NAVIGATION RECTIFIED: OR, THE COMMON CHART Proved to be the only TRUE CHART. By Peter Blackborow.  
LONDON: Printed by John Playford, and are to be Sold by Joseph Hindmarsh, Bookseller to His Royal Highness, at the Black Bull in Cornhill. 1684.   

TO ALL MEN That are Studious in NAVIGATION THIS TREATISE (Upon the Three Principal Kind's of Sailing, so called, wherein there is so great disproportion, that the Question may be asked, what is Truth in Navigation? Wherein is proved by Observation and Demonstration, that there is but one Truth in Navigation, and that is the Common Chart which is so much rejected by all Writers, so that all other Positions in Navigation must be rejected; if you will follow that which is Truth)  
Is DEDICATED By Peter Blackborow.   

TO THE READER.  
NAVIGATION must depend upon the Magnetic Compass, and the Latitude of places by Observation, otherwise you cannot shape your Course to any place in the World, by the Rumb, and the Right Lines of the Magnetic Compass, and whereas you have three principal kinds of Sailing so called: The Common Chart, Mercator, and the Arch of a Great Circle, wherein it will appear there is so great disproportion between the one and the other, that the Question may be asked, which of the three is Truth? There is but one Truth in Navigation, and tha● may be always Demonstrated to be so● so that although the Earth and Se● are Spherical, I shall prove by Observation, and Demonstration, that th● Magnetic Compass, as it is Horizontal, measures the Superficies of th● Earth and Sea by Right Lines, a● if the Superficies of the Globe wer● a Plain to the Horizon: Therefore ● shall first resolve the Question by th● Rumb and the Right Lines of th● Magnetic Compass, in a Right line● Triangle, which is in proportion to the Common Chart.  
Then I shall show you how the Meridian's and Parallels of places in the Common Chart, are transferred into the Meridian's and Parallels of the Globe.  
Then I shall resolve the same Question by Mercator as is given in the Common Chart, wherein I shall show you how great the disproportion is between the Meridian's of Mercator and the Meridian's of the Common Chart: And last of all, I shall show you wherein the disproportion appears to be a gross error in Mercator, in mistaking the Meridian's of the Magnetic Compass.  
Then I shall resolve the same Question by the Arch of a Great Circle, wherein it will appear the Arch of a Great Circle gives a nearer distance than is contained in the Parallels of Latitude, which altars the Angles of Position in the Arch of a Great Circle from the Angles of Position by the Magnetic Compass, that you cannot give the Question from the Practical part of the Sea, but you must alter the Meridian's and Parallels of places upon the Globe, so th● the Work of Navigation cannot b● performed by the Arch of a Grea● Circle, in proportion to the Rum● and the Right-lines of the Magnetics Compass.  
In the Conclusion it will appear by Observation and Demonstration, that the Common Chart that hath been so much rejected by all Writers, is the only Truth in Navigation. All which I would desire you to accept with the same Candour and Integrity, as it is offered by him, who is,  
A Lover of the MATHEMATICS, Peter Blackborow.   

NAVIGATION RECTIFIED.  
ALL the known part of the World hath been discovered by the Magnetic Compass, and the Latitude of Places by Observation, which doth produce the difference of Longtitude, and the distance of places in a Rightlined Triangle: Therefore the Magnetic Compass is of that singular use in Navigation, that all Questions must be resolved in proportion to the Rumb and the Right-lines of the Magnetic Compass, otherwise you have no Geometrical Demonstration in Navigation: So that all the known parts of the World are laid down in the Common Chart in proportion to the Rumb and the Right Lines of the Magnetic Compass.  
Then observe, the Superficies of the Earth and Sea being Spherical, the Lines of the Globe must be Obliqne, which differ from the Rumb and the Right-lines of the Magnetic Compass and of the Common Chart, therefore there is an absolute necessity to find such proportions from the Globe, as may be in proportion to the Meridian's and Parallels of the Common Chart; otherwise the Meridian's and Parallels of places in the Common Chart, cannot be transferred into the Globe according to the Meridian's and Parallels of the Globe: So that all the known part of the World is laid down in the Meridian's and Parallels of the Globe from the Meridian's and Parallels of the Common Chart, for the difference of Longitude in the Common Chart between places differing in Longitude and Latitude, is proportional to the difference of Meridian's in the middle Parallel between such places differing in Longitude and Latitude upon the Globe: And where places in the Common Chart differ in Longitude, being in one and the same Parallel, their distance is the same in the Globe as it is in the Common Chart, for the distance is transferred from the Common Chart, to the Globe, notwithstanding it will appear by the following demonstrations, that you cannot perform the Work of Navigation by the Oblique-lines of the Globe, in regard the Oblique-lines of the Globe differ from the Rumb, and the Right-lines of the Magnetic Compass, so that the Angles of Position in all Oblique-angles differ from the Angles of Position by the Magnetic Compass. Therefore I shall take this method: First, I shall resolve the Question by the Common Chart: Then I shall show you how the Meridian's and Parallels of places in the Common Chart are transferred into the Meridian's and Parallels of the Globe, and in the next place I shall resolve the same Question by Mercator: And last of all I shall resolve the same Question by the Arch of a Great Circle; all which are the three principal kinds of Sailing so called by all Writers; from whence I shall prove the Common Chart, that hath been so much rejected by all Writers to be the only Truth in Navigation, for you must Sail by the Rumb and the Right-lines of the Magnetic Compass, or the Magnetic Compass is of little use in Navigation.  

