NORTHWESTERN UNIVERSITY LIBRARY EVANSTON ILLINOIS % \ ICOSMOGRAPHIA, OR A VIEW OF THE Terreftrial and Coeleftial GLOBES^ I N A Brief Explanation 6 F T H fi PRINCIPLES Of plain and fblid geometry. Applied to Surveying arid Gauging of CASK. ^TheDodrine of the Primum Mobile. With an Account of the Joil^ & Gregorian Calendits , and the Computation of the Races of of thegjm Moon, ^nd Fixed Star!> from fuoh De- ciraal Tables of tneir Middle Motion, as fuppofeth the whole Circle to be divided into an hundred, Degrees or Parts. | To which is added an Introdudion untoj 6.E O G R A P H Yy ! Tobn Newton, London, Printed for ThomasPaffinger, at the Three Bibles on London-'Bridge "79-' TO THE Moft Honourable % HENRY SOMERSe!t, Herbert, Baron of Raglan, and Gower , Earl and Marquefs of Worcciter , Lord freftdent and Lord Lieutenant of Wales and the Marches, Lord Lieutenant of Gloucefter, Here- i"ord and Monmouth, and of the City and County of Bnd6\ ^Knight of the Moft Noble Order of the Garter, and one of His Majejiie''s Moji Honour able Friz/y Council, E that adventure^ upon any thing contrary tQ tltq General received prafticc, what ever his own" courage and refblutlons are, had need to be lupported, hot only by tjie moft Wife and Hongurable, but alio A 3 the The Epijile the moft Powerful Perfons that are in Nation or Kingdom •, For let the Propo- fals be never lb advantagious to the Pub- lick, they fhall not only be decried and neglebted, but it is well, if the Promoter be not both abufed and ruined .; Yet I, not- , withftanding all thele dilcouragements, have hot been filent, but in order to Child- rens better Education, have long fince pub- lifhed my thoughts,hxve ami do declare^ that the multitude of Schools for the learning of the Latine and Greek "Tongues, are dejlru - ii^i'ue both to our youth and the Commonwealth-^ and if the Opinion of Sir Francis Bacon in his Advice to YSm%^amis concerning Sut- tort s Hofpital, be not FaOicient to warrant my Alfertion, I could heartily wifli that no fuch Evidence could have been produ- ced, as the late unhappy Wars, in the Bowels of this Kingdom hath atfbrdcd us •, for what he faith there by way of Advice, we by woful Experience have found too true^ that by reafbn of the multitude of Grammar Schools , more Scholars arc dayly brought up, than all the Preferments in this Nation can provide for, and lb they become uncapable of other ProfeiTions, and unprofitable in their own, and at laft become, materia rerum novarum; whether this be aneffential or an accidental EfFedl, I will i Dedicatory, , I will not here difpute; the truth of it, I am hire, cannot be denied: but that is not all ,• by this means it comes to pals, that four or the feven Liberal Arts, are almoft wholly neglefted, as well in both Univer- hties, as in all Inferiour Schools; and let- ting afide the City of London, there are but few Places in this Nation, where a man can put his Son,to be well inftrufled in rithmetickfieometry^Mufick and /^fironomyi and even that Famous City was without a Publick School for Mathematical Learn- ing, till His pref^t Majel^ was pleaied to lay the Foundation ; nay fo averfe are men in the general to theie Arts (which are the fupport of all Trade) that withonta highhand, it willbe almoft impoflible, to make this People wile for dieir own good: I come therefore to your Honour, humbly to beg your Countenance and Aftiftance, that the Stream of Learning may be a lit- tie diverted, in thole Schools that are al- ready erected, and to be inftrumental for the creating more , when tliey lhall be wanting •, that we may not be permitted ftill to begin at the wrong end ^ but that according to the practice of the Ancient Philofbphers, Children may be inftrufted in ^rithmetick^ Geometry^ JiLttfick and A- flroncmy 5 before the Latine and Greek A 4 Gram^ the Epijile the moft Powerful Perfons that are in a* • Nation or Kingdom •, For let the Propo- fals be never lb advantagious to the Pub- lick, they fhall not only be decried and neglebled, but it is well, if the Promoter be not both abufed and ruined.: Yet I, not- , withftanding all thele difcouragements, have hot been filent, but in order to Child- reus better Education, have long fince pub- lifhed my thoughts,hxve md do decLtre, that the multitude of Schools for the learning of the Latine and Greek Tongues, are dejlru - ffive both to our youth and the Commonwealth-^ and if the Opinion of Sir Francis Bacon in his Advice to YSn^^amts concerning Sut- toms Hofpital,. be not luSicient to warrant my Affertion, I could heartily wifli that no fuch Evidence could Itave been produ- ced, as the late unhappy Wars, in the Bowels of this Kingdom hath adbrdcci us •, for what he faith there by way of Advice, we by woful Experience have found too truc^ that by realbn of the multitude of Grammar Schools , more Scholars arc dayly brought tip, than all the Preferments in this Nation can provide for, and lb tlicy become unjcttpable of other ProfeiTions, and unprofitable in their own, and at laft become, materia rerum novarum ; whether this be an elTential or an accidental EffeQ;, I . Dedicatory, , will not here difpute; the truth of it, I am lure, cannot be denied: but that is not all ,• by this means it comes to pals, that four or the feven Liberal Arts, are almoft wholly neglefted, as well in both Univer- fities, as in all Inferiour Schools; and let- ting afide the City of Loyidon, there are but few Places in this Nation, where a man can put his Son,to be w*ell inftrufled in rithmctickfieometry^Mufick and y^fironomyi and even that Famous City was without a Publick School for Mathematical Learn- ing, till His prefbit Maje% was pleaied to lay the Foundation ; nay fo averfe are men in the general to thefe Arts (which are the fupport of all Trade) that without a highhand, it willbe almofl: impoflible, to make this People wile for dieir own good: I come therefore to your Honour, humbly to beg your Countenance and AlTiftance, that the Stream of Learning may be a lit- tie diverted, in thofe Schools that are al- ready ere£l:ed, and to be inftrumental for the ereOiing more , when tliey fhall be wanting •, that we may not be permitted fl-iil to begin at the wrong end; but that according to the practice of the Ancient Philofbphers, Children may be inftrufted in ^rithmetick^ Geometry^ Jl-fujick and flroncmy • before the Latine and Greek A 4 Gram- The Epijlle Dedicatory. . Grammars are thought on, thefe Arts in.. themieives, are much more eafie to be learned, tend more to a general good, and will in a great mealure facilitate the Letirn- ing of the Tongues, to as many as fhall af- ter this Foundation laid , be continued at School, and provided for in either Univerr fitjes. Your Honour was inftrumental to enlarge the Maintenance for God's Mini- ffcer in the Place where I live, and perhaps it may pleale God to make you fo, not on- ly in making this Place in particular, but many other Places in this Land happy, by procuring Schools for thefe Sciences, and not only fb, but by your Loyal and Prudent managing the leveral Trufts committed to you, you may do much for God's Glory, your Countries Good,and the continuance of your own Honour to all Future Genera- tmns, which is, and ilaall be the Prayer of. Tour Honour's Obliged and Devoted Servant^ John nkvyron. TO THE READER. MT De(lgn in puhlifhing theft In- troduciions to Geometry and A- . ftronomy, is fo ive/I known by all the Epfiks^ to my other Treati- Jesof Grammar, Arithmetick, Rhetorick, and Logick , that I think it needle f to tell thee here, that it is my Opinion, that all the yJrts (hould be taught our Children in the Engltjb Tongue , before they begin to learn the Greek or Latin Grammar^ by which means many thoufands of Children would be fitted for all Trades J enabled to earn their own IJ~ vings, and made tfeful in the Commonwealth; and that before they attain to twelve years of age\ andby confequence the fvarming of Bees would be prevented , who being compelled to leave their Hives, for want of room, do fpread themfelves abroad^ and inflead of ga- thering of Honey, do fiing all that come in their way. fVt jhould not have fuch innu- mer able company of Gown-men to the lof and prejudice of themfelves and the Common- wealth ^ and thoje we had would probably be iyore learned^ and better regarded. His To the Reader. HU being f leafed to begin this fVork^ by His Bounty torvards a Alathemati- ^ xal School inChniVs Church London; /am not now without hopes-, to fee the fame effeHed in many other Places in this Kingdom ; and to this purpof I have to my Intro dull ions to the other Hrts, added thefe alfo to Geome- ti'y and Altronomy ; which I call by the name of Cofmographia ; and this I have divided into four Parts ; in the fir(l I have briefly laid down the frfi Principles belonging to the three kinds of Maonitude or continued «■ . f 1 • r Quantity, Lines., Plants and Solids ; whtch ought in fame meafure to be known , before we enter upon Aftronomy , and this part I caH. an Introdnciion unto Geometry, The ftcondani third Parts treat of Aftro- no my ^ the frjl of which fheweth the Do- Brine of the Primum Mobile, that is., the Declination, Right Hfcenfion, and Oblique ysfcenfions of the Sun and Stars , and fitch other Problems, as do depend upon the Do- Brine of S^dierical Triangles. The fecond Part of Aftronomy, treatetb of the motion^ of the Sun, Afoon and fixed Stars • in order where unto, J have firjl giv- en thee a brief account of the Civil Tear, with the caufe of thi difference between our Jtiiian and Gregorian Calendar, and of both fiora the true; for it m.tfl be acknowledged To the Reader. • '• that both are erroneous , though ours be the warfe of the t>vo ^ yet not fo bad, but that our D/(fenttng Brethren have / hope Jome better jdrgmnentsto jusbijie their Non-conformity^ than what I Jee publijhed in a little Book without any name to it^ concerning two £a(ltrs in one Tear; by the General Table^ faith this learned man, who owneth the Feajl of •Eafier WtU to be obferved Anna 1674. upon the i p day of A^n\f fo the Almanacks for that Tear^ cs well as the General Table fet before the Book of Common Prayer •, but by the Rule in the /aid Book of Common Prayer given^ the Peafl of Eafier fioould have been upon the twelfth of April , for Eafter-Day muft al- ways be the firft Sunday after the firft Full Moon, which happeneth next after the one and twentieth day of March , and if the Full Moon happen upon a Sunday,-Eafter- Day ■ is the Sunday after; Now in the Tear 1 674. the I p 0/'April being Friday was Full Moon^ therefore by th's Rule , Eafier-Day jhould be the twelfth., and by the Table and the Common Almanacks April the tenth but this learned man mufi knoWy th.tt the mifiake is in himfdfy and not in the Rule or Table fet down in the Book of Common Prayer ; for if he pleaf to look into the Calendary he will find that the Golden Number Three, (which was the Golden Number for that Tea f is To the Reader f ■laced againjl the Ufi day of March, and therefore according to the ftifpofed motion of the Moon^ that Day ivof New Moon\ and then the Full Moon will fall upon the fourteenth day of April, and not upon the tenth^ and fo by donfequence the Sunday following the fir(l Full Moon after the 21 day of March was the nineteenth of April and not the twelfth, ylnd thus the Mule and the Table in the Bookof Common Prayer for finding the Feajl of Fafler are reconciled 5 and when Authori- ty jhall think fit, the Calendar may be cor- reded and all the moveable Feafls be obftrved upon the days and times at frfi appoirited; but till that be, a greater difference than one f-Vtek will be found in the Feafi of Fafler be- tiveen the Obfirvation thereof according to the Moons true motion^ and that upon which the Tables are grounded ; for by the Fathers of the Nicene Council it was appointed.^ that the Feaff of Fafler fijo/ild be obferved upon the Sunday following the f^fl Full Moojt after the Ftrnal Fquinox^ which then indeed was the ^lof March ; but now the tenth^ and in the Tear 1674. CVednefday the \ \ of March was Full Moon^ and therefore by this Rule, Fajlcr-Day fhould have been upon March the fifteenth , whereas, according to the Rules we^go by^ it was not till April the nineteenth. The To the Reader. • The Tables of the Sun and, Moons middle • motions are neither made according to the u- fual Sexagenary Forms , nor according to the ufual Degrees of a Circle and Decimal Parts, hut according to a Circle divide-d into loo Degrees and Parts ^ and this J thought good to do, to give the fVorld a tafie of the excel- lency of Decimal JVu?nbers^ which if a Canon of Sines and Tangents were fitted 'to it, would be fomtd much better ^ as to the com- futing the Places of the Planets but as to the Primum Mobile, by reafon of the gene- ral dividing a Circle into 560 Degrees, I fhould think fuch a Canon with the Decimal Parts mojl convenient , and in fame cafes the common Sexagenary Canon may be very ufeful y and indeed fhould wijh and fball en- deavour to have all printed together, one Table of Logarithms will ferve them all, and two fuch Canons, one for the Study and an^ other for the Pocket, would be fujficient for all Mathematical Books in that kind; and then men may ufe them all or either of them as they jhall have occafion, or as every one is perfwaded in his own mind. /Vhat / have done in this particular, as it was for mine own jatisfaUiion , fo I am apt to believe, that it will be pleafing to many others j and although I fhall leave every one to abound in his own fenfe, yet I cannot think To the Reader. that Cujlom jhoidd be fuch a Tyrant^ as 'to force us always to uje the Sexagenary form^ iffo,I wonder that men did not always ufe the natural Canon • if no alteration may be admit- ted^ what rtafon can be given for the ufe of La- garithms ; and if that be found more ready 4han the naturaf in things of this kindjwhere none but fartieular Students are concerned^ / jhould think it reafonable, to reduce all things hereafter j into that form , which jhall be found mofl ready and exact •, now the Part Proportional in the Artifcial Sines and Tan- gents in the three frf Degrees cannot he well taken by the common difference, and the way of fnding theni otherwife will not be (o eajie in the Sexagenary Canon, as in either of the other, and this me thinks, jbould render that Canon which -divides each Degree into I oo Parts more accef table but thus to retain the ufe of Sines , 'Degrees, and Decimal Parts^ doth not to me fetm convenient y andtorec- konup, a Planets middle motion, by whole Circles will fometimes caufc a Divifon of Decrees by 60, which hath fome trouble in it aljo, but if a Circle he divided into too De- grecs, this inconvenience is avoided, and were there no other re of on to he given , this me thinks (hould make fuch a Canon to be dtf- Table ^ hut till I can fnd an .opportunity of fubli{hing fuch an one / jhall forbear to To the Reader. • further tifes of it, and for rvhat is I > * wanting here in this fuhje^i , I therefore re- fer thee to Mr.Strecfs Allronomia Caroli- . " na, and the feveral Books written in EngUfy by Mr. Wing. I The fourth Part of thisTreatife is an In- ' ' troduCiion unto Geography, in which / have p' given general Dire5iions^ for the underfiand- ' it\ habitable gart of the dVorld is 'p divided in refpebi of Longitude and Latitude ® in rejpecl of Climes and Parallels with fuch "■ other Particulars aswiUbe found uftfd unto IM fuch as fjallbe willing to underfiand Hiflory • mm which three things are required; The r^'f- time when and this depends upon Aftrono- my-, the place where, and this depends upon ertkt Geography; and the Perfon by whom any f() loo memorable jLcl was done , and this mujl be lintk had from the Hijlorical narratiou thereof'^ A*"//, and he that reads Hiflory without Jome know- '"'"ff" ledge in Aftronomy and Geography wiB pU( ffid himjelf at a lof^ and be able to give but f a lame account of what he reads ; but after tit it the learning of thefe Arts of Grammar, (/ fibt- mean fa much thereof ^ as tends to the un- derflanding of every ones Native Language) km Arithmetick, Geometry and Aftronomy ; tbf a Child may proceed proftablyto Rhetorick iq (j and Logick, the reading of Hifloryy and the if It learning of the Tongues; and fure there it jbes no To the Reader. ^0 fiidioiu md ingenious mm, but will (laiid in need of Jome Recreutiony and therefore if Iviafick in the JVnfljip and Service of God be not Argument enough to allow that a place among the drts , let that poor end of De- light and Pleafure. be her y^dvocate ; and al- though that all men have not P'oyces, yet I can hardly believey that he expells any Melodious Harmony inHeaveny that will not allow In- finrmental Mufi:k a place on Earth; and as for thofe that have Hoyces, furely the time of learning Eocal MufcLy mufl be in Eouthy and I am perjwaded th it the Hrts and Sci- ences to fome good degree may be learned by Children before they be full twelve years oldy and would our Grammar jWafers leave of their horrible feverityy and apply themfelves to fuch ways of teaching Touth , as the IVorld is not now unacquainted with, I am perfwaded that it is no d'fficult matter y in four years titne more to ft Children in fome good meajtire for the llniverftty. The great Obf ruction in this lEork , is the general Ignorance of Teachers, who be- ing unacquainted with this Learningy cannot teach others what they know not themfelves, I could propound a remedy for this , Sed Gyiidiiiis aurem vellit ; Therefore I will forbear and leave what I have writteny to he ■perufed and cenfured as thou jhall think ft^ John Newton. Pra6iical Geometry ; OR, THE Art of Surveying. ; \ . CHAP. I. Of the Definition and Di'z/ifionof Geometry, GEoraetry'is a Science explaining the kinds and properties of continued quantity or magnitude. 2. There are three Kinds or Spe- cies of Magnitude or continued Quantity, Lines, Superficies and Solids. 3. A Line is a Magnitude confifting only of length without either breadth orthicknefs. 4. In a Line two things are to be confidered^ the Terms dr Limits, and the feveral Kinds. 5. The term or limit of a Line is a Point. 6. A Point is an indivifible Sign in Magni- tude which cannot be comprehended by fenfe, but mufl be conceived by the Mind. 7. The kinds of Lines are two, Right and Ob- lique. B. A 2 i^iactical /T ii i t Dpgn a Right Litt.egiv.en.,.fO:inake d right-iintd Angle, equoil to an Angle£iven.. -.^ , <-). , . y ; v//-:',. : .... ' X. ' : _ - Let It be required upon the LineC D in Eig. 6. "r ' to 0?, t^e Tdtt of ^nrbeyui3. 7 to make an Angle,' equal to the Angle BAE 'm 5. From the Point Xas a Center,- at any ex- tent of the Compafles defcribe the Arch .jSC, between the fides of the Angle given,, and .with the fame extent defcribe the .Arch ML 'from the Point D, and then make ML equal to BG, then draw the Line D L, fo fhallthe Angle C D ib be equal to the Angle D..^£ given, as wasre- quired. i ^ C H A P. II. Of Figures in the general y more .farticHlarfy of a Circk andthe ajfe^Fions theriof - " Hitherto we havelpoken of the firil kind of Magnitude, that is, of Lines, as they are confidered of themfelves , or amongft them- felves. : 2. The. fecond kind of Magnitude is that which is m.ade of Lines, that is, a Figure con- lifting of breadth as well as length, and this is otherwife called a Superficies. 3. And in a Superficies there are three things to be confidered. ,i. The Term or Limit. 2. The middle of the Term. 3. The Thing or Figure made by the Term or Limit. 4. The Term or Limit is that which compile- hendeth aitd boundeth the Figure, it is common- ly called the Perimeter or Circumference. 5. The Term of a Figure is either Simple or various. ■ . 6. A Simple Term is that which doth confift of a Simple Line, and is properly called a Cir- 13 4 . cumference 8 5 16. Right Lines mlcribed within a Circle, are ^!i£- either fuch as do cut the Circle into two equal or unequal Parts,as the Diameter and lefler Chords, Jig ii or fuch as do cut the Diameter and lefler Chords u tlie into two equal or unequal Parts, as the Right and Kin vcrfed Sines. 17. A Diameter is a Right Line drawn Term through the Center from one fide of the Cir- kJj' cumference to the other, and divideth the Circle !!»!( into two equal Parts, 7. The Ri^ht Line GT) drawn through the Center B is the Diameter of rijilii- the Circle G E D L dividing the fame into the two e- if qual Parts G E D, and G L D: and this is alfo called IfX! the greatefi Chord or Suhtenfe. trjis 18. A Chord or Subtcnfe is a Right Line in- Is in- fcribed in a Circle, dividing the fame into two equal or unequal Parts-, if it divide the Circle into- •spk two equal Parts, it is the fame with the Diameter, nple but if it divide tlie Circle into two unequal Parts it is lefs than the Diameter,and is the Chord or Sub- iied tenfe of an Arch lefs than a Semi-circle, and alfo ircle of an Arch greater than a Semi-circle, ^s tn the former Fimre, the Ri. Propofition 0^3suet of 11 - Propofition I. ' ; •• V- The Arch of a Circle being given to defcribe the whole Periphery. • Ltt ABChe an Arch given, and let the Cit- cumferenceof that Circle he required. Let there l>e thr ee Points taken in the given'Arch at plea-' fure, as open your Gompailes td more than half the, diitance of i-'j ^iPd 'fetting on6 Foot in ^defcribe the Arch "of a. Circle, and the Compalfes remaining at the lame diftance, fet- ting one Foot in B, defcribe another Arch fo as it may cut the former in two "Points, fuppofe G, and//, and draw the Line HG towards that Part on which you fuppoie the Center of the Center of the Circle will fall. In like manner, opening -your CompafleTto more than half your diftaiico bf B, C, defcribi. two other Arches from the .'Points E and C, ictttrl ting each other in E and F, then.draw the Line E F till it interfcft the former Line HG, fo /hall the Point of Interfeftion be the'Center of the Circumference or Circle required,, as in Fig. roaybefeeno ■ • -n :r -m.T O -.....-Mjij. Propoutibn II.,' ' .,-"V The Conjugate Diameters of an E Uipfis being gintert, to draw the Ellipfis, ' '.'O . • ' t r • Let the given Diameter in Fia. 24.. be LB and E D, thegreatell Diameter. L F being bifefted in the Point of Bifedioii,.ereit the Perpendicular AD, 12 i^jactical (!5eometrp5 j4D. which let be half of the lefler Diameter. • E D, then open your Compares to the extent of AB, and fetting one Foot in X), with the other make a mark at M and N in the Diameter B X, then cutting a thred to the length of B L, faften the thred with your Compafles in the Points NM^ and with your Pen in the infide of the thred de- fcribe the Arch BFKL,iofhall you delcribe the one half of theEliipfis required, and turning the Thred on the other fide of the Compaffes, you may with your Pen in the like manner defer ibe the Other half of the Ellipfis G B HL. CHAP. III. Of Triangles. Hitherto we have Ipoken of the moft Simple Figure, a Circle. Come we now to thofe Figures that are Various or Angular. 2. Andean Angular Figure is that which doth confift of three or more Angles. 3. An angular Figure confifting of three Angles, otherwife called a Triangle, is a Super- ficies or Figure comprehended by three Right Lines including three Angles. 4. A Triangle may be confidered either in re- lped);of its Sides, or of its Angles. 5. ATriangleinrelpedof itsSides, is either Tfoplenron, Ifofceles, or Scdenum. 6. An Triangle, is that which hath three equal fides. An Ifofcecles hath two equal Sides. And a Scalenum hath dl the three Sides unequal. 7. A Triangle in refped of its Angles is Right or Oblique- 8. A Oh tbe 3irt of ^utbepi'ns. * 5 ~ S. A Right angled Triangle is that which Tiath one Right Angle and two Acute. 9, An Oblique angled Triangle , is 'either Acute or Obtufe. 1 o. An Oblique acute angled Triangle, is that which hath all the three Angles Acute. II. An Oblique obtufe angled Triangle , is that which hath one Angle Obtufe, and the other two Acute. Propofition I. Vfon a Right Line given to make an Ilbpleuron or an Equilateral Triangle. In Fig. 8. let it be required to make an Equila^ teral Triangle upon the Right Line AB. Open your Compalles to the extent of the Line given, and fetting one Foot of your Compalles in A, make an Arch of a Circle above or beneath the Line given, then fetting one Foot of your Com- palles in 5, they being full opened to the fame extent, with the other foot draw another Arch of a Circle eroding the former, and from the In- terfedion of thofe Arches draw the Lines AC and AB, fo fliallthe Triangle A CB be Equilatc« ral as was dehred. Propofition II. ZJpon a Right Line given to make Ilblceles Tri~ angle, or a Triangle having two Sides equal. "in Fig. 8. let AB be the Right Line given, from the Points A and B as from two'Centers, but at a lefler extent of the CompaHbthan AB ; 14 (l5eomctcp •, ifyoi! would hav4 5 the greateft Side, at a greater extent v if yon would have it to be the" leafb Side, deicribe two Arches cutting one ano- ther, as at F, and from the Interfedlion draw the Lines ^F, and F B, fo fliall the Triangle AFB have two equal Sides, as was required. Propofition 3. To fnalic a Scalcnum Triangk, or a Triangle,whofe three Sides are unequal. ' \nFig. 9. let the three unequal Sides be EFG make AB equal to one of the given Lines, fup- pofe G, and from ^ as a Center, at the extent of E defcribe the Arch of a Circle v in like manner from F at the extent of F defcribe another Arch jnterfeding the former, then lhall the Right Lines AC. CB ahdF^ comprehend a Triangle, whofe three fides fliall be unequal,as vvas requiredi CHAP. IV. Of 'Quadmngular and Multangular Figures.- WE have ipoken of Triangles or Figures con- filling ol three Angles, come we now to thofe that have more Angles than three, as the Quadrangle, Qmnquangle, Sexangle, &c. 2. A Qmdrangle is a Figure or Superficies, which is bounded with four Right Lines. 3. A Qmdrangle is either a Parallelogram or a TrafczAcm.' 4. A parallelogram is a Quadrangle wholc oppo- lite oj, tl)e Mtt of ^urbepmp:. 15 fitt Sides are parallel having equal diltances from ' one another in all Places. 5. k Parallelogram h either Right angled or Oblique. - 6. A Right angled Parallelogram, '\%'^Qmdran' de whole four Angles are all Right, and is either Sqyareor Oblong. 9. A Square Parallelogram doth confift of four equal Lines. The Parts of a Square are, the Sides of which the Square is made, and th"e Dia- gonal or Line drawn from one oppofite Angle to another through the middle of the Square. 8. An Oblong is a Right angled Parallelogram, having two longer and two Ihorter Sides. 9. All Oblique angled Parallelogram, is that whole Angles are all Oblique, and is either a Rhombpts or a Rhomboides. 10. A Rhombm is an Oblique angled and equi- lateral Parallelogram, 11. A Rhomboides is an Oblique angled and inequilateral Parallelogram. 12. A Trapezium is a Q^drangular Figure whofe Sides are not all paralel it is either Right angled or Oblique. 13. A Right angled hath.two op- polite Sides parallel, but unequal, and the Side between them perpendicular. 14. An Oblique, angled TrapezJum is a drangle, but not a Parallelogram, having at lealt two Angles Oblique, and none of the Sides pa- rallel. ' 15. Thus much concerning Qmdrangles of four lided Figures. Figures confilling of more than four Angles are almoft'infinite, but are re:-'^ ducible unto two forts , Ordinate and Regular,' or Inordinate and Irregular. , 16. Or- i6 ^^acticalseomtvp y 16. Ordinate and Regular Polygons are fuchj as are contained by equal Sides and Angles, as •' the Pentagon, Hexagon, and Tuch like. 17. Inordinate or irregular , arefuch as are contained by unequal Sides and Angles. The conftruftion of thefe Quadrangular and Multangular Figures is explained in the Propofi- tions following* Propofition I. U^on a Right Line given to defcribe a Right an- gled Parallelogram, whether Square or Oblong. In Fig. 10. let the given Line be AB, upon the Points ered the Perpendicular AD equal to A B you intend to make a Square, but long- er or fliorter, if you intend an oblong, and upon the Points D and B at the dillance oi A B and A D defcribe two Arches interfeding one ano- ther , and from the Interfedion draw the Lines £ D and £ B, fo lhall the Right angled Figure AE be aSquare, if AB and ADht equal, o- therwife an Oblong, as was defired. Propofition 11. To defcribe a'^ovc^vts, or RhomboideS. InF^'^. II. To the Right Line AB draw the Line AD Tit any Acute Angle at pleafure, equal ' toAB 'ii you intend a Rhombm, longer or fhort- er if you intend a Rhomboides, then upon your Compaflb to the extent of A D and upon B as a Center defcribe an Arch •, in like manner, at the extent of upon I> as a Center defcribe an- other 3!lrt of 17 ^driier Arch, interfering the former, then draw _• the Lines E D and E B, fo fliall be the Rhom^ hpu or Rhomboides^ as was required. Propofition III. Vpon a Right Line given to make a Regnktr Peri- tagon, or five fided Figure. In Fig. 12. Let the given Line be upon A and B as two Centers delcribe the Circles EBGH and CA G K, then open your CompalTes to the extent of B C, and making G the Ceii- tef, defcribe the Arch H^F K, then draw the Lines KFE and HFC: fo ftall ^E and BC be two fides of the Pentagon defired, and opening your Compalfes to the extent of B, upon E and C as two Centers defcribe two Arches inter- feding one another, and from the Point of Inter- fedlion draw the Lines E D and D C, fo fliall the figure u4B and D EhQ the Pentagon required. Propofition IV. To make a Pentagon and Decagon in a given Circle. InFig. 13. upon the Diameter cAB defcribe the Circle CDB L, from the Center bereft the Perpendicular AD., and let the Semidiameter AC be bifefted, the Point of Bifeftion is £, fet the diilance E D from £ to C, and draw the Line G D, which is the fide of a Pentagon, and A G the fide of a Decagon infcribed in the fame Circle. V Propo- i8 0iacttcal d^cometf? 5 c Propofition V. , In a C rcle given to deferibe a Regular HeXagOIl. The fide of a Hexagon is equal to the Radim ofa Circle, the Radim of a Circle therefore being fix times applied to the Circumference, will give you fix Points, to which Lines being drawn from Point to Point, will cbnftitutea Regular Hexagon, as wasdefired. Propofition V I. In a Circle given to defcrihe a Regular Hepta- gon or Figure confijling of feven equal fides. The fide of a Heptagon is equal to half the fide of a Triangle inlcribed in a Circle, having therefore drawn an Hexagon in a Circle , the Chord Line fubtending two fides of the Hexagon lying together, is the fide of a Triangle infcrib- ed in that Circle, and half that Chord applied feven times to the Circumference, will give fe- ven Points, to which Lines being drawn from that Point, will conftitute a Regular Heptagon, as in Fig. 14. is plainly fliewed. G H A P, 0^, ^tt of C H A P. V. Of Solid Bodies. f'l "jMrAvIng fpoken of the two firft kinds of 3 JlX Magnitude, Lines and Superficies, comewc r now to the third, a Body or Solid. 2. A Body or Solid is a Magnitude confifting of length, breadth and thicknefs. 3. A Solid is either regular or irregular. 4. That is called a regular Solid, whofe Bafes, Sides and Angles are equal and like. 5. And this either Simple or Compound. 6. A limple regular Solid, is that whithdotli confifl of one only kind of Superficies. .7. And this is either a Sphere or Globe, or a ". plain Body. 8. A Globe is a Solid included by one round and convex Superficies, in the middle whereof "f there is a Point, from whence all Lines drawn to ™ the Circumference are equal. I?*''' 9. A fimple plain Solid, is that which doth j confifl of plain Superficies. 10. A plain Solid is either a Pyramid, a Prifm, or a mixt Solid. 11. A Pyramid is a Solid, Figure or Body, contained by feveral Plains fet upon one right lin'd Bafe, and meeting in one Point. 12. Of all the feveral forts of Pyramids, — there is but one that is Regular, to wit a Tetrahe- dron.f or a Pyramid conlifting of four regular , or equilateral Triangles *, the form whereof ( as ^ it may be cut in Paftboard) may be conceived by Figure 15, € 1 • 13. A l^jactical (SJeometrp,, n. A Prifm is a Solid contained by fevera. Plains, of which thofe two which are oppolite, are equal, like and parallel, and all others are Par^ilcllo^ram. 14. APrifrais either aPwt^/?e(^ro», TiHexahe- dron^ or a Polyhedron, 15. h Pentahedron Prifm, is a Solid compre- hended of five Sides, and the Bafe a Triangle, as Fig. i6. 16. An Hexahedron Prifm , is a Solid com- prehended of fix Sides, and the Bafe a Qmdran- gle, as Fig. 17. Hexahedron Prifm, is diflinguifhed iptoa Parallelipipedon^md a Trapjeuum. 18. An Hexahedron Prifm called a Trapezium is a Solid, whofe oppofites Plains or Sides, are neither oppofite nor equal. 19. A Paralleiipipedon is either right angled or oblique. 20. A right angled Paralleiipipedon is an He.xa- hedron Prifm , comprehended of right angled Plains or Sides ^ and it is either a Cube or an Ob- long. 21. A Cube is a right angled Paralleiipipedon comprehended of fix equal Plains or Sides. 22. An Oblong Paralleiipipedon, is an Hexahe- dron Prifm, comprehended by unequal Plains or Sides. 23. An Oblique Paralleiipipedon , is an Hexahedron Prilm , comprehended of Oblique Sides. 24. A Polyhedron Prifm , is a Solid compre- hended by more than fix Sides, and hath a mul- tangled Bafe, as ^Qmncangle, Sexangle, o c. . 2 5. A regular compound or mixt Solid, is fuch a Solid 0?, tlje of ^tirii'epfng; 21 ,ti Solid as tiath its Vertex in the Center, and the feveral Sides expofed to view, and of this fort there are only three •, the OElohcdron, the Icofahe- dron, of both which the Bale is a Triangle; and the Dodecahedron^ whofe Bale is a Quincangle. 26. An Ociphedron is a Solid Figure which is contained by eight equal and equilateral Trian- gles, as in Fig. 18. 27. An Icofahedron is a Solid, which is con- tained by twenty equal and equilatefal Triaru- gles, 3S Fig. I g. 28. A Dodecahedron is a Solid^ which is con- tained by twelve equal Pentagons, equilateral and equiangled, as in Fig. 20. 29. A regular compound Solid, is fuch a So- lid as is Comprehended both by plain and circu- lar Superficies, and this is either a Cone or a Cylinder. 50. A Cone is a Pyramidical Body, whofe Bale is a Circle, or it may be called a round Py- ramis, as Fig. 21. 31. A Cylinder is a round Column every where comprehended by equal Circles, as Fig, 22. 32. Irregular Solids are fiich, which come not within thefe defined varieties, as Ovals, Fr«- finms of Cones, Pyramids, and fuch like. And thus much concerning the defcription of , the feveral forts of continued Quantity, Lines, Plains and Solids*, we will in the next place confider the wayes and means by which the Di- mentions ofthem may betaken and determined, and firft we will fhewthe meafurmg of Lines. C 3 • CHAP. 32 t^iacti'cal d^eometrp 5 ( CHAP. VI. Of the Meajurin^ of Lines both Right md Circular^ EVery Magnitude mufl: be meafured by feme known kind of Meafure as Lines by Lines, Superficies by Superficies, and Solids by Solids, as if I wiere to meiure the breadth of a River, or height of a Turret, this mult be done by a Right Line, vvhich being applied to the breadth or height defired to be meafured , fliall fhew the Perches, Feet or Inches, or by feme other known rneafure the breadth or height defired: but if the quantity of fome Field or Meadow, or any other Plain be defired, the number of fquare Perches mufl be enquired i and laflly , in meafuring of Solids, we mufl ufe the Cube of the meafure ufed, that we dilcover the number of thofe Cubes that are contained in the Body or Solid to be meafured. Firfl, therefore we will fpeak of the feveral kinds of meafure, and the making of fiich Inflruments, by which the quantity of anyMagni- tude maybe known. 2. Now for the meafuring of Lines and Su- perficies, the Meafures in ufe with us, are Inches, Feet, Yards, Ells and Perches. 3. An Inch is three Barley Corns in length, and is either divided into halves and quarters, which is amongfl Artificers mofl ufual, or into ten equal Parts, which is in meafuring the mofl ufeful way of Divifion. 4. A Foot containeth twelve Inches in length, vand is commonly fo divided •, but as for fiich things as are to be meafured by the Foot, it is far better ati tl^e %tt of ^urbcjPing. 23 idter for ufe, when divided into ten equal Parts, * and each tenth into ten more. 5. A Yard containeth three Foot, and is com- monly divided into halves and quarters,the which ^ for the meafuring of fuch things as are uliiaily' fold in Shops doth well enough, but in the mea- furing of any Superficies, it were much better to be divided into lo or loo equal Parts. 6. An Ell containeth three Foot nine Inches, aud is ufiially divided into halves and qrarters, and needs not be otherwife divided, becaufe we have no ufe for this Meafure, but in Shop Com- nwdities. 7. A Pole or Perch cotaineth five Yards and an half, and hath been commonly divided into Feet and half Feet. Forty Poles in length do make one Furlong, and eight Furlongs in length do make an Englifii Mile, and for thefe kinds of of lengths, a Chain containing four Pole, divided by Links of a Foot long, or a Chain of fifty Foot, or what other length you pleafe, is we.U enough, but in the meafuring of Land, in which the number of fquare Perches is reqiiired ■, the Chain called Mr. Gmters, being four Pole in length divided into loo Links, is not without julb reafon reputed the moll ufeful. 8. The making of thefe feveral Meafures is not difficult, a Foot may be made, by repeating an Inch upon a Ruler twelve times, a Yard is eight Foot, and foof the reft-, the Subdivifion of a Foot or Inch into halves and quarters, may be performed by the feventeenth of the firfl;,and into ten or any other Parts by the firfl; Propofition of the firfb Chapter, and all Scales of equal Parts, of what feantling you do defire. And this t C 4 thipk 24 i^jacti'cal(!5£omett^3 think is as much as needs to be faid concerning the dividing offucli Inftruments as are ufefolin the meafiiring Right Lines. p. The next thing to be confidered is the mea- furing of Circular Lines, or Perfedt Circles. ID. And every Circle is fuppofed to be divi- dedinto 360 Parts called Degrees, every Degree . into 60 Minutes, every Minute into 60 Seconds, and fo^forward this divifion of the Circle into 360 Parts is generally, retained, but the Sub- divifion of thofe Parts, lome would have be thus and 100, but as to our prefent purpofe either may be ufed , molt Inftruments not exceeding the fourth part of a Degree. II. Now then a Circle may be divided into 360 Parts in this maimer. Having drawn a Dia- meter tlirough the Center of the Circle dividing the Circle into two equal Parts, crofs that Dia- meter with another at Right Angles through the Center of the Circle alfo, fo lhall the Circle be divided into four equal Parts or Quadrants, each Quadrant containing 90 Degrees,-as inFz^. 7. GB. ED. D L and LG , are each of them 90 Degrees 3 and the Radius of a Circle being equal to the Chord of the fixthPart thereof, that is to the Chord of 60 Degrees, as in Fig. 14. if you fet the Radius G B from L towards G, and alfo from 6" towards A, the Quadrant G L will be fub- divided into three equal Parts , each Part con- taining 30 Degrees, G M. 30. MH 30 and HL 30, the like may be done in the other Quadrants alfo', fo will the whole Circle be divided into twelve Parts, each Part containing 30 Degrees. And becaufe the fide of a Pent agon a Circle is equal to the Chord of 72 Degrees, or ' ' " the oj, ti)e of ^utbepi'uo:. \ • the firft Part of 360, as in Fi^. 