riseth at 6 of the Clock and so there is no Difference of Ascension for he is then in the Aequinoctial which cuts the Horizon in the two opposite points of East and West In this Question the Difference of Ascension is O P and is a distance upon a line that goeth not through the Center therefore take half the length of the parallel I D and proceed as you did in the third Question and after you have found it in degrees and minutes convert it into hours and minutes of time and set it down I find it to be here 16 deg 18 min. which is 1 hour 5′ 3 15. By the Difference of Ascension thus found you may find the length of the day or night the hour of Suns Rising or the hour of uns Setting QUESTION VII For the time of the Suns Rising or Setting in this Example I Consider as much as the Sun riseth before 6 so much he sets after 6 here he riseth before 6 of the Clock 1 hour 5′ subtract that from 6 hours and you have the hour of the Suns Rising 4 h. 55′ add it to 6 hours and you have the time the Sun sets 7 h. 5′ double that and it is the length of the whole day which is here 14 deg 10 min. Subtract the length of the day from 24 hours and it leaves the length of the night 9 h. 50′ I omitted the Fractions But if the Sun riseth after 6 of the Clock and you have a desire to find these things as he doth when he is South Declination add the Difference of Ascension to 6 hours and it gives the time of the Suns Rising subtract it from 6 and it gives the time of Suns Setting and that doubled is the length of the whole day Again that subtracted from 24 hours is the length of the night QUESTION VIII To find the hour of the Suns being due East or West THe Sun is due East in this Example or any other when he goeth by the East point of the Horizon or West when he goeth by the West point In this Example the Suns parallel of Declination cuts the East and West Azimuth in S which is later in the morning than 6 of the Clock by the distance S P therefore see what S P is by The third way of Measuring and convert it into time and add it to 6 hours it shall give the hour of the Suns being due East I find it in this Example to be 6 h. 46′ Subtract this from 12 hours and it gives the time of the Suns being due West that day for as many hours and minutes as the Sun is due East before 12 of the Clock so many hours and minutes must it be due West after 12 According to this Example the Sun is due West at 5 of the Clock 14 min. If the Sun hath South Declination he passeth the point of East before he riseth and is set as long before he comes to the point of West provided the Latitude be Northerly as this is but if the Latitude and Declination be both one way the Sun is always up before he cometh to the point of East and the work is as I have shewed QUESTION IX To find the time of Day breaking and Twilight ending IT is an antient Observation and concluded Opinion That the Sun makes some shew of Day when he is 17 degrees under the Horizon therefore take 17 deg from your Scale of Chords and set it from both ends of the Horizon downwards and draw the line T r then fix your Compasses in r the place where the Suns parallel of Declination intersects that line and extend them to 6 of the Clock set it off by The third way of Measuring and convert it into time Subtract that from 6 of the Clock and it gives the time of Day breaking I find in this Example Day breaks at 3 of the Clock 6 min. â—8 15. Add it to 6 hours and it gives the time of Twilight ending 8 of the Clock 3 53 14 13. It may happen so sometimes that it may be past 6 of the Clock before the Day breaks in such a case you must add in the morning for break of Day and subtract for Night from 6 hours which is the contrary These things your own Reason will give you after you are used to it which makes me forbear to give any more reasons of it QUESTION X. To find the Continuance of Twilight THe Continuance of Twilight is the time between the Day breaking and Sun rising which is r o take it off by the Third way of Measuring and convert it into time I find it to be 1 hour 48 min. QUESTION XI To find the Length of the longest Day in that Latitude VVHen the Sun is nearest the Zenith in any Latitude that day must be the longest now in places near the Aequinoctial as between the Tropicks there is but little difference all the year long but in places nearer the Poles there is more The Sun is nearest the Zenith in this Latitude when he is in the Tropick of Cancer so that then must be the longest Day Imagine the Tripick of Cancer to be the parallel of the Suns Declination as indeed it is that day take the distance between Y and R which is between Suns Rising and 6 of the Clock that day and set it off by the Third way of Measuring the number of degrees and minutes convert into time and add it to 6 hours which makes the length of the Forenoon and that doubled is the length of the whole day as in Quest 7. which is equal to the longest Night in that Latitude I find it here to be 16 hours 10 minutes Subtract the length of this longest day from 24 h. 00 min. And it leaves the length of the shortest night 7 h. 50 min. Equal to which is the length of the shortest day 7 h. 50 min. But you may measure the length of the shortest day and subtract that from 24 hours and it will be the length of the longest night which is equal to the longest day Now the shortest day is when the Sun is in the Tropick of Capricorn for then it is latest before he riseth Now to measure it take the distance from 6 of the Clock which is T and the Suns Rising which is U and set it off by the Third way of Measuring convert the degrees into time and subtract that time from 6 hours it leaves the length of the Forenoon which doubled is the length of the shortest day as in Quest 7. I have spoke to that purpose I need say no more that 7 th Question is light sufficient I might do an Example of what is before done in a South Latitude but Reason gives that the same which the South Pole or Southern parts of the Heavens are deprest in a Northern Latitude the same will the Northern parts of the Heavens be deprest in a Latitude as far Southerly so that there will be no difference
THE Seaman's Companion BEING A Plain Guide to the Understanding OF ARITHMETICK GEOMETRY TRIGONOMETRY NAVIGATION and ASTRONOMY Applied Chiefly to NAVIGATION AND Furnished with a Table of Meridional Parts to every third Minute With excellent and easie ways of keeping a Reckoning at Sea never in Print before ALSO A Catalogue of the Longitude and Latitude of the principal Places in the World With other useful things The Third Edition corrected and amended By MATTHEW NORWOOD Mariner LONDON â—â—â—â—â—ed by Anne Godbid and John Playford for William Fisher at the Postern-Gate near Tower-Hill Robert Boulter at the Turks-Head and Ralph Smith at the Bible in Cornhill Thomas Passinger at the Three Bibles on London-Bridge and Richard Northcot next St. Peter's Alley in Cornhill and at the Anchor and Mariner on Fishstreet-Hill TO THE READER THE famous and ever to be admired Art of Navigation having been so learnedly handled and written of not only in all other Languages but also in our Mother-Tongue by so many learned and able men both of former and of our present Age that it may seem impossible to write any thing more thereof that hath not already been done by others yet in my experience which I have seen being at Sea in several Vessels where divers young Mariners have been I have heard this general Complaint among them that though the things that are chiefly useful for them in their Art may be found in several Books here and there dispersed yet they could wish that there were such a Book contrived that might be soley useful for them entire by it self which then would be more convenient for them and might be purchased at a more reasonable rate than otherwise they could be by buying of so many sorts of Books which they must be constrained to do if ever they intend to be able Proficients in that most noble Profession of Navigation at which they chiefly aimed This Complaint of theirs was one chief Motive which induced me to collect and compose the subsequent Treatise which I have endeavoured to handle in such a methodical manner as it ought to be read and practised by the young Seaman For First There is a Treatise of ARITHMETICK containing all the Rules thereof which are necessary for the Seaman to know and practise all or most of the Questions thereof being made applicable in one kind or other to Nautical Affairs Secondly There is a Treatise of GEOMETRY containing the first Grounds and Principles thereof with the making and dividing of the Mariners Scale and Compass with the projecting of the Sphere in Plano and the resolving of many Questions in Astronomy which are useful in Navigation thereby Thirdly you have a Treatise of the practick part of NAVIGATION wherein is shewed after a new experienced way used by the Author how to keep a Reckoning at Sea the making and use of the Plain Sea-Chart the Doctrine of plain Triangles made applicable to such Questions in Navigation as concern Course Distance Rumb Difference of Longitude and Departure likewise a Table of Meridional parts to every third Minute and the application and use thereof exemplified in