Geographie delineated forth in two bookes Containing the sphericall and topicall parts thereof, by Nathanael Carpenter, Fellow of Exceter Colledge in Oxford.

CHAP. XII. Of the distances of places compared one with another.

1 OF the simple and absolute distinction of distances, wee haue treated in the former Chapter: wee must in the last place handle it comparatiuely; that is to say, one place compared with another: whereof wee are to consider the Inuention and Expression.

2 The distance is the measured space betwixt two places: which is, either vniforme, or various: vniforme is in places different, either in Longitude onely, or in Latitude onely.

3 Those places differ in Longitude onely, which are situate vnder the same or like Parallels, but diuers Meridians; or at least vnder op∣posite

points of the same Meridian.

Of place•• differing onely in Longitude, there may bee three ca∣ses: For 1. they may be vnder the same Parallell, as the Iland of ••••int Thomas, and Summatra, which lie directly vnder the Equatour; or Noremberg and Hamberg, which hauing very neere the same Latitude, differ in Longitude, and lie in the same Parallell without the Equatour. 2. They may be vnder the like Parallels, that is, in points equidistant from the Equatour. As Siene in Egypt, vnder the Tropicke of Cancer; and Beach in the South continent, vnder the Tropicke of Capricorne. 3. They may be vnder the same Parallell and Meridian, but in opposite points of the said Parallell: such as are the Perioeci, spoken of in the 10. Chapter.

4 Places differing onely in Longitude, whose distance is here proposed to bee sought out, are seated in the same, or diuers Hemi∣spheares.

5 In the same Hemispheare, when both places haue either Easterne or Westerne Longitude. This againe may haue two cases; for either the places are vnder the Equatour, or with∣out it: in both which, the finding out of the distance shall bee opened in these Rules.

1 If two places vnder the Equatour in the same Hemispheare, differ in longitude: let the lesser longitude be subtracted from the greater, and the difference conuerted into Miles, and the di∣stance will be knowne.

As for example, wee will suppose of two places, whose di∣stance is to bee sought out, the former to be the Iland of Saint Thomas in Africa, the other the Iland Summatra in the East Indies, both situate directly vnder the Equatour; and there∣fore differing onely in Longitude. To expresse which, in this figure, let the first

[illustration]

Meridian from which the Longi∣tude is to be mea∣sured, be ABCD: the place where Saint Thomas I∣land is seated, K: and the place of Summatra, F. Thē subtracting AK, the Longitude of Saint Thomas I∣land, being lesser, out of the Longi∣tude of Summatra AE, the residue KE will shew the distance in degre••s: which being multiplied by 60, and so conuerted into Italian-miles, will shew how many miles the said places are distant the one from the other. As in this present example, wee finde the Lon∣gitude of Saint Thomas Iland to bee 32 degrees 20 minutes of Summatra, to bee 131 degrees: The lesser summe subducted from the greater; to wit, 32 degrees 20 minutes, out of 131; there will remaine 98 degrees 40 minutes: which being againe multiplied by 60, will produce 5920 Italian-miles, the true distance betwixt the said places.

2 Of two places in the same Hemispheare, situ∣ate without the Equatour; the distance may be knowne two wayes: either by the resoluti∣on of the Sphericall Triangle, or else by ta∣bles

of the miles answerable to the degrees of Latitude.

