The art of dialling by a new, easie, and most speedy way. Shewing, how to describe the houre-lines upon all sorts of plaines, howsoever, or in what latitude soever scituated: as also, to find the suns azimuth, whereby the sight of any plaine is examined. Performed by a quadrant, fitted with lines necessary to the purpose. Invented and published by Samuel Foster, professor of astronomie in Gresham Colledge.

THE DESCRIPTION OF THE QVADRANT, and the manner how the lines are inscribed and divided.

CHAP. I.

1. The description of the fore-side.

THe limbe is divided into 90 de∣grees, and subdivided into as ma∣ny parts as quantity will give leave. The manner of division, and di∣stinction of the subdivided parts is such as is usuall in all other Qua∣drants.

To describe the other Worke in the superficies; Take from the upper edge of the limbe about 3 de∣grees, and set off that space from the center R to A.

Then divide AE into seven parts, whereof let EB containe two. Or in greater instruments, if AE be 1000. let EB containe 285. Make SC equall to EB, and drawe the line BC. From C, draw CD

parallel, and of equall length to AB. Upon AB and CD, and BE also (as farre as it is capable) insert the 90 sines, from B towards A and E, and from C towards D, but let them be numbred from A un∣to B to 90, and so to E 113 degrees 30 minutes, from D to C unto 90 degrees.

Againe: Draw ES cutting CD at F; so shall BCFEB containe a parallelogram, whose opposite sides, being parallel, are divided alike, and in this manner. BE and CF as whole sines, doe containe the 90 sines, or as many of them as can distinctly bee put in: and from the divisions are drawne parallel lines, having every tenth, or fifth, distinguished from the rest. These serve for the 12 Signes and their degrees, and therefore you see upon every 30th degree, the characters of the 12 Signes inser∣ted, in such manner as the figure sheweth. And these lines may bee called, The Parallels of the Suns place.

In like manner, The lines BC, EF, being first bisected at X and Z, shall make 4 lines of equall length. These 4 lines XB, XC, ZE, and ZF, are each of them divided as a scale of Sines, begin∣ning at X and Z, and from each others like parts are parallel lines protracted, having every tenth and fifth distinguished from the rest. They are numbred; upon BC, from B to X 90, to C 180; upon FE, from F to Z 90, to E 180. These lines are called, The lines of the Sunnes Azimuth.

This done; Upon the center R describe the two quadrants VT, and BC, let their distance VC

bee one sixth part of Rc, or more if you will.

Divide them each into 6 equall parts, at e, o, y, n, s; and a, i, u, m, r, drawing slope-lines from each others parts, as Va, ei, on, ym, nr, sb: and these lines so drawne are to bee accounted as Houres. Then dividing each space into two equall parts, draw other slope-lines standing for halfe houres, which may be distinguished from the o∣ther, as they are in the figure.

Then from the points V and T draw the right line VT.

Lastly, Having a decimall scale equall to TR, you must divide the same TR into such parts as this Table here following alloweth, the numbers beginning at T, and rising upto 90 at R.

Vpon your instrument (for memory and dire∣ctions sake) neere to the line AB, write, The summe of the latitude and Sunnes altitude in Summer; The difference in Winter.

Over VT, write, The line of Houres.

Neere to CD write, The summe of the latitude and Sunnes altitude in Winter; The difference in Summer.

By TR, write, The line of latitudes for the deli∣neation of Dialls.

A Table to divide the line of Latitudes.

90	10000	62	9360	46	8259	30	6325	14	3325
85	9982	61	9311	45	8165	29	6169	13	3104
80	9924	60	9258	44	8068	28	6010	12	2879
78	9888	59	9203	43	7968	27	5846	11	2650
76	9849	58	9147	42	7865	26	5678	10	2419
75	9825	57	9088	41	7738	25	5505	9	2186
74	9801	56	9026	40	7647	24	5328	8	1949
72	9745	55	8962	39	7532	23	5146	7	1711
70	9685	54	8895	38	7414	22	4961	6	1470
69	9651	53	8825	37	7292	21	4772	5	1228
68	9615	52	8753	36	7166	20	4577	4	984
67	9378	51	8678	35	7036	19	4378	3	739
66	9519	50	86••••	34	6902	18	4176	2	493
65	9496	49	8519	33	6764	17	3969	1	247
64	9454	48	8436	32	6622	16	3758	0	••
63	9408	47	8348	31	6475	15	3543		S•…•…

2. The description of the backe-side.

Upon the backe-side is a circle only described, of as large extent as the Quadrant will give leave, noted with ABCD, divided into two equall parts by the Diameter AC.

The semicircle ABC is divided into 90 equall parts or degrees, every fifth and tenth being di∣stinguished from the rest by the longer line; They are numbred by 10, 20, 30, &c. unto 90. The same parts are also projected upon the diameter AC, by a ruler applyed to them from the point D. These are numbred also from A to C by 10, 20, &c. unto 90.