Questions of Sailing by Mercator's Chart with Tables of Longitude and Latitude of Places of right Ascension and Seasons of certain Fixed Stars with Rules to keep a Reckoning and to find the Latitude by the Meridian Altitude of the Sun or Stars Fourthly There is a short Treatise of ASTRONOMY wherein you have the Doctrine of Spherical Triangles applied to Questions in Astronomy and Navigation This is the brief Sum and Substance of the following Treatise which I commend to the Practice of all young Sea-men desiring their kind acceptance of these my first Labours which if I shall find to be kindly entertained by them it will encourage me to lanch farther into the more nice and critical part of this most noble Science In the mean time I commend this to them wishing good success in all their honest and laudable undertakings and in the interim bid them Farewel MATTHEW NORWOOD THE CONTENTS Of ARITHMETICK NUmeration 1 Addition 2 Subtraction 5 Multiplication 7 Division 10 Reduction 16 The Golden Rule or Rule of Three 19 The Rule of Interest and Interest upon Interest 24 The Rule of Fellowship 25 Of GEOMETRY Geometrical Definitions 28 How to raise a Perpendicular 30 How to divide a line into two equal parts 31 How to raise a Perpendicular on the end of a line ibid. From a Point given to let fall a Perpendicular 32 How to draw Parallel lines 33 How to make a Geometrical Square 34 How to make an Oblong or Square whose length and breadth is given 34 How to make a Diamond-figure whose Side and Angles shall be limited 35 To make a Rhomboiades whose Sides and Angles are given 36 To find the Center of a Circle 37 To find the Center of that Circle which shall pass through any three given points which are not situate in a streight line 37 How to divide a Circle into 2 4 8 16 32 equal parts 40 The Projection of the Mariners Compass 41 The Projection of the Plain Scale 42 The Projection of the Sphere in Plano 45 The Names and Characters of the Signs with the Months they belong to 48 How to project the Sphere 49 The meaning of certain Terms of Art 51 How to find the Suns Meridian Altitude 55 How to find the Amplitude of rising or setting ibid. How to find the Azimuth at six of the Clock 57 To find the Altitude at six 58 To find the Altitude the Sun being due East or West 59 How to find the Ascensional Difference ibid. To find the time of the Suns rising or setting 60 To find what hour it is the Sun being due East or West 61 To find the time of Day breaking and Twilight ending 62 To find the Continuance of Twilight ibid. To find the length of the longest day in any Latitude 63 To find the Suns Place and right Ascension 68 To find the Suns declination c. 69 Terms of Art used in Navigation explained 70 Prâ—positions of Sailing by the Plain Scale 72 Questions in Navigation resolved from page 73 to page 92 Of a Travis 92 The manner of keeping a Reckoning at Sea 97 Concerning the Variation of the Compass 102 The Use of a Plain Sea-Chart 108 Of Oblique TRIANGLES THe application whereof are in ten Questions perfectly explained all which Questions are applied to Questions in Sailing and are wrought both Geometrically by the Plain Scale and also by the Tables of Sines Tangents and Logarithms from page 113 to page 146 A Table of Meridional parts to every third minute 146 A Declaration of the Table 158 The Use of the Table of Meridional parts exemplified in five Questions appertaining to Navigation from page 160 to page 166 Of the Longitude and Latitude of Places A Table of them 167 The Use of them 171 How to keep a Reckoning of the Longitude and Latitude a Ship makes at Sea 172 The Names Declinations and Seasons
but in any case take this in general that if the Difference of Ascension be before 6 of the Clock subtract it if after 6 add it to 6 hours and you have the time of Sun Rising This stands to good reason forasmuch as the Difference of Ascension is the portion of time that the Sun riseth before or after 6 of the Clock In this Example you see the Difference of Ascension is before 6 of the Clock 1 hour 5′ 3 15 we will omit the part of a part of a minute I would know the time of the Suns Rising Example Subtract the Difference of Ascension 1 h. 5 m. 3 15 From 6 hours 5 60 The remainder is the time of Sun Rising 4 54 12 15 or â…˜ Thus I conclude the Sun riseth at 4 a Clock 54 min. â…˜ If you have a desire to find the time of Sun setting subtract the time of Sun rising from 12 hours and you have it for as many hours and minutes as the Sun riseth before 12 so many hours and minutes he sets after 12. Example Here the Sun riseth at 4 of the Clock 54 m. â…˜ which we express thus 4 h. 54 m. â…˜ This subtracted from 11 60 Leaves the time of Sun setting 7 05 â…• Which is 5 min. â…• past 7 of the Clock in the afternoon This doubled is the length of the whole day which is 14 hours 10 min â…– This subtracted from 24 hours is the length of the night 9 hours 49 min. 8 5. To find the length of the longest Day in that Latitude before proposed VVHen the days are at the longest in any North Latitude the Sun is in the Tropick of Cancer in a Southern Latitude in the Tropick of Capricorn This is a Northern Latitude therefore make the Tropick of Cancer the Parallel of the Suns Declination as here O R is the Parallel of the Suns Declination then must I 6 be the Difference of Ascension which is = to F ♈ In the right Angled Triangle I F ♈ right angled at F you have I ♈ F the Complement of the Latitude 40 deg given and I F the Suns Declination 23 deg 30 min. to find F ♈ find it as you was shewed to find the Difference of Ascension before convert it into time and find the length of the day as was shewed before This note that Sine s I F L is an Arch of the Meridian cutting the Horizon in that place of it where the Sun riseth Sine ♈ F + Radius is = to Tang. I F + Tang. comp I ♈ F. Tang. I F Suns Declination 23 deg 30 min. 9,638301 Tang. com I ♈ F Poles Elevation 40 deg 0 min. 10,076186 Sine F ♈ Difference of Ascension 31 d. 12 m. 39 sec 9,714487 But you may ask all this while how this Fraction is found it is thus found Admit I would find the sine of the Arch answering to 9714487 I look in the Sines and find the nearest less than it to be  9,714352 I take the next greater than it and find it to be 9,714560 I subtract the lesser from the greater the remainder is 208 Then from the figures that came forth 9,714487 I subtract the nearest in the Tables less 9,714352 And the Remainder is 135 Then say As the Difference between the nearest less and the nearest greater 280 comp arith 7,681936 Is to the Difference between the same and that less 135 2,130333 So is 60 min. 1,778151 To 39 sec 1,590420 The like is to be understood of any Fraction else if you have occasion for the artificial Sine or Tangent of an Arch that hath a Fraction to it Say As 60 seconds Is to the seconds in the Fraction which is here 39 So is the Difference between the nearest lesser and the nearest greater 208 To the Fraction 135 And this added to the lesser makes the artificial Sine of the Arch required 9,714352 9,714487 The like is to be understood of a Tangent  To find the hour of the Suns being due East or West Latitude 50 deg 0 min. Declination 13 deg 15 min. Northerly I demand the time of the Suns being due East or West And for the time of the Suns being due West you are to subtract it from 6 hours in this case and the remainder is your desire The reason is because as many hours as the Sun is due East after 6 of the Clock in the morning so long time is he due West before 6 of the Clock in the Afternoon I have left the working of this to your own practice only I have set down the Resolution of it The Sun is due East 45 min. 9 15 past 6 of the Clock The Sun is due West 45 min. 9 15 before 6 of the Clock Which is at 5 of the Clock 14 min. 6 15. Note That if the Suns Declination be Southerly then will he be East before 6 so that whereas here you add there you subtract from 6 hours to find the hour of the Suns being due East or add to 6 hours for the time of its being due West but your own Reason if you look well on the Scheme will guide you to know this and also to know that it is useless in such cases for then he is not above the Horizon Latitude 50 deg 0 min. Declination 13 deg 15 min. I demand the Continuance of Twilight AS I have noted before the Sun is accounted 17 deg under the Horizon when the day breaks therefore in the following Scheme let 17 I be an Arch parallel to the Horizon 17 deg under it Let D B C A be an Azimuth cutting the Aequinoctial and the line of 17 deg in the place of their intersection which is at C then in the Triangle r B C right angled at B you have given B C 17 deg the Angle B r C the Complement of the Poles Elevation 40 deg to find C r the continuance of Twilight for C r and B u are equal I leave it to your own Practice I find the continuance of Twilight to be 27 deg 4 min. which converted into time is 1 h. 48 min. 4 13 nearest So I conclude that between the Day breaking and Sun rising it is 1 h. 48 min. 4 15 which is u B Now if you add this to the Difference of Ascension before found B 6 which was 1 h. 5 min. 3 15 of a minute I omit the smaller Fraction you have the time between break of day and 6 of the Clock which may be termed u B 6 2 h. 53 min. 7 15. And because the day breaks so much before 6 of the Clock if you subtract it from 6 hours you have the hour and minute of day breaking which is at 3 of the Clock 6 min. â—8 15. Add it to 6 of the Clock and you have the time of Twilight ending which is at 8 h. 53 min. 7 15. To find the Suns Place and Right Ascension provided the Latitude and Declination be given Latitude 50
