having the latitude of the place, and the height of the sun at the hour of six, to find what azimuth he shall have at the houre of six.

FIrst, with the sine of the suns height at the houre of six, set one foot in the sine of the latitude, and with the other lay the threed to the neerest distance: then take the least distance thereunto from the cosine of the latitude, now with this distance setting one foot in the cosine of the alti­tude, with the other lay the threed to the neerest distance as before: so shall it shew upon the limbe, the azimuth of the sun from the East or West points.

So in the figure of the 13 chapter, the angle CLO being a right angle, and the angle L C O being 37 deg. 30 min. the cosine of the latitude, the angle L O C must be 52 deg. 30 min. the latitude of the place being the complement of the angle L C O, and C L being 15 deg 44 min. (as by the last chapter it did appeare) we shall find L O to be the sine of 12 deg. 30 min. for the azimuth of the sun from the East or West, at the hour of six as was required.

CHAP. XIX. Having the declination of the sun, to finde his Right Ascension.

FIrst, with your compasses take the sine of the suns de­clination given, and setting one foot in the sine of the suns greatest declination, with the other lay the threed to [Page 131] the neerest distance: then at the least distance from the co­sine of the suns greatest declination take it over: now again, with this distance lay the threed to the neerest distance from the cosine of the declination given, so shall it shew upon the limbe the right ascension of the sun.

So in the figure of the 13 chapter, C O the suns declinati­on being 20 deg. and the angle O K C being 23 deg. 30 min. the suns greatest declination, and the angle K C O being the complement of the angle O K C we shall find K O to be the sine of 56 deg. 50 min. for the right ascen­sion of the sun required.

CHAP. XX. Having the latitude of the place, and the declination of the Sun, to finde the Ascensionall difference.

FIrst, take the sine of the Suns declination, and setting one foot in the co-sine of the Latitude, with the other lay the threed to the neerest distance: then at the least di­stance take it over from the sine of the Latitude: with which, setting one foot in the co-sine of the declination, with the other lay the threed again to the neerest distance, so shall it shew upon the limbe the Suns Ascensionall dif­ference.

So in the figure of the 13 Chapter, the angle M C N being 52 deg. 30 min. and the angle C N M being the complement thereof, the one being the latitude, and the other the co-latitude, and C M being 20 deg. the sine of the Suns declination, we shall finde M N 28 deg. 19 min. for the difference of Ascensions, which being converted [Page 132] into time, maketh 1 houre, and somthing better then 53 min.

Now when the Sun hath North declination, if you take this difference of Ascension (which is 1 houre 53 min.) out of 6 houres, there will be left 4 hours 7 min. for the time of Sun rising, and if you adde it unto 6 hours, the same will be 7 houres 53 min. for the time of Sun setting.

And so contrarily, when the Sun hath South declinati­on, if you adde this ascensionall difference to 6 hours, you shall have the time of his rising, and if you take it away from 6 hours, that which is left shall be the time of Sun setting.

CHAP. XXI. The Latitude of the place, the Almicanter, and declination of the Sun being given, to finde the Azimuth.

IF the Suns declination be Northward, then by the 14 or 15 Chapters get his height in the Verticall Circle for the day proposed: from the sine of which take the distance unto the sine of the Suns altitude observed: with this di­stance, setting one foot in the co-sine of the Latitude, with the other lay the threed to the neerest distance; unto which (being kept still in this position) take the least di­stance from the sine of the Latitude, with this distance, set­ting one foot in the co-sine of the Suns altitude, with the other lay the threed again to the neerest distance, so shall it shew upon the limb the Suns Azimuth from the East or West, either Northward or Southward.

So in this figure, having N M the distance betwixt the [Page 133] sine of 14 deg. 33 min. (the height of the Sun in the Verticall circle) and the sine of 30 deg. 45 min. the height of the Sun at the time of observation, and 52 deg. 30 min.

the angle N O M the Latitude of the place, the comple­ment whereof is 37 deg. 30 min. the angle M N O, we shall finde M O to be the sine of 23 deg. 17 min. the Azi­muth from the East or West points Southward.

And here note, when the declination is Northward, that as when the latitude of the Sun given, and his height in the [Page 134] Verticall circle is equall, he is directly in the East or West, so when his altitude given is greatest, then is the Azimuth toward the South, and when his altitude given is least, then is the Azimuth towards the North.

But if the declination of the Sun be Southward, then by the 10 or 12 Chapters, finde the Amplitude for the day proposed.

