Speculum Nauticum.  
A Lookingglass FOR SEAMEN.  
Wherein they may behold, how by a small Instrument, called the PLAIN SCALE, all Nautical Questions, and Astronomical Propositions, are very easily and demonstratively performed.  
First set forth by John Aspley, Student in Physic, and Practitioner of the Mathematics in London.  
The Sixth Edition.  
Whereunto are added, many new Propositions in Navigation and Astronomy, and also a third Book, showing a new way of Dialling.  
By H. P. and W. L.  
LONDON, Printed by W. Leybourn, for George Hurlock, and are to be sold at his Shop at Magnus' Church-Corner, in Thamesstreet, near London-Bridge, 1662.   

TO THE WORSHIPFUL, THE MASTER, WARDENS, & ASSISTANTS OF THE TRINITY HOUSE; JOHN ASPLEY, IN TESTIMONY OF THE HONOUR HE BEARS TO THE GOVERNORS & PRACTISERS OF THE ART OF NAVIGATION, DEDICATES THESE HIS FIRST LABOURS.   

The Printer to the Reader.  
THis little book having been well accepted of among Seamen, being the first fruits of Mr. Aspley's Mathematical Studies, hath passed five Impressions, without any alteration; and so I doubt not might have done still: But because since that time there have been several books put out of this nature, I have procured this to be revised, and several alterations and additions to be made therein, So that here you have both the old, and a new book intermingled all in one, with a third part added thereto, concerning dialing▪ by a way not formerly published by any. All which I doubt not you will kindly accept of, and receive much delight and profit thereby.  
Your. G. H.   

ERRATA.  
PAge 34 line 26 read 360. Page. 45. l. 8. r. Distance I M. Page 50. line 13. for 14 etc. 〈…〉 which is just the length of the Gnomon. Page. 50 line. 28. for increase, read decrease, Page 52 line 4. r. H A I. line 18. r. point O. Page 57 line 11 r. point L.  
Also for some literal faults we shall desire your Pardon▪   

Speculum Nauticum, OR THE SEAMAN'S GLASS.  
The First Book.  

CHAP. I.  
The Explanation of certain Terms of Geometry.  
BEing intended in this Treatise of the plain Scale, to declare the manner of projection of the Sphere, in plano, I have thought fitting first, to show unto you some terms of Geometry which are necessary for the unlearned to know, (for whose sake chiefly I write this Treatise) before they enter into the definition of the Sphere. First therefore I intent to relate unto you, what a point or prick is, and afterward a Line both right and crooked, and such sorts thereof as are appertinent unto the operations and use of this Scale.  
Punctum, or a point, is the beginning of things, or a prick supposed indivisible, void of length, breadth, and depth: as in the Figure following is noted by the point, or prick A.  
Linea, or a Line, is a supposed length, or a thing extending itself in length, not having breadth nor thickness, as is set forth unto you by the Line BAD.  
Parallela, or a Parallel Line, is a line drawn by the side of another line, in such sort that they may be equidistant in all places. And of such parallels, two only belong unto this work of the plain Scale, that is to say, the right lined Parallel, and the circular Parallel.  
Right lined Parallels are two right lines equidistant one from another, which being drawn forth infinitely, would never touch or meet one another, as you may see in the Figure, where the line H I is Parallel unto the line CE, and the line GF is Parallel unto them both.  
A circular Parallel is a circle drawn either within or without another circle upon the same centre, as you may plainly see by the two circles BCDE, and XVYW. These circles are both drawn upon the centre A, and therefore are parallel the one unto the other. There is another kind of Parallel also, which is called a Serpentine Parallel, but because it is not belonging unto the use of this Scale, I will omit it, and so proceed unto the rest.  
Perpendiculum, or a Perpendicular is a line raised from, or let fall upon, another line, making equal Angles on both sides, as you may see declared in the figure, where in the line AC is perpendicular unto the line BAD, making equal ●ngles in the point A.  
Diameter circuli, or the Diameter of a Circle, is a right line drawn thorough the centre of any circle, in such sort that it may divide the circle into two equal parts, as you may see the line BAD is the Diameter of the circle BCDE, because it passeth thorough the centre A, and the two ends thereof do divide the circle into two equal parts, in the two extremes B and D, making the semicircle BCD equal unto the semicircle DEB.  
Semidiameter circuli, or the semidiameter of a circle is half of the Diameter, and is contained betwixt the centre, and the one side of the circle, as the line AD is the Semidiameter of the circle BCDE. This Semidiameter contains 60 degrees of the line of Chords, which we sometimes call the Radius.  
Semicirculus, or a Semicircle, is the one half of a circle, drawn upon his Diameter, and is contained upon the Superficies, or Surface, of the Diameter, as the Semicircle BCD which is half of the circle BCDE, and is contained above the Diameter BAD.  
Quadrants circuli, is the fourth part of a circle, and is contained betwixt the Semidiameter of the circle, and a line drawn Perpendicular, unto the Diameter of the same circle, from the Centre thereof, dividing the Semicircle into two equal parts, of the which parts, the one is the Quadrant, or fourth part of the same circle. As for example, the Diameter of the circle BCDE is the line BAD, dividing the circle into two equal parts: then from the centre A raise the Perpendicular AC, dividing the Semicircle likewise into two equal parts; so is ABC, or ACD, the Quadrant of the circle BCDE, which was desired.   

CHAP. II.  
The manner how to raise a Perpendicular from the middle of a line given.  
〈◊〉 first a ground line whereupon you would have a Perpendicular raised, then open your Compasses unto any distance (so it exceed not the end of your line,) placing one foot of the said Compasses in the point from whence the Perpendicular is ●o be raised, and with the other foot make a mark in the line on 〈…〉 removing your Compasses unto any other distance that 〈…〉 set one foot thereof in one of the marks, and with the 〈◊〉 foot make an Arch over the middle point, then with the same distance of your Compasses set one foot in the other mark upon the line, and with the other foot make another Arch of a Circle over the middle Point, so that it may cross the first Arch, and from the meeting of these two Arches, draw a right line unto the middle Point, from which the Perpendicular was to be raised, which line shall be the Perpendicular desired.  
Example, suppose your Base or ground line whereupon a Perpendicular is to be raised be the line FLK, and from L the Perpendicular is to be raised, set one foot of your Compasses in the Point L, and with the other, make the marks G and M on both sides of the point L, the● opening your Compasses wider, set one foot in the point M, and with the other draw the Arch S over the point L, then with the same distance of your Compasses, set one foot in G, and with the other make the Arch R, crossing the Arch S in the point T, then from T draw the line TL, which line is perpendicular unto the line FLK from the point L, which is the perpendicular desired.   

CHAP. III.  
To let a Perpendicular fall from any Point assigned, unto the middle of a line.  
LET the line whereupon you would have a Perpendicular let fall be the line LFK, and the point assigned to be the point T, from whence you would have a Perpendicular let fall upon the line FLK, first set one foot of your Compasses in T, and open your Compasses unto any distance so that it be more than the distance TL, which here we suppose to be the distance TM; then make in the line FLK the marks G and M, then with your Compasses take the one half of GM, which is the point L, then from L draw a line unto the point T, so the line TL shall be the Perpendicular, which was desired to be let fall from the assigned point T unto the middle of the line FLK.   

CHAP. IU.  
To raise a Perpendicular upon the end of a Line.  
SUppose the line whereupon you would have a Perpendicular raised, be the line FLK, and from the point F a Perpendicular is to be raised: first open your Compasses unto any distance, which here we put to be the distance FG, and set one foot of your Compasses in the point F, and with the other draw the Arch DEG, then set one foot of your Compasses in the point G, and with the other draw the Arch E; then placing one point of your Compasses in E, with the other draw the Arch DB; then place your Compasses in D, and with the same distance draw the Arch A, cutting the Arch DB in C, then draw a line from C unto the end of the line FLK, unto the assigned point F, so shall the line CF be a Perpendicular raised from the end of the line FLK, and from the assigned point F.   

CHAP. V.  
To let a Perpendicular fall from any point assigned unto the end of a Line.  
LET the line FLK be the Base or ground line, and from the point I a Perpendicular is to be let fall upon the end of the line at K, first from the assigned point I, draw a line unto any part of the Base, which shall be the line IHM, then find the middle of the line IN, which is at H; place therefore one foot of your Compasses in the point H, and extend the other unto ay, with which distance draw the Arch INK upon the Centre H, cutting the Base or ground-line in the point K, then draw the line KING, which line shall be the Perpendicular desired.   

CHAP. VI  
A right Line being given, how to draw another parallel there unto at any distance required.  
LEt the line given be AB, unto which it is required to draw another right line CD which shall be parallel to the former line AB, and at the distance AC.  
First open your Compasses to the distance AC, then set one foot in the point A, with the other describe the Arch C; again, place one foot in B, and with the other describe the Arch D; Then draw the line CD, so that it may only touch the two Arches C and D, so shall the line CD, so drawn, be parallel to AB, and at the distance required.   

CHAP. VII.  
A right line being given, how to draw another parallel thereunto, which shall also pass through a point assigned.  
LEt AB be a line given, and the point assigned be C: and let it be required to draw another line parallel thereunto, which shall pass through the given point C.  
NOw I doubt not but you understand the way to let fall, or to raise any manner of Perpendicular line, either from, or upon any part of a line: as also to draw lines parallel one to another at any distance required, therefore now I intent to proceed unto the main point here aimed at, which is, to declare, and make known unto you the several operations performed by the plain Scale, which though it be in use with very few, yet it is most necessary for Seamen, because all questions in Navigation are thereby easily and plainly wrought. And also all questions in Astronomy (belonging unto the expert, and industrious Seamen) may both speedily and easily be wrought by the same Scale: in regard whereof I have declared in this little Book, that knowledge (which God hath been pleased to bestow upon me) concerning the necessary use and practice thereof; hoping that you will as kindly accept it, as it is freely offered unto your courteous considerations.   

CHAP. VIII.  
Of the description of the Scale.  
The figure of the plane Scale.  
The second part of the Scale, is the single chord of a circle, or the Chord of 90, and is divided into 90 unequal divisions, representing the 90 degrees of the Quadrant: and are numbered with 10, 20, 30, 40, etc. unto 90. This Ghord is in use to measure any part or Arch of a circle, not surmounting 90 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 th●se Degrees are the Semidiameter of any Circle that is drawn upon that Radius.  
The third part of the Scale is divided into eight parts, representing the Points or Rumbes of the Mariner's Compass; which in all are 32 points: but upon the Scale there are only S reckoned, which is but one Quadrant or quarter of them, being to be reckoned from the Meridian of North and South both ways, as you may see more plainly by this figure, representing the order of the points of the Compass.  
It is usual also to have another line placed upon your Scale, to she● you how many leagues make a degree of longitude in every latitude, concerning which you shall have directions in the 14 Chapter following.   

