First, I shall resolve the Question by the Common Chart.

Example. I sail from the Meridian and Paral­lel of the Lyzard, in the Latitude of 50 d. 00 m. in an Angle of 71 d. 10 m. Southwestward into the Meridian and Parallel of Bermudas, in the Latitude of 32 d. 25 m. I demand the diffe­rence [Page]

[Page] [Page 5] of Longitude, and the distance between the Meridian and the Parallel of the Lyzard, and the Meridian and Parallel of Bermudas.

The Angle following being a Right-lined Triangle, and is in the like pro­tion in the Common Chart, so is L, the Meridian and Parallel of the Ly­zard, 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 Lyzard 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 Lyzard 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 Magnetick Compass, is in proportion in a Right-lined 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; [Page 7] 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 Mag­netick Compass, is in proportion in a Right-lined 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 Lyzard, and the Meridian and Parallel of Bermudas, for as it stands in the foregoing demonstration [Page 8] it is in the like proportion in the Common Chart, therefore I shall procee [...] 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 Magnetick Compass, and the diffe­rence of Longitude is in proportion in a Right-lined 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 [Page 9] 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 Mag­netick Compass, is in proportion in a Right-lined 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 [Page 10] 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 Magne­tick Compass, is in proportion in a Right-lined 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 Magnetick Compass, is in proportion to the difference of Latitude in the Common Chart.

Fourthly, I shall give the difference [Page 11] of Latitude L A, 1055 m. and the difference of Longitude A B, 3093 m. and demand the Rumb by the Magnetick Compass A L B.

To find the Rumb by the Magnetick 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 Mag­netick 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 [Page 12] demand the Rumb by the Magnetic [...] Compass.

To find the Rumb by the Magnetic [...] 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 Complement 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 Magnetick Compas [...] in the Common Chart. Add the Lo­garithm of the difference of La­titude to the Radius, and Subtrac [...] the Logarithm of the distance, and you have the Cosine of the Rumb 950897, A B L, 18 d. 50 m. whose complement 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 Magnetick Compass.

To find the Rumb by the Magnetick 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 [Page 14] 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 then 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 [Page 15] 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 Right-lined Triangle, in the Common Chart.

I have proved by the foregoing Questions, that the Rumb and the Right-lines of the Magnetick 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 Magnetick 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, then the Circulation of the Horizon must [Page 16] contain 360 d. To this I compare t [...] Magnetick Compass, which contai [...] 360 d. and is Horizontal to the H [...] rizon always, in all Meridians an [...] Parallels of the World; So that th [...] North and South, and the East an [...] West Points of the Magnetick Compass, point to the Celestial Poles o [...] North and South, and to the Celestia [...] Poles of East and West, in all Meridians and Parallels of the Earth and Sea, without giving any difference of observation for the SemiDiameter of the Earth and Sea; the [...] the Meridians of the Magnetick 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 Magnetick 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 Mag­netick Compass must be a false posi­tion [Page 17] in Navigation, for the work of [...]avigation must be performed by the [...]agnetick 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 shew you [...]ow the difference of Latitude and the [...]ifference of Longitude, and the distance of places in the Common Chart is transferr'd, 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 Magne­tick Compass doth differ from the Meridians 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 Right-lined 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 Meridians and parallels of the World.

The Demonstration follows, P the North Pole, D A the Latitude of Bermudus, A B a parallel of the La­titude of Bermudus, A P of the comple­ment of the Latitude of Bermudus, E F the difference of Longitude, found by the foregoing Right-lined Triangle, which is in proportion to the middle parallel upon the Globe; D L the [Page]

[Page] [Page 19] Latitude of the Lyzard, L G a parallel of the Latitude of the Lyzard L P, the complement of the Latitude of the Lyzard, P D is the Meridian of the Lyzard, P C is the Meridian of Ber­mudus, D P C is the difference of Lon­gitude of the Aequinoctial.

TO prove how the difference of La­titude, and the distance of places, and the difference of Longitude in the Common Chart is transferr'd by demonstration, to be in proportion to the difference of Latitude, and the distance of places upon the Globe.