13. therefore if you fet the Chord of the firft Part of the Circle given from G to Lor L to G, in Fig.'], you will have the Chord of -^2 Degrees, and the difference betweenCL72 andC//60is HP 12, which be-, ing bifefted, will give the Arch of 6 Degrees, and the half of fix will give three, and fo the Circle will be divided into 120 Parts, each Part con- taining three Degrees, to which the Choqd Line being divided into three Parts, the Arch by thofe equal Divifions may be alfo divided, and fb the whole Circle will be divided into 3 60, as was de- fired. 12. A Circle being thus divided into 3 60 Parts, the Lines of Chords, Sines, Tangents and Secants, are ib eafily made (if what hath been faid of them in the Second Chapter be but confidered) that I think it needlefs to fay any more concerning their ConftruftioB, but fhall rather proceed unto their Ufe. 13. And the ufe of thefe Lines and other Lines of equal Parts we will now fhew in circular and right lined Figures 3 and firft in the meafiiring of a Circle and Circular Figures. , CHAP. 2^ j^jaetical d^eometrp s • I \ CHAP. VII. t Of the Meafuring of a Circle. THe fquaring of a Circle, or the finding of a Square exadly equal to a Circle given,is that which many have endeavoured, but none as yet have-at^aincdiYet^rclj/wf^e^ that Famous Mathe- matician hath fiifficiently proved. That the Area of a Qrcle is equal to a Redanglemade of the Ra,r dim and half the Circumference: Or thus, The Area of a Circle is equal to aReftanglemadeof the Diameter and the fourth part of the Circum- ference. For Example , let the Diameter of a Circle be 14 and the Circumference 44 ■, if you multiply half the Circumference 22 by 7 half the Diameter, the Produd is 154-, or if you multiply 11 the fourth part of the Circumference, by 14 the whole Diameter, the Produd will ftill be . 154. And hence the Superficies of any Circle may be found though not exadly,yet near enough for any ule. 2. But Lndolphm Van Culen finds the Circum- ference of a Circle whole Diameter is i. 00 to be 5.14159 the half whereof i. 57095 being mul- tipliedby half the Diameter 50, &c. the Produd is 7.85395 v/hich is the Area of that Circle, and from thefe given Num.bers, the Area, Circumfe- rcnce and Diameter of any other Circle may be found by the Proportions in the jPropofitions folt lowing. Propofition Oh 3(iit of ^titfjepiug. 2 7 Propofition I. The Diameter of a Circle being givev to fmd the Circumference. "^As I.to 3.14159 : lb is the Diameter to the Circumference. Example. In Fig. 13. Let the Diameter IB be 13. 25. I Giy as i.to 3.14159. foIB. 13.25 10 41.626 the Circumference of that Circle. Propofition II. The Diameter of a Circle being given to find the Superficial Content. As I. to 78539-, Ibis the Square oftheDia- meter given, to the Superficial Content required. Example, Let the Diameter given be as before IB 13.25 theSquare thereof is 175.5625 therefore. As 1. to 78539: fo 175. 5625 to 137.88 the Superficial Content of that Circle. Propofition III. The Ctrcimjerefice of a Circle being given, to find the Diameter. This is but the Converle of the firll Propofiti- on: Therefore as 3.14159 is to i '• fo is the Circumference to the Diameter *, and making the Circumference- an Unite, it is. 3.14159. t : : 1. 318 308, and fo an Unite may be brought into the firft place. E.xample, Let the given Cir- ' " . (umference 28 ^jacti'cal (Seomctcy 5 cumfereiice be 41. 626, I fay , As 1. to 318308 : fb 41. 626 to 13. 25. the Diameter required. Propofition IV. The Circumference of a Circle being gii^en to find the Superficial Content. ( As the Square of the Circumference of a Cir- cle given is to the Superficial Content of that Circle; fo is the Square of the Circumference of another Circle given to the Superficial Con- tent required. Example , As the Square of 3.14i59ist07853938 : fb is i. the Square of another Circle to 079578 the Superficial Content required, and fo an Unite for the moll eafie work- ing may be brought into the firft place; Thus the given Circumference being 41.626. 1 fay. As I.to 0.79578: fbisthe Square of 41.626 to 137.88 the Superficial Content required. Propofition V. The Superficial Content of a Circle being given^ to find the Diameter. This is the Converfe of the fecond Propofitt- on, therefore as 78 5 3 9 is to 1. fb is the Superfici- al Content given, to the Square of the Diameter required. And to bring an Unite in the firft place: 1 fay. As 7853978. I : I. 1.27324, and there- fore if the Superficial Content given be 137.88, to find the Diameter; I fay, As 0?, t^e Irt of 29 * As 1. to 1.27324; fb 137.8810 175.5625 ■the whofe Square Root is 13.25, the Diameter fought. Propofition VI. ijmt The Superficial Content of a Circle being given) to find the Circumference. Cir- This is the Converfe of the Fourth Propofiti- that on, and therefore as 079578 is to 1: fo is the Su- :e!\C{ perficial Content given, to the Square of the Cir- Cm- cumference required, and to bring an Unite in the re of firft place: I fay, re of As 079578. 1 : i. 12.5664, and therefore im if the Superficial Content given be 137.88,10 ork- find that Circumference: Ifay, luis As I. to 12. 5664; foisthe 137.8810 1732.7 , whofe Square Root is 626 the Circumference. 626 . Propofition Vll. , The Diameter of a Circle being given to find the Side of the Square) which may be infcribed within the vti, fame Circle. The Chord or Subtenfe of the Fourth Part of ijiti. a Circle, whofe Diameter is an Unite, is 7071067, {(j. and therefore, as 1. to 7071067: fo is the Dia- (f; meter of another Circle, to the Side required, fjl Example) let the Diameter given be 13. 25 to find the fide of a Square which may be infcribed in that rg. Circle: 1 fay, • 8^ As 1.107071067; fb is 13.25 to 9.3691 the fide required. Is • Propofitioa 30 (Scometcp 5 Propofition VIII. The Circumference of a Circle being given, to fnd the Side of the Square which may be iufcrihed in the fame Circle. As the Circumference of a Circle whole Dia- meter is/an Unite, is to the fide infcribed in that Circle *, lb is the Circumference of any other Circle, to the fide of the Square that may be in- fcribed therein. Therefore an Unite being made the Cirqumference of a Circle., As 3.i4i59t0 707lo67: lb i.to 225078. And therefore the Circumference of a Circle being as before 41.626, to find the fide of the Square that may be infcribed ;* I fay, As I. to 225078 . fois41.626to 9. 3691 the fide inquired. Propofition IX^ . The Axis of a Sphere or Globe being given , to' fnd the Superficial Content. As the Square of the Diameter of a Circle, which is Unity , is to 3.14159 the Superficial Content, fo is the Square ol any other jixis given, to the Superficial Content "required. Ex- ample. Let 13.25 be the Diameter given, to find the Content of fuch a Globe; I fay, As I. to 3.14159: fo is the Square of 13. 25 to 5 51 .■ 54 the Superficial Content required. Propofition o|, tl^e Utt of jS^urtjeping. 31 Propofition X. To find the Area of m Ellipfis. As the Square of the Diameter of a Circle, is to the Superficial Content of that Circle-, fo is the Redangle made of the Conjugate Diame- ters in an ElHpfsyto the Area of that ElHpJ^s; And the Diameter of a Circle being one , the Area is 7853975, therefore in Fig. 26. the Diameters ACS and 5 i) 5 being given, tlie Area of the Ellipfis AB CD may thus be found. As 1. to 7853975; fo is the Redangle AC in 5 D 40 to 3.1415900, the Area of the£//;py;f required. CHAP. VIII. Of the Meafnring of Plain Triangles. HAving fiiewed the meafuring of a Circle, and Ellipfis, we come now to Right lined Figures, as the Triangle, Quadrangle, and Mul- tangled Figures, and firil of the meafuring of the plain Triangles. 2. And the meafuring of Plain Triangles is either in the meafuring of the Sides and Angles, or of their Area and Superficial Content. 3. Plain Triangles in refped of their Sides and Angles are to be meafured by two forts of Lines, the one, is a Line of equal Parts, and by that the Sides mult be meafured, the other is'a Line of Chords, the Conftrudion whereof hatk , . been 5 ^ (l5comctrp 5 been fhewed in the fixth Chapter, and by that thei Angles muft be mealured, the Angles may in- '" deed be meafured by the Lines of Sines^, Tan- gents or Secants, but the Line of Chords being not only fufficient, but molt ready, it ihall fuffice to fliew how any Angle may be protraded by a Line of Chords, or the Quantity of any Angle found, which is protraded. 4. And firft to protrad or lay down an Angle to the Quantity or Number of Degrees propo- fed, do thus, draw a Line at pleafure as ^ D in Figure 5, then open your Compafles to the Num- ber of 60 Degrees in the Line of Chords, and letting one Foot in j4y with the other defcribethe Arch B G, and from the Point A let it be requi- red to make an Angle of 3 6 Degrees; op6n your Compafles to that extent in the Line of Chords, arid letting one Foot in B, with the other make a mark at G, and draw the Line A G, fo lhall the Angle BAG contain 36 Degrees, as was re- quired. 5. If the Quantity of an Angle were re- quired, as liippofe the Angle ^^6", openyou'r CompalTes in the Line of Chords to the extent of <5o Degrees, and letting one Foot in A, with the other draw the Arch B G, then take in your Com- pafles the dillance of BG, and apply that extent to the Line of Chords, and it will Ihew the Num- ber of Degrees contained in that Angle, which in our Example is 36 Degrees, 6. In every Plain Triangle, the three Angles are equal to two right or 180 Degrees, there- fore one Angle being given, the lum of the other two is alfo given, and two Angles being given, tlie thirdjs given alfo. 7. Plaia Irt of ^uc^epmg. 3 3 , 7. Plain Triangles are either Right Angled or Oblique. 8. In a Right Angled Plain Triangle, one of the Acute Angles is the Complement of the other to a Quadrant or 90 Degrees. 9. In Right Angled Plain Triangles, the Side lubtending the Right Angle we call the Hyfotemfe, and the other two Sides the Legs, thus in Fig. 5. AE isthsHyptenufcy and AD and ED are'the Legs ^ thefe things premifed, the feveral cafes in Right Angled and Oblique Anglfed Plain Tri- angles may be refolved, by the Propohtlons foL lowing. Propofition L In a Right Angled PLiin Triangle, the Angles of ene Leg being given, to find the Hypotenufe and the ether Leg. In the. Right Angled Plain Triangle in Fig. 5. Let the given Angles bd DA E ^6, and DEA 54, and let the given Leg be AD\q6-., to find the Hypotenufe AE, and the other L,eg ED. Draw a Lihe at pleafure, as AD, and by your Scale of equal Parts fet from A to D /p-j6 the Quantity of the Leg given, then ereft a Perpen- dicular upon the Point D, and upon the Point A lay down your given Angle DAE by the fourth hereof, and draw the Line AE till it cut the Perpendicular DE, then meafure the Lines AE and DE upon your Scale of Equal Parts, fo fhally^JS 588.3 bethe Hypotenufe, and D E 345.8 tire other Leg. P • Propofitba i^jacticai d^comcttp, Propofition IL The Hypotenufe and Oblique Atgles given , to jind the Legs.- Let the given Hypotenttfe be 588, and one of the Angles 36 degrees, the other will then be 54 degrees, Draw a Line at pleafure,as AD, and upon the Point A by the fourth hereof lay down one of the given Angles fuppofe the lefs, and draw the Line A C, and from your Scale of equal Parts, fet off your ^Hypotemfe 588 from A JO E, and from the Point E to the Line AD Ictlfall the Perpendicular E D, then fhall AD be- ing raeafiired upon the Stale be 476 for one Leg, and£ D 345.8 the other. Propofition III. The Hypotenwik and one Leg given to find the An' gles and the other Leg. Let the given Hypotenufeht 588.and the given Leg 478. Draw a Line at pleafure as AD,i\^on which fet the given Leg from A to D. 476, and u]X)n the Point D, eredt the Perpendicular D E, then open your Compalfes in the Scale of Equal Parts to the Extent of your given Hypotenafe 588, and fetting one Foot of that Extent in A, move the other till it touch the Perpendicular D E, then and there draw AE, fb lhall E D he 345. 8 the Leg inquired, and the Angle DAE, will be found by the Line of Chords to be 36. whofe Comple- ment is the Angle DEA.^^. Propofition ntt of ^atfieping; 3^ Propofition IV. , Tlje Legs given to find the Hypoteriufe, and the Ohlitjue jingles. . Let one of the given Legs be 476, and the o- ther 545.8, Draw the Line AD to the extent of 476, and upon the Point Dy eredt the Perpendi- cular DE to the extent of 345.8, and draw the Line .y4£,folhall be the Hypotennfe 588, and the Angle DAE willbythe Lineof Chords be found to be 3 6 Degrees, and the Angle DEA$^y as before. i . Hitherto we have Ipoken of Right angled plain Triangles; the Propolitions following concern fuchas areObliqu'e. Propofition V. * Tm jinxes in an Oblique angled plain Lriatiglef being given, veith any one of the three Sides, to find the other tvpo Sides. In any Oblique angled plain Triangle, let one of the given Angles be 26.50 and the other 38. and let the given Side be 632, the Sum of the two given Angles being deduced from a Semi- circle, leaveth for the third Angle 115.50 De-. grees, fheii draw the Line B C632.and upon the, Points B and C protraft the. given Angles, and- draw the Lines BD and CD, which being mea- fured upon your Scale of equal Parts B D will be fou .dto be 3i2.43,and 431.05. D X Pfopofitiogr 3^ i^jactical^couietri^T K. PropoutLon .Vi. Two Sides if; An Oblique yingled Triangle being given.) with ari Angle offofite to one of thcm^ to find the other Angles and the third Side, if it be known rfhe- ther the Aigle Oppo/ite tp the-,other Sidp given be Acme or Ohmfe. ' , ' . ■ ■ : In sn' Obliqiie Angle4 Plain Triangle, let tfie given Angfe be sS DegregSiand let the Skle ad- jacent' to tliat Angle Te 6^2, and the Side oppo- - life'4- i I-, WIPQQ the' Line ^ C in f-VV. 2,5. prottadi, the given An^e 3^ Degrees uporitliePoint C, and draw the Line D C, thep. openyoOT Copipallcs to tSe Extent ,.df th'b; other,^^ide given 431. i' and ffftiiig-cnfe Pdot ih turn the other about till it touch the Line DC, which will be in two pla- ces, in the Points D and-£ *, if therefore the Angle at B be Acute the' third Side of the Triangle \viU he C£, according tlierefore to, the ^ecics of ithat Angle you mull draw the Line B D or 'B £ to cdmpleat the Triangle, and then you may meaiure the other Angles, and the third Sidehathbeeniliewed. r. r,. ^ I 'fr ',y ff '.". Propofition Vll. ,'J. ' Two 'Sidesof aii Ohliqm. AnglcdPlam Triangle be.- ing-gfv'cn') yvith the Angle,comprehended by them to fbidthe other Angles and the third Side. ' Let one of tlie^given Sides be 632, and the o- ther 431.1, arid let the Angle comprehended by them beDeg. 26:50, draw a Line pkafure, • .as o|, t\)z of 37 as B C, and by help of yonr Scale of Equal Parts* fet off one of yonr given Sides frpnv B to C6ii- then upon the Point B protracl the given Angle 26.50. and draw the Line B D, and from B to D, fet off your otlrir given Side 431. i. and draw the Ljne D C, fo have you conibituted the Triangle B D Q in which you may ineaiurc the Angles and the third Side, as hath been fhewed. Propofition VIIL The three Sides of an Oblique jirtsded frian^te being given., to find the Angles. Let the length of one of die given Sides be 6 3 2, the length of another 431. i, and the length of the third 312.4, and Draw a Line at pleafure, as 5 C in F.ig. 25, and by help of your Scale of E- qual Parts, fet off the greateft Side given 632 from B to C. then open your Compalles in the fame Scale to the extent of either of the other Sides, and fetting one Foot of your Compalles in B, with the other defcribe an occult Arch, then extend your Compalles in the fame Scale to the length of the third Side, and fetting one Foot in C with the other defcribe another Arch cutting the former, and from the Point of Interfedion draw the Lines B D and D C. to conilitute the T riangle B DC, whole Angles may be mealured, as hath heen hiewed. And thus may all die Cafes of Plain Triangles be relblved by Scale and Compafs, he that delires to refolve them Arithmetically, by my Trigome- tria Britannica, or my little Geometrical Trigo- flometry 3 only one Cafe of Right Angled Plain D 3 Idiangles sB i^iacti'cal iangles which I fhall have occafion to ufe, in . • the finding of the Areaoi the Segment of aCir- cle I will here ftiew how, to reiblve by Numbers. Propofition IX. In a Right Angled Plain Angle the Hypote- nufe and one Leg being given to fnd the other Leg. Take the Sums and difference of the iLypotenHje and Leg given, then multiply the Sum by the Dif- fefence, and of the Produft extraft the Square Root, which Square Rootffiall be the Leg inqui- red. ' ■ Example. In Fig. 5. Let the given Hypotenuse he AE 588.3, and'the given Leg^2)476, and jet DE be the Leg inquired. The Sum of AE and A D h\ 064.3, and their Difference is 112.3, i now then if yOu multiply 1064. 3 by 112. 3, the Produft will be 119520. 89, whofe Square R005 is the Leg 2) £. 345.8. \y. , ^. Propofition X. "The of a Right Angled Plain Triangle being gived, to find the Area or Superficial Content thereof. Multiply one Leg by the other , half the PioduLl fhall be the Content. Example, In the Right angled plain Triangle A D E, let the given Leg:, be AD ey-]6, and 2)£ 345, and let the Area of that Triangle be required, if you multiply 476 by 345 theProdud will be 164220, and the half thereof 82110is the Area o.rSuperficial Con- tent required. • - ■ Propofition o|, ti^e Urt of ^urijcpmg. 5; Propofition XI. Thf Sides of an Oblique angled plain Triatigle be- inggiven to fndtheAxc^ or Superficial Content there- Add the three Sides together, and from the half Sura fubtrad each Side, and note their Dif- ference j then multiply the liaif Sum by the faid Differences continually, the Square Root of the lafl; Produd, lhall be the Content required. Example. In FjV. p. Let the Sides of the Tri- angle JBChc AB 20. AC 13, and BC \ i the Sura of thefe three Sides is 44, the half Sum is 22, from whence fubtrading AB xo.^ the Dif- ference is 2 , from whence alfo if you fubtrad 13, the Difference is 9, and laMy, if you fubtrad BC \ i from the half Sum 22, the Diffe- rence will be 11. And the half Sum 22 being multiplied by the firff; Difference 2, the Produd is 44, and 44 being multiplied by the Second Dif- ference 9,the Produd is 396, and 396bemg mul- tiplied by the third Difference 11, the Produd is 4356, whole Square Root 66, is the Content re- quired. Or thus, from the Angle Ciet fall the Perpen- dicular D C, fo is the Oblique angled Triangle ABC^ turned into two right, now then if you meafure D C upon your Scale of Equal Parts, the length thereof will be found to be 6. 6, by which if you multiply theBafe AB 20, the Produd will be 132.0, whqfe half 66, is the Area of the Trh angle, as before. D 4 Propofiti- 40 |^jacti'cal(25eometrpj «., Propofition XII. The Sides of any Oblique angled Quadrangle he-^ tng given , to find the Area or Superficial Contenp thereof. , Let the Sides of the Oblique angled Quadran- dBE D in Fig. 11, be given, draw the Oiago- nal dE, and alio the Perpendiculars D C and B F, then meafuring AE upon the fame Scale by which the Quadrangular Figure was protracted, fuppole you find the length to be 652, the length of DC 312, and the length of BE 136, if you multiply AE 632 by the Flalf of DC$6j the Produdt will be 35392 the Area of AC ED. In like manner if you multiply AE 632, by the half of BE 6% , the Produd will be 42976 the Area of CE B, and the Sum of thele two Produfts is the A'^^a of AB E D ns was required. Or thus, take the Sum of F) C112 , and B F 136, the which is 248, and multiply £ 6 3 2 by half.that Sum, that is by 124, the Produd will be 78368 the Area of the C^adrangular Figure ABED) as before. rPropofition XIII. The Sides of a plain irregular mult angled Figure being given, to fnd the Content. In Fig. 26. Let the Sides of the multanglcd Fi- gure, A. B. C. D. E.F.G. FJ. be given, and let the Area thereof be required , by Diagonals drawn from the oppofitc Angles reduce the Figure given, into 1 oi> tl)c 3Irt of ^urbcjJing. 41 ■' kto Oblique angled plain Triangles, and thofe * * Oblique angled Triangles, into right by letting fall ot Perpendiculars,then meafure the Diagonals and Perpendiculars by the fame Scale, by which t the Figure it felf was protraded, the Content of thofe Triangles being computed, as hath been lhewed,niall be the Content required t thus by the Diagonals JIG. BE and £ C the mul- tangled Figure propounded is converted into three Oblique angled quadrangular Figures , AFGH. AFEB and B E DC, and each of thcfe are divided into four Right angled Triangles , whofe feveral Contents may be thus computed. Let 6"^ 94 be multiplied by half HL 27 more Half of KF 29, that is by 23, the Produd will Me be 21, be the Area bf AFiG F. Secondly, O B Ifci is 11, and FiV 13, their half Sum 12, by which jid if you multiply AE 132, the Produd will be G is 1584 the Area AF EB. Thirdly, let £ p be ]8 rnD 32, the half Sum is 25, by which if you Bf multiplyT4£C 125 the Produd will be 3125 the Area of BE DC, and the Sum of thefe ilk Produds is 6871 the Area of the whole irregu- lar Figure. AB CD EE G FJ, as was required, Proportion XIV, The N:'mber of Degree! in the Sedor of a Circle ■ffl heifjg given, to fnd the hxt?Lthcrcof. In Fig. 27. AD EG IS the Sc^or of a Circle, in 'i- which the Arch D EG, is Degrees. 23.50,andby Ik I. Prop, of A,'chimed, de Dimenfione Circuli, the ,-n length of half the Arch is equal to the Area bf n the ScVior of the doisblc Arch, there the length. of 42 i^jactical d^eometrp ? , of DE or £ (7 is equal to the Are4. of the SeUor, > AD EG: and the length or circumference of the whole Circle*whofe Diaraqtex is i according to Van Calen , is. 3. 14159265358979, thereiore the length of one Gentefine ,of a Degree, is. o. 01745 329259. Now then to find the length of any Number 01 Degrees and Decima,! Part^, you Hiuft: multiply the aforefaid length of one Gen- tefme by the Degrees and Parts given , and the Produft lhall be the length of thofe Degrees and Parts required, andthe of a containing twice thofe Degrees and Parts. Example, the half of DEG 23.50 is DE or EG 11.75, by whichif youmultiply 0.01745329259, thePro- duft will be 2050761879325, the length of the Arch p £, mid the Area of the Selior AD EG. Propofition XV. The Natnber of Degrees in the Segment of a Circle heinggiveny to find the Area of that Segment. In Fig. 27. Let the Area of the Segment DEG k be required, in which let the Arch DEG he. Degrees 23. 50, then is the Area of the | Sctlor ADEG 205076187932-5 by the lalt a- foregoing, from which if you dedudthe Area of the Tr iangle A DG,iht remainer vdll be the Area of the Segment DECK. And the Areao^ the Tri- angle ^DGmayfhusbefound. DK is the Sine of D E 11-75 ,which being fought in Gellibrand''s De- cimal Ganon is. 2036417511,and ^ AT is the Sine of DH -jS. 25, ortheCoiine of DJS. 9790454724, which being multiplied by the Sine of D £,the Pro- dud will be 199 3745 344-, or if you multiply A G Pentahedu^) F'lj.io • 0?, tl)c Urt of 43 . the Radius by half DF the Sine of the double Arch X> £ G", the Produft will be 19937453445 as be- fore, and this Produft being dedudled from the Areao^ the SeSlor ADEG 2050761879325, the remainer will be 57016434875 th.tAreA of the Segment J)EG L,^s ivas defired. Propofition XVL The Diameter of a Circle being cut into any Num- her of Equal Parts, to find the Area of any Segment made by the Chord Line drawn at Right Angles through any of thofe equal Parts of the Diameter. In Fig. 28. The Radius A Dhmt iuto five E- qual Parts, and the Segment EDF L is, made by the Chord Line ELF Right Angles to AD in the fourth Equal Part, or at eight tenths there- of; now then to find the Area of this Segment we have given AE Radius, and A £ 8, and there- fore by the ninth hereof E L will be found to be 600000, the Sine of £ £) 36,87, by which if you multiply o. o 1745 3 2, the Produft is the Area of the SeAor AE D F 64350286, and the Area of the Triangle AEF is 48, which being deducted from the Area of the SeBor, the Remainer 16350286 is the Area of the Sector EDF L, as was required. And in this manner was that Ta- ble of Segments made by the Chord Lines cutting i\\tRacUus into ico Equal Parts. Another way. In Fg. 28. Let the Radius AD he cut into 10.100 or 1000 Equal Parts, and let the Area ' • • the 44 i^iactical (i^cometrp *, the Segments made by tlie Chord Lines drawn at. Right Angles through all thofe Parts be required: firft find the Ordinates G K and Ad. PN. E L, the^ double of each Ordinate, will be the Chords of the feyeral Arches, and the Sura of thele Chords beginning with the leaft Ordinate, will orderly give you tlie Area of the fevcral Segments made by thofe Chord Lines, but the Diameter mull be be divided into looooo Equal Parts, becaufe of the unequal difierences at the beginning of the Diameter: but taking the Area of the Circle to be 5.1415926535, &c. as before, the Area of the Semicircle will be 1. 57079632, from which if you deduct the Chord GH 1999999 , the Chord anfwering to 999 Parts of thtRadim, the reraainer is. i. 56879632 the Area of the Seg- ment G DH. And in this manner by a conti- nualdedudion of the Chord Lines from the Area of the Segment of the Circle given, was made that Table fnewing the Area of the Segments of a Circle to the thoufandth part of the RadUu. And becaufe a Table fhewmg the Area of the Segments of a Circle to the thoufandth part of the Radim, whofe whole Area is Unity , is yet more ufeful in CommonPradlice, therefore from this Table, was that Table alfo made by this Pro- portion. As the Area of the Circle whofe Diameter is Unity, tp wit 3.14149 is to the Area of any part of that Diameter, lb is Unity the fuppofed Area of another Circle, to the like part of that Diameter. Example,anfwering to 665 parts of the Radim of a Circlewhofe Area is 5.14159 is 0.91354794 therefore, A? 3.14159265 is to 0.91354794-; So is i. to J- 45 the Table rts of the fference to nfequently ;ter: The ;n we come 'lances. )ter, how ured, not :s , but in of the A- t which is leof thofe d, Glafs, asalfothe eft we will ances» :s and Di- nks, each ave a Qua- the larger :he Angles z, four or divided in rees, and :,at every J 0.40 50. f the C^a- drant iiy ?F! iff' ; 1 li Nr : -I ■ i.Mf'l'' 44 the Segment Right Anglt firft find the double of e: the feyeral, beginning v give you tlv by thole Ch( be divided i the unequal Diameter; be 3.14159 the Semicin if you dedi Chord anfw reraainer is. ment GD h . nual dedudi of the Segr that Table i aCirdetot And bcc; Segments ol the Radim, more ufeful this Table, portion. As the Unity, tq ■ part of that Area of an( Diameter, parts of th 14*15 9 ' As 3.14 H' iQaum o;, tbc I^Ct of ^tltbcpiltj. 45 , to 2907pi,' the Area required •, and the Table being thus computed to the looo parts of the Radhu, we have enlarged it by the difference to the 5000 parts of the Radim, and confequently to the ten thoufandth part of the Diameter: The ufe of which Table fhall be (hewed when we come to the meafuring of Solid Bodies. CHAP.. IX. Of the Meafuring of Heights and Difiances. HAving (hewed in the former Chapter, how all plain Triangles may be meaiured, not only in relpcdof their Sides and Angles , but in refped of their Area, and the finding of the A- rta of all other plain Figures alio, that which is next to be confidered, is the pradical ufe of thofe Inllrudions, in the meafuring of Board, Glafs, Wainfcot, Pavement, and fuch like, asalfothe meafuring or furvey ing of. Land y and firft we will Ihew the meafuring of Heights and Diftances. 2. And in the meafuring of Heights and Di- ftances, befidesaChainof yoor looLinks, each Link being a Foot, it is neceflary to have a Qua- drant of four or five Inches-Radhts, and the larger the Quadrant is, the more exadly may the Angles be taken, though for ordinary Pradice, four or five Inches will be fufficient. Let fuch a Quadrant therefore be divided in the Limb into 90 Equal Parrs or Degrees, and numbred from the left hand to the right, at every tenth Degree, in this mamier 10.20.30.40 50. 60. 70.80.90. and v.'ithin the Limb of the Qua- drant 46 i^^acttcal (i^eometcp 5 drant draw another Arch, which being divided by ,. help of the Limb into two Equal Parts, in the Point of Interfeftion fet the Figure 1. reprefent- ing the Radius or Tangent of 45 Degrees, and from thence both ways the Tangents of 65.44 Deg. 71. 57 Deg# 75.97Deg. ;78.7oDeg.8o. 54 I Deg. that is, 2. 3.4.5 and 6 being fet alfo , your Quadrant will be fitted for the taking of Heights feveral ways, as fhall be explained in the" Propofitiohs following.. ProiX)fition I. To find the Height of a Torver, Tree ^ or other Object sUone Station. At any convenient diflance from the Foot of the Objed to be meafured, as fuppofeat Cm Fig. 30. and there looking through the Sights of your Quadrant till you efpie the top of the Qbjed at >^,obferte what Degrees in the Limb are cut by the Thread, thofe Degrees from the left Side or Edge of the Quadrant to the Right, is the Quan- tity of the Angle ACBi which fupjx)fe 35 De- grees-, then is the Angle 5'55 Degrees, be- ing the Complement of the former to po Degrees. This done with your Chain or otherwife mea- furethe diltance from B the foot of the Objed, to your Station at C, which fuppofe to be 125 Foot. Then as hath been (hewed in the i. Prop. Chaf. 8. draw a Line at pleafure as BC, and by your Scale of Equal Parts, fet off the diflance meafured from P to C 125 Foot, and upon the Point C lay down your Angle taken by obferva- tion 35 Degrees, then ered a Perperiditular upon the oj, t!)e Htt of ^urUepmg- 47 liedvl the Point 5, and let it be extended till it cut the in tt' Hypothenufal Line A C, fo ihall AB meafured on refen your Scale of Equal Parts, be 87. 5 Foot for the, I, zl Height of the Objeft above tlie Eye to which the Height of the Eye from the Ground being 80,5. added, their Sum is the Height required. , dnj ( Another way. dintl Let AB rcprcfent a Tower whofe Altitude you would take, go fo far back from it, that looking through the Sights of your Quadrant, to the top of the Tower at A the Thread may I cur juft 45 Degrees in the Limb, then lhall the diftance from the Foot of the Tower, to your Station, be the Height of the Tower above the FMtt Eye- Tj/,. Or if you remove your Station nearer and near- ofyji er to the Objed, till your Thread hang over the figures 2.3.4or 5 in the Quadrant, the Height J of the Tower at 2. will be twice as much as the diftance from the Tower to the Station, at 3. it will be thrice as much, &c. As if removing my \ Station from Cto D, the Thread Ihould hang o- J |j ver 2 in the Quadrant, and the diftance BD 62 Foot, then will 124 Foot be the Height of the T ower, above thefiy e. In like manner if you remove your Station ^ backward till your Thread fall upon one of thofe ^ Figures in the Quadrant, between 45 and 90 De- grees, the diftance between the Foot of the Tower, and your Station will at 2. be twice as jjj nmch as the Height, at 3. thrice as much, at 4. four times fo much, and fo of the reft. 43 (!5cometc^ 5 A Third yf^dy by a Station at Random. Take any Station atpleaftre fiippofe at C, and looking through the Sights of your Quadrant, obferve what Parts of the Quadrant theThread falls upon, and then meafure the diftance be- tween the Station, and the Foot of the Objeft, that diftance being multiplied by the parts cut in the Quadrant, cutting off two Figures from the Produd lhall be the Height of the Objed above the Eye. Example, Suppofe I ftanding at C, that the Thread hangs upon 36 Degrees, as alio upon 72 in the Quadrant which is the Tangent of the faid Arch, and let themealured diftance be 125 Foot, which being multiplied by 72, theProdud is 9000, from which cutting off his Figures be- caufe the Radimis fuppofcd to be 100, the Height inquired will be 90 Foot,' he that defires to per-' form this work with more exadnefs, muft make life of the Table of Sines and Tangents Natural or Artificial, this we think fufficient for our pre- Tent purpofe. Propolition II. To find an inaccejfihlc Hei^nt at two Stations. Take any Station atplcafiire as at D, and there looking through the Sights of your Quadrant to the, tbp of the Objed, obferve what Degrees are cut* bythe Thread in the Limb, which admit to be 68 Degrees, then remove backward, till the An- gle taken by the Quadrant, be but half fo much as 0?, tl^e Htt of ^urber>iii5. 49 ] as tlie former, that is 34 Degrees, then is the di- ' ftance between your two Stations equal to the Hypothenufal Line at your firft Station, viz.. ^D. if the diftance between your two Stations were ujfj 326 foot, then draw a Line at pleafure as B D, upon the Point D protrad, the Angle AD B 6^. ; Degrees, according to your firft: Obfervation, and OIju from your Line of equal parts fet off the Hy- pothenulal 326 Foot from D to A, and from the jij, J Point A let fall the Perpendicular AB which be- ing meafured in your Scale of Equal Parts, ftiail be the Altitude of the Objeft inquired. Or working by the Table of Sines and Tan- ' ; gents, the Proportion is. As the RadiHf:^ is to the meafured diftance or Hypothenufal Line A D-, lb is the Sine of the ^ Angle ^ D £, to the height A B inquired. 'CS i Another more General way, by any two Stations top pleafure. t ns . Admit the firft Station to be as before at D, '' and the Angle by obfervation to be 68 Degrees, and from thence at pleafure I remove to C, where obferving aim 1 find the Angle at C to be 32 De- grecs, and the diftance between the Stations 150 Foot. Draw a Line at pleafure as B C, and upon Clay down your laft obferved Angle 3 2 Degrees^ and by help of your Scale of Equal Parrs, fet off , your meafured diftance from C to D 150 Foot, then upon D lay down your Angle of 68 De- ^ grees, according to your firft Obfervation , and where the Lines AD and AC meet, let fall the Perpendicular A 5, which being meafured in your Scale of Equal Parts, fhall be the height of the Objeft as before. • E Ot 50 i^^acti'cal . 30. Then re- moving your Inftrument to D, lay the Ruler on the Diameter thereof, and then turn the whole In- ftrument about till through your Sights you cart ^ efpy the mark fet up at your firft Station at C, and there fix your Inftrument, and then upon the Cen- CiiE tre of your Circle turn your Ruler till through iitofi) the Sights you can efjiy the Objed whofe diftance iutO! is mqiiired, fuppofe at ^; and obferve the De- grees in the Circle cut by the Ruler, which let be ast 112, which is the Angle j4DC, and let the di- t;lfiance between your two Stations be DC 326 , Foot 3 fo have you two Angles and the fide be- 301' tween them, in a plain Triangle given, by which Up E i to 52 i^jactical (I5eometrp 5 to find the other fides, the which by protraftloii^ may be done as hath been fhewedjin the fifth Pro- pofition of Chapter 8. but by the Table of Sines and Tangents, the Proportion is. As the Sine of D AC, is to X» C-, fo is the Sine of ACD to the Side AD. Or, as the Sine o^DAC, is to the given Side DC. So is the Sine of ADC to the Side A C. 6. There is another Inftrument called the plain Table, which is nothing eire,but a piece of Board, in the fafhion and bignefs of an ordinary fheet of paper, with a little frame, to faften a fheet of pa- per upon it, which being alfo fet upc)n a Staff, you may by help of your Ruler, take a diftance there- with in this manner. Having meafiired the diftance between your two Stations at D and C, draw upon your paper a Line, on which having fet off your diftance place your Inftrument at yourfirft Station C, and laying your Ruler upon the Line lb drawn there- on, turn your Inftrument till through the Sights you can efpy the Station at D, then laying your Ruler upon the Point C, turn the fame about till through the Sights you can efpy the Objeft at A, and there draw a Line by the fide of your Ruler, and remove your Inftrument to D, and laying your Ruler upon the Line D C, turn the Inftni- ment about, till through the Sight you can efpy the Mark at C, and then laying your Ruler upon the Point D, turn the fame, till through the Sights you can efpy the Objeft at A, and by the fide of your Ruler draw a Line, which muft be extended till it meet with the Line AC, fo fhall the Line A D being rneafured upon your Scale of Equal Parrs, ^ oj, t^c 3rt of atljepi'ng. 55 . Parts, be the; diftance of the Objeft from Z), fand the Line A Cihall be the diftance thereof fromC. . 7. And in thismanner may the diftance of two, isfei three or more Objects be taken, from any two , . Stations from whence the feveral Objects may be S"®- feen, and that either by the plain Table, or The- odolite. itiiepi ' ~— ofBoJ C H A P. XI. i]k aeetO! Hovf to take the Plot of a Field at one Station^ from aSai whence the feveral Angles may he feen. sjcetk Although there are feveral Inftrumeats by which the Plat of a Field may be taken, larje yet do I think it fufficient to Ihew the ufe of thefe ■ dite two, the plain Table and Theodolite. onC 2,- In the ufe of either of which the lame chain ivdit!; which is ufed in taking of heights and diftances, ^ is not fo proper. I rather commend that which is known by the Name of Gunte-Fs Chain, which is .'jlJii four Pole divided into i oo Links ^ being as I con- ceive much better for the cafting up the Content ijj f; of a Piece of Ground,than any other Chain that I j(j jj; have yet heard of, whofe eafie ufe lhall be explain- jj lii ed in its proper place. gii( 3. When you are therefore entered the jig-f Field with your Inftrument, whether plain Table, ijSi; or Theodolite, having chofen out your Station, let vilible Marks be fet up in all the Corners thereof, and then if you ufe the plain Table, make j,j[ a mark upon your paper, reprelenting your Stati- on, and laying your Ruler to this Point, diredit fi E 3 your V '■ 11; r ■1(1 •' I i • ififpp » I'i. ■••■;■ ■ l-'r «:'■ 54 i^^actica! d^eomett:?, your Sights to the feveral Corners of the Field, where you nave caiifed Marks tobefetup, and ' draw Lines by the fide of the Ruler upon the paper to the point rcprefenting your Ration, then meafure the diftance of every of thefe -Marks from your Inftrument, and by your Scale fetthofe diftances upon the Lines drawn upon the paper, making linall marks at the end of every luch diftance, Lines drawn from Point to Point, lhali give you upon your paper, the Plot of the Field, by viihich Plot fo taken the content of the Field ' may ealily be computed. Example. Let Fig. 31. reprefent a Field whofe Plot is required,-, your Table being placed v/ith a Ihect of paper tliereupon, make a Mark about the middle of your Table, as at Ji. apply your Ruler from this Mark to B and draw the Line A By then with your Chain meafui'l the diftance there- of which fiippofe to be 11 Chains 3 6 Links, then take JI Chains 36 Links from your Scale, and fet thai diftance rfcm^ to By and at B make a mark. Ti .cn directing the Sights to C, draw a Line by the tide of your Ruler as before, and meafure the difta Kc A Cy whichiuppofeto be 7 Chains and 44 Links, this diftance mull betaken from your Scale, and fet from A to C upon your paper. An J in this manner you muft direftyour Sights from Mark to Mark, until you have drawn the Lines and fet down the diftances, between all the Angles in the Field and your ftation, which being done, you muft draw the Lines from one Point to another, till you conclude where you firft began, fo will thofe Lines BC. CD. DE. EG. and G By give yon the exadFigureof the Field. 