Selected quad for the lemma: day_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
day_n	hour_n	long_a	pole_n	7,371	5	12.8538	5	true

Text snippets containing the quad

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

in the beginning of Capricorn to which when the Sun cometh as about the 11th of December is the shortest day in the year and in the Astronomical account the beginning of Winter The Zenith is an imaginary point in the Heavens right over our heads 90d from the Horizon The Nadir is a point or prick in the Heavens under our feet opposite to the Zenith Of the Circles of the Sphere The 10 Circles are the Horizon the Meridian the Equinoctial the Zodiack the Colure of the Equinoxes the Colure of the Solstices the Tropick of Cancer the Tropick of Capricorn and the two Polar Circles The first six are called great Circles and the other four lesser Circles By the Center of a Circle is meant a Point or Prick in the middle of the Circle from whence all Lines drawn to the Circumference are equal and are known by the name of Radius resembling the spoak of a Cart-wheel That is said to be a great Circle which hath the same Center as the Sphere and divides it into two equal halfs called Hemispheres and that is called a lesser Circle which hath a different Center from the Center of the Sphere and divides the Sphere into two unequal Portions or Segments 1. Of the Horizon The Horizon is distinguished by the names of Rational or Sensible the Rational Horizon is a great Circle every where equidistant from the Zenith and divides the upper Hemisphere from the lower and by accident or chance is distinguished by the names of a right Oblique and parallel Horizon A right Horizon is such a Horizon as passeth through each Pole of the World and cuts the Equinoctial at right Angles whence the Inhabitants under the Equinoctial are said to have a right Horizon and a right Sphere An Oblique Horizon is such a one as cuts the Equinoctial obliquely or aslope A parallel Horizon is not such a one as cuts the Equinoctial but is coincident and is the same therewith and such is the Horizon under the Poles The sensible Horizon is a Circle dividing that part of the Heavens which we see from that part which see not thence called Finitor From the Accidental Scituation of the Horizon follows many consequences 1. Those that live in a right Horizon that is under the Equinoctial have their days and nights always of an equal length to them all the Stars both rise and set twice in a year the Sun passeth through their Zenith consequently they have two Summers and two Winters to wit Summers when the Sun passeth through their Zenith and Winters when he is in or near the Tropicks 2. In any right or Oblique Sphere the length of the day when the Sun is in the Equinoctial is equal to the length of the night 3. In any Oblique Sphere the nearer the Sun approacheth to the Visible Pole the longer are the Days more then the Nights some Stars always appear others never appear and the more remote from the Equinoctial the greater is the number of such Stars and the more inequality is there between the Days and Nights 4. To those that live under the Polar Circles their day once a year is 24 hours long and their Night nothing 5. Under the Pole one half of the Sphere doth always appear and the other half not appear and one half of the year is well nigh continually Day and the other half continually Night because the Equinoctial lies in the Horizon 'T is said well nigh for by reason of the Suns Excentricity the day under the North Pole is longer then the Night about eight days and on the contrary under the South Pole is shorter then the night as many days 2. Of the Meridian The Meridian is a great Circle which passeth through the Poles of the World the Zenith and Nadir and the North and South points of the Horizon and is so called because that at all times and places when the Sun by his daily motion cometh unto that Circle twice every 24 hours maketh the middle of the day and middle of the night all places that lie under the same Meridian bear North and South but places that lie East and West from one another have each of them a several Meridian 3. Of the Equinoctial It is a great Circle imagined in the Heavens dividing them into two equal parts or halfs called the North and South Hemisphere lying just in the middle between the two Poles being every where equi-distant from them and is called the Equator because when the Sun cometh unto it which is twice in the year at his entrance into Aries and Libra the days and nights are of an equal length throughout the whole World 4. Of the Zodiack The Zodiack alias Signifer is another great Circle that divides the Equinoctial into two equal parts the Points