EXAMPLE. The Lyzard 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 [Page 20] and Bermudus, then reduce the di [...] rence of Longitude found by the fo [...] going Right-lined Triangle, betwe [...] 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 Lyzard, to the Mer [...] dian of the Bermudus, in the paralle [...] of 41 d. 12 m. 30 sec. which is th [...] same distance in the Common Chart so that the Lyzard 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 shew you [...]w to find the distance by the Geo­ [...]etrical part of the Globe, to be in [...]oportion to the distance found by a [...]ght-lined Triangle between the Me­ [...]dian and parallel of the Lyzard, [...]nd the Meridian and parallel of [...]ermudus.

EXAMPLE. Reduce the distance found by a [...]ight-lined Triangle 3268 m. by 60 m. [...]nto degrees of the Aequinoctial, by [...]ividing 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 Lyzard L, to the Meridian and parallel of Ber­mudus B, so that the distance between the Meridian and parallel of the Lyzard, and the Meridian and parallel of Bermudus, is the same in the Globe as it is in a Right-lined Triangle.

Having proved how the difference of Latitude, and the distance of places, [Page 22] 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 Meridians 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 Meridians D C is 4200 m. or 70 d. 00 m. and the difference of Lon­gitude between those Meridians, in the parallel of Bermudus, A B, is 3545 m. or 59 d. 05 m. and the diffe­rence of Longitude, between those Meridians in the parallel of the Lyzard 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 [Page 23] and Parallel of Bermudus B, [...]nd the Meridian of the Lyzard A B, is [...]545 m. 59 d. 05 m. and the distance [...]y the Common Chart, between the Meridian and Parallel of the Lyzard, [...]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 Lyzard is 277 m. or 04 d. 37 m. more than the distance between the Meridian and parallel of the Lyzard, 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 Right-lined Triangle; for the difference of Longitude in Navigation, by the Magnetick 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 Magnetick [Page 24] Compass, as the Question hath b [...] transferr'd 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 Lyzard 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 diff [...] rence of Longitude, between the M [...] ridian and Parallel of the Lyzard, ar [...] the Meridian of Bermudus L G, t [...] proportion Arithmetical: As the R [...] dius 90 d. P D, 1000000, is to the L [...] garithm 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 Lyzard 40 [...] 00 m. P L, 980806, to the Logarithm of L G, 343130. Add the Logarithm of the difference of Longitude in [Page 25] [...]e Aequinoctial to the Co-sine of the [...]atitude, and Subtract the Radius, and [...]ou have the Logarithm of the diffe­ [...]nce of Longitude between the Me­ [...]dian and parallel of the Lyzard, and [...]e Meridiad of Bermudus, L G, 43130, or 2700 m.

To find the Logarithm of the diffe­ [...]ence of Longitude between the Meri­ [...]ian and parallel of Bermudus, and the Meridian of the Lyzard, B A, the pro­ [...]ortions Arithmetical: As the Radius [...]0 d. 00 m. P D, 1000000; is to the [...]ogarithm 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 Lyzard, 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 Lyzard 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 Longitu [...] in a Right-lined Triangle, and the L [...] ­titude of the Lyzard and Bermudu [...] and demand the difference of Lo [...] ­gitude in the Aequinoctial part of th [...] Globe.

EXAMPLE. Sail from the Lyzard in the Lati­ [...]e 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 Right-lined Tri­ [...]gle, 3093 m. I demand the diffe­ [...]ce of Longitude in the Aequinoctial [...]t of the Globe, between the Me­ [...]ian of the Lyzard and the Meri­ [...]n of Bermudus.

To find the difference of Longitude the Aequinoctial, the proportions [...]ithmetical: As the Co-sine of the [...]titude of the Lyzard, 40 d. 00 m. L, 980806; is to the Logarithm of [...]e difference of Longitude in the middle [...]rallel 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 [...]ifference of Longitude in the middle [...]arallel to the Radius, and Subtract [...]e Co-sine of the Latitude, and you [...]ave the Logarithm of 368231, [Page 28] which must be added to the n [...] proportion.