The other semicircle ADC, is first divided in∣to two Quadrants at D. And then upon these two

quadrants are inscribed 90 such parts as this Ta∣ble insuing doth allow. The inscription is made by helpe of a Quadrant of a circle equall to AD or CD, being divided into 45 equall degrees, out of which you may take such parts as the Table gi∣veth, and so pricke them downe, as the figure shew∣eth. Every fifth and tenth of these parts is distin∣guished from the rest by a longer line; they are numbred from A and C, by 10, 20, &c. unto 90 ending in D.

A Table to divide the upper and nether Quadrants of the Circle.

1	1.00	14	13.36	27	24.25	40	32.44	53	38.37	66	42.25
2	2.00	15	14.31	28	25.09	41	33.16	54	38.59	67	42.38
3	3.00	16	15.25	29	25.52	42	33.47	55	39.19	68	42.50
4	3.59	17	16 18	30	26.34	43	34.18	56	39.40	69	43.02
5	4.59	18	17.10	31	27.15	44	34.47	57	39.59	70	43.13
6	5.58	19	18.02	32	27.55	45	35.16	58	40.18	72	43.34
7	6.57	20	18.53	33	28.35	46	35.44	59	40.36	74	43.52
8	7.55	21	19.43	34	29.13	47	36.11	60	4054	75	44.00
9	8.53	22	20.32	35	29.50	48	36.37	61	41.10	76	44.08
10	9.51	23	21.21	36	30.27	49	37.03	62	41.27	78	44.22
11	10.48	24	22.08	37	31.02	50	37.27	63	41.43	80	44.34
12	11.45	25	22.55	38	31.37	51	37.51	64	41.57	85	44.53
13	12.41	26	22 40	39	32.1••	52	38.15	65	42.11	90	45.00

Thus have you both sides decribed. Besides all this, there are two sights added, with a threed and plummet like as in other instruments. The threed hath a moovable bead upon it for speciall use. The same threed passeth through the center R. quite behind the Quadrant, and is hung upon a pinne at the bottome of the Quadrant, noted with W. The

reason of the threeds length will be seene when wee come to the uses of the instrument.

CHAP. II.

The use of the Quadrant in generall.

FIrst upon the fore-side. The limbe serveth especially for observation of all necessa∣ry angles.

The lines AE, CD, with the Parallelo∣gram BCEF, are to find out the Suns Azimuth in any latitude whatsoever.

The slope-lines within the arkes VT, cb, by helpe of the threed and bead, doe serve artificial∣ly to divide the line of Houres TV, into its requi∣site parts; which together with TR the line of la∣titudes, doe serve to protract all plaine Dialls how∣soever scituated.

Secondly upon the back-side. Note that ABC is called the Semicircle: AC is called the Diame∣ter: AD the Vpper quadrant: CD the Nether qua∣drant.

The uses of these are to find out the necessary arkes and angles, either for preparation to the Di∣alls description, or serving after for the Dialls sci∣tuation upon the Plaine.

In all these uses the threed bearing part, and therefore having asufficient extent of length, that being loosed it may with facility reach over either side of the Quadrant.

CHAP. III.

To find the Azimuth of the Sunne in any Latitude whatsoever.

BEfore you can make any draught of your Diall, you must know the scitua∣tion of your plaine, both for declinati∣on and inclination. The best way to come to the plaines declination is by helpe of the Sunnes Azimuth.

By having the Latitude of the place; The place of the Sunne in the Eclipticke, and the altitude of the Sunne above the Horizon, you may find out the Azimuth thereof in this manner.

Adde the Sunnes altitude, and your latitude to∣gether, and substract the lesser of them from the greater; So shall you have the summe of them, and the Difference of them. With this summe and difference, come to your Quadrant, and according to the time of the yeare (as the lines will direct you) Count the said Summe and Difference respe∣ctively, and applying the threed unto them, find out the Sunnes place in the Parallels serving there∣to, and where the threed cuts this Parallel, ob∣serve the Azimuth there intersecting, for that is the Azimuth from the South, if you number it from the line whereon the summe was numbred.

Example 1. In the North latitude of 52 gr. 30 min. in the Summer-time the Sunne entring into 8, and the altitude being observed 30 gr: 45 min.

I adde the latitude 52 gr. 30 min. and the Sunnes altitude 30 gr. 45 min So I find the summe of them 83 gr. 15 min. and substracting the lesser of them from the greater, I find the difference of them 21 gr. 45 min. The summe I number in the line AE, and the difference in DC (because it is in Sum∣mer) and to the termes I apply the threed, and where it crosseth the parallel of the beginning of 8, there I meet with 66 gr. 43 min. which is the Azimuth from the South, being reckoned from the line AE whereon the Summe was counted.