Now first, take the sine of the Suns altitude, and setting one foot in the co-sine of the Latitude, with the other lay the threed to the neerest distance, which threed lying still in this position, take it over at the shortest extent from the sine of the Latitude, this distance adde to the sine of the Amplitude, by setting one foot in the sine of the Ampli­tude, and extending the other upon the line of sines, these two being thus joyned, take them betwixt your Compas­ses, setting one foot in the co-sine of the Suns altitude, and with the other lay the threed to the neerest distance: so shall it shew upon the limb the Suns Azimuth from the East or West, towards the South.

So in this figure, having V C or T N, 19 deg. 7 min. the Amplitude for the day proposed, and T V the sine of the Suns altitude being 13 deg. 20 min. and 52 deg. 30 min. the angle V X T, the latitude of the place; and the angle T V X, the complement thereof; we shall finde X N to be the sine of 40 deg. 11 min. the Azimuth of the Sun from the East or West points Southward, which was required.

CHAP. XXII. The latitude of the place, the declination and altitude of the Sun being given, to finde the houre of the day.

IF the declination of the Sun be Northward, finde the height of the Sun at the houre of six by the 17 Chapter, betwixt which sine, and the sine of the Suns altitude given, take the distance upon the line of sines, with which di­stance, setting one foot in the co-sine of the latitude, with the other lay the threed to the neerest distance, the threed lying still in this position, take it over at the shortest ex­tent from the sine of 90 deg. with this distance, setting one foot in the co-sine of the declination, with the other lay the threed again to the neerest distance: so shall it shew upon the limb the quantity of time from the houre of six.

So in this figure, having M N the distance betwixt the sine of 9 deg. 5 min (the height of the Sun at the houre of six) and the sine of 42 deg. 33 min. the height of the Sun given, and the angle M O N 52 deg. 30 min. the La­titude of the place, and his complement M N O, we shall finde N O to be the sine of 60 deg. the quantity of time from the houre of six, which 60 deg. is four hours of time. And here also note, that if the altitude given be greater then the altitude of the Sun at the houre of six, then is the time found to the Southward of the houre of six; but if it be lesser then is it to the Northward.

But if the declination of the Sun be Southward, finde his depression at the houre of six, by the 17 Chapter, for [Page 136] the day proposed, which will be equall to his height at six, if the quantity of declination be alike.

Now take the sine of this depression, and adde it to the sine of his altitude observed, by setting one foot in the sine of his altitude, and extending the other upon the line of sines: These two being thus joyned together in one, take them betwixt your compasses, and setting one foot in the co-sine of the latitude as before, and with the other, lay the threed to the neerest distance: which lying still in this po­sition, take it over at the shortest extent from the sine of [Page 137] 90 deg. with this distance, setting one foot in the co-sine of the declination as before, with the other lay the threed again to the neerest distance: so shall it shew upon the limb the quantity of time from the houre of six.

So in this figure, having the sine of 15 deg. 24 min. the altitude of the Sun given, and the sine of 9 deg. 5 min. his depression at the houre of six, joyned both together in one streight line, as T V, and having the angle T X V 52 deg. 20 min. the Latitude given, and the angle T V X the co-latitude, we shall find T X to be the sine of 45 deg. the quantity of time from the houre of six, which con­verted into time will make three houres.

CHAP. I. How to examine a plane for an Horizontall Diall.

IF your plane seem to be levell with the Horizon, you may try it by laying a Ruler thereupon, and applying the side A B of your Quadrant to the under side thereof, and if the threed with the plummet doth fall di­rectly upon his levell line A C; which way soever you turn it, it is a horizontall plane. Or if you set the side A B of your Quadrant upon the upper side of your Ruler, so that the centre may hang a little over the end of your ruler, and holding up a threed and plummet, so that it may play upon [Page 139] the centre, if it shall fall directly upon his levell line A C, making no angle therewith, it is an horizontall plane, as here you may see by this figure.

CHAP. II. Of the trying of planes, whether they be erect or inclining, and to finde the quantity of their inclination.

IF the plane seeme to be erect, you may try it by hold­ing the Quadrant, so that the threed may fall on the plumb line A C, for then if that side of the Quadrant shall lie close to the plane, it is erect, and a line drawn by [Page 140] that side of the Quadrant shall be a Verticall line; and the line which crosseth this verticall line at right angles, will be the Horizontall line, as here you may see in this figure, the plane D E F G being erect, and the line D E being verticall, the line F G must be horizontall. But if the plane shall incline, the quantity of inclination may be found out after this manner.

First, you must draw thereon the horizontall line, which you may doe upon the under face, by applying the side A B of your Quadrant thereunto, so as the threed and plummet may fall upon the plumb line A C, the side A B lying close with the plane, by which if you draw a line, it shall be parallel to the Horizon.