Then, As the Co-sine of the Latitu [...] 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 50d. 00m 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 [Page 29] difference of Longitude in the middle [...]arallel of the Globe, is the proper [...]ifference of Longitude 3093 m. in [...]roportion 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 Meri­dians and parallels of the common Chart differing in Longitude and La­titude, are transferr'd by Demonstra­tion, in proportion to the difference of Latitude, and the distance of places upon the Globe.

Last of all, I shall shew you how to transferr the difference of Lon­gitude in one and the same Lati­tude in the Common Chart, into the Meridians 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 [Page 30] the difference of Meridians in t [...] Aequinoctial part of the Globe.

To find the difference of Longitu­ [...] in the Aequinoctial part of the Glo [...] the proportions Arithmetrical: As t [...] Co-sine of the Latitude 40 d. 00 m. P 980806, is to the Logarithm of the diff [...] rence 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 oblique, do differ from th [...] Meridians of the Common Chart tha [...] are Right-lined; but the distance o [...] places in the Globe are the same in the Common Chart, in regard the distance of places is transferr'd from the Common Chart to the Globe, there­fore the superficies of the Common Chart in all Meridians and parallels, is in proportion to the superficies of the Globe in all Meridians and Pa­rallels.

AND whereas all Writers in Na­vigation, have condemned the [...]ommon Chart to be pestered with [...]otorious errours, it proves a great [...]istake; therefore I shall answer the [...]rincipal objection, and in that I [...]nswer all the rest.

Mr. Edward Wright in his Correction of Errours in the Common Chart, Fol. 5. [...]aith, ‘In shewing the distance of places, there is as great an errour 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, untill they come to the parallel of 60 d. Latitude, they should be there but only 50 Leagues distant, because at that Parallel, the Meridians are distant but half so much one from another, as they were at the Aequinoctial, as it may most mani­festly [Page 32] appear by the Globe, and y [...] the Chart will shew, that those t [...] Ships have the self-same distance 100 Leagues, being under the P [...] rallel of 60 degrees, which they ha [...] before when they were under t [...] Aequinoctial.

It is granted, that the difference [...] Meridians in the Common Chart, 100 Leagues in the Equinoctial, an [...] 100 Leagues in the Parallel of 6 [...] Degrees.

In regard it appears by observatio [...] by the Magnetick Compass, that th [...] East and West point of the Magnetic [...] 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 Magne­tick Compass, as it is Horizontal, does always point to the Poles of North and South in the Celestial Sphere, in all Meridians and Parallels of the Ter­restial Sphere, without making any difference of observation for the Semi-Diameter of the Earth: And whereas it appears by observation, that the [Page 33] [...]agnetick Compass in its true Poles [...]ill direct your the Circulation of the [...]arth and Sea, in any Parallel of [...]atitude, keeping an equal distance [...]om the Poles of North and South; [...] likewise it must be granted that the Meridians of the Magnetick Compass, [...]ust run Parallel to the Poles of [...]ast and West, the Circulation of the [...]arth in all Parallel Meridians, in [...]egard the Magnetick 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 Meridians and Parallels of the Terrestial Sphere, without making any difference of observation for the Semi-Diameter of the Earth; So that the Meridians do cross the Pa­rallels of the Magnetick Compass at Right-Angles, in all Meridians and Pa­rallels of the Earth and Sea, as it is by demonstration in the Common Chart. Therefore the Magnetick Compass as it is Horizontal, must measure the superficies of the Ter­restial [Page 34] Sphere by Right-lines, as if t [...] Superficies of the Terrestial 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, untill they come to th [...] Parallel of 60 d. Latitude, they mu [...] be 100 Leagues distant; in regard th [...] Meridians of the Magnetick Compas [...] must run Parallel to the Poles o [...] East and West, as the East and We [...] points of the Magnetick 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 Geometrica [...] demonstration, and the Arithmetical rules, how all places in the Common Chart are transferr'd into the Meri­dians 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 Magnetick 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 propor­ [...]ion 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 Meridians 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 [Page 36] of Meridians, in proportion to the di [...] ­rence of Meridians upon the Glo [...]

The Question to be resolved Mercator. The Lyzard, 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 Magnetick Compass.

