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ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A80170	Navigation by the mariners plain scale new plain'd: or, A treatise of geometrical and arithmetical navigation; wherein sayling is performed in all the three kindes by a right line, and a circle divided into equal parts. Containing 1. New ways of keeping of a reckoning, or platting of a traverse, both upon the plain and mercators chart ... 2. New rules for estimating the ships way through currents, and for correcting the dead reckoning. 3. The refutation of divers errors, and of the plain chart, and how to remove the error committed thereby ... as also a table thereof made to every other centesm. 4. A new easie method of calculation for great circle-sayling, with new projections, schemes and charts ... 5. Arithmetical navigation, or navigation performed by the pen, if tables were wanting ... By John Collins of London, Pen-man, accomptant, philomathet.	Collins, John, 1625-1683.	1659 (1659)	Wing C5378A; Thomason E1002_1; ESTC R207824	177,819	300

down every fifth degree from C to D then laying the beginning of your Tangents over P and every one of those graduations you may by the view see what Latitude the Arch passeth through at every 5 degrees difference of Longitude from the Perpendicular for which purpose the Tangents on your ruler may be double numbred both with the Arch and its complement to which they belong Example Laying the Ruler over 25 degrees it cuts the Arch at e and the extent P e being measured on the Tangents is 41d 51′ being the complement of the Latitude of the Arch at that diffe●ence of Longitude wherefore the Latitude of the Arch at 25d difference of Longitude from the Perpendicular is 48d 9′ In like manner the Latitudes of the Arch for 00d Difference of Longitude from the Perpendicular is 50d 56′ 05 50 49 10 50 30 15 49 57 20 49 10 25 48 9 30 46 51 35 45 16 40 43 20 45 41 4 50 38 23 and is the same for the first ten degrees difference of Longitude on each side the Perpendicular and also as far as the lesser Vertical Angle A second Example for two places in different Hemispheres In like manner if we suppose Trinity Harbour to be in as much South Latitude the difference of Longitude being the same to wit 68d 30′ place P L from P to I and draw I T continued and it shall represent the great Arch being continued from I upwards runs from the Lizard towards the Equinoctial and below T downwards runs also from the supposed place in South Latitude towards the Equinoctial where in this Projection it grows infinite Let fall the Perpendicular P Q equal to the Radius and prick off the Tangents from Q downwards beyond R parallel to H T and if you set P I from P to K so as to meet with H T then you may measure the Latitude of the Ark on each side the Equator the the same way by laying the beginning of the Tangents on the edge of the Ruler to P and laying the Ruler as before to every fifth degree of Longitude thus the Latitudes of the Arch at 55d Difference of Longitude from the Perpendicular is 44d 43′ 60 40 49 65 36 7 70 30 34 75 24 4 80 16 42 85 8 33 We have continued the Tangents but to 65d at R for want of room however without any great Excursions the Latitudes of it may be found near the Equinoctial by this Proportion As the Tangent of the Perpendicular Is to the Cosine of the Angle adjacent ∷ So is the Radius To the Tangent of the Arkes Latitude Example Let it be required to finde the Latitude of the Arch at 70 degrees difference of Longitude from the Perpendicular The complement thereof is 20d to the Tangent whereof draw a line from P then the nearest distance from Q to the said line is the Cosine of the said Angle which prick from H to S a ruler over P and S cuts this Tangent-line Q R at 30 degrees 34 minutes and so much is the Latitude required and at 90 degrees from the Perpendicular this Arch passeth over the Equinoctial A third Example for two places in the same Parallel to wit in the Latitude of 50 degrees difference of Longitude 92 degrees 46 minutes Thus we see this Projection will supply all the Cases that can be put and is in effect no other then a Scheme for the carrying on of Proportions and the Sphere being thus reduced into right lines we may thence raise Proportions for Calculating all that is required In the oblique-angled plain Triangle T P L we have the two Sides T P and P L given being the Tangents of the complements of both Latitudes and the Angle comprehended T P L given now we finde the other Angles at T and L by this Proportion As the sum of the two Sides Is