Example 2. The latitude and Sunnes place being the same if the altitude had beene 9 gr. 15 min. The summe of the latitude and altitude would bee 61 gr. 45 min. The difference 43 gr. 15 min. and so the threed applyed to these termes would have shewed 96 gr. 52 min. for the Azimuth from the South.

A third Example. In the same Latitude of 52 gr. 30 min. in the Winter-time, the Sunne entring the tenth degree of ♏, and the altitude being 9 gr. 30 min. I would know the Azimuth of the Sun from the South. I adde the Altitude 9 gr. 30 min. to the Latitude 52 gr. 30 min. and so find the summe of them 62 gr. 0 min. And substracting the Altitude out of the Latitude, I find the Diffe∣rence of them 43 gr. 0 min. The summe (be∣cause it is in Winter) I count upon the line DC in the Quadrant, and the Difference upon AE. So the threed applyed to these tearmes cutteth the

tenth of ♏, at 49 gr. 50 min. which is the Azi∣muth numbred from DC the South.

The Amplitude.

Note here by the way, That the threed applyed to the Latitude of your place numbred upon both lines AE, DC, will shew you, for any place of the Sunne, the due Amplitude of his Rising or Setting, or the Azimuth whereon hee riseth or setteth, if you number the same from the middle line no∣ted with XZ which here representeth the East and West Azimuths.

CHAP. IIII.

To find out the Declination of a Plaine.

THe declination of a Plaine is numbred from the South or North points to∣wards either East or West. And it is the arke of the Horizon comprehended be∣tweene the South-North, and a line infinitely ex∣tended upon the Horizon perpendicular to the horizontall line of the Plaine; which line may be called the Axis, and the extremity of it, the Pole of the Plaines horizontall line.

To find out this declination you must make two observations by the Sunne: The first is of the Di∣stance or angle made betweene the Axis of the ho∣rizontall line of the Plaine, and the Azimuth wherein the Sunne is at the time of observation. The second is of the Suns Altitude. Both these

observations should bee made at one instant, which may bee done by two observers, but if they bee made by one, the lesse distance of time betweene them, will make the worke to agree together the better.

1. For the Distance. Upon your Plaine draw a line parallel to the horizon, to this line apply the side of your Quadrant, holding it parallel to the horizon. Then holding up a threed and plummer, which must hang at full liberty, so as the shadow of the threed may passe through the center of the Quadrant, observe the Angle made upon the Quadrant by the shadow of the threed, and that side that lyeth perpendicular to the horizontall line, for that angle is the Distance required.

2. At the same instant as neere as may be, take the Sunnes Altitude; These two being heedfully done, will helpe you to the plaines Declination by these rules following.

When you have taken the Altitude, you may find the Sunnes Azimuth by the former Chapter. Then observe, whether the Sunne bee betweene the Pole of the horizontall line and the

• South
• North

point or not.

If the Sunne be betweene them, adde the Azi∣muth and Distance together, and the summe of them is the Declination of the plaine.

If the Sunne be not betweene them, subduct the lesser of them from the greater, and the difference shall be the Declination of the plaine.

¶By your observation you may know to what coast a Plaine declineth.

For if the

• South
• North

point bee in the midst be∣tweene the Sunnes Azimuth and the pole of the Plaines horizontall line, then doth the Plaine de∣cline to the coast contrary to that wherein the ☉ is: If otherwise, the declination is upon the same coast with the Sunne.

CHAP. V.

To find the Inclination of a Plaine.

THe Inclination of a Plaine is the angle that it maketh with the Horizon. When you have described your horizontall line upon a Plaine, as in this figure EF,

[illustration]

crosse it with a perpendicular GH, for the Verti∣call line.

And because the inclinations of the Upper and Under faces of the Plaine, are both of one quanti∣tie in themselves, if therefore you apply the side of the Quadrant noted with AB unto the verticall line of the under face, or to the under side of a Ru∣ler applyed to the verticall line of the upper face, as is here shewed in this figure; Then shall the de∣grees of the Quadrant give you CAD the angle of inclination required.

CHAP. VI.

Of upright declining Plaines.

THose Plaines are upright, which point up directly into the Zenith or verticall point of the Horizon, and may be tryed by a perpendicular or plumb-line. In these, as in the rest that follow, before the Houres can be drawne, two things must bee found; 1. The Rectifying arke; 2. The Elevation of the Pole above the Plaine.

1. To find the Rectifying arke.

Extend the threed from your Latitude counted in the upper Quadrant of the circle on the backe∣side, to the complement of the Plaines declination numbred in the Semi-circle; so shall the threed shew you on the Diameter the Arke required.

2. To find the Elevation of the Pole above the Plaine.

Extend the threed from the Rectifying arke numbred in he upper quadrant, to your Latitudes complement taken in the Semicircle; so shall the threed shew upon the Diameter, the Elevation of the Pole above the Plaine.

