CHAP. I. Certaine generall Advertisements concerning the use of this Instrument; together with the de­scription of such Circles as are newly added thereto, serving for Navigation.

WHen I penned the rules which have been The first part of this Chapter. formerly set out to shew the use of this Instrument, I was carefull to doe it with as much plainenesse and perspicuity, as might be in a subject not as yet obvious to vulgar knowledge, so that any one but moderately exercised in Arithmeticke and Geo­metrie, might (as I conceived) apprehend the workes and practices taught therein. But being since certi­fyed that some few difficulties seeme, or indeed rather are feared, to be in the manner of the delivery of those Rules: I thought it would not bee impertinent, and a­lien from this present purpose, if in the very beginning I shall endeavour to explane such doubts, for the satis­faction of any that shall sticke thereat.

The scruples, which chiefely seeme to cause their difficultie, are these two: ‘First, that the parts or fra­ctions [Page 2] are not set downe with their Numerator and Denominator, as is usually done; but are conteined with the whole Numbers, as it were in one summe, with a small rectangular line only between them to se­parate the parts from the Integers. And secondly, most of the examples are not wrought at large, but the sum­marie and finall resolution thereof briefely intimated.’ The former of which two scruples ariseth from the ig­norance of the true nature and manner of Decimall fra­ctions: and the latter, from want of rightly conside­ring the Rules, whereby the valure of the number emergent or found out by proportion, and other A­rithmeticall operations, is estimated: which are those that are delivered in the second and fifth Chapters of the first part of that booke.

That wee may the better conceive the nature of De­cimall fractions, let us imagine a line either straight or circular, of any length, bee it a foot, or a yard, or one degree, or many; or else an houre, or a day, or any o­ther continuity. This being considered in it selfe in­tire and undivided is an Vnite or one whole thing of that kind, as one foot, one degree, one houre, &c. Then imagine that Vnite or whole to bee divided to 10 equall parts, that whole shall bee 10. Againe imagine every one of those tenth partes to bee sub­divided into 10: the whole shall bee 100, and each first division shall bee 10: and these second divi­sions shall bee hundreth partes. Thirdly, imagine every one of those hundreth partes subdivided in­to 10: the whole shall bee 1000, and each first divi­sion shall bee 100: and each second division shall bee 10: and these third divisions shall bee thousanth parts. And so proceeding in this Decimall subdivision, you may in your imagination divide the Vnite or whole into ten-thousanth parts, and hundred-thousanth parts, [Page 3] and millioneth parts, and so infinitely. And so that segment which in the first division was 10, 20, 30, &c. shall in the second division bee 100, 200, 300, &c. and in the third division 1000, 2000, 3000, &c. As for example, 3/30 is 30/100 or 300/1000 or 3000/10000: and 45/1000 is 450/1000 or 4500/10000: and 374/1000 is 3740/10000 or 37400/100000: &c.

Hence followeth that you may encrease the Nume­rator of any Decimall fraction by putting thereto as many cyphers or circles as you please without altering the quantity thereof, so that also you joyne so many cy­phers to the Denominator.

Now therfore a Decimall fraction is that which hath for his Denominator the figure 1 with one or moe circles after it, as 10, 100, 1000, &c. And seeing the use of the Denominator in a fraction is to shew into how many such parts the whole or Vnite is divided: if other­wise by any convenient signe the Denominator may easily and certainely bee knowne by the Numerator onely, it will bee a needelesse labour still to set it downe.

The most fit and convenient signe to know the De­nominator of a Decimall fraction is by a separating Line. For if the number mixed of integers and parts be writ­ten together in one ranke, with a small rectangular line drawne next after the Vnite place, cutting off the parts from the Integers: the number of figures or places in the parts so cut off shall shew how many cir­cles or cyphers are to bee set after 1 in the Denomina­tor. [Page 4] As for example, 3700⌊6 is all one with 37006/10 that is 3700 Vnits, and sixe tenth parts. Againe 370⌊06 is all one with 370 6/100, because after the separating line follow two figures 06. Likewise 37⌊006 is all one with 37 6/1000, because three figures 006 follow the separating line. Also 3⌊7006 is all one with 3 7006/10000. And 0⌊37006 is all one with 37006/100000, that is no unite at all, but that fraction only. And 0⌊037006 is all one with 37006/1000000, because after the separating line are sixe places of figurss 037006. By all which diversities of placing the separating line it is apparant that the num­ber of circles in the Denominator of any Decimall fraction must bee equall to the number of places of figures following the separating line.

Wherfore though there be no Vnite, but that it be a pure fraction, yet it will be convenient to note the Vnite place with a circle before the separating Line; that so the value of the fraction, through the number of places therein may more plainely appeare.

And besides that the setting of Decimall parts thus in one line with the Integers, hath more concinnity and neatnesse with it, then either with a Denominator, or by noting (as some have done) with small figures the primes, seconds, thirds, and the rest. These fra­ctions both mixt and pure are ready without any fur­ther reduction, for any Arithmeticall operation.

For in Addition and Subduction, the numbers gi­ven, being fitted together by their separating lines, having the like places or degrees set under one ano­ther, each in their owne file, may be added or subdu­cted in the very same manner as if they were all whole numbers.

And in Multiplication the numbers given being mul­tiplyed [Page 5] one by the other, according to the usuall man­ner of whole numbers, the product found cut shall have so many places of parts, as are in both the num­bers multiplyed.

And in Division the ordinary manner of whole num­bers is to bee used; onely remembring that every fi­gure of the Quotient shall be of that degree, whereof that figure of the Dividend is, under which the Vnite place standeth in the finding our of it, is.

Thus have I with as much plainenesse and brevitie as possibly I could cleered the first scruple, by shewing the true reason of Decimall fractions.

The second conceived difficultie is for not setting downe at large the operation of most of the Examples, but onely of some few here and there.

It is true that in every worke I doe not say (as some have done) bring that hither, or remove this thither: But having first taught the manner of working propor­tions upon the Instrument, and also delivered proper rules for particular questions, and wrought at the full summe of the hardest, I would not in every Example shew the like punctuallnesse, that neither I might blunt the edge and industrie of the ingenious Practicer with too much easinesse, nor the Booke grow to an enormous bulke and greatnesse.

