NAVIGATION BY THE Mariners Plain Scale new plained: OR, A TREATISE OF Geometrical and Arithmetical Navigation; Wherein Sailing is performed in all the three kinds by a right Line, and a Circle divided into equal parts.  
Containing 

1. New Ways of keeping of a Reckoning, or Platting of a Traverse, both upon the Plain and Mercators' Chart, without drawing any Lines therein, with new ways for measuring of the Course and Distance in each Chart.  
2. New Rules for Estimating the Ships way through Currents, and for Correcting the Dead Reckoning.  
3. The refutation of divers Errors, and of the Plain Chart, and how to remove the Error committed thereby, with the Demonstration of Mercators' Chart from Proportion, and how to supply the Meridian-line of it Geometrically, albeit there is added to the Book a Print thereof from a Brass-Plate to go alone, or with the Book; as also a Table thereof made to every other Centesm.  
4. A new easy Method of Calculation for Great Circle-sayling, with new Projections, Schemes and Charts, giving the Latitudes of the Arch without Calculation.  
5. Arithmetical Navigation, or Navigation performed by the Pen, if Tables were wanting, with excellent new easy ways for raising of a Table of Natural Sins, which supplies the want of all other Tables.    
By John Collins of London, Penman, Accountant, Philomathet.  
London, Printed by Tho Johnson for Francis Cossinet, and are to be sold at the Anchor and Mariner in Tower-street, as also by Henry Sutton Mathematical Instrument-maker in Thread needle street, behind the Exchange. 1659.   


THE AZIMUTH COMPASS AND PLAIN SCALE.    

To the Honourable, The Governor, Deputy, and Committee of Merchant-Adventurers, Trading to the EAST-INDIA.  
THe Restitution and Establishment of your Honourable Society, promiseth and presageth no less than the future felicity of this Nation, the more or less disemboguing of Nilus could not be more infallible Signals of Egypt's Exuberance or Indigence, than the greater or lesser enlargement of your Trade, is of the increase or decay of our Fame, Riches, and Strength.  
You by penetrating those Oriental Coasts, will to our Clime in great measure restore the Golden Age; many poor being employed by your noble Adventures and Returns, will escape the miseries of complaint and want.  
Your Navy, as at present, it doth employ many able and experienced Seamen, who possibly might otherwise languish under many discouragements, so we may assuredly promise to ourselves the further increase of their number in your Service.  
That their Art and Knowledge might likewise be advanced and rendered more facile, as it should be the desire of all, so it hath been my particular aim and endeavour in the ensuing Treatise, which by that experience I have had amongst them, I guess not to be unsuitable to their desires and spare hours, and though intended chief for their benefit, yet may be of general use to every person, but especially to those that are Mathematically inclined. And now before whom can I more justly prostrate my Endeavours in this kind, than Your Honoured Selves, whose renowned Employments will exercise the choicest Results of this nature, which by how much the more serviceable they are rendered unto you, by so much the more will they become the Objects of Applause and Desire in others; Accept therefore this Offspring of some spare Hours improved, more with an intent for the Advancement of the Public good, then for any private benefit.  
I shall conclude this my humble Address with a Temporal and Spiritual Wish, viz. That the Increase of your Treasures may answer your Hazards and Desires, and that your Virtues and Graces may exceed your Treasures in this Life; and in that to come, may your Glories as far transcend BOTH, as heaven does earth: And for this you have not only the earnest Wishes, but the Cordial Prayers of  
Your Honour's most humble Servitor to be commanded, JOHN COLLINS.   

To the Courteous Reader, whether Mariner or Student.  
HAving formerly spent some part of my time at Sea, I perceived the Genius and Inclination of most Mariners to be affected with drawing such Paper Schemes and Delineations of the Sphere, as might not only represent the same to their fancy, but likewise suit the resolution of such Spherique questions, as commonly they had occasion of; and finding a complaint amongst them of the want of a good Treatise of that kind, I have often wished that some of our able Artists would display their endeavours to that purpose, wherein failing of my desire, and considering that some mean performances of this nature have formerly found good Acceptance with Seamen, as being suitable to the meanest Capacity, and performed by a Scale of small bulk and price, I thought fit to impart my Observations of that kind, and certainly to the most learned it will be matter of use and delight, to see that whatsoever they can perform by exact Calculation, if their Tables be free from error, may be confirmed and performed without them, by the sole assistance of a pair of Compasses and a bare Ruler, and what great ease ensues, both in Navigation and dialing, etc. by skill in carrying on of Proportions Geometrically, is scarcely as yet expected or apprehended by those who are strangers thereto.  
If thou art just entering upon the Mathematicals, this Treatise may serve as an Introduction, in which after thou hast read to page 20 of the first Part, thou mayest apply thyself to the Spherical Definitions in the second Part, and there proceed, or else where, as shall be most suitable to thy desires. Thou wilt easily perceive that the right Angled Cases from page 57 to 63, in the second Book, and also from page 2 to 11 in the third Book, might have been contrived in one general Scheme for each Book, but not so perspicuous to be understood: If this find acceptance it will encourage me to be more laborious hereafter, and if thou reap benefit thereby, I have attained my desire, and that thou mayest so do, is the hearty wish of thy Well-willer and Friend,  
John Collins.   

To his Worthy Friend Mr. JOHN COLLINS, ON HIS Mariners Plain Scale New Plained.  

I feared, good Friend, your Works would sole employ  
O ur Brass, not Wood, which time might putrify,  
H owe're I see your time not spent in vain,  
N or Labour lost the Scale in making plain.   

C apacious, useful, long before neglected,  
O bscured and tedious, therefore disrespected.  
L et those that work in Brass thy Worth admire,  
L et them applaud thee, they have their desire  
I n Wood, if any will be doing too,  
N or can they less in imitation do,  
S o they'll have work, and Students they will gain   

The use o'th' roughhewn Scale now it's made plain:  
And I for thee an Anagram have chose,  
S oh good a Work as this NO HIL INCLOES.  
N oted are the Accounts thou didst put forth,  
I ngenious are thy Quadrants of like worth:  
L et dialing Geometrical ne'er fail,  
L est Mariners forget to use the Scale;  
O h let not Momus with his inked thumbs,  
C ome near to slur thy Works, or try the Rumbs.   

N ever desist, but let's have more of thine,  
H ever's but a Tangent, but let's have a Sine,  
O r bosom full of thy industrious toil,  
I t will inform the weak, every our Soyl.   
Your loving Friend, Sylvanus Morgan.   

The CONTENTS of the First Book.  

In the Proportional Part.  
GEometrical Definitions. Page 1, 2  
To raise Perpendiculars. 2, 3, 4  
To draw a line parallel to another Line. 5  
To bring three points into a Circle. 6  
To find a right line equal to the Arch of a Circle. 9, 10  
Chords, Sins, Tangents, Secants, Versed Sines, etc. defined. 11, 12, 13  
The Scale in the Frontispiece described. 14  
Plain Triangles both right and obliqued Angled resolved by Protraction. from p 15 to 25  
Proportions in Sines resolved by a Line of Chords. p 21 to 25  
Proportions in Tangents alone so resolved. p. 25, 27, 28, 29, 30  
Proportions in Sines and Tangents resolved by a Line of Chords. p. 26, 31, 32, 33   

Particular Schemes fitted from Proportions to the Cases of Obliqne Angled Sphoerical Triangles.  
To find the Azimuth. p. 34, 35  
As also the Amplitude. p. 36  
The Azimuth Compass in the Frontispiece described. p. 38  
The Variation found by the Azimuth Compass. p. 39  
To find the Hour of the Day. p. 40  
As also the Azimuth and Angle of Position, p. 41  
To find the Sun's Altitudes on all Hours. p. 43, 46  
Also the Distances of places in the Arch of a great Circle. p. 44  
To find the Sun's Altitudes on all Azimuths p. 48  
The Latitude, Declination and Azimuth, given to find the Hour. p. 50 to 54  
☞ To find the Amplitude with the manner of measuring a Sine to a lesser Radius. p. 55  
To get the Sun's Altitude by the shadow of a Thread or Gnomon. p. 56   

The Contents of the Treatise of Navigation.  
OF the Imperfections and Uncertainties of Navigation. p. 1 to 5  
To measure a Course and Distance on the Plain Chart. p. 7, 8, 9  
Of the quantity of a degree, and of the form of the Logboard. p. 9, 10  
A Reckoning kept in Leagues, how reduced by the Pen to degrees and Centesmes. p. 11, 12  
Of a Traverse-Quadrant. p. 13  
A Traverse plaited on the Plain Chart, without drawing Lines thereon. p. 14 to 18  
A Scheme, with Directions to find what Course and Way the Ship hath made through a Current. p. 18 to 21  
Divers Rules for Correcting of the Dead Reckoning from p. 21 to 33  
Of the errors of the Plain Chart. p. 33  
And how such Charts may be amended. p. 34  
To find the Rumbe between two places. p. 35  
Proportions having one term, the middle Latitude how far to be trusted to. p. 35 to 38  
To find the Rumbe between two places by a Line of Chords only. p. 39 to 42  
The Meridian-line of Mercators' Chart supplied generally by a line of Chords. p. 42 to 47  
The Meridian-line divided from the Limb of a Quadrant, with the use thereof in finding the Rumbe. p. 48 to 51  
The error committed by keeping of a Reckoning on the Plain Chart removed. p. 52 to 54  
Of the nature of the Rumbe on the Globe. p. 55 to 57  
Mereators' Chart Demonstrated from Proportion. p. 58 to 60  
Objections against it answered. p. 60 to 63  
To find the Rumbe between two places in the Chart. p. 64  
Distances of places how measured on that Chart. p. 65 to 71  
Another Traverse-Quadrant fitted for that Chart, with a Traverse plaited thereby, without drawing lines on the Chart. p. 71 to 78  
To measure a Course and Distance in that Chart, without the use of Compasses. p. 79  
Of Sailing by the Arch of a great Circle. p. 81  
To find the Latitudes of the great Arch by the Stereographick Projection. p. 82 to 83  
Of a Tangent Projection from the Pole for finding the Latitudes of the great Arch p. 84 to 88  
With a new Method of Calculation raised from it p. 89, 90  
And how to measure the Distance in the Arch, and the Angles of Position. p. 91  
Another Tangent Projection from the Equinoctial for finding the Latitudes of the Arch. p. 93 to 100  
And how to find the Vertical Angles, and Arkes Latitudes Geometrically. p. 100 to 102  
To draw a Curved-line in Mercators' Chart resembling the Arch, with an example for finding the Courses and Distances in following the Arch. p. 102 to 104  
The Dead Reckoning cast up by Arithmetic. p. 106 to 108  
A brief Table of Natural Sins, Tangents and Secants for each point of the Compass and the quarters. p. 107  
The difference of Longitude in a Dead Reckoning found by the Pen. p. 109  
That a Table of Natural Sines supplies the want of all other Tables p. 110  
Many new easy Rules and Proportions to raise a Table of Natural Sins. p. 111 to 113  
And how by having some in store to Calculate any other Sine in the Quadrant at command. p. 114  
Of the contrivance of Logarithmical Tables of Numbers, Sins and Tangents, and how the want of Natural Tables, and of a Table of the Meridian-line, are supplied from them. p. 117  
The Sides of a Plain Triangle being given, to Calculate the Angles without the help of Tables two several ways. p. 118, 119  
An Instance thereof in Calculating a Course and Distance. p. 119    

CHAP. I. Containing Geometrical Definitions.  
A Point, is an imaginary Prick void of all length, breadth or depth.  
A Line, is a supposed Length without breadth or depth, the ends or limits whereof are Points.  
An Angle, derived from the word Angulus in Latin, which signifieth a Corner, is the inclination or bowing of two lines one to another, and the one touching the other, and not being directly joined together.  
If the Lines which contain the Angle be right Lines, then is it called a Right lined Angle.  
A right Angle, when a right Line standing upon a right Line, maketh the Angles on either side equal, each of these Angles are called Right Angles, and the Line erected is called a Perpendicular Line unto the other.  
An obtuse Angle, is that which is greater than a right Angle.  
An acute Angle, is that which is less than a right Angle, when two Angles are both acute or obtuse, they are of the same kind, otherwise are said to be of different affection.  
An Angle, is commonly denoted by the middlemost of the three Letters set to the sides, including the said Angle.  
The quantity of an Angle is measured by the arch of a Circle, described upon the point of Concurrence or Intersection, where the two sides enclosing the said Angle, meet.  
By the compliment of an Arch or Angle, is meant the remainder of that Arch taken from 90d, unless it be expressed the compliment thereof to a Semicircle of 180d.  
A Circle, is a plain Figure contained under one Line, which is called the Circumference thereof, by some the Perimeter, Periphery, or Limb, a portion or part thereof is called a Segment.  
The Centre thereof is a Point in the very midst thereof, from which Point all right lines drawn to the Circumference are equal, if divers Circles are described upon one and the same Centre, they are said to be Concentric, if upon divers Centres, they are in respect of each other said to be Eccentric.  
The Diameter of a Circle is a right Line drawn through the Centre of any Circle, in such sort that it may divide the whole Circle into two equal parts.  
The Semidiameter of a Circle, commonly called the Radius thereof, is just one half of the Diameter, and is contained betwixt the Centre and one side of the Circle.  
A Semicircle is the one medeity or half of a whole Circle, described upon the Diameter thereof.  
And a Quadrant is just the one fourth of a whole Circle.  
All Circles are supposed to be divided into 360 equal parts, called Degrees, consequently a Semicircle contains 180d, and a Quadrant 90d, and so much is the quantity of a right Angle.  
A Minute is the sixtieth part of a degree, being understood of measure, but in time a Minute is the sixtieth part of an hour, or the fourth part of a degree, 15 degrees answering to an hour, and 4 Minutes to a degree. A Minute is marked thus 1′, a second is the sixtieth part of a Minute, marked thus 1″.  
Where two lines or arches cross each other, the point of meeting is called their Intersection, or their common Intersection.   

CHAP. II. Containing some Geometrical Rudiments.  

Prop. 1. To raise a Perpendicular upon the end of a given Line.  
Let it be required to raise a Perpendicular upon the end of the Line C D, upon C as a Centre, describe the arch of a Circle as D G, prick the Extent of the Compasses from D to G, and upon G as a Centre, with the said Extent, describe the ark D E G, in which prick down the Extent of the Compasses twice, first from D to E, and then from E to F, then join the points F D with a right line, and it shall be the Perpendicular required.  

Otherwise in case room be wanting.  
Upon G with the Extent of the Compasses unvaried, describe a small portion of an Ark near F, than a Ruler laid over the Points C and G, will cut the said Ark at the Point F, from whence a line drawn to D, shall be the Perpendicular required.  
In this latter case the Extent C F becomes the Secant of 60d to the Radius C D, which Secant is always equal to the double of the Radius.  
As to the ground of the former Way, if the three Points C G F were joined in a right line, it would be the Diameter of a Semicircle, an Angle in the circumference whereof made by lines, drawn from the extremities of the Diameter, as doth the Angle C D F, is a right Angle by 31 Prop. 3 Book of Euclid.  
Otherwise:   

To raise a Perpendicular upon the end of a given Line.  
Set one foot of the Compasses in the Point A, and opening the other to any competent distance, let it fall in any convenient point at pleasure, as at D, then retaining that foot in D, without altering the Compasses, make a mark in the line A C, as at E: Now if you lay a Ruler from D to E, and by the edge of it from D, set off the Extent of the Compasses, it will find the point B, from whence draw the line A B, and it shall be the Perpendicular required: Thus upon a Dyal we may raise a Perpendicular from any point or part of a Line, without drawing any razes to deface the Plain.  
Otherwise upon the Point D, having swept the Point E, with the same Extent draw the touch of an Arch on the other side at B, then laying the Ruler over D and E as before, it will intersect the former Arch at the Point B, from whence a line drawn to A, shall be the Perpendicular required.  
The converse of this Proposition would be from a Point assigned, to let fall a Perpendicular on a line underneath.  
To be done by drawing a line from the Point B, to any part of the given line A C, as imagine a straight line to pass through B D E, and to find the middle of the said line at D, where setting one foot of the Compasses with the Extent D E, draw the touch of an Arch on the other side the Point E, and it will meet with the said line A C at A, from whence a line drawn to B, shall be the Perpendicular required.    

Prop. 2. To raise a Perpendicular from the midst of a Line.  
In the Scheme annexed, let A B be the line given, and let it be required to raise a Perpendicular in the Point C.  
First set one foot of the Compasses in the point C, and open the other to any distance at pleasure, and mark the given line therewith on both sides from C, at the point A and B, then setting one foot of the Compasses in the point A, open the other to any competent distance beyond C, and draw a little Arch above the line at D, then with the same distance set one foot in B, and with the other cross the Arch D with the arch E, then from the Point of Intersection or crossing, draw a straight line to C, and it shall be the Perpendicular required.  
The converse will be, to let a Perpendicular fall from a Point upon a given Line: Let the Point given be the Intersection of the two Arks D, E, setting down one foot of the Compasses there, with any Extent draw two Arks upon the line underneath, as at B and A, divide the distance between them into halfs, as at C, and from the given Point to the Point C draw a line, and it shall be the Perpendicular required; and if this be thought troublesome, upon the Points A and B with any sufficient Extent, you may make an Intersection underneath, and lay a Ruler to that and the upper Intersection, and thereby find the Point C.   

Prop. 3. To draw one Line parallel to another Line, at any distance required.  
In this Figure, let the Line A B be given, and let it be required to draw the Line C D parallel thereto, according to the distance of A C.  
First open the Compasses to the distance required, then setting one foot in A, or further in, with the other draw the touch of an Arch at C, then retaining the same extent of the Compasses, set down the Compasses at B, and with the former extent draw the touch of an Arch at D, then laying a Ruler to the outwardmost edges of these two Arks, draw the right line C D, which will be the Parallel required.  
This Proposition will be of frequent use in dialing, now the drawing of such pieces of Arks as may deface the Plain, may be shunned; for having opened the Compasses to the assigned Parallel distance, set down one foot opposite to one end of the line proposed A B, so as the other but just turned about, may touch the said Line, and it will find one Point: Again, find another Point in like manner opposite to the other end of the Line at B, and through these two Points draw a right Line, and it shall be the Parallel required.  
This way, though it be not so Geometrical as the former, yet in other respects may be much more convenient, and certain enough.   

Prop. 4. To draw the Arch of a Circle through any three Points, not lying in a straight Line.  
In the Figure adjoining let A B C be the three Points given, and let it be required to draw a Circle that may pass through them all.  
Set one foot in the middle Point at B, and open the Compasses to above one half of the distance of the furthest point therefrom, or to any other competent extent, and therewith draw the obscure Arch D E F H with the same extent, setting one foot in the point C, draw the Arch F H: Again, with the same extent, setting one foot in the point A, draw the Arch D E, then laying a Ruler to the Intersections of these Arches, draw the lines D G and G H, which will cross each other at the point G, and there is the Centre of the Circle sought; where setting one foot of the Compasses, and extending the other to any of the three points, describe the Arch of a Circle, which shall pass through the three points required.   

Prop. 5. Two lines being given, inclining each to other, so as they seem to make an acute angle, if they were produced, To find their angular point of concurrence or meeting, without producing or continuing the said Lines.  
And if they be multiplied the other way, from D to H, and from C to E, than the lines E H and F G being produced, find the same point I, without continuing either of the lines first given, and with much more certainty.  
An Oblong, a Rectangle, a right angled Parralellogram, or a Long Square, are all words of one and the same signification, and signify a flat Figure, having only length and breadth, the four Angles whereof are right Angles, the opposite sides whereof are equal.  
In Proportions the product of two terms or numbers, are called their Rectangle or Oblong, because if the sides of a flat be divided into as many parts as there are units in each multiplyer, lines ruled over those parts, will make as many small squares as there are units in the Product, and the whole Figure itself will have the shape of a long Square.  
A Rhombus (or Diamond) is a Figure with four equal sides, but no right Angle.  
But a Rhomboides (or Diamond-like Figure) is such a Figure whose opposite Sides and opposite Angles only are equal, either of these Figures are commonly called Obliqne Angled Parralellograms: Thus either of the Figures A B, F E, or B C, E D are Rhomboides, or Obliqne Angled Parralellograms.  
This foregoing Figure is much used in dialing, thereby to set off the Hour-lines: Admit the Sides A B and B E were given, and it were required on both sides of B E to make two obliqne Parralellograms, whose opposite sides should be equal to the lines given, this may be done either by drawing a line through the point E, parallel to A B C, and then make F E, E D and B C, each equal to A B, and through those points draw the sides of the Parralellogram, or continue A B, and make B C equal thereto, and with the extent B E upon A and C, draw the cross of an Ark at F and D: Again, upon E, with the extent A B, draw o●her Arks crossing the former at F, and those crosses or intersections limit the extremities of the sides of the Parralellogram.  
A line drawn within a four-sided Figure from one corner to another, is called a Diagonal-line.  
A Parralellipipedon, is a solid Figure, contained under six four-sided figures, whereof those which are opposite are parallel, and is well represented by two or many Dice set one upon another, or by the Case of a Clock-weight.  

To find a right Line equal to the circumference of a Circle given.  
Let the given Circle be B D C, divide the upper Semicircle B D C into halfs at D, and the lower Semicircle into three equal parts, and draw the lines D E, D F, which cut the Diameter at G and H, and make G I equal to G H, then is the length D I a little more than the length of the quadrant B D, neither doth the excess amount unto one part of the Diameter B C, if it were divided into five thousand, and four times the extent D I will be a little more than the whole circumference of the Circle.  
  

To find a right Line equal to any Arch of a given Circle.  
Let C D be an arch of a given Circle less than a Quadrant, whereto it is required to find a right line equal.  
Divide the Arch C D into halfs at E, and make the right line F G equal to the Chord C D, and make F H equal to twice C E, and place one third part of the distance between G and H, from H to I, and the whole line F I will be nigh equal in length to the Arch C D, but so near the truth, that if the line F I were divided into 1200 equal parts, one of those parts added thereto would make it too great, albeit the Ark C D were equal to a Quadrant, but in lesser Arches the difference will be less, and if the given Arch be less than 60 degrees, or one sixth part of the whole Circle, the line found will not want one six thousandth part of its true length; But when the given Arch is greater than a quadrant, it may be found at twice, thrice, or four times by former Directions: These two Propositions are taken out of Hugenius de magnitudine Circuli, Page 20, 21.  
In dialing, to shun drawing of Lines on a Plain, it may be of frequent use to prick off an Angle by Sines or Tangents in stead of Chords, it will therefore be necessary to define these kind of Lines.  
2. The right Sine of an Arch is half the Chord of twice that Arch: thus G F being the half of G L, is the Sine of the Arch, G A half of the Arch G A L, whence it follows that the right Sine of an Arch less than a quadrant, is also the right Sine of that Arks residue from a Semicircle, because, as was showed above, the Chord of an Ark is the same both to an Ark lesser than a quadrant, and to its compliment to a Semicircle.  
What an Arch wants of a quadrant, is called the Compliment thereof: thus the Arch D G is the Compliment of the Arch A G, and H G is the Sine of the Arch D G, or which is all one, it is the Cousin of the Arch A G, and the Line H G being equal to C F, it follows that the right Sine of the Compliment of an Arch is equal to that part of the Diameter, which lieth between that Arch and the Centre.  
From the former Scheme also follows another Definition of a right Sine, as namely, that it is a right Line falling from one end of any Arch perpendicularly upon the Radius, drawn to the other end of the said Arch, so is G F perpendicular to C A, being the Sine of the Arch G A; likewise A I falls perpendicularly on C G, therefore by the same definition is also the Sine of the said Arch.  
3. The Versed Sine of an Arch is that part of the Diameter which lieth between the right Sine of that Arch and the Circumference: thus F A is the Versed Sine of the Arch G A, and F B the Versed Sine of the Arch B D G.  
4. If unto one end of an Arch there be drawn a Radius, and to the other end a right line from the Centre, cutting the Circle, and if from the end of the Radius a Perpendicular be raised till it meet with the Line cutting the Circle, that Perpendicular is the Tangent of that Arch: thus A E is the Tangent of the Arch G A, and D M is the Cotangent of the said Ark, namely, it is the Tangent of the Arch H G, which is the Compliment of the former Arch.  
5. The aforesaid right Line cutting the Circle, is called the Secant of the said Arch: thus C E is the Secant of the Arch G A, and C M is the Cosecant of the said Arch, for it is the Secant of the Arch D G.  
6. Assigning the Radius C A to be an unit with cyphers at pleasure, to define or express the quantities of these respective lines, in relation to the Arches to which they belong, were to make a Table of natural Sins, Tangents and Secants; of which at large see Mr. Newtons' Trigonometria Brittanica, for abbreviate ways, and something we shall add about it in the Arithmetical part of Navigation.  
7. The Tables being made, their chief use was to work the Rule of Three, or Golden Rule Arithmetically, by multiplying the second and third terms of any Proportion, and dividing by the first, and thereby to resolve all Propositions relating either to Plain or Spherical Triangles, which in lines is performed by drawing a line parallel to the Side of a Triangle, and where four terms either in Sines, Tangents, Secants, Versed Sines, are expressed, as the first to the second, so is the third to the fourth, it implies a Proportion, and that the second and third term are to be multiplied together, and the Product divided by the first.  
The proportion or reason of two numbers, or reference of one to the other, is measured by the Quotient, the one being divided by the other.  
A Proportion is then said to be direct, when the third term bears such proportion to the fourth, as the first to the second: four numbers are said to be proportional when as often as the first and second are the one contained in the other, so often are the third and fourth the one contained in the other.  
Reciprocal Proportion is when the fourth term bears such Proportion to the third, as the first doth to the second.  
A Proportion is said to hold alternately, when the second and third terms thereof change places, and inversly, when the order of the terms are so altered, that one of the three first terms shall become the last.  
Divers Proportions are expressed in this following Book, which if the Reader would apply to Tables, he must understand that when a Side or an Angle is greater than a Quadrant, that in stead of the Sine, Tangent, or Secant of such an Ark, he must take the Sine, Tangent, or Secant of that Arks compliment to a Semicircle.  
That the words Cousin or Cotangent of an Arch given or sought, signify the Sine or Tangent of the Compliment of the Arch given or sought.  
That the Cousin or Cotangent of an Arch greater than a Quadrant, is the Sine or Tangent of the excess of that Arch above a Quadrant, or 90 degrees.  
8. What trouble the Ancients were at in resolving of Proportions by Multiplication and Division, is wonderfully abbreviated by an admirable Invention called Logarithmes, where by framing and substituting other numbers in stead of the former, Multiplication is changed into Addition, and Division into Substraction; of which also see the former Book Trigonometria Brittanica.  
9 What may be performed by either of the former kinds of Tables, may also with a Line of Chords and equal parts be performed, but not so near the truth, without them, and that either by projecting or representing the Sphere on a Flat or Plain, as we have handled in the second and third Part, or by Protracting and Delineating of such Proportions as may be wrought by the Tables: and this in some measure is the intended subject of our subsequent Discourse.  
10. Therefore before we proceed any further, it will be necessary to describe such Lines as are upon the Scale, intended to be treated of.   

The Description of the Scale in the Frontispiece.  
1. The first or uppermost line is a line of a Chords, numbered to 180 degrees, and is called the Lesser Chord, being a double Scale, the undermost side whereof being numbered with half those Arks, is a line of Sines and is called the greater Sins; at the end of this Scale stands another little Scale, which is called the lesser Sins, being numbered with 90 degrees, and both these Scales seem to be one continued Scale.  
2. The second Scale is another line of Chords, called the greater Chord, being fitted to the same Radius with the greater Sine, and numbered to 90 degrees.  
Annexed thereto is a single Line called the line of Rumbes or Points of the Compass, numbered from 1 to 8, in which each Rumbe is divided into quarters, having pricks or full points set thereto.  
3. The third Scale is a line of equal parts or leagues, divided into ten greater divisions, and each of those parts into ten smaller divisions, and each of those smaller divisions into halfs.  
Annexed thereto is another Scale of six equal parts, each of which parts is subdivided as the former.  
How to make a Line of Chords or Sins from the equal divisions of the Circle, is spoke to in the second Book, page 2. The Scale thus described is indeed a double Scale, for it hath two Lines of Chords, two Lines of Sines, and two Lines of equal Parts upon it, and this rather for conveniency then necessity, whereas indeed one of each kind had been sufficient, yea, the Sins might have been wholly spared; for throughout these Treatises nothing is more required of necessity, than a Circle divided into equal parts, as in the third Book, page 4. and a right Line divided into equal parts, which we suppose in every man's power to do if he have Compasses. The Schemes throughout these Books are fitted either to the Radius of the greater or lesser Chord before described, that is to say, they are drawn with 60 degrees of the one or the other of those Lines of Chords.     

CHAP. III. Showing how all the common Cases of Plain Triangles may be resolved by a Line of Chords and equal Parts.  
THis Chapter, from the very Title of it, will seem to many to be unnecessary, if not impertinent, as being in itself so easy that any person that knows any thing can perform, however I thought fit to add it for methods sake, that it might be said that all the usual Cases of Triangles are here performed with Scale and Compasses, which possibly some that are mere Beginners may be ignorant of.  
If a Figure be made of three right Lines, so joined that opposite to each there be an angular point or corner, it is called a right lined Triangle.  
If it have three equal sides, it is called an Equilateral Triangle.  
If it have but two equal sides, it is called an Isoceles, or Equicrural Triangle.  
If all the sides be unequal, it is called A Scalenon Triangle, or Obliqne plain Triangle.  
If it have one right angle, it is called A right angled Triangle, that side which subtends the right Angle, is called the Hipotenusal, the other sides are indifferently called Sides, or Legs, or one of them the Base, and the other the Perpendicular, those parts of the Six of a Triangle which are given or known, are termed the Data, and those unknown or sought, the Quesita.  

Cases of Plain Triangles being right angled.  

1. To find a Side. Given the Hipotenusal and one of the acute Angles, consequently both.  
Admit a Ship sail Southeast and by South ninety Leagues, In this case the Course is the Angle given, and the distance is the Hipotenusal.  
Having drawn E N and N S at right Angles one to another, and described the Quadrant E A with 60d of the Chords, therein set off A B equal to the Course from the south, being 3 points or 33d 45′ of the greater Chord, and this is called setting off of an Angle, and draw N B, wherein set off the distance 90 Leagues to D, the nearest distance from D to N A measured on the equal parts, shows the side sought, if B N A be the Angle given, which in this example is D C 50 Leagues, and so much is the Ships departure from the Meridian or Separation.  
But if B N E were the Angle given, the nearest distance from D to E N measured on the equal parts, is the side sought, in this Example D F 74 Leagues, and eight tenths more of another League, and so much is the difference of Latitude, called the Variation: Now to perform this without letting fall or drawing the pricked Perpendicular Lines D C, D F, which are added only for illustration, will be of excellent use, as shall follow hereafter.   

2. To find a Side. Given the Hipotenusal and the other Side.  
Admit the difference of Latitude were the given Side, and the Distance the Hipotenusal, and it were required to find the departure from the Meridian.  
Prick out of the equal parts the difference of Latitude from N to C 74, 8 Leagues, and from the point C raise the Perpendicular C D, then take the distance out of the equal parts ninety Leagues, and setting one foot in N, the other will somewhere meet with C D, as at D, and the line D C measured on the equal parts, is the Departure from the Meridian, to wit, 50 Leagues in this example, being the Side sought.   

3. To find an Angle. Given the Hipotenusal, and one of the Sides.  
Admit the Distance 90 Leagues were the given Hipotenusal, and the Difference of Latitude N C n were the other Side, and that it were required to find the Course, which is the Angle sought: Having proceeded so far as is directed in the second Case, draw N D, and upon N as a Centre, describe the Arch B A, which measured on the Chords or Rumbes, showeth the Course from that Coast of the world that N C represents, to be 3 points or 33 degrees 45 minutes.   

4. To find a Side. Given a Side and one acute Angle.  
Admit the Side given were the Difference of Latitude N C, and the Angle were the Course B A from the Meridian, and that it were required to find the Departure from the Meridian D C.  
In this case set off the Difference of the Latitude from the equal parts from N to C 74, 8 Leagues, and raise the Perpendicular D C. Upon N describe the Arch E B A, and therein out of the Rumbes, having set off the Course B A, draw the Line N B produced, and it meets with D C produced at D, and the Side D C measured on the equal parts is 50 Leagues, the Departure from the Meridian, as before.   

5. To find the Hipotenusal. Given one of the Sides and an acute Angle.  
In the former Triangle let D C represent a Tower of unknown height, perpendicular to the Horizon, and let N C be a distance measured off from it, 74 yards and eight tenths more, the line N C making right Angles with D C then standing at N. If by a Quadrant the Altitude of D C were found to be 33d 45′, upon N as a Centre, describe an Arch with 60d of the Chords, and therein set off 33d 45′ from A to B, and draw the line N B produced, till it concur with D C at D, than the line N D measured on the equal parts, shows the Distance between the eye and the top of the Tower, which is the Hipotenusal sought, and in this Example is 90 yards, and the line D C there measured, shows the height of the Tower 50 yards.   

6. To find the Hipotenusal. Given both the Sides.  
Admit in the former Triangle that D C being the Departure from the Meridian, namely 50 Leagues, and C N the difference of Latitude, to wit, 74, 8 Leagues were given, and it were required to find the Distance N D: Having drawn D C and N C at right Angles one to another, and therein pricked down from the equal parts the Variation and Separation, the distance between the Points N D measured on the equal parts, showeth the Hipotenusal sought, in this Example 90 Leagues.   

7. To find an Angle. Given both the Sides.  
If it were required to find the Course by what in the former Case was given: Having proceeded so far as is there expressed, draw N D, and upon N as a Centre with 60d of the Chords, draw the Arch B A, which Arch measured on the Chords, showeth the Course from that coast of the world that N A represents, in this Example 3 points, or 33d 45′ from the Meridian, and the Angle N D C is the compliment thereof; and the Ark might as well have been described on D as a Centre, if the Sides D N and D C had been produced far enough.    

Cases of Obliqne Plain Triangles.  

1. To find an Angle. Given two Sides with an Angle opposite to one of them, to find the Angle opposed to the other.  
Let the quantities of the two given Sides be 46, 6 and 30.  
Prick off that next the Angle given from A to B, and upon A as a Centre, describe the Ark D F, and therein prick down 30 degrees 58 minutes for the Angle given, and draw the line A F C, then from the end of the Side pricked off, prick the other Side from B to C, or E, and so the Angle B C A or B E A, is the Angle sought, but which of the two cannot possibly be determined, unless the affection be also given, to wit, whether it be obtuse or acute, though some of our Writers affirm it may be determined by drawing the Triangle as true as you can.  
Then upon the angular Point C, describe an Arch with 60d of the Chords, and measure the said Arch in the Chords, continuing the Sides if need be, and it shows the quantity of the said Angle to be 53d 6′, and the Compliment thereof to 180d, being 126d 54′ is the measure of the Angle B E A, because the Angle B E C is equal to the Angle B C E.  
How to frame such Triangles whose Sides shall be all whole numbers, is showed in our English Ramus, page 155, 156.   

2. From what is given, to find the third Side and the other Angle.  
In this case also, unless the quality of the Angle opposite to the greatest Side be determined, the third Side will be doubtful, to wit, it may be either A E or A C, which extents measured on the equal parts, show the Side accordingly, and the third Angle to be measured, as before.  
This Mr. Norwood doth not make a Case, because an Angle must be first found and determined before the third Side can be found, and then it will be resolved by the following Case.   

3. Two Angles with a Side opposite to one of them given, to find the Side opposed to the other.  
In this Case the third Angle is also given, as being the compliment of the sum of the two former Angles to a Semicircle. As if there were given the Side A B in the former Triangle, the Angle B A C, and the Angle A C B.  
First prick off the Side A B, then subtract the sum of the Angles B A C, and A C B, from 180d or a Semicircle, and there remains the Angle A B C, then upon A as a Centre, set off upon the Arch D F the measure of the Angle A.  
Also upon B as a Centre, the measure of that Angle must be set off, and lines drawn through the extremities of those Arks will meet, as at C, the point that limits the two Sides A C, and B C, which are to be measured on the equal parts: By this Case if A C were the Wall of a Town, and B a Fort shooting into the said Town, the distance of the said Fort might be found from any part of the Town wall, without going out to measure it; for first, with any whole circle or a compass, observe the Arch between A B and A C, and measure the distance C A, again at A observe the Arch between A B and A C, and protract as in this Example, and you may measure the distance between B and any point in the line A C; And so if B were a Tower or Mark on the land, and A C represent the Ships distance in her course, by observing how B bears both at A and C, the Ships distance therefrom might be measured.   

4, 5. Two Sides with the Angle comprehended given, to find the third Side and both the other Angles.  
Thus if there were given the two Sides A B and A C, with the Angle A between them, the said Angle must first be set off in the Arch D F, than a line joining the extremities of the two Sides, as doth B C, is the third Side, which being first found, upon the angular Points C and B, with 60d of the Chords Arks must be drawn, which being limited by the two Sides (produced when it is necessary) are the measures of the Angles sought.   

6. Three Sides to find an Angle.  
If the three Sides be joined in a Triangle which is easily done, first pricking down any one of the Sides, and from its extremities with the other Sides describe two Arks which will intersect at the Point where the other two Sides concur, then will the three angular Points be given, upon which Arks being described between the given Sides, are the measures of the Angles sought.  
7. Three Angles being given, are not sufficient to find any one of the Sides.     

CHAP. III. Showing how all Proportions relating to Spherical Triangles, may be performed by a Line of Chords.  

1. Proportions in Sines alone.  
LEt the Proportion be:  
As the Sine of 19d to the Sine of 25d, So is the Sine of 31d.  
To what Sine? to wit, the Sine of 42d.  
Otherwise:  
Place the extent wherewith the Ark F was described from V to E, and draw the line E G just touching the extremity of the said Ark, then with the extent V A, one foot of the Compasses resting in V, cross the aforesaid line at G, and a ruler over V and G will find the Point D in the Limb, as before.  
Here observe, that every Proportion without the Radius, may be made into two single Proportions, with the Radius in each, thus:  

As the Radius, Is to one of the middle terms,  
So is the other middle term, To a fourth Proportional.   
Again:  

As the first term, Is to the Radius,  
So is the fourth Proportional before found, To the true Proportional sought.   
From which consideration the former Scheme was contrived.  

Two other general ways for working Proportions in Sines.  
Let the Proportion be:  
As the Sine of 70d to the Sine of 50d, So is the Sine of 35d. To what Sine? Answer: 27d, 50′.  
Having drawn the Quadrant D E with 60d of the Chords, and its two Radii D C, C E at right Angles in the Centre, prick down from the Chords one of the middle terms, to wit, 35d from D to H, and draw a line into the Centre, and upon the said line from the Centre to A, prick down out of the Sins the other middle term, to wit, 50d, and through the Point A draw the line A B parallel to D C, then count the first term from D to G 70d, and draw a line from the Centre which passeth through A B at F, and the extent C F measured on the Sins, is 27d 50′ the fourth Proportional, and thus the first term may be varied as much as you please.  
Otherwise:  
Place the Sine of the first term, to wit, of 70d, which in this Example is the nearest distance from G to D C so from the Centre, that it may cross A B produced when need requires: In this Example it crosseth it at I, a ruler over the Centre and the point I, cuts the Limb at K, and the Arch D K being 27d 50′, is the measure of the fourth Proportional, as before.  
When it cannot be there placed, to wit, as when the Sine of the first term is shorter than C B, the fourth Proportional is more than the Radius, and the Arch of the first term being counted from D towards E, a line from the Centre meeting with A B produced, shall be the Secant of the fourth Proportional to the common Radius of the Quadrant.  
The Demonstration of this Scheme is inferred from varying the Proportion, which may stand thus, by changing the two first terms into their compliments, and altering their order they become Secants.  
As the Secant of 55d to Secant 20d, So is the Sine of 50d to the Sine of 27d 50′.  
If C B be made Radius, then is C A the Secant of 55d to the same Radius, and it is also by construction the Sine of 50d to the common Radius, then from the proportion of equality, because the first term is equal to the third, the second term C F being the Secant of 20d to the lesser Radius, is also the Sine of the fourth Proportional to the common Radius, this suitable to the former of those directions, the latter carries on this Proportion:  
As the Sine of 50d to Sine of 70d, So is the Secant of 55d to the Secant of 62d 10′.   

A third, being the common Geometrical way of working all Proportions in Lines, is by drawing a Line parallel to the Side of a Plain Triangle.  
Thus from the former Scheme make C A the Secant of 55d to the lesser Radius, and C H the Sine of 50d to the common Radius, and join A H; also make C F the Secant of 20d, and draw F G parallel to A H, then because A C is equal to H C, therefore F C shall be equal to G C, wherefore the Sine of the fourth Proportional is equal to the Secant of the second Proportional, which is the foundation of the former Scheme; which form of Operation holds generally, though by reason of excursions we shall pursue other forms of Operation more speedy.   

Another way for working Proportions in Sins with a Line of Chords only.  
Let the Proportion be:  
As the Sine of 30d to the Sine of 18d, So is the Sine of 23d. To what Sine? Answer is, 14d.  
If a Line of Sines be wanting, it may be observed that a Line of Chords will supply the defect thereof, by doubling the three first terms, and you will for the fourth term find the Chord of twice the Arch sought, the half whereof is the Arch sought.  
The three terms of the former Proportion doubled, are 60d, 36d, 46d.  
It will be convenient to describe the Ark C always with the least of the three terms, and this kind of Operation is best when the two first terms of the Proportion are fixed.  
If the Proportion had been of the less to the greater:  
As the Sine of 36d is to the Sine of 60d, So is the Sine of 14d to the Sine of 23d.  
In this case the Chord of 28 degrees must be so entered on the line A B, that on foot resting thereon, the other turned about should but just touch the line A C, the foot of the Compasses would rest at D, and the extent D A, being the Chord of 46 degrees, the half thereof 23 degrees, is the fourth Proportional sought.  
This way was chief added, to show the manner of Proportioning out of any line to a lesser Radius, for here A B being the common Radius, and the extent wherewith the ark C was described another lesser Radius, the nearest distance from D to A C, shall be the Chord of 46 degrees to the said lesser Radius.   

To work Proportions in Tangents alone by a Line of Chords.  
Let the Proportion be:  
As the Tangent of 40d: Is to the Tangent of 50d ∷ So is the Tangent of 20d: To the Tangent of 27d 20′.  
Double the three first terms, and they are 80d, 100d, 40d.    

Much after the same manner Proportions in Sines and Tangents jointly may be protracted.  
Example. Let the Proportion be:  
As the Sine of 20 degrees, Is to the Sine of 10 degrees,  
So is the Tangent of 31 degrees, To the Tangent of 17 degrees.  
Having drawn a Circle and its Diameter, as before, prick any extent from C to H, and from A to I, and draw the lines A H and C I, which will be parallel, then double the three first terms of the Proportion, and they are 40d 20d ∷ 62d.  
Prick the Chord of 40 degrees from A to B, and the Chord of 20 degrees from C to D, a ruler over B and D cuts the Diameter at E, and the Segment A E bears such Proportion to E C, as the Sine of 20 degrees doth to the Sine of 10 degrees, then prick the Chord of 62 degrees from A to F, a ruler over F and E, cuts the Circle at G, and the Arch C G measured on the Chords, is 34 degrees, the half whereof 17 degrees, is the Tangent sought.  
In like manner when a Sine is sought, the Diameter must be divided, according to the Ratio of the two first terms.  
If the Proportion were:  
As the Tangent of 17 degrees, Is to the Tangent of 31 degrees,  
So is the Sine of 10 degrees, To a forth Proportional, to wit, the Sine of 20 degrees.  
Here A H and C I being drawn parallel to each other, if C G be 34 degrees, the double of the first term and A F 62 degrees, the double of the second term, a ruler over G and F cuts the Diameter at E, and the Segment C E bears such Proportion to E A as the Tangent of 17 degrees doth to the Tangent of 31 degrees, then pricking the Chord of 20 degrees the double of the third term from C to D, a ruler over D and E cuts the other line at B, and the extent A B being the Chord of 40 degrees, the half thereof 20 degrees, is the fourth Proportional Sine sought.   

Otherwise to Delineate Proportions in Tangents alone within the limits of a Quadrant.  
In the performance whereof, it is required to take out one Tangent by help of the Chords, and to measure another, and so to regulate these two terms, as they may both be less than the Radius, thereby to shun Excursion, which requires knowledge in varying of Proportions.  
Example:  
As Radius, to Tangent 67 degrees, So is the Tangent of 70 degrees to a fourth Tangent. The answer is: 79 degrees 30 minutes.  
A Proportion wholly in Tangents may be all changed into their Compliments, without altering the order of their places, the former Porportion so changed, is:  
As the Radius is to the Tangent of 27 degrees, So is the Tangent of 20 degrees to the Tangent of 10 degrees 30 minutes.  
Operation.  
Otherwise:  
Any two Tangents may be changed into their compliments, if the other two terms of the Proportion do only change places.  
The former Proportion thus changed, is:  
As the Tangent of 63 degrees: To the Radius ∷ So is the Tangent of 20 degrees: To the Tangent of 10 degrees 30 minutes.  
This Proportion is also wrought by the former Scheme.  
The former Proportion lies evident, imagine a Perpendicular raised from the point A till it meet with D C, which being continued, is the Secant of the ark A D, it than lies:  
As the Radius C A to the said Perpendicular or Tangent:  
So is the other Tangent L C, equal to E C to L F, the Tangent sought.  
The other Proportion lies evident also.  
Imagine a Perpendicular erect from the point B, till it meet with C D continued, then:  
As the said Perpendicular the Tangent of 63 degrees: Is to its Base B C, the Radius ∷ So is the Perpendicular L C to its Base L F.  
These are Equiangled Triangles, for the Angle C F Z, is equal to the Angle D C B: what hath been said is sufficient for any Proportion whatsoever, if where the Radius is not ingredient, it be brought in according to former Directions.   

Another manner of Operation derived from the Proportion of Equality.  
Example.  
As the Tangent of 40d: To the Tangent of 50d ∷  
So is the Tangent of 62d 40′: To the Tangent of 70d.  
If this Proportion be changed into the compliments, it will be:  
As the Tangent of 50 degrees: To the Tangent of 40 degrees ∷  
So is the Tangent of 27 degrees 20 minutes: To the Cotangent of the Arch sought, namely, to the Tangent of 20 degrees.  
Draw the Quadrant A C B, and from A to D out of a Line of Chords, prick off the first term 50 degrees, and draw the line D C, then setting one foot in A, draw the Arch H C, and therein prick off 27d 20′ from C to I, a ruler laid from A to I, will help you to the Tangent of 27 degrees, 20 minutes C E, then either find the Point F by drawing a line through E parallel to A C, through which Point F draw the parallel M F, or take E C and enter it, so that one foot resting in D C, the other turned about may just touch A C, and then take the nearest distance from F to B C, which place from C to M, and through the Points F and M draw a right line, then prick off 40 degrees the other middle term from A to L, a ruler laid from the Centre to L will find the Point N, place M N from C to G, a ruler from A to G, will find the Arch K C, which measured on the Chords, is 20 degrees, the fourth Proportional sought in the latter Proportion, the compliment whereof 70 degrees, is the fourth Proportional in the former Proportion.  
The foregoing Scheme doth also represent a Proportion of the less to the greater.  
As the Tangent of 20 degrees, to the Tangent of 40 degrees,  
So is the Tangent of 27 deg. 20 min. to the Tangent of 50 degrees.  
Having pricked off 40 degrees to L, draw a Line from it into the Centre, then enter the Tangent of 20 degrees, so as one foot resting in L C, the other turned about may just touch A C, and it will find the Point N, through which draw a line parallel to B C, and therein prick down M F the third Tangent, then lay a ruler from the Centre over F, and it will cut the Circle at D, the Arch A D is the measure of the fourth Proportional, and by changing in many Cases the places of the second and third term, and admitting the work to be a little without the Quadrant, it may be performed after the same manner, only the parallel F M will be more remote from B C, and possibly the Operation thereby rendered more certain.  
If the Proportion were:  
As the Tangent of 5 degrees, To the Tangent of 10 degrees,  
So is the Tangent 65 degrees, To the Tangent of 77 degrees.  
This Proportion being changed wholly into their Compliments would not be inconvenient; or if the two latter terms were changed into their Compliments, it would be:  
As the Tangent of 10 degrees, To Tangent of 25 degrees,  
So is Tangent 5 degrees, To Tangent of 13 degrees.  
Either way might be commodious enough.   

Proportions in Sines and Tangents protracted from Equality of Proportion.  
Example:  
As the Sine of 10 degrees, To the Sine of 20 degrees,  
So is the Tangent of 59 degrees, To the Tangent of 73 degrees.  
Which by altering the two latter terms into their Compliments, and altering the place of the first and second term, will stand thus:  
As the Sine of 20 degrees, To the Sine of 10 degrees,  
So is the Tangent of 31 degrees, To the Tangent of 17 degrees.  
And the Radius might be introduced into either of these Proportions several ways, from whence might issue divers Varieties of Operation, either of which Proportions may also be performed with as much facility without the Radius.  
Example of the latter.  
Again, when a Sine is sought, I say the former Scheme serves to find it, if the Proportion were:  
As the Tangent of 31d: Is to the Sine of 20d ∷  
So is the Tangent of 17d: To the Sine of 10d.  
If the three first terms were given, the Scheme would find E G to be the measure of the fourth Proportional, the Sine sought.  
To sum or difference two Arks would be easily done, by applying the Chord of the lesser Ark both upwards and downwards: And thus to solve with a Line of Chords all the Cases of Spherical Triangles, will be a matter of no difficulty, but to prescribe Protractions for many particular questions, without intimation of the Proportion applied, were but to misled and puzzle the Reader, as hath been the vain affectation of some.   

Proportions in Sines and Tangents otherwise performed.  
Example: Let a Tangent be sought, and let the Proportion be:  
As the Sine of 40d: Is to the Sine of 25d ∷  
So is the Tangent of 39d: To another Tangent, to wit, 28d.  
Having drawn the Quadrant D L C, place the Sine of 40 degrees from C to A, and the Sine of 25 degrees to B, and draw B K and A I, parallel to C L, then prick the third term 39d in the Limb from D to E, and draw C E, it cuts the Line passing through the second term at F, place the extent B F upon the line passing through the first term from A to G, a rule● over C and G cuts the Limb at H, and the Arch D H being 28d, is the fourth Proportional sought.  
If a Sine were sought. Example.  
Let the Proportion be:  
As the Tangent of 28 degrees: Is to the Tangent of 39 degrees ∷  
So is the Sine of 25 degrees To what Sine? 40 degrees.  
Place 28 degrees and 39 degrees, the two Tangent Terms in the Limb at H and E, and draw Lines into the Centre, then place the Sine of 25 degrees the third term from C to B, and draw B K parallel to C L, and through the point where it cuts the line of the second term of the Proportion, as at F, draw a line parallel to C D, as is G M, and mind where it crosseth the line H G drawn through the first term of the Proportion, as at G, so is the extent G M the Sine of the fourth Proportional sought, to wit, 40 degrees, as it will be found being measured on the Sins.  
By the like reason Proportions in Tangents alone, or in equal parts and Tangents, might have been protracted, for the extents A C, B C, might as well have been Tangents or equal parts, as Sines.    

CHAP. IU. Containing other particular Schemes suited properly from Proportions to the resolution of the obliqne Cases of Spherical Triangles, having here laid a general Method for both, and have expressly handled the right Angled Cases in the second or Analemmatick part, wherein the Reader will meet with such Spherical Definitions as are here wanting.  

CASE I. Three Sides to find an Angle.  
THe particular Example shall be to find the Sun's Azimuth, the Proportion is:  
As the Cousin of the Altitude: Is to the Secant of the Latitude ∷  
So is the difference of the Versed Sins of the Ark of Difference between the Latitude and Altitude, and of the Sun's Distance from the Elevated Pole: To the Versed Sine of the Azimuth from the North in this Hemisphere.  
Example: 

Let the Latitude be 51d 32′.  
The Altitude— 41 34.  
The Sun's declination 23 31 North.    
Having first drawn the quadrant B F C, with the Radius, or 60 degrees of the Chords, then prick the Chord of the Latitude 51 degrees 32 minutes from B to L, and draw C L produced: prick the Chord of the Atitude, to wit, 41 degrees 34 minutes, from B to A, prick the Chord of the Declination 23 degrees 31 minutes in summer, from F upwards to D, but in winter downwards in the arch of the quadrant continued, and draw D E parallel to C F, the nearest distance from A to C F, is the Cousin of the Altitude, which place on the Latitude line, from C to H, so is C H the secant of the Latitude, the nearest distance from H to C F being Radius, which extent place from C to K.  
Then place the arch A L from B to G, and the nearest distance from G to E D is the difference of the versed Sins of the third term of the Proportion; it is also by reason of the Proportion of equality, the versed Sine of the Azimuth from the North, which place from C to M, then because it is greater than the Radius C K, the Azimuth is more than 90 degrees from the North, and K M is the Sine of the Azimuth from the East or West, wherewith on the point K describe the ark O, and a ruler from the Centre touching it, cuts the Limb at Q, and the arch F Q being 15 degrees, is the Sun's Azimuth to the Southwards of East or West, whereas if the point M had fallen as much on the other side K, it had been so much to the Northwards of it.  
This is an excellent Scheme, in regard it requires the drawing of no new Lines till the Declination vary.  
The Cousin of the Altitude may be multiplied, etc. being doubled, it reacheth to I, the Radius, to wit, C K doubled reacheth to L, the extent C M being doubled, reacheth to N, with L N describe the Ark P, and a ruler from the Centre, touching its extremity, finds the Point Q in the Limb, as before.  
Note also, that the Cousin of the Altitude, and the difference of the versed Sins aforesaid, may be taken from a line of Versed or Natural Sins on a Ruler of any Radius big enough, and therewith proceed as if they were taken from the Scheme.  
Moreover, the extent C R is the Sine of the Amplitude to the same Radius, wherewith the quadrant was drawn, and the extent E R is the Sine of the Ascensional difference, or of the time of Sun rising or setting from Six, the Radius to which Sine being D E.   

