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 THE 
 
 ELEMENTS 
 
 O F 
 
 NAVIGATION. 
 
 VOLUME THE FIRST,
 
 THE 
 
 ELEMENTS 
 
 O F 
 
 N A V I G AT I O N; 
 
 CONTAINING THE 
 
 THEORY and PRACTICE. 
 
 With the necefiary Table s, 
 
 And C o M P E N D I u M s for finding 
 The LATITUDE and LONGITUDE at SEA 
 
 To which is added> 
 
 A TREATISE 
 
 O F 
 
 MARINE FORTIFICATION. 
 
 Compofed for the Ufe of 
 
 The Royal MathExMatical School at Christ's Hospital, 
 
 The Royal Academy at Portsmouth, 
 
 And the G E N T L E M E N of the N A V Y. 
 
 In TWO VOLUMES. 
 
 By J. ROBERTSON, 
 
 r.ate Librarian to the Royal Society, and formerly Head-Mafler of 
 the Royal Academy, at Portfmouth. 
 
 The FIFTH EDITION, with ADDITIONS. 
 
 Carefully revifed and corredcd by 
 
 WILLIAM WALES, 
 
 Mafter of the P^oyal Mathematical School, Chrift's Hofpital, London, 
 
 LONDON, 
 
 Printed for C. N OURS E, in die Strand, 
 MDCCLXXXVI.
 
 'to "RS^ 
 
 v.j- 
 JOHN EARL OF 
 
 SANDWICH, 
 
 &c. &c. &c. 
 
 FIRST COMMISSIONER 
 
 O F T HE 
 
 
 
 \ Boards of Admiralty and Longitude. 
 
 M Y L O R D, 
 
 WHILE the public voice Is unanimous in applaud- 
 ing your humanity towards the Artificers, in general, of 
 His Majefty's Dock-yards, and your attention to reftore 
 the Royal Navy of Britain to the refpedlable Hate frotn 
 which it had been fuffered to decline lince the lafl 
 War, Philofophers not only admire thefe noble adls, 
 but likewife, your generous encouragement to improve 
 Geographical, Nautical, and Natural Knowledge. 
 
 A Such 

 
 D.^DICATION. 
 
 Such exertions of your Lordfliip's extraordinary 
 Mental and Official Abilities, will undoubtedly be tranf- 
 mitted with honour to the lateft poflerity : And your 
 laudable example muft infpire a regard for Works in-^ 
 tended to promote public utility* 
 
 The Author of ^be 'Bkmmts tif Navigation, not- 
 withftanding the favourable reception which the former 
 imprcffions have met with from Britifh Mariners, thinks 
 himfelf extremely happy that this improved Edition is 
 permitted to appear under your Lordfhip's Patronage. 
 
 That you may long enjoy the Opportunity as well as 
 Inclination of promoting ufeful Arts and Learning, is a 
 hope fincerely entertained by. 
 
 My Lord, 
 
 Your Lordship's 
 moll obedient 
 
 and humble Servant, 
 
 Nov. 1.1772. John Robertfon.
 
 T O T H E 
 RIGHT WORSHIPFUL 
 Sir ROBERT LADBROKE, Knt. Alderman, 
 PRESIDENT; 
 
 , THE 
 
 WorHiipful THOMAS BURFOOT, Efq. 
 TREASURER; 
 
 And Uie reft of the 
 
 WORSHIPFUL GOVERNORS 
 
 OF 
 
 Chrift's Hofpital, London: 
 
 This Book, containing the Elements of Navigation, 
 and a Treatife on Marine Fortification, firft pub- 
 liilied for the Ufe of the Children in the Royal Ma- 
 thematical School, when they were under my Care, 
 is, as a grateful acknowledgement for paft favours, 
 addrefled by 
 
 Your WORSHIPS' mofl humble Servant, 
 
 Nov. 1.1772. John Robertfon,
 
 A D V E R T I S E M E N T. 
 
 IN this Edition, the Editor has carefully correfled the er- 
 rorj which- had crept into the formtr; he has recomputed the 
 Tables in. Book V. Art. 308, 309, and 310, of the Sun's Lon- 
 gitude, Right Afcenfion, and Declination, and has alfo re- 
 vifed, as far as his materials extended, the Geographical Talkie, 
 and added the names of fuch places as his own obfervations, 
 or thofe of other perfons, have furnifhed him with ; fo that 
 he flatters- himfelf it- is the mofl: txtenfive and corredl of any 
 extant. On the whole, he prefumes, this Edition will be found 
 as worthy of the approbation of the public in general, and of 
 feamen in particular, as thofe which were printed under th? 
 Author's infpedion, .
 
 THE 
 
 P R E F A G E. 
 
 - / T having been part of my employment for tn any years pafl^ to 
 inflrul youth in the theoretical and fratfical farts of Navi^ 
 gation J 1 was naturally led to draw up rules and examples fit- 
 4ed to the years and capacity of the Jcholar : Jome of the precepts ^ 
 from time to time were alteredy according as L had obferved hois* 
 they were comprehended by the majority , of my pupils ; until ai 
 tength I had put together a Jet of materials y which I found fuffi- 
 ient for teaching this Art.. 
 
 Upon my being intrujied by the governors of Chrifl's Hofpital 
 (in the beginning of the year ij^^) zvith the care of the Royal 
 Mathematical fchool there ^ founded by King Charles the fecond, I 
 had a great opportunity of experiencing the method I had before. 
 iifed i and finding it fully anfwered my expe^ationt 1 determined 
 to print it for the uje of that fchool : but as thoje children are t 
 be infiru5led in the mathematical Jciences^ on which' the art of Na^ 
 vigation is fuundedy I judged it proper ^ on their account ^ to intro- 
 duce the fubjels of Arithmetic^ Geometry^ Trigcnon2etry, <^c. for 
 which reafony this treatife is dijiinguifloed by the title c/ Elements 
 cf Navi2:.'icion. 
 
 t> 
 
 After my appointment {in the year 1755^ to be lead tn offer of 
 the Royal Marine Academy at Portjmoulh, founded by King George 
 the fecondj I aljo found that this book teas fuffxiently inielligibls 
 to beginners of middling capi:c::-:ss ; and therefore.^ in the fecond 
 edition J in tie. year J 761, the :v.'inrer in which it was frft com- 
 pofcd zviJs CO ii tinned, except the i e^iimvi-ig of the book of Afir enemy y 
 from being the %ih. into the place of the c^th. ; whereby the books of 
 Plane Sailings Globular Sailing, and Days V/orks^ which logeth'er 
 nearly comprehend the drt cf Narjigationy foi^uiv in fuccefTion, 
 3 here were indeed feme variat:ons in the modes of exprefjion in a 
 jew places; hut the additions in every look ...',7v jmuk ra'ier to 
 extend the notions of learners, than tnJuppL a.r; Uifcimey i:anhng 
 in the Jorircr id:::cu -, except Jome of the additions 1^ the <^hh. book, 
 which we,e no^jc well knoi'^/t at the time cf the fffl imprejjion. 
 
 A 3 The
 
 X PREFACE. 
 
 The work is divided into ttn parts or hooks, each being a diftintf 
 treatife-t the preceding ones contain the necejfary elements which 
 are wanted in thoje that follow. 'The demonjiration of the feveral 
 proportions are given as concifel;^ as I could contrive^ to carry with 
 them a fufficient degree of evidence : Throughout the whole of the 
 elementary parts y brevity and per/picuity were confidered ; but the 
 practical parts are more fully treated on, and intended to include 
 every iifeful particular y worthy of the mariner's notice. 
 
 In the elementary parts, where it is not eajy to introduce neia 
 mattery there will be found the common principles treated in a man' 
 nevy which, it is apprehended, is better adapted to beginners j and 
 
 Jucb new lights thrown on Jeveral particulars, as will render them 
 more obvious than in the view wherein they have been commonly 
 
 Jee, 
 
 The treatije of Fortification annexed, is the rejult of many years 
 application ; and is delivered in a very different mode from what 
 other writers have taken ; for among the multitude which I have 
 Jeen, they generally begin with the fortifying of a town, the moji 
 difficult part of the art, and end with works the mofl eafy to con- 
 trive and execute : Herein the works of the fimplefl confiruStion 
 are begun with, and the learner gradually advanced to the fortify- 
 ing of a town : Indeed the limits chofen for this tra5i have caujed 
 fome articles to be briefly mentioned, and others to be totally omitted; 
 never thelefs, it is conceived that, in its prefent flate, it may be of 
 conftderable ufe to Marine Officers, ajid even furnifh fome hints not 
 altogether unworthy the notice of the Gentlemen of the Army. 
 
 The Maritime parts of theje Elements, contained in the vii^'', 
 vili'^, and ix'*" books, are alfo delivered in a manner fomewhat dif- 
 ferent from what is fern in other writers ; who, for the mofl part 
 copying from one another, have not much contributed towards per^ 
 feeling the art of Navigation ; the writers indeed have been many, 
 but the improvers have been very ftw ; Wright, Norwood, and 
 Hal ley, having done the mofl of what has been difcovcred fince a 
 Utile before the begifining of the ly th century : Hcivcvcr, in the 
 method here taken, it is apprehended that the proper judges will find 
 fome few improvements, as well in the art itfclf as in the manner 
 of communicating it to learners. 
 
 The common treat] fes of Navigation, which, on account of their 
 J mall bulk and ecijy price, are vended among the Britiflo Mariners, 
 jam net to be written with an intention to excite in their readers a 
 
 defire
 
 PREFACE. 3ll 
 
 deftre to ftirjue the Sciences^ farther than they are handled in 
 thofe books ; fo that it is no wonder our feamen in general had Jo 
 little mathematical knowledge \ for the perfon who could keep a 
 trite journal^ formed on the tnoji eafy occurrences^ has been reckoned 
 a g{iod artiji ; but whenever thofe occurrences have not happened^ 
 the JQurnalift has been at a lofs^ and unable to find the fhip^s place 
 with any tolerable degree of precijion j andjuch accidents havepro" 
 bably contributed to the dijlrejs which many fhips crews have expe^ 
 riencedi and which a little more knowledge among them might have 
 prevented J or at leafi have leffened^ 
 
 About the middle of the i (>th century ^ Navigation began to be 
 eonjidered as an art, in a great meajtlre dependent on the Mathe^ 
 matical fciences ; and on fuch a plan has it been cultivated by the 
 labours of the mofl judicious^ who have applied tbemf elves towa^'ds 
 its perfection ; and although the art has been enriched by the obfer-^ 
 vations offome learned men in different nations, yet it has Jo hap-^ 
 pened, that the chief of the improvements ^ and particularly the ma* 
 the matical ones *, werefirji publifkcd in Britain^ 
 
 Into this work are collected mofly if not ally of the tifeful and cU" 
 fious particulars relating to the art of Navigation ; there are aljo 
 interfperfed hijlorical remarks of inventiohSy with the 7tames of 
 many eminent men, and their works; thefe were intended as in* 
 centives to infpire learners and our feamen with a defire not only of 
 knowing the things herein treated of from their foundations^ but of 
 pufhing their inquiries into fuch other parts of the Jciences as ma;/ 
 procure to themjehes pleafure, profit y and refpe5ly and render thent 
 more ujrful to their country by the [kill rejulting from fuch ac- 
 quijitions. 
 
 I have always thought that the chief motives which ought to in-* 
 duce a perfon to appear as a writer fhould bCy either that he has 
 fomething new to publifhy or that he has arranged the parts of a, 
 known Jubjetl, in a method more regular and ujeful than had been 
 done before ; in either of thefe cafes he cannot be a proper judge, 
 unlejs he has Jeen the pieces extant on that fuhje5ly or at leafi y thofe 
 / the mofi eminent authors already publifhed : On thefe principles 
 I was led to examine what had been done by the different writers 
 0n Navigation ; and having perufed mofi of their books y of which X 
 
 * Sec the following difleftation. *'' 
 
 A 4 (euld
 
 xii PREFACE. 
 
 (ould gei inform ation^ 1 bad an op-portunity of dif covering the fief s 
 by whuh this art has fifen to its prefent perfeoliony and confequently 
 of knowing the mofi material parts of tjpe hiflory of its progrefs : 
 Among other things^ I could net avoid remarking a mi flake which 
 has crept into many of the modern books of Navigation ; which is, 
 that Wriglu'j invention of making a true fea-chart was Jiolen by 
 Mercator, and publifhed as his own. I fufpe^ this Jiory had its 
 rife in a book printed in the year 1675 by Edward Sherburne, in-' 
 titled *' The Sphere of Marcus Manilius made an Englifh Poem, 
 with Annotations and an Aftionomical Appendix." 
 
 My enquiries into thefe matters induced the late learned Dr, 
 
 James Wilfon to review and complete his obfervations on the fame 
 
 jubjeify and produced his Differtation on the Hiflory of the Art of 
 
 Navigation ; which he zvas pleafed to give me leave to puhlifh with 
 
 thejecond edition of this work, 
 
 There are few perfons^ however knowing and careful^ who may 
 not commit.^ and overlook.^ inadvertencies in their own compofitionSy 
 which may be dif covered by ethers : therefore at my requefi the 
 greateft part of the manufcript for the firji edition was read and 
 examined by two of my friends *, well acquainted with the theory 
 / nd pra^ice of Navigation ; who^ by their judicious obfervations, 
 enabled me to improve Jeveral articles : Some part of the additions 
 ;o the id Edition, received much elegance and perfpicuity through 
 :he friendly advice and communications of the late learned Dr, 
 Mtrnry Pemberton, F. R. S. 
 
 The Jccond Edition of thefe Elements having alfo been well re- 
 ceived by the Pullic ; Dr. Wilfon took the pains to revije his Dif- 
 fcriation, which he improved in many particulars : And I have 
 iilfo endeavoured to retain their fa.v our able opinions of my labours^ 
 ,'';.' giving Co7npendiums for performing the opcr aliens of the new 
 i.iithods of Jin ding the latitude and longitude of a fnp at fea j and 
 ''cine other alterations end additions which I couicivcd would ren- 
 ,':r this third Edition more generally ufefiil. 
 
 >T 
 
 r.ov. T, 1772. 
 
 * William Mour.laine, Efq. I'\ R. S. and iMr. William Payne.
 
 r .' 
 
 T H E 
 
 CONTENTS. 
 
 THIS treatife in two volumes contains ten books ; each is divided 
 into feveral fe6tions, and numbered with the Roman numerals. 
 
 The particular articles are numbered by the commpn figures, each 
 book beginning with the number i. 
 
 The references made from one article of the treatife to another is of 
 two kinds. 
 
 Firjl. When in the fame book. Then the number of the article re^ 
 ferred to, is put in a parenthefis. Thus (27), refers to the article num- 
 bered 27 in the fame book. 
 
 Second. When in another book. Then the number of the book in 
 Roman figures, and of the article in common figures, is put in a paren- 
 thefis. Thus (II. 160) refcis lo the i6oth article of the fecond book. 
 
 In the following contents, the fedions refer to the number of the page : 
 the particulars to the number of the article. 
 
 Vol. I. BOOK I. ARITHMETIC. 
 
 From Page i 
 
 S E C T 1 N I . Definitions and Prin- 
 ciples p. 1 
 Notation^ nutneration^ andfra^ions 
 
 at articles 17 to 21 
 
 Signs, and tables oy coin, weights, 
 
 Qc. 22, 23 
 
 Section II. Addition p. 6 
 
 Of whole numbers and fraolions 24 
 
 Section III. Subtraction p. 
 
 Of whole tiumhers and fractions 26 
 
 ^ueflions to exercife Addition and 
 
 Subtration 27 
 
 Section IV. Multiplication p. 10 
 
 Multiplication table art. 29 
 
 Multiplication of whole numbers 
 
 and fratiom art. 30 
 
 Section V. Divifton p. 12 
 
 Of whole numbers and fra^ions 32 
 
 Section VI. Reduction p. 14 
 
 Of feveral names to one name 35 
 
 ()f inferior name to fup. natne 36 
 
 f ulgar fralions to decimals 38 
 
 Inferior names to a decimal 40 
 
 '7o value decimal fraSlions 42 
 
 Prailiial quelliom 43 
 
 to Page 42. 
 
 Section VII. Of Proportion p. 20 
 Of the nature of proportional num- 
 bers - 44 
 The Rule of Three 46 
 A collection of 16 quejlions 47 
 Section Vill. Of powers and 
 roots p. 26 
 To raift numbers to a propofed 
 power art. 49 
 To extraSi the fquare root 54 
 To extraSi the cube root 56 
 Section IX. Of numeral fries 
 
 p. 3a 
 Of arithmetic progreffton art. 59 
 Of geometric progreffton 60 
 
 Some properties of fuch progrefft- 
 
 onal numbers from art. 62 to 67 
 Four prtblems in progreffions 
 
 68 to 71 
 Section X. Of logaritlmis p. 34 
 Of the nature of logarithms 
 To make logarithms 
 Of the tables of logarithms 
 (jf the indices of logarithm 
 Multiplication by logarithn 
 
 Diz 
 
 11 
 
 n 
 80 
 
 8s 
 
 fton
 
 CONTENTS. 
 
 t)ivifion hy logarithms 
 Of proportion by logarithmt 
 
 ZtsOfthe arith. comp. of logarithms 88 
 8 7 1 (Jf powers and roots by logarithms 89 
 
 BdOKiL GEOMETRY. 
 
 From Page 43 to Page 88* 
 
 Sectioit I. Definitions and Prin- 
 
 tipU p. 43 
 
 Definitions art. i to 42 
 
 PoJbtlaUt 43 
 
 Axioms 46 
 
 5bct. II. Geometrical Pfohiems ^. i^% 
 Gf right lined figures 55 to 69 
 
 Of the circle 70 to 74 
 
 Of proportional lines 75 to 80 
 
 Scales of equal parts and chords 
 
 81 lo 84 
 Of polygons 85 
 
 Section III. Geom. Theorems p. 58 
 Of lines and triangles art, 91 to 106 
 Of quadrangles and the circle 
 
 107 to 137 
 
 Section IV. Of Proportion p. 68 
 Definitions and Principles 1 38 to 147 
 General Properties 1 48 to 1 64 
 
 / fimilar triangles 1 6 5 
 
 / the circle and regular polygons 169 
 ^adrature of the circle i g I 
 
 Sect. V. Of planes and folids p. 82 
 Definitions and Principles art. 1 9^ 
 Properties of cutting planes lOC) 
 Seiion of pyramids and cones 2 1 1 
 
 Section VJ. Of the fpiral p. 86 
 Of the common fpiral art. 221 
 
 Of the proportional fpiral 222 
 
 Logarithmic fpiral 226 
 
 Of Napier* s Logarithms iic^ 
 
 BOOK III. PLANE TRIGONOMETRY. 
 From Page 89 to Page 124. 
 
 SfcCTiON I. Defnitions and Prin- 
 ciples P" ^9 
 Definitions art. J to 16 
 
 Section" II. Triangular canon p. 92 
 ConflruSiion of the plane fcale art. 1 7 
 Of fines offmall arcs 25 
 
 Of fines above 60 degrees 29 
 
 O/" //6^ relation of tangents, &c, 3 1 
 To compute the fines 38 
 
 Of the tables of fines ^ tangents^ 
 
 ^'' ... 39 
 
 Sect. III. Solution of triangles^. \oo 
 
 By oppofite fides and angles art. 45 
 
 Tivo fides and included right an- 
 gle 46 
 
 Two fides and included oblique an- 
 gle 48 
 
 By three fides 49, 50 
 
 Synapjis of the rules art. ^1 
 
 Twelve numeral examples 53 to 64 
 Section IV". Of the Gunter^s 
 fcale p. 1 14 
 
 Of natural and logarithmic fcale s 
 
 art. 65, 66, 67 
 Of the line of numbers 68 
 
 Of the line of fines 69 
 
 Of tangents and verfed fines 70, 71 
 Demonjlratim of their conflruhion 
 
 7^ 
 Sect.V. life of Gtmt^r^s fcale p. 118 
 
 Precepts art, 76 to 78 
 
 Exemplifications: 79 to 86 
 
 Section VI. Properties of plane 
 
 triangles p. 122 
 
 In oblique angled triangles art. 88 
 
 Rules for finding an angle 90 to 98 
 
 BOOK
 
 contents: 
 
 BOOK IV. OF SPHERICS. 
 From Page 125 to Page 193. 
 
 Section I. Principles p- 125 
 
 Definitions art. i 
 
 Axioms 16 
 
 Section II. Sieregraphie propo- 
 fttions p. 128 
 
 Principal properties art. 30 to 59 
 
 Sect. III. uph eric Geometry p. 135 
 The conJlruSion and demonjiration 
 6f 37 cafes art. 60 to 8 1 
 
 Section IV. Spheric Trigono- 
 metry P 145 
 Definitions and Principles art. 82 
 
 Section V. Spherical Theorems 
 
 Fifteen propofitions art. 90 to 1 10 
 Section VI. Solutions of right an- 
 gled fpheric triangles p. 152 
 By tiuo neiu rules art, 1 11 
 JBy Napier's general rule 1 19 
 Section VII. Solution of oblique 
 angled fpheric triangles p. 157 
 JVith a perpe7idicular 120 to 124 
 Given three fides or three angles 125 
 
 Synopfis of right angled fpher. 1 2 J' 
 Synopfis of oblique angled fpher, 144 
 Section VIII. Con/iruiiion and 
 calculation of the cafes of right 
 angled fpheric triangles p. 166 
 Numeral examples art. 156 to i6f 
 Exam, of a quadrantal triangle 162 
 Section IX. ConflruSfion and 
 calculation of the cafes of oblique 
 angled fpheric triangles p. 175 
 Numeral exa?nples art. 163 to 168 
 Section X. Goniometriclinesp. 179 
 Properties deduced art. 169 
 
 Table of goniometric relations 
 
 171 to 210 
 Variety of theoroms an to 224 
 
 Relation between the fides and a?!' 
 gles of fpheric triangles 225 to 236 
 Ruler when two fides and the in- 
 cluded angle are kncwn^ to find 
 the other angles 237 
 
 Rules when 'i^ fides are known 254 
 To find the 7iatural fines of given 
 arcs 5 and the contrary 256, 257 
 
 BOOK V. ASTRONOMY. 
 
 From Page 195 to Page 328. 
 
 Section I. Solar Ajlronomy p. 195 
 Of the Solar Syfiem art. i 
 
 Number of conjlellaiions andjiars 6 
 Primary and fecondary planets \1 
 Annual and diurnal motions 1 7 
 
 Planets orbits and nodes 21 
 
 Elliptic orbits of the planets 2 3 
 
 Kepler's laws 29 
 
 Conjun^ions and oppofitions 34 
 
 Table of the folar fyfiem 3 7 
 
 Of the fecondary planets 38 
 
 Of the figure of the planets 40 
 
 Sect. II. Terrejlrial Afironomy^.lo&^ 
 Of the direct^ fiationary^ and retro- 
 grade appearances art, 45 
 
 Phenomena of the inferior planets 46 
 Phenomtva of the fuperior planets 49 
 Apparent motions of the Sun 50 
 
 Obliquity of the ecliptic 55 
 
 Of the different feafons 56 
 
 Rifing and fitting of fan 65 
 
 art. 
 
 66 
 
 73 
 
 74 
 79 
 
 Of parallaxes 
 The meafure of the Earth 
 Of the Moon 
 
 Of folar and lunar eclipfes 
 Sectu N HI. Ajlronomy of the 
 fpher. p. 214 
 
 Definit ons and Principles art. 89 
 Ofi ajlr Gnomical tables 1 32 
 
 To convert time into degrees., and 
 
 degrees into time 133 
 
 Of the cubninating of ihejlars 1 34 
 Section IV. Oftheproje^ion of 
 the fphere p. 221 
 
 Four projeiiions on the four prin- 
 cipal circles art. 135 
 Sect. V. Problems of the fphere p. 225 
 The conJlruBion and numeral fo- 
 lution of 22 prob!e?ns 139 to l6g 
 Sect. VI. 7'o find the latitude p. 242 
 Stereographic folutinn of fifiteen 
 probUms of latitude 1 70 to j 84 
 Fsuriccfi
 
 CONTENTS. 
 
 Fourteen ether brobkms for find-' 
 
 ing the latitude 1JJ5 to 203 
 
 Nonagefimal degree 2O4 
 
 bfeCTioNVll. Pra6iical Ajlrono- 
 my p. 268 
 
 Of ajirononucal Injlrumenti art. 205 
 Of tlye Clock 206 
 
 Of the Telcfcepe 20" 
 
 Of the Micrometer 208 
 
 Of the ^adrant 209 
 
 Of the Tranfit Injlrument 2 1 o 
 
 Of the Ajlroncmical Sector 212 
 
 Of the equal altitude infiriiment liif. 
 *ro odjuji the clock 215 
 
 Of the change in the Sun's declina- 
 tion 2 1 6 
 Of Vernier's dividing plate 2 1 9 
 
 Section VIII. Elements of the 
 Earth's motion p. 280 
 
 Of mean motion and anomaly art. 222 
 Of the different years 229 
 
 Twelve problems relating to the 
 Earth's m'ition 234 to 255 
 
 Sect. IX. Equation of time p. 291 
 f)f the fyderial and equatorial day 
 
 art. 261 
 
 Of mean,and apparent time 263 
 
 To calculate the equation of time 268 
 
 Sect. X. To jnakefolar tables p. 293 
 
 For finding the Sun's place 
 
 art. 269 to 280 
 Of declinat. and right afcen, to 289 
 Equation of time 296 
 
 Nineteen ajlro7iomical tables to 319 
 
 BOOK VI. GEOGRAPHY, &c. 
 From Page 329 to Page 400. 
 
 Section I. Definitions and Prin- 
 ciples p- 329 
 Of the poles and circles art. 3 
 Of latitude and longitude 6, 9 
 Divifion of the Earth by %ones \ 3 
 Divifion of the Earth by climates i 7 
 Section il. Natural divifion of 
 the Earth P- 332 
 Definitions art. 18 to 31 
 Of continents and oceans 32, 33 
 Section III. Of the political /// 
 vifion of the Earth P- 334 
 Europe and its chief parts art. 35 
 Jfia ' " 3B 
 Jfrica 42 
 Anwrica 4.6 
 K c. I V . Geographical problems p. 342 
 Citicerning latitude and longitude 
 
 5c to 53 
 
 Sect. V. Theifeof the globes p. 346 
 Twenty -two problems on the celeflial 
 and terre/irial globes art. 54 to 76 
 
 Section VI. Of Winds p. 353 
 Air and Atrnofphere art. 78, 79 
 Trade-winds and Monf 00ns to 84 
 Dr. Halley's ohfervations 85 
 
 Section VII. Of Tides p. 358 
 Of gravitation and attra^fion to 89 
 Cuufe of the general tides art. 90 
 The chief propofitions on tides to loi 
 Of tj?e tides about Britain 103 
 
 Section VIII. Chronology p. 364 
 Of aras or epochs art. 1 1 1, 
 
 fulian account or old file 1 14 
 
 Gregorian account, or new file 115 
 Of the Kalendar 1 1 6 
 
 Chronological problems to 1 26 
 
 Geographical and tide table' ^ 137 
 
 Vol. II. BOOK VII. OF PLANE SAILING. 
 Erom P;'-ge I to Page i 30. 
 
 Section I. Of Principles p. 1 
 
 Definitions :ii't, i 
 
 Nccefjary elements 2 
 
 Nature of maps \ 
 
 Skct. II. Of the plane c-art p. 3 
 Cinj'.ru^lionof the plcr.e c'Duyt art. 5 
 
 /;'/ ufe, Jhcwn in 7 cafes 6 to 1 2 
 
 Section III. Plane failing . p. 6 
 Dcj-.nitions and Principles 
 
 art. 13 to 21 
 Rides for conjh-u6li'jns 22 
 
 Cfthc Travcrfc table ')-\
 
 CONTENTS. 
 
 Proportions 0/ ftdes and migles^ 
 
 making either fide the radius 25 
 
 Ganons of plane failing 26 
 
 ' Table of rhumbs 2 7 
 
 Sect. IV. Offingle courfes p. 1 1 
 
 The fix cafes of plane failingKozn, 33 
 
 ^eflions in plane failing 34 
 
 Section V . Compound tourfes p. 2 1 
 
 To work a traverfe art, 36 
 
 To confrut a traverfe 37 
 
 To conftruf a compound courf. 39 
 
 Twenty qiicflions in comp. courfes 40 
 
 Section VI. Oblique failing p. 35 
 
 - Bearitlg and feiting ofobjeSfs art. 41 
 
 Twenty quefiions and their fola- 
 
 tions ibid. 
 
 S E c t . V 1 1 . Sailing to windward p . 4 2 
 
 Definitions ' art. 42 
 
 General confiruSiion _ 46 
 
 Twenty quefiions and their fclutions 
 
 ibid. 
 
 Sect. Vlil. Current failing p. 51 
 
 Definkions and Principles art. 47 
 
 General conjlruciion 49 
 
 Twenty qujjiions and their folu- 
 
 tions ibid. 
 
 Section IX. Mifellaneous quef- 
 
 tions p. 60 
 
 Twejity complicated quefiions luith 
 
 their fohdioHs ' art. 51 
 
 Sect 10 NT X. Surveying of coafis 
 
 and harbours p. ^q 
 
 To furvey on jhip-hoard ' art. 52 
 
 To furvey mi fnore 53 
 
 Of colours and their ufe 55 
 
 Se'ct. XI. To eflimate difiances^. 76 
 
 By the motion of found art. 58 
 
 By the curvature of the earth 61 
 
 Tohle of di/iances feen at fca 6 5 
 
 Traverfe tables art, -66, 67 
 
 from p. 81 to 130" 
 
 BOOK VIII. OF GLOBULAR SAILING. 
 From Page 131 to Page 224. 
 
 Sec. V. Compound courfes p. 174 
 
 Section I. Definitions and Prin- 
 ciples P- 131 
 
 Of ajhip^s true place at fea art. 3 
 Of Mercator's chart 4 
 
 Of IVright's conflrudion 6 
 
 Section \\. Keckoning of longi- 
 tude^ in \\ propofiiions p. ] 34. 
 Meridion:il parts-iy the fecants art. 1 2 
 
 Principles and rules to art. 76 
 
 Four examples wrought by the fe- 
 ver al rules , to art, 8r 
 S EC T . V I . Of Mercator' s chart p, 1 8 2 
 Conflrudion art. 82 
 Of the lines of longitude and lati- 
 tude on Gunter's fcale 83 
 To conf met a true fca chart 131 Thirteen Problems on the ufe of 
 Truth of failing by the Mcrcator' s \ Mer^ator' s chart to art. 99 
 chart 1 ^''^PC.r.V II. Great circle failing p. igi 
 Of the jfpiral rhumbs on the globe 21 | The ufe art. 1 02 
 
 Jfferidional parts analogous to the 
 
 logarithmic tangents 24 
 
 Section' III. Parallel failing p. 146 
 Rules of computation art. 31 
 
 JHerid. dijl. of meridians 34 
 
 The cafes of parallel failing 37 
 
 Other examples 42 
 
 Section IV. Middle latitude and 
 
 Mercator s failing P-I52 
 
 Principles fr Mid- latitude art. 44 
 Principles for Mercator's 49 
 
 Tlnrtcen Problnns of globular 
 
 failings by Alid-lat. M creator 
 
 and logarith, tangents 
 
 The fix cafes in afngle courfe to 108 
 
 Concerning compound coinfes and 
 the principles to art. no 
 
 Three examples to 1 14 
 
 Section VlII. Of the figure of 
 the Earth p, 203 
 
 Of the experiments to find the fi- 
 gure nf the, Earth to art, 121 
 
 Of the raiio andtnagnitudeofthc 
 axes of the Earth to art. 126 
 
 Of the parallels of lat. degrees of 
 lat. and merid. parts on thefphe- 
 roid to 142 
 
 to art. 7 3' Tables of merid. parts p. 2 1 5 to 224 
 
 Book IX.
 
 CONTENTS. 
 
 BOOK IX, OF DAYS WORKS. 
 From Page 225 to Page 341. 
 
 Section!. Ofajhjp^srun p. 226 
 Oftht log. line and half min- glafs 
 
 ^rt. 3 
 Of their correSliont 4 
 
 Of M. Bougutr's log. 7 
 
 Sect. II. Ofthefea compafs p. 231 
 ConJlruSiion and properties art. 14 
 imperfeilions and amendments j 5 
 Difcovery of the variation j 6 
 
 Of the axifnuth compafi 17 
 
 \fo make artificial magnet $ lo 
 
 To obferve by the azinutth compafs 2 1 
 
 Section 111. Of amplitude p. 236 
 
 To work them \ \y %^^^"^^^; ;;t- II 
 I by Iraverfe table 20 
 
 Section IV. 0/'tfz/a//;i p. 239 
 
 1 by Gunter's fcale 29 
 
 Section V. Of the variation of 
 
 the compafs p. 242 
 
 What the variation is art. 30 
 
 To find the variatitn 3 1 
 
 To correct courfes by variation 32 
 
 Section VI. Of leeway p- 244 
 
 TVhat^ and how allowed for art. 33 
 
 To correSi the courfes 35 
 
 Section VII. Of quadrants p. 245 
 
 Of Davis's quadrant art. 36 
 
 40 
 
 41 
 
 42 
 
 50 
 51 
 52 
 53 
 
 54 
 255 
 
 Of the fore-flaf 
 
 Of Hadley's quadrant 
 
 To reSlify and obferve by it 
 
 Jfnprovements 
 
 Of an artificial horizon 
 
 Corrections^ by the dip 
 
 By ref ration 
 
 By parallax 
 
 Tables for cor reSf ions of alts, p 
 Section VIII. Of comparing and 
 cor re citing tim? p. 256 
 
 To find the time^ and correSi the 
 vjatch 258 
 
 S E c T 1 o N J X . To find the latitude 
 at Jea p. 261 
 
 iji. By decl. and %en. dijl. art. 60 
 
 id. By decl. and equal alts. 63 
 
 3^/. By 3 %en. dijl, at equal inter- 
 vals ^ near noon 64 
 
 j\.ih. By 3 zen. diji. at unequal in' 
 tervalsy near noon dy 
 
 <yth. By 3 %tn. dijl. at equal inter* 
 valsy not near noon 68 
 
 tth. By 3 alts, at unequal inter- 
 valf, not near noon 69 
 
 ']th. By 7. altitudes^ the interval 
 of tinie., declination f and latitude 
 by account 70 
 
 8/Z>. By declination^ two ahitudesy 
 the interval of time., lat. by ac- 
 count^ and courfe and dijlance run 
 bettveen the obfervfition 75 
 
 S E c T I o N X . To find the longitude 
 
 at fea p. 289 
 
 Method^ of correcting the long, 
 
 290 
 
 Method I. By a current art. 79 
 
 II. By the courfe and dijl. 8q 
 
 III. By the variation chart 85 
 IV.' By a perfeSi time-keeper 89 
 
 V. By the fun's declination 92 
 
 VI. By the moon' s culminating 93 
 VH. By cclipfes of Jupiter' s fa- 
 
 tellites 94 
 
 VI II. By eclipfes of the moon 97 
 
 IX. By occultation ofjlars 98 
 To obferve the angular dijlance of 
 
 celejlial objects 102 
 
 To correal the obferved angular 
 
 dijlance 107 
 
 Compendium for obtaining the true 
 angular dijlance^ with examples 
 
 108 
 Mr. JVitchelPs method 109 
 
 To find the Greenwich time aii- 
 Jwering to a given dijlance no 
 Method X. Longitude by obfer- 
 vations of dijl. and alts. 1 1 1 
 
 XI. Longitudes by obfervations 
 of the dijlance only 112 
 
 Sect. XI. Of a Jhip's reckoning 
 
 P'3'7 
 
 Of dead reckoning and the log-book 
 
 art. 113 
 
 General precepts 114 
 
 Four independent dap works to 120 
 
 Journal
 
 CONTENTS, 
 
 journal of a voyage from England 
 
 to Madeira in 27 days^ with each 
 
 dfifs work p. 324 to 341 
 
 Appendix p. 342 
 
 Theory of Davis's quadrant art. 124 
 
 Theory of Hadley's quadrcfnt 126 
 
 Dip of the horizon 128 
 
 Of refraiiion 130 
 
 OflatitHde 131 
 
 Facials problem *^^ 133 
 
 De la Caille on longitude 1 35 
 
 Of the Moon's ph^es 136 
 
 Of the time of high-water 1 37 
 
 Of the Moon's rifjng 14 1 
 
 Tables of ' logarithms of numbers^ 
 fines and tangents 
 
 from p. 353 to 393J 
 
 FORTIFICATION. 
 PART I, OF XANP FORTIFICATIONS, 
 
 From Page i 
 
 Section I. Of lines p. 2 
 ifVhat, and where ufed art. 6 to g 
 
 Dimenfions and conJlruSiion to 1 7 
 
 Section II. Of batteries p. 7 
 
 Several kinds art. 18 
 Of the ditch and parapet 
 Platform and magazine 
 
 Afafcine battery 
 Acofft 
 
 22, 23 
 28, 32 
 
 34 
 
 38 
 46 
 
 50 
 
 offer battery 
 To conJiruB embrafures 
 Profiles of batteries 
 
 Section III. Additional works p. 16 
 Of ramps and bar bets 51*56 
 
 Of cavaliers and traverfes 58, 59 
 Of pallifadts and barriers bi^di 
 Of redoubts and redan 63, 71 
 
 Section IV. Fortifying of towns 
 
 P-23 
 Definitions grt, 73 to 84 
 
 General principles 85 
 
 Table for regular works 86 
 
 To draw the majler-line J89 
 
 To make the rampart 93 
 
 To make the fofs and efplanade 98 
 
 104 
 J 09 
 112 
 114 
 
 ii5 
 
 to Page 52, 
 
 Of gates 
 
 Ofjireets and buildings 
 
 Of bridges 
 
 Orillon and retired flank 
 
 Double and cajemated flanks 
 
 Section V. Of works for the de- 
 fence of the fofs p. 38 
 Oftenails, caponiers^ and rams- 
 horns art. 117 to 12a 
 
 Section VI. Of outworks p. 39 
 Neceffary obfervations 
 Of the ravelin 
 Of Coehorn's ravelin 
 Of counter guards 
 Of the tenaillon 
 Of horn and crown-works 
 Of the covered-way, glacis, and 
 places of arms 139 
 
 Of traverfes and fally-ways 1 42 
 Of comTnunications 144 
 
 Of works beyond the glacis 145 
 
 Of profiles 148 
 
 Con/ir unions of profiles 152 
 
 Of irreg ular fortifications 160 
 
 122 
 123 
 
 129 
 
 131 
 132 
 13s 
 
 PART 11, OF MARINE FORTIFICATIONS. 
 
 From Page 52 to Page 76. 
 
 Of proper forts and their difpofl- 
 
 tion 187 
 
 Section III. Of booms P 58 
 
 To lay a boom art. 191 
 
 In ajlraight channel 193 
 
 In the bend of a river 195 
 
 General maxims 198 
 
 Section I. Of harbours P- 52 
 
 Artificial harbours art. 1 70 
 
 Natural harbours lyi 
 
 Section II. Of the fortifying of 
 
 roads and harbours p. 53 
 
 Exemplified in five forts 
 
 from art. 174 to 185 
 Of the figure andfize of forts J 8 6 
 6 
 
 Section
 
 CONTENTS. 
 
 Section IV. Of mooring flAps \>. 63 
 Of mooring in regard to the fi ream 
 
 art. 200 
 Of mooring in regard to the wind 
 
 . . \^ 204 
 Section V. Of gallant anions 
 
 p. 65 
 Battle ^f Santa Cruz. 1657 art. 207 
 Atlion at Londonderry i6^g 208 
 
 Action at Gibraltar 1 693 art.lOO 
 
 Difpofttion at Cadiz. 210 
 
 Difpofition ati St. fohn's 1 697 211 
 
 Battlq of Vigo 1 702 2 1 Z 
 
 A^ion at Carthagena 1741 213 
 
 Aoiion in river Hughly IT Si 214 
 
 Siege of Louijburg 1758 215 
 
 Taking of ^tebec 1759 216 
 
 Vol L 
 
 Plate Page 
 
 Spherics I. 125 
 
 fn.III.195 
 
 AjironomyX IV. 224 
 
 t V. 250 
 
 Geography VI. ' 329 
 
 In this Treatife are XVI . Plate. 
 
 Vol. II. 
 
 Plane 
 failing 
 
 Plate Page 
 
 Globular ^XU. i^i 
 
 Sailing I XIII. 202 
 
 DaysJVorksXW, 352 
 
 Fort if ca- f XV. 22 
 
 tm. tXVI. 76. 
 
 ERRATA. 
 
 VOLUME I. 
 
 P. 34, 1. 7 and 8, from the bottom, and p. 98, I. 9,/c?r Nepier, read Napier. 
 
 D C 
 P. 63,1. 28, 29, and 30, /or bc, r. ac. P. 71, 1. \,far--y r. . , 
 
 P. 77, I. 22, /or (art.) 145, r. 146. P. ^29, h 22, for (art.) 213, r. 212. 
 
 P. 132, 1. 9, /or ABS, r. ABL ; 1. 28, for af, r. fr, and /or ar, r. 
 
 FR ; 1. 39,/w- AR, r. FR ; and 1. 3i,yor abe, r. bae.^ P. 138, I. 10, 
 
 for a/, r. Af. P. 223, 1. 4, /or viii q ix, r. viii ^ iv. P. 225, 1- 
 
 40, /or tfii O, r. YO. P- 337> 1; 3' ? 388, 1. 37, and p. 389, 1. 27, 
 
 for Manilla, r. Manila. P. 35--, in the title, for Geograpay, r. Geo- 
 graphy. P. 395, catch-word, /or Siera, r. Siara. 
 
 V O L U M E II. 
 
 P. 144, 1. 2, 10, z^, 41, 42, and 44, for Napeir, r. Napier. P. 301, 1. 4 
 
 and 5,/r Regulus, r. Spica Virginis., -P. 307, I. 6,/or enlightened limb, 
 r..cenur.- P. 351, 1. 9 and i6,/orIV.,r. VI,
 
 DISSERTATION 
 
 ON THE 
 
 RISE AND PROGRESS 
 
 O F T H E 
 
 Modern ART of NAVIGATION. 
 
 IT has been much difputed to whom the world was obliged for the 
 mariner's compafs. A late Italian writer indeed contends, after 
 many *, that the honour of the invention is due to Flavio Gioja of 
 Jtnalfi in Campania^ who lived about the beginning of the I4.th cen- 
 tury t, though others fay it came from the Eail, and was earlier known 
 in Europe %- However that may be, it is certain, this wonderful dif- 
 covery gave rife to the prefent art of navigation ; which feems to have 
 made fome progrefs during the voyages, that were begun in the year 
 1420, by Henry Duke of Vifcd ||. This learned Prince, brother to Ed- 
 ward King of Portugal^ was particularly knowing in cofmography, and 
 fcnt for one mafter ^James from the ifland oi Majorca, to teach naviga- 
 tion, and make inftruments and charts for the fea. 
 
 Thefe voyages being greatly extended, the art v/as improved under 
 the fucceeding monarchs of that nation. For Roderic and yofeph, phy- 
 f.cians to King yohn the Second, together with onfe Martin de Bohemia, 
 a Pcrtuguefe native of the ifland oi Fayal, fcholar to RegiomontaKus, about 
 the year 1485, calculated tables of the Sun's declination, for the ufe of 
 the failors, and recommended the aftrolabe for taking obfcrvations at 
 
 The famous Chrijhphcr Colwnhus is faid, before he attempted the dif- 
 covcry of yfw^r/t(7, to have confukcd Martin d: Bohemia, v/irji other?, 
 and during the courfe of his voyage to have inftruited the Spaniards In 
 
 * Suitable to that verfe oS Panncrm:ta>:a, 
 
 P imn dedit na.it: s ujum ma^ncti Jma^phh. 
 \ S':eS]gnor Crr^orio Grif!:a/di's Diffcrtation on tliis fubjecl in the Mem:>!rs 
 of the Etri'/cuft Accidemy of Cor/ona, torn. iii. p. 193, printed at R.me ii 
 
 I Hl^'ioire des Mathettanq: es, par M. Mofttuch, a Pa' is, 175?). 
 
 i| MarianiT Hilt. Uijpan. lib. .xx. cap. 1 1, and lib. xxvi. C.ip. 1 7. Id gunt'.r, 
 
 Decados d'Afia par j. di Burnos, lib. xvi. i 5-z. 
 
 %\ Majj'ai Hillor. Indii:. lib, i. p. 6. printed at Fio'cnce ia 15^8. 
 
 y 'H., I. :i navigation ;
 
 ii DISSERTATION on the RISE, kc. 
 
 navigation *; for the improvement of which art, the Emperor Charht 
 the Fifth afterwards founded a led^ure at Seville f. 
 
 ThiT variation of the fea-compafs could not be long a fecret. ColumbuSy 
 on the 14th of September 149?., obferved it, as his fon Ferdinand affert^ if, 
 though others fcem to attribute that difcovery to Scbajiian (hbot ||. And 
 as this variation differs in different places, Gonzales d' Oviedi found there 
 was none jvt the jlx'-res^ ; where fome geographers have thought fit in 
 their maps to njake their firlf meridian to pafs through one of thofe iflands j 
 it not being then known, that the variation altered in time,' 
 ' The ufe of the Crofs Staff" now began to be introduced amongft the 
 Sailors. This very ancient inftrument being defcribcd by "Johrt IVerner 
 of Nuremberg^ in his Jnnc'atiotn on the firft book oi Ptolemy's Geography, 
 printed in 1514; he recoiuniends it for obferving the dilhmce between 
 the Moon and fome ftar, in order thence to determine the longitude. 
 Werner feems.to hive been the greateft geometer, as well as aftronomer, 
 of the time. In 1522, he publifhed a trat ^, containing a fpecimen of 
 the conies, with fome folid problems, and alfo he there determined the 
 preceflion of the equinox more cxacffly than it had been done. 
 
 But the art of navigation Hill remained very imperfect, from the con- 
 llant ufe of the plane-chart, the grofs errors of which mult h'ive often 
 mifled the mariner, efpecially in voyages far diflant from the equator. Its 
 precepts were probably at firft only fct down on the earlicft fea- charts, as 
 that cuftom is continued to this day ; and larger directions have been 
 \ifually premifed by the Dutch^ to colIe6iions of their charts called JVu' 
 goner's, from the name of the publifher : 1 he Dutch call thefe col- 
 lections alfo by many other afteded titles, fuch as Fiery-Columns, Sea- 
 Bcaccns, Mirrors, Atlajfcs, Sic. 
 
 At length, there v/erc publifhed in Spani/h two treatifes, containing a 
 fyftcm of the art, v/liich were in great vogue j the liril by Pedro de Aledina 
 at VaHadolid^ I 545? called Arte de Nai/e^ar ; the other at Seville, in 1556, 
 by Martin Cortes, with this title. Breve Compondio de la Sphera, y de la 
 Arte de Natiegar con nueuos Injirumcntos y Rcglas. The author of this lafl 
 tra(5l fays, he compofed it at Cadiv. in 154.5. 
 
 Thefe feem to have been the oklcft writers, who had fully handled 
 thisfubjectj for Medina, in his dedication to Philip Prince oi Spain, 
 
 '* La Kiiloria general y natural Je !.is Indlas par Gor.zalle? de Miedo, en 
 Scvilla, 1535. Aud Defcriptione de Jas IfiJias Occidentile, de Antonio de 
 lierrera, en Madrid, 1601. 
 
 f Hcicldtijt, in the dedication of his ^<:9i volnn-.e of Voyages, printed in 1599. 
 
 i In G;/.YW,i.vi's life v.'rinen in Spar.ifh, which is \ery fcarce, but it was 
 printed in Italian at Venice in 1571. 
 
 II See Livio Safiulo's Geographia, zt the fan-,e place in 1585. Dr. IFilUam 
 divert, de Maonite> Lordor., 1600; and Ptirchai'% Pilg'ivi, in 1625, vol. I. 
 
 % Cabot, ;i I'entlian by birth, firfc ferved our King Henry the feventh, then 
 the King of 5/.7/^7, and laftly, returning \.o England, he was conilituied grand 
 pilot by King Ed'-.vard \\iz lixth, with an annua! falary of above 160 pounds. 
 Of this famous navigator and his expeditions, many writers have made men- 
 tion, bo.h foreigners and Enghjh, as Peter Martyr, Ramufeo, H^nera, Hoiin- 
 JJ:fd. Lord Bacon, and'particulariy Hackhiyt and Purchas, in their Colkdions 
 of Voyages. 
 
 fi Opera Mathmatica at Nuremhrg, in quarto. 
 
 laments^
 
 OF NAVIGATION. iii 
 
 laments, that multitudes of ftiips daily perlfhed at fea, becaufe there were 
 neither teachers of the art, nor books by which it might be learnt; and 
 Cortes^ in his dedication, boafts to the Emperor, that he^was the firft who 
 had reduced navigation into a compendium^ enlarging much on what he 
 had performed *. 
 
 Mei'ina gave ridiculous dire<lions, how to guefs at the phce of the 
 horizon, v/hen it could not be feen ; as alfo he defended the errors of the 
 plane-chart, and advanced againft the variation of the magnetic needle 
 luch abfurd arguments, as Arijhtle and his followers had done to prove 
 the impoffibility of the Earth's motion. But Ci?r/^j briefly and clearly 
 made out the errors of the plane- chart, and feemed to refleft on what had 
 been faid againft the variation of the compafs, when he advifed the ma- 
 riner rather to be guided by experience, than to m.ind fubtle reafonings. 
 Bcfides he endeavoured to account for this variation, in imagining the 
 needle to be influenced by a magnetic pole (which he called the point 
 attractive) different from that of the world, and this notion has been far- 
 ther profecuted by others. 
 
 However, Medina % book, being perhaps the firfi: of its kind, was foon 
 tranflated into Italian^ French.^ and Flcmifn f, ferving for a long time as 
 a guide to navigators of foreign countries. 
 
 But Cortes was our favourite author, a tranflation of whofe work by 
 Mr. Richard Eden was, on the recommendation of that gre?.t navigator 
 A4r. Stephen Borrough^ and the encouragement of the Society for making 
 difcoveries at fea, publifhed at London in 1561 : v/hich underwent va- 
 rious impreffions J, whild that of Medina^ though tranflated within 
 twenty years after the other, feems to have been negleled, notwith- 
 flanding the encomiums beftowed on it bv Mr. "John Fra?npton^ the 
 tranflator. 
 
 A fyilem of navigation at that time confirted of fome fuch particu'ars as 
 thefe: An account of the P/s/Vww/V hypothecs, and the circles of the fphere; 
 of the roundnefs of the Earth, its longitudes, latitudes, climates, and 
 eclipfes of the luminaries ; a kalendar ; how to find the prime, cpadl, 
 ^c. and by the iaft the Moon's age, and thence the tides ; a defcription 
 ot die fea-compafs, not forgetting the loadftone, with fomething about 
 the variation, called its north-cafting and north-wefting, fir the difco- 
 vering of which, by night as well as by day, Cortes faid, an inlirument 
 might be eafily contrived j tables of the Sun's declination for four years ,- 
 
 * The learned T) on Nicdo Antcnlo, in his Bihliciheca HiJ^aiJca, printed at 
 Rome in 1672, torn. i. p. 323. puts down a bo>.k, intitlcd, 1 radado de la 
 ^<phera y ud mar ear on el 1 egiiaento de las alluras, written by Francijco Falero, 
 a Porti-geje, and printed at Seville in 1535 ; but pt-rhaps there ii a millake 
 in the (iate. Lie a!f.) mentions an eWtionoi' Cortes in ijp- 
 
 t \\\ii Italian and French tranflations were printt'd in 1554, the firl nt 
 i'ui'ce, the other at Ljo>:s ; the Flemijh edition, I have ken, vva^ at Jntiijsrp 
 in 1580; perhaps it had been printed before. 
 
 \ In the latter editions fome miftakes in the tranflation are corredcd. 
 
 Cortes fets down the places of the Surv for a twelvemonth, with an eqm- 
 t!On-table to corrcft thofs places, (ervivp^ for many year? to come ; a. id alfo 
 A:\oi'u(;i tabic to find the Sur-'s declinr.tion fro;n hi-s longitude being given. 
 
 a 2 ii^
 
 W DISSERTATION oh/ the RISE, &c. 
 
 in onler to find the latitude, from his meridian altitude ; to do the fcime 
 thing by thofe tailed the guard-ftars in the north, and the croliers in the 
 fouth ; of the courfe of the Sun and Moon ; the length of the days ; of 
 time and its divifions ; to find the hour of the day, and by the nocturnal 
 that of the night ; and laflly, a dcfcription of the fea-chart, on which to 
 difcover where the fhip is, they made ufe of a fmall table, that {hewed, 
 upon an alteration of one degree in the latitude, how many leagues were 
 run on each rhumb, together with the departure from the meridian. Be- 
 lidcs fome inftrumcnts were dcfcribed, efpeciaily by Cortes; as one to 
 find the place and declination of the fun, with the days and place of the 
 JVIoon ; certain dials, the aftrolabe and crofs-flalF, with a complex ma- 
 chine to difcover the hour and latitude at once. 
 
 And after this manner the art continued to be treated, though from 
 time to time improvements were made by the following authors. 
 
 As TVenier had propofed to find the longitude by obfervations on the 
 Moon ; (o Geir.ma Frifius^ in a tract: intitled De Principiis Ajh'onomia et 
 Cof?nographi^, printed at ^H/wrr/) in 1530, advifed the keeping of the 
 time by means of fmall clocks or watches for the fame purpofe, then, as 
 he fays, lately invented. He alfo contrived a new fort of crofs-ftaff, which 
 he defcribes in his treatifc jD^ Radio Jjironomico et Geometrico, printed at 
 the fame place 1545, and in his additions to Peier Jpicn'^ Cofmography, 
 gives the figure of an inftrument, he calls a Nautical Sluadrant^ as very 
 iifeful in navigation, promifing to write largely on the lubje6t j accord- 
 ingly, in an edition he made 9i553, [of '^'^ above-mentioned book 
 Dc Principiis Ajlronomice^ Sec. he delivers fevcral nautical axioms^ as he 
 calls them, which with fume alterations were repeated by his fon Cornelius 
 Gemrria^ in a .poflhumnus piece of his father on the Univerfal Ajirolabe., 
 publifhed in 1556. Gcunna FrifiUs died in 1555, aged 45 years. 
 
 With us Dr. IFilliam Cunningham^'ui his C'^Jinographical Glafs, printed in 
 1559, amongft other thmi^s briefly treats of navigation, efpeciaily fbew- 
 ing the ufe of the Nauticnl ^ladrar.t^ much praifing that inftrument. 
 
 But a greater senius than thefe undertook this fubje^l; for the famous 
 mathematician Pedro Nuvcx, or Nonius^ having fo early as 1=^37 pub- 
 liflied a book, written in the Poriuguefe language, to explain a difficulty 
 in navigation propofed to him -by the commander Don Martin Jlphonfo 
 de Sufa; which was thirty years after printed ;it Bafil., in Latin, with the 
 .iddition of a fccond book, the v.'hc;lc intitled ek Arte et Ratione Navi- 
 gandi; where \w. cxpofes, both truly and Icariicdlv, tiie errors of the plane- 
 charti and befides gives the folution oT ijvcral curious Aflronomical Pro- 
 blems, amongil v.'hich is that of determining tb;; latitude from two obfer- 
 f<:-r\'ations ot' the; Sun's altitude and the in.tcrmcdiate azimuth being given. 
 lie alio deli\'c;-s jiuny ufeful advices about the art of navigation, parti- 
 cularly how to pi'rform its operatioiis on the globe. He obfervcd, that 
 though the tiblique rhum.bs are fpiral li:K-3, yet theoircirt courfe of a fhip 
 wlil alv/ays be the arch of fome great circle, v.'hereby the angle v/ith the 
 meridians wi^l continually change ; al! that the fleerfman can here do 
 i'nr the preferviii.;; of the original rhumb ir. to correcSt thefe deviations, as 
 :. on as they appear fenfible. But thus the fnip will in reality defcribe a 
 cwiiric v.'ithout the rliumb-linc intended ; and therefore his calcuLniuns 
 h;; ;;;Tig!iing the latitude, wheje any rhumb-line crt'lles the fcvc.al me- 
 ridians,
 
 OF N A VI G AT I O N. v 
 
 ridians, will be m fome meafure erroneous. He alfo again fets dov;n his 
 method of divifion of a quadrant by concentric circles *, which he had 
 defcribed in his ingenious treatife de Crepufculis^ printed in 1542, ima- 
 gining it had been pra(Slifed by Ftohiny. There were alfo* added other 
 tracts of his, but the completelT: edition of his Latin works was made by 
 himfelf at Coimbra in 1573. His treatife oi Algebra^ written in Spamjh., 
 was printed zt Antwerp fix years before. 
 
 In 1577 Mr. William Bourne published his treatife f, intidcd, A Re^ 
 gimcnt for the Sea, which he defigned as a fupplement to Cortes, whom 
 he frequently quotes. Befides many things common with others. Bourne 
 gives a table of the places and declinations of thirty-two principal ftars, 
 in order to find the latitude and hour 3 as alfo a larger tide-table than 
 that publifhed by Mr. Leonard Digges, in 1556 |. He fliews, by con- 
 fidering the irregularities in the Moon's motion, the errors pf the failors 
 in finding her age by the epa61: ; and alfo in their determining the hour 
 from obferving upon what point of the compafs the Sun and Moon ap- 
 peared. He advifes in failing towards high latitudes to keep the rec- 
 koning by- the globe, as there the plane-chart errs moft. He defpairs of 
 our ever being able to find the longitude by any inllrument, unlefs the 
 variation of the compafs fhould be cauftd by feme fuch attratSlive point, 
 s Cortes had imagined. 'I'hough of this he doubts, and as he had fliewn 
 how to find the variation of the compafs at all times, he advifes to keep 
 an account of the obfervations, as ufeful to difcover thereby the place 
 of a fhip; which advice_the famous Simon Stevin profecuted at large in 
 a treatife pubiifhed at Leyden in 1599, intitled Portuum invejligandorum 
 Ratio Metaphrajfo Hugone Grotio ; the fubftance of which was the fame 
 year printed at London in EngUj]), by Mr. Edvjard Wright ^ intitled The 
 Haven- finding Art. 
 
 But the molt remarkable thing in this ancient tracl Is, the defcribing of 
 the way by which our fii'ors efiimatcd the rate a fhip made in lier courfe, 
 by an inftrument called the log. This was fo named from the piece of 
 wood, or Ic);-:, that floats in the water, while the time is reckoned during 
 which the line that is faflened to it is veering out. The author of this 
 device is not known, anil I find no farther mention of it till 1607, in aa 
 Lajl-Jndia voyage, publiflied by Purchas ; but from that time its name 
 occurs in other voyages, thiit are ainongft his coUeclions. And hencc- 
 torv/ard it became famous, being taken notice of, both by our own au- 
 t'w;-s, and by foreigners; as by Gu'ter in 1623, Snetlius in 1624, A4e- 
 tius in 1&31, Oughred \n 1633, Herigone in lb2-\-, Solton/lall 'ii\ i6'^6, 
 A'jrwQod in 1637, P^-'^'-i^i' i'l 1643; and indeed by almolt all the fuc- 
 ceeding writers on navigation, of every country. And it continues to 
 be Hill in ufe as at firir, tliough attempts have been often made to iin- 
 
 * The adiiiirabie (.liviuin, now fo much in ufe, is a very great impiovjnicu 
 of ihii ; fo thru when tlic famou?! Dr. E.l-niin<i Ha'ley, the Royal Allronoiner, 
 ic'ivcd that hy adaptl. g ic to hi. Muvnl Arch, fome body liere naiin-d it 
 A N-)n! cs, in v<hich he has been foMowed by many. 
 
 f He had ten years before puhlillioJ what he calls Rules of Xtzji^atisa, Sj 
 appears from !ii .J'manac, printed in '57i- 
 
 I In his treatile, intitled, A Prognojllcativi everlajiing, fol. 23. 
 
 " a 3 proT-
 
 vi DISSERTATION on the RISE, &c, 
 
 prove it, and other contrivances propofed to fupply its place. Many of 
 thefe have fuccecded in quiet waterj but proved uielefs in a troubled fca. 
 
 A following edition of this book was revifed by the author, Vhere, in 
 the preface, he fets forth the grofs ignorance of the old fliip-maftcrs, re- 
 peating fonie of the infipid jcrts they made ufe of to juRify their w%i\t of 
 knowlctlgc in their art. Amougfl: the additions, he enlarges on the ac- 
 count of tlie log-line. And at the end fubjoins an Hydrographical D'lf" 
 CQurJe- touching the jive fever al Pajfages into Cathay. 
 
 bourne publiilicd other trails, as one called Inventions or Devifes^ where 
 he defcribcs a mechod by wheel-work of nieafuring the velocity of a Ihip 
 at Sea, which artifice he attributes to one Mr. Humfrey Cole. 
 
 At Antwerp.^ in 15B1, Michael Coignet, a native of the place, publifiud 
 a fmall treatife, intitlcd, InJiruSiion noiivelle des Points plus excellcnts ^ 
 neceJJ'aires touchant V Art de Naviger*. This fervcd as a fupplemcnt to 
 JHedina^ whofe miflakes Coignct well expofcd. He there fhewed, that 
 as the rhumbs are fpirals, making cndlefs revolutions about the poles, nu- 
 merous errors muft arife from their being reprcfcntcd by flraight lines on 
 the fea-charts ; and exprefled his hopes of difcovering a rule to remedy 
 thofc errors ; faying, that moft of the fpcculations, delivered by the 
 great mathematician, Peter Nonius.^ for that purpofe, were fcarce pra6li- 
 cable"; and therefore in a manner ufelcfs to failors. In treating of the 
 bun's declination, he took notice of the gradual decreafe in the obliquity 
 of the ecliptic, a pointiong difputcd, but now fettled from the theory of 
 attralion. He alfo defcribed the crofs-flaft' with three tranfverfe pieces, 
 as it is at prcfent made, which he acknowledged to be then in common 
 ufe amongft the mariners ; but he preferred that of Gefnma Frifius. __ He 
 likewife gave fome inllruments of his own invention, which are now 
 quite laid afide, except perhaps his noilurnal. As the old fea-tablc, 
 mentioned above, erred more and more in advancing tovvards the poles ; 
 he fet down another to be ufed by fuch as failed beyond the 6otii degree 
 of latitude. At the end of the book is delivered a method of fiiiling on a 
 parallel of latitude by means of a ring dial, and a 24 hour glafs ; on v/hich 
 tiie author very much values himfclf. 
 
 The fame year Mr. Rolert JSlorman f publifhed a diicovcry, he had 
 Ion"- before made, of the dipping of the magnetic needle, in a fmall pam- 
 phlet called The Neive Attra^iue^ where he fhews how to determine its 
 quantity ; and in fpeaking of the loadftonc, he difputes againil: Cortes^ 
 notion, that the variation of the compafs was caufed by a point fixed Ui 
 tl)e heavens, contcjiding that it fhould be fought for in the earth, and 
 propofcs how to difcover its place. He alfo treats of tlie various forts of 
 compafics, fetcing forth at large the dangers that muft arife from the 
 then prevailing practice of not fixing, Oii account of the variation, t'le 
 wire directly under iht Jiotver-de-lucc ; as compaflcs niade in di^tcrcnt 
 
 * It had been publifhed in Flemijh, but the Fremh fdit'on i? th- fullcrt. 
 Co'gnet difd in 1623, Icavino mar.y mathematical manufcrij-ts, ice lahiH 
 Ar.dre^e Billiotheca Ltlgica, printed at Lowvai7i in 1643. 
 
 t He is commended for an excellent Artift by cur authors, zs Bou--7:e, 
 B irrou^h^ Sir Humjrfj Gillct, Hues, Pcttir, Blundewlle, Wngb!, and Dr. 
 Gilbert. ; 
 
 Q ' countries
 
 OF N A V I G AX I O N. 
 
 Vli 
 
 countries have It placed dIfFerently. Bourne indeed had warned againfl 
 this abufe, and there are many things common to both authors. 
 
 To Nnrman^i piece is always fubjoined, xVIr. JVilliajn Biirrough'' s Dif~ 
 courfe of the Variation of {he Cu/npafs or Magneticall Needle. The author 
 had been a famous navigator, having ufed the lea from fifteen years of 
 age, and for his merit promoted to be Controller of the navy by Qj^icen 
 Elizabeth*. He fhews how to determine the variation feveral ways, 
 fetting down many obfervations of it made by an azimuth-compafs of 
 Norman's invention, but improved by himfelf. He demonftrates the 
 falfhood of the rules commonly ufed, to tlnd the hititude by the guard- 
 ftars. He particularizes many errors in the then fea-charts, occalioned 
 by the neglecl of the variation ; adding. But of thcfe ccajles f towards the 
 north), and of the inivarde partes of the countries of Rullia, Alufcovia, &c. 
 / have made a perfect plat and defcription^ by myr.e oivue experience in furi- 
 drie voiages and travciles, hothe by fea and lande to and fro in thcfe partes, 
 which I gave to her MajcjVie in anno 157B. And laltlv, he juftly finds 
 fault with Coignsf=, inltrument, called a nautical hemifpbere \ but fpeaks 
 too fevcrely againft the writers of navigation, concluding thus. 
 
 But as I haue alreadie fiffcientlie declared, the cumpas Jlievueth not ahuaies 
 the pole of the tvorlde, but uarieth from the fame diver fy, and in fay ling de- 
 fcribcth circles accordyngly. JVhiche thing, if Pelvus Nonius, and the re/l 
 that haue ivritten of Navigation, had jointlie confidered in the traclation of 
 iheir rules and Injlruments, then might they haue been more auaileable to the 
 life of Navigation j but they perceivyng the difficultie cfthe thyng, and that 
 if they had dealt therev/tth, it would have utt.rly overX'chelmed their frmer 
 plaufble conceits, with Pedro de Medina (who, as it app.-'areth, hauyng 
 fome fmall fufficion of the matter, reafncth very clerkly, that it is not ne.ef- 
 fary that fiich an ahfurdity as the Variation, JJ)ould be admitted in fich an 
 excellent art as Navigation is) they haue all thought bejl to pafj'e it over with 
 flence. 
 
 The Spaniards too continued to publifli trcatifes of the art; particularly 
 at Seville was printed in 1585 an excellent Compendium by Roderico 'La- 
 7norano ; which is written clearly and with brevity, not being incumbered 
 with fuch idle fi)eculatjo;is as abound \n AIedi>ia and Cortes. Hie author 
 was Royal Leclurer at Seville, and contributed much to the reforming 
 the fca ch.rts ; as we arc told by his fucccflor, /hidres Garcia de Cefpedes^ 
 who alfo pubiiiiied a treatife of navigation -dt Af a arid, in 1606. 
 
 As globes may be very fcrvicca'ule for the mariner, Mr. Edward Mul~ 
 lineux fet forth in i 59?., at the charges of ivlr. IVilUam Sanderfon mer- 
 chant i-, a pair much larger than thofe the famous geographer (7i.7V7/-./ 
 Mercator publiflicd in 154.1. On the terrcCcrial one v.'cre dcfcribcd many 
 new difcovered countries, aud traced out tliC relpective voyages round 
 the world by Sir FraKcis Drake in 1577, and Mr. Thomas Can.:;])) \\\ 
 1 586, with tl'.e procrrels hir Adartin Frobijher had made towards the north 
 in 1^,76, to a place called t^.is Slraiis. . 
 
 * l/^cHjt'sVoysgcs, vol. i. p. 4 17, printed in 1599. 
 f ?.];. ca'i--/e'r/o,i was commended for his knowledge as well as gcnerofiif. 
 to ingenious ii;cn. 
 
 a 4 TiiclL-
 
 viii DISSERTATJON on the RISE, Sec, 
 
 Thefe globes were accompanied with a tra<Sl containing their ufes 
 written in Englijb ; but in 1594 Mr. Robert Hues publiflicd a more ela- 
 borate one in Latin ; wherein, amongft others, he folves by t)ic globe 
 the problem of determining the latitude from two heights of the Sun ob- 
 ferved with the intermediate time being given* ; and in the la(t part of 
 his book, he performs the ufual queftions in navigation, premifing a dif- 
 courfe on the rhumb-lines, where he attempts to refute what Gemma 
 Frifius had aflertcd, who fays, that they meet in the poles. At the con- 
 clufion he highly praifes a trcatife of Mr. Tho?nas Harioty hoping it would 
 be foon publiflied, in which that author had treated of this fubjc6t upon 
 geometrical principles, with great fagacity and judgment. But all tiie 
 manufcripts of that great mathematician were lolt, except his Artl^ Ano' 
 lyticte Praxis, which was publi(hed long after his death in 1631 ; wherein 
 is firft advanced that idea of algebraic equations, which has been ever 
 fnice followed. 
 
 Hues t was a perfon of letters, and befides had been far at fea. Amongfl 
 other curious particulars, he gives a good account of the attempts that 
 had been made at various times to meafure the Earth. In the Epijile to 
 Sir IVcAter Raivley he takes an occafion to enumerate the many difcovcries 
 of our mariners in very difFerer^t parts of the world. His book was re- 
 ceived with great applaufe, and has been indeed a pattern for fuch as 
 afterwards handled the fame fubject. It has been often printed abroad, 
 particularly in 1617 with the notes o^'John Ifaac Pontanus, who omitted 
 thtEplJIle and the mentioning of Hariot. However from this mutilated 
 edition it was tranflated into Englijh by one Mr. ^ohn Chilmead, and pub- 
 liflied in 1639. 
 
 Amongft our failors none were more famous than Captain yohn 
 Davis X-i who gave mime to the ftraits which he difcovcred ; and great 
 matters were expeled from his long experience and fkill. In 1594 he 
 publifhed a fmal! trcatife, intitled, The Seaman's Secrets. This is written 
 with brevity, though fomewhat pedantically, and was efteemed in its 
 time, an eighth edition being printed in 1657 ; fo that it fecms to have 
 fupplanted Cortes. Davis treats of plane failing, calling it horizontals 
 and fets down the form of keeping a reckoning at Sea. He jikcwife 
 fhews how to fail by the globe, and boails of what he intended to do ; 
 much commending great circle failing, without defcrihing it, as' alfo 
 
 This problem has been difcuiTed by Dr. Henry PcmhcrtoK, in the Philo- 
 fophical TranfaftioDS, vol. li. part. 2d. i;6o, p. 910, where he has alfo 
 given fome inifroveir.ents in trigonometry. 
 
 f There is an account of Hi-es and Harlot in /Anthony Wcoci\ Ath:n. Oxen. 
 vol. i. printed in 1721, as being both members of that Univerfit}'. 
 
 X Several of his voyages are in Hackluyt's zn'l P w chas' s co\\ei\on^. He 
 and Cap'.ain Alraham Ki ntial zte gTC3.t]y praifed by Sir Robert Dudley, in his 
 Jrcano (hi Ma.e, as keeping a perfed reckoning by the way of longitude :^nd 
 la.itude, where are given two of their 7<jffra//. This DudUy was a natural 
 fon of the threat Earl of Letcejler, and had commanded, in irc-), a fl et 
 a^^ainft the Spaniards ', but retiring to Florence, he a/Tumed the titles of I')a'se 
 cS Kcrihum' erland and Earl of i^Var^vick. His Arcano was printed at ihat 
 piace in two volumes in 1646 and 1647. 
 
 what
 
 OF N A V I G A T I O N. k 
 
 \vhat he calls paradoxal; that is, by a proje6tion on the plane of the 
 equator with fpiral rhumbs, faying, he will publifh a chart for that pur- 
 pofe. But above all, he extols the ufe of calculations in the cafes of na- 
 vigation, and promifes to handle that fubject. 
 
 At -the end of the book is given the figure of a ftafF of his contrivance, 
 to make a back obfervation. Of this the author is fo vain as to fay. Than 
 %vh'tch injlrument {in my opinion) the feajnan jhall not find e any fo good^ and, 
 in all cly mates of fo great certaintie^ the invention and demonjlration whereof^ 
 I tuav boldly challenge to appertain unto my felfe (as a portion of the talent 
 which God hath hejiowed upon me) I hope without ahufe cr offence to any. 
 
 This inftrument feems to have for fome time been in uie ; for Adrian 
 MetinSy in his treatife, intitled Jjlronomies Injlitutio, printed in 1605, gives 
 a figure of it from an original, in the pofl'eflion of M. Frederic Hautman, 
 governor of /hnboyna. But it foon yielded to one of a more comniodious 
 form, which is now commonly called Davis''s Quadrant ^ ; as if it was 
 alfo of his invention, and that perhaps only becauie a back obfervation is 
 made by both inftruments, fo the quadrant itfelf was at firil flyled a Staff 
 and Back- Staff. 
 
 The famous traveller Signor Pietro della Falle paHing, in 1623, frorji 
 Ormus to Surat aboard an EnglifJ? vcfld, where obferving this quadrant 
 much pracliftd by the feamen, as it was quite new to him, takes an oc- 
 cafion to fhew its ufe very diflinclly, and fays, ihey told him, that it had 
 been lately invented arid called David's, Staffs from its author. Alfo 
 Captain Charles Saltonjiall^ in his Navigation, defcribes it under the name 
 of a Back-Staff \ and in Captain Thomas fames's famous voyage for dif- 
 covcring a north-weft paflage, begun in 1631, amongft the many inftru- 
 ments, he carried along v/ith him, are meiitior.cd two of Mr. Davis's 
 Back-Jlaves, which were doubtlefs thefc quadrants. 
 
 Contemporary v/ith Davis was Mr. Richard Polter, v,'ho, it is faid, 
 had been a principal mafter aboard the Royal Navy. He wrote a very 
 fmall book intitled The Pathway to Perfctl Sailing, where, from an ob- 
 fervation he made in 15H6, he v/oukl infer 1", that difi'crent loadftones 
 commuiiicateJ difterent degrees of variation to the magnetic needle, and 
 therefore defpifes the publifliing obfcrvations of that kind, as necdicfs. 
 Plis book was not printed till 1644, nor did it defcrve to be publifhcd at 
 all, as it abounds with miftakes, and is written faritaftically, obfcurely and 
 arrogantly. 
 
 But all this while the plane-chart, notwithftanding its errors v/.tc fre- 
 quently complair;ed of, continued to be followed ; as its ufe is eafy, and 
 fcrvts tolerably well in fliort voyages, cfpecially near the equator. 
 
 It is called by the French ^uartier Anglms- 
 
 f" Da%\J Hiaf, che in lingua Iiigl.je njalc a liir legncdi David Vinggi ; Part 
 3. Litter I. a Rcma. This author not only praifcs tiie Captain Ni.balas JTood- 
 '"./, and other officers, but alio the common failor.--, for their care and (kill; 
 and fay?, tht Pcriugue/e lofe great number of fliipj for not being fo exact in 
 tiieir obf-Tvations as the Englijh. 
 
 I ro;ha;s he {hould have thence concluded the variation altered, as was 
 dilcovcied af;e:vvards. 
 
 However,
 
 DISSERTATIOK ON the RISE, &c. 
 
 However, a way to remedy thefe errors had, for fome time, been in- 
 'quired after. And Gtrrard Mercator fcems to be the fiift, who conceived 
 the means of efFeding this, in a manner convenient for fcamcn, bv con- 
 tinuing to reprcfent both the meridians and parallels of hititude by parallel 
 ftraight lines, as in the plane-chart, but gradually augmenting the dirtances 
 between the degrees of latitude in advancing from the equator towards 
 either pole, that the rhumbs alfo might be extended into ftraight lines, fo 
 that a ftraight line drawn between any two pjaccs, laid down in this chart 
 by their longitudes and latitudes, fliould make an angle with the meridi- 
 ans, exprefling the rhumb leading from one to the other. But though 
 Mercator^ in 1569, fct forth an univcrfal map thus coiiftrudled *, it does 
 not appear upon what principles he proceeded ; probably, by obfcrving iu 
 a globe furnifhcd with rhumbs, what meridians the rhumbs paflcd at each 
 degree of latitude. That he knew not the genuine principles, I fhall 
 make evident ; our countryman, Mr. Edward Wright^ was certainly the 
 firft who difcovcred them. 
 
 Wright infinuates, but without' fufficient grounds, that this enlarge- 
 ment of the intervals between the parallels had been fuggefted before by 
 Cortes t, and even by Ptolemy himfelf. 
 
 As to Cortesy he fpeaks of the number of the degrees of latitude, and 
 not the extent of them ; for his cxpreffion amounts to no more than this, 
 that the degrees of latitude are to be numbered from the equator, and 
 confequently both northwards and fouthwards from that line the numbers 
 affixed'to them muft continually increafe ; and from anyplace having la- 
 titude (fuppofe Cape St. Vincent in Spaiu^ which is his inftance) the de- 
 grees of latitude will be denoted by numbers increafing towards the pole, 
 and decreafing towards the equator. He had before exprcfsly dirctScd, 
 that they fliould be all equal by meafurement on a fcale of leagues adapted 
 to thfe map J. 
 
 The pafTage in Ptokiny^, referred to by Wright ^^ does indeed relate 
 to the proportion between the difl:ances of the parallels and meridians, 
 but contains no fhadow of Mercator % fcheme ; for initead of propofmg 
 any gradual enlargement of the diftances of the parallels in a general 
 chart J that pafTage relates only to particular maps, and is more diftinctly 
 explained in the flrft chapter of his laft book ; where he advifes explicitly 
 not to confine a f) lie m cf fuch maps to one nnd the fame fcalc, but to plan 
 them out by a different m'eafure, as occafion fliall require, with this only 
 caution ; th;it the degrees of longitude fliould ineach bear in fome mca- 
 fure that proportion to the degrees of latitude, which the magnitude of 
 the refpective parallels bear to a great circle of the fphere ; anS fubjoins, 
 that in particular maps, if this proportion be obferved in regard to the 
 
 * See liis life, written by his intimate friend, Gauherus Ghymmius, which 
 wOj prefixed to an enlarged edition of his Jilas, publilTied at Dujjhurg-, in 
 J 593' '^y ^-''rr^oldus his fon, a year after his father's death. Gerrard Mercator 
 was born in 15 1 2. 
 
 f See the 2d chapter of Wright's book. 
 
 X Fart 3d. cap. 2d. fol. 5S. II Gicgraph. lib. ii. cap. i. 
 
 % In an advertifement fet down on his univerfal map, at the end of h:s 
 fccond edition of his book; and in this niiilake he has been followed by 
 others. 
 
 middle
 
 OF N A V I G A T I O N. xl 
 
 tnldule parallel, the inconvenience will not be great, though the meri- 
 dians fhould be ftraight parallels to each other ; wherein his defign is 
 plainly no other, than that the maps fhould in fome fort reprefent the 
 figures of the countries they are drawn for. Mercator^ who drew maps 
 for Ptolemy's tables *, underftood him in no other fenfe, thinking it an 
 improvement not to regulate the meridians by one parallel, but by two ; 
 one diftant from the northern, and the other from, the fouthem extremity 
 of the map by a fourth part of ihe whole depth ; whereby in his maps, 
 though the meridians are ftraight lines, they are generally drawn inclining 
 to each other towards the pole. 
 
 But Alercator's univei-fal map, mentioned above, though the author de- 
 figned it for the benefit of failors, was fo far from being readily adopted, 
 thst fome of the moft (kilful amongft- them objedled to its ufefulnefs. 
 Thus Mr. Burrough fays of it -By augyncnt'ing his degrees of latitude 
 toivards the poles^ the fame is more ft te fr fuchs to heholde^ as fiudie in cof- 
 ynographie, by readyng authours upon the landc^ then to bee ifed in Navigation 
 at the fea. 
 
 And Mr. Thomas Blundcville, in Ins Bricfe Defcription of Univerfal 
 Alappes and Gardes, firft printed in 1589, gives an account of this map, 
 obferving that Barnardus Puteanus of Bruges had publifhed, in 1579, one 
 altogether like it. And though Blundeville is fo partictilar, as to fet 
 down numbers cxprefling the dillances between each parallel of latitude 
 in thofe maps, yet he feems to flight them, by faying, that no better rules 
 than ihofe given by Ptolemy can be devifed. But what is delivered by 
 this geographer about the conftruction of a general map, is a very indiffe- 
 rent performance, altogether unworthy the author of the Ahnagcfl, and 
 not in the leail correfponding with the f'gacity fhewn in two treatifcs on 
 the Planifphere and Andlemma, which the Arabians have handed dov/n to 
 us r.s Ptolc}ny's\. 
 
 jVIaginus alfo, at the end of his Gcographia Univerfa.^ (the former part 
 of which is a tranflation of Ptslemy's) firft printed at Fenice^ in 1596, 
 mentions tliis map oi' Afercator, and gives even a fketch of it j but feems 
 to have no diltindl conception of the author's defign. 
 
 That AIcr:ator\ map was not rightly dcfcrlbed, is manifcft from the 
 nwmbcrs given by Blundeville ; and tliat he was ignorant of a genuine 
 method of dividing the meridian, appears from a paffage in his life, wlierc 
 the writer fas AAcrcator often aflljred him, that this extending a fpherc 
 into a plane anfwered to the quadrature of the circle, as that nothing 
 fccmcd to be wanting but the demonflration. 
 
 Howcv tr, o'ar authors now began to entertain favourable thouirhts of it, 
 pcrhap-. fiOin the report that xMr. IVright was about to treat on that fub- 
 jcc^, Dr, Thomas Hood, to the firft editioii which he L^ave of Z*i/-; ;;,.'s 
 Regimirt in 1 592, added a Dialogue of his own, called The AL:riuei-''s 
 (j::ide, written only to (hew the ufc of the plane chart, wliere he acknow- 
 ledges and fets forth its errors, and highly praifes AIercato}-\, faying, lie 
 
 * Jr. nn ciition he made o( Piolftnyh Gsoe^raphy, in 15S4. 
 t Tht Te were publilhed by Ftd. Cummanainus, one at Vmhe, in 151?, the 
 oihcr at R<.nic, in 1562. 
 
 had
 
 xii DISSERTATION on the RISE, &c. 
 
 had compofed a treatlfe concerning it ; but the indiftin*^ account he gives 
 of jt, {hews it would not be this author's lot to render it fit for the life 
 of navigation. And Mr. Bluudtrjillc, in the following editioiis of his 
 above-mentioned traft, omitted the commendation he had given of P/c- 
 Jemy^s method of delineating an univerfal map. 
 
 Mercator'^ fcheme was not indeed conlriN'cd for rcprcfcnting the parts 
 of a country in a juft proportion to each other ; but is appropriated to the 
 ufe of mariners, who fail upon rhumbs by the guidance of the compafs ; 
 which our countryman, Mr. Edward Wright-^ perfeded *, by diicovering 
 a true way of dividing the meridian. An account of this he font frora 
 Cains college, in Cambridge^ where he was then a fellow, to his friend the 
 above-mentioned Mr. Blundcvi 'le^cont2.\mng a fhort table for that pur- 
 pofe, \vith a fpecimen of a chart fo divided, together with the manner of 
 dividing it. All which Blundeville publifhed, in 1594, amongft his Ex- 
 ercifest in that part-which treats of navigation f j where he has well de- 
 livered what had been before written on that art ; infomuch that his book 
 was long in great repute, a feventh edition having been printed in 1636. 
 To the fecond edition, j^nno 1606, and following ones, was added his 
 former difcourfe of univerfal maps. 
 
 In 1597, ^^^ Reverend Mr. fFlUlam Earkwe^ in his Navigator's 
 Supp/y^ g .ve a dcmonftration of this divifion as communicated by a 
 friend ; faying. This manner cf carde has hem puhliquely extant in print 
 ihefe thirtie yeares at Icajl J, hut a cloude (as it were) aud tbicke m'ljie of 
 ignorance doth keepe it hitherto concealed : And fo much the more^ hecaufe 
 fome who were reckoned for men of good knowledge ^ have hy glauncing 
 fpeeches [hut never by any one reafon of moment) gone about ivhat they could, 
 to difgrace it. 
 
 This book o^ Bar hive's contains defcriptions of fevcral inftruments for 
 the ufe of navigation, the principal of which is an azimuth compafs, with 
 two upright fights ; and as the author was very curious in making ex- 
 periments on the loadftone, he difcourfes well and largely on the fea- 
 compafs ; and Itill farther handles that fubjecl in a tract he publiftieU 
 fome vears after, intitled, JMugnetical jfdve. tifenients. 
 
 At length, in 1599, Mr. JFright\\\mk\i printed his famous treatifi-, 
 intitled, The CorreSfion cf certain Errors in Navigation, which had been 
 written many years before ; where he fhews the reafon of this divifion ;[, 
 
 * Some of our modern writers have faid, Mercatcr took the hint from 
 Wright, but that is a miflake ; for Alercatcr's map was publifhcd thirty years 
 before If'r'cbth book, who frequ nt!y refers to it. See Eanvard Sixrburna 
 tranflarinn of the firft book oi Man- 1: us, in 1675, P ^^* 
 
 -f CH.ip. 29. 
 
 J He ihould have faid 28 or.!y. 
 
 \ Minv n[ thL-fe inftruments are in the Arcano ael Mare, together with the 
 deinon.iratict ;.bove mentioned. 
 
 II Map- nil their meridians thus divided had been nubiiOied at Amfieraem 
 by joacc: s Hcnd: f, who, when in Lcnd(,n, wo:kii,g as an engraver, learnt the 
 jn?nner 01 uoing i" from M'. U'righ:''b Manufrtt ; the four:h chapter of which 
 he had tranfcnbed into one of his maps. Hokmus afterwards in his letters, 
 both 10 Mr. Briggs, and alfo to Mr. IVrighi, begged pardon for not having 
 
 acknow-
 
 OF NAVIGATION. xili 
 
 the manner of conftru^iing his table, and its ufes in navigation, with 
 other improvements : A book, aS Dr. Hallcy f.iys, well deferving the pc 
 rnfal cf all fuch as dcjlgn to ufe the fea*. 
 
 In the preface, Jy right complains of th^obftinacy of our mariners, 
 for>not liking an improvement in their art, faying, that they were like 
 thofe v/hofe ignorance Mafter Bourne had expofcd, repeating Bourne's 
 very words. 
 
 *' Though this great improvement in navigation by Wright has been 
 em-braced and followed by all proper judges ; yet fome undifcerning 
 perfons have of late, even amongft us, found fault with it, particularly 
 Henry JFilfon^ author of a Treatife on Navigation, by a propofal for a 
 curvilinear fea-chart^ in 1 720 ; 2nd the Rev. Mr. Wejl^ of Exeter^ in a 
 poflhumous piece, printed in 1762. But their cavih were fufficiently 
 obviated ; thofe of che firll by Mr. Hafeldcn^ in his Me,rcators Chart, 
 and in his Reply, both printed in 1722 ; and of the fecond, by Mr. 
 JVllliam Mowttaine^ in the Philofophical Tranfa6lions, vol. LIII. p. 69. 
 Anno 1763." V 
 
 In 1610 a fecond edition of Mr. Wright'' ?, book was publifhed, and 
 dedicated to Prince Henr-y, his royal pupil f, where the author inferred 
 farther improvements ; particularly, he propofed an excellent way of de- 
 termining the magnitude of the Eartli ; at the fame time recommending, 
 very judicioufly, the making our common meafures in fome fettled pro- 
 poi tion to that of a degree on its furface, that they might not depend on 
 tiic uncertain length of a barley-corn. 
 
 Some of his other improveme.'i.ts were ; The Table of Latitudes for 
 dividing the meridian, computed to m.inutes ; whereas before it was but 
 to every tenth minute, and the fliort table fent by him to Blundrollle to 
 degrees only : An inftrument, he calls the Sea-rings, by which the varia- 
 tion of the compaf-, alrituJe of the Sun, and time of the day, may be de- 
 termined rcndily at once in ar.y place, provided the latitude be known : 
 The correciing of the errors ariling trcm the excentricity of the eye in 
 obf.Tving by the crof^^-fiaff: A total amendment in the Tables of the 
 declinations and pL^.ces of the 'twn and ftars, iro:n his own obfcrvations, 
 ir,:K!e v.'itli a fix -foot quadrant, in the years 1 594, 95, 96, :md 97: A 
 f.a-qur.drant, to t;ikc altitudes by a forward or backv/ard obfervation, 
 and Iii-:cwirc v/irh a contrivance fi.>r the ready hnding the latitude by the 
 hf^'f^ht of the i'C:j-!lp.;-, when not upon the nicridian. And that his book 
 mi2;ht be the better underftood by bcr;inr.crs, in this edition is fubjoined a 
 tranflation of tiiC above-mentioned V^mncrano'i Corv.pcndiinr. ; he correift- 
 
 acl: )Oivlii!:.rcc! the obligriii-^r, S.c lyn-^o^'i preface, wiicre he complains cf 
 H'.iiai..s'i pi(,C-( d:rv ; ai.d farther relauj, hfivv his book, a copy c;f which 
 hiwi-.io hcc.-i prr! 'liud t.j tli-v I'rrl cf C^.:\lerlufJ, liad liked to jiave come rut 
 iMuU'i ih t,ri;ijc of a [anic:; rsv^.;to", whom, frcin fomc circuTiillances there 
 n-.tf;'inrfd, I iir.n;';iiL' to f avc bt-u Airaha-n Ki\d.iL 
 
 ' I'L'.Uj phica. l'iaiijuc:ic?is, for 1 696, IS z:g. 
 
 t In i^;j7 a V-' '^'ciition was ;r.-.!-iliIhed by TVir. 7-/cjJj ?-u'x-n, \v>,ctc <:' ? de- 
 dication i, m-advir.dly left our, nnd iU the c-.d is added by [i.c tn-.i or the 
 above-!!. ri,t.<-r,-:d Ilavfr. J.raincr A't, ?.% :ilfj ^' ri^[:\ ur.ivcii.il mnr, itr-.rovc4 
 bv the uiuxv.fic. rr.idc !i -cc i:ib time.
 
 xiv DISSERTATION on the RISE, &c, 
 
 ing fomc millakcs in the original, and adding a large tabic of the variation 
 of the compafs obicrved in very different parts of the vk^orld, to fhew it 
 is not occnlioncd by -.my mnj^netical pole. 
 
 This excellent perfon was allowed fifty pounds a year (no inccnH- 
 derable fum at that time) by the EaJI India Company, for reading a 
 leturc of navigation ; he alfo projected the conveying water to London^ 
 but was prevented from executing his fcheme by dcfigning men, which 
 is frequently the cafe. Whilil he led a ftudious and retired life, his repu- 
 tation was fo far known, that Qiicen Elizabeth granted, in 15S9, a dif- 
 tcnfation for his abfcnce from the univerftty, in order to accompany the 
 Earl of Cumberland in the expedition to the Azores \ as I am informed by 
 Sir Jamei Burroughs Matter of Cains College, vvhofe fine talle in archi- 
 tecture, part of the new buildings in Cambridge fhew, they rendering the 
 reft of thbfe buildings a difgrace to that famous feat of learning, which 
 has produced many great men, as, (to mention here only mathematicians) 
 jyright, Briggs, Oughtred^ Dr. Pcll^ Fojler^ Hcrrox, Bai?rbridge, Bifliop 
 TVardy Dr. JVaUisy Dr. Barrow, Rookcy Sir Ifaac Newton, Cotes, and 
 Dr. Brook Taylor, 
 
 Wright\ improvements on Mcrcator's chart became foon known 
 abroad. 
 
 In 1608 were piiblifhed the Hypomncmata Mathematica of the above- 
 mentioned Siinon Stevin, compoled for the ufe of Prince Adaurice. In 
 the part concerning navigation, the author, having treated of failing on 
 a great circle, and fliewn how to draw mechanically the rhumbs on a 
 globe, fcts- down JFrighfs two tables of latitude and of rhumbs, in or- 
 der to di'fcribe thofe lines more accurately ; and in an appendix he com- 
 mends i/i-i, fjicws a miftakc committed by Nonius in relation to the 
 rhumbs, and pretends to h:\ve diicovcrcJ an error in ^P'right's latter table ; 
 but IFright himfelf, in the fecond edition of his book, has fully anfwercd 
 all Strvin's objeflions, demonf^rating that they arofe from his grois way 
 of calculating. 
 
 And in 1624 the learned Willehrsrdus Snellius, ProfefTor of the I'.Iathe- 
 matics at Leydcn, publifhed his Typhis Batavus*, a treatife of navigation 
 en JVrigijC?' plan, written fomcwhat obfcurely. In the introduction are 
 praifed Nonius, Meycaior, Steviv, Hues, and IFright. But fince what 
 had been performed by our arlifts on this fubjedt, is not there particu- 
 larly declared, as are the improvements made by the others ; it has hap- 
 pened that fome have attributed JVright's priiicipal difcovery to this au- 
 thor. Thus Albert Girard, wlio in 1634 publifhed a French tranflation 
 c,f Sieving Works with note?, in one of them obfervcs, that Sncllius had 
 calculated, what he calls. Tabula Canouica Paralleloriim, to minutes as 
 far as 70 degrees ^ wliereas JPYight had fet forth in 16 10 fuch a' table fo 
 calculated to 89 degrees 59 minutes; notwithflanding which M. de 
 Lag;^, in the Memoirs of the I-loval Academy of Sciences at Paris for 
 1 yo3, treating of the Ccrrc^ed Chart^ fays, c'ejl IVillebrord Sn-elUus qui 
 
 * In 1617 had been publiflT^d his Eratofthtnes Batavus, where is given an 
 
 account of his meafuiing the earth.
 
 , Of N A V I G A T I O N. xv 
 
 eft eft l^nventeur. But the French writers now acknowledge our coun* 
 tryman to have been its author*. 
 
 SnelUus was followed in Holland by Adrian Melius^ in a treatife, in- 
 titled, Pri?nu7n Mobile, printed zt Amjierdam, in 1631 ; and in Francehy 
 the learned Peter Herigone, in his Curfus Mathematicus, where, in the de* 
 di cation of the fourth tome to the Marfhal BaJJhmpire, the author fays, 
 Artem navigandi in cenfu Alathematices non repofuere pleriqtie nojlrum, ncque 
 fan} in hunc ordlKem afcribi meruit^ quandiu cased tantiim nautarum praxi 
 cckbrata ejl ; riur.c vera cum inventis tabidis loxodromicis (quas nos primiim 
 Gallis exhibemus) fortnam certain jirmafque leges acceperit fine injuria omitti 
 non potefl. But to return to our countrymen. 
 
 Mr, Wright, in the 12th chapter, having fhewn how to find the place 
 of a fhip on his chart, obferved, the fame might be performed more ac- 
 curately by calculation ; b:it confidering, as he fays, that the latitudes, 
 and efpcciaily the courfes at fea, could not be determined fo precifely, he 
 forbore fctting down particular examples ; as the mariner may be al- 
 lowed to fave himfclf this trouble, and onlv mark out upon his chart, 
 when truly conftructed, the (hip's way after the manner then ufually 
 praiStifed. - 
 
 However, in i6i4t5 ^'f'*. Raphe Handfon, among his nautical quef- 
 tions fubjoined to a tranflation of Pitifcus's Trigonometry, folved very 
 diftinctlv every cafe of navigation, by applying arithmetical calculations 
 to Wright's, table of latitudes, or of meridional parts, as it has fince been 
 caiicd. 
 
 And bcfidcs, though the method JFright difcovered for determining the 
 change of longitude by a i^a^) failing on a rhumb, is the adequate means 
 of performing it 5 Har.dfon propofed two v/ays of approximation for that 
 purpoie, without the aflillance of Upright's divifion of the meridian line. 
 The hril was computed by the arithmetical niejan between the co-fines 
 of both latitudes j the otiier by the fame mejn betwceji their fccants, as 
 an alterriativc, v/hen [y'ri^ht's book was not at hand, thouirh this latter is 
 wider from the itu'lIi than the hrH ; and farther he fhewed by the afore- 
 l.'.id calcLili'lio:;?, how much each of thcfe cc?r:pendlums deviates from the 
 truth, and :\\U^ ho.v erroneouflv the computations on the principles of the 
 pLin- chart dilcr from them all. 
 
 There h another method of approximation, by what Is called The 
 A'TiddJe La^icuJe J, v/hich, thoujh it errs more than that by the arithme- 
 tical mean iKtwcea the co-Hncb ; yet being Icfs operofe, is that generally 
 ufed b.y our fiilors ; notv.ithilanoing the arithmetical mean between the 
 ]u_,arir!::nic co-li;/--.--, ccjuivr.lcnt to the geometrical mean between the co- 
 l;;-._s t'!)cn.l"Jvf. ;., !;ad !-.-.;i fmce propofed by Air. fobn Baffat j|, which in 
 },ig!i L.titUv'cjb ;> ;"j;::::\v;i..t prcterahle. 
 
 c' cj} qu 0". c-',' ;//'<! /, / cartes ViduitiS, inz-cn.'icn admirahle, de la qi.cdc 
 
 0': 1^'.' rcd:vab'i h EuOa: id Wr.'ijht, quoiquon /' ait j'ouvir.t atiribu'u a. MciCuLOr. 
 \\\\. d,- 1 Acid. Iloynle des Sciences, .\n. 1753, p. 275. 
 
 I V. wns r'.-f>,i:,i'jd in 1635. 
 
 \ f/.v ,..'<-!"? wnr(:s, !,:!l prnitcd in r'.z-:,. 
 
 ij .\b;u- I' .0, i:i a il::ilc.gi!e wh.ii h wr.s p'.'blifl'.cd after the ni-tbnr'r, ifeath, 
 in ar: ^p^c:r..'.>. ;c lii: Puibi^ny to fi^J\J S.u.'r^. F'-J^i' ^ '--''n a icjchcr 
 
 Cjf
 
 itvi DISSERTATION on the RISE, &c. 
 
 The computation by the middle latitude, will always fall fliortof th? 
 true change of longitude ; that, by the geometrical mean, will always 
 exceed ; but that, by the arithmetical mean^ fall fbort in latitudes above 
 45 degrees, nnd exceed in lefler latitudes. However, none of theft' me- 
 thods, when the change in latitude is fufficicntly fmall, will deviate 
 grtatly from the real change in longitude. 
 
 About this time logarithms* began to be introduced into the practice 
 of the mathematics ; and as they are of excellent ufc in the art of navi- 
 gation, we fhall here fay fomcthiug about their original. 
 
 Thefe were invented by 'John Napier, Baron of Marchijloun in Scot- 
 land, as appears from his treatife, intitled, Mirific'i Logarhhmorian Canonis 
 Defcript'id, firfl: printed in 16141. Soon after, the author communicated 
 to Mr. Henry Brings, Profeflor of geometry at Grejhatn college in Lon- 
 don Xt another form of logarithms ; with which Mr. Briggs was fo well 
 pleafcd, that he immediately fct about computing a very large table of 
 them, which he pubJifhcd in 1624, with his Arithynetica Logarilhmica )|. 
 But in the mean time, as a fpecimen, he printed in 1617 a fev/ copies for 
 his own ufe and that of his friends, of a very fmall one, not exceeding a 
 thoufand natural numbers. 
 
 From this table Mr. Edmund Gnntcr, Mr. Briggs""?, colleague in Aftro- 
 jiomy, computed one of artificial fines and tangents to every minute of 
 the quadrant, which he publiflied in '1620, being the firfl of its kind . 
 And when he made an edition of his works three years after, both thefe 
 tables v/crc fubjoined to his book. 
 
 of naviention at Chatham, and well made out what he undertook, that a (hip 
 would return to the place it departed from, by failing on the fame rhumb, 
 contrary to what Fuller and others had maintained. At the end of tliis 
 difcourfe, he ?.ppUes his compendium to the three principal problsyns in 
 
 failirg- 
 
 * Th foundation of logarithms is a pro-erty of two feries of numbers, 
 rn; in ^rirhr.ictica!, the other in geometrical proportion; which property ia 
 dec'ared by Archimedei ia his Arcnarius. 
 
 -t In 16:9 was made, after the author's death, a fecond edition, with his 
 farther improvements in Spherical l>i,'^onometry. 
 
 J He was in 1 619 appointed by Sir Henry Sa'ville, his profeiTor of geometry 
 
 at Oxford. 
 
 II Adrian Vla.q made an edition of this book at Tergou, in 1628, where the 
 table of logarithms was continued by him to one hundred thoufand number?, 
 thou-h the" logarithms themfelves are but to ten places, whereas in Briggs^s 
 book~they weie to fourteen. Seme conies of /'Vary's tables were purchafed 
 by our bookfellers, and published at LonJon, v/ith an EngUJh explanation pre- 
 ir.ifed, dated 1631. 
 
 f Vlacn alfo pubiilhed, at the {zms. place, in 1633, his Trigoncfnetria Artifi- 
 ciaiis, with tables of logarithmic fines and tangents to every tenth ftcond of 
 the quadrant. V'.arn'f, tables have a great reputation for their exafln.fs, as 
 Skirrcjin'z f.ifl editio'n in 1706, and Gardiner 5 in 1742, have amongil us. 
 M ce FcnteneUc, in the Hifiory of the Academy of Sciences for 17 17, com- 
 Ti ends an cclition of Via.q'i fmallcr table?, made at Lyons, in 1670, as does 
 M. de la Lande, in his Allionomv, printed at ?aiii, in 1761, tables publifhed 
 lir.re in ijLo. 
 
 There
 
 OF NAVIGATION. xvii 
 
 There he applied to navigation, according to Wright's table of meri- 
 dional partSj as well as to other branches of the mathematics, his ad- 
 mirable Ruler *, on which were infcribed the logarithmic lines for num- 
 bers and for fines and tangents of arcfies. He a!fo greatly improved the 
 jed^or f for the fame purpofes. And he fliewed how to take a back ob- 
 fervation by the crofs-ftafF, whereby the error, arifmg from the excen- 
 tricity of the eye, is avoided ; defcribing likewife an inffriiment of his 
 invention, named by him a Crofs-Bow, for taking altitudes of the Sun or 
 ftars, with fome contrivances for the more ready colledling the latitude 
 from the obfervation X. 
 
 The difcoveries relating to the logarithms were carried to France by 
 Mr. Edmund IVingate, who, going to Paris in 1624, publifhed in that' 
 city two fmall tradls in French |!, and dedicated them both to Gafion^ 
 the King's only brother. In the firft he teaches the ufe of Gunter's 
 ruler, and in the other, of the tables of logarithms and artificial fines and 
 tangents, as modelled according to Napier's laft form, attributed by IVm- 
 gate to Briggs^ which is a millake ; as appears from the dedication of 
 Napier's Rabdologia, printed in i6i6, and from what Mr. Briggs himfelf 
 faid in the preface of his Arithmetica Logarithmica. 
 
 The Reverend Mr. JVilliam Oughtred projected this ruler into a cir- 
 cular arch, fhewing fully its ufes in a treatife firft printed in 1633, 
 intitled. The Circles of Proportion ; where, in an appendix, are well 
 handled feveral important points in navigation. It has been made in 
 the form of a Sliding Ruler. See Scth Partridge's ufe of the double fcale 
 in 1662. 
 
 As by the logarithmic tables all trigonometrical calculations are great- 
 ly facilitated; fo the firft author, who, I find, has applied them to the 
 cafes of failing, was Mr. Thomas Addifon^ in his treatife, intitled. Arith- 
 metical Navigation^ printed in 1625. He alio gives two traverfe tables 
 with their ufes, the one to quarter points of the compafs, the other to 
 degrees. 
 
 Mr. Henry Gellibrand^ Mr. Gunfrrs fuccefibr at Grejham College, 
 publifticd his difcovery of the changes in the variation of the compafs in 
 a fmall qaarto pamphlet, intitled, A Difcourfe Mathematical on the Varia- 
 tion of the Magnetical Needle^ printed in 1035. This extraordinary //?<?- 
 nomenon he found out by comparing the obfcTvati;)ns made at different 
 times near the fame place by Mr. Burroughs Mr. Gunter^ and himfelf. 
 
 This Ruler is fo condantly i 1 the practice of our artifi.s that it has got 
 the name of The Gunter. 
 
 t The ufes of a Sedlor had been Ihewn by Yir. Robert Hoed, in a ti'ift he 
 publifhed in 1598. 
 
 X 1 his ingenious perfon died 1626, aged 45 years. His works have 
 been feveral times reprinted with fuccefiive additions; the fLCotul edition was 
 made in 1656 from hisown maiiufcript ; then from thofeof Mr. Samuel Fo'jh'- 
 ProteiTor of Altronomy at Grpfnam Coliegc, ac;ain by Mr. H-nry Bine, and 
 Mr. William Lobcum. 1", e Jullclt and !a"ll, being; the fifth, was in 167^. 
 
 11 Thefe were aftcrwajd* printed at London in EngVjh with itnprovcmentJ. 
 
 Vol. I. b all
 
 xviii DISSERTATION on the RISE, Sec, 
 
 all perfons of great (kill and experience in thefe matters. And this dlf- 
 covcry was foon known abroad *, for father Athanafiui Kircher, in his 
 treatile intitled Alagnes^ firft printed at Rome in 1641, fays our country- 
 man Mr. John Greaves had inforwied him of it, and then gives a letter of 
 the iwvc\ou% Alarinus Mcrfennusy containing a very diftindl account thereof. 
 Gcllihrand had been famousi for the part he bore in the Trigonometria 
 JBritnnnka of hh dcccafed friend Mr. Briggs, which was printed in 1633, 
 at Ttrgoti^ under the care of Jdrian Vlacq. Gellibrand alfo, in 1635, 
 publifhed in Englijh an Injlltut'ton Trigonometr'icall. 
 
 In 1631 Mr. Richa/d Norzvaod had publiflied an excellent treatife of 
 *rrigc>:o/>ietry, adapted to the invention of logarithms, particularly in ap- 
 plying Napt^\ general canons f . The author having, as he fays, ac- 
 quired his knowledge in the mathematics at fea Xi efpecially Ihewed 
 the ufe of trigonometry in the three principal kinds of navigation. And 
 towards the farther improvement of that art, he undertook a laborious 
 work for examining the divifion of the log-line. 
 
 As altitudes of the Sun are taken on Ihip-board, by obferving his ele- 
 vation above the vifible horizon ; to collecl from thence the Sun's true 
 altitude with corrciStncfs, Ji^rtght obfcrves it to be necefTary.^ that the 
 dip of the horizon below the obferver's eye ihould be brought into the 
 account, which cannot lie calculated without knowing the magnitude 
 of the earth. Hence he was led to propoie diiF^rent methods for finding 
 this ; but complains, that the mofl: effecflual was out of his power to 
 execute ; and therefore contented himfclf with a rude attempt, in fome 
 ineufure fufficient for his purpofe : and the dimenfions of the Earth de- 
 duced by him correfponded fo well with the ufual divifions of th^- log- 
 line, that as he wrote not an exprefs treatife on navigation, but oidy for 
 the corre^Ttiiig fuch errors, as prevailed in general practice, the lag -line 
 did not fall tinder iiis notice. But Mr. Nottvood^ for regulating this in- 
 flrumcnt upon genuine principles, put in execution the method Mr. 
 Wright recommends, as the moll: perfe6l for meafuring the dimenlion of 
 the Karth, with the true length of the degrees of a great circle upon it ; 
 and, in 1635, actually meafured the diftance between London and York ; 
 from whence, and the fummer-folftitial altitudes of the Sun obferved on 
 the meridian at both place?, he found a degree on a great circle of the 
 Earth to contain 367196 EngUjh feet, equal to 57300 French fathoms or 
 toifes, wliich is very exad ; as appears from many meafures, that have 
 been made fince that time. 
 
 In the Hiftory of the E.oyal Academy of Sciences at Paris for 1712, p. 
 19. it is faid by M. de Fontenelle, that the learned Peter GaJJ'endl was the prin- 
 cipal dft-OiCrer of this property; but GaJJhidi himfelf acknowledged that he 
 had before received informiition of Ge'///'^r;?<i''s difcoveries. Gajfend, Oper. 
 vol. ii. p. 152, L.gd. 1658. 
 
 f A very advantageous report of it was made by M. Mariotte at a meeting, 
 in 166S, of the Academy. Du Hamel, Hilt. /^cad. t^cient. p. 51. 1701. 
 
 J troxEahuIor he became a teacher, Ilyling himklf before his books, A 
 Header of the Mathematics in Londcn. 
 
 . Of
 
 OF NAVIGATION. 
 
 xix 
 
 Of this affair Mr. Norwood gives a full and clear account in his trca- 
 tife, called 77;^ Seaman's Pra^ice, firft publifhed in 1637. There, with 
 iinafFeiSed modelly, he apologizes for the hardincfs of a private perfon's 
 undertaking fo difficult a tafk ; and very cautioufly points out the true 
 reafon, how fo great a mathematician as Snellius had failed in his at- 
 tempt. He alfo fliews various ufes of his difcovery, particularly for cor- 
 recting the grofs errors hitherto committed in the divifions of the log-line.' 
 But fuch neceffary amendments have been .little attended to by the failors, 
 whofe obftinacy in adhering to inveterate miftakes has been always 
 complained of by the beft writers on navigation. This improvement has 
 at length, however, made its way into pra6lice : few navigators of repu- 
 tation ufing now the old meafure of 42 feet to a knot. 
 
 Farther, Mr. Norwood likewife there defcribes his own excellent me- 
 thod of fetting down and perfeSl'ing a Sea-Reckoning, ufing a traverfe- 
 table, which method he had followed and taught for many years j and 
 befides, fliews how to reftify the courfe, by the variation of the compafs 
 being confidered ; as alfo how to difcover currents, a.nd to make proper 
 allowance on their account. 
 
 This treatife, and that oi Trigonometry, were continually reprinted, as 
 the principal books for learning fcientifically the art of Navigation. What 
 he had delivered, efpecially in the latter of them, concerning this fub* 
 jeiSl, was contracted as a manual for failors, in a very fmall piece, called 
 his Epito?ne, which ufeful performance has gone through numberlefs 
 editions. 
 
 No alterations were ever made in T7;<? Seaman'' s FraSlice, till in the 12th 
 edition, printed in 1676, after the author's deceafe, there began to be 
 infertcd, at page 59, the following paragraph in a fmaller character 
 \_Jbotit the year 1672, Monfieur Picart has publijhed an account in French, 
 concerning the meafure of the Earth, a hreviate whereof may be feen in the 
 Philofop'iical Tranfadtions, Nii2, zvherein he concludes one degree ta 
 contain 365184 Englifh feet, nearly agreeitig to Mr. J>lorwood'5 expert' 
 ment.] And this advertifemcnt is continued in the fubfequent editions, 
 as I find it in one-printed fo lately as 1732. 
 
 Norwood's meafure therefore, though it was not knov.'n to the great 
 Sir Ifaac Neiuton in his youth, v/as not buried in oblivion, on account of 
 the confufions occafioned by our civil wars, as M. de Voltaire has been 
 pleafed to fay * ; on the contrary, it has been conftantly commended by 
 our writers on navigation : as by Mr. Henry Bond, foon after its pub- 
 lication, in a note at page 107 of the Seaman's Kalendar., which ancient 
 hook he reprinted and improved, v-'hofe ufc, through numberlefs editions, 
 is continued amongft our failors to this day ; by Mr. Henry Phillips in 
 his Gcotnelrical Seaman in 1652, and in his Advancement of Navigation m 
 1657 ; by Mr. John Collins in his Navigation by the Plane Scale, in 1659 ; 
 by the reverend Dr. Jchn Newton in his AIathc?natical Elements, in 1660 ; 
 Mr. John Seller in his PradVical Navigation, in 1669 j Mr. 'John Brown 
 in his Triangular ^ladrant in 1671. 
 
 * Elemen: de la Philofophie dt Newton, chap, xviii. printed at Paris in 
 17 8. 
 
 b X And
 
 XX DISSERTATION on the RISE, &c. 
 
 And in the PhihfophkalTranfaSiiom for 1676, N 126, there is given 
 a very partcular account of it. Nor had it cfcaped the royal notice;, 
 for when King Jame^y in 1690, honoured the obfervatory zt Paris with a 
 vifit, he informed the gentlemen, then prefent, of this nicafure of the 
 Earth J and upon their acquainting his Majcfty how that had been de- 
 termined by Mr. Picardy the King wifhcd the two meafurcs might be 
 compared together *. 
 
 But that it was not commonly known in France is no wonder, feeing 
 our books were not then fo much inquired after as at prefent by that polite 
 ami Itarned people. 
 
 In the yournal des S^avatn for Dcccfnher 1666, it was obferved of Dr. 
 Hooke's micrograph/ay qu'il eft ccr'it en une langue que peu de perfonnes 
 entendent-y but long after, in the fame Jourtial for February 1750, it is 
 faid of the Englijh tongue, that it was unc langue que tous les vra'is favam 
 drvroient fa^mr. And now, as Norivood is taken notice of in the latter 
 editions of Sir Ifaac Newton s Principm^ his name and merit indeed arc 
 become univerfally known. Infomuch that a particular account of his - 
 meafure is given by M. de Maupcrtuisy in the preface to his Treatifeof 
 the figure of the Earth, printed at Paris in 1738; wherein he defcribes 
 his method of determining the length of a degree on the Earth in Lap- 
 land y and Norwood is mentioned by two karned Spanift) fea officers,^ D. 
 yorge Juauy and D. Antonio (VVlloay 1. their voyage printed at Madrid 
 in 1748, which was undertaken, as they were appointed to accompany 
 the French mathematicians, fent to meafure a degree near the equator. 
 
 About the year 1645 Mr. Bond publlftied in Norwood's Epitome a very 
 o-reat improvement in TVright's method, by a property in his meridian 
 line, whereby its divifions are more fcientifically afllgned, than the au- 
 thor himfelf was able to-e'fFcdt; which was from this theorem. That thefe 
 divifions are analogous to the cxccfles of the logarithm.ic tangents of half 
 the refpcclive latitudes augmented by 45 degrees above the logarithm of 
 the radius. 
 
 This he afterwards explained fomcwhat more fully in the third edition 
 of Cw^/^^'s works, printed in 1653, where, after obferving that the lo- 
 garithmic tangents from 45 upwards increafe in the fame manner (as 
 he exprefles it) that the fecants added together do, if every half degree 
 be accounted as one whole degree of Mercator's meridional line ; his 
 rule for computing the meridional parts appertaining to any two latitudes 
 (fuppofed on the fame fide of the equator) is laid down to this effect ; To 
 take the logarithmic tangent, rejecfling the radius, of half each latitude 
 augmented by 45 degrees, and dividing the difference of thofc numbers 
 by the logarithmic tangent of 45 30', the radius being likewife re- 
 jedted, and the quotient will be the meridional parts required, exprefled 
 in decrees. And this nde is the immediate coiifequcnce from the ge- 
 neral theorem. That the degrees of latitude bear to 1 degree (or 60 mi- 
 nutes, whicli in TVright^s table ftands as the meridional parts for I de- 
 gree) the fame proportion as the logarithmic tangent of half any latitude 
 augmented bv 45 degrees, and the radius negle(fted, to the like tangent 
 
 * Dh liumel, Hill. Academ. Regal. Scient. p. 285.. 
 
 10 of
 
 OF N A V I G A T I O N. xxi 
 
 of half a degree augmented by 45 degrees, with the radius likewife re- 
 jected. 
 
 But here was farther wanting the demonftration of' this general theo- 
 rem, which was at length fupplied by that great mathematician, Mr. 
 yames Gregory of Aberdeen^ in his Exercitatianes Geometr'ica^ printed at 
 London in 1668 ; and fmce more concifely demonftrated, together with a 
 fcientific determination of the divifor, by Dr. Halley^ in the Ph'ilofophical 
 TranfaSiions for 1695, N 219, from the confideration of the fpirals into 
 which the rhurribs are transformed in the ftereographic projection of the 
 fphere upon the plane of the equinoctial ; which the excellent Mr. Roger 
 Cotes has rendered ftill more fimple, in his Logcmetrio^ firft publifhcd in 
 the Philofiphical TronfaSiions for 1714, N 388. 
 
 It is moreover added in Gu?2ter''s book, that if -*^ of this divifor (which 
 does not ienlibly differ from the logarithmic tangent of 45 i'' 30'''' cur- 
 tailed of the radius) be ufed, the quotient will exhibit the meridional 
 parts exprcilcd in leagues : and this is the "divifor fet down in Norwood's 
 Epitome, 
 
 After the fame manner the meridional parts will be found in minutes, 
 if the like logarithmic tangent of <:.5 o' 30^^ diminiflied by the radius be 
 taken, that is, the number ufcd by'-^thers* being 12633, when the loga- 
 rithmic tables confitT: of eight plac'^ befides the index. 
 
 This Mr. Bond, who introduce ') ufeful a difcovery into the art, was 
 a teacher of the mathematics in London, and employed to take care of 
 and improve the impreflions of the current treatifes of navigation. In an 
 edition of the Seaman'' s Kalendar, p. 103, he declared, he had difcovered 
 the longitude, by having found out the true theory of the magnetic varia- 
 tion ; and to gain credit to his afl'ertion, he foretold, that at London in 
 1657 there would be no variation of the compafs, and from that time it 
 would gradually incrcafe the other way, which happened accordingly. 
 Again, in the Ph'ilofophical TranjaBions for 1668, N" 40, he publiflied a 
 table of thv? variations for 49 years to come. 
 
 T'his joyhil news to all failors acquired Mr. Bond Bl. great reputation; 
 iiifomuch that the treatife he had compofcd, called The Longitude foundy 
 was in 1676 pubiilhed by the fpecial command of King Charles the Se- 
 cond, and ufhercd into the world with the approbation of fcveral of the 
 iTioit L-nunent mathematicians ot that time f. 
 
 But it was foon oppofed, there being publifhed at London a book in 
 1678, called The L'jngitude not founds written by one Mr. Beckborrow. 
 And in'iC'jd as FiOiid\ hypothcfis did not in any \vife anfwer its author's 
 fan2;uJ!ie expecflatiuns, the famous Dr. HalUy again undertook tliis affair ; 
 and trom a multitude of obfervations he would conclude, that the may-- 
 
 See \Ax. Pcrhns\ Treatife of Nwvigation in Vol. f. of Sir yonas Mcore*% 
 i^i'M : yjlem of the Mathematicks, p. 20.'J, printed at London in 16S1. Perhiiu'% 
 bo' k wai j ubiilhed by itfclf the )ear following, under the title of the ifean.ans 
 H':.t'jr. 
 
 t In the Phi'.ofophical l'raiiJac1io?u for the fame year, N 130, it is faiJ, the 
 Lord Brcuriker'i i-anie was inferted by millake. 
 
 b 3 neti
 
 xxii DISSERTATIpN pn the RISE, &c, 
 
 t -.' . " . 
 
 netic needle was influenced by four poles. His fpecqlations on this fub- 
 jedt are delivered in the Philofcph'ical TranfaSiions iox 1683, N 148, and 
 for 1692, "N 195. But this vvonslerful phenomenon feems to have hitherto 
 eludefi all our refcarches. 
 
 However, that excellent pcrfon in 1760 publifhed a general map, on 
 which were delineated curve lines exprefling the paths, where the mag- 
 netic needle had the fame variation. This was received with univerfal 
 applaufe*, as it may lead to feme difcovery in fo abftrufe an affair, and at 
 prefent be ufcful on many occafions in determining the longitude. The 
 pofitions of thcfe curves will indeed tontinually fufFcr alterations ; but 
 then they flioyld be corre(Sled from time to time j as they h^ve been 
 Tor the year 1744, and 1756, by tw-o ingenious perfons, Mr. IViUlnm 
 Mouutnlne and Mr. James Dodfon^ Fellows of the Royal Seciety. The 
 latter died not long after he had been chofen, for his merit, mathematical 
 riiafter, at ChrijTs Hofpital, ill London. 
 
 Dr. Hallcy alfo gave, in the Philofophical TranfaSlions for 1690, 
 N 183, a difTertation on the mojifoons, containing many obferva- 
 tions very ufelul for all fuch as fail to places that are fubjetfl to thof^ 
 winds. 
 
 The true principles of navigation having been fettled by Wright^ 
 Konvoody and Bond, many authors amongil us trod in their fteps, mak- 
 ing feme little improvements. It would be impo/nble to enumerate each 
 particular. Of the writers already mentioned, Phillips znd CcllinSy in 
 the title pages of their books, .declare what they aimed at j Phillips alfo, 
 in his tract called the Advancement of NavirutlQUy recommends a pen- 
 dulum inftead of a half minute glafs, to cfttraate the time the log-line is 
 running out. He alfo propofes to do the fame thing by wheel- work. 
 Befides, in the Philofophical Tranfa5ilms for 1668, N^ 34, ^le delivers a 
 berter method to detcifmine the tides than v/hat v/as commonly praclifed j 
 for which purpofe Mr. 'John Flamjieed, the Royal Aftronomcr, ftill gave 
 more perfecl dircftions in the fame Tranfalions for 1683, N 143; as 
 Jikev/ife he hrft ordered a giafs lens to be fixed on the fhade vane, in 
 what is c^.lled Davis's quadrants, v/hich contrivance Dr. Robert Hooky 
 Profcfibr of Geometry at Grejham College, had before thought of J. 
 
 Seller's PraSJical Navigation, though without demon Hration*, has the 
 rules of failing in the difiercnt kinds, as performed by calculation, by the 
 plane fcale, by the Gunt'cr, and by the finical quadrant* with various other 
 matters relative to the iu't ; as alio the ufe of the azimuth-comp;vfs as 
 now modelled, the ring-dial, the fea-ring, crofs-ftaff, Davis's quadrant. 
 
 II, 1.1.1 i^v^v^.-, ..-, ^.c, ..-. - . . 
 
 + Ste tHc ;ibove-mriition:_d f\-'kins'b Navigation, page 250. 
 
 X Sec Biihcp I'homai ^;.-at'i e.^cclient Hiitory of the Royal Society in 
 ' b , page 246 ; a: d Huok '^ i^uilhumoub Works, pubiillicd by Richard t^ alicrf^ 
 iq. in 1705, p. 557. 
 
 plough.
 
 ' o F N A V I G A T I O N. xxiii 
 
 plough, iio<3:urnal, incHnatory needle and globe, together with all the ne- 
 ceflary tables ; the whole being delivered in a manner {o well adapted to 
 the general humour of mariners, that it has undergone numberlefs edi- 
 tions : the lafl, I have feen, was in 1739 j but fome hte writers feem to 
 have abated the run of this book. 
 
 As in failing ^fpecial regard ought to be had to thf ke-way a fhip 
 makes, fo many authors have touched upon this point ; but tlie allow- 
 ances ufually made on that account are very particularly fet dov/n by Mr. 
 yo'h'n Buckler^ and publifhed in a fmall trail firft printed in 1702, intitkd, 
 ji New Compendium of the whole Art of Navigation^ written by iMr. 
 IVilUam Jones. 
 
 We ought not here to pafs over in filence the very ufcful invention of 
 Dr. Gowin Knight, which is the making artificial magnets, that are ot 
 greater efficacy than the natural ones. Though the Doctor has not 
 thought fit to reveal his fecret ; yet others have found it out, who have 
 made it public, particularly the Rev. J'jhn Mitchcl, and Mr. fchn Can- 
 ton; the firfl- in a treatlfe of Artificial Adagnets, printed in 175O; the 
 other in the Phihfophical Tranfia^ions, vol. XLV^II. Ann. 175 i. 
 
 The Earth being now univerfally agreed to be not a perfect globe, but 
 a fpheroid, whofe diameter at tiie poles is fliorter than any other ; the 
 Rev. Dr. Patrick Murdcch publifhed a tract in 1741, where he accom- 
 modated JVright\ failing to fuch a figure ; and Mr. Colin Maclaurin, the 
 fame year, in the Philofiophical TranfaBions, N 461, gave a rule to de- 
 termine the meridional parts of a fpheroid, which fpcculation he farther 
 treats of in his book oi Fluxions, printed at Edinburgh, in 1742. 
 
 Though Sir Jfaac Newton in his Principia, firft printed in 1686, had 
 demonftrated from the theory of gravity, that this muft be the real form 
 of the Earth, as it revolved about an axis ; yet in the year 171b M. Caf~ 
 fini again * undertook from obfervations to fhew the contrary, and that 
 the earth v/as a fpheroid, having its longcft diameter paffing through its 
 poles t ; and in 1720 M. de Alairan advanced arguments, fuppofed to be 
 llrengthened by gcoinetrical demonftrations, to confirm farther M. Cafi- 
 fini's aficrtion. But in the Philofiophical Tranfia^ions for 1725, N 3S'6, 
 387, 3S8, Dr. Z)^/^/zV;-j- publifhed a diflertation, wh'.^rein he made ap- 
 pear the weakncfs of the reafoning, and the infufficiency of the obferva- 
 tions, as they were managed, to fettle fo nice an afFair. He there alfo 
 propofcd a proper method for adjufting this point, when he fays. If any 
 confiequence ofi this kind could be drawn firom aSiiial meafiuring, a degree ofi 
 latitude /1)ould be meafiured at the equator, and a degree ofi longitude likewifie 
 meafiured there ; and a degree very northerly, as fior example, a whole degree 
 might be aStually meafiured upon the Baltic y^^, xvhen firozen, in the latitude 
 
 * In the Memoirs of the Royal Academy of Sciences at Parii, his father 
 in 1701, and he in 171 3, auempted to prove the Eaith was an oblong iphc- 
 roid. 
 
 t M. John Bernouilli in his EJfai d" nnt NowvelU Pliyfique Ctlcfte, printed at 
 Parti in I7?5i triumphs owr S,r lfaa< Nei^l-n ; vainly imagiri'M^ tlicle pic- 
 carioua obfervations could invalidate what Sir Ijaac had dcraoi.lt accd. 
 
 r.f 
 
 b4
 
 xxlv DISSERTATION on the RISE, &c. 
 
 cfjixty degrees. There, according to M. CaJJlni's laji fuppofition, a de- 
 gree would be 56653 toife$ \ whereas at the equator it would be of 58019 
 ioifes, the difference being 1^66 toifes, about the two and fortieth part of a 
 degree y which mujl be fenfible \ and like wife the degree of longitude would 
 according to him be of 56b* 1 7 toifes, lefs by 1202, or the forty-eighth party 
 than a degree e/" latitude at the fame place. 
 
 On this admonition, in 1735, there were ftnt from France two (ets of 
 mathematicians, nien)bers of tlic Royal Academy of Sciences ; one to- 
 wards the pole, the other to the equator, in order to mcafurc, at each 
 p'cTce, the length of a degree on the meridian. The report they brought 
 bomcj quite overfet what had been urged in favour of the oblong figure; 
 a degree towards the nocth, in the latitude of 66"^ ^o\ being found to con- 
 tain about 57438 toifes, and near the equator but 56750. 
 
 This unwelcome news caufcd a degree to be again meafured in Francey 
 which at length came out to be coiifonant with thofe which had been 
 brought from very dif^ant parts of the world, Thus ihefe mathemati- 
 cians confirmed oy painful -obfervaticns, what Sir Ifaac Newton had, as 
 M. dc Maupertiiis ufed to fay, deterniii'.ed in his elbcjw-chair ; Sir If'^ac 
 jr.aking the length of a degree under the pole to be 57382, and at the 
 equator 56637 toifes. And perhaps no obfervations can be exact enough 
 to determine this matter more precifely. 
 
 But let us mention fome of the foreign writers on navigation. 
 At F.'?ne^ in 1607, came forth a treatife, intitled, Nautica Mediterra- 
 nean written in Italian by Bartolomeiv Crefcenti^ the Pope's engineer. 
 '1 he author mifTes no opportunity of expofing the errors o{ Medina \ but 
 fcarce gives any thing of his own, except a machine for meafiiring the 
 W'iV a fhip made. 
 
 As the Jefuits have treated of moft branches of learning, fo this art 
 has not been beneath their confideration ; the three following authors 
 having been of their fociety. 
 
 At Paris, in 1633, Father George Fournier^ publiflied an Hydrography ^ 
 principally relating to navigation. The author would perfuade us, that 
 one of Dieppe had corrected the plane ch,;rt; and that the Hollanders 
 learnt of the French the making charts fo corre<3:td ; whereas this had been 
 engraved long before at Amjterdam^ by lodocus Hondius^ and others. 
 
 John Baptiji Riccioli, in his Geographia & Hydrographia Rcfar?}iata^ 
 printed at Bologna in 1661, inferts a treatife of navigation, collei!:ling his 
 materials from alrnoft every writer, as he does in his Almagejl and Chro- 
 r.Qhf\\ which is indeed the chief merit of his works. 
 
 Father Millet F>echalles wrote on this fubjeft after a more mafterly 
 manner, both in his Curfus Alathemattcus, firft printed at Lyons in 1674, 
 and in a Frmch treatife, publiihcd in 1677, intitled U Art de Naviger de- 
 montre par Principes. 
 
 Thcfe three authors, befuics treating of the different kinds of failing, 
 abound in methods for taking of altitudes, finding the variation, and 
 eilimatin^^, the way a fi^ip makes, &c. They alfo dcfcribe a niaehine re- 
 fem ng that of Cnfccnti. Riccioli gives a very faulty mtafure of the 
 Ear h, made by himfclf ; and Dechalles advifes (he ufe of a pendulum in 
 reckoning by the log-line, as alfo of wheel-work for the fame purpofe, as 
 P.;".'' '-i and Cole had done. 
 
 But
 
 OF NAVIGATION. 
 
 XXV 
 
 But there were writers in France between Fournier and Dechalles. For 
 in i656, and the following years, there were printed at Dieppe feveral 
 trails handling different parts of navigation, couipofed by M. G. Denysy 
 which have been often reprinted. 
 
 And in 1671 the Sieur Blondel S. Juhin publifhed a book called, 
 VArt de Naviger par le ^lartier de Redu^ion^ defcribing an inftrument* 
 much in ufe amongft the French failors, by which may be performed, as 
 by the finical quadrant, the operations of navigation, though not much 
 more fpeedily than by the traverfe table, and not at all fo accurately. 
 He alfo publifhed in 1673 his Trefor de la Navigation, where the art is 
 well treated of, particularly by calculations. 
 
 M. Saverien^ in his Marine Diiionary^ printed at Paris in 1758, fays, 
 that M. DaJJier f<rems to have been the firft of the French writers that 
 Ihewed the ufe of Gunter's Icales [cchei'es Angloifes"] in his Pilote expert^ 
 printed in 1683. 
 
 At Paris, ten years after, was publifhed the firfl part of a pompous 
 work, intitled, Le Neptune Franfois, by order of the French king, con- 
 fi{ftng of fea-charts, according to lVright\ fcheme, made from the latefl 
 obiervations, and reviewed by Meff. Pene^ Cajfini, and others. As this 
 contained the charts of Europe only, there were added others of different 
 parts of the world, printed the fame year 2XAmJlerdam. The whole was 
 preceded by a difcourfc of M. Sauveur, v/ho had formed fome of the 
 chart-S v/hcre he fliews how to perform the problems of aflronomy and 
 navigation by fcales ; which difcourfe had been publifhed by itfelf at 
 Paris, in ib^i. 
 
 M. John Bouguer compofed, by authority, his Traite Complct de la Na^ 
 vigation, firil printed in 1698, which was well received, as containing 
 molt of the practices then known ; and Father Pezenas, Jefuit and Royal 
 Profeffor of Hydrography at Marfeilles, publifhed there, in 1733, a trail-, 
 called, Elemens de Pilotage ; and at Avignon, in 1 741, a larger work, in- 
 titled, Pra^tique du Pilotage. This author fhews how to find the meri- 
 dional parts by the Jrtijicial Tangents, an old difcovery amongfl us, de- 
 clared {o long ago as 1645, in Norivood's Epitome ; he alfo has been 
 ir.dullrious ni tranflating feveral of our mathematical books into 
 French. 
 
 But in 1753 AI. Peter Boiiguer, fon of the former, publiflied a very 
 elaborate treat.fc on this fubject, intitled, Nouvcau Traite de Navigation, 
 which is written fcnfibly, the author being an excellent mathematician, 
 and famous for other produclions. He there gives a variation-compafs f 
 of his own invention, and attempts to reform the log, as he had done in 
 the Memoirs of the Academy of Sciences for 1747. He is alfo very 
 
 * It is Duly a kind of fkeletrn oi Wright'' % univerfal map. 
 
 f M;iny of thefc forts of compaffcs have been propofed at different times, 
 as by M. Buache, in the Memoirs of the French Academy of Sciences for 
 175:, f .u:c 377 ; Captain Ciriftopher Middleton, in the Pbilofophicai Tranf- 
 actua-.s, N 450, Ann. 1738; and Dr. Kntghi, as improved by the ingenious 
 Mr. "John SmcutOJii ibid, N 495, Attn. 1 750. 
 
 particular
 
 fecYi DISSERtATION on tHE RISE, &:c. 
 
 paniculaf in determining th6 lunations more accurately thdn by. the 
 corf nnon methods, and in defcribing the corredlions of the dead rec- 
 kojiings. 
 
 The excellent aftronomer, M. sit Ja Caille^ in 1760, made an edition of 
 M. Bouguey*s book, which he fomewhat abridged and improved. 
 
 In 1766, came, out at Paris, a treatifc, with this title, Abrege du Pi- 
 lot age divifi en deuic parties, ou on traitc prituipalemait des j^mplitudes, des 
 Loxodromi s, dans I'hypothefe de la Sphere et de Spberoide, des mareesy des 
 variations de iaiman. 
 
 The former part of this book was firft publifhed in 1693. Here the 
 whole is improved by M. le Monnier. 
 
 Though the Spaniards were the earlieft writers on navigation, yet they 
 trcre very backward to adopt its improvements. Indeed Antonio de Naie^ 
 ra publilned at Lijbon, in 1628, a treatife, intitled, Navegacion efpecula^ 
 iiva y praSiica ; where, though the author recflifies the tables of the Sun 
 and fixed ftars, from Tycho Brah/s obfervations, he proceeds no farther 
 in the theory of navigation than what had been advanced by Noniusy as 
 followed by Cefpedes. But of late, in 17 12, was printed at the fame place, 
 Jrte de Nr<vcgar por Manuel Pimental; where is fhewn the ufe of Wright' % 
 chart, which, in imitation of the Frenchy the author calls Charta Redu- 
 Scida. He likewife defcribes Davis's quadrant, and mentions Norwood 
 and Picard'5 meafurcs of the Earth. In 1757 a treatife was printed at 
 Cadiz, intitled. Compendia de Navigacion para el ufo de los Cavellcros 
 Guardias Marinas, written by the ingenious gentleman mentioned above, 
 Don "Jorge "JiMn. This is a good performance, delivering very diftincftiy 
 the feveral parts of the art, as now improved. Some things are here 
 omitted, that ufually occur in books on this fubjei ; but for the know- 
 ledge of fuch particular?, references are made to tra6ts compofed exprcfly 
 for the ufc of the fociety of gentlemen, dcflined for the fea-fcrvice. 
 
 Bougiier and Jorge yuan, defcribe and commend the method of divid- 
 ing inftruments for taking of angles, publifhed by Peter Fe'rnier, in a 
 treatife, intitled. La Co^iJiruSiion, i^c. du quadrant nouveau, printed at 
 BruJJUs, in 1631. This divillon is an improvement of that of Curtiusy 
 as that of Fe>erius is of the divifion by diagonals *, and readily follows 
 from the firft Lctnma of C'tavius's treatifc on the Jjiroiahe f, as has been 
 obk-rved by Pezenas, in a book he publifhed at A-vignon in 1 765, in- 
 titled, Ajirono7nie des Matins. 
 
 As to their tre.^,ting of Wright's chart, I mentioned above SnelUus and 
 Metitis. To an edition, in 1665, of T/acg's fmall tables of logarithms, 
 &c. is addc'I, by Abraham de Gruel, one of meridional parts, whofe ufe 
 he fhc'.vs, 'witli other parts of navigation, in his Courfe of MathematicSy 
 written !!> Ditch, and prin'-ed at Amfierdarn in 1676, as had been done 
 by J'^^-'n P'irct, in his Flambeau rehiijjant ou Threfor de la Navigationy at 
 the icUiC place, in 1677. 
 
 * See Mr. Kohins'''^ Mathematical Trafts, where thefe divifions are largely 
 trtstfd of. 
 
 i i-i;il printed at Rome, in 1693. 
 
 6 The
 
 OF NAVIGATION. xxvii 
 
 The Dutch are great navigators, and have been famous for their At- 
 lajfcs^ before which, are prcmifed treatifes of navigation, as has been al- 
 ready obferved. The oldeft I have feen of thefe, was publifhed at Ley den 
 in 1584, intitled, Spiegel der Zee-Vaert (or Mirror of Nevlf^ation,) by 
 Lucas "janjz, IVaghenaer. In their later Atlajfes there is defcribed an in- 
 firument to be ufed after the manner of Davis'% quadrant, but where in- 
 ftead of circular arches are fubftituted ftraight lines. 
 
 Notwithftanding all the improvements hitherto mentioned, th^ feA' 
 reckonings^ though kept by fuch as were deemed very fkilful mariners, 
 are often found widely different from the truth. But this often happens 
 through negligence, as I have heard Dr. Halley^ who had ufed the 
 fea, fay. 
 
 Thefe errors would be avoided, if from time to time the latitude and 
 longitude could be determined. The firft is generally obtained by the 
 meridian altitude and declination of the Sun being given. The declina- 
 tion is got by the help of tables of the Sun, with an eafy trigonometrical 
 operation. 
 
 But even the latitude could not be very exacl, before the hmoxxs Kephr 
 had determined the true form of the Earth's orbit . Hence were fabri- 
 cated his Tabula Rudolphina. Next, thofe of Mr. Thomas Street Were 
 in great rcqueft f. But they, in their turn, yielded to Dr. Halley'Sy and 
 his again to thofe of the accurate and elaborate Mayer \ which, howeve-, 
 will want to be corrected hereafter : For, as Sir Ifaac Newton has fliewn^ 
 that all bodies mutually attral one another, the Earth will be difturbed 
 in its motion by theadtions of fome of the other planets. 
 
 To find the longitude is a much more difficult affair. For this end, at 
 prefent, the focieties of learned men in Europe offer from time to time re- 
 wards to fuch as {hall beft treat oi particular fubjetSs in mathematicks or 
 phyficks. Some of thefe have been relating to navigation, when Pollenly 
 BcrnouiUiy Bouguer^ and others have obtained the prizes.- And it is hoped, 
 this inllitution may contribute to the advancement of the art. 
 
 Eclipfes of the moon were ufed of old } and Kepler recommended thofe 
 of the Sun as preferable jj. 
 
 The fatellites of Ju[iiter were no fooner difcovered by the great Gal- 
 lileo , than the frequency of their eclipfes recommended them for this 
 purpofe ; and amonglt thofe who attempted this fubjeit, none were more 
 fuccefsful than Signor Dontinic CMjfini. 
 
 This great aftronomer in 1688 publiflicd at Bokgna tables for calcu- 
 lating the r.ppearanccs of their eclipfes, with dirciSliorkS for finding thence 
 the loiv^iiudcs of places ; and being invited to France by Lewis the Four- 
 teenth, he th-jre publiflied correcter tables in 1693. But the mutual at- 
 tradlions of the fatellites on one another rendering their motiojis excef- 
 fively irregular, the tables foon run out ; infoniuch that they require to 
 be rencv.cd from time to tin:c, which has been pcitormcd by ingenious 
 
 * In his treatife Je Motu Martis, in 1609, 
 
 I In iiis Aflrunomia Carolina, in 1661. 
 
 II Tabui-f Rudolph, printed at Ul.-n in I'^z"', rap. xvi. .^ x.\xa. 
 In J'jo Sj(/er( :} ^'i;,n-ijr^ j^i(\ jrintcd ai /'aiLc in I' i ;-
 
 xxviii DISS'fiRTATION on t^he RISE, &c. 
 
 perfons, as Dr. James'Pcufi^y "Dr. jfamei Bradley *, ^f. Cajfin'i the Ton, and 
 M. Peter Wargentin \ \ fo that now many of the common Almanaa fet 
 down, when thcfe eclipfes happen throughout the year. 
 
 The Rev. Nev'il Majkelyne^ D. D. our prefent Royal Aftronomer, has 
 publifhcd annually, fince the year 1767, by order of the Commiflioners 
 of Longitude, a work entitled. The Nautical Almanac and Ajlronomical 
 Ephemeris^ containing not only the ecljpfc* of the fatellites, but alfo 
 many other tables, to enable the mariner to determine the longitude at 
 fea J particularly tables of the diftances which the moon's center will have 
 from that of the Sun, and from fixed flars, at every three hours, under 
 the meridian of the Royal Obfervatory at Greemuichy and which have 
 fince beeen copied into the Connoifance des Temps for thefe latter years by 
 the editor of that work. 
 
 The large reWard granted by the Parliament for a pra6lical way of 
 difcovering the longitude at fea, has put many upon the fearch : info- 
 much that feveral idle and abfurd fchemes have been offered by ignorant 
 and wrong-headed men. But the perfed:ing the methods propofcd long 
 ago by yohn Werner and Gemma Frijius, feems at prefent to engage the 
 attention of the public. 
 
 The theory of the moon, thbflgh much] a'njended by the noble Tycho 
 Brahe and Mr. Jeremy Hgrrox J, was found to be infufficient to anfwer 
 this end. But the caufes of her various irregularities having been difco- 
 vered by Sir Ifaac Neiuton, and _her theory thence improved beyond ex- 
 pectation, gave great hopes of fuccefs j which have fince been happily 
 fulfilled by means of the improvements which have fince been made in 
 the methods of computing the feveral quantities of thefe inequalities by 
 M. Elder, and Tobias Mayer of Gottingen ; The former of thefe gen- 
 tlemen having been happy in reducing Sir Ifaac Newton's theory into neat 
 analytical expreflions, of which the latter availing himfelf, was, by a very 
 fmgular addrefs of his own, enabled to bring out the greateil: quantities 
 of the equations with eafe and exacftnefs, and thence to conftrucl tables 
 
 * He fucceeded Dr. HalUy at Green-iuich, where he made a great number 
 cf Aftronomical Obfervations, which, as they are moft accurate, it is hoped 
 will not be loft. He became famous on obferving and accounting for an 
 apparent motion in the fixed ftars, and called their aberration, which was 
 immediately exhibited by the great mathematician Dr. Brook Taylor accord- 
 ing to the exift theory of the Earth's motion. See Mr. Robins'^ Mathemati- 
 cal Traits, vol. II. page 276. 
 
 f IFargeniin'^ tables are much eftcetned ; they were firfl publifhed at Stock' 
 holm in the Ada Societatis Regit Scientiarum Up/alenjis for the year 174', but 
 fmce more corrcft from a new copy of the aathor's at Paris in 1759. by M. 
 de la Lande. The ingenious author has rendered them yet more correiSt, and 
 his labours on this head may be feen in the Connoifance des Temps for 1766, and 
 xhe Nautical A'manacs for 177 i, and 1779. 
 
 X This great genius died in 1641, fcarce 23 years old. See his Opera 
 Po/fhuma, publilhed by the famous Dr. John ^VaHis at London, in 1673. 
 //orrcx firll obferved the Tranfit of A'^wBj over the Sun in 1639. ^'^ wrote 
 an sccount of this Phenomenon^ which was publifhed by the great ailronomer 
 JlcTiliust at Dant^ic, in 1661. 
 
 Cff/f. i>cciet, i?"^. Cdttingerf tom. IF. page 283. 
 
 agreeing
 
 OF NAVIGATION. 
 
 XXIX 
 
 agreeing to the moon's motion in every part of her orbit, with very fur- 
 prifing exadtnefs. And this ingenious perfon has left behind him tables 
 ftill more exacl *, for which the Britijb Parliament have rewarded his 
 widow with . 3000, as alfo Mr. Euler with . 300. Thefe tables were 
 publifhed in 1770, by Dr. Majkelyne. 
 
 As to the method of Gemma Frijius^ M. Huygem was perfuaded it 
 might be accompliflied by his inventions of pendulum clocks and watches; 
 a defcription of the firft he publiftied in a fmall trad, printed at the 
 Hague^ in 1658 ; and of the fecond, as improved, in the "Journal des S^a~ 
 vans for the month oi February^ 1675. And great expectations of fuc- 
 cefs had been raifed from forne trials made in a voyage with thefe watches 
 of the firft conftrudtion, by Major Holmes ; an account whereof is given 
 in the Philofophical Tranfa6lions, Ayin. 1669. But the various accidents 
 thofc movements arc liable to, foon caufed that way to be laid afide. 
 
 Notwithftanding which, the ingenious Mr. John Harrifon has for 
 many years paft employed himfelf in contriving a machine, that fhall 
 be free from all imaginable inconveniencics ; and his endeavours were 
 fo well approved of by gentlemen of the greateft knowledge in thefe fub- 
 jecls, that the commiffioners for the longitude thought fit to allow him 
 fome gratifications for his pains. He was afterwards farther confidered, 
 upon difclofing the internal ftrudure of his machine, and the whole re- 
 ward' has fince been given him by Parliament. 
 
 The difficulty of making obfervations at fea with fufficient exacf^nefs 
 for finding the longitude, was feared to be infurmountable ; but at- 
 tempts have not been wanting to overcome It. In the Hiftory of the 
 Royal Society, at page 24.6, we meet with the firft mention of an inven- 
 tion in thefe words : J new injlrument for taking angles by refie5iion^ by 
 which means the eye at the fame time fees the tivo objeSls both as touching the 
 fa?ne pointy though difiant almofl to a fcmicircle ; which is of great ufe for 
 making exaSl obfervations at fea. A figure of this inftrument, drawn by 
 Dr. Hook^ the inventor, is given in the DoClor's pofthumous works, with 
 a defcription, at page 5(^3. But here, as one reflection only was made 
 ufe of, it would not anfwer the purpofe. However, this was at l..ft 
 effected by Sir Ifaac Ntiuton, who communicated to Dr. Halley, about 
 the year 1760, a paper of his ow" writing, containing a defcription of 
 an inftrument with two reflection^, which loon after the Doftor's death 
 was found among his papers by Mr. fones, who communicated it to 
 the Royal Society, and it was publifhed in the Philofophical Tranf- 
 acStions, N465, Jnn. 1742. 
 
 How it happened that Dr. Halley never mentioned this in his life- 
 time, is very extraordinary ; feeing fohn Hadlcy, Efq. f had defcribcd, 
 
 See his fJogium in the Neva Aila Eruditorum, for March 1762. 
 
 t Mr. liadley being well acquainted with Sir Jfaac Nevoton, mi^ht have 
 he.ird him fay, Hak's [ropofal could be psrfefted by means of a double re- 
 fle:lion. However, Mr. Had!ey, being a very ingenious perfon, m'g' t have 
 hit on the fan; thc::j;!u; as well ai Mr. G^dfrty oi P^niyhania c when 
 the invention of this aJmirable indrument has been afcrilT-d by (omc entle- 
 incn of that colony : This is not ths only ca''e, wherein d ffercnt pcrfons have 
 prodac> d fimilar inventions.
 
 XXX DISSERTATION dN fttE RISE, &c. ' 
 
 mN**430, Ami. ifJTy an inftrument grounded on the fame principles, 
 which is fo well efteeincd, that our fhops abound with them, accom- 
 anodated with ^<rn/Vr's divifion, as they are made by our moft.fkilful 
 workmen ; and. are now in general ufe amongft the fkilful fcamen of 
 moft of the maritime nations. 
 
 Though Midina^s method for finding the place of the horizon was ab- 
 furd ; yet, for this end, feveral plaufible ones have been propofcd by in- 
 genious pcrfons, as Mefl'. Elton^ Hadley^ Godfrey^ and Leigh ; and that 
 chiefly by applying a level to Davis's quadrant. Their devices are de- 
 fcribed in the Philofophical Tranfadtions for 1732, 33, 34, and 37. 
 And, laftly, an Horizontal Top, invented by the late Mr, Serfon^ who 
 was unfortunately loft at fea aboard the ViSlory man of war, has been ap- 
 proved of, and publiihed by Mr. Smeaton in the Philofophical Tranfac- 
 tions, vol. XL VII. for 1752, part ii. page 352. 
 
 Some methods ufed for obtaining the place of the horizon, and of ob- 
 ferving with Mr. Hadley's RefieSling SeSior, are defcribed by Mr. Ro- 
 hertforty in his Elements of Navigation j which treatife has defervedly met 
 with the approbation of the public. 
 
 Thus have I endeavoured to trace out the principal fteps by which 
 the art of navigation has advanced to its prefcnt height ; nor without 
 hopesr that the*nttempt may not prove altogether unacceptable to thofe 
 whofe bufinefs or curiofity lead them to be acquainted with this very 
 ufeful branch of the mathematics : on the fuccefsful prailifmg of which 
 depends, in an efpecial manner, the flourifliing ftate of our country. 
 
 This Diflertation, written at firft by defire, is now reprijited with 
 alterations. Though I may be thought to have dwelt too long on fome 
 particulars not dire<^]y relating to the fubjecfl ; yet L hope that what is 
 lb delivered, will not be altogether unentertaining to the candid reader. 
 As to any apology for having handled a matter quite forfeign to my way 
 of life, I ftiall only plead, that very young, living in a fea-port town, I 
 was eager to be acquainted with an art that could enable the Mariner to 
 arrive acrofs the wide and pathlefs ocean at his defired harbour. 
 
 London. JAMES WILSON. 
 
 ADVER-
 
 [ xxxi J 
 
 ADVERTISEMENT. 
 
 As it may be expe<Eled that four kinds of readers will look into 
 this book, it was thought convenient to point Out to feme of 
 them, the places where they may meet with what they more particu- 
 larly want. 
 
 First. Thofe who having made a proficiency in the mathematics, 
 will, it is likely, examine in what manner the fubjets are here treated, 
 and whether any thing new is contained therein : it is conceived that 
 fuch readers will find feme things which may recompence them for 
 their trouble, in almoft every one of the books. 
 
 Secondly. Thoie learners, who are defirous of being infl:ru6led in 
 the art of Navigation in a fcientific mani>er, and would chufe to fep 
 the reafon of the feveral fteps they muft take to acquire it : To fuch 
 perfons, it is recommended that they read the whole book in the order 
 they find it ; or, if the learner is very young, he may omit the IVth 
 and Vth books till after he is mafter of the Vlth and Vllth. 
 
 Thirdly. That clafs of readers, which, with too much truth m.ay 
 be faid, comprehends moft of our mariners, who want to learn both 
 the elements and the art itfclf by rote, and never trouble themfelves 
 about the reafon of the rules they work by : As it is probable there 
 ever will be many readers of this kind, they may be well accommo- 
 dated in this work j thus, if they are nof already acquainted with 
 Arithmetic and Geometry, let them read the five firft rules of Arith- 
 metic, to page 20 ; thence proceed to the definitions and problems 
 in Geometry, from page 43 to 58. la the book of Trigonometry, 
 read pages 89, 90, 91, 92, 98, 99, and from 104 to 114: the whole 
 of book VI. In book the Vllth they may read to page 35, and as 
 much more as they pleafc. In book VIII, let them read the fecSlions 
 III, IV, V, VI, from page 146 to page 182. In book V, they may 
 read fcdtion III, and as many problems in the Vth and Vlth fcdlions 
 as they can ; and let them read the whole of the ninth book. 
 
 Fqurthlv.
 
 xxxii ADVERTISEMENT. 
 
 Fourthly. That fet of readers who will not be at the pains of 
 learning any thing more than how to perform a day's work ; fuch majr 
 herein meet with the pradlice almoft independent of other knowledge. 
 Let fuch perfons make thcmfelves acquainted with feilion IV. of book 
 VI, and the ufe of the table at page 374 ; then learn the ufe of the 
 Traverfe Table at the end of book VII, which they will find exem- 
 plified between pages 8 and 35, Vol. II j alfo they muft learn the ufe 
 of the Table of Meridional parts at the end of Book VIII. After 
 which, they may proceed to book IX, where they will find ample in- 
 flrudlions in all the particulars which enter into a day's work. But 
 with this fcanty knowledge of things, they will be obliged to omit 
 forae parts, which it is well worth their pains to be acquainted with.
 
 THE 
 
 ELEMENTS 
 
 O T 
 
 NAVIGATION. 
 
 BOOK I. 
 
 O F A R I T H M E T I C K. 
 
 SECTION I. 
 Dejifiitions and Principles. 
 
 I- J^ RiTHMETiCK is a fcience which teaches the properties of 
 /-\ numbers ; and how to compute or cftimate the value of things. 
 -^ -^ 2. An Unit or Unity, is any thing confidered as one. 
 
 3. Number, in general, is many units. 
 
 4. Digits or Figures are the marks by which numbers are de- 
 noted or expreiTed, and are the nine following. 
 
 Digits^ I. 2. 3. 4. 5. 6. 7. 8. 9. 
 
 Names^ One. Two. Three. Four. Five. Six. Seven, Eight. Nine. 
 And with thefe is ufed the mark o, called cypher, which of itfelf 
 ftands for nothing j but being annexed to a digit, alters its value. 
 Thus ^0 Jignifies forty j and ^00 J^ands for four hundred, is'c. 
 
 5. Integer, or Whole, Numbers, arc fuch as cxprefs a number 
 of things, each of which is confidered as an unit. 
 
 Thus four pounds, twelve miles, thirty-four gallons, one hundred days, 
 C c. are, in each cafe, called an integer number, or whole number. 
 
 6. Fractional Numbers, are thofe which cxprefs the value of 
 fomc part or parts of an unit. 
 
 Thu^ one half, one quarter, three quarters, is'c, are each the fra^ln?:al 
 parts of an unit. 
 
 Vol. I. B 7. Not a-
 
 % A R I T H M E T I C K. Book I. 
 
 7. Notation is the exprefling by digits or figures any number pro-t 
 pofed in words ; and the reading of any number that is^cxprefled by fi- 
 
 :gurcs, is called Numeration. 
 
 ,V 8. Decimal Notation is that kind ofnumbering in which ten units 
 of any inferior name are equal in value to an unit of the next fuperior. 
 
 9. Every number is faid to confift of as many places as it contains 
 figures. -^ 
 
 10. The value of every digit in any number is changed according to 
 the place it (lands in ; and the reading of any number confifts in giving 
 to each figure its right name and value. 
 
 11. The right hand place of an integer number is called the place of 
 units J and from this place all numbers begin, whether whole ox fra6lt- 
 cnal y the integers increafing in order from the unit place towards the 
 
 'left J and the fractions decreafing in order from the unit place towards 
 the right : and to diftinguifli decimal fraftions from integers, there is 
 always a point or comma ( , ) fet on the 'left hand fide of the fraftional 
 number ; fo that the integers {land on the left hand fide of the mark, 
 and the fra(lions on the right hand. 
 
 12. For the more convenient reading of numbers, they are divided 
 into periods of fijTplaces each, beginning at the unit place j and each 
 period into two degrees of three places each, the names and order of 
 which are as follow : where X ftands for the word tens, C for hun- 
 itrcds, and TK. for thoufands, 
 
 13, Integers Decimal fralions 
 
 _i f - -A^ , I -^ 
 
 > 
 
 Second period Firft period Firft period Second period 
 Degree Degree Degree Degree Degree Degree Degree Degree 
 
 
 s ^ s s o .0 ^ ^ <^ ^ ^ ^ ^ ^ '^ -^ o .0 ,0 : s ^ 
 
 
 543212 3 45678 9, 876543212345 
 
 Decimal Fiaclions are alfo 
 thus named, 
 
 The name of the firft pciiod is Units j of the fccond, Millions j of the 
 third, Billions; of the fourth, Trillions; &c. 
 
 In the above order it may be obferved, that each degree contains the 
 names of Units, Tens, Hundreds ; the firft degree of a period contains 
 the units of that period, and the fecond contains the thoufandths there- 
 of: fo that from hence it will be eafy to read a number confifting of 
 c\cr To many places by the following directions. 
 
 5 14. Rule,
 
 Book I. -ARITHMETICK. 3 
 
 14. Rule. ift. Suppofe the number parted into as many fets or de- 
 grees of three places each, beginning at the unit's place, as it will ad- 
 mit of; and if one or two places remain, they will be the units and 
 tens of the next degree. 
 
 2d. Beginning at the left hand, read in each degree, as many hun- 
 dreds, tens, and units, as the figures in thofe places of the degree ex- 
 prefs, adding the name thoufands, if in the fecond degree of a period ; 
 and adding the name of the period, after reading the hundreds, tens, and 
 Hnits in its firft degree. 
 
 Thus the integer number in the preceding table will be read. 
 - Five billions, four hundred thirty two thoufand, one hundred twenty 
 three millions, four hundred fifty fix thouf and, f even hundred eighty nine, 
 
 I5 All frailional numbers confifl of two parts, which are ufually 
 written one above the other with a line drawn between them : the num- 
 ber below the line, called the denominator, fhews into how many equal 
 parts the unit is divided ; the number above the line, called the nume- 
 rator, (hews by how many of thefe equal parts the value of that frac- 
 tion is exprefled. 
 
 Thus 9 pence, is q parts in twelve of a JhilUng ; and may he written 
 thus, t'i, when a fiiilling is the unit. 
 
 16. Thofe fra6lions, the denominators of which are 10, or 100, or 
 1000, or 1 0000, or 1 00000, &c. are called decimal fraSlions : but frac- 
 tions with any other denominators are called vulgar fraSlions* 
 
 The vulgar fra<^ions that moft frequently occur, are thefe : 
 
 I, which is read one fourth, or one quarter. 
 ^ _____ _ one third. 
 
 1^ - - - - - - one half. 
 
 .| _____ _ two thirds. 
 
 i _ _ _ ^ - _ three fourths, or three quarters. 
 
 17. As decimal fralions are parts of an unit divided into either lO, 
 100, loco, 1 0000, &c. parts, according to the places in the fradlional 
 number' ; therefore they are read like whole numbers, only calling them 
 fo many parts of lo, or of loo, or of looo, he, 
 
 !one -J r- 10, Ten. 
 
 ;r (places, will be fo j ^"' ^T^'^\ 
 three >^ . l \ lOOO, Thoufand. 
 
 r I many parts or I ' , ^,, ^ , 
 
 tour 1 J '^ \ 10000, Ten 1 houfand, 
 
 hz. J I &c. 
 
 18. Cyphers on the right hand 0/ Integers increafe their v.ilue; on 
 the left hand of a decimal frailion diminiih its value : but on the left 
 hand of integers, or on the right hand of fraclion?, do not alter their 
 value. 
 
 B 2 rhm
 
 4 A R I -F H M E T I C K. Book I. 
 
 f 8 is S units. I f ,8 is 8 parts in 10 1 ^ 
 
 Thus \ 80 8 tens. \ Jnd \ ,o8 8 parts in lOO f f J?" fj^'j 
 i 8oo 8 /?K^r^</j. I ( ,oo8 8 parts in looo \r divided^ 
 
 When a fraction has no integer prefixed, it is convenient to put o in 
 the place of units. 
 
 19. A Mixed NUiMBER, is when a fraction is annexed to a whole 
 number. 
 
 Thus five and a half is called a mixed number, and is written 5!, or 
 thusy 5,5 J which is thus read, five and five teyiths* 
 
 20. Like names in different numbers are fuch figures as ftand equally 
 diftant from the place of units j or have the fame denomination an- 
 nexed to them. 
 
 Thus all numbers of pounds Jler ling are like names, and fo are all num- 
 bers offi)illings J the like of any numbers of miles, ?V. 
 
 21. Befides the decimal notation explained in article 8, there are 
 other kinds in common ufe ; fuch as the duodecimal, in which every 
 fuperior name contains 12 units of its next inferior name : the Sexage- 
 nary, or Sexagefnnal, in which fixty of an inferior name make one of 
 its next fuperior. The former is ufed by workmen in the meafuring 
 of artificers works in building ; and the latter is ufcd in the divifion ox 
 a Circle, and of Time. 
 
 22. The following characters or marks are frequently ufed in Arith- 
 metical computations, briefly to exprefs the manner of operation. 
 
 The mark + (more) belongs to addition ; and (hews that the num- 
 bers it flands between are to be added together. 
 
 Thus 1 2 -}- 3 exprejfes the fum of 12 and 3 ; or that 3 is to be added 
 to 12, and is thus read, i2 more 3. 
 
 I'he mark (lefs) is for fubtraftion ; and fbews that the number 
 following it, or on the right hand, is to be taken from the number pre- 
 ceding it, or on the left hand. 
 
 Thus 12 3, exprejf'es the difference bitivcen 1 2 and 3 ; or that 3 is t 
 be fubtracfed from 12, and is thus read, i2 lej's 2, or 12 leJJ'ened by 3. 
 
 This mark x (into) for multiplication, jfhews that the numbers on 
 each fide of it are to be multiplied the one by the other. 
 
 Thus 12 X 3, denotes the product of 12 into 3 j or that 12 is to be 
 multiplied by 3. 
 
 Divifion is exprefTed by fettlng the divifor under the dividend with a, 
 line drawn between them, like a fraction. 
 
 Thus '/, exprejjei the quotient of 11 by 1; or that 12 is to he divided 
 
 This fign r=z (equal) fhews that the refult of the operation by the 
 numbers or quantities on one fide of it, is equal either to the numbers 
 or quantities on the other fide, or to the refult of the operation by thefe 
 numbers or quantities. 
 
 'H'z/i i2-f 3=15 ; and 12 3 = 9; ond 12x3 = 36; and V=4 J 
 
 fevcraUy Jheivs the value cf the preceding exprejfions, 
 
 23. Table's
 
 Book I. 
 
 A R I T H M E T I C K. 
 
 123. Tables of English Money, Weights, and Measures. 
 
 Money. 
 Farthings Pence Shill. Pound 
 960 zz 240 =: JO = I ;^. 
 48 =: 12 zz IS. 
 4 zz 16. 
 Nate, 1,^,3 farthings, arc thus writ- 
 ten - -' 1 
 
 AvoiRDUPOisE Weight. 
 Drams Ounces Pounds Hund.Ton 
 5734.40 1=35840 =2240 ZZ20 zzi 
 
 28672 := 1792 n: 112= I C. 
 256 = 16 zz zz. lib. 
 i5 = I 02. 
 
 iVo/*,Provillons, Stores, &c.areweighed 
 by the Avoirdupoife, orgreat weight. 
 
 Dry Measure. 
 Pints Gall. Pecks Bufli. Quarter 
 5i2rr64-=32:=:8::^i 
 64 = 8 z= 4=1 
 
 8 = I ~ 
 Note, ^haih. zz i Comb: loqrs. =:i 
 Wey : 12 Weys zr Laft of Corn. 
 36 Buihels =1 Chaldron f Coals. 
 
 Troy Weight. 
 Grains Pennyw'" Ounces Pourd 
 5760 =: 240 r: jz zz i lb. 
 480 zz: 20 =z I oz. 
 gr. 24 =: I dwt. 
 Note, Gold and Silver are weighed 
 by Troy Weight. 
 
 Wine Measure. 
 
 Solid inch. Pints. Gall. f-lGgfh.PipeTui) 
 5821 2z=2bi6r=252ir4zr2 m 
 29106=^1008= 1 26rr2=:i P. 
 4553= 504= 63 = 1 Hhd. 
 231=: 8= I 
 
 42 r= I Tierce. 
 
 84^=1 Puncheop. 
 
 Cloth Measure. 
 4 Nails rr 1 Quarter of aYard. 
 
 4 Quarters zz i Yard. 
 
 5 Quarters =: i Englifli ell, 
 3 Quarters = i Flemifli ell. 
 
 6 Quarters r= i French ell. 
 a Span zz 9 inches. 
 
 a Hand zz 4 inches. 
 
 Long Measure. 
 
 Yards 
 rr 1760 
 
 zz 220 
 
 = 54 
 
 Poles Furi. 
 320 = 8 
 
 40 
 I 
 
 
 Barleycorns Inches Feet 
 
 190080 == 63360 zz 5280 
 
 23760 r: 7920 zz 660 
 
 594 = 198 = i6| 
 
 108 = 36 = 3 
 
 36 := 12 I 
 
 3 = , 
 Alfo 3 miles make i league. 
 
 And 20 leagues or 60 Sea miles make a degree. 
 
 But a deo^ree contains about 69' miles of llatutc meafure. 
 
 A fathom rr 6 feet rr 2 yards. 
 
 Mile. 
 =: I 
 
 Time. 
 Seconds Minutes Hours 
 
 31556937 = 52594S = ^766 rr 
 
 86:100 ZZ 144 = 24 zz: 1 Uay 
 3600 zz 60 zz 1 hour. 
 
 60 zz I 
 
 D.nys Year 
 
 'J 
 
 Pence Table. 
 Pence Sh. Pence Fence Sh. Pence 
 
 20 ZZ I 
 y zz 2 
 40 ZZ 3 
 504 
 
 6d = 3 
 
 = 5.10 
 
 70= 5 
 
 80 =r 6 
 
 90 7 
 
 ICO = 8 
 
 1 10 r= 9 
 
 Even paits of a Pound 6terh 
 
 s. 
 
 10 
 6 
 
 5 
 
 4 
 3 
 2 
 
 
 >cn 
 
 2 . O -,' J '-.^ 
 
 ~2 
 
 d 
 
 6 is 
 4 
 3 
 2 
 
 >1 
 
 x1 
 
 
 J L 
 
 24. S E C-
 
 RITHMETICK. Bookt^^ 
 
 24. 
 
 SECTION II. ADDITION. 
 
 Addition is the method of collecting Jeveral numbers into 
 onejum. 
 
 Rule ift. Write the given numbers under each other, fo that like 
 names (land under like names j that is units under units, tens under 
 tens, &c. and under thefedraw a line. 
 
 2d. Add up fhe firft or right hand upright row, under which write 
 the overplus of the units of the fecond row, contained in that fum. 
 
 3. Add thefe units to the fum of the fecond row, under which write 
 the overplus of the units of the third row, contained in that fum. 
 
 And thus proceed until all the rows are added together. 
 
 Examples. ^ 
 
 Ex. I. J^dd2^ 7647 18 and 12 together. 
 
 Thefe numbers being written under each other will ftand thus. 28 
 Say 1 and 8 is 10, and 7 is 17, and 6 is 23, and 8 is 31 ; 76 
 
 then, becaufe lO units in the right hand row make an unit in 47 
 the next rowj therefore in 31 there are 3 units of the fecond 18 
 row, and an overplus of i; write down the i, and add the 3 to 12 
 
 the fecond row, faying, 3 that is carried and I is 4, and i iS 5, 
 
 and 4 is 9, and 7 is 16, and 2 is 18, in which is one unit of the 181 
 third row (had there been a 3d) and an overplus of 8 ; write - 
 down the 8, and add the i to the third row : but as there is no third 
 row, tiie 1 carried muft be written on the left hand of the 8 ^ and 181 
 will be the fum of the five given numbers. 
 
 Ex. II. Jild 476 378418329 
 290 75 7638 <7^4j6 to- 
 gether. 
 
 Ex. III. Add the numbers, 10768 
 3489 28764 289 6438 
 ic) and ^1*^ together. - 
 
 476 
 
 
 10768 
 
 ^\ . 37H 
 
 
 3489 
 
 The given numbers ^ 18329 
 
 The given numbers 1 
 
 28764 
 
 fet in order will Hand C 290 
 
 placed as the rule di- > 
 
 289 
 
 thus i 75 
 
 redls, Hand thus j 
 
 6438 
 
 7638 
 
 
 ^9 
 
 46 
 
 
 438 
 
 The Sum 30638 
 
 The Sum 
 
 Ex. V. Jdd the foUowin 
 
 50205 
 
 Ex. IV. Add thefe numbers together. 
 
 numbers' 
 
 
 together. 
 
 
 3720.4s 
 
 15836,071 
 
 
 25,0036 
 
 20,09 
 
 
 4179,802 
 
 34.7 
 
 
 3,6284 
 
 583,270 
 Sum 16474,131 
 
 08 
 
 Sum 7928,8840 
 
 08 
 
 In
 
 Book I. 
 
 A R I T H M E T I C K. 
 
 In the two laft examples, where there are both'integer and fracSlIonal 
 numbers, it may be obferved, that like integer places, and like fra<5lional 
 places, ftand under each other; and the manner of adding them together," 
 is the fame as explained in the ^rft example. 
 
 25. It frequently happens, that numbers arc to be added together, the 
 names of which do not incrcafe in a tenfold manner, as in the laft Ex- 
 amples ; fuch as in adding different fums of money, weights, or mea- 
 fures ; in which, regard is to be had to the number of thofe of a lower 
 name, contained in one of its next greater name, as fhewn in the pre- 
 ceding tables : Examples of which follow. 
 
 Ex. VI. Jdd the following fums of 
 money together. 
 
 353 14 
 276 10 
 
 89 17 
 
 34 12 
 
 d. 
 
 4 
 
 si 
 
 lol- 
 
 754 
 
 S 
 
 Ex. VII. Jdd the fo^owing fums of 
 money together. 
 
 7683 08 2| 
 
 95+ 19 
 682 10 
 
 63 5 
 
 9-i 
 
 9384 
 
 14 
 
 In thefe two examples the carriage is by 4 in the farthings; by 12 in the 
 pence; by 20 in the fhillings; and by 10 in the pounds. 
 
 Ex. VIII. Add the following Troy Ex. IX. Add the follcwing Avoir- 
 
 JVei*-hts together. 
 
 lb. 
 218 
 176 
 
 85 
 
 24 
 
 oz. 
 10 
 
 9 
 1 1 
 
 dwt. 
 13 
 
 '9 
 
 17 
 
 18 
 
 23 
 II 
 21 
 
 506 
 
 07 
 
 01 
 
 Carry for 24, 20, 12, 10. 
 
 Ex. X. Add the following parts of 
 Time together. 
 Weeks Da. Ho. 
 
 21 4 18 
 
 II 6 13 
 
 19 3 23 
 
 3H 4 cH 
 
 Min, 
 
 37 
 
 59 
 
 22 
 
 Sec. 
 
 59 
 
 47 
 28 
 
 39 
 
 9' 
 
 5 3 
 
 Carry for 60, 60, 24, 7, 10. 
 
 dupoife TVeights together. 
 
 Tons. Cwt. qrs. lb. oz. 
 
 535 17 3 22 '^ 
 
 94 19 I 27 \i 
 
 158 12 o 18 15 
 
 7 15 2 13 08 
 
 '97 
 
 15 
 
 Carry for 16, 28, 4, 20, 10. 
 
 Ex. XI. Add the folloiving parts of 
 
 a Circle together. 
 Deg. 
 
 176 32 59 
 
 8^ 59 27 
 
 28 45 
 12 3H 
 
 114 
 
 67 
 
 43 
 31 
 
 59 
 24 
 
 25 
 59 
 
 H 
 
 47 
 
 444 
 
 i) 
 
 Carry for 60, 60, 60, 60, lo 
 
 Explanation of Example VI. 
 
 Three farthings and i farthing is 4 farthings, and 2 fartliings is 6 far- 
 things ; which is a penny halfpenny ; fet down | and carry i. 
 
 Then \ and 10 is 1 1, and 5 is 16, and 4 is 20, and 8 is 28 pence ; 
 which is 2 ftiillings and 4 pence : fct down 4, and carry 2. 
 
 A^^iin, 2 and 12 is 14, and 17 is 31, and 10 is 41, and 14 is 55 
 fhiUings; which is 2 pounds 15 fhillings i fct down 15 fluHings, and 
 carry 2 pounds. The rcll is eafy. 
 
 B 4 aG. S E C
 
 ARITHMETIC K. 
 
 Book I. 
 
 \6. SECTION III. SUBTRACTION. 
 
 Subtraction ij the method of taking one number from an- 
 ether, and fhewivg the remainder^ or difference, or excefs. 
 
 T}nQ fubducend is the number to be fubtra^led, or taken zwzy. 
 
 The minuend is the number from which the fubducend is to be taken. 
 
 Rule ift. Under the minuend write the fubducend, fo that like 
 pames ftand under like names j and under them draw a line. 
 
 2d. Beginning at the right-hand fide, take each figure in the lower 
 line from the figure (landing over it, and write the remainder, or what 
 is left, beneath the line, under that figure. 
 
 3d. But if the figure below is greater than that above it, increafe the 
 upper figure by as many as are in an unit of the next greater name ; from 
 this fum take the figure in the lowcx line, and write the remainder un- 
 der it. 
 
 4th. To the next name in the lower line, carry the unit borrowed, 
 Ikpd thus proceed to the higheft denomination or name. 
 
 Examples. 
 
 Ex. I. From 436565874 the mhiuendy 
 Take Z49853642 the fubducend. 
 
 Remains 186712232 the difference. 
 
 i. " 
 
 Here the five figures on the right of the fubducend maybe taken from 
 fchofe over them : but the 6th figure, viz. 8, cannot be taken from the 5 
 above it. Now as an unit in the 7th place makes ic in the 6th place, 
 therefore borrowing this unit makes the 5, 15; then fay, 8 from 15 leaves 
 7, which fet down; and fay i carried and 9 is 10, 10 from 6 cannot be had, 
 but 10 from 16 leaves 6, fet it down; then i carried and 4 is 5, 5 from 13 
 leaves 8 \ fet it down ; then i carried and 2 is 3, 3 from 4 leaves i. 
 
 Ex. ir. Trom ' 7620908 
 
 Take 3^7509* 
 
 |lx. IV. From 
 Take 
 
 Remains 3745816 
 
 30007,29; 
 2536,876 
 
 Leaves 27470,419 
 
 C S' d. 
 
 Ex. VI. Borroived 24 14 6f 
 
 Paid 18 12 4| 
 
 Remains 
 
 6 02 zl 
 
 Ex. HI. 
 
 Ex. V. 
 
 From 
 Take 
 
 327.9563 
 49,8697 
 
 Remains 
 
 From 
 Take 
 
 278,0866 
 
 5000,0000 
 479,6378 
 
 Leaves 
 
 4520,36^2 
 
 ' d. 
 
 Ex. VII. Lent 294 15 9I 
 
 Received 89 18 io| 
 
 Remains 204 16 \o\ 
 
 Ex. VIIl.
 
 Book I. 
 
 A R I T H M E T I C It. 
 
 Ex. VIII. In Sexagejimah. 
 
 From 76 28 37 49 32 
 Ttfi/ 65 29 i6 53 45 
 
 Lea-vts 10 59 20 55 47 
 
 Ex. IX. In Sexagejimah. 
 
 IV 
 
 From 
 Take 
 
 Leaves 
 
 218 46 32 50 18 
 H9 Sg 47 53 29 
 
 68 53 44 56 49 
 
 27. Questions to exercife Addition and Subtradion. 
 
 Quest. I. The jhare of Jack's 
 priz.e money was 148^^. ijs. 6dj ; 
 and Tom received as much, he fide ']. 
 jZs.fmart money : How much money 
 did Tom receive ? 
 
 . s. d 
 Tom's prize money 148 17 6j 
 Smart money 7 180 
 
 Tom received 156 1$ 6^ 
 
 Quest. III. What yearwas King 
 George born in, he being 67 years old 
 
 in thfi vtnr T7/1Q ? 
 
 in the year 1749 f* 
 
 Current year 
 
 Age 
 
 1749^, 
 67 fubtr. 
 
 Year born in 1682 
 
 Qy E s T . V . A feaman who had re 
 eeived afo. ijs.bd. for wages, prize 
 money, <Jc. meeting with bad company 
 was tricked out of\% guineas : Now 
 "John had reckoned to pay his wife's 
 debts ofi'^f^. lbs. 6d. and his land- 
 lady's bill of lb . lis. Required 
 whether he can fulfil his intentions, 
 and what the difference will be P 
 
 Quest. II. The Spanifl} inva 
 fion was in the year 1588, and the 
 
 French attempted an invafion in the 
 year 1744.' How many years were 
 
 between thefe fruitlefs attempts ? 
 
 French 
 Spanifh 
 
 1744 
 
 1588 
 
 Years between 156 
 
 Quest. IV, Two Jhips depart 
 
 fro7n the fame port, one having failed 
 835 miles, is got 48 miles a-head of 
 the other: Required the aftermoji 
 Jhip's dijlance f 
 
 The firftfhip's diftance 835 
 Their difference 48 
 
 Second fliip's diftance 787 
 
 Quest. VI. Will and Frank 
 talking of their ages in the year IJ ^(^^ 
 Will faid he was born in the year 
 of the Rebellion, ? 1 7 1 5 ; and Frank 
 faid he remembered he was tm yean 
 old the year King George the fecond 
 was crowned in 1727 : Required the 
 age ofeachy and the difference of their 
 ages P 
 
 Money loft 
 Wife's debt 
 Landlady's bill 
 
 Total 
 Money received 
 
 He will want 
 
 c 
 
 18 
 
 >3 
 
 16 
 
 s. 
 
 18 
 16 
 
 12 
 
 46 
 
 6 
 
 17 
 
 6 
 6 
 
 2 
 
 9 
 
 
 
 Current year 
 Will was born 
 
 749 
 7iS 
 
 34 
 
 Will's age 
 
 Current year i749 
 
 King George crowned 17^7 
 
 Years fince 22 
 
 Frank's age then 'O 
 
 Frank's age 32 
 
 So Will was Oldcft by two years. 
 
 28, SEC-
 
 to 
 
 A R I T H M E T I C K. Book I. 
 
 fl8. SECTION IV. MULTIPLICATION. 
 ;, MvLTiPLiCATiON ts the method of finding what a given 
 number ivill amount to, when repeated as many times as is re- 
 fre/ented by another number. 
 
 54-*uoTbei^ta be multiplied, Is called the Multiplicand, 
 The number multiplied by, is called the Multiplier, 
 And the number which the multiplication amounts to, is called the 
 
 ProduSf. 
 
 Both multiplicand and multiplier are called FaSfors. 
 
 'Before any operation can be performed ir^ Multiplication, it is necef- 
 
 fary that the learner fhould commit to memory the following table. 
 
 29. The Multiplication Table. 
 
 times 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 II 
 
 12 
 
 .2 
 
 4 
 
 6 
 
 8 
 
 10 
 
 12 
 
 H 
 
 16 
 
 18 
 
 20 
 
 22 
 
 24 
 
 3 
 
 
 9 
 
 12 
 
 15 
 
 18 
 
 21 
 
 24 
 
 27 
 
 3^ 
 
 33 
 
 3^^ 
 
 4 
 
 
 
 16 
 
 20 
 
 24 
 
 28 
 
 32 
 
 36 
 
 40 
 
 44 
 
 48 
 
 <; 
 
 
 
 
 25 
 
 30 
 
 35 
 
 40 
 
 4^ 
 
 ?o 
 
 5? 
 
 60 
 
 6 
 
 
 
 
 
 i^ 
 
 42 
 
 48 
 
 H 
 
 60 
 
 66 
 
 72 
 
 7 
 
 
 
 
 
 
 49 
 
 16 
 
 63 
 
 70 
 
 77 
 
 84 
 
 8 
 
 
 
 
 
 
 
 64 
 
 72 
 
 80 
 
 88 
 
 96 
 
 9 
 
 
 
 
 
 
 
 
 81 
 
 90 
 
 99 
 
 108 
 
 16 
 
 
 
 
 
 
 
 
 
 100 
 
 no 
 
 120 
 
 II 
 
 
 
 
 
 
 
 
 
 
 121 
 
 132 
 
 li 
 
 
 
 
 
 
 
 
 
 
 
 144 
 
 Obferve, that in multiplying any figure in the upper line by any figure 
 in the left-hand column, the produ6l will ftand right againft the figure 
 ufed in the left-hand column, and under that ufed in the upper line. 
 Thus were 6 to be multiplied by 9, feelc the greater figure 9 in the upper 
 line, and right under it, againft 6 in the left hand, ftands 54 for the Pro- 
 du(5l. And fo of others. 
 
 The foregoing table being well known, the work of Multiplication 
 will be performed as follows. 
 
 To multiply any number, as 37256 
 
 By any Tingle figure, as by 7 
 
 Set them as in the margin, and proceed 260792 
 
 thus, 7 times 6 is 42, fet down 2 and carry 4; 7 times 5 is 35 and 4 
 carried is 39, fet down 9 and carry 3 ; 7 times 2 is 14 and 3 carried is 17, 
 fet down 7 and carry i ; 7 times 7 is 49 and i carried is 50, fet down o 
 and carry 5 j 7 times 3 is 21 and 5 carried is 26, which fet down, and the 
 work is done. But for compound Multiplication take the ff)llowing : 
 
 30. Rule ift. Write the Factors fo, that the right hand place of the 
 Multiplier ftands under the right hand place of the Multiplicand. 
 
 2d. Multiply the Multiplicand feverally by every figure of the Multiplier, 
 fetting the iiilt figure of each line under the figure then multiplying by. 
 
 3d. Add the feveral lines together; and their fum is the Produdl:. 
 
 4th. From the right hand of the Produ6l point off", for fradtions, as many 
 places as there are fra6lional places in both Fadors; and thofeto the left 
 of the mark of diftin<Stion arc integers ; thofe to the right are fraclions. 
 
 5th. If
 
 Book I. 
 
 A R I T H M E T I C K. 
 
 II 
 
 5th. If the number of places in the Produl are not fo many as the num- 
 ber of fra<SHonal places in both Faftors, make up that number by writing 
 cyphers on the left hand, and to thefe prefix the mark of diftindion. 
 
 Example I. Multiply 742 by 53. 
 
 The lefrerFa(9:or being written under 
 the greater Factor, as here fhewn, and a 
 line drawn under them ; fay 3 times 2 is 
 6, write 6 under the 3 ; then 3 times 4 is 
 1 2, write down 2 and carry i ; and3 times 
 7 is 21, andi carried is 22, write the 22 :- 
 Again, 5 times 2 is 10, write o under the 
 5, and carry i; and 5 times 4 is 20 and i ' 
 
 carried is 21, write down land carry 2; then5 times7 is 35, and~2 carried 
 is 37, write down the 37 : Now add the two lines together found by mul- 
 tiplying by 3 and by 5, and their fum 39326 is the produdl required. 
 
 Multiplicand 74.2 7 ^ 
 Multiplier 'jjjFadlors. 
 
 2226 
 3710 
 
 Produl 39326 
 
 Example II. 
 
 Multiply 28704 
 
 hj 8631 
 
 28704 
 86112 
 172224 
 229632 
 
 24.7744224 Produft. 
 
 Example IV. 
 
 Multiply 936,287 
 by 607, C2 
 
 1872574 
 6554C090 
 
 56177220 
 
 568344,93474 
 
 The cyphers in the Multiplierof this 
 example arc thus managed. Having 
 multiplied by the 2 as before, fay o 
 limes 7 is o,write o under the 0, and 
 proceed to the next figure 7,by which 
 i!ni';'ply as before, then coming to 
 the iL-cond o, fay, o times 7 is 0, 
 writf o under the place of the fc- 
 cc'Mcl o, and proceed to the next fi- 
 gure bj by which multiply as before. 
 
 Example III. 
 
 Multiply 3684,2795 
 
 h 7594 
 
 147371180 
 
 33i585'55 
 184213975 
 
 257899565 
 27978,4185230 
 
 Here, becaufe there are 4 fraftional 
 places in the Multiplicand, and 3 in 
 the Multiplier, which together make 
 7, therefore 7 places are pointed ofF 
 on the right of the produil for frac- 
 tions. 
 
 Example V. 
 
 Multiply 0,34796 
 
 ty 0,0258 
 
 278368 
 173980 
 69592 
 
 0,008977368 
 
 Here, becaufe there are 5 fradional 
 places in one Fa6tor, and 4 in the 
 other, there ftiould be 9 fractional 
 places in the ProduiSlj and there 
 arifing but 7, therefore two cyphers 
 are fct on the left hand to make 9 
 places. 
 
 31. S E C-
 
 i A R I T H M E T I C K. Book L 
 
 31. SECTION V. DIVISION. 
 
 DivisroN is the method of finding how often one number is 
 contained in another; or may be taken from another. 
 
 The number to be divided, is called the Dividend. 
 
 The number dividing by, is called the Divlfor. 
 
 The ^(otient is the number arifuig from the diviilon, and fhews how 
 many times the Divifor is contained in the Dividend. 
 
 The operations in Divifion are performed as follow. 
 
 32. Rule ift. On the right and left of the Dividend draw a crooked 
 line ; write the Divifor on the left fide, and the Quotient, as it arifes, on 
 the right fide of the Dividend. 
 
 2d. Seek how often the Divifor may be taken in as many figures 6r\ 
 the left hand of the Dividend, as are juft neceflary; write the number 
 of times it may be taken, in the Quotient ; and there will be as many 
 figures more in the""Quotient, as there are figures remaining in the Di- 
 vidend then not ufed. 
 
 3d. Multiply the Divifor by this Qitotient-figure, fet the Produfl un- 
 der that part of the Dividend ufed ; fubtrat, and to the right hand of the 
 remainder bring down the next figure of the Dividend : Divide as be- 
 fore ; and thus proceed until all the figures of the Dividend are ufed. 
 
 4th. If there is a remainder, to its right hand fide annex a cypher or 
 cyphers, as if brought down from the Dividend, and divide as before j 
 and thus it is that fradtions arife, viz. from the remainders in divifion. 
 
 5th. When any figure of the Dividend is taken down, or annexed, as be- 
 fore fhewn,and the Divifor cannot be taken in the number thus increaf- 
 ed ; put o in the Quotient, and take down, or annex, another figure; and 
 proceed in this manner,until the Divifor can be taken from the number. 
 
 6th. When fractions are concerned : From the number of fratSlional 
 places ufed in the Dividend, take thofe in the Divifor; count the num- 
 ber of remaining places from the right of the Qiiotient, put the mark 
 there ; and thofe to the left are integers, thofe to the right frad^ions. 
 
 7th. If there arife not fo many places in the Quotient as the 6th ar- 
 ticle requires, fupply the places wanting with cyphers on the left, and 
 to thofe prefix the fradional mark. 
 
 Ex. I. Divide 3656^. among % perfons. 
 
 Set the given numbers as in Art. ift. Now the two 8)3656(4.57 
 
 left hand figures contain 8; then fay 8 is contained in 36, 4 32 
 
 times ; fet 4 in the Quotient, and fay 4 times 8 is 32, fet 32 
 
 under 36, fubtrat, there remains 4, to which bring down 45 
 
 the next figure of the Dividend 5, makes 45 ; then fay 8 is 4 
 
 contained in 45, 5 times; fet 5 in the Quotient, and fay 5 ^ 
 
 times 8 is 40 ; write 40 under 45, fubtradt, and to the re~ ^^ 
 
 maindcr 5 take down 6, the next figure of the Divi<lend, ^ 
 
 makes 56; then fay 8 is contained in 56, 7 times; write ^ 
 7 in the Quotient, multiply 8 by 7 makes 56, which write 
 
 under the other 56, and fubtradling there remains o: So it may be 
 concluded, that 3656 contains 8, 4.57 times: Or, if 3656^^. be divide<i 
 among 8 pcrfons, the fhare of each will be 457 j^. 
 
 Ex. IL
 
 Book I. 
 