This Question cannot be resolved any Geometrical demonstration, regard I have not the difference Longitude in proportion to the Rum [...] by the Magnetick Compass, for Right-lined Triangle must be of equ [...] parts, otherwise the difference of L [...] titude and the difference of Longitud [...] cannot be in proportion to th [...] distance of places upon the Rum [...] by the Magnetick Compass, neithe [...] can this Question be resolved by th [...] Arch of a great Circle, in regar [...] the Angles of Position in all obliqu [...] Angles, differ from the Angles of posi­tion by the Magnetick Compass; s [...] that Mercator hath no way to find the Rumb from the Question given, but brief Rules in Arithmetick, which are [Page 37] [...]ear a proportion to the Rumb by [...]he Magnetick Compass, but when the [...]istance between the Meridian and Parallel of the Lyzard, and the Meridi­ [...]n and Parallel of Bermudus, and the difference of Longitude is required in a Right-lined Triangle, to be in pro­portion to the Rumb, by the Magnetick Compass, and the difference of Latitude then 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 [Page 38] 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 Ta [...] gent of the Rumb 71 d. 21 m. 104718 [...] So that Bermudus doth bear fro [...] the Lyzard in an Angle of 71 [...] 21 m. South-Westwards: Add th [...] Radius to the difference of Longitud [...] 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 Lyzard and Bermudus, in Meridional parts is 1417, and the Rumb is an Angle of 71 d. 21 m. South-Westwards, 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 Longi­ [...]ude in the Aequinoctial 4198 Miles, [...]62307: Add the Logarithm of the [...]eridional parts contained in the diffe­ [...]ence of Latitude, to the Tangent of [...]he Rumb, and Subtract the Radius, [...]nd you have the Logarithm of the [...]ifference 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 Lyzard, and the Meridian of Bermudus, 4200 m. The Rumb is an Angle of 71 d. 21 m. South-Westwards, 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, [Page 40] 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 Subtrac [...] 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 Right-lined 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 [Page 41] in a Right-lined Triangle, 3126 m. [...]9498; so is the Logarithm of the diffe­ [...]nce 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 [...]ght-lined Triangle, to the Logarithm [...] the difference of Latitude in Me­ [...]dional parts, and Subtract the Lo­ [...]arithm of the difference of Latitude [...]pon the Globe, and you have the [...]ogarithm of the difference of Lon­ [...]tude 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 Right-lined Triangle. To find the difference of Longitude in a Right-lined Triangle, by Mercators brief Rules: Say, As the Logarithm of the difference of Latitude [Page 42] 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 Logarith [...] the difference of Longitude in a Ri [...]-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 di [...] rence 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 Meridians any one Parallel.

EXAMPLE. There are two Meridians differi [...] in Longitude 70 d. 00 m. in the [...] ­quinoctial, Bermudus, Latitude 32 [Page 43] 25 m. I demand the difference of [...]idians in the Latitude of 32 d. [...] m.

[...]o find the difference of Meridians [...]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 Lo­ [...]rithm of the difference of Longi­ [...]de in the Aequinoctial, to the Co­ [...]e of the Latitude, and Subtract the [...]adius, and you have the Logarithm the difference of Meridians, 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 Meridians [...]iffering in Longitude 70 d. 00 m. in [...]he Aequinoctial; the Lyzard Latitude [...]0 d. 00 m. I demand the difference [...]f Longitude in the Latitude of 50 d.

To find the difference of Meridians [...]etween 70 d. 00 m. of Longitude, in [Page 44] 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 Lyzard 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 [...] ­zard, 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 proportio [...] to the distance between the Meridia [...] and Parallel of the Lyzard, 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 [Page 45] the distance of places; so that the [...]rk of Navigation cannot be per­ [...]med without those Rules are in [...]portion to a Geometrical demon­ [...]ation.

Then observe you cannot measure [...]e distance of places, differing in [...]ngitude and Latitude in Mercators [...]ani-Sphere, in proportion to the [...]stance of places upon the Globe: [...]herefore Mr. Edward Wright in his [...]orrection of Errours in Navigation, page 59. saith, if two places differ so in Longitude as well as in Latitude, [...]ok how many degrees the difference of [...]atitude 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 [Page 46] how many degrees are contained be [...] them, for so many score Leagues 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, untill 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 Lyzard and Bermud [...] Therefore Mercator doth grant t [...] the difference of Latitude, and distance of Places, must be measu [...] by a Scale of equal parts, to be in p [...] ­portion 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 [...] [Page 47] difference of Latitude, and the distance places, in a Right-lined Triangle, [...]erwise you must alter the Rumb the Magnetick Compass, and the [...]ance of places.

