to the Circumference are equal if divers Circles are described upon one and the same Center they are said to be Concentrick if upon divers Centers they are in respect of each other said to be Excentrick The Diameter of a Circle is a right Line drawn through the Center of any Circle in such sort that it may divide the whole Circle into two equal parts The Semidiameter of a Circle commonly called the Radius thereof is just one half of the Diameter and is contained betwixt the Center and one side of the Circle A Semicircle is the one medeity or half of a whole Circle described upon the Diameter thereof And a Quadrant is just the one fourth of a whole Circle All Circles are supposed to be divided into 360 equal parts called Degrees consequently a Semicircle contains 180d and a Quadrant 90d and so much is the quantity of a right Angle A Minute is the sixtieth part of a degree being understood of measure but in time a Minute is the sixtieth part of an hour or the fourth part of a degree 15 degrees answering to an hour and 4 Minutes to a degree A Minute is marked thus 1′ a second is the sixtieth part of a Minute marked thus 1″ Where two lines or arches cross each other the point of meeting is called their Intersection or their common Intersection CHAP. II. Containing some Geometrical Rudiments Prop. 1. To raise a Perpendicular upon the end of a given Line Let it be required to raise a Perpendicular upon the end of the Line C D upon C as a Center describe the arch of a Circle as D G prick the Extent of the Compasses from D to G and upon G as a Center with the said Extent describe the ark D E G in which prick down the Extent of the Compasses twice first from D to E and then from E to F then joyn the points F D with a right line and it shall be the Perpendicular required Otherwise in case room be wanting Upon G with the Extent of the Compasses unvaried describe a small portion of an Ark near F then a Ruler laid over the Points C and G will cut the said Ark at the Point F from whence a line drawn to D shall be the Perpendicular required In this latter case the Extent C F becomes the Secant of 60d to the Radius C D which Secant is always equal to the double of the Radius As to the ground of the former Way if the three Points C G F were joyned in a right line it would be the Diameter of a Semicircle an Angle in the circumference whereof made by lines drawn from the extreamities of the Diameter as doth the Angle C D F is a right Angle by 31 Prop. 3 Book of Euclid Otherwise To raise a Perpendicular upon the end of a given Line Set one foot of the Compasses in the Point A and opening the other to any competent distance let it fall in any convenient point at pleasure as at D then retaining that foot in D without altering the Compasses make a mark in the line A C as at E Now if you lay a Ruler from D to E and by the edge of it from D set off the Extent of the Compasses it will find the point B from whence draw the line A B and it shall be the Perpendicular required Thus upon a Dyal we may raise a Perpendicular from any point or part of a Line without drawing any razes to deface the Plain Otherwise upon the Point D having swept the Point E with the same Extent draw the touch of an Arch on the other side at B then laying the Ruler over D and E as before it will intersect the former Arch at the Point B from whence a line drawn to A shall be the Perpendicular required The converse of this Proposition would be from a Point assigned to let fall a Perpendicular on a line underneath To be done by drawing a line from the Point B to any part of the given line A C as imagine a streight line to pass through B D E and to finde the middle of the said line at D where setting one foot of the Compasses with the Extent D E draw the touch of an Arch on the other side the Point E and it will meet with the said line A C at A from whence a line drawn to B shall be the Perpendicular required Prop. 2. To raise a Perpendicular from the midst of a Line In the Scheme annexed let A B be the line given and let it be required to raise a Perpendicular in the Point C. First set one foot of the Compasses in the point C and open the other to any distance at pleasure and mark the given line therewith on both sides from C at the point A and B then setting one foot of the Compasses in the point A open the other to any competent distance beyond C and draw a little Arch above the line at D then with the same distance set one foot in B and with the other cross the Arch D with the arch E then from the Point of Intersection or crossing draw a streight line to C and it shall be the Perpendicular required The converse will be to let a Perpendicular fall from a Point upon a given Line Let the Point given be the Intersection of the two Arks D E setting down one foot of the Compasses there with any Extent draw two Arks upon the line underneath as at B and A divide the distance between them into halfs as at C and from the given Point to the Point C draw a line and it shall be the Perpendicular required and if this be thought troublesom upon the Points A and B with any sufficient Extent you may make an Intersection underneath and lay a Ruler to that and the upper Intersection and thereby finde the Point C. Prop. 3. To draw one Line parallel to another Line at any distance required In this Figure let the Line A B be given and let it be required to draw the Line C D parallel thereto according to the distance of A C. First open the Compasses to the distance required then setting one foot in A or further in with the other draw the touch of an Arch at C then retaining the same extent of the Compasses set down the Compasses at B and with the former extent draw the touch of an Arch at D then laying a Ruler to the outwardmost edges of these two Arks draw the right line C D which will be the Parallel required This Proposition will be of frequent use in Dyalling now the drawing of such pieces of Arks as may deface the Plain may be shunned for having opened the Compasses to the assigned Parallel distance set down one foot opposite to one end of the line proposed A B so as the other but just turned about may touch the said Line and it will finde one Point Again finde another Point in like manner opposite to the
