AN ADDITION UNTO THE USE OF THE INSTRUMENT CALLED THE CIRCLES OF PROPORTION, For the Working of Nautical Questions. Together with certain necessary Considerations and Advertisements touching NAVIGATION. All which, as also the former Rules concerning this Instrument are to be wrought not only Instrumentally, but with the pen, by Arithmetic, and the Canon of Triangles. Hereunto is also annexed the excellent Use of two Rulers for Calculation. And is to follow after the 111 Page of the first Part. LONDON, Printed by AUGUSTINE MATHEWES. 1633. OF NAVIGATION. CHAP. I. Certain general 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 show the use of this Instrument, I was careful to do it with as much plainness and perspicuity, as might be in a subject not as yet obvious to vulgar knowledge, so that any one but moderately exercised in Arithmetic and Geometry, might (as I conceived) apprehend the works and practices taught therein. But being since certified that some few difficulties seem, or indeed rather are feared, to be in the manner of the delivery of those Rules: I thought it would not be impertinent, and alien from this present purpose, if in the very beginning I shall endeavour to explain such doubts, for the satisfaction of any that shall stick thereat. The scruples, which chiefly seem to cause their difficulty, are these two: First, that the parts or fractions are not set down with their Numerator and Denominator, as is usually done; but are contained with the whole Numbers, as it were in one sum, 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 summary and final resolution thereof briefly intimated. The former of which two scruples ariseth from the ignorance of the true nature and manner of decimal fractions: and the latter, from want of rightly considering the Rules, whereby the valour of the number emergent or found out by proportion, and other Arithmetical operations, is estimated: which are those that are delivered in the second and fifth Chapters of the first part of that book. That we may the better conceive the nature of decimal fractions, let us imagine a line either strait or circular, of any length, be it a foot, or a yard, or one degree, or many; or else an hour, or a day, or any other continuity. This being considered in itself entire and undivided is an Unite or one whole thing of that kind, as one foot, one degree, one hour, etc. Then imagine that Unite or whole to be divided to 10 equal parts, that whole shall be 10. Again imagine every one of those tenth parts to be subdivided into 10: the whole shall be 100, and each first division shall be 10: and these second divisions shall be hundreth parts. Thirdly, imagine every one of those hundreth parts subdivided into 10: the whole shall be 1000, and each first division shall be 100: and each second division shall be 10: and these third divisions shall be thousanth parts. And so proceeding in this decimal subdivision, you may in your imagination divide the Unite or whole into ten-thousanth parts, and hundred-thousanth parts, and millioneth parts, and so infinitely. And so that segment which in the first division was 10, 20, 30, etc. shall in the second division be 100, 200, 300, etc. and in the third division 1000, 2000, 3000, etc. 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: etc. Hence followeth that you may increase the Numerator of any decimal fraction by putting thereto as many cyphers or circles as you please without altering the quantity thereof, so that also you join so many cyphers to the Denominator. Now therefore a decimal fraction is that which hath for his Denominator the figure 1 with one or more circles after it, as 10, 100, 1000, etc. And seeing the use of the Denominator in a fraction is to show into how many such parts the whole or Unite is divided: if otherwise by any convenient sign the Denominator may easily and certainly be known by the Numerator only, it will be a needless labour still to set it down. The most fit and convenient sign to know the Denominator of a decimal fraction is by a separating Line. For if the number mixed of integers and parts be written together in one rank, with a small rectangular line drawn next after the Unite place, cutting off the parts from the Integers: the number of figures or places in the parts so cut off shall show how many circles or cyphers are to be set after 1 in the Denominator. As for example, 3700 ⌊ 6 is all one with 37006/10 that is 3700 Vnits, and six tenth parts. Again 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 six places of figurss 037006. By all which diversities of placing the separating line it is apparent that the number of circles in the Denominator of any decimal fraction must be equal to the number of places of figures following the separating line. Wherefore though there be no Unite, but that it be a pure fraction, yet it will be convenient to note the Unite place with a circle before the separating Line; that so the value of the fraction, through the number of places therein may more plainly appear. And besides that the setting of decimal parts thus in one line with the Integers, hath more concinnity and neatness 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 mixed and pure are ready without any further reduction, for any Arithmetical 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 own 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 multiplied one by the other, according to the usual manner of whole numbers, the product found cut shall have so many places of parts, as are in both the numbers multiplied. And in Division the ordinary manner of whole numbers is to be used; only remembering that every figure of the Quotient shall be of that degree, whereof that figure of the Dividend is, under which the Unite place standeth in the finding our of it, is. Thus have I with as much plainness and brevity as possibly I could cleared the first scruple, by showing the true reason of decimal fractions. The second conceived difficulty is for not setting down at large the operation of most of the Examples, but only of some few here and there. It is true that in every work I do 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 sum of the hardest, I would not in every Example show the like punctuallnesse, that neither I might blunt the edge and industry of the ingenious Practiser with too much easiness, nor the Book grow to an enormous bulk and greatness. That therefore the studious Reader may not need such verbosity and tedious 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 several circles are declared: and also the second Chapter concerning the working of proportions, and of Multiplication and Division: and therein those four 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 quantity unto the Emergent number lieth the greatest difficulty of this operation, especially if the work be in the fourth Circle. In Signs and Tangents it is not altogether so hard, because all the revolutions or circuits of both are actually set down in several Circles. The Signs 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 four 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 contain all Degrees and places both of Integers and decimal parts. For the nine figures written in the spaces may signify unites, or ten, or hundreds, etc. or else tenth parts, or hundreth parts, or thousanth parts, etc. 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 point or place of the number proposed. As if 2 were proposed: seek the figure 2 in the spaces, & upon that line set one arm of the Index. Again if 375 be proposed: seek the first figure 3 in the spaces: and in the subdivisions from 3 towards 4 account 75: and at the end thereof set one arm of the Index. Likewise if 0 ⌊ 092 bee proposed: because the two Circles are not significant, seek the figure 9 in the spaces: and in the greater divisions thereof from 9 towards 1 account 2, and there set one arm of the Index. If any ratio be proposed to be taken on the Instrument: set the two arms of the Index upon the two terms of the ratio found out, as was even now taught. Then consider the distance or arch between those two terms, 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 term be less than the consequent: Or else backward, contrary to the order of the figures, if the antecedent be greater. This distance or arch between the places of the two terms in the Instrument (which is also the aperture of the arms of the Index) I may fitly call the Instrumental difference: but it is not evermore the real or true difference: which also is most needful to be known. 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 between the places of the two terms in the Instrument: the Instrumental difference shall also be the true and real. Secondly, if the numbers given be not of the same degree, and the line of the Radius fall not between the places of the two terms in the Instrument: look how many degrees the numbers differ one from the other, so many whole circuits of the fourth Circle shall be added to the Instrumental difference to make the real or true difference. Thirdly, if the numbers given be not of the same degree, and the line of the Radius doth fall between the places of the two terms in the Instrument: look how many degrees the numbers differ, so many whole circuits, wanting one, of the fourth Circle shall be added to the Instrumental difference to make the real. As in example: If the ratio of 375 to 2 be proposed: the same being taken upon the Instrument; the true difference between them, over and above the arch or angle of aperture, shall be 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 between them, over and above the arch or angle of aperture, shall, by the third rule be but three whole circuits, (although the terms differ four degrees) because the line of the Radius falleth within that arch, reckoning it from the antecedent arm to the consequent backward. Again, the antecedent term of any ratio being given, together with the real or true difference (that is both the due aperture of the Index, and also the number of circuits) between the terms, and whether of the two be the greater: it is also needful to know how to estimate the consequent term. The rules whereof are these two. Fourthly, if the true difference be less than one circuit, and the line of the Radius fall not between the places of the two terms; 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 contain one or more circuits, and the line of the Radius fall not between the places of the two terms; the numbers differ so many degrees as there are whole circuits. But if the line of the Radius fall between them they differ one degree more than 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 proportional consequent is required to be sought. Because the true difference of 0 ⌊ 092 unto his consequent in the Instrument is equal to the true difference of 375 to 2, that is two whole circuits more than the aperture: and the antecedent 0 ⌊ 092 is greater than the consequent sought for: set the antecedent arm of the Index upon 0 ⌊ 092, and the consequent arm reckoning backward, at the same aperture, will cut 49+. But of what value or degree this fourth number is, is yet uncertain. Now forasmuch as the real difference between the terms of the ratio proposed is two whole circles above the aperture, as was showed 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 arms: the difference of degrees shall also be two, by the fifth rule. Wherefore the first figure of 49+ shall be two