Or you may draw a horizontall line upon the upper face, by laying a Ruler thereupon, and applying the side A B of your Quadrant to the under side thereof, still moving your Rule, untill the threed and plummet doth fall directly upon the plumb line A C, the Rule lying thus close to the plane, you may thereby draw a line parallel to the Hori­zon.

Having drawn this horizontall line M N, crosse it at right angles with the perpendicular K D; unto which, if it be the under face, apply the side A B of your Quadrant, so shall the threed upon the limb give you the angle of in­clination required.

But if it be the upper face of the plane, then lay a Ruler to the perpendicular K D; unto the under side whereof, apply the side A B of your Quadrant, as is here shewed in this figure, so shall the degree of the Quadrant give you C A H, the angle of inclination required.

But if it be so, that you cannot apply the side of your Quadrant to the under side of your Ruler, then set it upon [Page 141]

the upper side thereof, so that the center thereof may hang a little over the end of the Ruler, and holding up a threed and plummet, so that it may fall upon the center A, and it shall shew upon the limbe, the inclination of the plane, which is the angle C A H, equall to the former angle.

Here you must be needfull that both edges of your Ru­ler be streight, and one parallel to the other.

CHAP. III. To finde the Declination of a plane.

TO finde out this declination you must make two ob­servations by the Sun: the first is of the angle made between the horizontall line of the plane, and the Azi­muth wherein the Sun is at the time of observation: the second is of the Suns altitude: both these observations should be made at one instant.

First, for the horizontall distance, having drawn upon your plane a line parallel to the Horizon, apply the side of your Quadrant thereunto; holding it parallel to the Ho­rizon, then holding up a threed and plummet at full liber­ty, so as the shadow thereof may passe through the centre of the Quadrant, observe the angle made upon the Qua­drant by the shadow of the threed, and that side with the horizontall line, for that is the distance here required.

Then at the same instant, as neer as may be, take the Suns altitude, that so you may finde the Suns Azimuth from the East or West points; by the 21 Chapter of the fourth Book.

Having thus gotten the horizontall distance, with the Azimuth of the Sun for the same time, describe a Circle as A B C D, representing the horizontall circle, and draw the diameter A C, which shall represent the horizontall line F G of the last Chapter. Now supposing the horizontall distance to be 38 deg. 30 min. the angle O A B of the last Chapter, place it from C Southward to E (that is from the same end of the horizontall line with which the angle was made upon the plane) and draw the line E Z: Then [Page 143] supposing the altitude of the Sun at the same time to be 30 deg. 45 min. with 11 deg. 30 min. North declination, and so by the 21 Chapter of the fourth Book, the Azimuth will be found to be 23 deg. 17 min. from the East South­ward, being the observation was made in the forenoon: this 23 deg. 17 min. I place from E (the place of the Sun at the time of observation;) unto R, (which is the true point of the East,) and draw the line H R representing the Verticall Circle, so shall the angle made between the ho­rizontall line of the plane, and the line of East and West, be the declination of the plane, which in this example is found to be 15 deg. 13 min. the angle C Z R.

Or you may observe the angle made between the sha­dow of the threed, and that side of the Quadrant which lyeth perpendicular unto the horizontall line of the plane, which in this example is 51 deg. 30 min. the complement of the former angle, and it is the angle O A C in the former Chapter upon the Quadrant.

Now having drawn your Horizontall Circle as before, and the diameter A C for the horizontall line of the plane, you may crosse it at right angles with the diameter B D, for the Axis of the planes horizontall line, from which as from D, you may set your horizontall distance on the same side thereof, as before you found it by your observa­tion, as here from D to E, and draw the line E Z for the line of the shadow, and having found the Azimuth of the Sun 23 deg. 17 min. from the East Southward, you may set it from E (the place of the Sun) Northward to R, and draw the line R Z H for the line of East and West, as be­fore.

Or if you take the Suns Azimuth from the South, which in this example will be 66 deg. 43 min. the com­plement of the former 23 deg. 17 min. you may set it from E (the place of the Sun) unto S Southward, and draw the line S Z N for the Meridian, so shall the arch S D or R C be 15 deg. 13 min. for the declination as before.

CHAP. IV. To draw the houre-lines upon the Horizontall, the full North or South planes, whether erect or inclining.

SEeing the making of these Dials are all after one man­ner, we will here proceed to make an Horizontall Diall, [Page 145] by help of the lines upon the Quadrant, fitted for that pur­pose.

Therefore having by the 10 Chapter of the third book found the elevation of the pole above the plane, we may proceed after this manner.