According to these rules, in the latitude of 52 gr. 30 min. supposing an upright Plaine to decline 55. gr. 30 min. I find the Rectifying arke to bee 28 gr. 36 min. And the elevation of the Pole a∣bove the Plaine to be 20 gr. 10 minutes.

CHAP. VII.

In East and West incliners.

THose plaines are called East and West incliners, whose horizontall line lyeth full North and South, and their inclina∣tion is directly towards either East or West.

1. To find the Rectifying Arke.

Extend the threed from your Latitudes comple∣ment taken in the upper quadrant of the Circle on the backside, to the complement of the Plains incli∣nation counted in the semicircle; so shall the threed shew upon the Diameter the Arke required.

2. To find the Elevation of the Pole above the Plaine.

Extend the threed from the Rectifying-arke

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counted in the upper quadrant, to your latitude ta∣ken in the Semicircle; so the threed upon the Dia∣meter gives the elevation of the Pole above the Plaine.

Thus in the latitude of 52 gr. 30 min. If a Plaine incline Eastward 40 gr. to the horizon, the Recti∣fying-arke will be 35 gr. 58 min. And the elevati∣on of the Pole 37 gr. 26 min. above the plaine.

CHAP. VIII.

In North and South incliners.

SUch Plaines are called North and South incliners, whose horizontall line lyeth full East and West, and their inclination is directly upon either North or South.

1. For the Rectifying-Arke.

There is no use of it in these plaines, because they all lye directly under the Meridian of the place.

2. To find the Elevation of the Pole above the Plaine.

If the inclination be toward the South, adde the inclination to your latitude; for the summe is the Elevation of the pole above the Plaine. If the summe exceed 90 degrees, take it out of 180, and the supplement gives you the Poles elevation.

If the inclination bee Northward, compare the inclination with your latitude, and subduct the lesser out of the greater: the Difference is the eleva∣tion

of the Pole above the Plaine, If there bee no difference, it is a Direct polar Plaine.

CHAP. IX.

In declining Incliners.

THose Plaines are called Declining incli∣ners, whose horizontall line declineth from the East or West, towards either North or South, and their inclination also deflecteth from the coasts of North and South towards either East or West.

The best way to find the Rectifying-arke, and the poles elevation for these Plaines, will be

First, to referre them to a New latitude, where∣in they may lye as East or West incliners. For which purpose you are first to find out an Arke, which in respect of its use may fitly be called, The Prosthaphaereticall arke, it is found by this rule:

¶Extend the threed from the complement of the Plaines declination counted in the upper qua∣drant, to the inclination numbred in the Semi∣circle; so the threed shall give you upon the Dia∣meter the Prosthaphaereticall-arke required.

This Prosthaphaereticall-arke is to be used as the Inclination was in the precedent Chapter. For,

If the Plaine doe incline towards the South, it must be added to your Latitude: and so the summe (if lesse then 90 degrees) gives you the New Lati∣tude: but if the summe bee greater than 90, then the residue, or supplement of it to 180 degrees will be the New Latitude required.

If the Plaine incline toward the North, com∣pare this Prosthaphaereticall-arke with your Lati∣tude, and subduct the lesser of them out of the greater; So the Difference shall give you the New Latitude. If there be no difference, it is a declining Polar plaine.

Secondly, it will be required to know what In∣clination these Plaines shall have in this their New latitude; and that is done by this rule:

¶Extend the threed from the Prosthaphaereti∣call-arke taken in the upper quadrant to the Plaines declination counted in the Semicircle: so the threed shewes on the Diameter, the New-inclinati∣on in their New latitude.

Being thus prepared, you may now proceed as in East and West incliners you did before.

1. To find the Rectifying-Arke.

Extend the threed from the New latitudes com∣plement taken in the upper quadrant, to the New-inclinations complement numbred in the Semi∣circle; so the threed upon the Diameter shewes the Arke required.

2. To find the Elevation of the Pole above the Plaine.

Extend the threed from the Rectifying-arke in the Vpper-quadrant to the New latitude in the Semi∣circle; so the threed upon the Diameter gives the Elevation of the pole above the plaine.

According to these rules, supposing a Plaine to in∣cline towards the North 30 degrees, and to decline from the South towards the West 60 degrees in the latitude of 52 gr. 30 min. First I find the Prosthaphi-arke

60 gr. 6 min. and because the Plaine inclineth toward the North; I compare this arke with the Latitude of the place, and taking it out of the La∣titude there remaineth 36 gr. 24 min. for the New Latitude. Then I find the New inclination to bee 25 gr. 40 min. and so the Rectifying-arke 59 gr. 8 min. and the Elevation of the Pole above the Plaine to be 32 gr. 20 minutes.