to their difference ∷ So is the Tangent of the half sum of the opposite Angles To the Tangent of half their difference ∷ which Proportion is demonstrated in many Books of Trigonometrie And so the two Sides of this Triangle are the Tangents of the Complements of the Latitudes But as the sum of the Tangent of any two Arkes Is to the difference of those Tangents ∷ So is the sine of the sum of those Arkes To the sine of the difference of those Arkes See this demonstrated in Mr. Newtons Trigonometria Brittanica Again the third tearm being the Tangent of the half sum of the opposite Angles in other Language is the Cotangent of half the contained Angle Lastly the half sum of the unknown Angles being added to their half difference the sum makes the greater and the difference the lesser of those unknown Angles In like manner if the said half difference be added to half the contained Angle the sum is the greater of those Angles next the Perpendicular and the difference is the lesser and these Angles we call the Vertical Angle The Proportion for finding the Perpendicular having the Hipotenusal or Colatitude and the Vertical Angle given is As the Radius Is to the Cosine of the Vertical Angle ∷ So is the Cotangent of the Latitude To the Targent of the Perpendicular ∷ Then ass gning the difference of Longitude from the Perpendicular we nave before set down the Proportion for finding the Arks Latitude and both these Proportions may be brought into one and so the Method of Calculation arising out of these Considerations suns thus 1. For a Parallel or East and West Course As the Cosine of half the difference of Longitude Is to the Tangent of the common Latitude ∷ So is the Cosine of the Ark of Longitude from the Perpendicular To the Tangent of the Arks Latitude answering thereto ∷ Again in all other Cases for the Vertical Angles As the Sine of the sum of the Complements of both Latitudes Is to the Sine of their difference ∷ So is the Tangent of the Complement of half the difference of Longitude To the Tangent of half the difference of the Vertical Angles ∷ which Ark thus found Adde to half the difference of Longitude the sum is the greater Vertical Angle which if it exceed the difference of Longitude the Perpendicular falls without and this Ark comprehends the whole Angle between the Perpendicular and both Colatitudes the difference between the fourth Proportional Ark and half the diffe●ence of Longitude is the Angle between the lesser Colatitude and the Perpendicular being the lesser Vertical Angle When one place is in South Latitude and the other in North the first and second tearms of this general Proportion change places and the Sine of the sum becomes the Sine of the difference and this that as will be found if you make one of the containing Sides greater then a Quadrant the Polar distance in stead of the Colatitude and then for the Sine of an Ark greater then a Quadrant take the Sine of that Arks complement to a

thy Works or try the Rumbs N ever desist but let 's have more of thine H ere 's but a Tangent but let 's have a Sine O r bosom full of thy industrious toyl I t will inform the weak enrich our Soyl. Your loving Friend Sylvanus Morgan The CONTENTS of the First Book In the Proportional Part. GEometrical Definitions Page 1 2 To raise Perpendiculars 2 3 4 To draw a line parallel to another Line 5 To bring three points into a Circle 6 To finde a right line equal to the Arch of a Circle 9 10 Chords Sines Tangents Secants Versed Sines c. defined 11 12 13 The Scale in the Frontispiece described 14 Plain Triangles both right and obliqued Angled resolved by Protraction from p 15 to 25 Proportions in Sines resolved by a Line of Chords p 21 to 25 Proportions in Tangents alone so resolved p. 25 27 28 29 30 Proportions in Sines and Tangents resolved by a Line of Chords p. 26 31 32 33 Particular Schemes fitted from Proportions to the Cases of Oblique Angled Sphoerical Triangles To finde the Azimuth p. 34 35 As also the Amplitude p. 36 The Azimuth Compass in the Frontispiece described p. 38 The Variation found by the Azimuth Compass p. 39 To finde the Hour of the Day p. 40 As also the Azimuth and Angle of Position p. 41 To finde the Suns Altitudes on all Hours p. 43 46 Also the Distances of places in the Arch of a great Circle p. 44 To finde the Suns Altitudes on all Azimuths p. 48 The Latitude Declination and Azimuth given to finde the Hour p. 50 to 54 ☞ To finde the Amplitude with the manner of measuring a Sine to a lesser Radius p. 55 To get the Suns Altitude by the shadow of a Thread or Gnomon p. 56 The Contents of