First, I shall resolve the Question by the Common Chart.   geometrical diagram 
Pag: 5.  
The Angle following being a Rightlined Triangle, and is in the like protion in the Common Chart, so is L, the Meridian and Parallel of the Lizard, and B is the Meridian and Parallel of Bermudas: Then is L, A, the difference of Latitude, 1055 Miles, and C L D, is the Rumb or Angle of Position between the Meridian and Parallel of the Lizard and the Meridian and Parallel of Bermudas 71 d. 10 m. and A B is the difference of Longitude, and L B is the distance between the Meridian and Parallel of the Lizard and the Meridian and Parallel of Bermudas.  
In the following demonstration, an Inch divided into ten equal parts is to be accounted for 1000 Miles, in regard the distance is great, and so many equal parts cannot be discerned in so small a space.  
By the Geometrical measure, the difference of Longitude is A B, 3093 Miles, and the distance is L B, 3268 Miles, all which is in like proportion to the Common Chart.  
To find the difference of Longitude, the proportions Arithmetical, As the Radius 90 d. A B N, 1000000; is to the Logarithm of the difference of Latitude 1055 Miles, L A, 302325: so is the Tangent of the Rumb 71 d. 10 m. A, L, B, 1046714. to the Logarithm of the difference of Longitude 349039, A B, 3093 Miles. Add the Logarithm of the difference of Latitude, to the Tangent of the Rumb, and Subtract the Radius, and you have the Logarithm of the difference of Longitude 349039, A B, 3093 Miles; so that the difference of Latitude and the Rumb by the Magnetic Compass, is in proportion in a Rightlined Triangle to the difference of Longitude in the Common Chart.  
To find the distance, the proportions Arithmetical: As the Co-sine of the Rumb 71d. 10 m. A, B, L, 950895; Is to the Logarithm of the difference of Latitude 1055 m. L A, 302325: So is the Radius 90 d. L, A, B, 1000000; to the Logarithm of the distance 351430, L B, 3268 Miles. Add the Logarithm of the difference of Latitude to the Radius, and Subtract the Co-sine of the Rumb, and you have the Logarithm of the distance 351430, L, B, 3268 Miles; so that the difference of Latitude and the Rumb by the Magnetic Compass, is in proportion in a Rightlined Triangle, to the distance in the Common Chart.  
Secondly, I shall give the Rumb 71 d. 10 m. A L B; and the difference of Longitude 3093 m. A B; and demand the difference of Latitude L A, and the distance L B.  
I have already given you the demonstration of the Geometrical part of all Questions, that are to be resolved between the Meridian and Parallel of the Lizard, and the Meridian and Parallel of Bermudas, for as it stands in the foregoing demonstration it is in the like proportion in the Common Chart, therefore I shall proceed to the Arithmetical proportion.  
To find the difference of Latitude the proportions Arithmetical; As the Tangent of the Rumb 71 d. 10 m. A L B, 1046714; Is to the Logarithm of the difference of Longitude 3093 m. A B, 349037: So is the Radius 90 d. A B N, 13000000; To the Logarithm of the difference of Latitude 302323, L A, 1055 m Add the Logarithm of the difference of Longitude to the Radius, and Subtract the Tangent of the Rumb, and you have the Logarithm of the difference of Latitude 302323, LA, 1055: So that the Rumb, by the Magnetic Compass, and the difference of Longitude is in proportion in a Rightlined Triangle, to the difference of Latitude in the Common Chart.  
To find the distance, the proportions Arithmetical: As the Sine of the Rumb 71 d. 10 m. A L B, 997610; Is to the Logarithm of the difference of Longitude 3093, L A, 349037: So is the Radius 90 d. L A B, 1000000; to the Logarithm of the distance 351427, L B; 3268 m. Add the Logarithm of ●he difference of Longitude to the Radius, and Subtract the Sine of the Rumb, and you have the Logarithm of the distance 351427, L B; 3268 Miles: So that the difference of Longitude, and the Rumb by the Magnetic Compass, is in proportion in a Rightlined Triangle, to the distance in the Common Chart.  
Third●y, I shall give you the Rumb A L B, 71 d. 10 m. and the distance L B, 3268 m. and demand the difference of Longitude A B, and the difference of Latitude L A.  
To find the difference of Longitude, the proportions Arithmetical: As the Radius 90 d. L A B, 1000000; is to the Logarithm of the distance, 3268 m. L B, 351428: So is the Sine of the Rumb 71 d. 10 m. A L B, 997610; to the Logarithm of the difference of Longitude 349038, A B, 3093 m. Add the Logarithm of the distance to the Sine of the Rumb, and Subtract the Radius, and you have the Logarithm of the difference of Longitude, 349038, A B, 3093 m. so that the Rumb and the distance by the Magnetic Compass, is in proportion in a Rightlined Triangle, to the difference of Longitude in the Common Chart.  
To find the difference of Latitude, the proportions Arithmetical: As the Radius 90 d. L A B, 1000000, is to the Logarithm of the Distance 3268 m. L B, 351428: So is the Co-sine of the Rumb 71 d. 10 m. A B L, 950896, to the Logarithm of the difference of Latitude, 302323, L A, 1055 m. Add the Longarithm of the distance to the Co-sine of the Rumb, and Subtract the Radius, and you have the Logarithm of the difference of Latitude 302323, L A, 1055 m. So that the Rumb, and the distance by the Magnetic Compass, is in proportion to the difference of Latitude in the Common Chart.  
Fourthly, I shall give the difference of Latitude L A, 1055 m. and the difference of Longitude A B, 3093 m. and demand the Rumb by the Magnetic Compass A L B.  
To find the Rumb by the Magnetic Compass, the proportions Arithmetical: As the Logarithm of the difference of Latitude 1055 m. L A, 302325, is to the Radius 90 d. A B N, 1000000; so is the Logarithm of the difference of Longitude 3093 m. A B, 349037; to the Tangent of the Rumb 1046712; A L B, 71 d. 10 m. Add the Radius to the Logarithm of the difference of Longitude, and Subtract the Logarithm of the difference of Latitude, and you have the Tangent of the Rumb, 1046712; A L B, 71 d. 10 m. So that the difference of Latitude, and the difference of Longitude, is in proportion to the Rumb, by the Magnetic Compss, in the Common Chart.  
Fifthly, I shall give the difference of Latitude L A, 1055 m. and the distance L B, 3268 Miles, and demand the Rumb by the Magnetics Compass.  
To find the Rumb by the Magnetics Compass, the proportions Arithmetical: As the Logarithm of the distance 326● m. L B, 351428, is to the Radius 90 d▪ L A B, 1000000; so is the Logarithm of the difference of Latitude 1055 m▪ L A, 302325; to the Co-sine of th● Rumb 950897, A B L, 18 d. 50 m whose Compliment is 71 d. 10 m. A L B▪ So that the difference of Latitude an● distance of places, is in proportion t● the Rumb, by the Magnetic Compas● in the Common Chart. Add the Logarithm of the difference of Latitude to the Radius, and Subtract the Logarithm of the distance, and you have the Cousin of the Rumb 950897, A B L, 18 d. 50 m. whose compliment is 71 d. 10 m. A L B, demanded.  
Sixthly, I shall give the difference of Longitude 3093 m. A B; and the distance 3268 m. L B. and demand the Rumb by the Magnetic Compass.  
To find the Rumb by the Magnetic Compass, the proportions Arithmetical: As the Logarithm of the distance 3268 m. L B, 351428; is to the Radius 90 d. L A B, 1000000: So is the Logarithm of the difference of Longitude 3093 m. A B, 349037; to the Sine of the Rumb 997609, A L B, 71 d. 10 m. Add the Radius to the Logarithm of the difference of Longitude, and Subtract the Logarithm of the distance, and you have the Sine of the Rumb 997609, A L B, 71 d. 10 m. So that the difference of Longitude, and the distance is in proportion to the Rumb in the Common Chart.  
Seventhly, I shall give the difference of Longitude 3093 m. A B, and the distance 3268 m. L B, and demand the difference of Latitude L A.  
To find the difference of Latitude, the proportions Arithmetical: Add the difference of Longitude 3093 m. to the distance 3268 m. and you have 6361 Mil. for the sum of the sides; then Subtract the difference of Longitude from the distance, and you hav● 175 Miles for the difference of th● sides; so take the Logarithm of th● sum of the sides A B, L B, 6361 Mile● 380352; and the Logarithm of th● difference of the sides 175 Miles 224303: Add them together, an● you have 604655, the half is 302327▪ which is the Logarithm of the difference of Latitude 1055 m. L A. S● that the difference of Longitude, an● the distance is in proportion in a Right lined Triangle, in the Common Chart▪  
Eightly, I shall give the differenc● of Latitude L A, 1055 m. and th● distance L B, 3268 m. and demand th● difference of Longitude.  
To find the difference of Longitude the proportions Arithmetical; Ad● the difference of Latitude 1055 m. to the distance 3268 m. and you have 4323 Miles for the sum of the sides than Subtract the difference of Latitude from the distance, and you have 2213 Miles for the difference of the sides: So take the Logarithm of the sum of the sides 4323 Miles, L A, L B, 363578; and the Logarithm of the difference of the sides, 2213, 334498: Add them together, and you have 698076, the half is 349038, which is the Logarithm of the difference of Longitude 3093 Miles, A B: So that the difference of Latitude, and the distance is in proportion in a Rightlined Triangle, in the Common Chart.  
I have proved by the foregoing Questions, that the Rumb and the Right-lines of the Magnetic Compass, agree with the Rumb and the Right-lines of the Common Chart, from hence may arise an objection, how the Right-lines of the Magnetic Compass can measure the Superficies of the Globe, which is spherical? To answer this objection, I will grant you to be in any Meridian and Parallel of the World, wherein you must see 180 d. of Heaven, for it is 90 d. from your Zenith every way to the Horizon, than the Circulation of the Horizon must contain 360 d. To this I compare t●● Magnetic Compass, which contain 360 d. and is Horizontal to the Horizon always, in all Meridian's an● Parallels of the World; So that th● North and South, and the East an● West Points of the Magnetic Compass, point to the Celestial Poles o● North and South, and to the Celestia● Poles of East and West, in all Meridian's and Parallels of the Earth and Sea, without giving any difference of observation for the SemiDiameter of the Earth and Sea; the● the Meridian's of the Magnetic Compass, must run parallel to the Poles of East and West, as the parallels of Latitude run parallel to the Poles of North and South; therefore the Magnetic Compass being of Right-lines, measures the Superficies of the Globe by Right-lines, as if the Superficies of the Globe was always a Plain to the Horizon; so that whatever position doth disagree with the Rumb and the Right-lines of the Magnetic Compass must be a false position in Navigation, for the work of navigation must be performed by the magnetic Compass, and the Latitude ● places by observation, by which ●eans all the known parts of the ●orld have been discovered, and laid ●own in a Plain in the Common Chart: Therefore I shall show you ●ow the difference of Latitude and the difference of Longitude, and the distance of places in the Common Chart is transferred, in proportion to the difference of Latitude, and the distance of places upon the Globe; wherein I shall prove by demonstration, that the Meridian of the Magnetic Compass doth differ from the Meridian's of the Globe, as they are drawn.  
To prove how the difference of Latitude, and the distance of places, and the difference of Longitude in the Common Chart, is transferred from the Common Chart, to be in proportion by Demonstration to the difference of Latitude, and the distance of places upon the Globe:  
In the foregoing Question, by th● Common Chart, the Lizard is in th● Latitude of 50 d. 00 m. the Bermudu● is in the Latitude of 32 d. 25 m. thei● difference of Latitude is 1055 m or 17 d. 35 m. and the difference o● Longitude is 3093 m. or 51 d. 33 m. and the distance is 3268 m. or 54 d. 28 m. before I proceed to the demonstration upon the Globe, you may observe the difference of Longitude in all Rightlined Triangles, is in proportion to the difference of Longitude in the middle parallel of the Globe, between such place differing in Longitude and Latitude, in all Meridian's and parallels of the World.   geometrical diagram 
Pag: 19  
TO prove how the difference of Latitude, and the distance of places, and the difference of Longitude in the Common Chart is transferred by demonstration, to be in proportion to the difference of Latitude, and the distance of places upon the Globe.  
EXAMPLE. The Lizard is in the Latitude of 50 d. 00 m. L. Bermudus is in the Latitude of 32 d. 25 m. B, their difference of Latitude is A L, 17 d. 35 m. the half thereof is 08 d. 47 m. 30 sec. which being added to D A, 32 d. 25 m. the Latitude of Bermudus, you have D E, 41 d. 12 m. 30 sec. the Latitude of the middle parallel between the Lizard and Bermudus, then reduce the di●rence of Longitude found by the fo● going Rightlined Triangle, between the Meridian and parallel of the Lyza● and the Meridian and parallel of B●mudus, 3093 m. into Degrees of t● Aequinoctial, by dividing 3093 by 60 m. and you have 51 d. 33 which distance being taken by you Compass, from the Aequinoctial pa● of the Globe, and measuring th● distance in the middle parallel E ● 41 d. 12 m. 30 sec. you will find tha● distance 51 d. 33 m. to reach from th● Meridian of the Lizard, to the Meridian of the Bermudus, in the parallel of 41 d. 12 m. 30 sec. which is th● same distance in the Common Chart so that the Lizard must be laid down upon the Globe, in the Meridian o● the middle parallel D E P, and in the Latitude of 50 d. 00 m. at L, and Bermudus must be laid down upon the Globe in the Meridian of the middle parallel C E P, and in the Latitude of 32 d. 25 m. at B.  
In the next place, I shall show you ●w to find the distance by the Geometrical part of the Globe, to be in coportion to the distance found by a ●ght-lined Triangle between the Me●dian and parallel of the Lizard, ●nd the Meridian and parallel of ●ermudus.  
EXAMPLE. Reduce the distance found by a right-lined Triangle 3268 m. by 60 m. ●nto degrees of the Aequinoctial, by dividing 3268 m. by 60 m. and you ●ave 54 d. 28 m. which distance being taken by your Compass, from the Aequinoctial part of the Globe, that distance will reach from the Meridian and parallel of the Lizard L, to the Meridian and parallel of Bermudus B, so that the distance between the Meridian and parallel of the Lizard, and the Meridian and parallel of Bermudus, is the same in the Globe as it is in a Rightlined Triangle.  
Having proved how the difference of Latitude, and the distance of places, and the difference of Longitude in th● Common Chart, is transferred from th● Common Chart, to be in propotio● by demonstration to the difference o● Latitude, and the difference of Longitude in the middle parallel, and th● distance of places on the Globe:  
In the next place, you may observ● the difference of Longitude in th● Globe, between the Meridian's of the middle parallel 41 d. 12. m. 30 sec E F, is 3093 m. or 51 d. 33 m. And the difference of Longitude in the Aequinoctial part of the Globe, between those Meridian's D C is 4200 m. or 70 d. 00 m. and the difference of Longitude between those Meridian's, in the parallel of Bermudus, A B, is 3545 m. or 59 d. 05 m. and the difference of Longitude, between those Meridian's in the parallel of the Lizard L G is 2700 Miles, or 45 d. 00 m. all which distances may be measured upon the Globe; from whence you may make this observation, that the difference of Longitude in the Globe, between the Meridian and Parallel of Bermudus B, ●nd the Meridian of the Lizard A B, is ●545 m. 59 d. 05 m. and the distance ●y the Common Chart, between the Meridian and Parallel of the Lizard, ●nd the Meridian and parallel of Bermudus, is 3268 m. or 54 d. 28 m. So that the difference of Longitude in the Globe, between the Meridian and parallel of Bermudus, and the Meridian of the Lizard is 277 m. or 04 d. 37 m. more than the distance between the Meridian and parallel of the Lizard, and the Meridian and Parallel of Bermudus, which is out of all proportion to the Rumb, and the difference of Longitude and the distance of places in a Rightlined Triangle; for the difference of Longitude in Navigation, by the Magnetic Compass cannot be greater than the distance: So that the work of Navigation cannot be performed by the Globe, before you have found such proportions upon the Globe, which are in proportion to the Rumb, and the Right-lines of the Magnetic Compass, as the Question hath b● transferred to the Globe from Common Chart.  
In the next place I shall give 〈◊〉 the Arithmetical proportions of Globe, which are in proportion the Geometrical part.  
The Lizard is in the Latitude 50 d. 00 m. L, Bermudus is in Latitude of 32 d. 25 m. B, their di●rence of Longitude in the Aequinoct● part of the Globe is 70 d. 00 m D I demand the difference of Longitu● in the middle parallel E F.  
To find the Logarithm of the difference of Longitude, between the Meridian and Parallel of the Lizard, ar● the Meridian of Bermudus L G, t●● proportion Arithmetical: As the R●dius 90 d. P D, 1000000, is to the Logarithm of the d●fference of Longitu● in the Aequinoctial, 70 d. 00 m. 4200 m. D C, 362324; so is the C● sine of the Latitude of the Lizard 40 ● 00 m. P L, 980806, to the Logarithm of L G, 343130. Add the Logarithm of the difference of Longitude in ●e Aequinoctial to the Co-sine of the latitude, and Subtract the Radius, and ●ou have the Logarithm of the difference of Longitude between the Meridian and parallel of the Lizard, and ●e Meridiad of Bermudus, L G, 43130, or 2700 m.  
To find the Logarithm of the difference of Longitude between the Meridian and parallel of Bermudus, and the Meridian of the Lizard, B A, the pro●ortions Arithmetical: As the Radius ●0 d. 00 m. P D, 1000000; is to the logarithm of the difference of Longitude ●n the Aequinoctial, 70 d. 00 m. or ●200 m. D C, 362324: So is the Co●ine of the Latitude of Bermudus 57 d. ●5 m. A P, 992643; to the Logarithm of A B, 354967: Add the Logarithm of the difference of Longitude in the Aequinoctial, to the Co-sine of the Latitude, and Subtract the Radius, and you have the Logarithm of the difference of Longitude, between the Meridian and parallel of Bermudus, and the Meridian of the Lizard, A B, 354967, or 3545 m.  
Then add the Logarithm of 34● found in proportion, to 70 d. 00 ● Longitude, in the parallel of 00 m. to the Logarithm of 354 found in proportion to 70 d. 00 ● Longitude, in the parallel of 25 m. and you have 698097, the in 3490481/2● which is the Logari●● of the difference of Longitude in middle parallel, between the Meri● and parallel of the Lizard and the ●●ridian and parallel of Bermudus, 3093 m. or 51 d. 33 m. So that the portions Arithmetical do agree w● the proportions Geometrical, wher● you have the middle parallel 3093 the proper difference of Longitude, which the difference of Latitude, a● the distance upon the Globe, is in li● proportion in the common Chart.  
Last of all, I shall give the Questi●● from the difference of Longitude in a Rightlined Triangle, and the Latitude of the Lizard and Bermudu● and demand the difference of Longitude in the Aequinoctial part of th● Globe.  
EXAMPLE. Sail from the Lizard in the Lative of 50 d. 00 m. L, between the ●ith and the West, in the Latitude 32 d. 25 m. B, I find my difference Longitude in a Rightlined Tri●●gle, 3093 m. I demand the difference of Longitude in the Aequinoctial ●t of the Globe, between the Me●●●ian of the Lizard and the Meri●●●n of Bermudus.  
To find the difference of Longitude the Aequinoctial, the proportions arithmetical: As the Co-sine of the ●●titude of the Lizard, 40 d. 00 m. L, 980806; is to the Logarithm of ●●e difference of Longitude in the middle parallel of the Globe, 3093 m E F, 49037: So is the Radius 90 d. 00 m. D, 1000000; to the Logarithm of 68231: Add the Logarithm of the difference of Longitude in the middle parallel to the Radius, and Subtract ●●e Co-sine of the Latitude, and you ●ave the Logarithm of 368231, which must be added to the n● proportion.  
Then, As the Co-sine of the Latitude of Bermudus, 57 d. 35 m. P A, 99264 is to the Logarithm of the differe●● of Longitude in the parallel of 〈◊〉 Globe, 3093 m. E F, 349037: So the Radius 90 d. oo m. P D, 100000● to the Logarithm of 356394: A● the Logarithm of the difference Longitude in the middle parallel, the Radius, and Subtract the Co-si●● of the Latitude, and you have t●● Logarithm of 356394:  
Then add the Logarithm of 36823 (found in proportion to 3093 m. Longitude in the parallel of 50 d. 00 m to the Logarithm of 356394, (found 〈◊〉 proportion to 3093 m. of Longitude, 〈◊〉 the parallel of 32 d. 25 m.) and yo● have 724625; the half is 362312 which is the Logarithm of the difference of Longitude in the Aequinoctia● part of the Globe, 4200 m. D C demanded: So that the proportions Arithmetical agree with the proportions Geometrical, therefore the difference of Longitude in the middle parallel of the Globe, is the proper difference of Longitude 3093 m. in proportion to the Rumb, by the Mag●etick Compass.  
I have done with first Questionn in the Common Chart between places differing in Longitude and Latitude, wherein I have proved how the Meridian's and parallels of the common Chart differing in Longitude and Latitude, are transferred by Demonstration, in proportion to the difference of Latitude, and the distance of places upon the Globe.  
Last of all, I shall show you how to transferr the difference of Longitude in one and the same Latitude in the Common Chart, into the Meridian's and parallels of the Globe.  
EXAMPLE. In the Common Chart in the Latitude of 50 d. 00 m. the difference of Longitude is 3093 m. L T, I demand the difference of Meridian's in t● Aequinoctial part of the Globe.  
To find the difference of Longitude in the Aequinoctial part of the Glo●● the proportion's Arithmetrical: As t● Co-sine of the Latitude 40 d. 00 m. P 980806, is to the Logarithm of the difference of Longitude, in the parallel 50 d. 00 m. L T, 3093 m. 349037 so is the Radius 90 d. P D, 1000000 to the Logarithm of 368231; which the Logarithm of D H, 4812 Miles, ● 80 d. 12 m. the difference of Meridian in the Aequinoctial part of the Glob● being obliqne, do differ from th● Meridian's of the Common Chart tha● are Rightlined; but the distance o● places in the Globe are the same in the Common Chart, in regard the distance of places is transferred from the Common Chart to the Globe, therefore the superficies of the Common Chart in all Meridian's and parallels, is in proportion to the superficies of the Globe in all Meridian's and Parallels.  
AND whereas all Writers in Navigation, have condemned the ●ommon Chart to be pestered with notorious errors, it proves a great mistake; therefore I shall answer the principal objection, and in that I answer all the rest.  
Mr. Edward Wright in his Correction of Errors in the Common Chart, Fol. 5. ●aith, In showing the distance of places, there is as great an error committed as in any of the former.  
For Example, If you imagine two Ships to be under the Aequinoctial 100 Leagues asunder, and that each of them should Sail from thence due North or South, under his Meridian, until they come to the parallel of 60 d. Latitude, they should be there but only 50 Leagues distant, because at that Parallel, the Meridian's are distant but half so much one from another, as they were at the Aequinoctial, as it may most manifestly appear by the Globe, and y● the Chart will show, that those t●● Ships have the selfsame distance 100 Leagues, being under the Parallel of 60 degrees, which they ha● before when they were under t●● Aequinoctial.  
It is granted, that the difference 〈◊〉 Meridian's in the Common Chart, 100 Leagues in the Equinoctial, an● 100 Leagues in the Parallel of 6● Degrees.  
In regard it appears by observation by the Magnetic Compass, that th● East and West point of the Magnetics Compass as it is Horizontal, does always point to the Poles of East and W. in the Celestial Sphere, likewise the North and South point of the Magnetic Compass, as it is Horizontal, does always point to the Poles of North and South in the Celestial Sphere, in all Meridian's and Parallels of the Terrestrial Sphere, without making any difference of observation for the Semi-Diameter of the Earth: And whereas it appears by observation, that the magnetic Compass in its true Poles ●ill direct your the Circulation of the ●arth and Sea, in any Parallel of latitude, keeping an equal distance ●om the Poles of North and South; ● likewise it must be granted that the Meridian's of the Magnetic Compass, ●ust run Parallel to the Poles of ●ast and West, the Circulation of the ●arth in all Parallel Meridian's, in ●egard the Magnetic Compass as it is Horizontal, makes the same position to the Poles of North and South in the Celestial Sphere, and to the ●oles of East and West in the Celestial Sphere, in all Meridian's and Parallels of the Terrestrial Sphere, without making any difference of observation for the Semi-Diameter of the Earth; So that the Meridian's do cross the Parallels of the Magnetic Compass at Right-Angles, in all Meridian's and Parallels of the Earth and Sea, as it is by demonstration in the Common Chart. Therefore the Magnetic Compass as it is Horizontal, must measure the superficies of the Terrestrial Sphere by Right-lines, as if t● Superficies of the Terrestrial Sphe●● was a plain to the Horizon; so that you imagine two Ships to be under t●● Aequinoctial 100 Leagues asunder, a● that each of them should Sail fro● thence due North, or due South, und●● his Meridian, until they come to th● Parallel of 60 d. Latitude, they mu●● be 100 Leagues distant; in regard th● Meridian's of the Magnetic Compas● must run Parallel to the Poles o● East and West, as the East and We●● points of the Magnetic Compass ru●● Parallel to the Poles of North an● South, I shall treat more at large o● this in its due place.  
I have given you the Geometrical demonstration, and the Arithmetical rules, how all places in the Common Chart are transferred into the Meridian's and Parallels of the Globe; therefore you cannot resolve any Question in Navigaton from the Globe, without you have such proportions from the Globe as are in proportion to the Rumb, and the Right-lines of the Magnetic Compass.   