CHAP. X.

To draw the Houre-lines upon the Horizontall, the full North or South plaines, whether standing upright or inclining.

IN the foure last Chapters we have seene the uses of the Circle on the backe-side of the Quadrant: in this and the next Chapter we shall shew the use of TR the line of latitudes, and of TV the line of Houres; which two lines with the helpe of the limbe VCTB, and of the threed and Bead, will serve to pricke downe any Diall, by the Precepts hereafter delivered. And first we begin with those Plaines which have no declination, whose Poles lye di∣rectly under the Meridian of the place; of which sort are the Horizontall, the Erect South and North plaines, with all Incliners looking directly North or South.

Having then by the former Precepts found the

Elevation of the pole above your Plaine, you may begin your draught in this manner.

First, Draw the line RAT of sufficient length, and out of the line of Latitudes in your Quadrant, take off the Elevation of the pole above the plaine, and pricke it downe from the point A, unto R and T both wayes.

2. Take the line of Houres TV also out of the Quadrant, and with that extent of your Com∣passes upon R and T as upon two centers, draw the arkes BV and CV, crossing each other in V; and draw the lines RV and TV: then comming to your Quadrant againe;

3. Apply the threed to every houre point in the limbe VT or CB, as first to s, or r, so shall it cutte the Line of houres TV in 1; Then take off with your Compasses T1, and pricke it downe here from V to 1, and from T to 7. Again, Apply your threed to the next houre in the limbe at n or m, it will cut the Line of houres TV in 2 take off T2, and prick it down here from V to 2, and from T to 8. So againe, the threed applyed to the third noure at y, or u, cuts the line TV, in 3; take off T3, and pricke it downe here from V to 3, and from T to 9. In like manner, the threed applyed to the fourth houre at o, or i, will cut the line TV in 4 take off T4, and pricke it downe here from V to 4, and from T to 10. So also the threed laid upon the fifth houre at e, or a, cutteth TV in 5; take off T5, and prick it downe here from V to 5, and from T to 11. Thus are all the Houres pricked downe.

[illustration]

[illustration]

An horizontull Diall to 52 gr: 30 m: lat:

Lastly then, laying your Ruler to the center A, through each of these points, you shall draw the houre-lines A7, A8, A9, A10, A11, AV which is 12, A1, A2, A3, A4, A5, RAT is the line of

the two sixes. So having 12 houres, which is halfe the Diall, drawne, you may extend the necessarie lines, as many as you will, beyond this center, as 5A5, 4A4, 7A7, 8A8, &c.

In the same manner may the halfe houres bee pricked downe and drawne, by applying the threed to the halfe houres in the limbe, &c.

And note also that in these Plaines before men∣tioned; As the extent from V to 1, is the same with that from T to 7, so likewise is it the same with V11, R5; And as V2 is the same with T8, so likewise is it the same with V10, R4: So likewise V9 and T9 are all one, and both equall to R3 and V3. So that the three first houres taken from the Quadrant, that is to say, T1, T2, T3, will give all the houres for these Dialls. T1, gives V1, V11, R5, T7. T2, gives V2, V10, R4, T8. T3, gives V3 or R3, V9 or T9. But in other Plaines it is not so, for which cause I have rather set downe this way before at length, as a direction for what comes after, for that is generall.

Here note againe, that if you desire to make your draught greater, you may in your description ei∣ther double or triple every length which you take in your Compasses. And so I proceed to all decli∣ning Plaines.

CHAP. XI.

To draw the Houres upon all sorts of declining Plaines, whether erect or inclining.

BY the former precepts you must first get the Rectifying-arke, with the Ele∣vation of the pole above the Plaine. After they are had, you may pricke downe the Houre points in this man∣ner following, little differing from the former.

[illustration]

[illustration]

A Plaine▪ inclininge Eastward 40 gr:

The horizointall line, parallel to the line of 12.

1. Asbefore; Upon the line RAT, set off the Elevation of the pole above the plaine, being taken out of the line of latitudes in the Quadrant, from A both wayes, to R and T.

2. Take the line of Houres TV out of the Qua∣drant, and with that extent upon R and T as upon two centers, describe the two arkes BV and CV crossing at V, and draw the lines RV, TV, and AV. Thus farre we goe along with the last Chapter.

3. If we take the example in the seventh chapter, that plaine is the upper face of an East incliner, whose Elevation is 37 gr. 26 min. and so much doth this line TA reach unto in the line of Lati∣tudes: the Rectifying arke is 35 gr. 58 min. This arke I number below in the limbe of the Quadrant ES, and thereto applying the threed I observe in the upper limbe Vcb T which of the Houres and where it cutteth, I find it to cut the slope line o u in the point P; to this point P I set the Bead, which by this meanes is rectified and fitted to the descrip∣tion of the Diall.