Another Scheme for finding the Azimuth.  
Example: 

Latitude 51d 32′.  
Declination 13d North.  
Altitude 31d 18′.    
Having with 60 degrees of the Chords, described a Semicircle A D B, draw the Diameter A C B, perpendicular thereto from the Centre raise C D, prick off the Latitude from A to L, and draw the Latitude line L M parallel to D C, prick off the height from D to H, and place the Sine thereof being the nearst distance from H to D C, from L to N place the Chord of the Sun's Polar distance, to wit 77 degrees, from N to P, and setting one foot on M, with the extent M P, describe an Ark, then take the Cousin of the Altitude being the nearest distance from H to A C, and upon C as a Centre, describe another Ark which crosseth the former at S, a Ruler from the Centre over S, cuts the Limb at V, and the Arch A V being 110d, is the Sun's Azimuth from the North.  
The like Operation holds for South Declination, only than the Chord of the Polar distance is greater than 90d by the Sun's declination.  
If the Sun have Depression, the Sine of it must be pricked upward above L in the Latitude Line, and then as before.  
Thus when three Sides are given to find an Angle, the compliments of those Sides may bear the names of Latitude, Altitude and Declination as here, and the Solution will be the same.   

To find the Amplitude.  
Prick 77d the Chord of the Sun's Polar distance from L to P, and setting one foot in M, with the extent M p, cross the Limb at K, and the arch D K being 21d 12′, is the measure thereof.  
In following Schemes we shall find the hour from Noon from the Point A, also the Azimuth from the South may be found from the same Point, and possibly with more convenience in regard the Intersections may not happen so obliqne, and that upon this consideration, that whatsoever Altitude the Sun hath in any Sign upon any Azimuth from the north, he hath the like Depression in the Opposite Sign upon the like Azimuth counted from the South.  
Wherefore retaining the former construction, L N the Sine of the Altitude, which in the former Scheme was placed downwards, in this following Scheme place upwards.  
This is a Scheme of much worth, in regard it requires the drawing of no new Lines till the Latitude vary. In South Latitudes when the Altitude is very great, the intersections of the pricked Arks will fall near the Centre, in that case let the Altitude and Latitude change Names, and fit the Scheme thereto: Here note, that the three points M C S are the angular points of the Sides of a plain Triangle, & if those Sides be doubled (doubling the sine M C outward beyond M) the intersection at S which now happens within the Semicircle, would happen without and beyond it: The foundation of this Scheme shall afterwards be suggested.   

The Azimuth Compass in the Frontispiece described.  
It is supposed to be made of a square Board, on which there is a Circle described which is cut out through: the Line N S represents a thread, upon the Point N as a Centre placed in the Circumference, there moveth a Label or Triangle represented by N S A, the Side N A is supposed to stand Perpendicular, and to have a Slit in it, the line S A is to be a thread extended from S to A, the other side of which Triangle resembles a Movable Tongue or Label, the Centre being in the Circumference, every degree is twice as large as it would be, if it were at the Centre, wherefore the quadrants S E, S W, are numbered with 45d on each side the Line N S, but are not divided with concentric Circles and Diagonals, nor can they be with truth when ever the Centre is placed in the Circumference, and this I call an Azimuth Compass, because though it be not so, yet it supplies the use of one, and if a right line be continued from N to E, and made a line of Sines; also a Tangent of 45d put through the Limb, it, or an Azimuth Compass, is rendered general, without the use of Paper-draughts, as I have showed in the Uses of the smallest Quadrant in my Treatise, The Sector on a Quadrant, Page 277 to 284. where the Reader will meet with ready Proportions for Calculating the Sun's Azimuth or true Coast, not before published.  
The Use of the said Azimuth Compass at Sea, is readily to apply it to any Compass in the Ship, and thereby find the true Coast of the Suns bearing by that Compass to which it is applied, and consequently the Variation thereof, and by my own experience I have often found, that by the thread which passeth through the Diameter of the said Azimuth Compass, it may very well by the View be placed over the Meridian line of the Compass, and then turning the movable Label towards the Sun, so that the shadow of the thread may pass through the Slit, the tongue of the Label, amongst the graduations of the Limb, shows how the Sun bears by the said Compass, in which the touched wires is supposed to be precisely under the Flower-deluce, & when the Sun is more than 45d from the Meridian either way, the thread in the ●iameter of the Circle must be placed by the view over the East or West point of the Compass, and the Sun's bearing accordingly reckoned from thence.  
Then admiting that the bearing of the Sun by the Compass and his true Azimuth or Coast of bearing, be found either by Calculation or the former Schemes, the Variation of the said Compass from the North (which all Needles are liable unto) with the Coast thereof may thus be found.  
Example:  
Let the bearing of the Sun by the Compass be 55d Eastward from the South, and his true coast in the Heavens be 43d ¾ from the South East-wards.  
Then admit it were required to steer the Ship away N E by E, being the fifth point from the North Eastward, it is desired to know how she must wind or steer by that Compass. Out of the Scale of Rumbes place five point from B to R, then measure the extent N R on the Rumbes, and it showeth four points, whence we may conclude th●t to make good the former Course, the Ship must be steered North-east by this Compass.  
The readiest wa● for fin●ing the Variation, is by those Sea Rings described by M●. Wright, but those are chargeable, are but in few ships, fixed but to one Compass, reserved for the Ship masters own peculiar Observations, so that the common Mariners can have no practice thereby.   

A Scheme for finding the Hour.  
Example. 

Latitude 51d 22′ North.  
Declination 23d 31′ North.  
Altitude 10d 28′, Compliment 79d 32′.    
Having described the Semicircle, and divided it into two Quadrants by the line D C, prick as before, the Latitude 51d 32′ from A to L, and draw L M parallel to D C: Prick off the Declination 23d 31′ from D to E, prick the Sine thereof being the nearest distance from E to D C in Winter, or South Declination from L to Q upwards, but in Summer or North Declination O N downwards, and with the Cousin of the Declination being the nearest distance from E to A C, upon C as a Centre, describe the Ark G W, so is the Scheme prepared for that Declination both North and South.   

To find the Hour in Winter.  
The Sun's height being 10d 28′, its Compliment is 79d 32′, prick the Chord thereof from Q to T, and setting one foot upon M, with the extent M T draw the arch I, a ruler from the Centre over that Intersection, finds the point K, and the Arch A K being 30d, the hour is either 10 in the Morning, or 2 in the Afternoon.   

To find the time of Sun rising.  
Prick the Chord of 90d from Q to O, and with the extent M O upon M as a Centre, cross the ark G W at P, a ruler from the Centre over P, finds the point R in the Limb, and the ark D R being 33d 12′, in time about 2 hours' 13′, is the time of the Suns rising or setting from Six to that Declination both North and South.   

To find the Hour of the Day in Summer to the same Declination, the Latitude being the same.  
Let the Altitude be 9d 30′, its Compliment is 81d 30′, prick the Chord thereof from N to H, and with the extent M H upon M as a Centre, cross the arch G W as at S, a ruler from the Centre over that Intersection, finds the point V in the Limb, and the ark D V being 15d, the true time of the day is either five in the Morning, or seven in the Afternoon.  
Having found the Hour first, than the Azimuth and Angle of Position may be easily found from the Proportion of the Sins of Sides to the Sins of their opposite Angles, as in the following Scheme.  
Example: 

Latitude 51d 32′.  
Declination 13d North.  
Altitude 37d 18′.    
By the former Directions the Hour will be found to be 45d from noon, either 9 in the morning, or 3 in the afternoon, the Intersection whereof happens at e, through e, draw e F parallel to A B, and prick the Altitude 37d 18′ from D to H, and draw H C, also join e C and make C O equal to C M, and through the point O, draw O Q parallel to A B, so is the extent C Q the Sine of the Angle of Position, and the extent C P the Sine of the Azimuth from the Meridian.   

Otherwise for the Azimuth.  
With the nearest distance from H to C B, setting one foot in C, cross the parallel e F at F, a ruler from the Centre cuts the Limb at I, and the arch B I is the Sun's Azimuth either from the North or South, in this Case 60d from the South.   

For the Angle of Position.  
With the former extent cross the parallel O Q at G, a ruler from the Centre cuts the Limb at K, and the arch B K being 33d 34′, is the measure of the Angle of Position, and this work might have been performed on the other side D C; but to avoid confusion when the Doubts about Opposite Sides and Angles may be removed, and when not, as when a double answer is to be given, I have showed in a Treatise, Entitled, The Sector on a Quadrant, page 139, 140. And how to find the points I or K without drawing the lines e F or O G, and that by help of a cross or Intersection like that at e, which may either happen within or without the outward Circle, the Reader may attain from the last Scheme for finding the Amplitude.  
The Converse of the former Scheme for finding the Hour, will find the Sun's Altitudes on all Hours, and the Distances of Places in the Arch of a great Circle.  
Example. 

Latitude 51d 32′.  
Declination 23d 31′ North.  
Hour 75d from Noon, that is either 7 in the Morning, or 5 in the Afternoon.    
Having drawn the Semicircle, its Diameter, and by a Perpendicular from the Centre divided it into two quadrants, and therein having pricked off A L the Latitude, and through the same drawn L M produced and parallel to D C, therein from L to M and Q prick off the Sine of the Declination.  
Then prick off the Hour from Noon from A to R, and laying the Ruler from the Centre, draw the line R E, and with the Co-sine of the Declination, namely, the nearest distance from F to D C, draw the Arch G E, and transfer the distance between the points M, and E from M to H.  
Lastly, the distance between the points N and H, is the Chor● of the Sun's distance from the Zenith for that Hour, namely, 62d 37′, the Compliment whereof 27d 23′ is the Altitude sought.  
Moreover, the distance between H and Q, is the Chord of the Sun's distance from the Zenith for the winter declination, namely, 99d 30′, which being greater than a quadrant, argues the Sun to have 9d 30′ of Depression under the Horizon, and so much is his Altitude for the hours of 5 in the Morning, or 7 in the Afternoon when his Declination is 23d 31′ North.   

Another Example for the same Latitude and Declination.  
Let the Hour from Noon be either 10 in the Morning or 2 in the Afternoon, prick off 30d from A to K, and from the Centre draw the line K G, place the distance M G from M to O, so is the distance O N the Chord of 36d 16′, the Compliment whereof 53d 44′ is the Summer Altitude for that Hour, and the distance O Q is the Chord of 79d 32′, the Compliment whereof 10d 28′ is the Winter Altitude for that Hour.  
Also for the Distances of places in the ark of a great Circle, the Case in Spherical Triangles is the same with that here resolved.  
So if there were two places, the one in North Latitude 51d 32′, the other in North Latitude 23d 31′, the difference of Longitude between them being 75d, their distance by the former Scheme will be found to be 62d 37′; but if the latter place were in as much South Latitude, than their distance would be 99d 30′.   

Another Example for finding the Distances of Places in the Arch of a great Circle.  
Example.  
Let the two Places be according to the Seaman's Calendar.  

Isle of Lobos, Longitude 307d 41′. Latitude 40d 21′ South.  
Lizard— 18 30. Latitude 50 10 North.  
Difference of Longitude 289 11. Compliment 70d 49′.   
Having described a Circle, make A M 70d 49′, M I the Latitude of the Island, B I the Sine thereof falling Perpendicularly on C M, A L the Latitude of the Lizard, L A the Sine thereof, make A E equal to the extent A B, and prick B I from L upwards to H, when the places are in different Hemispheres, but downwards to K, when in the same Hemisphere, and the extent H E or K E, is the Chord of the Ark of distance between both places in this Example 

H E is 109d, 41′.  
K E is 48d, 57′.     

Demonstration.  
This Scheme I first met with in a Map made in Holland, the foundation whereof was long since laid by Copernicus and Regiomontanus, who from a right lined plain Triangle, happening at the Centre of the Sphere, have prescribed a Method of Calculation for finding an Angle when three Sides are given: Here we shall illustrate the Converse, how from two Sides and the Angle comprehended, to find the third Side.  
From any two points in the Sphere, suppose Perpendiculars to fall on the plain of the Equator here represented by A L Q M, which Perpendiculars are the Sins of the Latitudes of those two Points, and the distance of the points in the Plain of the Equinoctial from the Centre of the Sphere, are the Cosines of those Latitudes, & the angle at the Centre between those points in the plain of the Equator, is equal to the arch of the Equinoctial between the two Meridian's, passing through the supposed Points in the Sphere: now a right line extended in the Sphere between any two Points, is the Chord of the Ark of distance between those Points.  
Understand then, that the three Points A, C, B, limit the Sides of a righ● lined Triangle in the Plain of the Equator, whereof the Angle A C B  at the Centre, than the extent A B is placed from A to E, if then we draw D E G perpendicular to A Q, and place B I from E to G and D, the extent L G shall be the Chord of the third Side when the places are in different Hemispheres, and the extent L D is the Chord of their distance when they are in the same Hemisphere.  
And if the extent E D, E G, be placed from L to H and K, the line D E G need not be drawn, because the extent L G and L D, if it be rightly conceived, are the two very Points at first supposed in the Sphere, the extent A B as to matter of Calculation, being one Side of a plain Triangle right angled. The sum or difference of the Sins of both Latitudes the other Side, and the Chord of the distance is the Hipotenusal, or third Side sought.  
Thus the Ancients by Calculation, and we by Protraction, having two Sides and the Angle comprehended given, find the third Side; or having three Sides given, find the Angle opposite to that Side which in the Scheme is measured by a Chord, as by result from the three Sides there will be got the two extents A E and C B, and consequently the Intersection at B, and thence the Angle A M, which before was insisted on in finding the Azimuth and Hour, by the like reason the Distances of Stars may be found from their Longitudes and Latitudes, or from their Declinations and right Ascensions. Divers other Schemes from other Proportions might be added for finding the Hour and Azimuth, etc. which which I am loath to trouble the Reader withal.  
I shall add another Scheme for this purpose, which carries on the same Proportion, by which this Case is usually Calculated: The Proportions are expressed in a Treatise, The Sector on a Quadrant, page 127.  
Example. 

Latitude— 51d 32′.  
Declination 23 31.  
Hour— 60 from Noon.    
Having drawn a Semicircle and the Radius Z D, prick the Latitude from C to L, and draw L A parallel to C E, and making A L Radius, place the Sine of 30d from A to 30d: how to do this, is showed in the Second Part, a ruler over D and 30d cuts the Limb at B, from B to F set off 66d 29′ the Sun's distance from the Elevated Pole, and draw a line into the Centre, and make D K equal to D 30d, and draw H K parallel to C E, and it is the parallel of Altitude required, the measure whereof being the Arch C H, is 36d 42′.  
Two Sides with the Angle comprehended to find both the other angles, & then the third side. Schemes may be fitted to Proportions which at two Operations find both those Angles, first according to the first Direction for operating Proportions in the Sins and Tangents, and then by another operation, the third Side may be found: also this Case is performed otherwise in the Geometrical dialing.   

To find the Sun's Altitudes on all Azimuths.  
Example. 

Latitude— 51d 32′.  
Declination 23 31 North.  
Azimuth 30 deg. from the Vertical or Point of East and West.    
Having described a Semicircle, and divided it into halfs by the Perpendicular H C, prick the Latitude from S to L, and draw L A parallel to S N, and proportion out the Sine of 30d to the Radius L A, and prick it from A to 30d, upon which point with the Sine of the Declination, describe a pricked Circle, and a ruler from the Centre just touching the outward extremities of it, cuts the Limb at T and K: Here note, that H C is the Horizontal line and any Ark counted from H towards K is Altitude, but towards N is Depression; so in this Example the Arch H K being 49d 56′, is the Altitude for 30d of Azimuth to the Southwards of the East and West for North Declination, and the Arch H T being 6d 36′, is the Sun's Depression under the Horizon for the same Azimuth, when his Declination is 23d 31′ South.  
It is also his Altitude for 30d of Azimuth to the North-wards of the East or West, when his Declination is 23d 31′ North, because if we place A 30d as much on the other side the Vertical at M, and with the Sine of the Declination describe an Ark at f, a ruler from the Centre will cut the Limb as much on this side H, as T is on the other side.  
If the pricked Circle had happened wholly on one side C H, a ruler from the Centre touching its extremities, and cutting the outward Circle in two points, that nearest unto H had been the Altitude to the assigned Azimuth for South Declination, and the remotest for North Declination.  
And if one Side of the pricked Circle happens below S C, as it may do both for the Sun or Stars, when their Declination is more than the Latitude of the place, than the Quadrant H S. must be continued to a Semicircle, and H C must be also produced, and the Altitude counted above the other end of the said Horizontal Line.  
If the ruler should touch the pricked Circle near the Centre, let it be noted that C 30d is the Radius, and the extent that described the pricked Circle, is a Sine of an Ark to that Radius; whence it follows, that a right angled Triangle may be framed, and the answer given in the Limb, as we have often done before. See the last Scheme for finding the Amplitude.  
The Demonstration of this Scheme, is founded not only on the Analemma, as I have said in the Second Part, but likewise on those Proportions, whereby the Sun's Altitudes are Calculated on all Azimuths.  
The first Proportion finds the Equinoctial/ Altitude proper to the given Azimuth, and is:  
As the Radius: To the Cotangent of the Latitude ∷  
So is the Sine of any Azimuth from the Vertical:  
To the Tangent of the Equinoctial Altitude ∷  
In the former Scheme L A is the Radius of a line of Sines, it is also the Cotangent of the Latitude, if we make C A Radius.  
Then because of the Proportion of Equality in the two first terms, it follows that the sine of the Azimuth from the Vertical A 30d, to the Radius L A, is also the Tangent of the Equinoctial Altitude, making C A Radius, wherefore a ruler over C 30d cuts the Limb at B, and the Ark H B being 21d 40′, is the Equinoctial Altitude sought.  
The next Proportion is:  
As the sine of the Latitude:  
Is to the Cousin of the Equinoctial Altitude ∷  
So is the sine of the Declination:  
To the sine of a fourth Ark ∷  
Get the sum and difference of the Equinoctial Altitude, and of this fourth Ark, the Sum is the Summer Altitude for Azimuths from the Vertical toward the South in this Hemisphere.  
The difference when this fourth ark is 

lesser  
greater   than the Equinoctial Altitude, is the 

Winter  
Summer   Altitude for Azimuths from the vertical towards the 

South.  
North.    
In the former Scheme if C 30d be Radius, 30d A is the sine of the Equinoctial Altitude, and C A is the Cousin thereof, which is also the sine of the Latitude to the common Radius, then by reason of the Proportion of equality in the two first terms, it follows that the sine of the Declination to the common Radius being the third term, is also the sine of the fourth Arch sought, C 30d being Radius, and the summing or differencing of the two Arks, is performed by describing the pricked Circle, and laying a ruler from the Centre touching its Extremities, after this m●nner a Scheme is contrived from Proportions, to find the third Side, having two Sides with an angle opposite to one of them given.   

Two Sides with an Angle opposite to one of them given, to find the Angle included.  
This of all other Cases is the most difficult either on the Analemma or from Proportions to suit Schemes unto, however we shall add a particular Question, with the Proportions and Scheme fitted thereto: 

Let there be given the Latitude,  
The Sun's Declination,    
And the Azimuth to the Sun or Stars from the East or West, and let it be required to find the Hour from Noon,  
The Proportion is:  
As the Tangent of the Azimuth from East or West:  
Is to the sine of the Latitude ∷  
So is the Radius:  
To the Cotangent of the first Ark ∷  
Again:  
As the Cotangent of the Declination:  
Is to the Cousin of the first Ark ∷  
So is the Cotangent of the Latitude:  
To the Cousin of the second Ark ∷  
The Declination being North, if the Azimuth and Angle of Position be both acute, the Sum of the first and second Ark is the Hour from Noon, if of different kinds, their difference.  
But when the Declination is South, if the Azimuth be to the Southwards of the East or West, the compliment of the Sum of the former Arkes to a Semicircle, is the Hour from Noon; but if the Azimuth be to the Northwards of the East or West, the compliment of their difference is the Hour from Noon.  
When the Angle of Position will be acute, and when obtuse. See my Treatise, The Sector on a Quadrant, page 117.  
These Proportions we shall carry on in the Scheme following.  
Example. 

Latitude 51d 32′ North.  
Declination 23d 31′  
Azimuth from the Vertical 30d    
Upon C as a Centre, describe a Semicircle, draw the Diameter N C S, and the Radius C E W perpendicular thereto, set off the Declination from S to D, and the Latitude to L, drawing lines into the Centre.  
Set off the Azimuth being 30d from the Vertical from E to A, the nearest distance from A to N C, prick on the Latitude line from C to I, and draw I Q parallel to C E, and where it intersects the Declination line (which may sometimes fall on the other side of L) set K; also draw I H parallel to S C, and H B equal to the nearest distance from A to C E, then with K Q upon the point B describe an Ark, and on both Sides of it lay a ruler from the Centre just touching the outward Extremity thereof, and you will find the points R and T in the Limb.   

The Declination being North.  
If the Azimuth be to the Northwards of the East, and Angle of Position acute, the Ark N T is the Hour from Noon, in this Example 110d 16′, but if the Azimuth be to the Southwards of the Vertical, and Angle of Position acute, or to the Northwards and the Angle of Position obtuse, the Ark S R is the Hour from Noon, in the former of these Cases 37d 26′.   

The Declination being South.  
If the Azimuth be to the Southwards of the East or West, the Ark S T is the Hour from Noon, in this Case 69d 44′.  
If it be to the Northwards, then is the Ark N R the Hour from Noon, to wit, 142d 34′, we speak in reference to the Northern Hemisphere, in the other Hemisphere let the words North and South change places.  
Thus when we have two Sides with an Angle opposite to one of them, we may find the Angle included by this Scheme, calling one of those Sides the Colatitude, the other the Polar distance, and the Angle given the Azimuth, and that sought the Hour; and thus for the Sun or such Stars as have two Altitudes on every Azimuth, we may find the respective times when they shall be on that Azimuth, by turning the Stars Hour into common time, as I have showed in a Treatise called, The Sector on a Quadrant.  
Here we have retained the former Scheme, but only continued some of the lines thereof, making the Cousin of the Azimuth, to wit, the nearest distance from A to N C to be Radius, equal to C I the sine thereof, to wit, the nearest distance from A to C E, will become the Tangent, which is equal to H B or C R, and making C I Radius, the extent C H equal to I Q, is the sine of the Latitude, than the first Proportion lies plain:  
As C R the Tangent of the Azimuth from the Vertical:  
Is to R B the sine of the Latitude ∷  
So is C S the Radius:  
To S G the Cotangent of the first Ark ∷  
Wherefore the point B is in a right line from the Centre with the first Ark.  
Then making C B Radius, H B is the sine of the first Ark, and C H the Cousin, than the second Proportion lies thus:  
As F H the Cotangent of the Declination:  
Is to H C the Cousin of the first Ark ∷  
So is M K equal to H I the Cotangent of the Latitude:  
To M C equal to K Q, the Cousin of the second Ark to the Radius C B. last, the summing and differencing of these Arks is performed by describing an Ark with K Q upon the Point B.   

Another Scheme for the same purpose, in which the Operations are Sinical, and the Latitude, Declination and Azimuth, the same as before.  
Having described a Semicircle, and divided it into two Quadrants by the Radius C W, prick the Declination from W to D, the Azimuth to A, drawing Lines into the Centre, also the Latitude from W to L.  
Then place the Cousin of the Latitude (being the nearest distance from L to N C) from C to I, and draw I K parallel to N C, then place the extent C O so at Q as a sine in the Limb, that it may fall Perpendicular on C E, as doth Q E, wherein make E B equal to K O, and upon the Point B as a Centre, with the sine of the Azimuth, being the nearest distance from A to C W, describe the pricked Ark, and lay a ruler from the Centre touching its extremities on both Sides, and cutting the Limb at R and T, and the Arkes 

S T  
S R  
N T  
N R   are agreeable in order to the four Cautions given in the former Scheme.  
This Scheme first finds the Angle opposite to the other of the given Sides, whereof C O is the sine, and then we have two Angles with a Side opposite to one of them given, to find the third Angle, which is the Angle included, and the Proportions are of the same kind with those for finding the Altitudes on all Azimuths, which may be found in this Scheme, if we place C I in the Limb as is Q E, and in it make the extent E B equal to I K, than an Ark described after the same manner, with the nearest distance from D to C W, shall in like kind give the Sun's Altitudes on all Azimuths, C W being the Horizontal Line.  
In page 60 of a Treatise of dialing by the Plain Scale or Line of Chords, for the more easy drawing the Parallels of Declination on the Horizontal Projection of the Sphere, I have intimated that a ready way should be showed for finding the Amplitude, and although that before prescribed be very ready, yet I shall add one more from the common Proportion:  
As the Cousin of the Latitude: Is to the Sine of the Declination ∷  
So is the Radius: To the Sine of the Amplitude ∷  
Example.  

Latitude— 62d 35′.  
The compliment is 27d 25′.  
Declination— 23d 30′.   
Having drawn the Quadrant C B F, double the compliment of the Latitude, and it is 54d 50′, prick the Chord thereof from C to R, then double the Declination, which will be 47d, with the Chord thereof upon the point R, describe the pricked Ark G, a ruler laid from the Centre just touching that Ark, cuts the Limb at A, and the Arch B A being 60d, is the measure of the Amplitude.  
Here note, That the extent wherewith G was drawn is the Sine of the Amplitude, if you make C R Radius, and when the Amplitude falls to be large, it may be better found the other way by help of the Cousin, thus: Prick the extent R G from C to D, then make C L and D E equal to C R, so is C E the Cousin of the Amplitude, with which extent upon the Point L, describe the Ark H, and a ruler from the Centre just touching it, cuts the quadrants Limb at A, as before.   

Otherwise with more certainty.  
Upon the Point E, with the extent C D, describe a little Ark near N: Again, upon the point D with the extent C E, describe another Ark crossing the former at N, than a ruler laid over the Centre and the cross at N, will cut the Limb at A, as before; Note this well, because it is wholly omitted in the Second Book, 
☝  page 18, where it should rather have been handled; and if you will double the extent C D, and the Radius C R, you will find the point E twice as far out towards B, and the intersection at N twice as remote from the Centre, as now it is, and by the like reason when a line is to be drawn unto the Limb from a Point that happens near the Centre, take the nearest distance from the point, first to C F, and triple it in the said line, than the nearest distance to C B, and triple it in that line, and find a point of intersection more remote from the Centre, through which and the Centre the line required is to be drawn, and the said extents may be doubled, or often multiplied.   

How to take the Sun's Altitude or Height, by the Shadow of a Gnomon.  
How to perform this, I have showed in the Second Part, but there the Shadow doth not immediately give the height, whereto notwithstanding a Gnomon may be fitted.  
Imagine the former Quadrant to lie flat upon the Plain of the Horizon, and draw the line F K perpendicular to C F of what length you please, which is supposed to be done upon a square piece of Wood  
Then to fit the Gnomon, prick the Chord of 45 degrees from F to M, and draw C K, then imagine the Triangle C K F to be a Gnomon or Cock standing perpendicularly upright over the line C E F, if then you turn this Quadrant so about towards the Sun, that the shadow of the straight edge of the Cock may fall in the line F K, the shadow of the slope edge of the Cock (which edge may be supplied by a Thread) will happen in the Limb, and there show the Sun's Height required, if counted from B towards F. This Conceit is taken from Clavius de Astrolabio.   
FINIS.   

THE MARINER'S PLAIN SCALE NEW PLAINED: OR, A Treatise showing the ample Uses of a Circle equally divided, or of a Line of CHORDS and equal Parts, Divided into Three Books or Parts.  
Being contrived to be had either alone, or with the other Parts.  
I. The first containing Geometrical Rudiments, and showing the Uses of a Line of Chords, in resolving of all Proportions relating to Plain or Spherical Triangles, with Schemes suited to all the Cases derived from Proportions, and the full Use of the Scale in Navigation, according to the particulars in the following Page, and in finding the Variation of the Compass.  
II. The second showing the Uses of a Line of Chords, in resolving all the Cases of Spherical Triangles, by Projecting the Sphere Orthographically, or laying down the Sphere in right Lines, commonly called, The Drawing or Delineating of the Analemma.  
With an everlasting Almanac in two Verses, etc.  
III. The third part showing the Uses of a Line of Chords, in resolving all the Cases of Spherical Triangles Stereographically that is on the Circular Projection with dialing, from three Shadows of a Gnomon on a regular Flat, or by two Shadows, etc.  
To which may also be added a Treatise of the Authors, of dialing by a Line of Chords, formerly published.  
Of great Use to Seamen, and Students in the Mathematics.  
By John Collins of London, Penman, Accountant, Philomathet.  
London, Printed by T. J. for Fr. Cossinet, at the Anchor and Mariner in Tower-street, 1659.   


The Greater Meridian Line.  
The Lesser    

CHAP. I. Of the Art of Navigation.  
THe end of this Art, is to show how a Ship is to be steered and guided, that at length she may arrive at the desired Port.  
The Imperfections and Defects of this Art are many, partly in the skill or theoric, partly in the practice.  
In the first Principles, or in the Theoric, a great defect is, that the true Longitudes of Places are not yet known, and unless the true Longitudes and Latitudes of Places be known, their true Courses and Distances cannot be found, whence it will unavoidably follow, that no true reckoning can be kept.  
After a long Voyage, when a Ship is supposed to be a competent distance, or not very far off the main Land, it is a usual custom of the Master to require from his Mates an account of their judgement concerning the bearing and distance of some Cape or Head-land which they may make, or gain sight of, and in the event he that gives the best judgement, is supposed to have kept the best reckoning; and this may be admitted of, provided they were all at first agreed concerning the true Course and Distance of the Place they shall first espy, from that place from which they were bound, otherwise a bad reckoning in the result may prove better than a good one, in respect of the uncertainty in the Longitude: I suppose Mr. Wright and others have made such grievous Complaints as these.  
That there is 150 or 200 Leagues error in the distance between the Bay of Mexico and the Azores, and in some Charts about 600 League's error in the distance between Cape Mendosino and Cape California, which great errors are partly to be ascribed to the uncertainty of Longitude, and partly to the Plain Chart, which makes Places far more distant than they would be, and situated much out of their true Rumbe.  
Another uncertainty there is in keeping the Ship steered upon the true Rumbe, albeit it were known, for the Wires of the Needle being touched by the Loadstone, are subject to be drawn aside by any Iron near it, and liable to variation, and doth not show the true North and South, which ought continually to be observed and allowed for, as I have said.  
And moreover in one and the same Region or Place, doth not exactly show the same Rumbe or Course, as is well known to Dyallists, by their Experiments in observing the Situation of a Wall; thereby Maetius saith, he sometimes found a degree or two difference, and Mr. Gunter's Experiments at Limehouse, for finding the variation, made it in some places of the same ground more by half a degree then in other places, and in some other places less, which may be thus illustrated.  
Let a Sea-Compass, or rather a Box and Needle, be fastened upon a Surveying Instrument, so that the Needle may exactly point to some mark or graduation in the Box, whereto it may be afterwards set, and then looking through the Sights of the Instrument, observe what mark, as Tree, Tower, Building, or the like, appears in view to the eye, and if there be no such mark, than a Staff may be set up at a competent distance, then move the Field-Instrument round, that the Needle may totter and be unsettled, but the Sights not altered, and afterwards in the same place let it settle itself to point out the very same Point or mark in the Box, as it did before, then if you look through the former sights, the mark set up will not appear in the same visual line to the eye, as before, but on one side or the other.  
And if there be so much uncertainty on the firm Land; how much more in the floating Sea, where the Ship may be carried away with secret and unknown Currents; for the finding out whereof it will be necessary for the Masters and Mates when they see any Island or Land, to observe which way the ripples of water use to set: It will also be very advantageous to observe the usual customs of the Winds, called A Trade Wind, which many times causeth the Sea to have a Course or Current therewith, as Maetius instanceth by an Example between Brasilia and Angola, in the opposite Coast of Africa.  
From the tenth of April to the tenth of July, the Current sets Northwest.  
From the tenth of July to the tenth of October, it seems to have no motion.  
From the tenth of October to the tenth of January, it sets Southwest.  
And from the tenth of January to the tenth of April, it seems to have no motion again.  
Another Instance of the uncertainty in a reckoning by reason of Currents, may be taken from Davis his straits, where the Sea flows with such a violent, secret, or undiscerned force, that those which steer from thence East by the Compass, do find they have made their Course Southeast, which is a sufficient Instance of the danger and uncertainty of Navigation, for when a Ship steers but one Rumbe, or 11d ¼ from the true Course, in the space of an hundred Leagues, she shall fall wide of expectation almost ten Leagues, or 1/10 of the distance, which in four Rumbes error will amoun● to about 70 Leagues.  
The Estimation of the distance sailed, for the most part depends upon ●he judgement, albeit when the Course is directly under the Meridian, it may be known by Observation of the difference of Latitude, if we suppose the true quantity of the measure of a degree on the Surface of the eatrh to be agreed upon, and hereby Estimation ought to be rectified, which otherwise is helped and furthered by the Log-line; the uncertainty whereof by Currents is so great, that Maetius instanceth, that an expert Master being bound to the Isle of St. Helen's in the midst of the Sea, in 16d of South Latitude, and having got into the parallel or Latitude thereof, thinking to make it Eastward, was notwithstanding carried by the hidden Motion or Current of the Sea about 800 Miles Westward, till he found himself near America or Brasilia, and yet notwithstanding stemmed the Current with a fair Wind: And S●ellius instanceth that a Navigator of good repute failing out of Holland, twice missed the Maderas, and returned home to the great damage of the Owners.  
And where such Currents are if you stem them, the distance run will be less than it is by Estimation, but if you sail with them, it will be much more, and proportionally it will be less or more, if you cross it either directly or obliquely, according as it sets with or against your intended Course.  
Lastly, the observation of the Latitude (being the sole help whereby to rectify an erroneous reckoning) cannot be performed so near the truth, but that divers minutes of error may be committed, as Suellius instanceth, that six several persons at the same time in one Ship, who were accustomed to Observations, and who for the most part of them had been more than once at the East and West Indies, observing the latitude, found it to be by their several observations: 48d 7′ 48 8 48 20 48 34 48 38 48 58 of which himself thinks that of 48d 38′ to be the truest, as relying most upon the ability of the Observer.  
Whereto we may add, that the Refraction of the Sun, especially in Winter, makes his Altitude or Height seem greater than it is, which is caused by the gross vapours and thickness of the Air near the Horizon, this we may allude to after this manner: Set down an empty Basin on a Stool, laying a shilling at the bottom thereof, and go so much backwards, till you bring the edge of the Basin and the shilling in a right line with your Eye, and then let another fill the Basin with fair water, and you may go a pretty way further backward, and still see the shilling and the edge of the Basin in a right line, the refraction of the water being the cause thereof; and from this Reason we conclude, That the Sun seems to appear above the Horizon when he is really set.  
This is confirmed also by many Experiments, we may recite one: A certain Dutch Ship being upon the Discovery of a North-east passage to the East-India, was constrained to winter in the Island of Nova Zembla, where after the Sun had been for divers months under the Horizon, the Mariners beheld the whole body of the Sun just above the Horizon 14 days sooner than according to his Declination he should appear, and by Computation was then at least 5 degrees under the Horizon; Also at other times, especially in winter, he appears to be higher than he is, for which there are Tables of Allowance, as also for the Parralax and height of the eye in Mr. wright's Correction of Errors in Navigation, wherefore those that come from the Southwards, expecting to fall in with the Landsend, or Lizard, are to be admonished that in trusting to the Observed Latitude, they do not estimate themselves more to the Southwards then really they are, and so incur danger through presumption; but this erro● is somewhat abated by the height of the eye above the water, observing the Sun with Davis his Quadrant backward in a Horizon more northwardly than the true one: Notwithstanding the imperfections and uncertainties that arise in the practic part, yet it should be our endeavour to render this excellent Art as easy and certain as we can, which is the thing I aim at, and the Instrument here used being the Plain Scale, is, as I said before, in every man's power, if he have Compasses.   

CHAP. II. Showing the Use of the Plain Chart, and of a Traverse-Quadrant, for the more ready keeping of a reckoning.  

Proposition I. The first Proposition is, how by having the Longitudes and Latitudes of two places, to lay them down on a blank or Plain Chart, in their Longitudes and Latitudes, and thereby to find their Rumbe and Distance.  
BY a Plain Chart, is meant a Chart drawn on Paper or Paste-board, lined with Meridian's and Parallels, making right Angles each with other, and numbered with degrees both of Latitude and Longitude, each equal to other, and what is commonly performed in casting up a Traverse on such a Chart, we shall perform on a Blank of Paper.  
By the Course, in a familiar sense, is meant that point of the Compass or Coast of the Horizon on which the Ship is to be steered from place to place.  
And the Rumbe is in effect the same thing with the Course, and is defined to be a Line, described by the Ships motion on the surface of the Sea, steered by the Compass, making the same Angles with every Meridian, the properties of which Line shall afterwards be handled; and whereas every Rumbe maketh with the Meridian both an acute and obtuse Angle, by the Angle of the Rumbe is meant the acute Angle.  
Henricus Sutton fecit 1659.  
We have here added in a Print of the Compass and its Winds, each Quadrant being divided into ninety degrees, and so may serve in stead of another Line of Chords, and those that are desirous may have these Prints upon lose Papers, and passed them on upon a Board, and so a quarter of it may serve for a Traverse-Quadrant, or in pricking down of Courses the common way, in stead of pricking down every Course from the Meridian, they may hereby prick off every following Course from the former Course last sailed upon, without any respect to the Meridian at all.   

Now to the Proposition:  
Let it be required to find the Course and distance between the Isle of Tenariff and St. Nicholas Isle, being one of the Hesperides, the Longitudes and Latitudes of these places I shall take from the late general Dutch Map, in two Hemispheres on the Stereographick Projection, being the best and largest general Map that ever was, in which the first Meridian or beginning of Longitude, takes its rise from the Isle of Tenariff, which having in it the highest mountain in the world, and lying in the Western Ocean, seems to be a very remarkable place for this purpose, and it were to be wished that the Longitudes and Latitudes of places in the Seaman's Calendar which were taken from the Globes of our Country man Jodocus Hondius, were corrected according to the said Map, or rather by those Tables by which it was made.  

Tenariff. Latitude 28d— Longitude 00.  
Isle St. Nicholas— 17— 352d.  
Difference— 11— 8d.   
Lastly, the extent T N measured on the same equal parts, showeth the distance to be 13d 59 Centesms, and when this distance is too large to be measured on the Scale, you may take ten degrees, and turn it over as many times as you can, and afterwards measure the remaining part of the distance by itself.   

Of the quantity of a degree.  
Here it is to be noted, that we have found the distance in degrees, and Centesimals or hundredth parts of a degree, which being a general common measure to all the world, may afterwards be reduced into the Leagues or Miles of any Country whatsoever. Now before we reduce this to our English measure, I will first show what is the usual custom of expressing the Course and Distance on a Ships Logboard at Sea, and then how in effect the same custom may be still retained or altered, and yet either way agree with the Truth.  
Time Course Knots Half knots Hours 2     4     6     8     10     12     2     4     6     8     10     12     
The first Column is for Time.  
The second for the Ships Course.  
The third for the Knots.  
The fourth for the Half-knots.  
Our English or Italian Mile by which we reckon at Sea, contains 1000 paces, and each pace five foot, and every foot 12 inches. The 120th part of that Mile is 41⅔ feet, and so much is the space between the Knots upon the Log-line: So many Knots as the Ship runs in half a minute, so many Miles she saileth in an hour; or so many Leagues, and so many Miles she runneth in a Watch or four hours, called A Watch, because one half of the Ships Company watcheth by turns, and changes every four hours.  
Example  
Six Knots in half a minute is six Miles in an hour, or six Leagues six Miles in a Watch, which is eight Leagues or twenty four Miles in all.  
Every Noon the Master and his Mates take the reckoning off the Logboard, and double the Knots run, and then divide the Product, which is the number of Miles run by three, the quotient is the Leagues run since the former Noon, and according to custom the Log is thrown every two hours, and I never knew the course nearer expressed on the Logboard, then to half a point of the Compass.  
But Mr. Norwood in his excellent Book, called, The Seaman's Practice, showeth, That according to a late exact Experiment of his, which seems to be very satisfactory, and confirmed by many other Experiments, that in ordinary Practice at Sea we cannot, if we will yield Truth the conquest, allow less than 360000 of our English feet to vary one degree of Latitude upon the earth, in sailing North or South under any Meridian, according to which account there will be in a degree of our Statute measure 68 2/1● Miles, each Mile containing 5280 feet, and of the common Sea-measure, allowing 5000 feet to a Mile, there will be 72 Miles, or 24 Leagues in a degree, which we shall take to be the truth.  
A reckoning being kept in degrees, and Centesms or hundredth parts of a degree, is a ready way, and well approved by Mr. Norwood in his said Seaman's Practice; as also by Mr. Phillip's in his Geometrical Seaman, who wholly useth this measure, and yet the Mariner may very well refrain this measure, if he will keep his reckoning in Leagues and tenths, as I shall afterwards show. Now to show this measure, the Logline must have a knot placed at every 30 foot length, and as many of those as run out in half a Minute, so many Centesms or hundredth parts of a degree the ship saileth in an hour, and for every three foot more you are to allow the tenth part of a Centesm, or the one thousandth part of a degree.  
But if you would have it to show the Miles of a true degree, allowing but 60 to a degree, the mile must be enlarged proportionally, and the distance between every one of the Knots must be 50 foot and as many of those as run out in half Minute, so many Miles or Minutes the ship saileth in an hour, and for every foot more you must allow the tenth part of a mile more and if Seamen be desirous to retain their former custom of reckoning the ships way in Leagues, then must either the said reckoning be reduced into degrees and Centesms of degrees by Arithmetic, because the degrees of Latitude on the Plain Chart, and of Longitude and Latitude on Mercators' Chart are divided, and that most conveniently into 10 parts, and each part supposed to be divided into 10 more, or else such a Scale of equal parts over and above must be added to each Chart, wherein a degree of Latitude in the Plain Chart, and of Longitude in Mercators' Chart, must be divided into 20 equal parts, and the labour of reducing by the Pen, will be saved; or the adding of any such Scale, may very well be spared and otherwise supplied, as I shall afterwards show.   

Of Reduction by the Pen.  
1. Miles and their Decimals parts, are reduced into Leagues and tenths of Leagues, by dividing by three: Here note, that the tenth part of any measure is expressed either with a Comma or point before it, or else with a separating line thus, 3 or ⌊ 3 either way, the figure 3 signifies three parts of any thing divided into ten, and ⌊ 35 signifies thirty five parts of any thing divided into one hundred, and generally in this Decimal Arithmetic, the Denominator is understood to be an unit, with as many cyphers following it, as there are units in the Numerator; thus ⌊ 003 signifies three parts of one thousand.  
Example:  
375 ⌊ 6 Miles divided by 3, makes 125 ⌊ 2 Leagues, and on the contrary, 135 ⌊ 2 Leagues multiplied by 3, makes 375 ⌊ 6 Miles.  
2. Leagues and their Decimals parts, are reduced into degrees and Centesmes, by separating one figure from the whole Leagues towards the left hand, and dividing the whole by 2.  
Example.  
271 ⌊ 8 Leagues separated, will stand thus, 27 ⌊ 18 and then divided by two, the quotient is 13 ⌊ 59, that is 13 degrees and 59 Centesms, or hundredth parts of a degree more.  
On the contrary degrees and centesms are deduced into Leagues, and their Decimal parts, by placing the Comma after the first Decimal part, toward the right hand, and then multiplying the whole by two: thus the former number of degrees are to be expressed 135, 9 which multiplied by two, makes 271 ⌊ 8 leagues, as before.  
3. Minutes, whereof there are 60 in a degree, are reduced into Centesimals by annexing a Cipher thereto, and dividing by 6: thus 27 Minutes with a Cipher, make 270, which divided by 6, giveth in the quotient 45 Centesms, here after 27 was divided by 6, the quotient was 4, and 3 remaining with the Cipher is 30, which divided by 6, the quotient is 5.  
On the contrary, the Centesmes of a degree are reduced into Minutes, by multiplying them by 6, and cutting off the last figure: thus, 75 Centesmes multiplied by 6, make 450, which is 45 minutes.  
This trouble, as I said before, is taken away by fitted Scales by the View only, if they stand back to back, as in the Scales of equal parts on the Plain Scale, where 75 Centesmes in the lesser Scale of equal parts, stands against 45 parts or Minutes in the greater Scale of equal parts, and thus when the Latitudes of two places are given in degrees and minutes, the minutes are to be turned or reduced into Centesmes.  
And so in the example of the former Chart, the line A B containing 10 degrees, is divided into 20 parts, and one of those parts into 10 others, which I call, The Scale of Leagues of a true degree, then because the whole distance T N is too long to be measured at once, I first measure 10 degrees or 200 Leagues from T to C, and then C N measured on the Leagues, is 71 leagues and eight tenths more, and on the centesimal degrees, is 3 degrees 59 Centesmes.  
So that the whole distance is 271 Leagues eight tenths, or 13 degrees 59 Centesmes, which is thus expressed 13d, 59: or thus, 13d ⌊ 59: or thus, 13d 59/100: and when we speak of minutes which are marked thus 1′, we always presume there are but 60 of them in a degree.   

Of the Traverse-Quadrant.  
Before we come to work any Traverse, it will be very ready and convenient to prepare a Traverse-Quadrant, which as you see by the Scheme of it is no other than a common Quadrant, having its Limb divided into 16 equal parts, for the points and half points of the Compass, which may be easily done by the pricked rumbes on the Scale, each whole point being numbered from the Meridian with Capital or letter figures.  
In casting up a Traverse by this Quadrant, we shall imita●● Mr. Norwood in his Seaman's Practice, who after the same manner plots the Survey of any Field, which he doth by help of Tables of Variation and Separation, but is here performed without them.  
Then we will suppose that the Ship looseth from Tenariff, and saileth 60 Leagues South South-West, afterwards she saileth 80 Leagues West South-West, then meeting with a contrary wind she sails 53 Leagues South and by East, half a point Eastwardly.  
Suppose the Course and Distance from the Ship to St. Nicholas Island were now required; as also that it were demanded what Course the Ship should steer, and distance she should run to bring herself about 23 League's East from St. Nicholas Island, we shall plot the whole Traverse, and resolve the Questions demanded without drawing any Lines on the Plat.  
1. The first Course is 60 Leagues, South South West, which is the second point from the Meridian, take 60 out of the Scale of Leagues, and place it in the Traverse-quadrant from C the Centre to a, the nearest distance from a to C W, place in the Chart, on the line T L from T to 1; also the nearest distance from a to C S in the Quadrant, place in the Court on the Line T D from T to 1.  
2. The second Course was 80 Leagues, West Southwest, which is the sixth point from the Meridian, take 80 out of the Scale of Leagues, and place it in the Traverse-quadrant from C the Centre to b, on the sixth Point, the nearest distance from b to C W place in the Chart on the line T L, from 1 to 2, and the nearest distance from b to C S in the Quadrant, place in the Chart on the line T D from 1 to 2, and thus the difference of Latitude or Variation, and the Departure from the Meridian or Separation, are added together, when they are of the same kind, that is, do still augment.  
3. The third Course is South and by East, half a point Eastwardly 53 Leagues, which is a point and a half from the Meridian, which enter in the Traverse-quadrant on the said Course from C to c, the nearest distance from c to C W, because the difference of Latitude doth still increase; place in the Chart on the line T L from 2 to 3 towards L (otherwise if it had decreased, it must have been pricked backwards towards T) and the nearest distance from c to C S in the Quadrant, place in the Chart on the line D T from 2 to 3 towards T, because the Departure from the Meridian decreaseth (otherwise if it had still increased, it should have been pricked from 2 towards D) having proceeded thus far, the point where the ship is in the Chart may be found to any one, or every one of the several Courses, without drawing any Lines in the Chart.  
1. Out of the West line T D take T 1, and with it upon the point 1 in the South line T L, describe a little Ark at a, also out of the South line take T 1, and with it on the West Line at 1 describe an Ark at a crossing the former, so the cross at the point a, shows the place where the ship is at the first Course.  
2. Again, take T 2 out of the West line, and setting one foot of the Compasses upon 2 in the South line, describe a small piece of an Ark near b; also take T 2 out of the South line, and setting one foot at 2 in the West line, describe an Ark crossing the former at b, which point is the place where the Ship was at the end of the second Course.  
3. Take T 3 out of the South Line, and setting one foot of the Compasses on 3, in the West Line describe a small Ark near c: Again, take T 3 out of the West Line, and upon 3 in the South Line, draw an Ark crossing the former at c, and c is the Point where the Ship is, according to this dead reckoning, and the former Point a and b need not have been found, for there is no question proposed concerning them.   

To find the Course and Distance from c to N.  
1. For the Distance, the extent c N measured on the Scale of Leagues, showeth it to be 114 Leagues and a half.  
2. For the Course, to find it without drawing lines on the Chart, lay a ruler over N and c, which we must suppose to cross som● Meridian or parallel in the Chart, here it crosseth the South line at e, and the Scale of Leagues at f, then place the Radius or 90d of the line of greater Sins (which before was pricked from T to R) from e to g, and the nearest distance from g to the edge of the ruler measured on the Sins, showeth the Course from the Ship to St. Nicholas Island, to lie 43d 17′ to the Westward of the South then if you look for 43d 17′ in the greater Chord, you shall find in the Rumbes against it, that this Course is 3 points and above 3 quarters more to the Westward of the Meridian, that is, almost Southwest, and if e g had been placed from f toward A, the nearest distance to the edge of the ruler, would have showed the compliment of the Course required.  
3. To find what Course the Ship must steer to bring herself about 23 Leagues or 22 Leagues 8 tenths East from St. Nicholas Island, draw a Line from N to L, and prick down 22 ⌊ 8 out of the Scale of Leagues from N to d, and this may be performed by the edge of a ruler without drawing any such Line, or otherwise it may be found by the intersection or crossing of two Arks, as the former Traverse Points were found. Now laying a ruler over d and c, it crosseth the South Line at h, than place T R from h to k, and the nearest distance from k to the edge of the ruler, measured on the greater Sins, showeth the Course to be 33d 45′ to the Westward of the South, that is three points, to wit, South-west and by South.  
4. The extent c d measured on the Scale of Leagues, showeth the distance to be 100 Leagues.  
Lastly, we suppose the Ship to sail this Course and distance till she come into the parallel or Latitude of St. Nicholas Island at d, and then she sails almost 23 Leagues West, and arrives at the Island, being her desired Port.  
If it were required to find the Ships Course and distance from the Point c to Tenariff.  
Lay a ruler over the Points T and c, the nearest distance from R to the edge of the ruler, measured on the greater line of Sines, showeth that the Course from Tenariff to the Ship is 30d 47′ to the Westward of the South, which is almost two points three quarters; and contrarily Tenariff bears from the Ship as much to the Eastward of the North, and the extent c T measured on the Leagues, showeth the distance of Tenariff from the Ship to be 158 Leagues.  
This I think sufficient to explain the use of the Plain Chart, upon which in the laying down of any two places in their Rumbe and distance, there is framed a right angled Plain Triangle, one side whereof T L is the difference of Latitude, the other side L N the difference of Longitude, and the third side N T the distance, And after the same manner a Traverse is to be plaited, when the Chart is made true as to the Latitudes of places, and as near the truth as to the Rumbe and distance as it can, waving the longitude; and as the whole Navigation of a Voyage is performed on this Chart without drawing any lines thereon to deface it, so after the same manner, and well nigh with as much ease, may it be performed on the true Sea Chartley, commonly called Mercators, as shall afterwards be showed.  
But before I part from this Discourse, I think it necessary to show how the former Scale of Leagues may very well be spared, and still account the distance run in Leagues and tenths, provided the degrees of Latitude on the Sides of the Chart be divided each of them into 10 equal parts, accounting each degree for 10 Leagues. Take 40 Leagues, and setting one foot in 80 leagues, draw the Ark G, which may be done by any other numbers in the same Proportion, and draw a line from H ●ust touching the said Ark, and the Scale of leagues A B may be spared.  
Suppose I would take out 60 leagues, take the nearest distance from six in the latitude side of the Chart, to the line H G, and it is the measure of 60 leagues, and so much is the distance at her first Traverse from T to a.  
But supposing the distance T a in the Chart were unknown, and I would measure it, take the said extent, and prick it twice in the side of the Chart from H, and it will reach to 60, and so many leagues is the distance required; and so any other extent turned twice over, shows the distance in leagues, accounting every degrees 10 leagues, and being turned but once over, shows the distance in degrees and Decimal parts.   