The same Question resolved by Mercator's Projection.  
MR. Edward Wright in his Projection of Mercators' Plain-Sphere, doth make the superficies of the Globe as a ●lain to the Horizon, and doth measure ●l the Parallels of the Globe, in proportion to the degrees of the Aequinoctial ●o the North and South Poles; so that ●ll the Parallels in Mercator's Plain-●phere, are of equal Diameter to the Poles. Then Mercator divides the Meridian Line into unequal parts, and doth increase the degrees and minutes of Latitude from the Aequinoctial to the Poles, to almost an endless proportion: Therefore at every point of Latitude in his Plain-Sphere, the difference of Meridian's doth so increase, that taking half a degree above a point of Latitude, and half a degree below the Latitude, that distance is to be a degree of Longitude in that Latitude; so that the design of Mercator's Plain-Sphere is to measure the difference of Meridian's, in proportion to the difference of Meridian's upon the Glo●●  
The Question to be resolved Mercator. The Lizard, Latitude 50 ● 00 m. Bermudus, Latitude 32 d. 25 ● their difference of Longitude in t● Aequinoctial 70 d. 00 m. I demand t● Rumb by the Magnetic Compass.  
This Question cannot be resolved any Geometrical demonstration, regard I have not the difference Longitude in proportion to the Rum● by the Magnetic Compass, for Rightlined Triangle must be of equ● parts, otherwise the difference of Latitude and the difference of Longitude cannot be in proportion to th● distance of places upon the Rum● by the Magnetic Compass, neither can this Question be resolved by th● Arch of a great Circle, in regard the Angles of Position in all obliqne Angles, differ from the Angles of position by the Magnetic Compass; s● that Mercator hath no way to find the Rumb from the Question given, but brief Rules in Arithmetic, which are ●ear a proportion to the Rumb by ●he Magnetic Compass, but when the distance between the Meridian and Parallel of the Lizard, and the Meridi●n and Parallel of Bermudus, and the difference of Longitude is required in a Rightlined Triangle, to be in proportion to the Rumb, by the Magnetic Compass, and the difference of Latitude than all Mercator's brief Rules in Meridional parts, and the difference of Longitude in the Aequinoctial are rejected.  
TO resolve the Question by Mercators brief Rules, you must first find the Meridional parts contained between the difference of Latitudes.  
EXAMPLE. The meridional parts contained in the Latitude of 50 d. 00 m. are 3475, the meridional parts contained in the Latitude of 32 d. 25 m. are 2058, then Subtract the lesser from the greater and you have 1417, the Meridional parts contained in the difference of Latitude; then multiply 70 d. 00 m. of Longitude by 60 m. and you have 4200 m. the difference of Longitude in the Aequinoctial. To find ● Rumb by Mercators brief Rules, ● As the Logarithm of the difference Latitude in Meridional parts, 1417 315136, is to the Radius 90 1000000; So is the Logarithm the difference of Longitude in the Aeq●●noctial 4200 m. 362324, to the Tangent of the Rumb 71 d. 21 m. 104718● So that Bermudus doth bear fro● the Lizard in an Angle of 71 ● 21 m. Southwestwards: Add th● Radius to the difference of Longitude in the Aequinoctial, and Subtract th● Meridional parts contained in th● difference of Latitude, and you hav● the Tangent of the Rumb 71 d. 21 m● 1047188.  
The second Question to be resolved by Mercators brief Rules; the difference of Latitude between the Lizard and Bermudus, in Meridional parts is 1417, and the Rumb is an Angle of 71 d. 21 m. Southwestwards, I demand the difference of Longitude in the Aequinoctial.  
To find the difference of Longitude the Aequinoctial, say, As the Radius ●0 d. 1000000, is to the Logarithm of ●e difference of Latitude in Meridional ●●rts 1417, 315136; so is the Tangent ● the Rumb 71 d. 21 m. 1047171, to ●he Logarithm of the difference of Longitude in the Aequinoctial 4198 Miles, ●62307: Add the Logarithm of the meridional parts contained in the difference of Latitude, to the Tangent of ●he Rumb, and Subtract the Radius, ●nd you have the Logarithm of the difference of Longitude in the Aequi●octial, 4198 m. 362307 demanded.  
The third Question to be resolved ●y Mercators brief Rules; the difference of Longitude in the Aequinoctial, between the Meridian of the Lizard, and the Meridian of Bermudus, 4200 m. The Rumb is an Angle of 71 d. 21 m. Southwestwards, I demand the difference of Latitude in Meridional parts.  
To find the difference of Latitude in Meridional parts, say, As the Tangent of the Rumb 71 d. 21 m. 1047171, is to the Logarithm of the difference Longitude in the Aequinoctial 4200 ● 362324; so is the Radius 90d. 100000● to the Logarithm of the difference ● Latitude in Meridional parts, 1417 315153: Add the Logarithm of th● difference of Longitude in the Aequ●noctial, to the Radius, and Subtract the Tangent of the Rumb, and yo● have the Logarithm of the differenc● of Latitude in Meridional parts, 1417 315153 demanded.  
The Fourth Question to be resolved by Mercator's brief Rules. The difference of Latitude as it is upon the Globe 1055 m. and the difference of Latitude in Meridional parts 1417; and the difference of Longitude in a Rightlined Triangle, according to the Rumb found by these brief Rules, 3126 m. I demand the difference of Longitude in the Aequinoctial.  
To find the difference of Longitude in the Aequinoctial, Say, As the Logarithm of the difference of Latitude upon the Globe 1055 m. 302325; is to the Logarithm of the difference of Longitude in a Rightlined Triangle, 3126 m. ●9498; so is the Logarithm of the difference of Latitude in Meridional parts, ●17, 315136; to the Logarithm of the ●●fference of Longitude in the Aequinoctial ●98, 362309: Add the Logarithm the difference of Longitude in a right-lined Triangle, to the Logarithm ● the difference of Latitude in Meridional parts, and Subtract the Logarithm of the difference of Latitude ●pon the Globe, and you have the logarithm of the difference of Longitude in the Aequinoctial, 4198 m. ●62309, demanded.  
The fifth Question to be resolved by Mercator's brief Rules; The difference ●f Longitude in the Aequinoctial ● 200 m. and the difference of Latitude ●s it is upon the Globe 1055 m. and the difference of Latitude in Meridional parts 1417; I demand the difference of Longitude in a Rightlined Triangle. To find the difference of Longitude in a Rightlined Triangle, by Mercators brief Rules: Say, As the Logarithm of the difference of Latitude in Meridional parts 1417, 31513 ● to the Logarithm of the difference of ●●●gitude in the Aequinoctial 420 ● 362324; So is the Logarithm of difference of Latitude upon the (● 1055 m. 302325; to the Logarithm the difference of Longitude in a Right-lined Triangle, 3127, 349513: ● the Logarithm of the differenc● Longitude in the Aequinoctial, to ● Logarithm of the difference of L●●●tude upon the Globe, and Subtract ● Logarithm of the difference of L●●gitude in the Aequinoctial, ● you have the Logarithm of the difference of Longitude in a Right-li●● Triangle, 3127, 349513; deman●●  
In the next place Mercator d● borrow the Rules of the Globe, find the difference of Meridian's any one Parallel.  
EXAMPLE. There are two Meridian's differing in Longitude 70 d. 00 m. in the ●●quinoctial, Bermudus, Latitude 32 25 m. I demand the difference of meridian's in the Latitude of 32 d. ● m.  
●o find the difference of Meridian's ●●●ween 70 d. 00 m. of Longitude, in Latitude of 32 d. 25. m. Say, As Radius 90 d. 00 m. 1000000; is the Logarithm of 70 d. 00 m. or 00 m. 362324; So is the Co-sine ●he Latitude of Bermudus 57 d. 35 m. ●2643; to the Logarithm of 354967, ●545 m. or 59 d. 05 m. Add the Logarithm of the difference of Longitude in the Aequinoctial, to the Co●●●e of the Latitude, and Subtract the ●adius, and you have the Logarithm the difference of Meridian's, in the ●●titude of Bermudus, 32 d. 25 m. ●4967; or 3545 m. or 59 d. 05 m. ●emanded.  
Last of all, there are two Meridian's differing in Longitude 70 d. 00 m. in ●he Aequinoctial; the Lizard Latitude ●0 d. 00 m. I demand the difference ●f Longitude in the Latitude of 50 d.  
To find the difference of Meridian's between 70 d. 00 m. of Longitude, in the Latitude of 50 d. oo m. Say the Radius 90 d. oo m. 1000000 to the Logarithm of 70 d. oo m 4200 m. 362324: So is the Co-sin● the Latitude of the Lizard 40 d. oo 980806; to the Logarithm of 3431 or 2700 m. or 45 d. oo m. Add Logarithm of the difference of L●●gitude in the Aequinoctial, to the C●-sine of the difference of Latitude, a● Subtract the Radius, and your ha● the Logarithm of the difference Merdians in the Latitude of the izard, 50 d. oo m. 343130; or 2700 or 45 d. oo m.  
I have given you a full account all the brief Rules, by which Mercat●●● doth pretend to perform the work Navigation, wherein you may observ● there is not one Question in all Merc●tors brief Rules, that is in proportion to the distance between the Meridian and Parallel of the Lizard, and th● Meridian and Parallel of Bermudus therefore Mercators' brief Rules are o● no further use in Navigation, in regard the brief Rules are not in proportion the distance of places; so that the ●●rk of Navigation cannot be performed without those Rules are in proportion to a Geometrical demonstration.  
Then observe you cannot measure ●e distance of places, differing in longitude and Latitude in Mercators ●ani-Sphere, in proportion to the ●●stance of places upon the Globe: ●herefore Mr. Edward Wright in his ●orrection of Errors in Navigation, page 59 saith, if two places differ so in Longitude as well as in Latitude, ●ok how many degrees the difference of latitude containeth; so many degrees of ●he Aequinoctial take with your Compass, ●nd leading one foot in the Aequinoctial, ●ove forward the other also Parallel-wise, ●eeping always that distance, till it cross the Rumb of those two places, in such sort, that one Foot resting in that crossing, the other carried about may but touch the AEquinoctial, then take with your Compass the Segment or part of the Rumb, between that crossing, and the Aequinoctial, set both Feet in the Aequinoctial, and see how many degrees are contained be● them, for so many score League's distance of those two places. So you are to take with your Com● the difference of Latitude from Aequinoctial, which is of equal p● and run that Parallel to the Ae●●●noctial, until the other point ● cross the Rumb, then take the dista● from the point that crosses the Ru●● to the place where the Rumb ● cross the Aeqinoctial, and meas that distance in the Aequinoct●●● which is of equal parts for the dista●●● between the Lizard and Bermud●● Therefore Mercator doth grant t● the difference of Latitude, and distance of Places, must be measure by a Scale of equal parts, to be in proportion to the Rumb by the Magneti●● Compass; this position agrees w● the Common Chart: Then in t● plainest Rules of Art, the differen●● of Latitude and the distance of Place being measured by a Scale of equ● parts, the difference of Longitude mu●● be of equal parts, in proportion to th● difference of Latitude, and the distance places, in a Rightlined Triangle, ●●●erwise you must alter the Rumb the Magnetic Compass, and the ●ance of places.  
But Mercator opposeth this position his Plani-Sphere, and from the arithmetical Rules of the Globe, ●erein the difference of Meridian's ●●●ween the Meridian and Parallel of ●ermudus, and the Meridian of the ●●zard is 59 d. 05 m. likewise Mer●●tor proves the difference of Meridian's his Plani-Sphere, between the Meridian and Parallel of Bermudus, and ●he Meridian of the Lizard, to be in proportion to the difference of Meri●ians upon the Globe, by taking half degree above the Latitude of Bermu●us, 32 d. 25 m. and half a degree below the Latitude of 32 d. 25 m. which distance is a degree of Longitude in the Latitude of Bermudus, therefore so oft as you can have that distance between the Meridian of Bermudus, and the Meridian of the Lizard, such is the difference of Longitude 59 d. 05 m. Do the like, and you sha●● have the difference of Meridian between places in any other Parallel upon Mercators' Plani-Sphere, in proportion to the difference of Meridian between places in any Parallel of th● Globe, so that the difference of Longitude between the Meridian an● parallel of Bermudus, and the Meridian of the Lizard, is greater tha● the distance, which is out of all proportion in a Rightlined Triangle● Therefore Mercator hath no Geometrical Demonstration, either in obliqne Angles or by Rightlined Angles, to prove the difference of Meridian's, as it is upon the Globe, to be in proportion to the difference of Latitude and the distance of places, and the Rumb by the Magnetic Compass.  
I have proved how the difference of Latitude, and the difference of Longitude, and the distance of places in the Common Chart, is transferred to be in proportion upon the Globe.  
I have proved how the difference of Longitude in all Rightlined Triangles, is in proportion to the middle Parallel of all places upon the Globe, therefore the Meridian's of the Globe, as they are drawn, must differ from the Meridian's of the Magnetic Compass; so that it is the principal point in Navigation, to inquire whether the North and South Points of the Magnetic Compass, run in the Meridian's of the Globe, as they are drawn, or whether the North and South points of the Magnetic Compass, run parallel to the Poles of East and West, as the East and West points of the Magnetic Compass, run parallel to the Poles of North and South: Therefore I shall proceed to lay a Foundation in Navigation, that men may no longer halt between three opinions, wherein I shall prove by observation and demonstration from the Magnetic Compass, as it is Horizontal, that the North and South Points of the Magnetic Compass, run Parallel to the Poles of East and West, as the East and West points of the Magnetic Compass run Parallel to the Poles of North and South; so that the Common Chart that is so much rejected, is th● only Truth in Navigation.  
In the first place, I shall prove the East and West Points of the Magnetic Compass, point to the Poles of East and West in the Celestial Sphere, in all Meridian's and Parallels of the Terrestrial Sphere; then I shall prove the East and West Points of the Magnetic Compass run Parallel to the Poles of North and South, in all Meridian's and Parallels of the Terrestrial Sphere.  
EXAMPLE. By Observation, if the Sun be upon the Horizon in the Aequinoctial, the Suns-Amplitude shall be East or West of you, by the Magnetic Compass, as it is Horizontal, in all Meridian's and Parallels of the Terrestrial Sphere; but if I was in the Latitude, 32 d. 25 m. or in any other Latitude, and I was to Sail by the Magnetic Compass East or West, the Circulation of the Earth and Sea, I shall find I have not altered my Latitude by observation; therefore the East and West course of the Magnetic Compass agrees with the Parallels of Latitude upon the Terrestrial Sphere by observation; this position cannot be proved, but from the Latitude of Places by observation, which agrees with the East and West course of the Magnetic Compass; so that the Parallels of Latitude upon the Globe, are of equal distance from the Poles of North and South, and in proportion to the Rumb by the Magnetic Compass.  
In the next place, I shall prove the North and South Points of the Magnetic Compass, point to the Poles of North and South in the Celestial Sphere, in all Meridian's and Parallels of the Terrestrial Sphere. Then I shall prove the North and South Points of the Magnetic Compass, must run parallel to the Poles of East and West, in all Meridian's and Parallels of the Terrestrial Sphere.  
EXAMPLE. By Observation, If the Sun or Star be upon their Meridian in my Meridian and Parallel of the Terrestrial Sphere, the Sun's or Stars Azimuth mus● be North or South, by the Horizonta● bearing of the Magnetic Compass to the Celestial Poles; so that the Magnetic Compass, as it is Horizontal, points to the Celestial Poles of North and South, and to the Celestial Poles of East and West, in all Meridian's and Parallels of the Earth and Sea; without giving any difference of observation for the Semi-Diameter of the Earth and Sea, and notwithstanding the Magnetic Compass points to the Celestial Poles of North and South, and to the Celestial Poles of East and West, in all Meridian's and Parallels of the Terrestrial Sphere, it is proved by observation, that you may run the Circulation of the Earth by the Magnetic Compass, in any Parallel of Latitude; therefore the Magnetic Compass must run Parallel to the Poles of East and West, as the Magnetic Compass runs Parallel to the Poles of North and South in the Terrestrial Sphere, in regard the Magnetic Compass makes the same position to the Poles of East and West, and to the Poles of North and South, in the Celestial Sphere; so that the difference of Meridian's by the Magnetic Compass, as it is Horrizontal upon the Terrestrial Sphere, is of equal distance from the Poles of East and West, as the difference of the Parallels of Latitude by the Magnetic Compass, as it is Horizontal, is of equal distance from the Poles of North and South.  
This position must be infallible; for by observation, the Magnetic Compass, as it is Horizontal, points to the Celestial Poles; so that the North and South points of the Magnetic Compass, do cross the East and West points of the Magnetic Compass, at Right-Angles, in all Meridian's and Parallels of the Earth and Sea: Therefore the Magnetic Compass, doth measure the superficies of the Glob● by Right-lines, as if the superficies o● the Globe was a Plain to the Orisons  
In the next place, if the Sun be i● the Aequinoctial, and that you were i● the Latitude of 60 d. or 70 d. or 80 d of North or South Latitude, the Su● upon the Horizon shall be East of you in the Morning, and West of you i● the Afternoon, by the Magnetic Compass, in its true Poles, as if you were in the Aequinoctial part of the Earth or Sea: The reason of this observation is this, the Terrestrial Sphere is spacious to us that are upon it, but it appears to be no other than a Centre to the body of the Sun or Stars, in regard the Magnetic Compass runs atwhart the Parallels of Latitude in the Terrestrial Sphere, to the Poles of East and West in the Celestial Sphere, without giving any difference of observation for the Semi-Diameter of the Earth and Sea: Therefore it had been impossible to find the Magnetic Compass should run Parallel to the Poles of North and South in the Terrestrial Sphere, if the Meridian Altitude of the Sun or Stars, by observation, did not produce the same Latitude or Parallel with the Magnetic Compass, the Circulation of the Earth, in regard the Magnetic Compass, as it is Horinzontal, points to the Poles of East and West in the Celestial Sphere, as the Magnetic Compass points to the Poles of North and South in the Celestial Sphere.  
Then observe, there is no reason in Navigation to suppose the Meridian, or the North and South Points of the Magnetic Compass, as it is Horizontal, to the Meridian Altitude of the Sun or Stars, in the Celestial Sphere, to run to the North and South Poles, as it is in the Terrestrial Sphere; but that you may likewise suppose Meridian's, or the East and West points of the Magnetic Compass, as it is Horizontal, to the Sun's Amplitude in the Aequinoctial, to run to the Poles of East and West in the Terrestrial Sphere; in regard the Magnetic Compass makes the same position, to the Poles of East and West, as it has to the Poles of North and South in the Celestial Sphere.  
In the next place it must be granted if the East and West points of the Magnetic Compass, in the Parallel o● 80 d. point to the Poles of East and West in the Celestial Sphere; without giving any difference of observation, for almost the Semi-Diameter of the Earth, in the Terrestrial Sphere, than the North and South Points of the Magnetic Compass, in the Meridian of 80 d. must point to the Poles of North and South in the Celestial Sphere, without giving any difference of observation, for almost the Semi-Diameter of the Earth, in the Terrestrial Sphere: Therefore the Meridian's of the Magnetic Compass, hath the same position in the Terrestrial Sphere, and must run Parallel to the Poles of East and West, as the Parallels of Latitude run Parallel by the Magnetic Compass, to the Poles of North and South.  
Then observe the Magnetic Compass, being always Horizontal to the Horizon, it doth measure the Superficies of the Earth and Sea by Right-lines, in the Parallel Meridian's to the Poles of East and West, and in the Parallels to the Poles of North and South, as if the Superficies of the Globe was a plain to the Horizon; this position you have at large in the Common Chart, wherein you have the difference of Meridian's, and the difference of Latitude, and the distance of places in proportion to the Rumb, and the Right-lines of the Magnetic Compass.  
If the position of the Common Chart be not the only truth in Navigation, there is no Geometrical Demonstration in proportion to the Rumb, and the Right-lines of the Magnetic Compass.  
Therefore it hath been a great mistake in all Writers, to suppose the Magnetic Compass to agree with the Meridian's of the Globe as they are drawn; whereas, there is no observation or demonstration to prove i● only the Meridian's of the Magnetics Compass, as it is Horizontal to th● Meridian's of the Sun and Stars, poin● to the Poles of the Celestial Sphere ● To this I must reply, that the Ea●● and West points of the Magnetics Compass, as it is Horizontal, poin● to the East and West Poles of the Celestial Sphere, from all Meridian's an● Parallels of the Terrestrial Sphere; s● that you may as well draw Meridian● in Navigation by the Magnetic Compass, from the Poles of East and West as from the Poles of North and South in the Terrestrial Sphere; therefore the work of Navigation cannot be performed by the obliqne Lines of the Globe.  
I have proved by observation, that the East and West points of the Magnetic Compass, run Parallel to the Poles of North and South in the Terrestrial Sphere, therefore the North and South points of the Magnetic Compass, must run Parallel to the Poles of East and West in the Terrestrial Sphere; in regard those Poles do cross at Right-Angles, in all Meridian's and Parallels of the Earth and Sea.  
Last of all, I shall show you the errors that attend Mercator.  
I have proved that Mercator does grant the difference of Latitude, and the distance of places to be of equal parts, and to be in proportion to the Rumb by the Magnetic Compass in a Rightlined Triangle; so that Mercator hath granted three parts in four in a Rightlined Triangle; then if the difference of Longitude in a Rightlined Triangle, be not the true difference of Meridian's, according to the Rumb by the Magnetic Compass, we have no Geometrical Demonstration in Navigation, therefore I shall proceed to find the distance between the Meridian and Parallel of Bermudus, and the Meridian and Parallel of the Lizard, in a Rightlined Triangle, from the Rumb sound by the brief Rules of Mercator, and the difference of Latitude.  
And last of all, I shall show you the disproportion between the Meridian's of the Common Chart, and the Meridian's of Mercator and the Globe, as they are drawn; so that whoever pretends to keep account of the Ships way, according to the difference of Meridian's in Mercator's Plani-Sphere, or according to the difference of Meridian's upon the Globe, they must commit gross errors.  
EXAMPLE. I Sail from the Meridian and Parallel of the Lizard, in the Latitude of 50 d. 00 m. in an Angle of 71 d. 21 m. Southwestward: As the Rumb is found by the brief Rules of Mercator, into the Meridian and Parallel of Bermudus, in the Latitude of 32 d. 25 m. I demand the distance and the difference of Longitude between the Meridian and Parallel of Bermudus, and the Meridian and Parallel of the Lizard, in a Rightlined Triangle.   geometrical diagram 
Pag: 61  
The Angles of position by Mercators brief Rules, do differ from the Angles of position by the Common Chart 11 m. however the distance between the Lizard and Bermudus cannot be known, but by a Rightlined Triangle.  
The Angle following being a Rightlined Triangle, is in like proportion to the Common Chart, so is L the Meridian and Parallel of the Lizard, and B is the Meridian and Parallel of Bermudus: Then is L A the difference of Latitude 1055 m. and C L D is the Rumb or Angle of position, between the Meridian and Parallel of the Lizard, and the Meridian and Parallel of Bermudus, then is A B and L T the difference of Longitude between the Meridian of the Lizard, and the Meridian of Bermudus, and L B is the distance between the Lizard and Bermudus, then is B H and L Q the difference of Longitude in Mercators' Plani-Sphere, and upon the Globe, which differs from the Meridian's of the Common Chart 419 m. and 426 Miles.  
To find the distance between the Meridian and Parallel of the Lizard and Bermudus, L B, the proportions Arithmetical: As the Co-sine of the Rumb 71 d. 21 m. A B L, 950485; is to the Logarithm of the difference of Latitude 1055 m. L A, 302325: So is the Radius 90 d. LABERNELE, 1000000; to the Logarithm of the distance 3299 m. L B; 351840 demanded: Add the Logarithm of the difference of Latitude to the Radius, and Subtract the Co-sine of the Rumb, and you have the Logarithm of the distance 351840, L B, 3299 m.  
Thus you see when Mercator hath wearied himself with finding brief Rules, to be in proportion between the difference of Longitude in the Aequinoctial, and the difference of Latitude in Meridional parts, to find the Rumb by the Magnetic Compass: Then Mercator doth grant that the distance of places differing in Longitude and Latitude, cannot be found by any Geometrical Demonstration, to be in proportion to the Rumb by the Magnetic Compass, and the difference of Latitude, but by a Rightlined Triangle; therefore Mercator doth grant three parts in four in a Rightlined Triangle; then if the difference of Longitude in a Rightlined Triangle, be not the true difference of Meridian's, there is no Geometrical Demonstration in Navigation, to find the difference of Meridian's in proportion to the Rumb by the Magnetic Compass.  
In the next place I shall proceed to find the difference of Longitude in a Rightlined Triangle, from the difference of Latitude, and the Rumb found by Mercators brief Rules.  
EXAMPLE. I Sail from the Meridian and Parallel of the Lizard, in the Latitude of 50 d. 00 m. in an Angle of 71 d. 21 m. Southwestward: As the Rumb is found by the brief Rules of Mercator, into the Meridian and Parallel of Bermudus, in the Latitude of 32 d. 25 m. I demand the difference of Longitude in a Rightlined Triangle.  
To find the difference of Longitude the proportions Arithmetical: As the Radius 90 d. A B L, 1000000; is to the Logarithm of the difference of Latitude 1055 m. L A, 302325: So is the Tangent of the Rumb 71 d. 21 m. A L B, 1047171; to the Logarithm of the difference of Longitude 349496, A B, 3126 m. Add the Logarithm of the difference of Latitude to the Tangent of the Rumb, and Subtract the Radius, and you have the Logarithm of the difference of Longitude, in proportion to the Rumb by the Magnetic Compass, and the difference of Latitude in a Rightlined Triangle.  
In the practical part of the Sea, the Rumb is always given, as in the first Question in the Common Chart, 71 d. 10 m. So that the brief Rules of Mercator do alter the Rumb 11 m. which altars the difference of Longitude 33 m. and the distance 30 m. otherwise this latter Rightlined Triangle would agree with the former. I shall pass by these small errors, and proceed to the greater in Mercators' Plani-Sphere, and upon the Globe, the difference of Longitude between the Meridian and Parallel of Bermudus, and the Meridian of the Lizard, is proved to be 3545 m. B H, or 59 d. 05 m. and the difference of Longitude is proved to be 3126 m. B A, according to the Common Chart; so that the difference of Longitude in Mercator's Plani-Sphere, and upon the Globe, is more than the difference of Longitude in the Common Chart, by 419 m. A H; therefore, if I was to Sail from B to H, I must outrun the Meridian of the Lizard in the Common Chart 419 m. before I can expect to arrive in the Meridian of Mercator, or the Meridian of the Globe: Then if I was to Sail from L to H by the Magnetic Compass, I must Sail in an Angle of 21 d. 40 m South-Eastward by the Magnetic Compass, to arrive in the Meridian of the Globe or Mercator; so that I have showed you how great the disproportion is between the Meridian's of the Common Chart, and the Meridian's of Mercator, or the Meridian's of the Globe; therefore the Meridian's of the Magnetic Compass were never rightly understood.  
Likewise in Mercators' Plani-Sphere, and upon the Globe, the difference of Longitude between the Meridian and Parallel of the Lizard, and the Meridian of Bermudus, is proved to be 2711 m. L Q, or 45 d. and the difference of Longitude is proved to be 3126, L T, according to the Common Chart; so that the difference of Longitude is Mercator's Plani-Sphere, and upon the Globe is less than the difference of Longitude in the Common Chart, by 426 m. Q T; so that if I was to Sail from L to T, I must run 426 m. before I can arrive in the Meridian of Bermudus, in the Common Chart: Then if I was to Sail from B to Q, by the Magnetic Compass, I must Sail in an Angle of 21 d. 59 m. Northeastward, before I can arrive in the Meridian of the Globe or Mercator; so that I have showed you how great the disproportion is between the Meridian's of the Common Chart, and the Meridian's of Mercator, or the Meridian's of the Globe; therefore the Meridian's of the Magnetic Compass were never rightly understood, in regard there are such various opinions in Navigation, and the disproportion between one position and the other is so great, insomuch that the Question may be asked, what is truth in Navigation?  
I have proved by Demonstration and Observation, that the Magnetic Compass, as it is Horizontal, doth point to the Poles of East and West in the Celestial Sphere, in all Meridian's and Parallels of the Earth and Sea, without giving any difference of observation for the Semi-Diameter of the Earth: But by observation from the Meridian Altitude of the Sun and Stars, the Magnetic Compass doth run Parallel to the Poles of North and South in the Terrestrial Sphere, the Circulation of the Earth; therefore it had been impossible to prove the Magnetic Compass should run Parallel to the Poles of North and South, in the Terrestrial Sphere; if the Magnetic Compass did not run Parallel to the Latitude of places by observation.  
Likewise, I have proved by Observation and Demonstration, that the Magnetic Compass, as it is Horizontal, doth point to the Poles of North and South in the Terrestrial Sphere, in all Meridian's and Parallels of the Earth and Sea, which Meridian's do cross the East and West Poles of the Magnetic Compass, as it is Horizontal to the Celestial Sphere, at Right-Angles, in all Meridian's and Parallels in the World: Therefore the Meridian's of the Magnetic Compass being Horizontal, must run Parallel to the Poles of East and West in the Terrestrial Sphere, as the Magnetic Compass, being Horizontal, doth run Parallel to the Poles of North and South in the Terrestrial Sphere.  
Last of all, the Magnetic Compass being of Right-lines, doth measure the Superficies of the Globe by Right-lines, as if the superficies of the Globe was always a plain to the Horizon ● so that whatever projection disagree● with the Rumb, and the difference o● Latitude, and the difference of Longitude, and the distance of places, as it is in proportion in a Rightlined Triangle, must be a false position in Navigation: therefore there is but one truth in Navigation, and that is the Common Chart, that hath been so much rejected by all Writers, for the Meridian's and Parallels of places cannot be known upon the Globe or Mercators' projection, but as it is transferred from the Common Chart, which is in proportion to a Rightlined Triangle.  
I shall proceed to the Arch of a great Circle, which is the third position, or principal kind of Sailing so called, wherein I shall show you, that the Question cannot be given from the practical part of the Sea, but must alter the Meridian and Parallel of all places upon the Globe.   