of certain Fixed Stars near the Equinoctial 179 The like for Stars near the North Pole 180 A Table shewing how much the North Star is above or below the Pole for every several Position of the former or greater Guard 181 Rules to find the Latitude by the Meridian Altitude of the Sun or Stars 182 Of ASTRONOMY VVHerein the Application and Use of the Doctrine of Spherical Triangles is exemplified in the Resolution of several Propositions of the Sphere which appertain to Astronomy and Navigation from page 185 to page 203 OF ARITHMETICK THE TABLE 7000000000000 Millions of millions 800000000000 Hundred thousand millions 90000000000 Ten thousand millions 6000000000 Thousand millions 500000000 Hundred millions 30000000 Ten millions 2000000 Millions 800000 Hundred thousands 40000 Ten thousands 5000 Thousands 700 Hundreds 10 Tens 1 Vnits The Use of the Table to number THis Table signifies thus much That if there be one Figure alone it is but so many units as there is in its name as 7 is seven ones or units If there be two Figures the first Figure towards the left hand is as many tens as there is units in its name and the other Figure is units As if it were 10 that is ten units if it were 23 the first Figure is two tens that is twenty units and the next Figure is 3 that is three units The next place is hundreds and consists of three Figures as 700 is seven hundred 723 is seven hundred twenty three And thus you may count any Number by observing the place of any Figure and in their places giving them their names as you count As suppose I have a Number namely 7864319 if you tell from 9 backwards to 7 you will find 7 in the place of millions then begin and say seven millions then if you count again you will find 8 in the place of hundred thousands 6 in the place of ten thousands and 4 in the place of thousands then to give them their proper Number you will read thus Seven millions for 7 eight hundred thousand for 8 sixty thousand for 6 four thousand for 4 that is Seven millions eight hundred sixty four thousand then for 3 three hundred for 1 ten and for 9 nine units that is then altogether 7 millions 8 hundred 64 thousand 3 hundred and ninteen which is the quantity of units that is in the given Number 7864319. And thus much for Numeration Addition ADdition teacheth how to bring several Sums into one total and is done as in the Example thus Suppose there be several Persons indebted to me and they owe as followeth Example A oweth 7272 B oweth 2732 C oweth 3399 D oweth 3999 Total is or debt 17402 I demand what the debt is Add the numbers together thus begin at the right hand and for every ten carry one to the next row adding them up in their several rows til you come to the last row on the left hand and as many tens as is in that after it is added up so many set down by it in the next place on the left hand Example In the first row towards the right hand say 9 and 9 is 18 and 2 is twenty and 2 is twenty two which should be expressed thus 22 but because the next place is the place of tens you set down 2 units and carry the other two which is two tens and add them to the next row of tens by saying 2 that I carried and 9 is eleven and 9 is twenty and so forwards still doing so till you come to the last row and as many tens as there is in that so many set down to the left hand as here it came but to 17 that is but one ten besides the 7 if they come to even tens as here one row came to thirty set down 0 and carry the tens to the next place except it be in the last roâ— to the left hand and then set down 0 and your number of tens to the left hand of it The Characters used in Arithmetick For pounds l. For shillings s. For pence d. For farthings q. For degrees deg For minutes ′ For seconds ″ For hours ho. Minutes of time ′ For hundreds C. For quarters qu. For ounces oun Addition of Degrees and Minutes NOte that sixty Minutes is one Degree three Miles is one League sixty Seconds is one Mile or Minute sixty Minutes of time is one Hour in time fifteen Minutes of a Degree is one Minute in time twelve Hours is an artificial Day twenty four Hours is a natural Day fifteen Degrees is an Hour in time four Minutes of time is one Degree three hundred and sixty Degrees is the Circumference of a Circle one hundred and eighty Degrees is half a Circle or a Semicircle ninety Degrees is a Quadrant or a quarter of a Circle and eleven Degrees fifteen Minutes is one point of the Compass two and thirty times eleven Degrees fifteen Minutes is the Circumference of a Circle so that there is two and thirty points of the Compass in every Circle one point of the Compass is three quarters of an Hour in time or 45 Minutes Admit I had kept an account of a Ships Difference of Longitude in Degrees and Minutes that had been out eight days and had made Difference of Longitude each day as followeth The first day 2 deg 20′ The second day 2 deg 45 The third day 1 deg 30 The fourth day 1 deg 11 The fifth day 2 deg 39′ The sixth day 1 deg 29 The seventh day 2 deg 19 The eighth day 2 deg 10 I desire to know if it be all one way what number of degrees and minutes it is in one sum Begin as you did in Addition and say 9 and 9 is 18 and 9 is 27 and 1 is 28 and 5 is 33 set down 3 and carry 3 then say 3 that I carried and 1 is 4 and 1 is 5 and 2 is 7 and 3 is â—0 and 1 is a 11 and 3 is 14 and 4 is 18 and 2 is 20 cast out all the sixes from 20 and there remains 2 set it down and as many sixes as you cast away so many units carry to the place of Degrees which was three sixes and say 3 that I carried and 2 is 5 and so forwards in that as in any other Sum in Addition it being thus added I find that it comes to 16 d. 23 m. the whole 2 deg 20′ 2 deg 45 1 deg 30 1 deg 11 2 deg 39 1 deg 29 2 deg 19 2 deg 10 16 deg 23 Now you may ask a reason why I cast away the sixes the reason is because 6 tens make a degree now with the three tens that I carried and the tens that where in the row added up last there were 20 tens that must be 3 degrees and 2 tens over the 3 degrees I carry to the place of degrees and the two tens I set down in the place of tens and thus in any other Sum. Addition of Hours and
Sum stands thus You might have said 3 times 3 is 9 from 8 I cannot but 9 from 18 and the Remainder is 9 and 1 for your borrowed one ten from 1 and there remains 0 which is most easie for memory Then set your Divisor on place further towards the right hand in this form How many times 2 can you have from 9 Four times then 4 times 2 is 8 from 9 and there remains 1 and 4 times 3 is 12 from 19 and the Remainder is 7 and 1 for the ten from 1 and the Remainder is 0 and then your Sum stands thus Then remove your Divisor to the next place and see how many times 2 you can have out of 7 which is 3 times 3 times 2 is 6 from 7 and the remainder is 1 and 3 times 3 is 9 from 9 and the remainder is 0 and your Sum stands thus Then remove your Divisor to the next place and see how many times 2 you can have from 10 which is 5 times now 5 times 2 is 10 from 10 and there remains 0 and 5 times 3 is 15 from 8 I cannot you see 8 is all that is left so that you have nothing to horrow from therefore you could not take 5 times 23 from 108 try for 4 times 4 times 2 is 8 from 10 and the remainder is 2 and 4 times 3 is 12 from 28 I can you see you may take 4 set 4 in the Quotient and say 4 times 2 is 8 from 0 I cannot but 8 from 10 and the remainder is 2 and 1 that I borrowed from 1 and the remainder is 0 cancel as you speak then 4 times 3 is 12 now 12 from 28 and the remainder is 16 but it is better to use your self to this method namely to take your units from the place of units and your tens from the place of tens as here 2 is in the place of units and 8 above therefore say 2 from 8 and there remains 6 and 1 for the ten from 2 and the remainder is 1 Now if this had fallen out so that the upper number had been 21 and the other 12 you know that 12 will be taken out of 21 but 2 would not be taken out of 1 now in such a case borrow one 10 and say as in Subtraction 2 from 1 I cannot but 2 from 11 and the remainder is 9 one 10 that I borrowed and 1 is 2 from 2 and there remains 0 and thus it is done at once The reason why it is done this way is because the Sum you are to take from may be big so that a man cannot tell readily what the remainder will be without Subtraction for this differs nothing from Subtraction as it is shewed before but only that you carry one number in your head whereas there both the numbers are before you The hardest Sum in Division hath but these Difficulties in them First Be sure take no more to set in the Quotient than your Divisor multiplied by it will come under the Number you are to take it from or equal to it Secondly Be sure you take the one as many times as you can from the other Thirdly Be careful in takeing one number from another to use this way