 A R r T H M E T r C K, 
 
 i 
 
 Ex. IL Divide 3125 ^_y 25. 
 25)3125(125 
 
 62 
 
 125 
 125 
 
 Ex. IV. Divide 5859 by 124. 
 124)5859(47,25 
 496 
 
 899 
 868 
 
 310 for the Remaind. 
 248 See precept 4th. 
 
 620 
 620 
 
 Ex. VI. Divide 2,3569 by 673,4. 
 
 673.4)2.3569(35 
 
 20202 Quot. 0,0035 
 
 33670 See precepts 
 33670 4ih, 6th, 7th. 
 
 Ex. Ill, Divide 95269 by 47^ 
 47)95269(2027 
 94 
 
 See precept 5 th. 126 
 94 
 
 339 
 329 
 
 Ex. V. Divide 337,27368 by 6,2%. 
 
 ^.28)337,27368(53,706 for 
 
 3140 the Quotient, 
 
 See precept 6th. 
 
 2327 
 1884 
 
 4433 
 4396 
 
 See precept 5th. 3768 
 3768 
 
 In Ex. VI. the 4 fracflional places 
 given in the Dividend, and the o 
 ufed with the Remainder, make 5 
 fradional places ; from which i 
 place in the Divifor being taken, 
 leaves 4 fradional places for the 
 Quotient j but in the Quotient are 
 nly the two places 35, therefore 2 cyphers are prefixed, and makes 
 ,0035, before which, for form fake, an o is fet for the place of units. 
 
 33, When the Divifor does not exceed the number i2, theDivifion 
 may be performed in one line ; by making the Multiplication and Sub- 
 traction mentally, or in the mind, and carrying the Remainder, as f* 
 many tens, to the next figure. 
 
 34. In all operations of Divifion, it muft be obferved, that the Pro- 
 dudt of the Divifor by the Quotient figure muft not exceed that part of 
 the Dividend then ufing ; and the Remainder, by fubtradling the Pro- 
 duct, muft ever be lefs than the Divifor. 
 
 As the Quotient multiplied by the Divifor makes the Dividend ; 
 
 So the Produdt of two numbers being divided by one of them, will 
 give the other ; that is, Divifion is proved by Multiplication, and Mul- 
 tiplication is proved by Divifion, 
 
 35. SEC-
 
 4 A R I T H M E T I C K. Book I. 
 
 3$; SECTION VI. REDUCTION. 
 
 Reduction is the method of reducing numbers from one 
 name, or denomination^ to another ; retaining the fame value, 
 Qasb.\,To reduce a number conftfiing of fever al name i^ to their leaf name. 
 
 Rule iQ. Multiply the firft, or greater name, by the parts which 
 an unit of that name contains of the next lefs name ; adding to the Pro- 
 duft the parts of the fecond name in the given number. 
 
 2d. Multiply this fum by the number of times that an unit of the 
 
 next lefs name is contained in one of the fecond name ; adding to the 
 
 ProduiSt the parts of the third name contained in the given number : 
 
 And thus proceed, until the leaft name ift the given number is arrived at. 
 
 Ex. I. In 22' 14^. 6f </. how Ex. II. In Sib. lOoz. of gold, how 
 
 many farthings ? many grains f 
 
 . ,/. d. lb. oz. 
 
 23 H 6i 
 
 20 
 
 474 Shillings. 
 12 
 
 5694 Pence. 
 4 
 
 Anfwer 22778 Farthings, 
 
 8 
 12 
 
 10 
 
 106 Ounces. 
 20 
 
 2120 Pennyweights. 
 24 
 
 8480 
 4240 
 
 ^0880 Grains. 
 
 Ex. IV. IH 36 deg. 48^ 2f^. Kfi>f\ 
 
 how many thirds ? 
 o / // /// 
 
 36 
 60 
 
 48 27 56 
 
 2208 Minutes. 
 60 
 
 132507 Seconds. 
 60 
 
 Ex. III. InacannonweighinglTons, 
 14(7, ^qfs. I gib. how many pounds f 
 T. C. Qrs. lb. 
 2 14 3 19 
 20 
 54 C. weight. 
 
 4 
 
 219 Qrs. 
 28 
 
 1771 
 
 438 
 
 61 5 1 Pounds. I 7950476 Thirds. 
 
 An explanation of the firft Ex. will make all the reft plain. Since 
 pounds is the greateft name in the given number, and an unit thereof 
 contains 20 of the next lefs name, or (hillings ; therefore multiply the 
 pounds by 20, faying O times 3 is 0, to which adding the 4 in the 14s. 
 makes 4 ; then 2 times 3 is 6, and the one, in the place of tens in the 
 (hillings, makes 7 ; then 2 times 2 is 4 : Now multiply 474s. by i2, 
 faying 12 times 4 is 48, and the 6 in the pence makes 54 ; write 4 and 
 carry 5 ; then 12 times 7 is 84 and 5 is 89, ^c. Laftly, multiply the 
 5694 pence by 4, faying 4 times 4 is 16, and the two farthings in the 
 jj^iven number is 18 j write 8 and carry i, ^c, 
 
 36. Case
 
 Book T. 
 
 JPti I T H M E TICK. 
 
 15 
 
 36. Case II. y^ number of an inferior name being given ; to find how 
 many of each fuperior denomination are contained in it. 
 
 Rule ift. Divide the given number, by the number of times that 
 one of its units is contained in an unit of the next fuperior name. 
 
 2d. Divide this Quotient by the parts malcing one of the next name. 
 
 3d. Divide this Quotient by the parts making one of the next name : 
 And proceed in |his manner, until the higheft name is obtained. 
 
 4th. Then the laft Qiiotient, and the feveral remainders, will be the 
 parts of the different names contained in the given number. 
 
 Ex. I. In 22778 farthings, how 
 many pounds, /hillings, and pence P 
 4)22778( z Farthings. 
 
 12) ^6g\{ 6 Pence. 
 
 2,0) 47, 4( 14 Sbillipgs. 
 
 23 Pounds. 
 Anfwcr 2-^ . i\s. 61^. 
 
 Ex. III. ^In 61 51 pounds, how many 
 T'ons, Hundreds, ^tarters. Pounds ? 
 28)6151(219 
 56 
 
 55 
 
 28 
 
 271 
 252 
 
 4)2i9( 3 Qrs. 
 2,o)5,4( 14 C. 
 2 Tons. 
 
 19 Pounds. 
 Anfwer 2T. 14C. 3Qxs. iglb. 
 
 Ex. II. In TcjSOJi^-jb thirds.of a.dt 
 greeyhov^ ?nany / ^^ ^'^:?. ^ 
 6,o)795047,6( 56TJliiTds.,i - 
 
 6,o)i32co,7( 27 Secondg. 
 
 6,0) 220,8 ( 48 Minutes, ' ;. 
 
 36 Degrees. "' ' 
 Anfwer 36, 48'. 27". 56'". ' 
 
 Ex. IV. In ^oSSo grains,'how m:aHy 
 Pounds, OunceSy Pennyweights, Grs, 
 24)50880(2120 
 
 48 
 
 28 
 24 
 
 2,0)212,0(0 dwt. 
 12) io6( 10 oz. 
 
 48 8 lb. 
 
 4 
 
 Anfwer 8lb. 10 oz. 
 
 Explanation of Ex. I. Since 4 of the g;ven number make one of the 
 next name, pence, then 22778 divided by 4, give 5694 pence, and a 
 Remainder of 2 farthings ; then 5694 pence divided by 12, the number 
 of pence in one of the next name, fhillings, the Qi_iotient is 474 fhil- 
 Hngs, and a Remainder of 6 pence ; then 474 fhillings divided by 20, 
 the number of fhillings in one of the next name, pounds, the Quotient 
 is 23 pounds, and a Remainder of 14 fhillings. And by the 4th pre- 
 cept, the anfwer is col'.ccled. 
 
 A like operation -will folvc the other examples, having regard to the 
 increafe of the different names. 
 
 37. In any Divifion, if the Divifor has one or more cyphers on the 
 right hand, thofe cypheis may be pointed off"; but then as manv places 
 mult be pointed off from the Dividend, wliicli places arc not to be di- 
 vided, but annexed to the r4;^ht hand of the Remainder. See the above 
 xamples. 38. Casf
 
 |6 
 
 A R I T H M E T I C K. 
 
 Book I. 
 
 38. Casb III. Ta reduce a vulgar fraSlion to its equivalent decimal 
 
 fraSlion. 
 
 Rule. To the Numerator annex one or more cyphers, divide this 
 by the Denominator, and the Quotient will be the fraction fought. 
 
 If the Divifion does not end when fix figures are found in the Quo- 
 ^ent, the work need not be carried any farther. 
 
 Exam. I. To reduce ^Vt io f*^ equivalent decimal /ration. 
 
 Here 423 the Denominator is 
 mAde the Divifor, and 15 the 
 Numerator is fet for the Dividend, 
 to which annexing a cypher or two 
 for fra^ional places, feek how often 
 the Divifor can be had in 15, the 
 integral part of the Dividend ; and 
 as it cannot be taken, put o in the 
 Quotient for the place of units : 
 Then taking in one fractional place, 
 feek how oft the Divifor can be had 
 in 150, fay o times, and put another 
 O in the Quotient for the place of 
 primes: Now taking in two fra<9:ional places to the 15, the Divifor 
 will be contained in it thrice, and thus proceed until the Divifion ends, 
 or till 6 places arife in the Qotient : But in this example,^ as the 6th 
 place would be o, it is omitted, becaufe cyphers on the right hand of 
 decimal fractions are of no fignification, as will evidently appear, No- 
 tation of Fraftions being well underftood. 
 
 423)15,00(0,03546 
 1269 
 
 2310 
 2115 
 
 1950 
 1692 
 
 2580 
 2538^ 
 
 420 
 
 Ex. II. Reduce \ to a decimal 
 
 fraifion, 
 
 2)1,0(0,5 Anfwer. 
 
 Ex. IV. Reduce ^ to a decimal 
 
 fraSfion. 
 
 4)3,00(0,75 Anfwer. 
 
 Ex. VI. Reduce y to a decimal 
 
 /ration. 
 
 3)1,00(0,33, &c. Anfwer, 
 
 Ex. III. Reduce \ to a decimal 
 
 frail ion. 
 
 4)1,00(0,25 Anfwer. 
 
 Ex. V. Reduce ^ to a decimal 
 
 fraSiion. 
 
 8)5,000(0,625 Anfwer. 
 
 Ex. VII. Reduce -l^ to a decimal 
 
 fraSiion, 
 12)7,0000(0,5833 &c. Anfw. 
 
 ^ 39. In the two laft Quotients, it may be obferved, that 3 would con 
 tinually arife j fuch decimal fradions are called circulating, or recur- 
 ring fradions : Thefe have a peculiar kind of operation belonging to 
 them, which the inquifitivc reader will find in a book intitled A Gene- 
 ral Treatife of Menfuration*, the third edition, publifhed in the year 
 1767 ; and alfo in other books. 
 
 ^y the Aathor of thefe Elementi. 
 
 40. Case
 
 Book L 
 
 A R I T H M E T I C K. 
 
 17 
 
 40. Case IV. To reduce a number conjijiing of different names, to a dc" 
 cimal fraSiion of iu greatejl name. 
 
 Rule rft. Write the given names orderly under one another, the leaft 
 name being uppermoft j and on their left fide draw a line : Let thefe be 
 reckoned as Dividends. 
 
 2d. Againft each name^ on the left hand, write the number making 
 one of its next fuperior name : And let thefe be the Divifors to the for- 
 mer Dividends. 
 
 3d. Begin with the upper one, and write the Quotient of each divifion 
 as fractions, on the right of the Dividend next below it j then let this 
 mixed number be divided by its Divifor, &c. 
 
 And the laft Quotient will be the decimal fraction fought. 
 
 Ex. I. Reduce 15X. 9!^. to the fra^ional part of a pcundj^erltng* 
 
 3 
 
 9.75 
 
 15.8125 
 
 0,790625 
 
 Firft fet the three farthings, the 9 penccj the 15 Ihillings 4 
 and O pounds under one another; and againlt the far- 12 
 things fet 4, againft the pence fet 12, and againft the 20 
 (hillings, 20 ; then the three with cyphers fuppofed to be 
 annexed, being divided by 4, the Quotient ,75 is written on 
 the right hand of the 9 pence; and the mixed number 9,75 with cyphers 
 annexed as they are wanted, being divided by 12, the Quotient ,8125 is 
 written on the right hand of the 15 j. then this mixed number 15,8125 
 being divided by 20, the Quotient 0,790625^^. is the anfwer. 
 
 Ex. II. Reduce is. 2\d. to the frac- 
 tional part of a pound Jitrling. 
 
 12 2,25 
 20 1,1875 
 |o05937; 
 Anfwer \t. 2$<i.=:o,059375j^. 
 
 Ex. IV. Reduce^oz. isdivt. i^gr. 
 to the fractional part of a pound troy. 
 
 \V 
 
 24' 
 
 20 
 12 
 
 (4.5 
 '5'75 
 
 o. 7^2291 
 
 Anf. 80Z. I5dwt. 18 grrro, 7322911b. 
 
 4.1. Here becaufe 24 is a number 
 too great to divide by in one line, 
 therefore it is broken into the parts 
 4 and 6, which multiplied together 
 make 24. 
 
 Ex. III. Reduce 48^ if'. Sl'^'- to 
 the fractional part of a degree % 
 
 60 
 
 60 
 60 
 
 53 
 
 17^883333 
 
 48,298055 
 
 0,804967 
 
 Anfw. 4S'. 17". 53". =0,804967 Deg 
 
 Ex. V. Reduce "^qrs. 19/^. 140Z. to 
 
 the fractional part of a C weights 
 
 14 
 
 (3,5 
 19,875 
 
 (496^75 
 3,7098.:! 
 
 0,9271?!; 
 
 1 
 
 23 
 
 4 
 
 Anfw. 3qr^ iplb. 140^=^,927455 C. 
 
 Here the 16 is broken into the 
 numbers 4 and 4 ; and 28 into 4 
 and 7 J and 14 is divided by 4 j and 
 the Quotient 3,5 by 4, &c. 
 
 42. Case
 
 la 
 
 A R I T H MET I C K. 
 
 Book t; 
 
 .42. Case V, To reduce a decimal /ration of a fup trior namey ib^its 
 value in inferior denominations, 
 
 : Rule ift.- Multiply the given fraftion by the number that en unit of 
 its name contains units of the next lefler name ; from the right hand' 6f 
 the Produdt point off as many places as there are in the given fradtion. 
 
 2d. Multiply the places, fo pointed off, by as many as an unit of this 
 name, contains of the next lefs name ; point off as before. 
 
 And thus proceed until the multiplication is made by the leaft name. 
 
 3d. Then the integers, or the numbers on the left of the diftinguifli- 
 ing marks in each Produ6t, will be the parts in each name, which to- 
 gether are equal to the given fradtion. 
 
 Example I. JVJjat number cf/hilUngs, pence, and farthings^ are equal in 
 value to Oy'jgo625,Jierling. 
 
 ' 
 
 Here an unit of the given name . 
 contains 20 of the next lefs name, 
 fhillings ; then multiplying by 20, 
 and pointing off 6 places on the 
 right, becaufe the given number 
 0,790625 contains 6 fradional pla- 
 ces, the Product is 15,812500 {hil- 
 lings J then the fractions of this 
 number, viz, 81 2500 multiplied by 
 12, the number that an unit of this 
 name contains of the next lefs name, 
 
 and the produ6l pointed as before, there arifes 9,750000 pence; the 
 fraftions of this number multiplied by 4, gives 3,000000 farthings; then 
 the parts pointed off on the left, viz, 15 s. 9-|d. are the value of the given 
 fraction. 
 
 0,790625 
 20 
 
 15,812500 
 12 
 
 9,750000 
 4 
 
 far. 3,ooc'ooo 
 
 Example II. IFhat is the value of 
 0,056285^. Jlerling ? 
 ^, 0,056285 
 20 
 
 s. i,i257Joo 
 12 
 
 d. 1,5084 
 4 
 
 far. 2,0336 
 
 Example IV. TFhat is th^ value of 
 0,732291//'. troy? 
 This example worked as above, 
 by multiplying by 12, 20, 24, the 
 value will be found to be 
 
 80Z. I5dwts. j8gr. nearly. 
 
 Example III. What is the value of 
 0,58695 degrees ? 
 Deg. 0,58695 
 60 
 
 Min. 35,217100 
 60 
 
 Sec. i3,02|o 
 60 
 
 Thirds 0,120 
 
 Example V. What is the value of 
 0,927455 part of a C. weight? 
 
 By operating as above, multiply- 
 ing by 4, 28, 16, the anfwer will 
 be 
 
 3qrs, 191b. 140Z. nearly. 
 
 43. Qj^EST,
 
 Book I. 
 
 ARITHMETIC K. 
 
 r^ 
 
 43- 
 
 Questions to exercife the preceding rules. 
 
 Quest. I. A Jkop with the cap- 
 tain and 26 hands take a prize which 
 fold for iSl^' of which each feaman 
 had \SL' ^'^^ '^^ captain the rejl : 
 How much was his Jhari ? 
 26 Men 
 ^^C' to each 
 
 fubtraa 
 from 
 
 130 
 104 
 
 iijOjC. the crew's fhare. 
 1578^^. the whole prize. 
 
 remains 408 /^. the captain's ftiare. 
 
 Quest. III. J feaman^ luhofe 
 wages are 35 J* dd. a months returns 
 heme at the erid of 29 months ; he hav- 
 ing taken up 12 f^. i8y. ; How much 
 has he to receive ? 
 s. d. 
 
 35 6 
 mult, by 12 
 
 426 pence a month, 
 mult, by 29 months, 
 
 3834 
 852 
 
 12) 12354 pence 
 
 2,0) 102,9 bd, 
 
 from 51;^- 9'- 6a'. ~ wages, 
 take 12^. 18s. cd. received. 
 
 remains 38 j^. i u. 6(/. to receive, 
 
 Quest. V. In 30b crowns^ how 
 
 many half crowns and pence ? 
 A r S 612 half crowns. 
 I 18360 pence. 
 
 Q_t;EST. VII. A feaman s fmre of 
 
 a p7i2.e ivas \\ guineas, 32 moidorcsy 
 
 12 thirty -fix /hillings pieces, and ^2 
 
 pifloks at i-] s. each: Hoxv much 
 
 Jhrling did the whole cofne to ? 
 
 Anfwer 123 ^. i\s. 
 
 C 
 
 Quest. II. A boat's crew of 1^ 
 men got by plunder 321 , Heiv muck 
 was the Jhare of each ? 
 
 5)32i(2i;C- 
 SO 
 
 21 
 IS 
 
 remains 6 ^. which 
 mult, by 20J. in 1 ,* 
 
 15)1201.(8/. 
 120 
 
 Anfwer 21^. %s. to each. 
 
 Quest. IV, Six mefs-matesy who 
 propofe to live well du?'ing an Eajl^ 
 India voyage of 22 ?nonths, agree to 
 expend among them $s. a day, hefides 
 the fhip' s allowance : Nozu one of 
 them having but 2$s. a month, hovj 
 will matters fland with him at the 
 end of the voyage ^ 
 
 Now 28 days, at 5 s. a day, makeS 
 i\cs. or 7j^. a month j which for 
 22 months, is i^^f,' 
 
 Then a fixth part of 1$^^^' ^S 
 25^. 13^, 4^. for each man. 
 
 Alfo 25 J. a month for 22 months 
 makes 27 j^. loj. for wages ; which 
 will overpay his expences, by i^, 
 lbs. 8d. 
 
 Quest. VI. In 30 chalders cf 
 coals, each of 36 hujhcls, how many 
 pecks ? 
 
 Anfwer 4320 pecks. 
 
 Quest. VIII. Suppofe a fip faih 
 
 5-|- miles an hour for 14 days : Hozo 
 
 7nany degrees and minutes has fve 
 
 failed in the %vhole ; 60 fa iniles 
 
 making one degree ? 
 
 Anfwer 30 cic";. 48 niin. 
 
 2 SEC-
 
 t9 A R I T H M E T I C K. Book tv 
 
 SECTION VII. Of PROPORTION: 
 Or, THE RULE OF THREE. 
 
 44. Four numbers arc faid to be proportional, when by comparing 
 them together by two and two, they either give equal Produ<Sts or equal 
 Quotients. 
 
 Suppofe thefe four numbers 3 8 12 32 
 
 In comparing them together by multiplication, 
 
 The Produd of 3 and 8 is 24 ; of 12 and 32 is 384, unequal, 
 of 3 and 12 is 36} of 8 and 32 is 256, unequal, 
 of 3 and 32 is 96 ; of 8 and 12 is 96, equal. 
 
 Therefore 3 8 12 32, are called proportional numbers. 
 
 Now let them be compared together by divifion. 
 
 The Quotient of 8 by 3 is 2,6$3'<r. of 32 by 12 is 2,6 &t;. equal, 
 of 12 by 3 is 4 of 32 by 8 is 4, equal, 
 
 of 32 by 3 is io,6^c. of 12 by 8 is 1,5, unequal. 
 
 Therefore by this comparifon, the numbers are faid to be proportional. 
 
 In this kind of comparing four numbers together, there is no need 
 to try for more equal Products, or Quotients, than one fet of either 
 fort^ for either cafe will determine the proportionality independent of the 
 other. 
 
 But it muft be obferved, that among four proportional numbers, there 
 v^jll be but one fet of equal Products, and two fets of equal Quotients, 
 the fmaller numbers being Divifors. 
 
 45. When four numbers are to be written as proportionals, they muft 
 be placed in fuch order, that the Produdl of the firft and fourth be equal 
 to the Product of the fecond and third. 
 
 A queftion is faid to belong to the Rule of Three, when three numbers 
 or terms are given to find a fourth proportional, which is the anfwer to 
 the queftion. 
 
 And in order to refolve fuch queftions, the three given terms muft be 
 firft placed in a proper order, which is called ftating the terms of the 
 queftion. 
 
 46. Qi^ieftions in the Rule cf Three are ftated, and refolved by the fol- 
 lowing precepts. 
 
 I ft. Confidcr of what kind the fourth term, or number fought, will be, 
 whether money, weight, meafure, time, If^c. and among the three num- 
 bers given in the queftion let that which is of the fame kind with what is 
 required be placed for the third term. 
 
 2d. From the nature of the queftion, determine whether the number 
 fought will bc.grcatcr or lefs than the number which is placed for the third 
 term. 
 
 3<^. If
 
 Book I. A R I T H M E T I C K. ai 
 
 3d. If the fourth term will be greater than the third, fet the greater of 
 the remaining two terms for the fecond, and the lefs for the firlh 
 
 But if the fourth term is to be lefs than the third, fet the greater of the 
 remaining two terms for the firft, and the lefs for the fecond. 
 
 Then in either cafe, the given three terms are ftated. 
 
 4th. Reduce thofe terms which confift of more names than one, to or>e 
 name ; and obferve that the firft and fecond terms are always to be of the 
 fame name. 
 
 5th. Multiply the fecond and third terms together, divide the produd 
 by the firft term, and the Quotient will be the fourth term, of the fame 
 name the third term was reduced to. 
 
 47. Quest. I. If 6^ yards of cloth coji i8j. what will i\ yards cojl? 
 
 Here it is plain, that the term 
 fought, or the worth of 24 yards, 
 
 will be money ; therefore the given yds* yds* s. 
 
 money 1 8 x. is fet for the third term ; j^' z\ 18 
 
 and as the worth of 24 yards mufl 8 
 
 be greater :han the worth of 4 yards, 
 
 therefore the 24 is fet for the 2d '9^ 
 
 term, and the 4 for the ift. Then ^'^ 
 
 the 2d term 24 being multiplied by ^ ^^ ,^g ^-^y 
 
 the 3d, 18, the Produdl is 432, _Il_ 
 
 which divided by the ift term 4, the 2^0) 10,8 fhillings. 
 
 Quotient or 4th term is 108, which .._ 
 
 are fhillings, the fame name of the 5 pounds. 
 
 3d term; then 108 fhillings divided 
 
 by 20, gives s' 8^. Anfwer 5^. 8/. 
 
 Quest. II. If I lend 200 . for 12 months^ how long ought I to have th 
 ufe of \ SOjC. to recoTnpence me ? 
 
 Here the anfwcr or 4th term is to C' ' 
 
 be time; therefore let 12 months, '50 200 12 
 
 the given time, be fet for the 3d '2 
 
 term : Now it is evident, that the 
 
 150/. being lefs than the 200^. 15,0)240,0(16 months. 
 
 mufi be kept a longer time, and fo ^^ 
 
 the 4th term will be greater than the 
 
 3d term : Therefore the 200 is put ^^ 
 
 for the 2d term, and the 150 for 
 
 the ift. Then the 2d term multi- Anfwer 16 months. 
 
 plied by the 3d, the Product will be 
 
 2400 ; which being divided by the ift term, the Q^iotient 16 is the 
 
 4th term ; and becaufe the 3d term was months, the 4th term will be 
 
 months. 
 
 Quest
 
 n 
 
 ARITHMETIC K. 
 
 Book I. 
 
 Quest. III. ff^jnt will 1836/*. 
 f raijins come to, at the rate of 6 s. 
 Bd./or ^Alb. ? 
 
 Here as money is the thing 
 fought, money muft be the 3d term : 
 And ^s 6 J. Sd. confifls of two 
 names, they muft be reduced to one 
 name, viz. pence. 
 it. lb. s. d. 
 
 ^^,..1836 -6 8 
 
 1836 
 So 
 
 6 
 12 
 
 8od'.r:3d term. 
 24)i4688o(6i20dr. r:4th term. 
 
 144 . 
 
 12)6120 
 
 Quest. V. Wbat'xvill 420 ^ards 
 of cloth come to, at i\s. 10 \d. for 
 I ell Englijh ? 
 
 The term fought being money, 
 the 14^. 10 J, d. muft be the 3d term, 
 and be reduced to farthings ; alfo 
 the ift and id terms are to be re- 
 duced to quarters of a yard. 
 Ell Eng. yds. /. , d. 
 
 1 420 1 4 
 
 420 
 
 4 
 
 1680 
 
 7'? 
 
 14 
 12 
 
 2Sk 
 io4 
 
 28 
 24 
 
 48 
 
 2,0)51,0 ( 10.'. 
 
 Here the 2d term being multi- 
 plied by the 3d, and the Product di- 
 vided by the firft, the quotient is 
 6120 pence; which being valued, 
 gives 2SL' ^0^* 
 
 Quest. IV. If 20 yards of cloth ^ 
 5 quarters wide, will ferve to hatig a 
 room: How many yards of \ quarters 
 wide vjHI feme to hang the fame 
 room ? 
 
 Here yards of length are required ; 
 then 20 yards muft be the ^d term. 
 qrs. qrs. yds, 
 
 45 20 
 
 5 
 
 4)ioo( 
 Anfwer 25 yards. 
 
 1.78 
 4 
 
 84CO 1 71'; far.=3d terra. 
 
 1680 ^ . , _ 
 11760 ^ 
 
 5)l20I2CO{ 
 
 4)240240 farthiBgsr=4th term. 
 
 12)60060 pence. 
 
 2,0)500,5 5 Ihillings. 
 
 250 pounds. 
 Anfwer 250;^. 5/. 
 The Divifor 5 being a fmgle digit, 
 the Quot. is written under the Divid, 
 
 Quest. VI. A owes to B A^zL' 
 hut compounds for Js. 6d. in the 
 pound: How much mujl B receive 
 for his debt ? 
 
 Here compofition money is the 
 thing fought; then the 3d term 
 muft be the compofition money, 
 w'z. "jS. bd. 
 
 ' j- 
 
 1 463 7 6 
 
 90 12 
 
 12)41670(6^. 90 pence. 
 
 2,0 ) 347 '2 (12/. 
 
 Anfwer \'}l,' 12s. 6d. 
 
 48. As it will be more convenient in moft cafes to reduce fuch num- 
 bers, or terms, which confift of feveral names, to the fraaional parts of 
 their greateft name, than to reduce them to their loweft name; therefore 
 in the folution of fome of the following queftions, the inferior parts of the 
 given terms are reduced by Cafe IV. of Rcduaion ; and the anfwers are 
 vduedbyCafeV. Q-U^^T.
 
 Book I. 
 
 A R I T H M E T I C K. 
 
 '^3 
 
 Quest. VII. If Hb. of pepper 
 (o/i 4 J. $d. : What will 7 G. '^qrs. 
 . 1 4 lb. come ta at that rate ?- 
 
 Ih. 
 
 C. 
 
 qrs. lb. s. 
 
 8- 
 
 7 
 
 3 H ^4 
 
 
 4 
 
 12 
 
 
 31 
 z3 
 
 50 
 882lb. 
 
 
 262 
 
 56 
 
 
 62' 
 
 5292 
 
 
 9.9.-> 
 
 .. 44'o 
 
 
 
 8)49392 
 12)6174 bd. 
 
 
 2,0)51,4 14/. 
 
 d. 
 
 25 o 
 
 Anfwer 25;^- 14 j 6^. 
 
 QjJEST. IX. /^^^^ is the inter eji 
 5/ ^"^^f^. for a year^ at 5 /)^r cent, 
 per annum: Or at the rate of $f. 
 for the ufe of 1 00 . for a year f 
 
 HeTe intereft is the term required ; 
 therefore s- the intereft of 100^. 
 is to be the 3d term : And as the 
 4th term, or the intereft of 584;^. 
 is greater' than the 3d term ; then 
 the 2d term is to be greater than the 
 ift. 
 
 ' ' ' 
 100 5S4 5 
 
 5 
 
 i,oo)29,2o( See Cnfc V. of 
 . Reduction. 
 20 
 
 4,00 
 
 Anfwer 29^. 4/. 
 
 Quest. XI. if^hat is the inter eJi 
 */542;^. lOs.for 219 days, at $ 
 per cent, per annum ? 
 
 To folvc this qucftion, find the 
 intereft for i year ; multiply this in- 
 tereft by 2ig, and divide the Pro- 
 duct by 365, the Quotient will be 
 the anfwer J anu is 16^^. 5;. 6c/. 
 
 QuST. VIII. One bought 4 
 Hhds, of fugar^ each containing 6 G, 
 iqrs. \\lb. at ^f. 8j, bd. for each 
 C. weight : TV hat did the whole come 
 to? 
 
 C, C. qrs. lb. . s. d. 
 1 6 2 14 286 
 
 Now I C. weight is ii2lb. 
 
 And 4 Hh. at 6C. 2q. I4lb=: 29681b. 
 
 Alfo 2. &s. 6d. is 582 J. 
 
 Then the Produ6l of the 2d and 3d 
 
 terms is 1727376. 
 
 Which divided by the ift term 112, 
 
 the Quotient is 15423 pence, whofe 
 
 value is 64^. fy. ^d. 
 
 Quest. X, fVhat is the inter ejl 
 of '^1 ' 12 s. for three years and 4 
 months^ at ^\ per cent, per annum ? 
 
 Find the intereft for i yearj then 
 thrice that, together with y of one 
 year, will be the intereft fought. 
 
 : ' r. 
 
 100 387,6 3,5 . 
 
 3.5 
 
 19380 
 1 1628 
 
 100) I 556, 6o( 
 
 13,506 for I year. 
 3 
 
 40,^:98 far 3 years. 
 -} of I year r:4.i;2 2 for 4 months. 
 The fum 45,220 is the intereft. 
 
 Anfwer 45 . 4/. ^d. 
 Qttest. XII. For how lo7tg mufl 
 487/. 10 1, he at fvnble- inter cj}, at 
 \\L' P*-"^ ^^"^* P^^' "^^-^'^f^* i<^ gain 
 
 I'ind what will he the intereft of 
 487^. 10 s. for I year; divide C)S' 
 \s. 3rt. by this intereft, and the 
 Qiioticnt will be 4! \'cais. 
 
 C 4 
 
 QUESV.
 
 24 
 
 A R I T H M E T I C K. 
 
 Book I. 
 
 % 
 
 Quest. XIII One bought l^ pipes 
 tfuuffte, and is allowed 6 months cre- 
 dit: But for ready money gets it 6d. 
 in a gallon cheaper ; How much did 
 hefave by paying ready money ? 
 Anfwcr 44;^* 2 * 
 
 Quest. XV. One bought 3 tons 
 f oil for I53;C' 9^* "^hich having 
 leaked 74 gallons ^ he would make the 
 frime-coji of the remainder : How 
 mu/l it be fold per gallon ? 
 Now I T. =25 2 Gall. And 3T.=756 
 SabtraA the gallons leaked = 74 
 
 Remains 682 
 
 G. G. . 
 
 Then 682 i 153,45 
 
 Anfwcr 4/. 6</. a gallon. 
 
 Quest. XVII. Ji 13/. for 
 
 JQOlh. of goods : What will^^$lb . 
 come tOy allawing ^Ib. upon (very 
 ZOO lb. for tret^ or wajle? 
 
 Since 41b. is to be allowed on the 
 100 lb. therefore rC4lb. is given 
 for 100. 
 
 lb. lb. /. 
 
 Then 104 895 13 
 
 Anfwer ixij^* 17 J. 6/. 
 
 Quest. XIX. If 100 pounds of 
 Jugar be worth 36 j. ^d. JVhat will 
 he the worth of Z']c^lb, rebating ^Ib. 
 upon every 100 lb. for tare ? 
 
 Here the buyer has 100 lb. on 
 paying for 961b. 
 
 lb. lb. lb. 
 
 Then loo : 96 -r ^875 
 And the 4th term will be 8401b. 
 Alfo 100 840- 11833333 
 
 Then the 4th term will be 15^^. 
 ?/. and fp much will the fugar come 
 to. 
 
 QjJEST. XIV. A clothier feld $0 
 pieces of kerfey^ each piece containing 
 34 ells Flemtjh^ at the rate of 8 s. 
 4</. per ell Englijh : What did the 
 whole come to ? 
 
 Anfwer 425 jf. 
 
 Quest. XVI. A broker fold^ of I 
 ofajhipfor 147 /. lis, '^d. ': How 
 much was the whole Jhip valued at ? 
 
 Nowfof|=^^|=^5--r'5byart.38. 
 For 2X3=6, a new numerator. 
 And 5 X4=:20, a new denominator. 
 Alfo 147 i^. lis. 3 </.= 147,5625. art. 
 40. 
 
 fhare fli^re . 
 
 Then 0,3 1 147,5635 art. 46. 
 
 Anfwer 49 1 j^. ijs. 6d. 
 
 Quest. XVIII. One has cloth 
 which co_^ 2 s. Sd. a yard : For how 
 much muji it be fold a yard on 3 
 months credit.^ to gain 25 f^. per cent, 
 per annum ? 
 
 mon. mon. j^, J^, 
 Firft 12 3 25 6,25 
 
 Secondly loo 6,25 0,133333 
 
 By multiplying and dividing, the 4th 
 term will be found zd. 
 Then zs. %d,-\-zd.-=;zs. |0</. ayard, 
 the felling price. 
 
 Quest. XX. A chapman bought 
 81 kerfeysfor 135;^. ; How mufi be 
 fell them per piece to gain i^, per 
 cent ? 
 
 Plnd how much 135;^. will be 
 advanced to, at is. per cent. 
 
 Then this fum divided by 81 will 
 be the felling price of each piece, 
 
 XT ^- ^' ^' ' 
 
 Now ICQ T 1 15 135 ' 155,25 
 Then 81) 155,25 (1,916666^. 
 Anfwer i. \2f, 4^/. a-piece. 
 
 (^PST,
 
 Book I. 
 
 ARITHMETIC K. 
 
 25 
 
 Quest. XXI, A merchant who \ Quest. XXII. One who had fold 
 is to receive a fum of money, is offered^a parcel of cloths at 2s. lod. a yard on 
 
 ducats at 6s. \d. which are worth but 
 6s. 2\d. or chequins at 8^. 2d. each, 
 that are worth but 8 j ; By which fpecie 
 will he fujiain the leaji lofs ? 
 
 6s. 2|^.=74, 5^. 7 
 g/. o d.-=.^6 J 
 6s. 4 d.'=z-j6 1 
 8i. 2 d.g% I 
 
 the real value, 
 the advan. val. 
 
 r. val. 
 Then 74,5- 
 
 r. val. ad. val. ad. val. 
 
 96 76 97,93 
 
 But the chequins are- valued at 98 d. 
 Therefore the ducats are molt advan- 
 tageous. 
 
 Quest. XXIII. Aperfon wants 
 750 pieces of foreign coin, each worth 
 1 1 J. \d. How mu^h will they come to, 
 allowing the broker the worth of 2 
 pieces upon every \Q0? 
 
 Now 100 102 750'~--"765- 
 
 He muft pay for 765 pieces, which 
 will come to 433 ^. \os. 
 
 Quest. XXV. A grocer bought 
 \\ ^- of pepper for isf. IV- 4^- 
 xuhich proving to he damaged, he is 
 willing to lofe I2\f^. per cent. For 
 hew -much jnujl he fell it a lb. ? 
 
 Since he Is to lofe i2f per cent, he 
 murt: take Hy^iT. \cs. for "S-Oof^. Now 
 diminifh the 15^. lyx. :^d. in this 
 proportion, and this fum divided by 
 the pounds in 4I C. will give 7^. for 
 what each pound is to be fold at. 
 
 3 months credit, found he had gained 
 '^S' per "nt. per annum : IVhat did 
 the cloth cojl per yard? 
 
 mo. . mo. J[^. 
 
 Now 12 25 -3 6,25 
 
 And ioo4-6,25 =: 106,25 j^. 
 
 ^. ^' ^' ^' 
 
 Then 106,25 1000,14166 
 
 The fourth term to which will be a 
 fraftion, the value of which will be 
 zs. 8d. which is the prime coil per 
 yard of the cloth. 
 
 Quest. XXIV. A gentleman 
 would exchange J2q pieces of ^s. 2d. 
 each into Jlerling money : Jiow Jnuch 
 will he receive for them, allowing the 
 broker i\f,. per cent. ? 
 
 P. P. C ' 
 
 Now I 729 0,208333 iS*>^75 
 the worth of the pieces. 
 Then 10 1,25 IOC 15 1,875-- 150;^. 
 He will receive iS- ^'^'^ them. 
 
 Quest. XXV^I. Suppofe \2 gal- 
 lons of honey be valued at 2^. and the 
 duty is is,- per cent, en this value, 
 and a drawback of ^. per cent, on 
 the duty for prompt payment : IVhat 
 will the ready money duty of b']2 gal- 
 lons come to ? 
 
 Now 42G. 
 
 : 672G. : 
 
 : 2. 
 
 3^C 
 
 And looj^. 
 
 : ^SC- 
 
 : r- 
 
 : 4M- 
 
 Alfo ioo2^. 
 
 9s--- 
 
 h^C 
 
 : 4.5 6>C 
 
 Anfwcr 4jC >i^ 2ld. 
 
 The Rule of Proportion is of almoft unlverfal ufc in all bufinefs where 
 coinputation is required ; as in buying and felling, values of flocks and 
 th(r!r dividends ; the intereft and difcount of money ; the cufloms and 
 duties on goods, (:fc. But the dcfigned brevity of this book will not per- 
 p^it farther illufUations, 
 
 SEC,
 
 t6 A R I T H M E T I C K. Book I. 
 
 SECTION VIII. OF THE POWERS OF J^JUMfepRS, 
 AND OF THEIR ROOTS. .,'.:;, 
 
 49, The Power of a number j is a produ5i arifing by multiplying 
 
 that number by itfelfy the froduEl by the fame number ^ this froduSl 
 
 by thejame number again, ^c. to any number of multiplications. 
 
 . . -, I f , . - 
 
 50. The given number is called the firft power or root. -> 
 
 The Produl of the ift power by itfelf, is the fecond power, or fquarc. 
 The Produdt of the ?d power by the ift^ is the 3d power^ or cube. 
 The Produ(3 of the 3d power by the ift, is the 4th power, &c. 
 
 51. Here follow the ift, 2d, and 3d powers of the nine digits. 
 Roots, or I ft power 1234 5 6 7 8 9 
 Squares, or 2d power 14 916 25 36 49 64 81 
 Cubes, or 3d power i 8 27 64 125 2r6 343 512 729 
 
 Ex. I. JVhat is the 2d power, or 
 
 Ex. II. What is the "^d power, or 
 cube of 38 .? 
 
 Now 38X38^:1444 the 2d power. 
 Then 1444X38 = 54872 the 3d power. 
 
 fqtiare of the number 24 ? 
 
 24X24=576 is the 2d power. 
 
 The figure, or number, fhewing the name of any power, is called the 
 index of that power. . 
 
 Thus I is the index of the firft power : 2 is the index of the 2d power ; 
 3 of the third power, &c. Alfo i is the index of the fquare root j 4> the 
 index of the cube root, tifc. 
 
 52. Any number may be confidered as a power of fome other number. 
 Thus 64 may be taken as the 2d power of 8, and the third power of 
 
 53. The root of a given number, confidered as a power, is a number 
 which being raifed to the index of that power, will either be equal to the 
 given number, or approach very near to it. 
 
 54. 'To extras the Square Root of a given number. 
 
 Rule ift. Begin at the unit's place, put a point over it, and ajfo over 
 every next figure but one, reckoning to the left for integers, and to the 
 right for fractions j and there will be as many integer places in the root, 
 as there are points over the integers in the given number. 
 
 The figure under a point, with its left-hand place, is called a period. 
 
 2d. Under the left-hand period write the greatefl fquare contained in it, 
 and fet the root thereof in the Q^iotient; fubtradl the fquare, and to the 
 remainder bring down the next period, as in Divifion. 
 
 3d. On the left of this Remainder write the double of the Root or 
 Quotient for a Divifor ; feek how often this may be had in the Remainder, 
 except the right-hand place j write what arifeth both in the Root, and on 
 the right of the Divifor. 
 
 4th. Multiply this increafed Divifor by the lall Quotient-<figure ; fub- 
 tra6l:, and to the Remainder bring down the next period j double the 
 Root for a Divifor, and proceed as before. 
 
 55. Fradtional places will arife in the Root, by annexing to the Re- 
 rnamders, periods of two cyphers each, and renewing -the operation. 
 
 2 ' x<
 
 Book I, 
 
 ARTTHMETICK. 
 
 ay 
 
 1444 (3^ Root. 
 9 
 
 Ex. I. What is the Square Root o/ij^/^f 
 Put a point over the units place 4, and alfo 
 over the place of ipos. Now the number con- 
 fills of ;i periods, and will have 2 integer places 
 in the Root : Then the greateft Square in 14, 
 the left-hand period, is 9, and its Root is 3 ; 
 write 9 under the period, and three in the Root ^ 
 
 now 9 from 14 leaves 5, to which annex the 
 
 next period 44 ; the Root 3 doubled makes 6, 
 
 which in 54 is contained 8 times, annex 8 to the 3 in the Quotient, and 
 to the Divifor 6, makes the Root 38 and the Divifor 68 j then 8 times 68 
 is 544 ; and there remaining 0, on fubtra<Slion, it may be concluded, 
 that 38 is the true Root. 
 
 68) J44 
 544 
 
 Ex. II. IVhat is the Square Root 
 ?/ 36372961.^ 
 
 36372961 (6031 Root. 
 36 
 
 1203) 3729 
 3609 
 
 12060 12061 
 1 2061 
 
 Ex. IV. Wl}at is the Square Root 
 c/ 2468 1024 F 
 
 Anfwer 4968. 
 
 Ex. VI. What is the Square Root 
 
 ''Z 76395820? 
 
 76395820(8740,4702 
 
 64 
 
 167 ) 1239 
 
 J 169 
 
 1744)7058 
 6976 
 
 Ex. III. What is the Square Root 
 5/1,0609? 3. 
 
 1,0609 C *>3 Root. 
 
 203 ) 0609 
 609 
 
 Ex. V. TVhat is the Square Root 
 5/911236798,794365.? 
 
 Anfwer 30186,699, t^c. 
 
 911236798,794366 ( 30186,6 
 601 ) 1123 
 
 6028 ) 52267 
 
 60366 ) 404398 
 
 174804 ) 822000 
 699216 
 
 17480S7 ) 122784C0 
 1 2236609 
 
 174809402 ) 4 1 79 1 0000 
 349618804 
 
 68291 196 
 
 603726 ) 4220279 
 
 597923 
 
 iffc. 
 
 Here the produds are omitted, the 
 multiplication and fubtr.iiSlion being 
 made in the mind. 
 
 In the Vlth Example, after all the 
 periods given were brought down, 
 there remained 8220, to which a pe- 
 riod of two cyphers was annexed, and 
 the operation renewed, r.nd con- 
 tinued until 4 dtcimal places were 
 obtained in the Root; every period 
 brought down giving one place. 
 
 56. Tf
 
 2 ARITHMETIC K. Book I. 
 
 56. To extras tpe Cube Root of a given Number, 
 
 Rule ift. Over the unit place of the given number put a point, and 
 alfo over every third figure from the unit place, to the left for integers, 
 and to the right for fra<5lions j and the root will have as many integer 
 places, as there are points, or periods, in the integral part of the given 
 number. 
 
 ad. Under the left hand period, write the greateft Cube it contains, 
 the root of which fet in the Quotient: Subtra^ the Cube from the pe- 
 riod, and to the Remainder annex the remaining periods j call this the 
 Refolvend. 
 
 3d. To the Quotient annex as many cyphers as there were periods 
 remaining ; call this the Root. 
 
 4th. Divide the Refolvend by the Root, add the Quotient to thrice the 
 Square of the Root, let the Sum be a Divifor to the Refolvend, and the 
 Quotient-figures annexed to the right of the firft Root, without the cy- 
 phers, will be the Cube Root fought. 
 
 5th. If the fecond figure of the Root be i, or O; then generally 3 or 
 4 figures of the Root will be obtained at the firft operation : But if the 
 iecond figure exceeds 2, it will be beft to find only two places at firft. 
 
 6th. To renew the operation ; fubtradl the Cube of the figures found 
 in the Root from the given number j then form a Divifor, and divide 
 as direted in the fourth precept ; and this will give the Root true to 5 
 or fix places : Tor each operation commonly triples the figures found in 
 the Uft Root. 
 
 Ex. I. JVloat is the Cube Root 5/9800344 ? 
 
 Put a point qver the unit place 4, 
 another over the place of thoufands, 9800344(2 
 
 and another over that of millions ; g 
 
 and becaufe there are 3 points, there ' 
 
 will be 3 places in the Root. Under 2,00) 18003,44 Refolvend. 
 the left hand period 9, write 8, the 
 
 greateft Cube in it, and its Root 2 9001 = Quotient, 
 
 write in the Quotient, then fubtraft- 1 20000 = thrice the Sq. of the R. 
 
 ing,the Refolvend is 1 800344: Now . . . 
 
 becaufe there are two periods remain- 129001 ji 00344^14 
 
 jng, therefore two cyphers annexed ^ 
 
 to the Root 2, make it 200, by which cio'?44. 
 
 dividing the Refolvend, the Qiio- c 16004 
 
 tient is 900 ij alfo the fquare of 200 , 
 
 \^ 40000, the triple thereof 1 20000 The Root is 214. 
 
 being added to 9001, makes 129001 
 
 for a Divifor, by which dividing 1800344, the Quotient is 14 nearly, and 
 is taken as 14, becaufe it is much nearer to it than to 13 j now 14 being 
 annexed to the former Root 2, makes 214, the Root fought, 
 J or 214x214x214=9800344. 
 
 Ex. II.
 
 Book I. A R I T H M E T I C K. 29 
 
 Ex. II. fVhat is tht Cube Root of 5 1 8749442^ 75? 
 * 
 
 518749442875(8 
 
 52f 
 
 8,000) 6749442,87s 
 
 843680 
 192000000 
 
 i9284368o( 6749442875 (035 
 
 S78531040 
 
 964132475 
 964218400 
 
 In this example, becaufe 3 periods were remaining, and confequently 3 
 piaces more to be found ; therefore in the laft divifion a point is put over the 
 3d place from the right hand, and the Divifor is firft to be tried in the Di- 
 vidend as far as this point, in which as it cannot be taken, is put in the 
 Quotient, iffc. here the laft figure 5 is too much, but it is much nearer to 5 
 than to 4 ; then 035 annexed to the firft Root 8, makes 8035 for the Root. 
 
 Ex. in. TVhat is the Cube Root of 1 14604290,028 P 
 
 114604290,028 48 
 
 64 48 
 
 4,00 ) 50604290 
 
 126510 192 
 
 480000 
 
 2304 
 
 606510) 50604290 (8 48 
 
 Here 480 is taken for the Root at the firft operation. 18432 
 
 9Z16 
 
 Then 114604290,028(48 
 
 110592 110592 
 
 480) 4012290 The work of the Divifion is fuppofed to be done 
 ' on a walle paper. 
 
 8358,9 the Quotient. 
 691200 zz triple the Squareof 480, viz. 230400x3. 
 
 Divifor 699558,9) 4012290,028 (5736 
 ' 34977945 
 
 To 480 the firft Root 
 5144955 Add 5,736 
 4896912 
 
 248043 
 209807 
 
 Sum 485(736 
 
 38176 
 57. Here, inftead of bringing down the figures of the Dividend to the 
 Remainders, the Divifor is leflened each time, by pointing off a place on 
 the right ; but regard is to be had to the carriage which will arife from 
 the places thus omitted. 
 
 5 SEO
 
 ja A R I T. Hr M E T I C K. Book I. 
 
 SECTION tx; ^^i^'MlNtEKAL Series. 
 
 58. J rank of three or more numbers that tncreaje or decreaje 
 By an uniform progreffion, is called a Numeral Series. 
 
 59. If the Progreffion is made by equal difFerenccs, that is by the con- 
 ftant addition or fubtradtion of the fame number j the feries is called an 
 Arithmetic Progrejfton. 
 
 fl 23456789 l^c, increafing by adding 1, 
 Thus< 3 6 9 12 15 18 21 24 27 i^c. increafing by adding 3, 
 
 C 49 43 37 31 25 19 13 7 ' ^^- decreafing by fubduaing 6, 
 are ranks of numbers in Arithmetic Progreffion : And of fucb ranka there 
 may be an infinite variety. 
 
 60. If the Progreffion is made by a conftant multiplication or divifion 
 with the fame number, the feries is called a Geometric ProgreJJion. 
 
 i 655! 
 
 L 16384 
 
 I 2 4 8 16 32 64 y^. increafing by 2, 
 
 . I 5 25 125 625 3125 15625 tiff, increafing by 5, 
 
 I 6561 2187 729 243 81 27 9 3 ^f. decreafmg by 3, 
 
 [6384 4096 1024 256 64 1 6 4 1 y<:. decreafing by 4, 
 
 are ranks of numbers in Geometric Progreffion : And of fuch ranks there 
 may be an infinite variety. 
 
 '61. The common Multiplier or Divifor is called the ratio. 
 Thus 2 is the ratio in the \Jl rank, 5 in the 2d rank, 3 is the ratio in the 
 ^d rank, and 4 in the ^th rank. 
 
 62. In any feries of terms in Arithmetic Progreffion, the fum of any 
 two terms, confidered as extremes, is equal to the fum of any two terms 
 taken as means equally diftant from the extremes. 
 
 T'irus in 3 terms (where the \Ji and '^d are extremes, and the other the mean J 
 t/k. 6 . 9 . 12 , then 6+i2r=9-|-9=i8. 
 And in \ terms, 'viz, 13. 19. 25. 31. 
 I'hen 134-31 = 19-1-25=44. 
 
 Alfo in the terms 49. 43. 37. 31. 25. 19. 13. 7. i. 
 y>^^49+i=43-|-7=37-i-i3=3' + >9=25+25=5<^- 
 
 63. In a feries of terms in Geometric Progreffion, the Product of any 
 two terms confidered as extremes, is equal to the Produ(3: of any two in- 
 termediate equidiftant terms confidered as means. 
 
 ^bus in 3 terms, viz. 5. 25 . 125 . Or 3 . 9 27- 
 
 Then 5X125 = 25 X 25=625. Jl/o 3X27=9X9 = 81. 
 
 And in 4 terms 4.8.16.32. 
 
 5r>&^ 32X4 = 16X8 = 128. 
 
 Alfo in the terms I . 4 . 16 . 64 . 256 . 1024 . 4096 . 16384 . 
 
 Then 16384X1=409.6x4=1024X16=256x64 = 16384. 
 
 64. In
 
 i! 
 
 Book I. ARITHMETIC K; 
 
 31 
 
 64. In iiny Arithmetic PrdgreiGlon, the fum of any two terms leflened 
 by the firft term ; or their difference increafed by the firft term, will be a 
 term alfo in that progreifion. 
 
 Tbut in the ProgreJJlon 1 . 3 . 5 , 7 . 9. 11 . I3 . 15 . 17 . 19 . 21 \^c. 
 Then j-]-ii:=iiS, and 18 izrrij is a term of the Progreffion. 
 
 'j^^ 1 1*^7:4, and 44-1 =5 is a term t)F the Progt-effion. 
 
 i> 
 
 65. In any Geometric Progrefllon, the product of any two terms di- 
 vided by the firft term ; or the Quotient of any two terms multiplied by 
 the firft term, will give a term alfo in that feries. 
 
 Thus in th0 ProgriJJjon 3 . 6 . 12 . 24 . 48 . 96 . 192 . 384 . 768 Cfff. 
 
 12 X96 , 192 ' ' . , . 
 
 Then =3'84; and X 3 =48, 'are terms in the Progreffion. 
 
 3 12 
 
 66. If over a feries of terms in Geometric Progreffion, be written a 
 feries of terms in Arithmetic Progreffion, the firft term of which is O, 
 and common difference is i, term for term ; then any term in the Arith- 
 metic Series, will fhew how far its correfponding term in the Geometric 
 Stries is diftant from the firft term, - i -r . V. 
 
 CT^, t o I 2 3 4 5 6 l^c. Arithmetic Series. 
 "' ( I 3 9 27 81 243 729 Cfff. Geometric Series. 
 Here 729 is dillant from the ift term, 6 terms; 243 is diftant 5 terms, 81 is 
 diftant 4 terms. 
 
 67. The terms of the Arithmetical Scries are called indices to the 
 terms of the Geometric Series. 
 
 Thus 5 is the index to 2^^ > 3 '^ *^^ index tb 2y ; 1 // tit index /o 3 ; ^c. 
 
 68. Problem I. In an Arithmetic ProgreJJlon : Given iht Jirfl Urm^ 
 the common difference^ and the number of terms. 
 
 Required the laji term. 
 
 Rule. Subtract i from the number of terms, multiply the remainder 
 by the common difference ; to the ^rodut add the firft term, and the fum 
 will be the laft term. 
 
 Ex. I. Suppoje 1 and 9 to be the Jirji and Jecond tertm, of an Arithmetic 
 Progrejfion o/" 1074 terms : IVhat is the lajl term F 
 
 Here 9 ir:8 is the com. difF. Now 1074 imo73. 
 And 1073x8 = 8584. Then 8584+ 1 ^SjSfrzlaft term. 
 
 Ex. II. A perfon agrees to difcharge a certain debt in a year , byweekly 
 payments^ viz. the jirJi week ^s. the 2d iveek 2)S. Sec, cmjiaritly incrtafing 
 each iveek by 3; . ; How much was the lajl payment ? 
 
 5mft. term. Now 52 1= 51. 
 
 3r:com. diff. And 51X3=153. 
 
 52 = Nof term?. Thn 1534-55:158 /.ss/^C' 18 /.=laft Pyinent- 
 
 69, Pro-
 
 3% A5 R; I T H M E T I C K* Book L 
 
 69. Problem II. In an Arltbmetu Progrejftsn : Given the firfi term., 
 taji term^ and the number of terms. 
 Required the fum of all the terms. 
 
 Rule. Add the firft and laft terms togetlicr, the fum multiplied by half 
 the number of terms, gives the ftim of all the terms. 
 
 Ex. II. A debt is to be Aifchargtd 
 in a year by weekly payments equally 
 increaftngi the \Ji to be 5^. andthelaji 
 7j^. 1 8 J. - Hovj much luas' the debt r 
 
 Here jJC* 18/ =158/.=: laft term. 
 52=::N'' of terms, its -| is 26. 
 Now 1584-5 = 163. 
 Then i6i'/ii6z:=.\z%^s.z=.i\\. 18/. 
 is the fum of the terms, or debt. 
 
 Ex. I. Required the fum of the 
 frjl 1000 numbers in their natural 
 4rder . 
 
 ' " 5* "- . ../';' 
 
 Here imftterm, i=com. diff". 
 ioco=:N** of terms, its { is 500. 
 Now ioOo-f-i=:iooi. 
 Then looi X5oo=5O05OO is the fum 
 required. 
 
 Ex. III. Suppofe d bajket and ^00 Jl ones were placed in ajlraight line., a 
 yard dijlant from one another": Required in what time a man could bring them 
 one by one to the bajket.^ allowing him to walk at the rate of 3 miles an hour ? 
 
 Between the baflcet and ftones are 500 fpaces, which is the number of terms. 
 Now 5004-1 = 501. Then 501 X25o=:i25250=rum of the terms. 
 But as he goes backwards and forwards, he walks 2505OO yards. 
 Which divided by 1760 (the yards in i mile) gives 142, 329. miles. 
 Which at 3 miles an hour, will take 47 h. 26 min. 35 feconds nearly. 
 
 70. Problem III. In a Geometric ProgreJJion : Given the firjl term, the 
 ratio and the lajl term. 
 
 Required the fum of all the terms. 
 
 Rule. Multiply the laft term by the common ratio, from the Produft 
 fubtradl the firft term for a Dividend. 
 
 Subtract i from the ratio for a Divifor j then divide, and the Quo- 
 tient will be the fum of all the terms. 
 
 Ex. I. Suppofe the firfi term of a feries to be 3, the ratio 3, and the lafl 
 t/rm^^6i: Required the fum of all the terms. 
 
 Now 6561=13(1 term. And 3S=ratio. 
 
 Mult, by 3r:ratio. Sub. i 
 
 ^ i9683=:Produa. Rem. 2=:Diviforj 
 
 Subtr. 3=:firft term. 
 
 Io68o=:Dividend. 
 
 Then 2)19680(9840 is the fum of all the terms. 
 
 vEx. II. Let the firfi term be 2, the fecond term lo, and the laji term 
 1562.50 ; Required the fum of all the terms. 
 
 Here 2) 10 (5 is the common ratio. 
 
 Now 156250X^=781250. And 781250 2=78i248=:DIvidend. 
 
 Alfo 51 =4 the Divifor. 
 
 Then 4) 781248 (195312 is the fum of all the terms. 
 
 71. Pro-
 
 Book 1. A R t T H M E T I C K. 33 
 
 71. Problem IV. In a Geometric Progrejfion: Given the jirjl terttiy 
 the ratioy and the number of terms. 
 
 Required the lajl term, 
 
 kuLfe ift. Write down 6 or 7 of the leading terms In the Geometric 
 Series, and over them their Indices. 
 
 2d. Add together the moft convenient indices to make an index lefs by 
 unity than the number expreffing the place of the term fought. 
 
 3d. Multiply together the terms of the Geometric Series, belonging to 
 thofe indices which were added j make the product a dividend. 
 
 4th. Raife the firft term to a power whofe indeji is one lefs than the 
 number of terms multiplied j make the refiilt a Divifor to the former Dl- 
 videndj and the Quotient will be the term fought. 
 
 Ex. I. Wl)at is the i-j-th tennof a Geometric Series y the firjl Urrn of which 
 ts 3, andfecond term is 6 P 
 
 ^ . . . 
 
 Now = 2 is the common ratio. 
 
 3 
 .^oi 23 4 5 6 isfc. Indices. 
 
 I 3 . 6 . 12 . 24 . 48 . 96 . 192 ifc. Geometric terms. 
 Then 6-|-5:r:ri, is the index to the 12th term. 
 And 192 X (^6= 1 8432, is the Dividend. 
 
 The number of terms multiplied together is 2 ; and i-^i rr i, the power 
 to which the firft term 3 is to be raifed ; but the firft power of 3 is 3. 
 
 18432 
 
 Then ir 6 1 44 is the 1 2ih term of the given ferifes. 
 
 3 
 
 Ex. II. APerfon being afked to difpofe of a fine horfe, fain he would felt 
 him on condition of having one farthing for the \Jl nail in his Jhoes, two far- 
 things for the id nail ; one penny for the ^d nail ; two pence for the ^.th ; 
 four pence for the ^th ; 8 pence for the 6th^ l5'c. ; doubling the price of every 
 tail to 32, the number of nails in the four Jhoes : Hoiv much would that 
 horfe he fold for at that rate ? 
 
 Here the firft term is i, the ratio 2, and the number of terms 32. 
 P irft. To find the laft term. 
 
 Kow 5'23 45 6 7 S^fff. Indices. 
 
 ( I . 2 . 4 . 8 . 16 . 32 . 64 . 123 . 256 t^f. Geometric terras. 
 And 31 is the index to the 32d term. 
 Then 8 + 8=16; i6 + 8=:24; 24 + 7 31. 
 
 The ift term being r, any power thereof is I; (o the 4th article of th 
 rule is ufeJefs in this queftion. 
 
 Now 2^6X2156^=65536 is the 17th term. 
 
 615536X2^6=16777216 is the 25th term. 
 i67772i6x 128 = 2147483648 is the ^zd term. 
 
 Then 2147483648 
 2 
 
 42^4967296 
 1 the ift term. 
 
 a 1=1) 4J94967295 the fumof the 
 
 term? : or tn ^jnce, in farthings, of 
 the horTc, 
 
 20 
 
 4254967291; 
 
 '073741S23 1/ 
 
 89478+^5 - 3</. 
 4.J7-Q24 - 5 /. 
 
 Anfwcr 4473924 ;C- S'- SU- 
 
 Vol. I. ' D SEC-
 
 Thus 
 
 or 
 
 34 A R I T H M E T I C K. Book I, 
 
 SECTION X. OF LOGARITHMS. 
 
 72. Logarithms /jrtf tf /<rr/>j of numbers Jo contrived, that 
 hy them the work of multiplication may be performed by addition j 
 and the operation of divifion may be done by JubtraSlion. 
 
 ' 73. Or, Logarithms arc the Indices to a feries of numbers in Geome- 
 trical Progreflion. "^ 
 
 01234 5 6 l^c. Indices'or ^.ogarithms. 
 
 I 2 4 8 16 32 64 i^c. Geometric Progreflion. 
 
 (012 3 4 5 t^c Indices or Logaiithms. 
 
 J I 3 g 27 8 1 243 i^c. Geometric Series. 
 
 012 3 4 5 l^c. Indices or Logarithms. 
 
 I 10 ICO 1000 locoo 1 00000 i^c. Geometric Series. 
 Where the fame Indices ferve equally for any Geometric Series. 
 
 74. Hence it is evident, there may be as many kinds of Indices or Lo- 
 garithms, as there can be taken kinds of Geometric Series. 
 
 But the Logarithms moft convenient for common ufes, are thofe adapted 
 to a Geometric Series increafmg in a tenfold Progreflion, as in the laft of 
 the examples above. 
 
 75. In the Geometric Series i . 10. 100. 1000. iffc. betw^een the 
 terms i and 10, if the numbers 2.3.4.5.6.7.8.9 were interpofed, 
 to them might Indices be alfo adapted in an Arithmetic ProgrclTion, fuited 
 to the terms interpofed betvk'een i and 10, confidered as a Geometric Pro- 
 greflion : Alfo proper Indices may be found to all the numbers that can 
 be interpofed between any two terms of the Geometric Series. 
 
 But it is evident that all the Indices to the numbers under lO muft be 
 lefs than i; that is, are fractions : Thofe to the numbers between 10 and 
 100 muft fall between i and 2 ; that is, are mixed numbers confifting of 
 I and fpme fradlion : And fo the Indices to the numbers between ico and 
 1000 will fall between 2 and 3; that is, are mixed numbers confifting of 
 two and fome fra(SI:ion : And fo of the other Indices. 
 
 76. Hereafter, the integral part only of thefe Indices will be called the 
 Index; and the fractional part will be called the Logarithm : And the 
 computing of thofe fractional parts is called the making of Logarithms ; 
 the moft troublefome part of this work is to make the Logarithms of the 
 prime numbers ; that is, of iuch numbers which cannot be divided by any 
 other number than by itfelf and unity. 
 
 77. To find the Logarithms of prime tnwibers. ^ 
 
 Rule ift. Let the fum^of the propofed number and its next lefs num- 
 ber be called A. 
 
 2d. Divide 0,868588963 * by A, refcrve the Quotient. 
 
 * The number 0,868588963 is the Quotient of 2 divided by 2,30258^093, 
 which is the Logarithm of 10, according to the firft form of the Lord Nepier, 
 who was the inventor of Logarithms. The manner by which Nepier^s Log. of 
 10 is found, may be feen in many books of Algebra ; but is here omitted, be- 
 caufe this treatife does not contain the elements of that fcience : However, 
 thofe who have not opportunity to enter thoroughly into this fubjeft, had 
 better grant the truth of one number, and thereby be enabled to try the accu- 
 racy of any Logarithm in the tables, than to receive thofe tables as truly 
 computed, without any means of examining the certainty thereof. 
 
 2 3d. Divide
 
 ^B6okI. .>sA. KvIlT H M E TAC:K, 3^ 
 
 3d. Divide the referved Quotient by the Square of. A, referve this 
 
 Quotient. 
 
 4th. Divide the laft referved Quotient by the Square of A, referving 
 the Quotient ; and thus proceed as long as divifion can be made. _ 
 
 5th. Write the referved Qi^iotients orderly under one another, the firft 
 being uppermpft. 
 
 6th. Divide thcfe Quotients refpeitively by the odd numbers 1.3.5. 
 7 . 9 . II, &c. that is, divide the firft referved Quotient by i, the 2d by 3, 
 the 3d by 5, the 4th by 7, ^c. let thefe Quotients be w^ritten orderly un- 
 der one another, add them together, and their fum will be a Logarithm. 
 
 7th. To this Logarithm, add the Logarithm of the next lels number, 
 and the fum will be the Logarithm of the number propofed. 
 
 Ex. L Required the Logarithm of the number 2t 
 
 Here the next lefs number is i, and 24-1=3 A. 
 
 And the Square of A is 9. Then ; ' j 
 
 0,868588963 ,289525654. 
 
 =,289529654. And jj =,289529654 
 
 0,289529654 ,032169962 
 =,032169902. & * =,010723321 
 
 0,032169962 . ,003^74440 
 
 ' =,003574440. & =,000714888 
 
 0,003574440 ,000397:60 
 =,000397160. & ^ - =,000056737 
 
 9 . 7 
 0,000397160 ,000044129 
 r:,cooo44i29. & =,000004903 
 
 9 9 
 0,000044129 ,000004903 
 '=,000004903. ::r, 000000445 
 
 0,000004903 ,000000545 
 =:,cooooo545. & :=, 000000042 
 
 o,oooooo:c4(; ,000000060 
 
 =,oocoooo5o. Si n:, 000000004 
 
 To this Log. o,3oio299(}4 
 
 Add the Log. of i=o,oocoooooo 
 
 Their fum is the Log. of 2:1^0,301029994 
 This procefs needs no other explanation than comparing it with the rule. 
 
 That the manner of computing thefe Logarithms may be familiar to 
 the Reader, the operations of making fcvcral of them arc here fubjoined. 
 
 D 2 Ex, n.
 
 3 
 
 A R I T H M E T I C K. 
 
 Book I. 
 
 Ex. II. Required the Logarithm of the number 1. 
 Here the next lefs number is 2 ; and 3 + 2=:5=A, whofe Square i 25,. 
 
 0,868588963 
 
 5 
 
 0,006948712^ 
 0,000277948 
 
 0,OOCO{ I I 18 
 
 0,000000445 
 
 =>i7377792. And 
 
 ^5 
 
 6948712. & 
 
 277948. & 
 11118. & 
 
 445- f 
 18. & 
 
 ,173717792 
 
 I 
 
 69^487 1 2 
 
 3 
 
 277948 
 
 5 
 iiiiS 
 
 =.17371779* 
 =:,O033ij63}57 
 
 =,000055590 
 
 7 
 445 
 
 9 
 
 i8 
 
 II 
 
 =,000001588 
 r=,oooooco49 
 
 To this Logarithm 
 Add the Log. of 2 
 
 The fum is the Logarithm of 3 
 
 =,000000002 
 
 ' ' ' ^ 
 0,176091258 
 0,301029994 
 
 0,477121252 
 
 78. Since the Logarithms are the Indices of numbers confidered in z 
 Geometric ProgrefTion j therefore the fums,or diiFerences of thefe Indices, 
 will be Indices or Logarithms belonging to the Produ<Sts, or Quotients, 
 of fuch terms in the Geometric Progreflion as correfpond to thofe Lo- 
 garithms which wer added or fubtraded (71). 
 
 Ex. IV. Required the Log. of 6, 
 
 Now 3 X2=6. 
 
 Then to the Log. of 3 0,477121252 
 Add the" Log. of 2 0,301029994 
 
 Ex. III. Required the Log. of a^. 
 
 N0W4 = 2X2. 
 
 Then to the Log. of 2 0,301029994 
 Add the Log, of 2 0,301029994 
 
 Sum is the Log, of 4 0,602059988 
 
 Ex. V, Required the Log. of \0. 
 In the original Series, 1 is aflumed 
 for the Logarithm of 1 o. 
 
 Ex. VII. Required the Log. of%. 
 
 Now 8 = 2x2X2. 
 
 Therefore the Log. of 2 taken thrice 
 
 gives the Log. of the number 8 
 
 The Log. of 2 is 0,301029994 
 
 "Which multiplied by 3 
 
 Gives the Log. of 8 
 6 
 
 0,903089982 
 
 Sum is the Log. of 6 0,778151246 
 
 Ex. VI. Required the Log. of 5, 
 Now 10 divided by 2 gives 5. 
 Then from Log. of 10 1,000000000 
 Take Log. of 2 0,301029994 
 
 Leaves the Log. of 5 0,698970006 
 
 Ex. VIII. Required the Log, ofq. 
 
 Now 9=3 X3. 
 
 Therefore the Log. of 3 doubled gives 
 
 the Log. of 9. 
 
 The Log. of 3 is 0,477121252 
 
 Which multiplied by z 
 
 Gives the Log. of 9 0,954242504 
 
 Ex. IX.
 
 Book I. A R I T H M E T I C K. 37^ 
 
 Ex, IX. Required the Logarithm of 'J , 
 
 Here 6 is the next lefs. Thea 7+6=i3=A ; and idgrrSquare of A. 
 
 0,86858806^ ..^ ' .'. ,066814536 
 
 2 =,066814536. And =,066814536 
 
 13 I 
 
 ff!!!:iif= 39535.. & 32531?=: ,3,^84 
 
 2:^^2^2131!^ ,335. & 1339^ ^58 _^ 
 
 169 5 
 
 0,000002? JO 14 
 
 2:=: 14. & -Irz 2 
 
 169 7 
 
 To this Logarithm 0,066946790 
 
 Add the Logarithm of 6 0,778151246 
 
 The fum is the Logarithm of 7 0,845058036 
 
 The Log. of 12 is equal to the fum of the Logs, of 3 & 4, or of 2 & 6. 
 The Log, of 14 is equal to the fum of the Logs, of 7 & 2. 
 
 of 1 5 of 3 & 5 . 
 
 of 16 of 4 & 4, or of 5 & 2. 
 
 ofi8 of 3 & 6, or of 9 & 2. 
 
 of 20 of 4 & 5, or of 10 & 2. 
 
 79. The Logs, of the prime numbers n, 13, 17, 19, are to be found 
 as in the examples L II. IX. and in like manner is the Log. of any other 
 prime number to be found ; but it may be obferved, that the operation is 
 ihortcr in the larger prime numbers ; for any number exceeding 400, the 
 firft Quotient added to the Logarithm of its next Icfler number, will give 
 the Logarithm fought, true to 8 or 9 places ; and therefore it will be very 
 eafy to examine any fufpeded Logarithm in the tables. 
 
 80. The manner of dlfpofing the Logarithms, when made, into tables> 
 is various : But in this treatife they are ordered as follows. 
 
 Any number under 100, or not exceeding two places., and its Logarithm^ 
 are found in the firfi page of the table., where they are placed in adjoining 
 columns ; and diftinguiflied by the title Num* for the common numbers ; 
 and by Log. for the Logarithms. 
 
 Thefe tables are at the end of Book IX. 
 
 A number of three or four places being given, its Logarithm is thus 
 found. 
 
 Seek for a page in which the given number fhall be contained betwcea 
 the two numbers marked at the top, annexed to the letter N : Thci\ 
 right againft the three firft figures of the given number, found in tlie 
 column figned Num, and in the column figned by the fourth, Itands tha 
 Logarithm belonging to that number of four places. 
 
 It the number confifted of 3 places only ; then thefe places found as 
 before directed, the Logarithm ftnads againft them in the columrt 
 figned 0. 
 
 D 3 Thus,
 
 3^ 
 
 A^ii I r^H m^e; tick. 
 
 Blt^i 
 
 Thus, tofnd the- Logarithm of 5738. Seek for a page in which ftands 
 at top N" 5200 to 5800 ; then in the column figned N find 573, right 
 againft which in the column figned 8- ad top or bottom ftands, 75876^' 
 which is the Logarithm to 5738, exchjfive of its Index. :_ ' _ 
 
 8r. j1 Logarithm being givni^ its number is thus found. 
 
 Seek for a page in which the three firfl: figures oV the given Logarithm 
 are found at top annexed to the letter L ; then in one of the columns 
 figned with the figures o, i, 2, 3, 4, 5, 6, 7, 8, 9, find a number the 
 neareft to the given Logarithm ; againft this number in the column 
 figned N, ftand three figures ; to the right of thefe annex the figure 
 with which the column was figned at top or bottom, and this will Be 
 the number correfponding to the given Logarithm, not regarding the 
 Index. 
 
 82. All numbers confifting of the fame figures, whether they be inte- 
 gral, fra6lional, or mixed, have the fradlional parts of their Logarithms 
 the fame. 
 
 If the following examples be well attended to, there will be no difficulty 
 in finding the Logarithm to a propofed number, or the number to a pro- 
 ppfed Logarithm, within the limits of the table of Logarithms here ufed. 
 
 Num. 
 
 Logarithms, 
 
 Logarithms. 
 
 Numbers. 
 
 5^7^ 
 
 3,7689^ 
 
 0,37295 
 
 2,360 
 
 587.4 
 
 2,76893 
 
 1,28631 
 
 19.33 
 
 58.74 
 
 1,76^93 
 
 2.51947 
 
 330.7 
 
 5,874 
 
 ' 0,76893 
 
 1.750^2 
 
 5632, 
 
 0,5874 
 
 _i^,76893 
 
 1''8397 
 
 0,001527 
 
 0,05874 
 
 2,76893 
 
 2,43020 
 
 0,02693 
 
 0,005874 
 
 3.76893 
 
 1,85962 
 
 0.7238 
 
 S3. A general rule to find the Index to the Log. of a given number. 
 'To the left of the Logarithm^ ivy it e that figure (or figures) ivhich ex~ 
 prejfes the dijlance from unity ^ of the higheft-place digit in the given num- 
 ^er, reckoning tl.e unit's pLce 0, the next place I, the next place 2, the next 
 place 3, Sec. 
 
 When there are integers in the given number, the Index is always 
 affirmative ; but when there are no integers, the Index is negative, and is 
 to bemarked by a little line drav/n above it : thus 7. 
 
 Thus a number having I .2.3.4.5 ^c, integer places. 
 
 The index of its Logarithm is o . i . 2 . 3 . 4 . i^c. 
 
 And a fraction having a digit in the place of Primes, Seconds, Thirds, 
 Fourths, ^'c. 
 
 Then the Index of its Logarithms will be 1 .2.3.4 i^c. 
 
 By the above rule, the place of the fractional comma, or mark of 
 diftiniStion, in the number anfwcring to a given Logarithm, will be 
 always known. 
 
 84. The more places the Logarithms confift of, the more accurate, in 
 general, will be the refult of any operation performed with them : But 
 for the purpofes of Navi-gatio.n, as five places, exclufive of the Index, are 
 fufficient, therefore the logarithmic tables in this treatife are not extended 
 any farther. 
 <' . ' 85. MUL^
 
 Book I. 
 
 ARITHMETIC K. 
 
 39 
 
 85. MULTIPLICATION BY LOGARITHMS; 
 
 Ory 7'wo or more numbers being given , to find their Frodu5l by 
 "Logarithms . 
 
 Rule. Add together the Logarithms of the P'aftors, and the fum is a 
 Logarithm, the correfponding number of which is the Product required. 
 
 Obferving to add what is carried from the Logarithm to the fum of the 
 affirmative Indices. 
 
 And that the difference between the affirmative and negative Indices 
 are to be taken for the Index to the Logarithm of the Produ6t. 
 
 Ex. 1. Multiply 86,25 h ^4S- 
 
 86,25 its Log. is 1,9^576 
 6,48 its Log. is 0,81157 
 
 Produft 558,9 
 
 2.74733 
 
 Ex. III. Multiply 3,768 by 2,053 
 and by 0,007693. 
 
 3,768 its Log. is 0,5761 J 
 2,053 0.3J259 
 
 0,007693 3,88610 
 
 Produft 0,0595 I 2,77460 
 
 The I carried from the left hand 
 column of the Logs, being affirm- 
 ative, reduces 3 to z. 
 
 Ex. 11. Midtiply 46,75 by 0,3275. 
 
 46,75 its Log. is 7,00978 
 
 , 0,3275 its Log. is 1,51521 
 
 Produfl 15,31 
 
 1,18499 
 
 Ex. IV. Multiply 27,63 by 1,859 
 and by 0,7258 and by 0,03591. 
 
 27,63 its Log. is 1,44138 
 1,859 0^6928 
 
 0,7258 1,86082 
 
 0,03591 2/5 >;2' 
 
 Produd 1,339 0,12669 
 
 PIcre 2 being carried to the In* 
 dex I, makes 3 ; which takes off 
 the 7 and ~l. 
 
 %G. DIVISION BY LOGARITHMS. 
 
 Or, 'T'vuo numbers being given, to find hozv often the one will 
 contain the other, by Logarithms. 
 
 Rule. From the Log. of the Dividend, fubtra(?t the Log. of the Divifor ; 
 then the number agreeing to the Remainder, will be the Quotient required. 
 
 But obferve to change the Index of the Divifor from negative to affir- 
 mative, or from affirmative to negative : And then let the difference of the 
 affirm, and neg. Indices be taken for the Index to the Log. of the Quotient. 
 
 When an unit is borrowed in the left-hand place of the Logarithm, add 
 it to the Index of the Divifor, if affirmative j but fubtraft it if negative ; 
 and let the Index arifmg be changed and worked with as before. 
 
 Ex. I. Divide 558,9 by 6,48, 
 
 Log. ofDivid. 558,9 is 2,74733 
 
 Log. of Divifor 6,48 is 0,81157 
 
 The Quotient is 86,2: 
 
 '.93576 
 
 Ex. III. Z)mV^o,o595 1/70,007693. 
 Log. ofDivid. 0,05951 is ^,77459 
 
 Ex. II. Divide 15,31 by 46,75. 
 
 Log. of Divid. 15,31 is 1,18497 
 
 Log. of Divifor 46,75 is 1,66978 
 
 The Quotient is 0,3275 
 
 ,51519 
 
 Ex. IV. Divide 0,6651 by 22,5, 
 Log. ofDivid. 0,665115 i,822j 
 
 Log. of Divifor 0,007653 is 7,88610 l^^og- of Divilor 22,5 is 1,35218 
 
 . . I '~^ 
 
 The Quotient is 7,735 0,88849 i ^^^ Qnotient is 0,02956 2,47071 
 
 D 4 
 
 87. 01
 
 40 
 
 ARITHMETICK. 
 
 Book I. 
 
 87. 
 
 OF PROPORTION. 
 
 Ill State the terms of the queftion (by 46) and let them be writtoa 
 ordijfiy under one another, prefixing to the firft term the word As^ to the 
 ftV^o.^w l"j, to the third So^ and under them fet the word To. 
 
 2d. Agajnft the firft t^rm, write the arithmetical complement of its 
 Logarithm. See Art. 88. 
 
 3J. Againft the fecond and third terms, write their Logarithms. 
 
 4.th, The fum of thofe three Logarithms, abating lO in the Index, will 
 b^ \\\t Logarithm of the 4th term ; which fought in the tables, the num- 
 ber anfwering to it is the anfwer or term fought. 
 
 88. The arithmetical complement of a Logarithm is thus found. Be- 
 ginning at the Index, write down what each figure wants of 9, except the 
 laft, or right-hand figure, which take from 10. 
 
 But if the Index is negative, add it to 9 j and proceed with the reft as 
 before. 
 
 Ex. I. Find a fourth proportional 
 number io 98,45 and 1,969 and ^^J^2 
 As 98,45 its * Ar. Co. Log. 8,00678 
 To 1,969 0,29425 
 
 So 347,2 2,54058 
 
 To 6,944. 
 
 0,84161 
 
 Ex. III. IFhat will a gunner's 
 pay amcunt to in a year at 2^, 12^. 
 bd. a month of 1% days ? 
 As 28 days its Ar. Co. Log. 8,55284 
 To 365 days 2,56229 
 
 So 2jC' >2/. 6d.=zz,62^. 0,41913 
 
 To 34^. 4/. 5 ^.=34,22^. 1,53^26 
 
 Ex. II. Find a third proportional 
 number to 9,642 and 4,821. 
 As 9,642 its Ar. Co. Log. 9,01583 
 To 4,821 0,6^314 
 
 So 4,821 0,68314 
 
 To 2,411 
 
 0,3821 1 
 
 Ex. IV. If I of a yard of cloth 
 co/i } of a guinea : Hoxv many elli 
 EngUjh for 3. \0s.? 
 Aj f Guin.m4j:. Ar. Co. 8,85387 
 f"o 3;^' 10/. =70/. 2,84510 
 
 So I ell =0,6 i778i5 
 
 To 3 ells 
 
 0,47712 
 
 Ex. V. What number will have Ex. VI. How many yards of Jhal- 
 the fame proportion to 0,8538 as Icon of I ell ivide will be enough to 
 
 0,3275 has to 0,01 31 f 
 
 As 0,0131 its Ar. Co. Log. m, 81^273 
 To 0,3275 _^'iJ52i 
 
 So 0,8538 1, 93 '36 
 
 *To 21,35 
 
 >3253<^ 
 
 line a coat containing 3-^- ells of \\ 
 yards wide? 
 
 As I x| yd. w. 0,9375 10,02803 
 To i]- yd. w. 1,75 0,24304 
 So Si^i yd. I-=4'37S 0,64098 
 
 To 8-g yd. longr:8,i67 0,91205 
 
 I where w. (lands for wide, 1. for long. 
 
 * At. Co. Log. fla.nd: for t':i Arithmetical Complement of the logarithm. 
 
 OF
 
 Book L A R I T H M E T I C K* ^ 
 
 OF POWERS AND THEIR ROOTS. 
 
 89. y^ number being given j to fnd any fropofed power of that 
 Number. 
 
 Rule ift. Seek the Logarithm of the given number. 
 
 jtd. Multiply this Logarithm by the Index of the propofed power. 
 
 3d. Find the number correfponding to the Produdt, and it will be the 
 ^wer required. 
 
 90. In multiplying a Logarithm havinga negative Index, theProdud 
 of that Index is negative. 
 
 But the carriage from the Logarithm is affirmative. 
 
 Therefore the difference will be the Index of the Produfl. 
 
 And is to be of the fame kind with the greater, or that which was 
 made the minuend. 
 
 ,^ 
 
 Ex. I. TVhat is the fecond power 
 of the number 3,874/* 
 To 3,874 its Log. is 0,58816 
 
 The Index is 2 
 
 The power fought is 15,01 1,17632 
 
 Ex: III. iVhat is the I2th power 
 tf the number 1,539 ^ 
 
 1,539 its Log. is 0;i8724 
 The index is 12 
 
 The power fought is 176,6 2,24688 
 
 Ex. II. What is the ^d power of 
 the number 2,768 F 
 
 The N" 2,768 its Log. is 0,44.217 
 The Index is 3 
 
 The power fought is 21,21 1,32651 
 
 Ex. IV. What is the 2^ ^^h power 
 of the number 1 ? 
 
 2 Its Log. is 0,30103 
 
 The Index is 365 
 
 150515 
 
 180618 
 
 In the IVth Ex. the Index of the Produ(fl: being 109, fhew 90309 
 that the required power will confiil -of no integer places ; of 
 
 -- \ ^_.._- __ - _Q__^ ^ 
 
 which no more than 4 places are found iji thefe tables ; there- *09'S7?95 
 
 fore the number fought may be thus exprelTed, 7515 [TFe] ; 
 That is, 7515 with 106 cyphers annexed. 
 
 Ex. V. IVhat is the id power of 
 the number 0,2^57 ^ 
 
 To 0,2857, its Log. is 
 The Index of 2d power 
 
 i,45S9> 
 2 
 
 The power 0,08163 
 
 2,91 182 
 
 Here, there being no carriage 
 fiom the Product of the Log. the 
 whole Product of the negative In- 
 dex is negative, vix. 7. 
 
 Ex. VI. TFhat is the ^d power of 
 
 the number 0,7916 .'^ 
 
 To 0,7916, its Log. is 
 The Index of 3d power 
 
 The power is 0,4961 
 
 1,89851 
 3 
 
 ""69553 
 
 Here the carriage from the Pro- 
 duct of the Log. is 2 ; then the 
 Produ6l of the negative Index 7, 
 viz. 7, being leflened hy 2, leaves 
 7, the Index of the Produ<ft. 
 
 91. A
 
 4^ 
 
 A R I T H M E T it K. 
 
 Book I. 
 
 51 . A number being given, to find any propofed Root of it. 
 
 Rule ift. Seek the Logarithm of the given number. 
 
 2d. Divide this Logarithm by the denominator of the Index of the 
 propofed root. 
 
 3d. The number correfponding to the Quotient will be the root. 
 
 When the Index to the Logarithm to be divided is negative, and Icfs 
 than the Divifor, or Denominator of the root. Then 
 
 Increafe the negative Index by as many units, borrowed, as fliall be 
 e^ual to the Divifor, and the Quotient will give 7 for the Index. 
 
 Carry the units borrowed as tens to the left-hand place of the Loga- 
 rithm, and then divide that Logarithm as in whole numbers. 
 
 Ex. I. What is the Square Root of Ex. II. What is the Cube Root of 
 
 the number 2121 ? 
 
 the nuniber 1501 .^ 
 Root fought is 38,74 
 
 2)3.17638 
 
 1,58819 
 
 Ex. III. What is the Root, of 
 which 176,6 is the 12th power? 
 
 12)2,24699 
 
 Root fought Is 1,539 
 
 0,18725 
 
 Ex. V. What is the Square Root 
 of the number 0,08162? 
 
 Log. of 0,081 62, div. by 2)2,91180 
 
 Gives Root 0,2857 to 1,45590 
 
 Here the Divifor 2 can be taken 
 in the Index a, and gives for the 
 Quotient T. 
 
 Root fought 12,35 
 
 3)3.32654 
 1,10885 
 
 Ex. IV. What is the Root, of 
 which 2 is the 365//' power ? 
 
 365)0,30103 
 
 Root fought Is 1,002 
 
 0,00082 
 
 Ex. Vr. What is the Cube Root 
 of the number 0,496 f 
 
 Log. of 0,496 div. by 3)1.69548 
 
 Gives Root 0,7916 to 1,89849 
 
 Here the Divifor 3 cannot be 
 taken in the Index 7 ; then T bor- 
 rowed makes with 1, 3; in which 
 the Divifor 3 will go 7 : the 2 bor- 
 rowed carried to the 6, i^c. makes 
 269548, which divided by 3, gives 
 89849. 
 
 END OF BOOK I.
 
 THE 
 
 defcrlptions. 
 Or of fuch 
 
 ELEMENTS 
 N A V I g"" A T I O N. 
 
 BOOK II. 
 OF GEOMETRY. 
 
 S E C T I O N I. 
 
 DeJi?iitions and Principles, 
 
 I. y^ EOMETR Y is a fcience which treats of the 
 \jr properties and relations of magnitudes in general : 
 things where length, or where length and breadth, or length, breadth, and 
 thicknefs, are confidered. 
 
 2. A Point is that which is without parts or dimenfions. 
 
 3. A Line is length without breadth : It is called a j^ 2j 
 
 Right Line when it is the fhorteft diftance between 
 
 two points, as ab : Or a Curved Line when it is 
 not the fhorteft diftance, as CD. 
 
 y/ litie is ufually denoted by two letters, viz. one at each 
 endy as ab or cd. 
 
 4. A Superficies or Surface is that magnitude 
 which has only length and breadth, and is bounded by 
 lines : as FG. 
 
 5. A Solid is that magnitude which has length, 
 brcadih and thicknefs. 
 
 6. A Figure is a bounded fpace, the limits or bounds 
 of which may be cither lines or furfaces. 
 
 7. A Plaki-, or a Plane Figure, is a fupcrficies 
 v/hich lies evenly, or perfedly flat, between its limits, 
 and may be bounded by one curve line ; but not with 
 Icfs than three right lines, as a, c, c, d, or c, ^V. 
 
 
 <^ 
 
 8. A
 
 44^ 
 
 GEOMETRY. 
 
 ' S. A Circle is a plain figure, bounded by an uni- 
 formly curved line, called the Circumference, as abd, 
 which is everywhere equally diftant from one point, as c 
 within the figure called the Center. 
 
 9. A Radius is a right line drawn from the center 
 to the circumference as ca, cd or CE. 
 
 All the radii of the fame circle are equal, 
 
 10. An Arc is any part of the circumference. 
 As^ the arc ab, or the arc ad. 
 
 11. A Chord is a right line joining the ends of an arc, as ab, and 
 is faid to fubtend that arc j it divides the circle into tv/o parts, called 
 Segments. 
 
 12. A Diameter is a chord paffing through the center, as de, and 
 divides the circle into two equal parts, called Semicircles. 
 
 13. The Circumference of every circle is fuppofed to be divided 
 into 360 equal parts, called Degrees ; each degree into 60 equal parts, 
 called Minutes j each minute into 60 equal parts, called Seconds, ^c. 
 
 14. A Plane Angle, is the inclination of two lines 
 on the fame plane meeting in a point, as acb. 
 
 A right lined angle is formed by two right 4ines. ^ 
 The point where the lines meet is called the angular 
 foifity as c. \A. 
 
 The lines which form the angle are called legs. 
 
 Thus CA and CB are the legs of the angle acb. 
 
 An Angle is ufually marked by three letter Sy viz. one at 
 the angular point, and one at the other end of each leg j hut 
 that at the angular point is always to be read the middle 
 letter, as acb, or bca. C 
 
 15. The raeafure of a right lined angle is an arc, 
 as BA contained between the legs cb, ca, including 
 the angle, the angular point c being the center of that 
 arc. 
 
 16. A Right Angle is that, the meafure of which 
 is a fourth part of the circumference of a circle, or 
 ninety degrees. Ihus the angle acb is a right angle, 
 
 i-j. A Perpendicular is that right line which 
 cuts another at right angles ; or which makes equal 
 angles on both fides. Thus dc is perpendicular to ab, 
 when the angles DC A and DCB are equal, or are right 
 angles. 
 
 18. An Acute Ancle, as acb, is that which is lefs 
 than a right angle dcb. 
 
 19. An Obtuse Angle, as efc, is that which is 
 grfater than a right ang-Je kfg. 
 
 Acute 2;:d obtufe ant^lcs are called Oblique Angles. 
 
 D 
 
 ^A. 
 
 20. Pa*
 
 Book n. 
 
 GEOMETRY. 
 
 4S 
 
 20. Parallel Lines, are right lines in the fame 
 plane, which do not incline to one another, as ab, 
 
 CD. 
 
 21. A Triangle is a plane figure bounded by 
 three lines. 
 
 22. An Equilateral Triangle is that in 
 which the three lines, or fides, are equal, as a. 
 
 23. An Isosceles Triangle is that which has 
 only two equal fidgs, as b or c. 
 
 24. A Right angled Triangle, as aSc, is that 
 which has one right angle b. 
 
 25. An Obtuse angled Triangle, as def, has 
 one obtufe angle e. 
 
 26. An Acute angled Triangle, as g, has all 
 ifs angles acute. 
 
 27. A Quadrangle, or Quadrilateral, is a 
 plane figure bounded by four right lines, or fides. 
 
 A ^adrangle is ufually expreffed by letters at the op" 
 foftte angles. 
 
 28. A Parallelogram is a quadrangle the op- 
 pofite fides of which are parallel and equal, as p. 
 
 29. A Rectangle is a parallelogram with right 
 angles : and <n which the length is greater than its 
 breadth, as r. 
 
 30. A Square is a parallelogram having four 
 equal fides and right angles, as s. 
 
 31. A Trapezium is a quadrangle the oppofite 
 fides of which are not parallel, as t. 
 
 32. The Diagonal of a quadrangle, is a line, as 
 AB, drawn from one angle, to its oppofite angle. 
 
 33. The Base of a figure, is the line it is fup- 
 pofed to ftand on. 
 
 34. The Altitude or Height of a figure, is the 
 perpendicular diftance ab, between the bafe and the 
 vertex, or part molt remote from the bafe. 
 
 35. Congruous Figures, are thofe which agree, 
 or correfpond, with one another, in every refpedt. 
 
 36. A Tangent to a circle is a right line, as ab, 
 touching its circumference, but not cutting ; and the 
 point c, where it touches, is called the point of 
 ionta^t 
 
 D 
 
 n 
 
 B 
 
 A 
 
 C B 
 F 
 
 D JE 
 
 ZI7 
 
 i\ 
 
 --B 
 
 A C B 
 
 37. An
 
 :^ G E O M.J ;T jP^ Y. JtoQk rr. 
 
 37. An Angle, bXc, ;>/ a S^^enty cadb, js, ^ 
 when the angular point is in the circ^mifercnce of the T)/'^^^/\\ 
 fcgmcnt, and the legs including the angle pafs through / X V\ 
 the ends B and c, of the .chord of the fegm?nt. 1/ \ ) 
 
 Such an angle is faid to he in a t'trcuir^erence ; and'tu ^\" '^C 
 
 fiand on the arcy BC,. included betuueen the legs, ab and ^^ 
 
 AC, of the angle, * :.-.'jj^\\ 
 
 38. Right lined figures, having more than four fides, are called Poly- 
 gons J and have their names from the number of their angles, or fides ; as 
 thofe of five fides are called Pentagons j of fix fides, Hexagons j of fcven 
 fides. Heptagons ; of eight fides, 0^agonSj i^c. ' 
 
 39. A regular Polygon is a figure with equal fideSj and equal, angles. 
 
 40. A figure is faid to be infcribed in a circle, when all the angles of 
 that figure are in the circumference of the circle. 
 
 41. A figure is faid to circumfcribe a circle, when every fide of the 
 figure is touched by the circumference of the circle. . 
 
 42. A Propofition is fomething propofed to be confidered ; and requires 
 either afolution or anfwer, or that fomething be made out, or proved. 
 
 - . . '' .1 , 
 
 A Problem is a praiSlical propofition, in which fomething is propofed 
 to be done, or efFe<5ted. 
 
 A Theorem is a fpeculativc propofition, or rule, in which fomething is 
 afiirmed to be true. ^ . 
 
 A Corollary is fome conclufion gained from a preceding propofition. 
 
 A Scholium is a remark on fome propofition; or an exemplification of 
 the matter which it contains. 
 
 An Axiom is a felf-evident truth, or principle, that every one aflents to 
 upon hearing it propofed. 
 
 A Poftulate is a principle, or condition,, requefted ; the fimplicity or 
 feafonablenefs of which cannot be denied. 
 
 In Mathematics, the following Poftulates and Axioms, are fome of the 
 principal ones that are generally taken for granted . 
 
 When a propofition, from fuppofed premifes, aflerts fuch and fuch con- 
 fequences ; and fubjoins, And the contrary : it is to be underftood, that 
 if the confequences be afTumed as premifes; then what were firft taken as 
 premifes, would become confequences. 
 
 Thus, in Article 95, it is premifed, that if two parallel right lines are 
 cut By another right line, there i efults this conlequence ; The alternate 
 angles are equal. And the contrary means ; that tuhere equal alternate 
 angles are made by a right line cutting two other right lines ; the right lines 
 fo cut, are parallel lines, 
 
 Poftulates,
 
 Book n. G E O M E T R Y. 47 
 
 . Poftulates. 
 
 43. I. That a right line may be drawn from any given point to an- 
 other given point. - 
 
 44. II. That a given right line may be continued, or lengthened at 
 pleafure. 
 
 45. III. That from a given point, and with any radius, a circle may be 
 defcribed. 
 
 Axioms. 
 
 46. I. Things equal to the fame thing, are equal to one another. 
 
 47. II. If equal things are added to equal things, the fums or wholes 
 will be equal. But if unequals be added, the fums are unequal. 
 
 48. III. If equal things are faken from equal things, the remainders, 
 or differences, are equal : but are unequal, when unequals are taken. 
 
 49. IV. Things are equal which are double, triple, quadruple, ^c. or 
 half, third part, &c. of one and the fame thing, or of equal things. 
 
 5c. V. Things which have equal meafures, are equal. And the con- 
 trary. 
 
 51. VI. Equal circles have equal radii. 
 
 52. VII. Equal arcs in equal circles have equal chords, and are the 
 mcafures of equal angles. And the contrary. 
 
 53. VIII. Parallel right lines have each the fame inclination to a right 
 line cutting them. 
 
 In what follows, it is to be underftood, that right lines (vtz. firaight 
 liiics) are drawn by the edge of a ftraight ruler : circles or arcs, are defcribed 
 with one foot of a pair of compafles, the other foot refting on the point 
 which is taken for the center j and the diflance of the feet, or points, of 
 the compafles is taken as the radius: alfo, that the point marked out by a 
 letter is to be undcrflood, when the reference is made to that letter. 
 
 54. It is alfo taken for granted, that a line or diflance can be taken be- 
 tween the compafles, and may be transferred or applied from one place to 
 another. Alio, that one figure can be applied to, or laid upon another, 
 or conceived to be fo applied. 
 
 In any problem, when a line, angle, or figure is faid to be given ; that 
 line, angle, or figure niufl be made, before any part of the operation is 
 performed. 
 
 SEC-
 
 t#l 
 
 GfiOMfetRY* 
 
 Book Hi 
 
 ^(P 
 
 S E C T 1 O N It. 
 Geometrical Problems* 
 
 S$, P R O B L E M I. 
 
 fo biJiSfy or dhide into two eqUal parts, a given line AiJ 
 Operation, ift. From the ends a and b *y 
 
 with one and the fame radius, greater than half 
 
 AB, defcribe arcs cutting in g and d. (45) 
 
 2d. A ruler laid by c and d, gives Ej the middle 
 
 erf" AB, as required* 
 
 The proof of this operation depends on articles 
 
 lOi, 99. 
 
 56. P R O B L E M IL 
 
 To bifeSf a gi'Oen right lined angle ABC* 
 
 Operation, ift. From b> defcribe an arc ac. 
 
 2d. From a and c*, with one and the fame radiusj 
 defcribe arcs cutting in t). (45) 
 
 3d. A right line drawn through B and d will divide 
 the angle into two equal parts, as required. 
 
 The proof depends on article lOi. 
 
 A 
 
 y 
 
 57' 
 
 PROBLEM Hi. 
 
 From a given point b, in a given right line af, to draw a right line per- 
 pendicular to the given line. 
 
 Case I. IFhen b is near the middle of the litter 
 
 Operation, ift. On each fide of b, take the 
 equal diftances bc, and be. (54) 
 
 2d. On c and E * defcribe, with one radius, 
 arcs cutting in d. (45) 
 
 3d. A right line drawn through b and d will be 
 theperpendicular required. (43) ^ 
 
 The proof depends on article lO^^ A ( 
 
 58. Case II. When b is at^ or near the end of the given Vinii 
 
 Operation, ift. On any convenient point c, 
 taken at pleafure, with the diftance, or radius cb, 
 defcribe an arc dbe, cutting af in d, b. (45) 
 
 id. A ruler laid by d and c will eut this arc in t. 
 
 3d. A right line drawn through b and e will 
 be the perpendicular required. 
 
 This depends on article 130. 
 
 t^ 
 
 '^ 
 
 9--''* 
 
 D 
 
 A 
 
 * That is, firft defcribe an arc from one point ; then defcribe an arc from 
 the Other point with the fame opening of the compaffea. 
 
 59. P R Of
 
 Book II. 
 
 GEOMETRY. 
 
 49 
 
 59- 
 
 PROBLEM IV. 
 
 To draw a line perpendicular to a given right line AB, from a point C 
 without that line. 
 
 Case I. When the point c ii nearly oppofite to the 
 middle of the given line. 
 
 Operation, ift. On c, with one radius, cut 
 AB in D and e. (45) 
 
 2d. On D and e, with one radius, defcribe arcs 
 cutting in F. (45) 
 
 A ruler laid by c and f gives g ; then draw CG, 
 and that will be the perpendicular required. 
 
 Thii depends on articles 10 1, 99. 
 
 t) 
 
 A. ^- 
 
 r? 
 
 Xe* 
 
 60. Case II. When c is nearly oppofite to one end of the given line AB, 
 
 Operation, ift. To any point d in ab, draw 
 the line cd. (43) 
 
 2d. Bifecl the line CD in E. (55) 
 
 3d. On E, with the radius EC, cut ab in o. 
 Then CG being drawn, will be the perpendicular 
 required. 
 
 This depends on article 1 30 . j 
 
 CI 
 
 
 iLi 
 
 61. P R O B L E M V. 
 
 To trifeSl^ or divide into three equal parts^ a right angle ABC. 
 
 Operation, ift. From b, with any radius ba, 
 defcribe the arc ac, cutting the legs ba, bc, in a, c. 
 
 2d. From A, with the radius ab, cut the arc ac in 
 E, and from c, with the fame radius, cut AC in d. 
 
 3d. Draw be, bd, and the angle abc will be 
 divided into three equal parts. 
 
 This depends on article 193. 
 
 62. 
 
 PROBLEM VI. 
 
 At a given point D, to make a right lined angle equal to a given right 
 lined angle abc. 
 
 Operatiok. ift. From d and b, with one ra- 
 dius defcribe the arcs kf, and ac, cutting the legs 
 of the given angle in the points a, c. 
 
 2d. Transfer the diflancc A c to the arc e f, 
 from F to E. frA\ 
 
 3d. Lines drawn from d, through e and f, will 
 form the angle kdf equal to the angle abc 
 
 This dept-nd: en article 10 1. 
 
 Vol. L 
 
 63. PROB-
 
 ^ 
 
 G E O M E T^RY. 
 
 Boole II. 
 
 \ 
 
 ^ 
 
 63. P R O B L E IMT VII. 
 
 To draw a lint paralkl to a ^ivtn right lint ab. 
 Case I. TVhen the parallel line is to pafs through a givtn point. 
 
 Operation, ift. From c, with aay convenient ,j. 
 
 radius, defcribe an arc df, cutting ab in d. ^V- 
 
 2d. Apply the radius cd from d to e; and from 
 E, with the lame radius, cut the arc df in f. jg 
 
 3d. A line drawn through f and c will be pa- a^ 
 rallcl to AB. 
 
 T7;u deptnds on article lOl, 95. 
 
 64. Case II. IVljen the parallel line is to be at the giveji dijlanu c fr 
 
 Operation, ift. From the points a and b, 
 with the radius c, defcribe arcs d and e. 
 
 ad. Lay a ruler to touch the arcs d and e, and a 
 line drawn in that pofition is the parallel required, 
 
 this operation is mechanical. 
 
 65. PROBLEM VIII. 
 
 Upofi a given Bne ab, to ynake an equilateral irimgle. 
 
 Operation, ift. From the points a and b,, .., 
 
 with the radius ab, defcribe arcs cutting in c. -'^''^ 
 
 AT. , U C ^^^^ 
 
 2d. Draw ca, cb, and the ngiire abc is the 
 triangle required. (43') ' 
 
 The truth of this operation is evident ', for the 
 fides are radii of equal circles. 
 
 c 
 
 D 
 
 ~ B 
 
 om ab. 
 
 A 
 
 66. "By a like operation, an Ifofceles triangle 
 DEF may be conftruded on a given bafe de, with 
 the given equal legs DF, EF, either greater, or Icfs, 
 than the bafe de. 
 
 67. PROBLEM IX. 
 
 T'o tvake a right lined triangle, the fides of%vhich Jhall he refpeclively equals 
 either t- thofe of a given triangle abc, or to three given lines, provided any 
 /Wi ''/ tl'em takeyi together are greater than the third. 
 
 v'prk.\TiON. ift. Draw a line de equal to the 
 line aj:^ _ (54) 
 
 .^d '/'n i->, with a radius equal to AC, defcribe 
 
 ' --- . . (45) 
 
 n P.., wch a radius equal to BC, defcribe 
 
 ^'i the former arc in F. (45) 
 
 nd the triangle dfe will be 
 
 E 
 
 68. PR OB-
 
 Book 11. G E O M E T R Y. $i 
 
 68. P R O B L E M X. 
 
 Upon a given line ab, to defcr'ihe a fquare. 
 
 Operation, ift. Draw bc perpendicular, and 
 equal to AB. (58) 
 
 2d. On A and c, with the radius ab, defcribe 
 arcs cutting in d. (45) 
 
 3d. Draw DC, da ; and the figure abcd is the 
 fquare required. 
 
 This depends on articles 10 1, 104, 95, 30. 
 
 69. P R O B L E M XL 
 
 To defcribe a reiangle whofe length jhall be equal to 
 a given line EF, and breadth equal to another given 
 line G. 
 
 Operationt. ift. At E and f ere6l two perpendi- 
 culars, EH and Fi, each equal to the given line g. 
 
 2d. Draw HI, and the figure efih will be the 
 reclangle required. 
 
 This depends on ar tides 29, lOj, 104, 95. 
 
 70. PROBLEM XIL 
 To find the center of a circle. 
 
 Operation, ift. Draw any chord ab. 
 2d. Bife6l.AB with the chord cd. (55) 
 
 3d. BifctSl CD with the chord ef, and their in- 
 terie6lion g will be the center required. 
 This depends on article 125. 
 
 71. P R O B L E M XIIL 
 
 To divide the circumference of a circle into tiuo, 
 ihlrty-two, (Jc. equal parts. 
 
 Operation, ift. A diameter ab divides the 
 circle into two equal parts. (12) 
 
 2d. A diameter de, perpendicular to ab, divides 
 the circumference into four equal parts. 
 
 3d. On A, D^ B, defcribe arcs cutting in <7, /> ; 
 then by the in'cerfecflions ^, b^ and the center, the 
 diameters fg. hi, being drawn, divide the circum- 
 lerencc into eight equal parts j and fo on by conti- 
 duual bii'eiStion. 
 
 For at each operation^ the intercepted arcs are hifeclcd^ and the parts dmhleJ, 
 
 72. PROBLEM XIV. 
 
 To defcribe a circle^ the circumfcroice of which fnaJl pafs through three 
 ^iven points A, u, c, provided they d') not lie in one r;gf:t line. 
 
 Operation, ift. Bifecl the dillance cb with 
 ihe line de. (55) 
 
 2d. Bife6\ the diilancc ab with the line fg. 
 
 3d. On H, tlie interfecV.on of thcfe lin.-s, wiili 
 the diftance to either of the given points, dclcnbc 
 tiie circle rcqiiiicJ. 
 
 This dipcndi on a^tide 1 2 5. 
 
 E 7 7.^ PROI3- 
 
 four^ eight, ftxtecn^
 
 5^ 
 
 GEOMETRY. 
 
 Book 11. 
 
 73. PROBLEM XV. 
 
 To draw a tangent to a given circle, that Jhall pafs through a given point A, 
 
 Case 1. If^hen a is in the circumference of the circle. R ^, _A_ l^ 
 
 Operation, ift. From the center c, draw the 
 radius ca. (43) 
 
 2d. Through a draw bd perpendicular to ca 
 (58), and BD is the tangent required. 
 
 This depends on article 126. 
 
 74. Case II. lPl)en the given point a, is without the given circle. 
 Operation, ift. From the center c, draw ca, 
 
 which bifedl in b. (55) 
 
 2d. On B, with the radius ba, cut the given 
 circumference in d. 
 
 3d. Through d, the line ae being drawn, will 
 be the tangent required. 
 
 This depends on articles 1 30, 126. 
 
 75. * PROBLEM XVI. 
 To two given right lines A, B, to find a third proportional. 
 
 Operation, ift. Draw two right lines mak- 
 ing any angle, and meeting in a. 
 
 2d. In thefe lines, take ^Zi^firft term, and ac, 
 ad, each equal to the fecond term. 
 
 3d. Draw hd., and through r, draw ce parallel a 
 jto bd\ then ae is the third proportional fought. B 
 
 And ah : ac : : ad : ae*. 
 
 This depends en article 165. 
 
 a 
 
 76. P R O B L E M XVII. 
 
 To three given right lines a, b, c, to find a fourth proportional. 
 
 Operation, ift. Draw two right lines mak- 
 ing any angle, and meeting in a, 
 
 2d. In thefe lines take <7Z'rrfirft term, aczzic- 
 ccnd term, and <7^rrthird term. 
 
 3d. Draw be, and parallel to it, through d, 
 draw de ; then ae is the fourth proportional re- 
 quired. 
 
 And ah : oc : : ad : ae. 
 
 This depc7uh on article 165. 
 
 * And is thus read : ah Is to ac, as ad is to ae. 
 The character ( : ) flandirg for ii lo^ and the charadler ( : : ) for as. 
 
 77. PROB.
 
 Book II. 
 
 GEOMETRY. 
 
 S3 
 
 77- 
 
 PROBLEM XVIII. 
 Between two given right lines A, B, to find a mean proportional, - 
 
 B 
 
 A 
 
 Operation, ift. Draw a right line, in which 
 take ac=iAy abz=B. 
 
 2. Bifecl he in f (55) ; and on F, with the ra- 
 dius Fby defcribe a femicircie bee. 
 
 3d. From a draw ae perpendicular to be (57) ; 
 then ae is the mean proportional required. 
 
 And ae : ae : : ae : ab. 
 
 This depends on article 171. 
 
 78. PROBLEM XIX. 
 
 To divide a given line BC in the fame proportion as a given line A is divided. 
 
 Operation, ift. From one end b of bc 4 1 r 1 1 
 draw BD, making any angle with bc. 
 
 2d. In BD apply from b the feveral divifions of 
 A 5 fo BD will be equal to A, and alike divided. 
 
 3d. Draw CD ; then lines drawn parallel to 
 CD through the feveral divifions of bd, will di- 
 vide the line bc in the manner required. 
 
 This depends on article 165. . 
 
 79. PROBLEM XX. 
 
 To divide a given right line ab into a propofed number of equal parts 
 (fuppofe -]). 
 
 Operation, ift. From a, one end of ab, 
 draw AE to make any angle with ab; and from 
 R, the other end, draw bf, making the angle 
 ABF equal to the angle bae. (62) 
 
 2d. In each of the lines ae, bf, beginning at 
 A and B, take, of any length, as many equal parts, 
 Icfsone, as ab is to be divided into, viz. i, 2, 3, 
 4 5. 6. 
 
 3d. Lines drawn from i to 6, 2 to 5, 3 to 4, 
 i5c. will divide ab as was required. 
 
 This depends on article 1 65 . 
 
 80. Another Method. 
 Operation, ift. Through one end a, draw a 
 
 line cc nearly perpendicular to Ac. 
 
 2d. Draw EF parallel to ab, at any convenient 
 diftance. 
 
 3d. In EF take, of any length, as many equal 
 ]\arts as ab is to be divided into ; as i, 2, 3, 4, 5, 
 
 6, to F. ; ",'^-r:>- * 
 
 4th. Through B and f, where the divifions n\'i-^>--'"'' 
 terminate, draw bc, meeting cc in c. 
 
 5th. Lines drawn from c through the feveral divifions of EF, will cut 
 AB into the equal parts required. 
 
 Note, ii the fum of the divifions from E to F chance to bc lefs than ab, 
 tlic ix)int c, and the line ef, will be on the famg fide of AB ; but if greater, 
 c falls on the contrary fide. 
 
 E 3 81. PROB- 
 
 C??;>. 

 
 ^4 
 
 GEOMETRY. 
 
 Book II. 
 
 81. PROBLEM XXI. 
 
 V To make Scales of equal parts. 
 
 Operation, ift. Draw three lines, a, b, c, parallel to one another, 
 and at convenient dillances, fuch as are here exprefled. 
 . 2d. In the line c, take the equal parts cb, be, cd, da^ $cc. each equal 
 to fome propofed length. 
 
 3d. Through c, draw the line de perpendicular to c^; and parallel to 
 DE, through the fcveral points b^ c, d, <?, Sec. draw lines acrofs the three 
 parallels a, b, c j then are the fpaccs or dillances c^, bcy cd, &c. called 
 the primary divifions. 
 
 4th. Divide the left-hand fpace into io equal parts (80), and through 
 thele points let lines, parallel to de, be drawn acrofs the parallels bc ; and 
 the primary divifio'n cb will be parted into i o equal fpaces, called fubdivifions. 
 
 5th, Number the primary divifions from the left towards the right, 
 beginning at c, and the fcalc is conflruted, 
 
 Such Scales are ufeful for.conflru^ing figures, the fides of which are ex- 
 prefled in meafurcs of length ; as feet, yards, mile?, tsc. Or to find the 
 meafures of the length of lines in a given figure. 
 
 Thus a line of 36, or 360, feet, yards, feV. is taken by fetting one 
 point of the compafies on the third primary divifion, and extending the 
 other point to the fixth fubdivifion : and fo of others. 
 
 In thefe Scales it is ufually fuppofed, that an inch is divided into fome 
 number of fuch parts, as are exprefied by the fubdivifions. 
 
 82. To fi7id the divifions of a Scale of equal parts ^ "when any given 7iu?)i- 
 Icr of them are to make an inch. 
 
 Operation, ift. Make a Scale c<3, where the primary divlfiojvs fhall 
 be each one inch; let DC be at right angles to C(7, and the left-hand fpace 
 Qb bedi\'ided into 10 equal parts, and draw dZ* from any point d. 
 
 2d. Draw dh, making v/ith DC any angle ; and make Dit^cZi. 
 
 3d. Take the number of parts propofed in an inch, from the Scale c^, 
 and apply them from d to ;;, in tlie line dc, continued if necefi'ary. 
 
 4th. Draw ?/l, and CG parallel to //i; and make rrrr bo, 
 
 5th. Through r draw rs parallel to co, cutting dZ- in / j then rt will bc 
 one of the primary divifions, containing xo of the parts the inch was to be 
 divided into. 
 
 6th. Lines drawn from d to the divifions of c7>, divide rt into 10 equal 
 Ep.rts. 
 
 ^ 83. PROB
 
 Book II. GEOMETRY. 55 
 
 83. . P R O B L E M XXII. 
 
 To divide the circumference of a circle into degrees i and thence to make a 
 Scale of chords. 
 
 Operation, ift. Defcribe the femicircle abd, the center of which 
 is c, and draw CD at right angles to ab. 
 
 2d. Trifedl the angles acd, dcb, in the points <?, i ; c, d (61) : then 
 by trials, divide the arcs Ai?, ab, Ad, Df, cd, ^b, each into three equal 
 parts, and the femicircuniference will be divided into 18 equal parts, of 
 10 degrees each. 
 
 3d. Thefe arcs being birc61^ed, will give arcs of 5 degrees. 
 
 4th. And thefe arcs being divided into 5 equal parts, by trials, Will 
 give arcs of i degree each. 
 
 5th. The chords of the arcs a^, ac, ad, aI., a^, and of every other 
 arc, being applied from a on the diameter ab, will divide ab into a Scale 
 of chords j which are to be numbered from a towards b. 
 
 80 -'-' 100 
 
 A Scale of chords is ufeful for conflru6lIng an angle of a given number 
 of degrees : And for finding the meafure of a given angle, 
 
 84. PROBLEM XXIir. 
 
 To make an angle of a propofed nu7nher of degrees. 
 
 Opf.r ation. ift. Take the firft 60 deg. from the 
 Scale of cliords, and with this I'adius defcribe an arc 
 Bc, the center of which is A. 
 
 id. Take the chord of the propofed number of de- 
 grees from the Scale, reckoning from its beginning, 
 and apply this diftancc to the arc bc, from b to c. 
 
 3d. Lines drawn from a, through the points b and c, will fqrm an 
 angle bac, the meafure of which is tlie degrees propofed. 
 
 With Scales of chords not exceeding 90 degrees, fuch as are gene- 
 rally put on rulers, angles greater than 90 degrees, fuppofe of 138 
 dcg. are to bc taken at twice, viz. 69 and 69 [\ of 138 1, or 93 
 and 48. 
 
 JVlicn a given angle .^ BAG, /x to he rncafured. 
 
 From the angular point A, with the chord of 60 degrees, defcribe the 
 arc nc, cutting the legs in the points b and c. 
 
 Then the (liftancc rc, ap()lied to the chords, from the beginning, will 
 flicw the decrees which meafure the angle cac. 
 
 E 4 
 
 85. PROB-
 
 5^ G E O M E T 1l Y. Book II. 
 
 85. PROBLEM XXIV. 
 
 / ag'tDen ctYck to infer i be a regular polygon y the number of fides being given. 
 
 Operation, ift. Divide 360 degrees by the 
 number of fides, and the Quotient will be the de- 
 grees which meafure the angle at the center of the 
 circle, fubtended by a fide of the polygon*. 
 
 2d. Draw the radius ch, make an angle BCD 
 equal to tliofe degrees (84), and draw the chord 
 DB ; then will DB be the fide of the polygon re- 
 quired ; .which applied to the circumference from b 
 to o, atobyb to f, &c. will give the points to which the fides of the po- 
 lygon are to be drawn. 
 
 If the polygon has an even number of fides, draw the diameter ab ; 
 and divide half the circumference, as before ; then lines drawn from thefe 
 points through the center, as ed, will give the remaining points, in the 
 other femlcircumference. 
 
 86. 
 
 PROBLEM XXV. 
 
 On a given right line ab, to corftru6l a regular polygon of any affigned 
 number of fides. 
 
 Operation, ifl-. Divide 360 degrees by the 
 number of fides; fubtrail the Quotient from 180 
 degrees, the remainder will be the degrees which 
 meafure the angle made by any two adjoining fides 
 of that polygon, and is called the angle of the poly- 
 gon f. 
 
 2d. At the ends A, b, of the line ab, make an- 
 gles ABC, bad, equal to the angle of the polygon. 
 
 3d. Make ad, bc, each equal to ab. 
 
 4th. At the points c, d, make angles equal to 
 that of the polygon as before ; and let the fides in- 
 cluding thofc angles be each equal to ab ; and thus 
 proceed until the polygon is conftrudted. 
 
 In figures of any number of fides, the two Jafl: 
 fides D, CE ; or eg, hg ; are readieft found by 
 defcribing arcs from c and D, or from e and H, v/ith 
 the radius ab, interfcdted in e, or in o. 
 
 In figures of an even number of fide?, having drawn half the number, 
 AD, AB, EC, CF, by means of the angles; the remaining fides may bc 
 found, by drawing through the points d and f the lines de, fh, parallel 
 and equal to their oppofite fides cF, ad ; and fo of the reft. 
 
 * For there ivill he as many equal angles at the center as there are equal Jides. 
 And the nvhole cirumference meafures the /urn of all the angles nt the center. 
 
 \ For each fide of the polygon nvith radii drawn to its ends form an Ifofcelis 
 triangle. 
 
 Tbeg the angje of the po^gon is derived from articles 05, 96, 104. 
 
 %-]. PROD,
 
 Book 11 
 
 GEOMETRY. 
 
 57 
 
 87. PROBLEM XXVI. 
 
 About a ghen regular Polygeny to circumfcribe a circle : or within thai 
 Polygon to infcrlbe a circle. 
 
 Operation, ift- Bifel any two angles fab, 
 CBA, with the lines ag, bg, (56) and the point g, 
 where they interfeft one another, will be the cen- 
 ter of the polygon. 
 
 2d. A circle defcribed from g, with the radius 
 Ga, will circumfcribe the given polygon. 
 
 TTjIs depends on article 85, 96, 104. 
 
 88. Again, ift. Bifeil any two fides fe, ed, 
 in the points h and i (55) ; and draw HG, ig, at 
 right angles to fe, ed (57) ; then the point G, where they intcrfedl 
 each other, will be the center of the Polygon. 
 
 2d. A circle defcribed from g, with the radius gh, will be infcribed in 
 the given Polygon. 
 
 T^/j depends on article 126. 
 
 89. PROBLEM XXVIL 
 
 On a given right line ab, to defer ibe a fegment of a circle^ that fhall 6n- 
 tain an angle equal to a given right lined angle c. 
 
 Operation, ift. Make an angle baf equal 
 to the given angle c. (62) 
 
 2d. From H, the middle of ab, draw hi at right 
 angles to ab (57), and from a draw ai at right 
 angles to af (58), cutting hi in i. 
 
 3d. From I, with the radius ia, defcribe a circle. 
 
 Then will the fegment agb contain an angle 
 agb equal to the given angle c. 
 
 This depends on articles 125, 126, 132. 
 
 90. PROBLEM XXVm. 
 
 To divide a right line in continued proportion, in the ratio of two glvtn 
 right lines ab, ac. 
 
 Operation, ift. From b, with the radius 
 AB (the antecedent) defcribe an arc ac. 
 
 2d. In that arc apply (the confequent) AC from 
 A to r ; draw Ar, and apply c.\ from c to d. 
 
 3d. Apply the following lines in the order 
 directed, viz. ad from A to d^ and from </ to e ; 
 AE from A to e, and from ^ to F ; af from a to 
 f and fromy to c ; ag from a to ^, and from 
 g to H, &c. Then will the proportional lines be 
 
 AB, AC, AD, AE, AF, AG, &C. And 
 
 ab : AC : : (ac = ) ac ; (A^/rz) ad. 
 ab : AC : : (a^=:) ad : (a^ z=) ae. 
 AB : AC : : (a^ :=) ae ; (Afzz) af. 
 bfc. 
 
 This depends on articles 104, 95, i65 
 
 A 
 
 SECTION
 
 si 
 
 91 
 
 GEOMETRY. Book U. 
 
 .S E C T 1 O.N III. 
 Geometrical T^heoretnu 
 
 Of R^kjht LirWEs and Planes. 
 
 THEOREM I. 
 
 IVlnn one right line cvt Jlands upon another ri^ht 
 line AB, they make twa ong/cs hCDy AQU which 
 together are equal to two right angles, .. , 
 
 \ . . A e . ." 
 
 Demonstration. For defcrlbinga feriiicircle adb, on c. (45) 
 
 Then the arc db meafurcs the angle Bcb. (15) 
 
 And the arc da meafurcs the angle acd. (15) 
 
 But the arcs db and ad together meafure two right angles. (l3v l6) 
 Therefore bcd and acd together, are equal to two right angles (50) 
 
 92. Corollary. Hence if any nurhber of right lines Qd ftands on 
 one point c, on the fame fide of another right line ab.; the fum of all the 
 angles are equal to two right angles ; or are meafured by 180 degrees. 
 
 93. T H E O R E M II. 
 
 If two right lines AC, eg inierfe^ egch other in 
 D, the oppojite angles an equaly viz, ^^ 
 
 * Z. CDEirZ-ADE, and Z.CDErr z. adb. 
 
 fDEM, For the angles At)E and ADB together make two right angles. (91) 
 And the angles cdb and adb together make two ri^ht angles. (91) 
 
 Therefore the fum of ade and ADBzrfum of cdb and adb. (46) 
 
 Confequentiy the Z. cdb is equal to the z. ade. (48) 
 
 94. X CoROL. Hence if any number of right lines crofs each other in 
 one point, the fum of all the angles which they make about that point, is 
 equal to four right angles ; or is meafured by 360 degrees, 
 
 95. THEOREM III. 
 
 If a right line FE cut two parallel right lines ab, 
 CD ; then is the outward angle a \\ equal to the inward ^ 
 and Gppofite angle d ; and the alternate angles f, d are 
 equal : and the contrary. j\_ 
 
 Dem. Becaufe cd and ab are parallel by fuppofition : 
 
 Then FE has the fame inclination to CD and ab. 
 
 And this inclination is exprefTed by the /. a or z_ d: 
 
 Therefore the outward z. a is equal to the inward and oppofite z. d. 
 
 Now the Z. ^ is equal to the Z. c ' (Q'X) 
 
 And /ince the /L a is equal to the jL d; 
 
 Therefore the alternate angles c ajid ^f are equal. (46) 
 
 * The mark Z. J^ands for the nvcrd angle. 
 
 f 'D^M-JlandsforDefHoufiratioji. % Cov.o\,. fands for Corollary. 
 
 \\ Angles are J'omttitnti marked hy a Jingle letter. Thus angle a is vfedfor angle vav>, 
 
 96. THEO- 
 
 (53) 
 (14)
 
 Book 11. 
 
 96. 
 
 GEOMETRY. 
 T H E O R E M IV. 
 
 59 
 
 In any right lined triangle abc, the fum of the 
 three angles a^ h^ c, is equal to two right angles : 
 -And if one fide BC he continued, the outward angle f 
 is equal to the fum of the two inward and oppofite an- 
 gles a, b. 
 
 Dem. Through a, draw a right line parallel to EC, (63) making with 
 AB the Z.^, and with AC the Z.d. 
 
 Now A^rrZ.^; and ^i= A f. being alternate. (95) 
 
 And two right angles meafure the JLe-\- jLa-\- jLd. (92) 
 
 Therefore Z.b-\-Z^a-{- Z.c i.e->r jLa-\- Z.d. (47) 
 
 Confequently Z./'4- A^z 4- Z.c=two right angles. (46) 
 
 Moreover Z-f-f-Z./" rrtwo right angles. (9^) 
 
 Therefore /Lb\-Z.a =: A.f ^ (4^) 
 
 97. Hence, if one angle is right or obtufe, each of the other is acute. 
 
 98. If two angles of one triangle are equal to two angles of another 
 triangle, the remaining angles are equal. And if one angle in one triangle 
 is equal to one angle in another triangle, then is the fum of the remaining 
 angles in one, equal to the fum of the remaining angles in the other. 
 
 99. 
 
 THEOREM V. 
 
 If two fides AB, AC and the included angle A in 
 one triangle ABC, are refpeSiively equal to two fides 
 DE, DF and the included angle D of another def, each 
 to each ; then are thofe triangles congruous, 
 
 Dem. Apply the point d to the point a, and the line de to ab. 
 
 Now as DErrAB (by fuppofition) ; therefore the point E falls on B. 
 
 But Z.D Z.A (by fup. ) ; therefore df will fall on AC. 
 
 And iincc df ac (by fup.) j therefore the point F falls on C. 
 
 Confcqucntlv fe will fall on CB. 
 
 Therefore the triangles acb, dfe, are congruous, fince every part agrees. 
 
 100. 
 
 THEOREM VI. 
 
 If two trian'rles abc, def have twv angles A, B and 
 the included fide AB in one refpeHively equal to two an- 
 gles D, E and the included fide DE in the other, each 
 to each j ttcn are thofe triangles congruous. 
 
 A. li D 
 
 Dei\i. Apply the point D to A, and the line DE to ab. 
 Now as OE Ab (by fup.) ; therefore the point e falls on b. 
 And as Z.D Z.A (by fup.) ; therefore the line df falls on AC. 
 Now if ihe line AC is lefs or greater than the line df ; 
 'i'hcn the line fe not falling on CB, makes the Z. B Icfs or greater thnn Z. K. 
 liut Z-i'.=::Z.E (by fup.) ; therefore AC is neither Icfs nor greater than df. 
 Or the line AC = nF ; confequently fe = cd. 
 '\ htrefore tiic triangles arc congruous, 
 
 xci. THEO-
 
 6o 
 
 GEOMETRY, 
 
 Book II. 
 
 lOI. 
 
 THEOREM VII. 
 
 Two triangles ABC, DEF are congruousy when the 
 ^ree fides in the one are equal to the three ftdes in the 
 other, each to each. 
 
 A 3 1) E 
 
 Dem. Apply the point d to a, and the line de to ab. (54) 
 
 Now as de=:ab (by fup.) j therefore the point E falls on B. 
 
 On A, with the radius AC, defcribe an arc. 
 
 Then as df=:ac, the point f will fall in that arc. 
 
 Alfo on B, with the radius EC, defcribe another arc, cutting the former in c. 
 
 And fince ef=bc, the point f will fall in this arc alfo. 
 
 But if the point f can fall in both thefe arcs, it can be only where they 
 
 interfedl, as in c. 
 
 Confequently the triangles are congruous. 
 
 K)2. THEOREM VIII. 
 
 Tkvo triangles abc, def are congruous, when two 
 angles a, B and a fide AC oppoftte to one of them, in 
 one triangle, are refpeSfively equal to two angles D, 
 E and a fide df oppoftte to a like angle in the other tri- 
 angle, each to each. 
 
 Dem. Apply the point d to a, and the line df to ac 
 
 Now as DFnAC (by fup.) ; therefore the point f falls on c. 
 
 And asZ.D=Z.A (by fup.) i the line de will fall on the line ab. 
 
 And if the point e does not fall on b, it muft fall on fome other point 
 
 G. Draw CG. 
 
 Then the angle agc is equal to the angle d-ef. (99) 
 
 And the angle ABC = (DEFr:) agc, which is not poffible. (96) 
 
 Therefore the point E can fall no where but on the point b. 
 
 Confequently the triangles are congruous. 
 
 103. 
 
 THEOREM IX. 
 
 lu the Ifofceles, or equilateral triangle acb ; a 
 line drazvn from the vertex c to the middle of the 
 bafe ab is perpendicular to the hafe, and bifeifs the 
 vertical angle : and the contrary *, 
 
 Dem. The triangles adc, bdc, are congruous. 
 
 Since ca cb (23) ; CDrrcD, and ad = db by fuppofition : 
 
 Therefore Z.a=:Z-B, Z.acd=:Z_bcd, ^adc = Z-Bdc. (lOi) 
 
 Confequently cd is at right angles to ab. (i?) 
 
 104. CoROL. Hence in any right lined triangle where there are equal 
 fides, or angles ; 
 
 The angles a, b, oppofite to equal fides bc, ac are equal. 
 And the fides bc, ac, oppofite to equal angles a, b, are equal. 
 
 That is, if a line drawn perpendicular to the bafe of a triangle biieds 
 the vertical angle; then that triangle muft be Ifofceles, and the perpendicu- 
 lar is drawn from the middle of the bafe. 
 
 105, THEO-
 
 Book IL 
 
 GEOMETRY. 
 
 61 
 
 105. 
 
 THEOREM X. 
 
 In every right lined triangle ABC th greater angle 
 c is oppofite to the greater ftde ab. 
 
 Dem. In the greater fide ab, take ad = ac; draw cd, and through, 
 E draw BE pareliel to cd. (63) 
 
 Then the angles adc, acd, are equal. (104) 
 
 And /.ADCrrZ-ABE. (95) 
 
 Therefore z.acd z.abe. (46) 
 
 That is, a part of the angle acb is greater than the angle abc. 
 Confequently the Z.C is greater than the Z.B ; and in the fame manner, it 
 may be proved to be greater than the L. a, if the fide ab be greater than cb. 
 106. CoROL. Hence in every right lined triangle, the greater fide is 
 oppofite to rfie greater angle. 
 
 107. 
 
 T H E O REM XI. 
 
 Parallelograms acdb, ecdf, eguf Jianding on 
 the fame bafe CD, or on equal hafes CD, GH, and 
 between the fame parallels, CH, AF, are equal. 
 
 C \> 
 Dem. For ABrzEF, being each equal to cd. 
 To each add be, and ae=:bf. 
 Nov/ AcrrBD, and ce = df. 
 The triangles ace, bdf, are therefore congruous. 
 Nov/ if from each of the triangles ace and bdf, be taken the triangle 
 BiE, the remaining trapeziums abic and feid are equal. (48) 
 
 Then if to each of the trapeziums abic, feid, be added the triangle cid, 
 their fum will be the parallelograms ad and dr, which are equal. (47) 
 And in like manner it may be fhcwn, that the parallelogram eh is equal 
 to the parellelogram edzzad. 
 
 c p c "D 
 
 108. THEOREM XII. "' ' ' 
 
 A triangle abc is the half of a parallelogram 
 AD, when they fl and on the fame bafe ab, and are 
 between the fame parallels AB, en. 
 
 Dem. AC is equal to de, and ab to dc. 
 
 Alfo Bc is a fide common to both the triangles abc and dcb. 
 
 Thcfc triangles are therefore congruous. 
 
 Confequently the triangle abc is half the parallelogram ad. 
 
 log. CoROL. I. Hence every parallelogram is bife<5^ed by its diagonal, 
 1 10. CoROL. II. Alfo, triangles {landing on the fame bafe, or on equal 
 
 bafes, and between the fame parallels, arc equal. 
 
 They being the halves of equal parallelograms under like circumftances. 
 
 Ill . THEO-
 
 6t GEOMETRY. .Book II. 
 
 rir. THEOREM XIIJ. 
 
 In every right angled triangle bac the fquare on 
 the fide BC oppojite to the right angle A is equal to 
 the fum of the fquare $ of the two fides AB, At: ton- 
 taining the right angle. 
 
 I>EM. On the fides ab, ac, bc, oonftrul the fquares ag, ae, cd 
 (6S) : draw ad, ce j and draw af parallel to bd. (63) 
 
 TThen the triangles abd, ebc, are congruous. ^99) 
 
 For the Z. ABE =,^CBD, being right angles. (30) 
 
 To each add flie angle abc, then Z.EBC and Z.ABD are equal. (47) 
 
 Thei^fore eb, bc, Z.ebc are refpedtively equal to ab, bd, Z-ABD. 
 Alfo the triangle EBC is half the pai-allelogram A E (108^ 
 
 For they ftand upon the fame bafe eb, and are between the fame parallels 
 EB and AC ; BA making right angles with b aj>d ca continued. 
 Likewife the triangle abd is half the parallelogram bf. (i^S) 
 
 For they fland upon the fame bafe bd, and are between the fame parallels 
 
 BD, AF. 
 
 Therefore, as the halves of the parallelograms ea and bf are equal, con- 
 fequently the parallelogram bf is equal to the fquare ae. (49) 
 
 In the fame manner may it be ihewn, that the parallelogram CF is equal 
 to the fquare ag. 
 
 But the parallelograms bf and cf together, make the fquare cd. 
 Therefore the fquare cd is equal to the fquares ea and ag/ 
 
 112, CoROL. I. Hence if any two fides of a right angled triangle are 
 known, the other fide is alfo known. 
 
 For Bcrr fquare root of the fum of the fquares of ac and ab. 
 
 Acrrfquare root of the difference of the fquares of bc and ab. 
 AB::=:rquare root of the difference of the fquares of bc and ac. 
 
 113. Or thus, making the quantities B(j% ab% ac^, to ftand for the 
 fquares made on thofe lines. 
 
 And the mark V to ftand for the fquare root of fuch quantities as ftand 
 under the line joined to the top of this mark. 
 
 Then bc=:v'acj+-ab^j 
 Acr: v'bc"' ab""; 
 
 ABrr v'bc'' AC . 
 
 Scholium. The lines of the lengths 5, 4, 3, (or their doubles, triples, 
 &;c.) will form a right angled trian< ' 
 
 igle. 
 For 5*=4*-J-3\ Or 25 = 16+9. 
 
 1:4. CoROL*
 
 Book Ih 
 
 0' d M E T R Y. 
 
 ^3 
 
 114. CoROL. II. Of all the lines drawn from a given point to a given 
 line, the perpendicular is the ihorteft, 
 
 115. CoROL. III. The fhorteft diftance between two parallel right 
 lines, is a right line drawn from one to the other perpendicular to both-. 
 
 116. CoROL. IV. Parallel right lines are equidiftant ; and the contrary. 
 For two oppofite fidgs of a redangular paraiklogram are eqtial {28) j and 
 each is the fhorteft diftance between the other fides. 
 
 117. 
 
 THEOREM XIV. 
 
 If a right line -ab i/e divided into any tzuo parts AC, CB ; then vfill 
 the fquare on the whole line he equal to the fum vf the fquar4S on the parts, 
 together ivlth two redangles under the two parts. 
 
 That is AB^'^IAC^ + CB^-i-SXACXCB.* 
 
 Dem.- Let AD, AF, be fquares on ab, ac. (68) 
 
 Then will fg, and gd, be each equal .to CB. (48) 
 
 Hence fd is a fquare on a line equal to CB, (30) 
 
 Alfo fb and fe are rectangles on lines equal to 
 
 AC, CB. 
 
 But the fquares af, fd, and the recflangles fb, fe, fill 
 up the fquare ad, or are equal to ad. E 
 
 118. CoROL. I. Hence the fquare of ac the difference between two 
 lines AB, CB, is equal to the fquare of the greater ab, lefTened by the 
 fquare of the lefs cb and by two rectangles under the lefler line CB and 
 the faid difference. 
 
 A. c 
 
 B' 
 G 
 
 
 J> 
 
 That is ab BC^'^AB* BC* 2EC X AC, 
 
 For AB ec=:ac. 
 
 Then ac''=:ab'' bU^ 2ec x ac. . 
 
 (48) 
 
 119. CoROL. II. The difference between the fquares on two lines 
 AB, AC is equal to the rectangle under die fum ab-j-bc and difference ab 
 EC of thofe lines. 
 
 Thatjs ah"" AC* = AB + BC X AB AC. 
 
 For AB^ ac^ = cb' 4- 2ac x cb. 
 
 r=CB+Ac:4-AC X CB. 
 
 ab+ac X (cb=:)ab AC. 
 
 (117,48) 
 
 The reJaugk under tivo lints is generally exprejjed hy 3 letters ; the f.rjl tnx<o 
 letters J} and for one line, and the lujl t wo for the other line. 
 Th:iS for AC XCB, ;'/ ivritten ACii. 
 for An Xcc, is ivritten aqc. 
 And for : X \CXcb, is iviitten 2AC15. 
 
 120. THEO-
 
 64 
 
 GEOMETRY. 
 
 Book II. 
 
 120. 
 
 THEOREM XV. 
 
 In every triangle, adc the fquare of a fide CD fubtending an acuta 
 tingle A, is equal to the fquares of the fides ad, AC about that angle^ 
 lefj'ened by two reSiangles under one of thofe fides AC and that part ab con^ 
 tained between the acute angle and the perpendicular db drawn to that fidt 
 AC from its oppofite angle D 
 
 That is, DC* = AD* + AC* 2cab. 
 
 DeM. DC* BC*=(dB*=)aD* AB*. (ln) 
 
 And AC* = 2AEC + BC*+AB*. (II7) 
 
 Then dc* bc* ac^ = ad* bc* 2 ab* 2 a bc . 
 
 (48) A 
 
 And dc*=ad* + ac* 2AB* 2ABC, by adding BC* + AC*. (47) 
 
 ( AB X 2AB BC X 2ab) 
 ( AB + BC X2AB ) 
 
 Then dc*=ad*+ac* (acX2ab = )2cab. 
 121. CoRoi,. Hence ab=: 
 
 2CA 
 
 122. 
 
 THEOREM XVI. 
 
 In an obtufe angled triangle acd, the fquare of the fide AD oppofite to the 
 ohtufe angle c is equal to the fum of the fquares of the fides AC, CD about the 
 obtufe angle ; together with two rectangles under one fide AC, and the ccnti^ 
 nuation CB of that fide to meet a perpendicular DB drawn to it from the op' 
 pofite angle D. 
 
 That is, ad*=2ac* + cd* + 2acxcb. 
 
 Dem. For^^ Ib*=(db' = )cd* cb*. (hi) 
 
 And ab*=2acb + ac* + cb% (117) 
 
 Then ad* =ac*+cd* + 2acb. (47) 
 
 123. CoROL, Hence cb=;- 
 
 AD' - AC" 
 2 AC 
 
 8 
 
 124. THEO.
 
 Book II. 
 
 GEOMETRY. 
 
 5 
 
 124. 
 
 THEOREM XVir. 
 
 In any circle, a diameter, ab, drawn perpendicular 
 to a chord, de, bifeSfs that chord and its fubtendkd 
 arc DBE. 
 
 Dem. From the center c, draw the radii CD, c e, 
 to the extremities of the chord de. 
 Then the triangles cfe, cfd, are congruous. (102) 
 For CF being at right angles to de, the /.CFDn Z-CFE. 
 And the triangle cde being ifofceles (23), the Z.d=Z.E (104). 
 CF is common. 
 
 Therefore df=fe: And the arc db = be : 
 
 For thofe arcs meafure the equal angles fce, fcd. (15) 
 
 1 25. CoRoL. Hence, in a circle, a right line drawn through the middle 
 
 of a chord at right angles to it, pafies through the center of that circle; 
 
 and the contrary. 
 
 126. T H E O R E M XVII [. 
 A tangent, ab, to a circle is perpendicular to a 
 
 diameter, dc, drawn to the point of contact, c, 
 
 Dem. If it be denied that dc is perpendicular to ab. 
 
 Then from the center d, let feme other line de, 
 
 cutting the circle in E,be drawn perpendicular to ab. 
 
 Now the angle dbg being right, the angle dcb is acute. (97) 
 
 Confequently dc is greater than db. (106) 
 
 But dcztde (9). Therefore de is greater than db, which is abfurd. 
 
 Therefore no other line pafling through the center can be perpendicular to 
 
 the tangent, but that which meets it at the point of contadl. 
 
 127. THEOREM XIX. 
 An angle, BCD, at the center of a circle, is double 
 
 cf the angle, bad at the circumference, when thofe 
 angles Jiand on the fame arc, bd. 
 
 Dem. Through the point a draw the diameter ae. 
 
 l^hen the angle ecd Z_cad+ Z.cda. (96) 
 
 But the Z_cad:=z.cda. (104) 
 
 Therefore the Z. ecd is equal to twice the angle cad. 
 
 In the fame manner it may be fhewn, that the angle bce is equal to 
 
 twice the angle bae. 
 
 Confequently the angle BCD (=rz.ECD+ * Z.bce) is equal to twice the 
 
 angle bad ( = Z.ead+Z.bae}. '^ (47,48) 
 
 128. CoROL. I. Hence an angle, bad, at the circumference is mca- 
 fured by half the arc, bd, on vhich it ftands. 
 
 For the angle at the center BCD is meafured by the arc bd. (15) 
 
 Confequently the angle BAD::=half the angle bcd, is meafured by half 
 the arc db. 
 
 129. CoRoL. II. All angles in the circumference, and ftandino- on the 
 fame arc, arc equal. 
 
 The inark + , iignifics the fum or difiVrcncr. 
 
 Vol. I. 
 
 130. THEO-
 
 66 
 
 130, 
 
 GEOMETRY. 
 THEOREM XX. 
 
 An angle i BAC, in a femicircle^ it aright one. 
 
 An angle i DAC, in a fegmentlefs than a femi circle , 
 is obtufe. 
 
 An angle ^ EAC, in afegment greater than a femi- 
 circUy is acute. 
 
 Book II. 
 
 Dem. For the angle bac is meafured by half the femiclrcular arc bec, 
 or is meafured by half of i8o degrees; that is, by 90 degrees. (128, 16) 
 -And Z.DAC is meafured by half the arc dec, greater than 180 *. 
 Alfo Z.EAC is meafured by half the arc EC, lefs than 180". 
 Therefore thefe angles are refpedlively equal to, greater, or lefs than 90 
 degrees. 
 
 131. CoR. Hence in a right angled triangle bag ; the angular point 
 A of the right angle, and the ends, b, c, of the oppofite fide, are 
 equally diftant from f, the middle of that fide ; that is, a circle will al- 
 ways pafs through the right angle, and the ends of its oppofite fide 
 taken as a diameter. 
 
 132. 
 
 THEOREM XXI. 
 
 TJie angle acd, formed by a tangent ab, to a 
 circle., cde, and a chords CD, drawn from the 
 point of contaSf., c, is equal to an angle, ced, 
 in the alternate fegynent ; and is meafured by half 
 the arc dc of the included fegment. 
 
 Dem. Draw the diameter fc, and join df. 
 The U acf and cdf arc both right. 
 Therefore the z.' dcf and dfc are together =: a right Z.. 
 But the Z.' acd and dcf are together = a right Z.. 
 Confequently the Z. dcf and dfc = the Z.' acd and dcf. 
 Take the Z. dcf from each, and the Z. dfc = the Z. acd. 
 But tiic Z.' dfc and dec are equal. 
 Conftquently the Z.' dec and acd arc equal alfo. 
 
 (126, 130) 
 (96) 
 
 (46> 
 
 (48) 
 
 (129) 
 
 (46) 
 
 * A fmall put above any figure, figriifies degrees. 
 
 133. THEO-
 
 i: Book II. 
 
 133- 
 
 GEOMETRY. 
 
 THEOREM XXII. 
 
 BeHveen a circular arc AHF, and its tangent ae, '' ^ "'' 
 no right line can be drawn from the point of contuSi a. ^ 
 
 Dem. For if any other right line can be drawn, let It be the right 
 
 line AB. 
 
 From D, the center of ahf, draw dg perpendicular to ab, cutting ab in 
 
 G, and the arc in h. 
 
 Now as z. dga is right ; therefore da is greater than dg. (io6) 
 
 But DA = DH (9). Therefore dh is greater than dg, which is abfurd. 
 
 Confequently no right line can be drawn between the tangent ae and the 
 
 arc AHF. 
 
 134. CoROL. I. Hence the angle dah, contained between the radius 
 DA, and an arc ah, is greater than any right lined acute angle. 
 
 For a right line ab muft be drawn from a, between the tangent ae and 
 radius ad, to make an acute angle. 
 
 But no fuch right line can be drawn between ae and the arc ah. (133) 
 
 135. CoROL. II. Hence the angle eah, between the tangent ea and 
 arc AH, is lefs than any right lined acute angle. 
 
 136. CoROL. III. Hence it follows, that at the point of contal the 
 arc has the fame dire6lion as the tangent, and is at right angles to the 
 radius drawn to that point. 
 
 137' 
 
 THEOREM XXIII. 
 
 If two right lines^ AB, CD, interfeSl any hozu (in 
 within a circle^ their inclination.^ aed, or CEB, is m 
 fured by half the fum of the intercepted arcs 
 
 ad, 
 
 Uem. For drawing db ; (43) 
 
 The L. ad=:z. edb+ Z. ebd. (96) 
 
 But the Z.EDB is meafurcd by \ arc CB. (1^8) 
 
 And the /_ebd is meafured by \ arc ad. (128) 
 
 Confequently the Z.AED is meafured by half the arc cb, together with 
 
 half the arc ad. ' (50) 
 
 F 2 
 
 SECTION
 
 68 . GEOMETRY., Book II. 
 
 SECTION IV. 
 Of Proportion, 
 
 Definitions and Principles. 
 
 138. One quantity A, h faid to he rneafured or divided by another quantity 
 B, when A contains hfonie number 0/ times, cxailly. 
 
 Thus if A = 20, and 8 = 5 ; then A contains b four times. 
 
 A is called a multiple of b ; and b is faid to be part of a. 
 
 139. If a quantity A (rz20) contains another b (=5) ^-^ many times as 
 a quantity c (=24) contains another d (=6) ; then 
 A and c are called like multiples of ^ and D. 
 B and D ar-e called like parts of a and c : And 
 a is faid to have the fame relation to B, as c has to D. 
 
 Or, like multiples of quantities are produced, by taking their Rectan- 
 gle, or Produ6l, by the fame quantity, or by equal quantities. 
 
 The Rectangle or Produ(3: of quantities, a and b, is exprefled by writ- 
 ing this mark x between them. Thusj axb, or b X a, exprelfes the 
 redlangle contained by a and b. 
 
 1 40 IVhen two quantities of a like kind are compared together^ the relation 
 ivhich one of them has to the other^ in refpeB to quantity^ is called Ratio. 
 
 The firft term of a ratio, or the quantity compared, is called the Ante- 
 cedent ; and the fecond term, or the quantity compared to, is called tlie 
 Confequent. 
 
 A ratio is ufually denoted by fetting the antecedent above the confe- 
 quent with a line drawn between them. 
 
 Thus- fignifies, and is thus to be read, the ratio of a to b. 
 
 B ^ 
 
 The multiple of a ratio-, is the product of each of its terms by the fame 
 quantity, or by equal quantities. Thus is the ratio - taken c times. 
 
 * X C u 
 
 The produft of two or more ratios, -, - , is exprefled by taking the pro- 
 dut of the antecedents for a new antecedent, nnd the proJud of the con- 
 fequents for a new confequent. Thus =:-x-. 
 
 ^ ^ B X D B D 
 
 141. Equal ratios are thofe ivhere the antacdcr.ts are like multiples or parts 
 cf their refpelive co7ifeqnents. 
 
 Thus in the quantities a, e, c, d : Or 20, 5, 24, 6. 
 
 In the ratio of a to b, or of 20 to 5, the antecedent is a multiple of its 
 confequent four times. 
 
 And in the ratio of c to d, or 24 to 6, the antecedent is a multiple of 
 its confequent four times. 
 That is, the ratio of a to b is the fame as tlie ratio of c to D. 
 
 And this equality of ratios is thus exprefled, - - , 
 
 142. Ratio
 
 Book II. 
 
 GEOMETRY. 
 
 69 
 
 142. Ratio of equality isy when the antecedent is equal to the confeqttent, 
 
 ^ , A A B . . - ,. " 
 
 Thus when a = b, then , or -, or -, is a ratio of equahty. 
 BAB ^ ' 
 
 143. Four quantities are faid to he proportional^ which^ when compared 
 together by two and two, are found to have equal ratios. 
 
 Thus, let the quantities to be compared be a, b, c, d : Or 20, 5, 24, 6, 
 Now in the ratio of a to b, or of 20 to 5 ; a contains b four times. 
 And in the ratio of c to d, or of 24 to 6 j c contains d four times. 
 
 __ A C 
 
 Then the ratios of a to b, and of c to d, are equal : Or =-. (141 ) 
 
 And their proportionality is thus exprefTed, a : b : : c : d. (75) 
 
 Alfo in the ratio of a to c, or of 20 to 24 ; c contains A, once and \. 
 And in the ratio of e to d, or of 5 to 6 j d contains b, once and |. 
 
 A B 
 
 Where the ratios are likewife equal, vi-z,, =-. 
 
 CD. 
 
 And thefe are alfo proportional, a : c : : b : d. 
 
 144." ^0 that when four quantities of the fa?ne kind are proportional, the 
 ratio between the firji and fecond is equal to the ratio between the third and 
 fourth ; and this proportionality is called Diref. 
 
 145. jllfo the ratio between the firfi and third is equal to the ratio between 
 the fecond and fourth ; and this proportionality is called Alternate, 
 
 146. Similar f sr like, right lined figures, are thofe which are equiangular, 
 (that is, the feveral angles of which are equal one to the other ;) and alfo, 
 the fides about the equal angles proportional. 
 
 Thus if the figures ac and eg are equiangular, 
 And ab : EC : : F : FG ; Or bc : CD : : fg : GH ; 
 
 Then are thofe figures called fimilar, or like figures. 
 And the like in triangles, or other figures. 
 
 147. Like arcs, chords, or tangents, in different circles, are thoft which 
 fuhtend, or are oppofite to, equal angles at the center. 
 
 Let F be the center of two concentric arcs aeb, 
 aeb, terminated by the radii f^a, rbs, produced ; 
 ab, ab, their chords, and CD, cd, their tangents. 
 Then as the angle cfd is mealured either by the 
 arc aeb, or aeb, thofe arcs are faid to be alike, 
 or fimilar ; that is, the arc aeb is the fame part 
 of its whole circumference, as the arc aeb is of 
 its whole circumference. 
 
 F 3 148. THEO.-
 
 70 GEOMETRY. Book II. 
 
 148. r H E O R E M XXIV. * , 
 
 ^iantlt'ies, and their like multiples^ have the fame ratio. 
 That is, the ratio of a to b is equal to the ratio of twice a to twice b, 
 
 A 2A ^A CXA 
 
 or thrice a to thrice b, ^c. Or thus = ^y ^c.zz. ; that is, 
 
 B 2B 33 C X B 
 
 equal to the ratio of c times a to c times b. 
 
 Dem. For the ratio of a to b muft either be equal to the ratio of like 
 
 multiples of a and b, or to the ratio of unlike multiples of them. 
 
 Now fuppofe the ratio of a to b is equal to the ratio of their unlike 
 
 A C X a 
 
 multiples, c times a, d times b ; that is, r: . 
 
 B D X B 
 
 Then A : B ; : ex A : DXB (143). And a : c x a : : b : dXb. (145) 
 
 A B 
 
 Therefore = ("144). Where the confequents are unequal 
 
 cxadxb^^^' 
 
 multiples of their antecedents, by fuppofition. 
 
 But is not equal to . (141) 
 
 CXA DXB \ -T / 
 
 Then i\:cxa:;b:qxb is not true. Alfo a:b;:cxa:dxb is 
 not true. 
 
 Confequently is unequal to . , 
 
 ^ B ^ DXB 
 
 Therefore the ratio of unlike multiples of two quantities, is not equal to 
 the ratio of thofe quantities. 
 
 Confequently the ratio of two quantities, and the ratio of their like mul- 
 tiples, are the fame. Or = . 
 
 *^ B ex b 
 
 149. Cor. I. In any ratio, if both terms contain the fame quantity 
 or quantities j the value of the ratio will not be altered by omitting, or 
 
 taking away thofe quantities. For -, by taking away c. 
 
 C ^ B B 
 
 150. Cor. II. Quantities, and their like parts, have equal ratios. 
 For a and b are like parts of c X a and c X b. 
 
 151. Cor. llf. Quantities, and their like multiples, or like parts, arc 
 proportional. For a:b: :c X a:c xb. And c X a: e xb : : a : b (148) 
 
 152. Cor. IV. If quantities are equal, their like multiples, or like 
 
 parts, are alfo equal. For if Ar:B : and rr ; 
 
 B ex B 
 
 Then are the antecedents and confequents in a ratio of equality. (141) 
 
 153. Cor. V. If the parts of one quantity are proportional to the parts 
 of another quantity, they are like parts of their refpective quantities. 
 
 For only like parts are proportional to their wholes, (iSi) 
 
 154. Cor. V^I. Ratios, which are equal to the fame ratio, are equal to 
 
 T- . p A e X A D X a , . , n X 
 
 one anotner. for tne ratio of ~ rr- , irr. fi4o) 
 
 B CX B DXB 
 
 155. CoRt
 
 BookIL GEOMETRY. 71 
 
 155. Cor. VII. Proportions, which are the fame to the fame propor- 
 tion, are the fame to one another. 
 
 If A : B : : c : D ; and a : b : : : f j Then c : d : : e ; f. 
 
 For-=-: and =- (144). Then - = - (4.6) 
 
 B c' B F D F ^^ ' 
 
 156. Cor. VIII. If two ratios or produ6ls are equal, their like multiples, 
 either by the fame or by equal quantities, or by equal ratios, are alfo equal. 
 
 ..A c _, AXE ex 
 That is, it-=:-: Then __ = -_. 
 
 B D BXE DXE 
 
 AXE CXF. 
 
 And if e=:f : Then 
 
 BXE D XF 
 
 And if - = -: Then rr . 
 
 FH BXFDXH 
 
 For in either cafe, the ratios may be confidered as quantities. 
 
 157. THEOREM XXV. 
 
 Equal quantities, a and b, have the fame ratio or proportion to another 
 quantity C. And any quantity has the fame ratio to equal quantities. 
 That is, if a = b : Then a ; c : : b : c. And c : a : : c : b. 
 
 Dem. Since a=:b; then c is the like multiple, or part of b, as it is of a. 
 And a:b::c:c(i5i). Therefore a : c : : B : c. (i45) 
 
 Alfo c : c : : a : B (151). Therefore c : a :: c : b. (^45; 
 
 158. Cor. I. Hence, when the antecedents are equal, the confequents 
 are equal ; and the contrary. 
 
 159. Cor. II. Quantities are equal, which have the fame ratio to an- 
 other quantity : or to like multiples or parts of another quantity. 
 Thus, if a : c : : B : c. Then a=:b. 
 
 160. CoR. III. Since a : c : : b : c ; and c : a : : c : b. Therefore, 
 when four quantities are in proportion. As antecedent is to confequent, 
 fo is antecedent to confequent : Then fhall the firft confequent be to its 
 antecedent, as the fecond confequent to its antecedent: and this is called 
 the inverfion of i alios. 
 
 161. THEOREM XXVI. 
 
 In two, or more, fets of proportional quantities^ the reiangles under the 
 like terms are proportional. 
 That is, if A : B : : c : D ; and e : f : : G : h. 
 Then axe:bxf::cxg: dxh. 
 
 A_C E_G 
 
 Dem. Smce - = -; and-=-. (144) 
 
 Therefore =r . fj^C) 
 
 BXFDXH VJ/ 
 
 Confequently axe:8Xf::cxg:dxh. ( 143} 
 
 F 4 162. THEO-
 
 72 
 
 l62. 
 
 GEOMETRY, 
 THEOREM XXVII. 
 
 Book II. 
 
 In four proportional quantities a : b : : c : d. Then the Re^angle or 
 Produ^ of the ttuo extremes is equal to the Rectangle or FroduSl of the two 
 means. That is, a X d=b X c. 
 
 A C 
 
 Dem. Since a : b ; : c : d by fuppofition. Therefore -=- (144) 
 
 . .aaxd,,..^ccxb . , . 
 
 And -= ^(156). Alfo '^^-;^^ (156} 
 
 B BX D' 
 
 D XB 
 
 A X D C X B 
 
 Therefore = (4.6), where the confequents are equal. 
 
 BXDDXB^^'' ^ ^ 
 
 Confequently aXd = c XB. C'SS) 
 
 163. Hence, if the RevSlangle or Produdl of two quantities is equal to 
 the Re(5langle or Produil of other two quantities} thofe four quantities 
 are proportional. 
 
 Thus, fuppofe the two Rectangles, X, Z, are equal ; 
 Where a, c, are their lengths, and b, d, their breadths. 
 
 Then axb = cxd by fuppofition. 
 Therefore a : c : : D : B, 
 
 7'hat is, As the length of X is to the length of Z. 
 So the breadth of Z is to the breadth of X. 
 
 B 
 
 X 
 
 D 
 
 In fuch cafes, the lengths are faid to be to one another reciprocally, as 
 
 their breadths. 
 
 Or that proportion a : c : : d : b is reciprocal, when a x b = c X d. 
 
 164. 
 
 THEOREM XXVIII, 
 
 If four quantities are proportional \ then will either of the extremes^ and 
 the ratio of the produSl of the means to the other extreme., be in the ratio of 
 tquality : And either nuan., and the ratio of the prcduSl of the extremes to tht 
 ether mean, will be alfo in a ratio of equality. 
 
 r^, . .r rx., BXC . , AXD 
 
 That IS, if A : B : : c : D. Then a=: And sr: - 
 
 D c 
 
 Dem. Since a : b : : c : d by fuppofition. 
 
 Therefore a x d:=b X c. 
 
 AXD B xc 
 
 And _-=-^(i57). 
 
 B X C 
 
 But B r^-- (149), 
 
 AXD 
 
 Therefore \ = b. 
 
 Alfo 
 And 
 And 
 
 bxc^axd 
 
 D ~ D ' 
 AXD 
 
 B X C 
 
 D 
 
 = A. 
 
 (16a) 
 (157} 
 
 (M9) 
 
 (46} 
 165. THEO-^
 
 Book II. 
 
 GEOMETRY, 
 
 73 
 
 165. THEOREM XXIX. 
 
 In any plane triangle^ ABC, any tiuo adjoining 
 fide y AB, AC, are cut proportionally by a line de, 
 drawn parallel to the other fide BC, -y/z. 
 AD : DB : : AE : EC. , 
 
 Dem. Through b and c draw b3, c^, at right an- 
 gles to BC, meeting ha^ drawn through A, parallel 'S > T^Q. 
 to BC : 'I'hrough />, q., the middles of A^, A<?, draw //, jy, parallel to b( 
 or c, meetinjr ab, ac, in the points d^ c ; and join dc. 
 Now the triai.gles a^/>, zdp^ and ac^, zcq^ are congruous. (95jIOo) 
 T'^'erefore Adzz^d^ aczzcc, pdzzpd^ qc'=.qc, 
 
 Bui f'f=:qq .'116) ; Therefore /)i/=:^r. (49) 
 
 And dc is parallel to BC. (116) 
 
 In the fame manner it may be fliewn, that lines parallel to b3, drawn 
 through the middles of a/>, pb ; a^, qa ; will alfo bife<Sl Ac/, Bd ; Af, cc ; 
 and that lines joining thefe points of bife<Stion will alfo be parallel to Bc : 
 And the fame may be proved at any other bifetions of the fegments of 
 the lines ab, ac : Alfo the like may be readily inferred at any other di- 
 vifions of the lines aZ-, Aa. 
 
 Therefore lines parallel to bc, cut off like parts from the lines ab, ac. 
 Then ab : ac : : Ad : A. And ab : AC : : bd : CE. (151) 
 
 Therefore ad : ae : : bd : ce. (^SS) 
 
 And by Alternation ad : bd : : AE : ce. - (i4S) 
 
 166. Cor. Hence, when the fides ab, ac, of a triangle are cut 
 proportionally, in d, e, the fegments ad, ae j db, ec; of thofe fides are 
 proportional to the fides : And the line de, drawn to thofe feilions, is 
 parallel to the other fide bc. 
 
 167. 
 
 THEOREM XXX. 
 
 Zi. 
 
 (95) 
 
 In equiangular triangles., ABC, abc, the fides about the 
 tqual angles are proportional ; and the fides oppofite to 
 equal angles are alfo proportional. 
 
 Dem. In ca, cb, take CD=f^, cE = cij 
 
 de. 
 
 Then the triangles cde, cab^ being congruous. (99^ \ 
 
 TheZ.CDE:r:^Z.a=^) Z.A. Therefore de is parallel to ab. 
 
 In the fame manner, taking ATzzac^ Aczzab; alfo BHzzbc, Blzzba\ 
 and drawing FG, hi, the triangles agf, ibh, abc, arc congruous j there- 
 fore FG is parallel to cb, and hi is parallel to cA. 
 Then {cv = ) ca : ca : : {ce =:) cb : cb. "l 
 
 {A-f ) ca : c A ::{ AG zz) ab : AB. > (165) 
 
 (BHzrJ bc : BC : : (bi ) ab : ab. j 
 
 168. Cor. Hence, Triangles have one angle in each equal, and 
 the fides about thofe ctjual angles proportional, thofe triangles are equian- 
 gular and fimilar. 
 
 J69. THEO-
 
 74 GEOMETRY. 
 
 169. THEOREM XXXr. 
 
 In a right angled triangle, ABC, if a line, BD, he 
 drawn from the right angle B, perpendicular to the 
 cppojite fide, AC ; then will the triangles abd, bcd, 
 en each fide the perpendicular, be fimilar to the whole 
 ABC, ar^ to one another. 
 
 Dem. For in the triangles abc, adb, the A a is common; 
 
 And the right angle abc =: right angle adb. 
 
 Therefore the remaining Ac= Z.ABD. (98) 
 
 In the fame manner it will appear, that the triangles ABC, BDC, are like. 
 
 Therefore the triangles abd, bcd, are alfo fimilar. 
 
 170. Cor. I. Hence, AC : ab : : ab : ad 
 
 AC 
 AD 
 
 BC : : bc : DC 
 DB : : DB : dc 
 
 \ 
 
 (167) 
 
 171. Cor. II. Hence a right line bd, drawn from a circumference 
 of a circle perpendicular to the diameter AC, is a mean proportional 
 between the fegments ad, dc, of the diameter. 
 
 And ADXDcrzDB*. (162) 
 
 For a circle, the diameter of which is ac, will pafs through a, b, c. (131) 
 
 Scholium. This corollary includes what is ufually called one of the 
 chief properties of the circle, namely ; 
 
 the fquare of the Ordinate is equal to the reSlangle under the tw 
 Abfciffas. 
 
 Here, the ordinate is the perpendicular bd ; and the two Abfciflas arc 
 the two fegments ad, dc, of the diameter ac. 
 
 172 
 
 THEOREM XXXIi. 
 
 In a circle, if two chords, AB, CD, interfeSi each 
 tther in E, either within the circle, or without, by pro- 
 longing them ; then the ref angle under the fegments, 
 terminated by the circumference and their interfeiion, 
 will be equal. 
 
 That is, AE X ehztce X ed. 
 
 Dem. Draw the lines bc, da. 
 
 Then the triangles dea, bec, are fimilar. 
 
 For the angle at e is equal (93), or common. 
 
 And the Z.Dr=z.B, as (landing on the fame arc 
 
 AC (129). Then the other angles are equal. (98) 
 
 Therefore ae : CE : : ED : eb. (^^7) 
 
 Confequently AE XEB=:cEXED. (1621) 
 
 173. THEO.
 
 I::: 
 
 Book II. G E O M E T R Y. 
 
 75 
 
 >m 173- 
 
 THEOREM XXXIII. 
 
 If with the leajl fide AB of a given triangle ABC, a femicircU he defcribtd 
 from the angular point A ; meeting the fide AC, produced in the points D, E j 
 and from b, the lines BE, bd, be drawn, and alfo bg perpendicular to i> E : 
 Then the values of the fever al lines AG, CG, GE, gd, be, bd, bg, may h< 
 exprefj'ed in terms of the fides of the triangle ABC, as follow. 
 
 -^JB 
 
 B 
 
 174. AG 
 
 Or AG 
 
 175- 
 
 CG 
 
 _BC 
 
 'AC* 
 
 AB 
 
 
 2AC 
 
 
 _AC 
 
 ^+Ib^ 
 
 ~bT* 
 
 
 2AC 
 
 
 _ AC 
 
 * Ti^ 
 
 +ic" 
 
 And ac4'ag=:ac + 
 
 2AC 
 
 BC* ^AC* AB 
 
 2AC 
 2Tc* -!- BC* AC* Tb 
 
 2AC 
 
 . (123) 
 
 . (121) 
 
 . For CG=:AC-f AG, or to AC AGi 
 
 (174) 
 
 (149} 
 
 176. 
 
 2HXH CB rr 
 
 GE=: ; Here 2H=:Ac4-AB-f BC. 
 
 2AC 
 
 ~ . I AC^'-f- AB* CB* 
 
 For ge=:ab(ae}+ag=:ab4-^^ "^^^ ~^ " . (174) 
 
 2AC 
 __ 2AC X AB-j-AC^-f-AB* BC* 
 ~ 2AC 
 
 But 2acxab4-ac*+ab*=(ac+ab*=;)(5*. (117) 
 
 CE^ Bc' 
 
 (149) 
 
 ThencE 
 
 \ 2AC / 
 
 CK-f^BCXCE BC 
 2AC 
 
 . (119) 
 
 _ CA+AB + RC X CA + AB BC 
 ~ 2AC 
 
 _2H X2H zBC_2H Xh BC 
 
 2AC 
 
 AC 
 
 177, CDS
 
 GEOMETRY. 
 
 Book 11. 
 
 H -AC X zH -lAB 
 
 177. GD=:- 
 
 AC 
 
 For OD = AD :j: AC = AB T AG = AB 
 
 . Here 2H = AC + AB + BC. 
 AC* AB* + BC* 
 
 2AC 
 _ 2ACX \B- AC* ^ar' + DC' 
 
 2AC 
 
 _AC AB* + BC* B-* CD* 
 
 (174) 
 
 2AC 
 
 2 AC 
 
 __BC CDXBC + CD 
 
 (119) 
 
 2aC 
 
 _BC + AB ACXBC+AO-AB 
 
 2AC 
 
 
 (118) 
 
 iH 2ACX2H zAB 
 2AC 
 
 178. BE = yr^X2HXH CB. 
 For BE* = DEXGE. (170) 
 
 CE BC , /i\ 
 
 = 2ABX-- ~. (176) 
 
 2AC 
 
 Therefore BE = -y^xcI^-Bc* = y^X2HXH::cB. (176) 
 
 179. 
 
 VTb ^ - 
 
 XH-.ACX2H lAc 
 
 For bd*=deXgd. (170) 
 
 :2ABX- 
 
 BC CD 
 
 2AC 
 
 (177) 
 
 Therefore BD=:*/^x 2^'' CD " ('77) 
 
 180. BGr= xVhX HCB X H AC X h3 
 
 AB. 
 
 For bg*:=:gexgd. Therefore bg=: v'ge X v'gd, (170) 
 
 AndGE = ;XHXH CB. (176) GD=-, XH ACXH AB. (177) 
 
 181. CoROL. Hence is derived the Rule ufually given for finding tht 
 area, or fuperficial content, of a Triangle, the three fides being known. 
 
 Rule. i. From half the fum of the three fides, fubtracl each fide feve- 
 rally, noting the three remainders. 
 
 2d. Multiply the faid half fum, and the three noted remainders con- 
 tinually. 
 
 3d. The fquarc root of the produdl is the area of the Triangle. 
 
 182. THEO-
 
 Book II. ' GEOMETRY. 77 
 
 182. THEOREM XXXIV. ^ ^ . 
 
 If a regular polygon, abcdef, be infcribed in a //r^-. -^\\*- 
 
 ctrcle\ and parallel to thefe fides if tangents to the f// \ / \\A 
 
 arcle be drawn, meeting one another in the points /vj(^' "XS ^j\/y 
 
 a, bj c, d, e, f; then Jhall the figure formed by ^'A / \ jU 
 
 thefe tangents circumfcribe the circle^ and befimilar - ^^. "^y^/ 
 
 to the infcribed figure. '^''^7~~^-'^ 
 
 Dem. Since the circle touches every fide of the figure abcdef by con- 
 
 ftrudlion ; therefore the circle is circumfcribed by that figure. (41) 
 
 Through a and b, draw the radii sa, sb, prolonged till they meet the 
 
 tangent ab, in a, b. 
 
 Then the triangles asb, asb, are equiangular. 
 
 For the z. at s is common ; and the other angles are equal, becaufe ae 
 
 and ab are parallel, by fuppofition. 
 
 AKo sa=:sb : For the triangles asb, asb, are ifofceles. (104) 
 
 And the fame may be proved of the other triangles ; and alfo, that they 
 
 are equal to one another. 
 
 Therefore the figure a^r^^/ has equal fides, and is equiangular to the 
 
 figure abcdef. 
 
 Now s A : sa :: AB : ab ; and sa : sa : : at : af ( 167) 
 
 Therefore ab : ab : : at : af And the like of the other fides. (i55) 
 
 Confequently the figures abcdef, abcdef are fimilar. (i45) 
 
 183. Cor. I. If two figures are compofed of like fets of fimilar tri- 
 angles, thofe figures are fimilar. 
 
 184. Cor. II. Hence, if from the angles a, b, of a regular polygon 
 circumfcribed a circle, lines as, bs, be drawn to the center s ; the 
 chords AB of the intercepted arcs will be the fides of a fimilar polygon, 
 infcribed in the circle : and the fides ah, ab, of the infcribed and cir- 
 cumfcribing polygons will be parallel. 
 
 185. Cor. III. The chords or tangents of like arcs in diff'erent circles, 
 are in the fame proportion as the radii of thofe circles. 
 
 For if a circle circumfcribe the polygon abcdef; then the fides of the 
 polygons abcdef ABCDEF, are chords of like arcs in their refpeitive cir- 
 cumfcribing circles. 
 
 And if a circle be infcribed in the polygon abcdef, the fides ab, ab, i^c. 
 arc tangents of like arcs alfo ; And thefe hav been (hewn to be propor- 
 tional to their radii sa, sa. 
 
 i86. Cor. IV. The Perimeters of like polygons (or the fum of their 
 fides) arc to one another as the radii of their infcribed or circumfcribed 
 circles. 
 
 For SA.sa: : ab : ab. (182) 
 
 And ab, ab, are like parts of the perimeters of their polygon:.. 
 Therefore sa ; sa : : perimeter abcdef : perimeter abcdef (iSO 
 
 187. THEO-
 
 7 
 
 GEOMETUY. 
 
 Book IL 
 
 187. 
 
 THEOREM XXXV. 
 
 K-1^. 
 
 If there be ttud regular and like polygons applied to 
 the fame circle., the one infer ibed and the other circum- 
 fcrtbed: Then will the circcumfcrence of that circle^ 
 and half the fum of the perimeters of thofe polygons, 
 approach nearer to equality^ as the number of fides in 
 the polygons increafe. 
 
 Dem. It is evident at fight, that the circumfcribing hexagon fghikl 
 is lefs than the circumfcribing fquare abed. 
 
 And alfo that the infcribed hexagon ^/^/i/ is greater than the infcribed 
 fquare abed. 
 
 And in both cafes, the difference between the hexagon and the circle 
 is lefs than the difference between the circle and the fquare. 
 
 Therefore the polygon, whether infcribed or circumfcribed, differs 
 lefs from the circle, as the number of its fides is increafed. 
 
 And when the number of fides in both is very great, the perimeters of 
 the polygons will nearly coincide with the circumference of the circle j 
 for then the difference of the polygonal perimeters becomes fo very fmall, 
 that they may be efteemed as equal. 
 
 And yet fo long as there is any difference between thefe polygons, 
 though ever fo fmall, the circle is greater than the infcribed, and lefs 
 than the circumfcribed polygons : Therefore half their fums may be 
 taken for the circumference of the circle, when the number of thofe fides 
 is very great. 
 
 Hence, the circumferences of circles are in proportion to one an- 
 other, as the radii of thofe circles, or as their diameters. 
 
 For the perimeters of the infcribed and circumfcribing polygons are to 
 one another, as the radii of the circles. (186) 
 
 And thefe perimeters and circumferences continually approach to 
 equality. 
 
 189. THEO-
 
 Book II. G E O M E T R r. 79 
 
 189. THEOREM XXXVI. 
 
 In a circle afb, if lines ^ ba, da, fa, he drawn 
 from the extremities of two equal arcSy BD, df, to meet 
 in that point A of the circutnference determined by one 
 of theniy BA, pajjing through the center ; then Jhall 
 the middU line ad, be a mean proportional between 
 the fum Ah + AY of the extreme lines y and the radius 
 BC ofthatcircle* 
 
 Dem. On D, with the diftance da, cut af produced in e. 
 
 Then drawing de, df, db, the triangles adb, edf, are congruous. (102) 
 
 For /LEFD =:(Z.FDA+Z.FAD (96) = ) Z.dba(i28). Becaufe the arc 
 
 DFA = DF-f-FA. 
 
 And z.e=:Z.fad (104) =Z_dab, by conftrulion j and de = da. 
 
 Therefore EFrrAB; and AE=:AB-f-AF. 
 
 Draw CD ; then the triangles acd, ade, are fimilar. 
 
 For they are Ifofceles and equiangular. 
 
 Therefore AC : AD ;: AD : (ae = ) AB + AF. (167) 
 
 190. Hence, whence the radius of the circle is exprefTed by i, and one 
 of the extreme lines, or chords, pafles through the center; then if the num- 
 ber 2 be added to the other extreme chord, the fquare root of that fum 
 will be equal to the length of the mean chord. 
 
 For fince ac : ad : : ad : ab + af ( 189. ) I'h. ad*=: ac X abxaf. ( 162) 
 Now if AC = i, then AB = 2; And ad''=:2 + af ; becaufe multiplying by 
 r is ufelefs here. Therefore AD=y/2+AF. 
 
 As the arcs bd and dfa make a femicircle, they are called the fupplc- 
 ments of one another: Therefore if the arc bd is any part (as, |, ^, -^V, 
 eft.) of the femicircumference ; then is the line da called the fupple- 
 mental chord of that part. 
 
 191. Remark. In the pofthumous works of the Marquis de k Hof- 
 pitaly (page 319, Englifh edition) this principle is applied to the do^lrine 
 of angular fedtions ; that is, to the dividing of a given arc into any pro- 
 pofed number of equal parts : Or the finding of the chord of any pro- 
 pofed arc. 
 
 For if bf was any aflumed arc, the chord of which had a known ratio to 
 the given radius bc ; then as bfa is a right angled triangle (130), the 
 fide AFr:'/^!^- bP (^13) will alfo be known. And by this Theorem 
 the mean chord ad will be known; and alfo db ( v'ab'^ Xd'') the 
 chord of half the arc bf will alfo be known. 
 
 And by bifefling the arc db in g, and drawing ag, gb, the mean 
 chord AG is known (189) ; and gb ( rrV^Aii'' ag"") is alfo given. 
 
 And in this manner, by a continual bifcction, the ch(^rJ of a very fmall 
 arc may bc obtained ; the pracSticc of which is facilitated by article ( 190) 
 deduced from page 330 of the faid work, 
 
 19^. Ex-
 
 8o 
 
 GEOMETRY. 
 
 Book II. 
 
 192. Example. Required the chord of the 
 Y cj ' i P^^^ ^f *^' circumference of a circle^ the ra- 
 dius of which is I . Or, required the fide of a re- 
 gular polygon 0^3072 fideSy infcribed in a circle, 
 the diameter of which is 2. 
 
 Let ADF be a femicircle, the diameter af = 2, and center c. 
 
 Take the arc AD = y of the femicircumfcrence, or equal to 60 degrees; 
 
 and draw dc, da, df. \ 
 
 Let d reprefent the point where the arc is bifefted ; dp the fupple- 
 mental chord to that bifedlion ; and let the marks d, d", d"\ </'% &c. ex- 
 prefs the bifedted points agreeable to the number of bifeilions. 
 
 193. Now fmce z.ACD=r6o'. 
 Therefore z.cad+z.adc = (i 80 60 = ) 1 20. 
 
 But z. CAD=Z.ADC (104) ; then z. cad= (~^=) 60**. 
 Therefore da = (dc;=aci=) i. 
 
 (98) 
 
 And as the triangle adf is right angled at d. (130) 
 
 Then df=(v'af* adM"3J=\/4 1 = ) v^3=i>73205o8o75688773 
 Therefore the fupplemental chord of the arc 
 AD, or of 
 
 ^of the femicircumfcrence is FD =v3 
 
 I of the fame (190) 
 
 fd 
 
 I 
 
 TT 
 
 t 
 -5-5 
 
 I 
 
 IT? 
 
 I 
 Ti'ff 
 
 is Yd' 
 
 Yd" ->IJ2_ 
 
 Yd'" ->J 2^-Yd" 
 
 Yd''' =ylij^ 
 
 Yd" ^ 2 + Yd'" 
 
 Yd"' ^ 2-fFi'^ 
 
 Yd"''' y i + Fr/"' 
 
 F^''' =v^2 + F^"" 
 
 = 1,7320508075688773 
 = 1,9318516525781366 
 = 1,9828897227476208 
 
 = 1,9957178464772070 
 =1,9989291749527313 
 
 = i9997322758i9i236 
 = i>999933o67834So22 
 = 1,9999832668887013 
 = 1,9999958167178004 
 = 1,9999989541791767 
 
 Now Fi'" the fupplemental chord ofTsV? being known, the chord 
 Ad'^ of TTj-5) P^i't of the femicircumference, or of -j^Vi? P^rt of the whole 
 circumference, is alfo known. 
 
 That is Ad'"^ = ( V^Af ^ F7ix* = ) yj ^j. + Yay'-^'- 0,0020453073606764 
 
 194. Confequentl}', the fide of a regular polygon of 3072 fides, in- 
 fcribed in a cirdc whofe diameter is 2, is 0,0020453073606764 
 
 195. The
 
 Book n. GEOMETRY, dt 
 
 195. The fide of a fimilar polygon c'lrcumfcrib'ing E "^ _ B 
 the fame circle^ the center of which is c, may be thus J^^ 
 
 found. \ 
 
 Let BE be the fide of the clrcumfcribed polygon ; \ 
 
 and draw bc, ec, cutting the circle in d and a. 
 
 Draw DA, and it will be the fide of the infcribed 
 polygon 3 and 15 parallel to BE. (184) ^' 
 
 Draw CI bifeiling the angle bce, and it will blfel be and DA at right 
 angles (103). , And DG r:( |DA=:) | A<af"'. 
 Then cg=:-/cd^ Do^=:/i_J.A:^r. 
 
 But f A</''^=:o,ooio2265368o338. And its fquare is 0,00000104582055 
 Which fubtraiSted from I leaves 0,99999895417945 
 
 Whofe fquare root, or CG, is equal to 0^99999947708959 
 
 Now the triangles cBi, CDG, are fimilar. 
 Then cc : ci : : 2dg : 2bi. (167, 151) 
 
 Therefore (2Bi = ) be=:( (164) = ) ; For ic=:i. 
 
 CG CG 
 
 ^ 0,002045107^606764 _ _ 
 
 Or BE=:-^ ~^-^^ ^=0,0020453084301895 
 
 which is the fide of a regular polygon of 3072 fides, circumfcribing a 
 circle the diameter of which is 2. 
 
 196. Scholium. TJie fide of a regular polygon of 3072 fides, in- 
 fcribed in a circle, the diameter of which is 2, is 0,0020453073606764. 
 ( 194). Which multiplied by 3072, will give the perimeter of that poly- 
 gon, which is 6,2831842119979622. 
 
 The fide of a fimilar polyg(*i, circumfcribing the fame circle is 
 
 0,0020453084301895. (195) 
 Which multiplied by 3072, will give for the perimeter of that po- 
 lygon ^ 6,28 3 1874973420925. 
 The fum of thcfe perimeters is 12,5663717093400547. 
 
 The half fum is 6,28318585, i^c. 
 
 Which is very nearly equal to the circumference of a circle, the diameter 
 of which is 2 (187), the difference between it 
 c and the infcribed polygon being only 0,00000164, &c. 
 
 \ and the circumfcribcd polygon being only 0,00000164, ^c. 
 
 ic)"]. Now the circumferences of circles being in the fame proportion, 
 as their diameters. fiS8) 
 
 Therefore the diameter of a circle being I, 
 
 The circumference will bc 3> H^ 592> ^C' which agrees with the 
 
 circumference as found by other metliods. 
 
 Vol. I. G :if:CTIOr
 
 6i 
 
 GEOMETRY. 
 
 Book II, 
 
 S E C T I O N V. 
 
 Of Planes and Solids, 
 
 Definitions and Principl,es. , 
 
 198. A line is faid to be in a plane, when it pafles through two or more 
 points in that plane; and the common fedtion of two planes is a line 
 which is in both of them. 
 
 199. The inclination of two meeting planes ab, 
 CD, is meafured by an acute angle gfh, made by 
 two right lines fg, fh, one in each plane, and both 
 drawn perpendicular to the common fc6tion de, of 
 thefe planes from f, fome point in it. 
 
 200. A right lljie de interfecSting two fides AC, 
 BC, of a triangle abc, fo as to make angles CDE, 
 CED, within the figure, equ^ to the angles cba, 
 CAB, at the bafe ab, but with contrary fides of the 
 triangle, is faid to be in a fubcontrary pofition to 
 the bafe. 
 
 201. If a circle in an oblique pofition be viewed, 
 it will appear of an oval form, as abcd ; that is, it 
 will feem to be longer one way, as ac, than another, 
 as db; neverthelefs the radii ea, eb, are to be 
 erteemed as equal. And the fame muft be under- 
 ilood in viewing any regular figure, when placed 
 obliquely to the eye. 
 
 202. If a line be fixed to any point c above the 
 plane of a circle adbe, and this line while ftretchcd 
 be moved^ round the circle, fo as always to touch 
 it ; then a folid which would fill the fpace palled 
 over by the line, between the circle and the point A 
 t, is called a Cone. 
 
 203. If the figure adbe had been a polygon, and the firetched line had 
 moved nlong its fides, the figure which would then have been defcribed, 
 is called a Pyramid. 
 
 So that Cones and Pyramids are folids which regularly taper from a 
 circle, or polygon, to a point. 
 
 The circle or polygon is called the Bafe ; and the point c the Vertex, 
 When the vertex is perpendicularly over the middle or center of the 
 bafe, then the folid is called a Right Cone, or a Right Pyramid j 
 cthcrwife an Oblique Cone, or Oblique Pyxamid. 
 
 204. If a Cone or Pyramid be cut by a plane pa/ling through the ver- 
 tex t, and cent^T of ihc bafe r, the fection abc, or tDC, is a triangle. 
 
 g 205. A
 
 Book II. 
 
 GEOMETRY. 
 
 % 
 
 205. A right line ab, is perpendicular to a plane CD, 
 when it makes right angles abe, abf, abg, with all 
 the right lines be, bf, eg, drawn in that plane to touch 
 the faid right line ab. # . 
 
 206. So that from the fame point b, in a plane, only 
 one perpendicular can be drawn to that plane on the 
 fame fide, 
 
 207. A plane ab, is perpendicular to a plane cd, 
 when the right lines ef, gh, drawn in one plane ab, at 
 right angles to fb, the common feftion of the two 
 planes, are alfo at right angles to the other plane cd. 
 
 208. So that a line ef, perpendicular to a plane cd, is in another plane 
 AB, and at right angles to fb, the common fedlion of the two planes. 
 
 209. 
 
 THEOREM XXXVII. 
 
 If two planes ab, cd, cut each other^ their common fe^ion 
 BD, will be a right line. 
 
 N 
 
 B 
 
 
 
 r 
 
 
 A 1^X1 
 
 c 
 
 Dem. P'or if it be not, draw a right line deb in the plane ab, from the 
 point d to the point b ; alfo draw a right line dfb in the plane bc. 
 
 Then two right lines deb, dfb, have the fame terms, and include a 
 fpacc or figure, which is abfurd. (n*^ 
 
 Therefore deb and dfb are not right lines : Neither can any other 
 lines drawn from d to e, befides bd, be right lines. 
 
 Confequently the line Dn, the common fedion of the planes, is a right 
 line. 
 
 210. 
 
 T H E O R E M XXXVIII. 
 
 ' If two planes ab, cd, which are both perpendicular 
 to a third plane ef, cut one another; their interfe^ion 
 HG is at right angles to that third plane ef. 
 
 Dem. For the common feaion of ab and CD Is a right line gh. (209) 
 Alfo HB, HD, arc the common fedions of ab, CD, with the plane ef. 
 
 Now from the point h, a line hg drav/n perpendicular to the plane ef, 
 muft be at right angles to HB, HD. (20^) 
 
 But HC muft bc in both planes ar, cd. (2cB) 
 
 Therefore it muft bc in the common fe6tion of thofe planes. 
 
 Confequently the feclion hg of the planes ab, cd, is at right angles to 
 the plane ef. 
 
 G 2 211. THEO-
 
 54 
 
 GEOMETRY, 
 
 Book 11, 
 
 211. THEOREM XXXIX. 
 
 The felionSy aebd, of a Cone or Pyramid caebd, 
 which ere parallel to the bafe aebd, are^imilar to that 
 bafe, 
 
 Dem. For let afbc, dfec, be fcftions through the 
 vertex Cj and center f of the bafe. 
 
 Then thcfe fe(5t^ions will cut one another in the 
 right line fc (209), and the tranfvcrfe fedion^^^^, 
 in the right lines ab^ and ed^ interfecting iny. 
 