SAILING by the Arch of a great Circle.  

The true distance between two places in the Arch of a great Circle, as in the following Rules.  
1. IF both places have no Latitude, as being under the Aequinoctial, and one of them no Longitude, than the Longitude of the other place is the distance: If it be not above 180 d. but if it be above 180 d. then Subtract it out of 360 d. the remainder is the distance.  
2. If both places have Longitude under the Aequinoctial, the difference of Longitude is the distance, so it be under 180 d. but if it be above 180 d. then Subtract it out of 360 d. and the remainder is the difference of Longitude.  
3. If both places have one and the same Meridian, and one of them have Latitude, than the Latitude is the distance.  
4. If both places have Latitude, as both Northerly or both Southerly, being under the same Meridian, the difference is the distance.  
5. If one place have North Latitude, and the other South Latitude, being under the same Meridian, the sum of both is the distance.  
The Arch of a great Circle in the Terrestrial Globe, doth give a nearer distance of places than the Common Chart; so that the Arch of a great Circle does not run in the Parallels of the Globe, without it be in the Aequinoctial: Likewise the Angles of position in the Arch of a great Circle, do differ from the Angles of position by the Magnetic Compass; insomuch that the Question cannot be given from the Rumb by the Magnetic Compass, in the practical part of the Sea, but you must alter the Meridian and Parallel of all places in the Globe.  
EXAMPLE. There are two places in the Parallel of 50 d. that differ in Longitude 70 d. in the Aequinoctial, I demand the Angle of position, and the distance.  
The Demonstration follows, E F D the Aequinoctial, L C B the Parallel of Latitude 50 d. L Q B the nearest distance, L Q half the nearest distance, E L the Latitude 50 d. L P the Compliment of the Latitude 40 d. E P D the Angle at the Pole, or the difference of Longitude in the Aequinoctial, 70 d. E P F the Angle of half the difference of Longitude 35 d.   

To find the distance.   geometrical diagrams 
Pag: 72.  
From hence you may observe that the Arch of a great Circle doth not run in the Parallel of Latitude by Observation, L C B, but doth run in an Arch of a nearer distance, as L Q B, for it is proved the difference of Longitude upon the Globe, in the Parallel of 50 d. L C B, is 45 d. or 2700 M. and by the Arch of a great Circle, the distance L Q B, is 43 d. 16 m. or 2596 m. So that the Arch of a great Circle doth differ from the distance of places in the Parallels of the Globe, I d. 44 m. or 104 Miles, which must alter the Latitude by observation, and the Rumb by the Magnetic Compass.  
In the next place I shall find the Angle of position by the Arch of a great Circle, the Question requires another demonstration, before I can resolve the Question as in the adjacent figure.   

To find the Angle of position.  
As the Sine of 50 d. L E, 988425; is to the Tangent of 55 d. E G, 1015477; So is the Radius 90 d. L H, 1000000; to the Tangent of 1027052, H L A, 61 d, 48 m. and is equal to the Angle B L P, 61 d. 48 m. which is the Angle of position, to the Arch of a great Circle L Q B. Add the Tangent 55 d. E G, to the Radius 90 d. L H, and Subtract the Sine of 50 d. L E, and you have the Tangent of the Angle H L A, 1037052, 61 d. 48 m. demanded.  
Then Observe, if you Sail in an Angle of 61 d. 48 m. North-West-ward, by the Magnetic Compass, from L, you cannot arrive at Q or B, in regard you cannot Sail from one place to another in the same Parallel, without you Sail East or West by the Magnetic Compass; so that the Arch of a great Circle runs in a nearer distance than the Parallel of Latitude: Therefore you cannot Sail by the Magnetic Compass from the Latitude of 50 d. L, into the Latitude of 50 d. B, in an Angle of 61 d. 48 m. in the Arch of a great Circle L Q B.  
But it may be Objected, that I may alter my Course to arrive at B. If so, than I must alter the Question. Therefore suppose the Question to be this, I have Sailed from the Latitude of 50 d. L, in angle of 61 d. 48 m. North-West-ward, 400 m. I demand in what Meridian and Parallel of the Globe I am in. I have proved I cannot Sail in an Angle of 61 d. 48 m. from the Latitude of 50 d. L, by the Magnetic Compass, and in the Arch of a great Circle LQB, so as to arrive in the Latitude of 50 d. at B; then is it impossible for me to know in what Meridian and Parallel of the Globe I am, in regard the Magnetic Compass does not run in the Arch of a great Circle: Therefore I must resolve the Question by a Rightlined Triangle, and transferr the Meridian and Parallel of places found in a Rightlined Triangle, into the Meridian and Parallel of the Globe; so that you cannot perform the work of Navigation by the Obliqne Lines of the Globe, in the Arch of a great Circle: In regard the Obliqne Lines of the Globe differ from the Rumb, and the Right-Lines of the Magnetic Compass, insomuch that you cannot give the Question from the practical part of the Sea, to be in proportion to the Angles of position, and the distance of places in the Arch of a great Circle, but you must alter the Meridian and Parallel of places in the Globe.   