That therefore the studious Reader may not need such verbosity and taedious instructions, he is to be ad­vised oftentimes (and that attentively) to peruse the first chapter of the first part, where the description and use of the severall circles are declared: and also the second Chapter concerning the working of proportions, [Page 6] and of Multiplication and Division: and therein those foure Considerations, or Rules for finding out the true value of the fourth or emergent number sought for: And thereto the fifth Chapter of the quadrating and Cu­bing of numbers. For in assigning a true quantitie un­to the Emergent number lyeth the greatest difficultie of this operation, especially if the worke bee in the fourth Circle.

In Signes and Tangents it is not altogether so hard, because all the revolutions or circuits of both are actu­ally set downe in severall Circles.

The Signes have two Circles, which in this new ad­ditament for Navigation are these; The tenth Circle from about 35 minutes, unto 6 Degrees; and the First from 6 Degrees, to 90, the end of the Quadrant.

The Tangents have foure Circles: namely the Ninth from about 35 minutes to 6 Degrees. The Second from 6 Degrees to 45. The Third from 45 Degrees to 84. And the Eighth from 84 Degrees till about 89 Degr. and 25 minutes.

But the fourth Circle being actually but one, doth potentially containe all Degrees and places both of In­tegers and Decimall parts. For the nine figures written in the spaces may signify unites, or tennes, or hun­dreds, &c. or else tenth parts, or hundreth parts, or thousanth parts, &c.

If any number be to be constituted upon the fourth Circle of the Instrument, take evermore one of those nine figures in the spaces for the first significant figure of that number: and among the subdivisions thereof reckon the true poynt or place of the number proposed. [Page 7] As if 2 were proposed: seeke the figure 2 in the spaces, & upon that line set one arme of the Index. Again if 375 be proposed: seeke the first figure 3 in the spaces: and in the subdivisions from 3 towards 4 account 75: and at the end thereof set one arme of the Index. Likewise if 0⌊092 bee proposed: because the two Circles are not significant, seeke the figure 9 in the spaces: and in the greater divisions thereof from 9 towards 1 account 2, and there set one arme of the Index.

If any ratio bee proposed to bee taken on the Instru­ment: set the two armes of the Index upon the two termes of the ratio found out, as was even now taught. Then consider the distance or arch betweene those two termes, counting from the place of the Antecedent to the place of the consequent forward, or according to the order of the figures, if the antecedent terme bee lesse then the consequent: Or else backward, contra­ry to the order of the figures, if the antecedent bee greater.

This distance or arch betweene the places of the two termes in the Instrument (which is also the aperture of the armes of the Index) I may fitly call the Instru­mentall difference: but it is not evermore the reall or true difference: which also is most needefull to bee knowne. The rules whereof are these three.

First, if either the numbers given be of the same de­gree: Or if they differ but one degree, and the line of the Radius fall betweene the places of the two termes in the Instrument: the Instrumentall difference shall also be the true and reall.

Secondly, if the numbers given bee not of the same degree, and the line of the Radius fall not betweene [Page 8] the places of the two termes in the Instrument: looke how many degrees the numbers differ one from the other, so many whole circuits of the fourth Circle shall bee added to the Instrumentall difference to make the reall or true difference.

Thirdly, if the numbers given be not of the same de­gree, and the line of the Radius doth fall betweene the places of the two termes in the Instrument: looke how many degrees the numbers differ, so many whole circuits, wanting one, of the fourth Circle shall be added to the Instrumentall difference to make the reall.

As in example: If the ratio of 375 to 2 be proposed: the same being taken upon the Instrument; the true diffe­rence betweene them, over and above the arch or angle of aperture, shall bee two whole circuits, by the second rule. And if the ratio of 375 to 0⌊092 be proposed: the same being taken upon the Instrument; the true diffe­rence betweene them, over and above the arch or angle of aperture, shall, by the third rule bee but three whole circuits, (although the termes differ foure degrees) be­cause the line of the Radius falleth within that arch, rec­koning it from the antecedent arme to the consequent backward.

Againe, the antecedent terme of any ratio being gi­ven, together with the reall or true difference (that is both the due aperture of the Index, and also the num­ber of circuits) betweene the termes, and whether of the two bee the greater: it is also needefull to know how to estimate the consequent terme. The rules where­of are these two.

Fourthly, if the true difference bee lesse then one cir­cuit, and the line of the Radius fall not betweene the [Page 9] places of the two termes; the numbers are both of the same degree. But if the line of the Radius fall between them they differ one degree.

Fiftly, if the true difference containe one or more cir­cuits, and the line of the Radius fall not betweene the places of the two termes; the numbers differ so many degrees as there are whole circuits. But if the line of the Radius fall betweene them they differ one degree more then there are whole circuits.

As in example: If the ratio of 375 to 2 be proposed: and also another antecedent 0⌊092: unto which a pro­portionall consequent is required to be sought. Because the true difference of 0⌊092 unto his consequent in the In­strument is equall to the true difference of 375 to 2, that is two whole circuits more then the aperture: and the antecedent 0⌊092 is greater then the consequent sought for: set the antecedent arme of the Index up­on 0⌊092, and the consequent arme reckoning backe­ward, at the same aperture, will cut 49+. But of what vallue or degree this fourth number is, is yet uncertaine. Now forasmuch as the reall difference betweene the termes of the ratio proposed is two whole circles above the aperture, as was shewed in the former example after the third rule; And in this present position of the In­dex the line of the Radius falleth not between the armes: the difference of degrees shall also bee two, by the fifth rule. Wherefore the first figure of 49+ shall bee two whole degrees backeward from the first significant fi­gure of 0⌊092 that is 0⌊00049+ (viz) somewhat better then 49 hundred thousand parts.