But Mercator opposeth this position his Plani-Sphere, and from the [...]ithmetical Rules of the Globe, [...]erein the difference of Meridians [...]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 Meridians his Plani-Sphere, between the Me­ [...]dian and Parallel of Bermudus, and [...]he Meridian of the Lyzard, to be in [...]roportion 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 Lyzard, such is the difference of Longitude [Page 48] 59 d. 05 m. Do the like, and you sha [...] have the difference of Meridian between places in any other Paralle [...] 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 Lyzard, is greater tha [...] the distance, which is out of all pro­portion in a Right-lined Triangle [...] Therefore Mercator hath no Geome­trical Demonstration, either in oblique Angles or by Right-lined Angles, to prove the difference of Meridians, 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 Magnetick 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 Right-lined Triangles, [Page 49] is in proportion to the middle Parallel of all places upon the Globe, there­fore the Meridians of the Globe, as they are drawn, must differ from the Meridians of the Magnetick Compass; so that it is the principal point in Na­vigation, to enquire whether the North and South Points of the Mag­netick Compass, run in the Meridians of the Globe, as they are drawn, or whether the North and South points of the Magnetick Compass, run parallel to the Poles of East and West, as the East and West points of the Magnetick 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 Magnetick Compass, as it is Horizontal, that the North and South Points of the Magnetick Compass, run Parallel to the Poles of East and West, as the East and West points of the Magnetick Compass run Parallel to the Poles of North [Page 50] 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 Magnetick Compass, point to the Poles of East and West in the Celestial Sphere, in all Meridians and Parallels of the Terrestial Sphere; then I shall prove the East and West Points of the Mag­netick Compass run Parallel to the Poles of North and South, in all Me­ridians and Parallels of the Terrestial 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 Magnetick Compass, as it is Horizontal, in all Meridians and Parallels of the Terrestial Sphere; but if I was in the Latitude, 32 d. 25 m. or in any other Latitude, and I was to Sail by the Magnetick Com­pass East or West, the Circulation of [Page 51] the Earth and Sea, I shall find I have not altered my Latitude by observa­tion; therefore the East and West course of the Magnetick Compass agrees with the Parallels of Latitude upon the Terrestial 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 Mag­netick 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 Magnetick Compass.

In the next place, I shall prove the North and South Points of the Mag­netick Compass, point to the Poles of North and South in the Celestial Sphere, in all Meridians and Parallels of the Terrestial Sphere. Then I shall prove the North and South Points of the Magnetick Compass, must run parallel to the Poles of East and West, in all Meridians and Parallels of the Terrestial Sphere.

EXAMPLE. By Observation, If the Sun or Star be upon their Meridian in my Me­ridian and Parallel of the Terrestia [...] Sphere, the Sun's or Stars Azimuth mus [...] be North or South, by the Horizonta [...] bearing of the Magnetick Compass to the Celestial Poles; so that the Mag­netick 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 Meridians 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 Magnetick Compass points to the Celestial Poles of North and South, and to the Celestial Poles of East and West, in all Meridians and Parallels of the Terrestial Sphere, it is proved by observation, that you may run the Circulation of the Earth by the Mag­netick Compass, in any Parallel of Latitude; therefore the Magnetick [Page 53] Compass must run Parallel to the Poles of East and West, as the Magnetick Compass runs Parallel to the Poles of North and South in the Terrestial Sphere, in regard the Magnetick 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 Meridians by the Magnetick Com­pass, as it is Horrizontal upon the Terrestial Sphere, is of equal distance from the Poles of East and West, as the difference of the Parallels of La­titude by the Magnetick 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 Magnetick Com­pass, as it is Horizontal, points to the Celestial Poles; so that the North and South points of the Magnetick Compass, do cross the East and West points of the Magnetick Compass, at Right-Angles, in all Meridians and Parallels of the Earth and Sea: There­fore the Magnetick Compass, doth [Page 54] measure the superficies of the Glob [...] by Right-lines, as if the superficies o [...] the Globe was a Plain to the Horizon [...]