The former way is performed in this manner: Let the Tri∣angle of two equall sides FBG in the figure before, bee re∣solued; in which the two equall sides FB, and GB are the complements of equall Latitudes; to wit, AF, and EG. The Angle FBG is the difference of Longitude, which Angle, whether it bee a Right Angle, or Oblique Angle, will easily bee knowne, if by letting a perpendicular line BI from B to I it bee parted into two Triangles FBI and IBG: for because those two Triangles according to the grounds of Geometry are equall; the Arch IG in the Triangle IBG being found out, the Arch also IF in the Triangle FBI will also bee knowne: which beingthus demonstrated, wee must proceed in this manner, according to the Golden Rule. As the Right angle BIG is to the complement of the Latitude BG, so is IBG the middle difference of Longitude to IG the middle distance: Pitiscus in his Trigonometry to this addes another manner of de∣monstration, expressible by the precedent figure: let the per∣pendicular IB be continued vnto K, that BK may make a whole Quadrant. Now will the Triangle IHK haue Right Angles at I and K, at I by supposition, at K by his 57 proposition demonstra∣ted in his first booke: because, If a greater circle of the Spheare passe by the Poles of a greater cîrcle, it will cut it at right Angles, and contrariwise: wherefore the sides IH and KH must bee quadrants: because, as hee shewes in his 68 proposition of his first booke; In a sphericall Triangle hauing more then one Right Angle, the sides subtending those Right Angles are Quadrants: Finally, because the Arches GH and EH, are the complements of the Arches IG and KE: by the 9 definition of the first booke▪ For as much as of any Arch lesse then a Quadrant, the complement is that which wants to make it vp 90 parts. We may by the helpe of the 57 proportion of his first booke, seeke out the comple∣ment of the third side GH; which will be the Arch GI: which will shew vs the probleme which wee sought, by redu∣ducing

it vnto the Table of signes, and Tangents, exactly se•• out by our forenamed Author and others. For an example of this, wee may take two famous cities of Germanie, Noremberg and Hamberg, which without any sensible difference haue the same Latitude, but differ in Longitude: For the Longitude of Noremberg is 31 degrees 45 minutes: of Hamberg 32 degrees 30 minutes: the difference of Longitude then is 0 degrees 45 minutes. These things supposed to be knowne, we will ima∣gine Noremberg to be in F, Hamberg in G▪ and therefore AF, or EG will haue 49 degrees 23 minutes: FB or GB will haue 40 degrees 37 minutes: FBG or AE will haue 0 degrees 45 minutes: KE 0 degrees 22 ½ minutes: EH 89 degrees 37 ½ mi∣nutes: if we worke by the Table of Signes, Tangent••, and Se∣cants, the knowledge whereof is required to this Probleme. But because the former way may seeme difficult to such as are not much acquainted with Trigonometry, some haue set downe ••n easier way, depending on the vse of a Table, wherein i•• cal∣culated the number of miles answering to euery degree of eue∣ry Parallell of the Spheare: in which working▪ we ought to bee directed by this Rule: If two places without the Equatour differ in longitude only, subtract the lesser number out of the greater, and multiply it by the n••mber of miles answerable to a d••gree of that Parallell, ••nd the product will giue the distance. As for exampl••, if you would know the distance betwixt London and Antwerpe, which haue in a manner the same Latitude, but differ in Longi∣tude: I finde them to differ in Longitude by 6 degrees, which number being multiplied by 37 miles answerable to 51 degrees of Latitude, these will arise to 247 miles, and 54 seconds of a mile.

A Table of Miles answerable to one Degree of euery seuerall Latitude.

1	2	3	4	5	6
D	M	S	D	M	S	D	M	S	D	M	S	D	M	S	D	M	S
1	59	59	16	57	41	3	51	26	46	41	41	61	29	5	76	14	31
2	59	58	17	57	23	32	50	53	47	40	55	62	28	10	77	13	30
3	59	55	18	57	4	33	50	19	48	40	9	63	27	14	78	12	28
4	59	51	19	56	4	34	49	45	49	9	22	64	26	18	79	11	27
5	59	4	20	56	23	35	49	9	50	38	34	65	25	21	80	10	25
6	59	40	21	56	1	36	48	32	51	37	46	66	24	24	81	9	23
7	59	33	22	55	38	37	47	55	52	36	56	67	23	27	82	8	21
8	59	25	23	55	14	38	47	17	53	36	7	68	22	29	83	7	19
9	59	16	24	54	49	39	46	38	54	35	6	69	21	30	84	6	16
10	59	5	25	54	23	40	45	58	55	34	25	70	20	31	85	5	14
11	58	54	26	53	6	41	45	17	56	33	33	71	19	32	86	4	11
12	58	41	27	53	28	42	44	35	57	32	41	72	18	32	87	3	8
13	58	2	28	52	59	43	43	53	58	31	48	73	17	33	88	2	5
14	58	13	29	52	29	44	43	10	59	30	54	74	16	32	89	1	3
15	57	57	30	51	58	45	42	26	60	30	0	75	15	32	90	0	0

6 The distance of places differing only in Lon∣gitude in diuerse Hemispheares is found out by this rule.

1 Let the greater longitude be subtracted from the whole circle, and vnto the residue added the lesser longitude, there will arise the distance betwixt those places.