of Subtraction This Sum is done and stands thus And it signifies that the 23. part of 78998 is 3434 units and 16 23 of a unit Here following is a Sum of four figures done and the way to prove any Sum. 34771262 2345 This Sum is set down in its several Operations From what hath been said already I suppose you may be able to examine it the like is to be understood of any Sum else in Division therefore I shall say no more only shew how to prove them when they are thus wrought The best way to prove Division is by Multiplication for if you multiply your Quotient by the Divifor and add in the Remainder that product will be equal to the Dividend if the Sum be right and it stands to reason it should be so for if the Quotient with the remainder be in this Sum the 2345th part of the Dividend then 't is evident that 2345 times that Quotient must be the same that the Dividend is with the remainder which is added in Example Quotient 14827 Divisor 2345  74135  59308 Remainder 7  44481 4  26654 9   1 Divid 34771262 Here you see the Example of it and the Product the same that the Dividend of the Sum divided was and this I say is the best way for young Practitioners to prove Division because this makes them perfect in Multiplication There is a shorter way to prove any Sum in Division and that is this Cast up your Quotient and cast away all the nines from it as you do in Multiplication and what remains set down upon one side of a Cross thus Then cast away your nines from your Divisor in like manner and there remains 5 set it down on the other side against 4 thus Multiply them one by the other and they produce 20 cast away all the nines from 20 and the remainder is 2 keep this 2 in mind and cast up the remainder of the Division and cast all the nines from it which if it be done in this Sum afterwards there will remain 3 which add to the 2 that was in your mind and it makes 5 which set down thus Then cast away all the nines from your Dividend and the remainder will answer to the last number you set down in the Cross if the Sum be right which here it is Reduction REduction sheweth how to bring gross or great Denominations into small or small into great Quest 1. Suppose a Man had been at Sea twenty five Years and for every Minute of that time he was to have on Farthing I demand how many farthings is due to him Here I see how many minutes is in that quantity of time and so many minutes so many farthings there is due which is 13140000 farthings Now to find this I first multiply the years by the days contained in one year which is 365 and that produceth 9125 days the number of days in that number of years this multiplied by 24 gives for its Product 219000 the number of hours in the whole time because 24 hours is a day That Product multiplied by 60 is the number of minutes in the whole time because every hour is 60 minutes which is the last Product or the thing required which I set down as the Answer to my Question thus Facit 13140000 farthings Quest 2. Pray tell me what these farthings come to in Pounds Shillings and Pence 1740120 Farthings Four farthings make a penny therefore I divide the whole Sum of Farthings by 4 and the Quotient is ½ of the number of farthings or the farthings reduced into pence 435030 d. of which Sum I take the 1 12 by dividing it by 12 which is the next Quotient the number of shillings that is the farthings which
a great Circle for by great Circles is the Sphere measured QUESTION I. To find the Meridian Altitude of the Sun IN this or any other Sphere M G E is that part of the Heavens that is visible the other half invisible to us for it is parted from our sight by the Earth and Sea and the furthest part of it which seems to mix as it were with the Heavens we call the Horizon which is the great Circle M E for it is a great Circle though here we are forced to represent it by a streight Line M is the South point of the Horizon Now the Suns Meridian Altitude is his distance between that point and the place he cuts the Meridian that day which is M I fix your Compasses in M and extend the other foot to I and apply it to your Scale of Chords and as many degrees as you find it there so many degrees is the Sun high when he is upon the Meridian that day which is the thing required Note that at O the Sun riseth at P it is 6 of the Clock and P I is equal to ♈ K when it is extended to a great Circle and both the Sine of 90 deg which is extended to a Chord must be the Chord of as many degrees which is 6 hours in time The time between 6 a Clock and 12 which proves that M I is the Meridian Altitude of the Sun and this measuring any Distance from the Meridian is called The first Way of Measuring QUESTION II. To find the Suns Amplitude of Rising and Setting FIx one foot of your Compasses in the place of the Suns Rising which is O and extend the other foot to the Center of the Sphere which is termed the East point of the Horizon and this Distance apply to your Line of Sines if you have any but if you have no line of Sines extend it to the Chord of the same Arch thus Fix your Compasses with that distance one foot in the Meridian so that the other may just sweep a line that goeth through the Center of the Sphere then say I the Arch between that foot of your Compasses that stands in the Circle and the place where this line you sweep cuts the Circle is the Chord of the thing required and will be the same number of degrees upon the line of Chords as O R would be upon the line of Sines and after this manner is the Chord of any distance taken from a line that goeth through the Center of the Sphere found and this is called The second Way of Measuring I find that the Sun riseth 20 deg 53 min. to the Northwards of the East for she hath North Declination and the Latitude is Northerly or sets so much to the Northwards of the West QUESTION III. To find the Suns Azimuth at six of the Clock THe Suns Azimuth at 6 of the Clock is the nearest distance between the Sun at 6 of the Clock and the East and West Azimuth which is P z Now if you mind it P z is taken off from a Circle which is not so great as the Horizon and yet is parallel to it as the line n w is parallel to the line M E and as many degrees as the Sun is from the nearest part of the East and West Azimuth in the little Circle so many he is from it if an Azimuth where drawn in the great Circle for P z is as many degrees in the little Circle as n ♈ is in the great one Now because your Scale is made by the great Circle therefore extend the distance taken from the lesser Circle to the measure of the greater which is done thus Take half the length of the pricked line which is z w and fixing one foot of your Compasses in the Center of the Sphere describe an Arch from some line that goeth through the Center of the Sphere as the Arch M l n then set the distance z P as a Sine upon that Arch for it is a Sine upon that Arch as well as o ♈ was a Sine upon the Meridian I find the Sine of it is the Sine of the Arch l M lay your Scale from the Center of the Sphere by l and draw the line l q then shall q M be the same quantity of degrees upon the great Circle as l M is upon the little one therefore take M q and apply it to the Scale of Chords and it answers your desire And thus is the Suns Azimuth at 6 of the Clock P z found to be 8 deg 36 min. to the Northwards of the East and this is called The third way of Measuring to measure any distance from a line that doth not go through the Center which must represent a small Circle And thus you may find the Suns Azimuth at any time of the day QUESTION IV. To find the Suns height at six of the Clock THe Sun is at P at 6 of the Clock fix one foot of your Compasses in P and extend the other to sweep the Horizon which is the same as though you let fall the Perpendicular P n set it off by The second way of Measuring as was shewed in Quest 2. and apply it to your line of Chords and the reason is because P n represents a part of a great Circle and so is to be understood to be of the nature of those lines that go through the Center of the Sphere for all Azimuths pass through the Zenith and Nadir which are two opposite points I find the Sun is 10 deg 7 min high at 6 of the Clock QUESTION V. To find the Suns height being due East or West VVHere the Suns parallel of Declination cuts the East and West Azimuth is the place the Sun is in when he is due East in the morning for you see he is then over the point of East in the Horizon which is ♈ or the Center therefore take the distance between that place and the Center which is S ♈ and apply it to the line of Sines but if you have no line of Sines extend it to a Chord after the manner of the second Question which I call the The second way of Measuring for it is the distance of a line which goeth through the Center I find the Suns height being due East or West is 17 deg 25 m. Note that the same height that the Sun is being over the East point so high he is being over the West point in the afternoon QUESTION VI. To find the Difference of Ascension THe Difference of Ascension is the portion of time that is between the Suns Rising and six of the Clock If the days be longer than the nights the Sun riseth before 6 but if shorter after 6 but whether it be before or after 6 that he riseth so many hours and minutes as it is from 6 so much is the half day or night longer or shorter than 6 hours from whence it is evident that if the Sun riseth due East he
