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 Cen­ter, describe the arch of a Circle as D G, prick the Extent of the Compasses from D to G, and up­on G as a Center, with the said Extent, describe the ark D E G, in which prick down the Extent of the Compasses twice, first [Page 3] from D to E, and then from E to F, then joyn the points F D with a right line, and it shall be the Perpendicular required.

Prop. 2. To raise a Perpendicular from the midst of a Line.

Though this may be per­med by the former Propo­sition, yet there remains an­other way of doing the same

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 streight 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 [Page 5] 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 troublesom, 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 finde the Point C.

Prop. 3. To draw one Line parallel to another Line, at any distance required.

A Line is then said to be parallel to another Line, when it is every where a­like, or equally distant from it, so that both those Lines produced or continu­ed, would never meet.

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 Dyalling, 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 finde one Point: Again, finde 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 streight Line.

In the Figure adjoyning 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 Center 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 finde their angular point of con­currence or meeting, without producing or continuing the said Lines.

Let the two Lines given be A B, and C D, and let it be required to finde the point of their meeting (as at I, if they [Page 7] were continued) without producing the said Lines: Over the two lines given, draw two other lines parallel to each other, as are G H, and F E, and prick the extent B D twice from B to G, also the extent A C twice from A to F, then if the line A B and F G be produced, they will finde the point I, being the point desired, and the extent A C and B D may be multiplied as oft as shall be thought convenient.

And if they be multiplied the other way, from D to H, and from C to E, then 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 signifie a flat Figure, having onely length and breadth, the four Angles whereof are right Angles, the opposite sides whereof are equal.

In Proportions the product of two tearms 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 it self 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 onely are equal, either of these Figures are commonly called Oblique Angled Parralel­lograms: Thus either of the Figures A B, F E, or B C, E D are Rhomboides, or Oblique Angled Parralellograms.

This foregoing Figure is much used in Dyalling, 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 oblique 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 extreamities 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 four­sided 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.

Cases of Plain Triangles being right angled.

Cases of Oblique Plain Triangles.

1. Proportions in Sines alone.

LEt the Proportion be:

As the Sine of 19^d to the Sine of 25^d, So is the Sine of 31^d.

To what Sine? to wit, the Sine of 42^d.

Double the two first Arks, and they are 35^d and 50^d, and ha­ving drawn the line V I, prick down the Chords of these Arks, to wit, of 38^d from V to A, and of 50^d from V to B, then on V as a

Center, draw the Quadrant I H and its Ra­dius H V per­pendicular to I V, and prick off the third tearm of the Proportion 31^d from I to C, and draw C V, from B the se­cond tearm of the Proportion take the near­est distance to C V, and with that extent upon A, the first tearm of the Proportion, describe the Ark F, a ruler from the Center just touching that Ark, cuts the Limbe at D, and the Arch I D being 42^d, is the measure of the fourth Proportional sought.

Otherwise:

Place the extent wherewith the Ark F was described from V to E, and draw the line E G just touching the extreamity 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 finde the Point D in the Limbe, 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 tearms,
• So is the other middle tearm, To a fourth Proportional.

Again:

• As the first tearm, Is to the Radius,
• So is the fourth Proportional before found, To the true Proportional sought.

From which consideration the former Scheme was contrived.

Much after the same manner Proportions in Sines and Tangents joyntly 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 tearms of the Proportion, and they are 40^d 20^d ∷ 62^d.

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 tearms.

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 fou [...]th 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 tearm and A F 62 degrees, the double of the second tearm, 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 tearm 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 tearms, as they may both be lesse then 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 Complements, 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.

Draw the Quadrant A B C, and set off the greater middle tearm 27 degrees from A to D, and draw the Line D C, then set­ting one foot in A, draw the Arch H C, and from the Chords prick off the other middle tearm from C to I, a ruler laid from A to I, helps you to the Tangent thereof E C, take that extent and enter it, so that one foot resting on the line D C, the other turn­ed about may just touch B C, the foot of the Compasses, will rest

at F, the nearest distance from F to A C set from C to G, and lay a ruler from A to G, and it cuts the Arch at K, the Arch K C measured on the Chords, is 10 degrees 30 minutes the fourth Proportional sought in the latter Proportion, the complement whereof is the fourth Proportional sought in the former Propor­tion, namely, 79 degrees 30 minutes.

Otherwise:

Any two Tangents may be changed into their complements, if the other two tearms of the Proportion do onely 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 lyes 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 then lyes:

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 lyes 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 40^d: To the Tangent of 50^d ∷

So is the Tangent of 62^d 40′: To the Tangent of 70^d.

If this Proportion be changed into the complements, 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 tearm 50 degrees, and draw the line D C, then setting one foot in A, draw the Arch H C, and there­in prick off 27^d 20′ from C to I, a ruler laid from A to I, will [Page 30] help you to the Tangent of 27 degrees, 20 minutes C E, then ei­ther finde 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 tearm from A to L, a ruler laid from the Center to L will find the Point N, place M N from C to G, a ruler from A to G, will finde the Arch K C, which measured on the Chords, is 20 degrees, the fourth Proportional sought in the latter Proportion, the complement 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 Center, 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 finde 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 Center 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 tearm, and admitting the work to be a little without the Quadrant, it may be performed after the same manner, onely the parallel F M will be more remote from B C, and possibly the Operation thereby ren­dred 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 Complements would not be inconvenient; or if the two latter tearms were changed into their Complements, 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 tearms into their Comple­ments, and altering the place of the first and second tearm, 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.

Having made the Quadrant A B C, making a right Angle at the Center, prick out of the Chords 31 de­grees from A to F, and draw F C, then take the Sine of 20 de­grees, and enter it so upon C F, that one foot resting thereon, as namely at D, the other turn­ed about may touch A C, through the point D draw a line paral­lel to B C, and therein from G to E out of the Sines, prick down 10 degrees, a line drawn, or a ruler laid over C and E, will cut the Quadrant at H, and the Arch A H measured on the Chords is 17^d, the measure of the Tangent sought, and this way may be per­formed on an Analemma ready cut.

Again, when a Sine is sought, I say the former Scheme serves to finde it, if the Proportion were:

As the Tangent of 31^d: Is to the Sine of 20^d ∷

So is the Tangent of 17^d: To the Sine of 10^d.

If the three first tearms were given, the Scheme would finde 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 Sphaeri­cal Triangles, will be a matter of no difficulty, but to prescribe Protractions for many particular questions, without intimation of the Proportion applyed, were but to mislead 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 40^d: Is to the Sine of 25^d∷

So is the Tangent of 39^d: To another Tangent, to wit, 28^d.

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 [Page 33] B K and A I, parallel to C L, then prick the third tearm 39^d in the Limbe from D to E, and draw C E, it cuts the Line passing through the second tearm at F, place the extent B F upon the line passing through the first tearm from A to G, a rule [...] over C and G cuts the Limbe at H, and the Arch D H being 28^d, 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 Tearmes in the Limbe at H and E, and draw Lines into the Center, then place the Sine of 25 degrees the third tearm from C to B, and draw B K parallel to C L, and thorough the point where it cuts the line of the second tearm 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 drawne thorough the first tearm 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 Sines.

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 equall parts, as Sines.

CASE I. Three Sides to finde an Angle.

THe particular Example shall be to finde the Suns Azimuth, the Proportion is:

As the Cosine of the Altitude: Is to the Secant of the Latitude ∷

So is the difference of the Versed Sines of the Ark of Difference be­tween the Latitude and Altitude, and of the Suns Distance from [Page 34] the Elevated Pole: To the Versed Sine of the Azimuth from the North in this Hemisphere.

Example:

• Let the Latitude be 51^d 32′.
• The Altitude —41 34.
• The Suns 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 Cosine 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 Sines of the third tearm of the Proportion; it is also by reason of the Proportion of equality, the versed Sine of the Azimuth from the North, which [Page 35] place from C to M, then because it is greater then the Radius C K, the Azimuth is more then 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 Center touching it, cuts the Limbe at Q, and the arch F Q being 15 degrees, is the Suns 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 Cosine of the Altitude may be multiplied, &c. 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 Center, touching its extre­mity, finds the Point Q in the Limbe, as before.

Note also, that the Cosine of the Altitude, and the difference of the versed Sines aforesaid, may be taken from a line of Versed or Natural Sines 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 51^d 32′.
• Declination 13^d North.
• Altitude 31^d 18′.

Having with 60 degrees of the Chords, described a Semicircle A D B, draw the Diameter A C B, perpendicular thereto from the Center 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 Suns 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 Cosine of the Altitude being the nearest distance from H to A C, and upon C as a Center, describe another Ark which crosseth the for­mer at S, a Ruler from the Center over S, cuts the Limbe at V, and the Arch A V being 110^d, is the Suns Azimuth from the North.

The like Operation holds for South Declination, onely then the Chord of the Polar distance is greater then 90^d by the Suns 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 finde an Angle, the complements of those Sides may bear the names of Latitude, Altitude and Declination as here, and the Solution will be the same.