Againe, if the ratio of 375 to 0⌊092 be proposed: and also another antecedent 2, unto which a proportionall consequent is required to bee sought. Because the true [Page 10] difference of 2 unto his consequent in the Instrument, is equall to the true difference of 375 to 0⌊092: and the antecedent 2 is greater then the consequent sought for. Set the antecedent arme of the Index upon 2, and the consequent arme reckoning backward at the same aper­ture will cut 494 as before. Now forasmuch as the re­all difference betweene the termes proposed is three whole circuits above the aperture of the Index, as was shewed in the latter example after the third rule. And in this present position of the Index the line of the Ra­dius falleth betweene the armes the difference of de­grees shall be one more then three, that is foure by the fifth rule; wherefore the first figure of 494 shall bee foure whole degrees backward from 2, that is 0⌊00049+. I will conclude this part, with a summary recapitula­tion of all the former rules into these two branches.

The termes of a ratio being proposed, to find the reall or true difference betweene their places in the fourth cir­cle of the Instrument. I. If either the numbers given be of the same degree: or else differ but one degree, the line of 1 falling betweene them: they differ lesse then a circuite. II. If the numbers bee not of the same degree: they differ so many whole ciruits as they doe degrees. But yet if the line of 1 fall betweene them: they differ one cir­cuit lesse.

The antecedent terme of a ratio being given, together with the reall or true difference of the termes in the Instru­ment: to find out the consequent terme. I. If there­all difference be lesse then one circuit, and the line of 1 fall not betweene the places of the two termes: the numbers are both of the same degree. But if the line of 1 fall be­tweene the places: they differ one degree. II. If the re­all difference containe one or moe circuits: the numbers differ so many degrees as there are whole circuits. But if [Page 11] the line of 1 fall betweene the places: they differ one de­gree more.

Thus have I with as much perspicuousnesse as I am able, explained the generall rules of working by this Instrument, which have beene delivered in the first, second, and fifth Chapters of the first part: and exem­plifyed the documents with as hard examples as any I could bethinke my selfe of. And now I suppose the solertious practizer will bee able easily to finde out a fourth proportionall unto any three numbers given, and certainely to estimate the value thereof: so that now he will not be troubled for want of working the Que­stions at large.

For the use of Navigation are added two circles, The second part of this Chapter. the sixth and the seventh: and a small alteration in the fifth. For the fifth circle is here divided also into 50 parts: and is conceived to have two circuits. The first circuit is unto 50: The second circuit from 50 unto 100. Wherefore the figures are doubly noted: on the neerer side of the long lines of tenth divisions are set 10, 20, 30, 40, 50, for the first circuit: And on the further side of those lines are set 60, 70, 80, 90, for the second circuit. And the ten subdivisions in every one of those 50 parts are the Decimall parts thereof.

The sixth and seventh circles are divided into degrees: and every degree into ten parts, containing 6 minutes, or rather 10 hundreth parts a piece, The sixth circle hath the degrees unto 44⌊5: and the seventh circle hath from 44⌊5 unto 70. And these degrees serve for so ma­ny severall Latitudes, or Elevations of the Pole.

The manner of using these circles is double. First, Two Latitudes being given in the same Hemisphere, that [Page 12] is both Northerne, or both Southerne, to find the summe of all the Secants betweene them. Set one arme of the Index upon one Latitude and the other arme upon the other; then remove the arme that stood upon the lesser Lati­tude unto the line of the Radius: and the other arme with the same opening, shall in the fifth circle give the number of Secants betweene the two Latitudes proposed. As if the number of Secants betweene these two heights of the Pole 48⌊3 and 56⌊7 bee desired. Set one arme of the Index upon 48⌊3 and the other arme upon 56⌊7: then remove that arme that stoood upon 48⌊3 unto the line of the Radius: and the other arme with the same opening, shall in the fifth circle give 13⌊853, the num­ber of Secants betweene the two Latitudes proposed.

Secondly, The summe of all the Secants between two Latitudes in the same Hemisphaere being given, together with one of the Latitudes, to find the other Latitude. Set one arme of the Index on the line of the Radius, and open the other arme unto the summe of Secants given (in the fifth circle): then remove the arme that stood on the line of the Radius to the Latitude given, if it be the lesser: or if the Latitude given be the greater, re­move that arme that stood at the end of the summe of the Secants, unto that greater Latitude: and the other arme at the same opening shall give the other Latitude. As if there be given 13⌊853 the summe of Secants from the Latitude of 48⌊3 to the Pole-ward: Set one of the armes of the Index on the line of the Radius, and the o­ther arme at 13⌊853 in the fifth circle. Then remove the arme that stood at the line of the Radius, unto the Latitude 48⌊3: and the other arme, at the same ope­ning shall point to 56⌊7 the degrees of the other Latitude sought for. Againe, if the same summe of Secants 13⌊853, with the greater Latitude 56⌊7 degrees, be given: set one of the armes of the Index on the line of the Radius, [Page 13] and the other arme at 13⌊853 in the fifth circle. Then re­move the arme that stood at 13⌊853, unto 56⌊7 deg: the greater Latitude, and the other arme, at the same opening shall cut 48⌊3 deg: which is the lesser Latitude sought for.

And if the two Latitudes be in the severall Hemisphaeres, that is one Northerne and the other Southerne, the manner of working differeth in effect but little from the former. As if the summe of the Secants betweene these two heights of the Pole, viz. 6⌊5 on the North side of the Aequino­ctiall, and 13⌊4 on the South side bee desired. Set one arme of the Index on the line of the Radius, and the o­ther arme on either of the Latitudes given, suppose on 6⌊5. Then bring that arme on 6⌊5 unto the line of the Radius: and where the other arme, at that opening, chanceth to light, there hold it fast: and open the arme that stan­deth on the line of the Radius, unto the other Latitude 13⌊4. Afterward bring the arme that stood on the for­mer Latitude 6⌊5 unto the line of the Radius, and the other arme, at the same opening, shall in the fifth circle cut 20⌊037, the summe of Secants sought for.

Lastly, the summe of all the Secants betweene two La­titudes, of which one is on the North side of the Aequino­ctiall, and the other on the South side, being given; toge­ther with one of the Latitudes, to find the other Latitude: As if the summe of the Secants be 20⌊037 and the Lati­tude degr: 6⌊5. Set one of the armes of the Index at the line of the radius: & open the other arme unto 20⌊037 in the fift circle: & keeping the same aperture, bring the arme that stood on the line of the radius unto the latitude 6⌊5: and the other arme shall shew 13⌊4, the other Latitude sought for.