CHAP. IX.  
Knowing the course any ship hath made, and the leagues she hath sailed: to find how much she hath raised or depressed the Pole; and likewise how much she is departed from her first Meridian.  
The Course is South-west and by South, the leagues sailed are 100, the difference of the Latitude, and the distance of the Meridian's is required.  
Now you must heedfully observe this point D, for this represents the place where your Ship is, and doth show both the differencefo the latitude of the place you are in, and also your distance or departure from your first meridian.  
First for the latitude, you see the line DF, being parallel to the line AB, cuts the Meridian line OF in the point F: So that if you take the distance FAVORINA with your Compasses, and apply it to the scale of equal leagues, you shall find it is just 83 leagues, which counting 20 leagues to a degree, makes 4 degrees 9 min. and so much you have altered your latitude by the said course, which degrees and minutes being added to, or substracted from the latitude of the place you came from, according as your course requires, shows you always the true latitude you are in.  
Likewise from this point D, take with your Compasses the distance DF, and you shall find it by your scale of equal leagues to be 56 leagues, and so much you are departed from your first Meridian to the Westward; which when you are near the Equinoctial, where the degrees of longitude are equal to the degrees of latitude, would show the longitude, by taking 20 leagues for one degree, etc. so it would be two degrees, and 48 min. for your difference of longitude, from your first Meridian AF. But in other places, you must first 〈◊〉 howmany leagues make a degree of longitude about that latitude where you are, and so turn your leagues of distance from the Meridian, into degrees and minutes of longitude, of which more hereafter, Chap. 14.  
I have been the larger in these two Propositions, because they are ●he first, for the better understanding of all the rest; and because they are most necessary, for thereupon depends the knowledge of the true Traverse point, and the keeping of your dead reckoning. Now because this cannot always be kept exactly, it is to be corrected by the observation of the latitude, according to this following proposition.   

CHAP. X.  
Knowi g the difference of latitude of two places, and the Rumb you have sailed upon, to find the leagues you have sailed, and the difference of Meridian's.  
The Pole depressed four degrees and the Rumb South-West by South or the third from the Meridian, to find your true Traverse point, viz. how far you have sailed, and how much you are departed from your first Meridian.  
In the first figure  
DRaw the lines as in the former Chapter, so that AKF may represent the Meridian line, and ACD may represent the third Rumb from the meridian; then because you have altered your latitude 4 degrees, which make 80 leagues, take 80 leagues with your Compasses out of your Scale, and set them upon the meridian line OF, from A to L: Then keeping the same distance of your Compasses, draw the line LM parallel to AB, (or else you may erect LM perpendicular to the line OF, in the point L) and mark where the said LM crosseth the Rumb line ACD, which is in the point M. This point M is the true Traverse point, the leagues sailed are showed by the line AM, which being measured in the Scale, will be found to be 96 leagues and an half, and the departure from the Meridian is LM, which is 54 leagues.  
Now by this Proposition (as I said) you may correct your dead reckoning; for suppose by the former proposition you reckon you had sailed 100 leagues upon the ●hird Rumb, then as you see there, you should have been at the point D, and have altered your latitude 83 leagues, and departed from your Meridian 56 leagues; but now suppose that by a good observation of the latitude, you find that you have altered the latitude only 80 leagues, from A to L, by drawing this line LM, which crosseth the Rumb or Ships way in M, you may conclude your true Traverse point to be at M, so that you have sailed only from A to M, which is 96 leagues ½, and departed from your Meridian 54 leagues. So that as you are short of the latitude you reckoned for 3 leagues or 9 min. you are also short of your way you reckoned 3 leagues ½, and two leagues less in your departure from the Meridian. And this you must account for your true reckoning, being thus corrected.   

CHAP. XI.  
By the difference of the latitudes of two places and the distance between their Meridian's, to find the Rumb by which you must sail from the one place to the other, and how far it is from the one place to the other?  
The difference of latitude between the two places is 4 deg. 9 min. and the distance between the two Meridian's is 56 leagues, and it is required to find the Rumb from the one place to the other.  
IN the former figure draw the quadrant AKCB, then turn your four degrees 9 min. of latitude into leagues, it maketh 83 leagues, which you must place upon the meridian line from A to F. And from the point F draw the line FD parallel to the line AB. Then open your Compasses to the distance of the meridians which is 56 leagues, and set it on the line FD, from F to D. Then lay your Ruler by this mark D and the Centre A, and draw the line ACD. Then mark where this line cuts the quadrant, which is in the point C, and setting one foot of your Compasses in the point C, open the other to K, and keeping your Compasses at that distance CK, measure it upon your Scale, either in the line of Chords, or in the line of Rumbs, you sh ll find it to be in the one 33 deg. 45 min. and in the other just the third Rumb from the meridian. So that the Rumb from A 〈◊〉 D ●s South-west and by South, and the Rumb from D to A is the Rumb opposite thereunto, which is North-east and by North.  
Then for the distance between the two places in the Rumb, ●et one foot of your Compasses in the one place at A, and open the other to the other place at D, and the length of the line A D ineasured in the Scale of leagues, shows the distance between them to be just 100 leagues.  
These three (or rather these six) Propositions, (for they are each of them double) are the most useful and necessary in the art of Navigation. By the first of these, knowing the point of the Compass you ●ail upon, and judging howmany leagues you have sailed thereon, you know and are able to give a reasonable account where you are, both in respect of latitude and longitude. By the second having a fair observation of the latitude at any time, you may more perfectly know where you are; and thereby correct your former account. And by this third you may know how to direct your course from any place to your desired haven. So that in effect you need no more, but yet for your better instruction by variety of cases and examples, I shall proceed.   

CHAP. XII.  
The difference of Latitude and the lea●●es sailed being given, to find the distanee from the Meridian, and the Rumb you have sailed upon.  
Sailing 100 leagues between South and West, until the Pole be depressed 4 deg. 9 min. the distance from the Meridian is demanded, and what Rumb you have sailed upon?  
IN the first figure draw the Quadrant AKCB, as in the former Chapters, and then reduce your degrees of latitude into leagues, so 4 deg. 9 min. make 83 leagues, which you must take with your Compasses out of your Scale of leagues, and set them off in this Meridian line from A to F. Then from the point F draw the line FD, parallel to the line AB, which you may do with the foresaid distance of your Compasses. Then open your Compasses unto your distance failed, which is 100 leagues, and setting one foot of your Compasses in the point A, with the other draw the little Arch HG, cutting the line FD in the point D. So the line FD measured in the Scale of leagues, shall show you the distance from the M ridian, which is 56 leagues, and if you draw the line ACD, it i● the Rumb line upon which you have sailed, and the Arch KC 〈◊〉ed in the Scale of Rumbs, shows it to be the third Rumb from the Meridian, or South-west by South.   

CHAP. XIII.  
To find the distance of any Island from you, that you may discern at two stati n●, knowing the polut of the Compass, the Island beareth unto each of the stations.  
Suppose, being at Sea you discover an Island bearing North-east off you, which place let it be your first station, and then sailing seven leagues full North you observe the Island to bear full East off you, which let be the second station; the aemand is to find the distance of the said Island from both the said stations?  
IN the second figure, or demonstration, let A be the first Station, and upon the Centre A draw the Quadrant ABDE; Then in regard you found the Island to bear North-East from you, take 4 of your 8 points of the Compass our of the Scale, and place them upon your Quadrant from B to D, then from the Centre A by the point D, draw the line ADF, representing the visual line passing between your sight and the Island, being at the first station A. Then seeing when y●● had sailed 7 leagues North, you observed the Island to bear full East off you, set off the said 7 leagues from A to C, (reckoning every 10 leag● s of your Scale to be but on●) and from this point C, which is the second station, draw the line C F parallel to A, and it will cut the line ADF in the point F: So shall the point F, be the place of the Island desired, and the distance OF, is the distance of the Island from the first station, viz. 9 leagues 90 parts or almost 10 leagues: Likewise the distance from C, to F, is the distance of the Island from the second station, which is just seven leagues. And by this manner of work, you may find the▪ distance of any Island or head land from you, or you may take the distances of as many places as you will or can see at any two such stations, and by the crossing of their visual lines, find their position and distances each from other.   

CHAP. XIV.  
To find how many leagues, miles, and parts do make one degree of longitude in every latitude.  
Note, All this while we have been sailing according to the Rules of the plain Chart, which supposith the degrees of longitude to be equal to the degrees of latitude, in all latitudes, but that is very false and erroneous; it being true only in places near the Equinoctial, where every degree of longitude contains 20 leagues, as the degrees of latitude do; But in places near the Poles it altar's very much, so that in the latitude of 60 degrees, 10 leagues make a degree of longitude: and in other latitudes the degrees of longitude alter, as in this little Table, which shows at what degree and minute of latitude, any nnmber of leagues make a degree of longitude, by which you may divide a Line upon your Scale for your ready use.  

Leagues in one Degree.  20 00 d 00 m 19 18 11 18 25 50 17 31 47 16 36 52 15 41 25 14 45 34 13 49 27 12 53 08 11 56 38 10 60 d 00 m 9 63 15 8 66 25 7 69 31 6 72 32 5 75 31 4 78 28 3 81 22 2 84 16 1 87 08  

Now to return to the Question, and show you by demonstration how to find how many leagues, miles, and parts, make a degree of longitude in any degree of latitude?  
The larger you make your Quadrant, the more exact will the work be, and show the leagues and miles more exactly, which you may make into a Table, as this following.  