First, draw the line D A F of sufficient length, out of the middle whereof let fall the perpendicular A B for the Meridian and Substile, then take the line D E or E F out of your Quadrant, and set it from A to B in the Meridian; through which point B draw the line E B C parallel to D A F, now supposing the elevation of the pole above the plane, to be 52 deg. 30 min. the latitude of the place; from the Sine thereof take the neerest extent unto A E the line of latitudes, and set it from A to D, and from A to F both wayes, and from B to C, and from B to E, and draw the lines D C and E F making the long square C D F E: the two angles whereof C and E shall be the points for the hours of 3 and 9 in all these kinde of planes that declines not from the North or South.

Then applying the threed to the first houre-point in the limbe B C or D F, as to m or g, it will cut the tangent line D E in 5, then take the distance D 5, and set it down here from D to 5, and from F to 7; with this distance setting one foot in the Sine of 90 deg. with the other lay the threed to the neerest distance; unto which take the shortest extent from the Sine of the elevation of the pole above the plane: this distance set from B to 1, and from B to 11, then again apply your threed to the next hour in the limbe as at n or h, and it will cut the tangent line D E at 4 there­fore take D 4 from your Quadrant, and set it from D to 4, and from F to 8, with this distance from the Sine of 90 deg. open the threed as before, and take it over from the [Page 146] Sine of the height of the Stile, this distance prick down from B to 2, and from B to 10, so have you all the houre-points pricked down; by which and the centre A you may

draw all the hour-lines, as here you see done, the line A B for 12, and the line D A F for the two sixes.

For the hour-lines before six and after, you may extend their opposite houre lines beyond the centre as was shewed in the 8 chapter of the third book.

What is here shewed concerning the hours, the like may be understood for the half hours, by applying the threed thereunto in the limbe.

CHAP. V. To draw a Diall upon a South or North erect declining plane.

IN the drawing of all these kinde of Dials by help of this Quadrant, when the Latitude of the place, and the de­clination of the plane is known, two things more is to be considered: First, the elevation of the Pole above the plane,: Secondly, the inclination of the Meridian of the plane, to the Meridian of the place; both which will spee­dily be found when you are ready for them.

First therefore, draw the line D A F as before, from the middle whereof let fall the perpendicular A B for the sub­stilar, and at the distance of the tangent line D E, draw the line C B E parallel to the line D A F.

Now to finde the elevation of the pole above the plane, lay the threed upon the co-sine of the Latitude counted on the limb, and take it over at the neerest extent from the co-sine of the declination, which distance shall be the sine of the elevation of the pole above the plane.

So the declination of the plane being 32 deg. in the La­titude of 52 deg. 30 min. the elevation of the Pole above the plane will be 31 deg. 5 min. from the sine of which take the neerest extent unto the line of Latitudes, this di­stance set from A to D and F both wayes, and from B to C and E, and draw the lines D C and E F, making the long square C D F E, as in the former Chapter.

For the inclination of Meridians, take the sine of the declination of the plane, and setting one foot in the co-sine of the stiles elevation, with the other lay the threed to the [Page 148] neerest distance, so shall it shew upon the limb the inclina­tion of Meridians to be 38 deg. 13 min.

The threed lying still in this position, observe which of the houres and where it cutteth; which will be the slope line n I in the point y; to this point y set the bead, which by this means is fitted to the description of this Diall: the threed lying still in the former position, you shall see it cut the tangent D E in the point S upon the Quadrant, there­fore take D S with your Compasses, and prick it down here from D to 6, with this same distance from the sine of 90 deg. open the threed to the least distance, and taking [Page 149] it over from the sine of the height of the stile; you shall have the distance from B to 12. And here note, that as in the 13 Chapter of the third Book, the substile is placed on that side the Meridian which is contrary to the planes de­clination; so here the Meridian is placed on that side the substile whereon the declination of the plane is.

The bead being thus fitted, apply it to every houre-line by removing the threed, as first, I remove it to the lines m h and B g, and it will cut the line D E upon the Qua­drant in r and q, therefore I take D r and D q, and prick them down from D to 5 and 4, with these same distances open the threed as before from the sine of 90; and by ta­king it over from the sine of the height of the stile, you shall have the distances B 11 and B 10: again, the bead being applyed to the line k E; the threed will cut the co-tangent line in T, therefore take F T from your Quadrant, and prick it down here from F to 1, with this fame di­stance, open the threed from the sine of 90 deg. as before, and by taking it over from the sine of the height of the pole above the plane, you shall have the distance B 7: then again, the bead being applyed to the lines l O and F P, the threed will cut the co-tangent line F E in V and W, there­fore take F V and F W, and prick them down here from F to 2 and 3, with these distances open the threed from the sine of 90 deg. and take them over from the sine of the height of the stile, so shall you have the distances B 8 and B 9, thus have you the twelve houres pricked down, by which points and the centre A, you may draw the houre-lines, as here you see.