 TTien are the following fets of triangles fimilar ; 
 namely, afc, afc j bfc, bfo. j dfc, dfc j efc, 
 tfc. 
 
 Wherefore fc :/c 
 
 FA ' fa^ 
 
 in 
 
 any 
 
 other 
 
 fedlions 
 (165) 
 
 FB:/^(And the like 
 FD :yW ^through c and f. 
 : : fe :fe i 
 
 Now in the Cone, fai=fb=fd=:fe; therefore fa -zzfb fd=:fe. (152.) 
 So that ail the right lines drawn from /to the circumference of the figure 
 adbe are equal to one another. 
 
 Confequently the figure adbe is a circle. (9) 
 
 And in the Pyramid, fc :yc : : dc : ^c : : db : (f^ : : da '. da. 
 YC : fc :: EC : ec :: EA : ca : : eb : eb. 
 Therefore in each pair of corrcfponding triangles in the bafe and tranfverfe 
 fcclion, the fides are refpccSlively proportional. 
 
 Confequently, as the bafe and tranfverfe fedlion are compofed of like fets 
 cf fimilar triangles j therefore they are alfo fimilar, (183,) 
 
 212, T H E O R E M XL. 
 
 If a Cone ablcK, the bafe of which is a circle 
 CBLCK, be cut by a plane in a fubcontrary poftion 
 it the bafe^ the fs^iion dieh will be a circle. 
 
 Dem. Through the vertex a, and center of the bafe, let the triangular 
 jedion abc be taken, fo as to be at right angles to the planes of the bafe 
 BKCL, of the fubcontrary fedion dieh, and of the fedlion figh, taken 
 parallel to the bafe, and cutting the fubcontrary fclion in the line ioh. 
 Theieforc ioh is perpendicular to de and fg (210) cutting one another 
 in o. 
 
 Now the fec^ion FIGH is a circle (211). Therefore fo X0G= 01*. (171J 
 Again the triangles goe, fod, are fimilar. 
 
 For iLGEo=: z.DFo=z.AEc by conftr. And agoe=:Z.dof. (93) 
 Therefore eg : oc : : fo : do (167.) And EoXDorzFO Xog(i62)=:oi'. 
 So that 01 is a mean proportional, either between Fo and OG, or do and 
 EG. But as the fame would happen wherever fg cuts de ; therefore all 
 the lines 01, both in the fc6tions figh and DIEH, are lines in a Circle. 
 Confequently the Icclion dieh is a circle, 
 
 213. If
 
 Book 11. GEOMETRY. 8| 
 
 213. If the fe<Slion cut both fides of the cone not in a fubcontrary pp- 
 fition to BC, the diameter of the bafe, then the fe6Hon (fuppofe it ftill) 
 DiEH, is called an Elliptic section, which though not a circle, will 
 be a bounded curve, longer one way than the other j and, like a circle, 
 return into itfelf. 
 
 ^ The curve dieh is called an Ellipfis. 
 
 . 214. The line de, the Transverse Diameter or Axij, 
 
 The line oh dr 01, is called an Ordinate. 
 
 115. The ordinate through the middle of de, is called the Conju- 
 gate Axi^ 
 
 The interfedion of the Tranfverfe and Conjugate Axes, is called the 
 
 Center of the Elliplis. , 
 
 2x6. If a circular arc be defcribed, with a radius equal to half the 
 Tranfverfe Axi?, from one end of the Conjugate Axis, its interfeftions 
 with the Tranfverfe Axis, are called Foci, one on each iide of the center 
 of the Ellipfis. 
 
 217. Every right line pafling through the center of the Ellipfis, and 
 terminated at each end by the curve, is called a Diameter. 
 
 a 1 8. The radius that would defcribe a circular arc of the fame curva- 
 ture witji the ellipfis at any point of it, is called the radius of curva- 
 ture. 
 
 A Tangent to any point in the Ellipfis, is a right line perpendicular 
 to the radius of curvature at that point. 
 
 219. Two Diameters being fo drawn, that one is parallel to a tangent, 
 and the other pafi^cs through the point of contacl ; thofe two Diameters 
 arc fuid to be Conjugate Diaimkters ; and have certain relations to 
 their Ordinates, Tangents, Radii of Curvature, and other lines belonging 
 to the Ellipfis. 
 
 220. A third proportional to any two Conjugate Diameters, is called 
 the Parameter. 
 
 G 3 SECTION
 
 tS' GEOMETRY. Book II. 
 
 S E C T I O N VI. 
 
 '^ - 0/ tie Spiral. 
 
 111. Suppofe the radius sf a circle to revolve with an uniform motion 
 round its center^ and while it is fo revolving^ let a point move along the ra- 
 dius ; then will the fucccjjive places of that point he in a curve, which is 
 called afpiral. 
 
 . This will be readily conceived by imagining a fly to move along the 
 fpoke of a wheel, while the wheel is turning round. 
 
 li" while the radius revolves once, the point has moved the length of 
 the radius ; then the fplral will have revolved but once round the center, 
 or pole J confequently the motion in the circumference is to the motion 
 in the radius, as the circumference is to the radius : And if the wheel re- 
 volves twice, thrice, or in any proportion to the motion in the radius ; 
 then the fpiral will make fo many turnsj or parts of a turn, round the 
 center. 
 
 222. Now fuppofe., while the radius revolves equably, a point from the 
 circumference moves towards the center, zuith a motion decreafmg in a geome- 
 tric prcgrejfion 3 then will a fpiral he generated, which is called a propor- 
 tional fpiral. 
 
 Let the radius ca be divided in any continued de- 
 creafmg geometric progreflion (90), as of 10 to 8 ; 
 then the feries of terms will be 10 ; 8 ; 6,4 ; 5,12 ; 
 4*096 ; 3,2768 ; 2,62144, ^c. Alfo let the cir- 
 cumference be divided into any number of equal 
 parts, in the points d, e,f, g, i^c. Then if the feveral 
 divifions of the radius ca be fucceffively transferred 
 from the center c, cutting the other radii in the 
 points D, E, F, G, Cffr. and a curved line be evenly 
 drawn tlirough thofe points, it will be a fpiral of the kind propofed. 
 
 223. From the nature of a decreafmg geometric progreffion, it is eafy 
 to conceive that the radius ca may be continually divided ; and although 
 each fucccfTive divifion becomes fhorter than the next preceding one, yet 
 if ever fo great a number of divifions, or terms, be taken, there will ftill 
 remain a finite magnitude. 
 
 224. Hence it follow*, tliat this fpiral winds continually round the 
 center, and does not fall into it till after an infinite number of re- 
 volutions. 
 
 Alfo, that the number of revolutions decreafe, as the number of the 
 equal partSj into which the circumference is divided, increafes. 
 
 225. T H E O-
 
 Book II. '^G E O M E T RY^ 7 
 
 J25. THEOREM XLL 
 
 Any proportional fpiral aits the intercepted radii at 
 tqual angles. .^ ^ - 
 
 Dem. If the dlvifions a</, de, ef^ fgy l^c. of the 
 circumference were very fmall, then would the fe- 
 veral radii be fo clofe to one another, that the in- 
 tercepted parts AD, DE, EF, FG, &c. of the fpiral, 
 might be taken as right lines. 
 
 And the triangles cad, cde, cef, ^c would be fimilar, having equal 
 angles at the point c, and the fides about thofe angles proportional.' (168) 
 Therefore the angles at a, d, e, f, l^c. being equal, the fpiral muft ne- 
 cefTarily cut the radii at equal angles. 
 
 226. THEOREM XLII. 
 
 If the radii of any proportional fpiral be taken as numbers, then will the 
 correfponding arcs of the circle, reckoned from their commencement , be as the 
 logarithm of thofe numbers, 
 
 Dem. As the lines ca, cd, ce, cf, cg, ^c. are-a feries of terms in 
 geometric progreflion ; and the arcs Ad, Ae, Af Ag, Sec. are a feries of 
 terms in arithmetic progrefHon ; therefore thefe arcs may ferve (I. 66) as 
 the indices to the geometric terms, and be thus placed ; 
 
 Radii of the fpiral c A, CD, CE, CF, CG, ^c. Geometric terms. 
 Correfponding arcs 0, Ad, Ae, Af, Ag, i^c, Arithm. terms, or indices. 
 
 In this difpofition, the firfl: term ca is not diftant from itfelf, therefore 
 its index is reprefented by o. 
 
 Then if the diftance of the fecond term CD from the firfl term ca be ex- 
 prefTed by the arc Ad; the diftance of the third term ce, from ca, will be 
 exprefied by tlie arc ac ; and fo of the reft. 
 
 Confequently, if the terms in the geometric feries be reprefented by 
 numbers, taken as parts of the radius, then the numbers of the fame kind, 
 expre/Ting the meafures of the arcs, or indices, will be as the logarithms of 
 the geometric terms. (I. 73) 
 
 227. CoROL. If the difference between ca and CB was indefiiiiteiy 
 fmall, or ca and cb were nearly in a ratio of equality; then mig/it the 
 number of proportional lines into which ca could be divided, be fo miny, 
 that any propofed number might be found among the terms of this feiics; 
 and if the number of parts in the circumference was incrcafed in like man- 
 ner, then would every term of the proportional divifion of the radius ca 
 have its correfponding index among the equal divifioas of the circumfe- 
 rence i and confequently would exhibit the logarithms of all numbers. 
 
 G 4. A28. T H E O-
 
 228. 
 
 'e E O M E T R Y. 
 THEOREM XLIII. 
 
 Book II, 
 
 ^ere may be almoji (m wfinite 
 variety of proportional fpirah^ and 
 as many different kinds of letaz 
 rithms, ' :\ 
 
 Dem. For with the fame equal divifions Ad^ de^ ef ^c. of the circum- 
 ference, every variation in the ratio of ga to cb, as of ca to c^, will pro- 
 duce a different fpiral, Konm. 
 
 And vi^ith the fame divifions of the radius c a, and difFerent fets of equal 
 parts, A^/, de^ ef Sec, and a^, pq^ qr, U'c pf the circumference, may be 
 formed difFerent fpirals adef, ahik. 
 
 Alfo, varying at the fame time both the divifions pf the radius and 'cir- 
 cumference, difFerent fpirals will be produced. 
 
 But the variations in thefe three cafes may be aknofl infinite : There- 
 fore the number of fuch fpirals are almoft infinite. 
 
 Now it is evident, that there is a peculiar relation between the rays of 
 any fpiral, and the correfponding arcs of the circle ; that is, bctwcf^n the 
 terms of a geometric progreflion, and its indices: l^herefore there maybe 
 as many kinds of logarithms, as there are proportional fpirals. 
 
 229. 
 
 THEOREM XLIV. 
 
 That proportional fpiral which interfeHs equidijlant rays at an angle of \^ 
 degrees, produces logarithms that arc (j/'Nepier'i kind, 
 
 Dem. Suppofe ab, the diftercnce between ca and cb, the firfl and fe- 
 cond terms of the geometric progreflion, to be indefinitely fmall, and take 
 A/), the logarithm of ce, equal to ab ; then may the "figure abh/> be taken 
 as afquare, whofc diagonal ah would be part of the fpiral ahik, and the 
 angle bah would be half a right one, or 45 degrees. 
 
 Therefore that fpiral which cuts its rays c a, ch, ^t. at angles of 45 
 degrees, has a kind of logarithms belonging to it, fo related to their cor- 
 refponding numbers, that the fmallefl variation between the fiift and fecond 
 numbers is equal to the logarithm of the fecond number. 
 
 But of this kind were the firfl logarithms made by Lord Nepier. 
 
 Therefore the logarithms to the fpiral which cuts its cquidiftant rays at 
 an angle of 45 degrees, are of the Nepierian kind. 
 
 END o F B O O K ir. 
 
 THE
 
 THE 
 
 ELEMENTS 
 
 NAVIGATIOR 
 
 BOOK III. 
 
 OF PLANE TRIGONOMETRY. 
 
 SECTION I. 
 
 Defi?ihions and Principles. 
 
 I. 
 
 iLANE TRIGONOAIETRY is an art which fhews how 
 
 J^ to find the meafures of the fides and angles of plane Triangles, 
 feme of them being already known. 
 
 It will be proper for the learner, before he reads the following Articles, 
 to turn to the definitions relative to a circle and angle, contained in th^ 
 Articles 8, 9, 10, ii, 12, 13, 14, 15, 16, 17, 18, 19, and 36, of 
 Book II . 
 
 2. A Triangle confifts of fix parts j namely, three fides and three 
 angles. 
 
 The fides of plane triangles are denoted, or efl:imated by meafures of 
 length J fuch as Feet, Yards, Fathoms, Furlongs, Miles, Leagues, b'f. 
 
 The angles of triangles are eftimated by circular meafures, that is, by 
 arcs containing Degrees, Minutes, Seconds, l^c. (II. 15) ; and for con- 
 venience thcfc circular meafures are repreiented by right lines, called 
 right fines, tangents, fecants, and verfed fines. 
 
 3. Thtt
 
 ^ 
 
 TRIGONOMETRY. Book III. 
 
 3. The Right Sine of an arc, is a right line drawn from one end of 
 6ie arc perpendicular to a radius drawn to the other end : Or it is half the 
 chord of the double of that arc. 
 
 Thus AH is the right fine of the arc ag -/ and 
 ^o of the arc dba. 
 
 4. The Tangent of an arc, is a right line 
 touching one end of the arc, and continued till it P 
 meets a right line drawn from the center through the 
 other end of that arc. 
 Thus GF is the tangent of the arc cat Z 
 
 5. The Secant of an arc, is a right line drawn through the center and 
 one end of the arc, and produced till it meets the tangent drawn from the 
 other end. 
 
 Thus CF is the fecant of the arc ag. 
 
 6.. The Versed Sine of an arc, is that partof the radius intercepted 
 between the arc and its right fine. 
 
 Thus HG is the verfed fine of the arc ag. . 
 
 7. The Complement of an arc, is what that arc wants of 90 
 degrees. 
 
 Thus if the arc cb=: 90*. Then ab is calledthc complement of ag j 
 and AG is the complement of ab. 
 
 8. The Supplement of an arc, is what that arc wants of 180 
 degrees. 
 
 Thus the arc abp is the fupplement of ag ; and ag of abd. 
 
 9. The Co-sine of an arc, is the right fine of the complefnent of that 
 arc. 
 
 The Co-tangext of an arc, is the tangent of that arc's complement. 
 
 The Co-SECANT of an arc, is the fecant of its complement. 
 
 The Co- VERSED Sine of an arc, is the verfed fine of its complement. 
 
 Thus Ai, BE, CE, Bi, being refpedlively the fine, tangent, fecant, and 
 verfed fine of the arc ab, which is the complement of ag ; therefore ai 
 fs called the co-fine, be the co-tangent, ce the co-fecant, bi the co-verfed 
 fine, of the arc ag. 
 
 The right lines, called fines, tangents, fecants, and verfed fines, are 
 ufed as well for the meafures of angles, as for the arcs which meafure 
 thefe angles : And it is as common to fay the fine, tangent, ^f. of an angle, 
 as the fine> tangent, i^c, of an arc, 
 
 10. The 
 
 I
 
 Book III. TRIGONOMETRY. ^t 
 
 10. The greateft right fine, is the fine of 90 ; and the fines to arcs 
 lefs than 90, ferve equally for arcs as much greater than 90". 
 
 Thus the fines of 80 and lOO^j of 60 and i20i of 40 and 1^0, t^c. 
 are refpedlively equal. 
 
 11. The fame tangent and fecantwill ferve to arcs equally diftant from 
 90 degrees ; that is, to any arc and its fupplement. 
 
 Thus if the arc bag 2=90, and bk.=:ba; then the arcs gn, ga, dk, 
 are equal ; and the arcs gak and gn, or dk, are fupplements to one an- 
 other : Then the fine km, the tangent gl, the fecant cl, of the arc gbk., 
 are refpedively equal to the fine ah, the tangent gf, the fecant of of the 
 arc GA. 
 
 12. When an arc is greater than 90% the fine, tangent, fecant, of the 
 fupplement is to be ufed. 
 
 13. The chord of an arc is equal to twice the co-fine of half the fup- 
 plemental arc. 
 
 Thus AN, the chord of the arc agn, = 2ci, the co-fine of the arc ab, 
 and AB is half of the arc abk, the fupplement of agn. 
 
 14. The verfed fine and co-fine together, HG-f CH of any arc ao, is 
 equal to the radius ; ch being equal to ai. 
 
 15. The fines, tangents, fecants, or verfed fines of fimilar arcs in dif- 
 ferent circles, arC in the fame proportion to one another, as the radii of 
 thofe circles. (II. 185) 
 
 16. The angles of two triangles may be refpeftively equal, although 
 their fides maybe unequal. 
 
 Therefore in a triangle among the things given, in order to find the 
 reft, one of them muft be a fide. 
 
 In Trigonometry, the three things given in a triangle muft be either, 
 I ft. Two fides and an angle oppofite to one of them. 
 2d. Two angles and a fide oppofite to one of them. 
 3d. Two fides and the included angle. 
 4th. The three fides. 
 
 In either cafe, the other three things may be found by the help of a few 
 Theorems, and a Triangular Canotiy which is a table where is orderly 
 infcrted every degree and minute in a quadrant or arc of 90 degrees ; and 
 againft them, the meafurcsof the lengths of their correfponding fines, tan- 
 gents, and fecants, eftimatcd in parts of the radius, which is ufually 
 iiippofed to be divided into a number of equal parts, as 10, 100, 1000, 
 1 0000, 1 00000, i^'c. 
 
 SECTION
 
 9* 
 
 TRIGONOMETRY. 
 
 Book III. 
 
 SECTION II. 
 
 Of the 7'ria?jgular Ca?ton, 
 
 70 
 
 G 
 
 J 
 
 PROPOSITION I. 
 
 17. To find the lengths of the Chords, Sines, Tangents, and Secants ta 
 arcs of a circle of a given radius. 
 
 Construction. Through each end of the 
 given radius CD, and at right angles to it (II. 
 60) draw the lines cf, dg : On c, with the 
 radius cd, defcribe the quadrantal arc da, and 
 draw the chord da. 
 
 16. For THE Chords. Trifeft the arc ad 
 (II. 61.), and (by trials) trifel each part ; then 
 the arc ad will be divided into 9 equal parts 
 of 10 degrees each; if thefe arcs are divided 
 each into 10 equal parts, the quadrant will be 
 divided .into ,90 degrees : Bur, in this fmall 
 figure, the divifions to every ip degrees only 
 are retained, as in (II, 83). 
 
 From D, as a center, with the radius to each 
 divifion, cut the right line DA; and it will 
 contain the chords of the leveral arcs into 
 vhich the quadrantal arc ad was divided. 
 
 For the diftances from d to the fevcral di- 
 vifions of the right line da, are thus made re- 
 fpedlively equal to the diftances or chords of 
 the feveral arcs reckoned from d. 
 
 19. For the Sines. Through each of the 
 divifions of the arc ad, draw right lines pa- 
 rallel to the radius AC ; thefe parallel lines will 
 be the right fines of their refpedlive arcs, and 
 CD will be divided into a line of fines, which 
 are to be numbered from c to d, for the right 
 iinesj and from d to c for the verfed fines. 
 
 For the diftance from c to the feveral divifions of the right line cd, arc 
 refpeclively equal to the fines of the feveral arcs beginning from A. 
 . 20. For the Tangents. A ruler on c, and the ieveral divifions of 
 the arc ad, will intcrfccl the line dg ; and the diftances from d to the 
 feveral divifions of dg, v/ill be the lengths of the feveral tangents. 
 
 21. For the Secants. From the center c, with radii to the divi- 
 fions of the tangents dg, cut the line cf ; and the diftances from c to the 
 feveral divifions of cf, will be the lengths of the fecants to the feveral arcs. 
 
 For thefe lengtiis are made rcfpectively equal to the fecants reckoned 
 frsm c to the feveral divifions of thg tangent do. 
 
 22. If
 
 ook in. TRIGONOMETRY, 
 
 9J 
 
 22, If the figure was fo large, that the quadrantal arc could contain 
 every degree and minute of the quadrant, or 5400 equal parts ; then the 
 chord, fine, tangent, and fecant to each of them could be drawn. Now a 
 fcale of equal parts being conftruded (II. 81), 1000 of which parts are 
 equal to the radius cd j then the lengths of the feveral fines, tangents, and 
 fecants may be meafured from that fcale, and entered in a table called the 
 triangular canon, or the table of fines, tangents, and fecants. 
 
 But as thefe meafures cannot be taken with fuificient accuracy to ferfc 
 for the computation to which fuch tables are applicable; therefore the 
 feveral lengths have been calculated for a radius divided into a much greater 
 number of equal parts j as is flicwn in the following articles. 
 
 ^3- 
 
 PROP. II. 
 
 In any circle the chord of bo degrees^ is equal to the radius : and the Jim 
 *f 30 degrees is equal to half the radius, 
 
 Dem. Let the arc cb, or z. cab =60 degrees 5 
 
 and draw the chord ce. 
 
 Now fince the radii AC and ab are equal ; (II. 9) 
 
 Therefore ^ err Z.B. (II. 104) 
 
 And the ^Lc + Z.b= (180*' (Z.a = ) 60 = ) 
 
 120 (II. 96) 
 
 Therefore z. c, or A B = (half 120 s or = ) 60'' 
 
 = AA 
 
 Confequently CB=ABrrAC. 
 
 From A, draw the radius ae perpendicular to cj. 
 
 Then ae bifedls the arc cb, and its chord. 
 
 And CD=:fine of (the arc cErrhalf of 60=) 30". 
 
 Confequently CD is equal to half the radius ab, 
 
 24. Hence, Twice the co-fme of 60 degrees is equal to the radius. 
 For 30* is the complement of 60, and twice the fine of 30 is equal 
 to the radius. 
 
 (II. 124) 
 (3) 
 
 25. 
 
 PROP. III. 
 
 To fnd the fine of one minute of a degree. 
 
 It is evident (II. 187), that the lefs the arc is, the lefs is the difFcrence 
 between the arc and its fine, or half chord ; fo that a very fmall arc, fuch 
 as that of one minute, may be reckoned to differ from its fine, by fo 
 fmall a quantity, that they may be cfteemcd as equal ; and confequently 
 may be expreflcd by the fame number of fuch equal parts of which the 
 radius is fuppofed to contain i,oo00o, t^c. which is readily found by the 
 following proportion. 
 
 As the circumference of the circle in minutes 21600 
 
 Tothccircumf. in equal parts of the radius (II. 196) 6,283185 
 
 So is the arc of one minute I, 
 
 To the correfponding parts of the radius 0,0002908881 
 
 So that for the fine of one minute, may be taken 0,0002908882 
 
 26. PROP.
 
 94 
 
 TRIGONOMETRY. Book III. 
 
 >6. 
 
 PROP. IV. 
 
 In a ffries of ares in arithmetic progrejjion, the fine of any one ofthem^ 
 taken as a mean, and the fum of the fines of any other two, taken as equt- 
 dijiant extremes, are ever in a confattt ratio, of ra^iusp tuticf the co-fme of 
 the common difference of thofe arcs, ' ^ ', ^, > 
 
 Dem. For in a circumference, the 
 center of which is c, and diameter ab, 
 kt there be taken a feries of arcs, arb, 
 
 ARD, ARE, ARF, ARC, ARH, i^fc. 
 
 the common difference of which is the 
 arc BD. 
 
 Then drawing the chords ab, ad, 
 AE, AF, AG, AH, ^c. their halves will 
 be the fines of half the arcs arb, ard, 
 (^c. (3) 
 
 Alfo half the arc bd, is the common difference of half the arcs arb, 
 ARD, are, b'f. (11. 150) 
 
 And the chord ad is twice the co-fine of half the fupplemental arc 
 
 D. {13) 
 
 From the points, d, e, f, g, ^c. with the radii da, ea, fa, ga, i^c, 
 cut AE, AF, AG, AH, ^c. produced in i, k, l, m, i^c. draw rD, ke, 
 
 LF, MG, iffc, and BD, DE, EF, FG, GH, &C. 
 
 Then by the firft part of the demonftration (II. 189), the following 
 triangles are congruous, namely, 
 
 ABD, lED; ADE, KFEj AEF, LGF ; AFG, MHG, (ffc. 
 
 Therefore IE = AB; kf=:ad ; lg = ae; mh af, &c. 
 
 Alfo the triangles ida, kea, lfa, mga, i^c. being each of them 
 ifofceles, and their angles refpeitively equal, are fimilar to dca. (II. 167) 
 
 Therefore ca : ad : : (ad : (ai =)ab +ae : :) |ad: iab +f ae. 
 : : (ae : (ak=:)ad+af : :) ^ae : ^ad +fAF. 
 : : (af : (al =)ae + ag : :) Iaf : f ae +f ag. 
 
 The halves being in the fame ratio as the wholes. (II. 150) 
 
 27. Confequcntly, in a feries of arcs in arithmetic progreflion, viz. 
 
 -JARB, f-ARD, vARE, ^c. the commou difference of which is half ths 
 
 arc BD, it will be, t (II. 164} 
 
 As (ac) radius. , 
 
 To (ad) twice the co-fine of the common difference ; 
 
 So is the fine of either arc taken as a mean. 
 
 To tlic fum of the fines of two cquidiftant extremes. 
 
 28, Hence, The fie of either extreme, f ultra 51 ed fro:>n the produSi of 
 the fine .of the mean by ttvice the co-fwe of the cojnmon difference, will give the 
 fine of the other extreme, (II. 164) 
 
 29. When
 
 Book m. TRIGONOMETRY. 
 
 95 
 
 29. When the common difference of three arcs is 60 degrees j then 
 twice the co-fine of that difference is equal to the radius. (24) 
 
 And with any fuch three arcs, as 30, 90, 1505 or 25, 85, 1453 or 
 20, 80, 140, tf^. it will be (27). 
 
 As R : (2 cof, 60=) R : 
 
 (s, 30 +s, I jo's) 8, 300 +s, 30. 
 
 : : s, 90 : (s, 30 +s, 150=) a, 300+8, 30. "> 
 
 : : s, 85 : (s, 25+s, 145*=) t. 250+3. 35. /^ 
 
 : : s, 80 : (s, 20+s, 140"=) s, 20+s, 40. ^ 5 
 
 : : s, 75 : (s, i5+8, 135=) s, i5'+s, 45. 1^ 
 
 Here; the firft and fecond terms in the proportions being equal, the 
 third and fourth terms are alfo equal. 
 
 30. Hence, The fine of an arc greater than 60 degrees, is eqjialto the fine 
 of an arc as much lefs than 60 degrees.^ added to the fine of its difference from 
 60 degrees. 
 
 Therefore the fines of arcs above 60 degrees are readily obtained from 
 thofe under 60 degrees. 
 
 31- 
 
 PROP. V. 
 
 The right fine of an arc being known, to find its co-fine ; and from thefe 
 to find the tangent, fecant, verfed fine ; and alfo the co-tangent, co-fecanty 
 and co-verfedfine. 
 
 Let AG be any arc, and let ah be its fine, ai its 
 ro-fine j GF the tangent, be the co-tangent ; cf 
 the fecant, CE the co-fecant j hg the verfed fine, bi 
 the CO- verfed fine. 
 
 Now if the fine ah be given, then the co-fine ai or 
 CH, will be known (II. 113) : For ca* ah^=:ch*. 
 Therefore the fquare root of the difference between the 
 fqiiares cf the radius and fine, will be the co-fine. 
 
 C Then the verfed fine HGr: CO ch; and co-verfedfine IB =:cb cr. 
 ^^'\ And fince the triangles cha, cgf, cbe, are fimilar, 
 
 C Therefore (II. 167) CH : ha : : cg : gf, the tangent. 
 ^^' I That is, As the co-fine to the fine, fo is radius to the tangent, 
 
 t And CH : CA : : CG : cf, the fecant. 
 ^^* \ That is. As the co-fine to the radius, fo is radius to the fecant. 
 
 f And CI ; CA : : CB : ce, the co-fecant. 
 ^^' \ That is, As the fine to radius, fo is radius to the co-fecant. 
 
 r Alfo ci : lA : : CB : be, the co-tangcnt. 
 
 36. \ That is, As the fine to the co-fine, fo is radius to the co-tangent. 
 
 L Or gf:cg::cb:be; that is. As tangent : rad. : : rad. : co-tangent. 
 
 37. Hence it is evident, that the tangent and co-tangent of an arc of \^ 
 are equal to %ne another, and to the radius, or fine of (^0 degrees. 
 
 And as the fquare of radius is equal to the rectangle of any tangent 
 and its co-tangcnt, 
 Therefore tan. x cot.z: tan. x cot. Therefore tan. : tan. : : cot. : cot. 
 
 (II. 163) 
 Or the tangents of different arcs are reciprocally as thcii co-tangents. 
 
 38. The
 
 5$ TRIGONOMETRY, Book III, 
 
 38. The principles by which the lengths of the fines, tangeiity,- fc- 
 cants, Uc. may be conftruaed, being delivered, the following examples 
 arc annexed to illuftrate this doclrine. 
 
 Required the co-fine of one minute. 
 
 The fine of i minute being 0,0002908882 (15) 
 
 Its fquare is 0,00000008461594 
 
 Which fubtradled from the fquare of /adius i, 
 
 Leaves 0,99999991538406 
 
 Whole iquare root 0,9999999577 is the 
 
 co-fine of I minute ; or the fine of 89 59'. 
 
 Now having the fine and co-fine of i minute, the other fines may be found 
 
 in the following manner, ^8) 
 
 twice the cof. i min. X fine of i m. =fum of the fines of 0' & i\ 
 twice the cof. i min. xfine of 2 m.r=fum of the fines of i' 6c 3^ 
 twice the cof. i min. x fine of 3 m. =:fum of the fines of 2' & 4^. 
 twice the cof. i min. xfine of 4 m.rrfum of the fines of 3' & 5'. 
 twice the cof. i min. xfine of 5 m.zrfum of the fines of 4'' k b\ 
 
 Proceeding thus in a progreilive order from each fine to its next, all the 
 
 fines may be found. 
 
 But as twice the co-fine of i minute, viz. 
 
 1,9999999154 is concerned in each operation, 
 
 therefore if a table be made of the produ6ts of 
 
 this number by the nine digits, as here annexed, 
 
 the computations of the fines may be performed 
 
 by addition only. 
 
 For the products by the digits in the given 
 multiplier, being taken from the table, and 
 written in their proper order, will prevent the 
 trouble of multiplication. 
 
 And even this operation may be very much 
 fhortened, by fetting under the right hand place 
 (viz. 4.) of the double co-fine of one minute, the unit place of the fine 
 ufed as a multiplier, and reverfing or placing in a contrary order, all its 
 other figures ; then the right-hand figure of each line arifing by the multi- 
 plication, is to be fet under one another ; and in thefe lines, the firft figure 
 to be fet down, is what arifes from the figure ftanding over the prefent 
 multiplying one ; obferving to add what would be carried from the places 
 omitted. 
 
 Now if the produ^s of the figures in the multiplier, thus inverted, be 
 taken from the above table of products, it is neceflary to remark what num- 
 ber of places will arife from each digit ufed in the multiplier ; then in the 
 produces of thofe digits in the table, take only the like number of places, 
 obferving to add i to the right-hand place, if the next of the omitted 
 ^gure$ exceed 5. 
 
 Required 
 
 Multi- 
 pliers. 
 
 Produds. 
 
 I 
 
 z 
 
 3 
 4 
 5 
 6 
 
 7 
 8 
 
 . 9 
 
 1,9999999154 
 3,9999998308 
 599"99997462 
 7,9999996616 
 
 9,999999577 
 11,9999994924 
 
 '3.9999994078 
 
 15,9999993232 
 
 17. 99999923^^6
 
 Book III. 
 
 TRIGONOMET R Y. 
 
 97 
 
 Required the fine of two minutes. 
 
 The fine of i min. placed in an inverted order 
 under the double cof. of i min. as in the mar- 
 gin j the right-hand figure 2 ftands under the 9 in 
 the 6th decimal place, therefore the firft 6 deci- 
 mal places of the product againft 2 in the table, 
 are to be ufed j but i being added, becaufe the 7th 
 place 8, exceeds 5, makes the produ(5l 40000CO : 
 Alfo for 9 the next figure in the multiplier, {land- 
 ing under the 5th decimal place, take 17,99999 
 from the table of products, and I being added to 
 the 5th place, becaufe the 6th exceeds 5, make 
 it 18,00000 : In like manner the product by 8, 
 adding i, is 16000^ &c. and the fum of thefe produdls 0,0005817764 
 is the fine of 2 minutes, as required. 
 
 This kind of operation will be very eafily conceived without farther 
 illuftration, by comparing the procefs in this and the following operations, 
 with what has been already faid. 
 
 Required the fines of 3', 4', 5', and 6 minutes. 
 
 1,9999999154 
 
 2888092000,0 
 
 40CO000 
 
 iScoooo 
 
 16000 
 
 1600 
 
 160 
 
 4 
 
 0,0005.817764 
 
 For 3 min. 
 
 For 4 min. 
 
 For 5 min. 
 
 For 6 min. 
 
 1,9999999' 54 
 
 1,9999999154 
 
 1,9999999154 
 
 1,9999999154 
 
 4677185000,0 
 
 5466278000,0 
 
 625^361 100,0 
 
 50+4454100,0 
 
 10000000 
 
 16000000 
 
 2O0OCO0O 
 
 20000000 
 
 160000Q 
 
 14OCO00 
 
 20G0000 
 
 8000000 
 
 2O00O 
 
 40000 
 
 1200000 
 
 lOGOOOO 
 
 I4CCO 
 
 I2C00 
 
 toioo 
 
 8cooo 
 
 1400 
 
 1200 
 
 ICOOO 
 
 8000 
 
 120 
 
 80 
 
 ICOD 
 
 800 
 
 8 
 
 10 
 
 40 
 I 2 
 
 10 
 
 0,001 !635;23 
 
 0,001 74 532'^'0 
 
 0,00232710^2 
 
 O,COiQ08S(^IO 
 
 0,0002908882 
 
 0,00058:7764 
 
 0;0?oS726w45 
 
 o.coi 1635526 
 
 0,0008726646 
 
 0,001 1635526 
 
 0,0014544407 
 
 0,0017453284 
 
 In each example, the fine of an arc which is 2 minutes lefs than that 
 required, (28) is fubtractcd. 
 
 The fines being made, the tangents, fecants, feV. are to be conflrucled 
 as before fliewn. (33, 24) 
 
 39. There are many methods by which the triangular canon may be 
 made ; but that which is here delivered was chofen as the moil eafv, the 
 beft adapted to this work, and what would give the learner a fufficietit 
 notion how thefe numbers are to be fuund : For at this time th-rc is no 
 occafion to conftrucl new tables of fines, and rarely to examine thofc al- 
 ready cxt int ; tiicy having paficd through the hand' of a great many care- 
 ful examiners, and for a long time have been reccwcd by the learned as a 
 work fufficiently corrcvl. 
 
 Vol. I. li Thefe
 
 ^8 TRIGONOMETRY. Book III. 
 
 Thcfe lines were firft introduced into mathematical computations by 
 Hlpparchus and Menelaus^ whofe methods of performance were contra6lcd 
 by Ptolemy y and afterwards perfected by Regiomontanus ; and fince his 
 time Rl?eticusy ClayiuSy PetifcuSy and many other eminent men, have 
 treated largely on this fubje^, and greatly exemplified the ufe of this 
 triangular Canon, or Tables ; which are now, by way of diftincSlion, 
 called Tables of jiatural CincSy tangents, &c. But the greateft improve- 
 r'lent ever made in this kind of mathematical learning, was by the Lord 
 Nepia-y Baron of Merchijion in Scotland ; who, being very fond of fuch 
 ftudies, where calculations by the fines, tangents, ^c, did frequently occur, 
 judged it would be of vaft advantage if thefe long multiplications and di- 
 vifions could be avoided j and this he effected by his happy invention of 
 computing by certain numbers, confidcred as the indices of others (I. 63}, 
 which he called logarithms; this was about the year 1614. 
 
 The tables now chiefly ufed in Trigonometrical computations, are the 
 logarithms of thofe numbers which exprefs the lengths of the fines, tan- 
 gents, ^r. and therefore to diftinguifli them from the natural ones, they 
 are called hcgarithm'ic fines, tangents, Cffr. (of by fome artificial fines, 
 l^c.') Only thofe of the logarithmic fines and tangents are annexed to 
 this treatife, becaufe the bufinefs of Navigation may be performed by 
 them ; neither are thefe tables carried to more than five places befide the 
 index, that being fufHciently exa6t for all nautical purpofes : But it muft 
 be allowed that, for general ufe, fuch tables arc the moil efteemed, as con- 
 flit of moll places. 
 
 40. Thefe tables afe at the end of Book IX. and are fo difpofed, that 
 each opening of the book, contains eight degrees ; four of which are num- 
 bered at the top, and four at the bottom of the page ; and thofe at the 
 top proceed from left to right, or forwards, from o degrees to 45 ; and 
 thofe at the bottom, from right to left, or backwards, from 45 to 90 de- 
 grees : To each degree there are four columns, titled fines, co-fines, tan- 
 gents, co-tangents ; and the minutes are in the marginal column of each 
 page, figned with m ; thofe on the left fide of the page belong to the de- 
 grees which are at the top, and thofe on the right-hand fide, to the degree^ 
 which are at the bottom of the page, 
 
 41. A fine, tangent, co-fine, co-tangent, to a given number of de- 
 grees, is found as follows : 
 
 For an arc lefs than 45 degrees, 
 
 Seek the degree at the top, and the minutes in the column figned m at 
 the top ; againlt which, in the column figned at the top with the propofed 
 name, ftands the fine, or tangent, i^c. required. 
 
 But when the arc is greater than 45 degrees. 
 
 Seek the degrees at the bottom, the minutes in the column with M at 
 the bottom, and the propofed name at the bottm. 
 
 Example
 
 Book III. 
 
 TRIGONOMETRY. 
 
 99 
 
 Example T. Required the logarithmic Jine ^28 37'. 
 
 P'ind 28 deg. at the top of the page j and in the fide column, marked with 
 M at the top, find 37 ; againft which, in the column figned at the top with 
 the word fme, ftands 9,68029, the log. fine of 28 37', as required. 
 
 Example II. Required the logarithmic tangent 0/(3"] 45', 
 
 Find 67 deg. at the bottom of the page; and in the fide column, titled 
 M at the bottom, find 45 ; then againft this, in the column marked tangent 
 at the bottom, ftands 10,38816, which is the log. tangent required. 
 
 42. But when a logarithmic fine or tangent is propofed, to find the de- 
 grees and minutes belonging to it, then. 
 
 Seek in the table, among the proper columns, for the neareft logarithm 
 to the given one ; and the correfponding degrees and minutes will be 
 found ; obferving to reckon them from the top or bottom, according as 
 the column is titled, where the neareft logarithm to the given one is found. 
 
 43. It may fometimes happen that a log. fine or log. tang, may be 
 wanted to degrees, minutes, and parts of minutes j which may be thus 
 found. 
 
 Take the difference between the logs, of the degrees and minutes next 
 lefs, and thofe next greater than the given number. 
 
 Then for i, take a quarter of this difference ; for -', take a third ; for 
 I, take a half; for \ take two thirds; for |, take three quarters, z^c. 
 
 Add the parts taken of this difference to the right-hand figures of the 
 log. belonging to the deg. and min. next lefs, and the fum will be the 
 log. to the deg. min. and parts propofed. 
 
 Example I. Required the log. 
 tang, to 60 56'-^. 
 
 Log. tang. 60 57' is 
 Log. tang. 60 56 is 
 
 The difF. is 
 
 Its half is 
 
 Add it to tang. 60 56' 
 
 10,2,-535 
 
 10,25506 
 
 2y 
 10,25506 
 
 Gives tang. 
 
 60 56^ 10,755: 
 
 Example II. Required the log. 
 fme to 32 I s'h 
 
 Log. fine 32 16' is 
 Log. fine 32 15 is 
 
 The difF. is 
 
 It:; three fourths i.s 
 Add it to fine 32" 15' 
 
 Gives fine 32 15I 
 
 9.72743 
 9,72723 
 
 5 
 
 9,72725 
 
 9,72738 
 
 In moft moft cafes the Avork may be done by infpcdlion. 
 
 44. And if a given lotr. fine or log. tangent falls between thofe in tie 
 tables: then the degrees and minutes anfuering maybe reckoned |, o -f, 
 or -^, y^. minutes more than thofe belonging; to the neareft lefs log. in 
 the tables, according as its difference from the given one is ', or-|, or i, 
 ijfc. of the difference between the logarithm next greater and next lefs than 
 the given leg. 
 
 H 2 
 
 SECTION
 
 100 
 
 TRIGONOMETRY. 
 
 Book in. 
 
 SECTION III. 
 Of the Solution of Plain 'Triangles. 
 
 45. P R o B L E M I. 
 
 In any plain triangle, abc, if among the things given there be a fide 
 and its oppofite angle, to find the reft. 
 Then fay, As a given fide ^ (ab) 
 
 To the fine of its oppofite angle j { ^ ) 
 
 So is another given fde^ " (ac) 
 
 To the fine of its oppofite angle. ( B ) 
 
 Therefore, to find an angle, begin with a fide oppofite to a known 
 angle. 
 
 Alfo, Js the fine cf a given angle, ( b ) 
 
 To its oppofite fide j (ac) 
 
 So is the fine of another given angle, ( C ) 
 
 To its oppofite fide. (ab) 
 
 Therefore, to find a fide, begin with an angle oppofite to a known fide. 
 
 Dem. Take BD=:CF=:radius of the tables. 
 
 Draw DEj AH, FG, each perpendicular toBC, 
 
 (11. 59) 
 Then de and fg are the fines of the angles b 
 and c. (3) 
 
 Now the triangles bde, bah, are fimilar, and 
 fo are the triangles cfg, cah. 
 Therefore bd : de : : ba : ah. (II. 167) 
 
 And (cf=:)bd : f g : : ca : ah. (II. 167) 
 
 But (bDXAH = ) de XBArZFCXCA. 
 
 Therefore de : ca : : fg : ba. Or, s, Z.B : ac : 
 
 c; c 
 
 (II. 162) 
 s, Z.C : ba. (II. 163) 
 Schol. Or, by circumfcribing the triangle with a circle, it will readily 
 appear, that the half of each fide is the fign of its oppofite angle. And 
 halves have the fame proportion as the wholes. 
 
 46. P R O B L E M II. 
 
 In a right-angled plane triangle, abc, if the two fides containing the 
 
 right angle B are known, to find the reft. 
 
 Then, Js one of the knotvn fides ^ (ab) 
 To the radius of the tables (or tangent of \S) ; (ad) 
 
 So is the other known fide, (bc ) 
 
 To the tangent of its oppofite angle. (de) 
 
 Dem. Take ADrrradius of the tables. 
 Then de, perpendicular to ad, is the tangent of 
 the angle a. (4) 
 
 And the triangles ade, abc, arc fimilar. 
 Therefore ab : ad : : Bc : de. (II. 167)
 
 Book III. TRIGONOMETRY. roi 
 
 47. PROBLEM III. 
 
 In any two quantities, their half difference added to their half fum^ gives 
 the greater. 
 
 The half diff. fuhtraSled from the half fum^ gives the lefs. 
 
 And if half the fum be taken from the greater y the remainder will be the 
 half difference of thofe quantities. 
 
 Dem. Let AB be the greater, and BC the lefs, of two quantities. 
 7"ake AD~BC ; then bd is their difference. 
 Bjfeil DB in E ; then de eb, is the half diff". 
 And AD + DEr=BC ^-BE (II. 47) ; therefore AE is the half fum. 
 NowAE + EB=AB, is the greater, j. jr; 
 
 And AE ed:i:(ad = )bc, is the lefs. 7- ' ' ' " ^ g 
 
 Alfo AB ae=:b, is the half diff. 
 
 48. P R O B L E M IV. 
 
 In any plane triangle, abc; if the three things known, be two fides, 
 AC, CB, and their contained angle c, to iind the reft. 
 
 Find the fum and difference of the given fides.. 
 
 Take half the ^iven angle fro?n 90 degrees^ and there remains half the fum 
 
 of the unkn-jwn angles. Then fay, 
 
 Js the fum of the given fides ^ A c -}- c B 
 
 To the difference of thofe fides ; AC cb 
 
 So is the tangent of half the fum of the unknown angles ^ t. IiTPa 
 
 To the tangent of half the difference of thofe angles. t. |b3a 
 
 Jdd the half difference cf the angles to the halffum^ and it will give the 
 
 greater angle ~ 1? . 
 
 cubtraft the half difference of the angles fro?n the half fum, and it luill give 
 
 Icffer angle r: a . 
 
 Dem. On c, with the radius cb, defcribe a circle, 
 
 cutting AC, produced, in e and d ; draw EB, and /' \ 
 
 bd j alfo draw DF parallel to eb. / ^1 
 
 Then ae = ac + cb, is the fum of the fidc-s. 
 And ADzzAC CB, is the difference of the fides. 
 
 Now Z.CDB=:Z.C!3D. (II. 104) ^ 
 
 And(CDB + CBD = )2CDB=:Z.CBA+ Z.A.(I1. 9^) A Y ^^ 
 
 Therefore f Z.cba4- Z.a= /LCDb, is half the fum of the unknown angles. 
 And BE (U. 123) is the tangent of cdb, to the radius db. (4) 
 
 Alfo (cba cbd=:)di5a=:-^Z.cba i/lA. (47) 
 
 Therefore | acba z.a:=Z.dba, is half the difFerencc of the unknown 
 angles. 
 
 And DP is the tangent of DBA, to the radius db. (\) 
 
 Now the triangles aeb, adf, are fimilar, df being parallel to v.w. 
 Therefore a : ad : : be : df. (II. 167) 
 
 Of AC + CB ; AC CB : ; t. |ZcbaTXa : t. i^tCA Z-a. 
 
 H3 49, PRO-
 
 I02 
 
 49 
 
 TRIGONOMETRY. Book III. 
 
 PROBLEM V. 
 
 In a plane triangle, abc, if the three fides are known, and the angles 
 required. 
 
 From the greateji angle ^ B, fuppofe a line BD drawn perpendicular to its 
 oppojite fide^ or baft ^ dividing it into tzvo fegrrientSy ad, CD, and the given 
 triangle into two right-angled triangles^ adb, cdb : Then fay. 
 
 Asthebafe., or fum of the fegments^ 
 
 Is to the f urn of the other two fides i 
 
 So is the difference of thofe fideSy 
 
 To the difference of the fegments of the hafe. 
 
 AC 
 
 AB + BC 
 AB BC 
 AD DC 
 
 jidd half the difference of the fegments to half the bafe^ gives the greater feg~- 
 ment ad. (47) 
 
 Subtract half the difference of the fegments from half the bafe^ there remains 
 the leffer fegment DC (47) 
 
 Then, In each of the triangles, ADB, cdb, there will be known two fides .^ 
 and a right angle oppofite to one of them ; therefore the angles will be found by 
 Problem L (45) 
 
 When two of the given fides are equal ; then aline drawn from the in- 
 cluded angle., perpendicular to the other fide, blfeSis the fide, (II. 103) 
 
 And the angles being fouyid in one of the right-angled triangles., will alfo 
 give the angles of the other. 
 
 Dem. of the foregoing proportion. ... .,_ 
 
 In the triangle ABC, the line bd, perpendicular to 
 AC, divides ac into the fegments ad, dc. 
 On B with the radius bc, defcribe a circle gce, 
 cutting ab, continued, in c, e j and ac in f j 
 draw BF. 
 
 C 
 
 Then dc =:df. 
 
 Now AC ( =:ad + dc) is the fum of the fegments. 
 And AF (=:ad fd) is the difference of the fegments. 
 Alfo AE (riAB 4-Bc) is the fum of the other fides. 
 And AG ( =r AB BC ) is the difference of thofe fides. 
 
 But AC X AFZrAEXAG. 
 
 Therefore ac : ae : : AG : AF. 
 
 Or AC : AB-j-Bc : : ab bc : ad dc j oizzad df. 
 
 (II. 103) 
 
 (".172) 
 (II. 163) 
 
 50. PRO.
 
 Book III. .TRIGONOMETRY. 
 
 103 
 
 50. 
 
 PROBLEM VI. 
 
 In any plane triangle, abc, the three fides being knov/n, to find either 
 of the angles. 
 Put E and F for the fides including the angle fought. 
 
 G for the fide oppofite to that angle. 
 
 T> for the difference between the fides E and F. 
 
 Find half the fum of G and D. 
 
 And half the difference of G and d. 
 Then write the fe four logarithms under one another.^ namely. 
 
 The Arithmetical complement of the logarithm of E; (I. 88) 
 
 The Arithmetical complement of the logarithm of 7 ; 
 
 The logarithm of the aforefaid half fum of G and d ; 
 
 The logarithn of the ciforefaid half difference of G and D. 
 Add them together., take half their (um ; which feek a7nong the leg. fines. 
 And the degrees and minutes anfiuering^ l?ting doubled, luill give the nieafure, 
 cf the angle fought. 
 
 Dem. In the triangle, ABC, let a be \', ^ 
 
 the angle fought. 
 
 Take ah = ah, draw bh ; and 
 through K, the middle of bh, draw 
 AP, which bifets the angle a, and 
 is perpendicular to bh. (II. 103) 
 
 Through k. draw kl, kq^, paral- 
 
 lei, to BC, AC ; which will bife61: ^ '' ^"^ '" 
 
 l^c, BC, in L and i ; then kl:z:ic, ki=:lc. (II. 28, 163) 
 
 And the difference between AC and ab is HC D ; then Ki = fD. 
 
 From I, with the radius ik, defcribe a circle cutting ap, bh, kq^, bc, 
 in p, o, Q^, M, N, and join CQ^; now iqriiK~LC=:LH ; therefo'-e 
 KQj=Hc, and CQjr:KH,asthc triangles cqr, khl, are congruous. (II. 99) 
 
 Therefore cq^ parallel to kh (II. 28.) being produced, will meet ap at 
 right angles (II. 53), in the point p, by the revcrfe of (II. 130). 
 
 Then pqjtko, as the triangles kqp, QOK, are congruous. (II. 95,103) 
 Now HMZz (Bi + iMrr{-EC + ^Hcn) ^u+d : AndfiNnfc D. 
 Alfo Boncp : For bk::z (khht) cq^, and KO vq^ 
 lyCt Ar=:radius of the tables ; then rn (parallel to bh) =:fmc of f Z. A. (3) 
 Then the triangles Ar, a KB, apc, are finiilar. 
 
 And Ar : r : : AB : BK ; alfo Ar : r : : ac : (cpnr ) bo. (II. 167) 
 
 Therefore a;-^ : rn^ : : ac x ai5 : (bk x bo:= j em X bn. (II. 161, 172) 
 Or (fq. rad. =r R^ : fq. fme f Z. A : : ac x ab : |- TTfu X fc D. 
 
 Therefore the fquare of the fine 
 
 ic + O X r,(;-_i) 
 ^ ACXAB 
 
 Therefore fme yZ-An V 
 
 IG + UX-^G D 
 E XF 
 
 (II. 164) 
 
 And R, the radius of the tables, being I. 
 
 H 4. 
 
 Or
 
 104 TRIGONOMETRY. Book III. 
 
 Log.icTD + Log. IoIId Log. e Log. f" 
 
 OrLog.s,|z.A=- 
 
 2 
 
 & 
 
 OrLog.s,UA= '^ + '- + Llc+D + L.ic-D 
 
 Where L. (lands for logarithm, and 1. for Arith. comp. of the logarithm. 
 
 5 1 . Every poflible cafe in plane Trigonometry may be readily folved by 
 the preceding Problems, obferving the following Precepts. 
 
 I. Make a rough draught of the triangle, and put 
 the letters a, b, c, at the angles. 
 
 II. Let fuch parts of this triangle be marked, as 
 reprefent the things which are given in the qucftion. 
 Thus, mark a given fide with a fcratch acrofs it ; 
 and a given angle by a little crooked linej as in the 
 figure; where the fide ab, and the angles a and c, 
 are marked as given. 
 
 in. If two angles are known, the third is always known. 
 
 For if one angle is 90 degrees, the other given angle (which (II. 97) 
 will be acute) taken from 90 degrees, leaves the third angle. 
 
 And if both the given angles are oblique j their fum taken from i8o 
 degrees, gives the other angle. (II. 96) 
 
 IV. Compare the given things together, and determine to which 
 Problem thequeftion propofed belongs. 
 
 V. Then according as the Problem dire(5ls, perform the preparatory 
 work; and write down, under one another, in four lines, (or more if ne- 
 cefTary), the literal J?ating; expreffing each angle bya letter, or by three j 
 each line by two letteiS] andthefams, or differences, of lines, by pro- 
 per marks. 
 
 VI. Againft fuch terms as are known, write their numeral value, as 
 given in the queftion, or as found in the preparatory work ; and againft 
 thefe numbers write their logarithms; thofe for the lines being found (by 
 I. 81) in the table of the logarithms of numbers; and thofe for the angles, 
 found (by 41) in the table of logarithmic fines and tangents: Obferv- 
 ing that an Arithmetical complement (fee I. 88) is always ufed in the 
 firft term : And that when an angle is greater than 90 degrees, its fuppje- 
 ment is ufed. 
 
 VII. Add thefe logarithms together, and feek the fum (I. 81) in the 
 log. numbers, when aline is wanted; or (42) in the log. fines or tan- 
 gents, when an angle is wanted. Then the number or degree, anfwering 
 to that logarithm which is the neareft to the faid fum, will be the thing 
 required. 
 
 A SY.
 
 {ook III. 
 
 52. 
 
 TRIGONOMETRY. 
 
 A SYNOPSIS. 
 Of the Rules in Plane Trigonometry, 
 
 105 
 
 Prob. 
 
 r. 
 
 fee 
 art, 
 45 
 
 Given. 
 
 Required. 
 
 SOLUTION. 
 
 All the 
 
 angles and 
 
 one fide. 
 
 Either 
 
 of the 
 
 jther fides. 
 
 Since two angles are known, the third is known. 
 And, As fin.ofZ.opp. to fide given, is to thatopp. fidci 
 So fin. of another angle, to its opp. fide. 
 
 Two fides 
 and an Z. 
 oppof. to 
 one fide. 
 
 The angle 
 oppof. to 
 the other 
 
 given fide. 
 
 As one given fide is to the fine of its opp. angle ; 
 So is the other given fide, to the fide of its opp. angle. 
 Then two angles being known, the third is known. 
 And the other fide is found as before. 
 
 II. 
 art. 
 46 
 
 Two fides 
 and the 
 included 
 
 right ^. 
 
 Either 
 
 of the 
 
 other 
 
 angles. 
 
 As one of the given fides, is to the Radius ; 
 So is the other given fide, to the tangent of its opp. Z. 
 Then two Z.s being known, the third is known. 
 The other fide is foui)d bv opp. fides and /Li, 
 
 III. 
 
 art. 
 48 
 
 Two fides 
 and the 
 
 included 
 oblique 
 angle. 
 
 The 
 other 
 angles. 
 
 Find the fum and difF. of the given fides. 
 
 Take J given Z. from 90 leaves 4 fum of the other Z.s. 
 
 Then, As fum fides, is to difF. of fides ; 
 
 So tan. 1 fum other Z.s, to tan. ^difF. thofeZ.s. 
 
 The 1 fum AS { 1 1 i difF. Zs, gives \ f^^^f" 
 
 Find the other fide by opp. fides and angles. 
 
 IV. 
 art. 
 
 49 
 
 The 
 
 three 
 
 fides. 
 
 All 
 
 the 
 
 angles. 
 
 Draw a line perpend, to the greateft fide, from the 
 opp. Z., dividing that fide into two parts. 
 Then, As the longeft fide is to fum other two fides ; 
 So is the difF.thofe fides, to the difF. p'* of longeft. 
 
 Thenilong.fide | + [ ^dif.p" gives the | f/^^^- { part. 
 
 Now the faidperp. cuts the triangleinto 2 right Z.d ones. 
 In both, are known the Hyp. a Leg. and the right Z. . 
 The angles are found by Problem I. 
 
 V. 
 
 art. 
 50 
 
 The 
 
 thise 
 fides. 
 
 Either 
 Angle. 
 
 Having chofe which angle to find ; call the fides in- 
 cluding that angle e and f. 
 The fide opp. that Z., call c. 
 Put D for the difference between e and f. 
 Find the half fum, and half difF qf g jnd d. 
 Then write thefe four Logs, under one another ; 
 y. j The Ar. Co. Log. of b. The Ar. Co. Log. of f, 
 '^'^l The Log. of A r,m, And the Log. of 'difference. 
 Add .he four logs, together, take half their fum. 
 iefk it among the log fines ; and the correfponding 
 dcg. and min. doubled, is the angle fought. 
 
 53. Ex.
 
 io6 
 
 TRIGONOMETRY. Book III. 
 
 53. Example I.^ In thi plane Triangle auc. 
 
 Given AJB=: 195 Poles. 
 
 Z.B= 90 00' 
 
 Z.A= 47 55. 
 Required the other parts* 
 
 For the linear Solution. 
 ift. Draw AB equal to 195 poles, taken from a fcale 
 of equal parts. 
 
 2d. From B, draw bc, making with ad an angle of 90. (!! 84) 
 
 3d. From A, draw ac, making with ab an angle of 47" 55'j and meet- 
 ing BC in the point c. 
 
 Then is the triangle ABC fuch, the parts of which correfpond with the 
 things given ; and the fides ca, cb, being applied to the fcale that ab 
 was taken from, their meafures will be found, viz, ac=29I j and 
 
 BC = 2l6. 
 
 For the numeral Solution, or Computation. 
 
 Since two angles are known j Therefore, 
 
 From 90 00' 
 Take 47 55 =^1 a 
 Remains 42 05=z.c 
 
 Now in this triangle, there are known all the angles and one fide ; there- 
 fore among the known things, there is a fide and its oppofite angle ; which 
 belongs to the firft problem. 
 Then to find the fide ac, begin with the angle c oppofite ab. 
 
 As the fine of Z. c. 
 To the oppofite fide ab ; 
 So the fine of the Z. b. 
 To the oppofite fide ac. 
 
 Or thus, As s, Z.c=:42 05' 
 To ABm95 po. 
 So s, Z-Brrgo co' 
 
 0,17379 Ar. Co. 
 2,29003 
 10,00000 
 
 To Ac:=29i po. 2,46382 
 
 And to find the fide bc, begin with the angle c oppofite ab. 
 
 As the fine of the Z. c, 
 To the oppofite fide ab ; 
 So the fine of the Z. a. 
 To the oppofite fide bc. 
 
 Or thus, As s, Z.c=:42 05' 
 To AB=: 195 po. 
 So Sj Z.Arz47 55' 
 
 0,17379 Ar. Co. 
 
 2,29003 
 
 9,87050 
 
 To Ecr;.2i6 po. 2,3343: 
 
 So that AC is 291 poles, and bc is 216 poles. 
 
 The letters Ar. Co. ftanding on the right of the firft line, fignify the 
 arithmetical complement of the log, fine of 42 05'. (I. 88) 
 
 54' Ex
 
 Book III. 
 
 TRIGONOMETRY. 
 
 107 
 
 5|f. Example If. In the plane Triangle abc. 
 
 Given AB =: 1 1 7 miles, 
 Z.B = i3446' 
 
 Z.A= 22 37. 
 
 Required the other parts. 
 
 For the linear Solution, or Construction. 
 Make ab=:ii7 equal parts ; at a make. an angle=:22'' 37' (II. 84); 
 nd at B make an angle of 134 46' ; then the lines which make with; a.b 
 thofe angles, will meet in c, and form the triangle propofe(J. 
 And the meafure of bc will be 117, and of ac 216. 
 
 By Computation., See art. 45. 
 Since two angles are known, 
 
 Namely, Z.b=:i34 46' 
 Z-Arz 22 37 
 
 Their fum =^157 23 
 
 Now from 
 Take 
 
 180 00 
 157 23 
 
 Since theangle crzZ. a 
 Therefore bc=:ab 
 
 Leaves Z.c:= 22 3.7 
 (II. 104) 
 
 And from 
 Take 
 
 i8o 00' 
 13+ 4^ 
 
 The fup'Z.BZ= 45 14 
 
 To find the fide AC. 
 
 As s, z.cz= 22 37' 0,41503 Ar. Co. 
 
 To ABzi:ii7 M. 2,06819 
 
 So s, z.B = i34 46' 9,85125 fup. 
 
 To Acrr2i6 M. 2,33447 
 
 55. Example III. In the plane Triangle x^q. 
 
 Given AB=4o8 yards. C 
 
 Z.B=:22'' 37' 
 
 z.A = 58 07. 
 Required the other parts. 
 
 ^ h 
 
 Construction. 
 
 Make AB=4o8 yards, or equal parts ; make the angle a 58 07' ; and 
 the Z.B = 22 37' ; then the lines forming thefe angles will meet in c j 
 and the meafure of ac is 159 yards, and of bc is 351. 
 
 Computation. See art. 45, 
 
 Two angles being known, viz A a 58 07' 
 
 Z.Bn-2 37 
 
 Their fum =80 44 
 
 From 
 
 Take 
 
 180 00' 
 80 44 
 
 Leaves 99 i6r:Z.c. 
 
 To find the fide AC. 
 As s,/Lr = 99'-' 16' 
 To AT. =408 Y. 
 
 S0,S,Z.B = 22 37' 
 
 0,00570 Ar. Co. 
 2,61066 
 
 To 
 
 : = .59 Y 
 
 2,201^,3 
 
 To find the fide B c . 
 Ass,Z.c = 99 16' 
 To A Mir: 408 Y. 
 So s,Z.Ar=58 07' 
 
 To nc =351 Y. 
 
 o,oo57oAr.Co- 
 
 2,61066 
 
 9,92:^97 
 
 2,54 1 33 
 
 In thele operations the- fupplemcnt of the angle c is ufcd. 
 
 56. Ex-
 
 loS 
 56. 
 
 TRIGONOMETRY. Book III. 
 
 Example IV. In the plane Triangle abc 
 
 ,LB=:90 00'. 
 Required the rejl. 
 
 Construction. 
 Make ab=:195 equal parts; draw bc, making an 
 angle at 8=90" 00'. From a with 291 equal parts 
 cut BC in c, and draw ac. 
 Then the z. a meafurcd on the fcale of chords will be about 48 degrees, and 
 Z.C about 42* : Alfo bc, on the equal parts, meafures about 216. 
 
 Computation. 
 Here being two fides, and an angle oppofite to one of them, the folution 
 falls under problem the firft. See art. 45. 
 
 7 find the angle c . 
 
 As Ac=:29i F. 7,53611 Ar. Co. 
 
 To s,/.BZ=9o co' 10,00000 
 
 So ABz:i95 F. 2,29003 
 
 To 3,^.0=142 05' 9,82614 
 
 Prom 90 00' 
 
 Take 42 05 =:Z.c, 
 
 Leaves 
 
 47 55 = -^*- 
 
 Tofindthcjide BC. 
 As s,Z.Br:90 00' 
 To Acrr29i F. 
 So s,ii.A=47" 55' 
 
 To Bc=:2i6F. 
 
 10,00000 
 
 2,46389 
 9,87050 
 
 ?viJ439 
 
 Here the fine of 90 00' or radius be- 
 ing the firft term, its Arith. Comp, 
 being o, is not taken. 
 
 57. Example V. In the plane Triangle ^.^z^ 
 
 Given ac = 2i6 7y ^j ^ 
 
 AB=:iI7 J V _^ 
 
 Z_C=:22 37'. \ "'"--..^ B 
 
 Required the rejl, ^ 
 
 A' 
 Construction. 
 
 Make AC = 216 yards ; the z.c=:22 37^^ and draw cb : Then from a, 
 with 117 yards, cut CB in b or in b; and either of the triangles ^zb or 
 ACB will anfwer the conditions propofed : But the triangle to be ufed is 
 generally determined by fome circumftances in the queltion it belongs to. 
 Thus if the angle oppofite to ac is to bc obtufe, the triangle is abc 
 
 Computation. 
 The folution belongs to problem the firlL See art. 45. 
 To find the angle B 
 
 As ABzriiy Y. 
 
 To S,Z.C = 22 37' 
 
 So Ac:=:2i6 Y. 
 To s,z.B 134." 46' 
 Z.c-f Z.B=:i57 23 
 
 7,93181 
 
 9>'i^497 
 2.3344"^ 
 
 9.85-23 
 
 From 
 Take 
 
 180= 
 157 
 
 CO 
 
 23: 
 
 Lc-\-A.-R, 
 
 Leaves 22 37Z.A, 
 
 And as Z. -irr^Lc, 
 Therefore bc:;=.^b, 
 
 (ir. 104) 
 
 If the angle required bc obtufe, fubtradl the deg. and min. correfponcjl- 
 ing to the fourth log. from 180 ; the remainder is the Z.B For the fourth 
 log. gives the ^^, which is the fupplement to the angles. (II. 104, 96). 
 9 ^8. ;<
 
 Book III. TRIGONOMETRY. 109 
 
 58. Example VI. In the plane Triangle K&c, 
 
 Given AC=4o8>p^^homs. ^ 
 
 AB-IS9\ 
 
 Z. err 22 37'. 
 Required the rejl. 
 
 Construction. 
 Make AC=4o8 fathoms ; thez.c = 22 37'; and drawc^ ; from a, with 
 159 fathoms, cut c^ in h^ or in b, and draw a^ or ab : 
 Then if the angle oppofite to ac is to be acute, the triangle ac^ is that 
 which is required j but if the angle is to be obtufe, acb is the triangle 
 fought. 
 
 Computation. See art. 45. 
 Here being a fide and its oppofite angle known, the folution falls under 
 problem the firft ; the Z.B is to be obtufe. 
 
 To find the angle b obtufe. 
 As ab=: I 59 Fr 
 
 To S,^C = 22 37' 
 
 So AC =408 F. 
 
 7,79860 
 
 9.58497 
 2,61066 
 
 To find BC. 
 As s,^cn22 37' 
 To AB = i59 F. 
 So s,ii.A=s8 04' 
 
 To Bc=350,9F. 
 
 0.41503 
 2,20140 
 9,92874 
 
 To s,Z.B=:99' 19' 
 
 9,99423 
 
 2,54517 
 
 Z.C + Z.BZZ12I 56 
 
 Taken from 1 80 00 
 
 
 
 Leaves/. A = 58 04 
 
 
 
 59. Example VII. In the plane Triangle a-rq 
 
 ""'""" tc^i?^ furlongs. 
 
 z.B=90 00'. 
 Required the rfjh 
 
 Construction. _ 
 
 A Ii 
 
 Make the angle ABC = 90; take ba= 195 equal parts, and bc=:2i6; and 
 draw AC; then abc is the triangle propofed j where the parts required 
 may be meafured by the proper fcales. 
 
 Computation. See art. 46. 
 As two fides and the contained right angle arc known, the folution be- 
 longs to problem the fecond. 
 
 To find the angle a 
 
 As ABr=i95 F. 
 
 To Rad. or tang. 45 00' 
 
 So BC = 2l6 F. 
 
 7.7=997 
 10,00000 
 
 2.33445 
 
 To find A C . 
 As s,Z.A=:47 55' 
 To Bcrr2i6 F. 
 So s, Z.Erz9o'' 00' 
 
 To AC=:29i F. 
 
 0,12950 
 
 2,33445 
 10,00000 
 
 To t.^A=:47'' 55' 
 
 10,04442 
 
 2,46395 
 
 90 00 
 
 
 
 ^c=:42 05 
 
 
 60. Ex-
 
 iid TRIGONOMETRY. 
 
 60. ExAMPL* VIII. In the plane Triangle abc. 
 
 G/V.AB=:ii7) Yards. ^ 
 BC=:II7 3 
 
 AB = i3446'. 
 
 Required the reji. 
 
 Construction." 
 
 Book III. 
 
 Make the z. abc = 134 46^ ; take ba and bc, each equal to 117 equal 
 parts, from the fame fcale, and draw ac j then is the triangle abc equal 
 to that propofed ; and the parts required, meafured on their proper fcales, 
 ivill give their values. 
 
 Computation. See art. 48. 
 
 Now as AB and bc are equal ; there- 
 fore the angles a and c are alfo^equal. 
 Ffom 180 00' 
 Take 134. 46=Ab. 
 
 Leaves 45 i4 = Z.a4-z.c. 
 Thehalf 22 37r=z.A=:Z.c. 
 
 To find AC. 
 As s,Z.A=:22'' 37' 
 To Bc=i 17 Y. 
 So s,z. 8 = 134 46' 
 
 To AC-:2i6 y. 
 
 0,41503 
 2,06818 
 9,85125 
 
 2,33446 
 
 '61. Example IX. In the plane Triangle abc. 
 
 Given AB=4o8? Yards. * 
 
 AC =159) f^- 
 
 AA = 58"'07^ 
 Required the reJl. 
 
 A Li 
 
 Construction. 
 Make an angle cab = 58 07' j take ac = i59, AB=4o8, from the fame 
 fcale of equal parts ; and draw CB ; then w^ill the triangle acb be equal 
 to that which was propofed. 
 
 Computation. 
 Here, there being two fides and their contained angle known, the folu- 
 tion belongs to art. 48. 
 
 ABZ=:4o3 
 /c=:i59 
 
 AB + A 0:13567 = fum of fides. 
 
 AB Acr=249:::difF. of fides. 
 
 To find the angles. 
 
 As AB+ AC = (;67 
 
 To AP, ACZ1249 
 So t.4z.c4-i!lB=6o'' 56|'(See43) 10,25520 
 To t . ^^c Z.B =38 19I 9,89781 
 
 Then (47) 
 
 And 22 37r:Z.B. 
 
 The half of 58 07' 
 
 Is 29 03', which 
 Taken from 90 00 
 
 Leaves 60 56[=lZ.c-f J/Lb. 
 
 7,24642 
 2,39620 
 
 99 i6=Z.c. (II. 105) 
 
 To find B c . 
 
 As s,Z.c=r99 16' 0,00570 
 
 To AB=r4o8 y. 2,61066 
 
 So s,/lA=58'' 07' 9,92897 
 
 To BC=:35i y. 2,5+533 
 
 62. Ex-
 
 j|Book III. TRIGONOMETRY. ,1^ 
 
 62. Example X. In the plane Triangle abc. 
 Given ab=:iq5 
 BG=:2IO 
 Acrrigi 
 ' Required the angles. 
 
 Construction. 
 
 ; Make ca=:29i equal parts ; from c, with 216, defcribe an arc b ; from A 
 with 195 cut the arc B in B j draw BC, ba, and the triangle is conftrudted, 
 then the angles may be meafured by the help of a fcale of chords. 
 
 Computation. 
 The three fides being given, the folution falls under either Problem V. or 
 Problem VI. But that the ufe of thefe Problems may be fufficiently illuf-. 
 trated, the folutions according to both of them are here annexed. 
 
 Solution by Problem V. (419) 
 From the angle b, draw bd perpendicular to ca, which will be divided 
 into the fegments cd, da, the fum of which ac is known. 
 
 Now Bc:r:2i6 
 And ABn:i95 
 
 BC +AE3:4i i=:rum of fides. 
 
 BC ae:^: 2irzdiff. of the fides. 
 
 To find the diff. of the figments. 
 ,AsAC = 2gi 7,53611 
 
 To BC + AB = 411 2,61384 
 
 So BC AB = 21 1,32222 
 
 Now the half of 291 is 
 And the half of 29,66 is 
 
 Therefore (47) the fum 
 
 '45'5 
 
 14,83 
 
 To CD AD ZZ 29,66 1,47217 
 
 i6o,33:;:cd; or cd r: 160,3 
 
 the difference i30,67r:AD ; or ad :=. 130,7. 
 
 Then in the triangle c db. 
 
 As BC =216 7*66555 
 
 To s, Z.CDB:r:9o co' 10,00000 
 So CD 1=160,3 2,20493 
 
 To s, Z.cbd=:47 55' 
 
 9,870^: 
 
 Wh. taken from 90 "00 
 Leaves Z. 0^=42 05 
 
 And in the triangle a D b . 
 As AB = 195 
 
 To s, Z. ADB r= 90 00' 
 
 So AD ZZ I 30,7 
 
 To 8, Z.ABD = 42 05 
 
 Wh. taken from 90 co 
 
 7*70997 
 
 10,00000 
 2,1 162B 
 
 9,82625 
 
 Leaves Z.A r= 47 55,&rZ.Br:90o'. 
 
 To find the angle C 
 Put Er:29ir:Ae 
 F:iz2i6r:BC 
 
 I) = 7 5 =E~P 
 
 G = 1 9 5 rr A B 
 
 2)2;c(i35 = ;g4-D 
 2) 120(60 .'g -u. 
 
 Solution by Problem VI. (50) 
 
 'IbenToAr.Co.log. e. 291 --- 7,53611 
 
 Add Ar. Co. log. F^ =216 --- 7,66555 
 
 Andthelog. -^+u= 135 --- 2,13033 
 
 Alfo the log. ^u uz= 60 - - - 1,77815 
 
 The half of this fura 2)19,11014 
 
 Is the log. fine of ni''02''(43) - - - o,j^-'<=7 
 
 Which doubled, gives 4 2 05'=: Z. c. 
 
 The angle c bein^ known, the other angles may be found by Prob. T. 
 
 63. tx-
 
 MtMMMM**te 
 
 112 
 
 TRIGONOMETRY. Book III.^ 
 
 A. L 
 
 63. Example XI. In the plane Triangle abc. 
 Given A 0=408 
 
 Bc = 35i 
 
 AB:ri59 
 Required the angles, 
 
 C0NSTRUCTI0^f. 
 
 The conftru(SHon and menfuration is performed as in thelaflExAMPLB< 
 
 Computation by Prob. V. art. 49, 
 
 From the Z. b draw the perpendicular bd ; and find the fegments aDj 
 
 DC ; which may be done without logarithms. 
 
 Thus Bc = 35i 
 
 AB=~I59 
 
 c 4- AB = ;io 
 
 BC AB=:i92 
 
 Then (I. 46), As 408 : 510 : : 192 : z^ozroc da. 
 For 192 X3ior=97920; which divided by 408 gives 240. 
 
 Now half of 408=204; and half of 240 is 120. 
 
 Then 2044-I20=:324=:dc ; and 204 i2o=84r=i>A. 
 
 Im the triangle a d b . 
 As AB=:i59 
 
 To s, ^.0=90 00' 
 So AD=:84 
 
 To S, Z.ABD=3I 53' 
 
 7,79860 
 
 16,06:00 
 
 1,92428 
 
 9,72288 
 
 And /I A ~^^ 07 
 
 In the triangle bdc. 
 As Bc=:35i 
 
 To s, Z.D=9o 00' 
 So 00=324 
 
 To s,Z.CBDri:67'' 23' 
 And 
 
 745469 
 
 10,00000 
 
 2,51054 
 
 9,96523 
 
 Z.CZZZ2 37 
 
 Then ^abd + Z.cbd=: Z.abc=:99 16' 
 
 Computation by Prob. VI. art. 50. 
 
 Then, to Ar. Co. log. e =408 738934 
 
 Add Ar. Co. log. f =351 7AS^^9 
 
 And the log. | G+D nio8 2,03342 
 
 Alfo the log. IG D= 51 >707S7 
 
 The half of this fum 2) 1 8, 58502 
 
 Is the log. fine of 1 1 1 8'!: 9,29251 
 
 To find the angle c. 
 
 Put = 40^ = ^0 
 h=35I=:bc 
 
 D= 57 = E F. 
 
 = 159 = ^8 
 2}2i6(io8=4^gTu 
 
 2)102(51 =. ^-G D Which doubled, is 22 37'izZ.c. 
 
 Now the angle c being known, the other angles may be found by Prob. I. 
 But for a farther illuftration of Prob. VI. the work for another angle is 
 here repeated. 
 
 Tefind the angle b. Then, to Ar. Co. log. E =351 7,45469 
 
 Put e=35i=bc Add Ar. Co. log. f = 159 7,79860 
 
 f = I59 = ab And the log. ^G + Dzr^oo 2,47712 
 
 Alfo the log. iG D=io8 2,03342 
 
 D=:i92~E F. 
 
 6=1403=: AC 
 
 2)600(300=^0 +D 
 
 2)2i6(io8=:-1g D 
 
 The half of this fum 2) . 9,76383 
 
 Is the log. fine of 49 38' 9,88191 
 
 Which doubled, is 99 i6'=Z.b. 
 
 64. x^
 
 Book III. TRIGONOMETRY, 
 
 113 
 
 64. 
 
 Example XII, In tht flgn^ Triangle abc, 
 
 = 1171 
 
 = 117 S-Milej. 
 
 = 2l6j 
 
 Q'lven AE = ii7 
 Bcr: 
 
 AC 
 
 Jie^uired the angles. 
 
 Construction", 
 
 The conftrution of this triangle, and the nieafurlng of thp angles, Is 
 performed as in the Xth and Xlth Examples, 
 
 Computation, 
 
 In the triangle abc, as ab=:bc ; therefore the angles a and c are equaj 
 (II. 104) ; and the perpendicular bd bifecls the fide ac ; fo that the right 
 angled triangles adb, cdb, are congruous ; confequently, the angles tie-s 
 ing found in one triangle, will give thofe of the other. 
 
 Now in the triangle adb, the fide ab=:ii7; the fide ad,;=:|ac, 15 
 ie8 ; and the z.d is 90 00' : Here, therefore, being a fide and i^s op^Q* 
 iite angle given, the folution belongs to Prob, I. 
 
 And 67" 23' doubled 
 
 Gives 134 46'=: 4. ABC. 
 
 A like procefs is to be ufed In every 
 triangle;^ in which 9fe tvyo e<|uaj 
 fides. 
 
 ^0 find the angle a 50. 
 
 As AB =117 793i8i 
 
 To s, ^ D =90" 00' 10,00000 
 
 So AD =ro8 2,03342 
 
 To s, /.ABD=67*' 23' 9,96523 
 
 Wh, t^ken from 90 00' 
 
 Leaves 22* 37'r:4.A =:Z.c. 
 
 The foregoing examples contain all the variety that can pofllbly happeji 
 in the folutions of plane triangles, confidered only with regard to their 
 fides and angles ; but befides the methods fhewn of refolving fuch tri- 
 angles by conftrudion and computation, there is another way to find 
 thefe folutions, called Inftrumental ; and this is of two kinds, vi%. eithep 
 by a ruler called a Senior, or by one called the Gunter's fcale : The method 
 by the feclor, the curious reader may fee in many books, particularly in 
 a treatife on Mathematical Inftmments publifhed in the year 1775, 3J 
 edition * ; But the other method by the Gunter's fcale being in greaj 
 life at fea, it will be proper in this place to treat of it. 
 
 J i .It il J ii U I UI 
 
 By the author of thefe Eleraenu. 
 
 Vol. I, 
 
 SEC-
 
 114/ T Jl I G O.N O M E T R Y. Book III. 
 
 SECTION IV. 
 
 Defcripiof}^ and ConJlruBion of the Gimters 
 
 Scale. 
 
 65 Mr. Edmund Gunttr^ Profeflbr of Aftronomy at Grcjkam College, 
 fpmetime about the year 1624, applied the Logarithms o( Numbers to a 
 iTat ruler : This he efFciSted by talcing the lengths exprcflt.d by the figures 
 in thofc logarithms from a fcale ef equal parts, and transferring their, to a 
 line, or fcale, drawn on fuch a ruler ; and this is the line which, from his 
 name, is called the Gunter's line : He alfo, in like manner, conftru6led 
 lines containing the logarithms of the fines and tangents ; and fmce his 
 time there have been contrived other logarithmic fcales adapted to va- 
 rious purpofes. 
 
 The Gunter's fcale is a ruler, commonly two feet long ; having on one 
 of its flat fides feveral lines or logarithmic fcales ; and on the other fide 
 various other fcales ; wliich, to diftinguifh them from the former, may 
 be called natural fcales. 
 
 While the reader is perufing what follows, it is proper he fhould have a 
 Gunter's fcale before him. 
 
 6$. Of the Natural Scales. 
 
 ' The half of one fide is filled with different fcales of equal parts, for the 
 convenience of conftrufting a larger, or fmaller figure : The other half 
 contains fcales of Rhumbs, marked Rhu ; Chords, marked Ch ; Sines, 
 marked Sin ; Tangents, marked Tan ; Secants, marked Sec ; Semi- 
 tangents, denoted by S. T. and Longitude diftinguiflicd byM. L. The 
 dcfcripiions and ufcs of thefe fcales will be confidcred hereafter, in the 
 places where they will be wanted. 
 
 67. Of the Logarithmic Scales. 
 
 On the other fide of the fcale are the following lines. 
 
 I. A line marked s. R. (fine rhumbs), which contains the logarith- 
 mic fines of the degrees to each point and quarter point of the compafs. 
 
 JI. A line figned t. R. (tan. rhumbs), the divifions of which cor- 
 refpond to the logarithmic tangents of the faid points and quarters. 
 
 Ill, A line marked Num. (numbers), where the logarithms of num- 
 bers are laid dov/n. 
 -IV. A liac-marked Sin. containing the log. fines, 
 V. A line of log. verfed fines, marked v. s. 
 VI. A line of log. tangents, marked Tan. 
 Vn, A meridional line figned Mer. 
 VIII. A line of equal parts, marked e. p. 
 
 9 68. I. Of
 
 i^ook fit. TRIGONOMETRt. 115 
 
 iS-8. \-r. ,jf .. I. Of the Line of Numbers. 
 
 Th^'whole length of this line, or fcale, is divided into two equal fpaces, 
 or intervals : the beginning, or left-hand end of the firft, is marked i ; 
 the end^of the firft interval, and beginning of the fecond, is alfo marked i; 
 and the end of the fecond interval, or end of the fcale, is marked with lOj 
 Both thdfe diflances are alike divided, beginning at the left-hand ends, 
 by laying down in' each the lengths of the logarithms of the num.bers 
 20, 30, 40, 50, 60, 70, 80, 90 J taken from a fcale of equal parts, 
 fuch that 10 of its primary divifions make the length of one interval : 
 And the intermediate divifions are found, by taking the logarithms gf like 
 intermediate numbers. 
 
 -From this conftruftion it is evident, that when the firft i ftands fox i, 
 die fecond i ftandsfofio, and the end lo denotes loo; 
 
 
 r '* 
 
 U5-T3 
 
 100, 
 
 \i!fc. 
 
 ^^ '^ ^ 
 
 1' -- 
 
 ^m t/i 
 
 10' 
 
 c - 
 
 < ( 
 
 
 " ,0 
 
 100, 
 
 w -g 
 
 1000; 
 
 1000, 
 
 oj 
 
 1 0000 ; 
 
 ^c. 
 
 iS^^ 
 
 yr.; 
 
 It 
 
 " G 
 
 10 ; 
 
 T5 
 
 c 
 
 I ; 
 
 tSfc. 
 
 " <-5 1 
 
 . i^c; 
 
 " An<i the pi'imary and Intermediate divifions in each interval muftbe 
 eftimated according to the values fet on their extremities, viz. at the be- 
 ginning, middle, and end of the fcale. 
 
 Now the examples moft proper to be worked by this fcale, are fuch 
 where the numbers concerned do not exceed looo, and then the firft i 
 ftands for lo, the middle i for lOO, and the lo at the end for lOOO : 
 The primary divifions in the firft interval, viz. 2, 3, 4, 5, 6, 7, 8, 9, 
 ftands for 20, 30, 40, 50, 60, 70, 80, 90, and the intermediate divi- 
 fions ftand for units. In the fecond interval, the primary divifions fign^d 
 2j 3> 4> 5 6, 7, 8, 9, ftand for 200, 300, 400, 500, 600, 700, 800, 
 900 ; each of thefe divifions are alfo divided into ten parts, which repre- 
 Icnt the intermediate tens. Between 100 and 200 the divifions for tens 
 are each fubdivided into five parts ; fo that each of thefe IcfTer diviPons 
 ftand for two units. The tens between 2C0 and 500 are divided into 
 two parts, each ftanding for five unit? : The units between the tens from 
 500 to 1000 are to be eftimated by the eye j which by a little praclicc is' 
 readily done. ^ 
 
 From this defcription it v/ill be eafy to find the divifion reprefinting a 
 given number not exceeding icoo: Thus the number 62 is the fecond 
 fmall divifion from the 6, between the 6 and 7 in the firft interval : The 
 number 435 is thus reckoned ; from the 4 in the fecond interval, count 
 towards the 5 on the right, three of the larger divifions, and one of the 
 fmallcr J and that will be the divifion exprelfing 435. And the Hke of 
 otlicr numbers. 
 
 I 2 69. II. For
 
 ii6 T R I G O N 0,M E T R Y. Book III, 
 
 69. n. For the Line of Sines, i 
 
 This fcalc terminates at 90 degrees, juft againft the 10 at the end of th 
 line of uumbfcrs ; ami from this termination the degrees are laid backwards, 
 or from thence towards the left : Now fecking in a table of logarithmic 
 fmes, for the numbers exprefling their arithmetic complements, witho;' 
 the index, take thofe numbers from the fcale of equal parts the logs, '\ 
 the numbers were taken from, and apply them to the fc^le of fines (rC^ 
 90, and they will give the feveral divifions of this fcale. "^ 
 
 Thus the arith. comp. of the log. fines (or the co-fecants) abating t^ 
 index, of 10, 20, 30", 40, i^fc. are the numbers 76033, 46595, 3010 t 
 J9193, ifc. then the equal parts to thofe numbers, laid from 90, wijf 
 give the divifions for jo", 20", 30, 40, ^c. and the like for the inter- 
 mediate degrees. 
 
 Proceeding in this manner, the arith. comp. of the fine of 5 45' will bc 
 about equal to 10 of the primary divifions of the fcale of equal parts, or to 
 one interval in the log. fcale ; fo that a decreafe of the index by unity, 
 anfwers to one interval ; then a decreafe of the index by 2 anfwers to two 
 intervals, or the whole length of the log. fcale; and this happens about 
 the fine of 35 min. and the divifions anfwering to the fine of a little above 
 3 min. I'iz. 3' 26^^. will be equal to 3 intervals ; and the fine of about 
 26''' will be 4 intervals, &c. fo that the fine of 90 being fixed, the be- 
 ginning of the fcale is vaftlydiftant from it. 
 
 It is ufiial to infert the divifions to every 5 minutes, as far as lO de- 
 grees ; from 10 to 30, the iVnall divifions are of 15 minutes each ; froin 
 30 to 50", contains every half degree ; from 50 to 70", are only wholq 
 jjegrees j the reft are eafily reckoned, 
 
 yo. III. For the Lwe of Tangents, 
 
 As the tangent of 45 degrees is equal to the radius, or fine of 90; 
 therefore 45 on this fcale, is terminated diredtly oppofite to 90 on the 
 fines ; and the feveral divifions of this fcale of log. tangents are con- 
 ftru(fted in the fame manner as thofe of the fines, by applying their arith. 
 comp. backwards from 45, or towards the left-hand. 
 
 The degn^es above 45, are to be counted backwards on the fcale : 
 Thus the divifion at 40^ rcprefents both 40 and 50 ; the divifioH 30 
 ferves for 30 and 60" j and the like of th.e other divifions, and their inter- 
 n^.ediates, 
 
 71. IV. For th-e Line of Veifed Sines. 
 
 This line begins at the termination of the numbers, fines, and tangents : 
 But as the nimibers on thofe lines defcend from the right to the left, fo 
 thcfe afcend in the fame direction : Now having a table of logarithmic 
 verfcd fines to 180 degrees, let each log. verfcd fine bc fubtra6led from 
 that of I <So degrees ; then the remainders being fucceffively taken from 
 the faid fcale of equal parts, and laid on the ruler backv/ards from the 
 coniinou termination, the feveral divifions of this fcale will be obtained. 
 
 The
 
 look III. T iFt I G O N O M E T R Y. 
 
 The numbers for each lodeg. are in the following table. 
 
 "7 
 
 D 
 
 iu 
 mc 
 
 Numb. 
 
 0,0033 
 0,0133 
 0,030) 
 
 D. Numb. 
 
 400,0540 
 500,08^4 
 600,1219 
 
 Numb- 
 
 0,173; 
 
 0.23 '5 
 0,301c 
 
 Dep. 
 
 100 
 
 lie 
 
 120 
 
 Num b. 
 
 0,3^39 
 
 o 4828 
 o,6c2i 
 
 Deg, 
 
 130 
 
 140 
 150 
 
 Numb. |Deg, 
 
 0,74^1: 
 0,9319 
 1,1740 
 
 I bo 
 
 180 
 
 Numb. 
 
 1,5207 
 
 2,2194 
 
 10,3010 
 
 3(i 
 
 k. 
 
 
 
 Now co-fine: rad, 
 
 fine : rad. 
 tan. : rad. 
 
 The other fcales will be deCcribed in their proper places. 
 
 Demonftration of the foregoing confiru^ions. 
 Xhatof the log. numbers, is evident from the nature of logarithms, 
 
 Ftr the Sines and Tangents^ 
 
 rad. : fecant {l4-)>. Then co-f. x fecant =1 
 rad. : co-fee. (35). fine X co-fee. i 
 
 rad. : co-tan. (36). tan. x co-tan. =1 
 
 The radius of the tables being fuppofed equal to i. 
 Hence it is evident, that in either cafe, one of the quantities will be 
 equal to the quotient of unity divided by the other. 
 
 But divifion is performed by fubtraclion with logarithms* 
 And to fubtradl a log. is the fame as to add its arith. comp. 
 Confequcntly, the logarithmic co-fine and fecant of the fame degrees 
 arc the arithmetical complements of one another. 
 
 And fo are the logarithmic fines and co-fecants : Alfo the logarithmic 
 tangents and co-tangents are the arith. comp. of one another. 
 
 73. Now as the arith. comp. of any number is what that number 
 wants of unity in the next fuperior place j 
 
 Therefore every natural fine and its arith. comp. together make the ra- 
 dius. 
 
 And the fines begin at one end of a radius, and end in 90" at the other 
 end. 
 
 Therefore in a fcale of fines, the arith. comp. of any fine, or its co- 
 fecant, laid backwards from 90", gives the divifion for that fine. And 
 the like muft happen in a fcalcof log. fines. 
 
 74. Alfo, as the logarithmic tangents and co-tangentS are the arith* 
 comp. of one another; therefore in a fcale of log. tangents, the divifions 
 to the degrees both under and above 45, are equally diftant from the di- 
 vifion of 45". 
 
 Confequcntly the divifions fcrving to the degrees under 45, will fervcj 
 by reckoning backwards, for thofe above 45. 
 
 75. For the Verfed Sines. 
 
 Although the numbers in the line of verfed fines afcend from right to 
 left, yet they are only the fupplements of the real verfed fines, which arc 
 numbered in the fame order as the fines, that is, from left to right : But 
 as the beginning of the verfed fines falls without the ruler, therefore it is 
 moft convenient to lay down the divifions from the point where the verfed 
 fines terminate at 180 degrees, that is, againft 90'^ on the fines. 
 
 Now it is evident that the divifions laid ofF from this termination mufl 
 be thediftcrcnces between the log. verfed fines of the fevcral degrees, ^V. 
 and that of 180 degrees, 
 
 I 3 SECTIO>f
 
 it8 trigonometry. Book 11^^ 
 
 S E C T I O N y. 
 
 The life of the Gunter's Scale in Plane Triggmtnetry. 
 
 76. When a Trigonometrical Q^ieftion is to be folved by the Guntcr's 
 fcale, it niuft- fijft be ftated by the precepts to that problem under which 
 the queftion falls, whether it be by oppofite fides and angles, or by two 
 fides and their included angle, or by three fides. 
 
 77. In all proportions WTOught by the Gunter's fcale, when the firft and 
 fecond terms arc of the- fame kind, then , 
 
 The extent from the firjl term to the fecond, will, r^ach from the third term, 
 to the fourth. 
 
 Or, when the firft and third terms arc of the fame kind. 
 
 The extent from the firfl term to the third will reach from the fecond, 
 term to the fourth. 
 
 That is, fct one point of the compafTes on the divifion exprefllng the firf^ 
 term, and extend the other point to the divifion exprcffing the fecond (or 
 third) term ; then, without altering the opening ofthe compafics, fet one 
 point on the divifion reprefenting the third term (or fecond term), an^ 
 the other point will fall on the divifion fhewing the fourth term or 
 anfwer. 
 
 In working by thefe directions, it is proper to obfcrve, 
 
 78. Firft. The extent from one fide to another fide, is to be taket> 
 from the fcale of numbers ; and the extent from one angle to another is 
 to be taken from the fcale of fines, in working by oppofite fides and angles- 
 or from the fcale of tangents, in working by two fides and the included 
 angle. . 
 
 Secondly. When the extent from the firft term to the fecpnd (or third) 
 is decrcafing, or is from the right to the left, then the extent from the third 
 term (or fecond) muft be alfo decrcafing ; that is, applied from the right 
 towards the left : And the like caution is necellary when the extent i^ 
 from the left towards the right. 
 
 Thcfc precepts being carefully attended to, what follows will be readily 
 imdexftood. 
 
 79' ^.
 
 Book III. T R I GO N O M E T R Y. 119 
 
 79. In Example I. See article 53. 
 
 As s,Z.c : AB : : s,z.b : AC Or 8,42 05' : 195 : : s, 9 00'; Q^, 
 where Q^ftands for the number fought. 
 
 Now the extent from 4.2 05' to 90" 00', taken on a fcale of fines, and 
 applied to the fcale of numbers, will reach from 195 10291. See art 69. 
 
 Alfo. As s,z.c: ab:: s,<^a:bc. Or s, 42* 05' : 195 : : 5,470 55' ; q. 
 
 Then the extent from 42 05' to 47 55' on the fines, being applied to 
 the numbers, will reach from 195 to 2i6. See art. 68. 
 
 In each of thefe operations, the firft extent was from the left to the right, 
 or increafing j therefore the fecond extent muft be from left to right alfo, 
 
 80, In Example IV. See art. 56. 
 
 As AC : 3,Z.B : : AB : s,z.c. Or 291 : 5,90 00' : : 195 : q. 
 
 t 
 
 Here the extent from 291 to 195, taken on the numbers, and applied 
 to the fines, will reach from 90 00' to 42'' 05'. 
 
 The firft extent being from the right towards the left, or decreafing j 
 therefore the fecond extent muft be alfo from the right to the left. 
 
 In Example VII. See art. 59. 
 
 As AB : Rad. : : bc : t,z. a. Or 195 : t, 45 00' : : 216 : q^ 
 
 Then the ej^tent from 19S to 216 on the numbers, will reach from 
 45" 00' to 47 55' on the tangents. 
 
 Here the firfl extent being from left to right, or increafing, therefore 
 the fecond extent muft alfo be increafing : Now on the tangents, this in- 
 creafe above 45 does not proceed from left to right, but from right to 
 left, the fame way that the decreafe proceeds (70} j confequently the di- 
 vifion which the point falls on for the fourth term, muft be eftimated ac- 
 cording as the firft extent is increafing or decreafing. 
 ' Thus had the proportion been. 
 
 As EC : Rad. : : ab : t,z.c. Or, 216 : t,45 00' : : 195 : q. 
 
 Then the extent from 216 to 195 on the numbers, will reach from 
 45 00' to 42 05', eftimated as decreafing. 
 
 81. When two fides and the included angle are given, and the tangent 
 of half the difference of the unknown angles is required. 
 
 Then, on the line of numbers take the extent from the fum of the 
 given fides to their difference j and on the line of tangents apply this ex- 
 tent from 45 downwards, or to the left; let the point of the compailL-s 
 reft v/hcrc it falls, and bring the other point (from 45") to the divifion 
 anfwering to the half fum of the unknown angles ; then this extent ap- 
 plied from 45 downwards, will give the half difference of the ujiknovvn 
 angles : Whence tbc angles may be found. (47) 
 
 I 4 In
 
 tid ^TRIGONOMETRY. Book III. 
 
 In Example IX; S^tthe art. 6i.' 
 
 As Afe + AC i AB AC : : t,iiic+ Z.B : t,|z.c Z^b, 
 Or 567 : 249 : : t, 60" 56' : q^ 
 
 . Kow the extent from 567 to 249 on the numbtl-s, being applied to thd 
 langcntSi will reach from 45" to about 23" 40' : Let one point of the com*- 
 J)afie6 reft on this divifion, and bring the other to 60 56' ; then this ex- 
 tent vrill reach from 45' to 38 19^, the half difference fought. 
 
 And this method will always give the half difference, whether the half 
 fum of the angles is greater or lefs than 45* 
 
 82. But when the half fum and half difference are greater than 45*; 
 then the extent from the fum of the fides to their difference on the fcale of 
 humbers, will (on the tangents) reach from th? half fum of the angles to 
 kheir half difference, reckoning from left to right. 
 
 Ahd when the half fum and half difference are both lefs than 45 ; then 
 the extent from the fum of the fides to their difference, taken from the 
 humbers and applied to the tangents^ will reach from the half fum of the 
 iangles downwards to their half difterence. 
 
 83. When the three fides are given to find an angle, and a perpendi- 
 cular is drawn from an angle to its oppofite fide. See Ex. X. art. 62. 
 
 As AC : BC + AB : : bc ab ; cd AD. 
 Or 291 : 411 : : 21 : (^ 
 
 ^Tow the extent from 291 to 411 on the fcale of numbers, will reach 
 from 21 to 29,6 on the numbers alfo. 
 
 Then the extent for the angles is performed in the fame manner as 
 fiiewrt in Ex. L (78) 
 
 84. Or an angle may be found by Problem VI. as follows* 
 
 In the fcale of numbers, take the extent from the half fum (of G and d) 
 to either of the containing fides (as e) ; apply this extent from the other 
 containing fide fas f), to a fourth term : Let one point of the compafles 
 tei\ on this fourth teVni, and extend the other to the half difference (of 
 n and >) ; then this extent applied to the Verfed fines from the begin* 
 hingv will give the fupplement of the aiigle fought. 
 
 In Example X. See art. 62* 
 
 =291; F = 2i6; half fum =135; half difT. =: 60. 
 
 Thert oh the numbers, the extent from 135 to 291, will reach from 
 -216 to 465 ; let the point reft there, and extend the other to 60 ; thert 
 this extent applied to the verfed fines, will reach from the beginning tc 
 ^37'* i^''i Whih taken fronr i^C, leaves 4.2'- 04^ for the angle' foaght* 
 
 B5. In
 
 Book IIL TRIGONOMETRY, 121 
 
 8|. In Example XI. See art. 63. 
 
 HereE=4o8; f = 35i ; halffum of g and d=:io8; half difF.=5i 
 Then on the numbers, the extent from io8 to 408, will reach from 
 351 in the fecond interval, to a fourth number : But as the point of the 
 tbrnpaifTes falls beyond the end of the fcale, therefore let the extent from 
 108 to 408 be applied in the firft interval, which will reach from 35,1 
 to 1 32,6 ; let one point reft on 1 32,6, and ektend the other point of the 
 compafles to 5 1 . Now as this extent of the Compafles is lefs than it 
 ought to be, by one inter/al^ or half the length of the fcale of numbers ; 
 therefore the laft extent, when applied to the verfed fines, muft be from 
 that divifion, on the verfed fines, oppofite to the middle of the fcale of 
 numbers, which is nearly at 143 ; and it will reach from thence to the 
 verfed fine of 157 23'; which taken from 180, leaves 22 37' for the 
 angle fought* 
 
 86. Moft 6f the writers on Plane TrigOiiometry treat of right angled, 
 and of oblique angled triangles feparately ; making feven cafes in the 
 former, and fix cafes in the latter : But as every one of thefe thirteen cafes 
 fall under one or other of the foregoing Problems, therefore fiich diftinc- 
 tions are here avoided, it being conceived, that they rather tend to per- 
 plex than inftrudl a learner : Alfo in the generality of the treatifeson this 
 fubjcft, it is ufually ftiewn how the folutions of right angled triangles 
 are performed, by making (as it is called) each fide radius ; that is, by 
 comparing each fide of the triangle with the radius of the tables : And 
 although thefe confiderations are here omitted, yet the inquifitive reader 
 will find them in Book VII. near the beginning* 
 
 87. Befide the demonftration of Problem IV. at art. 48. it has been 
 thought proper to give another demonftration ; becaufe there arifes from 
 it a Theorem ufeful on fome occafions : Moreover, there is alfo added 
 methods of deriving other rules for the folution of the cafe where the 
 three fides are given to find an angle ; which, if they fhould be found of 
 no other ufe, will perhaps be agreeable cxercifes of Geometry to thofc 
 who are delighted with thefe ftudies. 
 
 SECTION
 
 i2e 
 
 TRIG ON O M ET R Y. 
 
 Sbbk III. 
 
 SECTION VI. 
 
 Propertied of Plane 'Triangles, 
 
 ^, in ahy plane triingle abc. 
 
 Given CA, ^B, and ^c. 
 llequired the angles Band a. 
 
 SoLtJTiON. Take cd = cb, and draw pb. 
 Bife^l DB in f, da in e, draw cfg, and ef, 
 ^hich is parallel to Ab. (II. 165) 
 
 Kow DE or AE is equal to half the difference 
 ofcAandcB. 
 
 And CE ( zrcA ae) is equal to the half fum 
 of CA and CB. 
 
 The fum of the equal angles cbd, Cdb (II. 104) is equal to the fum 
 of the unknown angles cba, cab. (II. 08) 
 
 Then the angle cbd is the half fum, and the angle abd is the half 
 4iifefence of the unknown angles ci^a, cab. (47} 
 
 And as cfg is at right angles to db (II. 103) ; cf is the tangent of 
 4. CBD, and OF is the tangent of Z-ABF to the rad. bf. (4) 
 
 Thtn CE : ea: : cf : of (II. 165} : Or 2ce : 2ea : : cf : fg. (II. 151) 
 
 That is, ca + cb : ca cb : : tan. |Z.cba+cab : tan. f iicBA cab. 
 
 9. Again. From h, the middle of cd, draw hi at right angles, and 
 <fqual to CH ; draw Di, and EK parallel to Di, meeting ci produced iii 
 K ; and join IE. 
 
 Now he=:(jCD + |da)|ca; and eH = |cB. 
 And HEcztangent of the angle hie to the radius, HirtHC. 
 Then CB : CA : : (2hc : 2he : : hc : he : : ) Radius : tan. Z.hie. 
 And Z.HIE (Z.hid=:) 45=Z.die = akei (II. 95) is known. 
 
 CK : Ki; becaufe ilKCE=:Z.KEc. 
 
 CE : ED. (II. i66) 
 
 (I. 151) 
 
 : : ca4cj^:ca cb:: ?,f fumZ.5:/,IdifF.z.j(48) 
 
 Ponfequently rad. : tan. Z.kei : : tan. |:Z.cba+cab : tan. |Z.tBA cab. 
 
 This rule is often ufeful in Aftronomical calculations, when the loga- 
 rithms of the fides ab, bc are only known, and the angles bac, bca, arQ 
 required, without finding, from thofe logarithms, the fides themfclves. 
 
 For the difference between the logarithms of the fides, increafed by 
 radius, gives the tangent of an arc^ which arc leffened by 45^ leaves a 
 fecond arc. 
 
 Then as rad. : tan. fecond arc : : tan. ' fum ^r ; tan. \ diff. /Ls. 
 
 Then rad. 
 
 : tan./iKEi : 
 
 : EK 
 
 Ki : 
 
 But 
 
 
 : CK 
 
 Ki : 
 
 
 
 :2CE 
 
 2ED. 
 
 90, In
 
 Pook IIL 
 
 TRIGONOMETRY. 
 
 123 
 
 90. In any plane triangle abc, where the three fides are known, the 
 meafure of either angle (as the angle a, included between the fides aB| 
 Ac) may be found feveral ways, as {hewn in the following articles. 
 The letters {,/, "y, ftand for fine, tangent, verfed fine. 
 s\ /", ftand for co-fine, co-tangent. 
 v\ ftands for the verfed fine of the fupplement. 
 Alfo ss^ it, ftand for the fquares of the fine and tangent. 
 Alfo jV, /'/', ftand for the fquares of the co-fine and co-tangent, 
 Radius=zRr:AB=:AE = ADj and u=iAB + |AC+iBC. 
 
 B 
 
 E c 
 
 91. J,|/lA=Ryy "-^^^^"-^^; Here^E=|/.A. 
 
 ' V AC X AB 
 
 For R : J, iz.A : : (DEr=)2AB ; BQ. 
 And RR : 5^, ^Z-A t : 4ab^ : bd\ 
 
 XT . BD^ 
 
 Mow JijT^ArrRRX . 
 
 4AB^ 
 
 2AB 2H 2ACX2H 2BC 
 
 77^^ ' xrr: 
 
 4AB 2AC 
 
 Then s, l^ArzRX V"-- ^^^"-^.g. 
 
 V ACXAB 
 
 f45) 
 (II. 161) 
 
 (II. 164) 
 (II. 179) 
 
 2R 
 
 92. ^'^ A_-^^^^-^Xy'HXH-CBXH ACXh~AB' 
 
 For AB : BG : ; R : j,Z.a. 
 And ab^ : bg* : : rr : ss, Z. A. 
 
 tvT ^^ 
 
 JNOW SSj Z.A= rXBG\ 
 
 AB f 
 
 rr 4. 
 
 "T^ ^ T7t X H X H CB X H AC X H AB. 
 AB AC 
 
 (45) 
 (II. 161) 
 
 (11. 164) 
 (II. 180) 
 
 93- ^Mz.A=RX./!L^ilz?. Herez.E=Jz.Ajandz.D = comp.Z.E. 
 
 V AC X AB 
 
 For R : j\ |z.A : : (de=:) 2ab : be. 
 And RR : i^\ IZ-A : : 4AB* : be\ 
 
 No'A^ jV\ iZ. A=:RR X 
 
 4AB'- 
 
 4AB"X2AC 
 
 2AB X4XH Xh CK 
 
 =:rrx ^ 
 
 (45) 
 (II. 161) 
 
 (11. 164) 
 (II. 178) 
 
 94. '.!
 
 i4 T k I G O N P M E T R Y. 
 
 Book III. 
 
 
 
 P HXHCB 
 
 *:/, ^2.a::be:bd. 
 
 R x^H ACXH AB 
 
 =RRX 
 
 kxhUbc 
 
 95. '\UA=RxV'r=^^iir^,^ 
 
 ir - H .ACXH~AB 
 
 For BD* : BE* /'/ ^ / A . \ 
 Tfeefli>/^,.^A=RRxiC 
 
 BD* 
 
 A 
 
 (46) 
 (II. ibi) 
 
 ^^ (II. 164) 
 (11. 178, 179) 
 
 (36) 
 (11. 178, 179} 
 
 9- t;jZ.Ar:2RX^i=^^i^^t-^. 
 ACXab 
 
 AB : GD. 
 GB 
 
 Then ^, ^A=Rx-=Rx!ilili:iilll? 
 
 -'^S ACXAB 
 
 $7. vVa=2Rx''-^^^^. 
 . ACXAB 
 
 For R : ^;\ A : : AB : ge. 
 
 rhcnv\ Z.A=:RX^r:Rxi^l!12iEEH?. 
 
 ^^ ACXAB 
 
 98. j' Z- A = ' R X 
 
 CB* AC* AB 
 
 P ^ AC X AB 
 
 ror n.:s, /.a:: ab: ag. 
 
 7-hens\ / A-p v^^^ ^ Bc'-^Ac^-AB* 
 
 ^fi 2ACXAB 
 
 (45) 
 (" ^77) 
 
 (45) 
 (II. 176) 
 
 (45) 
 (11. 174; 
 
 END OF BOOK III. 
 
 mt
 
 ''' J //'ly-' <>>:uOi 
 
 at^a/ntit/ii/i/i- lu/i
 
 THE 
 
 ELEMENTS 
 
 of' 
 
 NAVIGATION, 
 
 
 BOOK IV. 
 
 
 
 OF SPHERICS. 
 
 
 SECTION!. 
 
 Definitions and Principles^ 
 
 j.PjPHERICSis that part of the Mathematics which treats of the 
 (^ pofition and magnitude of arcs of circles defcribed on the furface 
 of ji fpherc. 
 
 2. A Sphere is a folid contained under one uniform round furface, 
 fuch as would be formed by the revolution of a circle about its diameter, 
 that diameter being immoveable during the motion of the circle. 
 
 Thus the circle A f.bd revolving about the diameter h^^^ loill generate a fphcre^ 
 the furface of which will be formed by the circumference Aebd. See Plate I. 
 
 3. The Center and Axis of a fphere are the fame as the center and 
 diameter of a generating circle : And as a circle has an indefinite num- 
 ber of diameters, fo a fphere may be confidered as having alfo an indefinite 
 number of axes, round ^ny one pf which the fphere may be conceived to 
 be generated. 
 
 4. Circles of the Sphere are thofe circles defcribed on its furface 
 by the motion of the extremities of fuch chords in the generating circle as 
 arc at right angles to the diameter, or to the axis of the fpherc. 
 
 Thus by the motion of the circle aebd about the diameter A P., the extre- 
 mities of the chords ED, GF, IH, at right angles to ab, will defer ibe circles 
 the diameters of which are equal to thofe chords refpe^ively, Plate I. 
 
 5. The
 
 Ud^ ^^ F H E R I C S. Book IV. 
 
 5. The Poles of a circle on the fphercj are thofe points on its fuN 
 facc equally diftant from the circumference of that circle. 
 
 Thus A anfi b are the poles of the circles defcribed on the fphere by the 
 etati of thf chords EOy gf, ih. Plate I. 
 
 6. A Great Circle of the fphere, is that circle which is equally 
 diftant from both its poles. 
 
 Thus the circle defcribed hj^ }he extremities E, D, of the diameter ED, at 
 ri^hf angles /<? ab, being equally difiant from its poles A and b, is called a 
 great circle, 
 
 7. Lesser Circles of the fphere, or fmall circles, are thofe circles 
 which are unequally diftant from!both their poles. 
 
 TTjus the cir^Us of which FG, i^i, are diameters^ having thiifr polfSM. and B 
 unequally diflanti Jrotn iheniy are cilled lejfer^ circlt^,> \ v IT 
 
 8. Parallel Circles of the fphere, are thofe circles, the planes of 
 which are confidered as parallel to the plane of fome great circle. 
 
 Thus the circles htivinfthi diantetcrs FG, Hi, are caltedfarallei circlei iH 
 rff^e^ of the ^r eat Arcleof%vhich*i^ is the diameter. ' ^ -"- 
 
 9. A Spheric Angle is the- inclination of two great circles of the 
 Iphere meeting one another; 
 
 10. A Spheric Triangle_ is a figure formed on the furface of a 
 Iphere by the mutual interfet^tions of three great circles. 
 
 11. The Stereographic Proje'ction of the fphere, is fuch a re- 
 prefentation of its circles, upofl th<J plane of one of therti'pafling through 
 the center, and called the Plane of Projection, as would appear to 
 an ye placed in one of the poles-of that great circle,- and thence viewing 
 the circles on the fphere. 
 
 12. The place of the Eye is called the Projecting Point, or lower 
 pole : and the point diametrically oppofite is called the remoteft, or op- 
 pofite, or upper pole. 
 
 Alio, the projedlion of any point on the fphere, is that point in the 
 plane of projedion, through which the vifual ray pafles to the eye. 
 
 13. The Primitive Circle is that great circle, on the plane of 
 which the reprefcntations of all the other circles are fuppofcd to be drawn. 
 
 14. An Oblique Circle is one which has its plane obHque to the 
 eye. 
 
 15. A RiGT Circle is that which is perpendicular to the plane of 
 the primitive circle, and if it be a great circle, its plane pafles through the 
 eye, and it is feen edgewife ; confequentiy it is reprefented by a ftraight 
 line drawn through the center of the primitive circle. 
 
 AXIOMS.
 
 Book IV. SPHERICS. ra; 
 
 AXIOMS. 
 
 i6. The diameter of every great circle pafles through the center of the 
 fphere J but the diameters of fmall circles do not pafs through. the fame 
 center : Alfo the center of the fphere is the common center of all its 
 great circles. 
 
 17. Every feiion of a fphere, by a, plane paffing thpugh its circum- 
 ference, is a circle. 
 
 18-. A fphere is divided into two equal parts by the plane of every great 
 circle ; and into two unequal parts by the plane of every fmalj circle. 
 
 19. The pole of every great circle is at 90 degrees diftance from it oijr 
 the furface of the fphere; And'no two great circles can have a common 
 pob. 
 
 20. The poles of a great circle are the extremities of that diameter, or 
 axis, of the fphere, which is perpendicular to the plane of thatcircle, 
 
 21. Lines flowing to the projeling point, or place of the eye,^ from 
 every point in the circumference of a circle which it, view.s fwm. tho 
 convex furface of a Cone. 
 
 22. A plane pafling through three points on the furface of a fphere,. 
 equally diftant from the pole of a great circle, will be parallel to the 
 plane of that circle. 
 
 23. The (horteft diftance between two points on the furface of a fphere, 
 js the arc of a great circfe paffiiij; through thofe points. 
 
 24. If one great circle meets another, the angles on either fide are fuppie- 
 nients to one another j and every fpheric angle is lefs than 180 degrees. 
 
 25. A fpheric angle is meafured by an arc of a great circle intercepted 
 between the legs of that angle, 90 degrees diftant from the angular point, 
 
 26. If two circles interfedl one another, the oppofite angles are equal. 
 
 27. Two fpheric triangles are congruous, if two fides and their con- 
 tained angle in one, arc equal to two fides and their contained angle in 
 the other, each to each : Or if two angles and the contained fide in the 
 one, are equ,il to two angles and their contained fide m the other, each 
 to each : Or if the three fides in the one are rcfpe6lively equal to the 
 three fides in the other. 
 
 28. All parallel circles have the fame pole, and maybe conceived to be 
 concentric to the great circle which they are parallel to. 
 
 29. All parallel circles on the fphere, having the fame pole, arc cut into 
 fimilar arcs by two great circles pafling through that pole. 
 
 SECTION
 
 12% SPHERICS. Book IV, 
 
 is E C T I O N 11. 
 Stereographic Propojitiom, 
 
 30. PROPOSITION!. 
 
 Great circles of a fphere mutually cut one another into two equal parts. 
 
 Dem. Any two great circles have the fame common center, (i6) 
 
 And their planes interfeil in a right line. (II. 209} 
 
 Now the center muft lie in the line of their interfedlion. 
 Therefore this right line is a diameter common to both. 
 But ?very circle is bifeled by its (jiameter. 
 Therefore the circles mutually bifecl one another. 
 
 31. CoROL. I. The circumferences of any two circles InterfecSling one 
 smother twice, make the angles at both fedlions equal. 
 
 For the planes of thofe circles have the fame inclination at both ends of 
 their interferon, or wh?re the circumferences interfedt, 
 
 32. CoRoL. II. Two great circles of the fphere will cut each other 
 twice at the diftance of 180 degrees, or in oppofite points of the fphere. 
 
 33, PROP. II. 
 
 The dijlance of the poles of two great circles, is equal to the angle formed 
 by the inclinatisn of thofe circles. Tlate I. 
 
 Dem. Let aeb, ced, be two great C'/cles of the fphere, their planes 
 paffing through its center f ; and let a, b^ be the poles of the circle 
 AEB, and r, d^ the poles of the circle ced. 
 
 Then is the ar^ A(2=arc crzrgo'*. (19) 
 
 And the arc ca is common to both the arcs Aa and cc. 
 Therefore the arc ac^ meafqring the inclination cfa of the circles, is 
 equal to the arc ac meafuring the diftance of the poles, (II. 48.) 
 
 34. CoROL. I. Two great circles are at right angles to one another, 
 when they pafs through each other's poles. 
 
 35. CoROL. 11. The pole of a great circle is 90 degrees diftant from 
 it, taken in another great circle, or in an arc of it, drawn perpendicular 
 tQ the former circle, 
 
 36; CoRoL. III. Two or more great circles, at right angles to an-^ 
 other great circle, interfecSl one another 90 diftant from it, or in the 
 pole of the latter circle. And the like of arcs of great circles, 
 
 37. CoROL. IV". If feveral great circles Interfel: one another in the 
 pole of another great circle ; then art; the former circles perpendicular to 
 ?he jatter, 
 
 8 38. PROp. 
 
 1
 
 Book lY; S P H E R I: C-S. i^ 
 
 38. PRO p: m: 4^ 
 
 />; thejfereograpjy'ic proje5iion of thefphere^ the reprefentationi oftilTctfthSt 
 nit paffwg through the projiSlmg point, will be circles. Plate I.' xhj^i'i'' 
 
 * .-. '^V \-'j . Aw ^>T 1 VnK 
 
 Let ACEDB reprefent a fphere, cut by a plane rs, pa fllngl through th^f^ 
 center i, at right angles to the diameter eh, drawn from e, the place of 
 the eye. . _ . . I 
 
 And let the fe<5Hon of the fphere (17) by the plane rs, be the circkf\ 
 CFDL, its poles being H, and e. ^. 
 
 Suppofe AGB is a circle on the fphere to be projefted, its pole, mo ft 
 remote from the eye, being p : And the vifual rays from the circle aB3 
 meeting in E, form the cone agbe (21 ) of which the triangle aeb, is ar 
 fediion through the vertex e, and diameter of the bafe ab. (II. 204.)- 
 
 Then will the figure agbf, which is the projection of the circle bg \, 
 be a circle. 
 
 Demonstration.. Since the z. e<x^ is n^eafured. by |iJjrc AC 4< (faro 
 EDxz) I arc CE. i oin'^.tr; n j.-':a4^I.' 137^ 
 
 And the /leba is meafured by | arc AC + f acftiCftt-.:: ia:-: (II. 128) 
 Therefore the angle eba rr angle E^i^. . .-. (II. 50} 
 
 And fo the triangles eab, Eba are fimilar, the Z.E being comnjqu. , 
 Therefore ab cuts the fides ea and eb of the cone, in a fubcontrary pbfi- 
 tion to AB ; and confequently the fecStion afbg is a circle. (II. 213) 
 
 Now fuppofe the plane rs to revolve on the line' to, till it coincides 
 with the plane of the circle aceb ; , 
 
 Then it is evident, that the point L Will. fall, in H, the point F in e, 
 and the circle cfdl will coincide with the circle cedh, which now be- 
 comes the primitive circle^ where the point F or e, iS the projecting 
 point : Alfo the projected circle afbg will become the circle atibK. 
 
 39. CoROL. I. Hence the middle of the projected diameter is the cen- 
 ter of the projected circle, whether it be a great. circle or a fmall one. 
 
 40. CoRoL. II. Hence in all circles parallel to the plane of projedti^Ln, 
 their centers and poles will fall in the center of the projection. (> 
 
 41. CoRoL. III. The centers and poles of circles, inclined tp the 
 plane of projection, fall in that diameter of the primitive circle which Is 
 at riglit angles to the diameter drawn through the projecting point j but 
 at different diftances from its center. 
 
 42. CoROL. IV. All oblique great circles cut the primitive circle.in 
 two points diametrically oppofite. 
 
 Vol.. I. K 43. PROP.
 
 i^ SPHERICS. Book IV. 
 
 43. PRO P. IV. 
 
 f}i( mtdfjurt of^ tbg- angle which the preJeSIed diameter of arty ctrclt fuh' 
 ttndi at the eye^ is equal to the diflance of that circle from its pole^ which is^ 
 mo/i remote from the projeSfing pointy taken on the furface of the fphere. 
 And that angle is bifehed by a right line joining the projeSting point and that 
 
 : ' ' > .i" V, /: . 
 
 Let the plane rs cut the fphere hfeg, as in the laft. 
 And let ABC be any oblique great circle, the diameter of which AC is 
 projeif^ed into ac \ and kol any fmall circle parallel to abc, the diameter 
 dr which KL is irojeted into kl. 
 
 ' The diftances of thofc circles from the pole P, being the arcs ahp, 
 KHP, and the angles a'Lc^ kEly are angles at the eye fubtended by their 
 projected diameters acy kl. 
 
 Then is the angle ai^c meafured by the arc ahp, the angle k&l is mea- 
 fured by the arc khp ; and thofe angles are bifefted by ep. 
 
 I>SM* I^of arc p^ilA^arc PC ; and arc PHK=arc PL. ($) 
 
 And'the ,^aec is meafured by \ arc APC=arc pha. (II. 128) 
 
 Alft) the /.KEL is meafured by | arc KPLrrarc phk. (il. 128) 
 
 Tivereifoi^e the angles akc, kel, are refpeftivcly meafured by the arcs" 
 
 ?HA, PHK^ ' 
 
 And; it i^ evident tbofc angles are bifcked by the line ep. 
 
 44. CoROL.'I. Hence as the line ep projels thq pole p in ^ ; fo the 
 fame line refers a projedled pole to its place on the fphere, in the cir 
 ciimferi^ntie of the primitive circle. 
 
 45* C<>tt.ot. II. Hence, on the' plane of the primitive circle, maybe 
 defcribed the reprefentation of any circle whofe diftance from its pole, 
 and the projected place of that pole, are given. 
 
 For PA and pc are projedted into pa and pc ; and the bife6lion of ac 
 gives the center of the circle fought. 
 
 c 46. C0R6L. III. Hfence every proje<Eled oblique great circle cuts the 
 primitive circle in an angle equal to the inclination of the plane of that 
 oblique circle to the plane of proje<lion. 
 
 For ra is equivalent to fa the inclination. 
 And ja meafures the angle fh^, fince fh, h<7, are each 90*, (9) 
 
 47. CoROL. IV. The diftance between the projedions of a great circle 
 and any of its parallels is equivalent to their diftance on the fphere. 
 Thus the projeftion ok is equivalent to ak. 
 
 4S. PROP.
 
 Book. IV. S P H ERICS. f^ 
 
 48. P R O P. V. 
 
 Any point of a /there Jlenographtcally projeSfed, is dijiant from the centtr 
 of projeSlion^ hy the tangent of half the arc intercepted between that point 
 and the pole oppoftte to the eye : The femidiameter of the fphere being made 
 radius. Plate 1. 
 
 Let c^EB be a great circle of the fphere, the center of which is e, gh tho 
 plane of projefiHon cutting the diameter of the fphere in ^, b j e, c, the 
 poles of the fcdion by that plane j and a the proje(Stion of a. 
 Then is ca equal to the tangent of half the arc AC. 
 
 Dem. Draw qf, a tangent to the arc cd=:| arc ca, and join fF. 
 Now the triangles CFf, raE, are congruous : For cczzcEy /L c=:Z.E^4 
 =right Z., /.ccF=cEa (II. 128) : Therefore ca=cF, 
 Confequently ca is equal to the tangent of half the arc ca* 
 
 49. PRO P. VI. 
 
 The angle made by the interfeSlion of the circumferences of two circles in 
 the fame plane, is equal to the angle made by tangents to thofe circles in the 
 point of felion \ and alfo is equal to the angle made by their radii drawn to 
 that point. Plate I. 
 
 Let CE, CD, be two arcs of circles in the fame plane cutting in the 
 point c } AC, BC, their radii j gc, fc, tangents at the point c. 
 Then is the curve-lined angle ecd=i Z.GCF Z. acb. 
 
 Dem. Since the radii ac, bc, are at right angles to their tangents GC, fc 
 
 (II. 126) ; and are alfo at right angles to the arcs cE, CD. (II. 136) 
 
 Therefore the pofition of the tangents and arcs at the point c are the 
 
 fame ; and confequently the z.ECDr= Z.GCF. 
 
 Alfo the /1ACB+ Z.BCG=:(right/L = )Z.FCG+ Z.BCG. 
 
 Therefore the angle acb is equal to the angle f CG, by taking away the 
 
 common angle bcg. (II. 48) 
 
 Confequently /.ecd=:gcf=:Z.acb. 
 
 50. Scholium. If the arcs ce, cd, were in different planes, the 
 fame would hold true with regard to their tangents. 
 
 For fuppofc the circle CD to revolve on the fixed radius BC, ftill cutting 
 the circle ce in c : Then the tangent CF revolving wuh ir, has (till the 
 fame inclination to bc : And as the inclination of the planes of the circles 
 vary, fo much will the inclinaiion of the tangents vary. 
 
 Therefore the angle made by the tangents, in all poflcion's of the cir- 
 cular planes, is the fame as the ani^le made by their circumferences. 
 
 51. CoROL. Hence if a plane touches a fphere, at the point where two 
 circles of it interfecl one another, the tangents to both circles will lie ini 
 that plane ; and confequently, in all oblique pofitions, a ri j,ht line per- 
 pendicular to one tangent will cut the other tangent, 
 
 K 2 5a. PROP.
 
 I5A S P ,H ERICS, Book IV^ 
 
 52, PRO P. Vll/ ; , 
 
 The angle which any two circles make^ whehjlereagraphlcally proJelctLjr 
 equal to the angle which thofe circki make on the fphere. Plate I. !, ."^^ 
 
 Suppofe DAEL a fphere to be projedled on the plane sbrcf, and alde 
 a great circle paffing through the projeting point e. Let lba be any 
 other circle, cutting the former in l and a, under the angle bae, which 
 wiirbe fepYefented by the circle sbr, (38) as the circle elda is by the 
 right line sc (15). The angle bae is equal to the angle brc. 
 
 From the point a draw ac, af, to touch the circles ael, abs in a, and" 
 meet the plane of proiedHon in c ami f ; aWb draw rk and cf, which will* 
 be in that plane ; and, in the plane of the great circle ael, draw ad pa- 
 rallel to 9, and join ed. , 
 
 I>EM. The angular pornt a is proje(led into r ; {iz) confequently ac 
 is projeled into RC, and af into rf. And ftnce sc is the eommoft 
 fc6lion of the plane of proje6tion with that of the great cireJe eld*a 
 (II. 210) the lines ac, sc, ad, ed, ae, lie all in the plane of that circle; 
 Alfo becaufe ad is parallel to 3C, the Z. arc = Z.dae = Z. ADE=r Z.RAC 
 (II. 94, 104, 132) confequently AcrzRc (II. 104). Now the plane 
 pailTng through ac and af touches the fphere in a, (51) it is therefore 
 perpendicular to the plane of the circle aed i and fc its common fecflion 
 with the plane of projeftion, is at right angles to that plane (II. 210) j 
 FC is therefore at right angles both to the lines ac and cR (II. 205) : 
 Hence, the triangles acf, rcf, being right angled at c, having the fide 
 FC common, arid AcrrcR, are coijgruous (II. 99), and the /1caf:=: 
 Z-CRF. Confequently the z.EAB= Z.FAC (51) =; Z.FS.C. Now it is ma- 
 flifeft that as af touches the bafe, abl, of the cone eabl, in the point a, 
 a. plane pafling through af and ae will touch the fide of the cone in the- 
 line AEj but af is alfo in that plane (II. 198} ; therefore ar touches, 
 the cone in the line ae ; and as ar lies alfo in the plane of the circle 
 SBR, it muft touch that circle alfo j confequently (50) brc=:Z.frc=: 
 Z.fac=:Z.abe. 
 
 53. PRO P. VIII. 
 
 The dljlance between the poles of the primitive circle and an oblique great 
 circle^ injlcreog'ophic projeSiions^ is equal to the tangent of half the inclina- 
 tion of thofe circles ; and the dijlance of their centers is equal to the tangent 
 af their inclination : The femidiameter of the primitive circle being made 
 radius, Plate I. 
 
 Let AC be the diameter of a circle, the poles of which are p and c>^, 
 and inclined to the plane of projection in the angle aif. 
 And let a, r, />, be the projections of the points a, c, p. 
 Alfo let uai. be the projeCted oblique circle, the center of which is q. 
 Now when the plane of jTrojection becomes the primitive circle, the 
 pole of which is i. 
 
 Then is i^=:tangent of half Z.aif, or of half the arc af. 
 And i^=:tangent of AF, or of the /.FH^rr aif. 
 Hem. For ah4-hp = ah + af. Therefore hp = af. 
 But i^= tangent of half hp, or of half af. (48) 
 
 Again,
 
 IBook IV. S P H E R 1 C S. iJS 
 
 Again. As AC is projedcd in ac^ then q, the middle of ac. Is the center 
 of the projected circle, and of its rtprefentative h^e. (45) 
 
 Draw Ej' produced to r : Then as qrigE j the AqEA^/CqaE. (II. 104) 
 But the Zjf^/E is meafured by half the arc Fa. (II. 137) 
 
 Therefore the arc AHrrzarc afe (fl. 50) : And as the arc ahp^fqe ; 
 Therefore prrrAFrrHP ; and HPr=:twicc the arc af. 
 Therefore (II, 127) the 2Lie^ = aif, the inclination of the circles. 
 But iq is the tangent of the Z.iE^, ei being the radius, 
 
 54. CoRoL. Hence the radius of an oblique circle is equal to the 
 iecant of the obliquity of that circle to the primitive. 
 For E^ is the fecant of the angle ie^, to the radius ei. 
 
 55. P R O R IX. 
 
 Jf through any given point In the primitive circle an oblique circle he defer il- 
 ed; then the centers of all other oblique circles paffing through that pointy will 
 be in a right line drawn through the center of the firjl oblique circle at right 
 angles to a line paffmg through the given pointy and the center of the primitive. 
 
 Let gace be the primitive drcle, adei a great circle defcribed through 
 D, its center being b. 
 
 HK is a right line drawn through b, perpendicular to a right line ci 
 parting through d, and the center of the primitive circle. 
 
 Then the centers of all other great circles fdg palling through d will 
 fall in the line hic. 
 
 Dem. For'lf E be the projcding point, the circle edai will be the 
 proje(5tion of a circle, the diameter of which is nm. (3^) 
 
 Therefore D and i are the projedtions of n, m, which are oppofite 
 points on the fphere ; or of points at a femicirclc's diftance. 
 
 Therefore all circles palling through d and i muff be the projections of 
 great circles on the fphere. 
 
 But Di is a chord in every circle parting through the points d, r. 
 
 Confequcntly the centers of all thofe circles will be lound in hk drawn 
 perpendicularly through n, the middle of di. (II. 125^ 
 
 56. P R O P. X. 
 
 Equal arcs of any tivo great circles of the fphere, will he intercepted he' 
 tween two other circles drawn on the fphere through the remoteji poles of thofe 
 great circles. Plate I. 
 
 LetPBEA be a fphere, on. which agb, cfd, are two great circles, the 
 remoteft poles of wliich are f, p ; and through ihcfe poles let the great 
 circle pbec, and fmall circle pge, be drawn, interfedting the great 
 circles agb, cfd, in the points b, g, and d, f. 
 'I hcii are the intercepted arcs bg and df equal to one another. 
 
 X)em. For the arcs ED4-DB=:arcs pb + db ; therefore ed = pb. 
 
 And the arcs ef + FG=:arcs pg + FG f iq) ; therefore EFirpG. 
 
 For the points f and g are equally diftant from their poles p, K. 
 
 Alfo the Z.DFr=npG j for interfedling circles make equal angles at the 
 
 fe<5^ions. (ji) 
 
 Therefore the triangles efd and pgb are congruous. ' (27) 
 
 Therefore the arc BG=arc dp. 
 
 K3 57. PROP.
 
 Hl^f SPHERICS. Book IV. 
 
 :|7. PRO P. XI. 
 
 If lints b ^ratvn from the projeHed pole of any great circle, cutting the 
 peripheries of the projeSied circle and plane of proje^ioUi the intercepted arcs 
 cfthofe circumferences are equal. Plate I. 
 
 On the plane of projelion, agb, let the great circle cfd be pro- 
 jedled into cfd^ and its pole p in ^ ; moreover, draw the lines pd, pf: 
 the arcs GB and yi/ ate equal. ^; 
 
 Since pd lies both in the plane AC B and apbe it is their common fac- 
 tion. (II. 198) 
 But the point b is in their common fclion : (56) 
 Therefore pd pafles through the point b. 
 
 And in this manner it may be proved that ^/pafles through c. 
 Now the points d and f are projetSted into d andy. (3^) 
 
 Therefore the Tixcfd is equivalent to the arc FD. 
 
 But the arc fd is equal to the arc gb : (56) 
 
 1'herefore the arc gb is equivalent to the arc fd, (II, 46) 
 
 58. PRO P. XII. 
 
 Tl)e radius of any fmall circle , the plane of which is perpendicular to that 
 tf the primitive circle^ is equal to the tangent of that leJJ'er circle's dijlance 
 from its pole I and the fecant of that dijiance, is equal to the dijlance of the 
 tenters of the primitive and leer circle. Plate I. 
 
 Let P be the pole, and ab the diameter of a lefler circle, the plane 
 being perpendicular to the plane of the primitive circle, the center of 
 which is c : Then d being the center of the projected lefler circle, dA. is 
 equal to the tangent of the arc pa, and ^c= fecant of pa. 
 
 Dem. Draw the diameter ed parallel to ab, and through p draw cb. Now 
 
 E being the projedling point, the diameter ab is proje(fied in ak, (22) 
 
 And d^ the middle of ab, is the center of a circle on ab. (39) 
 
 Then a right line drawn from d through a, will meet b : (II. 130) 
 
 And draw ca, dA. 
 
 Now the right-angled triangles dc^, dae, having the angle d common ; 
 
 the z.d3c=Z.dea. (II. 98) 
 
 But z.DEA=iz,DCA; and z.D^c = f Z.A</c : (II. 127) 
 
 Therefore z.dca= A A^c. 
 
 Now Z.DCA+ l.Acdzzz right angle. 
 
 Then jLAdc-{- jLaqg z right angle: Therefore Z.C a ^/ is right. (II. 96) 
 
 Confequently rfA,= radius of the circle a^b, is the tangent of the arc pa, 
 
 to the radius ca. (II. 126) 
 
 And^c, the diftance of the centers, is the fecant of the arc ap. (III. 5) 
 
 59. CoROL. Hence the tangent and fecant of any arc of the primitive 
 circle, belongs alfo to an equal arc of any oblique circle ; thofe arcs be- 
 ing reckoned from their interfe6lion. 
 
 For the arc ?c of every oblique circle intercepted between p and the 
 arc of the fmall circle a^b, is equivalent to the arc pa of the primitive 
 circle: Becaufe the arc aab is equally diftant from its pole p. (5) 
 
 SECTION
 
 Book IV. 
 
 S P H E R I C . 
 
 ygi 
 
 S E C T I O tSt it. 
 Spherical Geometry. 
 
 Spheric Geometry ^ or fpheric projection ^ is the art of dejcribingy 
 or reprejentingj Juch circles or arcs of circles as are u/ually drawn 
 upon ajphere on the plane of any one of them j and of meafuring 
 fuch arcs, and their pofttions to one another, when pro]e6led. 
 
 60. 
 
 PROBLEM I. 
 
 7o defcrtbe a great circle that Jhall pafi through two given points in the 
 primitive circle, or plane ef projeSiion, 
 
 Let the given points be a, bj and c the center of the prim, circle. 
 
 Case i. When one point, a, is the center $ftht 
 primitive circle. 
 
 Const. A diameter drawn through the given 
 points A, B, will be the great circle required. ( 15) 
 
 Case 2. When one point a, is in the circum^ 
 ference of the primitive circle. 
 
 Const. Through a draw a diameter ad. 
 
 Then an oblique circle defcribed through the 
 three points a, b, d, (IL 72) will be the great circle 
 required. (42) 
 
 61. Case 3. When neither point is at the center, 
 or circumference of the primitive circle* 
 
 Const. Through one point a, and the center 
 C, draw AG, and draw CE at right angles to AG. 
 
 A ruler by e and a gives d ; by d and C gives 
 F i and by e and f gives g, in AC continued. 
 
 Through the three points g, b, a, defcribe a circumference (IL 72) 
 cutting the primitive circle in H and i. 
 
 Then the oblique circle hbai will be the great circle required. 
 
 For AG maybe taken as the projccStion of the great circle fd. (12) 
 Therefore a and g arc the projections of oppofite points on the fphere. ( 32) 
 
 Confequently, all circles pafT.ng through c and a will be the repre- 
 fentativcs of great circles on the fphere. 
 
 K 
 
 62. P R O B-
 
 15^ 
 
 S P H E R I e S. 
 
 Book IV. 
 
 6t. 
 
 .T P:^9)Bi^f M ^, .' 
 
 jfbeut any ^heft point as a pole ^ Jo defer ibe a gre^ circle in a given pri^ 
 mitfve circle. ./\\^'X\' 
 
 .> 
 
 Let V be the. given point, and | the center ^of the primitive xJrcJe, 
 
 Case i, When the given poky P, is in th( fe?ii(r 
 tf'the primitive circle. 
 
 Const. The primitive pircle yi\\\ be the great 
 circle required, (13) 
 
 -^Gase 2. ff^en the givjn pole, p, is in th? cirfumV .'^* ^ 
 ferencf of the primitive circle. -^ .^ ^ > ^ ^\ < ku.\<\ x- 
 
 CoSST. Through the given pole p, draw PE a 
 difimeter to the primitive circle. a 
 
 Then another diam. Ab, drawn at right angles 
 to p^j will be the great circle required. (20, 15) 
 
 63. CASBr^..ff^jen the given pole, p, is neither in 
 ihf (ent^r or cirfumferenc^ of the primitive circle. 
 
 Const. Through p draw a diameter bd, and another be at right angles 
 to bd-^ then a ruler by e apd p gives p. 
 
 Make the arc pA^=:go; a ruler by e and a gives 
 # in the diameter bd. 
 
 Then a circle defcribed thrpugh the three points 
 Bj ('i ^3 is the great cjrcle required. 
 
 Or thus. Make the ^rc ^d arc p^ ; a ruler on 
 J and T) gives c in db produced. 
 
 Then on c, with the radius ca defcribe b^e. 
 
 J^'or, As E is the projecting ooint, and p the projecled pole ; 
 Therefore p is the pole of the circle af to be projedcd, (44) 
 
 And B6E is the projedion of the circle af. (38) 
 
 Now Z-CflE is meafured by half the arc a^e. (I I. 137) 
 
 But arc ABD=arc a^e: For Ap=:(^Bdzz)dE ; and ^d a^ by con- 
 ftrudion, 
 
 Therefore /L aec= Z.c<7E ; and cE=:ca. (II. 104) 
 
 Cenfequentiy c is the center required. 
 
 ^4- PROB*
 
 Book IV. 
 64. 
 
 S P H E R i C 
 
 # 
 
 37 
 
 PROBLEM III. 
 
 A projeSied circle -being given j to find its poles. 
 
 Case i. TVhen the given circle aeb is the pri- 
 mitive. 
 
 Const. Find the center c, (II. 70) and it Is 
 jhe pole fought. 
 
 Case 2. When the given circle acb is a right 
 circle. 
 
 Const. Draw a diameter ed at right angles 
 to ab, and the ends or points, D, E, of that dia- 
 meter are the poles required. 
 
 65. Case 3. JVljen the given circle abe is oblique. 
 
 Const. Through the interfetions of the pri 
 mitive and oblique circles draw a diameter ae, 
 and another at right angles to ae, cutting the 
 given oblique circle in b. 
 
 A ruler by e and b gives b ; make hp., bq, each 
 zz:&n arc of 90. 
 
 A ruler by and p gives, in the diameter through b, the point p, which 
 is the pole required. 
 
 And a ruler by e and q gives, in cb continued, the point Q^for the 
 other, or oppofite or exterior pole. 
 
 Make pD-=:pA ; then a ruler by e and p gives, in bc continued, the 
 point F, which is the center of the oblique circle abe. 
 
 The rcafon of this operation is evident from that of the laft Problem. 
 
 66. 
 
 PROBLEM IV. 
 
 About any given projccied pole, to defcribe a circle at a given dijlance from 
 that pole. 
 
 Or^ at a propofad dkfiance from a given great circle, to defcribe a parallel 
 circle. 
 
 Let p be the given pole, belonging to the given great circle dfe. 
 
 General Solution, l^hrough the given pole p, and c the center 
 of the primitive circle, draw a diameter, and another de at right angles 
 to it, 
 
 A ruler on e and p gives p in tlic primitive circle. 
 Make p\ and /)B, each equal to the propofed diftancc from the pole. 
 A ruler on e and a, and then on e and B, will cut the diameter cp in 
 /ind b. 
 
 Bifedt ab in c ; and on c as a center defcribe a circle pafTing through ct 
 and b^ which will bc the circle required. 
 
 But
 
 *s% 
 
 ^ P^ H E R I C S. 
 
 Book IV 
 
 But when the parallel circle is to be at a propofed 
 diftance from the given great circle dfe. 
 
 Find p as -before ; and make ^a=:j>b, equal to 
 the complement of the propofed diflance j the reft 
 as before. 
 
 For p is the pole, the projcdlion of which is p. 
 
 \ (44) 
 
 But^ IS the poleof a circle, the diameter of which 
 AB is projefled in ab. (*^) 
 
 Therefore r, the middle of ab, is the center of 
 the proje<^d circle. ( 39) 
 
 Sy.Thefirft cafe is readily done,bydefcribing the 
 ftnall circle about the center of the primitive circle 
 with the tangent of half its diftance from the pole f., 
 
 68. The fecond cafe is fooneft performed thus. 
 
 From the points a, b, (found as above) with the 
 tangent of their diftance from p, the pole of the right 
 circle, defcribe arcs cutting in c, which is the center 
 of a fmall circle parallel to the right circle dfe. 
 
 For Ap is the tangent of the arc a p. 
 
 69. P R O B L E M V. 
 
 TTje primittve circle., and the proje6iton of a fmall circky being given j U 
 fnd the pole of that fmall circle. 
 
 (58) 
 
 Let c be the center of the primitive circle, and abd a projefted fmall 
 drcle, the center of which is r, and radius cb. 
 
 General Solution. Through c the center of the fmall circle, and 
 c the center of the primitive, draw a diameter CF, and another, ce, at 
 right angles to it. 
 
 Find the proje<Sed diameter b3=2^b. 
 
 Lines drawn from e through b and ^, cut the primitive circle fn tf, d\ 
 then bife<a the arc ad in p. 
 
 A ruler by e and p cuts the diameter b^ in p, the pole fought. 
 
 The truth of this conftrudtion is evident by that of the laft Prob. 
 
 70. PRO B-
 
 3ook IV. 
 
 SPHERICS. 
 
 HIJ9 
 
 70. P R O B L E M VI. 
 
 To meafttre any are of a projected great circle: Or, in a given projeIed 
 great circle, to take an arc of a given number of degrees^ 
 
 General Solution. Find the pole of the given circle. (64) 
 
 From that pole draw lines through the ends of the propofed arc, cut- 
 ting the primitive circle. 
 
 Then the intercepted arc of the primitive circle applied to the fcale of 
 chords will give the meafure fought. 
 
 Thus, if ab be the arc to be meafured, and p the pole of the given 
 circle daf. 
 
 Then lines drawn from p through a and b, give the arc ah in the pri- 
 mitive circle, correfponding to ab in the projebsd circle. 
 
 Now if an arc of a given number of degrees was 
 to be taken from a given point a, in the given pro- 
 je^ed circle daf. 
 
 Draw, from the pole p, through a, the line ptf 
 to the primitive circle. 
 
 Apply the given number of degrees from a to b. 
 
 Draw P^, and the intercepted arc ab will con- 
 tain the degrees propofed. 
 
 71, Any number of degrees is readily applied to 
 a right circle by the fcale of half-tangents. Thus 
 
 When the diftance of the point a from the cen- 
 ter c is known, and the given quantity of the arc 
 is to be laid from a towards f ; 
 
 To the known diftance ca add the propofed are 
 ab, the degrees in the fum taken from the fcale of 
 half-tangents, and laid from c to b, will make the 
 arc AB equal to the degrees propofed. 
 
 But when the arc ab is to be laid from a to- 
 wards D ; 
 
 Then the difference between the arcs ab and 
 AC, taken from the fcale of half-tangents and laid 
 towards D from c to b, will make the arc ab equal 
 to the degrees propofed. 
 
 The reafon of all thefe operations is evident P 
 
 from art. 57. 
 
 Note, V he half, or femi-tangents, are only the tangents of half the 
 arcs the fcale of tangents is made to; their conftrudion depends on art. 48. 
 On the plane fcale they are put under the Tangents, and marked s. T. 
 
 72. PROBLEM VII. 
 
 To meafure any projefed fpherical angle. 
 
 General Solution. Find the poles of the two circles which form 
 the angle (64) ; and from the angular point draw lines through thofe 
 poles to cut the primitive circle. 
 
 Then the meafure of that angle, if acute, will be the intercepted arc 
 of the primitive circle ; or the fupplement of that arc will be the meafuri 
 of the angle when obtufe. 
 
 Let the propofed angle dab, formed by the great circles ad, ab, the 
 poles of which are c and p j and lines drawn from the angular point a, 
 
 through the poles c and p, cut the primitive circle in E arid p. 
 
 xft.
 
 S P H E R^I G S. 
 
 ift. 'Syhen the angle informed by the primitive and oblique circles. 
 
 Then the arc^E meafures the acute angle dab. 
 
 But the obtufe angle ^af is meafured by the fupplemcntof j>E.- 
 
 2d. When the angle is forrned by right and oblique circles meeting in 
 the primitive's circumference. ' 
 
 Then the arc pE meafures the angle dAb. . 
 
 3d. When the angle is formed by right and oblique circles meeting 
 within the primitive circle. ' -i :i- Tic'} .v-'*. i'l 
 
 Then the arc pE meafures the acute angle dab. = ' ' ' " 1 
 
 But the obtufe angle daf is meafured by the fupplement of pE. 
 
 4th. When the angle is formed by two oblique circles meeting within 
 the primitive circle; 
 
 Then the acute angle dab is meafured by the arc pE, 
 
 But the fupplement of^E meafures the obtufe angle daf. 
 
 For, as the angular point a is in both circles, and 90 diftant from 
 their poles c and p (19). Therefore a great circle dcfcribed about a, 
 as a pole, will pafs through the poles c and p. 
 
 And lines drawn from a through c and p, cut off, in the circumference 
 of the plane of projeilion, an arc equal to the diftance of the poles c 
 ar.d p. (57) 
 
 But the meafure of the diftance of the poles c, p, is equal to the in- 
 clination of the planes of the circles ad, ab ; (33,) 
 
 And confequently meafures the angle dab. 
 
 73. PROBLEM VIII. 
 
 Through a given point in any projeSfed great circle^ to dffcribe another 
 great circle at right angles to the given one. 
 
 General Solution. Find the pole of the 
 given circle. (64) 
 
 Then a great circle defcribed through that pole 
 and the given point will be at right angles to the 
 given circle. 
 
 Let the given projected great circle be bad j 
 and A the given point. 
 
 I ft. When BAD ii the primitive circle , the pole 
 eftuhich is p. 
 
 A diameter through a will be perpendicular to 
 BAD. (II. 136) 
 
 2d. ff^en BAD is a right circle^ the poles of 
 which are p and c. 
 
 An oblique circle defcribed through the points 
 C, A> Pj (II. 72} will be at right angles to bad. 
 
 JO 3d. JVhen.
 
 Book IV. 
 
 SPHERICS; 
 
 141 
 
 3d. Jf^hen BAD is an oblique circle^ the pole of 
 which is P. ,\-" 
 
 Through the points p and a, a great circle 
 PAG being defer ibed (61), will be at right angles 
 
 to BAD. 
 
 The truth of thefe operations is evident from 
 art. 34. 
 
 74. 
 
 PROBLEM IX. 
 
 trough any ajjigned point in a given proje^ed great circle^ to defcrihe an- 
 Bther great circle cutting the former in an angle of a given number of de- 
 grees. 
 
 Let p be a given point in any great circle apb. 
 
 ift. When APB is the primitive circle.^ 
 
 Through the given point p draw^ a diameter 
 P, and draw the diameter ab at right angles to 
 P. 
 
 Draw PD cutting ab in d, fo that the angle 
 CPD be equal to the angle propofed. 
 
 On D with the radius dp defcribe the great 
 circle pfe. 
 
 Then will the angle apf contain the given de- 
 grees. 
 
 For the Z-FPAnangle made by the radii pc, pd. 
 
 And D being equally diftant from p and e, is the center fought. 
 
 (49) 
 
 75. Or thus. Make cd equal to the tangent of the given angle to 
 the radius CP. 
 
 Or, Make pd equal to the fecant of that angle. 
 
 76, 2d. JVhen APB is a right circle. 
 
 Draw a diameter GH at right angles to apb. 
 
 Then a ruler by g and p gives a in the primi- 
 tive circle. 
 
 Make \ibz:.i.\a\ a ruler by c and h gives c in 
 ab. 
 
 Draw CD at right angles to ab. 
 
 Draw PD cutting cd in d, fo chat the Z.cpd = 
 complement of the degrees given. (IL 84) 
 
 On d with the radius dp defcribe a circle fpe, which will be a great 
 circle making with apb the angle apf as required. 
 
 For, c is the center of a great circle gph, by the demonftration to 
 
 art. 63. 
 
 And the centers of all great circles through p, will be in cd. (55) 
 Now z.DPE= 90, (II. 136) 
 
 Thercroic <i. apf or z.bp (26}, the compl. of cpd, is the angle fought. 
 
 77- 3^'
 
 I4t' 
 
 -i P H E R I C S. 
 
 Book IV, 
 
 A 
 
 A'"'. 
 
 ^<-9 
 
 ,.-1 ! 
 
 ..d//g 
 
 77. 3cl. H^iit APB Is an oblique circlt* 
 
 From the given point p, draw the lines PG, PC, 
 through the centers of the primitive and given ob- 
 lique circles, and through c the center of Apsdraw 
 CD at right angles toPG. (II. 59) 
 
 - Draw PD, making the Z.cPd= given degrees, 
 and cutting cd in d. (II. 84) 
 
 From D with the radius dp, a circle fpe being "" 
 
 defcribed, will be a great ^rcle cutting apb in the angle propofed. 
 
 For c, the center of apd, is in a line perpendicular to pg, drawn 
 throoigh p and the center of the primitive, by conftrudtion. 
 
 And the centers of all great circles through p will be in cd. (55) 
 Now the .cpd made by the radii PC, pd, contains the given degrees. 
 Therefore the angle apf is equal to the angle required. (49) 
 
 78. 
 
 PROBLEM X. 
 
 Through any point in the plane of projefion^ or primitive circle^ to de^ 
 feribe a great circle that Jhall cut a given great circle in any angle propofed : 
 Provided the meefure of that propofed angle is not lefs than the dijiancs be- 
 tween the given point and circle. 
 
 Let the given point be a, through which a circle 
 is to be defcribed to cut a given great circle bdc, 
 the pole of which is P, in an angle equal to a pro- 
 pofed number of degrees. 
 
 General Solution. About the given point a, 
 as a pole, defcribe a great circle gf. (62) 
 
 About p, the pole of the given circle bdc, de- 
 fcribe a fmall circle at a diftance equal to the given 
 angle, cutting the great circle egf in G. (66) '. 't ' 
 
 About the point g, as a pole (62), defcribe a great citol* cutting the 
 given circle bdc in d. Then will aDC be the angle re<|trtr^.: 
 
 Note^ When the given angle is equal to the diftance beif^een the giveiV 
 point and circle, the problem is limited to one anfwer only: When the 
 meaflire of the angle is greater, the problem has two folutions by the circle 
 defcribed cutting the given one in two points : But when the meafure of 
 the angle is lefs, the problem is impoffible. This conftru<Si6a is thus 
 proved. 
 
 p aad G arc the poles of bc and ad. 
 
 And
 
 Book IV. 
 
 S^P H E R I C S, 
 
 HJ 
 
 ' And the diftance of p and g iV equal to the degrees in the prepofed 
 angle, by conftru<ftion. But z. ADcrrdiftance of p and a, * ^^'^' ^^2) 
 Therefore the Z.ADC is the angle required. ' ' 
 
 79. ]^en the required circle is te make a given angle^ith the primitive. 
 
 Then, from the center of the primitive, with the tajig. of the given 
 angle, defcribean arc; and from the given point a, with the fecant of 
 the given angle, cut the former arc. 
 
 On this in^rfedion, a circle being defcribed through the given point 
 A, will cut the primitive circle in the angle propofed. 
 
 This depends on art. 75. --^ 
 
 80. P R O B L E M XI. 
 
 jiny great circle, cutting the primitive, being given, to defcribe another 
 great circle, which Jball cut the given one in a propofed angle, and ham a' 
 given arc intercepted between the primitive and given circles. 
 
 Let ABC be the primitive circle, the center of which is pj and the* 
 given great circle be ad,c, the center of which ts e. 
 
 Solution. Draw a diameter ebi> at right an- 
 gles to ADC ; and make the angle bdf equal to 
 the complement of the given angle; fuppofe = 
 complement of 35, 
 
 Make df equal to the tangent of the given arc 
 (fuppofe 58**) ; and from Pj with'the fecant of that 
 arc, defcribe an arc cg 
 
 Now when adc is an oblique circle ; from e 
 the center of adc, with the radius ef, cut the 
 arc c^ in g. 
 
 But when ajjc is a right circle ; through f draw 
 FG parallel to adc, cutting the arc eg in g. 
 
 From G, with the tangent df, defcribe an arc, 
 m, cutting adc in i ; and draw ci. 
 
 Through g and the center p draw GK, cutting 
 the primitive circle in H, K; draw PL perpendi- 
 cular to GK ; and LL atright angles to ig, cutting PL in l. 
 
 And l will be the center of a circle pafling through H, l> 
 will be the great circle required. 
 
 Then the Z. aih=:35'^ ; and arc IH = 58*', as was proposed. 
 For GP is the fecant, and Gi is th^ tang, of the arc HI. 
 
 And as the triangles egi, efd, are congruous; the z.eic=: z.edf. 
 
 (II. lOl) 
 
 But the Z.EIG made by the tangent of the arc hi and the radius of 
 the arc ai, is the complement of the angle made by thofe arcs. (49) 
 
 Confcquently the ^laih is the comp ement of the ^^.edf. 
 
 The center of the right circle AC being fuppofed at an infinijfe.din 
 fiance, tiierefore any circle fg ds^fciibed from that center, will be pa- 
 rallel to AC. 
 
 (59) 
 
 When
 
 144 
 
 SPHERICS. 
 
 Book IV. 
 
 When the given arc is more than 90% the tangent and fecant of its 
 fupplement is to be applied on the line df the contrary way, or towards 
 the right i the former conftrudtion being reckoned to the left. ., ^ 
 
 tu 
 
 PROBLEM Xn. 
 
 
 Jny great circle in the plane ofprojelion ietng given j to defcribe another . 
 great circlej which Jhall make given angles with the primitive and given 
 circles. 
 
 ?n c J r.rf* Let the given great circle be Abe, and its pole q. 
 
 J . . ' ihj ',1. ''m^'- ' r'f' 
 
 Solution. About p, the pole of the primitiYeru ni> eLfi::^ hiO^ 
 circle, defcribe an arc mn^ a-t the diftance of as 
 many degrees as are in the angle which vthe re-; 
 quired circle is to make with the primitive : Sup- 
 pofe62. (67) 
 
 About Q^, the pole of the other given circle, 
 and at a diftance equal to the meafure of the angle 
 which the required circle is to make with the given 
 circle adc (fuppofe 48), defcribe an arc on, cutr-, 
 ting mn in n, (66) 
 
 About , as a pole, defcribe the great circle 
 EDF, cutting the given circles in e ando., (62) 
 
 Then is the angle aed=: 62 j and ade=48''. 
 
 For the diftance of the poles of any two great 
 circles, is equal to the angle which thofe circles 
 make with one another. (33) 
 
 Remark. The nth Problem, which is particularly. ufeful in con-' 
 ftru6ling a fpherical triangle, in which are given two angles and a fide 
 cppofite to one of them, includes only two cafes of a more general Prob- 
 lem, viz. 
 
 Any two great, circles being given in pofitlon; to defcribe a third, which 
 Jhall cut one of thofe given in an angle propofed, and have a given arc inter ^ 
 cepted between the given circles. 
 
 Alfo the 1 2th Problem, ufed when the three angles are given, con- 
 tains only two cafes of another Problem ; viz. 
 
 Any two great circles being given in pofition, to defcribe a third which 
 Jhall cut the given circles in given angles. 
 
 The folution of thefe two general Problems not being wanted in any 
 part of this work ; it was not thought neceflary here to annex them ; 
 more having been already delivered in the preceding pages than it is ufual 
 to meet with on this fubje6l. However, their folution is recommended 
 ^s cxercifes to fpeculative learners. 
 
 SECTION
 
 Book IV. , SPHERICS. .MS 
 
 SECTION IV. 
 
 Spheric 7'rigonometry, 
 
 Definitions, 
 
 82. Spheric Trigonometry is the art of computing the meafures 
 of the fides and angles of fuch triangles as are formed on the furface of 
 a fphere, by the mutual interfecSions of three great circles defcribed 
 thereon. 
 
 83. A Spheric Triangle confifts of three fides and three angles. 
 
 The meafures of unknown fides or angles of fpheric triangles are efti- 
 mated by the relations between the fines, or the tangents, or the fecants, 
 of the fides or angles known, and of thofe that are unknown. 
 
 84. A Right Angled Spheric Triangle has one right angle : 
 The fides about the right angle are called Legs j and the fide oppofite to 
 the right angle is called the Hypothenufe, 
 
 85., A QuADRANTAL SpHERic Tri ANGLE has One fide equal 
 to ninety degrees, 
 
 86. An Oblique Spheric Triangle has all its angles oblique. 
 
 87. The Circular Parts of a triangle, are the arcs whicli meafure 
 its fides and angles. 
 
 88. Two fpheric triangles aie faid to be fupplcments to one another, 
 when the fides and angles of the one are rcfpcctive fupplements of the 
 angles and fides of the other : And one, in regard to the other, is called 
 the fupplemental triangle. 
 
 89. Two arcs or angles, when compared together, are faid to be alike, 
 or of the fame kind, wlien both are acute, or lefs than 90, or when both 
 are obtufc, or greater than 90 : But when one is greater and the other 
 lefs than 90, they are faid to be unlike. 
 
 The lefier circles of the fphere do not enter into TrigOiiometrical com- 
 putations, bccaufe of the diverfity of their radii. 
 
 Vol. T. 1 SECTION
 
 14^ 
 
 SPHERICS. 
 
 Book IV. 
 
 S E C T I O N V. 
 Spherical 'theorems. 
 
 90. .THEOREM!. 
 
 In every fphcnc tiiangky abc, equal angles^ b, c, are oppofite to equal 
 fides^ AC, AS : And equal ftdes, AC, Ac, are oppofite to equal angles, c, B. 
 
 Dem. Since ab = ac, makeAE=:ADj and draw 
 
 BD, CE. 
 
 ThenisBD = CE; and Z. AF.crr z. adb. 
 For the triangles aec, adb arc congruous, 
 Since ADrrAC ; ADriAE ; Z. A common. 
 Alfo, the triangles bec, cdb are congruous ; 
 Therefore Aebcz:z.dcb. 
 
 P'orEcrrBD; eb=:(ab AEr= ) dc ( rr AC ad^. ^-^. 
 
 And /.BECrr Z.cdb, they being the fuppl. of equal angles AEC, adb. 
 Again, if Z. ABcr: z.acb : llien is ab = ac. 
 Tor take be=zcd ; and defcribe the arcs ce, bd. 
 
 Then is EcriDB, Z.bec=: Z-CDB ; Z.ECE=r Z. CBD. (27) 
 
 For^* BCE = AcBD ; fmce BC is common, BE z:::CD, Z.EBC=: Z-DCB. 
 Alfo the triangles abd, ace are congruous ; 
 
 Since eczzdb, Z. ACEr:( Z.eca BCE=:) Z.ABD (rrz-CBA cbd). 
 
 (II. 48) 
 And Z.AEC (flip. Z.BEC=r) Z.ADB (^rfup. Z-Cdb). (II. 48} 
 
 Therefore ae:::::ad; and ab:i= (AE + Enrr) Ac (=:ad + DC). (II. 47) 
 
 91. CoROL. A line drawn from t!ie vertex of an ifofcclcs r^iheric tri- 
 angle, to the middle of the bafc, is perpendicular to the bafe. 
 
 (II. 48) 
 
 Tliis is ealily proved from art. 90, 27. 
 )2. T H E O R E M 
 
 II. 
 
 Either fide of a fphcrlc triangle is kfs than the fu?n of the ether two fides, 
 
 Dem. For on the furface of tiie fphere, the fhorteft diftance between two 
 points, is an arc of a great circle paffmg tlirough thofe point?. (23} 
 
 But each ficic of a fpheric triangle is an arc of a great circle. ( 10) 
 
 Therefore either fide being the fhorteft diftance between its extremi- 
 ties, is lefs than the fum of die other two fides. 
 
 93. T I-] E OR E M III. 
 
 Each fide of a fpheric triaiigle is Ufs than a femicircle, or 180 degrees. 
 Dem. Two' great circles interfcct each other twice at the diftance of 180 
 
 degrees. 
 
 (32) 
 (10) 
 
 The Hdes about any fpheric angle are arcs of two great circles. 
 But a fpliefic tiia!-.;jle has three hiics. 
 Therefore every two fides before their fecond meeting muft be inter- 
 fecl:ed by the third iide. 
 
 Confcquenily each Hue is lefs t'lan a femicircle. 
 
 The maik A Uancilfoi- the word triangle. 
 
 94. T H E O-
 
 Book IV. 
 
 SPHERICS. 
 
 147 
 
 94. 
 
 THEOREM IV. 
 
 In every fphefic triangle^ ABC, the greateji fide, BC, Is eppofite the 
 greateji angle ^ A. 
 
 Dem. Make Z. bad =Z. ABC. 
 Then aditbd (90) ; and bc:ad + dc. 
 But AD + DC is greater than AC. 
 Therefore bc is greater than ac. 
 
 95. T H E O R E M V. 
 If from the three angles of a fphcric triangle.^ ABC, as poks, he defcribcd 
 
 three arcs of great circles^ fornmig another fpheric triangle^ f.df ; then ui'II 
 the fides of the latter ,' and the oppofte angles of the former^ he the fup pigments 
 ef one another : Alfo the angles in the latter -^ and their eppofite fides in the 
 former^ are the fupplcments of one another. 
 
 That is, FE and Z-CAB, fd and Z-ABC, de 
 and Z.ACB, are fupplcments to one another. 
 Alfo Z.E and AC, Z.D and CB, Z.F and ab, are 
 the fupplcments to one another. 
 Dem. The interfeilion e of the arcs about the 
 poles A and c, being 90^ diflant from them, is 
 the pole of the arc AC. (19) 
 
 And for the fame reafon, D is the pole of ce, 
 and F of AB. 
 
 Let the fides of the triangle abc be produced to meet the fides of ths 
 triangle def in G and H, i and L, m and n. 
 
 Then fi =: dl = 90 : Therefore (dl + Fi = dl -f- fl + li ~ ) 
 DF + LI =: 180". (II. 47) 
 
 Therefore df and Li are fupplcments to one another. 
 
 Eut LI meafures tht; angle abc. (9) 
 
 Therefore l_AP.c and df are the fupplcments to one another. 
 And in the fame manner it may be demonftrated, that the Z. BAC and 
 >E, Z.ACB and DE, are the fupplcments of one another. 
 
 Again, fmce Ei~ ah = 90 degrees j (ic)) 
 
 Therefore (iB f AHrriB -I- eh + AB ) IH 4- AB 180 degrees. 
 
 Eut IM meafures the angle f ; (o) 
 
 Therefore ab and Z_F are the fupplcments of one another. 
 
 And the lame maybe ilicwn of ac and /.e, cb and /_d. 
 
 96. T H E O R E M VI. 
 
 The fum of the three fides cf every fphsric triangle, ABC, is lefs than a cir- 
 cnnfercnee, or 360 degrees. 
 
 Df.m. Continue the fides AC, AB, till they meet 
 in r;. 
 
 Then the arcs acd, ard, are each i8c. (p.) 
 JBut nc 4 DB is greater than bc. (92) d 
 
 Thcrefo.e ac4-ab4dc4db is greater than 
 AC I- An 4 EC 
 
 Or the lemicircles acd 4 abd is Jireatcr than ac 4 ae 4 r.i'. 
 
 'I h.it 1?, 360 ii> greater than the three fides of the trian-Ic Anr. 
 
 L 2 97. Til EG-
 
 14? 
 
 SPHERICS. 
 
 Book IV* 
 
 (24) 
 
 97. THEOREM VII. 
 
 The fttm of the three angles of afpheric triangle, abc, is greater than two 
 right angles, and lefs than fix j or will always fall between 180 and 548 
 degrees. < 
 
 Dem. Since Z.A and fe, Z.b and fd, Z.c and 
 DE, are fupplenients to one another. (95) 
 
 Therefore the three angles a, b, c, together with 
 the three fides fe, fd, de make thrice 180% or 540. 
 
 Now the fum of the three fides fe + fd + de, is 
 lefij than twice 180, (96) 
 
 Therefore the fum of the three angles a + b + C 
 is greater than 180. 
 
 Again, as a fpheric angle is ever lefs than 180 ; 
 
 Therefore the fum of any three fpheric angles is ever lefs than thrice 
 180% or 540 degrees. 
 
 98. THEOREM VIII. 
 
 If one fide, ab, of a fpheric triangle, abc, he produced, then the outward 
 angle, cbd, is either equal to, lefs, or greater than the inward oppofite 
 angle A, adjacent to that fide \ according as the fum of the other txvo ftdes^ 
 ca + cb, is equal to, greater, or lefs than 180 degrees. 
 
 Dem. Produce ac, ab, to meet in d. 
 
 Then arc Acoriarc ABDmSo". (32) 
 
 And Z.D=:Z-A. (31) 
 
 Nowif AC-f CBis equal toi8oi then cb = cd. 
 And z.CBD== (Z-Dzr) Z.A. (90) 
 
 If AC+CB is greater than 180; then cb is 
 greater than cd. 
 And Z.CBD is lefs than (z.d = ) z. a. 
 
 If AC-f CB is lefs than iSo; then cb is lefs than cd. 
 And /.cbd is greater than (z.,d=:)z.a. j 
 
 99. T H E O R E M IX. 
 
 /;/ right angled fpheric triangles, the oblique angles and their oppofite ftdes 
 are of the fa?ne kind: That is, if a leg is lefs or greater than 90", its oppofite 
 
 angle is alfo lefs or greater than cp". 
 
 In the right angled fpheric triangle ABC, right 
 angled at a. 
 
 If AC 15 greater than 90 ; then Z. ABC is greater 
 than 90". 
 
 If AC is Icfs tlun C)Z ; then 1. abc is lefs than 90. 
 
 Deal Let the leg ac be lefs, ad equal, ac 
 greater, than co", and defcribe the arc db. 
 
 Now D being the pole of ab (37). Therefore 
 Z.DBA is right. 
 
 Confequently if ac is lefs than ad, the /.cba is lefs than Z.DBA, 
 
 But if AC is greater than ad, the Z.CBA is greater than Z.DBA. 
 
 And the fame may be proved of the kg ab and its oppofite angle. 
 
 100. THE O- 
 
 (94) 
 (94)
 
 Book IV. 
 
 SPHERICS. 
 
 149 
 
 A 
 
 D 
 
 100. THEOREM X. 
 
 In right angled fpheric triangle s^ bac, the hypothenufe^ BC, is lefs than 
 90, when the legs^ ab, AC, are of a like kind: But the hypothenufe is 
 greater than 90, when the legs are of different kinds, 
 
 ift. When the legs ab, ac, areboth lefs than 90. 
 Dem. In BA, AC produced, take bd, af equal to j^\, 
 
 quadrants ; through f and d de'fcribe an arc fd \\ 
 
 meeting BC produced in E. : '. 
 
 Now F being the pole of bd (19). Therefore b 
 is the pole of ED. 
 
 Confequently bc is lefs than (beiz) 90. 
 
 2d. When the legs ab, ac, are both greater 
 than 90". 
 
 Produce AC, ab, till they meet in d. 
 
 Now the hypothenufe cb is common to both the 
 right angled triangles bac and bdc, 
 
 Afld the legs DC, db, being both lefs than 90". 
 
 Therefore the hypothenufe bc is lefs than 90% 
 by the firft cafe of this Theorem. 
 
 3d. When the legs ab, ac, are one greater, the 
 other lefs than 90. 
 
 In AB, and ac produced, take bd, af each of 
 90, and dcfcribe the arc fed. 
 Then b is the pole of fd ; and fince f is the pole 
 of BA, and fd is at right angles to bd. Therefore 
 BE = 9o. (37) 
 
 Confequcntly Rc is greater than 90 degrees. 
 
 ici. CoROL. I. The hypothenufe is lefs or greater tha 90, accord- 
 ing as tlie oblique angles are of a like, or different kinds, n 
 
 For if legs are like, or unlike, the angles are like or unlike. (99) 
 
 And if legs arc like, or unlike, the hypoth. is acute or obtufe. (100) 
 Therefore if the angles are like, the hypothenufe is acute, or lefs than 
 90 ; but if unlike, the hypothenufe is obtufe, or greater than 90. 
 
 rc2. CoROL. II. The Icj^s and their adjacent angles are like, or un- 
 likc, as the hypothenufe is lefs, or greater than 90 degrees. 
 
 For like legs, or like angles, make the hypothcnufj acute (by ift and 
 2d of 100). 
 
 And unlike legs, or unlike angles, make the hypothenufe obtufe (by 
 3 J of 130 and by 101). 
 
 1C3. CoROL. III. A \c'i^ and its oppofite angle are both acute, or botli 
 obtu.'c, according as the hypotlienufe and other leg are like, or unlike. 
 'ibis i.-, evident from the three cafes of this Tlieorem. 
 
 104, CoRoi,. W . Either angle is acute, or obtufe, as the hyfothcnuf* 
 and the other angle are like, or unlike. 
 
 This follows from cafe ift and 2d of this Theorem. 
 
 JL 3 105. TIIEO-
 
 i5d 
 
 SPHERICS. 
 
 Book IV. 
 
 105. T H E O RE M XI. 
 
 Jn every fphcrlc triangle, abc, if the angles adjacent to either fide, ab, be 
 aliks, then a perpendicular, CD, drawn to that fide from the other angle, will 
 fcAi ivithin the triangle : But the perpendicular CD falls without the triangle, 
 xvhen the angles adjacent to the fide ab it falls on are unlike, 
 
 Demqnst. Since in all right angled triangles 
 the perpendicular and its oppoiitc angle are of the 
 fame kind. (99) 
 
 Therefore the z.s cad, cbd, are each like cd. 
 
 Nov/ in Fig. i. the angles cad, cbd, or cab, 
 CBA, are angles adjacent to the bafe ab within 
 the triangle, and are tncrcfore alike. 
 
 Therefore the perpendicular falling between a 
 and B, falls within the triangle. 
 
 In Fig. 2. the angles cad and cab are the fup- 
 plenients of each ether, and are therefore unlike, 
 as CA falls obliquely on ab. 
 
 Therefore Z-Cab is unlike to Z.cba. 
 
 Confequcntly the perpendicular cd cannot fall 
 between a and b : Therefore it muft fall without. 
 
 106. THEOREM XII. 
 
 If the tzvo leffer fides, CA, CB, cf a fphcric triangle, ABC, are of the fame 
 
 kind; then an arc, CD, drawn from their included angle, acb, perpendicular 
 
 to the oplofte fide, AR, vjHI fall within the triangle. 
 Demonst. In ab take af=:ac ; draw CF 
 
 and AH at right angles to cF. 
 
 Then chzzhf (91) are each lefsthan 90. (93) 
 Aifo take beztbcj draw ce and bg at right 
 
 angles to ce. 
 
 Then cgitge (91) are each lefs than 90. (93) 
 
 Now in the right angled triangles fha, egb \ if the hypothenufes af 
 
 (riAC), and be (~bc), are acute, or like fh and eg ; 
 
 T>jen the angles af?i and beg are acute, and like AC and bc ; (103) 
 Therefore the perpendicular cd falls on ef, within the triangle. (105) 
 Aifo if the hypothenufes af and be are obtufe, or unlike to fh and ge ; 
 Then the angles afh and beg are obtufe, and aifo like ca and cb.( 103) 
 Confequcntly the perpendicular will fall on ef. (105) 
 
 Therefore in either cafe the perpendicular falls within the triangle. 
 
 1S7. THEOREM XIII. 
 
 Jn all right angled fphcric triangles, 
 
 yfs hju- h'jpoth. : Rad : : fine of a leg : fine of its oppofite angle. 
 ic8. Ji'df:?is a leg : Rad. : : ian. other leg : tan. of its oppof.te angle. Pi. I. 
 
 DrMON'-'T. Let fdafg rcprclent the eighth part of a fphere, where 
 the quadrantal planes edfg, fdbc, are both perpendicular to the qua- 
 d.antal plane adfb ; and the qtiadrantal plane adgc is perpendicular lo 
 the quadrantal plane edfg : and the fphcric triangle abc is right angled 
 at B, where ca i> the hvpcithcnufe, and ba, bc, are tiie legs. 
 
 '1 o the arcs r. f, ch, draw the tangents hf, ob, and the fines GM, cr, 
 on the radii df, ci? ; alio draw r,L, the fine of the arc ab, and CK, the 
 fine of AC ; then join IK and OL. 
 
 Now
 
 Book IV. 
 
 SPHERICS. 
 
 J 5; 
 
 . Now HF, OB, GM, CI, are all perpendicular to die plane adfe. 
 
 Ant^ HD, CK, OL, lie all in the fame plane adgc. 
 
 Alfo FD, IK, BL, lie all in the lame plane adfb. 
 
 Therefore the right angled triangles hfd, cik, obl, having the equal 
 angles hdf, cki, olb, (II. 199) are funilar. (II. 167) 
 
 Therefore CK : do : : ci : cm. 
 
 That is. As fin. hyp. : Rad. : : Hn. of a leg : fm. opp. angle. 
 
 For GM is the fmeof the arc gf, which meafures the angle cab. (9) 
 
 Alfo, As LB : df : : bo : fh. 
 
 That is, Asfni.ofaleg : Rad, : : tan. of other leg : tan. opp. angle. 
 
 109. T H E O R E M XIV. 
 
 In right angled fpheric triangles, abc, if about the oblique angles^ A, c^ 
 as poles, at 90" dijlance, there be defcribed arcs, de, fe, cutting one another 
 in E ; and the fidiS ab, ac, bc, of the triangle be produced to cut thofe arcs 
 in D ; G, F ; H, I ; there will be con/Iituted two other triangles, CGK, HIE, 
 the parts of which are either equal /-;, or are the complements of the parts 
 of the given triangle, ABC. PI. I. 
 
 Demokst. Now fince a is the pole of ed (19}. Therefore ad, ag 
 are at right angles to ed ; and fo is ed to ad. (37) 
 
 And fince bi and d are at right angles to ad, their interfc<5lion h is 
 the pole of AD (36). Therefore Ku, hd are quadrants. (35) 
 
 Then in the triangle CGH, rigb.t angled at g. 
 
 CGrrccniplcment of AC. 
 
 HGzrcomp. Z-Aj For no is tlie comn. of gd, which mea. Z.A. (9) 
 
 HC the hypoth. is the comp. of cb. 
 The Z-HCGrzZ-ACB. (26) 
 
 1 he ACHGircomp. ab : For ed, the comp. of ab, meafures Z.chg. 
 
 Alio in the triangle eiii, right angled at i : Becaufe CF, ci are at 
 right angles to ef ; and ef, eg being alfo at right angles to af ; therefore 
 ]: is the pole of af ; (36) confequcntly ef and eg are quadrants. (35) 
 
 Then the hypoth. ETirrZ-Aj For GHziicomp. of em and gd ; and 
 CD meafures tlie angle a. 
 
 Hi=rcB ; for Hcrrcomp, of in ai-.d en. 
 
 Ei^rcomp. Z. c ; For Ei^comn. of if, v/hicii me:;riires z.c. 
 The Z. H::::conip.. ab ; P'or bd, the corr.pof A!5, meafures Z.H. 
 The z. Err AC ; For gf, the meafurc of Z.E, is equal to Z..\c. 
 no, T II E C) R E M XV. 
 
 /;.' every fpheric triii-igle, ii -vsill !<', 
 As the fine of cither angle, is to ihefne cf its of.fci. 
 t'o is the file of ancther awh, to the fne of its 
 
 ^ppon: 
 c a fpheric tii.'nglc, where bd is 
 to AC prduccd ; forniing the tv\o 
 r!L;i',t a:'.;,L-J tri.u;gks .\dp., <:iyi. 
 1);,M. "NcA f.:i. A.: : rail. : : fin. i,n : fm, z. A.(lC7) 
 K i'.c : rad. : : fm. v.D : fm. Z_c. 
 
 Let Ai'c I 
 perpendicular 
 
 And f:; 
 
 There: ore fi:-;. 
 
 Aiui tin. 
 
 Therefore !in. 
 
 'Jlicrtf'ji'c lin. 
 
 AH X an. A. A \-:A. x fin. r d. 
 ncxfin. ACz::rad. Xfui. \-.\). 
 A'sxHu. z.A~ru). r::xiin. z.c. 
 Z.A : hn. iic : : fm. /_ c : iin. ab,
 
 152 SPHERICS. Book IV. 
 
 SECTION VI. 
 Of the Solution of right angled fph eric Triangles. 
 
 In every cafe of right angled fpheric triangles, three things befide the 
 radius enter the proportion, of which two are given, and the third is 
 fought. 
 
 Now the folution of every cafe will be obtained by the application of 
 the two following rules to Theorem XIII. and XIV. ( 107, io8, 109.) 
 
 iir. Rule I. If of the three things concerned, or their complements, 
 two are oppofitc to one another, and the third is oppofite to the right 
 nnglc, in one of the triangles marked i, 2, 3, in the fig. toTheo. XIV. 
 PI. I. Then the thing fought will be found by the firft proportion (107) 
 cither diredtly, or by inverfion, 
 
 112. Rule II. If of the three things concerned, or their comple- 
 ments, two are fides, and the third is an oblique angle, in either of the 
 three triangles marked i, 2, 3, in fig. to Theo. XIV. PI. I. Then the 
 thing fought will be found by the fecond proportion (108) either diredly, 
 or by inverfion. 
 
 113. PROBLEM I. 
 
 In the right angled fpheric triangle abc, Plate I. Theorem XIV. 
 
 Given the hvpothenufe ac 7 j ^u n. 
 
 J {-*^, , > required the reft, 
 
 and one ot the legs ab j ^ 
 
 I ft. T^o find the angle ACB oppofite the given leg ab. 
 
 Here the things concerned are ac, Z.B, ab, Z_c ; which are found in 
 the triangle, N I, to be oppofite ; and fo fall under Rule I. (m) 
 
 Then fin. AC : rad. : : fin. ab : fin. Z-ACB. (107) 
 
 Or fin. hyp. : rad. : : fin. g. leg : fin. op. z.. Like the g. leg. (99) 
 
 2d. To find the angle cab adjacent to the given leg ab. 
 
 Here the things concerned are ac, Z.B, ab, Z. a. 
 
 Now trying in the triangle, N I, I find the things concerned will 
 fall under neither of the Rules. 
 
 But trying in the triangle, N 2, the things concerned, or their com- 
 plements, fall under Rule II. (^12) 
 
 Then fin. HG : rad. : : tai}. GC : tan. Z.CHG. (108) 
 
 Orco-f Z.CAB : rad. : : co-t, ac : co-t. ab. 
 
 Or co-f. Z.CAB : co-t. ac : : (rad. : co-t. ab) : : tan. ab : rad. (III. 36) 
 
 Therefore rad. : co-t. hyp. : : tan. g. leg : co-f. adj. angle. (II. 145) 
 
 Like, or unlike the given leg; as the hyp. is acute, or obtufe. (102) 
 
 3d. To find the other leg BC. 
 
 Here the things concerned are ac, Ab, ab, bc ; which in tlie triangle, 
 K I, do not fall under either Rule : But in N 2 they will be found to 
 fall under the firft Rule. (m) 
 
 Then fin. he : rad. : : fin. CG : fin. id.CHG. (107) 
 
 Or co-f. CB : rad. : : co-f. AC : co-f. ab. 
 
 Therefore co-f. g. leg, ab, : rad. : : co-f. hyp. AC, : co-f. rcq. leg cb. 
 
 And is acute, if hyp. and given leg are like 3 but obtufe, if unlike. (103) 
 
 114. PROB^
 
 m 
 
 Book IV, SPHERICS. 153 
 
 114. PROBLEM II. 
 
 In the right angled fpheric triangle abc. PI. I. Theorem XIV. 
 Given the hypothenufe ac J 
 
 And one of the oblique angles a 3 ^ 
 
 ift. To find the leg CB oppofite to the given angle A. 
 
 In the triangle, N i. the things concerned fall under Rule I. ( n i) 
 
 Then rad. : fin. ac : : fin. ^cab : fin. cb. (1C7) 
 
 Or rad. : fm. hyp. : : fin, given angle : fin. opp. fide. 
 
 And is like the given angle. (99) 
 
 2d. To find the leg ab adjacent to the given angle a. 
 In the triangle, N 2. the things concerned fall under Rule IT* (112) 
 Then fin. hg : rad. : : tan. CG : tan. Z-Chg. (io^) 
 
 Or cof. Z.BAC : rad. : ; (co-t. ac : co-t. ab : :) tan. ab : tan. ac. 
 
 (III. 37) 
 Therefore rad. : tan. ac : : co-f. Z.bac : tan. ab. 
 Or rad. : tan. hyp. : : co-f. given angle : tan. adjacent leg. 
 And is acute, if hyp. and given angle are alike i but obtufe if unlike. (104) 
 
 3d. To find the other angle acb. 
 
 In the triangle. No 2, the things concerned fall under Rule II. (112) 
 Then fin. cg : rad, : ; tan. ch : tan.z.HCG. (10^) 
 
 Or co-f AC : rad. : : co-t. z. bag : tan.Z.BCA : :) co-t. ^l bca. : tan. 
 
 Z-Bac. (III. 37) 
 
 Therefore rad. : tan. Z.bac : : co-f. ac : co-t. Z.bca. (IL 145) 
 
 Or rad. : co-f. hyp. : : tan. given angle : co-t. req. angle. 
 
 And is acute, if hyp. and given angle are alike ; but obtufe, if unlike. 
 
 (104) 
 
 115. PROBLEM in. 
 
 In the right angled fpheric triangle abc. Plate I. Theorem XIV. 
 
 Given one of the lees ab 7 n j .u /i 
 A ,- r. 1 > Required the relt. 
 
 And Its oppoiite angle acb ) ^ 
 
 ift. To fitnd the hypothenufe AC. 
 
 In the triangle, N'' i. the things concerned fall under Rule I. (m) 
 
 Then fiji. Z. ACB : fin. AB : : rad. : fin. AC. (^07) 
 
 Or fin. given angle : fin. given leg : : rad, : fin. hyp. 
 
 And is cither acute or obtufe. 
 
 2d. To find the other leg CB. 
 
 In the triangle, N" 1. the things concerned fall under Rule II. (112) 
 
 Then fin. CB:(rad. ::)tan.AB(: tan.Z. ACi;):: co-t. z. acb: rad. (II 1. 36) 
 
 Or rad. : co-t. given angle : : tan. given leg ; fin. req. leg* 
 
 And is cither ucutc or obtufe. 
 
 3d. To fiind the otJjcr angle CAC. 
 
 In the trianglj, N' 3. the things concerned fall under Rule I. (m) 
 
 Then fin. liii : rail. : : fui. f.i : fm z.nn-. (^07} 
 
 Or fin. ABAC : rad. : : co-f. z.Acr, ; co-1'. Ai;. 
 
 Or co-f given I'.'g : co-f. given angle ; ; rad. ; fin. required angle. 
 
 And is cither acute or obtuiv. 
 
 116. PROB-
 
 M4 
 
 SPHERICS. Book IV 
 
 116. P R O B L E M IV. 
 
 In the right angled fpheric triangle abc, Plate I. Theorem XIV. 
 
 Give one of the legs AB 7 . , ^, 
 A J- J- i . C Required the reft. 
 And Its adjacent angle bAc j ^ 
 
 ift. To find the other angle nc A. 
 
 In the triangle N" 3. the things concerned fall under Rule I. (m) 
 
 Then rad. : fin. eh : : fin. Af.hi : fin. Ei. (107) 
 Or rad. : fin. Z-Bac : : co-f. ab : co-f, Z.acc. 
 
 Therefore rad. : co-f. given leg : : fin. given angle : co-f. req. angle. 
 
 And is like the giveij leg. (99) 
 
 2d. To find the other leg -e.c. 
 
 In the tri.ingle,N i. the things concerned fall under Rule II. (112) 
 Then fin. ab : rad. : : tan. bc : tan. A cab. (108) 
 
 Or rad. : fin. ab : : tan. Z-CAB : tan. bc. 
 
 Therefore rad. : fin. given leg : : tan. given angle : tan. req. leg. 
 And is like the given angle, (99) 
 
 3d. To find the hypothenufe AC. 
 
 In the triangle, N 2. the things concerned fall under Rule II. f 112) 
 Then fin. gh : rad. : : tan. cg : tan. Z.chg. (loS) 
 
 Or co-f. /Lcab : rad. : : co-t. ac : co-t. ab. 
 
 Therefore rad. : co-f. given angle : : co-t. given leg : co-t. hpothenufe.- 
 And is acute, if the given leg and angle are alike y but obtufe, if unlike. 
 
 (102) 
 
 117. P R O B L E M V. 
 
 In the right angled fpheric triangle abc, Plate I. Theorem XIV. 
 Given both the legs ab, bc. 
 Required the reft. 
 
 ift. To find either of the ohl'ique angles^ as BAG. 
 
 In the triangle, N" i. the things concerned fall under Rule II. (112) 
 Then, as fin. ab : rad. : : tan. nc : tan. Z.bac. (id^) 
 
 Or rad. : fin. ab : : (tan. Z-BAC : tan. bc : : ) co-t. ec : co-t. Abac. 
 
 (HI. 37} 
 
 Therefore rad. : fin. one leg : ; co-t. oth. leg : co-t. opp. angle. 
 And is like its oppofite leg. x (99) 
 
 2d. To find the hypothenufi; AC 
 
 In the triangle, N^^ 2. the things concerned fall under Rule I, (m) 
 Then, As rad. : fin. hc : : fin. Z-CHG : fin. cg. {'^^l) 
 
 Or rad. : co-f. hc : : co-f ab : co-f. AC. 
 
 Therefore rad. : co-f. omc leg : : co-f. oth. leg : co-f. hypothenufe. 
 And is acute, if the legs are alike ^ but obtufe, if unlike. (10) 
 
 118. PROI3-
 
 Book IV. SPHERICS* 
 
 ^5S 
 
 ii8. P R O B L E M VI. 
 
 In the right angled fpherical triangle abc, Plate I. Theorem XIV. 
 Given both the angles cac, bca. 
 Required the reft. 
 
 I ft. To find either of the legs, as &c. 
 
 In the triangle, N 2 or 3. the things concerned fall under Rule I. ( 1 1 1 ) 
 
 Then, rad. : fin. hc : : fjn. z.hcg : fin. hg. (io?) 
 