A Table showing how many leagues, miles, and hundred parts of a mile make one degree of longitude in any latitude.  Latitude Leagues Miles Parts Difference  Latitude Leagues Miles Parts Difference  Latitude Leagues Miles Parts Difference 0 20 0 0 —  30 17 0 96 —  60 10 0 0 — 1 19 2 99 1  31 17 0 43 53  61 9 2 09 91 2 19 2 96 3  32 16 2 88 55  62 9 1 17 92 3 19 2 92 4  33 16 2 32 56  63 9 0 24 93 4 19 2 85 7  34 16 1 74 58  64 8 2 30 94 5 19 2 77 8  35 16 1 15 59  65 8 1 36 94 6 19 2 67 10  36 16 0 54 61  66 8 0 40 96 7 19 2 55 12  37 15 2 92 62  67 7 2 44 96 8 19 2 42 13  38 15 2 28 64  68 7 1 47 97 9 19 2 26 16  39 15 1 63 65  69 7 0 50 97 10 19 2 09 17  40 15 0 96 67  70 6 2 52 98 11 19 1 90 19  41 15 0 28 68  71 6 1 53 99 12 19 1 69 21  42 14 2 59 69  72 6 0 54 99 13 19 1 46 23  43 14 1 88 71  73 5 2 54 100 14 19 1 22 24  44 14 1 16 72  74 5 1 54 100 15 19 0 96 26  45 14 0 43 73  75 5 0 53 101 16 19 0 68 28  46 13 2 68 75  76 4 2 52 101 17 19 0 38 30  47 13 1 92 76  77 4 1 50 102 18 19 0 06 32  48 13 1 15 77  78 4 0 48 102 19 18 2 73 33  49 13 0 36 79  79 3 2 45 103 20 18 2 38 35  50 12 2 57 79  80 3 1 42 103 21 18 2 1 37  51 12 1 76 81  81 3 0 38 104 22 18 1 63 38  52 12 0 94 82  82 2 2 35 103 23 18 1 23 40  53 12 0 11 83  83 2 1 31 104 24 18 0 81 42  54 11 2 27 84  84 2 0 27 104 25 18 0 38 43  55 11 1 41 86  85 1 2 23 104 26 17 2 93 45  56 11 0 55 86  86 1 1 18 105 27 17 2 46 47  57 10 2 68 87  87 1 0 14 104 28 17 1 98 48  58 10 1 80 88  88 0 2 09 105 29 17 1 48 50  59 10 0 90 90  89 0 1 05 104 30 17 0 96 52  60 10 0 0 90  90 0 0 0 105    

CHAP. XV.  
The difference of latitude, and the Rumb or distance sailed being known, to find the distance of the Meridian's, and thereby to find the degrees and minutes of the difference of longitude in any latitude.  
Sailing from the North parallel of 56 degrees and 5 min. latitude, 100 leagues upon the third Rumb from the Meridian▪ viz. South-west and by South until I find the Pole is depressed 4 deg. 9 m. and the Meridional distance 56 leagues; the longitude is desired thereby?  
I● the first figure  
Now to reduce this 56 leagues into degrees of longitude, you must consider from what latitude you have sailed, and to what latitude you are come, viz. from latitude 56 d. 5 m. to 4 deg. 9 min. less, which is 51 d. 56 m. and take the middle latitude (or somewhat more) between the two places, which in this example falls out to be 54d. 01 m. Then by the Table in the former Chapter, find out howmany leagues and miles in the said middle latitude make one degree of longitude, and you shall find in that Table, that in the latitude of 54 d. there is but 11 leagues, and 2 miles, and 27 parts in one degree of longitude; Therefore open your Compasses upon your Scale of leagues, to this 11 leagues, 2 miles, 27 parts, and keeping your Compasses at that distance, set one foot of them at 56 leagues in your Scale of leagues, or in the line DF in the figure, (or upon the like line in your Chart at any time) either at F or D, and measure howmany times you find that distance either to the end of your Scale coming backward, or in the line DF, for so many degrees is the difference o● longitude, and if any odd part remain, you may proportion i● by your eye, judging it to be a quarter, a third, an half, or any part more or less of a degree, which you may either reckon by parts, or 15, 20, 30 etc. minutes,  
Thus this line DF being 56 leagues, opening your Compasses to 11 leagues 2 miles 27 parts, you will find this distance in it, 4 times and 3 quarters; so that the difference of longitude is 4 deg. 45 min.  
Or you may reduce it into miles and work by the rule of proportion, so you shall find As 11 leagues, 2 miles, 27 parts, that is 35 miles 27 parts. 35,27 To one degree of longitude in the latitude of 54 d. 01,00 So is 56 leagues, or 168 miles. 168,00 To 4 degrees, 76 parts. 04 76  
But if your Scale be large, the other way with your Compasses will give you the degrees and parts of longitude as exactly as you need for most uses.  
Also if the latitude fall not out in equal parts, you may find out for your odd minutes by proportion, for which purpose I have set the differences between each degree in the Table.  
So that as one hundred parts or 60 minutes being one degree, to the difference in the Table between the two next degrees;  
So the odd hundred parts or minutes of latitude, to the parts and minutes proportional to be allowed.   

CHAP. XVI.  
Sailing from the South latitude of 60 degrees, 51 min. and from longitude 25 degrees, 24 min. 99 leagues, upon a South-west course: the latitude and longitude of the second place is demanded.  
IN the second demonstration, draw the Quadrant ABCDE, as is formerly taught, then in regard you sail South-west, take 4 points of the Compass from your Scale, and place them from B unto D, then by the point D draw the line ADF, then place your ninety nine leagues upon the line ADF, from A unto F, so shall F be the place of your Ship. Then from F draw the line FC parallel unto A, cutting the line ABC in C, so shall the distance CA be the leagues you have run South, which is seventy leagues, or 3 deg. 30 minutes, which being added to the latitude from whence you dearted, makes 64 deg. and 21 minutes for the latitude of the second place: then take the distance CF, and apply it unto the line of equal parts, and you shall find it likewise 70 leagues: Then finding the middle latitude 62 degrees 36 minutes in the Table, Chap. 14. you shall find that 9 leagues and 0 miles, and 61 parts, do alter a degree of longitude in that latitude. Then opening the feet of your Compasses to 9 leagues 0 miles 61 parts, in the Scale of equal leagues, and keeping the Compasses at that distance, see howmany times that distance is in the line CF, which is seven times and somewhat above an half, the true difference of longitude being 7 deg. 36 m. which being substracted from the longitude from whence you departed, leaves 17 degrees and 48 minutes for the longitudeof the second place.   

CHAP. XVII.  
A Ship sailing from the North Parallel of fifty degrees, having an hundred leagues to sail South-west, and by West, by the way is enforced by contrary winds to sail upon several points of the Compass, first sailing thirty leagues upon a direct course, than West Northwest twenty leagues, than South sixty leagues, the question is to find the latitude of the second place, how far it is to the place wherewto you are bound, the distance of the Rumb that is betwixt them, the distance that you are from your first Meridian, and thereby the difference of longitude.  
IN the third demonstration, draw the line AD, and from the point A, raise the perpendicular AB, then open your Compass unto the Radius of your Scale, and place one foot thereof in the centre A. and with the other draw the Quadrant BCD, then take three points of the compass & place them upon the Quadrant from D. unto C, then from the Centre A, by the point C, draw the line ACL, 100 Leagues in length, which is the true course you are to sail, Then in regard you sailed thirty leagues direct, take thirty leagues from your Scale of equal parts, and place them upon the line AEC, it extends from A unto E: then in regard you turned your Course, West, Northwest, from the Centre E, draw the Line EGLANTINE parallel unto A. D. and again from the centre E draw the line EH perpendicular to EGLANTINE, and parallel to AB, then witn the distance of the Radius, set one foot of your compasses in the centre E, and with the other draw the Quadrant GMH, and in regard you sailed West, Northwest, which is two points from the West Northward, take from your Scale two points of the Compass, and place them upon the Quadrant GMH, from G unto M, then from the centre E unto the point M, draw the line EFM, then take 20 Leagues with your Compasses from the Scale of equal parts, and place them upon the line EFM, from E unto F, then is your Ship in the point F. Lastly, in regard you run South 60 Leagues from F, draw a Line Parallel unto the Meridian AB, which is the line FI, then take from your Scale of equal parts sixty Leagues, and place them from F, unto I, then is your Ship in the point I: then last of all is to be found how far it is to the place where unto you were bound, the distance of the Rumb that is betwixt you, the degrees and minutes you have raised the Pole, the distance of departure from the first Meridian, and thereby t●e difference of Longitude: and that you may so do, first draw the line OIK, Perpendicular unto the line IF in the point I, and with your Compasses opened unto the distance of the Radius, set one foot of your Compasses in the Centre I, and with the other draw the Quadrant KNF, then in regard your ship is in the point I, and the place whereunto you are bound is the point L, therefore from I, thorough the point L draw the line ILN, cutting the Arch KNF, in the point N, therefore let IL, be the Leagues you have unto the place whereunto you are bound, which is forty one Leagues and a half, and the Rumb the distance KN, which is West, and by North, and three degrees unto the Northward, so likewise is the line AO, the number of Leagues you have run due South, which is sixty eight Leagues and one mile, or three degrees and twenty five minutes, which being taken from fifty degrees, the parallel from which you departed, leaves forty six degrees and thirty five minutes for the Parallel you are in. Last of all, shall the line IO, be the Leagues that you have departed your first Meridian, which are forty two leagues and one mile, Then take the middle latitude which is forty eight degrees seventeen minutes and in the Table chap. 14 you shall find that thirteen Leagues 0. mile, 92 parts, do answer unto a degree of Longitude in that Parallel; then setting one foot of your Compasses in thirteen Leagues, and ninety two parts, extending the other to the beginning of the Scale, keeping the Compasses at that distance, turn them over the line I O, and you shall find it contains that distance three times and almost a quarter, So the difference of longitude is three degrees eleven minutes.   

CHAP. XVIII.  
Two Ships departing from one Parallel, and Port, the one in sailing eight Leagues betwixt the North, and the West, hath raised the Pole two degrees, the other in sailing a hundred Leagues betwixt the North, and West, hath raised the Pole four degrees, I demand by what Rhumbs the said Ships have sailed, and the Rhumb and distance that is betwixt them?  
IN the fourth Demonstration, draw the Quadrant ABCDE, then in regard the first Ship hath raised the Pole two degrees, which is forty leagues, take forty Leagues off your Scale, and apply them unto the Meridian line AGL, from A unto G: then from the point G, draw the line GF, parallel unto AB, then opening your compasses unto 80 Leagues, set one foot in the Centre A, with the other make a mark in the line GF, which will be at F, so shall F be the place of the first ship; the second Ship hath raised the Pole four degrees, which is 80 Leagues, therefore place 80 leagues upon the Meridian line AGL, from A unto L, and from the point L draw the line LM, parallel unto GHF, then open your Compasses unto the distance of a hundred leagues, which are the Leagues the second ship did run, and set the foot of your Compasses in the Centre A, and with the other make a mark in the line LM, which will be at M, then draw the line MA, which is the course of the second Ship, and the line FAVORINA, is the course of the first ship, then from F let a Perpendicular fall, being Perpendicular to the line GF, which is the line FK, then opening your Compasses unto the Radius of your Scale, set one foot in the Centre F, and with the other draw the Quadrant HIK, likewise from F, the place of the first Ship, draw a line by the point M, the place of the second, cutting the Quadrant KHI, in I, so let IK, be the course that is betwixt them, that is, if you will sail from the first ship unto the second, you must sail North and by East, and one and forty minutes to the Eastward, likewise let F M, be the distance that is betwixt them, which in this Demonstration is forty Leagues, two miles, so shall BC, be the course of the first ship from the West Northward, which here is found to be thirty degrees and one minute from the West Northward, or Northwest by West, and three degrees and forty four minutes to the west ward. Lastly the Arch ED, is in the distance of the course that the second Ship made from the North Westward, which is found by this Demonstration to be Northwest and by North, and three degrees five minutes to the Westward.   