of Intersection being called Aries and Libra the one half of it doth decline into the North the other half into the South as much as the Poles thereof are distant from the Poles of the World namely 23d 31′ and likewise passeth through the two Solstitial Points it 's ordinary Breadth or Latitude is 12 degrees but late Writers make it 14 or 16d by reason of the wandrings of Mars and Venus A Line dividing the breadth thereof into two halfs is called the Ecliptick Line because the Eclipses of the Sun and Moon are always under that Line it 's Circumference is divided into 12 parts called the 12 Signs whereof each containeth 30d. The Names and Characters of the 12 Signs are Aries ♈ Taurus ♉ Gemini ♊ Cancer ♋ Leo ♌ Virgo ♍ Libra ♎ Scorpius ♏ Sagittarius ♐ Capricornus ♑ Aquarius ♒ Pisces ♓ The six former are the Northern and the six latter the Southern Signs Of the Colures These are two great Circles and are no other then two Meridians passing through both the Poles of the World crossing one another therein at right Angles and divide the Equinoctial and the Zodiack into four equal parts making thereby the four Seasons of the year The Colure of the Equinoxes is so called because it passeth through the Equinoctial points of Aries and Libra shewing thereby the beginning of the Spring and Autumn when the days and nights are equal The other Colure passeth through the two Solstitial or Tropical Points of Cancer and Capricorn shewing the beginning of the Summer and Winter at which two times the days are longest and shortest The very beginning of Cancer where the Colure crosseth the Ecliptick line is called the Point of the Summer Solstice to which place when the Sun cometh he can approach no nearer the Zenith but returneth towards the Equinoctial again the Arch of the Meridian or Colure contained betwixt the Summer Solstice and the Equator is called the greatest Declination of the Sun Of the four Lesser Circles The Tropicks are two lesser Circles parallel to the Equinoctial limiting the Suns greatest Declination towards both the Poles that towards the North Pole is called the Tropick of Cancer because the

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

the Side N A is supposed to stand Perpendicular and to have a Slit in it the line S A is to be a thread extended from S to A the other side of which Triangle resembles a Moveable Toung or Labell the Center being in the Circumference every degree is twice as large as it would be if it were at the Center wherefore the quadrants S E S W are numbred with 45d on each side the Line N S but are not divided with concentrick Circles and Diagonals nor can they be with truth when ever the Center is placed in the Circumference and this I call an Azimuth Compass because though it be not so yet it supplies the use of one and if a right line be continued from N to E and made a line of Sines also a Tangent of 45d put through the Limbe it or an Azimuth Compass is rendred general without the use of Paper-draughts as I have shewed in the Uses of the smallest Quadrant in my Treatise The Sector on a Quadrant Page 277 to 284. where the Reader will meet with ready Proportions for Calculating the Suns Azimuth or true Coast not before published The Use of the said Azimuth Compass at Sea is readily to apply it to any Compass in the Ship and thereby finde the true Coast of the Suns bearing by that Compass to which it is applyed and consequently the Variation thereof and by my own experience I have often found that by the thread which passeth through the Diameter of the said A●imuth Compass it may very well by the View be placed over the Meridian line of the Compass and then turning the moveable Label towards the Sun so that the shadow of the thread may pass thorough the Slit the tongue of the Label amongst the graduations of the Limbe shewes how the Sun bears by the said Compass in which the toucht wires is supposed to be precisely under the Flower-deluce when the Sun is more then 45d from the Meridian either way the thread in the ●iameter of the Circle must be placed by the view over the East or West point of the Compass and the Suns bearing accordingly reckoned from thence Then admiting that the bearing of the Sun by the Compass and his true Azimuth or Coast of bearing be found either by Calculation or the former Schemes the Variation of the said Compass from the North which all Needles are lyable unto with the Coast thereof may thus be found Example Let the bearing of the Sun by the Compass be 55d East-ward from the South and his true coast in the Heavens be 43d ¾ from the South East-wards Then admit it were required to steer the Ship away N E by E being the fifth point from the North East-ward it is desired to know how she must wind