other end of the Line at B and through these two Points draw a right Line and it shall be the Parallel required This way though it be not so Geometrical as the former yet in other respects may be much more convenient and certain enough Prop. 4. To draw the Arch of a Circle through any three Points not lying in a streight Line In the Figure adjoyning let A B C be the three Points given and let it be required to draw a Circle that may pass through them all Set one foot in the middle Point at B and open the Compasses to above one half of the distance of the furthest point therefrom or to any other competent extent and therewith draw the obscure Arch D E F H with the same extent setting one foot in the point C draw the Arch F H Again with the same extent setting one foot in the point A draw the Arch D E then laying a Ruler to the Intersections of these Arches draw the lines D G and G H which will cross each other at the point G and there is the Center of the Circle sought where setting one foot of the Compasses and extending the other to any of the three points describe the Arch of a Circle which shall pass through the three points required Prop. 5. Two lines being given inclining each to other so as they seem to make an acute angle if they were produced To finde their angular point of concurrence or meeting without producing or continuing the said Lines And if they be multiplied the other way from D to H and from C to E then the lines E H and F G being produced find the same point I without continuing either of the lines first given and with much more certainty An Oblong a Rectangle a right angled Parralellogram or a Long Square are all words of one and the same signification and signifie a flat Figure having onely length and breadth the four Angles whereof are right Angles the opposite sides whereof are equal In Proportions the product of two tearms or numbers are called their Rectangle or Oblong because if the sides of a flat be divided into as many parts as there are units in each multiplyer lines ruled over those parts will make as many small squares as there are units in the Product and the whole Figure it self will have the shape of a long Square A Rhombus or Diamond is a Figure with four equal sides but no right Angle But a Rhomboides or Diamond-like Figure is such a Figure whose opposite Sides and opposite Angles onely are equal either of these Figures are commonly called Oblique Angled Parralellograms Thus either of the Figures A B F E or B C E D are Rhomboides or Oblique Angled Parralellograms This foregoing Figure is much used in Dyalling thereby to set off the Hour-lines Admit the Sides A B and B E were given and it were required on both sides of B E to make two oblique Parralellograms whose opposite sides should be equal to the lines given this may be done either by drawing a line through the point E parallel to A B C and then make F E E D and B C each equal to A B and through those points draw the sides of the Parralellogram or continue A B and make B C equal thereto and with the extent B E upon A and C draw the cross of an Ark at F and D Again upon E with the extent A B draw oâ—her Arks crossing the former at F and those crosses or intersections limit the extreamities of the sides of the Parralellogram A line drawn within a four-sided Figure from one corner to another is called a Diagonal-line A Parralellipipedon is a solid Figure contained under six four-sided figures whereof those which are opposite are parallel and is well represented by two or many Dice set one upon another or by the Case of a Clock-weight To finde a right Line equal to the circumference of a Circle given Let the given Circle be B D C divide the upper Semicircle B D C into halfs at D and the lower Semicircle into three equal parts and draw the lines D E D F which cut the Diameter at G and H and make G I equal to G H then is the length D I a little more then the length of the quadrant B D neither doth the excess amount unto one part of the Diameter B C if it were divided into five thousand and four times the extent D I will be a little more then the whole circumference of the Circle To finde a right Line equal to any Arch of a given Circle Let C D be an arch of a given Circle less then a Quadrant whereto it is required to finde a right line equal Divide the Arch C D into halfs at E and make the right line F G equal to the Chord C D and make F H equal to twice C E and place one third part of the distance between G and H from H to I and the whole line F I will be nigh equal in length to the Arch C D but so near the truth that if the line F I were divided into 1200 equal parts one of those parts added thereto would make it too great albeit the Ark C D were equal to a Quadrant but in lesser Arches the difference will be less and if the given Arch be less then 60 degrees or one sixth part of the whole Circle the line found will not want one six thousandth part of its true length But when the given Arch is greater then a quadrant it may be found at twice thrice or four times by former Directions These two Propositions are taken out of Hugenius de magnitudine Circuli Page 20 21. In Dyalling to shun drawing of Lines on a Plain it may be of frequent use to prick off an Angle by Sines or Tangents in stead of Chords it will therefore be necessary to define these kinde of Lines 2. The right Sine of an Arch is half the Chord of twice that Arch thus G F being the half of G L is the Sine of the Arch G A half of the Arch G A L whence it follows that the right Sine of an Arch less then a quadrant is also the right Sine of that Arks residue from a Semicircle because as was shewed above the Chord of an Ark is the same both to an Ark lesser then a quadrant and to its complement to a Semicircle What an Arch wants of a quadrant is called the Complement thereof thus the Arch D G is the Complement of the Arch A G and H G is the Sine of the Arch D G or which is all one it is the Cosine of the Arch A G and the Line H G being equal to C F it follows that the right Sine of the Complement of an Arch is equal to that part of the Diameter which lieth between that Arch and the Center From the former Scheme also follows another Definition of a right Sine as
namely that it is a right Line falling from one end of any Arch perpendicularly upon the Radius drawn to the other end of the said Arch so is G F perpendicular to C A being the Sine of the Arch G A likewise A I falls perpendicularly on C G therefore by the same definition is also the Sine of the said Arch. 3. The Versed Sine of an Arch is that part of the Diameter which lieth between the right Sine of that Arch and the Circumference thus F A is the Versed Sine of the Arch G A and F B the Versed Sine of the Arch B D G. 4. If unto one end of an Arch there be drawn a Radius and to the other end a right line from the Center cutting the Circle and if from the end of the Radius a Perpendicular be raised till it meet with the Line cutting the Circle that Perpendicular is the Tangent of that Arch thus A E is the Tangent of the Arch G A and D M is the Cotangent of the said Ark namely it is the Tangent of the Arch H G which is the Complement of the former Arch. 5. The aforesaid right Line cutting the Circle is called the Secant of the said Arch thus C E is the Secant of the Arch G A and C M is the Cosecant of the said Arch for it is the Secant of the Arch D G. 6. Assigning the Radius C A to be an unit with Ciphers at pleasure to define or express the quantities of these respective lines in relation to the Arches to which they belong were to make a Table of natural Sines Tangents and Secants of which at large see Mr. Newtons Trigonometria Brittanica for abbreviate ways and something we shall adde about it in the Arithmetical part of Navigation 7. The Tables being made their chief use was to work the Rule of Three or Golden Rule Arithmetically by multiplying the second and third tearms of any Proportion and