whole degrees backward from the first significant figure of 0 ⌊ 092 that is 0 ⌊ 00049+ (viz) somewhat better than 49 hundred thousand parts. Again, if the ratio of 375 to 0 ⌊ 092 be proposed: and also another antecedent 2, unto which a proportional consequent is required to be sought. Because the true difference of 2 unto his consequent in the Instrument, is equal to the true difference of 375 to 0 ⌊ 092: and the antecedent 2 is greater than the consequent sought for. Set the antecedent arm of the Index upon 2, and the consequent arm reckoning backward at the same aperture will cut 494 as before. Now forasmuch as the real difference between the terms proposed is three whole circuits above the aperture of the Index, as was showed in the latter example after the third rule. And in this present position of the Index the line of the Radius falleth between the arms the difference of degrees shall be one more than three, that is four by the fifth rule; wherefore the first figure of 494 shall be four 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 terms of a ratio being proposed, to find the real or true difference between 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 between them: they differ less than a circuit. II. If the numbers be not of the same degree: they differ so many whole ciruits as they do degrees. But yet if the line of 1 fall between them: they differ one circuit less. The antecedent term of a ratio being given, together with the real or true difference of the terms in the Instrument: to find out the consequent term. I. If thereall difference be less than one circuit, and the line of 1 fall not between the places of the two terms: the numbers are both of the same degree. But if the line of 1 fall between the places: they differ one degree. II. If the real difference contain one or more circuits: the numbers differ so many degrees as there are whole circuits. But if the line of 1 fall between the places: they differ one degree more. Thus have I with as much perspicuousnesse as I am able, explained the general rules of working by this Instrument, which have been delivered in the first, second, and fifth Chapters of the first part: and exemplifyed the documents with as hard examples as any I could bethink myself of. And now I suppose the solertious practizer will be able easily to find out a fourth proportional unto any three numbers given, and certainly 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 nearer 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 decimal 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 several Latitudes, or Elevations of the Pole. The manner of using these circles is double. First, Two Latitudes being given in the same Hemisphere, that is both Northern, or both Southern, to find the sum of all the Secants between them. Set one arm of the Index upon one Latitude and the other arm upon the other; then remove the arm that stood upon the lesser Latitude unto the line of the Radius: and the other arm with the same opening, shall in the fifth circle give the number of Secants between the two Latitudes proposed. As if the number of Secants between these two heights of the Pole 48 ⌊ 3 and 56 ⌊ 7 bee desired. Set one arm of the Index upon 48 ⌊ 3 and the other arm upon 56 ⌊ 7: then remove that arm that stoood upon 48 ⌊ 3 unto the line of the Radius: and the other arm with the same opening, shall in the fifth circle give 13 ⌊ 853, the number of Secants between the two Latitudes proposed. Secondly, The sum of all the Secants between two Latitudes in the same Hemisphere being given, together with one of the Latitudes, to find the other Latitude. Set one arm of the Index on the line of the Radius, and open the other arm unto the sum of Secants given (in the fifth circle): then remove the arm 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 arm that stood at the end of the sum of the Secants, unto that greater Latitude: and the other arm at the same opening shall give the other Latitude. As if there be given 13 ⌊ 853 the sum of Secants from the Latitude of 48 ⌊ 3 to the Pole-ward: Set one of the arms of the Index on the line of the Radius, and the other arm at 13 ⌊ 853 in the fifth circle. Then remove the arm that stood at the line of the Radius, unto the Latitude 48 ⌊ 3: and the other arm, at the same opening shall point to 56 ⌊ 7 the degrees of the other Latitude sought for. Again, if the same sum of Secants 13 ⌊ 853, with the greater Latitude 56 ⌊ 7 degrees, be given: set one of the arms of the Index on the line of the Radius, and the other arm at 13 ⌊ 853 in the fifth circle. Then remove the arm that stood at 13 ⌊ 853, unto 56 ⌊ 7 deg. the greater Latitude, and the other arm, at the same opening shall cut 48 ⌊ 3 deg. which is the lesser Latitude sought for. And if the two Latitudes be in the several Hemisphaeres, that is one Northern and the other Southern, the manner of working differeth in effect but little from the former. As if the sum of the Secants between these two heights of the Pole, viz. 6 ⌊ 5 on the North side of the Equinoctial, and 13 ⌊ 4 on the South side be desired. Set one arm of the Index on the line of the Radius, and the other arm on either of the Latitudes given, suppose on 6 ⌊ 5. Then bring that arm on 6 ⌊ 5 unto the line of the Radius: and where the other arm, at that opening, chanceth to light, there hold it fast: and open the arm that standeth on the line of the Radius, unto the other Latitude 13 ⌊ 4. Afterward bring the arm that stood on the former Latitude 6 ⌊ 5 unto the line of the Radius, and the other arm, at the same opening, shall in the fifth circle cut 20 ⌊ 037, the sum of Secants sought for. Lastly, the sum of all the Secants between two Latitudes, of which one is on the North side of the Equinoctial, and the other on the South side, being given; together with one of the Latitudes, to find the other Latitude: As if the sum of the Secants be 20 ⌊ 037 and the Latitude degr: 6 ⌊ 5. Set one of the arms of the Index at the line of the radius: & open the other arm unto 20 ⌊ 037 in the fift circle: & keeping the same aperture, bring the arm that stood on the line of the radius unto the latitude 6 ⌊ 5: and the other arm shall show 13 ⌊ 4, the other Latitude sought for. Or else peradventure you may more easily find out the sum of the Secants between any two Latitudes given, thus: Set the edge of the Index upon one of the Latitudes: and look what division it cutteth in the fifth circle: keep it in mind. Again, set the edge of the Index upon one of the Latitudes: and look what division it cutteth in the fifth circle: keep that in mind also. These two numbers kept in mind are the sums 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 divers Hemisphaeres. Also in like manner, The sum 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 look what division it cutteth in the fifth circle. To this number add the sum of the Secants, if the lesser of the two Latitudes be given: Or else out of it subduct the sum 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 sum of the Secants. And so shall you have the other Latitude. CHAP. II. Of the Latitude, and Longitude of places in genetal: and of keeping the account of time at Sea. THe care and skill of the perfect Seaman is to guide the ship at sea unto any port that shall be desired: which cannot be done unless he be 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 known 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 known how fare any place upon the Globe of the earth is wide of the Equinoctial unto either Pole: and also how fare the Meridian of the same is distant from the Meridian of any known place: the true situation thereof is said to be had. The Latitude or distance of the place wherein the Ship is from the Equinoctial (which is all one with the height of the Pole there) is taken by observation of the Meridional altitude, either of the Sun by day, or of any Star by night: as is not unknown to almost every common Mariner: Or also by the 47 proposition of the 12 Chapter of the first Part. And therefore being so vulgarly known, and taught of most that write of Navigation, I shall not need to spend time about it: 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 difficulty, and Masterpiece of Nautical science: Which hath set on work the wits and inventions of many men, proceeding therein on divers grounds. For some have laboured to find the reason thereof by the variation of the Magnetical needle, supposing certain 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 preciseness, especially at Sea, where the chiefest use of Longitude is: the conceit is only imaginary, without the warrant of any natural principle. Some considering the swifnesse of the motion of the Moon, which is every day above 13 degrees, have supposed that either by the true place of the Moon, to be observed by exact Instruments; or else by the moment of the Moons coming into the Meridian, the Longitude might be obtained. But neither the true motion of the Moon is so exactly known, nor observation can at Sea be so precisely made, that any certain truth in so subtle a business may be argued thereby. Some have thought to observe the Longitude with automata or artificial motions of long continuance: but not without great error and hallucination. Some by Sandglasses, or Waterglasses: but both oblioxious to the divers alterations and temperatures of the air and climate wherein they are, especially that of sand. The other by water is more probable: wherein I should, in my judgement, prefer some chemical spirit or liquor: because it is not so subject to the impression of the air. And that there should be three glasses used, one to run, and two to receive successively: That which runneth to be open above, to pour in the liquor, and to let in the air, that the issue of the water be not hindered for want of air to supply the vacuity: The receivers to be cylindrical, with marks set on the outside distinguishing hours and parts: and that there be two of them, that when the liquor is come to the just height, another may instantly be substituted, without loss 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 means the true time in the place where the account began being known; and the time by observation of the Sun or some Star in the place, whither the Ship is come, being found; the difference of those times resolved into degrees of the Equinoctial will show the difference of Longitude between the place of beginning the account, and the place where the Ship is, Eastward, if the excess be of the time in the former place: or Westward, if the excess be of the time in the present place of the Ship. And in this manner of keeping the reckoning of Longitude it will be expedient to make as frequent observations as the serenity of the sky will permit: that thereby your account may the rather be freed from such subreptious errors, which else will be very incident. This or any such way of keeping the time, which shall by experience be found most certain (until it shall please God to open a more natural and proper 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 Compass and log-line, it also together with the Latitude observed be discreetly called into consultation. CHAP. III. Of the Mariner's Compass, and Rumbes or points thereof: and of finding the circuit of the earth in miles. THere be four things therefore whereof a Seaman should be most careful & circumspect, that he may happily with prosperous success and a good conscience perform his intended voyage: First the angle of inclination with the Meridian, on which the Ship maketh her course: which angle is directed by the Compass: and is commonly called the Rumbe or point of the Compass. For the ordinary Mariners (by a rude and gross division of the Horizon into 32 parts) observe 32 points, whereof four are cardinal; other four half points; eight are quarter points; and sixteen are by points. Others more curiously divide each point into four parts making in all 128, which they denominate by a quarter, an half, 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 therefore is degr: 2 13/16, that is degr: 2, min: 48 ⌊ 75, or degr: 2 ⌊ 8125. By the continual addition of which number this table of Rumbes ensuing is composed. THE TABLE OF RUMBES. Rumbs. Rumbs. Grad. Gr. min. Rumbs. Rumbs. NORTH. The Meridian Line. SOUTH. 