the Treatise of Navigation OF the Imperfections and Uncertainties of Navigation p. 1 to 5 To measure a Course and Distance on the Plain Chart. p. 7 8 9 Of the quantity of a degree and of the form of the Log-board p. 9 10 A Reckoning kept in Leagues how reduced by the Pen to degrees and Centesmes p. 11 12 Of a Traverse-Quadrant p. 13 A Traverse platted on the Plain Chart without drawing Lines thereon p. 14 to 18 A Scheme with Directions to finde what Course and Way the Ship hath made through a Current p. 18 to 21 Divers Rules for Correcting of the Dead Reckoning from p. 21 to 33 Of the errors of the Plain Chart. p. 33 And how such Charts may be amended p. 34 To finde the Rumbe between two places p. 35 Proportions having one tearm the middle Latitude how far to be trusted to p. 35 to 38 To finde the Rumbe between two places by a Line of Chords onely p. 39 to 42 The Meridian-line of Mercators Chart supplied generally by a line of Chords p. 42 to 47 The Meridian-line divided from the Limbe of a Quadrant with the use thereof in finding the Rumbe p. 48 to 51 The error committed by keeping of a Reckon●ng on the Plain Chart removed p. 52 to 54 Of the nature of the Rumbe on the Globe p. 55 to 57 Mereators Chart Demonstrated from Proportion p. 58 to 60 Objections against it answered p. 60 to 63 To finde the Rumbe between two places in the Chart. p. 64 Distances of places how measured on that Chart. p. 65 to 71 Another Traverse-Quadrant fitted for that Chart with a Traverse platted thereby without drawing lines on the Chart. p. 71 to 78 To measure a Course and Distance in that Chart without the use of Compasses p. 79 Of Sailing by the Arch of a great Circle p. 81 To finde the Latitudes of the great Arch by the Stereographick Projection p. 82 to 83 Of a Tangent Projection from the Pole for finding the Latitudes of the great Arch p. 84 to 88 With a new Method of Calculation raised from it p. 89 90 And how to measure the Distance in the Arch and the Angles of Position p. 91 Another Tangent Projection from the Equinoctial for finding the Latitudes of the Arch. p. 93 to 100 And how to finde the Vertical Angles and Arkes Latitudes Geometrically p. 100 to 102 To draw a Curved-line in Mercators Chart resembling the Arch with an example for finding the Courses and Distances in following the Arch. p. 102 to 104 The Dead Reckoning cast up by Arithmetick p. 106 to 108 A brief Table of Natural Sines Tangents and Secants for each point of the Compass and the quarters p. 107 The difference of Longitude in a Dead Reckoning found by the Pen. p. 109 That a Table of Natural Sines supplyes the want of all other Tables p. 110 Many new easie Rules and Proportions to raise a Table of Natural Sines p. 111 to 113 And how by having some in store to Calculate any other Sine in the Quadrant at command p. 114 Of the contrivance of Logarithmical Tables of Numbers Sines and Tangents and how the want of Natural Tables and of a Table of the Meridian-line are supplied from them p. 117 The Sides of a Plain Triangle being given to Calculate the Angles without the help of Tables two several ways p. 118 119 An Instance thereof in Calculating a Course and Distance p. 119 CHAP. I. Containing Geometrical Definitions A Point is an imaginary Prick void of all length breadth or depth A Line is a supposed Length without breadth or depth the ends or limits whereof are Points An Angle derived from the word Angulus in Latine which signifieth a Corner is the inclination or bowing of two lines one to another and the one touching the other and not being directly joyned together If the Lines which contain the Angle be right Lines then is it called a Right lined Angle A right Angle when a right Line standing upon a right Line maketh the Angles on either side equal each of these Angles are called Right Angles and the Line erected is called a Perpendicular Line unto the other An obtuse Angle is that which is greater then a right Angle An acute Angle is that which is less then a right Angle when tvvo Angles are both acute or obtuse they are of the same kinde othervvise are said to be of different affection An Angle is commonly denoted by the middlemost of the three Letters set to the sides including the said Angle The quantity of an Angle is measured by the arch of a Circle described upon the point of Concurrence or Intersection where the two sides inclosing the said Angle meet By the complement of an Arch or Angle is meant the remainder of that Arch taken from 90d unless it be expressed the complement thereof to a Semicircle of 180d. A Circle is a plain Figure contained under one Line which is called the Circumference thereof by some the Perimeter Periphery or Limbe a portion or part thereof is called a Segment The Center thereof is a Point in the very midst thereof from which Point all right lines drawn