In the like manner may the half hours be supplyed.

The Diall being thus drawn on paper, you must place it so upon the plane, that the twelve a clock houre-line may [Page 150] be perpendicular unto the Horizon, according to the 11 or 15 Chapters of the third Book.

Note, that if the inclination of Meridians shall be more then 45 deg. so that the threed doth cut the co-tangent line F E, then you must take the distance from F to the threed, and prick it down either from D or from F upon the line D C or F E for the twelve a clock point, according as the plane shall decline either Eastward or Westward, and his parallel taken from the sine of the height of the stile, shall give the distance from B to 6, and so of the rest.

CHAP. VI. To draw a Diall upon an East or West Inclining plane.

IN these planes, as in the former, when we have the lati­tude of the place and the inclination of the plane, we have two things more to consider before we can draw the houre-lines upon the plane.

First, the elevation of the Pole above the plane: Se­condly, the inclination of both Meridians.

For the elevation of the Pole above the plane, lay the threed upon the Latitude counted in the limb, and take it over at the neerest extent from the co-sine of the inclina­tion, which distance shall be the sine of the elevation of the Pole above the plane.

So in the Latitude of 52 deg. 30 min. if a plane shall in­cline 40 deg. to the Horizon, the height of the stile will be 37 deg. 26 min. with which you may proceed to make your parallelogram or long square, as in the former Chap­ters.

Then for the inclination of Meridians, take the sine of the inclination of the plane with your Compasses, and setting one foot in the co-sine of the the Poles elevation above the plane, with the other lay the threed to the nee­rest distance, and it will shew upon the limb the inclination required.

Thus in the latitude of 52 deg. 30 min. if a plane shall in­cline 40 deg. to the Horizon, the inclination of both me­ridians,

[Page 152] will be 54 deg. 2 min. The threed lying still up­on this inclination of Meridians, you may see both which and where it cutteth the houre-lines, and so accordingly rectifie the bead as before was shewed.

And you may also see where the threed cutteth the co­tangent line E F, that so you may take the distance from the point F upon the Quadrant, unto the point of intersection of the threed with the co-tangent line; this distance you must set here from D to 12, and with the same, open the threed from the sine of 90 deg. as before, and take it over from the sine of the height of the stile, which shall be the distance from B to 6, and so you may proceed to prick down the rest of the hours as in the last Chap was shewed.

In all these planes you must place the line of twelve parallel to the Horizon, according to the 17 Chapter of the third Book, in which Chapter is fully shewed the true scituation of this Diall upon the plane.

CHAP. VII. To draw a Diall upon a declining inclining plane.

IN the making of these kinde of Dials by this Quadrant, when the latitude of the place, and the declination, and inclination of the plane is known, there is six things more to be considered before we can come to the drawing of this Diall upon the plane.

• 1 The inclination of the plane to the Meridian.
• 2 The Meridians ascension.
• 3 The elevation of the Meridian.
• 4 The position latitude.
• [Page 153]5 The elevation of the pole above the plane.
• 6 The inclination of Meridians.

All these six may speedily be found out upon the Qua­drant after this manner.

CHAP. VIII. In any erect declining Diall, having the distance of the substile from the Meridian, in a known Latitude; how thereby to get the Cocks elevation, and the declination.

IN any of these Dials, if the Cock be lost, you may here­by get the height thereof again, and make it anew, for though it be gone, the substile where it stood will remain.

First, therefore get the quantity of the angle of deflexiō by the second Chapter of the second Book, which is the angle included between the line of 12 and the substilar, which admit to be 22 deg. 8 min. as in the Diall of the 15 Chap. of the third Book.

This being found, take the sine of the Latitude betwixt your Compasses, and setting one foot in the co-sine of the angle of deflexion found, with the other lay the threed to the neerest distance: so shall it shew upon the limb the ele­vation of the Pole above the plane.

So in the Latitude of 52 deg. 30 min. the distance of the substile from the Meridian, being 22 deg. 8 min. as in the said 15 Chap. the elevation of the Pole above the plane will be 31 deg. 3 min. by which you may fashion a new Cock to the Diall at your pleasure.

To finde the Declination.

Take the sine of the height of the stile, and setting one foot of your Compasses in the sine of the complement of the Latitude, with the other lay the threed to the neerest distance: so shall it shew upon the limb the complement of the declination of the plane. So shall you finde the de­clination of the former plane to be 32 deg.