I shall add one Example more from the foregoing Question, between the Lizard and Bermudus.  
The Lizard in the Latitude of 50 d. E L, B●rmudus in the Latitude of 32 d. 25 m. F B; their difference of Longitude in the Aequinoctial 70 d. E P F. I demand the Angles of position E L B, and L B P, and their distance L B, in an Arch of a great Circle.  
The demonstration follows, E F the Aequinoctial, C B the Parallel of the Latitude of Bermudus, L A the Parallel of the Latitude of the Lizard, E P F the Angle at the Pole, or the difference of Longitude in the Aequinoctial 70 d. L P the Compliment of the Latitude of the Lizard 40 d. B P the Compliment of the Latitude of B●rmudus 37 d. 35. m. L B the distance between the Meridian and Parallel of the Lizard, and the Meridian and Parallel of Bermudus, then is the Angle E L B, the Angle of position between the Lizard and Bermudus, and the Angle L B P, is the Angle of position between Bermudus and the Lizard.  
This Question requires a further Demonstration, before we can find Arithmetical proportion, as in the adjacent figure, and in the first place we are to find the side P A; then say, As the Sine of 20 d. G E, 953405; is to the Tangent of 50 d. E L, 1007618: So is the Radius 90 d. G F. 1000000; to the Tangent of 1054213, F A, 73 d. 59 m. whose Compliment is 16 d. 01 m. A P: Then Subtract A P, 16 d. 01 m. out of P B, 57 d. 35 m. and you have 41 d. 34 m. A B. Add the Tangent of 50 d. E L, to the Radius 90 d. G F, and Subtract the Sine of the 20 d. G E, and you have the Tangent of 73 d. 59 m. F A, whose Compliment is A P, 16 d. 01 m. demanded.  
Now to find the distance you have the side A P, 16 d. 01 m. and the side L P, 40 d. and the side A B, 41 d. 34 m. I demand the distance L B.  
As the Co-sine of 16 d. 01 m. A P, 998280; is to the Co-sine of 40 d. L P, 988425: So is the Co-sine of 41 d. 34 m. A B, 987400; to the Co-sine of 977545, L B, 53 d. 24 m. the distance between the Meridian and Parallel of the Lizard, and the Meridian and Parallel of Bermudus by the Arch of a great Circle, but by the Common Chart the distance is 3268 m. or 54 d. 28 m. which is 64 m. or 01 d. 04 m. more than the distance by the Arch of a great Circle.   geometrical diagrams 
Pag: 78  
As the Sine of the side LB, 53 d. 24 m. 990461; is to the Sine of the Angle LPB, 70 d. 00 m. 997298: So is the Sine of the side P B, 57 d. 35 m. 992643; to the Sine of the Angle of position between the Lizard and Bermudus 999480, C L B, 81 d. 10 m. Southwestwards. So that the Angle of position in the Arch of a great Circle, doth differ from the Angle of position by the Magnetic Compass in the Common Chart 10 d. 00 m. Then if I should Sail in the Angle of 80 d. 10 m. by the Magnetic Compass, from the Lizard L, towards Bermudus B, I cannot arrive at Bermudus, in regard the Angle of position by the Arch of a great Circle, doth differ from the Angle of position by the Magnetic Compass 10 d. 00 m.  
Last of all, I shall find the Angle of position from Bermudus to the Lizard, P B L, the side P B is 57 d 35 m. the Angle C L B is 81 d. 10 m. the side P L is 40 d. 00 m. I demand the Angle of position from Bermudus to the Lizard P B L.  
As the sine of the side P B, 57 d. 35 m. 992643; is to the Sine of the Angle C L B, 81 d. 10 m. 999481: So is the Sine of the side L P, 40 d. 980806; to the Sine of the Angle of position from Bermudus to the Lizard 987644, P B L, 48 d. 48 m. North-Eastwards. Add the Sine of the Angle C L B, 81 d. 10 m. to the Sine of the side L P, 40 d. and Subtract the Sine of the side P B, 57 d. 35 m. and you have the Sine of the Angle of position from Bermudus to the Lizard, in the Arch of a great Circle 987644, P B L, 48 d. 48 m. North-Eastwards; so that the Angle of position in the Arch of a great Circle, differs from the opposite Angle of position by the Magnetic Compass, in the same Question 39 d. 58 m. for the Angle of position from the Lizard to Bermudus is 81 d. 10 m. Southwestward, and the Angles of position from Bermudus to the Lizard is 48 d. 48 m. Northeastward; then by the Common Rules in Navigation, if I Sail by the Magnetic Compass from the Lizard to Bermudus, in an Angle of 81 d. 10 m. Southwestward, 3204 M. and was to return to the Lizard, I must Sail in an Angle of 08 d. 50 m. East-Northwards by the Magnetic Compass, in regard 08 d. 50 m. is the opposite point to 81 d. 10 m. Likewise if I Sail by the Magnetic Compass from Bermudus to the Lizard, in an Angle of 48 d. 48 m. Northeastward 3204 M. and was to return to Bermudus, I must Sail in an Angle of 41 d. 12 m. West-Southward.  
Then Observe, such is the disproportion in the Angles of position in the Arch of a great Circle, that it is impossible to Sail by the Magnetic Compass in the Arch of a great Circle, and in the Angles of position found in the Arch of a great Circle, into the same Meridian and Parallel of places upon the Globe, in regard the Angles of position in the Arch of a great Circle, does differ from the Angles of position by the Magnetic Compass.  
But it may be Objected, that I may alter my Course to arrive at my Port. If so, you must alter the Question, for by the Angles of position in the Arch of a great Circle, you cannot tell where you are upon the Globe; in regard the Angles of position in the Arch of a great Circle, does differ from the Angles of position by the Magnetic Compass, for the Angle of position in the Arch of a great Circle, is 81 d. 10 m. Southwestward from the Lizard to Bermudus, and it is 48 d. 48 m. Northeastward from Bermudus to the Lizard; and by the Magnetic Compass I must Sail in an Angle of 71 d. 10 m. Southwestward from the Lizard to Bermudus, and I must Sail in an Angle of 18 d. 50 m. East-Northward from Bermudus to the Lizard; so the opposite point of the Magnetic Compass, must be the Angle of position to return to the same Port, or the Magnetic Compass is of no use in Navigation; so that I cannot give the Question from the Practical part of the Sea, to be resolved by the Arch of a great Circle, but I must alter the Meridian and Parallel of places in the Globe, therefore you must resolve the Question from the Angle of position by the Magnetic Compass, and the difference of Latitude in the practical part of the Sea, by a Rightlined Triangle, and transferr the Meridian and Parallel of places found in a Rightlined Triangle, into the Meridian and Parallel of the Globe, as I have already showed; so that the work of Navigation must be resolved by the Rumb, and the Right-lines of the Magnetic Compass in a Rightlined Triangle, before the Meridian and Parallel of places can be known upon the Globe.  
Notwithstanding the Arch of a great Circle will not answer the design of Navigation by the Magnetic Compass; it must be granted, that the Arch of a great Circle in the Terrestrial Sphere, and in the Celestial Sphere is a true position, in regard the Angles of position in all Obliqne Angles, are in proportion to the several sides of all Obliqne Angles, which is the nearest distance of places.  
EXAMPLE. If I observe the distance of two Stars, in one and the same Parallel, I do measure the nearest distance between those two Stars, but I cannot measure the extreme Arch of that Parallel by observation; so that the nearest distance is the Arch of a great Circle, by observation and demonstration: Likewise all observations in the Arch of a great Circle gives the nearest distance; so that the Arch of a great Circle does not run in the Parallels of the Globe, without it be in the Aequinoctial, to the Poles of North and South, and in the Ecliptic, which is as the Aequinoctial to the Poles Ecliptic, and upon the Horizon, which is as the Aequinoctial to the Zenith; therefore the Angles of position found in the Arch of a great Circle, in any Parallel of Latitude, is not to be measured by any Geometrical measure upon the Terrestrial or Celestial Sphere, without it be at the North or South Poles the measure of those Angles are in the Aequinoctial, or at the Poles Ecliptic the measure of those Angles are in the Ecliptic Line, or at the Poles of your Zenith, or point over your head, the measure of those Angles are in the Horizon. In this case the Angles make a Right-Angle in the Aequinoctial, and in the Zodiac, and in the Horizon, but in all other Angles of position, they are not to be measured Geometrically upon the Globe; so that such Angles, are only in proportion to the sides of an Obliqne Angle, which is the nearest distance of places upon the Globe; therefore those Angles are out of all proportion to the Rumb, and the Right-Lines of the Magnetic Compass; so that the work of Navigation, cannot be performed by the Arch of a Great Circle.  
To Conclude, I have proved how all the known part of the World is laid down in the Meridian's and Parallels of the Common Chart, in proportion to the Rumb, and the Right-lines of the Magnetic Compass.  
I have proved how the Meridian's and Parallels of all places in the Common Chart are transferred into the Meridian's and Parallels of the Globe; in regard the difference of Meridian's upon the Globe, does increase from the Poles to the Aequinoctial, and the difference of Meridian's in the Common Chart, which is in proportion to the Rumb, and the Right-lines of the Magnetic Compass, are of equal distance, so that the difference of Meridian's upon the Globe being Obliqne, are out of all proportion to the Rumb and the Right Lines of the Magnetic Compass, without it be in the middle Parallel of places differing in Longitude and Latitude.  
It is proved, Mercator divides his Plani-Sphere in proportion to the difference of Meridian's upon the Globe: Therefore the Meridian's of Mercator does not run in the Meridian's of the Magnetic Compass, or the Meridian's of the Common Chart, in regard the difference of Meridian's upon Mercators Plani-Sphere, does increase from the Poles to the Aequinoctial, and the difference of Meridian's in the Common Chart, which is in proportion to the Rumb, and the Right-lines of the Magnetic Compass are of equal distance, which leads me to this consideration; all points of the Magnetic Compass, as they are drawn, are of Right-lines, therefore the North and South point of the Magnetic Compass, does cross the East and West point of the Magnetic Compass at Right-Angles; so that it is proved by Observation, that the East and West point of the Magnetic Compass, will direct you the Circulation of the Earth and Sea, in all Parallels of Latitude; then observe the North and South point of the Magnetic Compass as it is drawn, doth cross the East and West point of the Magnetic Compass at Right-Angles, in all Meridian's and Parallels of the Earth and Sea. Therefore the difference of Meridian's by the Magnetic Compass, must run Parallel in all Meridian's and Parallels of the Earth and Sea, in proportion to the difference of the Parallels of Latitude, otherwise you may as well say the North and South point of the Magnetic Compass, as it is drawn, doth not cross the East and West point of the Magnetic Compass at Right-Angles.  
I have proved by Observation, that the Meridian's of the Magnetic Compass, doth cross the East and West points of the Magnetic Compass at Right-Angles, in all Meridian's and Parallels of the Earth and Sea; therefore the Meridian's of the Magnetic Compass, must run Parallel to the Poles of East and West, as the Parallels of the Magnetic Compass run Parallel to the Poles of North and South, so that the Common Chart is the only Truth in Navigation, in regard it is in proportion to the Rumb, and the Right-lines of the Magnetic Compass, therefore the Magnetic Compass doth measure the superficies of the Globe by Right-lines, as if the superficies of the Globe was a plain to the Horizon.    
FINIS.   

ADVERTISEMENT.  
THere will be Published the Practical part of the Sea, with some Questions in Astronomy, useful in Navigation, by the same Author.   

APPENDIX.  
THE foregoing Treatise is only upon a single Question in the three principal kinds of Sailing, so called, proving that the work of Navigation cannot be performed by the Arch of a great Circle, or by Mercator's Position; and that the Common Chart that hath been so much rejected, is the only truth in Navigation, in regard it is in proportion to the Rumb and the right lines of the Magnetic Compass; therefore I shall Illustrate this Position of the Common Chart by a double Question in the practical part of the Sea, comparing it with the same Question by Mercator's Position; wherein the defect of Mercator's Position will more plainly appear, than in a single Question.  
Before I proceed to give this Question, I shall show you by what Practical Questions at Sea the work of Navigation is performed by the Common Chart, and by Mercator.  
First, by the Common Chart; if you have daily Observation, than you have the difference of Latitude and the Rumb rectified to all impediments to find the distance and the difference of Longitude in a rightlined Triangle.  
By Mercator; if you have daily Observation, than you have the difference of Latitude in Meridional parts, and the Rumb rectified to all impediments to find the difference of Longitude in the Equinoctial.  
The Second Question in the Practical part of the Sea by the Common Chart; if you have no Observation for many days, and that you have run several traverses, than you have your daily distance run by your Log, and the Rumb rectified to all impediments upon each traverse, to find the difference of Latitude, and the difference of Longitude; this traverse is called a dead reckoning; but as soon as you have Observation, having as before allowed for all impediments of the Rumb in your daily traverse, and that you find the Ship before your Observation, or astern of your Observation, than you must add or subtract by a due proportion to each distance run by your Log, so as to fall into your Latitude by Observation: The several traverses being thus resolved, you must find the difference of Longitude, and the distance, and the Rumb upon which you have made your way from your last Observation in a rightlined Triangle, which is in proportion to the Rumb and the right lines of the Magnetic Compass.  
The Second Question in the Practical part of the Sea by Mercator. If you have no Observation for many days, and that you have run several traverses, than you have your daily distance run by your Log, and the Rumb rectified to all impediments upon each traverse, to find the difference of Longitude in the Equinoctial; this traverse is called a dead reckoning. Here you may observe Mercator's brief Rules are not in proportion to the Rumb and Distance in a rightlined Triangle; therefore Mercator must take the Latitude of places as they are found in each traverse in a rightlined Triangle, than Mercator has the Meridional parts contained in the difference of Latitude, and the Rumb in each rightlined Triangle, to find the difference of Longitude in the Equinoctial, which must be added or subtracted, as you increase or lessen the Degrees of Longitude in the Equinoctial, until you come to shape a direct Course to your Port, according to Mercator's Position, by a rightlined Triangle.  
In the last place, having made several traverses in the Sea, if you desire to know a direct Course from your Ship to your Port in the Common Chart, you have the difference of Latitude between the Ship in the Sea and your Port; and the difference of Longitude between your Ship in the Sea and the Port you are to sail to, to find the Rumb and Distance upon a direct Course to your Port by a rightlined Triangle.  
By Mercator, having made several traverses in the Sea, if you desire to know a direct Course to your Port, you have the difference of Latitude in Meridional parts between your Ship in the Sea and your Port, and the difference of Longitude in the Equinoctial between your Ship in the Sea, and the Port you are to sail to, to find the Rumb and Distance upon a direct Course to your Port by a rightlined Triangle.  
Having given you this general account of the Practical part of the Sea by the Plain Chart, and by Mercator's Projection, I shall proceed to the Question.  
Two Ships Sail from the Lizard in the Latitude of 50 d. the first Ship Sails in an Angle of 71 d. 10 m. Southwestwards 3268 miles; the second Ship Sails West Northwest 2400 miles. I demand the Rumb or Course, and the distance that the second Ship must sail to arrive at the Port of the first Ship.  