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 Magnetick 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 Terrestial Sphere is spacious to us that are upon it, but it appears to be no other than a Center to the body of the Sun or Stars, in regard the Magnetick Compass runs atwhart the Parallels of Latitude in the Terrestial 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 Magnetick Compass should run Parallel to the Poles of North and South in the Ter­restial [Page 55] Sphere, if the Meridian Alti­tude of the Sun or Stars, by obser­vation, did not produce the same La­titude or Parallel with the Magnetick Compass, the Circulation of the Earth, in regard the Magnetick Com­pass, as it is Horinzontal, points to the Poles of East and West in the Ce­lestial Sphere, as the Magnetick 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 Magnetick 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 Terrestial Sphere; but that you may likewise suppose Meridians, or the East and West points of the Magnetick Compass, as it is Horizontal, to the Suns Ampli­tude in the Aequinoctial, to run to the Poles of East and West in the Ter­restial Sphere; in regard the Mag­netick Compass makes the same [Page 56] 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 Magnetick 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 Terrestial Sphere, then the North and South Points of the Magnetick 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 Terrestial Sphere: Therefore the Meridians of the Magnetick Compass, hath the same position in the Ter­restial Sphere, and must run Parallel to the Poles of East and West, as the Parallels of Latitude run Parallel by the Magnetick Compass, to the Poles of North and South.

Then observe the Magnetick Com­pass, being always Horizontal to the Horizon, it doth measure the Super­ficies of the Earth and Sea by Right-lines, in the Parallel Meridians 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 Meridians, and the difference of Latitude, and the distance of places in proportion to the Rumb, and the Right-lines of the Magnetick 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 Magnetick Compass.

Therefore it hath been a great mistake in all Writers, to suppose the Magnetick Compass to agree with the Meridians of the Globe as they are drawn; whereas, there is no obser­vation [Page 58] or demonstration to prove i [...] only the Meridians of the Magnetic [...] Compass, as it is Horizontal to th [...] Meridians 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 Magnetic [...] Compass, as it is Horizontal, poin [...] to the East and West Poles of the Ce­lestial Sphere, from all Meridians an [...] Parallels of the Terrestial Sphere; s [...] that you may as well draw Meridian [...] in Navigation by the Magnetick Com­pass, from the Poles of East and West as from the Poles of North and South in the Terrestial Sphere; therefore the work of Navigation cannot be per­formed by the oblique Lines of the Globe.

I have proved by observation, that the East and West points of the Mag­netick Compass, run Parallel to the Poles of North and South in the Terrestial Sphere, therefore the North and South points of the Magnetick Compass, must run Parallel to the Poles of East and West in the Ter­restial [Page 59] Sphere; in regard those Poles do cross at Right-Angles, in all Meridians and Parallels of the Earth and Sea.

Last of all, I shall shew you the errours 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 Magnetick Compass in a Right-lined Triangle; so that Mercator hath granted three parts in four in a Right-lined Triangle; then if the difference of Longitude in a Right-lined Triangle, be not the true difference of Meridians, according to the Rumb by the Magnetick 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 Lyzard, in a Right-lined Triangle, from the Rumb sound by the brief Rules of Mercator, and the difference of Latitude.

And last of all, I shall shew you the disproportion between the Meridians of the Common Chart, and the Me­ridians 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 Meridians in Mercator's Plani-Sphere, or according to the difference of Me­ridians upon the Globe, they must commit gross errours.

EXAMPLE. I Sail from the Meridian and Pa­rallel of the Lyzard, in the Latitude of 50 d. 00 m. in an Angle of 71 d. 21 m. South-Westward: 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 Lyzard, in a Right-lined Triangle.

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 Lyzard and Bermudus cannot be known, but by a Right-lined Triangle.

The Angle following being a Right-lined Triangle, is in like proportion to the Common Chart, so is L the Me­ridian and Parallel of the Lyzard, 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 Lyzard, and the Meridian and Parallel of Bermudus, then is A B and L T the difference of Longitude between the Meridian of the Lyzard, and the Me­ridian of Bermudus, and L B is the distance between the Lyzard 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 Meridians of the Common Chart 419 m. and 426 Miles.