Or else peradventure you may more easily find out the [Page 14] summe of the Secants betweene any two Latitudes given, thus: Set the edge of the Index upon one of the Lati­tudes: and looke what division it cutteth in the fifth circle: keepe it in minde. Againe, set the edge of the Index upon one of the Latitudes: and looke what di­vision it cutteth in the fifth circle: keepe that in mind also. These two numbers kept in mind are the summes of the Secants for the two Latitudes given: And are to be subducted one out of the other, if the Latitudes are both in the same Hemisphere: or else to be added together, if the Latitudes are in diverse Hemisphaeres.

Also in like manner, The summe of the Secants and one of the Latitudes being given, you may find out the o­ther Latitude thus: Set the edge of the index upon the Latitude given; and looke what division it cutteth in the fifth circle. To this number adde the summe of the Secants, if the lesser of the two Latitudes be given: Or else out of it subduct the summe of the Secants, if the greater of the two Latitudes be given. But if the two La­titudes are in the contrary Hemisphaeres, the number found in the fifth circle is to be subducted out of the summe of the Secants. And so shall you have the other Latitude.

CHAP. II. Of the Latitude, and Longitude of places in genetall: and of keeping the account of time at Sea.

THe care and skill of the perfect Sea-man is to guide the ship at sea unto any port that shal be desired: which cannot be done unlesse he bee able to find out in in what place the ship is at any time.

The place of the ship at sea is estimated and understood by comparing it with any knowne place: that is how much the same is situated from the place, where the Ship is, either toward the North or South, which is called the difference of Latitude: or else toward the East or West, which is called the difference of Longitude. For it being once knowne how farre any place upon the Globe of the earth is wide of the Aequinoctiall unto either Pole: and also how farre the Meridian of the same is distant from the Meridian of any knowne place: the true situation thereof is said to be had.

The Latitude or distance of the place wherein the Ship is from the Aequinoctiall (which is all one with the height of the Pole there) is taken by observation of the Meridionall altitude, either of the Sunne by day, or of any Starre by night: as is not unknowne to almost every common Mariner: Or also by the 47 proposition of the 12 Chapter of the first Part. And therefore be­ing so vulgarly knowne, and taught of most that write of Navigation, I shall not need to spend time about it: [Page 16] Especially my intent here being to teach the use of my In­strument only, in tracing the Ships course.

The Longitude of the place wherein the Ship is, that is the Easterly or Westerly distance of the Ship from the place whence the Voyage began, is the difficultie, and Master-piece of Nauticall science: Which hath set on worke the wits and inventions of many men, pro­ceeding therein on diverse grounds.

For some have laboured to find the reason thereof by the variation of the Magneticall needle, supposing cer­taine Poles or points, unto which the ends of the needle doth in all places exactly respect. But besides that the Meridian is difficultly to be had with sufficient precise­nesse, especially at Sea, where the chiefest use of Longitude is: the conceipt is only imaginary, without the warrant of any naturall principle.

Some considering the swifnesse of the motion of the Moone, which is every day above 13 degrees, have sup­posed that either by the true place of the Moone, to be ob­served by exact Instruments; or else by the moment of the Moones comming into the Meridian, the Longitude might bee obtained. But neither the true motion of the Moone is so exactly knowne, nor observation can at Sea bee so precisely made, that any certaine truth in so subtile a bu­sinesse may be argued thereby.

Some have thought to observe the Longitude with automata or artificiall motions of long continuance: but not without great errour and hallucination.

Some by Sand-glasses, or Waterglasses: but both ob­lioxious to the diverse alterations and temperatures of the aire and climate wherein they are, especially that of [Page 17] sand. The other by water is more probable: wherein I should, in my judgement, preferre some chymicall spirit or liquor: because it is not so subject to the impression of the aire. And that there should be three glasses used, one to runne, and two to receive successively: That which runneth to be open above, to poure in the liquor, and to let in the aire, that the issue of the water be not hindred for want of aire to supply the vacuitie: The receivers to be cylindricall, with markes set on the out­side distinguishing houres and parts: and that there bee two of them, that when the liquor is come to the just height, another may instantly bee substituted, without losse of any liquor or time. This manner of observing the time is, in my opinion, the most likely of any that I know in use to conduce to the attaining of the difference of Longitudes of places. For by this meanes the true time in the place where the account beganne being knowne; and the time by observation of the Sunne or some Starre in the place, whither the Ship is come, being found; the difference of those times resolved into de­grees of the Aequinoctiall will shew the difference of Longitude betweene the place of beginning the account, and the place where the Ship is, Eastward, if the excesse be of the time in the former place: or Westward, if the excesse be of the time in the present place of the Ship.

And in this manner of keeping the reckoning of Longi­tude it will bee expedient to make as frequent observati­ons as the serenity of the skye will permit: that there­by your account may the rather bee freed from such subreptious errours, which else will bee very in­cident.

This or any such way of keeping the time, which shall by experience bee found most certaine (untill it shall please God to open a more naturall and proper [Page 18] way for the discovery of Longitude) I would advise were carefully, and with a kind as it were of religious diligence practised in all, specially long voyages: and that in computing and tracing the course of the Ship by the Compasse and log-line, it also together with the Latitude observed be discreetly called into consultation.

CHAP. III. Of the Mariners Compasse, and Rumbes or points thereof: and of finding the circuit of the earth in miles.

THere be foure things therfore whereof a Sea-man should be most carefull & circumspect, that he may happily with prosperous suc­cesse and a good conscience performe his intended voyage: First the angle of in­clination with the Meridian, on which the Ship maketh her course: which angle is directed by the Compasse: and is commonly called the Rumbe or poynt of the Compasse. For the ordinary Mariners (by a rude and grosse division of the Horizon into 32 parts) observe 32 points, whereof foure are cardinall; other foure halfe points; eight are quarter points; and sixteene are by points. Others more curiously divide each point into foure parts making in all 128, which they denominate by a quarter, an halfe, and three quarters of a point. A point containeth degr: 11¼, that is degr: 11, min: 15, or degr: 11⌊25: and a quarter of a point therfore is degr: 2 13/16, that is degr: 2, min: 48⌊75, or degr: 2⌊8125. By the conti­nuall addition of which number this table of Rumbes en­suing is composed.