As for example, Lisbone in Spayne hath in East longitude 13 de∣grees: and Cap de Los Slanos in America, hath in West longi∣tude 334 degrees: to know the distance betwixt those places, you must first subduct 334, which is the greater Longitude out of 360 the whole circle, and there will remaine 26 Degrees, to which if wee adde the East longitude of Lisbone, which is 13 degrees, it will make 39 degrees, which is the true difference of those longitudes: which being multiplied by the Number of miles in the Table going before, answerable to the latitude of the said places (if they differ only in longitude) there will arise the number of miles contained in the Distance.

7 Distant places which differ onely in latitude, are such as lye vnder the same Meridian, but di∣uerse Parallells: These are supposed to be either in One, or in Diuerse latitudes or Hemispheres.

8 In One, when both the places haue either North latitude, or both South Latitude: The finding out of which distances depends on these Propositions.

1 If the latitude of each place be towards the same P••le, subtract the lesser from the greater lati∣tude, and the residue conuert into miles.

The reason may bee explained in this Figure: wee will imagine

[illustration]

EF to bee the lesser, EG the greater la∣titude. There will remaine an Arch of the Meridian FG: which being multi∣plied by 60 (being part of a great circle, will make the nūber of miles answerable, to that distance. For an example we will take two Citties of England, Oxford and Yorke. The latitude Oxford, we take to be 31 degrees 30 minutes: of Yorke 54 degrees 30 minutes. The lesser latitude subtracted from the greater, there will remaine three degrees, which being multiplied by 60, will render 180 Italian-miles, the Distance of thse two places.

2 If two places in latitude only distant, be situate in diuerse kindes of latitude, adde the latitude of the one to the other, and the whole summe shall be the distance.

As for example, in the former Diagram, imagining as in the former case BD to be the Meridian of those distant places, and AC the Equatour, we will suppose the one place to bee situate towards the North Pole, as G, the other towards the South, as in H: then as appeares by sense, will the distance bee the Arch of the Meridian GH, whereof GE, and EH, are parts, whereof it is compounded: wherefore it must needs follow that those parts added together make the whole distance: for example we will take Bellograde in Europe, and the Cape of good hope in Afri∣ca, which haue neere the same longitude, to wit, 48 degrees 30

minutes: but they differ in latitude in such sort, as the former hath of the Northerne latitude 44 degrees 30 Minutes; the o∣ther of Southerne latitude about 35 degrees 30 minutes. These two numbers added together, will make 80 degrees, which be∣ing multiplied by 60 will produce 4800 miles the distance of those places.

9 Hitherto of the distances of places which are Vniforme, that is to say, of such as differ either only in longitude, or onely in latitude: wee are next to consider of such distances as are vari∣ous, wherein the places differ both in longitude and latitude.

10 The Inuention of such a distance, may bee performed two wayes, either Abstractiuely by the resolution of Triangles, or else Mechanically by instruments. The former againe may bee two wayes, either by the Right-line Triangle, or by the Sphericall: The inuention of the di∣stance by the Rigt-line Triangle depends on these following Propositions declaring two wayes of inuention.

11 The first is by a Rectangle Triangle barely considered by it selfe, according to this Theo∣reme.

1 The square Root of the number made of the diffe∣rences of longitude and latitude of two places di∣stant, will shew the distance of those places.