To finde the Amplitude.

Prick 77^d the Chord of the Suns Polar distance from L to P, and setting one foot in M, with the extent M p, cross the Limbe at K, and the arch D K being 21^d 12′, is the measure thereof.

In following Schemes we shall finde 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 oblique, and that upon this consideration, that whatsoever Altitude the Sun hath in any Signe 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.

Also prick the Chord of 103^d the Suns di­stance from the de­pressed Pole, from N to P, and upon M as a Center, with M P, de­scribe a prickt Ark al­so upon C, with the nearest distance from H to A C, describe another Ark crossing the former at S, a Ru­ler over C and S, cuts the Limbe at V, and the Arch B V being 110^d, is the Suns Azi­muth from the North, and the Arch A V 70^d, his Azimuth from the South in this Hemisphere.

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 prickt Arks will fall near the Center, 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 Center placed in the Circum­ference, there moveth a Labell 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 Moveable Toung or Labell, the Center being in the Circumference, every degree is twice as large as it would be, if it were at the Center, wherefore the quadrants S E, S W, are numbred with 45^d on each side the [Page 38] Line N S, but are not divided with concentrick Circles and Dia­gonals, nor can they be with truth when ever the Center 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 45^d put through the Limbe, it, or an Azimuth Compass, is rendred general, without the use of Paper-draughts, as I have shewed 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 Suns 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 finde the true Coast of the Suns bearing by that Compass to which it is applyed, 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 A [...]imuth Compass, it may very well by the View be placed over the Meridian line of the Compass, and then turning the moveable Label towards the Sun, so that the shadow of the thread may pass thorough the Slit, the tongue of the Label, amongst the graduations of the Limbe, shewes how the Sun bears by the said Compass, in which the toucht wires is supposed to be precisely under the Flower-deluce, & when the Sun is more then 45^d 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 Suns 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 lyable unto) with the Coast thereof may thus be found.

Example:

Let the bearing of the Sun by the Compass be 55^d East-ward from the South, and his true coast in the Heavens be 43^d ¾ from the South East-wards.

Having upon C with 60 degrees of the Chords, described the Circle S W N E, prick 55 degrees the Suns coast by the Compass

from S to ⊙, and draw ⊙ C, then place 43^d ¾ the Suns true coast from ⊙ to A, and A is the true South Point, and B opposite thereto the North Point, and the Arch S A sheweth that the Compass varieth 11 degrees and a quarter from the South West­ward; or, which is all one, the Arch B M sheweth it to vary as much from the North East-ward, because the Point N falleth to the Eastward of B.

Then admit it were required to steer the Ship away N E by E, being the fifth point from the North East-ward, 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 sheweth 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 re [...]diest 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 Ob [...]ervations, so that the common Mariners can have no practise thereby.

A Scheme for finding the Hour.

Example.

• Latitude 51^d 22′ North.
• Declination 23^d 31′ North.
• Altitude 10^d 28′, Complement 79^d 32′.

Having described the Semicircle, and divided it into two Quadrants by the line D C, prick as before, the Latitude 51^d 32′ from A to L, and draw L M parallel to D C: Prick off the Declination 23^d 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 Cosine of the Declination being the nearest distance from E to A C, upon C as a Center, describe the Ark G W, so is the Scheme prepared for that Declination both North and South.

To finde the Hour in Winter.

The Suns height being 10^d 28′, its Complement is 79^d 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 Center over that Intersection, finds the point K, and the Arch A K being 30^d, the hour is either 10 in the Morning, or 2 in the Afternoon.

To finde the time of Sun rising.

Prick the Chord of 90^d from Q to O, and with the extent M O upon M as a Center, cross the ark G W at P, a ruler from the Center over P, finds the point R in the Limbe, and the ark D R [Page 41] being 33^d 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 finde the Hour of the Day in Summer to the same Declina­tion, the Latitude being the same.

Let the Altitude be 9^d 30′, its Complement is 81^d 30′, prick the Chord thereof from N to H, and with the extent M H upon M as a Center, cross the arch G W as at S, a ruler from the Center over that Intersection, finds the point V in the Limbe, and the ark D V being 15^d, the true time of the day is either five in the Morning, or seven in the Afternoon.

Having found the Hour first, then the Azimuth and Angle of Position may be easily found from the Proportion of the Sines of Sides to the Sines of their opposite Angles, as in the following Scheme.

Example:

• Latitude 51^d 32′.
• Declination 13^d North.
• Altitude 37^d 18′.

By the former Directions the Hour will be found to be 45^d 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 37^d 18′ from D to H, and draw H C, also joyn 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 [Page 42] 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 Center cuts the Limbe at I, and the arch B I is the Suns Azimuth either from the North or South, in this Case 60^d from the South.

For the Angle of Position.

With the former extent cross the parallel O Q at G, a ruler from the Center cuts the Limbe at K, and the arch B K being 33^d 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 shewed in a Treatise, Entituled, 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 finde the Suns Altitudes on all Hours, and the Distances of Places in the Arch of a great Circle.

Example.

• Latitude 51^d 32′.
• Declination 23^d 31′ North.
• Hour 75^d 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 Center divided it into two quadrants, and therein having prickt off A L the Latitude, and thorough 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 Center, draw the line R E, and with the Co­sine 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 Suns distance from the Zenith for that Hour, namely, 62^d 37′, the Complement whereof 27^d 23′ is the Altitude sought.

Moreover, the distance between H and Q, is the Chord of the Suns distance from the Zenith for the winter declination, namely, 99^d 30′, which being greater then a quadrant, argues the Sun to have 9^d 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 23^d 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 30^d from A to K, and from the Center draw the line K G, place the distance M G from M to O, so is the di­stance O N the Chord of 36^d 16′, the Complement whereof 53^d 44′ is the Summer Altitude for that Hour, and the distance O Q is the Chord of 79^d 32′, the Complement whereof 10^d 28′ is the Winter Altitude for that Hour.

Also for the Distances of places in the ark of a great Circle, the Case in Sphaerical Triangles is the same with that here resolved.

So if there were two places, the one in North Latitude 51^d 32′, the other in North Latitude 23^d 31′, the difference of Longitude between them being 75^d, their distance by the former Scheme will be found to be 62^d 37′; but if the latter place were in as much South Latitude, then their distance would be 99^d 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 Sea-mans Calendar.

• Isle of Lobos, Longitude 307^d 41′. Latitude 40^d 21′ South.
• Lizard —18 30. Latitude 50 10 North.
• Difference of Longitude 289 11. Complement 70^d 49′.

Having described a Circle, make AE M 70^d 49′, M I the Lati­tude of the Island, B I the Sine thereof falling Perpendicularly on C M, AE 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 109^d, 41′.
• K E is 48^d, 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 Center 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 AE L Q M, which Perpendiculars are the Sines of the Latitudes of those two Points, and the distance of the points in the Plain of the Equino­ctial from the Center of the Sphere, are the Cosines of those Lati­tudes, & the angle at the Center between those points in the plain of the Equator, is equal to the arch of the Equinoctial between the two Meridians, 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 is at the Center, then 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 Sines 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 [Page 46] and C B, and consequently the Intersection at B, and thence the Angle AE 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, &c. which which I am loth to trouble the Reader withal.

I shall adde 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 —51^d 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 30^d from A to 30^d: how to do this, is shewed in the Second Part, a ruler over D and 30^d cuts the Limbe at B, from B to F set off 66^d 29′ the Suns distance from the Elevated Pole, and draw a line into the Center, and make D K equal to D 30^d, 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 36^d 42′.

Two Sides with the Angle comprehended to find both the other angles, & then the third side. Schemes may be fitted to Proportions [Page 47] which at two Operations find both those Angles, first according to the first Direction for operating Proportions in the Sines and Tangents, and then by another operation, the third Side may be found: also this Case is performed otherwise in the Geometrical Dyalling.

To finde the Suns Altitudes on all Azimuths.

Example.

• Latitude—51^d 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 30^d to the Radius L A, and prick it from A to 30^d, upon which point with the Sine of the Declination, describe a prickt Circle, and a ruler from the Center just touching the outward extreamities of it, cuts the Limbe 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 49^d 56′, is the Altitude for 30^d of Azimuth to the South­wards of the East and West for North Declination, and the Arch H T being 6^d 36′, is the Suns Depression under the Horizon for the same Azimuth, when his Declination is 23^d 31′ South.

It is also his Altitude for 30^d of Azimuth to the North-wards of the East or West, when his Declination is 23^d 31′ North, because [Page 48] if we place A 30^d 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 Center will cut the Limbe as much on this side H, as T is on the other side.

If the prickt Circle had hapned wholly on one side C H, a ruler from the Center touching its extreamities, 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 prickt Circle happens below S C, as it may do both for the Sun or Stars, when their Declination is more then the Latitude of the place, then 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 prickt Circle near the Center, let it be noted that C 30^d is the Radius, and the extent that described the prickt Circle, is a Sine of an Ark to that Radius; whence it followes, that a right angled Triangle may be framed, and the answer given in the Limbe, as we have often done before. See the last Scheme for finding the Amplitude.