[Page 20] The second is the measure of the Ships way on the Rumbe or point, which is ordinarily reckoned in miles; supposing a mile on earth to answer to a minute of a de­gree; and that 60 miles on a great circle give the diffe­rence of one whole degree. But I rather reckon the way of the Ship in hundreth parts of a degree, and have framed my rules of Navigation thereto: because this hath a more easy and convenient calculation then that by sexa­gesme parts: and as I beleeve (for so I would have it) will hereafter grow into publike use. This measure or quantity of the Ships way is found by the Logg-line and mi­nute-glasse.

The other two are, The observation of Latitude as oft as it may be for the weather: and the keeping of time: Of both which I spake sufficient for my purpose in the former chapter. The two former, that is the Rumbe and way of the Ship, more properly fall within my present consideration. For these are the continuall companions and faithfull guides of the Sea-man, which must direct him still in shaping his course: unto these therefore hee must applie his studie, and acquaint himselfe most familiarly with them.

And first for his compasse he must be carefull or rather scrupulous that it be exactly made, and not bungled up, as those usually are, which are made for sale: but that they be framed by some skillfull and conscionable Arti­ficer. The manifold cautions which are fit to bee had therein, are very gravely advertised by that reverend Divine and learned Mathematician Master William Barlow in his Navigators supply neere the beginning.

And as he is in the making of his compasse to shew his care, so specially in the using thereof he must exercise all industrie and diligence, that the course be steered aright, [Page 21] and kept to the just point or Rumbe: and not to commit his owne and all his companies safety, and the good suc­cesse of the voyage to the negligence of a loose and idle Steeresman: whereby it cannot be but that the account of the Ship shall be much confounded, and made uncertaine.

Againe for measuring of the quantity of the ships way, It must first be knowne how many English feet of 12 inches to the foot, answer to one degree of [...] great circle upon the earth. For if this be enormously mistaken, it cannot bee that the computation of the Ships course shall agree with the observations: but must needes make a maine difference, to the amazement of the Sea-man, and the casting of the whole Ship and company into unforeseene dangers.

Now an English mile by statute is the length of 8 fur­longs: and every furlong is 40 perches: and a perch is feet 16½: so that by this reckoning a mile containeth 5280 feet in length: though it be usually taken, or rather mistaken, that 60 of such miles make a degree (which would bee very strange, that our English mile drawne from Barley cornes should so happily fall out to an­swer to one minute) yet the truth is that above 66 of our miles answer to a degree, as by the observations of the most diligent enquirers is found out: so that in voyding of every ten degrees above one degree is lost: which is a maine enormity. But of this enquirie it will not bee amisse from our purpose if we shall a little discourse.

Diverse wayes by diverse Artists have beene practised for finding out the true compasse of the earth: And I know not whether any have given full satisfaction therein: but either the grounds they have wrought on have beene un­certaine; or the distances of the places of observation too short; or the dilligence of the practiser to bee suspected. That way which is by the height of an hill, and a tangent [Page 22] line from thence to the superficies of the sea, is rather a phantasie, then a thing of actuall performance. For nei­ther the perpendicular h [...]ight of the hill above the levell of the water can with any certainty bee obtained: nor such a tangent line by reason of the refraction of the va­pours continually rising out of the sea can be estimated.

But it would for the performance hereof be an excel­lent worke, if the height of the Pole at two townes of this Land, distant North-ward one from the other some scores, or rather hundreds of miles, being with Instruments of suf­ficient magnitude by some learned Artists exactly observed: there were also imployed certaine skilfull Surveyors (such as are indeed lovers of art and truth) to take the true di­stances and positions from place to place betweene the said townes. Which survey I could wish were made with good plaine tables, and with the same scale, which should not be lesse then a foot by standard for 10 miles and that these measures of a foot according to a standard were all made in brasse by the same Workman: and their chaines exactly fitted thereto: and that the measure bee taken not along the High-wayes, but by side stations where Steeples and other places eminent and of note may bee seene. If the two townes of the observations were London and Edenborough, it would be precisenesse suffi­cient: nay if they were but London and Cambridge, it would yield a greater certainty then any that I know hath yet beene used. This I say were an excellent work, and worthy the heroicall magnificence of some great man: and yet not of any very chargeable performance: but it would bring a marveilous light and furtherance to Navigation and unto all Astronomie.

In the meane time till it shall please God to stirre up some truely noble spirit for the effecting thereof, I will make bold to propose away, which any ingenious stu­dent, whose sight both of his eyes and understanding is [Page 23] quicke and perspicacious, may himselfe privately with much facility practise: the reason whereof consisteth upon these three principles.

The I. is, that if with a levelling Instrument set up in any place parallel to the Horizon a man take a true levell unto another place: the visuall line by which he levelled, shall be a tangent to such an arch of a great circle on the earth, as is contained betweene the station and the marke: Because that the visuall line, together with the two lines imagined, out of the center of the earth, doe include a right angled Triangle▪ having the right angle at the levell.

The II. is, that if the same Instrument he set just even with the former mark, and you levell backward to the former sta­tion, this last visuall line shall overshoot the former place of the Instrument: and shall inclose a new and greater right-angled triangle, hauing the right angle at the second station.