The ground of this Proposition is taken from the 27 Proposition

of the first booke of Euclide: where it is demonstrated that the square of the Hypotenusa, or greatest side of a Rectangle Trian∣gle is equall to the two squares made of the two other sides: which being well vnders••ood, will lend an easie light to this pro∣position. To performe which we must first take the difference of longitude, which is imagined to make one side of this Triangle. Then wee must obserue also the difference of latitude, which is supposed to make another side. Then are we sure by the former Proposition of Euclide that the squares of these two sides, are e∣quall to the square of the Hypotenus••, or third side; which is to be sought out, and expresses the distance betwixt those places: wherfore we must first multiply these two sides in themselues, whence they will become squares. 2. We must adde them toge∣ther. 3 We must out of the totall extract the quadrat root, which will shew the distance: as suppose according to this Figure, two Cities d••stant and differing both in longitude and latitude: wher∣of the one shall haue in longitude 21 degrees, in latitude 58: the other is supposed to haue in longitude 26 degrees, in latitude

[illustration]

52. Here first I subtract the lesse lon∣gitude out of the greater, to wit, 21 out of 26, and the residue will bee 5, which I suppose to be one side of the Rectangle Triangle. Then likewise I subtract the lesse latitude as 52 out of 58, the residue will be 6, which I make the other side of my Triangle, which done, I multiplie 6 in it selfe, and it makes 36, which is the square of one side: Then I multiply 5 in it selfe, and the product will be 25, the square on the other side. These two squares added together by the foresaid Proposition must be equall to the square of the Hypo∣teneus▪ orthird side 61, whereof the square root being extracted, will shew the side it selfe, which will be 7 7/25 which is the di∣stance: If any man desire to know this distance according to miles, he must reduce the degrees of longitude and latitude into miles according to our former rules, before he begin to worke:

because (as wee haue shewed) the degrees of longitude being measured in the Parallells are not alwayes equall, the Parallels being somewhere great••r, somewhere lesser. This way must needs bee more exact, in that a Mile is a smaller part then a De∣gree, and (as Pitiscus notes) the Fractions which fall out in ex∣traction of roots can hardly bee reduced to any proportion. Ne∣uerthelesse this way of finding out the distance by a Right-line Triangle, howsoeuer common and receaued, is very vnperfect and subiect to great errour, especially in places far distant: for as much as it supposeth the Meridians with the Parallels on the Globe to make true squares, whereas indeed all the Meridians meet in the pole, and so by consequence cannot make true squares: But yet this errour is far lesse in a lesser distance; be∣cause in a small space of earth, the roundnesse and conuexity of the Earth is insensible, or at least of very small importance: so that this way cannot be altogether vnusefull.

12 Another is found out more exact then the former by the tables of Signes, Tangents, and Secants. This is performed by finding out the numbers: whereof the former is called Inuen∣tum primum, or the first found number. The se∣cond Inuentum secundum, or the second found number. The working of which Probleme depends on these rules.

1 Multiply the Right Signe of the difference of the longitude into the summe of the complement of the lesser latitude, and diuide the product of that multiplication by the totall summe, & then by the rules of Signes and Tangents the Arch of that Quotient found out will giue the first found number.

2 Multiply the right signe of the lesser latitude by the totall signe, and hauing diuided the product thereof by the signe of the complement of the first number, subtract the Arch of that quotient out of the greater latitude which giues the second found number.

3 Then multiply the signe of the complement of the first found number into the signe of the complement of the second found number, and hauing diuided the product by the Totall Signe, Let the Arch of the quotient be sought out by the Tables, which Arch subtracted out of the whole quadrant, will giue the degrees of a distance in a great circle.

To expresse the practice and manner of working according to our former Rules, we will suppose the two cities, whose di∣stance is here sought out to be Ierusalem and Norimberge in Ger∣many. Ierusalem hath in longitude 66. degrees. 0 min. and in latitude 31 degrees, 40. minutes. Againe Norimberge hath in longitude 28. degrees. 20. minutes, and in latitude 49 deg. 40. min. The difference of their longitude is 37. deg. 40. minutes. The right signe whereof is 36664: (for here wee make 60000 to be the totall signe, rejecting the two last figures on the right hand in the tables of Regiomontanus.) Now you must multiply 36664: into the signe of the Complement of the lesser latitude, which is 51067: the product of which two signes being multi∣plied the one by the other, there will arise 1872320488: which if you diuide by the totall signe 60000, the quotient will giue you 31205, whose Arch is 31 deg. 20 min. and this must be your first found Number.