in the work at all And indeed my desire to be brief makes me omit things that I think may be understood without treating of them I 'll only touch upon an Example in a Sphere which hath South Declination Latitude Northerly 50 deg 00 min. Declination 13 deg 15 min. Southerly Also if you subtract this Amplitude V ♈ from ♈ H 90 deg the remainder is the Suns Azimuth from the South V H or if you add V ♈ to ♈ A 90 deg it is V A the Suns Azimuth of rising from the North forasmuch as V A is as much above 90 deg as V H wants of it The same is to be understood in any Sphere that the Amplitude and Azimuth of the Sun are Complements one of an other to 90 deg in that Quarter When the Suns Declination is Southerly in any Northern Latitude the days are not so long as the nights so that the Sun cannot be up at 6 of the Clock which is at R therefore the Suns Azimuth at 6 of the Clock is of no use Neither is he up when he passeth the East point of the Horizon which is at P so that you cannot take his height but you may measure how far he is under the Horizon at either of those times or you may find his true Azimuth at 6 which is of no value to us that are Seamen because we cannot have a Magnetical Azimuth they are measured as before The Difference of Ascension is here after 6 of the Clock as much as it was before 6 in the last Example And the reason is because the Latitude is the same and the Declination of the Sun is just as far Southerly as before it was Northerly so that the Sun will now be just as long before he riseth after 6 as before he rose sooner it is V R take it off by the third way of Measuring as you did before and set it down I find it to be 1 hour 5 min. 3 15 it serves for the same uses that it did in the other Example For the length of the day subtract the Difference of Ascension V R from 6 hours which is R f and the remainder is f V the length of the Forenoon which doubled is the length of the whole day that subtracted from 24 hours is the length of the night For the hour of the Suns being due East it is P R which is from the place where the Sun cuts the East and West Azimuth to 6 of the Clock and set it off by the third way of Measuring convert your degrees and minutes into time and subtract that time from 6 hours and it gives the due time of the Morning that the Sun cuts the East point of the Compass the reason why you subtract is because the Sun is due East so long before 6 of the Clock Subtract the hour of the Suns being due East from 12 hours and it will give the true time of the Suns being due West For the time of day breaking draw the line 17 ⊙ parallel to the Horizon and 17 deg under it as was shewed before and measure from P to R that is from the place where the Suns Parallel of Declination cuts that Circle of 17 deg under the Horizon to 6 of the Clock convert it into time and subtract it from 6 of the Clock and it leaves the time of Day breaking namely the time in the morning that the Sun is in P you subtract because the day breaks so long before 6 of the Clock If it where so that the Sun were in the Tropick of Capricorn you must take the Distance S q. You see the Sun is past the hour of 6 before it is break of day now in this case you must add whereas before you subtracted For the continuance of Twilight it is measured as before from the place where the Parallel of the Suns Declination cuts the Circle of 17 deg to the place where it cuts the Horizon convert it into time and set it down In this Example it is P V. What I have not mentioned here I have given sufficient instructions about in the other Example before this Latitude 51 deg 30 min. Northerly Declination 11 deg 10 min. Northerly I demand the Suns place and right Ascension BEfore we can find the Suns place in the Ecliptick you are to consider that the Ecliptick is divided into 12 equal parts by the 12 Signs six of these Signs divide that half of the Ecliptick which is to the Northwards of the Aequinoctial and are called Northern Signs and the other 6 divide the half that is to the Southwaads of the Aequinoctial and are called Southern Signs as I have set them down in this Book before with their Names and Months to their Characters and have set the Northern Signs by themselves and the Southern Signs by themselves This plain Superficies can shew but one part of the Globous Body but you must know it is round which makes the Ecliptick to be divided on both sides from ♈ to ♋ is 90 deg from ♋ to ♎ is 90 deg from ♎ to ♑ is 90 deg and from ♑ to ♈ is 90 deg every one of these quarters containing 3 Signs The Ecliptick being thus divided into Signs we may find the Suns place and right Ascension THe Suns place in the Ecliptick is the nearest Distance between the next Aequinoctial point and the Sun in the Ecliptick by Aequinoctial points is meant ♈ and ♎ the two points of intersection that the Aequinoctial and the Ecliptick make Now in this case the Sun must be nearest the Equinoctial point at ♈ because the Month is belonging to a Sign nearer ♈ than ♎ Therefore fix your Compasses in the point of ♈ and extend the other foot to the Sun in the Ecliptick which is I and apply it to the line of Sines or if you have no line of Sines convert it to a Chord by the second way of Measuring and as many degrees as it is so far is the Sun distant from the nearest Aequinoctial point if it exceed 30 deg the Sun must be in that degree of ♉ that is above 30 deg if it exceed 60 deg the Sun must be in that degree of ♊ above 60 but here I find it in less than 30 namely 29 deg 3 min. therefore I conclude the Sun is in 20 deg 3 min. of ♈ But suppose it were time of year that the Sun were returning from the Tropick towards the Aequinoctial and were the same Declination then would the Suns place be the same from entring into ♎ that it now is in ♈ namely in 57 min. of ♠which wants 29 deg 3 min. of entring into ♎ Thus much for the Suns place The Suns right Ascension is measured from the place where the Suns parallel of Declination cuts the Ecliptick to 6 of the Clock which is I 6 set it off by the third way of Measuring to extend it to a Great Circle for the Sine of I
Angle at E and so for the Course As D E Dist run and Depar 197 8 10 which is 1978 tenths comp arith 6,7037736 Is to Radius   So is the Difference of Latitude D B 600 tenths 2,7781512 To Tangent E 16 deg 53 min.  9,4819248 The Complement of which to 90 deg is E B D 73 deg 7 min. Now because the Angle P E A and P B A is equal subtract the Angle E B A from E B D and the Remainder is A B D the Course Example E B A 16 deg 53 min. equal to A E P subtracted from E B D 73 deg 7 min. the Remainder is A B D 56 d. 15 m. the Course The reason that I take 3 from 7 and set down 5 is because if you look to the absolute minute and part of a minute of the Arch that answers to this artificial Tangent 9,4819248 it will fall to such a Fraction or to be the 15 part of a minute less than 3 as will bring it out but it is needless to be so exact in cases or Questions of this nature I will not shew how to work that till I come to a work that requires such exactness here you see if you take it without allowing for the Fraction it is but 1 min. less or more which is but the 675 th part of a point of the Compass but to go to the exactness of it it is the same For the Departure from the Meridian A D. As Sine comp the Course A B D 56 d. 15 m. co ar 0,2552609 To the Difference of Latitude 600 tenths B D 2,7781512 So is the Sine of the Course 56 deg 15 min. A B D 2,9198464 To the Departure from the Meridian A D 898 tenths 9,9532585 Which divided by 10 brings out 89 miles 8 tenths of a mile as it was in the former Examples To find the Distance run A B. As Sine com A B D the Course 56 d. 15 m. co ar 0,2552609 To the Difference of Latitude 600 tenths B D 2,7781512 So is Radius to the Distance A B 108 miles 3,0334121 A TRAVIS Admit I set from Sommars Islands lying in the Latitude of 32 deg 20 m. North Lat. and sail S E 40 leagues from thence S S E till I alter my Latitude 1 d. 0 m. from thence upon some point between the North and the East 40 leagues till I alter my Lat 1 d. 45 m. I demand what Lat. I am in my Distance run upon a streight line my Departure from my first Meridian and my Course made good FOr the doing this Geometrically I have the Distance run and the Course given to lay down the Triangle A C B make A the place you set from and lay it down as was shewed in an Example of that nature before Quest 3. The departure from the first Meridian is F R 56 leagues The Course is the measure of the Arch s e O which is E S E 9 deg 8 min. Easterly The Distance that I am from the Port I set is A F 57 leagues Diff. of Lat. made since I set out is R A 13 leagues or 00 d. 39 m. Which subtracted from the Latitude I set from 32 20 The Remainder is the Latitude you are now in 31 41 And thus you see you sailed from A to C and then altered your Course and sailed from C to e and then from E to F the like is to be understood of any Travis else And thus if you had a Travis of 24 hours you might find what Distance your Ship is from the