CHAP. X●X.  
Two Ships departing from one Parallel and Port in the Parallel of 47 deg. 56 min. the first in sailing 80 leag. betwixt the North and West, hath raised the Pole two degrees, I demand by what course the second ship must run, and how much she shall alter in her first Meridian or longitude, to bring herself 40. leagues and two miles' North and by East, and 41. minutes to the Eastward of the first ship?  
IN the fourth Demonstration draw the Quadrant ABCDE, then multiply your two degrees you have altered your latitude by twenty and it maketh forty Leagues; which forty Leagues set upon the line AEL, from A unto G, then from the point G draw the line GF, parallel unto AB, then open your Compasses unto the distance of 80 Leagues, which are the Leagues your first ship did run, and place one foot of your Compasses in the Centre A, and with the other make a mark in the line GF, which will be at the point F, then from the Centre A unto the point F draw the line OF, representing the distance of the Course of the first Ship 80 leagues: Then from F let fall a Perpendicular FK, and upon the Centre F, with the Radius of the Scale draw the Arch HIK, Then in regard you must bring the second ship North and by East, and 41 minutes Eastward of the first ship, take 11 degrees 56 minutes from your Scale of Chords, and place them from Kunto I, upon the quadrant KIH. Then from F draw the line IF, and upon the line, FI, place the distance that you must bring the second ship from the first (which is forty leagues and two miles) from F unto M. So is M the place of your second ship. Then from M draw the line ML parallel unto FG, cutting the line AGL in L, then draw the line MA, cutting the Quadrant BDE in D. So shall the Arch DE be the course that the second ship must run, to bring herself forty leagues and two miles' North and by East, and 41 minutes East of the first ship. Then to know what you have altered the latitude, first take the distance LA and apply it unto the Scale of equal parts, and you shall find it to be 80 leagues, which is just 4 degrees, which you have altered your latitude, or Poles elevation: which 4 degrees added unto the latitude you departed from, it makes 51 degrees 56 min. for the latitude that your second Ship is in, then take the distance LM and apply it to the Scale, it gives 60 leagues; then open your Compasses unto the distance of the middle latitude, which is 40 deg. 5● min. of the Chord, and apply it unto the Table of longitudes, and it gives 12 leagues, and 2 miles, and 62 parts, to alter one degree of longitude in that Parallel: Then set one foot of your Compasses in 12 leagues 2 miles, and 62 parts, and open the other to the beginning of the line, and with that distance measure the line L M, being 60 leagues, and you shall find that it is contained there in four times and two thirds, so the longitude is 4 degrees 40 minutes.   

CHAP. XXI.  
Of the Ebbing and Flowing of the Sea, and of the Tides, and how to find them in all places.  

A general Table for the Tides in all places.  The Moon's age. Hours and minutes to be added. Hours and minutes to be added.  The Moons age. Hours and minutes to be added: Hours and minutes to be added: Days. Degrees: Minutes:  Days. Degrees: Minutes: 1 0 48  16 0 48 2 1 36  17 1 36 3 2 24  18 2 24 4 3 21  19 3 12 5 4 0  20 4 0 6 4 48  21 4 48 7 5 36  22 5 36 8 6 24  23 6 24 9 7 12  24 7 12 10 8 0  25 8 0 11 8 48  26 8 48 12 9 36  27 9 36 13 10 24  28 10 24 14 11 12  29 11 12 15 0 0  30 0 0  

The use of the Table of the Tides.  
FIrst it is to be understood, that by the swift motion of the first Mover, the Moon and all the rest of the Stars and Planets, are turned about the World in four and twenty hours, upon which swift motion of the Moon, the daily motions of the Sea, do depend, which motion of the Sea falleth not out always at one hour, the reason thereof is, because of the swift motion of the Moon in regard she goeth almost thirteen degrees in four and twenty hours, and the Sun moveth scarce one degree, which gives every day twelve degrees, that the Moon cometh slower to any point in the Heaven than the Sun: which twelve degrees makes forty eight minutes of time for the difference of every full Sea, according unto the middle motion of the Moon, which difference is here set down in this Table for every day of the Moon's age. Therefore if you would know the full Sea at any place in the World, first you must know at what hour it is full Sea at the new or full Moon; which hours and minutes keep in mind, then seek the age of the Moon as is before taught, and with the number of her age enter this Table, under the Title of the Moon's age, and having found her age in the Table, against it you shall find the hours and minutes which are to be added unto the time that the Moon maketh full Sea in any place, and the whole number of hours and minutes is the time that the Moon maketh full Sea in that place upon the day desired. As for example, I desire to know the full Sea at London Bridge upon the 13 of July 1624. the age of the Moon being found as before, is eight days, then in the Table I find eight days, and against it 6 hours, and 24 minutes, which being added unto 3 hours, the full Sea upon the change day gives 9 a clock 24 minutes for the time at the full Sea upon the 13 day of July 1624.     

THE SEAMAN'S GLASS.  
The Second Book.  
Wherein is declared the Definition of the Sphere, a Description of the six great Circles, and also of the four lesser Circles, last of all, certain Questions Astronomical, performed by the said Scale.  

CHAP. I.  
Of a Sphere, and the Circles thereof.  
The figure of the plain Scale.  
A Sphere according to the Description of Theodosius, is a certain solid Sup● ficies, in whose middle is a Point, from which all lines drawn unto the Circumference are equal; which Poi●● is called the Centre of the Sphere, by which C●●●er a right Line being drawn, and excending himself on either side unto that part of the Circumference whereupon the Sphere is turned, is called Axis Spherae, or the Axletree of the World.  
A Sphere accidentally is divided into two parts, that is to say, in Sphaeram rectam & Sphaeram obliquam.  
Sphaera recta, or a right Sphere, is only unto those that dwell under the Equinoctial, Quibus neuter Polorum magis altero elevatur: that is, to whom neither of the Poles of the World are seen, but lie hid in the Horizon.  
Sphaera obliqua, or an oblique Sphere, is unto those that inhabit on either side of the Equinoctial, unto whom one of the Poles is ever seen, and the other hid under the Horizon.  
The Circles whereupon the Sphere is composed are divided into two sorts: that is to say, in Circulos majores & minores.  
Circuli majores, or the greater Circles, are those that divide the Sphere into two equal parts: and they are in number six, viz. the Equinoctial, the middle of the Zodiac, or the Ecliptic line, the two Colours, the Meridian, and the Horizon.  
Minores vero Circuli, or the lesser Circles, are such as divide the Sphere into two parts, unequally, and they are four in number; as the Tropic of Cancer, the Tropic of Capricorn, the Circle Arctic and the Circle Antarctic.   

CHAP. II.  
Of the six greater Circles.  
I. THE Equinoctial is a Circle that crosseth the Poles of the World at right Angles, and divideth the Sphere into two equal parts, and is called the Equinoctial, because when the Sun cometh unto it, (which is twice in the year, viz. In principio Arietis, & Librae, that is, in March and September) the days and nights are equal throughout the whole World, whereupon it is called Equator diei & noctis, the equal proportioner of the day and night artificial: and in the figure is described by the line CAESAR.  
II. The Meridian is a great Circle passing thorough the Poles of the World, and the Poles of the Horizon, or Zenith point over our heads; and is so called, because that in any time of the year, or in any place of the World, when the Sun (by the motion of the Heavens) cometh unto that Circle, it is noon, or twelve of the Clock. And it is to be understood, that all Towns and places that lie East and West one of another, have every one a several Meridian: but all places that lie North and South one of another, have one and the same meridian. This Circle is declared in the figure following by the Circle BCDE.  
IV. The two Colours, Colurus Solstitiorum, or the Summer Colour, is a Circle passing by the Poles of the World, and by the Poles of the Ecliptic, and by the head of Cancer and Capricorn, whereupon, the first scruple of Cancer, where the Colour crosseth the Ecliptic Line, is called Punctus solstitiae aestivalis, or the point of the Summer Solstice: to which place when the Sun cometh, he can approach no nearer unto our Zenith, but returneth unto the Equator again. Arcus vero Coluri, The Ark of the Colour contained betwixt the Summer Solstice and the Equator, is called the greatest declination of the Sun, which Ptolemy found to be 23 degrees, 31 minutes: but by the observation of Copernicus it was found to vary, for ●e found the declination sometimes to be 23 degrees 52 minutes, and in the process of time to be but 23 degrees 28 minutes. And in these our days (by the observation of Tycho de Brahe, and that late famous Mathematician, Mr. Edward Right) it is found distant from the Equinoctial 23 degrees, 31 minutes, 30 seconds.  
V. The other Colour passeth by the Poles of the World, & by the first point of Aries and Libra, whereupon it is called Colurus distinguens Equinoxia. These two Colours do cross each other at right Angles in the Poles of the world, whereupon these, verses were made. 
Haec duo Solstitia faciunt Cancer Capricornus,  
Sed noctes aequant Aries & Libra diebus.    

CHAP. III.  
Of the four lesser Circles.  
THe Sun having ascended unto his highest Solstitial Point doth describe a Circle, which is the nearest that he can approach unto the North Pole, whereupon it is called Circulus Solstitii aestivalis, the Circle of the Summer Solstice, or the Tropic of Cancer, and is noted in the figure before, by the line H Y I  
The Sun also approaching unto the first scruple of Capricornus, or the Winter Solstice, describeth another Circle, which is the utmost bounds that the Sun can depart from the Equinoctial Line towards the Antarctic Pole, whereupon it is called Circulus solstitii hyemalis, sive Tropicus hyemalis, vel Capricorni: the Circle of the Winter Solstice, the Winter Tropic, or the Tropic of Capricorn, and is described in the figure by the line GXF.  
So much as the Ecliptic declineth from the Equinoctial, so much doth the Poles of the Ecliptic decline from the Poles of the World, whereupon the Pole of the Ecliptic, which is by the North Pole of the World, describeth a certain Circle as it passeth about the Pole of the World, being just so far from the Pole as the Tropic of Cancer is from the Equator, and it is the third of the lesser Circles, and is called Circulus Arcticus, or the Circle of the North Pole, and is described in the Diagram, in the second Chapter by the line PO.  
The fourth and last of the lesser Circles is described in like manner, by the other Pole of the Ecliptic, about the South Pole of the world, and therefore called Circulus Antarcticus, the Antarctick Circle, or the Circle of the Antarctick or South Pole, and is demonstrated in the former figure, by the line NM.   