2 ⌊ 8125 2 48 ⌊ 75 5 ⌊ 625 5 37 ⌊ 5 8 ⌊ 4375 8 26 ⌊ 25 NbE NbW 11 ⌊ 25 11 15 SbW SbE 1 14 ⌊ 0625 14 3 ⌊ 75 16 ⌊ 875 16 52 ⌊ 5 19 ⌊ 6875 19 41 ⌊ 25 NNE NNW 22 ⌊ 5 22 30 SSW SSE 2 25 ⌊ 3125 25 18 ⌊ 75 28 ⌊ 125 28 7 ⌊ 5 30 ⌊ 9375 30 56 ⌊ 25 NEbN NWbN 33 ⌊ 75 33 45 SWbS SEbS 3 36 ⌊ 5625 36 33 ⌊ 75 39 ⌊ 375 39 22 ⌊ 5 42 ⌊ 1875 42 11 ⌊ 25 NE NW 45 45 00 sweet SE 4 47 ⌊ 8125 47 48 ⌊ 75 50 ⌊ 625 50 37 ⌊ 5 53 ⌊ 4375 53 26 ⌊ 25 NEbE NWbW 56 ⌊ 25 56 15 SWbW SEbE 5 59 ⌊ 0625 59 3 ⌊ 75 61 ⌊ 875 61 52 ⌊ 5 64 ⌊ 6875 64 41 ⌊ 25 EVEN WNW 67 ⌊ 5 67 30 WSW EASE 6 70 ⌊ 3125 70 18 ⌊ 75 73 ⌊ 125 73 7 ⌊ 5 75 ⌊ 9375 75 56 ⌊ 25 EbN WbN 78 ⌊ 75 78 45 WbS Ebbs 7 81 ⌊ 5625 81 33 ⌊ 75 84 ⌊ 375 84 22 ⌊ 5 87 ⌊ 1875 87 11 ⌊ 25 East West 90 90 00 West East 8 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 believe (for so I would have it) will hereafter grow into public 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 spoke 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 continual companions and faithful guides of the Seaman, which must direct him still in shaping his course: unto these therefore he must apply his study, and acquaint himself most familiarly with them. And first for his compass he must be careful 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 skilful and conscionable Artificer. The manifold cautions which are fit to be had therein, are very gravely advertised by that reverend Divine and learned Mathematician Master William Barlow in his Navigators supply near the beginning. And as he is in the making of his compass to show his care, so specially in the using thereof he must exercise all industry and diligence, that the course be steered aright, and kept to the just point or Rumbe: and not to commit his own and all his companies safety, and the good success of the voyage to the negligence of a lose and idle Steeresman: whereby it cannot be but that the account of the Ship shall be much confounded, and made uncertain. Again for measuring of the quantity of the ships way, It must first be known 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 be that the computation of the Ships course shall agree with the observations: but must needs make a main difference, to the amazement of the Seaman, 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 be very strange, that our English mile drawn from Barley corns 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 voiding of every ten degrees above one degree is lost: which is a main enormity. But of this enquiry it will not be amiss from our purpose if we shall a little discourse. Divers ways by divers Artists have been practised for finding out the true compass of the earth: And I know not whether any have given full satisfaction therein: but either the grounds they have wrought on have been uncertain; or the distances of the places of observation too short; or the diligence of the practiser to be suspected. That way which is by the height of an hill, and a tangent line from thence to the superficies of the sea, is rather a fantasy, than a thing of actual performance. For neither the perpendicular h●ight of the hill above the level of the water can with any certainty be 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 work, if the height of the Pole at two towns of this Land, distant Northward 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 employed certain skilful Surveyors (such as are indeed lovers of art and truth) to take the true distances and positions from place to place between the said towns. Which survey I could wish were made with good plain tables, and with the same scale, which should not be less than a foot by standard for 10 miles and that these measures of a foot according to a standard were all made in brass by the same Workman: and their chains exactly fitted thereto: and that the measure be taken not along the Highways, but by side stations where Steeples and other places eminent and of note may be seen. If the two towns of the observations were London and Edinburgh, it would be preciseness sufficient: nay if they were but London and Cambridge, it would yield a greater certainty than any that I know hath yet been used. This I say were an excellent work, and worthy the heroical magnificence of some great man: and yet not of any very chargeable performance: but it would bring a marvellous light and furtherance to Navigation and unto all Astronomy. In the mean time till it shall please God to stir up some truly 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 quick and perspicacious, may himself 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 level unto another place: the visual line by which he leveled, shall be a tangent to such an arch of a great circle on the earth, as is contained between the station and the mark: Because that the visual line, together with the two lines imagined, out of the centre of the earth, do include a right angled Triangle▪ having the right angle at the level. The II. is, that if the same Instrument he set just even with the former mark, and you level backward to the former station, this last visual line shall overshoot the former place of the Instrument: and shall enclose a new and greater rightangled triangle, having the right angle at the second station. diagram illustrating the use of a leveling instrument for measuring distances 528. 0 ⌊ 0138:: 100000,00. 2,61: the tangent of the arch Min, 0 ⌊ 09+ Say again, 0 ⌊ 09+ the number of feet answering to a degree upon the earth. Thus have I set down the rule, and illustrated it with an example. But in the practice (by reason of the weakness of our sight, not able to discern a thing distinctly at any great distance, we are constrained to take but short stations, whereby the overshooting of the second line of level above the first is but very small) there is required great preciseness. For the performance whereof it will not be amiss to set down some directions, both concerning the Instrument, place, and time. The levelling Instrument to be used in this work, I would not have to be either with a channel for water; nor with sights. For the water, besides that it doth continually exhale vapours, hath a certain tenacity, whereby to avoid any dryness near to it, it will rather collect itself, and stand in a heap, then mix with its enemy: and contrariwise very gladly diffuseth itself in pursuit of any moisture. And as for Sights, if the sight-hole be very small, it hindereth our seeing: if any whit large, it admitteth too many visiverayes; which dilating themselves cannot fix on the true and individual point of the object. But I would have it only with a ledge, one inch thick, and three inches broad: and so broad also I would have a black stroke to be in a square white board, for the mark to level at, that having set the ledge of the Instrument by the plumbe-line parallel to the Horizon in one station, you standing aloof off, and guiding your eye along the two edges of the ledge, and your companion at the other station raising up or letting down the marke-board, as you shall direct him, you may see the upper line of the black stroke level with the upper edge, and the lower line level with the lower edge. The place for the trial of this experiment, I would have to be a plain 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 discern 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, dry, and calm weather: when, having set up your levelling Instrument in the Eastern station, you may take your first level about eleven a clock in the forenoon. Which being done, you may remove your Instrument to the Western station, and about one a clock in the afternoon (when the Sun is gone so fare past the Meridian) take your back level. These are the most necessary and accurate cautions that I can devise: and all little enough for so curious and subtle an inquiry. I have also here set down the forms of the levelling Instrument and of the mark. leveling instrument and mark CHAP. FOUR 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 hour; or in any space of time assigned. And also of certain necessary reductions. We shall therefore come near the matter if we take mile's 66¼, that is 349800 feet 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 hour) which the Watch-glasse runneth, by the letter G: and the number of feet vered in the Logg-line while the glass is running, by the letter F. Which grounds being thus laid, we may find out a rule to know how many hundreth parts of a degree the Ship saileth in one hour; after this manner. Say G. F:: 3600. 3600F/ G: so many feet gone in an hour Say again 349800. 100:: 3600F/ G. 360000F/349800G: Or by reduction into parts having the Denominator one Unite 1 ⌊ 092F/ G: which are so many centesimes of a degree gone in an hour. Hence ariseth this general rule for Centesimes. As the number of seconds in the Watch-glasse, is to the number of feet vered in the Log-line: So is 1 ⌊ 029, to the number of hundreth parts of a degree, which the Ship runneth in one whole hour. But to know how many minutes of a degree the Ship saileth in one hour: Say again 349800. 