dividing by the first and thereby to resolve all Propositions relating either to Plain or Spherical Triangles which in lines is performed by drawing a line parallel to the Side of a Triangle and where four tearms either in Sines Tangents Secants Versed Sines are expressed as the first to the second so is the third to the fourth it implyes a Proportion and that the second and third tearm are to be multiplyed together and the Product divided by the first The proportion or reason of two numbers or reference of one to the other is measured by the Quotient the one being divided by the other A Proportion is then said to be direct when the third tearm bears such proportion to the fourth as the first to the second four numbers are said to be proportional when as often as the first and second are the one contained in the other so often are the third and fourth the one contained in the other Reciprocal Proportion is when the fourth tearm bears such Proportion to the third as the first doth to the second A Proportion is said to hold alternately when the second and third tearms thereof change places and inversly when the order of the tearms are so altered that one of the three first tearms shall become the last Divers Proportions are expressed in this following Book which if the Reader would apply to Tables he must understand that when a Side or an Angle is greater then a Quadrant that in stead of the Sine Tangent or Secant of such an Ark he must take the Sine Tangent or Secant of that Arks complement to a Semicircle That the words Cosine or Cotangent of an Arch given or sought signifie the Sine or Tangent of the Complement of the Arch given or sought That the Cosine or Cotangent of an Arch greater then a Quadrant is the Sine or Tangent of the excess of that Arch above a Quadrant or 90 degrees 8. What trouble the Ancients were at in resolving of Proportions by Multiplication and Division is wonderfully abbreviated by an admirable Invention called Logarithmes where by framing and substituting other numbers in stead of the former Multiplication is changed into Addition and Division into Substraction of which also see the former Book Trigonometria Brittanica 9. What may be performed by either of the former kindes of Tables may also with a Line of Chords and equal parts be performed but not so near the truth without them and that either by projecting or representing the Sphere on a Flat or Plain as we have handled in the second and third Part or by Protracting and Delineating of such Proportions as may be wrought by the Tables and this in some measure is the intended subject of our subsequent Discourse 10. Therefore before we proceed any further it will be necessary to describe such Lines as are upon the Scale intended to be treated of The Description of the Scale in the Frontispiece 1. The first or uppermost line is a line of a Chords numbred to 180 degrees and is called the Lesser Chord being a double Scale the undermost side whereof being numbred with half those Arks is a line of Sines and is called the greater Sines at the end of this Scale stands another little Scale which is called the lesser Sines being numbred with 90 degrees and both these Scales seem to be one continued Scale 2. The second Scale is another line of Chords called the greater Chord being fitted to the same Radius with the greater Sine and numbred to 90 degrees Annexed thereto is a single Line called the line of Rumbes or Points of the Compass numbred from 1 to 8 in which each Rumbe is divided into quarters having pricks or full points set thereto 3. The third Scale is a line of equal parts or leagues divided into ten greater divisions and each of those parts into ten smaller divisions and each of those smaller divisions into halfs Annexed thereto is another Scale of six equal parts each of which parts is sub-divided as the former How to make a Line of Chords or Sines from the equal divisions of the Circle is spoke to in the second Book page 2. The Scale thus described is indeed a double Scale for it hath two Lines of Chords two Lines of Sines and two Lines of equal Parts upon it and this rather for conveniency then necessity whereas indeed one of each kinde had been sufficient yea the Sines might have been wholly spared for throughout these Treatises nothing is more required of necessity then a Circle divided into equal parts as in the third Book page 4. and a right Line divided into equal parts which we suppose in every mans power to do if he have Compasses The Schemes throughout these Books are fitted either to the Radius of the greater or lesser Chord before described that is to say they are drawn with 60 degrees of the one or the other of those Lines of Chords CHAP. III. Shewing how all the common Cases of Plain
prick down 30 degrees 58 minutes for the Angle given and draw the line A F C then from the end of the Side prickt off prick the other Side from B to C or E and so the Angle B C A or B E A is the Angle sought but which of the two cannot possibly be determined unless the affection be also given to wit whether it be obtuse or acute though some of our Writers affirm it may be determined by drawing the Triangle as true as you can Then upon the angular Point C describe an Arch with 60d of the Chords and measure the said Arch in the Chords continuing the Sides if need be and it shews the quantity of the said Angle to be 53d 6′ and the Complement thereof to 180d being 126d 54′ is the measure of the Angle B E A because the Angle B E C is equal to the Angle B C E. How to frame such Triangles whose Sides shall be all whole numbers is shewed in our English Ramus page 155 156. 2. From what is given to finde the third Side and the other Angle In this case also unless the quality of the Angle opposite to the greatest Side be determined the third Side will be doubtful to wit it may be either A E or A C which extents measured on the equal parts shew the Side accordingly and the third Angle to be measured as before This Mr. Norwood doth not make a Case because an Angle must be first found and determined before the third Side can be found and then it will be resolved by the following Case 3. Two Angles with a Side opposite to one of them given to finde the Side opposed to the other In this Case the third Angle is also given as being the complement of the sum of the two former Angles to a Semicircle As if there were given the Side A B in the former Triangle the Angle B A C and the Angle A C B. First prick off the Side A B then substract the sum of the Angles B A C and A C B from 180d or a Semicircle and there remains the Angle A B C then upon A as a Center set off upon the Arch D F the measure of the Angle A. Also upon B as a Center the measure of that Angle must be set off and lines drawn through the extreamities of those Arks will meet as at C the point that limits the two Sides A C and B C which are to be measured on the equal parts By this Case if A C were the Wall of a Town and B a Fort shooting into the said Town the distance of the said Fort might be found from any part of the Town wall without going out to measure