By the Common Chart.  
The Question of the first Ship is resolved by demonstration in the following Figure: The Lizard Latitude 50 d. I sail in an Angle of 71 d. 10 m. Southwestwards, A L B, 3268 miles L B. I demand the difference of Latitude, and the difference of Longitude? I answer, I find my difference of Latitude to be 1055 miles in the Latitude of 32 d. 25 m. L A. and my difference of Longitude to be 3093 miles, A B, which is the Meridian and Parallel of Bermudus in the Common Chart: I have resolved the Arithmetical part of this Question in the first part of this small Treatise, which is in like proportion with the Geometrical part.   geometrical diagram 
The Question of the second Ship in her first traverse resolved Arithmetically, to find the difference of Longitude, say, As the Radius 90 d. L D C, 1000000, is to to the Logarithm of the distance L C, 2400 miles, 338021; so is the sine of the Rumb 67 d. 30 m. D L C 996561, to the Logarithm of the difference of Longitude 334582, or 2217 miles demanded. Add the second and third Numbers in all Questions, as I have given them, and you shall have the Sine, or the Tangent, or the Logarithm of the Numbers required.  
The Question of the second Ship in her first traverse resolved Arithmetically, to find the difference of Latitude, say, As the Radius 90 d. L D C 1000000, is to the Logarithm of the distance 2400 miles L C, 338021, so is the co-sine of the Rumb 67 d. 30 m. D L C 958283, to the Logarithm of the difference of Latitude 296304, or 918 miles L D. which agrees with the former demonstration.  
The Question of the second Ship in her second traverse is resolved by demonstration in the following Figure. In the first place, you must subtract the difference of Longitude the second Ship has made upon a West Northwest Course D C 2217, from the difference of Longitude between the Lizard and Bermudus A B 3093 miles, and you have E B 876 miles, that your Ship is to the Eastward of the Meridian of Bermudus; then add the difference of Latitude L D 918, to the difference of Latitude between the Lizard and Bermudus L A 1055 miles, and you have 1973 miles D A, equal to C E, the difference of Latitude between the Ship and the Latitude of Bermudus. Then the Question stands thus in proportion, you have the difference of Longitude between the Ship and Bermudus E B 876 miles, and the difference of Latitude between the Ship and the Latitude of Bermudus 1973 miles C E. I demand the Rumb and Distance I have to sail to Bermudus? I answer, I find the Rumb to be in an Angle of 23 d. 56 m. E C B Southwestwards, and I find the distance to be 2159 miles CB; so that the second Ship is arrived in the Meridian and Parallel of Bermudus by Demonstration, demanded.  
The Question of the second Ship in her second traverse resolved Arithmetically; to find the Rumb, say, As the Logarithm of the difference of Latitude 1937 miles C E 329512, is to the Radius 90 d. C B E 1000000; so is the Logarithm of the difference of Longitude 876 miles E B 294250, to the tangent of the Rumb 964738 E C B 23 d. 56 m. So that your Course from your Ship to Bermudus is in an Angle of 23 d. 56 m. Southwestward, demanded.  
The Question of the second Ship in her second traverse resolved Arithmetically, to find the distance, say, As the Sine of the Rumb 23 d. 56 m. E C B. 960817; is to the Logarithm of the difference of Longitude 876 miles E B 294250; so is the Radius 90 d. C E B 1000000 to the Logarithm of the distance 333433 C B 2159 miles, the distance between the Ship in the Sea and Bermudus, demanded.  
Then Observe, I have resolved the Question of the first Ship sailing from the Lizard to Bermudus Geometrically and Arithmetically, all which positions are in proportion in a rightlined Triangle to the Rumb and the right lines of the Magnetic Compass and the Common Chart. Likewise I have resolved the Question of the second Ship in a traverse from the Lizard to Bermudus Geometrically and Arithmetically; all which Positions are in proportion in rightlined Triangles to the Rumb, and the right lines of the Magnetic Compass and the Common Chart. So that the second Ship is arrived in the Meridian and Parallel of Bermudus. Therefore the second Ship in her traverse has not altered the difference of Longitude in the Equinoctial of the first Ship, as it was transferred from the Common Chart to the Globe, or Merca●or.   

The same Question resolved by Mercator.  
I have given you a general account of the practical part of the Sea by the Common Chart, and by Mercator; so that all traverses must be resolved in proportion to the Rumb and the right lines of the Magnetic Compass, I shall proceed to the Question.  
Two Ships sail from the Lizard in the Latitude of 50 d. the first Ship sails in an Angle of 71 d. 10 m. Southwestwards 3268 miles; the second Ship sails West North west 2400 mil●s. I demand the Rumb and the Distance that the second Ship must sail to arrive at the Port of the first Ship.  
Before I can resolve this Question by M●rcator's brief Rules, to find the difference of Longitude in the Equinoctial, I must see what Latitude is in proportion to the Rumb and Distance of the first Ship as it is found in a rightlined Triangle, in regard Mercator's brief Rules are not in proportion to the Rumb and the Distance of places, than the Question stands thus.  
The Question of the first Ship to be resolved by Mercator's brief Rules, the Lizard, Latitude 50 d. Bermudus, Latitude 32 d. 25 m. the Rumb is an Angle of 71 d. 10 m. Southwestwards: I demand the difference of Longitude in the Equinoctial?  
To find the difference of Longitude in the Equinoctial, say, As the Radius 90 d. 1000000, is to the difference of Latitude in Meridional parts 1417: 315136, so is the tangent of the Rumb 71 d. 10 m. 1046714, to the Logarithm of the difference of Longitude in the Equinoctial 4155, 361850, demanded.  
The Question of the second Ship in her first traverse resolved by Mercator's brief Rules; before I can resolve this Question, I must see what Latitude is in proportion to the Rumb and Distance of the second Ship in her first traverse in a rightlined Triangle, in regard Mercator's brief Rules are not in proportion to the Rumb and Distance; then the Question stands thus:  
The Lizard Latitude 50 d. the second Ship in her first traverse, Latitude 65 d. 18 m. the Rumb West Northwest, or in an Angle of 67 d. 30 m. I demand the difference of Longitude in the Equinoctial?  
To find the difference of Longitude in the Equinoctial, say, As the Radius 90 d. 1000000, is to the Logarithm of the difference of Latitude in Meridional parts, 1747, 324229; so is the tangent of the Rumb 67 d. 30 m. 1038277, to the Logarithm of the difference of Longitude in the Equinoctial 362502, or 4217 miles, demanded.  
The Question of the second Ship in her second traverse resolved by Mercator's brief Rules: In the first place, you must observe, by the first traverse of the second ship the difference of Longitude in the Equinoctial is 4217 miles; therefore you must subtract the difference of Longitude in the Equinoctial (of the first ship) 4155 miles, and you have 62 miles, that the second ship in her second traverse is to the Westward of the Meridian of Bermudus in the Equinoctial, than the Question stands thus:  
The second ship in her second traverse is in the Latitude of 65 d. 18 m. she is to sail into the Latitude of 32 d. 25 m. the whole difference of Longitude in the Equinoctial between the second ship in her second traverse and Bermudus is 62 mile's East. I demand the Rumb or Course, and the distance the second ship in her second traverse must sail, to arrive at the Port of the first ship?  
To find the Rumb, say, As the Logarithm of the difference of Latitude in Meridional parts 3164, 350023, is to the Radius 1000000, so is the Logarithm of the whole difference of Longitude in the Equinoctial 62 miles, 179239, to the tangent of the Rumb 829216, or 01 d. 07 m. So that I am to sail in an Angle of 01 d. 07 m. South-Eastward from the place of my ship in her last traverse, to arrive at Bermudus: Then observe, Mercator has altered the Rumb from that of the Common Chart 25 d. 03 m. To prove it, add 01 d. 07 m. that the second ship in her second traverse sails by Mercator South-Eastward into the Latitude of Bermudus, to 23 d. 56 m. that the second ship in her second traverse has sailed Southwestward upon the Common Chart to Bermudus, and you have 25 d. 03 m. that the Rumb differs between the Common Chart and Mercator; these are gross errors in Navigation, therefore I shall show you wherein Mercator commits them.  
In the next place, I shall resolve the remaining part of Mercator's Navigation by rightlined Triangles in proportion to the Rumb found by Mercator's brief Rules; for all traverses must be resolved by the Rumb and the right lines of the Magnetic Compass, in regard Mercator's brief Rules are not in proportion to the Rumb and the distance sailed; otherwise you cannot tell upon what Rumb or Coarse you have sailed from your first Port upon a traverse, or upon what Rumb or Coarse you are to return to your Port, or to any other part of the World: Then observe, if the Rumb and the difference of Latitude found by Mercator in a traverse by the second Ship, does not make out the same Rumb and distance that the first Ship made from the Lizard to Bermudus, than it must be granted that Mercator's brief Rules have led you into gross Errors; this I shall prove from the Practical part of the Sea.  
I shall not be tedious, therefore I shall give the Questions resolved.  
By Mercator, the first Ship sails from the Lizard in the Latitude of 50 d. at L, in an Angle of 71 d. 10 m. A L B, Southwestwards 3268 miles L B; I demand the difference of Latitude, and the difference of Longitude in a rightlined Triangle? I answer, I find my difference of Latitude 1055 miles, and to be in the Latitude of 32 d. 25 m. L A; and I find my difference of Longitude 3093 miles A B, demanded.  
By Mercator, the second Ship in her first traverse sails from the Lizard in the Latitude of 50 d. at L, in an Angle of 67 d. 30 m. D L C, Northwestward 2400 miles L C. I demand the difference of Latitude, and the difference of Longitude in a rightlined Triangle? I answer, I find my difference of Latitude 918 miles, and to be in the Latitude of 65 d. 18 m. L D; and I find my difference of Longitude 2217 miles D C, demanded.  
By Mercator, the second Ship in her second traverse sails from the Latitude of 65 d. 18 m. at C, in an Angle of 01 d. 07 m. South-eastwards Q C E, into the Latitude of 32 d. 25 m. Q C. I demand the distance sailed, and the difference of Longitude in a rightlined Triangle? I answer, I find the distance C Q 1974 miles; and I find my difference of Longitude Q E, 39 mile's East, demanded.  
Then subtract the difference of Longitude of the second Ship in her second traverse Q E 39 miles, from the difference of Longitude of the second ship in her first traverse D C 2217 miles, in regard the second Ship in her second traverse is gone to the Eastward, and you have 2179 miles Q A, the difference of Longitude the second Ship has made from the Lizard into the Latitude of Bermudus.  
Then subtract the difference of Longitude the second Ship has made from the Lizard into the Latitude of Bermudus Q A 2179 miles, from the difference of Longitude the first Ship has made to Bermudus in the Latitude of 32 d. 25 m. A B 3093 miles, and you have 914 Q B, or 15 d. 14 m. that Mercator is to the Eastward of the Meridian and Parallel of Bermudus, when Mercator expects to be in the Meridian and Parallel of Bermudus.  
In the last place, you have the difference of Latitude between the Lizard and Bermudus L A 1055 miles, and the difference of Longitude 2179 miles A Q; I demand the Rumb and the distance the second Ship has sailed from the Lizard into the Latitude of Bermudus upon a direct Course? I answer, I have sailed in an Angle of 64 d. 10 m. A L Q. Southwestward; and I find my distance sailed 2421 miles L Q demanded; at which place Mercator expects to be in the Meridian and parallel of Bermudus; then subtract the Rumb of the second ship from the Lizard upon a direct Course A L Q 64 d. 10 m. when Mercator expects to be in the Meridian and parallel of Bermudus, from the Rumb of the first ship from the Lizard to Bermudus A L B, 71 d. 10 m. and you have 07 d. 00 m. that the Angle of the position does differ between the first and second ship; then subtract the distance of the second ship L Q 2421 miles, from the distance of the first ship L B 3268 miles, and you have 847 miles, that the second ship does differ from the first ship; likewise subtract the difference of Longitude at the second ship 2179 miles A Q, from the difference of Longitude of the first ship 3093 miles A B, and you have 914 mile●, or 15 d. 14 m. that the second ship is to the Eastward of Bermudus in the Latitude of 32 d. 25 m. These are the gross Errors that attend Mercator's Position; for all traverses by Mercator in the practical part of the Sea, must be resolved by rightlined Triangles, otherwise you cannot give any account of the Rumb and the distance, and the difference of Longitude your second ship has made from her first Port, or that you can give any account upon what Rumb you are to return to your Port, or to direct your course to any other Port, if you do not resolve all traverses by the Rumb and the right lines of the Magnetic Compass in rightlined Triangles; therefore it must be granted, that I am to find the difference of Longitude in the Equinoctial that the second ship has made in her traverse from the Lizard to be in proportion to the Rumb upon a direct Course by Mercator.  
In the last place, I must find the difference of Longitude in the Equinoctial that the second ship has made upon a direct Course from the Lizard into the Latitude of 32 d. 25 m. In the last Question, the Rumb is found to be in an Angle of 64 d. 10 m. Southwestward that the second ship has made in her traverse from the Lizard upon a direct Course into the Latitude of 32 d. 25 m. then you have the Latitude of the Lizard 50 d. and the Latitude of 32 d. 25 m. and the Rumb 64 d. 10 m. Southwestward; I demand the difference of Meridian's in the Equinoctial that the second ship has made from the Lizard into the Latitude of 32 d. 25 m? I answer, the difference of Meridian's in the Equinoctial is found to be 2926 miles, when Mercator expects to be in the Meridian of the Equinoctial equal to the first ship; then subtract the difference of Meridian's in the Equinoctial of the second ship 2926 miles, from the difference of meridians of the first ship in the Equinoctial 4155 miles, and you have 1229 miles, or 20 d. 29 m. that the second ship is East of the first ship in the Equinoctial, when Mercator expects to be in the meridian of the Equinoctial with the first ship; so that Mercator hath altered the Rumb, and the distance, and the difference of meridians of the second ship in a rightlined Triangle, and in the Equinoctial, from the Rumb, and the distance, and the difference of meridians of the first ship in a rightlined Triangle, and in the Equinoctial: So that to know the difference of Longitude in the Equinoctial is no way useful in Navigation, either upon the Globe, or upon Mercator's Plani-Sphere.  
I have proved by the Common Chart, that the second ship making a traverse, is arrived in the meridian of the first ship, and has not altered the meridian of the first ship in the Common Chart, or in the Equinoctial, as it is transferred to the Globe: I have satisfied the World that the Common Chart in the Practical part of the Sea, is the only true Chart in Navigation; if otherwise, I desire to be better informed: Therefore it must be granted, that the Magnetic Compass as it is Horizontal, runs parallel to the Poles of East and West, as it runs parallel to the Poles of North and South, and is as much in proportion by Observation and Demonstration to those Poles, as the Poles Ecliptic are in proportion to any Question in Astronomy.   
FINIS.   

Addenda.  
The following Treatise will prove Mercator's practical Rules in Navigation, to be notoriously false.  
Likewise Longitude not to be found by Observation, from the Celestial Bodies, without it be when Eclipses appear in the Equinoctial.  
Last of all, the Earth is proved to be the Centre of the Starry Heaven.  
All which is proved by Observation and Demonstration.  
THE last Appendix to the foregoing Treatise was a Question given me, the Answer I gave to it, I thought sufficient to have given satisfaction to any reasonable Men, wherein I prove the Work of Navigation cannot be performed by Mercator's brief Rules; but finding Mr. Flamsteed and other of our Shore Navigators to discourse in Company in derision of what I have offered publicly to the World; I must tell him, it is not becoming a Man of his Quality, but I hope this following Tratise may induce Mr. Flamsteed and the rest, to let the World see some of their skill in Navigation, and to do me so much right as to allow or disprove what I have offered that Men may not halt between Opinions; however I shall make some small alteration in the Question in the foregoing Appendix, and give the Question more Practical in Navigation, wherein all Men that are Studious in Navigation may give an Answer to all Objections any man shall propose, in opposition to what I have offered: for the Rules of Navigation are Demonstrative, and the Arithmetical Rules must be in proportion, otherwise such Rules must be rejected; therefore I shall resolve the following Question by the Plain Chart, and by Mercator, wherein I shall show you how far Mercator's brief Rules are in proportion in each Traverse to the Rumb, and the Right-lines of the Magnetic Compass: then I shall show you in the following Question, wherein Mercator's Practical Rules in Navigation do disagree with the Rumb and the Right-lines of the Magnetic Compass, by which Mercator pretends to Navigate; which will prove Mercator's Position in the Practical Parts of Navigation to be notorious false; as likewise are all those that are of his persuasion, without they will satisfy the World to the contrary. Lastly, as the Meridian and Parallel of all the known part of the World cannot be found in the practical part of Navigation, by the Arch of great Circle, or by Mercator's brief Rules, so likewise the Meridian and Parallel of places cannot be transferred from the Common Chart to the Globe, but only from one Meridian and Parallel of the Common Chart by such Demonstrations; you shall have the same Latitude and Distance of places to all the known Parts of the World, upon the Globe as it is transferred from only one Meridian and Parallel of the Common Chart to the Globe: otherwise the Meridian and Parallel of places cannot be known upon the Globe as they are found by the Practical part of Navigation. All which I shall prove at large.  