60:: 3600F/ G. 216000F/349100G: Or by reduction into parts having the Denominator one Unite 0 ⌊ 6175F/ G: which are so many sexagesimes of a degree gone in an hour. Hence also ariseth this general rule for sexagesimes. As the number of seconds in the watch-glasse, is to the number of feet vered in the Log-line: So is 0 ⌊ 6175, to the number of minutes of a degree sailed in one hour. These two numbers 1 ⌊ 029 and 0 ⌊ 6175 (or whether of them you mean to follow) being of most frequent, and indeed continual use, it were fit to note in the fourth circle of your Instrument with some apparent mark: 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 general rule for any other number of feet contained in a degree upon earth, both for the decimal parts of a degree, and also for the Sexagesimes wherein only the third terms in every of the second proportions will be changed. Because the true finding out of the way, which the Ship maketh in an hour, estimated in the parts of a degree, is the main ground and principle, by which the place of her being both for longitude and latitude is argued and computed: I will set down 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 contain 40 sec: and that in the running out thereof the Ship hath gone 175 feet 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 arm of the Index on 40 in the fourth circle, taking the figured divisions 1, 2, 3, etc. for so many ten: and open the other arm unto 175, taking the same divisions for so many hundreds: the distance between the arms will be above half that circle. Then remove the antecedent arm unto the third term 1 ⌊ 092, taking the same divisions for so many unites: and the consequent arm shall point at 450, which shall be 4 centesimes and a half, 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 swiftness shall go in an hour 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 arm at 40, and the other at 512: the distance between them exceedeth one whole circuit. Then remove the antecedent arm to the third term 0 ⌊ 6175: and the consequent arm shall point out 7902: which because the distance exceeded one circuit shall be Min: 7 ⌊ 902. Which in centesimes would have been degr: 0 ⌊ 1317. The proportion of the Ships sailing for one hour being thus given either in centesimes or sexagesimes of a degree: multiply the same by the whole time of the continuance at the same swiftness reckoned in hours and decimal parts of hours: and the product shall give the whole way the Ship hath made, either in degrees or minutes accordingly. As for Example; If the Ship sailing after degr: 0 ⌊ 045 in an hour, continue so for Ho. 29, Min: 37, that is Ho: 29 ⌊ 617: Multiply 29 ⌊ 617 by 0 ⌊ 045, and the product shall be degr: 1 ⌊ 333, the whole way that the Ship hath made. Or if the Ship for so long continuance hath sailed after Min: 2 ⌊ 7 in an hour: Multiply 29 ⌊ 617 by 2 ⌊ 7 and you shall have Min: 7 ⌊ 9966, which being divided by 60, will give degr: 1 ⌊ 333, as before. Now follow certain reductions, which are of frequent use. I. To convert degrees or hours 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 hours, is to divide the minutes by 60. And to reduce seconds into degrees or hours, is to divide them by 3600. III. To convert minutes of degrees or hours into centesimes or hundreth parts: Say, As 60, is to 100: so is the number of minutes, to the number of hundreth parts. And, FOUR To reduce centesimes of degrees or hours 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 sailing into circular and spiral. Two fundamental theorems. Of sailing, by one of the four Cardinal Rumbes: and certain Questions belonging thereto. THe motion of the Ship upon a Rumbe is either circular, or winding with a kind of spiral line. If the ship sail upon one of the four cardinal 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 Equinoctial, it describeth by the motion thereof an arch of a great circle. But if it sail directly East or West wide of the Equinoctial on either side, it describeth a lesser circle, according as the parallel in the heavens is, under which it moveth. All great circles are equal one to another, and have equal degrees: but the parallels are greater or lesser one than another; and consequently have greater or lesser degrees, as every one is nearer or farther distant from the Equinoctial. And because in computing the motion of the ship we shall have continual occasion to speak of degrees both of the greater and lesser circles, let this be advertised, that as oft as I shall mention Just 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 fundamental Theorems for the computation of the motion of the ship: and are therefore faithfully to be imprinted in our memory. The second is but the converse of the first: and are so familiar, that they shall need no demonstration. Theor. I. As the Radius, is to the sine of the compliment of the parallel: So is an arch of the Equinoctial in Just Degrees, to the number of Just Degrees contained in a like arch of the same parallel. Theor. II. As the sine of the compliment of the parallel, is to the Radius: Or As the Radius, is to the secant of the parallel: So is the number of Just Degrees contained in an arch of the same parallel, to a like arch of the Equinoctial. If a Ship sail under a Meridian, that is upon the North or South Rumbe, it varyeth not the longitude at all: but only changeth the Latitude: and that just so much as the number of degrees it hath run 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 sailed from the Aequinoctiall-ward towards either Pole: Or else to be subducted out of the latitude of that place, if you have sailed towards the Equinoctial. Again if the Ship sail under the Equinoctial upon the very line itself 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 run in that whole time amounteth unto. Which number is to be added to, or subducted from the longitude of the place wherein you began your account, according as you have sailed East or West. And thirdly if the Ship sail directly East or West under any parallel circle, that is upon the East or West Rumbe, be it in the Northern or Southern Hemisphere, it there also changeth not the Latitude at all, but only the Longitude: yet not according to the number of Just Degrees it hath gone, as under the Equinoctial: but more than so many, according as the proportion is between that parallel and the Equinoctial. For the lesser every parallel is, the greater must needs be the difference of the Longitude in sailing so many Just Degrees under it. Quest: I. By the way of a Ship upon a parallel being given in Just Degrees, to find how many degrees the Longitude is varied. This is done at one operation by Theor: I. As the sine of the compliment of the parallel, is to the Radius: So is the way of the ship upon that parallel in just degrees, to the degrees of the difference of longitude. An Example. A ship making her course upon the parallel distant from the Equinoctial degr: 51, min: 32, by the estimation of the way hath sailed 9 ⌊ 4 in Just degrees: how many proper degrees of that parallel hath she gone? The compliment of 51°, 32′ is 38°, 28′, the sine whereof is 62206. Say therefore. The difference of longitude sought is degr: 15 ⌊ 111+: Which arch so found is to be added to, or subducted from the longitude of the place where you began your account, according as you have sailed either East or West. Quest: II. How many English miles change one degree of longitude in going Eastward or Westward at the elevation of the Pole degr: 51, min: 32. It was supposed in the beginning of Chapt: IIII, that miles 66¼ do answer to one degree of a great circle upon the earth. The compliment of 51°, 32′ is 38°, 28. Say therefore by Theor: I. Wherefore miles 41 ⌊ 211 make a degree on the parallel 51°, 32. Keep this number 41 ⌊ 211 in mind for the resolving of the two questions following. Quest: III. There are two places having the same latitude of degr: 51, min: 32: and the difference of their longitudes is degr: 15 ⌊ 111+: How many miles are they distant by the parallel? First find out the number of miles answering to one degree in the parallel 51°, 32, by Quest: II. which you shall find 41 ⌊ 211. Then multiply the same by the degrees of the difference of longitude 15 ⌊ 111+: thus, 1. 41 ⌊ 211:: 15 ⌊ 111+. 622 ⌊ 74. Their distance is miles 622 ⌊ 74. Quest: FOUR There are two places having the same latitude of degr: 51, min: 32: and they are distant by the parallel miles 622 ⌊ 740: how many degrees are they distant in longitude? First find out the number of miles answering to one degree in the parallel 51°, 32′, by Quest: II. Which you shall find 41 ⌊ 211. Then by the same number found divide the sum of the miles given, that is 622 ⌊ 74: thus, 41 ⌊ 211. 1:: 622 ⌊ 74. 15 ⌊ 111+. The distance of longitude is degrees 15 ⌊ 111+. CHAP. VI Of the obliqne Rumbes between the Meridian, and that of East and West; what they are, and how composed: of finding out certain fundamental Theorems for obliqne sailing. THat circular sailing upon any one of those four cardinal points, whether it be a great circle, or a parallel, hath (as we have seen) no great difficulty in understanding or computing: so that you be 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 difficulty in obliqne sailing when the Ship runneth upon some Rumbe between any of the four cardinal points, making an obliqne 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 near the Equinoctial, where the Meridian's are almost parallel; and in those Rumbes which are near the Meridian, where the longitude is but little altered; there is no such lubricity and propenseness to err. schematic depiction of navigational method for oblique sailing These Helices' or spiral lines (which are the obliqne 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 carried down from the true place whither the Ship is gone, towards the Equinoctial: neither can you return by the Rumbe you came. For imagine in the former Scheme two Meridian's PAC, and PBK, and that AB and CK be like segments of two parallels, so that ABCK shall be a kind of spherical rightangled 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, showeth that you are fallen from your Rumbe into another point; and had need to bear up the Ship again into the Rumbe BD, making with the Meridian an angle P B D equal to that other A C B. Again, the diagoniall arch CB cutteth the quadrangle into two triangles unequal one to the other: for though in both the sides AC and BK (which we will call the catheti) be equal, and the hypotenusa CB be the same: yet the bases AB & CK, and likewise their angles, are unequal: yea though the distance of the parallels AC and BK be but one scruple of a degree. But yet the less you take the distance of the parallels, that inequality will also be the less. So that if by any artifice it may be 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 be the millioneth part of a minute, scarce exceeding one fifteenth part of an inch (which thing by the help of God the giver of all light I have discovered, and am able to perform in tables unto the Radius 10000000, yet nothing at all differing either in their form 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 plain rectangled triangles: and the spiral line of the ships course be recalled to a precise exactness. 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 useful science (which is as it were the band and tie of most disjunct countries, and the consociation of nations farthest remote) willingly to relinquish their inveterate errors, and to use thankfully and conscionably, without envy and self-conceited stubbornness, such light and helps as the due and mature study of true art shall afford. In the mean 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 been before showed in the second part of the first Chapter. 