or steer by that Compass Out of the Scale of Rumbes place five point from B to R then measure the extent N R on the Rumbes and it sheweth four points whence we may conclude th●t to make good the former Course the Ship must be steered North-east by this Compass The re●diest wa● for fin●ing the Variation is by those Sea Rings described by M● Wright but those are chargeable are but in few ships fixed but to one Compass reserved for the Ship masters own peculiar Ob●ervations so that the common Mariners can have no practise thereby A Scheme for finding the Hour Example Latitude 51d 22′ North. Declination 23d 31′ North. Altitude 10d 28′ Complement 79d 32′ Having described the Semicircle and divided it into two Quadrants by the line D C prick as before the Latitude 51d 32′ from A to L and draw L M parallel to D C Prick off the Declination 23d 31′ from D to E prick the Sine thereof being the nearest distance from E to D C in Winter or South Declination from L to Q upwards but in Summer or North Declination O N downwards and with the Cosine of the Declination being the nearest distance from E to A C upon C as a Center describe the Ark G W so is the Scheme prepared for that Declination both North and South To finde the Hour in Winter The Suns height being 10d 28′ its Complement is 79d 32′ prick the Chord thereof from Q to T and setting one foot upon M with the extent M T draw the arch I a ruler from the Center over that Intersection finds the point K and the Arch A K being 30d the hour is either 10 in the Morning or 2 in the Afternoon To finde the time of Sun rising Prick the Chord of 90d from Q to O and with the extent M O upon M as a Center cross the ark G W at P a ruler from the Center over P finds the point R in the Limbe and the ark D R being 33d 12′ in time about 2 hours 13′ is the time of the Suns rising or setting from Six to that Declination both North and South To finde the Hour of the Day in Summer to the same Declination the Latitude being the same Let the Altitude be 9d 30′ its Complement is 81d 30′ prick the Chord thereof from N to H and with the extent M H upon M as a Center cross the arch G W as at S a ruler from the Center over that Intersection finds the point V in the Limbe and the ark D V being 15d the true time of the day is either five in the Morning or seven in the Afternoon Having found the Hour first then the Azimuth and Angle of Position may be easily found from the Proportion of the Sines of Sides to the Sines of their opposite Angles as in the following Scheme Example Latitude 51d 32′ Declination 13d North. Altitude 37d 18′ By the former Directions the Hour will be found to be 45d from noon either 9 in the morning or 3 in the afternoon the Intersection whereof happens at e through e draw e F parallel to A B and prick the Altitude 37d 18′ from D to H and draw H C also joyn e C and make C O equal to C M and through the point O draw O Q parallel to A B so is the extent C Q the Sine of the Angle of Position and the extent C P the Sine of the Azimuth from the Meridian Otherwise for the Azimuth With the nearest distance from H to C B setting one foot in C cross the parallel e F at F a ruler from the Center cuts the Limbe at I and the arch B I is the Suns Azimuth either from the North or South in this Case 60d from the South For the Angle of Position With the former extent cross the parallel O Q at G a ruler from the Center cuts the Limbe at K and the arch B K being 33d 34′ is the measure of the Angle of Position and this work might have been performed on the other side D C but to avoid confusion when the Doubts about Opposite Sides and Angles may be removed and when not as when a double answer is to be

a parallel of Altitude called an Almicanterath The prickt Arches Z ⊙ and Z G K being Ellipses represent the Azimuths or Vertical Circles And the other prick't Arches Represent Meridans or hour Circles which are also Ellipses the drawing whereof would be troublesome and therefore is not mentioned and how to shun them in the resolution of any Proposition of the Sphaere by Chords shall afterwards be shewed Any line drawn parallel to AE E as is f p F D R Q will represent parallels of Declination And any Line drawn parallel to F Y will represent a parallel of Latitude in the Heavens Fifthly divers Arches relating to the Suns Motion such as are commonly found by rhe Globes or Calculation are in the same Scheme represented in right Lines 1. The Suns Amplitude or Coast of rising and setting from the East or West is there represented C W in North Signes and by C g in South Signes 2. His Ascensional difference or time of rising from six in Summer by G W in Winter by g h. 3. His Altitude at six in Summer by H C his Depression at six in Winter by C