Here you see the use of the Bead, and the reason why this arke counted upon the limbe is called the Rectifying arke: and here bee carefull that you stretch not the threed.

4. The threed and Bead being thus placed and rectified, you shall see the threed to cut the line TV at a upon the Quadrant; take T a in your Com∣passes, and pricke it downe here from V to 12, and from R to 6.

Here by the way observe, that because this

plaine is an Eeast-incliner, the face of it loo∣keth toward the West, and then if you ima∣gine the true scituation of this Diall upon the plaine whereon it must stand, you will easily conceive that the line of 12 is to stand on the right hand from the line AV. and so the line of 6 on the left hand, whereas if this plaine had faced toward the East, the line of 12 must have stood on the left hand, and 6 on the right hand. Your owne conceit, together with the precepts of the chapter following, must helpe in this, and in other things concer∣ning the right scituating of the lineaments of your Diall.

To proceed then,

In the same manner must you apply the Bead to every houre line, as in the next place I remove it to the line y m in the Quadrant, and then I see it to cut the line TV in b; I take 1 b in my Compasses, and with it doe pricke downe from V to 1, and from R to 7. Againe, the Bead being applyed to the lines nr, sb, the threed will cut the line TV up∣on the Quadrant in c and d; I take the points TC, Td, in my Compasses, and pricke them downe from V to 2 and 3, and from R to 8 and 9. Then againe, the Bead applyed to the lines ei, Va, the threed will cut the line TV in the points e and o; I take then Te and Tf, and pricke them downe from U ••o 11 and 10, and from R to 5 and 4.

5. Lastly, lay your rule to A, and draw A10, A11, A12, A1, A2, A3, A4, A5, A6, A7, A8,

A9. Thus have you twelve houres, and if you ex∣tend these beyond the Center, you shall have the whole 24 houres, of which number you may take those that shall bee fit for the Plaine in this scitua∣tion.

The halfe houres may thus bee pricked on and drawne also, by applying the Bead to the halfe houres pricked downe in Vcb T the upper limbe of the Quadrant, for so the threed will give you the halfe houre points upon the line TV, which may be taken off, and set downe upon the Diall as the houres themselves were.

CHAP. XII.

How to place the Diall in a right Scituation upon the Plaine.

AFter the houre-lines are drawne by the last Chapter, they are to be placed in a right scituation upon their Plaine. Which to doe, upon some Plaines is more difficult than the Description of the Diall it selfe. To give some directions herein, I have added this Chapter, where you have 9▪ Que∣stions with their Answers, giving light sufficient to what is here intended and required: but first be ad∣monished of three things.

1. That the inclination mentioned Chap. 8. is the very same in Use with the Prosthaphaereticall arke mentioned Chapter 9. And therefore when I

mention the Prosthaphaereticall arke, because it is of most frequent use, you must remember I meane both the Prosthaph: arke, Chap. 9, and the Incli∣nation, Chap. 8.

2. That these rules, though given primarily for places of North-latitude, lying within the Tem∣perate, Torrid, and Frigid Zones, yet are also as true, and may bee applyed to all places of South-latitude, if we exchange the names of North and South, for South and North.

Here by the way note, that the North part of the Torrid Zone extendeth from 0 degrees of la∣titude to 23 gr. 30 min. the Temperate Zone rea∣cheth from 23 gr. 30 min. to 66 gr. 30 min. the Frigid Zone extendeth from 66 gr. 30 min. to 90 gr. of latitude. And so I come to the 9 Questi∣ons.

1. What Pole is elevated above the Plaine.

Upon all Upright plaines declining from the North: Upon the upper faces of all East or West incliners: Upon the upper faces of all North-incliners, whose Prosthaph: arke is lesse than the latitude of the place: On the under faces of all North-incliners, whose Prosthaph: arke is greater then the Latitude of the place; and on the upper faces of all South-incliners, The North pole is e∣levated. And therefore contrarily,

Upon all upright Plaines declining from the South: On the under faces of all East and West, and South incliners: On the under faces of all North-incliners, whose Prosthaphaereticall arke is

lesse than the Latitude of the place: On the upper faces of all North-incliners, whose Prosthaph: arke is greater than the Latitude of the place, The South pole is elevated.

2. What part of the Meridian ascendeth or de∣scendeth from the Horizontall line of the Plaine?

In all Upright plaines the Meridian lyeth in the Verticall line, and if they decline from the South it descendeth, if from the North it ascendeth.

Upon both faces of East and West Incliners the Meridian lyeth in the Horizontall line.

In all North-incliners, the North part of the Meridian ascendeth, the South part descendeth: in all South incliners the South part of the Meridian ascendeth, the North part descendeth: upon both upper and under faces. And if these North and South incliners be direct, then the Meridian lyeth in the Verticall line, and so maketh a right angle with the Horizontall line: but if they decline, then the Meridian on the one side maketh an acute angle with the horizontall line.