CHAP. IU.  
Definitions of some peculiar terms fit to be known by such as intend to practise the Art of Navigation or Astronomy.  
THe Zenith is an imaginary point in the Heavens over our heads, making right Angles with the Horizon, as the Equinoctial maketh with the Pole.  
The Nadir is a prick in the heavens under our feet, making right Angles with the Horizon under the earth, as the Zenith doth above, and therefore is opposite unto the Zenith.  
The declination of the Sun is the Ark of a Circle contained betwixt the place of the Sun in the Ecliptic, and the Equinoctial, making right Angles with the Equinoctial. But the declination of a Star is the Ark of a Circle let fall from the Centre of a Star, perpendicularly unto the Equinoctial.  
The Latitude is the Ark of a Circle contained betwixt the Centre of any Star, and the Ecliptic Line, making right Angles with the Ecliptic, and counted either Northward, or Southward, according to the situation of the Star, whether it be nearer unto the North or South Pole of the Ecliptic.  
The Latitude of a Town or Country, is the height of the Pole above the Horizon, or the distance betwixt the Zenith and the Equinoctial.  
The Longitude of a Star is that part of the Ecliptic which is contained betwixt the Stars place in the Ecliptic, and the beginning of Aries, counting them from Aries according to the succession or order of the signs.  
The Longitude of a Town or Country are the number of degrees, which are contained in the Equinoctial, betwixt the Meridian that passeth over the Isles of Azores, (from whence the beginning of longitude is accounted) East wards, and the Meridian that passeth over the Town or Country desired.  
The Altitude of the Sun or Star is the Arch of a Circle, contained betwixt the Centre of the Sun, or any Star, and the Horizon.  
The Amplitude is that part of the Horizon which is betwixt the true East or West points, and the point of the Compass that the Sun or any Star doth rise or set upon.  
Azimuth's are Circles, which meet together in the Zenith, and cross the Horizon at right Angles, and serve to find the point of the Compass, which the Sun is upon at any hour of the day, or the Azimuth of the Sun or Star, is a part of the Horizon contained betwixt the true East or West point, and that Azimuth which passeth by the Centre of the same Star to the Horizon.  
The right ascension of a Star is that part of the Equinoctial that riseth or setteth with the Star, in a right Sphere: or in an oblique Sphere, it is that portion of the Equinoctial, contained betwixt the beginning of Aries, and that place of the Equinoctial, which passeth by the Meridian with the Centre of the Star.  
The oblique ascension is a part of the Equinoctial, contained betwixt the beginning of Aries, and that part of the Equinoctial that riseth with the Centre of a Star, in an oblique Sphere.  
The difference ascensional, is the difference betwixt the right and oblique ascension: or it is the number of degrees contained betwixt that place of the Equinoctial that riseth with the Centre of a Star, and that place of the Equinoctial that cometh unto the Meridian, with the Centre of the same Star.  
Almicanterahs' are Circles drawn parallel unto the Horizon, one over another, until you come unto the Zenith: these are Circles that do measure the elevation of the Pole, or height of the Sun, Moon, or Stars above the Horizon, which is called the Almicanter of the Sun, Moon, or Star: the Ark of the Sun or Stars Almicanter, is a portion of an Azimuth contained betwixt that Almicanter which passeth thorough the Centre of the Star, and the Horizon.   

QUESTIONS ASTRONOMICAL, performed by the plain Scale.  
CHAP. V.  
The true place of the Sun being given, to find his declination. The Sun being in the head of Taurus, his declination is desired.  
BY the seventh Demonstration, draw the line AD, then upon the Centre A raise the Perpendicular AB, then opening your Compasses to the Radius of your Scale, place one foot in the Centre A, and with the other draw the Quadrant BCD, then opening your Compasses unto the greatest declination of the Sun, place it upon the Quadrant, from D unto K, then from the point K draw the line KH, parallel to DA, cutting the line AB in H, then with the distance AH draw the small Quadrant GEH, and in regard the Sun is in the head of Taurus, which is 30 degrees from the beginning of Aries, let AD be the Equator, and D the beginning of Aries, DC 30 degrees, or longitude of the Sun, then from the point C draw the line CA, cutting the Quadrant GEH in E, then from E draw the line EI parallel to AD, cutting the Quadrant BCD in I, so shall the Arch ID be the declination of the Sun desired, which in this demonstration is found to be eleven degrees, and thirty one minutes.   

CHAP. VI  
The declination of the Sun, and quarter of the Ecliptic that he possesseth, being given, it is desired to find his true place.  
The Declination is 10 deg. 31 min. the first quarter that he possesseth, is betwixt the head of Aries and Cancer.  
FIrst, by the seventh Demonstration, draw the Quadrant ABCD, as is taught in the former Chapter, than set the greatest declination of the Sun upon the Chord from D unto K, which is 23 deg. and 31 min. then from K draw the line KH parallel unto the Equator DA, cutting the line BASILIUS in the point H. So shall HA be the sign of the Sun's greatest declination, then with the distance AH draw the Quadrant GEH, then from D upon the Quadrant DBC set the declination of the Sun, which is 11 degrees 31 minutes from D unto ay, then draw the line IE parallel unto AD, cutting the Quadrant GEH in E. Then from the Centre A by the point E, draw the line AEC, cutting the Quadrant BCD in C. So shall the Ark CD be the distance of the sun from the head of Aries, which is here found to be just 30 degrees, which is in the beginning of Taurus.   

CHAP. VII.  
By the elevation of the Pole, and declination of the sun, to find the amplitude of the sun, or his distance of rising, or setting from the true East or West point.  
The elevation of the Pole is 51 deg. 32 min. the declination of the sun is 14 deg. 52 min. North.  
BY the eight Demonstration, first draw the line BD, then upon the Centre A draw the Circle BCDE, then from A raise the Perpendicular CAESAR, then is your Circle divided into four equal parts: then suppose the elevation of the Pole to be 51 degrees, 32 minutes, which must be placed upon the Circle, from D unto F, then from the point F, by the Centre A, draw the line FAG, representing the Pole of the World, F being the North Pole, and G the South Pole, then subtract 51 deg. 32 min. from 90 deg. and the remainder is the height of the Equinoctial, which is 38 deg. 28 min. which must be placed upon the Circle from the Horizon B, unto the point I, then from ay, by the Centre A, draw the line JAH, representing the Equinoctial Circle. Then from I unto M set the declination of the Sun, being here supposed 14 deg. 52 minutes North, then from the point M draw the line, or Parallel of declination MTN, parallel unto the Equator I A H, cutting the Horizon BD in T, then from T raise the perpendicular TV, cutting the Circle BCDE in V, so shall the distance CV be the true amplitude of the sun desired, which here is found to be 24 deg, 21 minutes North.   

CHAP. VIII.  
By the Amplitude of the Sun, to find the variation of the Compass.  
HAving found the Amplitude of the Sun by the last Chapter, first observe with a Compass, or rather with a Semicircle, upon what degree and minute the Sun riseth or setteth, beginning to reckon from the East or West, and ending at the North or South at 90 degrees: and when you have diligently observed the Magnetical rising or setting, by the Semicircle, or by some other like fitting Instrument: and also the true Amplitude found, as is declared in the last Chapter, the difference of these two Amplitudes, is the variation of the Compass: But when the Sun riseth upon the same Degree of the Compass, as is found by the Scale, the variation is nothing, but the Needle pointeth directly unto the Poles of the World, which by M. Mulinux was affirmed to be at the Westernmost part of S. Michael's, one of the Islands of the Azores, from whence he will have the Longitude reckoned. Secondly, when the Sun is in the Equinoctial Circle, where he hat● no Amplitude, look what distance the Compass maketh the Sun to rise from the East or West of the Compass, the same distance is the Compasses variation, from the North or South. Thirdly, if the Sun rise more to the South of the Compass, or setteth more to the North of the Compass, than is showed by the Scale, the difference betwixt the Amplitude given by the Scale, and the Amplitude given by the Needle, is the variation of the Compass from the North Westward. Fourthly, if the Compass showeth the Sun to rise more Northward, or set more Southward, than is showed by the Scale, the difference is the variation of the Compass, from the North Eastward. Fifthly, if the Scale show the Amplitude of the Sun rising Southerly, and the Compass show it to be Northerly, add both the Amplitudes together, and they show you the variation Westernly.   

CHAP. IX.  
The place of the sun being given, to find his declination, by a whole Circle.  
The sun's place is the tenth degree of Taurus.  
ACcording unto the eighth Demonstration, first draw the Circle BCDE, then draw the Horizon BAD, and then the Equinoctial JAH, as is before taught: and then the Tropic of Cancer KL, twenty three degrees and a half from the Equinoctial: then draw the Tropic of Capricorn PO, of like distance from the Equinoctial, and after from K to O draw the Ecliptic line KAO. And when you have thus laid down the Sphere, suppose the Sun to be in the tenth degree of Taurus, at which time his declination is desired. And in regard the Sun is more near unto the Tropical point Cancer, than unto Capricorn; first find how many degrees he is from the Tropic of Cancer, and you shall find him to be 50 degrees; therefore take with your Compasses 50 degrees from the Chord, and apply it from the Tropical point Cancer at K, unto V, upon one side, and unto P on the other side: then draw the Line UP, cutting the Ecliptic KO in the point R, then from R draw the Line MRN parallel unto the Equinoctial JAH, and cutting the Quadrant BC in the point M. So shall the ark MI be the declination of the Sun desired, which being applied unto your Scale, gives you 14 deg. and 52 minutes.   

CHAP. X.  
The elevation of the Pole, and declination of the sun given, to find his height in the vertical Circle.  
The Pole is elevated 51 degrees 32 minutes, the declination of the sun is 14 degrees 52 minutes North, his height in the Vertical Circle is found as followeth.  
FIrst, according unto the former Chapter, draw the Circle BCDE, than the Horizon BAD, and after the vertical line CAESAR, than the Axis of the World FG, and likewise the Equator JAH, this being done, place the declination of the Sun 14 degrees 52 minutes, upon the Circle from I unto M, and also from H unto N, then draw the line MN, cutting the line CAESAR in S, then from S draw the line SW, parallel unto the Horizon BAD, cutting the Meridian Circled BCDE in VV: so shall the distance DW be the height of the Sun in the vertical Circle, for the time demanded, which by this proposition is found to be 19 degrees and 8 minutes.   