How in estimating the Ships Course and Distance, to allow for known Currents.  
This subject is handled by Mr. Norwood in his Seaman's Practice, at the end, and by Mr. Phillip's in his Advancement of Navigation, page 54 to 64. As also how to how find them out by comparing the reckoning outwards between two places, with that homewards, wherefore I shall be very brief about it, and perform that with Scale and Compasses, which is by them done with Tables.  
1. If you stem a Current, if it be swifter than the Ships way, you fall a stern; but if it be slower, you get on head so much as is the difference between the way of the Ship, and the race of the Current.  
Example: If a Ship sail 7 Miles North in an hour, by estimation against a Current that sets South 4 Miles in an hour, than the ship makes way three Miles or a League in an hour on head, but if the Ships way were 4 Miles in an hour by estimation North against a Current that sets 7 Miles in an hour South, the Ship would fall 3 Miles or a league a stern in an hour.  
But suppose a Ship to cross a Current that sets S S E about 3 miles an hour, and first the Ship in 4 hours' sails 8 leagues East and by South by the Compass, then in 8 hours more she sails 12 leagues East Southeast, by the Compass: Now it is demanded what Course and Distance the Ship hath made good from the first place where this reckoning began: In resolving this we shall account every ten leagues of the former Scale of leagues in the Plate to be but one.  
Draw two lines at right Angles at the Centre A, thus: First draw the line A F, then with 60 degrees of the Chords describe the pricked quadrant F I L, and therein set off one point from the East to I, and draw a line to A, this is the line of the Ships first Course, wherein prick off 8 leagues from A to B. Prick off the course of the Current, being 6 points to the Southwards of the East, from F to E, and draw the line L A, then because the Current in 4 hours sets 4 leagues forward in its own race, draw the line B C parallel to A L, by the former directions, and prick down 4 four leagues from B to C, and the line A C shows what course and distance the Ship hath made good the first Watch, or four hours.  
Then for the second Course, draw C H parallel to the line A F, and with the Radius upon C as a Centre, draw the Arch H D, wherein prick two points for the Ships second Course from the East from H to D, and draw D C, wherein prick down C D the Ships distance run in that Course, and draw D G parallel to A L, as you did B C; then because the Current sets 8 leagues in 8 hours, prick down 8 leagues from D to G, and join A G, so the extent A G being measured on the Scale of leagues, showeth that the Ships direct distance from the first place, is about twenty eight leagues and a half; then measure the Arch F M on the Rumbes, and you will find it to be a little above 3¼ points from the East, so that the Ship hath made her way good Southeast and by East, and above a quarter of a point more Southwardly, and is now at the point G, whereas if there had been no Current, she had been but at N in the same line with D G, and distant from it equal to B C.  
And what we have done here with drawing many Lines, after the old fashion of keeping a reckoning on the Plain Chart, may be done by help of the Traverse-quadrant by Intersection and points only, without drawing any other Lines at all, but the two Lines making right angles at A, in which are plotted the Ships two Courses and distances; as also the Currents, Course, and distances proper to each of the Ships Courses, and the Variation and Separation proper to each Course, hath the figures 1, 2, 3, 4, on the South and East-line truly set down, whereby may be found the 4 points B, C, D, G, after the same manner, as the reckoning was plaited between Tenariff and St. Nicholas Island before, and in stead of laying down the Current at the end of each Course and distance, it may be done at once for many courses and distances, if you bring them first into one right line from the first place, and cast up into one Sum how much aught to be allowed to each Course and distance during the time of its continuance, and from the Traverse Point of the Ships place so found, set off the allowance for the Current in a Rumbe line that runs the same way with the Current.   

How to Rectify the Account when the dead Latitude differs from the Observed Latitude.  
By the Dead Latitude, is meant the Latitude in which the Ship is by the Dead Reckoning or Estimation, and this is always given, as in the former Chart between Tenariff and St. Nicholas Island, by the points 1, 2, 3, in the South-line, if they be measured in the other side of the Chart amongst the degrees of Latitude.  
The whole Practice of the art of Navigation in keeping a due reckoning, consists chief of three Members or Branches.  
1. An experienced judgement in estimating the Ships way in her Course upon every shift of Wind, allowing for Leeward-way and Currents.  
2. In duly estimating the Course or Point of the Compass on which the Ship hath made her way good, allowing for Currents and the Variation of the Compass.  
3. In the frequent and due Observing the Latitude.  
The reckoning arising out of the two former Branches, is called the Dead Reckoning, and of these three Branches there ought to be such an harmony and consent, that any two being given, a third Conclusion may be thence raised with truth.  
As namely, from the Course and distance, to find the Latitude of the Ships place.  
Or by the Course and difference of Latitude, to find the distance; or by the difference of Latitude and distance, to find the Course: But in the midst of so many uncertainties that daily occur in the practice of Navigation, a joint Consent in these three particulars is hardly to be expected, and when an error ariseth, the sole remedy to be trusted to, is the observation of the Latitude, or the known Soundings when a Ship is near land, and how to rectify the Reckoning by the observed latitude, we shall now show.  
Those that will not yield unto truth in this particular, that a bout 24 of our common English Sea leagues are to be allowed to vary a degree of latitude under the Meridian, do put themselves into a double incapacity.  
First, in sailing directly North or South under the Meridian where there is no Current, finding their Reckoning to fall short of the observed latitude, they take it to be an error in their judgement in concluding the Ships way by Estimation or guests to be too little.  
And secondly, if there be a Current that helps set them forward, that there is a near agreement between the observed and the dead latitude, they conclude there is no such Current.  
Or lastly, if they stem the Current, they conclude it to be much swifter than in truth it is, and thus one error commonly begets another, but supposing a conformity to the truth, we shall prescribe four Precepts for correcting a simple or single Course.  
Prec. I. First therefore, if a ship sail under the Meridian, if the difference of latitude be less by estimation than it is by observation, the Ships place or Variation only is to be corrected and enlarged under the Meridian, and the error i● to be imputed either to the judgement in guessing at the distance run, in making it too little, or if the said distance be guessed at by a sound and experienced judgement, you may suppose you stem some Current.  
So if a Ship sail from A in the latitude of 28d directly South 48, such leagues whereof 20 are a degree, the difference of latitude is 2 degrees 24 minutes, each league being 3 minutes, by this estimate the Ship should be at B, but if the observed latitude be but 25 degrees, the reckoning being amended, the Ships place is at the point D, and in this Scheme we have made every degree of latitude to be twice as large as it was in the former Chart, and so the degrees of latitude in that Chart will become a Scale of leagues to this.  
But if the difference of latitude be more by Estimation, than it is by observation, either the judgement errs in supposing the distance run to be too much: in this case the distance is to be shortened, and the Ships place corrected, according to the observed latitude under the Meridian; so if a ship sail South from A in the Latitude of 28d, till she hath varied her latitude 2d 24′ by Estimation, being at the Point B in the latitude of 25d 36′, if the observed latitude be 26d, the Ships place being corrected by this Observation, will be in the Point E, and not at B.  
Or if you can presume upon the truth of your judgement, and find it so to happen, commonly that the distance run is more than you can make it by observation, you may well suppose you cross some Current which sets either Eastward or Westward, but cannot tell which until you come near some Land, to see which way the ripples of Water set, or are informed by the experience of others.  
Precept II. But supposing no Current, if the ship sail upon one of the five Rumbes next the Meridian, if the Dead Latitude differ from the Observed Latitude, the error is in misjudging the distance run, which is either to be enlarged or shortened, as the case requires.  
In the former Scheme, suppose a ship sail from A South-west and by South 48 leagues, being by estimation at C in the latitude of 26d, but if the latitude by observation be but 25d 36′, suppose at B, than a line drawn through B, parallel to A W, crosseth the line of the ships course at M, which is the corrected Point where the ship is, and hereby the Distance is enlarged, the extent A M being 57, 7 leagues, that is 57 leagues, and a little above two mile more.  
In like manner if the Ship had sailed about 72 leagues on that course, and were by estimation at the Point G in the latitude of 25d, and by the observation the latitude were found to be 25d 36′, in this case the ships distance is to be shortened, by drawing the aforesaid line B M parallel to A W, which crosseth the line of the ships course at M, which is now the corrected Point of the ships place.  
Either of these Instances may be performed on the Chart, by finding out the separation or westwardly distance proper to the corrected latitude without drawing any lines, and that by help of the Traverse-Quadrant, page 13. in which C B there, is made equal to A B the difference of latitude here, then in the said Quadrant either raise a Perpendicular from that point, cutting the third Rumbe, or enter the said extent on the third Rumbe, so that one foot resting therein, as at D, the other turned about will but just touch C W, than the nearest distance from D to C S, is the separation required, which in this Chart is to be placed from A to L, being ●he separation corrected, whereas the dead separation need not be known or found at all.  
But in these cases, if the judgement suppose there is some Current, and can depend upon the observed difference of Latitude and dead Distance, as both true, than the Departure from the Meridian may be found, as in the last caution of the former Precept, whereby the error will be imputed to the Rumbe, which altars by reason of the supposed Current.  
Precept III. But in Rumbes near the East or West, if the dead latitude differ from the observed latitude, the error is to be imputed either wholly to the Rumbe, or partly to the Rumbe, partly to the Distance.  
If wholly to the Rumbe, then retain the observed difference of latitude, and the estimated distance, and find the Separation or Departure from the Meridian, as was done in the last Caution of the first Case or Precept.  
But if the judgement would allot the error partly to the Rumbe, partly to the Distance, retain the observed difference of latitude; and for the Departure from the Meridian, let it be the same as it was made by the Dead Reckoning.  
Example:  
So if a ship sail West and by South half a point Southerly 68, 9 leagues, that is 68 leagues and nine tenths, from the latitude of 28d from A to H, and by the Dead Reckoning should be in the latitude of 27d; if the latitude by observation be 27d 20′, which will happen at the point I: In this case, if the error were wholly imputed to the distance, the line I K being drawn parallel to A W, would cut off and shorten the distance as much as the measure of K H, which is 23 leagues, which because it seems absurd and improbable, is not to be admitted of; wherefore imputing the error to the Rumbe only, place one foot of the extent A H in I, and with the other cross the line A W at T, and so is A T the Departure from the Meridian required, whereby the Rumbe line, if it were drawn, would be altered to pass through the cross at a.  
But according to the last Caution of this Precept, if you judge the error to be partly in the Rumbe, and partly in the Distance, prick the Departure by the Dead Reckoning F H, from A to V, and so by help of the points of Variation I, and Separation V, you may draw a new Rumbe line, and find the quantity thereof; as also of the shortened distance by the direction for platting a Traverse.  
Precept IU. If a Ship sail directly East or West, and the dead and observed Latitude do both agree, the Reckoning cannot be corrected.  
But if they differ, the error is either partly in the Rumbe, and partly in the Distance, in which case retaining the Separation or Westwardly Distance the same, the difference of Latitude is the Variation required, whereby a new Rumbe-line might be drawn if it were needful.  
But if by frequent observation you find the Ship is still carried from the East or West, either Northwards or Southwards, you may conclude some Current to be the cause thereof, in which case retaining the distance to be the same it was by the Dead Reckoning, and the difference of Latitude to be the same you found it by observation, find the Departure from the Meridian by former directions, whereby the Ships Rumbe or Course might be drawn if it were needful: It is not necessary to press Examples, if what before is written be well understood, especially in this case where all directions are slippery.  
Thus in imitation of Maetius a Hollander, though a Latin Author, we have prescribed several rules for the correction of a single Course, which Mr. Phillips in his Geometrical Seaman makes but one rule, retaining always the same Course, and correcting the distance run therein, by drawing a parallel through the observed Latitude, and so for many Courses they are first brought all into one line, and the distance corrected by the same rule.  
But concerning it, we must give a double Caution: First, that no three places can be laid down true in their Courses and Distances from each other on the Plain Chart, as shall afterwards be handled, however the error in small distances will be inconsiderable.  
And secondly admitting they could, the said general Direction is unsound, but the nearer the truth the nearer the Courses are to the Meridian, and when all the Courses do either increase or diminish the Latitude, but very erroneous when some Courses increase and others lessen the Latitude; in all which Cases it is most safe to allot to the Variation or dead Difference of Latitude, of every Course its proportional share of the whole error between the Dead and Observed Latitude, and then to correct each course by the former directions.  
First therefore in the following Chart, let us suppose a Ship to sail from A in the Latitude of 28d South South-west, almost 65 leagues to B, this Course is set off in the Arch E F, and by the Dead Reckoning she should now be in the Latitude of 25d.  
Again, from B she sails South-west and by West 72 leagues to C, which Course being three points from the West, is set off in the Arch G H, and now by the dead Reckoning she should be in the Latitude of 23 degrees, whereas by a good observation she is found to be in the Latitude of 23d 30′; wherefore to correct this Reckoning draw the line C A, which is the compound Course arising from the two former Courses, and through the parallel of observed Latitude, draw L K parallel to A W, so is the point K the corrected point of the Ships place, according to Mr. Phillip's, and agreeing with the truth, as we have fitted the Example.  
But now as to the other way of correcting a compound Course, it is to be done by this Proportion.  
First find the Variation or Difference of Latitude proper to each Course, than it holds:  
As the sum of all the Variations or Differences of Latitude:  
Is to the whole error between the Dead and Observed Latitude ∷  
So is each particular Difference of Latitude:  
To its proportional share of the whole error ∷  
Than if the differences of Latitude fall all the same way, if the the estimated difference of Latitude be too much, you must abate out of each dead Latitude its proportional error, so in this case the said error is pricked from E to N.  
But when the estimated difference of Latitude is too little, the proportional error must be added to each difference of Latitude, then prick the second dead difference of Latitude being equal to E S, from N to O, and place the said extent from A the Centre, to Y, and take the nearest distance to M A as before, and prick it from O to L, being the second error: this is needful when there are more Courses than two, but for the last Course not at all necessary, neither is it for this; then through the point N draw the line N F parallel to A W, so is F the corrected Point of the Ships place at the first Course, then draw F K parallel to B C, and where the parallel of Latitude cuts it as at K, is the corrected Point of the Ships place at the second Course, being the same we found it before the other way.  
But in stead of the second Course and distance, which was 72 Leagues South-west and by west, let us now suppose the Ship sails the same distance from the point B Northwest and by west, which Course being as much on the other side the west, make G I equal to G H, and draw the Course B I, therein pricking off the former distance to D, so is D the point of the Ships dead reckoning in the Latitude of 27d; and now supposing the observed Latitude to be 27d 30′, the error and difference of Latitude are as much now as they were before; wherefore draw D A the compound rumbe: draw Q T parallel to A W, & where it cuts the compound rumbe as at P, by M●. Phillip's his reckoning, is the corrected point of the Ships place at the end of the second Course, whereas in truth it should happen at T, and so P bears from A, in this example 76d 43′ from the Meridian, and is distant from it 43 Leagues and a half, whereas the Ships true Course from A to T, is 83d 32′ from the Meridian, and the distance almost 89 leagues, which is very considerable.  
Now for as much as the Sum of the differences of Latitude A E and E f, in this latter example, is equal to A S in the former example, also the error f Q here, is equal to S L there, therefore the proportional part of each error will be the same as before.  
Then if some Courses decrease the Latitude Southwardly, and others increase it North-wardly, if the dead Latitude be too little, as in this example, consider that to place the Ship more North-wardly, so as to allot to each difference of Latitude its proper error, that the South-wardly differences of Latitude must be decreased or lessened, and the North-wardly increased; wherefore the proportion of the error is placed from E to N, and the Point F found as before.  
In like manner, if the dead Latitude were too much to bring the Ship more South-wardly, the Southern differences of Latitude must be increased, and the Northern decreased, now the point T is found by drawing a line from F the corrected point of the first Course, parallel to B D, and so the line F V being equal to B D, is the Ships second Course and distance from the corrected point F, then in regard part of the error in the Latitude is supposed to be committed, as well in the latter as in the former Course, which error being too little, the distance F V is to be enlarged, and where the parallel of observed Latitude cuts it, as at T, is the corrected point of the Ships place at the end of the second Course.  
And though what we have here performed be done by the drawing of many Lines, yet by help of the Traverse-quadrant it may may be inserted into the Chart, without drawing any Lines therein at all, for in each the Course and corrected Difference of Latitude is given; and that two things are sufficient to dispatch the work, we have showed before.  
Those that are prompt in Plain Triangles, may exercise their knowledge in calculating the things here required  
1. In the Triangle A B E, there is given the Angle at A, and the Side A B, whence find the Side B E.  
2. In the Triangle B D X, there is the like given, to wit, the Angle at B, and the Side B D, whence find the Side X D.  
3. In the Triangle A f D, the Side f D is equal to E B more X D, and the Side A f is given, whence find the Angle f A D the Rumbe required.  
4. In the Triangle Q A P, the Angle will be the same, and the Side Q A is given, whence may be found the Side A P the distance required.  
Again:  
In the Triangle A N F, find N F, afterwards in the Triangle F R T there is given F R, and the Angle R F T, whence find the Side R T; then in the Triangle A Q T, we have the Sides A Q and Q T given, whereby may be found the Angle Q A T the Rumbe desired, and the Side A T the distance sought, the Side Q T being equal to N F more R T.   

How to correct the Dead Reckoning, when the Account is kept upon Mercators Chart.  
How to keep a reckoning on this Chart, shall afterwards be showed, and there in stead of the 

Easterly  
Westerly   distance, Separation or Departure from the Meridian, we must find the difference of Longitude, which is not the same with the Departure from the Meridian, but differs as much from it, as any parallel of Latitude doth from the Equinoctial; so in 60d of Latitude 30 true Miles or minutes Departure from the Meridian, altars 60 minutes, or a whole degree difference in Longitude.  
In finding out the difference of Longitude, we shall always have the corrcted difference of Latitude, and the Rumbe, either absolutely or consequently given, and how the difference of Latitude and Rumbe being given, to find the difference of Longitude will there be showed, the difference of Longitude by the dead reckoning in most cases need not be known, or else the rumbe will be consequently given; as namely, when the distance and corrected difference of Latitude is given as in the latter part of the first and third Precepts, or when the Departure and difference of Latitude are given, as in the third & fourth Precepts, for these are sufficient to draw it, as we have before handled; but when the Course or Rumbe varies, as in these two Cases, the difference of Latitude and Departure from the Meridian are given, and the difference of Longitude may be found without drawing any new Rumbe line in the Traverse Quadrant, by this Proportion:  
As the Variation or difference of Latitude:  
Is to the Separation or departure from the Meridian ∷  
So are the Meridional parts between both Latitudes:  
To the difference of Longitude required ∷  
And the same Proportion will serve to find the difference of Longitude for many short Traverses near the same Rumbe, making the two first terms the whole Variation and Separation caused by those short Traverses.  
Example:   

To find the Difference of Longitude, by drawing the Rumbe-Line.  
To draw the Rumbe-line, setting one foot in S, with the extent A E, describe an Ark at C: Again, setting one foot in E, with the extent A S, describe another Ark at C, crossing the former, a line drawn through that cross to the Centre A, is the Rumbe required; now placing one foot of the extent A M so in the Rumbe-line, that the other turned about may just touch A L, the nearest distance from the resting point at a, being taken to the line A M, is the difference of Longitude required, being equal to the pricked line M a.  
Having before shown the Use of the Plain Chart, it now remains to show how far it is to be trusted to, in regard the Meridian's therein are all parallel, whereas in the Globe, which it is conceived to represent, they all grow narrower and narrower, till they meet in the Pole point.  
I shall therefore propose an example of the true Course and distance between Horn Sound in the Island of Spitsberge, Latitude 76d 40′, Longitude 35d 30′, and Fog-bay in Newfoundland, Lat. 50d, Longitude 329d, & so the difference of longitude is 66d 30′, the true Rumbe between these two places, according to Mercators' Chart, which we shall demonstrate, is 45d 38′ from the Meridian, and the true distance in the Rumbe 762 Leagues and a half.  
1. First therefore, I say that no two places can be laid down true upon the Plain Chart, in respect of Longitude, Latitude, Course, and distance, unless they be under the same Meridian, or under the Equinoctial.  
Horn Sound and Fogg bay being laid down true as to the difference of Latitude and Longitude, the Course between them by the Plain Chart is 68d 11′ from the Meridian, and the distance 1424 Leagues.  
2. If they be laid down true (as they may be) in their Course, distance, and difference of Latitude, than the difference of Longitude will be false, consequently the difference of Longitude between those two places thus laid down, should be but 27d 17′, whereas in truth it should be much more, to wit, 66d 30′.  
3. No two places can be laid down true in these three respects, as to their Course, Distance, and Difference of Longitude, but if you would lay them down true in two of these respects, whereof there are three Cases: First, in their Course and Distance, than the Difference of Latitude will be true, and the Difference of Longitude false, as in the second Case. Secondly, in their Course and Difference of Longitude, then will the Distance and Difference of Latitude be much more than it should; the Distance thus found will be 1860 leagues, and the difference of Latitude 65d 3′, and should be but 26d 40′. Thirdly, to lay them down true in their Distance and Difference of Longitude, in many Cases is impossible, and in this Example; but yet where the Difference of Latitude is more than the Difference of Longitude, this may be done, but then the Difference of Latitude will always be too little, and the Rumbe too wide from the Meridian.  
4. Three places cannot be laid down true, as to their Latitudes, Courses and Distances from each other, though we wave the Longitude, as shall afterwards be showed; from which premises we may raise these consequences.  
1. That in regard three places cannot be laid down true in their Courses, Distances and Latitudes, although we wave and admit the Difference of Longitude to be false, that it is no safe way to make Charts, as generally they are, to be true as to the Latitudes of places, and as near the truth as they can as to the Courses and Distances, for out of such a false Chart a true one cannot be made, nor the true rumbe and distance between most places found.  
2. That it were best to make these Charts true, as to the Longitudes and Latitudes of all places, in regard the true rumbe and distance may be thence easily found, as we shall show; and another true Chart or Globe graduated from them, and that in stead of putting in such abundance of Compasses and Rumbe-lines, both in the Plain and Mercators' Chart, it were better to leave them out, and to put into some spare place the Traverse-quadrant with points and half points, and a Limb divided into degrees, with a Line of Sines, or else the Sins of the points, halfs, and quarters only.  
3. That being so made near the Equinoctial, they are very near the truth, and may there very well serve, as also under the Meridian, and for short Distances or Voyages: and how to take away the Error of the Chart, and make them serve for long and remote Voyages shall be handled; in order whereto the first Proposition in the use of the Plain Chart must be repeated.  

Propos. I. The Longitudes and Latitudes of any two places being given, to find the true Course or Rumbe between those places, and the Distance in the Rumbe.  
This, of all other Propositions in Navigation, is the most useful, and withal the most difficult, and hath, as learned Snellius well observes, been often attempted by the Learned, but in vain; and to give our own Nation its due repute, was never generally and satisfactorily performed by any man, till our late famous Countryman Mr. Edward Wright invented that excellent Chart, called Mercators' Chart, but aught more properly to be called wright's Chartley, the Meridian-line whereof requires a Table to be made by the perpetual Addition of Secants, without which Table as yet there are no Proportions known that will serve to calculate the Rumbe generally between any two places, by help of the Natural Tables of Sines, Tangents, and Secants only, and whatsoever may be done by those Tables, may be also done geometrically by Schemes: True it is, this Proposition may be performed by the differences of the Logarithmical Tangents, having a Table of them, as in Mr. Norwoods' Epitome, without the help of a Table of the Meridian-line, but as yet we have no geometrical way known for making the Logarithmical-lines of Tangents, nor Sins and numbers.  
Before I proceed any further, it may be objected, That we have Proportions in our English Books, delivered for calculating the Rumbe between two places, which Proportions may be performed by the Natural Tables of Sines and Tangents only; as namely, in Mr. Gunter's Works, both in the former and latter Editions: in the third Edition in page 90.  
As the difference of Latitude:  
Is to the Cousin of the middle Latitude ∷  
So is the difference of Longitude:  
To the Tangent of the Rumbe from the Meridian:  
A precious Proportion if it were true: the true Proportion is,  
As the Meridional parts between both Latitudes:  
Is to the Radius ∷  
So is the difference of Longitude:  
To the Tangent of the Rumbe ∷  
The Meridional parts are to be taken out of a Table of the Meridian-line, by substracting the Meridional parts of the lesser Latitude, from the Meridional parts of the greater Latitude; by comparing these two Proportions together, because the third and fourth term are alike in each; it would follow that we might calculate the Meridional parts required, without the perpetual addition of Secants by this Proportion, raised out of the two former terms of each Proportion:  
As the Cousin of the middle Latitude:  
Is to the difference of Latitude, if in one Hemisphere, or the sum of both Latitudes if in different Hemispheres ∷  
So is the Radius: To the Meridional parts ∷  
And so by this Proportion the Meridional parts answering to the Latitude of 50d should be 55d, 157 but are in truth 57d, 909. 70d 85d, 459 by the Table 99d, 431.    Difference 41, 522.    Sum 157,340. which sufficiently refutes the truth of Mr. Gunter's Proportion, in calculating a Rumbe from the Equinoctial.  
But now, as for finding the Meridional parts between the two former Latitudes, in one or both Hemispheres, the middle Latitude in one Hemisphere will be 60d, and the Meridional parts 40d, 00, and in both Hemispheres if found at once by the middle Latitude, which is 10d, is 121d, 85; or if found severally at twice, 140d, 616, which varying from the truth, as above expressed in the Sum and Difference, we may conclude, that the Proportion is very unsound and intolerable for any great difference of Latitude, but Mr. Gunter's Works deliver no Caution about it: Before we can find the distance, the Rumbe must be calculated, and if that be false, a small error therein may cause a considerable error in the distance.  
Where the difference of Latitude is not above five degrees, it may serve very well near the truth from the Equinoctial to 60d of Latitude, and afterwards to 80d; it will not serve for three degrees difference of Latitude, and in all Cases the Cousin of the middle Latitude is a term too great, the middle Latitude being too small, and I think no certain Rule can be given to correct it.  
From what hath been said, the Reader may take due Caution how far to depend upon such Proportions, whereof one term is the middle Latitude: such are,  
As the Cousin of the middle Latitude: To the Radius ∷  
Or, As the Radius: Is to the Secant of the middle Latitude ∷  
So is the Departure from the Meridian:  
To the Difference of Longitude:  
And to this Proportion Mr. Phillip's his late Table of Secants are fitted, in the use whereof the middle Latitude must always be taken to be a whole degree, unless you will by proportion find the difference required.  
And I taking it to be what it truly happened, did cast up the Courses and Distances Mr. Norwood expresses in his Trigonometry, home from the Bermudas to the Lizard, by his Tables in the said Book, and found I had gotten almost half a degree of Longitude in the whole too much, by reason the Proportion is not sound, of which Mr. Norwood makes no use: those Courses and Distances are truly expressed in Mr. Phillip's his Geometrical Seaman, pag. 31. whereas in the last Impression of Mr. Norwoods' Book they are misprinted.  
Another Proportion of this kind, is:  
As the Cousin of the middle Latitude:  
Is to the Sine of the Rumbe from the Meridian ∷  
So is the Distance sailed:  
To the Difference of Longitude ∷  
So also there may be two Proportions for finding the enlarged Distance:  
As to the Radius:  
To the Secant of the middle Latitude ∷  
So is the distance run:  
To the enlarged distance on Mercators' Chart ∷  
Otherwise,  
As the Sine of the Rumbe from the Meridian:  
Is to the Secant of the middle Latitude ∷  
So is the Departure from the Meridian:  
To the enlarged Distance ∷  
Note when these Proportions are used, the measure of the enlarged distance must be taken from the degrees of Longitude in Mercators Chart.  
Another Proportion for finding the Rumbe, may be gathered from Mr. Hansons' Additions to Pitiscus his Trigonometry, and is:  
As the difference of Latitude:  
Is to the half Sum of the Cosines of both Latitudes ∷  
So is the difference of Longitude:  
To the Tangent of the Rumbe ∷  
This he doth not insist upon as absolutely true, but yet leaves it uncertain when to use it, and when not, the half Sum of the Cosines is a term somewhat near in many Cases to the Cousin of the middle Latitude, and by the like reason this Proportion may be refuted, as that was; the half sum of the Cosines is a term that may very well serve near the truth for 10 degrees difference of Latitude, or more between the Latitudes of 20d and 60d, near the Poles it is too great, in which case the geometrical mean is truer, near the Equinoctial, and generally in both Hemispheres it is too small; and this I thought fit to add, that those that use geometrical Schemes derived from these Proportions, may not be misled, and after a troublesome manner the Course and distance in the Seaman's Calendar is laid down between two places, in effect from this latter Proportion: and for the performance of what is there done, the Scale of Longitudes was formerly added to the Plain Scale, that thereby we might find how many miles in every Latitude would answer to one degree of Longitude, and thence by Proportion obtain the distance between two places that differ only in Longitude, which way is tedious. And for finding how many miles in any Latitude vary one degree of Longitude (if it were needful) is easily performed by the Plain Scale, as I have now contrived it; take the Compliment of the Latitude from the greater Scale of Sines, and measure it in the greater Scale of leagues or equal parts.  
Example.  
So if the Latitude were 52d, the Compliment thereof is 38d, and the Sine thereof measured in the greater Scale makes 37, and so many miles or minutes in that Latitude alter one degree of Longitude, which on the other Scale of equal parts, shows 61 Centesmes of a degree.  
And before I apply myself to small Distances, I think it not amiss to express how these Proportions find the Rumbe between Horn Sound and Fog Bay.  
By the Cousin of the middle Latitude, the Rumbe is 48d 14′ from the Meridian, and the Distance 800 leagues. In one Hemisphere the middle Latitude is the half sum of both Latitudes.  
By the half sum of the Cosines of both Latitudes, the Rumbe is 47d 28′ from the Meridian, and the Distance about 789 leagues.  
But supposing these places to be in different Hemispheres, the middle Latitude will be the half difference of both Latitudes: And by the Cousin of the middle Latitude, the Rumbe is 27 degrees 2 minutes. and the Distance is 2842 leagues.  
By the half sum of the Cosines of both Latitudes, the Rumbe is 12 degrees 54 minutes, and the Distance 2598 leagues, whereas in this latter Case the truth by Mercators' Chart is 20 degrees 11 minutes from the Meridian for the Rumbe, and 2699 leagues for the Distance.  
And the Distance between those places in the Arch of a great Circle, is near 729 leagues, and supposing them in different Hemispheres, it is 2667 leagues.    

Now to the Proposition for finding the Rumbe geometrically, as far as the former Proportions hold true.  
Each of the former Proportions may be made two, as in the first Part, by bringing in the Radius, whereof we need only work one, and that will be in the first Proportion:  
As the Radius: Is to the Cousin of the middle Latitude ∷  
So is the difference of Longitude:  
To the whole Departure from the Meridian, in the Course between the two places proposed ∷  
And in the second Proportion:  
As the Radius: Is to the half sum of the Cosines of both Latitudes ∷  
Or rather for Geometrical Schemes.  
As the Diameter: Is to the sum of the Cosines of both Latitudes ∷  
So is the difference of Longitude: To the Departure from the Meridian, in the Course between the two places ∷  
The latter Proportion of this Division, of which we make no use, is:  
As the difference of Latitude:  
Is to the aforesaid Departure from the Meridian ∷  
So is the Radius: To the Tangent of the Rumbe ∷  
An Example of the former Proportion.  
Let the Rumbe be required between Cape Finisterrae, Latitude 43 degrees, Longitude 7 degrees 20 minutes, and St. Nicholas Isle, Latitude 38 degrees, Longitude 352 degrees, the middle Latitude is 40d 30′, the compliment is 49 degrees 30 minutes, and the difference of Longitude is 15d 20′, or 33 Centesms.  
Out of the lesser equal parts, prick down 15d, 33 Centesmes from C to L, and describe the Arch B D with 60d of the Chords, and make it equal to 49d 30′, and draw C D continued further to A, from L take the nearest distance to A C, which is equal to L M, and make it one Leg of a right angled Triangle: Make the other Leg the difference of Latitude 5d, which prick from the equal parts from L to F, than the extent M F measured on the said parts, showeth the distance to be 13d 39 Centesmes, which allowing 20 Leagues to a degree, is almost 268 Leagues: with the Radius C B setting one foot at M, cross the Rumbe Triangle at G and H, which extent measured on the greater Chord is almost 22d, the Compliment whereof is 68d, and so much is the Rumbe from the Meridian between these two places, which is 6 points and about 30 minutes more, wherefore St. Michael's Isle bears from Cape Finister west south west, half a degree more westwardly.  
If two places had been both in the Latitude of 40d 30′, having the same difference of Longitude, to wit, 15d 20′, than had the extent L M been their distance, to wit, 11d 68 Centesmes, at 20 Leagues to a degree, is 233 Leagues and a half, and thus we supply the want of the Scale of Longitudes in finding the distance of Places that bear East and West, as those that are in the same Latitude must need do.   

An Example of the latter Proportion.  
Let it be required to find the true Rumbe and distance between the Lizard and the Bermudas, Mr. Norwood in his Seaman's Practice page 110, maketh the Latitude of the Lizard to be 50d, and of the Bermudas 32d 25′, or 32d, 41 Centesmes, and the difference of Longitude between these places to be 55d.  
Draw the lines A C and C D at right angles, now for want of room I use the lesser Chord, and with 60d thereof I describe the Quadrant H I, and prick the Radius from I to D, so is C D the Diameter, then count both Latitudes from H to F and G, the nearest distance from F to C I, is the Cousin of Bermudas Latitude, which prick from C to E: Again, the nearest distance from G to C I, is the Cousin of the Lizards Latitude, which place from F to S, so is C S the Sum of both Cosines; draw D S and prick down 55d the difference of Longitude from C to V, out of the greater equal parts, and draw V B parallel to D S, so is C B the Departure from the Meridian in the Course between both places, then making that one Leg of a right angled Triangle, prick down 17d, 59 Centesmes, the difference of Latitude between those places out of the same equal parts from C to L, and draw B L which represents the Course and distance truly between the Lizard & Bermudas, and the extent L B measured on the same equal parts, shows the distance to be 44d 31 Centesmes, which allowing twenty Leagues to a degree, is 886 Leagues.  
Then to find the Course with 60d of the Chords, setting one foot in L, with the other make a mark at Y and Z, then the extent Z Y measured on the Chords, showeth the Rumbe to be 66d 37′ from the Meridian, which is almost 6 points, and in this example the Proportion doth not err any thing from the truth, according to Mercators' Chart, whereas if you use the former Proportion by the middle Latitude, the Rumbe would have been 67d 2′ from the Meridian, and the distance 902 leagues, if you make C A equal to C V, than a line joining L A should be the course and distance according to the same Longitudes and Latitudes laid down on the Plain Chart, and thereby the Course should be 72d 17′ from the Meridian, and the distance 1155 leagues, however when two places are laid down true at first in their Rumbe, distance and Latitudes on the Plain Chart if you sail home, in, or near the same Rumbe, the Plain Chart will very well serve to keep the reckoning upon, and to sail by in the greatest Voyage.   

How Geometrically to supply the Meridian-line of Mercators' Chart generally.  
That the finding of the true Rumbe between two places, might not be a Proposition out of the reach of Geometry, or not to be performed by Scale and Compasses, the supply thereof became the Contemplation of the late learned Mr. Samuel Forster Professor of Astronomy in Gresham College, who in his Treatise of a ruler, Entitled, Posthuma Fosteri, makes a Meridian-line out of a Scale of Secants: this we shall be brief in, and show how near it comes to the truth. Let it be required to make a Meridian-line of such a scantling, that one degree of Longitude may be half an inch, which is of the same size with the Print thereof at the end of the Book.  
And so if it were required to make the Meridian-line from 43d to 50d of Latitude, the former extent C F out of the quadrant shall reach in this line F L from 43d to 44, in like manner the Secant of 44d 30′ so taken out, shall reach from 44d to 45.  
And thus may the whole degrees be taken out, now for dividing them into Centesmes, they may be equal divisions, and yet very near the truth; or rather first divide the half degrees true by the middle Secant, thus: the half of the Secant of 49d 15′, shall reach from 49d to 49d and a half, and instead of halfing the Secant, it may be taken out to half the Radius, and afterwards the parts of each half degree may be divided equally, and so if you would divide a degree into ten parts, each part will be 6 minutes; and if it were required to find the true length of the Meridian-line from 49 degrees to 49 degrees 6 minutes, the tenth part of the middle Secant, to wit, of 49 degrees 3 minutes, shall be the length required, and so on successively. And so if it were required to find the length of 49d 7 tenths, or 42′, I say that 7/10 of the middle Secant, to wit, of the Secant of 49 35 Centesms, or 49d 21′, is the length required.  
Now it may be doubted that the making of the Meridian-line by whole degrees, is not near enough the truth, in regard the Tables at first were made by the adding of the Secants of every minute successively together; and the learned Mr. Oughtred in stead of adding the Secants of every minute, would have a minute divided into a hundred, a thousand, ten thousand, or rather a million of parts, and the Secants of every one of those parts added together.  
To this I answer, That the making of the Meridian-line by whole degrees, and in the whole, doth not breed any error at all to be regarded, compared with the making of it up successively by every minute, and for each particular degree it doth not breed any sensible error, compared with the best Tables.  
To the end of this Book we have added a Table of the Meridian-line to every second Centesm, wherein we have supplied the vacuity that is in Mr. Gunter's Table, by which Table it doth appear that the Meridional parts between the Latitudes of 50 degrees and 60 degrees, are 17d ⌊ 542, which other Tables make more; and by the adding up of the Secants of 51 degrees 30 minutes, 52 degrees 30 minutes, and so successively to 59 degrees 30 minutes they will amount to 17d, 547, the difference being only in the thousandth parts of a degree; and I suppose there are no Mercators Charts made, wherein a degree of Longitude is an inch, the biggest, I have seen is but half an inch, and if they were an inch it could scarcely be divided into one hundred parts, much less into a thousand, and so in every part of the Meridian-line as far as it can be used, the difference will be inconsiderable, and so also in the whole.  
 Deg. Parts The Meridional parts for 70d of Latitude, by our Tables and Mr. Gunters, are 99, 431. By adding up the Secants of 30′, then of 1d 30′, 2d 30′, successively to 69d 30′, they are 99, 426. By Mr. wright's Table reduced to degrees 99, 444. Calculated by the Logarithmical Tangents, are 99, 436. By the Foreign Tables of Maetius and Snellius, which are not extended to 80d 99, 416.  
And for the latter part of the Objection, I hear Mr. Oughtred was making a New Table of them, according to his own mind, wherein it is probable he attained the manner of adding up those Secants by some new Proposition, in regard it would be extremely tedious to make a Table of Natural Secants to all those parts and then add them up, wherein he long since desisted upon this consideration, that the decrease would happen in the decimal parts remote from the degrees; for it must be conceived that a Table of the Meridian-line consists of the sum of all the Secants divided by Radius; and thus the Meridional number for the first thousand Centesms, or ten degrees, added up from a Table whose Radius is 1000, amounts to 1005079, and divided by the Radius is 1005, 079, then because we would have a Table to express the Meridian-line in degrees, allowing 100 centesms to a degree, we must divide the former Number by 100, and the Quotient is 10, 05079, which is the very Number in our Tables, saving that our Table is not continued so far by two places; and for the last two figures which are omitted, we added in an Unit in the third place of Decimals, and so when a minute is divided into 100 parts, there will be ten thousand of them in a degree, and the sum of those Secants divided by Radius, must be again divided by ten thousand, so that the decrease, as I said before, will be in the remote, and in a manner useless Decimal parts: for it is to be noted, that a Table added up from whole minutes of Secants, as was Mr. wright's, makes the Meridional parts somewhat too great, as himself grants.  
Now in regard the former way of making the Meridian-line, though true enough, may seem to be tedious, as not to be performed to 80 degrees, without 80 several additions of those portions or pieces by the Compasses, I thought fit to supply 8 points on the Scale of equal parts in the Frontispiece, which have only pricks or full points set to them, being indeed the Meridian-line for every 10d from 0 to 80d; and thus the Reader may supply them, making 10d of those lesser parts to be Radius, the distances of these Points from 10d to 60, are the Secants of the middle Latitudes between them increased by 15′.  
From 10 to 60, add to the middle Latitude 15 minutes.  
For 70d 25′, or rather 21′. For 80d 35′ For the first 10d 45′ which the Reader may easily lay up in memory, then for the intermediate or middle degrees between every ten degrees, they may be supplied by a single proportion. We shall give an Example of both.  
To supply these 8 Points, draw the Quadrant C A B, and prick off 10 out of the lesser equal parts from C to R, and raise the Perpendicular R D, then prick off from A towards B in the Limb, 5d 45′ 
which Arks are numbered with the Figures 1, 2, 3, etc. and laying a Ruler over every one of those Divisions and the Centre, mark the line C R with 1, 2, 3, and so to 8, and you are prepared to prick down these Points: Here note, that the Secant of a great Arch may be more certainly found in lines, by working this Proportion:  
As the Cousin of that Ark: Is to the Radius ∷  
So is the Radius: To the Secant ∷  15 15 25 15 35 15 45 15 55 15 65 21 75 35 withal in lines multiplying or increasing the first and third terms as often as is convenient.  
Take the distances in the line R D, from 1 to the Centre C, and prick it in the single A B from A to 1, also take the distance 2 C, and prick it in the single line from 1 to 2, and so for all the rest; and thus may those Points be put upon any Plain Scale on which they are wanting: and thus, or from the Tables they were put upon the Plain Scale in the Frontispiece of this Book, and though they be but small, yet they may thereby be made to a great scantling.   

An Example of their Use.  
Let it be required to find the Rumbe between the Lizard and the Berbadoes, in North Latitude 13d 20′, Longitude 315d 40′.  
Lizard. Latitude 50d, Longitude 11d, 00.  
The difference of Latitude is 36d 40′, or 36d, 66 Centesms.  
The difference of Longitude is 55d 20′, or 55d, 33 Centesms.  
As the Cousin of the middle Latitude: Is to the Radius ∷  
So is the difference of Latitude: To the Meridional parts ∷  
The middle Latitude is 11d 40′, and the Compliment thereof 78d 20′.  
With 60 degrees of the Chords draw the Arch C D, making it to be 78 degrees 20 minutes, and draw lines from C and D into the Centre at A, then taking 3 degrees 33 Centesms out of the former equal parts, enter it so that one foot resting on A C, the other turned about may but just touch A D, the foot of the Compasses will rest at B, and make A I in the line before equal to A B, here so is L I the true Meridional parts between the Latitude of the Berbadoes and the Lizard: upon the Point I raise a Perpendicular, and therein prick down the difference of Longitude 55 Degrees 33 Centesms out of the former equal parts from I to B, and draw the line L B, which shall truly represent the Rumbe required, which measured as we did, in the Bermudas Example, is 51 degrees 13 minutes from the Meridian.   

To measure the Distance.  
Prick down 36 Degrees 67 Centesms, the difference of Latitude out of the lesser equal parts from L to V, and draw V D parallel to I B, so is D L measured on the same equal parts, is 58d 54′ at 20 leagues to a degree, is 1171 leagues.  
When two places are on different sides of the Equinoctial, that is, in different Hemispheres, the Meridian-line A B must be conceived to be the same on the other side A, on the left hand towards the South Pole, as it was on this side towards the North Pole.  
Thus we may put the Meridian-line both into our memory and power, if there be none graduated, nor any Tables thereof at hand; and thus by the Natural Tables of Sines only (for Tangents and Secants may be made out of them) the Rumbe may be calculated very near the truth in all necessary Cases.  
And for Varieties sake the rather, in regard we have no Geometrical Way yet known for making the Logarithmical Tangents, I have endeavoured to force the Meridian-line so, as to divide it off from the equal degrees of a Quadrants Limb near the truth, behold the Scheme in which the several lines B, C, D, E, F, contain the Divisions of the Meridian-line of Mercators' Chart from 0 to 48d, divided from the equal degrees of an Arch of a Circle G H, being 25 degrees of a Quadrant, as commonly divided into 90 degrees; and the Radius of this Arch A G is five times the Radius of the lesser Chord on the Scale, equal to A I, for the bigger the better; and the degrees of Longitude to which this Meridian-line is thus divided, are the equal parts of the greater Scale, which are here also divided on the line A H.  
The Distances of these leaning Lines from the Centre at A, taken out of the same Scale of equal parts, with the Angle they make with the Sine A G, are:  
Distance. Angle. On which the Meridian-line is divided. B 57, 90d from 0d to 10d. C 58, 5 89, 15′ from 10 to 30. D 65, 3 77, 40 from 30 to 55. E 77, 2 51, 45 from 55 to 75. F 92, 6 25, from 75 to 84 or 85d.  
The first line B is Perpendicular, and the rest all lean outward, and these Angles and Distances the Reader will find the same as we have above expressed, if he goes about to measure them.  
The manner of forcing the Meridian-line was thus: Mr. Sutton Mathematical Instrument-maker, having one well made, and divided on the edge of a square rod, fitted to half an inch Radius, that is, every degree of Longitude was half an inch, and a larger than that he never made (a Chart made thereto from the Equinoctial to 80 degrees of Latitude towards one of the Poles, will require five foot and above nine inches breadth.)  
I desired him to passed white Paper over a large Table, and to divide the Degrees of a Quadrant thereon, drawing lines into the Centre, which he accordingly did, and then moving the Meridian-line rod too and again till we found it would fit, it seemed to both our judgements, that it might very well be divided from the degrees of the said Quadrant, according to the Distances and Angles here set down, if carefully performed; and though this be not Geometrical, yet it may very well serve for Seaman's use; and those that may doubt of the truth hereof till they have tried, having one already made, may make another carefully after this manner upon Paper, and then folding the Paper backwards in the lines B, C, D, E, F, lay the folded edges to their graduated line, and find a good agreement, if performed well.  
The Meridian-line is of such general use in Navigation, that indeed without it, or a Table thereof, the true Rumbe between two places cannot be known, a good Reckoning cannot be kept, and thereby as Maetius well observes, the Navigator is in a capacity to sail unto all the coasts or corners of the earth, wherefore that the Reader might be every way supplied therewith, we have not only added in a Table thereof, made to every second Centesm, but likewise a Print from a Brass Plate of a Meridian-line, made to half an inch Radius, so that by doubling or folding that Print (which is made to lie without the Book) upon a Paper or Blank, the Reader may by his Pen easily, without the use of Compasses, graduate the Meridian-line of a Chart for any Voyage: And supposing that Seamen will not go to Sea without it, that it may not be cumbersome, they may have it put upon the edges of a square rod of a foot length to put in their pockets, upon which the other Scales of the Plain Scale may be likewise graduated; or if the Reader would graduate a Meridian-line of a lesser size on his Chart, he may prepare himself a line of equal parts on the sloap edge of a thin Ruler, and laying it by the line in his Chart, on which he would graduate the Meridional Divisions, easily and conveniently perform it by his Pen, without Compasses, by help of the Meridional Table.   

An Example for finding the Rumbe by the forced Meridian-Line.  
Let it be required to find the Rumbe between the Isle of St. Helen's, Latitude 16d South, Longitude 14d, and the Berbadoes, Latitude 13d 20′ North, Longitude 315d 40′.  
The Meridional parts may be pricked down at thrice, first set 10d out of the Meridian-line B from A in the following Scheme, both upwards and downwards, then take from the Meridian-line C to 13d 20′, and prick from 10 upwards to B in the former Triangle, also take from C in the Meridian Scheme to 16d, and prick it beneath from 10 to S downwards, then raise S H perpendicular to S B, and make S H equal to 58d 20′, or 33 Centesms taken out of the line A H of the former Scheme, and draw B H, and it shall be the Rumbe required from the Berbadoes to St. Helen's, to wit, the Angle B 63d 3′ from the Meridian, if measured by former directions, the sum of both Latitudes is 29d 33 Centesms, which prick from the same equal parts from B to L, and draw L K parallel to S H, and the extent B K measured on those equal parts, is 64d, 7 the distance required, which in leagues allowing 20 to a degree, is 1294 leagues; or without drawing the line L K, if you enter the extent B L so that one foot resting in the line B K, as at M, while the other turned about will but just touch S H, then shall H M be the distance as before.  
Thus when places are in both Hemispheres, the sum of the Meridional parts in both Latitudes, is the Meridional Leg; but when in the same Hemisphere, the difference of the Meridional parts is the Meridional Leg, and the difference of Longitude in both Cases the other Leg. The reason of this manner of measuring a Distance in Mercators' Chart, both in this and the former Example, shall afterwards be handled: And note, if you prick down the difference of Latitude in the Meridian in leagues or miles, and draw a Parallel through it, cutting the Rumbe continued, when need requires, you will then find the measure of the Distance in leagues or miles accordingly.   

Another Example, serving to explain the Use of the Table of the Meridian-line, in finding the true Rumbe by any Scale of equal Parts.  
Lizard, Latitude 50d. St. Michael's Isle, Latitude 38d 00, the difference of Latitude is 12d, and the difference of Longitude 19 degrees.  
Whether this be the true difference of Longitude and Latitude or no, between these places, is not material, as to the explaining of what I am about to say: Let us then suppose these places to be laid down true on a Plain Chart, as to their Latitudes, Course, and Distance, the difference of Latitude is 12d, and the Distance 18d, 12, that is about 362 leagues and a half.  
Out of any equal parts prick down 12d, the difference of Latitude from L to M, and raise the Perpendicular M I, and take 18d 12 Centesms, the Distance out of the same equal Parts, and setting one foot in L, cross the Perpendicular at I, and draw I L, so are these places laid down true in their Distance on a Plain Chart or Blank, as also in their Rumbe the Angle at L, which is 48d 34′ from the Meridian.  
The Ship at A hath made 3d 88 Centesems, the difference of Longitude in that Course, which taken out of 19d, the whole difference of Longitude there, rests 15d 12 Centesms.  
 Deg. Parts The Latitude of the Lizard is 50d, Meridional parts 57, 909. The Latitude of the Ship is 45d, Meridional parts 40, 499. The Difference is 7d, 410.  
Take 7d, 41 Centesmes out of the Scale L C, and prick them from L to C, and draw C D parallel to M I, and therein out of the same equal parts, prick down 15d 12 Centesmes from C to D, through the point A draw the parallel A E, and from D draw a line to L, so doth the line O L truly represent the Ships Course and Distance to the Lizard; hereby the Lizard will be found to bear from the Ship 63d 53′ from the Meridian, and is distant from it 227 leagues, whereas in the Chart from the point A, the Lizard bears 64d 57′, and is distant from it 236 leagues: Thus the Chart in so short a run as 151 leagues, about 2¼ points out of the direct Course, begets 1d 4′ error in the Rumbe, and 9 leagues error in the Distance, which in a long run, as between the Bermudas and the Lizard, may very well amount to 160 leagues error, as Mr. Norwood showeth, notwithstanding those places were laid down true at first, as to their Latitudes Rumbe, and Distance.  
What is here accomplished by the Tables, may readily be performed by a Meridian-line, out of which with Compasses take the distance between both Latitudes, and prick it from L towards M, at the end whereof raise a Perpendicular, and therein prick down out of the Equinoctial degrees or equal parts, the difference of Longitude, drawing a line from it to L, which shall pass through the former point O, whatsoever be the Radius whereto the Meridian-line is fitted; and after the same manner the error of the Plain Chart is to be removed, when places are at first laid down in it, according to their Longitudes and Latitudes, which is most easily and suddenly done, especially if a Meridian-line on a rod fitted to the degrees of Longitude on the Plain Chart, and the manner of measuring a westwardly distance will be the same as in Mercators Chart.  
Before we proceed to the Demonstration of Mercators' Chart, it will be necessary to search into the Nature of the Rumbe-line, on the Globe, and to collect what we find observable in Foreign Authors to this purpose:  
1. They define it to be such a Line that makes the same Angles with every Meridian, through which it passeth, and therefore can be no Arch of a great Circle; for suppose such a Circle to pass through the Zenith of some place not under the Equinoctial, making an obliqne Angle with the Meridian, then shall it make a greater Angle with all other Meridian's then with that through which it at first passeth, therefore the Rumbe which maketh the same Angles with every Meridian, must needs be a Line curving and bending, which some call a Helix-Line, and others, because it is supposed to be on the Surface of the Sphere, a Helispherical-line.  
2. Another Property of the Rumbe-line, is, that it leads nearer and nearer unto one of the Poles, but never falleth into it, and near the Pole it turneth often round the same, in many Spires and turn.  
If by any Rumbe (but not under the Meridian) it were supposed possible to sail directly under the Pole point, it would follow that the Meridian-line of Mercators' Chart should be finite, and not infinite, and that one and the same line should cut infinite other Lines as such, at equal Angles in a mere point, which is against the nature and definition of a line, and then seeing that the Rumbe derives its definition from the Angle, it makes with the Meridian: the nature and use of a Meridian under the Pole ceaseth, and consequently the Rumbe ceaseth, for under the Poles no star or other  point of the heavens, doth either rise or set, in reference whereto the word Meridian hath its original, being used to signify the Midday time, or high noon of the Sun, or Star, relating to their proper Motion; moreover under the Pole point the sides of the Rumbe-triangles in the Sphere cease, wherefore the Angles must needs do so too.   