3. To which part of the Meridian is the style with the substyle to be referred, as making with it an acute angle?

The style is the cocke of the Diall; the substyle is the line whereon it standeth, signed out in the for∣mer descriptions by the letters AV.

In all Plaines whereon the North pole is eleva∣ted, it is referred to the North part of the Meridi∣an, and maketh an acute angle therewith.

In all Plaines whereon the South pole is elevated, it is referred to the South part of the Meridian, and is to make an acute Angle therewith.

Except here only those South-incliners, whose Prosthaph: arke is more than the complement of your Latitude: for on these plaines the substyle standeth on that part of the Meridian, whose de∣nomination is contrary to the Pole elevated above the Plaine. For on the upper faces the North pole is elevated, but the substyle standeth toward the South end of the Meridian: and on the under 〈◊〉〈◊〉 the South pole is elevated, but the substyle lyeth toward the North end of the Meridian.

Note here, that in South-incliners whose Pro∣sthaphaereticall arke is equall to the complement of your Latitude, the substyle lyeth square to the Meridian upon the line of 6 a clocke; which line in such plaines alwayes lyeth perpendicular to the Meridian line. Amongst these falleth the Equino∣ctiall plaine.

4. On which side of the Meridian lyeth the substyle?

In all direct plaines it lyeth in the Meridian. In all Decliners it goeth from the Meridian toward that coast which is contrary to the coast of the plaines declination.

And so doe all houres also goe upon the Plaine to that coast which is contrary to the coast whereon they are; As all the morning or Ea∣sterne houres goe to the Westerne coast of the Plaine, and all the Evening or Westerne houres

goe to the Easterne coast of the Plaine. Which being observed will bee a great helpe to place them aright.

5. What plaines have the line of 12 upon them, and which not?

All upright Plaines, in all latitudes whatsoever, declining from the South have the line of 12; and decliners from the North in the temperate Zone have it not, but in the other Zones they also have it.

The upper faces of East and West incliners in all Latitudes have it, the underfaces have it not.

The upper faces of all North incliners whatsoe∣ver have it; their under faces in the Temperate Zone want it, in the Frigid Zone have it, and in the Torrid Zone likewise if the Prosthaph: arke bee greater than the Sunnes least North Meridian alti∣tude, but if it be lesse they want it also.

For South incliners, consider the Sunnes greatest and least Meridian altitude upon the South coast. For if the Prosthaphaereticall arke bee such as fal∣leth betweene them, that is, if it be greater than the least, or lesse than the greatest, then have bothsides the line of 12 upon them; but if it be lesse than the least, then doth the Underface want it universally, and the upper face alone hath it▪ if greater than the greatest, then doth the Upper face want it, and the under face alone hath it: Except in the Frigid Zone where the upper face hath it also, by reason of the Sunnes not setting there for a time.

6. Whether the North or South part of the Meri∣dian serveth for the line of 12?

In those Plaines that have the line of 12, where the North pole is elevated, there the North part of the Meridian serveth for 12. and where the South pole is elevated, there the South part of the Meridian serveth for the line of 12 or mid∣day.

Except, in all Latitudes, the under faces of those South incliners, whose Prosthaphaereticall arke falleth betweene the Sunnes greatest and least Me∣ridian altitudes, for in them the South pole is ele∣vated, but the North part of the Meridian serveth for the line of 12.

Except in speciall those Upright Plaines in the Torrid-zone which looke toward the North, and the Under faces of North-incliners also, whose Prosthaphaereticall arke is greater than the least North-meridian-altitude; for these have the South or lower part of the Meridian serving for 12, though the North pole be elevated.

7. Which way the style pointeth, and how it is to bee placed?

In Plaines where the North pole is elevated, it pointeth up towards it; and where the South pole is elevated, it pointeth downe towards it.

The style lyeth perpendicularly over the sub∣style, noted in the former figures with AV, and is to be elevated above it to such an angle as the Ele∣vation of the pole above the Plaine shall be found to be by the 6, 7, 8, and 9 Chapters.

8. When is it that that part of the Meridian next the substyle, and the line of twelve doe goe contrary wayes?

In all Latitudes, Upon the upper faces of South-incliners, whose Prosthaphaereticall arke is greater than the complement of the Latitude, but lesse than the Sunnes greatest South Meridian altitude: And on the Under faces of those South-incliners also, whose Prosthaph: is lesse than the complement of the Latitude, but greater than the Sunnes least South meridian altitude: In the Torrid-Zone a∣lone you must adde hither also, North upright Plaines, and those North-incliners on the Under∣face, whose Prosthaphaereticall-arke is greater than the least North-meridian altitude of the Sun; for these have the line of midday standing on that coast which is contrary to the coast of that part of the Meridian next the substyle, and none else. The line of 12. I call herethe line of midday be∣cause in the Frigid-zone, where the Sunne setteth not in many dayes together, there are two twelves, the one answering to our midday, and the other to our midnight: and so all Upper faces of South-incliners, whose Prosthaphaereticall arke falleth betweene the least and greatest South me∣ridian altitudes, have there two 12 a clockelines upon them.