The Demonstration of this Scheme, is founded not onely on the Analemma, as I have said in the Second Part, but likewise on those Proportions, whereby the Suns 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 tearmes, it followes that the sine of the Azimuth from the Vertical A 30^d, to the Radius L A, is also the Tangent of the Equinoctial Altitude, making C A Radius, wherefore a ruler over C 30^d cuts the Limbe at B, and the Ark H B being 21^d 40′, is the Equinoctial Altitude sought.

The next Proportion is:

As the sine of the Latitude:

Is to the Cosine of the Equinoctial Altitude ∷

So is the sine of the Declination:

To the sine of a fourth Ark ∷

Get the sum and diffe [...]ence 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

then the Equino­ctial Altitude, is the

• Winter
• Summer

Altitude for Azimuths from the ver­tical towards the

• South.
• North.

In the former Scheme if C 30^d be Radius, 30^d A is the sine of the Equinoctial Altitude, and C A is the Cosine 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 tearms, it follows that the sine of the Declination to the common Radius being the third tearme, is also the sine of the fourth Arch sought, C 30^d being Radius, and the summing or differencing of the two Arks, is performed by describing the prickt Circle, and laying a ruler from the Center touching its Extreamities, 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 finde 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 adde a particular Question, with the Proportions and Scheme fitted thereto:

• Let there be given the Latitude,
• The Suns Declination,

And the Azimuth to the Sun or Stars from the East or West, and let it be required to finde 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 Cosine of the first Ark ∷

So is the Cotangent of the Latitude:

To the Cosine 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 kindes, their difference.

But when the Declination is South, if the Azimuth be to the South-wards of the East or West, the complement 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 complement 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 51^d 32′ North.
• Declination 23^d 31′
• Azimuth from the Vertical 30^d

Upon C as a Center, 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 Center.

Set off the Azimuth being 30^d 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 Center just touching the outward Extreamitie thereof, and you will finde the points R and T in the Limbe.

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 110^d 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 37^d 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 69^d 44′.

If it be to the Northwards, then is the Ark N R the Hour from Noon, to wit, 142^d 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 finde 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 finde the respective times when they shall be on that Azimuth, by turning the Stars Hour into common time, as I have shewed in a Treatise called, The Sector on a Qua­drant.

Here we have retained the former Scheme, but onely conti­nued some of the lines thereof, making the Cosine 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, then the first Proportion lyes 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 Center with the first Ark.

Then making C B Radius, H B is the sine of the first Ark, and C H the Cosine, then the second Proportion lyes thus:

As F H the Cotangent of the Declination:

Is to H C the Cosine 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 Cosine of the second Ark to the Radius C B. Lastly, 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 Center, also the Lati­tude from W to L.

Then place the Cosine 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 Limbe, 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 Center, with the sine of the Azimuth, being the nearest distance from A to C W, describe the prickt Ark, and lay a ruler from the Center touching its ex­treamities on both Sides, and cutting the Limbe 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 findes 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 finde the third Angle, which is the Angle included, and the Proportions [Page 54] 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 Limbe as is Q E, and in it make the extent E B equal to I K, then an Ark described after the same manner, with the nearest distance from D to C W, shall in like kind give the Suns Altitudes on all Azimuths, C W being the Horizontal Line.

In page 60 of a Treatise of Dyalling by the Plain Scale or Line of Chords, for the more easie drawing the Parallels of Declination on the Horizontal Projection of the Sphere, I have intimated that a ready way should be shewed for finding the Amplitude, and al­though that before prescribed be very ready, yet I shall adde one more from the common Proportion:

As the Cosine of the Latitude: Is to the Sine of the Declination ∷

So is the Radius: To the Sine of the Amplitude ∷

Example.

• Latitude—62^d 35′.
• The complement is 27^d 25′.
• Declination—23^d 30′.

Having drawn the Quadrant C B F, double the complement of the Latitude, and it is 54^d 50′, prick the Chord thereof from C to R, then double the Declination, which will be 47^d, with the Chord thereof upon the point R, describe the prickt Ark G, a ruler laid from the Center just touching that Ark, cuts the Limbe at A, and the Arch B A being 60^d, 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 Cosine, 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 Cosine of the Amplitude, with which extent upon the Point L, describe the Ark H, and a ruler from the Center just touching it, cuts the qua­drants Limbe at A, as before.

Otherwise with more certainty.

Upon the Point E, with the extent C D, describe a little Ark neare N: Again, upon the point D with the extent C E, describe another Ark crossing the former at N, then a ruler laid over the Center and the cross at N, will cut the Limbe 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 finde the point E twice as far out towards B, and the intersection at N twice as remote from the Center, as now it is, and by the like reason when a line is to be drawn unto the Limbe from a Point that happens near the Center, take the nearest distance from the point, first to C F, and triple it in the said line, then the nearest distance to C B, and triple it in that line, and find a point of inter­section more remote from the Center, through which and the Cen­ter the line required is to be drawn, and the said extents may be doubled, or often multiplyed.

How to take the Suns Altitude or Height, by the Shadow of a Gnomon.

How to perform this, I have shewed 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 lye 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 streight 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 Limbe, and there shew the Suns Height required, if counted from B towards F. This Conceit is taken from Clavius de Astrolabio.

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 finde their Rumbe and Distance.

BY a Plain Chart, is meant a Chart drawn on Paper or Paste­board, lined with Meridians and Parallels, making right An­gles each with other, and numbred 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.

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 loose Papers, and paste 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, [Page 7] they may hereby prick off every following Course from the for­mer Course last sayled upon, without any respect to the Meridian at all.

Now to the Proposition:

Let it be required to finde 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 Seamans Ka­lendar 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 28^d—Longitude 00.
• Isle St. Nicholas—17—352^d.
• Difference —11—8^d.

Having made the following Scheme like a Square, the Sides L T and T D being at right angles, let T represent Tenariff, prick the difference of Latitude, which is 11^d out the greater Scale of equal part [...], which is graduated on the Sides of this Scheme from T to L, and the difference of Longitude, which is 8^d from T to D, and with that extent upon L, describe a prickt Ark near N, and with the extent T L upon D, describe ano [...]her prickt Ark, crossing the former: where these Arks cross as at N, is the place of St. Ni­cholas Island, draw the line T N, and the Angle L T N shewes the Course from the Meridian, to measure it, with 60^d of the Chords draw the arch R S, which measured on the Scale of prickt Rumbes, annexed to the greater Chord, shews that St. Nicholas Isle bears from Tenariff 3 points and a little more then a quarter to the Westward of the South, that is, South West and by South, a quarter Westwardly, and this prickt Ark need not be drawn, all that is required is the distance between the points R, S, which may be found without drawing any such Arch at all, for the nearest di­stance

[Page 9] from R to S T measured on the greater Sines, sheweth it to be 36^d 2′ from the Meridian, which in like manner may be found from the West when the Rumbe falleth more that way; and in stead of drawing the line N T, you may do the work by laying the slope edge of a Ruler over those points, on which Ruler, if the Scale of Leagues be graduated, you may measure the distance by view without Compasses.

Lastly, the extent T N measured on the same equal parts, sheweth the distance to be 13^d 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 it self.

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 Countrey whatso­ever. Now before we reduce this to our English measure, I will first shew what is the usual custom of expressing the Course and Distance on a Ships Log-board at Sea, and then how in effect the same custom may be still retained or altered, and yet either way agree with the Truth.

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 120^th part of that Mile is 41⅔ feet, and so much is the space betweene the Knots upon the Log-line: So many Knots as the Ship runs in half a minute, so many Miles she sayleth in an hour; or so many Leagues, and so many Miles she runneth in a Watch or [Page 10] 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 Log-board, 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 Log-board, then to half a point of the Com­pass.

But Mr. Norwood in his excellent Book, called, The Sea-mans Practise, sheweth, That according to a late exact Experiment of his, which seemes to be very satisfactory, and confirmed by many other Experiments, that in ordinary Practise at Sea we cannot, if we will yield Truth the conquest, allow less then 360000 of our English feet to vary one degree of Latitude upon the earth, in sayling 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 Sea-mans Practise; as also by Mr. Phillips 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 shew. Now to shew 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 sayleth 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 shew 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 [Page 11] 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 Sea-men 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 Arithmetick, 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 labor 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 shew.

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 Arithmetick, 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 multiplyed 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, [Page 12] 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 multiplyed 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, gives 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 multiplyed by 6, make 450, which is 45 minutes.

This trouble, as I said before, is taken away by fitted Scales by the View onely, 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 13^d, 59: or thus, 13^d ⌊59: or thus, 13^d 59/100: and when we speak of minutes which are marked thus 1′, we alwayes 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 [Page 13] by the Scheme of it is no other then a common Quadrant, having its Limbe divided into 16 equal parts, for the points and half points of the Compass, which may be easily done by the prickt rum­bes on the Scale, each whole point being numbred from the Meri­dian with Capital or letter figures.