b. 4. His Azimuth at the hour of six by H G in Summer equal to h b in Winter 5. His Vertical Altitude or Altitude of East and West by I C his Depression therein in Winter by C q. 6. His hour from six being East or West in Summer by G I in Winter by h q. 7. His Azimuth from the East and West upon any Altitude is represented in the parallel of Altitude where it intersects the parallel of Declination here by M ⊙ 8. The hour of the day from six to any Altitude is represented in the said point of Intersection but in the parallel of Declination here by G ⊙ and all these Arches thus represented in right Lines are the Sines of those Arches to the Radius of the parallel in which they happen being accounted from the midst of the said parallel Now how to measure the quantities of these respective Arches by a Line of Chords and consequently thereby to resolve all the cases of Sphaerical Triangles is the intended subject of some following Pages The former Arches thus represented in right Lines many whereof fall in parallels or lesser Circles when Calculation is used are all represented by Arches of great Circles namely such as bisect the Sphaere and the former Scheme doth represent the Triangles commonly used in Calculation Thus the right angled Triangle C d y right angled at d supposing the Sun at y is made of C y The Suns place or distance from the nearest Equinoctial point C d his right Ascension Y d his Declination d C y the angle of the Ecliptick and Equinoctial C y d the angle of the Suns Meridian and Ecliptick In the right angled Triangle W O P right angled at O supposing the Sun at W. O P is the poles Elevation P W the complement of the Suns Declination W O the Suns Azimuth from the North. W P O the hour from Midnight or complement of the Ascensional difference P W O the angle of Position that is of the Suns Meridian with the Horizon and of the like parts or their complements is made the Triangle C m W. In the right angled Triangle C K G right angled at K supposing the Sun at G. C G is his Declination G K his height at the hour of six C K the Suns Azimuth from the East or West at the hour of six K C G the angle of the Poles Elevation C G K the angle of the Suns position In the right angled Triangle C k I right angled at k supposing the Sun at I. I k is the Suns Declination C k his hour from six C I his height being East or West k C I the Latitude k I C the Angle of the Suns position In the oblique Angled Triangle Z ⊙ P if the Sun be at ⊙ Z P is the Complement of the Latitude P ⊙ his distance from the elevated Pole in this Case the complement of his Declination Z ⊙ the Complement of his Altitude or height Z P ⊙ the Angle of the hour from Noon P Z ⊙ the Suns Azimuth from the North or midnight Meridian Z ⊙ P the Angle of the Suns Position Thus we have shewed how the former Scheme represents the Sphaerical Triangles used in Calculation whereby of the six parts in each Triangle if any three are given the rest may be found by Calculation from the Proportions and that either by Multiplication and Division when the natural Tables of Sines and Tangents are used or by Addition and Substraction when the Logarithmical are used and what is performed by either of those sorts of Tables we shall here perform by Scale and Compass from which performances the like measure of exactness is not attainable as from the Tables CHAP. III. Shewing how to know upon what day of the Week any day of any Moneth happens upon for ever 1 TO perform this Proposition there must be a general Rule prescribed to find on what day of the Week the first of March will happen upon for ever which take in the following Verses To number two adde year of our Lord God And a fourth part thereof neglect the odd Remainder if such be the sum divide By seven lay your quotient aside The Rest when your Divisions finished Will number shew day of the Week you need On which the first of March doth chance to be Still counting Lords day first if you do see That nothing do remain then you may say The day you seek's the seventh and Saturns day Example Let it be required to find on what day of the Week the first of March will happen in the year of our Lord 1687. Operation Divisor 7 2 The even fourth of the Year The remainder neglected 301 Quotient 1687 421 2100 21 10 7 3 remains Because three remains the first of March in that Year happens on a Tuesday in the Year 1679 nothing remains therefore it happens on a Saturday Proposition 2. The day of the Week on which the first of March happens on any Year being known and remembred To find on what day of the Week any day of any Moneth in the said Year hapneth To perform this Proposition the following Verse being in effect a perpetual Almanack is to be recorded laid up in Memory All evil chances grievous evils bring Fierce death attends foul chances governing In this Verse are twelve words relating to the number of the twelve Moneths of the Year accounting March the first wherefore the word proper to that Moneth is All and so in order Fierce is the seventh word and therefore belongs to the seventh Moneth or September That which is to be observed from these Words is what letter the word beginneth withal and to count the number of that letter in the order of the Alphabet which will never exceed seven and the number of the said letter shews what day of the Moneth proper