4. To do this by the llieodolite, in ftead of drawing 0^, tlbe Mtt of ^utbeping. 5 s . drawing Lines upon your paper in the Field > P, aa! • you muft have a little Book, in which the Pages ion tk mult be divided into five Columns, in the firll in,tliai Column whereof you mult fet feveral Letters to tM fignifie the feveral Angles in the Field, from kH which Lines are to be drawn to your place of lepa^j. Handing, in the fecond and third Columns the ery Ik degrees and parts taken by your Inllrument, and nt, Ik' the fourth and fifth, to fet down your diftances he Fid: Chains and Links, this being in readinefs, and ;lieFid have placed your Inftrument dired your Sights to the firll mark at B, and obferve how many De- eldvik grees are comprehended between the Diameter lacedtii of your Inllrument, and the Ruler, and fet them farhk in the fecond and third Columns of your Book jplfje againfl the Letter B, which Hands for your firH mM Mark, then meafure the diHance AB as before, ccthcR' and fet that down, in the fourth and fifth Co- " fojtte lurans, and lb proceed from Mark to Mark, until jandfe .you have taken all the Angles andDiHances in atraii. the Field, which fuppofe to be, as theyare ex- I linti preifed in the following Table. ■skd ■ IMS® oDiyoc r. irSigk fawntk aaDik dikic Poiiitit :beg2i ?. aiii , I 5. Havingthustakenthe Angles andDiHances in the Field, to pcotiad the fame on Paper or * £4. Parchment, Defr. Fart Chains Links B 39 75 11 56 C 40' .75 7 44 I) 96 GO 7 48 E 43- 25 8 92 F 80 CO <5 08 G 59 9 $6 ?^?actfcai d^eomettp ^ Parchment, cannot be difficult •, for if you draw _ a Line at pieafure as £ 5 reprefenting the Dia-' meter of your Inflrument about the middle there- of, as at mark a Mark, and opening your Com- paffiesto 60 Degrees in your Line of Chords, up- on ^ as a Center delcribe a Circle, then lay your Field book before you feeing that your firft Ob- fervattion cut no Degrees, there are no Degrees to be marked out in the Circle, but the Degrees at Care 40.75 which being taken from your Line of Chords, you muft fet them from H to 7, and draw^ the Line ^ I. the Degrees at D are 96 which muft in like manner be fet from / to K, and fo the reft in order. This done obferve by your Field-book the length of every Line, as the Line at your hrft Oblervation was 11 Chains and 36 Links, which being by your Scale fet from ^ will give ■the Point B in the Paper, the fecond diftance be- ing let upon ^7 will give the Point C, and fo proceediug with the reft, you will have the Points B CD E F and C,by which draw the Lines B C. CD. D E • EF. F G and G B, and lb at laft yoii have the Figure of the Field upon your Paper, as was required; And what is here done at one ftation, may be done at two or more, by meaiiiring one or two diftances from your firft ftationi, taking at every ftation, the Degrees and diftances to as many Ai- gles, as gre vilible at each ftation. And as jfor taking the the Plot of a-Field by In- terfedion of Lines, he that doth but confiderhow the diftances of feveral Objeds may be taken at two ftations, will be able to do the other alfo, and therefore I think it needlefs, to make anyil- luftration by example. CHAP. ph t^e ^rt of ^urfjeprng. 5 7 CHAP. XII. fifovp to take the Plot tf a Wood, Parkjir other Cham- port Plain, by going round the fame , and making Obfervatiron at every ^ngle. By thefe Direftions which have been already given,may the Plot of any Field or Fields be taken, when the Angles may be leen alone or more ilations within the Field, which though it is the cafe of fonie Grounds, it is not the cafe of all; now where obfervation of the Angles cannot be obferved within, they multbeobferved without, and although this may be done by the plain Ta- ble, yet as I judge it may be more conveniently done by the Theodolite, in thefe cafes thereof I chiefly commend that Inflrument, I know fome ufe a MarinersCompafs, but the working with a Needle is not only troublelbra , but many times uncertain, yet ifaNeedle be joyned with the Theodolite the joynt Obfervations of the Angles may ferve to confirm one another. 2. Suppofe the Fig. 32. to be a large Wood whofe Plot you defire to take; Having placed your Inflrument at the Angle A, lay your Ruler on the Diameter thereof, turning the whole Inflru- ment till through the Sights you efpy the j Angle at K, then fallen it there, and turn your Ruler up- on the Center, till through the Sights you efpy your fecond Mark at B, the Degrees cut by the Ruler do give the quantity of that Angle?5^iif, fuppofe 125 Degrees,and the Line AB 6 Chains, 45 Links,which you mull note in your Field-book, as wasfliewed before. 3. Then 58 ia?articdl (geometer ^ 3. Then remove your Inftrument to'5, and | laying your Ruler upon the Diameter thereof, turn it about, till through the Sights you can efpy your , third mark atC, and there fallen your In- ftrunient, then turn the Ruler backward till through the Sights you fee the Angle at the Degrees cut by the Ruler being 106. 25 thequjm- tity of the Angle AB C, and the Line B C contain- ing 8 Chains and 30 Links, whicli note in your Field-book, as before. 4. Remove your Inftrument unto C, and laying the Ruler on the Diameter thereof, turn the In- llrument about till through the Sights you fee the Angle at D, and fixing of it there, turn the Ruler upon the Center till you fee your lafl llation at By and oblerve the Degrees cut thereby, which lup- pofe to be 134 Degrees,and the Line CD 6 Chains 65 Links, which mull be entered into your Field- book alfo, and becaufe the Angle 5 CD is an in- ward Angle, note it with the Maik > for your better remembrance. 5. Remove your Inftrument unto D, and laying the Ruler on tlie Diameter, turn the Inftrument a- bout , till through the Sights, you fee the Angle at jB, and there fixing your Irftlrument, turn your Ruler backward till you efpy the, Mark at C, where the Degrees cut are, fuppofe 68.0 and the Line DE 8 Chains and 23 Links. 6. Remove your Inftrument unto £, and laying the Ruler on the Diameter, turn the Inftrument about, till tlirough the Sights you fee the Angle at F, and there fix it, then turn the Ruler back- ward till you-fee the Angle at D, where the De- grees cut by the Ruler fuppofe to be 125 and the Line £ F 7 Chains and 45 Links. , 7. Re- \ o^> tlje of ^urbcfitig. 59 7. Removeyour InftrumentuntoF, and laying your Ruler upon the Diameter, turn the Inftru- ment about, till through the Sights, you fee the Angle at Gpwhere fix the fame, and turn the Ruler backward till you fee the Angle at £, where the Degrees cut by the Ruler are 70,and the Line F G 4 Chains 15 Links, which mult be fet down with this > or the like Mark at the Angle. 8. Remove your Inftrument unto G, and lay- ing your Ruler upon the Diameter, turn the In- ftrument about, till through the Sights you fee the Angle at //, where fix the fame, and turn the Ru- ler backward till you fee the Angle at F , where the Degrees cut by the Ruler are 65. 25, and the Line G H Chains 50 Links. 9. Remove your Inftrumient in like manner to H and K-, and take thereby the Angles and Di- ftances as before, and having thus made obferva- tion at every Angle in the Field, fet them down in your Field-book, as was before direded, the which in our prefent Example will be as follow- cth. A 151. OQ. 6.45 B 106. 25' 8.30 0 I 34. GO 6. 65 D 68.00 8.23 E 125.00 7-45 F> 70.25 4. 15 6' 65.25 — S-50 H 130.00 — 6. 50 K 140.00 —. II.00 The taking of the inward Angles B CD and JBFG was more for Conforipity fake than any neceftity. |&^acticaliitbeping. 6b CHAP. XIV. flovp to take the Plot of Mountainous and uneven Groundsf and how to find the Content. "V TT THen you are to take the Plot of any V V Mountainous or uneven piece of Ground, fuch as is that in Figure 3 3, you mult firll place your Inftrument at J?, and dired your Sights to i, meafuring the Line AB., obferving the Angle GAB, as was (liewed before, and fo proceed from 2? to C,and becaufe there is an aH cent from C to D, you mult meafure the true length ther-eof with your Chain , and fet that down in your Book, but your Plot mult he drawn according to the length of the Horizontal Line, which mult be taken by computing theBafe of a right angled Plain Triangle, as hath been lltevved before, and fo proceed from Angle to Angle until you have gone round the Field, and having drawn the Figure thereof upon your Paper, reduce into Triangles andTrapezias, as ABC. CDE-ACEF and AFG. then from the Angles B. C. D.F and G-, let fall the Perpendiculars,, 5 7C C-V. DZ,. F M. and G Fi. This done you mult meafure the Field again from Angle to Angle, fetting down the Diitance taken in a Itr.lightLine over Hill and Dale, and fo likewifo the feveral Perpendiculars, which will be much longer than the ftreight Lines meafured on your Scale, and by thefe Lines thus meafured with your Chain cait up the Content i which will be much more than the Horizontal Content of that Field according to the Plot, but if it fhonld be otherwife plotted, than by the Ho- rizontal ^4 i^jacticai ^eometrp? rizontal Lines, the Figure thereof could not be • ■ contained within its proper limits, but being laid down among other Grounds,would force fbme of them out of their places , and therefore fuch Fields as thefe muft be fliadowed off with ILlIs, if it be but to Ihew that the Content thereof is computed according to the true length of the Lines from Corner to Corner, and not according to their Diftance meafiired by Scale in the Plot. CHAP. XV. lioVv to reduce Statute Meafure into Cuflomary, and the contrary, T TT THereas an Acre of Ground by Statute V V Meafure is to contain i6o fquare Perches, meafured by the Pole or Perch of fixteen foot and a half-.tin many places of this Nation, the Pole or Perch doth by cuftom contain 18 foot, in fbme 20. 24. 28 Foot; it will be therefore re- quired to give the Content of a.Field according to luch feveral quantities of the Pole or Perch. 2. To do this you muff confider how many fquare Feet there is in a Pole according to thefe feveral Quantities. In 16. 5 tothe Pole, there are 272.25 fq. feet. In 18 to the Pole there are 324 fquare feet. In 20 to the Pole there are 400 fquare feet. In 24 to the Pole there are 576 fquare feet. In 28 to the Pole there are 784 fquare feet. Now then if it were defired to reduce 7 Acres, ' 3 Roods,27 Perches,according to Statute Meafure, into Perches of 18 Foot to the Perch 3 firft re- duce Oh t^e of ;^oU'DiB:. 6$ ■ .4uceyour given quantity, 7 Acres. 3 Rods, 27 •' Poles into Perches, and they make 1267 Perches* Then lay, as 324. to 272. 25. fo is 126710 106^.6. that is 1065 Perches, and 6 teiithsof a Perch. But to reduce cuftomary Mealiire into fta- tutenaeafure, fay as272.25. isto 324 lb is 1267 Perches in cuftomary meaiure, to 1507. 8 that is 1507 Perches and 8 tenths of a Perch in ftatute meaiure, the like may be done, vrith the cufto- mary meafures of 20.24 and 28 or any othes mea- fure that lhall be propounded. CHAP. XVi. Of the Meaftiring of folid Bodies. HAving Ihewed how the content of all plains may be computed, we mre now come to the meafuring of Iblid Bodies, as Prifms, Pyramids and Spheres, the which ihail be explained inthb Propofitions following. Propofition. I. The hafe of a Brifm or Cylinder being given, to find the folid content. . The bale of 4Pn7»»iseithcr Triarignlar, as the Pentahedron , Quadrangular, as the Hexahedron^ or Multangular,or the Po/yWmz Prifm, all which muft be computed as hath been (hewed,which done if you multiply the bafegiven by the altitude, the pi oducft fi^all be the folid content required. Example. In an Hexahedron Prifm, whc^e bafi F , is 66 i^jactical d^eametcp •, is quadrangular, one fide of the Bafe being 65 foot and the other 43, the Superficies or Bafe will be 27.95. Which being multiplyedby the Alti- tudejluppofe 12.5. the produd. 359-37S* is the lb- lid content required. In like manner the Bale of a Cylinder being 45. 6. and the altitude 15.4. the content will be 702. 24. And in this manner may Timber be meafurcd whether round or lquared,be the fides of the fqua- red Timber equal or unequal. Example. Let the Diameter of a round piece of Timber be 2. 75 foot. Then, As 1 it to 785 3 P7' fi> is the fquare of the Diameter 2.75. to 5.9395 the Superficial content of that Circle. Or if the circumference had been given 8.64. then,As 1 is to 079578, fo is tlie Iquare of 8.64. to 5.9404 the fuperficial content. Now then if you multiply this Bafe 5.94. by the lengthjfuppofe 21 foot, the content will be 124.74. If the fide of a piece of Timber perfedly fquare be 1.15 this fide being multiply ed by it felf, the produd will be 1.3225 the fuperficial content, or content of the Bale, which being multiplyed by 21 tlielength, the content will be ;i7.7745. • Or if a piece of Timber were in breadth i. 15. in depth 1.5 the content of the Bafe would be 1. 725 which being multiplied by 21 the length, the content willbe.36.225. Propofition. IT. The Bafe and Altitude cf a Pyramid or Com being given) to fad the [olid eontent. Multiply o^t^e^dtucmgof 67 Multiply the Altitude by a third part of the Bafe, or the whole Bafeby athird part of the Al- titude, the Produftlhallbe the folid Content re- quired. Example. In a Pyramid having a Quadrangu- lar BafeasinF»g.22. The fide Cf 17. CD 9. 5, the Produd is the BafeCDjEF. 161. 5,which being multiplyed by 10.5 the third of the Alti- tude^F 3i.5theft-odudis 1695.75 con- tent. Or the third of the Bafe. -viz^ 53-& 3 being multiplied by the whole Altitude 11.$ the Produdt will be the content as before. 2. Example. InF«^. 21. Let there be given the Diameter of the Cone F3.5. The Bafe will be 96.25.whoIe Altitudelet be CD 16.92 the third part thereof is 5.64 & 96.25 being multipli- ed by 5.64, theProdudt 542.85 is the folid con- tent required. Propofition. III. The Axis of a S'^herebeinggiveriy to find the fo' lid content. If you multiply the Cube of the Axis given by 523 598 the folid content of a Sphere whofe At:- is is an unite,the Produd lhallbethe folid conteqc required. Example. Let the Axis given be 3, the Cube thereof is27,by which if you multiply. 523598, theProdud 14.137166 is the folid content re-c quired. F 2 ■ Propofition. t^jactfcal (5comettp 5 Propofition. IV. The Bafis and Altithde of thefruHiWClof a Tyro- mid orCone bein^giverty to fnd the content. If the aggregate of both the Bafes of the Frufi- an and the mean proportional between them, fhall be multiplied by the third' part of the Altitude, the Produd Ihall be the folid content of the Fruftum. Example. \nFig. 22. Let CDEF reprefent the greater Bafe of a Pyramid, whofc fuperhci- alcontent let be 1,92, and let the leffer Bafe be F/GLKO.S^ the mean proportional between them is. r. 2775 and the aggregate of thefe three numbers is.4.0475. Let the given Altitude be 15. the thirdpartthcreof is.yby which if you muj- t'ipIy4.©475theProdud 20.2-375 content of the FruHnm Pyramid. And to nnd the content of the Frufinm Cone. 1 fay. As. I. 1078539. fo 20.23 to,15. 884397, the content of the Cone required. Eutif the Bafes of the Pyramid lhali fquare, you may find the content in this man- *r. Multiply each Diameter by it fclf and by one another,and the aggregate of thefe Produds, by the third part of the altitude, the lafl Produd (hall be the content of the Frnjfum Pyramid. Example. Let the Diameter of the greater Bafe be 144,the Diameter of thcIeflerBafe 108, aird the altitude 60. filn 1 tl)c #.ea(.unn8 of ^9 * 'VheSquare of 144is 20736 ThcrSquare of 108 is 11664 ThePredudt of 144+108 is 15552 TheSumofthefe 3 Produdsis 4795-2^ Which being multiplyed by 20 the third pait of the Altitude, the Product 959040 is the con- tent of the Fmftum Pyramid. And this content being multiplied by. 785 39 the content of the Frufinm Cone will be. 75 3.228. Another vcay. Find the content of the whole Pyramid of the greater and lelTer Diameter, the leflcr content de- ducted from the greater, the remain (ball be the content of the To find the content of the whole Pyramid, you mull firft find their fe- yeral Altitudes in this manner. As the difference between the Diameters, Is to the lelfer Diameter. So is the Altitude given, to the Altitude cut off. Example. The difference between the former Diameter. 144.and 108 is 3 6,the Altitude 60. now then As 36.108:: 69.108. the altitude cut off Now then if yon mnlciply the lelfer Bafe 11664 60 the third part of 180 the Produdt 699840 is the content of that Pyramid. And adding 60 to 180 the Altitude of the great- er Pyramid is 240, the third part whereof is 8p, by which if you multiply the greater Bafe before found, 7P736,thcrrodudl is the content of the F J . greater 70 i^jacticd! ? greater Pyramid. 1658880, from which if you." ■ dednft the lefler 699840 the remainer 959040 is the coijtent of the Frufium Pyramid as before. And upon thefe grounds may the content of Taper Timber,whether round or fquare, and of Brewers Tuns, whether Circular or Elliptical, be computed, as by the following Propofitions Ihall be explained, Propofition. V. Tk)! breadth anddefth of aTaper piece of Stjuared Ttmbery both ends being given together with tho lengthy to find the content. Let the given Dimenfions, At the &)ttom be 5.75 and A 2.34 AttheTop. C2.i6and D.-i.Si And let the given length be 24 Foot, According to the lafl Propofition, find the A- rca or Superficial content of the Tree at both 1 ends thus. Multiply the breadth 3.75 0.574031 1 By the depth 2,34 0.369215 , The Product 8.7750 0.943246 7. Multiply the breadth 2.16 0,334453 By the depth 1.82 0.262451 TheProduftis 3-9528 0.596904 3, Multi- oi, tiijc or 71 'V Multiply the I. Content. 87750 0.945246 by the fecondcontent. i-PS^S 0.596904 And find the fquare root 5.8986 1.540150 0.770075 The Sum of thefe 18.6264 being raultiplyed by 8 one third of the length, the content will be found to be 149.0112. Thus by the Table of Logarithms the mean proportional between the two Bafes is eafily found, and without extracting the fquare Root, may by natural Arithmetick be found thus. ^4t Cx AhalfCmultiplyed by B-. And C more half ^multiplyed by D being added toge- ther and multiplyed by 30,the length lhall give the content. Example. 3.75 l-C. 1.08 Sum 4.8 3 B- 2.34 1932 1449 966 11.3022 C2.16 1^1.^75 Sum. D. 4.035 1.83 12105 32280 4035 7-38405 11.30220 The fum of the Products 18.68625 Being multiplyed by 8 thethirdof the length, the content will be. 149.490CO. The like may be done for any other. F. 4 6. Pro- 73 |3^actical d^eomett? 5 Propofition. VI, The Diii>ncters of a piece of Timber bein^given at the T3p and and Bottom^ together with the length, to. find the content. The Propofition may be relblved either by the Squares of the Diameters, or by the Areas of the Circles anlwering to the Diameters given,for which purpofe I have here annexed not only a Ta- ble of the Squares of all numbers under a thou- fand, but aTable fharing the third part of the Areas of Circles in fullmeafure, to any Diame- ter given under 3 foot. And therefore puttiug 5 = The Sum of the Tabular numbers anfwering to the Diameters at each end. X = The difference between thefe Diameters. Z.= the length of the Timber, C~ The content. Then i\S~[-XX. .^L. — C. If you work by the Table of the fquaresof Numbers, youmuft multiply the lefs fide of the Equation, by 0.26179 the third part of 0.78539 the Produd being mnltiplyed by the length, will give the content. But if you work by the Table of the third parts of the Areas of Circles ,in full meafure, the ta. bular Numbers being multiplyed by the length will give the content. Only infteadof the fquare of the difference of the Diameter, you inufb qike half the Tabular number anrwerin0,to that Dif- ference, and you fhaU have the content as be-- fore. Example. oi, tlje ^eafan'tig of 7.^ >. Let the greatefl; Diameter by 2.75, and the *lefs 1.93. Their difference is o.S 5 The fqnareof 2.75 is The fquare of i .9 3 is I The Sum of the Squares The half Sum The Sum of them is Halfthefquareof C.82 dedudt. The Difference is ^Vhich being multiplyed by TfeProduawillbc. 7.5625 3.7249. 11.2874 5.6437 16.9311 0.3362 16.5949 26179 1493541 •1161643 165949 995694 331898 4.344378871 Oy 74 i^^actCcaid^eometrp; Or by the Table of Areas. The Area of 2.75 is i-979^57 TheAreaof 1.95 is 0.975176 TheSum 2.955033 The half Sum x .477 516 TheSum of them 4.432549 Half the Area of 0.82 deduift 0.0S8016 The formerProdudl 4.344533 Which being multiplyed by 24 17378132 8689066 The content is 104268792 But becaufe that inmeafuring of round Tim- bcr the circumference is ufually given and not the Diameter, lhave added another Table by which the circumference being given, the Diame- ter may be found. Example. Let the circumference of a piece of Timber be 8 325220 lookingthis Number in the fecond column of thatTable,l find the next lefs to be 8.168140 and thence proceeding in a ftr eight' Line, Ifind that in the feventh Column the Num- ber given, and the Diameter anfwering thereun- to to be 2.65. and thus may any other Diameter be found not exceeding the three foot. The Proportion by which the Table was made, isthus. As I . to 3.i4i59fois the Diameter given, to the circninference required, Or • 75 ind the ;ircum- round being I Tree, t third reas of ider to imbers, in the mher at the con- iven,the (hewed, firftPro- 't of the purpofe )y which 7957747 Itiplying nference bers, the fferences. ed by ad- f the cir- rithra of numbers the 02, tl)e ^eafa ting of j^oiiDiS. 7 5 '. Or the Circumference being given, to find the Diameter,fay: As. i.to.o.3i83,fois thcCircum- ferencc given to the Diameter required. And although bythcfc two Tables all round Timber may be eafily mcafured, yet it being moreufual to take the Circumference of a Tree, then the Diameter, I have here added a third Table, (hewing the third part of the Areas of Circles anfwcring to any circumference under i o foot, and that in Natural and Artificial numbers, theufeof whichTable (hall be explained in the Propofition following. Proportion. Vll. The Circumference of a piece of round Timber at both ends, with the length heinggiven, to find the con- tent. TheCircumferencc of a Circle being given,thc Area thereof may be found as hath been fhewed, in the 7Chaptcr,Propofition 4.and by the firftPro- pofition of this-, and to find the third part of the Area, which is more convenient for our purpofe Itookathird part ofthe number given by which to find the whole,that is a third part of 07957747 that is 0.0265 2 582 and having by the multiplying this number by the fquareof the Circumference computed three orfourof the firft numbers, the reft were found by the firft and feconddifferences. ^ The Artificial numbers were computed by ad- ding the Logarithms of the Squares of the cir- cumference, to 8.42966891 the Logarithm of 0.02652582. And by thcfe Natural and Artificial numbers the 7^ j^^scti'cal (!50omettf5 *he content of round Timber may be found two- ways By the Natural numbers in the lame manner as the content was computed, the Diameters being given, and by the Natural and Artificial numbers b»th,by finding a mean proportional between the two Areas at the top and bottom of the Tree, as by Example fliall be explained. Let the given DImeniions, or Circumferences be At the Bottom . ,.0- • , AttheTop "5 Their difference IS 6.20 The tabular Numbers. Natural Artificial. Anfwerlng to 9.95 2.626162 0.4.18931 Andto 3.75 0.37301^ 9.571731 TheSumoftheLogarith. 9.990662 Thehalf Sumor Logarith.9S9300 9-99S33I The Sum of the Number is 3.988481 The Sum of the Natural Numbers is 2.9*91S i The half Sum 499190 The Sum of them 4.498771 Half the number anfwer. to.6..2o is 0.509826 The remainer is 3 • 9 8 8 945 Which being multiplyed by the length 24, the content will be 95.7 3468. Mr. Darling in his Carpenters Rule made eafie, doth propound a fhoi ter way, but not fo exaft, which is by theCircumference given in the middle of the piece to find the fide of the Square, name- ' 17 ^^eaftirCns of 77 'ly by multiplying the Circumference given by 28209, or 2821. whichfideof the Square being computed in Inches, and lookt in his Table of Timber meafure, doth give the content of theTree not exceeding 51 foot in length, the which way of meafuring may be as eafily performed by this T able. Example. The circumference at the top and bottom of the Tree being given 9.95 and 3.75 the Sum 13.70 IS The half thereof isthemeancircumfer, 6.85 Which fought in the Table,the Numbers are. The Natural number is 1.2446 5 7,which being multiplyed by 3 the Product is 3.733971, which multiplyed by the length 24, the content 89.615304. IS The Artificial number is TheLogaritlim of 24 is 0.095049 1.380211 The AbfoluteNumber 29.871 1.475260 Which multiplyed by 3)the Produft is 89613 Propofition. VIII. "The Diameters of a Brewers Tun at top and bottom- being given with the height thereof to fnd the con- tent. In Fig.ig. Let the given Diameter. Altlt-S. IncWs. At the bottom. KG 152 HF 144 ^ The which by the 5 Propofition of diis Chap, may tIiUsbecornp. by which the content may be found in Ale Gallons in this manner. The former Produd 5.999944 The Logarithm of 0.00092846 6.967710 Thecontent in Gallons 928.24 2.967663 Prcpofition. IX. The Diameters of a clofe Cask^, at head and bang mth the length given, to find the content. In the refolving of this Propofition, we are to conilder the feveral forms of Casks, as will as the kind of the Liquor, with which it is filled, for one and the fame Rule .will not find the con- tent in all Cask. And oj, tlje S^cafurmg of doling, 79 'Wii And a Coopers Cask is commonly taken, ei- Mi * ther for the middle FnifiHm of a Spheroid, the middle Frujiimot a Parabolical Spindle, the mid- Vb die Frnfium of two Parabolick ConoidSjOr for the middle Frujhm of two Cones abutting upon one common Bafe. And the content of thefe leveral Casks may be 2304 found either by equating the Diameters, or by e- — quating the Circles, for the one, a Table of pppjii Squares isneceirary,and a Table (hewing the third 4119^ part of the Areas of a Circle to all Diameters. — The making of the Table of Squares, every one 541^,. knows, to be nothing elfe but the Produd; of a Number multiplyed, by it felf, thus the Square liiiii of i is 9. the Square of 8 is 64 and fo of the reft. )ero;t And the Area of a Circle to any given Diame- lieK tcr may be found, as hath been (hewed, in Chap.7 Proportion 2. But here the Area of a Circle in In- iSili ches, will not fufiice, it will be more fit for ule, if cone the third part of the Area be found in Ale and ct, Wine Gallons both, the which may indeed be donebydividingthewhole Area in Inches by j and the quotient by 282 to make the Table for p(5- Ale-meafure, and by 231 to make the Table for y6-;: Wine-mcafurebut yet thefe Tables may be more readily made in this manner. The Square of any Diameter in Inches, being divided by 3.81972 will give the Area of the Cir- cle in Inches •. And this Divifion being multiply- edby 282 will give you 1077.161 for a common Divifion, by which to find the Area in Ale-Gal- jjitj Ions, or being multiplyed by 231 the Produft, 882.355 will be acoramou Divifion by which to , ^ find the Area in Wine-Gallons. But becaufe it is caller to multiply then divide: ,t if ■Kr ■*■ •!l> «!^ %■ "..V. a -4'..! '41 ■ Jll"! I •, I ■ / * ! ' ifllU'i?] ' l-i.. «l So d^eometr^ 5 If you multiply the feveral Squares by 26178'" the third part of 78 5 3 9 the Produft will give the > Area in Inches, or if you divide. 26179 by '282 the quotient will be.00092886 for a common Mul- tiplicator, by which to find the Area in Ale-Gal- Ions, or being divided by 231 the quotient wiH be00II333a commonMultiplicator, by which to find the content in Wine-Gallons. An Exam- pie ortwowiilbefufficient for illuftration. Let the Diameter given be 32 Inches, the Square thereof 1024being divided by 3.81970 the quo- tientis268.o83, and the fame Square 1024 be- ing multiplyed by 261799, the Produft will be 268.082. Again if you divide 1024 by 1077.161 the quotient will be 9508, or being multiplied by 00092836,the Produd will be 9508. Lallly if you divide 1024 by 88 2.7 5 5,the quo- tient will be 1.1605, or being multiplied by CO 113 3 3 3 the Produd is 1.1605, And in this manner may the Tablesbe made for Wine and Beer-meafure, but thefecond differen- CCS in thefe Numbers being equal, three or four Numbers in each Table being thus computed,the reft may be found by Addition only. Thus the Scjuares of i. 2. 3. and 4 Inches arc. 1.4.9 and 16 by which ifyou multiply 00113 3 3 3, the ieveral Produds will be third part of the Area, of the Circles anfwcririgto thofe Diameters in Wine-Gallons. Or 00092836 being multiplied by thofe Squares, the feveral Produds, will be the thirdpartof the Areas of the Circles anfwering to thofe Diameters in Ale-Ga'ldns-, the which v.'ith their firft and fctond difierences are as fol- Icwcth. The oj,t^e^ea{udt?8of 8i by 'lliiy'i * Produdls or Areas in Wine-Gallons* ®oa!' mU 00113333 _ 1«B. :^-Coo4SSVi^ "MM rWr 3-COIOI9997 ^ ^ 226666 4.>"8.„23 79653. The Produds in Ale-Galloas. I?5 '!•;) 00092836 278tfoS ' ' lX-.\lll< ^ T JJ/ wtipk And by the continual addition of the fecoud^ differences to the firft, and the firft differences to jf,® 5. produdts before found,the Table may be con- iltipM tinned as far as you pleafe, J . The conftrudtion of the Tables being thus jbem. ftiewed: We will now fhew their life in finding- omdiE the content of any Cask, ireeoi Let5—the Sum of the Tabular Numbers an- oapott- fweringto the Diameters at the Head and Bung. D == their differ ence X = the difference of the Indie*; Diameters themfelves. i= the length of the 0011!:* Veffel,andC = the content thereqfj . 1,, oftbet If a Cask be taken for the middle liaffiKf; of a Spheroi^ intercepted between two Planes iiiltipk' parallel, cutting the Axis at right Angles: Then dk; ji5 4V*D«X=C. . i oiifi'-- 2. If a Cask be taken for the middle FrH^um " die «'• of a parabolical Spindle, intercepted between are 21 two planes parallel cutting the Axis at right , Angles. Then 11 S 4 a jD * L = €. G • 3- If TJ i| m f, . % \c ■^ 'f *> , "■;»■; ;,!^i fJ:,:* '2 irf". ii?; ^ ■ -■ -jiioi I ,>. tj.-■ U!^ ■«i!: ..0. I !!>•»' MT't; I H '1 i!!r ■ 't 82 )^|actical (0comett;^ ? 3. If a Cask be taken for the middle Frnftnm '^ of twoParabolick Cbndids, abutting upon one common Bafe, intercepted between two Planes- parailely cutting the Axis at right Angle: Then I : « A == C, 4- if a Cask be taken for the middle FnifiHm. of two Cones, abutting upon one common bafe, intercepted between two Planes parallel cutting the Axis at Right Angletv Then i s-S-— 5 AT. * L In all thefe four Equations, if you work by the Table of Squares of numbers, you muft muki- ply the lefs fide of the Equation by 262, if you would have the coutentin Cubical Inches; by C01133 if you would have the content in Wine- Gallons-, afldbyooopiSjif ydu Would have the content in Ale-Gallons. But if you work by the Tables of the third parts of the Areas Circle> the Tabular Numbers being multiplyed by the length only will give the content fequired, only in the fourth Equation inftead of half the Square of the Diftetence of the Diameters, take half the T:^ulai Number ahlWeringtOthfitdlflerence, and you lhall have the cxintent required; as by the following Exam- pies willbetter nppear, then by many words. Examples, in VV ine-mcafureby the Table of the Sqiiares'of Numbers. - • ■ The Diameter of aVellel At the Bung being 32 Inches. At the Head 22 Indies. The difference of the Diameters to Inches. And the kngthof the VeHH 44 Inches. Spheroid I! |A;|, f •'* 'V Vi*-' • sK'*' .'' , fht, 04 i^jaftica! d^cometty i Parabolick Conoid Cone. 1024 1024 ' 484 484 1508 1508 754 754 r 50 2262 2212 2262 2212 6786 ' 6636 • 6786 6636 6786 6636 25635246 25068596 44 44 102540984 200274384 102540984 100274384 J 12.79508241 110.30182224 u?, tlje of 8% This which hath been done by the Tabic of Squares may be more eafily performed, by the Table of the third part of the Areas of Circles, ready reduced to VVine-Gallons. , ^ Spheroid Parabolick Spindle, 1.16053 1.16053 0.54853 C.54853 1.70906 1.70906 85453 85453 30600 ' 61200 2.86959 2.624790 44 44 1x47836 1049916 1147836 10499160 126.26196 . ! H 115.490760 G 3 Parabolick * Examples in Ale-meaf^re by tlie Table of the Squares of Numbers. Spheroid. Parabolick Spin^e. 1024 1024 484 484 1508 2508 7S4 754 270 540 2532 2316.0 00092836 • 00092836 22758* 20844* 5064* 4632* 20256* 18528* 759<5* 6948* 15192 138960 { 235660752 2.150081760 44 44 948623008 860032704 940643C08 860032704 103.22673088 94.60359744 G 4 Parabolick p^, tbe ^eafutttig of ^oitp^. 8p By the Areas of Circles. Spheroid. Parabolick Spindle. 0.95052 0.44930 1.39982 .69991 .25061 2.35034 H 940136 940136 0.95052 0.44930 1.39982 69991 050122 2.149852 44 8599408 8599408 103.41496 94.593488 Parabolick 90 . content 126.25 Gallons, andlet the wet part of the Diameter be 12 Inches, firftHay. As the whole Diameter 32. is to the wet part 12. fb is 10.000 to 3750, which being fought in the Table, Ifind, the Area of tliat Segment to be.34.2518 which being muftiplyed by the whole content of the Cask 126.25, the Produdt is 43.2428975oand therefore there is remaining in ^e Cask 43 fere. 2. Example. Inthe fame Cask let the wet part ' of the Diameter be 18 Inches. Ifay. As 3 2.18 10000.5625 whofe Complement to 10000 is 43 75 which being fought in the Ta- ble, I find the Area anfwering thereto to be 4206305 now then Ifay. Asthe whole Area of the Circle 1000000 is to the whole content of the Cask 126.25. So is the Area of the Segment fought.420630, to the content 53.10443 75 which is in this cafe the content of the Liqnor that is wanting, this therefore being deducted from the content of the whole Cask, 136.25. the part remaining in the Vefldis.73ii455625. Thus may Casks be gauged in whole or in part, in which a Table of Squares is fometimes necellary, as Ixiing the Foundation, from whom the other Tables are deduced*, fuch a Table therefore is here exhibited, for all Numbers un- der 1000, by help whereof the Square of any Number under 10.000 may eafily be found in this manner. j The Reftnngle made of the Sum and Dificrence of any two Numbers, is equal to the Difference of the Squares of thefe Numbers. Example, Let the given Numbers be 3 6 and 8 5 their ^ea(unns of i^oitDia:. 9 5 P® their 5um Is 121,their difference 49, by which if •you multiply 12 i,the Produft will be 5929. The Square of 36 is 1296,andtheSquareof 85^7225, ^ the difference between which Squares is 5929 as ;iiiK before. ifii And hence the Square of any Number under rA 10.000 may thus be found, the Squares of all aink Numbers under 1000 being given. Example. Let the Square of 5715 be required, ffet; The5quareof571 bytheTableis326o4i,there- fore the Square of 5710 is 3 26 04100: the Sum of ipte 57ioand57i5is ii425,andthedifference5, by tk; which ifyou multiple i i425,theProdu!ff: is 52125 ;oi5 which being added unto 32604100 the Sum 32656325 is the Square of 5715. The like may )coc; be done for any other. zaJ; iisc itoi; —— — t J « TABLES ill iffli ofi Elil ' "crti i; TABLES FOR THE Meafuring O F TIMBER^ AND THE GAUGING CASKS AND Brewers Tuns. LONDON, Printed for Thomas Pajfmger at the three Bibles on London-Bridge,' 1679. ) 3 Cable of Square's?. 99 I I 54 j 1156 69 67 4489 1351 2 4 35 1225 71 68 4624 137 3 09 36 1296 73 69 4761 139 4 16 7 37 1369 75 7Q 4900 HI S 25 9 3^ 1444 77 71 5041 143 6 36 11 ?9 1521 79 72 5184 145 7 49 13 40 1600 81 73 5329 147 8 64 15 41 1681 83 74 5476 149 9 81 17 42 1764 85 75 5625 151 10 100 19 43 00 87 76 5776 '53 11 I2I 21 44 1936 89 77 5929 15s 12 144 23 45 2025 91 78 6084 157 ' 3 i6q 25 46 21 I 6 93 79 6241 159 14 196 27 47 2209 95 80 6400 161 225 29 48 2304 97 81 6561 163 i 6 256 31 49 2401 99 82 6724 165 17 287 33 5^ 2500 lOl 83 6889 167 18 324 35 51 2601 103 84 7056 169 19 361 37 52 2704 105 05 7225 171 20 400 39 53 2809 107 06 7396 173 21 441 41 54 2916 109 87 7569 '75 22 484 43 55 3025 111 88 7744 177 23 s;29 45 56 3136 113 89 7921 179 24 576 47 57 3249 115 90 8100 181 2<> 625 49 5« 3364 117 91 8281 183 26 676 51 59 3481 119 92 8464 185 27 729 53 60 3600 121 93 8649 187 28 784 55 61 3721 123 94 8836 189 29 841 57 62 3844 125 95 9025 191 30 900 59 63 3969 127 96 9216 193 31 961 61 64 4096 129 97 9409 195 32 1024 63 <55 4225 131 98 9604 197 33 1089 <^5 66 4356 133 99 9801 199 34 1156 67 67 4489 135 i i po 10000 201 H- 2 ; n cable of ^quareg. Iff; fl.'l.ti;* Vt J; ■ I ' tlis-i * ij, .!|-i 'S'f mk lOl 10201 203 ^34 17956 269 102 10404 205 135 18225 271 103 10609 207 136 18496 773 104 10816 209 137 18769 275 105 I 1Q2^ 211 138 19044 277 106 1 1236 213 139 19321 279 107 II449 215 140 19600 281 108 I 1664 217 141 19881 283 109 11881 219 142 20164 285 110 I 2100 221 143 20449 287 111 1 2321 223 144 20736 289 1 12 12544 225 H5 21025 291 I M 12769 227 146 21316 293 114 12996 229 '47 21609 295 1^5 13225 231 148 21904 297 I ?6 13456 233 149 22201 299 '17 1 3689 235 150 22500 301 118 13924 237 151 OC 0 303 119 14161 239 152 23104 305 120 14400 241 '53 73409 307 121 14641 243 154 23716 309 122 14884 245 '55 24025 3" 123 15129 247 156 24336 313 124 15376 249 '57 24649 315 12^ 15625 251 '58 24964 3'7 126 i is87<5 -53 '59 25:81 319 127 ' 16129 255 160 25600 321 128 163 84 257 161 25921 323 129 16641 ^59 162 26244 325 530 169,00 261 163 26569 327 13 i 17167. 263 164 26896 329 132 I7J,24 265 1.65 27225 .331 133 T.76S9 267 1 166 27556 333 134 17956 76.9! ! '67 27889 335 31 caWe of lot 167 27889 168 28224 169 28561 170 28900 171 29241 172 29584 173 29929 174 30276 175 30625 176 30976 177 31329 178 31684 179 32041 180 32400 181 32761 182 33124 183 33489 184 33856 185 34225 186 34596 187 34969 188 35344 189 35721 190 36100 191 36481 192 36864 193 37249 '94 37636 195 38025 196 38416 197 38809 198 39204 199 39601 20c 40000 335 337 339 341 343 345 347 349 351 353 355 357 359 361 363 365 367 369 371 373 375 377 379 381 383 385 387 389 391 393 395 397 399 401 201 40401 403 202 40804 405 203 41209 407 204 41616 409 205 42025 411 206 42436 413 207 42849 4^5 208 43264 417 209 43681 419 210 44100 .421 211 44521 423 212 44944 425 213 45369 427 214 45796 429 215 46225 431 216 46656 433 217 47089 435 218 47524 437 219 47961 439 220 4-8400 441 221 48841 443 222 49284 445 223 49729 447 224 50176 449 225 5062? 451 226 51076 453 227 51529 455 228 51984 457 229 52441 459 230 52900 461 231 53361 463 232 53824 465 233 54289 467 234 54756 469 H 3 ii •Hi' fik ■ ''.M I''' <* . {, •111 w !! fii' i. ' .;twji.!.! ■; .i i'C 41 U ■ ;, 'ill (<** lA'i-. ■Wilf-l t I ;• ^' ■ 'i^ ' *' :» . ' - ' 'W ' 'C.' .>ri/ •' i ' "f I - L'' !!► • /<[( -6 - :■ i i {■t!'