In casting up a Traverse by this Quadrant, we shall imita [...] Mr. Norwood in his Sea-mans Practise, who after the same man­ner plots the Surveigh 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 sayleth 60 Leagues South South-West, afterwards she sayleth 80 Leagues West South-West, then meeting with a contrary winde she sayls 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 her self about 23 Leagues East from St. Nicholas Island, we shall plot the whole Traverse, and resolve the Questions demanded without drawing any Lines on the Plat.

[Page 15]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 Center 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 Center 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 prickt 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, shewes 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 [Page 16] 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 finde the Course and Distance from c to N.

1. For the Distance, the extent c N measured on the Scale of Leagues, sheweth 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 90^d of the line of greater Sines (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 Sines, sheweth the Course from the Ship to St. Nicholas Island, to lye 43^d 17′ to the Westward of the South then if you look for 43^d 17′ in the greater Chord, you shall finde 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 shewed the complement of the Course required.

3. To finde what Course the Ship must steer to bring her self 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, then place T R from h to k, and the nearest distance from k to the edge of the ruler, measured on the greater Sines, sheweth the Course to be 33^d 45′ to the Westward of the South, that is three points, to wit, South-west and by South.

[Page 17]4. The extent c d measured on the Scale of Leagues, sheweth the distance to be 100 Leagues.

Lastly, we suppose the Ship to sail this Course and distance till she come into the parrallel or Latitude of St. Nicholas Island at d, and then she sailes almost 23 Leagues West, and arrives at the Island, being her desired Port.

If it were required to finde 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, sheweth that the Course from Tenariff to the Ship is 30^d 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, sheweth 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 platted, when the Chart is made true as to the Latitudes of places, and as neare 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 Chart, commonly called Mercators, as shall afterwards be shewed.

But before I part from this Discourse, I think it necessary to shew 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 numbe [...]s 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, shewes the distance in leagues, accounting every de­grees 10 leagues, and being turned but once over, shewes 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 Sea-mans Pra­ctice, at the end, and by Mr. Phillips 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 then 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, then 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 sayls 8 leagues East and by South by the Compass, then in 8 hours more she sayls 12 leagues East South-east, by the Compass: Now it is demanded what Course and Distance the Ship hath made good from the first place where this recknoning began: In resolving this we shall ac­count every ten leagues of the former Scale of leagues in the Plat to be but one.

Draw two lines at right Angles at the Center A, thus: First draw the line A F, then with 60 degrees of the Chords describe the prickt 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 shews 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 Center, 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 joyn A G, so the extent A G being measured on the Scale of leagues, sheweth 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 finde it to be a little above 3¼ points from the East, so that the Ship hath made her way good South-east 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 onely, 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 [Page 21] platted 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 ought 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 Rectifie 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 Practise of the art of Navigation in keeping a due reckoning, consists chiefly 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 finde the di­stance; or by the difference of Latitude and distance, to finde the Course: But in the midst of so many uncertainties that daily occur in the practise of Navigation, a joynt 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 [Page 22] the known Soundings when a Ship is near land, and how to rectifie the Reckoning by the observed latitude, we shall now shew.

Those that will not yeild 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 guess 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 then 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 therfore, if a ship sail under the Meridian, if the dif­ference of latitude be less by estimation then it is by observation, the Ships place or Variation onely 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 saile from A in the latitude of 28^d 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, then it is by observation, either the judgement erres in supposing the distance run to be too much: in this case the distance is to be shortned, and the Ships place corrected, according to the observed [Page 23] latitude under the Meridian; so if a ship sayle South from A in the Latitude of 28^d, till she hath varied her latitude 2^d 24′ by Estimation, being at the Point B in the latitude of 25^d 36′, if the observed latitude be 26^d, 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 finde it so to happen, commonly that the distance run is more then 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.

In this Case, supposing the distance and difference of Latitude to be true, you are to conclude the Ship hath made her way good not North or South under the Meridian, but upon some other point of the Compass, either Eastward or Westward: let us sup­pose Westward, the Rumbe it self need not be drawn in the Chart, wherefore in the Chart or Traverse-quadrant, having

[Page 24] made A E equal to the observed difference of latitude, with the distance equal to A B, set one foot of the Compasses in E, and with the other cross A W at S, so is A S the Depa [...]ture from the Meridian or Separation required, whereby you may draw the Rumbe line required A C, by finding the Intersections at C; or upon B, with the extent A S, describe an Ark, and draw the line A C [...]ust touching the outward edge therreof, and it shall be the Rumbe required; which being measured by former directions, will be found to be the third point from the Meridian, so that the ships course was South-west and by South, and not South as it seemed to the appearance: In altering the Rumbe, in this case be cau­tious, for a small mistake in the estimating of the distance, will cause a considerable alteration in the Rumbe.

Precept II. But supposing no Current, if the ship sayl upon one of the five Rumbes next the Meridian, if the Dead Latitude dif­fer from the Observed Latitude, the error is in mis-judging the di­stance run, which is either to be enlarged or shortned, as the case requires.

In the former Scheme, suppose a ship sayl from A South-west and by South 48 leagues, being by estimation at C in the latitude of 26^d, but if the latitude by observation be but 25^d 36′, suppose at B, then 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 sayled about 72 leagues on that course, and were by estimation at the Point G in the latitude of 25^d, and by the observation the latitude were found to be 25^d 36′, in this case the ships distance is to be shortned, 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 [Page 25] 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, then 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, then 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 al­ters 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 finde 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 sayl West and by South half a point Southerly 68, 9 leagues, that is 68 leagues and nine tenths, from the latitude of 28^d from A to H, and by the Dead Reckoning should be in the latitude of 27^d; if the latitude by observation be 27^d 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 onely, 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 [Page 26] 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 finde the quantity thereof; as also of the shortned distance by the direction for platting a Traverse.

Precept IV. If a Ship sayl 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 finde 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, finde 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 Latine Au­thor, we have prescribed several rules for the correction of a single Course, which Mr. Phillips in his Geometrical Sea-man 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 [Page 27] 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 erronious 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 sayl from A in the Latitude of 28^d 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 25^d.

Again, from B she sayls 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 23^d 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. Phil­lips, 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 finde the Variation or Difference of Latitude proper to each Course, then 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 ∷

In this case A S is the sum of all the Differences of Latitude, and S L is the whole error between the dead and observed Lati­tude, therewith upon S as a Center, describe an Ark at M, and draw the line A M just touching the outward edge of the said

Ark, then the nearest distance between E and the said line, is the error in the dead latitude at the first Course.

Then 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 prickt 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 Center, [Page 29] 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 then 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 sayls the same distance from the point B North-west 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 27^d; and now supposing the observed Latitude to be 27^d 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 [...]. Phillips 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 76^d 43′ from the Meridian, and is distant from it 43 Leagues and a half, whereas the Ships true Course from A to T, is 83^d 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 shewed 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 finde 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 finde 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 finde 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, finde N F, afterwards in the Triangle F R T there is given F R, and the Angle R F T, whence finde 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 shewed, and there in stead of the

• Easterly
• Westerly

distance, Separation or Departure from the Meridian, we must finde 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 60^d of Latitude 30 true Miles or minutes Departure from the Meridian, alters 60 minutes, or a whole degree difference in Longitude.

In finding out the difference of Longitude, we shall alwayes 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 finde the difference of Lon­gitude will there be shewed, 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 finde the difference of Longitude for many short Traverses near the same Rumbe, ma­king the two first tearms the whole Variation and Separation cau­sed by those short Traverses.

Example:

In this Scheme let A E be two degrees difference of Lati­tude between 26^d and 28^d of La­titude, as it was in the Scheme of the first Precept for correcting the Dead Reckoning, and let A S be the Departure from the Me­ridian, here the same as it was there, then if you joyn E S with a right line, it is the distance in the Plain Chart, supposing E A S to be a right Angle: Now to finde the difference of longitude you must take the difference of latitude, being the distance be­tween 26^d and 28^d out of the Meridian line of Mercators Chart, and prick it from A to M, and draw the line M L parallel to E S, so is A L the difference of longitude required, which is to be increased or diminished by adding or substracting other differences of longitude to it or from it, as was done by the Departure from the Meridian in the Plain Chart, and here M L is the enlarged distance in the Rumbe on Mercators Chart.

To finde 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 Center 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 prickt line M a.

Having before shewed the Use of the Plain Chart, it now re­mains to shew how far it is to be trusted to, in regard the Meri­dians 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 76^d 40′, Longitude 35^d 30′, and Fog-bay in Newfound-land, Lat. 50^d, Longitude 329^d, & so the difference of longitude is 66^d 30′, the true Rumbe between these two places, according to Mercators Chart, which we shall demonstrate, is 45^d 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 68^d 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, then the difference of Longitude will be false, consequently the difference of Longitude between those two places thus laid down, should be but 27^d 17′, whereas in truth it should be much more, to wit, 66^d 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, then 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 then it should; the Distance thus found will be 1860 leagues, and the difference of Latitude 65^d 3′, and should be but 26^d 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 then the Difference of Longitude, this may be [Page 34] 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 shewed; 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 shew; 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 Limbe divided into degrees, with a Line of Sines, or else the Sines of the points, halfs, and quarters onely.