Of the Nature of the Triangles made upon the Globe by the Rumbe-Line.  
And in the whole Rumbe-triangle A t m, the whole distance is A m, the whole difference of Latitude is A t, but the whole Departure from the Meridian is not given in any one Triangle, but in divers Triangles, and is by supposition as much in one Triangle as another, to wit, the whole Departure from the Meridian is the sum of 

a b  
c d  
e f  
g h  
i k  
l m   each of which Portions are supposed to be equal each to other, and the whole difference of Longitude is the Arch A Q, the Segments whereof Q L, Q s, s r, cannot be equal to each other, because the Arks or Departures l m, i k, etc. are supposed to be equal each to other, not in the same, but in different Parallels of Latitude.  
Now then for all uses in Navigation, we suppose the Segments of the Rumbe m k, k h, and the rest, to be a right Line, and if we give the Rumbe, to wit, the Angle l k m, and the Side k l, we may by the Doctrine of Plain Triangles find the Side k m, the distance by the common Proportion:  
As the Cousin of the Rumbe from the Meridian: Is to the Radius ∷  
So is the difference of Latitude: To the distance ∷  
And from hence we may observe another property in the Rumbe, as namely, that the Segments or Pieces thereof contained between two Parallels, having the like difference of Latitude, are also equal to each other, so that the like distance being sailed in the same Rumbe in several parts of the earth, shall cause the like difference or alteration of Latitude in each of those parts.  
For Example: If a Ship sail from the Latitude of 10d North-East, till she be in the Latitude of 30d, and then sail from thence on the same Rumbe till she be in the Latitude of 50d, the latter distance shall be equal to the former distance, and the like shall hold from thence to 70d, and the same also should hold from 70d to the Pole, if it were possible to sail thither by any Rumbe, and the reason is, because in the Triangle k l m the Angles at k and l, with the Side l k, are equal by construction to the Angles at h and i, and the Side i h in the Triangle i h k, and the like in any other Triangle, wherefore if these were given to find the Side m k or k h, it must needs be found the same in both.  
Again, if in the former Triangle we should give the Rumbe, to wit, the Angle l k m, and the difference of Latitude l k, we might find the Departure from the Maridian l m by this Proportion:  
As the Radius: Is to the Tangent of the Rumbe, viz. Tang. l k m:  
So is the difference of Latitude, 1 minute, to wit, k l:  
To the Departure from the Meridian l m ∷  
Now such Proportion as one Circle hath to another such Proportion, have their Degrees, Semidiameters, and Sins of like Arches one unto another.  
As in the former Scheme, A C is the Radius of the Equinoctial, and E D is the Radius of a Parallel or lesser Circle, whose Latitude from the Equinoctial is A E, but E D is the Sine of the Ark E B, which is the compliment of the Latitude A E.  
It therefore holds:  
As the Cousin of the Latitude: Is to the Radius ∷  
Or rather with other terms in the same Proportion.  
As the Radius: Is to the Secant of the Latitude ∷  
So is a Mile or any part thereof, or number of Miles:  
To the difference of Longitude answering thereto ∷  
And because the two first terms of the Proportion vary not, it will hold after the manner of the Compound rule of three.  
As the Radius: Is to the Sum of the Secants of all the parallels or Latitude between any two places: (and we allow a parallel to pass through every minute or Centesme of a degree)  
So is a Mile or any number or part thereof allotted to every Parallel:  
To the whole difference of Longitude, that is, to the Sum of all the differences of Longitude proper to each Parallel ∷  
Now as we shown before, the Departure from the Meridian is equally scattered or shared, and allotted the like portion thereof to every parallel, wherefore in stead of the third term above which is the distributed part, we may take in a term equivalent thereto that finds it Tang. Rumbe/ Radius, as we shown before, then in working the rule of three in mixed Numbers, where the second or third term stand in form of a Fraction, the other terms not being mixed, the Product of the first term and Denominator, is the Divisor or fi●st term in another Proportion, and the Numerator and other term are the second and third terms, or their product the Dividend, wherefore by this Composition it follows:  
As the Square of the Radius:  
Is to the Sum of the Secants of all the Parallels between both Latitudes ∷  
So is the Tangent of the Rumbe: To the difference of Longitude ∷  
And in stead of the two first terms, we may say:  
As the Radius: Is to the sum of all the Secants between both Latitudes divided by the Radius: (which as I have showed is the very term given by the Meridional Table)  
So is the Tangent of the Rumbe from the Meridian:  
To the difference of Longitude ∷  
The four terms of this Proportion being all of a different kind, may be so inverted in their order, that any three of them being given, the fourth may be found, and so if it were required to find the Rumbe, the difference of Longitude and Latitude being given, it will hold backward:  
As the Meridional parts between both Latitudes:  
Is to the difference of Longitude ∷  
So is the Radius: To the Tangent of the Rumhe ∷  
Wherefore if we make the Meridional parts one Leg of a right angled Triangle, and the difference of Longitude the other, the Meridional parts being assumed or made Radius, the difference of Longitude becomes the Tangent of the Rumbe, and consequently the Angle between the Meridian and Rumbe-Line, or Hipotenusal, measureth the quantity of the Rumbe from the Meridian; and this is the very case and thing done in Mercators' Chart between all places, wherefore the truth of that Chart is as much in every respect above all contradiction, as the best Calculation that is as every was published, and we may very well conclude with Mr. Norwood in his Trigonometry, page 119. that all or any parts of the world may be set down therein according to their Longitudes, Latitudes, Courses and Distances, as truly and far more conveniently for the Mariners use, then upon the Globe itself, with the which elsewhere he saith it agrees without sensible error; And in the words of its renowned Author Mr. Wright, we may say it agrees therewith without sensible or considerable error.  
Now let us see what can be objected against it: Mr. Speidel in his Brief Treatise of Spherical Triangles, page 48 saith, That by Mercators' Chart in the bearing of two places, there will appear a manifest error of whole degrees, being compared with the Globe.  
And I add, if this were true, there would also arise a great error in the distance, seeing we have nothing but his bare word for it, it is as easily here denied, as there affirmed, and it were to be wished he had left us his mind concerning it; if by comparing with the Globe he means it should be the same with the angle of Position, this we deny; for the Rumbe makes the same Angles with both the Meridian's of the places between which it is extended, whereas the Angle of Position at one of the places differs from what it is at the other place: And here note, that a Course sailed under the Meridian, or in a Parallel, is not properly called a Rumbe, for under the former the Ships motion describes a great Circle, and in a parallel a lesser Circle, whereas a Rumbe is no Circle at all, but rather resembles a Spiral-line.  
Another Objection may be raised from learned Mr. Oughtreds Treatise of Navigation, page 38. at the end of his Circles of Proportion, wherein he makes this one property of the Rumbe, that a Ship sailing therein from one place to another, cannot return back to the first place by the opposite Rumbe: behold the former Scheme wherein in the Triangle k l m, suppose a Ship to sail from k to m, then is l k m the Angle of the Rumbe first kept, being subtended by the Side l m; now if she would sail back again from m to k, the Angle of the Rumbe will be n m k, a bigger Angle than the former, because the Side k n that subtends it, is a bigger Side, and so that if the Ship should sail back by the very opposite Angle, the Ship would not arrive at k, but fall wide of it nearer unto n.  
To which I answer, That this is true in nicety of speculation, for in the right angled Triangle k l m, if I give the Side k l the difference of Latitude, and the Side l m the difference of Longitude, reduced to the measure of that Parallel, I may by these find the Distance k m, and the Angle of the Rumbe l k m. In like manner in the Triangle m n k, the Side m n is equal to l k, but the Side k n is greater than l m, and is the difference of Longitude in the given Parallel, whereby we may find m k greater than it was found before, and the Angle of the Rumbe n m k greater than the Angle l k m; And so in the whole number of Rumbe-Triangles in finding the Rumbe, get the Meridional parts between both Latitudes, by substracting the less from the greater, and thereby find the Rumbe, this finds the lesser Angle, and includes the most Northwardly, and excludes the most Southwardly Parallel.  
Again, subtract a Centesm or Minute from each Latitude, and then find the Meridional parts, whereby calculate the Rumbe: this finds the bigger Angle, and excludes the most Northwardly, and includes the most Southwardly Parallel; and thus we may find how much the inward Angle of the Rumbe differs from the outward, between both which the true Rumbe is a mean; but if the case be put in different Hemispheres, the sum of the Meridional parts in both Latitudes will find the lesser Angle, then subtract a minute from the greater, and add a minute to the lesser Latitude, and with the sum of the Meridional parts proper to both Latitudes thus altered, you may also find the greater Angle.  
The Meridional parts thus found, either way will be so inconsiderably different, that there will scarce be any sensible difference between the Rumbe found by each operation, and therefore though the Objection be true in speculation, yet cannot be sensible in practice, and what difference may thus arise, may be removed to all possible nearness, if in stead of making a Rumbe triangle to every minute, we should divide a Minute or Centesm into a hundred, a thousand, ten thousand, or a million of Triangles, and add up the Secants of them according to Mr. Oughtreds opinion; but we may say with Mr. Wright, page 114. that this were a matter more curious than necessary, a Table made to every Minute or Centesm being so near the truth, that it is not possible by any Rules or Instruments of Navigation to discover any sensible error in the Sea Chartley, so far forth as it shall be made according thereto: Secondly, were the Objection more considerable than it is, yet to me it seems a gross absurdity in practice, to affirm that a Ship cannot return by the opposite Rumbe.  
Suppose therefore that a Voyage were to be performed from Tenariff, Latitude 28d, Longitude 0, to St. Nicholas Isle, Latitude 17d, Longitude 352d, the difference of Longitude being 8d, this Example being the same we used in the Plain Chart, we here retain it with the same Courses and Distances, that the Reader comparing both together, may see how much error is committed by the Plain Chart; to fit a particular Chart hereto, I draw the line T W, and divide it into 8 equal parts or degrees, being in this Chart half an inch each, and then divide every one of those degrees into ten lesser parts, then draw the line T S perpendicular, and by folding the printed Cut of the Meridian-line thereto, I graduate the same with a Pen from 28 to 17d of Latitude; and now to shun the turning of leagues into degrees and Centesms, I divide another line of equal parts W N, making each degree therein twice as large as a degree of Longitude, and so that line becomes a line of inches, and may be made on a lose Ruler; each degree thereof is divided into ten parts, and numbered with as many degrees of Latitude of the Meridian-line, as the Chart would suffer for want of room.  
Draw a line from T to N, to represent a Ruler laid over the two places Tenariff and Nicholas Island, for when we have a Ruler that line need not be drawn.   

To find the Rumbe.  
And now let it be required to find the Rumbe or Course between Tenariff and St. Nicholas Island without drawing any lines in the Chart.  
Example:  
Prick 60d of the Chords from T to o, and also prick it down by the edge of the Ruler from T to R, and take the distance R o, which measured in the greater Scale of Chords is 33d 49′, and so much doth St. Nicholas Isle bear from Tenariff to the Westward of the South, which measured on the Scale of Rumbes, is 3 points (and 4 minutes more) and the Rumbe is South-west and by South; Also the nearest distance from R to the Ruler T X, measured on the Sins, showeth the same Arch as before, to wit, 33 degrees 49 minutes.  
Another Example.  
Let us suppose two Islands in the Sea, the one situated in this Chart at f, the other at d, and let it be required to find the Rumbe or true Course between them.  
To perform this, in regard we suppose the Chart to be made without any Compasses, Winds, or Rumbes drawn, it will be necessary to have some few Meridian's and Parallels drawn therein, here we have drawn one through 20d of Latitude, and another through 25d, lay a ruler over f and d, and it cuts the line r B at E, then place the extent T R from E to r, the nearest distance from r to f d, which resembles the edge of the ruler measured on the greater Line of Sines, is 11d 55′, and so much to the Southwards of the East, doth the Island f bear from the Island d, which is East and by South, and 40′ Southwardly.  
The ruler must cut some Meridian or Parallel in the Chart, and if it should so happen that it cuts none, than some Meridian or Parallel must be drawn that it may cut.   

To Measure the Distance between any two Places in Mercators Chart.  
Herein also we shall forsake the common Road, and make no use of the Meridian-line at all, which as it is not used to this purpose in Calculation, so neither need it in any operation upon the Chart, on which we shall observe those Proportions that are used in Calculation, which are but two.  
Case 1. When places differ both in Longitude and Latitude.  
As the Cousin of the Rumbe: Is to the difference of Latitude ∷  
So is the Radius: To the Distance ∷  
Suppose the points R & Q in the Chart were two Islands, whereof I would measure the distance, supposing these places to be truly placed in the Chart according to their Latitudes and Longitudes, it is necessary to find in what Latitudes both places are: thus, if you take the nearest distance from Q to some Parallel, to wit, K L, and place it in the Meridian-line T S, the right way from that Parallel you will find that place to be in the latitude of 22d 5 tenths, and the point R will be found to be in the latitude of 24d 42 centesms, get the difference of Latitude, which is one degree and 92 Centesms, which take out of the Scale of Inches, to wit, one inch, and 92 hundred parts of another, and lay a Ruler over the two places R Q, and it will cross the Parallels at the Points K and X, then enter the former extent so by the edge of the Ruler, that one foot resting by it, the other turned about may but just touch some one of the Parallels in the Chart; thus one foot of the Compasses will rest at Y, and the other being turned about will just touch K L, then is the extent K Y the distance in degrees, if measured in the Scale of Inches, to wit, 2d 31 Centesms; and if you measure the said extent in the line T W, it gives you the distance in leagues, to wit, 46 leagues and a quarter.  
And if you have the Scale of leagues graduated on the sloap edge of your Ruler, you may see the distance without measuring any extent; and if the Line of Inches were wanting, you might double the difference of Latitude, and take it out of the Scale T W, and find the distance the same.  
In like manner, if the former extent were so entered, that one foot turned about should just touch r X, the Compasses would rest at Z, and the extent Z X would be the same distance as before.  
We would have measured the whole Distance between Tenariff at T, and Nicholas Island at N, but that we are confined to a narrow room, and therefore will only measure one tenth part thereof: The whole difference of Latitude is 11d, whereof a tenth part is one degree and a tenth, take one inch and a tenth out of the Scale of Inches, and entering it so that one foot resting by the edge of the Ruler T N, the other turned about may just touch T W, the resting foot will happen at P, and the extent T P measured on the Scale T W, is 26 leagues and a half, wherefore the whole distance is ten times as much, namely, 265 leagues.   

Demonstration.  
The Angle that the edge of a ruler (or right Line) laid over two places, makes with any Parallel in the Chart, is the Compliment of the Rumbe between those places, so the Angle Q K L measureth the bearing of the Ports R Q from the West, and the Angle Q X r measureth the same thing from the East, and the entering of the difference of Latitude taken out of some Scale of equal parts, as above, makes it to become the Cousin of the Rumbe, wherefore the Radius thereto K Y, or X Z becomes the distance in the same measure wherein the difference of Latitude was taken, and so the Proportion before delivered is observed.  
Case 2. The second Case is when two places are both in the same Latitude, and differ only in Longitude, the Proportion holds:  
As the Radius: Is to the Cousin of the Latitude ∷  
So is the difference of Longitude: To the Distance ∷  
I said before that we might very well spare the Points of the Compass and Rumbes, wherewith most Chards are filled, and that the Limb of a Quadrant in some spare place of the Chard would be of good use, in particular for the resolving of this Proposition, yet the defect thereof we thus supply.  
Prick down 60d of the Chords from T to H, which Point may be always in a readiness, and now let it be required to measure the distance r X in the Latitude of 20d, the Compliment whereof is 70d, with the Chord of 70d draw an Arch at g, and setting one foot in T, with the Radius T H cross the former Ark at g, over which and the point T, lay the edge of a Ruler, then take the extent r X, and place it from T towards W, it reaches to G, the nearest distance from G to the edge of the Ruler, is the distance sought, if measured in the line T W, to wit, 3d 45 Centesms, where turning it twice over (making every degree to be ten leagues) I shall find it to be 69 leagues.  
Otherwise:  
Suppose G to be a place in the Latitude of 28d, and it were required to measure the distance thereof from T, with the Sine of 28d upon o as a Centre, describe the Ark q, and lay the edge of the Ruler so as it may lie over the outward edge of that Ark and T, then take the nearest distance from G to the edge of the Ruler, which being turned twice over in the line T W, you will find the distance sought to be 65 leagues.  
Now we may apprehend how to lay the Ruler without drawing the Ark q, so as the Compasses turned about may just touch the edge thereof, and then setting down one foot of the Compasses by the said edge, turn the Ruler about that the edge may be towards G, and take the nearest distance as aforesaid to it, and in Latitudes above 40d or 45d, the trouble of turning about the Ruler may be best spared in that Case, with the Sine of the Latitudes Compliment, setting one foot at H, lay the ruler over T, so that the other foot turned about may touch the thin sloap edge thereof, and then take the nearest distance thereto, as before.  
For places that differ both in Longitude and Latitude, if they bear one from another East or West within a quarter or half a point, it may be truly alleged, that the way of measuring delivered in the first Case will be very uncertain, because the Compasses will run along almost in a Parallel to the edge of the ruler, and so cut it very obliquely.  
As in Page 6 of the first Part, Prop. 5. where C I may represent a Westwardly Course and Distance, and C A the difference of Latitude, then doth the Perpendicular A I cut the line C I in the Point I very obliquely, and with uncertainty; for the finding of which with more certainty, the e●tent C A was tripled to F, and the extent D B was tripled to G, as was declared in that Scheme.  
To which I answer, The distance may be also found as here in the second Case, by the Cousin of the middle Latitude between both places.  
And thus if it were required to find the distance between the points d and f in the former Chart, by this rule it would be found to be 2d 98 Centesms, which turned twice over in the Scale T W, is 59 leagues 7 tenths, which is not the true distance, because the Rumbe between these places is above a Point from the East or West, whereas the true distance is 60 leagues 9 tenths; and in this Example the difference of Latitude is 63 Centesms, and though it were four degrees or more, we might first find the distance by the Cousin of the middle Latitude, and then enlarge it by entering it so by the edge of the ruler laid over the two places, that one foot resting by the sloap edge, the other turned about might but just touch some Meridian, in stead of a Parallel in the former way.  
Thus a ruler laid over d and f cuts the Meridian-line at u, and the former extent so entered, one foot of it will rest at m, while the other turned about will but just touch T S, than the extent m u turned twice over and measured in the line T W, will be 60 leagues 9 tenths the Distance required, the enlargement thus found being but very small as to matter of extent.  
Now we may take a view of what Rules have been formerly delivered for the Measuring of Distances, here we need not recite Mr. Gunter's way, which is not only troublesome, as not to be performed without two Pair of Compasses, but also uncertain, if the Distance be great.  
The Way in use amongst Seamen is to take half of the whole extent between any two places in the Chart, and either to set one foot of the Compasses down in the Meridian-line, at the middle Arch, which is the middle Latitude between both places, or at the middle space or just half between the Latitudes of both places, which always falls somewhat nearer the Pole then the middle Latitude, and then turn the other foot of the Compasses both upwards and downwards, and mind what Arks it crosseth in the Meridian-line, than they count the whole Arch contained between the places where the movable foot so crossed, and take that for the whole distance sought.  
Example:  
Latitude 13d, 33 Centesms North. 16, South.  
Difference of Longitude is 180d, the extent in the Chart between two places so situated, is 182d 42, the half whereof is 91d 21, which measured in the Meridian-line from the middle Latitude or Arch, which is 1d 33, being half the difference of both Latitudes, and counted towards the greater Latitude, reacheth Northward to 66d 15 one way, and Southward to 67d 32 the other way, and so by this reckoning the distance between these places should be 133d, 47 by the middle Latitude, and 133d, 98 by the middle space, whereas in truth by Calculation the distance in the Rumbe is much more, to wit, 180d, 54.  
But supposing two places to be in the same Latitude, and to have but 58d, 33 Centesms difference of Longitude, the example will be the same with one of those before put between the Berbadoes and St. Helen's, and the distance found by the middle Arch or Latitude, is 62d, 00, and the same by the middle space 61d 38 centesms, but should be in truth 64d, 7.  
Also in the former Example between the Berbadoes and the Lizard, the true distance is 58d, 54, and by the middle Latitude or Arch is 58d, 7, but by the middle space it is 57d, 66.  
Also in the Example between the Bermudas and the Lizard, the true distance was found to be 44d, 31 Centesms, by the middle Arch or Latitude it is 45d, 13 Centesms, and by the middle space it is 44d, 97 Centesms.  
Where places are in the same Latitude or Parallel, the Compasses must be set down in the Meridian-line at the Latitude given, and the half extent applied both upwards and downwards, as before; and if the distance be large, the measure thereof will be much more erroneous than when the Rumbe lies nearer the Meridian.   

Some Examples of a parallel Distance.  
Let there be two places in the Latitude of 35d.  
  True Distance. Distance by the Meridian-line. Difference of Longitude 180d 147d, 45 123d, 93. 18 14, 74 13, 98.  
Another Example of two places in the Latitude of 50d.  
  True Distance. Distance by the Meridian-line. Difference of Longitude 180d 115d, 8 111d, 87. 18 11, 58 11, 59  
A third Example of two places in the Latitude of 70d.  
  True Distance. Distance by the Meridian-line. Difference of Longitude 180d 61d, 57 76d, 41. 18 6, 15 6, 18.  
From which Examples we may observe, that a large Distance cannot be so certainly measured in the Meridian-line as a small one, whereof Mr. Wr●ght was very sensible, and therefore prescribes Rules for the measuring of a small part of the Distance at a time, and argues for the truth thereof; but where the whole extent between two places is not above ten of the degrees of Longitude, I see nothing to the contrary, but that it may well enough be measured in the Meridian-line; and so for a great distance we may measure a tenth, or a twentieth part of the whole, and by multiplying the known part, find the whole; for the ready performing whereof another Scale of equal parts, whereof the degrees are twice as large as those in the Scale of Longitude, will be of much conveniency and ease.  
Example: Suppose it were required to measure the distance between the Points f and d in the former Chart, take the same distance, and measuring it in the inches, find how much it is, to wit, 1 Inch 61 Centesm●, then take the same number out of the Scale of Longitude, and setting one foot at the middle Latitude, to wit, 19d 18 Centesms, the other will reach Northwards to 20d 69 Centesms, and Southwards to 17d 64 Centesms, the difference of which two Arks is 3d, 05 the distance sought, which allowing 20 leagues to a degree, is 61 leagues, as before, and this is more easily done, then to take the half of any extent, and by the same reason you may find the middle space between both Latitudes. So also when you would measure the tenth part of a great distance, measure the whole extent in the Scale of Longitudes, and take the twentieth part found by the pen or memory, out of the said Scale, and set it at the middle Latitude, or middle Space, turning the other foot in the Meridian-line, both upwards and downwards, and the degrees so intercepted, will be the tenth part of the whole distance. Now the taking of the twentieth or fortieth part of an Extent is easily done, by help of these two Scales of equal parts; Suppose I would find the twentieth part of 3 inches or degrees in the greater Scale, I say 3 of the small parts in the lesser Scale is the length required, and so the twentieth part of 3 inches 5 tenths is three and a half of the smaller parts in the lesser Scale, and the half of that is the fortieth part of the whole. I need not insist further upon these ways of measuring, seeing I have before delivered others, which as they are more ready in the practice, so also they are built upon better foundations.   

To keep a Reckoning on the true Chart.  
Here I shall insist upon a new Method never before published, which will render this Chart very easy and acceptable to Seamen, and having made our Example, that before laid down being the same with that in the Plain Chart, we shall here also retain the same Traverses.   

The first Operation is to find the Latitude.  
The first Course the Ship sails is South South-west, 60 leagues from Tenariff, to protract this Traverse I shall make use of another Traverse-quadrant bigger than that, which was used before, which may be made upon state: Take 60 leagues out of the Scale of Longitudes T W, and enter it in the Traverse-quadrant on the second point from C to A, the nearest distance from A to C W prick in the Scale of Inches from W to A +, and it shows me now that the Ship is in the Latitude of 25d, 23 Centesms, in the Meridian-line T S set the figure 1 to this Latitude.   

Secondly, to find the difference of Longitude.  
Take the extent T 1 in the Meridian-line, and enter it so in the Traverse-quadrant on the second Rumbe, that one foot resting thereon as at a, the other turned about may but just touch C W, then is the nearest distance from a to C S, the difference of Longitude required, to wit, 1d 24 Centesms, which prick in the Scale of Longitude from T towards W, and set the figure 1 at it.   

Thirdly, to plot the Traverse Point.  
Set one foot of the said extent at 1 in the South line of the Chart, and with the other draw a small Ark at a, then take the extent T 1 out of the South line, and setting one foot at 1 in the West line, with the other cross the former Ark at a, and there is the point where the Ship is at the end of this first Traverse.   

Demonstration.  
The Proportion for finding the difference of Latitude, we have before handled, the Proportion for finding the difference of Longitude is:  
As the Radius: Is to the Meridional parts between any two Latitudes ∷  
So is the Tangent of the Rumbe:  
To the difference of Longitude ∷  
The extent T 1 being taken out of the South line, is the parts of the Meridian-line between the Latitude of 28d, and the Latitude of the Ships place, namely, 25d 23 Centesms, which being entered as before, in the Traverse-quadrant at a, becomes the Radius to the Tangent of that Rumbe, and so the Tangent of the said Rumbe to that Radius, being the nearest distance from a to C S becomes the difference of Longitude required, the Proportion for finding it being duly observed.  
The Ships second Traverse is 80 Leagues West South West, which I take out of the West line T W on the Chart, and place it in the Traverse-quadrant from C to B upon the sixth Point, the nearest distance from B to C W, I place in the line of Inches from the Point A to the Point B, and thereby I see that the Ships Latitude now is 23d, 7, at which in the South Meridian I set the figure 2, and take the distance in the said line between it and the figure 1, and enter this extent so upon the sixth Point in the Traverse-quadrant at b, that the Compasses turned about will but just touch C W, and the nearest distance from b to C S, is the difference of Longitude required, which I prick in the West line from 1 to 2, and upon the point 2 in the South line, with the extent T 2 taken out of the West line, draw a small Ark near b, and then taking the extent T 2 out of the South line, setting one foot upon 2 in the West line, with the other I cross the former Ark at b, and there is the point of the Ships second Traverse.  
The third Traverse is South and by East, half a point Eastwardly 53 leagues, this distance I take out of the West line Scale, and enter it in the Traverse-quadrant from C to D, on a point and a half from the Meridian, the nearest distance from D to C W, I should prick in the Scale of Inches from B to C, but because I have not room, I have continued the said Scale of Inches apart by itself, in page 73. and this extent reaches from B to C, and now I see I am in latitude 21d 17 centesms, at which I set the figure 3 in the South line of the Chart T S, and having taken the distance between it and the figure 2, I enter it so in the Traverse-quadrant on the Rumbe sailed, that one foot resting thereon as at d, the other turned about will but just touch C W, than the nearest distance from d to C S, is the difference of Longitude required, which I prick in the West line T W from 2 to 3 Eastwardly, because the Course run Eastwardly; with T 3 of the West line, setting one foot at 3 in the South line, I draw an Ark at c: again, wi●h T 3 of the South line, setting one foot at 3 in the West line, I cross the former Ark at c, and there is the point of the Ships place from which it is desired to know how St. Nicholas Island bears, which by former directions is 38d 37′ to the Westward of the South, and the Distance thither will be 106 leagues and 6 tenths.  
Now it is desired to know what Course and Distan●e must be steered to bring the Ship 23 Leagues East from St. Nicholas Island; The Latitude of that Island is 17d, and the Compliment of it is 73d, lay the Ruler over N so as to make such an Angle with the Line N S, which it will do if it be laid over the point N, and the cross p found in the same manner as it was at g, the former distance turned into degrees, is 1 degree 15 Centesmes, which I take out of the West Scale T W, being the very half of the leagues there accounted, which I enter so by the edge of the ruler, that one foot resting on the Line N S, the other turned about may just touch the ruler; in this manner it will rest at the point e, which is the point of such distance sought, being but the Converse of finding a distance between two places in the same Latitude; then find the Course and distance between the point c and e, by former direction, and you answer the question propounded.  
The Course will be found to be 29d 23′ to the Westwards of the South.  
And the distance about 95 Leagues 74 Centesmes more.  
From the point c, the Ship sails directly South 33 Leagues 4 Tenths, which I take out of the west-line, and prick in the lose Scale of Inches from C to D, whereby I see the Ship is now in the Latitude of 19d and a half, or 5 tenths, at which in the Meridian line of the Chart, I set the figure 4, and because the Longitude is not altered, I writ 4 in the West-line also over the figure 3, and by help of these two Points plate the point d, as was done before.  
And now lastly, it is desired to know what Course and Distance I should steer to bring the Ship to bear from the Point S in the Chart, 45 Leagues Northwest and by North.  
To perform this, we must find a point in the Chart that shall have the bearing and distance required, the Course assigned is 3 Points from the Meridian, wherefore take 45 Leagues out of the West Scale, and enter it in the third Point of the Traverse-quadrant from C to F, and the nearest distance from F to C W measured on the Scale of Inches, shows that the difference of Latitude is 1d 87 Centesms, which I count in the South or Meridian-line from S Northward, and set the figure 5 thereto, from which take the distance to S, and enter it in the Traverse-quadrant on the thi●d point, that one foot resting thereon as at f, the other turned about may but just touch C W, and the nearest distance from f to C S, is the difference of Longitude, which prick in the line S N from S to 5, and with it upon 5 in the South line, describe an Ark at f: again, with S 5 from the South line, setting one foot at 5 in the line S N, describe another Ark at f crossing the former, and there is the point which hath the bearing and distance required, and that which is to be done is to find the Course and Distance between d and f, which we have performed before, the Course was 11d 55′ to the Southwards of the East, and the Distance about 61 leagues, and the greater Scale of equal parts might as easily have been shunned in this Chart, as in the Plain Chart, but then the Scale of Longitudes, or West Scale, must have supplied both, and the Latitude would have been transferred from the lesser to the greater, which is uncertain.  
The manner of finding the difference of Longitude for a Course that lies East or West, was instanced in finding the point e, or it may be found with more certainty for Courses near the East or West, after the manner shown in Page 32. Or having found the Departure from the Meridian first, make use of the Cousin of the middle Latitude, and with the Departure find the difference of Longitude required, as you found the point e, or do it by the pen, as we shall afterwards show, and then prick it down; or lastly it may be done in a Scheme apart, by this Proportion:  
As the Secant of the Rumbe from the East or West:  
Is to the Distance run ∷  
So is the Secant of the middle Latitude:  
To the difference of Longitude ∷  
Admit a Ship sail West and by South 3 degrees or 60 leagues from Latitude 41d, now by the dead reckoning she is in Latitude 38d 6 Centesms, the middle Latitude is 39d 53 Centesms: And let it be required to find the difference of Longitude proper to this Course.  
This we insist upon, because when a Course lies nearer the East or West then a point, it cannot be so certainly found by the Traverse-quadrant, because the Compasses will cross that Rumbe very obliquely, having drawn the Quadrant A B C, prick the Course 11d 15′ from B to H, and draw C H, on it prick down 3d out of the West Scale in the Chart from C to F, and draw F D parallel to C A, then count the middle Latitude 39d 53 Centesms, or 31′, from B to E, and draw C E, then is the extent C G the difference of Longitude required, to wit, if measured in the former Scale 3d 81 Centesms, and when the line C E falls above D, continued I D far enough.  
Illustration.  
C I being Radius, C F is the Secant of the Rumbe from the East or West to that Radius, and it is also the Distance run, than also by the proportion of equality, C G being the Secant of the middle Latitude, it is also the difference of Longitude required; So also the Course being East or West, if C I being the distance be made Radius, then is C G the Secant of the Latitude to that Radius, the difference of Longitude required, but in this Scheme C I is the Departure from the Meridian in the former Course, which being given, neither the Rumbe nor Distance (both placed in the line C H) are necessarily required. And this I think sufficient to explain the Use of Mr. wright's excellent Sea-Chart, commonly called Mercators, because Mercator a Dutchman, having obtained the making of the Chart from Mr. Wright, was the first that made Maps of the World upon that kind of Projection.  
We have hitherto kept a Reckoning upon both Charts, which might as well have been done upon some fitted Traverseboard, and thence removed into the Chart, as oft as it should be thought needful; And for the Preservation of the Chart it may also be noted, that a Course and Distance on either Chart may be measured without the use of Compasses, by the joint help of a Ruler, having equal parts on the edge, of the same size as those we use for a Scale of Leagues, and also of a Semicircle (or Quadrant) of the Compass, printed in page 6, being cut through the Centre, and near to the outward Divisions of the Limb, fo● then laying the Ruler over any two places in the Chart, bring the Diameter of the Semicircle to lie along by the edge of the Ruler, so that the Centre thereof may lie upon some Meridian, and the said Meridian passing under the Semicircle, it will be easily seen by view what Arch of the Semicircle is contained between the edge of the Ruler and that Meridian, which measureth the quantity of the Rumbe.  
Then for the Distance count the difference of Latitude in some Meridian, suppose W N of the latter Chart above some Parallel drawn therein, as namely, above r B, suppose it ends at N, there set the Centre of the Semicircle, so that the same degree of the Limb thereof may lie over the Meridian underneath, as it did before in measuring the Rumbe, then lay the Ruler along by the Radius of the Semicircle, so that the beginning of the Rulers divisions may be at the Centre, and the distance from the Centre of the Ruler to the Parallel (r B) above which the difference of Latitude was counted, being measured by view on the Ruler, showeth the Distance required.   

For East or West Distances.  
Lay the edge of a cut sheet of Paper, Paste-board, etc. over the Centre of a Quadrant, drawn in some corner or spare place of the Chart, and also over the compliment of the Latitude counted from the East or West, and holding it so, count from the Centre of the said Quadrant in the parallel the difference of Longitude, and laying the edge of the graduated Ruler to it, slide it so that the Ruler may lie over the difference of Longitude in the Parallel, and that the graduated beginning of it may make right Angles, or be perpendicular to the straight edge of the sheet of Paper before laid over the Colatitude, and where the parallel in which the difference of Longitude was counted cuts the Ruler, it shows the Distance required; and the converse of this manner of work will also find the Distance for any other Course, if you lay the cut Paper to the Rumbe from the Meridian, and place the difference of Latitude upon the sliding Ruler, and the Distance will be found in the Parallel, in which the Difference of Longitude was counted.  

OF Sailing by the Arch of a GREAT CIRCLE.  
THE foundation of this kind of sailing supposeth a great Circle, whose Centre is the same with the Centre of the Sphere, to pass through any two places propounded, the Arch of which great Circle on the Surface of the Sphere, is always somewhat less than the distance between the two places in the direct Rumbe that leads from the one to the other: Moreover, this great Arch makes several and different Angles with every Meridian, whereas the Rumbe always maketh the same Angle. Now the chief benefit of this kind of sailing, is, that in an Eastwardly or Westwardly Course we may often shift the Latitude with good advantage, both in respect of time and way, and thereby be the better enabled to correct the Dead Reckoning: Now as he that on a Globe or a true Projection of the Sphere, would draw a line resembling the Rumb that passeth between any two places, must find the Longitudes and Latitudes of many points through which it passeth, and then by the best diligence he can draw a bending line through all those Points, so also they that would draw a line in Mercators' Chart resembling the great Arch which passeth between any two places, must find the Longitudes and Latitudes of many Points through which it passeth, and then through all those Points in the Chart draw a curved line, which shall represent the great Arch, and being drawn, are to sail as near it as they can, finding the respective Courses and Distances by which they are to sail before the Rumb, doth vary considerably, whereas in truth it doth vary continually, though not very much in a reasonable run.  
Now as Mercators' Chart doth give all the Longitudes and Latitudes through which the Rumbe between any two places laid down thereon doth pass, which may also be found by this Proportion:  
As the Meridional parts between both Latitudes:  
Is to the whole difference of Longitude ∷  
So are the Meridional parts between one of the Latitudes first given, and any other Latitude between both places:  
To the difference of Longitude answering to the assigned intermediate Latitude ∷  
And by altering the order of the Proportion, if the difference of Longitude were given, we might find the Latitude answerable thereto; So likewise there are Projections of the Sphere which will give the Longitudes and Latitudes through which the great Arch doth pass, and from whence also we may raise Proportions for finding the same exactly by Calculation or Instruments, if the Geometrical performance be thought either troublesome, or not exact enough.  
Because the Stereographick Projection doth very plainly represent the Triangles on the Sphere, we shall therefore put one instance upon that Projection.  
Let there be two places, the one is 50d North Latitude, as is the Lizard, the other in 36d 00′ South Latitude, and let the difference of Longitude between them be 68d 30′.  
First it is required to draw the great Arch in that Projection, and then to find the greatest Latitude or Obliquity thereof, and the Distance in the said Arch, and thirdly to find what Latitude it passeth through at 10d difference of Longitude from the place in South Latitude.  
Now to measure the greatest Latitude of it, a Perpendicular must be let fall from the Pole upon it, therefore where this Arch crosseth the Equinoctial as at H, is the Pole of the said Perpendicular: now to find the Centre wherewith to draw it, lay a ruler over P and H, and it cuts the Limb at K, prick S K upward from K to a, and a ruler over P and a, will cut the Equator, being continued at the Centre of the said Perpendicular, where setting one foot, with the other describe the pricked Ark P e S.  
1. A ruler over H and e, cuts the Limb at O, and the Arch Q O being 59d 56′, is the greatest Latitude of the Arch, being the obliquity required.  
2. From the Centre of the great Arch, which happened at V, draw a line to C, and where it crosseth the Perpendicular as at r, is the Pole of the great Arch, a ruler over r and L cuts the Limb at t, and the Arch T t being 105d 53′, is the Distance in the Arch required.  
3. Prick 10d from S to n, and again to u, a ruler from P laid over n, and u, cuts the Equator at x the Pole, and z the Centre of the Meridian to be drawn, upon which describe it, namely, the Arch S f P.  
A ruler over x and f cuts the Limb at m, and the Arch A m being 22d 8′, is the Latitude of the great Arch required, to the difference of Longitude assigned.    

Some Observations from the former Scheme.  
1. That if those places be supposed to be both in one Hemisphere, and to have the Compliment of the former difference of Longitude to a Semicircle, one & the same great Arch is still common, and the distance is the Compliment of the former distance to a Semicircle, as is evident in the Triangle P L A, in which the Angle L P A is the Compliment of the former difference of Longitude to a Semicircle, and the Arch L A is the compliment of L B the former Distance, and the lesser Vertical Angle is common to both Triangles.  
2. That the right Angled Triangle A Q E hath its Sides and Angles equal to the Sides and Angles of the Triangle B A H, and therefore the Arch A E passeth through the like Latitudes in the North Hemisphere, that the Arch H B doth in the South.  
3. That in any right Angled Spherical Triangle if it have one obtuse Angle, it hath also the Leg opposite to that Angle, and the Hipotenusal greater than Quadrants, and the contrary, and the other Leg will be less than a Quadrant, and subtend an acute Angle.  
And that in stead of resolving such a Triangle we may, and actually do in Calculation, resolve a right Angled Triangle, in which all the parts besides the right Angle, are less than Quadrants; This is evident in the two Triangles f W S and f e P, in which the right Angle at e and W, the Angle at f and the Perpendicular P e, or S W, is common to both Triangles.  
But the Sides 

f S  
W f   are the compliments of the Sides 

f P  
f e   to a Semicircle, and the Angle f S W, is the compliment of the Angle f P e to a Semicircle, for it is equal to the two Angles A P e, and B P f, which together are equal to the compliment of the Angle f P e to a Semicircle.  
This being forenoted, because these Arkes are tedious and troublesome to draw, we shall handle in the next place a projection in Tangent lines, on the which, albeit we cannot project an entire Hemisphere, yet all may be very well thence supplied.  
First therefore if we suppose a Plain to be raised Perpendicular to the Axis of the world, and to pass through one of the pole points, and then place the eye at the Center-rayes issuing from the sight, through the supposed graduations of any Meridian, unto the Plain, shall be the Secants of the arches from the pole point, through which the sight passeth, and on the Plain meeting therewith shall project Tangent lines which because they grow infinite, it follows that one entire Hemisphere cannot be thus projected; moreover, the sight there will project any great Circle in a right line as follows from 91 Prop. 6 Book of Aguilonius: this Projection is applied by Mr. Phillip's to great Circle sailing, divers things by him not handled we shall add, and then proceed to show how the same may be otherwise performed.   

First Example of two places in one Hemisphere.  
Now let it be required to find the Longitudes and Latitudes of the great Arch between the Lizard Latitude 50d, and Trinity Harbour in Virginia, Latitude 36d, difference of Longitude 68d 30′, draw G F and upon P as a Centre, describe a Semicircle, prick the difference of Longitude 68d 30′ from F to N, draw P N.  
Now for speedy Operation it will be convenient to have a line of natural Tangents on the sloap edge of a ruler, which, as before in the first Part, may be divided from a Quadrants Limb, or rather made by an Instrument-maker, and such a one we suppose here used, out of which prick down P L 40d, the compliment of the Lizards Latitude, and P T 54d the compliment of the Latitude of Trinity Harbour, and draw T L, and it shall represent the great Arch between these two places; let fall a Perpendicular upon it from P, to do which make P W equal to P L, and in the middle between L and W, as at A, let fall the Perpendicular P A, the Arch F O shows how far it will happen from the Lizard. Now if it were required to find the Latitudes of this Arch for every 5d difference of Longitude from the Perpendicular, then must the Arch O N be divided into every fifth degree from o, but more certainly prick the Radius in the Perpendicular from P to C, and draw C D parallel to A T; and thereto laying the Tangents on your ruler, prick down every fifth degree from C to D, then laying the beginning of your Tangents over P, and every one of those graduations, you may by the view see what Latitude the Arch passeth through at every 5 degrees difference of Longitude from the Perpendicular, for which purpose the Tangents on your ruler may be double numbered both with the Arch and its compliment to which they belong.  
Example. Laying the Ruler over 25 degrees, it cuts the Arch at e, and the extent P e being measured on the Tangents, is 41d 51′, being the compliment of the Latitude of the Arch at that difference of Longitude, wherefore the Latitude of the Arch at 25d difference of Longitude from the Perpendicular, is 48d 9′.  
In like manner the Latitudes of the Arch for 00d Difference of Longitude from the Perpendicular, is 50d, 56′. 05 50, 49 10 50, 30 15 49, 57 20 49, 10 25 48, 9 30 46, 51 35 45, 16 40 43, 20 45 41, 4 50 38, 23 and is the same for the first ten degrees difference of Longitude on each side the Perpendicular, and also as far as the lesser Vertical Angle.   

A second Example for two places in different Hemispheres.  
In like manner, if we suppose Trinity Harbour to be in as much South Latitude, the difference of Longitude being the same, to wit, 68d 30′, place P L from P to I, and draw I T continued, and it shall represent the great Arch, being continued from I upwards, runs from the Lizard towards the Equinoctial, and below T downwards, runs also from the supposed place in South Latitude towards the Equinoctial, where in this Projection it grows infinite.  
Let fall the Perpendicular P Q equal to the Radius, and prick off the Tangents from Q downwards beyond R, parallel to H T, and if you set P I from P to K, so as to meet with H T, than you may measure the Latitude of the Ark on each side the Equator the the same way by laying the beginning of the Tangents on the edge of the Ruler to P, and laying the Ruler as before to every fifth degree of Longitude, thus the Latitudes of the Arch at 55d Difference of Longitude from the Perpendicular, is 44d, 43′ 60 40, 49 65 36, 7 70 30, 34 75 24, 4 80 16, 42 85 8, 33  
We have continued the Tangents but to 65d at R for want of room, however without any great Excursions, the Latitudes of it may be found near the Equinoctial by this Proportion:  
As the Tangent of the Perpendicular:  
Is to the Cousin of the Angle adjacent ∷  
So is the Radius:  
To the Tangent of the Arkes Latitude:  
Example:  
Let it be required to find the Latitude of the Arch at 70 degrees difference of Longitude from the Perpendicular.  
The compliment thereof is 20d, to the Tangent whereof draw a line from P, than the nearest distance from Q to the said line, is the Cousin of the said Angle, which prick from H to S, a ruler over P and S, cuts this Tangent-line Q R at 30 degrees 34 minutes, and so much is the Latitude required, and at 90 degrees from the Perpendicular this Arch passeth over the Equinoctial.   

A third Example for two places in the same Parallel; to wit, in the Latitude of 50 degrees, difference of Longitude 92 degrees 46 minutes.  
Thus we see this Projection will supply all the Cases that can be put, and is in effect no other than a Scheme for the carrying on of Proportions, and the Sphere; being thus reduced into right lines, we may thence raise Proportions for Calculating all that is required. In the obliqueangled plain Triangle T P L, we have the two Sides T P and P L given, being the Tangents of the compliments of both Latitudes, and the Angle comprehended T P L given, now we find the other Angles at T and L by this Proportion:  
As the sum of the two Sides: Is to their difference ∷  
So is the Tangent of the half sum of the opposite Angles:  
To the Tangent of half their difference ∷  
which Proportion is demonstrated in many Books of trigonometry; And so the two Sides of this Triangle are the Tangents of the Compliments of the Latitudes.  
But as the sum of the Tangent of any two Arkes:  
Is to the difference of those Tangents ∷  
So is the sine of the sum of those Arkes:  
To the sine of the difference of those Arkes:  
See this demonstrated in Mr. Newtons' Trigonometria Brittanica.  
Again the third term being the Tangent of the half sum of the opposite Angles, in other Language is the Cotangent of half the contained Angle.  
Lastly, the half sum of the unknown Angles being added to their half difference, the sum makes the greater, and the difference the lesser of those unknown Angles. In like manner if the said half difference be added to half the contained Angle, the sum is the greater of those Angles next the Perpendicular, and the difference is the lesser, and these Angles we call the Vertical Angle.  
The Proportion for finding the Perpendicular, having the Hipotenusal or Colatitude, and the Vertical Angle given, is:  
As the Radius: Is to the Cousin of the Vertical Angle ∷  
So is the Cotangent of the Latitude:  
To the Targent of the Perpendicular ∷  
Then ass gning the difference of Longitude from the Perpendicular, we nave before set down the Proportion for finding the Arks Latitude, and both these Proportions may be brought into one, and so the Method of Calculation arising out of these Considerations, suns thus:  
1. For a Parallel or East and West Course.  
As the Cousin of half the difference of Longitude:  
Is to the Tangent of the common Latitude ∷  
So is the Cousin of the Ark of Longitude from the Perpendicular:  
To the Tangent of the Arks Latitude answering thereto ∷  
Again, in all other Cases for the Vertical Angles.  
As the Sine of the sum of the Compliments of both Latitudes:  
Is to the Sine of their difference ∷  
So is the Tangent of the Compliment of half the difference of Longitude:  
To the Tangent of half the difference of the Vertical Angles ∷  
which Ark thus found.  
Add to half the difference of Longitude, the sum is the greater Vertical Angle, which if it exceed the difference of Longitude, the Perpendicular falls without, and this Ark comprehends the whole Angle between the Perpendicular and both Colatitudes, the difference between the fourth Proportional Ark and half the difference of Longitude, is the Angle between the lesser Colatitude and the Perpendicular, being the lesser Vertical Angle.  
When one place is in South Latitude, and the other in North, the first and second terms of this general Proportion change places, and the Sine of the sum becomes the Sine of the difference, and this that, as will be found if you make one of the containing Sides greater than a Quadrant, the Polar distance in stead of the Colatitude, and then for the Sine of an Ark greater than a Quadrant, take the Sine of that Arks compliment to a Semicircle, which may sometimes happen when both places are in one Hemisphere, if the sum of the Compliments exceed 90 degrees.  
Secondly, the Latitudes of the great Ark are found by this Proportion, which is in a manner the same as for a Parallel Course.  
As the Cousin of either of the Vertical Angles, but rather the lesser:  
Is to the tangent of the latitude of that place to which it is adjacent ∷  
So is the Cousin of the Ark of difference of Longitude from the Perpendicular:  
To the Tangent of the Arks Latitude sought ∷  
Thus by this excellent Method of Calculation we dispatch that at two Operations, which Master Norwood and others do not attain under seven or eight, which rendered the Sailing by the great Arch so difficult and laborious, that none cared to practise it. And this latter Proportion having two fixed terms in it, will be performed on a Serpentine-line, or Logarithmical Ruler, without altering the Index or Compasses, which Proportion being varied, is carried on in the former Scheme:  
As the Secant of either of the Vertical Angles:  
Is to the Cotangent of the Latitude adjacent thereto ∷  
So is the Secant of the difference of Longitude from the Perpendicular:  
To the Cotangent of the Arks Latitude sought ∷  
In the former Scheme making the Perpendicular P A Radius, P T becomes the Secant of the Vertical Angle T P A to that Radius, and is also by construction the Tangent of the Compliment of the Latitude next that Angle, to another Radius, then by reason of the Proportion of Equality, which we have formerly handled, the Secant of any other Angle from the Perpendicular to the former Radius, as is P e, shall be also the Cotangent of the Arks Latitude to the latter Radius.  
Also the former Scheme shows us how to delineate Proportions in Sines and Tangents, by framing of rightlined obliqne Triangles; for as the Sine of the Angle at L is to its opposite Side T P a Tangent, so is the Sine of the Angle at T to its opposite Side P L another Tangent; and by the like reason Proportions in Sines alone, or in Equal parts and Sins, etc. may be carried on in the Angles and Sides of Plain Triangles.  
Of the six parts of a rightangled Spherical Triangle, no more but four at a time can be laid down in right Lines and Angles.  
As in the rightangled Triangle P A T, to wit, the right Angle at A, the Hipotenusal P T, the Perpendicular P A, and the Vertical Angle between T P A, so that the Angle at T being the Compliment of the Vertical Angle, is not the Angle of Position in the Sphere, nor is the Side A T the measure of the Distance on that side the Perpendicular, however both these Arks may be easily found.   

Example for the Distance.  
Prick the Radius of the Tangents from A to B, and place the extent, B P, being the Secant of the Perpendicular from A to E, from whence lines drawn to L and T, shall contain an Angle equal to the Distance between the Lizard and Trinity Harbour in the great Arch; To measure it, with 60d of the Chords, setting one foot at E, draw K M, which extent measured in the Chords showeth the Distance to be 50d 9′ at 20 leagues to a degree, is 1003 leagues, the Rumbe between these two places is 74d 17′ from the Meridian, and the Distance of the Rumbe 1034 leagues, and after the same manner any part of it may be measured; and so likewise might the parallel Distance I V, which if we had room would be found to be 55 degrees and a half; Also the Distance I T would be found to be 74d 7′, the Compliment whereof to a Semicircle being 105d 53′, would be the Distance between the Lizard and Trinity Harbour, as we supposed them, the one in South, the other in North Latitude.  
The Proportion carried on to find the Distance, is:  
As the Radius: Is to the Sine of the Perpendicular ∷  
So is the Tangent of the Vertical Angle:  
To the Tangent of the Distance ∷  
The Perpendicular A P is a Tangent to the Radius A B, and if we make the extent B P Radius, it than becomes a Sine, as is evident if you describe an Ark from P therewith; which R●dius if we should place from P outward beyond C, and suppose a Tangent erected thereon parallel to A T, then doth the extent A T become the Tangent of the fourth Proportional to that Radius, which that it might be measured, the Radius was pricked from A to E.   

To find the Angle of Position.  
If the Hipotenusal and one Legg be given, the Proportion to find the Angle next that Leg, would be:  
As the Radius: Is to the Tangent of the Hipotenusal ∷  
So is the Tangent of the given Legg:  
To the Cousin of its adjacent Angle ∷  
And so in the former Scheme, if we make T P Radius, which is also the Tangent of the Hipotenusal, then doth P A being the Tangent of the given Leg, become the Sine of the Angle P T A, which is the Compliment of the Angle T P A; whence we may observe, that if a Perpendicular be raised at the end of the Tangent of the given Leg, the Tangent of the Hipotenusal being from the other end of the said Leg made the Hipotenusal opposite to the right Angle, the Angle included shall be the Angle sought; therefore having the Distance or Leg A T given, with its Radius A E, you may proportion out the Hipotenusal to that Radius, and with it setting one foot in T, cross A E with the other, from whence drawing a line to T, the Angle between it and A T, shall be the Angle of Position required.   

Otherwise more readily:  
Take the nearest distance from C to T P, and it shall be the Cousin of the Angle of Position, if we make E A Radius; wherefore upon A describe the pricked Ark n, a ruler from the Centre E just touching it, cuts the Ark x n at n, which Ark measured on the Chords is 38d 49′, the compliment whereof being 51d 11′ is the Angle of Position required, and so may all the other Angles of Position be found, if there were any need of them. The Proportion carried on, is:  
As the Radius: Is to the Sine of the Vertical Angle:  
So is the Cousin of the Perpendicular:  
To the Cousin of the Angle of Position ∷  
If we make B P Radius, then is A B the Cousin of the Perpendicular to that Radius, and the nearest distance from C to T P, is the Cousin of the Angle of Position to the Radius E A.  
Thus we have showed the use of this Pro●ection in more particulars than the Seaman shall have occasion to use.  
Lastly it may be noted, that in stead of pricking down a new Line of Tangents from the Perpendicular in every question that shall be put, that if the Perpendicular be placed in the Side P L, and thence the great Ark raised perpendicularly, as also a Line of Tangents to the common Radius, at the distance of the Radius from P, that there needs no new Tangents be p ickt down to every several question about the Arch, in other Cases whe●e the difference of Longitude, as also the Latitudes of both places vary.  
THe former Pro●ection for places near the Equinoctial grows infinite and inconvenient, if therefore, as before, we suppose the eye at the Centre of the Earth, and place a plain Parallel to the Axis, touching the Equinoctial in some point, then will the Equinoctial itself from that point be projected on each side in a Tangent-line, all the Meridian's in that plain will be parallel to each other, and are Tangent-lines of several Radii, the Radius of each of which is the Secant of that Meridian's distance from the touch point, any other great Circle passing through the Eye, w●ll on that Plain be projected in a right line. The Angle that every great Circle makes with the Equinoctial, I call the Obliquity of that Circle: thus the Angle of the Ecliptic and Equinoctial being measuted by the Sun's greatest declination, is called the Obliquity of the Ecliptic; if this Plain toucheth the Equinoctial at the Meridian of the greatest Obliquity of any other Arch, then will the eye project the said Arch on this Plain in a right line, parallel to the Equinoctial, which being measured on their Secants, will all fall in the Circumference of a Circle; And though all this might be handled as a Projection, yet for brevity I shall only show you that a Scheme so made doth carry on the former Proportion for Calculating the Latitudes of the great Arch, and therefore one Quadrant only may, and will supply all Cases whatsoever.  
In the following Scheme, the line Q P is divided into a Line of Tangents, describe the Quadrant A R, and through every fifth degree of it draw lines from the Centre, o● in stead of diving such a Quadrant, you may by the edge of the Ruler divide the lines A B and R B, each of them into a Tangent of 45d.   