For the finding of the second Number, you must proceede in this māners: Multiply the right signe of the lesser latitude, which

is 31498 by the totall signe 60000, and the produ••t will bee 1889880000: which summe, if wee diuide by the signe of the Complement of the first-found Number, which is 51249, wee shall finde in the quotient 36876; the Arch whereof is 37 de∣grees, 55 min: which Arch subtracted out of the greater lati∣tude, there will remaine 11 degrees, 29 min: and this is our se∣cond-found Number. These things thus supposed to bee found out, wee must multiply the fore-said signe of the Complement of the first-found Number, which is 58798, and the product will arise to 3013338702, the Arch whereof is 56 deg. 50 min: which being subtracted out of the whole quadrant, viz: 90 de∣grees, there will remaine 33 degrees, 10 min: of the greater circle. These 33 degrees if we multiply by 60, there will arise 1980 miles, whereunto if we finde the 10 miles answerable to the 10 min. wee shall finde the distance betwixt these places to be 1990 Italian-miles. This example is vsed by Appian, and wrought according to his owne Tables, and farther explained by our countryman Blundeuill in his Exercises. The same way of working hath been deliuered by Clauius, Iunctinus, and others, although not according to the same Tables. This was of measu∣ring the distance by the Signes and Tangents according to these Authors, may be warranted more exact than the other, because it admits of smaller parts in the calculation; yet will it come far short of truth.

10 Another way of finding out the Distances of places, differing both in longitude & latitude, is by the Resolution of a Sphericall triangle.

This way of all the rest must needs be most certaine: for as much as this kind of triangle best expresseth the sections of the Globe. The methode of which working I finde no-where better taught then in Pitiscus his Trigonometry: of whose ingenious industry notwithstanding little vse can bee made, except the Learner first acquaint himself with his principles, because in his Geographicall Problemes, he briefely referres his Reader to his former grounds and Axiomes, accurately demonstrated in his former books: For mine owne part it might perhaps seeeme as absurd in this Trea∣tise,

to intermixe all his preparatory demonstrations, being meerely Geometricall, and without the limites of my subject, as by leauing out so necessary a way to mangle my discourse. Wher∣fore intending a middle way, I will (God-willing) in such sort set downe these propositions, that I may giue some light to this excellent inuention, and referre my Reader to Pitiscus his Axi∣omes for farther Demonstration.

14 The Distances proposed to be measured by Sphericall triangles, admit of two cases: 1 When two places are so situate, that the one is vnder the Equatour, the other without. Secondly, when both are without the Equa∣tour.

15 The former againe is three-fold: For either the difference of longitude betwixt those pla∣ces is Equall to a quadrant, or Lesse, or Grea∣ter. The seuerall wayes of inuention on shall bee directed by these Rules.

1 If the Difference of longitude be Equall to a qua∣drant, the distance will also be a quadrant.

As for example in this present figure wee will imagine the circle ABCD to bee the first Meridian: the places whose di∣stance is sought out A and G: whose Distance AG will bee a quadrant. For A will be a pole of a Greater Circle BGD, by the 56 prop. of the 1 of Pitiscus: wherefore all the Arches drawne from thence to BGD will bee quadrants by the same propositi∣on. For a more familiar instance wee will take the Iland Suma∣tra, which hath in longitude 131 degrees, but no latitude, be∣ing sited vnder the Equatour: and the city Buda the Metropolis of Hungary, which hath in longitude 41 degrees, in latitude 47 degrees; The difference of longitude is 90 degrees; for 41 be∣ing

[illustration]

[illustration]

subducted out of 131, there will remaine 90, wherefore the distance betwixt those places will be 90, which being multipli∣ed by 60, will produce 5400 Italian-miles.

2 If the difference of longitude bee lesse then a qua∣drant as AF: the Triangle AEF here is to bee resolued into his parts, according to the 4^th Axi∣ome of Pitiscus.