.r:'"v ■■] ■ ■)t I.; *1 ■ !:. liih 102 ^ Table of ^nttate^. V 234 54756 469 267 71289 535 23 5 55225 471 268 71824 537 236 55696 473 269 72361 539 237 56169 475 270 72900 541 238 56644 477 271 73441 543 239 57121 479 272 73984 545 240 57600 481 273 74529 547 241 58081 483 274 75076 549 242 58564 485 275 75625 551 143 59049 487 276 76 T 76 553 244 59536 489 277 76729 555 245 60025 491 278 77284 557 246 60516 493 279 77841 559 247 61009 495 280 78400 561 248 61504 497 281 78961 563 249 62001 499 282 79524 565 250 62500 501 283 80089 567 251 63001 503 284 80616 569 252 63504 505 285 81225 571 253 64009 507 286 81796 5-73 254 64516 509 287 82369 575 255 65025 511 288 82944 577 256 65536 513 289 83521 579 257 66049 515 290 84100 581 258 66564 517 291 84681 583 259 67071 519 292 85264 585 260 67600 521 293 85849 587 261 68121 523 294 86436 589 262 68644 525 29) 87025 591 263 69 i6g 527 296 '87616 593 264 69696 529 297 88200 595 265 70225 531 298 88804 597 266 70756 533 299 89401 599 2 71289 V 535 jOC 90COQ 601 n cable of ^quateiE?; *03 301 ?02 303 304 305 306 307 308 309 3 10 311 312 313 314 315 3 16 3i7 3 18 319 320 090601 091204 091809 092416 093025 093636 094249 094864 095481 096109 096721 97344 97969 98596 99325 99856 100487 101124 101761 102400 603 605 607 609 611 613 61S 617 619 621 623 625 627 629 631 633 645 637 639 641 334 335 336 337 338 111556 112225 112896 113569 114244 669 671 673 675 677 339 340 341 342 343 I14921 115600 116281 116964 117649 679 681 683 685 687 344 345 346 347 348 118336 119025 119716 120409 121104 689 691 693 695 697 349 350 351 352 353 1218011 122500 123201 123904 124609 699 701 703 705 707 321 322 323 324 325 103041 103684 104329 104976 105625 643 645 647 649 651 354 355 356 357 358 125316 126025 126736 127449 128164 709 711 713 715 717 326 327 328 329 330 106276 106929 107584 108241 1089OO 653 655 657 659 66 i 359 360 361 362 363 128881 129600 138321 131044 131769 719 721 723 725 727 331 332 333 .334 109561 110224 110889 111556 663 665 667 669 364 365 366 367 132496 133225 133956 134689 729 731 73 j H4 T04 ^ Cable of i§>quarefli. . ".f I ' ' i : «!i> ''ill': mi •',!'t :i V- i . 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I iif 01 oz 03 04 05 06 07 08 09 1.0 1.1 1.1 »-s 1.4 i.j I.'5 1-7 1.8 1.9 2.0 z.l 2.2 2.4 1.5 2.6 ^•7 2.8 2.9 3.0 o. 002617 o. 010471 O.023 561 O. 041887 O. 065449 o. 094247 o. 128281 O.167551 o.212057 o. 261799 o. 316777 o. 376991 o. 442440 o. 513126 o. 589048 o. 670206 o. 756600 O.848230 0. 94 ^'3 97 1. 047197 I- lUfM 1.267109 1. 384918 I. 507964 I.636246 1. 769763 1. 908517 2. 052507 Z.20I75Z 2.356194 I 2 3 003167 011545 025158 044008 068093 003769 011671 026808 046181 0JOJ90 004424 013849 028509 048406 073^39 097415 1J197? 171766 216796 267061 100631 135716 176033 211586 272376 103908 139512 180353 226430 277742 322563 383300 449273 520483 596928 328401 1389662 '456159 151789^ 604861 334i9i 396076 463096 ^353^3 612846 678610 7^66^7 8 57680 955070 057^96 687066 774f07 867184 965097 068246 695574 783539 876739 975176 078849 I<55657 278654 396987 5^0^17 649362 176631 290252 409109 533201 662530 187757 501902 421282 545899 675751 783403 922680 067194 216943 797095 036896 081933 232206 8IO?40 951164 096725 247521 005131 015079 030264 050684 076340 I07ZJ2| 143361 184725 231325 283162 340134 402542 470086 542867 620883 704135 792613 886348 983308 089504 198936 315604 433508 558648 689014 814637 965485 111569 262889 in Foot Mtafure and Decimal parts of a Foot. 135 6 7 8 9 005890 016361 OllO-jQ 053014 079194 006701 017697 033919 055396 081100 007566 01908 5 03 5840 057831 085058 008481 OZ0525 037803 060318 088069 009450 022017 039819 062858 091132 110610 147261 189149 236273 2 8 8 6-3 3 114039 151115 193616 141274 194157 117521 155120 198155 246316 199734 121055 159278 201737 251431 305361 124642 163388 207371 256589 311043 346119 409061 477129 550433 618973 351177 415632 484224 558051 637114 558377 421156 491371 565712 645309 364529 481932 498570 573445 ^53555 370734 435660 505822 581220 661855 712748 801760 896008 995492 XOOll1 721414 810949 905721 005728 110971 730132 820191 915486 016017 121784 738902 829485 915303 016358 132648 747725 838831 935173 036751 143565 \i 210167 3^5359 445787 57'45O 701350 221451 337166 458117 584305 715718 232787 349016 470500 597111 729158 244175 360937 481936 610170 7^2641 255616 372902 495424 623182 756176 838486 979857 116465 278309 852387 994183 141414 293781 866341 008760 156415 309306 880347 023 290 171^^68 314883 S94406 037872 186574 340512 ) i P 'M t'T^.: ^ •>»rii;, -j. 1, 'I'l ''' lf?l f-. ^ ^,a.,. * "I "■•v'M- 134 ./i TdWe JhewiHg the Dimeter of iiny\ Circle under r, . '1 ■<' \ j' "1 ;j. 11,'. -jfi -:. •!:■•» ,^•1 ^ v> r ' 'Id '(■""f'Vt •K • ■' 'W'-'"" '1 -. ' ! ■ ;. • • • -1. ■V ■'■t,#? fl ♦; ,f'i 'I' '• ■ V i - ii • .,' ■ - '■ "T'S ■ ,• , i h»J j 'liJ.t i , 0 1 3 3 4 lO zo 3° 43 yo 0.314159 0. 628318 0. 94M77 1. 256^37 I. ^70796 345575 659734 974893 288052 602212 376991 691150 006309 319468 633628 408407 722566 037725 3 50884 665044 439822 753982 069148 382300 696460 6o 70 80 90 I.00 1. 884955 2. 199114 2. 5i32'74 2. 827433 3. 14159^ 916371 230530 544690 858849 X73008 947787 261946 576105 890265 204424 979203 293362 607521 921681 235840 010619 324778 638937 953097 267256 10 io 3° 40 50 3- 455751 3.769911 4. 084070 4. 398229 4.712388 5. 026548 5.340707 5.654866 5, 969026 6. 2 8 310 5 487167 801317 115486 429645 743804 518583 832743 146902 461061 775220 549999 864158 178318 492477 806636 581415 895574 209734 523893 838052 60 70 80 90 2.00 057964 37^1^3 686282 000441 314601 089380 403539 717698 031857 346017 120796 434945 749114 063273 377433 152211 466371 780530 094689 408849 10 20 30 40 10 597344 6.911503 7. 225663 7. 539822 7.853981 628760 942919 257079 57123B 885397 660176 974336 288494 602654 916813 691592 005751 319910 634070 948229 723008 037167 351326 665486 979645 [60 7° So 90 8. 168140 8.482300 8.796458 9. 110618 199556 513716 827875 142034 230972 54513^ 859291 173450 262388 576547 890707 204866 295804 607965 922125 236282 3.00 9-4^4777 5 foot, tht Circmfennce being given-, mi the Contrary. 135 6 7 8 9 471138 783398 100557 413716 7X7875 502654 816814 131973 445131 759291 534070 848230 162389 476548 790707 565486 879645 193805 507964 822123 596902 91x061 225221 539380 853539 041035 356194 670353 984513 298671 073451 387610 yoij69 015928 330088 104867 419026 733185 047344 36x504 X 36183 450442 764601 078760 392920 167-698 481858 796017 110176 414335 613831 926990 241150 555309 869468 644147 958406 272566 586725 900884 675663 989822 303981 618x41 931300 707079 021238 33^397 649557 963716 738495 052654 366813 680973 9^5131 18^627 497787 811946 126105 440164 215043 529203 843362 157521 471680 246459 560618 874778 188937 503096 277875 591034 906194 120353 534511 309291 623450 937610 2 517^9 565928 754424 06858 J 382741 69690X OH06I 785840 099999 414158 728317 042477 817256 1314x5 445574 759735 073893 848671 162831 476990 791x49 X05309 880087 194247 508406 822565 136724 J25220 639379 953539 267798 356636 670795 984954 299114 388051 701211 016370 330530 419468 733617 047786 36x946 450884 765043 07^202 393361 1 4 ig(5 A Table Jhewiitg the third, part of the Area of any Natural I Artificial 0,01 ,or 0.000010; .03 0.000013 ,04 o,oooo4Z D.O J 0,000066 .06 0.000092 .07 0,0001 29 .oslo.oooi ' 69 6 Op 0,0002 14 'I o.iop.0002 .11 .12 •I? .14 0.15 6S 0.0003 ^°| 0.000382 0.000448 0,000519 0.000 ^96 ,i6jo,ooo(?7p 0.000766 '0.0008 J9 0.000957 0.001061 •*7 .18 •19 0.10 .21 .2 2 .24 D.25 .26 .^7 .28 .29 P.30I •51 •3- •34 .0011 69 ,001283 .001405 ,001 527 .0016J7 ,001793 .001933 4.423668 f.025:728 ^3779" 5.627788 5.821608 f-97997» 6.113864 .229848 6.332153 .423668 6.506454 6.582031 6.651555: 6.715924 6.775851 6.831908 6.884566; 6.934213 [6.981176 7.025728 17,068 107 7.108514I 7.147114, 7.184091 7.219548 7.253615 7.286396 OO2OI267'5 00123O'7.34^464 0023 53[7.377911 002549 7.406392 09271617.433968 002 88817.460695 003066I7.486626 0031491,7.511804I p. 3 6 0.37 [0.38 0.39 0.40 0.41 0.41 0,43 0.44 0.45 0.003437 0.003631 0.003 8 3o|7 0.004034 0.0041441 0.004458 0.004679 0.004904 0.005135 0.005371 0.46 0.47 0.48 0.49 0.50 0.51 0.52 P.53 0.54 I0.55 p.56 0.57 0.58 0.59 0.60' 0.61 0.62 0.63 0.64 ,0.65 Natural , Artificial 0.005612 0.005 89 2| 0.006111 0.006368 0.006631 0.0068991 0,007172 0.007451 0.007734 0.008024I 0.008318 0.008618 0.008923] 0.009233 7.536173 7.560071 .583236 7.605798 7.627788 7.649236 7.670167 7.690605 7 710574 7.730093 7.749184 7.767864 7.786151 7.804C60 7.821608 7.838809 7.855675 7.872220 7.888456 7.905394 7.710044 7.935418 7.950514 7.965371 0.009549 7.979971 0.009870 0.010196 0.010528 0.010864 ^O.OI I 207 o.66|0.oi 1554 7.994318 8.008452 8.022358 I 8.036028 ; 8.049495 i 8.062756 j 0.67 0.011907 8.075818 I 0.68 0.012265 8.c8b686 j lc.69 0.01262 8 8.101367 0.7010.012997j8.i 13864 Circle in Toot meafure^ not exceeding lo foot circumf. 137 0.71 0.72 0.7: 0,74 0.75 0.76 0.77 0.78 C.79 0.80 Natural Artificial 0.013571 ■).0I :).oi4i 0.01451 O.OI [37508 358 14920 8 8.126175 •138383 150314 5 8.162132 173791 0.81 0.81 083 0.84 0.85 0.86 0,87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 0.95 O.OI ^521 O.OI 572 0.016138 8 0.016554 8 0.016976 8 8.185296 7 8.190991 2078 58 .218927 .229848 0.017405 0.017835 8 0.018275 0.01871 0,0191 i6i^ 8 0.019618 8 0.020077 0.020541 8 0.02I0I I 0.021485 0.021999 ® 0.0224 51 0.022942 0.023438 8 0.023939 ' 0.96 0.024446 8.38B211 0.9 0.98 099 I,0C 1.01 1.02 I.O 1.04 0.02495s 0.025475 0.025997 0.026^25 240638 .251296 7.261825 6 8.27 2227 .282506 292665 8.302707 ■ 312634 8.322448 8.332153 341751 8.351244 8.360634 *.369924 8.379116 0.027597 0.028141 0.028690 8.4567.5 5 I.05!O.O29:I}4 8.466047 8.397212 3.406121 8.414939 8.423668 0.027058 8.432311 8.442369 ^•459343 1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 Natural Artificial 0.029804 0.030369 0.030939 0.03151 0.032096 8.474280 8.482437 8.490516 5 8.498521 8.506454 0.031682 0.033173 0.03387c 0.034472 0.03 508 0.035697 8.552584 0.036324 8.560040 0.036934 8.567431 0.037563 8.574762 1.20 0.038197 8.582031 1.21 1.22 1.23 1.24 1.25 1.16 ^7 1.28 .29 0.038836 0.039481 0.040130 0.040786 OaOi|. 1 44^ 0.0421 12 8.624410 0.042783 8.631176 0.043459 8.638088 0.04414I 8.644848 1.300.044828 8;65i55y 1.31 1.31 0.045520 8.658211 0.046118 8.664816 1,33 0.046921 8.671372 1.34O.047629 8.677878 1.35 0.048343 3.514314 8.522104 8.528825 8.537478 8.545064 8,589239 8.596388 8.603471 8.610512 8.617488 8.684336 1.36:0.049062 1.37,0.049786 1.38,0.050515 8.703427 i.39[o.05i25o 8.689746 8.697110 8.709698 1.40,0.051990'8.715924 153 A Table fhevfing the third Part »f the Area »f any 1.41 1.4Z •I.4J *•44 *•45 1.46 *•47 1.48 *•49 1 i.fo *•5* *•5* *•5? *•54 *•55 Natural .Artificial ^•o5*7?5 <*•055453 o.oj4i4Z 8 0,05;4007 8 *•055770 8,7Ziio7 [8.71824 J 7545401 740595 [8.746404 [0.056^41 8.7^2374 0.0573198.758303 10.058102 8.764192 0.058889 8.770071 [0.059687 8.775851 f.56 *•57 1.58 *•59 1.60 0.0645568.809918 0.065383 8.815468 0.066219 8.820983 0.067059 8.826463 0.067906 8.831908 1.61 1.62 1.63 1.64 1.65 1.66 1.67 1.68 *.^19 1.70 0.060481 8.781622 0.061285 8.7933 56 0.062094 ^•79505* [0.062908 8.798710 0.063728 8.804332 0.068754 8.837320 0.069614 8.842698 0.070476 8.848044 [0.071545 8.853556 0.072216:8.858636 •71 *•7* [ *.75 *.74 1.75 ^•°7509i 8,8638.85 [<^•<*75944 8.869101 *•074553 ^•874595 ©•075760 8.879442 0.076659 8.884566, ©•077 5 64.8 •889661 0,078440] 0.079555 8.894725 8.899761 0,08030^18,9047^7 .081235 8.909745 *•7^1 1.77 1.78 1.79 1.80 1.81 *.82[ 1.83 1.84 *•85! Natural | Artificial 0.0821661 0.083102] 0.084044 0.084991 0.085943 0.086901 0.087864I8 0.088832 0.089805 0.090784 1.86.0.091768 1.87 1.88 1.89 1.90 0.095748 1.91 1.92 *•93 *•94 0.092758 0.093752 0.09441 0.096768 8 0.09778418 0.0988061" 0.099832 1.95 0.100864 9^OO3738 1.96 97 98 1.99 1.00 2.0 I 2.02 ,05 04 2.05 8.914694 8^919615 8.924508 8^929374 8.934213 8.939016 943811 8.948571 8.955291 3.958012 3.961694 3.967551 13.97*984 813.^76^91 8.981 176 985755 .990271 8.994783 8.999272 O.IOI9OI o. 102944I 0.103981 0.105044 0.106103 O.IO7166 0.10823 5 0.1093 10 O.I 10389 o. 111441 2.06 |o.112564 9.051403 0.1 136(^0 9.055609 *•**44*7 9.059795 0.115867 9.063961 0.116978 9.068107 2.07 2.08 2.09] 2.10 9.008181 9.012601 9.016999 9.021375 9.025728 9.030061 [9.034571 9.038660 9.042929 9.047176 circle in Foot meafure, not exceeding lo foot Clrcumf. 139 i.i I i.l» 1.13 2.14 2.15 2,1 6 2.17 2.18 2.19 2.20 2.21 2.22 2.2J 2.24 2.25 2.26 2.27 2. 2,2p 2.JO 28 o. 2.J1 2.J2 2-35 2.54 2.35 2.41 Natural Artificial 0.118095 0,1 19210 0,120^45 0,121477 0.122614 0.123758 0.124907 0.126061 0.127220 0.128584 9.072255 9.076540 9.080428 9.084496 9.088545 0.129554 0.1 5072 0.151910 0.133095 0.154286 99 0.135485 0.156684 9 1378919 0.159104 9 0,140^2 I 0.141544 9. 0.142 0.144006 9' 0.145244 0.146488 9 1.56 0.147758 9.169492 2.57 0.148992 9.I75165 2.5 8 2.59 .40 0.1 52788 9.184091 9.092576 9.096588 9.10058 1 9.104557 9.108514 9.111463 .116574 9,120278 9.1 24164 9.128055 9.131885 135720 139538 ^43339 9.1471^4 .150892 .154644 158580 9.162100 165804 0.150252 9.176822 0,151518 9.180464 0.154064 2.42 0.155545 2.43,0.156652 2.44 2.45 0.157924 0.1 59221 9.187702 9.191299 9.194881 9.198448 p.2.02001 2.46 2.47 2.48 2.49 2.50 2.51 2.52 ^•53 2.54 ^•55 2.56 '■•57 2.58 2.59 2.60 2.61 2.62 2.65 64 2.65 2.66 2.67 2.68 2.69 1.70 2.71 2.72 '•73 2.74 '•75 1.76 2.77 2.79 1.80 Natural Artificial 0.160523 0.161 851 9.205559 9.209062 0.165144 9.''12571 O.I64462'9.2I6O67 0.165786 9-219548 0.167115 9.222016 0.168449 9.226469 o.I69789'9.2299O9 0.171133I9.233536 0.1724849.256749 o*i73839l9-24oi48 0.1752009.245555 176566 0.177957 0.1795149.255615 0.180685 9 0.182 0.185476 9. 0.184874 0.186277 0.187686 9, 0.189099 9 0.190519 191943 9 0.195573 9 9.246908 9.250268 .256949 .260271 .265 580 2-6687^ 9.270160 '7343^ 276691 9.279958 ^83173 .286596 194808 9.289607 0.196248 9.292806 o. 197694 0.199145 0.200601 0.20265 0.205 5''9 2.78 0.205002 0.206475 0.207562 9.295994 9.299170 9.302554 9.305486 9.308628 9.311758 9.314877 9.517984 140 Tahlt Jhevfing the third Part ef the Area »[ any 1.81 i.8z 1.83 1.84 1.8 5 1.8^ 1.87 1.88 i.8p 1.90 1.91 1.91 1.93 1.94 Z.95 1.96 2.97 *.9'^ J.00 3.01 3.01 3.03 3.04 3-Of 3.06 3.07 3.08 3.09 3.10 o 3.11 3.12 Natural Artificial 0.109430 0.210943 0,2,12442. 0.113946 0.113433 0.116970 0.218490 0,22001 0.121 346 0.223082 0.11772 I 9.321081 9.324167 9.3I7I4I 9.330306 9.333358 9.336400 9-33944^ 39-341453 9.344464 9.348464 0.22462 J 9.351454 0.226169 54434 9.357404 0.229278 9.360363 0.230840 9.363312 O.2324O8'9.36623I 0.23398 i'9.369I8I 0-135559 9-371101 2.99 0.237143.9.373011 O.238732'9.3779II 0.240526,9.380801 0.241926 9-383682 .1435309.386334 243I4I!9-3894I6 246736 9.391168 0.248377 9.395111 0.249003 0.231634 153171 236627 ,238212 ,239870 9.397945 9.400770 9-403383 2^4713 9.406392 3.14 0.261334 5.1 3 0.26320: 9.4091 89 9.411978 9-414757 9.417518 j;,4201901 3.16 3.17 3.18 3.19 3.10 Natural 0.2^4876 0.26633 3 0.268139 269926 0.271624 9.433968 Artificial 9.413043 9.415787 9.418513 9.431150 3.11 3.220.273030 3.13 3.14 3.15 3.26 3.17 3.28 3.19 3.30 3.31 31 3.33 3. 5.35 340 3.3^ 3.38 3-39 3.40 0.273324 9.436678 9.439380 9.441073 9.444758 9.447435 0.276741 0.178457 0.280179 0.281905 0.283604 0.285375 0.28711 0.288866 9.450104 9.451764 9.455416 4 9.458060 9.460695 0.290619 0.292^78 0.294142 195911 0.297686 0.299465 3.37,0.301251 3-41 3.41 3.43 3-44 3.45 0.303041 0.304504 0.306638 0.308444 0.310256 9-491711 9.463324 9.465945 9-468557 9.471161 9.473758 3.476347 9.478918 9.481502 9.48406S 9.486626 9.489177 0.312073 0.313895 0.315716 9.494157 9.496785 9.499307 3.460.3175^9.501821 3.47|0.3 19394 9.504317 5.480.321238 9.506782 3.490.323087 9.509319 3.500,324941 9.511804 ctYtUinTootmafm,Mtexcteding lo footclrcumf 14s Natural | Artificial | 3-51 3-^3 3-U 3-55 0.516800 9 0.318665 9 0.330501 0.352411 0.334191 3.56 3-^7 3.58 S-f9 3.60 3.6 3.61 3-<53 3.64 3.65 3.66 3.67 3.6b 3.65 3-7' 3-7 3-7- 3-73 3'7^ 3-75 3-7^ 3-77 3.7!; 3-79 3.80 382 3.83 3.8 3.85 514283 516754 9.519218 9.511675 9.5141^5 0.336177 9 0.338068 9 0.339965 9, 0.3418 °*343774 9 0.345687 9 0.347604 9, 0.349518 9 0.351456 o 353390 0.355329 0.357273 0.359223 9 0.36117b O.36313S 0.36510^ 3.36707^ 0.36905 1 0.371032 3.373019 .526568 518005 f3M54 533857 53^^»73 .538683 .541086 543481 9-H587I 9.547154 9.550631 9.553001 555364 9.55772.I 9.560071 9.562416 9.564754 9.567086 9.569412 9.571731 3.375011 0.377008 0-3 79011 o 38101c o. 383032 0.385051 0.387075 0.389104 3.391139 9.574044 9.576351 3.578652 9.580947 9.583236 9.585518 9.587795 9.590066 9.591JJ 1 ^•393179/9.594590 Natural Artificial 3.86 3.87 3.88 3.89 3.90 0.395224 0.597174 0.399330 0.401391 0.403457 9.596743 9.599090 9.601332 9.603 568 9.605798 3-91 5.92 3-93 3-94 3-95 0.405529 0.407606 0.409688 o#4i 177^ 0.413869 9.608022 9.610241 9.612454 9.614661 9.616863 3.96 3-97 3.98 3-99 4.00 0.415967 0.418070 0.420179 0.422293 0.424413 9.619059 9.621249 9.623435 9.625614 9.627788 4.01 4.02 4.03 4.04 4.05 0.426504 0*428667 0.430803 0.432943 0.435089 9.629957 9.632121 9.634279 9.636431 9.638578 4.06:0.43724119.640720 4.07jo.439397'9.642857 4.0810.4415599.644989 4.09 0.443726 9.647115 4.10I0.445899 9.649236 4.11.0.448076 4.120.450259 4.13I0.45 2448 4.14 4.15 4.16 4.17 4.18 4.19 4.20 0.454642 9.651352 9.653463 9.655569 9.657669 0.45684019.659765 0.459045 9.661855 0.461254 9.663941 0.463469 9.6.v6O2I 0.465690 9.668056 0.467915 9.670167 Ill' •' J!i| 1 142 A Tabic JhcTving the third Part of the Arei of any 4,zi 4.2Z 4.Z3 4.24 4.z^ 0.470146 0.472^ bz 0.474623 0.476870 0.479122 4.26 4.17 4.28 4.29 4.30 4.31 4-3^ 4.33 4.34 4-35 4.36 4.37 4.38 4.39 4.40 4.41 4.4 i 4.43 4.44 4-45 4.46 4-47 4.48 4-49 4.90 4.51 4.52 4-53 4-H 4-J5 Natural Artificial 0.481380 0.483642 0.485910 9.672233 9,674293 9.676349 9.678400 9.680446 9.68248^ 9.684524 9,686556 0.488183 9.688583 0.490462 9.690605 0.492746 0.495032 0.497330 9-692623 9.694636 9.6966 34 0.499629 9*698648 0.501934 9.700647 .504245 9.702641 .506560 9.704631 ,508882 511208 9.706617 9.708597 5135399.710574 0.515876,9.7''^546 0.5i82i9!9.7i45i3 0.52056619.716476 0.52291919,7^.8434 0.527277:9.7^0388 0.527641 9.712.338 530009^9.714183 .532383I9.716224 5344299-718161 ,537147]9.73°093 .539f57'9.73ioii ,54193319.733945 .54433319-735865 ,54673919-737780 ,54915019.759691 4.56 4-57 Natural 1 Artificial 0.551567 0.553989 4.580-556416 4.59 4.60 4.61 4.62 4.63 4 64 4-65 0.5 58848 0.561286 4.66 4.67 4.68 4.69 4.70 4.71 4.71 4-73 4.74 4-7 5 4.76 4-77 4.78 4.79 4.80 4.81 4.82 4.83 4.84 4*8 5 0.563729 0.566177 0.56863 1 0.571090 0-573554 0.576024 0.578499 0.5 80979 0.583464 0.585955 9.741598 9.743501 9-745399 9.747194 9.749184 9.751070 9.751951 9-754830 9-7 5 6704 9-758574 0,588451 0.590952 0-593459 0.595971 0.598488 0.601011 0.603539 0.606072 0.608611 0.611154 9.760440 9.762302 9.764160 9.7^6014 9.767864 9-769710 9-771551 9.773411 9.775125 9.777056 9.778882 9-780705 9-7S2524 9-784359 9.786151 4.86 4.87 4.88 4.89 4.90 0.613704I9.787959 0.61625819.789761 0.618818 9-791563 0.621383 9-793359 0.623953 9-795151 0.626529 0.629110 9.796941 9.798716 .631696.9.800508 .634288 9.8022S6 .636885 9.804060 Circli in Fo$t mi'ajnrc, net txceeding lo footclrcumf. 14.5 Natural \ Artificial 1 4.91 ( 4.91 491 4-94 4-91 3.639487 3.642094 3.644707 0.647325 0.649948 9.805831 9.807599 9.809362 9.811122 9.812879 4.96 4-97 4.98 4-99 J.oo J.OI 5.01 5.0J J.04 5-0$ 0.652577 0,655211 0.657851 0.660495 0.663145 9.81463 2 9.815638 9.818127 9.819869 9.821608 0.665800 0.668461 0.67 II27 0.673798 0.676474 9.813344 9.815076 9.816804 9.828529 9.830451 5.06 5.07 5.08 ^09 J.io 0.679156 0.681843 0.684536 0.687267 0.689936 9.831979 9.833684 9 835396 9.837104 9.838809 J. 11 5.12 5.14 5-M 0.692644 0.695358 3.698077 0.700801 0.703531 9.840516 9.842208 9.843903 9.845595 9 847183 5.16 5-17 5.18 5.19 S.to 0.706165 0.709006 0.711751 0.71450; 0.717158 9.848968 9.850649 9.852328 9.854003 9.855675 f.l2 ^23 5.24 5-M O.7ZOOI5) 0.722786 0.725558 .728335 o-7JIU8 9.857344 9:8 59009 9.860671 9.862_53I 9.863987 Natural Artificial 5.16 5.27 5.28 5.29 5.30 0.733905 0.736699 0.739497 0.741301 0.745110 9.865640 9.867290 9.868936 9.870580 9.872220 0.747924 0.750744 0.753569 0.756399 0.759235 9.873857 9.875492 9.877223 9.878751 9.880376 5.36 ^37 5.38 5-?9 5.40 0.76 2076 0.764922 ^^7774 0.770630 0.773493 9.881998 9.883617 9.885233 9.88684^ 9.888456 5.41 5.41 5-4? 5-44 5-45 0.776360 J.779133 0.782111 0.784994 0.78788 . 9.890063 9.891667 9.892268 9.894866 9.896461 5.4f 5-47 5.48 5-49 5.5c 0.790777 0.793676 0796581 0.79949c 0.802406 9.S98054 9.899643 9.900230 9.901813 9.905394 5-51 J-n 5-54 5-55 0.805316 0.808252 0.811283 0.814129 0.817061 9.905972 9.907547 9 909119 9.910688 9.911254 5.56 5-57 5.58 f-5S 5.6c 0.8 20008 0.812994 0.815918 0.828881 0.831845 9.915818 9-915379 9 916937 9.918492 9.910044 144 ^ Jhming the third varr of the Area of any .66] 67 68 73; Natural 0.854813 0.837801 0.840786 Artificial! 9.9iif94 9.913141 9.91468 J 0.84577? 9.926117 0.846770 0.849770 jo.8 51776 i0.8??786 0.858801 0.861814 7" ■7^ '71 74 '75 ■76I ■77 .78 .791 ,80 ,81 .81 .83 .04 jo.8648 50 [0.867881 0.870919 0.873963 0.877010 b.886185' .86 .87 ,88 ,891 9- 91 9^ 9? 4 5-9)' 9.917765 9.919301 9.930835 9.931365 9.933893 9.935418 9,936941 9.9384601 9.939978 9.941491I 9.943004 0.880069 9.944513 ,0.8831 ii| 9.946O1'0 9.947514 ,0.889154 9.949016^ b.891318 9.950514 0.895408)9-951011 0.898493,9.953 5i4j 0.901586^9.95 5006 ;o.904679 9.956494 jo.907779, 9.957980 0.910886 9.959464 0.913997I9.960945 0.917114 9.961413 0.910236,9.963908 o.923363;9.965372j 0.916496 9.966843 0.929967 9.968311 0.931777 9.969778 0.93 5926 9.971141 0.939080 9.972701 Natural l.'Arcificial 5.96 0.941139 5.97 0.945404' 5.98 0.948574 5.990.951749 6.00 0.954929 6.01 0.95811 6.oijo.96i 306 6.03 0.964501 6.0410.967704 6.05 0.970911 5 9.981417 ' 9-98i86I 9.984305 9-98 5741 9.987179 16.06 0.97411^ 6.07'O.977341 6.08 0.980564 [6.09 0.983791 6. IO'O.9870I5 6.13 o.9902 64j 0.995508 0.996758 6.I4'i.ooooii 6.15|i.003271' ^.16j 1.0065 5 Sjo.002 8^0 6.i7| 1.009808,0.004239 9.974161 9.975617 9.977071 9.978512 9.979971 3I9.9S8614 9.990046 9.991476 9.991903 9.994318 9-995751 9.997171 9.998589 0.000005 o.001419 '6,18 16.19 6.101 6.11 6.22 6.13 6.24 ;6.i5 Li6l ^6.17 !6.i8 ,6.19) '6.30' i.oi3o84j 1.016366 1.01965 2 I 021944! 1.016241 1.0 29 5441 1.031851 1.036164 0.005645 0.007050- 0.008451 '0.009851 0-0 I I 249 0.011645 0,01 4038 5.01 5428 1.039450 1.041807 1.04613 6 1.049470! 1.051809 0.016817 0.01810; 0.019588 0.020970 0.CI13 50 cifcte in Foot meajire, netexcteding ro foot clumf. 6.JI 6.51 6.5J 6.J4 tu 6.56 6.37 6.^8 6.^9 6.40 Na-rural Artificial 1 6.41 6.4Z 6.43 6.44 ^•45 ^.46 6.47 6.48I <^•49 6.?o! 6.,:! 6.51, 6.33' "5.34 Z76 6.5r 6.38 6.391 6.60 6.6} 6.6z 6.61 6.6:< 16.63 .056i340.oZ37i7| .039305 o.oi5ior .061860 0.0264761 .066351 0.017847I .069387 0.019116, .071938 0.030383 ,076333 0.031947 .079717 0.033^ lO .083103^0.034670 .086497 O.''36O28 J, .089899 o Q37384I .o93298'o.o38738j .096707 C)'040090 .100121 0.0414401 .103 340 0.041788 .106965 .110394 .113819 117 270I 0.044133 0.043477^ 0.046818 0.048138 110716 0,049493 ,iz4I67 o.6yo8JO 12761310.031164 131084! ^134331 .138024; 141301 .i4J9''4 ' ^ 4^47^ .131966 .135464 0.053493 0.054814 0.036130 0.037476 0.0387991 o,o5oi >o| 0.061439 0.061736 ,1585)6810.064071 .i62478jo.063438] .163991I0.066895 .16693 ijo 068003 .173038,0 069311 6.66 6.67 6.68 6.69 6.70 6.71 6.72 6.73 6.74 6.73 6.76 6.77 6.78 6.79 6.80 6.81 6.81 6.83 |6;8, 6.83 6.86 6.87 6.88 6.?9' 6.90 6.91 6.92 5.9. 6.94 6.95 Natural ) Artificial ,175688 .1S0104 ,183643 187191 .190744 0.070617 0.071910 0.07 3111 b.0743ii ^.67,3813 194301 197863 101431 105084] 108381 111166 "3733 "9349 221949 126334] .130164 "33779! .237073b .2410161 I44637I .248294 .152936 .233587 .239136, ;2 6I€94 077113 0 O784O7 0.079699 0.080988 0.032276 0.083364 0.0S4846 o;o86ii8 o.087468 0.088686 0.089963 ^.091137 ,092310 6.693781 10.093049 .266337 270216J ,274900 ^7757 .28126; 6.96] 6-97 6.98 6 99 ,7.00 096317 ,097382 098843 100167 101367 102343 H03881 105133 106387 107638 1 2S493 0.108877 i.i.,8648jo.110134 1.191348 0.1 "379 1.296054 0.112623 1.499765 0.113864 L- 14^ A Tabu fhwing the third part of the Areas of any 7.01 7.01 7.03 7.04 7.05 Natural l Artificial J03481 0.114104 1.307103 0.116343 1.310930 0.117479 1.314661 0.118814 ^•318399 0.110047 7.06 7.07 7.08; 7.od 7.10' 7.1« 7.12 7-1 7.14 7.15 1.314890 1.319644 1.333401 337166 0.111178 0.111407 0.113734 0.114961 0.116174 1.340936 I.344710 348490 1.346066 0.117408 0.118618 0.119747 0.131064 0.131180 1.349861 .363667 .367469 1.371181 1.374098 1.378911 1.381748 1.38648 I.390419 1.394163 |0-i39339l 0.140743 [0.141946 0.143146 [0.144344 7.16 7.17 7.18 »! 7'i'9| ;l7;io 'j7.11 l7.ii| 17.13 [7.16 '7.17 ,|7.1« ' 7.1>'| 1^7.3^ ';7.3ili.4i7436io.i 41503 7,311.411317:0.14169 7.3 3p-41510310.153^76 117.34)1,419094I0.14 4060 •7.35j'-43i95"i°.'56i43 0.133494 0.134767 0.134917 0.137116 P.13S333 1.39811 1.401966 1.404816 1.409691 1.413461 0.145441 jo.146737 0.14793 0.149115 0.1 40314 7.36 7.37 38 7.39 7.40 Natural Artificiil .436893 1.440800 1.444713 1.448630 1.441554 |7.4' 7.41 7.43 7.44 7.45 0.157414 0.158603 0.149781 0.160947 0.161131 1.460416 1.464355 1.468199! 1.471149 7.46 17-47 ,7.4« 7-49 i7.5o 7.51 7'SS .53 .541 .55 56 7.57 7.58 7.59 7.60 1.4464820.163304 0.164476 0.164646 0.166814 0.167981 1.476104 1.480164' .484130 1.488101 1.491077 3,169146 I0.170310 ,0.171471 [0.171631 0.173791 1.496O49|O.I7 4948 1.400044I0.176104 1.404038 0.177118 1.40803 4'o 178411 1.411038 0.179442 1.416046 0.180711 1.410049 0.181860 1,414078 o.183007 I.4I8ioi[O,I84I 41 1.43^13 1 4)ii84i96 - ^—T-; 7.6 ji.43616610.186458 !7.6ri 1.540160:0.187 578 7.6311.44414110.188717 l7.64!i.5483oilo.i89854 |7-65|I.4413^5 0.190991 [7.66 I7.67 7.68 17.69 17.70 1.546418 [1.460418 1.464446 1.468633 1.471716 0.191116 I 0.193149! 0.194391 0.194411 0.196640 I ChcU in Foot mafm, net excteding io footclrcumf. 147 7.71 7.72 7-7i 7.74 7.75 7.78 7'77 7.78 7'79 7.80 7.81 7.82] 7.83 7.84I 7.3 J 7.86; 7.87 7.88 7.89 7'9° ! 7-91 7-9 7-9S 7-94 7.95 Natural I Artificial 1.576803 1.580896 1.584994 1.589098 1.593207 1.597321 1.601440] 1.605 565 1.609695 1.613831 0.197777 0.198903 0.200027 0.201150 [0.202272 1.617971 1.622117 1.626269I 1.630425' 1.634587^ 7'9(>\ 7.97! 7.98 7-991 8.oo| 8.01 8.02 8.03 8.04 I 8.05 1.638754; 1.642927 1.647105 1.651288! 1.655476] 0.203402 jo. 2 04 5 00 jo.205628 iO.206744 0.207858 jO.208970 0.210082 0,21 1 192 0.212^ 11 0.21340S 1.659670; 1.663869! 1.668073 1.67*183 I'.676498| 1.680718 1.684944I 1.689175 1.693411 1.697652 1.7O1899 1.706151 1.710468 1.714671 0.214514 6.215618 0.216721 0.217822 0.218923 0.-2 20021 6.-2 21119 0.22221 5 0.223309 0.224403 0:225495 0.226585 0.22767 4I 0.228762 0.229848 0.230933 0.232017 0.23 3 100 0,23418 I.7I89J9;O,2^516*0 8.06 8.07 |8.o8 j8.09 .lol 8,1 I 8.12 18.13 18.14, |8.i6 8.17 8.18 8.19I 20 Natural (Artificial 1.72 3212 1.727491 751775 1.736064 740359 I1.744659 1.748964 |'*7f3i74 '•757590 761911 ,766237 1.770569 1.774906 ,779248 1.783596 • 21 8.2 2 8.13 8.24 8.2 5 8.26 8.27 8.28 8.29 8.30 8.31 8.32 8.33 8.34 8.35 I.7S7949 II.792307 .796670 1.801039 1.805413 0.236338 0.237415 0.238491 6.239565 0.24063 8 241710 0.242780 6.243850 244917 0.245984 0.247049 0.148113 0.249175 [0.250236 0.251296 .809793 I.8I4I77 .818568 .822963 1.827363 0.252355 [0.253412 0.254468 [0.255523 6.2 56576 8.36 8.37 8.38 8.39 8.40 831769 1.836181 1.840597 1.845019 1.849480! 1.853879 1.8 58316 1.8 617 60 1.867208 1,871662 [0.257629 0.258679 [0.2 5 97 29 !O.26O777 [0.261825 6.262870 0.26391 5 0.164958 0.266001 0.267041 0.268081 0.269119 0.2701 56 0.271 0.272227 L. 2 148 A TMt [hewing the third pm oj the Areas of any 8.41 8.41! 8 43 8.44] 8.45 1.876121 1.880585 1.885054 1.889515 1.894010 8.46 8.47! 8.48 8.49 8.50 I 8.51 8.52 8-55, 8.54 8.55 8.56| 8.57 8.58 8-551 8.6c' 8.61 8.62 8.63 8.64] 8.65 Natural I Artificial j 0.173250, 0.174293' 0.175324 0.176353 1.898495 901886 1.907481 1.911983 1.916490 1.911001 t.925510 1.930042 1.934570 1.939104 1.943642 1.948186 •^775 82 0.178409 0.179435 0.280450 0.281484 0.281506 0.183518 0.183748 0.185566 0.286584I 0.187601 0.188616 0.189630 '■5517350 150^45 '•557250/0.191655 1.96184910.191665 1.966414! 1.970985 1.975561 1.980141 1.9847181 8 66] 8.67 8.68 8.6; 8.70] 0.293675 0.294683 O.I956;,O o. 196696 0.1977 01 8-71 8.7 2 I 2 7.3 1 8.74 1.9893^0,0,29^704 1.993916^0.199707 1.958519:0.30(5708 i.oo3ii6io.3017081 2.Q07406 0.3027071 1.012357,0.3037051 ,016981 0.304701! I„.o2i6O5 0.30,5697! 0.7.5,2.026243 o-3o''^5'[ g;75|i.o3o883 o 30/685'. 8.761 77' 8.78 879 8.8o| 8.81 8.82 8.83 2.03 5518 0.30S677 1.040177 0.309668 2.041.333 0.310657 2.049493 0.311646 1.0541550.312634 8.85 058830 0.3 1^620 1.063 507 0.314606 ;.O68I890.3I559O 1.072876:0.316573 077568 5-317555 8.86] 8.87 8.88 8.89) 8.50 8.91 8.92 8-93 8.54] S.55 8.96 8.9; 8.98 8.5c, 9.0CI Natural Artificial 2.082166 0.318536 1,086969 0.319516 1.091677 0.310494 2.096391 0.31147! 2.101110.0.312448 1.1058340.313424 I.iio564'O.324498 1.115299 1.110039; 2.124784 0.32537' 0.326343 0.317314 2.1^9535 2.134191 1.159053 1.143 8 '5 1.148591 0.318184 0.319153 I0.3301H 0.331178 0.331153 9.0 J 9.01 9.03 9.04 19.05 9.06 9.07 |9.O8 9.09: 2.1533681.353II8J 2.158151 0.334081 2.16253910.33504+ 1.167752®.336005 -53' 0.337966 -•^7733 5|0•538925 i.LSI i44;o.3 38883 I.I86'958^O.3 3984J j 1.1917T8.0.340796 j 2.I566^3;O 341751; S|f«| 'ik; ii^r m- 4;.] I76C.; .lii:.; jq6OO»'I 16,-t nop.ji. i£ Clrtlt in Foot mufun^ not txciedlng to foot droit mf. 14 9 9.11 19.11 9.13 9.14 9.15 9.16 9.17 9.18 9.20 Natural .Artificial z.iOi4^ ^ 2.ao6i69 Z.2I 1 110 2.215956 2.220808 2.225665 0.342705 0.343658 0,344610 0.345561 0.346511 0.347459 2.2305279-348407 ^•1.35394 9.1912.240267 2.245145 0-349354 0,350299 0.351244 9,21 2,250029 9,222.254917 I 9.2312.259811 I 9.24'2.2647I I 1 9.25 2.26961 5 0.552881 0.353130 0.354072 0.3 55OIZ 0.355872 ,, , 9.26'2.2745 25 0.3 56890 WH I 9.27U.27944010.357828 ! 9.28 2.28436110.3 58764 Aj'.; 19.29 2.189287 0.359700 hW '9.30 2.294218 0.360634 9.31 2.2991540.361568 9.32 2.3040^60.362500 9-33 ^•309043!O.363432 9-54 1-3 >3996 9,35 2.318953 19.36 2.323916 .37 1.3 28885 ,38 J.3 338 58 .392.358847 ..,502 343821 9.41 2,348144 0.570848 9.42 2.5538060.371770 9.43 2.5 58806 0.37269 9.-.4 2.763478 o. 373612 9.45 2.368489,0.374532 0.36436 0.365292 0,366220 *0.367148 0.368074 0.369000 0.369924 9.46 9.48 9.48 9.49 9.50 9.51 9.12 9.5' 9.54 9.55 9.56 9.57 9.58 9-59 9.60 9.61 9.62 9.63 9.64 9.65 9.66 9.67 Natural 'Artificial 2.375858 2.378893 2.383886 2.388918 1.898955 0.375451 0.376368 0.377285 0.378201 0.379116 2.398998 2.404045 2.409099 2.414157 2.419221 2.424290 2.429364 ».4J4444 2.439529 2.444629 1.449715 2.454816 2.459922 2.465034 2.4701 50 ^.475173 2.480400 9.68j2.485533 9.69! 2.490671 9.70,2.495814 9.7112.500963 9.72 2.506117 9.7 5'^.51^1^43 9.742.516441 9.75 2.5216x1 9.76'2,526786 9-77 9.78 9.79 9.80 2.551966 I.5?7M^ 2.542343 2.5475400 0.580029 0.380942 381854 0.382765 0.383675 0.384584 0,385492 0.386399 0.387306 0.388211 0.389115 0.390019 0.390911 0.391822 0.392715 0.393623 0.394521 0.395419 0.396316 0.397212 0.398107 0.399001 0.399894 0.400786 0.401678 0.402568 0.403458 0.404346 0.405254 .406121 150 A Tible jhtwing the third part of the hra of any See. Natural Artificial 9.811.551745 9.811.557948 9.8j 1.565164] 9.841.568578 9.8 5I1.573601 1.578819] 1.584065 1.589501 0.407006 0.407891 0.408775 1O.409659 0.410541 0.41 142.1 0.411305 I0.415181 1.5945460.414061 I.^9979SIO-4I4959 Natural Artificial 9.91 1.605050 0.415816 9.91 1.610310 0.416691 9.95 1.615575 0.417567 9.94 1.610846 0.418441 9-95 1.616111 0.419315 9.96 1.651404 0.410187 9.97 1.636690 0.411059 9.98 1.641981 0.411919 9.99 1.647179 0.411789 10.00 1.651581(0.413668 A Table A Table for the fpeedy find' ing of the Length or Cit' cumference anfwering to a-' ny Arch in Degrees and Decimal Parts. . • . ! L 4 A Table ✓ 4'* • J'? I ] : •»/ 'M--' V^Sif CM C t . • V ,' ; *t|^ I f'-f ' !&' >|iit I ^ r: ^ : I f Tf it '•t,» 11 I? I r 152 A Table for the fpeedy find infi[ of the Length or Circiimfe- rcnce anfvvering to any Arch, in Degrees and Decimal Parts. 11191^9 .987 7f 57 II o. 1919 li o. 1094 Ij|0. zi6 140, 1445 o. 1617 16 17 18 I? 20 11 21 i4 ^5 o. 2791.5168 o. 2967.0597 : o. 5i4il59ir o, 3ji6|i255 o. 5493 6585 o. 3665 1914- o. 38^9:7245 ■ o. 4014 2972; o. 4iS8|79S2 i o. 436313231 ; 26 0. 45 J7 I S5<5o I 5495 27 0. 47113 3885 8013 28 o. 4886 1 821^ j 0531 29'o. 5061 i 4548 I 5051 31 3^ 52'35; 4877 ( o. 5410;5206 o. 5585 ? 0536 0.5759 t 5865 34P- 5934 i 1194 55 0.6108 1 6523 56 57 58 59, 40 o. ^-83 6457 6632 6806 6981 I 41 o. 4^2 O. 43 o. 4 + i°' 45 0. 460. 47,0. 48 o. 49iO. 5010. 7155 7350 7504 7679 7853 8028 8103 8377 855^ 8726 }?53 7128 2511 7840 5573 8089 0608 5i»7 5646 8165 0604 3203 5713 8241 3170 0760 8499 3828 3^79 5798 9157 8317 4487 0836 3355 9816 3145 5874 0474 ■;8393 5804 '091 1133 | 343' 6462 49)0 1- li *5? A Table for the fpcedy find- ing of the Length or Circumfe- rcncc anrvvering to any Arch, in Deg rees and Decimal Parts. 51 53 u 55 5? 60 61 6z <53 D, 890111793^846^ I 3. 9O7S.7J2.I 3. 9150^1450 94-4 7779 3. 9599J3108 0. 9775,84^8 994*1^767 J t.Oiizlpop66 1.02974415 " 1. 0471 9755 1. 0646 5084 I. 0821 04I2 '• 099515741 64!'- I I70ji072 6 43 1350 8848 4079 44 1382 3007 6731 45 1413 716^ 9385 46 1445 1326 2038 47 1476 5485 4691 48 1507 9644 7344 49 >539 3803 9997 50 1570 7963 2659 IS 5 A Common Divilbr for the fpeedy converting of the Table , fhewing the Jrta of the Segments of a Circle whole Diameter is a. oooo &c. into a Table Ihewing the Jrta of the Segment of any other Circle whole Jrea, is given. SI i6oi 2122 ^5=5 76 2387 6104 1628 Si I6?5 6281 7956 77 2419 0263 4281 s^ i66s 0441 0609 78 2430 4422 ^954 S4 1696 4600 3262 79 2481 8381 9387 ss 1717 «759 5915 80 ^515 2741 2240 56 1759 2918 6568 81 i544 6900 4895- S7 1790 7078 1221 82 2376 1039 754^ S8 1822 1237 5874 8? 2607 3119 0199 59 1855 6327 84 2638 9878 2832 60 1884 9555 9180 85 2670 5557 5305 61 1916 3715 1833 86 2701 7696 8138 6z 1947 7874 4486 87 i755 1836 0811 1979 2033 7159 88 2764 6013 3464 64 2010 6192 979 i 89 2796 0174 6117 65 i042 0352 i445 90 2827 4555 8770 66 1073 4SII 5098 91 2838 8495 1425 67 1104 8670 7751 9i 2890 2632 4076 68 21 2830 0404 91 2921 6811 6729 69 z I 67 6989 5057 94 i955 0970 9581 70 2199 1148 3710 95 2984 3130 2053 71 2250 55=7 8565 96 3015 9289 4688 7i 2261 9467 1016 97 5047 5448 7541 55 t 229J 3626 5669 98 3078 7607 9994 74 1 2324 7785 6322 99 5110 1767 2647 75 ! 2356 1944' 897.5 100 3141 5926 3500 I i 5T3I-*'.v'E5' ■ r •it>, A -1 ■ ri ' j4 «i 1: I *' -.M* •4. t Itil ; :;:p)f ■i aJilKi , I s:m i iit:! »5^ ■|>j' 1: m i 4'- A Table rtiewing the Ordi-'- natcs^ Arches and Areas of the Segments of a Circle, whofe Diameter is 2000, ^c. to every Hundredth hi- 'i: ii} Part of the liaJm, t I'h Hi n' r-\ i: Ordinates i'l M J, rtf Areas and Ordinates to every looo part o/Radius. 157 Ordinates Deg.Sc Dcc.p. Areas 100 99 98 97 96 10000000000 9999499:71 9997999799 999U^^9h 999199^7974 90,00000000 89.^1704196 38.8 .400799 8,28987110 87.70756124 1.57079652 1.55079682 1.53079890 1.51080558 I...•9081774 9^ 94 9? 92 91 9987492177 998198^776 997 J4^9915 99^7948655 9959417655 .87.15402020 '86.56018749 8 5.98601 58 1 >5.41145529 84.85659515 1.47085808 1.45086857 1.45091081 1.41096718 1.59105966 90 89 88 87 S6 9949874571 9939315871 9927758916 9915139938 9901515055 84.26085018 85,68468641 ,85.10789860 82.55040793 81.95215479 1.57113017 1.55124084 1.55157360 1.51153053 1.29171572 8f 84. 85 81 9S868 59966 987117015S 98544 10625 9836666101 9817840905 81.57907468 80.79310474 80.21 218180 79.65024050 79.04721672 1.27192518 1.25216697 1.25244118 I 21274989 1.19509522 80 79 78 77 76 9797958971 9777013859 9754958718 9731906288 9707718879 78.46504188 77.S7761112 77.29096755 76.70291905 76.11545681 ' ■ 1.17347924 1.1 5'3"9056I 1.13457189 1.11488481 1.09544458 7S 74 7J 72 7.1 9681458565 9656086163 9628605221 9600900000 9570266454 75.52248845 74.91996014 74.33573392 73-73979456 75.14102474 1.07605462 1.05-671627 1.05743102 i.oi 820220 0.99905143 70 69 58 67 9559592014 950 365565 9474175425 9439809519 72,542397:7 71.94076969 71.35707564 70.75 122476 0.P7992192 0.96087497 0.94189525 0.9-2 297905 \ 158 Iht Areas and Ordiistts to mry 1000 ^irt of Radius. Ordinates DegStDec.p. Areas 67 66 61 64 9459809J19 9404^545 5'5 9567496997 95Z95»505i 9190517540 70.75122476 70.12512662 69.51268522 68.89980401 68.28458526 0.91197905 0.90415479 0.88556285 0.86666560 0.84804557 6z 61 60 S9 58 9249864864 9208148564 9165151589 9120855222 9075241045 67.66651784 ^7-04550117 66.42182524 65.79516567 65.16541298 0.82950517 0.81104695 0.79267545 0.77438711 0.75619089 57 5^ 55 54 55 9028288874 8979977718 8950285549 8879189152 8826664149 64,55 244020 65.89612058 65.25651645 62.61289754 61.96570587 0.75808715 0.72007866 0.70216884 0.68455845 0.66665254 5i 5« 50 49 48 8772684879 8717224755 8660254057 8601744009 8541662601 61.51459858 60,65941181 60.00000000 59.35^17061 58.66774875 0.64905275 0.65156249 0.61418485 0.59692260 0.57977891 47 46 45 44 45 847997641.5 8416650165 8551646544 8284926070 8216446926 . . 57.99454553 57.31636147 56.65507065 55.94410256 55.14977433 0.56275702 0.54586011 0.52909299 0.51245467 0.49595300 41 41 40 39 58' 8146264741 807405 2449 8000000000 7924014154 7846018098 54.54945741 53.84199105 55.15010257 52.41049708 51.68586597 0.47959008 0.46336957 0.44715221 0.43137885 0.41560051 57, 56 55 54 77'5595 I31J 7685749084 7599541076 7 51265 59'88 50.94987748 50.20810657 49.45851012 48.70012721 0.599988 rS 0.58453*585 0.56925512 0.5551411- V, The Areas and Ordinatis to every looo part o/Radius. 159 1 Ordinates ] Deg.Sc Dcc.p.^ Areas t 34 7512655988 48.70012721 0.55414227 33 7425610981 47-93^93539 0.55920561 : 3^ 7552121U1 47.15655717 0-52444946 • 3' 7258095671 46.56989115 0.50987884 30 7141428428 45.57^99618 0.29549884 7042016756 44.76508489 0.28151495 28 6959740629 43-94551977 0.26755268 ^7 6854471449 45.11560615 0-15355796 26 6726068688 42.268584$!. 0-23999689 6614578277 41-4096156? 5j 0.22665594 24 6499230725 40.55 580228 0.21354168 13 6580458856 59.64611152 0.20066158 22 6i5779!fi58 38.75942400 0.18802248 11 6151068422 57.81448867 0.1756529a 20 6000000000 5 6.86989765 0.165 50' I ^ 19. 5864298764 55.90406875 0.15165601 18 5723655208 54.91520640 0.14004712 »7 5577655906 55.90125515 0.12874491 16 5425865986 52.85988059 0-11774055 15 5267826876 51.78855069 0-10704574 >4 5102940528 50.68541722 0.09^67379 M 4950517214 29.54156121 0.08665902 I 2 4749756854 28.55775666 0.076757 28 1 1 4559605246 27.12675521 0.067646 29 10 4558898945 25.84195282 0.05872590 09 4146082488 24-49464857 0 05021866 08 5919185 588 25-07391815 j 0.0421 5095 07 5675595189 21-56518547 i ^-03455513 06 5411744421 19-94844563 1 0,02746204 05 5122498999 18.19487244 ' 0.01091302 04 2800000000 16.260104,1 0.01499411 03 2451049156 14.06986184 0.0097556-5 02 1989974874 11.47854097 0.005 5175c 5 01 1410675597 1 8.10961446 0,0018827 3 i6o rtf Areas and OrdiaaYcs to every looo partofRidms. Ordinaces ■ Deg & Dcc.p. Areas OlO 009 008 007 1410673597 1338618691 1261378707 "81143513 8.10961446 7.69281247 7.25224680 6.78528891 0.00188278 0.00160779 0.00134761 0.00110317 00 6 ' 10PJ80071J 00y'0998749117 004 o89j5jrji6 ooj 0774015503 6.27958064 5.73196797 5.12640010 4.43922228 0.00087554 0.00066616 0.00047674 0.00030969 1 00 2 001 O632I39225' 0447101778' |2<.56Z5 5874 0.00016860 0.00005961 • > o ] Ue } * - I ■ V Ihe Areas and Ordhates to every iooo pan of Radius. 161 999 998 I.570796}! 199999 I. 568796}! 199999 I. 566796}! 983 98! 981 I- 53^79796 199969 I. 53479817 199965 I.5}!7986I , 997 996 199999 1.564796}? 199998 I.562796}4 199997 980 979 199962 1.53079899 199957 1. 52879941 199953 99S 994 991 I. 56O796}6 199996 I. 558796}? 199995 I.55679644 978 977 976 I. 52679988 199949 I. 52480039 199944 I. 52280095 992 991 199994 I- 55479^49 199992 1.55299657 199991 975 974 199939 r. 52080156 199934 I. 51880222 199929 990 989 988 I. 55079666 199988 I. 54879677 199986 I. 54679690 1 973 972 971 1.51680293 199924 I. 5I48O}69 199918 I.51280451 987 986 199984 I. 5447970^ 199981 '•,^4279714 199978 C 970 969 199912 1.51080539 199906 I. 50880633 199909 ' 985 984 1-54079745 ■ 19997^ I. 5}879769 199972 I.5}679796 968 967 96( 1. 50680735 199894 I. 50480839 199887 I. 40280955 «i': ■' 51.^ If ';r«s V; .■It:? V .,i A ff- 1. (VI. M i6z Hit Areas and Ordinalesto every looo pirt 0/Radius. 1 966 9^4 [,50180952 199880 [.50081071 19987J 1.49881199, ( ?49 748 947 I.46884065 199734 1.466843 28 199714 I. 46484604 ' 963 961 199866 1.49681355 199859 1.49481474 199851 946 945 199713 I.46284890 199701 1.46085187 199691 961 960 959 1.49181615 199845 1.49081774 199835 I.4S881958 1 944 943 942 1.45885496 199680 1.45685815 199669 I. 45486146 958 957 199817 I.486S1110 199819 I.48482291 199810 ! 