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.

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 onely work one, and that will be in the first Proportion:

As the Radius: Is to the Cosine 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 40^d 30′, the complement is 49 degrees 30 minutes, and the difference of Longitude is 15^d 20′, or 33 Centesms.

Out of the lesser equal parts, prick down 15^d, 33 Centesmes from C to L, and describe the Arch B D with 60^d of the Chords, and make it equal to 49^d 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 5^d, which prick from the equal parts from L to F, then the extent M F measured on the said parts, sheweth the distance to be 13^d 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 22^d, the Complement whereof is 68^d, and so much is the Rumbe from the Meridian between these two places, which is 6 points and about 30 minutes more, wherefore St. Michaels Isle bears from Cape Finister west south west, half a degree more westwardly.

If two places had been both in the Latitude of 40^d 30′, having the same difference of Longitude, to wit, 15^d 20′, then had the [Page 41] extent L M been their distance, to wit, 11^d 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 beare 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 finde the true Rumbe and distance be­tween the Lizard and the Bermudas, Mr. Norwood in his Sea­mans Practise page 110, maketh the Latitude of the Lizard to be 50^d, and of the Bermudas 32^d 25′, or 32^d, 41 Centesmes, and the difference of Longitude between these places to be 55^d.

Draw the lines A C and C D at right angles, now for want of room I use the lesser Chord, and with 60^d 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 Cosine of Bermudas Latitude, which prick from C to E: Again, the nearest distance from G to C I, is the Cosine 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 55^d the difference of Longitude from C to V, out of the [Page 42] 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 17^d, 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, shewes the distance to be 44^d 31 Centesmes, which allowing twenty Leagues to a degree, is 886 Leagues.

Then to finde the Course with 60^d 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, sheweth the Rumbe to be 66^d 37′ from the Meridian, which is almost 6 points, and in this exam­ple the Proportion doth not erre 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 67^d 2′ from the Meridian, and the distance 902 leagues, if you make C A equal to C V, then a line joyning 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 72^d 17′ from the Me [...]idian, 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 sayl home, in, or near the same Rumbe, the Plain Chart will very well serve to keep the reckon­ing upon, and to sayl 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 Astronomie in Gresham Colledge, who in his Treatise of a ruler, Entituled, Posthuma Fosteri, makes a Meridian-line out of a Scale of Secants: this we shall be brief in, and shew 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.

Make a quadrant as A C B, wherein prick down half an inch from C to D, and raise the Perpendicular D E: Sup­pose I would take out the Se­cant of 43^d 30′ to that Ra­dius, prick this arch down from B in the Limbe towards E, and lay a ruler over it, and it will cut the Line D E at F, then is D F the Tangent of that Arch to half an inch Ra­dius, and C F the Secant thereof. After this manner the Secant of any Arch is to be taken out, I say then that the Secant of 30′ thus taken out, shall be the length of the first degree of the Meridian-line, and the Secant of 1^d 30′ so taken out, shall reach from the first to the second degree of the Meridian-line, and the Secant of 2^d 30′ shall be the distance between the second and third degree of the Meri­dian-line, and so on successively through every degree to 80^d or 85^d, and further we do not need it.

And so if it were required to make the Meridian-line from 43^d to 50^d of Latitude, the former extent C F out of the quadrant shall reach in this line F L from 43^d to 44, in like manner the Secant of 44^d 30′ so taken out, shall reach from 44^d 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 49^d 15′, shall reach from 49^d to 49^d 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 finde 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 [Page 44] length required, and so on successively. And so if it were requi­red to finde the length of 49^d 7 tenths, or 42′, I say that 7/10 of the middle Secant, to wit, of the Secant of 49 35 Centesms, or 49^d 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. Gunters Table, by which Table it doth ap­pear that the Meridional parts between the Latitudes of 50 de­grees and 60 degrees, are 17^d ⌊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 17^d, 547, the difference being onely 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.

And for the latter part of the Objection, I hear Mr. Oughtred was making a New Table of them, according to his own minde, wherein it is probable he attained the manner of adding up those Secants by some new Proposition, in regard it would be extreamly tedious to make a Table of Natural Secants to all those parts and then adde 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. Wrights, 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 onely pricks or full points set to them, being indeed the Meridian-line for every 10^d from 0 to 80^d; and thus the Reader may supply them, making 10^d of those lesser parts to be Radius, the distances of these Points from 10^d to 60, are the Secants of the middle La­titudes between them encreased by 15′.

From 10 to 60, adde to the middle Latitude 15 minutes.

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 Limbe,

withal in lines multiplying or increasing the first and third tearms as often as is convenient.

Take the distances in the line R D, from 1 to the Center 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 [Page 47] 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 finde the Rumbe between the Lizard and the Berbadoes, in North Latitude 13^d 20′, Longitude 315^d 40′.

Lizard. Latitude 50^d, Longitude 11^d, 00.

The difference of Latitude is 36^d 40′, or 36^d, 66 Centesms.

The difference of Longitude is 55^d 20′, or 55^d, 33 Centesms.

Here I take the Distance between the first and fifth point, being in effect the Meridional parts between 10 degrees and 50 degrees of Latitude, and prick them down from L to A, and then because the Latitude of the Berbadoes is 30 de­grees 20 minutes, or 33 Centesmes more North­wardly then 10 degrees, we must by Proportion finde the Meri­dional parts between 10 degrees and 13 degrees 20 minutes of Latitude, thus:

As the Cosine of the middle Latitude: Is to the Radius ∷

So is the difference of Latitude: To the Meridional parts ∷

The middle Latitude is 11^d 40′, and the Complement there­of 78^d 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 Center 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 [Page 48] 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 58^d 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 onely (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 wee have no Geometrical Way yet knowne 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 Limbe 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 48^d, 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 Center at A, taken out of the same Scale of equal parts, with the Angle they make with the Sine A G, are:

The first line B is Perpendicular, and the rest all lean outward, and these Angles and Distances the Reader will finde 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 then 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 paste white Paper over a large Table, and to divide the Degrees of a Quadrant thereon, drawing lines into the Center, 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 Sea-mens 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 finde 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 bee 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 [Page 51] that the Reader might be every way supplyed therewith, we have not onely 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 lye 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 finde the Rumbe between the Isle of St. Helens, Latitude 16^d South, Longitude 14^d, and the Berbadoes, Latitude 13^d 20′ North, Longitude 315^d 40′.

The Meridional parts may be pricked down at thrice, first set 10^d out of the Meridian-line B from AE in the following Scheme, both upwards and downwards, then take from the Meridian-line C to 13^d 20′, and prick from 10 upwards to B in the former Triangle, also take from C in the Meridian Scheme to 16^d, and prick it beneath from 10 to S downwards, then raise S H perpen­dicular to S B, and make S H equal to 58^d 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. He­lens, to wit, the Angle B 63^d 3′ from the Meridian, if measured by former directions, the sum of both Latitudes is 29^d 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 64^d, 7 the distance required, which in leagues allowing 20 to a degree, is 1294 leagues; or without drawing [Page 52] 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 finde 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 50^d. St. Michaels Isle, Latitude 38^d 00, the difference of Latitude is 12^d, 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 12^d, and the Distance 18^d, 12, that is about 362 leagues and a half.

Out of any equal parts prick down 12^d, the difference of Lati­tude from L to M, and raise the Perpendicular M I, and take 18^d 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 48^d 34′ from the Meridian.

Now let us suppose a Ship to sayl from I to A 151 leagues and a half N N E, and by this reckoning is in the Latitude of 45^d, and now desires to know how the Lizard bears, and how far it is di­stant from the Ship, I say the prickt line A L doth not represent

truly neither the Course nor Distance from the Ship to the Li­zard: Again, suppose the Ship to sayl from A to B East South-East 114 leagues and a half, and is now by this Account in the Latitude of 43^d, I say a line drawn from B to I, shall not shew the true Course and Distance from the Ship back to St. Mi­chaels, neither shall a line drawn from B to L, shew the true [Page 54] Course and Distance from the Ship to the Lizard: And now he that would remove the Error that is thus committed by reason of the Chart, must for every Course and Distance finde the Differ­ence of longitude as well as of latitude, as shall follow, and then by having the Meridian line, or a Table thereof, he may remove the Error of the Chart when he pleaseth after the same maner, and as easily as to finde the Rumbe and Distance between the two places at the first.

The Ship at A hath made 3^d 88 Centesems, the difference of Longitude in that Course, which taken out of 19^d, the whole dif­ference of Longitude there, rests 15^d 12 Centesms.