it for first with any whole circle or a compass observe the Arch between A B and A C and measure the distance C A again at A observe the Arch between A B and A C and protract as in this Example and you may measure the distance between B and any point in the line A C And so if B were a Tower or Mark on the land and A C represent the Ships distance in her course by observing how B bears both at A and C the Ships distance therefrom might be measured 4 5. Two Sides with the Angle comprehended given to finde the third Side and both the other Angles Thus if there were given the two Sides A B and A C with the Angle A between them the said Angle must first be set off in the Arch D F then a line joyning the extreamities of the two Sides as doth B C is the third Side which being first found upon the angular Points C and B with 60d of the Chords Arks must be drawn which being limited by the two Sides produced when it is necessary are the measures of the Angles sought 6. Three Sides to finde an Angle If the three Sides be joyned in a Triangle which is easily done first pricking down any one of the Sides and from its extreamities with the other Sides describe two Arks which will intersect at the Point where the other two Sides concur then will the three angular Points be given upon which Arks being described between the given Sides are the measures of the Angles sought 7. Three Angles being given are not sufficient to finde any one of the Sides CHAP. III. Shewing how all Proportions relating to Sphaerical Triangles may be performed by a Line of Chords 1. Proportions in Sines alone LEt the Proportion be As the Sine of 19d to the Sine of 25d So is the Sine of 31d. To what Sine to wit the Sine of 42d. Otherwise Place the extent wherewith the Ark F was described from V to E and draw the line E G just touching the extreamity of the said Ark then with the extent V A one foot of the Compasses resting in V cross the aforesaid line at G and a ruler over V and G will finde the Point D in the Limbe as before Here observe that every Proportion without the Radius may be made into two single Proportions with the Radius in each thus As the Radius Is to one of the middle tearms So is the other middle tearm To a fourth Proportional Again As the first tearm Is to the Radius So is the fourth Proportional before found To the true Proportional sought From which consideration the former Scheme was contrived Two other general ways for working Proportions in Sines Let the Proportion be As the Sine of 70d to the Sine of 50d So is the Sine of 35d. To what Sine Answer 27d 50′ Having drawn the Quadrant D E with 60d of the Chords and its two Radii D C C E at right Angles in the Center prick down from the Chords one of the middle tearms to wit 35d from D to H and draw a line into the Center and upon the said line from the Center to A prick down out of the Sines the other middle tearm to wit 50d and through the Point A draw the line A B parallel to D C then count the first tearm from D to G 70d and draw a line from the Center which passeth through A B at F and the extent C F measured on the Sines is 27d 50′ the fourth Proportional and thus the first tearm may be varied as much as you please Otherwise Place the Sine of the first tearm to wit of 70d which in this Example is the nearest distance from G to D C so from the Center that it may cross A B produced when need requires In this Example it crosseth it at I a ruler over the Center and the point I cuts the Limbe at K and the Arch D K being 27d 50′ is the measure of the fourth Proportional as before When it cannot be there placed to wit as when the Sine of the first tearm is shorter then C B the fourth Proportional is more then the Radius and the Arch of the first tearm being counted from D towards E a line
C B the Departure from the Meridian in the Course between both places then making that one Leg of a right angled Triangle prick down 17d 59 Centesmes the difference of Latitude between those places out of the same equal parts from C to L and draw B L which represents the Course and distance truly between the Lizard Bermudas and the extent L B measured on the same equal parts shewes the distance to be 44d 31 Centesmes which allowing twenty Leagues to a degree is 886 Leagues Then to finde the Course with 60d of the Chords setting one foot in L with the other make a mark at Y and Z then the extent Z Y measured on the Chords sheweth the Rumbe to be 66d 37′ from the Meridian which is almost 6 points and in this example the Proportion doth not erre any thing from the truth according to Mercators Chart whereas if you use the former Proportion by the middle Latitude the Rumbe would have been 67d 2′ from the Meridian and the distance 902 leagues if you make C A equal to C V then a line joyning L A should be the course and distance according to the same Longitudes and Latitudes laid down on the Plain Chart and thereby the Course should be 72d 17′ from the Meâ—idian and the distance 1155 leagues however when two places are laid down true at first in their Rumbe distance and Latitudes on the Plain Chart if you sayl home in or near the same Rumbe the Plain Chart will very well serve to keep the reckoning upon and to sayl by in the greatest Voyage How Geometrically to supply the Meridian-line of Mercators Chart generally That the finding of the true Rumbe between two places might not be a Proposition out of the reach of Geometry or not to be performed by Scale and Compasses the supply thereof became the Contemplation of the late learned Mr. Samuel Forster Professor of Astronomie in Gresham Colledge who in his Treatise of a ruler Entituled Posthuma Fosteri makes a Meridian-line out of a Scale of Secants this we shall be brief in and shew how near it comes to the truth Let it be required to make a Meridian-line of such a scantling that one degree of Longitude may be half an inch which is of the same size with the Print thereof at the end of the Book And so if it were required to make the meridian-Meridian-line from 43d to 50d of Latitude the former extent C F out of the quadrant shall reach in this line F L from 43d to 44 in like manner the Secant of 44d 30′ so taken out shall reach from 44d to 45. And thus may the whole degrees be taken out now for dividing them into Centesmes they may be equal divisions and yet very near the truth or rather first divide the half degrees true by the middle Secant thus the half of the Secant of 49d 15′ shall reach from 49d to 49d and a half and instead of halfing the Secant it may be taken out to half the Radius and afterwards the parts of each half degree may be divided equally and so if you would divide a degree into ten parts each part will be 6 minutes and if it were required to finde the true length of the Meridian-line from 49 degrees to 49 degrees 6 minutes the tenth part of the middle Secant to wit of 49 degrees 3 minutes shall be the length required and so on successively And so if it were required to finde the length of 49d 7 tenths or 42′ I say that 7 10 of the middle Secant to wit of the Secant of 49 35 Centesms or 49d 21′ is the length required Now it may be doubted that the making of the Meridian-line by whole degrees is not near enough the truth in regard the Tables at first were made by the adding of the Secants of every minute successively together and the learned Mr. Oughtred in stead of adding the Secants of every minute would have a minute divided into a hundred a thousand ten thousand or rather a million of parts and the Secants of every one of those parts added together To this I answer That the making of the Meridian-line by whole degrees and in the whole doth not breed any error at all to be regarded compared with the making of it up successively by every minute and for each particular degree it doth not breed any sensible error compared with the best Tables To the end of this Book we have added a Table of the Meridian-line to every second Centesm wherein we have supplied the vacuity that is in Mr. Gunters Table by which Table it doth appear that the Meridional parts between the Latitudes of 50 degrees and 60 degrees are 17d ⌊ 542 which other Tables make more and by the adding up of the Secants of 51 degrees 30 minutes 52 degrees 30 minutes and so successively to 59 degrees 30 minutes they will amount to 17d 547 the difference being onely in the thousandth parts of a degree and I suppose there are no Mercators Charts made wherein a degree of Longitude is an inch the biggest I have seen is but half an inch and if they were an inch it could scarcely be divided into one hundred parts much less into a thousand and so in every part of the Meridian-line as far as it can be used the difference will be inconsiderable and so also in the whole  Deg. Parts The Meridional parts for 70d of Latitude by our Tables and Mr. Gunters are 99 431. By adding up the Secants of 30′ then of 1d 30′ 2d 30′ successively to 69d 30′ they are 99 426. By Mr. Wrights Table reduced to degrees 99 444. Calculated by the Logarithmical Tangents are 99 436. By the Forreign Tables of Maetius and Snellius which are not extended to 80d 99 416. And for the latter part of the Objection I hear Mr. Oughtred was making a New Table of them according to his own minde wherein it is probable he attained the manner of adding up those Secants by some new Proposition in regard it would be extreamly tedious to make a Table of Natural Secants to all those parts and then adde them up wherein he long since desisted upon this consideration that the decrease would happen in the decimal parts remote from the degrees for it must be conceived that a Table of the Meridian-line consists of the sum of all the Secants divided by Radius and thus the Meridional number for the first thousand Centesms or ten degrees added up from a Table whose Radius is 1000 amounts to 1005079 and divided by the Radius is 1005 079 then because we would have a Table to express the Meridian-line in degrees allowing 100 centesms to a degree we must divide the former Number by 100 and the Quotient is 10 05079 which is the very Number in our Tables saving that our Table is not continued so far by two places and for the last two figures which are omitted we added in an Unit in the
more to wit 180d 54. But supposing two places to be in the same Latitude and to have but 58d 33 Centesms difference of Longitude the example will be the same with one of those before put between the Berbadoes and St. Helens and the distance found by the middle Arch or Latitude is 62d 00 and the same by the middle space 61d 38 centesms but should be in truth 64d 7. Also in the former Example between the Berbadoes and the Lizard the true distance is 58d 54 and by the middle Latitude or Arch is 58d 7 but by the middle space it is 57d 66. Also in the Example between the Bermudas and the Lizard the true distance was found to be 44d 31 Centesms by the middle Arch or Latitude it is 45d 13 Centesms and by the middle space it is 44d 97 Centesms Where places are in the same Latitude or Parallel the Compasses must be set down in the Meridian-line at the Latitude given and the half extent applyed both upwards and downwards as before and if the distance be large the measure thereof will be much more erronious then when the Rumbe lyes nearer the Meridian Some Examples of a parallel Distance Let there be two places in the Latitude of 35d.   True Distance Distance by the Meridian-line Difference of Longitude 180d 147d 45 123d 93. 18 14 74 13 98. Another Example of two places in the Latitude of 50d.   True Distance Distance by the Meridian-line Difference of Longitude 180d 115d 8 111d 87. 18 11 58 11 59. A third Example of two places in the Latitude of 70d.   True Distance Distance by the Meridian-line Difference of Longitude 180d 61d 57 76d 41. 18 6 15 6 18. From which Examples we may observe that a large Distance cannot be so certainly measured in the Meridian-line as a small one whereof Mr. Wrâ—ght was very sensible and therefore prescribes Rules for the measuring of a small part of the Distance at a time and argues for the truth thereof but where the whole extent between two places is not above ten of the degrees of Longitude I see nothing to the contrary but that it may well enough be measured in the Meridian-line and so for a great distance we may measure a tenth or a twentieth part of the whole and by multiplying the known part finde the whole for the ready performing whereof another Scale of equal parts whereof the degrees are twice as large as those in the Scale of Longitude will be of much conveniency and ease Example Suppose it were required to measure the distance between the Points f and d in the former Chart take the same distance and measuring it in the inches finde how much it is to wit 1 Inch 61 Centesmâ— then take the same number out of the Scale of Longitude and setting one foot at the middle Latitude to wit 19d 18 Centesms the other will reach Northwards to 20d 69 Centesms and Southwards to 17d 64 Centesms the difference of which two Arks is 3d 05 the distance sought which allowing 20 leagues to a degree is 61 leagues as before and this is more easily done then to take the half of any extent and by the same reason you may finde the middle space between both Latitudes So also when you would measure the tenth part of a great distance measure the whole extent in the Scale of Longitudes and take the twentieth part found by the pen or memory out of the said Scale and set it at the middle Latitude or middle Space turning the other foot in the Meridian-line both upwards and downwards and the degrees so intercepted will be the tenth part of the whole distance Now the taking of the twentieth or fortieth part of an Extent is easily done by help of these two Scales of equal parts Suppose I would finde the twentieth part of 3 inches or degrees in the greater Scale I say 3 of the small parts in the lesser Scale is the length required and so the twentieth part of 3 inches 5 tenths is three and a half of the smaller parts in the lesser Scale and the half of that is the fortieth part of the whole I need not insist further upon these ways of measuring seeing I have before delivered others which as they