To find the distance between the Meridian and Parallel of the Lyzard 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. LAB, 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 Magnetick 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 Mag­netick [Page 63] Compass, and the difference of Latitude, but by a Right-lined Tri­angle; therefore Mercator doth grant three parts in four in a Right-lined Triangle; then if the difference of Longitude in a Right-lined Triangle, be not the true difference of Meridians, there is no Geometrical Demonstra­tion in Navigation, to find the diffe­rence of Meridians in proportion to the Rumb by the Magnetick Com­pass.

In the next place I shall proceed to find the difference of Longitude in a Right-lined Triangle, from the diffe­rence of Latitude, and the Rumb found by Mercators brief Rules.

EXAMPLE. I Sail from the Meridian and Pa­rallel of the Lyzard, in the Latitude of 50 d. 00 m. in an Angle of 71 d. 21 m. South-Westward: As the Rumb is found by the brief Rules of Mercator, into the Meridian and Parallel of Bermudus, in the Latitude of 32 d. [Page 64] 25 m. I demand the difference of Lon­gitude in a Right-lined 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 Magnetick Compass, and the difference of Latitude in a Right-lined 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 alters the difference of Longitude 33 m. and the distance 30 m. otherwise this latter Right-lined Triangle would [Page 65] agree with the former. I shall pass by these small errours, 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 Lyzard, 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 out-run the Meridian of the Lyzard 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 Magnetick Compass, I must Sail in an Angle of 21 d. 40 m South-Eastward by the Magnetick Compass, to arrive in the Meridian of the Globe or Mer­cator; so that I have shewed you how [Page 66] great the disproportion is between the Meridians of the Common Chart, and the Meridians of Mercator, or the Me­ridians of the Globe; therefore the Meridians of the Magnetick 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 Lyzard, 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 Magnetick Com­pass, I must Sail in an Angle of 21 d. 59 m. North-Eastward, before I can arrive in the Meridian of the Globe [Page 67] or Mercator; so that I have shewed you how great the disproportion is between the Meridians of the Common Chart, and the Meridians of Mercator, or the Meridians of the Globe; therefore the Meridians of the Magnetick 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 Magnetick Compass, as it is Horizontal, doth point to the Poles of East and West in the Celestial Sphere, in all Meridians 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 Magnetick Compass doth run Parallel to the Poles of North and South in the Terrestial Sphere, the Circulation of the Earth; therefore it [Page 68] had been impossible to prove the Mag­netick Compass should run Parallel to the Poles of North and South, in the Terrestial Sphere; if the Magnetick Compass did not run Parallel to the Latitude of places by observation.

Likewise, I have proved by Obser­vation and Demonstration, that the Magnetick Compass, as it is Hori­zontal, doth point to the Poles of North and South in the Terrestial Sphere, in all Meridians and Parallels of the Earth and Sea, which Meri­dians do cross the East and West Poles of the Magnetick Compass, as it is Horizontal to the Celestial Sphere, at Right-Angles, in all Meridians and Parallels in the World: Therefore the Meridians of the Magnetick Compass being Horizontal, must run Parallel to the Poles of East and West in the Terrestial Sphere, as the Magnetick Compass, being Horizontal, doth run Parallel to the Poles of North and South in the Terrestial Sphere.

Last of all, the Magnetick 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 Right-lined 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 Meridians 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 Right-lined 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 shew you, that the Question cannot be given from the practical part of the Sea, but must [Page 70] alter the Meridian and Parallel of all places upon the Globe.

SAILING by the Arch of a great Circle.

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 Twylight appears upon the Horizon; the proof of which is very sutable in this Pa­rallel or Latitude.

EXAMPLE. London, Latitude 51 d. 30. North, the Suns Declination 23 d. 30 m. North, the Suns Depression 15 d. 00 m. I demand the Time of Twylight? By the Work you will find the Sun be­ing [Page 152] depressed 15 d. at Midnight, the 11th day of June, in the Latitude of 51 d. 30. you will have Twylight 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.