Take 7^d, 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 15^d 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 63^d 53′ from the Meridian, and is distant from it 227 leagues, whereas in the Chart from the point A, the Lizard bears 64^d 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 1^d 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 sheweth, 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 [Page 55] 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 maner 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 finde observeable in Forreign 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 oblique Angle with the Meridian, then shall it make a greater Angle with all other Meridians 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 turnings.

If by any Rumbe (but not under the Meridian) it were suppo­sed possible to sayl 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 meer 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 Meridia [...] under the Pole ceaseth, and con­sequently the Rumbe ceaseth, for under the Poles no star or other immoveable point of the heavens, doth either rise or set, in refer­ence whereto the word Meridian hath its original, being used to signifie the Mid-day 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, wherefo [...]e the An­gles must needs do so too.

Of the Nature of the Triangles made upon the Globe by the Rumbe-Line.

The Rumbe of it self is a curved or bending Line, and therefore how smal a portion soever thereof we assume to be a right Line, in its own nature differs something therefrom, and can by no arti­fice be precisely reduced thereto, yet the smaller the Portion be thereof, the nearer as to use, it doth approach to a right Line: Let us therefore on the Globe suppose Circles that are parallel to the Equinoctial to pass through every minute of the Meridian, as in the following figure, wherein P may represent the Pole, and all Lines drawn from thence to AE Q, may signifie Meridians, and the Arks a b, c d, e f, &c. may represent the Parallels supposed: Then suppose a Rumbe to be traced from the Equinoctial at AE towards

the Pole, being the prickt curved Line, and through the Points where this Line crosseth the former Parallels, as at AE, b, d, f, h, k, m, draw Meridians from the Pole to the Equator AE Q, and there will be divers small Triangles represented to the fancy; for Example, the Triangle AE a b right angled at [...], in which the [Page 57] side AE b is the distance in the Rumbe, AE a is the difference of Latitude supposed one minute, and a b the Departure from the Meridian, which Triangle we take to be a right lined, not a Sphae­rical Triangle, supposing the sides thereof to be extended into right lines.

And in the whole Rumbe-triangle AE t m, the whole distance is AE m, the whole difference of Latitude is AE 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 AE 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, &c. 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 finde the Side k m, the distance by the common Proportion:

As the Cosine 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 sayled 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 sayl from the Latitude of 10^d North-East, till she be in the Latitude of 30^d, and then sayl from thence on the same Rumbe till she be in the Latitude of 50^d, the latter distance shall be equal to the former distance, and the like shall [Page 58] hold from thence to 70^d, and the same also should hold from 70^d to the Pole, if it were possible to sayl 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 finde 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 finde 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 because the Unit in the third place doth neither multiply nor divide, it follows that if you divide the Tangent of the Rumbe by the Radius, the Quotient is the Departure from the Meridian, or without such division it may be expressed like a Fraction, thus: Tang. Rumbe / R, which signifies that the Tangent of the Rumbe is to be divided by the Radius; and now having the Departure from the Meridian given in any Parallel, we may by another Proportion finde the difference of Longitude agree­able thereto, and if the same Departure be often scattered amongst many Parallels, as we have before supposed, we may by compoun­ding that Proportion, finde the sum of all the differences of Lon­gitude proper to each Parallel, which is the whole difference of

Longitude required, and must be sup­posed to be added into one sum before the fancy apprehends by what Legs ei­ther the common Rumbe to all those Triangles, and the Hipotenusal or Di­stance AE m is subtended.

Now such Proportion as one Circle hath to another such Proportion, have their Degrees, Semi-diameters, and Sines 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 complement of the Latitude A E.

It therefore holds:

As the Cosine of the Latitude: Is to the Radius ∷

Or rather with other tearmes 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 tearms 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 shewed 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 tearm above which is the distributed part, we may take in a tearm equivalent thereto that findes it Tang. Rumbe / Radius, as we shewed before, then in working the rule of three in mixt Numbers, where the second or third tearm stand in form of a Fraction, the other tearms not being mixt, the Product of the first tearm and Denominator, is the Di­visor or fi [...]st tearm in another Proportion, and the Numerator and other tearm are the second and third tearms, or their product the Dividend, wherefore by this Composition it followes:

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 tearms, 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 shewed is the very tearm given by the Meridional Table)

So is the Tangent of the Rumbe from the Meridian:

To the difference of Longitude ∷

The four tearms of this Proportion being all of a different kinde, 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 finde 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 it self, 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 Sphaerical 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 adde, 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 denyed, as there affirmed, and it were to be wished he had left us his minde 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 Meridians of the places between which it is extend­ed, [Page 61] 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 sayled 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 sayling 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 sayl 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 sayl back again from m to k, the Angle of the Rumbe will be n m k, a bigger An­gle then the former, because the Side k n that subtends it, is a big­ger Side, and so that if the Ship should sayl 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 finde 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 then l m, and is the difference of Lon­gitude in the given Parallel, whereby we may finde m k greater then it was found before, and the Angle of the Rumbe n m k grea­ter then 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 finde the Rumbe, this findes the lesser Angle, and includes the most Northwardly, and excludes the most Southward­ly Parallel.

Again, substract a Centesm or Minute from each Latitude, and then finde the Meridional parts, whereby calculate the Rumbe: this findes the bigger Angle, and excludes the most Northwardly, and includes the most Southwardly Parallel; and thus we may [Page 62] finde 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 finde the lesser Angle, then sub­stract a minute from the greater, and adde a minute to the lesser Latitude, and with the sum of the Meridional parts proper to both Latitudes thus altered, you may also finde 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 adde 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 then 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 Chart, so far forth as it shall be made according thereto: Secondly, were the Objection more considerable then it is, yet to me it seems a gross absurdity in practice, to affirm that a Ship cannot return by the opposite Rumbe.

Admit a Ship sayl 1000 leagues up a River all the way North-East, he were no less then frantick that should deny that she could return down the same River back again, and then by supposition the Compass did truly point all the way up North-East, how can it be conceived that the other end of the Rumbe-line on the Com­pass, if we make it a streight line so as to agree with the opposite Coasts of the Horizon, did point at the same time other ways then South-West? or can it be apprehended that the meer winding or tacking about of the Ship to return, should all the way back cause the Compass to point otherwise then it did all the way up? and notwithstanding the Objection, Mr. Oughtred grants, and pleads as much for the truth of Mercators Chart, as he doth for the truth of the Calculation delivered by himself and others. Now having shewed the making, and confirmed the truth of that Chart, we [Page 63]

[Page 64] proceed to the Uses thereof, and the manner of keeping a Rec­koning thereon.

Suppose therefore that a Voyage were to be performed from Tenariff, Latitude 28^d, Longitude 0, to St. Nicholas Isle, Lati­tude 17^d, Longitude 352^d, the difference of Longitude being 8^d, 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 17^d 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 loose Ruler; each degree thereof is divided into ten parts, and numbred 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 finde the Rumbe.

And now let it be required to finde the Rumbe or Course be­tween Tenariff and St. Nicholas Island without drawing any lines in the Chart.

Example:

Prick 60^d 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 33^d 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 Sines, sheweth the same Arch as before, to wit, 33 degrees 49 minutes.

Another Example.

Let us suppose two Islands in the Sea, the one scituated in this Chart at f, the other at d, and let it be required to finde 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 Meridians and Parallels drawn therein, here we have drawn one through 20^d of Latitude, and another through 25^d, 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 11^d 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, then 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 Cosine 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 22^d 5 tenths, and the point R will be found to be in the latitude of 24^d 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, [Page 66] 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, 2^d 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 finde the distance the same.

In like manner, if the former extent were so entred, 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 onely measure one tenth part there­of: The whole difference of Latitude is 11^d, whereof a tenth part is one degree and a tenth, take one inch and a tenth out of the Scale of Inches, and entring 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 Complement 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 en­tring of the difference of Latitude taken out of some Scale of equal parts, as above, makes it to become the Cosine of the Rumbe, wherefore the Radius thereto K Y, or X Z becomes the distance [Page 67] 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 onely in Longitude, the Propor­tion holds:

As the Radius: Is to the Cosine 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 Limbe 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 60^d of the Chords from T to H, which Point may be alwayes in a readiness, and now let it be required to measure the distance r X in the Latitude of 20^d, the Complement whereof is 70^d, with the Chord of 70^d 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, 3^d 45 Centesms, where turning it twice over (making every degree to be ten leagues) I shall finde it to be 69 leagues.

Otherwise:

Suppose G to be a place in the Latitude of 28^d, and it were re­quired to measure the distance thereof from T, with the Sine of 28^d upon o as a Center, describe the Ark q, and lay the edge of the Ruler so as it may lye 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 finde 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 40^d or 45^d, the trouble of turning about the Ruler [Page 68] may be best spared in that Case, with the Sine of the Latitudes Complement, 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 alledged, 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 rep [...]esent 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 Cosine of the middle Latitude between both places.