The Question given is as following.  
EXAMPLE. Two Ships sail from the Lizard in the Latitude of 50 d. the first Ship sails in an Angle of 71 d. 10 m. Southwestward 3268 miles; the second Ship sails in an Angle of 67 d. 30 m. Northwestward 2400 miles, than the second Ship sails in an Angle of 23 d. 56 m. South-west-ward 2159 miles, I demand the difference of Longitude in each right lined Triangle, and the Latitude the first and second Ship was in upon each Rumb and distance sailed; and the difference of Longitude in the Equinoctial contained in each right lined Triangle.  
This Question must be resolved by right lined Triangles, which are in proportion with the Common Chart, otherwise the Latitude upon each distance run cannot be known, in regard Mercator's brief Rules are not in proportion to the Rumb, and the distance of places, therefore I shall give you the Question resolved by right lined Triangles in the Practical part of the Sea, according to the Common Chart, and proceed to Mercator's brief Rules in the Practical part of the Sea.  
I. The first Ship sails from the Lizard in the Latitude of 50 d. L in an Angle of 71 d. 10 m. South-west-ward, A L B, 3268 miles, L B. I find my difference of Latitude 1055 miles, South, and to be in the Latitude of 32 d. 25 m. L A, likewise I find my difference of Longitude to be 3093 miles, West A B, which is the Meridian and Parallel of Bermuda upon the Common Chart, to which the Arithmetical Rules are in proportion.  
II. The second Ship in her first Traverse sails from the Lizard in the Latitude of 50 d. L, in an Angle of 67 d. 30 m. D L C, 2400 miles L. C. I find my difference of Latitude to be 918 miles, North L D, and to be in the Latitude of 65 d. 18 m. at C, likewise I find my difference of Longitude to be 2217 mile's West D C, demanded, to which the Arithmetical Rules are in proportion.  
III. The second Ship in her second Traverse sails from C, in the Latitude of 65 d. 18 m. in an Angle of 23 d. 56 m. Southwestwards E C B, 2159 miles C B, I find my difference of Latitude to be 1973 mile's South, and to be in the Latitude of 32 d. 25 m. at B, likewise I find my difference of Longitude to be 876 mile's West, E B, demanded, to which the Arithmetical rules are in proportion.  
To sum up the Practical part of Navigation, according to the Common Chart, and to give you the Meridian and Parallel of the second Ship in her first and second Traverse; first you must add the difference of Longitude the second Ship made in her second Traverse, 876 miles, E B, to the difference of Longitude the second Ship made in her first Traverse 2217 miles, D C, and you have 3093 miles, A B, the difference of Longitude of the second Ship in her first and second Traverse from the Lizard, where she meets the first Ship in the Latitude of 32 d. 25 m. which is the Meridian and Parallel of the second Ship in her first and second Traverse. See the Demonstration which is according to the Common Chart.  
Having found the Latitudes and the difference of Longitudes in each right lined Triangle that the first and second Ship was in, upon each Traverse according to the plain Chart, I shall proceed to show you how far Mercator's brief Rules are in proportion to the Rumb and the right lines of each right lined Triangle, in the foregoing Question, Then I shall show you wherein Mercator's Practical Rules in Navigation disagree.  

By MERCATOR.  
In the first place, I shall show you that the Rumb and the difference of Latitude in Meridional parts is in proportion to the difference of Longitude in the Equinoctial in each right lined Triangle: In the next place, I shall show you that the Meridional parts contained in the difference of Latitude and the difference of Longitude in the Equinoctial, and the difference of Latitude is in proportion to the difference of Longitude in each right lined Triangle, of the foregoing Question: In the third place, I shall show you that Mercator's Practical Rules in Navigation do disagree with the foregoing Rules of Mercator, in each Traverse; In regard the difference of Latitude in Meridional parts from the Lizard, to the second Ship in her second Traverse, and the difference of Longitude in the Equinoctial found according to Mercator's Practical Rules in Navigation, and the difference of Latitude between the second Ship in her second Traverse from the Lizard, doth not give the same difference of Longitude, in a right lined Triangle from the Lizard to the second Ship in her second Traverse, in proportion to the difference of Longitude, found in each single Traverse in a right lined Triangle. This being proved, Mercator's Practical Rules in Navigation must be notorious false.    

The Questions to be Resolved by Mercator's brief Rules, are as followeth.  
EXAMPLE. 1. To find the difference of Longitude in the Equinoctial, in proportion to the foregoing Question, The first Ship sails from the Lizard, in the Latitude of 50 d. L, in an Angle of 71 d. 10 m. A L B, South-west-ward into the Latitude of 32 d. 25 m. at B. I demand the difference of Longitude in the Equinoctial.  
To Resolve this Question, say as the Radius 90 d. 1000000 is to the Logarithm of the difference of Latitude in Meridional parts 1417, 315136, so is the Tangent of the Rumb, 71 d. 10 m. 1046714, to the Logarithm of the difference of Longitude in Equinoctial 361850 or 4155 miles, or 69 d. 15 m. demanded.  
2. To find the difference of Longitude in the Equinoctial, in proportion to the foregoing Question, the second Ship sails from the Lizard in the Latitude of 50 d. L in an Angle of 67 d. 30 m. D L C North-west-ward into the Latitude of 65 d. 18 m. at C. I demand the difference of Longi. in the Equinoctial.  
To Resolve this Question, say as the Radius 90 d. 1000000 is to the Logarithm of the difference of Latitude in Meridional parts, 1748, 324254, So is the Tangent of the Rumb, 67 d. 30 m. 1038277, to the Logarithm of the difference of Longitude in the Equinoctial 362531, or 4220 miles or 70 d. 20 m. demanded.  
3. To find the difference of Longitude in the Equinoctial, in proportion to the foregoing Question, the second Ship in her second Traverse sails from C, in the Latitude of 65 d. 18 m. in an Angle of 23 d. 56 m. E C B, Southwestwards, into the Latitude of 32 d. 15 m. I demand the difference of Longitude in the Equinoctial.  
To Resolve this Question, say as the Radius 90 d. 1000000 is to the Logarithm of the difference of Latitude in Meridional parts 3165, 350037. So is the Tangent of the Rumb, 23 d. 56 m. 964722, to the Logarithm of the difference of Longitude in the Equinoctial, 314759, or 1405 miles, or 22 d. 25 m. demanded.  
To sum up the Practical part of the second Ship, in her first and second Traverse, according to Mercator, you must add the difference of Longitude in the Equinoctial; the second Ship made in her second Traverse 23 d. 25 m. to the difference of Longitude in the Equinoctial; the second Ship made in her first Traverse 70 d. 20 m. and you have 93 d. 45 m. the difference of Longitude in the Equinoctial of the second Ship from the Lizard, in the Latitude of 32 d. 25 m. Thus I have given you a full account of Mercator's Practicai Rules in Navigation.  
Having given you a brief account of the Practical part of Navigation, according to the Common Chart, and according to the brief Rules of Mercator, I shall proceed to show you how the Meridional parts, contained in the difference of Latitude, and the difference of Longitude in the Equinoctial, and the difference of Latitude in each single Traverse, according to Mercator, is only in proportion to the difference of Longitude, contained in each single Traverse in all right lined Triangles; Then I shall show you that the like Rules of Mercator are out of all proportion with the difference of Longitude, in a right lined Triangle from the Lizard according to his Practical Rules in Navigation, the Questions to be Resolved by Mercator's brief Rules are as follow in Example.  
1. The first Ship sails from the Lizard, Latitude 50 d. into the Latitude of 32 d. 25 m. I find my difference of Latitude in Meridional parts 1417 the difference of Longitude in the Equinoctial is 4155 miles; the difference of Latitude is 1055 miles. I demand the difference of Longitude, in a right lined Triangle that the first Ship made according to the Common Chart from the Lizard, A B.  
To find the difference of Longitude of the first Ship in a right lined Triangle, according to Mercator's brief Rules, say as the Logarithm of the Meridional parts contained in the difference of Latitude 1417, 315136, is to the Logarithm of the difference of Longitude in the Equinoctial 4155 miles, 361857. So is the Logarithm of the difference of Latitude 1055 miles, 302325, to the Logarithm of the difference of Longitude, in a right lined Triangle, 349046, or 3093 miles, which is equal to the difference of Longitude, the first Ship made in a right lined Triangle. See the foregoing Demonstration, according to the Common Chart, A B.  
2. The second Ship sails from the Lizard, Latitude 50 d. into the Latitude of 65 d. 18 m. I find my difference of Latitude in Meridional parts 1748. The difference of Longitude in the Equinoctial, is 4220 miles, the difference of Latitude is 918 miles, I demand the difference of Longitude, in a right lined Triangle, that the second Ship made according to the Common Chart, from the Lizard, D C.  
To find the difference of Longitude of the second Ship, in her first Traverse, in a right lined Triangle, according to Mercator's brief Rules, say as the Logarithm of the Meridional parts contained in the difference of Latitude, 1748, 324254; is to the Logarithm of the difference of Longitude in the Equinoctial; 4220 miles, 362531, so is the Logarithm of the difference of Latitude, 918 miles, 296284 to the Logarithm of the difference of Longitude, in a right lined Triangle 334561 or 2217 miles, which is equal to the difference of Longitude, the second made in a right lined Triangle in her first Traverse. See the foregoing Demonstration, according to the Common Chart, D C.  
3. The second Ship sails from C. in the Latitude of 65 d. 18 m. into the Latitude of 32 d. 25 m. I find my difference of Latitude in the Meridional parts, 3165. The difference of Longitude, in the Equinoctial is 1405 miles, the difference of Latitude is 1973 miles; I demand the difference of Longitude, in a right lined Triangle; that the second Ship made in her second Traverse, according to the Common Chart from E to B.  
To find the difference of Longitude, the second Ship made in her second Traverse, in a right lined Triangle; say as the Logarithm of the Meridional parts, contained in the difference of Latitude 3165, 350037, is to the Logarithm of the difference of Longitude in the Equinoctial 1405 miles, 314767. So is the Logarithm of the difference of Latitude 1973 miles 329512 to the Logarithm of the difference of Longitude, in a right lined Triangle, 294242 or 876 miles which is equal to the difference of Longitude, the second Ship made in a right lined Triangle, in her second Traverse: See the foregoing Demonstration according to the Common Chart; then add the difference of Longitude the second Ship made in a right lined Triangle in her first Traverse, 2217 miles found according to Mercator's brief Rules to the difference of Longitude, the second Ship made in a right lined Triangle in her second Traverse 876 miles, found according to Mercator's brief Rules; and you have 3093 miles equal to the difference of Longitude, the first Ship made in a right lined Triangle 3093: So that Mercator's brief Rules are in proportion to the Rumb, and the difference of Latitude and the difference of Longitude in all single Questions, in a right lined Triangle. See the foregoing Demonstration which proves the Practical part of Navigation, performed by the second Ship, according to the Common Chart, to be in the Meridian and Parallel of the first Ship.  
In the last place, I shall prove that the like Rules in the practical part of Mercator's Navigation, will not give like difference of Longitude contained in the Traverse of the second Ship, in a right lined Triangle from the Lizard 3093 miles, which will prove Mercator's practical Position in Navigation to be notoriously false, for in the plainest Rules of Art where propositions shall disagree, we are to reject them. The Question is as following.  
EXAMPLE. According to Mercator's Practical Rules in Navigation, the second Ship in her second Traverse, is in the Latitude of 32 d. 25 m. the Lizard in the Latitude of 50 d. The difference of Latitude in Meridional parts, 1417; the difference of Longitude in the Equinoctial, according to Mercator's practical Rules in Navigation is 93 d. 45 m. or 5625 miles. The difference of Latitude is 1055 miles. I demand the difference of Longitude in a right lined Triangle, between the Lizard and the Ship in her second Traverse.  
To find the difference of Longitude, in a right lined Triangle, say as the Logarithm of the Meridional parts, contained in the difference of Latitude 1417, 315136 is to the Logarithm of the difference of Longitude in the Equinoctial, 5625 miles 375012 So is the Logarithm of the difference of Latitude 1055 miles, 302325 to the Logarithm of the difference of Longitude in a right lined Triangle, 362201 or 4188 miles between the Lizard and the Ship, in the Latitude of 32 d. 25 m. demanded.  
Then subtract the difference of Londitude found by the like Rules of Mercator, in each Traverse in a right lined Triangle 3093 miles from the Lizard to the Ship, in the Latitude of 32 d. 25 m. from the difference of Longitude, found by the like Rules of Mercator, in a right lined Triangle, according to Mercator's practical Rules in Navigation 4188 miles, and you have 1095 miles or 18 d. 15 m. that Mercator's practical Rules in Navigation doth differ from Mercator's practical Rules in each Traverse, in a right lined Triangle in the Latitude of 32 d. 25 m. So that Mercator doth alter the Rumb and the distance and the difference of Longitude in the practical part of Navigation, from the Lizard 1095 miles or 18 d. 15 m. more West in the Latitude of 32 d. 25 m. Therefore Mercator's Position in the practical part of Navigation is notoriously false; for in the plainest Rules of Art where propositions shall disagree we are to reject them.  
To sum up the foregoing Treatise, In the first place, I have proved the plain Chart, that has been rejected by all Writers, to be the only true Chart in Navigation, by Observation and Demonstration to which the Arithmetical Rules are in proportion. Secondly, I have proved Mercator's Arithmetical Rules, to be out of all proportion in the practical part of Navigation; and Geometry, In regard the whole Dependants of Mercator's Navigation, is upon his brief Rules in Arithmetic; which Rules disagreeing, disappoints the Design of Geometry: Therefore the work of Navigation cannot be performed by Mercator: Thirdly, I have proved the Art of Navigation cannot be performed by the Arch of a great Circle; In regard the Arch of a great Circle doth measure the nearest distance of places in North and South Latitude without having any regard to the Parallels of Latitude; so that the Angles of Position in the Arch of a great Circle doth differ from the Angles of Position, by the Magnetic Compass. See the foregoing Demonstrations: So that you cannot give the Question as it stands in the practical part of Navigation to find the Meridian and Parallel the Ship is in by the Arch of a great Circle; neither can the Meridian and Parallel of places be known upon the Globe, but as it is transferred to the Globe from the Common Chart, of which I shall give you a farther Account in the following Discourse. The chief intent of this Treatise, is principally for Information, that all Men may give an Account of the Certainty of Navigation: Therefore I desire Mr. Flamsteed, and the rest of our Shore Navigators, to let the World see something of their skill in Navigation, and that they will not hid their Talon in a Napkin, but allow or disprove what I have made public to the World, which I expect from them.   

To transfer the Meridian and Parallel of places from the Common Chart, to the Globe.  
As the Meridian and Parallel of places, cannot be found by Mercator's practical Rules in Navigation, or by the Arch of a great Circle; so likewise you cannot transfer the Meridian and Parallel of places from the Common Chart, to the Globe; but it must be transferred only from one Meridian and Parallel of the Common Chart.  
Then by the following Demonstration, by which I transfer the Meridian and Parallel of places from the Common Chart to the Globe: You shall have the same Latitude and distance of places upon the Globe, as is upon the Common Chart, from that Meridian and Parallel from whence it is transferred, otherwise the Meridian and Parallel of places cannot be known upon the Globe, as it is found by the practical Part of Navigation.   