Examples for finding the Latitudes of the great Ark. 1. For a Parallel Course.  
Admit there were two places in the Latitude of 50d, and their difference of Longitude were 70d, and it were required to find the Latitudes of the great Ark between these places.  
And to save the trouble of taking out these Extents, and then measuring them, having the line of Tangents on the sloap edge of a ruler, you may slide it along perpendicularly to Q R, keeping the beginning of the Tangents always upon that line, and as it meets with the Ark at every fifth degree difference of Longitude from the Perpendicular, you will see without trouble what Latitudes the Arch passeth through; and if no Line of Tangents be at hand, than the nearest distances from every fifth degree in the Arch to the Equinoctial line A Q, may be placed on the Perpendicular Q P, and setting one foot in R, if with R Q you describe an Arch, as in page 28 of the First Part, by a ruler over R, and each Tangent placed on the Perpendicular, you may measure the Arks Latitude by a Line of Chords.  
Thus the Latitudes of the Arch in the former Case for 00d Difference of Longitude from the Perpendicular, is 55d, 29′ 5 55, 23 10 55, 4 15 54, 33 20 53, 48 25 52, 49 30 51, 33 35 50, 00  
The Distance in the Arch is 846 leagues, but in the Rumbe 900 leagues, so following the Arch there is saved in the Distance 54 leagues, besides the altering of the Latitude 5d 29′, which is a great convenience in this East or West Course.  
Another Example.  
The Lizard and Fog-Bay in Newfoundland are both in the Latitude of 50 degrees, and the difference of Longitude between them, according to the  late Dutch Map, is 42 degrees: through 21 degrees of the Quadrant draw a line meeting with the parallel of Latitude at F, and with the Extent Q F draw the Arch F P, which represents one half of the great Arch between these places, the Latitudes whereof for 0d Difference of Longitude from the Perpendicular, are 51d, 50′. 5 51, 49 10 51, 30 15 50, 58 20 50, 11  
Another example for places that differ both in latitude and longitude.  
In this case, if the greatest Obliquity be attained by the former Projection, or any other ready way, being the compliment of the Perpendicular there found, there needs no more trouble but with the Tangent of the said Obliquity to describe an Arch upon Q as a Centre, and it will show all the Latitudes of the Arch as in the former case, also the Parallels of both Latitude meeting with this Arch, do therein show you the quantities of the Vertical Angles, beyond which the Latitudes of the Arch are not required.  
Let the Example be the same as in the Projection first handled, namely, between the Lizard and Trinity Harbour in Virginia, the Perpendicular A P measured on the Tangents, is 39d 4′, wherefore the compliment thereof 50d 56′, is the Obliquity of the Arch, with the Tangent of the said Ark, setting one foot in Q, describe an Ark, in this latter Projection as T L, and it represents the great Ark between both places. Because the Perpendicular in this example fell within the arch e L, is supposed to happen as much on the other side Q P, the lesser Vertical Angle e L ●s 14d 38′ greater e T 53 52.  
And the Latitudes of this Arch measured as before, are the same as we found them in the other Projection, only here I have expressed the minutes in Centesimal parts of a degree.  
00d Difference of Longitude from the Perpendicular, is 50d, 93′ 05 50, 82 10 50, 50 15 49, 95 20 49, 17 25 48, 15 30 46, 85 35 45, 27 40 43, 33 45 41, 07 50 38, 38  
In like manner, if these places be supposed to be in different Hemispheres, the Perpendicular P H in the former Projection measured on the Tangents, is 30d 4′, the compliment whereof 59d 56′, is the Obliquity or greatest Latitude of the Arch, with the Tangent whereof draw the arch A E t l p, and it represents the arch between these places and the lesser vertical angle p l is 46d 23′ greater p t 65d 8′ being the Compliment of the greater to a Semicircle, because▪ should have happened as much below A, as it doth above.  
Part of this Arch A t, is supposed to fall below the Equinoctail Line Q A in the other Hemisphere, where it would pass through the like Latitudes as it doth here above it, and the Latitudes of this Arch measured as before, will be the same as we found, and before expressed them, in the use of the other Projection.  
If one place be under the Equinoctial, and the other towards one of the Poles, the Latitude of the said place is the obliquity required; and therefore with the Tangent of the Latitude the Arch is to be described.   

Places otherwise laid down.  
THis way of laying down the great Arch by it greatest obliquity, is universal, and when the first Projection is inconvenient in finding the Perpendicular, and thence the Obliquity, one of the Vertical Angles must be found, which being drawn in the Quadrant from the Centre, where it meets with the parallel of Latitude, gives a point passing through which the great Arch is to be drawn upon the common Centre Q: Now this Vertical Angle may be either found on the former Scheme, or in an obliqne plain Triangle, one side whereof sought shall be the Tangent of the half difference of the Vertical Angles, or in a Circular Scheme, as shall afterwards follow, by carrying on the Proportion before delivered for Calculating the same either of these three ways; it would be too long to insist upon all, and by varying the Proportion used for finding the distance in the former Projection, it may also be found in this.  
The former Scheme, albeit it may be demonstrated from Projecting the Sphere, yet it may be more easily and speedily done by showing that it carries on the Proportions in Sines and Tangents before delivered for Calculating the Latitudes of the Arch, we need only give one instance for places in the same Latitude.  
The Arch G E in the former Scheme, is one of the Vertical Angles, or half the difference of Longitude for two places in the Latitude of 50d, whose difference of Longitude is 70d: the nearest distance from E to Q R, is the Cousin of half the difference of Longitude, making Q G Radius, which by construction is made equal to the Tangent of 50d, the common Latitude to another Radius, wherefore by reason of the Proportion of Equality, the Cousin of any Ark from the Perpendicular to the former Radius, to wit, the nearest distance from a to Q R, being the Cousin of 5d difference of Longitude from the Perpendicular, is therefore equal to the Tangent of the Arks Latitude sought, to the latter Radius, proper to the given difference of Longitude, wherefore the Proportion is duly observed.  
When the Obliquity or greatest Latitude is very great, it will be inconvenient to draw it the former way, this commonly happens when places differ much in Latitude, but little in Longitude, however generally places may be laid down otherwise on the latter Projection.  
First if their difference of Longitude be less than 90 or 85 degrees, they may be laid down without finding either of the Vertical Angles, by supposing this Plain to touch the Equinoctial for most conveniency at the Meridian of the greatest Latitude; and if they be in different Hemispheres, then may the Parallels be continued below A Q.  
But if their difference of Longitude be more than 90d, they may after the same manner be laid down by the difference of their Vertical Angles, whether they be in the same or both Hemispheres, as in the Projection first handled, the Angle V P T being the difference of the Vertical Angles, doth as well help us to draw the great Ark T V I, as if the Angle I P T were given, being the difference of Longitude greater than a Quadrant.  
Example: Suppose I would find the Latitudes of the great Arch between the Lizard in 50d of Latitude, and the Bermudas in Latitude 32d 25′, being both in the North Hemisphere, the difference of Longitude being 55d.  
To lay down these places, draw the Parallel A B passing thorough the Tangent of 55 degrees, on which making the Secant of that Ark Radius, the Tangent of 32d 25′ is to be pricked down, to take out that Tangent to the said Radius.  
Note, That the line Q k passing through 55d of the Quadrant, counted from C, being continued till it meet with the Radius C D continued, is the Secant of 55d, and by Parallels drawn through the Tangents in the line Q C, will also be divided into a line of Tangents, wherefore through the Tangent of 32d 25′, draw a Parallel meeting with the Line or Secant of 55d drawn through the Quadrant at k, then is the extent Q k the Tangent to that Radius, which prick from A to B, and draw L B, and it represents the great Arch between these places, and the Latitudes thereof from the Equinoctial are measured by the Parallel Meridian's, which are of several Radiusses, being Parallels drawn through the Tangents of 10d, 20d, 30d, etc.  
Now to find the Latitudes of the Arch from the Lizard for every 10d difference of Longitude.  
In the first Parallel take the extent 1, 10, and place on the Secant of 10d from Q to a, also take the extent 2, 20, and place on the Secant of 20d from Q to b; or if these extents be too long, the extent f 2 being pricked on the Secant of 20d, shall reach from the Arch of the Quadrant to b, as before.  
Now if you take the nearest distances from the points a b c d e to the Equinoctial A Q, and measure them in the Meridian-line Q L, they there show you the latitudes of the Arch required: thus the latitudes of the Arch or the points a for 10d Difference of Longitude from the Lizard, is 49d, 19′. b  20 47, 43 c  30 45, 4 d  40 41, 11 e  50 53, 47 or the nearest distance from a to Q A measured on the Scale of Tangents, is 49d 19′, the latitude of the Arch for 10d difference of longitude as before.  
To these former ways, I shall lastly add the Geometrical way, which is void of all Caution or Excursion, and carries on the Proportions before delivered for Calculation.  
Let the Example be the same as before, between the Lizard and Trinity Harbour.  
Upon the Centre V with the lesser Chord, describe a Circle, and draw A S and B I parallel, the sum of the Compliments of both Latitudes is 96d, the Sine whereof is the Sine of 84d its Compliment to a Semicircle, which prick from A to S, and the difference of those Compliments is 14d, make B D the Sine thereof, a ruler over S and D cuts the Diameter at E.  
Out of the lesser Chords prick 68d 30′, the difference of Longitude from B to H, a ruler over H and E cuts the Limb at K, and the Arch B K is half the difference of the Vertical Angles, which prick from H to G and L, then if you divide the Semicircle B A into 90 equal parts, the Arch B G is the greater Vertical Angle, to wit, 53d 52′, and the Arch B L the lesser, to wit, 14d 38′. Now by help of this Scheme you may easily describe the great Arch in the former Quadrant, take the Extent B E from hence, being the Sine of the lesser Vertical Angle, which belongs to the greater Latitude, and upon A in the same Quadrant describe an Ark therewith at n, a ruler from the Centre cuts the parallel of Latitude there at L, through which point the great Arch was drawn.  
In the former Scheme, the Vertical Angles being the Angles at the Pole on each side the Perpendicular, are measured from the Point B, and the Chords of those Angles from the Point A, are the Cosines of those Angles; and so the extent A L is the Cousin of the lesser Vertical Angle, which prick from A in the line A S, the point found we may call the Vertical point; then prick down the Sins of every fifth degree from 90, as of 85d 80d &c, in the line I B, from I towards B, and laying a ruler over the Vertical Point, and all those Sine Points divide the Diameter into as many points, then prick down the Sine of the Latitude, to wit, 50d in the Semicircle A B, from A upwards towards B, and a ruler over the said Latitude point, and all the former points found in the Diameter, will cut the under Semicircle B A in as many points or Arkes more, which being counted or measured from B downwards towards A, are the respective Latitudes of the great Arch sought.  
Nota, if a line of sins equal to the diameter be graduated on the floap edge of a ruler, the sins in the line B I may be easily pricked down with the pen, by the edge thereof without Compasses, and the Arkes in the under Semicircle B A, may be readily measured by view, by laying the beginning of that line of Sins to the point B, and moving the edge of the ruler to each respective Ark before found in the said Semicircle.  
Here note, that a Semicircle being divided into 90 equal parts, the distances of each degree from one end of the Diameter, are Sins of those Arkes, the whole Diameter being their Radius, wherefore the use of Sines is as Geometrical as the use of Chords.  
In the same Scheme the Reader may first find the Perpendicular, and then the Distances on each side of it, by Proportions before set down.   

Now for the Latitudes of the great Ark.  
Having some of these ways found the Latitudes of the Arch, make a Mercators Chart for your Voyage, laying down the two places, suppose the Lizard at L in the following Chart, in the Latitude of 50 degrees, and Trinity Harbour at T in the Latitude of 36 degrees, the difference of Longitude, to wit, L M, being 68 degrees and a half, then because the Perpendicular falls between both places, count off the lesser Vertical Angle 14 degrees 63 Centesms from L to P, and there raise the Perpendicular P R.  
This following Chart we have made as large as the page would admit, and having a Meridian-line fitted to the degrees of Longitude in that Chart, by the edge thereof on the Perpendicular P R, you may prick down the Latitudes of the Arch before found, and 5 degrees difference of Longitude on each side of the Perpendicular at a time, and thereby graduate the respective Points or Crosses through which the curved pricked line is drawn.  
Example.  
Now to the apprehension the right Line L T will seem nearer than to sail along by the pricked Arch, also it may seem improbable that a Ship bound from the Lizard to Trinity Harbour in Virginia, being a place nearer the Equinoctial by 14 degrees of Latitude, should yet run into a more Northwardly Latitude than the Lizard.  
To which I answer, That Mercators' Chart being no Projection of the Sphere, doth not represent things to the fancy as they are in the Sphere, but as they are accommodated to that Chart, by which Trinity Harbour in that Chart bears from the Lizard 74d 17′ from the Meridian, which is above 6½ points to the Westwards of the South, to wit, West and by South ¼ of a point, and one degree 39 minutes more Southwardly, and the Distance in the Rumbe is 1034 leagues, whereas the Distance in the Arch, if it could be precisely followed, is 1003 leagues, and following it from point to point, through every 5 degrees difference of Longitude from the Perpendicular, it is 1020 leagues almost, being less than the Distance in the Rumbe by 14 leagues, as appears by Calculation or the Chart itself. The respective Courses and Distances to be sailed.  
  Course.  Distance.   Deg. Min.  Leagues. Parts. From L to 1 80, 25 to the Westwards of the North 60, 06 1 2 84, 14½ 63, 86 2 3 88, 00½ 61, 02 3 4 88, 00½ to the Westwards of the South 61, 02 4 5 84, 14½ 63, 86 5 6 80, 14½ 64, 88 6 7 76, 28 66, 66 7 8 72, 51 69, 18 8 9 68, 56 72, 32 9 10 65, 30 69, 20 10 11 61, 32 81, 40 11 12 59, 00 87, 76 12 13 55, 02 93, 86 13 T 52, 20 77, 88    Whole Distance 1019, 96.  
This Example serves fully to explain the Sailing by the great Arch, though it might not be safe to follow it, by reason of haling too near the Coast of Ireland.    

OF Arithmetical Navigation.  
ALl performances by the Pen require the help of some Tables, by which the Question proposed may be speedily resolved, if such Tables be at hand, and if no other Table be at hand but that of Natural Sins, it will serve to do the whole work; and how such a Table may be made, or the Sine of any Ark at command, we shall handle.  
For conveniency and dispatch, there follows a Brief Table of Natural Sins, Tangents and Secants, to every one of the eight Points of the Compass, and their Quarters from the Meridian, and such a kind of Table (but false printed) is in the Works of Maetius, the use whereof he explains in some of the following Propositions.  
Given. Sought.  1. The Longitudes and Latitudes of two places Rumbe which is required at the beginning of a Voyage Distance 2. The Rumb and Distance given Difference of Latitude. whereby Dead Reckoning kept.  Departure from the Meridian. 3. Difference of Latude and Rumbe Departure from the Meridian Whereby the Dead Reckoning is corrected.  Distance. 4. Difference of Latude and Distance Rumbe Departure   
To give the difference of Latitude and Departure, whereby in some Cases to correct the Dead Reckoning, is the same with the first Proposition, and that we shall handle last.  
5. Rumbe. Difference of Latitude. Departure. Distance. 6. Departure. Rumbe. Distance. Difference of Latitude.  
These two last Propositions are of small use.  

Some Examples of the Use of the said Table. 1. In keeping the Dead Reckoning.  
As the Radius: Is to the Distance run ∷  
So is the Cousin of the Rumbe: To the Difference of Latitude:  
The Table of Sines is numbered with the Points of the Compass, on the left side in the Variation Column from the bottom upward, and show the difference of Latitude to 1000 leagues sailing upon any Rumbe, taking the figures that stand against the Point of the Compass as far as the Comma, and the other two figures beyond it may be taken in, if more preciseness be required; so if a Ship sail 1000 leagues South South West, being the second Point from the Meridian, count the same in the Variation Column upward, and against it in the Sins you will find the difference of Latitude is 923 leagues and 87 parts more of another league, divided into 100 parts.  
But if the difference of Latitude be required for any other Distance; multiply the Sine of the given Point by the Distance run, cutting off three figures from the Product.  
Example.  
If the Ship sails 60 leagues on that Course, the difference of Latitude is 55 leagues and 38 Centesms, multiply 923 by 60 and the Product, cutting off three places, is 55, 380.  
To find the Departure from the Meridian, the Proportion is,  
As the Radius: Is to the Distance run:  
So is the Sine of the Rumbe: To the Departure ∷  
Example.  
If a Ship sail 1000 leagues South South West, count the Point sailed in the Separation Column downward, being the third Column and on the right hand of the Sins, and you will in the Column of Sines find the Departure required to be 382 leagues 68 Centesms; But if a Ship sail but 60 leagues on that Course, multiply 382 thereby, cutting off the three last places, and you will find the Separation or Departure required to be 22 leagues, 92 Centesms, or 22, 920 leagues.  
Points Sines Separ. Tangents Secants Angles Sins quintupled deg. parts ¾ 49, 07 ¼ 49, 12 1001, 20 2d, 48m m ¾ 2, 45 ½ 98, 02 ½ 98, 48 1004, 84 5, 37 ½ 4, 90 ¼ 146, 73 ¾ 148, 32 1010, 94 8, 26 ¼ 7, 33 7 195, 09 1 198, 91 1019, 59 11, 15 9, 75 ¾ 242, 98 ¼ 250, 48 1030, 89 14, 3 ¾ 12, 14 ½ 290, 28 ½ 303, 36 1044, 99 16, 52 ½ 14, 51 ¼ 336, 89 ¾ 357, 80 1062, 08 19, 41 ¼ 16, 84 6 382, 68 2 414, 21 1082, 39 22, 30 19, 13 ¾ 427, 55 ¼ 472, 96 1106, 21 25, 18 ¾ 21, 37 ½ 471, 39 ½ 534, 52 1133, 88 28, 7 ½ 23, 56 ¼ 514, 10 ¾ 599, 36 1165, 96 30, 56 ¼ 25, 70 5 555, 57 3 668, 17 1202, 68 33, 45 27, 77 ¾ 595, 70 ¼ 741, 65 1245, 28 36, 33 ¾ 29, 78 ½ 634, 39 ½ 820, 68 1293, 64 39, 22 ½ 31, 71 ¼ 671, 56 ¾ 906, 34 1349, 61 42, 11 ¼ 33, 57 4 707, 10 4 1000, 00 1414, 21 45, 35, 35 ¾ 740, 96 ¼ 1103, 32 1489, 08 47, 48 ¾ 37, 04 ½ 773, 01 ½ 1218, 48 1576, 32 50, 37 ½ 38, 65 ¼ 803, 21 ¾ 1348, 36 1678, 68 53, 26 ¼ 40, 16 3 831, 47 5 1496, 60 1799, 95 56, 15 41, 57 ¾ 857, 73 ¼ 1668, 37 1945, 14 59, 3 ¾ 42, 88 ½ 881, 91 ½ 1870, 73 2121, 36 61, 52 ½ 44, 09 ¼ 903, 99 ¾ 2114, 20 2338, 88 64, 41 ¼ 45, 19 2 923, 87 6 2414, 21 2613, 12 67, 30 46, 19 ¾ 941, 53 ¼ 2794, 76 2968, 34 70, 18 ¾ 47, 08 ½ 956, 94 ½ 3296, 50 3445, 69 73, 7 ½ 47, 84 ¼ 970, 03 ¾ 3992, 24 4115, 56 75, 56 ¼ 48, 50 1 980, 78 7 5027, 33 5125, 83 78, 45 49, 03 ¾ 989, 17 ¼ 6741, 44 6808, 52 81, 33 ¾ 49, 45 ½ 995, 18 ½ 10153, 66 10202, 32 84, 22 ½ 49, 75 ¼ 998, 79 ¼ 20355, 48 20380, 15 87, 11 ¼ 49, 93 Variation        
Here note, that though the distance sailed be given in Leagues, yet the table of Sines may be so altered that the difference of Latitude (and Departure) may be found in degrees and Centesmes, and that only be multiplying all the figures in the Column of Sines by the Number 5, for which purpose we added the last Column of quintupuled Sins, in which the whole degrees are distinguished from the Decimal parts.  
Example:  
If a Ship sail 1000 Leagues South South-west, against the second Point counted upward, we shall find the difference of Latitude to be 46d 19 Centesmes, but counted downward we shall find the Departure from the Meridian to be 19 degrees 13 Centesmes, which Numbers multiplied by 60, cutting off five places towards the right hand, give the difference of Latitude and Departure in that Course for 60 leagues to be 

2d, 77140 Parts Diff. of Lat.  
1, 14780 Departure   of which results we need take no more places than 

2, ᵈ 77.  
1, 14 or 1d, 15.    
By these two Propositions the Dead Reckoning (to be applied to the Plain Chart) may be kept after the form prescribed by Mr. Norwood, and accordingly the difference of Latitude and Departure from the Meridian is here expressed in degrees and Centesmes for the three Traverses from Tenariff towards Nicholas Island, which we made our former Example.  
Course Distance in leagues North South Deg. Cent. East West ᵈ Cent. S S W 60,  2, 77  1, 15 W S W 80,  1, 53  3, 69 S b E ½ E 53,  2, 53 0, 77 0, 00    6, 83 0, 77 4, 84  
When some Courses increase the Latitude, and others decrease it, the difference between the North and South Column (found by substracting the lesser from the greater) gives the difference of Latitude, so likewise the difference of the East and West Columns give the Departure, which in this Example is 4d 07, this difference of Latitude and Departure substracted out of the whole difference of Latitude and of Longitude between the said two Islands, there rest 4d 17 Centesmes difference of Latitude between the Ship and St. Nicholas Island, and 3d, 93 difference of Longitude, by which the Course from the Ship to the said Island, will be found to be 43d, 28 Centesms to the Westwards of the South, and the distance 5d, 73 being about 114 leagues and a half, as shall afterward follow.  
The difference of Latitude and the Rumbe given, to find the Departure from the Meridian and distance.  
As the Radius: Is to the difference of Latitude ∷  
So is the Tangent of the Rumbe:  
To the Departure from the Meridian ∷  
And so is the Secant of the Rumbe: To the distance ∷  
Example. For the Departure.  
Let the instance be the same as in page 24, where we suppose the Course to be 3 points from the Meridian, to wit, S W b S, and the corrected difference of Latitude to be 2 degrees 24 minutes, or 2 degrees 4 tenths, or 40 Centesms, by which multiplying 668, the Tangent belonging to 3 Points in the former Table, the result cutting off 5 figures, is 1d 60320, wherefore the Departure from the Meridian is 1 degree 6 tenths▪  
For the Distance.  
In like manner the Secant of three points, to wit, 1202 being multiplied by the difference of Latitude 2d, 40, cutting off 5 figures is 2d, 88480, wherefore the distance is 2 degrees 88 Centesmes at 20 Leagues to a degree, is 50 leagues, and three quarters, or 50, 76.  
For keeping a reckoning by Longitude (which is appliable to Mercators' Chart and removes the error of the Plain Chart) the Table of Meridional parts were added at the end of the Book for working of this Proportion.  
As the Radius: Is to the Tangent of the Rumbe ∷  
So are the Meridional parts between both Latitudes:  
To the difference of Longitude ∷  
Which Proportion requires the difference of Latitude to be first found, and then by help of the Table of Meridional parts at the end of the Book, the difference of Longitude may be found, which Table being made but to every second Centesm, hath half the difference set down at the bottom of the Page, which added to the Meridional parts of any even Centesm above it, makes the Meridional parts for each odd Centesm, thus the Meridional parts for 25d 22 is 26, 076, whereto adding 11, the sum being 26, 087 are Meridional parts for 25d 23 Centesms, in the first Example of the Traverse the difference of Latitude is 2d 77 Centesms.  
The Latitude of Tenariff is 28d Meridional parts 29d, 186 The Latitude of the Ships place is 25, 23 26, 087 The Meridional parts between both Latitudes, are 3, 099  
Which being multiplied by the Tangent of the second Point, to wit, 414, cutting off 6 places (three by reason the Radius is 1000, and three more for the Decimal parts of the Meridional Number) the amount being 1d, 282986, shows that the difference of Longitude is 1 degree 28 Centesms, of which there may be a Column of East and West kept, like as was done for the Departure; and this casting up of the Longitude may be readily done also on the Logarithmical Ruler, or by Mr. Phillip's his late Tables for that purpose without Calculation, whereof we made mention in Page 37.  
We have before said, that a Table of Sines is sufficient to supply all Calculation, though o●●er Tables may be more ready for dispatch when they are at hand, as we have showed concerning the Table of Meridional parts in Page 48.  
Out of it the Secants are made by this Proportion:  
As the Cousin of any Arch proposed: Is to the Radius ∷  
So is the Radius: To the Secant of the given Arch ∷  
Which Proportion holds backwards to find the Arch in the Table of Sines, if a Secant were given at adventure, and the Arch required.  
Also out of it the Tangents are made by this Proportion:  
As the Cousin of an Arch: Is to the Sine of the said Arch ∷  
So is the Radius: To the Tangent of the Arch proposed ∷  
Which Proportion doth not hold backwards to find the Arch if a Tangent were given, in which Case the Secant may be found, and thereby the Arch, for the square of a Tangent more the square of the Radius, is equal to the square of the Secant. In this Case we have also the Proportion of the Cousin to the Sine given (being the same with that of the Radius to the Tangent) and the sum o● their squares given (being equal to the square of the Radius) which very Case is reduced to a double equation in the 33d Question of Mr. Moor's Algebraick Arithmetic, and by which the Radius and a Tangent being given, either the Sine or Cousin may be found, without finding the Secant.  
If such a Proportion as this were proposed:  
As the Radius: Is to the Tangent of an Arch ∷  
So is the Tangent of another Arch:  
To the Tangent of a fourth Arch ∷  
It might be resolved thus, without making a Table of Tangents: Make the Product of the Radius, and of the Cousin of the second Ark the Divisor, and the product of the Sins of the second and third Ark, and of the Cousin of the third Ark the dividend, and the Quotient will be the Tangent of the fourth Ark sought.  
In like manner, if all those terms were Tangents, the Product of the Radius Sine and Cousin of the first Ark would be the Divisor.  
And the Product of the Sins, and Cosines of the second and third Ark, the Dividend whereby might be found, the Quotient being the Tangent sought, and consequently the Arch answering thereto; after the like manner with due regard to the Proportion for making the Secants, any Secant might be supplied, and a Proportion wholly in Secants by turning those Arkes into their compliments, is changed wholly into sins, or two terms being Secants are changed into Sins by altering the places or order of those terms, and their Arkes into their compliments.   

How to Calculate a Table of Natural Sins.  
How to make the Sine of any Ark at pleasure, Snellius in his Cyclometria hath showed, without making many such Sins as are not required, in which he went beyond all fo●mer Writers, one way he hath holds true as far as 30d of the Quadrant, out of which Sins their Cosines may be made, and so the Quadrant filled from 60d to 90d, and then by finding the Sine of the double Ark it may be supplied from 30d to 60d, but in regard the performance is very tedious, and therefore not to be prosecuted in the Construction of a whole Table, we shall omit to mention it; another way he hath which doth not hold true to above 1/12 part of the Quadrant so that when a great Ark near the end of the Quadrant is given, he finds the Cousin thereof first, and thereby the Sine, or if the Ark be remote or about the middle of the Quadrant, he finds the Sine of the eighth or some such part of it, and then by finding the Sine of the double Ark, etc. backwards he finds the Sine sought; but how to Calculate a whole Table by Proportion with ease, is attained unto by our worthy Countryman Mr. Michael Darie, who discovered the same in the year 1651, and communicated unto me the Proportions for that purpose long since, whose method we shall now insist upon.  
1. The readiest way is to assume the Proportion of the Diameter of a Circle to its Circumference to be known, and to be in the Proportion of 113 to 355 (of which see Maetius his Practical Geometry) which two Numbers are easily remembered, being the three first odd figures of the Digits, each of them twice wrote down and cut asunder in the middle.  
2. It is most convenient that the Radius or whole Sine should be an Unit with Ciphers, and so the Proporrtion of the Diameter to the Circumference, found by the two Numbers above, is such as 2000000, to 6283185, so far true; and to find it true to as many places of figures as you please, consult Dr. Wallis his Writings, or Hungenius de Magnitudine Circuli.  
3. The Sine of one Minute or Centesm doth so insensibly differ from the length of the Arch to which it belongs, that the length of the Arch of one Minute or Centesm may be very well taken to be the Sine thereof, and in the largest Tables that ever were published, differs nothing therefrom, wherefore a Minute being the 21600 part of the Circumference of a whole Circle, the like part of the number 6283185 being 290, is the Sine of 1 Minute.  
4. You may find the Cousin of one Minute by substracting the square of the Sine of one minute from the square of the Radius, the square root of the remainder is the Cousin of one minute.  
5. The Sine and Cousin of one Minute being thus given, all the rest of the Sins in the Quadrant may be calculated by Mr. Daries excellent Sinical Proportion, which runs thus:  
If a rank of Arches be equi-different,  
As the Sine of any Arch in the said rank: Is to the sum of the Sins of any two Arks alike remote from it on each side ∷  
So is the Sine of any other Arch in the said rank: To the sum of the Sins of the two Arches next it on each side, having the like common difference ∷  
And in a Semicircle we may make the Radius the middlemost of the three Arches, and the two Arches on each side of it shall be Arches one of them less by a Minute, and the other of them greater by a minute then a Quadrant, the same Sine being common to both Arches, by reason whereof it will follow from the former Proportion, that  
As the Radius: Is to the double of the Cousin of one minute:  
So is the Sine of one minute: To the Sine of two minutes and of 00 ∷  
And so is the Sine of 2 minutes: To the sum of the Sins of 3 minutes and of one minute ∷  
From which sum taking the Sine of one minute, the remainder is the Sine of 3 minutes; And so also by the former Proportion,  
Is the Sine of 3 minutes: To the sum of the Sins of 2 minutes and 4 minutes ∷ and so on successively.  
In which excellent Proportion, because the first term is the Radius, Division is shunned, and because the s●cond term varies not, having multiplied it by all the 9 Digits, the whole Calculation will then be performed by Addition and Substraction; But if you will find the Sins of Arks near the end of the Quadrant first, the same Proportion continues.  
As the Radius: Is to the double of the Cousin of one minute ∷  
So is the Cousin of the said minute; To the sum of the Cosines of two minutes, and of 00, the Cousin whereof is the Radius:  
And so is the Cousin of two minutes: To the sum of the Cosines of three minutes, and of one minute ∷  
And so is the Cousin of 3 minutes: To the sum of the Cosines of 4 minutes, and of 2 minutes, and so on successively.  
Note also, that having made the Sins of the five first and last degrees, and their quarters, the intermediate Sins may well enough in a Table not very large be supplied, by dividing the differences proportionally, especially for the five first degrees of the Quadrant, in which the differences do but very little differ from each other: If besides the Sins of the five first and last degrees, and their Minutes or Centesms, there be moreover given in store the Sine of every tenth degree in the Quadrant, which may be found by the former Proportions (after the Sine of the fifth degree is found) the Sine of any other Arch whatsoever without making any more Sins in a successive order, may by help of those in store be readily calculated, by reason no Arch in the Quadrant can be given, but it will be within five degrees or less of some known Sine, the Arch of that Sine we must make the middle Arch, and the Sine and Cousin of the Arch of difference between the middle Arch, and the given Arch, is by supposition already in store, and it will hold:  
As the Radius: Is to the Cousin of the Arch of Difference ∷  
So is the Sine of the mean or middle Arch: To the half sum of the Sins of the extreme Arches on each side of it alike remote, whereof the given Arch is one ∷  
Again, As the Radius: Is to the Sine of the Ark of difference ∷  
So is the Cousin of the middle Arch: To the half difference of the Sins of the said extreme Arches ∷  
The half sum of those Sins added to their half difference, makes the Sine of the greater of those extreme Arches, and their difference makes the Sine of the lesser of those extreme Arches, one of which is the Sine of the Arch sought.  
Admit in this latter Proportion it were required to produce the fourth term so, as that it should be the whole difference of the Sins of the Extreme Arches, then must half the Radius be the first term, being equal to the Sine of 30d, which is the Cousin of 60d, and therefore if we make the middle Arch 60d, by reason of the Proportion of equality in the two first terms, it follows that the Sine of the Ark of Distance of either of the extremes from 60d, is equal to the difference of the Sins of those extreme Arches, and consequently having all the Sins under 60d made, all the rest above it are made by Addition only: for instance, the Sine of 10d is equal to the difference of the Sins of 50d, and of 70d, and therefore being added to the Sine of 50d, shall make the Sine of 70d.  
So likewise in the former of those Proportions, if you would have the last term to be the sum of the Sins of the Extremes, half the Radius must be the first term, and then making the mean or middle Arch to be 30d, by reason of the Proportion of equality between the Sine thereof and of half the Radius, it follows that the Cousin of the Ark of Distance of either of the extremes from 30d, is equal to the sum of the Sins of those extreme Arches, from which taking the lesser of those Sins, the remainder is the greater; hence it follows that if the Sins of the first and last 30d of the Quadrant be given, from them all the Sins between 30d and 60d may be made by substraction. Let the three Arks be 20D D, 30d, 40d, the Sine of 80d being the Cousin of 10d, the distance of these Arks from the middlemost, is also equal to the sum of the Sins of 20d and of 40d, from which sum Substracting the sine of 20d the remainder is the sine of 40d; or the Sins of the last 60d of the Quadrant being given, by the same reason the Sins of the first 30d may be found by Substraction, as in the present instance, if out of the sine of 80d you subtract the sine of 40d, the remainder is the sine of 20d.  
From the first Sinical Proportion, making the Radius the Sine of the middlemost of three equi-different Arks, it follows:  
That as the Radius: Is to the sum of the Sins of the two arches next it on each side ∷ Or rather by two other terms in the same Proportion, As half the Radius: Is to the sine of either of those arks ∷  
So is the sine of the said arch: To the sum of the Radius, and of the Sine of a fourth arch, having the same common difference ∷   

Consectary.  
Wherefore the Rectangle of half the Radius, and of the sum of the Radius, and of the sine of any other Arch, is equal to the square of the sine of the middle Arch, from thence it follows, that the sine of any Arch in the Quadrant being given, we may by this Rule, and extracting the square root, perpetually bisect or find the sine of the middle Arch between it and 90d, and so run up very speedily towards the end of the Quadrant; of which sine so found, the Cousin may be found by another extraction, and then you have the sine of as small an Arch as you please, near the beginning of the Quadrant, from which by Proportion, in regard the length of the sine and of the Arch to which it doth belong, have no sensible difference, you may find the sine of one minute or Centesme, and thence by the former Proportions raise a whole Table of Sines to as large a Radius as you please, albeit the Proportion of the Diameter to the Circumference were wholly unknown, which notwithstanding hereby might be found, for having made a sine to a very small part of the Quadrant, the double thereof is the side of a Polygon inscribed, and by Proportion:  
As the Cousin so found: Is to its Sine ∷ So is the Radius: To the Tangent of the Arch to which the small Sine belongs ∷  
The double of which Tangent is the side of the like Polygon circumscribed, the length of the Arch of the Circle contained between the sides of these two Polygons, being greater than the side of the inscribed Polygon, and lesser than the side of the circumscribed Polygon.  
In finding the Sine of any middle Arch by the former Consectary, to shun the trouble of multiplying, we may assume the Radius to be 2, with a competent number of Ciphers, than the Sine of 30d, because it is half the Radius, will be an Unit with as many Ciphers, then to find the Sine of 60d to that Radius, to the Sine of 30d, prefix to the left hand the number 2, and annex the Ciphers in the Radius to the right hand, and the square root of the number so made, shall be the Sine required, and the like for finding the Sine of the middle Arch between the Radius and the Sine of any other Arch given, and retaining the Radius 2 with Ciphers, the difference between the Sine of any Arch given and the Radius, having the figures of the Radius annexed thereto, the square root of the number so composed, shall be the Cousin of the middle Arch; or we may deliver it as a Consectary from former Proportions:  
That the Rectangle of half the Radius, and of the difference between the Sine of any Arch given and the Radius, shall be equal to the square of the Cousin of the middle arch, to any Radius whatsoever.  
We forbear to suit Arithmetical Examples to the foregoing Proportions and Consectaries, supposing the Reader furnished with so much Arithmetic, as that he can extract the square root, and work the Golden Rule, or Rule of Three. See page 12 of the First Part.   

Of Books of Tables.  
Such Tables as have a degree divided into 100 parts or Centesms, are to be preferred before those that divide the degree but into 60 Parts, Minutes or Sexagesms, because they are more exact in calculation, and more speedy in finding the part proportional; and it were convenient for a Book of Tables to be so contrived, that the Natural Tables of Sines, Tangents, and Versed Sins to 90d, might stand against the Logarithmical Tables of Sines and Tangents in a portable Book, to be had by itself, to which might be added a Table of the Meridian-line to each, or every second Centesm, with Tables of the Sun's Declination, Right Ascension, and of the Longitudes, Latitudes, Declinations, and Right Ascensions of some of the principal fixed Stars, and of the Longitudes and Latitudes of places; but those that have the Logarithmical Tables of Numbers, Sins and Tangents without the Natural, may supply the want of them.  
Example. 1. Let the Natural Sine of 38d be required, the Logarithmical Sine of that Ark is 9, 789342, because we may make the Radius of the Natural Sins to be an Unit, and all the rest to be Decimal parts thereof, reject the first figure of the Logarithm or Characterisk, being 9, and seek the remaining figures of the Logarithmical Sine, to wit, 789342, amongst the Logarithms of absolute Numbers, and you will find the absolute Number answering thereto to be 61566 nearest, and that is the Natural Sine of 38 degrees required.  
2. The Table of the Meridian-line may be supplied by this Proportion, raised out of Mr. Bonds Additions to Mr. Gunter's Works  
As 75795: Is to 

60 ∷  
100 ∷   So is the difference of the Logarithmical Tangents of 45d, and of an Ark compounded or made of 45d, and of half the given Ark:  
To the Meridional parts belonging to the given Ark ∷  
If you use the number 60, you will make such a Sexagesimal Table as Mr. wright's or Mr. Norwoods', but if you use the number 100, than you will produce a Centesimal Table like Mr. Gunters, or that in Mr. Roes Tables, in which the Arks to which the Meridional Tables are fitted, are degrees and every other minute, but the Tabular Numbers are degrees and decimals, being 〈◊〉 very good Table of that kind, and much fuller than either Mr. Gunters or Mr. Norwoods'; the Table at the end of this Book is of the same kind, but fitted to every second Centesm of a degree, in stead of every second minute.  
The ground of the former Proportion is, that the Logarithmical Tangents above 45d, accounting every half degree for a whole one, are in the same Ratio or Proportion with a Table of the Meridian-line, whence also it follows for Instrumental use, that a Line of Logarithmical Tangents will supply the defect of the Meridian-line.  
The number above used, to wit, 75795, is not the difference between the Logarithmical Tangents of 45d and 45½ ᵈ, though near it, being the difference of the Logarithmical Tangents of 45d, and of 45d and one Centesm more, multiplied by 50; and for to avoid the trouble of Division in working this Proportion, it were convenient either to have a Table of the said Number, multiplied by all the nine Digits, or rather to alter the Proportion so as an Unit might be the first term, and then making such a Table for the second term, the said Proportion in a manner would be wrought wholly by Addition and Substraction.  
What former Ages performed by Tables, this latter Age hath endeavoured in some respects to perform without them: Snellius in his Cyclometria, shows us how the Sides of a Plain right Angled Triangle being given, we may without Tables find the Angles of that Triangle.  
In the right angled Plain Triangle C E A, with the Radius C A, describe a Semicircle, produce the Diameter, and therein make D R ●qual to the Radius, and draw R B passing through A till it meets with the Tangent I B at B; now that which Snellius asserts, is, That the Tangent cut off, to wit, I B, is somewhat shorter than the Arch I A, though near it in length, when the Arch is not above 1/12 part of a Quadrant; and this Hugenius demonstrates in his Book De Magnitudine Circuli, where he finds fault with Snellius his Demonstration thereof. Now the Proportion for finding an Angle raised from that Scheme, lies thus:  
As R E the sum of the double of the Hipotenusal C A, and of the side C E: Is to E A the lesser side ∷  
So is R I the triple of the Radius or Hipotenusal ∷  
To I B the length of the Arch required ∷ propé verum.  
This Proportion finds the length of the Arch, making the Hipotenusal always Radius, whereas in Calculation we always retain such a Radius whereof the Circumference of the Circle is 360d: now the Diameter of such a Circle will be found by the numbers in page 112, in which the Proportion of the Circumference to the Diameter is expressed to be 114, 5915, wherefore the Radius of such a Circle is 57, 2957, and the triple thereof is 171, 8871, then retaining the two first terms of the former Proportion, we may make this number the thy d term, and by one single work find the Angle; or rather taking the halfs of all four terms, the Proportion will hold:  
As the sum of the Hipotenusal and of half the greater Leg of a Plain right Angled Triangle: Is to the lesser Leg thereof ∷  
So is 86: To the Angle opposed to the lesser Leg:  
The half of 171, 8871 is 85, 9435, which because we have taken it to be 86, the Proportion if the Angle be less than 30d, finds the Angle to be about one Centesimal part of a degree too much, but if the Angle be above 35d, by reason the Scheme is not absolutely true, must have these additions made to the Angle found thereby, from 35d to 38d add one Centesm, from 38 to 40d add two Centesmes to it, and afterwards to 45d, for every degree it exceed; 40d, add one Centesm more besides the former two Centesms, and thus we may always find the lesser acute Angle, and consequently the greater, being the Compliment thereof within one Centesm of the truth, which is nearer than any Mechanic way.  
Example. In the former Triangle let there be given the Sides 

C E 4, 17,  
A E 3, 93,   by extracting the square root of the sum of the squares of these two Numbers, we shall find the side C A to be 5, 73, to which adding the half of C E, the sum is 782 the Divisor, then multiplying 393 the lesser Leg by 86, the Product is 33798, to which you may annex cyphers at pleasure to find the Decimal parts of a degree, and dividing by 782, you will find the quotient to be 43d, 22 Centesmes, to which if you add 5 Centesmes error, the Angle sought is 43d 27 Centesms.  
Readily to find what allowance must be made in respect of the Arch found, you may repair to a Table of Natural Sins, and take the two Legs of the right Angled Triangle to be the sine and cousin of any Arch, and by the last Proportion find how near you can recover the Arch again, whereby you will find what allowance must be made.  
The Example here used is that mentioned in page 16 and 109, so that hereby we find a Course and Distance on the Plain Chart, without the help of Tables, and by the like reason the height of a Gnomon, and the length of its shadow being given, the Sun's height may be got without Tables.  
Hugenius not thinking this way of Snellius to be exact enough, propounds another of his own upon this consideration, that the Chord of an Arch being increased by one third part of the difference between the Sine and the Chord of the said Arch, shall be very near equal in length to the Arch itself, yea so near, that in an Arch of 45d, it shall not err or fall short above 1/18000 part of a degree, and in an Arch of 30d, not above the 1/21600 part of a degree, whereby the Sins may be examined, and an Angle found without Tables, as if the sides of the former Triangle be given, by taking C E out of C A there remains E I, the square whereof added to the square of E A, the Sine is equal to the square of the line A I the Chord, whereby may be found the length of the Arch I A to the given Radius C A, and then by another Proportion the measure of the said Arch to the Radius of such a Circle, whose Circumference is 360d.  
In like manner, if the sides of an obliqne Plain Triangle were given, the Angles thereof might be found, if you first reduce that obliqne Triangle to two right angled Plain Triangles, which is performed in every Book of Trigonometry without Tables.  
But for such Cases of Plain Triangles, in which but one Side with two Angles are given to find the other Sides, in regard the Proportions for such Cases require Sins, and that we have not attained any ready way to make the Sine of any arch at command, forbearing to mention such ways as are both troublesome and uncertain, we must suppose that the Reader is furnished with a Table of Sines, which most Mariners have in their Seaman's Calendar.   
FINIS.   

ERRATA.  
PAge 1 line 31, for would read should, p. 48. l. 30. for 48 degrees r. 84 degrees, p. 49 l. 3. for Angle r Angles, l. 4 for Sine r. Line, p. 51. l. 25. for following r former, p. 59 l. 14. for or, r. of p. 60. l. 21. for as every, r. or ever. p. 81 l. 16. for one is, r. one in. page 88 l. 26. for Tangent r. Tangents.   