place she was the day before and he other things as Difference of Latitude the Latitude you are in the Departure the Course made good upon a streight distance so that you need not set down every thing to every alteration in your Reckoning which is endless and fruitless By the Tables To give an answer to the demands in the Travis you need find nothing but the Difference of Latitude and Departure that is made in every Angle and note which way it is either North or South for Latitude East or West for Longitude and taking this notice after you have done see which is greatest in the Difference of Latitude either the Northing or Southing or for Longitude the Easting or Westing and that which is greatest that way it is and so much as it is above the other We will shew an Example of this when we have wrought the Travis I mind the Travis and I find I have the Course and Distance run in the first Triangle to find the Difference of Latitude and Departure For the Departure As Radius  To the Distance run A C 40 leagues 400 tenths 2,6020600 So is the Course B A C 45 deg 9,8494800 To the Depar BC 283 tenths or 28 leag 3 tenths 28 leagues 3 tenths East Departure 2,4515450 For the Difference of Latitude Forasmuch as the Angle opposite to the Difference of Latitude is equal to the Angle opposite to the Departure namely both 45 deg the Sides will be also equal therefore is A B the Difference of Latitude South 28 leagues 3 tenths For the Second Triangle Here I have given the Difference of Latitude and Course to find the Departure from the Meridian which is all I need As Sine com the Course D C E 22 d. 30 m. co ar 0,0343846 Is to the Diff. of Latitude 20 leagues or 291 tenths 2,3010300 So is the Sine of the Course D C E 22 deg 0 min. 9,5828307 To the Departure D F 8 8 10 leagues or 88 tenths 1,9182543 The Departure is 8 leagues 8 tenths East because the Course is Easterly For the third Triangle In the third Triangle I have given the Difference of Latitude and Distance run to find the Departure For the Course As the Distance run G F 40 leagues comp arith 7,3979399 Is to Radius  So is the Difference of Latitude G e 35 leagues 3,5440680 To the Sine comp of the Course s G F e the comp of which is the Course G e F 28 deg 58 min. 9,9420079 For the Departure As Radius  To the Distance 40 leagues 2,6020600 So is the Course 28 deg 58 min. 9,6851151 To the Departure 19 4 10 leagues 2,2871751 The Departure is East because the Course is Easterly Thus now in every Angle you know the Difference of Longitude and Difference of Latitude in leagues and also which way either North or South East or West set them down apart thus Lon. East Lon. West Northing Southing 28 3 10 00 00 28 3 10 8 3 10 00 00 20 19 4 10 00 35 00 56 1 10 00 35 48 3 10 Here I have added them up and I find no West Longitude so that the Departure from the Meridian is East 56 leagues 1 10 the Difference of Latitude North I find to be 35 leagues the Difference of Latitude Southerly is 48 3 10 subtract the Northing which is least from the Southing and the remainder is
the Difference of Latitude in the whole run 13 leagues 3 10 if you had West Longitude you must have subtracted thus likewise and that which was greatest the remainder belongs to OF A RECKONING FRom what hath been already shewed you understand that if the Latitude of two places be known and the Difference of Longitude between them you may find the bearing of them one from the other their distance asunder or any thing you desire and this is all so plain that if I set from a place and am bound to a place whose Latitude is known and the difference of Longitude also known between them and keep a reckoning of the departure both East and West that I make setting them down in two distinct Columns it is but subtracting one from the other and the remainder will be the East or West Longitude that I have made that of the two which was most also if I know what Latitude I am in and what Latitude the place lies in it is but subtracting the lesser from the greater and the remainder will be the difference of Latitude between your ship and that place provided the Latitudes be both one way Also subtract the Difference of Longitude that you have made provided that it be that way which shortens your Longitude between the place you set from and are bound to from the whole difference of Longitude between the two places and the remainder will be what is still between you and the place you are bound to but if your departure which you have made hath increased the difference of Longitude between the place you set from and are bound to add the remainder after your subtracting the Columns one from the other to the whole difference of Longitude between the places the like for Latitude Example in the Reckoning following Suppose the Island of Ditiatha lies so that it hath been found to have 950 Leagues departure West from the Meridian of the Lizard according to the Courses that be ordinarily sailed by Plano the Latitude of it is 16 deg 16 min. North Latitude the Lizard lies in the Latitude of 50 deg 00 min. Now upon some Occasion that fell in the term of our Voyage namely the 29th of January on Tuesday the Captain demands of me what Distance Ditiatha was from me by Plano how it bears what Leagues of Departure I have to it and the like I look in my Reckoning and I find in the West Column 434 Leagues I look in the East Column and see 60 Leagues I subtract 60 my whole East Column from 434 the whole West Column and the Remainder is 374 Leagues which signifieth the Ship hath departed from the Meridian of the Lizard 374 Leagues West for the West Column was greatest Subtract this West departure 374 Leagues from 950 the whole Departure and I find there is still wanting 576 Leagues This 576 Leagues is the Departure that is from the Meridian that my Ship is in and Ditiatha Subtract also the Latitudes one from the other namely the Latitude your Ship is in and the Latitude of Ditiatha and the Remainder is the Difference of Latitude in degrees and minutes between the Ship and the Place you are bound to which in this Example I find to be 6 degrees 24 minutes Take the Latitude out of the Column of Latitude for that day in the Reckoning and you will find it so if you subtract 16 deg 16 min. from it Thus you have the Difference of Latitude between you that day and your Port and the Departure that the Port hath from the Meridian that you are in that day given you to find the things demanded I will not work it for it hath beeen shewed sufficiently already I have here following set down a Reckoning and afterwards how I work it and lastly the reason that makes me use this way of keeping my whole Reckoning in the last line so that I am not troubled to add it up when I give account of it January the 3d Anno Dom. 1655 we departed from the Lizard lying in the Latitude of 50 d. 00 m. being bound for Ditiatha which lies in the Latitude of 16 d. 16 m. it having West Departure from the Meridian of the Lizard 950 Leagues according to the courses which we then steered at 4 of the clock the Lizard bore from us N N E about 6 leagues distance by estimation EVery day at Sea you set down at noon that 24 hours work Now I advise you to set it down as it is here In the first Column on the left hand I set the day of the Month in the next the day of the Week in the third the Latitude I am in by dead Reckoning or by Observation only I make a distinction between them by the letter E which stands for Estimation and is set down only when I do not observe that so you may know where to begin to correct In the 4 th Column I set down the Easting in the fifth column the Westing The manner that I set my Easting and Westing down is thus Every day there is the Ships course minded and an observation whereby you have the difference of Latitude and course to find the departure or else if you cannot observe you have the distance run guessed at and the course to help your D. M. Week days Latitude Dep. E. in leag Dep. W. in leag deg min. 4 Friday 49 00 000 021 5 Saturday 47 55 000 059 6 Sunday 46 20 000 085 7 Munday 45 00 000 088 8 Tuesday 43 E 54 000 108 9 Wednesday 42 4 000 147 10 Thursday 41 E 19 000 262 11 Friday 41 10 000 264 12 Saturday 40 46 000 274 13 Sunday 40 45 000 178 14 Munday 39 E 49 000 179 15 Tuesday 37 E 24 000 200 16 Wednesday 35 59 000 229 17 Thursday 34 42 000 249 18 Friday 33 15 000 278 19 Saturday 31 15 000 322 20 Sunday 29 5 000 364 21 Munday 28 E 20 000 400 22 Tuesday 27 5 012 400 23 Wednesday 27 30 022 400 24 Thursday 27 22 031 400 25 Friday 26 E 49 049 400 26 Saturday 25 36 060 400 27 Sunday 24 31 060 406 28 Munday 23 28 060 406 29 Tuesday 22 E 40 060 434 30 Wednesday 22 08 060 434 31 Thursday 21 53 060 444 February 1 Friday 21 40 060 450 2 Saturday 20 47 060 456 3 Sunday 19 22 060 460 4 Munday 19 53 060 472 5 Tuesday 18 23 060 485 D. M. Week days Latitude Dep. E. in leag Dep. W. in leag deg min. 6 Wednesday 17 19 060 518 7 Thursday 16 47 060 556 8 Friday 16 54 060 606 9 Saturday 16 29 060 643 10 Sunday 16 19 060 678 11 Munday 16 21 060 726 12 Tuesday 16 28 060 784 13 Wednesday 16 35 060 840 14 Thursday 16 27 060 889 15 Friday 16 24 060 929 16 Saturday 16 2 060 973 17 Sunday 16 44 060 1003 guess is the Log-line every 2 hours your course and distance is set