 Or rad. : co-f. bc : : fin. Z.acb : co-f. /.bag. . 
 
 Therefore fine of one angle : rad. : : co-f. oth. angle : co-f. oppofite fide. 
 
 And is like its oppofite angle. (09) 
 
 2d. To find the hypothenufe AC. 
 
 In the triangle, N" 2. the things concerned fall under Rule II, (112) 
 Then, As fin. CG :rad. : : tan. gh : tan. z.hcg. (ig8) 
 
 Or co-f. AC : (rad. : :) co-t.Z.BAC (: tan. /.bca) : : co-t.Z.BCA : rad. 
 
 (III. 36) 
 Therefore rad. : co-t. one angle : : co-t. oth. angle : co-f. hypothenufe. 
 And is acute, if the angles are like. - (loi) 
 
 But obtufe, if unlike. 
 
 In thefe fix Problems are contained fixteen proportions, which are ap- 
 plicable to the like number of cafes ufually given to right angled fpheric 
 triani.;,!cs ; ar.d thefe proportions being collected and difpofed in a Table, 
 will readily fhevv, by ir.fpeftion, how any of the cafes are to bc folved. 
 
 The celebrated Lord Nepier, the inventor of logarithms, contrived a 
 general rule, eafy to be remembered, by Avhich the folution of every cafe 
 in right angled fpheric triangles is readily obtained, where the table of 
 proportions is wanting ; which rule is as follows. 
 
 General Rule. 
 
 119. Radius multiplied by the fine of the rniddle party is cither equal to the 
 prodiief of the tatigcnis of extremes iO>ijunSf. 
 
 Or to the produSi of the co-fines of extremes diijunSl. 
 Ohfcrving ever to ufc the con.plevients of the hypcth. and angles. 
 
 Lord Nepier called the five parts of every right angled Iphcric triangle, 
 omitting the right angle, circular parts ; which he thus diilinguilhed ; 
 the tzvo legs, the compleynents of the tiuo angles, and the ccmpletnent of the 
 hypothen-fe ; and any two of thefe circular parts being given, the others are 
 to be found by this rule, as is fticwn in v.'hat follow?. 
 
 Now, In all the proportions about right angled fpheric triangles, there 
 are, befKiCs the radius, three things coiicerncJ ; one of which maybe 
 called the middle term in refpecft of the otl^r two; and thefe two, in 
 refpcft of the middle term, may be called extremes. 
 
 When the two extremes are joined to the middle, they are called ex- 
 tremes co!;juncl: But when each of them is disjoined I'rom the middle, by 
 i'li intermediate term (not concerned), they are then called extremes dil- 
 junct- ; t;:king notice that the right analc does not disjoin the lc^';s. 
 
 If the three parts under confideration do all join, the mitldic one of 
 thofe tliree is readily fccn, and the otiier two are extremes coiijiinLt. 
 
 But it only two of tlie three parts are jr)incd, t'lofe two are extremes 
 disjunii, and the other term is the middle part. 
 
 Thefe
 
 156 SPHERICS. Book IV. 
 
 Thefe things duly obferved, the pra(SUce of the Rule will appear in the 
 following examples. 
 
 Example I. When the hypothenufe and the angles are eoneerned. 
 
 The hypoth. is the middle term, and the two angles are extremes con- 
 junl ; then by the rule. 
 
 Rad. xfin. hyp. = tan. one angle X tan. other angle. 
 But the comp. of the hypoth. and angles are always to be ufed. 
 Therefore rad.x co-f. hyp.=rco-t. one angle xco-t, other angle. 
 Hence rad. : co-t. one angle : : co-t. other angle :co-f. hypoth. (II. 163) 
 From whence are deduced the 6th and 15th cafes. 
 
 
 
 Exam. II. IVI^en the hypothenufe and legs are under canfideration. 
 
 The hypothenufe is the middle term, and the two legs are extremes 
 disjuncSt, having the angles between them and the hypothenufe. 
 
 Then by the rule. Rad. x fin. hyp. r= co-f. one leg X co-f. other leg. 
 But the complement of the hypothenufe is to be uled. 
 Therefore rad. X co-f. hypoth. = co-f. one leg X co-f. other leg. 
 Hence rad. : co-f. one leg : : co-f other leg : co-f. hypoth. (II. 163) 
 From whence are deduced the 3d and 13th cafes. 
 
 Exam. III. The legs and an angle under confideration. 
 
 Here the angle and its oppofite leg are extremes conjunct j and the 
 other leg is the middle part. 
 
 And thefe being refolved into a proportion by the rule, will produce 
 the 8th, nth, and 14th cafes. 
 
 Exam. IV. The angles and a leg under confideration. 
 
 Here one angle is the middle, and the other angle and leg are extremes 
 disjunct, the hypothenufe and other leg intervening. 
 
 Now thefe being refolved into a proportion, give the 9th, 12th, and 
 1 6th cafes. 
 
 Exam. V. The hypothenufe., a Icg^ and the angle between them., being 
 under confideration. 
 
 Here the angle is the middle term, and the hypothenufe apd leg are 
 extremes conjun6l. 
 
 And thefe being refolved into a proportion, will give the 2d, 4th, and 
 loth cafes. 
 
 Exam. VI. The hypothenufe ^ a leg., and its oppofite angle., being under 
 confideration. 
 
 Here the leg is the middle term, and the hypothenufe and angle are ex- 
 tremes disjunct, the other leg and other angle falling between them and 
 the middle. 
 
 And thefe being converted into a proportion, from thence the ill, S^'""* 
 anu jth cafes are deduced. 
 
 SECTION
 
 Book IV. 
 
 S P H E R I C St 
 
 ^57 
 
 SECTION VII. 
 
 Of the Solution of oblique angled ff her ic T'ri- 
 
 angles. 
 
 120. All the cafes in oblique angled fpheric triangles, except where the 
 three fides, or the three angles are given, are moft conveniently refolved 
 by drawing a perpendicular from one of the angles to its oppofite fide, 
 continued if neceflary ; which perpendicular will either divide the given 
 triangle into two right angled triangles, or make two that are right 
 angled, by joining a right angled one to the given triangle. 
 
 In drawing this perpendicular, obferve, 
 
 ift. It muft be drawn from the end of a given fide, and oppofite to a 
 given angle. - 
 
 2d. It mull be fo drawn, that two of the given things in the oblique 
 triangle may remain known in one of the right-angled triangles. 
 
 3d. This perpendicular is to be ufcd as a known quantity ; and being 
 drawn as here diredled, will either fall within or without the triangle, as 
 the angles, next the fide on which it falls, are of the fame or of dif- 
 ferent kinds, (105) 
 
 121. 
 
 PROBLEM I. 
 
 In the oblique angled fpheric triangle abc. 
 Given two fides ca, cb | ^ j^ ^j^^ ^^^ 
 
 And the angle opp. to one, Z.C AC 3 ^ A 
 
 ifl. To find the angle oppof.te to the other given fide 
 
 { Z.CBA). 
 
 As fin. EC : fin. ^cab : : fin. ac : fin. Z.cba. (no) ^ 
 
 Or, As fin. one fide \fin. oppcfte angle : :f.n. ether fide :fn. oppof.te angle. 
 Whleh may be either aeiite or oltufe. 
 
 l\. To find the angle between the given fdes ( Z. ACb). 
 Now rad. : tan.Z.CAB : : co-f. ac : co-t. (acd, call it) m. (3d 114) 
 Or rad. : tan. given Z- : : co-f. adj. fide : co-t. {of a fourth r^.') m. 
 And is acute, if AC and z_cab are like ; but ootuf., if unlike. 
 But rad. : tan. cd : : co-t. ac : co-f. (Acor;) m. (2d 113) 
 
 rad. : tan. CD : : co-t. cb : co-f. (rcd, call it) n 
 Therefore co-t. ac : co-t. ci3 ; : co-f. m : co-f n, (II. i55) 
 
 0/ co-t. fide adj. given /_ : co-t. other fide : : co-f. m : co-f. n. 
 And IS like the fide oppfite the given angle., if that angle is acute. 
 But unlike that fide ^ if the given angle is obtufe. 
 
 C7-, w //-/./ \ urn m and n., tt A.* talls'ivitiJin 
 
 I r:en the angle fought-, \\z. l_AZ?, \ '.-.r "> , -V , > //. ..r,hn,jf 
 
 ^ J ^ ^ I diff. of m and //, // juils ivithout 
 
 The n\v.k X H^niflcs the perpendicular. 
 
 3d. To
 
 158 
 
 SPHERICS. 
 
 Book IV, 
 
 3d. To find the other ftde AU. 
 
 Now rad. co-f. Z.CAB : : tan. AC : tan, (ad, call it) M. (2d 114) 
 
 Or rad. : co-f. given angle : : tan. adj. fide : tan. (of a fourth :zi) m. 
 Acute, if the angle and its adj. fide are like j but obtuj'e^ if unlike. 
 But co-f. CD : rad. : : co-f. AC : co-f. (ad = ) m. 
 
 co-f. CD : rad. : : co-f. cb : co-f. (db call it) N. (3d 113) 
 
 Therefore co-f. ac : co-f. cb : : co-f. m. : co-f. n. (II. 155) 
 
 Or co-f.ftde adj. given angle : co-f. other fide : : co-f. M. : co-f. N. 
 Like the fide oppofite the given angle, if that angle be acute ; 
 But unlike that fide, if the angle be obtufe. 
 
 --, ,' ri r 1 \ fum ofu and N, if the falls within. 
 
 Then the fide fought ab= ^d^^, if^, and r^, if the Al falls without. 
 
 But if CA = CB, or if cazziSo'' cb, or if ca is between bc and 
 180 BC ; 
 
 Then z. B is like bc only, 
 like 
 
 And if BC 
 
 : is "I 
 
 unlike 
 
 Z-A 
 
 5 ThenZ.ACB = mn only i and abztm 
 1 N only. 
 
 122. PROBLEM II. 
 
 In the oblique angled fphcric triangle abc. 
 Given two angles cab, cba 1 Required 
 
 And a fide oppofite one of them ac 5 the reft. 
 
 ift. To find the fide oppofite the it her given angle, 
 viz. cb. 
 
 Then, As fin.z.CBA : fin. ac : : fin. Z-Cab : fin. ^ 
 
 Or fin. one angle '.fin. oppofite fide : : fin. other angle : fin. oppofite fide. 
 Which may be either acute or ohtifc. 
 
 2d. To find the fide included by the given angles, viz. AB. 
 
 Now rad. : co-f. Z.cab : : tan. ac : tan. (ad, call it) m. (II. 114) 
 Orrad. : co-f A.adj . given fide : : tan. thegivenfide:ran.(ofafourth'=:)M. 
 Like the angle adj. the fide given^ if that fide is acute; but unlike, if obtufe. 
 But rad. : tan. cd : : co-t. Z-Cab : fin. (ADrr) M 
 
 rad. : tan. CD : : co-t. Z.cbd : fin. (db, call it) N. (2d 115) 
 
 Therefore co-t. Z.CAB : co-t. Z.CBD : : fin. m : fin. N. (II. 155) 
 
 Or co-t.L.adj. given fide : co-t. other angle : : fin, M : fn. K. 
 JFhich may be either acute or cbtufe. 
 
 cTL .; r' r 1^ i fum of M and s, if the p;iven angles are alike. 
 
 1 hen tie fue fought AB=. i J ..yr--^ _r j -i- .i r vi 
 
 J J '^ \ dijf. ofm and N, if the given angles are unlike. 
 
 3d. To find the other angle, viz. Z.ACB. 
 
 Now rad. : tan. z. cad : : co-f. AC : co-t. (z. acd, call it) m. (3d 114) 
 Orrad. : tan. A.adj . fide given : : co-f of given fide: co-t. (ofafourthzz.'jm. 
 Like Z. adj. fide given, if that fide is acute \ but unlike, if obtufe. 
 But co-f. CD : rad. : : co-f. z.cab : fin. ( Z. ACDr:)m. 
 
 co-f. CD : rad. : : co-f. Z.ABC : fin. (Z_RCD, call it) n. (3d 115) 
 Therefore co-f. z.cab: co-f. 2Labc : : fin. m. to fin. n. (11. 155) 
 
 Or co-f. /_adj. fide given : co-f. other angle : : fin. m : fn. n, 
 IVhich may be either acute or obtufe. 
 
 Then
 
 feook IV. SPHERICS. 
 
 Th 
 
 B 
 
 Then BC cannot be unlike its oppofite angle. 
 Neither can db, or theZ.BCD be obtufe. 
 
 159 
 
 123. 
 
 PROBLEM III. 
 
 In the oblique angled fpheric triangle abc. 
 Given two fides ac>, ab 1 r* j ^1. n 
 
 And their contained angle bac } ^^^"^^^^ "^^ '^^' 
 
 ift. To find either of the other angles, as A ABC. j^l """7^^-^ 33 
 
 As rad. : co-f. Z-Cab : : tan. ac : tan. (ad, call it) m. (2d 114) 
 
 Or rad. : co-f. given /_ : : tan. fide oppofiit: Z-fiought : tan. {ofi a fourth m^w. 
 
 Like the fide oppoj-.te L. fought, if the given Z. is acute ; 
 
 But unlike that fide, if the given Z. is obtufe. 
 
 Take the diff. between K^, fide adj. /L fought, and {a'D'=:)i.\ ; call it a. 
 
 Now rad. : co-t. cd : : fin. ( ad=:) m : co-t. z.cab. -(iftiiyj 
 
 rad. : co-t. cd : : fin. (db=:) n : co-t. z.abc. 
 Therefore ^m, n : fin m. : : co-t. /.abc : co-t. Z-Cab. (II. 155) 
 
 : : tan. z.cab : tan. z.cba. (ill. 37) 
 
 Or fin, N \ fi.n. M : : tan. given L. '. tan. /_ fought. 
 
 Like the given angle, bag, if m is lefs than ab, the fide adjacent the 
 angle fiought ; but unlike, if M is greater. 
 
 2d. To find the other fide cb. 
 
 As rad : co-f. Z-CAB : : tan. ac : tan. (ad, call it) m. (2d 114) 
 
 Or rad. : co-f . given i_ : : tan. of either given fide : tan. {^':f a fourth :=.)y\.. 
 
 Like the fide ufcd in thii proportion, if the given Z-is acute ; 
 
 But unlike that fide, if the angle is obtufe. 
 
 Take the difference bctvoeen the other fide, AB, and (ADr:) M ; call it v. 
 
 Now rad. : co-f en : co-f (Anr=) m : co-f ac. (2d 117) 
 
 rad. : co-f cd : : co-f (or, ) n : co-f. cb. 
 Therefore co-f m : co-f N : : co-f AC : co-f cb. (II. 155) 
 
 Or co-f. M : cof. N : : co-f. fide ufid in fir ft proportion : cc-fi.fiide required. 
 Like N, if the given ^is acute; but unlike N, if that L.is obtufe. 
 
 124. PROBLEM IV. 
 
 In the oblique angled fphcric triangle arc. 
 
 (aivcn two ar.tilcs Z-Cab, Z-ACB \ Y^ 1 .1 a 
 
 . , , . . ,&, fz , ' S Required the rclt. 
 
 And their included hde ac J ^ 
 
 I ft. To fijid either cf the other fides, as cv>. 
 
 As r.iJ. : co-f ac : : tan. Z_ cab : co-t. (Z_ acd, call it^ m. {^A 1 14) 
 Or rad. : co-f. given fide : : tan. /_ oppofite fde fought : co-t. fofufui ihzz )ni. 
 Like the angle oppofte fide fought, if the given ude Is acute ; 
 But unlike that angle. If the given fdc be obtife 
 
 Take the dljf. betiveen A. .\CBy adj. fdc fought, and ( Z Acn } //;, call It n. 
 Then rad, : co-t. en : : co-f (Z-Acd ) 17-1 ; co-t. ac. (3d 116) 
 
 rad, : co-t, cd : : co-f ( z ncD~ ) n ; co-t. c;:. 
 
 Therefore
 
 i6o 
 
 SPHERICS. 
 
 Book IV. 
 
 Therefore co-f. n. : co-f. m : : co-t. cb : co-t. ac. (II. 155) 
 
 : : tan. ac : tan. cb. (III. 37) 
 
 Or co-f. n : co-f. m. : : tan. given fide : tan. fide required. 
 
 Like n, if the angle oppofite the fide fought be acute , 
 
 But unlike , if the angle is obtufe. 
 
 2d. To find the other angle abc. 
 
 As rad. : co-f. ac : : tan. Z. cab : co-t. (z.acd, 
 call it) m. (3d 114) 
 
 Or rad. '.co-f. given fide ; : tan. either given ^ ; co- 1. 
 (of a fourth zz.) jn. 
 
 Like L. nfed in this proportion, if the given fide, 
 AC, is acute ; 
 
 But unlike that Z., if the given fide is obtufe, 
 
 Take the difference between the other Z-, acb, and 
 Z. ( acd = ) m, call it n. 
 
 Now rad. : co-f. CD : : fin. (/.Acorr) m '. co-f. Z.CAB. (ift 116) 
 
 rad. : co-f. cd : : fin. (z.BCD=r) n: co-f. Z-Abc. 
 
 Therefore fin. m : fin. n : : co-f. z.cab : co-f. z.abc, (II, 155) 
 
 Or fin. m : fm. n : : co-f. Z. ufed in firfl prop. : co-f. ^fought. 
 
 Like the Z. ufed in both proportions, tf m. is lefs than the others j 
 
 But unlike, if m is greater than the other angle. 
 
 125. PROBLEM V. 
 
 In the oblique angled fpheric triangle abc. Plate I, Problem V. 
 
 Given the three fides ab, bc, ac 3 Required the angles. 
 
 To find the angle abc. 
 
 Let HBKLM reprefent the quarter of a fphere, the center of which is o. 
 Where the femicircular fedions hbk, hlk, are at right angles to one 
 another ; and cb is perpendicular to hk. 
 
 Then, continuing the fide bc to l, the arc hml meafures Z. abc. (9) 
 
 And HQjrf chord hl, will be the fine of \ (arc HMcrrf )z. abc. 
 
 Draw the radius oqm ; and draw LP, qN at right angles to hk. 
 
 Then LPrzfine, HP =verfed fine, ofz. abc ; And HNrzNP. (II. 165) 
 
 But as HQO is a right-angled triangle ; OQ^being perp. to hl. (II. 125) 
 
 Therefore oh : nq^: : hq^: hn. (II. 170} 
 
 And OH XHN=::Hq^ (II. 162]= fquare of the fine of fZ-ABc. 
 
 Make bd = be=zbc; and AF::r:AG=: ac 
 
 Then the femicircular plane dce, which is parallel to hlk (23), will 
 
 "be cut by the femicircular plane fcg, drawn at right angles to the plane 
 
 hbk, in the line ci (II. 209) at right angles to de. (II. 210) 
 
 And the arc dc, and its verfed fine di, are fimilar to the arc hl and 
 its verfed fine hp. (29. III. 15) 
 
 r^. , , /PH \2HN 
 
 Then rad. ch : rad. ds : ; ph : iDtrl Xds= / Xds. 
 
 \oh oh 
 
 Draw OR parallel to fg ; then arc ARr:(90 = ) arc bk, and rk=:ab. 
 
 Therefore Z. DiF rz ( Z.KOR=r arc RKrr) arc ab. 
 
 Now DS =:(SE= fine arc be:^:) fine arc bc 
 
 And ad=:(bd EAr^EC BA)=difF. fides about Z. fought. 
 
 AlfoZ-DFirri arc(DG= AG + AD =) ac+ad, the fine of which is 
 
 I id. Schol. to art. III. 45. 
 
 And arc fd=:(af t^\i ) AC AD, the fine of which is \_ df. 
 
 5 Now
 
 Book IV. ' SPHERICS. i6i 
 
 Now fin. /.DIP : fin.z.DFi : : (fd : id ::) jFd:|id. (Schol. III. 45) 
 Or fm.Z-DiF : fin.ADFi : : I FD : xds. 
 
 OH 
 HN 
 
 Therefore fin. Z.DIF XDS X =fin. z.dfi x|fd. (II. 163) 
 
 HN 
 
 Therefore fin. /.difXdsx XOH = fin./LDFi xfFDXoH (II. 156) 
 
 Or fin.Z-DiF xds XHN=:fin.z.DFix|FDXOH. (II. 149) 
 
 Theref. fin.z.DiFXDS : fin.z.i>Fi x|fd : : oh : hn. (II. 163) 
 
 : : OH xoH : (hnxoh=:)hq2,. (H. 155) 
 Therefore fin. Z. dif X ds : fin. /l dfi x |fd : : oh* : hq^- 
 Or fin. ABxfin.BC : fin. iAC + ad xfin. f ac ad : : Rad"" : lin. az-Abc* 
 
 fin. I AC + ADXfin. I AC AD. 
 
 Thereffqu.fin.|z.ABCrr ^z r- xfqu. Rad.(II.i64) 
 
 ^ * fin. ABxfin. BC 1 \ T/ 
 
 Now fuppofing Rad.zri, and L. to fland for logarithm. 
 
 Then 2l, fin. f Z.abc=:l. fin. |ac+ad + l, fin. ac ad l. fin. ab 
 
 L. fin. BC. (I. 90, 85, 86) 
 
 And putting /for the arithmetic complement of a logarithm. 
 
 /. fin. AB + /. fin. BC + L. fin. jac + ad + l. fin. |ac ad 
 
 Then l, fin. Jz. abc=: 
 
 * 2 
 
 That is, having determined which angle to find. 
 
 To the arithmetic complement of log. Jin. of one containing fide.. 
 
 Add the arithmetic complement of log. fin. of the other containing fide .^ 
 
 And the log.fm, of the \ fum of '^dficle atid difference of the containing fdes^ 
 
 Alfo the log. fm. of the \ difference of i^d fide and diff. of the containing fides., 
 
 Then the degrees anfwering to half the fwn of thefe four logarithms, found 
 
 among the fines , being doubled., will give the angle fought. 
 
 126. P R O B L^E M VI. 
 
 In the oblique angled fpheric triangle abc. F 
 
 Given the three angles a, e, c ; Requ. the fides. 
 
 To find the fide ab. 
 
 About the given angles as poles, defcribe arcs of 
 
 great circles meeting one another, and forming the 
 
 triangle fde. 
 
 Then are the fides of fde, the fupplements of the 
 
 angles a, b, c. (95) 
 
 Continue fd, fe, the fupplements of the angles b, 
 
 A, adjacent to the fide ae required, till they meet in G. 
 
 Then in the triangle dge, the fides gd, ge, arc the meafures of th 
 angles b and a, adjacent to the fide fought. 
 
 The fide de is the fupplementof Z.c oppofite the fide ab. 
 
 NowZ-G ( Z.F, by 31) is the fupplement of ab. 
 
 Therefore thcz.G being found in the triangle dgb byPaoB. V. (125) 
 will give the fupplement of the fide ab required. 
 
 That is., Let the given angles he taken as the fides of another triangle , 
 thferving to ufe the fupplement of that angle oppol.te to the fide required. 
 
 In this new triangle find l^hy Frob. V.) the angle oppofite to that fide xuhtrt 
 the fupplement is ufcd. 
 
 Then will the fuppkmcnt of the angle thui found he tie p.de required. 
 
 M 4 Table
 
 i6i 
 
 SPHERICS. 
 
 Pook IV. 
 
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 54 
 
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 165 
 
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 SECTION
 
 i66 
 
 SPHERICS. 
 
 Book IV. 
 
 SECTION VIII. 
 
 'The ConJlruSiion and numerical Solution of the 
 cafes of right angled fpheric Triangles, 
 
 156, Exam. I. In the right-angled fpheric triangle abc. 
 G.ven the hypoth. ac=64^ 40' J r j^^j ^he reft. 
 And one leg = 42 12 J 
 
 CONSTR UCTI ON S. 
 
 1 ft. To put the given leg on the primitive circle. 
 Defcribe the primitive circle, and draw the 
 right circle ac. 
 
 Apply the given leg (42'' ii') to the primitive 
 circle from b to c. 
 
 About c, as a pole, at a diftance equal to the 
 hypothenufe (64 40') defcribe (68) a fmall circle 
 aa^ cutting the right circle ad in A ; and draw^ the 
 right circle CD. 
 
 Thro'igh c, a, d, defcribe an oblique circle. 
 .And A3C is the triangle fought. 
 2d. To put the required leg en the primitive circle. 
 Defcribe the priiv.itive circle, and draw the right 
 r'yr\' CH ; on which lay the given leg (42 12') 
 -Tur, .0 c. _ (70) 
 
 About c, as a pole (66), at a ciftance equal to 
 the hypothenufe (644o'j defcribe a fmall circle 
 cutting the primitive in A ; and draw ad. 
 
 Through a, c, d, defcribe an oblique circle. 
 
 (II. 72) 
 Then ABC is the triangle required : Whofe fides and angles 
 fured by art. 70, 72. 
 
 Computation. 
 
 To find/^adj, the given leg 
 As Rad. =90 co' 
 
 To co-t. hyp. =^^4 40 
 So tan. gn. leg. ^=42 12 
 
 arc mea- 
 
 TofindjLoppof. the given leg. (127) 
 As fin. hyp. ^=64" 40' 0,04391 
 To R;'.d. "izzgo 00 10,00000 
 
 So fill. gn. leg ^=-\2 12 9,82719 
 
 To fia. op. Z. ~4S OQ 9,87110 
 
 This angle :s acute, becaufe it is to be 
 like cue given leg, which is acute. 
 To find the ether leg. (129) 
 
 As co-f. gn. leg ^=4^'' io' c, 13030 
 To Rad. =90 00 io,ODcoo 
 
 Soco-f. hyp. =64 40 9,63133 
 
 (128) 
 
 10,00000 
 
 9,67524 
 
 9.9574^ 
 
 To co-f.ri 
 
 54 43 9,-6163 
 
 Toco-f. adjz. =64 35 9,63272 
 
 This angle is acute, becaufe the hyp. 
 and given leg are of like kinds. 
 
 This leg is acute, becaufe the hyp. 
 arid given leg are of like kinds. 
 
 Note, In thefe operations, and in all the follov/ing ones, although the 
 
 word co-fmc, or co-tangent, h ulcd in the proportions, yet the degrees 
 
 * and r/iiniucs fet down, arc not the complements, but the real fides or angles. 
 
 157. Ex-
 
 Book IV. 
 
 SPHERICS. 
 
 167 
 
 157. Example II. In the right-angled fpheric triangle ABC. 
 
 Given the hyopth^AC = 64 40' 1 R i,^^ ^he reft. .^^ 
 
 And one angle acb =: 64 35 j ^ 
 
 Constructions. 
 ift. To put the leg adjacent to the given angle on the primitive circle. 
 
 Through any point c, in the primitive circle, 
 defcribe (75) the oblique circle cad, making with 
 the primitive circle the angle bca, equal to the 
 given angle 64 35'. 
 
 In the oblique circle cad, take CA equal to the 
 given hypothenufe 64"^ 40'. (70) 
 
 Through A defcribe the right circle AB. 
 
 And cab is the triangle required. 
 
 2d. To put tJpe leg oppofite the given angle on the primitive circle. 
 
 Having defcribed the primitive circle, and 
 drawn the right circle on ; 
 
 Defcribe (80) an oblique circle acd, cutting 
 the right circle ob in c, with the given angle 
 64 35^, and having the part AC intercepted be- 
 tween the right circle ob and the primitive 
 circle, equal to the given hypothenufe 64 
 40' ; 
 
 Then abc is the triangle required. 
 
 The fides required are meafured by art. 70. 
 
 And the required angle by art. 72, 
 
 Computation. 
 
 
 To find the leg opp. the gij. /L (^3^) 
 As Rad. rrgO 00' iO,oOoco 
 
 To fin. hyp. =64 40 
 So fin. given Z. =^64 35 
 
 To fin. op. leg =154 43 
 
 Like the given angle. 
 
 9,9,-609 
 99)579 
 
 To find the leg adj. the giv. Z. ( 1 3 1 )' 
 As Rad, =90 00' lo.ooooo 
 
 To tan. hyp. =64 40 10,32476 
 So co-f. givenZ. =64 35 9,63266 
 
 9,91 188 j To tan. adj. leg =42 12 9,95742 
 
 Acute, as the hypothenufe and given 
 ! angle are of like kind. 
 
 To fi)id the other angle. (132) 
 
 As Rad. 
 
 To co-f. hyp. 
 
 So tan. eiven angle 
 
 rrgo co 
 =64 40 
 
 = 64 35 
 
 To co-t. required ang.e =148 co 
 
 10,00000 
 
 9'63'33 
 10,32313 
 
 9,9^446 
 
 AnJ is acute, as the hypothnufe and given angle are of like kind. 
 
 M 4 158. Ex:^^
 
 i6S 
 
 SPHERICS. 
 
 Book IV. 
 
 158. 'Example III. In the right angled fpheric triangle abc. 
 Given one leg 
 And its opp. an 
 
 gle CAB = ^8 00 } Required the reft. 
 
 Constructions. 
 I ft. To put the rtqu'tred leg on the primitive circle, 
 
 Defcribe an oblique circle acd (75)> making 
 with the primitive circle the angle cab, equal to 
 the given angle 48 00'. 
 
 About the center o of the primitive circle de- 
 fcribe (67) a fmall circle at the diftance of the 
 complement of the given leg 42 12', cutting acd 
 in c. 
 
 Draw the right circle ocb, and acb is the tri- 
 angle fought. 
 
 2d. To put the given leg on the primitive circle. 
 
 Drav/ the right circle gab, and another oe at 
 right angles. 
 
 Make BC equal to the given leg 42 12' ; draw 
 the diameter cd, and another op at right angles. 
 
 About F, the pole of ab, defcribe a fmall circle 
 (68), at the diftance of the given angle 48" 00', 
 cutting OP in p. 
 
 About P, as a pole (62), defcribe the oblique 
 circle cad, cutting ab in a. Then cea' is the 
 triangle required. 
 
 The fides are meafured by art. 70, and the angles by art. 72. 
 
 Computation. 
 
 To find the hypothenufe. 
 Asfin.giv.Z. =48 00' 
 To fin. giv. leg =42 12 
 So Rad. =90 CO 
 
 Tohn.hyp. =164 40^ 
 
 (133) 
 o, 1 2893 
 9,827 19 
 
 lO.CCOOO 
 
 9,95612 
 
 And is eiiher acute or obtufe. 
 
 To find the other leg. ( 1 34 ) 
 
 As Rad. =90 00' lo.oocoo 
 
 To co-t. giv.Z. =148 00 9>95444 
 
 So tan. giv. leg =142 12 9,95748 
 
 To fm. req. leg =54 44 9,91192 
 
 And is either acute or obtufe. 
 
 To find the other angle. ( 135) 
 
 As co-f. given leg 
 To co-f. given Z. 
 So Rad. 
 
 r: 42 12' 
 =: 48 00 
 
 =r 90 00 
 
 To fin. required A =: 64. 
 And is either acute or obtufe. 
 
 35 
 
 0,13030 
 
 9,82551 
 
 J 0,00000 
 
 9,95581 
 
 159- E-^-
 
 Book. IV. 
 
 SPHERICS. 
 
 169 
 
 159. Example IV. In the Tight angled fpheric triangle abc. 
 Given a leg ab = 54 43^7 r, j .u n. 
 
 And its ad^ angle CAB=48 So j ^^"^^''^^ '^^ '^^' 
 
 Constructions. 
 ift. To put the given kg on the primitive circle. 
 
 Having defcribed the primitive, and right circle 
 OB ; 
 
 Make ba equal to the given leg 54 43' 
 
 Dravir the diameter ad. 
 
 Through a defcribe the oblique circle acd (75) 
 making with the primitive the given angle bac 
 48" 00', cutting OB in c. 
 
 Then is acb the triangle required. 
 
 2d, 71? put the required leg on the primitive. 
 
 In the right circle ob, take (71) ab, equal 
 to the given leg 54 43'. 
 
 Through the point a, defcribe (76) the ob- 
 lique circle cad, making with ab the angle 
 BAC, equal to the given angle 48"^ oo', cutting 
 the primitive circle in c. 
 
 Then is abc the triangle fought. 
 
 The fides required are meafured by art. 70. 
 
 And tbe required angle by art. 72. 
 
 Computation. 
 
 To find the other angle. 
 As Rad. irgo" 00' 
 
 Toco-f.giv. Iegrz54 43 
 Sofin.givenZ. =:48 co 
 
 {136) 
 
 10,00000 
 9,76164 
 9,87107 
 
 To co-f. req./. :=z6\ 35 9,63271 
 
 And is like the given angle. 
 
 To find the other leg. C ^ 3 7 ) 
 
 As Rad. ==90 00' xo,ooooo 
 
 To fin. giv. leg =54 43 9,91185 
 
 Sotan.giv. Z. =48 00 10,04556 
 
 To tan. req. leg ^342 12 9,95741 
 
 And is like the given leg. 
 
 To find the hjpothenufe. ( 1 38) 
 
 As Rad. 
 
 To co-f. given Z. 
 So co-t. given leg 
 
 = 90 
 
 00 
 
 I0>00000 
 
 = 48 
 
 00 
 
 9,82551 
 
 = 5 + 
 
 43 
 
 9,84979 
 
 To co-t. hypoth. zz 64 40 
 
 9,67530 
 
 And is acute, as the given leg and angle are of a like kind. 
 
 160, Ex-
 
 lyo 
 
 SPHERICS. 
 
 Book ly. 
 
 160. Example V. In the right angled fpheric triangle abc. 
 Given one leg ^^ = S4- 4-3'^ 1 v> j .u n. 
 And the other leg ^c=ll A P^q"'^^^ the reft. 
 
 Construction. 
 To put either leg on the primitive circle, 
 
 Defcribe the primitive circle, and draw the 
 right circle ob. 
 
 Then, let the given legs 54 43', and 42 12', 
 be applied, one from b to a, and the other from b 
 to c (74) ; and draw the diameter ad. 
 
 Through the points a, c, d, defcribe an oblique 
 circle, (II. 72) 
 
 Then is abc the triangle required. 
 
 The angles a and c may be meafured by art. 72. 
 
 And the hypothcnufe ac by art. 70. 
 
 Computation. 
 
 To find the angle a. ( i 39 ) 
 
 As Radius - rr 90 00' 10,00000 
 
 To fin. of leg ab = 54 43 9,91185 
 
 So co-t. other leg bc r= 42 12 10,04251 
 
 To co-t. op. angle a rr 48 00 9,95436 
 And is acute, as the oppofite leg cb is acute. 
 
 To find the angle c . (139) 
 
 As Radius ' rz 90 00' io,oooco 
 
 To fin. of leg cb =: 42 12 9,82719 
 
 So co-t. other leg A B ::z. 54 43 9,84979 
 
 To co-t. op. angle c r= 64 35 9,67698 
 
 And is acute, becaufs the oppofite leg ab is acute. 
 
 To find the hypothennfe AC. (140) 
 
 As Radius r: 90 00' io,oocoo 
 
 To co-f either leg AB = 54 43 9,76164 
 
 So co-f. other leg cb zz. 42 12 9,86970 
 
 To co-f. hypoth. ac 
 
 64 40 
 
 9,63134 
 
 And is acute, as the legs are of the fame kind. 
 
 161. Ex-
 
 Book IV. 
 
 SPHERICS. 
 
 i6i. Example VI. In the right angled fpheric triangle ABC. 
 Given one angle a =48 00' 1 p j ^l n 
 And the other angle 0=64 35 } ^'^"^''^^ '^^ ''^' 
 
 Const r y c t i o n. 
 
 T'o put either leg, as CB, on the primitive circle. 
 
 Having defcribed the primitive circle, and 
 drawn the right circle ob ; 
 
 Then (81) defcribe the oblique great circle 
 CAD, cutting the primitive circle in the given 
 angle c, and the right circle ob in the given 
 angle a. 
 
 The fides are to be meafured by art. 70. 
 
 Computation. 
 To find the leg ce. ( 141 ) 
 
 As fin.Z. adj. req. leg 
 
 c 
 
 = 640 35 
 
 0,04421 
 
 To Radius 
 
 
 :=. 90 00 
 
 lOjOCOOO 
 
 So co-f. other angle 
 
 A 
 
 = 48 CO 
 
 9,82551 
 
 To co-f. of its op. leg cb =42 12 
 
 9,86972 
 
 And is acute, becaufe the oppofite angle is acute. 
 
 To find the leg ab. ( 141 ) 
 
 As fin.Z. adj. req. leg a = 48 00' 0,12893 
 
 To Radius ' zz 90 00 10,00000 
 
 So co-f. Other angle c 1= 64 35 9.63266 
 
 To co-f. of its op. leg ab zz 54 43 976i59 
 
 And is acute, becaufe the oppofite Z. is acute. 
 To find the hypothemife AC (142) 
 
 As Radius zz 90 00' 
 
 To co-t. either angle as a zz 48 go 
 So co-t. other angle as c zz 64 35 
 
 To co-f. hypoth. 
 
 AC =: 64 40 
 
 10,00000 
 
 9'95444- 
 9,67687 
 
 9.63131 
 
 171 
 
 And is acute, becaufe the angles are both acute, or like. 
 
 162, Ex-
 
 172 
 
 SPHERICS. 
 
 Book IV. 
 
 162. Example VII. In the quadrantal triangle abc. 
 Given the quadrantal fide AC = 90 00' 1 
 
 an adjacent angle a r: 42 12 > Required the lefl. 
 And the oppofitc angle b r: 64 40 j 
 
 Construction. 
 To put the quadrantal ^de on the primitive circle. 
 
 Having defcribed the primitive circle, and 
 drawn the diameters ai>, .bc, at light angles ; 
 
 Dcfcribe the oblique circle abd, making with 
 AC an angle of 42 12% (75) 
 
 Through c defcribe a great circle cbe, cut- 
 ting the circle abd in an A. of 64 40'. (74) 
 
 Then is abc the triangle fought. 
 
 The angle c is to be meafured by art. 72. 
 
 And the fides ab, cb, are meafured by art. 70. 
 
 C o M P U.T AT ION. 
 
 Imagine the given triangle abc to be changed into a right angled 
 triangle, where the fupplement of the angle b is to reprcfent the hypothe- 
 nufe, and the angle a to be one of the legs. 
 
 Then v/ill the folution fall under art. 127, 128, 129, in the table j and 
 the numerical computations will be the fame as in Example I. Obferving 
 that the angles there found are, in this example, the meafures of the fides 
 AB, CB ; -and the fide ab in that example ftands for tlie angle c in this. 
 
 Now in determining the value of the parts of this triangle, as they 
 arife in the computation, the words like and unlike are to be changed one 
 for the other, where the hypothenuTc is concerned in the determination : 
 Thus the leg ab is taken acute, bccaufe the fupplement of the angle 
 oppofite to the quadrantal fide, which is here ufed as the hypothenufc, 
 is unlike the other given angle ; and its oppofite angle c is to be acute 
 for the fame rcafon : But, the kind of the fide nc being known by the 
 kind of its oppofite angle a, it muft be taken acute, as the oppofite angle 
 is acute. 
 
 In the confiructlon there arifes two triangles, either of which v/ill an- 
 fwer the conditions in the example For the finull circle defcribed about 
 p, the pole of the oblique circle abd, cuts the diameter ad in the points, 
 rt, b ; and either of thefc points may be taken for the pole of the oblique 
 circle wanting to complete the triangle. 
 
 Now if a he taken for the pole, then in the triangle abc, the meafure 
 of the things fought, will be equal to thofe arifing from the computation : 
 But the angle b is the fiipi^'lemcnt of what was given. 
 
 And if b is taken for the pole ; tlien the triangle abc will arife from 
 the' conftruction ; wherein the angles A and b are refpedtively equal to 
 what is propounded : But then the fide ab, and the angle c, will both 
 be obtulc. 
 
 SECTION
 
 Book IV. 
 
 SPHERICS. 
 
 I7J 
 
 S E C T I O N IX. ' 
 
 The ConJlruEiion and nmnerical Solution of the 
 cafes of oblique angled fph eric Triangles, 
 
 163. Example I. In the oblique angled fpheric triangle abc. 
 Given the fide ab = 114 30^1 
 
 the fide BC = 56 40 1- Required the reft. 
 
 And an angle oppofite to one fide, RCA =125 20 j 
 Construction. 
 To put the given fide^ adjacent to the known angle, on the primitive circle. 
 
 Defcribe the primitive circle, and draw the 
 diameter bd. 
 
 Make BC equal to the fide adjacent to the given 
 angle = 56 40^ (70) 
 
 Defcribe the great circle CAE, making the angle Df 
 BCA equal to the given one, =1125 2.0'. (75) 
 
 Through B defcribe a great circle bad, cutting 
 AE in A, at the diftance of ab, the other given 
 fide from B,= ii4 30. (68) 
 
 Then ABC is the triangle fought. 
 
 And the parts required are meafured by art. 70, 72. 
 
 Computation. 
 To find the angle A, oppofite to the other given fide. (144) 
 As fin. one fide abii: i 14 30' 0,04098 "J 
 
 1 o iin. op. Z. cm25 20 9,91158 / Which may be either acute or ob- 
 So fin. oth. fide cb= 56 40 9,92194 Mufe from the things given : But the 
 
 Iconftrudion fhevvs it to be acute. 
 
 To fin. op. /L. A=: 4S 30 9,87450 J 
 
 To find the angle B between the given fides. (145) 
 As Rad. rr 9oco' 10,00000 
 
 To tan. giv. Z. c=:i25 20 10,14941 
 Soco-f. adj fid.Bc^^: 56 40 9>73'y97 
 
 To co-t. m 
 
 = 127 47 9,88938 
 
 And is obtufe, as the given angle 
 and its given adjacent fide are unlike. 
 
 Asco-t.S.ad.g./.ncrr 56^" 40' 0,1 8 197 
 Toco-t.oth.fide ABz::i 14. 30 9,65870 
 So co-f. m ZZ127 47 9,78723 
 
 To co-f. n 
 
 = 64 53 9,62790 
 
 Which^s acute, being unlike fide op- 
 pofite given Z., that 2. being obtufe. 
 
 Then as tiie given fides arc unlike, the difF. of m and n, or 62" 54'= Z.B. 
 To find the other fide AC. ( 1 4^ ) 
 
 As Rad. rr go'' co' io,oocoo 
 
 To co-f giv. Z. cm 25 20 9,76218 
 So tan. adj.fid. Bt := 56 40 10,18197 
 
 To tan. M 
 
 = 138 40 9,94415 
 
 And is obtufe, as ^c and cb are unl. 
 
 As co-rS.ad.g.Z.Bcr:56''4o'o,26oo2 
 Toco-foth. fide ABr=:ii4 30 9,61773 
 So co-f. M ==138 40 9,87557 
 
 To co-f N 
 
 == 55 299,75332 
 
 And is acute, being unl. a n as above. 
 
 Then as uc and cA) are unlike the diff. of m and k, or li" 1 1 :=:ac. 
 
 164, Ex-
 
 174 
 
 SPHERICS. 
 
 Book IV. 
 
 164. Example II. In the oblique angled fpheric triangle abc. 
 Given the angle BAcr: 48 3c/ 1 
 
 the angle bca = i25 20 > Required the reft. 
 
 And the fide oppofite to one angle, ab = ii4 30 J 
 
 Construction. 
 To put the given Jide Ah on the primitive circle, 
 
 Defcribe the primitive circle ; draw the diame- 
 ter da ; and through a defcribe the great circle 
 ACD, making the given angle bac=48 30^. (75) 
 
 Make the arc ab equal to the given fide:=: 
 114" 30' (70) J and draw the diameter be. ^ 
 
 Through b, defcribe the great circle bce, 
 cutting ACD in an angle equal to the given angle 
 bca = i25 ?.o'. (78) 
 
 Then is acb the triangle fought. 
 
 And the parts required are to be meafured by art. 70, 72 
 
 Computation. 
 
 To find the fide oppofite the other given angle, ( 147 ) 
 
 As fin. one Z. en 125 20' 0,08842 
 To fin. op. fide AB=: 1 14 30 9,95902 
 So fin. other Z. a= 48 30 9,87146 
 
 To fin. op. fide bc=: 56 40 9,92190, 
 
 Which may either be acute or ob- 
 tufe from what is given. Butthecon- 
 ftrudion Ihews it to be acute. 
 
 To find the fide AC between the given angles. (148) 
 
 As Rad. :=9o'* 00' 10,00000 
 
 Toco-f Z.ad.g.S,A=48 30 9,82126 
 So tan. gn. S. ABrzi 14 50 10,34130 
 
 To tan. 
 
 r=:i24 31 10,16256 
 
 And isobtufe, being unlike Z.A, as 
 AB is greater than 90 
 
 Asco-t.Z-ad.g.S.Ar: 48 30' 0,05319 
 To co-t. other Z. cm 25 20 9,85059 
 So fine M =:i24 31 9,91591 
 
 To fine N 
 
 =: 41 19 9,81969 
 
 Which may be either acute or ob- 
 tufe; either 41 20' or 138 4c'. 
 
 Then as the given angles are unlike, the difference of m and n, or 83 12', 
 is the fide ac. Or the fum of 138** 41', and 124 31', lefTcned by 180", 
 leaves 83"^ 12'. 
 
 71? find the other angle abc. ( H9) 
 
 As Rad. =90oo' io,oooco 
 
 Totan.Z-ad.g.S. An:48 30 10,05319 
 So co-f. gn. S. ab:=i 14 30 9.61773 
 
 To co-t. m 
 
 1 15 07 9,67092 
 
 And is obtufe, being unlike Z.A, 
 as its adj. fide ab is greater than 90' 
 
 Asco-f.Z.ad.g.S.A= 483o'o,i7874 
 To co-f. other Z. cm25 209,76218 
 
 So fine m 
 To fine n 
 
 =:ji5 07 9,95686 
 = 52 13 9,89778 
 
 Which may be either acute or ob- 
 tufe, viz. 52 13', or 127"^ 47' 
 
 Then as the given angles are unlike, the difference of m and n, or 62" 54 , 
 is the angle b required. Or the fum of 1 15*' 07', and 127 47', leffened by 
 1 80, leaves 62* 54', 
 
 8 165. Ex-
 
 Book IV. 
 
 SPHERICS. 
 
 165. Example III. In the oblique angled fpheric triangle abc 
 Given the fide ab = 114" 30'! 
 
 the fide Bc = 56 40 > Required the reft. 
 
 And the contained angle abc = 62 54 3 
 
 Construction. 
 
 To put either sf the given fides ^ as bc, on the primitive circU. 
 Defcribe the primitive circle ; drav/ the dia- 
 meter BD ; and through B defcribe a great circle 
 BAD, making the given angle abc =62 54'. 
 
 (75) D 
 On the circles bcd, bad, take the arcs bc, 
 
 BA, refpeclively equal to the given fides, viz. 
 
 Bc = 56 40', and ba=:ii4 30^ (70) 
 
 Draw the diameter ce, and through c, A, E, 
 defcribe the great circle cae ; then abc is the triangle fought. 
 
 The required parts of abc are meafured by art. yo, 72. 
 
 Computation. 
 
 To find the angle c. (150) 
 As Rad. r= 90^00' 10,00000 
 
 Toco-f. j^ivenZ-Bn 62 54 9,65853 
 So t.S.op.re.Z. AB=:i 14. 30 10,34130 
 
 175 
 
 To tan. M 
 
 = 135 CI 9,99983 
 
 Obtufe, being like fide op, req. Z., 
 the given angle being acute. 
 
 Take the difference between m and 
 BC, and it is 78^ 21'; call it n. 
 
 As fir.e N rz 1"^ z\ 0,00904 
 
 To fine M ^='35 o' 98493^ 
 
 So tan. given Z.Brr 62 54 10,29096 
 
 To tan. req. Z- crzi25 20 10,14936 
 
 And is obtufe, being unlike the given 
 
 >angle, becaufe m is greater than bc, 
 
 the fide adjacent to the required angle. 
 
 To find the angle a. (150) 
 As Rad. =:90"oo' 10,00000 " 
 
 To co-f. given Z. B=rj2 54 9,65853 
 So t. S. op. re.Z.ucrr56 40 10,18197 
 
 Acute, being like fide op. req. /., 
 the given angle being acute. 
 
 Take the difference between m and 
 BA, and it is 79" 48' j call it n. 
 
 To tan, of M :z:34 42 9,84050 
 
 As fme N zr7948' 0,00692 '^ 
 
 To fine M ^^3+ 42 9.75533 (And Is acute, being like the given 
 
 So tan, given Z. bz=:62 54 10,29096 >angle, as m is lefs than ab, the fid 
 
 . I adjacent to the required angle. 
 
 To tan. req. Z. Arr48 30 10,05321 J 
 
 __-____ ___^__.^- ^ 
 
 To find the other fide KQ. (151) 
 
 As Rad. =: go^oo' 10,00000 
 
 To co-f. given Z.B~ 62 54 9,65853 
 So tan.eith. S. AB:rzii4 30 10,34130 
 
 To tan. M 
 
 = 135 01 9,9993 
 
 Obtufe, being like a r, the fide ufed, 
 becaufe the given angle is acute. 
 
 The tliff. of M and bc, or 78* 21' n. 
 
 As co-f. M r:i35oi'o, 15039 
 
 To co-f N r= 78 21 9,30521 
 
 So co-f. S. ufed ABni 14 30 9,61775 
 
 Toco-f S. req. Acn 83 12 9,07533 
 
 And is acute, being like n, becaufe 
 he given angle ii acute. 
 
 i66. Ex-
 
 iy6 
 
 SPHERICS. 
 
 Book IV. 
 
 i66. Example IV. In the obHque angled fpheric triangle abc. 
 Given the angle bca =r 125 20' 1 
 
 the angle bac = 48 30 > Required the reft. 
 
 And the included fide ac = 83 12 j 
 
 Construction. 
 To put the given fide on the primitive circle. 
 Defcribe the primitive circle ; draw the diame- 
 ter AD J and through a defcribe the great circle 
 abd, making the given Z. bag =48 30^ (75) 
 Make ac equal to the given fide r: 83 ii'. 
 
 Draw the diameter ce, and through c defcribe 
 the great circle cbe, making the given angle 
 BCA=i25 20' (75), cutting abd in b. 
 
 Then is abc the triangle fought. 
 
 And the parts required are meafured by art. 70, 72. 
 
 Computation. 
 To find the fide ab. (152) 
 As Rad. r: 90^00' 10,00000 "1 
 
 Toco-f.gn.fideAC= 83 12 9,07337 lObtufe, being like Z. op. fide req. the 
 So ta.Z.op, r.S. c:=i25 20 10,14941 f given fide being acute. 
 
 I Take the diff. between m and Z. 
 To co-t. M n 99 29 9,22278 I A, and it is 50 59'; call ic n. 
 
 As co-f. n =: 50'^ 59' 0,20097"^ 
 
 To co-f. m =: 99 29 9,21615 I And is obtufe, being unlike n, be- 
 
 So tan. gn. fide ac= 83 12 10,92357 >caufe the angle oppofite to the fide 
 
 1 required is obtufe. 
 
 Totan.req.fid.ABn: f i43o' 10,34139 J 
 
 To find the fide ^C' (^52) 
 As Rad. rz:gooo' 10,00000 "1 
 
 Toco-f.gn.fid. Ac=:83 12 9,07337 I Acute, being like Z. op. fide re- 
 So tan.Z.op.r.S. A:r:48 30 10,05319 I quired, the given fide being acute. 
 
 ' f Take the difF. between m and Z. 
 
 To co-t. of m rr82 22^ 9, 12656 i c, and it is 42" 57'i; call it n. 
 
 As co-f n 4257'i 0,13558 
 
 To co-f. m =:82 22^ 9,12283 
 
 So tan. gn. fid. Acr=83 12 10,92357 
 
 To tan.req.fid. Bcrr56 40 io,i8i( 
 
 And is acute, being like n, becaufe 
 >the angle a oppofite to bc, the fide 
 required, is acute. 
 
 As Rad. =: 90^00' 10,00000 
 
 To co-f.gn.fid.AC=z 83 12 9,07337 
 So tan. either Zc= 1 25 20 10,14941 
 
 To find the other angle B. (i53) 
 
 To co-t. m 
 
 rr 99 29 9,22278 
 
 Obtufe, being like Z c here ufed, 
 becaufe the given fide is acute. 
 
 Take difference of m and Z a, 'viz. 
 42^ 57' J and call it n. 
 
 As fine m r: 99 29' 0,00598 
 
 To fine n r: 50 59 9,89040 
 
 So co-f. Z ufed, c=:i25 20 9,76218 
 
 To co-f. req. Z b= 62 54 9,65856 
 
 And is acute, being unlike the angle 
 c here ufed, as m is greater than the 
 other angle a, 
 
 167. Ex-
 
 ^ook IV. 
 
 SPHERICS. 
 
 177 
 
 167. Example V. In the oblique angled fpheric triangle abc. 
 
 Given the fide ab =: 1 14 30' 1 
 
 the fide ac =: 83 13 > Required the reft, 
 the fide bc = 56 40 3 
 
 Construction. 
 
 *ro put either Jide, as ac, on the prhnitive circle. 
 
 Defcribe the primitive circle, and from any 
 ^oint in the circumference, as a, fet off one of 
 the given fides, as AC,r:83 13' (70) j and draw 
 the diameters ad, ce. 
 
 About c, as a pole, and at a diflance equal to 
 the given fide bc, =r56 40', defcribe a final! 
 circle rrs. (68) 
 
 About A, as a pole, and at a diftance equal to 
 the given fide ab (when ab is lefs than 90^ de- 
 fcribe another fin.Jl circle we (68), cutting the former in B : But when 
 the fide, as AB,=rii4 30', is greater than 90; then about d, the oppo- 
 fite pole to A, defcribe a fmall circle with the fupplement of ab, as #iB, 
 cutting the former fmall circle ?zb in b. 
 
 Thro' the points A, i, d, and c, b, e, defcribe the great circles abd, cbe. 
 Then is abc the triangle fought, and the angles are meafured by art. 72. 
 
 C O M i* U T A T I O N. 
 
 To find the angle c. ( 1 54) 
 Ar. Co fine e 
 
 Here ac=:e~ 83 13 
 cb=:f=: 56 40 
 
 E- Frrorr 26 
 ABi=G = i 14 
 
 33 
 30 
 
 + D= 141 03 
 
 87 
 
 70''3il=ifum 
 43 58^=idifF. 
 
 Ar. Co. fine F 
 Sine f lum 
 Sine I diff. 
 
 =:83 
 = 56 
 = 70 
 =43 
 
 0,00305 
 
 0,07806 
 
 13 
 40 
 
 3'i 9'9744i 
 58;- 9,8458 
 
 Sum of the four Log, - 19,89710 
 I fum is fin. of 62 39I' - 9,94855 
 Which doubled gives 12^ 19'rrZ.c. 
 
 Here AB"E 1 14 30' 
 AC Kr= 83 13 
 
 To f.nd th^ mwle A. ( ? 54) 
 
 i. Frrn: 
 Bcr:::G: 
 
 31 17 
 56 40 
 
 G + D= 87 57 
 
 c Drr 25 23 
 
 43-58'^=ifum 
 12 i,x\-\m. 
 
 Ar. Co-, fine v. =:ii4 30' 0,04098 
 
 Ar. Co. fine f := H3 13 0,00305 
 
 Sine I fum rz 43 ^%\ 9,^4158 
 
 Sins J diff. := 12 41 1 9,34184 
 
 Sum of four Log. - - 19,22745 
 
 \ fum is fin^ of 24 15'^ - 9,61372 
 
 y 
 
 Which doubled pivp<; 4^' 31'rrZ.A. 
 
 To f.nd the angle v.. ( I 54) 
 Here ab=:e 114 30' 1 Ar. Co. ;.ne h m 4' 30' 0,04098 
 
 RcrziFr^ 56 40 
 
 S F = n 57 50 
 
 At G 83 13 
 
 Ar. wO. fine f m ,0 40 0,0- S06 
 Sine ^- ium 70 3i- 9,97441 
 
 Sine -J diff. =z 12 4IV 9,34184 
 
 G + Dm.^i o 
 
 '3' 
 
 i^fum 
 
 Sum of four Lop-. 
 
 J 9.43 5 29 
 
 J fum Is fin. of V 28' - 9,71764 
 C-^D~ ?5" 23! 12 41 ^ ' (liiF. , 'A'hich t'.ouiicd j^iwi 62" 56'=::Z.b.
 
 le ABC. 
 
 178 SPHERICS. Book IV. 
 
 168. Example VI. In the oblique angled fphcrlc triangle 
 Given the angle A = 48 31' 7 
 
 the angle b =: 62 52 ^-Required the reft. 
 
 the angle c =125 20 3 
 
 Construction. 
 jTo put either two angles, as c and b, at the primitive, 
 
 Defcribe the primitive circle, draw the diameters 
 CD and EF at right angles to one another; and thro' 
 C defcribe a great circle cad, making the angle 
 BCA equal to the given angle cm 25 20'. (75) 
 
 Defcribe a great circle bag, cutting the given 
 great circles cfd, cad, in the given angles Brr 
 62 52^ and A =48 31'. (81) 
 
 Then is abc the triangle fought. 
 
 Where the Jides are meafured by art. 70. 
 
 Here 2113 ~E=: 62 5 2' 
 
 E FrzDzr 14 21 
 Sup. Z.c=:g=: 54 40 
 
 Computation. 
 
 To find the fide ab. (155) 
 
 Ar. Co. fine e 
 Ar. Co. fine f 
 Sine I fum 
 Sine I difF. 
 
 C + D= 69 O 
 
 -Dz: 40 15 
 
 j43o|'=r|fum 
 
 lo C9|=:|difF. 
 
 = 62''52' 0,05064 
 = 48 31 0,12543 
 = 3+ 30I 9.7532= 
 = 20 09! 9.53733 
 
 Sum of the four Log. 19,46662 
 
 I fum is fin. of 32' 45I;' 9.7333 
 The fup. of its double is i I4''29'::r ab. 
 
 HereZ. c = e =z 1 2 5 20' 
 jL:\:=.v=- 48 31 
 
 E Frrorr 76 49 
 Sup.iiBrrc.:r:ii7 08 
 
 To find the fide AC (155) 
 
 G + D = i93 57 
 G D=: 40 19 
 
 965Sl-'=:|fum 
 
 20 09! I difF. 
 
 Ar. Co. fine e ==125" 20' 0,08842 
 
 Ar. Co. fine f ~ 48 31 0,12543 
 
 Sine I fum 96 58^ 9,99677 
 
 Sine i difF. == 20 09I 9,53733 
 
 Sum of four Log. 
 
 ^9'7^79i 
 
 X fum Is fin. of 48 25!' 9.87397 
 The fup. of its double is 83 09'rrcA. 
 
 Here 2.0 = 8 = 125 20' 
 Z.Br=F=: 62 52 
 
 E rrror: 6 28 
 Sup. A A =0 = 131 29 
 
 To find the fide bc. (155) 
 Ar. Co. fine e 
 Ar. Co. fine v 
 Sine Y fum 
 Sine I difF. 
 
 c + D = i93 57 
 
 C D= 69 01 
 
 96^ 5 8 ['=f fum 
 
 34 3c|=Hiff. 
 
 Sum of four Log. 
 
 = 125^20' 0,08142 
 rz 62 52 0,05064 
 = 96 58|- 9,99677 
 = 34 30? 9.75322 
 
 19,88905 
 
 I fum is the fin. of 61 39' 9,94452 
 The fup. of its double is 56 42^ = 8 c. 
 
 S F. C T I o >r
 
 n 
 
 Book IV. 
 
 SPHERICS. 
 
 179 
 
 -SECTION X. 
 
 Ht .Z'j - 
 
 169. The principles already delivered have been fliewn fufficient for 
 deriving methods for the folution of all the cafes in fpherical .Trigono- 
 metry : yet as there are many other ufeful and curious particulars which 
 appertain to the fubjeift, it was thought proper to add fomc of them for 
 the entertainment of fpccubtive readers. The chief q( thefe relations- 
 cannot, perhaps, be better inveftigated, than by imitating the method of 
 the late William Jones, Efq. who publifhed in the year 1747, in the 
 Philofophical Tranfadlions, N 483, fome properties of Goniometrical 
 lines ; which properties are moftly derived from a general figure which 
 Mr. Jones improved from one communicated to him by the great Dr.' 
 Hallcy. See Synopfis Palmar iorum Mathejeosy p. 245. 
 
 Let AB, AD ; or a^, a^/, be any two arcs, each Icfs than QO di'grees, 
 B' and nE, or ^e and he^ be the fum and difference of their riglu fines. 
 K and DE, tlic fum and difference of their co-f lies. 
 
 The arcs c*/, bd ; or bo, hd; exprefs the fum and difference of the arcs 
 AB, AD. </o, DL, arc fines of the arcs Be/, no, the fum and difference of 
 arcs AB, AD : lio, El,, the verfed fines of that linn aiui difference. 
 o, aL:si./^ the vcjfcd fines of the fupplenients of tlicir 'i\w\\ and ui.'l. 
 
 N 2 Lst
 
 ff 
 
 ,80 SPHERICS. Book IV 
 
 Let the arcs b/, b^, be the half fum, and half diff. of the arcs ab, aD 
 BF, BG, the fines 
 CF, CG, the co-fines 
 
 c".: l"', ZTd^T iof thehalf fum andhalf diff. of ab and ad. 
 
 B</, BD, twice the fines 
 
 KB, K^, twice the co- fines 
 
 PB, PC, the co-tangent and the co-fecant of the half fum of the arcs ab, ad. 
 
 Now the following fet of triangles being fimilar, 
 
 VtZ. CBG, B^^, KBE, Khl, DBL, CHB, BHG, Kdb, 
 
 _, CB __B^__KB_K^__BD CH__^BH__K^ 
 CG~'b^~KE~'k/""dl'~'cB '~BO~K^ 
 CB_^Brf_KB__KZ'__BD__CH BH _K^ 
 BG~ J^ ~BE "" ^/ "" Bl"~'bH~"hG~~ ^/^ 
 CG__B^__KE__K/_DL CB__BG___K^ 
 
 s,G'~'de "~BE ~'bl """bl ""bh'~hg'~^^ * 
 
 The following fet of triangles being alfo fimilar,- 
 
 viz. CBF, BDE, kZ'E, Z^/o, ^BO, GIB, BIF, PCB, PIC, K^B. 
 
 There will refult, 
 
 CB_BD__K^ __'^d _Bd _^Cl __BI __PC__PI __YLd 
 CF~^BE~'KE~Zo"~io~"cB BF~"pb"~PC~ EK. 
 CB BD V^h xd B</___CI_BI_CP_PI_^K 
 
 bf'~"de~eZ'"~^o~bo~bi~if~*cb~'ci~^^b 
 
 CF BE KE_ZO '5?0_CB_BF_PB__PC_BIC 
 
 bf~'pe~ ^ ""(^cj '~bo~'bi ~"if ""cb~ci ~B^. 
 
 170, Nor
 
 (AVi- 
 
 SPHERICS. 
 
 i8i 
 
 SfQ. Now the feveral values of the radius cb being colle<Sed, are placed in 
 nnexed table; where the letters s, t,f, v, ftand for the fine, tangent, fecant, 
 A fine ; and the letters /, t\ /, the co-fine, co-tangent, co-fecant, of the arcs 
 a J or of the arcs |^?+^, ^a^ j and v\ the verfed fine of the fupplempnt. 
 
 (175) 
 
 Goniometrical Properties. 
 
 
 , J-i-s,a 
 
 (179) 
 
 
 (72: 
 
 s\J+s\a 
 
 , J ;: :; >^ 'ji^a. 
 
 s, A s^ a 
 
 (176) 
 
 j-X/, k/l^a. 
 
 s^,A-{-s\a 
 
 s, J+Sy a 
 
 (iSo) 
 
 (1S3) 
 
 /, a /, A , 
 
 : ^ ---,y^ty\A^a 
 
 J, v^ .f, ' 
 
 (184) 
 
 s\a[\A , J 
 
 ^ X J ) 2, A a. 
 
 S ixA+a 
 
 S, -iA^u 
 
 (187) 
 
 X U 1 35+7. 
 
 (191) 
 
 
 X/", ^.^+^. 
 
 <-jXt,l:A- 
 
 2s , i^/ a 
 
 ^,7/+J, 
 
 (173) 
 
 X J, -f 35+7. 
 
 (177) 
 
 s,J+s,a ^ 
 
 Z ^ i - X/j z^+<- 
 
 {181) 
 
 2 f, -iA^a 
 
 (188) 
 
 ^, V .^ a 
 
 X /, \a^. 
 
 \ j\y/ 
 
 r-vXJ, l^-h. 
 
 . ^ (174) 
 
 
 25,A^y_a 
 
 (182) 
 
 2J, l^^ , 
 
 X S, , f J+a. 
 
 J, A .r,rt 
 
 
 2J, tv^ J 
 
 ti86) 
 
 s^A--s^a . 
 
 1==- x/,|.^+^. 
 
 25, T_da 
 
 , (189) 
 
 r=rr- X J, l^+fl. 
 
 5, y^+a 
 
 (19=^) 
 
 -rrrr X /, kAa. 
 
 2.f, j,4+^ 
 
 (igsi* 
 
 ^, ^+1 
 
 y,.i+a 
 
 X /, \ A+a. 
 
 J, ^ a 
 
 (199; 
 
 ^-^.7-t-a ^, 
 
 -r r: X/,{A^a. 
 JizA+a 
 
 (103) 
 
 (IC7) 
 
 j->'iA+a 
 
 (196) 
 
 VyA a 
 
 Xt,{A-3. 
 
 (200) 
 
 -4^x/, -^2=7. 
 
 i-, -.A a 
 
 (204: 
 
 -^X/fvT 
 
 y^yi + a 
 
 (193) 
 X S-) iA-^-a. 
 
 (97 
 
 2s\lA+d . J 
 
 . , ^XS,{A+a. 
 
 (201) 
 
 
 (190) 
 
 2Sy\A-^ , 
 
 =r-XS,lAa. 
 
 s, A a 
 
 (194) 
 
 2S,iA- 
 
 VyAa 
 
 XSyiAa. 
 
 , (198) 
 
 2SylAa . J 
 
 y ,^ <! 
 
 (105) 
 
 (208) 
 
 ; c 1 -; X /, lA^a. 
 
 I ' 11 A a 
 
 j,v,^+,. 
 
 r- ^ /vi-^+'. 
 
 (209' 
 
 Sj_+5 
 
 X/ + /,iW+^ 
 
 (i02) 
 
 xy;^w+.. 
 
 (^-06) 
 
 - xf,l-j+.. 
 
 (2I0)f 
 
 s\lAa -r-, 1 
 
 r-^=x/ + ^,iW+^. 
 
 /a:-i+ 
 
 N 3 
 
 From
 
 iZi 
 
 S P li S'k I C S. Book IV. 
 
 <i-> 
 
 ,tih Wj , t From the preceding table a very great number of properties are readily 
 ^if' ;i; deduced j fomc of which are here annexed, as examples ot its ufe; where 
 V -injilogies are, in general, exprefled by equal ratios, 
 
 rj., fum of the fines of two arcs __tan. of half the fum of thofe arcs 
 |, * ' difF. ox the lines of thoic arcs~taii. of half the diff. of the arcs* . 
 
 I (i7i> 172) 
 
 , ' ^, funi of the co-fin. of two arcs _ CO- tan. of half the fum of the arcs 
 
 h"/ ' ditf.of the co-fin.ofthoie arcs"" tan. of halt the diff. of the arcs' 
 
 ' .^.- ' (175,180) 
 
 ' / 
 
 _ fineofthe furn of two arcs__rum of the tan. of thofe arcs 
 
 '^' fine of the diff. of thofe arcs ~ diff. of the tan. of thole arcs* 
 
 _ 5,A + J,^ ^iaT^, . - .S,A-{-S,A + S,A S^a_ /,f A +a -f /, | A J 
 
 For = j-=l2li.) And ; T~r= TIT^^ 
 
 S^A-S,a t,iAa^ S,A-{-S,aS,A-i-SjA if,i-A + Ma 
 
 by Compolition. 
 
 Here the arcs a, ^, arc the fum and diff'. of the arcs ^A+at I a a, 
 
 _, cof ofthc fum of two arcs^diff. of tan, of one and cot, of other 
 '^* ^cofTofthe diff. of the arcs "" fum of tan. of one and cot. of other" 
 taking the tan. of the fame arc, 
 
 AvAiP s\a-\-s\(2 . f\iT?^ t.\Aa Aa + /, /,^ + j\a 
 
 For -;._Z_ - : (2i2.And-^-j - ^ =- , , ^ ^ 
 
 tijAa S,aS,A^ ^,lA+d + /,2.A d i , A + i ,rt + i j J , A 
 
 rru t\lA+^--tAJ=Z_f2s'A_\s\A 
 Then-TT rr r-= --- ( -rT- } "T" 
 
 Here the arcs A, <?, are the fum and diff. of the arcs f A+a> \.\a, 
 
 215. The fine of the fum of two arcs, into radius; is equal to thet 
 fum of the producSls, of the fine of the greater by the co-fine of the Icfs, 
 and the fine of tiie IcTs by the co-fine of the greater. And, 
 
 The fine of the difference of two arcs, into radius ; is equal to the 
 difference of the produiSh, of the fine of the greater by the co-fine of the 
 lefs, and the fine of the lefs by the co-fine of the greater. 
 
 F-- iRx|7;Xr^^;.::r.,I=:x.,fZ+:(i82).l Here U+. and |A--, 
 arc tlic arcs. 
 
 y, 5 R x^,A(the fum) .f,f.^-f., x/,'^ \_,,-f ,f,? a ^ x/,|~-fT 
 
 '^^^^^ 1 R X .f//;the diff.) = J, 11+;; X s ,1 A^ J,i A^^ X Af'M^ 
 
 216. The
 
 feooklV. SPHERICS. 1S3 
 
 216. The co-fine of the fum of two arcs, into radius; is equal to 
 the difference, between the producSl of the co-fines, and produ6l of the 
 fines, of thofe arcs. 
 
 The co-fine of the difference of two arcs, into radius ; is equal to 
 the fum, of the produdl of the co-fines, and produd of the fines, of 
 thofe arcs. 
 
 Tor's 7 V T > T I / o \ f lA+a and fA j tieins 
 
 the arcs. 
 
 ^, f RXAA(the fum) 5^,1 A+^ X Af.'wl J,|a+^ XX,|A--a. 
 
 I R X jV(the diff.)=Zs\^A+aXs\lA^a + S,l7+^y.S,lAa. 
 
 Radius, lefs the co-fine of an arc _ Square the tan, of half that arc 
 ^' 'Radius, more the co-fine of an are"" Square of the Radius 
 
 R /,aT^ = (^'jA+^=)-^^^ ^X7,A+Z (195) 
 
 For^ 
 
 ^ R 
 
 K-{-s\A+a=:{'u\A+a = ) ,^ XS,A+a, (iQi; 
 
 R^s\A + a tt.^A+a. 
 
 Then . . - j. = ^ 
 
 R + AAfa RR. 
 
 A ^, fum of the fine & co-fine of an arc Radius 
 
 difF. of the fine & co-fine of that arc~tan. of difF.of that arc &:45" 
 
 _, fum of Rad. and tan. of an arc__ Radius 
 
 diff. of Rad. ajid tan. of that arc tan.of diff. of thatarc &45^* 
 
 For if A + ^=90"; then|A=45 1<7; and |^=45^ ^-a. 
 
 AKo s\az=:s,A : s,a=s\A: and f,A=:/jA X^'jA-j-Rt (III. 33) 
 
 s\A + s\a t\lA+7^ 
 Then-;^ -=--7^= (212; 
 
 S\a r,A t,lA-.a. 
 
 o 
 
 i,A /,A~' V /,A 45"+ -^ A J ""/,: 
 
 A (^45 
 
 ^ . -SA + Aa /,_ /,AXi,A-^R + AA \/,AXAa + AaxR 
 
 Agam, r-=l (111- 33) 1 r-~= ) ;^ 
 
 S,A s\a \^ -^^ t,AXs\A~-R S,A /^jAxAa ^,AXR' 
 
 Tl / ^A-f i\A _ ^X:^xAA __\R-f /,A_ R 
 
 V J,A i'jA^/jA R X Aa~/ R yi /,A~/,Ac/3 45>> 
 
 f 
 
 This mark v3 (hews the difference of the values it (lands between. 
 
 N 4 219. The
 
 i84 SPHERICS. Book IV, 
 
 219. The dlfFerence of" the co-fines of two arcs, is equal tp the dif- 
 ference of the vcifcd fines of thofc arcs. 
 
 220. The produ(5l of the fines of two arcs, is equal to the produl of 
 half the radius into the difference of the co-fmes, of the fum and dif- 
 ference of ihofe arcs. 
 
 Or j,z X J,x=iR X j',z+x j',/. X. Putting z=:|A+d i xnlAa. 
 
 221. The produ<Sl of the fines of two arcs, is equal to the produt of 
 half the Radius into the difference between the veiled fines, of the fum, 
 and difference of thofe arcs. 
 
 Thatis,j,z xj,x = (|r xs\z.+x s\z~-x(220)=z)v,z+K v,z^xx\k^ 
 
 TT ir V T. V fqunre of the fine of an arc 
 
 222. Half the Radius ri-i-p-j-r -. . p (iQ7\ 
 
 verled line of twice that arc ^ ^^^. 
 
 fquare of the co-fine of an arc 
 "^fup-verlcd line of twice that arc* ^ "' -^ 
 
 prorhj6t of the fines of two arcs 
 "^difF. of ver. fines of the fum and diff. of thofe arcs* 
 
 (221) 
 
 ,^, ^ , ^ , prod, of the fquares, of the fine and cot. of an arc 
 
 223. The fq. of Rad. -^ -. ^ --^ r -r 
 
 "^ ^. Iquaie ol the co-line of that arc 
 
 prod, of the fquares of the co-fine & tan. of an arc 
 "~ Iquare ot the line ol that arc 
 
 _ S^AXt\A AaX^A, ^ _, SS.AXtYyA iV,AX//,A 
 
 For Rr= = 1187.) Then rr ^tt =- ' 
 
 /,A S,A ^ ' ' i i\A SS,A 
 
 224. The product of Radius, and the co-fine of an arc, is equal to 
 the difference of the fquares, of the fine and co-fine of half that arc, 
 
 ^Qf ,, . - -1^^193^197)=) , .-[ ' ?UtZ = A + a. 
 
 ^^''KTs',..ir^-i;^ = 77:i^T^' ^>' ^<'^P^^''^on and d.v.fion, 
 
 25\Z /x\vZ 55, ^Z V A , 
 
 -^= '-^^^^' "') And RxAzrr/Az~/5,-^z. 
 
 =^h'z^-i-hi'^>'^s\lzSjlz. (II, 119) 
 
 In
 
 Book IV. SPHERICS. 185 
 
 In any fpheric triangle abc, if in the fide cb produced, be taken BE, bd, 
 
 each equal to ba, and ?g be drawn at right angles to ca. 
 
 Then CE = BC + BA is the fum 7 r ^^ , 1 j- u 1 ^ 
 
 ^ ' . c-.u J AT > of the legs including the angle at B. 
 CD=:bc BA IS the ditt. 3 ^ - *= ^ 
 
 CG and AG are the fegments of the 
 
 bafe. or fi^e oppofite to the angle 
 
 p. 2. A and Z.C arc called bafe 
 
 angles. ACBCrz^, ^abg=:c are 
 
 the vertical angles. 
 Now a very great number of relations 
 fnay be formed between .the fides and 
 angles ; fome of which are here enu- 
 merated. 
 
 225. The fines of the legs, are as the fines of the oppofite bafe angles. 
 That 4s, j,BC : j,B A : : j,a : j,c. { 1 10) 
 
 fum of the fines of the legs fum of the fines of the bafe angles 
 
 difF. of the finesofthc legs"~difF, of the fines of the bafe angles 
 by compofition. 
 
 226. The co-fines of the bafe angles, are as the fines of the vertical 
 angles. 
 
 That is, s\c : /,A : : s^a : s^c. (3d of 122, and 2d of 124) 
 
 fum of co-fines of bafe angles fum of the fines of vertical angles 
 xience ^ . . s ^ . h 
 
 diff. of co-fines of bafe angles diiF. of fines of vertical angles 
 by compofition and divifion of ratios, 
 
 227. The co-fines of the legs, are as the co-fines of the adjacent feg- 
 ments of the bafe. 
 
 That is, /,BC : /,BA : : /,CG : /,AG. (3d of 121, and 2d of 123) 
 
 fum of co-fines of the legs fum of co-fines of bafe ferments 
 
 Hence ' '^^-r-. 
 
 diff, of co-fines of the legs cITff'. of co-fines of bafe fegments 
 
 by compofition and divifion of ratios. 
 
 228. The co-tangcnts of the legs, are as the co-fines of the adjacent 
 vertical angles. 
 
 That is, /,BC : /\ra : : s\a : s\c. (2d of 121, and 2d of 124) 
 
 fi^mof co-t- of the legs fum of co-fines of vcrr. angles 
 
 difF. of co-t. of the ]cgs~~diff. of co-fines of vert, angles v 
 pofition and divifion of ratios. 
 
 229. The tangents of the legs, arc as the co-f. of the adjacent vertical 
 ingles reciprocally. 
 
 fum of tan. of the legs fum of co-f. of vert, angles 
 
 dii*i'. of tan. of the legs ~ difF. of co-f. of vert, angles ^ comp. 
 
 230. The
 
 iS6 3 P H J R I C S. 3ooklV. 
 
 230. The fiilQS of the bafe fegments, arp as the tangents of the ad- 
 jacent bafe angles reciprocally, 
 
 Thatis,x,CG :j,ag : : (/',c:/\a: :)/,a ; /,c.{2d of 122, and iftof 123) 
 
 fum of fines of bafe fegments fu,m of tan of bafe angles 
 
 Xjcnce "" ^ 
 
 difF. of fines of bafe fegments ~d iff. of tan, of bafe angles 
 by compofition and divifion. 
 
 231. The tangents of the bafe fegments, are as the tangents of the op.- 
 pofite vertical angles. 
 
 That is, /,CG : f,A.c : r/,fl : /,r. (ic^) 
 
 fum of tan. of b^fc fegments fum of tan. of vert, angles 
 
 Hence 1 . . '' . ' "^ 
 
 diff". of tan. of bjife fcgments dili". of tan. of vert, angles 
 
 by compofitiori and divifion of .ratios, 
 
 _, tan.ofhalf thfumofthelco:s tan.ofhalfthefnmofthebafeang. 
 
 n'^2,. T^he~ *= ~ ^ ' ' - 
 
 ^ ' tan. ofhalf the diff, of the legs tan.of half the diff. of thebaic ang, 
 
 jjBCjB.A ~J,A -^,C ^ ''^^^tyia^BA~iiiA^ {^^^4 
 
 _ tan. of i fum of bafe fegments tan. of f diff. of the legs ' 
 
 tan. of i fum of the legs ""tan. of idiff. of thebafe fegments' 
 
 /,BA+/,nc /,GA + j\gC, . /\vBC4-BA t^Jvc+GA 
 
 For- . = ^ (227)=- , ,. f2i2) 
 
 S ,BAS yhC SyGAS^GC^ '' ?, tBC BA ^ f -G--GA ^ ^ 
 
 -ri ^Jc c BA /^\Ibc4-ba ^,.n_Vv^ -G4-oa 
 
 Then - { 7~i (ii- i45j==y r r ' . (HI. 37) 
 
 fine of fum of legs co-tan. of | fum of vert, angles 
 
 ' fine of diff. of legs tan. of -^- diff, of vert, angles 
 
 co-tan. i diff. of vert, angles 
 
 tan. off fum of vert, angles 
 
 ,, J.RC-fi?A //,BC-f-/,BA 
 
 tor -^= 
 
 (212) 
 
 ,, cot. off furn of vert. an2;les tan. of \ diff. of bafe angles 
 tan. of I fum of the bafe angles~~tan. of \ diff. of vert, angles* 
 
 For -rr-r=~[- ^(212)=: (226)] , (211) 
 
 Ilence rT=====: 7-r====(II. 145}= J ~~-. (III. 37) 
 
 236. The
 
 Book IV. SPHERICS. 1S7 
 
 fine of fum of the legs _ fquare of co-t. off Turn of vert, angle? 
 ^ ' fine of diff. of the legs'" tan. i fum, into tan. ^ ^iff. of bafez.* 
 
 For -TY^ =/,|.-.. {235) 
 
 rr, , ^ /,vA + CXM'a3c , . 
 
 1 hen j,BC+BA : JjBc ba : : / ,|a+c : -= 7T= (234J 
 
 T : t^t\l7+c : MaTc xMirU. (II. 151) 
 
 _,, fine of I the fum of legs co-tan. of | the fum of vert, angles 
 
 ^37. The ^ ^ ~ . 
 
 ^ fine of ~ the diiF. of legs tan. of | the difF. of the bafe angles 
 
 jor-,^^^':^M{2z(>). And^4SI='4^. (232) 
 
 Then tt\^~fci<t,\A+c __j,Bc +BAX^f rc-i-b^ - 
 
 /,|A + CX/,Amixr,|;A^ J,BC B A X ^|bC BA* ^ * '5 J 
 
 And ;_--- = ^ (195, 193). Then- I ^- V", - 
 
 cof. off fum of the legs _ co-t. off the fum of vertical angles 
 cof. of I diff. of the legs tan. of ^ the fum of the bafe angles' 
 
 For ^ ^l^. - (236). And--p== c^-= -^ . (232) 
 
 /,A4-C:X^iA_C J-jBC BA^ ^ Ma+c ?,.bc ba ^ ^ ^ 
 
 rp, fY,f7 i7xM A c __ j,Bc + :iA x^Mbc+ba .,- ,. 
 
 '^^ ^j^T+Tx /,a7^ X /,|aTc JjBCBa X /,|bc_ba ' ( ^5 ; 
 
 And- =:-^-T-7r==rr (191, 107). Then-rT===Tl= =. 
 //, tA+c 2i\;,|Bt>-BA ^ ^ ' ''' JjIbcba />|a+c 
 
 The two lad propofitions folve the problem where two fides, andthe in- 
 cluded angle, of a fpheric triangle, are given to find the other angles. 
 
 Or where two angles and the included fide are given, to find the other 
 fides, ufing the word angles for legs ; the given fide for fum of vertical 
 angles j the other fide for bafe angles. 
 
 In art. 237, 238, the conclufions were gained from this principle, namely, 
 Vhat tbejides of proportional Jq-iures, art in tht fame proportion as tboj'e Jquares. 
 
 239. The
 
 jBS spherics. Book IV. 
 
 239. The tio-fmc of an angle, is to Radius ;^ 
 
 As the Radius into co-f. of the oppofite fide, lefs the produil of 
 
 the co-fines of the including fides. 
 To the produl of the fines of the including fides. 
 
 For /,CG = (/,AC AG=:)/,ACX/,AG+J,ACXJ,AG-r R. (^I^) 
 
 And (/,CG-) ''jv^j (227)=/,ACX/AG+J,ACXJ,AG-r-R.(II.46) 
 
 Therefore /,bc x s\\c x^ , a cxi\ ag + -^ x f,AC X ^>ag. 
 
 -- ^ RxAbC ^j\abxAaC , 5\AB'X.f,AC 
 
 Therefore xsjAC ^^r" -XJjAG. 
 
 _, RX/,BC AaBXAaC /i, AG ._^ . , ^^AG 
 
 Then ^ >= ( V :II. 16-?) =: ) 
 
 ijAiiXi,CA VJjAO^ -^^ J R 
 
 ^ Aa , > a j'vAG /i\a V''^ ;,AnxR 
 But^AG:=: X/jABfni). And [ X/,AErr ) x ; . 
 
 * R ' V J / J), y RR ' >' RR J,AB 
 
 (187) 
 _., Aa ^r AR RX5\bC j\abX/,AC , ,5\a Ry5\BC /,ACX5\aB 
 
 ihcn X- .And =; ^ ' , 
 
 a J,AB f,ABX5,Ae R .^jACXj-jAB 
 
 /a X5,AC X J,AI^ 
 
 240. Hence R X5,BC:=: ^ 4-ijACX^,ABi 
 
 R 
 
 2.'. I. The verfcd fine of an angle, is to the fquare of Radius : 
 
 As the difF.of the vcrfed fines of op. fide, and difi". of including fides. 
 To the produdl of the fines of the fides including that angle. 
 
 For (239^ =: I =: ) i_. 
 
 J'heiefore rr X i\ nc R X i\ ac X j\ ab = r x >', ac X .', AB - j, AC x ^5 Mk 
 
 X i',A, 
 
 Therefore jir x /,rc + .<, ac y s, ^B x ^',a 5\Acxi,AB + j, ac a s\ab x r. 
 
 :r /, AC AB X RR . (216) 
 
 Then T,AC X;,ar x z',a=:(rr x ^cq rr xs\bcz=]s\ cii i\Rc x R R . 
 
 ^W /s\cv-s\v(: \v,Qr.-v,cr> 
 And- -{ -J . (no) 
 
 RR \5,ACX;jAfJ /o,ACXijAii K^'^^J 
 
 242. Hencs
 
 Bodk IV'.' SPHERICS. 1S9 
 
 '^/Li Hence -^-^ ^ Ori'y.An ,wnenR=:i, 
 
 PR 
 
 For ( j,AC X X, AB rr )Ir x -yjCE -yjCD ( 222) =i^,cb i/jCD X . (241) 
 
 TV. ^><^B'-1^,CD _/ |RX^,A __\ t;,A , 
 Z/jCE 'y,CD~V RR / 2R* 
 
 ^4^. The verfed fine of the fup. of an angle, is to the fquare of Radius ; 
 As the difF. of the verfed fines of the oppofite fide, and fum of the 
 
 including fides, 
 To the produa of the fines of the fides including that angle. 
 
 , RX/,BC 5\ aCXj\aB __ fs\A\ t;, A R 
 
 ^or(239) ,,Acxi,AB "-Vr""/ r 
 
 Therefore rrxAbcRx/,acx/,ab=:5,acx J, ABX-y'jARX/jACXJjAB. 
 
 Therefore rr x^\bc j,ac Xj,ab Xi/^ArrR x/,ac x j\ab 5,ac xi,ab 
 
 = (rR X/,AC + AB = ) RR X/,CE. (2l6} 
 
 Then RRXi\Ec rr x/,ce=J5ACXj,abxv,a. 
 
 J ^^A_ /s\ bC r\cE _'\ ^,CE y,BC 
 " Ilk" \s] ACXS,AB~J i,ACXi,AB* 
 
 v\a ^;,ce 7;,cb ^ , v^CE VjCZ 
 
 144.. Hence ^ =^~ Or f y , a =: j when R =: i . 
 
 . R^ / ^ 
 
 For ( J, AC X ^, AB=: )|-R X t-jCE V^ii-D (222) ^l^jCE X'jCB X J^ * (H3) 
 
 _, T',CE V^QV. /iRX^',A \ 'y,A 
 
 Then = { = =. 1 . 
 
 yjCE V,CD \ RR / 2R 
 
 245. The fquare of the fine of half an angle, is to the fquare of the Ra- 
 dius ; 
 As \ Radius into the difF. of the verfed fines of the fide oppofite, 
 
 and dift'. of the fides including that angle. 
 To the produ(5l of the fines of the fides including that angle. 
 
 for (t;,A = ) ^^ (222) = 3 XRR. (^40 
 
 ^ ' 2'<- i,ACXi,AB ^ ' 
 
 ,^ ^f,vA 'y,cn 7;, en .f\cD /c b , , . 
 
 Then =-! ^ x!Rrr-^ x I-r. (219 
 
 KR -SACXyjAB " ijACXXjAB 
 
 246. Hence
 
 ,jo 5 P H E R I C S4 BookiV, 
 
 ^ &R X,ACX/,AB 
 
 AB 
 f,ACAJ,AB 
 
 c:-^- Putting 2h=:ac + ab + bc. 
 
 247. The fqu. of the co-fine of half an angle, is to the fqu. of Radius ; 
 As i Radius into the difF. of verfed lines of the fide oppofite, and 
 
 fum of the included fides, 
 To the produdt of the fines of the fides including that angle* 
 
 Then -^=^ ^^ x|R=: ^ x|r. (II9) 
 
 RR J,ACX5,AB * ^, ACXfjAB \ 7/ 
 
 ^8. Hence^J^^'-^^"^^^^^-"^^^^^- (i77) 
 
 ^ RR JjACXijAB 
 
 I J,|'AC4-AB + CBXX,f AC + AB CB ^ 
 
 :zi ^ -^ -^-* ForcE=AC + AB. 
 
 JjACXijAB 
 
 J,H X ^,H IB _ 
 
 Putting 2H=AC + AB + BC. 
 
 J, AC X^,AB 
 
 249. The fquare of the tan, of half an angle, is to the fquareof Radius* 
 As the diff. of ver. fines of the fide op. and difF. of including fides. 
 To the difF. of ver. fines of the fide op. and fum of including fides. 
 
 t;, A ss,Ia ^^|-A VyCn -z/jCd 
 
 ^'>'^=17T^ ("^) = 1F ("3)= ^,cE-..CB - (241.243) 
 
 ajo. Hence 'J^=b^^^J:^^. ^ .8,) 
 
 RR -SvCE + CBXi,i(j CB V 7J / 
 
 __. r,fcb+A <J AHXj, fcH AC + AB 
 i-i-C. + AC + AB X ^,iAC + AC .CB 
 
 ?,H Ai: X5,H AC 
 S,H Xi,H 110 
 
 From the articles in the three hfl pages may be deduced many rules 
 for folving the problem, where the three fides arc given to find an angle ; 
 and thence, from the three angles given to find a fide, 
 
 ^i. When
 
 Book IV. SPHERICS. t9t 
 
 251. When two fides and the included angle are given to find the 
 third fide : Or when the three fides are given to find an angle ; Tor thefe 
 particular cafes there have been given compendiums by Sir Jonas Moore, 
 in his Mathematics, Vol. II. page 383 : Alfo by Nicholas Facio Duillier, 
 in a fmali tract of his, which is very fcarce -y and by the learned Dr. Pem- 
 berton, in the Philofophical Tranfa<5^ions for the year 1 760 : The prin- 
 ciple employed by each of them is the fame as in Article 245 ; which will 
 be here illaftrated on account of its utility in fome aftronomical fubjedts. 
 
 In the triangle abc, where cd=:ac w ab 
 
 Given AB, AC, Z.A ; required bc, ^^^""l^K 
 
 Now^H^=^'''"-^^^. - (245) X \ 
 
 R^ J,ACX.JjAB ^ ^'^ / \ 
 
 . , ;^,fZ.A X5,ACX i,AB ,^ B ^"^ -....^ 
 
 And ^3 =is\cE c f5,cD=N \D 
 
 Then 2n /,cd=:/,cb. \ 
 
 Hence the following practical Rules. ^ 
 
 252. I. To twice the log. fine of half the gi-ven angle^ 
 Add the log. fines of the tivo containing fides ; 
 
 The f urn ^ abating three radii in the index^ leaves the log. fine of an arc. 
 From twice the nat, fine of that arc ; take the nat. co-fine of the diff". of tht 
 given fides., 
 
 Leaves the nat. co-fine of the third fide ^ or of its fupplement. 
 
 253.11. But the fide required may be found without the ufe of natural 
 fines. Thus 
 
 To twice the log. fine of half the given angle^ 
 Add the log. fines of the tivo containing fides ; 
 
 From half the fum of thefe logs, fuhtracl the log. fine cf half the diff, if 
 the fides. 
 
 And the remainder is the log. tangent of an arc ; 
 
 The log. fine of which arc, fuhtr act cd from the f aid ha f fum of logs y 
 
 Leaves the log. fine cf half the required fide. 
 
 Take the Example ufed in the fix cafes of oblique fp!;evic triangles. 
 
 Where ac 83'' 11', ABrrsG-" 40'; Z.A 125 20'. 
 
 Required bc, which was there found to bc 114 30. 
 
 Given z. =: 125 20' 
 
 . ,r "7 - Log. f.ne|9>94S-^ 
 
 Its half == 62 40 ^ C9'9485S 
 
 One fide r: 83 1 1 - '" I^og. fine 9,^9692 
 
 ether fide 56 40 - Log. fine 9,9zj94 
 
 Are found r:: ^jo 53^ its nat. fine 65467 (256) 9,81602 
 
 its double 1 30934 
 difF. fides ~ 26 *o the nat. co-finc 89480 
 
 65 30 the nat. CO fine 41454 i^SJ) 
 
 The fide required 1 1+ 30 
 
 The
 
 t^i SPHERICS. Book IV*. 
 
 The faid Example wrought by the fecond Rule is as follows 5 
 Given Z. = iS 20 
 
 log. fineJ 9.94858 
 ^ I 9*94858 
 
 log. fine 9,99692 ^ 
 
 log. fine 9,92194 
 
 39,81602 
 
 its half 
 
 One fide = 
 
 Other fide = 
 
 62 40 
 83 11 
 56 40 
 
 Sam logs = 
 
 half fum 
 
 i difF. fides = 
 
 >3 5 
 
 Arc = 
 
 74 o 
 
 19,90801 half fum logi 19,90801 
 
 9*3^^48 
 log. tan. 10,54753 Log. fine 74 10 9,98326 
 Log.fineof57 15 9,92^81 
 The required fide = i'4 3 
 When the three fides are given to find an angle ; 
 2544 I. To the Mat co-f. of the fide oppofite the required anghy add the nat. cb-f. 
 ofthe diff. of the fides about that angle ; half the fum is the nat.fineofan arc. 
 Vo the Tog. fine of that arc, add the arith. comps. ofthe log. fines of ihefidei 
 about the required ajigle and alfo the radius. 
 The half of this fum is the log. fine of half the angle fought. 
 Or without ufing the natural fines. 
 255. II. To the log. fine of half the diff. of the fides about the angle^ add the 
 arith. comp. of the log. fine of half the bafe\ the fum is the log. fine of an arc. 
 To the log. co-fme of this arc, add the log. fine of half the bafe ; rejet ra- 
 dius from the fum^ and to the double of what will then remain add tht 
 arith. comps. ofthe log. fines of the containing fides* 
 Half the fum is the log. fine of half the angle. 
 Exam. Let the three fides be bc = i 14 30', ac=:8o' 1 1', ab=:56'' 40'. 
 Required the angle a. 
 
 By I. Rule 
 Bafe Bc = 1 14 30" 
 
 Its fupl. 63 30 nat. co-fine rr 41469 (25^) 
 DifF. fides 26 31 nat. co.f. = 89480 
 
 Sum 130949 Rad. io,ocdoo 
 
 Half fum is the nat. fine. (217) 65474 arc 40 54' log. fine 9,81607 
 
 Ar. CO. log. fine 83 11 0,00308 
 
 Ar. CO. log. fine 56 40 0,07806 
 
 19,89721 
 Half the angle fought 62* 40' log. fine 5,9^860 
 By II. Rule. 
 -|oc = 13" i5i' Log. fine 9,36048 
 
 |bc = 57 15 Ar.Co.L.fin. 0,07^18 log. fine 9,92482 
 
 Log. fine 15 49V an arc 9,43560 leg. co-f. 15 49I' 9,98322 
 
 9,90804 
 
 2 
 
 19,81608 
 Ar. Co. Log. fine ac 0,00308 
 
 Ar. Co. Log. fine ab 0,0780 6 
 
 Sum ' - 19,89722 
 
 Half fum 13 L.og. fine of 62 40' 9,94oi 
 
 Angle fought is 1^5 2cr:^.^, 
 
 h%.
 
 Book. IV. SPHERICS. 193 
 
 As the natural fines of arcs are not contained in this work, and arc 
 on fome occafions neceflary, it will be proper to Ihew how they may be 
 found from the Logarithmic tables contained herein. 
 
 256. Firji. An arc being given., to find its natural fine to five places of figures. 
 Rule. Take out the Log. fine of the arc, rejedling the Index j 
 
 Seek thefe figures among the logarithms of numbers ; 
 
 The correfponding number is the natural fine of the given arc ; 
 
 which is to be reckoned as a decimal fradtion of the radius, or 
 
 unity : 
 
 Prefixing the decimal comma (,) if the index of the log. fine was 9 ; 
 
 But if the index was 8 ; 7 ; or 6 ; prefix ,0 ; ,00 ; or ,000 ; by 
 
 which means the left hand digit of the natural fine will ftand in the 
 
 place of the firfts, feconds, thirds, or fourths. (L 18) 
 
 Ex. I. Required the natural fine and co-fine of d^ ll' ? 
 Log, fines fine '8,88 i6i Co-fine ^,(:)<)%-j\ 
 
 Num. or nat. fines 0,07614 0,99710 
 
 Ex. II. Required the natural fine and co-fine of 2^'^ is' P 
 Log. fines fine 9,67982 Co-fine 9.94355 
 
 Num. or nat. fines 0,47844 0,87812 
 
 If a given log. fine is found in the table of logs, of numbers, its na- 
 tural number confifl:ing of four places is feen at fight; and its right hand 
 place is o when the index of the log. fine was 9. 
 
 But if a given log. fine is not found to every figure in the tables of 
 log. numbers, its 5th, or right-hand place is thus found. 
 
 Take the difF. between the log. num. next greater and lefs, than the 
 given log. fine ; and alfo the difF. between the given log. fine and its 
 next lefs log. numb. 
 
 Then, As ift diff. is to 2d difF. fo is 10, to the digit for the right 
 hand place. 
 Thus to 4" 22', the nat. fine is 0,07614 ; and co-fine is 0,99710. 
 But to 28^ 35', the log. fine and co-f. does not appear exactly among 
 the log. numb. 
 
 And the above-mentioned two difFerences, for fine, arc 9 and 3 ; for 
 co-f. are 5 and i. 
 Then 9 : 3 : : 10 : 3, the 5th place. And 5 : i : : 10 : 2, the 5th place. 
 
 257. On the contrary. A natural fine being given., its correfponding arc may 
 be thus found. 
 
 In the tables of num. and logs, enter with the natural fine as a num. 
 and take out its log. 
 
 Seek this log. in the table of log. fines, and the correfponding de- 
 grees and minutes fhew the arc required. 
 
 Prefixing the index 9, 8, 7, 6 ; according as the left hand digit 
 ftood in the place of firfls, feconds, thirds, or fourths. 
 What has been faid of the nat. and log. fines of arcs, Is alfo appli- 
 cable to the nat. and log. tangents of arcs. 
 
 END OF BOOK IV. 
 Vol. L O THE
 
 THE 
 
 ELEMENTS 
 
 O F 
 
 NAVIGATION. 
 
 B O O K V. 
 
 OF ASTRONOMY. 
 
 S E C T I-^ O N I. 
 
 Of Solar AJlrG7to77ty, 
 
 A 
 
 and dilhinces of the heavenly bodies, and of the appearances 
 thence arifing. 
 
 There have been great variety of opinions among the philofophers of 
 preceding ages concerning the fituation of the great bodies in the univerfe, 
 or of the pofitions of the bodies which appear in the heavens : But the 
 notion now enihrriccd by the mofl judicious Aflronomcrs is, that the uni- 
 vcric is compoied of an infinite number of lyderns, or worlds j in every 
 fyft:.'m there are certain bodies moving in free fpace, and revolving atdif- 
 icrcnt diflances around a Sun, placed in, or near, the center of the 
 fyflem ; and that thel'c fufis and other hdies arc the ftars which arc fecn in 
 the heavens. 
 
 ?.. The Sor.AR Systf.m, fo called by Aftronomcrs, is that in which 
 our Earth is placed ; and in which the Sim is fuppofed to be fixed in or 
 near the center, with fcveral bodies fimilar to our Earth revolving round 
 him at different diflanrcs. This hypothefis, which is confirmed by all 
 the obfcrvations hitherto made, i> tailed tiic C'^rtRiVXCAN Syste;/I| 
 
 U 2 1 frr ;ii
 
 196 
 
 A S T 11 O N O M Y. 
 
 BookV. 
 
 from Nicholas Copernicus^ a Polifh Philofopher, who about the year 1500 
 
 revived this notion from the oblivion it had been buried in for many ages. 
 
 Stars arc diftinguifhed into two kinds, namely, yf*-^^/ and watitiering. 
 
 3. The Fixed Stars are the funs, in the centers of their fyftems, 
 fhining by their own light ; and arc obferved to prcfcrve always the fame 
 fituation in refpc<Sl to one another. 
 
 4. The fixed ftars appear of various fixes, which is doubtlcfs occa- 
 fioned by their different diftances ; thefe fizes are ufually diftinguiihcd into 
 fix or feven clafles, called Magnitudes : The largefl, or brighteft, are faid 
 to be of the first magnitude; thofe in the next clafs or degree of 
 brightnefs, are called of the second magnitude; and fo on to the 
 leaft, or thofe juft difcernible to the naked eye. But befides thefe, there 
 is fcattered throughout the heavens a great number of other ftars, called 
 Telescopic Stars, becaufe they cannot be feen except through a tele- 
 fcope. And indeed, it is to the afliftance of that moft admirable inftrument, 
 that a great part of the modern Aftronomy owes its rife ; which will un- 
 doubtedly tranfmit with the greateft honour to the lateft pofterity the name 
 of Galileo, among whofe many inventions the telefcope is ranked, 
 
 5. Although the vifible fixed ftars are probably at very unequal dif- 
 tances from the center of the folar fy/iem, yet Aftronomers, for their eafe 
 in computation, confider them as equally diftant from our Sun, forming 
 the furface of a fphere which inclofes our fyftem, and is called the Ce- 
 lestial Sphere. This fuppofition, with regard to the Solar Syftem, 
 may be ftriilly admitted, confidering the immenfe diftance even of the 
 ncareft fixed ftars from the center of the fyftem. 
 
 6. A Constellation is a number of ftars lying in the neighbourhood 
 of one another on the furface of the celejlial fphere., which Aftronomers, 
 for the fake of remembering them with greater eafe, fuppofe to be circum- 
 fcribed by the outlines of fome nnimal, or other fgure : by this means the 
 motions of the wandering ftars are more readily dcfcribed and compared^ 
 
 The number of thefe conftellations is 80, each containing feveral ftars 
 of different magnitudes. The number of ftars of each magnitude, and 
 alfo the conftellation in which they are ranged, are contained in the fol- 
 lowing table ; where it may be obferved, that the conftellations are diftin- 
 guifhed under three heads ; namely, in the zodiac, and in the northern, 
 and fouthern hemifpheres. 
 
 7. Constellations in the Zodiac. 
 
 Names. 
 N^orthern. 
 
 2 
 
 J. 
 
 s 
 3 
 
 0- 
 
 n 
 
 Magnitudes. 
 
 Names. 
 Southern. 
 
 2 
 
 7T 
 
 z 
 
 c 
 3 
 cr 
 
 Magnitudes. 
 
 I 
 
 n 
 
 111 
 
 IV 
 
 7 
 
 V 
 
 VI 
 
 ' 
 
 2 
 
 111 
 I 
 
 IV 
 
 8 
 
 v|vi 
 
 1 
 319 
 
 Aries. 
 
 r 
 
 46 
 
 
 
 I 
 
 I 
 
 S 
 
 36 
 
 Libra. 
 
 u^ 
 
 33 
 
 
 
 Taurus. 
 
 
 
 109 
 
 I 
 
 I 
 
 3 
 
 9 
 
 24 
 
 71 
 
 Scorpio. 
 
 '11 
 
 44 
 
 J 
 
 3 
 
 6 
 
 M 
 
 5 '5 
 
 Gemini. 
 
 n 
 
 94 
 
 I 
 
 2 
 
 4 
 
 b 
 
 3 
 
 66 
 
 Sagittarius. 
 
 s 
 
 48 
 
 
 
 I 
 
 5 
 
 9 
 
 I Ij22 
 
 Cancer. 
 
 Si 
 
 7> 
 
 c 
 
 
 
 
 
 6 
 
 8 
 
 61 
 
 Capricornus. 
 
 vs 
 
 5 
 
 
 
 
 
 2 
 
 S 
 
 942 
 
 Leo. 
 
 a 
 
 9^ 
 
 2 
 
 2 
 
 6 
 
 '3 
 
 u 
 
 >7 
 
 Aquarius. 
 
 A^^ 
 
 9^ 
 
 
 
 c 
 
 4 
 
 7 
 
 2854 
 
 V' irgo. 
 
 'Of 
 
 93 
 
 I 
 
 
 
 5 
 
 1 1 
 
 ^45^ 
 
 Pifces. 
 
 K 
 
 I 10 
 
 G 
 
 
 
 I 
 
 7 
 
 274 
 
 8. Northern
 
 Book V. 
 
 ASTRONOMY. 
 
 197 
 
 8. 
 
 Northern Constellations. 
 
 Names. 
 
 2; 
 
 c 
 3 
 
 Magnitudes. 
 
 Names. 
 
 
 Magnitudes. 
 
 
 
 
 
 
 
 3 
 
 
 
 
 
 
 
 
 
 I 
 
 
 11 
 
 2 
 
 III 
 
 
 
 I 
 
 IV 
 
 4 
 
 V 
 
 VI 1 
 
 i 1 
 
 
 
 I 
 
 " 
 
 II! 
 
 
 IV 
 
 S 
 
 V, 
 
 VI 
 
 II 
 
 Little Bear. 
 
 12 
 
 Camelopardalus. 
 
 23 
 
 c 
 
 
 
 Great Bear. 
 
 'OS 
 
 
 
 S 
 
 S 
 
 16 
 
 30 
 
 49^ 
 
 Serpent. 
 
 5c 
 
 c 
 
 1 
 
 7 
 
 6 
 
 5 
 
 31 
 
 Dragon. 
 
 4<i 
 
 
 
 I 
 
 7 
 
 X 
 
 IC 
 
 -i i 
 
 Serpentarlus. 
 
 by 
 
 c 
 
 I 
 
 6 
 
 12 
 
 17 
 
 31 
 
 Cepheus. 
 
 4c 
 
 
 
 
 
 ->, 
 
 7 
 
 10 
 
 20; 
 
 Sobieiki's Shield. 
 
 8 
 
 c 
 
 
 
 
 
 2 
 
 3 
 
 3 
 
 Greyhounds. 
 
 24 
 
 
 
 
 
 
 
 I 
 
 7 
 
 16; 
 
 Eagle. 
 
 29 
 
 I 
 
 
 
 5 
 
 I 
 
 /I 
 
 18 
 
 Bootes. 
 
 S^ 
 
 I 
 
 c 
 
 7 
 
 10 
 
 12 
 
 ^^ 
 
 Antinous. 
 
 34 
 
 
 
 
 
 S 
 
 2 
 
 7 
 
 20 
 
 Mons Maenalus. 
 
 1 1 
 
 
 
 
 
 
 
 I 
 
 c 
 
 10 
 
 Dolphin. 
 
 18 
 
 c 
 
 
 
 6 
 
 c 
 
 2 
 
 10 
 
 Berenice's Hair. 
 
 24 
 
 c 
 
 
 
 
 
 6 
 
 J 
 
 10 
 
 Colt. 
 
 12 
 
 
 
 
 
 
 
 4 
 
 I 
 
 7 
 
 Charles's Heart. 
 
 ? 
 
 
 
 I 
 
 
 
 
 
 
 
 2 i 
 
 Arrow. 
 
 n 
 
 
 
 
 
 
 
 
 
 I 
 
 8 
 
 Northern Crown. 
 
 1 2 
 
 
 
 I 
 
 c 
 
 6 
 
 
 I 
 
 Andromeda. 
 
 66 
 
 
 
 ^ 
 
 2 
 
 IC 
 
 16 
 
 3'> 
 
 Hercules. 
 
 q2 
 
 
 
 
 
 12 
 
 12 
 
 28 
 
 40 
 
 Perfeus. 
 
 67 
 
 1 
 
 I 
 
 S 
 
 IC 
 
 14 
 
 36 
 
 Cerberus. 
 
 9 
 
 
 
 
 
 c 
 
 1 
 
 I 
 
 S 
 
 Pegafiis. 
 
 81 
 
 
 
 3 
 
 4 
 
 9 
 
 II 
 
 S4 
 
 Harp. 
 
 24 
 
 I 
 
 
 
 > 
 
 
 8 
 
 10 
 
 Auriga. 
 
 46 
 
 1 
 
 I 
 
 I 
 
 9 
 
 9 
 
 25 
 
 Swan. 
 
 73 
 
 c 
 
 I 
 
 S 
 
 'S 
 
 20 
 
 12 
 
 Lynx. 
 
 ss 
 
 c 
 
 
 
 I 
 
 b 
 
 21 
 
 25 
 
 Fox. 
 
 2q 
 
 c 
 
 
 
 
 
 6 
 
 II 
 
 12 : 
 
 Little Lion. 
 
 20 
 
 c 
 
 c 
 
 I 
 
 6 
 
 S 
 
 8 
 
 Goofe. 
 
 10 
 
 
 
 
 
 c 
 
 
 
 2 
 
 i 
 
 Great Triangle. 
 
 IC 
 
 c 
 
 c 
 
 
 
 
 I 
 
 6 
 
 Lizarcf. 
 
 12 
 
 
 
 
 
 c 
 
 3 
 
 S 
 
 41 
 
 Little Triangle. 
 
 s 
 
 c 
 
 
 
 
 
 c 
 
 
 
 S 
 
 Cafliopea. 
 
 5^ 
 
 
 
 
 
 5 
 
 7 
 
 7 
 
 33 jlMufca. 
 
 6 
 
 c 
 
 
 
 I 
 
 - 
 
 2 
 
 I 
 
 Southern Constellations. 
 
 Names. 
 
 Z 
 
 Magnitudes. 
 
 Names. 
 
 Z 
 
 c 
 
 Magnitudes. 
 
 d 
 
 
 
 
 
 
 
 =! 
 
 
 
 
 
 
 
 - 
 
 cr 
 
 I 
 
 II 
 
 III 
 
 IV 
 
 V 
 
 VI 
 
 
 0- 
 
 1 
 
 II 
 
 III 
 
 IV 
 
 V 
 
 VI 
 
 Whale. 
 
 Sc 
 
 
 
 2 
 
 8 
 
 13 
 
 10 
 
 47 
 
 Pe;^cock. 
 
 14 
 
 
 
 I 
 
 3 
 
 S 
 
 4 
 
 I 
 
 Eridanus. 
 
 72 
 
 I 
 
 c 
 
 10 
 
 24 
 
 19 
 
 18 
 
 Southern Crown. 
 
 12 
 
 
 
 c 
 
 
 
 ) 
 
 3 
 
 8 
 
 Phenix. 
 
 13 
 
 
 
 1 
 
 
 S 
 
 
 
 2 
 
 Crane. 
 
 14 
 
 
 
 2 
 
 I 
 
 2 
 
 9 
 
 
 
 American Goofe. 
 
 9 
 
 
 
 c 
 
 
 2! 3 
 
 
 
 Southern Fifh. 
 
 IS 
 
 I 
 
 
 
 2 
 
 9 
 
 2 
 
 I 
 
 Orion. 
 
 93 
 
 2 
 
 4 
 
 
 i9;'5 
 
 50 
 
 Hare. 
 
 = 5 
 
 
 
 
 
 4 
 
 9 
 
 4 
 
 8 
 
 Monoceros. 
 
 
 
 
 c 
 
 
 lo'io 
 
 II 
 
 Noah's Dove. 
 
 10 
 
 
 
 2 
 
 
 
 I 
 
 6 
 
 
 
 Little Dog. 
 
 K 
 
 I 
 
 
 
 
 oj 2 
 
 10 
 
 Charles's Oak. 
 
 13 
 
 
 
 I 
 
 2 
 
 6 
 
 4 
 
 
 
 Hydra. 
 
 53 
 
 
 
 I 
 
 
 i4i'3 
 
 22 
 
 Ship Argo. 
 
 48 
 
 I 
 
 6 
 
 II 
 
 '3 
 
 14 
 
 ^ 
 
 Sextans Uraniae. 
 
 4 
 
 
 
 c 
 
 
 
 0! 
 
 4 
 
 Great Dog. 
 
 2q 
 
 i| S 
 
 I 
 
 4 
 
 10 
 
 8 
 
 Cup. 
 
 1 1 
 
 
 
 
 
 
 
 8 2 
 
 I 
 
 Bee. 
 
 4 
 
 0[ 
 
 
 
 
 2 
 
 
 
 Crow. 
 
 ? 
 
 
 
 
 
 2 
 
 21 2 
 
 3 
 
 Swallow. 
 
 1 1 
 
 o| 
 
 
 
 4 
 
 3 
 
 4 
 
 Centaur. 
 
 3f 
 
 2 
 
 6 
 
 6 
 
 14 8 
 
 
 
 Indus. 
 
 12 
 
 Qi 
 
 
 
 4 
 
 6 
 
 2 
 
 Wolf. 
 
 36 
 
 
 
 c 
 
 3 
 
 618 
 
 9 
 
 Charrielion. 
 
 10 
 
 Oi 
 
 
 
 c 
 
 9 
 
 I 
 
 Altar. 
 
 c 
 
 
 
 
 
 I 
 
 6 , 
 
 I 
 
 Flying Fifh. 
 
 7 
 
 
 
 
 
 c 
 
 ^ 
 
 I 
 
 Southern Triangle. 
 
 5 
 
 
 
 I 
 
 2| o| 2 
 
 
 
 ' Sword Fifh. 
 
 7 
 
 c 
 
 2 
 
 2 
 
 ! 
 
 z 
 
 Conftellations in the s^odiac 12, contain 
 Northern conftellations 36, contain 
 
 Southern condellations 32, contain 
 
 Number of ftars in the 80 conftellations 
 
 Stars 
 
 1 
 
 II 
 
 III 
 
 IV 
 
 V 
 
 VI 
 
 S94 
 
 6 
 
 12 
 
 3 
 
 100 169 
 
 569 
 
 243 
 
 5 
 
 21 
 
 92 
 
 200 291 
 
 034 
 
 706 
 
 9 
 
 32 
 
 ''^ 
 
 i5lu;8 
 
 217 
 
 2843 
 
 20 
 
 65 
 
 205 
 
 485- 6+S 
 
 1420 
 
 As thcfe flars are found not to alter their fituation in rcfpevH: to one 
 another, they ferve Ailronomers as fixed points, by which the motions of 
 
 O 3 other
 
 't93 ASTRONOMY. Book V. 
 
 other bodies may be coppared ; and therefore their relative pofitions 
 have been fought after with great care for many ages, and Catalogues of 
 their places have fn^m time to time been publifhcd by thofe, who have 
 been at the pains to make the obfervations. Among thefe catalogues, 
 the moft copious, and, as generally cfieemed, the bclT, is that called the 
 Hiftoria Celeji'n of our countryman Flamsteed. v 
 
 10. The pofitions of the flars being obtained, their relative places 
 may be delineated on a fphere or plane ; and thus are the maps or charts 
 of the heavens made, and the conftcllations drawn Inclofing their re- 
 fpecflive flars. There are two maps, ufually called Celeftial hemifpheres, 
 which are prefixed to this book ; by the help of which a perfon may 
 readily become acquainted with the pofitions and names of fomc of the 
 principal fixed^ ftars, thus : 
 
 On a clear night, let rhcfe prints be laid fo as to correfpond to the 
 north and fouth parts of the heavens j then the obferver looking on the 
 flars, and then on the hemifphcres, will with a little praclice know 
 fome of the ftars in the heavens, the like pofitions and names of which 
 he has obferved on the prints. 
 
 11. The Wandering Stars are thofe bodies within our fyftem, 
 or celeftial fphere, which revolve' round the Sun; they appear luminous 
 or bright, only by reflcdling the light they receive from the Sun ; and are 
 of three kinds, nzm^lv^ primary plaiiets^ feconda^-y planets^ and comets. 
 
 12. The Primary Planets are tjiofe bodies, which in revolving 
 round the Sun refpe6l him only as the center of their courfes ; the mo- 
 tions of which are regularly performed in tracks, or paths, that are found 
 by obfervations to be nearly circular and concentric to one another. 
 
 13. A Secondary Planet, commonly called a Satellite or 
 Moon, is a body, which, while it is carried round the Sun, does alfo re- 
 volve round a primary planet, which it refpecls as its center. 
 
 14. Comets, vulgarly called blazing Jiars^ are bodies which alfo 
 revolve round the Sun ; probably in as regular Order as the planets, 
 but in much longer periods of time, from what is hitherto known of 
 them. They are in number many more than all the planets, and their 
 tracks or courfes pafs among the paths of the planets in a great variety 
 of directions. 
 
 15. The Orbit of a planet or comet is that track or path along 
 which it moves. 
 
 There are fix primary phmf.ti ; and reckoned in order from the Sun, 
 their names and niprks are, Mercury 5, VeK^us ?, the Earth J 
 or 0, Mars rf, Jupiter ^i, Saturn I?. 
 
 Mars, yijpiier^ and Saturn, arc called Superior Planets, as their 
 orbits include that of the Earth: but Venus and Mercury, the orbits of 
 which are contained v/ithin the Eaith'i^, are called Inferior Planets. 
 
 16. It has been difcovcred by the help ol' tclcfcopes, that there are ten 
 feandary planets; the Earth being attended by one, called the Moon, 
 
 Jupiter by four, and Saturn by livr. 
 
 Saturn is alfo obferved to have a kind of circle, called his Ring, 
 which furrounds the plarict at fomc ciflaiice from his furfacc : and Ju- 
 piter hai; certain appearances, which fecrn like zone or girdles round 
 him 3 aijd thele are calkd Jupiter's Belt^. 
 
 Every
 
 BookV. ASTRONOMY. 191 
 
 Every primary planet is fuppofed to have two motions, namely, annual 
 and diurnal. 
 
 17. The Annual Motion of a planet is that whereby the planet i'5 
 carried in its orbit round the Sun j which in every one is found by obfer-' 
 vation to be from weft to eaft. 
 
 This motion is difcovered by the planets changing their places in th 
 celeftial fphere ; upon the furface of which they appear to move am un-j; 
 the fixed ftars ; and in certain times to return to the fame ftars from whic i 
 they were feen to depart ; and fo on continually. 
 
 18. The Diurnal Motion of a planet is that by which it turns or 
 Tpins about its axis, and is alfo from weft to eaft. 
 
 This motion is difcovered by the fpots that are feen by telefcopes on the 
 furfaces of the planets. The fpots appear firft on the eaftern margin, or 
 fide of the planet, and gradually move from thence acrofs it, till they dif- 
 appear on the weftern fide, or limb ; after a certain time they appear 
 again on the eaftern fide, and fo on. 
 
 19. Each planet is obferved always to pafs through the conftellations 
 Aries^ Taurus^ Gemini^ Cancer., Leo., Virgo., Libra., Scorpio., Sagittarius, 
 Capricornus, Aquarius., Pifces ; and it alfo appears, that every one has a 
 track peculiar to itfelf ; hence the paths of the fix planets form amoj-.g the 
 ftars a kind of road, which is called the Zodiac, the middle path of 
 which, called the Ecliptic, is the orbit defcribed by the Earth, \^it!i 
 which the orbits of the other planets are compared. 
 
 As the ecliptic runs through twelve conftellations, it is fuppofed to be 
 divided into twelve equal parts, of 30 degrees each, called figns, having 
 the fame names with the twelve conftellations which they run through. 
 
 20. I'he Equinoctial Points are thofc two points of the Ecliptic, 
 oppof.teto one another, through which the Eaith palies in its annual mo- 
 tion, when the length of the day and night is equal in all parts on the Earth. 
 Oncof thefc points, called the Vernal Equinox, anfwers nearly to the 
 20th of March ; and the other, called the Autumnal Equinox, nearly 
 to th(i 22d of September. 
 
 21. 'I'he Plane of the Ecliptic is fuppofed to divide the celeftial fpherj 
 into two equal parts, called the northern ^x\i\ fouthern celejllal bcmifphcres j 
 and any body in cither of thefe hemifphercs is faid to have north or fouth 
 latitude, according to the hemifphere it is in: So that the Latifuds 
 of a celeftial objci^t is its neareft diftance from the ecliptic, taken on the 
 fphere. 
 
 The Planes of the other five orbits are obferved to lie partly in th? 
 northern, and partly in the fouthcrn hemifphere ; fo that every one cuts xhz 
 ecliptic in two oppofite poinf;, called Nodes. Or.e called the Ascend- 
 fNG Nonr, marked thus, Q^, is that through which the planet pafics 
 v/hcn it moves out of the loiithern into tiio northern hcmiipherc ; and th.; 
 other called the DrsCENDiNC Nodi:, marked thus, y, is that tijrough 
 which the planet nuift pafs in going out of the northern mto the fouthcrji 
 hcmiipherc. 
 
 The right line joining the two Nodes of any planet, is called the Line 
 CF THE Nodes. 
 
 ? ?.. 'I'he names of moft of the conftellations were given by the ancient 
 Allronomcrs, who reckoned that ftar in Aries, now maikg-d y, (according 
 
 O 4 \K>
 
 aoo ASTRONOMY. Book V. 
 
 to Bayer's maps) to be the firfl: point in the ecliptic, this ftar being next 
 the Sun when he entered the Vernal Equinox ; and at that time each con- 
 ilellation was in the fign by which it was called. But obfervations Ihew, 
 that the point marked in the heavens by the Veinal Equinox has been con- 
 ftantly going backward by a fmall quantity every year, from which caufe 
 the ftars appear to have advanced forward as much ; fo that the conftel- 
 lation Jries is now removed almoft into the fign Taurus^ the faid firft ftar 
 y having got almoft 30 degrees forwards from the equinox ; which dif- 
 ference is called the Precession of the Equinoxes, and the yearly 
 alteration is about 50 feconds of a degree, or about a degree in 72 years. 
 
 23. It was faid in art. 12, that the planets revolved round the Sun i^ 
 orbits nearly circular and concentric ; for the feveral phaenomena arifing 
 from their motions (hew they are not ftridlly fo ; and the only curve they 
 can move in, to reconcile all the various appearances, is found to be an 
 Ellipfis : . So that the orbits of the primary planets and comets are Ellipfes 
 of different curvatures, having one common focus, in which the Sun is 
 fixed : But every fecondary planet refpedls the primary planet round which 
 it revolves, as the focus of its elliptic motion. For as no other fuppofitions 
 can folve all the appearances that are obferved in the motions of the pla- 
 nets, and as thefe agree with the ftri(5teft phyfical and mathematical reafon- 
 ing, therefore they are now received as elementary principles. 
 
 24. The line of the nodes of every planet pafles through the Sun : For 
 as the motion of every planet is in a plane paffing through the Sun, con- 
 fequently the interfedions of thefe planes, that is, the lines of the nodes, 
 muft alfo pafs through the Sun. 
 
 25. All the planets, in their revolutions, are fometimes nearer to, fome- 
 times farther from the Sun : This is the confequence of the Sun not being 
 placed in the center of each orbit, and of their being ellipfes. 
 
 26. The Aphelion, or Superior Apsis, is that point of the orbit 
 where the planet is fariheft from the Sun. The Perihelion, or Infe- 
 rior Apsis, is that point where it is neareft to the Sun : And the tranf- 
 verfe diameter of the orbit, or the line joining the two apfes, is called the 
 Line of the Apses, or Aspides. 
 
 27. The planets move fafter as they approach the Sun, or come nearer 
 to the perihelion, and flower as they recede from the Sun, or come nearer 
 to the aphelion. This is not only a confequence from the nature of the 
 planets motions about the Sun, but is confirmed by all good obferva- 
 tions. 
 
 28. "^f a right line drawn from the Sun through any planer, ufually 
 called the Radius Velor^ is fuppofcd to revolve round the Sun with the 
 planet, then this line will defcrihe or pafs through every part of the plane 
 of the orbit j fo that the Radius Velor may be faid to defcribe the area of 
 the orbit. 
 
 29. There are two chief laivs obferved in the Solar Syftem, which re- 
 gulate the motions of all the planets j ni m.^Iy, 
 
 I. T})e planets defcrihe equal areas in equal times : That is, in equal por- 
 tions of time the Radius VeSior defcribes equal areas, or portions of 
 the fpace contained within the planet's orbit. 
 
 12 II. Th^
 
 Book V. ASTRONOMY. 201 
 
 II. The fquares of the periodical times of th^ planets are as the cubes of 
 their mean defiances from the Sun : That is, as the fquare of the time 
 which a planet a takes to revolve in its orbit, is to the fquare of 
 the time taken by any other planet B to run through its orbit ; fo is 
 the cube of the mean diftance of A from the Sun, to the cube of the 
 mean didance of 6 from the Sun. 
 
 ' 30. The Mean Distance of a planet from the Sun is its diftance 
 from him, when the planet is at either extremity of the conjugate diame- 
 ter ; and it is equal to half the tranfverfe diameter. 
 
 31. The foregoing laws are the two famous laws of Kepler, a great 
 Aftronomer, who flourifhed in Germany about the beginning of the 
 17th century, and who deduced them from a multitude of obfervations : 
 But the firlt who demonftrated thefe laws was the incomparable Sir 
 Isaac Newton. 
 
 By the fecond law, the relative diftances of the planets from the Sun 
 are known ; and were the real diftance of any one known, the abfolute 
 diftances of all the others would be obtained by it. 
 
 32. Every thing already faid of the planets is found in a great mea- 
 fure to be applicable alfo to the comets, as well from the obfervations 
 that have been made of them, as from the phyfical and mathematical con- 
 fiderations of their motions. 
 
 33. Were the motions of the planets to be obferved from the Sun, 
 each of them would be ever feen to move the fame way, though with 
 different velocities ; thofe nearer to the Sun running their courfes through 
 the Zodiac in lefs time than thofe at greater diftances : And hence it 
 would happen, that fome of them overtaking the others would in pafling 
 by them appear to be fometimes above, fometimes below, and fometimes 
 as if they touched one another, according to the parts of the orbits in 
 which thofe planets happened to be with refpe^l to their nodes. 
 
 34. When two planets are feen together in the fame fign equally 
 advanced, they are faid to be in Conjunction : But when they 
 are in diredl oppofite parts of the Zodiac, they are faid to be in Oppo- 
 sition. 
 
 35. As the planes of the orbits are inclined to one another, therefore 
 when two planets happen to be in conjunction at the time they come 
 near a node of one of them, they would be feen from the Sun apparently 
 to touch one another ; and the fartheft of thofe planets from the Sun 
 would fee the neareft moving over the face of the Sun like a black fpot, 
 being then dire(5tly between the Sun and the remoter planet j fo the pla- 
 net Venus was obferved from the Earth in the tranfits of the years 1761 
 and 1769. Alfo, fliould an oppofition of two planets happen near a node 
 of one of them, the Sun, being then directly between them, would hide 
 the light of one from the other. Thefe obfcurations, or interceptions of 
 the light of the planets one from the other, are called Eclipses. 
 
 36. The place that any planet appears to occupy in the celeftial fphere, 
 when feen by an obfcrver fuppofed to be in the Sun, is called its Helio- 
 centric place : And indeed all celeftial appearances, as k^ix from the Sun, 
 are called Hdiocentric phanomina 
 
 37. The
 
 20V A S T R O^N^'d K)I Y. Book V. 
 
 37. The following tabic exhibits at once fome of the moft tr<ateria| 
 conclufions that have been deduced from the obfcrvatlons hitherto 
 made, the mean diftance of the Earth from the Sun being reckoned 
 at 1000. ' 
 
 Ta kle of the Solar System. 
 
 5;; 15 
 
 2 
 
 comp< 
 that 
 Earth 
 
 2 
 
 3 
 
 n 
 
 s 
 
 
 
 S- 
 
 n 
 
 
 
 sr ST. 
 
 n s 
 
 
 Ti 
 
 of 
 
 peric 
 
 revolu 
 
 ^* 
 
 3 3 
 
 a. 
 
 2 
 
 c 
 
 
 
 9 2 
 00 ^ 
 
 7r 
 
 ST ^ 
 
 
 0. 
 
 3. 
 
 -a' 
 
 K 
 
 0-0 
 
 
 c ci. 3, 3 
 
 
 
 
 
 
 
 
 3 
 
 
 Mercury. 
 
 ^ 
 
 387 
 
 
 81 
 
 6 
 
 52' 
 
 3 
 
 m. or 8/'' 23*' 
 
 uncertain 
 
 C,^2 
 
 Venus. 
 
 ? 
 
 724 
 
 
 <; 
 
 3 
 
 23 
 
 8 
 
 m. or 224. 17 
 
 2^h qIh^jS 
 
 0,87 
 
 
 
 Earth. 
 
 6 
 
 I COO 
 
 
 17 
 
 
 
 00 
 
 I 
 
 y. or 365 6 
 
 23 56 4 
 
 1,00 
 
 I 
 
 Mars. 
 
 $ 
 
 ^24 
 
 
 .41 
 
 I 
 
 52 
 
 2 
 
 y. or 6S6 23 
 
 24 40 
 
 0.73 
 
 0! 
 
 Jupiter. 
 
 '4 
 
 5201 
 
 
 250 1 
 
 20 
 
 12 
 
 y. or JJ332 12 
 
 9 56 
 
 7.70 4 
 
 Saturn. 
 
 h 
 
 9533 
 
 
 5432 
 
 20 
 
 30 
 
 y. or 10759 8 
 
 uncertain 
 
 4.J9 5 
 
 1 
 
 38. By all the obfervations made on the fecondary planets, it ap- 
 pear?, 
 
 ift. That the flitellltes revolve round their fuperior planets from wcH: 
 to eaft, in curve-lined orbits like ellipfes, the primary planet being in tiie 
 focus, ?nd one of the orbit's diameters diredted towards the Sun. 
 
 2d. That the plan-s of the orbits of the fatellites are inclined to the 
 plane of the orbit of their rcfpedtive planet. 
 
 3d. That, like the primary planets, they defcribe equal areas in equal 
 limes ; and the fquares of the times of their revolutions are as the cubes 
 of the mean diflan.ces from their primary planet. 
 
 In every revolution of a moon round its primary planet, there muft be 
 two conjunflions betv/ixt the planet, moon, and Sun: namely, once, 
 when the moon is in that part of its orbit neareft to the Sun ; and once, 
 v/hcn in that part of its orbit farthefl from the Sun : 'And an eclipfe 
 may happen at either conjunction, according as the moon's nodes hap- 
 pen to be pofited at thofe times. For the plane of a moon's orbit is in- 
 clined to that of its primary, and fo makes two nodes : And whenever 
 the S.i:i, planet, and moon happen to be at the fame time in the line of 
 the node?, there muft be an eclipfe ; which would occur at every con- 
 junction, but for the inclination of the orbit. 
 
 One of the conjunclions, namely, that made by the nioon's going 
 beyond the primary, from the Sun, is called the Superior Con- 
 junction; and the other, made by the fatellite on the fide of the 
 planet next the Sun, is called the Inferior Conjunction. 
 
 The
 
 Book V. 
 
 Astronomy. 
 
 203 
 
 The following table fhews the time taken by each fatellite in its re- 
 volution, and alfo its mean diftance from the primary in femtdiameters 
 of it. 
 
 39 
 
 Saturn's fatell. 
 Dift. from Sat. 
 
 Jupiter's fatell. 
 Dilh from Jup. 
 
 The Moon 
 
 I 
 
 II 
 
 Ill IV V 
 
 d h ni 
 
 I 21 i8i 
 
 8f f.diam. 
 
 d h ni 
 2 17 41} 
 
 I i|f.diam_ 
 
 d h , ni 
 
 3 '2 254- 
 15 f. diam. 
 
 d h m 
 15 22 41I 
 
 36 f. diam. 
 
 d h m 
 
 -9 7. 48 
 108 f.diam. 
 
 I 18 28| 
 5|f.diam. 
 
 3 n.i8^ 
 
 9 f. diam. 
 
 7 3.59T 
 14! f.diam. 
 
 16 18 5-j'o 
 25I f' diam. 
 
 
 29 12 44-^'^ 
 
 and is dill, from the Earth 60-5- femidiameters. 
 
 40. 
 
 Of the figure mid light of the Planets. 
 
 That the Sun and planets are fpherical bodies is evident from all the 
 obfervations that h'ave been made of them ; and that the Earth is of like 
 figure is nut only deduced by analogy, but fufFiciently confirmed by ob- 
 fervation-*. Now although Aftronomcrs generally fay that the planets 
 are fpherical, yet they do not mean a Geometricnl fphcrc, but a figure 
 called an oblate fpheroid, which is fomething like the figure that a flexi- 
 ble fphere would be formed into by gently prelling it at its poles* Ob- 
 fcrvations have determined this in Jupiter, and it is known that the Earth 
 is o^ this figure both from obfervations and actual menfuration. 
 
 That the planets mufl have this oblate figure, is evident from this 
 confideration ; that as they are of matter, and violently whirled on their 
 xes, all the parts v/ould endeavour to fly off, like water from a trundled 
 mop ; thofe in the equator moving fwifteff, have the greateft tenccncy to 
 depart: And although the parts are retained in the fphere by the fuperior 
 force of gravity, yet the equatorial diameters will be fomcwhat increafed, 
 and the j^olar leflcncd. 
 
 41. 'Fhc planets are all opake or dark bodies, and confequcntlv fliine 
 only by the light they receive from the Sun : This is knov/n by ob- 
 ferving, that thofe bodies are not vifible, when they are in fuch parts of 
 their orbits as are between the Sun and Earth, or partly {o. Now, as 
 all the planets fomctimes appear with a ftrong light, therefore the rays 
 they receive from the Sun mulf convey to them a degree of warmth pro- 
 portional in fome meafure to their diltance ; which proportion is recipro- 
 cally as the fquares of the diftances ; and this mull be readily inferred 
 from the heat v^-hich the inhabitants of tlie Earth receive from the Sun, 
 
 42. As a planet revolves on it^ axis, every part of its furface will be 
 turned towards the Sun, and fo enjoy its liglit and heat. 
 
 * Obfervations (hew, that in cclipf-'s cf the moon the darkened part ij 
 bounded by a circular curve ; and confequcntlv the body, which cnlts the 
 fliade, or O'jliruiJb the light, muil be bounded by a like curve: but as thefe 
 obfcurations are caufcd by different parts of the ['".arth, tonfcqcuniiy its lur- 
 face muft be limited by ;i circular ligure, that iy, it mull be globular. 
 
 S J: C IT f ^ N
 
 ao4 ASTRONOMY. Book V. 
 
 S E C T I O N II. 
 
 Of Terreftrial Agronomy, 
 
 43. Terrestrial Astronomy is that which confiders the mo- 
 tiohs of the celeflial bodies as feen from the Earth, which is alfo in mo- 
 tion. 
 
 The motions defcribed in the preceding fection were fuch, as would 
 appear to an obfer\'er viewing the heavens from the Sun : But were he 
 placed in one of the planets, fuppofe the Earth, and there obferved the 
 motions of the reft, the Sun, and other pianets, would appear to him 
 to revolve round the Earth as a center ; but the Sun would be the only 
 one that moved uniformly the fame way : For the other planets would 
 feem to move fometimes from weft to eaft, and then to ftand ftill j 
 then they would ieem to run from eaft to weft ; and after ftanding 
 Ibme time, they would again move from weft to eaft, and fo on conti- 
 nually. 
 
 44. The place in the celeftial (phere that any planet appears in, 
 when feen from the center of the Earth, is called its Geocentric 
 Place. 
 
 45. The Direct Motion of a planet is that by which it appears to 
 move from weft to eaft, and this motion is faid to be according to the or- 
 der ofthefignsy or in confequentia. When the planet appears to ftand ftill, 
 it is faid to be Stationary ; and when its motion is apparently from 
 eaft to weft, it is then called Retrograde, or has a motion in antece- 
 dentia^ or contrary to the order of the figns. Thefe different appearances 
 follow partly in confequence of the obferver being himfelf in motion 
 while he is viewing the motions of the planets, and partly becaufe he is 
 not in the center of the motions which he obferves. 
 
 46. TheThanomeyia cf the Inferior Planets. See Fig. i. Plate IV. 
 
 Let ABC reprefent an arc of the celeftial fphere ; eop the Earth's or- 
 bit ; LNIG the orbit of an inferior planet, as of Venus -, and s the'Sun : 
 Let the Earth at firft be fuppofed to ftand ftill in its orbit at E : Now it 
 is evident that the Sun will appear at the point B, and the planet always 
 within the arc ac. Whilft the planet moves in its orbit from i through 
 <^ to N, it will feem to move from B to a in confequentia : But pafTmg 
 from N to L, it will feem to an eye at E to return back from a to b, or 
 be retrograde. While the planet is at, or near, the point K, and moving 
 as it were in a right line towards the Earth, it will for fomc time feem to 
 ftand ftill near a, and it is then faid to he^Jlationary. 
 
 47. When the planet is in that part of its orbit n or o, which is 
 contiguous to the tangent ea or ec, it will then appear at a or c, 
 its greateft diftance from the Sun, and is faid to have then the greateft 
 Elongation : This elongation is meafured by the angle seg. The 
 rrore diftant a planet is from the Sun, the greater will its angle of elonga- 
 tion
 
 Book V. ASTRONOMY. 205 
 
 tlon be ; that of Venus is about 48 degrees, and that of Mercury about 28 
 degrees. 
 
 In the fpace of a revolution, the two inferior planets will, with refpe<3 to 
 the Earth, undergo two conjunctions ; one when it is beyond the Sun at 
 ij the other when it is at L between the Sun and the Earth ; the former is 
 called the fupericr, and the latter the inferior conjun^isn. 
 
 48. Whilft Venus goes from her fuperior conjundlion, fhe appears in 
 the arc ba always to the eaftward of the Sun, and therefore fets fome tinac 
 after the Sun, and is called the evining Jiar. But during the time fhe is 
 going from her inferior to her fuperior conjunction, Ihe will be feen fome- 
 where in the arc BC to the weftward of the Sun, and fo will fet before him 
 in the evening, and rife before him in the morning ; hence fhe is called the 
 morning Jiar. 
 
 Hitherto the planet only has been fuppofed to move while the Earth 
 flood ftill i but when both move, the foregoing phenomena will be much 
 the fame, only the planet will be more direct in the fartheit part of the 
 orbit, and lefs retrograde in the neareft ; the former arifing from the fum 
 of their motions, and the latter from the difference. 
 
 49. Of the Tbancmev.a cf the Superior Planets, 
 
 The dire J?, Jlatisnary-y and retrograde appearances of the fuperior pla- 
 nets are explained much after the fame manner as thofe of the inferior ones, 
 but with thele differences. 
 
 ill. The retrograde motions of the fuperior planets happen oftener the 
 flower their motions are, as the retrogradations of the inferior planets hap- 
 pen oftener the f.vifter their angular motions are : Becaufe the retrograde 
 motions of the fuperior planets depend upon the motions of the Earth, bux 
 thofe of the inferior on their own angular motions. A fuperior planet is 
 retrograde once in each revolution of the Earth ; an inferior one, in every 
 oi:c of its ov.-n revolutions. 
 
 2d. The fuperior planets ^o not alwavs accompany the Sun as the in- 
 ferior do, but are often in oppcfition to him ; which neceffarilv follows 
 irom the orbit of the Earth being included in the orbits of the fuperior 
 planets. 
 
 5c. Cf the u^^^rent }^[cticn cf the Sun. 
 
 As a rjKclator in the Sun would kc the Earth revolve through the figns 
 in the ecliptic ; fo to a i'pecfator in the Plarth the Sun apparently revohcs 
 the lame v.ay, but is always in the oppofite point of the ecliptic: (For it is 
 well :r.o\vii to every one, efpeciallv to thole who ufe the fea, that hxed ob- 
 jects appear to change their place by the motion of the obl'erver :) So that 
 the hci.occntric place of the Earth, and the geocentric place of Lhe Sun, 
 are always in direct oppof.te points of the ecliptic. 
 
 Now although it is the motion of the Earth that really cautcs a great 
 variety ir. the apparent motions of the other planets, vet as the motion of 
 the Suii bcin:; known i:i\e5 that of Lhe Earth, Lhcrefore Altrcnomcrs fpeak 
 
 of
 
 2o6 ASTRONOMY. ^ok V. 
 
 of the motion of the Siin ; and in their computations ufe the quantkies of 
 tiiofe motions as if they were real. 
 
 51. Bciide the various appearances that arife from the annual motion 
 of the Earth, there arc many refuhing from its diurnal motion : For that 
 the Earth has a daily motion round its axis muft neccflarily be inferred 
 from the moft ftricl rcafoning on the motions of the planets ; and the no- 
 tion, that bodies fo immcnfely diftant as the Itars are, really revolve 
 round the earth in 24 hours, is now treated as a great abfurdity by every 
 one who has rightly confidered thefe things : However, as the motions 
 are apparent, and the fpcaking of them as real is cuftomary and no way 
 affecSts the concluiions j therefore Aftronomers treat of thofc motions as 
 they appear. 
 
 52. Any fphere revolving as on an axis, muft have two points on its 
 furface at the extremities of its axis, that do not revolve at all ; thefe 
 points, with rcfpedt to the Earth, are called its poles. 
 
 53. By the Earth's rotation on its axis from weft to eaft in a day, the 
 furface of the celcftial fphere appears to move from eaft to weft in the fame 
 time ; and all the celeftial objects appear to defcribe circles in the heavens, 
 which are greater or lefs according as they are farther from, or nearer 
 to, the apparent centers of thofe motions : For there are two points in 
 the heavens which are apparently fixed, and the nearer any ftars are to thefe 
 points, the ilowcr are their motions. Thefe points are called the Celes- 
 tial Poles ; the right line joining them is called the Axis of the 
 Sphere, and paflcs through the poles of the Earth ; the circle in the 
 heavens, equally diftant from the poles of the celeftial fphere, is called the 
 Equinoctial ; the correfponding circle on theEarth is called the Equa- 
 tor, which is equally diftant from both the poles of the Earth. 
 
 54. As the Sun's rays falling on any fphere enlighten one half of its 
 furface ; therefore one half of the Earth is always illuminated at once, and 
 confequently the enlightened part is bounded by a great circle, which 
 may be called the Terminator, from its proj:erty of terminating, or 
 bounding, the verges of light and darknefs. Now, by the rotation of the 
 Earth on its axis once in 24 hours, there will be a conftant fucceffion of 
 light on all parts of its furface as they are turned towards the Sun, and of 
 darknefs in thofe parts as they move out of his rays j and hence,arife the 
 viciflitudes of Day and Night. 
 
 55. If the plane of the equator coincided with the plane of the ecliptic, 
 and the axis of the Earth (tood perpendicular to it, the terminator v.ould 
 iilways pafs through the poles of the Earth,. and there would be a conftant 
 equality of day and night in every part of its furf.ce, except at the two 
 poles, where there would be conftant day. But the contrary of this is 
 known to every one, and obfervations fiicw, that the Earth's axis is in- 
 clined to the plane of the ecliptic in an angle of about 65| degrees; there- 
 fore the poles of the ecliptic and equator are about 23^ degrees diftant 
 from one another ; confequently the ecliptic and equinoctial, which in the 
 heavens interfeil: one another in the oppofitc points of Aries and Libra, 
 make at thofe interfcclioxis angles of about 23-^ degrees (IV, 33) : This 
 angle is called the Obliquity of the Ecliptic. 
 
 The axis of the Earth being thus inclined to the plane of the ecliptic, 
 and moving parallel to itfclf in all points of the AnnCal Orbit, or 
 
 2 ecliptic,
 
 Book V. A S T R O N O M Y. 207 
 
 ecliptic, is the occafion of the inequality of days and nights, and of the 
 different feafons of the year j which two phaenomena are explained as 
 follows. 
 
 56. It muft be obfervcd, that the Sun will appear to be vertical to that 
 part of the Earth, which is cut by a ftraight line joining the center? of the 
 Sun and Earth. . 
 
 57. Now when the Earth^is at VS, Fig. 3- Plate IV. the Sun appear- 
 ing then in ffi will be vertical to that point of the terreftrial ecliptic, it ly- 
 ing in the Hght line joining thg Centers of the Sun and Earth. And this 
 point being in tHe Earth's northern hemifphere, all thofs who live there 
 will enjoy fummer, or the hotteft time of the year, the folar rays falling 
 more copioufly, and more perpendicularly, upon their hemifphere at that 
 time. ^ 
 
 58. At the fame time the inhabitants of the fouthern hemifphere will 
 have winter, the rays of the fun falling more obliquely, and in lefs quan- 
 tity, on them, and confequently affording them lels heat- 
 
 59. Again, the inhabitants of the northern hemifphere will have their 
 days longer than their nights, in proportion as they are more diitant from 
 the equator ; while thofe who live under the equator will have an equal 
 (hare of day and night all the year round. For in this pofition the termi- 
 nator, which is always at right angles to the plane of the ecliptic, will pafs 
 53I degrees beyond the north pole, and confequently will cut all the 
 circles parallel to the equator which it meets with into two unequal parts : 
 thole tnat are in north latitude will have the greater portions of thofe pa- 
 rallels in the enlightened hemifphere : but the terminator being a great 
 circle, will cut the equator into two equal parts; therefore half the equator 
 is always illuminated. 
 
 60. tlcnce it necefiarily follows, that thofe who live under the equator 
 will have their days and nights equal : thole who live within the limits of 
 23I degrees round the north pole, will have no night ; and the inhabi- 
 tants between this limit and the northern neighbourhood of the equator 
 will have their nights fhorter tlian their days. In the mean time thofe 
 who live in the fouthern hemifpnere will have their nights longer than their 
 days, in proportion as they approach nearer to the fouth pole ; and the 
 regions contained within the limits of 23!^ degrees round the fouth pole' 
 will have no day. 
 
 61. Suppofe the Earth now to move in its orbit from "^ through the 
 figns c, K to Y, the Sun will feem to run through the figns SI, ^^^ to : ; 
 and this will be th. piace of the Sun in autumn. 
 
 While the P!aith is in T, the days and nights will be equal in both 
 jiemifpheres, and the feafon is a medium between fummer and winter : 
 I" or at that time the Sun will appear vertical to the equator, becaufc a right 
 line joining the centers of the Sun and Earth will tlien cut the furface of 
 the Earth in the equator ; lb that the terminator^, the plane of which is al- 
 ways at riuht angles to the faid line, will pafs through the poles ; confe- 
 quently, all the Earth will then have an equal fhare of day and night. And 
 becaufe the rays of the Sun then fall perpendiculary upon the axis of the 
 Earth, It will then follow, that they muli- fall with an equal obliquity, and 
 with equal number, upon either hemifphere ; therefore they muft enjoy an 
 equal degree of heat and cold, 
 
 Now
 
 2oS ASTRONOMY. Book V. 
 
 Now fuppofe the Earth to move from T to ffi, the Sun will feem to 
 move from s to Yf, where it will be in its neareft approach to the fouth 
 pole ; and at this time of the year it will be winter in the northern hemi- 
 fphcre. For to this hemifphere the like phasnomena will now happen, 
 which did before to the fouthern, when the Earth was in VS ; and by a 
 parity of reafon, when the Earth has got as far as a, and the Sun is ap- 
 parently in Ti the northern hemifphere will enjoy fpring, and the fouthern 
 will have autumn. 
 
 62. The four points of the ecliptic. In which the Earth has been con- 
 fidered in fummer, autumn, winter, and fpring, are called the four Car- 
 dinal Points ; V^and are called Solstitial Points j id and T, 
 Equinoctial Points. 
 
 63. The firft point of Cancer is called the Summer Solstice ; be- 
 caufe when the Sun enters it, which is about the 21ft of June, he has then 
 got to the greateft extent northwards, and being about to return towards 
 the equator, he feems for a day or two to be at a ftand. And for the 
 fame reafon, the firft point of Capricorn, which the Sun enters about the 
 2ift of December, is called the Winter Solstice, withrefpedl to the 
 northern hemifphere. 
 
 64. The firft points of Aries and Libra are called the Vernal and 
 Autumnal Equinoctial Points, from the equality of days and nights 
 all over the furface of the Earth, when the Sun enters thofe points. 
 
 6^. Of the Rifwg and Setting of the Stars. 
 
 There is only one half of the celeftial fphere vifible at one time to any 
 obferver on the furface of the Earth, the other half being hid by the Earth 
 itfelf. Now the apparent plane on which the obferver ftands, feems to 
 be extended to the heavens, and there marks out a circle that divides the 
 vifible from the invifible hemifphere ; this circle is called the Horizon, 
 above which all the celeftial motions are feen. When this horizon is a 
 great circle of the celeftial fphere, it is called the Rational Horizon : 
 but when by the particular fituation of the obferver, he fees more or lefs 
 than half the celeftial fphere, then the circle bounding his view is called 
 tht; Sensible Horizon. 
 
 The horizon is one of the moft ufeful circle~^s in Aftronomy ; for to this 
 circle, which is the only apparent one, almoft all the celeftial motions are 
 referred. It is the common termination of day and night; it marks out 
 the times of the rifing and fetting of the Sun and ftars, and many other 
 particulars, of which hereafter. 
 
 6 6 . Of Parallaxes . 
 
 The Parallax of any objecl is the difference between the places that 
 object: is referred to in the celeftial fphere, when feen at the fame time from 
 two different places within that fphere : Or, it is the angle under which 
 any two places in the inferior orbits are feen from a fuperlor planet, or 
 even from the fixed ftars : But the parallaxes which are moft ufed by 
 Aftronomers are thofe which arife from feeing the obje<Sl from the cen- 
 ters
 
 Book V. ASTRO N O M Y. 209 
 
 ters of the Earth and Sun ; from the Surface and center of the Earth ; and 
 from all three compounded. 
 
 67. The difFerence between the heliocentric and geocentric place of 
 a planet, is called the parallax of the annual crhit (namely, that of the 
 Earth) ; that is, the angle at any planet, fubtended by the diftance be- 
 tween the Sun and Earth, is called the parallax of the Earth'?, or annual 
 orbit. 
 
 68. The difference in the two longitudes is called the parallax of 
 longitude j and that of the two latitudes is called the parallax of lati- 
 tude. 
 
 69. In the Syzicies, that is, in the eppofitions or conjurii^hns, the Sun 
 and planet being equally advanced in the fame fign, or in like places in op- 
 pofite figns, the parallax of latitude is then greateft. 
 
 70. And when the planet is in its Quadratures, that is, Vv'hen it 
 is 90 degrees diftant from the Sun, the parallax of longitude is then the 
 greateft. 
 
 71. To explain the parallaxes which refpe(1: the Earth only. Fig. 2. 
 Plate IV. 
 
 Let Hsw reprefent the Earth, where T is the center; or part of the 
 iMoon's orbit, ?rg part of a planet's orbit, and za.A. part of a great circle 
 in the celeftial fphere. Now to a fpeftator at s upon the furface of the 
 Earth, let the Moon appear in G, that is in the fenfil le horizon of s, and it 
 will be referred to a; but if viev/cd from the center r, it will be referred 
 to the point d, which is its true place. 
 
 The arc ad will be the Moon's parallax ; the angle sgt the parallacSlfC 
 nngle : Or the parallax is exprefled by the angle under which the femidia- 
 rneter ts of the Earth is {etn from the Moon. 
 
 If the parallax is confidered with rcfpedt to different planets, it will 
 be greater or lefs as thofe objecls are more or lefs diftant from the 
 Earth. Thus the parallax ad of G is greater than the parallax Ael oi g. 
 If it is confidered with rcl'pecl to the fame planet, it is evident that the 
 horizontal parallax (or the parallax when the objecl is in the horizon) 
 is greateft of all ; and diminifhes gradually as the lody rifes above the 
 horizon, until it comes to the zenith, where the parallax vanifhes, or be- 
 com.es equal to nothing. Thus ad and Ar/, the horizontal parallaxes 
 of G and gy are greater than a?, and ah^ the parallaxes of R and r ; and 
 the obje<5ls o or p, fecn from s or T, appear in the fdme place z, or the 
 zenith. 
 
 72. By knowing the parallax of any celeftial objccf, its diftance from 
 the center of the Earth may be eafily obtained by Trigonometry. Ehus, 
 if the diftance of g from t is fought ; in the triangle stg, the fide st be- 
 ing known, and the angle sgt determined by obfervation, the fide tg is 
 thence known. 
 
 The parallax of the Moon may be determined by two perfons obfcrving 
 her from difterent ftations at the fame lime, ftie being vertical to the one, 
 and horizontal to the other : and it is generally concluded to be about 57 
 minutes of a degree ; confcqucntly her mean diftance tg is about 60 femi- 
 diametcrs of the Earth, or 60 times ts. 
 
 But the parallax molt wanted is that of the Sun, by which his ab- 
 folutc diftance from the Earth would be known j and thejicc the abfo- 
 
 P lute
 
 lio A _S T R O N O M Y. Book V. 
 
 lute diftanccs of all tlic other planets would be obtained from their relative 
 diftanccs found bv the fccond Kcplcrian Laxv. 
 
 Before the year 1761 fome Altronomers reckoned the Sun's parallax 
 at I2j feconds, others at 10 ; thefe different parallaxes gave very dif- 
 ferent diftance between the Sun and the Earth j the former making the 
 dillances near 8270 diameters of the Earth, and the latter 1031 3 dia- 
 meters. 
 
 But in the years 1761 and 1769 the planet Venus pafl'ed between the 
 Earth and the Sun, and was feen like a black fpot moving over the face 
 of the Sun. Thefe phenomena (which had not happened in more than 
 ICO years before) were obferved by many Aftronomers from different 
 parts of the Earth, and the rcfult of their obfervations make the Sun's 
 mean parallax about 8} feconds, and hence the mean diftance between 
 the ^un and Earth comes very nearly to 11900 diameters of the Earth : 
 And from what was fhewn many years ago by the excellent Dr. Hallcy, 
 if thclc obfervations were made with the accuracy he fuppofed, the dif- 
 lance between the Sun and the Earth might be obtained to kfs than a 
 5C0th part of the whole diftancc. 
 
 73. Of the Meafure of the Earth. 
 
 The relative diftances of the planets are difcovercd by the 2d Kgpla-'um 
 LaiVy and their relative magnitudes are gathered from the angles which 
 they appear under (when viewed with very accurate inftruments) com- 
 pounded with their diflances. Nov/ as tliefe difhuices and magnitudes 
 can by means of the parallaxes be compared with the diameter of the 
 Earth, confequeritly this diameter being accurately known wou!d fervc as 
 a meafure with which the magnitudes and diftances of all the other planets 
 might be compared. 
 
 To find the meafure of the Earth is a problem of fuch importance in 
 Aftronomy, that it has been attempted by fome of the moft confiderable 
 men in almofl all the preceding ages. But its folution was not brought to 
 any degree of accuracy till the year 1635, v.'hen it was very nearly afcer- 
 tained by our countryman Richard Norwood, an eminent mathema- 
 tician at that time. The principle he proceeded upon was this, that as 
 ;^o degrees were contained in e\'ery great circle, both of the celeftial Cphcre 
 and of the Plarth, and as thefe circles are conildcred as concentric to the 
 center of the Earth ; therefore, were the meafure of a degree known on a 
 great circle of the Earth, correfponding to a degree of a great circle of 
 the licavens, then, by analogy, the whole circumference of a great circle 
 of the Earth would be known in that meafure, and confequentjy its dia- 
 meter would be obtained. (II. 197) 
 
 Norwood folved this problem in the following manner : He chole 
 two dillant places which were know to lie nearly north and fouth one 
 of the other, as London and York ; and by a method like that of 
 Traverfe failing (explained in Book VII.) he found their difference of la- 
 titude, or, the diftance between the parallels of latitude palTuig through 
 thnfc places ; or, which is the fame thing, the length of that arc of the 
 tcrreflriaJ meridian. He alfo with a good inftrumcnt found the diftancc 
 
 between
 
 Book V. ASTRONOMY. ill 
 
 between the zeniths of thofe places, and confequently he theiice k:ne\<r 
 the quantity of the celeftial arc anfwering to the meafured terreflrial 
 one. Then faying, As that celeftial arc is to a great circle of the ce- 
 leftial fphere, or 360 degrees ; fo is the arc of the terreftial great circle 
 meafured in feet, to the circumference of a great circle of the Earth ih 
 feet meafure. 
 
 And thus he found that about 69! Englifli miles anfwefe'd to one degree; 
 hence the circumference of the Earth appears to be 25020 miles, and its 
 diameter about 8000 miles. 
 
 By the fame kind of reafonlng, the diftancea found In art; 72. from the 
 parallaxes, were obtainedi 
 
 For I2|": 360" : : 1 femi-diam. : 103680 1 f Circumference of the 
 10 : 360 : : i femi-diam. : 129600 > the < Earth's orbit in femi- 
 8j : 360 : : 1 femi-diam. : 149538} {^ diameters of the Earth, 
 
 And 6,283185 : 103680 : : I : 16539,5 1 f Mean dift. of the Earth 
 6,283185 : 129600 : : I : 20626,4 > the ^ from '^^ Sun, in femi-d. 
 6,283185 : 149538 : : I : 23799,8 ) (of the Earth. (II. 197) 
 
 Then 16539,5 ^ 4000 rr 66158000 f f Mean dillance of the 
 2o>26,4 X 4000 =: 82505600 > the < Earth from the Sun, in 
 23799 8 X 40G0 rr 95 199^00 3 ^ miles. 
 
 74. 0/ the Momt: 
 
 The Moon revolves in her orbit from weft to eaft round the Earthy 
 and is carried perpetually with it through the annual orbit round the 
 Sun, making in the fpace of one year 13 periodical^ and \^ fynod'ical revo->- 
 lations. 
 
 75. A Periodicai. Month, or Revolution, is the time the Moon 
 takes up in revolving from one point of her orbit to the lame point again, 
 and confifts of 27 d. 7 h. 43 m. 
 
 76. A Synodical Month, or ReVolutiokj is the time the Moon 
 fpends in pa/fing from one conjuniStion with the Sun to anotiicr, which is 
 2g d. 12 h. 44m.; being 2d. 5h. i m. longer than the Periodical 
 Month. For whilft the Moorl is paftlng from her forrricr conjun6tion with 
 the Sun round to it again ; the Earth has proceeded forwards in its an- 
 nual courfc, as it were leaving the Moon behind it ; (o that, in order to 
 complete her next conjunction with the Sun, flic muft not only come 
 round to her former point again, but alfo go beyond it. 
 
 77. Bcfidcs this monthly motion of the Moon round the Earth, fhe 
 has alfo a motion round her axis, v/hich is performed exactly in the fame 
 time with her periodical revolution : Hence it comes to pafs, that the 
 fame face of the Moon is always turned towards the Earth, her diurnal 
 motion turning juft as much of her face to us, as her periodical motion 
 turns it from us. 
 
 7 3. Though the fame fide of tiic Moon is ever turned towards us, yet 
 it is not always vifiblc, but fccms daily to put on dili'crcnt appearances, 
 
 P 2 called
 
 ,'4ti ^ ASTRONOMY. Book V. 
 
 billed Phases : For the Moon being an opake ^ody like the reft of the 
 planets, borrows its light from the Sun, having always'onc hemifphcre en- 
 lightened by the folar rays. 
 
 When the enlightened hcniifpherc is wholly turned from the Earth, as 
 at her change or time of new-moon, the planet then being betwixt us 
 and the Sun, the Moon's whole enlightened face, or dijk't muft needs 
 |}C invifiblc to the Earth. When flie paflcs from this Itatc, and turns 
 fome little portion of the illuminated half to us, fhe muft appear horned, 
 the Cusps or points being turned from the Sun towards the ealK When 
 the Moon is in her quadratures, or at 90 degrees from the Sun, then half 
 Xhe ^lli^fTiinated face becomes vifible : She afteiAvards continues to fhcw 
 more than half the enlightened diflc, until fhe comes in oppofition to the 
 Sun or time of full-moon, when the whole of the illuminated orb is prc- 
 faited to us i from whence receding, fhe muft put on the like phafes as 
 before, but in an iiiverfe order, the cufps being now turned towards the 
 weft. 
 
 79. Of Solar and lunar Eclipfes. 
 
 Eclipfes of the Sun and Moon can only happen about the times of the 
 conjunctions and oppofitions : thofe of the Sun fall out at the conjunc- 
 ^ tions, when the Moon intercepts the light of the Sun from the Earth ; and 
 thofe of the %vm- occur in the oppoGtions, when the Earth getting be- 
 tween the Sun and Moon, the latter lofes her light during the time of that 
 interpofition. 
 
 The caufe why there is not nn eclipfe in every fyzigie is the inclination 
 cf the plane of the A4oon's orbit to that of the ecliptic, which is about 
 5^ 18' : for it is certain, that unlefs the Sun, Earth, and Moon, arc all in 
 ^he plane of the ecliptic, or nearly fo, the fliadows of the Earth and Moon 
 can never fall on one another, but muft be directed either above or be- 
 low. Now they can never be in the fame plane, and in one right line, 
 except when the Moon is in her nodes, the nodes and Sun's center being 
 %n t})e fame right line. 
 
 %o. The folar and lunar eclipfes do not happen every year in the fame 
 places of the zodiac, but in fuceeding years they fall in places gradually 
 removed backwards, or towards the antecedent figns : For fmce the nodes 
 are found to go continually backwards, the eclipfes muft alfo oblerve the 
 liime order. 
 
 81. Eclipfes of the Moon are either total or partial : the total happen 
 u'licn the node falls in or near the center of thafhadow : and the partial, 
 when the node happens to be on either fide the center, within or with- 
 out the (hadow. Now the longer the duration of a partial eclipfe is, fo 
 much the greater is that part of the Moon which enters into the fliadow 
 pf the Earth. 
 
 ^2. Hence it is ufual to conceive the Moon's diameter as divided into 12 
 parts, called Digits, by which the greatnefs of partial eclipfes is mea- 
 fured, they being faid to be of fo many digits as they arc parts covered by 
 the Earth's ftiadow : Thus iff of tlie 12 parts are covered, it is called aiv 
 eclipfe of 5 digits. 
 
 83. As the planet Mars is never eclipfcd by the Earth, it is plain 
 tJie Ihadow of the latter does not reach fo far as the orbit of the former, 
 
 bu$
 
 Book V. ASTRONOMY. 213 
 
 but tapers to a point at a lefs diilance ; and confequently the Farth's fha- 
 dow inuft be a cone, the vertex of which is extended beyond the orbit 
 of the Moon. It nilturally follows from hence, that the Sun is a much 
 larger body than the Earth j it is indeed, in diameter, above lOO times that 
 of the Earth. 
 
 84. If a perfon was placed juft: at the vertex, or point of this fhadow, he 
 would fee nothing of the Sun but a fmall rim of light round iiis diflc ; and 
 the farther the obfervcr was removed from the vertex, the larger would the 
 rim of light appear, and confequently the fewer rays would be intercepted 
 by the opake body, till at laft it would appear only as a fpot in the Sun ; 
 in like manner as the planets Venus and Mercury appear when they are 
 feen to pafsover the Sun's difk. 
 
 85. What has hitherto been faid of the fhadow of the Earth includes 
 that of the atmofphere furrounding the Earth : for in lunar eclipfes the 
 {hadow of the atmofphere is to be confidered. And hence it is that the 
 Moon is vifible in eclipfes, the fliadow caft by the atmofphere being not 
 near fo dark as that caft by the Earth. 
 
 86. The Moon always enters the weftern fide of the fhadow with her 
 caftern limb, and quits it with her weftern limb; and in her approach to 
 and recefs from the fhadow, fhe muft pafs through a Pekumbra, or im- 
 perfe<St fhade, which is caufed by the Earth itfelf. 
 
 87. In the fame manner, in which it has been fliewn that the A'loon muft 
 come into the fhadov/ of the atmofphere, Vv^hen fhe is at full and at or near 
 a node, it may alfo be fhewn, that her fhadow muft fall upon the Earth at 
 the time of new Moon, provided fhe is in or near a node : But the pen- 
 umbra of the Moon's fliadow is much more fenfible in folar eclipfes, than 
 that accompanying the fhadow of the Earth in lunar ones. 
 
 88. It is obferved, that to determinate parts of the Earth folar eclipfes 
 are not fccn fo oft as lunar ones ; which is owing to the fhadow of the 
 Moon being Icfs than that of the Earth : For the Earth's fhadow often 
 covers all the Moon ; but that of the Moon cannot cover all the Earth ; 
 and as it foDietimes falls on one part, fometimes on another, it caufes folar 
 cclipfe?, in general, to be more frequent than lunar ones ; yet to any de- 
 terminate place on the P'.arth there are more eclipfes of the Moon vifible 
 than of the Sun. 
 
 What has hitherto been fixid, may fuffice to give beginners a general idea 
 of the motions of the bodies in the folar fyftcm, and of fome of the pheno- 
 mena thence nrifing; thofe who defire to be farther acquainted with parti- 
 culars, may find them fully treated of in M.de la Caille's Elements ofAftro- 
 nomv, publifhcd in Englifh a few years fince*^ and alfo in the works of 
 f ircgory, Kcil, and others. 
 
 * Tranflated by the Author of thefs Elements. 
 
 P3 SECTION
 
 514 ASTRONOMY. Book V, 
 
 SECTION III. 
 ^he AJlronomy of the Sphere, 
 
 r?EFiNjTioN5 and Principles. 
 
 89. By the Aftronomy of the fphere is meant the finding, from proper 
 things given, the meafure of certain arcs and angles formed on the fur- 
 faces of the celeiHal and terreftrial fpheres, by the apparent motions of the 
 bodies which are feen in the heavens. 
 
 The furfaccs of thofc fpheres are fuppofed to be concentric to the center 
 pf the Earth, and to have correfpondent circles defcribed on both fpheres. 
 
 90. Great Circles are thofe which divide either fphere into. two 
 pqua) parts. 
 
 Lesser Circles, thofe which divide the fphere into unequal parts. 
 
 The Poles of a circle are the points on the fphere equally diftant fron; 
 tliat circle. 
 
 An Axis is a right line fuppofed to connect the poles. 
 
 The Celes Ti-'V.L Axis is that right line about which the heavens feem 
 tP revolve. 
 
 The North and South Poles of the world are thofe two points where 
 the ajfis cuts the celeftial fphere. 
 
 91. The Equin'octial or Equator, is tlie great circle of the fphere 
 equally diftant from the poles of tlic world. 
 
 92. Meridians, or Hour Circles, or Circles of Right Ascent- 
 sign, or Circles of Terrestrial Lonc-.itude, are great circles 
 perpendicular to the equator, and pailing through the poles of the world. 
 
 93. The Ecliptic is a great circle inclined to the equator in an angle 
 of about 23%'', aiid cutting it in two points diametrically oppofite. 
 
 'I'he ecliptic is fuppofed to be divided into 12 equal part.-, called 
 Signs, beginning from one of its interfections with the equator ; each 
 fign containing 30 degrees, named ajid noted thus : 
 
 Aries Taurus Gemini Cancer Lro ^ i' g'> 
 
 r 8 -n f?p SI ^^jC 
 
 J^ibra Scorpi-j Sagittarius Capriccrnus Aquarius Pi/ccs 
 
 ~ 111 / Vf :tz U. 
 
 The firft fix are called ncrii-ern, and the latter fix fouthern figns. 
 
 94. The Cardinal Points of the ecliptic arc the four firlt points 
 of the figns Y"> S, :i= , VS ; thofe of T and (t:^ arc called Eqj:inoctial 
 Points, and thofe of S nwd \f are called SoLsTiriAL Points. 
 
 95. The Equinoctial Colure 1^ a meridian palfmg through the 
 iquinoifial points ; and the Solstitial Colure is another meridian 
 paffmg through the foljlitial points. The colourcs cu]t one another at 
 right angles in the poles of the world, 
 
 q6. C^ir
 
 BookV. ASTRONOMY. - 215 
 
 96. Circles of Celestial Longitude are great circles perpendi- 
 cular to the ecliptic. 
 
 97. The Latitude of any point in the heavens is an arc of a circle of 
 longitude intercepted between that point and the ecliptic, and is called 
 north or fouth latitude, as the point is on the north or Ibuth fide of the 
 ecliptic. 
 
 98. Parallels of Celestial Latitude are fmail circles parallel 
 to the ecliptic. 
 
 99. The Longitude of any objetEl in the heavens is an arc of the eclip- 
 tic intercepted between the fiift point of Aries and a circle of longitude 
 palT!i;ig through that point, 
 
 100. The Right V\scensiov of any objeil: is an arc of the equator, 
 contained between the firfl point of Aries and a meridian paffing through 
 that point : Or, it is the angle formed by the equinoctial colure, and the 
 meridian pafling over that point. 
 
 ici. 'ihe Declination of any objccl; is an arc of a meridian con- 
 t.iined between that point and the equinoiilial : If the point is on the north 
 jule of the equinoctial, it is called north declination; but if on the foutli 
 lide, it is called yiw//; declinalion. 
 
 102. The Obliquity of the Ecliptic i? the angle made by the 
 interfection of the equator and ecliptic, and is meafured by the Sun's 
 grcatcft declination ; which, according:; to modern obfervations, is about 
 23-^28'. 
 
 icj. Parallels of Declination are fniall circles parallel to the 
 equinoctial. The Tropic of Cancer is a parallel of declination at 23"* 
 28^. dilbnt from the equinoctial in the northern hemifphere ; and the 
 Tropic of Capricorn is the parallel of declination as far diftant in the 
 fouthcrn hcmilphcrc. 
 
 104. The Arc nc Polar Circle is a parallel of declination at 23'' 
 28^ diitant from tiic north pole; and tiie Antarctic Polar CiRCLi: 
 is the par;.llcl of iicclination as far diftant from the fouth pole. 
 
 IC5. The Zknitii is the point of the heavens dirccHy over a place ; 
 and tlic N ' niR is the pcint directly underneath. 
 
 106. Tiic HoRl/ON is that great circle of the I'phcrc which is equally 
 dilhmt from the zenith and nadir of any place, and di\idcs the fphcre into 
 tiie upper and lower hemifphcrcs. 
 
 1 07. The Rising of a celeflial objccl: is when its center appears in the 
 caftern part of the horiz,on ; and its Setting is when its center difap- 
 pears in ti'.e weilern quarter of the horizon. 
 
 108. Azimuth, or Vr.R'irc al Circles, are great circles perpcntli- 
 cular to the horizon, pailing throui^h its pole?, whicii are the zenith ar-d 
 nadir. 
 
 IC9. The Prime Vertic al Is that vertical circle wliich pafTes through 
 the call: ajid wcfi: points of the horizon, and is at right angles to the 7Kc~ 
 ridian of the place, which is a vertical circle pafling through the north and 
 fouth points of the horizon. 
 
 110. As the meridian of a place Is called the twelve o'clce^ l<our cire!r^ 
 fo the hour circle at right angles to the meridian is called thcy/.v o'clock 
 }>our circle. 
 
 111. The AziMurn of any celeflial objccl: Is an angle at the zer.ith 
 formed by the meridian vfany place, and li vtrticd circle palling through 
 
 V 4 thiL
 
 ai6 ASTRONOMY. - Book V. 
 
 that obje<^ when it is above or below the horizon : AnJ it is meafured by 
 the arc of the horizon intercepted between thofe vertical circles. 
 
 III. The Amplitude of any object in the heavens is ufually taken 
 a> an arc of the horizon contained between the caftern point of it, and the 
 center of the obje6l at its rifing, or between the wcftern point of it and 
 the center of the objedl at its fetting ; or it may be taken as an angle at 
 the zenith, included between the meridian of a place and a vertical circle 
 palTuig through the object at its rifing or fetting. 
 
 1 13. The Altitude of anyobjeflin the heavens is an arc of a vertical 
 circle intercepted between the center of that objedl and the horizon. 
 
 1 14. The Zenith Distance of any object is an arc of a vertical 
 circle contained between the center of that object and the zenith. 
 
 The altitude and zenith diltance are complements one of the other, 
 
 115. The Meridian Altitude, or Meridian Zenith Dis- 
 tance, is the altitude or zenith diltance when the object is on the meri- 
 dian of the place. 
 
 1 16. The Culminating of any celeftial object, is the time it tran~ 
 ftSy or conies to the Meridian. And the Medium Coeli, or Mid- 
 Heaven, to any place, is that degree of the ecliptic, or part of the hea- 
 vens, over the meridian of that place, at any time. Or the Mid-Heaven 
 is the diltance of the meridian from the firlt point of Aries, reckoned on 
 the equinoctial. 
 
 117. The Nonagefimal degree is the 90th degree of the Ecliptic, rec- 
 koned from its interfection with the eaftern point of the horizon, at any 
 given time. 
 
 Confequently the altitude or height of the nonagefimal degree above 
 the horizon is equal to the diltance of the poles of the Ecliptic and Ho* 
 rizon ; and is the mcafure of the angle which the ecliptic makes with the 
 horizon. 
 
 118. Almicanthers, or Parallels of Altitude, are fmall 
 circles parallel to the horizon. 
 
 119. A Parallel Sphere is that pofitioli of the fphere in which 
 the circle?, npparcntly defcribed by the diurnal rotation, are parallel to the 
 horizon ; which can happen only at the poles. 
 
 120. A Right Sphere is that in which the diurnal motions are 
 at right angles to the horizon : Thus it appears in all places under the 
 equator. 
 
 121. An Oblique Sphere has all the diurnal motions oblique to the 
 horizon : And thus the motions appear to all parts of the Earth, except 
 under the poles and equator. 
 
 12?., Diurnal Arcs are thofe parts of the parallels of declination of 
 celeftial objcdts whicli are apparently defcribed between the times of the 
 rifing and fetting of thofe objects : And Nocturnal Arcs are the parts 
 of thofe parallels apparently defcribed from the- time of fetting to the time 
 of rifing. 
 
 123. Semi-diurnal and Semi-nocturnal Arcs, or the halves 
 cf diurnal and nocSturnal arcs, are the parts of the parallels intercepted 
 between the meridian and the horizon. The correfponding part of the 
 equator anfwering to the feaii-diurnal arc, gives the times between noon 
 '.::u :r.': ilfir.g or letting j and the equatCH'ial j^art anfv/cring to the fenii- 
 
 Rofturnal
 
 Book V. A S T H O N O M Y. 217 
 
 nocturiKtl arc, (hews the time between midnight "and the time of fetting 
 or rifing. 
 
 124. The Oblique Ascension of any obje6l in the heavens, is an 
 arc of the equinoctial intercepted between the firft point of Aries and the 
 c;in:ern part of the horizon when that objecSl is rifing ; and the Oblique 
 Desceksion' is an arc of the equinoftial intercepted between the iirft 
 point of Aries and the weftern part of the horizon at its fetting. 
 
 125. The Ascensional Difference belonging to any celeftial ob- 
 ]sci is an arc of the equinocSlial intercepted between the horizon and the 
 hour-circle which the object is on when it rifes or fets ; or it is the dif- 
 ference between the right and oblique afcenfion of that objecSl. In the 
 Sun, it is the time that he rifes or fets before or after the hour of fix, 
 
 126. The Latitude of any place on the Earth is an arc of a ter- 
 reilrial meridian coiitained between that place and the equator ; or it is 
 an arc of a celeftial meridian intercepted between the zenith of the place 
 and the equinoctial ; being north or fouth, according to the fide of the 
 equator it is on. 
 
 127. The Longitude of any place on the Earth is an arc of the 
 equator contained between the meridian of that place and the meridiarji 
 which is chofcn for the firft, where the reckoning of longitude begins : 
 Or, it is the angle at the pole formed by the firft meridian and that of 
 the place. 
 
 128. Refraction, in an aftronomical fenfe, is the difference be- 
 tween the true aiid apparent altitudes of celeftial objects ; they appear- 
 ina; more elevated above the horizon than they really are, on account 
 of the denfity of the Earth's atmofphere, or air and vapours furround- 
 ing it. ^ _ 
 
 129. The Twilight is that medium between light and darknefs, 
 v/hich hi^ppcns in the morning before fun-rife, and in the evening after 
 fun-fet. 
 
 This is occafioncd by the atmofphcrc's rcfrating the folar rays upon 
 any [)lace, iilthough the Sun is below the horizon of that place, and by 
 obiervation it is found to begin and end when the Sun is about 18 below 
 the horizon. 
 
 130. The Crepusculu.m is a fmall circle parallel to the horizon at 
 iB'' below it, where the twilight begins and ends. 
 
 131. The Laiitude of a Place is expreficd by an arc of the me- 
 ridian, fticwing the diftance between the zenith of that place and the 
 equinoctial ; or, by an arc of the meridian, (hewing the height of th 
 pole above the horizon. 
 
 For under the pole, or in the latitude of go degrees, the pole is in the 
 zenith, or is 90 degrees above the horizon j fo that, in this cafe, tiic ho- 
 rizon coincides with the cquinoLtial. 
 
 And as many degrees as the obfervcr goes from the pole towards the 
 equator, fo many degrees does his horizon go below the equator on one 
 iidc, and approach the pole on the other fide. 
 
 Therefore the pole approaches the horizon juft as much as the zenith 
 approaches the equator ; that is, the height of the pole above the hori- 
 zon, is equal to the diftance of the zenith from the equinoclia), which is 
 equivalent to the diftancQ of the obfervcr from the equator, or is equal to 
 tU latitude. 
 
 8 IJ2. A5-
 
 2ig A S T R O N O M Y. Book V. 
 
 132. AsTROtioMTCAL Tables in general contain numbers (hewing, 
 either the nicafure of the diftaiices of the heavenly bodies from certain 
 kmits which are ufed to rcprefcjit remarkable t'lTnes and places ; or, the 
 times when thofe bodies had, or will have, given pofitions relative to thofe 
 limits. 
 
 Some of the chief aftrononiical tables are. 
 Solar and lunar tables for linding the places of thofe luminaries at given 
 times. 
 
 Tables for finding the places of the other planets. 
 Stellar tables for finding the places of the liars. 
 
 'J'ables fhewing the Sun's place, declination, and right afccnfion for 
 given times. 
 
 Tables of refractions for correcting obfervations on altitudes. 
 Tables of the equation of time j or the difference between the tinic5 
 Ihewn by a fun-dial and a well-regulated clock. 
 
 The aftrononiical tables chiefly wanted in this work arc placed at the 
 end of this book ; and are preceded by an account of their conftrudion 
 aiid ufe. 
 
 133. As the Earth makes one revolution on its axis in a common day 
 of 24 hours ; therefore every point of the equator will defcribe the circle 
 of 360 degrees in 24 hours j and confequently, if 360 degrees give 24 
 hours, any other number of degrees will give its proportional hours : 
 And if 24 hours give 360 degrees, any other numb.er of hours will give 
 its proportional number of degrees. 
 
 And hence are derived methods for con\'erting arcs of circles into 
 meafures of time, and meafures of time into arcs of circles. 
 
 To reduce degrees^ minutes^ l^c. to tune. 
 Multiply by 24, and divide by 360 ; or multiply by 4, and divide by 60 : 
 Or, Divide the given degrees by 15 for hours ; multiply the remainder 
 by 4 for minutes, adding to the produ61: i minute for every 15^ of a de- 
 gree J the overplus minutes of a degree, multiplied by 4, give feconds of 
 time, ^c. 
 
 Or thus : Let the quotient of the given degrees by 60 ftand for the firft 
 name ; the remaining degrees for the fecond name ; and the other given 
 names in order following : Then this number multiplied by 4, will give 
 the hours, minute's, feconds, Sic. in order, 
 
 7o reduce time into degrees. 
 Multiply the given hours by 15 gives degrees, to which add 1 for every 
 4 minutes of time; for every overplus minute reckon 15^ of a degree; 
 and for every fecond of time take 15'^ of a degree. 
 
 Or thus : Divide the time by 4, carrying by fixties, the quotient will 
 be in order, fixties of degrees, degrees, minutes, feconds, he. : Then 
 fjxties of degrees snd degree's being reduced, will give the degrees, ^"c, 
 required. 
 
 ExAM;
 
 ^'. I Book V. 
 
 A S T R O N O' M Y. 
 
 ^219 
 
 Exam. I. Reduce 69 20', 45^^, 
 to its correfponding time. 
 
 J5) 6jo' 45" (4" 37- 23' 
 9x4.+ 1 =37 
 
 5X + + 3=^23 
 
 Exam. II. Rednee 4'', 37", 23% 
 /(? zVj correfponding degrees* 
 
 4^ 37"' 23 
 
 60'^ o' o" for 4 hours. 
 915 o for 37 minutes. 
 5 45 for 23 feconds. 
 
 69 20' 4j' 
 
 Exam. III. Redi/ce 237% 44', Exam. IV. i^^^wr^ 15^,50^,58% 
 37 % /a /Vj equivalent time. 
 
 60) 237^ 
 
 3 57 4+ 37 
 4 
 
 Anfwer 15*^ 50'" 58' 28c 
 
 28', to its equivalent degrees. 
 4) 15^ 50'" 58' 28c 
 
 Or 
 
 3' 57 44 37' 
 
 237 44' 37" Anfwer. 
 
 As the Sun is conftantly changing his place, the tables of his right 
 afcenfion fhew for every day at noon (when he comes to the meridian of 
 the place for which the tables are made) what part of the equator is in- 
 tercepted between that meridian and the equinoctial point Y. The ta- 
 bles for the ftars fhew the equatorial arcs contained between the point V^ 
 and the fedtion of circles of right afcenfion, paffing through thofe ftars: 
 The meafures of the arcs of right afcenfion are reduced to time. 
 
 There are few days when one or more ftars do not come to the meri- 
 dian with the Sun, and then they have the fame right afcenfion with 
 bim : Alfo, at fome time of the year, the Sun muft have the fame right 
 afcenfion which any propofed ftar has j though at other times he may 
 have a Icfs, and fo precedes, or comes to the meridian before that ftar; 
 or a greater, and fo follows the ftar, and comes to the nieridian later. 
 And hence is dtrjved the following method. 
 
 Of Finding the Culminating of the Stars, 
 
 J 34. To find the time tvhen any Jlar in the talk ivill be on the meridian. 
 
 Rule. Subtra6l the fun's right afcenfion for the propofed day from 
 ihc right afcenfion of the given ftar j the difference will be the time of 
 the ftar's culminating nearly. Say as 24** is to the daily change of the 
 fin's right afcenfion, fo is the time of culminating, nearly, to a fourth 
 number ; which being fubtra6ted from the time of culminating nearly, 
 will give the true time of the ftar's culminating. If this time be lefs 
 than 12'' it happens in the afternoon ; but if more than twelve hours, 
 ihe cxcefs above 12'* will fhew the time next morning. 
 
 N. B. 24*^ muft be added to the ftar's right afcenfion, if the fun's right 
 -fcgnfion be grcatvft,
 
 A S T R O N O M Y. 
 
 20 
 
 Exam. I. At what time will the 
 flar Ariurus come to the ffieridian of 
 'Loudon on the \Jl of September^ 1 780 ? 
 
 10 44 
 
 U 
 
 Right afcen. of Ardlurus 
 
 Sun's right afcenfion 
 
 Time of culmin. nearly 
 
 And 3** 21 1'" give 
 
 True time of ftar's culm. 3 20 38 
 
 BookV. 
 
 Ex. II. On the 26th of Feb. 1780, 
 i7t what hour will the Jiar Firgin's 
 Spike he on the jneridian of London f - 
 
 14" 5' 42" Virg. Spike's right afc. 
 
 8 
 30 
 
 Sun's right afcenfion 
 I Time of culm, nearly 
 I And 14'^, 36I'" give 
 
 True time of ftar's culm. 
 
 of 
 22 
 
 H 
 
 '3 3S 
 37 \o 
 
 28 
 18 
 
 36 
 
 2 
 
 H 3+ o M 
 
 If the time of the ftar's culminating be wanted for any other meridian 
 than that of Greenwich, or London, add the longitude in time to the 
 time of culminating nearly, if the longitude be weft, or take their dif- 
 ference if it be eaft, and ufe that ilim or difference inftcad of the time of 
 Culminating nearly : obferving, only, in the latter cafe, that if the longi- 
 tude in time be greater than the time of culminating nearly, that the 
 min. and fee. refulting from the proportion, muft be added to the time 
 of culminating nearly, inftead of being fubtracled from it. 
 
 Exam. On the ibth of February., 1784, what time will Syrius he en the 
 meridian of a place which is in longitude 166 30^ E. of London ? 
 
 Rt. afcen. Syrius, 1780 
 Preceffion for 4 years 
 
 6 35' 
 + 
 
 zS 
 11 
 
 JRt.afc, of Syrius, 1784 6'' 35' 
 Sun's right afcen. 22 37 
 
 39' 
 10 
 
 Time of culm, nearly 
 
 7 5^ 
 
 36 
 
 Time of culm, nearly 7 58 
 
 29 
 
 Long, in time 
 
 11 6 
 
 00 
 
 And 3^ 7"', 4 give + 
 
 29 
 
 Difference 
 
 3 7 
 
 24 
 
 True time of ftar's culm. 7 5S 
 
 5^ 
 
 To find if any Jlar in the table will be on^ or near the tneridian at a given 
 time^ reckoned from the preceding noon. 
 
 Rule. To the given time add the Sun's right afcenfion for that time ; 
 the fum (rejedling 24 hours, if above) is the right afcenfion of the mid- 
 heaven ; which being fought among thofe of the ftars, will fliew what 
 ftar will be on, or near the meridian at the time propofed. 
 
 Exam. I. What Jlar will he on the 
 meridian of London about 10 0^ clock at 
 night en the 25/^ January .^ 1784.'' 
 
 Given time 10 hours P. M. 
 Sun's right afcenfion at noon 
 And for 10 hours more 
 Sum (abating 24 hours) 
 anfwers to Syrius. 
 
 10" 
 20 
 
 o" 
 
 3' 
 2 
 
 ^ 33 
 
 Ex. IT. On the icth May., 1784, 
 tvhatjiar will be on the mcrid. ofLond. 
 about 30 min, after 4 in the morning ? 
 
 Given time 16^ 30' 
 
 Sun's right afcenfion at noon 3 iz 
 
 And for 16 hours more 3 
 
 Right afcenfion mid-heaven 19 45 
 anfwers nearly to Altair. 
 
 SECTION
 
 Book V. ASTRONOMY. !:2i 
 
 SECTION IV. 
 
 Of the ProjeBio?! of the Sphere. 
 
 135. ' P R O B L E M I. 
 
 To projeSi the fphere upon the plane ofthefoljVittalcoliire^ or upon the plant 
 if the 7neridian of any place^ thofe planes being fuppofed to coincide. 
 
 For this projetSlion, the eye is fuppofed to be in the firft point of Aries, 
 or the common interfeiflion of the equator, ecliptic, and equino6lial co- 
 lure ; that being the pole of the plane of projeflion, or primitive circle. 
 PI. IV. Fig. 4. 
 
 ifl. With the chord of 60 degrees defcribe a circle PESQ^to reprefent 
 the folflitial colure, the center of which T is its pole. (IV. 62) 
 
 2d. A diameter EQ^will be the equator, and another ps at right angles 
 to it will fhew the equinoctial colure (IV^. 60), or the axis of the world, 
 the extremities of which p, s, will be the north and fouth poles. 
 