shall shew and another true Chart or Globe graduated from them and that in stead of putting in such abundance of Compasses and Rumbe-lines both in the Plain and Mercators Chart it were better to leave them out and to put into some spare place the Traverse-quadrant with points and half points and a Limbe divided into degrees with a Line of Sines or else the Sines of the points halfs and quarters onely 3. That being so made near the Equinoctial they are very near the truth and may there very well serve as also under the Meridian and for short Distances or Voyages and how to take away the Error of the Chart and make them serve for long and remote Voyages shall be handled in order whereto the first Proposition in the use of the Plain Chart must be repeated Propos I. The Longitudes and Latitudes of any two places being given to finde the true Course or Rumbe between those places and the Distance in the Rumbe This of all other Propositions in Navigation is the most useful and withal the most difficult and hath as learned Snellius well observes been often attempted by the Learned but in vain and to give our own Nation its due repute was never generally and satisfactorily performed by any man till our late famous Countrey-man Mr. Edward Wright invented that excellent Chart called Mercators Chart but ought more properly to be called Wrights Chart the Meridian-line whereof requires a Table to be made by the perpetual Addition of Secants without which Table as yet there are no Proportions known that will serve to calculate the Rumbe generally between any two places by help of the Natural Tables of Sines Tangents and Secants onely and whatsoever may be done by those Tables may be also done geometrically by Schemes True it is this Proposition may be performed by the differences of the Logarithmical Tangents having a Table of them as in Mr. Norwoods Epitome without the help of a Table of the Meridian-line but as yet we have no geometrical way known for making the Logarithmical-lines of Tangents nor Sines and numbers Before I proceed any further it may be objected That we have Proportions in our English Books delivered for calculating the Rumbe between two places which Proportions may be performed by the Natural Tables of Sines and Tangents onely as namely in Mr. Gunters Works both in the former and latter Editions in the third Edition in page 90. As the difference of Latitude Is to the Cosine of the middle Latitude ∷ So is the difference of Longitude To the Tangent of the Rumbe from the Meridian A precious Proportion if it were true the true Proportion is As the Meridional parts between both Latitudes Is to the Radius ∷ So is the difference of Longitude To the Tangent of the Rumbe ∷ The Meridional parts are to be taken out of a Table of the Meridian-line by substracting the Meridional parts of the lesser Latitude from the Meridional parts of the greater Latitude by comparing these two Proportions together because the third and fourth tearm are alike in each it would follow that we might calculate the Meridional parts required without the perpetual addition of Secants by this Proportion raised out of the two former tearms of each Proportion As the Cosine of the middle Latitude Is to the difference of Latitude if in one Hemisphere or the sum of both Latitudes if in different Hemispheres ∷ So is the Radius To the Meridional parts ∷ And so by this Proportion the Meridional parts answering to the Latitude of 50d should be 55d 157 but are in truth 57d 909. 70d 85d 459 by the Table 99d 431.       Difference 41 522.       Sum 157,340 which sufficiently refutes the truth of Mr. Gunters Proportion in calculating a Rumbe from the Equinoctial But now as for finding the Meridional parts between the two former Latitudes in one or both Hemispheres the middle Latitude in one Hemisphere will be 60d and the Meridional parts 40d 00 and in both Hemispheres if found at once by the middle Latitude which is 10d is 121d 85 or if found severally at twice 140d 616 which varying from the truth as above expressed in the Sum and Difference we may conclude that the Proportion is very unsound and intolerable for any great difference of Latitude but Mr. Gunters Works deliver no Caution about it Before we can finde the distance the Rumbe must be calculated and if that be false a small error therein may cause