down upon the Log-board every 2 hours whereby you may see it and work it every 24 hours to find the departure and difference of Latitude Now whatsoever I find the departure to be I add it from the first day all along thus the departure when I set the Lizard which was N N E 6 Leagues off is 2 Leagues West I keep mind of that till the next day at noon and I find what departure I have made from the Meridian that I was in the day before at noon by the things I have given me and here I found it was 19 Leagues I add it to my departure from the Meridian West yesterday 2 Leagues and it makes my whole departure West 21 Leagues Again the next day at noon I work my Ships Travis as hath been shewed and I find that she hath departed from the Meridian she was in yesterday 38 Leagues West which I add to what I had before which was 21 and it makes 59 Leagues and thus I go forwards adding my last days departure in its true Column to all the rest which is in one sum and by this means the last line is the whole Reckoning 21 38 59 In the East Column I set Ciphers to fill it up till I have some East departure to put in it and as that increaseth so I add it as the other every day Also I carry the West departure along in its full number of Leagues in the same line without increase or decrease till such time as I have more to increase it and then I increase that and carry the whole Easting with it in the same line and thus you have need to look upon nothing but the last line to resolve you any thing that you desire either of your course made good since you set out your distance upon a straight line that you have failed your Ports bearing from you what departure is yet between you and your Port your distance to it upon a straight line for there you have your whole Westing and your whole Easting and the Latitude you are in as also the Latitude of the place you are bound to whereby as I have formerly shewed you may find them If there be Longitude but one way you will see nothing but Ciphers in the other Column this way you see is done without a Plat. I hold a Plat to be necessary only to shew what dangers lie in your way that so you may shape a course clear by it and I should use a Plat for nothing else except it be for Coasting where it is really useful I commonly set down the Easting or Westing between the place I set and the place I am bound to at the beginning of my reckoning as also the Latitude of each place as I have done here You may ask why I do not set down the course that I made good every day indeed that is not unnecessary but the reason why I omit it is First because I find it of little use in my Reckoning and secondly I find it can be better expressed in a Journal which is kept with my Reckoning and indeed there I set it with the Winds and the reasons for steering upon such Courses but I leave every one to their own Judgment for that as also for Distances and Winds for Difference and Variation It would have been necessary here to have set down some Tables to have worked Triangles by for as I have said sometimes in 24 hours you have a Travis of 4 or 5 several courses and to work them this way may seem tedious I commend you to my Fathers Practice where there is as good Tables as can be to every degree of the Compass its use is easie and works to the tenth part of a Mile or League CONCERNING the VARIATION OF THE COMPASS THere is always two things given to find the Variation of the Compass that is the true Amplitude of the Suns rising and the Magnetical Amplitude of the Suns rising or setting the true Amplitude of the Suns rising is as I have said before in another place the true and absolute quantity of degrees that the Sun riseth from the East either Northwards or Southwards or sets from the West and is found as I have before shewed in the use of the Sphere The Magnetical Amplitude of the Sun is what the Sun riseth from the East or sets from the West by the Compass Now because the first gives the Truth how far the Sun riseth from the East or sets from the West therefore whatsoever difference there is between them so much is the variation As now Suppose I find in a certain Latitude such a day of the Month by the Sphere that the Sun riseth to the Northwards of the East 17 degrees that is the true Amplitude but I observe the Sun at her Rising with an Azimuth Compass which is made for that purpose and find that she riseth but East 10 degrees Northerly then I conclude the Variation is 7 deg 0 min. or the Compass is false so much so that whereas if I direct my Course East 10 degrees Northerly by the Compass I do not go on that Course but I go East and by North 5 deg 45 min. Northerly which is just 7 d. from my expectation or East 17 deg Northerly the true Amplitude Now that is a gross error and in long runs may deceive a man much and perhaps be a means to lose a Ship when one little thinks of it and therefore it ought to be looked to An Azimuth Compass is no other in effect but a Compass fitted for the exact taking of the Sun at her Rising or Setting or upon other certain times of the day as you may have occasion In like manner you may find the Variation of the Compass at other times by taking the Suns Azimuth at any time of the day Example Suppose I were in the Latitude of 33 deg 20 min. Northerly and upon the 8 th day of November the Suns Declination is Southerly 19 deg 20 min. I demand the Suns Azimuth at 8 of the Clock The Suns Azimuth at her Rising as I have shewed is the Complement of the Suns Amplitude but after the Sun is up it may be also the Distance of the Sun from the East and West Azimuth Now in this Proposition you desire to know how many Degrees the Sun is from the East and West Azimuth which is that Part of the Heavens that is distant from the Sun parallel to the Horizon over the East Point of it or if it had been at 4 of the Clock in the Afternoon it would have been required from the West Point For the resolving this Question project a Sphere as hath been taught for the Latitude proposed with the Parallel of the Suns Declination drawn as it is given Divide this Parallel of Declination into the hours of the day from 6 to 12 or which serves from 12 to 6 in the Afternoon the way to divide it is thus From 6 of the
demonstrated it as evidently as can be For Mercator use the Tables here inserted The Vse of the foregoing Tables of Meridional Parts When you work this way find first what Latitude you are in if you cannot observe do it by the things given that 24 hours Then take notice what Latitude you were in the day before and look in the Tables what answers to each Latitude setting them down subtract the lesser from the greater and the remainder are the Meridional Parts contained between the two Latitudes which number make for the length of the side which contains the Difference of Latitude and so work according to the Directions following I will be brief Example To find the Meridional Parts answering to any Latitude Let the one Latitude be 50 deg 9′ in the Tables 3488 3274 214 answers it Let the other Lat. be 47 deg 48′ in the Tables answers it Subtract the lesser from the greater and the Remainder is the Meridional Parts contained in that Difference of Latitude The same is to be understood of any Latitude else QUESTION I. The Latitude of two Places being given and the Rumb to find the Distance and Difference of Longitude SUppose I set sail from a Place in the Latitude of 39 deg 12 min. North Latitude and am bound to a Place in the Latitude of 13 deg 12 min. North Latitude which bears S W by W from the other I demand the Distance between those Places and Difference of Longitude Let A represent the Northermost place C the Southermost then is A and B the Latitudes of each place C B the Difference of Longitude B A C is the Course from the Northermost to the Southermost place For the Distance between the Places A C. As Sine comp A 56 deg 15 min. comp arith 0,25526 To A B the Difference of Latitude in miles 1560 3,19312 So is Radius  To the Distance between them C A 2808 miles 3,44838 For the Difference of Longitude The Meridional parts answering to 39 deg 12 min. are 2560 The Meridional parts answering to 13 deg 12 min. are 0799 The Meridional parts contained between the two Lat. 1761 As Radius  To B A 1761 parts 3,245759 So is Tan. the Rumb A 56 deg 15 min. 10,175107 To C B the Diff. of Longitude in minutes 2635 m. 3,420866 The Quotient is the number of degrees and the Remainder is the number of minutes contained in the Difference of Longitude 43 deg 55 min. QUESTION II. Course and Distance run with one Latitude given to find the other Latitude and the Difference of Longitude SUppose I set from the Latitude of 49 deg 9 min. North and sail S W b W 2808 miles I demand the Latitude I shall then be in and the Difference of Longitude Let A be the Latitude of 49 deg 9 min. let A C be the Distance run 2808 miles then is A B the Difference of Latitude and C B the Difference of Longitude For the Difference of Latitude A B. To A B 1560 miles Which being divided by 60 The Quotient and the Remainder is the Difference of Latitude in degrees and minutes which is 26 d. 0 m. That subtracted from 49 d. 9 m. Leaves the Latitude you are now in 23 d. 9 m. The