The III. is, that the former of the two visuall lines shall cut this latter and greater right-angled triangle into two right angled triangles like to it self and one like to the other: by the 8 prop: of the 6 book of Euclide. As in the scheme, the center of the earth is C, the first place or station of the levelling instrument is A, and the visuall line thence is A B to the marke B, which is al­so the place of the instrumēt in the second station, from whence the visuall line back­ward is B D, over-reaching the first place A. Here are 3 like right-angled triangles, namely the greatest C B D, cut into two other C A B, and BAD, with the line AB. Wherfore A B. AD:: AC. AB: that is; as the distance

[Page 24] between the two Stations (for by reason of the vast great­nesse of the earth, and the exceeding small distance be­tweene the two stations in comparison thereof, the vi­suall line AB shall be the same with the ground line AE) is to the over-shooting of the second line of levell: so is the Radius to the tangent of the arch A E, intercepted be­tweene the two stations. The quantity of which arch be­ing sought out in the Canon of tangents, either in sexa­gesime or Decimall parts of a degree, say againe, As the same arch in sexagesime or Decimall parts of a degree is unto a degree in the like parts; so is the distance betweene the two stations in feet, to the number of feet answering to a degree upon the earth. As for example, suppose the distance between the two stations to be 528 feet, which is the tenth part of a mile: and that the second line of levell over-shooteth the former 138/10000 of a foote: or 0⌊0138, which you shall finde will bee neere about the matter. Say,

528 . 0⌊0138 :: 100000,00 . 2,61 : the tangent of the arch Min, 0⌊09+

Say againe, 0⌊09+ the number of feet answering to a degree upon the earth.

Thus have I set downe the rule, and illustrated it with an example. But in the practise (by reason of the weake­nesse of our sight, not able to discerne a thing distinctly at any great distance, we are constrained to take but short stations, whereby the over-shooting of the second line of levell above the first is but very small) there is re­quired great precisenesse. For the performance where­of it will not bee amisse to set downe some directions, both concerning the Instrument, place, and time.

The levelling Instrument to be used in this worke, I would not have to bee either with a channell for water; [Page 25] nor with sights. For the water, besides that it doth con­tinually exhale vapours, hath a certaine tenacity, whereby to avoyd any drynesse neere to it, it will rather collect it selfe, and stand in a heape, then mixe with its enemy: and contrariwise very gladly diffuseth it selfe in pursuit of any moysture. And as for Sights, if the sight-hole be ve­ry small, it hindreth our seeing: if any whit large, it ad­mitteth too many visiverayes; which dilating themselves cannot fixe on the true and individuall point of the object. But I would have it onely with a ledge, one inch thicke, and three inches broad: and so broad also I would have a blacke stroke to be in a square white board, for the marke to levell at, that having set the ledge of the Instrument by the plumbe-line parallel to the Horizon in one station, you standing aloofe off, and guiding your eye along the two edges of the ledge, and your companion at the other station raysing up or letting downe the marke-board, as you shall direct him, you may see the upper line of the blacke stroke levell with the upper edge, and the lower line levell with the lower edge.

The place for the tryall of this experiment, I would have to be a plaine field, wherein you are to have for your use ready measured out by the foot, directly East and West, such a distance, as you can discerne distinctly thereat: which to a good and perfect sight may be 1000 feet, or to an indifferent sight 528 feet, which is the tenth part of a mile. And at both ends of that distance (which are to be your stations) the ground to be handsomely plained and beaten, for the more exact setting up of your Instruments thereon.

The time for making your observation I would have about Midsummer, in a seasonable, constant, drye, and calme weather: when, having set up your levelling In­strument in the Easterne station, you may take your first [Page 26] levell about eleven a clocke in the forenoone. Which be­ing done, you may remove your Instrument to the Westerne station, and about one a clocke in the after­noone (when the Sun is gone so farre past the Meridian) take your backe levell.

These are the most necessary and accurate cautions that I can devise: and all little enough for so curious and sub­till an inquiry. I have also here set downe the formes of the levelling Instrument and of the marke.

CHAP. IIII. The manner how to measure the Ships way; or how many degrees, and parts of a degree, either centesimes, or sexagesimes, the Ship moveth in one houre; or in any space of time assigned. And also of certaine ne­cessary reductions.

WEe shall therefore come neere the matter if wee take miles 66¼, that is 349800 feete to answer to a degree upon the earth.

Now because the measure of the Ships motion or way is observed by the watch-glasse and Log-line: let us for brevity sake call the number of seconds (whereof there are 3600 in an houre) which the Watch-glasse run­neth, by the letter G: and the number of feet vered in the Logg-line while the glasse is running, by the letter F. Which grounds being thus layed, wee may find out a rule to know how many hundreth parts of a degree the Ship sayleth in one houre; after this manner.

Say G . F :: 3600 . 3600F / G: so many feet gone in an houre

Say againe 349800 . 100 :: 3600F / G . 360000F / 349800G: Or by reduction into parts having the Denominator one Vnite 1⌊092F / G: which are so many centesimes of a degree gone in an houre.

Hence ariseth this generall rule for Centesimes.

But to know how many minutes of a degree the Ship sayleth in one houre: Say againe 349800 . 60 :: 3600F / G . 216000F / 349100G: Or by reduction into parts having the Denominator one Vnite 0⌊6175F / G: which are so many sexagesimes of a degree gon in an houre.

Hence also ariseth this generall rule for sexagesimes.

These two numbers 1⌊029 and 0⌊6175 (or whether of them you meane to follow) being of most frequent, and indeed continuall use, it were fit to note in the fourth circle of your Instrument with some apparant marke: that you may not be still searching them out, when you have occasion to use either of them.

And after this very manner you may find a generall rule for any other number of feet contained in a degree upon earth, both for the Decimall parts of a degree, and also for the Sexagesimes wherein onely the third termes in every of the second proportions will bee changed.

[Page 29] Because the true finding out of the way, which the Ship maketh in an houre, estimated in the parts of a degree, is the maine ground and principle, by which the place of her being both for longitude and latitude is argued and compu­ted: I will set downe the practice thereof at large in two Examples: the first for centesimes of a degree: and the second for sexagesimes:

Example I. Suppose the Watch-glasse to containe 40 sec: and that in the running out thereof the Ship hath gone 175 feete by the Log-line. The rule is, As 40▪ to 175: so is 1⌊092, to the number of hundreth parts of a degree sought. Set therefore the antecedent arme of the Index on 40 in the fourth circle, taking the figu­red divisions 1, 2, 3, &c. for so many tens: and open the other arme unto 175, taking the same divisions for so many hundreds: the distance betweene the armes will be above halfe that circle. Then remove the antecedent arme unto the third terme 1⌊092, taking the same divisi­ons for so many unites: and the consequent arme shall point at 450, which shall be 4 centesimes and a halfe, or 45 thousanth parts of a degree, (viz) degr: 0⌊045, in the same circuit of that circle: because the distance from 40 to 175 out reacheth not the line of I. Wherefore the Ship at that swiftnesse shall goe in an houre degr: 0⌊045. Which in sexagesimes will be found to be Min: 2⌊7.