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 will in certain Theorems demonstrate the ground of obliqne sailing: And then in the next Chapter apply the same foundations to the answering of the several questions in Navigation. And because those triangles are all supposed to be equal (or rather the same triangle so often multiplied) 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 Meridian's: and the bases B A being all on the several parallels: and the hypotenusae C B are the motion of the ship upon the Rumbe. The Theorems are set down in these proportions. Theor: I. As the Radius, is to the sine of the compliment of the Rumbe: So is the way of the Ship in degrees upon that Rumb, between any two places on the earth, to the difference of latitude between those two places. For R. s co:: BC. CA:: many BC. so many CA And so conversely. Theor: II. As the Radius, is to the sine of the Rumbe from the Meridian: So is the way of the ship in degrees upon that rumb, etc. to the sum of the bases of all the triangles intercepted between the parallels of those two places. For R. sC:: BC. BASILIUS:: many BC. so many BASILIUS. Theor: III. As the Radius, is to the tangent of the Rumbe from the Meridian: So is the difference of latitude between any two places, to the sum of the bases of all the triangles intercepted, etc. For R. tC:: CA. BASILIUS:: many CA so many BASILIUS. Theor: IIII As the Radius, is to the sum of the secants of all the parallels between any two places upon the earth: So is the base of one of those triangles, to the difference of longitude between those two places. For by Theorem TWO, Chap: 5. R. sec: parall:: base AB. diff: of long: in base BASILIUS:: many sec: parall. diff: of long: in so many bases BASILIUS. Again because by the last Theorem R. sum: sec. parall:: BASILIUS. diff: long: in BASILIUS. and by Theor: III. R. tC:: CA. BASILIUS. and because that CA is but 1, be it sexagesime or centesime, etc. therefore by composition of those two proportions ariseth, Theor: V. As the quadrat of the Radius, is to the sum of the secants of all the parallels between any two places upon the earth: Or, As the Radius, is to the sum of the secants of all the parallels between any two places, divided by the Radius: So is the tangent of the Rumb from the Meridian, to the difference of longitude between those two places. CHAP. VII. Of the several questions which are incident unto obliqne sailing. IT is needful to be advertised: First, that in working the questions following upon the Instrument, the degrees of the ships way (found out by Chapt: FOUR) and of the differences both of latitude and longitude, and also the sum 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 down in degrees and decimal parts of degrees. But the Sins and Tangents are to be accounted in their own circles. That hereafter we may not need evermore to be telling unto what circle every number or term doth belong, And secondly, that if you please to work these questions with your pen: you may do it by the tables for the division of the Meridian line according to Mercator: Which tables are nothing else but a perpetual addition of secants. And are to be found both in Master wright's Errors of Navigation, and in Willibrordus Snellius his Tiphys Batavus, for every minute: and in Master Gunter's Book for every tenth part of a degree. Which last for more readiness sake I do herein make use of. But in using the tables of Master Wright or Snellius, you must reckon the latitudes in degrees and minutes; with decimals of Minutes, and not in decimals of Degrees. In the examples I have set down the numbers so, that you may work them either by the Instrument, or with the pen. The manner of working the decimal parts with the pen 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 diligently studied and practised. And now I come to the Questions. QVEST. I. By the Rumbe and way of the Ship given, to find the difference of latitude between two places. This is done at one operation by Theor: I. in the former Chapter. As the Radius, is to the sine of the compliment of the Rumbe: So is the way of the ship in degrees upon that rumb, between any two places on the earth, to the difference of latitude between those two places. An Example. A Ship beginning her course in the latitude of degr: 50 ⌊ 7, that is 50°, 42′, hath sailed on the NWbN Rumbe degr: 9 ⌊ 36: into what latitude is she come? Here the angle of inclination which the NWbN Rumbe maketh with the Meridian is (by Chap: III.) 33°, 45′: the compliment of which is 56°, 15′: and the sine thereof 83147. Say therefore, . the difference of latitude which being added to the Latitude 50 ⌊ 7 given (because the greater latitude is sought) giveth degrees 58 ⌊ 482. that is, 58°, 29′. But if the lesser latitude had been sought: the said difference should have been subducted out of the latitude given. And if the difference of latitude found (the Ship sailing toward the Equinoctial) chance to exceed the latitude given, subduct the latitude given out of the said difference found: and the remains shall be the second latitude, but in the contrary Hemisphere. For if the two latitudes be in the contrary hemisphaeres, the sum of both is the difference between them. QVEST. II. By the way of the Ship and the difference of latitude between two places given, to find the Rumbe leading from one place to the other. This is done also at one operation by the said Theorem I. As the way of the Ship in degrees upon the Rumbe sought between any two places, is to the difference of latitude between those two places: So is the Radius, to the sine of the compliment of the Rumbe sought. An Example. A Ship beginning her course at the latitude of degrees 50 ⌊ 7, that is 50°, 42′, hath sailed up to the latitude of degrees 58 ⌊ 482, that is 58°, 29′: in which space it hath gone degrees 9 ⌊ 36 upon one Rumb▪ What Rumb was it that she followed? Here the difference of latitude is degrees 7 ⌊ 782. say, the compliment of which arch found is 33°, 45′: which is the angle of inclination of the Ships course to the Meridian: and is (by Chapt: III.) the NWbN Rumbe. QVEST. III. By the Rumbe, and difference of latitude between two places, to find the quantity of the Ships way in degrees. This is done at one operation by the converse of the said Theor: I, As the sine of the compliment of the Rumbe, is to the Radius: So is the difference of latitude between any two places to the measure of the Ships way in degrees. An Example. A Ship beginning her course at the latitude of degr: 58 ⌊ 482, hath sailed upon the SEbS Rumbe unto the height of degr: 50 ⌊ 7: how many degrees hath she gone upon the Rumbe? Here the difference between the two latitudes given is degr: 7 ⌊ 782. And the angle of inclination of the SEbS Rumbe is 33°, 45′, by Chapt: III: the compliment of which is 56°, 15′: and the sine thereof 83147. Say therefore. s 56°, 15′. Rad:: 7 ⌊ 782. 9 ⌊ 36: which is the measure of the Ships way in degrees. QVEST. FOUR By the Rumbe, and difference of latitude, to find the difference of longitude. This is done at one operation by Theor: V, in the former Chapter. As the Radius, is to the tangent of the Rumbe: So is the sum of the secants of the parallels between any two places, divided by the Radius: as they are set down in the tables, to the difference of longitude between the Medians of those two places. An Example. A Ship beginning her course at the latitude of degr: 38 ⌊ 2, saileth upon the WbN Rumb, unto the latitude of degr: 50 ⌊ 5: how many degrees of longitude hath it varied in that course? Here the angle of inclination of the WbN Rumbe with the Meridian is 78°, 45′: the tangent whereof is 502734. And the sum of the secants for 50 ⌊ 5 is 58 ⌊ 691: and the sum of the secants for 38 ⌊ 2 is 41 ⌊ 392: the difference of which is 17 ⌊ 299, the sum of the secants of the parallels between those two latitudes: which else by the Instrument is found out by the second part of the first Chapter. Say therefore, Which is the difference of longitude between the Meridian's of the two places. But because this question is of excellent and very frequent use, it will not be amiss to set down at large the manner of working this Example upon the Instrument. Thus, Set one of the arms of the Index upon 38 ⌊ 2 in the sixth circle, and open the other arm unto 50 ⌊ 5 in the seventh circle, according as hath been taught in the second part of the first Chapter. Then move the arm of the Index, which stood on 38 ⌊ 2 to the line of the 1: and the other arm at the same opening shall in the fifth circle cut 17 ⌊ 299, the sum of the secants. Again set one of the arms of the Index upon the line of the Radius, and open the other arm unto the tangent of 78°, 45′. Then move the antecedent arm of the Index, which stood at the line of the Radius, unto 17 ⌊ 299 in the fourth circle: and the consequent arm shall in the same fourth circle cut 87 ⌊ 927, which are the degrees of the difference of longitude sought for. QVEST V By the latitude and longitude of any two places given, to find what Rumbe leadeth from the one place to the other. This is done at one operation by the same Theor: V. As the sum of the secants of the parallels between those two places, is to the difference of longitude between them: So is the Radius, to the tangent of the Rumbe sought. An Example. There are two places, the one having the latitude of degr: 50 ⌊ 5: and the other the latitude of 38 ⌊ 2. And the difference of longitude between their Meridian's is degr: 87 ⌊ 927. By what Rumbe shall a Ship sail from one place to the other? Here the sum of the secants of the parallels between the two latitudes given is 17 ⌊ 299, as was found out in the example of Quest. FOUR Say therefore, Which is the angle of the inclination of the Rumb leading between those two places, with the meridian: and is therefore (by the third Chapter) the WbN or Ebbs Rumb: if the latitudes be on the North side of the Equinoctial. QVEST. VI By the Rumbe, and difference of longigitude between two places, whereof one is given, to find the difference of their latitudes. This is done at one operation by the converse of Th. V. As the tangent of the Rumbe, is to the Radius: So is the difference of longitude between the Meridian's of those two places, to the sum of the secants of the parallels between those two places. An Example. A Ship beginning her course at the latitude of degr: 38 ⌊ 2, saileth upon the WbN Rumbe until it hath changed the longitude degr: 87 ⌊ 927: Into what latitude shall she then be come? Here the angle of inclination of the WbN Rumbe with the Meridian is 78°, 45′. Say therefore, t78°, 45′. Rad:: 87 ⌊ 927. 17 ⌊ 299: Which is the sum of the secants of the parallels between the latitude degr: 38 ⌊ 2 given, and the latitude of the place wherein the Ship is. Wherefore if unto the sum of the parallels for degr: 38 ⌊ 2 found out by the second part of Chap: I. namely 41 ⌊ 392, you add the fourth term found 17 ⌊ 299: the sum 58 ⌊ 691 shall be the sum of the parallels for the latitude sought: which by the said second part of Chap: I. you shall find to be degrees 50 ⌊ 5. QVEST. VII. By the Rumbe, and measure of the way of the Ship in degrees, to find the difference of longitude between two places, whereof one is given. This is done by two operations. The first is, By the Rumbe, and way of the Ship given, to find the difference of latitude: which is Quest. I. The second is, By the Rumbe, and difference of latitude given, to find the difference of longitude, which is Quest. FOUR An Example. A Ship beginning her course in the latitude of degr: 50 ⌊ 4 hath sailed upon the WNW rumb deg. 13 ⌊ 7: how much hath she changed the longitude? Here the angle of inclination of the WNW Rumbe with the Meridian is 67°, 30′: the compl: of which is 22°, 30′ the fine whereof is 38268. Say first by Quest. I. Which is the difference of latitude between the beginning and place where the Ship is. Now because the Ship sailing toward the Pole increaseth the latitude: Add degr: 5 ⌊ 24 to deg. 50 ⌊ 4 the latitude given: and the sum deg. 55 ⌊ 64 shall be the latitude of the place whither he Ship is come. Seek the sum of the secants of the parallels for both those places, by the second part of Chap: I, which will be found to be 58 ⌊ 534, and 67 ⌊ 259: the difference of which two numbers is 8 ⌊ 725, the sum of the secants between those parallels. Also the tangent of the Rumbe, (viz) of 67°, 30′; is 241421. Say therefore again by Quest. FOUR Which arch of degr: 21 ⌊ 064 is the difference of longitude sought for. QVEST. VIII. By the difference of latitude, and measure of the way of the Ship in degrees: to find the difference of longitude between two places, whereof one is given. This also done by two operations. The first is, By the difference of latitude, and the way, to find the Rumbe: which is Quest: II. The second is, By the