And thus if it were required to finde the distance between the points d and f in the former Chart, by this rule it would be found to be 2^d 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 finde the di­stance by the Cosine of the middle Latitude, and then enlarge it by entring 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 entred, one foot of it will rest at m, while the other turned about will but just touch T S, then 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. Gunters way, which is not onely troublesom, 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 minde what Arks it crosseth in the Meridian-line, then they count the whole Arch contained be­tween the places where the moveable foot so crossed, and take that for the whole distance sought.

Example:

Difference of Longitude is 180^d, the extent in the Chart be­tween two places so scituated, is 182^d 42, the half whereof is 91^d 21, which measured in the Meridian-line from the middle Lati­tude or Arch, which is 1^d 33, being half the difference of both La­titudes, and counted towards the greater Latitude, reacheth Northward to 66^d 15 one way, and Southward to 67^d 32 the other way, and so by this reckoning the distance between these places should be 133^d, 47 by the middle Latitude, and 133^d, 98 by the middle space, whereas in truth by Calculation the distance in the Rumbe is much more, to wit, 180^d, 54.

But supposing two places to be in the same Latitude, and to have but 58^d, 33 Centesms difference of Longitude, the example will be the same with one of those before put between the Berba­does and St. Helens, and the distance found by the middle Arch or Latitude, is 62^d, 00, and the same by the middle space 61^d 38 cen­tesms, but should be in truth 64^d, 7.

Also in the former Example between the Berbadoes and the Lizard, the true distance is 58^d, 54, and by the middle Latitude [Page 70] or Arch is 58^d, 7, but by the middle space it is 57^d, 66.

Also in the Example between the Bermudas and the Lizard, the true distance was found to be 44^d, 31 Centesms, by the mid­dle Arch or Latitude it is 45^d, 13 Centesms, and by the middle space it is 44^d, 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 applyed both upwards and downwards, as before; and if the distance be large, the measure thereof will be much more erronious then when the Rumbe lyes nearer the Meridian.

Some Examples of a parallel Distance.

Let there be two places in the Latitude of 35^d.

Another Example of two places in the Latitude of 50^d.

A third Example of two places in the Latitude of 70^d.

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, finde the whole; for the ready per­forming whereof another Scale of equal parts, whereof the de­grees [Page 71] 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, finde 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, 19^d 18 Centesms, the other will reach Northwards to 20^d 69 Centesms, and Southwards to 17^d 64 Centesms, the difference of which two Arks is 3^d, 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 finde 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 finde 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 practise, 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 easie 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 finde the Latitude.

The first Course the Ship sayls is South South-west, 60 leagues from Tenariff, to protract this Traverse I shall make use of another [Page 72] Traverse-quadrant bigger then 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 shews me now that the Ship is in the Latitude of 25^d, 23 Centesms, in the Meri­dian-line T S set the figure 1 to this Latitude.

Secondly, to finde 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, 1^d 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 28^d, and the Lati­tude of the Ships place, namely, 25^d 23 Centesms, which being entred 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 23^d, 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 it self, in page 73. and this extent reaches from B to C, and now I see I am in latitude 21^d 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 sayled, that one foot resting thereon as at d, the other turned about will but just touch C W, then 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 38^d 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 [Page 76] steered to bring the Ship 23 Leagues East from St. Nicholas Island; The Latitude of that Island is 17^d, and the Complement of it is 73^d, 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 29^d 23′ to the Westwards of the South.

And the distance about 95 Leagues 74 Centesmes more.

From the point c, the Ship sailes directly South 33 Leagues 4 Tenths, which I take out of the west-line, and prick in the loose Scale of Inches from C to D, whereby I see the Ship is now in the Latitude of 19^d 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 write 4 in the West-line also over the figure 3, and by help of these two Points plat the point d, as was done before.

And now lastly, it is desired to know what Course and Distance I should steere to bring the Ship to beare from the Point S in the Chart, 45 Leagues North-west and by North.

To perform this, we must finde 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, shews that the difference of Lati­tude is 1^d 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 [Page 77] 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 finde the Course and Distance between d and f, which we have perfo [...]med before, the Course was 11^d 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 sup [...]lied both, and the Latitude would have been transferred from the lesser to the greater, which is uncertain.

The maner 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 shewed in Page 32. Or having found the Departure from the Meridian first, make use of the Cosine of the middle Latitude, and with the Departure finde the difference of Longitude required, as you found the point e, or do it by the pen, as we shall afterwards shew, 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 sayl West and by South 3 degrees or 60 leagues from Latitude 41^d, now by the dead reckoning she is in Latitude 38^d 6 Centesms, the middle Latitude is 39^d 53 Centesms: And let it be required to finde the difference of Longitude proper to this Course.

This we insist upon, because when a Course lyes 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 11^d 15′ from B to H, and draw C H, on it prick down 3^d 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 39^d 53 Cen­tesms, [Page 78] 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 3^d 81 Centesms, and when the line C E falls above D, continue 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, then 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. Wrights 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 kinde of Projection.

We have hitherto kept a Reckoning upon both Charts, which might as well have been done upon some fitted Traverse-board, 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 joynt 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 Center, and near to the outward Divisions of the Limbe, fo [...] then laying the Ruler over any two places in the Chart, bring the Diameter of the Semicircle to lye along by the edge of the Ruler, so that the Center thereof may lye 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 Center of the Semicircle, so that the same degree of the Limbe thereof may lye 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 Center, and the distance from the Center of the Ruler to the Parallel (r B) above which the difference of La­titude was counted, being measured by view on the Ruler, sheweth the Distance required.

For East or West Distances.

Lay the edge of a cut sheet of Paper, Paste-board, &c. over the Center of a Quadrant, drawn in some corner or spare place of the Chart, and also over the complement of the Latitude counted from the East or West, and holding it so, count from the Center 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 lye over the difference of Longitude in the Paral­lel, and that the graduated beginning of it may make right Angles, or be perpendicular to the streight edge of the sheet of Paper be­fore [Page 80] laid over the Colatitude, and where the parallel in which the difference of Longitude was counted cuts the Ruler, it shews the Distance required; and the converse of this manner of work will also finde 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.

Some Observations from the former Scheme.

1. That if those places be supposed to be both in one Hemi­sphere, and to have the Complement of the former difference of Longitude to a Semicircle, one & the same great Arch is still com­mon, and the distance is the Complement 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 Complement of the former difference of Lon­gitude to a Semicircle, and the Arch L A is the complement 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 AE 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 then Quadrants, and the contrary, and the other Leg will be less then 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 then 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 complements of the Sides

• f P
• f e

to a Semicircle, and the Angle f S W, is the complement of the Angle [Page 84] 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 complement of the Angle f P e to a Semicircle.

This being fore-noted, 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 supplyed.

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 followes that one intire Hemisphere cannot be thus projected; moreover, the sight there will project any great Circle in a right line as followes from 91 Prop. 6 Book of Aguilonius: this Projection is applyed by Mr. Phillips to great Circle sailing, divers things by him not handled we shall adde, and then proceed to shew how the same may be otherwise performed.

First Example of two places in one Hemisphere.

Now let it be required to finde the Longitudes and Latitudes of the great Arch between the Lizard Latitude 50^d, and Trinity Harbor in Virginia, Latitude 36^d, difference of Longitude 68^d 30′, draw G F and upon P as a Center, describe a Semicircle, prick the difference of Longitude 68^d 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 Limbe, or rather made by an Instrument-maker, and such a one we suppose here used, out of which prick down P L 40^d, the complement of the Lizards Latitude, and P T 54^d the complement 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 shewes how far it will happen from the Lizard. Now if it were required to finde the Latitudes of this Arch for every [Page 85] 5^d 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 numbred both with the Arch and its complement 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 41^d 51′, being the complement of the Latitude of the Arch at that diffe [...]ence of Longitude, wherefore the Latitude of the Arch at 25^d difference of Longitude from the Perpendicular, is 48^d 9′.

In like manner the Latitudes of the Arch for

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, 68^d 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 [Page 86] 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, then 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

We have continued the Tangents but to 65^d 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 Cosine of the Angle adjacent ∷

So is the Radius:

To the Tangent of the Arkes Latitude:

Example:

Let it be required to finde the Latitude of the Arch at 70 de­grees difference of Longitude from the Perpendicular.

The complement thereof is 20^d, to the Tangent whereof draw a line from P, then the nearest distance from Q to the said line, is the Cosine 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.

The Angle I P K is equal to the difference of Longitude, wherefore the Perpendicular P H is the same as before, from [Page]

Thus we see this Projection will supply all the Cases that can be put, and is in effect no other then 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 oblique-angled plain Triangle T P L, we have the two Sides T P and P L given, being the Tangents of the complements of both Latitudes, and the Angle comprehended T P L given, now we finde 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 Trigonome­trie; And so the two Sides of this Triangle are the Tangents of the Complements 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 tearm 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 Cosine 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 Cosine of half the difference of Longitude:

Is to the Tangent of the common Latitude ∷

So is the Cosine 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 Complements of both Latitudes:

Is to the Sine of their difference ∷

So is the Tangent of the Complement of half the difference of Longitude:

To the Tangent of half the difference of the Vertical Angles ∷

which Ark thus found.