A Table of Meridional parts.  
D 0 1 2 3 4 5 6 7 8 9 0 0, 000 1, 000 2, 000 3, 001 4, 003 5, 006 6, 011 7, 017 8, 02● 9, 037 2 020 020 2 020 0●1 023 029 031 037 046 057 4 040 040 04● 041 043 046 051 057 066 077 6 060 060 060 061 063 066 071 077 086 097 8 080 080 080 081 083 086 091 097 106 117 10 , 100 1, 100 2, 100 3, 101 4, 103 5, 106 6, 111 7, 118 8, 127 9, 138 12 120 120 120 121 123 126 131 138 147 158 14 140 140 140 141 143 146 151 158 167 178 16 160 160 160 161 163 166 171 178 187 198 18 180 180 180 181 183 186 191 198 207 218 20 , 200 1, 200 2, 200 3, 201 4, 204 5, 207 6, 212 7, 219 8, 228 9, 239 22 220 220 220 221 224 227 232 239 248 259 24 240 240 240 241 244 247 252 259 268 279 26 260 260 260 261 26● 267 272 279 288 300 28 280 280 280 281 284 287 292 299 308 320 30 , 300 1, 300 2, 300 3, 301 4, 304 5, 307 6, 312 7, 319 8, 329 9, 341 32 320 320 320 321 324 327 332 339 349 361 34 340 340 340 341 344 347 352 359 369 381 36 360 360 360 361 364 367 372 379 389 401 38 380 380 380 381 384 387 392 399 409 421 40 , 400 1, 400 2, 400 3, 402 4, 404 5, 408 6, 413 7, 420 8, 430 9, 442 42 420 420 420 422 424 428 433 441 450 462 4● 440 440 440 442 444 448 453 460 470 482 46 460 460 460 462 464 468 473 48● 490 502 48 480 480 480 482 484 488 498 500 510 522 50 , 500 1, 500 2, 500 3, 502 4, 504 5, 508 6, 514 7, 521 8, 531 9, 543 52 520 520 520 522 524 528 534 541 551 563 54 540 540 540 542 544 548 554 561 571 583 56 560 560 560 562 564 568 574 581 591 603 58 580 580 580 582 584 588 594 601 611 624 60 , 600 1, 600 2, 601 3, 602 4, 605 5, 609 6, 61● 7, 622 8, 632 9, 641 62 620 620 621 622 625 629 634 6●2 652 665 64 640 640 641 642 64● 649 654 661 672 685 66 660 660 661 662 665 669 674 682 692 705 68 680 680 681 682 68● 689 694 702 712 725 70 , 700 1, 700 2, 701 3, 702 4, 705 5, 709 6, 715 7, 713 8, 733 9, 746 72 720 720 721 722 725 729 735 743 753 766 74 740 740 741 742 745 749 755 763 773 786 76 760 760 761 762 765 768 775 783 793 806 78 780 780 781 782 785 789 795 803 813 827 80 , 800 1, 800 2, 801 3, 80● 4, 805 5, 810 6, 816 7, 824 8, 834 9, 8●8 82 820 8●0 8 1 823 825 830 836 844 854 868 84 840 840 841 8●3 845 850 856 864 874 888 86 860 860 861 863 865 870 876 884 894 908 88 880 880 881 883 885 890 896 904 915 928 90 , 900 1, 900 2, 901 3, 903 4, 906 5, 910 6, 916 7, 925 8, 936 9, 949 92 920 920 921 923 926 930 936 945 956 969 94 940 940 941 943 9●6 950 956 965 976 989 96 960 960 961 962 966 970 976 985 996 10, 009 98 980 980 982 983 986 990 996 8, 005 9, 016 030 h d 10 10 10 10 10 10 10 10 10 10  
D 10 11 12 13 14 15 16 17 18 19 0 10, 051 11, 068 12, 088 13, 112 1●, 141 15, 174 16, 21● 17, 255 18, 303 19, 356 2 071 088 108 133 16● 194 232 275 324 377 4 091 108 128 153 183 215 253 296 345 398 6 111 129 1●9 174 203 235 274 317 366 419 8 131 150 170 194 244 256 295 338 387 441 10 10, 152 11, 170 12, 16● 13, 215 14, 2●4 15, 277 16, 31● 17, 359 18, 4●8 19, 463 12 172 190 210 236 264 297 337 380 429 484 14 192 210 231 257 285 318 358 401 450 505 16 212 231 251 277 30● ●39 379 422 471 526 18 233 251 272 298 326 360 400 443 492 547 20 10, 254 11, 272 12, 293 13, 318 14, 347 15, 381 16, 420 17, 464 18, 513 19, 569 22 274 292 313 339 367 402 441 484 534 590 24 294 312 334 359 388 412 461 505 555 611 26 314 333 354 380 409 44● 482 526 576 632 28 335 35● 375 401 430 464 5●3 547 597 653 30 10, 355 11, 374 12, 395 13, 421 14, 450 15, 48● 16, 524 17, 568 18, 619 19, 675 32 375 394 415 441 471 508 514 589 630 696 34 395 414 436 461 492 526 565 600 651 717 36 415 435 456 481 512 546 586 621 672 738 38 436 456 477 502 523 567 607 642 693 759 40 10, 457 11, 476 12, 497 13, 523 14, 553 15, 588 16, 628 17, 673 18, 724 19, 781 42 477 496 517 543 574 609 648 694 745 802 44 497 516 538 564 595 630 669 715 766 8●3 46 518 537 558 585 615 651 690 736 787 845 48 538 557 579 605 636 671 711 757 809 866 50 10, 559 11, 578 12, 600 13, 626 14, 656 15, 692 16, 732 17, 778 18, 830 19, 887 52 579 598 6●0 646 677 713 752 799 851 908 54 599 618 641 667 698 734 773 820 872 929 56 620 639 661 687 7●8 755 794 841 893 951 58 641 66● 681 708 739 775 815 861 914 972 60 10, 661 11, 680 12, 702 13, 729 14, 760 15, 796 16, 836 17, 883 18, 935 19, 993 62 681 700 722 750 780 817 857 904 956 20, 014 64 701 721 743 771 801 837 878 928 977 035 66 722 731 764 791 8●1 858 899 916 998 057 68 7●2 752 785 812 842 879 930 967 19, 019 079 70 10, 76● 11, 782 12, 805 13, 832 14, 863 15, 900 16, 941 17, 988 19, 041 20, 100 72 782 802 825 852 883 920 96● 18, 009 062 121 74 802 822 846 873 904 941 983 030 083 142 76 8●3 843 866 893 925 962 17, 003 051 104 163 78 844 864 887 91● 916 983 024 072 125 185 80 10, 864 11, 884 12, 907 13, 935 14, 967 16, 004 17, 045 18, 093 19, 146 20, 206 82 884 904 928 956 987 024 066 114 167 227 84 11, 005 924 948 976 15, 008 045 087 135 188 248 86 025 945 969 997 028 065 108 156 209 269 88 045 966 990 14, 017 049 086 129 177 230 281 90 10, 966 11, 986 13, 010 14, 038 15, 070 16, 107 17, 150 18, 198 19, 251 20, 312 92 986 12, 006 030 059 090 128 171 219 272 333 94 11, 00● 026 051 079 111 149 192 240 293 355 96 027 04● 071 100 13● 160 213 261 314 376 98 048 068 092 120 133 180 234 282 335 397 h d 10 10 10 10 10 10 10 10 10 10  
D 20 21 22 23 24 25 26 27 28 29 0 20, 419 21, 486 22, 561 23, 643 24, 73● 25, 833 26, 941 28, 058 29, 186 30, 324 2 430 507 583 664 756 855 96● 080 208 346 4 451 528 605 686 778 877 984 103 230 369 6 473 550 626 708 8●0 899 27, 007 126 253 392 8 494 571 648 730 822 9●1 029 148 276 415 10 20, 525 21, 593 22, 669 23, 752 24, 844 25, 943 27, 052 28, 171 29, 299 30, 438 12 546 614 690 773 865 965 074 193 322 461 14 567 635 711 795 887 987 097 214 344 484 16 588 657 733 8●7 909 26, 009 120 238 367 507 18 610 679 75● 839 931 031 142 261 39● 530 20 20, 632 21, 701 22, 777 23, 861 ●●, 953 26, 054 27, 164 28, 283 29, 413 30, 553 22 653 722 798 882 975 076 186 305 435 575 24 674 743 820 904 997 098 208 327 457 598 26 695 765 841 926 25, 019 120 230 350 480 621 28 7●7 786 863 948 041 142 252 373 503 644 30 20, 738 21, 808 22, 885 23, 970 25, 063 26, 164 27, 275 28, 396 29, 526 30, 667 32 759 829 9●6 991 085 186 297 418 548 690 34 780 850 928 24, 013 107 208 319 440 571 713 36 8●● 872 9●0 035 129 230 342 463 594 736 38 8●3 893 97● 057 151 252 364 485 617 759 40 20, 845 21, 915 22, 993 24, 079 25, 173 2●, 275 27, 387 28, 508 29, 640 30, 782 42 866 9●6 23, 0●4 090 194 297 409 530 662 805 44 887 957 036 112 216 319 A 1 552 985 828 46 909 979 058 134 238 341 454 575 708 851 48 93 22, 001 08● 156 260 363 477 598 731 874 50 20, 952 22, 023 23, 101 24, 188 25, 282 26, 3●6 27, 499 28, 611 29, 753 30, 897 52 973 044 1 2 209 304 408 521 643 775 9●0 54 995 065 144 231 326 430 543 665 798 ●43 56 21, 16 087 166 253 348 452 565 688 811 966 58 0●7 109 188 275 360 474 587 721 844 989 6● 21, 059 22, 130 23, 210 24, 297 25, 392 26, 497 27, 610 28, 734 29, 867 31, 012 6● 080 151 231 3●8 414 519 632 756 890 035 64 102 173 252 340 436 541 654 778 913 058 66 1●3 195 274 362 458 563 676 801 936 081 68 144 ●17 296 384 480 585 699 824 958 104 70 21, 165 22, 238 23, 318 24, 406 25, 502 26, 608 27, 722 28, 847 29, 981 31. 127 72 186 260 330 427 524 630 744 869 30, 003 150 74 207 281 351 449 546 652 766 891 026 173 76 229 303 373 461 568 674 789 914 049 196 78 251 324 395 483 590 696 812 936 072 219 80 21, 272 22, 345 23, 417 24, 515 25, 613 26, 719 27, 834 28, 959 3●, 095 31. 242 82 293 366 448 536 635 741 856 981 127 265 84 315 387 469 558 657 763 878 29, 003 140 288 86 336 409 481 580 679 785 901 026 163 311 88 358 431 503 602 701 807 923 049 186 334 90 21, 379 22, 453 23, 535 24, 624 25, 723 26, 830 27, 946 29, 072 30, 209 31. 357 92 400 474 556 646 745 852 28, 068 095 231 380 94 422 495 578 668 767 874 090 117 254 403 96 443 517 600 690 789 896 113 140 277 426 98 465 539 621 712 811 918 136 163 301 449 hd 10 11 11 11 11 11 11 11 11 11  
D 30 31 32 33 34 35 36 37 38 39 0 31, 473 32, 633 33, 806 34, 992 36, 191 37, 405 38, 633 39, 877 41, 137 42, 415 2 496 656 829 35, 015 215 429 657 902 162 44● 4 519 679 852 039 23● 453 682 927 187 4●6 6 54● 712 876 063 263 477 707 952 202 492 8 565 736 900 087 287 4●2 732 977 228 518 10 31, 589 32, 750 33, 924 35, 111 36, 312 37, 527 38, 757 40, 002 41, 26● 42, 544 12 612 773 647 135 336 551 781 027 289 569 14 635 796 970 159 36● 573 80● 052 314 595 16 658 829 994 183 33● 599 83 077 340 621 18 681 843 34, 018 2●7 408 SIXPENCES 855 102 366 647 20 31, 704 32, 867 34, 042 3●, 231 36, 43● 37, 649 38, 800 40, 127 41, 392 42, 673 22 79● 890 065 254 457 673 9●4 152 417 698 24 8●0 9 3 089 278 481 697 929 177 442 7●4 26 843 936 11● 302 505 7●1 9TH 201 467 750 28 877 960 137 326 529 746 9TH 227 493 776 30 31, 82● 32, 984 34, 161 35, 350 36, 5●4 37, 771 ●9, 004 40, 253 41, 519 42, 802 32 843 33, 007 184 374 578 795 029 278 544 827 34 866 030 207 398 602 816 054 30● 569 853 36 8●9 053 231 422 626 8●4 07● 3●8 5●4 879 38 912 077 255 446 650 869 10● 35● 6 0 ●05 40 31, 936 33, 101 34, 279 35, 470 36, 675 37, 894 39, 129 40, 379 41, 646 42, ●31 42 959 124 302 494 699 918 153 4●4 6 1 957 44 982 147 325 518 723 942 178 429 696 983 46 3●, 005 170 349 512 747 967 205 454 ●22 43, 009 48 028 194 373 566 771 99● 28 479 758 035 50 32, 052 33, 218 34, 397 35, 590 36, 7●6 38, 017 39, 253 40, 505 41, 774 43, 061 52 075 231 420 614 820 041 27● 030 799 087 54 098 251 444 638 84● 065 302 055 8●4 113 56 121 278 468 662 868 090 327 080 850 139 58 144 302 492 6●6 892 115 352 105 876 165 60 32, 168 33, 336 3●, 516 35, 710 36, 917 38, 130 39, 377 40, 631 41, 902 43, 191 62 181 357 539 734 941 164 402 656 927 217 64 204 378 563 758 965 188 427 681 952 243 66 227 399 587 782 989 213 452 706 978 269 68 251 431 611 806 37, 011 238 477 781 42, 004 295 70 32. 284 33, 453 34, 635 35, 830 37, ●39 38, 263 39, 502 40, 757 42, 030 43, 321 72 307 476 658 854 063 287 527 782 055 347 74 330 500 682 87● 087 311 552 807 080 373 76 353 524 7●6 902 112 336 577 8●2 106 399 78 376 547 730 926 137 36● 602 858 132 425 80 32, 40● 33, 57 34, 754 35, 950 37, 16● 38, 386 39, 627 40, 884 42, 158 43, 451 82 42● 596 777 971 185 410 652 909 183 477 84 445 611 801 993 209 434 677 934 209 503 86 468 646 825 36, 022 233 459 702 959 235 529 88 48● 671 849 046 258 48● 727 985 261 555 90 32, 516 33, 688 34, 873 36, 071 37, 283 38, 509 39, 752 41, 011 42, 287 43, 581 92 539 701 896 095 307 533 777 036 312 607 94 562 724 9 0 119 331 558 802 061 337 633 96 585 748 944 143 355 583 827 086 363 659 98 609 772 963 167 380 608 852 111 389 685 hd 11 11 11 12 12 22 12 12 12 13  
D 40 41 42 43 44 45 46 47 48 49 0 43, 711 45, 026 46, 362 47, 718 49, 097 50, 499 51, 927 53, 380 54, 860 56, 369 2 737 052 388 745 124 527 955 409 890 399 4 763 078 415 772 152 555 984 438 9 0 429 6 789 105 4●2 799 18● 583 52, 003 467 950 460 8 815 132 469 827 ●08 612 032 496 980 491 10 43, 842 45, 159 46, 496 47, 855 49, 236 50, 641 52, 071 53, 526 55, 010 56, 522 12 868 185 523 882 263 669 109 555 040 552 14 894 211 550 909 291 697 1 8 584 070 5●2 16 920 238 577 936 319 725 167 6 3 100 613 18 946 265 604 964 347 754 196 643 130 644 20 43, 973 45, 292 46, 631 47, 992 49, 375 50, 783 52, 215 53, 673 55, 160 56, 675 22 999 318 652 48, 019 403 811 244 702 190 705 24 44, 025 314 685 046 431 839 273 731 220 735 26 051 371 712 073 459 867 302 760 250 766 28 0●7 398 739 101 487 896 331 790 280 797 30 44, 104 45, 425 46, 766 48, 129 49, 515 50, 925 52, 360 53, 820 55, 310 56, 828 32 130 451 793 156 543 953 389 849 340 858 34 156 477 820 183 57● 981 418 878 3●0 888 36 182 504 847 210 599 51, 010 4●7 908 400 919 38 208 531 874 238 627 039 476 938 430 950 40 44, 235 45, 558 46, 902 ●8, 266 49, 65● 51, 068 52, 505 53, 968 55, 460 56, 981 42 261 584 9●9 293 683 096 534 997 490 57, 011 44 287 610 956 320 711 124 563 54, 026 520 042 46 313 627 983 348 739 153 592 ●56 500 073 48 339 664 47, 010 376 767 182 621 086 580 104 50 44, 366 45, 691 47, 037 48, 404 48, 795 51, 210 52, 650 54, 116 55, 61● 57, 135 52 392 717 064 431 823 238 679 145 641 165 54 418 744 091 458 8●1 266 708 174 671 196 56 444 771 118 486 879 295 737 204 701 217 58 471 798 145 514 907 3●4 766 231 731 248 60 44, 498 45, 825 47, 173 48, 542 49, 935 51, 353 5●, 795 54, 264 55, 762 57, 289 62 524 851 200 569 963 381 824 293 792 320 64 550 888 227 597 981 409 853 323 822 351 66 576 915 254 625 50, 009 ●38 882 353 852 382 68 603 932 281 653 037 467 911 383 882 413 70 44, 630 45, 959 47, 309 48, 680 50, 076 51, 496 52, 941 54, 413 55, 913 57, 444 72 656 985 336 707 104 524 970 442 943 474 74 682 46, 012 363 735 132 552 999 472 973 505 76 7 8 029 390 763 160 581 53, 028 502 56, 003 536 78 735 056 417 791 188 610 057 532 034 567 80 44, 763 46, 093 47, 445 48, 819 50, 217 51, 639 53, 087 54, 562 56, 065 57, 598 82 788 119 472 846 245 667 116 591 095 630 84 814 146 499 874 273 696 145 621 125 661 86 8●0 173 526 902 301 725 174 651 155 692 88 867 200 553 930 329 754 203 681 186 223 90 44, 894 46, 227 47, 581 48, 958 50, 358 51, 783 53, 233 54, 711 56, 217 57, 754 92 920 254 608 985 386 811 262 7●0 247 785 94 946 281 635 49, 013 414 840 291 770 277 816 96 972 308 663 041 442 869 320 800 307 847 98 999 335 691 069 470 918 350 830 338 878 hd 13 13 14 14 14 14 14 1● 15 15  
D 50 51 52 53 54 55 56 57 58 59 0 57, 909 59, 481 61, 088 62, 730 64, 412 6●, 134 67, 900 69, 711 71, 572 73, 486 2 940 512 120 763 446 168 935 747 609 524 4 971 544 152 796 480 203 971 784 647 563 6 58, 002 576 185 830 514 238 68, 007 8●1 685 602 8 033 608 218 864 5●8 283 043 858 623 641 10 58, 065 59, 640 61, 250 62, 897 64, 582 66, 308 68, 079 69, 895 71, 761 73, ●80 12 096 672 282 920 616 343 114 932 798 719 14 127 704 314 953 650 378 150 ●79 836 758 16 158 736 347 986 684 413 186 70, 016 874 797 18 189 768 380 63, 019 718 448 222 053 912 836 20 58, 221 59, 800 61, 413 63, 063 64, 753 66, 483 68, 258 70, 079 71, 950 73, 875 22 252 832 445 096 787 518 284 115 988 914 24 283 864 478 129 821 553 ●20 152 72, 026 953 26 314 896 511 163 855 588 366 199 064 992 28 345 928 544 197 889 623 402 236 102 74, 031 30 58, 377 59, 960 61, 577 63, 230 64, 924 66, 659 68, 438 70, 263 72, 140 74, 070 32 408 992 609 263 958 694 474 300 17● 119 34 439 60, 024 641 296 992 729 510 337 216 158 36 470 056 674 330 65, 026 764 546 37● 254 197 38 502 088 707 364 061 799 582 411 292 237 40 58, 534 60, 120 61, 7●0 63, 398 65, 096 66, 835 68, 618 70, 449 72, 331 74, ●67 42 565 152 77● 431 130 870 654 486 369 306 44 596 184 805 464 164 905 690 523 407 315 46 628 216 838 498 198 94● 726 560 445 384 48 659 2●8 871 532 233 975 762 597 ●83 424 50 58, 691 60, 280 61, 904 63, 566 65, 268 67, 011 68, 799 70, 635 72, 5●2 74, 464 52 722 312 937 599 302 046 835 ●72 560 503 54 753 344 970 632 336 081 871 719 598 542 56 784 376 62, 003 666 370 116 907 756 636 581 58 816 408 036 700 405 152 944 793 675 621 60 58, 848 60, 441 62, 069 63, ●34 6●, 440 67, 188 68, 9TH 7●, 821 72, 7●4 74, 661 62 879 473 102 7●7 474 223 69, ●17 8●8 752 790 64 910 505 135 801 508 ●58 ●53 895 790 739 66 942 537 161 8●5 513 ●●3 089 932 828 779 68 974 569 201 869 578 339 126 970 867 819 70 59, 006 60, 601 62, 234 63, 9●3 65, 61 67, 65 69, 163 71, ●08 72, 609 74, 859 72 037 633 267 936 6●7 ●00 199 045 944 898 74 068 665 300 970 681 435 235 082 982 937 76 100 697 333 64, 00● 716 471 271 119 73, 021 977 78 132 7●0 366 038 751 507 308 157 06● 75, ●17 80 59, 164 6●, 763 62, 399 64, 072 65, 786 67, 543 67, 345 71, 195 73, 099 75, 057 82 195 795 422 106 820 578 381 232 137 096 84 226 827 455 140 855 613 417 269 175 136 86 258 860 488 174 89● 6●9 454 307 214 176 88 290 892 521 208 925 685 491 345 253 216 90 59, 322 60, 925 62, 564 64, 242 65, 960 67, 721 62, 528 71, 383 73, 292 75, 256 92 353 957 597 276 994 756 564 420 330 296 94 385 979 630 310 66. 029 792 600 458 369 336 96 417 61, 012 663 344 064 828 637 496 408 376 98 449 045 696 378 099 864 674 534 447 416 hd 16 16 61 17 17 18 18 19 19 20  
D 60 61 62 63 64 65 66 67 68 69 0 75. 456 77. 487 79. 583 81. 749 83. 99● 86. 3●3 88 7●5 91. 2●2 93. 8●6 96. 575 2 494 528 625 793 84. ●3● 360 77● 283 899 6 0 4 533 5●9 667 837 081 407 823 331 952 686 6 571 6 1 710 881 127 451 872 385 9●. 00● 7●2 8 607 653 753 925 173 502 921 437 059 798 10 75. 650 77. 694 79. 796 81. 970 84. 219 86. 550 88 971 91. 489 91. 113 96. 854 12 691 735 838 82. 014 264 597 89. 020 540 166 910 14 732 776 881 058 310 644 069 59 220 969 16 773 818 924 1●2 356 692 119 643 284 97. 022 18 815 860 967 146 402 740 16● 694 338 078 20 75. 857 77. 901 80. 010 82. 191 84. 448 86. 788 ●9. 219 91. ●46 91. 38● 97. 135 22 897 94● 053 2●5 494 835 268 797 436 191 24 937 983 096 279 5●0 8●3 317 8 9 490 247 26 977 78. 025 139 323 586 931 367 901 544 304 28 76. 018 067 182 368 632 979 417 953 5●8 361 30 76. 059 78. 109 80. 225 82. 413 84. 678 87. 027 89. 467 92. 005 94. 652 97. 418 32 099 150 268 457 714 075 516 056 706 474 34 139 191 311 501 760 123 566 108 760 530 36 179 233 354 545 806 171 61● 160 814 597 38 210 275 397 590 852 219 666 212 868 654 40 76. 260 78. 317 80. 441 82. 635 81. 909 87. 267 89. 716 9●. ●64 9●. 923 97. 701 42 301 358 484 680 955 315 766 316 977 758 44 341 400 527 725 85. 001 363 816 368 95. ●31 815 46 382 442 570 770 ●47 411 866 420 085 872 48 423 484 613 815 094 459 916 472 140 929 50 76. 464 78. 526 80. 657 82. 860 85. 141 87. 508 89. 967 92. 525 95. 195 97. 986 52 50● 568 700 905 187 556 90. 017 577 249 98. 043 54 515 600 743 950 2●3 604 067 619 303 100 56 586 642 786 995 280 652 117 680 358 157 58 62● 684 830 83. 040 327 700 167 732 413 214 60 76. 667 78. 736 80. 874 83. 084 85. 374 87. 749 90. 218 92. 787 95. 468 98. 272 62 707 778 917 129 420 797 268 839 513 329 64 748 820 960 174 466 845 3●8 891 568 386 66 789 862 81. 004 219 513 894 868 944 623 444 68 830 904 047 264 . 560 943 419 997 678 502 70 76. 871 78. 947 81. 091 83. 3●0 85. 607 87. 992 90. 470 93. 050 95. 743 98. 560 72 912 989 134 355 654 88 040 520 102 798 617 74 953 79. 031 178 400 701 ●98 570 155 853 675 76 994 ●73 222 445 748 147 621 208 908 733 78 77. 035 115 266 490 795 196 672 261 963 791 80 77. 076 79. 158 81. 310 83. 536 85. 842 18. 235 90. 723 93. 314 96. 019 98. 849 82 117 200 353 581 889 284 774 367 074 907 84 158 242 397 626 936 333 815 420 129 965 86 199 284 411 671 983 381 876 473 185 99 023 88 240 337 485 717 86. 030 431 927 526 341 081 90 77. 281 79. 370 81. 529 85. 763 86. 077 88 480 90. 978 93. 579 96. 296 99 139 92 322 412 572 808 124 529 91. 028 632 351 197 94 363 454 616 853 171 578 079 685 407 255 66 404 497 660 898 218 627 130 738 463 313 95 446 540 704 944 265 676 181 79● 519 372 hd 21 21 22 23 23 24 25 26 27 28  
D 70 71 72 73 74 75 76 77 78 79 0 99, 431 102, 42 105, 58 108. 90 112. 43 116. 17 120. 16 124. 45 129. 08 134. 09 2 499 48 64 97 50 24 24 54 18 19 4 557 54 71 109. 04 58 32 32 6● 28 30 6 616 6● 77 11 65 40 40 72 38 41 8 675 66 84 18 72 48 49 81 48 51 10 99, 724 102, 73 105, 90 109. 25 112. 79 116. 56 120. 58 124. 90 129. 56 134. 62 12 785 79 97 32 86 63 66 99 65 72 14 846 85 106, 03 38 93 71 74 125. 08 74 82 16 9●7 91 10 45 113. 01 79 82 17 84 93 18 968 97 17 52 08 87 91 26 94 135. 04 20 100LS, 02 103, 04 106, 23 109. 59 113. 16 116. 95 121. 00 125. 35 130. 04 135. 15 22 07 10 30 66 23 117. 02 08 44 14 25 24 13 16 36 73 30 10 16 53 24 36 26 19 22 43 80 38 18 24 62 34 47 28 25 28 50 87 45 26 33 71 44 58 30 100, 31 103, 35 106, 56 ●09. 94 113. 52 117. 34 121. 42 125. 80 130. 54 135. 69 32 37 41 62 110. 01 60 42 50 89 64 79 34 43 47 69 ●8 67 50 58 98 73 90 36 49 53 76 15 74 58 66 1●6. 08 83 136. 01 38 55 60 83 28 81 66 75 17 9● 12 40 100, 61 103, 67 106, 89 110. 29 113. 89 117. 74 121. 84 126. 26 131. 03 136. 23 42 67 73 95 36 96 82 92 35 13 3● 44 73 79 107, 02 43 114. 03 90 122. 0● 45 23 45 46 79 85 09 5● 11 98 09 54 33 56 48 85 91 16 57 19 118. 06 18 63 43 67 50 100, 91 103, 98 107, 22 110. 64 114. 27 118. 14 122. 27 126. 72 131. 53 136. 78 52 97 104, 04 28 71 34 22 35 81 63 89 54 101, 03 1● 35 78 42 30 43 90 73 137. 00 56 09 1● 41 85 49 38 52 127. 0● 83 11 58 15 22 48 92 56 46 61 09 93 22 60 101, 21 104, 23 107, 55 110, 99 114. 64 118. 54 122. 70 127. 18 132. 03 137. 33 62 27 35 62 111. ●6 71 62 78 27 13 44 64 3● 4● 68 13 78 70 86 37 23 55 66 39 48 75 20 86 78 95 45 33 66 68 45 54 82 27 94 8● 123. 04 55 43 77 70 101, 51 104, 61 107, 89 111. 35 115. 02 118. 94 123. 13 127. 65 132. 54 137. 88 72 57 67 95 42 09 119. 02 22 74 64 99 74 63 73 108, 02 49 17 10 30 83 74 138. 10 76 69 80 0● 56 24 18 39 92 84 21 78 7● 86 16 63 32 26 48 1●8. 02 95 33 80 101, 81 104, 93 108. 22 111. 71 115. 40 119. 35 123. 57 128. 12 133. 06 138. 45 82 87 99 29 78 47 43 65 21 16 56 84 9● 105, 05 35 85 55 51 74 30 26 67 86 99 12 42 92 6● 59 83 40 37 78 88 102, 0● 18 49 99 70 67 92 50 47 89 90 102, 1● 100L, 25 108. 56 112. 06 115. 78 119 76 124. 01 128. 60 133. 57 139. 01 92 18 31 64 13 85 84 09 69 67 1● 94 24 37 70 21 93 92 18 79 77 23 66 30 4● 77 28 116. 01 120. 00 27 88 87 3● 95 36 51 84 35 09 ●● 36 98 98 47 hd 3 ● 3 3 4 4 4 5 5 139. 5●   

THE MARINER'S PLAIN SCALE NEW PLAINED: THE SECOND BOOK.  
Showing how by a Line of CHORDS only, to resolve all the Cases of Spherical Triangles Orthographically, that is, By projecting or laying down the Sphere in right Lines, commonly called, The Drawing or Delineating of the ANALEMMA.  
Of great Use to Seamen and Students in the Mathematics.  
Being contrived to be had either alone, or with the other Parts.  
Written by John Collins of London, Penman, Accountant. Philomathet.  
LONDON: Printed by Tho. Johnson for Francis Cossinet, and are to be sold at the Anchor and Mariner in Tower-street, and by Hen. Sutton Mathematical Instrument-maker in Thread-needle-street, behind the Royal Exchange. 1659.   

Courteous Reader,  
THis Book being the Second of that which is Entitled, The Mariners Plain Scale new Plained, I thought fit so to contrive (for the ease of the Buyer, and advantage of the Stationer, who find small Bulks more convenient for Sale then great) that it might be had alone by itself at an easy rate: What Definitions or Rudiments of Geometry are here wanting, the Reader will be supplied withal in the first Part, and though performances of this nature are not so exact as Calculation, yet they will be a good Introduction to the ignorant, a Confirmation to the Studious, who use Tables that may be liable to Mistakes at the Press, and may very well serve where better helps are wanting: The Demonstration of the Analemma, and other ways of performance by it, are passed by here, and intended to be handled in a small Treatise of the use of it as a Mathematical Instrument, to be cut in Brass, with Paper Prints fitted up for Sale, which will be of excellent use to Seamen, Surveyors, and all that are Mathematically Studious: I remain thy friend, desirous of the Advancement of Knowledge,  
JOHN COLLINS.   

THE CONTENTS.  

A Double Scale of Chords used in this Book, and described. Page 1, 2.  
Spherical Definitions from page 2 to 9  
All the Points, Arks and Circles defined, represented to the view in a Scheme of the Analemma. p. 10 to 14  
A general Almanac in two Verses p. 15  
The manner of measuring and proportioning out Sins by a Line of Chords. p. 18, 19  
A general Rule in two Verses for finding the Sun's place. 21  
To find the Sun's Declination and right Ascension. 23, 24  
To find the Sun's amplitude, height at Six, Vertical height, time of rising, etc. 25, 26  
The sixteen right angled Cases resolved by proportions of four several kinds. p. 27 to 32  
To find the Sun's height without Instrument. 32  
To find the Hour, Azimuth, and Angle of Position. p. 33 to 39  
To find the Sun's Altitudes on all Hours. 40, 41  
To find the Distances of Places, etc. 42, 43  
All the Obliqne Cases solved. p. 46. p. 53  
To find the Altitudes on all Azimuths. p. 46 to 52  
All the sixteen Cases of right angled Spherical Triangles projected, and otherwise resolved. p. 57 to 63  
The Longitude and Latitude of a Star given, to find its Declination and right Ascension p. 64  
Two Azimuths and two Altitudes given, to find the Latitude and Declination by Projection. p. 65  
With Proportions to find the same by Calculation. p. 66, 67   
Page 2. Line 17. Obliterate Superficies.   

OF THE SCALE Used in this Book.  
THough this Treatise bears the name of the Mariners Plain Scale new Plained, yet the Scale intended thereby, is cut in the Frontispiece of the first Book.  
Nothing more is necessarily required in the performances of this Book, then from the commonly known Division of a Circle into 360 equal parts, called Degrees, to prick down any number of Degrees less than 180d, and a quadrant divided into nine equal parts, and one of those parts for convenience below the Diamater into 10 Sub-divisions, called Degrees, may very well serve the turn, which the Readers ingenuity will furnish himself withal in any place, if he have but Compasses, yet for expedition a line of Sines is often made use of.  
In the following Diagram, the equal Divisions of the Semicircle, are transferred into the line of Chords in the Diameter, by setting one foot of the Compasses in A, which is called the lesser Chord.  
Those Chords being numbered by the half Arches, to which they belong, become a line of Sins of the same Radius with the Diameter of the Semicircle, and are called the greater Sins.  
To that Radius there is fitted a Chord of 60d, called the greater Chord, that the Reader might be supplied with both, for the lesser Chord will not serve to prick off an Arch in a Circle of twice the Radius whereto it is fitted, unless the said Chord be doubled in a right line before it be pircked into the Circumference.  
The Sins to the Chord in the Diameter, are graduated on the Radius or line C B, by drawing lines through each degree of the quadrants AB, DB, and are called the lesser Sins.  
There is also added a Scale of equal parts and Rumbes for other Protractions, but we use neither of them in this Book.  
The Schemes in the Book are fitted either to the lesser or greater Chord here described.  
Every degree of a Line of Sines, or Chords, we suppose to be divided into 60 parts, which we call minutes, which in the following Operations are guessed at, for a small Instrument will not admit of so many Sub-divisions.  

CHAP. I. Spherical Definitions.  
BEfore we proceed to the Resolution of any particular questions, it will be necessary to premise the common Spherical Definitions, and to show how the Analemma repesents them.  
The word Sphere, though, as Herigonius showeth, it be taken in a fourfold sense, yet I think it not necessary to define above two of them.  
1. Therefore a Sphere may, according to Theodosius, be understood to be a solid Superficies, or round Body, contained under one Surface, in the middle whereof there is a point whence all right lines drawn unto the Circumference, are equal, and is made by the turning round of half a Circle, till it end where it began.  
2. It is taken for a certain round Instrument consisting of divers Circles, whereby the motions of the Heavens, and the Situation of the whole World, is most conveniently represented.  
For the better explanation whereof, Astronomers do imagine, 10 Principal points, and 10 Circles to be in the hollow inside of the first movable Sphere, which are commonly drawn upon any Globe or Sphere, besides divers other Circles which are not delineated, but only apprehended in the fancy.  
The Points are the two Poles of the World, the two Poles of the Zodiac, the two Equinoctial points, the two Solstitial points, and the Zenith and Nadir.  
The Poles of the World are two points, which are Diametrically, or directly opposite to one another, about which the whole frame of Heaven moveth from the East into the West, whereof one is perpetually seen by us, and is called the Arctic, or North Pole.  
The other being hid from us, and directly opposite to the former, is called the Antarctick, or South pole, a right line imagined to be drawn from the one Pole to the other, is called the Axis, or Axletree of the World.  
The Axis differs from the Diameter, because that every right line drawn through the Centre of the Sphere, and limited on each side of the Surface of the Sphere, is a Diameter; but not an Axis, unless the Sphere move round about it.  
The Poles of the Zodiac, are two points Diametrically opposite, upon which the Heavens move from the West into the East, one of them is towards the North, distant from the Arctic or North Pole, 23 degrees 31 minutes; the other is towards the South, and is as much distant from the South Pole.  
A Degree is the 360th part of any whole Circle, and a Minute is the 60th part of a Degree, but of late some divide a Degree into 100 parts, which are called Centesmes, or Centesimal Minutes, in defining some of the Points we must refer to Circles afterwards to be defined.  
The Equinoctial Points are in the beginnings of Aries and Libra, to which when the Sun cometh, he makes the day and night of an equal length throughout the whole World, to wit, in the beginning of Aries about the 11th of March, which is accounted the beginning of the Spring, and in the beginning of Libra, about the 13 of September, which is the beginning of Autumn.  
The Point of the Summer Solstice, is in the beginning of Cancer, to which when the Sun cometh, as about the 12th of June, is the beginning of Summer, and the longest day in the year.  
The Point of the Winter Solstice, is in the beginning of Capricorn, to which when the Sun cometh, as about the 11th of December, is the shortest day in the year, and in the Astronomical account, the beginning of Winter.  
The Zenith, is an imaginary point in the Heavens, right over our heads 90d from the Horizon.  
The Nadir, is a point or prick in the Heavens under our feet, opposite to the Zenith.  

Of the Circles of the Sphere.  
The 10 Circles are the Horizon, the Meridian, the Equinoctial, the Zodiac, the Colour of the Equinoxes, the Colour of the Solstices, the Tropic of Cancer, the Tropic of Capricorn, and the two Polar Circles.  
The first six are called great Circles, and the other four lesser Circles.  
By the Centre of a Circle, is meant a Point, or Prick in the middle of the Circle, from whence all Lines drawn to the Circumference, are equal, and are known by the name of Radius resembling the spoke of a Cartwheel.  
That is said to be a great Circle which hath the same Centre as the Sphere, and divides it into two equal halfs, called Hemispheres, and that is called a lesser Circle, which hath a different Centre from the Centre of the Sphere, and divides the Sphere into two unequal Portions or Segments.  

1. Of the Horizon.  
The Horizon is distinguished by the names of Rational or Sensible, the Rational Horizon is a great Circle every where equidistant from the Zenith, and divides the upper Hemisphere from the lower, and by accident or chance, is distinguished by the names of a right Obliqne and parallel Horizon.  
A right Horizon is such a Horizon as passeth through each Pole of the World, and cuts the Equinoctial at right Angles, whence the Inhabitants under the Equinoctial, are said to have a right Horizon, and a right Sphere.  
An Obliqne Horizon is such a one as cuts the Equinoctial obliquely, or aslope.  
A parallel Horizon is not such a one as cuts the Equinoctial, but is coincident, and is the same therewith, and such is the Horizon under the Poles.  
The sensible Horizon is a Circle dividing that part of the Heavens which we see, from that part which see not, thence called Finitor.  
From the Accidental Situation of the Horizon, follows many consequences.  
1. Those that live in a right Horizon, that is under the Equinoctial, have their days and nights always of an equal length, to them all the Stars both rise and set, twice in a year the Sun passeth through their Zenith, consequently they have two Summers and two Winters, to wit, Summers when the Sun passeth through their Zenith; and Winters when he is in or near the Tropics.  
2. In any right or Obliqne Sphere, the length of the day when the Sun is in the Equinoctial, is equal to the length of the night.  
3. In any Obliqne Sphere, the nearer the Sun approacheth to the Visible Pole, the longer are the Days more than the Nights, some Stars always appear, others never appear, and the more remote from the Equinoctial, the greater is the number of such Stars, and the more inequality is there between the Days and Nights.  
4. To those that live under the Polar Circles, their day once a year is 24 hours long, and their Night nothing.  
5. Under the Pole, one half of the Sphere doth always appear, and the other half not appear, and one half of the year is well nigh continually Day, and the other half continually Night, because the Equinoctial lies in the Horizon; 'Tis said well nigh, for by reason of the Sun's Excentricity, the day under the North Pole, is longer than the Night, about eight days; and on the contrary under the South Pole, is shorter than the night as many days.   

2. Of the Meridian.  
The Meridian is a great Circle, which passeth through the Poles of the World, the Zenith and Nadir, and the North and South points of the Horizon, and is so called, because that at all times and places when the Sun by his daily motion cometh unto that Circle twice every 24 hours, maketh the middle of the day and middle of the night; all places that lie under the same Meridian, bear North and South, but places that lie East and West from one another, have each of them a several Meridian.   

3. Of the Equinoctial.  
It is a great Circle imagined in the Heavens, dividing them into two equal parts, or halfs, called the North and South Hemisphere, lying just in the middle between the two Poles, being every where equi-distant from them, and is called the Equator, because when the Sun cometh unto it, which is twice in the year, at his entrance into Aries and Libra, the days and nights are of an equal length throughout the whole World.   

4. Of the Zodiac.  
The Zodiac, alias Signifer, is another great Circle that divides the Equinoctial into two equal parts, the Points of Intersection being called Aries and Libra, the one half of it doth decline into the North, the other half into the South, as much as the Poles thereof are distant from the Poles of the World, namely 23d 31′, and likewise passeth through the two Solstitial Points, it's ordinary Breadth or Latitude is 12 degrees, but late Writers make it 14 or 16d by reason of the wander of Mars and Venus.  
A Line, dividing the breadth thereof into two halfs, is called the Ecliptic Line, because the Eclipses of the Sun and Moon are always under that Line, it's Circumference is divided into 12 parts called the 12 Signs, whereof each containeth 30d.  
The Names and Characters of the 12 Signs, are  
Aries ♈ Taurus ♉ Gemini ♊ Cancer ♋ Leo ♌ Virgo ♍ Libra ♎ Scorpius ♏ Sagittarius ♐ Capricornus ♑ Aquarius ♒ Pisces ♓  
The six former are the Northern, and the six latter the Southern Signs.    

Of the Colours.  
These are two great Circles, and are no other than two Meridian's passing through both the Poles of the World, crossing one another therein at right Angles, and divide the Equinoctial and the Zodiac into four equal parts, making thereby the four Seasons of the year.  
The Colour of the Equinoxes is so called, because it passeth through the Equinoctial points of Aries and Libra, showing thereby the beginning of the Spring and Autumn, when the days and nights are equal.  
The other Colour passeth through the two Solstitial or Tropical Points of Cancer and Capricorn, showing the beginning of the Summer and Winter, at which two times the days are longest and shortest.  
The very beginning of Cancer where the Colour crosseth the Ecliptic line, is called the Point of the Summer Solstice, to which place when the Sun cometh, he can approach no nearer the Zenith, but returneth towards the Equinoctial again, the Arch of the Meridian or Colour contained betwixt the Summer Solstice and the Equator, is called the greatest Declination of the Sun.   

Of the four Lesser Circles.  
The Tropics are two lesser Circles, parallel to the Equinoctial, limiting the Sun's greatest Declination towards both the Poles; that towards the North Pole is called the Tropic of Cancer, because the Sun being in the very point of entrance into Cancer, which is the nearest he can approach unto the North Pole, is then in the point of the Summer Solstice, and by his diurnal or daily motion, describes a parallel, from thence called the Tropic of Cancer.  
The Tropic of Capricorn, likewise limiteth the Sun's greatest Declination Southward, and is a lesser Circle parallel to the Equinoctial, and hath that Denomination put upon it, because it passeth through the beginning of Capicorn, and hath the like reference to the South Pole, as the Tropic of Cancer, hath to the North Pole.   

Of the two Polar Circles.  
These are two lesser Circles, distant so much from the Poles of the World, as is the Sun's greatest Declination from the Equinoctial; in these Polar Circles, are the Pole points of the Zodiac, which moving round the Poles of the World, describe by their motion the said two Circles; that about the North Pole, is called the Arctic Circle, and that about the South Pole, the Antarctic Circle.   

Of other Circles imagined, but not described in a material Sphere or Globe.  
Such are the Azimuths, Almicanteraths, parallels of Latitude and Declination.  
Azimuths are all great Circles bisecting the Sphere which meet together in the Zenith, and may be imagined to pass through every degree and minute of the Horizon at right Angles thereto, and serve to find the true coast of bearing of the Sun or Stars at at any time, in respect of the four chief Coasts of the Horizon, East, West, North, South. By some they are termed Vertical Circles, because they pass through the Zenith, but then they call the Azimuth of East, or West, the prime Vertical.  
The Sun or any Star having Elevation or Depression above or below the Horizon, are then properly said to have Azimuth; But if they be in the Horizon, either Rising or Setting, the Arch of the Horizon, between the Centre of the Sun or Star, and the true Points of East or West, is called Amplitude.  
Almicanteraths are Circles parallel to the Horizon, continued up even to the Zenith, and serve to measure the Altitude, or height of the Sun, Moon, or Stars above the Horizon, which is no other than a portion or Arch of an Azimuth contained betwixt that Almicanter which passeth through the Centre of the Sun or Star and the Horizon.  
Parallels of Declination are lesser Circles, all parallel to the Equinoctial, and may be imagined to pass through every degree, & part of the Meridian, and are described upon the Poles of the World.  
Those parallels which in respect of the Sun or Stars, are called parallels of declination in respect of the Situation of the earth, are called parallels of Latitude.  
The Latitude of a Town or Place, is measured by the Arch of the Meridian, between the Zenith of that place, and the Equinoctial, or which is equivalent thereto, by the Arch of the Meridian of the place between the Elevated Pole and the Horizon.  
In like manner the Declination of the Sun or any Star, is measured by the Arch of the Meridian, between the Sun or Star, and the Equinoctial.  
Parallels of Latitude in the Heavens, are all lesser Circles described upon the Poles of the Zodiac or Ecliptic, and serve to define the Latitude of a Star, which is the Arch of a Circle, contained betwixt the Centre of any Planet or Star, and the Ecliptic Line, making right Angles therewith, and counted either towards the North or South Poles of the Ecliptic; the Sun never passing from under the Ecliptic Line, is said to have no Latitude.  
Longitude in the Heavens, is measured by the Arch of the Ecliptic, comprehended between the Point of Aries, and a supposed great Circle or Meridian of Longitude, passing through the Centre of the Sun or Stars, and the two Poles of the Ecliptic, but counted according to the order or succession of the Signs.  
Longitude on the Earth, is measured by an Arch of the Equinoctial, contained between the primary or first Meridian of any place where Longitude is assigned to begin, and the Meridian of any other place, but counted Eastward from the said first place, according as the right ascension is counted in the Heavens.  
Right Ascension, is an Arch of the Equinoctial (counted from the beginning of Aries) which cometh to the Meridian with the Sun, Moon, or Stars, or any portion of the Ecliptic; this is so useful, that Tables thereof are made both for the Sun and Stars, whereby is known the true time when they come to the Meridian, also by help of the Stars hour, the true time of the Night.  
Olique Ascension, is an Arch of the Equinoctial, between the beginning of Aries, and that part of the Equinoctial that riseth with the Centre of a Star, or any portion of the Ecliptic, in an Obliqne Sphere.  
Ascensional Difference, is the Ark of difference between the right Ascension and the Obliqne Ascension, and thereby is measured the time of the Sun or Stars rising before, or after six.    

CHAP. II. Showing how the Analemma represents the Points and Circles before described.  
IN this Scheme are represented the Points and Circles of the Sphere before described, fitted for the Latitude of London.  
Upon the Centre C, with 60d of a Line of Chords, draw the Circle S Z O N.  
Draw the Diameter S C O, and perpendicular thereto cross it with another Diameter Z C N.  
From S to A, as also from Z to P, prick off 38d 28′ out of a Line of Chords, and draw A C E and P C A.  
From A to F and X, also from E to D and Y, prick off 23d 31′, with Chords, do the like from P to R and Q, as also from A to T and V, and through those Points draw the Lines R Q, F D, X Y, and V T.  
From F through the Centre, draw the Line F C Y.  
Parallel to S O through the point G, draw H G also parallel thereto, at any other distance draw L M B.  

In this Scheme are represented the Points before defined.  
P the North pole or pole Arctic, A the South pole or pole Antarctick.  
Q the North, and V the South pole of the Ecliptic.  
C both the Equinoctial points of Aries and Libra.  
F the point of the Summer Solstice.  
Y the point of the Winter Solstice.  
Z the Zenith, N the Nadir.   

Secondly, the greater Circles are there represented.  
S C O the Horizon, and Z C N the Axis thereof, or Azimuth of East and West.  
S Z O N the Meridian, it represents also the Colour of the Summer and Winter Solstices.  
A C E the Equinoctial.  
F C Y the Ecliptic.  
A C P represents the Colour of the Equinoxes, as also the Axis of the World, and the hour Circle of six.   

Thirdly, the lesser Circles are there represented.  
F D the Tropic of Cancer, X Y the Tropic of Capricorn.  
R Q the Arctic or Polar Circle about the North pole.  
V T the Antarctick Circle, or Circle about the South pole.   

Fourthly, other Circles not described upon Globes are there represented.  
L B represents a parallel of Altitude called an Almicanterath.  
The pricked Arches Z ⊙, and Z G K being Ellipses, represent the Azimuths or Vertical Circles.  
And the other pricked Arches,  
Represent Meridans or hour Circles, which are also Ellipses, the drawing whereof would be troublesome, and therefore is not mentioned, and how to shun them in the resolution of any Proposition of the Sphere by Chords, shall afterwards be showed.  
Any line drawn parallel to A E, as is f p, F D, R Q, will represent parallels of Declination.  
And any Line drawn parallel to F Y, will represent a parallel of Latitude in the Heavens.  
Fifthly, divers Arches relating to the Sun's Motion, such as are commonly found by the Globes or Calculation, are in the same Scheme represented in right Lines.  
1. The Sun's Amplitude or Coast of rising and setting from the East or West, is there represented C W in North Signs, and by C g in South Signs.  
2. His Ascensional difference or time of rising from six in Summer by G W, in Winter by g h.  
3. His Altitude at six in Summer by H C, his Depression at six in Winter by C b.  
4. His Azimuth at the hour of six by H G in Summer, equal to h b in Winter.  
5. His Vertical Altitude, or Altitude of East and West by I C, his Depression therein in Winter by C q.  
6. His hour from six being East or West in Summer by G I, in Winter by h q.  
7. His Azimuth from the East and West upon any Altitude, is represented in the parallel of Altitude, where it intersects the parallel of Declination here by M ⊙.  
8. The hour of the day from six to any Altitude, is represented in the said point of Intersection, but in the parallel of Declination here by G ⊙, and all these Arches thus represented in right Lines, are the Sins of those Arches to the Radius of the parallel in which they happen, being accounted from the midst of the said parallel.  
Now how to measure the quantities of these respective Arches by a Line of Chords, and consequently thereby to resolve all the cases of Spherical Triangles, is the intended subject of some following Pages.  
The former Arches thus represented in right Lines, many whereof fall in parallels or lesser Circles, when Calculation is used, are all represented by Arches of great Circles (namely such as bisect the Sphere) and the former Scheme doth represent the Triangles commonly used in Calculation.  
Thus the right angled Triangle C d y, right angled at d supposing the Sun at y is made of  
C y The Sun's place or distance from the nearest Equinoctial point.  
C d his right Ascension.  
Y d his Declination.  
d C y the angle of the Ecliptic and Equinoctial.  
C y d the angle of the Sun's Meridian and Ecliptic.  
In the right angled Triangle W O P, right angled at O, supposing the Sun at W.  
O P is the poles Elevation.  
P W the compliment of the Sun's Declination.  
W O the Sun's Azimuth from the North.  
W P O the hour from Midnight, or compliment of the Ascensional difference.  
P W O the angle of Position, that is, of the Sun's Meridian with the Horizon, and of the like parts or their compliments is made the Triangle C m W.  
In the right angled Triangle C K G, right angled at K, supposing the Sun at G.  
C G is his Declination.  
G K his height at the hour of six.  
C K the Sun's Azimuth from the East or West, at the hour of six.  
K C G the angle of the Poles Elevation.  
C G K the angle of the Sun's position.  
In the right angled Triangle C k I, right angled at k, supposing the Sun at I.  
I k is the Sun's Declination.  
C k his hour from six.  
C I his height, being East or West.  
k C I the Latitude.  
k I C the Angle of the Sun's position.  
In the obliqne Angled Triangle Z ⊙ P, if the Sun be at ⊙.  
Z P is the Compliment of the Latitude.  
P ⊙ his distance from the elevated Pole, in this Case the compliment of his Declination.  
Z ⊙ the Compliment of his Altitude or height.  
Z P ⊙ the Angle of the hour from Noon.  
P Z ⊙ the Sun's Azimuth from the North or midnight Meridian.  
Z ⊙ P the Angle of the Sun's Position.  
Thus we have showed how the former Scheme represents the Spherical Triangles used in Calculation, whereby of the six parts in each Triangle, if any three are given, the rest may be found by Calculation from the Proportions, and that either by Multiplication and Division, when the natural Tables of Sines and Tangents are used, or by Addition and Substraction when the Logarithmical are used, and what is performed by either of those sorts of Tables, we shall here perform by Scale and Compass, from which performances the like measure of exactness, is not attainable as from the Tables.    

CHAP. III. Showing how to know upon what day of the Week, any day of any Month happens upon for ever.  
1 TO perform this Proposition, there must be a general Rule prescribed, to find on what day of the Week, the first of March will happen upon for ever, which take in the following Verses.  

To number two, add year of our Lord God,  
And a fourth part thereof, neglect the odd  
Remainder, if such be, the sum divide  
By seven, lay your quotient aside,  
The Rest when your Divisions finished,  
Will number show day of the Week you need,  
On which the first of March doth chance to be,  
Still counting Lords day first, if you do see  
That nothing do remain, than you may say,  
The day you seek's the seventh, and Satur's day.   
Example.  
Let it be required to find on what day of the Week the first of March will happen, in the year of our Lord 1687.  

Operation.  Divisor 7) 2 The even fourth of the Year, The remainder neglected, (301 Quotient. 1687 421 2100 21 10 7 3 remains.  
Because three remains, the first of March in that Year happens on a Tuesday, in the Year 1679 nothing remains, therefore it happens on a Saturday.  
Proposition. 2. The day of the Week, on which the first of March happens on any Year, being known and remembered.  
To find on what day of the Week, any day of any Month in the said Year happeneth.  
To perform this Proposition, the following Verse being in effect a perpetual Almanac is to be recorded, & laid up in Memory.  

All evil chances, grievous evils bring,  
Fierce death attends, foul chances governing.   
In this Verse are twelve words, relating to the number of the twelve Months of the Year, accounting March the first, wherefore the word proper to that Month is All, and so in order Fierce is the seventh word, and therefore belongs to the seventh Month, or September.  
That which is to be observed from these Words, is what letter the word beginneth withal, and to count the number of that letter in the order of the Alphabet which will never exceed seven, and the number of the said letter, shows what day of the Month proper to the said word, shall be the same day of the Week the first of March happened upon.  
Example.  
The word Fierce belongs to the Month September, and the first letter of the said word being f, is the sixth letter of the Alphabet, a, b, c, d, e, f, wherefore the sixth day of September, is the same day of the Week that the first of March happened upon, which in the year 1687 will be on Tuesday, and then by adding perpetually seven, we may find on what days of the Month all the Tuesdays in that Month will happen:  
Thus the 

6  
13  
20  
27   days of September in the Year 1687 will be all of them Tuesdays, then if it were required to know on what day of the Week the 29 of September, or quarterday would happen, we might conclude it to be on a Thursday, because the 27th day happened on a Tuesday; and the like for the Month of December, the word proper whereto is Foul, and then in that Year because the 27th day is Tuesday, we may conclude the 25th day being Christmas day, to happen on the Lord's Day.  
The day of the Week being given, to find what day of what Month it is.  
The Month must be given, and also the number of the Week in the said Month, as whether it be the first, second, third or fourth Week in the Month, otherwise no Almanac whatsoever can resolve this Proposition.  
Wherefore let it be required to know what day of the Month Friday, in the third Week of January, happens in the year 1687, the word proper to January, is Chances, beginning with the third letter of the Alphabet, wherefore the 

3  
10  
17   days of that Month are Tuesdays, wherefore Friday, in the third Week of that Month, happens on the 20th day of that Month, because the 17th was Tuesday.  
The foundation of which Almanac is this, that the 

1 March  
5 April  
3 May  
7 June  
5 July  
2 August  
6 September  
4 October  
1 November  
6 December  
3 January  
7 February   Do for ever happen in the same year (which always beginneth the first of March, and ends on the last of February) on the same day of the Week, which every year varieth, as doth the first of March, for the more ready finding whereof, this Proposition may be added.  
The day of the Week on which any day of any Month happened, being known; To find on what day of the Week, the first day of March happened that year.  
Example: In the year 1660, admit it were known, or should be remembered that the 21 of May happened on Monday, the third Word of the Verse being the Word proper to May, is Chances, the first letter whereof is C, the third letter of the Alphabet, wherefore the 

3  
10  
17  
24   days of the same month happen on the same day of the week that the first of March happened upon; now the 24th day of that month will be Thursday, because the 21 day was Monday, wherefore the first of March happened that year on a Thursday.   

CHAP. IU. Showing the nature of Sines.  
AN Arch being given to take out the Sine or Cousin thereof to the common Radius, whereby the defect of a Line of Sines is supplied, if one be not at hand.  
Draw the two Radii A C, and C P, making right Angles at C the Centre, then with 60d of the Chord upon the said Centre, describe the Quadrant A P, and thereon prick off the Ark propounded: admit 50d from A to F, the nearest distance from F to A C, is the Sine of 50d, and the nearest distance from P C is the Cousin, to wit, the Sine of 40 degrees.  

To prick off an Arch or Angle by Sines in stead of Chords, or to find what Arch belongs to any Sine proposed.  
Admit I would prick off an Arch of 50 degrees by a Line of Sines, first with the Sine of 90 degrees the Radius, draw the Quadrant A P C, then with the Sine of the Arch proposed, upon A as a Centre, describe the Arch A B D, a Line drawn from C the Centre into the Limb so as to touch the extremity of that Ark at B, shall make an Angle of 50 degrees with the Line A C, the Arch A F being an Arch of 50d, and would be found so much, if measured on the greater Chord.  
But when an Arch to be pricked off by Sines is great, it were best to prick off the compliment thereof from the other Radius, thus upon P as a Centre, with the Sine of 40d, describe the Arch K, a line from the Centre touching that Arch, finds the Point F in the Limb, as before.  
Otherwise,  
Prick the Sine of 50d from C to E, and with the said extent upon A, as a Centre, describe an Arch at A, a line drawn from E, touching the extremity of the Arch at A, will pass through the point F in the Limb, as before, and the same point might have been found by pricking the Sine of 40d on the other Radius C A, after the same manner.  
Thirdly, the extent C E, so entered in the quadrant, that one foot resting thereon may but just touch A C, finds the point F as before.   

To proportion out a Sine to a lesser Radius.  
Admit I would make F E a line of 90d Sines, and to that Radius would prick down the Sine of 20d; First, prick the Radius E F from C to R, then prick off 20d from the Chords, from A to H, and draw the line H C into the Centre, the nearest distance from R to the said line, is the Sine of 20d to the Radius F E, and is equal to E L.   

Otherwise without drawing li●es from each respective Arch into the Centre.  
Draw the line F C into the Centre, and from F set off a quadrant each way to M, and N, and from those Points prick off the Arch proposed to S and T, a ruler laid over those Points cuts the line F C at G, and the nearest distance from G to C E, is equal to E L, as before.  
But when a line of Sines is at hand, these quadrants and Arches need not be pricked off, only set off the Sine of 20d to the common Radius from C to G, and the nearest distance from G to C E, is the Sine required, equal to E L, as before.   

A Sine being given in a parallel or lesser Radius, to reduce it to the common Radius, and thereby find to what Arch it belongs.  
Let E L be the Sine of an Arch in the parallel F E, prick F E from C to R, and upon R as a Centre, with the extent L E, describe the Arch V, a ruler laid from the Centre, touching the extremity of that Arch, finds the Point H in the Limb, and the Arch A H measured on the Chords is 20d, being the Arch proper to L E, and this is the best way when an Arch is not very great, and is the same used in finding the Sun's right ascension.   

Otherwise, by Sines.  
Prick E L from C to I, a ruler laid from I to L cuts the line F C at G, and the extent C G measured on the Sins, is 20d, as before.   

Another way to find it in the Limb.  
Through the point L, draw L I parallel to E C, then with the Radius of the parallel F E, setting one foot of that extent in C, with the other cross the line L I, as at y, a ruler laid from C to y, finds the Point X in the Limb, and the Arch X P is 20d, the measure of L E required, here having a Sine, and the Radius I y, is the Cousin according to the Definitions of Sines, and therefore subtends its opposite Angle, the Arch whereof is measured by X A elsewhere in a Treatise of dialing, we have by the converse hereof, having the Sine, and Cousin of an Arch given, found the Radius thereto, by this way to Porportion out a Sine to any Radius, is no other in effect then the same with the first way, for y C here, is equal to R C there.    