 3d. Fo7- the parallels of declination. On the primitive circle, beginning 
 at E and q^, apply the chords of the given degrees of declination, fuppofe 
 every 10 degrees, and alfo the diftances of the tropics and polar circlei 
 from the equator, namely, 23!-" and 66f. Then from the center T in 
 the axis ps produced, apply the refpecStive fccants of the complements of 
 the degrees laid on the primrtive (IV. 58), and thefe will give the cen- 
 ters of the corrcfponding parallels of declination ; from which centers, 
 with the extents to the feveral divifions in the circumference, defcribe 
 the fmall circles 10, 10 ; 20, 20 ; difc. and thefe will be the parallels of 
 declination required : Among which a 55, h VS, are the tropics of Can 
 ccr and Capricorn ; and rr, dd^ the arctic and antarctic polar circles. 
 
 4th. For the circles of right afceufiony or h'Air circles. In the diameter 
 EQ^ produced lay off from the center T both ways the tangents of 15, 
 30", 45"^, 60, 75% refpe(5tively, and they will give the centers of circles 
 to be defcribed through p and s, and cutting the equator in the point?; 
 reprcfcnting the 24 hours ; the folftitial colure being the 12 o'clock, and 
 the equinoctial colure, ps, the fix o'clock hour circles. And in like 
 manner may any other of this kind of circles be drawn. (^V. 75) 
 
 5th. The ecliptic S3 VS is drawn, making with the equator an angle oi 
 231^; the poles of which r, d^ are the interfetftions of the polar circlei 
 with the folftitial colure. 
 
 6th, Parallels of ccleflial latitude are drawn parallel to the ecliptic, in 
 the fame manner as the circles of declination are drawn parallel to the 
 equator. 
 
 7th. Circles of celcflial longitude are defcribed through r, r/, the poles of 
 the ecliptic, in the fame manner as the circles of right afcenfion were de- 
 fcribed through p, s, the poles of the equator ; and thus were the divifions 
 of the ecliptic found that are marked with the figns. 
 
 8th. T}}e horizon is reprefcnted by drawing a diameter hr, making an 
 angle with the axis ps, equal to the latitude of the place ; and the poles of 
 the horizon z, N, the zenith and nadir, are at 90'-' dift. from the circle hr. 
 
 9th. Jzimuthy or vertical circles^ making any angle with the meridian, 
 are defcribed like circles of right afcenfion : I'hus zn' is the prime ver- 
 tical, and 'AN li another azimuth, 45' froui the fouth. 
 
 icth, /^/w-
 
 112 A S' T-R O N O M y. Book V. 
 
 loth. jllmicanthersy or parallels of altitude^ arc in this projection drawn 
 parallel to the horizon, in like manner as the circles of dcelination were" 
 drawn parallel to the equator. 
 
 m 
 
 136. PROBLEM II. 
 
 I0 projei the fphere upon the plane of the horizon. 
 
 In this proje<5lion, the eye is fuppofcd in the nadir, one of the poles of 
 tlie horizon, or plane of projed^ion. Plate IV. Fig. 5. 
 
 ift. The horizon is reprefcnted by the primitive circle, where the upper 
 XII is the north, the lower xii the fouth, e the weft,- and Q^the eaft points. 
 
 2d. The azimuth circles are reprefented by diameters drawn through z^ 
 the center or pole of the horizon : Thus the diameter xii, xil is for the 
 meridian, and ezq^ for the prime vertical 5 and other azimuth circlesj 
 forming any angle with the meridian, are readily drawn by laying off their 
 diftances in the primitive from the north or fouth points. 
 
 3d. Parallels of altitude are concentric to the primitive, and are de- 
 fcribed about the pole z with the half tangent of their diftance from it ; 
 Thus the fmall circle, the diameter of which is ab^ is a parallel of altitude 
 10 above the horizon, or at 80^ diftant form its pole z. 
 
 4th, The diftance of the equinoSlial from the zenith is equal to the la- 
 titude of the place, and therefore this circle makes with the horizon an 
 angle, which is meafured by the complement of the latitude ; then fetting 
 off from the center z in z xn continued, the tangent of 50 (the latitude 
 in this example being 40), it will give the center of the circle eaq^, re- 
 prefenting the equinodlial ; and the half tangent of 50"', fet the fame way 
 from z, will give p, the pole of the world. 
 
 5th. The fix o'clock hour circle paffes through the poles of the world, 
 making with the horizon an angle equal to the meafure of the latitude ; 
 therefore taking in the meridian, from z towards a, the tangent of the 
 latitude 40, it gives g, the center of the fix o'clock hour circle epq^ 
 
 6th. The hour circles pafs through the poles of the world, and make 
 with one another angles of 15 degrees : Therefore (IV. 55) in a line de, 
 drawn through o, at right angles to the meridian, (cX off on both fid^s of 
 G the tangents 15% 30"", 45'^, 60, 75, to the radius pg, and they will' 
 give the centers of the feveral hour circles palling through p, cutting the 
 horizon and equinocilal in the hour points. 
 
 7th. The polar circles^ tropics^ and other circles of declination^ are de- 
 fcribed parallel to the equinoctial, about its pole p, at given diftances 
 from it, either by finding the centers of fuch parallels j as fliewn in B. IV. 
 66 ; or by fetting off on each fide of z the half tangents of their greatelt 
 and leaft diftances from z ; then the middles of thufe intervals are the 
 centers fought. Thus ; the arclic circle is diftant from p 23! ; then to, 
 and from zP=:5r)% add and take 23! ; there remain 73^ and 26^ ; the 
 half tangents of thcfc fet off from z give p and q \ then a circle defcribed 
 on the diameter pq is the artic circle. 
 
 In like manner will the centers of the tropic of Cancer c 05 r, and of 
 Capricorn d VS d, be obtained. 
 
 8th. The northern portion of the ecliptic 7" S i^ defcribed from a- 
 center diftant from z towards p, the tangent of 7 3!'', = A the ecliptic 
 makes with the hgrizon, 
 
 9th. C/''-
 
 Book V. ASTRONOMY. 223 
 
 gth. Cirdes of longitude T p =^, p b% / U, /> SI, /> "R, are defcribed thro* 
 ^, the pole of the ecliptic, from centers in the line bfc ; in like manner 
 :as the hour circles were defcribed through p, the pole of the equator. 
 
 loth. Circles of cdeftial latitude^ww q ix, are defcribed about p^ as the 
 circles of declination were defcribed about p, the pole of the equino<5lial. 
 
 137. PROBLEM Iir. 
 
 2I7 projccl the fphcre upon the plane of the equator. 
 
 In this projection the eye is fuppofcd to be in one of the poles of the 
 equator, fuppofe in the fouth pole, and projcdling the north hemifpherc. 
 Plate IV. Fig. 6. 
 
 ill. The equator is reprefented by the primitive circle, the center and 
 pole of which is p. 
 
 2d. The hour circles are expreffcd by diameters making angles of 15^ 
 with one another ; of which xii p xii is the meridian, or folflitial colure, 
 and VI p VI the 6 o'clock circle, or equinoctial colure. 
 
 3d. Circles of declination are circles parallel and concentric to the equa- 
 tor, defcribed from its center,with radii equal to the half tangents of their 
 icvcral diftances from the pole p, or half co-tangents of their degrees of 
 declination : Thus pq the arctic circle, and a ss th'e tropic of Cancer, 
 .:rc defcribed with the half tangents of 23!" and 66| refpe(5lively j and 
 fo of the others. 
 
 4th. The ecliptic making an angle of 23 p with the equator ; the tan- 
 cent of thefe degrees laid from p towards a will give the center for de- 
 fcribing tlic ecliptic Ysr^, the pole ol' which p is in the polar circle. 
 
 5th. Circles of longitude are defcribed through py the pole of the eclip- 
 tic, in like manner as the hour circles were defcribed through p, the pole 
 of the equator in the lad problem ; and thus were the divifions ^, ix> 
 a, 'W, obtained. 
 
 6th. Circles of celejlial latitudt are projected in the fame manner as the 
 circles of declination in the laft problem. 
 
 7th. The horizon of any place, fuppofe of London, being inclined to the 
 equator in an angle equal to the co-latitude, 38 1^' ; the tangent of this 
 laid from p towards S, and the half tangent laid from p to z, will give 
 the center, and z the pole, of the horizon hor. 
 
 8th. The prime vertical m.K making an angle with the equator equal 
 to 51 32^, the latitude of the place, its center is found by laying the tan- 
 gent of 51 32^ from p towards o. 
 
 9th. Azimuth circlesy making given angles with the meridian zo, are 
 thus defcribed : \.\\ a line drawn through the center of the prime vertical, 
 at right angles to the meridian, take diftances from that center, equal to 
 the tangents of the propofed azimuth angles, the fcmidiametcr of the 
 prime vertical being the radius, thofe diftances give the centers fought j 
 and thus was the azimuth circle za defcribed. 
 
 icth. Parallels of altitude arc defcribed about z, the pole of the hori- 
 zon, at the diftances of the co-altitudes, in the fame manner as the cir- 
 cles of declination were defcribed about p, the pole of the equator in the 
 laft: problem ; and thu> was the fmall circle v b vii defcribed at iC' dif- 
 taucc from the horizon, or 80 diftant from its pole z. 
 
 138. P RO-
 
 444 ASTRONOMY. Book V. ' 
 
 138. P R O B L E M IV. 
 
 To project the fphtre upon the plane of the ecliptic > 
 
 The eye is here fuppofed to be in one of the poles of the ecliptic, aiici 
 thence viewing the northern hemifphete* Plate IV. Fig. 7. 
 
 ift. The ecliptic is here rcprefented by the primitive circle, the center 
 of which p is its pole. 
 
 2d. Circles of longitude are here rcprefented by diameters ; thofc that 
 make angles of 30 with one another, being drawn through the divifions 
 marked with the Hgns of the zodiac. 
 
 3d. Parallels of celcftial latitude are circles defcribcd about ^, concen- 
 tric to the ecliptic ; fuch is the fmall circle, the diameter of which is ab^ 
 reprefenting the parallel of 10 of latitude. 
 
 4th. The equator making an angle with the ecliptic of 23! ; therefore 
 the tangent of this inclination laid from p towards S will give the center 
 of the equator T xii ti ; and the half tangent of 235; laid from p the 
 fame way, gives p for the pole of the equator. 
 
 5th. Tlie equinoSiial colure^ which here makes xhzfix o"" clock circle^ makes 
 an angle with the ecliptic of 66^ ; therefore the tangent of 66| laid 
 from p towards Yf, gives the center of the 6 o'clock circle Y" p i. 
 
 6th. Hour circles pafling through p, and making angles of 15 with 
 one another, are defcribed from centers, found in a right line pafTing 
 through the center of Y p =^, and drawn at right angles to the folftitial 
 colure T /) kf ; by laying ofF in that line the tangents of 15, 30^, 45% 
 60", 75^, reckoned from the center of T p ^, on both fides, the femi- 
 diameter of this circle being the radius. Thefe hour circles cut the 
 equator in the hour points. 
 
 7th. Parallels of declination^ fuch as the tropic of Cancer, and the arctic 
 circle, the diameters of which are 12, 12, and pq^ are defcribed by laying 
 off from * the half tangents of their greateft and leaft diftances : Thus q 
 being diftant from p 47, makes pq:=.\ tangent of 47% the middle oi pq 
 will be the center of the polar circle. 
 
 8th. The horizon HOR is to make an angle with the ecliptic equal to 
 the difference between the co-latitude and the obliquity of the ecliptic, 
 when p is projected to the -north of p j otherwife that angle is equal to 
 the fum of thofe quantities. And for London, where the faid difference 
 = (38 28' 23 28'=) 15 00', the tangent of 15 00^, gives the cen- 
 ter of HOR, and the half tangent gives z the zenith. 
 
 9th. The prime vertical hzr is defcribed by laying from />, towards o, 
 the CO- tangent of /)Z for a center. 
 
 icth. A%imuth circles are defcribed through z, making given angles 
 with the meridian zo, by finding their centers in a line drawn through 
 the center of hzr, in the manner defcribed for the hour circles, Prob. II. 
 nth. Parallels of altitude are rcprefented by defcribing fmall circles 
 parallel to the horizon hor, at given diftances from it ; or, which 
 comfes to the fame, defcribing fmall circles about the pole z, at diftances 
 equal to the complements of the given altitudes : And thus the circle 
 cdi was defcribed for a parallel of 33** of altitude. 
 
 SECTION
 
 Book V. A S T R O N O M 1f . -25 
 
 'S E C T I O N V. 
 
 Problems of the Sphere* 
 
 139. ^ PROBLEM V. Pl.V. 
 
 Given the Sun's longitude, and the obliquity of the ecliptic; 
 Rquired the Sun's right afcenfion and declination. 
 
 Exam. Let the obUqiiky of the ecliptic, or the Sun's greatejl declinatisni 
 ie%'i iW, mid the Sun's place 13 lb' in Taurus : Required the rejl. 
 
 Construction. 
 In the primitive circle pesc^, reprefenting the folflitial colure, the center 
 ofv/hich is y, drav/ a diameter EQ^for the equator, and at right angles to 
 EQ^draw a diameter ps for the equinoctial colure : Make E S =23'' 28'', 
 and draw a diameter 05 vy for the ecliptic, in which (IV. 71.) take T-,; = 
 43 16' for the Sun's diftance from the point T : Through pQs defcribe 
 a circle of right afcenfion. 
 
 Computation. See Book IV. art. 130, 131. 
 In the right angled fpheric triangle y b. 
 
 Given Sun's longitude T = 43*^ 16'' f Req. right afcen. VB. 
 
 Obliquity of the Eclip. Z. T B =: 23 28 | declin. Bi- 
 
 To find the declination. | To f.nd the right afcenfion. 
 
 As Radius 1= r iOj-'ooooI As Radius r io.coooo 
 
 To f. Sun's Ion. = 43 6' 9,53594 fo t. Sun's Ion. := 43 16' 9>9737t 
 Sof. ob. cclip. = 23 28 9.60012 So co-f. obi, eel. 23 28 9.962^1 
 To f. Sun's decl.rz 15 50 c;, 43006 : To t. rt. afcen. =. 40 48 9, 93-^22 
 
 140. While the Sun is moving from T to 55, or is in the finl quadrant 
 of the ecliptie, the given longitude is the hypothcnufe in the triangley b, 
 the declination BQis north, and T B is the right afcenfion. 
 
 When the Sun has pail: the folftice S), and is defcending towards --, he 
 is then faid to be in the fecond quadrant, and his longitude or diftance from 
 T being taken from 180, the remainder tc>. becomes the hypothenule, 
 and the declination is ftill north ; but the arc b tii found for the right 
 afcenfion is only the fupplement, and mud therefore be taken from lSo. 
 
 The Sun having paft the point t:^, and defcending towards yp has got 
 into the third quadrant ; the longitude then, reckoned from 7", will ba 
 greater thaii 180 : In this cafe the exccfs above 180% or the diftancc the 
 Sun is removed from :, will be the hypothcnufe r^ ; the declinatioti 
 will be fouih ; and the a;c =^ a, found for the right afcenfion, muit bs 
 added to l8c, to give the right afcenfion eftimatcJ from y. 
 
 When the Sun has paft the foiftice \t -, and is afcending towards Yj '^3 
 is tlKii in the fourth qualrant; ther^iorc tlie longitude is greater thin 
 ajc, and muft be taken from 3&':^, to give :!ie hyp<;thcn'.ife ;^ . Here 
 the declination is fotr.h, arJ tlw right ;dcc-nfi(;ii ^: a, found by the pr .-"por- 
 tion, muit be taken from 360'% to give the ;:,y;lit afccnfioii from >". 
 
 At cqu^l diftaiices from the e-quinocciai points Y ox- ro , ih- ^Sun v.ili 
 havetqu;tl quantities of declination j but Vviil be of t'lfForcnt i.aiiies, ac-- 
 cording ai it is on th: r.Oitii oi fouth fjJ.es ci iiic scuir.jcti^i. 
 
 Vcj- I, - Q, 141. PilQ.
 
 226 
 
 ASTRONOMY. 
 
 Book V. 
 
 I4i< 
 
 PROBLEM VI. 
 
 Pl.V. 
 
 Given the obliquity of the ecliptic, and the Sun's declination j 
 Kcquired the Sun's longitude and right afcenfion. 
 
 Exam. The ohliqulty of the ecliptic^ being 2^" 28'', wheit is the Sun*s Ungi- 
 iudc and right afcenfion when he has 20 43^ of north declination F 
 
 Construction. 
 Having defcfibcd the folflitial colurc, and drawn the equator f.q^, the 
 Jucis PS, and the ecliptic S yf, as before j make e, q, cquid to the given 
 declination, and (3d 133) defcribe the parallel of declination n, its in- 
 fcrfelion with the ecliptic' gives O the Sun's place i through P, 0> s, 
 defcribe the circle of right afcenfion p s. 
 
 C O M T* U T A T I O N. 
 
 In the right-angled fpheric triangle y O C 
 Given the ob. eclip. Z. ( T b = 23 28'' 1 Required Sun*s long. Y 0. 
 the Sun's decl. O B = 20 43 j rt. afcen. Y b. 
 
 To find the Swi's longitude. 
 As f. obiK]. eclip. 23 28' 0,39988 
 Tofjn Sun'sdecl.rrzo 43 9,54.869 
 J>o radius zzs, 10,00000 
 
 To fin.01ongu.=:62 40 9,94857 
 
 To find the Sun^s right afcenfion. 
 A? Radius ira i6,ocec<J- 
 
 To co-t.obl. eclip. =23 28' 10,36239 
 So tan. Sun's dee. rzzo 43 '^t^'llfz 
 
 To fin. ft. afcen. =60 36 9,94ori 
 
 Therefore the Sun is in H 2 40^, or in 27 20'', according as the 
 time of the year is before or after the fummer folitice. 
 
 142. PROBLEM VIL PI. V. 
 
 Given the obliquity of the ecliptic, and the Sun's right afcenfion j 
 Required the Sun's longitude and declination. 
 
 Exam. JVhen the Suns right afcenfion is 60 31^, what is the longitude 
 end prefent declination^ the obliq. of the eclip. being 23" 2.W I' 
 
 Construction. 
 The folflitial colure, equator, axis, and ecliptic being defcribed as 
 before, niake Y b =: given right afcenHon (4th 133), and defcribe ther 
 circle PBS, cutting the ecliptic in O the Sun's place. 
 
 Computation. 
 In the right-angled fpheric triangle Y" b j the leg V B and Z O T 
 being known, the hvpoth. y O, and other leg Ob, are found as in art. 
 137, 138. Book I V/ 
 
 As Rad. . co-f. ob. eclip. ; : co-t. rt. 
 
 [;if. : co-t. () long. 
 
 As Rad. : co-f. xj" 2b' : : co-t. 60" 3 1 ' 
 
 [: co-t. 62 35' 
 
 .As Rad. : tan. obi. eclip. : : fin. rt. 
 
 [af. : tan. decl. 
 
 As Rad. : tan. 23'* z8' : : fin. 60" 31' 
 
 [: tan. 20 42' 
 
 Three other problems may be formed out of the four things concerned, 
 or obliquity of the ecliptic, declination, longitude, and right afcenfion : 
 But thefe being of little more importance than as an exercife for right- 
 *ji;rlcd fpheric triangles, they are tliereforc omitted, 
 
 14^. PRO-
 
 Book V. ASTRONOMY. 
 
 227 
 
 143. PROBLEM Vlir. ' PI. V. 
 
 Given the latitude of the place, and the Siki's declination ; 
 Required the Sun's altitude and azimuth at 6 o!clock. 
 
 Exam. At London^ in lat. 51 32^ M, on the longeji day^ when the Sun's 
 declination is 23 28^; Required the Sun's altitude and azimuth at 6 o'clock 
 in the morning or evening. 
 
 Construction. 
 Defcribe the meridian, draw the horizon HR, and prime vertical ZN ; 
 make RP=:Iatitude 51 32' N. ; draw the 6 o'clock hour circle PS, the 
 equator eq^, the 23 2W N. parallel of declination n m, cutting the 6 
 o'clock hour circle ps in O ; and through z, O, N, defcribe the azimuth 
 circle z O Nj cutting the horizon in a ; then the things given and re- 
 quired fall in either of the triangles z O P or T Q A, they being fupple- 
 mental triangles one to the other. 
 
 Computation.. 
 In the fpheric triangle z O p, right-angled at p. 
 
 Given the co-latit. zp = 38 28' l Required the co-altitude z . 
 the co-dccl. Gp 66 32 3 the azimuth z. O zp. 
 Or in the fpheric triangle T A O, right-angled at a. 
 
 Given the latit. A T 0=5i 32'* Required the altitude aQ. 
 
 the dccl. Y O =23 28 3 the co-azimuth T a. 
 
 To find the altitude A O . 
 As RadJLM rr r 10,00000 
 
 To fm. dec). = 23 28' 9,60012 
 Sofin.lar. =51 32 9.89375 
 
 To fin. alt. i3 10 9,49387 
 
 To find the azimuth Aiu 
 As Radius zr R. 10,00000 
 
 To co-f. lat. = 5i32' 9,79383 
 So tan, decl. :r: 23 28 9,63761 
 
 Toco-t. azim. zr 74 53 9,4314.4 
 
 For the arc ar mcafurcs the /LRZa, the azimuth. (IV. 9) 
 
 I4.4. On the fhorteil day at London, the parallel of S. declination cuts 
 the 6 o'clock hour circle below tie horizon ; and as the triangles TaO, 
 Y^Oj arc congruous, the deprcifion hclovv the horizon, on the fhorteft 
 dav at 5 o'clock, will be equai to the altitude at the fame hour on the 
 longeft day ; and the azimuth will alfo be equal, if eflrimated iVom the 
 fouth. 
 
 So that on the 211I of June, at London, the Sun will bear N. 74 
 53' E. at 6 o'clock in the morning, and N. 74 53'' ^V^ at 6 in the 
 evening; but on tlie 2ifl: of December, at the fame hours, it will bear 
 S. 7453'E., andS.7V^53'W. 
 
 rrom a due connderr.tinii o\ this Problem it is evident, that as the de- 
 clination increafos, tiie altitude incrcafe*: and the azimuth jcd'er.s ; ani 
 the contra.'-y happens whi!j the declination \% diminifhing : So that oii 
 the day!", of the cqunioxes, on which the Sun has no declination, the ahi- 
 tudc at 6 o'clock v.'ill be nothinq', or the Su!i will be in the liorizon , snd 
 the azimuth being then f;o degrees, the Sun will be due eaft in rhc morn- 
 ing, and v/clt in' the evening -, that is, on the davs of the equinoxes the 
 Sun riils "nd fcts at hx, in the eafi; and v.'eft points of the horizon. 
 
 0,2 i45> PRO-
 
 aaS AST R 0,N O M Y. Book V, 
 
 1^5. P R O B L E M IX. PI. V. 
 
 Given the latitude of the place, and the Sun's declination ; 
 Required the altitude and hour when the Sun is due eaft or weft. 
 
 Exam. At London^ in latiiudc 51 32^ N.^ what is the Sun's altitude, 
 and the hour zvhen he if due eajl or ivej}^ en the iongej} day^ or when ths- de- 
 ciinattjn is 23" 28' N. ? 
 
 Construction. 
 Dcfcribe the primitive circle to reprefcnt the meridian ^of London, 
 dravvr the hnri/.on iiu, and the prime vertical ZN j make Rpn5i32', 
 the given latitude, draw the 6 o'clock hour circle ps, the equator eq^, 
 the parallel of declination nm (3d 135), cutting the prime vertical in O, 
 and through pQs dcfcribe (11. 72) the hour circle pQs-, cutting the 
 equator in a. 
 
 Htrc the things concerned in the Problem fall in either of the triangles 
 pzO or Tag. 
 
 Computation. 
 In the fpheric triangle PzQj right-angled at z. 
 . Given the co-latit. rz = 3S iW r Required the co-altitude zo 
 
 the co-decl.po ^6 32 j the hour fr. noon ZiZPQ. 
 
 . Or in the fpherical triangle TaQ, right angled at A. 
 Given the hatit./LATG =51 32'' 1 Required the altitude y' . 
 
 the decl. aO =^23 28 \ the hour after 6 T a. 
 
 Tafind the altitude TO- [ To find the hour after 6. 
 
 To fin. decl. aO=23 2S 9,60012 
 So Radius =:r. io,cooco 
 
 To fin. sit. 7^0=30 34. 9,7063; 
 
 As Radius -=.9. 10,00000 
 
 Toco-t. lat. aYO ==5 1*^32' 9,90009 
 So tan. decl. a=:z3 28 9,63761 
 
 To f. h. fr. 6. ATr:2o 11 9,53770 
 
 Which 20 II'' converted into time (132), gives i h. 20 m. 44 s. for 
 the time after 6 in the morning, and before 6 in the evening, when tiie 
 Sun will appear due eaft or welt ; which will be at 7 h. 20 m. 44s. in 
 the morning, and 4 h. 39 m. 16 s. in the afternoon. 
 
 Or, the compl. of 20 ii', viz.. 69 49^ put into time, which gives 
 4 h. 39 m. 16 s., (hews the time before and after noon, when the Sun 
 will be due eaft or weft.. 
 
 146, This Problem worked for the fhorteft day, namely in the A V^0> 
 which is congruous to 7" A, would give the Sun's depreffion at the 
 time when' he v/as eaft or weft, which would be before 6 in the morn-' 
 ing, and after 6 in the evening, by as much as was found above, viz. 
 I h. 20 m. 44 s. 
 
 By this Problem it appears, that when the latitude of the place, and 
 the Sun's declination, have the fame name, then, the greater the declina- 
 tion and latitude, the greater the altitude and time from 6 : and having 
 contrary names, the fame things- happen ; but with this diftercnce, that 
 in the former cafe the days lengthen on account of the incrcai'e of the la- 
 titude and decliiiationi whereas in the latter cafe the days ftiortcn on 
 that account, 
 
 147. PRO-
 
 Book V. 
 
 ASTRONOMY. 
 
 CL29 
 
 147. 
 
 PROBLEM X. 
 
 PI. V* 
 
 Given the latitude of a place and the Sun's declination: 
 Required his amplitude and afcenfional difference. 
 
 Exam. Jt London, lot. 51 -p.' N. on the lift of June, bcln^ the longefi 
 day, xvhen the Sun's decl'nation is 23 1%' N. How far from the north does 
 the Sun rife and ft^ at what time, and what is the length of the day and 
 night F 
 
 ' Construction. 
 
 Let the primitive circle reprefent the meridian.of the place, and the 
 diameter hr the horizon ; from R, the north point, take Rp:=5i<' 32' for 
 the latitude, draw the axis, or 6 o'clock hour circle ps, and at right angles 
 to it draw the equator Ec^^; make En, 0^/2 = 23 2S', the declination, and 
 (3d 135) dcfcribe the parallel of declination ;; ot, cutting the horizon in 
 (^, the place of the Sun at its rifing and fetting 3 through which delcnbe 
 (II. 72) the hour-circle p s. 
 
 Computation. 
 
 Now as the arc qr co-latitude, meafures the Z.QjrR. 
 
 In the fpheric triangle T ,*) A, right-angled at a. ^ 
 
 Given Sun's dccl. a3 2328^ ^Required* the ami^litude T J) 
 co-latit; 4- AT" =r 38 28 3 the afcen. dift. y- a. 
 
 As fin.A Y'O.co-l'^SS" 28' 0,20617 
 Toiin.decl. aO '^^3 28 9,60012 
 So RaJias zza. io,ODOoo 
 
 To fin. amp. '\fQzz7g 48 9,80629 
 
 To find the amplitude f Q . 
 
 This 39 48'' is the amplitude rec- 
 koned from the eaft or weft points of 
 the horizon : But its complement 
 50 I2''fliews how far from the north 
 the Sun rifes or fcts on the longeil 
 d.vf at London, 
 
 To find the afcenfional difi'crcnce T \. 
 
 As Radius v, 10,00000 ! Which 33 o-f converted into time 
 
 Tot. Iat.z.p-|r0=5i32' 10,09991 (132) cives 2h. 12 m. 28 s. for the 
 So tan. dec], a Q =23 28 9.63761 t,ir:e which the Sun rifes before, and 
 
 , ^ ..._,, fets after, the hour of lix on the 
 
 101. al.dilT. K^A "^-XX 07 c,7^7C2 1 ,,ri j 
 
 ii I y/j/? iongeit day, 
 
 Suppofe r% to be a parallel of declination as far fouth, as m n is north ; 
 tlic;! the hour circle pies, pafTuig thioagh (i) the place of the fun at its 
 rifing- or fctting, will form a triangle Y Cv Br= AT A, where^the ampli- 
 tuJe lb to the fouthward of the eaft and weft poiiits. 
 
 148. Hence It is evident, that when the latitude and drclinatiyn hav^ the 
 fur.': i:afnc, i'ce Sun rifes htfore, and fits after 6 . Bat ,'.h,nt hey arc of coti^ 
 trury Hur^iij t'jc Sun rijes afery and ft^ bcfr-: i\ 
 
 0.3 MO- ^'>^
 
 2 
 
 12 
 
 o. 
 
 28 
 
 8 
 
 12 
 
 28 
 
 3 
 
 47 
 
 32 
 
 i6 
 
 H 
 
 S6 
 
 7 
 
 35 
 
 04 
 
 230 A S T R O N O ,M Y. Book V. 
 
 149. And as the Sun defcribes the parallel of declination w in 24 
 hours, being at n when it is noon, and at m when it is midnight ; therefore 
 the time in paffing from 772 to (', or the time of rifmg being doubled, 
 gives the length of the night j and the time of fetting being doubled, muft 
 give the length of the day. 
 
 Then to, and from 
 Add aud fubcradl the afcen. diff. 
 
 Sum, gives fetting 
 
 DifF. gives rifing 
 
 Length of day is 
 
 Length of night is 
 
 But when it is the fhortcfl: day at London, which is, when the Sun has 
 23 28'' fouth declination ; then the lengths of the day and night change 
 places J the day being 7 h. 35 m. 04s. long, and the night 16 h. 24m. 
 56 s. ^ 
 
 150. When the latitude and declination have the fame name, the difFe- 
 rcnce between the right afcenfion and the afcenfional difference, is the ob- 
 lique afcenfion } and thtir fum is the oblique defccnfion.. 
 
 But when they are of contrary names, their fum is the oblique afcenfion, 
 and their difference is the oblique defccnfion, 
 
 151. When the declination is equal to the co-latitude of anyplace 
 (which can only happen to places v/ithin the polar circles), than the pa- 
 rallel of declination will not cut the horizon, and confequently the Sun 
 will not fet in thofe places during the time his declination exceeds^the co- 
 latitude : And the fame may be laid of all thofc liars, the polar diftance 
 of which is lefs than the latitude of the place ; or, which is the fame 
 thing, that have declinations lefs than the co-latitude, for thofe ftars will 
 never defcend below the horizon of that place. But this is to be under- 
 flood only when the Sun or ftars arc in the fame hemifphere with the given 
 pkce ; for when the Sun or ftars are in a contrary hemifphere to any place, 
 the co-latitude of which does not exceed the declination of thofe celeilial 
 objects, then they v/ill never rife above the horizon of that place, and con-^ 
 ftquently are never vifible there. 
 
 J52. PRO-
 
 ^ook V. 
 
 ASTRONOMY. 
 
 231 
 
 152. P R O B L E M Xr. PI. V. 
 
 Given the latitude of a place, the Sun's declination and altitude; 
 , Required the hour from noon, and the Sun's azimuth. 
 
 Exam. In the latitude of $1 'pf N. the Sun's altitude was obfervedto he 
 46*' 20', when his declination was 23 lW N. What was the Suns azimr^th, 
 and the hour when the obfervation was jnade ? 
 
 Construction. 
 Let the primitive cirtie zrnh reprefent the meridian of London, hR. 
 the horizon, zn the prime vertical ; make rp = 5i 32^ the height of the 
 pole at London ; draw the axis ps, and the equator EQ^j lay off the declina- 
 tion E, qw, 23" 28^ N. the altitude Hr, rj, 46 10' \ and (IV. 68) dc- 
 fcribe the parallel of declination n m, and the parallel of altitude rj, cu 
 ting one another in , the place of the Sun at that time ; through z, O , n, 
 defcribe an azimuth circle zGn, and through p, O, s, defcribe an hour 
 circle p s : Then the angles Qzp, 0pz, being meafured (IV. 72), will 
 give the azimuth and hour from nobn required. 
 Computation. 
 In the oblique-angled fpheric triangle p z. 
 Given the co-latitude - zpr:38' 28''7 ^^fj'J'^^fl t^s ^''^Im. A (? zp 
 the co-alt. or zen. difl:.z0r:43 40 ^ajid the h. fr. noon A ^ PZ, 
 the co-dec. or pol. difl:.(i^p=:66 32 3 See art, i6y. Book IV. 
 
 To find the azi?nuth Z. zp. 
 
 Then Co-ar. fin. co-!at. ==38'' 28' 0,20617 
 
 Co-ar. fin. co alt. ^^43 4 0.16086 
 
 Sin. ^ fum co-decl. & d rr^j 52 
 Sin, ^ difF. co-decl. & orijo 40 
 
 Here 20=43" 4' 
 - zrr=38 28 
 
 jV) zp= 5 
 
 I 2ZZX) 
 32 
 
 2)- 
 
 71 41 
 
 61 
 
 3> 52 
 30 40 
 
 :6782 
 
 '076I 
 
 The fum of the four logs. 
 The |- fum gives 56 3i{' 
 
 19,84246 
 
 9,92125 
 
 Which doubled, gives 113 03' for the azimuth 
 
 fought, reckoning from the north. 
 
 To find the hour from noon., Z. pz. 
 Here p =66 32' 
 pz=38 28 
 
 r0 pzr:23 4 = 
 02=43 40 
 
 2)- 
 
 44 
 
 5 3 
 
 35 52 
 
 Then Co-ar. fin. co-decl. 
 
 Co-ar. fin. co-lat. ^=^38 28 
 
 Sin. I fum co-alt. & d ^^35 52 
 
 Sin. I diff. co-alt. & d = 7 48 
 
 :r:66 32' 0,03749 
 0,20617 
 9,76782 
 9,13263 
 
 The fum of the four logs. 
 The I fum gives 21 55' 
 
 19,14411 
 
 9,5-206 
 
 This doubled, gives 43 50' for the meafure of the 
 hour from noon, which is 2 h. 55 m. 20 s. 
 
 Hence it appears, that the obfervation was made either at 9 h. 4 m. 40 s. 
 in the morning, or at 2h. 55 m. 20 s. in the afternoon. 
 
 The azimuth being firlt found, the hour from noon might have been 
 found by the proportion between oppofite fides and angles. 
 
 0.4 Had
 
 34 
 
 A S T R ON O M Y. 
 
 Book V. 
 
 Had the declination and latitude been of contrary names, the fame kind 
 of operation would have been ufed to find the things required, only the 
 fide (5) P would have been obtufe ; by adding the declinat. to 90, inftcad of 
 fabtrading it, a? in the cafe of the lat. and dccl. having like names. 
 
 J53- 
 
 PROBLEM XII. 
 
 PI. V. ] 
 
 Given the latitude of the place, and the Sun's declination ; 
 Required the time when the twilight begins and ends. 
 
 Exam. At what time does the ftvi light brgin and end at London, when the 
 Sun's declinaiivn is 15" 12^ N. the latitude of the place being 51 32^ A'"* 
 
 Construction. 
 Let the circle iRNH reprefent the meridian of the place, HR the hori- 
 zon, zn the prime vertical, and ts the Crepufculum, orfmall circle parallel 
 to the horizon defcribed at 18 degrees below it (IV. 68) ; lay oft' the lati- 
 tude RP, draw the axis ps, the equator eq., and defcribe the parallel of 
 declination n m, and where n m cuts / j in 0, is the Sun's place at the time 
 of the beginning or end of the twilight; through defcribe (11. 72) the 
 vertical circle z n, and the hour circle P s ; then the/izp beingj mea- 
 fured (IV. 72) mW give the time before or after noon as required. 
 
 Computation, 
 In the oblique-angled fpheric trianglii z p. 
 
 " Given the co-lat. zv 38 28' Rcq. the hour from noon=:Z.zP 
 the polar dift. p=: 74 48 ? The manner of folution is the 
 the zenith dift.z=:io8 CO J fime as in lafl Problem. 
 
 Here p?r: 74 48' 
 ri=z 38 28 
 
 FT pzrr 36 20: 
 
 "^ (Jz 108 00 
 
 D 
 
 144 20 
 
 -> , ,_ 
 
 72 10' 
 
 71 4c 
 
 ^5 50 
 
 Then Co-ar. fin. polar dift. =74"4S' 0,01 547 
 Co-ar. fin. co-latit. '^^3^ 28 0,20617 
 
 Sine I fum. of zen. d. & 0=^:72 10 9,97861 
 Sine I difF. of zen. d. & D 35 50 9,76747 
 
 The fum of thefe four logs. 19,96772 
 
 The half fum gives 74 28 1;' 9,98386 
 
 Which doubled, gives 148 57' forZ.zp. 
 
 And 148" 57'' reduced to time gives 9h. 55 m. 48 s. either before or after 
 noon ; that is, the twilight begins at 2h. 04m. I2s, in the morning, and 
 ti.uj at 9h. 55 ni. 48 f.ja the evening on the given day, at London. 
 
 154. When the declination becomes greater than the difference be- 
 tv/cen the co-latitude and 18 degrees, then fhe parallel of declination n fn 
 will not cut the parallel^ i 18 degrees below the horizon, and confe- 
 ijuently i^ that time there will be no night at that place, but the twi- 
 light will continue from Sun-fetting to Suji-iifing ; and on this account iif 
 is, tliat from the 22d of May to the 2iil: of July nearly, there is no total 
 da;kn.rs at London, the Sun's (icelinatioiuiuring that interval being greater 
 th.m 2c^ 28^, which is the diflcrencc between 18'^ and 38" 28', the comple- 
 ji.uut of the latitude. 
 
 155. PRO.
 
 Book V. 
 
 ASTRONOMY. 
 
 ^33 
 
 153. PROBLEM XIII. PI. V 
 
 Given the time of the year, the latitude of a place, and the- altitude o 
 
 3 known fixed ftar ; 
 Required the hour of the night when the obfervation was ipade. 
 
 Exam. Some time in the nighty en the I/? of September 1780, fuppofe 
 
 ihejlar Ariiurus^ the declination of which is 20^30'' N.JhouU he obferved at 
 
 Lo7ii}on to be 2"^ li! abiVe the horizon : At what hour would tin obfcrua- 
 
 tion be made f . - ' 
 
 Construction. 
 
 Defcribe the meridian of the place, draw the horizon hr, the zenith 
 and nadir of which are z and N, and defcribe the parallel of altitude rs at 
 27 12' above the horizon ; take p the north pole 51 32' above the ho- 
 rizon for the latitude of the place, and s the fouth pole as much below the 
 horizon ; draw the equator eq^, and defcribe (3d 135) the liar's parallel of 
 declination n m ; and where this parallel n m cuts the former r j in #, ii 
 the pofition of the ilar at the time of obfervation ; defcribe (11. 72) the 
 vertical circle z# n, and the hour circle p ^ s, and the angle zp # beiyg 
 meafured (IV. 72) gives the hour from, or to, ths time of the liar's cul- 
 minating. 
 
 Computation. 
 In the ollique-angled fpheric triangle P -^ z. 
 
 Given the co-latitude pz = 38- 28'') d j / w l i. 
 
 . ,.-. J 2 o f Kcqmred [_ zP %, or the hour 
 
 the co-altitude z*rz:62 4.0 > > 1 . 
 
 , , j-n c \ from culmmatmu;. 
 
 the polar dilt. * p =09 30 j ^ 
 
 Here p# =59' 
 
 30 
 
 pz=:38 
 
 28 
 
 F^ Pz=r3i 
 
 2 rr 
 
 p:^ =r62 
 
 48 
 
 2) 
 
 3 
 
 5c'-i^ 
 
 46':,- 
 
 Then Co. ar. fin. co-lat. =38" 
 
 Co. ar. fin. pol. diil. rfiQ 
 Sin. V fum zcn. dill. & 0=346 
 Sin. V difF. zen. dift. & 0:1:15 
 
 The fum of the four logs. 
 The \ fum gives 35 51' 
 Which doubled, gives 71 42'r 
 
 28' 
 30 
 55 
 S3 
 
 0,20617 
 0,02841 
 9,86354 
 9.43724 
 
 1903536 
 
 9,7676s 
 
 This ']v 42' turned into time (132) gives 4h. 46 m. 48 s. for the time 
 
 which has clapfcd fuice the flar was on the meridian. 
 
 Now, at the time of obfervation, September ifl, 1780. 
 The right afccnricn of Ar6\urus was I 4'' 
 
 The right afcen-lon of the fun at noon f 10 
 
 Time of culminating nearly 
 
 And 24*" is to 3' 37' as 3*^ i\\ is to 
 
 The Par fputh.s or culminates at 
 
 'i he time that the ftar has paffed the meridian 
 
 (133) 
 
 5 
 
 4+ 
 
 4.- 
 
 34 
 
 The fum is the hour of the night 
 
 + A.lronomical ubles at the end of Bock V. 
 
 3 
 
 21 
 
 08 
 
 30 
 
 3 
 4 
 
 2D 
 
 46 
 
 3 
 
 4 s' 
 
 8 
 
 07 
 
 26 P. iVf. 
 
 
 
 
 And
 
 ft34 
 
 ASTRONOMY. 
 
 Book V. 
 
 And whether to fubtrail or add will always be known by the ftar's 
 being in the caftern part of the horizon, or afcending ; or by being in the 
 Yveileru part of the horizon, or delccnding. 
 
 156. PROBLEM XIV. Pl.V. 
 
 Given the obliquity of the ecliptic, and a ftar's right afcenfion and de- 
 clination ; 
 
 Required its latitude and longitude. 
 
 Exam. What is the latitude arid longitude of a jiar^ its right afcenfion 
 heing 16 h. 14 ;., its declination 25*^ 51' A^., and the obliquity of the ecliptic 
 23 28^? 
 
 C O N S T R UX T I O N. 
 
 Let the primitive circle reprefent the folftitial colure, in which dra\* 
 the equator eq^, mark its poles p, s, and defcribe (3d 135J the parallel of 
 the ftar's declination n m. 
 
 The right afcenfion 16 h. 14 m.=:243 30', which being 63*" 30' above 
 180, falls in the third quadrant ; therefore make (IV. 75.) Y ^=63 30', 
 defcribe (4th 135) the circle of right afcenfion, cutting the parallel n ; 
 in -Jf , the point of the heavens reprefenting the flar. 
 
 Make =23 28', the obliquity of the ecliptic, draw the ecliptic 
 ffi VS, find its poles p, q^ and through p, i(-, q defcribe a circle of longi- 
 tude ; then the arc /> 5^ meafured (IV. 70) will give the co-latitude, and 
 lheZ.P/> ^ will fliew the longitude. 
 
 Computation. ' 
 
 In the oblique-angled fpheric triangle pv^. 
 Given the obliq. ecliptic pv r= 23 28' 1 Required the co-lat. p ^, 
 the co-declination P^ rz 64 og [ and the longit. Z. P/) *. 
 the right afcen.iL/>P3^ =243 30 J See art. 150, 151. B. IV. 
 
 To T"nd the latitude. 
 
 As co-f. 4th arcr=6i 
 
 As Radius = r 
 
 To co-f. rt. afc. =26 30' 
 
 So tan. co-decl.=:64 09 
 
 'J 
 
 lO.OOOCO 
 
 9.95179 
 10,31471 
 
 To tan, 4th arc =61 
 Obi. edipt. 2^ 
 
 34 
 28 
 
 10,26650 
 
 Fifth arc =:38 
 
 06 
 
 To find th 
 
 As fine 5th arcr: 38 
 To fine 4th3rc=: 61 
 So tan. rt. afc. =. 26 
 
 06' 
 34 
 30 
 
 20969 
 9>9447 
 
 9'69774 
 
 34 0,32227 
 To co-f. 5th arc rr 3 8 06 9)89594 
 So fine decl. =25 51 9,63959 
 
 To fin. lat. =46 c6^- 985771 
 
 Here the ftar's latitude Is 46 o6-j-'N. 
 becaafe the declinat. is N., and greater 
 than the obliquity of the ecliptic. 
 
 Here the longitude 144 36' being 
 added to 90, gives 234 36' for the 
 ftar's longitude, reckoning from the 
 firft point of Aries. 
 To tan. longit. = 234 36 9,85160 
 
 For the right afcenfion being in the third quadrant, the ftar is there alfo. 
 Now in 234 36', are 7 figns 24 36'' ; that is the flar's place, or longi- 
 tude is 24" 36' in the 8th fign, or 24*^ 36' in lU. 
 
 By the precefllon of the equinoxes, the fixed flars, although they al- 
 ways keep the fame latitudes, yet are contiaually altering their longitude, 
 
 right
 
 Book V. 
 
 ASTRONOMY. 
 
 ^3S 
 
 right afcenfion, and declination ; the alteration in longitude Is uniformly 
 50 feconds and 3-ioths yearly (22)^ but that of the right afcenfion and 
 declination is conftantly varying : So that many ftars, which once had 
 north declination, come to. have fouth j while others change from S. to 
 N. declination. 
 
 157. PROBLEM XV. PI. V. 
 
 Given the right afcenfions and declinations of two fixed ftars j 
 Required their diftanbe. 
 
 Exam. Wlyat is the dijlance between the fixed Jiars^ Betelguefe in theeajl 
 JhoulderofOrioHy and Aldebaran in Taurus; the former- having 72i'iv. 
 declinat. with 57j. 43?^. 16 j. right afcenfion , and the other id" o^f N, 
 declinat, with 4 h. 23 m. 20 s. of right afcenfion. 
 
 Construction. 
 
 As Aldebaran precedes Betelguefe in right afcenfion, let the primitive 
 circle reprefent the circle of right afcenfion paffing through Betelguefe; 
 defcribe the circle of right afcenfion pas, making with pbs an angle of 
 19 59', (IV. 75) equal to the difference between the given right afcen- 
 fions. 
 
 Defcribe the parallels of declination ew, rs, at the given diftances 16* 
 03' N. and 7 21' N, (3d 133); and the interfedtions a, of Aldebaran's 
 declination, and B, of Betelguefe's, with their refpedlive circles of right 
 afcenfion, will be the pofitions of thofe ftars from one another : Then 
 draw a great circle bag, through b and a, and the intercepted arc ba (mea- 
 fured by art. 70. Book IV.) {hews the diftance of thofe two ftars. 
 
 Computation. (IV* iSi} 
 
 In the oblique-angled fpheric triangle pab. 
 Given the co-decl. of Aid. pb = 73 57''} 
 
 the co-dec. of Betelguefe pa = 82 39 C Required their difl. AB. 
 
 diff. of right afcen. z. ape =19 39 j 
 
 Radius =:r. 1 0,00000 1 Co-f. 4th arc . 82 11' 0,86645 
 
 To co-f. difF rt. af. 19 59 9,97303 To co-f. 5th arc S 14 9*99550 
 
 Asco-t. Betelg. dec. 7 21 10,88944 As fin. Bctelg. dec. 7 21 9,10697 
 
 To tang. 4th arc 82 11 10,86247 
 
 Aldeb. co-dec. 
 Remains 5 th arc. 
 
 158. 
 
 73 57 
 
 H 
 
 To co-f. dift. 
 
 21 25 9,96892 
 
 The fame refult would have come 
 out, had the declination of Aldebaran 
 been ufed in the proportions. 
 
 PL V. 
 
 PROBLEM XVI. 
 
 Given the latitudes and longitudes of two known fixed ftars j 
 
 Required their diftance. 
 
 Exam. Aliath, in the Great Bear, Lon. "^ 5'' 49' Lot. 54 18' .V. 
 Ariturus, in Bootes, Lon. -^ 21 lO Lat. 30 54 A^. 
 
 The conftruction of this Problem is like that of the laft; only inftead 
 of circles of right afcenfion read circles of longitude, and ufe parallels of 
 latitude inftead of parallels of declination. _*
 
 ^^6 ASTRONOMY. Book V.- 
 
 The compotarion Is alfo like that of the laft,' there beiHg given two co- 
 latitudes and the included angle, which is the difference between the given 
 longitudes : Thus Ar<ftupus's long, is 6' 21 lo', and Aliath's is 5* 5 49'; 
 their diftcrence is I* 15 21', or 45 21'. 
 
 Hence the diftance will be fouiid to be 39 45'. 
 
 159. . PROBLEM XVir. Pi. V. 
 
 Given the latitude and longitude of a fixed ftar, and alfo the obliquity 
 
 of the ecliptic ; > 
 
 Required the right gfcenfion and declination of that ftar. 
 
 Exam'. Suppoje the hiltude of ajlar is 7 09' N.jts longitude Y aq^Ol' : 
 JVhat is the right afcenfion and declination of that Jiar, the obliquity of the 
 ecliptic being 2-^"" 2%^ I* 
 
 The conftrudlion of this Problem Is much Hke that of Prob. XIV. ; 
 only-here the interfe6lIon of a parallel of latitude cb with a circle of longi- 
 tude ^ Ay, will give the place of the flar. 
 
 The computation is alfo as in Prob, XIV; for here are given p/ = 
 23 28^ />A=:82 51', and thez.P/)A=r6o 59', the longitude from the firft 
 point of 55, to find pa the co-declination, andz./>PA the right afcenfion. 
 
 The declination of the ftar will be found to be J7''49' N. (IV. 151) 
 
 And the right afcenfion will b? 24 19^ (1^ ^5^) , 
 
 160. ' . PROBLEM XVIIL PI. V. 
 
 Given the meridional altitude of any celcftial obietS', fuppofe a comet, 
 
 its diftance from a known ftar, and the latitude of the place j 
 Required the declination and right afcenfion of that comet. 
 
 Exam. Suppofe a comet was ohferved on the tneridian at London, when its 
 altitude ivas 51^ 55', and its dijiance from the far Ardurus was 59 47'; 
 fp^at was the declination and right afcenfion of i!)^ comet at that time ? 
 
 Construction. 
 
 In the primitive circle, reprefenting the meridian of the place, draw the 
 horizon hr and prime vertical zx ; lay ofF the given latitude Rpzrfi'^ 32'', 
 draw the axis ps, the equator eq^, and (3d 135.) n ??i ArClurus's parallel 
 of deciiriation=::20 21^ From the fouth poiiit of the hori2,on lay oft" the 
 given altitude of the cometz::5i' 55^ from H to o : About the poiiu o as 
 a pole, at the given diftance between the comet and Arclurus, defcribe 
 (IV. 68) a fmall circle ^^/ cutting the' parallel //; 'in "^^ the pofition of 
 Ardturus at that time : Defcribe the circle of right afcenfion p % s, and 
 a great circle thro^igh o and "Jf . 
 
 Computation. 
 
 Since HF.r=38" 28', the co-l.it. and the alt. H0=:5i'' 55', then Eo=: 
 (Ho-^HEr:) 13" 27', i? the dec!, (biiglu ; which is north, as the altitude 
 e<>,';ccds the co-Ut. ; confcqucntly the- polar diftance o'p~'](f 33^ 
 
 "' - ' 6 Then
 
 Book V. A S T R O N O M Y. 237:. 
 
 Then in the trjangle P-Jf o are given the three fides to find the Z.op -Jf , 
 the difference betWecii the right afcenfions of the comet and Arcturus. 
 And (IV. 154) theZ.OP* will be foundrr62 24'=:4h. 9 m. 36 s, which 
 is the difference of their right afcenfions : Now if Arcturus had pafled the 
 meridijgin, the right afcenfion of the comet was 18 h. 15 m. 16 s. but if 
 Arlurus had not pafled the meridian, the right afcenfion of the comet was 
 9 h. 56 m. 4 s. ; it being, in the former cafe, equal to the fum, and in the 
 latter to the diff. of Ardurus's right afcen. and theZ.OP ifr' ' 
 161. PROBLEM XIX. , ; PLV. 
 
 Given the latitude of a place, the Sun's declination and Azimuth ; 
 
 Required his altitude and the time of the obfervation. 
 
 Exam. In the latitude <?/ 13 30' N, and when the Sun has 23' 28' A''. 
 declination : What is the Sun's altitude and time of the day, when he is feen 
 9n the ENE azimuth circle P 
 
 Construction. 
 
 Let the primitive circle reprefent the meridian of the place, in which 
 HR reprefents the horizon, and ZN the prime vertical ; make rp equal to 
 the latitude, draw the 6 o'clock hour circle ps, the equator eq^, and (3d 
 153^ the parallel of 23'^ 28' of declination n rn : The tangent of 67 30' 
 being laid from the center a towards h, gives the center of the vertical 
 circle ZDN, which cuts the parallel n m in the points a and B ; and fhews 
 that at two diftant times in the forenoon the Sun will have the azimuth 
 propofed : Through the point A and b defcribe the hour circles pas, pbs 
 (II. 72); the angles zpa, zpb, fhewing the times from noon, may ber. 
 meafured by art. 72. Book iV. ; and the altitudes da, db, by art. 70, 
 Book IV^ 
 
 Computation. See art. 146. Book IV". 
 
 In the fphcric triangle pza, or pzb, there are known. 
 
 The co-lat. pz = 76'3o'i the co-decl. pa, or pb 1=66 32'^ the azim^ 
 Z-vzo by" 30'. 
 
 To find zA, or zB, and the Z-ZPA, or A zpb. 
 
 As Iladius : co-f. azimuth ;: co-t. lat. : to tan. of a 4th arc m=z^j'^ 54'. 
 
 And as fin. lat. : fin. dec!. : : co-f m to co-f of a 5th arc k=:24 56'. 
 
 Then m + n, or 57''54'-j-24"56' 82''5o^=za, is the comp. of lead alt. 
 
 And M N,or 57 54' 24''5t) 32'^58'~zB, is the comp. of thegr. alt. 
 
 Therefore, when the Sun nas 7^ 10'', or 57^02', of alt. he is on the given 
 azimuth. 
 
 Again, As co-fdec. : fin.azim. :: co-f. leafl alt. : fin. hour fr. noon 87 55'. 
 
 And as co-f dec. : fin. azini. :: co-1". great .r alt. : iin. h.fr. noon 33 14^. 
 
 But 8755' = 5'^ 51"' 40' ; and 33-' i^^ 2^ 12^'' 56% the refpcct. times 
 fr. noon. 
 
 Confcqucntly the Sim v.-;:I be f^.'cn on the ENE azimuth at 6h, 08m. 
 2':s. ar,d again at 9 h. 47 r.i 4 s. both in the morning: Alio, in the? 
 afternoon he will be on the "vVN'.V azimuth at 2 h, 12 m. 56 s. and at 
 5h. sun 40s. 
 
 lO?.. Nov/ to find at what tinic, and at what altitude, the greatcfl azi- 
 muth will happen at that place on th'j fi;u d.'.y ; 
 
 As the azinuith rir(.lo in tliis ca(c is to touch tlie parallel k w, tlicrc- 
 forc t!ij grcatcfl diltancc of the aziniutli tViini the equator will be 23 28' ; 
 and as th'.ir pdcs mull t:c at the fame d;lt jicc (IV. 73} therefore a finall 
 circle r, dcfcriJvJ ab',,-.;t tiic ^ u!c i, at lui; diitancc of 23' 28 (lV^ 66), 
 
 its
 
 4^8 AS T k O N O M Y. Book V. 
 
 its interfelIon / with the horizon, is the pole of the circle zcn j then dc 
 
 fcribe an hour circle pes through p. 
 
 In the fpheric triangle zpc right angled at e. (IV. 34) 
 
 Given the co-lat. pz=:76 30^ and tiie co-decl, pcr:66 32^ 
 Required the ^reateft azim.Z.Pzc=:7o 37^ the dift. 2c=:54'' 08'', and 
 
 the hour from noon =56" 26'= 3 h. 45 i?i. 44 s. 
 
 So that the azimuth is altered only 3 7'' in 2 h. 6 m. ; and confe- 
 
 quently the variation of the compafs may be oblerved with more certainty 
 
 in the torrid zone than elfewhcre. 
 
 163. P R O B L E M XX. PI. XIV. Fig. i. 
 
 Irt the latitude of 20 OO'' N.Jlandi a horizontal dial, the gnOTnon of which 
 is perpendicular to the plane of the horizon : It Is required to know at what 
 hour tn the afternoon on the longefl day, the Jhadow of that gnomon Jhall Ji and 
 jVill -y and how many degrees ftjall the Jhadow rUn back. 
 
 Construction. Let the circle iRNH be the meridian of the place, MR 
 the horizon, the poles of which are z, n ; rp=::2o, the latitude p and s 
 the north and fouth poles : About p defcrlbe the tropic of Cancer dr^, cut- 
 ting the horizon in l j about s, a fmall circle being defcribed at the dift.of 
 23 28^, the complement of the dift. of aa from p, its interfeition p with 
 the horizon, is the pole of the azimuth circle which will touch the parallel 
 aa in 0, the place of the Sun when he has the greateft azimuth that day : 
 Through z, 0, n, defcribe a vertical circle cutting the horizon in K j and 
 through O and l defcribe the hour circles pOs, pls. 
 
 Then will thez.zP0 be equivalent to the hour when the fhadow will 
 {land ftill ; and kl, the difference between the meafures of the azimuth 
 and amplitude, will fliew how much the fliadow will run back. 
 
 Computation. In the right-an- 
 gled fpheric triangle prl. 
 
 Given Z.R = 90 00' 
 
 lat. rr Rp =:: 20 00 
 
 co-decl. =: PL rz 66 32 
 
 In rt. angl. fpheric trianglfc P0b. 
 Given Z. 90 00'' 
 
 co-lat. =: ZP =70 00 
 
 co-dec. m P0 = 66 32 
 
 Required hour:=: z.zp = 33 06 
 azim. =z.zp =: 77 28 
 
 Requir. ampl. = rl = 64 56 
 
 Then 33 02 = 2 h. 12 m. 08 s, : And 77 28^ 64 56^=12 32^ 
 
 So that the fhadow will ftand ilill at 2 h. 12 m. 08 s. and will run back 
 
 12 32^ 
 
 164. PROBLEM XXr. PI. XIV. Fig. 2, 
 
 A comet, the declination of which ivas 47^ 00' N. zuas ohfcrved to he dljlant 
 fro7n ajlar, to the ea/huard of It, 49 00''; the fiar' s declination was 36 00' 
 N. and Its right rfcenjion 45 00^ ; Jl'hat was the latitude and longitude of 
 that comet P 
 
 Construction. 
 On the plane of the folftitial colurc, where p and E are the poles of the 
 equator and ecliptic, put the ftar at a by its right afcenfion and declina- 
 tion : About p and A defcribe fmall circles, at the dillances of the comet 
 from thofe points, their interfec^ion ^ives the place of the comet ; de- 
 fcribe great circles throughA O , P O , e O , then E G will be the co-latitude, 
 and Z.PE O the co-longitude ; and their meafures may be obtained from 
 articles 70 and 72 of Duck IV, 
 
 CoMPU-
 
 Book V. ASTRONOMY. ^^j^ 
 
 In the triangle ep * 
 Given comet's co-decl. p3=:43 oo' 
 obliq. of ecliptic pe=:23 28 
 comet's rt. ar.Z.EP 2=^69 12 
 
 Computation. In the triangle 
 
 AP. 
 
 Given the -Jf-'s co-dec. pa = 54'' 00' 
 comet's co-dec. . P^=43 00 
 their diftance Ar--49 00 
 
 Req.cora.rt.af.fr.^2LAP:^:=:65 48 
 Then 65 48^45 00=20'^ 48^ 
 Andgo^oo^ 2O''48'=:69i2''=r0PE 
 
 Req. comet's co-lat. 3 = 3.9 54 
 
 and co-longit. z. PHC?ir:83 42 
 Which being taken from 90"^, leaves- 
 Y 6 18' for the long, required. 
 
 155. PROBLEM XXII. PI. XIV. Fig. 3. 
 
 At London^ on the lotb of Decernber 1780, at what time of the night wilt 
 the flan Aldeoaran and Rigel be on the fame azimuth circle f 
 
 Aldeb. decl. = i6" 03^ N. ; right afc.=:4 h. 23 m. 19s. Rigel's decL 
 = 8" 28^ S. ; right afc. = 5 h. 03 m, 58 s. 
 
 Their difference of right afcenfions is 40 m. 39 s. or 10 10'. 
 
 Construction. On the plane of the equinoclial put Aldebaran at a, 
 and Rigel at b, by their right afcenfions and declinations, then a great 
 circle through b and a will be the azimuth they are on at the time fought; 
 and the parallel of London's lat. defcribed about p, will cut the azimuth 
 circle BA in zz the zenith, through which draw the meridians pz, pz. 
 
 Here the neareft interfeilion to the ftars is taken for the zenith, for as 
 the ftars are both above the horizon, the grcateft zenith diftance is lefs 
 than 90 degrees. 
 
 Computation. In the A apb. 
 
 Given b's co-decl. pb= 98^28^ 
 
 a's co-decl. PAzr 73 57 
 
 difF. rt. afc. A APB = 10 10 
 
 In the triangle apz. 
 Given a's co-decl. ^'^"^T^ S7^ 
 the co-lat. pz = 38 oJi 
 
 the fuppl. of BAPorz.PAz:=23 02 
 
 Required the z.bap=:i56 58 Req. z. apz = 142 35^ or =24 01 
 
 Now the ftar Aldebaran comes to the meridian at iih. 8 m. 27 s. in the 
 evening; which leflencd by i h. 36 m. 4s. (24 oi'') gives 9h. 32 m. 23 s, 
 for the time in the evening when thofe ftars will be on the fame azimuth. 
 
 166. PROBLEM XXin. PI. XIV. Fig. 4. 
 
 At ivhat time in the evening will tJicflars Betelguefe and Pollux have one 
 common altitude above the horizon of London, on the 10th of December 1 780 ? 
 
 Betelguefe right afcen. = 5 h. 43 m. 16 s. ; decl.i=7' 21'' N. Pollux 
 right afcen. = 7 h. 31m. 51 s. ; decl. =28" 32'' N. 
 
 Their difference of right afcenfion is i h. 48 m. 35 s. = 27'' 09', 
 
 Construction. On a primitive circle, where any point p reprefents 
 the pole of the equinodl:ial, put the ftar Pollux at B, and Betelguefe at A, 
 by their declination and difference of right afcenfions ; through A, b, de- 
 fcribc a great circle ; through c, the middle of ab, defcribe a great circle 
 at right angles to ab, and cutting PA in d ; then a fmall circle defcribed 
 about p, at the diftance of 38'' 28^, the co-lat. will cut the circle CD in z, 
 the zenith. 
 
 For the ftars a and e having a common altitude, are equally diftant 
 from z, 
 
 I COMPU-
 
 240 A S T k O N O M y. Book V. 
 
 Computation. In the triangle apb. 
 Given a's co-dccl. pa =82 39' ^Required the Z.BAP = 47*01' 
 b's co-decl. PB =61 28 C AB = 33 14^ 
 
 diff. rt, afc. z. apb = 27 09 J Its half ac = 16 37 
 
 In the triangle acd. 
 Given the Z.c =90** 00^ ^ Required the Z.D rr 45 30' 
 
 Z.A =47 Oi > AD = 23 38 
 
 AC =16 37 J Theref. pa ad=::pd = ^cj 01 
 In the triangle pdz. 
 Given the co-1 at, pz = 38" 28'' 7 Required / Z.ZPD = 48 56 
 
 PD=:59 01 r Or it is := 75 46 
 
 Z.PDzr=45 30 J 
 Now the ftar Retelguefe comes to the meridian at I2h. 28m. 8s. that 
 is between twelve and one o'clock in the morning (133) } from which 
 take 3 h. 15 m, 44 s. as the ftars are to the eaft of the meridian, and it 
 leaves 9h. 12 m. 24 s. in the evening, for the time when thofe ftars have 
 the fame altitude. 
 
 167. PROBLEM XXIV. PI. XIV. Fig. 5. 
 
 JVantlng to know the latitude and longitude of a comet C, its dijlance from 
 two known Jlars a and B were obferved^ an^d are as follows : 
 
 A*s lat. =49 12^ A^. Lon. 16 30^ > j dijlance from 0=49" 05^ 
 b*jA7/. = 30 05 N. Lon.zz. 2 48 H ; dijlance from QzzJ^S 57. 
 Hence the place of the comet c is required. 
 
 Construction. On the plane of the folftitial coloure, where e is the 
 pole of the ecliptic, pat the ftars a and b by their latitudes and longi- 
 tude?, and defcribe a great circle through a and b; then fmall circles de- 
 fcribed about a and b as poles, at the refpedlive diftances of the comet, 
 their interfection c will give its place ; defcribe great circles through a, 
 c; B, c ; E, c ; and E c will be the co-latitude, and fromZ.AEC will be 
 
 obtained the longitude. 
 Computation. IntheAABE. 
 Given a's co-lat. ae=:4048^ 
 b's co-lat. BE 59 55 
 difF.longit. Z. aeb = 76 09 
 
 In the triangle abc. 
 Given the diftance ab= 59 01'' 
 
 diftance Acrr 49 05 
 
 diftance bc3z 45 57 
 
 Required the z.bac= 56 27 
 
 ThcnZ.EAB + Z.EAC = Z.B^AC= i345!5 
 
 Required ab = 59 oi 
 
 and Z.EAB =78 31 
 
 In the triangle cea. 
 Given a's co-lat. ae r= 40 48^ ^ Required the co-lat. ec=: 8i33'' 
 diftance ac =: 49 05 C difF. long. AEcrr 32 43 
 
 and the z.eac =134 58 3'Hencelat.is827'N.lon.i922'inY" 
 
 168. P R O B L E xM XXV. Pi. XIV. Fig. 6. 
 
 The dijlance of the flar c Icing chfervcd from two Jlars A a7td B, the lati- 
 tude and dijlance of which are known^ and alfo the longitude of one of them j 
 thence to find the lat, and long, cf c. 
 
 Suppofe a's lat. to be 5^ 30'' N. ; its dift. from c 39 40^ : b's lat. 
 9'' 57^N. its long. Taurus, 18" ib\ and dift. from c io'^7|-^: And the 
 diftance cf ae 44" 43' : Required ths lo.ng. and lat. of c. 
 
 Con-
 
 Book V. 
 
 ASTRONOMY. 
 
 241 
 
 Construction. On a circle of longitude, where e Is the pole of 
 the ecliptic, put the ftar b by its lat. ; about the points b and e defcribe 
 circles at the diftances of a from thofe points, their interfeftion gives the 
 place of A : alfo circles defcribed from a and b, at the diftances of c re- 
 fpe6tively from them, their interfeflion is the place of c : Then defcribe 
 
 the great circles ea, eg ; ac, ab ; 
 ^bec the longitude, of c from b. 
 
 Computation. IntheAAEB 
 
 Given a's co-lat. ae=:84 30^ 
 
 b's co-lat. be =80 03 
 
 the diftance ae=:44 43 
 
 H 
 
 bc : and ec will be the co-lat. and 
 
 In the triangle abc. 
 Given the diftance ab=4443' 
 diftance AC = 39 40 
 diftance bc=:io o 
 
 71 
 
 Required the 
 
 ilABC=:55 22 
 
 Required the AABEirgi 
 
 In the triangle beg. 
 
 Given b's co-lat. be = 80 03' "^Required c's co-lat. ce = 72oi' 
 
 the diftance bc = 10 07^ i c's Ion. fr. b,z.bec= 6 22 
 
 (Aabe Z-ABCn) z.cbe = 36 52 3 And its abfolute long. '^11 54 
 
 169. PROBLEM XXVI. PI. XIV. Fig. 7. 
 
 From the altitudes of two known fixed Jiars, and the altitude of a planet 
 when in the fame azimuth with one ofthefejiars ; to find the place of the planet. 
 
 Example. Obferved the Moon and Cor Leonis in the fame azimuth, 
 when the Moon's zenith diftance was 36 37'. 
 
 Cor Leonis's zen. dift.=:45oo'; decl. 13'' 02^ N, ; rt. af. 9'' 56"' 39', 
 
 Cor Hydra's zen. dift. =49 16 ; decl. 7- 43 S. ; rt. af. 9^ 16"" 47*. 
 
 Construction. On the plane of the equinoctial, the pole of which 
 is p, draw the colures, and in the folftitial, take E for the pole of the 
 ecliptic ; put the given ftars at B and a by their declinations and right 
 afcenfions : About b and a as poles, with their refpecflive zenith diftances, 
 defcribe circles cutting in z the zenith ; through z and B defcribe an 
 azimuth circle, and making z D equal to the ])'s zenith diftance, it gives 
 her place : Then defcribe the great circles za, ab, e]) ; and the arc ED 
 will be the co-latitude, and ape]) the longitude from the firft point 
 of S. 
 
 Computation. In the A aep. 
 
 Given a's co-decl. PAr=:97 43^ 
 b's CO decl. PB = 76 58 
 diff". rt. af. /1bpa=: 9 58 
 
 Required the z.ABPm53 
 the fide ABr:: 22 
 
 In the triangle b p p 
 C Iven n's co-decl. 
 
 (bz z 1) =) fide E i) r: 8 
 Z.pbD=:64 
 
 p:^76= 
 
 Sll 
 59^ 
 
 58^ 
 
 23 
 
 19 
 
 Requ. l)'s co-decl. pI) 73 ib\ 
 li's rt. af. fr b, Z.up})=: 7 53 
 
 Then 90'^ + Z. =0= pi; 4 /. dp 1) = 
 Z-EP])=:i28 4''/. 
 
 Vol.. I. ] 
 
 In the triangle azb. 
 Given a's zen. dift. ZA=r49" 16' 
 b's zen. dift. ZB:r:45 00 
 the fide ab=:22 59 f 
 
 Required the z.abz=:89 38^ 
 
 Then z. abp abz =: z. zbp = 
 64'' 19^=/. Dbp. 
 
 In the triangle P D E. 
 Given obi. of eclip. pe=: 23 28' 
 ])'s co-decl. Pi) =: "Jl^ 26^- 
 D'sco-rt. af. Z.EP]) = I28 43 
 
 Requir. ])'s co-lat, ]) e =: 88 41 
 
 D's Ion. fr. 53, APr.:)=r 48 25 
 
 And its abfolute long, is a i8 25 
 
 SECTION
 
 44? ' A S T R O N Q M Y.^ Book V-. 
 
 S E C T I O N VI. 
 
 Of various methods to fmd the Latitude. 
 
 The ufual way at Tea to find the latitude is from the Sun's meridional 
 altitude and declination j the manner of doing this will be particularly 
 fhcwn in Book IX. But as it frequently happens at fea, that the meri- 
 dian altitude cannot be taken, therefore the mariner fhould be furniflied 
 with other means to come at the knowledge of this moft ufcful article. 
 To help him in this point, and as a farther exercife in the Aftronomy of 
 the Sphere, the following problems are coUedled together, 
 
 170. PROBLEM XXVII. Pi. V. 
 
 Given the Sun's declination and his amplitude j 
 
 Required the latitude of the place. 
 
 Exam. Being in a place where the compafs had no variation., on a day 
 ivhen the Sun's declination tvas 15 l2f N., I obferved him to rife 62 30' 
 frc/n the north towards the eaji : Required the latitude of that place. 
 
 Construction. 
 Having defcflbed the primitive circle, drawn the horizon hr, and 
 (IV. 71) taken ROzr62* 30''; then about o as a pole defcribe (IV. 66) 
 a final! circle, at the diftance of 74 48'=:co-decI., cutting the primitive 
 in r, the place of the north pole : Draw the axis ps, the equator eq^, 
 and the circle pos, cutting the equator in a. 
 
 Computation*. 
 T.n the fpheric triangle YoA right angled at /, 
 
 Given the co-amp. Y 27 30' 
 the declin. a = 15 12 
 Required the co-latitude aYo. 
 
 Thenf.Vo : rad. : : f. AO : f. Z. A Vo. 
 Hence the latitude will be 55"* 
 24^ N. 
 
 ijr. PROBLEM XXVIII. PL V. 
 
 Given the Sun's declination, and his afccnfional difference ; 
 
 Required the latitude of the place. 
 
 Exam. IVhen the Sun had 20 01'' of declination S.^ he zvas clferved ts 
 ft Gt 4 h, 30 m. : Required the latitude of the place. 
 
 As the aicenfional difference is the time that the Sun rifes or fets be- 
 fore or after 6 o'clock; therefore 6 h. 4 h. 30 m. i h. 30 m.r:22'^ 
 30'= afcenfional difference. 
 
 CoxsTRUCTioN'. In the primitive circle rcnrefenting the meridian of 
 the place, draw the equator eq^, the axis ps, the parallel ot declination 
 ro, 20'' 01^ S. : Make -"f r = 22 30^, the afccnfional difference ; defcribe 
 the circle of right afccnilon I'es, cutting ro in o ; then a diameter hr 
 through o will be the horizon, and rp the lat. fought. 
 
 Computation. In the fpheric triangle Ybo, right angled at b. 
 
 Given the afc. diff, T b = 22 
 
 the declin. ob=:20 01 
 Required the co-lat. Z.oVb 
 
 Thcnrad.:cot.oR::r.TE:cot. Z-oy'R. 
 Or rad,:cot.dcc. iifm.af.diff. : tan. lat. 
 Hence the lat. will be 46 25^ N. 
 
 bciii''- contrary to the dccl. when the afc. diff, falls between noon and fix. 
 
 172. PRO-
 
 Sook V. ASTRONOMY. 243 
 
 172. PROBLEM XXIXi PL V. 
 
 Given the Sun's declination, and altitude at fix o'clock ; 
 Required the latitude of the place. 
 
 Exam. Being at fea^ on a day when the Sun's dcctlndtien was-2o 0\' iV". 
 his altitude at fix o'clock in the evening was iS'' 45^." IVlxit was the lati- 
 tude of the place of ohfervation ? 
 
 Construction. Having defcribed the meridian, drawn the horizon 
 HR, the prime vertical zn, and the parallel s t oi iS 45' of altitude ; from 
 the center r, with the half tangent of the declination = 20 04^, cut the 
 parallel st in o : Through o draw the axis ps, and the azimuth circle zoM 
 (II. 72), and the meafure of rp will give the latitude fought. 
 
 Computation. In the fpheric triangle Tao, right-angled at A. 
 
 Given the decl. 7^0 = 200 04^ N. I Then fni.To : rad. ;: (. ao : f. /loTA* 
 
 the akit. Aorz iS 45 Or fin. decl. : rad. : : fin. alt. : fiii. lat. 
 
 Required the lat.= Z.oYa j Which is 69" 37/N.,as the decl. is N. 
 
 173. PROBLEM XXX. PI. V^ 
 Given the Sun's declination, and his altitude when due eafl or weft j 
 Required the latitude of the place. 
 
 Exam. In a place ziihere the compafs had no variation^ the Sun was oh' 
 fcrvcd to be due eajl xvhen hi^ declination ivas 16' 38'' iV^., and his altitude 
 20" I 2' : IVhat is the latitude of that place 'f 
 
 Construction. In the meridian h2rn, draw the horizon hr, the 
 prime vertical zn, and make Y or:the half tan. of. the alti 20 12' : About" 
 o as a pole, at the diftance of 73" 22', the co-dccl., defcribe (IV, 66) a 
 fmall circle, cutting the meridian in v the elevated pole; draw the axis 
 PS, equator eq^, and through i', o, s, defcribe an hour circle po s 3 th^i 
 the meafure of pr fhcws the latitude. 
 
 Computation. In the fpheric triangle Y^ ? right-angled at a. 
 fjivcn th'i alt. "1^0 2C 12^ I Then 'in\.To: rad, : : fin, ao : fin, Z. ^T o* 
 
 the decl, ao 16 38 N. I Or fin, alt. : rad, : : fin. decl. : fin. lat. 
 Required the latit. rr Z. a Y o, j Which is 56''oo''N,, as the decl. is K. 
 
 But had the declination been S,, the other intcrlbiflion of the parallel 
 circle and meridian mufl have been taken for the elevated pole, and the 
 latitude would be fouth. 
 
 ;7|, PR O J^ L E AI XXXI. PI, V, 
 
 ( jiven the ^un'^ altitude and the hoiu- of the day on cither equinox ; 
 Required trie latitude of the [jLice. 
 
 Kx AM, On the d'.iy the Sun aitrrcd tij.- v. nir.l t-.juinax., his alt. tvas fni:.!- 
 <9. 56'' at 9 '/clock in the morniuif. In iihai lat. ,vas-that ohfervation made '' 
 
 C( NoTRUction. Defcribe the meridian. Jiaw the horizon, the prime 
 -.crtical, and (IV, 68) the parallel > t of 2?." 56^orahilude ; from the cen- 
 t-^rv, with the half t.;n. of 450 3 ii., the time from 6 o'cloclr, cut st lU 
 c, r.nd defcribe trie vertical circle zoN, cutting the horizo;. in 1:. 
 CoMi'UTATio;;. In the fpheric triani'Jc V B o, righl-anglod at K. 
 { jivcn the time after 6, Yo 4-5" co'i As \. T o : rad, : -A. vo : f. ,-:. f. To. 
 tlie altitude uo rj 56 \0: f. tunc ..6 : ra.l. :: f. all. ; cu-f. 1- 
 
 Kcq-iir-J the co-Ia;;tudc e -r o. ! Which '^ ^^.^ \V, 
 
 R': T7C, PKO^
 
 m 
 
 244 ASTRONOMY. Book V. 
 
 J75. PROBLEM XXXII. Pl.V. 
 
 Given the Sun's altitude, declination and azimuth ; 
 Required the latitude of the place. ** 
 
 Exam. Be! fig at fea in a place where theeompafi had no 'Variation, in the 
 afternoon when the Sun was 42 30^ high, his bearing was S, 57 45'' /F. 
 and his declination 22^ 30'' N. : TVhat is the latitude of that place ? 
 
 Construction. Draw the meridian, the horizon hr, the prime ver- 
 tical zN, ajid the parallel x/at 42* 30^ above the horizon (IV. 68) : The 
 tangent of 57^*45'' (ct from a towards R gives the center of the azimuth 
 circle zon, cutting the parallel of altitude st'mo: About o as a pole 
 (IV. 66), at the diftance of 67"^ 30', equal to the co-declination, defcribc 
 a fmall circle, cutting the meridian in p, the place of the pole j then the 
 meafure of rp gives the latitude fought. 
 
 Computation. In the oblique-angled fpheric triangle zop. 
 Given the zenith dift. zo= 47 30'' 7 
 
 the polar dift. pori 67 30 [-Required the co-latitude pz, 
 
 the azimuth Z-Pzom 22 15 3 
 As rad. rr r 10,00000 { As fin. alt. r= 42 30' 0,17032 
 
 To co-f. azim. n izz" 15' 9.72723 
 tSo co-t. alt. n: 42 30 10,03795 
 
 To fin. decl. r: 22 30 9,58284 
 So co-f. 4th. =30 13 9.93658 
 
 To tan. 4th. =: 30 13 9,76518 To co-f. 5th. = 60 42 9,68974 
 Then the difference between the 5th and 4th arcs, that is 30 13^ taken 
 from 60^ 42^, the remainder 30 29'' i the co-lat. Therefore 59 31^ N. 
 3S the latitude fought. 
 
 ?76. PROBLEM XXXIII. PI. V. 
 
 Given the Sun's declination, his altitude and the hour of the day ; 
 Required the latitude of the place. 
 
 Exam. Being at fea ^ the Sun's altitude was ohferved to be 37 20^ at 9 h. 
 4-5 m. in the iuorning^ his declination at that time being 22 30' iV. ; What is 
 the latitude of the place of obfervation ? 
 
 Construction. In the meridian pesq^, draw the equator eq^, axis 
 rs, and parallel of declin. n w, 22** 30' dift. from the equator (3d 135)* 
 Set off" from the center a towards q_the tang, of 33 45''=2 h. 15 m., the 
 diftance between the time of obfervation and noon, which gives the cen- 
 ter of the hour circle Pos, cutting the parallel nm'\no\ about the point o 
 as a pole, defcribe (IV. 66) at the dift. of 52^40^, the zen. dift., a fmall 
 circle, cutting the meridian in z the zenith ; through z o defcribe an azi- 
 muth circle zon ; then the meafure of z will give the lat. fought. 
 
 Computation. In the oblique-angled triangle zop. 
 Given the zenith diftance zo zz.'^i'^ \o' "^ 
 
 the polar diftance po r:67 30 > Required the co-Iat. zp, 
 
 the hour from noon/. zporr 33 45 J 
 As rad. : co-f. hour A. M, : : co-t. decl. : tan. 4th arc =63 31^ 
 As fin. decl. : fin. alt. : : co-f. 4ih arc : co-f. 5th arc 45 02. 
 Their difference is the co-latitude tS'' 29^ Therefore the lat. is 71*^ 
 31'N. 
 
 377. PRO-
 
 Book V. 
 
 ASTRONOMY. 
 
 245 
 
 177, 
 
 PROBLEM XXXIV. 
 
 PI.V. 
 
 Given the altitude of one of two known fixed ftars, when they have the 
 
 fame azimuth ; 
 Required the latitude of the place. 
 
 Exam. Being at fea in an unknown latitude, I ohjerved thejlar Schedar 
 in CaJJiopeia, and Almaach in Andromeda^ to have the fume azimuth, vuhen 
 ibe altitude of Schedar was 37 I $' : What is the latitude of that place P 
 
 Construction. Let the primitive circle reprefent the equator, the 
 pole of which is p, and any point T the place where the right afccnfion 
 begins, from whence lay ofF 7-(2=:27 37' for Almaach's right afcenfion, 
 and Yizzy" 1' for Schedar's ; draw the circles of right afcenfion p^, vb : 
 Defcribe (3d 137) Almaach's and Schedar's parallels of declination, cut- 
 ting Pa, ph, in A, B J A being Almaach, and b Schedar. A great circle 
 pafling through A and b (IV. 61) will be the azimuth they are on. About 
 B at the diftance of 52^ 45^, Schedar's zenith diftance, defcribe (IV. 66) 
 a fmall circle, cutting the faid azimuth circle in z, the zenith of the 
 place ; draw Pz, which meafured on the half tangents gives the co-lati- 
 tude of the place of obfervation. 
 
 Computation. 
 
 ift. In the oblique-angled fpheric triangle abp. 
 
 Given Almaach's co-dec. PAr:4844^ I Required the angle of pofi- 
 
 Schcdar's co-dec. pb = 34 40 | tion abp. 
 
 their difF. of r. afc.Z.APB 2033 jFor the folution,feeIV. 165. 
 
 As rad. 
 
 To CO-f. Z. APB 
 
 So is tan. ap 
 
 To tan. 4th arc 
 The fide BP 
 
 90^ 
 20 
 
 46 
 34 
 
 00 
 
 35 
 
 44 
 
 5 
 
 40 
 
 lO.OCOOO 
 
 9.97135 
 
 10,05676 
 
 10,0281 1 
 
 12 I I 
 
 As fin. 5 th qrc 
 ' o fin. 4th arc 
 So is tan. Z.apb 
 
 To tan. jL b 
 
 120 
 
 46 
 
 20 
 
 1 1 
 
 35 
 
 0,67563 
 9,86306 
 9,57466 
 
 52 23 10,1133^ 
 
 The 5th arc 
 
 2d, In the oblique fpheric triangle pbz. 
 
 Given Schedar's co-dec. pb =: 34 40^! Required the co-lat. pz. 
 
 Schedar's co-alt. bz =r 52 45 j For the folution, fee IV. 16^. 
 angle of pofition pbz = 52 23 j Here /_ PB2 = fUp. z. pba. 
 
 As rad. 
 
 r= R 10,00000 
 
 To co-f. z. pofit. n 92'23' 9,785'' o 
 So co-t. Sch. decl.rr 55 20 9,83984 
 
 To tan. 4th arc r= 22 53 9,62544 
 Which taken from 52 4.5 bz 
 
 Leaves 5th arc 29 52 
 
 As co-f. 4th arc rr 22** 53' o,03i;6o 
 
 To co-f. 5th arc =: 29 52 9,93811 
 
 So fin. Sch. dec. =: 55 20 9.91512 
 
 To fin. lalit. =: 50 44 c,,8d8S| 
 
 R3 
 
 J78. PRO
 
 54^ ASTRONOMY. Book V, 
 
 ,-8. PROBLEM XXXV. PI. V. 
 
 Given the difFcrencc of time between the rifing of two known ftars j 
 Required the latitude of the place. 
 
 Exam. Being at fea in an unknown place^ the Jiar Aldeharan ivas ob^ 
 ferved to rife 3/;. 15 ;/2. later than the bright Jlar in Aries : Required the 
 latitude of that place. 
 
 Bright ftar in T decl. 22" 25^ N. ; right afccn. i h. 54111. 49 s. Alde^ 
 baraa's dec], i6 03' N, ; right afccnfion 4h. 23 m. 19 s. 
 
 Construction. 
 
 Let the primitive circle reprefent the equator, defcribe (3d 137) the 
 parallels of declination of the two ftars, that of Aries being 22 25'' N, 
 and of Aldebaran 16 03^ N. : Draw va for the circle of right afcenfion 
 pafling through the ftar in T, which fuppofe in a : From a lay off" ab^ 
 " 37 yi' diff. of right afccnfions ; and aczz\% \</z=.T,h. 15 m., the 
 difF. of time between their rifmg ; draw ^p, rp, cutting the parallels of 
 declination in E, c : Through the points b, c, defcribe (IV. 61) the 
 great circle hor ; draw po at right-angles to HRj then the meafure of pq 
 will give the latitude fought. 
 
 Computation, 
 In the oblique-angled fpheric triangle PBC. 
 
 Given b's co-decl. pb = 7357' 
 a's co-decl. pc-rzby 35 
 
 Given c's co-dccl. pc=:6735^ 
 and the Z. PCor:6l 54 
 Required the latitude po. 
 
 Rad. :co-f. Z. BPC : : t. PB : t.M=:73''38'. 
 
 From M take pc, leaves n =6 03^ 
 Z. Apc z. APB= Z.BPc=:ii 371 As fln.N : f. M : : tan. Z.CPB : t. z.c::= 
 Required the angle pcb. jHence Z_ Pco=;6i 54''. [118 o6^ 
 
 In the fpheric triangle pco, right-rangled at o. 
 
 rivpn /-'c nn-Ar'r] nr f^n o e^ As Tad. : fui. PC : l fui. Z. PCO : fin. 
 
 [Po=:5438^ 
 Therefore the latitude is 54 38^. 
 
 379. PROBLEM XXXVI. PI. V, 
 
 Tv/o known fixed ftars being obfcrved to have the fame altitude j 
 Required the latitude of the place of obfervation. 
 
 Exam. In the evening, the frars Capella and Procyon %vere obferved at 
 ile fame tiiiie to have each 38' 00^ of altitude : Reqiured the latitude of the 
 fJace %vhere that obfervation ivas taken. 
 
 . Capella's decl. 11:45 45' N. right afcen. 5 h. o m. 28 s. Procyon's decl. 
 :^ ^' 47' N. right afcen. = 7 h. 27 m. 48 s. 
 
 Construction. 
 On the plane of the equator, rcprefented by the primitive circle, de- 
 fcribe (3d 137) the parallels of the given declinations ; take ab ^b" 50'. 
 the diff. of the given right afcenfions ; draw pa, vb, then A reprefents 
 Procyon, and b Cr.pc!!;'. 
 
 . About A and b as poles, defcribe (TV. 66) arcs of fmall circles, at the 
 diftance of 52, the co-altitude, and their interfection gives z the zenith of 
 the place : 'ilirough the points a, b ; 7, a ; z, b ; defcribe great circles 
 (IV. 61; and draw zp, the meafure of which vvill give the co-latitude 
 fef the jJacc of obrcr\'aUon. - 
 
 C M F f --
 
 Book V. 
 
 ASTRONOMY. 
 
 ^47 
 
 t.Bp:t. M=:3757 
 PA leaves Nr:46 16 
 : tan.Z.P : tan. 2L bap 
 [=32^31 
 
 Computation. 
 In the-oblique angled fpheric triangle apb. 
 Given a's co-decl. APn:84 13''; Rad. : cof. z.p 
 b's co-decl. BP=44 15 I And M talc. fr. 
 
 difF. rt. afc. z.apbzz:36 50 ! Sin. N ifin. M : 
 Required the ftars diftance ab. 
 
 And the angle BAP. j Cof.Mtcof.N-.cof.BPicor.BAnfi 06 
 
 In the oblique angled fpticFic triangle azb. 
 Given a's co-a!t, za = 52 00^1 The angle baz vi^ill be found 
 
 b's co-alt. zc=:52 00 i 68 04''. 
 
 ftars diilance abi:::5i 06 .Then Z.BAZ Z. bap niZ. PAzr: 
 Required the angle zab. | 35 33^ 
 
 In the fpheric triangle azp. 
 Given a's co- alt. zaz::52oo^ Rad. : cof. z. a :: t. za : t.M=47''29' 
 
 a's co-declin. ap:z:84 13 
 the angle zap=::35 30 
 
 Required the co-lat. zp. 
 
 M taken from pa leaves N = 36 44 
 Cof.M:cof.N::cofz.\:cof.pz 43 c6 
 Therefore the latitude is 46 54'' N. 
 
 180. PROBLEM XXXVII. PI. V. 
 
 Given the altitudes of two known ftars ; 
 
 Required the latitude of the place. 
 
 Exam. The altitude of the Hydra's heart zuas obferved to he 40'^ 44'', 
 and of the Lion's heart 45 00' ; TFhat is the latitude of the place of obfer- 
 vation ? 
 
 Hydra's heart, decl 7 43^ S. ; right afccn. = 9 h. 16 m. 47 f. 
 Lion's heart, decl. ::=i3^ ^~' ^' '> ^^S^t afcen. 9 h. 56 m. 39f. 
 
 CON'STRUCTIOK. 
 
 If this problem is conftructcd on the plane of the equator, it will be in 
 every refpecl like the laft ; only the fmall circles, defcribed about a and 
 E, arc to be unequally diflant from their rcfpedive poles a, b. 
 
 C o M P U T A T I o N. 
 
 Here, as iu the lafl, there will be three fpheric triangles to work in j 
 namely, the triangles apb, zar, and zpb. 
 
 \\\ the triangle apb, where a P =197'^ 43'', BPr:76'^ 58^, z.APBrr 
 cf 58'. 
 
 As rad. : cof. z.apb : : tan. r,p : tan. Mrr76 46'!. Then ap m=im 
 = 20 57^1. 
 
 As fm. N : fm. m : : tan. Z-APb : tan. Z.bap = 25'' 3l^. 
 
 And as cof. m : cof. n : : cof. bp : ccf. ba = 22 59. 
 
 In the triangle baz, where Azrz49^ 16^, r.z 45' co', 
 
 The angle uaz will be found equal to 68" 56^ 
 
 'i'hcn ZLHAZ Z.BAP=:Z.PAZ=43'' 22^. 
 
 In the triangle apz, where Ap=:g7'43 , az = 49'' 16',/. 
 As rad. : cof. z.i'Az : ; tan, az : tan. 1^1=40 \z' . 'i 
 N-57' 3.V. . 
 
 j\\v\ as co(. M : cof. n : : cof. a/. : cof. p/, O2 
 Hwiicc t'.i'J kuilud'v fought is 27'' lO' N. 
 
 il4 
 
 39 
 
 h.v. iz^ 59 
 
 A7,=r43'22. 
 
 Cll AP MZ: 
 
 181. P R O.
 
 24? 
 
 A S T R ON O M Y. 
 
 Book V. 
 
 i8i. 
 
 PROBLEM XXXVIII. 
 
 PI. V. 
 
 Given the Sun's declination, two altitudes, and the time between the 
 
 obfervations ; 
 Required the latitude of the place. 
 
 Exam. On a day when the Sun's declination was 20 oo' N., in th* 
 forenoon the Sun's altitude was obfcrved to he 18" 30', and 3 hours afier^ 
 his altitude was 44 00^ ; JVhat was the latitude of the place ? 
 
 Construction. 
 Let the primitive circle reprefent that hour circle on which the Sun 
 was at the firft obfervation, eq^ being the equator, then AOy the parallel 
 of 20 of declination, gives a the Sun's place at firft; and as cq^is the 
 tangent of 45, Q^will be the center of the hour circle pbs three hours 
 diftant from the former, its interfedtion b with the parallel of declination, 
 is the Sun's place at the fecond obfervation : About a as a pole, at the 
 diftance of 71 30', the firft zenith diftance, defcribe (IV. 66) a fmall 
 circle ; about b, as a pole, at the diftance of 46 00', the fecond zenith 
 diftance, defcribe (IV. 66) another fmall circle, cutting the former in z 
 the zenith : Through z, a ; z, b ; a, b ; p, z ; defcribe (IV. 61} great 
 circles j then pz is the co-latitude required. 
 
 Computation. 
 Here are three triangles to work in j namely, abp, abz, epz. 
 
 In the ifofceles fpheric triangle apb 
 
 Given AP = 70**oo^ 
 
 BP=:70 00 
 
 z.APE=45 00 
 
 Req.Z-ABPand ab. 
 
 Suppcfe the perpendicular vh is drawn. 
 Rad. : tan, hvb : : cof. pb : co-t. z,pba=:8i'' 56'', 
 Rad. : fin. PB : : fin. Z.bp3 : fin. b^=:21 041^ 
 Then hb double gives abzz4209''. 
 
 In the oblique angled fpheric triangle abz. 
 
 Given az = 7i30 
 bz=:46 oo 
 AB 42 09 
 
 Required /.abz. 
 
 Then working with the three fides, the angle abz 
 
 will be found n 1 14 1 1\ 
 And Z.ABZ Z.PBAnPBz=:32 15^ 
 
 In the oblique angled fpheric triangle pbz. 
 
 Given PBr=70"00 
 
 87,^:46 00 
 
 Z. PBZ ==32 15 
 
 Rcq. theco-l.:t. Pz. 
 
 As rad. : cof. ^i. pbz : : tan. Bz : tan. Mr:4i*' 13', 
 And PB m=:n=:28 47'. 
 As cof. M : cof. N : : cof. bz : cof. pz=:30 59''. 
 Therefore the latitude is 54 01^ N. 
 
 182. If the Sun's altitude can be taken both before and afternoon, 
 when he has equal heights, then the time between thefe two obfervations 
 being bifecfled, will give the time when the Sun was on the meridian: 
 Now the co-declination, the co-altitude, ajid the time from noon at 
 cith.r obfervation being known, the latitude may be readily computed in 
 one oblique angled triangle, in which arc known two fides, and an angle 
 oppofite to one of them to find the other fide, which is the co-latitude ^ 
 for which fee the problem, art. 176. 
 
 I 183. PRO-
 
 Book V. ASTRONOMY* 249 
 
 183. problem' XXXIX. Pl.V. 
 
 Given the Sun's declination, two altitudes, and the difference of the 
 
 magnetic azimuths ; 
 Required the latitude of the place. 
 
 Exam. On the i\Ji of May ^ the Sun's declination being 20 16'' N.^ in 
 the morning when the Sun was on the ESE, point of the compafs, his altitude 
 tvas j^.2'^ 30''; and when he bore S, 20 30' . his altitude was 58 30'; 
 ll^hat is the latitude of the place of obfervation f 
 
 Construction. 
 
 Let the primitive circle reprefent the azimuth circle which the Sun 
 was on at the greater altitude, 58 30^ a being the Sun's place at that 
 time, HR the horizon, and z the zenith ; draw aa a parallel of 43 30^ of 
 altitude, and (IV. 75) defcribe a vertical circle, making an angle with az, 
 of 47 00', the difference of the obferved azimuths ; the place where this 
 cuts the parallel of altitude aa^ gives b the Sun's place at the firft obferva- 
 tion : Then fmall circks being defcribed about a and b, as poles at the 
 diflances of 69" 44^, the co-declinatlon (IV. 66), their interfeiftion will 
 give p the place of the pole : Through a, b ; p, a ; p, b ; and z, P, de- 
 fcribe great circles (IV. 61} j then pz is the co-latitude fought. 
 
 Computation. 
 
 I ift. In the fpheric triangle azb : Given az r= a's co-alt. = 31 30' 
 
 Bz = b's co-alt. =: 46 30 
 azb r: diff. of az, z=. 47 00 
 
 Required Z. abz 
 
 AB 
 
 ^d. In the ifofceles fpheric A apb : Given ap =: a's co-decl. rr 69 44 
 
 BP = b's co-decl. = 69 44 
 
 AB = distance r: 32 17 
 
 Required z. abp = 83 52 
 
 3d. In the fpheric triangle z^P : Given bz r: b's co-alt. =46 30 
 
 PB = b's co-decl. r= 69 44 
 
 /.ZBP = 38 II 
 
 Requir. zp = co-lat. =: 39 21 
 Therefore the latitude of the place is 50 39' N. 
 
 184. PRO-
 
 d^Q 
 
 ASTRONOMY. Book V. 
 
 184. P R O B L E M XL. PI. V. 
 
 Two known ftars being obferved on the fame azimuth, and two other 
 known ftars being obferved on another azimuth, and the time between 
 the obfervation being known j to find the latitude of the place. 
 
 Exam. The -Jlars Aldebaran in Taurus , and Rigel in Orion^ were ob- 
 ferved on the fame azimuth ; and 2 h. 35 min. after^ theflars Cafior in Ge- 
 jnini and the Hydra's heart were alfo obferved on another azimuth : IfJjat 
 was the latitude of the place of obfervation ? 
 
 Construction'. 
 
 On the plane of the equator put the ftars Aldebaran and Rigel at a, b, 
 alfo the ftars Caftor and Hydra at r, </, by means of their right afcen- 
 fions and declinations (2d and 3d of 137) : Let the ftars f, d, be removed 
 forwards 2*^ 35"", or 38 45^ to c, D ; through A, B, and D, c, defcribe 
 (IV. 61) great circles interfedling in z, the zenith of the place; dravi^ 
 the great circles AC and pz j then the meafure of pz gives the co-latitude 
 required. 
 
 Computation. 
 
 Aldebaran's declin. = i603'N. right afccn.r: 4''23'"i9' = 65' 50'. 
 
 Rigel's = 8 28 S. = 5 03 58 = 75 591. 
 
 Caftor's = 32 "21 N. = 7 20 33 =110 08. 
 
 Hydra's heart zz 7 43 S. z= 9 16 47 =1139 12. 
 
 I ft. In the triangle pab, where pb=:98 28^, pa = 73'' 57^, ^lAPsr: 
 ioo9{--. 
 Then the z. pab will be 156" 59^, and thez.PAz, the fuppl.=23 oi''. 
 
 2d. In the triangle pcd, where porrgy'' 43^, pc=:57'' 39'', z.cpa 
 s=2904^ 
 
 Then thez_PCD will be 140 09^ and theZ_PCZ, the fuppl.rr39 51^ 
 
 3d. In the triangle apc, where Ap 73 57^, cpz=57' 39'', z. a PC = 5" 
 33i^ which is the difference between 2 h. 35 m. and the difference of the 
 right afcenfions of a and c. 
 
 Then thez.PAC will be 16' I2^ Z.pca=z 161 30^ and ac=:i703^ 
 N0V/APAC+ /Lpaz=z.caz 39 13'' J andz.PCA Z.pcz = z.ACZ. 
 r::i2i 39^ 
 
 4th. In the triangle Acz, where AC 1 7"" 03' J Z.CAzr:39' 13''^ ^acz 
 ^121" 39''. 
 
 Then cz will be found equal to 28 26^ 
 
 5th. In the triangle cpz, where cz = 28^ 26', Q?~Kf 39^ ^Pcz 
 
 1 hen PZ will he found equal to 38' 49'. 
 
 And the latitude of the place of obicrvation is 51 li\ N. 
 
 5 There
 
 Tol I 
 
 ASTRONOMY 
 
 H.V. 
 
 Tn^:j.6.;.^ 
 
 Frai'.S 
 
 I\vt:fi. 2, 
 
 Trtri:*o . 
 
 Twt.tj.tS. p 
 
 Ihf^:/S. 
 
 J'roi':^- -B z 
 
 Fyii>i>:zj 
 
 Pn'^ a. 
 
 Twf> 2^. 2 
 
 Jh?^3C3i.^ 
 
 7iv^J2C>,f- 2 \ 
 
 /h'/-:if 
 
 /^'/ .-?.< 
 
 Pnjhs6 
 
 I^^hj-j. 
 
 / 
 
 /h'/-.i^ 
 
 r 
 
 /h-/',;o 
 
 " <J 
 
 rn>/'^c. 
 
 '//. i/^^Ufr'.. 
 
 
 4, A/-'' ^ii*. 2;f<'- 
 
 nr\
 
 Book V. 
 
 ASTRONOMY, 
 
 251 
 
 There might be given a great variety of other problems to find the la- 
 titude from various circumftances ; but the trouble of folving them, as well 
 as fome of the foregoing ones, is too great to render them of general ufe ; 
 And indeed fome of them were only inferted as trigonometrical exercifes 
 for young fludents ; it being generally allowed that the fciences are moft 
 readily learned by working many examples : And on this account it was 
 judged, that the few following queftions might not only be entertaining 
 to thofe who have a love for thefe matters j but on fome occallons might 
 be ufefully applied at fea. 
 
 185. 
 
 P R O B L E xM XLI. 
 
 Given the Sun's meridian, or mid- day altitude rr62 oo'. 
 
 Jnd its mid-night deprejjlon, below the horizon, =22 00, 
 Required the latitude of the place, and the Suns declination. 
 
 Solution. Let the circle hzr be the meri- 
 dian ; 
 arc h;, the meridian alt. j its 
 
 fine Y?n j 
 arc Rn, the mid-night deprcf- 
 
 fion ; its fine nf; 
 7nn, the parallel of declination ; 
 Qjl, parallel to ni n the equator ; 
 PS, at right angles to Q^^, the 
 axis, or 6 o'clock circle. 
 
 NoWHQj-(v = Hm-)p^^ and cvn-qn. 
 
 And HQj Q/=:R i -s^ 2 ' -^ I 
 
 Then ho =1 co-latitude =42 00'. 
 
 ^^ 2 
 
 HW R .... , 
 
 And q;;z =:dcchnationr:20o CO , 
 
 In the following problems, as it was the method of computation which 
 was chiefly intended for the information of beginners, the conftru61:ion is 
 fuppoft'd to be done : And the lines and letters, as here dcfcribed, are to 
 be undcrllood to rcprcfcnt the fame things in eacli figure. 
 
 j86. PRO^
 
 ^5* 
 
 ASTRONOMY. 
 
 Book V. 
 
 i86. PROBLEM XLII. 
 
 Same time in the mtntb of May ^ 1780, at place in the wejiern ocean y the 
 Sun's meridian altitude was obferved to be 62 00'; and i^ 48"' 14' after^ 
 the altitude was found to be 54 30' ; Required the latitude of that place^ and 
 the Sun's declination. 
 
 Let m and a be the Sun's places at the given 
 times. 
 
 UJF, AD, the fines of the obferved alti- 
 tudes. 
 
 mB the difference of thofe right 
 
 fines. 
 
 Z.QP^, the given interval of time, 
 
 and Q^, the verfed fine of that interval. 
 
 NowQfJ : Q.y : mA'. mn (II. 182). And mA : mn: :mii: mk. (II. 167) 
 Therefore q^ : Qjj : : ms : mK:=:m + nf. 
 
 _Q.^xmB 2 X radius x diff. of fines of alts. 
 
 And zK TTr TT r 
 
 qji verfed Ime of hour rrom noon 
 
 But verfed fine of an arcr=twice fquare of the fine of half that arc. 
 
 (IV. 193) 
 
 ^. ^ 2XRxdiff. fines of alts. diff. of fines of alt. 
 
 Therefore fMK =. ^ - = :: . Radius 
 
 2SS, I hour a noon * ss, f hour a noon 
 
 being i. 
 
 Or L, diff. fines of alts. 2L5, f hour a noon = L, fum fines, mT + nf. 
 
 Herei*>48'" i^'^rz;" 3' 30", (131) | Alt. 62" 00' nat.fine=:o,88295(iv.256) 
 
 I Alt. 54 30 nat. fine =0,8 141 2 
 
 And I hour a noon=:i3''3i'45"' I Diff- of fines of alts. =1:0,06883 
 Now diff. fines alts. =0,06883 ; its log. 8,83778 f 
 
 I hour a noon =13 31' 45"; twice its log. fine 8,73822 J 
 =11,2574. the number to log. 10,09946 || 
 
 =0,8829 the nat. fine of 62 00' the meridian altitude. 
 
 MK 
 
 mv 
 
 nf 
 
 =0,3745 the nat. fine 21 59^ 36^^ the mid-night depr. 
 Sum 83 ;9 36, its |4i59*48^^rrco-lat. 
 Diff. 40 00 24 its I 20 00 12 =:dec. 
 Latitude 48 o'' 12^^ N. obfervations made on the XQth of May. 
 
 * The mark a is ufed for the word /rom 
 
 t Here 8, is the index; becaufe 6, the left-hand digit of 0,0688 ij is in 
 the place of 2ds. 
 
 t The log. fin. of 13 3 i' is 9,36871; and of 13 32' is 9,36924 ; their diff. 
 is 53 ; then 60": 53 :: 45": 40; and 9,368714-40=9,36911 ; its double 
 is 8,73822, rejedling 10 in the doubled index. 
 
 In fubtrafting 8,73822 from 8,83765 ; the index of the minaend is to be 
 incrcafed by 10 for a radius ; or augment o, the index of the remainder, by 10. 
 
 () The log. 10,09946, having 10 for its index, fliews that the left-hand 
 place of its correfponding number Hands in the place of units. 
 
 ^0 find the degrees J minutes, and feconds to a gi-ven natural right fine. 
 
 Now ^=0,3745 its log. is ,57345 ; which fought among the log. fines, 
 falls between thofe of 21 59' and 22 00'; the difference of their logs, is 31; 
 and the difference between the given log. and that of 21 59' is 19 ; then 
 31:19;: 60" : 36" i fo that w/"anfwers 10 2\ 59' 36". (iv. 257.) 
 
 187. PR Or
 
 Book V. 
 187. 
 
 ASTRONOMY. 
 PROBLEM XLHI. 
 
 25a 
 
 Being at fea^Jome time in July^ in North lat. the Sun was olferved to rife 
 at 4 h, 24 tn. 36 s. A. M. ; and in the fame place, his altitude at noon was 
 62 00' : Required the latitude of that place, and day of the month. 
 
 Let OT, V, , be places of the Sun, at noon, 
 at rifing, and at midnight. 
 
 mY, nfy fines of mer. alt. and midnight 
 
 depr. 
 ov the fine of the afcenfional diiFerence. 
 vq the verfed fine of the time of fetting 
 from midnight. 
 
 qvriradius + fme of the afcenfional dif- 
 ference. 
 
 Now <^ '. vq : : {mv ivn'. :) mY xnf. 
 
 Then nf:=: ; or Lx, nfzzL , qp + L;, mF + LV, Z.VP R. 
 
 Or, l\ Qv+LSy merid. alt. + 2L/, | time midnight + L, 2=1;, mid- 
 night depreflion. 
 
 Hereo^.=(i,I h.35m. 24f.=:;', 235i'=)o,40435 ; and 0^=1,4043, 
 Time from midnight =4 h. 24 m. 36 $.=66 09' j its half =33 4i'- 
 
 ^=1,4043 
 
 mFZZij 62 
 
 its L^ 
 its Lt, 
 2L/, 
 
 its h 
 its Lf, 
 
 
 9,85254 
 9>94593 
 9'47397 
 0,30103 
 
 
 ^/=2i59'4o" 
 
 9>S7347 
 
 
 Merid. alt. 
 Midnt. depr. 
 
 =62 
 = 21 
 
 0' 0" 
 
 59 40 
 
 
 Sum 
 
 =83 
 
 59 
 
 40; its|=:4i 
 
 59*" 50" 
 
 DifF. 
 
 =40 
 
 
 
 20 ; its 1=20 
 
 10 
 
 Latitude =48" o' 10" N. 
 
 Declination =20" o' io" N, on July 23d, 
 
 {185.) 
 
 i85. PRO-
 
 '^; 
 
 188. 
 
 ASTRONOMY. 
 PROBLEM XLIV. 
 
 Book V. 
 
 Jt a place in the northern hemifphere^ fofne time in the month of May ^ the 
 Sun was obferved to have 14 43^' altitude at 6 h. P. M. and to fet at 
 7 h. 35 m. 24 s. : Required the latitude of that place, and day of the fnonth. 
 
 Let w, N, V, n, be places of the Sun, at noon, 6 o'clock, fctting, and 
 midnight. See the fig. to Problem 43. 
 
 ;F, N/, fines of the altitudes at m and n. 
 
 ov, vq, and c^, the fame as in the laft problem. 
 
 Now oz; '.vq : '. {^v : v : :) n/ : nf. Then Ljw/" rrL^ov+L, nZ + Lj^^. 
 Alfo oT/ : Qp : : (nv : v/w : :) N/ : WF. And l,/f=:l^ ot' + L, n/ + l, qy. 
 Here o^!=J, 23 5i''=o,40434 ; and (^ = 1,4043. 
 
 qvzzv^bif 00)'2s5, 33 41'. 
 
 ThenoT/r:/, 23 51' Its l^j, 0,39325 
 N/=/, 14" 43I' Lj, 9,40514 
 
 . ^^^.33 ^i' 2U, 9,47397 
 0,30103 
 
 f^^lz 
 
 w/'j, 2 2' O' 
 
 9*57339 
 
 And o&rr;, 23<'5i' its lV, 0,39325 
 N/ =/, 14 43I; its Ls, 9,40514 
 Ci_<L':r:i,40 43 its l, 10,14746 
 
 mrzzs, 6z o 
 
 hf, 9-945^5 
 
 62^*4-22'* 
 Then ^ = 42 the co-latitude. 
 
 And 
 
 2 
 6222 
 
 = 20'^ the declination anfwering to May 19th. (185.) 
 
 189. 
 
 PROBLEM XLV. 
 
 j^t a place in the zvejlern ocean, fame tiine in July, the Sun^s altitude 
 was obferved to be 36 at 3 A. 51 m. 49 s. P. M. ; and was feen to Jet at 
 7 h, 35 7n, 24 1< P. M> : Required the latitude of the place, and day of the 
 month. 
 
 Let ;,v, c, V, n, be the Sun's places, at noon, 
 at 3h. 51m. 49 f. at fetting, and at mid- 
 night. 
 
 mf IF, the fines of the altitudes at 7;;, c ; 
 77/* the fine of the midnight dcprefTion. 
 or, ov, the fines of the times from 6 h, and cv, 
 their fum. 
 
 Now
 
 Book V. 
 
 ASTRONOMY. 
 
 ^SS 
 
 Now cv.Qv: :(cv ; vm : :) if : mF. Or Ls,mF=L\cv-i-ts,iT-\-L^qv. 
 
 And cv: qv : :{cv : vn : :) IF : nf. Or ts,nf=L\cv-{-LSjiF + LVyqpv 
 
 Here o. =s,2h. 8m. ns. =.,32 2' 45'^= 0,53059 7 ^^_ _ ..^^^ 
 OT/rrj, ih.35m. 24S.=^,23 51 00=0,40434]" ^''-- ^'vJ49J > 
 (^ = 1,4043; ^^;=:^.,66V = 2JJ,33V3o"i orL,j^=2LJ,334l^ + L2. 
 
 ThcDL^ fz/^io, 93493 
 L/, iFrz36" co' 
 
 L,q^ =1,4043 
 tjOTFrrS^" od' 
 
 0,02922. 
 9,76922 
 10,14746 
 
 9,94590 
 
 And lVi' o>93493 
 Lj,iF=:36 oq' 
 
 L/,/^2 2 00' 
 
 0,02922 
 9,76922 
 
 9'47397 
 0,30103 
 
 957344 
 
 _, 62'+ 22 
 
 Then = 42 the co-latitude. 
 
 2 ^ 
 
 62 22 
 And = 20 the declination, anfwering to July 23, 
 
 (185.) 
 
 J 90. 
 
 PROBLEM XLVI. 
 
 So?ne time in the ?nonth of May^ at a place in the wejiern ocean^ the day 
 broke at \ h. 45 m. 36 s. A. M. ; and at 8 h. 8 in. \\ s. A. M. the Sun's 
 altitude zvas obferved to he 36 : Required the latitude and day of the month. 
 
 Let ;/;, c, r, , be the Sun's places, at noon, at 8 h. 8 m. ii s., at the 
 beginning of twilight, and at midnight. 
 
 /F, IF, the fmes of the altitudes at mc. 
 
 FT, FK, the fines of the depreffions at r, n. 
 
 oc, Of, the fines of the times from 6 h,, and c s their fum. 
 
 Now cs : (^ : : (cr : mr : :) it : ?nT. And mT Fxrr^F. 
 Alfo cs : qc : : (cr : c : :) it : IK. And ik iY-=znf 
 Here IK =.,36^^ ^0=0,58778 7 .^^g^g^ 
 
 FT = i,l8 00=0,309025 ^ 
 
 ocs, 2h. 8m. I IS. = 5, 32'' 2' 45"=o,53oS9 ; and ^i:= 1,5306 
 o; J, 4h.i4m. 24?. =5,63 36 00 =0,89571; and 01=1,8957 
 
 Then cs 9,84579 
 IT 9,95269 
 
 q^ 10,27777 
 
 mr 10,07623 
 
 f^'zz 1,1919 
 
 9,84579 !T FTr:wF=o,8829 
 IK IF :z:/"=:o,37453 
 Hence the lat. 48 N. 
 
 cs 
 
 IT 9,95269 
 qc 10,18486 
 
 IK 9,9^3^4 
 
 IK ZZ::, 96237 
 
 the 
 
 the 
 
 /, 62" 
 /, 22 
 
 00 
 
 00 
 
 Decl, 22 00' N., aufwerlcg to M-ay 19th, 
 
 TIT. PRO.
 
 4^6 ASTRONOMY. Book V. 
 
 191. PROBLEM XLVII. 
 
 In the month of May, at fame place in the wejiern Bcean, the Sun*s altitudt 
 atb*" J. M. was 14 43^'' ; and at Sh. Sm. lis. its altitude was 360 . Re- 
 quired the latitude of the place and day of the month. 
 
 Let w, c, N, , be the Sun's places at Roon, Q 
 at 8 h. 8m. 1 1 s., at 6 h., and at midnight. 
 mYy IF, LF, the fines of the alts. atz, c, N. 
 ff/"the fine of the midnight depreffion. 
 oc the fine of 2 h. 8 m. J is. \ j^ 
 
 Now f : oQj. : (nc : vim::) il : mh. Then mL + LVzzmT. 
 And oc:cij : : (nx : c : :) il : ik. Then ik~if=/; 
 
 Hence Lj?L=:L\of + rad.-t-L,iL. And L,iK=:L\of-f-Lf^ + LjiL. 
 
 Here Of =/, 2 h. 8 m. IIS. =^30 2'45"=o,53059 ; and f^=ri,53o6. 
 
 if=:j,36'^o'=:o,58778; andLF=/, i4''43i'=:o,254i8 ; foiL=o,3336 
 0^=32 2' 45" its L/, 0,27524 Of =32" 2' 45" its h^s, 0,27524 
 
 iL=:o,3336 itJ i. 9.S2323 11=0.3336 its l. 9,53323 
 
 Rad. 10,00000 ff =1,5306 its L. 10,18486 
 
 OTi =0,62874 
 
 9,79847 iK=o,96234 
 
 Then WF =0,88292 the fine of 62 00^ the meridian altitude. 
 And w/"=o,37456 the fine of 22 00 the midnight deprefTion, 
 Hence the latitude is 48 N, decl. 20 N. on May 19th. 
 
 9'98333 
 
 (185). 
 
 192. 
 
 PROBLEM XLVIIL 
 
 j^t a place in the wejiern ocean, in the month of July, the Sun's altitude 
 VJ as found to he 46 at 2^- 49"' 9' P. Al. ; and to be 36 high at '^ 51"" 49" 
 F. M. : Required the latitude of the place and day of the month. 
 
 Let ?n, B, c, ?7, be the Sun's places at noon, at 2 h. 49 m. 9 s., at 3 h, 
 51 m. 49 s., and at midnight j and niY, HF, If, the fines of the alts, at /t?, 
 j;, c. o/', Of, the co-fines of the time from noon j or the fines of tlie 
 time to 6 o'clock. 
 
 Now ^f : Qf : : (bc ; ;c : ; ) hi : mi. Then wi + if=/72F. 
 And bc : bq : : {nc : -zn : :) in : hk. Then hk hf = fk=/. 
 
 Hereo^=.f, 3 h. lom. 51 8.=i,47 42'45"=o,73q7S 7 ,. ,_ 
 
 Of rr/, 211. 8m. iis.rzj,32 2 45 =o,53oc9 j ' ^ ^ 
 
 HF=J46rro,7i934.; ii-m, 36*^=0,58778; h 1=0, 131 56; ^^=1,739?. 
 Alfo qr=z[\^r. fine of 57" 57' 15"=} 2;/;5:5757' I5":=2/j, 28 58' 37". 
 
 Then
 
 Book V. 
 
 ASTRONOMY. 
 
 Then bcr=. 0,20919 
 Hi= 0,131^6 
 
 _ f,28''58'37' 
 
 qc: 
 
 aii=, 29520 
 
 0,67947 
 9,1 1912 
 9,37050 
 0,30103 
 
 9,47012 
 
 And ^f=:o,209T9 
 Hi=:6,i3i56 
 ^^=1,7398 
 
 HKm, 0942 
 
 257 
 
 0,67947 
 9,119/2 
 
 10,24050 
 
 10^03909 
 
 Then otf=o, 88298 the fine of 62 the mer. alt. 7 Hence Iat.=:48 N. decl. 
 And w/r:o,37486 the fine of 22 the mid. depr. J 20 on July 23. 
 
 193- 
 
 PROBLEM XLIX. 
 
 Being at fea in the ive/lern ocean^ the Sun was obferved to have 27 24' of 
 altitude when due IV. ; and to have 14*431' ^^^- ^ ^ ^^' ^' ^^' ' ^(({wed 
 the latitude of that place ^ and the Sun's declination. 
 
 Let C) N, be the Sun's places at W., and at 6 h. 77) 
 
 oc, N/, the fines of their altitudes. Q 
 
 Cf= ON, the arcs of declination. 
 
 Z. cof = z. NO/=latitude. 
 
 s cc 
 In Aocc. As s,oc:R::SiCc:s,jLcoc::^- XR. 
 
 j,oc 
 
 In Ao/N. As i-,oN:R::x,N/:j,z.No/= XRi 
 
 i hen = =1 And ss,ccz=s.ocxs,iit, 
 
 i,OC \J,ON / S)CC ' ' ' 
 
 L^, alt. W. + Ls, alt. at 6 
 
 Or i, alt. W. X Sy alt. at 6=:jj,decl. Or- 
 
 =:L5,decl, 
 
 oc-:/,27 
 
 24 Its L,t 
 
 ^/= 14 
 
 43i its L,j 
 
 
 Si 
 
 cctz 2C 
 
 00' 
 
 o.-rr 27 
 
 24 its l' 
 
 And LV,alt.W. -f L5,decl.=:Lx,lat. 
 
 9,66295 
 9.40514 
 
 19,06809 
 
 9,53404 
 
 0,33705 
 
 Z.qi>7-= 4 
 
 194. 
 
 :i= 48 00 
 
 9,87109 
 PROBLEM 
 
 L. 
 
 At a place in the wejlern ocean^ the Sun at rifing was chferved to he 59* 
 15^ 40''' from the true north point of the horizon ; and at 6 h. A. M.^ the 
 altitude was obferved 14 43^^.' Required the latitude and declination. 
 
 Let V, and N, be the places of the Sun at rifing and at 6 h. A. M. 
 N/, the fine of the alt. at 6. OKjVz; are arcs of declination, 
 ox-, the afcenfional difF. Z.No/=lat. i Z. Yoyz^co-lautudc. 
 ov, =:co-amplitudc. 
 
 N 
 
 ow
 
 2S8 
 
 ASTRONOMY. Book V. 
 
 J,OVXi,VOV 
 
 How in A OVt/. As R : s^v : : s,vov : s^vv 
 in A NO/. As j,Nor : R : : j,Nr : j,on= 
 
 R 
 R X J,N/ 
 
 ^. J,OV X JVOW J,Nt X R , J,N/ 
 
 Then = i and /jVoux/jNO/r: Xrr. 
 
 R JjNO/ SyOV 
 
 That IS, =j,No/ X s f^oti hence =:(2j,no/ X ; ,no/=);,2ko/. 
 
 (IV, 189) 
 
 Therefore lV, ampl. + Lj, alt. at 6+ l,2 = lx, double the latitude. 
 
 If the latitude is lei's than 45" j othcrwife it is double the co-latitude. 
 lV, ampl. 59 15' 40" 0,29147 Rad. r 10,00000 
 
 I/, alt. 6 14 43 30 9,40514 /.ampl. 59 15' 40" 9,708^3 
 
 I, 2 0,30103 /, lat. 47 59 9>^256s 
 
 1/, 84* 02' 
 
 Co-lat. is 42 01 
 
 195. 
 
 9,99764 
 
 tf decl. 20 00 
 
 9.53418 
 
 PROBLEM LI. 
 
 Being at fed in the ivejlern ocean^ feme time in the night, the dijkince of 
 iwojiars when both were on the meridian, was ohferved to be 20*; and l*^ 49"^ 
 after, the difference of their altitudes was 14 35|-^ and the difference of 
 their azimuths 30 C)\': Requir-'d the latitude of the place. 
 
 As the flars were on the meridian when firft -.0^ 
 obferved ; their diftance, difference of declina- 
 tion, and difference of altitudes, at that time, 
 are equal : If they are firft at d, b, and in the 
 diffeience of time revolve to d, b; then is 
 known Z.P, Z.DZB, db, and ZB ZD. 
 Let ZB ZDz:;Nj and find ZB + zDr^M. 
 
 Now (IV. 239) J,ZB X^,ZD xADZB4-i\zD X;ZB /,DB. 
 
 Or (IV, 181, 174) i/jN f/,MxADZB + |AN-f |5\m=:/>db. 
 
 Or /,DZB X {j\n /,DZB X {'/m 4- { j'n + 5/ Vi =/,DB. 
 
 Or s\dzb-\- I X i/,N4- 1 J,DZi; x i/,Mrr/,DB, 
 
 Th ' _ / 2/,DB I -f i'pZB Xi\N__ '\ 2/,UB T;\dZ3 X j\n 
 ' ~\ I S^ DZB ~' -i/jDZB " 
 
 ^ /2/,DB </, DZBXi^N \ 2i\DB 2jViDZBXAf 
 
 Or j,M=:(- =) Ll. 
 
 V^'jCZB VjUZB /2J;,-DZB 2.Vi,iDZB 
 
 (iV. 193, 197) 
 
 Let2L\',|DZB f-Lr,DB=:L,A ; and 2L'j,|DZB.L2LV,|DZB + Lf-,KrrL,3. 
 Then j\Mr:A p.. 
 Here de:^ 20^00' 
 
 ADZB^IjO 9 
 |Z.D7,B = I5 4| , 
 
 aD=:N:^i4 35 
 
 Then 
 
 2lV 
 
 {dzb 1,16952 
 
 
 2l\,|dzB 
 
 1,16952 
 
 l/ 
 
 ,l)E 9.Q72QQ 
 
 
 21.S ,\Z& 
 
 9,969-60 
 
 Ar: 
 
 13,884 1,14251 
 
 
 l/.N 
 
 g,98c76 
 
 B=: 
 
 I^.?-?! 
 
 
 8=13,331 
 
 1,12488 
 
 i,M = 
 
 0,553 ; and 2bH 
 
 -2D 
 
 =56" =5 r. 

 
 Book V. 
 
 ASTRONOMY. 
 
 i59 
 
 Then f fum + jdifF.=:zB = 35 30^''; and f fum | diff. =izd=:20^55^ 
 Now x,DB : J,DZB : : s^xb : s^pdz j and j,zpd : 5,pdz ; : j,zd : x,zp 42 i\ 
 Hence the latitude is 47 59^ N, 
 
 196. PROBLEM Lir. 
 
 By obfervattons made at a place In the wejlern oceariy it was found that 
 the Sun's altitude zuas 36 5., when bis azimuth zuas N. 100 5^ E. and 
 his alt. was 46 5., vchen his azimuth was N. 114 28' E, IVhat was 
 the latitude of the place and the Sun's declination f 
 
 Let A, B, he the Sun's places when obferved. 
 In'the triangles paz, pbz, where pa::::pb. 
 By IV 2-^0 J ^zPXi'zA APZAXx,zPXr,ZA=:APAi 
 
 / Jy' i ^'jZPXi\zB ApZBX5,ZPXJ,ZB=:i\pAi 
 
 Then/,pzBXx,ZBXi^,zp j\pzAXx,ZAXJ,zprr5\zBXj\zp j\zax/,!8P. 
 
 Or 
 
 Then 
 
 S ,PZB XX,ZB J ,PZA X J,ZA XSt'^V s'iT.Q J ,ZA Xj\zAXi,ZP, 
 ApzB X f,ZB CO /,PZ A X 
 
 X ,ZB X ,ZA 
 
 the latitude. 
 
 Let Lx\ PZB 4- L^, ZB 
 
 L.A 
 
 And L/,pzA-}-Ly,zA : L>Bt Then 
 
 A o B 
 
 :f ,RP. 
 
 J ZB S ,ZA 
 
 Here A~o,i4i64 ; b=:o,28770; f\zB ft 0,71933 j j\za 2= 0,58779. 
 
 Then^^^^~'-,Q005irf,42. Hence lat^ is 48 N. decl. 20^ N 
 0,14606 7 J '-r 
 
 197. PROBLEM Llir. 
 
 Given two altitudes of the Sun and the time from noon when th'ofe oltitucUi 
 I'sej-e taken ; thence to find the latitude and declination. 
 
 Exam. At 8 /;. 8 ot. i i s. A. M.<, the alt. zvas 36 ; and at 9 h. 10 w. 
 51 J. the alt. was 46" ; at a place in th< zve^/lcrn ocean ^ fame tim in 
 May, 1763. 
 
 Let B, A, be the Sun's placci 
 obferved ; 
 
 w, , thofe of noon and midnight; 
 Ff, Fi'/, F7W, FK, reprcfcnt the fines 
 of thcdiftances of thofc pbces from 
 the horizon HR, to the dian;eter 
 *^q\ and d: Yd Fe. 
 
 On mn dcfcrihe the femicirclc 
 Pibn, rcprefenting half the parallel 
 of declination, and let a^, b^, be 
 at right angles to mn : Then will 
 the angles mEb, m'E.a, reprefent the 
 times from noon, at the obferva- 
 tions B, a ; and mQ, mA, arc as 
 the verfcd fines of thofe times. 
 
 And m^ mA'^^AB. 
 Then AB : d( : : ms : me 'y and mt + TezzFrnj the fmc of am. 
 
 i 2 Alio
 
 a6o 
 
 ASTRONOMY. 
 
 Book V. 
 
 Alfo AB '.de'.'. mn : otk i and Km rwrrpK, the fine of R. 
 
 Hence the latitude and declination are found. (^^5^ 
 
 -, , , , , mb-\-ma nib ma , 
 
 Here ab = ( J ,OTa s'ytnbzzsy x 2^, ;; = )j,M X 2f,N. 
 
 ji J J / t , ^ 2d+ift 2d ift , 
 
 And ^^=(j,2d alt. /,ift alt.=j\ X 2;, =j\w x 2/,v 
 
 (IV. i8i) 
 
 X2/,V. 
 
 (IV. 182) 
 
 r\ow L,- X ;wrrL,wK=:u\M + lj ,n + lj,w + l,^v,l,2. 
 
 AB 
 
 And L, xmB L,77UZZLymK.-i-ZtSiljLmzb. 
 
 In this example, the z.mEbr=2 h. 51 49=57*' 57!'; its 1=28 sS^. 
 iC. ;;;= 2 h. 49 9=42 17!. 
 
 42 
 
 7i 
 
 
 36 
 
 100 
 
 Hi 
 
 50 ll'-U 
 
 82 
 
 41 o'rrw. 
 
 '5 
 
 40 
 
 7 50 = N 
 
 10 
 
 5 =v. 
 
 i,36^ 
 
 me 
 
 ttn 
 x.m 
 
 = 0,58778 
 = 0,29521 
 
 
 
 = 0,88299 = 
 = 1,25810 
 
 162 0' 19" 
 
 
 = 0,37511 = 
 
 :22 I 51 
 
 
 
 84 2 10 
 
 
 39 
 
 58 28 1 
 
 L J,U 
 
 50" 71' 
 
 o,ii49t 
 
 L /,N 
 
 7 50 
 
 0,86553 
 
 Lj ,W 
 
 41 
 
 9.8777 
 
 LJ.V 
 
 5 
 
 8,94030 
 
 h,Z 
 
 1.25785 
 
 0,30103 
 
 L,Km 
 
 10,09962 
 
 2Lj,^mi 
 
 28" 58f 
 
 9,37050 
 
 L,me 0,29521 
 42 o' 55" the co-Iatitudc. 
 
 9,470 1? 
 
 19 59 14 the decl. on May 20tb. 
 
 19S. The following is another folution, on difFerent principles. 
 In the triangles bpz, apz, are given, (fee the foregoing figure.) 
 
 Bz, Az ; BPZ, APZ, hour angles ; and P3 = pa ; 
 To find PB, rz; for whoie fum and diff. put M and N. 
 
 Now rlV 2-2Q) J^'PBXX,PZxAbPZ + ApBxApz=AbZ. 
 V iyJ i j,PA XX,PZ x/,APZ + /,P'A x/,PZ=r/,AZ. 
 
 Or (IV. 181. 174) I f ;^ (i,M xA 
 
 BPZ4-|i\N'-- y/,M=/,BZ. 
 APZ + IAn' + i^ ,M =/, AZ. 
 
 Therefore (f AK-i/,M = )^^-'^^--^^-'^^^^^>^r.^^--^''^5^-^-^^^^-^' 
 
 S ,CPZ 
 
 S ,APZ 
 
 And (|x%N4-iy\M = )/,BZ-^|/,N i;,M Xi^B^Z=J\AZ |/,N |i\M 
 X j\aPZ. 
 
 Hrnce /,Bzxi\APZ ls\s + |/,MA;\AFZ = ;\Ai/, j\BPZ |i\N -fi/,M 
 X/,BPZ. 
 
 Therefore
 
 Book V. ASTRONOMY. 261 
 
 Therefore /,bz x/,apz j\ az X ^^dpz n/ yAPz /,bpz x |/_^>M-J/Vm. 
 Again /,az /,bz r=/,APz j\bpzX|/,N {s\m, 
 
 /,BPZX/,AZ /jAPZX/jBzl 
 
 1 herefore Is ,n + 1 J ,m =: v ^ J 
 
 JjAPZ fBPZ. f 
 
 >HenceIV. 216- 
 
 . , . J,AZ J,BZ I 
 
 And |j,N -Jj,M=:-t ^ l 
 
 * ' - ' '/,APZ /,BPZ 
 
 //,AZ + /,BPZ X /, AZ j\bz /,APZ X j\bz \ 
 ' V J,APZ J,BPZ J 
 
 I +j\bpzx /,Az 1 4 /,APz xAbz 
 
 /,APZ /'bPZ 
 
 //jBPZ X/,AZ /,AZ /,APZ X/,BZ+/, BZ__ Y 
 
 ^'^"V /,APZ /,BPZ '"' 
 
 1 /,BPZ X 
 
 /,APZ- 
 
 /,AZ I 
 J ,APZ j\bPZ 
 
 .f\APZ X j\bz 
 
 ,APZ X j\bZ_ \ 
 -,_._^ ,BPZ ~J 
 
 'fBPZ Xi\AZ 2jV,|aPZ x/,BZ 
 /,APZ /,BPZ 
 
 , /^\bpzXj\az v\. 
 ;r, JM = ^ c- 
 
 V S yAPZ S fl 
 
 25VfBPZ Xi\AZ 2S 
 
 s\aPZ /,BPZ 
 
 . , X /T<, BPZX/,AZ 'Z;,ABZX/,RZ \ 
 
 And^jKrzf ^ ; = 1 
 
 ' V jjAPz j,BPz y 
 
 2/J,fBPZ X/,AZ 2^J,iAPZ Xx\bZ 
 j\aPZ /jBPZ 
 
 Here BPzr=57 57i-'. '^^ |r:28 58.^/; apz=42 ijl\ its f = 21'' 8|'. 
 Bi=:54; Az=:44; /,BPzno,53c6o ; /,APz = 0,73978, their 
 difference =0,209 1 8. 
 
 21.//, 
 Ls\ 
 
 l.B 
 
 I.,A .B 
 
 2S 585-' 
 
 44- 
 1,10104 
 
 21 8^' 
 54- 
 
 1,02260 
 0.0784.4 
 
 ( 0,30103 , 
 
 } 9,88384 ^^"' ^^ i^f 
 9,85693 
 
 10,04180 h,a 
 
 { 0,301c 
 i 9.939^ 
 
 103 
 
 , . .39+6 
 9,76922 
 
 10,00971 
 
 8,89454 
 
 Lj\aP7. j\ BI'Z 0,20918 9,320^ 
 
 I./,M 112* 1^ 
 
 9,57402 
 
 l/. 
 
 2L//, 
 
 l/. 
 
 44- 
 0,33764 
 
 2I 8f 
 
 54 
 
 L,3 0,15298 
 
 L,6i h 0,18466 
 Lj\ APZ S, BPZ 
 
 l/,n' 28 o' 
 
 J 0,30103 
 
 I 9>3705 
 9,^5693 
 
 9,52846 
 
 C 0,3010} 
 J 9,11438 
 
 9,76922 
 
 9' '8463 
 
 9,26637 
 0,20918 9,32052 
 
 9.945^5 
 
 FTence the latitude is 48^ 00^ N. ; declination 19" 59^, which aurwcrs 
 (Q tUe 2.0th of iVIaj'. 
 
 S 3 J9Q. And
 
 jgj ASTRONOMY. Book A^ 
 
 iQg. And hence is readily derived the invcftigation of that method, 
 publiihed in the year 1759, and then ufed by fome for finding the true 
 latitude at fea, by knowing the latitude by account (or dead reckoning), 
 the Sun's declination, two altitudes of the Sun, and the time between the 
 obfervations. Thus. See the laft figure. 
 
 Let M and N reprefent- the half fum and half difF. of the times from 
 noon ; W and v, the half fum and half difF. of the two altitudes. 
 AB the diff. of the co-fines of the times from noon, to the radius Em. 
 AD the difF. xjf the fines of the altitudes. 
 /.PAD rcprcfents the latitude ; ^m the co-f. of the declination. 
 
 Now (i97)^,MX2j,N = AB reduced to the rad. I c^yj and j\wxJi, 
 t:;=ad. But in the triacgle abd. 
 
 A lat. : R ; : AD : ABrr-r-i-t- x ad, to rad. ei. 
 ' Si lat. 
 
 And /,decl. : R : : ab : ABX;r^^=^X;r^ X ad = ab reduced 
 
 to the radius f Q.y. 
 
 Then .,M X 2^>N^=7;d^ Xtt^ X Aw X 2.,v. 
 
 And j,M=:-T-] rx-v-j x Xf,wxj,v. 
 
 jjdeci. s ,Iat. ^, n 
 
 Then M + Nnz.7E^, the time from noon at the leaft altitude. 
 
 And M N=:z.iE^, the time from noon at the greateft altitude. 
 
 Hence the vcrfed fines of the arcs 7nb or 7;/^, are known to rad. f <^^. 
 
 Now R : Ew : : v^ma : ?.v A ~ ? x v^ma j or wB zz Em X v^mb. 
 
 And R : s^mhd : : rtiA : md { : : mB : me. ) 
 
 Then m Fd+dm, or to Fe-i-enij is the fine of the mer. alt. 
 
 Hence the two operations, 
 ill. lV, decl. + LV, lat. -f lV,n + l/,w + Lj,vrrLi,M. 
 
 Hence the times are known ; viz. arcs ma^ mb. 
 ad. Ls\ decl. + L/, lat. -}-L,2 + 2Lj,iOTfl~L,W. 
 
 Then by the merid, alt. and declination the latitude may be found. 
 If this latitude and that affijmed are the fame, then the latitude by ac- 
 count, or dead reckoning, may be taken as the true latitude. 
 
 But if they diffbr, it is plain that the /Lmhd^ the co-latitude, is lefs or 
 greater than was aflumed. 
 
 200. PROBLEM LIV. 
 
 Given three dtfcendiug (or afcendlng) altitudes of the Sun, taken on the 
 fame day at unequtil known intervals of time \ thence to find thofe times, the- 
 latitude of the place of obfervation, and the Sun^s declination. 'Thus, fuppofe 
 tn July, 1 763, the altitudes were 54 30'!, 46"^, and 36'^ j and the iniervaU 
 ef tl!i:c 60 tn. 55 s. and 62 ;;:. 40 s. 3 required tht rejl^ 
 
 Ui
 
 Fook'V. 
 
 ASTRONOMY. 
 
 Let ma ft he the parallel of de- 
 clination defcribed on m n, and a, 
 h^ Cy the places' of the Sun when ob- 
 ferved ; a, 3 b, and c c the fines 
 of the times from noon to the ra- 
 dius Em; /n, w, the places at noon 
 and midriight ; and let/" F, ^ F, ^ F, 
 ?n F, F K, reprefcnt the fines of the 
 diftances of thofe places from the 
 horizon hr. Draw c g parallel to 
 A c ; then if the Z. ^^E, which is 
 equal to the Z. mEc, the time from 
 noon when the greateft altitude was 
 taken, could be found, the times 
 from noon when the other two were 
 taken would be known alfo. Now 
 ch being one interval, and ba the other, & c and b <?, the chords of thefe 
 arcs, will be known, as alfo ca, which is the chord of their fum. Draw 
 h perpendicular to ac. Now in the right angled /S.cbo, theZ.^ cozz\ 
 the arc ba (II. I20), and the fide c b = twice the fine of i the arc be, 
 confcquentiy K : s,b c o { s arc b a) : : b c (= 2 Sy i; arc be) : ba; and 
 R : SyC bo (s\ { arc b a) : : b c {2 $,{ arc be) : c o. Now d f^ d e^ thp 
 difF. of th fines of the altitudes, are known ; and the lines f d^ ac, a c, 
 are cut proportionally in ^, b and r\ confcquentiy, df\ ^<f : :(ca : CB : :) 
 e a \ c r^ and c o c r=:ro : In the triangle bar, (III. j^6) r o : b o : : 
 K : t^y A r b ozz /_r eg. NowZ. e e ( rrcompt. of \ whole interval a f) 
 Z.r<r^~Z.^ f E = Z.C Ewr: time from noon when greateft altitude was 
 obferved ; therefore the time is known, as well as cm the verfed fine of tfeat 
 time from noon. 
 
 Again R : s-,acg '.: ae : e grzAC. 
 
 And AC : mc : : f d : dm. Then Fd-i-dm Fm, the fine of um. 
 And AC : mn : : fd : ctk. Then kw FwrrpK, the fine of r//. 
 
 This Problem has, at times, for near a century pall, exerclfed the ta- 
 lents of many ingenious perfons, as well in Ruflia, Germany, Holland, 
 and France, as in England j perhaps, on account of its apparent uie at 
 fea : And among the diftcrent folutions there fcems none fhorter or more 
 intelligible, particularly to beginners, than that above ; howeycr, for the 
 fake of the more inquilltive, another folution, which has been commonly 
 given, is here fubjoined. 
 
 20I. In the laft figure, the three triangk-s cpz, cpz, APZ, are tholf 
 concerned ; in which the fame two fides are common in each. 
 /, CPZ X^,PC Xi,PZ + /,PC X j\pz- .j\zc 1 
 /,BPz XI, PC Xi,pz + j\pc xi\pzm\z6 >hy IV. 239. 
 
 i\APZ X;,PZ Xi,PC -|-/,PC X j\j'/.~/,ZA J 
 
 Then j\ci'z i\bpz X j,pc }<.s,pz=:[s \%c 5\zn = ) .'/. (II. 4S.) 
 
 And i\cp/. - -/. APZ >r f,PC X j,Pz:=:(i\''.C /,ZA::=) D. 
 Hence nx /,.!'/, j,r:' ' =Vxi\cpz i\Ai'z. (II. 14?) 
 
 Then PXfjCP.-, - ..'//,CPz:^j:) x/,BPZ ^/X/,APZ. 
 
 b 4 Or,
 
 f^S^ 
 
 astronomy: 
 
 Book V. 
 
 Or, DXAcPz </x/,cpz (=UIIjXj\cpz) =:dxAbpz dxs\A2z. 
 
 ^ut, BPZ=:CPZ + BPC, and ;^pz = cpz + APC. 
 
 r^ r .1 5 ABPZ = j\cPZXi\BPC J,CPZ X/,BPC. 7 ,., ^..\ 
 
 Confequcntly, 1 .^l^p^^/^cPzx Aapc^Acpzx/,apc. i(^^- ^^^'^ 
 f lence d / X A j d X j\cpz x /,bpc d x j,cpz x j,bpc I u v f bftitu 
 c pz = . ( d xi',cpzxs\APC -i-d Xi,cp zxj.,APc. j y " 
 
 Or, D DX j\BPC dri-dXs\APC X /, CPZ = dx S, APC D X S, BPC X 5, 
 
 CPZ. Wherefore, 
 D DXS^yBVCtn d dxi'yAPC 
 
 c/XJ,A PCco D XJ,B P C. 
 ^', BPC X Deo -y. APCXd 
 
 /_i j\bp( 
 \~ dxl 
 
 BPC X D c/3 1 j\apc xd 
 
 ) 
 
 J, CPZ 
 
 :-t, CPZ, 
 
 APC CA)D X^,BPC 
 
 the meafure of the time 
 
 "~J, APC X^'-/3 J,BPCXD ""ijCPZ 
 
 from noon when the greatcfl altitude w^s obferved. 
 |Jere f^=;j,54. 3o'=o,8i4I2 l 
 
 e=s,/{.6 00 =0,71934 (.Then ^=Oj22633 J des:0,o<):\.'jS., 
 f/=j,36 00 =0,58779 3 
 Thearc r ^, orz.cPB,= i h. o m. 55 f. = 15 i3i'j its f = 7" 36^''. 
 arc a by OTjCBPAy i h. 2 m. 40 f. = 15 40 ; its |= 7 50. 
 arc cq, or4.cPA, = 2 h. 3 m. 35 f.=30 53J ; its 1^1$ 26|. 
 
 To find thi hour by the id method. 
 
 *To find the ho^ur by thi \Jl method. 
 10,00000 
 
 9,13447 
 9 12224 
 
 Iladiu 
 
 Is to s, f ba 
 
 As s, \bc 
 
 To i ob 
 
 Radius 
 
 li to J, I be 
 
 As / 1 ba 
 
 To \ CO 
 
 Is to de 
 
 So is I ac (j,|rEa) 
 To \cr "' 
 
 I CO 
 
 A? f ro 
 Is to f ^ 
 So is Rad. 
 To tyjLrbo 
 
 Z.a c E 
 
 90^00 
 
 7 50 
 
 7 36J 
 ,01806 
 90 00' 
 ? 36I 
 7 50 
 '3i27 
 22633A. 
 ,09478 
 i5''26f 
 ,11154 
 >327 
 ,01973 
 ,0180^ 
 C)0 00 
 
 47 32'- 
 74 33f 
 
 Time of 2d obfefvation 
 Tinie of 3d obferyation 
 
 8,25671 
 
 lOjOOCOO 
 
 9,12224 
 
 9-99593 
 
 q,i 1817 
 
 CO, 64526 
 
 8,97672 
 
 9,42547 
 
 0,04745 
 
 1,70480 
 
 8,25671 
 
 lo.orooo 
 
 o,q6i5i 
 
 h m 
 : I 4S 4 
 
 2 48 59 
 
 3 51 39 
 
 D {~fd) 
 
 .',Z.BPC 
 
 DX/.BPC 
 
 d[,-de) 
 
 /,Z.APC 
 yXi.APC 
 
 D (=/'^) 
 fiZ-BPC 
 
 D Xl/,BPC 
 
 d{zzde') 
 
 i;,APC 
 
 </Xz/,APC 
 
 DX V.BPC 
 
 (fXfjAPCDXfjBPC ,00550 
 
 DXijBTC i/XJjArc ,01078 
 
 2633 9.35474 
 
 15 Ui' 94'943 
 
 ,05945 ^774'7 
 
 ,09478 8,97673 
 
 3053F 9-71052 
 
 ,04867 8,68724 
 
 '22633 9,35474 
 
 i5'3r _8j^54i5J 
 
 ,00795 7,90026 
 
 ,09478 8,97673 
 
 3053f 9-'S'9 8 
 o'345 
 ,00795 
 
 8, 1 2870 
 
 17.74036 
 8,0326 2 
 
 tang. 27 i|' 9,707^4 
 
 Hence the h.fr.n. of iftob.is i''48' n"' 
 Of the fecond obfervation 2 49 z 
 Of the third obfervation 3 51 42 
 Confequently from any two of thele, 
 with their correfponding ahitudcs the 
 latitude and declination may be found 
 by Problem 53. 
 
 Here the. altitudes are fuppofgd to be defcending pnes, or in the after- 
 noon. ' 
 
 202. If
 
 Book V. ASTRONOMY. 26$ 
 
 202. If the altitudes were taken at equal intervals of time ; the two 
 firft proportions are ufelefs : For a b and b c being equal, the point falls 
 in the middle of the chord c a ; and b Oy the verfed fine, is known. 
 Then df'. de: : (ca : cb : :) ca : c r ; and c cr=:ro. 
 And bo : ro : : Rad. : t^r b o^zza eg ; and atE acg-=.gcE=:cEm. 
 Hence the times from noon are known j and alfo mc, the verfed fine of 
 cEm. 
 
 Again, in /^ a c g. n : s^,acg : '. ac. cg:= ac. 
 And AC : ;wc : :fd : dm. Then Fd+dmzzFm. 
 Alfo AQ : mn : /fd : mvi. Then k/k fot=fk.. 
 
 Here dvzz:o,%\^\z, fFzro,7i934 ;/f=o, 58779; dfzz.oiz^'i^i ; <jVr:o,09478. 
 Alfo, arc f^=:i h. i m. 47I f.nij'^ 26^'; <-o:=:o,26636 ; ^0^:0,03613. 
 Then df '. de : : ca : rrr:o,22309 ; and ro fr=ro,04327. 
 And bo -.re:: Rad. : t,rboz=^o S}' ; then 74 33'^ 50 S'{=^J^? 24!'. 
 
 Iv^ rn s 
 
 Then 24 24!'=: 1 37 39 from noon at firft alt. 
 
 And 24 24|-'4-i5' 261=39 5l|:=:2 39 26y from noon at 2d alt. 
 Alfo 39 51I+15 261=55 i8j=:3 41 14 from noon at 3d alt. 
 Here the altitudes are fuppofed to be afcending ones, or in the forenoon. 
 Again, rad. : s,ca : : ac : Acr=o,34i42. 
 
 And AC : OTC : lyi' : </ot=:o, 05927 ; then mF=o,87338=:T, 60 51'. 
 Alfo AC : wa ::/!/: OTKm, 32581 ; then fk=o,45 243=:/, 26 54'. 
 Hence the latitude is 46 vj N. and the declin. 16 ^8^' ; or on May 7th. 
 
 203. But the operation In this cafe maybe much contracted. Thus, 
 
 fmce [df: de : : ca :j cr= ; thence roz=:lcG -jy =J 
 
 df-^2dexco 
 
 /ro nxco \ -Dxico 
 And (bo : ro : : k:) t.rb ol 7 =-77 j 1 = j-rTZ J r 
 
 where n-zidf ^d e. 
 
 '&\x\.ViXv^cb si^\cbz:L\s^cbxt^\cb. (IV. 193, 195) 
 
 Then t^rbQ-rr \ r; or L/,r 3 <7 = LV,^rZ'4-L\^/+L,D. 
 
 dfx t^\cb -^ 
 
 Alfo (r : i^acg : : a c :) AC {s^^a c gXaczz) s\a c gxis^cb. 
 
 . ,, r, s ftnnxfd IX fd \ fd 
 
 And ( AC : m n : : f d :) mYizz[ ~=r 7=: j j. 
 
 ^ > - ' \ AC s ^a c gX2SyC b ys,acgxs,cb 
 
 . ,, ^, X , I'mcxfd fdx2ss,lmT.c \ fd 
 
 hnd (ac: mc:: fdAmdi^-',-^ ^- , ~H ^^-3 7 
 ^ ' ^ \ AC s\a cgX2S,cb /s ^acg xsjcb 
 
 XsslmEc. 
 
 Hence Ls^^a c g-\-Ls,cb + L,fd rrL,wK. 
 
 And Li\a c g-Y l'JjC b + L,fd + 2Ls,lmEc = L, W. 
 
 Here ^==0,2263^; aj'f^o, 18955 ; and vzz^df- zdt:^:) 0,036/8. 
 Alfo i cb=zj 43' 26"; and a(k=:y^ 33 j'. 
 
 i,r
 
 ^66 
 
 x,r, 7* 43' 26" 
 
 l,D =0,03678 
 
 t,f,ricZ2i;0'> 09' 
 f* =7+ 33t 
 
 ASTRONOMr. 
 
 0,86764 
 0,64526 
 8,56561 
 
 10,07851 
 
 mic- 24 24^; half is 12" 12tV 
 
 L J ,<fr^ rr 50* 09 
 Cj,6e = 15 2t^ 
 it/J =: 22633 
 
 L,rK, =2 1,3260a 
 2Ls,lmtc zz 12 I2tV 
 
 L,fflr </ rr 0,05923 
 
 Book r: 
 
 0,19329 
 0,57452 
 
 9-3547 
 
 * ii 
 
 10^12255 
 8,64998 
 
 8.77253 
 
 P R O B L E M LV. 
 
 204. Given the obliquity of the ecliptic, the latitude of a place, and 
 the apparent time at that place. To find the longitude of the nonagcfi- 
 mal degree* 
 
 Construction. 
 
 The given time applied to the fun's right 
 afcenfion, gives the right afcenfion of the mid- 
 he.iven. 
 
 Then let the primitive circle reprefent the fol- 
 ftitial colure, where p, />, reprefent the poles of 
 the equator V b, and ecliptic T c, the center Y 
 being their interfe(lion. In the equator apply the 
 right afcenfion of the mid-heaven from T to r, 
 and the circle pzb, being defcribed, is that nieri- 
 ridian, the interfeclion of which by a parallel of latitude defcribed about 
 p, gives z, the place of the zenith ; through z defcribe a circle of lon- 
 gitude pzc, and the point c is the nonagefimal degree, and Tc itss 
 longitude. 
 
 Computation. 
 
 In the triangle p/ z. Given p p the obliquity of the ecliptic^ 
 
 pz the co-latitude. 
 /. z p p the right af. of the mid-heaven. 
 Required z /> nV.237,23H, 
 
 Z. P/> z 
 Determination. 
 
 and 144. 
 
 When the right afcenfion of the mid-heaven falls in the firfl quadrant, 
 its quantity in degrees, increafed by 90, gives the angle z p />, and the 
 acute angle p /> z is the complement of the longitude of the nonagefimal 
 degree. 
 
 When in the fecond quadrant, the faid right afcenfion in degrees taken 
 from 270, leaves the angle x p p ; and the acute angle p /> z, increafed by 
 90 degrees, is the longitude of the nonagefimal degree. 
 
 When the faid right afcenfion falls in the third quadrant, its degrees, 
 taken from 270, leaves the angle z p />, and the angle ? p z, increafed 
 by 90 degrees, is the longitude foi-'gUt, 
 
 _5 Whijjj
 
 BookV. 
 
 ASTRONOMY. 
 
 a57 
 
 When in the fourth quadrant, the Hiid right afcenflon in degrees, lef- 
 fened by 270, leaves the angle z p/> ; and the fupplement of the angle, 
 Pdz, fo long as it continues to Hie obtufe, being added to three right 
 angles, or 270 degrees, gives the longitude of the nonagefimal degree j but 
 after it becomes acute, its complement is the longitude required. 
 
 Note. In thefe determinations the latitude of the place is fuppofed to be 
 jiorth, and lefs than the diftance of the tropic from the neareft pole. 
 
 Example. At Greemvich, in latitude 51* 2^\^ N. the obliquity of the 
 icliptic being 23 28': IVhat is the longitude of the nonagefimal degree on the 
 l^thofMay 1780, at i h. 2/\.m. 245. P. M. at ^ h. 4.0 m. is. P, M, at 
 13 h. 22 m. 26 s. P. M. and at i^h. 38 m. 4 s. P. M. P 
 
 The feveral right afcenfions of the mid-heaven are thus found : 
 
 1780. May i4tli, at 
 The Sun's right afcen. 
 
 Right afc. mid. hea. in time 
 
 Right afc. mid hea. in degrees 
 
 The angle z p /, or 2 p / 
 
 The two following operations are wrought by IV. 237, and 238. 
 
 h m 5 
 1 24 24 
 
 3 27 36 
 
 
 
 h m s 
 3 40 2 
 
 3 27 58 
 
 h m 8 
 
 13 22 26 
 3 29 34 
 
 h m s 
 15 38 4 
 
 3 29 56 
 
 4 52 
 
 7 8 oji6 52 c 
 
 19 8 
 
 73 00' 
 90 00 
 
 107 00' 
 270 00 
 
 253 00' 
 270 00 
 
 287 co' 
 270 00 
 
 163 00 
 
 163 00 
 
 17 CO 
 
 17 00 
 
 r z 
 
 '/ 
 Sum 
 DifF. 
 l-tvp 
 
 38 31}' 
 23 28 
 
 61 59-|., halfis 30" 59]' Ar.co. /. 0,28823 
 
 15 34., half is 7 31^- J. 9.H729 
 
 163 00, halfis 81 30 t" . 9.i74>o 
 
 Ar. CO. 
 
 Half diff./.3Z and/ 
 
 Halffum 
 
 The angle / 
 
 The different e 
 The fum 
 
 Sum 61" 59'/, half 
 
 iff. 1 5 3-^, half 
 
 /. z p/ 17 00, half 
 Half diff. A'sxand'/ 
 Half fum 
 The angle at p 
 
 Long, nonag. in 3d qd. 
 
 2 
 
 IC 
 
 9 
 
 4r 
 
 1 1 
 
 57 
 
 90 
 
 00 
 
 /. 0,5^002 
 
 /. 0,06591 
 s . 9,99624 
 t\ Q>I74';''^ 
 
 /. 9,23765 
 
 78 
 lOI 
 
 30 
 7 
 
 01 
 59 
 59t' 
 
 8 30 
 
 is the long, nonag. in ifl; quad, 
 is the long, nonag. in 2d quad. 
 
 Ar. CO. i. o,28r'2 5 Ar. co. /. 0,06691 
 /. 9,11729 /. 9,99624 
 
 /^.io.82:;;o /' io,825!;o 
 
 59 
 82 
 
 34 
 
 38 
 
 142 
 
 go 
 
 12, 
 00 
 
 232 
 
 12 
 
 /. 10, :3I02 
 
 its fupplement is 
 
 Long, nonag. in 4th qd. 
 
 /. 10,88865 
 
 37" 48' 
 
 270 O") 
 
 ^07 
 
 lS 
 
 *** The altitude of the nonagefimal being equal to z />, the diflance 
 of the zenith from the pole of the ecliptic, it is found by the fines of op- 
 pofite fides and angles in the fphcric triangle z p p : that is, fin. long, 
 pofla^. ; cg-f, lat. ; ; fm. i, z t p : fm. alt. nonagefimal. 
 
 S E C 1' I O N
 
 s63 ASTRONOMY. Book V. 
 
 SECTION VII. 
 Of Praciical AJlronomy. 
 
 205. Description and Use of Astronomical Instruments. 
 
 By Practical Astronomy is meant the knowledge of obfcrvinj 
 the celelHal bodies with refpe5l to their pofition, and time of the year j and 
 of deducing from thofe obfervations, certain conclufions ufeful in calcu- 
 lating the time, when any propofed pofition of thofe bodies fhall happen. 
 
 For this purpofe the Aftronomer, or Obferver, (hould have an obfer- 
 vatory properly furnifhed. 
 
 An Observatory is a room, or place, conveniently fituated, con- 
 trived, and furnifhed with proper aflronomical inftruments for obferving 
 the motions of the heavenly bodies : it Ihould have an uninterrupted view, 
 from the zenith, down to (or even below) the horizon, at leaft towards 
 its cardinal points ; and for this purpofe that part of the roof which lies ia 
 the direction of the meridian, in particular, fhould have moveable covers, 
 which may be eafily removed and put on again : by which means an in- 
 rtrument may be directed to any point of the heavens between the horizon 
 and zenith, as well to the northward as fouthward. 
 
 The furniture fhould confifl of fome, if not all, of the following in- 
 ftruments. 
 
 III. A Pen'dulum Clock for fhewing equal time. 
 - 2d. An Achromatic Refracting Telescope, or a Reflecting 
 Q-ii^^ of two feet at leaft in length, for obferving particular phaeno- 
 mena. 
 
 3d. A Micrometer for meafuring fmall angular diftances. 
 
 4th. An Astronomical Quadrant for obferving meridian alti- 
 tudes of the celeftial bodies. 
 
 5th. A Transit Instrument for obferving objeds as they pafs over 
 the meridian. 
 
 6th. An Equatorial Sector to obferve angular diftances of feveral 
 degrees, and the differences of right afcenfion and declination. 
 
 7th. An Equal Altitude Instrument for finding when an ob- 
 ject has the fame altitude on both fides of the meridian. ^ 
 
 It is not intended to give in this work any other than a general ac- 
 count of thefe inftruments, moft of which have met with confiderabie im- 
 provements (if they were not contrived) by the late Mr. George Graham, 
 F. R. S. one of the moft eminent artifts in mechanical contrivances that 
 this, or any other nation has produced : thofe readers who are curious to 
 fee a minute defcription of fuch, and other, inftruments, together with 
 their ufe fully exemplified, may confult the fccond volume of Dr. Smith's 
 complete Treatife of Optics, Stone's Treatife of Mathematical Inftru- 
 ments, the Philofophical Tranfadtions, and the works of many writers 
 who have trpated on iiich fubjccts, 
 
 5106. 0/
 
 BookV. ASTRONOMY. ^^69 
 
 2,06. Of the Pendulum Clock. 
 
 A clock which fhews time in hours, minutes, and feconds, fiiould be 
 chofen ; with which the obferver, by hearing the beats of the pendulum, 
 may count them by his ear, while his eye is employed on the motion of 
 the celeftial obje6l he is obferving. 
 
 Juft before the obje<3: arrives at the pofition defired, the Obferver fnoulJ 
 look on the clock and remark the time ; fuppofe it 9*^ 15"* 25' ; then fay- 
 ino- 25, 26, 27, 28, &c. refponfive to the beats of the pendulum, till he 
 fees through the inftrument the objeft arrived at the poQtion expec^d, 
 which fuppofe to happen when he fays 38 ; he then writes down 9^^ 
 15"' 38' for the time of obfervation, annexing the year and day of the 
 month. 
 
 If two perfons are concerned in making the obfervation, one may read 
 the time audibly, while the other obferves through the inflrumei^t, the Ob- 
 fer%''er repeating the laft fecond read, when the defircd pofition happens. 
 
 207. Of the 'Telefcope. 
 
 The Refracting Telescope is an inftrument with which almofl 
 every pcrfon is acquainted, efpecially the marine gentlemen ; it will 
 therefore be fufficient to remark here, that an ajircnornical telefcope has 
 only two convex glafTes ; viz. the eye-glafs, or that which is ufed next to 
 the eye ; and one at the other end, ufually called the objeft-glafs, which 
 has much the longer focal diftance : fuch an inftrument, although it in- 
 verts all objects, is yet as ufeful for viewing thofe in the heavens, as if it 
 fhewed them eredl ; the Obferver knowing that the motions are in an op- 
 pofite direcliion to thofe he fees through this telefcope : But the Achro- 
 matic Refracting Telefcope, which has been lately invented by Mr. 
 Holland, has its object-glafs compounded of three glafles, and combined 
 with two eye- glafles placed near each other. This inftrument, whicli ftiews 
 ohje6h in their true pofition, need not exceed three feet and a half in 
 length. 
 
 The Reflf.ctikg Telescope, as is generally well knov/n, (hews 
 objedls in their true pofitions ; and as it is much ftiortcr thiin the old rc- 
 fracler, it is therefore in much greater cftcem by feme. 
 
 A telefcope^ ufed in aftronomical obfcrvations, fhould have a metal 
 frame fixed in the focus of its objc(5t-glaf?, carrying fine filver wires 
 ftrctchcd at right angles to one another; one of them is to be vertical, 
 and the other hori/,onral ; the interfecSHon of thofe v/ires ought to be ex- 
 actly in the middle of the focus of the objcdl-glafs ; a line paiTing through 
 this interfcction and the center of the objeSt-glafs, is called the line ot 
 fight, or line of colUmation. 
 
 208 . Of the Micrometer. 
 
 A MiCROMETE;. is an inftrument ufed to meafure fin:;tl ang;ular di- 
 ftances by being placed in the focus of a telefcope. This is cftccted Sy 
 turning a fcrew, which moves a fine wire in a pofition paralLi to itfclf, ;,:.. 1 
 .nlfo parallel to a fixed wire ; both being in a plane at right aiij^ic to '.'jo 
 line of collimation : the dlftance of thcfe parallel v/ircs is ir.cafurcd hy :[) 
 number of t'irn<; the fcrew has taken to caufc their re-.c-fs ; 'A.hicb; iiiinibcr 
 of turns i: fl:cvvn on a gradiiatcd circular plate (like that oi a ';!o^;k} by ?v.
 
 ayd ASTRONOMY. Book V. 
 
 4 
 
 index, or hand, which revolves by the turntng of the fcrcw : now the di- 
 vifions on the plate, anfwering to a known angle or arc intercepted be- 
 tween the parallel wires, being known by experiment, any other diifance, 
 to which the wires can recede, may be known by* proportion ; and fo a 
 table of angles anfwering to every divifion on |he circular plate may b 
 formed, by which the oblerved angles will be readily known. 
 
 Thus in obfcrving the diameter of a planet ; when the wires are re- 
 moved fo far afunder, as to become parallel tangents at the fame time to 
 oppofite points of the planet, the meafure of the recefs of the wires will 
 fhew the diameter of the planet in minutes and feconds. 
 
 There is another micrometer publifhed by the late very ingenious Mr. 
 Dollond *, an account of which was given to the Royal Society by Mr. 
 James Short, F. R. S. and publifhed in the Philofophlcal TranfaiSlions fof 
 die year 1^53, which is thus. 
 
 Let a good circular objeft-glafs be neatly cut into two femicircles ; and 
 each femicircle fitted in a metal frame, fo that their diameters Aiding on 
 one another (by the means of a fcrcw) may have their centers fo brought 
 together as to appear like one glafs, and fo form one image ; or by their 
 centers receding may form two images of the fame objedl : it being a pro- 
 perty of fuch glaflbs, for any fegment, to exhibit a perfe;5l image of an 
 obje<Sl, although not fo bright as the whole glafs would give it. 
 
 Now proper fcales being fitted to this inftrument, to fhew how far the 
 centers recede, relative to the focal length of the glafs, will alfo fhew 
 how far the two parts of the fame object are afunder relativ? to its diftance 
 from the obje6l-glafs ; and confequently give the angle under which the 
 diftance of the parts of that objeil are feen. 
 
 209. Of the AJlronomical ^drant. 
 
 An Astronomical Quadrant Is an inftrument in the form of a 
 quarter of a circle, contained under two radii at right angles to one an- 
 other, and an arch equal to one fourth part of the circumference of ths 
 circle, and confifts of the following parts. 
 
 I ft:. Its frame. This Is ufually compofed of Iron or brafs bars, fet at 
 right angles to one another in as ftrong and neat a manner as a workman 
 can contrive, to preferve the face of the Inftrument in the fame plane, and 
 be as little affcfted by heat and cold as is pofllble. 
 
 2d. Itt center. This center, which Is a very fine point, fhould be con- 
 tained in a feparate piece of work fcrewed to the bars; and fo contrived, 
 that if the index, or telefcope, by frequent motion in a length of time, 
 fhould become irregular in its rotation, bv the parts wearing, a new col- 
 lar and focket may be fitted to the firft ccjitcr, and the inftrument reftored 
 to its original accuracy. 
 
 The firft notion of fuch a micrometer was given by Roemtr, a Dane, in 
 the year \t~f^ ; Mr. Sa-very, an Englifliman, alfo thought of fuch a con- 
 trivance, which he communicated to the Royal Society in the year 1743 ; Mr. 
 Bouguer, a Frenchman, alfo propofed it in the year 1748 ; and Mr. Dollond., 
 an Englilhman, publiOied it in the year 1753 : but the public are obliged to 
 Mr. ^hort for putting the theory into execution, othervvife it might fiJll have 
 continued only as an ingenious thought. 
 
 3d. lu
 
 Book V. ASTRONOMY. T271 
 
 ' 3d. Its limb. This is a brafs arch of about two inches broad, welt 
 fixed to the faid frame-work, and generally continued a little farther at 
 ach end than the extent of go degrees ; the two perpendicular ^ladii may 
 be ^Ifo covered with plates of brafs fcrewcd to them ; and the whole face 
 , of the inftrument is to be worked fmooth, and brought into the fame 
 'plane with the greateft care. 
 
 4th. Its dlvijions. The arcs of 60, 30, and 90 degrees, and alfo the 
 intermediate degrees, together with fuch fubdivifions as the fize of tUc 
 degrees will conveniently contain, are laid down by accurate methods 
 well known to good workmen. 
 
 5th. Its index or ulefcope. This, which is ufually a brafs tube con- 
 taining the proper glafles and crofs wires, is fixed near the objeil: end to 
 SI brafs plate, a little above a circular hole, or focket, in tlie plate : this 
 focket goes rou;id a collar concentric to the center, and fixed to the cen- 
 ter-piece : fo that, although the axis of the telefcope does not, as a ra- 
 dius, pafs through the center, yet it always keeps at the fame diftancc 
 from it in every pofition : to the eye -end of the tube is fcrewed a flat 
 plate, which Aides along the limb with the telefcope \ this plate, called 
 the Vernier, contains certain divifioiis, which, ufed with thofe on the 
 limb, give the angle to minutes or feconds of a degree, according to the 
 fize of the inftrument : tfee beginning of the divifions called the index, 
 on the Vernier^ is as far diftant from the axis, or line of coUimation, as 
 the center is ; and therefore the pofition of an obje6i is givezi as truly, as 
 if the line of collimation coincided with a radius, 
 
 6th. Its -pedejlal. This part, which fhould, by its conftrudtion, be 
 very fteady, may be either moveable or fixed : the moveable pedeftal is 
 commonly a ftrong pillar (landing on a tripod, or three-footed ftand ; 
 with holes through each foot, either to fcrew them to a floor, or to pin 
 them to the ground : the fixed pedeftal may be either a ftrong timber 
 frame, or the wall of the obfcrvatory, or a ftone fhaft built from the 
 ground through the middle of the floor of the obfervatory. On the top 
 of the pillar, of either fort of pedeftal, may be fixed a piece of machinery 
 called the army which is attached by fcrews to the midd/e of the plane of 
 the quadrant, on the under fide. The arm is contrived to give to the 
 inftrument, either an horizontal, vertical, or oblique motion ; which mo- 
 tions fhould be fteady, and free from jerks, or fhakcs : but when a wall, 
 or ftone ftiaft, is ufed as the fupporter, the quadrant is then fixed to the 
 wall, or fhaft (without its arm attached) and is called a Mural Arch ; 
 its plane is adjuftcd to that of the meridian, and this is the beft method 
 of fixing the quadrant for taking the mcridiaja altitude of tlic ftars, or 
 planets. 
 
 7th. //; plummet. This is a fufficient ball, or vi'cight, hanging to one 
 end of a very fine filver wire, the upper end being fixed in the radius 
 continued above the center. Now when the face of the quadrant is fet 
 in the plane of an azimuth circle, one of its radii is brought into a verti- 
 cal pofition by the help of the plummet, the wire being made to bif^'(it 
 the center-point and the divifion of 90 on the arch ; and to diftinguifli 
 thcfc bifecflions with accuracy, they arc to be examined with a ('mall 
 profpcil, or magnifying-glafs : the tjall fhould hang freely in a veflel of 
 
 Waiter to check its vibrations, 
 
 210. Of
 
 aya ASTRONOMY. Book \ 
 
 110. Of the tratifit Injirument. 
 
 This inftrumcnt confifts of a telefcope fixed at right angles to art hd- 
 rizontal axis, which axis muft be fo fupported, that the line of coUima- 
 tion of the telefcope may move in the plane of the meridian.* 
 
 The axis, to the middle of which the telefcope is fixed, fliould gra- 
 dually taper towards its ends, and terminate in cylinders well turned and 
 fmoothed : and a proper balance is to be put on the tube, fo that it may 
 ftand at any elevation when its axis rcfts on the fupporters. 
 
 Two upright pofts of wood or ftone, firmly fixed at a proper diftance, 
 are to fuftain the fupporters of this inftrument : thefe fupporters are two 
 thick brafs plates, having well fmoothed angular notches in their upper 
 ends to receive the cylindrical arms of the axis : each of the notched 
 plates are contrived to be moveable by a fcrew, which Aides them upon 
 the furfaces of two other plates immoveably fixed to the two upright 
 pofts ; one plate moving in a vertical, and the other in an horizontal, di- 
 re(5lion, to adjuft the telefcope to the planes of the horizon and meridian : 
 to the plane of the horizon, by a fpirit level hung in a parallel pofition to 
 the axis, and to the plane of the meridian in the following manner. 
 
 Obferve the times by the clock when a circumpolar ftar, feen through 
 this inftrument, tranfits both above and below the pole : and if the times 
 of defcribing the eaftern and weftern parts of its circuit are equal, the te- 
 lefcope is then in the plane of the meridian ; otherwife, the notched 
 plates muft be gently moved till the time of the ftar's revolution is bi- 
 fe<Sled by both the upper and lower tranfits, taking care at the fame time 
 that the axis remains perfedlly horizontal. 
 
 When the telefcope is thus adjuftcd, a mark muft be fet at a confi- 
 derable diftance (the greater the better) in the horizontal direction of 
 the interfedtion of the crofs-wires, and in a place where it can beillumi- 
 nated in the night-time by a lanthora hanging near it ; which mark be- 
 ing on a fixed object, will ferve at all times afterwards to examine the 
 pofition of the telefcope by, the axis of the inftrument being firft adjufted 
 by means of the level. 
 
 21 1. To adjuji the Clock by the Sim's Tranfit ever the Meridian, 
 
 Note the times by the clock, w^hen the preceding and following edges 
 of the fun's limb touch the crofs wires ; the difference between the mid- 
 dle time and I2 hours, ftiews how much the mean, or time by the clock, 
 is fafter or flower than the apparent, or folar time for that day; to which 
 the equation of time being applied, will fhew the time of mean noon for 
 that day, by which tjie clock may be adjuffed. 
 
 212, Of the Equatorial Setfor. 
 
 This is an inftrument contrived for finding the difference in right af- 
 ccnflon and declination between two objects, the diftance of which is too 
 
 great
 
 ^ook V- ASTRONOMY. 275 
 
 great to be obferved by means of a micrometer. It confifts of the fol- 
 lowing particulars. 
 
 ift. A brafs plate called a fedlor, formed like a T, having the fliank 
 (as a radius) of Jlbout 2f feet long, and 2 inches broad, and the crofs 
 ])iece (as an arch) of about 6 inches long, and i| inch broad; upoa 
 which, with a radius of 30 irichcs, is deicribed an arch of 10 degrees, 
 each being fubdivided into as fmall parts as are convenient. 
 
 id. Round a fmall cylinder^ containing the center of this arch, and 
 fixed in the fliank, moves a plate of brafs, to which is fixed a telefcope, 
 having its line of collimation parallel to the plane of the fector, and 
 pafling through the center of the arch and the index of a Fernier's divid- 
 ing plate, which Aides on the arch, and is fixed to the eye end of the te-* 
 lefcope. This plate, with the telcfcOpe and Vernier, are moved on the. 
 cylinder, by means of a long fcrew which is at the back of the arch, and 
 communicates with the Vernier through a flit cut in the brafs work, pa- 
 rallel to the divided arch. 
 
 3d. A circular brafs plate, of 5 inches diameter, round the center of 
 which there moves a brafs crofs, which has the oppofite ends of one bar 
 turned up perpendicularly about 3 inches. 'I'hefe ferve as fupporters to 
 the fefior, and are fcrewed to the back of its radius, fo that the plane of 
 the feclor is parallel to the plane of the circular plate, and revolves round 
 the center of that plate in this parallel pofltion. 
 
 4th. A flat axis of 18 inches long is fcrewed to the back of the circu- 
 lar plate, along one of its diameters ; (o that the axis is parallel to the 
 plane of the fe(itor : the whole inftrument is fupported on a proper pe- 
 deftal, in fuch a manner that the faid axis is parallel to the axis of the 
 earth ; and proper contrivances are annexed for fixing it in that pofition. 
 
 Now the inflrument, thus fupported, can revolve round its axis, pa- 
 rallel to the earth's axis, with a motion like that of the ftars; the plane 
 of the fector being always parallel to the plane of fom.e hour circle, and 
 confequcntly every point of the telefcope defcribes a parallel of declina- 
 tion : and if the fector be turned round the joint of the circular plate, its 
 graduated arch maybe brought parallel to an hour circle; and confe- 
 cjuently any two liars, between vv-hich the difference of declination is not 
 greater than the number of degrees in that arch, may be obferved by the 
 inftrument. 
 
 213. To obferve their pajfa^e. Dlre6l the telefcope to the preceding 
 flar, and fix the plane of the fe<?^or a little to the v/cflv/ard of it ; move 
 tlie telefcope by the fcrew, and obfcrve the time fliewn by the clock at 
 the tranfit of each flar over the crofs wires, and alfo the diviuon Ihewn 
 by the index ; then is the difierence of the arches the difference of decli- 
 nation ; and that of the times fhews the difference of right afcenfion of 
 thofe ftars. 
 
 214. Of the Equal- Altitude Injirument. 
 
 An Equal-Altitude Instrument is that ufed to obfcrve a cc- 
 leftial objed, when it has the fame altitude on berth the eaft and weft fide* 
 of the meridian, or in the morning and afternoon ; and confifts of a tele- 
 fcope of about 30 inches long (v/ith 2 vertical, and 3 or 5 horizontal, 
 wires in its focus) fupported on the end of an iron bar, or axi^ of 30 
 inches long, a:id ^ibjuj an inch in diameter. The axis is fuil^ined 
 
 Vol. I. T iu
 
 ^74 ASTRONOMY. Book V. 
 
 in a vertical pofitlon by pafling through a hole in the upper end of abrafs 
 box, whilft its lower end fupports the lower point of the axis. The box, 
 which is abou^ai inches long, with ends about 4 inches fquare, has only 
 two fides, which are fixed at right angles to each other. To one of 
 thcfe fides are fixed four flat arms, with a hole in each, by which the box 
 IS fixed in a vertical pofition to an upright port with fcrews. On the 
 lower end of the box lies a brafs plate, which Aides in grooves, and can 
 be moved gently backwards or forwards by means of a fcrew. In this 
 plate a fine hole is punched to receive the fmooth conical point, which 
 the lower end of the axis is formed into. On the upper end of the box 
 are two plates, which Aide alfo in grooves ; and, by the means of 
 fcrews, can be moved gently fideways, till their angular notches embrace 
 the axis j which, in this part, is made perfedlly cylindrical, and very 
 Smooth. 
 
 To the upper part of the axis is fixed, by its radius, a brafs fextant 
 
 (or arch of 6g, to a radius of feven or eight inches) with the arch 
 
 tiownwards, fo that the center is juft above the top of the axis : alfo a 
 
 fpirit level is fixed at right angles acrofs the axis, juft under the arch, fo 
 
 . as to be clear of the upper end of the box. 
 
 To the under part of the tclefcope is fixed a brafs femicircle, of the 
 fame radius with the fextant, both arches having a common center-pin. 
 In the femicircle is a groove cut through the plate parallel to its limb, to 
 receive two fcrew-pins, which go iiUo the fextantal arch near its ends ; 
 by thefe fcrew-pins the two arches may be preficd clofe, and the tele- 
 fcope fixed in any defired elevation ^ which might be nearly afcertained, 
 by graduating the femicircle, and putting -a Vernier's fcale on the 
 fextant. 
 
 To vfe the Injlrtiment. Fix the box to thr poft, put the axis into the 
 box, letting the conical point drop into tb.e punched hole, fcrew on the 
 level, and anjiex the tplefcope, ohferving to infert the center and arch 
 pins J then, by the help of the fcrevz-platcs at the bottom and top ends 
 of the box, corre(5l the vertical pofition of the axis, fo that the fame end 
 of the air-bubble in the level may ftand at the fame point throughout the 
 whole revolution of the axis, which will thereby be known to be then 
 truely vertical, fo that the tcLTcope will defcribe a parallel of altitude : di- 
 rect the tube to the fun, or ftar, and fix it at the defired elevation by 
 preifrng the two arches together with the tvv^o fcrew pins. 
 
 Some infiruments have been contrived to anfwer both kinds of obfer- 
 vation ; viz. either a tranfit, or equal altitude^. 
 
 
 'To o.ilji'Jl the ClocK 'hy equal altitudes of the Sun. 
 
 Having rpclified the innrunicnt by the level, and being provided with 
 a piece ot tranf'parcjitly c<jlourcd or fmoked glafs to prcferve the eye; then 
 at any convenient time from about 6 to 3 hours before noon, dire^ the 
 f&lel'cope to the fun, and fix it by the arch, fo that the whole body of the 
 fun fhall be above the upper wire (the afcent of the fun appearing through 
 die telefcope' as a defcent) : mark the times fliewn by the clock, when 
 tlicv) receding edge of the fun touclies each ot the vsirvs j and alfo when 
 
 . tiic
 
 Book V. ASTRONOMY* .471 
 
 the following edge touches thofe wires, writing down thofe tinies ; the 
 inftrument being turned horizontally on its axis to follow the Sun, arid 
 keep his center in the middle of the telefcope between the vertical wires* 
 About t^ie fame time after noon (taking care to be early enough) tura 
 the inftrument on its vertical axis, the telefcope remaining fixed at the 
 fame elevation as in the morning, and redifying its horizontal fituation 
 by the level, obferve the Sun in its defcent, which through the telefcope 
 apparently afcends, and write down the times, when the preceding edge' 
 touches each wire, and alfo when they are touched by the following edge, 
 keeping the Sun in the middle of the telefcope j and the fets of obferva- 
 tions are made. 
 
 There are as many fets of obfervations, as there are horizontal wires : 
 for the fore and afternoon contacts of the fame edge of the Sun with the 
 fame wire, make one fet ; and the fame edge which precedes in the fore- 
 noon, follows in the afternoon ; and that which follows in the forenoon, 
 precedes in the afternoon j therefore the " 
 jft, 7 f laft, 
 
 Jit, 1 flalt, 7 
 
 ad, S-A. M. preceding, and thes laft but one, C. p. m. following, 
 
 3d, &c. 3 > tlaft but2,&c. i 
 
 flafl, 1 
 
 . following, and the ^ laft but one, S. P. M. preceding, 
 i laft but 2, &c. i 
 
 make a fet. 
 
 Then to each fet, or pair of obfervntions, Ifind the middle 'time, which 
 added to the time of the morning obfervation, gives the time fhewn by 
 the clock when the Sun was on the meridian, if the obfervations were 
 tnade within two or three days of the fclftice, when the Sun's declination 
 would not fenfibly alter bctv/een the fore and afternoon obfervations ; 
 but on other days, this time muft be correcSted, by applying an equation 
 to it, fhewing the alteration in time, ariling trcm the alteration in decli- 
 nation between the fore and afternoon obfervations. 
 
 The time, hy the clock, of tiic foiar or apparent noon being thus ob- 
 tained, the time of the mean noon may be had by applying the proper 
 equation of time. 
 
 When the time of noon is fought from two or more pairs of obicr- 
 vations, if they give different times, it is beft to take the medium be- 
 tween them, which is found by dividing the fum of all the times by their 
 number. 
 
 2j6. problem LV. 
 
 Given the latitude^ the declinatiofi of the Sun, and interval of thne hetween 
 the Sun's having equal altitudes before and after noon, to find the dijianee 
 from noon of the middle point of time between the obfervations. 
 
 ift. Find the change made in the Sun's declination during the in- 
 tcrvul between the obfervations ; which will nearly bear the fame pro- 
 portiyn to the change nvide between the noon of the day, oa which 
 
 T 2 the
 
 37 
 
 ASTRONOMY. 
 
 Book V. 
 
 the ohfervailons are made, and the noon of the day immediately prc- 
 <;ding or loUowing, as the interval of time between the obfcrvations to 
 24 hcMirs. 
 
 adiy. Add the co-tangent of tbe latitude to the co-fine of half the in- 
 terval of time reduced to degrees and minutes of the equator; the fum, 
 rcje5ling the radius, is the tangent of an arc to be taken lefs than a quad- 
 rant, when the interval of time is Icfs than twelve hours, and jrrcater 
 than a quadrant, ftiould the interval of time exceed twelve hours. 
 
 3d]y. Add together the arithmetical complements of the fine of thii 
 arc, and of the fine of the Sun's diftance from the pole at noon, the lo- 
 garithmic fine of the difFercnce of thefe arcs, the logarithmic co-tangent 
 of half the interval of time in degrees, and the logarithm of half the 
 change in the Sun's declination during the interval between the obferva-' 
 tions ; the fum, rejecting twiee the radius, is the logarithm of an arc, 
 uKich, divided by 15, gives the dillance of the middle point between the 
 obfervations from noon, in feconds of time. 
 
 4thly. When the Sun's drftance from the elevated pole increafes, this 
 middle point of time precedes noon, otherwife it falls beyond. 
 
 DEMONSTRATrON. 
 
 Let p be the pole of the equator hd^ z the 
 lenith, a the morning place, c the after- 
 noon place, ABD a parallel of declination, 
 ABC a parallel of altitude. 
 
 Then the afternoon hour angle, zpc, 
 differs from the morning hour angle, zpa, 
 by the hour angle bpc, the points a and 
 B having the fame declination and diftance from the zenith ; and the arc 
 p 5 bifecting the Z.Bpc, the L.2.V0 will be half the interval of time, 
 which being increafed or diminifhed by half the /.bpc, will give the po- 
 rtion of the meridian to a or c ; alfo vo will be the Sun's diftance from 
 the pole at noon. 
 
 Now here tne dif^^rence between pu, p^, pc being but fmall, the 
 /.BP!? wi:l be to the difference between vo and pb, or pc, that is, half 
 fz c/v FC, nearly as t, Vo't to s^vo *. 
 
 A2'ii"i i" the triangle apz, an arc M being taken, that as rad. : s\ivo 
 :: t,r-/. : t.M the arc M to be lefs than a quadrant, when the 2Lzp 
 i-- r.rutr, but greater, Ihould the z.zvo be obtufe ; then (IV. 123) 
 
 j,rjyojvi : 5,M : : t\vcz : t\7.vsy and confc^uently 
 
 i,iO C/. -.1 Xi\7.P0=:S,JA Xt\l07.. 
 I'B CO PC 
 
 Lut Lbt-o: : : : / , poz : s, Po : : .-, m x / , voz : ;, m x j, ?*? j 
 
 2 
 
 PH C/i PC 
 
 therefore Lv.vs 
 
 to the rule above laid down. 
 
 : : ;,r cr. m X/^zpj : x,m x 5,p^ conformably 
 
 See Cot^. JEjlimat, Ernr, in Mixt, Math, Thtoren 23. 
 
 2r 
 
 Ex.
 
 lii 
 
 ^, (Book V. ASTRONOMY. 277 
 
 I217. Example. 
 
 f^ I In the latitude 50^ N. on the 27th of 0*5lober, 1780, the Sun was ob- 
 ' i ferved to have equai altitudes at 9 h. 1 1 m. 50 s. A. M. and at 2 h. 22 m. 
 22 s. P. M. by a clock adjufted nearly to the true raeafure of time, to find 
 what correiion may be wanted to fet this clock to the true hour of the 
 ,,. I day, the Sun's diftance from the pole on the 27th day at noon being ISJ** 
 7 I (/ 34'', and on the iBth 103 26' 38 ''. 
 
 Here the interval of time is 5 h. 10 m. 32 s. its half is 2 h. 35 m. 16 s-, 
 (in degrees and minutes of the equator 38 49^, and the difference in dc- 
 ' I clination in one day is 2& 4''''. 
 
 I* ! Then 24h. : 5 h. 10 m, 32 s. : : 20^ 4'''' : 4^ 20''=: 260''^, the akeratioii 
 ' ' in declination, the half of which is 2'' 10^=130^ 
 i Then 
 1 Latitude 50 , log. /", 9,9233 1 
 
 L k timezr ^b 49' log. A 9,89162 
 
 Sf jo'32"log. r. 9.81543 
 
 33 ^ 
 103 06 
 
 69 56 G2 
 
 38 49 
 130 
 
 3^" 
 
 L,S, 
 
 0,26185 
 
 34 
 
 L /, 
 
 0,01 147 
 
 G2 
 
 log. /, 
 
 9,97280 
 
 
 
 log. / , 
 
 10,09447 
 
 
 log. 
 
 ?,U394. 
 
 . corr. = 285:' _ log. 2.45453 
 
 or 19 iec. of time. 
 
 Hence, for fetting the clock to the true hour of the day, add the half 
 interval of time 2 h. 35 m. 16 s. to 9 h. 11 rn. 50 s. ; the fum 11 h. 47 m, 
 6 s. is the middle time between the obfcrvations, as noted by the clock. 
 
 And 1 1 h. 47 m. 6 s, 4- 19 s. r: II h. 47 m. 25 s. will be the time 
 pointed out by the clock, when the Sun pailbs the meridian, and fhevvs 
 the clock to be 12 m. 35 s. behind the Sun. 
 
 Though the clock Ihould not keep time with peffetfl: exaftncf^, yet if*' 
 the de^'iation is but fmall, the correction computed will not differ much 
 from the truth ; and the clock being examined- again within a few days-, 
 will flicw whctiie;" it keeps time truly, or moves too fall or too flow, and 
 Its rate of going may be cciictited accordingly. 
 
 218. The method here direl:ed fuppofes the fnip to be flntionary : But 
 the Abbe de la Caille propofes a method of correcting a watch at Tea, even 
 while the fhip is in motion, by taking two equal altitudes of the Sun with 
 a quadrant, one before and tlic other after noo;;. 
 
 His method is this. With the common altitude ohferved, together 
 with the latitudes at the time of each obfervation, and the Sun's correft 
 declination to thofe t!;r.es, the times from noori are to he computed at both 
 obfervations, which tim.es being applied to the two times of obf::rvation, 
 give the rcfpcctive times of noon : then the mean of the two noons being 
 taken, will give the time fiiewn by the watch wlien it was tlic true m.id- 
 duv. 
 