Rumbe, and difference of latitude, to find the difference of longitude: which is Quest FOUR An Example. A Ship beginning her course in the latitude of degr: 55 ⌊ 64, hath sailed degr: 13 ⌊ 7 upon one and the same Rumbe, even unto the latitude of degr: 50 ⌊ 4: how many degrees of longitude hath she changed? Here the difference between the two latitudes given degr: 5 ⌊ 24 Say first by Quest. TWO, the compliment of which arch (viz) 67°, 30′, is the angle of the Rumbe: And the tangent thereof is 241421. Seek also the sum of the secants of the parallels for both those places, by the second part of Chap. I: which will be found to be 58 ⌊ 534, and by 67 ⌊ 259: the difference of which two numbers is 8 ⌊ 725, the sum of the, secants between the parallels. Say therefore again by Quest. IIII, Which arch of degr: 21 ⌊ 064 is the difference of longitude sought for. QVEST. IX. By the differences of latitude and longitude between two places given, to find the measure of the way of the Ship in degrees. This is also done by two operations. The first is, By the difference of latitude and longitude to find the Rumbe leading between those two places: which is Quest. V The second is, By the Rumb, and difference of latitude, to find the measure of the Ships way in degrees: which ss Quest: III. An Example. A Ship beginning her course in the latitude of degrees 50 ⌊ 4, saileth still following one and the same Rumbe until she cometh to the latitude of degr: 55 ⌊ 64: in which time she hath changed the longitude degr: 21 ⌊ 064: How many degrees hath the Ship gone upon that Rumbe? Here the sum of the secants of the parallels for both the places proposed, by the second part of Chap. I, will be found to be 58 ⌊ 534, and 67 ⌊ 259: the difference of which two numbers is 8 ⌊ 725, the sum of the secants of the parallels between those two latitudes. Say first by Quest. V, Which is the angle of inclination of the Rumbe, with the meridian: the compliment of which is deg. 22, min: 30: the sine whereof is 38268. And the difference between the two latitudes degr: 55 ⌊ 64, and degr: 50 ⌊ 4, is degr: 5 ⌊ 24. Say therefore again by Quest. III, Which is the measure of the Ships way in degrees. QVEST. X. By the Rumbe, and difference of longitude between two places, whereof one is given, to find the quantity of the way in degrees between those places. This is also done by two operations: The first is, By the Rumbe, and difference of longitude, to find out the difference of latitude: which is Quest. VI The second is, By the Rumbe, and difference of latitude, to find out the measure of the way of the Ship in degrees: which is Quest. III. An Example. A Ship beginning her course in the latitude of degr: 55 ⌊ 64, saileth upon the EASE Rumbe so long till it hath changed the longitude degr: 21 ⌊ 064: How many degrees hath the Ship gone upon that Rumbe? Here the angle of the EASE Rumbe with the Meridian is degr: 67, min: 30; the tangent whereof is 241421. Say first by Quest. VI t67°, 30′. Rad:: 21 ⌊ 064. 8 ⌊ 725: Which is the sum of the secants of the parallels between the latitude of degr: 55 ⌊ 64, and the other latitude sought. Now the sum of the secants of the parallels for the latitude of degr; 55 ⌊ 64 is 67 ⌊ 259, by the second part of Chap. I. Out of which number if you subduct 8 ⌊ 725 last found (because the course is towards the Equinoctial) the remains shall be 58 ⌊ 534, the sum of the secants of the parallels for the other latitude of degr: 50 ⌊ 4, by the same second part of Chapt. I. So that the difference of the latitudes is degr: 5 ⌊ 24. And the fine of 22°. 30′, the compliment of the EASE Rumbe is 38268. Say therefore again by Quest. III. Which is the quantity of the Ships way in degrees. QVEST. XI. By the way of the Ship and the difference of longitude between the Meridian's of any two places, whereof one is given, to find out the Rumbe leading from one place to the other. & QVEST. XII. By the way of the Ship, and the difference of longitude between the Meridian's of any two places, whereof one is given, to find out the difference of their latitudes: by which the other place may be had. These two Questions, as they are of little or no use in Navigation; so also they have no direct and immediate solution. But are performed after the manner of the rule of false position, by supposing reasonably either a Rumbe, or another latitude: and then according to Quest. VII, and Quest. VIII, to find the difference of longitudes: which if it chance to fall out to be the same that is given in the Question; you have your desire. If not: suppose the second time. And lastly by comparing of both errors argue the truth. These two Questions are not so material, that I should spend more time in setting down Examples thereof. I will leave that work to the studious practiser. QVEST. XIII. If it be required to know the distance upon the Rumb between any two places, the measure of the way being known in degrees. You may multiply that measure of the way in degrees by miles 66 ⌊ 25, which is the number of miles contained in one Just degree upon the earth, as was before assumed in Chapt. FOUR And thus have I shown the use of the Instrument in the solution of all nautical Questions: which thing I specially in this small tractate aimed at. Which if it shall give any light and satisfaction to such as are studious in that most noble and useful art, I have my desire: which indeed only is, that the society of mankind may be benefited, and God glorified, by every poor ability he hath been pleased to bestow upon me. I was also in part minded to have annexed hereunto certain problems, how by reasonable conjecture the course of the Ship may be most probably rectified, when the reckoning thereof by the Compass and way estimated, shall be found to disagree from the celestial observations: Wherein I should have occasion to speak of the currents or hidden motions of the Seas, how they are to be observed, and how to be considered of in computing the motion of the Ship: And also of the deflexion (or as I may call it, the bias) of the Ship bending and wheeling itself about continually to the one side; how, and what allowance may most reasonably be made for it. But because these do not properly belong to the Instrument, and are to me only in speculation (which by reason of my want of experience in Nautical affairs, I cannot so well direct and ordain for practice at sea) I will for this present praetermit, contenting myself with what hath been already delivered. And if the Masters of Ships and Pilots will take the pains in the journals of their voyages diligently and faithfully to set down in several columns, not only the Rumbe they go on, and the measure of the Ships way in degrees, and the observations of latitude, and variation of their compass; but also their conjectures and reasons of the correction they make of the aberrations they shall find, and the quality or condition of their Ship, and the diversities and seasons of the winds, and the secret motions or agitations of the Seas, when they begin, and how long they continue, how fare they extend, and with what inequality; and what else they shall observe at Sea worthy consideration, and will be pleased freely to communicate the same with Artists, such as are indeed skilful in the Mathematics, and lovers and inquirers of the truth: I doubt not but that there shall in convenient time be brought to light many necessary precepts, which may tend to the perfecting of navigation, and the help and safety of such, whose vocations do enforce them to commit their lives and estates in the vast and wide Ocean to the providence of God: to whom be all praise, honour, and glory: And this is The End. I have at the request of Master Elias Allen, given way that Master Gunter's Table of the division of the Meridian line after Mercator, should be here inserted, for the use of such as will take the pains to enter into a unmerary calculation of the former Problems. The other Tables of natural Sins and Tangents are every where to be had. A Table for the division M Gr Par M Gr Par M G Par M Gr Part M Gr Part 0 0 0 3 3 001 6 6 011 9 9 037 12 12 088 100 3 101 0 111 9 138 12 190 200 3 201 6 212 9 239 12 293 300 3 301 6 312 9 341 12 395 400 3 402 6 413 9 442 12 497 501 3 502 6 514 9 543 12 600 600 3 602 6 614 9 645 12 702 700 3 702 6 715 9 746 12 805 800 3 803 6 816 9 848 12 907 900 3 903 6 916 9 949 13 010 1 1 000 4 4 003 7 7 017 10 10 051 13 13 112 1 100 4 103 7 118 10 152 13 215 1 200 4 204 7 219 10 254 13 318 1 300 4 304 7 319 10 355 13 421 1 400 4 404 7 420 10 457 13 523 1 500 4 504 7 521 10 559 13 626 1 600 4 605 7 622 10 661 13 729 1 700 4 705 7 723 10 762 13 832 1 800 4 805 7 824 10 864 13 935 1 900 4 906 7 925 10 966 14 038 2 2 000 5 5 006 8 8 026 11 11 068 14 14 141 2 100 5 106 8 127 11 170 14 244 2 200 5 207 8 228 11 272 14 347 2 300 5 307 8 329 11 374 14 450 2 400 5 408 8 430 11 476 14 553 2 500 5 508 8 531 11 578 14 656 2 601 5 609 8 632 11 680 14 760 2 701 5 709 8 733 11 782 14 863 2 801 5 810 8 834 11 884 14 967 2 901 5 910 8 936 11 986 15 070 3 3 001 6 6 011 9 9 037 12 12 088 15 15 174 M Gr Par M Gr Par M Gr Par M Gr Part M Gr Part 15 15 174 18 18 303 21 21 486 24 24 734 27 28 058 15 277 18 408 21 593 24 844 28 171 15 381 18 513 21 701 24 953 28 283 15 485 18 619 21 808 25 063 28 396 15 588 18 724 21 915 25 173 28 508 15 692 18 830 21 023 25 282 28 621 15 796 18 939 22 130 25 392 28 734 15 900 19 041 22 238 25 502 28 847 16 004 19 146 22 345 25 613 28 959 16 107 19 251 22 453 25 723 29 072 16 16 211 19 19 356 22 22 561 25 25 833 28 29 186 16 316 19 463 22 669 25 943 29 299 16 420 19 569 22 777 26 054 29 413 16 524 19 675 22 885 26 164 29 526 16 628 19 781 22 993 26 275 29 640 16 732 19 887 23 101 26 386 29 753 16 836 19 993 23 210 26 497 29 867 16 941 20 100 23 318 26 608 29 981 17 045 20 206 23 427 26 719 30 095 17 150 20 312 23 535 26 83● 30 300 17 17 255 20 20 419 23 23 643 26 26 941 29 30 324 17 359 20 525 23 752 27 052 30 438 17 464 20 632 23 861 27 164 30 553 17 568 20 738 23 970 27 275 30 667 17 673 20 845 24 079 27 387 30 782 17 778 20 952 24 188 27 499 30 897 17 883 21 059 24 297 27 610 31 012 17 988 21 165 24 406 27 722 31 127 18 093 21 272 24 515 27 834 31 242 18 198 21 379 24 624 27 946 31 357 18 18 303 21 21 486 24 24 734 27 28 058 30 31 473 M Gr Part M Gr Part M Gr Part M Gr Part M Gr Part 30 31 473 33 34 992 36 38 633 39 42 415 42 46 362 31 588 35 111 38 757 42 544 46 496 31 704 35 231 38 880 42 673 46 631 31 820 35 350 39 004 42 802 46 766 31 936 35 470 39 129 42 931 46 902 32 052 35 590 39 253 43 061 47 0●7 32 168 35 710 39 377 43 191 47 173 32 284 35 830 39 502 43 320 47 309 32 400 35 950 39 627 43 451 47 445 32 517 36 071 39 752 43 581 47 581 31 32 633 34 36 191 37 39 877 40 43 711 43 47 718 32 750 36 312 40 002 43 842 47 855 32 867 36 433 40 128 43 973 47 992 32 984 36 554 40 253 44 104 48 129 33 101 36 675 40 379 44 235 48 267 33 218 36 796 40 505 44 366 48 404 33 336 36 917 40 631 44 498 48 542 33 453 37 039 40 757 44 630 48 681 33 571 37 161 40 884 44 762 48 819 33 688 37 283 41 011 44 894 48 958 32 33 806 35 37 405 38 41 137 41 45 026 44 49 097 33 924 37 527 41 264 45 159 49 236 34 042 37 649 41 392 45 292 49 375 34 161 37 771 41 519 45 425 49 515 34 279 37 894 41 646 45 558 49 655 34 397 38 017 41 774 45 691 49 795 34 516 38 140 41 902 45 825 49 935 34 635 38 263 42 030 45 959 50 076 34 754 38 386 42 158 46 093 50 217 34 873 38 509 42 287 46 227 50 358 33 34 992 36 38 633 39 42 415 42 46 362 45 50 499 M Gr Part M Gr Part M Gr Part M Gr Part M Gr Part 45 50 499 48 54 860 51 59 