Example II. Suppose the same watch-glasse of 40 sec: and that in the running out thereof the Ship hath gone 512 feet. The rule is, As 40 is to 512: so is 0⌊6175, to the number of sexagesimes or minutes of a degree sought. Set therefore the antecedent arme at 40, and the other at 512: the distance betweene them exceedeth one whole circuit. Then remove the antecedent arme to the third terme 0⌊6175: and the consequent arme shall point out 7902: which because the distance exceeded [Page 30] one circuit shall bee Min: 7⌊902. Which in centesimes would have beene degr: 0⌊1317.

The proportion of the Ships sayling for one houre being thus given either in centesimes or sexagesimes of a de­gree: multiply the same by the whole time of the con­tinuance at the same swiftnesse reckoned in houres and Decimall parts of houres: and the product shall give the whole way the Ship hath made, either in degrees or mi­nutes accordingly. As for Example; If the Ship sayling after degr: 0⌊045 in an houre, continue so for Ho. 29, Min: 37, that is Ho: 29⌊617: Multiply 29⌊617 by 0⌊045, and the product shall bee degr: 1⌊333, the whole way that the Ship hath made. Or if the Ship for so long con­tinuance hath sayled after Min: 2⌊7 in an houre: Multi­ply 29⌊617 by 2⌊7 and you shall have Min: 7⌊9966, which being diuided by 60, will give degr: 1⌊333, as before.

Now follow certaine reductions, which are of frequent use. I. To convert degrees or houres into Minutes, is to multiply them by 60. And to convert them into seconds, is to multiy them by 3600. And contrariwise.

II. To reduce minutes into degrees or houres, is to di­vide the minutes by 60. And to reduce seconds into degrees or houres, is to divide them by 3600.

III. To convert minutes of degrees or houres into cen­tesimes or hundreth parts: Say, As 60, is to 100: so is the number of minutes, to the number of hundreth parts. And,

IIII. To reduce centesimes of degrees or houres into minutes: Say, As 100, is to 60: so is the number of cen­tesimes or hundreth parts, to the number of minutes.

CHAP. V. The division of sayling into circular and spirall. Two fundamentall theorems. Of sayling, by one of the foure Cardinall Rumbes: and certaine Questions belonging thereto.

THe motion of the Ship upon a Rumbe is ei­ther circular, or winding with a kind of spirall line. If the ship saile upon one of the foure cardinall points▪ it describeth a circle: which is either a great circle or lesser, according as the circle of the hea­vens is, under which it moveth. For if the Ship saileth directly North or South under some Meridian, or di­rectly East or West under the Aequinoctiall, it describeth by the motion thereof an arch of a great circle. But if it saile directly East or West wide of the Aequinoctiall on either side, it describeth a lesser circle, according as the parallel in the heavens is, under which it moveth.

All great circles are equall one to another, and have equall degrees: but the parallels are greater or lesser one then another; and consequently have greater or lesser degrees, as every one is neerer or farther distant from the Aequi­noctiall. And because in computing the motion of the ship we shall have continuall occasion to speake of de­grees both of the greater and lesser circles, let this be adver­tised, that as oft as I shall mention Iust Degrees, I under­stand the measure of so many degrees of a great circle; else speaking of lesser degrees, I call them proper degrees of such a parallel.

These two proportions following are the fundamen­tall [Page 32] Theoremes for the computation of the motion of the ship: and are therefore faithfully to bee imprinted in our me­mory. The second is but the converse of the first: and are so familiar, that they shall neede no demon­stration.

If a Ship saile under a Meridian, that is upon the North or South Rumbe, it varyeth not the longitude at all: but onely changeth the Latitude: and that just so much as the number of degrees it hath runne in that whole time amounteth unto, which number is to be ad­ded to the latitude of the place, where the account began, if you have sayled from the Aequinoctiall-ward towards either Pole: Or else to be subducted out of the latitude of that place, if you have sayled towards the Aequinoctiall.

Againe if the Ship sayle under the Aequinoctiall upon the very line it selfe Eastward or Westward: it varieth not the Latitude at all: but only changeth the Longitude: and that just so much as the number of degrees it hath runne in that whole time amounteth unto. Which num­ber is to be added to, or subducted from the longitude [Page 33] of the place wherein you beganne your account, accor­ding as you have sayled East or West.

And thirdly if the Ship sayle directly East or West un­der any parallel circle, that is upon the East or West Rumbe, be it in the Northerne or Southern Hemisphaere, it there also changeth not the Latitude at all, but only the Longitude: yet not according to the number of Iust De­grees it hath gone, as under the Aequinoctiall: but more then so many, according as the proportion is betweene that parallel and the Aequinoctiall. For the lesser every parallel is, the greater must needes bee the difference of the Longitude in sayling so many Iust Degrees under it.

CHAP. VI. Of the oblique Rumbes betweene the Meri­dian, and that of East and West; what they are, and how composed: of finding out certaine fundamentall Theoremes for oblique sayling.

THat circular sayling upon any one of those foure cardinall points, whether it bee a great circle, or a parallel, hath (as wee have seene) no great difficulty in under­standing or computing: so that you bee sure of the true measure of the ships way: because that therein either only the latitude, or only the longitude is altered.

But there is greater difficultie in oblique sayling when the Ship runneth upon some Rumbe between any of the foure cardinall points, making an oblique angle with the Meridi­an: because therein the ship continually changeth both latitude and longitude, And the difficulty is so much the greater by how much the voyage is more distant from the Aequinoctial towards either Pole: and upon a Rumb more remote from the Meridian. For neere the Aequi­noctiall, where the Meridians are almost parallel; and in those Rumbes which are neere the Meridian, where the longi [...]ude is but little altered; there is no such lubricity and propensenesse to erre.