are more ready in the practise so also they are built upon better foundations To keep a Reckoning on the true Chart. Here I shall insist upon a new Method never before published which will render this Chart very easie and acceptable to Seamen and having made our Example that before laid down being the same with that in the Plain Chart we shall here also retain the same Traverses The first Operation is to finde the Latitude The first Course the Ship sayls is South South-west 60 leagues from Tenariff to protract this Traverse I shall make use of another Traverse-quadrant bigger then that which was used before which may be made upon state Take 60 leagues out of the Scale of Longitudes T W and enter it in the Traverse-quadrant on the second point from C to A the nearest distance from A to C W prick in the Scale of Inches from W to A + and it shews me now that the Ship is in the Latitude of 25d 23 Centesms in the Meridian-line T S set the figure 1 to this Latitude Secondly to finde the difference of Longitude Take the extent T 1 in the Meridian-line and enter it so in the Traverse-quadrant on the second Rumbe that one foot resting thereon as at a the other turned about may but just touch C W then is the nearest distance from a to C S the difference of Longitude required to wit 1d 24 Centesms which prick in the Scale of Longitude from T towards W and set the figure 1 at it Thirdly to plot the Traverse Point Set one foot of the said extent at 1 in the South line of the Chart and with the other draw a small Ark at a then take the extent T 1 out of the South line and setting one foot at 1 in the West line with the other cross the former Ark at a and there is the point where the Ship is at the end of this first Traverse Demonstration The Proportion for finding the difference of Latitude we have before handled the Proportion for finding the difference of Longitude is As the Radius Is to the Meridional parts between any two Latitudes ∷ So is the Tangent of the Rumbe To the difference of Longitude ∷ The extent T 1 being taken out of the South line is the parts of the Meridian-line between the Latitude of 28d and the Latitude of the Ships place namely 25d 23 Centesms which being entred as before in the Traverse-quadrant at a becomes the Radius to the Tangent of that Rumbe and so the Tangent of the said Rumbe to that Radius being the nearest distance from a to C S becomes the difference of Longitude required the Proportion for finding it being duly observed
it were needful It is not necessary to press Examples if what before is written be well understood especially in this case where all directions are slippery Thus in imitation of Maetius a Hollander though a Latine Author we have prescribed several rules for the correction of a single Course which Mr. Phillips in his Geometrical Sea-man makes but one rule retaining always the same Course and correcting the distance run therein by drawing a parallel through the observed Latitude and so for many Courses they are first brought all into one line and the distance corrected by the same rule But concerning it we must give a double Caution First that no three places can be laid down true in their Courses and Distances from each other on the Plain Chart as shall afterwards be handled however the error in small distances will be inconsiderable And secondly admitting they could the said general Direction is unsound but the nearer the truth the nearer the Courses are to the Meridian and when all the Courses do either increase or diminish the Latitude but very erronious when some Courses increase and others lessen the Latitude in all which Cases it is most safe to allot to the Variation or dead Difference of Latitude of every Course its proportional share of the whole error between the Dead and Observed Latitude and then to correct each course by the former directions First therefore in the following Chart let us suppose a Ship to sayl from A in the Latitude of 28d South South-west almost 65 leagues to B this Course is set off in the Arch E F and by the Dead Reckoning she should now be in the Latitude of 25d. Again from B she sayls South-west and by West 72 leagues to C which Course being three points from the West is set off in the Arch G H and now by the dead Reckoning she should be in the Latitude of 23 degrees whereas by a good observation she is found to be in the Latitude of 23d 30′ wherefore to correct this Reckoning draw the line C A which is the compound Course arising from the two former Courses and through the parallel of observed Latitude draw L K parallel to A W so is the point K the corrected point of the Ships place according to Mr. Phillips and agreeing with the truth as we have fitted the Example But now as to the other way of correcting a compound Course it is to be done by this Proportion First finde the Variation or Difference of Latitude proper to each Course then it holds As the sum of all the Variations or Differences of Latitude Is to the whole error between the Dead and Observed Latitude ∷ So is each particular Difference of Latitude To its proportional share of the whole error ∷ Then if the differences of Latitude fall all the same way if the the estimated difference of Latitude be too much you must abate out of each dead Latitude its proportional error so in this case the said error is prickt from E to N. But when the estimated difference of Latitude is too little the proportional error must be added to each difference of Latitude then prick the second dead difference of Latitude being equal to E S from N to O and place the said extent from A the Center to Y and take the nearest distance to M A as before and prick it from O to L being the second error this is needful when there are more Courses then two but for the last Course not at all necessary neither is it for this then through the point N draw the line N F parallel to A W so is F the corrected Point of the Ships place at the first Course then draw F K parallel to B C and where the parallel of Latitude cuts it as at K is the corrected Point of the Ships place at the second Course being the same we found it before the other way But in stead of the second Course and distance which was 72 Leagues South-west and by west let us now suppose the Ship sayls the same distance from the point B North-west and by west which Course being as much on the other side the west make G I equal to G H and draw the Course B I therein pricking off the former distance to D so is D the point of the Ships dead reckoning in the Latitude of 27d and now supposing the observed Latitude to be 27d 30′ the error and difference of Latitude are as much now as they were before wherefore draw D A the compound rumbe draw Q T parallel to A W where it cuts the compound rumbe as at P by M◠Phillips his reckoning is the corrected point of the Ships place at the end of the second Course whereas in truth it should happen at T and so P bears from A in this example 76d 43′ from the Meridian and is distant from it 43 Leagues and a half whereas the Ships true Course from A to T is 83d 32′ from the Meridian and