9. How much the Meridian line ascendeth or descendeth from the Horizon∣tall line?

The quantity of the Angle is to be found upon

the circle on the back-side of your Quadrant, in this manner;

Extend the threed from the complement of the Plaines inclination taken in the lower Quadrant, to the complement of the Plaines declination counted in the Semicircle, and the threed will shew you upon the Diameter, the degrees and minutes of the Meridians Ascension or Descension.

In the example of the 9. Chapt. taking the Up∣per face of that Plaine, I find the Meridian to a∣scend above the Horizontall line 33 gr. 41 minutes.

¶These directions are sufficient for the bestowing of every line into its proper place and coast. As may bee seene in the Example of the ninth Chap∣ter. For,

First, upon the upper face of that North incli∣ner, because his Prosthaph: arke 16 gr. 6 min. is lesse than 52 gr. 30 min. the Latitude of the place, therefore the North pole is elevated above it: by the Answer to the first Quest.

2. Because it is a North-incliner, therefore the North part of the Meridian ascendeth above the Horizontall line, by the answer to the second Que∣stion.

3. Because the North pole is elevated, therefore the Style with the substyle maketh an acute angle with the North end of the Meridian, by the An∣swer to the third Question.

4. Because this Plaine declineth toward the

West, therefore the substyle lyeth on the East-side of the Meridian, and so doe the houres of the after∣noone: by the Answer to the fourth Question.

5. This Plaine, being the Upper face of the North-incliner, will have the line of 12 to bee drawne upon it, by the Answer to the fifth Que∣stion.

6. Because the North Pole is elevated, there∣fore the North part of the Meridian serveth for the line of 12: by the Answer to the sixt Question.

7. Because the North pole is elevated, therefore the style pointeth upward toward the North pole; by the Answer to the seventh Question.

8. That part of the Meridian next the Substyle, and the line of 12 are both one, and so therefore goe both one way: by the Answer to the eight Question.

9. By the second the Meridian line ascendeth, and the quantity of the ascent is 33 gr. 41 min. a∣bove the Horizontall line: by the Answer of the ninth Question.

Thus you see every doubt cleared in this ex∣ample: the like may be done in all others.

CHAP. XIII.

The making and placing of Polar Plaines.

POlar Plaines are those North incliners, whether direct or declining, that lye in the Pole, and so have no Elevation of the Pole above them, as the direct North

[illustration]

Place this Diagram betweene folio 32. and 33.

The horizontall line of the Plaine.

and South polar, the East and West upright Plaines, and infinite others declining betwixt both. You may know them by the 8 and 9 Chapters, as is there intimated.

These Plaines may have Dialls described upon them by this Quadrant, but the better way is the common way, to protract them by an equinocti∣all circle, for otherwise the style will be alway of one distance from the Plaine, be the Diall greater or lesser.

The Polar plaines that decline, before they can be described, must have their New-inclination known, and then their delineation will be easie, the manner of it may be seene in this Example.

Suppose the upper face of a North-inclining Plaine, lying in the Latitude of 52 gr. 30 min. to decline from the South toward the East 68 gr. and to incline towards the North 73 gr. 57 min. you shall find by the ninth Chapter, the Prosthaph: arke to be 52 gr. 30 min. the same with the Latitude of the place, and therefore you may conclude this plaine to be Polar. By the same Chapter you shall find the New inclination to be 63 degrees.

When you have these you may draw your Se∣micircle AB4, and divide it into 12 equall parts for the houres: so signing the new-inclination 63 de∣grees from A to B, draw CB: and supposing the altitude of your style to be CD, through D draw the perpendicular D 12; and where the lines drawne from C through the divisions of the semi∣circle doe cut the line D 12, there raise perpendi∣culars

for the houres, and so finish it up as the man∣ner is. The style lyeth directly over and parallel to the substyle CB, & the distance of it from the plain is CD, and in this Example the substyle CB stan∣deth from the line of 12 Westward, because the plaine declineth Eastward, according to the rules in the former Chapter, and so doe the morning houres also.

For the placing of the Diall in a true site upon the Plaine, you shall find by the answer to the 9 Quest. in the former Chapter, that the Meridian ascendeth 55 gr. 38 min. for other necessaries, the precepts of the former Chapter will direct you. Onely observe, that in Upright East and West plaine, the line of 6 is alwayes the substyle, and it ascendeth above the North end of the Hori∣zontall line, as much as the Latitude of the place commeth to.

FINIS.