Proportion of the same nature and after the same manner as we found the Azimuth at six before by the Analemma was the said Proportion protracted yet here it is to be suggested that in the Analemma there are three Proportions in Sines wrought instead of the one in Sines and Tangents before expressed 1. As namely to finde the Suns Altitude at Six As the Radius is to the Sine of the Latitude So is the Sine of the Declination To the Sine of the Suns height at six 2. To finde his Azimuth in that parallel of Altitude As the Radius is to the Cosine of the Latitude So is the Sine of the Declination to the Sine of the Azimuth in the said Parallel 3. To reduce it to the common Radius As the Cosine of the Altitude at six Is to the Radius So is the Sine of the Azimuth in that parallel To the Sine thereof in the common Radius The two latter Proportions in Sines may be brought into one as I have shewed in a Treatise the Sector on a quadrant Pag. 111 114. and that will be As the Cosine of the Altitude at six Is to the Cosine of the Latitude So is the Sine of the Declination To the Sine of the Azimuth sought And thus in effect the Analemma performs that single Proportion intermingled with Tangents after a more laborious manner in Sines or if you will the Altitude at six being found it holds As the Cotangent of the said Altitude Is to the Radius So is the Cotangent of the Latitude to the Sine of the Azimuth sought and this Proportion lies visible in the Analemma By these Directions derived from the Analemma together with the Proportions for each Case all the 16 Cases of right angled Sphaerical Triangles may be resolved some whereof seem to require the drawing of an Ellipsis as namely if the Suns place and right Ascension were given to find his greatest Declination which notwithstanding according to these Directions is easily shunned CHAP. VIII Shewing how to come by the Suns Altitude or Height without Instrument UPon any Flat or Plain that is level or parallel to the Horizon erect or set up a Wire without inclining or leaning to either side admit in the Point C and when you would finde the Suns Altitude or height make a mark at that instant in the very end or extreamity of its shadow suppose at G the shadow be-being the Line C G. Then upon the same Flat draw the quadrant of a Circle C A F with 60d of your Chords and make C E equal to the height of the Wire and through the point E draw the line E D parallel to C A and therein prick down the length of the shadow from E to D a Ruler laid from the Center to D cuts the quadrant at B and the Arch B A measured on the Chords sheweth the height required in this Example 20d 25′ in like manner if the length of the shadow were E K the height would be N A 38d 16′ Otherwise This may be performed otherways by drawing the quadrant C A F upon any plain board whatsoever then stick in a Pin at the Center C and hold the board so towards the Sun that the shadow thereof may fall upon the line C A then imagine C G to represent or supply the use of a Thread and Plummet hanging upon the Pin in the Center at liberty and mark where it cuts the Arch of the quadrant F A suppose at H measure the Arch F H on the line of Chords and it shews the height requi●ed By the next Chapter we shall finde the Suns Azimuth belonging to the Altitude 20d 25′ according to the Latitude and Declination there given to be 31d 19′ from the Meridian admit to the Westward of the South then doth the shadow happen so much to the Eastward of the No●th wherefore if 31d 19′ be set off in the quadrant from the line of shadow from H to N a line drawn into the Center as N C shall be a true Meridian Line or line of North and South CHAP. IX Shewing the Resolution of such Propositions wherein the Suns Altitude i● either given or sought THe Latitude of the place Declination of the Sun and his Altitude given to finde the hour of the day and the Azimuth of the Sun Example Declination 13d South Altitude 14d 40′ With 60d of the Chords draw the Circle S Z N E the Center whereof is at C and draw the Diameter S C N and Perpendicular therto Z C prick off the Latitude 51d 3′ from N to P and from