94» 940 1996^7 I. 45186489 199645 1.45086857 199653 i 956 955 954 1.48181480 199801 1.48081678 199791 1.47882885 939 938 937 1.44887104 199611 1.44687583 199608 I.44487975 •■Iv"'! 1 953 952 199783 I.47685101 199774 1.47485518 199764 ^956 U: 1 199596 1.44188379 199585 1.44088794 199570 1 ] ■ ■ ■ q 951 ! 95c 945 1.47185563 199754 1.47085808 199744 ' ih 4688406: • i j935 93; 1 93' 1.43889114 199557 I. 45689667 199543 i.4349oiiii I . -i: !! WW \< IT': li mm !» 1* f ^18,1.0 I r'^ ! I ■'ill jk. . ■ The Areas and Ordinaus to rjtry looo pan of Radius. 165 9$2 9JI 9io 9^9 918 9^7 926 91J I* 4J490I14 199530 '•43290594 199516 I.4J091078 199501 I. 42891 ^78 199488 I. 4x^92090 '99473 1.4249x^17 '99459 1.42x95158 199444 1.42095714 924 923 922 9XJ 9x0 919 918 199429 1.4189430^ '99413 I. 41694892 199398 1.41495494 199582 [. 41x96112 19956'6 . 41096746 199350 .40897596 '99334 .40698062 199318 917,' n" } Jj(r 916 1 ,jrt9i5''. ,40498744 I99J01 • 40299443 199x84 40100159 915 914 9'3 912 911 910 909 908 907 906 905 904 903 902 901 900 M 2 • 40100X5^ 199267 .}9900891 199250 .59701642 199x52 ,59502410 199215 ■39305195 199197 .59105998 199178 .389048x0 199160 ,58705660 199143. .58506518 199123 .38307595 199104 .58108x91 199085 37909206 199066 .57710140 199047 37511093 199027 57512066 199007 57115059 I P^r i. 'w^ '.v. Hi-; JH 64 Thi Areai and Ordinatts ta ever J 1000 fart of Radius. 897I1.36516117 198915 896:1. J6317192 198904 895'!. 56118288 j 198885 8941.35919405 j 198861 895 1.55720544 ] 198859 X.J5511705 198818 1.355228871 198797 I. 55124090 198775 88911. 54925515 I *98752 88811. 54726565 198729 887 886 885 I. 34527834; 198707 1.54529127 •198684 1. 34130443 198661 884' I. 55951782 j 198658 ■ 7. J 7722 J4.4 885 882| 88'i 880 879 878 877 Is 76] 875 8741 873 87 871 870 869 33733144 198619 ■ 335345251 198590 •33335935 198566 • 331373^0 198541 . 31958819I 198517 1.52740502 198499] I. 52541805 198480] I. 32545525 198449 I. 52144874! 198418 1.31946456 198595 1.51748065 198567 I. 31549696 198541 1-3135*355 868 867 198515 t. 31153033 198189' 1.50954764'^ 1982621 I. 30756502 198255] 1. 50558267 198209 !■•! ii. Tit Areas and Ordinates to n/ery lOoo part of Radius. i^f 866 86f 198209 I. 303600(8 198182 1. J0161876 198154 » 850 849 848 1.27152518 197721 1.26994797 197691 1.26797106 864 863 862 I. 19963722 198127 I. 297^5595 198100 I. 29(67495 1 i 847 846 197660 1.26599446 197629 I.26401817 197598 861 860 198072 I. 29369423 198044 1.29171379 198015 845 844 843 1.26204219 197561 1.26006658 197534 1.25809124 8$y 858 89784? r. ?7785872 197811 I. 27588061 837 836 ^97341 1. 24624641 197308 1. 24427333 197275 ' 851 850 197781 1.27390280 197751 1.27192529 835 r 1.24230058 197241 1.24032817 197212 M 3 ^r' i66 Thi Areas and Ordinatts to rviry looo part of Radius. * ::'ri t * *'i> V •4 ¥m J ,4; a-iwl ^ Iv'^ ' 197171 00 1.23835605 197173 832 r. 23638432 197139 831 1,23441293 197105 830 I.23244118 197072 829 1. 23047046 197036 00 t* 00 • 00 0 0 0 19700I f 827 1.22653009 196966 1 826 1.22456043 196930 825 I. 22259113 196895 824 I. 22062218 196861 823 1.21865357 196825 822 I.21668532 196787 821 I.2I471745 196750 820 I. 21274989 196714 819 1.21078275 196677 8IS I.20881598 196640 ?'7 1.20684958 817 816 815 814 81J 1.106849^4 I. 204885^5 IS1656J I.20291790 I.2009526j 196479 t.19898774 1964T1 812 811 810 1.19702J2J 196413 I.19505910 196375 I.19J09525 196347 8091.19113554 ' 196298 808 I, 13916956 196258 8071. 18720698 I 196119 806 1.118524479 1 j 196188 805 I. 18328291 196148 804 I. 18132143 196100 1. 17936043 196060 803 802 801 800 I.. 17739983 196019 1.17543964 195978 1. 17347986 %t Areas and Ordinatts to every looo part o/Radius. 167 800 799 ■ 798 1.17547924 195938 I.17151986 195897 1.16956089 784 783 782 I,14217966 195256 I. 14022710 195211 I. 13827499 797 195855 1.16760234 195814 1.16564420 195775 ) 781 780 195166 I. 15652333 195122 I.13437211 195076 ■79T 1.16368647 1 1 : 195751 1 794 I. 161729I6 1 1 195689 i 795 I- 15977247 779 778 777 1.13242155 195031 J.13047102 194985 1.12852117 1 195646 791 I. 15781581 I 195605 7911.15585978 1 195561 776 775 194959 1.12657178 194893 I. 1246228^ 194847 1 7901. 15390417 1 ->95518 789 I. I 5194899 1 195472 788 I.14999427 774 775 772 1. 12267438 .194801 I. 12072637 , 194755 1.11877882 ' 1 195429 787 1.14803998 1 195588 786 I. 14608610 1 771 770 19470S I.11683174 194661 I.11488487 194614 I • f -" i 1 785 ! 784J 195544 1.14415266 195500 1,14217966 1 769 768 1. 11293867 194566 I. 11099501 194518 M4 168 The fiTCSti snd Ordwtis to every looo part $j ^^^d\\xs, V . I lift -I I t ''7 ■ i.'l :i-U '-'i 1 ^r#| ' I ill® ■Jij- ■•' v M-k'\ ■ I iSiH ji. -?=-v. I 1945'8 7°7 I. 10904783 A 194471 76° 1,10710J12 I 194413 76J1.10,1^889 i 194574 764 I.IOJ21515 I 194515 763 I.10127190 I 194176 701r. 09931914 I 194117 761.1. 09738687 194173 7601.09544314 194129 759.1.09350385 I 594079 758^1. 09156306 757 756 194029 1. 08961277 193980 I. 08768297 19393d 755 754 755 1.108574367 193878 1.08380489 193817 r. 0818666 2 193777 752J1.07991885 193716 751 1.07799159 751 750 749 1.07799159 193674 1.07605485 193611 1.07411863 748 747 746 745 744 745 741 741 740 755 758 757 736 735 193570 I. 07118193 195518 1.07024775 195466 I.06831309 193414 1.06637895 193561 1. 06444534 193508 I. 06^51226 195^55 1. 06057071 193201 I. 05864770 193147 I. 05671625 193093 r. 05478530 193039 I. 052S5491 192985 r. 05091506 191931 1.04899575 191876 I.04706699 191821 Tht Areas and Ordinatts to every looo fart of Radius, I ipiSzij 7J41.04SIJ878 I 192766 73J i.o4jini2 192710 7 J2'l. 0412840* j 192655 7JJ I.°J5'5 5747 I J9?6O® 73OI.Q?74?I47. 1 '92543 729 1.0J550604 I 192486 728 1.P3358118 I i9*43o_ 727 1.03165688 I '92373 7261.02973115 I 192316 7251.02780999 I 17*259 7241.02588740 I 192213 723 I. 02396527 I 192155 7221.02204372 j 192086 721r.02012286 192029 72J:i. 01820221 I 191970 7I9 1. 01628251 191911 7181.01436340 7I8 717 716 715 714 713 712 711 I.'6i436340 191853 1.01244487 191794 1.01052693 191734 I.00860959 191674 1.00669285 191615 1.00477670 191556 1. 00286114 191505 I. 00094609 191444 710 o. 99903165 191374 709 o. 99711791 '91313 708 O. 99520478 191252 707 0. 99329226 191191 7060.99138035 705 704 703 702 701 191129 O. 98946906 191067 0.98755839 191005 0. 98564834 190943 O. 98273891 190881 o. 98183010 Jf^ .Iht htcss tnd Ordimtts tt ivtry looo, firt fl/Radiu$. L 70o\o. 9799^19^ 190755 <5pJo. 97801457 I 190692 i ^ I . IMII 698o« 97^i®74i 190629 o.97420116 19OJ66 o.97i»9i5Q 697 696 695 694 190502 o.97059048 190458 o.96848610 19(3576 695 o.96658254 I 190504 •692 o. 96467950 190244 691 o. 96277686 690 689 687 I90I79 o.96087497 190115 0.95897584 190048 o.957®7JJ6 189985 o. 95 5I7JSJ 189917 686'o. 95527456 I 189851 6850.95157585 ■ j 189784 684 o, 94947801 I 189717 189717 68510.94758084 189651 o.94568455 189584 682 681 680 679 o.?4J78848 189516 o,94189524 189448 P. 9J999876 678 <677 189581 D.93810495 189515 o.95621182 189244 676 675 674 672 671 670 669 668 667 o. 9|4JI9J8 189176 0.95242762 189107 0.93058655 189038 o. 92864617 188969 0. 92675648 188899 0.9*486749 188825 O. 92297905 188769 o. 92109156 188696 O.9192044O 188619 O.91751821 188549 'X Thi Areas Md Ordinttts to tvtry looo fart oj Radius. 171 1 1 1 188545 666 o-9» 545272 188478 6650.S>'J54794 1 188407 . 650 649 648 0,88536284 187311 0.88348973 187237 0. 88161736 > 664 66j 661 0.51166387 I88jj6 0.50578051 188164 0. 90789787 647 646 187163 0.87974573 187087 0. 87787486 187010 1 188191 661 0.9060»595 1 1881Z0 66oO« 9®4IJ479 1 188048 645 644 645 0.87600476 186534' ,0. 87413541! 186858' 0, 87226684I 655 658 657 0.902254J1 187973 0.50037458 187900 0. 89849558 642 641 186782 0.87039902 186705 0.86853157 186628 656 ^55 654 «;3 651 187815 0. 85661719 •87757 0-89475972 18768 5 640 639 638 0,26666^60 186551 0,86480005 186473 0.86293536 0. 85286x87 187610 0. 85058677 187555 0. 88911141 657 636 186595 0.86107141 186317 0.85920824 186235 651 650 187461 0. 88723681 187386 0. 88536255 65 5I0.85754585' 1 186161', 6260. 84061978 1 184288 6110. 81288911 1 184204 6100. 81104695 1 184119 1 1854441 6250,83876534' 1 18536? 0240. 83691 l7ii j 185281' 609b. 80920576 1 1840?^ 608 0,80736541 1 183949 607 0.80552592 6230.83505890 1 185200; 622 0. 83 320690 1 13-5119 6210.83135571 1 183865 606 0. 80368727 183780 6050. 80184947 1 183693 1 185038 6200.82950517 j 184956 6190.82765561 j 184873 604 0. 80001254 j 183606 603 0. 79817548 183519 6,02|o. 79634029 618 0. 82580688 1 184790 617 0.82395898 j 184707 1 183433 601 0. 79450596 183346 600 0. 79267I5Q Areas And Oxdi^nutis to svtiy looo pArt Radius, 6000.79267345 183258 O,79084087 183170 O. 7 8900917 $99 798 597 596 595 594 59? 592 59» 183082 0.787x7835 182994 0,78534841 182906 o-78351935 182818 o.78169117 182729 O, 77986388 182640 O. 77803748 182551 77621197 182461 590O. 7743873^ I 182371 5890.77256365 I 182281 588 c. 77074084 I 182191 5 87 0. 76891893 I 181100 586 o.76709793 ] 182009 5850.76527784 I 181918 584 o. 76345866 131826 j 583 582 581 0.76164040 181734; 0. 75982306, 181639! 0. 758006671 580 579 181543 0. 75619124 181458 0.7$437^7° 181365 578 577 576 0. 75256305 181271 o-75075934 181173 0. 74893856 575 574 181085 0.74712771 180991 0.74531780 180897 573 572 571 o» 74350883 180802 0.74170081 180707 0. 73989374 570 5^9 180611 0. 73708713 180516 0. 75628197 180422 568 567 o.73447775 180326 0. 73267449 180230 174 Tht htcii tnd Ordhitum fvtry looo firc»j Radius, Ml' I' ml 1 1 566 565 18013c 0. 73087219 180134 0.72907085 180037 564 5^3 562 0. 72727048 179940 0.71547108 279845 0.71367265 561 560 179745 0. 71187520 '79647 0.72007866 '79548 5j9jo. 71828318 179450 558c. 71648868 '79353 557jo. 71469515 1 179254 5560.71290261 1 179155 555 7"I"O6 1 179056 5540. 70931050 j 178956 553 0. 70753094 j 178856 5520. 70574138 1 178755 551,0. 70395483 1 178654 5500. 70116829 f JO 0. 70216834 I78J53 o. 70038281 178431 o. 69859829 T49 548 547 546 545 544 543 54^ 54« 178352 o. 69681477 178250 o. 69503127 178149 o. 69315078 178048 0. 69147030 177943 o. 68969087 177841 o.68791246 177738 o. 68613 508 177634 540 539 538 o. 68435845 177518 0.68258317 '77423 o. 68080894 537 536 535 177318 3. 67903576 177218 0. 67726358 177I14 o. 67549244 j 177009 5340.67372235 176903 The Areas and. Ordinatts to ntry looo fare e/Radius, 17 y I 535 0.67195332 176799 5?2 0.67018533 176693 55» 0. 66841840 176585 550 0. 66665234 176479 5»9 0.66488755 176372 528 0.66311383 527 176265 0.66136118 516 176158 0. 65959960 176050 525 0.65783910 524 175942 0.65607968 175834 525 0. 65431134 '75725 512 0. 65256409 t7^6zi 521 0. 65080787 175512 520 0.64905175 175398 519 0. 64729877 1752891 51810. 64554588 175179! 517 0.64379409 175068^ yi6 175068 o. 64304341 »749f7 515 0.64029384 174846 5»4 512 509 508 )07 506 505 504 ■503 o. 63854538 '747JS 0.63679803 174624 0.63505179 17451a 5ii|o. 63330667 174400 5100.63156249 174187 0. 61981962 174174 0.62807788 174062 o.62633716 173948 o. 62459778 173855 o. 62285943 173721 o.61112122 173607 o.61938615 173491 5o2jo. 61765123 501 500 173377 0.6159174^ 173262 0.61418484 I: I *'1^'' 'Jf'l w f: ■■fjii'.i.i " Jifp , i ^ ' it' J '■ f t h'^ "1 i' . i'l :;i 'uk 1-ltW' ■'i"^ I ii! m lij ^ .''.l 176 Thi Atda and Ordinatetta every 1000 fart of Radiui. 500 499 498 '497 ^496 o,61418485 173147 o.61245338 173031 o. 61072307 171914 o. 60899393 171798 o.60726595 172681 495 o. 60553914 171564 494 o. 60381350 178447 I49J o« 60208903 49* 491 172319 o. 60036574 172211 o.59864363 171093 490 o. 59692160 171977 4890.59520285 1 17I856 488,0.59348419 I7I736 4870. ^917669^ I 171617 4860. 59005076 I X7I498 485!®. 78833578 I7IJ77 484I0. 58662201 171256 483 482 481 480 479 478 477 476 477 474 47 J 472 471 475 469 468 467 o. 58490948 171136 o. 58319809 171015 o. 58148794 170893 0.77^77892 170771 O. 57807III 170649 o. 57636472 170517 o. 57467947 170406 O' 57195SJ9 170281 o.57125258 170158 o. 56955100 170034 o.56785066 169910 o. 5661 5156 169786 o. 76445370 169661 o.56275701 169736 o.56106166 169411 o.77956755 169285 O- 77767470 169159 "X tht hrei% and Ordinates to evtry looo parN^ Ridiui. 177 i 1 1 m l?K: IH'-W 466 46 s 169159 0. 55598311 169035 0.'55429178 168901 45o'o. 52909299 1 166966 4490. 52742333 j 166834 448 0.52575499 [■ji. 464 46} 462 0. 55260377 168779 0; 5 5091594 168652 0. 54922946 447 446 1 166702 0. 524087^7 1665 70 0. 52242227 166437 V nm 461 460 1,68524 o* 54754,4" , 168397 oi 5458601'! 168268 445 444 443 0. 520757'9C 166302 0. 51909488 166168 0. 517.43-3«^ - 1 m 458 m 457 oi 5441774} 0. 54249604 ' i'68,oio o- 54o8r5'94 442 441 16(^935 0. 51577^85 165900 0.51411385 1657^ 1 f-f »53i 1635; 6131 iS)^ 'W!' 456 455 .167881 0. 53913713 \67751 0- 53745962 167621 I 440 439 438 0. 51245467 165634 o. 51079833 i 165494 o- 5.0914135^ ' j I'f ijif 15^1 -^3' 454 45J 452 0. 53578J41 167491 0'.- 53410850 167360 0, 5324H9G 1 1 t 437 436 d. 5074^981 16522: 0.50583759 16 5o8f_ .14 j 1 1 Ijv 451 ;j;: 450 167229 0. 53076261 167098 0. 51909163 i 435 434 or. 5041867 r 16494; d. 50253724 1 164811 N lyS Thi Areas and Ordinitiste tvtry iooq fan of Radius. r 1 1 j4J3 0- 5008891J 1 1^4^73 4J2 0.49934140 1 164533 4310^49759705 4160 4150 162422 .47308470 162278 47146192 162134 1 164397 4300i. 49595308 1 ' 164159 429 0^49431049 1 ! 154120 ! 4140 4130 4120 46984058 161989 ,46822069 161844 i 46660225 427oL49102949 1 163835 4250^48939114 j — —. 1 ! i 4110 410c 161699 346498526 161570 ,46336957 161410 153700 4250.48775414 163^60 4240^48533854 1 ,5-163419 • J 1' ■! 409 0,4517$ 547 161260 408 0.46014287 161113 40710.4^5^853174 48448435 jt 15^277 4220,482851^8 1 ,163135 : 421 q. 48122013 ■^1 406 405 160966 5.4569/208 160818 Ot'4f?3i39o ,160670 420c 419c ^ 152998 447959025 i 152843 J. 4779615$ ' ^"^152708 404 403 40Z 3t 4tJ707io 160522 0, 45116198 j 1^0375 0345049825 418 4I7< 1 >' 47635457 162565 \ 47470891 162422 ^401 I400 160223 O; 44889602 16007; 0.44729529 Willi , . ■ ■ —, Tht Areas and Ordinates to tvny looo part of Radius. 179 «« . II _ 11% liiii' dil 4OO!o. 44729522 I *59913 599,0. 44569599 J 159775 39o|o. 44409826 I *59615 597,0. 44250205 J *59472 599 0. 44090751 I *59520 ifc iJr J^j9j a-r«iiS 595|0.45951411 I 159168 594 0.45772243 159016 0.45615227 W i *58405 \ 389 O. 42978685 <«1' ! 1,98248 388^0. 42820455' ^ a I »5®°93 3!>7 o. 42662542 ^ R/5' 0.42 5Q 4404 ^ I 157782 ^10.41346622 *57616 "!.584 0.42188996 17''' *57470 583 o.42051526 *573*3 582 0. 41874215 *57*56 5810.41717057 580 379 *56999 41560058 156841 4*403217 156682 378^0. 377,0. 376jo. 41246535 156522 41090013 156565 409,35650 375:0. I 374 o, 373 372 37* 370 3690 I 56204 40777446 156044 40611402 -155885 -404655*9 *55722 40309797 155561 Ji<:42 56 ""*'55399' 59998818 155258 39843586 155025 3680, J 367,0 39688555 154911 L 39533644 154788 i V . < * -i- V- t' lU I f i iiA k ■■■f i W' &«■ ;.: ' .ft mplf ■ r-'r' ■;r¥m ■' -i. •«;:!»,_'» ,:i: ■ff.' I »*■ h' Sf •.#. r/i '«' f 5 1 .> ; '!i' t L, > I go 7t;{ Areas and Ordinatis to ettnj looo part (^Radius. 154788 o.39578896 154584 o.39224312 154419 640.39069893 I 154254 63'O. 38915^39 j 154089 6i'o. 38761550 I M39i3 61o.38607627 I >53757 600 3,8453633 >53591 590* j8}coop^ i I5J414 590. 38 46668] I >53256 57 O. 37993411I I 153088I 560.37840324 151910 o.37687404 152751 55 54°-37534653 I 152582 53,0. 37381071 I 152443 jio.37129653 I 152242 510. 37077426 I 152075 500, 36925315 550 349 348 o,36915312 151905 o.36773407 151718 o.36611679 347 1346 ■345 I 344 343 151556 0.36470113 151384 o.36318739 151211 I 342 341 340 339 338 3 37 336 o. 36167518 151038 o. 36016490 150865 o. 35865625 150690 O.35714935 150515 O.35564420 __ 150340 o.35414127 150164 o. 35264063 149988 o. 35114075 149811 0.34964164 149634 o. 34814630 149457 335,0. 34665173 149279 334,0.34515894 149100 -c •V:.! the Areas tnd Oydimtej tv every loco part »/Radms. 13 r 353 o- 54366794 I 148921 ,,20.34217875 I 148742 5,110.5406915 J i 148562 550 o- 3592056] I 148581 519 0,55772180: . j 148200 327 jzSjO-33615980 ' 148024 o. 5347^956 147842 5I6|o-35328124 32s 147665 o. 551804,51 147480] 5i4;0- 33032971 147288 323'o. 52885685 j 147104 3220.52758579 146919 5210. 3 2591660 1 146755 320 2.51444946 1 146550 5190. 52298596 j 146362 - 3180.52152034 1 '46175 517 0. 52005859 , j 145990 I 145990, 5160. 51859869 145803 o. 51714066 145614 315 314/0. 51568452 145425 3I30' 51425027 , 145256 5i2jo. 51277791 I 145047 511,0. 51152744 1448561 5100. 50987884I I 144665 . 508452191 144474 . 50698745 144282 50555463: 144090, 50410573 143897, 30266476 145705 502 o, 50122775 145508 29978265 143315 29855950 301,0 5oo'o. 1 1 N T I 4^ I 2.0j 29692850 142926I 29549904 . 4' t' ■, M 1. f £ 'V n:.; V ■: -f! J' ■..■ •«(# ■:y\'35»87 163 o. 24404243 135069 261^0.14269174 I 134850 i6i o. 14134324 '34553 2000.23999689 I '343^3 2590. 23865356 134189 158 0. 23731165 133968 o. 1359719^ 257 256 255 133746 O. 23463451 133523 o. 2332^)928 133300 2540^23196628 133076 253 0.23063552 132801 252 o.22930751 251 250 13^-575 o. 22798176 132399 O. 22665777 250 249 248 247 246 »45 244 243 242 241 239 238 o. 22665594! 132173 o. 22533421 131946 o. 22401475' 131718 O. 22269757 131488 o. 22138269 131259 o,22007010 131029 o. 21875981 130799 o. 21745182 130567 0.21614615 130334 O 21484281 130101 240 O. 21354168 129867 O.2122430I 129632 o. 21094669 '37 236 129396 O. 2096527) 12916c O. 20836113 128924 Z3 5 o. 20707189 128688 134:0.20578501 12844s N 4 184 Areas tnd Ordiaates to eviry 1000 pii't ij Radius 1 1 255 0.20450052 128208 2520. 20521844' j 1259^8 251 0. 20195876 I t^77^9 250 0. 20066158 1 • 127488 229 0.19958650 1 117245 2280. 19811405 j 127002 227 0.19684405 ( 126758 2260. 19557*545 1 126514 2250.1945115I j 116269 2240. 19304862 j 126025 2250. 19178859 115776 lll'o, 19055065 1 125528 22Ip.18927555 110 219 125279 0. 18802248 125027 0. 18677221 124777 218 117 0. 18552444 114519 0.18427915 124278 114178 216 o. 18305657 1Z4025 21^0.18179611 125771 2140. 1805 5841 UJ5«7 1150. 17932^^4 115162 112 o.17809062 125006 ^ 11 o. 17686056 I 112749 2100. 17565291 122490 1090. 17440805 208 122252 0. 175185^9 122974 107|0. 17196595 121713 206 t 0. 17074882 12I45I .205 0. 16955431 1 121189 204 0. 16832242 120926 205 0. 16711316 120665 '202 0. 16590655 120599 O. 16470254 110155 2000. 16550121 Yai Areas tnd Ordinxtts to every looo part of Radius. 18 j 1 1 200 0. I6J50II1 j 119866 1^90. 1643014? 1 119598 ip8|0.16110647 183 182 181 ■ 0. 14349427 115084 0.14234143 114900 0. 14119345 197 196 119549 0. 15991518 119959 0.15874159 118789 180 '79 114615 0. 14004728 114348 0. 13890400 114040 i9?'o. 15715470 1 U8518 1940.156549^4 1 118^46 19J0.155J6706 178 '77 176 0-13776354 113751 0.13611601 163461 0. 15549140 1 117972 19*0.1559875 J I 117698 191:0.15181055 1 117422 175 J74 113164 0.15435916 112873 0. 13513103 111587 190 189 188 0.15165596 117146 0. 15046450 116869 0. 14919581 1 '78 171 171 0.13110516 112292 0. 15098114 111996 0.12986128 00 00 116591 0.14812990 1i6j11 0.14696678 116032 170 169 111700 0. 11874498 II1403 0.21763088 111105 185 184 0. 14580646 115751 0.14464895 115468 i68 167 0.12651983 110805 0,12541178 110503 . ■I. i4>(.' • f A VI, 1 ■; m • ■ i !(.fc: I a ,. im <:!''. ■■'' "'X : ' • .ui r r t !|iy'x I'Mn: ■:,:4,.i^:s;.N i-'> C'T iX";': 18 6 Thi Arrts and drdinatis te 164 16} i6t I I1056J 1660, 114^607^ 110206 o. 11520475 10989^ O. 12210579 109592 o.12160987 109287 O. I 1991700 x6i 1^0 108980 O.11881720 108671 o. 1177405} io8}6i IJ90.11665692 j 108047 ^580.11557645 107755 o. 11449910 *57 156 *55 107425 o. 11542485 107110 O. **iJ5?75 106794 *54 *55 151 o.11128581 106478 o. 11022105 106159 o. 10915944 105858 0. 10810106 105517 1500.10704589 *5* vffy 1.000 fart oJf Radius. 45 44 43 o, IO704589 105194 O.10599595 104870 o,10494525 104545 o.10389980 104218 o. 10285762 105889 o, 10181873 103560 o. 10078515 105229 0.09975084 102895 0.09872199 102561 0.09769638 102ZI} o.09667579 101876 o.09565503 10I550 0.09465953 10I210 o.09562745 100869 O.09261874 100526 55.0. 09161548 j 100181 34 o. 09061167 S'9834 •Jill Areas and Ordiutts to every looo part of Radius. 187 «33 o. 08961;?? 99461 >3^0.08861872 99112 o.08762760 i;i ;'3= 129 128' 127 126 >2? 124 12; 122 111 120 119 118 117 98786 o. 08665902 98433 0.08765469 98078 o. 08467391 97722 o. 08369669 97364 o.08272305 97004 o. 08175301 96643 0.08078658 96280 O.C»7982378 94915 O. O7886463 95548 o. 07790915 95>79 0. 07695736 94811 0.07 60092 5 94458 o, 07506487 94061 o. 07412426 93685 Ii6 IIS M4 113 112 HI 93685 o.07318741 93307 o.07225434 92901 0.0713253? 92524 o. 07040009 92161 o. 06947848 91774 O. 06856074 91386 I lo'o. 06764629 90944 IO90.06673685 I 90551 1080.06583134 I 90208 '07 0. 0649*2926 I 89811 0.06403115 I 89412 '05o.06313703 I 89011 '04 o. 06224692 I 88608 '0)0. 061 36084 I 83202 '0* o.06047882 ,101 87793 o. 05960089 -I 87382 [100 o. 05872707 I'CJ'-lil ■ v i itHt a- j/ 'I. ,'4 t"* ' [;■ ■ ' '' ;*. V'., i 5 ' i" ' ' ,'* (l I . ' ^ " /I ^ • J ■ <, ' t r •Hill..,, :' • ^ ♦ K« 'ji ■ "i^'. ; il'lHiiH i^; I " : ' ■ ', ,1'.' i:*' i ''■*% I' I.''-: S m .. itii I i s88 7hi Areas and Ordinates to evory looo fart oj Radius. loo 0.05872590 869^9 J^'o. 05785611 I 98 86554 o.05699067 97 56 86157 o. 05611950 85717 0.05517113 85295 95 94 9i o. 05441910 84867 0.05557055 84440 o. 05172613 92 9J 90 89 88 87 86 85 84 84010 0.05188605 85666 o. 05104957 85129 o. 05111866 81700 o. 04959166 8i 159 3. 04856907 81814 o.04775093 81366 3. 04693717 80916 0. 0461281 I 89461 0.04551549 8000 5 83 o. 04452544 79545 810. 04372799 79083 81 80 79 78 77 76 75 74 73 72 7» 70 69 0.04293716 786x7 o.04115095 78147 o.04136948 77674 o.0405 8274 77297 O- O59-S1077 76707 o.05905370 76214 o,05819146 75748 0.03753398 75150 O.03678140 74764 o. 03603376 54165 O. 05 529! II 73752 0.03455515 73246 o,05 381067 72746 68)6.05309311 72131! O. 03 1570891 71716 67 1 66 65 71716 0.05165575 71195 0,05094180 70664 [ 64 ^3 62 0.05025516 70152 0, 02955584 69595 0. 02885789 61 60 69054 0.028147 5 5 68 508 0.02746204 67961 —1 59 57 0. 02928245 67405 0.02610858 66840 0.02545998 56 55 66175 0. 02477725 6,-701 0.02412024 65125 54 53 52 0. 02546901 64539 0. 02282561 65950 0.02218412 5« 50 63353 0. oil 55059 61750 0.01091509 48 50 0. ozopi^oi 490. 0203.0159 61528 o.01968651 60906 o. 01907725 60277 0.01847448 59640 O. 01787808, 58996 O. 01728812 58J44 04 0167046^ 47 46 45 44 43 42 57685 0.01611784 57016 410.01555768 56540 40 39 o. 01499411 55655 0.01445756 54960 O. 01588796 37 54256 C.01554540 53540 560.0x281000 52815 35 34 0,01228185: 51079! 0.011761061 5»33' f 'V Hm ■f^}' 1 T*.i - ''X I C/'f; . ,^;,''. ii; ' I' ■ ■'n.r'i* %'■* •' ! ■ 'l|lt; . I ' ' i 'M.' ' i>i! ) l« . . !•! r'! 11*- l'»!i :v;-- "5 ipo !2^i< Areas and Ordinatts td every looo fare oj naaius, ~n 350.01124776 1 5057^ 320. PI074104 1 49801 3*0. 01024403 1 49016 3°o.00975364 1 48217 ^■90.'00927147 1 i 47405 28 0.00879742 1 46578 270.00833164 j 45734 *60.00787430 : i • 1 44874 250.0D742556 1 43997 240.00698559 J 43*02 23 0.P0655457 1 42*85 22 0.90613272 1 4*244 21 0. 00572028 ' 1 40273 ' 200. 00531730 1 3929* *90.00492439 1 38297 *8 0.00454*42 37248 *7 0. 00416894 j 36176 01^6 i6o- O03?O7I8 JS07I Iflo. 00J45647 I 31929 140. 003; 1718 5*746 IJ 0.P0278972 5»JJ7 120.002474^5 i 30236 II 0.00217219 28897 o. 00188178 27441 lo 90.90160836 j iy??? 8 o.00134877 I 24434 7 0. 00110443 22749 60. 09087694 20925 5 0. 00066769 18922 4 0. 00047847 16675 3 0. 00031172 14061 0- 00017I11 10792 I 0. 00006519 63*9 0 0. oooeoooo I ' • A m TABLE SHEWING THE A R E A O F T H E SEGMENTS OF A CIRCLE ' W'HQSE Whole Area is Unity, to the ten Thoufandth part of the Diameten '"i r i T' .'ti" '}) •'I'- I'' ,v • p- H I I" y* : Hi t; m s •' f ^ '-i ■, i'y'' I,'* ) '1 ■ • ' ,, V / • ;.i 'K "c f' k-.' Ip2 , u*' ' r If n . ; V ' 'V f t* . V- ' ■ I# ' I 'S'?' 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The Fir/1: Part. Of the Primtm Mobile' CHAP. I. Of the GefjeralSnhjeli of Afironomy. Astronomy, is a Science concerning the Meafure and Motion of the Spheres and Stars. 2. Aftronomy hath two parts, the firft is Abfolute, and the other Comparative. 3. The Abfolute part of Aftronomy is that which treateth of the Meafure and Motion of the Orbs and Stars abfolutely without refpeft; to any diftui(ftiouof Time, 4. 2-2^ " 3^n UntroBiicttoit 4. The Comparative part of Aftronomy is , ic.1" that, which treateth of the Motion of the Stars, in reference to feme certain diftindtion of Time. 5. The Abfolute pai t of Aftronomy treateth of the Primum Mobile, or Diurnal Motion of all the iffrrt oil Celeftial Orbs or Spheres. ^ .'iete- 6. The or Diurnal Motion of aitistt: the Heavens, is that Motion, by which the fe- iblii veral Spheres arc moved round the World in a fe ic Day"^.apd a Nigfit,' that is, in 24 hours from ±3! EafttowardsW^, apdlo forward, from Weft ii.Tk towards Eaft, and fo continually returning to the , ial®c 'lame point from whence they began their Mo- MjI tiohi" " ■ :|iole& 7. This firih and cqmipon Motion of the Hea- idaiic vens, will be bell: uhderflood, by help of an In- Itr-ument called a Globe, which is an Artificial re- iir®ai prefentation of the Heavens,or the Earth and Wa- ters under that Form and Figure of Rounduefs which they are fiippofed to have. Hi This Reprefentation of Defer Iption of the Vifible World is by Circles, gyeat and fmall,feme ^ouiielt of which are expreffecl upon," and others are fra- med without the Globe. 1,, i]|.| 9. The Circles without the Globe are chiefly two-, the Meridian and the Horiz.of7, the one of jf.u, Brals; and titc other of WoodAndthefetwo Circles are variable or mutable ■, for although j.,!, thereis butone Hm'z.oa5nd one in re- - fpedt of the whole World, or in relpcdt of the whole Heaven and Earth, yet in refpedl of the particular parts.of Heaven, or^ rather in refped ofthedivcrfq Provinces, Countries and Cit^s on the Earth, there are diverle both Hori^ns and Mcridiims. ;.,j to 33ftconomp. 227 romB; The Meridian then is a great Circle with- itkSc' i-}^g Globe, dividing the Globe, and confe- ^ , quently the Day and Night into two equal parts, [tei fj-om the North and South ends whereof a Itrong ®0i;; Wyre of Brafs or Iron is drawn or fuppofed to be drawn through the Center of the Globe repre- 1* Pent ing the Axis of the Earth, by means whereof liclitl: the whole Globe turneth round within the faid Circle, fo that any part may be brought direftly omir under this Brafs Meridian at pleafure. iciB W II. This Brafs Meridian is divided into 4 e- qual parts or Quadrants, and each of them are iQcii; fubdivided into 90 Degrees, that is 360 for the whole Circle. The reafoh why this Circle is not offc divided in 360Degrees throughout, butftiilllop- Ipof: ingat 90, beginneth again with 10.20.30&C. is, for that the ufe of thisMeridian.in reference to its tim& Divifionin Degrees,requireth no more than that' toi Number. 12. The Horizon is a great Circle without the piioif Globe, which divides the upper part of Heaven Iffflii from the lower, fo that the one half is always above that Circle, and the other under it. 13. The Poles of this Circle are two, the one ^ fCJ. direftly over our Heads,and is called the Zenith y I tks the other is under feet, and is called the Na- Idiltf dir. foiiSs 14. The Horizon is either Rational or Senfi- 1*14* blc* 'jpji; 15. The Rational Horizon is that, which divi- deth the Heavens and the Earth into two equal ,jiii!? parts, which though it cannot be perceived and jjdi diftinguiflied by the eye, yet may be conceived iJ' our minds, in which refpcd all the Stars maybe conceived to rife and fet as in our viewn 0^2 IC5, 228 3a 16. The Vifible Horizon is that Circle which "• ' the eye doth make at its farthelt extent of fight, when the body in any particular place doth turn itfelf round. Of thelc two Circles there need- xfef'F- eth no more to be faid at prefent, only we may ob- ferve, that it was ingenioufly deviled by thole, who firft thought upon it, to let one Meridian '®lt® and one Horizon without the Globe, to avoid thcconfufion,if nottheimpoffibilityjof drawing ;iii. a feveral Meridian and a feveral Horizon for eve^ 'i ' ry place, which mull have been done if this or the like device had not been thought upon. :! 17. BelidesthefetwogreatCircleswithoutthe tctoctib Globe, there arc 4 other great Circles drawn up- ieto on the Globe it felf belldes the Meridian. 2].Jk J. TheT.quatororEquinodlialCircle. 2. The JtWte Zodiack. 3. The Squinoftial Colure. 4. Sol- iw-te ftitial Colure. And thefe four Circles are immu- table, that is,in whatfoever part of the World :50KPoi! you are,thefe Circles have no variation, as the 0- xWoi'i!, ii ther two have. ii.W 18. The iTquator is a great Circle drawn upon fflcsPc the Globe, in the middle between the two Poles, "'ilijci'ijjlii and plainly dividing the Globe into two equal ijIfeK parts, 7, ' 19. The iTqnator is the meafure of the Moti- on of the PrimHm Mobile, for 15 Degrees of this Circle do always arife in an hours time-, the which doth clearly lliew, that the whole Heavens, are turned round by equal intervals in the fpace of one day or 24 hours. , 20. In this Circle the Declinations of the Stars ; are computed from the mid-Heaven towards the hsnoDg North or South. •j,]],; '■••it® 21. toMtonomp' 239 2T. This Circle gives denomimtion to the jEquinox, for the Sun doth twice in a Year and ■^1; no more crofs this Circle, to wit, whenheenters •'i®?, the firft points of Aries yXid Libra, and then he **50. maketh the Days and the Nights equal: His en- ^ !i)D; trance into Aries is in March, and is called the e M Vernal Equinox ; and his entrance into Libra, is I to,- in September, and is called the Autumnal Equi- ■fdrjir nox. ofrf 22. And from one certain point in this ifl'ffi Circle, the Longitude of Places upon the Earth J®, are reckoned and the Latitude of Places are msa reckoned from this Circle towards the North, ste or the South Poles. ; life 23. .The Zodiack is a great Circle drawn up- on the Globe, cutting the TEquinodial Points at f, r Oblique Angles: for although it divides the ijrcr whole World into two equal parts, in reference to its own Poles; yet in reference to the Poles of 1;: the World, it hath an Oblique Motion, 24. The Poles of this Circle are as far diftant i-ipr from the Poles of the World, as the greateft Obliquity thereof is from the Equinoftial, that jujjj is 23 Degrees, and 31 Minutes or therea- bouts, 25. This Circle doth differ from all other Qr- jjgj ties upon the Globe in this; other Circles(to fpeak ■si' properly J have Longitude affigned them, but no jjdiij Latitude-, but this hath both. Whereas other Cir- •vit cles are in reference to their Longitude or Rotun- dity only divided into 360 Degrees, this Circle 0 in refped ofits Latitude is fuppofed to be divided 0 into 16 Degrees in Latitude. 26. The Zodiack then in refped of Longi- ■ tude is commonly divided into 3 60 Degrees as o- 3 ther ..7 2 30 Un 3Introliuctiou ther Circles are: but more peculiarly in refpedl of its felf it is divided into 12 Parts called Signs, and each Sign into 30 Degrees, and 12 times 30 do make 360. _ ^ 27. The 12 Signs into which the Zodiack is divided, have thefe Names andCharadters. Jrks T. Taurus'6. GeminiTL. Concert. Leo Virgo nj;. Libra ss. Scorfio ni. Sagittarius Capricor- rats vy. Aquarius sssi. and Fifces >£. 28. Thefe two Circles of the Equator and Zo- diack are crofled by two other great Circles,which are called Colurcs; They are drawn through the Pe^lesof the World, andcutpne another as well as the Equator at Right Angles. One of them palTeth througli the Interfedions of the Equi: noftial poiiits, and is called the Equinodial Co- lure. Tlie other pafleth through the points of the great;fi; diftance of the Zodiack, from the Equator, and is called the Solftitial Co- lure. 29. The other great Circles defcribed upori the Globe are the Meridians: Where v/e mult not think much to hear of the Meridians again! That of Brafs without the Globe is to lerve all turns, and the Globe is framed to apply it felf ^ereto. The Meridians upon the Globe, will eafily be perceived to be ot a new and another ufe. k: 30. The Meridians upon the Globe are either tlte great or the lels; Not that the great arc any greater than the lefs, for they have all one and the fame center .and equally pafs through the Poles of the Earth •, But thofe which are called lefs,are pf lefs ufe than tliaf which is called the great. 31- to Mtonomp' 231 ■f?-:: 31. The great is otherwife called thefixt and ft, firft Meridian, to which the lefs are lecond, and 5 );, refpedively moveable. The great Meridian is as it were the Landmarks)^ the whole Sphere, from Diii , whence the Longitude of the Earth, or any part ; thereof is accounted. And it is the only Circle " " which palling through the Poles is graduated or Cf. divided into Degrees, not the whole Circle but tlie half, becaufe the Longitude is to be reckoned round about the Earth. s,rai 32. The lefier Meridians are thofeblack lines, which you fee to pafs through the Poles and fnc- m. ceedingthe great at loand ioDegrees,as in moft of; Globes; or at 15 and i 5 Degrees difference, as in Ik!, fome. Every place never fo little more Eaft or 6 Weft than another, hath properly a feveral Me- t jx! ridian, yet becaufe of the huge diftance of the ■s,k Earth from the Heavens, thereisnofenfibledif- ,aaU ference between the Meridians of places that are lefs than one Degree of Longitude afunder, and id: therefore the Grographers as well as the Aftrono- k: mers allow a new Meridian to every Degree of jifif the Equator 3 which would be 180 in all; but ex- : cept the Globes were made of an extream and an ;;ii unufual Diameter, fo many would ftand too thick Gi, for the Defcription. Therefore moft common- ly they put down but 18,that is, at 10 Degreesdi- fiance from one another *, the fpecial ufe of the lef^ 2;- fcr Meridians being to make a quicker difpatch, j;,-;: iu the account of the Longitudes.Others fet down ! cj: but I z ,at 15 Degrees diference •, aiming at this, j:'. That the Meridians might be diftant from one ic another a full part of time, or an hour; for fee- ing that the Sun is carried 15 Degrees of the E- " qiiinoftial every hour, the Meridians fet at that 0^4- diftance 232 3in Jntroliucti'on ^iftancemirit make an hours difference in the ri- fing or fetting of the Sun in thofe places which differ 15 Degrees in Longitude. And to thispurpofe alfo upon the North end of the Globe, without the Brafs Meridian, there is a fmall Circle of Braft let, and divided into two ■ equal parts, and each of them into twelve, that is, twenty four all ^ to fhew the hour of the Day and Night,in any place where the Day and Night exceed not 24 hours •, for which purpofe it hath a little Brafs Pin turning about upon the Pole, and pointing to the feveral hours, which is therefore the Index Horariusy or Hour Index. •33. Having dcfcribed the great Circles fra- raed without and drawn upon the Globe, we will now defcribe the lefler Circles alfo •, And thefe leffer Circles are called Parallels, that is, fuch as are in all places equally diftant from the Equator-, and thefe Circles how little foever, arefoppofed to be divided into 360 Degrees: but thefe De- grees are not fo large as in the great Circles, but do proportionablydecreafe according to thei?^- dins by which they are drawn. 34. Thefe leffer Circles are either the Tro- pick 5 or the Polar Circles. 3 5. The Tropicks are two fmall Circles drawn upon the Globe,one beyond the Equator towards the North Pole, and the other towards the South, Shewing the way which the Sun makes in his Di- iirnal Motion,when he is at his greateft diftance fromtheEquatoreitherNorth or South, Thefe Circles are called Tropicks cctto rtii; rtjovni?, that is, from the Suns returning; for the Sim coming to thefe Circles, he is at his greateft diftance from the Equator, and in the fame Moment of time hoping to 3C(ltonomp. 235 {1 oping as it were his courfejhe returns nearer and nearer to the Equator again. 36. Thefe Tropical Circles do fhew the point of Heaven in which the Sun doth make either the longeft Day, or the Shorteft Day in the Year, ac- ■ cording as he is in the Northern or the Southern Tropick ; And are drawn at 23 Degrees and a half diftant from the Equator. 37. The Polar Circles are two lefier Circles drawn upon the Globe at the Radius of 2 3 De- grees and a half diftant from the Poles of the World, lliewing thereby the Poles of the Zodi- ack, which is fo many Degrees diltant from the Equator on both fides thereof. 38., Thefe Polar Circles are 6 6 Degrees and a half diftant from the Equator, and 43 Degrees diftant from his neareft Tropick. They arc called the Ardick and Antardick Circles. 3 p. The Ardick Circle is that which isdefcri- bed about the Ardick Pole, and paileth almolb through the middle of the Head of the greater Bear. It is called the Ardick Circle aTW t&v a^RT(i)v from the two confpicuous Stars towards the North, called the greater and the lelTer Bear. 40. The Antai'dick Circle is that which is de- fcribed about the Antardick or South Pole. It is fo called adi tm.; that is, from behig oppo- fite to the greater and Idfer Bear. Having thus defcribed theGlobe or Afttonomi- cal Inflrumcnt by which the Frame of the World i-; reprefcnted to cur view, 1 will proceed to fhew t' e ufe for which it is intended. CHAP. Jnttotuction chap. ii. ' Of the Dijlinciions and y^ffeB 'tons of Sfher rical Lines or jirches. The ufes of the Globe as to pradlice, are ei- ther fuch as concern the Heavens or the Earth, in either of which, if we Ihould defcend unto particulars, the ufes would be more in num- ber, than a IhortTreatife will contain: Seeing therefore that all Problems which concern the Globe, may be beft and molt accurately refol- ved by the Doftrine of Spherical Triangles, we willcontradthefe ufes of the Globe(which other- wife might prove infinite) to fiich Problems as come within the compafs of the 28 Cafes of Right and Oblique angled Spherical Triangles. 2. And that the nature of Spherical Trian- gles may be the better underllcod, and by which of the 28 Cafes the particular Problems may be beft refolved, I will fet down feme General De- finition? and Affections, which do belong to fuch Lines or Arches of which the Triangle muftbe framed, with the Parts and Affedions of thofe Triangles, and how the things given and requi- red in them, may be reprefented and refolved upon and by the Globe, as alfohow they may be reprefented and refolved by the Projection of the Sphere, and by the Canon of Triangles. 3. ASpherical Triangle then is a Figure con- fifting of three Arches of the greateft Circles upon the Superficies of a Sphere or Globe, eve- ry one being lefsthan a femicircle, 4' to 3i(!ronom^. 255 4. A grcnt Circle is that which dividcth the Sphere or Globe into two equal parts, and thus the Horizon, Equator, Zodiack and Meridians before ddcribed are all of them great Circles: And of thefe Circles or any other, there mull: be three Arches to make a T rianglc, and every one of thefe Arches feverally mult be lefsthan afe- niicircle: To make this plain. In F-!^. I. The ftreightLine HAR dothrepre- fentthe Horizon, F F the height of the Pole a- bove the Horizon, P a Meridian, and thefe three Arches by their interfefting one another do viftbly conllitnte four Spherical Triangles. I. PMR. 2. PMH. 3. SHM. 4. SMR. And every Arch is lefs than a femicircle, as in the Triangle P A/P, the Arch P P is lefs than the Se- micircle P PS, the Arch MR is lefs than the Se- micircle AMR^ and the Arch PM is lets than the Semicircl cP MS, the like may be Ihewed in the other Triangles. 5. Spherical or circular Lines are Parallel or Angular. 