a considerable error in the distance Where the difference of Latitude is not above five degrees it may serve very well near the truth from the Equinoctial to 60d of Latitude and afterwards to 80d it will not serve for three degrees difference of Latitude and in all Cases the Cosine of the middle Latitude is a tearm too great the middle Latitude being too small and I think no certain Rule can be given to correct it From what hath been said the Reader may take due Caution how far to depend upon such Proportions whereof one tearm is the middle Latitude such are As the Cosine of the middle Latitude To the Radius ∷ Or As the Radius Is to the Secant of the middle Latitude ∷ So is the Departure from the Meridian To the Difference of Longitude And to this Proportion Mr. Phillips his late Table of Secants are fitted in the use whereof the middle Latitude must always be taken to be a whole degree unless you will by proportion finde the difference required And I taking it to be what it truly happened did cast up the Courses and Distances Mr. Norwood expresses in his Trigonometry home from the Bermudas to the Lizard by his Tables in the said Book and found I had gotten almost half a degree of Longitude in the whole too much by reason the Proportion is not sound of which Mr. Norwood makes no use those Courses and Distances are truly expressed in Mr. Phillips his Geometrical Sea-man pag. 31. whereas in the last Impression of Mr. Norwoods Book they are mis-printed Another Proportion of this kinde is As the Cosine of the middle Latitude Is to the Sine of the Rumbe from the Meridian ∷ So is the Distance sayled To the Difference of Longitude ∷ So also there may be two Proportions for finding the enlarged Distance As to the Radius To the Secant of the middle Latitude ∷ So is the distance run To the enlarged distance on Mercators Chart ∷ Otherwise As the Sine of the Rumbe from the Meridian Is to the Secant of the middle Latitude ∷ So is the Departure from the Meridian To the enlarged Distance ∷ Note when these Proportions are used the measure of the enlarged distance must be taken from the degrees of Longitude in Mercators Chart. Another Proportion for finding the Rumbe may be gathered from Mr. Hansons Additions to Pitiscus his Trigonometry and is As the difference of Latitude Is to the half Sum of the Cosines of

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 o●●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 Propo●tion of the Cosine to the Sine given being the same with that of the Radius to the Tangent and the sum o● 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 fo●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

other Arch is equal to the square of the sine of the middle Arch from thence it followes that the sine of any Arch in the Quadrant being given we may by this Rule and extracting the square root perpetually bisect or finde the sine of the middle Arch between it and 90d and so run up very speedily towards the end of the Quadrant of which sine so found the Cosine may be found by another extraction and then you have the sine of as small an Arch as you please near the beginning of the Quadrant from which by Proportion in regard the length of the sine and of the Arch to which it doth belong have no sensible difference you may finde the sine of one minute or Centesme and thence by the former Proportions raise a whole Table of Sines to as large a Radius as you please albeit the Proportion of the Diameter to the Circumference were wholly unknown which notwithstanding hereby might be found for having made a sine to a very small part of the Quadrant the double thereof is the side of a Polygon inscribed and by Proportion As the Cosine so found Is to its Sine ∷ So is the Radius To the Tangent of the Arch to which the small Sine belongs ∷ The double of which Tangent is the side of the like Polygon circumscribed the length of the Arch of the Circle contained between the sides of these two Polygons being greater then the side of the inscribed Polygon and lesser then the side of the circumscribed Polygon In finding the Sine of any middle Arch by the former Consectary to shun the trouble of multiplying we may assume the Radius to be 2 with a competent number of Cyphers then the Sine of 30d because it is half the Radius will be an Unit with as many Cyphers then to finde the Sine of 60d to that Radius to the Sine of 30d prefix to the left hand the number 2 and annex the Cyphers in the Radius to the right hand and the square root of the number so made shall be the Sine required and the like for finding