Z to AE and draw the Axis B P and the Equator AE C Q prick off the Declination from AE to D and from Q to E being 13d from the Chords and draw the parallel of Declination D B E then from S to A and from N to O out of the Chords prick off 14d 40′ the Altitude and draw the parallel of Altitude A O so is B ⊙ the hour from six towards Noon in the parallel of Declination and M ⊙ the Azimuth of the Sun from the East or West Southwards To measure the Hour Set the extent B D from C to F and upon F as a Center with the extent B ⊙ draw the Arch G a Ruler laid from C just touching that Arch findes the point H the Arch N H measured on the Chords sheweth 45d and so much is the hour from six to wit in time three hours either nine in the forenoon or three in the afternoon To measure the Azimuth In like manner set M A from C to R and upon R with the extent M ⊙ draw the Arch I a Ruler laid thereto from C cuts the Limbe at L the Arch L S measured on the Chords is 44d 35′ and so much is the Sun to the Southwards of the East or West To finde the Angle of Position Place the complement of the Altitude Z A from P to g then place the extent A g from g to f a Ruler laid over A and f cuts the parallel of the Suns Declination at t and D t is the versed Sine of the Angle of Position To measure it Thorough the point t draw k t y parallel to the Axis B P and draw D C then the extent C y measured on the Sines sheweth the complement of it to wit 62d 57′ therefore the said Angle is 27d 3′ Otherwise With the extent B D setting one foot in the Center at C with the other cross the former parallel line k t at k a Ruler laid from C to k cuts the Limbe at X and the Arch AE X is 27d 3′ the measure thereof as before the Complement whereof is the Arch V E and might be measured from the South Pole V if the Equinoctial AE Q were not drawn Three sides given to finde an Angle Another Example the Declination being North Latitude 51d 32′ Declination 13d North Altitude 37d 18′ In the following Scheme Upon the Center C with

60d of the Chords describe a Circle prick one of the sides namely the Colatitude 38d 28′ from Z to P in the Limbe and from P prick the Suns polar distance 77d to D and E and draw the parallel of Declination D E from Z prick off the complement of the Altitude to A and O to wit 52d 42′ and draw the parallel of Altitude A O so is the hour and Azimuth found in these two parallels without drawing any more lines the drawing of the Axis C P was onely to divide the parallel D E into halfs at B Likewise the drawing of C Z divides the parallel A O into halfs at M which may be found without drawing lines by laying a Ruler from C to Z and P or if you will draw the Horizontal line S N passing thorough the Center and above it the Altitude may be set from S to A and from N to O in like manner if the Equinoctial be drawn the Declination being North is to be set above it The manner of measuring is the same as before To finde the Hour C F is made equal to B D then upon F with the extent B ⊙ draw the Arch G a Ruler from the Cen●er touching G findes H and the Arch H N being 45d is the hour from Noon For the Azimuth C R is made equal to M A then upon R with M ⊙ draw I a Ruler from the Center touching it findes L and the Arch L S measured on the Chords is 30d and so much is the Azimuth to the Southwards of the East or West To finde the Angle of Position Here also Z A is placed from P to g and then g f is made equal to g A a Ruler from A to f cuts the parallel of Declination at t and D t is the versed Sine of the angle of Position which being measured as in the former Example for finding it will be found to be 33d 34′ Another Example for South Declination 13d retaining the former Latitude A new Scheme need not be drawn the Declination being as much South let the Altitude be 20d 25′ prick the Altitude from S to Q and from N to X and draw the parallel Q X where it crosse●h the parallel of Declination set V joyn C X and draw V r parallel to W C. To finde the Hour Upon F as a Center with the extent B V describe the Arch K a Ruler from the Center touching it findes T the Arch N T being 60d is the hour from six to wit either ten in the morning or two in the afternoon To finde the Azimuth The extent C r measured on the Sines is 58d 41′ and so much is the Suns Azimuth to the Southward of the East or West in this our Northern Hemisphere A third Example for finding the Azimuth In the former Examples it may be observed that the Azimuth is alwayes found in the parallels of Altitude which towards the Zenith grow very small and consequently this way of finding the Azimuth in Latitudes between or near the Tropicks and sometimes in our own Latitude when the Sun hath much Altitude will be very inconvenient for remedying