CHAP. XI.  
The elevation of the Pole, and the Amplitude of the sun, being given, to find the declination.  
The elevation of the Pole is 51 degrees 32 minutes, the sun's amplitude is 24 degrees 21 minutes, the declination is found as followeth.  
 
FIrst, as in the eight demonstration, upon the Centre A, draw the Circle BCDE, then draw the Line BAD, representing the Horizon: dividing the circle into two equal parts then draw the Line CAESAR, perpendicular to BAD, representing the East and West points of the Compass, then placing the elevation of the Pole 51 degrees and 32 minutes, from D unto F, from F, by the centre A▪ draw the Line FAG, which let be the Pole or Axletree of the world, then from B unto ay, and from D unto H, set the compliment of the Poles elevation: which shall represent the Equinoctial, in regard it maketh right Angles with the Pole of the world, in the centre A. Then from C unto V place the amplitude of the Sun, which is 24 degrees and 21 minutes: then from V let fall the perpendicular VT, cutting the Horizon BAD in the point T, then from the point T, draw the Line MTN parallel unto the Equinoctial JAH, and cutting the Circle BCDE in the points, M and N, so shall the distance, M, or HN, be the declination of the Sun, which was desired: which being applied unto your Scale, gives you fourteen degrees and fifty two minutes.   

CHAP. XII.  
The elevation of the Pole, the declination of the Sun, and hour of the day being given▪ to find the Almicanter.  
The elevation of the Pole is thirty degrees, the declination of the Sun is twenty degrees North, the hour is nine in the morning, at which time the Almicanter is found, as followeth.  
BY the ninth demonstration, first upon the Centre A, draw the Circle BCDE, then draw the line BD for the Horizon, then place your Poles elevation, which is thirty degrees, upon the Circle from D unto R, then from R by the centre A, draw the Line RAS, representing the Axis of the World, then from B unto F place the compliment of the Poles elevation, which is ●0 degrees, and from the point F, by the Centre A, draw the line FAH, representing the Equinoctial line, and then set the declination of the Sun from F unto L▪ and from L draw the Line LPO parallel unto the Equator FAH, cutting the Axis of the World in the point P, then set one foot of your Compasses in the point P, and extend the other either unto L or unto O, and with the same distance of your Compasses, upon the Centre P, draw the circle LNOQ, which is called the hour circle: so shall L be the point of twelve a clock at noon, N the place of six a clock after noon, O the place of twelve a clock or midnight, and Q the place of six a clock in the morning: Every one of the four quarters must be divided into six equal parts, or hours, making the whole Circle to contain twenty four parts, representing the twenty four hours of the day and night, then in regard the hour of the day was nine of the clock, which is three hours before noon, take three of those twenty four hours, and place them upon the circle LNOQ, from the Meridian point L unto K, the nine a clock point in the morning, and unto M the point of three a clock after noon, then draw the line MK, cutting the parallel of the Sun LO in the point I, then from I draw the line IG parallel unto the Horizon BAD, which shall cut the Meridian Circled BCDE in the point G, so shall the distance of G and B be the Almicanter the Sun, which was desired, which in this demonstration is found to be forty eight degrees and eighteen minutes.   

CHAP. XIII  
The elevation of the Pole, the Almicanter, and declination of the Sun, being given, to find the hour of the day.  
The elevation of the Pole is thirty degrees, the declination of the Sun, is twenty degrees, the Almicanter of the Sun, is forty eight degrees, and eighteen minutes, the hour of the day is found as followeth.  
FIrst, as in the ninth demonstration, upon the Centre A, draw the Circle BCDE, then draw the Diameter BD, representing the Horizon, then from D unto R, set 30 degrees, the elevation of the Pole, then from R unto the point A, draw the line RAS, representing the Pole of the World, then draw the line FAH, crossing the Pole in A, at right Angles, cutting the Meridian circle in F, then from F, set twenty degrees, the declination of ●he Sun unto L, and then from the point L, draw the line LPO, representing the parallel of the Sun, and cutting the Pole of the World in P, then placing one foot of your Compasses in P, extend the other unto L, with which distance of your Compasses, draw the hour Circle LNOQ, then from the Horizon at B, place the Sun's Almicanter: (which is forty eight degrees, and eighteen minutes▪) upon the Quadrant BGL, from B unto G, then from the point G, draw the line G● parallel unto the Horizon BAD, cutting the Line LO, in I, then from the point I, draw the line KIM, parallel to the Pole of the World QAN, cutting the Circle LNO, in M, then let LN, be divided into six hours, whereof LM, are there: whereupon I conclude, that is is three hours from noon, that is, at nine a clock in the morning, or three in the after noon.   

CHAP. XIV.  
The Latitude of the place, the Declination of the Sun, and the Altitude of the Sun being given, to find the Hour of the day: By a n●w way differing from that in the former Chapter.  
 deg. min.  deg. m The Sun's Altitude is 48 18    The Lat●ude of the place is 30 00 its Comple. 60 00 The Sun's declination is 20 00 N. 70 00    Sum 130 00    difference 10 00   

The Compliment of any arch less than 90 degrees, is so much as the arch wants of 90 degrees, as the Compliment of 20 degrees is 70 degrees, etc.  
FIrst, find the sum and difference of the Compliment of the Sun's declination, and the Compliment of the Latitude, as above is done, where the sum is 130 deg. and the difference 10 deg. Then your Compasses being opened to the Radius of your line of Chords: describe the Semicircle ABC, and divide it into two Quadrants by the perpendicular BD, then out of your line of Chords; take 48 deg. 18 min. the Sun's Altitude, and set it from B to E, and draw E F parallel to B D: Then from your line of Chords take 130 deg. the sum, and set it from A to G, (or its Compliment to 180 deg. which is 50 deg. from C to G) and draw the line GH also parallel to BD. Again, out of your line of Chords, take 10 deg. (which is the difference) and set that distance from A to K, and draw K L parallel to OF or BD.  
This done, take with your Compasses the distance from F to H, and setting one foot in A, with the other describe the Arch MP, likewise take the distance from F to L, and setting one foot in C, with the other describe the arch NQ. Lastly draw the straight line PQ▪ which only touching the two former arkes will cut the line AC in O, Upon the point O, therefore, erect the perpendicular OR, cutting the Semicircle in R, so will CR being measured upon your line of Chords, give you the degrees of the Sun from the South part of the Meridian, which here you will find to be 45 degrees, which make 3 hours, allowing 15 degrees for an hour, for 15 degrees make one hour, and one degree makes 4 minutes of an hour, so that it is either 9 of the clock in the morning, or 3 in the afternoon.   

CHAP. XV.  
The Almicanter, or height of the Sun being given, to find the length of the right shadow.  
The Almicanter is 45 degrees.  
ACcording unto the tenth Diagram, draw the line OF, and upon the centre A, raise the perpendicular AC, then upon the centre A, draw the Quadrant CDF, then suppose the height of your Gnomon, or substance yielding shadow be the Line, AB, which is to be divided into 12 equal parts, which Gnomon, I have here made just 12 degrees of the equal Leagues of the Scale, then from B, to the top of the Gnomon draw the Line BE, parallel unto OF, then set the Almicanter which is forty five degrees from F, unto D, and from the point D, draw the Line DA, cutting the Line BE in the point G, so shall BG, be the length of the right shadow desired, which here is found to be fourteen degrees and eighteen minutes, which is but just the length of your Gnomon, and 2/12 and ⅓ of a twelfe over: Note that the right shadow, is the shadow of any post, staff, or steeple, that standeth at right Angles with the Horizon, the one end thereof respecting the Zenith of the place, and the other the Naedir.   

CHAP. XVI.  
The Almicanter, or height of the Sun being given, to find the length of the contrary shadow.  
The Almicanter given is 70 deg.  
BY the verse or contrary shadow, is understood the length of any shadow, that is made by a staff or Gnomon, standing against any perpendicular wall, in such a manner that it may l●e parallel unto the Horizon, the length of the contrary shadow, doth increase as the Sun riseth in height, whereas chose the right shadow doth increase in length, as the Sun doth increase in height: the way to find the verse shadow is as followeth. First, draw your Quadrant as is taught in the last Chapter, wherein let AB, be the length of the Gnomon, likewise from B, draw the line BE, parallel unto OF, as before, then set your Almicanter from C upon the Quadrant which is given to be seventy degrees and it will extend from C unto H, then from the point H draw the line HA, cutting the line BE, in the point K, so shall KB, be the length of the contrary shadow, which here is found to be thirty four degrees and eight minutes, or twice so long as your Gnomon, and ●0/●2 about ½ part of a twelfth more.   

CHAP. XVII.  
The latitude of the place, the Almicanter, and declination of the Sun being given, to find the Azimuth.  
The latitude of the place is fifty one degrees, thirty minutes, the declination of the Sun twenty degrees North, the Almicanter thirty eight degrees thirty minutes, the true Azimuth of the Sun is desired.  
 
FIrst as in the eleventh Demonstration upon the Centre A, draw the Circle BCDE, then draw the Diameter BAD▪ and from D unto F, set the Elevation of the Pole, which is one and fifty degrees, and thirty minutes, whose compliment is eight and thirty degrees and thirty minutes, which must be placed from B unto H, then from H, draw the line HAL, representing the Equinoctial line, and from F, draw the line FAG, representing the Pole of the World, then from H unto P, and from I unto Q, set the declination of the Sun, which is twenty degrees, and by those two points draw the line PQ, for the Parallel of the Sun's declination; then upon the Circle from B unto H▪ set the Sun's Almicanter, thirty eight degrees, and thirty minutes, then from H, draw the line HR▪ parallel unto the Horizon cutting the Sun's parallel POQ in O, then draw the Line TVAE Perpendicular unto the line BAD, in the Centre A, and cutting the line HUR, in V, then setting one foot of your Compasses in the point V, extend the other unto R, and with the same distance draw the Semicircle HLR, then draw the Concentricke Circle upon the Radius of the Scale MTN, and where the Line POQ, and the line MON do meet in the point V, raise the Perpendicular OL, cutting the Semicircle HLR in L, then lay the Scale from the Centre A to the point L, and draw the line LK, cutting the Semicircle MTN, in K, so shall M K, be the true distance of the Sun from the East, or West point Southward, or the Sun's true Azimuth, which is here found to be seventy two degrees, and forty minutes from the South part of the Meridian.   