6. Parallel Arches or Circles, are fuch as arc drawn ijpon the fame Center within, without, or equal to another Arch or Circle. Thus in Fig. 1. The Archcss M^ and vj' o V3' are though leifer Circles, parallel to the Equinodial vE A ^and do in that Scheme reprefcntthe Tropicks of Cancer and Capricorn. The manner ofdefcribing them *or any other Pardlel Circle is thus, fet off their dihance from the great Circle,to which you are to draw a parallel with your Corapailes, by help of your Line of Chords, which in this Example is 2? Degrees and a half fi om iEto !S, then draw tiic l-ine A 'b:^, and upon the point % cred a Per- pendi- fl«': ti'., ) » ■ . ..41 I Ir' Mi'tlli m [-! |Jii«Mj','|| I'ilii i:itl 235 3In 31ittcot)ucti'oii pendicular, where that Perpendicular lhall cut the Axis P^S extended,is the Center of that Pa- railel- 7. A Spherical Angle, is that which is con- teincd by two Arches of the greateft Circles up- on the Superficies of the Globe interlefting one , another; Angles made by the Interfedtion of two little Circles, or of a little Circle with a great, , wo take no notice of in the Dodlrine of Spheri- cal Triangles. 8. A Spherical Angle is either Right or Ob- lique. 9. A Spherical Right Angle is that which is conteined, by two Arches of the greateft Circles in the Superficies of the Sphere cutting oneano- ther at Right Angles, that is, the one being right or perpendicular to the other : thus the Brafs Meridian cutteth the Horizon at right An- gles; and thus the Meridians drawn upon the Globe, as well as the Brafs Meridian, do all of them cut the Equator at Right Angles. 10. An Oblique Spherical Angle, is that which is conteined by two Arches of the greateft Circles in the Superficies of the Sphere, not be- ing right or perpendicular to one another, 11. An Oblique Spherical Angle is Obtiife, or Acute. An Obtufe Spherical Anglc,is that which 12. is greater than a Right Angle. An Acute is that| which is Icfs than a Right Angle. 13. If two of the greateft Circles of the Sphere fliail pafs through one anothers Poles, thole two great Circles fliall cut one another at Right An- glcs: Thus the Brazen Meridian doth interfecl; the Equinoftial and Horizon. i+. I III M I llWIUi, q.11 toBflronotnp. 237 14. If two of the greateft Circles of the Sphere fliall interfeft one another, and pals through each others Poles, theyihall interfedt one another at unequal or Oblique Angles, the Angle upon the one lide of the interfedion being Obtufe, or more than a Right, and the Angle upon the other fide of the interfedion being Acute or lefs than a Right. Thus in F>^.I. The Arch PylFdoth interfedthe Meridian and Horizon, but not in the Poles of either, therefore the Angle HPM upon one fide of the interfedion of that Arch with the Meridian, is more than a Right Angle-, And the Angle MP R upon the other fide of the Inter- fedion is lefs. And fo likewife the Angle P MH upon the one fide of the interfedion of the Arch P M with the Horizon H R, is greater than a right Angle; and the Angle P upon the o- therfideof the Interfedion is lefs than a Right. 15. A Spherical Angle ismeafuredbythe Arch of a great Circle defcribed from the Angular point between the fides of the Angle, thofe fides being continued unto Quadrants. Thus the Arch of the Equator in F;V. i. is the meafiire of the AngleyJ/PP, or TP the fides P T and P i^being Quadrants. And the nieafure thereof in the Projedion may thus be found; lay a Ruler from P to T,and it will cut the Primitive Circle in F, and the Arch F jQ_ being taken in your Compaflb and applyed to your Line of Chords, will give the Quantity of the Angle propounded. 16. The Complement of a Spherical Arch or Angle, is fo much as it wanteth of a Quadrant, if the Arch or Angle given be lefs than a C^a- drant ^ or fo much as it wanteth of a Semicir- file. 258 2En ^fntrotjuctioit cle, if it be more than a Quadrant. 17. An Arch of a great Circle cutting the Arch of another great Circle, lhali interfe^t one another at Right Angles^ or make two Angles, which being taken together, ihail be equal unto twoRight. Thusinfi>.i. The Axis Por Equinoftial Colure doth cut the Equator at Right Angles, but the Meridian ? MS doth cut the Horizon HMR at Oblique Angles, ma- king the Angle P/^Plefs than a Right, and the Angle SMR more than a Right, and both to- gether equal to a Semicircle. 18. Fromthefe general Definitions proper to Spherical Lines or Arches, the general Atfecdi- ons of thefe Arches may eafily be difcerned I mean the various Politions of the Globe of the Earth, in relpeftof all and lingular the Inhabi- tants thereof. 19. And the whole Body of the Sphere or Globe, in refped of the Horizon,is looked upon by the Earths Inhabitants, either in a Parallel, a Right, or an Oblique Sphere. 20. A Parallel Sohere is, when one of the Poles of the World is elevated above the Horizon to" the Zenith, the other deprefled as low as the Na- dir, and the Equinodial Line joyned with the EIo- rizon. They which there inhabitc (ii any fuch be) fee not the Sun or other Star riling or fetting, or higher or lower in their diurnal revolution. And feeing that the Sun traverfeth the whole Zo- diack in a Year, and that half the Zodiack, is a- bove the Horizon and half under it, it cometh to pafs, that the Sunfetteth not with them, for the fpace of fix Months, nor givcth them any Lightforthefpace of other fix Months, and fo maketh to ItHconoinp- 239 maketh but one Day and Night of the whole Year. • 21. A Right Sphear is, when both the Poles of the World do lie in the Horizon, and the E- quinodtial Circle is at his grcateft diftance from it, paffing through the Zenith of the place. And in this pofition of the. Sphere, all the Cccleftial Bodies, Sun, Moon, and other Planets, and fix- ed Stars, by the daily turning about of the Hea- ven, do diredly afcend above, and alfb diredly defcend below the Horizon, becauie the Moti- ons which they make in their Daily motion do cut the Horizon Perpendicularly, and as it were at Right Angles.ln this Pofition of the Sphere, all the Stars may be oblerved to rife and fet in ane- qua! fpace of time, and to continue as long above the Horizon, as they do under it, the Day and Night to thole Inhabitants, being always of an equal length. 22. An Oblique Sphere is, when the Axis of the World f being neither Dirednor Parallel to the Horizon J is inclined obliquely towards both fides of the Horizon, as in Fig. i. Whence it co- meth to pais, that fo much as one of tlie Poles is elevated above the Horizon, upon the one fide fo much is the other deprefled under the Hori- zon, upon the other fide. And in this Pofition of Sphere,the Days arc fomc- times longer than the Nights, fometimes ihorter, and fometimes of equal length .• When the Sun is in either of the Equinodial Points, the Days and Nights are equal-, but when he declineth from the Equator towards the elevated Pole, the Days are obferved to encreafe and when he declineth from the Equator towards the oppofite Polcjor the Pole depreffed, 240 Sn ^fntcoliiictiott deprefled, theDaysdodecreafe,as is manifeftin Fig. I. For when the Sun rifeth at M, the Line Af S above the Horixon is the Semidiurnal Arch of the longeft day. When he rifeth at C,the Arch C above the Hor izon, is the Semidiurnal Arch of the IhorteftDay ; And when he rifeth at the Days and Nights are of equal Length, the Semi- diurnal Arch being equal to the Semino- durnal Arch AQ^ CHAP. IIL Of the kind and parts of Spherical Trian- gles 5 and how to project the fame upon the Plane of the Meridian. HAving (hewed what a Spherical Triangle is, and of what Circles it is compofed,vvith the general Affedtions of fuch Lines: I will now (hew how many feveral forts of Triangles there are, of what Circular parts they do confift, and llich Afiedions proper to them as will render the foliitionof them more clear and certain. 2. Spherical Triangles arc either Right or Ob- llque. 3. A Right Angled Spherical Triangle, is that which hath one or more Right Angles. J. A Spherical Triangle which hath three Right Angles, hath always his three (ides Qua- drants. As in Fig.x. The Spherical Triangle AZR^ the Angles R A Z txA AZ R Tae. right Angles, and the three fides ZRtsA A R are Quadrants alfo. S. A to Sflcoiiom?^ 241 Ui 5. A Triangle that hath two right Angles,hath 1: tlie fides oppolite to thofe Angles (Quadrants, and the third lide is the nieafure of the third Angle. AsinFi^.i. The fides of the Spherical Trian- dh. gle TPjQ^, namely TP and P ^ are Quadrants, al. and the Angles oppofite to thefe fides,to wit,PT<^ I tltk and T Qj are Quadrants alfo, and the third kSa. Angle Ti^is the meafure of the third Angle TPBut the Right Angled Triangle which hath one Right and two Acute Angles, is that which cometh moll commonly to be reiol- ved. 6. The Legs of a right Angled Spherical , V Triangle are of the fame Affedion with their op- polite Angles-, as in the Triangle Fig.i. ^ T he fide is a Quadrant,anothe Angle at Z is right, becaufe Z is the Pole of the Arch and ZZ is perpendicular thereunto. And in the Tri- Tiffi angle R A 4 to s and another perpendicular thereunto from 2S it will cut the Axis S ^ P extended in the Gen- ter of that Tropick, by which extent of thecom- palTes the other Tropick maybe drawn alio. Or thus the Co-tangent of eydEz fet from $ to the Axis extended will give the Center as before, and thus may all other Parallels be defer ibed. 17. In the fecond and third figures, the two extream points given in the Meridians are not e- qnidiftant from the third, for the drawing of which Circles, if the common way of bringing three points into a Circle be not liked*, you may do thus, from the given point at F and the Cen- ter ^ draw the Diameter T ^5, and crofs the lame at Right Angles with the Diameter P a Ruler laid from c? to P will cut the primitive Cir- cle in L,make £ L = 5 L a Ruler laid from G to E will cut the Diameter S AT \n F the Center of the Circle Which Circle doth cut the Diameter H AR in the Pole of Z F, and tlie Di- ameter AQ^ in D in the Pole of PFX, and a Ruler laid from Z to C will cut the Primitive Circle in £, and making T 0 equal to T a Ruler laid from Zto 0 will cut the Diameter H A Ry extended in the Center of the Circle Z F. 18. Having drawn the Circle Z£/, inF?^.i3. The Circle P £X, or any other palling through the point F, may eafily be defcribed. Draw at right Angles to P T,a Ruler laid from Cunto ( e j Will cut the Primitive Circle in (m) inakew«=. a Ruler laid from G" to « R 3 fliaU 24^ ^ fliall cut the Diameter TF Sin p make F ^ == Fp fofhallF^be the Radius, and the Center of the Circle P F X as was defired. ^ ip. Thepreceedingdiredions are fufficicnt for the projeding of feveral Circles of the Globe n 3. before defcribed upon the Plane of the Meridian, and the parts of thofe Circles fo defcribed may . thus be meafnred. In Fig.HZ = CZ=^ AZ 90 Degrees. Whence it followeth, that the Quadrant CX is divided into Degrees from its Pole M, by the Degrees of the Quadrant HZ^ that is a Ruler laid from M to any part of the ^ ^ Quadrant HZ will cut as many Degrees in C Zas it doth in the Quadrant H Z, and thus the Arch ■ ■ ■ fF = HK the Arch CB — HL, and the Arch .a , BF = LK. 20. Thatwhichisncxttobeconfideredisthe projcding or laying down the Angles of a Trian- gle, and the meafuring of them being projeded, , and the Angles of a Triangle are either fuch as ' ^ are conteincd between two right l-incs as the Am gle Z in the Triangle PZP; or fuch as are con- ^ teined by a ftreight and a Circular Line, as the ; Angle P MR. Fig. i. Or fuch as are conteined by two circular Lines, as the Angles FZPoxZFP inF^>.3. The projefting or meafuring the firlt ' 'j fort of thele Angles, needs no diredfion. 21 .To projedt an Angle conteined by a ftreight ' ® and a circular line as the Angle c/FEZ in Figh Do thus, lay a Ruler from Nto C, and it v;ill cut the Primitive Circle inXniake ZX = P/X,aRu- ' ler laid from ATo Twill cut the Diameter H AR ■'? ijn the point M the Pole of the Circle ZCN^ a Ruler laid from Mtop the Angular point pro- ■' poundedjwillcutthe primitive Circle in I, make ™ - • - ■ to Hffrditomp. 247 j;5 f. NT — HL a Ruler laid from N to T will cut the Circle Z CNinW,a Ruler laid from B to W ■ will cut the Primitive Circle in ^,make ^qual to the Angle propounded, and draw the Diame- ter 5^^, then is the Angle . 3. You mult firlt find the Pole of one of the i®. two Circles conteining the Angle propounded, if'- fuppofe Z F 7, a Ruler laid from C the Pole there- of to f,the Angular point propounded, will cut the Prim'tive Circle in a make ab equal to the Angle propounded, a Ruler laid from F to b will If ' cut the Diameter i'l ^ the Pole of the k/' Circle FFX, a Ruler laiJ from G to e will cut ici^ the Primitive Circle in m, make m ri = B m ij/;' R 4 aRu-' 24® 3(ln 3lntrotucti'oti ler laid from Gto n will cut the Diameter T AS in />, make A ^ — ^pfolhallFpbetheRadi- jociF us and the Center of the Circle PFX and the An- '. aJS' ZF P= ab,i[s was propounded. T ' asffi ' 26. If the Angle had been projedled and the ^ meafure required ; through the point F draw the ' Diamerer "TF S Md the Diameter B AG at right . osffd Angles thereunto, a Ruler laid from 6'to F will cut the Primitive Circle in K, and making KE = d T B K a Line drawn from 6'to £ will cut the Dia- Goiie, rneter TAS in the Center of the Circle G DB sjiiic cutting the Diameter HA R in Cthe Pole of the- M Circle ZB I, and the Diameter tA A,Q^ in £, iS. the Pole of the Circle P E Xand a Ruler laid from edf F to C and D will cut the Primitive Circle in ^ areilT and ^ the meafure of the Angle required. comii; Or a Ruler laid from F to K and M will cut the Tg Primitive Circle in Deg. the meafure of the Angle ■ • propounded as before. arf Or thus a Ruler laid from Cand H to F will a,Hi cut the Primitive Circle in to I will cut PE X in K. This done a Ruler laid from f jf, to K and Afwill cut the Primitive Circle in^ and 1 ] d the mealure of the Angle as before. \ AndinFi^.2. The quantity of the Angle ZFf j may thus be found. ' A Ruler laid from Cthe Pole --[(j; pf the Circle Z F / to F the angular point will cut the Primitive Circle in a, fet off a Quadrant from . ] a to L a Ruler laid from C to b will cut the ■ ^ Circle ZF/ in the point M. in like manner a jfj Ruler laid from D the Pole of the Circle P £ J, will cut tiie Primitive Circle in Z>, fetoffaQua- ^rant from A to h, a Ruler laid from D to P'will cut to 3!lftronoitty. 24^ Gut the Circle P F X'lnK: Laftly a Ruler laid from F to and .4/will cut the Primitive Circle mNS the meafure of the Angle KF M or ZFP^ as was propounded. 27. Having lliewed how a right or oblique An- gled'Spherical Triangle may be projefted up- on the Plane of the Meridian, as well as delinea- ted upon the Globe, we will now confiderthc le- veral Triangles ufually reprefented upon the Globe, with the feveral Allronomical and Geo- graphical Problems conteined in them, and re- folved by them. 28. The Spherical Triangles ufually reprefent- ed upon the Globe are eight, whereof there are five Right angled Triangles, have their De- nom'inrition (rom their Flypotemfas. The firfl is called the Ecliptical Triangle,whole Hypotenufa is Arch the Ecliptick, tlie Legs thereof arc Arches of thcTquator and Meridi- an, this is reprefented upon the Globe, by the Triangle^ D F, inF^^.i, In which the fiveCit- cular parts, befides the Right Angle are-, 1. The Hypotenuse or Arch of the. Ecliptic!; AF. ' 2. The Leg or Arch of theTquator, AH. 55, The Leg or Arch of the Meridian Z) F. 4. The Oblique Angle of the Equator with the Ecliptick and the Suns greateik Declination HAF. 5. The Oblique Angle of the Ecliptick and Meridian, or the Angle of the Suns pofition AfH. The two next I call Meridional, becaufe the Hypottnufas in them both, are Arches of a Meri- dian. One of thefe is noted with the Letters ■ ' MPR 350 3n 3Inttoliucti'oti MP RxnFi^.i, In which the five Circular parts 1. The Hypoteftufa or Arch of a Meridian PM. 2. The Leg or Arch of the Horizon the Suns Azimuth North. 3. The Leg or Arch of the Brafs Meridian, Keprefenting the height of the Pole P R. 4. The Oblique Angle of the Meridian upon the Globe, with the Brafs Meridian, or Angle of the Hour from Midnight. P. _ 5. The Oblique Angle of the Suns Meridian with the Horizon, or the Complement of the Suns Angle of Pofition P MR. The other Right Angled Meridional Triangle is noted with the Letters AEG 'm Fig. 1. In which the 5 Circular parts are. 1. The Hypotemfa OT preient Declination of the Sun, AE. 2. The Leg or Suns Amplitude at the hour of fix, AG. 3. The other Leg or Suns height at the fame time E G. 4. The Angle of the Meridian with the Ho- rizon, or Angle of the Poles elevation, EAG. 5. The Angle of the Meridian with the Azi- muth, or the Angle of the Suns pofition, AEG. The fourth Right Angled Spherical Triangle, Italian Azimuth Triangle, becaufe the Hypote- Knfa doth cut the Horizon in the Eaft; and Weft Azimuths, as is reprefented by the Triangle ADr. inF^v.i. In which the 5 Circular parts are, 1. The to 3(l(lrottomp. 351 1. The Hypote«i^a, or Arch of the Sun or Stars Altitude^ r. 2. The Leg or Declination of the Sun or Star, DP^. 3. The other Leg, or Right Afcenfion of the Sun or Star, AD. 4. TheObiique Angle or Angle of the Poles elevation,!) Jr. 5. The other Oblique Angle or Angle of the Sun or Stars Pofition, D F J. The fifth and lafl; Right Angled Spherical Tri- angle, that I (ball mention, 1 call an Horizontal Triangle, becaufe the Hypotenufa thereof is an Arch of the Horizon, and is reprefcnted by the Triangle AMTinFig.i. In which the 5 Circular parts arei 1. Th^Hypotennfa zndi Krc\\ of the Horizon, or Amplitude of the Sun at his rifing or letting, AM. 2. The Leg conteining the Sun or Stars Decli- nation T M. 3. The other Leg or Afcenfional difference AT, that is, the difference between DT the Right Afcenfion and D v^the Oblique Angle. 4. The Oblique Angle of the Horizon and E-s qiiator, or height of the Equator T AAF. 5. The other Oblique Angle, of Angle ofi the Horizon and Meridian AMT. The Oblique Angled Spherical Triangles u- fually reprcfented upon the Globe are three. The firft I call the Complemental Triangle, be- caufethefides thereof are all Comple-ments, and this is reprefentcd by the. Triangle FZP'vx Fig. i.Whofe Circular parts are ; I. The Complement of the Poles elevation ZP. ' 2. The 252 3n UtttroDucKoit 2. The Complement of the Suns Declination, FP. 3. The Complement of the Suns Altitude or Almicantar FZ. 4. The Suns Azimuth or Diflance from the North FZ P. 5. The hour of the day or diftance of the Sun from Noon ZPF. 6. The Angle of the Suns Pofition ZF P. The fecond Oblique Angled Spherical Trian- gle, I call a Geographical or Nautical Triangle, becaufe it ferveth to refolve thofe Problems, which concern Geographie and Navigation, and this is alfo reprefented by the Triangle F Z P in FjV.i. Whofe parts are. "1. The Complement of Latitude as before Z P. 2. The diftance between the two places at Z and F or fide F Z. 3. The Complement of the Latitude of the place at F or fide F P. 4. The difference of Longitude between the two places at Z and F or the Angle FBZ. 5. The point of the compafs leading from Z to F or Angle F Z P. 6. The point of the Compafs leading from F toZ,or An^IeZFP. The third Oblique Angled Spherical Trian- gle is called'a Polar Triangle, becaufe one fide thereof is the diftance between the Poles of the World, and the Poles of the Zodiack. This Triangle is reprefented upon theCoeleftial Globe, by the Triangle F S P in Fig.j\. In which the Cir- ciilar parrs arc •, 1. The diftance between the Pole of the World, to SIftronomp. 255 World, and the Pole of the Ecliptick, or the Arch S P. 2. The Complement of the Stars Declination, FP. 3. The Complement of the Stars North La- titude, froratheEcliptickor the Arch F 5. 4. The Angle of the Stars Right Afcenfion FPS. 5. The Complement of the Stars Longitude FSP. 6. The Angle of the Stars Pofit ion S F F. 29. And thus at length I have performed, what was propofed in the 15 of this Chapter, that is, I have Ihewed how the feveral Circles of the Globe, may be projected upon the Plane of the Meridian, the feveral ufeful Triangles that are defcribedbyfuchprojeJite thereunto being given., to find the other AnHe. nJ . , ■ In the Right angled Spherical Triangle yl/PP,. let there be given, 1 he An- IPS and the IMR J to find the ? M. 3^1 Leg JPP 1 Angle 5 P. By the i o. Redlangle, cs APR. Rad ;: cs P.cs M. I pf By the i. Rectangle, cs PR, Rad :; csAP^ s P. ' P - CASE 15, m rt.i! •'im :ry{hyi s' v^!';? ^66 3n lntrot)dct(oti CASE 1^. Tht ObliqKe Atgles given, to find a Leg, In the Right angled Spherical Triangle MVt, Ut there be given, the Angles at M and P, to find the Legs MR and P R. By the lo. Redangle, %M. Rad ;: cs P- cs MR. By the 6. Redangle, s P. Rad;; cs M. cs P R. CASE i6. The Hypotemfe and one Leg given, to find-the other Leg. In the Right angled Spherical Trianglc MPR, let there be given, By the 8. Re-c cs PR. Rad :: cs MP- cs MR. dlangle, i cs MR. Rad :; cs MP. cs P R. Thus I have given you the Proportions by which the i6 Gales of a Right angled Spherical Triangle may berefolved, In which there are contained 30 Aftronomical Problems. Two in every Cafe except the Second and the Eighth. In both which Cafes there are but two Problems. And thus I have done with Right angled Sphcri- cal Triangles. " 4. In Oblique angled Spherical Triangles there ii A f 7 t to Itftronom?. 16 j there are twelve Cafes, ten whereof may be rc: folved by the Cathplick Propofition If the Spherical Triangle propounded be firft convert- jed into two right, by letting fall of a Perpendi- cular, fonietimcs within, fometimes without the Triangle. 5- If the Angles at the Bafe be both acute or both obtufc, the Perpendicular Ihall fall with- in the Triangle-, "but if one of the Angles of the Bafe be acute and the other obtufe , the Perpendicular fhall fall without the Triangle. 6. However the Perpendicular falleth, it muft be always oppofite to a known Angle, for your better diredion, take this General Rule. from the end of a Side pven, heino adjacent to an yin^le given, let fall the Verfendicular. As in the Triangle FPS in fig. 4. If there were given the Side fS and the Angle at S, the Perpendicular by this Rule mufl: fall from f upon the Side 5 f extended, if need require. But if there were given the Side P5 and the Angle at S, the Perpendicular mult fall from P upon the Side FS. 7. To divide an Oblique angled Spherical Triangle into two Right, by letting fall a Per- pendicular upon the Globe itfelf, is not necelfa- ry, becaufe all the Cafes may be refolvcd with- out it, but in projcdion it is convenient to in- form the fancy: and feeing the reafon by which it is done in projedion doth depend upon the na- ture of the Globe, I will here Ihew it both ^ays, firft upon the Globe, and then by pro- jedicn. 8. An '6a sun fnttoliucti'oii An Oblique angled Spherical Triangle may be divided into two Right, by letting fall a Per- pendicular upon the Globe it lelf, in this manner. In the Oblique angled Spherical Tri- angle BPS in 4. let it be required to let fall a Perpendicular from P upon the Side F 5. Sup- pofethe Point P to ftand in the Zenith, where the Arch F 5 fliall cut the Zodiack, which in this Figure is at FT, make a mark , andfromthis Point of Interlcdion of the Circle upon which the Perpendicular is to fall with the Zodiack, reckon 90 Degrees,whichfuppofctpbeat P-, a thin Plate of Brafs with a Nut at one end thereof, whereby to fallen it to the Meridian, as you do the Quadrant of Altitude , being graduated as that is, but of a larger extent (for that a Qua- drant in this cafe will not fuffice) being faflnedat P and turned about till it cut the Point L in the Zodiack, wnll defcribe upon the Globe the Arch of a great. Circle PEL, interfeding the Side FS at Right Angles in the Point £, becaufe the Point L in the Zo Hack is the Pole of the Circle SFK, now all great Circles which palfing through the Point L , fliall interled the Cifclp FFTG"", fliall interfed it at Right Angles-, by the 13. of the 2. Chapter. 9. And hence to divide an Oblique angled Spherical Triangle into two Right by projedi- On is cafie, as in the Triangle F P 5, the Pole of the Circle S F K is L, therefore the Circle BLP lhall cut the Arch FS at Right Angles in the Point E. And becaufe the Point AFis the Pole of the Circle BFP , therefore the Circle GMS fliall cut the Circle PFP at Right An- gles in the Point D, the Side FP being extend- to 3lffroiiom?. td. Come we now to the feveral Cafes which af- terthispreparationmaybe refolved, bytheCa- tholick Propolition. CASE I. Two Sides with an Angle oppojite to one of them be-^ ing given, to find the Angle oppofite to the olher. In the Oblique angled Spherical Triangle TPS, in Fig. 4. the Sides and Angles given and required will admit of fix Varieties; all which may be refolved by the Catholick Propofition, at two operations, bur thofe two may be re- duccd to one, as by the following Analogies to every Variety will plainly appear. •tbet igkt ^ ■c die 02^ Given Required 1 FP 1. PS PFS PSF Rad. s PS ::s PSF. s PE | jPF.Rad:;sPE.jPf5 sPF.sPS::sPSF.sPFs\ FP '1. PS PSF PFS K^d.sFP-.-.sF.sVE 1 /PS. Rad;;sPE. s PSF s. PS. sFP wsPFS.sPSF PS 3. FS FPS PFS Rad./SF;;jF. sDS / PS.Rad:;sDS./SPP) /. PS. s SF ;: / PFS. s PSF PS 4- FS PFS FPS Rad./PS;: / SPD.sDS /FS.l^ad:; sDS.sSF /. FS. / PS ;: / SPF. / SF FS S. FP FPS FSP Rad. / FS :: / S. s FC i/FP. Rad-.-.sFC. /FPC s.FP.sFS -.-.sPSF.s FPS FS nn Untcotiuctiort FS Rad.jPP::^PPC.sFG 6. FP FSP jpS.Rad:: sFC. sS FPS s. FS. sFP:: FPS. s PSF. Given Required FP i. PS FPS PSF EPS^FPF==FPS I; cot PSF,Rad:: cs PS. ct EPS 2. ct PS. Rad:: cs EPS. t EP 3. Rad. t EP :: ct FP. cs FPE ct PS. cs EPS:: ct FP: ct FPE FP 2. PS FPS PFS FPSJ,EPF=FPS I .cot PFS.Rad:: cs PF.ct EPF 2. ct PF. Rad:: cs EPF. t EP 3. Rad. t EP;: cot PS. cs EPS cot PF.CS EPF:: ct PS ct EPS PS 3. FS PSF\ PFS FSD~PSD~PSF\ I .tot PFS.Rad:: cs FS.ct FSD 2. ctFS.cs FSD:' Rad. t DS 3. Rad. t DS:: cfP5.cj PSD ct FS.cs FSD;: ct PS.cs PSD CASE 2. Tm Sides with m Angle appofite to one of them being given J to fnd the contained Asgle, In this Cafe there are fix Varieties, all which may be refolved by the Catholick Propofition, ,. ■ according to the Table following. ', , ^ PSF I .cot FPS.Rod:; cs PS.ct PS D ct PS.cs PSD:'.R^d.-tiyS 3. Rad. t DSPS. cj PSD ct PS.cs PSD: -ct FS. cs FSD cot FSP. Rad:: cs FS. ct Sf C| 2. Ct FS. a SPC;; Rad. tFC j Rad. tFC;: a PP. csPFC, ct FS.cs SFC:: ct FP.cs Pft PS to 30ronomp. 2/r FS 6. FF PFS FPS SFC~PFC=PFS 1. cot FPS.Rod:: cs FPxt PFC 2. cot FP. cs PFC:: Rad, t FC 3. Rad. t FC ;: cr FS. cs SFC ct FP.CS PFC:: ct FS.cs SFC. CASE 3. Two Sidts and an Atgle oppojiteto one of them being given, to find the third fide. The Varieties in this Cafe, with their refolu- tion by the Catholick Propofition, are as follow- eth. Given Required| FP I. PS FS PSF ES^FE — FS 1. ct PS :: csPSF. t ES 2. cs ES. cs PS:: Rad. cs EP 3. Rad. cs EP:: cs FP. cs FE cs ES. cs PS:: cs FP. cs FE FP I. PS FS PFS SE^FE = FS 1. cot FP. Rad:: cos PFS.t FE 2. cosFE.cos FP;; Rad.cosEP 3. Rad .cos EP:: cos PS.cos SE cos FE.cos FP:: cos PS.cos SE PS 3. FS FP PFS EL—PD=zFP 1. cot FS. Rad:; cos PFS.t FE> 2. cos FD.cos FS;; Rad. cs SD 3. Rad.cos SD;; cos PS.cs PD cos FD. cos FS;; cs P$.cs PD PS f FS FP FPS ED~PD^FP 1. cot PS. Rad:: cos FPS. t PD 2. cosPD.cosPS:; RadxosSD, 3. R)td.cos SD :: cos FS.csFD, cos PD.COS PS:". cos FS cs FD^ t w- *■ ■ y (I : *1 ■f I'" ■. V' > -*■■■ ' ■< .1,' J « ' I '■ I U-, , ■; • x; . ' : , ♦ i FS 2Bn 3IntroDuct!oii FS FP FSP SC—PC=PS FS 6. FP FPS SC—PC=PS 1. cot FS. Pad: : cos FSP. t SC 2. COS SC.COS FS;: Rad. cos FC S.Rad.cos FC;; cos FP.cos PC cos SC. cos FS. cos FP. cos PC 1. cot FP. Red:; cos FPS. t PC 2. cos PC.COS FP;; Rad.cos FC 3. Rad.cos FC;; cos FS. cos SC cos PC. cos FP:; cos FS.cos SC Tvto j4ngks with a Side oppo/ite to one of them being < given, to find the Side oppojite unto the other. ■■jjg-i,; The Varieties in this Cafe, with their Refolu- tion by theCatholick Propofition, areas follow- • , eth. Given Required PFS 1. FPS PS Rad. J PS : ■.sDPS.sS'Q s. FP. Rad :: sSD. sFS s. PFS. s PS -. mFPS. jFS 2. FPS Rad. .f FS::s PFS. s SD J. FPS. Rad:; sSD. j PS J. FPS. sFSv.sPFSisPS FPS 3. PSF FP Rad. sFP-.-. s FPS. s FC r. PSF. Rad;: sFC. sFS 's. PSF. sFP sFPS. sFS ^ ■ fjs to Sfironomi'' 275 1 FPS '4. PSF FS FP 1 R^d. sFS:: t PSF. sFC V.fP5.Rad :: sFC. s FP j.FPSsFS:: sPSF. sFP PSF 5. SFP PS 1 Rad. s PS ;-i sPSF, s PE SFP. Rad;: sPE. s FP ■s. SFP. sPS'.-.sPSF.sFP. FP FSF 6. SFP !■■ . FP PS ' Rajd. s FP f. s PFS. s PE PSF. Rad :: sPE: J ^5 \s.PSF. sFP::sPFS. sPS C .A S E 5. ^ Tm Angles and a fide ofpofite t'O otte of them hein^ given, to find the Side between them. The Varieties and Proportions, are as follow-' ethi i iOiven Required PFS I. FPS FP PS FD—PD — FP 1. ct PS, Rad:; cs DPS. PD 2. ct DPS. s PD :; Rad. t DS 3. Rad. t DS •: ct PFS. s FD ct DPS. s PD: : ct PFS. s FD .'PFS i:FPS FP FS Pd~pd=fp 1. ct DFS. Rad:: cs PFS. t FD 2. cot PFS. s FD:: Rad. t DS i.Rad.tDS::ct FPS.sPD '-FPS 3. PSF PS : fp SC-PC=^PS I I. cot FP. Rad:: cs FPC. t PC I. cot FPC. //'C;:Rad.t FG 3. Rad.tFC.: ctPSF. s SC cot FPC. s'PC-.: ctPSF.CS FF^ 274 ^nttoDuctton ■FPS 4. PSF PS FS ■ 'sc-^pc=ps 1. cot FS. Rad;: cs PSF. t SC 2. cot PSF. J SC:: Rad. t FC 3. Rad. t FC:: cot FPS. s PC cot PSF. sSC'.i cot FPS. s PC PSF 5. SFP - FS PS f£-f-.S£=^f5 1* cot PS. Rad:: ct PSF. t SE 2. cot PSF. sSEi: Rad. t PE 5. Rad. t PE-:: cot SFP. s FE cot PSF. sSE cot SFP. s FE PSF 6. SFP FS FP FE+SE = FS 1. cot fP.Rad:: cs SFP. tFE 2. cot SFP. s FE t: Rad. tPE 3. Rad. t PE:: cos PSF. s SE cot. SFP.s FE:: cs PSF. s SE CASE 6^ Two yitigles and a Side oppefite to one of them being given^to find the third Angle. ' - The Varieties and Proportions are as follow- eth. Given Required] PFS \\.ctDFS.Rad\'.csPS.ctPSD 1. FPS PSF2. s PSD.CS DPS:; Rad.csDS PS IB. cs DS.Rad;; cs DFS.s FSD FSD-PSD=:PSFfs DPS.S PSDvxs DFS.sFSD PFS U .ct PFS. Rad ::cs FS.ct FSD 2. FPS PSFiZ. s FSD. cs PFS : ;Rad.csDS FS U.csPDS.K^dixsDPS.csPSD FSD-PSD—PSFs^ PFS.sFSD\ cs DPS.csPSD to mtrondntp* 275 ' fPS 3. rsf PFs FP SFC—PFC^ PFS I»a FPC- Pjtd:: cs Fl'.ct FFC Z. s PFCi cs FPC:: Rad. c$ FC J. csFC.Rad::cx jP5F. jfc cs FPCJ PFC:: cs PSF.s SFCj FPS 4. PSF PFS FS 'SFC-^PFCd^PFS \.cot PSF.Rod':: cosFS-ctSFC, z. s ^FC. cs PSF:: Rad. cs FC ( 3. cs FC. Rad:: cs CPF. s PFCj cs PSF.S SFC:: cs CPF.i PFq PSF 5. SFP FPS PS PPE-{-SPM^PS 1. cot PSF.Rod;.: cs PSxt SPl^ 2. s SPF. cs PSF'.: Radi csPE 3. cs PE. Rad:: cs SFP. s FPE cs PSF.S SP£:: cs SFP.s FPE PSF 6. SFP FPS FP FPF-\-SPE^FP$j I .cot SPP.Rad:: cs FP.ctpPE Z. s FPE. cs SEP:; Rad.cs PE 3. COS PE Rad:: cs PSF^sSP^ cs SFP.S FPEcs PSF is SPfij . C A S E 7. ^ Txfi Sides Mi their contdhtd Angle Being given ^ t4 find Hther ef the ether Angles. eth The Varieties and Prdppriions are as follow-' ,0lfl Given Required FS liFP FSP PFS FS-FE=ES i. ct FP. Rad: :cs PFS. t FE zietPFS.s FEi:9ad.t?Z 3. t PE. Rad. ::sES.ct PSF sEf. ctPFS::sES.ctPSF T its 2;^ ' _ 35113Inttol)iTCtf6n FS 1 2. Fp ' FPS . FFS FD—FP—FD , {.cotFS. Pad:;cs PFS.t DF I. cot PFS. s. DF;: Rad .'t DS 3. tDS, Rad;: J PX).«SPi) 5 DF. ct PFS:: s PD.ct SPD '' * 1 FP i 3. PS .. PSF .:fps . P$^PC=^CS/ . I. cot FP. Rad :: cos FPC. t PC z.jCetFPC. s PC:: Rad. t FC 3. t FC. Rad:: s CS. cot FSP sPC.ctFPC\:iCS.ctFSP FP ; ^fs -5'ff :.ffs, • ... . FP_j^PX)*= fD" I. cot PS.Pad:: cos SPD.t PD \x,cotSPD. s PD: -.Rad.tDS 3 . t DSRad ; .i FD. cog SFP 1 i PDict SPH C: s ED.cot SFP 'AC~Q^ = ZF. TheAnglc^ ACD^r M~Hf—Z ^e=ZF. lA DM= s K—^l=F h^FF. And thus the Sides of the Triangle Z P F ate equal to the Angles of the Triangle ACD. The Ccmplenient of the greateft Side P F to a Semicircle being titken for the greateft Angle ADC. And in this Cafe therefore,as in the preceding, there are three Varieties which make up fixty Pro- blems in every Oblique angled Spherical Trian- gle; which adually to refolve in fo many Tri- angles, as have been mentioned, would be both tedious, and to little purpofe •, I will therefore fcled fome few, that are of moft general ufe in the Dodlrine of the Sphere, and leave the reft to thine own praftice. CHAP. V. %(i UnttoDuctioti C H A p. V. 0/ fuch Spheric^tl Problems as are of mofi' General 1X[e i» the Doolvine of the Pri- mum Mobile or Diurnal Motion of the Sun and Stars. PROBLEM I. The greate/f' Declination of the Sunheinr given-,t9, fnd^e Declination of any Foittt of the Jfcliftickz THe Declination of the Sun or other Star, is his or their diftance from the Equator, and as they decline from thence either Northward or Southward*, fo is their Declination reckoned North or South. 2. The Sun's grcateft Declination, which in this and many other Problems is fuppofed to be given, with the Diftance of the Tropicks, Ele- vation of the and Latitude of the Place, may thus be found. Take with a Quadrant, the Sun's greateftand leaft Meridian Altitudes, on the longeft and ihorteft days of the year, which fuppofe at Loncbn tobeasfoliqweth. The to Hfltronomp. 9^9 -iM c ^ 5 grcateft? Meridian IHs. <51.9016 IhcSon's-Jf^^ }• Altitude Their difference is the diftance ? ^ oftlieTropicks ^S.W,+7. 650 Half that Difference, is the ? ^ Sun'sgreateft Declination ^ 2,3, 525 Which dedudt from the Sun's^ greateft Altitude, the remainer is tlie height of ^e Equator J The Complement is the? „ height of the Pole f"f2orP^.s.. S33 Now then in the Right angled Spherical Tri-i angle AD? va Fig. i.there being given. 1. The Angle of the Sup's greateft Declina- tionD^F. 23. 525. 2. The Sun's fuppofcd diftance from r to sj Af. 6odeg. The Sun's pr^fent Declination DF may be found, by the 10 Cafe of Right angled Spherical Triangles. As ^he Radius- IstotheSinepf DAF. 23.525. 9.60113517 So is the Sine of ^ F 60. 9.93753065 To the Sine of D F. 20.22. 9.53866580 PROBLEM 2- 2^4 Zn Jntroliucffoii PROBLEM 2. The Suti's groatefi Dcclwation, with his Difiance fiom the next n^quimHtialVoint being given, to fnd his Eight Afcenjion. In the Right angled Spherical T riangle ADF in Fig. I. Having the Angle of the Sim's great- eft Declination DF. 23.525. And his fup- poled diftance from T or;^, the Hypotennfa AF. 60. The Right Afcenfionof the Sim, or Arch of the zAiquator, AD maybe found, by the ninth Cafe of Right angled Spherical Triangles, As tile Cotang. of the Hypot. } , JF.60. r S'-7«'4?9!7 Isto the RnJias 10. ooooocoo SoistheCofneofD^F.23.525. p 96231533 To the Tang, of 57.80. 10.20087596 PROBLExM 3. To find the Declination of a Vianet or Fixed Star with Latitude. , In the Oblique angled Spherical Triangle FFS in Fig. y. we have given, i. F the greateft IDeclination of the Ecliptick, 2. The Side F S the Complement of the Stars Latitude from the Ecliptick atii::'. 3. The Angle PSF the Complement of the Stars Longitude. To find P F the Complement of Declination. By the eighth Cafe of Oblique angled Splierical Triangles, the Proportions are. As to Bltronomp- 285 AstheCot. of PS. 23. 525. 10. ?6i 1802 Is to the Raditu. i o. ooooooo So is theCof.of PS F. 20cleg. 9.9729858 To the Tang, of S£. 22.25. 9.6x18056 FS. 86 deg—ES. 22. 25.= F £. 63.75. As the Cof.of ES,22.2$.Comp.Aath. o. 03 36046 TotheCofineofPS. 25. 525. 9. 9623154 SotheCof.F£. 63.75. 9.6457058 To the Cof. P F.64,01. 9.6416258 Whole Complement, is FT. 25.99. the Dc- clination fought. PROBLEM 4. To find the Right Afccnfion of a Planet, or other Star with Latitude. The Declination being found by the lall Pro- blem, we have in the Oblique angled Spherical Triangle P F 5 in Fig. 4. All the Sides with the Angle F S P 20 deg. or the Complement of the Stars Longitude. Hence to find F PS by the firlt Cafe of Oblique angled SphericalTriangles, Hay As the Sine of P F.S^.oi.Camp.Arith. o. 0363059 Is to the Sine of F S P. 20. 9.5340516 So is the Sine of F S. 86. 9. 9984407 To the Sine of F PS. 22. 28. 9.5787982 Whcfe Complement 67,^72. is the Right Ale. of a Statu. 10. North Lat.4, PRO- 2^6 lin jntvdDucti'dii PROBLEM 5. iTje teles Blevatioriy Sm^s great'efl DeelinMion and Meridian j4ltitude being given, to find his trh'e place in the Zodiack- Tf the Mericjian Altitude of the Sun be lels than the height of the ey£qHator, dedud the Me- ridian Altitude from the height of the tor, the Remainer is the Sun's Declination to- Wards the South Pole : but if the Meridian AI- titude of the Sun be more than the height of the iy^qaatTr , dedud the height of the v^qnator from the Meridian Altitude, what remaincth,is the Sun's Declination towards the North Pole, in theft Northern Parts of the World : the con- trary is to be obferved in the Southern Parts. Then in the Right angled Spherical Triangle j4DF'u\Fig. I, we have given the Angle the Sun's greateft Declination. The Leg D F the Suit's prefent Declination, To find y4F the Sun's diftarice from the next quinodtial Point. Therefore by the Cafe of Right angled Spherical Triangles. As the SincofF AD.2'ij'^2$.Comp.j4r.o. 598864^ Isto the Sineof Z) F. 23. 5. 9. 5945468 So is the Radios. 10.0009000 TotheSincof AF. 80. 04. 9*99541 PROBLEM 6. to Hlftronomp. aS; PROBLEM 6. The Poles Elevation and Sari's Declination being given, to find his ^Amplitude. The Amplitude of the Sun's fifing or fetting is an Arch of the Horizon intercepted betwixt the t/Equator and the place of the Sun's rifingor fetting i and it is either Northward or South- ward , the Northward Amplitude is when he rilethor fetteth on this Side of the t/Eqaator to^ wards the North Pole', and the Southern when he rilethor fetteth on that Side of the tAlqnat. 1. is the Afcenfionaldif- ■ ffcrence between D/I the Suifs Afcenfion hi a < Right Sphere, and jDTthe Siih's Afcenfion in an i\ Oblique Sphere. Now thenln the Right angled I Spherical Triangle we have given. U The Angle MAT. 38. 47. the Complement < of the Poles Elevation, x^nd MT. -23. 15-. To / had A T the Afcenfional difference. I As Had. TorheCot.ofMAT.^S.^'j.Com.Ar.id.o^^()i-}6 SoisTang.A/T.23. 55. 9.63100^1 To the Sine of ^T. 32.56. 9.7 309187 . v 45KAi PROBLEM 8. • 5?!Dc ; ' Having the Right Afcenfion and Afcenfional Diffe* F '' renccy to find th: Oblique Afcenfion and ^^ ■, - Defcenjion. - . _ : . ■. '.r liijji Tn F»^. 1. pT reptefents the Right Afcenfion, AT the Afcenfional D ifference. DA the Oblique Ixaf Afcenfion which is found by dediidling the Afcen-' .iIks fionalDiffe: cnce x^T. from the Right Afcenfion ^3150] D T. according to the Diredioa following. f;;!^ Tn. fflCf, 1 o; JD afcn: c SlKcr .2 '•M n aiiOtffi" in fitlofflli 0 ienCoDii D fflta Xi tet?- Jffilis 4 IK ■ 3.0955' to ^Kronomri Subt. 289 North ^ Add Add 'The Afcentional Difference from the Right, and it giveth the Oblique A- fcenfion. The Afcenfional Difference to the Right,and it giveth . the Oblique Defcenlion. 'The Afcenfional Difference to the Right,and it giveth the Oblique Afcenfion. South ' The Afcenfional IDiffefence I from the Right, and it I giveth the Oblique De- LSubt. L ftenfion. ^ Right Afcenfion of H. o deg. 57. 80 Afcenfional Difference 27. 62 Oblique Afcenfion H. o deg. 50. 18 ObliqueDefcenfionH. o deg. 85.42 P R O B L E M 9. To f fjd the time of the Sun's rifng and fittings veith the length of the Day and Night, The Afcenfional Difference of the Sun being ^ added to the Semidiurnaf Arch in a Right Sphere, that is, to 90 Degrees in the Northern Signs, or fiibftradted from it in the Southern, their Sum or ''■v DifFerense will be the Semidiurnal Arch, which V doubled i 3lti 3fntrotiuctjoti doubled is the Right Arch, which biieded is the time of the Sun rifing, and the Day Arch bifeded is the time of hisfetting. As when the Sun is in o deg^ H. his Afcenfio- nal Difference is 27. 