the distance almost 89 leagues which is very considerable Now for as much as the Sum of the differences of Latitude A E and E f in this latter example is equal to A S in the former example also the error f Q here is equal to S L there therefore the proportional part of each error will be the same as before Then if some Courses decrease the Latitude Southwardly and others increase it North-wardly if the dead Latitude be too little as in this example consider that to place the Ship more North-wardly so as to allot to each difference of Latitude its proper error that the South-wardly differences of Latitude must be decreased or lessened and the North-wardly increased wherefore the proportion of the error is placed from E to N and the Point F found as before In like manner if the dead Latitude were too much to bring the Ship more South-wardly the Southern differences of Latitude must be increased and the Northern decreased now the point T is found by drawing a line from F the corrected point of the first Course parallel to B D and so the line F V being equal to B D is the Ships second Course and distance from the corrected point F then in regard part of the error in the Latitude is supposed to be committed as well in the latter as in the former Course which error being too little the distance F V is to be enlarged and where the parallel of observed Latitude cuts it as at T is the corrected point of the Ships place at the end of the second Course And though what we have here performed be done by the drawing of many Lines yet by help of the Traverse-quadrant it may may be inserted into the Chart without drawing any Lines therein at all for in each the Course and corrected Difference of Latitude is given and that two things are sufficient to dispatch the work we have shewed before Those that are prompt in Plain Triangles may
Meridian count the same in the Variation Column upward and against it in the Sines you will finde the difference of Latitude is 923 leagues and 87 parts more of another league divided into 100 parts But if the difference of Latitude be required for any other Distance multiply the Sine of the given Point by the Distance run cutting off three figures from the Product Example If the Ship sayls 60 leagues on that Course the difference of Latitude is 55 leagues and 38 Centesms multiply 923 by 60 and the Product cutting off three places is 55 380. To finde the Departure from the Meridian the Proportion is As the Radius Is to the Distance run So is the Sine of the Rumbe To the Departure ∷ Example If a Ship sayl 1000 leagues South South West count the Point sayled in the Separation Column downward being the third Column and on the right hand of the Sines and you will in the Column of Sines finde the Departure required to be 382 leagues 68 Centesms But if a Ship sayl but 60 leagues on that Course multiply 382 thereby cutting off the three last places and you will finde the Separation or Departure required to be 22 leagues 92 Centesms or 22 920 leagues Points Sines Separ Tangents Secants Angles Sines quintupled deg parts ¾ 49 07 ¼ 49 12 1001 20 2d 48 m ¾ 2 45 ½ 98 02 ½ 98 48 1004 84 5 37 ½ 4 90 ¼ 146 73 ¾ 148 32 1010 94 8 26 ¼ 7 33 7 195 09 1 198 91 1019 59 11 15 9 75 ¾ 242 98 ¼ 250 48 1030 89 14 3 ¾ 12 14 ½ 290 28 ½ 303 36 1044 99 16 52 ½ 14 51 ¼ 336 89 ¾ 357 80 1062 08 19 41 ¼ 16 84 6 382 68 2 414 21 1082 39 22 30 19 13 ¾ 427 55 ¼ 472 96 1106 21 25 18 ¾ 21 37 ½ 471 39 ½ 534 52 1133 88 28 7 ½ 23 56 ¼ 514 10 ¾ 599 36 1165 96 30 56 ¼ 25 70 5 555 57 3 668 17 1202 68 33 45 27 77 ¾ 595 70 ¼ 741 65 1245 28 36 33 ¾ 29 78 ½ 634 39 ½ 820 68 1293 64 39 22 ½ 31 71 ¼ 671 56 ¾ 906 34 1349 61 42 11 ¼ 33 57 4 707 10 4 1000 00 1414 21 45 35 35 ¾ 740 96 ¼ 1103 32 1489 08 47 48 ¾ 37 04 ½ 773 01 ½ 1218 48 1576 32 50 37 ½ 38 65 ¼ 803 21 ¾ 1348 36 1678 68 53 26 ¼ 40 16 3 831 47 5 1496 60 1799 95 56 15 41 57 ¾ 857 73 ¼ 1668 37 1945 14 59 3 ¾ 42 88 ½ 881 91 ½ 1870 73 2121 36 61 52 ½ 44 09 ¼ 903 99 ¾ 2114 20 2338 88 64 41 ¼ 45 19 2 923 87 6 2414 21 2613 12 67 30 46 19 ¾ 941 53 ¼ 2794 76 2968 34 70 18 ¾ 47 08 ½ 956 94 ½ 3296 50 3445 69 73 7 ½ 47 84 ¼ 970 03 ¾ 3992 24 4115 56 75 56 ¼ 48 50 1 980 78 7 5027 33 5125 83 78 45 49 03 ¾ 989 17 ¼ 6741 44 6808 52 81 33 ¾ 49 45 ½ 995 18 ½ 10153 66 10202 32 84 22 ½ 49 75 ¼ 998 79 ¼ 20355 48 20380 15 87 11 ¼ 49 93 Variation       Here note that though the distance sayled be given in Leagues yet the table of Sines may be so altered that the difference of Latitude and Departure may be found in degrees and Centesmes and that onely be multiplying all the figures in the Column of Sines by the Number 5 for which purpose we added the last Column of quintupuled Sines in which the whole degrees are distinguished from the Decimal parts Example If a Ship sayl 1000 Leagues South South-west against the second Point counted upward we shall finde the difference of Latitude to be 46d 19 Centesmes but counted downward we shall finde the Departure from the Meridian to be 19 degrees 13 Centesmes which Numbers multiplyed by 60 cuting off five places towards the right hand give the difference of Latitude and Departure in that Course for 60 leagues to be 2d 77140 Parts Diff. of Lat. 1 14780 Departure of which results we need take no more places then 2 d 77. 1 14 or 1d 15. By these two Propositions the Dead Reckoning to be applyed to the Plain Chart may be kept after the form prescribed by Mr. Norwood and accordingly the difference of Latitude and Departure from the Meridian is here expressed in degrees and Centesmes for the three Traverses from Tenariff towards Nicholas Island which we made our former Example Course Distance in leagues North South Deg. Cent. East West d Cent. S S W 60  2 77  1 15 W S W 80  1 53  3 69 S b E ½ E 53  2 53 0 77 0 00    6 83 0 77 4 84 When some Courses increase the Latitude and others decrease it the difference between the North and South Column found by substracting the lesser from the greater gives the difference of Latitude so likewise the difference of the East and West Columnes give the Departure which in this Example is 4d 07 this difference of Latitude and Departure substracted out of the whole difference of Latitude and of Longitude between the said two Islands there rest 4d 17 Centesmes difference of Latitude between the Ship and St. Nicholas Island and 3d 93 difference of Longitude by which the Course from the Ship to the said Island will be found to be 43d 28 Centesms to the West-wards of the South and the distance 5d 73 being about 114 leagues and a half as shall afterward follow The difference of Latitude and the Rumbe given to finde the Departure from the Meridian and distance As the Radius Is to the difference of Latitude ∷ So is the Tangent of the Rumbe To the Departure from the Meridian ∷ And so is the Secant of the Rumbe To the distance ∷ Example For the Departure Let the instance be the same as in page 24 where we suppose the Course to be 3 points from the Meridian to wit S W b S and the corrected difference of Latitude to be 2 degrees 24 minutes or 2 degrees 4 tenths or 40 Centesms by which multiplying 668 the Tangent belonging to 3 Points in the former Table the result cutting off 5 figures is 1d 60320 wherefore the Departure from the Meridian is 1 degree 6 tenths For the Distance In like manner the Secant of three points to wit 1202 being multiplyed by the difference of Latitude 2d 40 cuting off 5 figures is 2d 88480 wherefore the distance is 2 degrees 88 Centesmes at 20 Leagues to a degree is 50 leagues and three quarters or 50 76. For keeping a reckoning by Longitude which is applyable to Mercators Chart and removes the error of the Plain Chart the Table of Meridional parts were added at the end of the Book for working of this Proportion As the Radius Is to the Tangent of the Rumbe ∷ So are