As for example the places, whose longitude is here sought out, shall be A and F; The Triangle here to be knowne is AEF; whose Resolution depends on our Authors 4^th Axiome. For the Diffe∣rence of longitude is ABF; because the measure of a Sphericall Triangle being taken in a great circle, is an Arch of a greater circle, described from the Angular point, and comprehended be∣twixt the two legges of the Triangle so farre as a quadrant, as is taught in the 58 proposition of his first Booke. For a more speci∣all instance we will take two places; whereof the one shall bee the Iland of S. Thomas before mentioned, which hath in longi∣tude 32 degrees and 20 minutes. The other Amsterdam in Hol∣land,

which hath in longitude 26 degrees, 30 minutes. The for∣mer we imagine in A; the later in F. The Difference of longi∣tude ABF will be 5 degrees, 50 min: Then the distance sought out must be AF: so working according to the fourth Axiome of Pitiscus, we shall find the Arch AF, which is the distance, to be 54 degrees, 19 minutes.

3 If the Difference of longitude bee greater then a quadrant, as of the two places of F and C, the Triangle to bee resolued will bee FCE, being a Rectangle at E.

Here because the Triangle FCE hath his two sides FC, and EC, greater then quadrants, insteed of it you may worke on the Triangle AEF, adioyned to the Triangle FEC: and the whole worke will be dispatched: for by the resolution of the Triangle AEF, you shall find out the Arch FG, which being added to the quadrant CG, there will be produced the Arch FC, which is to be ••ought out. As for example, we will imagine Heidelberge as it were placed at F, to haue in longitude 30 degrees, 45 mi∣nutes, in latitude 49 degrees 35 min: Then wee will suppose Summatra, as placed at C, to haue in longitude 131 degrees, but no latitude: The difference of longitude will be EC, of 100 de∣grees, 15 minutes: and the complement AE 79 degrees, 45 minutes. Then working according to the Rules of Trigonome∣try, we shall find the signe of the Arch FC, to be 6 degrees, 37 ½ minutes; which being added to FC, being 90 degrees, will pro∣duce 96 degrees, 37 ½ minutes, to which Arch there will ans∣wer 1449 German-miles.

16 The second Case is, when both places are si∣tuate without the Equatour: This is againe twofold: For either the two places are vn∣derstood to be situate towards the same Pole, or else one place toward the Northerne, the

other towards the Southerne Pole. Both which Cases shall be taught in these Rules.

1 If both places whose distance is sought, be situate towards the same Pole, there will arise a Trian∣gle, whose sides and Angles will be knowne by the fourth Axiome of Pitiscus in Trigonometry the fourth Booke.

As for example, in this present figure, let the two places giuen bee FG, the

[illustration]

Triangle to bee knowne, will be FBG, whose a∣cute Angle will be at B. Let the places giuen bee as FH; the Tri∣angle to bee re∣solued & known will bee FBH, hauing a right Angle at H. Fi∣nally, if the pla∣ces suppos••ed to be giuen, are as FI, the Triangle to bee knowne will bee FBI, with an obtuse Angle at I.

2 If the one place be situated towards the North-pole, and the other towards the South-pole, there will arise a Triangle, whereof the one side about the Angle which is giuen, will be greater then a quadrant.

As in the former figure, let the places giuen be as G and K, al∣so

H and K, also I and K: There will still fall out a Triangle▪ whose one side containing the Angle giuen, will be greater then a quadrant, as BK: wherefore for the side BK, you must take his complement to the Semi-circle BF, that is, for the Triangle GBK, you must worke by the Triangle GBF: and insteed of the Triangle HBK, you must take the Triangle HBF: and for the Triangle IBK, you must worke by the Triangle IBF, ac∣cording to the fourth Axiome of the fourth booke of Pitiscus, to which I had rather referre my Reader, then intermixe our Geographicall discourse with handling the Principles of Geome∣try, which here are to be supposed so many precedent propositi∣ons; which, expressed as they ought, would transcend the bounds of my intended journey.