reason why there is no inequality between degrees of Latitude is because all Meridians are great Circles so that 60 miles is a degree of them always For the Difference of Longitude The Meridional parts contained between these two Latitudes I find to be 3396 1428 1968 pts As Radius  To A B in parts 1968 3,294025 So is Tang. A the Rumb 56 deg 15 min. 10,175107 To the Difference of Longitude 2945 m. 3,469133 Now you may ask a reason why this Difference of Longitude should differ from the other in the last Question seeing the Difference of Latitude Course and Distance in each Question are alike the reason is because this Question runs between two less parallels than the other did and therefore the same things must give a greater part of this Parallel than it did of the other though the number of miles of Departure be alike Divvide 2945 min. by 60 and you have the Difference of Longitude in degrees and minutes 49 deg 5 min. QUESTION III. The Latitudes of two places and the Distance between them given to find the Course and Difference of Longitude LEt one place namely A be in the Latitude of 49 deg 9 min. North let the other place namely C be in the Latitude of 23 deg 0 min. North and let the Course be between the South and the East the Distance between them is A C 2808 miles For the Course A. As A C Dist 2808 mil. 6,551602 Is to Radius  So is the Diff. of Lat. A B reduced into miles 1560 miles 3,193124 To the Sine C whose Comp. is A 9,744726 Which is 56 deg 15 min. to the Eastwards of the South For the Difference of Longitude B C. As Radius  To B A in parts 1968 3,294025 So is Tangent the Course A 56 deg 15 min. 10,175107 To the Difference of Longitude in minutes 2945 3,469133 Which is as it was in the last Question 49 deg 5 min. QUESTION IV. The Latitude of two places and their Difference of Longitude given to find the Course and Distance provided the Course be between the North and the West LEt the Latitudes be as before and the Difference of Longitude 49 deg 5 min. or 2945 parts or minutes For the Course As the Diff. of Latitude in parts A B 1968 co ar 6,70597 Is to Radius  So is the Diff. of Longitude in minutes or parts B C 2945 3,46908 To the Tangent of the Course A 56 deg 15 min. 10,17505 To the Westwards of the North. As Sine C 33 deg 45 min. comp arith 0,25526 To the Difference of Latitude in miles A B 1560 3,19312 So is Radius  To the Distance run C A 2808 miles 3,44839 QUESTION V. The Difference of Longitude Course and one Latitude given to find the other Latitude and the Distance in the Triangle CBA LEt C B be the Difference of Longitude 49 deg 5 min. or 2945 min. A the Latitude of 23 deg 9 min. North Latitude C A B the Angle of the Course North-westerly 56 deg 15 min. let it be required to find A B the Difference of Latitude which added to the Latitude of A 23 deg 9 min. will be the Latitude of the other place B and A C the Distance between them Here you must find your Difference of Latitude in parts first which will be reduced into miles after the Latitude of the place at B is found and then find your Distance 49 deg 5 min. reduced into minutes as I said in the Question is 2945 min. the Difference of Longitude given For the Difference of Latitude in parts As Radius  To the Difference of Longitude in parts 2945 parts 3,469085 So is Tan. comp
Difference of Longitude between them which things may be taken from such a Catalogue as this is after the manner following Suppose I were bound from the Lizard to Barbadoes I look in the Column of Latitude against the Lizard and find 50 deg 0 min. for its Latitude and 25 deg 20 min. for its Longitude from Floures and Corves I look in the Column of Latitude against Barbadoes and find 13 deg 12 min. for its Latitude and 327 deg 42 min. for its Longitude from Floures and Corves Here you have the Latitude of each place but to find the Difference of Longitude between these places I consider that Barbadoes hath 327 deg 42 min. of East Longitude from these Islands but if I subtract that from 360 deg the whole Circumference there must remain the West Longitude that is between them which I find to be 32 d 18 m. Circumf 360 00 East Long. 327 42  32 18 To it add the longitude between the Lizard and Floures and Corves 25 d. 20 m. That Sum is 57 d. 38 m. The Difference of Longitude between the Lizard and Barbadoes the Meridian of Barbadoes being to the Westwards of the Lizard so many degrees And thus any Difference of Longitude between two places is taken provided that one of them is nearer than 19 deg to the Westwards of the place you begin your Longitude from and the other nearer than 90 deg to the Eastwards And the reason of this is because the Longitude of all places is set down from Floures and Corves to the Eastwards so that if a place were but 1 deg to the Westwards of Floures and Corves it would be set down in the Catalogue to lie in the 359 deg of Longitude For to know the Difference of Longitude between any two places consider which way they are nearest and that take Suppose there were two places one lieth in 20 deg 0 min. of Longitude the other lieth in 195 deg of Longitude I consider that the difference between them is not 180 deg and therefore I will subtract 20 deg 0 min. from 195 deg and the Remainder is the Difference of Longitude between them 175 deg but if the Difference between two places be above 180 deg as they are laid down in the Catalogue Subtract that difference from 360 deg and the Remainder is the nearest Difference of Longitude Your own reason will guide you in this and therefore I will proceed How to keep a Reckoning of the Longitude and Latitude a Ship makes at Sea HAving found the Difference of Longitude between the places I set from and am bound to as also the Latitude of each place I will begin my Reckoning thus Imagine it were between the Lizard and Barbadoes and I set from the Lizard the 3 d. of January 1658. From the Lizard we departed January 3 1658 being bound to Barbadoes lying in Latitude North 13 d. 12 m. Longitude being West from the Meridian of the Lizard 57 d. 38 m.  Mo. Day Week Days Latitude East Long. West Long. D. M. D. M. D. M. January 4 Friday 49 00 00 00 03 00  5 Saturday 47 24 00 00 06 27  6 Sunday 46 09 00 46 06 27  7 Munday 45 00 01 05 06 27  8 Tuesday 43 01 01 05 06 27  9 Wednesday 41 10 01 05 09 00  10 Thursday 40 00 01 05 11 02  11 Friday 38 11 01 05 13 20  12 Saturday 36 12 01 05 14 40  13 Sunday 34 10 01 05 16 00  14 Munday 32 20 01 05 18 47  15 Tuesday 30 11 01 05 20 00  16 Wednesday 27 00 01 05 21 40  17 Thursday 25 01 01 05 22 30  18 Friday 23 10 01 05 24 00  19 Saturday 21 45 01 05 26 10  20 Sunday 20 45 01 05 28 44  21 Munday 19 00 01 05 31 00  22 Tuesday 18 30 01 05 33 32  23 Wednesday 18 00 01 05 36 00  24 Thursday 17 27 01 05 38 20  25 Friday 16 49 01 05 40 40  26 Saturday 16 10 01 05 42 50  27 Sunday 15 40 01 05 44 59  28 Munday 15 00 01 05 47 00  29 Tuesday 14 30 01 05 49 02  30 Wednesday 13 45 01 05 50 57  31 Thursday 13 12 01 05 52 00 February 1 Friday 13 14 01 05 54 30  2 Saturday 13 12 01 05 54 40  3 Sunday 13 12 01 05 54 40  4 Munday 13 12 01 05 55 30  5 Tuesday 13 12 01 05 57 30  6 Wednesday 13 12 01 05 58 35 Here in this Reckoning I have set down the Longitude in degrees and minutes as before I set down the Departure in leagues carrying all in the last line the last line of my Reckoning saith that they were in the Latitude of 13 deg 12 min. and the whole East Longitude that we have made since we set from the Lizard is 1 deg 5 min. the whole West Longitude is 58 deg 35 min. subtract the Easting from the Westing and the Remainder is 57 deg 30 min. which lacks but 8 min. of 57 deg 38 min. the Meridian of the Place Also the same line saith that we are in the Latitude of 13 deg 12 min. which is the Latitude of the place therefore I conclude I am but 8 min. from it so that I expect to see Land There is no Difference in the setting down of this Reckoning from that before but only this is degrees and minutes that leagues or miles neither is there any other difference in their casting up for there your Easting from your Westing in leagues here in degrees That there be nothing wanting to make me be understood I will here following do the three first days works in three Examples SUppose the first day I set out I sailed S W b W 6 deg 25 min. Westerly till I come into the Latitude of 49 deg 0 min. which was the next day at noon I consider that I have the two Latitudes given me and the Course to find the Difference of Longitude The Meridional part answering to 50 deg 0 min. is 3475 The Meridional part answering to 49 deg 0 min. is 3382 The Meridional parts contained between them is 93 For the Difference of Longitude As Radius  To the Difference of Latitude in parts 93 1,96848 So is the Tangent of the Course 62 d. 40 m. 10,28661 To the Difference of Longitude in minutes 180 2,25510 This 180 divided by 60 produceth 3 deg 0 min. the Difference of Longitude made that 24 hours set it down in your Reckoning with the Latitude you were in filling up the East Column with Ciphers I need not shew how to find the Distance run because I suppose by what hath been said before you can do it For the second days Work Suppose till the next day at noon we sail S W 10 deg 11 min.