 Or, Half the difference of the computed noons being ap-licd to cither 
 of tlicm, will alio give tiic true time of noon. 
 
 And although the latitudes ufed in the com<putations be fomcth'ng er- 
 roneous, yet the altitudes being equal, the error in each of t!:c computed 
 diftanccs from noon will be nearly the fame, if the change in l.uiiudc and 
 longitude bctwccji the obfervations be duly attended to." 
 
 T 3 219 0/
 
 ft7 ASTRONOMY. Book V, 
 
 a 1 9, Of the Vtrnier's dividing plate. 
 
 When the relative unit of any line is to be divided into many fmall 
 equal parts, thofe parts may be too many to be conveniently introduced, 
 or if introduced, they may be too clofc to one another to be readily efti- 
 mated j and on thcfe accounts there has been a variety of methods con- 
 trived for eftimating the aliquot parts of the fmall divifions, into which 
 the relative unit of a line may be commodioufly divided ; among thofe 
 rnethods that is mofl juftly preferred which was publiflied by Petei^ 
 Vernier (a gentleman of Tranche Comte) at Brufl'els, in the year 
 1631 ; and which, by fome ftrange fatality, is moft unjuftly, although 
 commonly, called by the name of Nonius : for Nonius's method is not 
 pnly very different from that of Fcrriier's, but much lefs convenient. 
 
 Vernier's method is derived from the fallowing principle. 
 
 If two equal right lines, or circular arcs a, b, are fo divided, that the 
 number of equaldivifions rn B is one lefs than the number of equal divi- 
 sions of A ; then will the excefs of one divifion of b above one divifion of 
 A be compounded of the ratios of one of a to a, and one of b to b. 
 
 For let A contain 11 parts j then one of a to a, is as i to 11 ; or 
 
 Let B cpntain 10 parts i then one of b to b, is as i to lO; or r 
 
 10, 
 
 II /I XII I XII ,^^ ^ II 10 I \ 
 
 Now - ( _ (II. 148) = - J 
 
 10 II \ioxi| ii?<io^ ^ '' 10X11 10X11 y 
 
 I I 
 
 Gr. If B contains parts, and a is of -f i parts j| 
 
 and is one part of 
 
 xTfi ly^n n-i I- 
 
 Then is one part of b, and is one part of a, 
 
 . n * n+i ^ 
 
 , , 1 I /IX-fi ix 
 And ~ = ( = =-= (H. 148) ___ _ 
 
 r. X ;T+T ) n j2-\-i 
 
 Or thus. Let a and b be unequal right line?, or circular arcs; and 
 let any part of A, confidcrtd as the relative unit, be divided into n 
 parts J and a part of b, equal to m + 1 parts of a, be divided into m 
 
 parts : then will th of b th of a =: th of bx th of a. 
 *^ m n m n 
 
 r r . m\-\ , 
 
 Put n parts of a : i unit o\ k : : m -\- i parts of a : units of a.. 
 
 But IT, parts of B = (-f I parts of a =) units of a. 
 
 rt-i J m-^ I 
 
 Then m parts of 3 : units pf a : : i part cf B : units of a* 
 
 ^ rj ^ mnn 
 
 J 2 Therefore
 
 Book V. ASTRONOMY. ^ 279 
 
 m f I 1 (yn\\ ?nxi ni-\- 1 m __ i __ \ 
 Therefore I "=' 3 ~r~~) 
 
 II 
 X 
 
 m n 
 
 The moft commodious divifions, and their aliquot parts, into which 
 the degrees on the circular limb of an inllrument may-be fuppofed to be 
 idivided, depend on the radius of that inftrument. 
 
 Let R be the radius of a circle in inches ; and a degree to be divided 
 
 m n parts, each degree being th of an inch. 
 
 Now the circumference of a circle in parts of its diameter, 2R inches, 
 is 3,1415926 X2R inches. (il. 197) 
 
 Then 360'' : 3,1415926 X2R :': i : ^-^ ^X2r inches. 
 
 Or, 0,01 745379 XR is the length of one degree, in inches. 
 
 Or, 0,01745379 XR X/) is the length of 1, in ^th parts of an Inch. 
 
 But as every degree contains n times fuch parts, 
 
 Therefore =o,oi 745379 XRX/>. 
 
 The moft commodious perceptible divifion is -rr or of an inch. 
 ^ ^ b 10 
 
 ExAi.l. Suppofe an inftrument cf 30 Inches radius : into how many con' 
 venient parts may each degree be divided? how many of thofe parts are to ga 
 to the breadth of the Vernier^ and to what parts of a degree may an obferva . 
 tion be made by that injlrianent f 
 
 Now 0,01745 X R = 0,5236 inches, the length of each degree. 
 And if ^ be fuppofed about -^ of an inch for one divifion. 
 
 Then 0,5236 x/)rr4, 1 88, (hews the number of fuch parts in a degree. 
 But as this number muft be an integer, let it be 4, each being 15^. 
 And let the breadth of the Vernier contain 31 of thofe parts, or 71, and 
 be divided into 30 parts. 
 
 Here n-=. ; 7/zrz 1 then X = of a dejrrce, or 30''', 
 
 4' 30 4 30 120 ^ ' -^ 
 
 Which is the Icafl part of a degree that inftrument can fliew. 
 
 If 77 rz , and mr then x - = 7- of a minute, or 20'^. 
 
 5 3^ 5 36 5 X 3^ 
 
 220. The following table, taken as examples in the inftruments 
 commonly made from 3 inches to 8 feet radius, fhews the dM'ifions oi 
 the limb to ncareft tenths of inches, ft) as Lo be an aliquot of 6c's, and 
 wb.at parts of a degree may be eftimatcd by tijc Vernier, it being divided 
 into fucti equal parts, and containing fuch degrees, as tlicir columns 
 Ihcw. 
 
 T 4 Rad.
 
 1^3q 
 
 ASTIIONOMY. 
 
 Book V. 
 
 Bad. 
 
 Parts ii; 
 
 farts in 
 
 Breadth 
 
 Parts 
 
 Rad, 
 
 Parts \n 
 
 Part:, in 
 
 Bicadth 
 
 Parts 
 
 4achei 
 
 2 dcg. 
 
 Veroier. 
 
 o.'-Vcr. 
 
 obferved. 
 
 inches 
 
 a dug. 
 
 Vernier. 
 
 c.fVcr. 
 
 obferved. 
 
 3 
 
 I 
 
 s 
 
 i^i*^ 
 
 4' 0" 
 
 30 
 
 5 
 
 30 
 
 20 
 
 6 
 
 I 
 
 20 
 
 'l 
 
 3 
 
 36 
 
 6 
 
 30 
 
 ^i 
 
 20 
 
 9 
 
 2 
 
 20 
 
 io| 
 
 1 30 
 
 4 
 
 8 
 
 3 
 
 3^- 
 
 P '5 
 
 12 
 
 2 
 
 2 + 
 
 I2i 
 
 I 15 
 
 48 
 
 9 
 
 40 
 
 4i 
 
 10 
 
 5 
 
 3 
 
 20 
 
 6i 
 
 I 
 
 60 
 
 10 
 
 36 
 
 3/^ 
 
 10 
 
 i8 
 
 3 
 
 30 
 
 ici 
 
 40 
 
 72 
 
 12 
 
 30 
 
 2.\- 
 
 JO 
 
 21 
 
 4 
 
 30 
 
 7t 
 
 30 
 
 84 
 
 '5 
 
 40 
 
 r 
 
 6 
 
 24 
 
 4 
 
 36 
 
 9I 
 
 25 
 
 96 
 
 15 
 
 60 
 
 4 
 
 4 
 
 By altering the number of divifions, cither in the degrees or in the 
 Vernier, or in both, an angle can be obferved to a dift'erent degree ^f 
 accuracy. Thus to a radjus of 30 inches, if a degree be divided into 
 12 parts, each being five minutes, and the breadth of the Vernier be 
 
 I I ;* 
 
 21 fuch parts, or i|:, and divided into ?0 parts, then X ? = 
 
 r the breadth of the Vernier of 2j\, and divided intq 
 II 1 .. I I i^ 
 
 = 15^ 
 
 30 parts; then -X =:-- , or 10^'': Or x 
 
 ^ ^ . 12 30 3^0' 12 50 
 
 \vhere the breadth of the Vernier is 4-^. 
 
 600 
 
 SECTION 
 
 Vllf, 
 
 PraSiiccil ^Jlronomy 
 
 The ELEMENTS of the Earth's Motiok. 
 
 ^21. By the theory of the Sun, or Earth, is meant the knowledge of 
 all the requiiites, or elements, necelfary for determining its place in the 
 ecliptic at any prcpofed time, 
 
 222. Mean Motion', or Mean Angular Velocity, is a motion 
 made uniformly in the circumference of a circle, the center of motion 
 being the center of that circle. 
 
 Ti)e mean motion of a planet is the degree and parts fhewing its 
 diftancc from the firit point of Aries, reckoned in the order of the 
 
 figns- 
 
 ^2^, Anomaly, or True Anomaly, is an angle made by two lines 
 drawn from the center of motion, one to the Aphelion, or Apogee, and 
 the other to the place of the revolving body, or planet : Or, Anomaly 
 is the angular diilance of a planet from its Apheiionj the angular point 
 being the center of .motion, 
 
 224. Mean Anomaly is that made by an uniform circular motion 
 about the center, aiid 15 th,c fame as meaA niptioiii beginning at the 
 Aphelion. 
 
 225. Ex-
 
 Book V, 
 
 ASTRONOMY. 
 
 tfi 
 
 225. Eccentric Anomaly is an angular diftance from the aphelion, 
 determined in a circle gn the tranfverfe axis by a normal to that axis, 
 gaffing through the planet's place in its elliptical orbit. 
 
 226. The Equation of the Center, fometimes called the pro/lbo" 
 fhisrefi^y is the diftance between the mean and true anomalies. 
 
 227. The moticn of the equinoxes is the fame as xhe prece^ion of the equi-m 
 noxesy which is backwards, or contrary to the order of the ligns j by which 
 the ftars appear to have advanced forwards from the equinodtial point 
 Aries : this motion is about 50 feconds of a degree in a year. 
 
 228. The motion of the apfides is a flow motion of the Earth's orbit 
 around the Sun in the order of the figns ; difcovered by the apogeon change 
 ing its place among the fixed ftars : this motion is found, by comparing 
 diftant obfervations together^ to be about 16 feconds of a degree in % 
 year, in refpecl to the fixed ftars j and about 66 feconds (=:50'^-|- 16^''} 
 with refpecl to the equinoxes. 
 
 229. A Tropical or Solar Year is the time elapfed between two 
 fucceflive pafiages of the Sun through the fame Equinoctial or Solftitial 
 points of the ecliptic. 
 
 230. A Siderial Year is the time the Sun takes between his depar* 
 ture from any fixed ftar to his next return to that ftar. 
 
 231. An Anomalastic Year is the interval c^F time between two. 
 fucceeding pafliiges of the Sun through the fame apfis, 
 
 232. By the annexed figure the foregoing ar- 
 ticles may be eafily comprehended. 
 On the line of the apfides ap defcribe a circle 
 A DP, called the excer.tric ; and an ellipfis aep 
 forthe Earth's orbit, having the exceutricitycs. 
 Let s be the place of the Sun, c the center of 
 the orbit, a the aphelion, p the perihelion ; 
 SA the aphelion, or apogeon diftance; SP the 
 perigeon diftance. 
 
 Let E be a true place of the earth in it? orbit ; 
 D a correfpondiiig place in the excentric, in FE 
 continued, normal to ap. 
 Let the z, acb reprcfent the mean anomaly j 
 the l_ ACD is the eccentric anomaly ; and the 
 Z.ASB is the true anomaly; the diftereuce be- 
 tween /.ACB andz. AS is the equation of the 
 center. 
 
 When the Earth is in the apfides, then B and 
 ^ fall together in a r/nd p, and here is no equation of the center, the mean 
 and true anomalies being equal ; but the greateft equation of the center 
 muft be, when the Earth is at its mean diffence from the Sun. 
 
 ^33* Obfervations fhew, that in this age the Earth pafles the apogee 
 on the 30th of June, when its daily motion is 57^ 12" -, and pafles the 
 perigee on the 30th of December, when the daily motion is 61'' 12'^: anj 
 15 at the mcun diftance about the 28th of March and 301)1 of September, 
 ^hcn its daily motion is 59^ ^'\ 
 
 234. PRO^
 
 -t82 ASTRONOMY. Book Y. 
 
 *34. ^ PROBLEM LVI. 
 
 To find the Latitude of a Place. 
 
 Solution. Selcft a ftar, the diftance of which from the pole ftar does 
 not exceed 8 or lo degrees ; and obferv6 with a quadrant the greatcft and 
 leaft meridional altitudes j then 
 
 If both obfervations are on the fame fide of the 'zenith ; 
 Half the fum of the alts, is the latitude, on the fame fide of the zenith. 
 
 If the obfervations are on different fides of the zenith ; 
 Half tlie difference of the altitudes is the co-latitude, on the fame fide of 
 the zenith, with the lefTer altitude. 
 For let HZR be the meridian, HR the horizon, 
 z the zenith, rb, ra, two altitudes on the fame 
 fide of the zenith ; h^, kg, two altitudes on con- 
 trary fides of the zenith. 
 Then, the arc ab, or aby being bife<Sled, will 
 
 five p the pofition of the elevated pole, 
 'or a flar is equally diftant from p in its revo- 
 lution. 
 
 Therefore PArrpBj or Fa Fb; and RP equal 
 to the latitude. 
 
 ,, / 2RA 2PA RA -f RA + AB \ RA + RB 
 
 Hence rp -r { ra + pa = ^ -:= = ) 
 
 \ 2 2 2 / 2 
 
 . . / Q.v.a zvb Ha -h na -i- ab npb + Ra 
 And rp = \^Ka + razz^-i-'Y' = ^ = :^ 
 
 l8o HicRA \ o H^cflRrti 
 
 = -^ =;=> 
 
 235. Remarks, i. There will be about 12 hours between the two 
 obfervations. 
 
 2. This method is fubjeif^ to a fmall error, on account of the lefler al- 
 titude being more affeded by refra6tion, than the greater, 
 
 236. PROBLEM LVn. 
 
 To find the Obliquity of the Ecliptic. 
 
 Solution. Let the meridian altitude of the Sun's center be obferved 
 on the days of the fummer and winter folflice ; the difference of thofe 
 altitudes will be the diftance of the tropics 3 and half that diftance will 
 ftiew the obliquity of the ecliptic. 
 
 Or. The meridian altitude at the fummer folftice, leflened by the co- 
 latitude of the place, will give the obliquity of the ecliptic. 
 
 From good obfervations the obliquity of the ecliptic, about the time of 
 the vernal equinox 1772, was found to be 23 28^. 
 
 Diftant obfervations compared together, Ihew that the obliquity is de.. 
 crcaniig at the rate of about one minute in 120 years. 
 
 237. Remark,
 
 Book V. ASTRONOMY. -2B3 
 
 237. Remark. By the fecond method the declinations oFthe fixei 
 ftars, or of any other celeftial phenomenon, may be found j obferving that 
 their declination is of the fame name, viz. north or fouth, with the lati- 
 tude of the place, when its complement is lefs than the altitude j other- 
 wife, of a contrary name with the latitude, 
 
 238. PROBLEM LVIII. -^ 
 To find the Time of an Equinox, 
 
 Solution. In a place the latitude of which is known, let the Sun's 
 meridian altitude be taken on the day of the equinox, and on the day 
 preceding, and that following it. Then the difference between thofe 
 altitudes and the co-latitude will be the Sun's declinations at the times 
 of obfervations. (^3^) 
 
 If either of the altitudes is equal to the co-latitude, that obfervation 
 was made at the time of the equinox. 
 
 But if the co-latitude is unequal to either of the altitudes, proceed 
 thus. Let DG reprefent the equator j AC 
 the ecliptic, e the equinoclial point j the 
 points A, B, c, the places of the Sun at the 
 times of obfervation ; the arcs ad, bf, cg, 
 the correfponding declinations. 
 
 Now ufing either the two firft, or two 
 laft obfervations, fuppofe the latter, in the 
 right-angled fpherical triangles ceg, bef, 
 
 in which there are known the obliquity of the ecliptic, and the declina- 
 tions ; find fc, EB : then bc, the fum or difference of EC, eb, is the 
 ecliptic arc defcribed in 24 hours. Then fay, 
 As BC to BE, fo 24 hours for bc, to the time correfponding to be. 
 And this time fhews the diftance of the equinox from the time of the 
 middle obfervation. 
 
 239. PROBLEM LIX. 
 
 To find the length of the tropical^ periodical^ and anomalijlical revolutions 
 of the earth. 
 
 Solution. Let two obfervations be chofen, among the mofl authentic 
 >{ thofe on record, of the time when the Sun had like pofitions, viz. 
 ift. In regard to his longitude, or place in the ecliptic. 
 2d. In regard to the right afcenfion of fome noted ftar. 
 3d. In rcfpedl to the line of the apfides. 
 
 The greater the interval (fuppofe 80 or 100 years) between each two 
 obfervations, the more accurate will be the refult : then that interval 
 being divided by the number of revolutions made during that time, will 
 give the time of one periodical revolution. 
 
 According to Mayer's tables the numbers arc thefe, 
 
 A tropical year is made in. 365** 5*^ 48'^ 42% 
 
 A periodical, or fiderial revolution 365 697, 
 
 An anomaliftic revolutio:> 365 6 15 29, 
 
 3:40. Remarks,
 
 ag4 ASTRO N O M Y. Book V. 
 
 >40. Remarks, r. The tropical year being fhorter than the fideriaJ 
 ty ao m. 25 s., (hews that the Sun has returned to tlie fame point of the 
 ecliptic, before he has made one complete revolution witli regard to the 
 jft*fs ; and confequently every point of the ecliptic mull have moved in 
 enUcedentia during that tropical period, and fo have produced what is 
 called the preceffion of the equinoxes. 
 
 Now 365 d. 6 h. 9 m. 7 s. : 360' : : 20 m. 25 s. : S0''^,3, or nearly %q'\ 
 for the preceflion in one year. 
 
 If there was no preceflion, the tropical and fiderial years would be 
 equal. 
 
 241. 2. A fiderial revolution being performed fooncrby 6 m. 22 s. 
 than the anomaliftic, (bews tliat the line of the apfides has a motion in 
 conftquentia : now 365d. 15 h. 29 m. : 36 o"^ : : 6 m. 22 s. : iS^'yJy the 
 yearly quantity by which the Sun's apogee is advanced in refpeft to the 
 ffers : and as the equinoxes move in antecedent'ia^ and the apfides in confe- 
 qmrttint t}c\^n fum 66"' ( 50,3 -f 15,7) fhe^ the motion of the apfides 
 from the equinoxes. 
 
 242. 3d. From the comparifon of many obfervations it appears, that 
 the length of the folar year, deduced from two very dtftant obfervations 
 made at the time when the Sun was in the fame point of the ecliptic 
 near its apogee, differs by many feconds from the length of the year de- 
 iluced by like obfervations, when the Sun was in another part of the 
 ecliptic, near its perigee ; thofe made near the apogee giving the revo- 
 lutions lefs, and thofe made near the perigee making them greater, than 
 the revolutions deduced from obfervations taken at the Sun's mean dif- 
 tance j. this alfo ftiews, that the line of the apfides has a motion in confe-> 
 qnenUa ; and that the length of a tropical revolution fhould be deter- 
 rnined from very diftant obfervations, made at the times when the Sun is 
 at its mean diftance from the Earth ; or that the mean revolution fhould be 
 taken between thofe deduced from obfervations made on the Sun's place, 
 vrhen he is in both the apogee and perigee. 
 
 243. P R O B L E \i LX. 
 
 To find the right afcenfion of fome' noted fixed Jiar. 
 
 Having a good clock well regulated to mean or equal time, a large 
 aftronomical quadrant fixed in the plane of the meridian, and an equal 
 altitude or tranfit inftrument : then, on fome day a little before or after 
 the vernal equinox, when the daily alteration of the Sun's declination 
 is about 18 or 20 minutes, obferve the Sun's meridian latitude; and 
 by equal altitudes find the times when both Sun and fiar come to the 
 meridian j the difference of thefe tin)GS is their difference of right af- 
 cenfion. 
 
 Again, At iow.e time a little after or before the autumnal equinox, 
 before the Sun has pafied the faid declination, obferve his meridian al- 
 titude ; and by equal altitudes find the times of the Sun and fame fl:ar*s 
 coming to the meridian, the difference of thofe times is alfo the difference 
 of their right afcenfions. 
 
 If the vernal and autumnal meridian altitudes are the fame, then 
 th5f: obfervations were made, ^Yhe^ the Sun was on the fj;mc parallel 
 
 ^*
 
 Book V. astronomy; 2S5 
 
 of declination : now the fum, or diff. of the two obferved differences of 
 right afcenfion, fhews the equatorial arc defcribed by the Sun between 
 thofc times ; which arc, being bife(^edy fhews thediftance of the nearefl 
 folfticc from the Sun, at the time when the obfervations have equal alti- 
 tudes; and tlaat diftance correcled and taken from 90, (hews the Sun's 
 right aicenfion at the vernal oblervation, or its complement to 360 de- 
 grees. 
 
 From hence, and the firft difference of right afcenfion between the 
 Sun and ftar, the ilar's right afcenfion will be obuined. 
 
 244. If the ^vVO meridian altitudes .of the Sun are not the fame, their 
 difference fhews the difference of the mid-day declinations, when thofe 
 obfervations were taken : now from fom-e tables of right afcenfion and 
 declination take the Sun's daily alteration in declination and rij^ht afcen- 
 fion on the day the leffer altitude was taken ; then fay, /Is the daily change 
 of (UJ. is to that cf right afcen. ; fo is the diff. of the altitudes^ to the cor^ 
 recti on in right afcenfion. 
 
 This correction being added to the vernal, or fubtrated from the au- 
 tumnal difference of right afcenfion, as cither is leafl, reduces that dif- 
 ference of right afcenfion to what it would be when the declination is the 
 fame with the other ; and then the difference between thofe two dif- 
 ferences of right afcenfion, fo reduced, gives the equatorial arc, as before 
 xecited. 
 
 245. At the Royal Obfervatory at Greenwich, in the year 1770, ob- 
 fervations were made on the Sun and the flar Aqiiils. 
 
 March 15, Sun's mcr. zcn. difl. cleared of refra<5i:ion and parallax, waft 
 
 53" 28^ ^<)" ; and their diff. of rt. afc. was 60 30'' ']fi". 
 
 Sep. 28th. Sun's mer. zen. difl. cleared of refraction and parallax, vva& 
 
 53* 36' 7h" ; and their diff. of rt. afc. was 109 59' 7.^^W . 
 
 Thf^n 7' 57^' is the diff. of zen. difts. or the alteration in declination. 
 
 Alfo 23' 40^^ and (3^" 39'' or) 54^ 45^^ are the diffs. of decl. and rt. afc. 
 
 between the 15th and i6th days of March 1770. 
 
 Nov/ 23' 40''' : f '^f : : 54' 45^^ : iW 23,5''^ the increafe of the dif- 
 ference of right afcenfion after the noon of the 15th nf March. 
 Then 60" -p' 07,8'^ 18^ 23,5'''r:6o \i' 44,3'^ which is the firfldiff. rt. 
 afc. when the Sun had the fame decl. as at the fccond obfervation. 
 Here, the times of the two obfervations fall nearell the winter folftice. 
 Then 60" 11'44,3'^-f- 109 59^ 22,8^''= 170 \\' "f' -, its half 85" 5^33,5''' 
 is the dillancc of the winter folitice from the Sun. 
 Hence 270"*+ 85'^ 5' 33,5'^ -hiB^ 23,5''^=: 355" 23' 57^^ is the Sun's rt 
 afc. on March 15th. 
 
 Alfo 355 23' 57'^ 60 3^ 7, 8''^ =2 94 53^ 49,2^^ is the rt. afc. of 
 Aquil:c. 
 
 246. The right afcenfion of one ftar being known, the right afcen- 
 fions of all the reft are found by noting the times fhev.'n by the clock, 
 when thofe ftars come to the meridian : for the differences of thofe times, 
 from the tranfit of the chofen ftar, are the differences of right afcenfion ; 
 by which the right afcenfion of all the obferved ftars will be known ; 
 taking care to augment or diminifli the right afcenfion of the chofen ftar 
 ly thofc differences, according as the chofcr. ftar is preceded, or followed 
 by the other obferved ftars. 
 
 247. PRO-
 
 aU ASTRONOMY. Book V 
 
 iLfy, PROBLEM LXL 
 
 To find the Sun's Place, 
 
 Solution. Let the time be obfervtd both when the Sun, and a ftar 
 (the right afcenfion of which is known) paffcd the meridian, and hence the 
 Sun's right afcenfion is known. 
 
 With that right afcenfion, and the obliquity of the ecliptic, compute 
 (142) the longitude, and thus his place in the ecliptic will be known. 
 
 548. PROBLEM LXIL 
 
 To find the greatejl Equation of the Center. 
 
 Solution. At the times when the Sun is near his mean diftance, let 
 his longitude be found j their difierence will fhew the true motion for 
 that interval of time. 
 . Find alfo the Sun's mean motion for that interval of time. 
 
 Then half the difference between the true and mean motions will fheW 
 the greateft equation of the center. 
 
 Obfervation made at the Royal Obfervatory at Greenwich, fhews that 
 
 1769 O6lober ift. at 23'' 49"" 12' mean time, long, was 6* 9 32' o^b" 
 
 1770 March 29th. at o 4 50 mean time, long, was o 8 50 27,5 
 
 ThedifF. oftimei78d. o 15 38 j True difF. long. 5 29 18 27 
 
 The tropical year =365 d. 5 h. 48 m. 42 s. = 365,2421527 
 The obferved interval = 178 o 15 38 178,01085648. 
 
 Then 365,2421527 : 178,01085648 ::36o: 175,455948 mean motion. 
 So 175"" 27' 21^^ oF mean motion, anfwers to 179 18^ 27" true motion. 
 Their^Iiff.=:3 51^ (^" ; its half 1 55' 33^^ is the greateft equation of the 
 ceriter according to thefe obfervations. 
 
 249. 
 
 PROBLEM LXIIL 
 
 To find the eccentricity of the Earth's orlit. 
 
 . Solution. Say, As the diameter of a circle in degrees. 
 To the diameter in equal parts j 
 So the greateft equation of the center in degrees. 
 To the eccentricity in equal parts. 
 
 The greateft equation of the center 1^55' 33"=i,9258333, &c. 
 
 The diam. of a circle being i, its circumf. is 3,1415926. (II. 197) 
 
 Then 3,1415926 : i : : 360 : ii4,59i56o9 equal to the diameter. 
 
 And 114,591609 ; i,oo&c. : : 1,9258333 : 0,0168061 the eccentricity. 
 
 Hence 1,016806 (m, 000000 + 0,016806) aphelion diftance. 
 
 And 0,983194 ( = ij000C0Sh- 'O,oi68c6)=:perihelion diftance. 
 
 150. PRO.
 
 Book V. ASTRONOMY. 187 
 
 250. P Jl O B L E M LXIV. 
 
 To find the time and place of the Sun's Apogee* 
 
 Solution. On each day of two fucceffive apfides let the Sun's place 
 and the time be obferved. 
 
 Then if the interval of thofe times and places is equal to the halves of 
 365d. 6h. 15 m. 29 s. and 36001^6'"; thofe obfcrvations v^^ere made 
 when the Sun was in the apfides. 
 
 For fuch intervals of time and place belong to no other points of the 
 Earth's orbit. 
 
 But if thofe obferved intervals of time and place differ from the faid 
 halves, take the difference between the interval of place and i8o o' 33^'. 
 
 Then to the daily motion of the Sun's apogee (233), the faid diff. and 
 24 h. find the proportional time ; which proportional time and difference, 
 being applied to the time, and places, of the apogeon obfervation, gives a 
 time and place when it is r8o o^ 33^^ diftant from the obferved perigeon 
 place: now if the interval of thele times is equal to 182 d. 15 h. 7 m, 
 44t s. the times and places of the apfides are known. 
 
 But if the interval of time differs from 182 d. 15 h. 7 m. 44I s., fay, 
 jts ths diff. bettvcen the perigeon and the apogeon daily /notions, is to the daify 
 motion of the apogee ; fo is the diff. of the interval of time .^ to a fecohd cor- 
 region of the time of the apogee. 
 
 This correftion applied to the apogeon time, corrected as above, will 
 give the true time of the Sun's apogee. 
 
 Alfo, to the laft corredlion of time find the proportional motion of the 
 Sun's apogee; and apply it to the laft correiled place of the apogee, and 
 the true place of the apogee will be obtained. 
 
 ^^1 obfervations made at the Royal obfervatory at Greenwich in the 
 year 1769. 
 
 jfuly iff. at o'' 3"^ 20' meantime O long. =3' 90 46'" 38,5'''' 
 
 December 29th. at o 2 49 mean time O long. =9 8 10 58,1 
 
 Interval I Sod. 23 59 29. Interval of place :=5 28 24 19,6 
 The Sun's motion in half an anomaliffic year 6 o o 33 
 
 The Sun's place at firft obfervation is too forward by i 36 13,4 
 
 Then 57^ \i" : l 36^ I'iW^ ' : 24 h. : 40 h, 22 m. 24s. to be taken from 
 the time of July iff, to make the diftance of the times anfwer to the half 
 of 360'^ 1' 6'^ ; and it leaves June 29 d. 7 h. 40 m. 56 s. ; at which time, 
 the Inn was in 3^ 8" lo', 25,1^^, which is diftant from the December ob- 
 fervation by 180 o' 33'^: But here the interval of time is i82d. i6h 
 21 m. 53 s. ; which is greater than 182 d. 15 h. 7 m. 44 f s. the half ano- 
 nialiftic revolution, by i h. 14 m. 8|; s. ; therefore the Sun has fome time 
 to run before he comes to the apogee. 
 
 Now 4''o"': 57'i2'' : : ih. 14m. S^s.: i7h. 13 m. 14s. corre6lionof time. 
 And 24h. : 17 h, 13 m. 14 s. : : 57' 12'' : 42'' 6, 8''^ corredion of place. 
 Then June 29 d. 7 h. 4 m. 565.4-17 h. 13 m. 14 s. gives 
 
 June 30 d. oh. 21 m. 10 s. for the time of the apogee. 
 'Mid 3' 8 20' : 25,1^^ + 42' 6,8'' gives 3' 8' 52' 33'' for the place of the 
 
 251. PRO.
 
 m A 3 T R O K O M y. Book V* 
 
 ssu PROBLEM LXV. 
 
 At any glvitt tirne 1 6 find the Sufj*t mean ittl^ndly. 
 
 SotUTiON, Let an epocha of the Sun's pafTage throo<;h its aphdioh be 
 accurately determined. Then fay 
 As the time of a tropical revohition, or folar ycnr. 
 To the interval between the aphelion and given time ; 
 So is 360 degrees, 
 
 To the degrees fhcwing the mean anomaly. 
 
 Or. From the tables of mean motions hnd the Sun*s moaii motion for 
 the given time, and this will be the mean anomaly. 
 
 152. If the Sun's motion in the ecliptic was uniform, his true place fc(Y 
 any time could be found by the tables of his mean motion ; but the Sun's 
 longitude found by thofc tables, called his mean longitude, muft be cor- 
 rc^ed on account of his irregular motion. 
 
 As the Earth revolves in an elliptical orbit about the Sun, placed in one 
 of its foci, its angular motion round the Sun will difter from the angular 
 f&otion it would have, were the Sun in the center of the ellipfis. 
 
 Now the table of mean motions gives the angular motions from the 
 center of the ellipfis in a circle dcfcribcd on the line of the apfides, and 
 reckoned from the firft point of Aries ; this motion, leflened by that of 
 the apogee, gives the Sun's mean longitude, or mean anomaly, from the 
 aphelion point. 
 
 But the motion of the Earth being in an elliptic orbit, its true ano- 
 maly will differ from its mean; this difference, called the equation of 
 the center, is the corrc5lion wanted to reduce the mean motions to the 
 true ones* 
 
 253. To find the equation of the center, or to folve (what is called) 
 the Kcplcrian problem, is the moll dilKcult operation, particularly in or- 
 bits the eccentricity of which bears a confuierable proportion to the mean 
 didancc : how to do this has been fhewn by Newton, Gregory, Kcil, 
 La Caillc, and many others, by method? little differing from one another: 
 it confifts cliicfly in finding an ijitcrmedtatc angl(, called the eccentric 
 anomaly, as (hewn in the loUowing problem, 
 
 254. PROBLEM LXVL 
 
 Tbt Sur.'s 7fifan ammoJy hfin^ knoivn., atid tJ'e dimtiijiom cf its erhity tr^ftni 
 tht <'cccnttic i-mtmal^ 
 
 Si^l.UTlON,
 
 ASTRONOMY. 
 
 a8 
 
 Book V. 
 
 Solution. Say, 
 As the ' aphelion dif- 
 tance. 
 To the perihelion dif- 
 
 tance -, 
 
 So is the tan. | the 
 
 mean anomaly. 
 
 To the tan. of an arc. 
 
 Which arc added to 
 
 half the mean anoma- 
 ly gives the excentric 
 
 anomaly. 
 
 For let ADPB be the 
 
 excentric. 
 
 A EP the Earth's orbit, 
 
 c the center, s the Sun 
 
 A the aphelion, p the 
 
 perihelion, e the true 
 
 place, D the corre- 
 sponding place in the 
 
 excentric, and b the 
 
 mean place. 
 
 Now it is evident, that the lefs the eccentricity is, the nearer will the 
 elliptic orbit approach the excentric circle ; the nearer will the true and 
 mean places, e and b, approach one another ; and the lefs will be the dif- 
 ference between the mean, the excentric, and the true anomalies ; alfo 
 the nearer will the lines cd, sb, approach to parallelifm, or coincidence : 
 fo that in orbits of fmall eccentricities cd and SB may be taken as pa- 
 rallel lines, particularly in the Earth's orbit, where cs is only about -J- 
 of CP. 
 
 Therefore z. asb = z. acd the excentric anomaly. 
 
 Then in the triangle Bcs, where the fum of the fides bc + CS sA ; the 
 difF, of the fides bc csrrsp, and /_ bcs (= fupplement of acb) are 
 known; the z. CSB may be found. (III. 48) 
 
 Thus SA : sp : : tan. | (fum z.;, csB-f Brr:)z.ACB : tan. of an arc. 
 Then z. ACB + that arcr:z.csjB (III 47,) the excentric anomaly.' 
 
 255- 
 
 PROBLEM LXVII, 
 
 The Sun's excentric anomaly -^ and the dimcv.fions of its crbit being knoivriy ti 
 find t-he true anomaly. 
 
 Solution. Say, As the fquare root of the aphelion di{}ance. 
 To the fquare root of the perihelion diftancc. 
 So tangent of half the excentric anomaly, 
 To tangent of half the true anomaly. 
 For let a femicirclc be dcfcribed from E through the other focus x, 
 cutting AP in y, i, and SE, produced, in G, H. 
 
 T-, ,TT , /SlXSi bX + JiXSf \?.CFX2CS 
 
 Then (II. 172) SH ; si : ; s; : sc=:| zz- = ) . 
 
 Vol. I. U "Or
 
 %^ ASTRONOMY. Book V. 
 
 OrCFXCS=:(|sG=:)|sE |ej; ca(=|se + [eO being radius, and=ti. 
 Therefore SI = ( I + CSX cr (III. 47)=:)i + csx/,acd. (III. 9) 
 
 A T . X /SF \ SC + /,ACD 
 
 Ag.,n. InAsri. AssE:R::sF:,,ASE = (-=)j:j^^3^^r^j5. 
 Theni+.-.ASi:i-.-,ASE::H- ^'^ + V^" .._ j^'^ + Aacd _ 
 
 I+SCXJ,ACD I + SCXJ,ACD 
 
 O t j\aSE _/I -{ -$ X/ ,ACD SC /,ACD 
 / I + f'jASfi "* V I + SC X /, ACD + SC + /,ACD 
 _ 1 SC + SC Xj\aCD s\ACtf 
 
 *~i + sc + sc xj\acd + j\acd 
 
 _ S P + Cs3i X/,A C D ^S P j\a C D X S p \ 
 
 ~" ""sA + cs+iX/'jAC d~sa + /,a c dxs a/ 
 
 I/, A CD S P 
 
 I+/,ACD^SA* 
 
 ^"^7XrTr7=">^A^ and ^ X =,fACDX . (IV. 117) 
 
 l+^jASE " i I4-J%ACD SA '* SA ^ " 
 
 Then^x/MACD=//,iAS. Or l^=z^-^li^. 
 
 SA ' '* SA W,|aCD 
 
 , r /v/SP /, |aSE _ \ , 
 
 ^^^''^^^'^ i77X=7;rX^* Or j/sA : /s p ; : /,f acd : ^f ase, 
 
 256. Remark. The iLCQsr=z.ACB en z.ase {\l.()$), h the equation 
 of the center^ to he applied to the mean anomaly ; and is fubtr active from the 
 aphelion to the perihelion^ or in the firjl fix figns of anomaly ; and additive 
 from the perihelion to the aphelion^ or in the laji fix ftgns of anomaly. 
 
 For the lines cb, se ; cb, se, which coincide in sa, sp, will in every 
 other pofition crofs one another ; in q^ while revolving from a to P, 
 and in q while revolving from p to a : in the firft half revolution, the 
 mean anomaly, or the external Z. acb, exceeds the Z. ase, the true ano- 
 maly, by theZ-CQs (II. 96) : in the latter half, the true anomaly, or ex^ 
 tcrnal/L ps<?, exceeds the mean anomaly, or A pc/-, by the A s^c. 
 
 SECTION
 
 Book V. ASTRONOMY. A^l 
 
 S E C T I O N IX. 
 
 PraSlical AJlronomy, 
 
 Of the E Q^U AT I O N of Tl M E. 
 
 257. Time, which of itfelf flows uniformly, has its parts meafured by 
 the motion of fome vifible objedl; and the Sun being the moft confpi- 
 cuous moving objedt in the heavens, its motion has been chofen as the 
 moft proper meafure of the parts of time, as well for the day, as for the 
 year. 
 
 258. The aftronomical day, at any place, begins when the Sun's 
 center is on the meridian of that place ; and is divided into 24 hours, 
 reckoned in a numeral fuccefllon from i to 24 : the firft 12 are fome- 
 times diftinguiflied by the mark P. M., fignifying pojl meridiem^ or after- 
 noon ; and the latter 12 are marked A. M., fignifying ante meridiem,, or 
 before noon : but aftronomers generally reckon through the 24 hours, 
 from noon to noon ; and what is by the civil, or common way of reckon- 
 ing, called morning hours, is by Aftronomers reckoned in the fuccefllon 
 from 12, or midnight, to 24 hours. 
 
 Thus 5 o'clock in the morning of April the loth, is by aftronomers 
 called April the 9th, at 17 h. 
 
 259. The Sun's daily motion in longitude is the arc of the ecliptic run 
 through in tJuit day ; and his daily motion in right afcenfion is the corre- 
 fponding arc of the equator ; and the mean daily motion in either circle is 
 meafured by 59^ W nearly. For 365 d. : i d. : : 360 : 59' 8'^ 
 
 260. An Astronomical or Solar Day is the interval of time be- 
 tween two fucceflive tranfits of the Sun's center over the fame meri- 
 dian ; and is meafured by the fum of the whole equator, and an arc of it 
 equal to the daily motion in right afcenfion. 
 
 For at the end of a diurnal rotation, which by obfervations is known 
 to be uniform, the meridian has returned to the fame ftar, or point of the 
 ecliptic, which it was againft at the preceding noon ; but the Sun, during 
 this rotation, has removed from that ftar to another, which has a greater 
 right afcenfion : therefore, before the meridian can be again oppofite to 
 the Sun, fo much of another rotation muft be defcribed, as is equal to 
 the daily motion in right afcenfion. 
 
 261. A SiDERiAL Day is the interval between two fucceflive returns 
 of the lame meridian to the fame fixed ftar, is lefs than the folar day, and 
 is meafured by 360. 
 
 262. A Mean or Equatorial Day is the time elapfcd between 
 two fucceflive tranfits of the Sun over the meridian, and is meafured by 
 360' 59^ 8" nearly. 
 
 263. Mi; AN or Equal Time is that fhewn by a clocks whofe 24 
 hours meafure the time which the Sun takes to defcribe an equatorial arc 
 equal to 360 59' 8''^ nearly. 
 
 264. The difference between the meafures of a mean folar day and 
 a fiderial day, viz. 59'' 8''', reduced to time (132), gives 3 m. 56 s. ; 
 which ihews, that a ftar which was on the meridian with the Sun on 
 
 U 2 one
 
 ^9^ ASTRONOMY. Book V. 
 
 one noon, will return tp that meridian 3 m. 5<9S. before the next noon ; 
 therefore a clock, which meafures mean or equal days by 24 hours, will 
 give 23 h. 56 ni. 4 s. for the length of a fiderial day. 
 
 265. Apparent, or True Time, is that mewn by z fun- dial -y 
 where 24 hours, or a day, is mcafured by the fum of 360, and that' day's 
 motion in right afccnfion. 
 
 266. The folar days are unequal to one another, for obfcrvations flicw 
 that the fun's daily motion in right afcenfioix is continually varying. 
 
 The true and mean folar days are never equal, but when the Sun's 
 daily motion in right afcenfion is 59^ 8'^; which happens about February 
 lith. May 14th, July 26th, and November ift: at all other times the 
 lengths of the true and mean days differ. The accumulation of thefe dif- 
 ferences produces the equation of time; and fometimes the apparent noon 
 will precede the time of the mean noon, and fomctirrfes fall after it; their 
 dilfercnce amounting to above 16 minutes at the beginning of November. 
 267. The Equation of Time is the difference between the times 
 ihcwn by a deck and z fun-dial ; or between the fnean and true noons ; or 
 between the Sun's right afcenfion and his mean longi^tude when turned 
 into time at the rate of 15 to an hour. 
 
 This difference arifes on two accounts. Firft, becaufe of the obli- 
 quity of the ecliptic the daily motions in longitude and right afcenfion arc 
 unequal. Secondly, becaufe of the unequal motion of the Earth in an 
 elliptic orbit. 
 
 In the firft and third quadrants, or between the figns Y"S, ^"V^, the 
 right afcenfion being lefs than the longitude (140), or the rnean motion 
 taken in the equator ; the point of right afcenfion is to the weft, and 
 therefore the apparent noon precedes, or comes in confequentia to the 
 meridian before the mean noon : but in the 2d and 4th quadrants, or be- 
 tween the figns ^^, '^T> the right afcenfion being greater than the 
 longitude or mean motion, taken in the equator, the mean noon is weft- 
 ward, and therefore precedes, or comes in confequentia to the meridian 
 before the apparent noon. 
 
 From the aphelion to the perihelion, or in the firft fix figns of anomaly, 
 the mean noon precedes the apparent ; and in the laft fix figns of ano- 
 maly the apparent noon precedes the true ; their difference in either cafe 
 is the equation of the center, which convert into time. 
 Now becaufe. the points of Aries, and cf the Sun's apogee, the places 
 where the two parts of the equation of time commence, do conftantly re- 
 cede from one another ; therefore the whole equation of time made up 
 of thofe two parts will ferve only for a few yeari,, and requires to be cor- 
 rected from time to time. 
 
 268. Tg calculate the equation^ or difference between the mean and appa- 
 rent noons^ for any propofed day. 
 
 Find the mean and true anomalies for that time (255) ; their diffe- 
 rence, or the equation of the center, is one part. 
 
 The true anomaly gives the Sun's longitude; with v/hich, and the 
 obliquity of the ecliptic, compute the right afcenfion (39); the diffe- 
 rence between the longitude and right afcenfion gives the other part. 
 
 The fum, or^iff, of the CVr'O parts, turned into time, gives the equa- 
 :oii ibiight. 
 
 SECTION
 
 Book V, ASTRONOMY. 
 
 293 
 
 S E C T I O N X. 
 
 PraEikal Aflrono7ny, 
 
 To " make Solar Tables. 
 
 260. I. Tables of the mean motiotis of the Sun. ( 30^> 3 2,\ 
 ^ / J \303>3O4./ 
 
 Divide 360 degrees by a folar revolution, the quotient fhews the mean 
 motion for one day o* 59' oS^^&rc. 
 
 Take the multiples of one day's motion from i to 365 for every day in 
 the year ; and thefc properly dil'pofed, according to the month days, will 
 give the mean motions for every day of each month. (304') 
 
 The 24th part of one day's motion vvilJgive that for one hour, and its 
 multiples to 24 times will fhev/ the mean motions anfwering to each hour : 
 from hence, thofe for the minutes of an hour, the feconds of a minute, 
 ^c. are eafily obtained. ( 303) 
 
 The mean motion of a year of 365 days (viz. for the laft of December) 
 teing doubled, tripled, and quadrupled, thofe for i, 2, 3, and 4 years 
 will be obtained, adding one day's mean motion to the 4th year, it 
 being leap-year, and containing 366 days : the motion for leap-year 
 being increafed by thofe of i, 2, 3, and leap-Jears, give thofe for 5, 6, 
 7, and 8 years : the mean motion for 8 years being increafed by thofe 
 for I, 2j 3, and 4 years, give thofe for g, 10, 11, and 12 years : and 
 thus increafing the mean motion for the laft leap year by thole of i, 2, 3, 
 and 4 years, the mean motions may be continued for any number of ab- 
 iblute years. iZ^^) 
 
 270. In the following tables the numbers ufed were, 
 
 Length of the year S^^d. 5h. 48m. 54!?. 
 
 Yearly motion of the apogee, o" i' ^"', 
 
 Place of the apogee, beginning the year 1^60, 3' 8 47 25. 
 
 Greateft equation of the Earth's urbit. I 55 39. 
 
 271. Now 365 d. 5h, 48 m. 54! r.rr 365,2423003472 days. 
 Then 365,2423003472 d. : 360" : ; r d. : 0,9856470613 degrees. 
 Hence the mean motion for i day =: 0= 0^59^ 8''^ 19^^^ 45'' 54" 50'' 
 January 5th 5 ^^^ys^ o 4 55 4^ 3^ 49 34 ^o 
 January 30th 30dayszr o 29 34 9 52 57 25 O 
 March 31ft 90 daysr: 2 28 42 29 38 52 15 o 
 June 29th i8odaystr: '5 27 24 59 17 44 30 o 
 December 26th 360 days 11 24 49 58 35 29 00 
 December 31(1 365daysmi 29 45 40 i^ 78 34 10 
 
 U 3 NOVT
 
 294 ASTRONOMY, Book V. 
 
 Now I year's mean motion = i i(.2(f 45' 4cy'i4'''j8'^34'' lo''^ 
 
 2 years 
 
 = 11 29 31 20 28 
 
 37 20. 
 
 'i 
 
 3 years 
 
 = 11 29 17 42 
 
 55 42 30. 
 
 ft 
 
 4, or leap year 
 
 = I 49 17 
 
 II 3oz:3y. + iy. + id. 
 
 
 5 years 
 
 = 11 29 47 29 31 
 
 18 45 40 = 4y. + iy. 
 
 1 
 
 6 years 
 
 = 11 29 33 9 45 
 
 37 19 50 = 4y. + 2y. 
 
 7 years 
 
 = 11 29 18 49 59 
 
 55 54 o=4y. + 3y. 
 
 1 
 
 8 years b 
 
 =0 3 3 34 
 
 23 0::=4y. X2. 
 
 1 
 
 20 years b 
 
 = 9 6 25 
 
 57 3o=4y-x5. 
 
 
 100 years b 
 
 =0 45 32 5 
 
 4 47 3o=2oy. X5. 
 
 
 1000 years b 
 
 =0 7 35 20 50 
 
 fV "1 1 
 
 47 55 o=iooy. xio. 
 
 -rf 
 
 Where b ftands for biflextile, or leap-year. 
 
 .^72. But to find the mean motions for the years related to any par- 
 ticular epocha, the mean motion for fome particular time in that epocha 
 muft be known. Thus, 
 
 Let the mean motion of the Sun be determined by obfervation (or 
 otherwife) when the Sun is in fome noted point of the ecliptic, fuppofe 
 near Aries : or let the time of its entrance into the fign Aries be well 
 afcertained. Take the difference between the time of that ingrefs and 
 the 31ft of December at noon, in days, hours, minutes, and feconds 
 (reckoning the end of the 31ft of December to be the beginning of 
 January at noon,) and find the mean motions for thofe days, hours, 
 minutes, and feconds, and it will (hew the motion from Aries, for 
 the 31ft of December, or the mean motion at the beginning of the 
 year propofed j or the radix for that year with relation to the propofed 
 epocha. 
 
 The relative mean motions for one year being known, thofe for any 
 number of fucceeding years belonging to that epocha, may be had, by 
 adding fuch of the before found abfolute years to the firft relative year, 
 as will make the number wanted : and the mean motions for any paft 
 year of that epocha will be found by leflening the radical years by 
 fuch a number of the abfolute years, as will produce the relative years 
 required. (302) 
 
 And in this manner are tables conftruled, by which the mean motions 
 of the Sun for anytime, paft, or to come, may be computed. 
 
 273. Suppofe in the year 1760, the Sun entered Aries on the 20th of 
 March, at 13 h. 42 m. 3|s. P. M. : required the Sun's mean motion for 
 the beginning of the year 1760. 
 
 Now 1760 being leap-year, February has 29 days, and from the equi- 
 nox to the commencement of the year is 80 d. 13 h. 42 m. 3^ s. 
 Then i d. : 0,9856470613 deg. : : 80 d. 13 h. 42 m. 3I s. : 79,41444 
 &c. degrees. 
 
 Therefore at the beginning of the year 1760, the Sun's mean longitude 
 was 2* 19" 24'' 52''^ Inort of Aries ; or his mean longitude was 9* 10 35'' 
 d^\ which is the radical mean place for the year 1760. 
 
 274. II. Of the mean motions of the Sun's apogees. (301, 302) 
 
 The yearly motion of the apogee being determined (241) j the mo- 
 tion for any number of abfolute years will be that multiple of one 
 9 year's
 
 Book V. ASTRONOMY. 29^ 
 
 year's motion, and fo for any part of a year : the monthly motions wiU 
 be 5 feconds for fome months, and 6 for others, to make 65 in the 12 
 months. 
 
 Let the time of the Sun's pafl^ge through the aphelion be accurately 
 determined by obfervation (250), and alfo its place in the ecliptic j then 
 the diftance of the place of the apogee from Aries will be known at that 
 time : let this diftance be leflened by the apogee's motion from the laft 
 day of the year preceding the propofed epocha to the time of the apogeon 
 paflage, and the mean motion of the apogee will be known for the be- 
 ginning of that year, taken as a radix. 
 
 Then that radical mean motion, increafed by the multiples of the yearly 
 motion, will give thofe for fucceeding years : but being diminifhed by 
 thofe multiples will give them for paft years. 
 
 The Sun's mean motion for any time, leflened by that of tlie apogee for 
 that time, gives the Sun's mean anomaly. 
 
 275. III. Of the equation of the Sun*s center. {Z'^S^ 
 
 To every degree of the firft fix figns of mean anomaly afTumed, find 
 the true anomaly (253, 254) : the difference between the mean and true 
 anomalies will be the equations of the center to thofe degrees of mean 
 anomaly; which ferve alfo for the degrees of the laft fix fines j as equal 
 anomalies are at equal diftances on both fides of either apfide. 
 
 Set the equations of the center orderly to their figns and degrees of 
 anomaly, the firft fix being reckoned from the top of the table down- 
 wards, and figned at top with the title fubtrat^ ; the laft fix, for which 
 the fame equations ferve, but taken in a contrary order, viz. from the 
 bottom of the table reckoned upwards, are figned at b(Atom with the 
 iit]e add ; and let the difference between every adjacent two equations, 
 called tabular differences, be fet in another column. 
 
 From thefe equations of the center, augmented or diminifhed by the 
 proportional parts of their refpedive tabular differences for any given 
 minutes and feconds, arc deduced equations of the center to any given 
 mean anomaly. 
 
 276. Aftronomical tables are ufually computed to anfwer to two given 
 denominations only ; as to figns and degrees : degrees and minutes ; 
 months and days ; &c. : for if made to more names, fuch tables would 
 fwcll into a bulk fo great, as to be tedious to compute, expenfive to print, 
 and of no great advantage in the ule ; but it generally happens in calcu- 
 lations, that numbers are wanted from tables to anfwer to given nuqibers 
 of three, or more denominations as to figns, degrees, minutes, and fe- 
 conds ; months, days, hours, minutes, and feconds ; &c. : and to obtain 
 from the t.ibles numbers anfwering to all the given names, the tabular 
 numbers are to be increafed or diminifhed by a proportional part of their 
 difference. 
 
 Thus. To find the equation of the center to 4' 2i 44' 36'''' ? 
 
 Now the equation to this number will fall between thofe belonging 
 4 21" and 4' 22"; which equations are 1 13^ 59^^ and 1 12^ 24'' (205) 
 Their diff. is 1' 35^^=95^'j the pro. pt. of which is to be taken for 44^ 36'^ 
 
 U 4 And
 
 a^d A S T R O N O M Y. Book V: 
 
 And as tKe difT. i or 60' : dift'. 95'^ : : 44'',6 : 70^^6 = 1' 1 1^'' the proper 
 tioial part. 
 
 Now to 4* 21* o' o^\ 1 13' 59^' is the equation of the center. 
 And to o o 44 36 , I 1 1 is the prop, part to be fubtradec 
 
 Then to 4 21 44 36 , I 12 48 is the equation of the center. 
 
 When the tabular numbers are increafing, the proportional part is to be 
 added j but when decreafing, the proportional part is to be fubtrailed. 
 
 277. IV. Tables of ibe Sun's true place. (308) 
 
 The Sun's true place at any propofcd time is thus found. 
 
 Colle*Sl together the mean motions of the Sun, and alfo thofe of the 
 apogee, for the given year, month, day, (hour, minute, and fecond, 
 if given) J and their fum will be the mean motions of the Sun and its 
 apogee. 
 
 The Sun's mean motion, leflened by that of the apogee, gives the mean 
 anomaly ; to which find the proper equation of the center by proportion- 
 ing for the minutes and feconds. 
 
 Then the Sun's mean motion, augmented or diminished by the equation 
 of the center, as the title of its table diredls, gives the Sun's true longi-, 
 tude, or place, for that given time. 
 
 The Sun's place thus found to every day for four fucceilivc years, viz. 
 for leap-year, and i, 2, 3 years after; and thofe places ranged under 
 their proper years', according to their refpeclive months and days, confti- 
 lute the tables of the Sun's place. 
 
 Thefe tables find the Sun's place at noon only ; but the place for any 
 intermediate time is found by applying to the noon-piacc the proportional 
 part of the daily difference at that time. 
 
 278. To find the Sun's kngltude^ P^PPfi " -^^^y 4> 1788, at the time of 
 
 apparent nocn ? 
 
 In the table of the Sun's longitude (308) for 1788, againft May 4th, 
 ilands 1= 14" 36' 02'^, which fhews that the Sun's longitude, reckonca 
 from Aries, is 44*^ 36' Qi" ; or that his place is in ^ 14 36^ 02''. 
 
 But to find the Sun's place at any other hour, fuppofe on M-jy ^tli, at. 
 7 h. 24 m. 36 s. apparent time, proceed thus. 
 
 The difFerence betv/een the noon places of the 4th and 5th of May i> 
 57' 59'^ 3479^'', anfwering to 24 hours in time. (.io8} 
 
 Then 24 h. : 7 h. 24 m. 36 s. : : 3479'^ : 1074'"= 17^ 54", the propor- 
 tional part. 
 
 And I' /4 36' 02^^4-17^ 54'''=!' H" 53' 5^ '> ^'^'^ ^^'"''- longitude at 
 that time. 
 
 279. Or, the Sun's pli.ce may be found by ti)e tvihles of mean mo- 
 
 iion-i. 
 
 Fro;?, J
 
 ook V. ASTRONOMY. 297 
 
 .^rom the apparent time 'j^ 24' 36^/ take the equation of time (316) 
 al to 3^ 35'^ and the remainder, y"* 21^ i'\ is the mean time. 
 
 Nffot. ap. 
 
 ?. O's m. mot. (302) 9 
 
 r^4 (305)4 
 
 > Mjrs (303) 
 
 inutes (303) 
 
 oond (303) 
 
 's mean longitude 
 quat. center + 
 
 Sun's true longitude ' ^4 53 57 
 
 9^ 
 4 
 
 10 
 
 2 
 
 4753 
 J3 3 
 17 15 
 
 
 
 I 
 
 13 
 
 I 
 
 19 '3 
 
 34 44 
 
 3 
 
 9 
 
 17 
 
 45 
 22 
 
 m. 
 nt. 
 
 (302) 
 (307) 
 
 3 
 
 9 
 
 18 
 
 07 
 
 apogee. 
 
 10 
 
 4 
 
 I 
 
 06 
 
 anom. 
 
 DIE 
 
 Now to 10* 4 the equation of the 
 center is 1 34' 4^^" (305) 
 
 And thedifF. isyo^'decreafing. 60': 70" 
 : : i' 06" : \" the proportional part. 
 And i** 34' 45" I'^i 34 44 equa- 
 tion of the center. 
 
 280. To find the Sun^s longitude at any given time and place. 
 
 Seek, in the table of the longitudes of places, at the end of Book VI. 
 for the difference of longitude between London and the propofed place 5 
 and convert the diff. of longitude into time. 
 
 If the propofed place is to the eaftward of London, take the diff. be- 
 tween the propofed time and diff. of longitude, and this will fhew the cor- 
 refponding time at London ; after noon, if the propofed time is greater 
 than the diff. of longitude j but before noon, if the propofed time is 
 lead. 
 
 If the propofed place is to the weflward of London, the fum of the 
 propofed time and diff. of longitude will be the correfponding time at 
 London. 
 
 The Sun's place found to the correfponding time at London, will be 
 the Sun's longitude fought for the propofed time and place. 
 
 Thus. In a place 6 h. to the eaft of London, when it is 8 h. P. M. 
 at that place, it is 2 h. P. M. at London ; and when it is 4h. P. M. at 
 that place, it is 2h. before noon at London. 
 
 For when vt is noon at London, it is 6 h. P. M. at the other place. 
 
 Alfo, in a place 6 h. to the weft of London ; when it is 8 h. P. M. at 
 that place, it Is 14 h. P. M. at London. 
 
 For when it is noon at the propofed place, it is 6h, P. M. at 
 London. 
 
 23 1. V. Tables of the Suns declination. (309) 
 
 With each of the Sun's longitudes, already found, and the obliquity of 
 ihe ecliptic, find the declination to each day of the four years. (139) 
 
 Or thus. To each degree of the three firft figna of the ecliptic, 
 tukcn as longitudes, find the declination ( 1 39 ) : and of thefe declinations, 
 regularly ranged to their fign and degree, take the difference of each ad- 
 jacent two, which fet againft them in another column ; and this auxiliary 
 table is prepared, anfwering to each fign and degree of longitude (306J : 
 for to equal longitudes, taken on both fides of each equinox, belong 
 equal declinations. 
 
 Now thefe auxiliary declinations' augmented, or diniinifhed (accord- 
 ;;'J as thty arc incr'.,afing or dctreafing, by the proportional part of their 
 
 difference.
 
 59 ASTRONOMY. Book V. 
 
 difference, for the minute and feconds in any given longitude, will give 
 the declination for that longitude. 
 
 And this being done for every day in the four years, ufmg the longi- 
 tudes already computed, will give the declinations fought : which are to 
 be ranged according to their year, month, and day. (309) 
 
 282. To find the Sun's declination. Suppofe on May 4, 1708, at 
 noon. In the table of the Sun's declination (309) for 1788, againft May 
 4, (lands 16 14' 13^^ for the Sun's declination, which is N. as being be- 
 tween the vernal and autumnal equinoxes. 
 
 283. But if the declination vt^s wanted on May 4, 1788, at 7 h. 24 m* 
 36 f. P. M. proceed thus. 
 
 The difFerence between the noons of May 4 and 5, is i5' 59'', which 
 anfwcrs to 24 h. Then 24 h. : if o" : : 7h. 24 m. 36 f. : 5'' 15^'', the 
 proportional part. 
 
 And as the declination is increafing ; then 16 14' 13^' + 5' 15^'' gives 
 16 19^ 28^^ for the Sun's dccl. at the propofed time. 
 
 284. But art. 311 is a table for finding the proportional part at fight, 
 for fitting the noon declination to any other time. Thus. 
 
 Seek in the left-hand column for a daily diiFerence, neareft to the given 
 one J againft which, in a column marked at top with hours, neareft to thofe 
 given, ftands the proportional part fought. 
 
 Thus againft if o'^ of daily diiF. and to 7 h. 24 m, time, ftand 5^ \^'\ 
 the proportional part fought. 
 
 Although this table goes no farther than 8 h., yet it may be applied 
 quite to 12 h. or 180 degrees. 
 
 285. Exam. What will be the Sun's declination at London^ on the i^tb 
 sf Augujl^ 1788, at lob, 35 OT. F. M. ? 
 
 In 1788, the daily difF. between the noons of the 25th and the 26th of 
 Auguft is 10' S^" decreafing. (309) 
 
 Now 10 h. 35 m. is equal to 2 h. 35 m. + 8 h. o m. 
 To the difF. 10/ 58'^, and to 2h. 35 m., anfwcrs 2^ i^" . (3^1) 
 
 To the diiF. 20^ 58''', and to 8 h. cm., anfwcrs 6'' 59^''. 
 The fum 9^ 1$" is the proportional part, by which the decl. lO** 28' 
 29'" to Auguft 25th, is to be diminilhed ; fo 10 19^ \\'' is the decl. 
 fought. 
 
 Here 20' 58'^, is taken as if it was 21' o" , 
 
 And 2 h. 35 ni. is |, the interval between 2 h. 20 m. and 2h. 40 m. 
 Now x\' gives ^' %" for 2^ 20^, and 2'' ^o" for 7^ 40^ ; the difF. is i%'\ 
 three fourths of which is idf" '. and this being added to l' l" gives 
 1' lb" for i\' with 2^ 35^ Moreover 21'' with 8h. gives f 00''^; but 
 I take one fecond lefs bec^ufe the daily dift'. in declination is 7." lefs 
 than 21'' 00'"^ 
 
 286. From the table of declination, fitted to the meridian of Lon- 
 don, or Greenwich, the declination may be found at any time, under 
 ?-ny other nvcridian, at a given diiFerence of longitude from London. 
 Ttj?. 
 
 Required
 
 Book V. A S T R O N O M T. 
 
 ^99 
 
 Required the Sun's declination at neon under a mtridian 1 10 to the weji of 
 London^, on the 2J^th of February, 1788. 
 
 (311) Now at 110 to the weft of London, it is noon 7 h. 20 m. after 
 it is noon at London ; that is, when it is 7 h. 20 m. P. M. at London, 
 it will be noon at the propofed place j fo the declination found to that 
 time at London (285) will be the declination fought. , 
 
 In 1788, the difF. between the declinations of the 24th and 25th of Fe- 
 bruary, is 22^ 13''' decreafing (309) : and againft 22'' 1o^' of daily difF,, 
 and under 110, or 7 h. 20 m., is h' 49^^ in table, art. 311, which taken 
 from 9" 27'' 33'', leaves 9 20^ 44^', the declination fought. 
 
 Exam. IL tfhat is the Sun*s declination on September id, 1788, ct 
 10 h. 30 w., under a meridian I00 to the eajiivard of London? 
 
 Now under a meridian 100^ to the eaftward of London, it is noon 6h, 
 40 m. before it is noon at London (311) ; or when it is noon at Lon- 
 don, it is 6h. 40 m. after noon at the propofed place ; and when 20 h. 
 30 m. after noon at that place, it is 13 h. 50m. after noon at London; 
 fo the declination found at that time {285), will be the declinatioa 
 fought. 
 
 In 1788, the daily difF. at September 2d, is 22'' h" (309), againft 
 which (in tab. art. 311), and under 8 h. and 5 h. 5 m., fland -j' i\" and 
 5' i\"-i their fum \i' \i" taken from the dec), to September 2, viz. 7* 
 36' 30^^, leaves 7 23^ 48^'', the declination fought. 
 
 Here 5 h. 50 m. fall in the middle between 5 h. 40 m. and 6 h. o m. ib 
 5^ 2i"t the middle between ^' 11" and 5^ y:/\ is taken. 
 
 287. VI. Tables of the SurCs right ajcenfion. (jio) 
 
 To the obliquity of the ecliptic, and each degree in the three firft 
 figns of longitude, find the right afcenfions (i39)> and of each take 
 the fupplement. 
 
 Range the right afcenfions according to their fign and degree for the 
 three firfl figns ; and for the three next figns, range the fupplements, (b 
 that the 4th fign begins with the leafl fupplement, and the 6th fign ends 
 with the greateft : becaufe the right afcenfions in the 2d and 4th 
 quadrants are the fupplements of thofe in the firfl and third. 
 
 Let the differences of thefe right afcenfions, viz. each adjacent two 
 through the fix figns be taken, and fet in other columns. (307) 
 
 Then this auxiliary table, ufed like that of declination, will give the 
 right afcenfion to each day in the four years. 
 
 288. To find the Sun*s right afcenjion, fuppofe on June 12 at noon, in tl^c 
 y(ar 1788, at London, 
 
 In the table of the Sun's right afcenfion (310) for 1788, againfl Juac 
 12, flands 5 h. 25 m. 19 f,, which is the right afcenfion fought, and fhcw? 
 how much later the Suh paffcd the meridian -ef London than the equi- 
 noctial point Aries,
 
 500 
 
 A Sr T R O N O M Y. Book V. 
 
 Exam. II- Required the Sun's right afcenfton at London on the id of 
 November, 1788, at qh. 30W. P. M.? 
 
 Between the 21I and 31I of November, 1788, the daily difF. is 3 m. 58 f. 
 which anfwcrs to 24. h. Then 24 h.: 3 m. 58 f. : : 9 h. 30 m. : i m. 34 f.,' 
 the proportional part. 
 
 Then the right aCcenfion on the 2d at noon, i + h. 33 m. 2of. + i m. 34 C 
 gives 14 h. 34111. 54 f. for the right afcenfioii at the time required. 
 
 By this table the right afcenfion may alfo be found at any time in places 
 that arc-Nift the caflward, or wcliward, of London, the difference of longi- 
 tude of thofe places being known ; by finding the time at London cor- 
 refponding to the given time at the propoied place, and feeking the right 
 afcenfion to that correi'ponding time at London. 
 
 The table at art. 311, pages 222, 223, maybe applied to the tables of 
 the Sun's longitude and right afcenfion, as well as to thofe of the declina- 
 tion, for finding the proportional parts of the difference between the noon 
 of adjoining days, which (hall anfvver to any intermediate hours. 
 Thus in the Ex. page 296. To find the pro. pts. of 58^ to 7h, 24m. 36 f. 
 Nov/ |th of 58^ is 14^ 30^'' ; which falls between 14^ lo/^ and 14^ 40''^ 
 And the time 7 h. 24 m. 36 f. falls between 7 h. 20 m. and 7 h. 40 m. 
 The mean of the equatrons under 7 h. 20 m. and 7h. 40 m. and againft 
 14'' 20''^ and 14^ 40^', are 4' ^b'' and 4' 38'^, their difF. is 12^'. 
 And 20 : 12 : : 4,6 : 2| : and 4^ 26'''' + 2|=4' 28^ the pro. pts. to \ of 
 
 Then the proportional parts to 58^, are 17' 54^^. 
 
 Again. In the Exam, above. To find the parts proportional to 3 m. 
 
 57 f. as 9 h. 30 m. is to 24 h. 
 
 Here 3 m. 57 f. being taken as 4m. ; and 4h. 40 m. as the half of 9 h. 30 m. 
 
 The equation is 47^^ j which doubled gives 1^ 34^^ for the proportional 
 
 parts required". 
 
 28^. VII. Of the right ajcenjions and declinations of the fixed 
 
 ^Stars, (312) 
 
 This table, which contains 120 of the principal fixed ftars, viz. 60 
 naving north declination, and 60 with fouth declination, are fitted to the 
 year 1780 ; and are fclefled partly from the catalogue which is given in 
 fhe Nautical Ahnanac for 1773, ^^ deduced from Dr. Bradley's Obfer- 
 vations ; and partly from that given by M. de La Caille, which, he fays, 
 "' are all derived from his own obfervations made, during len years attcn- 
 *' tion ro this bufinefs, either at Paris, or at the Cape of Good Hope ; 
 *' that the pofitions are afcertaincd with all the accuracy that could be de- 
 *' rived from the modern Aftronomy ; and that he had all proper helps, 
 ' v/ith regLird to inftruments, affiftarts, and convenience, and neither 
 "* care or pains were vvrnting to perfect the work. 
 
 ** The right atceniions were determined by a multitude of correfpond- 
 ** ing altitudes of each, taken with a quadrant of three feet radius, to have 
 *' their paflage over the meridian with the greatell exaclncfs. Almoft all 
 *' the ftars in the northern hemifphere have been compared with the 
 *' bright ftar in the Harp; and thofe in tl:e fouthern hemifphere, widi 
 *' Syrius ; that is to iay, 011 each day that the t^me of the ftar's pafling 
 *' the meridian had been found by equal altitudes, that of t Lyras and 
 " Svjius were foiuid in like manner . r^.e ri^'iit af^cnfions oi thcfe iv/o 
 
 *' liars.
 
 Book V. ASTRONOMY. jci 
 
 ** ftars having been fettled by a great many obfervations taken when 
 *' they were in the propereft fituation for this purpofe. 
 
 *' The declinations have been deduced from a fufficient number of 
 *' obfervations of their zenith diftances, taken with an inftrument of Ci^i 
 " feet radius, made with great care for this purpofe." 
 
 The table confifts of nine columns ; that on the left hand contains the 
 name of the conftellation ; the next fhews in what part of the conftella- 
 tion the ftar is ; in the 3d are the names by which certain ftars are dif- 
 tinguifhed } the 4th column fhews the Greek charadlcrs by which the 
 ftar is marked in the coelcftial charts, or maps of the conftellations ; the 
 5th fliews the magnitude of the ftars ; the 6th and 7th contain the right 
 afcenfion in time, reckoned from Aries, and the yearly variation in right 
 afcenfion ; the 8th and 9th contain the declinations and the yearly varia- 
 tion in declination ; where thofe which are marked + are augmented by 
 the yearly variation ; but thofe which have the mark annexed, are to be 
 diminiftied by the variation : by the help of thefe yearly variations the 
 right afcenfions and declitiations of thefe ftars may be fitted for any 
 diftant year. 
 
 Precepts for finding the culminating of the ftars are at articles 133, 1 34. 
 
 290. VIII. I'ables of the Equation of Time. 
 
 In page 318 are three tables, articles 313, 314, 315 : Article 313 is 
 a table of the Sun's right afcenfion in degrees, to each degree of longi- 
 tude in the firft quadrant of the ecliptic ; and alfo the differences between 
 thofe longitudes and right afcenfions. The table, art. 314, contains the 
 faid differences turned into time (132), of minutes, feconds, and the 
 tenth part of feconds : the numbers in this table are the differences be- 
 tween the mean and true noons, arifing from the obliquity of the ecliptic 
 (267) ; and the table, art. 315, is nothing more than the equations of 
 the center, table art. 305, converted into time ; and are the differences 
 between the times of the mean and true noons, arifing from the eccen- 
 tricity of the Earth's orbit : thefe two equations of time, properly put 
 together, conftitute another table, art. 316, of the abfolute equation of 
 time with relation to the place of the Sun's apogee. 
 
 291. To conJlruSl the talk 316, of the abfolute Equation of Time. 
 
 I ft. To the given time find the Sun's true place, or affume a place. 
 
 2d. The difference between that place, and the place of the apogee, 
 gives the Sun's true anomaly. 
 
 3d. From the true anomaly find the mean. . (294) 
 
 4th. In table I. 314, feck the equation of time to the Sun's place. 
 
 5th. In table II. 315, feek the equation of time to the mean ano- 
 maly. 
 
 6th. The fum, or difference of thefe equations, according as their 
 titles, or figns direct, will be the abfolute equation of time to the Sun's 
 place foimd at firft, or to the corrcfponding time. 
 
 The tables, articles 314, 315, are made only to whole degrees of lon- 
 gitude and anomaly; the proportional parts of the diffcrcncrs are to be 
 taken for minutes or feconds, above whok decrees of the Sun's longitude 
 aad anomaly, 
 
 292. EXA.M.
 
 -02 ASTRONOMY. Book V. 
 
 392. Exam. I. If^bat is tht Equation ofTimt^ when the Sun's longitude 
 
 is 2* li**.^ 
 
 In table, art. 316, againft 12 in the outfide column, and under ni, or 
 J f,, ftands 16 m. 12 f. ; which fhews that 16 m. i2 f. is to be fub- 
 tra<3ed from the apparent time ; to give the mean time of apparent noon, 
 or the time which fhould be fhewn by a good cloclc, when the Sun's 
 center is on the meridian. 
 
 293. Exam. II. IVhat is the Equation 9/ Time when the Sun's longitude 
 j4. 243o'4Z^'f 
 
 The difference between the equation in table, art. 316, to 4' 24* and 
 4* 25, is 13* decreafing ; and 30' 42''= 30,7'. 
 
 Then bo' : 30,7' : ; 13* : 6,65 or 7% the proportional part decreafing. 
 And + 3 m. 46 f. 7 fl=: + 3 m. 39 f-, -the equation fought. 
 So 24 h. the apparent time of folar noon, increafed by 3 m. 39 f. will 
 give the mean time of noon. 
 
 If the time was given^ viz, the months day^ hour, t^c.y to find the Equation, 
 
 To the given time find the Sun's longitudie, (-^78) 
 
 Then to this longitude find the equation of time, as above. 
 
 294. To find the mean anomaly from the true being given. 
 
 Solution, Say, As the fquare root of the perihelion diftance. 
 To the fquare root of the aphelion diftance j 
 So the tangent of half the true anomaly. 
 To the tangent of half the excentric anomaly. 
 
 And As radius, to the fign of the excentric anomaly, 
 
 So the degrees in an arc equal in length to the eccen- 
 tricity. 
 To the degrees, &c. in the arc of correction. 
 
 The corrclion added to the excentfic anomaly gives the mean 
 anomaly. 
 
 295. Remarks, ift. The greateft equation of the center being 
 taken at i 55^ 39''^, the eccentricity (249) will be 0,01682 j the aphe- 
 lion diftance will be 1,01682, and the perihelion 0,98318. 
 
 Hence the ratio of the fquare root of the perihelion diftance to the 
 fquare root of the aphelion diftance will be exprefled by the logarithm 
 0,00731 ; which conftant logarithm, added to the logarithmic tangent of 
 \ ihe true anomaly, will give the logarithmic tangent of \ the excentric 
 anomaly. 
 
 296. 2d. In the 2d proportion, the arc equal to the length of the ec- 
 centricity 1682 is a conftant quantity. 
 
 Now the radius, or mean diftance, is equal to the length of an arc 
 of 57^2i9578 (249) ; then looooo : 1682 : : 57,29578 : o,96375, the 
 length of the eccentricity in degrees ; the conftant logarithm of which is 
 9,98396, which added to the logarithmic fine of the excentric anomaly, 
 abating lO in the index of the fum, gives the logarithm of an arc, the de- 
 grees, minutes, and feconds of whih being added to the excentric ano- 
 maly, give the mean anomaly, 
 
 297.
 
 Book V. 
 
 ASTRONOMY. 
 
 J03 
 
 297. IX. 'Table of corre^ions for the middle tinu between eq^uai 
 altitudes of the Sun, Art. 317. 
 
 This table, which is fitted to the latitudes of 30*, 40% 50*, and 60% 
 will alfo ferve, nearly, to all latitudes between 25*^ and 65; by entering 
 the table with the neareft latitude to that given, and the given declina- 
 tion in degrees. It is conftrudled by art. 216. 
 
 Exam. I. In latitude 50 iV., when the Sun's declination is 16 N., 
 snd the interval betzveen the morning and afternoon obfervation is 5 hours : 
 what correction muji be applied to the middle time, to give the time of appa^ 
 r^nt neon ? 
 
 In the table, art. 3^7, againft 16*" of declination taken in the outfide 
 column, and under 50* latitude, and 5 hours, with N. declination, ftand 
 12 feconds ; which 12 f. applied to the middle time between the obfer\'a- 
 tions, give the time when the Sun was on the meridian. 
 
 The corre<5tion is applied to the middle time by the precepts at the 
 biottom of the table. 
 
 298. Exam. II. In latitude 50"^ AT., on November ibth, 176 1, obferva- 
 tions at equal altitudes of the Sun were taken at the following times Jbewu by 
 clocky the equal altitude injlrument having three horizontal wires. 
 
 Morning obfervations. Afternoon obfervations. 
 
 preceding limb O following limb O preceding limb G following limb 
 
 9^28"'5s" 9*'3;"'2!i" ih46"43i" 1*^53"' SOi" 
 
 9 39 23f 
 
 9 32 46? 
 
 9 36. 44f 
 
 50 53t 
 1 54 56 
 
 , .-- 9 43 30 
 
 Now i^ ^ ^-- =11*' 45=" 7f 
 
 9 32 46! +1 57 285 +>2 _ _j 
 11 45 /z- 
 
 9 36 44f +1 53 3of +12 
 
 1 =11 45 7. j 
 
 Again^-il-i^t-4i^^^-il=M 45 K ^ 
 
 1 57 28i 
 
 2 I 20[ 
 
 the mean mi'' 45"^ 
 )-7,6" by the preced- 
 ing limb. 
 
 9 39 23 +1 50 53f +12 
 
 2 
 9 43 30 +1 46 4Vf +12 
 
 the mean=:i i** 45=^' 
 
 II 45 8f ^8" by the following 
 
 limb. 
 
 = 11 45 6{ 
 
 The mean time of obfervation from both limbs is 11 h. 45 m. 7,^ f. 
 The declination on the day of obfervation is 19 S. nearly ; the intcrvul 
 between the obfervations is about 4 hours j and thefe give + 14 feconds 
 for the correction of the middle time. 
 
 So the Sun was on the meridian when the clock fhcwed iih. 45m. 21, Sf. 
 The Sun's place, at that time, was 7' 24 26' 46" nearly. 
 la table 316, to 7' 24' the tabular difference is 12', dccrcafing. 
 
 Then
 
 304 ASTRONOMY. BookV. 
 
 Then 60' : 26,75' : : 12 f. : 5,35 f. ; and + 14 m. 56 f. Si f- = + 
 14 m. 50} f., which is the equation of time : hence 1 1 h, 45 m. 21,8 f. + 
 14 m. 5o,6f. = i2h. om. I2,4f. 
 Which fhews that the clock was 12 f., nearly, too fail. 
 
 299. X. tables of RefraSiion and cf the Sun' 5 parallax. (318,319) 
 
 Thefe tables are the refult of the experience of fome of the moft emi- 
 nent Aftronomers. By the refradlion of the atmofphere, objects appear 
 more elevated than they really are, and therefore the apparent altitude is 
 to be diminiflied by the refradlion, which is greateft near the horizon, 
 and gradually diminifhes towards the zenith, where there is no refra(5tion. 
 The parallax in altitude is the difference between the altitude of an ob- 
 jet, as feen from the cente rand furface of the Earth, that from the center 
 being the true altitude, and the greateft, except at the zenith, where pa- 
 rallax vanifties j therefore the apparent altitude is to be augmented by the 
 parallax. 
 
 Exam. The Sun's apparent altitude was ohferved to be 18 34''48'''i 
 ivhat was his true altitude ? 
 
 Apparent altitude 18 34' 48^^ 1 Refradion 2' 47'' 
 Correction is 2 38 J Parallax 9 -f 
 
 Sun's true altitude 18 32 10. 
 
 ''"XXXX'*'" 
 
 300. ASTRO.
 
 feook V, ASTRONOMY. 305 
 
 300. 
 
 ASTRONOMICAL TABLES, 
 
 Fitted, in general, to the meridian of Greenwich. 
 
 Art. 
 
 I. MeaH motions of the Sun and Apogee for years. (301) 
 
 II. Mean motions of the Sun and Apogee for radical years. (3^) 
 
 III. Mean motions of the Sun for hours, minutes, and feconds. (303) 
 
 IV. Mean motions of the Sun and Apogee for months and days. (304) 
 
 V. Equations of the Sun's center. (3^5) 
 
 VI. Sun's declination to figns and degrees. (306) 
 
 VII. Sun's right afcenfion in time to figns and degrees. (3*^7) 
 
 VIII. Sun's longitude to each day for the years 1792, 1793, 
 
 1794, and 1795. (308) 
 
 IX. Sun's declination to each day for the fame four years. (3^9) 
 
 X. Sun's right afcenfion to each day for the fame four years. (310) 
 
 XI. To fit the tables VIII. IX. X. to any meridian. (311) 
 
 XII. Right afcenfions and declinations to 120 fixed flars. (3^21) 
 
 XIII. Sun's right afcenfion in degrees, &c. to figns and de- 
 
 grees of longitude. (313) 
 
 XIV. Equation of time on the obliquity of the ecliptic. (314) 
 
 XV. Equation of time on the eccentricity of the Earth's orbit. (315) 
 
 XVI. Abfolute equation of time^ to the Sun's longitude. (316) 
 
 XVII. Correction of the middle time between equal altitudes. (317) 
 
 XVIII. Correction of altitude for refraClions. (31^) 
 
 XIX. Correction of the Sun's altitude for his parallax. Is^O) 
 
 The tables of the Sun's place, declination, and right afcenfion, are 
 fitted to the years 1792, 1793, 1794, 1795 > and will ferve in moft nau- 
 tical operations as well for the lour years preceding, viz, 1788, 1789, 
 1790, 1 79 1, and alfo for the four years following, viz. 1796, 1 797, 
 1798, 1799, as is mentioned at the heads of thofe tables. But as a 
 Nautical Almanack is publifhcd yearly under the direction of the com- 
 miflioners of longitude, the tables contained therein (hould be confulted 
 in cafes where the utmgft prccifion is neceiTary, 
 
 X Tables
 
 305^ 
 
 AST A Or^ O MY. 
 
 Hook V, 
 
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 M 
 
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 2 
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 2: 
 
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 o ^ 
 
 PI U-l M Tt- IN TJ- 
 
 O OO \o * o 
 
 M c<^ ro ^ 
 
 1-. ui rn n a< -, 
 
 
 r< \ t^ 1/1 ro "^ 
 
 
 " r N N ri 
 
 fn ro r<i ro xl- vj- 
 
 vnoo O rrt looo 
 1^ ^ lO irl vo u^ 
 
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 A S T R O N O MY. 
 
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 3ot 
 
 ASTRONOMY. 
 
 Book V. 
 
 o 
 
 c 
 
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 1
 
 - BookV. 
 
 ASTRONOMY.. 
 
 309 
 
 Deg. 
 
 i" H ro - o\o 1 t^oo Ox 11 (4 1 i^"i^ mvo l^oo 1 a. O ^ cJ t*i +( vio t-,00 c^ O 
 
 Q 
 
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 n t^ M tJ-wj rt 
 
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 cortrtrtrtrtrtlrtrtrtrtM.>-i|MMMMMM|Mi-i | 
 
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 rt r^ xoxo 
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 rt to to Ti- <}- xo 
 
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 rt + xn (^ t-^00 
 rt rt M M M rt 
 
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 rt rt rt rt rt M rt 
 
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 rt .! CO M CO CO 
 
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 rt tJ- M to xn rt 
 
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 rt ri- M CO xo M 
 
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 '
 
 3*9 
 
 ASTRONOMY. 
 
 Bopk V. 
 
 Uays.l 
 
 - n *> 'I- '^' 
 
 r-oo On O < r 
 
 to ^ nvo c^oO 
 
 M M M M M 11 
 
 ON O " N to 4- 
 
 1 d M d d 
 
 miO t^OO On O 1 
 d d d d N to to 
 
 
 
 J 
 
 u^ u-> in w^ w^ 
 
 m r^ m ON 'I- 
 m in 11 
 
 O l-~ tJ- M o oo 
 r< M o * in m 
 
 "1 d CO ^ n 
 
 00 O O d o moo 
 d CO ^in n 
 
 
 ja 
 
 
 r-i n <if nvO t- 
 
 CO e -" to .^ 
 
 mvO e~oo 0\ O 
 
 d CO T- m\o t- 
 d d H d d d 
 
 00 oi o M d to 
 d d 
 
 ON 
 
 ON K d to mvo 
 d to to to t*^ CO CO 
 
 * iOnO r>.0O On O 
 
 
 E 
 
 Q 
 
 c 
 
 O M N ro t1- u^ 
 
 NO r-.oo ON O w 
 
 d M N H c< 
 
 
 oo 
 
 
 
 C 
 
 ^ 
 
 O ^ in o >o *n 
 m + n . M + 
 
 vo CO tooo in ir 
 
 * t^ O m to -. 
 4- inrt M- 
 
 d * ON Th M 
 d ^ d - 
 
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 m to d d 1 
 
 
 E 
 
 - 
 
 m in \f^\c \o lO 
 
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 <*+* T- 4- Th 
 
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 to rf- rJ- ln\0 1^ 
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 mvo t^oo ON 
 
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 t^oo oi o d 
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 % 
 
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 1 
 
 
 
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 8 
 
 - 
 
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 in vn in ui 
 
 
 c 
 
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 mo i-^M ON 
 
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 r^OO On O I" d 
 d cl d 
 
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 ^ 
 
 . 
 
 m 
 
 
 
 
 
 5 
 
 t- >* ro -i- rJ-vO 
 r) in n m N m 
 
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 to Tj- >-i in to 
 
 m r- o * ^ t^ 
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 tJ- CO tl rl n II 
 
 OnOO on > no d O 
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 3 
 < 
 
 ^ 
 
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 to t> t) M M n 
 
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 m in in m 
 
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 m ii- tJ- ^ Tj- Tl- t- 
 
 
 ,0 
 
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 'n\o i^oo Ov O 
 
 1 c) to -^ tnvD 
 r* rt M rl 
 
 r-^oo On O M 
 d d d d 
 
 tn 
 
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 <, 
 
 * 
 
 
 
 
 1 
 
 ^ 
 
 r< >nvO O 'H t'T 
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 to m d ^ to 
 
 
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 - 
 
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 1 
 
 o 
 
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 r^oo On O M d 
 d d d 
 
 CO ^ ^ mvo t^(3o 
 
 
 
 - 
 
 PO 
 
 
 
 <* 
 
 
 
 
 
 C4 t~- ri m, r^ o^ 
 
 1 Th >- to in 11 
 
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 Ti- r) to in 11 
 
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 to Tj- M to ^^ 
 
 M NO O Ti- mso 
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 " to 4- m c 
 
 
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 to -, r n t - 
 
 m to o t-- -^ f I 
 
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 in m m in ^ ^ 
 
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 d d n " 
 
 
 1 1 
 
 
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 rl to li- mio K 
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 d d 
 
 to 
 
 Tj- mvo t^oo On 
 
 
 
 
 
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 to rhvo m yo o t^ 
 
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 >, 
 
 ^ 
 
 rt" to to to to t^. 
 
 On t^ in to M ON 
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 r-. * r> O 00 m, 
 
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 m m in m 
 
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 ^ 
 
 
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 to ^ m,vo t^oo 
 ri rl n r> n rl 
 
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 d 
 
 
 
 
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 ::= 
 
 = 
 
 m Th to ti M o 
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 oo t^vD in to rl 
 
 " ONOO t^ m rj- 
 
 d W ON t^NO tj- 
 
 m m m u- 
 
 d M ON t^ in to 
 m m^j. rj- ^ ^ 
 
 
 
 
 
 <; 
 
 ' 
 
 tJ to + in\o t-. 
 O 
 
 u; c?i O n rl to 
 M M ti r) rl M 
 
 + mvo l^oo O^ 
 rl tl M cl M 
 
 w M tl CO ^ 
 
 mvo r~-oo ON O 
 
 
 
 
 fOOO Ov o.oo -> 
 to to to to to f- 
 
 v:. " -g w t~~ 11 
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 NO * O -"t-NC t^ 
 ^ 11 tJ. rl i1- 
 
 m - lOvO -t- m 00 
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 -= 
 
 
 cc oc eo eo oo i>o 
 
 c; c'. r'-. CO r^ t-^ 
 
 fi CO (O r*" to t^ 
 
 r-^O '-O ND m m 
 to to to to to t*- 
 
 rh + r<i ro d *- 
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 ^ 
 
 r 
 
 -. r> to r^ I'-r'O 
 
 ^~fm o - fl 
 
 to ^ mo r^co 
 cl rl cl r) N rl 
 
 ON w d CO tJ- 
 
 mvo t^OO On O n 
 w i 
 
 
 tt, 
 
 <. 
 
 :: 
 
 
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 V4^ 4- i'-' t O i*" 
 
 -1 in *;j- to K^ 
 
 CN -. ti " a-.'O 
 'n -l- tl to 
 
 1 ^nD t--NCi rl 
 
 m f 1 in rl m H 
 
 r-- t. J- m, to o 
 tJ- M to in n c^ 
 
 v^OO O O CO 
 tJ- m. n M r> 
 
 
 z 
 
 
 ti r tl ti t< rl 
 
 C/- ON O "I 1 N 
 N ti "^ to to f- 
 
 rl to to -:J- Ti- i/- 
 
 ro CO to to to to 
 
 mvo NO NO r^ t^ 
 to to to to to t^ 
 
 i~^ r-ec OO OO 
 
 to ro d CO to 
 
 
 -C 
 
 C 
 
 r\ to -"^ Lr.\o t- 
 
 OO C? C -. rt t- 
 
 - " cl ti rl rl 
 
 Kt- miO t^OO CN 
 
 n n rt rl cl rl 
 
 n d CO tJ- u- 
 
 o r-00 ON o 
 
 
 
 
 
 O 
 
 
 
 lU 
 
 
 
 
 ON o >- M "^ ir 
 
 tl to t1- in 
 
 " -^i -J- ^ to ^ 
 1 M to Tj- vn 
 
 QO in M oo in ^ 
 1 M tl to i- 
 
 'noo O to tl- to 
 ^t-L^ m m m in 
 
 rl - r- cooo d <*- 
 in in li- -^ CO CO d 
 
 
 >% 
 
 
 tl 'O + m r-, 1 c-. c- o - rt * 
 
 mvS t--0C On 
 
 1 d to ij- mvo 
 
 1^00 ON O >i d to 
 
 
 c 
 
 ' 
 
 O ~ to ^ i<- 
 
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 to 4- m.vo t^oo 
 d cl ti rl cl 
 
 ON d CO Tj- 
 ti 
 
 
 
 m,vO l-^OO On O 1 
 
 
 ON 
 
 
 
 IJ 
 
 
 t " '" ^J- " - 
 
 I c/3 J. o " r; 
 1 , . 
 
 CO Tj- ij-.o r^c/o 
 
 1 ON O 1 tl to <*-| mvc t^O'; On Cj " 
 
 ^ tl rl M n d 1 rl rl n rl N f t^ 
 
 i
 
 1^, 
 
 Book V. 
 
 ASTRONOMY. 
 
 311 
 
 
 
 enco C^OO ON O H, 
 d d cl d d CO CO 
 
 
 ^ 
 
 
 VO HI 00 U-, VI- 
 
 t^oo "ll-OC CO 
 
 ^ HI w r 
 
 00 en CO c cr.NO 
 
 f^ to Id en en 
 
 * er i^ .;)- in ^ 
 IH d to -cj- cr- 
 
 CO oo a\ o HI c. -j-^ 
 HI d to en HI HI 
 
 
 E 
 
 , ^ 
 
 00 w H t^ 
 
 ^ ^ U-l L(-, lO U- 
 
 cm c>) cT) en crj en 
 
 C5 HI cl to li-vo 
 
 t-^00 On HI to 
 
 ^ HI W 
 
 4- e^NO t^ O HI 
 
 HI W W M H4 d d 
 
 
 Q 
 
 c 
 
 a HI n ^/^ .^ 
 
 cmva t^oo o\ 
 
 c to r^ envo (^ 
 d c< d d d d 
 
 00 On w d to 
 d d 
 
 On 
 
 tJ- envo C-.00 ON u 
 
 HI 
 
 
 00 
 
 
 
 
 
 ro -it- r^ M 00 -X. 
 
 r-00 c) 00 <i- M 
 
 IS ^ HI CO CO 
 
 cl "l-os d Onoc 
 to ^ M en 
 
 t^ ON d r-^ r)- 
 co HI en 4 d HI 
 
 d HI o HI en O 
 n Tj* to d d 
 
 
 e 
 
 - 
 
 M M M HI r) 
 
 tJ d CO CO T^ r}- 
 
 CO CO to to to C'l 
 
 en enco CO r^ t^ 
 
 CO to CO CO CO cr 
 
 00 On ON HI d 
 to to to T^ tJ. rl- 
 
 TO CO -^ enco t^ 
 * + 4 "J- cj- <4- 
 
 
 z 
 
 
 
 0^ HI H ro )- 
 
 cnco t--00 Cs 
 
 Hi d CO + enco 
 d d d d d d 
 
 t--0 Cn HI d 
 d d d 
 
 00 
 
 to <- loO t^OO 
 
 
 t- 
 
 
 
 . 
 
 ' 
 
 00 c^ to r^ "^ 
 
 CO en ^ HI CO 
 
 en en.oo <} M 
 en HI CO CO 
 
 d CO M CO On Hi 
 to ^Ti- er, ^ 
 
 en HI 00 0> c^ 
 d HI en ^ -cj- Cl 
 
 On p~.co On d r^ en 
 d d d d to CO i^ 
 
 
 ^ 
 
 
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 Tj- + ro cx^^ ^ 
 
 CO vo en en 5J- T^ 
 CO CO CO to to to 
 
 CO CO d d HI H. 
 
 to CO CO HI CO CO 
 
 - HI 
 
 to CO to CO to CO 
 
 O O O O O O 
 to to to to CO cn CO 
 
 
 
 
 
 
 00 0> w r) n- 
 
 cj- cneo t--oo ON 
 
 >- d to ,J- en 
 
 d d ri d d d 
 
 NO t-~00 ON HI 
 
 d d d d 
 
 d ^ tJ- u^nO txoo 
 
 
 
 
 ^O 
 
 
 
 
 ^ 
 
 -- 
 
 Ti- * r-. H. r-- -^ 
 
 to .^ VO HI H Tj- 
 
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 tocc Ti- to tr 
 + " "i- d * 
 
 en en tJ- enoo 
 d HI en >* to cr 
 
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 d HI HI W HI C) 
 
 
 JQ 
 
 ^ 
 
 t 00 t-~ m to 
 
 c> 00 r- en Tj- 
 
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 w. - -. - - H. I -, rj^ d cl d d 1 rl d cl rl M CO CO 1
 
 jii 
 
 ASTRONOMY. 
 
 Book V^ 
 
 Days.l 
 
 * <^^ 
 
 t^OO ON -" t 
 
 to tJ- vovo t--00 ON O >i t to *! 
 
 VONO r^oo On O T 
 d d d d d ro to . 
 
 
 
 
 5 
 
 oo 1^ O oo r^vo 
 r H r - - - 
 
 VO 1^ -^ Tj- 0> * 
 < !- H t< M to 
 
 O VO ~ O 00 VO 
 
 ^ VO to to cl to 
 d to 1^ VO >< 
 
 to VO r-~oo o " to 
 d to tJ- VO " d to 
 
 
 fo c*i M r^ r^ c^ 
 
 ON O - t^ to ^ 
 tOTj-^^^^ 
 
 tovo t^ On O 1- 
 >* * * + o VO 
 
 d to * VO r-oo 
 
 VO VO VO VO o 
 
 ON O -< C tJ- tovo 
 VO 
 
 Q 
 
 o 
 
 a\ H <^ * 
 
 vovO ' t^oo 0\ O 
 
 n t4 to Th Vi^vO 
 
 r H M M H M 
 
 t-^OO On O M d 
 
 M d d 
 
 ON 
 
 to VOVO t^OO On 
 
 00 
 
 
 tl 
 
 i 
 
 
 t^ ON to r^ to to 
 
 Tj- to VO t) VO 
 
 - rovo o 00 NO 
 H u-i M to 1-1 
 
 to tooo o "^ O 
 to N ^ to 
 
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 " o t1- t1- to 
 
 a 
 
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 vo vo \o o ^ t^ 
 
 r^OO oo 00 On On 
 
 O O " M t to 
 tl H n d W M 
 
 >- t* to ^ VONO 
 
 tl tl n M n tl 
 
 to -^ VOVO NO t^ 
 
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 t^oo ON O T d 
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 00 ON On O 1-1 d 
 d d d to to to 
 
 to tJ- VONO t^oo 
 
 
 
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 OVO t^OO ON O 
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 to VO I to to 
 
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 r^vo o^ ^ tn m 
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 d to x^ VOVO t^OO 
 
 
 VO 
 
 
 
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 ^ 
 
 <y\ (j\ ^ W-) o oo 
 
 OO OO O "^ t< O 
 VO 1-1 ^ to 
 
 O tovo tl O O 
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 d NO 1- On O - 
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 to n On On -1 vo 
 
 VO VO 
 
 Jo 
 
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 00 VO 1- to w On 
 
 t^^NO * to 1-1 O 
 
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 -4- + 'I- ^ * 'i- 
 
 oo t-^vo 'o * 
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 to d O ON OnOO 
 to ro to d d d 
 
 a. 
 
 
 a> - f< ro to 
 
 ^h VONO f-OO ON 
 
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 NO 
 
 d to tJ- tovo l^ 
 
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 U-1 
 
 
 
 
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 OnvO rj- w^ vr> t^ 
 to ro to 
 
 On iooO ^ ii O 
 to rl- r Ti- 
 
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 r--vo r^ O "^ On 
 
 tJ- to d d " 
 
 NO TJ- tJ-no O tJ- 
 
 M H tl 
 
 3 
 
 ' 
 
 r^ "^ t< O t^ t/^ 
 
 Pl o I^ to to o 
 
 VO VO to to 
 
 OO VO to >- ONVO 
 
 >1- <1- -^ <* to to 
 
 4- d O oo VO Ti- 
 to to to d d d 
 
 d O 00 NO tJ- d O 
 
 rl M M n 11 
 
 D 
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 CT\ - t to r^ 
 
 votO VO t^OO ON 
 
 O -< tl to ^ VO 
 M tl tl tl M 
 
 NO t^OO ON O IH 
 
 d d d d 
 
 to 
 
 d to Tl- VONO t^oo 
 
 
 '* 
 
 
 
 lo u-ioo o ^^ 
 <n - to -"^ u^ 
 
 1^00 t) tooo O 
 
 11 to ^ VO 
 
 tl 4- >^ " tl Ti- 
 
 oo VO to d -1 -1 
 
 VO 1-1 to to 1-1 to 
 
 to -1- vooo d r-. d 
 VO M to to d tI- n 
 
 
 
 -^ 0>^ to O l^ 
 tJ- to to to to rt 
 
 VO tl ONNO to - 
 
 t< r< tl -1 1-. -. 
 
 00 VO M O t^ + 
 VO VO 
 
 ONNO to ~ 00 
 " Tl- t}- Tl- Tl- to 
 
 lo to t^ VO d O 
 to to to d d d d 
 
 
 -' 
 
 ON " fl to + 
 
 tovo t^OO ON O 
 
 - N to tJ- r^ to 
 tl tl tl tl tl tl 
 
 VO t^OO On i-i 
 d d d d 
 
 Tl- 
 
 d to Ti- lovo t^oo 
 
 to 
 
 
 o r r^ " t> "^ 
 CO H >^ -> to 
 
 VO r^. t^ VO t}- rl 
 
 to 1-1 to VO CO 
 
 ON >o pi r^ o -51- 
 ^ tl to to 
 
 00 O to to t^ On 
 1-* to t1- to 
 
 d Ti-vo r^ 
 to ^ to 11 d 
 
 C 
 3 
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 - 
 
 CO oo vo r^ o 
 u- u^ to vn 
 
 r.- to H ON t^ + 
 
 ^ ^ Tt- to to CO 
 
 -< OnnO to o oo 
 
 to M n tl tl " 
 
 >^ d ONnO t}- 1-1 
 
 oo VO d O I^ t4- 
 to VO VO to t1- tJ- 
 
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 " f t\ to ,)- lO 
 
 vO f^oo ON 1- 
 
 tl to t1- IOnO t^ 
 M tl tl tl tl tl 
 
 oo On w d to 
 d d 
 
 to 
 
 CO ^ VOVO t^OO 
 
 
 a 
 
 rJ 
 
 
 
 
 5 
 
 ^- -^^ VO to lo v^ 
 
 VO tooo o O 
 to to t1- to c< M 
 
 onvo 4-oo " to 
 
 lO ^ tl 1-1 lO 
 
 to to d ON t}- ON 
 
 to M to d to 
 
 d tI- vovO no t1- o 
 
 >- ^ - Tl- Tl- 
 
 
 ^ 
 
 CO - On t-. VO to 
 
 ^ ON t^ to to K^ 
 
 VO to *o to to 
 
 OnvO * rl t-- 
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 to to O 00 VO to 
 to to to d d d 
 
 00 NO to -1 OQ VO 
 
 d 11 -< " 11 
 
 
 - f) to ^ lOvO 
 
 r^ t^V> On - 
 
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 Tl- VOVO t^oO On 
 
 
 
 
 
 
 
 
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 O OO On t- >! ON 
 CO to to CO to N 
 
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 d oo tovo t^oo 
 to Tl- H, rl to 
 
 a- 
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 \D >o Tj- to ri - 
 lO to W-i to to to 
 
 O OnOO VO to tJ- 
 
 "^ 'I- * -"l- -"l- ^ 
 
 r^ iH o 00 h^ VO 
 si- T^ Tl" CO to to 
 
 t1- d " On t--VO 
 to CO to tl d d 
 
 tJ- d n ON r^ to 
 d d d I 1 " 
 
 C 
 
 tl to ^ voto 
 
 r^CO On o - tJ 
 
 to Tj- tovo r^oo 
 n tl tl tl n tl 
 
 ON - d to t1- 
 d 
 
 v'lNO t-sOO ON 
 
 
 V. 
 
 o 
 
 
 
 
 
 CO CT\ -* CO c4 On 
 t< t< to to to p( 
 
 ':J-CO On OnvO t^ 
 t< VO it- to 
 
 to I-- r-~ Tl- to 
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 to Tl- - C-. ON >^ 
 to d VO 11 CO 
 
 O t^ " to to to tl 
 d to to tl tl to 
 
 J= 
 
 ^ 
 
 ^ 0> ON 0> On ON 
 
 ON ON ONOO 60 OC 
 
 00 c^ t^ r^NO NO 
 
 VO VO -4- t}- to to 
 
 d O O Onoo t-- 
 
 VO VO VO 
 
 s 
 
 
 
 - tl to ^ vovo 
 
 I--0O ON O ~ N 
 " M >-i C< N t< 
 
 to t}- vovO t^OO 
 
 tl tl tl tl tl tl 
 
 ON " d to Tt- 
 d 
 
 O 
 
 "^vo 1-^.00 00 ON o 
 
 t- 
 
 ~ 
 
 tl tl tl CT\ to o 
 to c -> *o -^ to 
 
 to t^ ^ to O NO 
 
 to to to tl 
 
 1-1 tJ- to tJ- 1-. OO 
 
 to to d 
 
 tJ- t^oo oo tJ- o 
 
 VO n to VO n to 
 
 to r-oo t^ 
 Tl- to >1 
 
 ^ 
 
 tov) r^ i-^co ON 
 
 O " tl M to 
 
 t1- t}- VO VOVO VO 
 
 vo t^ t^ r-.oo oo 
 
 00 00 On On 
 
 J3 
 
 c 
 
 ci t i- too t^ 
 
 Ov O " t) to ^ 
 
 VONO I-~00 On 
 tl tl d tl tl 
 
 - d to ^ >ovo 
 
 r--oo ON o 
 
 n 
 
 O 
 
 
 
 5 
 
 toco CO O - 
 to ^ to M to 
 
 N O O NO 
 <* "1 - t< M 
 
 Tl- d O to Ti- 
 to ^ VO to 
 
 oo d + to Ti- 
 
 d ~ NO rovo O d 
 
 MM VO VO Tl- 
 
 >N 
 
 
 rj to ^-vo t~.oo 
 t< t) t< t) tt f< 
 
 On O t< to ^ VO 
 tl to to CO to to 
 
 VO I--.00 On w r< 
 
 to CO to to rl- xj- 
 
 to Tl- VOVO r--oo 
 
 Tl- ^ Tl- Tl- ^ id- 
 
 ON O >- d d to ^ 
 
 <1- VO VO VO VO to VO 
 
 c 
 
 
 
 - t< to ^1- to\o 
 
 t^oc ON q - rl 
 
 co Tl- VOVO r^oo 
 d d d d d d 
 
 ON 1-1 d m t1- 
 d 
 
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 ovo t^oo On O 11 
 
 OS 
 
 
 D 
 
 jyc 1 - r> f^ + "^>0 
 
 t^oo On O - r^ 
 
 to t1- v/-,o t^oo ON o >- d to rl- 
 
 _ ^ _ _ rl rl rt rl rl 
 
 tovo t-^oo ON O - 
 d n d d tl to ro 
 
 
 
 
 
 

 
 Boojc y. 
 
 ASTRO NO M Y. 
 
 31J 
 
 " 
 
 Days.l 
 
 ^ d CO Tj- u-ivo 
 
 r^OO On " C4 j 
 
 CO tt- voivo t-.eo 1 
 
 CT\ >- d CO rj-l w-ivo t^oo iJN - 11 
 '- d d d d d 1 d d d d d CO CO II 
 
 i 
 
 L^ 
 
 I 
 
 >*- - a^ f-CO CO ON ci +00 c^lsO w-> t~~ r^ ^l 
 rcocoMcici c1CCOCOCO^|tJ-^ -ci| 
 
 n 
 CO ti- m n 
 
 d ti- uoo r^ ON d 
 
 CO -^ u- w d -l- 
 
 * 
 
 1 
 
 E 
 
 - 
 
 50 ON " H CO 
 
 " " c< CO d c< 
 
 tJ- mvo t^oo ON 
 d n fi c< cJ H 
 
 'I CO ^ io\o 
 
 t^oo On - d CO 
 CO CO CO -^ tj- rj- 
 
 ti- mNO 00 o\ -^ 
 ^ ti- t*- ti- ti- in, LO 
 
 ^ 1 
 
 u 
 
 
 
 On " M CO r^ 
 
 v^vo r^oo ON 
 
 - H CO rl- lONO 
 C> H~ C< M d C 
 
 r-co ON -< d 
 d d d 
 
 On 
 
 CO ti- vnvo t-.00 On 
 
 i ' 
 
 a 
 
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 X 
 
 
 
 -1 
 
 C 
 
 ^i- U-) <-. CO VN 
 
 ON M 00 -i- 
 
 CO ui "<* >-i 
 
 " tJ- t-~ CO -^ 
 -l- " ^ - >^ CO 
 
 Ol ON -> COOC * 
 
 ^ CO *- m tj- 
 
 d 0*00 On d'NO 
 CO -1 u-i u^r}- 
 
 u 
 
 g 
 
 
 
 C 
 
 ^ - n ^ ci n 
 
 Cn - c CO -^ 
 
 CO ro CO i- + in 
 iO\> t-,)0 On 
 
 v^vO NO f- t^OO 
 - CJ CO r^ WNNO 
 
 d d d M d ci 
 
 ON ON - 1- d 
 
 CO t}- mt u-NvO r-* 
 
 5 
 
 r^oo ON "* d 
 d d d 
 
 so 
 
 CO tj- NOVO t->00 
 
 3 
 
 r- 
 
 
 s 
 
 uT 
 
 5 
 
 - 00 t^ 4- 0\ 
 I/-, vr, H CO rt- 
 
 rt w O^ao 
 00 0\ " f CO 
 
 00 CO - tJ- w 
 <1- /N\f5 t-^oO C 
 
 CT\ d 00 CO CO ^i- 
 co ri- d !- 
 
 ^ sh CO CO CO d 
 
 >-< d CO -i- m 
 d d d d d d 
 
 NO d t^sO I^ ON 
 
 d i-i w-> ti- CO d 
 
 tj- - d t^ CO 
 d d d d d d CO 
 
 
 
 
 
 
 
 NO t^oo c^ H. 
 d d d d 
 
 r-. 
 
 d CO ^ irivo tvoo 
 
 bO 
 
 0, 
 
 VO 
 
 
 
 53 
 
 !- 
 
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 -/- w, (^ 0^ -i- - 
 CO Tj- v/~, ~ CO 
 
 CO ir^ -. 
 
 vy-^ .-. CO l/N c< ^^ 
 
 - tJ- NO NO 
 
 d i*^ d CO -. 
 
 r^ -! vo CO CO to 
 w-i c^ d " \o 
 
 ON t*- d "-NO 
 + <3- ti- ti- ti- tt- 
 
 Q 
 
 
 E 
 
 
 UN .y^ WN Tj- t}- ^ 
 
 CO d 30 t^ 'J- 
 4- "+ Ni- CO CO CO 
 
 4- d - 00 t-^ 
 CO CO CO CO d d 
 
 10 i- CO d On 
 d d d d d - 
 
 00 P-VO 10 + CO 
 
 
 c 
 
 M 0-. " N c- 
 
 rj- ^/^vO t~~oo ON 
 
 - d CO ^ u- 
 d d d d d d 
 
 NO t^eo ON '-' 
 d d cl d 
 
 NO 
 
 CO NO 00 CO o\ 
 ti- to cl d 
 
 d CO -i- vn.vjs t^ 
 
 
 
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 VN 
 
 
 
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 vO N - - " d 
 
 ^ 1^ H 00 W-, c^ 
 
 to tJ-NO >^ Cl 
 t) -1- CO - 
 
 > CO CO covo t^ 
 
 c 
 
 
 
 CO 00 VO CO ~ 
 w> iy-1 WN vn 
 
 0<5 vO CO " ONNO 
 ^ tJ- Ti* ^ CO CO 
 
 ^ rt On t^ ^ CO 
 CO CO d d d d 
 
 00 tj- d 
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 ir, m 
 
 
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 a\ - ci CO 
 
 >i- i^NO t^OO On 
 
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 d d d d d d 
 
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 d CO ti- vr.vo VO r^ 
 
 
 '^ 
 
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 CO ^50 ov n 
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 CO ^t- .-. CO tn 
 
 CO NO to CTN 
 
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 - -i- r~- tj- 1 
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 ir 
 
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 ti- - ONNO CO 
 
 in U-) -^ rj- rt- tj- 
 
 co NO d t^ Ti- 
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 d ONNO tj- w 00 NO i 
 
 d -. - - , 
 
 
 
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 d CO tJ- K-. 
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 3 
 
 
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 t-, m d ON t-~ -4- 
 
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 + ^i- CO CO CO CO 
 
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 ASTRONOMY, 
 
 3^5 
 
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 5i6 
 
 ASTRONOMY. 
 
 Book V. 
 
 ?f 
 
 D'ays.| 
 
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 ASTRONOMY. 
 
 319 
 
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 ^20 
 
 ASTRONOMY. 
 
 Book V^ 
 
 
 Rita-* 
 
 
 
 
 
 
 
 1 
 
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 Vol. I.
 
 ASTkONOMY. 
 
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 fo (O TO m -i^ ^ T^ -vj- cr, U-. c/- w, " ^ t-i t-. c^ r r) N to ro to CO -^ ^ -^ tJ- U-, CO crt co; 
 ._-.< flclrjTTfl on ririr<rlc<c<r)flrlwc<Mc)MNpr)l 
 
 ^ 
 
 O to .0 l^ 0-. - 'O i^c/S to CO r~ ON - ro ;/- or; Q n tj- t- ON -< tov; C<5 G r! -J- t- ON - f \0 ON 
 
 fl N rl M rJ c. c^ to to -^ ^ .i- .^J- t- CO cr, cr, to -< " -. M r< c) M tl to to to f . 
 
 jr. 
 
 CO 
 
 
 _ 
 
 ^ 
 
 tl -l-NC !<. r) -^ i-- I-- o-N . to V l~- CN - to .0. r-. ON -. to >o r^ ON cT -"l-NO 00 M -4- t^ ON 
 .,HC(tltlrc^rototototOti-'=}--4--;t-^coco, w-icovn -, 
 
 
 
 
 
 tl to ^. l-->2 ro 1/-.-0 OC fi t^. 10 I c^- rl to C/-VO c/; rl t-^ co r-. On O CI to i^ i^, ^-^ 
 ^ H ^ H- .^ M c-1 C4 c; C) cl cl to to to CO to to ti- ^ ^ ^ ^ ^- CO CO, to CO CO CO 
 
 1 
 
 
 __ 
 
 fcjj 
 
 
 - 
 
 C T 
 
 -c " 
 
 A 
 
 ; 
 
 rl .0 -t- lo. t-,c.c On M rO >^ r-OO - cl + 'f ^ 'y: CN ~ CO co,nC l~~ CN O N CO tJ-vO OC ON 
 cr, 10 CO CO cr, to CTi CO - j) r) cl C< M C< CJ CO CO CO to to to t' 
 
 W M 
 
 c 
 
 ^ cl to W- to, r VD On " rJ to -r^-'wC t >: on *- cl to rj- cr, i ,co ON - N to -^ c^-, f-or, ' 
 
 
 
 < 
 
 
 S ^ 
 
 
 - - t to r- coo r- l^OG C. - - rl to t4- cr.o r. |~.oO On O O -^ to ^ t<-ivC. 1 -vj On 1 
 c-: to to to CO CO to to to to t*-. CO ri- ^ -^ ^ T^- T^ ^ ^ ! ^ ^ ^ CO CO CO to CO CO crt CO, 10, CO cr, ; 
 
 h 
 
 
 
 C -4- 
 
 A 
 
 c 
 
 J= 
 
 
 00 f* H to to Tt" ^ 'o. t/-,sO r- I vj o ctn t; ci n to to -i- 'o /-. a r 1 -0 v: "on 
 
 cl cl cl C4 cl t M cl Ct cl d cl d N tl d cl to to to to to to to to to to to to to to to lO to to ^ 
 
 '"' 
 
 " - - - fi t" cl Cl t-. to ro -J- .4. -^ -j. >,, ,r .,- .0 .c vr, t . 1 . 1 - r -^ y- -./; yi ct> on on 
 
 ro 
 
 
 
 - 
 
 > 
 
 '^ 'J 00 > J 1 r, ) iJ , ' ^ ) ij !.' ; ', .J j I- rt- 
 c ^ tl tl- Cl -1- tl + c -4- c -J- tl tt- (1 >- tl rJ- tl * Cl 'I- N c. 
 
 r< f ^ 10, CO 1 ~ c/:i CN " cl ''^ 
 - - - - - ., - _ rl rl M cl 
 
 
 
 
 ' 

 
 3M 
 
 ASTRONOMY. 
 
 Book V. 
 
 312. Table of the Right Ascensions and Declinations of fixty 
 Stars in the Northern Hemifpherej for the Year 1780. 
 
 Pegalus 
 
 Cafl[i)pea 
 
 Pv)1p Star 
 
 Andromeda 
 
 Aries 
 
 .\n.t->meda 
 
 ConftclUtions. 
 
 Places of the Stars in tin 
 Conftcllations. 
 
 Ending oi the Wing 
 Brealt 
 
 Girdk 
 
 : receding Horn 
 
 Foot 
 
 Names. 
 
 Scliedar 
 
 Alruccabah 
 
 Mii.;ch 
 
 Almaach 
 
 Oiion 
 
 Auriga 
 Gemini 
 
 L-ttlc DAg 
 Gemini 
 Great Bear 
 Cancer 
 Great Bear 
 Leo 
 
 Great Bear 
 
 Leo 
 Great B.v.r 
 
 Dragon 
 
 Bootes 
 
 I'Crowii 
 liSerpent 
 ' Ilctcuks 
 
 0;.'h)ucu5 
 
 Vrics 
 
 Follow' ng Horn 
 
 \Vhalc 
 
 Following in tlic Check 
 
 
 Jaw 
 
 VIedufa 
 
 Head 
 
 Pcrfcus 
 
 Briglitcft 
 
 Taurus 
 
 Brit'hteft in 
 
 Pcrfeus 
 
 K.nee 
 
 Taurus 
 
 Firft of the 
 
 
 Northern Eye 
 
 
 Soutliern Eye 
 
 Auriga 
 
 In the Goat 
 
 Taurus 
 
 Northern Horn 
 
 Menkar 
 ,\igol 
 Algenib 
 I'kiades 
 
 Hyades 
 
 Aldcbaran 
 CapcUa 
 
 Weft Shoulder 
 E.'.ft Shoulder 
 In the Hand 
 Foot of Pollux 
 Knee of i'oUux 
 Brighteft in the Head 
 
 Bcllalrix 
 Betel{,uefe 
 
 Caftor 
 
 F>riphtfft 
 Head of Pollux 
 North Paw 
 In the Claw 
 Preceding Knee 
 North in the Head 
 
 In the Heart 
 
 Lower Pointer 
 
 Upper Pointer 
 
 In the Tail 
 
 S f-jllov.-iriij in the Square 
 
 !>;iil in tlic Square 
 
 Procyon 
 Pollux 
 
 Acubens 
 
 Re^ulus 
 Dub he 
 
 Rt.Afccn. 
 in time. 
 ;. m. f. 
 
 O T 56 
 
 O 28 S 
 
 o 47 45 
 
 57 29 
 
 1 42 32 
 I 50 29 
 
 Vear. 
 Var. 
 f. 
 
 Dcclinat. 
 North. 
 
 Yearly 
 
 Var. 
 
 3.08^ J 3 
 3.31 '55 
 io,o5|S8 
 
 30I34 
 
 57 3620,04+ 
 19 46ji9,9i + 
 
 7 57119.69 + 
 26 56119,45 + 
 
 3.^^,19 43 32|^.'o + 
 3,62141 I"; 54117,80+ 
 
 1 54 49 
 
 2 31 56 
 
 2 50 48 
 ^ 53 56 
 
 3 ^ 43 
 3 34 i7 
 
 3.34,-2 
 3,12 2 
 
 3.'3 3 
 3.8540 
 4,2049 
 
 3.551^3 
 
 24 52 
 
 7 59 
 12 56 
 
 5 37 
 3 44 
 
 17.^4+ 
 15,86 + 
 14,80+ 
 14,63 + 
 1 3.72.+ 
 
 24 38 12,00+ 
 
 3 43 oS 
 
 4 7 18 
 4 JS 48 
 
 4 -3 19 
 
 5 o 28 
 5 12 24 
 
 5 13 21 
 5 43 16 
 
 5 44 43 
 
 6 25 QC 
 
 6 51 
 
 7 20 33 
 
 ft in th.: Tail 
 Mid lit; of the Tail 
 L.ilt in liic Tall 
 In the Tad 
 Skirt of the Coat 
 
 the. folkiwinp Thigh 
 
 Alioth 
 Benetnach 
 
 Arfturus 
 Mirach 
 
 The briuhteil 
 In the Neck 
 Preceding Shoulder 
 Follr,-,vi;i;; ;'.t the Side 
 Th" He.ul 
 TIk- Head 
 
 Aiphacca 
 
 Rif. Algcthi 
 R.-.f. A'haciie 
 
 7 ^7 
 
 7 31 51 
 
 8 44 
 
 8 46 27 
 
 9 18 5 
 9 40 13 
 
 3.94 
 3.39 
 
 3.4S 
 
 3?43 
 
 4,40 
 
 3.79 
 
 38 21 24 
 
 15 4 5 
 18 40 32 
 1636 
 
 45 44 48 
 28 24 13 
 
 3,22 
 
 3.^5 
 4,09 
 
 3.47 
 3.58 
 3,88 
 
 11,41 + 
 9,60 + 
 8,90 + 
 
 ?,32 + 
 
 5.30 + 
 4,26 + 
 
 682 
 7 20 56 
 37 10 36 
 16 34 9 
 20 52 31 
 32 21 8 
 
 3,20 5 46 
 3,7528 32 
 48 53 
 
 4.5 
 
 3j*4 
 4.2.3 
 3.47 
 
 12 41 
 52 40 
 27 I 
 
 4.15 + 
 1,58 + 
 1,40 + 
 2,10 
 
 4.33 
 6,78 
 
 7,40 
 
 7.7 
 1 3,00 
 
 13.30 
 
 15,. r 
 16,33 
 
 56 39 
 48 27 
 50 c 
 
 37 50 
 42 
 
 4 ^7 
 
 44 17 
 15 
 38 52 
 58 ^-7 
 5 40 
 35 28 
 
 25 2;j 
 
 33 
 
 20 4e 
 
 51 ';2 
 
 4 38 
 
 =4 43 
 
 3,24] I 3 2 
 
 3.74,57 33 
 
 3,88 62 56 
 
 3,ic 15 48 
 
 3>24 54 55 
 
 3.05'58 15 
 
 17,16 
 19,05 
 19,091 
 
 8I19.95 
 4 19,99 
 
 25120,05 
 
 2.6957 9 
 
 2.44ls6 4 
 
 2.4i|5o 25 
 
 1^63 65 25 
 
 2,82 20 20 
 
 2,63128 o 
 
 2,54j27 28 
 -.94 7 7 
 
 2,59|2I 58 
 
 2,2931 15 
 2,74,14 39 
 2,75 12 44 
 
 19,69 
 19,01 
 ,24 
 1 7.46 
 17,16 
 I 5,67 
 
 1 2,60 
 
 12,03 
 8,51 
 5,96 
 4,87 
 3.>5 
 
 Di:-.gon 
 
 C-iic'is 
 
 In ti-.e Head 
 The b:ij;' Tt 
 FoHowinj; m Lozange 
 Preceding Wing 
 The brighteft 
 ri-,e Bret 
 
 Ruitaben 
 Vega 
 
 Atair 
 
 3- 
 
 ^.e T.iil 
 Preceding Shoulder 
 The Neck 
 
 # Thigh 
 Wing 
 The Herd 
 
 Dcneb 
 
 Aiderain:;n 
 
 Sch-at 
 M.ukab 
 
 5' 31 
 29 
 46 49 
 
 14 
 40 oa 
 14 20 
 
 20 33 56 
 
 21 13 10 
 
 22 30 29 
 22 53 8 
 
 22 53 49 
 
 23 57 3 
 
 I.-37I5I 31 
 ^,o^\^^ 35 
 2.1 1 36 37 
 3,02, 2 41 
 2,9c 8 17 
 2,]6|-;9 33 
 
 0,78 
 2,52 + 
 3.97 + 
 6,31 + 
 8,40 + 
 1 1,00 + 
 
 2,05^44 30 
 i.44|6i 39 
 2,99l 9 41 
 2,88 26 53 
 2,q8 14 I 
 3,0727 52 
 
 12,44+ 
 
 '4.95 + 
 18,46 + 
 19,18 + 
 19,20 + 
 
 22 20,05 +
 
 Book V. 
 
 ASTRONOMY. 
 
 3^ 
 
 312. TABLEofthe Right Ascensions and Declinations of fixty 
 Stars in the Southern Hemifphere j for the Year 1780. 
 
 Conftellations 
 
 i 
 
 Places of the Stars in the 
 Con;lellations. 
 
 Names, 
 
 S glRt.Afcen. 
 ^.in in time. 
 f\ ? h. m. f. 
 
 Year. 
 Var. 
 f. 
 
 Ueclinat. Yearly 
 South. Var. 
 
 Fhcnix 
 
 The Head 
 
 
 a 
 
 2 
 
 
 
 IS 22 
 
 3,01 
 
 43 
 
 29 43 
 
 20,oo 
 
 iWhale 
 
 Brighteft in the Tail 
 
 
 
 
 a 
 
 
 
 32 32 
 
 3,01 
 
 19 
 
 II 50 
 
 I9,86 
 
 Phcnix 
 
 Thigh 
 
 
 e 
 
 3 
 
 
 
 56 15 
 
 2.73 
 
 47 
 
 54 58 
 
 19.46 
 
 1 
 
 Following Wing 
 
 
 y 
 
 3 
 
 I 
 
 18 49 
 
 2,67 
 
 44 
 
 26 52 
 
 18,90 
 
 jEridanus 
 
 Source of the River 
 
 Achernar 
 
 a 
 
 I 
 
 1 
 
 29 31 
 
 2,25 
 
 58 
 
 ii 35 
 
 18,56 
 
 iWhale 
 
 Preceding Jaw 
 
 
 J 
 
 _3 
 
 2 
 
 28 13 
 
 3.07 
 
 
 
 37 50 
 
 1 6,00 
 
 Eridaniis 
 
 Near the Whale 
 
 
 K 
 
 3 
 
 T 
 
 5 ' 
 
 2,91 
 
 ~9 
 
 38 55 
 
 13,92 
 
 
 The following 
 
 
 J 
 
 3 
 
 3 
 
 32 44 
 
 2,88 
 
 10 
 
 31 30 
 
 12,08 
 
 
 The fourth Bend 
 
 
 y 
 
 3 
 
 3 
 
 47 46 
 
 2,80 
 
 14 
 
 8 47 
 
 11,01 
 
 Goldfifh 
 
 In the Tail 
 
 
 a 
 
 3 
 
 4 
 
 29 15 
 
 1,28 
 
 55 
 
 30 20 
 
 7,76 
 
 Orion 
 
 Bright Foot 
 
 Rigel 
 
 e 
 
 I 
 
 5 
 
 3 58 
 
 2,85 
 
 8 
 
 28 11 
 
 4.94 
 
 
 Preceding in Belt 
 
 
 J 
 
 2 
 
 _5. 
 
 20 47 
 
 3.07 
 
 
 
 28 40 
 
 3,50 
 
 - - 
 
 Middle of Beit 
 
 
 c 
 
 2 
 
 5 
 
 25 4 
 
 3.05 
 
 I 
 
 21 30 
 
 3.13 1 
 
 
 Lafr in the Belt 
 
 
 r. 
 
 2 
 
 5 
 
 29 41 
 
 3>04 
 
 2 
 
 4 ^7 
 
 2,77-! 
 
 Dove 
 
 Preceding of the brighteft 
 
 
 a 
 
 2 
 
 5 
 
 31 43 
 
 2,2c 
 
 34 
 
 12 6 
 
 2,s6-j 
 
 Orion 
 
 In the Knee 
 
 
 X 
 
 3 
 
 5 
 
 37 io 
 
 2,85 
 
 9 
 
 45 35 
 
 2,10 
 
 Oove 
 
 Following of brighteft 
 
 
 s 
 
 3 
 
 5 
 
 43 13 
 
 2,11 
 
 35 
 
 51 5' 
 
 1.65 
 
 Arco 
 
 The biigiiteft 
 
 Canopus 
 
 a 
 
 il 6 
 
 '9 5 
 
 i>34 
 
 52 
 
 34 57 
 
 1,60 + 1 
 
 Great Dog 
 
 The brighteft 
 
 Syrius 
 
 a 
 
 I 
 
 6 
 
 35 ^8 
 
 2,69 
 
 TT 
 
 25 8 
 
 3,io + , 
 
 
 In the Back 
 
 
 J 
 
 2 
 
 6 
 
 59 *7 
 
 2>45 
 
 26 
 
 3 37 
 
 5.08 + 
 
 
 In the Tail 
 
 
 n 
 
 2 
 
 7 
 
 15 24 
 
 2,38 
 
 28 
 
 53 5 
 
 6,42 + : 
 
 Argo 
 
 In the Poop 
 
 
 ? 
 
 2 
 
 7 
 
 55 52 
 
 2,12 
 
 39 
 
 23 28 
 
 9,62 -f- 
 
 
 Preceding in the Hull 
 
 
 y 
 
 2 
 
 8 
 
 2 46 
 
 1,85 
 
 46 
 
 41 40 
 
 10,16+ 
 
 1 
 
 Rri^htcft in the Middle 
 
 
 J 
 
 2 
 
 8 
 
 38^6 
 
 1,61 
 
 53 
 
 54 24 
 
 >2.73 + 
 
 1 
 
 Briglit among the Oars 
 
 
 
 
 I 
 
 9 
 
 10 44 
 
 0,75 
 
 68 
 
 48 50 
 
 14.79 + 
 
 Female Hydra 
 
 The Heart 
 
 Alphard 
 
 a 
 
 2 
 
 9 
 
 16 47 
 
 2,96 
 
 7 
 
 42 51 
 
 5,'3 + 
 
 jArgo 
 
 Northern in Section 
 
 
 D 
 
 2 
 
 10 
 
 36 34 
 
 2,27 
 
 58 
 
 31 59 
 
 18,68 + 
 
 Gentaur 
 
 Preceding '.n the Crupper 
 
 
 J 
 
 3 
 
 '' 
 
 57 3 
 
 3,06 
 
 49 
 
 29 40 
 
 20,04+ 
 
 Grofs 
 
 Preceding Arm 
 
 
 J 
 
 3 
 
 12 
 
 3 35 
 
 3,10 
 
 57 
 
 31 31 
 
 20,04-}- 
 
 
 Th;- Foct 
 
 
 a 
 
 I 
 
 12 
 
 14 33 
 
 _h^ 
 
 61 
 
 52 48 
 
 20,01 + 
 
 1 
 
 The Hea 1 
 
 
 y 
 
 2'l2 
 
 19 4 
 
 3=4 
 
 55 
 
 52 42 
 
 19.98 + 
 
 iCentiur 
 
 Top of the Crupper 
 
 
 y 
 
 212 
 
 29 30 
 
 3.27 
 
 47 
 
 44 5<^ 
 
 19,89 + 
 
 iCrofs 
 
 Following Am; 
 
 
 B 
 
 2|T2 
 
 35 a 
 
 3.42 
 
 58 
 
 29 3 
 
 '9.83 + 
 
 Female Hydra 
 
 The Tail 
 
 
 y 
 
 3'i3 
 
 7 00 
 
 3,22 
 
 22 
 
 22 
 
 19,22 + 
 
 Virgo 
 
 The Sheaf 
 
 Virgins Spike 
 
 u 
 
 J;'3 
 
 13 38 
 
 3,! 5 
 
 10 
 
 24 
 
 19,00 + 
 
 Centaur 
 
 Preceding L?ir 
 
 
 p 
 
 J:H. 
 
 48 30 
 
 4,09 
 
 S9_ 
 
 17 59 
 
 17.91-I- 
 
 
 S>u'.h i.i the Sliictd 
 
 
 >) 
 
 3,' 4 
 
 21 37 
 
 3.75 
 
 41 
 
 10 42 
 
 16,43 + 
 
 
 Bright in t;.c Foot 
 
 
 a 
 
 lll^ 
 
 25 2 
 
 4.41 
 
 59 
 
 55 17 
 
 16,26 + 
 
 Libra 
 
 Southern Scple 
 
 Zubenefch 
 
 a. 
 
 2; 14 
 
 38 45 
 44 56 
 
 3.3' 
 
 '5 
 
 6 55 
 
 '5.5" + 
 
 Centaur 
 
 Following i:i the Head 
 
 
 K 
 
 3;h 
 
 3.84 
 
 41 
 
 12 21 
 
 5,'7 + 
 
 Libra 
 
 Northern S.:dc 
 
 Zubenclg. 
 
 s 
 
 rJ.s 
 
 5 li 
 
 3,22 
 
 8 
 
 33 31 
 
 '3.93 + 
 
 Southern A 
 
 ThcVe-"x 
 
 
 e 
 
 _2il_ 
 
 35 56 
 
 5,12 
 
 62 
 
 43 31 
 
 11,884 
 
 Scorpio 
 
 Middle ol th-: Fo.chcad 
 
 
 T 
 
 3''-5 
 
 47 21 
 
 3.53 
 
 21 
 
 58 42 
 
 ii,co-r 
 
 
 N. in tlu- Forchotd 
 
 
 ,3 
 
 215 
 
 52 41 
 
 3=47 
 
 '9 
 
 1 1 14 
 
 10,70 + 
 
 Ophiucus 
 
 Preceding Hund 
 
 
 J 
 
 3,6 
 
 2 50 
 
 3.14 
 
 3 
 
 6 45 
 
 9.89 + 
 
 Scorpio 
 
 The Heart 
 
 Antarcs, 
 
 a 
 
 I 16 
 
 15 57 
 
 3.66 
 
 25 
 
 55 3^ 
 
 8,90+ 
 
 
 Firft Joint in the Tail 
 
 
 I 
 
 316 
 
 35 5S 
 
 3.90 
 
 33 
 
 52 21 
 
 7.34+ 
 
 Ophiucus 
 
 Folly>vin,; Knee 
 
 
 1 
 
 z 16 
 
 57 47 
 
 3.44 
 
 l 
 
 26 14 
 
 5.52 + 
 
 Altar 
 
 In the Mi^Jk: 
 
 
 a. 
 
 3~ 
 
 14 52 
 
 4,61 
 
 49 
 
 40 36 
 
 4,12+ 
 
 Scorpio 
 
 Bright at Tail's End 
 
 
 k 
 
 3;i7 
 
 18 42 
 
 4,oS 
 
 36 
 
 55 22 
 
 3.77 + 
 
 
 Sixth Knot in the Tail 
 
 
 I 
 
 3 '7 
 
 32 13 
 
 4,1 S 
 
 40 
 
 I 7 
 
 2,60 + 
 
 Sagittarius 
 
 S.EncI of the Bow 
 
 
 
 2 18 
 
 9 35 
 
 4,0c 
 
 34 
 
 27 58 
 
 C,72 
 
 
 Prerpiling Shoulder 
 
 
 <r 
 
 3,8 
 
 4' 37 
 
 3.73 
 
 26 
 
 32 57 
 
 3.54 
 
 
 Foilowin,: !;\ tin- Head 
 
 
 w 
 
 -I'll 
 
 56 41 
 
 3,58 
 
 21 
 
 21 17 
 
 4.80- 
 
 CiprlL.iru 
 
 In the toiiowii)^ Horn 
 
 
 ~ 
 
 3 20 
 
 5 54 
 
 5.35 
 
 ~ 
 
 12 45 
 
 1 0,40 
 
 ircacoc'< 
 
 Tiic Ey 
 
 
 n. 
 
 2'20 
 
 8 X 
 
 A:h 
 
 57 
 
 25 !4 
 
 1 0,44 
 
 ;C;ipri'''ni 
 
 !! tiic Forciicad 
 
 
 3 
 
 320 
 
 8 37 
 
 3.35 
 
 '5 
 
 27 43 
 
 ic,6o 
 
 iCr.uic 
 
 I'rcc'-.lini' Wing 
 
 
 a. 
 
 2,21 
 
 54 18 
 
 3.92 
 
 48 
 
 5' 
 
 i7,co 
 
 A'i'i.ii iy. 
 
 f'lWov-'.f.i^- Arm 
 
 
 y 
 
 3,22 
 
 IQ 17 
 
 3.H 
 
 2 
 
 29 23 
 
 17,76 
 
 Sontli-n Flili 
 
 T!:r r.rii,^htra IFomMIiiMt 
 
 JL 
 
 1 22 
 
 45 ^7 
 
 3.33'3o 
 
 46 5<; I8,97
 
 3-6 
 ei 
 
 Owe 
 
 -'^ .5 
 cr o o 
 
 u. g c 
 
 O O I, 
 
 C 3 
 
 1-1 < ** 
 
 -< ^ I- 
 
 . !i 
 
 J t' ts 
 
 pa c/3 CJ 
 
 < W t) 
 
 H-5-S 
 
 u% o O 
 
 A. T R O N O M Y. 
 
 BookV 
 
 lo|4 
 
 ij^s-sy!? 
 
 COM >< Q a>o 
 
 ^>VO 
 
 in * CO m" 
 
 cso t^e 
 
 m ^ en T* 
 
 
 
 "' " " 
 
 " 
 
 
 vO c ii- r^ Wn r 
 W1 ^ ^ m M - . 
 
 VI CO tn m Cl M Ml 
 
 * W r - H v> 
 ri -^ .^ ro c< 
 
 to c) M N t< C 
 
 oo O 
 t< H 
 
 rt VJ-O 00 
 in tJ- to 
 
 CO cl "< m ^t- 
 
 - TO m\o 00 
 ^cor< - 
 
 000000 
 
 C 
 
 CO 
 
 *^ 
 
 
 
 a *- 
 v5 6 
 
 o-, r. c4 op ti iv 
 ^ J- CO H t< -^ 
 
 S ^ S? ^ S? SP 'A 
 
 c^ o\ (> ^ cv 
 
 vO vO \B W v> w^ 
 
 tooo rt ^0 O "i- 
 
 J- CO CO c< M -' 
 
 lo vn ui u-> vn 
 
 40 r o CO r^ 
 vn m ! to 
 
 -1 to\o coio 
 to t - m 
 
 ^^^^^co 
 
 e 
 
 1 e '^ 
 
 N to rl M rt . 
 ++<**'*-** 
 
 f^ t^ r^ (^ 1^ t~ t^ 
 
 Q On4o t^vo .j- 
 
 ^ CO to CO CO CO 
 
 r~ 1^ t^ f^ r^ r^ 
 
 CO 
 CO CO 
 
 t^ 1^ 
 
 o\ 1-^ >n CO 
 M M N r 
 
 t^ 1-^ r^ r~ 
 
 00 n cl Ovo 
 
 (4 i-i 
 
 t-- t~~ t>~ r-- t^ t^ 
 
 to 10 On vn 
 
 inin-<j- * 
 
 t~. t^vO 10 VO IC 
 
 c 
 
 CO 
 
 C "-^ 
 
 CO ^ 
 
 C O 
 60 - . 
 
 CO ^ + ^ irt m vo 
 
 ^ \0 VO vp ^O ^ ^ 
 
 ^o "Mr* 
 to ro ,1- ^ * vf ^ 
 
 t^ t^ ^ t^ t^ t^ 
 
 ^ O vb ro""^riri 
 CO ^ -1- vo "-1 
 
 <*** Tj- ^ "1 
 
 O D 
 r t^ 
 
 'i-^o 00 o 
 
 M M N to 
 
 t^ t- t^ t^ 
 
 N 00 to ON 
 C< O CO CO 
 
 t >! n t^OO On 
 
 CO CO to C^l CO CO 
 
 r^ t^ t^ (^ t^ t^ 
 
 +0 vn vn 
 ^ vn m n 
 
 - -> r) N c< 
 Ni- Nj- ,}- * tJ- ^ 
 
 t^ t^ t^ t^ t>. t^ 
 
 w X c) t to to 
 
 lO NO 10 IC 10 10 
 
 B 
 
 CO 
 
 2I 
 CO 
 
 Vrtirt-, _, , -. 
 
 in m in\o v vo 
 
 60O . 
 CO E 
 
 'J4 
 
 " rt to to <* 
 
 C o o 
 
 irv to t-i cOsO' 5^ 
 Vr> n t) t4 CO 
 
 ?s, 
 
 t-^ vn ro o 
 vn H< cJ 
 
 40 in CO I^ in 
 H to ^ m m 
 
 N t t> f t^ to 
 
 t ON\D to q t^ 
 
 I M r< to ^ '^ 
 
 CO CO to to CO CO 
 
 _ a 
 CO 
 
 
 " " " " " 
 
 - K CO ^ tr,^ 
 
 t-00 C\ O - c> 
 
 H( 
 
 CO Ti- v%tfl t-~9!3 
 
 - kH - l-t M "! 
 
 ON -> M CO ^h 
 
 - N N r< t< t< 
 
 mvo t^oo ON 
 M f< tl H CO 
 
 
 3 (U < 
 
 H u '-^^ 
 
 ^ - 
 
 f)cg. 
 
 "^ i?Noo r-^vO in + 
 CO c) r< c< N c< c< 
 
 00^ is V; tj^ tj^ r; 
 ^ ^ to cT d t^^ CO 
 i- to c> -< !- CO 
 
 00 00 00 00 00 ^ r^ 
 
 CO r ^ onoo 
 
 l-^O Vf-, rj- CO N 1 -. ONOO t^VO 
 
 vr, ,j- 
 
 CO M >- 
 
 jUeg. 
 
 
 
 t^ -^ On rj^ r^ tj^ 
 ON in cS. I? t^ '^ 
 
 A jj- CO 
 
 r> r~-sc> NO \0 
 
 On f4 - m ij"i -. 
 
 coviToo" ON d - 
 
 rj- t ^ CO , 
 
 wn in ^ ^ 4- 
 
 ^ il^ "^O ON C;;- 
 
 ^ S ^ 6^00 
 m CO - vn d 
 
 CO CO CO cl N fl 
 
 m r) 00 CO t^ 
 
 1^^ * CO -T d 
 + 1< t- tl 
 
 e^ 
 
 (6 r^ 
 
 ^0^ 
 
 
 t-^ -^ CO c^ Tj^ 
 
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 1
 
 Book V, 
 
 ASTRONOMY. 
 
 3V 
 
 316. Table of the Abfolute Equation of Time, fitted to 
 each Sign and Degree of the Ecliptic. 
 
 Place of the Apogee S 9. Obliquity of Ecliptic ^3 28'i 
 
 a\r 
 
 
 
 b ' 
 
 it 2 
 
 3 
 
 - 4 
 
 '91 
 
 s 
 
 :^ 6 
 
 1% 7 
 
 / 8VS 9 
 
 ;tio 
 
 h I I 
 
 C 
 
 
 
 + 
 
 
 
 . 
 
 + 
 
 + 
 
 + 
 
 + 
 
 ~ 
 
 
 
 
 
 - !~ + 
 
 + 
 
 + 
 
 w 
 
 
 m 
 
 f. 
 
 m. f. 
 
 m. f. 
 
 Ai. f. 
 
 m. f. 
 
 rr. 
 
 f. 
 
 m. i. 
 
 m. f. 
 
 m. f , 1 m. f. 
 
 m. r. 
 
 TO. i 
 
 
 7 
 
 36 
 
 I 9 
 
 3 51 
 
 I J3 
 
 5 57 
 
 2 
 
 20 
 
 7 38 
 
 IS 31 
 
 13 33 I II 
 
 II 28 
 
 14 19 
 
 0| 
 
 ' 7 
 
 '7 
 
 I 23 
 
 3 47 
 
 I 26 
 
 5 59 
 
 2 
 
 4 
 
 7 5 
 
 15 39 
 
 13 17: 42 
 
 II 45 
 
 14 13 
 
 I 
 
 2 6 
 
 5 
 
 I 36 
 
 3 41 
 
 I 40 
 
 6 
 
 I 
 
 481 8 19 
 
 15 46 
 
 13 ' 12 
 
 IZ I 
 
 14 6 
 
 2 
 
 3 fa 
 
 39 
 
 I 4X 
 
 3 37 
 
 I 53 
 
 b I 
 
 I 
 
 31 
 
 8 40 
 
 15 52 
 
 12 42 + 17 
 
 12 17 
 
 13 59 
 
 3 
 
 4 
 
 6 
 
 20 
 
 2 
 
 3 32 
 
 2 7 
 
 6 I 
 
 r 
 
 14 
 
 9 I 
 
 15 57 12 231 46 
 
 12 32 
 
 13 SI 
 
 4 
 
 f 
 
 6 
 
 I 
 
 2 11 
 
 3 26 
 
 2 20 
 
 6 Q 
 
 
 
 56 
 
 9 21 
 
 1 5 212 4j I 16 
 
 12 46 
 
 13 43 
 
 5 
 
 6 
 
 S 
 
 4* 
 
 2 22 
 
 3 19 
 
 2 33 
 
 5 59 
 
 
 
 3 
 
 9 41 
 
 16 6 11 44J 1 45 
 
 12 59 
 
 13 34 
 
 6 
 
 7 
 
 5 
 
 24 
 
 2 32 
 
 3 12 
 
 2 45 
 
 5 57 
 
 
 
 20J10 I 
 
 16 9 II 231 2 14 
 
 13 12 
 
 13 24 
 
 7 
 
 8 
 
 5 
 
 5 
 
 2 42 
 
 3 4 
 
 2 5 
 
 5 54 
 
 + 
 
 i!io 20 
 
 16 II 11 il 2 43 
 
 '3 24 
 
 13 14 
 
 8 
 
 9 
 
 4 47 
 
 2 51 
 
 2 56 
 
 3 11 
 
 5 51 
 
 
 
 18 
 
 10 39 
 
 16 13 
 
 10 39 3 11 
 
 13 35 
 
 13 3 
 
 9 
 
 10 4 
 
 2 
 
 3 
 
 2 47 
 
 3 23 
 
 5 47 
 
 
 
 37 
 
 10 57 
 
 16 13 
 
 10 16 3 39 
 
 13 45 
 
 12 51 
 
 10 
 
 II 4 
 
 9 
 
 3 
 
 2 38 
 
 3 35 
 
 5 42 
 
 
 
 57 
 
 n 15 
 
 lb 13 
 
 9 53! 4 7 
 
 13 54 
 
 12 39 
 
 XI 
 
 12 
 
 3 
 
 5c 
 
 3 ,6 
 
 2 29 
 
 3 46 
 
 5 37 
 
 I 
 
 17111 33 
 
 16 12 
 
 9 29 4 35 
 
 14 2 
 
 12 27 
 
 12 
 
 '3 
 
 3 
 
 32 
 
 3 23 
 
 2 19 
 
 3 58 
 
 5 31 
 
 1 
 
 38111 51 
 
 16 10 
 
 9552 
 
 14 c^ 
 
 12 14 
 
 14 
 
 3 
 
 13 
 
 3 30 
 
 2 8 
 
 4 9 
 
 5 24 
 
 I 
 
 58:12 8 
 
 16 7 
 
 8 40 5 29 
 
 14 16 
 
 12 
 
 '4 
 
 IS 
 
 2 
 
 5? 
 
 3 36 
 
 I 57 
 
 4 19 
 
 5 17 
 
 2 
 
 19 12 25 
 
 lb 4 
 
 8 14I 5 56 
 
 14 22 
 
 11 46 
 
 '5 
 
 16 
 
 2 
 
 57 
 
 3 41 
 
 I 46 
 
 4 29 
 
 5 9 
 
 2 
 
 40 12 41 
 
 lb 
 
 7 48: 6 22 
 
 14 27 
 
 II 31 
 
 16 
 
 17 
 
 2 
 
 '9 
 
 3 46 
 
 I 35 
 
 4 39 
 
 5 I 
 
 3 
 
 112 57 
 
 IS 55 
 
 7 22! 6 48 
 
 14 31 
 
 II 16 
 
 17 
 
 i 
 
 2 
 
 I 
 
 3 50 
 
 I 2-; 
 
 4 4J< 
 
 4 52 
 
 3 
 
 22|l3 12 
 
 15 49 
 
 6 55'' 7 13 
 
 14 35 
 
 II I 
 
 18 
 19 
 
 19 I 
 
 43 
 
 3 53 
 
 I 1 1 
 
 4 57 
 
 4 43 
 
 3 
 
 44ii3 27 
 
 15 42 6 28 7 37 
 
 14 38 
 
 10 46 
 
 20 I 
 
 26 
 
 3 5^' 
 
 59 
 
 5 "^ 
 
 4 33 
 
 4 
 
 5J13 42 
 
 15 35 6 0' 8 I 
 
 14 40 
 
 10 3c 
 
 20 
 
 21 I 
 
 9 
 
 3 5 
 
 46 
 
 5 13 
 
 4 22 
 
 4 
 
 26 13 56 
 
 15 26 5 32 8 24 
 
 14 41 
 
 10 14 
 
 21 
 
 22 
 
 52 
 
 4 
 
 34 
 
 5 20 
 
 4 II 
 
 4 47|i4 9 
 
 15 17 5 4! 8 47 
 
 14 42 
 
 9 5^ 
 
 22 
 
 23 
 
 3b 
 
 4 I 
 
 21 
 
 5 27 
 
 3 59 
 
 5 
 
 9 14 21 
 
 15 7 4 36 9. 9 
 
 14 41 
 
 9 41 
 
 23 
 
 24I 
 
 20 
 
 4 I 
 
 8 
 
 5 33 
 
 3 46 
 
 5 
 
 3014 33 
 
 14 56 4 8, 9 31 
 
 14 40 
 
 9 24 
 
 -1 
 
 ^s|+ 
 
 4 
 
 4 > 
 
 5 39 
 
 3 33 
 
 5 
 
 52114 44 
 
 14 44 3 39 9 53 
 
 '4 39 
 
 9 6 
 
 2faU 
 
 1 1 
 
 4 c 
 
 19 
 
 5 44 
 
 3 19 
 
 b 
 
 i3|i4 53 
 
 14 31 3 10 10 14 
 
 '4 37 
 
 b 4^, 
 
 26 
 
 27I 
 
 26 
 
 3 50 
 
 31 
 
 5 4^ 
 
 3 4 
 
 b 
 
 35ii5 5 
 
 14 17 2 41 10 34 
 
 '434 
 
 I 3c. 
 
 ?-7 
 
 28J 
 
 40 
 
 3 57 
 
 46 
 
 5 52 
 
 2 50 
 
 b 
 
 56,15 14 
 
 14 3 2 Jijio 53 
 
 .4 30 
 
 8 12 
 
 28 
 
 29, 
 
 53 
 
 3 54 
 
 59 
 
 5 55 
 
 2 35 
 
 7 
 
 17 15 23 
 
 13 48 I 41 II II 
 
 14 25 
 
 7 5'< 
 
 iq 
 
 3c! I 
 
 9 
 
 3 51 
 
 I 13 
 
 5 57 
 
 2 2C 
 
 7 
 
 31I5 31 
 
 13 33! I ii'ii 28 
 
 14 10 
 
 7 3t- 
 
 ^o| 
 
 The equations with +, are to be added to the apparent time, to have 
 the mean time ; thofe with , are to be fubtradlcd from apparent for 
 mean time. 
 
 The preceding mark, whether + or , at the head of any column, be- 
 longs to all equations in that column until the fign changes ; and thof^? 
 columns having two figns at the head, fhew that the preceding fi<,ii 
 changes to the followng fomcwhere in that column, 
 
 II 
 
 317. Ta3L E
 
 J28 
 
 ASTRONOMY. 
 
 Book V. 
 
 317, Table ofCorrcftions for the Middle Time between the 
 Equal Altitades of the Sun. 
 
 
 
 j Latitude 30 D.N. 
 
 Latitude 40 D.N. 
 
 Latitude 50D. N. 
 
 Latitude 
 
 60D.N. 
 
 C 
 
 
 
 1 Hours btc.Obfcr. 
 
 Hours bet, Obfer. 
 
 Hours bet. Obfer 
 
 Hours bet. Obfer. 
 
 CR 
 
 
 
 -i 
 
 1 6| 5l 4 
 
 6| 5|4 
 
 f>\ 5l4 
 
 6| 5|4 
 
 6| 5| 41 6| 5| 4 6| 5| 4 
 
 6| 5l4 
 
 
 
 
 ^r' 
 
 N.decl. 
 
 S. decl. 
 
 N.decl. 
 
 S. decl. 
 
 N. decl. 1 S. decl. 
 
 N.decl. 
 
 S.decl. 
 
 
 
 
 
 fr 
 
 ~ 
 
 rT" 
 
 fr 
 
 ^ 
 
 fT 
 
 n 
 
 fi* 
 
 S 
 
 rt" 
 
 ^ 
 
 7?" 
 
 ~> 
 
 ^1?^ 
 
 <t 
 
 ^ 
 
 " 
 
 ~ 
 
 ^ 
 
 ST" 
 
 ^ 
 
 ^ 
 
 
 
 
 
 H 
 
 1' 
 
 
 
 9 
 
 9 
 
 " 
 
 4 
 
 3 
 
 '3 
 
 H 
 
 3 
 
 13 
 
 20 
 
 
 20 
 
 '9 
 
 71 
 
 28 
 
 78 
 
 27 
 
 7? 
 
 .'^ 
 
 fi 
 
 3 
 
 
 
 
 
 9 
 
 9 
 
 g 
 
 19 
 
 \'^ 
 
 28 
 
 27 
 
 
 
 
 
 I 
 
 9 
 
 9 
 
 8 
 
 9 
 
 9 
 
 9 
 
 '4 
 
 3 
 
 12 
 
 '4 
 
 '3 
 
 13 
 
 '9 
 
 19 
 
 18 
 
 20 
 
 '9 
 
 18 
 
 28 
 
 27 
 
 27 
 
 29 
 
 28 
 
 27 
 
 I 
 
 
 
 2 
 
 9 
 
 8 
 
 8 
 
 9 
 
 9 
 
 9 
 
 13 
 
 '3 
 
 12 
 
 4 
 
 '4 
 
 3 
 
 '9 
 
 18 
 
 18 
 
 20 
 
 i 
 
 18 
 
 28 
 
 *7 
 
 2b 
 
 29 
 
 28 
 
 27 
 
 2 
 
 
 
 ? 
 
 K 
 
 8 
 
 7 
 
 10 
 
 9 
 
 9 
 
 13 
 
 12 
 
 12 
 
 14 
 
 '4 
 
 '3 
 
 18 
 
 18 
 
 17 
 
 20 
 
 '9 
 
 18 
 
 28 
 
 ^7 
 
 2b 
 
 29 
 
 28 
 
 *7 
 
 3 
 
 
 
 
 4 
 
 8 
 
 8 
 
 7 
 
 10 
 
 10 
 
 9 
 
 3 
 
 12 
 
 12 
 
 '4 
 
 4 
 
 '3 
 
 18 
 
 17 
 
 17 
 
 20 
 
 19 
 
 9 
 
 27 
 
 2b 
 
 *S 
 
 29 
 
 29 
 
 28 
 
 4 
 
 
 
 y 
 
 8 
 
 7 
 
 7 
 
 10 
 10 
 
 10 
 10 
 
 10 
 10 
 
 12 
 
 12 
 
 12 
 
 II 
 
 1$ 
 '5 
 
 1 
 14 
 
 1 
 4 
 
 18 
 
 '7 
 
 7 
 
 20 
 20 
 
 12 
 
 2C 
 
 12 
 
 '9 
 
 27 
 27 
 
 2b 
 
 2S 
 
 11 
 24 
 
 29 
 
 29 
 
 28 
 
 5 
 
 
 
 6 8 
 
 7 
 
 6 
 
 11 
 
 II 
 
 7 
 
 17 
 
 16 
 
 29 
 
 29 
 
 28 
 
 6 
 
 
 
 7 
 
 7 
 
 7 
 
 6 
 
 10 
 
 10 
 
 10 
 
 12 
 
 II 
 
 II 
 
 15 
 
 '4 
 
 '4 
 
 17 
 
 lb 
 
 lb 
 
 20 
 
 20 
 
 >9 
 
 2b 
 
 2^ 
 
 24 
 
 29 
 
 28 
 
 27 
 
 7 
 
 
 
 8 
 
 7 
 
 7 
 
 6 
 
 10 
 
 10 
 
 10 
 
 II 
 
 II 
 
 10 
 
 '5 
 
 14 
 
 14 
 
 '7 
 
 lb 
 
 lb 
 
 20 
 
 20 
 
 '9 
 
 2b 
 
 24 
 
 23 
 
 29 
 
 28 
 
 27 
 
 8 
 
 
 
 9 
 
 7 
 
 6 
 
 b 
 
 10 
 
 10 
 
 10 
 
 II 
 
 10 
 
 10 
 
 15 
 
 15 
 
 14 
 
 lb 
 
 '5 
 
 M 
 
 20 
 
 20 
 
 '9 
 
 25 
 
 ^3 
 
 23 
 
 28 
 
 28 
 
 27 
 
 <* 
 
 
 
 10 
 
 7 
 
 6 
 
 S 
 
 10 
 
 10 
 
 10 
 
 II 
 
 10 
 
 10 
 
 '5 
 
 '5 
 
 '4, 
 
 lb 
 
 '5 
 
 15 
 
 20 
 
 10 
 
 '9 
 
 ^4 
 
 23 
 
 22 
 
 28 
 
 28 
 
 27 
 
 10 
 
 
 
 II 
 
 12 
 
 6 
 
 6 
 
 b 
 6 
 
 _5 
 
 5 
 
 10 
 
 10 
 
 10 
 10 
 
 10 
 
 10 
 
 10 
 9 
 
 _9 
 9 
 
 1 
 14 
 
 14 
 
 ,4! 
 
 ^5 
 
 '.5 
 
 11 
 14 
 
 14 
 
 12 
 19 
 
 12 
 
 '9 
 
 18 
 78 
 
 11 
 22 
 
 22 
 21 
 
 22 
 21 
 
 
 
 27 
 27 
 
 27 
 26 
 
 II 
 12 
 
 
 
 10 
 
 10 
 
 
 
 13 
 
 b 
 
 5 
 
 5 
 
 10 
 
 10 
 
 9 
 
 9 
 
 9 
 
 8 
 
 14 
 
 14 
 
 14 
 
 14 
 
 14 
 
 13 
 
 19 
 
 9 
 
 18 
 
 21 
 
 21 
 
 20 
 
 
 2b 
 
 26 
 
 I? 
 
 
 
 '4 
 
 5 
 
 5 
 
 4 
 
 10 
 
 9 
 
 9 
 
 9 
 
 8 
 
 8 
 
 '4 
 
 14 
 
 14 
 
 13 
 
 13 
 
 12 
 
 18 
 
 18 
 
 17 
 
 20 
 
 20 
 
 19 
 
 
 25 
 
 2S 
 
 J4 
 
 
 
 '5 
 
 5 
 
 5 
 
 4 
 
 9 
 
 9 
 
 9 
 
 8 
 
 8 
 
 7 
 
 14 
 
 '4 
 
 13 
 
 13 
 
 12 
 
 12 
 
 18 
 
 18 
 
 17 
 
 19 
 
 '9 
 
 18 
 
 
 
 24 
 
 IS 
 
 
 
 16 
 
 5 
 
 4 
 
 4 
 
 9 
 
 9 
 
 9 
 
 8 
 
 7 
 
 7 
 
 13 
 
 13 
 
 13 
 
 12 
 
 12 
 
 II 
 
 17 
 
 17 
 
 '7 
 
 18 
 
 18 
 
 17 
 
 
 
 23 
 
 16 
 
 
 
 17 
 
 4 
 
 4 
 
 3 
 
 _9 
 9 
 
 9 
 
 8 
 
 8 
 ~8 
 
 7 
 
 7 
 
 6 
 
 12 
 
 12 
 
 12 
 
 II 
 
 II 
 
 10 
 
 lb 
 
 5 
 
 lb 
 
 15 
 
 16 
 
 'S 
 
 17 
 16 
 
 11 
 
 lb 
 
 IS 
 
 
 
 
 
 22 
 21 
 
 17 
 
 
 
 (8 4 
 
 3 
 
 3 
 
 6 
 
 6 
 
 5 
 
 12 
 
 12 
 
 12 
 
 10 
 
 10 
 
 9 
 
 
 
 19 3 
 
 3 
 
 2 
 
 8 
 
 8 
 
 8 
 
 b 
 
 5 
 
 5 
 
 II 
 
 II 
 
 II 
 
 9 
 
 9 
 
 8 
 
 14 
 
 '4 
 
 14 
 
 >5 
 
 14 
 
 3 
 
 
 
 19 
 
 19 
 
 
 
 20 3 
 
 2 
 
 2 
 
 8 
 
 7 
 
 *T 
 
 5 
 
 5 
 
 4 
 
 10 
 
 10 
 
 10 
 
 8 
 
 8 
 
 7 
 
 13 
 
 3 
 
 '3 
 
 14 
 
 3 
 
 12 
 
 
 
 '7 
 
 20 
 
 
 
 21 2 
 
 2 
 
 I 
 
 7 
 
 7 
 
 7 
 
 4 
 
 3 
 
 3 
 
 9 
 
 9 
 
 9 
 
 7 
 
 b 
 
 b 
 
 
 1 1 
 
 II 
 
 12 
 
 II 
 
 10 
 
 
 
 14 
 
 ZI 
 
 
 
 22 I 
 
 I 
 
 
 
 6 
 
 
 
 5 
 
 3 
 
 2 
 
 2 
 
 7 
 
 7 
 
 7 
 
 5 
 
 5 
 
 5 
 
 
 9 
 
 9 
 
 8 
 
 8 
 
 8 
 
 
 
 II 
 
 22 
 
 
 
 23 c 
 
 
 
 
 
 1 
 
 3 
 
 3 
 
 2 
 
 I 
 
 
 
 4 4 
 
 4 
 
 3 
 
 2 
 
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 5 
 
 5 
 
 5 
 
 5 
 
 4 
 
 1 
 
 7 
 
 ^3 
 
 
 
 The correflton is fubtradive from December 22 tojun 
 
 e 2 1 , er 
 
 
 ' 
 
 in afcending figns; and additive from June 21 to 
 
 Decem- 
 
 
 
 her 22, or in defcending figns. 
 
 
 
 
 318. 
 
 A Table of Refractions. 
 
 > 
 
 Rt 
 
 fr. 
 
 S> 
 
 R 
 
 fr. 
 
 > 
 
 Re 
 
 fr. 
 
 > 
 
 Refr. 
 
 > 
 
 R 
 
 =fr. 
 
 > 
 
 R 
 
 sfr. 
 
 > 
 
 Refr. 
 
 
 
 
 
 rf 
 
 
 
 rr 
 
 
 
 i-r 
 
 
 
 -f 
 
 
 
 r-r 
 
 B 
 
 
 rt 
 
 
 a 
 
 1. 
 
 ft' 
 
 n 
 
 3 
 
 n 
 
 r; 
 
 3. 
 
 fT' 
 
 n 
 
 3. 
 
 fT' 
 
 C 
 
 3. 
 
 ^' 
 
 
 
 fT' 
 
 r 
 
 i. ^ 
 
 
 
 
 
 3 
 
 
 rt 
 
 
 
 
 
 
 
 
 
 
 
 
 a ' 
 
 CTQ 
 
 
 
 
 
 
 tp 
 
 
 
 U5 
 
 
 
 
 
 
 
 
 r^ 
 
 
 
 
 33 
 
 oc 
 
 4 
 
 II 
 
 51 
 
 14 
 
 3 
 
 45 
 
 iz6 
 
 
 56 
 
 38 
 
 
 Uilso 
 
 
 
 48 
 
 I62 
 
 ?o 
 
 oi 
 
 3^ 
 
 35 
 
 4i 
 
 10 
 
 48 
 
 5 
 
 3 
 
 3*^ 
 
 127 
 
 
 51 
 
 39 
 
 
 10 
 
 S' 
 
 
 
 46 
 
 f'S 
 
 29 
 
 oi- 
 
 28 
 
 22 
 
 5 
 
 9 
 
 54 
 
 lb 
 
 3 
 
 17 
 
 |28 
 
 
 47 
 
 40 
 
 
 8 
 
 52 
 
 
 
 44 
 
 b4 
 
 28 
 
 o>- 
 
 2b 
 
 21 
 
 si 
 
 9 
 
 8 
 
 17 
 
 3 
 
 4 
 
 29 
 
 
 42 
 
 41 
 
 
 5 
 
 S3 
 
 
 
 43 
 
 |<'5 
 '66 
 
 26 
 
 I 
 
 24 
 
 20 
 
 b 
 
 8 
 
 28 
 
 18 
 
 2 
 
 54 
 
 30 
 
 
 3 
 
 42 
 
 J 
 
 3 
 
 IS4 
 
 
 
 41 
 
 25 
 
 'i 
 
 22 
 
 48 
 
 7 
 
 7 
 
 20 
 
 '9 
 
 2 
 
 44|3> 
 
 
 35 
 
 43 
 
 
 I 
 
 S5 
 
 
 
 40 
 
 167 
 
 24 
 
 I-i 
 
 21 
 
 1 > 
 
 3 
 
 6 
 
 29 
 
 20 
 
 2 
 
 35 
 
 I32 
 
 
 3' 
 
 44 
 
 
 
 59 
 
 56 
 
 
 
 3S 
 
 6S 
 
 23 
 
 'i 
 
 19 
 
 5> 
 
 9 
 
 5 
 
 48 
 
 21 
 
 2 
 
 27 
 
 |33 
 
 
 28 
 
 45 
 
 
 
 57 
 
 57 
 
 
 
 37 
 
 jb9 
 
 22 
 
 2 
 
 18 
 
 35 
 
 10 
 
 5 
 
 15 
 
 22 
 
 2 
 
 20 
 
 34 
 
 
 24 
 
 46 
 
 
 
 55 
 
 5^' 
 
 
 
 35 
 
 70 
 
 21 
 
 2| 
 
 lb 
 
 24 
 
 1 1 
 
 4 
 
 47 
 
 23 
 
 2 
 
 14 
 
 35 
 
 
 2l| 
 
 47 
 
 
 
 S3 
 
 S9 
 
 
 
 34 
 
 71 
 
 19 
 
 3, 
 
 14 
 
 3b 
 
 12 
 
 4 
 
 23 
 
 24 
 
 2 
 
 7 
 
 36 
 
 
 i8j 
 
 148 
 
 
 
 5' 
 
 bo 
 
 
 
 33 
 
 72 
 
 18 
 
 5I |i3 
 
 b 
 
 13 
 
 4 
 
 3 
 
 25 
 
 2 
 
 2 
 
 j37 
 
 
 ibj49 
 
 
 
 49 
 
 ^ 
 
 
 
 32 
 
 11 
 
 17 
 
 319. Of the Sun's Parallax. 
 
 Altitudes 0. 10 20 30'' 40 ;o 60 jo^ 80" 90. 
 Parallax 9988653210. 
 
 END OF BOOK V.
 
 T H 
 
 ELEMENTS 
 
 OF 
 
 NAVIGATION. 
 
 B O O K VI. 
 
 OF GEOGRAPHY. 
 
 SECTION I, 
 
 Defi?jitions and Principles, 
 
 t . ^^ E O G R A P H Y is the art of dcfcribing the figure, magnitude, 
 \JX antl pofitions of the feveral parts of the furface of the Earth. 
 
 2. The Earth is a fpherical or globular figure *, and is ufually called 
 the terraqueous globe. 
 
 3. There are two points on the furfaceof the terraqueous globe, called 
 the Poles of the Earth, which are diametrically oppofite to one an- 
 other ; one is called the north pole .^ and the other, the Joutb pole. 
 
 * For in fhips at lea the firll parts of them that become vifible are ih.e up- 
 per fails; and as they approach nearer, the lower fail^ appear; and fo on until 
 they (hew their hulli. 
 
 Alfo fViips in failing from high capes, or head lands, lofe fight of thofe emi- 
 nences gradually from the lower parts, until the top vanifhes. 
 
 Now as thcfe appearances are the objects of our fenfes in all parts of the 
 Earth, 
 
 Therefore the farface of the Earth muft be convex. 
 
 And this convexity is, at fea, obferved to be every where uniform. 
 
 But a body, the furfacc of which is every where uniforaily convex, is. a globe. 
 
 Therefore the figure of the liarth is globular. 
 
 Voi.. I, A a In
 
 330 GEOGRAPHY. Book VL 
 
 In order to Jcfcribe the pofitions of places, Geographers have found It 
 twccflary to imagine certain circles drawn on the furface of the Earth, t> 
 vvhjch they have given the names of Equator, Meridian, Horizon, Pja^ 
 raliels of latitude, &c. 
 
 4. The EqtTAl-OR is a great circle on the Earth, equally diftant frin* 
 each pole, dividing the terraqueous globe into two equal parts ; one called 
 the northern hemifphere, in which is the north pole ; and the other, con- 
 taining the fouth pole, is called the Jsuthern hevtifphers. 
 
 5. KIeridians are imaginary circles on the Earth pafllng through 
 both the poles, and cutting the equator at right angles. 
 
 Every point on the furface of the Earth has its proper meridian. 
 
 "6. Latitude is the' drftance of a- place from the equator, reckoned 
 in degrees and parts of degrees on a mei;idia4i. 
 
 On the north fide of the equator it is north latitude ^ and on -the foutb 
 fide it is fouth latitude. 
 
 As latitude begins at the equator, where it is hothhig ; fo it ends at the 
 poles, where the latitude is greateft, or 90 degrees. 
 
 7. Parallels of Latitude are circles parallel to the equator. 
 Every place on the Earth has its parallel of latitude. 
 Difference of Latitude is an arc of a meridian, or the Icafl: dif- 
 
 tance of the parallels of latitude o^ two places ; fhewing how far one of 
 shcm is to the northward, or fouthward, of the other. 
 The difference of latitude can never exceed 180 degrees. 
 
 8. In north latitudes, if about the middle of the months of March and 
 Scptcn.bcr a perfon looks towards the Sun at noon, the fouth is before- 
 h'.m, tbc north behind, the weft on the right hand, and the eaft on the 
 left : and in fouth latitudes,, if the face is turned toward the Sun at the 
 fame times, the north is before,, the fouth behind, the eaft to the right, 
 and the weft on the left. 
 
 In latitudes greater than 23I degrees,.thefe poll dons, found at noon, will 
 hold good on any day of the year. 
 
 9. Lokgitude of any place on the Earth is exprefled by an arc of 
 the equator, fl^ewing the eaft or weft diftance of the meridian of that 
 place from fome fixed meridian, where longitude is reckoned to begin. 
 
 10. Difference of Longitude is an arc of the equator, inter- 
 cepted between the meridians of two places, fliewing how far one of them 
 is to the eaft ward, or weft ward, of the other. 
 
 As longitude begins at the merldan of fonie place, and is counted from- 
 thence both eaftvvard and weftward, till they meet at the fame meridian, 
 on the cppofite point of the equator > therefore the difference of longitude 
 can nc\er exceed 180 degrees. 
 
 11. When two places have latitudes both north, or both fouth ; or have 
 longitudes both eaft, or both weft, they are faid to be of the fame, or of 
 like name : but when one has north latitude, and the other fouth; or if 
 one lias eaft longitude, and the other weft, then they are faid to have con- 
 trary, or difterent, or unlike names. 
 
 12. The Horizon is that apparent circle which limits, or bounds^ 
 the view of a fpectator on the fea, or on an extended plain ; the eye of the 
 fpcclator being always fuppofed in the center of his horizon,. 
 
 Whm
 
 Book VL G E O G R A I^ H T. 331 
 
 t 
 
 When the Planets or Stars come above the eaftern part of the horizon, 
 they are faid to rife ; and when they defcend below the weftern part, they 
 are faid to fet. 
 
 When a Ihip is under the equator, both the poles appear in the horizon ; 
 and in proportion as (he falls towards either, or increafes her latitude, 
 that pole is feen proportionally higher above the horizon, and the other 
 difappears as much : but when a fhip is failing towards the equator, or 
 decreafes her latitude, fhe depreiles the elevated pole ^ that is, its difl:ance 
 from the horizon decreafes. 
 
 Of the div'ifton of the Earth into Zones, 
 
 13. A ZcNE is a broad fpace on the Earth, included between two pa- 
 rallels of latitude. 
 
 There are five zones : namely, one Torrid^ two Frigid^ and two Tem^ 
 perate ; thefe names arife from the degree of heat or cold, to which their 
 fituations are liable. 
 
 14. The Torrid Zone is that portion of the Earth, over every p^rt 
 of which the Sun is perpendicular at one time of the year or other. 
 
 This zone is about 47 degrees in breadth, extending to about 23I de- 
 grees on each fide of the equator ; the parallel of latitude terminating the 
 limits in the northern hemifphere, is called the Tropic of Cancer; and in 
 the fouthern hemifphere, the limiting parallel is called the Tropic of Ca- 
 pricorn. 
 
 15. The Frigid Zones are thofe regions about the poles, where the 
 Sun does not rife for fome days, nor fet for feme days, of the year. 
 
 Thefe zones extend round the poles to the diRance of about 23! de- 
 grees : that in the northern hemifphere is called the north frigid zone, 
 and is bounded by a parallel of latitude, called the Ariiic polar circle : and 
 the other, in the fouthern hemifphere, the foutb frigid zone ; the parallel 
 of latitude bounding it, being called the Antarctic polar circle. 
 
 16. The Temperate Zones are the fpaces between the Torrid 
 and the Frigid zones. 
 
 Of the divifion of the Earth by CUfnates. ^ 
 
 17. A Climate, in a geographical fcnfe, is that fpace of the Earth 
 contained between two parallels of latitude, wbcn the difference between 
 the longed day in each parallel is half an hour.^^ 
 
 Thefe climaf:es are narrower the farther they are from .|,^e equator; 
 therefore, fuppofinj^ the equator to be the beginning of the Hrft climate, 
 the polar circle v/ill be the end of the 24th climate ; for afterwards the 
 longeft day does dot increafc by half hours, but by days and months. 
 
 Ar. 2 SECTION
 
 3J2 GEOGRAPHY. Book VI. 
 
 
 
 SECTION II. 
 
 Of the natural divijion of the Earth, 
 
 18. By the natural divifion of the Earth is ineant the parts on its fur* 
 fice formed by nature j fuch as ContlnoitSi Oceans^ IJIands^ Seasy Rivers^ 
 Movniains^ he. 
 
 The furfacc of the Earth is naturally divided into Land and Water; 
 
 Land is divided into 
 
 Water is divided into 
 
 \t 
 
 Continents. 
 
 4. 
 
 Ifthmufes, 
 
 Iflands. 
 
 5- 
 
 Promontaries. 
 
 Peninfulas. 
 
 6. 
 
 Mountains. ' 
 
 Oceans. 
 
 4. 
 
 Straits. 
 
 Seas. 
 
 5- 
 
 Lakes. 
 
 Gulfs. 
 
 6. 
 
 Rivers. 
 
 ig. A Continent, or, as it is frequently called, the main land, is a 
 rery large track, comprehending feveral contiguous Countries, King- 
 doms, and Statts. 
 
 20. An Ocean is a vaft colle3Lion of fait water, feparating the conti- 
 ricnts from one another. 
 
 21. Aii li^i.AND is apart of dry land, furrounded with water. 
 
 22. A Sea is a branch of the Ocean, flowing between fome parts of 
 the Continent, or feparating an Uland from the Continent. 
 
 23. A PENINSULA is a part of dry land encompalled by water, except 
 a narrow neck which joins it to fome other land. 
 
 24. An Isthmus is the neck joining the peninfula to the adjacent 
 iand, and forms the pafl'age between them. 
 
 25. A Mountain is a part of the land more elevated than the adjacent 
 country, and to be {^-cn at a greater diltance than the neighbouring lower 
 lands. 
 
 26. A Prctviontory is a mountain ftretihing itfelf into the fea j the 
 extremity of which is called a Cape^ or Head-land. 
 
 27. A Hill is a fmall kind of mountain : A Cliff is a fteep fhore, hill, 
 *^r mountain : And Recks are great llones, riling like hills above the dry 
 land, or abcive the bottom of the fea. 
 
 28. A (3UI K, or Bay, is a part of the Ocean, or Sea, contained be- 
 tween two^orts : and is every where environed with land, except at its 
 entrance, vrfiere it conimunicates with other Bays, Seas, or Oceans. 
 
 29. A Strait is a narrow pall'age, by which there is a communication 
 between a Gulf and its neighbouring fea, or which joins one part of the 
 ic:!, or ocean, v/ith another. 
 
 30. A Lake is a cclledKion of Waters contained in fome hollow or 
 c.i\ ity, in an inland place, of a large extent, find ev ery where furrounded 
 with land, having no vifible communicunon with the Ocean. 
 
 31. Rivers are dreams of Water, flowiiig chiefly from the Moun- 
 t.iin?, and running in long narrow channels, or cavities, through the land, 
 'till \)hc)' fall into the fea, or into other rivers, which at lafl run into the 
 
 32. There
 
 Book VI. G E O G R A P H Y. 333 
 
 32. There are generally reckoned four Continents, namely, Europe, 
 Asia, Africa^ and America. ' 
 
 To thefc may be added the Terra arSiica, or nothern continent, and 
 the Terra antarilica, or lands detached from Jfia, towards the fouth. 
 
 The continent oi America is ufually divided into two parts, called North 
 and South America ; they are joined together by the Ijihmtts of D^r'ten^ 
 Alfo the continents of Afia and Africa are joined together by the IjVomui 
 of Sues. 
 
 The Terra arnica, Europe^ and A^ay lie all within the northern hemi- 
 fphcre ; and alfo part of Africa and America : 'J'he other parts of thc-fe 
 two continents, together with the Tprra antar^ica^ lie in the fouthtrh 
 hemifphere. 
 
 33. There are five Oceans, namely, the Northern', the Atlan- 
 tic, the Pacific, the Indian, and the Southern. 
 
 The Atlantic ocean is ufually divided into two parts, one called the' 
 nq,rth Atlantic ocean^ and the other xht fouth Atlantic, or Ethiopic ocean. 
 
 The Northern ocean ftretches to the northward of Europe, Au.i, and 
 America, towards the north pole. 
 
 The Atlantic ocean lies between the continents of Europe and Africa 
 on the eaft, and America on the weft. 
 
 That part of the north Atlantic ocean, lying between Europe and 
 America, is frequently called the Wefiern ocean. 
 
 The Pacific ocean, or, as it is fometimes called, the South Sea, is bounded 
 by the weftcrn and north-weft fliores of America, and by the eaftern and 
 north-eaft fliorcs of Afia, 
 
 The Indian ocean wafties the fhorcs of the eaftern coafts of Africa, and 
 the fouth of Afia ; and is bounded on the eaft by the Indian iflands, New 
 Holland, and New Zeeland. 
 
 The Scuthern ocean extends to the fouthward of Africa and Americii 
 tov/ards the fouth pole. 
 
 The northern and fouthern continents not being fufEciently known to 
 Geographers, all that nct^d be faid of them is, that the Terra Ardica, or 
 land to the northward of Hudton's Bay and Greenland, is in general too 
 cold for the refidence of mankind and that the lands formerly fuppofed 
 to be parts of the fouthern continent, are found to be very'largo ifiands ." 
 viz. New Zeland is much Ir.rgcr than Great Britain, and has a ftrait di- 
 viding it into two iflands. New Holland is an ifland as large as Europe, 
 New Guinea is a very iarj^e ifland ;- and New Britain is a cjlufter of largo 
 and I'mail iAands, and ae thought by fume tQ be the iflands hitherto culled 
 the Solomon's iflands, 
 
 A ;i 1 SECTION
 
 334 
 
 GEOGRAPHY. 
 
 Book Vli 
 
 SECTION III. 
 Of the Political divijton of the Earth. 
 
 34. By the political divifion of the Earth is meant the clifFcrent 
 Countries, Empires, Kingdoms, States, -and otllcr denominations cfta- 
 blifhed by men, either from the ambition of tyrants, or for the fake of 
 good government, 
 
 OF EUROPE. 
 Europe is bounded on the north by the northern, or frozen, ocean ; on 
 the eaft by Afia ; on the fouth by the Mediterranean Sea, feparating 
 Europe from Africa ; and by the north Atlantic, or weftern ocean, on 
 the weft. It lies between the latitudes of 36 and 72 degrees of north 
 latitude ; and between the longitudes of 10 degrees weft, and 65 degrees 
 caft from London ; is about 3000 miles long, reckoning from the N. . 
 tp the S. W. and about 2500 miles broad. 
 
 35. The countries, their pofition, with regard to the middle parts of 
 Europe, the chief cities, principal rivers, with their courfes, and the moft 
 noted mountains, and what quarter of the country they arc in, are ex- 
 hibited in the following table ; where E. ftands for empire, K. for king- 
 dom, R. for republic, Nd. for northward, y<:. 
 
 Countries. 
 
 Pofuion. 
 
 Chief Cities. 
 
 Rivers. 
 
 Courfe. 
 
 Mountains. 
 
 E. Turky 
 
 s. p. 
 
 Conftantii^oplf 
 
 Danube 
 
 E. 
 
 Argentum Nd. 
 
 K.Poland 
 
 Mid. 
 
 Warfaw 
 
 Viftula 
 
 N. N. W. 
 
 Carpathian Sd. 
 
 . Mufcovy 1 
 E. Kuflia J 
 
 N. E. 
 
 ( Mofcow 
 
 '\ Peterlburg 
 
 Volga 
 
 E. to S. 
 
 Boglowy Sd. 
 
 Niaper 
 
 s. 
 
 Riphean Wd. 
 
 K. Sweden 
 
 N. 
 
 Stockholm 
 
 Dalecarlia 
 
 E. 
 
 Dofrine Wd. 
 
 K. Norway 
 
 N. N.W. 
 
 Bergen 
 
 Glama 
 
 S. 
 
 Dofrine Ed. 
 
 K. Denmark 
 
 N. W. 
 
 Copenhagen 
 
 Eyder 
 
 W. 
 
 
 K. Hungary 
 
 Mid. 
 
 Pre fbu rg 
 
 Danube 
 
 S. E. 
 
 CarpathianNd. 
 
 E. Germany 
 
 Mid. 
 
 Vienna 
 
 Danube 
 
 E. 
 
 Alps Sd. 
 
 Italy 
 
 S. 
 
 Rome 
 
 fPo 
 
 \ Tyber 
 
 E. 
 S. 
 
 Alps Nd. 
 ApennineMid. 
 
 R. Switzerland 
 
 Mid. 
 
 Bern 
 
 Rkine 
 
 W. 
 
 Alps Sd. 
 
 Netherlar.di 
 
 w. 
 
 Bruffels 
 
 Maefe 
 
 N. 
 
 
 R. Holland 
 
 w. 
 
 dmfterdam 
 
 Rhine 
 
 N. N. W. 
 
 
 K. France 
 
 w. 
 
 Paris 
 
 t Loire 
 } Rhone 
 
 N. to W. 
 S. 
 
 Pyrenees S. W. 
 Alps Ed. 
 
 X. Spain 
 
 s. w. 
 
 Madrid 
 
 Tagus 
 
 W. 
 
 Pyrenees N. E. 
 
 K. Portugal 
 
 s. w. 
 
 Lifpon 
 
 Tagus 
 
 W. 
 
 C. Rocca W. 
 
 K. England 
 
 w. 
 
 London 
 
 Thames 
 
 E. 
 
 Malvern N.W. 
 
 K.- Scotland 
 
 vv. 
 
 Edinburgh 
 
 Forth 
 
 E, 
 
 Grampian Nd. 
 
 K^. Irelaad 
 
 w. 
 
 Dublin 
 
 Shannon 
 
 S. W. 
 
 KnockpatiicW 
 
 There arc in Europe four Kingdoms befide thofe enumerated above j 
 but they are contained in the forenamed Countries. 
 
 The Kingdom of Pruflla, which is part of Poland j the King's refi- 
 dence is at iJerlin, a city in Germany^ 
 
 The
 
 Book VI. 
 
 GEOGRAPHY. 
 
 33i 
 
 The Kingdom of Bohemia, a part of Germany; the chief city is Prague. 
 
 The Kingdom of Sardinia, an Italian ifland ; the King refides at Tu- 
 rin, a city in Italy. 
 
 T'he Kingdom of the Sicilies, appending to Italy ; the King refides at 
 Naples, a city in Italy. 
 
 In fome of the forenamed countries are feveral dominions independent 
 one of the other ; particularly in Germany and Italy. 
 
 The principal ftates in Germany are the following I2 j where D. ftands 
 for duchy, El. for elediorate, P. for principality. 
 
 States j D. Auftria J K. Bohemia | EI. Bavaria J El. Brandenburg 
 Ch. cities Vienna Pratjue Munich Berlin 
 
 States 
 Ch. cities 
 
 El. Saxony 
 Drefden 
 
 El. Hanover 
 Hanover 
 
 El. Palatine 
 Manheim 
 
 El. Mentz, 
 
 Alentz 
 
 States 
 Ch. cities 
 
 El. Triers 
 Triers 
 
 El. Cologne 
 Cologne 
 
 P.HefTeCafl'el 
 CafTel 
 
 D.Wurtembupg 
 
 Stutirard 
 
 The principal ftatcs in Italy are the following 12. 
 
 States I D. Savoy 
 Ch. cities Chamberry 
 
 P. Piedmont 
 Turin 
 
 D. Milanefe 
 Milan 
 
 D. Parmefan 
 Parma 
 
 States 
 Cli. cities 
 
 D.Modenefe 
 Mode n a 
 
 D. Mantuan 
 Mantua 
 
 R. Venice 
 Venice 
 
 R. Genoa 
 Genoa 
 
 States j D. Tufcany 
 Ch. cities ] Florence 
 
 Patriarchate 
 Rome 
 
 R. Lucca 
 
 Lucca 
 
 K. Naples 
 Naples 
 
 36. The principal SeaSj Gulfs and Bays in Europe., are 
 
 The Mediterranean Sea, having Europe on the N. and Africa on the S. 
 
 The Adriatic Sea^ between Italy and Turky. 
 
 The Euxine^ or Black Sea, in Turky, between Europe and Afia. 
 
 The White Sea, in the N. N. W. parts of Mufcovy. 
 
 The Baltic Sea, between Sweden, Denmark, and Poland. 
 
 The German Ocean, or Sea, between Germany and Britain,. 
 
 The Englijh dhmnncl, between England and France. 
 
 St. George's Channel, between Britain and Ireland. 
 
 The Bay of Bifcay, formed between France and Spain. 
 
 7'he Gulf of B'jthnia, in the N. E. parts of Sweden. 
 
 The Gulf of Finland, between Sweden and RulTia. 
 
 The Gulf of Venice, the N. W. end of the Adriatic Sea 
 
 37. The principal If ands in Europe, are 
 
 The Britijl) Ifes, viz. Great Britain, Ireland, Orkneys, and Wcftern 
 Ifles. 
 
 The Spanif Ifes ; Majorca, Minorca, Ivica, in the Medit. Sea. 
 Turkijh Ifes \ Sicily, Sardinia, Corfica, Lipari, in the Medit. Sea. 
 Italian Ijles ; Candia, Archipelago Ifles, in the iVIedit. Sea. 
 Swedijlj Ifles i Gothl.uid, Oeland, Alan, Rugen, inthc Baltic Sea. 
 Danift) Ifes ; Zeland in the Baltic Sea ; and Iceland, Faro Ifles, E. and 
 
 W. Greenlands in the Northern ocean. 
 A'mOi es Ifes in the Atlantic ocean, belonging to Portugal. 
 
 A a 4 OF
 
 33^ 
 
 GEOGRAPHY. 
 
 Book VI. 
 
 O F A S 1 A. 
 
 38. The continent of Afia is bounded on the north by the Northern 
 or frozen ocean, on the eaft by the Pacific ocean, on the fouth by the In- 
 dian ocean, and by Africa and Europe on the weft. , It lies, including its 
 Iflands, between the latitudes of lO degrees fouth, and yi degrees north ; 
 and is between the longitudes of 25 and 148 degrees eaft of London ; its 
 length, exclufive of the ifles, being about 4800 miles, and breadth aboiit 
 4300 miles. 
 
 39. The pofitions and names of the chief countries, cities^ rivers, and 
 mouiitains, are contained in the following table. 
 
 Countries. 
 
 Pofition. 
 
 ChiefCities. 
 
 Rivers. 
 
 Courfe. 
 
 Mountains. 
 