CHAP. V Showing how to resolve those Propositions that require the knowledge of the Sun's place.  
THe Day of the Month being known, to find the Sun's true place.  
This Proposition is propounded in the first place, because many others depend upon it, for that being given, his Declination will be easily attained, and this is necessary to be insisted on, because a Table thereof may not be at hand.  
Here, according to the Hipothesis of Tycho, it is to be suggested that there is ascribed to the Sun a triple motion; First, A motion upon his own Centre, whereby he finisheth one revolution in 26 days time.  
2. A daily motion from the East into the West, whereby he causeth the day and Night.  
3. An opposite motion from the West into the East, called his annual motion, whereby he runs once round in a year through the whole Ecliptic, moving near a degree in a day, and thereby causeth the several Seasons of the year: these two latter motions are fancied out unto us by a man turning a Grindstone 365 times round, while a worm struggling against, and contrary to that motion, creeps once round the contrary way, but obliquely and aslope, that is, from the further side of the Grindstone towards the hithermost, and by this motion the Sun is supposed to describe the Ecliptic Line, and continually to insist in this course; the other Planets, except the Moon, moving round him, and following after him like Birds flying in the air, being subject to his motions, and divers of their own besides, many of which motions are removed by the Copernican supposition of the earth's motion, which is a subject of much controversy among the learned, however be it either the one or the other, the Propositions hereafter resolved, vary not by reason thereof.  
And so the Sun being supposed not to vary from under the Ecliptic, in respect of Latitude, the Proposition or query in effect is, what Longitude he hath therein; which for the present purpose need only be known from the nearest Equinoctial point, now this may be found within a degree by a following verse newly framed for this purpose, which may serve for some ages, the old Latin Distich being now out of date and erroneous.  
Evil ♈ attends ♉ its ♊ object, ♋ unveiled ♌ vice ♍ Vain ♎ villains ♏ jest, ♐ into ♑ a ♒ Paradise. ♓  
In which are twelve words, to represent the twelve Months of the year, the first March, the second April, and so forward, and over the respective words are the Characters of the twelve Signs of the Zodiac, thereby denoting that in the Month to which the word belongs, the Sun is in that Sign over head; And if it be required to know the day of the Month in which the Sun enters into any of those Signs, if the first letter of the word proper to the Month be a Consonant, the Sun enters into the Sign thereto belonging on the eighth day of the said Month, as in the word Paradise belonging to February, in that Month he enters Pisces the eighth day, but if it be a Vowel, as all the rest are, add so many days unto eight as the Vowel denotes; now the Vowels are but five in number, which almost every Child knows how to number.  
By this Rule we shall find that the Sun enters into the respective Signs, 

♈ Aries, March 10  
♉ Taurus, April 9  
♊ Gemini, May 11  
♋ Cancer, June 12  
♌ Leo, July 13  
♍ Virgo, August 13  
♎ Libra, September 13  
♏ Scorpio, October 13  
♐ Sagittarius, November 11  
♑ Capricornus, December 11  
♒ Aquarius, January 9  
♓ Pisces, February 8    
Now knowing on what day of the Month, the Sun enters into any Sign, it will be easy afterwards to know in what degree of the said Sign he is in for any other day.  
1. If the number of the day of the given Month, exceed the number of that day in which the Sun enters into any Sign, subtract the lesser number from the greater, and the remainder is the degree of the Sign the Sun possesseth.  
Example.  
On the 21 of April, I would find the Sun's place, by the verse it appears, the Sun enters into Taurus on the ninth of that Month, which taken from 21, there remains 12, showing that the Sun is in the twelfth degree of Taurus, the second Sign, that is, 42d from the next Equinoctial Point.  
2. But if the number of the day of the given Month, be less than the number of that day in which the Sun enters into the beginning of any Sign, the Sun is not yet entered into the said Sign, but is still in the Sign belonging to the former Month, in this case subtract the given day, from the day of his entrance into the next Sign, and again subtract the remainder from 30, and the remainder shows his place in the Sign of the former Month.  
Example.  
Let it be required to know the Sun's place the fifth of August, on the 13th day of that Month the Sun enters into Virgo, 5 from 13 rests 8, and that taken from 30, there remains 22, showing that the Sun is in the 22th degree of Leo, the fifth Sign, and consequently his distance from the Equinoctial point Libra, is 38 degrees; Having compared the Sun's place, found by the former verse with a new Table of the same in the Seaman's Calendar, it doth not at any time differ a degree from the truth, and seldom half so much.  
In the Propositions following, we shall assume the Sun's Declination to be given, in regard there are tables thereof in almost every Mathematical Book.  
1. Proposition. The place of the Sun in the Ecliptic, and his greatest Declination being given, to find his Declination.  
In the general Scheme of the Analemma we observed that the nearest distance from the Sun's place to the Equinoctial, was equal to the Sine of the Sun's Declination thereto, having drawn A C and P C, perpendicular to each other, with 60d of the Chords, setting one foot of the Compasses in C, draw the quadrant or Arch A P in it, set off the Chord of 23d 31′, the Sun's greatest Declination from A to F, and draw the line F C, the Scheme is fitted to the greater Chord.  
Then out of the line of Sins in the Diameter, prick down the Sine of the Sun's place, or distance from the next Equinoctal point, which for the 21 of April was 42d, and it reaches to y, the nearest distance from y to A C, measured on the line of Sins in the Diameter, shows the Declination sought in this Example, 15 degrees and a half, to measure it without Sines, prick the said nearest distance from C to p, and draw the parallel p y f, and the Arch A f is the measure thereof on the greater Chord.  

The Sun's place or Declination being given (as before) to find his right Ascension.  
In the former Scheme having drawn the parallel of the Sun's Declination passing through his place at y, the extent y p, is the Sine of the Sun's right Ascension, from the nearest Equinoctial Point to the Radius of the parallel f p, wherefore place the extent p f from C to R, and upon R as a Centre, with the extent y p, describe an Arch at B, a Ruler laid from the Centre just touching the Extremity of that Arch, finds the point A in the Limb of the quadrant, and the Arch A A measured on the greater Chords, is 39d 33 minutes, and so much is the Sun's right Ascension in the first quarter of the Ecliptic, when he is departing from the Equinoctial in Spring.  
In the second quarter of the Ecliptic in Summer, the Sun returning towards the Equinoctial, and having the like Declination, his right Ascension is the compliment of the Arch before found, to a Semicircle, to wit, 140d 27′.  
In Autumn, or the third quarter, the Sun having the like Declination towards the depressed Pole, the right Ascension found by the Scheme, must have a Semicircle added thereto, and would in this Example be 219d 33′.  
In the Winter, or last quarter, the right Ascension is the compliment of the Arch found to a whole Circle, and would be in this Example 320d 27′, the uses hereof we mentioned in the Definitions.    

CHAP. VI Showing how to resolve those common Propositions relating to the Sun's Motion, that require the Latitude of the place to be known, as also the Sun's true place, or his Declination to be given.  
 
WIth 60d of a line of Chords upon the Centre C, describe the Semicircle S Z N, and draw the Diameter S C N representing the Horizon, and raise Perpendicular thereto Z C, and from Z to A, as also from N to P, out of the Chords prick off 51d 32′ the given Latitude, and draw A C representing the Equinoctial, and C P the Axis, then if the Sun's place be given, prick from the Chords from A to F 23d 31′ his greatest Declination, and draw the line F C, wherein prick down from C to y out of the Sins, his distance from the nearest Equinoctial point, as in the first Scheme, admit 60d, and through the point y draw the line y W, parallel to A C, and it shall be the parallel of Declination, but in this Example we suppose the Sun's Declination to be given 20d 12′, prick the Chord of it from A to G, then prick the Sine of it from C to A, or which is all one, set the nearest distance from G to A C, from C to A, as before, and through the point A, draw the line G A W, and it represents the Parallel of the Sun's Declination; and through the point A, draw the line H A B parallel to the Horizontal line C N, and so is the Sphere Orthographically (in right lines) projected, and fitted for the resolution of many questions.  

1. To find the Amplitude.  
Measure the Extent, C W on the line of Sines, and it will reach to 33d 42′, and so much doth the Sun rise or set to the Northward of the East and West in the Latitude of London, when his Declination is 20d 12′ North, but he rises and sets so much to the Southward of the East and West, when his Declination is as much South, and this Proposition is of use to find the variation of the Compass.  
By comparing the Coast, or bearing of the Sun, observed at his rising or setting by an Azimuth Compass with his true Coast, or bearing found by this Proposition or Calculation, the difference showeth the Variation sought.  
If the Sun's parallel of Declination G W, doth not meet with the Horizontal line S N, as in Regions or Latitudes far North, the Sun doth neither rise nor set.   

2. To find his Altitude at the hour of six.  
The nearest distance from A to C W equal to H C, measured on the Sins, showeth it to be 15d 41′, and so much is his Depression under the Horizon at six, when his Declination is 20d 12′ South.   

3. His Altitude or Height, being East or West.  
Measure the Extent C I, on the line of Sines, and it reaches to 26d 11′, and so much is the Altitude sought in Summer, but so much is his Depression under the Horizon in Winter, to the like Declination when he is East or West. If the Sun's parallel of Declination G A doth not meet with the Vertical Circle C Z, the Sun cometh not to be East or West, as it often happeneth in small Latitudes or in Countries between the Tropics.   

4. His Ascensional difference or time of rising and setting from six.  
This is represented by A W in the parallel of Declination, and is therefore to be reduced to the Common Radius.  
Take the Radius of the parallel A G, and prick it from C to R, then take the Extent A W, and setting one foot upon B, with the other draw the touch of an Arch at E, lay a Ruler from C, so as it may but just touch the outwardmost verge of the said Arch, and it cuts the Circle at D, take the Chord or extent D N, and measure it on the line of Chords, and it reaches to 27d 35′ which being converted into time, is one hour 46 minutes, and so much doth the Sun rise before and set after six in Summer, but so much doth he rise after and set before six in Winter, when his Declination is as much South.   

5. The time when the Sun will be due East and West.  
The hour from six is represented by A I in the parallel of Declination, with that extent upon the point R, draw the Arch O, a Ruler laid from C to the extremity of the said Arch, cuts the Circle at P, and the distance P N measured on the Chords, showeth 16d 18′, which converted into time, is one hour 5′, and so much after six in the Morning, or before it in the afternoon, will the Sun be due East and West.   

6. The Sun's Azimuth at the hour of six.  
This is represented by H A, in the parallel H A B, prick H B from C to K, and with H A upon the point K, draw the Arch L, a Ruler laid from C, just touching the said Arch, cuts the Circle at M, the distance S M measured on the Chords, sheweth 12d 53′, and so much is the Sun to the Northward of the East, at the hour of six.    

CHAP. VII. Showing how all the 16 Cases of right Angled Spherical Triangles, may be reduced to four heads of Proportion, and resolved by light from the Analemma.  
1. IN Sines alone of the greater to the less, wherein the Radius leads, whereof there are three such Cases.  
2. Proportions in Sines alone of the less to the greater, wherein the Radius may be in the second or third places, whereof there are five such Cases.  
3. Proportions wherein the Radius is first, a Sine last, and two Tangents in the middle, whereof there are three Cases.  
4. Proportions wherein the Radius is first, a Sine second, and two Tangents, the two other Terms, according to which Distribution, the Proportions for all the Cases may be collected from my Treatise the Sector on a quadrant.  
Now the Analemma, if it be minded, shows us how to resolve all these Cases.  
In finding the Sun's Declination from his given place, and greatest Declination, or his Altitude at six, it shows how to work Proportions in Sines alone of the first kind, and the Converse, or in finding the Amplitude or Vertical Altitude, it shows the second kind, suitable to each Variety, we shall add one Example.  

1. An Example of the first Variety.  
Let the Proportion be,  
As the Radius to the Sine of 38d 49′.  
So is the Sine of 31d 27′.  
To the Sine of 19d 6′.  
Having drawn the quadrant P C E, prick one of the middle Terms of the Proportion from P to O, to wit, 38d 49′, and draw the line O C, then out of the line of Sines prick the other middle Term from C to W, to wit, 31d 27′, and the nearest distance from W to C P, being equal to W G, is the Sine of the fourth Proportional, and being measured on the Sins, showeth 19d 6′.   

2. An Example of the second kind.  
Let the Proportion be,  
As the Sine of 51d 11′, Is to the Radius,  
So is the Sine of 24d.  
To what Sine? 31d 27′.  
Having drawn the quadrant C E P, prick the middle Term (not the Radius) from E to B 24d, and draw B G parallel to E C, then prick the first Term from E to O, a Ruler laid over O, from the Centre, finds the point W, and the Extent W C measured on the Sins, showeth 31d 27′, the fourth Proportional Sine sought.   

3. An Example of the third kind.  
As the Radius to the Tangent of 38d 49′.  
So is the Tangent of 24d.  
To what Sine? Answer: The Sine of 21d.  

Operation.  
With 60d of the Chords, draw the quadrant P C E, and then prick off one of the middle Terms being a Tangent in the Limb with the Chords, here 38d 49′ is set from P to O, and the other middle Term being a Tangent, prick down the Chord thereof, namely, of 24d from E to B, and through the point B, draw B G parallel to C E, so is G W the Sine sought, but found in a parallel set G B from C to A, and upon A, with the extent G W, draw the Arch F; a Ruler laid from C, just touching the extremity of that Arch, cuts the Circle at D, the distance P D being measured on the Chords, showeth 21d, and so much is the sine sought.  
If it be desired to measure it in a Sine, draw a line from B to C, and through the point W, draw a line parallel to G C, the distance between C, and the point of Intersection, or crossing being measured on the Sins, will show 21d as before, but this is not necessary unless G W be large.   

Demonstration.  
The Proportion that find●● the Ascensional difference is,  
As the Radius, is to the Tangent of the Latitude:  
So is the Tangent of the Declination,  
To the Sine of the Ascensional difference.  
The Proportion before wrought, is a Proportion of the same nature with this, and after the same manner as the Ascensional difference was found in the Analemma, was the said Proportion protracted, the truth of the Analemma being out of all doubt, many hundred years since invented, and demonstrated, which I shall not make my present task to repeat.   

Otherwise.  
It may be demonstrated from Proportion solely, the former Proportion may be Varied to stand thus.  
As the Tangent of 66d to Radius,  
So the Tangent of 38d 49′,  
To the Sine of 21d.  
Making G C Radius, B G becomes the Tangent of 66d, and W G the Tangent of 38d 49′, than it holds, making B G also t●e Radius of a line o● Sines.  
As B G the Tangent of 66d to B G the Radius,  
So W G the T●ngent of 38d 49′.  
To W G the Sine of the fourth Proportional to that Radius, which was reduced to the common Radius, by the prescribed Directions, whence it may be observed, that if two Terms of such a Proportion be fixed, a line placed against the Limb of a quadrant as B G, with a Thread from the Centre, will operate Proportions in Sines and Tangents.    

Otherwise taking the Proportion as at first propounded.  
Raise a Perpendicular from P, meeting with C O produced, and the said Perpendicular shall be the Tangent of 38d 49′, and then it holds.  
As the Radius C P, to the said Tangent,  
So is C G the Sine of the third Term, to G W, the fourth Proportional, the● because the third Term C G, being a Tangent, was changed into a ●i●●, the Cousin thereof G B must become Radius, for as the Tangent o●●n Arch, is to the Sine of an Arch, so is the Radius to the Cousin of th●●●id Archippus   

4. An Example of the fourth kind.  
Let the Proportion be,  
As the Radius to the Sine of 38d 49′,  
So is the Tangent of 31d 27′,  
To what Tangent? 21d.  

Operation.  
With 60d of the Chords upon the Centre C, draw the quadrant P E, and draw the Radiu● P C, with 90d of the Chords prick off P E, and draw E C, so 〈◊〉 the quadrant finished.  
Out of the Chords prick off P O, 38d 49′, the second Term of the Proportion, and draw the line O C.  
Then out of the Sins from C to V, prick off 31d 27′, the third Term of the Proportion, and through the point W, draw G W B, parallel to C E, and place G ● from C to A, upon which as a Centre, with the extent G W, describe the Arch F, a Ruler from C touching it, finds the Point D in the Limb, and the Arch P D, is the measure of the fourth Proportional.   

Demonstration.  
The Proportion that finds the Sun's Azimuth at the hour of six, is,  
As the Radius is to the Cousin of the Latitude,  
So is the Tangent of the Declination,  
To the Tangent of the Azimuth from East or West.  
The Proportion before protracted, is a Proportion of the same nature, and after the same manner as we found the Azimuth at six before by the Analemma, was the said Proportion protracted, yet here it is to be suggested, that in the Analemma there are three Proportions in Sines wrought, instead of the one in Sines and Tangents before expressed.    

1. As namely to find the Sun's Altitude at Six.  
As the Radius is to the Sine of the Latitude,  
So is the Sine of the Declination,  
To the Sine of the Sun's height at six.   

2. To find his Azimuth in that parallel of Altitude.  
As the Radius is to the Cousin of the Latitude,  
So is the Sine of the Declination, to the Sine of the Azimuth in the said Parallel.   

3. To reduce it to the common Radius.  
As the Cousin of the Altitude at six,  
Is to the Radius,  
So is the Sine of the Azimuth in that parallel,  
To the Sine thereof in the common Radius.  
The two latter Proportions in Sines, may be brought into one, as I have showed in a Treatise, the Sector on a quadrant, Pag. 111, 114. and that will be,  
As the Cousin of the Altitude at six,  
Is to the Cousin of the Latitude,  
So is the Sine of the Declination,  
To the Sine of the Azimuth sought.  
And thus in effect, the Analemma performs that single Proportion intermingled with Tangents after a more laborious manner in Sines, or if you will the Altitude at six being found, it holds:  
As the Cotangent of the said Altitude,  
Is to the Radius,  
So is the Cotangent of the Latitude, to the Sine of the Azimuth sought; and this Proportion lies visible in the Analemma.  
By these Directions derived from the Analemma, together with the Proportions for each Case, all the 16 Cases of right angled Spherical Triangles may be resolved; some whereof seem to require the drawing of an Ellipsis, as namely, if the Sun's place and right Ascension were given, to find his greatest Declination, which notwithstanding, according to these Directions, is easily shunned.    

CHAP. VIII. Showing how to come by the Sun's Altitude or Height, without Instrument.  
UPon any Flat or Plain, that is, level or parallel to the Horizon, erect or set up a Wire, without inclining or leaning to either side, admit in the Point C, and when you would find the Sun's Altitude or height, make a mark at that instant in the very end or extremity of its shadow, suppose at G the shadow be-being the Line C G.  
Then upon the same Flat draw the quadrant of a Circle C A F, with 60d of your Chords, and make C E equal to the height of the Wire, and through the point E, draw the line E D parallel to C A, and therein prick down the length of the shadow from E to D, a Ruler laid from the Centre to D, cuts the quadrant at B, and the Arch B A measured on the Chords, showeth the height required in this Example 20d 25′, in like manner if the length of the shadow were E K, the height would be N A 38d 16′.  

Otherwise.  
This may be performed otherways, by drawing the quadrant C A F, upon any plain board whatsoever, then stick in a Pin at the Centre C, and hold the board so towards the Sun, that the shadow thereof may fall upon the line C A, then imagine C G to represent, or supply the use of a Thread and Plummet hanging upon the Pin in the Centre at liberty, and mark where it cuts the Arch of the quadrant F A, suppose at H, measure the Arch F H, on the line of Chords, and it shows the height requi●ed: By the next Chapter we shall find the Sun's Azimuth belonging to the Altitude, 20d 25′, (according to the Latitude and Declination there given) to be 31d 19′ from the Meridian, admit to the Westward of the South, then doth the shadow happen so much to the Eastward of the No●th, wherefore if 31d 19′ be set off in the quadrant, from the line of shadow, from H to N, a line drawn into the Centre, as N C, shall be a true Meridian Line, or line of North and South.    

CHAP. IX. Showing the Resolution of such Propositions wherein the Sun's Altitude i● either given or sought.  
THe Latitude of the place, Declination of the Sun, and his Altitude given, to find the hour of the day and the Azimuth of the Sun.  
Example.  
Declination 13d South,  
Altitude 14d 40′.  
With 60d of the Chords draw the Circle S Z, N E, the Centre whereof is at C, and draw the Diameter S C N and Perpendicular thereto Z C, prick off the Latitude 51d 3′ from N to P, and from Z to A, and draw the Axis B P, and the Equator A C Q, prick off the Declination from A to D, and from Q to E, being 13d from the Chords, and draw the parallel of Declination D B E, then from S to A, and from N to O, out of the Chords, prick off 14d 40′ the Altitude, and draw the parallel of Altitude A O, so is B ⊙ the hour from six towards Noon in the parallel of Declination, and M ⊙ the Azimuth of the Sun, from the East or West Southwards.  

To measure the Hour.  
Set the extent B D, from C to F, and upon F as a Centre, with the extent B ⊙, draw the Arch G, a Ruler laid from C, just touching that Arch, finds the point H, the Arch N H measured on the Chords, showeth 45d, and so much is the hour from six, to wit, in time three hours, either nine in the forenoon, or three in the afternoon.   

To measure the Azimuth.  
In like manner set M A, from C to R, and upon R with the extent M ⊙, draw the Arch I, a Ruler laid thereto, from C cuts the Limb at L, the Arch L S measured on the Chords, is 44d 35′, and so much is the Sun to the Southwards of the East or West.   

To find the Angle of Position.  
Place the compliment of the Altitude Z A, from P to g, then place the extent A g, from g to f, a Ruler laid over A, and f cuts the parallel of the Sun's Declination at t, and D t is the versed Sine of the Angle of Position.   

To measure it.  
Through the point t, draw k t y parallel to the Axis B P, and draw D C, then the extent C y measured on the Sins, showeth the compliment of it, to wit, 62d 57′, therefore the said Angle is 27d 3′.  
Otherwise: With the extent B D, setting one foot in the Centre at C, with the other cross the former parallel line k t at k, a Ruler laid from C to k, cuts the Limb at X, and the Arch A X is 27d 3′, the measure thereof as before, the Compliment whereof is the Arch V E, and might be measured from the South Pole V, if the Equinoctial A Q were not drawn.   

Three sides given to find an Angle.  
Another Example, the Declination being North,  
Latitude 51d 32′,  
Declination 13d North,  
Altitude 37d 18′. In the following Scheme,  
Upon the Centre C, with 60d of the Chords, describe a Circle, prick one of the sides, namely, the Colatitude 38d 28′ from Z to P in the Limb, and from P prick the Sun's polar distance 77d to D and E, and draw the parallel of Declination D E, from Z, prick off the compliment of the Altitude to A and O, to wit, 52d 42′, and draw the parallel of Altitude A O, so is the hour and Azimuth found in these two parallels without drawing any more lines, the drawing of the Axis C P was only to divide the parallel D E into halfs at B; Likewise the drawing of C Z, divides the parallel A O into halfs at M, which may be found without drawing lines by laying a Ruler from C to Z and P, or if you will draw the Horizontal line S N passing through the Centre, and above it the Altitude may be set from S to A, and from N to O, in like manner if the Equinoctial be drawn, the Declination being North, is to be set above it.   

The manner of measuring, is the same as before, To find the Hour.  
C F is made equal to B D, then upon F with the extent B ⊙, draw the Arch G, a Ruler from the Cen●er touching G finds H, and the Arch H N being 45d, is the hour from Noon.   

For the Azimuth.  
C R is made equal to M A, then upon R with M ⊙ draw I, a Ruler from the Centre touching it, findes L, and the Arch L S measured on the Chords, is 30d, and so much is the Azimuth to the Southwards of the East or West.   

To find the Angle of Position.  
Here also Z A is placed from P to g, and then g f is made equal to g A, a Ruler from A to f, cuts the parallel of Declination at t, and D t is the versed Sine of the angle of Position, which being measured as in the former Example for finding it, will be found to be 33d 34′.  
  

Another Example for South Declination 13d, retaining the former Latitude.  
A new Scheme need not be drawn, the Declination being as much South, let the Altitude be 20d 25′, prick the Altitude from S to Q, and from N to X, and draw the parallel Q X, where it crosse●h the parallel of Declination, set V, join C X, and draw V r parallel to W C.   

To find the Hour.  
Upon F as a Centre, with the extent B V, describe the Arch K, a Ruler from the Centre touching it finds T, the Arch N T being 60d, is the hour from six, to wit, either ten in the morning, or two in the afternoon.   

To find the Azimuth.  
The extent C r measured on the Sins, is 58d 41′, and so much is the Sun's Azimuth to the Southward of the East or West in this our Northern Hemisphere.   

A third Example, for finding the Azimuth.  
In the former Examples, it may be observed that the Azimuth is always found in the parallels of Altitude, which towards the Zenith grow very small, and consequently this way of finding the Azimuth in Latitudes, between or near the Tropics, and sometimes in our own Latitude, when the Sun hath much Altitude, will be very inconvenient; for remedying whereof let it be noted, that the Azimuth may be always found in the parallel of Latitude.  
Example: For the Latitude of the Barbadoss 13d, ⊙ Declination 20d North, Altitude 52d 27′.  
In the first Scheme, having drawn the Horizon S N, its Axis C Z, the parallel of Altitude B A 52d 27′, set off the Latitude from N to P 13d, and draw the Axis C P, prick off the compliment of the Declination 70d from P to D, and E, and draw the parallel of Declination D E, then is F ⊙ the Sine of the Azimuth, from East or West Northwards F A being Radius, which being placed from C to G, and the Arch I described with F ⊙, a Ruler from the Centre touching it finds S L 16d the measure of the Azimuth in the Limb, but thus to transfer from a less to a greater will breed much incertainty, and the parallels of Altitude near the Zenith decrease very much, for remedying this inconvenience, change the names Latitude and Altitude, and fit the Analemma thereto, for the bare changing of the names of the two containing sides of a Triangle, doth not alter the quantity of the angle comprehended:  
Then will the new Latitude be 52d 27′,  
And the new Altitude 13d 00.  
I say, the Azimuth is the same as it is in the Latitude of 13d when the Altitude is 52d 27′. Also the hour from noon in the new Latitude, is equal to the angle of Position in the old Latitude, and the angle of Position in the new Latitude, is equal to the hour from noon in the old Latitude.  
The old or first Scheme may very well serve if you set off 70d the compliment of the Declination each way from A, the end of the parallel of Altitude, and then through the point P, draw the parallel of Altitude parallel to the Horizon.  
But here we have drawn a second Scheme, where N P is 52d 27′, P D and P E 70d, through which points are drawn the parallel D E; Also S B and N A is 13d, through which is drawn the parallel B A, then is F ⊙ the Sine of the Sun's Azimuth to the Northwards of the East, to the Radius F A which is placed from C to G, upon which with F ⊙, was the Arch I described, and the Azimuth S L found to be 16d as before.  
Nota, also in the first Scheme (or in both) that the Radius C G might have been doubled, and pricked upon N S produced, then also must the extent F ⊙ have been doubled, and the Arch I therewith described; also, the hour of the day might after the same manner be found in the parallel of Latitude which may be convenient for Stars that have much Declination in that Case, the Declination and Latitude must change Names.  
And thus when three sides are given to find an angle, we may find it in the Analemma, by calling the compliments of those sides the Declination, the Latitude, and the Altitude, or Depression at pleasure.  
Also when three Angles are given to find a side, those Angles must be changed into sides, by taking the compliment of the greatest Angle to a Semicircle, and writing it, and the other Angles down to their opposite sides in another Triangle, as in the Scheme following, wherein as the angles are changed into sides; so are the sides changed into angles, and then the Case will be the same as before.   

Two sides with the angle comprehended to find the third side, and both the other angles.  
By this Case may be found the Sun's Altitudes on all hours, and the distances of places in the Arch of a great Circle.  
First, the Sun's Altitudes on all hours, thereby is meant that if the hour of the day, the Declination and Latitude be given, the Sun's Altitude proper to that hour (or his depression) may be found.  
Upon the Centre C, with 60d of the Chords, describe the Arch S D P E, the Diameter or Horizontal Line S C N, and from N to P prick off by the Chords the Latitude 51d 32′, and from P to D and E, set off 66d 29′ the compliment of the Sun's declination, and draw the Parallel of Declination D E, and the Axis C G P. Draw the Radius D C, and therein out of the Sins prick down 15d for the hours of 1 from 6 before it and after it,, then take 30 2 45 3 60 4 75 5 the nearest distances from 15d to C G, and prick it from G to 5 and 7, likewise take the nearest distances from 30d to C G, and set it from G to 8 and 4, and the like for the rest, then will the nearest distances from 4, 5, 6, 7, 8, 9, 10, 11, to the Horizontal Line S C N, be the Sins of the respective Altitudes sought, and are accordingly to be measured on the line of Sines, so the Altitudes for the hours 4 in Summer will be 1d 34′ 5 9 30 6 18 12 and so much is the Sun's depression under the Horizon at the hours of 8, 7, and 9 in Winter: the Summer Altitudes for the hours 7 are 27d 23′ 8 36 42 9 45 42 10 53 45 11 59 42 And the Winter Altitudes for the hours 9 are 5d 13′ 10 10 28 11 13 48 and so much are the Summer depressions for the hours 3, 2, 1, from midnight, and are found in the Line H E, by taking the nearest distances in that line from the points 9, 10, 11, unto the line H N, ●or if this Scheme be turned upward, then is H E the parallel of ●eclination above the Horizon for 23d 31′ of South Declination; ●nd if you will, the parallels of Altitude may be drawn, as is 8 A, ●nd the arch A N measured on the Chords, is 36d 42′ as before; ●nd now having the Declination, Latitude and Altitude, the Azimuth and Angle of Position may be found according to former directions, and are the two other Angles required.  
I have before suggested, that the names of Declination and latitude might change places, the angle remaining the same, hence it follows that a Star that hath 51d 32′ of north declination in the latitude of 23d 31′, shall have the like altitudes on all hours, or horary distances from the Meridian, as the Sun having 23d 31′ of north declination, hath in the latitude of 51d 32′, whereof no Example is needful.  
Now to apply what hath been said to the finding of the distances of places in the arch of a great Circle, let it be noted that this Case of finding the Sun's Altitudes on all hours, and that for finding of distances are both of them one and the same case of Spherical Triangles, namely, two sides with the Angle comprehended given, to find the third side, in the following Triangle B L P.  
Let L signify London, P the North pole, I Java major, and let it be required to find the distance between Java major and London in the arch of a great Circle, their difference of longitude being 114d 10′. To resolve this Proposition, call the difference of Longitude the hour from noon, and the side I P the Sun's polar distance, and the side L P the compliment of the Latitude, and fit the Analemma to the resolution thereof, as before, for finding the Sun's Altitudes or Depressions.  
Example.  
Java major, Lat. 9d. South Long. 140d. London, Lat. 51d 32′. North Long. 25d 50′. Diff. Long. 114d 10′.  
Upon C as a Centre, describe the arch G L F. In the Limb from L to P prick the compliment of the Latitude of London 38d 28′, from P set off 99d to D and E, being the distance of the other place from the North pole, and draw D M E, draw L C, and C P continued, and the Radius D C, therein from C to K out of the Sins prick down 24d 10′, the excess of the difference of Longitude above a quadrant, the nearest distance from K to M C, place from M to I, so is D I the versed Sine of the difference of Longitude in that parallel; and if from L and P you draw the pricked arks meeting at I, the Triangle is represented: through I draw F B G perpendicular to L C, and L B is the versed Sine of the distance of these two places, to wit, 111d 58′, found by measuring C B on the Sins, to wit, 21d 58′, whereto the quadrant L C is to be added. The distance thus found, is the ark of a great Circle passing between those places, which is to be converted into Leagues or Miles, allowing 60 Miles to a degree, if it be assented to, that so many raise or lay the Pole one degree under the Meridian; Also F I is the versed Sine of the Angle of Position P L I, to measure it prick the Radius B F from C to k, and upon it with B I, describe the Ark O, a Ruler from the Centre finds Q, and the Ark L Q is 13d 41′, being substracted from 90d, rests 76d 19′, and so much is the Angle P L I the Angle of Position at London, thereby is meant the Angle between the Meridian of London, and the Ark of the great Circle passing between both places.   

To find the Angle of Position at Java major, that is, the Angle between the Meridian of that place, and the aforesaid Arch of a great Circle.  
Place L G from P to F, and make f h equal to G f, a Ruler laid from h to G cuts D M at t, parallel to M P draw t y, than measure y C on the Sins, and it is 52d 17′, the compliment whereof is 37d 43′, being the Angle of Position at Java major.  
After the same manner are the distances of Stars to be found, their Longitudes and Latitudes being given, or their Declinations and right Ascensions.  
Thus when two sides with the Angle comprehended are given, may the third side be found, and that being found, then either of the other angles may be found.  
If two Angles with the side between them, or their interjacent side were given to find the third Angle, or one of the other sides, in both these cases there must be a conversion or changing of the given angles into sides, and of the given side into an angle, by taking the compliment of the greatest angle, (or of the side if that be greatest) to a Semicircle, and then drawing a new Triangle, writ them down to their opposite sides, and the case will be the same as before.  
Thus we have finished the six certain obliqne Cases of Spherical Triangles, to wit,  
1. Three sides to find an angle.  
2. Three angles to find a side.  
3. Two sides with the angle comprehended, to find the third side.  
4. Or to find one of the other angles.  
5. Two angles with the side between them, to find the third angle  
6. Or one of the other sides.  
And here it may be observed that the fourth case cannot be resolved by this Projection without finding the third side first, which is the thing required in the third case, yet it may be resolved from proportions immediately, without the knowledge of the third side, as I have explained in a Treatise of Geometrical dialing; as also in the first Book of the Mariners plain Scale new Plained. There remains yet Six other Cases, to wit,  
7. Two sides with an angle opposite to one of them, to find the ●hird side.  
8. From the same Data or parts of the Triangle given, to find ●●e angle included, that is, the angle between the two given ●●des.  
9 To find the Angle opposite to the other side.  
The eighth and ninth Cases also requires the third side to be found first, before the angle included, or the angle opposite to the other side can be found from this Projection (as to my knowledge) yet are easily resolved from proportions without finding the said third side at all, as in the said first Part.  
10. Two Angles with a side opposite to one of them, to find the third Angle.  
11. To find the interjacent side, or side between them.  
12. To find the side opposite to the other Angle.  
And these three Cases are the same with the seventh, eighth, and ninth, when the Angles are changed into sides, and the side into an Angle.  
These six latter Cases I call the doubtful Cases, because sometimes a double answer must be given, and both true, and when these Cases will happen, and when not, I have showed at large in a Treatise, called The Sector on a quadrant.  
3. Two sides with an angle opposite to them, to find the third side.  
By this Case if the Sun's true Azimuth and Altitude thereto be observed at any time off the Meridian, and his Declination given, the Latitude may be found.  
Example given.  
Sun's Declination, 23d 31′ North.  
Azimuth, 20d to Southwards of the Vertical.  
Altitude, 44d 39′  
Demanded the Latitude, Z A, 51d 32′.  
In every Scheme of the Analemma, observe that the nearest distance of the point of the Sun's place in his parallel of Altitude from the Equinoctial Line, is equal to the Sine of his Declination, from whence the resolution of this Proposition easily follows.  
Having with 60 degrees of the Chords upon the Centre C, drawn the Semicircle S Z N, the Diameter S C N, the Axis of the Horizon Z C, out of the Chords prick off the given Altitude from S to A, and from N to B, and draw the parallel thereof A F D B, then ptoportion out the Sine of 20 degrees to the Radius A D, draw the line A C, and therein from the Sins prick off 20 degrees from C to 20 degrees, and take the nearest distance to C D, and set that extent from D to the point F, then take 23 degrees 31 minutes from the Sins, and setting one foot in F, with the other draw the arch L, a right Line drawn from C, just touching the outward extremity of that arch, as doth A C, shall be the Equinoctial, and so the arch S A is the compliment of the Latitude, and Z A the Latitude sought, in this Case 51d 32′.  
By this Case of Triangles, the Declination and Latitude being given, the Sun's Altitudes (or Depressions) to each assigned Azimuth may be found  
This case may be done without the drawing of any Ellipsis, and that divers ways, though it hath not hitherto been so reputed; in order whereto observe that any two containing sides of a Triangle, may change names, as in the former Example: if the Latitude be 44d 39′, the Altitude to that Azimuth will be 38d 28′, consequently by the same Scheme and manner of work, if the Azimuths be pricked off in the parallel of Latitude, the respective Altitudes thereto may be found; but this Scheme I have further insisted upon in the first Part.   

Another manner of Operation.  
Let the Latitude be 51d 32′ North.  
The Sun's Declination 23d 31′ North.  
And let it be required to find the Sun's respective Altitudes to the Azimuths of 20d and 60d, on each side from the Vertical Circle. Upon the Centre C, describe an arch bigger than a Semicircle, and draw the Horizontal Line S C N, and its Axis Z C at right angles thereto, and set off the Latitudes from S to A, and draw the parallel thereof L A, set off the Declination likewise from N to E, and draw the parallel thereof D E, which is easily done by pricking off the like arch above S.  
Now to proportion out the Sins to the Radius A L, draw the Line A C, and there prick down the Sins of 20d and 60d, and take the nearest distances therefrom to the line C Z, and transfer or place those respective extents into the parallel of Latitude L A, pricking them down from L, then one foot of the Compasses resting in C, take the distance to 20d and 60d in the parallel of Latitude L A, and transfer or place them into the parallel of Declination D E, and there number them, as before, then take the distance between the Zenith at Z, and 20d in the parallel of Latitude L A, and the said extent shall reach from 20d in the parallel of Declination D E, to F upward, and to H downwards.  
The arch N F being 44d 39′, is the Sun's height for 20d of Azimuth to the southwards of the East or West, and the arch N H 14d 15′, is the Sun's height for the like Azimuth to the northwards of the East or West.  
Also the distance between 60d in the parallel of Latitude and the Zenith Z, reacheth from 60d in the parallel of Declination to Q upwards, and to R downwards.  
The ark N Q being 59d 21′, is the Sun's height for 60d of Azimuth to the southwards of the East or West, and the ark N R being 9d 43′, is the Sun's depression in Summer for 60d of Azimuth to the northwards of the East or West; It is also his Winter Altitude for 60d of Azimuth from East or West southwards, when his Declination is 23d 31′ south, which is as much as we supposed it to be north.  
The ground of the former Scheme, is, that the Sun in his parallel of Altitude hath the like distance from the Zenith, as in his parallel of Declination, also his distance from the Centre in his parallel of Altitude, is the same with his distance from the Centre in his parallel of Declination, whence follows another Delineation of the Analemma suitable to this case, for we may place the parallel of Declination parallel to the Horizon, which now must represent the Equinoctial, then if the Sine of the Azimuth be pricked down in any given parallel of Altitude, the distance of that point from the Centre and Zenith are both given, then making the Sun to have the like distance in his parallel of Declination from the Centre, as he had in his parallel of Latitude: and moreover, if from the said point his distance from the Zenith in his parallel of Altitude be applied upward, it will give the Latitude or Zenith sought, from the Horizontal line representing the Equinoctial, and by the bare change of names, if the Sins of the respective Azimuths be pricked down in the parallel of Latitude, the respective Zeniths will become the Altitudes sought. See page 38 to this purpose. And the same Zenith distance and Centre distance, is common to Azimuths as much remote on the other side the Vertical, wherefore the Scheme finds both.  
In like manner if it were required to find what Altitudes the middlemost Star in the Bears-tail should have upon the Azimuth of 40d to the northwards of the East or West. Having drawn the Parallel of Latitude A L 51d 32′, and therein proportioned out the Sine of 40d L B, to the Radius L A, prick off the Declination of that Star from N to E, being 56d 45′, and place the nearest distance from E to N C, from C to D, and draw the Parallel D E, than one foot of the Compasses resting in C, place the distance C B into the Parallel of Declination at G, than the extent Z B will reach from G to F upwards, and to H downwards.  
The arch S F being 80d 50′, is the greater Altitude that the Star shall have upon that Azimuth, and N H 45d 3′ is the lesser Altitude: and here observe, that such Stars as have more declination than the compliment of the Latitude, never rise nor set; if their declination be also more than the latitude of the place, they will have two altitudes or heights on every Azimuth from the Meridian, except the remotest. See Page 115, 130, 132 of my Treatise The Sector on a Quadrant about this subject.   

To find the Altitudes on all Azimuths.  
To these ways I shall add one more, purposely invented by my loving Friend Mr. Thomas Harvie, to shun the drawing of an Ellipsis, occasioned by a Question sent out of France by Jean Montfert, proposed to all the English Mathematicians, which the said Mr. Harvie speedily resolved, and hath since published his Resolution in Latin in a single Sheet, sold by Henry Sutton Mathematical Instrument-maker in Thread-needle street behind the Exchange, from which the Reader may be furnished with the Demonstration of what follows, the Geometrical performance ensuing being the same with that in the Author's Sheet, from whence, and by conference with the Author, I attained it.  
Given Latitude 51d 32′ North.  
Declination 23d 31′ North.  
Azimuth 30d from the Vertical, that is, from the east or west Required the Sun's Altitudes to that Azimuth on each side the Vertical.  
Having described a Circle S Z N, and drawn the Horizon S C N, the Axis or Vertical Z C B, prick off the Poles height or latitude from N to P, and from P set off 66d 29′, the Sun's Polar distance being the compliment of the Declination to D and E, and draw the Parallel D E. Prick down the Sine of the given Azimuth 30d out of the Sins from C to G, and from G draw B G parallel to D E, and join S B produced or continued, then through the Point F, draw the Line H A parallel to S B, then is the arch S A 49d 56′, the Sun's Altitude for the Azimuth of 60d from the South, and the arch N H being 6d 36′, is the Sun's height for the Azimuth of 60d from the North. If H had fallen below N, then had the arch H N been the ark of the Sun's depression under the Horizon, proper to the Azimuth given from the East or West northwards.  
It had also been the Sun's height for the like Declination towards the south Pole, proper to the like Azimuth counted from the East or West southwards, so that there needs no Scheme for South Declinations.  
Nota, The excellency of this way further appears in this, that if the Azimuth we●e required, and the Latitude, Declination and Height given, the very contrary work would find G C the Sine of the Azimuth to the common Radius.  
The same manner of Operation also holds for the Sun or Stars when their Declination is more than the Latitude, in which case the Parallel of Declination will not cross the line C Z within the Circle, but must be produced to cut the line C Z continued without the Circle above Z, and then a line from the Intersection F happening without, being drawn parallel to S B, will pass through the quadrant Z N in two points, which shall be the two Altitudes required, when it so happens. And here note, that if C G had been pricked towards N, then B would have fallen as much above C, as it doth in the former Example below it, from which point B a line must be drawn to N in stead of S, which will be parallel to S B; and therefore a line drawn parallel thereto passing through F, shall as well give the Altitudes required, as it did before; wherefore when the distance between A and F is but small, though the point F happen within, yet the operation will be most certain to place C G on the other side the Vertical.  
But here we must prevent an inconvenience, for in small Latitudes when the Declination towards the elevated Pole is more than the Latitude, the parallel of Declination will not meet with the Vertical or Axis of the Horizon, but at a remote or inconvenient distance. I shall only mention how this Inconvenience may be shunned, omitting an Example thereof, in regard other ways before spoke to may prove more convenient.   

Two Sides with an Angle opposite to one of them, to find the Angle included, that is, the Angle between those Sides, and the Angle opposite to the other given Side.  
Example. Admit there were given the Latitude of the place and the Sun's Declination, (from which ark are got the two sides) and the Azimuth, and it were required to find the hour, that is, the angle included, and the angle of Position, that is, the angle opposed to the other side.  
The first work will be to find the Altitude or Height according to the directions before given, and then we have the Latitude, Declination and Height given to find the hour, and the angle of Position, which are the angel's required, and we have largely before shown how to find them.   

Two Angles with a Side opposite to one of them, to find the Side between them and the Side opposite to the other Angle.  
These angles being changed into sides, and the side into an angle, as in the second Triangle, we have then two sides with an angle opposite to one of them to find the third side, and both the other angles, and the case is the same as before.  
The side A B lying next the given angle, we call the compliment of the Latitude, the side A C the Sun's Polar distance, and the angle A B C the Azim th' from the North, and then fit the Analemma to ●●nde the Altitude or height, the compliment whereof is C B, and thereby find the Angle C A B, calling it the hour, and the Angle A C B, calling it the Angle of Position, by former directions.  
Having found the angle C A B, it is equal to the side Z P in the first Triangle, which is the compliment of the Latitude required. And the compliment of the angle A C B in the second Triangle, being taken to 180d is equal to the side ⊙ P in the first Triangle, the Sun's distance from the elevated Pole required.  
Example in this Scheme.  
The Latitude S T 30d, whose compliment is Z T 60, equal to the side of the Triangle A B.  
Z Y is the Polar distance 69d 46′, the compliment whereof is N Y. L G is the Sine of 6d 36′ in that parallel the excess of the Azimuth above a quadrant from the north.  
C F is equal to C G, the extent G Z reaches from F to A, and the arch Z A being 35d 59′, is equal to C B in the second Triangle, or to the angle Z ⊙ P in the first Triangle, being there the angle of Position.   

To find the Hour.  
Make N P equal to S T, drawing C P, and set off 69d 46′ from P to D and E, and draw the parallel D E. Then draw the parallel of Altitude H A, and B ⊙ is the Sine of the hour from six, C R is equal to B D, upon R with B ⊙ describe K, a Ruler from the Centre just touching it finds M, and the arch M O being 38d 28′, is the angle C A P in the second Triangle, equal to the side Z P in the first Triangle, which is the compliment of the Latitude there required.   

To find the Angle of Position.  
Place Z H from P to f, and make f g equal to f H, a Ruler from H to g cuts D E at I, upon R with B I draw Q, a Ruler from the Centre touching it finds V, and the ark O V being 66d 29′, is the measure of the angle A C B in the second Triangle, which is equal to the side ⊙ P in the first Triangle, the Sun's distance from the Pole required, and thus this question hath exercised most of the Rules before delivered; and thus solely upon the consideration of the Projection of the Sphere, I have showed how all the twenty eight common Cases may be resolved Orthographically.    

CHAP. X. Showing how to project all the Cases of right angled Spherical Triangles.  
ANd because Projections of the Sphere do not depend upon Proportions for the resolving of any Case propounded, but on the contrary Proportions are derived from Projections, I shall therefore further enlarge in showing how the right angled Cases are resolved without the knowledge of the Proportions.  
In Projecting the right angled Cases Orthographically, the Triangle V A ⊙ right angled at A, shall be the Example used, in which the arch S A is equal to the Compliment of the Latitude at London 38d 28′, equal to the Angle A V ⊙.  
The Hipotenusal V ⊙ is the hour from Six.  
The leg A ⊙ the Altitude or Sun's height at the time proposed.  
The leg V A the Sun's Azimuth from the Vertical.  
The angle A ⊙ V, the angle between the Sun's Azimuth Circle and the Equinoctial.  

Case 1, 2, 3. Given a Leg and its adjacent Angle, to find the rest.  
Data A V 31d.  
Angle A V ⊙ 38d 28′.  
Having drawn the Quadrant S Z, and S V, and Z V at right angles, in the Centre place the given angle 38d 28′ from S to A, and draw A V, draw A L parallel to Z V, then prick the Sine of the given Leg from V to A and F, the nearest distance from A to A V, prick from L to B.   

1. To find the other Side, or Leg.  
A ruler over V and B, cuts the Limb at D, and the arch S D being 22d 16′, is the measure of the side A ⊙.   

2. To find the Hipotenusal.  
Through D draw D ⊙ G parallel to S V, and the extent V ⊙ measured on the Sins, is 37d 30′ the Hipotenusal sought.   

3. To find the angle V ⊙ A.  
Through F draw F E parallel to V S, and make V C equal to V ⊙, a Ruler from the Centre over C, cu●s the Limb at L, and the arch S L being 57d 55′, is the measure of the angle sought.  
Otherwise: Make V I equal to V C, then take the distance between C and Z, and the said extent shall reach from I to L in the Limb as before; after the same manner the point D might have been found: Also the extent V F may be doubled or tripled upwards, and at the end thereof a Perpendicular raised, accordingly multiply or increase the extent V ⊙, and one foot resting in the Centre, cross the said Perpendicular, and a ruler laid over the point of Intersection or crossing, and the Centre V shall pass through the Limb at L, as before.   

Otherwise:  
Place V ⊙ from V to Q and upon Q with V A, describe the ark H, a ruler from the Centre just touching the outward extremity of it, cuts the Limb at L, as before, the arch S L being the angle sought.  
Or to keep the said arch more remote from the Centre, prick V ⊙ twice from V to P, then double the extent V A, and therewith upon P describe the ark K, a rule from the Centre just touching the outward extremity of that ark, cuts the Limb at L, as before.  
These Cases being the most difficult (as requiring an Ellipsis to be drawn, which we have shunned, and may be otherwise avoided, according to the Example in Page 51) I thought fit first to handle.  
The Triangle resolved is not represented, unless there be an Elliptical a k drawn from ⊙ to A.  
The finding of the Point D was carried on from a proportion of this kind:  
As the Radius A L to the Tangent of the ark S A, being the measure of the given angle,  
So is the Sine of the given Leg L B proportioned out to that Radius, to the Tangent of the ark S D, being the measure of the Leg sought.   

Case 4, 5, 6. Given a Side and its opposite angle, to find the rest.  
Data the Side A V 31d.  
The Angle V ⊙ A 57d 55′.  
Place the given Angle at the Centre of the quadrant, wherefore prick it, to wit, 57d 55′ from S to L, and draw L V.  
Then place the given Leg from S to D, to w●t, 31d, and draw D ⊙ G parallel to S V.   

To measure the Arks sought.  
1 The Hipotenusal.  
The Extent V ⊙ applied to the Sins, is 37d 30′, being the measure thereof.  
2. The other Leg.  
Place D G from V to R, and upon R with the extent G ⊙, draw the ark H, a ruler from the Centre touching the extremity of it, cuts the Limb at E, and the arch S E being 22d 16′, is the measure of the other Leg.  
3. The other Angle.  
Draw E F parallel to S V, and make V C equal to V ⊙, a ruler from the Centre over C, cuts the Limb at A, and the arch S A being 38d 28′, is the measure of the other obliqne angle. Lastly, the Triangle resolved is represented by the Eliptical pricked arch ⊙ A, and by the sides ⊙ V and V A.  
Note from the first Scheme, that if ⊙ V by being produced, have a quadrant added to it, and if ⊙ A be continued to the Nadir, and so have a quadrant added to it, the other side between the Nadir and ⊙ V produced, will be equal to A Z, and so there will be another right angled Triangle right angled in the Limb, and if there were given a Leg in the first Scheme, to wit, A Z, and its opposite angle Z ⊙ A, there would be given the same Leg and its opposite angle in the lower Triangle, wherein the other Leg and Hipotenusal are greater than quadrants, and the other obliqne angle obtuse, and so a double resolution will arise, unless the affection of some part of the Triangle unknown be likewise given. And how to resolve such a Triangle wherein any of the Data are bigger than quadranrs, the Reader may observe from the said Triangle and its opposite, by resolving the Complimental Triangle near the Centre, arising from the other Triangles.   

Case 7, 8. Given both the Legs.  
Having drawn the Quadrant and its two Radii, place one of the given Legs from S to D 22d 16′, and draw D G parallel to S V, and make V R equal thereto.  
Prick the other given Leg from S to E, and draw a line into the Centre, and from R take the nearest distance to the said Line, which place from G to ⊙, and draw V ⊙ A  
The Arch S A measured on the Chords, is 38d 28′, being the measure of the obliqne angle at the Centre.  
The extent V ⊙ measured on the Sins, is 37d 30′, and so much is the Hipotenusal.  
Draw E F parallel to S V, and therein make V C equal to V ⊙, a ruler from the Centre over C, cuts the Limb at L, and the arch S L being 57d 55′, is the measure of the angle V ⊙ A; by which Letters the Triangle is represented. Nota, in finding this obliqne angle we have the Hipotenusal and Leg that subtends, it given, which joined in a right angled Triangle, the angle thereby subtended, is the angle sought.   

Case 9, 10, 11. Given the Hipotenusal and its adjacent Angle, to find the rest.  
Data Hipotenusal V ⊙ 37d 30′.  
Angle V ⊙ A 57d 55′.  
Having drawn the quadrant S Z, and S V, and Z V, at right angles in the Centre, prick the given angle 57d 55′ from S to L, and draw L V: Prick the Sine of the Hipotenusal from V to ⊙, and through the point ⊙, draw D G parallel to S V, so is S D the Chord of the Leg opposite to the angle placed at the Centre, to wit, 31d.  
Place D G from V to R, and upon R with the extent G ⊙, describe the arch H, a ruler from the Centre touching the extremity thereof, cuts the Limb at F, and the arch S E being 22d 16′, is the measure of the other Leg.  
From the Point E, draw E F parallel to S V, and therein make V C equal to V ⊙, a ruler over C from the Centre, cuts the Limb at A, and the arch S A being 38d 28′, is the measure of the other obliqne angle.  
Lastly make V A equal to V F, and draw the Ellipticall arch V ⊙, and the Triangle is epresented by V A ⊙.   

Case 12, 13, 14. Given the Hipotenusal and one of the Legs, to find the rest.  
Data Hipotenusal ⊙ V 37d 36′.  
The Leg ⊙ A 22d 16′.  
Place F E from V to R, and upon R with F ⊙, draw the Arch H, a ruler from the Centre touching it, cuts the Limb at D, and the arch S D being 31d, is the measure of the Leg V A.  
Make V C equal to V ⊙, and a ruler from the Centre over C, cuts the Limb at L, and the arch S L being 57d 55′, is the measure of the angle V ⊙ A.   

Case 15, 16. Given both the obliqne Angles.  
In the first of these six Schemes, if the Angles V ⊙ A, and A V ⊙, be given in the Triangle ⊙ V A, then there is also given the Leg A Z, and the Angle Z ⊙ A in the Triangle Z A ⊙, and that Triangle being resolved, the compliments of the arks sought in the Triangle in which the case is put, are also found, and the operation will be the same with Case 4, 5, 6.    