whereof let it be noted that the Azimuth may be alwayes found in the parallel of Latitude Example For the Latitude of the Barbadoes 13d ⊙ Declination 20d North Altitude 52d 27′ In the first Scheme having drawn the Horizon S N its Axis C Z the parallel of Altitude B A 52d 27′ set off the Latitude from N to P 13d and draw the Axis C P prick off the complement of the Declination 70d from P to D and E and draw the parallel of Declination D E then is F ⊙ the Sine of the Azimuth from East or West Northwards F A being Radius which being placed from C to G and the Arch I described with F ⊙ a Ruler from the Center touching it findes S L 16d the measure of the Azimuth in the Limbe but thus to transfer from a less to a greater will breed much incertainty and the parallels of Altitude near the Zenith decrease very much for remedying this inconvenience change the names Latitude and Altitude and fit the Analemma thereto for the bare changing of the names of the two containing sides of a Triangle doth not alter the quantity of the angle comprehended Then will the new Latitude be 52d 27′ And the new Altitude 13d 00. I say the Azimuth is the same as it is in the Latitude of 13d when the Altitude is 52d 27′ Also the hour from noon in the new Latitude is equal to the angle of Position in the old Latitude and the angle of Position in the new Latitude is equal to the hour from noon in the old Latitude The old or first Scheme may very well serve if you set off 70d the complement of the Declination each way from A the end of the parallel of Altitude and then through the point P draw the parallel of Altitude parallel to the Horizon But here we have drawn a second Scheme where N P is 52d 27′ P D and P E 70d thorough which points are drawn the parallel D E Also S B and N A is 13d thorough which is drawn the parallel B A then is F ⊙ the Sine of the Suns Azimuth to the Northwards of the East to the Radius F A which is placed from C to G upon which with F ⊙ was the Arch I described and the Azimuth S L found to be 16d as before Nota also in the first Scheme or in both that the Radius C G might have been doubled and pricked upon N S produced then also must the extent F ⊙ have been doubled and the Arch I therewith described also the hour of the day might after the same manner be found in the parallel of Latitude which may be convenient for Stars that have much Declination in that Case the Declination and Latitude must change Names And thus when three sides are given to finde an angle we may find it in the Analemma by calling the complements of those sides the Declination the Latitude and the Altitude or Depression at pleasure Also when three Angles are given to finde a side those Angles must be changed into sides by taking the complement of the greatest Angle to a Semicircle and writing it and the other Angles down to their opposite sides in another Triangle as in the Scheme following wherein as the angles are changed into sides so are the sides changed into angles and then the Case will be the same as before Two sides with the angle comprehended to finde the third side and both the other angles By this Case may be found the Suns Altitudes on all hours and the distances of places in the Arch of a great Circle First the Suns Altitudes on all hours thereby is meant that if the hour of the day the Declination and Latitude be given the Suns Altitude proper to that hour or his depression may be found Upon the Center C with 60d of the

next the rump Declination 57d 51′ shall have upon the Azimuth of 22d from the North. Having drawn the primitive Circle H Z B N with its two Axes H V B and Z V N at right angles place the complement of the Latitude 48d from Z to P and from H to AE and draw P V F and AE V Q then place 32d 9′ that Stars polar distance from P to D and twice from Q to G by a ruler from AE to G finde M the extent V M will reach from D to C the Center of the parallel in the extended Axis therewith describe the parallel D * S then place 22d from N to O and from O to e a ruler over o and e from Z findes the pole of the Azimuth Circle at d and its Center at f now having the Center describe the said Circle Z S N. Which because it passeth through the parallel of Declination in two places at * and S that Star will have two Altitudes on that Azimuth and a double Solution must be given Through the three points F * P and F S P draw two Meridians To measure the Arks required In the Triangle Z * P. 1. A ruler laid from P to h cuts the Limbe at K and the arch AE K being 12d 19′ is the measure of the angle Z P * being the Stars hour from the Meridian 2. A quadrant placed from K towards N and a ruler laid over the Intersection found and P will finde the pole of that hour Circle at a then a