 
 
 f Pekn 
 
 Yellow R. 
 
 S. to E. 
 
 Ottorocoran Nd. 
 
 E. China 
 
 S.E. 
 
 < Nr.okin 
 
 Kiam 
 
 E 
 
 Damafian Wd. 
 
 
 
 (_ Canton 
 
 fa 
 
 S.E. 
 
 
 1^. Korea 
 
 E. 
 
 Kingkitau 
 
 Yaiu 
 
 S. 
 
 Shanalin 
 
 ChinefeTartar)' 
 
 E. 
 
 Chynian 
 
 Yamour 
 
 N.E.toE. 
 
 Fongwanflian 
 
 Mengalia 
 
 Mid. 
 
 Kudak 
 
 Yellow R. 
 
 N. toW. 
 
 
 K. Thibet 
 
 Mid. 
 
 L^flterdu 
 
 Yaru 
 
 E. 
 
 Kantes 
 
 Bukharia, or 
 
 tlfix-'cs 
 
 I^Mid. 
 Mid. 
 
 Samarkand 
 
 Amu 
 
 N.W. 
 
 Belurlag 
 
 Karazm 
 
 Urjenz 
 
 Amu 
 
 S. W. 
 
 Irder 
 
 Kalmucks 
 
 Mid. 
 
 
 fekis 
 
 
 Pubratubufluk 
 
 E. Siberia 
 
 N. 
 
 j Tobolfki 
 ( Aftracan 
 
 Oby 
 jenifka 
 
 N. 
 N.N.W. 
 
 Stolp 
 
 E. Turk/ 
 
 w. 
 
 Smryna 
 
 tuphrate; 
 
 S. E. 
 
 Taurus ' 
 
 K. Syria 
 
 . W. 
 
 Aleppo 
 
 Euphrates 
 
 S. 
 
 Lebanon 
 
 Arabia 
 
 s. w. 
 
 Medina 
 
 [Euphrates 
 
 S. E. 
 
 Gabel el arcd 
 
 E. Perfia 
 
 s. 
 
 Ifpahan 
 
 f Oxus 
 1 A raxes 
 
 W. 
 
 s. VV. 
 
 Caucafus 
 Taurus 
 
 India Weft of 
 
 : s. 
 
 Agra 
 
 Indus 
 
 s. w. 
 
 Caucafus 
 
 the Ganges 
 
 Delli 
 
 Ganges 
 
 s. w. 
 
 Bal agate 
 
 
 f 
 
 Ava 
 
 Domea 
 
 s. 
 
 J 
 
 India Eaft of 
 the Ganges 
 
 Is. 
 
 Pegu 
 
 Siam 
 
 Mccon 
 Menan 
 
 s. 
 s. 
 
 \ Damafcene 
 
 
 L 
 
 Cambodia 
 
 Ava 
 
 s. 
 
 i 
 
 Son-ic of thefc countries contain feveral others 
 
 Afiatic Turky contains 
 Countries 
 
 Ch. cities 
 
 Georgia N. | Turcomania E. 
 
 I or Armenia 
 Tcfiis I Erzerum 
 
 Countries j EyracaS.E. j Arabia defeitS. 
 Ch. cities | Bagdat 
 
 Curdiftan E. 
 
 or AfTyria 
 
 Betlis 
 
 Natolia W. 
 Smyrna 
 
 Diarbec E. 
 Moufol 
 
 Syria W. 
 Aleppo 
 
 Indi4
 
 Book VI. 
 
 GEOGRAPHY 
 
 \SJ 
 
 India weft of the Ganges contains 
 
 Countries I Indoftan N. 
 
 Northern 
 Parts 
 
 Ch. pities Delli 
 
 Cambaya S. W. 
 or Guzarat 
 Surat 
 
 BengalaS.E. 
 Patna 
 
 Malabar 
 Coaft 
 
 Countries 
 
 Ch. cities 
 
 Decan 
 or Vifapour 
 Goa 
 
 Bjfnagar W. 
 
 or Carnate 
 CalcutandCochin 
 
 Coromandel 
 Coaft 
 
 Countries 
 
 Ch. cities 
 
 Bifnagar 
 CarnatCjE.fide 
 Madras 
 
 K. Goiconda 
 Golcorda 
 
 K. Orixa 
 Orixa 
 
 India eaft of the Ganges contains 
 CountrieslK. Ava N. W.iK. Pegu 
 Ch. cities) Ava | Pegu 
 
 WlK.SiamS.IK. Malacca 
 Si am i Malacca 
 
 CountrieslK. Cambodia S.jK. Cochin China E.jK. Laos N. 
 Ch. cities! Cambodia Thoanoa | Lanchan 
 
 K. Tonquin N. E. 
 Keccio 
 
 40. The principal Seas, Gulfs and Bays in Afia^ are 
 
 Cafpian Sea^ quite furrounded by Siberia on the north, Korazm eaft, by 
 
 Perfia on the fouth, and by Georgia on the weft, 
 Korean Sea^ between Korea and the iflands of Japan. 
 I'i/Io'iv Sea, between China and the Japan ifles. 
 Gulf of Cochin China, on the borders of Tonquin and Cochin China, 
 Bay of Siam, formed by the countries of Siam and Malacca. 
 Bay of Bengal, between India eaft, and India weft of the Ganges. 
 Gulf of Perfia, having Perfia on tl^e N. E. and Arabia on the S. W. 
 
 41. The principal Iflands helonging to ^fa, are 
 Ladrone, or Marian IJlcs, whofe chief iiland is Guam. 
 
 jfles I Japan 
 ities I Jeddo 
 
 Of . rn i Ch. jfles 
 Japan Jfles < ^>,, 
 J ^ J I Ch. citu 
 
 Bongo 
 Bongo 
 
 Tonfii 
 Ton fa 
 
 Pi.;}:^^:,..r f Ch. ifles Luconia 
 
 '^ I Ch. cities Manil a 
 
 Mindanao 
 Mindanao 
 
 Saniar 
 
 Ch. iiles 
 ies 
 
 Chtnefc Ifles \ ^^^ ^^^ 
 Moluccos 
 
 Formofa 
 Taywanfu 
 
 Ainan 
 T'an 
 
 Makao 
 Malcau 
 
 5 Ch. if 
 1 Ch. c 
 
 Acs 
 ities 
 
 Celebes 
 MacafTcr 
 
 Gilolo 
 
 Gilolo 
 
 Ceram 
 Ambay 
 
 ^ J m S ^h. ifles 
 ^unda,fl?s I Ch. cities 
 
 Borneo 
 Banjar 
 
 I Sumatra 
 1 Achin 
 
 Java 
 Batavia 
 
 The Andaman Ijles to the weft of Siam. 
 
 Niobar Iflands weft of Malacca. 
 
 Maldive Iflands to the S. W. of Bifnagar. 
 The Ifland of Ceylon S. E. of Bifnagar; ihc chigf ci.ty is Candy, or 
 Candy L'ta, 
 
 OF
 
 51* 
 
 GEOGRAPHY. 
 OF AFRICA. 
 
 Book VL 
 
 4a. This large continent is a peninfula, joined to Afia by the Ifthmus 
 of Suez. On the N. E. it is feparatcd from Afia by the Red Sea j it has 
 the Indian Ocean on the eaft, the Southern on the louth, the Atlantic on 
 the weft, and the Mediterranean Sea on the north, which feparates it 
 from Europe. luJs fituated between the latitudes of 37 degrees N. and 
 35 degrees S ; and between the longitude of 18 degrees VV. and 50 de- 
 grees E. from London ; is about 4300 miles long, and 4200 miles broad. 
 
 43. The pofitions and names of the chief countries, cities, rivers, and 
 mountains, are contained in the following table. 
 
 Ccuntries. 
 
 Pofiticn. Chief Cities. 
 
 Rivers. 
 
 Courfe. 
 
 Mountains. 
 
 E. Morocco 
 
 N. W. 
 
 Fez 
 
 Mulvia 
 
 N. 
 
 Atlas 
 
 K Algiers 
 
 N. 
 
 Algiers 
 
 SaiFran 
 
 N. 
 
 Atlas 
 
 K. 1 unis 
 
 N. 
 
 Tunis ' 
 
 Megrada 
 
 N. 
 
 Atlas 
 
 K. Tripoli 
 
 N. 
 
 Tripoli 
 
 Salines 
 
 N. E. 
 
 Atlas 
 
 K. Barca 
 
 N. 
 
 Docra 
 
 
 
 Mcies 
 
 K. Egypt 
 
 N.E. 
 
 Cairo 
 
 Nile 
 
 N. 
 
 Gianadel 
 
 E. Abyflinia 7 
 or Ethiopia J 
 
 E. 
 
 Ambamarjani 
 
 Nile 
 
 N. 
 
 
 Ajan 
 
 E. 
 
 Adea 
 
 Madadoxa 
 
 S. 
 
 
 Zanguebar 
 
 E. 
 
 Melinda 
 
 Cuama 
 
 E.S.E. 
 
 
 Sofala 
 
 S. E. 
 
 Sofala 
 
 Amara 
 
 E. 
 
 Amara 
 
 Terra de Natal 
 
 S. E. 
 
 Natal 
 
 St. Efprit 
 
 E. 
 
 Amara 
 
 Cafraria 
 
 s. 
 
 Care Town 
 
 St.Chriftopher 
 
 E. 
 
 Table 
 
 Mataman 
 
 3. S.W. 
 
 
 Angri 
 
 W. 
 
 Sunda 
 
 K. Renguela 
 
 S.S.W. 
 
 Bengucla 
 
 Negros 
 
 W. 
 
 Sunda 
 
 K. Anf;ola 
 
 s. w. 
 
 l.oando 
 
 Coan7a - 
 
 w. 
 
 Sunda 
 
 K. Congo 
 
 S. W. 
 
 St. Salvador 
 
 Zaara 
 
 s. w. 
 
 Sunda 
 
 K. Loango 
 
 S. W. 
 
 Loango 
 
 Zette 
 
 s. w. 
 
 St. Efprit 
 
 Biafara 
 
 s. w. 
 
 Biafara 
 
 Camerones 
 
 s. w. 
 
 St. Eiprit 
 
 K. Benin 
 
 s. w. 
 
 Benin 
 
 Formofa 
 
 s. w. 
 
 
 Guinea 
 
 s. w. 
 
 Cape Coaft 
 
 Volta 
 
 s. 
 
 Sierra Leon 
 
 Mandinga 
 
 w. 
 
 Janaes Fort 
 
 Gambia 
 
 w. 
 
 C. Vc;d 
 
 Sanhaga 
 
 w. 
 
 Sanhaga 
 
 Senegal 
 
 N. W. 
 
 
 Biidulgerid 
 
 Mid.N. 
 
 Dara 
 
 Dara 
 
 s. 
 
 Atla s 
 
 Zara 
 
 Mid. 
 
 Zucnzega 
 
 Nubia 
 
 E. 
 
 
 Nubia 
 
 Mid. 
 
 Nubia 
 
 Nubia 
 
 E. 
 
 
 Negroland 
 
 Mid. 
 
 Toir.bute 
 
 Niger 
 
 W. 
 
 
 Ethiopia inter 
 
 Mid. 
 
 Chaxunio 
 
 Niger 
 
 W. 
 
 Luna 
 
 MonoDHipi 
 
 Mid. S. 
 
 Merango 
 
 Cuama 
 
 E. S. E 
 
 Luna 
 
 E.Monomotapa 
 
 Mid. S. 
 
 Morgar 
 
 Amara 
 
 S. E. 
 
 Amara 
 
 Many parts of the co?A^ of Africa are fubjecl: to the European nations : 
 Thus the Kingdoms of Algiers, Tunis, Tripoli, Barca, and Egypt, are 
 either fubject to the Ottomauj oi Tur^ifli empire, or acknowledge them- 
 fclves under its protection. 
 
 AbyiTuiia
 
 Book VI. GEO G Jl A P ,H :^, 
 
 339 
 
 Abyfllnia is governed by its own Emperor. 
 
 Ajan or Anian is peopled by a few wild Arabs. 
 
 In Zanguebar and Sofala, the Portuguele have many black Princes 
 tributary to them. . 
 
 Cafraria, or the country 6f the Hottentots, belongs to the Dutch. 
 
 The fea coafts of Guinea are ufually diftinguifhed by the names of the 
 S/ave Co^Jfy Geld Coaji^ Ivory Caaft^ Grain Coaji, and Sierra Leon. . 
 
 The Englifh, Dutch, French, Portuguefe, and others, have feveral 
 fettlements along thcfe coafts, and even many miles up the country, par- 
 ticularly the Englifh on the rivers Gambia and Senegal. 
 
 In the general table the countries are taken in a very large fenfe ; for 
 many of them contain a great number of ftates independent one of the 
 Other, the particulars of v^-hich are not known to Geographers. 
 
 44. The principal Seas, Gulfs, and Bays in Africa, are 
 
 The Red Sea, between Africa and Afia : It wafhes the coaft of Ara- 
 bia on the Afiatic fide, and the coafts of Egypt and 
 Abyllinia on the African fide. 
 
 Mofamhiqne Sea, between Africa and tlie ifland of Madagafcar caftward. 
 
 Saldanna Bay in Cafraria, on the Ethiopic Ocean. * 
 
 Bight of Benin on the coaft of Guinea, in the Ethiopic Ocean, 
 
 45. The principal African Ijlands, are 
 
 Madeira ijles 
 Canary ijles 
 . CVerd ijles 
 Ethiopian ijles 
 Komora ijles 
 Sekotora ijles 
 Almirante ijles 
 
 The ifland of Madagafcar, one of the largeft in the world, lies in the 
 Indian Ocean : It is divided into a multitude of little ftates ; feme of 
 them formed by the European privateers, and their fucccflbrs defcended 
 from a mixture with the natives. 
 
 The iflands of Bourbon and Mauritius lie in the Indian Ocean, to the 
 eaft of Madagafcar : thefe belong to the PVench. 
 
 The Madeiras, and Cape de Verd Ifles, belong to the Portuguefe. 
 
 The Canary Ifles to Spain \ St. Helena to England. 
 
 . Chief ifle. 
 
 Chief Town. 
 
 Situation. 
 
 Madeira 
 
 Funchal 
 
 N. Atlantic Ocean 
 
 Canaria 
 
 Pal ma 
 
 N. Atlantic Ocean 
 
 St. Jago 
 
 St. Jago 
 
 N. Atlantic Ocean 
 
 St. Helena 
 
 
 Ethiopic Ocean 
 
 Johanna 
 
 Demani 
 
 Indian Ocean 
 
 Zocotora 
 
 Calanfia 
 
 Indian Ocean 
 
 But little 
 
 known 
 
 Indian Ocean 
 
 O F
 
 34 
 
 GEOGRAPHY. 
 
 Book VI; 
 
 O F 
 
 
 46. This vaft continent, called by fomc tfce new world, having been 
 difcovercd by the Europeans fince the year 1492, is ufually divided into 
 two pa.rts, one called North, and the other South America, being joined 
 to one another by the Ifthmus of Darien. 
 
 North America lies between the latitudes of 10 degrees and 80 de- 
 grees north ; and chiefly between the longitudes of 50 degrees and 130 
 degrees weft of London ; is about 4200 miles from north to fouth, and 
 3bout 4800 from eaft to weft. It is bounded on the caft by the north 
 Atlantic Ocean, by the Gulf of Mexico on the fouth, on the weft by 
 the Pacific Ocean, and by the Northern continent and ocean to the 
 northward. 
 
 South America is bounded on the eaft by the fouth Atlantic Ocean, by 
 the Southern Ocean to the fouth, by the Pacific Ocean on the weft, and 
 on the north by the Caribbean Sea. It lies between the latitudes of 12 
 degrees north, and 56 degrees fouth ; and between the longitudes of 45 
 degrees and 83 degrees weft from London j is about 4200 miles long, 
 and abouj: 2200 miles in breadth. 
 
 47. The pofitions and names of the chief countries, cities, rivers, and 
 mountains, in North America, are in the following table. 
 
 Countries. 
 
 Pofition. 
 
 Chief Cities. 
 
 Rivers. 
 
 Courfe. 
 
 Mountains. 
 
 California 
 
 W. 
 
 St. Juan. 
 
 
 
 
 New Mexico 
 
 s. 
 
 Sante Fe 
 
 N. River ^ 
 
 S. S. E. 
 
 
 Old Mexico 
 
 s. w. 
 
 Mexico 
 
 Panuco 
 
 E. 
 
 
 Louifiana 
 
 s. 
 
 New Orleans 
 
 Mifliffipi 
 
 S. 
 
 
 Florida 
 
 s. 
 
 S{. Auguftine 
 
 St. John 
 
 N. to E, 
 
 Apalachian 
 
 Georgia 
 
 S. S. E. 
 
 Savannah 
 
 Alatamaha 
 
 E. S. E. 
 
 Apalachian 
 
 Carolina 
 
 S. E. 
 
 Charles Town 
 
 Afhly 
 
 S. E. 
 
 Apalachian 
 
 Virginia 
 
 E. 
 
 James Town 
 
 Powtomack 
 
 S. E. 
 
 Apalachian 
 
 Maryland 
 
 E, 
 
 Annapolis 
 
 Powtomack 
 
 S. E. 
 
 Apalachian 
 
 Penfylvania 
 
 E. 
 
 Philadelphia 
 
 Delawar 
 
 S. 
 
 Apalachian 
 
 Jerfeys 
 
 E. New York 
 
 Albany 
 
 s. 
 
 
 New England 
 
 E. BoHon 
 
 Connedicut 
 
 s. 
 
 
 Nova Scotia 
 
 N. E. Halifax 
 
 St. John 
 
 S. S. E. 
 
 Ladies 
 
 Canada 
 
 Mid. Qnehec 
 
 St. Lawrence 
 
 N. E. 
 
 
 New Britain 
 
 N. N. E. Fort Rupert 
 
 Rupert 
 
 W. 
 
 
 New Wales 
 
 N. lYork Fort 
 
 Nelfon 
 
 W. 
 
 
 California, Old Mexico, and New Mexico, belong to Spain. 
 
 Louifiana, to the weft of the river Miflillippi, was polTeflcd by the 
 French at the end of the late war ; but is now transferred to Spain. 
 
 Ail the other countries are in the hands of the Enf^lifli. 
 
 hi
 
 Book VI. 
 
 GEOGRAPHY. 
 
 J4S 
 
 in South America the pofition and names of the chisf countries, cities, 
 livers, and mountains, are as follow i 
 
 Con'ntrics. 
 
 / 
 Pofition. 
 
 Chief Cities. 
 
 Rivers. 
 
 Courfc. 
 
 Mountains. 
 
 Terra Firma. 
 
 N. 
 
 Panama 
 
 Oronoque 
 
 N. E. 
 
 
 Peru 
 
 W. 
 
 Lima 
 
 Chuquimayo 
 
 W.N.W. 
 
 Andes 
 
 Chili 
 
 S. W. 
 
 St. Jago 
 
 Valparifo 
 
 W. 
 
 Andes 
 
 Patagonia 
 
 s. 
 
 
 Defaguadero 
 
 S. 
 
 Andes 
 
 La Plata 
 
 S. E. 
 
 Buenos Ayres 
 
 La Plata 
 
 S. 
 
 Andes 
 
 Paraguay 
 
 Mid. 
 
 Aflumption 
 
 Paragaa 
 
 S. 
 
 
 Brafil 
 
 E. 
 
 St. Salvador 
 
 aio Real 
 
 N. E. 
 
 
 Amazonia 
 
 Mid. 
 
 
 Amazons 
 
 E. 
 
 
 Guiana 
 
 N. E. 
 
 Surinam 
 
 Efquebe 
 
 N. N. E. 
 
 
 
 Terra Firma, Peru, Chili, La Plata, and Paraguay, are in thepofleffion 
 of the Spaniards. 
 
 Brafil belongs to the Portuguefe. 
 
 Patagonia, Amazonia, and Guiana, are poflefled by the native Indians, 
 except fome parts of the coalts of Guiana, in the hands of the Dutch and 
 French. 
 
 48. The principal Seas, Gulfs^ and Bays in America. 
 
 The Caribbean Sea^ bounded by Terra Firma on the fouth, and a rang 
 
 of iflands on the north and eaft. 
 Gulf of Mexico^ formed by Old Mexico, Louifiana, and Florida, 
 Bay of Campeacky, part of the Gulf of Mexico, on tjie Mexican coafl. 
 Bay of Honduras^ part of the Caribbean Sea, next to Mexico. 
 Bay of Panama^ in the Pacific Ocean, next the Ifthmus of Darien. 
 Bay of California, in the Pacific Ocean, having "alifornia on the weft. 
 Bay of Fitndy^ in Nova Sotia, north Atlantic Ocean. 
 Gulf of St. Laiorence, in the North Atlantic Ocean, bounded by Nova 
 
 Scotia, New Britain, and fome iflands eaftward 
 
 and fouth-caftward. 
 IliiJfons Bay, between New Britain E. and New Wales W. 
 
 49. The chief American Ifaruh in the Atlayitic Ocean. 
 Newfoundland, iind Cape Breton, ealt of the Gulf of St. Lawrence. 
 Bermudas, or Summer If.ands, eaft of Carolina. 
 
 Bahama Ifcs, fouth eait of Florida. 
 
 f Cuba ch. town Havanna 
 
 Great Antilles}. Ilifpaniola ch. town St. Domingo 
 
 t Jamaica ch. town Kingfton 
 Caribbce Ifes, bounding the Caribbean Sea on the E. and N. E. 
 LrJJer Antilles, on the N. N. E. of Terra Firma, in the Caribbec Sea, 
 Terra del Fuego on the fouth of Patagonia, in the Southern Ocean, 
 Galiipago Jjles, lying N. W. of Peru in the Pacific Ocean. 
 
 lying E. of ihr 
 Mexican Gulf, and 
 N. of the Car. Sea, 
 
 SECTION'
 
 3+i 
 
 GE O G R A P H rj 
 
 Bookvr. 
 
 SECTION IV. 
 
 Geographical Problems^ 
 
 50. Problem I. Given the latitudes of two places : 
 
 Required their difference ojf latitude. 
 
 Case' I. When the latitudes of the 
 given phces have the fame name : 
 
 Rule. Subtract the lefTer latitude 
 from the greater, the remajxuier is 
 the difference of latitude. 
 
 Exam. I. fVhat is the difference of 
 latitude between London and Rome ? 
 LoDdon's lat. 51* 32'N. 
 
 Rome's lat. 41 54 N. 
 
 Diff. lat. 
 
 9 38 
 60 
 
 578 miles. 
 
 Exam. II. What is the difference 
 tf latitude between the Lizard and the 
 IJland of Madeira ? 
 
 Lizard's lat. 49 57' N. 
 
 Madeira's lat. 32 36 N. 
 
 DifF. lat. 
 
 17 2iz::i04im. 
 
 Exam. III. IVloat is the difference 
 vf latitude between the Ijland of St. 
 Helena and the Cape of Good Hope f 
 
 C.GoodHope'slat. 34 29' S. 
 
 St. Helena's lat. 15 55 S. 
 
 DIfF. lat. 
 
 18 34n:iii4m, 
 
 CASElI.Whenthelatitudesofthe 
 given places have contrary names : 
 
 Rule. Add the latitudes together^ 
 and the fum w^Ul be the difference 
 of latitude. 
 
 Exam. I. Required the diff. of lat. 
 between C. Finijlerre andC.St.Roque. 
 C. Finifterre lat. 42 57' N. 
 C. St. Roquc lat. 5 00 S. 
 
 Diff. lat. 
 
 47 57 
 60 
 
 2877 miles. 
 
 Exam. II. TVhat is the difference 
 of latitude between the JjJand of Bar- 
 tadoes and C. Negro ? 
 
 I.of Barbadoes'slat. 130 co' N. 
 
 C. Negro's lat. 16 30 S. 
 
 Diff. lat. 
 
 29 30=: 1770m.' 
 
 Exam. III. Required the diffe- 
 rence of latitude between Cape Horn 
 and Cape Corientes in Mexico. 
 
 Cape Horn's lat. 55 59' S. 
 
 C. Corientes's lat. 20 18 N, 
 
 Diff. lat. 
 
 76 i7=:4577m. 
 
 Exam. IV. A Jhip from the lati- 
 tude of if-^ \%' N. is come to the lat. 
 cf 34 49^ N. Required the diff. of 
 Luitude. 
 
 Lat. from 43 18' N. 
 
 Lat. in 34 49 N. 
 
 Diff. lat. 
 
 8 
 
 29=5091 
 
 Exam. IV. J Jhip from the lat. 
 of 8 28^ 5. has Jailed north to the 
 lat. 6 45^ N. Required the diff. of 
 latitude. 
 
 Lat. from 8" 28' S. 
 
 Lat. in 6 45 N 
 
 Diff. lat. 
 
 15 i3=9i3m. 
 
 The fituation of about 1500 particular places are contained in a Geo- 
 graphical Table, art. 137, at the end of this book ; where the latitudes 
 and longitudes of places arc to be fought, as they follow in alphabetical 
 order. 
 
 51. Pro-
 
 'ftook VI. 
 
 GEOGRAPHY. 
 
 ^4-3 
 
 51. Problem II. Given the latitude of one place and the differenu of 
 latitude between it and another place : 
 Required the latitude of the latter place. 
 
 Case I. When the given latitude 
 and difference of latitude have the 
 fame name ; 
 
 Rule. To the given latitude add 
 the degrees and minutes in the diiF. 
 of latitude, that fum is the other 
 latitude of the fame name. 
 
 Exam. I. A Jhip from the latitude 
 of 38* \\' N. fails north till her dif- 
 ference of latitude is 1 7P 32^ .* fVhat 
 latitude is Jhe come to ? 
 
 Lat.from 38'' 14' N. 
 
 Diff. lat. 12 32 N. 
 
 Case IL When the given lati- 
 tude and difFerence of latitude have 
 contrary names : 
 
 Rule. Take the difference be- 
 tween the >given latitude and the 
 degrees and minutes in the diff. of 
 latitude, the remainder is the dthec 
 latitude, of the fame name with the 
 greater. 
 
 Exam. I. A Jhip from the latitude: 
 of 38 14^ N. fails fouth till her dif- 
 ference of latitude is 12 32'' : What 
 latitude is jhe come to ? 
 
 Lat. from 38 i^'N. 
 
 Diff. lat. 12 32. S. 
 
 Lat. In 
 
 50 46 N. 
 
 Exam. II. A Jhip from the ifland 
 of Afcenfion runs fouth till her diff. 
 of latitude is 5 37^ . Wl:)at is the 
 prefent latitude of the Jhip P 
 
 1. Afcenfion's lat. 
 
 Diff. lat. 5 
 
 37 S. 
 
 Ship's lat. 
 
 13 36 S. 
 
 Exam. III. A Jhip from the ijland 
 of Madeira fails N. 675 miles : What 
 lat. is Jhe in ? 
 
 I. Madeira's lat. 32"^ 36' N. 
 
 Diff. lat. 
 
 675 
 60 
 
 (I. 22) 1 1 15 N. 
 
 Lat. in 
 
 25 42 N. 
 
 Exam. II. A Jhip from the ijland 
 of Afcenfion runs north till her diff^ 
 lat. is 5 37^ : JVhat is the prefent 
 latitude of the Jhip ? 
 
 I. Afcenfion's iat. 7 S9 S. 
 
 Diff. Iat. 5 37 N, 
 
 Ship's lat. 
 
 2 22 S. 
 
 Ship's lat. 
 
 43 51 N. 
 
 Exam. IV. Tljree days ago we 
 vuere in the latitude of the Cape of 
 Good Hope., and have run each day 
 92 miles d'-ireily S. What is our pre- 
 fent latitude ? 
 
 C. Good Hope's lat. 34 29' S. 
 
 Diff. lat. ^i^(I. 22)= 4 36 S. 
 
 60 
 Ficfeat latitade 
 
 39 o> S. 
 
 Exam. III. A fnip from Sierra 
 Leon fails S. 839 miles : What latir- 
 tude is Jue in P 
 
 Sierra Leon's lat. 8' 30' N. 
 
 Diff. lat. 
 
 CO 
 
 Ship's lat. 
 
 5 ^9 S, 
 
 Exam. IV. Four day; ago we 
 were in the latitude of the ijland of 
 St. AlatthezUj and Jailed due north 6 
 miles an hmir : What latitude is thf 
 Jhip in P 
 
 St. Matthew's lat. i* 23'S. 
 
 6x24x4. 
 
 Diff. lit.- 
 
 00 
 
 t(L2 2)=9 36 M; 
 
 Ship's latitude 
 
 8 13 N. 
 52. Pro?
 
 344 
 
 GEOGRAPHY. 
 
 Book VI. 
 
 52. Problem. III. Given the longitudes of two places : 
 
 Required their difference of longitude. 
 Rule. If the longitudes are of the fame name, their difference is the 
 difference of longitude required. 
 But if the longitudes are of different namesj their fum gives the 
 
 difference of longitude. 
 And if this fum exceeds 180 degrees, take it from 360 degrees, and 
 there remains the difference of longitude. 
 Exam. I. Required the difference \ ExAM. IV . Required the difference 
 
 cf longitude between London and Na 
 
 pies ? 
 London's long. 00" 00' 
 
 Naples' long. 14 19 E. 
 
 of longitude hctiveen St. Chrijlopher'i 
 and Cape Negro. 
 
 St. Chriftopher's long. 62" 54' W. 
 
 C. Negro's long. 11 30 E. 
 
 DifF. longitude 
 
 14 19 
 60 
 
 85-9 
 
 Exam. II. A Jhip in longitude 14 
 
 Diff. longitude 
 
 74 24 
 60 
 
 4464 miles. 
 
 Exam. V. J Jhip in longitude 
 
 45^ IV, is bound to a port in longitude 140 20'' IV. is bound to a place in 
 J^W' iW wejl : What diff. of longitude longitude 139 25' E. what diff. of 
 
 ihujl Jhe make? 
 Ship's long. 
 Long, bound to 
 
 DiS, longitude 
 
 14 
 48 
 
 W. 
 W. 
 
 - 33 33 
 60 
 
 ^013 miles. 
 
 longitude muji Jhe ?nake f 
 
 Ship's long. 140 20' W< 
 
 Longi bound to 139 25 E. 
 
 DlfF. longitude 
 
 279 
 
 360 
 
 55 
 00 
 
 80 
 
 05 
 
 Exam. III. iVhat is the difference Exam. VI. TVhat is the difference 
 of longitude between Cape Gardafuir of longitude between Cape Horn and 
 
 and Cape Comorin ? 
 C. Gardafuir's long 
 C. Comorin's long. 
 
 Diff. longitude 
 
 50^25'E. 
 78 17 E. 
 
 27 
 60 
 
 5= 
 
 1672 miles. 
 
 Manila ? 
 
 Cape Horn's long. 
 Manila's long. 
 
 67 26' W. 
 izo 25 E, 
 
 187 51 
 360 00 
 
 Diff. longitude 172 09 
 
 Sometimes the diff. Ion. between two places is eflimated by the diff. of 
 time, allowing an hour to every 15 degrees of longitude, and one min. of 
 time for every 15 min. of a deg. or a deg. for every 4 min. of time. 
 
 Exam. At b h. 48^. P. M. having obferved at fea a certain appearance 
 in the heavens., which I knew was feen the fame infiant at 3 h. 2^m. P. M. 
 in London : Required the diff, longitude betivecn the places of ohfervationi 
 From 6h. 48 m. 3 h. =: 45 deg. 
 
 Take j 3^ 13 m. cr 315' 
 
 Remain 3 13 = diff. time. Sum 48 1 ,-=rDlff. long. 
 
 And becauie the hour of appearance at London was leaft, therefore I know 
 iryfclf to be to the eaftward of London, 53. Pro-
 
 Book VI. 
 
 GEOGRAPHY. 
 
 345 
 
 53. Problem IV. Given the longitude of one place, and the difference 
 of longitude between that and another : 
 Required the longitude of the fecond place, 
 
 RuikB. If the given longitude and difference of longitude are of a con- 
 trary name, their difference is the longitude required j and i 
 of the fame name with the greater. 
 
 But if the given longitude and difference of longitude are of the fame 
 name, the fum is the longitude fought, of the fame name with 
 the given place. 
 
 And if the fum is greater than 180 degrees, take it from 360 de- 
 grees, remains the longitude required, of a contrary name to 
 that of the given place. 
 
 Exam. I. A pAp fro7n the Ion 
 gltude of fl 12' E. fails wejlward 
 until her difference of longitude is 
 IS'^ 47^ ." 0%at is her prefent longi- 
 tude ? 
 
 Ship's lARgltude 
 DiiF. long. 
 
 Pref. long. 
 
 IS 
 
 12' E. 
 47 W. 
 
 25 
 
 E. 
 
 Exam. II. A JJnp from Cape 
 (Iharles in f^irginia fails eaflivard 
 until Jhe has altered her longitude 11 
 53' ; IVbat longitude is Jhe in ? 
 
 C. Charles's long. 76 07' W. 
 
 DifF. long. 22 53 E. 
 
 Ship's long. 
 
 53 14 W- 
 
 Exam. III. Four days ago I de- 
 parted from C. St. Sebajiian in Aia- 
 Jiigafcar, and I have made each day 
 1 ^ nAlcs of eafl longitude: Required 
 the longitude the Jhlp is in ? 
 
 75 
 4 C. Sebaf. long. 49* 13' E. 
 
 DifF. long. 5 CO E. 
 
 6,o)3c,o - 
 
 - Ship's long. 54 13 E. 
 
 ExAi,!. IV. A Jhlp from Cape 
 Finiflerre fails wejlward, and finds 
 Jhe has altered her longitude 5^7 
 miles : IVtjat longitude is Jhe arrived 
 in? 
 
 C. Finlfterre's long. 
 
 60 , 
 
 DiiF. Ion. 
 
 9 36' W. 
 9 47 W- 
 
 Long. in. 19 23 W. 
 
 Exam. V. A Jlnp from Cape St. 
 Lucar in California has made 87 
 1 8^ cfwejl longitude : JVhat longitude 
 is Jhe in t 
 
 C. St. Lucar's long. 109 4'/ W. 
 
 DifF. long. 87 18 W. 
 
 196 58 W. 
 ^60 00 
 
 Ship's longitude 163 02 E. 
 
 Exam. VI. Seven days ago my 
 longitude was 172"^ 17' JV. and I 
 have made each day 1^2 ?niles of weji 
 hngiludc : Required 7ny prej'ent lon^ 
 pitiide ? 
 
 132 
 / 
 
 6,o)r;c,4 
 IS 24 
 
 Departed long, 172 17' W. 
 
 Diff. long. 15 24 W. 
 
 I i. 1 4 1 
 
 360 CO 
 
 Prefent long, iz 19 E- 
 
 VoL. I, 
 
 fi b 
 
 SECTION
 
 34^ . GEOGRAPHY. Book VI. 
 
 SECTION V. 
 
 ^4. Of the Ufe of the Globes, 
 
 By tl) globes are here meant two fpherlcal bodies, called the Tor* 
 rcftrial and Ccleflial Globes, the convex furfaccs of which are fuppofcd to 
 to jiivc a true rcprcfentation of the earth and heavens. 
 
 The Terrestrial Globe has delineated on its convexity the whole 
 furlaceof the earth ^u^ fca in their relative fizc, form, and fituation. 
 
 The Celestial Globe has drawn on its furface the images of the 
 fcvcral conflellations and^flars ; the relative magnitude and pofition which 
 the Itars are obfcrvcd to have in the heavens, being preferved on this 
 globe. 
 
 The globes are fitted up with certain machinery, by means of which a 
 great variety of ufeful problems arc neatly folved. 
 
 The Brazen Meridian is that ring, or hoop, in which the globe 
 hangs on its axis ; which is reprefented by two wires pafling through its 
 poles. This circle is divided into four quarters, of 90" each ; in onefemi- 
 circle the divifions begin at each pole, and end at 90% where they meet ; 
 In the other fcmicircle, the divifions begin at the middle, and proceed 
 thence tov/ards each pole, where they end at 90 degrees. The graduated 
 fide of this brazen circle ferves as a meridian for any point on the fur- 
 face of the earth, the globe being turned about till that point comes un- 
 der the circle. 
 
 The Hour Circle is a fmall circle ofbrafs, which is divided into 24 
 hours, the quarters r.nd half quarters. It is fixed on the brazen meridian, 
 equally diftant from the north end of the axis, to which an index is fitted, 
 that points out the divifions of the hour circle as the globe is turned about. 
 
 The Horizon is reprefented by the upper furface of the wooden cir- 
 cular frame encompafllng the globe about its middle. On this wooden 
 frame is a kind of perpetual calender, contained in feveral concentric 
 circles : The inner one is divided into four quarters, of 90 degrees each j 
 the next circle is divided into the twelve months, with the days in each 
 according to the new llyle ; the next contains the 12 equal figns of the 
 zodiac, each being divided into 30 degrees: the next is the 12 months and 
 days according to the old ftyle ; and there is another circle, containing 
 the 32 winds, with their halves and quarters. Although thefe circles are 
 on all horizons, yet their diipof.tion is not always the fame. 
 
 The Ql'adrant of Altitude is a thin ftreight flip ofbrafs, one 
 edge of which is graduated into 90 degrees and their quarters, equal to 
 thole of the meridian. To one end of this is fixed a brafs nut and fcrew, 
 by which it is put oti, and faftcned to the meridian : and if it is fixed to 
 the zenith, or pole of the horizon, then the graduated edge reprefents a 
 vertical circle pafling through any point. 
 
 Kefidcs thefe, there are feveral circles defcribed on the furfaces of both 
 clobes ; fuch as-riie equinoctial, ecliptic, circles of longitude and right 
 afccnfion, the tropics, polar circles, parallels of lat. and decl., on the ce- 
 leftial globe; and on the terreftrial, the equator, ecliptic, tropics, polar 
 circles, parallels of latitude, hour circles, or meridians to every 15 degrees, 
 and thefpiral rhumbs flowing from rfvc;\il centers, called I'lics. 
 
 55. PRO.
 
 Book VI. GEOGRAPHY; j47 
 
 ^5. P R O B L E M I. 
 
 To find the latitude and longitude of any place on the terrejlrial globe, 
 
 I ft. Bring the given place under that fide of the graduated brazen meri- 
 dian where the degrees begin at the equator, by turning the globe about. 
 2d. Then the degree of the meridian over it fhews the latitude. 
 3d. And the degree of the equator under the merid., fliews the long. 
 
 On fome globes the longitude is reckoned on the equator from the 
 meridian where it begins, eaftward only, until it ends at 360 : On fuch 
 globes, when the Ipngitude of a place exceeds 180, take it from 360? 
 and call the remainder the longitude v/eftward. 
 
 56. PROBLEM II. 
 
 To find any place on the glebe .^ the latitude and longitude of which are given, 
 
 ift. Bring the given longitude, found on the equator, to the meridian. 
 2d. Then under the given latitude, found on the meriuianj is the place 
 fought. 
 
 57. PROBLEM in. 
 
 To find the dijhince and bearing of any tivo given places on the globe. 
 
 ift. Lay the graduated edge of the quadrant of altitude over both 
 places, the beginning, or o degree, being on one of them, and the de- 
 grees between them fhew their diftance ; thefc degrees multipled by 60 
 give fea miles, and by 70 give die diflance in land miles nearly ; or 
 multiplied by 20 give leagues. 
 
 2d. Obfcrve, v/hilc the quadrant lies in this pofition, what rhumb of 
 the nearcft fly, or co-r.pafr., runs moflly parallel to the edge of the qua-* 
 drant, and that rhumb fhcws the bearing fought, nearly, 
 
 58. P R O B L E M IV. 
 
 To find the Sun's place and d&ctlnation on any day, 
 
 I ft. Seek the given day in the circle of months on the horizon, and 
 right againft it in the circle of figiis is the Sun's place. 
 
 Thus it will be found that the Sun enters 
 The fpring figns, Aries, March 2C. Taurus, April 20. Gemini, May 21. 
 The fummerfigns, C'dnccr. June 21. Leo, July 23. Virgo, Aug. 23. 
 Autumnal fi^ns^ Libr.i, Sept. 22. Scorpio, Oct. 23. Sagittar. Nov. 22. 
 The winter fign^,Co.pilc. Dec. 21, Aquarius, Jan. 20. Pifces, Feb. 18. 
 
 2d. Seek the Sun's place in the ecliptic on the globe^ bring that place 
 ^o the meridian, and the divifion it ftands under is the Sun's declination 
 on the given day. 
 
 On the globes, the ecliptic is readily diftinguifhed from the equator, 
 not only by the different colours they arc fi:ain;d with, but alfo by the 
 ecliptic's approaching towards the pole?, after its interfedtion with the 
 equator. The marks of the figns are alfo put along the ecliptic, one at 
 the beginning of every fucccflive 30 degrees. 
 
 B b 2 59. PRO*
 
 34ft GEOGRAPHY. Book VI. 
 
 59. P R O B L E M V. 
 
 To reSlify the globe for the latitude, zenith, and noon. 
 
 I ft. Set the globe upon an horizontal plane with its parts anfwering to 
 thofc of the world ; move the meridian in its notches, by raifmg or de- 
 prefling the pole, until the degrees of latitude cut the horizon ; then is 
 the globe rc<Stified for the latititude. 
 
 2d. Reckon the latitude from the equator towards the elevated pole, 
 there fcrew the bevil edge of the nut belonging to the quadrant of alti- 
 tude, and the rcdlification is made for the zenith. 
 
 3d. Bring the Sun's place (found by the laft problem) to the meridian, 
 fet the index to the xii at noon, or upper xii, and the globe is redtified 
 for the Sun's fouthing, or noon. 
 
 60. P R O B L E M VI. 
 
 To find where the Sun is vertical, at any given time, in a given place. 
 
 I ft. Bring the Sun's place, found for the given day (58), to the meri- 
 dian, and note the degree over it. 
 
 2d. Bring the place, for which the time is given, to the meridian, and 
 fct the index to the given hour. 
 
 3d. Turn the globe till the index comes to 12 at noon, then the place 
 under the faid noted degree has the Sun in the zenith at that time. 
 
 4th. All the places that pafs under that degree, while th globe is 
 turned round, will have the Sun vertical to them on that day. 
 
 61. PROBLEM VII. 
 
 7"? /fid en ivhat days the Sun will he vertical, at any given place, in the 
 torrid xoric. 
 
 liK Note the latitude of the given place on the meridian. 
 
 2d. 'i'urn the globe, and note \\hat two points of the ecliptic pafs un- 
 der the latitude noted on the meridian. 
 
 3d. Seek thofe points of the ecliptic in the circle of figns on the ho- 
 rizon, and ri^ht againit them, in the circle of months, ftand the days re- 
 quired. 
 
 In this manner it will be found, that the Sun will be vertical to the 
 I Hand of St. Helena on the 6th of November, '^n<\ on the 4th of February, 
 And at Barbadoes on the 24th of April, and the i8th of Auguft. 
 
 62. P R O B L E M VIII. 
 
 y// any g'rccn hour in a given place, ts find what hour it is in any othe^ 
 place. 
 
 ifl. Bring the place where the time is given to the meridian, and fet 
 the index to the given hour. 
 
 2d. Bring the other given place to the meridian; and the index fhcws 
 the liour correfponding to the i^iven tune. 
 
 63. P R 0-1
 
 Book VI. GEOGRAPHY.. 349 
 
 63. PROBLEM IX. 
 
 At any given tune to find all thofe places of the Earth where the Sun is 
 thfn rifing or fetting, and where it is mid-day or midnight. 
 
 Find the place where the Sun is vertical at the given time (60), rectify 
 the globe for the latitude of that place, and bring it to the meridian. 
 
 Then all thofe places, that are in the weftern half of the horizon, have 
 the Sun rifing ; and thofe in the eaftern half have it fetting. 
 
 Thofe under the meridian, above the horizon, have the Sun culminating, 
 or noon ; and thofe under the meridian, below the horizon, have midnight. 
 
 Thofe above the horizon have day j thofe below it have night. 
 
 64. PROBLEM X. 
 
 Tg find the angle of pofttion of two places^ or the angle made by the meridian 
 pf one place ^ and a great circle pajfing through both places. 
 
 Rectify for the latitude of one of the given places, and bring it to the 
 meridian ; there fix the quadrant of altitude, and fet its graduated edge to 
 the other place : then will that edge of the quadrant cut the horizon in 
 the degree of pofition fought. 
 
 Thus, the angle of pofition at the Land's End to Barbadoes is fouth 
 715; wellerly : but the angle of pofition at Barbadoes to the Land's End 
 is north 37! degrees eafterly. 
 
 Hence neither of thofe pofitions can be the true bearing ; for the rhumb 
 paffing through both places, will be oppofite one way to what it is the 
 other. 
 
 65. PROBLEM XI. 
 
 The latitude of any place net within the polar circle being given ^ to find the 
 time of fun-rlfing and fetting^ and the length of the day and night. 
 
 Rectify for the latitude and the noon ; bring the Sun's place to the 
 eaflcrn fide of the horizon, and the index fliews the time of rifing ; the 
 Sun's place being brought to the weflcrn fide of the horizon, the index 
 gives the fetting. 
 
 Or, the time of rifing taken from 12 hours gives the time of fetting, 
 
 Thx time of letting being doubled gives the length of the day. 
 
 And the time of riilng being doubled gives the length of the night. 
 
 Thus, at London, on April i5tli, the day is 13^ lioars j the night lOf 
 hours. 
 
 66. PROBLEM XII. 
 
 To find the length of the longefi and fi)orteJl days in any given place. 
 
 Redify for th latitude -, bring the folflitial point of tnat hemifphere 
 to the cailcrn part of the horizon, fet the index to 12 at noon, turn the 
 globe till the lolititial point comes to the weftern fide of the horizon, 
 the hours paft over by the index give the length of th.- longcil day, or 
 iiijiht ; and its complcnicm to 24 hours gives the length oi the Ihortclt 
 night, or d.iy. 
 
 B b 3 67. P R O-
 
 3S0 G E O G R A P A Y. Book VI. 
 
 67. PROBLEM XIII. 
 
 A place being given In either frigid or frozen zone^ to find the time when 
 the Svn begins to appear at, or depart from, that place : alfo how tnany fuC" 
 cejfive days he is prcfent to, or ahfent from, that place. 
 
 Rectify for the latitude, turn the globe, and obferve what degrees in 
 the firft and fecond quadrants of the ecliptic arc cut by the north point of 
 tlie horizon, the latitude being fuppofed to be north. 
 
 Find thofc degrees in the circle of iigns on the horizon, and their cor- 
 refponding days of the month ; and all the time between thofe days the 
 t)un will not fet in that place. 
 
 Again. Obferve what degree in the third and fourth quadrants of the 
 ecliptic will be cut by the fouth point of the horizon, and the days an- 
 iwcring : then the Sun will be quite abfent from the given place during 
 the intermediate days ; that day in the third quadrant fhews when he be- 
 gins to difappear ; and that in the fourth quadrant fhews when he begins 
 to fhine in the place propofed. 
 
 Thus at the North cape, in lat. 71, the Sun never fets from May 15 
 to July 28, which is 74 days; and never rifes from November 16 to 
 January 24, which is 69 days. 
 
 68. PROBLEM XIV. 
 
 To find the antceci, pcrlceci, and antipodes of any place. 
 
 Bring the given place to the meridian, tell as many degrees of latitude 
 on the contrary fide of the equator, and it gives the place of the antaeci ; 
 that is, of thofc who have oppofite feafons of the year, but the fame times 
 of the day. 
 
 The given place being under the meridian, fet the index to 12 at 
 noon, tun; the globe until the index points to I2 at night, and the point 
 under the meridian in the given latitude is the place of the pcriceci ; that 
 is, of thofe who have the fame feafons of the year, but oppofite times of 
 the day. 
 
 The globe remaining in this pofition, feek on the contrary fide of the 
 equator for the degrees of latitude given, and the point under the meri- 
 dian, thus found, v.iil be the antipodes to the given place j that, is, there 
 the feafons of the year aiid times of the day are diredtly oppofite to thofc 
 of the given place. 
 
 69. P R O E L E M XV. 
 
 To find the hc^innin^ and end of the tivilight in any place. 
 
 Rectify the globe fur the latitude, zenith, and noon. (59) 
 
 Seek the point of the ecii[)Lic oppofite to the Sun's place, turn the 
 globe and quadrant of altitude, till the laid oppofite point of the ecliptic 
 iiands againft 18 degrees on the quadrant of altitude; then will the in- 
 di.x fliew the bcginiiing or end of tl:c twilight; that is, the beginning 
 in the morning, v*'hen thofe points meet in the weRern hemifpht-re ; or 
 the end in the evening, when the i'aid poiifts meet in the eaflern he- 
 niifpijcie. 
 
 , 70. P R O.
 
 Book VI. G E O G R^A P H Y. .^si 
 
 70. PROBLEM XVI. 
 
 *rhe latitude of a place and day of the month being given, to find the Sun's 
 declination, meridian altitude^ right afcenfion, amplitude., oblique afcenfiouy 
 afcenfional difference j and thence the time of rifmg-, fetting.^ length of the day 
 and night* 
 
 Redify for the latitude and noon. Then, 
 
 The degree of the meridian over the Sun's place is the declination. 
 * The meridian altitude is fhewn by the degrees the Sun is above the 
 horizon ; and is equal to the fum or difF. of the co-lat. and decl. 
 
 The Sun's right afcen. is that degree of the equator under the meridian. 
 
 Bring the Sun's place to the eaftern part of the horizon. Then, 
 
 The amplitude is that degree of the horizon oppofite the Sun. 
 
 The oblique afcenfion is that degree of the equator cut by the horizon. 
 
 The afcen. difF, is the diiF. between the right and oblique afcenfions. 
 
 The afcen. difF. converted into time, will give the time the Sun rifes 
 before or after the hour of fix, according as his amplitude is to the north- 
 ward or fouthward of the eafl point of the horizon. 
 
 ^i. PROBLEM XVir. 
 
 Given the latitude of the place and day of the month, to find the Sun's al- 
 titude and azimuth, either when he is due eafi or wefi, at 6 o'clock, or at any 
 other hour while he is above the horizon. 
 
 Rcdify the globe for the latitude, zenith, and noon. 
 
 Sef the quadrant of altitude to the eaft point of the horizon, turn the 
 globe till the Sun's place comes to the quadrant's edge, and it (hews tii;; 
 altitude, his azimuth being now 90, and the index fhevvs the hour. 
 
 Turn the globe till the index points at 6, there ftay it, and move the 
 quadrant until its edge cuts the Sun's place ; then the degrees at the Sun 
 fhew its altitude, and the degrees cut by the quadrant in the horizon 
 fhew the azimuth, reckoning from the north. 
 
 In lilcc manner, the globe being turned till the index is againft any 
 other hour, fuppofc 10 in the forenoon ; then. 
 
 The graduated edge of the quadrant of altitude being turned to cut 
 the Sun's place, will give both the altitude and azimuth at that time. 
 
 -2. PROBLEM XVIIL 
 
 / 
 
 Given the latitude, day of the month, and Suns altitude, to find the azi- 
 
 7nnth and hour of the day. 
 
 Rccftify the globe for the latitude, zenith, and noon. 
 
 Turn the globe and quadrant, until the Sun's place coincide with the 
 altitude on the graduated edge of the quadrant. 
 
 Then will that edge of the quadrant cut the degrees of azimutli on the 
 horizon, reckoned from tiic north ; and the index will flicw the hour of 
 the day. 
 
 Bb4 73-I'RCJ.
 
 55* 
 
 GEOGRAPHY. Book VI. 
 
 73. PROBLEM XIX. 
 
 ITo reprefitit the appearance of the heavens at any time in a given place, 
 
 Recflify the ccleftial globe for the latitude, zenith, and noon, and turn 
 the globe till the index points at the given hour. Then, 
 
 The Itars in the eaftern half of the horizon are rifing ; thofe in the 
 weftern are fctting : and thofe on the meridian are culminating. 
 
 The quadrant being fet to any given flar will fhew its altitude, and z% 
 the fame time its azimuth, reckoned on the horizon. 
 
 Now by turning the globe round it will readily appear, what ftars never 
 fet in that place, and what never rife : thofe of perpetual apparition 
 never go below the horizon, thofe of perpetual abfence never come 
 above it. 
 
 74. PROBLEM XX. 
 
 To find the latitude and longitude ofanyjlar. 
 
 Put the center of the quadrant of altitude on the pole of the ecliptic, 
 and its gr.iduated edge on the given ftar. Then, 
 
 The latitude is fhewn by the degrees between the ecliptic and ftar. 
 The longitude is the degrees cut on the ecliptic by the quadrant. 
 
 75. PROBLEM XXI. 
 
 To find the decli-nation and right afcenfjon ofi ajlar. 
 
 Bring the ftar to the meridian, the degree over it is the dpcHnation j and 
 the degrt;e of the eqqato.r under the meridian is the right afcenfion. 
 
 76. PROBLEM XXIL 
 
 On any day., and in any given place^ to find ivhen a propofedflar rifes^ fets^ 
 
 or culminates. \ 
 
 Retify tlic globe for the latitude and noon. 
 
 Bring the ftar to the eaftern fide of the horizon, anfi the index fhews 
 the time of its rifing. 
 
 Turn the globe till the ftar cornes to the meridian, and the indeji fhews 
 the time of its culminating ; and in like manner when it fets, the time 
 will be fticwn by the index. 
 
 Its meridian altitude, oblique afcenfion, and afcenfional difference, are 
 found in the fame manner as for the Sun, at art. 70. 
 
 SECTION
 
 Book VI. GEOGRAPHY. 353, 
 
 SECTION VI. 
 
 Of Winds. 
 
 77. A Fluid is a body, the particles of which readily give way to any 
 impreffed force j and by this readinefs of yielding, the particles areeafily 
 put into motion. 
 
 Thus, not only liquids, but flreams or vapoiys, fmoke or fumes, and 
 others of the like kind, are reckoned as fluids. 
 
 From all parts of the Earth vapours and fumes are conftantly arrfing to 
 fome diftance from its furface. 
 
 This is known by obfervation j it is caufed chiefly by the heat of the 
 Sun, and fometimes by fubterraneous fires arifing from the accidental 
 mixing of fome bodies. 
 
 78. Air is a fine invifible fluid furrounding the globe of the Earth, 
 and extended to fome miles above its furface. 
 
 The Atmosphere is that colledion of air, and of bodies contained in 
 it, which circumfcribes the Earth. 
 
 79. From a multitude of experiments, air is found to be both heavy and 
 fpringy. 
 
 By its weight it is capable of fupporting other bodies, fuch as vapours 
 and fumes, in the fame manner as wood is fupported by water. 
 
 By its fpringinefs or elafticity a quantity of air is capable of being ex- 
 panded, or of fpreading itfelf fo as to fill a larger fpace * ; and of being 
 comprelTed or confined in a fmaller compafs f . 
 
 80. Air is comprelled or condenfed by cold, and expanded or rarefied 
 by heat. 
 
 This is evident from a multitude of experiments. 
 
 An alteration being made by heat or cold in any part of the atmofphere, 
 its neighbouring parts will be put in motion, by the endeavour which the 
 air always makes to reftore itfelf to its former Irate. 
 
 For experiments fhcv/, that condenfed or rarefied air will return to its 
 natural flate, when the caufe of that condenfation of rarefadtion is re- 
 moved. 
 
 81. Wind is a flream or current of air which may be felt ; it ufually 
 blows from one part of the horizon to its oppofite part. 
 
 82. The horizon^ befide being divided into 360 degrees, like all other 
 circle^, is by mariners fuppofed to be divided into four quadrants, called 
 the north-cad, north- well, fouth-eafl and fouth-well quarters; each of 
 thefe quarters they divide into eight equal parts, called points, and each 
 point into four equal parts, called quarter-points. 
 
 So that the horizon is divided into 32 points, which are called rhumbs^ 
 or vj'mds ; to each wind is afTigned a name, which fhews from what poiiK 
 of the horizon the wind blows. 
 
 The points of North, South, Eaft, and Weft:, are called cardinal 
 points ; and are at the diftancc of 90 degrees, or 8 points, from one an- 
 other. 
 
 * Near 14000 times. ^/'a/Z/j's Hydrof. p. 13. 
 + Into the A part. Fhil. Tranf. N*" iSt. 
 
 ll. Winds
 
 j^^ GEOGRAPHY. Book VL 
 
 J??. Winds are either coiiftant or variable, general or particular. 
 
 Conftant tvInJs are fuch as blow the fame way, at lead for one or niorc 
 days ; and variable ivirUls are fuch as frequently ftiift within a day. 
 
 A rentful iv'md is that which blows the fame way over a large tral of 
 the Earth, almoft the whole year. 
 
 A particular wind is that which blows in any place, fometimcs one 
 way, and fomctimes another, indifferently. 
 
 If the wind blows gently, it is called a breeze ; if it blows harder, it 
 is called a gale, or a flift' gale j and if it blows very hard, it is called a 
 ftorm . 
 
 84. The following obfervations on the wind l^ave been made by fkilU 
 ful feamen ; and particularly by the great Dr. Halley. 
 
 i(J. Between the limits of 60 degrees, namely, from 30 of north lati- 
 tude to 30 of fouth latitude, there is a conftant eafterly wind throughout 
 the year, blowing on the Atlantic and Pacific Oceans j and this is called 
 the trade-wind. 
 
 For as the Sun, in moving from eaft to weft, heats the air more im- 
 mediately under him, and thereby expands it ; the air to the eaftward is 
 conftantly rufhing towards the weft to reftore the equilibrium, or natural 
 ftate of the atmofphere ; and this occafions a perpetual eafterly wind in 
 thofe latitydes. 
 
 2d. The trade-winds near the northern limits blow between the north 
 and eaft i and near the fouthern limits they blow between the fouth and 
 eaft. 
 
 For as the air is expanded by the heat of the Sun near the equator ; 
 therefore the air from the northward and fouthward will both tend to- 
 wards the equator to reftore the equilibrium. Now thofe motions from 
 the north and fouth, joined with the foregoing eafterly motion, will pro- 
 diice the motions obierved near the faid limits between the north and eaft, 
 and between the fouth and eaft. 
 
 3d. Thofe general motions of the wind are difturbed on the continents, 
 and near their coafts. 
 
 For the nature of the foil may caufc the air to be either heated or 
 cooled ; and hence will arife motions that may be contrary to the fore- 
 going general one. 
 
 4th. In fome parts of the Indian ocean there are periodical winds, 
 caKed Monsoons ; that is, fuch as blow one half the year one vv'ay, and 
 the other half-year the contrary way. 
 
 For air that is cooi and denfe, will force the warm and rarefied air in 
 i continual ftream upwards, where it muft fpread itfelf to preferve the 
 equilibrium : So that the upper coiirfe or current of the air (hall be con- 
 trary to the under current ; for the upper air muft move from thofe parts 
 where the greatcft heat is ; and fo, by a kind of circulation, the N. E. 
 trade-wind bclo-.v will be attended with a S. W, above ; and a S. E. be- 
 low w^Lh a N. \'/. above : And this is confirmed by the experience of 
 Teamen, who, as focn as they get out of the trade-winds, immediately 
 hnd a wind blowing from the oppofite quarter. 
 
 * The r.viftnefs of the wind in a florm is not more than 50 or 60 miles in 
 in hour ; art! a '.omiftcn brifk g^Jc i: about J 5 miles an hour. 
 
 5th. In
 
 Book VI. GEOGRAPHY, 355 
 
 5th. In the Atlantic Ocean near the coafts of Africa, at about loo 
 leagues from the fhore, between the latitudes of 28 and 10 north, fca- 
 men conftantly meet with a frefh gale of wind blowing from the N E. 
 
 6th. Thofe bound to the Caribee Iflands, acrofs the Atlantic Ocean, 
 find, as they approach the American fide, that the faid N. E. wind be- 
 comes eafterly ; or feldom blows more than a point from the eaft, either 
 to the northward or fouthward. 
 
 Thefe trade- winds, on the American fide, are extended to 30, 31, or 
 even to 32 of N. latitude ; which is about 4 farther than what they 
 extend to on the African fide. Alfo to the fouthward of the equator, the 
 trade-winds extend 3 or 4 degrees farther towards the coaft of Brafil on 
 the American fide, than they do near the Cape of Good Hope on the 
 African fide. 
 
 7th. Between the latitudes of 4*^ North, and 4 South, the wind al- 
 ways blows between the fouth and eaft : On the African fide the winds 
 are neareft to the fouth ; and on the American fide neareft to the eaft. 
 In thefe feas Dr. Halley obferved, that when the wind was eaftward, the 
 weather was gloomy, dark, and rainy, with hard gales of wind : but when 
 the wind veered to the fouthward, the weather generally became ferene 
 with gentle breezes next to a calm. 
 
 Thefe winds are fomewhat changed by the feafons of the year ; for 
 when the Sun is far northward, the Brafil S. E. wind gets to the fouth, 
 and the N. E. wind to the eaft ; and when the Sun is far fouth, the S. E, 
 wind gets to the eaft, and the N. E. winds on this fide of the equator veer 
 more to the north. 
 
 8th. Along the coaft of Guinea, from Sierra Leon to the IHand of St. 
 Thomas (under the equator) which is above 500 leagues, thefoutherly and 
 fouth-weft winds blow perpetually. For the S. E. trade-wind having 
 pafled the equator, and approaching the Guinea coaft within 80 or lOO 
 leagues, inclines towards the fliore, and becomes fouth, then S. E. and by 
 degrees, as it comes near the land, it veers aibout to fouth, S. S. W. and 
 in with the land it is S. W. and fometimes W. S. W. This tra<5i is 
 troubled with frequent calms, violent fudden gufts of wind, called Torna- 
 does, blowing from all points of the horizon. 
 
 The reafon why the wind fets in weft on the coaft of Guinea, is, in 
 all probability, owing to the nature of the coaft, which being greatly 
 heated by the Sun, rarefies the air exceedingly ; and confcquently the cool 
 air from off" the fea will keep ruftiing in to rcftorc the equilibrium. 
 
 9th. Between the 4th and lOth degrees of north latitude, and between 
 the longitudes of Cape Verd, and the caftcrmoft of the Cape Verd ifles, 
 there is a tract of fca which feems to be condemned to perpetual calms, 
 attended with terrible thunder and lightnings, and fuch frequent ruins, 
 that this part of the fea is called the Rains. Ships in failing thcf 
 6 degrees have been fometimes detained whole months, as it is re- 
 ported, 
 
 'I'he caufe of this feems to be, tliat the wcftcrly winds fetting in on 
 this coaft, and meeting the general eafterly wind in this trai^t, balance 
 each other, and io caufe the calms ; and the vapours carried ihiihcr by 
 each wiiid meeting and condcnfing, occalion the iiimoft coiiilajit lains. 
 
 The
 
 25^ . G E O G R A P H Y. Book VL 
 
 The laft three obfervations Ihcw the reafon of tvi^o things which 
 mariners experience in failing frona Europe to India, and in the Guinea 
 trade 
 
 > Firft. The difficulty which fliips in going to the foathward, efpccialiy 
 in the months of July and Auguil, find in pafling between the coaft of 
 Guinea and Brufil, notwithftandi'.g the width of this fea is more than 
 500 leagues. This happens, becaufc the S. E. winds at that time of the 
 year commonly extend fome degrees beyond the ordinary limits (jf 4^ N. 
 latitude ; and befides come fo much foutherly, as to be fometime? <bi!tlV, 
 fometimes a point or two to the weft ; it then only remains to piy :o 
 windward. And if, on the one fide^ they, fteer W. S. W. they get a 
 wind more and more eafterly ; but then there is a danger of falling in 
 with the Brafilian coaO, or fhoals ; and if they fteer E. S. E. they fall 
 into the neighbourhood of the coaft of Guinea, from whence they cannot 
 depart without running eafterly as far as the ifland of St. Thomas j and 
 this is the conftant pra6lice of all the Guinea fhips. 
 
 Secondly. All fhips departing from Guinea for Europe, their dire6l 
 courfe is northward j but on this courfe they cannot go, becaufe the 
 coafl: bending nearly caft and weft, the land is to the northward : 
 Therefore as the winds on this coaft are generally between the S. and 
 W. S. W. they are obliged to fteer S. S. E. or fouth, and with thefe 
 courfes they run off the fhore ; but in fo doing they always find the 
 winds more and more contrary ; fo that when near the fhore, they can 
 lie fouth, at a greater diftance they cm make no better than S. E., and 
 afterwards E. S. E. ; with which courfes they commonly fetch the ifland 
 of St. Thomas and Cape Lopez, where finding the winds to the eaft- 
 ward of the fouth, they fail wefteriy with it, until they come to the 
 latitude of 4 degrees fouth, where they find the S. E. wind blowing per- 
 pctualJy. 
 
 On account of thefe general winds, all thole who ii{e the Weft India 
 trade, even thofe bound to Virginia, reckon it their beft courfe to get as 
 foon as they can to the fouthv/ard, that (o they may be certain of a fair 
 and frcfh gale to run before to the weftward. And for the fame reafon 
 the homeward bound fhips from America endeavour to gain the latitude 
 of 30 degrees, where they firft find the winds begin to be variable j 
 though the moft ordinary winds in the north Atlantic Ocean come from 
 betvv'cen the fouth and weft. 
 
 icth. BetweeJi the fouthern latitudes of 10 and 30 degrees in the In- 
 dian Ocean, the general trade-wind about the S. E. by S. is found to blow 
 all the year long in the fame manner, as in tlie like latitude in the Ethi- 
 opic Ocean : and during the fix months from May to December, thefe 
 v/inds reach to within 2 degrees of the equator; but during the other fix 
 months, from November to June, a N. W. wind blows in the tra<5t lying 
 between the 3d and i oti^vdegeees of fouthern latitude, in the meridian of 
 the rorth end of Madagafcar : and between the 2d and i2th degree of 
 fouth latitude, nsar the longitude of Sumatra and Java. 
 
 iith, jfrii
 
 Bock VL GEOGRAPHY; 357 
 
 I ith. In the trat between Sumatra and the African coaft, and from 3 
 degrees of fouth latitude quite northward to the Afiatic coafts, including 
 the Arabian Sea and the Gulf of Bengal, the Monfoons blow from Sep- 
 tember to April on the N. E. ; and from March to Odlober on the S. W- 
 lii the former half year the wind is more fteady and gentle, and the 
 weather clearer than in the latter fix months ; and the wind is more 
 flrong and fteady in the Arabian Sea than in the Gulf of Bengal. 
 
 1 2th. Between the Ifland of Madagafcar and the coafl: of Africa, and 
 thence northward as far as the equator, there is a trail, where from 
 April to October there is a conftant frefti S. S. W. wind ; which to the 
 northward changes into the W. S. W. w^ind, blowing at that time in the 
 Arabian Sea, 
 
 13th. To the eaftward of Sumatra and Malacca, on the north of the 
 equator, and along the coafts of Cambodia and China, quite through 
 the Phillippines as far as Japan, the Monfoons blow northerly and 
 foutherly, the northern fetting in about October or November, and the 
 fouthern about May ; -the winds are not quite fo certain as thofe in the 
 Arabian Seas. 
 
 14th. Between Sumatra and Java to the weft-, and New Guinea to the 
 eaft, the fame northerly and foutherly winds are obferved ; but the firlt 
 half year Monfoon inclines to the N. W. and the latter to the S. E. 
 Tbefe winds begin a month, or fix weeks after thofe in the Chinefe Seas 
 fet in, and are quite as variable. 
 
 15th. Thefe contrary winds do not fhift from one point to its oppofite 
 all at once; in fome places the time of the, change is attended with 
 calms, in others by variable winds. And it often happens on the fhores 
 of Coromandel and China towards the end of the Monfoons, that there 
 are moft violent ilorms, greatly refembling the hurricanes in the Weft 
 Indies ; when the wind is fo valtly flrong, that hardly any thing can rcTiif: 
 its force. 
 
 All navigation in the Indian Ocean muH: necelKirily be regulated bv 
 thefe winds ; for if mariners fhould delay their voyages till the contrary 
 Monfoon begins, they mull either fail back, or go int-j harbourj aud ws\% 
 for the return of the irade-wind. 
 
 5 K C T I G X
 
 75^ GEOGRAPHY. Bb6k VI. 
 
 * SECTION VII. 
 
 Of the Tides. 
 
 S5. A Tide is that motion of the waters in the fcas and rivers, by 
 which they are foun^ regularly to rife and fall. 
 
 The general caufe of the tides was difcovercd by Sir Ifaac Newton, 
 and is deduced from the following confidcrations. 
 
 86. Daily experience fhews, that all bodies thrown upward^s from the 
 tarth, fall down to its furface in perpendicular lines ; and as lines per- 
 pendicular to the furface of a fphere tend towards the center, therefore the 
 lines, along which all heavy bodies fall, are dire6led towards the Earth's 
 center. 
 
 As thcfe bodies apparently fall by their weight, or gravity ; therefore 
 the law by which they fall, is called the Law of Gravitation. 
 
 87. A piece of glafs, amber, or fealing-wax, and fome other things, 
 being rubbed againft the palm of a hand, or againft a woollen cloth, until 
 they are warmed, will draw bits of paper, or other light fubftances, to- 
 wards them, when held fufficiently near thofe fubftances. 
 
 Alfo a magnet, or loadftone, being held near the filings of iron or fteel, 
 or other fmall pieces of thefe metals, will draw them to itfelf j and a piece 
 of hammered iron or fteel, that has been rubbed by a magnet, will have a 
 like property of drawing iron or fteel to itfelf. And this property is 
 called Attraction. 
 
 88. Now as bodies by their gravity fall towards the Earth, it is not 
 improper to fay the Earth attracts thofe bodies ; and therefore in refpe<5t 
 to the earth, the words gravitation and attraction may be ufed one for the 
 other, as by them is meant no more than the power, or law, by which 
 bodies tend towards its center. 
 
 And it is likely, that this is the caufe, why the parts of the Earth adhere 
 and keep clofe to one another. 
 
 89. The incomparable Sir Ifaac Newtjon, by a fagacity peculiar to 
 himfelf, difcovered from many obfervations, that this lawof gravitation or 
 attradiiou v/as univerfally dift'ufcd throughout the folar fyftem ; and that 
 the regular motions obfcrved among the heavenly bodies were governed 
 by this fame principle ; fo that the Earth and Moon attracted each other, 
 and both of them are attracted by the Snn. He difcovered alfo that the 
 force of attraction, exerted by thelc bodies one on the other, was lefs and 
 Icis as the diftance incrcafed, in proportion to the fquares of thofe dif- 
 tanccs that is, the povv'cr of attrachon at double the diftance was four 
 times kls, at triple the diftance nine times lefs, at quadruple the diftance 
 lixteen times lefs, and fo on. 
 
 90. Nov/ as the Earth is attraclcd by the Sun and Moon, therefore all 
 the parts of the Earth will not gravitate towards its center in the fame 
 maiuicr, as if thofe parts were not afteclcd by fuch attractions. And it 
 is very evident that, were the Earth entirely free from fuch actions of the 
 Sun and Moon, the ocean being equally attracted towards its center, on 
 all fides, by the force of gravity, would continue in a perfect ftagnation 
 without ever ebbing or flowing. But fmcc the cafe is otherwife, the water 
 ill the ocean muft needs rife higher in thofe places where the Sun jmkI 
 
 IXIoon.
 
 Book VI. GEOGRAPHY* 55^^ 
 
 Moon diminifh Its gravity, or v^here the Sun and Moon hare the greatef^ 
 attra<51;ion. 
 
 As the force of gravity muft be diminifiied nioft in thofe parts of the 
 Earth to u^hich the Kloon is neareft, that is, where he is in the Zenith, 
 or vertical, and, confequently, where her attracftion is moft powerful ; 
 therefore the waters in fuch places will rife higheft, and it will ht full fea 
 ox flood in fuch places. 
 
 9 1 . T^he parts of the Earth dlreSfly under the Moon^ and alfo thofe in her- 
 Nadir, viz. fuch places as are diametrically oppoftte to thofe where the Moon 
 is in the Zenith^ will have the flood^ or high ivater^ at the fame time. 
 
 For either half of the Earth would gravitate equally towards the other 
 half, were they free from all external attradion. 
 
 But by the adtion of the moon, the gravitation of one half-earth towards 
 its center is diminiflied, and of the other is increafed. 
 
 Now in the half-earth next the Moon, the parts in the zenith being 
 moft attrafled, and thereby their gravitation towards the Earth's center 
 diminiflied, the waters in thefe parts muft be higher than in any other 
 part of this half-earth. 
 
 And in the half-earth fartheft from the Moon, the parts in the nadir 
 being lefs attraded by the Moon than the parts nearer to her, gravitate 
 lefs towards the Earth's center, and confequently the waters in thefe 
 parts muft be higher than they are in any other part of this half-earth. 
 
 92. Thofe parts of the Earth, wht'rs the Moon appears in the horizon, or 
 is 90 degrees dljiant from the zenith and nadir, ivill have the ebbs or louvji 
 waters. 
 
 For as the waters in the zenith and nadir rife at the fame time, 
 the waters in their neighbourhood will prcfs towards thofe places to 
 maintain the equilibrium ; and to fupply the places of thefe, others will 
 move the fame way, and fo on to places of 90^ diftant from the faid 
 zenith and nadir j confequently, in thofe places where the Moon 
 appears in the horizon, the -waters will have more liberty to defccnd 
 towards the center ; and therefore in thofe places they will be the 
 ioweft. 
 
 93. Hence it plainly follows, that the ocean, if it covered the furfacc 
 of the Eaith, muft put on a fpheroidal, or egg-like figure ; in which 
 tlie longeft diameter pafles through the place \vhcre the Moon is vertical ; 
 and the fhorteft diameter, will be, in the horizon of that place. And 
 as the Moon apparently (hifts her pofition from eaft to well: in going 
 round the Earth every day, the longer diameter of the fj^hcroid follow- 
 ing her motion, will occalion the two floods and ebbs obf'crvable in about 
 every 25 hours, which is about the length of a lunar day, or the time 
 fpcnt between the Moon's leaving the meridian of any place, and com:;ig 
 to it again. 
 
 94. Hence the greater the Moon's meridian ahitudc is at any {^iace, 
 tlic greater thofe tides v/ill be v/liich happen when ^hc is above the ho- 
 rizon J and the greater her meiidian depreilian is, the greater will tliofc 
 tides be which happen while ftie is below f'lc horizon. 
 
 Moreover the funimcr day, and the winter niL^ht-tilcs have a ten- 
 dency to be higheft, bocaufc the Sun's fummcr aititufic and his wiriter 
 dcnrcilion are t rcatcft , but this i> mor'.: ci':-.eci:;llv to lie noted when- the 
 
 :'> ' Moo:i
 
 269 GEOGRAPHY; Book VL 
 
 ^oon has north declination in fummer, arid fouth declination in 
 winttr. 
 
 95. The time of high water is not precifely at the time of the MoorCs 
 coming to the meridian^ but about an hour after. 
 
 For the moon ads with fome force after fhe has paft the meridian, 
 and by that means adds to the llbratory, or waving motion, which fhe 
 had put the water into, whilft fhe was in the meridian ; in the fame 
 manner as a fmall force applied upwards to a ball, already raifed to fome 
 height, will raife it flill higher. 
 
 96. The tides are greater than ordinary twice every month ; that isj about 
 ihe times of new and full Moon : thefe are called Spring-Tides. 
 
 For at thefe times the actions of both Sun and Moon concur to draw 
 in the fame right line j and therefore the fea muft be more elevated. In 
 conjuniion, or when the Sun and moon are on the fame fide of the Earth, 
 they both confpire to raife the water in the zenith, and confequently in 
 the nadir. And when the Sun and Moon arc in oppofition^ that is, when 
 the Earth is between them, whilft one makes high water in the zenith 
 and nadir, the other does the fame in the nadir and zenith. 
 
 97. The tides are lefs than ordinary Hvice every month ; that is, about the 
 times of the fir Jl ayid laji quarters of the Moon : and thefe are called Neap 
 Tides. 
 
 Because in the quarters of the Moon, the Sun raifes the water where 
 the Moon deprefl'es it; and <ieprefles where the Moon raifes the watery 
 k> that the tides are mad^ only by the difference of their actions. 
 
 It mufl be obferved, that the fpring tides happen not direcStly on the 
 new and full moons, but rather a day or two after, when the attractions 
 of the Sun and moon have confpired together for a confiderable time. 
 In like manner the neap-tides happen a day or two after the quarters, 
 when the moon's attraction has j^een leflencd by the Sun for feveral days 
 together. 
 
 98. JVhen the Mcon is in her Perig^um, or nearejl approach to the 
 Barth.^ the tides increafe more than in the fame circumjiances at other times, -' 
 
 For according to the laws of gravitation, the Moon muft attract moft 
 when fhe is nearcft to the Earth. 
 
 99. The fpring-tidei are greater about the time of the EQUINOXES, that 
 is, about the latter ends of March and September^ than at tther times of the 
 ytar ; and the neap-tides then are lefs. 
 
 Because the longer diameter of the fpheroid, or the two oppofite 
 floods, will at that time be in the Earth's equator j and confequently 
 will defcribe a great circle of the Earth ; by the diur-nal rotation of 
 which thofc floods will move fwifter, defcribirig a great circle in the 
 lame time they ufcd to defaribe a leifer circle parallel to the equator, and 
 confequently the waters being thrown more forcibly againft the fhores, 
 muft rife higher. 
 
 100. The follov/In2: obfcrvations have been made on the rife of the 
 tides. 
 
 ^ ift. The morfiing tides generally differ in their rife frem the evening- 
 tides. 
 
 2d. The new and full Moon fpring-tides rife to different heights. 
 
 ^. In wimer the morning-tides are higheft, 
 
 4th. In
 
 Book VI. I G E O G R A P H Y. 561 
 
 ' 4th. In futnmer the evening-tides are higheft. 
 
 So that after a perioB of about fix months the order of the tides are 
 
 inverted ; that is, the rife of the morning and evening-tides wiW chang':? 
 
 places, the w^inter morning high-tides becoming the fummer evening 
 
 high-tides. 
 
 Some of thefc effects arife from the different diftances of the Moon 
 from the Earth after a period of f.x months, when flic is in the fame 
 fituation with refpecl to the Sun ; for, if fhe is in perigee at the time of 
 new m.oon, in about fix months after {he v;ill be in perigee about the time 
 of full Moon. 
 
 T'hefc particulars being known, a pilot may chufe that time,'v/hich i 
 moft convenient for conducting a ihip in or out of a port, w";:cre there is 
 not fufficient depth at low-water. 
 
 Small inland feas* fuch as the Mediterranean and Baltic, are little 
 fubjedt to tides ; becaufc the acrtion of the Sun and Moon is always 
 nearly equal at both extremities of fuch feas. In very Wigh latitudes the 
 tides are alfo very inconfiderable. For the Sun and Moon acting towards 
 the equator, and always raifing the water towards the middle of the torrid 
 zone, the neighbourhood of the poles muft confequently be deprived of 
 thofe waters, and the fea muft, within the frigid zoneSj be low, with re- 
 lation to other parts. 
 
 iot. All the things hitherto explained would exa<tly obtain, were the 
 whole furface of the Earth covered v/ith fea. But fince it is not fo, and 
 there being a multitude of iflands, beHdes continents, lying in the way of 
 the tide, which interrupt its courfc ; therefore in many })laces near the 
 fliores, there arifes a ereat variety of other appearances befide the forego-^ 
 ing ones which require particular folutions, in wliich the htuations of the 
 fliorcs, ftraits, Ihoiils, winds, and other things, muft neceflarily be confi- 
 dered. For inltance : 
 
 102. As the lea* has no vifj^je paffage between Europe and Africa^ 
 let them be fuppofed to be une continent, extending from 78 degrees 
 north to 3.^ de^^^rces fouth, the middle between thefe two would be in lati-^ 
 tudc 19 degrees north, mar Cape Blanco, on the welt coaft of Africa. 
 But it is impoffible that the flood-tide fhould fet to the weltv/ard upon 
 the v/c'ftern coaft of Africa (for the general tide following the courfe of 
 tiic Moc n nnut fet from eaft to w.ft), becaufe the continent, for above 
 50 degree^, both northward and fouth ward, bounds that fea on the eaft ; 
 therefore if any regular tide, proceeding from the motion of the fea, from 
 ,ilt to w^il, ftiould reach this place, it muft come either from the north 
 of Europe fouthward, or from the fouth of Africa northward, to the faiJ 
 latitude upon the weir coaft of Africa. 
 
 i::2- 'i'his opinion i~, further corroborat:?d, or rather fully confirmed 
 by common experience, whicii flicws that the flood-tide fcts to the fouth- 
 ward r'.org the wc^ coa{l of Nozvay, from the north cape to the Nc:zcy 
 o: cnlr.-'nc:- of the Bultic .?c.v, and 10 proceeds to the fouthward along the 
 c: r. c(^ 1^ ". (irc.it i^viL'.in^ and in it': p.'lT:,^? fupplics rd) rhofc ports wi'Ji 
 the tiJ: :; :.i;(ur:cr, rhj coafl of Scotldvtl hrw'-;;!; t'; tido firll, b- 
 
 * rai{':i.s''-i Gc^[-r. p. ^\;\. lidit. 173^.
 
 361 GEOGRAPHY. Book VI. 
 
 caufc it prcccfds from the northward fb the foutbward ; and thus on the 
 days of the full, or change, it is high-water at Aberdeen at i2h. 45 m. 
 but at Tinmcuth Bar^ the lame day, not till 3h. From thence rolling to 
 the fouthward, it makes high-water at the Spurn a little after 5h. ; but 
 not till 6 h. at Htilly by reafon of the time required for its paflage up the 
 river j from thence pafling over the TVelbank into Tartnouth Road, it 
 makes high- water there a little after 8 h. but in the Pier not till 9 h., and 
 it requires near a.n hour more to make high-water at Tarmouth town ; in 
 the mean time fetting away to the fouthward, it makes high-water at 
 Harivich at lOh. 30 m., at the ]>Jore at 12, at Gravcfend i h. 30 m., and 
 at Lcndon at 3h. all On the fame day. And although this may fecm to 
 contradi<St that hypothefis of the natural motion of the tide being from 
 eaft to v^ert, yet as no tide can flow weft from the main continent of Nor" 
 tvay or Holland^ or out of the Baltic, which is furrounded by the main 
 continent, except at its entrance, it is evident that the tide we have been 
 row tracing by its feveral ftages from Scotland to London, is fupplied by 
 tke tide, the original motion of which is from eaft to weft ; yet as water 
 always inclines to the level, it v/ill in its paflage fall towards any other 
 point of the compafs, to fill up vacancies where it finds them, and yet not 
 contradid, but rather confirm, the iirft hypothefis. 
 
 104. While the tide, or high-water, is thus gliding to the fouthward 
 along the eaft coaft of England^ it alfo fets to the fouthward along the 
 weft coafts of Scotland and Ireland, a branch of it falls into St. George's 
 channel, the flood running up north-eaft, as may be naturally inferred 
 from its being high-water at IVatcrford above three hours before it is 
 high-water at Dublin, or thereabout, on that coaft ; and it is three 
 quarters of an hour ebb at Dublin before it is high-water at the Jjle of 
 Alan, Sic. 
 
 But not to proceed farther in particulars than to our own, or the 
 Britijl) channel, we find the tides fct to the fouthward from the coaft of 
 Ireland, and in their paflage a branch falls into the Britijl) channel be- 
 tween the Lizard and JJfliaiit \ this progrefs to the fouthward may be 
 ealily proved, by its being high-water on the day of the full and chajiges 
 at Cafe Clear a Uttle after 4h., and at Ujliant about 6h., and at the 
 Lizard after 7 h. The Lizard and Ujhant may properly be tailed the 
 chaps of the Britifh channel, between which the flood fets to the eallward 
 along the coafts o{ England d.x\A France, till it comes to the Goodwin or 
 Galloper, where it meets the tide before mentioned, which fets to the 
 I'outhward along the caftcrn coaft of England to -the Doivns, where thefe 
 two tides meeting, contnbute very much towards fending a powerful tide 
 i:p the river Thames to Lcndon. And when the natural courfe of thefe 
 two tides has been interrupted by a fudden ftiift of the wind, by which 
 means tlrat tide was accelerated which had before been retarded, and that 
 driven h:xk which was before hurried in by the wind, it has been known 
 to occafion twice high-water in 3 or 4 hours, which, by thofe who did 
 nor confider this natural caufe, was looked upon as a prodigy. 
 
 105. But now it maybe objefted, that this courfe of the flood-tide, 
 e: If, or eafl-north-caft, up the channel, is quite coiUrary to the hypo- 
 ii'.'S.b of tije gcncriil motion of the tides being from eaft to weft, and con- 
 kc'.:cntly of its being high-water where the moon is vertical, or any 
 v.r.erc LJle ia the meridian. 
 
 In
 
 B<H5k VI. GEOGRAPHY, 363 
 
 In anfvirer. This particular diretion of any branch of the tide does 
 not at all contradifl the general direction of the whole ; a river, with 
 a weftern courfe, may fupply canals which wind north, fouth, or even 
 eaft, and yet the river keep its natural courfe ; and if the river ebbs and 
 flows, the canals fupplied by it would do the fame, although they did 
 not keep exadl time with the river, becaufe it would be flood, and the 
 river advanced to fome height, before the flood reached the farther part 
 of the canals ; and the more remote the canals are, the longer time it 
 would require; and it maybe added, that if it was high-water in the 
 river juft when the moon was on the meridian, (he would be far paft it 
 before it could be high-water in the remoteft part of thofe canals, or 
 ditches, and the flood would fet according to the courfe of thofe canals 
 that received it, and could not fet weft up a canal of a different pofition ; 
 and as St. George's channel, the Britijh channel, &c. are no more in 
 proportion to the vaft ocean, than thefc canals are to a large navigable 
 river ; it will evidently follow, that among thofe obftruftions and con- 
 finements the flood may fet upon any other point of the cotnpafs as weU 
 as weft, and may make high- water at any other time as well as when 
 the Moon is upon the meridian, and yet no way contradicl the genera!- 
 theory of the tides before aflerted. 
 
 106. When the time of high-water at anyplace is, in general, men- 
 tioned, it is to be underftood on the days of ihe fyzygies, or days of new 
 and full Moon, when the 3un and Moon pafs the meridian of that place 
 -at the fame time. Among pilots it is cuftomary to reckon the time 
 of flood, or high-water, by the point of the compafs the Moon is on at 
 that time, allowing | of an hour for each point : Thus on the full and 
 change days, in places where it is flood at noon, the tide is faid to flow 
 north and fouth, or at 12 o'clock ; in other places, on the fame days, 
 where the Moon bears i, 2, 3, 4, or more points to the eaft or weft of the 
 meridian, when it is high-water, the tide is faid to flow on fuch point : 
 Thus, if the Moon bears S. E. at flood, it is faid to flow S. E. and N.W. , 
 or 3 hours before the meridian, that is, at 9 o'clock ; if it bears S. W., it 
 flows S. W. and N. E., or at 3 hours after the meridian ; and in like 
 manner for other points of the A'loon's bearing. 
 
 107. In fome places it is high water on the fhore, or by the ground, 
 while the tide continues to flow in the ftream, or offing ; and according 
 to the length of time it flows longer in the ftream than on the ftiore, it is 
 faid to flow tide, and fuch part of tide ; allowing 6 hours to a tide ; 
 Thus 3 hours longer in the offing than on the ftiore, make tide and half- 
 tide ; an hour and half longer makes tide and quarter-tide ; three quar- 
 tees of an hour longer make tide and half-quarter tide ; &c. 
 
 108. Along an extent of coaft next to the ocean, iuch as the weftern 
 coaft of Africa, and the eaftern and weftern coafts of South America, 
 it is generally high-water about the fame hour. But ports on the coafts 
 of narrow feas, or within land, have the times of their higli-vvatcr 
 fooncr or later on the fame day, according as thofe ports arc farther re- 
 moved from the tide's way, or have their entrances more or iefs con- 
 tracted. 
 
 109. The times of high-water, in anyplace, fill about the fame hours 
 after a period of 15 days nearly* which is the time between one fpring- 
 
 C c 2 ude
 
 3^4 
 
 G E O G R A P H Y, 
 
 Book VI 
 
 tide and another: and during that period, the times of high-water fall 
 each day later by about 48 minutes. 
 
 110. From the obfervations of different pcrfons, the times \Vhen it is 
 high-water on the days of the new and full Moon, on moft of the fea- 
 coafts of Europe, and many other places, have been colleted. Thefe 
 times are ufually put in a table againft the names of the places, digefted 
 in an alphabetical order ; the like is followed in this work, only they are 
 not given in a table by themfclves, but make a column in the table of 
 the latitudes and longitudes of places ; which column is not filled up 
 againll many of^the names in that table, for want of a fufficient cullcL^iiori 
 of obfervations. This may be fupplied by thofe who have oppoitunity 
 and inclination. 
 
 The ufe of fuch a table is to find the time when it is high-water at 
 any of the places mentioned in it. But as this depends upon knowing 
 the time when the Moon comes to the meridian, and this on the Moon's 
 age, and this on the knowledge how to find fome of the common notes 
 in the Calendar ; therefore it was thought convenient to introduce ini 
 this place a compendium of Chronology, containing the feveral article* 
 above mentioned. 
 
 SECTION VIII. 
 
 Of Chro7iology 
 
 III. Chronology is the art of cftimating, and comparing together,, 
 the times when remarkable events have happened, fuch as are leiutcd in 
 hiflory. 
 
 An tT'ra, or Epoch A, is a time wlicn fome memorable tranfadion 
 occurred ; and from which feme nr.tior.s d:;tc and i.icafure thuii com- 
 putations of time. 
 
 , Years of j Years | 
 [theJuriiMij before | 
 i Period. Chril'-. 
 
 Some h-\ve dated their events from tli? crratiDn "l 
 
 f the world, and fupnofc it to h.vc h;.pj;cned j 
 Other?, from the delude, or Hood 
 
 Tlic (jreeks, from their Olympiad,-; of 4 yer.rs 1 
 
 each i 
 
 'I'he Romans, from tlie buiidin^r of Rome 
 
 The Afi:ronorner&, fror.i NaboiiaQ.r king of ) 
 
 Babylon ) 
 
 Some Hiflorians, from the death of Alexaiu'cr ) 
 
 the Cjrcut - ) 
 
 The Chridians, from the birth of Chrif!: 
 
 Tht; Mi^hi metans, from tlie flight of iMaho- 7 
 
 mci, called the Hegira \ 
 
 710 
 
 4004 
 
 5.366 
 
 23-, 8 
 
 3938 
 
 776 
 
 3961 
 
 753 
 
 3967 
 
 747 
 
 439'^ 
 
 324 
 
 4713 
 
 A. D. 
 
 5335 
 
 622 
 
 In
 
 Book VI. GEOGRAPHY. , 365 
 
 In order to afllgn the diftance between thefe, and other events, the an- 
 cients found it necefrary to h?.ve a large meafure of time, the limits of 
 which were naturally pointed out to thciii by the return of the feafons 
 and this interval they crJled a year. 
 
 U2. The moft natural divifion of the year appeared to them to be 
 the returns of the new Moon; and as they obfwTved 12 new moons to 
 happen w^ithin the time of the general return of the feafons, they there- 
 fore firft divided tiie year into 12 equal parts, which they called months'; 
 and as they reckoned about 30 returns of morning and evening between 
 the times of tiie new moon and new moon, therefore they reckoned their 
 month to confiil of 30 days, and their year, or 12 months, to contain 
 360 days ; and this is what is generally underftood by the lunar year of 
 the ancients. 
 
 113. But in length of time it was found, that this year did not agree 
 with the courfe of the fun, the feafons gradually falling later in the year 
 than they had been formerly obferved ; this put them upon correcting the 
 method of eftimating their year, which they did from time to time, by 
 taking a day or two from the month, as often as they found it loo long 
 for the courfe of the moon ; and by adding a month, called an intercalary 
 month, as often as they found 12 lunar months to be too fhort for the 
 return of the four feafons and fruits of the Earth. This kind of year, fo 
 corrected from time to time by the priefls, whofe bufmefs it was, is what 
 is to be underftood by the htni-folar year ; v/hich was anciently ufed ia 
 moft nations, and is ftill amoiig the Arabs and Turks. 
 
 As a great variety of methods was xiiCid in different countries to corre^ 
 the length of the year, fome by intercalating days in every year, and 
 others by inferting months and days in certain returns, or periods of years ; 
 and thefe different methods being obferved by fome eminent men to 
 create a confiderable difference in the accounts of time kept by neigh- 
 bouring nations, introducing a confufion in the chronological order of 
 times, they therefore invented certain periods of years, called cycles, with 
 which they compared the moft memorable occurrences. 
 
 114. At length Julius Ctrfar ohicvvxng the confufion which this va- 
 riety of accounts occafioned, and knowing that his order, as Emperor of 
 the Romans, would be followed by a very confiderable part of the world y 
 he therefore, about 40 years before the birth of Chrift, decreed that 
 every fourth year fhould confift of 366 days, and the other three of 
 365 days each. 'J'his he did in confcquence of the information given 
 bim by Sofigcncs^ an eminent mathematiciiui of Alexandria in E;^^,ypt ; for 
 at that time the philofophers of the Alcxandrinian fchool knew from 
 a length of experience, that the year conliftcJ of about 365 days and a 
 quarter; and this v/as the rcafon of ordering every fourth year to confift 
 of 366 days, thereby compcnfating for ihe quarter day omitted in each 
 of the preceding three years. 'I'iiis nxthod, called the Julia AcL.unt, 
 or Old utile, continued to be ufed in moft Clirifiian llatei> until tne year 
 IS82. 
 
 The Aftronomcrs, fince tli:; tiinc'of y./Z/.v; Cucfir, h?.vc found that the 
 true length of the folar yec.r, or co:v...-on year, is 365 Aw^-, 5 liours, 
 4S minutes, 55 fcconds, n.ar'.y ; bein^ \-S^. than tiic Julian, of 365 days, 
 C hours, by about \\ nii-.uics, 5 f.conds, which is ab^ut the 130th 
 
 C c 3 part
 
 ^66 ' GEOpRAPHY. Book VI. 
 
 part of 86400, the feconds contained in a day; (b that in 130 Julian 
 years thcic would be one day gainecl above 130 folar years j and in 400 
 Julian years there would be gained 3 days, i hour, 53 minutes, 2,0 
 feconds ; confequently one day omitted in ^very 130 common years, 
 would bring the current account of time to agree very nearly with the 
 motion of the Sun. 
 
 115. In the year of our Lord 325, when the Council of Nice fettled the 
 day for the celebration of Eafter, the Vernal Equinox (that is, the day 
 in the fpring whtu the Sun rofe at fix and fct at fix) happened on the 
 2ift of March ; but about the year 1580 the Vernal Equinox fell on the 
 Xllh of. March, making a difference of about 10 days. Now Gregory 
 the Xlllth, who was Pope at that time, obferving that this difference of 
 time in the falling out of the Equinox would affe<ft the intention of the^ 
 Nicene Council^ concerning the time of the year appointed, by them for 
 the celebration of Eafter j he therefore, in the year 1581, publifhed a 
 Bully ordering that in the year 1582, the 5th of Odober fhould be called 
 the 15th, and fo on ; thus the 10 days taken off would caufe the time of 
 the Vernal Equinox to fall on the 21ft of March, as at the time of the 
 Nicene Council', and becaufe a little more than three days were gained 
 in every 4C0 years by the 'Julian account j therefore to prevent any 
 future difference, every century, or aumber of 100 years, not divifible 
 by 4, fuch as 17 hundred, i8 hundred, 19 hundred, &c., fhould con- 
 tain only 365 days, which, by the Julian account, fhould have con- 
 tained 366 ; and the centuries divifible by 4, luch as 16 hundred, 20 
 hundred, 24 hundred, &c. fhould be leap-years of 366 days ; and thus 
 the three days would be omitted, which the anticipation of the equinoxes 
 would gain in 400 years ; the fmall excefs of 1 hour, 53 minutes, 20 
 feconds, not amounting to a whole day in lefs than 5082 years, being 
 rejected as inconfidcrable ; the intermediate years to be reckoned as they 
 tifed to be in the Julian or Old Stile. This Pope's alteration, called the 
 Gregorian or Neiv Stile, was received in moft of the Chriflian flates : 
 But fomc at that time chofe to continue the Julian;, among whom were 
 the Englifli ; and they, in the year 1752, reformed their account, and 
 introduced among themfelves a new one, that nearly correfponds with the 
 Gregorian. 
 
 A certain length of the year being once fettled, and a regular account 
 of time in confcqucnce of it being fo fitted, as to conform invariably to 
 the feafons ; it was natural for the States who received fuch account, to 
 fit to it a regifler of the days in each month, and to note the days when 
 any remarkable occurrence was to be commemorated ; and fuch regifler 
 l)as obtained the name oi Calendar. 
 
 lib. The Calendar now in ufe among mofl of the Chriflian flatci 
 confifls of 12 months, called January, February, March, Apr il.y May ^ 
 June, July, Augujl, September, Oilober, November, December; thefe 
 months are called civil, the number of days in each may be readily re- 
 membered by the following rule. 
 
 117. Thirty days has September, April, June, and November; 
 
 February has twenty-eight alone : All the reft have thirty-one ; 
 
 When the year confifts of 365 days : But in every fourth, which con- 
 
 fifts cf 366 days, February has 29. This additional day was interca- 
 
 9 lated
 
 Book VI. GEOGRAPHY. 367 
 
 laled after the 24th of February, which in the Old Roman Calendars was 
 called thcfixth of the calends of March, and being this year reckoned twice 
 over, the year was called Biffextilis, or Leap-Year. 
 
 Befide the months, time is alfo divided into weeks, days, hours, mi- 
 nutes, &c, a year containing 52 weeks, a week 7 days, a day 24 hours, an 
 hour 60 minutes, &c. 
 
 In the Calendars it has been ufual to mark the feven days of the week 
 with the feven firfl letters of the alphabet, always calling the firft of Ja- 
 nuary A, the 2d B, the 3d C, the 4th D, the 5th E, the 6th F, and the 
 7th G, and fo on, throughout the year : and that letter anfwering to all 
 the Sundays for a year, is called the Dominical Letter. 
 
 According to this difpofition, the letters anfwering to the firft day of 
 every month in the year, will be known by the following rule : 
 
 118. At Dover Dwells George Brown, Efquire, 
 Good Caleb Finch, and David Frier j 
 
 Where the firft letter of each word anfwers to the letter belonging to the 
 firft day of the months in the order from January to December. 
 
 119. A year of 365 days contains 52 weeks and i day j and a leap- 
 year has 52 weeks and 2 days j therefore the firft and laft days of a com- 
 mon year fall on the fame week-day, fuppofe it Monday, then the nexr 
 year begins on a Tuefday, the next year on IVednefday, and fo on to the 
 eighth year, which would be on Monday again, did every year contain 
 365 days ; alfo the Dominical letter would run backwards through all 
 the feven letters. But this round of feven years is interrupted by the 
 leap-years ; for then February having a 29th day annexed, the firft Do- 
 minical letter in March muft fall a day fooner than in the common year ; 
 fo that leap-year has two Dominical letters, the one (fuppofing G) ferv- 
 ing for fanuary and February, and the other, which is the preceding letter 
 (F) ferves for the reft of the Sundays in that year. 
 
 120. The Solar Cycle, or cycle of the Sun, is a period of 28 years, 
 in which all the varieties of the Dominical letters will have happened, and 
 they will return in the fame order as they did 28 years before. '^At thQ 
 birth of Chrift 9 years had paft in this cycle. 
 
 For the changes, were all the years common ones, would be 7; 
 
 But the interruptions by leap-year being every fourth year ; 
 
 Therefore the changes will be 4 times 7, or 28 years. 
 
 This return of the Dominical letter is conftant in the fulian account. 
 But in the Gregorian, where among the complete centuries, or hundredth 
 years, only every fourth is leap-year ; the other three hundred years, 
 which according to the Julian, would be leap-years, are by the Gre- 
 gorian only common years of 365 days : in thefe all the letters muft be 
 removed one place forwards in a diredl order ; and either year, inftead of 
 having two Dominical letters (as fuppofe D, C), will have only one 
 (as D), the Dominical letters moving retrograde. 
 
 12 1. The Lunar Cycle, or cycle of the Moon (and fomctimes 
 called the Metonic Cycle, from Melon, an Athenian who invented it 
 .ibout 432 years before the time of Chriji), is a period of nineteen years. 
 containing all the variations of the days on which the new and full 
 Moons happen j after which they fall on the fame days they did 19 years 
 before. 
 
 C c 4 The
 
 36S 
 
 GEOGRAPHY. Book VI. 
 
 The PrimE) or Golden Number, is the number of years elapfed in ^ 
 this cycle. v 
 
 At tb.c birth of C/t/// the golden rttimbcr was 2. .-^ 
 
 For many years after the Kiwuc Council^ it was thought that 19 folar % 
 years, or 228 folar months, were cxav^tly equal to 235 fynodical, or lunar, , 
 jr.ontlis : and tliat the fame yearly golden nwiiler fet in their calendars 
 a>:ainii the days when the new Moons happened throughout one lunar 
 cycle, would invariably ferve for the new Moons of corrcfponding years 
 thrcui';hout every fucceiuve lunir cycle. But l.ter obfcrvations fhew, 
 that this cycle is Icfs th.an 19 years, by :-- little more than one hour, 
 tvvcnty-cight'' minutes i therefore, the new Moons will, in a little lefs 
 than 311 years, hap-pen a d^y earlier than by the Metonic account ; and 
 confequciitiy al! the feliivals depending on ilie new Moons, will in 
 tline \^e removed into other feafons of the ye?r than thofc which they fell 
 in nt t!icir hrli inftitution : thus the new Moon^. in the year 1750 hap- 
 petieJ above 4^ days earlier than the times fhewn by the calendar. But 
 were the golden numbers, when once prefixed to the proper new Moon 
 c'.iys in a Alctonic period, to be fet a day earlier at the end of every 310,7 
 Vviar^i, a pretty regular correfpondcnce mi^^ht be prefcrvcd between the 
 Ibhr and lunar years. 
 
 122. 7"he EpACT of any year is the Moon's T.^e ^'^^ beginning of diat 
 year > that is, the days pail fmce the lad new Moon. 
 
 The time between new Moon and new Moon is in the ncareft roimd 
 r.'.imbcrs 29^ days ; therefore the lunar year confifting of 12 lunations 
 inuft be et]ual to 354 days, wliich is 1 1 days lefs than the folar year of 
 365 days. Now fuppofing the folar ai'id lunar years to begin together, 
 iho enact is o ; the bcgiiming of the next folar year, the epacf is 1 1 ; 
 ti\-c 3d year the cpacSt is 22 ; the fourth 33, 5cc. But vvi)en the epact 
 c.\ceeds 30, an intercalary month of 30 days is added to the lunar year, 
 m-iking it conliit of 13 months ; fo that the cpa6t at the beginning of the 
 4111 year is only 3, the 5th 14, the 6th 25, the 7th 36, or only 6, on ac- 
 count of the intercalary month ; and fo on to the end of the cycle of 19 
 vears ; at tlie expiration of which the fame enacts would run over again, 
 were the cycle perfeJt j ar.d the epact v/ould alv, ays be 1 1 times the 
 prime. 
 
 123. By the Niccne Couiicil it v/as enacted, 
 
 lit. 'I'hat Lafter-day fliould be celebrated after the vernal equinox, 
 v>hich at that time happened on the 2i{t: of March. 
 
 2d. That it fhould be kept after the full, or 14th day of that Moon 
 v'lhich happened firft after the iift of I\ larch in common years, and firfi: 
 iftcr the 2Cth of March in leap-years. 
 
 3d. l"i;at the Sunday next fo!lov.-i!,:; t!ic I4.th, or day o full iMoon, 
 ihould be Eafler-Sunday : which malt always fall betv,-een the 20th or 
 21 if of March, and 25th of April. 
 
 124. The Moon's Southin'g at any place is the time when fhc 
 comes to the nieridian oi that place, v/liic!. is every day l.Uer by about ^ 
 cA an hour ; becaufe 24, the hours in a day, being divided by 30, the ."lum- 
 ber of times which file pafils the meridir.n between new I.loon and new 
 Moon, will give t =48' fur the retard;.tion of her pah'sge oyer the meri- 
 -un \\ one day. 
 
 The
 
 Book VI. 
 
 GEOGRAPHY. 
 
 3^9 
 
 The Sun and Moon come to the meridian at the fame time on the day 
 of the change, or at new Moon ; alfo the Moon comes to the oppofite 
 part of the lame meridian, when fhe is in oppofition, or at full Moon. 
 Hence between new and full fhe comes to the meridian in the afternoon ; 
 at full fhe comes to the meridian at mid-night j and when paft the full, 
 after mid-night, or in the morning. 
 
 125. The Roman Indiction is a cycle of 15 years, ufed by the an- 
 cient Romans for the times of taxing the provinces. Three years of this 
 cycle were elapfed at the birth of Chrift. 
 
 The DioNYSiAN Period is a cycle of 532 years, arifing by multi- 
 plying together 28 and ig, the folar and lunar cycles ; it was contrived 
 by Dionyfms Exiguus^ a Roman abbot, about the year of Chrift 527, as a 
 period for comparing chronological events. 
 
 The Julian Period contains 7980 years; it arifes by multiplying 
 together 28, 19, 15, the cycles of the Sun, Moon, and Indidtion. Tiiis 
 was alfo contrived as a period for chronological matters ; and its begin- 
 ning falls 710 years before the ufual date of the creation. 
 
 On the principles laid down in the preceding articles depend the folu- 
 tion of the following problems. 
 
 126. Problem I. 7o fir.d whether any given year is leap-year. 
 Rule. Divide the given year by 4 ; if o remains, it is leap-year j if r, 
 
 2, or 3 remains, it is fo many yearr. after. 
 Obferving that the years iSoo, 1900, 21CO, &c. ?rc common years. 
 
 Exam. I. Is 1788 Lap-year? 
 4)1788(447 
 
 Remains o, io it is leap-year. 
 
 Exam. II. Is i-'^l leap-year? 
 4)1787(446 
 
 Remains 3 years pr.fl leap-year. 
 
 127. Pro DL EM II. To fi:ul ihe years :f iJ:: p-Jar^ liir.ar, and indiSiidn cycles. 
 Rule. To the given year add 9 for the fo'ar, i for the lunar, 3 for the 
 indiction: Divide the fums in order b;; 28, 19, 15; the remainder in 
 tacii fhevvs the year of its refpcLiivc cycle. 
 
 Exam. Required the years of the folar ^ lunar ^ and indlrtion cycles for 
 
 ihc- year 1 1'o'] ^ 
 1/C7 
 9 
 
 1707 
 I 
 
 28) I -;()L>{6\ 
 Remains 4rrr:>lar cycle. 
 
 19)17-8(94 
 
 i:S7 
 3 
 
 :Iunar eye. or ";olden N '. ^rrindic"!:. cycle. 
 '*"''' "" :e the 
 
 of 
 
 Whereby it appears f 4th year of*thc 65th loj-.r cvclc ^ fince tli 
 
 dut the year J 787 s 2d )c:-.r of thj 95f.!i iu'ir. '-vcIj ^ birth c 
 
 i^ fix C 5th ye:.r of the :2'.t;, i;:aiwti-M cycle j Ci..ift. 
 
 {28. Pr(*-
 
 370 
 
 GEOGRAPHY. 
 
 Book VI. 
 
 1-28. Problem III. To jhid the Dominical Utter till the year 1800. 
 
 Rule. To the given year add its fourth part, divide the fum by 7 ; 
 the remainder taken from 7 leaves the index of the letter in 
 common years, reckoning a i, b 2, c 3, &c. 
 But in leap-year, this letter and its preceding one (in the retrograde order 
 which thefe letters take), are the Dominical letters. 
 
 Exam. I. For the yinxr 1787. 
 
 4) '787 
 446 
 
 7)2233(3 '9 
 
 Remains o. Then 7 orr7=:G. 
 So G is the Dominical letter. 
 
 Exam. II. For the year 1788. 
 4)1788 
 447 
 
 7)2235(319 
 
 Remains 2. Then 7 2 = 5 = 5 
 
 So F and e are the Dominical letters. 
 
 And in this manner vi^ere the follovi^ing numbers computed. 
 For the Dominical letters during the i8th century. 
 
 Solar cycles i 
 Pom. letters dc 
 
 5 
 
 FE 
 
 Solar cycles 15 16 17 18 
 Dom. letters g f ed c 
 
 19 
 
 B 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 D 
 
 c 
 
 B 
 
 AG 
 
 F 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 A 
 
 GF 
 
 E 
 
 D 
 
 C 
 
 E 
 
 12 
 
 D 
 
 CB 
 
 14 
 
 A 
 
 25 
 BA 
 
 26 
 G 
 
 27 
 F 
 
 28 
 
 
 The year 1800 being a common year, flops the above order, and the 
 follovv^ing are the Dominical letters for the 19th century. 
 Solar cycles i 2 3 4 5 6 7 8 9 10 11 12 13 14. 
 Dom. letters ed c b a gf e d c ba g f e dc b 
 
 Solar cycles 15 16 17 18 19 20 21 22 23 24 25 26 27 28 
 Dom. letters agfed c bagf edcba g f 
 
 129. Problem IV. To find the Epa^ till the year 1(^00, 
 
 Rule. Multiply the golden number for the given year by n, and di- 
 vide the produd by 30 i from the remainder take 11, and it 
 will leave the epadl. 
 
 If the remainder is lefs than 11, add 19 to it, and it gives the epa6l 
 
 Ex. I. To find the epaSi for 1783 
 The Golden number is 17. (127) 
 Multiply by 1 1 
 
 Remains 
 Add 
 
 30)187(6 
 180 
 
 7 
 19 
 
 Ex. II. To find the epaSl for 1786. 
 The Golden number is 01 (127) 
 Multiply by 11 
 
 Subtrafl 
 
 II 
 II 
 
 Remains the Epad =00 
 
 Confequently the Epal is 26 
 
 And thus might the following numbers be found. 
 Ciold.N^i 2 345 6 7 8 9 10 II 12 13 14 15 16 17 18 19. 
 Epadts 29 II 22 3 14 25 6 17 28 9 20 I 12 23 4 15 26 7 18. 
 The epafts here proceed by the diftcrence 11, rejecting thirties. 
 
 130. Pro-
 
 Book VI. 
 
 GEOGRAPHY. 
 
 37 
 
 13Q. PR03I.EM V. To findthe Masn's agt. 
 
 Rule. To the epaft add the number and day of the month j their 
 fum, if under 30, is the Moon's age ; 
 but if it be above 30, take 30 from it, and the remainder will be the 
 Moon's age, or days fince the laft conjun(5lIon. 
 
 The numbers of the months, or monthly epaidls are the Moon's age 
 at the beginning of each month, when the folar and lunar years begin to- 
 gether J 
 
 ^021 23 4 56 78 9 10. 
 
 ( Jan. Feb. Mar. Apr. May, June, July, Aug. Sept. Oa. Nov. Dee. 
 
 And are 
 
 Exam. I. TPl?at Is the Moon's age 
 on the \^th of O^ober^ '^I'^l ^ 
 The epaft is 2 (129) 
 
 The N' of month 8 
 
 The day of the month 1 4 
 
 The fum is 
 
 24 the Moon's 
 age. 
 
 E X A M . 1 1 . TVhat is the Moon 's age 
 on the igth of March, 1786 ? 
 The epad is o. ('29) 
 
 Then o+i +29=3018 the fum of the 
 epaft, number and day of the month. 
 And 30 30=0 is the Moon's age. 
 
 131. The day of next new Moon is readily found by taking her age 
 from 30. 
 
 The day of new Moon in any month is equal to the difference between 
 the fum of the year's and month's epa(5ts, and 30. Thus.; 
 
 On March 29, the Moon is o days old. 
 
 So that new Moon is on the 29th. 
 
 Now c + I = I , is the fum of the epafts. 
 
 Then 30 1 =229, the day of new Moon, as it fhould be. 
 
 132. 
 
 Problem VI. The day of the ?nonth in any year being givetiy to 
 know on what iveek-day it will fall. 
 
 Rule. Find the Dominical letter (128) : alfo the week day on which 
 the firft of the propofed month falls (118) ; and hence the name 
 of the propofed day of the month will be known ; obferving 
 that the ilt, 8th, 15th, 22d, and 29th days of any month fall 
 on the fame week-days. 
 
 Ex. I. On ivhat day of the iveek does 
 the \\th of Oct. fall, in I'j'i-j P 
 The Dominical letter is c. 
 The iRof Oaoberis a, (u8) 
 
 Therefore October 7th is Sunday. 
 Confctjucnily i^th is alfo Sunday 
 
 Ex. IT. In 1788, on what week~ 
 day does the 20th of March fall F 
 (128) jThe Dominical letter is b. (128) 
 
 The ift of March is d, (m8) 
 
 Then March 2d is Sunday. 
 And fo March 20th is Thuifday. 
 
 133. Pro.
 
 372 
 
 GEOGRAPHY. 
 
 Book VI. 
 
 133. Problem VII. To find when Eajier-day will fall in any year be- 
 tween lyoo and iSgg. 
 
 JR.ULE. Find what day that new Moon falls on which is neareft to the 
 21ft of March in common years, or to the 20th in leap-years ; 
 then tiie Sunday next after the full, or 15th day of that new 
 Moon, will be Eaftcr-uay. 
 If the 15th day fall on a Sunday, the next Sunday is Eafter-day. 
 
 Ex. I. lV>}en dees Eajler- day fall 
 in the year 1787 ? 
 
 The Dominical letter is c. (128) 
 
 March 21, Moon's age is 3. (130) 
 New Moon on March 18. 
 The 15th day is .April 2. 
 April the ill is g, on Sunday. 
 Then Eaftcr-Sunday is April 8th. 
 
 Ex. II. Required the time of Eajler- 
 day in the year 1 788 ? 
 The Dominical letter is e. (128) 
 
 March 20, Moon's age 13. ('3) 
 
 New Moon on March 7 th. 
 full Moon on March 22. 
 March ift is d, on Saturday. 
 Then Ealler-Sunday is March Z3d. 
 
 134. Eaflcr-day is always ^o^wd by the Pafchal full Moons, and thefe 
 are readily found in the following curious table, which was communi- 
 cated to the Royal Society in the yc.ir 1750, l^y the Earl of Macclesfield, 
 and pubiilhed ia the Philofophicr.l Tranfaclious for the fame yearj and 
 its uic fhewn in the following precepts. , 
 
 " To find the day, on which th^ Pafchal limit, or full Moon, falls in 
 " any given year ; look, iji tlie column of golden numbers belonging to 
 ** that period of time wherein the given year is contained, for the golden 
 " number of that year ; over-againil which, in the fame line continued 
 " to the column intitled Pafchai full Moons, you will find the day of the 
 " month, on whicii the Pafchnl limit, or full Moon, happens in that 
 *' year. And the Sundry next after that day is Eai!er-day in that vear, 
 *' according to the Grcp^orian accounc." 
 
 His Lordfhip al:b [.;;ave witli the t'oll.owing table an account of the prin- 
 ciples upon which he conilriictca it ; :.:vi v/hich the more inquifitivc 
 jr^aders may confui:, if thvy nl-- *-.'':, 
 
 A Table,
 
 Book VI. 
 
 GEOGRAPHY. 
 
 373 
 
 ' A Table, fhewlng, byimeans of~the Golden Numbers, the fcveral dy'.ys on 
 which the Pafchal limits, or full Moons, according to the Gregorian ac- 
 count, have already happened, or will hereafter happen ; from the Re- 
 formation of the Calendar in the year 1582, to the year 4199, inclufive. 
 
 Golden Numbers from the year 1583 to 1699, and fo on to 4199, 
 
 all inclufive. 
 
 Palchal 
 full Moons. 
 
 IS8^ 
 
 1700 
 
 1900 azoc 
 
 23002400250c 2600J2900I3100 
 
 3400 
 
 3500 
 
 3600 
 
 3700 
 
 3800 
 
 4100 
 
 Days of the 
 
 to 
 
 to 
 
 to 1 to 
 
 to 1 to 1 to 
 
 to to 
 
 to 
 
 to 
 
 to 
 
 to 
 
 to 
 
 to 
 
 to 
 
 month and 
 
 1699 
 
 i8q9 
 
 21992299 
 
 239924992599 
 
 2899!3099!3399 3499 
 
 3599 
 
 3699 
 
 3799 
 
 4099 
 
 4199 
 
 Srun. letters. 
 
 -? 
 
 14 - 
 
 6 
 
 17 
 
 6 
 
 17 
 
 9 
 
 
 
 I 
 
 12 
 
 I 
 
 12 
 
 
 
 4 
 
 March 21.C 
 
 
 -? 
 
 14 
 
 
 
 & 
 
 
 
 6 
 
 17 
 
 9 
 
 
 
 ^ 
 
 
 
 I 
 
 12 
 
 " . 
 
 22 D 
 
 11 
 
 
 ^ 
 
 14 
 
 
 
 14 
 
 
 
 6 1 17 
 
 
 
 9 
 
 
 9 
 
 
 
 I 
 
 12 
 
 23 E 
 
 
 
 11 
 
 
 3 
 
 14 
 
 3 
 
 14 
 
 6 
 
 17 
 
 
 
 9 
 
 
 
 9 
 
 
 
 I 
 
 , ?f r 
 
 19 
 
 
 
 II 
 
 
 
 3 1 - 
 
 3 
 
 14 
 
 b 
 
 17 
 
 
 
 17 
 
 
 
 9 
 
 
 ] .\g 
 
 8 
 
 IP 
 
 
 
 1 1 
 
 1 1 
 
 
 
 3 14 
 
 
 
 6 
 
 17 
 
 6 
 
 17 
 
 
 
 9 
 
 ' kA 
 
 _ 
 
 8 
 
 iq 
 
 
 
 1 1 
 
 II 
 
 3 
 
 14 
 
 
 
 b 
 
 
 
 b 
 
 17 
 
 
 
 27 B 
 
 16 
 
 _ 
 
 8 
 
 19 
 
 19. 
 
 
 
 II . 
 
 3 
 
 14 
 
 
 
 14 
 
 
 
 b 
 
 17 
 
 28 C 
 
 S 
 
 16 
 
 
 
 8 
 
 19 8 
 
 '9 
 
 1 1 
 
 
 
 3 
 
 14 
 
 3 
 
 14 
 
 
 
 6 
 
 29 D 
 
 
 5 
 
 16 
 
 
 
 818 
 
 19 
 
 1 1 
 
 
 
 3 
 
 
 
 3 
 
 14 
 
 
 
 30 B 
 
 M 
 
 _ 
 
 s 
 
 16 
 
 _ ,6|_ 
 
 8 
 
 '9 
 
 
 
 II 
 
 
 
 i I 
 
 
 
 3 
 
 14 
 
 31 V 
 
 2 
 
 "i 
 
 
 s 
 
 16 5 
 
 j6 
 
 
 
 8 
 
 '9 
 
 
 
 II 
 
 
 
 11 
 
 
 
 3 
 
 April 1 G 
 
 
 
 2 
 
 n 
 
 
 
 5 
 
 5 
 
 16 
 
 
 
 8 
 
 '9 
 
 
 
 '9 
 
 
 
 1 1 
 
 
 
 2 A 
 
 10 
 
 , 
 
 2 
 
 1-; 
 
 , n 
 
 
 
 S 
 
 16 
 
 
 
 8 
 
 19 
 
 8 
 
 19 
 
 
 
 1 1 
 
 3 B 
 
 
 
 10 
 
 
 
 2 
 
 13 1 2 
 
 13 
 
 
 
 5 
 
 16 
 
 
 
 8 
 
 
 
 s 
 
 19 
 
 
 
 4C 
 
 10 
 
 
 
 10 
 
 
 
 2 
 
 2 
 
 13 
 
 
 
 5 
 
 id 
 
 
 
 16 
 
 
 
 8 
 
 19 
 
 5^ 
 
 7 
 
 iS 
 
 
 
 10 
 
 10 
 
 
 
 2 
 
 13 
 
 
 
 5 
 
 16 
 
 S 
 
 lb 
 
 
 
 8 
 
 6E 
 
 
 7 
 
 18 
 
 
 10 
 
 10 
 
 
 
 2 
 
 13 
 
 
 
 S 
 
 
 ; 
 
 lb 
 
 
 
 7F 
 
 15 
 
 
 7 
 
 18 
 
 _ 18 
 
 
 
 10 
 
 
 
 2 
 
 13 
 
 
 
 13 
 
 
 
 S 
 
 16 
 
 8G 
 
 4 
 
 '5 
 
 
 7 
 
 18 7 18 
 
 
 
 10 
 
 
 
 2 13 
 
 2 
 
 13 
 
 
 5 
 
 9A 
 
 
 
 4 
 
 5 
 
 
 
 7 1 7 
 
 18 
 
 .0 
 
 2 
 
 
 
 2 
 
 '3 
 
 
 
 10 H 
 
 12 
 
 
 4 15 
 
 15 1 
 
 7 18 
 
 
 
 10 
 
 10 
 
 
 
 2 
 
 11 
 
 11 C 
 
 I 
 
 12 
 
 4 
 
 15 4 J >5 
 
 - 7 , 18 
 
 10 
 
 
 
 10 
 
 
 
 2 
 
 12 D 
 
 
 
 I 
 
 12 
 
 414 
 
 15 ; 
 
 7 
 
 18 
 
 18 
 
 
 
 10 
 
 
 
 13E 
 
 9 
 
 
 
 1 12 
 
 1 12 , 
 
 4 1 15 
 
 
 
 7 18 
 
 7 
 
 18 
 
 
 
 iO 
 
 14 F 
 
 __ 
 
 f) 
 
 1 i 
 
 12 1 1 ( 12 
 
 - j 4 
 
 'S 
 
 1 7 
 
 
 
 7 
 
 18 
 
 
 
 15 G 
 
 I 7 
 
 
 9 {- 
 
 i ; ' 
 
 12 , 
 
 4 
 
 '5 
 
 
 
 '5 
 
 
 
 7 
 
 18 
 
 16 A 
 
 6 
 
 1-^ 
 
 '7 1 9 
 
 9 1 
 
 I iZ 
 
 12 
 
 4 
 
 '5 
 
 4 
 
 15 
 
 15 
 
 7 
 
 17B 
 
 '4 
 
 6 
 
 6 .- 
 
 9 17 1 9 
 
 9 I 
 
 I 
 
 .2 1 + 
 
 12 
 
 4 
 
 4- 
 
 I ; 
 
 18 C 
 
 135. Proj;i,em VIII. To find the time of the j^oofi' s fcuthing on a givtn 
 
 day. 
 
 Rule. The Moon's age in days, mukiplicd by 0,8, gives the time of her 
 foil thing, nearly, in hours and tenth parts. 
 That time, if lefs than 12 hours, is the time after mid-uay. 
 But if greater, the excefs is the time after laii. midnight. 
 
 Y.x. I. ylt what lime does the Monn Ex. II. Required the time of the 
 
 come to the meridian of London, on th 
 ij^th cfO^cbery i']'6'] ? 
 The Moon'i age is 3 days. 
 
 Which muhipiicd by o,S 
 
 Moon So. 2'' 24"' n 2,^ 
 
 Moon i fonthing on the loth of Aaarch^ 
 
 1788.^ 
 
 The Moon's age 13 days. (i3) 
 
 Which multiplied by o,B 
 
 Mocn So. lo'- 24" z= 10,4. 
 
 Farh tenth part of an hour being 6 minutes, any nutr.bcr of fuch tenth 
 parts multiplied by 6, produces minutes.
 
 374 
 
 GEOGRAPHY. 
 
 Book VI. 
 
 136. Problem IX. To /i/td the tmf / high-water at any place. 
 
 Rule. To the time of the Moon's fouthing add the time the Moon 
 has pafled the meridian on the full and change days to make 
 high-water at that place ; the fum (hews the time of high- 
 water on the given day. 
 The time of high-water, on the full and change days, is found in the 
 right-hand column of the geographical table, art. 137, againft the name 
 of the place. 
 
 Ex. I. On the l\fh of Oiiober 
 j-jS-jy at what time will it be high- 
 water at London ? 
 
 MoonfoDthsat 2h. 2401. 035) 
 
 H.W. atLond. 3 o oa fyzygics 
 
 Sam 5 24 
 
 H.W. at 5h 24m.P.M. on the 
 
 day propofed. 
 
 Ex. II. Required the ti?ne when it 
 will he high-water atUjhant on March 
 20th, 1788. 
 
 Moon fouths at loh. 24m. (13?) 
 High-water at Uftiant 4 30 P.M. 
 
 H 54 
 
 Subtra^ 12 oo 
 
 High-water at 
 the day propofed. 
 
 2 54 A.M. on 
 
 The V. VIII. IX. problems preceding have folutions, fuch as are com- 
 mon in books of pilotage, and which in fome cafes will produce con- 
 clufions confiderably wide of the truth ; it has therefore been judged ne- 
 ceflary to confider thefe articles in a more accurate manner in Book IX. 
 of Days works. 
 
 T^37* 
 
 SECTION IX. 
 
 ^ Geographical T*able, 
 
 Containing the latitudes l&nd longitudes of the chief towns, iflands, bays, 
 capes, and other parts of the fea-coafts in the known world, colle6led 
 from the moft authentic obfervations and charts extant; with the times 
 of high-water on the days of the new and full Moon. 
 
 The longitudes are reckoned from the meridian of London. By the 
 latitude and longitude of an ifland, or harbour, is meant the middle of 
 that place. 
 
 ^ote. B. ftands for bay; C. for cape ; R. for river; P. for port; 
 Pt. for point; I. for Ifle ; St. for faint; G. for gulf; Al. fo;- 
 mount ; Eu. for Europe ; Am. for America ; Atl. for the Atlan- 
 tic ; Ind. for Indian ; Med. Sea for Mediterranean Sea ; Wh. 
 Sea for White Sea ; Archip. for Archipelago ; Nov. Sco. for 
 Nova Scotia ; Phil. I. for Philippine Ifles ; Adriat. for Adria- 
 tic ; Eng. for England ; D. Ncth. for the Dutch Netherlands : 
 Befides other contractions which will be eafily underftood. 
 
 Names
 
 Book VI. 
 
 GEOGRAPHY. 
 
 375 
 
 
 Names of Places. 
 
 Cont. 
 
 Countries. 
 
 Coaft. 
 
 Latitude. 
 
 Longitude. 
 
 H.Water. 
 
 
 
 A 
 
 I. Abacco, 5 N. point 
 or Lucayos I S. point 
 
 Am. 
 
 Bahama I. Ad. Ocean 
 
 C 27 12 N. 
 ;'26 isN. 
 
 77 05 w: 
 
 77 01 W. 
 
 
 
 Abbreviak 
 
 Eu. 
 
 France 
 
 ing. Channel 
 
 48 32 N. 
 
 4 MW. 
 
 4h. 3oin. 
 
 
 
 St. Abb/head 
 
 Eu. 
 
 Scotland 
 
 Germ. Ocean 
 
 55 55 N. 
 
 1 56 W. 
 
 
 
 
 I. Abdeleur 
 
 Africa 
 
 Anian 
 
 [ndian Sea 
 
 II 55 N. 
 
 5. 45 E- 
 
 
 
 
 Aberdeen 
 
 Eu. 
 
 Scotland 
 
 Germ. Ocean 
 
 57 06 N. 
 
 01 44 W. 
 
 45 
 
 
 
 Abo 
 
 Eu. 
 
 Finland 
 
 Baltic Sea 
 
 60 27 N. 
 
 22 15 E. 
 
 
 
 
 Abrolhos Bank 
 
 Am. 
 
 Brafil 
 
 Atl. Ocean 
 
 18 22 S. 
 
 38 45 W. 
 
 
 
 
 Abrollo Bank, N. part 
 
 Am. 
 
 Sahama 
 
 Atl. Ocean 
 
 21 33 N. 
 
 69 50W. 
 
 
 
 
 Achen 
 
 Afia 
 
 [. Sumatra 
 
 [ndian Ocean 
 
 5 22 N. 
 
 95 40 E. 
 
 
 
 
 Aden 
 
 Afia 
 
 Arabia 
 
 Indian Sea 
 
 12 55 N. 
 
 45 35 E- 
 
 
 
 
 I. Admiralties 
 
 Eu. 
 
 Nova Zem. 
 
 North Ocean 
 
 75 05 N. 
 
 52 50 E. 
 
 
 
 
 Adventure Iftand 
 
 Afia 
 
 Soc. Ifles 
 
 Pacif. Ocean 
 
 17 6 S. 
 
 144 18W. 
 
 
 
 
 I. Agalega, or Gallega 
 
 Africa 
 
 Madagafcar 
 
 Indian Ocean 
 
 10 15 S. 
 
 54 46 E. 
 
 
 
 
 C. St. Agnes 
 
 Am. 
 
 Patagonia 
 
 S. Atl. Ocean 
 
 53 55 S. 
 
 66 29 W. 
 
 
 
 
 Agra 
 
 Afia 
 
 India 
 
 Mogul's 
 
 26 43 N. 
 
 76 49 E. 
 
 
 
 
 I. St. Agufta 
 
 Eu. 
 
 Dalmatia 
 
 Adriatic Sea 
 
 42 40 N. 
 
 18 57 E. 
 
 
 
 
 C. Ajuga 
 
 Am. 
 
 Peru 
 
 Pacif. Ocean 
 
 6 38 S. 
 
 80 50 W. 
 
 
 
 
 B. Alagoa 
 
 Africa 
 
 Carters 
 
 Indian Ocean 
 
 25 30 S. 
 
 33 33 E. 
 
 
 
 
 If. Aland 
 
 Eu. 
 
 Sweden 
 
 Baltic Sea 
 
 60 20 N. 
 
 21 30 E. 
 
 
 
 
 R. Albany 
 
 Am. 
 
 NewS. Wales 
 
 Hud. Bay 
 
 52 35 N. 
 
 85 18W. 
 
 
 
 
 I. Alboran 
 
 Africa 
 
 Algiers 
 
 Medit. Sea 
 
 36 CO N. 
 
 2-27W. 
 
 
 
 
 Aldborough 
 
 u. 
 
 England 
 
 Germ. Ocean 
 
 52 20 N. 
 
 I 25 E. 
 
 9 45 
 
 
 
 I. Aldcrney 
 
 Eu. 
 
 England 
 
 Eng. Channel 
 
 49 48 N. 
 
 2 iiW. 
 
 i 00 
 
 
 
 Aleppo 
 
 Afia 
 
 Syria 
 
 Mcdit. Sea 
 
 35 45 N. 
 
 37 25 E. 
 
 
 
 
 Alexandretta 
 
 Afia 
 
 Syria 
 
 Medit. Sea 
 
 36 35 N. 
 
 36 20 E. 
 
 
 
 
 Alexandria 
 
 Africa 
 
 Eg)-pt_ 
 
 Medit. Sea 
 
 31 II N. 
 
 30 17 E. 
 
 
 
 
 I. Algeraoca 
 
 Africa 
 
 Canaries 
 
 Atl. Ocean 
 
 29 23 N. 
 
 15 S3W. 
 
 
 
 
 Algiers 
 
 Africa 
 
 Algiers 
 
 Mcdit. 5ea 
 
 36 49 N. 
 
 2 18 E. 
 
 
 i 
 
 
 Alicant 
 
 Eu. 
 
 Spain 
 
 Medit. Sea 
 
 38 34 N. 
 
 07 W. 
 
 
 
 I. Alicur, Lipari If. 
 
 Eu. 
 
 luly 
 
 Medit. Sea 
 
 38 3, N. 
 
 14 37 E. 
 
 
 
 
 Alkofir 
 
 Africa 
 
 Egypt 
 
 Red Sea 
 
 26 20 N. 
 
 34 41 E. 
 
 
 1 
 
 
 B. All Saints, or ? 
 Todos Sanftos J 
 
 Am. 
 
 Brafil 
 
 Atl. Ocean 
 
 13 05 S. 
 
 38 45W. 
 
 
 i 
 
 
 Almcrla 
 
 Eu. 
 
 Spain 
 
 Medit. Sea 
 
 36 51 N. 
 
 2 15W. 
 
 
 
 
 If. Almirantc, 7 
 limits i 
 
 Africa 
 
 Zanguebar 
 
 Indian Sea 
 
 S 5 45 S. 
 
 2 4 30 s. 
 
 51 30 E- 
 55 40 E. 
 
 
 
 
 St. Alphonfo'i If. 
 
 Am. 
 
 T. delFuego 
 
 Pacif. Ocean 
 
 55 51 S. 
 
 69 28 W. 
 
 
 
 
 Altur 
 
 Afia 
 
 Arabia 
 
 Red Sea 
 
 28 20 N. 
 
 34 19 E. 
 
 
 
 
 R. Amazons, ]' 
 mouths ' 
 
 Am. 
 
 Terra Firma 
 
 Atl. Ocean 
 
 30 S. 
 
 C47 35W. 
 249 20 W. 
 
 6 CO 
 
 
 
 I. Amboyna 
 
 Afw 
 
 Molucca I. 
 
 Indian Ocean 
 
 425N. 
 
 127 25 E. 
 
 
 
 
 Anibrym 
 
 Afia 
 
 N. Hebrides 
 
 Pacif. Ocean 
 
 16 10 S. 
 
 168 12 E. 
 
 
 
 
 If. Ambrofa 
 
 Am. 
 
 Chili 
 
 Pacif. Ocean 
 
 26 40 S. 
 
 82 30 W. 
 
 
 
 I. Amcyland 
 
 Eu. 
 
 D. Neth. 
 
 Germ. Ocean 
 
 53 30 N. 
 
 6 20 E. 
 
 7 30 
 
 
 
 I. Amoy 
 
 Afia 
 
 China 
 
 Pacif. Ocean 
 
 24 30 N. 
 
 118 45 E. 
 
 
 
 
 Amftcrdam 
 
 Eu. 
 
 D. Neth. 
 
 Germ. Ocean 
 
 52 23 N. 
 
 C4 52 E. 
 
 3 00 
 
 
 
 I. Amftcidjm 
 
 Afia 
 
 Madagafcar 
 
 Indian Ocean 
 
 37 55 S. 
 
 75 15 E. 
 
 
 
 
 I. Amitcrdam, or i' 
 Tonga-'I'abu [] 
 
 Afia 
 
 Friendly If. 
 
 Pacif. Ocean 
 
 21 09 S. 
 
 174 41 w. 
 
 8 30 
 
 
 
 I. Anabona 
 
 Afiica 
 
 Eth. Coaft 
 
 -Atl. Ocean 
 
 2 36 s. 
 
 5 35 E. 
 
 
 
 
 Ancona 
 
 Lu. 
 
 Itniy 
 
 Meditcrran. 
 
 43 38 N. 
 
 13 31 E. 
 
 
 
 If. Andaman, 7 
 liniiti ^ 
 
 Afia 
 
 India 
 
 B. Bengal 
 
 C 14 00 N. 
 
 I 10 08 N. 
 
 93 03 E. 
 93 35 E. 
 
 
 
 
 I. Andaro 
 
 Afia 
 
 India 
 
 Indian Ocean 
 
 10 CO N. 
 
 73 40 E- 
 
 
 
 
 I.St.AnderojSotovcnti 
 
 Am. 
 
 Mexico 
 
 Atl. Ocean 
 
 12 30 N. 
 
 81 35W. 
 
 
 
 
 C. St. Andrea 
 
 Africa 
 
 Madagafcar 
 
 Indian Ocean 
 
 15 46 S. 
 
 45 22 E. 
 
 
 
 
 St. Andrew s 
 
 Eu. 
 
 Scotland 
 
 Germ. Ocean 
 
 56 i8 N. 
 
 2 37 W. 
 
 2 15 
 
 
 
 If.Androf. J^.poin 
 I S.pom 
 
 Am. 
 
 Bahama I. 
 
 Atl. Ocean 
 
 C25 CO N. 
 ^23 30 N. 
 
 77 5SW. 
 77 coW. 
 
 
 
 
 If. Angafay 
 
 Afriia 
 
 M:idagafcar 
 
 Indian Ocean 
 
 17 00 S. 
 
 58 40 K. 
 
 
 
 
 C. St. Angelo 
 
 Eu. 
 
 Furlccy 
 
 Archip''!ag<) 
 
 36 27 N. 
 
 33 38 E. 
 
 
 
 Mount
 
 37^ 
 
 GEOGRAPHY. 
 
 Bo6k VI. 
 
 Names of Places. 
 
 Cont. 
 
 Countries. 
 
 Coaft^ 
 
 Latitude. 
 
 Longitude. 
 
 H. Water. 
 
 Mnint St. Angelo 
 
 tu. 
 
 Italy 
 
 Mcdit*. Sea 
 
 O / 
 
 41 42 N. 
 
 6 16 E, 
 
 
 R. d'Angra 
 
 Africa 
 
 Ethiopia 
 
 N. Ati, Ocean 
 Indian Ocean 
 
 01 00 N, 
 
 9 35 E. 
 
 
 C. d'Anguilhas 
 
 Africa 
 
 Cafters 
 
 34 SO S. 
 
 20 06 E. 
 
 
 1. Anguilla 
 
 Am. 
 
 Antilles If. 
 
 Atl. Ocean 
 
 18 15 N. 
 
 62 57 W, 
 
 
 C. Anguillc 
 
 Am. 
 
 STewfoundl. 
 
 Ad. Ocean 
 
 47 55 N. 
 
 59 iiW. 
 
 
 1. Anholt 
 
 Eu. 
 
 [Denmark 
 
 Sound 
 
 56 40 N. 
 
 12 00 E. 
 
 oh.oom. 
 
 C. Anne 
 
 Am. 
 
 Mew Eng. 
 
 Weft. Ocean 
 
 42 50 N. 
 
 70 27 W. 
 
 
 C. Queen Anne 
 
 Am. 
 
 Greenland 
 
 North Ocean 
 
 64 J5 N. 
 
 50 30W, 
 
 
 Q^ Anne's Foreland 
 
 Am. 
 
 N. Main 
 
 Hudfon's Str. 
 
 64 08 N. 
 
 74 36 w. 
 
 
 Annamocka, or 7 
 Rotterdam 5 
 Annapolis Royal 
 
 Afia 
 
 Friendly If. 
 
 Pacif. Ocean 
 
 20 16 S. 
 
 174 30W, 
 
 
 Am. 
 
 Mova Scotia 
 
 B. Fundy 
 
 44 52 N. 
 
 64 00 W. 
 
 
 I. Aiitego 
 Antibes 
 
 Am. 
 
 Caribbce If. 
 
 Atl. Ocean 
 
 .6 57 N. 
 
 61 56W, 
 
 
 Eu. 
 
 France 
 
 Medit. Sea 
 
 43 35 N. 
 
 7 09 E, 
 
 
 I. Ante- J W. point 
 coft I E. point 
 
 Ami. 
 
 Canada 
 
 ;; B. St.Lau- 
 
 49 52 N, 
 
 64 04 W, 
 
 
 
 
 l rence 
 
 49 10 N. 
 
 61 42 W. 
 
 
 Antiochetta 
 
 Afia 
 
 Syria 
 
 Medit, Sea 
 
 36 oS N. 
 
 36 17 E, 
 
 
 C. d'Antifer 
 
 Eu. 
 
 France 
 
 Eng. Channel 
 
 49 47 N. 
 
 34 E, 
 
 
 C. Antonio 
 
 Am. 
 
 Ifie Cuba 
 
 Atl. Ocean 
 
 21 45 N. 
 
 84 05 W. 
 
 
 I. St. Antonio 
 
 Africa 
 
 Cape Verd 
 
 Atl. Ocean 
 
 17 00 N. 
 
 25 02 W. 
 
 
 C. St. Antony 
 
 Am. 
 
 Magellan 
 
 Atl. Ocean 
 
 5446.S, 
 
 63 42 W. 
 
 
 Antwerp 
 
 Eu. 
 
 Flanders 
 
 R. Scheld 
 
 51 13 N. 
 
 4 24 E. 
 
 6 00 
 
 B. Apaiaxy 
 
 Am. 
 
 Florida 
 
 G. Mexico 
 
 30 00 N. 
 
 83 53W. 
 
 
 I. Apalioria 
 
 Afia 
 
 India 
 
 Indian Ocean 
 
 9 08 S. 
 
 79 40 E, 
 
 
 Aquapi'.lco 
 
 Am. 
 
 Mexico 
 
 Pacif. Ocean 
 
 17 10 N, 
 
 101 40 W, 
 
 
 Aquatul'co 
 
 Am. 
 
 Mexico 
 
 Pacif, Ocean 
 
 15 27 N. 
 
 96 03 W. 
 
 
 Archangel 
 
 Eu. 
 
 Ruffia 
 
 Whke Sea 
 
 64 34 N. 
 
 38 59 E. 
 
 6 00 
 
 I. d' Areas 
 
 Am. 
 
 Mexico 
 
 G. Mexico 
 
 20 45 N. 
 
 92 35W. 
 
 
 Arica 
 
 Am. 
 
 Peru 
 
 Pacif. Ocean 
 
 18 27 S, 
 
 71 05 W. 
 
 
 I. Arran 
 
 Eu. 
 
 Ireland 
 
 St. Geo. Ch. 
 
 54 48 N, 
 
 8 59W. 
 
 II 00 
 
 |l. Afccnfion 
 
 Am. 
 
 Brafil 
 
 Ati. Ocean 
 
 7 56 S, 
 
 14 iSW, 
 
 
 jl. AlVmaria, 7 
 
 1 Sardinia ', 
 
 Eu. 
 
 Italy 
 
 Medit. Sea 
 
 41 06 N. 
 
 8 36 E. 
 
 
 'r. Aflilcy 
 
 Am. 
 
 Carolina 
 
 Atl. Ocean 
 
 33 i* N, 
 
 79 50 W, 
 
 45 
 
 :R. AUene 
 
 Africa 
 
 Guinea 
 
 Atl. Ocean 
 
 5 30 N- 
 
 2 20 W. 
 
 
 I. Altorcs 
 
 Africa 
 
 MaJagafcar 
 
 Indian Ocean 
 Archipelago 
 
 10 22 S, 
 
 53 25 E, 
 
 
 JAthens 
 
 Eu. 
 
 Tuikey 
 
 38 sn: 
 
 23 52 E. 
 
 
 lAtkin's Key 
 
 Am. 
 
 Bahama Ifles 
 
 Atl. Ocean 
 
 22 07 N. 
 
 74 26 w. 
 
 
 'Atwood's Kevs 
 
 Am. 
 
 Bahama Ifies 
 
 Atl. Ocean 
 
 21 22 N. 
 
 72 04W. 
 
 
 C. Ava 
 
 Afia 
 
 Japan 
 
 F'acif. Ocean 
 
 34 45 N- 
 
 141 00 L. 
 
 
 If. Avcs, Sotovento 
 
 Am. 
 
 Terra Firma 
 
 Atl, Ocean 
 
 15 26 N. 
 
 66 i5\V. 
 
 
 C. St. Auguftine 
 
 Am. 
 
 Brafil 
 
 Atl, Ocean 
 
 8 .18 S. 
 
 35 coW. 
 
 
 ,C. St. Auguftine 
 
 Afia 
 
 Mindanao 
 
 Pacif. Ocean 
 
 6 40 N. 
 
 126 25 E. 
 
 
 ,St. Auguftine 
 
 Am. 
 
 Florida 
 
 Atl. Ocean 
 
 30 10 N. 
 
 81 29 W. 
 
 7 30 
 
 'Auiora 
 
 Afia 
 
 N. Hebrides 
 
 Pacif. Ocean 
 
 15 S S, 
 
 168 17 E. 
 
 
 ;Aydhab 
 
 Africa 
 
 Egypt 
 
 Red Sea 
 
 21 53 N. 
 
 36 26 E. 
 
 
 :Avlah 
 Babelmondel Straits 
 
 Afia 
 
 Arabia 
 
 Red Sea 
 
 29 08 N. 
 
 35 41 ^^ 
 
 
 Africa 
 
 Abyflinia 
 
 Red Sea 
 
 12. 50 N. 
 
 <-3 50 E. 
 
 
 |C. Ba:a 
 
 Afia 
 
 Natolia 
 
 Archipelago 
 
 39 33 N. 
 
 26 22 E, 
 
 
 .1. Bacliian 
 
 Afia 
 
 Moluccalfics 
 
 Pacif. Ocean 
 
 00 40 N, 
 
 123 00 E. 
 
 
 I. Bahama 
 
 Am. 
 
 Bahama I lies 
 
 JAtl. Ocean 
 
 Z.6 45 N, 
 
 78 35W, 
 
 
 .Bahama Bank, N. pt. 
 
 Am. 
 
 Bahama] lies 
 
 lAtl. Ocean 
 
 27 50 N, 
 
 78 43 W. 
 
 
 'C. Bajador 
 
 Africa 
 
 Ncgroland 
 
 Atl. Ocean 
 
 26 29 N. 
 
 14 36W. 
 
 00 
 
 Baker's Dozeix 
 
 Am. 
 
 Labrador 
 
 Hudfon's Bav 
 
 57 N, 
 
 
 
 Baijfor 
 
 Afia 
 
 India 
 
 B. Bengal 
 
 21 20 N. 
 
 86 00 E. 
 
 
 Buldivia 
 
 Am. 
 
 Chili 
 
 Pacif. Ocean 
 
 .' 39 jS 'i- 
 
 73 20 W. 
 
 
 I, Bali 
 
 Afia 
 
 Sunda Ifles 
 
 Indian Ocean 
 
 S 05 S, 
 
 114 30 E, 
 
 
 'Baltimore 
 
 Eu. 
 
 Ireland 
 
 Weft, Ocean 
 
 51 16 N, 
 
 9 26 W. 
 
 4 30 
 
 i, CS. end 
 
 Afia 
 
 Sunda Ifles 
 
 Indian Ocean 
 
 \ ^'5 S. 
 
 , 107 10 E. 
 
 
 ; '^^"'--' 2^<-Vv-end 
 
 
 
 
 I I -50 S. 
 
 ' 105 30 E. 
 
 
 1. B.^nda 
 
 Afia 
 
 MohiCfi-.Ifle', 
 
 Indian Ocom 
 
 4 30 N, 
 
 ' 127 25 E. 
 
 
 
 Eujijir
 
 Sook VI. 
 
 GEOGRAPHY. 
 
 377 
 
 Names of Places. 
 
 <%nt. 
 
 Banjav 
 BanJcs's Ifle 
 Bantam 
 
 B. Bantry 
 I. Barbadoes 
 
 Bridge-town 
 
 C. Barbas 
 I. Barbuda 
 C. Barcam 
 Barcelona 
 C. Barfleur 
 Bargazar Point 
 I. Bardfey 
 
 |C. Barlb 
 
 (I. Bartholomew 
 
 I. de Bas 
 
 Baflora 
 
 C. BalTos, or Baxos 
 I BafTos dc Panhos 
 ; Baflbs de Chagos 
 jl. BafTus dcs Indes 
 
 Batavia 
 
 Bayonnj 
 
 B.iyoua Ifles 
 
 Bcachy Head 
 I B'ar-bay 
 !N. Bear ? 
 jS. Bear J 
 jl. Bccrenberg 
 'R-^lchcr'sines 
 i Bc-lfaft 
 iBL-i.lid-i 
 Belliil': 
 
 BcmLridgP f'oint 
 I Straits ot Eellliie 
 I I'.ell Sound 
 iBtncolin 
 |Bcng..l 
 I Bergen 
 
 Berlin 
 
 [. Ecrmtid.T; 
 
 I. Bcrni.ija 
 I Berwick 
 Jerrv l'.jjnt 
 
 Bir i' Illar.d 
 
 Bllboa 
 !l. du Bic 
 
 Bl iLknrv 
 
 Hlaek P';,ur 
 : Black Ille 
 iC Blanco 
 
 C. Blanco 
 
 C. Blinto 
 
 C. Biarico 
 
 ; I. Blanco, S -tnver.to 
 
 Blanchnrt Rat'; 
 
 :Bl.fq.:'-s 
 
 ! Bia vet, or Port Lo'jii 
 
 Dociciil!.:! 
 
 lB..ial,ola 
 
 "R. B'.'.ih.-.va 
 
 1. Rornt.v,' 
 
 Afia 
 Afia 
 Afia 
 Eu. 
 
 Am. 
 
 Am. 
 
 Eu. 
 
 Am. 
 
 Eu. 
 
 Eu. 
 Am. 
 Eu, 
 Am, 
 
 V ,, _ 
 
 Alia 
 
 Alia 
 
 Eu. 
 
 E J. 
 
 Am, 
 
 Am. 
 
 r.u, 
 
 Eu. 
 
 Am. 
 
 Ell, 
 
 Am. 
 
 Eu. 
 
 Eu. 
 
 Eu. 
 
 Africa 
 
 A}n. 
 
 Eu. 
 
 Am. 
 
 Am. 
 
 Eu. 
 
 Ej. 
 
 Eu. 
 
 Am. 
 
 Afia 
 
 Afia 
 
 Afia 
 
 Countries. 
 
 I. Borneo 
 N. Zealand 
 I. Java 
 Ireland 
 
 Caribbeelfles* 
 
 Africa 
 
 Sanaga 
 
 Am. 
 
 Caribbeelflcs 
 
 Eu. 
 
 Greenland 
 
 Eu. 
 
 Spain 
 
 Eu. 
 
 France 
 
 Eu. 
 
 Iceland 
 
 Eu. 
 
 Wales 
 
 Eu. 
 
 Ruffia 
 
 Am. 
 
 Caiibbeelfles 
 
 Eu. 
 
 France 
 
 Afia 
 
 Arabia 
 
 Africa 
 
 Anian 
 
 Africa 
 
 Zanguebar 
 
 Afia 
 
 India 
 
 Africa 
 
 Zanguebar 
 
 Afia 
 
 I, Java 
 
 Eu. 
 
 France 
 
 Eu. 
 
 Spain 
 
 Eu. 
 
 England 
 
 Eu. 
 
 Greenland 
 
 Labi-adore 
 
 Labradore 
 Ireland 
 France 
 N'ev.fi.und. 
 Ifie Wight 
 Nfwfoundl. 
 Grceijh nd 
 i. .Sumatra 
 Inj;a 
 Ncrv.sv 
 Gc.rmanv 
 iJahsma'ifiet 
 Mexico 
 Engl.md 
 England 
 Aca^^ia 
 Siain 
 Acidia 
 Eri;!.(nd 
 Greenland 
 Nova Zem. 
 N' gr(^liinJ 
 Pat.-g';nia 
 "Greenland 
 Mexico 
 Terra Firnria 
 France 
 Irrland 
 Fiance 
 Terra Firma 
 
 Soticty in, 
 
 Sbciia 
 India 
 
 Coaft. 
 
 Indian Ocean 
 Pacif. Ocean 
 Indian Ocean 
 Atl. Ocean 
 
 Atl. ,Ocean 
 
 N. Atl. Ocean 
 Atl. Ocean 
 North Ocean 
 Mcdit. Sea 
 Eng. Channel 
 North Ocean 
 St. Geo. Cha. 
 White Sea 
 Atl. Ocean 
 Eftg. Channel 
 Perfian Gulf 
 Indian Sea 
 Indian Ocean 
 Indian Ocean 
 Indian Ocean 
 Indian Ocean 
 B. Bifcay 
 Atl. Ocean 
 Eng. Channel 
 North Ocean 
 
 Hudfon's Bay 
 
 North Ocean 
 Hudfon's Bay 
 Irilli Sea 
 B. Bifcay 
 Atl. Ocean 
 Eng. Channel 
 Atl. Ocean 
 North Ocean 
 Indian Ocean 
 B. Bengal 
 Weftcrn Oc. 
 R. Elbe 
 Atl. Ocean 
 G. Mexico 
 Germ. Ocean 
 En^. Channel 
 (/. St. Lawr. 
 15. Bifcay 
 R.St. Ldv\r. 
 Gfrm. Ocean 
 North Ocean 
 North Ocr.-.n 
 Atl. Ocean 
 Atl. Ocean 
 Norcli Ocean 
 Pacif. Ocean 
 Atl. Ocean 
 Erg. Channel 
 Atl. Ocean 
 B. Bifcay 
 Carib. Sra 
 Pncif. Ocean 
 P.cif. Oce '.n 
 kdi^r. Occ:u. 
 
 Latitude. 
 
 2.27 S. 
 43 45 S. 
 
 6 IS S. 
 51 45 N. 
 
 13 05 N. 
 
 21 50 
 
 N 
 
 17 46 
 
 N 
 
 78 18 
 
 N 
 
 41 26 
 
 N 
 
 49 38 
 
 N 
 
 66 30 
 
 K 
 
 5^ 44 
 
 N 
 
 66 30 
 
 N 
 
 17 56 
 
 N 
 
 48 50 
 
 N 
 
 29 45 
 
 N 
 
 4 12 
 
 N 
 
 5 00 
 
 S 
 
 6 42 
 
 S 
 
 21 19 
 
 s 
 
 6 12 
 
 S 
 
 43 30 
 
 N 
 
 41 45 
 
 N 
 
 50 44 
 
 N 
 
 79 10 
 
 N 
 
 54 40 
 
 N 
 
 54 ^5 
 
 N 
 
 71 45 
 
 N 
 
 56 
 
 N 
 
 54 43 
 
 N 
 
 47 21 
 
 N 
 
 51 55 
 
 N 
 
 50 41 
 
 N 
 
 SI 4 
 
 N. 
 
 77 IS 
 
 N 
 
 3 49 
 
 S 
 
 22 00 
 
 N 
 
 60 10 
 
 N 
 
 52 33 
 
 N 
 
 3^ 35 
 
 N 
 
 21 40 
 
 N 
 
 55 45 
 
 N 
 
 5^ 37 
 
 N 
 
 47 44 
 
 N 
 
 43 -6 
 
 N 
 
 48 30 
 
 N 
 
 53 " 
 
 N 
 
 7S CO 
 
 N 
 
 72 52 
 
 N 
 
 20 55 
 
 N 
 
 47 
 
 S 
 
 77 5 
 
 N 
 
 9 1^ 
 
 N 
 
 1 1 42 
 
 N 
 
 49 42 
 
 N 
 
 C2 00 
 
 N 
 
 47 45 
 
 N 
 
 ]^ iO 
 
 N 
 
 16 33 
 
 I, 
 
 <;2 48 
 
 t.' 
 
 ,8 57 
 
 M 
 
 Longitude. 
 
 113 50 E. 
 
 172 40 E. 
 
 106 25 E. 
 
 10 46 W. 
 
 59 36W. 
 
 16 26 W. 
 61 47W. 
 20 06 E. 
 
 2 18 E. 
 I 16W. 
 
 17 12 W. 
 5 coW. 
 
 38 00 E. 
 
 63 iiW. 
 
 4 coW. 
 
 47 40 E. 
 
 47 07 E. 
 
 48 08 E. 
 68 20 E. 
 41 43 E. 
 
 106 45 E. 
 I 30 W. 
 9 oiW. 
 
 25 E. 
 24 15 E. 
 
 ^80 oW. 
 
 4 33 E. 
 83 4W. 
 
 5 52 W. 
 
 3 13 W. 
 
 55 25 W. 
 
 1 5W. 
 
 56 coW. 
 
 12 40 E. 
 102 5 E. 
 
 92 45 E. 
 
 6 14 E. 
 
 13 26 E. 
 
 63 23W. 
 02 53 W. 
 
 1 50 W. 
 3 49 W. 
 
 60 24 W. 
 3 i3W. 
 
 68 36 V/. 
 o 55 E. 
 10 50 E. 
 52 35 E. 
 17 5W, 
 
 64 37 W. 
 20 04 E. 
 85 55W. 
 64 20 W. 
 
 2 C.nV. 
 
 10 56W. 
 
 3 ^3W. 
 75 30W. 
 
 151 52 W. 
 157 CO E. 
 72 43 E. 
 
 H.Water. 
 
 yh. 30m. 
 
 3 45 
 
 3 30 
 
 o 00 
 
 12 00 
 
 10 o 
 3 30 
 
 7 CO 
 2 30 
 
 2 O 
 
 6 CO 
 
 9 45 
 
 o 00 
 3 o 
 
 Vol. I 
 
 P d 
 
 B?na
 
 37 
 
 G"E O G R A P H Y. 
 
 Book VI, 
 
 1 Name? at PUcrs. 
 
 Cont. 
 
 Countries. 
 
 Coaft. " 
 
 Latitude. 
 
 Long 
 
 itude. 
 
 H. Water.' 
 
 I 
 Bona 
 
 Africa 
 
 Tunis 
 
 Mediterran. 
 
 o 
 37 
 
 08 N. 
 
 
 
 7 
 
 10' E. 
 
 C Bona 
 
 Africa 
 
 Tunis 
 
 Mcditerran. 
 
 37 
 
 10 N. 
 
 10 
 
 00 E. 
 
 
 C Pons vift 
 
 Am. 
 
 Newfoundl. 
 
 Atl. Ocean 
 
 48 
 
 54 N. 
 
 Si 
 
 33W. 
 
 
 I. Bcna villa 
 
 Africa 
 
 C.VerdKkb 
 
 Atl. Ocean 
 
 16 
 
 05 N. 
 
 22 
 
 42 W. 
 
 
 C. Boni forfur.a 
 
 Eu. 
 
 Ruflia 
 
 White Sea 
 
 65 
 
 35 N. 
 
 38 
 
 25 E. 
 
 
 I.Bnnayrc,Sotovento 
 
 Am. 
 
 Terra Firma 
 
 Atl. Ocean 
 
 II 
 
 52 N. 
 
 67 
 
 20W. 
 
 
 B. Honaventura 
 
 Am. 
 
 Terra Firma 
 
 Paclf. Ocean 
 
 3 
 
 18 N. 
 
 76 
 
 50 w. 
 
 
 ,C. Hon Efperancc 
 
 Africa 
 
 C afters 
 
 Indian Ocean 
 
 34 
 
 29 S. 
 
 18 
 
 23 E. 
 
 3h. oom. 
 
 Bo'j -deaux 
 
 Eu. 
 
 France 
 
 B. Bifcay 
 
 44 
 
 50 N. 
 
 
 
 30 w. 
 
 3 00 
 
 ! f Eaftpnint *) 
 
 
 
 
 r I 
 
 12 N. 
 
 117 
 
 10 E. 
 
 
 , ) Weft point (. 
 |ce 1 North point f' 
 
 Afia 
 
 
 Indian Ocean 
 
 [^ 
 
 15 N. 
 oc N. 
 
 108 
 "3 
 
 57 E. 
 40 E. 
 
 
 ;-J t_ South point J 
 
 
 
 
 L 3 
 
 32 s. 
 
 112 
 
 05 T. 
 
 
 I. B,)r- 7 Borneo 7 
 neo J Succadano l' 
 
 Afia 
 
 
 Indian Ocean 
 
 h 
 
 00 N. 
 50 S. 
 
 112 
 
 108 
 
 IS E. 
 35 E. 
 
 
 I. B irriholin 
 
 Eu. 
 
 Sweden 
 
 Baltic Sea 
 
 55 
 
 12 N. 
 
 15 
 
 50 E. 
 
 
 Boftin" 
 
 Eu. 
 
 England 
 
 Germ. Ocean 
 
 53 
 
 10 N. 
 
 
 
 25 E. 
 
 
 Bofton 
 
 Am. 
 
 Mew Eng. 
 
 Atl. Ocean 
 
 42 
 
 25 N. 
 
 70 
 
 32 w. 
 
 
 Bnt..ny Ifle 
 
 Alia 
 
 M.Caledonia 
 
 PacJf. Ocean 
 
 22 
 
 27 s. 
 
 ,67 
 
 12 E. 
 
 
 Bot \ny Bay 
 
 Aha 
 
 N. Holland 
 
 Pacif. Ocean 
 
 34 
 
 00 S. 
 
 iS 
 
 28 E. 
 
 
 Boulogne 
 
 Eu. 
 
 France 
 
 Eng. Channel 
 
 50 
 
 44 N. 
 
 I 
 
 40 E. 
 
 10 30 
 
 I. Bourbon, Sr. Den. 
 
 Africa 
 
 Mad.igafcar 
 
 Indian Ocean 
 
 20 
 
 52 S. 
 
 55 
 
 35 E. 
 
 
 1. St. Brr-ndon 
 
 Africa 
 
 Madagafcar 
 
 Indian Ocean 
 
 16 
 
 45 S- 
 
 64 48 E. 
 
 
 B. Br.indwvns 
 
 Eu. 
 
 Greenland 
 
 North Oct- an 
 
 79 
 
 50 N. 
 
 26 
 
 20 E. 
 
 
 I. Bravas 
 
 Africa 
 
 C. Verd 
 
 Atl. Ocean 
 
 '4 
 
 hN. 
 
 24 45 W. 
 
 
 Bremen 
 
 Eu. 
 
 Germany 
 
 R. Wcfer 
 
 53 
 
 30 N. 
 
 9 
 
 00 E. 
 
 6 ' 00 
 
 Breefound, a fand 
 
 Eu. 
 
 D. Neth. 
 
 Germ. Ocean 
 
 'i3 
 
 12 N. 
 
 5 
 
 ,5 E. 
 
 4 30 
 
 Brtrflau 
 
 Eu. 
 
 Sileiia 
 
 R. Oder 
 
 51 
 
 03 N. 
 
 17 
 
 13 E. 
 
 
 Breft 
 
 Eu. 
 
 France 
 
 B, Bifcay 
 
 48 
 
 23 N. 
 
 4 
 
 26 w. 
 
 3 45 
 
 . Brcft 
 
 Am. 
 
 New Brltnin 
 
 Weih Ocean 
 
 5^ 
 
 10 N. 
 
 5- 
 
 30 w. 
 
 
 Cape Bret 
 
 Afia 
 
 N. Zealand 
 
 Pacif. Ocean 
 
 35 
 
 07 S. 
 
 173 
 
 52 E. 
 
 
 Bririgc Town 
 
 Am. 
 
 1. BarbadoCo 
 
 Atl. Oceun 
 
 3 
 
 05 N. 
 
 59 
 
 36W. 
 
 1 
 
 Bridlington Bay 
 
 Eu. 
 
 England 
 
 Germ. Ocean 
 
 54 07 N. 
 
 00 
 
 04 E. 
 
 3 45 
 
 Brill 
 
 Eu. 
 
 D. Neth. 
 
 Germ. Ocean 
 
 5' 
 
 56 N, 
 
 4 
 
 10 E. 
 
 I 30 
 
 Bri.'.n Ifie 
 
 Am. 
 
 Acadia 
 
 G . St. Lawr. 
 
 47 
 
 50 N. 
 
 60 
 
 47 W. 
 
 
 Briftol 
 
 Eu. 
 
 England 
 
 St. Geo. Ch. 
 
 51 
 
 28 N. 
 
 2 
 
 30W. 
 
 6 45 
 
 C. Briftol 
 
 Am. 
 
 Sandwich L. 
 
 Atl. Ocean 
 
 59 
 
 2 S. 
 
 26 
 
 46 W. 
 
 
 r_.c r Loulfbourgh 
 
 
 
 
 f45 
 
 54 N. 
 
 59 
 
 55W. 
 
 
 ;j 5 J I. Scatcri 
 
 Am. 
 
 Acadia 
 
 Atl. Ocean 
 
 ^46 
 
 01 N. 
 
 61 
 
 57W. 
 
 
 (S t f^'O'th Cape 
 
 
 
 
 I47 
 
 05 N. 
 
 60 
 
 8W. 
 
 
 - r I. Mathias 
 
 
 
 
 r " 
 
 00 S. 
 
 147 
 
 50 E. 
 
 
 ':; I North Doint 
 
 
 
 
 ' 
 
 30 S. 
 
 148 
 
 40 E. 
 
 
 ^1 S.W/pV,nr 
 . j Strait Dumpier 
 
 Afi.i 
 
 New Guinea 
 
 Pacif. Ocean 
 
 
 CO S. 
 15 S. 
 
 146 
 
 146 
 
 37 E. 
 15 E. 
 
 
 :y 1 C. St. George 
 
 
 
 
 5 
 
 30 S. 
 
 150 
 
 55 . 
 
 
 l'^ L I- St. John 
 
 
 
 
 L 4 
 
 20 S. 
 
 152 
 
 40 E. 
 
 
 r-uchuriefs 
 
 Eu. 
 
 Scotland' 
 
 Germ. Ocean 
 
 57 
 
 29 N. 
 
 I 
 
 23W. 
 
 3 00 
 
 Fucr.os Avres 
 
 Am. 
 
 Brafil 
 
 Atl. Ocean 
 
 34 
 
 35 S. 
 
 53 
 
 26W. 
 
 
 C. Bulier' 
 
 Am. 
 
 S. Georgia 
 
 Atl. Ocean 
 
 53 
 
 58 S. 
 
 37 
 
 40 W. 
 
 
 Burgaford point 
 
 Eu. 
 
 Iceland 
 
 North Ocean 
 
 66 
 
 03 N. 
 
 16 
 
 34 W. 
 
 
 Burgco.Iilos 
 
 Am. 
 
 Newfoundl. 
 
 Atl. Ocean 
 
 47 
 
 36 N. 
 
 57 
 
 TlW. 
 
 
 Burlings, rock.3 
 
 Eu. 
 
 Portus^a! 
 
 Atl. Ocean 
 
 39 
 
 20 N. 
 
 9 
 
 32 W, 
 
 
 B-.irlint'ton 
 
 Eu. 
 
 England 
 
 (ierm. Ocean 
 
 54 
 
 00 N. 
 
 
 
 08 E. 
 
 
 Button's Illes 
 
 Am. 
 
 New Britain 
 
 Hudf. Straits 
 
 60 
 
 35 N. 
 
 65 
 
 20W. 
 
 6 50 
 
 Cape Byron 
 
 Afia 
 
 N, Zcaluiid 
 
 Pacif. Ocean 
 
 28 
 
 39 S. 
 
 153 
 
 31 E. 
 
 
 Byron's Iflc 
 C 
 I. C.ihrera 
 
 Afia 
 
 
 Pacif. Ocean 
 
 I 
 
 iS S. 
 
 170 
 
 6W. 
 
 
 Eu. 
 
 Italy 
 
 Mediterran. 
 
 43 
 
 10 N. 
 
 9 
 
 11 E. 
 
 
 jc^-.d;3 
 
 Eu. 
 
 Spain 
 
 Atl. Ocean 
 
 36 
 
 31 N. 
 
 6 
 
 07W. 
 
 4 30 
 
 |Cr,en 
 
 !:u. 
 
 France 
 
 Eng. Channel 
 
 49 
 
 II N. 
 
 
 
 17W. 
 
 9 00 
 
 jCugliari, T. Sardln'r. 
 
 J'U. 
 
 Iraly 
 
 Mcdit. Sea 
 
 39 
 
 25 N. 
 
 9 
 
 38 E. 
 
 
 !C jj'i'-binng 
 
 lEu. 
 
 Finland 
 
 Baltic Sea 
 
 64 
 
 13 N. 
 
 . 27 
 
 SI E. 
 
 1 
 
 lilCo
 
 Book VI. 
 
 GEOGRAPHY. 
 
 379 
 
 Names of Places. 
 
 Cont. 
 
 C.untries. 
 
 Coaft. 
 
 Latitude. 
 
 Longitude. 
 
 I 
 H. Water. 
 
 
 
 1 
 
 o 
 
 
 
 
 
 
 Illes Calcos, or '' 
 Cankrofs, from ' ', 
 
 Am. 
 
 Bahamalfles 
 
 Atl. Ocean 
 
 \" 
 
 27 
 5 
 
 N. 
 N. 
 
 71 
 
 72 
 
 24 W. 
 15W. 
 
 
 ^'"^" {Z } 
 
 Africa 
 
 Guinea 
 
 Eth. Ocean 
 
 r^ 
 
 30 
 00 
 
 N. 
 N. 
 
 8 
 
 7 
 
 10 E. 
 00 E. 
 
 
 C. Calaberno 
 
 Afia 
 
 Natolia 
 
 Archipelago 
 
 38 
 
 42 
 
 N. 
 
 26 
 
 44 E. 
 
 
 Calais 
 
 Eu, 
 
 France 
 
 Eng. Channel 
 
 50 
 
 58 
 
 N. 
 
 01- 
 
 51 E. 
 
 iih. 30m. 
 
 C. Calamadon 
 
 Afia 
 
 India 
 
 B. Bengal 
 
 10 
 
 22 
 
 N. 
 
 80 
 
 40 E. 
 
 
 Calcutta 
 
 Afia 
 
 India 
 
 B. Bengal 
 
 22 
 
 35 
 
 N, 
 
 88 
 
 54 
 
 
 Caldera 
 
 Aha 
 
 I. -Mindano 
 
 Pacif. Ocean 
 
 7 
 
 CO 
 
 N. 
 
 121 
 
 25 E. 
 
 
 I. Caldy 
 
 Eu. 
 
 England 
 
 St. Geo. Ch. 
 
 51 
 
 33 
 
 N. 
 
 5 
 
 14W. 
 
 5 15 
 
 Calecut 
 
 Afia 
 
 India 
 
 Indian Ocean 
 
 II 
 
 IS 
 
 N, 
 
 75 
 
 39 E- 
 
 
 Cairo 
 
 Africa 
 
 Egypt 
 
 R. Nile 
 
 30 
 
 02 
 
 N. 
 
 31 
 
 26 E. 
 
 
 Callao 
 
 Am. 
 
 Peru 
 
 Pacif. Ocean 
 
 12 
 
 2 
 
 S. 
 
 76 
 
 53W. 
 
 
 1. Great Camanis 
 
 Am. 
 
 Weft Indies 
 
 Atl. Ocean 
 
 19 
 
 18 N. 
 
 80 
 
 29 W. 
 
 
 I. Little Camanis 
 
 Am 
 
 Weft Indies 'Atl. Ocean 
 
 19 
 
 42 
 
 .N. 
 
 .79 
 
 20 W. 
 
 
 Camboida 
 
 Afia 
 
 India 
 
 Indian Ocean 
 
 10 
 
 35 
 
 N. 
 
 104 45 E. 
 
 
 Cambridge 
 
 Eu. 
 
 England 
 
 
 
 5^ 
 
 '3 
 
 N. 
 
 
 
 9 E. 
 
 
 Cambridge 
 
 Am. 
 
 N. England 
 
 
 42 
 
 25 
 
 N. 
 
 71 
 
 sw. 
 
 
 
 C. Cambron, or 7 
 Carbon J 
 
 Africa 
 
 Algiers 
 
 Medit. Sea 
 
 37 
 
 18 
 
 N, 
 
 4 
 
 58 E. 
 
 
 C. Cameron 
 
 Am. 
 
 New Spain 
 
 Atl. Ocean 
 
 15 
 
 35 
 
 N 
 
 S3 
 
 29 W. 
 
 
 
 Africa 
 
 Guinea 
 
 Atl. Ocean 
 
 3 
 
 30 
 
 N. 
 
 9 
 
 10 E. 
 
 
 B. Camsrones 
 
 Am. 
 
 Magellan 
 
 Atl. Ocean 
 
 44 
 
 50 
 
 S. 
 
 67 
 
 loW. 
 
 
 Camfer, a land 
 
 Eu. 
 
 D. Neth. 
 
 Germ. Ocean 
 
 53 
 
 33 
 
 N. 
 
 5 
 
 30 E. 
 
 I 30 
 
 Camin 
 
 Eu. 
 
 Germany 
 
 Baltic 
 
 54 
 
 04 
 
 N. 
 
 15 40 E. 
 
 
 C. Campbell 
 
 Alia 
 
 N. Zealand Pacif. Ocean 
 
 41 
 
 51 
 
 S. 
 
 174 41 E. 
 
 
 Compeachy 
 
 Am. 
 
 Yucatin 
 
 Atl. Ocean 
 
 '9 
 
 36 
 
 N. 
 
 90 
 
 55W. 
 
 
 I. Canaria 
 
 Africa 
 
 Canaries 
 
 Atl. Ocean 
 
 2S 
 
 01 
 
 N. 
 
 15 
 
 oV/. 
 
 3 
 
 C. Candcnofe 
 
 Eu. 
 
 Ruffia 
 
 North Ocean 
 
 69 
 
 25 
 
 N. 
 
 45 
 
 30 E. 
 
 
 r C. St. John, w. 
 ^ 1 end 
 
 
 
 
 r35 
 
 12 
 
 N. 
 
 23 
 
 54 E. 
 
 
 1 =. -l Cindia 
 
 Eu. 
 
 Turkey 
 
 Medlt. Sea 
 
 1^5 
 
 19 
 
 N. 
 
 25 
 
 23 E. 
 
 
 i^^ j C. Solomon, E. 
 !- L_ tr.d 
 
 
 
 
 [34 
 
 57 
 
 N. 
 
 27 
 
 06 E. 
 
 , 
 
 |Candia 
 
 Afia 
 
 I. Ceylon 'Indian Ocean 
 
 7 
 
 54 
 
 N. 
 
 81 
 
 53 E- 
 
 
 Candlemis Illes 
 
 Am, 
 
 Sandwich L. Atl. Ocean 
 
 57 
 
 10 
 
 S. 
 
 27 
 
 13W. 
 
 
 1. Ca-iu 
 
 Afia 
 
 India ;Indian Ocean 
 
 7 
 
 30 
 
 S. 
 
 77 
 
 55 E. 
 
 
 C. Cani'a 
 
 Am. 
 
 N'o\a Scotia 
 
 Atl. Ocean 
 
 45 
 
 18 
 
 N. 
 
 60 
 
 48 W. 
 
 
 Canfo Fafi'age 
 
 Am. 
 
 Nova Scotia 
 
 Atl. Ocean 
 
 45 
 
 30 
 
 N. 
 
 61 
 
 coW. 
 
 
 :C. Cantin 
 
 Africa 
 
 Barhary 
 
 Atl. Ocean 
 
 32 
 
 41 
 
 N. 
 
 9 
 
 01 W. 
 
 00 
 
 .Cap.tirc, Mul 
 
 L.U. 
 
 Scotland 
 
 Weft Ocean 
 
 55 
 
 22 
 
 N. 
 
 5 
 
 45W. 
 
 
 , Canton. 
 
 Afia 
 
 China 
 
 Pacif. Ocean 
 
 23 
 
 08 
 
 N. 
 
 113 
 
 07 E. 
 
 
 .Cape Town 
 
 Africa 
 
 CafTers 
 
 Atl. Ocean 
 
 33 
 
 55 
 
 s. 
 
 18 
 
 23 . 
 
 2 30 
 
 1. C.pri 
 
 Eu. 
 
 Italy 
 
 Mcdit. Sea 
 
 40 
 
 34 
 
 N. 
 
 H 
 
 II E. 
 
 
 I. Carr.T.a 
 
 Eu. 
 
 Italy 
 
 Medit. Sea 
 
 43 
 
 '^3 
 
 N. 
 
 10 
 
 15 E. 
 
 
 C. Ci:,.^';r 
 
 Afia 
 
 India 
 
 B. Beng.d 
 
 '9 
 
 22 
 
 N. 
 
 86 
 
 05 E. 
 
 
 Car.icc! , 
 
 Am. 
 
 Terra Firma'Atl, Ocean 
 
 10 
 
 06 
 
 N. 
 
 66 
 
 4,-W. 
 
 
 |C,iri!-.,orIi:<cliii:siriC. 
 
 Am. 
 
 Grf-nl.ind 
 
 Baffin's Bay 
 
 77 
 
 15 
 
 N. 
 
 62 
 
 00 W. 
 
 , 
 
 |C.irin,i i'<..-i: 
 
 Am. 
 
 California 
 
 P.icif. Ocean 
 
 i^ 
 
 24 
 
 N . 
 
 124 
 
 25 U'. 
 
 
 |C,i-;ekrf<on 
 
 Eu. 
 
 Sweden 
 
 Baltic 
 
 5''' 
 
 20 
 
 N. 
 
 15 
 
 31 V. 
 
 1 
 
 'Cirliil'- 
 
 Eu. 
 
 Enrjland 
 
 Iriih Sr-a 
 
 54 
 
 47 
 
 K. 
 
 2 
 
 35W. 
 
 j 
 
 C. C-.::r.cl 
 
 -^lia 
 
 Syria 
 
 Levant 
 
 33 
 
 08 
 
 N 
 
 35 
 
 35 E. 
 
 I 
 
 C.-oiinc Illc;, limits 
 
 Afia 
 
 
 Pacif. Ocean 
 
 ^ 7 
 
 10 
 
 N. 
 
 '37 
 
 2S. E. 
 
 i 
 
 
 i!'^ 
 
 00 
 
 N. 
 
 127 
 
 2> i~.. 
 
 
 'C. C.irtli.ije 
 
 Africa 
 
 Barbary 
 
 Mcdit. Sea 
 
 3i 
 
 5^ 
 
 N. 
 
 ;o 
 
 1' 1 
 
 
 ,C .irtii ijj'!! < 
 
 Am. 1 
 
 Terra I'lrma 
 
 Caribbean Sc.i 
 
 10 
 
 7.7 
 
 N'. 
 
 7 5 
 
 2; V,'. 
 
 
 Cuih.uync 
 
 Eu. 
 
 Spain 
 
 Mtdit. Scu 
 
 37 
 
 57 
 
 N. 
 
 1 
 
 03 w. 
 
 
 ,r..>:(T<-\ in<- 
 
 Afi:. 
 
 New Britain 
 
 Pacif. 0.c.ni 
 
 S 
 
 .'.0 
 
 s. 
 
 '^9 
 
 .4 1;. 
 
 
 :V n\\ .Swanb Nc:t 
 
 Am. 
 
 
 
 Hudfon\ B.:\ 
 
 62 
 
 :o 
 
 X. 
 
 ^3 
 
 30 w. 
 
 
 ,C/.!ki-t-. 
 [C. Cinn.lcr 
 
 Eu. 
 
 I. {iucrr.fey 
 
 Eng. Ch mr.ol 
 
 49 
 
 53 
 
 N 
 
 
 zGVv. 
 
 8 15 
 
 Ku. 
 
 Tuikjy 
 
 Archipcl.wjo 
 
 40 
 
 i^2 
 
 N. 
 
 27 
 
 4i E. 
 
 
 ;,!. St Citii'-riiir'' 
 
 A-n. 
 
 Br.ifil 
 
 Atl. Oco:.n 
 
 27 
 
 .5'> 
 
 N, 
 
 49. 
 
 12W. 
 
 
 D d a 
 
 C<t
 
 38o 
 
 GEOGRAPHY. 
 
 Book VI. 
 
 Nines of Places. 
 
 Cont. 
 
 Countries. 
 
 Coaft. 
 
 Latitude. 
 
 Longitude. 
 
 H. Water. 
 
 Catir.c5^.poi"t 
 I 4> pont 
 
 Am. 
 
 Bahama 
 
 Atl. Ocean 
 
 o , 
 C 24 50 N. 
 I Z2 48 N. 
 
 7S 3W. 
 75 3SW. 
 
 
 .Cathnefi Point 7 
 I Dinnet Head 5 
 
 Eu. 
 
 Scotland 
 
 Weft. Ocean 
 
 58 42 N. 
 
 3 '7W. 
 
 9h. com. 
 
 Ca::nea 
 
 Eu. 
 
 I.Sicily 
 
 Medit. Sea 
 
 42 40 N. 
 
 20 30 E. 
 
 
 C. Catocha 
 
 Am. 
 
 New Spain 
 
 Caribbean Sea 
 
 20 48 N. 
 
 86 35W. 
 
 
 I. Cayenne 
 
 Am. 
 
 Terra Firms 
 
 Atl. Ocean 
 
 4 56 N. 
 
 52 loW. 
 
 
 TN. E. point 
 
 
 
 
 r I 48 N. 
 
 124 37 E. 
 
 
 3 1 W. point 
 
 
 
 
 I 3 00 s. 
 
 117 55 
 
 
 'S J S. W. point Ma- 
 's s ^ 
 
 Afia 
 
 Spice Ides 
 
 Indian Ocean 
 
 \ 5 " S. 
 
 117 50 E. 
 
 
 (J j c.iuer 
 
 
 
 
 1 
 
 
 
 L; / S. point 
 
 
 
 
 / 5 40 S. 
 
 119 55 E. 
 
 
 1 l_S. E. point 
 
 
 
 
 L 5 20 s. 
 
 121 58 E. 
 
 
 I. Cephaloniii 
 
 Eu. 
 
 Turkey 
 
 Medit. Se.x 
 
 38 20 N. 
 
 20 II E. 
 
 
 Ceuta 
 
 Africa 
 
 Barbary 
 
 Medit. Sea 
 
 35 49 N. 
 
 5 25W. 
 
 
 ^" Infanapatam, N. 
 = I point 
 
 
 
 
 r 9 47 N. 
 
 80 55 E. 
 
 
 f-.) Trinquemale, S. 
 ;j) E. end ' 
 
 Afia 
 
 India 
 
 Indian Ocean 
 
 i S 40 N, 
 
 Si 40 E. 
 
 
 / C. Gallo,S.Weft 
 
 
 
 
 1 6 27 N. 
 
 8z 10 E. 
 
 
 I. end 
 
 
 
 
 L 6 15 N. 
 
 80 20 E. 
 
 
 Chain I Hand 
 
 Afia 
 
 Society Ifles 
 
 Pacif. Ocean 
 
 17 25 S. 
 
 145 30 W. 
 
 
 Chmdeiiagar 
 
 Afia 
 
 Bengal 
 
 River Ganges 
 
 22 51 N. 
 
 S8 34 E. 
 
 
 Charles Town 
 
 Am. 
 
 Carolina 
 
 .'\fliley River 
 
 33 22 N, 
 
 79 50 W. 
 
 3 
 
 !C. Ch.-.rles 
 
 Am. 
 
 Virginia 
 
 Atl. Ocean 
 
 37 II N. 
 
 76 07 W. 
 
 
 I. ot ?Eaft end 
 Cbarlesi Weft end 
 
 Am. 
 
 Labradore 
 
 Hudfon's Str. 
 
 C 62 46-^N. 
 I 62 48 N' 
 
 74 15W. 
 
 75 30 w. 
 
 10 15 
 
 C. Charles 
 
 Am. 
 
 New Britain 
 
 Weft Ocean 
 
 51 50 N. 
 
 51 loW. 
 
 
 Charlotte's Ifles 
 
 Afia 
 
 Guadalcanal 
 
 Pacif. Ocean 
 
 11 S. 
 
 164 E. 
 
 
 C. Charlotte 
 
 Am. 
 
 S. Georgia 
 
 Atl. Ocean 
 
 54 32 S. 
 
 36 12W. 
 
 
 0^ Charlotte's Sound 
 
 Afia 
 
 N. Zealand 
 
 Pacif. Ocean 
 
 41 6 S. 
 
 174 19 E. 
 
 9 CO 
 
 '(^Charlotte's Foreld. 
 |l. Charlton 
 
 Afia 
 
 N.Caledonia 
 
 Pacif. Ocean 
 
 22 15 S. 
 
 167 iS E. 
 
 
 Am. 
 
 New Wales 
 
 Hudfon's Bay 
 
 52 03 N. 
 
 79 00 W. 
 
 
 Chittt-aux Bay 
 
 Am. 
 
 Labradore 
 
 Atl. Ocean 
 
 5Z I N. 
 
 55 50 W. 
 
 
 IB. Chfbudto 
 
 Am. 
 
 Nova Scotia 
 
 Atl. Ocean 
 
 44 45 N. 
 
 63 18 W. 
 
 
 Chcignecio 
 
 Am. 
 
 Nova Scotia 
 
 B. Fundy 
 
 46 15 N. 
 
 63 iiW. 
 
 45 
 
 Chjrlioiirgh 
 
 Eu. 
 
 France 
 
 Eng. Channel 
 
 49 38 N. 
 
 01 33 W. 
 
 7 30 
 
 C Kerry Lie 
 
 Eu. 
 
 Greenland 
 
 North Ocean 
 
 74 35 N. 
 
 18 05 E. 
 
 
 Che.ler 
 
 Eu. 
 
 England 
 
 IriiTi Sea 
 
 53 10 N. 
 
 2 2SW. 
 
 
 Chiddocic "" 
 
 Eu. 
 
 England 
 
 Eng. Channel 
 
 50 47 N. 
 
 3 00 W- 
 
 
 C. Chidlsy 
 
 Am. 
 
 New Britain 
 
 Hudf. Straltb 
 
 60 22 N, 
 
 65 00 W. 
 
 
 I.Chi!oc5f P^^--'^ 
 ^ S. Doint 
 
 Am. 
 
 Patagonia 
 
 Pacif. Ocean 
 
 C41 45 S. 
 
 243 50 S. 
 
 73 05 W. 
 
 
 C. Chick^f,.ago 
 
 Afia 
 
 Siberia 
 
 North Ocean 
 
 64 00 N. 
 
 174 45 W. 
 
 
 Chriftiana 
 
 Eu. 
 
 Norway 
 
 Sound 
 
 59 25 ^' 
 
 10 30 E. 
 
 
 Chriitianople 
 
 Eu. 
 
 Sweden 
 
 Baltic Sea 
 
 55 55 N. 
 
 15 10 E. 
 
 
 Chrjftianftadt 
 
 Ea. 
 
 Sweden 
 
 G. Bothnia 
 
 62 47 N. 
 
 22 50 E. 
 
 
 Chiift.Tiaf Sound 
 
 Am. 
 
 T. del Fuegr> 
 
 Pacif. Ocean 
 
 55 22 S. 
 
 70 01 W. 
 
 2 30 
 
 jl. St. Chriftopher's 
 
 Am. 
 
 Carib. Ifles 
 
 Atl. Ocean 
 
 17 15 N. 
 
 62 38W. 
 
 
 IR. St. Ch-.iftoplitr's 
 
 Africa 
 
 Caft'crs 
 
 Indian Ocean 
 
 32 47 S. 
 
 30 00 E. 
 
 
 IC. Chukchcnle 
 
 Afia 
 
 Siberia 
 
 North Ocean 
 
 66 30 N. 
 
 171 loW. 
 
 
 jC. C!u;rchi:i I 
 |R. Churchill 5 
 
 Am. 
 
 New Walc. 
 
 Hudfon's Bay 
 
 C 58 48 N. 
 I 58 47 JN^ 
 
 93 loW. 
 
 94 03 W. 
 
 7 20 
 
 1. Chilian 
 
 Afia 
 
 China 
 
 Chinefe Se.i 
 
 30 CO N. 
 
 i2i 50 E. 
 
 
 Civita Vecchia 
 
 Eu. 
 
 Italy 
 
 Medit. Sr-a 
 
 42 5 N. 
 
 II s' 
 
 
 C. Clear 
 
 Eu. 
 
 Ireland 
 
 Weft. Occp.n 
 
 51 18 N. 
 
 9 50 w. 
 
 4 30 
 
 :iirk-s Lies 
 
 Am. 
 
 S. Georgia 
 
 Atl. Ocean 
 
 S5 6 S. 
 
 34 S7V^. 
 
 
 (. Cloate 
 
 Afia 
 
 India 
 
 Indian Occnn 
 
 21 CO S, 
 
 95 4"^ E. 
 