481 54 64 412 57 69 711 50 641 55 010 59 640 64 582 69 895 50 783 55 160 59 800 64 753 70 080 50 925 55 3●0 59 960 64 924 70 263 51 068 55 460 60 120 65 096 70 449 51 210 55 611 60 280 65 268 70 635 51 353 55 762 60 441 65 440 70 811 51 496 55 913 60 602 65 613 71 008 51 639 56 065 60 763 65 786 71 195 51 783 56 217 60 925 65 960 71 383 46 51 927 49 56 369 52 61 088 55 66 134 58 71 572 52 071 56 522 61 250 66 308 71 761 52 215 56 675 61 413 66 483 71 9●0 52 360 56 828 61 577 66 659 22 140 52 505 56 981 61 740 66 835 72 331 52 650 57 135 61 904 67 011 72 522 52 795 57 289 62 069 67 188 72 714 52 941 57 444 62 234 67 365 72 906 53 087 57 598 62 399 67 543 73 099 53 233 57 754 62 564 67 721 73 292 47 53 380 50 57 909 53 62 730 56 67 900 59 73 486 53 526 58 065 62 897 68 079 73 680 53 673 58 221 63 063 68 258 73 875 53 821 58 377 63 231 68 438 74 071 53 968 58 ●34 63 398 68 618 74 267 54 116 58 691 63 566 68 799 74 464 54 264 58 848 63 734 68 981 74 661 54 413 59 006 63 903 69 163 74 859 54 562 59 164 64 072 69 345 75 057 54 7●1 59 322 64 242 69 528 75 256 48 54 860 51 59 481 54 64 412 57 69 711 60 75 456 M Gr Part M Gr Part M Gr Part M Gr. Part M Gr. Part 60 75 456 63 81 749 66 88 725 69 96 575 72 105 579 75 656 81 970 88 971 96 854 105 904 75 857 82 191 89 219 97 135 106 230 76 059 82 413 89 467 97 418 106 558 76 261 82 635 89 716 97 701 106 888 76 464 82 860 89 967 97 986 107 220 76 667 83 084 90 218 98 272 107 553 76 871 83 310 90 470 98 560 107 888 77 076 83 536 90 723 98 849 108 226 77 281 83 763 90 978 99 139 108 565 61 77 487 64 83 990 67 91 232 70 99 431 73 108 906 77 694 84 219 91 489 99 724 109 249 77 901 84 448 91 746 100 018 109 594 78 109 84 678 92 005 100 314 109 941 78 317 84 909 92 264 100 612 110 290 78 526 85 141 92 525 100 910 110 641 78 736 85 374 92 787 101 211 110 994 78 947 85 607 93 050 101 513 111 349 79 158 85 842 93 314 101 816 111 707 79 370 86 077 93 579 102 121 112 066 62 79 583 65 86 313 68 93 846 71 102 427 74 112 428 79 796 86 550 94 113 102 735 112 792 80 010 86 788 94 382 103 044 113 158 80 225 87 027 94 652 103 356 113 526 80 441 87 267 94 923 103 668 113 897 80 657 87 508 95 195 103 983 114 270 80 874 87 749 95 468 104 229 114 645 81 091 87 992 95 743 104 616 115 023 81 310 88 235 96 019 104 936 115 403 81 529 88 480 96 296 105 257 115 786 63 81 749 66 88 725 69 96 575 72 105 579 75 116 171 M Gr. Part M Gr. Part M Gr. Part M Gr. Part M Gr. Part 75 116 171 78 129 075 81 145 650 84 168 947 87 208 705 116 559 129 558 146 292 169 912 210 649 116 949 130 045 146 942 170 893 212 668 117 342 130 536 147 600 171 891 214 745 117 737 131 031 148 265 172 907 216 909 118 135 131 530 148 937 173 941 219 158 118 536 132 034 149 618 174 994 221 498 118 639 132 542 150 307 176 067 223 938 119 345 133 055 151 003 177 160 226 486 119 755 133 572 151 709 178 275 229 153 76 120 166 79 134 094 82 152 423 85 179 411 88 231 950 120 581 134 620 153 147 180 569 234 891 121 000 135 151 153 878 181 752 237 991 121 420 135 687 154 620 182 960 241 268 121 843 136 228 155 372 184 194 244 744 122 270 136 775 156 132 185 454 248 445 122 700 137 326 156 903 186 743 252 402 123 133 137 883 157 685 188 062 256 652 123 570 138 445 158 478 189 411 261 243 124 009 139 012 159 281 190 793 266 235 77 124 452 80 139 585 83 160 096 86 192 210 89 271 705 124 898 140 164 160 922 193 661 277 753 125 348 140 748 161 761 195 151 284 517 125 801 141 339 162 612 196 680 292 191 126 258 141 936 163 475 198 251 301 058 126 718 142 538 164 352 199 867 311 563 127 182 143 147 165 242 201 529 324 455 127 649 143 763 166 146 203 240 341 166 128 121 144 385 167 065 205 005 365 039 128 596 145 014 167 999 206 825 408 011 78 129 075 81 145 650 84 168 947 87 208 705 90 Infinite. The Declaration of the two RULERS for Calculation. THe Rulers are so framed and composed, that they may not only be applied to the Calculation of Triangles, and the resolution of Arithmetical Quaestions': but that they may also very fitly serve for a Crossestaffe to take the height of the Sun, or any Star above the Horizon, and also their distances. In which regard I call the longer of the two Rulers the Staff, and the Shorter the transversary. And are in length one to the other almost as 3 to 2. The Rulers are just four square, with right angles: and equal in bigness: they are thus divided. The transversary at the upper end noted with the letters S, T, N, E, on the several sides, hath a pinnicide or sight: at the lower edge of which sight is the line of the Radius, or Unite line, where the divisions begin. On the left edge of one of the sides are set the Degrees from 0 to 33 degrees. And on the right edge of the same side is set the line of Sins from 90 to 1 degree. In the next side are set two lines of Tangents, that on the right edge goeth upward from 1 to 45 degrees: and that on the left edge goeth downward from 45 to 89 degrees. In the third side, on the right edge is set the line of Numbers, having these figures in descent, 1, 9, 8, 7, 6, 5, 4, 3, 2, 1, 9, 8, 7, etc. In the fourth side on the right edge is set the line of Equal parts: And on the left edge are divers chords for the dividing of Circles. The Staff at the further end of it hath a socket with a pinnicide or sight: at which beginneth the 30 degree, and so goeth on to 90 degrees at the end of the Staff next your eye: which degrees from 30 to 90 are set on the right edge of one of the sides of the Staff. Then applying your transversary to the Staff with the lower end set to 90, mark on the four sides of the Staff the line of the Radius or Unite: at which on every left edge must begin the single line of Sines, Tangents, and Numbers, the very same which were in the transversary (that of the Sins being on that side where the degrees are) only the line of Tangents, and numbers are continued beyond the line of the Radius, to the further end of the Staff. And on the fourth side of the Staff in the middle are double divisions: that on the right hand is a line of Equal parts to 100, reaching the whole length of the Staff: And on the left hand contiguous to the former, is the line of Latitudes or Elevations of the Pole unto 70 degrees marked with the letter L. The degrees both of the Staff, and transversary, and also of the Sins and Tangents may be divided into 6 parts which contain 10 minutes apiece: or rather into 10 parts containing 6 minutes apiece: for so they may serve also for Decimals. Thus have you on the two Rulers the very same lines which are in the Circles of Proportion: and whatsoever can be done by those Circles, may also as well be performed by the two Rulers: and the Rules which have been here formerly set down for that Instrument, may also be practised upon these: so that you be careful to observe in both the different propriety in working. It will not therefore be needful, to make any new and long discourse, concerning these Rulers, but only to show the manner, how they are to be used, for the calculation of any proportion given. In working a Proportion by the Rulers, hold the Transversary in your left hand, with the end at which the line of the Radius or Unite line is, from you ward: turning that side of the Ruler upward, on which the line of the kind of the first term is, whether it be Number, Sine, or Tangent: and therein seek both the first term, and the other which is homogene to it. Then take the Staff in your right hand with that side upward, in which the line of the kind of the fourth term sought for is: and seek in it the term homogene to the fourth. Apply this to the first term in the transversary: and the other homogene term shall in the Staff show the fourth term. As if you would multiply 355 by 48: Say 1. 355:: 48. 17040. For if in the line of Numbers on the Staff you reckon 355, and apply the same to 1 in the line of Numbers on the transversary; then shall 48 on the transversary show 17040 on the Staff. Again if you would divide 17040 by 48: Say 48. 1:: 17040. 355 For if in the line of Numbers on the transversary you reckon 48, and to the same apply 1 in the line of Numbers on the Staff: then shall 17040 on the transversary show 355 on the Staff. The true value of the fourth term found, may be had by the 5 & 6 sect: 2 chap: 1 part. Some Examples of working Proportions we will borrow out of 3 chap: 1 part Example I. If 54 elnes of Holland be sold for 96 shillings: for how many shillings shall 9 elnes be sold? the work shall be thus 54 el. 96 sh:: 9 el. 16 sh. for if in the line of Numbers on the transversary you seek the first term 54 elnes, and in that line on the Staff you seek 96 shillings: and apply one to the other: then shall 9 elnes sought out on the transversary point out 16 shillings on the Staff. Example FOUR There is a Tower whose height I would measure. I take two stations in the same right line from the Tower: and at either station having observed the height by the sights of the Staff, I find the nearer station 28 deg. 7 min: almost: and the further station 21 degr: 58 minutes almost: and between both the Stations the distance was 76 feet. The rule of measuring heights by two stations is contained in these Theorems. THEOR. As the difference of the Tangents of the arches cut in either Station, is to the distance between the stations: so is the Tangent of the lesser arch, to the nearer distance from the Tower. Again THEOR. As the Radius is to the Tangent of the greater arch; so is the nearer distance found, to the height. And therefore because according to 6 sect: 1 chap: 1 part, by application of the line of Numbers to the line of Tangents (that is by applying the Unite line of the Staff, to the Tangent in the transversary, if the arch be less than 45 degr: but if the arch exceed 45 degr: by applying the said Unite line, unto the arch itself, or the compliment thereof, which in the transversary is all one) the Tangent of 28°, 7′ is 5343, and the tangent of 21°, 58′ is 4033: whose difference is 1310: the Proportions will in the lines of Numbers be thus First, 1310. 76:: 4033. 234 wherefore 234 feet is the nearest distance Second, Radius. tang: 28°, 7′:: 234. 125 wherefore 125 feet is the height sought for. Or else you may resolve it at one operation thus, THEOR. As the difference of the tangents of the compliments of the arches cut in either station, is to the Radius; So is the distance between the stations, to the altitude. Because accordingly as was before showed, the tangent of the compliment of 28°, 7′ is 18715: and the tangent of the compliment of 21°, 58′ is 24792: whose difference is 6077. the proportions will in the line of Numbers be thus 6077. 10000:: 76. 125. And these Rules may be also applied to find out the distances of objects. Example V To find the declination of the Sun the ninth day of May. Because upon the ninth day of May the place of the Sun is in ♉ 29: which is 59 degrees distant from the next Equinoctial point. Say in the line of Sines Radius. sine 59°:: sine 23°, 30′. sine 19°, 59′. And so much is the declination sought for. If the distance of the Sun from the next Aequinoctial point exceed not degrees 2°, 30′. Break that arch into minutes, or decimal parts of a degree: and by the lines of Sines and numbers say As the Radius is to the Sine of 23°, 30′; So is the distance (of the Sun from the next Aequinoctial point) in minutes, or Decimals, to the declination in minutes, or Decimals. As if the declination of the Sun being in ♍ 27°, 45′ be required: the distance of it from the next Aequinoctial point is 2°, 15′ that is minutes 135, say therefore Rad. sine 23°, 30′:: 135′. 