CHAP. XI. The Longitude and Latitude of any Star or Point of the Heavens being given, to find the declination and right ascension thereto.  
Example. Medusas-head, a Star of the third Magnitude, Longitude, as set down in M. Wings Harmonicon Coeleste, is 20d 37′ ●, North Latitude 22d 2●′.  
Having with 60d of the Chords drawn the outward Circle with the two Diameters A V Q, and P V, making right angles in the Centre, prick the obliquity of the Ecliptic 23d 31′ from P to F, and from A to E, and draw A V Q and F V, then prick the Latitude of that Star from E to L, and draw L A parallel to E C, and join L V, wherein prick down from V to I the Sine of 50d 37′, that Stars distance from the Equinoctial Intersection of Aries, and take the nearest distance from I to V F, which place from A to O, through the point O draw D H parallel to A Q, and the arch E D being 39d 22′, is the measure of that Stars Declination.  
Make V R equal to G D, and upon R as a Centre, with the extent G O, draw the ark K, a Ruler from the Centre V, touching the extremity of that ark, cuts the Limb at B, and the arch A B being 40d 37′, is the right Ascension of that Star from the Equinoctial Point Aries, in time is 2 hours 42 minutes and a half.  
The Triangle resolved will be represented by drawing two Eliptical Arks from O, to the extremities of the side placed in the Limb, to wit, to P and F, in which there is given the Side P F, the Angle of the Ecliptic and Equinoctial equal to the distance of the Poles of those Circles.  
The side O F the compliment of that Stars North Latitude, with the Angle between them, P F O the Longitude of that Star from the Colour of the Solstice.  
To find the Side P O, the compliment of the Declination required, and the angle F P O, the difference between which angle and a quadrant, is the right Ascension of the Star sought, from one of the Equinoctial Points.  
For a Conclusion I shall add one Proposition more.   

CHAP. XII. Two Altitudes of the Sun, and the two Azimuths thereto belonging being given, to find the Latitude of the place and the Sun's Declination.  
Example. Let one Altitude be 49d 56′, Azimuth thereto 30d to Southwards of the East; the other Altitude 14d 15′, Azimuth thereto 20d to the northwards of the East.  
In the Scheme annexed make S A the greater Altitude, 49d 56′, and draw the Parallel A F, which being Radius, make F B the Sine of 30d the Azimuth thereto.  
Again, Make N G the lesser Altitude 14d 15′, and draw the Parallel thereof H G, which being Radius, prick down the Sine of 20d the given Azimuth thereto from H to I, because it was to the northwards of the East.  
Through the points B I, draw the line D E, and it shall be the Parallel of Declination.  
Divide the ark D E into halfs at P, and the arch P N is the Latitude 51d 32′.  
Place Z P from S to A, and the arch A D is the Declination 23d 31′. Moreover, the Amplitude, Hour, etc. are given by the Scheme, though not required.  

To Calculate the Latitude from the said Scheme.  
From the point I draw I L parallel to H F, and produce B F to L, then is I L equal to H F, the difference of the Sins of both the given Altitudes, and B L is equal to the Sum of B F and H I, and the angle B I L is equal to the Latitude of the place the first work is to find B F and H I in the Parallels of Altitude: The Proportion will be,  
As the Radius is to the Cousin of the given Altitude (being the Radius of the Parallel)  
So is the Sine of the Azimuth from East or West belonging to the said Altitude, to the Sine of the said Azimuth in its parallel of Altitude, whereby at two Operations may be found B F and H I  
When the Azimuths are on different sides the Vertical, take the Sum, when on the same side the difference between the fourth terms found by each Operation, than it holds:  
As the difference of the Sins of both the given Altitudes,  
Is to the aforesaid Sum or difference:  
So is the Radius to the Tangent of the Latitude.  
The two first terms of this latter Proportion in this Scheme, are as I L to L B.  
Or it may be expressed in two Proportions, thus:  
As the difference of the Sins of both the given Altitudes,  
Is to the Cousin of the greater Altitude:  
So is the Sine of the Azimuth (from the East) thereto belonging, to a fourth number.  
Again:  
As the difference of the Sins of both the given Altitudes,  
Is to the Cousin of the lesser Altitude:  
So is the Sine of the Azimuth thereto belonging, to a seventh number.  
The Sum of the fourth and seventh Numbers, when the Azimuths fall on different sides the Vertical; but their difference when they happen on the same side, is equal to the Tangent of the Latitude.  
Or it may be thus delivered:  
Multiply the Cousin of each Altitude by the Sine of the Azimuth thereto belonging, and there will arise two Products, the sum of these Products when the Azimuths happen on different sides, but the difference of them when they happen on the same side the Vertical, divide by the difference of the Sins of both the given Altitudes, and the Quotient will be the Tangent of the Latitude sought. This supposeth the Operations to be performed by the natural Tables of Sines and Tangents.  
Now having the Latitude, either of the Altitudes with the Azimuth theteto belonging, we have two Sides and the Angle comprehended given, to find the third Side, whereby is found the Declination required.  
How from three Shadows happening the same day on a Horizontal Plain, to Calculate the Latitude, etc. as also to perform it by Scale and Compass, with another way for finding the Azimuth, I have showed in a Treatise of Geometrical dialing, whereto the Reader is referred.    
FINIS.   

Errata.  
Page 47. Line 28. for 38d 28′, read 51d 32′.  
Page 49. Line 23. for Latitude, read Altitude.   

THE MARINER'S PLAIN SCALE NEW PLAINED: The Third PART or BOOK.  
Showing the Uses of a Line of CHORDS only, in Resolving or Projecting all the 28 Cases of Spherical Triangles on the Stereographick or Circular Projection.  
Also how from three Shadows of a Wire on a Horizontal Plain, taken the same day, to find the Latitude of the Place, the Sun's Declination, etc.  
Whence follows a general Method of dialing from three Shadows, or by help of two Shadows, etc. With other Propositions of excellent Use to Seamen and Practitioners in the Mathematics.  
Being contrived to be had either alone, or with the other Parts.  
Written by John Collins of London, Penman, Accountant. Philomathet.  
LONDON: Printed by Tho. Johnson for Francis Cossinet, and are to be sold at the Anchor and Mariner in Tower-street, and by Hen. Sutton Mathematical Instrument-maker in Thread-needle-street, behind the Royal Exchange. 1659.   

Courteous Reader,  
THis part of the Treatise is also contrived to go alone, wherein we use no other Scales but only a Line of Chords, or rather the equal Divisions of a Quadrant of a Circle, and accordingly have divided the same in the second Scheme of this Book, having shunned the use of Tangents, Semitangents, and Secants, which others use upon Scales or Sectors for this purpose; the Demonstration of this Projection the Reader will find sufficiently handled in the Sixth Book of Aquilonius his Optics, in Guidoubaldus his Theoric of Planisphaeres, see also Clavius his Astrolabe; and the two former Authors have also handled the Orthographick Projection: The Spherical Definitions necessary to be understood by the Reader, are handled in the Second Part. Not willing further to detain thee, I rest a Well-willer to the Advancement of Knowledge.  
JOHN COLLINS.   

THE CONTENTS.  

CErtain Definitions and notions about Spherical Triangles. Page 1  
The 16 right angled Cases projected. from page 2 to 12  
The Scheme for Chords used in this Book. 4  
How quadrantal Triangles arise from the right angle● 5  
How otherwise in less room than a Semicircle, to project the right angled Cases. 12  
The 12 obliqne Cases projected. from page 13 to 22  
How it ariseth from projection, that the Angles of one obliqne Triangle are equal to the sides of another. 16  
How it ariseth from projection, that in some Cases the Sun and Stars have two Altitudes upon one and the same Azimuth. 24  
How from three shadows on a Horizontal Plain, to find the Latitude of the Place, the Sun's Declination, Amplitude, and the Azimuths of those shadows. from page 25 to 28  
A general Method thence derived for Inscribing the Style and Substile into such Dial's as have Centres. 29  
Also a Scheme for placing the Meridian line in such Dial's. 31  
The Sun's Declination, two Altitudes, with the difference of Azimuth being given, how by projection to find the Latitude of the place, and the Azimuths of these shadows, etc. 33, 34  
A Method of dialing thence issuing. 34  
How from the former Data, only altering the difference of Azimuth into the difference of Time, to find the Latitude, etc. 35  
After the same manner the Declinations and right Ascensions of two fixed Stars being given, as also the Altitudes of them both, being at any period of time observed, to find the Latitude, etc. 36    

SPHERICAL TRIANGLES Solved by PROJECTION.  
A Spherical Triangle is supposed to be described on the Surface or Convexity of the Sphere.  
The sides of a Spherical Triangle, are the Arches of three great Circles of the Sphere mutually intersecting each other.  
Those are said to be great Circles which bisect or divide the Sphere in halfs.  
A Spherical Angle (being the arch of Inclination between two Circles of the Sphere) is measured by the arch of a great Circle described on the Angular point, as a Centre between the sides, being extended to Quadrants.  
Every Spherical Triangle is composed of three sides, and three angles opposed thereto, any three of which six parts being given, the rest may be discovered by Projection, and is the subject of the following Discourse.  
The Pole of a Circle is a point in the Surface of the Sphere, distant always 90d from the Circumference of that Circle to which it belongs, from whence it follows that all right lines produced to the Circumference of the said Circle, are equal.  
Any two Lines so drawn, contain an Angle between them, measured by the Arch or Circumference to which they belong. By help of this Pole point, the sides of Spherical Triangles in all the following Propositions are measured.  
A Pro●ection of the Sphere, is the Representation of the Circles of the Sphere on some Plain or Flat, as they appear to the eye, according to its supposed situation.  
By helps of such Representations of the Sphere on a Plain, the same Conclusions are performed on a Flat, as upon the Sphere or Globe itself.  
A Spherical Triangle is either right angled, or obliqne angled; a right angled Spherical Triangle hath one right angle at the least, the measure of a right Angle being always a quadrant or 90d, the Sides about or including the right Angle, are called Sides or Legs; and the Side subtending the same, is called the Hipotenusal.  
Because these Triangles are more easily resolved then the obliqne, we shall first handle the sixteen Cases thereof, and then proceed to the twelve remaining Cases of obliqne angled Spherical Triangles.  

CASE I, TWO, III. Given a Leg and its adjacent Angle, to find the Hipotenusal, the other Leg and its adjacent Angle.  

Given the Leg E Z, 46d 43′.  
Angle, E Z ⊙ 42d.  
To find the Hipotenusal Z ⊙.  
The other Leg E ⊙.  
The Angle Z ⊙ E.   
With 60d of a Line of Chords, or of a quadrant divided into 90 degrees, upon V as a Centre, describe the Circle Z S N B, and draw the two Diameters S V B and Z V N, making right Angles in the Centre, which is to be observed in the following Schemes, relating to the right angled Cases, prick 46d 43′ the given Leg, from Z to E, and from B to L, and draw the Diameter E V I  
Prick 42d the given Angle, from N to G, and from G to H, a Ruler from Z on H, cuts the Diameter at C, the Centre of a Circle to be described with the extent C Z, to wit, the Circle Z A N.  
Nota, when the point H falls in the upper quadrant Z B, the Diameter S B must be extended, and the point C will happen to be found therein without the Circle; and a remedy for the finding it with more certainty than this way affords, shall afterwards be prescribed.  
If S F be made equal to N G, a Ruler from Z on F, finds the point A in the Diameter through which the Circle was drawn, which may be found in many Cases, not for necessity, but for more certainty: this Circle, in regard we project upon the plain of the Meridian, is an Azimuth Circle.  
A Ruler from Z on G, finds the point P in the Diameter, being the Pole point of the Circle Z A N, a point that must of necessity be found, without which an Arch assigned in the said inward Circle cannot be measured in the Limb, nor can an Arch given in the Limb be transferred into the said Circle.  

The Arks sought measured.  

1. The Hipotenusal.  
A Ruler laid from P over ⊙, cuts the Limb at D, and the arch D Z, being 55d, is the measure of the Hipotenusal ⊙ Z.   

2. The Leg.  
A ruler from L on ⊙, finds the Point K in the Limb, and the arch K E, being 33d 14′, is the measure of the Leg E ⊙.   

3. The other obliqne angle.  
A ruler laid from ⊙ on P, cuts the Limb at R, and the arch L R being 62d 41′, is the measure of the angle Z ⊙ E  
Otherwise to measure this angle.  
Set off a quadrant from D to M, or which is all one, make N M equal to S D, a Ruler from P on M, cuts the Azimuth Circle at Q, than a Ruler from ⊙ on Q, cuts the Limb at T, and the arch I T being 62d 41′, is the measure of the angle I ⊙ N, which being the opposite angle to Z ⊙ E, is equal thereto, the compliment whereof to a Semicircle, is the angle Z ⊙ V.  
Any right angled Spherical Triangle, having the Sides thereof continued to quadrants, there will arise out of it five other Triangles, all made of the same parts or their compliments of the first Triangle; wherefore any Case propounded, may be varied and variously projected within the primitive or fundamental Circle, as also without it, but in the following Cases we shall keep where we began.  
To illustrate this, extend V E, and through the three Points O P L, describe a Circle, the Centre whereof happens in the extended Diameter I X, to find it upon the Points P L with any extent, make a cross with two arks near C, also the like with any other extent near Y, a ruler laid through these Intersections or Crosses, cuts the extended Diameter at X, upon which, as a Centre, describe the ark W L P O, the pole of which ark is the point ⊙, and if K Y be made equal to K O, a ruler from L on Y, finds the Centre X as before, the six several Triangles thus arise:  
1. The Triangle Z E ⊙ right angled at E.  
2. The Triangle Z W L right angled at W, whereto is equal the opposite Triangle N Q O.  
3. The Triangle ⊙ A V right angled at A.  
4. The opposite Triangle thereto V W P, right angled at W.  
5. The Triangle P B L right angled at B.  
This second Scheme is to show how all these Triangles arise out of the Triangle first proposed: the arch N B divided into 9 equal parts, being 10 degrees each, with the ten small divisions or degrees beyond it, above B is all the Scale required or used in this part of the Treatise, from whence any ark required may be taken, or any ark given may be measured.  
There are also included in the said Scheme as many quadrantal Triangles, the quadrantal Sides being pricked lines, the sides of each of which quadrantal Triangles are limited by the Pole points of that right angled Triangle from which it doth arise.  
In the quadrantal Triangle V L P, the points V, L, P, are the three Pole points to the three sides of the right angled Triangle E Z ⊙, and the sides of this quadrantal Triangle, are equal to the angles of that right angled Triangle; Also the angles of this quadrantal Triangle, are equal to the Sides of that right angled Triangle, only the right Angle and the Hipotenusal, being the greatest Side opposed thereto, are changed into their compliments to a Semicircle.  
The like also holds, if any other Angle and its opposite Side were changed into their compliments to a Semicircle.  
If the Side E Z and its opposite Angle E ⊙ Z, were so changed, and the other parts of that Triangle not altered, then would the Sides of the quadrantal Triangle O V P, be equal to the Angles of the right angled Triangle Z E ⊙, and the Angles of the said quadrantal Triangle, equal to the Sides of the right angled Triangle, and the like holds in the other quadrantal Triangles.  
The quadrantal Triangle O P V, ariseth from the right angled Triangles Z W L, or O Q N.  
So also do the quadrantal Triangles 

⊙ Z V  
Z L P  
Z L ⊙   arise from the right angled Triangles 

P B L.  
⊙ A V.  
V W P.    
From the consideration whereof, may follow much variety in the projecting of any Case proposed.  
Note in the first Scheme, that the arch N G being 42d, the extent V P is the Semitangent of that ark, that is to say, it is the tangent of 21d, and the arch N H being 84d, to wit, the double of N G, the extent V C is the tangent of 42d, and the extent Z C the Secant thereof, the extent Z V being Radius, and to take out thus the tangent or secant of any ark, because the Centre happens in the circumference of the Circle, the ark itself must be doubled, or twice pricked down in the circumference; and after this manner may we take out the Semitangent, Tangent or Secant of any ark needful for the drawing of this Projection; which how to do by Scales of Tangents and Secants on a small Ruler, is handled in a Treatise, Entitled, Posthuma Fosteri.     

CASE IV, V, VI Given a Leg and the opposed Angle thereto.  

Given in the former Scheme E ⊙ the Leg, 33d 14′: the Angle E Z ⊙ 42d.  
To find the other Leg Z E, the Hipotenusal ⊙ Z, the Angle E ⊙ Z.   
Having drawn the primitive Circle, prick 42d for the given Angle, from N to G, and from G to H, and by a ruler from Z over P, and H, find the pole point at P, and the Centre at C, and with the extent C Z, describe the Circle Z ⊙ N; then prick 33d 14′ for the given Leg from S to K, a ruler from Z finds the point F, with the extent V F cross the Azimuth circle, this Intersection will happen at ⊙, through which Point and the Centre, draw the Diameter E I  
The Arch E Z measured on the Chords, is the other Leg sought, to wit, 46d 43′.  
A ruler laid over P and ⊙, cuts the Limb at D above, and at R, in the opposite quadrant.  
The Arch Z D measured on the Chords, is 55d, being the measure of the Hipotenusal ⊙ Z.  
Place the extent B R from N to T, and the arch I T is the measure of the angle I ⊙ N, equal to the angle Z ⊙ E, to wit, 62d 41′.  
Note in the resolving of these three Cases, we may observe that the three given parts of the Triangle, to wit, the right angle, a Leg and the angle opposed thereto, are not sufficient without the quality of one of the other unknown three parts, to determine the affection of the Leg, the Angle, or Hipotenusal sought, for the extent V F will as well meet with the Azimuth Circle at O, through which point draw the Line V Q, and there ariseth the right angled Triangle Z Q O, right angled at Q, the side Z Q being 133d 17′, the Hipotenusal Z O 125d, and the Angle Z O Q 117d 19′, the compliments to a Semicircle of the Leg, Hipotenusal and Angle found in the upper Triangle, and so a double solution is to be rendered, unless the affection of the arks sought be determined from one of the parts not given.   

CASE VII, VIII. Given both the Legs.  
The Leg Z E 46d 43′.  
The Leg E ⊙ 33d 14′.  
To find either of the Angles at Z or ⊙, and the Hipotenusal Z ⊙.  
Prick 46d 43′ from Z to E, and from B to L, and draw the Diameter E I, then prick the other given Leg from E to K 33d, 14′, Lay a Ruler from L to K, and it cuts E V at ⊙, through the points Z ⊙ N describe a Circle, the Centre whereof will be easily found by two Intersections, the one above Z ⊙ at g, the other beneath it at y, the arks of those Intersections being described upon the points Z and ⊙, with any two extents at pleasure, but alike on each; A Ruler laid through the Crosses g y, cuts the Diameter at C the Centre required, and when the Centre doth not fall within, V B must be extended without.  
Having described the Circle Z A N, to find the Pole thereof lay a Ruler from Z to A, and it cuts the Limb at F, place S F from N to G, a Ruler on it from Z finds the Pole point required at P, a Ruler laid over ⊙ and P, cuts the Limb at D and R.  
The Arch Z D measured on the Chords is 55d, the measure of the Hipotenusal Z ⊙.  
The Arch S F being 42d, is the measure of the angle E Z ⊙.  
The Arch L R being 62d 41′, is the measure of the angle Z ⊙ E  
Though we find three Requisites, yet we have but two Cases, because the finding of either of the obliqne Angles, is but one Case.   

CASE IX, X, XI. Given the Hipotenusal and its adjacent Angle.  
The Angle E Z ⊙ 42d.  
The Hipotenusal Z ⊙ 55d.  
To find the 

Side Z E.  
Side E ⊙.    
And the Angle Z ⊙ E.  
Prick off 42d, the given angle from N to G, and from G to H, a Ruler from Z on G, and H, finds the Pole and Centre at P and C; Upon C with the extent C Z, describe the Circle Z A N.  
Prick off 55d the given Hipotenusal from Z to D, lay a Ruler from D to P, and it cuts the Azimuth Circle at ⊙; through ⊙ and the Centre draw the Diameter E I  
The Arch E Z measured on the Chords is 46d 43′, being one of the Legs sought.  
Place Z E from B to L, then a Ruler laid over L and ⊙, cuts the Limb at K, and the arch E K measured on the Chords is 33d 14′, the measure of the other Leg.  
A Ruler from ⊙ on P, cuts the Limb at R, and the arch L R being 62d 41′, is the measure of the Angle Z ⊙ E.   

CASE XII, XIII, XIV. Given the Hipotenusal and one of the Legs.  
The Hipotenusal Z ⊙ is 55d.  
The Leg Z E 46d 43′.  
To find the other Leg E ⊙, the Angle E Z ⊙, and the Angle Z ⊙ E.  
Having drawn the fundamental Circle Z S N B, and the two Diameters S V B & Z V N, prick 55d the given Hipotenusall from Z to D and H, from B to O, and from O to Q, a Ruler laid from S to Q, cuts the Diameter N Z extended at M, then place the extent S M from V to Y upwards, and upon Y as a Centre, with the extent Y D, draw the Circle D W H: otherwise with the extent V M, place one foot at D, and the other will reach to Y.  
Otherwise a Ruler laid from S to H, cuts Z V at W, and then it is required to find the Centre of those three Points, which may be easily found by two Intersections, because it happens in V Z extended: or lastly, setting one foot in S, opening the other to B, describe the ark B X, then double the extent Z D in a right line, and it reaches from Z to T, which extent place from B to X, and a Ruler laid from S to X, finds the point M as before; by the like reason an extent might be tripled or quadrupled, provided the Radius be increased after the same rate, and this remedy must be used if the point Q chance to fall too near unto S.  
Prick 46d 43′ from Z to E, and draw the Diameter E V I, which intersects the Circle D W H at ⊙, having the points Z, ⊙ and N, through them draw a Circle, the Centre whereof C, is found by applying a Ruler through the two crosses within, as before was instanced.  
Prick Z E from B to L, a Ruler on L and ⊙, cuts the Limb at K, and the arch E K being 33d 14′, is the measure of the Side or Leg E ⊙.  
A Ruler laid from Z over A, cuts the Limb at F, and the arch S F being 42d, is the measure of the Angle E Z ⊙.  
Place S F from N to G, a Ruler laid over Z and G, finds the Pole point of the Azimuth Circle at P, a Ruler laid over ⊙ and P, cuts the Limb at R, and the arch L R being 62d 41′, is the measure of the Angle Z ⊙ E, and all this may be otherwise performed; for the Case propounded in this Triangle in the Triangle P B L of the first Scheme, is another Case before handled, to wit, the fourth, fifth and sixth, and then having resolved that Triangle, by consequence this is also resolved.   

CASE XV, XVI. Given both the obliqne Angles to find (both) either of the Legs, being one Case, and the Hipotenusal the other Case.  
Let there be given the Angle E Z ⊙ 42d.  
The Angle Z ⊙ E 62d 41′.  
This Case cannot be resolved by projecting the Triangle E Z ⊙, but some other of the Triangles thence arising.  
Repair to the first Scheme. In the Triangle Z W L right angled at W, we have the Leg L W given, being 27d 19′, the compliment of the given Angle Z ⊙ E; Also in that Triangle we have the Angle opposed thereto W Z L given 42d, and the resolution of that Triangle would be the same with the fourth, fift, and sixth case, as before. Or in the Triangle P L B, we have the Hipotenusal P L given, and one of the Legs P B, and the resolution would be the same with the 12, 13, 14 Cases.  
Which may be also resolved in the Triangles ⊙ A V or V P q, the position of any of which Triangles may be the same with that we have hitherto handled, and having resolved any of them other Triangles, the Triangle proposed is also resolved, because the parts of the one are the same or the compliments of the other.  
Sun's place 60 degrees from Equinoctial point, greatest declination 23 degrees, 31 minutes.  
We have before said, that any of the right angled Cases might be projected without the primitive Circle: take one example thereof:  
Let the Sun's place and greatest Declination be given, to find his right Ascension and Declination.  
Draw the Diameter A V Z, and upon V as a Centre, describe the quadrant V Z B, and prick 23d 31′ (the Angle between the Equinoctial and Ecliptic) from Z to F, and from F to D, a Ruler from A finds the Pole at P, and the Centre at C. Upon C, with the extent C Z, draw the arch Z ♋, then prick down 60d the Suns given place from Z to e, a Ruler laid from P to e, cuts the outward Circle at ⊙, draw ⊙ V, the arch Z a measured on the Chords, is 57d 48′, and so much is the Sun's right ascension.  
Place the extent V ⊙ from V to f, a Ruler laid from Z to f, cuts the inward Circle at o, and the arch B o being 20d 12′, is the Sun's Declination required.  
To measure the other obliqne Angle, would require the continuance of these Circles about; for if the Circle Z B were continued round, a Ruler laid from ⊙ to P cutting it on the other side, the ark of distance of that Intersection from ⊙ V produced till it also cut it, would be equal to the compliment of the Angle sought; and this agrees with Case 9, 10, 11, only there the work is performed within the primitive Circle, here without, by continuing the inward Circle about, there represented by the Azimuth Circle Z A N.   

CASE I. Of obliqne angled Spherical Triangles.  

Three Sides given to find an Angle.  
THis being the most useful Case, I thought fit to handle first, because of some Observations from it, which will further us in the solving of some following Cases.  
We refrain drawing a Triangle apart, because the Scheme will serve, though the construction thereof is afterwards prescribed.  
In the precedent Triangle ⊙ Z P, let there be given the three Sides, ⊙ Z the compliment of the Sun's height, 37d 53′, Z P the compliment of the Latitude, 38d 28′, and ⊙ P the Sun's distance from the visible Pole 66d 29′, and let it be required to find the three Angles, Z P ⊙ the hour from noon, P Z ⊙ the Sun's Azimuth from the North in that Hemisphere, and the Angle Z ⊙ P, being the Angle of Position.  
Having upon V as a Centre, described the fundamental Circle S Z B N, and drawn the two Diameters S V B and Z V N, making right Angles in the Centre, and produced without the Circle, prick any one of the given Sides into the Limb from Z to P, and from S to A in this Scheme, 38d 28′ the colatitude, and draw the Axis P V F, and the Equator A V Q, and upon A as a Centre with A Q, describe the ark Q X.  
From one extremity of the Side placed in the Limb as at P, prick off one of the other given Sides, to wit, 66d 29′ from P to D, and if you will also to E; prick it likewise twice in the Axis from F to t, the said extent F t place from Q to X, a Ruler laid from A to X, cuts the extended Axis at M, and the extent V M will reach from D to Y, the Centre of the parallel of Declination D E, wherewith describe it.  
A Ruler laid from Q to D, cuts the Axis at W, a third point through which that Circle is to pass, if it be thought needful to limit it.  
From the other extremity of the Side placed in the Limb as at Z, prick off the third Side 37d 53′ to A, also prick it twice from B to G and H, a Ruler laid over H and S, cuts the Diameter Z N at I: One foot of the extent V I being placed at A, the other will meet with the Diameter N Z produced at L, the Centre of the parallel of Altitude A ⊙, then describe it.  
Through the three points Z ⊙ N draw a Circle, the Centre whereof happens at C, and may be found by applying a Ruler through the two pricked Intersections, which are found as before; also through the three points P ⊙ F draw a Circle, the Centre whereof will be found at g, then is the Sphere projected for the resolution of the arks sought, which are thus found:  

1. The Angle Z P ⊙ being the Hour from Noon.  
Lay a Ruler from P to h, and it cuts the Limb at K, the ark A K being measured on the Chords, is 33d 15′, and so much is the hour from noon, in time 2 hours 13 minutes, either so much in the afternoon, or 9 hours 47 minutes in the morning.   

2. The Angle ⊙ Z P being the Sun's Azimuth from the North.  
Lay a Ruler over Z and n, and it cuts the Limb at O, and the arch S O being 55d, is the measure of the Sun's Azimuth from the South, the compliment whereof to a Semicircle being 135d, is the Angle ⊙ Z P.   

3. The Angle of Position Z ⊙ P.  
Place S O from N to o, and a Ruler over it and Z, finds the Pole of the Azimuth Circle at R, also place A K from F to a, a Ruler laid over P and a, cuts the Equator at T the Pole of the Hour-circle, then apply a Ruler over ⊙ and R, and it cuts the Limb at c, apply it also over ⊙ and T, and it cuts the Limb at d, the arch c d measured on the Chords, is 33d 45′, and so much is the Angle of Position Z ⊙ P.  
Otherwise:  
Set off a quadrant in the Azimuth Circle from ⊙ to f, by help of its Pole R, also set off a quadrant in the Hour-circle from ⊙ to e, by help of its Pole T, as was showed in the first Case of right angled Triangles, a Ruler applied from ⊙ to e and f, cuts the Limb at j and k, and the arch j k measured on the Chords, is 33d 45′, the measure of the Angle of Position, as before.  
From the former Scheme in the Triangle F ⊙ N, we may observe that every Spherical Triangle hath opposite to each angular Point another Triangle, having the side subtending the said Angle common to both, and the Angle opposite thereto equal the other parts of it, are the compliments of the former parts to a Semicircle; so that if a Question were put in that Triangle, it might be conveniently resolved in this we have handled.     

CASE II. Three Angles given to find a Side.  
This Case is in effect the same with the former, after the Angles are changled into Sides; and how that is to be done ariseth from the former Scheme.  
From the Centre draw a Line passing through the Point ⊙, as doth V m, in that Line doth happen the Centre of a great Circle that passeth through the two Pole points R, and T, as also through the Points f e, the Centre whereof at m may be found by two Intersections described on T and R, but here we have made use of two Intersections describedd on f and e.  
This pricked Circle being drawn, look for the three Pole points of the Triangle ⊙ Z P, which are V, R, T, and they limit the three sides of another Triangle, which sides are equal to the Angles of the former Triangle, only one of them is changed into its compliment to a Semicircle.  
Thus the Side V R in the under Triangle being 55d, is the compliment of the Angle ⊙ Z P in the upper Triangle to a Semicircle.  
The Side V T in the lower Triangle being 33d 15′, is equal to the Angle ⊙ P Z in the upper Triangle.  
And the Side T R in the lower Triangle being 33d 45, is equal to the Angle Z ⊙ P in the upper Triangle.  
And as the Angles are changed into Sides, so are the Sides into Angles, by writing them down in two several Triangles opposite one to the other.  
This mutual conversion is most convenient in Calculation, to avoid recourse to the opposite Triangle, when the greatest Angle and its opposite Side are changed into their compliments to a Semicircle, but is true of any Side and its opposite Angle so changed; for any other of the three Sides, may be placed in the Limb as well as that here instanced, and in many Cases with more convenience, as when one of the three Sides is near a quadrant, and the other two under 70 degrees each.  
Having changed the Angles into Sides, the Case is the same as before.   

CASE III, IV. Two Sides with the Angle comprehended, given to find the third Side, and either of the other Angles.  
Admit in the former (double) Triangle, there were given the Sides 

⊙ Z 37d 53′,  
Z P 38d 28′,   and the Angle between them ⊙ Z P 125d the Sun's Azimuth from the North, to find the rest.  
Having drawn the primitive Circle with its two Diameters, making right angles in the Centre, from the extremity of one of them, as at Z, prick off one of the Sides from Z to P, and draw P V. Prick off 125d the given angle from B to M, a ruler over Z and M finds A, through the points Z A N describe a Circle, S M being placed from N to O, and from O to G, a ruler from Z finds the pole of that Circle at R, and the Centre at C.  
Prick the other given Side from Z to A, a ruler from R cuts Z A N at ⊙, through the points P ⊙ F describe a Circle, by applying a ruler through the two Intersections at p and q (made upon the points ⊙ and P, on one side with any extent at p, and on the other side with the same, or any other extent, as at q) the Centre thereof will be found at g, and the Sphere is projected.  

To find the angle Z P ⊙.  
Apply a ruler from P to h, and it cuts the Limb at K, and the arch A K, being 33 degrees 15 minutes, is the hour from noon.   

To find the Side ⊙ P.  
Place A K from P to a, a ruler from P finds the pole of the hour Circle at T, a ruler applied over T and ⊙, cuts the Limb at D, and the arch D P being 66d 29′, is the measure of the Side ⊙ P.  
The angle of Position Z ⊙ P is found in the Limb below B, as before, by applying a ruler from ⊙ over R and T, being 33d 45′.    

CASE V, VI Two Angles with the Side between them given, to find the third Angle and the other Sides.  
In the former distinct Triangle V R T, if there were given the angles V R T 37d 53′, the angle R V T 38d 28′, and the side between them V R 55d, by changing these angles into sides, and that side into an angle, by taking its compliment to 180d, there would be given, the same two sides and the angle between them now resolved, and having found the third side to be 66d 29′, because the angle opposed thereto, was the compliment of the given side in the lower Triangle to a Semicircle, therefore the compliment to a Semicircle of the third side found in the upper Triangle, to wit, 113d 31′, is the third Angle sought in the lower Triangle, and the other Angles found in the upper Triangle, are equal to the sides sought in the lower Triangle.   

CASE VII, VIII, IX. Two Sides with an Angle opposite to one of them given, to find the third Side, the angle included, or between them and the Angle opposed to the other Side.  
Admit there were given in the former (distinct) Triangle, the Side Z ⊙ the compliment of the Altitude 37d 53′, the Side ⊙ P 66d 29′, the compliment of the Declination, and the Angle P Z ⊙ the Sun's Azimuth from the North 125d, to find the rest.  
Having drawn the primitive Circle S Z B N, and the two Diameters S V B, Z V N, place the side next the given angle in the Limb from Z to ⊙, to wit, 37d 53′; also place it from S to A, and draw the two Diameters ⊙ V F and A V Q. Place the other given Side 66d 29′, from ⊙ to D and E, also place it twice from Q to I and K, a ruler over A and K cuts the extended Axis at M, the extent V M placed from D, reaches to y in the extended Axis, the Centre of the parallel of Declination D E, wherewith describe it.  
Then place 55d the compliment of the given Azimuth to a Semicircle from N to G, and from G to H, a ruler from Z finds the pole of the Azimuth Circle at R, and the Centre at C in the extended Horizon S V B, thereon describe it, to wit, Z P N.  
Through the three points ⊙ P F, draw the arch of a Circle, the Centre whereof will be found by help of the two pricked Intersections at g, and the Sphere is projected.  
A ruler laid over R and P, cuts the Limb at A, and the arch Z A being 38d 28′, is the compliment of the Latitude sought, being the measure of the side P Z.  
A ruler applied from ⊙ to h cuts the Limb at f, and the arch A f being 33d 45′, is the measure of the angle Z ⊙ P, the angle of Position sought.  
Place A f from F to a, then laying a ruler from it to ⊙, it cuts V Q at T.  
A ruler laid from P over R and T, cuts the Limb at c and d, and the arch c d measured on the Chords, is 33d 15′, and so much is the hour from noon, being the measure of the angle Z P ⊙.   

CASE X, XI, XII. Two Angles with a Side opposite to one of them given, to find the third Angle, the Side between the given Angles, and the Side opposite to the other Angle.  
Admit in the double Triangle, in the undermost there had been given the angle V R T 37d 53′, the angle R T V 113d 31′, and the side V R 55d, this by changing the angles into sides, and the side into an angle, had been no other than the Cases now resolved.  
These six Cases I have often called the doubtful Cases, though in the former example there happens no Doubt, yet a question may be propounded in the same Triangle, whereto there will arise a double Solution, and both true.  

Two Sides with an Angle opposite to one of them given, to find the rest.  
As if in the Triangle oft  there were given  
The Side ⊙ P 66d 29′ the compliment of the Declination.  
The Side ⊙ Z 37d 53′ the compliment of the Altitude.  
The Angle Z P ⊙ 33d 15′ the hour from noon, and it were required to find the rest.  
In the following Scheme, the Side next the given angle ⊙ P, to wit, 66d 29′, is placed in the Limb from ⊙ to P, and from B to A, draw P V F and V A  
Place 33d 15′ the given Angle, from F to a, and from a to e, and thereby find the pole at R, and the Centre at C, by a ruler applied from P, and draw P Z F  
Also place 37d 53′ the other given side from ⊙ to A, and twice from B to G and H, and by a ruler over H find M, and with V M placed from A to Y, describe the parallel of Altitude A L, which passeth through the hour Circle at two places, to wit, at z Z (and therefore a double Solution must be given) through both those points draw the Circles ⊙ Z N and ⊙ z N, and find the pole of ⊙ Z N, which will happen at u.   

The Arks sought measured.  
1. Apply a ruler from R to Z, and it cuts the Limb at m, and the arch P m measured on the Chords, is 38d 28′ the compliment of the Latitude sought.  
2. Apply a ruler from ⊙ to k, and it cuts the Limb at n, and the arch B n measured on the Chords, is 33d 45′ the angle of Position sought.  
3. A ruler laid from Z over R and u, cuts the Limb at o and t, and the ark o t being 55 degrees, is the compliment of the angle ⊙ Z P to a Semicircle, wherefore the Sun's Azimuth is 125d, being the angle ⊙ Z P.   

In the other Triangle ⊙ z P.  
1. Lay a ruler from R to z, and it cuts the Limb at x, and the arch P x being 86d 32′, is also the compliment of the Latitude sought.  
2. Apply a ruler from ⊙ to b, and it cuts the Limb near a, the Ark between the Intersection found, and B being 116d 58′, is the measure of the angle of Position P ⊙ z.  
3. The acute Angle ⊙ z P is 55d, as we found it in the former Triangle.  
Whence 'tis evident, that unless the quality of the said Angle be also known, a double answer must be given, as here we have done, which I have demonstrated in a Treatise, Entitled, The Sector on a Quadrant.  
In like manner, if in the former Triangle there were given the compliment of the Latitude 38d 28′, the compliment of the Altitude 37d 53′, and the hour from Noon 33d 15′, to find the rest, a double solution would arise.  
First,  
The angle of Position would be— 33d 45′.  
The compliment of the Declination would be-66, 29.  
The Azimuth from the North would be— 125, 00.  
Secondly,  
The angle of Position would be— 146d 15′.  
The compliment of the Declination would be-00, 43.  
The Azimuth from the North— 00, 38.   

A third Example.  
Suppose in the Latitude of Rome 42d, it were required to find what Altitude the first Star in the great Bear's tail next the rump, Declination 57d 51′, shall have upon the Azimuth of 22d from the North.  
Having drawn the primitive Circle H Z B N, with its two Axes H V B and Z V N at right angles, place the compliment of the Latitude 48d from Z to P, and from H to A, and draw P V F and A V Q, then place 32d 9′ that Stars polar distance from P to D, and twice from Q to G, by a ruler from A to G, find M, the extent V M will reach from D to C, the Centre of the parallel in the extended Axis, therewith describe the parallel D * S, then place 22d from N to O, and from O to e, a ruler over o and e from Z, finds the pole of the Azimuth Circle at d, and its Centre at f, now having the Centre, describe the said Circle Z S N.  
Which because it passeth through the parallel of Declination in two places, at * and S, that Star will have two Altitudes on that Azimuth, and a double Solution must be given.  
Through the three points F * P and F S P, draw two Meridian's.   

To measure the Arks required.  

In the Triangle Z * P.  
1. A ruler laid from P to h, cuts the Limb at K, and the arch A K being 12d 19′, is the measure of the angle Z P *, being the Stars hour from the Meridian.  
2. A quadrant placed from K towards N, and a ruler laid over the Intersection found and P, will find the pole of that hour Circle at a, than a ruler laid from * to d and a, cuts the Limb at k and u, and the arch k u being 31d 32′, is the measure of the angle Z * h, wherefore the angle Z * P is 148d 28′.  
3. A ruler from d over *, cuts the Limb at t, and the arch Z t being 17 degrees 39 minutes, is the measure of the Side Z *, wherefore the greater Altitude is t B, 72 degrees 21 minutes.   

Again in the Triangle Z S P.  
1. A ruler over P and E cuts the Limb near N, the ark between the said Intersection and A being 137d 25′, is the measure of the angle Z P S.  
2. A ruler from d over S, cuts the Limb at i, and the arch Z i being 74d 3′, is the measure of the Side Z S, wherefore the lesser Altitude is i B 15d 57′.  
3. The arch k u being 31d 32′, is the measure of the angle Z S P, being the angle of Position.    

From three shadows of a Gnomon or Wyre on a Horizontal Plain, to find a true Meridian-line, and thereby the Azimuths of those shadows, the Latitude of the place, the Sun's Declination, Amplitude, Altitudes, and the Hour of the Day.  
The Gnomon is supposed to be perpendicular to the Plain it stands upon, or at least a point in the said Plain must be found, through which a perpendicular let fall from the tip would pass, and from the said Point the lengths of the three shadows must be measured.  
In the whole Circular Scheme following, let C B represent a Wire or Stile standing erect on a Horizontal Plain, and let the three shadows thereof be C F, C E, C D.  
Upon C as a Centre, with 60d of a Line of Chords, describe the Circle O S Z N, and produce the three lines of shadow beyond the Centre towards H I G.  
Then in another Scheme, upon B as a Centre, describe the Semicircle A G N with its Diameter A B N, which divide into two quadrants with the perpendicular B G, then make B C equal to the height of the perpendicular Style, and draw C F parallel to B G, and therein prick down the three lengths of shadows from C, to E, to D, and F, and from those points draw lines into the Centre, cutting the Limb at E, I, H, and the arks between those points and G, are the respective Altitudes or heights of the Sun, but if measured from A, they are the compliments of those Altitudes.  
Then lay a ruler from N to the three points E, I, H, and it will cut the Radius B G at K, L, M, the distances of which points from B, are the Semitangents of the compliments of the Suns three Altitudes.  
Then repair to the following Circular Scheme, and place B K on the shadow E, produced from C to I, also make C G on the second shadow equal to B L; likewise make C H on the third shadow equal to B M in the Scheme above.  
Then through the three points H I G draw a Circle, the Centre whereof will be found at V; to find it with any extent upon G, describe an ark at a, with the same extent upon I, cross the former ark, do the like with the same or any other convenient extent at e; also upon the points H and I, do the same at o, also beneath at u, then lay a ruler over the Intersections a e, and draw a line near V, do the like through the other Intersections at o and u, where these lines cross as at V, is the Centre of the arch Q H I G R, then describe it, and from V draw the line V C S passing through the Centre, and it is a true Meridian-line, or line of North and South, from which the Azimuths of any of the shadows may be measured, perpendicular thereto draw O Z passing through the Centre.  
The arch Q O measured on the Chords, showeth the Sun's amplitude.  
A Ruler laid from O to A, cuts the Limb at M, and the arch S M is the Sun's Meridian Altitude.  
The nearest distance from Q to O C, is the Sine of the Sun's Amplitude, which place from C to K, when it happeneth on that side of the Vertical, and draw M K produced, and it shall be the parallel of the Sun's Declination in the Analemma, if we make S N the Horizontal line; from C draw a line parallel to K M, and it will cut the Limb at A, the arch M A is the measure of the Sun's Declination, and the arch S A is equal to the compliment of the Latitude, consequently the arch Z A is the measure of the Latitude of the place.  
Place the said Extent from N to L, then a ruler laid over it and O, cuts the Meridian at P, which is the projected pole Point, through which point and the points H, I, G, if there be arks of great Circles drawn, the angles that the said arks make with the line S P, shall be the measure of the respective hours from Noon proper to each shadow: How to draw such arks, I have explained in a Treatise of Geometrical dialing, page 49.— See also Clavius de Astrolabio, Liber secundus, Prop. 13. who handles it largely.  
This manner of finding the Hours I confess is troublesome, and may be sooner resolved by the Analemma: now we have the Latitude, Declination, and all the Altitudes given.  
In northwardly Regions where the Sun for some competent season doth not rise nor set, he hath no amplitude in that case, the Circle to be drawn through the three points, will fall altogether within the outward Circle, and may very well be described, and it will cut the Meridian S N in two points, the one towards the South, the other towards the North, a ruler from the East point O over each of the former points, will give the Sun's south and north Meridian Altitudes in the Limb; And if the ark between these Meridian Altitudes be divided into halfs, the point in the middle so found, is the Latitude point L, the ark between which and N, shows the Latitude required; after the same manner might the point L be found in all other Cases, which notwithstanding to avoid the drawing of a whole vast Circle we shunned, though when the Sun is near the Winter Tropic, the drawing of part thereof cannot be shunned, without the help of a Steel-bowe or some such like remedy, whereby to describe part of a Circle thorough three points given, which is the great and only inconvenience of the Stereographick Projection.  
Clavius handles this Proposition after another manner, but not so convenient, and makes no further use of it.  
But here we shall also apply it to the making of all Dial's that have Centres, except the Equinoctial. In a Treatise of Geometrical dialing from page 72 to 82, I have showed a general method both by Calculation, and with Scale and Compasses, how from three shadows to make all Dial's with Centres, from whence those that have leisure, may calculate the Arks found in the former Scheme, whereto I now further add, that the former Proposition here insisted on performeth the same; the three shadows C D, C E, and C F here, are the same in length with those there, and make the same angles at the Centre here, as those did at the foot of the perpendicular Style there; and supposing the former Plain to be a Declining Leaning Plain in some other Latitude, the Line S C N is the Substilar Line, and the Arch Z A is the Styles height, the same as we found it in that Treatise: Suppose this be a Plain that looks southward, not much leaning from the Zenith, if the former shadows happened in the Summer half year in north Declinations, the Scheme will describe a Winter parallel of as much south declination; also if the Sun had south declination, the Scheme would describe a parallel of as much north declination; but this is only true when the elevated Pole is elevated above the northern or other face of the Plain, to illustrate this.  
In the Scheme above let S C N represent the Horizon of a certain place, the Latitude whereof is Z A, or N L, the Equator is represented by A Q, and D E represents a parallel of North Declination, let Z C represent a Vertical or upright South Plain in that Latitude, but if the said Vertical Plain become an Horizontal Plain, the height of the North Pole will be Z L, and the height of the South Pole H K, I say then, in respect of the first Horizon, the Sun being in the parallel of North Declination between D and A, his Altitudes on any hour from Noon (being the Angles between the Wall and the Sun) above the Vertical Plain Z A in the first Horizon, are in respect of the second Horizon, if the North Pole be elevated, his depressions under the same upon the like hour from midnight, which are equal to his Winter Altitudes above the second Horizon Z C H, the Sun having as much South Declination being between B and G in the parallel F G; or, which is all one, retaining the Sun's Declination the same, they are his Winter Altitudes upon the like hour from Noon, when the South Pole K is elevated above the Horizon H C Z, which is the reason why the Scheme describes a parallel of South Declination, when his Declination was as much North.   

To place the Meridian Line.  
This, as I have showed in the  Treatise, may be projected by the eye on the Plain, by help of a Thread and Plummet, and if the Plains 

Re clination  
In clination   be given, together with the Substiles distance from the Plains perpendicular (both which may be got without dependence on the Sun) the Meridian's place may be Calculated, or at least the Inclination of Meridian's as was there suggested, either will serve; But for the placing of the same (and for finding the Latitude of the place, and the Plains Declination, though not required in the making of the Dyal) the Converse of the dialing Scheme there used will perform it. We shall take the Example there used.  
Let the Substiles distance from the Plains perpendicular be— 32d 16′.  
The Styles height— 41 30.  
The Plains Inclination be— 15.  
Upon V as a Centre, describe the Circle S E N W, and draw the Diameters S V N and W V E, making right angles in the Centre, then assuming V N to represent the Plains perpendicular, set off the Substiles distance from it the same way it happened from N to Y, to wit, 32d 16′, and from Y prick off the Styles height to K, from the point K let fall a perpendicular on the Substile at I, and draw I C parallel to N V, from which point let fall I Q produced perpendicular to the Plains perpendicular V N, and make Q T equal to I K, and upon V as a Centre, with the extent V T, describe an ark in the other quadrant of the same Semicircle, as at B, then from the same end of the Horizontal Line prick the Inclination from E to R upwards, and draw V R (but if the Plain recline, it must be pricked downward towards N) then with the extent I K draw a Line parallel to R V, and it will pass through the former ark at B, which here we found by entering one foot of that extent in that ark, so that the other turned about, would but just touch V R: Having discovered the point B, through it draw B O parallel to E C, then take the nearest distance from C to R V, and place it in the Line O B from P, on that side thereof which is farthest from R V, here we placed it from P to M, then from the Centre draw a line passing through the point M, as doth V X, and it shall be the Meridian Line required.  
Through the point O draw V D, and the arch N D shows the Declination of the Plain to be 40d, and the extent V O is the Cousin of the Latitude; Or through the point B draw G B L parallel to V N, and the arch N L is the measure of the Latitude of the place, to wit, 51d 32′. Now having placed the Substile, Stile, and Meridian, the Hour-lines are easily inscribed either in a Circle or Paralellogram, as in that Treatise is largely showed.  
For the Readers recreation and practice of what is here delivered, and for trial how well this kind of dialing agrees with other kinds, he may make a Dyal true to any known Latitude and Situation, and from any point in the Style let fall a Perpendicular to the Substile, the said Line is called the Perpendicular Style, and then by help of Mr. Leybourns Appendix to Mr. Stirrups dialing, inscribe any parallel of Declination into his Dyal, and from the foot of the perpendicular Style draw three Lines, making any angles, limiting them in the said described parallel of Declination, the lengths of which three Lines and their angles retained, he may assume the same to be the three shadows of the forementioned perpendicular Style, and therewith proceed to the finding of the place of the Substile from those shadows, and of the height of the Style above the same, as if all things else were unknown, and then giving or finding the true Situation of the Meridian line, proceed to the finishing of the Dyal, and find it the very same as was at first made.   

By two Shadows on a Horizontal Plain, and the Sun's Declination given, to find a true Meridian line, the Latitude of the place, the Amplitude, etc.  
Or, in regard the lengths of shadow give both the Altitudes and the Angle between them is the difference of Azimuth between those shadows, we may propound it otherwise.   

Two Altitudes of the Sun, with the difference of Azimuth between them, and his Declination given, to find the Latitude of the place, and the Sun's true Azimuth.  
In the following Scheme, let the first shadow be Z Y, and the Altitude thereto 63d 51′, and the second shadow Z K, and the Altitude thereto 41d, 34′, therefore the difference of Azimuth between these shadows, is the ark Y K.  
Upon Z as a Centre, with 60d of a Line of Chords, describe a Circle, and produce the said shadows through the Centre to the opposite side far enough; set off a quadrant from each shadow, to wit, from Y to Q, also from K to q, then prick the Altitude belonging to the first shadow, to wit, from y to A; also prick the Altitude belonging to the other shadow from k to a, then prick the Sun's polar distance, to wit, 66d 29′ from A to D and E, also place it from a to d and e, then apply a ruler over Q, and D, and E, and it cuts the line of shadow y Y, at B and G, which extent divide into halfs at H, and upon H as a Centre with H B, describe the arch of a Circle.  
Again, a ruler over q, and d, and e, cuts the other shadow extended at L and M, the middle of which extent is at C: upon C as a Centre with C L, describe the arch L P the Northern Intersection of these two Arks happeneth at P, and there is the Pole point, through which draw the line Z P N, and it shall be the Meridian line required.  
It may so happen that the point M may fall very remote, in that case we may find the Centre C, without finding the whole Diamater, draw a line from a into the Centre at Z, and prick down the versed Sine of the Polar distance from a to R, or which is all one, prick down the Sine of the Declination 23d 31′, from Z to R, the taking out of a Sine is easily done, as I have showed in the Analemma. A ruler from q over R, cuts the Limb at V, make a T equal to a V, than a ruler from T over q, cuts the shadow Line at C, the Centre sought. See Clavius de Astralabio, Liber 3. Canon 12. who propounds the Case here resolved.  
And what hath been here performed for finding the Meridian line, and Latitude of the place by the help of two shadows on a Horizontal Plain, will find the Substilar line and Styles height on any other Plain.  
Another Proposition like this, and performed after the same manner, is:   

To give the Sun's Declination, two Altitudes with the difference of Time between them, to find the Latitude of the place, and the respective times of the day, answering to those Observations.  
Declination 23d 31′ North.  
Altitudes 

the lesser 27d 23′.  
the greater 53d 45′.    
Difference of time 7 hours, or 105d.  
Again, prick off the compliment of the lesser Altitude, to wit, 36d 15′ from d to a and l, a ruler over q and a, cuts K P at R, also a ruler over q and l, cuts P K produced at F, the half between F and R is at G, with the extent G R, upon G and a Centre, describe the ark R Z; where these arks intersect, as at Z, is the Zenith, draw the line S M passing through the said Intersection and the Centre, and it shall represent the hour-line of 12, the arch S K shows the hour to the greater Altitude to be 30d from the Meridian, or 2 in the afternoon, and the arch S V shows the hour proper to the first observation to be 75d, or 7 in the morning.  
Set off a quadrant from S to E, a ruler over E and Z, cuts the primitive Circle at B, and the arch S B being 51d 32′, shows the Latitude of the place to be 51d 32′.  
When the Declination is South (being in our Northern Hemisphere) it must be pricked below the Hour-lines V P and K P, whereas in this Example being North, it was pricked above them; Or if the Declinations, the Difference of right Ascensions (which is the difference of Time) and the Altitudes of two Stars were given at any time, though off the Meridian, the Latitude and true time might be found after the same manner; for those Cases of Triangles (though the two Declinations are different) are the same with those here resolved, as I may have occasion elsewhere to calculate, whereto this will be a good check; which Propositions may be of good use at Sea, for finding the Latitude by Observations taken off the Meridian.    
FINIS.