ruler laid from * to d and a cuts the Limbe at k and u and the arch k u being 31d 32′ is the measure of the angle Z * h wherefore the angle Z * P is 148d 28′ 3. A ruler from d over * cuts the Limbe at t and the arch Z t being 17 degrees 39 minutes is the measure of the Side Z * wherefore the greater Altitude is t B 72 degrees 21 minutes Again in the Triangle Z S P. 1. A ruler over P and E cuts the Limbe near N the ark between the said Intersection and AE being 137d 25′ is the measure of the angle Z P S. 2. A ruler from d over S cuts the Limbe at i and the arch Z i being 74d 3′ is the measure of the Side Z S wherefore the lesser Altitude is i B 15d 57′ 3. The arch k u being 31d 32′ is the measure of the angle Z S P being the angle of Position From three shadows of a Gnomon or Wyre on a Horizontal Plain to finde a true Meridian-line and thereby the Azimuths of those shadows the Latitude of the place the Suns Declination Amplitude Altitudes and the Hour of the Day The Gnomon is supposed to be perpendicular to the Plain it stands upon or at least a point in the said Plain must be found through which a perpendicular let fall from the tip would pass and from the said Point the lengths of the three shadows must be measured In the whole Circular Scheme following let C B represent a Wyre or Stile standing erect on a Horizontal Plain and let the three shadows thereof be C F C E C D. Upon C as a Center with 60d of a Line of Chords describe the Circle O S Z N and produce the three lines of shadow beyond the Center towards H I G. Then in another Scheme upon B as a Center describe the Semicircle A G N with its Diameter A B N which divide into two quadrants with the perpendicular B G then make B C equal to the height of the perpendicular Stile and draw C F parallel to B G and therein prick down the three lengths of shadows from C to E to D and F and from those points draw lines into the Center cutting the Limbe at E I H and the arks between those points and G are the respective Altitudes or heights of the Sun but if measured from A they are the complements of those Altitudes Then lay a ruler from N to the three points E I H and it will cut the Radius B G at K L M the distances of which points from B are the Semitangents of the complements of the Suns three Altitudes Then repair to the following Circular Scheme and place B K on the shadow E produced from C to I also make C G on the second shadow equal to B L likewise make C H on the third shadow equal to B M in the Scheme above Then through the three points H I G draw a Circle the Center whereof will be found at V to find it with any extent upon G describe an ark at a with the same extent upon I cross the former ark do the like with the same or any other convenient extent at e also upon the points H and I do the same at o also beneath at u then lay a ruler over the Intersections a e and draw a line near V do the like through the other Intersections at o and u where these lines cross as at V is the Center of the arch Q H I G R then describe it and from V draw the line V C S passing through the Center and it is a true Meridian-line or line of North and South from which the Azimuths of any of the shadows may be measured perpendicular thereto draw O Z passing through the Center The arch Q O measured on the Chords sheweth the Suns amplitude A Ruler laid from O to A cuts the Limbe at M and the arch S M is the Suns Meridian Altitude The nearest distance from Q to O C is the Sine of the Suns Amplitude which place from C to K when it happeneth on that side of the Vertical and draw M K produced and it shall be the parallel of the Suns Declination in the Analemma if we make S N the Horizontal line from C draw a line parallel to K M and it will cut the Limbe at AE the arch M AE is the measure of the Suns Declination and the arch S AE is equal to the complement of the Latitude consequently the arch Z AE is the measure of the Latitude of the place Place the said Extent from N to L then a ruler laid over it and O cuts the Meridian at P which is the projected pole Point through which point and the points H I G if there be arks of great Circles drawn the angles that the said arks make with the line S P shall be the measure of the respective hours from Noon proper to each shadow How to draw such arks I have explained in a Treatise of Geometrical Dyalling page 49. See also Clavius de Astrolabio Liber secundus Prop. 13. who handles it largely This manner of finding the Hours I confess is troublesom and may be sooner resolved by the Analemma now we have the Latitude Declination and all the Altitudes given In northwardly Regions where the Sun for some competent season doth not rise nor set he hath no amplitude in that case the Circle to