Adde 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 diffe [...]ence 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 tearms 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 then a Quadrant, the Polar distance in stead of the Colatitude, and then for the Sine of an Ark greater then a Quadrant, take the Sine of that Arks complement to a Semicir­cle, which may sometimes happen when both places are in one He­misphere, if the sum of the Complements 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 Cosine 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 Cosine 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 attaine under seven or eight, which rendred the Sayling by the great Arch so difficult and laborious, that none cared to practise it. And this latter Proportion having two fixed tearms 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 ad [...]acent 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 Comple­ment 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 shews us how to delineate Proportions in Sines and Tangents, by framing of right-lined oblique 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 Sines, &c. may be carried on in the Angles and Sides of Plain Triangles.

Of the six parts of a right-angled Spherical Triangle, no more but four at a time can be laid down in right Lines and Angles.

As in the right-angled Triangle P A T, to wit, the right Angle [Page 91] 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 Complement of the Vertical Angle, is not the Angle of Position in the Sphaere, 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 60^d of the Chords, setting one foot at E, draw K M, which extent measured in the Chords sheweth the Distance to be 50^d 9′ at 20 leagues to a degree, is 1003 leagues, the Rumbe between these two places is 74^d 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 74^d 7′, the Complement whereof to a Semicircle being 105^d 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 finde 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 then 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 finde the Angle of Position.

If the Hipotenusal and one Legg be given, the Proportion to finde the Angle next that Legg, would be:

As the Radius: Is to the Tangent of the Hipotenusal ∷

So is the Tangent of the given Legg:

To the Cosine 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 Complement 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 Cosine of the Angle of Position, if we make E A Radius; where­fore upon A describe the prickt Ark n, a ruler from the Center E just touching it, cuts the Ark x n at n, which Ark measured on the Chords is 38^d 49′, the complement whereof being 51^d 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 Cosine of the Perpendicular:

To the Cosine of the Angle of Position ∷

If we make B P Radius, then is A B the Cosine of the Per­pendicular to that Radius, and the nearest distance from C to T P, is the Cosine of the Angle of Position to the Radius E A.

Thus we have shewed the use of this Pro [...]ection in more parti­culars then the Sea-man 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 [Page 93] 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 va [...]y.

THe former Pro [...]ection for places near the Equinoctial growes infinite and inconvenient, if therefore, as before, we suppose the eye at the Center of the Earth, and place a plain Parallel to the Axis, touching the Equinoctial in some point, then will the Equinoctial it self from that point be projected on each side in a Tangent-line, all the Meridians 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 Meridians 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 Ecliptick and Equinoctial being measuted by the Suns greatest declination, is called the Obliquity of the Ecliptick; 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 onely shew you that a Scheme so made doth carry on the former Pro­portion for Calculating the Latitudes of the great Arch, and therefore one Quadrant onely 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 Center, 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 45^d.

Examples for finding the Latitudes of the great Ark. 1. For a Parallel Course.

Admit there were two places in the Latitude of 50^d, and their difference of Longitude were 70^d, and it were required to finde rhe Latitudes of the great Ark between these places.

In all Cases we take the Line p Q to represent the Perpendi­cular, through 50^d of the Tangents, draw the parallel of Latitude L E parallel to the Equinoctial Q AE, and in the Quadrant count half the difference of Longitude from A, to wit, 35^d, and draw a line meeting with the parallel of Latitude at E, with the extent Q E draw the Arch E G, and it shall represent half the great Arch required; the nearest distance from a to Q E measured on the Tangent-line Q p, is 51^d 49′, and so much is the Latitude of

[Page 95] the Arch for 5^d difference of Longitude from the Perpendicular on each side; also the nearest distance from b to Q R, is the La­titude of the Arch for 10^d difference of Longitude on each side the Perpendicular.

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, then the nearest distances from every fifth degree in the Arch to the Equinoctial line AE 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

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 5^d 29′, which is a great convenience in this East or West Course.

Another Example.

The Lizard and Fog-Bay in Newfound-land are both in the Latitude of 50 degrees, and the difference of Longitude be­tween them, according to the before-mentioned 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 [Page 96] great Arch between these places, the Latitudes whereof

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 complement 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 Center, and it will shew all the Latitudes of the Arch as in the former case, also the Parallels of both Latitude meeting with this Arch, do therein shew 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 39^d 4′, where­fore the complement thereof 50^d 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

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.

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 30^d 4′, the complement whereof 59^d 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

be­ing the Complement of the greater to a Semicircle, because▪ should have happened as much below AE, as it doth above.

Part of this Arch AE t, is supposed to fall below the Equinoctail Line Q AE 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 Center, where it meets with the parallel of Lati­tude, gives a point passing through which the great Arch is to be drawn upon the common Center Q: Now this Vertical Angle may be either found on the former Scheme, or in an oblique 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 Sphaere, yet it may be more easily and speedily done [Page 98] by shewing that it carries on the Proportions in Sines and Tan­gents before delivered for Calculating the Latitudes of the Arch, we need onely 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 50^d, whose difference of Longitude is 70^d: the near­est distance from E to Q R, is the Cosine of half the difference of Longitude, making Q G Radius, which by construction is made equal to the Tangent of 50^d, the common Latitude to another Radius, wherefore by reason of the Proportion of Equality, the Cosine of any Ark from the Perpendicular to the former Radius, to wit, the nearest distance from a to Q R, being the Cosine of 5^d 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 then 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 AE Q.

But if their difference of Longitude be more then 90^d, they may after the same maner 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 then a Quadrant.

Example: Suppose I would finde the Latitudes of the great Arch between the Lizard in 50^d of Latitude, and the Bermudas in Latitude 32^d 25′, being both in the North Hemisphere, the difference of Longitude being 55^d.

To lay down these places, draw the Parallel AE B passing tho­row the Tangent of 55 degrees, on which making the Secant of that Ark Radius, the Tangent of 32^d 25′ is to be pricked down, to take out that Tangent to the said Radius.

Note, That the line Q k passing through 55^d of the Quadrant, counted from C, being continued till it meet with the Radius C D continued, is the Secant of 55^d, 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 32^d 25′, draw a Parallel meeting with the Line or Secant of 55^d drawn through the Quadrant at k, then is the extent Q k the Tangent to that Radius, which prick from AE 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 Me­ridians, which are of several Radiusses, being Parallels drawn through the Tangents of 10^d, 20^d, 30^d, &c.

Now to finde the Latitudes of the Arch from the Lizard for every 10^d difference of Longitude.

In the first Parallel take the extent 1, 10, and place on the Se­cant of 10^d from Q to a, also take the extent 2, 20, and place on the Secant of 20^d from Q to b; or if these extents be too long, the extent f 2 being prickt on the Secant of 20^d, 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 AE Q, and measure them in the Meridian-line Q L, they there shew you the latitudes of the Arch required: thus the latitudes of the Arch or the points

or the nearest distance from a to Q AE measured on the Scale of Tangents, is 49^d 19′, the latitude of the Arch for 10^d 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 Center V with the lesser Chord, describe a Circle, and draw A S and B I parallel, the sum of the Complements of both Latitudes is 96^d, the Sine whereof is the Sine of 84^d its Complement to a Semicircle, which prick from A to S, and the difference of those Complements is 14^d, make B D the Sine thereof, a ruler over S and D cuts the Diameter at E.

Out of the lesser Chords prick 68^d 30′, the difference of Lon­gitude from B to H, a ruler over H and E cuts the Limbe 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, 53^d 52′, and the Arch B L the lesser, to wit, 14^d 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 Center 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 Cosine 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 Sines of every fifth degree from 90, as of 85^d 80^d &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, 50^d 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 sines equal to the diameter be graduated on the floap edge of a ruler, the sines in the line B I may be easily pricked [Page 102] 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 Sines 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 Sines 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 finde 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 prickt line is drawn.

Example.

The Latitude of the Arch at 30^d difference of Longitude from the Perpendicular, is 46^d 85 Centesmes, at which in the Meridian line P R is set the letter a; with the extent P a setting the Com­passes at 30 in the line L M, describe a small cross at 9: Again, with the extent P 30, setting one foot at a, describe another cross or little Ark at 9: thus may all the rest be found, through all which

you may draw a curved line which shall run in the Chart near the great Arch; And now the work to be done, is to finde either by Calculation, or upon the Chart, all the respective Courses and distances from Point to Point, and to sayl as near the said Arch as you can.

Now to the apprehension the right Line L T will seem nearer then to sayl along by the prickt 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 then 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 Harbor in that Chart bears from the Lizard 74^d 17′ from the Meridian, which is above 6½ points to the Westwards of the South, to wit, West and by South ¼ of [Page 104] 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 then the Distance in the Rumbe by 14 leagues, as appears by Calculation or the Chart it self. The respective Courses and Distances to be sayled.

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.