CHAP. XVIII.  
The Latitde of the place, the Declination of the Sun, and the Altitude of the Sun being given to find the Azimuth: By a new way differing from that in the former Chapter.  
 deg. min. S. deg. m. The Sun's Declination is 20 00    The Latitude of the place is 51 30 its Comple. 38 30 The Sun's Altitude is 12 00  78 00    Sum 116     difference 39 30  
HAving found the sum and the difference of the compliment of the Sun's Altitude, and the compliment of the Latitude as above is expressed where you find the Sum of them to be 116 deg. 30 min. and their difference 39 deg. 30 min.  
Secondly, take 116 deg. 30 min. the sum out of your line of Chords, and set it from C to G, and draw the line GK parallel D to B,  
Thirdly take 39 deg. 30 min. the difference, out of your line of Chords, and set it from C to H, and draw the line HL parallel also to BD.  
Fourthly Take in your compasses the distance from F to K, and setting one foot in A, with the other describe the arch S.  
Fifthly, Take the distance from F, to L, and setting one foot in C, with the other describe the arch R.  
Sixthly, Lay a ruler, that it may only touch these two arches, S, and R, and by it draw a line as SIR, cutting the line AC in N.  
Lastly, upon the point N, erect the perpendicular NM, than the distance AM, measured upon your Line of Chords, is the Azimuth from the South part of the Meridian, which in this example will be found to be 34 deg. MC the Azimuth from the North 146 deg. And MD, the Azimuth from the East or West, 56 deg.   

CHAP. XIX,  
The place of the Sun being given, to find the right ascension, Suppose the Sun be in the twentieth degree of Taurus, his right ascension is found as followeth.  
FIrst, as in the 12 demostrastion, draw the line BAF, for the Pole of the World, the ● upon the Centre A draw the Circle BCDE, then from the Centre A, raise the Perpendicular CAESAR, for the Equator, then place your greatest declination from C unto Q, and from E unto P, then daw the line QAP, which doth represent the Ecliptic line, then in regard the Sun is in the twentieth degree of Taurus, which is forty degrees, from the head of Cancer, which forty degrees, place from Q unto L, and unto K, then draw the line KL, cutting the Ecliptic in I, then from the point I draw the line HI, parallel unto CAESAR, cutting the Pole of the World in O than set one foot of your Compasses in O, and extend the other unto G, with which distance draw the Semicircle HDG, then opening your Compasses unto the Radius of the Scale, and upon the Centre O, likewise draw the Circle HNFG, then draw the line IN, parallel unto AOD, cutting the Semicircle HMDG, in M, then lay your Scale from the Centre O, unto the point M, and draw the Line NM, cutting the Concentricke Circle in N, so shall the distance NF, be the right ascension, which is here found tobe two and forty degrees, seven and twentieminutes.   

CHAP. XX.  
The elevation of the Pole, and declination of the Sun given, to find the difference of the ascensions.  
The Poles elevation is 51 degrees, 32 minutes, the declination of the Sun is 21. degrees.  
FIrst, as in the 13th. demonstration, draw the Line BACK, representing the Horizon, then upon the Centre A, draw the Circle BCDEF, Then from K unto D, set the elevation of the Pole which is 51 degrees, and thirty two minutes: then from the point D, by the Centre A, draw the Line DAF, representing the Pole of the World, then from B unto C, set the Compliment of the Poles elevation which is thirty eight degrees, and 28 minutes: then from C by the centre A, draw the line CAESAR, representing the Equinoctial Line; then from C unto G▪ and likewise from E unto H, for the declination of the Sun, which is 21 degrees, then from G unto H, draw the parallel of the Sun's declination, cutting the Pole of the world in L, and he Horizon in I, then set one foot of your Compasses in the point L, and extend the other unto G, then with that distance of your Compasses draw the Semicircle GMNH, then opening your Compasses unto the Radius of your Scale, upon the same Centre draw the Concentricke Circle, GXOH, then from ay, where the declination of the Sun doth cut the Horizon, draw the Line IN, parallel unto the Pole of the World AM, cutting the Circle GMH in N, then lay your Ruler from the point I unto the point N, and so draw the line NO, cutting the Concentricke Circle GXOH, in O, so shall the distance of O and X, be the difference of the ascensions, which is here found to be eight and twenty degrees, and four and fifty minutes.   

CHAP. XXI.  
The right ascension of the Sun or of a Star being given, together with the difference of their ascension, to find the oblique ascension or descension.  
The Sun is in the 4th. degree of Sagitarius, his right ascension is 242 degrees, or 16 hours 8 minutes, the difference of ascension is 1 hour 53 min. or 28 deg. 28 min. the oblique ascension or descension is required.  
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 Star is situate in the Northern or Southern Signs: As for example, if the Sun be in any of these six Signs, Aries, Taurus, Gemini, Cancer, Leo, or Virgo, than the difference of the ascensions is to be substracted from the right ascension, and the remainder is the oblique ascension. Suppose therefore the Sun to be in the fourth degree of Gemini, where the right ascension is found to be four hours, and 8 minutes, or 62 degrees, and the difference of ascension where the Pole is elevated 51 degrees, is found to be one hour 53 minutes, otherwise 28 degrees 50 minutes, which being taken from the right ascension, leaves two hours and 16 minutes, or 33 degrees and 42 minutes, 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 Equinoctial, either in Libra, Scorpio, Sagitarius Capricornus, Aquarius, or Pisces, than the difference of the ascensions is to be added unto the right ascension, and the Product will be the oblique ascension. Suppose the fourth degree of Sagitarius is given, for which Sign and degree the oblique ascension of the Sun is desired, his right ascension being then found to be 242 degrees, or 16 hours 8. min. the difference of the ascensions is one hour, 53 minutes, or 28 degrees, 18 minutes: which being added unto the right ascension, makes 18 hours, and one minute; or in degrees 270 degrees, and 18. minutes: which is the oblique ascension of the Sun, when he is in the fourth degree of Sagitarius. And if you would find the oblique descension, you must add the difference of the ascensions unto the right ascension, when the Sun is in these six Signs. Aries, Taurus, Gemini, Cancer, Leo, Virgo: and chose, when the Su●n is in the other six Signs, you mnst subtract the difference from the right ascension, and you shall have the oblike descension of the Sun or any Star, whose right ascension and difference of ascensions is known. But it is to be understood, that this manner of operation, doth serve no longer than you are upon the North side of the Equinoctial. For if the South Pole be elevated, the work is contrary: for so long as the Sun is in any of the Northern Signs, the difference of the ascensions is to be added unto the right ascension, to find the oblique ascension. And chose, substracted to find the oblique descension. Likewise if the Sun or Star be in the Southern Signs, then is the difference of ascensions, substracted from the right ascension, to find the oblique ascension, and added, to find the oblique descension.  
The end of the Second Book.    

THE SEAMAN'S GLASS: The Third Book.  
Showing how by the Plain-Scale, to delineate Houre-lines upon all kind of Upright Plains, either Direct or Declining, in any Latitude. 
The figure of the plain Scale.   

CHAP. ay  
How to draw hour lines upon an Horizontal Plain, in any Latitude.  
With the Radius of your line of Chords, upon E as a Centre, describe the Circle ABCD, and cross it with he diameters AB, and CD. This done, out of the line of Chords take the compliment of the Latitude of your place (which we here suppose to be London, whose latitude is 51 deg. 30 m. and its compliment 38. deg. 30 m.) which set from B to G, from G to N, and from D to M; then lay a ruler from A to G, and it will cut the line CD in H, and from A to N it will cut C D in O, and from A to M it will cut the same line in F.  
This done, upon O (as a centre) place one foot of your compasses, and extend the other foot to F, and with this distance describe an arch of a circle, which (if the rest of your work be true) will fall just in the points A and B, and so constitute the arch AFB, representing the Equinoctial Circle, and so we shall hereafter call it.  
Having drawn the Equinoctial AFB, divide the Semicircle ADB, into 12 equal parts in the points ***, etc. Then laying a ruler to the Centre E, and every one of these marks *** etc. it will divide the Equinoctial circle into 12 unequal parts in the points ●●●● etc.  
Again, Lay a ruler to H, and every of these unequal parts ●●●●, etc. it will cut the semicircle ADB in the points 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5 and 6.  
Lastly, If you lay a ruler on the centre E, and from thence draw right lines to the several points 7, 8, 9, 10, etc. they shall be 12 of the true houre-lines belonging to an horizontal dial for the latitude of 51 degrees, 30 minutes.  
But for the hours before 6 in the morning, and after 6 at night, do thus; draw the hour lives of 4 and 5 in the evening, quite through the centre E, and they shall be the hours of 4 and 5 in the morning; also, 7 and 8 in the morning drawn through the centre, shall give the hours of 7 and 8 at night, as in the figure.   

CHAP: II.  
Concerning direct South Dial's.  
A Direct South dial is no other than an horizontal dial, the making whereof is before described, the difference consisting only in the numbering of the hours, and in the placing of it, the one being to be fixed on a post or the like, and the other to be fixed to a Wall which exactly beholds the South, I say here is no other difference: for  degrees  degrees An horizontal Dial for the Latitude of 10 Will be a direct South Dial in the Latitude of 80 20 70 30 60 40 50 50 40 60 30 70 20 80 10  
And the like in any other Latitude, as 15, 16, 33, etc.   

CHAP. III.  
Of driect North dials.  
A Direct North dial, is the same with a direct South dial; for, i● you take a South dial and turn it upside down, causing the Sc●le or cock to point upwards, as the Cock of the South doth down wards; and leaving out the hours near the Meridian, in these Northern Latitudes; as the hours of 9, 10, 11, and 12 at night, and 1, 2 and 3 in the morning, all which time the Sun is under the Horizon. I say a South dial so disposed, and fixed against a direct North Wall, shall give you the true hour of the day.   

CHAP. IU.  
How to draw the hour lines on a direct East or West plain.  
This done, upon the point G, with the radius of your Chord, descirbe an occult arch of a Circle H I, and set thereon 15 degrees from H to I, then from G, through I, draw the line G K, cutting N Min K, On K, as a centre, with the radius of your Chord, describe the quadrant K S T, which divide into 6 equal parts in the points ●●●●, through which points and K, draw the lines, K●, K●, etc. cutting the Equinoctial EBB in **** etc. Through these points ***, etc. draw right lines quiet through your plain perpendicular to the equinoctial, which will be parallel to your lines of VI, and XI, and will be the true hours of VII, VIII, IX, and X, than the like distances of VII and VIII, set above VI, on the other side, and drawn parallel thereto, shall be the true hours of FOUR and V. and thus have you all the hours of an East dial truly drawn, which is from Four in the morning, till Eleven at noon, and is the same with a West dial only naming the hours contrary: for, in the East dial 4, 5, 6, 7, 8, 9, 10, 11, in the morning, are in the West dial 8, 7, 6, 5, 4, 3, 2, 1, h in the evening.  
The Style of either of these dials, is a t in plate of brass, made directly of the breadth of the distance between the hours of VI, and IX, and must be placed directly perpendicular upon the line of VI, and so is your dial finished.   