the Sine of the middle Arch between the Radius and the Sine of any other Arch given and retaining the Radius 2 with Cyphers the difference between the Sine of any Arch given and the Radius having the figures of the Radius annexed thereto the square root of the number so composed shall be the Cosine of the middle Arch or we may deliver it as a Consectary from former Proportions That the Rectangle of half the Radius and of the difference between the Sine of any Arch given and the Radius shall be equal to the square of the Cosine of the middle arch to any Radius whatsoever We forbear to suit Arithmetical Examples to the fore-going Proportions and Consectaries supposing the Reader furnished with so much Arithmetick as that he can extract the square root and work the Golden Rule or Rule of Three See page 12 of the First Part. Of Books of Tables Such Tables as have a degree divided into 100 parts or Centesms are to be preferred before those that divide the degree but into 60 Parts Minutes or Sexagesms because they are more exact in calculation and more speedy in finding the part proportional and it were convenient for a Book of Tables to be so contrived that the Natural Tables of Sines Tangents and Versed Sines to 90d might stand against the Logarithmical Tables of Sines and Tangents in a portable Book to be had by it self to which might be added a Table of the Meridian-line to each or every second Centesm with Tables of the Suns Declination Right Ascension and of the Longitudes Latitudes Declinations and Right Ascensions of some of the principal fixed Stars and of the Longitudes and Latitudes of places but those that have the Logarithmical Tables of Numbers Sines and Tangents without the Natural may supply the want of them Example 1. Let the Natural Sine of 38d be required the Logarithmical Sine of that Ark is 9 789342 because we may make the Radius of the Natural Sines to be an Unit and all the rest to be Decimal parts thereof reject the first figure of the Logarithm or Characterisk being 9 and seek the remaining figures of the Logarithmical Sine to wit 789342 amongst the Logarithms of absolute Numbers and you will finde the absolute Number answering thereto to be 61566 nearest and that is the Natural Sine of 38 degrees required 2. The Table of the Meridian-line may be supplied by this Proportion raised out of Mr. Bonds Additions to Mr. Gunters Works As 75795 Is to 60 ∷ 100 ∷ So is the difference of the Logarithmical Tangents of 45d and of an Ark compounded or made of 45d and of half the given Ark To the Meridional parts belonging to the given Ark ∷ If you use the number 60 you will make such a Sexagesimal Table as Mr. Wrights or Mr. Norwoods but if you use the number 100 then you will produce a Centesimal Table like Mr. Gunters or that in Mr. Roes Tables in which the Arks to which the Meridional Tables are fitted are degrees and every other minute but the Tabular Numbers are degrees and decimals being 〈◊〉 very good Table of that kinde and much fuller then either Mr. Gunters or Mr. Norwoods the Table at the end of this Book is of the same kinde but fitted to every second Centesm of a degree in stead of every second minute The ground of the former Proportion is that the Logarithmical Tangents above 45d accounting every half degree for a whole one are in the same Ratio or Proportion with a Table of the Meridian-line whence also it follows for Instrumental use that a Line of Logarithmical Tangents will supply the defect of the Meridian-line The number above used to wit 75795 is not the difference between the Logarithmical Tangents of 45d and 45½ d though near it being the difference of the Logarithmical Tangents of 45d and of 45d and one Centesm more multiplied by 50 and for to avoid the trouble of Division in working this Proportion it were convenient either to have a Table of the said Number multiplied by all the nine Digits or rather to alter the Proportion so as an Unit might be the first tearm and then making such a Table for the second tearm the said Proportion in a manner would be wrought wholly by Addition and Substraction What former Ages performed by Tables this latter Age hath endeavoured in some respects to perform without them Snellius in his Cyclometria shews us how the Sides of a Plain right Angled Triangle being given we may without Tables finde the Angles of that Triangle In the right angled Plain Triangle C E A with the Radius C A describe a Semicircle produce the Diameter and therein make D R ●qual to the Radius and draw R B passing through A till it meets with the Tangent I B at B now that