In this kind of oblique sayling, the ship is so directed by the Compasse, and guided by the helme, that the line of the ships length is every where kept firmely in one and the same angle with the Meridian, according to the di­stance of that Rumbe from the North and South line. And because the Compasse is as it were a moveable Hori­zon: and the lines of direction thereupon are the inter­sections [Page 37] of Azumiths or verticle circles with the same Horizontall plaine, dividing it into so many parts, which are called Rumbes: it commeth to passe that in such ob­lique sayling towards the apparent pole, the place where­unto the Compasse leadeth is evermore betweene the parallel through the place wherein you are, and the Pole. Wherefore the line of the Ships oblique course is a helix or spirall line, approaching neerer and neerer to the Pole, but never falling into it. As in the Scheme, suppose the center of the circle P to be the Pole of the world; and all the concentric circles to be parallels described at equall distance one from the other; and the streight lines

out of the Pole, PAC, PEB, PID, POF, &c. intersecting those parallels in the points C, B, D, F, &c. to be Meridi­ans: [Page 38] so that all the segments of the Meridians CA, BE, DI, &c. be equall: and all the segments of the parallels KC, BA, DE, &c. bee of equall length, though not of equall degrees; every one of those arches containing foure or more degrees, according as every circle is grea­ter or lesse then another. Suppose also that the ship kee­ping a just North-East course describeth the crooked line CBDF, which therefore must needes be the North-East Rumbe: and in the continuation of it doth ap­proach unto the Pole neerer and neerer: but can never fall into the Pole: because it still keepeth the same di­stance upon the Compasse betweene the Meridian, and the parallel in which it is, and maketh with the Meridian an angle of 45 degrees.

These Helices or spirall lines (which are the oblique Rumbes) ought to consist of most minute and insensible, yea indivisible parts: for if they be any whit great, the ac­count of the Ships motion will be confounded, and car­ryed downe from the true place whither the Ship is gone, towards the Aequinoctiall: neither can you re­turne by the Rumbe you came. For imagine in the for­mer Scheme two Meridians PAC, and PBK, and that AB and CK be like segments of two parallels, so that ABCK shall bee a kind of sphaericall right-angled qua­drangle: draw therein diagoniall-wise the arch of a great circle CBL, in which the ship is supposed to have gone from C to B: first the outward angle PBL being (as may easily be demonstrated) greater then the inward angle A C B, sheweth that you are fallen from your Rumbe into another point; and had neede to beare up the Ship againe into the Rumbe BD, making with the Meridian an angle P B D equall to that other A C B. Againe, the diagoniall arch CB cutteth the quadrangle into two triangles unequall one to the other: for though in both the sides AC and BK (which we will call the [Page 39] catheti) be equall, and the hypotenusa CB be the same: yet the bases AB & CK, and likewise their angles, are unequall: yea though the distance of the parallels AC and BK be but one scruple of a degree. But yet the lesse you take the distance of the parallels, that inequality will also bee the lesse. So that if by any artifice it may bee brought about that the arch AC be not one minute of a degree, which on the face of the earth answereth to a­bove an English mile, but the hundred-thousanth, or if need bee the millioneth part of a minute, scarce excee­ding one fifteenth part of an inch (which thing by the helpe of God the giver of all light I have discovered, and am able to performe in tables unto the Radius 10000000, yet nothing at all differing either in their forme or manner of working from those that are now commonly in use) all that inequality will be taken away, and those most small triangles will indeed, and unto all use, become plaine rectangled triangles: and the spirall line of the ships course be recalled to a precise exactnesse. By what artifice this is done, together with other secrets of that nature, I may peradventure hereafter be induced to declare; if so be I shall first see the practisers of this most noble and usefull science (which is as it were the band and tye of most disjunct countries, and the consociation of nations farthest remote) willingly to relinquish their inveterate errours, and to use thankfully and conscionably, without envy and selfe-conceited stubbornenesse, such light and helpes as the due and mature studie of true art shall afford.

In the meane time we will here make use of the or­dinary canon of the Meridian divided according to Mer­cator: which I have therefore set upon the sixth and seventh Circles of this Instrument, unto 70 degrees: as hath beene before shewed in the second part of the first Chapter.

[Page 40] And first out of the inspection of the Rumb in the last diagram compacted of the hypotenusae of an infinite num­ber of those minute rightangled triangles, I wil in certain Theoremes demonstrate the ground of oblique sayling: And then in the next Chapter apply the same foundati­ons to the answering of the severall questions in Navi­gation.

And because those triangles are all supposed to bee equall (or rather the same triangle so often multiplyed) let them be also noted with the same letters A B C, as the lowest of them is: the catheti C A being all on the Meridians: and the bases B A being all on the severall parallels: and the hypotenusae C B are the mo­tion of the ship upon the Rumbe.

The Theoremes are set downe in these proportions.

CHAP. VII. Of the severall questions which are incident unto oblique sayling.

IT is needfull to bee advertised: First, that in working the questions following upon the Instrument, the degrees of the ships way (found out by Chapt: IIII.) and of the differences both of latitude and longitude, and also the summe of the secants of parallels, are all to be taken on the fourth circle, after the manner of absolute num­bers: for which cause they are still to be set downe in degrees and Decimall parts of degrees. But the Sines and Tangents are to be accounted in their owne circles. That heereafter wee may not neede evermore to bee tel­ling unto what circle every number or terme doeth belong,

And secondly, that if you please to worke these questi­ons with your pen: you may doe it by the tables for the division of the Meridian line according to Mercator: Which tables are nothing else but a perpetuall addition of secants. And are to be found both in Master Wrights Errours of Navigation, and in Willibrordus Snellius his Tiphys Batavus, for every minute: and in Master Gun­ters Booke for every tenth part of a degree. Which last for more readinesse sake I doe herein make use of.

But in using the tables of Master Wright or Snellius, you must reckon the latitudes in degrees and minutes; with Decimalls of Minutes, and not in Decimalls of Degrees.

[Page 43] In the examples I have set downe the numbers so, that you may worke them either by the Instrument, or with the pen. The manner of working the Decimall parts with the penne you shall find in my Clavis Mathe­matica, Chapt. 1, 2, 3, 4, 6. But by the Instrument, in the first Chapter of this present tractate at large: which I could wish were dilligently studied and practised.

And now I come to the Questions.