53 ⌊ 7′. which is the declination of the Sun in that place. Example VI To find the right ascension of the Sun upon the ninth of May. Because upon the ninth of May the Sun is 59 degr: distant from the next Equinoctial point: say in the line of Sins on the transversary, and the line of tangents on the Staff Rad. sin: compl: 23°, 30′:: tan: 59°. tan: 56°, 46′. which is the Sun's right ascension upon the same day. Or else (because the Radius is the mean proportional between the tangent of an arch and its compliment) the same proportion might have been thus set down. t: come: 59°. Rad:: sin: come: 23°, 30′. t: 56°, 46′, In which manner of proposure happening only when there is in the proportion the Radius and two tangents, because the two homogenes of the one kind are both extreme terms, and the two homogenes of the other kind are both middle terms: the tangent is to be turned into the tangent of the compliment: and must change places with the Radius. As by comparing the two former proportions doth plainly appear. Because that the greatest difficulty of working by these Rulers falleth out in the tangents, when the arches are in the second mediety of the Quadrant, it will be convenient to set down some cases wherein the work differeth from the ordinary manner. Case I. If the four proportionals being all tangents, the arches of two of the terms given exceed 45 deg. and the arch of the third be less than 45 degr: as in this Example : here the tangent of 31° on the transversary being applied to the tangent of 56° on the Staff, the tangent of 79° on the Staff will outreach the Radius or end of the transversary. Wherefore to find out the fourth proportional, mark what point of the Staff, the line of the Radius on the transversary doth touch, and to it (turning the transversary) set the other end of the line of the Radius, and so shall the tangent 79° in the Staff give you tangent 64°, 24′ in the transversary. Case II. If the four proportionals being all tangents, the arches of the three terms given exceed 45 deg. as in this Example . turn the transversary, and set the tangent of 56° therein to tangent 79° on the Staff: and because the Radius or end of the transversary reacheth not to the tangent 64°, 24′ on the Staff: to find out the fourth proportinal, mark what point of the Staff the line of the Radius of the transversary doth touch, and to it (turning the transversary) set the other end of the line of the Radius, and so the tangent 64°, 24′ in the Staff, will give you tangent 31° in the transversary. These two Cases, being nothing else but a supplying of the shortness of the transversary, may serve as a rule, and direction for all other works of the same kind. Concerning the manner of working by Quadrats and Cubes upon the line of Numbers. And of duplicated and triplicated proportions. The difference of a Quadrat from a Quadrat is double the difference between their sides. And the difference of a Cube from a Cube is triple the difference between their sides. Example TWO, chap: 6, part I. How many acres of Wood-land measured with a Perch of 18 feet, are there in 73 acres of Champane-land measured with a Perch of 16 ⌊ 5 feet? The measures given, 18, 16 ⌊ 5 being reduced into the least terms, are as 12 to 11, and the proportion is reciprocal. Say therefore, Q: 12, Q: 11:: 73. 61 ⌊ 34 Which is thus wrought: In the line of Numbers apply 11 on the Staff, to 12 on the transversary, then shall 73 on the transversary give 67— on the Staff: which 67— being reckoned on the transversary (the Rulers standing as they did) shall on the Staff give 61 ⌊ 34 the number of acres in Wood-land measure. Example III, chap 6, part 1. If pounds 0 ⌊ 43 of gunpowder suffice to charge a Gun whereof the concave Diameter is yaches 1 ⌊ 5: how many pounds of powder will suffice to charge a Gun whose concave Diameter is inches 7? The capacities are one to the other as the Cubes of the Diameters. And the proportion is direct. Say therefore C: 1 ⌊ 5. C: 7:: 0 ⌊ 43. 43 ⌊ 7 Which is thus wrought: In the lines of Numbers apply 7 on the Staff, unto 1 ⌊ 5 on the transversary, then shall 0 ⌊ 43 on the transversary give 2 ⌊ 01— on the Staff: which 2 ⌊ 01— being reckoned on the transversary (the Rulers standing as they did) shall on the Staff give 9 ⌊ 333: and again the same 9 ⌊ 333 being reckoned on the transversary shall on the Staff give 43 ⌊ 7 the quantity of pounds of powder sufficing. Example in 46 pag. of Navigation. A ship beginning her course at the Latitude of deg. 38 ⌊ 2, saileth upon the WbN Rumbe, unto the Latitude of deg. 50 ⌊ 5: how many degrees of Longitude hath it varied in that course? Here the angle of Inclination of the WbN Rumbe with the Meridian is 78°, 45′, the tangent whereof is 502734. And by the double divisions on the fourth side of the Staff, the Sum of the Secants for the Latitude of 50 ⌊ 5 is 58 ⌊ 691: and the sum of the Secants for the Latitude of 38 ⌊ 2 is 41 ⌊ 392: the difference of which is 17 ⌊ 299; the sum of the Secants of the parallels between those two Latitudes. Say therefore. Rad. tang: 78°, 45′:: 17 ⌊ 299. 86 ⌊ 068 which is the difference of Longitude between the Meridian's of the two places. But because this question is of excellent and very frequent use, it will not be amiss to set down at large the manner of working this Example upon the Rulers. Look the two Latitudes 50 ⌊ 5, and 38 ⌊ 2 given, in the line of Latitudes or elevations of the Pole on the fourth side of the Staff: and either mark what number each of them showeth in the line of Equal parts there, which you shall find to be 58 ⌊ 691, and 41 ⌊ 392, the difference of which is 17 ⌊ 299, as was before said: or else more easily, set one foot of your Compasses on one of the Latitudes given, and open the other foot to the other Latitude given: then keeping that aperture, set one of the feet in the beginning of that line of Equal parts, and the other foot shall upon the same line show the difference of Secants between the said two Latitudes given, that is 17 ⌊ 299. Then in working the Proportion; because the angle of inclination of the Rumbe 78°, 45′ is more than 45 deg. turn that edge of the transversary on which the tangents of arches above 45 are, toward the Staff in your right hand: and to the line of the Radius apply 17 ⌊ 299 sought out on the line of Numbers on the Staff: and so shall tang: 78°, 45′ on the transversary: show 86 ⌊ 97— on the Staff. The Use of the Crossestaffe. FOr the more ready use of the Crossestaffe, you are to remember that the degrees serving for the Crossestaffe are placed both on the Staff and transversary, on the same side on which the line of Sines is. And that in framing thereof the transversary is to be set in the Socket so that it may stand on the right hand of the Staff. The degrees on the transversary are only the first 30. and serve to show an angle not exceeding 30 degrees. Yet it would not be unuseful if both the Transversary and Staff were made somewhat longer that the transversary might contain 5 deg. after 30; and the Staff 5 degrees before 30. To find an angle less than 30 degrees▪ between any two objects. Place the Socket at 30 deg. on the Staff; and screw it fast there: then setting the end of the Staff to your eye, draw the transversary up and down through the Socket, till you may see with your eye the two objects upon the two sights of your Crossestaffe: and so shall the degrees cut on the transversary show you the angle of their distance, if it be not above 30 degrees. To find an angle greater than 30 degrees, between any two objects. Place the Socket at 30 degrees of the transversary, and screw it fast there: then setting the end of the Staff to your eye, draw the Socket up and down along the Staff till you may see with your eye the two objects upon the two sights of your Crossestaffe: and so shall the degrees cut on the Staff, show the angle of their distance, if it be above 30 degrees. And thus much, together with that which hath been before taught in Example IIII: will be sufficient for the Use of the Crossestaffe: especially seeing so many men have already written upon this Argument. Soli Deo gloria. FINIS. The Translator to the Reader. Gentle Reader, by reason of my absence, whilst this Book was in the Press, it is no marvel though some faults have escaped, which you will be pleased to amend thus. Pag. 3, lin. 1, the third circle p. 8, lin. ult. so 240/320 is 0 ⌊ 75. pag. 14, lin. 14, 2▪ 0413927 pag. 15, lin. 1, the first term of a progression p. 16, lin. 24, 108 ⌊ 33+; pag. 17, lin. 17, the antecedent arm lin. 28, 4 chap. pag. 18, lin, 19, term given from pag. 19, lin. 11, in the fift circle pag. 20, lin. 19, lie hid. As in this lin. 20, D. rat. multa— 1 in R:: α. Z. lin. 28, and Rat. multa in R in α— R in α, lin. ult. and Rat. multa in R— R, in α, pag. 21, lin. 1, and Rat multa in α— α, in R, And also in the Aequations pag. 21, 24, 26, which have a magnitude equal to a fraction: the same magnitude together with the note of equality, aught to be set right against the line that is between the Numerator and Denominator of the fraction, as in these, Rat. multa— 1 in R in α/ D = Z. And ZD/ α = Rat. multa in R— R. And so of the rest. pag. 24, lin. 25, Rat. multa— 1 in R in α/ Rat.. multa in D pag. 25, lin. 20, arm at 71 ⌊ 382: lin. 23, (for it is Rat. multa— 1 in R in α) pag 26, lin. 16. Ratiocination▪ pag. 29, lin. 29, number of figures pag. 35, lin. 5, 61 49/144 pag. 36, lin. 11, 43 ⌊ 7. 17 ⌊ 48. lin. 14,:: 17 ⌊ 48. 3 ⌊ 26+ pag. 37, lin. 11, 3 ⌊ 1416, pag. 39, lin. 15, 339 ⌊ 2928 pag. 41, lin. 19, or as 1 is to 1 ⌊ 0472: pag. 44, lin. 8, is a roof lin. 26, thereof pag. 45, lin. 10, feet 529 ⌊ 175. pag. 46, lin 3, more sides than four pag. 53, lin, 10, Cylindrical vessel. pag. 53, lin. 18, if false, why lin. 20, error? pag. 57, lin. 14, common opinion is, that at London a Cylindrical lin. 28, 16 ⌊ 5 pag. 74, lin. 28, the Sum, pag. 78, lin. 6, third houses lin. 28, Add 90 degrees pag. 79, lin. 22, and the 90th degree pag. 82, lin. 5, 26 pag 94, lin. 21, a circle, or 90 degrees. pag. 95, lin. 34, angle D be obtuse, pag. 96, lin. 4, sign + lin. 21, √ q: Z + X: pag. 100, lin. 3, and then the side DC In the VIII diagramme of rightlined Triangles the letter A is wanting at the perpendicular. And in the VI the angle B aught to have been marked with a little line. pag. 113, lin. 7, the delineation pag. 127, lin. 1, the sun goeth not under pag. 131, lin. 19, in the paper pag. 132, lin. 10, 10 the tangent of the arch. pag. 134, lin. ult. North or South direct inclining: pag. 135, lin. 1, North or South direct reclining. lin. 8, either face of the Plain looketh: p. 143, lin. 3, In North reclining and South inclining pag. 144, lin. 19, instrument through the Pole of the Aequinoctial, is one of these three. pag 15●, lin. 30, North dyal declining Eastward 35 degrees In the Additament of Navigation. pag. 2, lin. 11, for valour read value pag. 6, lin. 9, & 12, for signs read sins pag. 29, lin. 20, 21, 24, for 45 read 48 lin. 20, for an half read eight thousanth parts & lin. 25, for min: 2 ⌊ 7 reed min: 2 ⌊ 88 pag. 46, lin. 20 pag. 47, lin 9, & 28, pag 48, lin. 17 & 21, for 87 ⌊ 927: read 86 ⌊ 968 pag. 47, lin. ult. for 205734, read 502734. pag. 28, lin. 8, 216000F/349800G: and keeping the same aperture, bring this latter arm unto the Latit: 6 ⌊ 5: and where the former arm shall light, there hold it fast, drawing in the latter arm to the line of the Radius. Lastly with this new opening bring the other arm to the line of the radius: and so shall you find 13 ⌊ 4, the other Latitude sought for.