62. which being added to 90 degrees, becaufe the Declination is North, the Sum will be 117.62 the Semidiurnal Arch. The double whereof is 235.22 the Diurnal Arch , which being converted into time makes 15 hours 41 minutes-.for the length of the Day, whofe Complement to 24*, is 8 hours 19 minutes the length of the Night ^ the half whereof is 4 hours 9 minutes 30 Seconds the time of the Sun's rifing. PROBLEM 10. 7'he Poles Elevation and the Sun^s Declination beint given) to find his Altitude at any time ajjigned. In this Problem there are three Varieties. 1. When the Sun is in the t^qnator, that is,in the beginning of T and in which cafe fbppo- ling the Sun to be at 5, 60 degrees or four hours diftant from the Meridian , then in the Right angled Spherical Triangle 5 Z , in Fig. 1. we have given, e^Z, 51.53. the Poles Elevation, and B t/f 60 degrees, to find B Z. Therefore by the 2 Cafe of Right angled Sphe- rical Triangles. As As the Radius lisMts TotheCofineof t/£Z. 51. 53. p. 7938635 igaW: So is the Cofine of 5 60. 9.6989700 . iialAa TothsCofuieof 5 2. 71. 88. 9.4928335 tlicDa « • . . ttwiB Whofe Complement S C 18. 12. is the: © Ai- titude required. The fecond Variety is when the Sun is iii the Northern Signs,that is, in r. y. H. $. si. iiE. in which Cafe fuppofing the Sun to be at F in Fij, i. Then in the Oblique angled Spherical Triangle F2 P, we have given, i, PZ 38.47 the Com- plement of the Poles Elevation. 2. FP 67.97 the Compilemertt of Declination. 3, ZPF 45 ,r.^ the Diftance of the® from the Meridian, To findPZ. Therefore by the eighth Cafe of Oblique ^ jljj angled;Spherical Triangles. AstheCotang.of ZP. 38. 47.. 10.0997059 htox!ciS Radim. 10.0000000 " So is the Cofine oi ZPF. 45. 9. 8494850 To the Tang, of 5 P. 29.33. 9<74977S'i g;, Then from E P. 67. 97 Dedua5P. 29.33 There refts FS. 38. 64 V 2 Ai ^it 3|nttottuctton JC. AstheCoCneof5P. ig.^-^.Comf.Ar. 0.0595768 TotheCof1neofPZ.38.47. 9.8937251 SoistheColineofP S. 38.64. 9.8926982 To the CofineofFZ. 45.45. 9.8460001 Whofe Complement FC. 44.55 is the © Alti- tude required. The third Variety is when the Sun is in the Southern Signs as in is. lU. vy.k. And in this Cafe fuppo(ingthe©to be >?' 10 degrees, and his Declination South Db 22. 03. and his Di- ftance from the Meridian 45 as^fore, then in the Oblique angled Spherical Triangle Z bP in Fig. I. we have given ZP. 38.47. The Side hP 112.03. and the Angle ZP^ 45. To find Zh. Therefore by the 8 Cafe of Oblique angkd Spherical Triangles. As the Cotang. of ZP. 38.47. ro. 0997059 Is to the Padius. 1 o. ocoocoo So istheColineof Z P^. 45. 9.8494850- TotheTang.of SP. 29. 3 3. 9.7497791 Then from ^ p. n'2. 03 Dedudt 5 P. 29. 33 There refts b S. 82. 70 AstheCofineof P5. 29.3 3.C&wp.Z7'.' 0.0595768 To theCofine of ZP. 38.47. 9.8937251 So the Cofne of bS. 82.70 9.1040246 To the Cofne of Z^. 83.45. 9-o5732,65 - ' Whofe Complement 6.55 is the © Altitude required. PROB'LEM 11.- to aiarouottt^. PROBLEM n; ■ Having the Altitude of the San , his Difl'ance from the Meridian, and Declination, to find his Azimuth. The AzJmuth of the Sun is an Arch of die Horizon intercepted between the Meridian and , the Vertical Line palfing by the Sun, being un- derftood by the Angle HZC \n.Fig. i. or Arch HC. And in all the Varieties of the laft Pro- hlem,inay be found, bythefirft Cafe of Oblique angled Spherical Tdangles. Thus in the T r iaiigle Z BP. As the Sine BZ.'^i .SS.Comp.Ar. o. 022090 j Is to the Sine of 5 P Z. 60. 9-937S3od So is the Sine of S P. 90. 10.0000000 To the Sine of P Z P. 65. 67. 9.9596209 In the Triangle ZF P. I fay. s.ZF- s. ZPF :: s.FP. s.FZP. In the Triangle ZbP. I fay. SinpZl'. Sine Z P ^ : Sine ^ p. S'mc bZP^ V 3 PRO- 994 SntroDttctien PROBLEM 12. ^0' The Poles plevatioH, veith the Siuis Altitude and Declination given^ to find his Azimuth. 3^ of tt In the Oblique angled Spherical Triangle fZP in Fig. I. let therfe be given. ficiitiiil ttlitN' 1. FP.6^. 97 the Complement of the © De. clination. 2. ZP. 38.47 the Complement of the Pole? Elevation. 3. F.2.45.46 the Complement of the© Al- titudc. And let the Angle FZP the © Azimuth be required. By the 11 Cafe of Oblique Angled Spherical Triangles. As the Sine ZP* Sine FZ, Is to the Square of ciiol So is the Sine ^ Z of the Sides ZP*iZcr — ZF, ^ To the Square of tlie Sine of half the Angle , ' FZF. _ ^ The Sum of the three Sides is 151. 89 The half Sum is 7^.945 from which dedudt FZ 38.47. The difference is 37.475 And the Difference between 75. 945 and FZ is 30.495. . 2i£[t Tl Sine ■M to Hftronomp. SineofPZ. ■^%.\'i.Com^.Ar. 0.2061365 Sine of FZ- 45. 45.o. 1471308 s.\Zcr~PZ.V-l.\-i is the © AzJmuth from the South. 1 PROBLEM 13. "To find the To'mt of the EcUptick^ Cfdtfittitt itt^, and its Altitude, Before we can know what Sign and Degree of the Ecliptick is in the Medium Coeli\ we muft find the Right Afcenfion thereof, to do which, we muft add the Sun's Right 'Afcenfion to the time afternoon, being reduced into Degrees and Minutes of the tyEquator^ the Sum is the Right Afcenfion of tht Medium CcelL Example. Let the time given , be March the 20. 1674. at one of the Clock ^ in the After- noon. At which time the Sun's place is in V. 10 deg. 23 Centefms. To find the Right Afcenfion thereof, in the Right angled Spherical T jiangle ^ D F in Fi£. 1. we have given 3 The Angle of the Sun's great- eft Declination DAF 23. 525 and the Sun's di- ftance from the next Equino&al Point ZF 10.23. Therefore by the ninth Cafe of Right angled Spherical Triangles. V 4 As y'.j Ji-'i 296 ^it JntwBucti'oft As the ct. AF. 10. 23, Is to 'Radius. So iscs DAF22. S25r TQ t AD g. 39- 10-7435974 10.0000000 ikf'" 19.9623154 £<^.^6 nfsi 9.2187180 To which adding the Equinodial Degrees an- fweringtoone hour, wt. 15. the Sura is 24.39 ,tbe Right Afcenfion of the Mid Heaven. Hence to find the Point culminating; in the Right angled Spherical Triangle ADF in Fi^. 1. we have given24.39and P>4F 23.525 to find^F. Therefore by the fifth Cafe of Right Angled Spherical Triangles. As t AD 2:^.. 39. ]s to Radtifs. poises D AF 23,. $25. 10. 6564908 10.OOCQOOO 9.9623154 ,Tpc^ ^F.2<5;.3-I. 10.3058246 7; ; Therefore the Point culminating is r 26. 31. ' To find the Altitude thereof above the Hori- jflqn we have given in the lame Triangle P^f :?,3. 525. andYf F26. 31. to find P F. Therefore by the tenth Cafe of Right angled Spherical Triangles. As Radius. Is to J ^F—26, 31. So is s DAF 23.525. 10. 0000000 9.6466268 9. 6011352 To the s DF 10.19. p. 2477628 Which to 3^^ll;onom^ 297 Which is the North Declination of the Point of theEcliptick culminating, and being added to the height of the c/dFquator at 38.47the Sum is 48.66 the Altitude of the Mid Heaven as was required. PROBLEM 14. "'H Having thegreatefi obliquity of the Ecliftick^together ^ jfje Dijbance of the Point given from the EquinoElial, to fnd the Meridian Jingle^ or InterfeBion of the Meridian voith the Ecli^ticki Having drawn the Primitive Circle HZ R N inF?^. 5. reprefentingthe Meridian, and the two Diameters H Aand Z AN, fet off the height of the Pole from ^ to P. 51.5 3, and from A'to S, and draw the Diameters P AS for the Axis of the World, and t^y^^for thttALquaior-^ this, done, the Right Afcenfion of the Mid Heaven being given, as in the laft Problem 24.3 9 with the Point culminating. T. 26.31, and the Declination thereof 10.19, if you fet 10 cleg. xpCentefmes from cy£ to F and e to X, you may draw the Dia- meters F^X and .slat Right Angles thereun- to, and becaufe the Imam Casli is diredlly oppofite to the Point culminating, that is,in 26.31 ,if you fet26.3r fromZtoAa Ruler laid from cto I'will cut the Diameter FJT in 6", and then making XhZXb you have the three Points bGh, by which to draw that Circle, which will cut the fAS- ^uator ^ AQjn Ci, and lb you have the three Points X^F by which to delcribe the Arch of the EtliptickrPs=3 A". And 2^3 ain llntroUuttion And in the Right angled Spherical Triangle Te/^F we have given. The Angle ^TF. 23. 525 the Sun's greateft Declination, and rF. 26. 31, the Point culminating,to find the Angle TF^. Therefor® by the feventh Cafe of Right ang- led Spherical Triangles, Asthecfe/^FTF. 23. 525, 10.3611802 Is to the Radim. 10.0000000 So is the « tF. 26. 31. 9.9525062 To thecof. rFe/F. 68.60. ' 9.5913260 Which is the Angle of the Ecliptick with the IVleridian. PROBLEM 15, To find the Angle Orient, or Altitude of the Nomh gefime Degree of the Eclifticki In Fig. 5. the Pole of the Ecliptick TF^sX h at m, and fb you have the three Points ZmN to draw the Vertical Circle Z kN cutting the Ecliptick at Right Angles in the Point a: And then in the Right angled Spherical Triangle FaZ, we have given; FZ 4.i• 34 the Coraple- racnt o( F H the Altitude of the Mid Heaven •, And the Angle aFZ 68. 68 the Angle of the Ecliptick with the Meridian. To find Z^. Therefore by the tenth Cafe of Right angled Spherical Triangles. H As ^ to SEfttonom?. 399 As the Radius. To theSineof F^.41. 34, p.SipSSp® So is the Sine of ^F R is the lame with HB R, and the height of the Pole a- bove HT R is the fame with HK R. Having found the Afcenfions of the fevcral Houles together with the Elevation of the Pole above their Circles of Pofition, in the Oblique angled Spherical Triangle T B S, we have given. 1. The Angle r 5 S the Complement of HB 2. The Angle 5 r5< 23; The Sun's greateft Declination. 3. Their included SideTS. 54.39 the Af- ccnlion of the eleventh Honfe^ To find T S the Point to Hfftdnoitf^. 303 j?ointof the Ecliptick, which isrcfblvable by the ninth Cafe of Right angled Spherical Triangles. But in my Tngommetria Britannica, Problem, y, for the refolving of Oblique angled Spheric^ Triangles, I have Ihewed how this Cafe as to our prcfent purpofe may be refolved, by thefe Pro- portions fcAlowing. 1. s iZ Ang.j ^A'Ang-.rflTiS.fiTCru. 2.aiZAng.cj|TAng;:f^r^.rjZ Cm. f ZCru4-1ZCru= T S the Arch of the E- cliptick defired. For the Cujp of the Eleventh Houfe. TB Arch r5.4439.the half whereof is 27.195. TBS. 122. 18374. BTS. 23. 525. Z 145.70874—^Z 72. 85437. X. 198.65874 — iX.49. 32937. s\Z. 72.85437.CCWP.o. 01977589 J i X 49. 32937. 9.88000800 t^T B. 9.71081089 jCru. 22. 192. 9.61059478 2. Operation. cs. i Z. 72. 85437. Comp. Arith. o. 53012277 ss IX. 49. 3'^9'il' 9.81395860 t \ T B, 27. i95.( 9- 7ic'8ia89 r|ZCru. 48. 611. 10.05489226 1. Arch. 22.192. Their Sura is 70.803 the Point of the Ecliptick. V, I '. ^ ■ i I. 304 3lntroDuct(on cSiiZ. 82. %\^\6.CompAnsh. 0.88517901 cs i X. 59.00416. 9' 71164753 i. i t a. 57. 195* 19072348 ^ t^Cru; 78. 397- 10.68754999 1, Arch—53. 296. Their Sam 121. 693 is the Point of the Ecliptick for the Afcendant. ki For the CUjp of the Second Houfi. .ni'f! jitelsT In the Oblique angled Spherical Triangle rTy. we have given, . rT. 144.39. The half whereof is 72.195. 2. vTy. 122.18374? To findTy. The An- :j. Tyy. 23.525 f glesare the fame with thofe of the Twelfth Houfe. Therefore. 'jiSm sEIot J. f Z. 80. 49596. Comf- Arith. six. 56. O.C0601663 9.92351651 ttlii Their Sum t i tT. 72.195. rj XCru. 69. 306. 9.92953314 10.49327695 le. 422 81009 -Uflii 2. Operation^ cs IZ 80. 49596. Comp. Arith. cs i X56. 97096. o. 78170174 9.73628614 ■I'lu; f ^ Their Sum t i -i,T. 72. 195. t|ZCru. 84. 34. I. Arch. 69.306. 10.5179878S IG. 49327695 II. 01 126483 Their Sum is 53.740 is the Point of the Ecliptick for the Second Houfe. ■i, Fer •lw> !.68";'V to ^Hdrononip. For the dtjp of the 1%ird Honfe. 305 In the Oblique angled Spherical Trljuiglc rv 6, we have, ■ • I ^ I. r V. 174- 39. The half whereof is 87. i95. The Angls rv fi and vr6 are the fame with thbfe of the Eleventh Houfe. 1 Tm IS"!.:; Hi; .' lelkr Tim cc6oi( JI Z. 72. 85437. Comp. Arithl sjy 49.32937- Their Suni tjXv- 87.195. b.oi97758j;> b.88000800 9.89978389 11. 30984054 For the Eleventh Houle. For the Cajp of the Twelfth Houfe. In the Oblique angled S|)herical Triangle TKFjYte have given. 1.vii:. 84.39. Thehalf whereof is. 42.195. 2. TiCM37-4<5692? ^ P , i.KTt. 23.525 J-ioMcirf. ; 2.160. 99I92 iZ. 80.49596 T. 113. 94192 fx 56. 97096 '0^' J f Z. 80.49596.' Comf. Arith. 6.00601665 xfX 56.97096. 9.92351651 0112^. t\YK. 42.195. 9.95740882 ffTCru, 37.625. 9.88694196 ' X 2. Operation. So6 %n Unttotuction 2. Operatkn. cs. { Z. 80.4959<5- Comp. Aritk o. 78170174^'^ cs iX. 56.97096. 9.73628614' r i r X. 42-195- 9- 95740S82 j-sj! tiZCru.71.496. . 10.47539670 1. Arch. 37.625. Their Sum 113.6691 is the[jj.j,.2 Point of ^eEcliptickfor theTwelfthHoufe. ^'si^ For the Cnfft of the Afcendmt, In the Oblique angled Spherical Triangle Afijj, r^C7wehave, 1. TA.i 14. 39. The half whereof is 57.195. 7.. T AZ. 141. 5353. The Complement of.jjjjjj, 38.46667. 3. ^v_y. 23.525. :Z. 165.05833 iZ. 82.51916 7 X' 118.00833 1X59.00416 J.i Z.82.5191 6. Comp.Arith. s.i Xr 59.00416. tirA.^'j. 195. f ^ X. 53.296. 2. Operation. t^XCru. 86.468. 0.00371629.1^, 9-93 3I3477:^_ 10.19072348;;^' 10. 12757454,,^. \k ■ 4t! 11.20962043-'®^ . .. ,v , ■ «W ■ '2. OperMion. I to ^tffortortip* 3a7 2. Operation. 72.85437. Comp.Arith. o. %'ioia.lrif 32937* 9.81395860 ij J;; Their Sum 10.3 440813 7 1,1,1« i r V. 87. 19 5. 11..39984054 wlit^» 11.65392191 S. Arch. 86.468. Their Sum 175. 197 is the Point of the Eciiptick for the Third Houfe. Mt ,,, And thus we have not only eredied a Figure for the Time given, but compofed a Table for the general eredling of a Figure in that Latitude 3 ., for by adding together the fir ft and fecond Num- bers in each Proportion for the firft, fecond and wpm Houfes there is compofed two Numbers for each Houfe, to each of which the Artificial Tan- gent of half the Afcenfioa of each Houfe being added, their Aggregates are the Tangents of two Arches, which being added together, do give the diftance of the Cufp of the Houfe, ftom the firft Point of Aries, as in the preceding Ope- rations hath been fhewed. 5# Only note,That if the Aftenfion of any Houfe fee more than a Semicircle, you muft take the iC'if Tangent of half the Complement to a whole Circle. And to find the Cufp of the Houfe, you muft alfo take the Complement of the Sum of the Arches added together. 11.^''' The Numbers according to the former Ope- rations which do conftitutea Table of Houfes for the Latitude of London. are as follow- cth. 'iJjh X 2' x.Oper. 3oS Ifn UntroBuctiotl (• ■ ( ■ -k" };' »'i I' ■; <'f' 11 ani 3 Houfes Afcendant 12 and 2 Houfes 1. @per. 2. Ofer. 9.89978389 10.34408137 9.93685105 10.$9682551 9- 92953?i4 10. 51798788 The Six Oriental HottfeSj by the preceding Oferationt, ■4 Houje 25. 5 Houfe 10. 80} 9 Houfc yp 23. 691 7 Houfe 11. 69} 8 Houfe X 3. 740 9 Houfe X 25, 197 10 Houfe T 25. 311 11 Houfe H 10.803 12 Houfc $ 23.591 AfccndancSl 11.693 2 Houfe ni 3. 740 3 Houfe ni 2j. 197, Figure of the Twelve Ccelefiial Houfes. Aftronomj ASTRONOMY- THE Second Part: OR, AN ACCOUNT O F T H E Civil Year J With the Reafbn of the Difference Between the Julian & Gregorian Calendars, ^.nd the manner of Compu- ting the Places of the S 'VN and MOON. L 0 N D 0 Nf I'inted for Thomas Taffmger, at the Three Bibles O'O. London-Bridge, 1679. to isaroitomi?. 311 A N INTRODUCTION ;r The Second Book. C H A p. h of the Tear Civil and y^JlronomicaL HAving Ihewed the Motion of the Fri^ mum Mobile , or Doftrine of the Sphere , which I call the Ablblute Part of Aftronomy^ I come now un- to the Comparative, that is, to Ihew the Motion of the Stars in reference tolbmecer^ tain Diftinftion of Time. ^ 2. And the Diftindion of Time is to be con- fidered either according toNature, or according to Inftitution, X 4 3. The 3^12 ijn Snftobnctiott 3. The Diftinftion of Time aecording to Na- st) turc, is that fpace of Time,in which the Planets do fiHifh their Periodical Revolutions from one certain Point in the Zodiack, to the fame again, to®' and this in reference to the Sun is called a "Vear, XH reference to the Moon a Month. (ti 4. The Sun doth pafs through the Zodiack in 3 6 s Dpys, 5 Hours, and 49 Minutes. And the to® Moon doth finilh her courfe in the Zodiack, and cWeel return into Conjunftion with the Sun, in 29 Days, 12 hours, 44 Minutes, and 4 Seconds. And from the Motion of thefe two Planets, the Civil ^ yeaf'iin'every Nation doth receive its Inftitu- tion. V :i!!rs,tli2 5. Twelve Moons or Moneths is the meafure wotoo of the Common Year, in Turkey in every Moneth lanie^ they have 29 or 30 Days, in the whole Year 354 ' mW Days, and in every third Year 3 55 Days. 6. The Perfians md Egyptians do^ltoaccomt iofti 12 Moneths te their Year "3 but their moneths lofoir are proportioned to the Time of the Suns con- :;2lKa tinuance in every of the Twelve Signs; in their tk \ Year" therefore which is Solar, there are always jttja 3 6 y Days, that is eleven Days more than the Lu- :;iist7o narYear. , 'I •7. And the Year which is the Account ef all Chriftendom, dothdifer from the other in ioiocJ this; that by reafonof the Sun's Excefs in Moti- on above 365 Days, which is 5 Hours,49 Minutes, it hath a Day intercalated once in 4 Years, and by this intercalation, it is more agreeable to the ~ Motion of the Sun, than the former, and yet ®ijy^ there is a confiderable difference between'them, which hath occafioned the Church of Rome to jnake fome further aniendment of the Solar inij, to 913 Year, but hath not brought it to that exadtnefi, which might be wifhed. »tet 2. This intercalation of one Day once in 4 kme;^ Years, doth occafion the Sunday Letter ftill to al- olldife ter till 28 Years be gone about •, The Days of the Week which ufe to be figned by the feven firfl: ' h Letters in the Alphabet, do not fall alike in eve- KB. k'. ry Common Year, but becaufe the Year confifteth tWiai; of 52WeeksandoneDay,Sunday this Year will J fciii i:; fall out upon the next Year's Monday, and fo for- M k ward for feven years, but every fourth year con- filling of 52 weeks and two days, doth occafion y.cit the Sunday Letter to alter, till four times feven years, that is till 28 years be gone about. This iis!k:!r Revolution is called the Cycle of the Sun, taking erajlt its name from the Sunday Letter , of which it (iioltfa; flieweth all the Changes that it can have by rea- iJjp, fon of the Bifiextile or Leap year. To find oiacw which of the 28 the prefent is, add nine to the kiiMK year of our Lord, (bccaufe this Circle was fo far ticfe; gone about, at the time of Chrifts Birth) and di- saiS;i3 vide the v/holeby 28, what remaineth is the prefent year, if nothing remain the Cycle is out, and that you mull call the laft year of tlie Cycle, or 28. iitkfc" 9- 'This Intercalation of one day in four years, doth occafion the Letter F to be twice repeated in February, in which Moneth the day is added, that is, the Letter F is fet to the 24 and ig days J of that Moneth, and in fuch a year SMatthiat day is to be obferved upon the 25 day, and the next Sunday doth change or alter his Letter, from which leaping or changing, fuch a year is called J Leap-year , aud the number of days in each f ,1,; Moneth is well expreHed by thefe old Verfes. Thirty 314 2ln UtttTDbncttdit Thirty days hath September, April, June and No- vember. February hath 28 alorte^AU the refi have thirty and one. But when of Leap-year cometh the Time^ Then days hath February twenty and nine. That this year is fomewhat too long, is ac- knowledged % the molt skilful Aitronomers, , as for the number of days in a year the Emperours Mathematicians were in the right, for it is cer- tain, that no year can confiji of more than 365 days, but for the odd hours it is as certain that they cannot be fewer than five, nor yet fo many as fix-, fo then the doubt is upon the minutes, 60, whereof do make an hour , a finall matter one would think, but how great in the confequence we (hall fee. The Emperours year being more than 10 minutes greater than the Suns, will in 134 years rife to one whole day, and by this means the Vernal or Spring Equinox, which in Juhtu Cafar\ time was upon the 24 of March,, is now in our time upon the 10 of Mtrch, 13 days backward, and fomewhat more, and fo if it be let alone will go back to the firft of March, and firft of February, and by degrees more and more backward ftill. : 10. To reform this difference, fome of the late Roman Biflrops have earneftly endeavoured. And the thing was brought to that perfeftion it now ftandeth^ by Gregory the Thirteenth, in the year 1582. His Mathematicians, whereof Lili- Hi was the Chief, advifed him thus: That con- fidering there had been an Agitation in the Council toaaroncmp. . 3*^ Council of Nice fomewhat coaceraed in this matter upon the motion of that Queftion, about the Celebration of Eaftcr. And that the Fa- thers of the Aflembly , after due deliberation with the Aftronomers of that time, had fixed the Vernal Equinox at the 21 of March^ and con- fidering alfo that fmce that time a difference of ten whole days had paft over in the Calendar, that is, that the Vernal Equinox, which began upon the 21 of March, had prevented fo much, as to begin in Gregorie's days at the 10 of the fame, they advifed, that lo days fliould be cut off from the Calendar, which was done, and the LO days taken out of OEloher in the year 1582. as being the moneth of that year in which that Pope was born •, fothat when they came to the fifth of the moneth they reckoned the 15, and to the Equinox was come up to its place a- gain, and happened upon the 21 of March, as at the Council of Nice. But that Lilipu Ihould bring back the begin- ning of the year to the time of the Nicene Council and no further, is to be marvelled at, he Ihould have brought it back to the Emperours own time, where the miftake wasfirll entered, and inffeadof 10, cut off 13 days-, however this is the reafon why thefe two Calendars differ the fpace of 10 days from one another. And thus I have given you an account of the year as it now ftands with us in England, and with the reft of the Chriftian World in relpeft of the Sun, Ibme o- ther particulars there are between us and them which do depend upon the motion of the Moon,, as well as of the Sun, and for the better under- derftanding of them, I will alfo give you a brief account ^fntrobucttoti account of her revolution. But firfti willlliew you, how the day of the moneth in any year pro-, pounded in one Couutry, may be reduced to its correfpondent time in another. 11. Taking therefore the length of the year, to be in feveral Nations as hath been before de- dared, if we would find what day of the moneth » in one Country is correfpondent to the day of that moneth given in another , there muft be fome beginning to every one of thefe Accounts, and that beginning mull be referred to fome one, as to the common meafure of the reft. 12. The moft natural beginning of all Ac- counts, is the time of the Worlds Creation, but they who could not attain to the Worlds Begin- sing, have reckoned from their own, as the Ro- hjans from the building of the Greeks from their Olympicks, the Ally rians from Nabonajfary andall Chriftians from the Birth of Chrift ; the beginning of which and all other the moft nota- hhEpochaesy we have afcertained to their corre- Ijxtndent times in the Julian Period, which Sea- contrived by the continual Multiplication of thofe Circles, all in former time of good ufe, and two of them do yet remain; the Circles yet in u(e are thofe of the Sun and Moon, the one, to wit, the Sun, is a Circle of 28 years, and the Cir- cle of the Moon is 19, aslhallbe Ihewed here- after. The third Circle which now ferves for no other ufe than the conftituting of the Julian Period , is the Roman Indidion, or a Circle of 15 years;, if you multiply 28 the Circleof die Sun, by 19 the Circle of the Moon, the Produdt is 53 2, which being multiplied by 15, the Circle of the Roman Indidion;the Produdt is 7980,the Number to iiaronomi^. ^17 Number of years in the Julian Period: whofc admirable condition is to diftinguiih every year within the whole Circle by a feveral certain Cha- radler, theyearoftheSun, Moon, and Indidion being never the fame again until the revolution of 7980 years be gone about, the beginning of this Period was 764 Julian years before the molt reputed time of the Worlds Creation which being premifed, we will now by Example fliew you how to reduce the years of Forreigners to our Julian years, and the contrary. I. Example. I defire to know at what time in the Turkilh Account, the fifth of Jme in the year of our Lord 1649. doth fall. The Julian years complete are 1648 , and are thus turned into days, by the Table of days in Julian years. 1000'Julian years give days 365250 600 Julian years give days 219150 40 Julian years give days 14610 8 Years give days 2922 complete 151 Days 5 The Sum is 602088 Now becaufe the Turkifii Account began fuly 16. Amo chrijli. 611. you muft convert theie years into days alfo. 600 ^i8 Un SnttoDUction 600 Julian years give days 219150 20 Years give days 7305 1 Year giveth days 365 June complete 181 Days 15 The Sum is 227016 Which being fubftraded from 602088 There rcfteth days 3750?^ 900 Turkilh years give days 318930 There refteth 56142 150 Turkilh years give days 53155 There refteth 02987 8 Turkilh years give days 2835 There refteth 152 G'mmddi. 4. 148 There refteth 4 Therefore the fifth of June 1649- out* glifh Account doth fall in the year 1058. of Ma- , hornet, or the Turkifh Hegira, the fourth day of, the moneth Giumadi. 11. 2. Exarnflcm I defire to know upon what day of our Julian year the 17 day of the moneth in the 1069 year complete of the Perlian Account from JeJJjagUe doth tail. The to 3lffronomp. 319 The beginning'of t\[\^Efocba isfrotnthe£;»tf^ cha of Chrill in complete days 2 3 06 3 9 1000 Perfian years give 365000 60 Yearsgive 21900 9 Yearsgive 3285 Chortal complete 90 Days complete 16 The Sum 620930 1000 Julian years Subftradled 365250 There refts 255680 7C0 Julian years 2,55675 There refts 5 Therefore it falls out in the Julian year from Chrift 1700. the fifth day of Jamary. . He that underftands this may by the like me- thod convert the years of other Epochs, into our Julian years and the contrary. The Anticipation of the Gregorian Calendar ismoreeafily obtained, for if you enter theTa- ble with the years of Chrift complete, you have the days to be added to the time in the Julian Account, to make it anfwer to the Gregorian, which will be but ten days difierence till the year 1700. and then the differeneewillbeaday more, until the year 1800. and fo forward three days difference more in every 400 years to come, mi- lefs our year lhall be reformed as well as theirs. CHAP. II. 320 3ln 3ffntvotiU(ttott CHAP. It Of the Cycle of the Moony what it is, horv placed in the Calendar y and to what pur- pofe, THat the Civil Year in ufe with us and all Chriftians, doth confift of 365 days, and every fourth year of 3 6 6,hath been already (hew- ed, with the return of the Sunday Letter in 28 years. In which time the Moon doth finidi her courfe in the Zodiack nolefs than twelve times, which twelve Moons, or 354 days, do fallfhoit of the Sun's year, eleven days in every common year, and twelve in the Bilfextile or Leap-year. And by Obfervation of Afemi Athenian y it was found out about 43 2 years before Chriit/ that the Moon in nineteen years did return to be in Conjundtiort with the Sun on the felf, fame day, and this Circle of nineteen years is called the Cy- cleof the Moon, which being written in the Ca- lendar agaiaft the day in every Moneth, in which the Moon did change,in Letters of Gold, was alfo called the Golden Number, or from the excellent ufe thereof, which was at firft, only to find the New ^oons in every Moneth for ever, but a- mongft Chriftians it ferveth for another purpofe alio, even the finding of the time when the Feaft of Eafter is to be obferved. The New Moons by this Niimber are thus found. In the firft year of the Circle, or when the Golden Number is 1, where the Number i was fet in the Ca- lendar in any Moneth, that day is New Moon, in the i 15 u: cttk; to 3iftronomr. 321 the Eleventh Year twice 11 Days being added to the 9 Days refervedjdo make 31 Days, that is, one Month of 3 o days and one day over,which be- ing added to the fupernumerary days in the four- teenthYear do make another Month of 30 Days and 4 Days over, andthefe being added to the fupernumerary Days in the fevententh Yeardo make another Month of 30 and 7 Days over, and thefe 7 Days being added to the 22 fupernu- nierary Days in the Ninteenth Year of the Moons Cycle do make another Montji of 29 Days. 4. But becaufe there are 6939 Days and 18 Hours in 19 Solar Years, that is, 4 Days 18 Hours more then in the common and EmbolifmicalLu- nar Years, in which the excefs between the Lu- nar and the Solar Year is fuppofed to be no more thed 11 Days in each Year, whereas in every fourth Year the excefs is one Day more, that is, 12 Days, that is, in 16 Years 4 Days, and in the remaining 3 Years three fourths of a day more. And that the new Moons after 19 Lunar Years or 235 Lunations do not return to the fame days again, but want almoft 5 days, it is evident'that the civil Lunations do not agree with the Aftro- nomical and that there rtiuft be yet fome kind of intercalation ufed; 5. Now therefore in diftributing the golden Number throughout the Calendar. If the new Moons Ihould interchangeably confilt of 30 and 29 days, and fobut 228 Lunations in 19 Years, we mightproceed in the fame order in which wc have begun, and by which as hath been fnewed the third Year oftheGolden Number falls upon the Calends of Janmry. But for as much as there arc firfi fixLynatioiisof 30 days apiece and Y 2 CDC im • V r; K' M 6^2 3[tit rot action one of 29 days to be iiiterpofed, therefore thefe mufl; be 6 times 2 Lunations together confiiling . jpj of 30 days and once three Lunations of 29 days. And that relped may be alfo had to the BilTextile days, although they are not cxpreft in the Calen- dar, that Lunation which doth contain the Bif- fertile day, if it (liould have been 29 days, it mull be 30, if it Ihould have confifted of 3c days it -jjjji mult confill of 31. Tjiiij;) 6. Andbecaule it was thought convenient, as hath been Ihewed, to begin with the thirdYearof the Cycle of the Moon, becaufe the Golden Nura- bcr 3 is fet to the Calends of Jamary^ therefore in this Cycle the Embolifmical Years are, 2, 5, 8, II, 13, 16, 19. But yet that it may ap- pear, that thefe Years are in effedl the fame, as ' if we had begun with thefirft: Year of the Gob den Number, fave only that die eighth Year in- Head of the ninth is to be accounted Embolifmb , cal, 1 have added the Table follvving, in which it is apparent that the former Embolilinical years 'i _ do agree with thefe lafl; mentioned. ■ ; 7. But as I faid before, it was thought more ^ ' convenient to begin the account from the num-, r' ber 3 fet to the Calends of January, becaufe by lb reckoning 30 and 29 days to each Lunation interchangeably, the fame Number 3 falls upon January 31. March i, and 31. Ayril 29. May " "■ 29. June 27. July 27. Au^ufi 2<^. September Ocloher 23. November 22. December 21. As . if the L unar years were compleated u}X)n the 20 v of December there remain juft 11 Da^s, which the Solar years doth exceed the Lunar. ' 8. And by ranking on and accounting 4 for ' "i' the Golden Number of the next year, you will ' ' fmd * 16 to ^fironom^ 32^ find It fet on January 20, February 18, March 20 , Jfril 18, May 18 , June 16 , July Augufi 14, September 13, Oclob. 12, iVb- vemb. I l^Decemh.io, 9. Bnt in going on, and talcing 5 for the Golden Number in the third year, we miifl; remember that that is an Embolif- mical Year , and therefore that fome- where there mult be 2 Months together of 30 days. And for this reafon the Gol- den Number 5, is fet to January 9, Febru- ary 7, March 9, April 7, May 7, June 5, July 5, Augiijj i,Se- ptemher 2, as alio up- on the fecond day of OEtobcr, and not up- on the firft, that lb there may be 2 Luna- tions together of 30, and the fame Num- ber 5 is alio fet to the thirty firlt of Othher^ to make the Lunation to coniift 0 n Z p. 0 tn 3 a 0 0 0 Hsi i- cr a> •-n s a nr 0 nr 0 i-j • 3 0 0 0 0 0 n D p yi I 3 354 2 4 3H 3 5 Embol. 384 4 6 3 54 5 7 354 6 8 Embol. 384 7 9 354 8 10 354 9 11 Embol. 384 10 .12 354 II 13 EmboL 384 12 14 354 13 15 3 54 14 16 Embol. 384 15 17 354 16 18 354 17 19 Embol. 384 18 I 3 54 19 2 Embol. 384 of 29 days, and to the thirtieth of in- Head of the twenty ninth, thatfo a Lunation of Y 3 30i BH 5llti JnttoDucti'on 50 may again fncceed as it onght. 10. In like manner in the fixth Year, having gone through the fourth and fifth as common years, you may fee the Golden Number 8 let to the fifth of which fliould have been upon the fourth , and in the ninth Year the Golden Number 11 is fet to the fecond of February which jhould have been upon the fir ft. And there is a particular reafon, for vvhich thefe numbers are otherwife placed from the eighth of March to the fifth of namely,that all the pafchalLunations may confift of 29 days: For thus from the eighth of March to the fixth of April^ to both which days the Golden Number is 16, there are but 29 days.And frpm the ninth of MarchX.o thefeventh of Aprils to both which days the Golden Number is 5, there are alio 29 days, and fb of the reft till you come to the fifth of Aprils which is the laft Pafchal Lunation, as the eighth of March is the firft, but at any other time of the Year, the length of the Month in the Embolifmical Year, maybe fixed asyoupleafe. 12. And in this manner in the 17 years, in which the lunations of the whole Circle are fi- nifhed, and in which the Golden Number is 19, the Month of July is taken at pleafure,to the thir- tieth day whereof is fet the Golden Number 19, which (hould have been upon the thirty firft, and the fame Number being notwithftanding placed upon the twenty eighth of Augufi, that by the two Lunations of 29 days together, it might be underftood,that the feventh Embolifmical Month xronfiftmg of 29 days is there inlerted, inftead of a Month of 5 o days. In which place the Era- bblifmical or leaping Year of the Moon may plainly to 3I(ltono'mp. 335 plainly be obferved for that year is one day lefs than the reft, which the Moon doth as it were pais over. Tlie which one day is again added to the 29 days of the laft Month, that we may by that means come,as in other Years, totheGol- den Number, which fheweth the New Moon in January following. And for this reafon the E- pad then doth not confift of 11 but of 12 days. And thus you fee the reafon, for which the Gol- den Numbers are thus fet in the Calendar as here you fee. In which we may alfo obferve, that every following Number is made by adding 8 to the Number preceding,and every precedingNum- ber is alfo made by adding 11 to the Number next following, and calling away 19 when the addition, ihali exceed it. for Example, if you add 8 to the Golden Number 3 fet againft the firft of January Jx. ma- keth II, to which add 8 more and it maketh 19, to which adding 8 it maketh 27, from which fubftrading x 9 the remainer is 8, to which again adding 8,the fum is 19,to which adding 8 the lum is 24, from which deduding 19 the remainer is 5, and fo of the reft. In like manner receding backward, to the 5 add 11 they make 16, to the 16 add 11 they make 27, from which deduding 19 the remainer is 8, to which 11 being added the fame is 19, to which 11 being added the fum is 30, from which deduding 19 the remainer is 11, to which 11 being added the fum is 22, from which deduding 19 the remainer is 3.And by this we may fee that every following number will be in ufe 8 years after the preceding, and every prece- ding Number will be in ufe 11 years after thefol- lowing,that is,the famewill return to be in ufe after Y 4 8 326 Un in'rouncttoit 8 Years and 11, and the other after 11 Years and 8, or once in 19 years. CHAP. III. Of theVfs of the Golden Number in finding the Feajl of Eaiter. , '' THe Cycle of the Moon or Golden Number is a circle of 19 years, as hath been faid al- ready, which being diftributed in the Calendar as hath been IhAvn in the lafl; Chapter, doth (liew the day of the New Moon for ever ^ though not, ttd. exaflly; Bat the ufe for which it was chiefly ih~ tended, was to find the Pafchal New Moons, that is,thofe new Moons on which the Fealt of £afer »jI, and other moveable Feafts depend. To this purpofe we muft remember, t: 1. That the vernal Equinox is fuppofed to be fixed to the twenty firfl; day of March. . 2. That the fourteenth day of the Moon on which the Feafl: of E.fer doth depend, can ne- ver happen before the Equinox:, though it may fall upon it or upon the day following. 5. That the Feaft of Eafier is never obferved upon the fourteenth day of the Moon, but upon the Sunday following:, fo that if the fourteenth day of the Moon be Sunday, the Sunday follow- ingisEafter day. 4. That the Feafl: of Eafier may fall upon the fifteenth day of the Moon, or upon any other day unto the twenty firfl, incluflvely. 5. That the Falchal Sunday is difcov'ered by the to ^0ro!!om?.' 527 the proper and Dominical Letter for every Year The which may be found as hath been already do- dared, or by the proper Table for that purpole. — Hence it follovvcth, I. That the New Moon immediately preccr ding the Feaft of Eafle^y cannot be before the eightli day of Anarch, for if you fuppofe it to be March 6, the Moon will be 14 days old March-ig, whicli is before the Equinox, contra- ry to the fecond Rule before given, and upon the iVmler! feventh day of A'fnrch there is no Golden Number saHal fixed •, and therefore the Golden Number 16 , kCdii which ftandeth againit March 8, is the firft by ifcii which the Pafchal New Moon may be difco- te:=; vered. isaV; . 2. It followeth hence, That the lalt Palchal lloflir>,E New Moon cannot happen beyond the fifth day of Jprily becaufe all the 19 Golden Numbers are To til cxprefled from the eighth of March to that day. And if a New Moon fhould happen upon thefixth of ^^r;7,there would be two Pafchal NeWMoons i that year, one upon the eighth of March and a- tk.ltei nother upon the lixth of a^pril, the fame Golden Number 16 being proper to them both, but this dtGi is abfui d becaufe cannot be obferved twice ))" in one year. Fffofc 3. It followeth hencc,That the Feaft of l)jti can never happen before the twenty fecond day ifojtiS nor after the twenty fifth dayof^/jn'/.- For if the firft New Moon be upon the eighth of Ma-chy and that the Feaft of Eafier muft be upon .jjupe' the Sunday following the fourteenth day of the jjicj. Moon; it is plahi that the fourteenth day of the Moon muft be Anarch 21 at the fooneft: So that ftppofii'S the next day to be Sunday, Ea/cr can- C not 52 8 3fttfrotiucti'on not be before M-^rrch the twenty fecond. And becaufe the fourteenth day of the laft Moon fal- leth upon the eighteenth day of Jpril, if that day be Saturday, and the Dominical Letter D, Eafier Ihall be upon the nineteenth day, but if it be Sun- day, Eafier cannot be till the twenty fifth. 