To transfer the Meridian and Parallel of Places, from the Common Chart to the Globe.  
EXAMPLE. 1. In the Common Chart, the first Ship sails from the Lizard, L in the Latitude of 50 d. into the Latitude of 32 d. 25 m. B, her distance sailed is 3268 miles or 54 d. 28 m. L B, then take with your Compass 54 d. 28 m. L B, from the Equinoctial part of your Globe; placing one Foot of your Compass in the Latitude of 50 d. of your Globe for the Meridian and Parallel of the Lizard, then turn the Globe East, in regard your Ship is gone to the Westward, until the other point doth cut under the Brass Meridian of the Globe, in the Latitude of 32 d. 25 m. there make a Mark which is the Meridian and Parallel of the first Ship upon the Globe. To prove the Truth of this Demonstration, observe the difference of Longitude between the Meridian of the Lizard and the Meridian of your Mark, in the Equinoctial part of your Globe is 69 d. 15 m. as it was found by the brief Rules of Mercator. So that you have three certain distances upon the Globe, as is in a right lined Triangle; that is the difference of Latitude and the distance of places, and the difference of Longitude in the middle Parallel. I shall proceed to the second Ship.  
2. To transfer the Meridian and Parallel of the second Ship, in her first Traverse, from the Meridian and Parallel of the Lizard, according to the Common Chart, to the Globe. Example in the Common Chart, the second Ship sailed from the Lizard L in the Latitude of 50 d. into the Latitude of 65 d. 18 m. C her distance sails is 2400 miles or 40 d. L C, then take with your Compass 40 d. from the Equinoctial part of your Globe, placing one Foot in the Latitude of 50 d. as you did before for the Meridian and Parallel of the Lizard: Then turn the Globe Eastward, in regard your Ship is gone to the Westward, until the other point doth cut under the Brass Meridian, in the Latitude of 65 d. 18 m. there make a Mark which is the Meridian and Parallel of the second Ship in her first Traverse, from the Lizard, upon the Globe. To prove the Truth of this Demonstration, observe the difference of Longitude between the Meridian of the Lizard and the Meridian of your mark in the Equinoctial part of your Globe is 70 d. 20 m. as it was found by the brief Rules of Mercator; so that you have Three certain distances upon the Globe, as is in a right lined Triangle, that is the difference of Latitude and the distance of places, and the difference of Longitude in the middle Parallel. I shall proceed to the second Ship in her second Traverse.  
To transfer the Meridian and Parallel of the second Ship, in her second Traverse, from the Meridian and Parallel of the Lizard, according to the Common Chart, to the Globe, Example the second Ship in her second Traverse, sails from C in the Latitude of 65 d. 18 m. in an Angle of 23 d. 56 m. South-west-ward. 2159 miles to B and meets the first Ship, in the Latitude of 32 d. 25 m. at B. so that the distance of the second Ship from the Lizard, B L is 3268 miles, and is equal to the distance of the first Ship from the Lizard, B L, 3268. Therefore the Latitude of the Lizard 50 d. L and the Latitude of the second Ship in her second Traverse 32 d. 25 m. B and the distance of the second Ship in her second Traverse from the Lizard 3268 miles B L or 54 d. 28 m. is in the like proportion to the Meridian and Parallel of the first Ship in the Common Chart, as it is transferred to the Globe, from the Common Chart, and is in the like proportion to the difference of Longitude in the Equinoctial of the first Ship from the Lizard 69 d. 15 m. This Demonstration must be Infallibly True, in regard Mercator's brief Rules, and the Rules of the Globe are in proportion to the difference of Longitude in all right lined Triangles; as in the foregoing Triangle between A and B.  
In the last place, I shall prove the Meridian and Parallel of all the known parts of the World, must be only transferred from one Meridian and Parallel of the Common Chart to the Globe, otherwise you will alter the Meridian and Parallel of places upon the Globe, as they are transferred from only one Meridian and Parallel of the Common Chart: In regard the Meridional parts contained in the difference of Latitude, and the Rumb, according to Mercator, are only in proportion to the difference of Longitude in the Equinoctial, in a single Traverse, but not in the practical part of Navigation, by which the Meridian and Parallel of all the known parts of the World is found.  
EXAMPLE. The second Ship, in her second Traverse, sails from C in the Latitude of 65 d. 18 m. in an Angle of 23 d. 56 m. 2159 miles or 35 d. 59 m. into the Latitude of 32 d. 25 m. Then take with your Compass 35 d. 59 m. the distance sailed from the Equinoctial part of the Globe, placing one Foot in the Meridian and Parallel of the second Ship, in her first Traverse, as it was transferred from the Common Chart the Globe; then turn the Globe Eastward, in regard your Ship is gone to the Westward until the other Point doth cut under the Brass Meridian of the Globe, in the Latitude of 32 d. 25 m. there make a mark upon the Globe, and you shall have 23 d. 25 m. the difference of Longitude in the Equinoctial, as it was found by Mercator's brief Rules in that single Traverse, which being added to the difference of Longitude in the Equinoctial, that the second Ship made in her first Traverse from the Lizard, 70 d. 20 m. You have 93 d. 45 m. the difference of Longitude in the Equinoctial, West of the Meridian of the Lizard, then subtract 69 d. 15 m. the difference of Longitude in the Equinoctial, the second Ship made in proportion to the difference of Longitude in each single Traverse in a right lined Triangle, from the Lizard, according to the practical Rules of Mercator in each Traverse, and you have 24 d. 30 m. that you must alter the Meridian and Parallel of the second Ship in the Equinoctial part of the Globe, which proves Mercator's practical Rules in Navigation to be notoriously false. In regard Mercator's Rules doth disagree in the practical Part, with the Rumb and the right lines of the Magnetic Compass. Therefore as the Meridian and Parallel of places cannot be found by the practical part of Navigation by the Arch of a great Circle, or by the brief Rules of Mercator; so likewise you cannot transfer the Meridian and Parallel of places from the Common Chart to the Globe, but only from one Meridian and Parallel of the Common Chart, by such Demonstrations you shall have the same Latitudes and distance of places to all the known parts of the World, upon the Globe, as it is transferred from any one Meridian and Parallel of the Common Chart, otherwise the Meridian and Parallel of places cannot be known upon the Globe, as they are found by the practical part of Navigation; therefore the Common Chart, is the only True Chart in Navigation.   

In the last place.  
There have been some that pretend to find the difference of Longitude or the difference of Meridian's, by Observation from the Celestial Bodies; first I shall prove it is not to be found by observation from the Stars: Secondly, I shall prove it is not to be found by observation from the Planets, upon their greatest inclination or declination into the Equinoctial, without it be when the Sun or Moon be Eclipsed in the Equinoctial.  
1. In the first place, I shall prove the difference of Meridian's, or the difference of Longitude cannot be found by observation from the Stars.  
EXAMPLE. Suppose yourself to be in any Meridian and Parallel of the Terrestrial Sphere, there you take the Meridian altitude, of one or more Stars in the Equinoctial, or out of the Equinoctial, the Meridian altitude of those Stars: or the altitude of the guards upon their Angles of Position with the North Star is the same the whole Circulation of the Earth, in that Parallel you made your Observations in, so that you have no difference of Observation to give the difference of Longitude, or the difference of Meridian's. Therefore the difference of Longitude, or the difference of Meridian's cannot be found by Observation, from any of the Stars, without you have difference of Observation, to give the difference of Meridian's.  
2. In the next place, I shall prove the difference of Longitude or the difference of Meridian's cannot be found by Observation, from any of the Planets upon their greatest inclination or declination into the Equinoctial, or day after day.  
EXAMPLE. If you observe in any Meridian and Parallel of the World, the Meridian altitude of any of the Planets, upon their greatest inclination or declination into the Equinoctial, or day after day, you will find the difference in your Observations, in any one Meridian and Parallel you observe in, not to have a proportion to the difference of Longitude or the difference of Meridian's with the circulation of the Earth, therefore the difference of Longitude or the difference of Meridian's cannot be found by Observation from the Meridian altitude of any of the Planets without you have difference of observation. In proportion to the difference of Longitude or the difference of Meridian's the circulation of the Earth.  
3. In the last place, I shall show you how the difference of Longitude or the difference of Meridian's may be found by Observation when Eclipses shall happen to be in the Equinoctial and not otherwise.  
EXAMPLE. If the Sun or Moon be Eclipsed in the Equinoctial, and that several persons in several Meridian's and Parallels of the World, doth observe the altitude of the Sun or Moon, at the beginning or end of the Eclipse, these Observations being compared, the difference of Observation between such places, shall give the difference Longitude or the difference of Meridian's between those places such Observations were made in, In regard the Sun or Moon being Eclipsed in the Equinoctial doth appear at that instant of time those Observations were taken at the beginning or end of the Eclipse as a Pole in the East or West, so that the difference of Longitude or the difference of Meridian's must have the like proportions by Observation from such Eclipses, to prove the Meridian's doth run Parallel to the East and West part of the Terrestrial Sphere, as the Parallels of Latitude are proved to run Parallel by Observation to the the Equinoctial and to the North and South Pole of the Terrestrial Sphere, therefore it is a very strange Opinion in any Men to grant the Magnetic Compass rectified to his true Poles to run East or West in all Parallels of Latitude to the Equinoctial, and to the Poles of North and South, and not to grant the Meridian's of the Magnetic Compass to run Parallel to the East and West part of the Terrestrial Sphere; in the foregoing Treatise, I have proved by Observation, that the Magnetic Compass in all Meridian's and Parallels of the World doth point to the East and West part of the Celestial Sphere, as it points to the North and South part of the Celestial Sphere; therefore as the Magnetic Compass makes the same Position to the East and West part of the Celestial Sphere as it doth to the North and South part of the Celestial Sphere, in all Meridian's and Parallels of the World: therefore it must be granted that the Magnetic Compass runs in Parallel Meridian's, as it runs in Parallels of Latitude upon on the Terrestrial Sphere; to which the Observations of Eclipses in the Equinoctial, and the Observations from the Poles of the North and South doth agree.  
4. In the last place, If the Sun or Moon be not Eclipsed in the Equinoctial, than the altitude of the Sun or Moon being observed in several Meridian's and Parallels at the beginning or end of the Eclipse, the difference of such Observations being compared cannot give the difference of Meridian's or the difference of Longitude in proportion with the Parallels of Latitude; In regard the Sun or Moon so Eclipsed out of the Equinoctial, such Observations cannot cross the Parallels of Latitude at right Angles: For such Observations are as much out of proportion with the Parallels of Latitude, as the Parallel Meridian's of the Poles Ecliptic, would be out of proportion with the Parallel Meridian's, of the North and South Poles, upon the Terrestrial Sphere. So that the difference of Longitude or the difference of Meridian's upon the Terrestrial Sphere, cannot be found by Observation from the Stars, or from the Planets, without it be when the Sun and Moon be Eclipsed in the Equinoctial.    

To Prove the Earth the Centre of the Starry-Heaven, and not to have any Inclination towards the Poles, as Copernicus would have it.  
THE Earth by Observation keeps its Parallels with the Starry-Heaven all the Year, without alteration; for by Observation, that Star that is in the Equinoctial part of Heaven, is always in the Equinoctial part of the Earth; so likewise, take all the Stars in their several Parallels to the Poles from the Equinoctial, and you will find they keep their Parallels with the Earth for ever.  
We need not go to the Equinoctial part of the Earth, to prove the Earth to keep her Parallels with the Equinoctial part of the Starry Heaven: For, observe in this Parallel or Latitude of London 51 d. 30 m. the Amplitude of any Star in the Equinoctial, either upon his Rising or Setting, and you shall find Amplitude to be East or West of you for ever, in this Parallel or any other.  
Likewise observe the Meridian Altitude of any Star in the Equinoctial, in this Parallel or in any other Parallel or Latitudes, and you shall find his Meridian Altitude to be the Elevation of the Equinoctial for ever, in this Parallel, or Latitude, or in any other Parallel or Latitudes.  
And by daily Observations we find the Sun to alter his Amplitudes, and Meridian Altitudes, and Parallels with the Starry Heaven and the Earth. And we find the Starry Heaven to keep its Parallels with the Earth always, in regard the Stars keep their Meridian Altitudes and Amplitudes with the Earth, without alteration.  
But if we should admit the Sun the Centre of the Starry Heaven, and the Earth should have her Declination towards her Poles; then the Sun must be always in the Equinoctial part of the Starry Heaven; and the Sun must have the same Amplitudes, and Meridian Altitudes, with the Stars in the Equinoctial, in all Parallels! And then the Sun, and all the Stars in Heaven, should have a daily Calculation of the Declination of the Earth, as the Earth should alter her Parallels, by Inclining or Declining towards her Poles.  
But it is proved by Observation, that the Sun cannot be the Centre of the Starry-Heaven; in regard the Sun is not always in the Equinoctial part of that Heaven, and the Sun hath not the same Meridian Altitude, and Amplitude, and Parallels, with the Starry-Heaven in the Equinoctial, but twice in the Year, and that is as the Sun Inclines and Declines from one Tropic to another.  
And it is proved by Observation, that the Equinoctial part of the Starry-Heaven is always in the Equinoctial part of the Earth, for the Meridian Altitude of the Stars in Equinoctial is the Elevation of the Equinoctial in all Parallels. Now there is a necessity, that the Declination of the Sun should be Calculated for every day in the Year, in regard of his Declination towards his Poles 23 d. 30 m. which is the cause the Sun altars his Parallels, and Amplitudes, and Meridian Altitudes every day.  
But for Stars in the Starry-Heaven, their Declination or distance from the Equinoctial is the same for ever, and keeps its Parallels with the Earth, by Observation.  

Another Example from the Sun, to prove the Earth the Centre of the Starry-Heaven.  
MOST Mathematicians hold, that when the Sun is depressed below the Horizon 15 d. that Twilight appears upon the Horizon; the proof of which is very suitable in this Parallel or Latitude.  
EXAMPLE. London, Latitude 51 d. 30. North, the Sun's Declination 23 d. 30 m. North, the Sun's Depression 15 d. 00 m. I demand the Time of Twylight? By the Work you will find the Sun being depressed 15 d. at Midnight, the 11th day of June, in the Latitude of 51 d. 30. you will have Twilight appear in the North all Night: At which time we find by Observation in the Equinoctial, and in this Parallel, and in all other Parallels, the Sun to be distant from the North Poles 66 d. 30 m. Now if the Sun was the Centre of the Starry-Heaven, the Sun would be always distant from the Poles 90 d. as the Stars in the Equinoctial are, and as the Sun is the 10th day of March, or the 10th day of September, when he is in the Equinoctial.  

I shall add one Observation more from the Magnetic Compass, as it is Horizontal to prove the Earth the Centre of the Starry-Heaven.  
It is granted by all Astronomers, that Stars of the least Magnitude, are bigger than the Earth, so that the least Star appears as a Centre to us upon the Earth; In like manner I shall prove the Earth by Observation by the Magnetic Compass, appears but as a Centre to the Stars, and that the Earth is the Centre of the Starry-Heaven.  
EXAMPLE. The Magnetic Compass as it is Horizontal in all Meridian's and Parallels of the World rectified to its true Poles, where there is occasion, doth point to the Stars in the Equinoctial, and to the East and West, and to the North and South parts of the Celestial Sphere, crossing the Celestial Sphere at right Angles, without giving any difference of Observation for the Diameter of the Earth, so that the Earth appears as a Centre to the Stars and is the Centre of the Starry Heaven, by the Horizontal Observations of the Magnetic Compass; in regard you have no difference of Observation, for the whole Diameter of the Earth; but if the Earth should incline or decline towards its Poles as Copernicus supposed it doth, 23 d. 30 m. then the Horizontal Observations of the Magnetic Compass cannot point to the Stars in the Equinoctial, or to the East and West part of the Celestial Sphere, for if the Earth, as a Centre should move upon its Axis 23 d. 30 m. to each Tropic, than the Earth must alter the Equinoctial part of the Starry Heaven, as the Sun and the rest of the Planets doth, therefore Copernicus is under a very great mistake, to suppose the Sun the Centre of the Starry Heaven.   

In the next place, I shall prove the foregoing position by Observation and by Demonstration, the outward, Sphere representing the Starry-heaven the lesser Sphere the Earth.  
1. To demonstrate the Earth to be the Centre of the Starry Heaven. If the Earth should move upon its Axis 23 d. 30 m. towards the North or South Poles, than there is a necessity the Earth must alter its Parallels with the Sun and the Stars in the Equinoctial part of Heaven, 23 d. 30 m. but on the contrary it is proved by Observation in all Meridian's and Parallels of the World, that A is always in the Equinoctial part of the Starry-Heaven, and that B is always in the Tropic of Cancer; and that C is always in the Tropic of Capricorn, so that the Earth always keeps its Parallels with the Starry-Heaven without alteration; therefore the Earth is the Centre of the Starry-Heaven.  
2. If the Sun should be the Centre of the Starry-Heaven, according to Copernicus' supposition; then the Sun by Observation should be always in the Equinoctial part of the Starry-Heaven, as it is the tenth day of March, and the tenth day of September, at which time the Sun is by Observation distant from the Poles, 90 m. but on the contrary the Sun is by Observation when he is in the Tropics 66 d. 30 m. distant from the Poles; therefore the Sun doth alter its place in the Ecliptic with the Constellations of the Starry-Heaven, by inclining from the Tropic of Capricorn, to the Tropic of Cancer 47 d. so that the Sun cannot be the Centre of the Starry-Heaven but by supposition.   geometrical diagram 
between pag. 156 and 157.  
What I have offered to the World in this small Treatise, I have proved by Observation and Demonstration; therefore whatever Men may pretend to, if they do not accord with the like Rules, it is but Fancy or Imagination.    
FINIS.