the Meridional parts between both Latitudes To the difference of Longitude ∷ Which Proportion requires the difference of Latitude to be first found and then by help of the Table of Meridional parts at the end of the Book the difference of Longitude may be found which Table being made but to every second Centesm hath half the difference set down at the bottom of the Page which added to the Meridional parts of any even Centesm above it makes the Meridional parts for each odd Centesm thus the Meridional parts for 25d 22 is 26 076 whereto adding 11 the sum being 26 087 are Meridional parts for 25d 23 Centesms in the first Example of the Traverse the difference of Latitude is 2d 77 Centesms The Latitude of Tenariff is 28d Meridional parts 29d 186 The Latitude of the Ships place is 25 23 26 087 The Meridional parts between both Latitudes are 3 099 Which being multiplied by the Tangent of the second Point to wit 414 cutting off 6 places three by reason the Radius is 1000 and three more for the Decimal parts of the Meridional Number the amount being 1d 282986 shews that the difference of Longitude is 1 degree 28 Centesms of which there may be a Column of East and West kept like as was done for the Departure and this casting up of the Longitude may be readily done also on the Logarithmical Ruler or by Mr. Phillips his late Tables for that purpose without Calculation whereof we made mention in Page 37. We have before said that a Table of Sines is sufficient to supply all Calculation though oâ—â—er Tables may be more ready for dispatch when they are at hand as we have shewed concerning the Table of Meridional parts in Page 48. Out of it the Secants are made by this Proportion As the Cosine of any Arch proposed Is to the Radius ∷ So is the Radius To the Secant of the given Arch ∷ Which Proportion holds backwards to finde the Arch in the Table of Sines if a Secant were given at adventure and the Arch required Also out of it the Tangents are made by this Proportion As the Cosine of an Arch Is to the Sine of the said Arch ∷ So is the Radius To the Tangent of the Arch proposed ∷ Which Proportion doth not hold backwards to finde the Arch if a Tangent were given in which Case the Secant may be found and thereby the Arch for the square of a Tangent more the square of the Radius is equal to the square of the Secant In this Case we have also the Propoâ—tion of the Cosine to the Sine given being the same with that of the Radius to the Tangent and the sum oâ— their squares given being equal to the square of the Radius which very Case is reduced to a double equation in the 33d Question of Mr. Moores Algebraick Arithmetick and by which the Radius and a Tangent being given either the Sine or Cosine may be found without finding the Secant If such a Proportion as this were proposed As the Radius Is to the Tangent of an Arch ∷ So is the Tangent of another Arch To the Tangent of a fourth Arch ∷ It might be resolved thus without making a Table of Tangents Make the Product of the Radius and of the Cosine of the second Ark the Divisor and the product of the Sines of the second and third Ark and of the Cosine of the third Ark the dividend and the Quotient will be the Tangent of the fourth Ark sought In like maner if all those tearms were Tangents the Product of the Radius Sine and Cosine of the first Ark would be the Divisor And the Product of the Sines and Cosines of the second and third Ark the Dividend whereby might be found the Quotient being the Tangent sought and consequently the Arch answering thereto after the like manner with due regard to the Proportion for making the Secants any Secant might be supplyed and a Proportion wholly in Secants by turning those Arkes into their complements is changed wholly into sines or two tearms being Secants are changed into Sines by altering the places or order of those tearms and their Arkes into their complements How to Calculate a Table of Natural Sines How to make the Sine of any Arke at pleasure Snellius in his Cyclometria hath shewed without making many such Sines as are not required in which he went beyond all foâ—mer Writers one way he hath holds true as far as 30d of the Quadrant out of which Sines their Cosines may be made and so the Quadrant filled from 60d to 90d and then by finding the Sine of the double Ark it may be supplyed from 30d to 60d but in regard the performance is very tedious and therefore not to be prosecuted in the Construction of a whole Table we shall omit to mention it another way he hath which doth not hold true to above 1 12 part of the Quadrant so that when a great Ark near the end of the Quadrant is given he findes the Cosine thereof first and thereby the Sine or if the Ark be remote or about the middle of the Quadrant he findes the Sine of the eighth or some such part of it and then by finding the Sine of the double Ark c. backwards he findes the Sine sought but how to Calculate a whole Table by Proportion with ease is attained unto by our worthy Countrey-man Mr. Michael Darie who discovered the same in the year 1651 and communicated unto me the Proportions for that purpose long since whose method we shall now insist upon 1. The readiest way is to assume the Proportion of the Diameter of a Circle to its Circumference to be known and to be in the Proportion of 113 to 355 of which see Maetius his Practical Geometry which two Numbers are easily remembred being the three first odde figures of the Digits each of them twice wrote down and cut asunder in the middle 2. It is most convenient that the Radius or whole Sine should be an Unit with Cyphers and so the Proporrtion of the Diameter to the Circumference found by the two Numbers above is such as 2000000 to 6283185 so far true and to finde it true to as many places of figures as you please consult Dr. Wallis his Writings or Hungenius de Magnitudine Circuli 3. The Sine of one Minute or Centesm doth so insensibly differ from the length of the Arch to which it belongs that the length of the Arch of one Minute or Centesm may be very well taken to be the Sine thereof and in the largest Tables that ever were published differs nothing therefrom wherefore a Minute being the 21600 part of the Circumference of a whole Circle the like part of the number 6283185 being 290 is the Sine of 1 Minute 4. You may finde the Cosine of one Minute by substracting the square of the Sine of one minute from the square of the Radius