17 Of the Astractiue way of finding out the Di∣stance of places, we haue spoken: The Me∣chanicall depends on the vse of Instruments and Mechanicall operation, whereof wee will shew one way in this Theoreme.

1 By the working with a Semi-circle, the Di∣stance of two places may be found out.

This inuention by M^r Blundeuill, seemes to be ascribed to Ed∣ward Wright, yet hath it beene taken vp of forreine Writers as their owne, and vsed in their Charts and Mappes. The manner of operation is thus: First, let there be drawne a semi-circle vpon a right Diameter signed out, will be the letters ABCD, where∣of D shall be the center, as you find it deciphered in this present figure. The greater this Semi-circle be made, so much the more easie will be the operation; because the degrees will be larger. Then this Semi-circle being drawne, and accordingly diuided, imagine that by the helpe of it, you desire to find out the distance betwixt London and Ierusalem, which cities are knowne to di∣ffer both in longitude and latitude. Now, that the true distance betwixt these two places may bee found out, you must first sub∣tract the lesser longitude out of the greater, so shall you finde the

Difference of their longitudes, which is 47 degrees. Then rec∣kon that Difference vpon the Semi-circle, beginning at A, and so proceed to B; and at the end of that Difference, make a marke with the letter E, into which point by your Ruler, let a right line be drawne from D the center of the Semi-circle. This being in this sort performed, let the lesser latitude be sought out, which is 32 degrees in the foresaid Semi-circle, beginning your accompt from the point E, and so proceeding towards B, and at the end of the lesser latitude, let another point bee marked out with the letter G: from which point let there be drawne a per∣pendicular, which may fall with right Angles vpon the former line, drawne from D to E; and where it chanceth to fall, there marke out a point with the letter H: This being performed, let the greater latitude, which is 51 degrees, 32 minutes, be sought out in the Semi-circle, beginning to reckon from A towards B, and at the end of that latitude, set downe another point, signed out by the letter I: from whence let there bee drawne another perpendicular line, that may fall with right Angles vpon the Diameter AC, and here marke out a point with the letter K: This done, take with your Compasse the distance betwixt K and H; which distance you must set downe vpon the Diameter AC, placing the one foote of your compasse vpon K, and the other towards the center D, and there marke out a point with the let∣ter L: Then with your compasse take the shorter perpendicular line GH, and apply that widenesse vpon the longer perpendicu∣lar line IK, placing the one foot of your compasse at I, which is the bounds of the great latitude, and extend the other towards K, and there make a point at M. Then with your compasse take the distance betwixt L and M, and apply the same to the semi∣circle, placing the one foot of your compasse in A, and the other towards B, and there marke out a point with the letter N. Now the number of degrees comprehended betwixt A and N, will expresse the true distance of the two places, which will be found to be 39▪ degrees: which being multiplyed by 60, and so con∣uerted into miles according to our former Rules, will produce 2340, which is the distance of the said places.

17 The expression of the Distance of two places may be performed either by the Globe or Map according to these Rules.

1 The distance betwixt two places in the Globe, be∣ing obserued by the quadrant of Altitude, and ap∣plied to the degrees of the Equatour, or any great circle, will shew how many miles such places are distant.

The practise hereof is very easie, as shall be taught in this ex∣ample: we wil for instance take Tolledo in the middest of Spaine, and the Cape of Good Hope in the South Promontory of all Afri∣ca: The space taken by a quadrant of Altitude, or any threed ap∣plyed to the Equatour, will be found to bee about 82 degrees, which being multiplyed by 60, and so conuerted into miles, will render 4920, the true distance betwixt these two places.

2 The distance betwixt any two places in the Chart, obserued by a compasse, and applyed to the degrees of a greater Circle, will shew how many miles such places are distant one from the other.

For an example, we will take the city Seuill on the Southmo•••• part of Spaine, and Bilbao on the North-side: the space betwixt those places being taken with a thre••d or a compasse, and apply∣ed to one of the greater Circles, will containe about 6 degrees; which being multiplyed by 60, and so conuerted into Italian-miles, will produce 360: and so many miles those Cities are to be esteemed distant the one from the other.