CHAP. V.  
Of upright declining Plains.  
BEfore we come to draw the hour lines upon a declining plain, two things are first to be discovered, viz.  
First. The height of the pole above the plain, which is the height of the Cock or Style.  
Secondly, The deflexion, or distance of the substile from the Meridian or line of Twelve a Clock.   

1. To find the height of the Pole above a declining Plain.  
With the tadius of your line of Chords, upon A, as a centre, describe the Quadrant AB C, than your Latitude being 51 deg. 30 min. take it out of your line of Chords, and set it from B to F, and draw the line ED parallel to AB, cutting the line AC in D, then with the distance DE, on the centre A, describe the Quadrant GHR. Then supposing your plain to decline 30 deg. set 30 deg. from B to F, in the Quadrant BEC, and draw the line FAVORINA cutting the Quadrant GHR in H, through which point H, draw the line SHN parallel to CA, and cutting the Quadrant BE C in N, so shall the arch CN be the height of the Pole above the plain, and in this example contains 32 deg. 37 min.   

2. To find the Deflexion, or the distance of the Substile from the Meridian.  
Out of this figure, take the distance HIS, and set it in the line DE, from D to K; through which point K, draw the line AKL, cutting the Quad ant BC in L; so shall the arch CL be the distance of the Substile from the Meridian: and in thls' Example will be found to be 21 degrees 42 minutes.   

CHAP. VI  
How to draw the Houre-lines upon an upright Plain declining from the Meridian towards the East or West.  
WE will here take for Example a South erect plain, declining Eastward 30 deg.  
Having (by the Fifth Chapter of this Book) found the Defl●xion of such a plain to be 21 deg. 42 min. And the height of ●he ●●ile (by the same Chapter) to be 32 deg. 37 min. we may proceed to draw the Dial in manner following.  
With the radius of your line of Chords, on the Centre C, describe the Circle XNSW; and in it, draw SN through the Centre C, for the Meridian, or line of 12. Then the deflexion being found to be 21 deg. 42 min. set that from N to E, and draw the line ●C through the centre to G▪ This line representeth the Substilar line of your Dial, upon which line the Style or Co●k must stand▪ Also, out from your line of Chords take 32 deg. 37 min. the height of the S●ile, and set that distance from E to H, and draw the line CH for the Style of your Dial; so shall the Triangle EACH, be the true pattern for the Cock of your Dial.  
The Substilar line EGLANTINE being 〈◊〉 ●●aw the line XW through the centre C, and perpendicular to EGLANTINE. This done, take the distance EH, (which is equal to the Styles height) and set that distance from A to B, and from W to D. Likewise, take the distance from W to B, and set it from B to I. These three points I, B and D, being found in the circumference of the Circle XNSW, lay a ruler from X to I and it will cut the substilar line EC being extended in the point G, which is the centre upon which the equinoctial Circle must be described. Again, a ruler laid from X to B, will cut the substilar line in F, and a ruler laid from X to D, will cut the substilar in O. Now, if you set one foot of your Compasses in G, and extend the other to X or W, you may describe the Equinoctial circle XOW, which (if you have not erred in your former work) will pass exactly through the point O in the substilar line before found. In the next place, if you lay a Ruler from F to N, it will cut the Equinoctial circle in P, and a ruler laid from C to P, will cut the Dial circle in V.  
These things being performed, the next thing is to draw the hour lines, which will be easily effected if you 〈◊〉 the former directions.  
First, from the point V last found, begin to divide your hour circle into 24 equal parts (or only one half of it into 12 parts) which you may do by taking 15 deg. out of your line of Chords and set that distance on both sides of V at the marks ⚹ ⚹ ⚹ etc. so many times as the plain is capable of hours.  
This done, If you lay a ruler on the centre C, and every of these points **** etc. you shall divide the equinoctial Circle into 12 unequal parts in the points ●●●● etc. Now a ruler laid from F to every of these unequal points ●●●●, etc. will divide the hour circle into 12 other unequal parts marked with 4. 5. 6. 7. 8.▪ 9 10. 11. 12. 1. on the one side of V, and with 2. 3▪ ●n the other side of V.  
Lastly, a ruler laid from C to the several points 4. 5. 6. 7. 8. 9 10. 11. 12. 1. 2. 3. and lines drawn by the side thereof they shall be the true hour lines belonging to such a declining plain of 30 deg. in the Latitude of 51 deg. 30 min. But if you desire more hours than 12, the equinoctial may be divided into more unequal parts, being continued beyond X and W, and if you will, quite round the whole Circle, but that is needless without you would make 4 dials in the making of one as you may easily do. For,  
The hours that are on the West side of the Meridian of a South East dial, being drawn through the Centre, will make a North West dial of the same declination. And the hours on the east side of the Meridian of a South West dial; being drawn through the centre, will produce a North East dial of the same declination. And Again, the real hour lines of a South East dial being drawn on the other side of the paper, and the hours named by their Compliments to 12, that is, 10 for 2, 9 for 3, 8 for 4, etc. will make a South West dial of the same declination.   

CHAP. VII.  
How to place any upright dial truly.  
ALL upright dials, in what oblique latitude soever have the Meridian perpendicular to the horizon, wherefore to set your dial exact, hang a line with a plummet at the end thereof, and with a nail fixed in the line of 12 towards the top thereof, to hang the plummet upon, apply the dial to the place where it is to be fixed, so that the line and plummet may hang just down upon the line of 12, neither inclining on one side or the other, the dial thus fixed if the declination were truly taken, and the dial rightly made, by the former directions, shall at all times (the Sun shining upon it) give you the true hour of the day.   

CHAP. VIII.  
How to insert the halve and Quarters of hours in all dials.  
THe halves and quarters of hours are drawn in all plains by the same rules, and the like reason, that the hours are inserted. Therefore take notice that if you would insert the half hours into any dial, you must divide your Equinoctial Circle into 24 equal parts instead of 12, and if you would insert the quarters, than you must divide it into 48 parts, and then proceed in all respect, as you did for the whole hours.   

CHAP. IX.  
How to find the declinatioon of any upright Wall.  
THe declination of a plain is an arch of the horizon comprehended between the pole of the plains horizontal line, and the meridian of the place.  
To find this declination, two observations must be made, the Sun shining, and both at one instant of time (as near as may be.) The first is the horizontal distance of the Sun from the pole of the plain. The second is the Sun's Altitude.  
First, to find the horizontal distance. Apply the side of a Quadrant to your plain, holding it (as near as may be) horizontal, that is to say, level, Then holding up a third and plummet, which must hang at full liberty, so that the shadow of the third may pass directly through the centre of the Quadrant, then diligently note ● through what degree of the Quadrant the shadow passed, and count those degrees from the side of your Quadrant which is perpendi●cular to the plain, for those degrees are the horizontal distance.  
Secondly, At the same instant, take the Sun's a●●itude, these two being heedfully taken, will help you to the plains declination by th' rules following.  
By the 17 or 18 Chapters of the Second Book find the Sun's Azimuth. Then observe whether the Sun be between the pole of the plains horizontal line and the North or South points, or not.  
If the Sun be between them, add the Azimuth and horizontal distance together, and the sum of them is the declination of the plain.  
If the Sun be not between them, subtract the lesser of them from the greater, and the difference shall be the declination of the plain. These rules sh●w you the quantity of your plains declination. But,   

CHAP. X.  
Showing how to know whether your plain declin from the Meridian towards either the East or West.  
YOu must take notice in your observation, that if the Meridian point fall between the Azimuth and the pole of the plains horizontal line, then doth the plain decline to the Coast contrary to that wherein the Sun is, that is to say, if the Sun be to the Eastward of the Meridian, the plain declines to the Westward, But if the Meridian point be not between the forementioned distance and the pole of the plain, then doth the plain decline to the same Coast in which the Sun was at the time of observation.   

CHAP. XI.  
Concerning Polar Dial's.  
A Polar dial is made in all respects as an East or West Dial is made, only the line of 6 a clock in the East or West Dial, is 12 a clock in the Polar Dial, the hour of 7 is 1, of 8 is 2, of 9 is ●, of 10 is 4, and of 11 is 5. Also the hour of 5 in the East or West Dial, is 11 in the Polar, of 4 is 10, of 3 is 9, of 2 8, of ● is 7, etc. The Cock of this Dial is a plate of Iron or Brass made of the breadth between 12 and 3 a cloock, and set perpendicular upon the line of 12, as in the East or West Dial it is upon the line of 6. In these dials the Equinoctial line is to lie parallel to the Horizon, and not to be elevated according to the compliment of the Latitude of the place, as in the East or West Dial it is.   

CHAP. XII.  
Concerning Equinoctial dials.  
AN Equinoctial Dial is of all other dials, the most easy to make, for if you describe a Circle, and divide it into 24 equal parts, and draw lines from the centre through eve●● one of those equal parts, the lines so drawn shall be the true hour lines.  
For the Style of these dials, it is no other but a straight Wire of any length set perpendicular in the Centre of the Circle, whose shadow shall give the true hour of the Day.   

CHAP. XIII.  
Of such Plains as decline very far from the East or West towards the Meridian as 75, 80, or 85, deg▪ above which plains the Pole hath small Elevation.  
SUch plains as decline above 60 degrees the hour lines will come very close together, so that if they be▪ not extended very far from the centre, there will be no sensible distance between hour and hour▪  
To remedy this inconvenience, there are several ways, I will instance only in one which is familiar and easy, and that is this.  
When 〈◊〉 have 〈…〉 your dial on a large sheet of paper, fix it on some large Table or smooth Floor of a Room, if the Dial you are to make be very large, as 5, 6, or 7▪ foot square, then by the side of a long ruler laid to the Centre and every hour line, as also to the Style and Substile, draw lines to the full extent of the Table or Flour, and you shall find them to be of a competent largeness. Then according to the bigness of your plain, cut off the hours. Style and Substile, leaving the centre quite ou●, and your work is finished.   

CHAP: XIIII  
Concerning Declining Reclining and Inclining Dial's.  
WE should now show the manner of drawing hour lines upon declining reclining and inclining plains, of which there are several varieties, and many cautions, which in this place and at this time, would be too many to ennumerate: but if this which hath been already delivered concerning Upright decliners shall be kindly accepted, it shall animate me to do the like for all other plains whatsoever.  
FINIS.    

ADVERTISEMENT.  
NOte, that this Scale and all other Instruments for the Mathematics, are made by Walter Hayes, at the Cross daggers in Moor, Fields next door to the Pope's head Tavern, London.