 
 ''och::i 
 
 Afia 
 
 India 
 
 Indian Ocea:. 
 
 9 50 N. 
 
 76 05 E. 
 
 
 '. Cccos 
 
 Afia 
 
 India 
 
 Indian Ocean 
 
 12 20 S. 
 
 98 10 E. 
 
 
 . Coco:. 
 
 Am. 
 
 Mexico 
 
 Pacif. Ocean 
 
 5 cc N. 
 
 SS 4sW. 
 
 
 '. C:-i 
 
 Am. 
 
 Kf'v Eng. 
 
 AtJ. Oer,,n 
 
 42 15 N. 
 
 69 27 V/. 
 
 
 Colchefter
 
 Book VI. 
 
 GEOGRAPHY. 
 
 J8i 
 
 Names of Places. 
 
 Cont. 
 
 Countries. 
 
 Coaft. 
 
 Latitude. 
 
 Longitude. 
 
 H. Water, 
 
 Colchefter 
 
 Eu. 
 
 England 
 
 Germ. Ocean 
 
 / 
 
 52 00 N. 
 
 . , 
 58 E. 
 
 
 C. Cold 
 
 Eu. 
 
 Greenland 
 
 North Ocean 
 
 79 00 N. 
 
 10 00 E. 
 
 
 C. Colnet 
 
 Afia 
 
 N.Calcdonia 
 
 Pacif. Ocean 
 
 20 30 S. 
 
 164 56 E. 
 
 
 R. Colerado 
 
 Am. 
 
 New Spain 
 
 G. California 
 
 31 40 N. 
 
 115 25 W. 
 
 
 I. Colgau 
 
 Eu. 
 
 Ruffia 
 
 North Ocean 
 
 69 20 N. 
 
 45 00 E. 
 
 
 CoUioure 
 
 Eu. 
 
 Spain 
 
 Medit. Sea 
 
 4Z 31 N. 
 
 3 10 E. 
 
 
 C. Colone 
 
 Eu. 
 
 Turkey 
 
 Archipelago 
 
 37 43 N- 
 
 24 41 E. 
 
 
 C. Colone 
 
 Afia 
 
 Natolia 
 
 Archipelago 
 
 39 JO N. 
 
 27 04 E. 
 
 
 C. Colonni 
 
 Eu. 
 
 Italy 
 
 Medit. Sea 
 
 38 50 N. 
 
 18 05 E. 
 
 
 C. Colville 
 
 Afia 
 
 N, Zealand 
 
 Pacif. Ocean 
 
 36 27 S. 
 
 174 48 E. 
 
 
 Comana 
 
 Am, 
 
 Terra FirmalAtl, -Ocean 
 
 10 00 N. 
 
 65 07 W. 
 
 
 C. Comarin 
 
 Afia 
 
 India 
 
 Indian Ocean 
 
 7 55 N- 
 
 78 7 E. 
 
 
 C. Comfort 
 
 Am. 
 
 New Wales 
 
 Hudfon's Bay 
 
 64 45 N. 
 
 S2 30 W, 
 
 
 Concarncau 
 
 Eu. 
 
 France 
 
 B. Bifcay 
 
 47 54 N. 
 
 3 50 w. 
 
 jh, 00m. 
 
 C. Conception 
 
 Am. 
 
 Calefornia 
 
 Pacif. Ocean 
 
 35 40 N. 
 
 126 01 W. 
 
 
 B. Conception Entra 
 
 Am. 
 
 Nev/foundl. 
 
 Atl. Ocean 
 
 48 25 N. 
 
 50 07 W 
 
 
 Conception 
 
 Am. 
 
 Chili 
 
 Pacif. Ocean 
 
 56 43 S. 
 
 - 73 13W. 
 
 
 R. Congo 
 
 Africa 
 
 Congo jEth. Ocean 
 
 5 45 S. 
 
 II 53 E, 
 
 
 I. Ccningen 
 
 Afia 
 
 N. Zealand Pacif. Ocean 
 
 34 30 ^ 
 
 164 25 E. 
 
 
 Coningfburgh 
 
 Eu. 
 
 Poland Baltic Sea 
 
 54 44 N. 
 
 21 S3 E. 
 
 
 Conquet 
 
 Eu. 
 
 France Eng. Chancel 
 
 48 30 N. 
 
 4 35^- 
 
 a 15 
 
 C, Conquibaco 
 
 Am. 
 
 Terra FirmaAtl. Ocean 
 
 12 15 N. 
 
 69 57 W. 
 
 
 Ccnftantinople 
 
 Eu. 
 
 Turkey 
 
 Archipelago 
 
 41 00 N. 
 
 28 53 E. 
 
 
 Ccoic's Straits 
 
 Afia 
 
 N. Zealand 
 
 Pacif. Ocean 
 
 41 6 S. 
 
 174 30 E. 
 
 
 Cooper's ifle 
 
 Am. 
 
 S. Georgia 
 
 Atl. Ocean 
 
 54 57 S. 
 
 36 W. 
 
 
 Copciihrigen 
 
 Eu. 
 
 Denmark 
 
 Baltic Sea 
 
 55 41 N. 
 
 12 40 E. 
 
 
 CoDerwic 
 
 Eu. 
 
 Norway 
 
 Sound 
 
 59 20 N. 
 
 10 lo E. 
 
 
 1l. Copland 
 
 Eu. 
 
 Ireland 
 
 iriih Sea 
 
 54 40 N. 
 
 6 40 W. 
 
 
 L Coquet 
 
 Eu. 
 
 England 
 
 Germ. Ocean 
 
 55 20 N. 
 
 I 25 W. 
 
 3 00 
 
 R. Coquimbo 
 
 Am. 
 
 Chili 
 
 Pacif, Ocean 
 
 29 54 S. 
 
 71 10 W. 
 
 
 C. Corbau 
 
 Afia 
 
 Natolia 
 
 Archipelago 
 
 38 03 N. 
 
 26 58 E. 
 
 
 Cordouc 
 
 Eu. 
 
 France 
 
 B. Bifcay 
 
 45 30 N. 
 
 01 10 W, 
 
 
 Corel, South limit 
 
 Afia 
 
 China 
 
 Pacif. Ocean 
 
 34 5 ^ 
 
 J 124 25 E. 
 2 127 25 E, 
 
 
 I. Corfu 
 
 Eu. 
 
 Turkey 
 
 Mcditerran, 
 
 39 50 N. 
 
 19 48 E. 
 
 
 iC. Corientes 
 
 Africa 
 
 Ca.'tVcs 
 
 Indian Ocean 
 
 24 08 S. 
 
 36 49 E. 
 
 
 JC. Corientes 
 
 Am. 
 
 Mexico 
 
 Pacif. Ocean 
 
 20 18 N. 
 
 ic8 00 W. 
 
 
 lo.rinth 
 
 Eu. 
 
 Turkey 
 
 Archipelago 
 
 37 30 N. 
 
 23 00 E. 
 
 
 Corke 
 
 Eu. 
 
 Ireland 
 
 St. Geo. Ch. 
 
 SI 54 N. 
 
 8 30 W. 
 
 6 30 
 
 jC. Coronation 
 
 Afia 
 
 N.Caledonia 
 
 Pacif. Ocean 
 
 22 5 S. 
 
 167 8 E. 
 
 
 'C. Corfc 
 
 Africa 
 
 Guinea 
 
 Edi. Sea 
 
 5 12 N, 
 
 23 W. 
 
 3 30 
 
 { ^ r C. Corfo, North 
 
 
 
 
 
 
 
 ' \ pcint 
 
 . S ^ BonKicio, Sou'.h 
 
 i (. point 
 
 Eu. 
 
 Italy 
 
 Meditcrran. 
 
 C42 53 N. 
 
 ^41 22 N. 
 
 9 40 E. 
 9 26 E, 
 
 
 
 
 
 
 
 
 I, Corvo 
 
 Eu. 
 
 Azorc; - 
 
 Atl. Ocean 
 
 39 4^ N. 
 
 31 C2W. 
 
 
 II. Cofmolcdo 
 
 Africa 
 
 Mad.igafcar 
 
 Indian Ocean 
 
 10 23 S. 
 
 51 40 E. 
 
 
 I. Cou.irc 
 
 Am. 
 
 Canada 
 
 R. St. Lawr. 
 
 47 3 N. 
 
 69 2 W. 
 
 
 Cow and C df 
 
 Eu. 
 
 Ireland 
 
 Well. Ocean 
 
 51 TZ N. 
 
 10 30 W. 
 
 
 I. Co2u:r.'jl 
 
 Am. 
 
 Yucatan 
 
 Atl. Ocean 
 
 19 36 N. 
 
 86 35 V/. 
 
 
 R. Cocci 
 
 Afi.i 
 
 China 
 
 B. Nankin 
 
 34 -6 N. 
 
 J20 10 E. 
 
 
 Cromer 
 
 Eu. 
 
 England 
 
 Germ. Ocean 
 
 53 05 N. 
 
 56 E. 
 
 7 00 
 
 C'rnf)k(fi I. N, point 
 
 Am. 
 
 Bahama 
 
 Atl. Ocean 
 
 "22 47 N. 
 
 73 5'^^^'- 
 
 
 Crof-; Ifc 
 
 Eu. 
 
 Ruir.a 
 
 White Sea 
 
 66 31 N. 
 
 3<' 3 3 E. 
 
 
 Cro'.': I'v int 
 
 Eu. 
 
 Nova Zcm. 
 
 North Ocean 
 
 72 00 N. 
 
 S3 !2 1:. 
 
 
 '".. Cr-17. 
 
 Africa 
 
 Barbary 
 
 Atl. Ocean 
 
 30 36 N. 
 
 9 3^W, 
 
 
 'I. S;. Crvz 
 
 Am. 
 
 Antilles I. 
 
 Atl. Ocean 
 
 '7 53 ^^ 
 
 64 55 \V. 
 
 
 _ C C. Antonio, W. 
 
 
 
 
 
 
 
 :-2 p'.lnt 
 
 
 
 
 21 45 N. 
 
 S4 05W. 
 
 
 -^l I". .Ic Mais, E. 
 -J point 
 
 Am. 
 
 Antillc: I. 
 
 Atl. Ocean 
 
 20 03 N. 
 
 74 5^-W. 
 
 
 1 I H'.il',v. C-ip- 
 
 
 
 
 
 I .9 41 >J. 
 
 77 =sw. 1 
 
 .... , fc . , . 1 
 
 10.
 
 382 
 
 GEOGRAPHY. 
 
 Book VI. 
 
 Names of Places. 
 
 Cont. 
 
 Countries. 
 
 Coaft. 
 
 Latitude. 
 
 Longitude. 
 
 |H. Water. 
 
 
 
 
 
 
 o , 
 
 , 
 
 
 
 ^ r St. Jago 
 St. Mary 
 
 
 
 
 r 2o 03 N. 
 
 7S 51 W. 
 
 
 
 
 
 
 31 26 N. 
 
 78 loW. 
 
 
 
 U -^ Le St. Efprit 
 
 Am. 
 
 Antilles I. 
 
 Atl. Ocean 
 
 i 21 56 N. 
 
 79 50 W. 
 
 
 
 ^ Havannah 
 
 
 
 
 23 12 N. 
 
 81 4sW. 
 
 
 
 -" [.B. Hondy 
 
 
 
 
 L22 54 N. 
 
 82 40 W. 
 
 
 
 Cubbs IHes 
 
 Am. 
 
 New Wales 
 
 Hudfon's Bay 
 Indian Ocean 
 
 54 IS N. 
 
 82 34 W 
 
 - 
 
 
 Cubello 
 
 Ada 
 
 Ind. Malab. 
 
 7 so N. 
 
 71 55 E. 
 
 
 
 C. Cumberland 
 
 Afu 
 
 N. Hebrides 
 
 Pacif. Ocean 
 
 14 4c S. 
 
 166 47 E. 
 
 
 
 Cumberland Ides 
 
 Afia 
 
 Socicoy Ifles 
 
 Pacif. Oeean 
 
 19 18 S. 
 
 140 36 E. 
 
 
 
 B. Cumberland 
 
 Am. 
 
 North Main 
 
 Davis's Str. 
 
 66 40 N. 
 
 65 20 W. 
 
 
 
 Curaflba 
 
 Am. 
 
 Terra Firma 
 
 Atl. Ocean 
 
 II s6 N, 
 
 68 20 W. 
 
 
 
 I.Cuzzola 
 
 Eu, 
 
 Turkey 
 
 Medit. Sea 
 
 4^ 50 N. 
 
 16 55 E. 
 
 
 
 Cufco 
 
 Am. 
 
 Peru 
 
 Inland 
 
 12 2S S. 
 
 73 35 W. 
 
 
 
 f-C. Baffii, W. end 
 
 
 
 
 r 35 04 N. 
 
 33 04 E. 
 
 
 
 ^ 1 C. St. Andr. E. 
 
 
 
 
 I 
 
 
 
 
 a \ end 
 
 
 
 
 1 35 40 N. 
 
 35 08 E. 
 
 
 
 ^S:^ C. de Gaffe, S. 
 
 Afia 
 
 Syria, 
 
 Medit. Sea 
 
 i 
 
 
 
 
 '"^ j point 
 
 
 
 
 / 34 35 N. 
 
 33 41 E. 
 
 
 
 - 1 C. Grego, S. E. 
 
 
 
 
 / 
 
 
 
 
 L point 
 D. 
 
 
 
 
 L 34,57 N. 
 
 34 36 E. 
 
 
 
 Dabul 
 
 Afia 
 
 India 
 
 Arabian Sea 
 
 18 24 N. 
 
 73 33 E- 
 
 
 
 iDihlak 
 
 ACm 
 
 Arabia 
 
 Red Sea 
 
 15 50 N. 
 
 41 44 E. 
 
 
 
 jlHes ot" Danger 
 
 Aha 
 
 Society Ifles Pacif. Ocean 
 
 10 15 S. 
 
 165 50 W. 
 
 
 
 il. D.i^eroor' > 
 j Light-houfe J 
 
 Eu. 
 
 Livonia Baltic Sea 
 
 58 55 N. 
 
 22 32 E. 
 
 
 
 Dantzic 
 
 Eu. 
 
 Poland Baltic Sea 
 
 54 22 N, 
 
 18 36 E. 
 
 
 
 |Str. Dardanek 
 
 Eu. 
 
 Turkey Archipelago 
 
 40 10 N. 
 
 26 26 E. 
 
 
 
 Gulph Durieu 
 
 Am. 
 
 Terra Firma Caribbean Sea 
 
 3 45 N. 
 
 76 35 W. 
 
 
 
 iDartmo-irh 
 
 Eu. 
 
 England 
 
 Eng. Oiannel 
 
 50 27 N. 
 
 3 36 w. 
 
 6h. 30m. 
 
 
 I. Dauphin 
 
 Am. 
 
 Loulfiana 
 
 G. Me)<ico 
 
 29 40 N. 
 
 87 53 W. 
 
 
 
 St. Dsvid'i Head 
 
 Eu. 
 
 Wales 
 
 St. Geo. Ch. 
 
 51 55 N.. 
 
 5 22 W. 
 
 6 00 
 
 
 iOrt St. David'o 
 
 Alia 
 
 India 
 
 Corom. Coaft 
 
 12 05 N. 
 
 80 55 E. 
 
 
 
 I. Dcfeada 
 
 Am. 
 
 Carib. Ifies 
 
 Atl. Ocoin 
 
 16 36 N. 
 
 61 loW. 
 
 
 
 C. D.:fenda 
 
 Am. 
 
 T. delFucgo,'Pacif. Ocelli 
 
 53 4 S. 
 
 74 J3W. 
 
 
 
 C, Detire 
 
 Eu. j 
 
 Nova Zcm. 
 
 North Ocean 
 
 77 45 N- 
 
 79 20 E. 
 
 
 
 iC. Dclbiatlon 
 
 Am. 1 
 
 Greenland 
 
 North Ocean 
 
 61 45 N. 
 
 47 00 W. 
 
 
 
 iDeviPi. Idei 
 
 Eu. 
 
 Greenland 
 
 North Ocean 
 
 80 00 N, 
 
 II 43 E, 
 
 
 
 Devi'point 
 
 Alia 
 
 India 
 
 B. Bengd 
 
 16 07 N. 
 
 81 47 E. 
 
 
 
 I. Diego R lyes 
 1 
 
 Afia 1 
 
 1 
 
 Ind. Malab. 
 
 Indian Ocean 
 
 C 45 s. 
 
 I 30 N. 
 
 70 25 E. 
 
 
 
 jl. Diego Garcia 
 
 Africa ' 
 
 [ndia 
 
 Indian Ocean 
 
 8 45 S. 
 
 63 10 E. 
 
 
 
 jStr. Diemen 
 
 Afia 1 
 
 Japan Ides Pacif. Ocean 
 
 31 12 N. 
 
 130 55 E. 
 
 
 
 jl. Dicu 
 
 Eu. 
 
 France BayofBifcay 
 
 46 26 N. 
 
 2 20 W. 
 
 
 
 |D:eppe 
 
 Eu. 
 
 Fr ince Eng. Channel 
 
 49 55 N. 
 
 I 12 E. 
 
 10 30 
 
 
 pigge's, or Dudley's 7 
 
 1 Cioe 5 
 
 Am. \ 
 
 Greenland 
 
 Baffin's Bay 
 
 76 48 N. 
 
 59 07 W. 
 
 
 
 C. Dig;s . 
 
 Am. ! 
 
 L,abr?.dorc 
 
 FTudfnn's Bay 
 
 62 45 N. 
 
 78 50 W. 
 
 
 
 jl'o. Diu 
 
 Afia 
 
 India 
 
 [ndian Ocean 
 
 21 37 N. 
 
 70 28 E. 
 
 
 
 Ic. DoSbi 
 
 Am. 
 
 NortiiWalos Hudfon's Bay| 
 
 65 10 N. 
 
 86 25 W. 
 
 
 
 Dotarc 
 
 Afia 
 
 Arabia 
 
 [ndian Ocean 
 
 16 24 N. 
 
 53 40 E. 
 
 
 
 'St. DTiJneo 7 
 
 j Hiipunrru 5 
 
 Am. 
 
 Antilles 
 
 Atl. Ocean 
 
 18 25 N. 
 
 69 30 W. 
 
 
 
 I. Don;ini' J 
 
 Am. 1 
 
 C-iribb:c 
 
 Atl. Ocean 
 
 IS 18 N. 
 
 6i 28 W. 
 
 
 
 iDordrecht 
 
 Eu. 
 
 D. Ncth. 
 
 R. Macs 
 
 52 00 N, 
 
 4 26 E. 
 
 
 
 iC Dorfui 
 
 A!:ica' 
 
 -^jan 
 
 juiian Ocean 
 
 10 15 N. 
 
 50 44 E. 
 
 
 
 :c. Doro 
 
 Eu. ' 
 
 Turkey 
 
 Archipelago 
 
 38 02 N. 
 
 25 12 E. 
 
 
 
 lUort 
 
 Eu. ! 
 
 D. Neth. 
 
 Germ. Ocean 
 
 5' 47 N, 
 
 4 40 E. 
 
 3 00 
 
 
 jl. Dofcl 
 
 H.], i 
 
 .ivonia 
 
 Baltic Sea 
 
 58 20 N. 
 
 23 00 E. 
 
 
 
 1-. Do. Banlios 
 
 Ar'rica | 
 
 Zanguebar 
 
 ndian Ocean 
 
 5 >5 N, 
 
 49 2} E. 
 
 
 
 Do->er 
 
 Eu. 1 
 
 ngland 
 
 Er.g. C.har.i:d 
 
 5' 07 N. 
 
 I 19 E. 
 
 I 30 
 
 
 i'^::!-""^. -, ^ 
 
 Eu. ; 
 
 F.n-cind 
 
 Gem:. Oc^an 
 
 51 25 N. 
 
 I 27 E. 
 
 I J 5 
 
 
 Po.
 
 Book VI. 
 
 GEOGRAPHY. 
 
 3S5- 
 
 1 - 
 
 Names of Places. 
 
 Cent. 
 Africa 
 
 Countries. 
 
 Coaft. 
 
 Latitude. 
 
 Longitude. 
 
 H.Water.'' 
 
 1 
 
 Po. Dradate 
 
 Egypt 
 
 Red Sea 
 
 i9 56 N. 
 
 / 
 37 40 E. 
 
 
 
 Po. Drake, fir Francis 
 
 Am. 
 
 California 
 
 ^acif. Ocean 
 
 38 45 N. 
 
 128 35W. 
 
 
 
 Drontheim 
 
 Eu. 
 
 Norway 
 
 lTorth Ocean 
 
 63 26 N. 
 
 II 08 E. 
 
 
 
 Dublin 
 
 Eu. 
 
 Ireland 
 
 !riih Sea 
 
 53 21 N. 
 
 6 5W. 
 
 9h 
 
 , 15m 
 
 Dunbar 
 
 Eu. 
 
 Scotland 
 
 jerm. Ocean 
 
 55 58 N. 
 
 2 22 w.- 
 
 2 
 
 SO 
 
 Dundalk 
 
 Eu. 
 
 [reland 
 
 [rifli Sea 
 
 53 57 N. 
 
 6 28W. 
 
 
 
 Dundee 
 
 Eu. 
 
 Scotland 
 
 Germ. Ocean 
 
 56 26 N. 
 
 2 48 W. 
 
 a 
 
 15 
 
 Dungarven 
 
 Eu. 
 
 [reland 
 
 Atl. Ocean 
 
 51 57 N. 
 
 7 55W. 
 
 4 
 
 30 
 
 Dungenefs 
 
 Eu. 
 
 England 
 
 ing. Channel 
 
 50 53 N. 
 
 59 E. 
 
 9 
 
 45 
 
 Duncanlby Head 
 
 Eu. 
 
 Scotland 
 
 Germ. Ocean 
 
 58 40 N. 
 
 2 57 W. 
 
 
 
 Dunkirk 
 
 Eu. 
 
 France 
 
 Germ. Ocean 
 
 51 02 N. 
 
 2 27 E. 
 
 
 
 00 
 
 Dunnofe 
 
 Eu. 
 
 I. White 
 
 Eng. Channel 
 
 50 34 N. 
 
 I 15W. 
 
 9 
 
 45 
 
 Durazzo 
 
 Eu. 
 
 Turkey 
 
 Medit. Sea 
 
 41 58 N. 
 
 25 00 E. 
 
 
 
 Dufky Bav 
 E 
 C.Eaft 
 
 Alia 
 
 N. Zealand 
 
 Pacif. Ocean 
 
 45 47 S. 
 
 166 23 E. 
 
 10 
 
 57 
 
 Am. 
 
 Statenland 
 
 9tra. le Mai re 
 
 54 54 S. 
 
 64 47 W. 
 
 
 
 Eafter in. 
 
 Am. 
 
 Chili 
 
 Pacif. Ocean 
 
 27 7 S. 
 
 109 42 W. 
 
 2 
 
 00 
 
 Edinburgh 
 
 Eu. 
 
 Scotland 
 
 Germ. Ocean 
 
 55 58 N. 
 
 3 7W. 
 
 4 
 
 30 
 
 Edyftone 
 
 Eu. 
 
 England 
 
 Eng. Channel 
 
 SO 8 N. 
 
 4 20 W. 
 
 5 
 
 30 
 
 Egmont Ifle 
 
 Aiia 
 
 Society Ifles 
 
 Pacif. Ocean 
 
 19 20 S. 
 
 138 30W. 
 
 
 
 C. Egmont 
 
 Alia 
 
 M. Zealand 
 
 Pacif. Ocean 
 
 39 20 S, 
 
 173 45 E. 
 
 
 
 I. Elba 
 
 Eu. 
 
 Italy 
 
 Mediterran. 
 
 42 52 N. 
 
 10 38 E. 
 
 
 
 R. Elbe mouth 
 
 Eu. 
 
 Germany 
 
 Germ. Ocean 
 
 54 18 N. 
 
 7 10 E. 
 
 
 
 00 
 
 Elbing 
 
 Eu. 
 
 Poland 
 
 Baltic Sea 
 
 54 12 N. 
 
 20 35 E. 
 
 
 
 EUingburgh 
 
 Eu. 
 
 Sweden 
 
 Bald': Sea 
 
 56 00 N. 
 
 13 35 E. 
 
 
 
 Elfinore 
 
 Eu. 
 
 Denmark 
 
 Baltic Sea 
 
 56 00 N. 
 
 13 23 E. 
 
 
 
 I. Elutheria 5 J'-Pi"'^ 
 I b. point 
 
 Am. 
 
 Bahama 
 
 Atl. Ocean 
 
 f25 45 N. 
 
 I 24 57 N. 
 
 76 42 W. 
 75 53 W. 
 
 
 
 EmbJen 
 
 Eu. 
 
 Germany 
 
 Germ. Ocean 
 
 53 05 N. 
 
 7 26 E. 
 
 
 
 00 
 
 R. Em2<; mouth 
 
 Eu. 
 
 Germany 
 
 Germ. Ocean 
 
 53 10 N, 
 
 7 20 E. 
 
 7 
 
 30 
 
 Enchuyfen 
 
 Eu. 
 
 D. Ncth. 
 
 Zuyder Sea 
 
 52 43 N. 
 
 5 06 E. 
 
 
 
 00 
 
 Endeavour R. 
 
 Afia 
 
 N. Holland 
 
 Pacif. Ocean 
 
 15 26 S. 
 
 145 12 E. 
 
 
 
 I. Enguno, or 7 
 Trompoufe J 
 
 Am, 
 
 Sumatra 
 
 Indian Ocean 
 
 6 00 S, 
 
 102 35 E. 
 
 
 
 B. Eiihorn 
 
 Eu. 
 
 Greenland 
 
 North Sea 
 
 78 45 N. 
 
 26 05 E. 
 
 
 
 Ephcfus 
 
 Alia 
 
 Nacoiia 
 
 Archipelago 
 
 38 00 N. 
 
 27 53 E. 
 
 
 
 Erramangi 
 
 Mia. 
 
 N. Hebrides 
 
 Pacif. Ocean 
 
 18 44 S. 
 
 169 20 E. 
 
 
 
 Eltaplcs 
 
 Eu. 
 
 Frar.cc 
 
 Eng. Channel 
 
 50 34 N, 
 
 I 42 . 
 
 II 
 
 00 
 
 Euilatia 
 
 Am. 
 
 Caribbee 
 
 Atl. Ocean 
 
 17 30 N. 
 
 63 14W. 
 
 
 
 I. Exuma 
 F 
 
 Falrhcad 
 
 Am. 
 
 Bahama 
 
 Atl. Ocean 
 
 23 25 N. 
 
 75 35W. 
 
 
 
 Eu. 
 
 Ireland 
 
 Weft. Ocean 
 
 55 19 >'^- 
 
 6 20 W. 
 
 
 
 C. Falcon 
 
 Africa 
 
 Barbary 
 
 Medit. Sea 
 
 36 03 N. 
 
 14W. 
 
 
 
 r E. end 
 
 
 
 
 551 05 s, 
 
 I 52 27 S. 
 
 56 40 W. 
 
 
 
 I. F.ilkland^ 111. A- 
 (_ nil an I 
 
 Am. 
 
 Patagonia 
 
 Atl. Ocean 
 
 61 53W. 
 
 
 
 Falmouth 
 
 Eu. 
 
 F-nghad 
 
 Eng. Channel 
 
 50 8 N. 
 
 5 oW. 
 
 i 
 
 30 
 
 C. Falo 
 
 Eu. 
 
 Turkey 
 
 Archipelago 
 
 40 12 N. 
 
 24 27 E. 
 
 
 
 C. F,i!fo 
 
 Africa 
 
 Catfcrs 
 
 Indian Ofcan 
 
 34 '6 S. 
 
 18 44 E. 
 
 
 
 C. Falfrj 
 
 Africa 
 
 Zaugucbar 
 
 Indian Ocean 
 
 8 52 S. 
 
 59 55 E. 
 
 
 
 FaiftTbom 
 
 Eu. 
 
 Sweden 
 
 Baltic Sea 
 
 55 20 N- 
 
 13 36 E. 
 
 
 
 I. Fana 
 
 Eu. 
 
 Turkey 
 
 Medit. Sea 
 
 40 14 K. 
 
 19 32 E. 
 
 
 
 R. Fame 
 
 Africa 
 
 F-syft 
 
 Red Sta 
 
 21 40 N. 
 
 36 29 . 
 
 
 
 IFaro Heal 
 
 Eu. 
 
 Scotland 
 
 Weft, (Jccan 
 
 58 40 N. 
 
 \ SOW. 
 
 
 
 C. F,ir<-wcll 
 
 Afia 
 
 N, Zealand 
 
 Pacif. Ocean 
 
 40 35 S. 
 
 172 47 E. 
 
 
 
 ic. F,.rr:v.-:l 
 
 Am. 
 
 Greenland 
 
 North Occau 
 
 59 37 ^' 
 
 42 -iyVv. 
 
 
 
 C. Firtack 
 
 Af,. 
 
 Arabia 
 
 Indian Occuri 
 
 15 41 N\ 
 
 5- s- I-:. 
 
 
 
 C, Fchr 
 
 Am. 
 
 Carolina 
 
 Atl. Occ;ui 
 
 34 C4 N. 
 
 7S oyW. 
 
 
 
 I. Fciinnd : N'..i.j-J I 
 
 Am. 
 
 Brafil 
 
 Ati. Ocean 
 
 i 5'' .- 
 
 32 23W. 
 
 
 
 I. F.licui, L:yjii llic. 
 
 Eu. 
 
 i:.-ily 
 
 Medit. Sea 
 
 3i-; r> 
 
 14 51 E. 
 
 
 
 1. Fcrmiia 
 
 Eu. 
 
 Turkey 
 
 Aichipcl:igo 
 
 37 24 N. 
 
 2f 05 E. 
 
 
 
 ,1. F.-riii ,.'-p . 
 
 Africa 
 
 Guinea 
 
 -Vtl. Ocean 
 
 3 C2 N. 
 
 3 i5 ^ 
 
 
 
 1. rc::o
 
 HH 
 
 GEOGRAPHY. 
 
 fiook vr* 
 
 1 Names of Places. 
 
 Cent. 
 
 Countries. 
 
 Coaft. 
 
 Latitude. 
 
 Longitude. 
 
 H. Water. 
 
 ... 
 I. Ferro 
 
 Africa 
 
 Canariis 
 
 Atl. Ocean 
 
 27 48 N. 
 
 17 40 W. 
 
 
 C. Finiftenft 
 
 Eu. 
 
 Spain 
 
 Atl. Otean 
 
 42 52 N. 
 
 9 16W. 
 
 
 I. Fironda 
 
 Afia 
 
 Corea 
 
 Pacif. Ocean 
 
 33 30 N. 
 
 127 25 E. 
 
 
 Flamborough Head 
 
 Eu. 
 
 England 
 
 Germ. Ocean 
 
 54 08 N. 
 
 M E. 
 
 4h. oom; 
 
 I. Flores 
 
 Eu. 
 
 Azores 
 
 Atl. Ocean 
 
 39 34 N. 
 
 30 59 W. 
 
 
 C. Floiida 
 
 Am. 
 
 Florida 
 
 G. Mexico 
 
 25 so N. 
 
 80 20W. 
 
 7 30 
 
 Flu/hihg 
 
 Eu. 
 
 D. Neth. 
 
 Germ. Ocean 
 
 S' 33 N. 
 
 3 20 w. 
 
 4S 
 
 I. Fly 
 
 Eu. 
 
 D. Ncth. 
 
 Germ. Ocean 
 
 53 ,6N. 
 
 5 35 E. 
 
 7 30 
 
 Fnrbi/her's Straits 
 
 Am. 
 
 Greenland 
 
 Atl. Ocean 
 
 62 05 N. 
 
 47 18W. 
 
 
 korth Foreland 
 
 Eu. 
 
 England 
 
 Germ. Ocean 
 
 51 28 N. 
 
 1 25 E. 
 
 9 45 
 
 South Foreland 
 Foreland Fair 
 
 Eu. 
 
 England 
 
 Eng. Channel 
 
 51 12 N. 
 
 I 24 E. 
 
 9 45 
 
 F.u. 
 
 Ireland 
 
 North Ocean 
 
 55 OS N. 
 
 6 30W. 
 
 
 Foreland Fair 
 
 Em. 
 
 Greenland 
 
 North Ocean 
 
 79 18 N, 
 
 10 50 E. 
 
 
 Forsland Mcrcliants 
 
 Am. 
 
 Greenland 
 
 fJorth Ocean 
 
 6-5 20 N. 
 
 17 osW. 
 
 
 I. Formcntaria 
 
 Eu. 
 
 Spain 
 
 Medit. Sea 
 
 39.33 N. 
 
 I ,5 E. 
 
 
 I. Formigas 
 
 Eu. 
 
 Azores 
 
 Atl. Ocean 
 
 37 17 N. 
 
 24 43 W. 
 
 
 C. Formola 
 
 Africa 
 
 Guinea 
 
 Eth. Sen 
 
 4 22 N. 
 
 5 43 E. 
 
 
 R. Formoft 
 
 Africa 
 
 Guinea 
 
 Eth. Sea 
 
 6 10 N. 
 
 4 49 E- 
 
 
 I. Formofa^J^-r'^' 
 
 Afia 
 
 China 
 
 Indian Ocean 
 
 521 25 N. 
 
 121 25 E. 
 120 40 E. 
 
 
 I S. point 
 
 
 
 
 I 22 00 N. 
 
 
 I. Forteventura, S. 7 
 W. end 1 
 
 Africa 
 
 Canaries 
 
 Atl. Ocean 
 
 28 35 N. 
 
 14 04 W. 
 
 
 Fouinefs 
 
 Eu. 
 
 England 
 
 Germ. Ocean 
 
 52 57 N. 
 
 58 E. 
 
 6 45 
 
 Foulfound 
 
 Eu. 
 
 Greenland 
 
 North Ocean 
 
 77 30 N. 
 
 t2 50 E. 
 
 
 Fowey 
 
 Eu. 
 
 England 
 
 Eng. Channel 
 
 SO 25 N. 
 
 4 30 W. 
 
 5 15 
 
 I. France, P. Louis 
 
 Africa 
 
 Madagafcar 
 
 Indian Ocean 
 
 20 10 S. 
 
 57 33 E- 
 
 
 C. St. Francis 
 
 Am. 
 
 Peru 
 
 Pacif. Ocean 
 
 30 N. 
 
 80 35 W. 
 
 1 
 
 I. St. Francifco 
 
 Africa 
 
 Zangucbar 
 
 Indian Ocean 
 
 6 23 S. 
 
 53 22 E. 
 
 
 R. St. Francifco 
 
 Am. 
 
 Brafil 
 
 Atl. Ocean 
 
 10 55 S. 
 
 36 30 W. 
 
 
 C. Francois 
 
 Am. 
 
 Domingo 
 
 Atl. Ocean 
 
 19 47 N. 
 
 72 15W. 
 
 
 IFrcderickftadt. 
 
 Eu. 
 
 Norway 
 
 Sound 
 
 59 00 N. 
 
 II 10 E. 
 
 
 [French Keys 
 
 Am. 
 
 Bahama 
 
 Atl. Ocean 
 
 21 30 N. 
 
 72 loW. 
 
 
 '.Fretum Borough 
 
 Eu. 
 
 Ruflia 
 
 North Ocean 
 
 70 00 N. 
 
 61 20 E. 
 
 
 C. Frio 
 
 Am. 
 
 Brafil 
 
 Atl. Ocean 
 
 23 00 S. 
 
 40 iiW. 
 
 
 JR. Fugor 
 
 A frica 
 
 Zanauebar 
 
 Indian Ocean 
 
 00 10 N. 
 
 42 OS E. 
 
 
 jl. Fuego 
 
 Africa 
 
 Di Verd 
 
 Atl. Ocean 
 
 14 55 N. 
 
 24 28 W. 
 
 
 jFurncaux Ifland 
 
 Afia 
 
 Soc. Ifics 
 
 Pacif. Ocean 
 
 17 11 s. 
 
 143 07 W. 
 
 
 |B. Fulhan 
 
 Afia 
 
 China 
 
 Pacif. Ocean 
 
 23 CO N. 
 
 I12 35 E. 
 
 
 I. Fyal 
 
 1 G 
 
 I. Galla 
 
 Eu. 
 
 Azores 
 
 Atl. Ocean 
 
 38 32 N. 
 
 28 36 W. 
 
 2 20 
 
 Am. 
 
 Terra Firma 
 
 Pacif. Ocean 
 
 2 40 N. 
 
 79 3SW. 
 
 
 R. Gallcga 
 
 Am. 
 
 Patagonia 
 
 Atl. Ocean 
 
 51 37 S. 
 
 65 3sW. 
 
 
 I. Gallego 
 
 Am. 
 
 Terra Firma 
 
 Pacif. Ocean 
 
 I 40 N. 
 
 104 35W. 
 
 
 jCallipoli 
 
 Eu. 
 
 Italy 
 
 Mcdit. Sea 
 
 40 19 N. 
 
 18 08 E. 
 
 
 Gp.liiiaoly 
 
 Eu. 
 
 Turkey 
 
 Archipelago 
 
 40 36 N. 
 
 27 02 E. 
 
 
 'l. Gillita 
 
 Africa 
 
 Barbary 
 
 Medit. Sea 
 
 37 4^ N. 
 
 9 03 E. 
 
 
 :C. Gallo 
 
 Ana 
 
 I. Ceylon 
 
 Indian Ocean 
 
 6 15 N. 
 
 80 20 E. 
 
 
 Is. Gallepago 
 
 Am. 
 
 Peru 
 
 Pacif. Ocean 
 
 5 2 CO N. 
 I 2 00 S. 
 
 89 00 W. 
 
 
 jGally Head 
 
 Eu! 
 
 Ireland 
 
 Weft. Occ:jn 
 
 52 40 N. 
 
 9 30 W. 
 
 
 Cal'.vay 
 
 Eu. 
 
 Ireland 
 
 Weft. Occsn 
 
 53 10 N. 
 
 10 03 w. 
 
 
 |R. Gambia 
 11. Gamo 
 
 Africa 
 
 Ncgroland 
 
 Atl. Ocean 
 
 13 00 N. 
 
 14 s8W. 
 
 
 Afia 
 
 India 
 
 Indian Ocean 
 
 3 '55 S. 
 
 77 25 E. 
 
 
 :C, Gardafui 
 
 Africa 
 
 Anian 
 
 Indian Ocean 
 
 II 48 N. 
 
 50 25 E. 
 
 
 jR. Garronnc 
 
 Eu. 
 
 France 
 
 E. Bifcay 
 
 45 30 N. 
 
 I 05W. 
 
 3 CO 
 
 iCafpcy Bay 
 
 Am. 
 
 Acadia 
 
 G. St. La-A-r. 
 
 48 49 N, 
 
 63 3d.W. 
 
 I 30 
 
 iC. dc Gate 
 
 Eu. 
 
 Spain 
 
 Medit. Sea 
 
 36 32 N. 
 
 2 05 W. 
 
 
 ;C. Gear 
 
 A frica 
 
 Barbary 
 
 At). Ocean 
 
 30 3S N. 
 
 10 01 W. 
 
 
 Genoa 
 
 Eu. 
 
 Italy 
 
 iMcdit. Sea 
 
 44 2,- N. 
 
 8 41 E. 
 
 
 Ic. St. George 
 
 Am. 
 
 Nevvf lur.Jl. 
 
 Atl. Ocean 
 
 48 28 N. 
 
 57 43 W. 
 
 
 ;C. Gro;-- 
 
 Am. 
 
 S. Georgia 
 
 Atl. Ocean 
 
 54 17 S- 
 
 36 33W. 
 
 
 :B. <vr. G"-,rgc 
 
 Am. 
 
 Ncwfoui;vi!. 
 
 Atl, Ocean 
 
 48 19 N. 
 
 57 30 W. 
 
 
 
 
 
 
 
 I. 
 
 St. Georgg
 
 Book VI. 
 
 GEOGRAPHY. 
 
 385 
 
 Names of Places.' 
 
 Cont. I Countries, 
 
 |I. Gorgoni 
 
 I.Goth- V^' 
 land 1^,^ 
 
 I. St. George 
 
 I. St. George 
 St. Georgc'j Fc 
 
 Gibraltar 
 Gilbert's Ifland 
 
 Giiolo 5?-p^.'"; 
 
 I S. point 
 Glafgow 
 
 Gh,u center Ides 
 doucefter Ifles 
 Goa 
 Goes 
 
 Galfe triile 
 Gombroon 
 I. Gomjro 
 C. Gondcwar 
 C. Good Hope 
 
 I. Gorea 
 Gorgoni 
 
 N. end 
 end 
 iiby 
 I. Goto 
 Gottcnberg 
 Gottingen 
 Gosvcr'i Iflc 
 R. Grand 
 Granville 
 C. D- GrLit 
 I. Gittiola 
 
 I I. Gr.itiofa 
 C. G ratios a Dios 
 
 1 Gravtline 
 
 Gravell-nd 
 
 I. Grenada 
 
 Grocnvvlch 
 i C. Grtm a 
 iGripi.vald 
 
 GriiKcrt.y 
 t. Groin, or C.Corunn;^ 
 1 1. Gdv 
 
 I. Giuii iloupe 
 
 Gua\ .iqui 
 
 I. Gii.-rnkv 
 
 Afia 
 
 IE. I. 
 Afia 
 u. 
 Am. 
 
 Afia 
 
 Eu, 
 
 Afn 
 
 Ana 
 
 Afu 
 
 Eu. 
 
 Am-. 
 
 Afii 
 
 Africa 
 
 Afia 
 
 Africa 
 
 Africa 
 
 Eu. 
 
 Eu. 
 
 Afia 
 
 E^. 
 
 Eu. 
 
 Afia 
 
 Aiiu 
 
 Ea. 
 
 Am. 
 
 Africa 
 
 Eu. 
 
 Am. 
 
 Eu. 
 
 Eu. 
 
 Am. 
 
 Eu, 
 
 Eu. 
 
 Eu. 
 
 Eu. 
 
 Eu, 
 
 Am. 
 
 Am, 
 
 .'Vm. 
 
 Eu, 
 
 Eu. 
 
 Afia 
 
 Coaft. 
 
 N a toll a 
 
 Aiore* 
 
 India 
 
 Spain 
 
 T. delFueg 
 
 Spice Ifland 
 
 Scotland 
 Society Ifles | 
 Society Ifles 
 India 
 D. Ncth. 
 Terra Fij-ma 
 Peru a 
 Canaries 
 India 
 Caffers 
 Negrolani 
 Italy 
 
 Sweden 
 
 Corea 
 
 Sweden 
 
 Germany 
 
 N. Britain 
 
 Paraguay 
 
 France 
 
 Newfoundl, 
 
 Canaries 
 
 Azores 
 
 New Spain 
 
 France 
 
 England 
 
 Cairibbce 
 
 England 
 
 Turkey 
 
 Germ iny 
 
 EngU'id 
 
 Spain 
 
 KcwtOundl, 
 
 Carriilcc 
 
 I'cru 
 
 F.nt'land 
 
 Eni-imJ 
 
 Aihacm 
 
 Latitude. 
 
 Archipelago 
 Ad, Ocean 
 B. Bengal 
 Medit. Sea 
 Pacif. Ocean 
 
 Indian Ocean 
 
 R. Clyde 
 Pacif. Occrjn 
 Indian^ Ocean 
 Malabar 
 Germ. Ocean 
 Carrib. Sea 
 Perfian Gulf 
 Atl. Ocean 
 B, Bengal 
 Indian Ocean 
 Atl. Ocean 
 Medit. Sea 
 
 Baltic Sea 
 
 Pacif. Ocean 
 Sound 
 Inland 
 
 Pacif. Ocean 
 Atl. Occin 
 Eng. Channel 
 .Atl. Ocean 
 Atl, Ocean 
 Atl. Ocean 
 Carribbe. Sea 
 Eng. Channel 
 R, 'riiamcs 
 Atl. Ocean 
 R, Thames 
 Archipelago 
 Baltic Sea 
 Germ. Ocean 
 B, Bifcay 
 AU, Ocean 
 Atl. Ocean 
 Pacif, Ocean 
 Er.g. Channel 
 St. Ceo. Cti. 
 Cafpian Sea 
 
 North Ocean 
 
 I;idian Ocean 
 
 Weftcrn Oc 
 AM. Ocean 
 >.'orth Ocean 
 R, Elbe 
 R.St, I awr. 
 Grrm. Ocean 
 Btilloi Ci-aii. 
 Cicrm. Oecan 
 tji'rm. Oceji) 
 At!. Occ.'.a 
 At!, Oc-.-ui 
 
 55 
 
 19 
 
 Longitude.' 
 
 38 47 N. 
 
 35 39 N. 
 15 05 N. 
 
 36 05 N. 
 55 J 3 S. 
 
 C 2 30 N. 
 i 1 30 S. 
 
 52 N. 
 
 n S. 
 
 36 S. 
 
 15 31 N, 
 51 j9 N. 
 10 10 N. 
 
 27 40 N. 
 
 28 c6 N. 
 
 16 55 N. 
 34 ^9 S, 
 14 40 N. 
 43 21 N, 
 
 55 00 N, 
 
 56 58 N. 
 
 57 40 N, 
 34 25 N. 
 57 42 N, 
 ei 32 N. 
 
 7 56 S, 
 31 58 S. 
 4S 50 N. 
 51 36 N. 
 29 15 N. 
 
 39 oi N. 
 48 N, 
 59 N. 
 35 N. 
 52 N, 
 
 51 29 N, 
 
 40 33 N. 
 54 04 N- 
 53 30 N- 
 43 2S N. 
 50' 56 N, 
 16 CO N. 
 
 H. Water. 
 
 14 
 5= 
 5' 
 
 2 10 
 
 49 30 N. 
 
 50 c6 N. 
 
 47 7 N, 
 
 79 55 N. 
 
 ; 19 45 N. 
 
 l2 22 N. 
 
 44 4(J N- 
 63 5O N. 
 
 b4 30 N. 
 
 53 34 N. 
 
 48 CO N, 
 52 24 N< 
 
 51 c6 N. 
 
 r ^4 
 23 I'.'. 
 
 4 'I V 
 
 25 07 E, 
 2S 00 W 
 80 34 E 
 
 5 iVV*' 
 7. 4W 
 
 12S CO E. 
 
 129 25 E. 
 
 4 loW. 
 
 140 4W. 
 
 146 7W, 
 
 73 50 E 
 
 4 05 E 
 
 67 40W 
 
 tj 20 E. 
 
 17 03W 
 82 55 E. 
 
 18 28 E, 
 17 zoW. 
 
 9 II E 
 20 15 E. 
 
 19 37 
 19 50 E. 
 
 25 50 E, 
 11 44 E. 
 
 58 E. 
 
 56 E. 
 50 35W. 
 
 1 32 W. 
 55 33 W, 
 
 13 07 W. 
 27 53 V/, 
 
 S2 15 V/. 
 
 2 i: E. 
 
 o 20 E. 
 
 61 39 V>/, 
 
 o 05 E. 
 
 26 io E. 
 13 43 E, 
 
 o 56 E, 
 
 9 2oNV, 
 55 35W. 
 61 55W. 
 81 C5W, 
 
 2 47 \V. 
 
 6 00 W. 
 52 02 E. 
 
 oh.oom*' 
 
 9 
 
 158 
 
 1 10 
 
 ic'S 
 
 63 
 
 -14 
 27 
 
 9 
 
 63 
 
 4 
 
 00 
 30 
 
 Vof.. I, 
 
 v: c
 
 J8 
 
 GEO G R A P H r. 
 
 Book VI. 
 
 Nimes 0/ Places. 
 
 Cont. 
 
 ' " 
 Countries. 
 
 Coaft. 
 
 Latitude. 
 
 Longitude. 
 
 H. Water. 
 
 
 Hwke's Bay 
 
 Atia 
 
 N. Zealand 
 
 Pacif. Ocean 
 
 4 
 39 30 S. 
 
 b , 
 177 6 E. 
 
 
 
 I. St. Helena 
 
 Africa 
 
 Caffers 
 
 Atl. Ocean 
 
 *5 55 S. 
 
 5 44 W. 
 
 
 
 jHelie's Sound 
 
 Eu. 
 
 Greenland 
 
 North Ocean 
 
 79 15 N- 
 
 12 50 E. 
 
 
 
 jls. Heligh's Land 
 
 Eu. 
 
 Norway 
 
 North Ocean 
 
 38 4I n1 
 
 9 30 E. 
 
 
 
 C. Hcnlopen 
 
 Ain. 
 
 Maryland 
 
 Atl. Ocean 
 
 75 08 W. 
 
 
 
 C. Henrietta Mkrh 
 
 Am. 
 
 ^few Wiles 
 
 Hudfon's Bay 
 
 55 10 N. 
 
 --84 00 w. 
 
 I zh. 00m. 
 
 
 C. Henry 
 
 Am. 
 
 Virginia 
 
 Atl. Ocean 
 
 37 00 N. 
 
 76 23 W. 
 
 II 15 
 
 
 Hervcy's Iflc 
 
 Afia 
 
 Society illes 
 
 Pacif. Ocean 
 
 19 17 S. 
 
 158 43 W. 
 
 
 
 I. Heys 
 
 Eu. 
 
 France 
 
 B. Bifcay 
 
 46 24 N. 
 
 2 14W. 
 
 
 
 High Mount 
 
 Eu. 
 
 Greenland 
 
 North Ocean 
 
 83 23 N. 
 
 26 40 E. 
 
 
 
 Hinchingbrtiok I. 
 
 Afu 
 
 N. Hebrides 
 
 Pacif. Ocean 
 
 17 IJ S. 
 
 168 38 E. 
 
 
 
 
 -C. TiberoOB, W. 
 
 
 
 
 
 
 
 
 M 
 
 . P^- 
 
 
 
 
 18 17 N. 
 
 74 24 w. 
 
 
 
 
 
 S. Louis 
 
 
 
 
 18 19 N. 
 
 73 iiW. 
 
 
 
 1 
 
 C. St. NichoL 
 
 
 
 
 
 
 
 
 :^< 
 
 N. W. pt. 
 
 Am. 
 
 Antilles 
 
 Atl. Ocean 
 
 19,50 N. 
 
 73 18 W. 
 
 
 
 X 
 
 Po. Grave 
 
 
 
 
 18 28 N. 
 
 72 42 W. 
 
 
 
 ^ 
 
 St. Domingo 
 
 
 
 
 18 25 N. 
 
 69 30 W. 
 
 
 
 
 C. Rapliael N. 
 E. pt. 
 
 
 
 
 19 05 N. 
 
 68 30 W. 
 
 
 
 Hogfties 
 
 Am. 
 
 Bahama 
 
 Atl. Ocean 
 
 21 41 N. 
 
 73 25W. 
 
 
 
 ^ fW. limit 
 
 
 
 
 25 30 S. 
 
 in 10 E. 
 
 1 
 
 
 * g J N. Ditto 
 2 =3 1 S. Ditto 
 
 Afia 
 
 
 Indian Ocean 
 
 tz 35 S. 
 43 38 S. 
 
 141 31 E. 
 146 00 E. 
 
 
 
 ffi L E. Ditto 
 
 
 
 
 27 10 S. 
 
 153 39 E. 
 
 
 
 Holy Cape 
 
 Afia 
 
 Siberia 
 
 North Ocean 
 
 72 32 N. 
 
 179 45 E. 
 
 
 
 Holy Head 
 
 Eu. 
 
 Wales 
 
 Iri/h Sea 
 
 53 23 N. 
 
 4 40 W. 
 
 I 30 
 
 
 C. Honduras 
 
 Am. 
 
 New Spain 
 
 Caribbean Sea 
 
 16 18 N. 
 
 85 23W. 
 
 
 
 B. Hondy, I. Cuba 
 
 Am. 
 
 Antilles 
 
 Atl. Ocean 
 
 22 54 N. 
 
 82 40 W. 
 
 
 
 Honfleur 
 
 Eu. 
 
 France 
 
 R. Seine 
 
 49 24 N, 
 
 20 E. 
 
 9 00 
 
 
 Hood's Ifle 
 
 Afia 
 
 Marqusfas 
 
 Pacif. Ocean 
 
 9 26 S, 
 
 138 47W. 
 
 
 
 Hope Ifle 
 
 Eu. 
 
 Greenland 
 
 North Ocean 
 
 76 22 N, 
 
 23 40 E. 
 
 
 
 C. Horn 
 
 Am. 
 
 T. del Fuego 
 
 Pacif, Ocean 
 
 55 59 S. 
 
 67 21 W. 
 
 
 
 Hornfound 
 
 Eu. 
 
 Greenland 
 
 North Ocean 
 
 76 41 N, 
 
 13 36 E. 
 
 
 
 La Hogue 
 
 Eu. 
 
 France 
 
 Eng. Channel 
 
 49 45 N. 
 
 I 52W. 
 
 
 
 Howe's Iflc 
 
 Afia 
 
 Society's If. 
 
 Pacif. Ocean 
 
 16 46 S. 
 
 154 2W. 
 
 
 
 C. How 
 
 Afia 
 
 N. Holland 
 
 Pacif. Ocean 
 
 37 24 S. 
 
 150 00 E. 
 
 
 
 R. Hughly 
 
 Afia 
 
 India 
 
 B. Bengal 
 
 21 45 N. 
 
 89 15 E. 
 
 
 
 Hull 
 
 Eu. 
 
 England 
 
 R. Humber 
 
 S3 50 N- 
 
 28 W. 
 
 6 00 
 
 
 'R. Humber, Ent. 
 
 Eu. 
 
 England 
 
 Germ. Ocean 
 
 53 55 N. 
 
 24 E. 
 
 5 13 
 
 
 L Hyneag9 
 Jado 
 
 Am. 
 
 Bahama 
 
 Atl. Ocean 
 
 21 27 N. 
 
 73 29 W. 
 
 
 
 Afia 
 
 Japan 
 
 Pacif. Ocean 
 
 36 CO N. 
 
 139 40 E. 
 
 
 
 C. Jaffanapatan 
 
 Ana 
 
 I. Cevlon 
 
 Indian Ocean 
 
 9 47 N. 
 
 8o 55 E. 
 
 
 
 L Tago 
 
 JakutiTioi 
 
 Africa 
 
 C. Verd 
 
 Atl. Ocran 
 
 15 07 N. 
 
 23 30 W. 
 
 
 
 Afia 
 
 Siberia 
 
 Pacif. Ocean 
 
 62 2 N. 
 
 129 52 E. 
 
 
 
 p r We.t end 
 
 
 
 
 18 45 N. 
 
 78 00 W. 
 
 
 
 ^^ Port Roval 
 
 Am. 
 
 Weft Indies 
 
 Atl. Ocean 
 
 18 00 N. 
 
 76 4.0 W. 
 
 
 
 ^ i Ea;> End 
 
 
 
 
 17 58 N. 
 
 76 05 W, 
 
 
 
 ijair.es Town 
 
 Am. 
 
 Virginia 
 
 B. Chcfapeik 
 
 37 30 N. 
 
 76 03 w. 
 
 
 
 R. Janeiro 
 
 Am. 
 
 Brafil 
 
 Atl. Ocean 
 
 22 54 S. 
 
 42 4>W. 
 
 
 
 Japan I lies 
 
 Afia 
 
 
 Pacif. Ocean 
 
 f 40 40 N. 
 
 h' 45 N. 
 
 141 25 E. 
 126 10 E. 
 
 
 
 ^ r C. Delo, W. pt. 
 
 
 
 
 r 6 50 s. 
 
 105 15 E. 
 
 
 
 ~,(Eaft limit 
 
 Afia 
 
 Slam 
 
 Indian Ocean 
 
 -? 7 00 s, 
 I 8 30 S. 
 
 115 55 E. 
 
 
 
 Ice. Cove 
 
 Am. 
 
 N. Main 
 
 Hudf. Straits 
 
 62 20 N. 
 
 69 coW. 
 
 10 00 
 
 
 Ice Point 
 
 Eu. 
 
 Nova Zem . 
 
 North. Orran 
 
 77 40 N 
 
 C9 10 F-. 
 
 
 
 Ice Sor.nd 
 
 Eu. 
 
 Greenland 
 
 North Occ.in 
 
 78 ,3 N. 
 
 12 00 E. 
 
 
 
 1. Jerfcy 
 
 Eu. 
 
 Engb.nd 
 
 ii'- g. Channel 
 
 49 07 N. 
 
 2 26W. 
 
 
 
 .[crufnlem 
 
 Afia 
 
 I^alcfline 
 
 In land 
 
 31 55 N- 
 
 35 25 E. 
 
 
 
 i.H 
 
 ay, S. pt. I 
 
 Eu. 
 
 .'^rothnd 
 
 Wc'r. Ocf^an 
 
 55 39. r^'- 
 
 6 20 W. 
 
 
 
 R. Indui
 
 .BCok VI. 
 
 tS E' O G R A P H T. 
 
 387 
 
 Names of Places. 
 
 Cont. 
 
 Countries. 
 
 Coaft. 
 
 Latitude. 
 
 Longitude. 
 
 H. Water. 
 
 ' 
 
 R. Indus 
 
 Afia 
 
 India 
 
 Indian Ocean 
 
 / 
 25 50 N. 
 
 
 66 
 
 33 E. 
 
 
 Invernefs 
 
 Eu. 
 
 Scotland 
 
 Germ. Ocean 
 
 57 33 N. 
 
 4 
 
 02 W. 
 
 
 
 I. Joanna 
 
 Africa 
 
 Zanguebar 
 
 Indian Ocean 
 
 12 05 S. 
 
 45 45 E- 
 
 
 
 Juddah 
 
 Afia 
 
 Arabia 
 
 Red Sea 
 
 22 00 N. 
 
 39 
 
 27 E. 
 
 
 
 C. St. John 
 
 Am. 
 
 Newfoundl. 
 
 Atl. Ocean 
 
 .50 08 N. 
 
 55 
 
 32W. 
 
 
 
 C. St. John 
 
 Africa 
 
 Maiumba 
 
 Eth. Ocean 
 
 I 17 N. 
 
 9 
 
 34 E- 
 
 
 
 St. John's 
 
 Am. 
 
 Newfoundl. 
 
 Atl. Ocean 
 
 47 34 N. 
 
 5* 
 
 18W. 
 
 6h. om. 
 
 
 i.st.johj^-p;; 
 
 Am. 
 
 Canada '| 
 
 Bay St. Lau- 
 rence 
 
 46 30 N. 
 
 47 07 N- 
 
 62 
 64 
 
 03 w. 
 05 w. 
 
 
 
 I. St. John de Nova 
 St. Jonn do Luz 
 
 Africa 
 
 Madsgafcar 
 
 Indian Ocean 
 
 17 00 S. 
 
 44 
 
 02 E. 
 
 
 
 Eu. 
 
 France 
 
 B. Bifcay 
 
 43 10 N. 
 
 I 
 
 38 W. 
 
 3 30 
 
 
 Cape Jones 
 
 Am. 
 
 New Britain 
 
 Hudfoh's Bay 
 
 58 50 N. 
 
 79 
 
 00"^ 
 00 . 
 
 
 
 Joppa 
 
 Afia 
 
 Syria 
 
 Levant 
 
 3a 45 N. 
 
 36 
 
 
 
 Jones Sound 
 
 Am. 
 
 Greenland 
 
 Baffin's Bay 
 
 71 07 N. 
 
 91 
 
 30 'W. 
 
 
 
 St. Jofeph 
 
 Am. 
 
 California 
 
 Pacif. Ocean 
 
 23 3 S. 
 
 109 
 
 35 W. 
 
 
 
 Ipfwich 
 
 Eu. 
 
 England 
 
 Germ. Ocean 
 
 52 14 N. 
 
 I 
 
 00 E. 
 
 
 
 Ifpahan 
 
 AIJa 
 
 Perfia 
 
 R. Zenduro 
 
 32 2; N. 
 
 5^ 
 
 55 E. 
 
 
 
 C. St. Juan 
 
 Am. 
 
 Statenland 
 
 Atl. Ocean 
 
 54 47 S. 
 
 63 
 
 42 W. 
 
 
 
 I. Juan Fernandez 
 
 Am. 
 
 Chili 
 
 Pacif. Ocean 
 
 33 45 ^ 
 
 78 
 
 37 W. 
 
 
 
 Port. St. Julian 
 
 Am. 
 
 Patagonia 
 
 S. Atl. Ocean 
 
 49 10 S. 
 
 66 
 
 loW. 
 
 4 4S 
 
 
 I. Ivica 
 
 K. 
 
 Kalmer 
 
 Eu. 
 
 Spain 
 
 Mcdit. Sea 
 
 38 54 N. 
 
 I 
 
 15 E. 
 
 
 
 Eu. 
 
 Sweden 
 
 Baltic Sea 
 
 56 40 N 
 
 17 
 
 25 E. 
 
 
 
 Kambaya 
 
 Afia 
 
 India 
 
 Indian Ocean 
 
 23 36 N. 
 
 72 
 
 50 E. 
 
 
 
 Kamtfchatka?!;''''"' 
 
 Afia 
 
 Liberia 
 
 Pacif. Oct an 
 
 5 55 II N. 
 
 155 
 
 25 E. 
 
 
 
 I Upper 
 
 
 
 
 I 54 4? N. 
 
 157 
 
 25 E. 
 
 
 
 I, Karaghinfkoy 
 
 Afia 
 
 Siberia 
 
 Pacif. Ocean 
 
 58 00 N. 
 
 162 
 
 10 E. 
 
 
 
 I. St. Katharine's 
 
 Am, 
 
 Brafil 
 
 Atl. Ocean 
 
 27 35 S. 
 
 4'> 
 
 12 W. 
 
 
 Keco 
 
 Afia 
 
 Tonquin 
 
 Indian Ocean 
 
 21 55 N. 
 
 !0b 
 
 10 E. 
 
 
 
 Kegcr 
 
 Eu. 
 
 Mufcovy 
 
 Nortli Ocean 
 
 70 18 N. 
 
 34 
 
 00 E. 
 
 
 
 R. Kenncbeck 
 
 Am, 
 
 N. England 
 
 Atl. Ocean 
 
 44 CO N. 
 
 69 
 
 45W. 
 
 
 
 Kentiih Kiiock, a 7 
 
 
 
 
 
 
 
 
 
 fand i 
 
 Eu, 
 
 England 
 
 Germ. Ocean 
 
 51 42 N. 
 
 I 
 
 45 E. 
 
 00 
 
 
 I. St. Kilda 
 
 Eu. 
 
 Scotlar.d 
 
 Weft. Ocem 
 
 57 44 N. 
 
 8 
 
 18W. 
 
 
 
 I. Kilduin 
 
 Eu. 
 
 Laplar.d 
 
 North Ocean 
 
 6g 'o N. 
 
 31 
 
 20 E. 
 
 7 3^ 
 
 
 Kinfale 
 
 Eu. 
 
 Irdand 
 
 Atl. Oc-:.n 
 
 51 32 N. 
 
 9 
 
 01 W. 
 
 5 5 
 
 
 Klip 
 
 Eu. 
 
 Grccnhnd 
 
 North Ocean 
 
 80 10 N, 
 
 12 
 
 22 E. 
 
 
 
 R. Kcli 
 
 Eu, 
 
 Liij-lanJ 
 
 North Ocean 
 
 6S C3 N. 
 
 33 
 
 08 E. 
 
 
 
 C. KA 
 
 Eu. 
 
 Sweden 
 
 Sound 
 
 56 N. 
 
 11 
 
 15 E. 
 
 
 
 Port Komol 
 
 Africa 
 
 AbylTinia 
 
 Red Sea 
 
 22 30 N. 
 
 36 
 
 17 E. 
 
 
 
 IComero Iflcs 
 
 Africa 
 
 ZingucLar 
 
 Indian Ocean 
 
 C 10 48 S. 
 2 13 10 S. 
 
 44 
 
 46 
 
 4.5 E. 
 22 E. 
 
 
 
 R. Kov.imia 
 L 
 
 Afia 
 
 Siberia 
 
 North Ocean 
 
 70 40 N, 
 
 '59 
 
 CO E. 
 
 
 
 Ladronc, or Marian } 
 
 ines I 
 
 A fill. 
 
 
 Pacif. Ocean 
 
 Til 00 N. 
 i 13 15 N. 
 
 I4.V 
 142 
 
 CO E. 
 55 E. 
 
 
 
 \C. L'AlguUc 
 
 Africa 
 
 CruTraiia 
 
 InJian Ocean 
 
 34 50 ^ 
 
 20 
 
 c6 E. 
 
 
 
 !Lin:.-iftei 
 
 Eu. 
 
 EngUad St. G::o. Ch, 
 
 54 4^ N. 
 
 4 
 
 36 W. 
 
 
 
 I. Lancjrota 
 
 Africa 
 
 Cauariis Atl. Ocean 
 
 29 10 N. 
 
 13 
 
 20 W. 
 
 
 
 Land's End 
 
 Eu. 
 
 Enr.land >S:. Geo. Ch. 
 
 50 o5 N. 
 
 c 
 
 50 W. 
 
 7* 30 
 
 Langericfs 
 
 Eu. 
 
 Nova Zcm. North Ocean 
 
 74 4'^ -' 
 
 53 
 
 36 2. 
 
 ! 
 
 
 jl. LamLi)- 
 
 Eu. 
 
 Ir-hnd 
 
 Irifh S:a 
 
 r; 2.-. N 
 
 7 
 
 3c W. S I.-; i 
 
 
 |I. Limpa.lofa 
 
 Africa 
 
 Tuiiis 
 
 -Mcdit. Sea 
 
 r^ V^ -'' 
 
 12 
 
 45 E. 
 
 
 
 in. St. LaZarus 
 
 Am. 
 
 PaC'i^onia 
 
 Pacif. Ocean 
 
 ^6 42 5. 
 
 73 
 
 35 Vv. 
 
 
 
 ,C. Lclo 
 
 Africa. 
 
 Anjoia 
 
 Atl. Occ-'.n 
 
 c; 24 S. 
 
 12 
 
 r; E. 
 
 
 
 iLcith 
 Lcghjrn 
 
 Eu. 
 Eu. 
 
 Scotland 
 kaiy 
 
 CicriTi. Ocran 
 Meiit. S:;a 
 
 55 5- '' 
 4? 3 3 ^' 
 
 10 
 
 cc,W. 4 30 
 
 
 1. Lcrnn'ii 
 
 Ail a 
 
 Nitolii 
 
 Archijiclr.'o 
 
 /\r, OZ N. 
 
 25 
 
 5fi ?.. 
 
 
 C. Lci^ua 
 
 Eu. 
 
 Turkey 
 
 Mcill. -jJa 
 
 AG /i4 N. 
 
 ; u 
 
 36 V. 
 
 
 :Lcoftj/i 
 
 Eu. 
 
 Enxl ind 
 
 r,-rrr. Oocrr. 
 
 52 3,S N^ 
 
 1 
 
 S4 '^- 9 45 
 
 
 Lcpinta 
 
 Eu. 
 
 Turkey 
 
 Mr-iic. Sn 
 
 2'< 20 iV. 
 
 7.1 
 
 C3 E.! 
 
 
 'L-:pcr'i Idf 
 
 .\n.i 
 
 N. HcbriJ;-, 
 
 Pac;:\ (.\' ':. 
 
 ' C - " '' 
 
 J '[; 
 
 J 
 
 1 
 
 
 i. L:fcsv
 
 l38 
 
 GEOGRAPHY. 
 
 Book VI. 
 
 F 
 
 r.c? -;" t'li'-es. {Cont.j 
 
 Countries. 
 
 Coaft. 
 
 Latitude. 
 
 Longitude. I 
 
 L Water. 
 
 
 :. t.LlrAT I 
 
 ru. 
 
 Denmark 
 
 Sound 
 
 o , 
 
 57 05 N. 
 
 / 
 
 11 06 E. f 
 
 
 Livorpo.")! ] 
 
 lu. 
 
 ingiand 
 
 Irilh Sea 
 
 53 2a N. 
 
 3 loW. iih.i^n^.y 
 
 
 I. Lewes, N. point ] 
 
 lu. 
 
 Scotland 
 
 Weft. Ocean 
 
 58 35 N- 
 
 6 37 W. 
 
 6 30 
 
 
 Liampo, or fJingpa i 
 
 \fia 
 
 China 
 
 ?acif. Ocean 
 
 29 58 N 
 
 110 23 E. 
 
 
 
 Lima t 
 
 \m. 
 
 i'cru 
 
 Pacif. Ocean 
 
 12 01 s. 
 
 76 44 W. 
 
 
 ; 
 
 Lime 
 
 la. 
 
 England 
 
 Eng. Channel 
 
 50 45 N. 
 
 3 15W. 
 
 7 00 
 
 1 
 
 Limerick 
 
 Eu. 
 
 Ireland 
 
 R.. Shannon 
 
 51 2a N. 
 
 10 oo,W. 
 
 
 r* 
 
 I. Limofa 
 
 Eu. 
 
 Italy 
 
 Medit. Sea 
 
 36 oS N. 
 
 13 01 E. 
 
 
 
 L Lipari 
 
 Eu. 
 
 Italy 
 
 Medit. Sea 
 
 33 35 N. 
 
 5 3 E- 
 
 
 
 L Liquto 
 
 Alia 
 
 Japan 
 
 Pacif. Ocean 
 
 28 00 N. 
 
 127 30 E. 
 
 
 1 
 
 LiAon 
 
 Eu. 
 
 Portugal 
 
 R. Tagus 
 
 38 42 N. 
 
 9 4W. 
 
 a IS 
 
 i 
 
 Lifb.in Rock 
 
 Eu. 
 
 Portugal 
 
 Weft. Ocean 
 
 38 42 N. 
 
 9 25W. 
 
 
 I 
 
 C. Liiburnc 
 
 Alia 
 
 N. Hebrides, 
 
 Pacif. Ocean 
 
 15 41 S. 
 
 166 C7W. 
 
 
 
 j '.. Lifla 
 
 Eu. 
 
 Dalmatia 
 
 Adriatic Sea 
 
 42 56 N. 
 
 18 32 t. 
 
 
 
 1 Lizard 
 
 Eu. 
 
 England 
 
 Eng. Channel 
 
 49 57 N. 
 
 5 loW. 
 
 7 30 
 
 
 ilbs y S. W. end 
 LofTout iN.E. end 
 
 Eu. 
 
 Norway 
 
 North Ocean 
 
 ';63 15 N. 
 
 ;[ 69 00 N. 
 
 10 20 E. 
 12 00 E. 
 
 
 
 R. Loire, Eiit. 
 
 Eu. 
 
 France 
 
 B. Bifcay 
 
 47 07 N- 
 
 2 05 W. 
 
 3 00 
 
 
 London 
 
 Eu. 
 
 England 
 
 R. Thames 
 
 51 3z N. 
 
 CO 
 
 3 CO 
 
 
 New London 
 
 Am. 
 
 N. England 
 
 Weft. Ocean 
 
 4r 50 N. 
 
 72 14 W. 
 
 1 30. _ 
 
 
 Londonderry 
 
 Eu. 
 
 Ireland 
 
 Weft. Ocean 
 
 55 01 N. 
 
 7 -JiW, 
 
 
 
 Long Ifle 
 
 Am. 
 
 N. England 
 
 Weft. Ocean 
 
 41 CO N. 
 
 571 59 W. 
 2 74 20 W. 
 
 3 00 
 
 
 L Longo 
 
 Eu. 
 
 Dalmatia 
 
 Adriat. Sea 
 
 43 45 N- 
 
 17 33 E. 
 
 
 
 Longfand Head 
 
 Eu. 
 
 Englmd 
 
 Germ. Ocean 
 
 51 47 N. 
 
 J 41 E. 
 
 10 30 
 
 
 Lookout Tuinc 
 
 Eu. 
 
 Greenland 
 
 North Ocean 
 
 76 4c N. 
 
 16 i^ E. 
 
 
 
 C. Lopas 
 
 Africa 
 
 Loango 
 
 Atl.- Ocean 
 
 47 S. 
 
 8 30 E. 
 
 
 
 B. St. Louis 
 
 Am. 
 
 Loulfiana 
 
 G. Mexico 
 
 28 50 N. 
 
 97 08 W. 
 
 
 
 Louifbourg 
 
 Am. 
 
 C. Breton 
 
 B. St. Lav,-. 
 
 45 54 ^'' 
 
 59 50 W. 
 
 ' 
 
 
 Luhcc 
 
 Eu. 
 
 Germany 
 
 Bnltic Sea 
 
 54 00 N. 
 
 . I 40 E. 
 
 
 
 C. St. Lucar 
 
 Am. 
 
 California 
 
 Pacif. Ocean 
 
 23 15 N. 
 
 IC9 40 W, 
 
 
 
 R. Lucia 
 
 AfVica 
 
 Caffers 
 
 Lidian Ocean 
 
 27 s^ s. 
 
 33 23 E. 
 
 
 
 L St. Lucia 
 
 AiVic-i 
 
 C.deVerd 
 
 Atl. Ocean 
 
 16 43 N. 
 
 24 33 W. 
 
 
 '; 
 
 L St. Lucia 
 
 Am. 
 
 Caribbee 
 
 Ad. Ocean 
 
 13 25 N. 
 
 60 46 w. 
 
 
 
 _ r N, E. point 
 
 
 
 
 19 25 X. 
 
 121 45 E. 
 
 
 
 S C. Baj.idor 
 
 
 
 
 18 50 N. 
 
 120 25 E. 
 
 
 
 |< Maniiia 
 
 Afn 
 
 Phil, incs 
 
 Pacif. 0<?ean 
 
 14 36 N. 
 
 120 58 E. 
 
 
 
 J S. W. point 
 
 
 
 
 13 30 N. 
 
 119 35 E. 
 
 
 
 l^ E. point 
 
 
 1 
 
 
 14 CO N. 
 
 124 CO E. 
 
 
 
 jLundrn 
 
 Eu 
 
 lS,ve:!'"i 
 
 !Bal;ic Soa 
 
 55 42 N'. 
 
 n 26 E. 
 
 
 
 '<l. Lundy 
 
 Eu. 
 
 ^England 
 
 iSt. Geo. Ch. 
 
 :;l 20 K 
 
 4 04 W. 
 
 5 '5 
 
 
 jLupis's Head 
 
 |Eu. 
 
 ll,e;and 
 
 jWrft. Ocean 
 
 ?2 24 N. 
 
 10 I5W. 
 
 
 
 Lvnn 
 
 M 
 
 C. M;ih) 
 
 Eu. 
 
 England 
 
 {Germ. Ocear 
 
 52 46 N. 
 
 32 E. 
 
 6 45 
 
 
 Alia 
 
 Nc-."Guir.c 
 
 \;P".cif. Occin 
 
 1 40 s. 
 
 130 05 E. 
 
 
 
 iMacao, or Malc.iu 
 
 lAf.a 
 
 Ch.na 
 
 ipacif.Occ.jii 
 
 j 22 M N. 
 
 I. -3 46 E. 
 
 
 
 !M:ical'iar 
 
 Alia 
 
 I. Celebes 
 
 il'acif. Ocean 
 
 5 9 ^ 
 
 I iq 50 E. 
 
 
 
 C. Machinn 
 
 Eu. 
 
 Spain 
 
 ;B. Bifcay 
 
 AT, 44 ^ 
 
 3 05 W. 
 
 
 
 
 'C. St. Mary, S. 
 
 
 
 i 
 
 25 24 S. 
 
 45 53 
 
 
 
 
 p int 
 
 
 
 ' 
 
 
 
 
 
 ^ 
 
 1!. St. Ausuftinc 
 
 
 
 i 
 
 ^3 3.^ S- 
 
 4-. J 3 E- 
 
 
 
 
 
 Terra dc Gad a 
 
 
 
 
 10 sr. s. 
 
 4 . 4.6 E. 
 
 
 
 ^'' 
 
 C. St. Andre'v 
 
 
 
 1 
 
 i^ 46 s. 
 
 .. ..2 E. 
 
 
 
 
 C. St. Scba;ii;\n 
 
 au-:l 
 
 
 jlndlan Occn: 
 
 1 
 
 n -r, S 
 
 ^u -o E. 
 
 
 
 .; > 
 
 C. dc Arr.hre / 
 
 N. point 5 
 
 
 1 J, ^C' .J. 
 
 
 
 c 
 
 
 
 
 ; I- '5 ^ 
 
 1 en ic E. 
 
 
 
 1 
 
 B. d'Antonjil 
 
 
 
 1 
 
 , J 6 c- s. 
 
 j 4(, 4-- E. 
 
 
 
 Antavare 
 
 1 . 
 
 
 ' 
 
 1 ~o 5- S. 
 
 1 ->- 4^ E. 
 
 
 
 1 
 
 J-'.,. I)aun!iin 
 
 
 
 * 
 
 ; 24 4>' S. 
 
 1 4S <A: i.. 
 
 ' 
 
 
 ;L Ma-^ Kun;ii.U 
 dc'ra 1 W. Odd 
 
 Afric 
 
 \'Can-:r;c5 
 
 :a;;. Cr-.-n 
 
 i 
 
 i s 3^ 3-; N^ 
 
 1 1- .t w 
 
 12 4 
 
 12 Q 
 
 i 
 I 
 
 M.idnfi 
 
 ( f. 1 : -c: _ 
 
 Alii 
 
 ' \i.<i]i 
 
 i]r.!M:. Occ::r 
 
 :^11J ^ 
 
 i ^. 34i-:. 
 
 
 1 
 
 Madrid
 
 Book VI. 
 
 GEOGRAPHY. 
 
 3^ 
 
 Names of Places. 
 
 Cont. 
 
 Countries. 
 
 Ceaft. 
 
 Latitu de. 
 
 Longitude. 
 
 > , 
 
 H. Water. 
 
 Madrid 3 
 
 Eu. Spain 
 
 I. Man z ana 
 
 O r 
 
 40 25 N. 
 
 o3 2'iW.^ 
 
 
 Madura ' 
 
 A.fia India 
 
 Indian Ocean 
 
 10 15 N. 
 
 7S 35 E. 
 
 
 R. Maes, Mouth ] 
 
 Eu. D. Neth. 
 
 Germ. Ocean 
 
 52 06 N;- 
 
 3 50 E. 
 
 ih. 3om 
 
 Str. Le Mairc 
 
 Am. Patagonia 
 
 Atl, Ocean 
 
 54 51 S. 
 
 65 00 W. 
 
 
 Magadoxa 
 
 Africa Zanguebar 
 
 Indian Ocean 
 
 2 53 N. 
 
 45 25 E. 
 
 
 Str. Ma- C E. ent. 
 gellan I W. ent. 
 
 Am, Patagonia 
 
 Atl. Ocean 
 Pacif. Ocean 
 
 52 30 S. 
 
 52 55 s. 
 
 67 50 W. 
 74 18 W. 
 
 
 iMagifiland 
 
 Afia 
 
 ndia 
 
 Vlalabar Coa. 
 
 12 10 N. 
 
 74 14 E. 
 
 
 I. Maguana 
 
 .Am. 
 
 Bahama I. 
 
 Atl. Ocean 
 
 22 36 N. 
 
 72 25 W, 
 
 
 P. Mahon, Ifle 7 
 Minorca S 
 
 Eu. 
 
 Spain 
 
 Medit. Sea 
 
 39 51 N- 
 
 3 53 E. 
 
 
 Majorca, W. Ma- 1 
 
 Eu. 
 
 Spain 
 
 Medit. Sea 
 
 39 35 N. 
 
 2 35 E. 
 
 
 jorca J 
 
 
 
 
 
 
 
 C. Mala 
 
 Eu. 
 
 Turlcey 
 
 Archipelago 
 
 37 20 N, 
 
 24 07 E. 
 
 
 M4acca 
 
 Afia 
 
 India 
 
 Str. Malacca 
 
 2 12 N. 
 
 102 10 E. 
 
 
 Malaga 
 
 Eu. 
 
 Spain 
 
 Medit. Sea 
 
 36 43 N. 
 
 4 02 W. 
 
 
 Ides Mai- 7 N. end 
 
 Afia 
 
 India 
 
 Indian Ocean 
 
 ;; 7 20 N, 
 t[ 20 s. 
 
 73 03 E. 
 76 10 E. 
 
 
 dive 5 S. end 
 
 
 
 
 
 Maleilroom Whirl- 7 
 pool 5 
 I. M.dique 
 
 Eu. 
 
 Norway 
 
 Weft. Ocean 
 
 68 oS N. 
 
 10 40 E, 
 
 
 Afia 
 
 Maldive I. 
 
 Indi.in Ocean 
 
 7 45 ^' 
 
 72 40 E. 
 
 
 St. Maloes 
 
 Eu. 
 
 France 
 
 Eng. Channel 
 
 48 39 N. 
 
 I 57 W. 
 14 2 E. 
 
 6 00 
 
 I. Malta 
 
 Eu. 
 
 Italy 
 
 Medit. Sea 
 
 35 54 ^^ 
 
 
 I. Man, W. end 
 
 Eu. 
 
 England 
 
 I rift Sea 
 
 53 45 N. 
 
 5 oc \V, 
 
 9 00 
 
 Mangalore 
 
 Afia 
 
 India 
 
 Indian Ocean 
 
 13 02 N. 
 
 7^ 10 E. 
 
 
 (Manilla 
 
 Afia 
 
 I. Luconla 
 
 Pacif. Ocean 
 
 14 36 N. 
 
 120 58 E. 
 
 
 I. Mansfield, N. pt. 
 
 Am. 
 
 New Britain 
 
 Hudfon's Bay 
 
 62 3S N. 
 
 80 33W, 
 
 
 I. Manila 
 
 Africa 
 
 Zanguebar 
 
 Indian Ocean 
 
 8 -,6 S. 
 
 40 40 E. 
 
 
 I. r.Iardou 
 
 u. 
 
 Nor'.vay 
 
 Sound 
 
 58 H N. 
 
 8 S5 E. 
 
 
 1 I. Margarita 
 
 Am. 
 
 Terra Firm a 
 
 Atl. Ocean 
 
 II 15 N. 
 
 63 35 W. 
 
 
 1 R. Maragnon 
 
 r Margate 
 
 Am. 
 
 Brafil 
 
 Atl. Ocean 
 
 I 48 S. 
 
 44 17W. 
 
 
 Eu. 
 
 England 
 
 Eng. Channel 
 
 51 29 N. 
 
 1 10 E. 
 
 1 1 15 
 
 j C. St. Maria 
 
 Eu. 
 
 Portugal 
 
 Atl. Oci^^an 
 
 56 45 N. 
 
 7 45 W. 
 
 
 C. St. Maris, or Luci.i 
 
 Eu. 
 
 Italy 
 
 Medit. Sea 
 
 40 04 N, 
 
 .8 31 E. 
 
 
 Marian or 1 N. lijn. 
 
 
 
 
 (-21 00 N, 
 
 144 00 E, 
 
 
 Lidrone > 
 
 Afia 
 
 
 Pacif. Occin 
 
 ^ 
 
 
 
 llles }S. Wqn. 
 
 1 
 
 
 
 in 15 ^^ 
 
 142 55 E. 
 
 
 A. St. Mario 
 
 Eu, 
 
 Ar.O'.tt'i 
 
 -Atl. Ocean 
 
 37 00 N. 
 
 25 coW. 
 
 
 jSt. Mulie^ 
 
 Eu. 
 
 1. Sciily 
 
 Eng. Channel 
 
 49 57 ^' 
 
 (; 3SW. 
 
 
 r. Marlgallante 
 
 Am. 
 
 Wert Indiei 
 
 Atl. Ocean 
 
 16 00 N. 
 
 , 61 iqW. 
 
 
 |1, Maritime, Sicily 
 
 Eu. 
 
 Inciii 
 
 Medit. Sea 
 
 38 04 N. 
 
 12 33 E. 
 
 
 Marqiicfa If. 
 C. Martt-lo 
 
 Afia 
 Eu. 
 
 
 Pacif. Ocean 
 Medit. Sea 
 
 9 56 N. 
 
 38 00 N. 
 
 I 39 00 W. 
 
 26 00 E. 
 
 z 30 
 
 Turkey 
 
 'St. M.irtru 
 
 Atn, 
 
 Terra Firnja 
 
 .Atl. Ocean 
 
 ir 26 N. 
 
 74 00 W, 
 
 
 I. St. Martin 
 
 jAlTl. 
 
 Weft Indic> 
 
 Atl. Ocean 
 
 iS 06 N. 
 
 63 c6W. 
 
 
 C St. Martin 
 
 Africa 
 
 Catfcrs 
 
 Atl. Ocean 
 
 31 08 S. 
 
 .y 58 E. 
 
 
 C. St. Mjrtin 
 
 Lu. 
 
 Spiin 
 
 Medit. Sea 
 
 38 44 N. 
 
 u 25 E. 
 
 
 I. M.irtliij>;iic, Port i 
 ' R.-'.il 
 
 1 
 jAni. 
 
 {Wtlt Indies 
 
 Atl. Ocean 
 
 14 36 N. 
 
 Oi 04 W. 
 
 
 Marfe'illc. 
 
 Eu. 
 
 IF; ance 
 
 Medit. Sr, 
 
 43 iS N. 
 
 1 5 ^7 E. 
 
 
 C. S'. M,.ry 
 
 Am. 
 
 N'cwfnundl. 
 
 Atl. Ocean 
 
 46 52 N. 
 
 i 54 Gl W. 
 
 i 
 
 C. St. M.,ry 
 
 Am. 
 
 Biafil 
 
 Atl. Ocr-.in 
 
 34 5^ ^ 
 
 1 51 55 W. 
 
 t 
 
 C. St. M,.ry 
 
 Aha 
 
 Natolia 
 
 .Archipelago 
 
 37 4^ 1^- 
 
 27 21 E. 
 
 1 
 
 ('. St. M,-v 
 
 i;.i, 
 
 Spain 
 
 N. Atl. Ocean 
 
 36 46 N. 
 
 7 49 W. 
 
 \ 
 
 C. Vir/iu Vl.iry 
 
 Am. 
 
 P;.t.igonia 
 
 S, Atl. Ocean 
 
 52 23 S. 
 
 68 lo'vV. 
 
 
 ;M .1 t ;tr.: 
 
 Am. 
 
 Chili 
 
 P.'fif. Oe.:.i;i 
 
 35 45 ^ 
 
 80 34 W. 
 
 
 ; I. M-.!i .,..,!.- 
 
 Afri..a '/, iU-^'i'-\,H 
 
 Iii.lian Ocean 
 
 20 52 S. 
 
 55 35 E. 
 
 
 1 I. M.,K u.l; 
 
 JAni. 
 
 :I\mu 
 
 P. .it'. O.ean 
 
 1 20 s. 
 
 : 8S 50 w. 
 
 
 i'M>:, .. 
 
 lAli.. 
 
 Aubia 
 
 I^kIw. Oci:ir 
 
 23 10 N. 
 
 1 57 40 E. 
 
 
 j \\.:k'\:r.r- Ifc:. 
 
 ;.vi>., 
 
 X. llcl.ri.ic 
 
 . Pa.I! 0.-.Vn 
 
 16 ^2 S. 
 
 i >'>7 59 E. 
 
 
 [M ,!hlUDl 
 
 li.. 
 
 Sv.vJ,-,, 
 
 iSr.u-ui 
 
 57 'y' ^ 
 
 1 12 c. r. 
 
 
 |A1 ,(;;!;: ,t.r^ 
 
 !.);.. 
 
 It.dia 
 
 'B. Bfp^-.l 
 
 16 28 N. 
 
 1 81 40 . 
 
 
 
 
 
 
 

 
 59^ 
 
 GEOGRAPHT. 
 
 Book VI. 
 
 
 Names of Places. 
 
 Cont 
 
 Countries. 
 
 Coaft. 
 
 Latitude. 
 
 Longitude. 
 
 H. Water. 
 
 1 
 
 
 C. Matapan 
 
 Eu. 
 
 Turkey 
 
 Archipelago 
 
 36 25 N. 
 
 , 
 22 40 E. 
 
 
 
 I. Matbare 
 
 Afia 
 
 Japan 
 
 Pacif. Ocean 
 
 26 30 N. 
 
 137 00 E. 
 
 
 
 
 1, St, Mathcw's 
 
 Ai\<u 
 
 k Guinea 
 
 Eth. Ocean 
 
 I 23 s. 
 
 6 iiW. 
 
 
 
 
 1. Mauritius 
 
 Afric 
 
 1 Madagafcar 
 
 Indian Ocean 
 
 20 10 S. 
 
 57 33 E- 
 
 
 
 
 Maurua 
 
 Afia 
 
 Scckty liles 
 
 Pacif. Ocean 
 
 16 26 S. 
 
 152 33 w. 
 
 
 
 
 1. May 
 
 Afiic, 
 
 > C. Ver d 
 
 Atl. Ocean 
 
 15 10 N. 
 
 23 00 W. 
 
 
 
 
 C. May 
 
 Am. 
 
 Penfilvania 
 
 At). Ocean 
 
 39 ^5 N. 
 
 74 43 W. 
 
 
 1 
 
 
 I. Mayettc 
 
 Afric. 
 
 I Madag^kfcar 
 
 Indian Ocean 
 
 12 53 s. 
 
 46 10 E. 
 
 
 
 Mecca 
 
 Afia 
 
 Arabia 
 
 Red Sea 
 
 21 40 N. 
 
 41 00 E. 
 
 
 
 Medina 
 
 Aiia 
 
 Arabia 
 
 Red Sea 
 
 24 58 N- 
 
 39 53 E- 
 
 
 
 
 I. Mclads 
 
 Eu. 
 
 Dalmatia 
 
 Adriat. Sea 
 
 42 40 N. 
 
 19 34 E. 
 
 
 
 
 JMclindc 
 
 Africa 
 
 Zangucbar 
 
 Indian Ocean 
 
 3 07 S, 
 
 39 40 E. 
 
 
 
 
 I. Mclo 
 
 Eu. 
 
 Turkey 
 
 Archipelago 
 
 36 41 N. 
 
 25 05 E. 
 
 
 
 
 Mcmel 
 
 Eu. 
 
 Courlani 
 
 Baltic Sea 
 
 55 48 N. 
 
 22 23 . 
 
 
 
 
 Mcmiiran 
 
 Eu. 
 
 France 
 
 B. Bifcay 
 
 44 20 N. 
 
 I 23 w. 
 
 3h.3oni, 
 
 
 
 |I. Menado 
 
 Afia 
 
 I. Celebe^s. 
 
 Pacif. Ocean 
 
 I 36 N. 
 
 122 25 E. 
 J30 15W. 
 
 
 
 
 jC. Mend'jzin 
 
 Am. 
 
 California 
 
 Pacif. Ocean 
 
 41 20 N. 
 
 
 1 
 
 
 Mercury Pay 
 
 Afia 
 
 N. Zealand 
 
 Pacif. Ocean 
 
 36 50 S. 
 
 175 12 E. 
 
 
 
 
 R. Metaparvous 
 
 Am. 
 
 fiahama 
 
 Atl. Ocean 
 
 21 58 N. 
 
 74 13W. 
 
 
 
 
 Meflina 
 
 Eu. 
 
 I. Sicily 
 
 Medit. Sea 
 
 38 21 N. 
 
 16 21 E. 
 
 
 
 
 iC. Mcfurato 
 
 Africa 
 
 Tripoli 
 
 Mcdit. Sea 
 
 32 18 N. 
 
 16 36 E. 
 
 
 
 
 I.Mcty-?J;f^f= 
 
 Afia 
 
 Natolia 
 
 Archipelago 
 
 39 3ti N. 
 39 n N. 
 
 26 08 E. 
 26 ^7 E. 
 
 
 3 
 
 
 ^^"'^ ^Po.pliviea 
 
 
 
 
 39 00 N. 
 
 26 50 E. 
 
 
 
 
 I. Meua 
 
 Eu. 
 
 Denmark 
 
 Baltic Sea 
 
 55 00 N. 
 
 13 15 E. 
 
 
 
 
 [Mexico 
 
 Am. 
 
 Mexico 
 
 Inland 
 
 19 54 N. 
 
 ICO 01 W. 
 
 
 
 
 Miatea 
 
 Alia 
 
 Society Ifies 
 
 Pacif. Ocean 
 
 17 52 s. 
 
 148 I W. 
 
 
 
 
 1. St. Michael 
 
 Eu. 
 
 Azores 
 
 Atl. Ocean 
 
 37 45 N- 
 
 25 38 w. 
 
 
 
 
 Middleburgh 
 
 Eu. 
 
 D. Neth. 
 
 Germ. Ocean 
 
 51 37 N- 
 
 3 58 E. 
 
 
 
 
 Middleburgh, or 7 
 Eaoowe J 
 
 Afia 
 
 Friendly Ifl. 
 
 Pacif. Ocean 
 
 21 21 S. 
 
 174 34 W. 
 
 
 
 
 Mitfo:d 
 
 Eu. 
 
 Wales 
 
 St. Geo. Ch. 
 
 51 45 N- 
 
 5 'SW. 
 
 5 '5 
 
 
 
 MHo, I. Milo 
 
 Afia 
 
 Tuikcy 
 
 Archipelago 
 
 36 41 N. 
 
 25 05 E. 
 
 
 
 
 Mill Ifies 
 
 Ara. 
 
 North Main 
 
 Hudfon's Bay 
 
 64 36 N. 
 
 80 30 W. 
 
 
 ^ 
 
 
 o r N. point 
 
 S I S. E. pt. C, St. 
 
 
 
 
 9 40 N. 
 
 124 25 E. 
 
 - 
 
 
 
 -S ) AuguilJne 
 J| S. W. pt. Cal- 
 ^ 1 dera 
 "* L S. point 
 
 Afia 
 
 Spice Ifland. 
 
 Pacif. Ocean 
 
 6 40 N. 
 
 7 00 N. 
 3 so N. 
 
 126 25 E. 
 
 121 25 E. 
 124 43 E. 
 
 
 
 
 I. Mindora 
 
 Afia 
 
 Phl'.ip. Ifies 
 
 Pacif, Ocean 
 
 13 oc> K. 
 
 119 37 E. 
 
 
 
 
 I. Mi- 7 N. W. pt, 
 norca J S.E. pt. 
 
 Eu. 
 
 Spain 
 
 Mediterran. 
 
 S 39 58 N. 
 ^ 40 24 N. 
 
 3 54 E. 
 
 4 18 E. 
 
 
 
 
 G. Miquclcn 
 
 Am. 
 
 Nevvf.andU 
 
 Atl. Ocean 
 
 47 3 I^^ 
 
 56 13W. 
 
 
 , 
 
 
 L. Micuclon 
 
 Ani. 
 
 Nev.found. 
 
 Atl. Ocean 
 
 46 50 N. 
 
 56 13W. 
 
 
 
 
 I. Mifco 
 
 Am. 
 
 Nova Scctia 
 
 G.St. Lawr. 
 
 48 04 N. 
 
 64 19 W. 
 
 
 
 
 C. Mif^rata 
 
 Africa 
 
 Guinea 
 
 Ul. Ocean 
 
 6 25 N. 
 
 9 35"W. 
 
 
 
 
 R. Milliflippi, nioutli 
 
 Aiti. 
 
 I.ouifana ^ 
 Ireland 
 
 G.'Mcjcico 
 
 29 00 N. 
 
 89 17 w. 
 
 
 
 
 Mizen Head 
 
 Eu. 
 
 Atl. Ocean 
 
 51 i5 N. 
 
 10 20 W. 
 
 
 
 
 Mocha 
 
 Afia 
 
 Ar.abia 
 
 Red Sea 
 
 13 4=; N, 
 
 44 04 E. 
 
 
 
 
 Modon 
 
 Eu. 
 
 Tuikcy 
 
 Medit. Sea 
 
 36 55 N. 
 
 21 03 E. 
 
 
 
 
 
 I. Mohilla 
 
 Afric. 
 
 Zangucbar 
 
 Indian Ocean 
 
 II 55 S. 
 
 45 00 E. 
 
 
 
 
 I. Monfirnt 
 
 Am. 
 
 Wcit Ir.dks 
 
 Atl. Ocean 
 
 16 4S N. 
 
 62 12W. 
 
 
 
 
 Montagu Ifle 
 
 Afia 
 
 N. Hebrides 
 
 Pacif. Ocean 
 
 17 26 S. 
 
 16S 36 E, 
 
 
 
 
 Montreal 
 
 Am. 
 
 Canada 
 
 K. St. Lawr. 
 
 45 52 N- 
 
 73 iiW. 
 
 
 
 
 I. Monte Chrifto 
 
 Eu. 
 
 iuly 
 
 Mcdit. Sea 
 
 42 17 N. 
 
 10 2S E. 
 
 
 
 
 C. Monte San^o 
 
 Eu. 
 
 Turkey 
 
 Arthip'iSago 
 
 40 27 N. 
 
 24 39 E. 
 
 
 
 1 
 
 Monument 
 
 Afia 
 
 M. Hebrides 
 
 -acif. Ocean 
 
 17 14 S, 
 
 168 38 E. 
 
 
 
 
 Mount St. Michael 
 
 Eu- 
 
 France 
 
 litig. Channel 
 
 48 39 N. 
 
 I 35 W. 
 
 
 
 
 I. Mcrgo 
 
 Afia 
 
 Natolia 
 
 Archipelago 
 
 36 55 N. 
 
 26 30 E 
 
 
 
 
 Morlrtix 
 
 Eu. 
 
 France 
 
 ing. Channel 
 
 48 30 N. 
 
 3 50W. 
 
 
 
 
 Mort Point | 
 
 Eu. 
 
 Fcgland 
 
 it, Geo.Ch. 
 
 51 12 N. 
 
 4 40 W. 
 
 
 
 
 
 
 
 
 
 M 
 
 ofambi^uc 

 
 Book VI. 
 
 GEOGRAPHY. 
 
 39^ 
 
 Names of I*laces. 
 
 Cont. 
 
 Countries. 
 
 Coaft. 
 
 Latitude. 
 
 1 Longitude. 
 
 H. Water. 
 
 Mofamblque 
 
 Africa 
 
 Zanguebar 
 
 Indian Ocean 
 
 O / 
 
 15 CO S. 
 
 a , 
 41 40 E. 
 
 
 Molcow 
 
 Eu. 
 
 Ruffia 
 
 R. Mofcow' 
 
 55 45 N. 
 
 37 5^ E- 
 
 
 Mofquitos Bank 
 
 Am. 
 
 Mexico 
 
 Atl. Ocean 
 
 14 45 N. 
 
 80 C5W. 
 
 ' 
 
 C. Mount 
 
 Africa 
 
 Guinea 
 
 Atl. Ocean 
 
 7 13 N. 
 
 10 4.4 w. 
 
 
 Mount's Bay 
 
 Eu. 
 
 England 
 
 Eng. Channel 
 
 50 05 N. 
 
 5 45 W. 
 
 4h. 3om< 
 
 Moufe River 
 
 Am. 
 
 New Wales 
 
 Hudfon's Ba\ 
 
 51 zs N. 
 
 83 I 5 V/. 
 
 
 C. Mufaldoh 
 
 N 
 C. Nabo 
 
 Afia 
 
 Arabia 
 
 Perfian Gulf 
 
 26 04 N. 
 
 55 22 E. 
 
 
 Afia 
 
 Japan 
 
 Pacif. Ocean 
 
 40 35 N. 
 
 141 25 E. 
 
 
 Nangafack 
 
 Afia 
 
 Japan 
 
 Pacif. Ocean 
 
 32 32 N. 
 
 128 50 E. 
 
 
 Nankin 
 
 A 111 
 
 China 
 
 Pacif. Ocean 
 
 3z C7 N. 
 
 118 35 E. 
 
 
 Nantes 
 
 Eu. 
 
 Frar.ce 
 
 B. Bifcay 
 
 47 13 ^' 
 
 I 29 w. 
 
 3 00 
 
 Nantucket ICe 
 
 Am. 
 
 New Eng. 
 
 Weft. Ocean 
 
 41 34 N. 
 
 69 40 W. 
 
 
 Naples 
 
 Eu. 
 
 Italy 
 
 iMcdit. Sea ' 
 
 40 51 N. 
 
 14 19 E. 
 
 
 Narhonne 
 
 Eu. 
 
 France 
 
 Medit. Sea 
 
 43 II N. 
 
 3 05 E- 
 
 
 Narfjnga 
 
 Afia 
 
 India 
 
 B. Bengal 
 
 18 OS N. 
 
 85 20 E. 
 
 
 Narva 
 
 Eu. 
 
 Livonia 
 
 G, Finland 
 
 59 c8 N. 
 
 29 iS E. 
 
 
 I. NalTau 
 
 Afia 
 
 Sumatra 
 
 Indian Ocean 
 
 3 00 S. 
 
 IOC 25 E. 
 
 
 C. Naflbu 
 
 Am. 
 
 Terra Firma 
 
 Atl. Ocean 
 
 7 53 N- 
 
 58 07 W. 
 
 
 NafTau Str. 
 
 Eu. 
 
 Rulfia 
 
 North Ocean 
 
 69 55 N. 
 
 57 30 
 
 
 B. Natal 
 
 Africa 
 
 Ca.Ters 
 
 Indian Ocean 
 
 29 25 S. 
 
 33 10 E. 
 
 
 I. Naxos 
 
 Eu. 
 
 Turkey 
 
 Archipelago 
 
 37 06 N. 
 
 25 58 E. 
 
 
 Naze 
 
 Eu. 
 
 Norway 
 
 Weft. Ocean- 
 
 57 5^ ^' 
 
 7 32 E. 
 
 II 15 
 
 Needles 
 
 Eu. 
 
 England 
 
 Eng. Channel 
 
 50 AJ N. 
 
 I 28 W. 
 
 10 15 
 
 C. Negrailles 
 
 Alia 
 
 Pegu 
 
 B. Bengal 
 
 16 20 N. 
 
 94 15 E- 
 
 
 C Negro 
 
 Afi-ica 
 
 Caffers 
 
 Atl. Ocean 
 
 16 30 S. 
 
 II 30 E. 
 
 
 C. Negro 
 
 Africa 
 
 Barbary 
 
 Medit. Sea 
 
 37 17 N. 
 
 9 09 E. 
 
 
 Negropont 
 
 Eu. 
 
 Turkey 
 
 Archipelago 
 
 38 30 N. 
 
 24 05 E. 
 
 
 Port Nelfon 
 
 Am. 
 
 New Wales 
 
 Hud Ion" 3 Bay 
 
 57 07 N. 
 
 92 37W. 
 
 
 Port Nelfjn's Shoals 
 
 Am. 
 
 New Wales 
 
 Hudfon's Bay 
 
 57 35 N. 
 
 92 07 W. 
 
 8 10 
 
 I. Nevis 
 
 Am. 
 
 Caribbeelllcs 
 
 A-rl. Ocean ' 
 
 17 II N. 
 
 62 52W. 
 
 3 15 
 
 Newcaftle 
 
 Eu. 
 
 England 
 
 Germ. Ocean 
 
 55 03 N. 
 
 I 28 W. 
 
 R. Nicaragua 
 
 Am. 
 
 New Spain 
 
 At!. Ocean 
 
 11 40 N. 
 
 82 47 W. 
 
 
 Nice 
 
 Eu. 
 
 Italy 
 
 Medit. Sea 
 
 43 42 N. 
 
 7 22 E. 
 
 
 if. N'icobar 
 
 Afia 
 
 Siam 
 
 B, Bengal 
 
 7 22 N. 
 
 94 40 E. 
 
 
 I. Sr. Nicholas 
 
 Africa 
 
 C. Vcrd I. 
 
 Atl. Ocean 
 
 16 35 N. 
 
 24 06 W. 
 
 
 Nlcotera 
 
 Eu. 
 
 Italy 
 
 Medit. Sea 
 
 33 33 N. 
 
 16 30 E. 
 
 
 Nieuport 
 
 Eu. 
 
 Flanitcrs 
 
 Germ. Ocean 
 
 51 cB N. 
 
 2 50 E. 
 
 12 00 
 
 Ninhay 
 
 Afia 
 
 China 
 
 Pacif. Ocean 
 
 37 10 N. 
 
 122 25 E. 
 
 
 Ningpfi, or Liampo 
 
 Af.a 
 
 China 
 
 Pacif. Ocea,n 
 
 29 58 N. 
 
 120 23 E. 
 
 
 I. Nio 
 
 Eu. 
 
 Turkey 
 
 Archipelago 
 
 36 48 N. 
 
 26 02 E. 
 
 
 I. Noel 
 
 Afia 
 
 India 
 
 Indian Ocean 
 
 10 30 S. 
 
 105 25 E. 
 
 
 C. Noir 
 
 Am. 
 
 T. del Fuego 
 
 Pacif. Oce.in 
 
 54 32 S. 
 
 73 3W. 
 
 
 Norfolk Ifle 
 
 Afia 
 
 N. Holland 
 
 Pacif. Ocean 
 
 29 2 S. 
 
 168 15 E. 
 
 
 C. de Non 
 
 Africa 
 
 Barbary 
 
 Atl. Ocean 
 
 28 04 N. 
 
 10 32W. 
 
 
 Nonibrc d;; Dios 
 
 Am. 
 
 Terra Firma 
 
 Carribbe, Sea 
 
 9 43 ^ 
 
 73 35W. 
 
 
 Norc 
 
 Eu. 
 
 Eng'tnd 
 
 R. Thames 
 
 51 28 N. 
 
 48 . 
 
 00 
 
 Norlton 
 
 Am. 
 
 I'cnfylv.inia 
 
 Inland 
 
 40 10 N. 
 
 75 '7W. 
 
 
 C. North 
 
 Am. 
 
 Tcria Fi.ma 
 
 Atl. Ocean 
 
 I 45 ^' 
 
 49 coW. 
 
 
 C. Nn.th 
 
 Am. 
 
 C. Breton 
 
 Atl. Ocean 
 
 47 5 ^ 
 
 60 8W. 
 
 
 C. North 
 
 Am. 
 
 S. Georgia 
 
 Atl. Ocean 
 
 54 5 '' 
 
 38 loW. 
 
 
 N.Ca]>e, I. Maggoro'; 
 
 E.J. 
 
 Lapland 
 
 North Ocean 
 
 71 10 N. 
 
 26 02 E. 
 
 3 CO 
 
 Nr.rth I'-.i.;- 
 
 Eu. 
 
 N'a|-u,iy 
 
 North Ocean 
 
 62 15 N. 
 
 6 15 E. 
 
 
 N ,:t-n El'ir^- 
 
 Am. 
 
 North Main 
 
 Jludfon's Str. 
 
 62 30 N. 
 
 70 50 W. 
 
 
 1. No'in^him, E. p'. 
 
 Am. 
 
 New Britain 
 
 Hudf-jn's St;r. 
 
 63 35 N. 
 
 77 4''^W. 
 
 1 00 
 
 O.ilri I'.lia Rjy 
 
 Afia 
 
 Orahe'te 
 
 Pacif. Oeean 
 
 17 46 S. 
 
 149 9VV. 
 
 
 Ov7Ao-.v 
 
 E'u 
 
 1 urkcy 
 
 Black Sea 
 
 45 .7-N. 
 
 Vx 4'5 E. 
 
 
 ' S^-.i 
 
 Evu 
 
 Swcd'-a 
 
 Baltic S.-a 
 
 5 56 .5 N. 
 
 i^ 5S E. 
 17 ^5 E. 
 
 
 ,0';n:r, .n"-. , B :y 
 
 Afia 
 
 Society l/les 
 
 U! la tea 
 
 J 6 46 "s! 
 
 I -, 1 \\ w. 
 
 1 1 20 
 
 ;;>l' H-:', ..! '^inr.l- 
 
 Eu. 
 
 lix-l.i::d 
 
 Ati. Oc^n 
 
 51 -,o N. 
 
 S : ; V/. 
 
 
 I. Oleioa
 
 39^ 
 
 GEOGRAPHY. 
 
 Book VL 
 
 Names of Places. 
 
 Cent. 
 Eu. 
 
 Countries. 
 
 France 
 
 Coaft. 
 
 Latitude. 
 
 Longitude. 
 
 H.Water.' 
 
 I. Oleron 
 
 B. Bi^ayk^ 
 
 46 03 N. 
 
 I 20 W. 
 
 
 O'inde 
 
 Am. 
 
 Brafil 
 
 S.Atl. Ocean 
 
 $ 13 s. 
 
 3? 00 W. 
 
 
 Oliva 
 
 Eu. 
 
 Germany 
 
 Baltic Sea " 
 
 r4 20 N. 
 
 18 30 E. 
 
 
 Ollone 
 
 Eu. 
 
 France 
 
 B. Bifcay 
 
 46 34 N. 
 
 I 36 W. 
 
 3h.45m; 
 
 OMgKa 
 
 Eu. 
 
 Italy 
 
 Medit. Sea 
 
 43 57 N. 
 
 7 52 E- 
 
 
 Oporto 
 dan 
 
 Eu. 
 
 Portugal 
 
 Atl. Ocean 
 
 41 10 N. 
 
 8 22 W. 
 
 
 Africa 
 
 Barbary 
 
 Mcdit. Sea 
 
 35 45 N"' 
 
 00 
 
 
 C. Orange 
 
 Am. 
 
 Terra Firma 
 
 Atl. Ocean 
 
 47N. 
 
 50 50 W. 
 
 
 Oibitdlo 
 
 Eu. 
 
 Italy 
 
 Medit. Sea 
 
 42 30 N. 
 
 12 00 E. 
 
 
 I. Orchillo ^ 
 
 Am. 
 
 Terra Firma 
 
 Carribean Sea 
 
 II 32 N. 
 
 65 25W. 
 
 
 Orenburg 
 
 Aiia 
 
 Aitracan 
 
 Inlanii 
 
 51 4 N. 
 
 55 14 E- 
 
 - 
 
 Orfoidnefs 
 
 Eu. 
 
 England 
 
 Germ. Ocean 
 
 52 17 N' 
 
 I U E. 
 
 9 "45 
 
 Orkney Ifles, limits 
 
 Eu. 
 
 Scotland 
 
 Weft. Oceafl 
 
 'JS9 24 N. 
 ,[58 44 N. 
 
 3 23W. 
 z iiW. 
 
 3 00 
 
 New Orleans 
 
 Am. 
 
 Louifiana 
 
 R. Miffiflipi 
 
 30 00 N. 
 
 89 54W. 
 
 
 I. Ormus 
 
 Afia 
 
 Pcrf.a 
 
 G. Perfia 
 
 27 30 N. 
 
 55 17 E. 
 
 
 C, delOrOjOrOlerada 
 
 Africa 
 
 N'egroland 
 
 Atl. Ocean 
 
 :i3 30 N. 
 
 14 31W. 
 
 
 
 R. OronoCjUS 
 
 Am. 
 
 Terra Firma 
 
 Atl. Ocean 
 
 8 08 N. 
 
 59 50 W. 
 
 
 C. Oropeib 
 
 Eu. 
 
 Spain 
 
 Medit. Sea 
 
 40 20 N. 
 
 49 E. 
 
 
 Orik 
 
 Afia 
 
 Aiiracan 
 
 Inland 
 
 51 12 N. 
 
 58 37 E. 
 
 
 C. Ortcpal 
 
 Eu. 
 
 Sr>ain 
 
 B. Bifcay 
 
 43 47 N. 
 
 8 3^W. 
 
 
 Ortona " 
 
 Eu. 
 
 Italy 
 
 Medit. Sea 
 
 42 19 N. 
 
 14 37 E. 
 
 
 I. Ortiba 
 
 Am. 
 
 I'erra Firma 
 
 Caribbean Sea 
 
 i: 03 N. 
 
 69 03 W. 
 
 
 tofnaburg IHc 
 
 A:i3 
 
 Srciety Ifles 
 
 Pacif. Ocean 
 
 22 00 S. 
 
 141 34 w. 
 
 
 }Oft;-nd 
 
 Eu. 
 
 Fl.mders 
 
 Germ. Ocean 
 
 51 14 N. 
 
 3 00 E, 
 
 12 CO 
 
 iC. Otranto 
 
 Eu. 
 
 tAy 
 
 Medit. Sea 
 
 40 23 N. 
 
 17 41 E. 
 
 
 lOwharre Bay 
 
 lifia 
 
 I-lL:ahc!ne 
 
 Pacif. Ocean 
 
 16 44 S. 
 
 151 3W. 
 
 
 I'Ozaca 
 
 ft P 
 !,C. Padron 
 
 Afia 
 
 Japan 
 
 Pacif. Ocean' 
 
 35 10 N. 
 
 134 05 E. 
 
 
 Africa 
 
 Congo 
 
 Atl. Ocean 
 
 6 00 S. 
 
 ir 40 E. 
 
 
 Paira 
 
 Am. 
 
 Peru 
 
 Pac'.f. Ocean 
 
 5 20 S. 
 
 Sq 35W. 
 
 
 C. yiilio'jrl 
 
 Eu. 
 
 Turkey 
 
 Archipelago 
 
 39 59 N. 
 
 24 03 E. 
 
 
 Palcrinc, I. Sicily 
 
 Eu. 
 
 It^ay ' 
 
 Medit. Sea 
 
 38 10 N. 
 
 13 43 E- 
 
 
 P..!ia:-:ate 
 ' Pdlifsr^ Ides 
 
 AT.a 
 
 India 
 
 B. Bengal 
 
 13 40 N. 
 
 So 50 E. 
 
 
 Ai'a 
 
 Society Lies 
 
 Pacif. Ocean 
 
 15 38 S. 
 
 146 2SW. 
 
 
 C. ?2)iif?r 
 
 Afa 
 
 N. Zealand 
 
 Pacif. Ocean 
 
 41 40 S. 
 
 17.; 28 E. 
 
 
 I. Fal.-na 
 
 A t ric:i 
 
 Ciinarics 
 
 At'. Ocean 
 
 23 36 N. 
 
 17 45W. 
 
 
 il. PAlnj,\ria 
 
 Eu. 
 
 Italy 
 
 Medit. S<ra 
 
 41 00 N. 
 
 13 03 E. 
 
 
 Fai.Tic.rjto;'': IHe 
 
 Afia 
 
 Society IHes 
 
 Pacif. Ocean 
 
 18 00 S. 
 
 162 52W, 
 
 
 :C. Fiimiras 
 
 Aha 
 'Africa 
 
 Inula 
 
 B. Bengal 
 
 20 40 N. 
 
 87 35 E. 
 
 
 ;C. r'aimas 
 
 Guinea 
 
 Atl. Ocean 
 
 4 26 N. 
 
 5 56 W. 
 
 
 Pdnama 
 
 Am. 
 lu. 
 
 Mexico 
 
 Pacif. Ocean 
 
 8 45 N. 
 
 80 16V/. 
 
 
 I, Panaria 
 
 Italy 
 
 Medit. Sea 
 
 38 40 N. 
 
 15 41 E. 
 
 
 PiTiornia 
 
 Ku. 
 
 Turkey 
 
 JVledit. Sea 
 
 40 05 N. 
 
 21 40 E. 
 
 
 I. Pancalari^ 
 
 ENj. 
 
 Italy 
 
 Mcdit. Sea 
 
 36 55 N. 
 
 12 31 E. 
 
 
 R. Panuco 
 
 Am. 
 
 Mexico 
 
 G. Mexico 
 
 24 02 N. 
 
 100 13W. 
 39 50W. 
 
 
 iR. P^rai^a 
 
 Am. 
 
 Drafil 
 
 Atl. Ocean 
 
 21 26 S. 
 
 
 ^aris 
 
 Eu. 
 
 France 
 
 R. Seine 
 
 48 50 N. 
 
 2 25 E. 
 
 
 C. PafTcro 
 
 Ej. 
 
 I. Sicily 
 
 Medit. Sea 
 
 36 35 N. 
 
 15 22 E. 
 
 
 |C.^a:ani 
 
 Afia 
 
 M.iUcta 
 
 Indian Ocean 
 
 7 27 N. 
 
 lOI 20 E. 
 
 
 I. Pat-nos 
 
 Alia 
 
 Natolia 
 
 Archipelago 
 
 37 22 N. 
 
 26 48 E. 
 
 
 R. Pitrajan 
 
 Alia 
 
 I. Sumatra 
 
 Str. Mal-itca 
 
 28 N. 
 
 loz 25 E. 
 
 
 C. Paul 
 
 Ea. 
 
 Sp^in 
 
 Mcdit. Sea 
 
 37 5 ^ 
 
 ; 5 W. 
 
 
 I. St. Paul 
 
 Am. 
 
 Newfoundl. 
 
 B. St. I.a.'.r. 
 
 47 12 N. 
 
 59 59 W. 
 
 
 J. St. Paul 
 
 Afi.: 
 
 Midajjafcur 
 
 Indian Ocean 
 
 37 51 s. 
 
 77 S3 
 
 
 iSt. Paul dc Leon 
 
 Eu. 
 
 Franc..- 
 
 Eng. Cliannel 
 
 48 41 N. 
 
 3 55W. 
 
 4 00 i 
 
 !l. Paxeros 
 
 Am. 
 
 CAjifoinia 
 
 Pacif. Ocean 
 
 30 18 N. 
 
 120 45 w. 
 
 1 
 
 I. Pearl, or Scran:, 
 
 Am, 
 
 Weft indies 
 
 AtL Ocean 
 
 14 55 N. 
 
 79 coW. 
 
 t 
 
 jP.gu 
 
 Afia 
 
 India 
 
 B. Bengal 
 
 17 00 N. 
 
 96 58 E. 
 
 
 jPckin 
 
 Alia 
 
 China 
 
 Inland 
 
 39 55 ^J- 
 
 116 29 E. 
 
 
 I|. PdugC'fa 
 
 Eg. 
 
 Italy 
 
 Adriatic Sea 
 
 42 20 N. 
 
 iS 32 E. 
 
 
 I.-Pembi 
 
 A Trie 
 
 Z.inguebar 
 
 Indi.iii (.)c',.ia 
 
 r.~.'i -. 
 
 5 3S S. 
 
 40 09 E. 
 
 ' r 
 
 C. Pembroke
 
 Book Vr. 
 
 GEOGRAPHY. 
 
 ^^3 
 
 Names of Places. 
 
 Cont. 
 
 Countries. 
 
 Coaft. 
 
 Latitude. 
 
 Longitude. 
 
 H.Water. 
 
 C. Pembroke 
 
 Am. 
 
 Uew Wales 
 
 Hudfon's Bay 
 
 63 12 N. 
 
 82 54 W. 
 
 
 I. Pengwin 
 
 Am, 
 
 Newfoundl. 
 
 Atl. Ocean 
 
 47 23 N- 
 
 56 56 w. 
 
 
 Penmark 
 
 Eu. 
 
 France 
 
 B. Bifcay 
 
 47 50 N. 
 
 4 20W. 
 
 
 R. Penoblcot 
 
 Am. 
 
 New Eng. 
 
 Atl. Ocean 
 
 44 40 N. 
 
 68 52 W. 
 
 
 Pei'nambuc^ 
 
 Am. 
 
 Brafil 
 
 S. Atl. Ocean 
 
 8 30 S. 
 
 35 07 W. 
 
 
 Petapoli 
 
 Afia 
 
 India 
 
 B. Bengal 
 
 16 14 N. 
 
 81 10 E. 
 
 
 I. St. Peter 
 
 Am. 
 
 Newfoundl. 
 
 Atl. Ocean 
 
 46 46 N. 
 
 56 5W. 
 
 
 Peterfburg 
 
 Eu. 
 
 Ruffia 
 
 Baltic Sga 
 
 59 56 N. 
 
 30 24 E. 
 
 
 
 C. Petra 
 
 Afia 
 
 Natolia 
 
 Archipelago 
 
 37 6z N. 
 
 27 38 E. 
 
 
 Peverel Point 
 
 Eu. 
 
 England 
 
 Eng. Channel 
 
 50 34 N. 
 
 I 22 W. 
 
 
 Philadelphia 
 
 Am. 
 
 Penfilvania 
 
 R. Delawar 
 
 39 57 N. 
 
 75 8W. 
 
 
 St. Philip 
 
 Africa 
 
 Benguela 
 
 Atl. Ocean 
 
 12 22 S. 
 
 13 20 E. 
 
 
 I. Pianofa 
 
 Eu. 
 
 Italy 
 
 Medit. Sea 
 
 42 46 N. 
 
 10 34 E. 
 
 
 IHe of Pines 
 
 Afia 
 
 N.Caledonia 
 
 Pacif. Ocean 
 
 22 38 S. 
 
 167 43 E. 
 
 
 I. Pico (Pike) 
 
 Eu. 
 
 Azores 
 
 Atl. Ocean 
 
 38 29 N. 
 
 28 19W. 
 
 
 C. Pinas 
 
 Eu. 
 
 Spain 
 
 B. Bifcay 
 
 43 51 N. 
 
 6 14W. 
 
 
 Pickerfglll's I. 
 
 Am. 
 
 S. Georgia 
 
 Atl. Ocean 
 
 54 42 S. 
 
 36 53 w. 
 
 
 Mo. Pintados, or J 
 St. Martin J 
 
 Am. 
 
 California 
 
 Pacif. Ocean 
 
 27 30 N. 
 
 117 15W. 
 
 
 Pifcadore Iflcs 
 
 Afia 
 
 China 
 
 Pacif. Ocean 
 
 23 30 N. 
 
 119 25 E. 
 
 
 Pitcairn's Ifles 
 
 Am. 
 
 Chiii 
 
 Pacif. Ocean 
 
 25 2 S. 
 
 133 21W. 
 
 
 Placentia 
 
 Am. 
 
 Newfoundl. 
 
 Atl. Ocean 
 
 47 15 N. 
 
 53 43 W. 
 
 gh. 60m. 
 
 R. Plata 
 
 Am. 
 
 La Plata 
 
 Atl. Ocean 
 
 36 00 S. 
 
 57 40 W. 
 
 
 R. Platewrack 
 
 Am. 
 
 Bahamalfles 
 
 Atl. Ocean 
 
 20 04 N. 
 
 68 37W. 
 
 
 Plymouth 
 
 Eu. 
 
 England 
 
 Eng. Channel 
 
 50 22 N. 
 
 4 loW. 
 
 6 00 
 
 Policaftro 
 
 Eu. 
 
 Italy 
 
 Medit. Sea 
 
 40 18 N. 
 
 15 45 E. 
 
 
 Is. Polfapate 
 
 Afia 
 
 Cambaya 
 
 Indian Ocean 
 
 9 45 N; 
 
 109 55 E. 
 
 
 I. Poma 
 
 Eu. 
 
 Dalmatia 
 
 Adriat. Sea 
 
 42 57 N. 
 
 18 14 E. 
 
 
 Pondichcrry 
 Pontorfon 
 
 Afia 
 
 India 
 
 B. Bengal 
 
 n 42 N. 
 
 79 58 E. 
 
 
 Eu. 
 
 France 
 
 Eng. Channel 
 
 48 33 N. 
 
 I 27 W, 
 
 
 Ponoi 
 
 Eu. 
 
 Lapland 
 
 North Sea 
 
 67 s N. 
 
 38 48 E. 
 
 
 I. Ponza 
 
 Eu. 
 
 Italy 
 
 Medit. Sea 
 
 40 53 N. 
 
 13 09 E. 
 
 
 Pool 
 
 Eu. 
 
 England 
 
 Eng. Channel 
 
 51 00 N. 
 
 I 50W. 
 
 
 ! Porto Port 
 
 Eu. 
 
 Portugal 
 
 Atl. Ocean 
 
 41 10 N. 
 
 8 22 W. 
 
 
 Port Mahon 
 
 Eu. 
 
 Spain 
 
 Medit. Sea 
 
 39 51 N. 
 
 3 53 E- 
 
 
 i Portland 
 
 Eu. 
 
 England 
 
 Eng. Channel 
 
 50 30 N, 
 
 2 48 W. 
 
 8 15 
 
 Port rOrlent 
 
 Eu. 
 
 France 
 
 B. Bifcay 
 
 47 47 N. 
 
 3 nw. 
 
 
 i Porto Bello 
 
 Am. 
 
 New Spain 
 
 Carrib. Sea 
 
 9 33 N. 
 
 79 45W. 
 
 
 ! Porto Praya 
 
 Africa 
 
 Cape Verd 
 
 Atl. Ocean 
 
 14 54 N. 
 
 23 24 W. 
 
 II 00 
 
 ; 2 o 7 E. point 
 
 
 
 
 i8 35 N. 
 
 65 58 w. 
 
 
 ji^ .ri J- Port Rico 
 1^"=^ iW. point 
 
 Am. 
 
 Antilles 
 
 .A.t!. Ocean 
 
 18 29 N. 
 
 66 35W. 
 
 
 
 
 
 iS 34 N. 
 
 67 siW. 
 
 
 il. Porto Sanfto 
 
 Africa 
 
 Cunaries 
 
 Atl. Ocean 
 
 32 58 N. 
 
 16 20W. 
 
 
 Portfail 
 
 Eu. 
 
 France 
 
 Eng. Channel 
 
 48 36 N. 
 
 4 43 W. 
 
 
 Portfmouth.R.Aca. 
 
 Eu. 
 
 England 
 
 Eng. Channel 
 
 50 48 N. 
 
 I 01 W. 
 
 n 15 
 
 Prakcn 
 
 Afia 
 
 Coth. Chi, 
 
 Indian Ocean 
 
 17 15 N. 
 
 106 15 E. 
 
 iPrcnau 
 
 Eu. 
 
 Livoni* 
 
 Baltic Sea 
 
 58 26 N. 
 
 24 58 E. 
 
 
 1. Princes 
 
 Africa 
 
 Guinea 
 
 Atl. Ocean 
 
 1 47 N. 
 
 6 39 E. 
 
 
 C. Prior 
 
 Eu. 
 
 Spain 
 
 Atl. Ocean 
 
 43 29 N. 
 
 8 15W. 
 
 
 I. Providence 
 
 Am. 
 
 Bahama 
 
 Atl. Ocean 
 
 24 5' N. 
 
 77 01 W. 
 
 
 .1. Providence, or 1 
 ! St. Catheiine J 
 
 Am. 
 
 Mexico 
 
 Atl. Ocean 
 
 13 26 N. 
 
 So 42 W. 
 
 
 'Pudyoui 
 
 Afia 
 
 N.Caledonia 
 
 Pacif. Ocean 
 
 20 18 S. 
 
 164 46 E. 
 
 6 00 
 
 ,1. Pulo condor 
 
 .(|uaqua, or Ivory ' 
 Coaft 5 
 
 Afii 
 
 Cambaya 
 
 Indian Ocean 
 
 8 40 N. 
 
 107 25 E. 
 
 
 Africa 
 
 Guinea 
 
 Etii. Sea 
 
 5 00 N. 
 
 4 00 W. 
 
 
 'Quebec 
 
 Am. 
 
 Canada 
 
 R. Sr. Lawr. 
 
 46 55 N. 
 
 69 48 W. 
 
 7 30 
 
 ((jueda 
 
 Afia 
 
 Malaya 
 
 B. Bengal 
 
 6 15 N. 
 
 100 12 E. 
 
 
 I. Quelpcrt 
 
 Afu 
 
 Korea 
 
 Pacif. Ocean 
 
 33 3* N. 
 
 128 04 E. 
 
 
 'QuUoa 
 
 Africa 
 
 1 
 
 Zangucbar 
 
 Indian Ocean 
 
 9 30 S. 
 
 39 09 E. 
 
 
 
 
 
 
 
 
 
 Vol. I. 
 
 F f 
 
 Quimpw
 
 394 
 
 .GEOGRAPHY. 
 
 Book VI. 
 
 Names of Tlaccs. 
 
 Cont. 
 
 Countries. 
 
 Coaft. 
 
 Latitude. 
 
 Longitude. 
 
 H.WaterJ 
 
 i 
 
 Quimpcr 
 
 Eu. 
 
 France 
 
 B. Bifcay 
 
 47 58 N. 
 
 4 02 W. 
 
 
 Qiiinam 
 
 Afia 
 
 Cbth. Clii. 
 
 Indian Ocean 
 
 12 52 N. 
 
 109 10 E. 
 
 
 Quiraba Ifici 
 
 Africa 
 
 Zanguebar 
 
 IndianOcean 
 
 11 00 N. 
 
 41 39 E. 
 
 
 , C. Quiros 
 
 Afia 
 
 N. MuMdes 
 
 Pacif. Ocean 
 
 14 56 S. 
 
 167 15W. 
 
 
 Quito 
 
 R 
 C. Race 
 
 Am. 
 
 Peru 
 
 Inland 
 
 13 S. 
 
 77 50W. 
 
 * 
 
 Am. 
 
 Ncwfoundl. 
 
 Atl. Ocean 
 
 46 40 N. 
 
 52 38 W. 
 
 
 Ragufa 
 
 Eu. 
 
 Dalmatia 
 
 Medit. Sea 
 
 42 45 N. 
 
 20 00 E. 
 
 
 Raj;vpour 
 
 Afia 
 
 India 
 
 Indian Ocean 
 
 17 19 N, 
 
 73 50 E- 
 
 
 Ramfgatc 
 
 Eu. 
 
 England 
 
 Downs 
 
 51 20 N. 
 
 1 22 E^ 
 
 
 Ramhead 
 
 Eu. 
 
 Englaitd 
 
 Eng. Channel 
 
 50 19 N. 
 
 4 i5^V. 
 
 
 C. Rafalgatc 
 
 Afia 
 
 Arabia 
 
 Indian Ocean 
 
 22 46 N. 
 
 58 48 E. 
 
 
 Ravenna 
 
 Eu. 
 
 Italy 
 
 Mcdit. Sea 
 
 44 26 N. 
 
 12 21 E. 
 
 
 C.Ray 
 
 Am. 
 
 Nevvfoundl. 
 
 Ati. Ocean 
 
 47 37 N. 
 
 59 8W. 
 
 
 I. Rhee 
 
 Eu. 
 
 France 
 
 B. Bifcay 
 
 46 15 N- 
 
 1 28 W. 
 
 3tr.oom. 
 
 Regio 
 
 Eu. 
 
 Italy 
 
 Meditcrran. 
 
 38 22 N. 
 
 16 37 E. 
 
 
 Cape Refolutlon 
 
 Am, 
 
 N. Main 
 
 iludfon's Str. 
 
 6i 29 N. 
 
 65 iqW. 
 
 
 Refolution Bay 
 
 Alia 
 
 Marqucfas 
 
 Pacif. Ocean 
 
 9 55 S. 
 
 139 4W. 
 
 
 Resolution I/land 
 
 Afia 
 
 Society I lies 
 
 Pacif. Ocean 
 
 17 23 S. 
 
 141 40W. 
 
 
 Revel 
 
 Eu. 
 
 Livonia 
 
 Baltic Sea 
 
 59 22 N. 
 
 25 33 E. 
 
 
 .. r Rhodes, N. 
 -a V 3 end 
 i S S C. Tranqv.il, 
 e^ L S. end 
 
 Afia 
 
 Natolia 
 
 Archipelago 
 
 36 27 N. 
 
 35 55 N. 
 
 28 36 E. 
 
 28 23 Y^ 
 
 
 Riga 
 
 Eu. 
 
 Livonra 
 
 Baltic Sea 
 
 56 55 N. 
 
 24 51 E. 
 
 
 Ri(>raps, a fand 
 
 P.u. 
 
 England 
 
 Straits Dover 
 
 51 53 N. 
 
 I 25 E. 
 
 
 Robin Hood's Bay 
 
 Lu. 
 
 England 
 
 Germ. Ocean 
 
 54 ~S N. 
 
 08 W. 
 
 3 00. 
 
 I. Rocca 
 
 Am. 
 
 Terra Firma 
 
 Atl. Ocean 
 
 II 21 N. 
 
 66 17W. 
 
 
 jRochcfort 
 
 Eu. 
 
 Francs 
 
 B. Bifcay 
 
 46 03 N. 
 
 c 54 W. 
 
 4 15 
 
 ;Rc(hel 
 
 Eu. 
 
 France 
 
 Bay Bifcay 
 
 46 10 N. 
 
 I 5W. 
 
 3 45 
 
 Rocheftcr 
 
 Eu. 
 
 Engl.ind 
 
 R. Medway 
 
 SI 26 N, 
 
 30 E. 
 
 45^ 
 
 I. Rndiiguc 
 
 Afia 
 
 Madagafcar 
 
 Indian Ocean 
 
 19 41 S. 
 
 62 45 E< 
 
 
 C. Romaiii 
 
 Am. 
 
 Terra Firma 
 
 Atl. Ocean 
 
 II 40 N. 
 
 69 05W. 
 
 
 ;Romc 
 
 Eu. 
 
 Italy 
 
 Medit. Sea 
 
 41 54 N. 
 
 12 34 E. 
 
 
 ;I. Roncadorc 
 
 Am. 
 
 Mexico 
 
 Atl. Ocean 
 
 13 30 N. 
 
 78 53W. 
 
 
 JRood Bay 
 
 Eu. 
 
 Greenland 
 
 North Ocean 
 
 79 53 N. 
 
 14 CO E. 
 
 
 IC. Roque 
 
 Am. 
 
 Brafil 
 
 Atl. Ocean 
 
 5 00 S. 
 
 35 43 W. 
 
 
 jl. Roqueniz 
 
 Africa 
 
 Madagafcar 
 
 Indian Ocean 
 
 9 51 S. 
 
 64 30 E. 
 
 
 iG. Rofes 
 
 Eu. 
 
 Spain 
 
 Mcdit. Sea 
 
 42 10 N. 
 
 3 18 E. 
 
 
 Ro.lock 
 
 Eu. 
 
 Germany 
 
 P;utic Sea 
 
 54 10 N. 
 
 12 50 E. 
 
 
 'I. Rotterdam 
 
 Afia 
 
 Fi'icndly Is. 
 
 Pacif. Ocean 
 
 20 16 S. 
 
 174 25 W- 
 
 
 ^Rotterdam 
 
 Eu. 
 
 D. Neth. 
 
 Germ. Ocean 
 
 SI 56 N, 
 
 4 33 E- 
 
 3 00 
 
 Rouen 
 
 Ell. 
 
 France 
 
 R. Seine 
 
 49 27 N. 
 
 I 10 E. 
 
 I 15 
 
 !jC. Roxant; 
 
 Fu. 
 
 Portugal 
 
 Atl. Ocean 
 
 38 45 N. 
 
 9 30 W. 
 
 
 C. Roxo 
 
 Africa 
 
 N'egroland 
 
 Atl. Ocean 
 
 II 42 N. 
 
 14 33W. 
 
 
 Po, Roval 
 
 Am. 
 
 I. Jamaica 
 
 Caribbean Sen 
 
 17 40 N. 
 
 76 37W. 
 
 
 !C. Rcziei- 
 
 Am. 
 
 Novii Scotia 
 
 G. St. Lav,-. 
 
 48 S5 N. 
 
 63 36 W. 
 
 
 1. R-ig^n 
 
 ta. 
 
 Germany 
 
 Baltic .Sea 
 
 54 32 N- 
 
 14 30 E. 
 
 
 'I. Rum Key, or 7 
 Saniana 5 
 
 Am. 
 
 Bahama 
 
 Atl. Ocean 
 
 23 00 N. 
 
 74 20 W. 
 
 
 JR. Rupert 
 
 Am. 
 
 New Britain 
 
 Hudfon's Bay 
 
 51 45 ^' 
 
 78 40 w. 
 
 
 'C. Ruii'to 
 
 Africa 
 
 Barca 
 
 Medit, Sea 
 
 32 53 N. 
 
 20 41 E.. 
 
 
 'Ruft Ii1(.s 
 
 Eu. 
 
 Norw^ty 
 
 North Sea 
 
 67 40 N. 
 
 JO 25 E. 
 
 
 ,Ry= 
 
 1 s 
 
 C. Sable 
 
 u. 
 
 England 
 
 Eng. Channel 
 
 51 03 N. 
 
 45 E. 
 
 " '5 
 
 Am. 
 
 .Vnva Scot'a 
 
 Atl, Ocean 
 
 41 24 N. 
 
 65 35W. 
 
 
 il. .Sjb!e, W. end 
 
 Am. 
 
 Nova Scotia 
 
 Atl. Ocean 
 
 U 9 ^z 
 
 60 29 W. 
 
 1 
 
 ,1. SacdJ? b:)clt 
 
 Am. 
 
 North Main 
 
 Hu if. Strait. 
 
 62 07 N. 
 
 68 13W. 
 
 10 00 
 
 !.Saffi;t 
 
 Africa 
 
 Barbiry 
 
 Atl. Ocean 
 
 :i: 70 N. 
 
 8 50 W. 
 
 
 11. Sarr.'al b'u:'. 
 
 AflicH 
 
 Egypt 
 
 Red Sea 
 
 27 0^ N. 
 
 34 40 E. 
 
 
 IB. Saldanna 
 
 AtriLa 
 
 Carters 
 
 Atl. Ocean 
 
 32 35 S- 
 
 19 30 E. 
 
 ' J^ 
 
 LSaI
 
 Book vr. 
 
 GEOGRAPHY. 
 
 3H 
 
 Names of Places. 
 
 Cent. 
 
 Countries. 
 
 Coaft. 
 
 Latitude. 
 
 Longitude. 
 
 H. Water. 
 
 
 I. Sal 
 
 Africa 
 
 C. Vcrd 
 
 Atl. Ocean 
 
 i6 3'8 N. 
 
 22 s'lW. 
 
 
 
 Salerno 
 
 u. 
 
 Italy 
 
 Mcdit, Sea 
 
 40 39 N. 
 
 14 48 E. 
 
 
 
 I. Salini, Liparl 111. 
 
 Eu, 
 
 Italy 
 
 Medit. Sea 
 
 3S 39 N. 
 
 15 24 E. 
 
 
 
 I. SalUbiii-y 
 
 Am. 
 
 N Main 
 
 Hudfon's Bay 
 
 63 29 N. 
 
 76 47 W. 
 
 
 
 Sallee 
 
 Africa 
 
 Barbaiy 
 
 Atl. Ocean 
 
 33 58 N. 
 
 6 20 W. 
 
 
 
 Solomon Illes 
 
 Afia 
 
 
 Pacif. Ocean 
 
 C 5 so S. 
 
 Ill 15 S. 
 
 171 05 W, 
 178 35W. 
 
 
 
 
 
 Salonechi 
 
 Eu, 
 
 Turkey 
 
 Archipelago 
 
 40 41 N. 
 
 23 13 E. 
 
 
 
 I. Salvages 
 
 Africa 
 
 Canaries 
 
 N.AU. Ocean 
 
 3 00 N. 
 
 15 49 w. 
 
 
 
 ""'"^'{Si:;} 
 
 Am. 
 
 North Main 
 
 Hudf. Straits 
 
 561 48 N. 
 I62 32|N. 
 
 66 20 W. 
 70 48|W 
 
 gh. oom. 
 II 10 
 
 
 I. Samos 
 
 Afia 
 
 Natolia 
 
 Archipelago 
 
 37 46 N. 
 
 27 13 E. 
 
 
 
 C. Sambrough 
 
 Am. 
 
 Nova Scotia 
 
 Weftcrn Oc. 
 
 44 33 N. 
 
 63 20 W. 
 
 
 
 Sandwich 
 
 Eu. 
 
 England 
 
 Downs 
 
 51 20 N. 
 
 I 20 E. II 30 1 
 
 
 Sandwich Ifland 
 
 Afia 
 
 N. Hebrides 
 
 Pacif. Ocean 
 
 17 41 S. 
 
 i65 38 E. 
 
 
 
 Sandwich Harbour 
 
 Afia 
 
 Malicola 
 
 Pacif. Ocean 
 
 16 25 S. 
 
 167 58 E. 
 
 
 
 Sandwich's Bay 
 
 Am. 
 
 St. Georgia 
 
 Atl. Ocean 
 
 54 42 S. 
 
 36 4W. 
 
 
 
 I. Sanguin 
 
 Afia 
 
 Philip, Illes 
 
 Pacif. Ocean 
 
 3 5^ N. 
 
 122 30 E. 
 
 
 
 I. Sanlen 
 
 Eu. 
 
 Norway 
 
 North Ocean 
 
 69 3c N, 
 
 14 30 E. 
 
 
 
 Santa Cruz 
 
 Africa 
 
 Barbary 
 
 Atl. O.ccaii 
 
 30 30 N. 
 
 9 35W. 
 
 
 
 2 r N. limit 
 
 
 
 
 41 15 N. 
 
 9 31 E- 
 
 
 
 .'S S. pt. C. Tavo- 
 ^ < laro 
 
 Eu. 
 
 Italy 
 
 Medit. Sea 
 
 38 54 N. 
 
 9 '5 E- 
 
 
 
 '^ Cagliari 
 
 
 
 
 39 25 N. 
 
 938E. 
 
 
 
 -< l_Oriftagni 
 
 
 
 
 /. 39 53 N. 
 
 9 01 E. 
 
 
 
 Sarcna 
 
 Am. 
 
 Chili 
 
 Pacif. Ocean 
 
 29 40 S. 
 
 71 15W. 
 
 
 
 Saunders's Ifle 
 
 Am. 
 
 Sandwich L. 
 
 Atl. Ocean 
 
 58 00 S. 
 
 26 53 w. 
 
 
 
 C. Saunders 
 
 Am. 
 
 St. Georgia 
 
 Atl. Ocean 
 
 54 6 S. 
 
 36 53 W. 
 
 
 
 Scanderoon 
 
 Afia 
 
 Syria 
 
 Levant 
 
 36 35 N. 
 
 36 25 E. 
 
 
 
 Scarborough head 
 
 Eu. 
 
 England 
 
 Germ. Ocean 
 
 54 18 N. 
 
 00 00 
 
 3 45 
 
 
 I. Scarpanto 
 
 Afia 
 
 Natolia 
 
 Arcliipelago 
 
 35 45 N. 
 
 27 40 E. 
 
 
 
 1. Scatarie, N. E. pt. 
 
 Am. 
 
 Acadia 
 
 Wert. Ocean 
 
 46 01 N. 
 
 61 57W. 
 
 
 
 Scaw 
 
 Eu. 
 
 Denmark 
 
 Sound 
 
 57 34 N- 
 
 10 54 E. 
 
 
 
 I Schclling 
 
 Eu. 
 
 D. Neth. 
 
 Germ. Ocean 
 
 53 27 N. 
 
 5*30 E. 
 
 
 
 .2 r C. St. Nicholas 
 
 
 
 
 r 38 is N. 
 
 26 12 E. 
 
 
 
 ^ -i Scio 
 
 M4 
 
 Natolia 
 
 Archipelago 
 
 i 38 24 N. 
 
 26 29 E. 
 
 
 
 >^ {_C. Blanco 
 
 
 
 
 I 3'S 08 N. 
 
 26 20 E. 
 
 
 
 Scilly I/lcs 
 
 Eu, 
 
 Engl.'.nd 
 
 St. Geo. Ch. 
 
 50 00 N. 
 
 6 45 W. 
 
 3 45 
 
 
 Scolt Head 
 
 Eu. 
 
 England 
 
 Germ. Ocean 
 
 53 00 N. 
 
 44 E. 
 
 6 20 
 
 
 Scats Sctilemcnt 
 
 Am. 
 
 Terra Firma 
 
 Carribbc. Sea 
 
 8 45 N. 
 
 76 3sW. 
 
 
 
 1. Sea 
 
 Eu. 
 
 Turkey 
 
 Archipelago 
 
 37 3 N. 
 
 24 53 E. 
 
 
 
 ScanicT 
 
 Eu. 
 
 France 
 
 B. Bifcay 
 
 48 00 N. 
 
 4 51 W. 
 
 
 
 I. ScbaWes 
 
 Am. 
 
 Patagonia 
 
 S. Atl. Ocean 
 
 50 53 S. 
 
 59 35 W. 
 
 
 
 C. Scbajtian 
 
 Am. 
 
 California 
 
 Pacif. Ocean 
 
 43 00 N. 
 
 126 00 W. 
 
 
 
 C. St. ScbafKan 
 
 Africa 
 
 Mad.igafcar 
 
 Indian Ocean 
 
 12 30 S. 
 
 46 30 E. 
 
 
 
 St. Stbarti an 
 
 Eu. 
 
 Spain 
 
 B. Bifcay 
 
 43 16 N, 
 
 2 05 W. 
 
 
 
 i'ort Scgura 
 
 Am. 
 
 Brafil 
 
 Atl. Ocean 
 
 16 57 S. 
 
 39 45 W. 
 
 
 
 R. Senegal 
 
 Africa 
 
 Negroland 
 
 Atl. Ocean 
 
 15 53 N. 
 
 16 26W. 
 
 10 30 
 
 
 I, Scrnnilha 
 
 Am. 
 
 Weft Indies 
 
 Atl. Ocean 
 
 16 20 N. 
 
 79 40 w. 
 
 
 
 I. Serigo 
 
 Eu. 
 
 Turkey 
 
 Artliipcla'jo 
 
 36 09 N. 
 
 23 34 E. 
 
 
 
 I. St-rtcs 
 
 Africa 
 
 Canarlei 
 
 Atl. Ocean 
 
 32 35 N. 
 
 16 20 W, 
 
 
 
 R. .Scftos 
 
 Africa 
 
 Giilnca 
 
 Atl. Ocean 
 
 5 4 N. 
 
 8 13W. 
 
 
 
 Seven Capc; 
 
 Africa 
 
 Barbaiy 
 
 McJii. Sea 
 
 37 30 J^. 
 
 6 15W. 
 
 
 
 Seven Stones, or Iflcs 
 
 Eu. 
 
 England 
 
 St. Geo. Ch. 
 
 50 ID N. 
 
 6 40 W. 
 
 4. 30 
 
 
 R. Severn, Ent. 
 
 Eu. 
 
 England 
 
 St. Geo. Ch, 
 
 51 4' N. 
 
 3 osW. 
 
 6 
 
 
 R. Severn 
 
 Am. 
 
 New Wales 
 
 Hudfon's Bay 
 
 56 12 N. 
 
 88 57 W. 
 
 
 
 R. Scyn, Ent. 
 
 Eu. 
 
 France 
 
 Eng.Cliannel 
 
 49 36 N. 
 
 30 5. 
 
 9 00 
 
 
 SeynhcaJ 
 
 u. 
 
 France 
 
 Eng. Channel 
 
 49 44 N. 
 
 34 E, 
 
 
 
 Shecrnel", 
 
 Eu. 
 
 England 
 
 K. Tliamc 
 
 51 25 N. 
 
 50 F..1 Q CO 
 
 
 ShcphcrJ'b Illes 
 
 Afia 
 
 N. Hebrides 
 
 Pacif. 0>.ean 
 
 17 CO S. 
 
 16S 47 E. 
 
 
 liam 
 
 Afia 
 
 Indln 
 
 Bay Slum 
 
 14 iS N. 
 
 1 100 Si E. 
 
 
 K.ISiarri Enr. 
 
 Afia 
 
 India 
 
 Bay Slum 
 
 13 IS N. 
 
 1 100 4- H. ! 
 r mi.mt- IT a. if , a- 
 
 Ff 2 
 
 s;c!4
 
 39^ 
 
 GEOGRAPHY. 
 
 Book VI. 
 
 
 Names of Places. 
 
 Cont. 
 
 Countries. 
 
 Ceaft. 
 
 Latitude. 
 
 Longitude. H. Water. 
 
 
 
 Siara 
 
 f E. end, Mcflina 
 
 1 Catanca 
 >:> 1 Syracufe 
 
 ^m. 1 
 
 3rafil 
 
 \tl. Ocean 
 
 / 
 3 18 S. 
 
 "38 lo N. 
 37 42 N. 
 37 04 N. 
 
 39 50 W. 
 
 15 58 
 
 15 21 E. 
 15 31 E. 
 
 
 
 
 ;5 J S. end, C, Paf- i 
 .> fara 
 
 Eu. 
 
 taly 
 
 Mcdit. Sea 
 
 36 35 N. 
 
 37 M N 
 
 15 22 E. 
 
 
 
 
 c: 
 
 Alicata 
 
 
 
 
 14 07 E. 
 
 
 
 
 
 W. end, C. Bocco 
 
 
 
 
 37 5' N. 
 
 21 43 E. 
 
 
 
 
 
 Palermo 
 
 
 
 
 ^38 10 N. 
 
 13 43 E. 
 
 
 
 
 Sierra Leona 
 
 Africa 
 
 Guinea 
 
 Atl. Ocean 
 
 8 30 N. 
 
 12 07W. 
 
 8h. 15m. 
 
 
 
 Sillabar Road 
 
 Afia 
 
 t. Sumatra 
 
 [niian Ocean 
 
 ' 4 cx) S. 
 
 102 50 E. 
 
 
 
 
 Str. Sincapore 
 
 Afia 
 
 Malacca 
 
 Indian Ocean 
 
 1 00 N. 
 
 104 30 E. 
 
 
 
 
 R. Sindn, or Indus, > 
 mouth 3 
 
 Afia 
 
 India 
 
 Indian Ocean 
 
 I 24 30 N. 
 i25 45 N, 
 
 63 10 E. 
 62 40 E. 
 
 
 
 
 Po. Shabak 
 
 Africa 
 
 Abyflinia 
 
 Red Sea 
 
 18 5S N. 
 
 38 24 E. 
 
 
 
 
 Shark, or Sahorfe ? 
 
 point * ' ] 
 
 Am. 
 
 New Wales 
 
 Hudfon's Bay 
 
 64 05 N. 
 
 82 12W. 
 
 V 
 
 
 
 
 Shields 
 
 Eu. 
 
 England 
 
 Germ. Ocean 
 
 55 oz N. 
 
 I 20 W, 
 
 
 
 
 Shelvock's Ifle 
 
 Am. 
 
 California 
 
 Pacif. Ocean 
 
 23 15 N. 
 
 117 35W. 
 
 
 
 
 Shiilocks 
 
 Eu. 
 
 Ireland 
 
 Weft. Ocean 
 
 51 30 N. 
 
 11 05 w. 
 
 5 
 
 
 
 1. Shetland, limits 
 
 Eu. 
 
 Scotland 
 
 Weft. Ocean 
 
 '[60 47 N. 
 \[ 59 54 N. 
 
 loW. 
 J 31 W, 
 
 3 00 
 
 
 
 Shoreham 
 
 Eu. 
 
 England 
 
 Eng. Channel 
 
 50 55 N. 
 
 00 17 E. 
 
 10 30 
 
 
 
 I. Sky {^-r?'"' 
 
 ' I S. pomt 
 
 Eu. 
 
 Scotland 
 
 Weft. Ocean 
 
 :> 57 50 N. 
 
 1:57 15 N. 
 
 6 30 W. 
 6 16W. 
 
 5 30 
 
 
 
 Sleepers Ifles T 
 
 Am. 
 
 New Britain 
 
 Hudfon's Bay 
 
 f 60 00 N. 
 < 58 35 N. 
 
 ?8i 30 W. 
 
 
 
 
 Great Sleeper 5 
 
 
 
 
 (.60 10 N. 
 
 82 00 W. 
 
 
 
 
 The Sleepers lie in 
 
 
 
 
 
 
 
 
 
 a chain from the 
 
 
 
 
 
 
 
 
 
 Great Sleeper down 
 
 
 
 
 
 
 
 
 
 to Lat. 58 50' N 
 
 
 
 
 
 
 
 
 
 & Long. 82 20' W. 
 
 
 
 
 
 
 
 
 
 Sline Head 
 
 Eu. 
 
 Ireland 
 
 Weft. Ocean 
 
 53 20 N. 
 
 2 iSW. 
 
 
 
 
 R. Slude 
 
 Am. 
 
 New Britain 
 
 Hudfon's Bay 
 
 53 H N. 
 
 78 50 w. 
 
 
 
 
 Sluyce 
 
 Eu. 
 
 D. Neth. 
 
 Germ. Ocean 
 
 51 19 N. 
 
 3 50 E- 
 
 
 
 
 C. Smith 
 
 Am. 
 
 Labradore 
 
 Hudfon's Bay 
 
 60 48 N, 
 
 80 55W, 
 
 
 
 
 Smyrna 
 
 Afia 
 
 Natolia 
 
 Archipelago 
 
 38 28 N. 
 
 27 25 E. 
 
 
 
 
 i. Socatora 
 
 Africa 
 
 Anian 
 
 Indian Ocean 
 
 12 15 N. 
 
 52 55 E. 
 
 
 
 
 C. Solomoji 
 
 Eu. 
 
 I. Candia 
 
 Medit. Sea 
 
 34 57 N. 
 
 27 06 E. 
 
 
 
 
 R. Sommc 
 
 Eu. 
 
 France 
 
 Eng, Channel 
 
 50 18 N. 
 
 1 40 E. 
 
 n 00 
 
 
 
 Sound Royal 
 
 Eu. 
 
 Iceland 
 
 North Ocean 
 
 66 22 N, 
 
 15 I5W. 
 
 , 
 
 
 
 Southampson 
 |C. Southampton 
 
 Eu. 
 
 England 
 
 Eng. Channel 
 
 50 55 N. 
 
 I 00 W. 
 
 00 
 
 
 
 Am. 
 
 New Wales 
 
 Hudfon's Bay 
 
 61 54 N. 
 
 86 14W. 
 
 
 
 
 South Cape 
 
 Afia 
 
 IDiemen's la. 
 
 Pacif, Ocean 
 
 42 40 s. 
 
 130 05 E. 
 
 
 
 
 'C. Spartivento 
 
 Ku. 
 
 iltaly 
 
 Medit. Sea . 
 
 37 50 N, 
 
 16 41 E. 
 
 
 
 
 ,C. Spartel 
 
 Africa Barbary 
 
 Atl. Ocean 
 
 35 46 N. 
 
 5 5SW. 
 
 
 
 
 :I. Spirito SanSo 
 
 Am. 
 
 iBrafil 
 
 Atl. Ocean 
 
 20 24 S. 
 
 39 55 W. 
 
 
 
 
 Spurn 
 
 Eu. 
 
 England 
 
 Germ. Ocean 
 
 53 35 N 
 
 30 E. 
 
 5 '5 
 
 
 
 i. Stnir.pp.lli 
 
 Afia 
 
 Natolia 
 
 Archipelago 
 
 36 25 N. 
 
 26 55 E. 
 
 
 
 
 I. S^ancho 
 
 Afia 
 
 Natolia 
 
 Archipelago 
 
 36 50 N, 
 
 27 30 E. 
 
 
 
 
 IStart point 
 
 Eu. 
 
 England 
 
 Eng, Channel 
 
 50 09 N. 
 
 346W. 
 
 6 45 
 
 
 
 ; 2 r C. St. John 
 
 
 
 
 r 54 45 S. 
 I 55 08 S. 
 
 60 35W. 
 
 
 
 j|- ^ C. St. Bartho- 
 
 Am. 
 
 Patagonia 
 
 Atl. Ocean 
 
 60 45 W. 
 
 
 
 
 jStavenger 
 
 Eu. 
 
 Norway 
 
 Weft. Ocean 
 
 58 47 N. 
 
 6 4sE. 
 
 
 
 
 jC. Strphcn* 
 
 Afia 
 
 N. Zealand 
 
 Pacif. Ocean 
 
 40 36 S. 
 
 174 05 E. 
 
 
 
 
 Sietin 
 
 Eu. 
 
 Germany 
 
 Baltic Sea 
 
 53 36 N. 
 
 15 25 E. 
 
 
 
 
 C. Stiilo 
 
 Eu, 
 
 Italv 
 
 Medit. Sea 
 
 38 23 N. 
 
 17 07 E. 
 
 
 
 
 Port Steven 
 t 
 
 Am. 
 
 Chili 
 1 
 
 Pacif. Ocean 
 
 \ 
 
 46 50 S. 
 
 82 36W. 
 
 
 
 
 
 q 
 
 
 
 
 
 
 Stoekholt 
 
 B
 
 Book VI. 
 
 GEOGRAPHY. 
 
 397 
 
 
 Names of Places. 
 
 Cont. 
 
 Countries. 
 
 Coaft. 
 
 Latitude. 
 
 Longitude. 
 
 H.Watcr. 
 
 
 Stockholm 
 
 Eu. 
 
 Sweden 
 
 Baltic Sea 
 
 < 
 59 22 N. 
 
 18 12 E. 
 
 5h. 1511). 
 
 
 Stockton 
 
 Eu. 
 
 England 
 
 Germ. Ocean 
 
 54 33 N. 
 
 I isW. 
 
 
 Straelfund 
 
 Eu. 
 
 Germany . 
 
 Baltic Sea 
 
 54 23 N. 
 
 14 10 E. 
 
 
 
 Strangford Bay 
 
 Eu. 
 
 Ireland 
 
 IriA Sea 
 
 54 23 N. 
 
 5 40 W. 
 
 10 30 
 
 
 I. Stromboli 
 
 Eu. 
 
 Italy 
 
 Medit. Sea 
 
 38 42 N. 
 
 IS 48 E. 
 
 
 
 Succefs Bay 
 
 Am. 
 
 T. del Fuego 
 
 Atl. Ocean 
 
 54 50 S. 
 
 65 20 W. 
 
 
 
 Suez Town 
 
 Africa 
 
 Egypt 
 
 Red Sea 
 
 29'5o N. 
 
 33 17 E. 
 
 
 
 Sukadana 
 
 Afia 
 
 I. Borneo 
 
 Indian Ocean 
 
 I 00 s. 
 
 110 40 E. 
 
 
 
 I, Suma- C NW. end 
 tra I SE. end 
 
 Afia 
 
 India 
 
 Indian Ocean 
 
 J 5 IS N. 
 
 i 5 07 S. 
 
 95 55 E. 
 106 20 E. 
 
 
 
 Sunderland 
 
 Eu. 
 
 England 
 
 Germ. Ocean 
 
 54 55 N; 
 
 I 00 W. 
 
 3 30 
 
 
 Str. Sunda 
 
 Afia 
 
 Siam 
 
 Indiaji Ocean 
 
 6 10 S. 
 
 105 35 E. 
 
 
 
 Surinam 
 
 Am. 
 
 Terra Firma 
 
 Atl. Ocean 
 
 6 30 N. 
 
 55 30 W. 
 
 
 
 Surat 
 
 Afia 
 
 India 
 
 Indian Ocean 
 
 21 10 N. 
 
 72 25 E. 
 
 
 
 I. Surroy 
 
 Eu. 
 
 Lapland 
 
 North Ocean 
 
 71 00 N. 
 
 22 00 E. 
 
 
 
 Swaken 
 
 Africa 
 
 Abyflinia 
 
 Red Sea 
 
 19 30 N. 
 
 37 38 E. 
 
 
 
 Swally Road 
 
 Afia 
 
 India 
 
 Arabian Sea 
 
 21 55 N. 
 
 72 CO E. 
 
 
 
 Swanfey 
 
 Eu. 
 
 Wales 
 
 St. Geo. Cha. 
 
 51 40 N. 
 
 4 25 W. 
 
 
 
 Sweetnofc 
 
 Eu. 
 
 Lapland 
 
 North Ocean 
 
 68 08 N. 
 
 34 4 E. 
 
 
 
 Swin, a fand 
 
 Eu. 
 
 England 
 
 Ent. Thames 
 
 5' 37 N. 
 
 I 12 E. 
 
 12 00 
 
 
 Syracufe 
 
 Eu. 
 
 I. Sicily 
 
 Medit. Sea 
 
 37 04 N. 
 
 IS 31 E. 
 
 
 
 Syriam 
 
 T 
 Tadoufac Fort 
 
 Afia 
 
 Pegu 
 
 B. Bengal 
 
 16 00 N. 
 
 96 40 . 
 
 
 
 Am. 
 
 Canada 
 
 R. St. Lawr. 
 
 48 00 N. 
 
 67 35W. 
 
 
 
 I- Tamarica 
 
 Am. 
 
 Brafil 
 
 Atl, Ocean 
 
 7 56 S. 
 
 35 05W. 
 
 
 
 Tamarin Town 
 
 Africa 
 
 I. Socatora 
 
 Indian Ocean 
 
 12 30 N. 
 
 53 14 E. 
 
 9 00 
 
 
 B. Tananarin 
 
 Afia 
 
 Malacca 
 
 B. Bengal 
 
 12 00 N. 
 
 98 48 E. 
 
 
 
 I. Tandoxima 
 
 Afia 
 
 Japan 
 Barbary 
 
 Pacif. Ocean 
 
 30 30 N. 
 
 130 40 E. 
 
 
 
 Tangier 
 
 Africa 
 
 Atl. Ocean 
 
 35 55 N. 
 
 545W. 
 
 
 
 Tanna 
 
 Afia 
 
 N. Hebrides 
 
 Pacif. Ocean 
 
 19 32 s. 
 
 169 45 E. 
 
 3 op 
 
 
 Taoukaa 
 
 Afia 
 
 Society Ifles 
 
 Pacif. Ocean 
 
 14 31 s. 
 
 145 4W. 
 
 
 
 Tarento 
 
 Eu. 
 
 Italy 
 
 Medit. Sea 
 
 40 43 N. 
 
 17 31 E. 
 
 
 
 C. Tat'nam 
 
 Am. 
 
 New Wales 
 
 Hudfon's Bay 
 
 57 35 N. 
 
 91 30W. 
 
 
 
 R. Tees, mouth 
 
 Eu. 
 
 England 
 
 Germ. Ocean 
 
 54 36 N. 
 
 52 W. 
 
 3 00 
 
 
 Tegoantepec 
 
 Am. 
 
 Mexico 
 
 Pacif. Ocean 
 
 14 45 N. 
 
 96 23W. 
 
 
 
 Tellichery 
 
 Afia 
 
 India 
 
 Malabar Coaft 
 
 II 42 N. 
 
 75 30 E. 
 
 
 C. Telling 
 
 Eu. 
 
 Ireland 
 
 Weft. Ocean 
 
 54 40 N. 
 
 10 07 W. 
 
 
 
 I. Tenedos 
 
 Afia 
 
 Natolia 
 
 Archipelago 
 
 39 57 N. 
 
 26 14 E. 
 
 - 
 
 
 1. Teneriff (Peak) 
 
 Africa 
 
 Canaries 
 
 Atl. Ocean 
 
 28 13 N, 
 
 16 24 W. 
 
 3 00 
 
 
 C. Tenes 
 
 Africa 
 
 Barbary 
 
 Medit. Sea 
 
 36 26 N. 
 
 1 53 E. 
 
 
 
 I. Tercera 
 
 Eu. 
 
 Azores 
 
 Atl. Ocean 
 
 38 45 N. 
 
 27 01 W. 
 
 
 
 Terra Nieva 
 
 Am. 
 
 N Main 
 
 Hudf. Straits 
 
 62 4 N. 
 
 67 2W. 
 
 9 50 
 
 
 Tervere 
 
 Eu. 
 
 D. Neth. 
 
 Germ Ocean 
 
 SI 38 N. 
 
 3 35 E. 
 
 45 
 
 
 Tetuan 
 
 Africa 
 
 Barbary 
 
 Medit. Sea 
 
 35 27 N. 
 
 4 50 w. 
 
 
 
 I. Texel 
 
 Eu. 
 
 D. Neth. 
 
 Germ. Ocean 
 
 53 10 N. 
 
 4 59 E. 
 
 7 30 
 
 
 C, St, Thadau! 
 
 Afia 
 
 Siberia 
 
 North Ocean 
 
 62 10 N. 
 
 175 05 E. 
 
 
 
 R. Thames, mouth 
 
 Eu. 
 
 England 
 
 Germ. Ocean 
 
 51 28 N. 
 
 1 10 E. 
 
 1 30 
 
 
 C. St. Thomas 
 
 Africa 
 
 Catfcrs 
 
 Atl. Ocean 
 
 24 54 s. 
 
 15 25 E. 
 
 
 
 I. St. Thomas 
 
 Africa 
 
 Guinea 
 
 Atl. Ocean 
 
 00 00 
 
 I CO E. 
 
 
 
 St. Thomas 
 
 Afia 
 
 India 
 
 B. Bengal 
 
 13 00 N. 
 
 80 00 E. 
 
 
 
 C. Tlirec Points 
 
 Am. 
 
 Terra Firma 
 
 Atl. Ocean 
 
 10 51 N. 
 
 62 41 W. 
 
 
 
 C. Three Points 
 
 Africa 
 
 Guinea 
 
 Atl. Ocean 
 
 4 48 N., 
 
 I 2lW. 
 
 
 
 South Thule 
 
 Am. 
 
 Sandwich la. 
 
 Atl. Ocean 
 
 59 34 S. 
 
 27 45 W, 
 
 
 
 I. Tidore 
 
 Afia 
 
 Molucca Is. 
 
 Indian Ocean 
 
 35 N. 
 
 126 40 E. 
 
 
 
 , ^. CNE. pt. 
 I.T.mor Jsw.pt. 
 
 Afia 
 
 Molucca Is. 
 
 Indian Ocean 
 
 ': 8 20 s. 
 
 i 10 23 s. 
 
 127 40 E. 
 123 55 E. 
 
 
 
 Tinmouth 
 
 Eu. 
 
 England 
 
 Germ. Ocean 
 
 55 03 N- 
 
 1 17W. 
 
 3 00 
 
 
 I. Tino 
 
 Bu. 
 
 Turkey 
 
 Archipelago 
 
 37 33 N. 
 
 25 43 P.. 
 
 
 
 I. Tobago 
 
 Am. 
 
 Carribbce 
 
 Atl. Ocean 
 
 n 15 N. 
 
 60 27W. 
 
 
 
 ToboKki 
 
 Afia 
 
 Siberia 
 
 Inland 
 
 58 12 N. 
 
 68 20 E. 
 
 
 
 B. Todo? Sanfloj 
 
 , 
 
 Am. 
 
 Brafil 
 
 Atl, Ocean 
 
 13 05 S. 
 
 38 45 W. 
 
 
 Tontjuirj
 
 39 
 
 G E O G R A P H Y* 
 
 Book VL 
 
 < 1 
 
 =" 
 
 'ii.'. 
 
 * 
 
 
 
 
 =SKS: 
 
 
 
 Names of Places. 
 
 Cont. 
 
 Countries. 
 
 Coaft. 
 
 Latitude. 
 
 Longitude. 
 
 H.W?tr. 
 
 Tonquin 
 
 Afia 
 
 India 
 
 Pacif. Ocean 
 
 O ( 
 
 20 50 N. 
 
 
 
 los 
 
 55 E- 
 
 
 Tonfcerg 
 
 Eu. 
 
 Norway 
 
 Sound 
 
 58 50 N. 
 
 10 
 
 osE. 
 
 
 Topfljam 
 
 Eu. 
 
 England 
 
 Eng. Channel 
 
 50 37 N. 
 
 3 
 
 27 w. 
 
 6h.oom. 
 
 Torbay 
 
 Eu. 
 
 England 
 
 Eng. Chartnel 
 
 50 34 N. 
 
 3 
 
 36 w. 
 
 S 'j 
 
 Tome* 
 
 Eu. 
 
 Sweden 
 
 G. Bothnia 
 
 65 51 N- 
 
 24 
 
 16 E. 
 
 
 R. Tortof* 
 
 Eu. 
 
 Spain 
 
 Medit. Sea 
 
 40 47 N. 
 
 I 
 
 03 E. 
 
 
 I. Tortola 
 
 Am. 
 
 Antl. Ifle 
 
 Atl. Ocean 
 
 18 24 N. 
 
 65 
 
 00 W. 
 
 
 I. Tory 
 
 Eu. 
 
 Ireland 
 
 Weft. Ocean 
 
 55 09 N' 
 
 8 
 
 30 w. 
 
 5 30 
 
 Toulon 
 
 Eu. 
 
 France 
 
 Medit. Sea 
 
 43 07 N. 
 
 6 
 
 oz E. 
 
 
 C. Trefalga 
 
 Eu. 
 
 Spain 
 
 Atl. Ocean 
 
 36 08 N. 
 
 5 
 
 58 W. 
 
 
 I. Tremiti 
 
 Eu. 
 
 Italy 
 
 Medit. Sea 
 
 42 09 N. 
 
 15 
 
 40 E. 
 
 
 C. dc Tics forcas 
 
 Africa 
 
 Barbary 
 
 Medit. Sea 
 
 35 30 N- 
 
 2 
 
 iiW. 
 
 
 I. Trailles 
 
 Afia 
 
 India 
 
 Indian Ocean 
 
 19 30 S. 
 
 101 
 
 25 E. 
 
 
 I. Trinity 
 
 Am. 
 
 Brafil 
 
 AtUOcean 
 
 20 25 s. 
 
 23 
 
 35 W. 
 
 
 I. Trinidada, E. pt. 
 
 Am. 
 
 Terra Firma 
 
 Atl. Ocean 
 
 10 38 N. 
 
 60 
 
 27 W. 
 
 
 Trinity Bay, Ent. 
 
 Am. 
 
 Newfound). 
 
 Atl. Ocean 
 
 48 30 N. 
 
 52 
 
 35W. 
 
 
 Trieft 
 
 Eu. 
 
 Carniola 
 
 Adriat. Sea 
 
 45 51 N. 
 
 14 
 
 03 E. 
 
 IS 
 
 Trinquemali 
 
 Afia 
 
 I. Ceylon 
 
 Indian Ocean 
 
 8 so N. 
 
 83 
 
 24 E. 
 
 
 Tripoli 
 
 Afia 
 
 Syria 
 
 Levant 
 
 34 53 N. 
 
 36 
 
 07 E. 
 
 
 Tripoly 
 
 Africa 
 
 Barbary 
 
 Medit. Sea 
 
 32. 54 N. 
 
 13 
 
 10 E. 
 
 
 G.Trifte 
 
 Am. 
 
 Terra Firma 
 
 Atl. Ocean 
 
 10 19 N. 
 
 67 
 
 41 W. 
 
 
 I. Triftian d'Acunha 
 
 Africa 
 
 Cafters 
 
 S. Atl. Ocean 
 
 37 i S. 
 
 13 
 
 23 W. 
 
 
 If. Tromfound 
 
 Eu. 
 
 Lapland 
 
 North Ocean 
 
 70 20 N. 
 
 19 
 
 00 E. 
 
 
 Truxilla 
 
 Am. 
 
 Peru 
 
 Pacif. Ocean 
 
 8 00 S. 
 
 78 
 
 35W. 
 
 
 Tunder 
 
 Eu. 
 
 Denmark 
 
 Weft. Ocean 
 
 <;5 00 N. 
 
 9 
 
 35 E. 
 
 
 Tunis 
 
 Africa 
 
 Barbary 
 
 Medit. Sea 
 
 36 47 N. 
 
 10 
 
 16 E. 
 
 
 Turin 
 
 Eu. 
 
 Italy 
 
 R. Po 
 
 45 05 ^' 
 
 7 
 
 45 E. 
 
 
 I. Turks 
 
 Am. 
 
 Bahama 
 
 Atl. Ocean 
 
 21 18 N. 
 
 7> 
 
 05 W. 
 
 
 Turtle Illand 
 
 Afia 
 Eu. 
 
 
 Pacif. Occam 
 Medit. Sea 
 
 19 49 S. 
 
 39 30 N. 
 
 177 
 
 
 52 W. 
 
 40 W. 
 
 
 V 
 
 Valcncij 
 
 Spain 
 
 
 St. Valery 
 
 Eu. 
 
 Franc* 
 
 Eng. Channel 
 
 50 II N. 
 
 t 
 
 42 E. 
 
 10 30 
 
 Valoni 
 
 Eu. 
 
 Turkey 
 
 Medit. Sea 
 
 40 55 N. 
 
 21 
 
 15 E. 
 
 
 Valparifo 
 
 Am. 
 
 China 
 
 Pacif. Ocean 
 
 33 03 S. 
 
 J^ 
 
 14W. 
 
 
 Van Diemen's land 
 
 Afia 
 
 N. Holland 
 
 Indian Ocean 
 
 43 38 S. 
 
 146 
 
 27 E. 
 
 
 Vanncs 
 
 Eu. 
 
 France 
 
 B. Bifcay 
 
 47 39 N. 
 
 2 
 
 41 W. 
 
 3 45 
 
 C. Vela 
 
 Am. 
 
 Terra Firma 
 
 At). Ocean 
 
 12 15 N. 
 
 71 
 
 20 w. 
 
 
 P. Venus 
 
 Afia 
 
 Otahcite 
 
 Pacif. Ocean 
 
 17 29 S. 
 
 149 
 
 31 w. 
 
 10 38 
 
 Venice 
 
 Eu. 
 
 Italy 
 
 Medit. Sea 
 
 45 27 N. 
 
 13 
 
 9 .. 
 
 
 Vera Cruz 
 
 Am. 
 
 New Spain 
 
 G. Mexico 
 
 19 12 N. 
 
 97 
 
 25 W. 
 
 
 C. Verd 
 
 Africa 
 
 Negroland 
 
 Atl. Ocean 
 
 14 45 N. 
 
 J7 
 
 28W. 
 
 
 Vhma 
 
 Eu. 
 
 Sweden 
 
 G. Bothnia 
 
 63 45 N. 
 
 21 
 
 10 E. 
 
 
 Vicegapatam 
 
 Afia 
 
 India 
 
 B. Bengal 
 
 17 30 N. 
 
 H 
 
 02 E. 
 
 
 C. Viftory 
 
 Am. 
 
 Patagonia 
 
 Pacif. Ocean 
 
 52 IS S. 
 
 74 
 
 28 W. 
 
 
 Vienna 
 
 Ea. 
 
 Germany 
 
 R. Danube 
 
 48 11 N. 
 
 16 
 
 28 E. 
 
 
 Vigo- 
 
 Eu. 
 
 Spain 
 
 Atl. Ocean 
 
 42 14 N. 
 
 3 
 
 23 W. 
 
 
 B. St. Vincer.t 
 
 Am. 
 
 Paraguay 
 
 Atl. Ocean 
 
 23 55 S. 
 
 45 
 
 iiW. 
 
 
 C. St. Vincent 
 
 Eu. 
 
 Portugal 
 
 Atl. Ocean 
 
 37 01 N. 
 
 8 
 
 58 W. 
 
 
 I. St. Vincent 
 
 Africa 
 
 C. Verd 
 
 Atl. Ocean 
 
 17 47 N. 
 
 24 
 
 44W. 
 
 
 I. St. Vincent 
 
 Am. 
 
 Carribbee 
 
 Atl. Ocean 
 
 13 OS N. 
 
 6i 
 
 05 w. 
 
 
 R. St. Vincent 
 
 Africa 
 
 Guinea 
 
 Eth. Ocean 
 
 4 SO N. 
 
 n 
 
 41 w. 
 
 
 C. Virgins 
 
 Am. 
 
 Patagonia 
 
 Atl. Ocean 
 
 52 23 S. 
 
 67 
 
 50W. 
 
 
 I. Virgins 
 
 Am. 
 
 Antil. Ific 
 
 Atl. Ocean 
 
 18 18 N. 
 
 64 
 
 I4W. 
 
 
 Virgin Rocks 
 
 Am. 
 
 Newfound!. 
 
 Atl. Ocean 
 
 46 30 N. 
 
 51 
 
 30 w. 
 
 
 Umba 
 
 Eu. 
 
 Ruffia 
 
 Inland 
 
 65 40 N. 
 
 34 
 
 IS E. 
 
 
 C. Vo'.o 
 
 Eu. 
 
 Turkey 
 
 Archipelago 
 
 39 07 N. 
 
 23 
 
 23 E. 
 
 
 R. Voltas 
 
 Afiica 
 
 Guinea 
 
 Atl. Ocean 
 
 5 52 N. 
 
 I 
 
 10 E. 
 
 
 C. Voltas 
 
 Africa 
 
 Caffers 
 
 Atl. Ocean 
 
 28 04 S. 
 
 16 
 
 j8 E. 
 
 
 Upf^il 
 
 Eu. 
 
 Sweden 
 
 R. Sala 
 
 59 5" N. 
 
 17 
 
 47 E. 
 
 
 Lramlberg 
 
 Eu. 
 
 Denmark 
 
 Baltic Sea 
 
 55 54 N. 
 
 12 
 
 57 E. 
 
 
 i. Ufhant 
 
 Eu. . 
 
 France 
 
 Eng. Channel 
 
 48 30 N. 
 
 5 
 
 qW. 
 
 4 %*> 
 
 J. Vitlca
 
 Book VI. 
 
 <j E O G R A P H T. 
 
 399 
 
 1 
 
 
 
 
 
 -~~ 
 
 1 
 
 
 
 i -Names of Places. 
 
 Cont. 
 
 Countries. 
 
 Coaft. 
 
 Lat 
 
 tude. 
 
 Longitude. 
 
 H.W.iter. 
 
 I. Uftica 
 
 Eu. 
 
 Italy 
 
 Medit. Sea 
 
 o 
 38 
 
 43 
 
 N. 
 
 , 
 13 33 E. 
 
 I. Vulcano 
 
 W 
 Priii. Wales's Lies 
 
 Eu, 
 
 Italy 
 
 Medit. Sea 
 
 38 
 
 29 
 
 N. 
 
 15 33 E- 
 
 - 
 
 Afia 
 
 New Guinea 
 
 Endeavour St. 
 
 10 
 
 26 
 
 S. 
 
 141 00 E. 
 
 
 JR-. Wager 
 
 Am, 
 
 New Wales 
 
 Hudfon's Bay 
 
 65 
 
 28 
 
 N. 
 
 87 25W, 
 
 6h. 00m. 
 
 Wallii's Ifle 
 
 Afia 
 
 
 Pacif. Ocean 
 
 13 
 
 18 
 
 S. 
 
 176 20W. 
 
 
 'C. Wallinshara 
 
 Am. 
 
 New Britaia 
 
 Hudfon's Str. 
 
 62 
 
 39 
 
 N. 
 
 77 48 W, 
 
 12 00 
 
 iWardhus 
 
 Eu. 
 
 Lapland 
 
 North Ocean 
 
 70 
 
 23 
 
 N. 
 
 51 12 E. 
 
 
 jWarfaw 
 
 Eu. 
 
 Poland 
 
 R. Viftula 
 
 5^ 
 
 14 
 
 N. 
 
 21 5 E. 
 
 
 Watcrford 
 
 Eu. 
 
 Ireland 
 
 St. Geo. Ch. 
 
 52 
 
 07 
 
 N. 
 
 7 42 w.- 
 
 6 30 
 
 Watling Ifle 
 
 Am. 
 
 Bahama 
 
 Atl. Ocean 
 
 ^3 
 
 4^ 
 
 N. 
 
 74 22 w. 
 
 
 Wells 
 
 Eu. 
 
 England 
 
 Germ. Ocean 
 
 53 
 
 07 
 
 N. 
 
 1 00 E. 
 
 6 00 
 
 Weftcrn Ifles 
 
 Afia 
 
 Diemcn's la. 
 
 Pacif. Ocean 
 
 43 
 
 36 
 
 S. 
 
 147 00 E. 
 
 
 Welleni J S. point 
 Ifles I N. point 
 
 Eu, 
 
 Scotland 
 
 Wert. Ocean 
 
 55646 
 hs 35 
 
 N, 
 N. 
 
 7 40 w, 
 6 37 W. 
 
 
 If. Welimania 
 
 Eu. 
 
 Iceland 
 
 Weft, Ocean 
 
 63 
 
 55 
 
 N. 
 
 17 30W. 
 
 
 If. Weltroi 
 
 Eu. 
 
 Lapland 
 
 North Ocean 
 
 69 
 
 15 
 
 N. 
 
 24 00 E. 
 
 
 Wexford 
 
 Eu. 
 
 Ireland 
 
 St. Geo. Ch, 
 
 52 
 
 13 
 
 N, 
 
 6 56 W. 
 
 
 Weymouth 
 
 Eu. 
 
 England 
 
 Eng, Channel 
 
 52- 
 
 40 
 
 N. 
 
 2 34 w. 
 
 7 20 
 
 Whale's Back 
 
 Eu. 
 
 Iceland 
 
 Weft. Ocean 
 
 63 
 
 44 
 
 N 
 
 17 05 W. 
 
 
 IWhalc's Head 
 
 Eu. 
 
 Greenland 
 
 North Ocean 
 
 77 
 
 18 
 
 N, 
 
 21 30 E. 
 
 
 Whale Rock 
 
 Eu. 
 
 Azores 
 
 Atl. Ocean 
 
 38 
 
 50 
 
 N, 
 
 24 41 W. 
 
 
 Whitby 
 
 Eu. 
 
 England 
 
 Germ. Ocean 
 
 54 
 
 30 
 
 N. 
 
 50W. 
 
 J 00 
 
 Whitehaven 
 
 Eu. 
 
 Ensland 
 
 Irifli Sea 
 
 54 
 
 25 
 
 N. 
 
 3 15W, 
 
 
 iWhitfuntide I. 
 
 Afia 
 
 N.Hebrides 
 
 Pacif. Ocean 
 
 16 
 
 44 
 
 S. 
 
 168 25 E. 
 
 
 Wicklow 
 
 Eu. 
 
 Ireland 
 
 St. Geo. Cha. 
 
 52 
 
 50 
 
 N. 
 
 6 30 W. 
 
 
 Wilhs's Ifles 
 
 Am. 
 
 S. Georgia 
 
 Atl, Ocean 
 
 54 
 
 00 
 
 S. 
 
 38 25W. 
 
 
 Wind aw 
 
 Eu. 
 
 Courland 
 
 Baltic Sea 
 
 57 
 
 oS 
 
 N. 
 
 22 20 E. 
 
 
 1 t; fN. end ") 
 
 
 
 
 fSo 
 
 47 
 
 N, 
 
 I iiW. 
 
 
 P.S^j S, end / 
 ^ i E. end f 
 
 Eu. 
 
 England 
 
 Eng. Channel 
 
 J 50 
 
 / 5 
 
 34 
 41 
 
 N. 
 N, 
 
 1 loW. 
 I 00 w. 
 
 
 
 ULW.end J 
 
 
 
 
 1-50 
 
 41 
 
 N, 
 
 I 23W. 
 
 
 If. Prince William 
 
 Afia 
 
 
 Pacif. Ocean 
 
 16 
 
 45 
 
 S. 
 
 177 55 E. 
 
 
 IjWiJiam Henry I. 
 
 Afia 
 
 Society Ifles 
 
 Pacif, Ocean 
 
 >9 
 
 CO 
 
 s. 
 
 141' 6 E. 
 
 
 Winchelfea 
 
 Eu. 
 
 England 
 
 Eng. Channel 
 
 50 
 
 58 
 
 N. 
 
 50 E. 
 
 45 
 
 Wintertrtr.cfs 
 
 Eu. 
 
 England 
 
 Germ. Ocean 
 
 53 
 
 02 
 
 N. 
 
 I zz E. 
 
 9 00 
 
 |Wiibiiy in I. Gotland 
 
 Eu. 
 
 Sweden 
 
 Baltk Sea 
 
 57 
 
 40 
 
 N. 
 
 19 50 E. 
 
 
 jC. Wrath 
 
 Eu. 
 
 Scotland 
 
 Weft. Ocean 
 
 58 
 
 40 
 
 N. 
 
 4 50 E. 
 
 
 Wjhourg 
 
 y 
 
 Eu. 
 
 Finlana 
 
 C. Finland 
 
 60 
 
 55 
 
 N. 
 
 30 20 E. 
 
 
 Yamhoa 
 
 Afia 
 
 Arabia 
 
 Red Sea 
 
 24 
 
 25 
 
 N. 
 
 38 54 E. 
 
 
 ! Yarmouth 
 
 Eu. 
 
 England 
 
 Germ. Ocean 
 
 5^ 
 
 55 
 
 N, 
 
 I 40 E. 
 
 9 45 
 
 |Yas dc Amber 
 
 Africa 
 
 Zan^uebar 
 
 Indian Ocean 
 
 
 
 00 
 
 
 47 J5 E. 
 
 
 jVtUmv Rivxr 
 
 Afia 
 
 China 
 
 Pacif, Ocean 
 
 34 
 
 06 
 
 N. 
 
 120 10 E. 
 
 
 |Y1o 
 
 Ani. 
 
 Peru 
 
 Pacif. Ocean 
 
 17 
 
 36 
 
 s. 
 
 71 08 w. 
 
 
 !Capc York 
 
 Afia 
 
 N, Holland 
 
 Endeavour St. 
 
 10 
 
 41 
 
 s. 
 
 141 39 E. 
 
 
 York For: 
 
 Am. 
 
 New W lies 
 
 HudfouS Bay 
 
 57 
 
 02 
 
 N. 
 
 92 47 W. 
 
 9 10 
 
 York, Xe// 
 
 Am. 
 
 N. England 
 
 Atl, Ocean 
 
 4a 
 
 43 
 
 N. 
 
 74 04 w. 
 
 3 
 
 Youghal! 
 
 z 
 
 Eu. 
 
 Ireland 
 
 St. Geo, Ch. 
 
 51 
 
 46 
 
 N. 
 
 8 06 W. 
 
 4 30 
 
 Zacan.;^ 
 
 Am. 
 
 Mexico 
 
 Pscif Ocean 
 
 y 
 
 ir 
 
 N. 
 
 105 ooW. 
 
 
 Zaclce 
 I. /ant 
 I. Za,,7'bar 
 jZuro 
 
 Am. 
 
 Antilles 
 
 Atl. Ocran 
 
 18 
 
 24 
 
 N. 
 
 67 52 w. 
 
 
 Fu. 
 
 Italy 
 
 Adiiatic Sea 
 
 37 
 
 5'J 
 
 N. 
 
 21 30 E, 
 
 
 Africa 
 
 Zang'icbar 
 
 Indian Ocean 
 
 6 
 
 55 
 
 S, 
 
 40 10 E, 
 
 
 r.u. 
 
 Daln.itia 
 
 Medit. Sea 
 
 44 
 
 '5 
 
 N. 
 
 16 65 E. 
 
 
 
 
 
 C34 
 
 23 
 
 s. 
 
 172 44 E. 
 
 
 Afin. 
 
 
 Pacif. Ocean 
 
 ] 
 
 
 
 
 
 iz- ( r-i"'- 
 
 
 
 
 I47 
 
 20 
 
 s. 
 
 167 50 E. 
 
 
 /.enan ) 
 
 Afn 
 
 Arab; I 
 
 Inh.nd 
 
 16 
 
 J.O 
 
 N. 
 
 47 44 E. 
 
 
 'Zujit Stj 
 
 I-.u. 
 
 1>. Nr.lh. 
 
 Germ. Occ-iTi 
 
 
 
 
 
 3 00 
 
 -tr^
 
 400 GEOGRAPHY. Book VI. 
 
 Bcfides the times of high-water in the preceding table, the following 
 times ferve for eoafts of confiderable extent, and will ferve nearly for tht 
 places on thofe eoafts. 
 
 Finmark, orNNW. coaft of Lapland, ih. 30m. Jutland Iflesoh.om. 
 
 Friefland coaft yh. 30m. Zealand coaft ih. 30m. 
 
 Flanders coaft oh. cm. Picardy and Normandy eoafts lOh. 30m. 
 
 Bifcay, Gallician, and Portugal eoafts 3h. com. 
 
 Irifli W. coaft 3h. com. Irifh S. coaft 5h. 15m. 
 
 Africa W. coaft 3h. cm. America W. coaft ^h, om. 
 
 America . coaft 4h. 30 m, 
 
 END or BOOK VI. a^d or VOL. L
 
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