4. It followeth hence, That although there are but 19 days, on which the fourteenth day of the Moon can happen, as there are but 19 Golden Numbers, yet there are 3 5 days from the twenty fecond of March to the twenty fifth of Jpnl, on which the Feafl of Eafier may happen, becaufe there is no day within thole Limits, but may be the Sunday following the fourteenth day pf the Moon. And although the Feaft of Eafier can ne- trer happen upon 22, but when the four- tecBth day of the Moon is upon the twenty firft, and the Sunday Letter D, nor upon the twenty fifth of y^pril, but when the fourteenth day of the Moon is upon Jpril 18, and the Dominical Letter C. Yet Eafier may fall upon March 23, not only when the fourteenth day of the Moon is upon the twenty fecond day which is Saturday, but alfo if it fall upon the twenty firft which is Friday. In like manner Eafter may fall upon Jpril 24, not only when the fourteenth day of the Moon is upon the eighteenth day which is Monday, but alfo if it happen upon the feventeenth being Sunday. And for the fame reafon it may fall oftner upon other days that are further diffant from thefaid twenty fecond of March and twenty fifth of u4pril. 5. It followeth hence, That the Feaft of Eafier may be eafily found in any Year propounded: For tire Golden Number in any Year being given, if you toUtfronomp. 329 you look the fame between the eighth of March apcl fifth of both inclufively, and reckon 14 days from that day, which anfwereth to the Golden Number given, where your account doth end is the fourteenth day of the Moon ; Then confider which is the Dominical Letter for that Year,and that which followeth next after the four- teenth day of the Moon is Eafter day. Example, In the year 1674 the Golden Number is 3 , and the Sunday Letter E>, which being fought in the Calendar between theaforefaid limits, the four- teenth day of the Moon is upon A^ril the thir- teenth, and the D next following is Jfril 19. And therefore Eafierds.^ that Year is ^pril 19. , Qtherwife thus. /« March after the firfi C, Loak.the Prime wherever it he, the third Sunday after Eafter day jhall be. aindtf the Prime on Sunday hey Reckon that for one of the Three. 6. Thus the Feaft of Eajler may be found in the Calendar,and from thence a brief Table fhew- ingthe fame, maybe extraded in this manner. Write in one Column the feveral Golden Num- bers in the Calendar from the eighth of March to the fifth of Jprily in the fame order obferving the fame diftance. In the fecond Column let the Dominical Letters in number 35 fo difpofed , as that no Dominical Letter may ftand againft the Golden Number 16, but fetting the Letter D againft' the Golden Number 5, write the reft in this 55Q 3IH ^nfrotiucffoit this order. £, F, (7, A, B, &c. and when you come to the Golden Number 8, fet the Letter G, and there continue the Letters till you come to C again, becaufe when the Golden Number is i6, which in the Calendar is fet to the eighth day of March, is new Moon, and the fourteenth day of that Moon doth fall upon the twenty firft, to which the Dominical Letter is C, upon which the Feail; of Eajhr cannot happenand therefore in the third Column containing the day in which the Feallof is to be obferved, isallbvoid.But in the next place immediately following, to wit, ngainftthe letter DhCct March 21, becaufe if che fourteenth day of the Moon liiall fall upon the twenty firft of March being Saturday, the next day -being Sunday, ftiall be the Feaft of Eafler. To the Letters following, E, F, G, A, B, &c, are let 23, 24, 25, and lb orderly to the laft of March,znd fo forward till you come to the twen- ty fifth of ApH, l>y which Table thus made, the Feaft of Eafier may be found until the Calendar ftiall be reformed. For having found the Golden Number in the firft Column , the Dominical Letter fo? the Year next after it, doth (hew the Feaft of Eafier, as in the former Example, the Golden Number is 3 and the Dominical Letter D, therefore day is upon April 19. The other moveable Feafts are thus found. Achem Sunday is always the neareft Sunday to -;St. Andrews, whether before or after. Septaa-t to 15(trunomp. 331 tthtlt;; lOBtoat' «eigter; ffltentbi' - r c A • Septfi^e/ima Smday is Nine Weeks before Eafler. Sexagepma Sunday is Eight Weeks before Eufler. ■ QAwqHageJima Sunday is Seven Weeks be- fore Eafier. rtj,K . Q,t^Afji they thought it fufficient to bring the Equinox 'Mt back to that time, by cutting off 10 days in the Calendar as hath been dec lared, and to prevent i)lsi any anticipation for the time to come, have ap- ii. pointed, that the Leap-year fliall be thrice omit- 1, Tk ted in every 4C0 Years to come, and for memory fake, appointed the fird omiilion to be account- cafot ed from the Year 1600 , ijot from 1582, in iof which the reformation was made, becaufe it was ^ not only near the time, in which the emendation was begun, butalfo becaufe the Equinox has not fully made an anticipation of 10 days from the to SOronomp. ^ 335 place thereof, at the tim c of the Nkme Council which was 21. fffe The Years then 17C0, 1800, 1900, which Ihould have been Bifiextile Years, are to he ,ac- counted common years, but the Year 2000 muft be a Biflextile; In like manner the Years 21C0, feCih 2200, 2300, fhall be common years, and the Mffb Year 2400 Bilfextiie, and lb forward. mdiof/A 2. Again, becaule it was fuppofed that the Bofe Cycle of the Moon, or Golden Number was lo JihiiCi:! fixed, that the new and full Moons would in eve- ESlki! ry 19 years return to the fame days again; where- as their not returning the fame hours,but making j®:; an anticipation of one hour 27 minutes or there- leiiajr abouts, it muft needs be that in 17 Cycles or lit- feojjl;:: tie more than 300 Years, there would bean anti-' •£3[5jiir cipation of a whole day. And hence it is evi- jullip dent that in 1300 Years fince the Nicene Council, the New and Full Moons do happen more than cftte 4 fooner than the Cycle of the Moon or G0I-, ritjic den Number doth demonftrate: Whence alfo ic comes to pafs, that the fourteenth day of the Moon by the Cycle is in truth the eighteenth day, . jr^! and fo the Feafl of Eafier Hiould be obferved not from the fifteenth day of the Moon to the twenty jilp; firft , but from the nineteenth to the twenty iiwiu'' fifth. Moon therefore being one? brought into order, might not make any antici- ,^0 pation for the time to come, it is appointed that ^ J. a Cycle of 30 Epafts fhould be placed in the Ca- lendar inftead of the Golden Number, anfwering to every-day in the Year •, to fhew the New Moons . jj;; inthefedays, not only for 300 Years or there- abouts, but that there might be new Epadts with- .i out 334 out altering the Calendar, to perform the fame thing upon other days as need fnall require. 4. For the better underflanding whereof, to the Calendar in ufe with us, we have annexed the Gregorian Calendar alfo ; In the firft Column whereof you have 30 numbers from i to 30, faVe only that in the place of 30 you have this Afterisk *, But they begin vvitii the Calends of Jdnmry, and we continued and repeated af- ter a Retrograde order in this manner, *29, - 28, 27, &c. and that for this caufe efpccially, that the number being given which Iheweththe New Moons in every Month for one Year, you might by numbi ing 11 upwards exctulively find the number which will (liew the New Moons the Year following, to wit, the Number which fal- leth in the eleventh place. 5; And thefe Numbers are called Epadts, be- taufethey do in order fltew thofe i r days, which are yearly to be added to the Lunar Year conlift- ing of 3 54 days, that it may be in conformity with the So'lar Year conlifting of 365 days. To' this purpofe, as hatli been laid concerning the GoldeiiNumber, thefe Epadtsbeing repeated 12 times, and ending upon the twentieth day of De- cemheri thelanie Numbers mull be added to the 11 remaining days, which were added to thefirft 11 days in the IVlonth of Jamary. 6. And becaufe 12 times 30 do make 360, whereas from the fu ll oi January to the twentieth of December inclufively, there are but 354 days, you mufl know that to gain the other iLx days,the numbers 2 5 and 2 4 are in every other Month both placed againlt one day, namely, to February 5, 5, Jme 3, A-igufi I, September 29, and Novem- 5ii to 3!C(lconomp. 555 oraitE?;^ November 27. But why tbefetwo Numbers are iiquitt, chofen rather then any other, and why inthefe 6 Months the number 2 5 is Ibmetimes writ to XVI, nearffi;::; fometimes to XXV in a common charader, and ieSft(j^ why the number 19 is fet to the lall day of De- from r,/ cemher in a common Charader, (liali be declared ;yoite!s- hereafter. 7- Here only note that this Afl:eris.k * is fet in-j' id ffjocii ftcadofthe Epad 30,-becaufe the Epad Ihew- ing the Number of days which do remain after the Lunation in the Month of December j it may fome- times fall out that 2 Lunations may fo end, that the one may require 30 for the Epad, and the other o, which would, if both were written, caule fome inconveniences, and therefore this * Afte- risk is there fet, that it might indifferently ferve to both. And the Epad 29 is therefore fet to Mj-; the fecond day of betaufe after the com- f •-kofa W,' EliPMcL tdafs, slii skffliS COB- pleat Lunation in the fecond of December there are 29 days, and for the like rcafon the Epad 28 is fet againft the third o^Jami.try, becaufe after the com pleat Lunation in the third of December ' 5 ',,1. there are then 28 days over, and fo the reft in order till you come to the thirtieth of January^ where you find the Epad i. becaufe after the compleat Lunation on the thirtieth day there is only one day over. 8. And befides the (licwingof the New hloons ^ in every Month, which is and may be done by the do®' Golden Number, the Epads have this advan- tage, that they may be perpetual and keep the te jjr lame place in the Calendar in all future ages,which ^erliisr can hardly be effeded with the Golden Number, fflte • for ia lirrle more then 700 years, the New Moons iof^[ do make an anticipation of one day, and then it Z will Si6 ^ti Jutrotjuction willbenecefTary tofet the Golden Number one degree backward, and fo the Golden Number which at the time of the Nicene Council was fet to the firit of January^ (hould in 3 co years be fet • i. i* to December, and fo of the reft, but the Epadts being once fixed lhall not need any fuch retraction or commutation. For as often as the New Moons do change their day either by f}i(}|c Anticipation or by Suppueflion of the Biftextile (jotii: year, you lhall not need to do any more than to ^121x11 take another rank of 19 EpaCts, infteed of thofe jr jjd which were before in ufe. Forinftance,theEpads which are and have been in ufe in the Church of jjjsj; ^owffmce the year of reformation 1582, and will continue till the year 1700, arethefe lofol- lowing 1. 12. 23 4. 15. 26. 7. 18. 29. 10. 21. 2. 13. 24. 5. 16. 27. 8. 19. Andfrom jiiuj the year 1700 ti e EpaCts which will be in ufe jjgjrj arethefe. * 11. -22. 3. 14. 25. 6. 17. 28. 9. 20, I. 12. 23. 4. 15. 26. 7. i8.andniallcon- tinue not only to the year 1800, but from thence g ..| until the year 1900 alfo •, and although in ,5! the year 1800 the Biifcxtile is to be fupprefled, yet is there a compenfation for that Suppreftion, by the Moons Anticipation. To make this a little more plain, the motion oftheMoon,whithjjgj. doth occafion the change of theEpaCf, muft be jji;? more fully ccnfidered. .. 'Whbi- '■IkN '■ ■ iilie; CFIAP. V. tolftronomp. 337 CHAP. V. fc.re, I]®., dy the Moons mean Alotion^ and how the aintici- fation of the New Moons may he dtfcovered by the EpaEis. "01 rj lay A: ^-pHe Moon according to her middle motion tkfe J|[ ^oth finilh her coiirfe in the Zodiack in 29' days, 12 hours 44 minutes,, three feconds or there- about, and therefore a common Lunar year doth confift of 354 days, 8 hours, 48 minutes, 38 feconds and fome few thirds, but an Embolifmi- cal year doth confift of 383 days, 21 hours, 32 isik;;.. minutes, 41 feconds and fomewhat morej and '■18 - therefore in 1 9 years it doth exceed the motion ?• ofthe Sun I hour, 27 minutes, 33 feconds rfi* ■- 2. Hence it cometh to pals, that although tlte ,1-, 28, New Moons do after 19 years return to the fame .sAtt days^ yet is there an Anticipation of i hour, 27 JiSiOiiiE minutes,5 3 ieconds. And in twice 19 years, that 1 iws is, in 3 8 year--, there is an Anticipation of 2 hour s, minutes, 6 Ieconds, and after 312 years and a half, there is an Anticipation of one whole day hast' and fome few Minutes. And therefore after 312 k5S((!' years no new Mooit can happen upon the fame Epj.';,® day it did 19 years before, but a day fooner. Hence it comes to pafs that in the Julian Galen- dar, in which no regard is had to this Anticipa- tioii, the New Moons found out by the Golden' Number muft needs be erroneous, and from the time ofthe Nrcene Ccruncil 4 days after the New CEi-' Moons by a regular Computation. 3. And hence it follows alio, that if the Gcl- den Number, after 312 were upon due confide- Z 2 ration 33^ 3In 31ntrol)uction ration removed a day forwarder or nearer the be- 211 w ginning of the Months, they would Ihew the tfeis New Moons for 312 years to come. And being again removed after thofe years,a day more would la by the like reafon do the fame again. But it was thought more convenient fo to difpofe 30 iafc; Epadts, that they keeping their conftant places, «•. 19 of them ihould perform the work of the Gol- den Number, until by this means there (houldbe - -■ an Anticipation of one day. And when fuch an . -Jt; Anticipation Ihould happen, thofe 19 Epadlsbe- ing let alone, other 19 Ihould be ufed, which do belong to the preceding day, without making^ any alteration in the Calendar. jjj- 4. And if this Anticipation would do the j whole work, nothing were more plain, then to make that commutation of the 19 Epad once in tjg. 312 years; but becaufe the detradion of the Bifiex- tile days doth varioufly interpofe and caufe the 19 y,. Epads fomctimes to be changed into thefe that do precede, fomctimes into thefe that follow, y, fometimes into neither, but to continue ftill the fame •, therefore feme Tables are to be made, by which we may know, when the commutation was to be made and into what Epads. 4. Fii lt therefore therewas made a Table cal- Icdl^ahffla Epa^arum Expanfj, in this manner. Firfton the top were placed the 19 Golden ■" Numbers in order, beglrming with the Number 3, which in the old Calendar is placed againll; the Calends oijunuary, and under every one of ihele Golden Numbers there are placed 30 Epads ^ all confiituted rromthclowefl; numlrer in thefirft .rank in which the Epad is i, and in that raiik ti.e Golden Number is 3, the reft from thence"'' fo Biftcoiiom^ 35P thence towards the right Hand are made by the conftantaddition of it,andthecafl:ingaway of 30, ' as often as they ihall exceed that number, only when you come to the J7, the Epail under I'"!:,® the Golden Numbtr 19, there muft be added 12 inftead of 11, thatfo theEpad following may be g» not 8, for the Reafons already given in this soiK Dilcourle concerning the Golden Number and WK Embolifmicnl years. And this rank being thus mfc tiig other Epads are difpofed in their na- rural order afcending upwards, and the number ^ntl, once again refumedafter theEpad 30 or rather this Afterisk * fet in the place thereof: only ob- fcrve that under the Golden Number 12. 13. fWi. j5. jy. jp, in the place of XX pky. there is yet 25 in the common Charader. And Ej®®, to the Epads under the Golden Number 19,12 ofttt muft ftill be added to make that Epad under the Golden Number i. As was faid before concern- Kikfi ingthe loweft Rank. 5. And on the left hand of thele Epads before thofe under the Golden Numbers, are fet 30 toktsi Letters of the Alphabet, 19 in a fmall Charader, and 11 in a great,in which fome arc pafted by,for 5 no other realbn fave only this, that their fimili- tilde with feme of the fmall Letters, fliould not tliib- occafion any miftake in their ufe, which lhall be (hewed in its place, li 6. Befidcs this Table there was another Table plxfi!- made which is called Tabula ^yT:mnoms Epa£la- fit!!'; mm, in which there is a/enVj of years, in which icedji". the Moon, by reafon of her mentioned anticipa- kriih tion doth need ^Equation, and in which the num- linife' her ofEpads figned with the letters of the Al- liefr phabet, are to be changed j being othervvife Z 3 quated / ?4o JntroUucti'on quated where it needeth, by the fjpprefTion of the BifTextile days. \'j, 7. Bnt it fiippofeth, that it was convenient to v fupprefsthe Biirextiles once only in 10c years i and the Moon to be aequated, or as far as concerns | . her felf, the rank of Epafts to be changed, once f'f only in 300 years, and the 12 years and a half J ' more, to be referred till after the years 2400, they do amount unto 100 years, and then an x- ' quation to be made: but then it mull be made ^ by reafonof the interpofing this hundred not in i™'' the three hundredth but the hundredth year, p"" Moreover this equation is to be made as inrefe- rece to the Moon only, becaufc as the fupprellion . '?■' of the Billextiles intervene, the order of chang- W ing the ranks of Epads is varied, as lliall be fiiewed hereafter, 8. Again this Table fuppofeth, that feeing the New Moon at the time of the Nicene Council was upon the Calends of January, the golden 7® Number 3 being there placed, that it would have been the fame iftheEpad* had been fet to the fame Calends, that is if theEpacT;s had been '"W then in ule. And therefore at that time the higheft or lail rank ofEpads was to be ufed, whofe Index is P, and then after 3 00 years, the loweft or firft rank fhould fucceed, whofe Index is.nction year, totl-ieycar 17C0 therankofEpaftsisftill to be ufed, which do belong to that Letter. And for as much as the Letter C is fet to the year 1700, . it flievvcth that that rank of Epafts is then tq beufcd, which do belong thereto, and lb of the reft. II. The rcafon why thefe Letters in the firft Column are Ibmetimes changed in too years, foractimcs in 200, fomctimes not in lefs then 300 Years, and that they are feme- times taken forward , fometimes backward, according to the order of the Alphabet, is- becaufe the liippreftion of the Bilfextilcs do intervene with the lunar Equation; for if the ■ ^iftextilc were only to be fupprefied, in thefe 300, or fometimes 400 years, in which the Moon necdeth Equation, the rank of Epadls in that cale would need no commutation, but would continue the fame for ever-, andthegol--. den Number would have been fufficient, if the fupprelfion of the BilTcxtile, and anticipation of the Moon, did by a perpetual compenfati- on caufe the new Moons ftill to return to the fame days: but beraufe the Biifextile is ofttimcs fuppreffed, when the Moon hath noxquafoh, the Moon hath fometimes an Equation vyhen the Bif- fextile is not fupprelled, fometimes alio both are to be done and fometimes neither-, all which varieties may yet be reduced to thpfe three Rules. I. As often as the Billcxtile is fnpprelTcd without any Equation of the Moon, then the let- ter whieli ferved to that time fiiai) be changed to the next below it contrary to the order of the Alphabet. Amd the new Moons Ihall be removed one to sa^:o^om^ 345 I'P'^iif; one day towards the end of the Year. Utis.i; 2. As often as the Aloon needeth Jequation, without fLipprcffion of the BilTextile, then the "5 isfc; Letter which was in ufe to that time fhall be chan- aciiirf ged to the next above it according to the order of the Alphabet, that the New Moons may a- gain return one day towards the beginning of the year. tBiKtJ 3. As often as there is a SupprelTion and an equation both, or when there is neither, the s tec Letter is not changed at all but that which fer- yed for the former Centenary, fhall alfo conti- febi*' nue in the fucceeding", becaufe the compenfation ®;[o:k .. fo made, the New Moons do neither go for- ciiiiii 'a ward nor backward, but happen in the compafs , ofthe fame days, aioffp 1. And this is enough to fhew for what rea- tsmt fon the letters are fo placed in the Table, as jiiliicgi there you fee them; for in the year 1600 the idait: Biffextile being neither fuppreflcd,nor theMoon imip, sequated, the letter D ufedin the formerCen- i tenary or in the latter part thereof from the year KtitJii® . 15^ 2, is ftill the fame. dIjiijrBi, In the year 1700, becaufe there is a fupprefli- jotsti' .on, but no equation, the commutation is . nird: to the Letter C defcending. , In the Year 180c, becaufe there is both a fup- pr:;r preffion and an Equation, the fame letter C doth liiOtliv ftill continue. In the Year 2400, becaufe there is an Equation and no fuppreiTion, there is an afcenfion to the Letter And thus yon fee not only the conftruftion of d this Table, but how it may be continued to any Year, as long as the World fhall lafl. ; 12. And 244 3ntcot)uctioit 12. And by thefe two Tables we may eafily know which rank of the 30 Epafts doth belong to, .or is proper for any particular age: for as in our age,that is,from the Year 1600 to the Year 1700 exclufively, that feries is proper whofe hdex is Z). Namely, 23>4, iS>26, &c. lo in the two Ages follow ing, that is, from the Year 1700 to the Year ^900 exclufively, that fenes is proper whofe /«- a: is C, namely thefe, 22, 3,14,25. and in the three ages following thence, that is from the Year 1900 to the Year 2100 exclufively, that feries is proper whofe Index is 5, namely thefe,' 21,2, i3> 24, &c. And fo for any other. Hence alfo it may be known, which of the 19 doth belong to any particular Year, for which no more is rteceflary,than only to know the Gol- den Nnnber for the year given, which being fought in the head of the Table, and the Index of that Age in the fide, the common Angle, or meeting of thefe two, will fncw you the Epaft defired; As in the year 16 74 the Golden Number is 3 and the Index D •, therefore m the common Angle 1 find 23 for theEpaT that year, and (hew- eth the New Moons in every Month thereof. And here it will not be unfeafonable to give the reafon, for which the Epadt 2 5 not XXV is writ- ten under the Golden Numbers 12,13, 14, 15, 16, 17, 18, 19. namely, becaiife the ranks of E- pafts, which under thele greater Numbers hath this' Epaet 25, hath alfo XXIV, it would follow that in thefe Ages in which any of thefe Ranks , wereinufe, the New Moon in 19 years will hap- pen twice upon the fame days, in thole fix Months in which the Epadls XXV and XXIV are fet to the fame day: Whereas the New Moons do not hap- pen ■ fesijiil tlietB!2 'MtOSt! -J' dJlllil^i jfwjitt- !ife I sliidioicii ■ •. 'ffliSxnJ; • taBi." ■' ifiidlxf Blthek BiMde,! , otofe laiii'jES p.0' jtiri'sif,' Tears to 3ftrottomp- 345 Pen on the fame day till _i 9 years be gone about. To avoid this inconvenience, the Epad 25 not XXV is fet under thefe great numbers, and the Epad 25 is in the Calendar, in thefe Months fet with the Epad XXVI, but in the other Months vv ith the Epad XXV. 14, Hence it Cometh to pafs, i. Thatinthele Years the Epads 25 and XXIV do never meet on the fame day. 2. That there is no danger that the Epads 2 5 and XXVI fliould in thefe 6 Months caule the fame inconvenience, feeing that the E- pads 25 and XXVI are never both found in the fame Rank. 3. That the Epad 25 may in other Months without inconvenience be fet to the fame day with the Epad XXVI, becaufe in thefe there is no danger of their meeting with the Epad XXIV on the fame days. 4. That there is no fear that the Epads XXV and XXIV being fet on the fame days, fljould in future Ages caule the fame Inconvenience,becauie the Epads XXV and XXIV are not found together in any of the other Ranks. But that either one or both of them are wanting. BefideSjWhen one of thefe Epads is inufe, the o- ther is not, and that only which is in ufe is pro- per to the day. As in this our Age until the Year 1700 the Epads iif ufe are thofe in the rank whole /Wfx is D. In which thefe two XXIV and XXV are not both found. And in the two following Ages, becauie the rank of Epads in ufe is that whofe Index h C, in which there is the Epad XXV, not XXIV, the New Moons are flrewed by the Epad XXV not by XXIV. But becaufe in three following Ages, the rank of Epads in ufe is that whofe Index is 5, in which 25 and XXIV are both found, the New Moons life T* if'IS I'Wi If hp' ■Tig! 34^ 3ln 3fntrol)uction Moons are fhcwed by the Epati; XXIV when thegolden Number is 6. And by the Epadt 25 when the golden Number is 17, and not by the EpadXXV. 15. And if it be asked why the Epadt 19 in the common Charadler isfetwith the EpadlXX againft the laft day of December •, know that for the rcafons before declared, the laft Embo- lifinical Month within the fpace of 19 years, ought to be but 29 days and not 30, as the reft are-, and dierefore when the EpacT: 19 doth concur with the golden Number 19, the laft Month or laft Lunation beginning the fecond of December^ (hall end upon the 3c and not up- on the 31 of that Month, and the New Moon ftiould be fuppofed to happen upon the 31 un- der the fame Epadt 19, that 12 being added to 19 and not 11, you may have one for the Epadf of the year following, which may be found up- on the 30 of Janmry, as if the Lunation of 30 days had been accomplifhed the Day be- fore. CHAP. vr. fjovc to find the Dominical hotter and Fea(l of Eafier according to the Gregorim account. HAving (liewed for w^hat reafon, and in what manner the Epadts are fubftituted in the place of the golden Number, -and how the New Moons may be by them found in the Calendar for everi, Ifnall now fliew you how to find the Fcaft of Eaftcr and the other moveable Feafts- tu ^Utonomp. 347 Feafts according to the Gregorian or new ac- count •, and to this purpofe I mull firlt ihcw you how to find the Dominical Letter, for that the Cycle of 28 years will not ferve the turn, be- caufe of the fupprelfiGr), of the BilTextile once in a hundred years, but doth require 7 Cycles of 28 years apeice. The firft whereof begins with CB, and endeth in X). The fecond begins with DC, and endeth in E. Thethird begins with E D, and endeth in F &c. The firft of thefe Cycles began to be in ufe 1582, in vhich year the dominical Letter according to the Julian ac- count was G, but upon the fifteenth day of OHober, that Year was changed to C; for the fifth of OiEober being Friday and then called tlie fifteenth, the Letter became Friday, XSatur- day, and C Sunday, the remaining part of the year, in which the Cycle of the Sun was 23, and the fecond after the BilTextile or leap Year, and fo making C, which anfwereth to the fifteenth year of that Circle, to be 23, the Circle will end at and confequently CB, which in the old account doth belong to the 21 year of the C;rcle, hath ever fince been called the firft, and fbfhall continue until the year 1700, in which the BilTextile being fuppreflcd, the next Cycle will begin with X) Cas hath been faid already. Under the firft rank or order of Dominical Letters are written the-years 1582 and 1600, under thefe- cond 1700, under the third 1800, under the fourth 1900 and 2000, under the fifth 2100, un- derthefixth 2200 and under the feventh 2300 and 2400. And again under the firft Order, 2^00, under the fecond 2600, under the third 2700 and 2800, .and fo forward as far as you pleafe, p 348 JntroDuttion pleaie, always obferving the fame order, that the 100 Bilfextile years may ftill be joyned with the not Biflextile immediately preceding. I. And hence it appears, that the feven or- ders of Dominical Letters, are fo many Tables, lucceffively ferving all future Generations. For as the firft Order ferveth from the year 1582 and 1600 to the year 1700 exclufively, and the fecond Order from thence to the year 1800 ex- clufively, lb fliall all the reft in like manner which here are fet down, and to be fet down at plea- fure. And hence the Dominical Letter or Let- ters may be found for any year propounded, as if it were required to find the dominical Letter for the year 1674, becaufe the year given i^ contained in the centenary 1600. 1 find the Cycle of the Sun by the Rule already given to be 3. In the firft order againft the number 3, I find G for the Sunday Letter of that year, in like manner becaufe the year 1750 is contained under the Centenary i -00, the Cycle of the Sun being 27, Ifind in the fecond rank the Let- ter D anfwering to that Number, and that is the Dominical Letter for that year, afnd fo of the reft. 3. Again for as much as the fifth Order is the fame with that Table, which ferves for the old account, therefore that order will ferve the turn for ever where that Calendar is in u(e, and fothis.laft will be of perpetual ufe to both'the, Calendars. 4. Now then to find the time in which the Feaftof£<«y?f?-istobeobferved,there is but little to be added to that which hath been already faid concerning the Julian Calendar. For the Paf- chal i spl^j to Sftronomp. 349 chal Limits are the fame in both, the difference is only in the Epadts, which here are ofed in- ftead of the golde.n Number. 5. For the terms of thePafchal New Moons are always the eighth of March and the fifth of Jpril: but whereas there are 11 days within theft Limits to which no golden Number is affixed, there is now one day to which an Epad is not appointed, becaufe there is no day within thofc Limits, on which in procefs of time a New Moon may not happen. And the reafon for which the two Epads XXV and XXIV are both fet to the filth of is firlt general, which was fhew- ed before, namly that by doing the lame in 5 other Months, the 12 time 30 Epads might be contraded to the Limits of the lunar Year which conliftsof 3 54 days; but there is a particular rea- fonalfo for it,that the Antients having appointed that all the Pafchal lunations fliouldconiiftof 29 days, it was necelfary that fome two of the E- pads (hould be fet to one of thefe days in which the Pafchal lunation might happen, the Epads being 33 in number. And it was thought con.- venientto choofe the lalt day, to which the E- pad XXV belonging, the Epad XXIV lliould alfo be fet; and hence by imitation it comes to pals, that thefe and not other Epads are fet to that day in other Months, in which two Epads are to be fet to the lame days. ' 6. The uieofthefe Epads in finding the Fetill of Eafier, is the fame with that which hath been lliewed concerning the golden Numbers. For the Epad and the Sunday Letter for that year propounded being given, the Fealt of Eafier may be found in the Calendar after the fame manner. 350 3ln(rol)ucttoit manner. Thus in the year 1674, the Epadt is 23 and the Sunday Letter G, and therefore reck- oning fourteen days from the eighth of March to which the Epadt is fet, the Sunday following is March 25, which is the day on which the Feaft of Eajter isobferved. 7, And hence as hath been llrewed in the third Chapter concerning the Julian Calendar, a brief table may be made to fhew the feaft of Eafierand the other moveable Feafts for ever, in which there is no other difference, fave only that the Epadfs as they are in this new Calendar, are to be ufed as the golden Numbers are, which ftand in the old Calendar. And a Table having the golden Numbers of the old Calendar fet in one Column, and the Epadts as they are in the new Calendar fet in another, will indifferently fhew the movable Feafts in both accounts, as in the Year 1674, the golden Number is 3 and the Sunday Letter according to the Julian ac- count is X), according to the Gregorian G", and the Epadt 23, and therefore according to this TablcoMv Eafier is Jpril 19, and the Other, to wit, the Gregorian, Is March 15. The like may be done for any other year paft or to come. CHAP. VJ. to inilronomp. CHAP. VII. tlovp to reduce Sexagenary nHmhtrs into Decimal, and the Contrary. EVefy Circle hath antiently, and is yet ge- nerally fuppofed to be divided into 3 60 dc- grees, each tiegree into 60 Minutes, each Mi- nute into 60 Seconds, and fb forward as far as need fhall require. But this partition is forae- what troublelbm in Addition and Subtradtion, much more in Multiplication and Divifion-, and the Tables hitherto contrived to eafe that man- net of computation, do fcarce fufficiently p«f- form the work, for which they are intended. And although the Canon publilhed by the lear- ned H. GelUhrmd, in which the Divifion of the Circle into 36 b degrees is retained, but every degree is divided into 100 parts, is much bet- ter than the old Sexagenary Canon, yet fomc are of opinion, that if the Antientshad divided the whole circle into 100 or 1000 parts, it would have proved much better then either \ only they think Cuftome fuch a Tyrant, that the alteration of it now will not be perhaps fo advantagious *, leaving them therefore to in joy their own opinions, they wiU not I hope be or- fended if others be of another mind: for their fakes therefore, that do rather like the Deci- malway of calculation. Having made a Canon of artificial Signs and Tangents for the degrees and parts of a Circle divided into 100 parts, Ilhallhere alfo Ihew you, how to reduce fexa- genary Numbers into Decimal, and the contra. A a < ry as2 2in UnttoDucUott rjf as well in time as motion. 2. The parts of a Circle confifting of 360 degrees, may be reduced into the parts of a cir- xle divided into 100 degrees or parts, by the rule of Three in this manner. As 360 is to 100, lb is any other Number of degrees, in the one, to the correfpondent dc- md grees and parts in the other. fir tic D But if the fexagenary degrees have Minutes dsdciit .and Seconds joyned with them, you muft reduce fcialpt the whole Circle as well as the parts proponfl- sstte .ded into the leaft Denomination, and fo proceed te to according to the rule given. itdtm Example. Let it be required to convert 125 mjc degrees of the Sexagenary Circle, into their -'tko correfpondent parts in the Decimal. I fay, as uissj 360 is to ICQ, lb is 125 to 34, 722222, ly, Tl that is, 34 degrees and 722222 Parts. 2. Sample. Let the Decimal of 238 de- ^jti^ grees 47 Minutes be required. In a whole Cir- cle there are 21600 Minutes, and in 238 de- grees, there are 14280 Minutes, to which 47 being added the fum is 143 27. Now then I fay if 21600 give 100, what (hall 14327. The gjjj Anfw. is 66,3 287 c^c. In like manner if it were required to convert the Hours and Minutes of a Day into decimal Parts, fay thus, if 24 Hours ^ give lOo, what (hall any other number of Hours give. Thus if the Decimal of 18 hours were required^ the anfwer would be 75, and the De- .j[^ cimal anfwering to 16 Hours 30 Minutes is ■ VS* • , _ Sg[j9 Butifitbe required to convert the Decimal J'arts of a Circle into its correfpondent Parts ' in Sexagenary. The proportion is j as 100 is J to . fo siftwnomp; sil to the Decimal given, fo is 36b to the Sex^e- ' nary degrees and parts required. Example. Let the Decimal given be itSiifc 722222, ifyou multiply this Number given by 360, the Produft will be 12499999921 thtitls tctSa: cutting off7Figures, 124 degreesand 9999992 pmfci parts of a degree. IfMinutesbereqUirediffluL tiply the Decimal parts by 60 ^ and froin the wSis produft cutolfas many Figures, asWfere in the wlrdi Decimal parts given, the reft IMlLbe the Mi- iwpfflj nutes defiredi ■ ' x ifep But to avoid this trouble, I have here exhi- bited two Tables, the one for converting feXr CK!J ftgenary degrees and Minutes in td DeciiA^lsj :,i®S and the contrary. The other for corivettilig li iSi! Hours and Minutes into Decimals, abd thecOn- trary. The ufe of which Tables IWill explain s, by exam ple. if 2}5 ( Let it be required to convert 258 degtees 44*., 47", into the parts of a Circle deCirtially divi- ffli'jli ded. lOKfc ■ The Table for this purpofe doth cOHfift ai otiwil two Leaves, the firft Leaf is divided into 21 j|i-, H Columns, of which the i. 3. 5. 7, 9. 11. Ktiw 13. 15. 17. 19 doth contain the degrees in a fexagenary Circle, the 2. 4^ 6. 8. 10. 12. 14! jnftJ 16. 18 and 20 doth contain the degrees of n ituiilltJ Circle Decimally divided, anfwering to the fot» lioij' iner, and the laft Column doth contain the pe- jjiii! cimal parts, to be annexed to the Decimal de- grees. Thus the Decimal degrees anfwering teafures of the year in feveral Nations, there . ^ yet oqe thing wanting, which is, byanobfer- vation made of a Planets place in one Country to find when the Planet is in that place in refe- ^ rence to another; asfiippofe the © by oblerva- tion was found ?LtVrambHrgXo be inT. 3"^. 13^ " 14". the fourteenth 1583 at what time - was the Sun in the fame place at London ? To re-" fblvethis and the like qmftions, the Longitude of places froiP fome certain Meridian muftbe known-, to which purpofe I have here exhibi- 'ip® ted a Table fhewing the difference of Meridian? iq flours and Minutes, of molt of the eminent places in England from the City of London, and of fome places beyond the Seas alfo, Theufe whereof is either to reduce the time given under the Meritfian pf London to fome other Meridfi an, or the time given inlbme other Meridian to the Meridian or London. I . |f it he required to reduce the time given under the Meridian of London to fome other Meridian, feek the place defired in the Catalogue, ®o[tl and the differertce of time there found, cither « add to or fubtrafl from the times given at Lon-.iwk don, according as the Titles of Addition or Sub- tofew tradion (hew, fo will the time be reduced to the wlOil Meridian of the other place as was required, -icoa Example, Tfie fame place at London was in the ■ Ani firft Point of«, 6 Hours f.M. and it is required 4 ast |o reduce the fame to the Meridian of i therefore feek y^Vramhurg in the Catalogue of-in thf places, againft which 1 find 50' with the Letter-'then 4^ annexed, therefore I conclude, that the Sunshlii W?s that day at Vrmiburg in the firft point of dj !• Hei ^ l^purs 5q'. jP. H li to lllfronomp. 357 lioss, tk 2. If the time given be under fbme other Me- bjfflolj; ridian, and it be required to reduce the fame to. omCt the Meridian of Londony you muftfeek the place )ljccii![ given in the Catalogue, and the difference of ;bj(iiiS): time there found, contrary to the Title is to be added or fubtraded from the time there gi- yen. Example, Suppofe thej place of the Sun had been at VranibHrg., at 6 Hours 50'. P. M. and I would reduce the fame to the Meridian of Lon- don j againft Vraniburg as before I find 50' therefore contrary to the Title I Subtraft 50* and the remainder 6 Hours is the time of the Suns place in the Meridian of London, (krWET; C H A p. IX. ■Iriiffl . Of the Theory of the Sards or Earth's Motion. r N the firft part of this T reatife we have fpok- en of the primary Motion of the Planets and Stars, as they are wheeled about in their di- "Zjl, urnal motion from Eaft to Weft, but here we are to fhew their own proper motions in their feveral Orbs from Weft to Eaft, which we call their fecond motions. 1. And thefe Orbs are iiippoled to be Ellip- tical, as the ingenious Repler, by the help of Tycho's accurate obfervations, hath demonftra- ted in the Motions of Mars and Mercury^ and may therefore be conceived to be the Figure in which the reft do move. 2. Here then we are to confider what an liffsis, howitjnaybe drawn, and by what Me- Aa ^ thod r-y-'- jf 358 3in Inttoliucti'oti rhod the motions of the Planets according to that Figure may be computed. .'3. What an ElUpfis is ylpUoniui Perg<£usin Conkisy Claukm Mjdorgim and Others have well Ss'f® defined and explained, but here I think it fuffici- ent to tell the Reader, that it is a long Circle, or a circular Line drawn within or without along Square*, or a circular Line drawn between two ®tiieSi Circles of different Diameters. 4. The nfual and Mechanical way of drawing tKisEllipfis isthusi firft draw a line to that length ^ofthi which you would have the greateit Diameter to dW be, as the Line j4P in Figure 8, and from the :tt is&oi middle of this Line at X, fetoffwith your com- km pafles the Equal diltance XM and XEf. .stkgi 5. Then take a piece of thred of the fame ' i Now length with the Diameter ^P and fallen one middle end thereof in the point M and the other in i!,AeIi the point //, and with your Pen extend the thred thus faflened to the point A, and from sPriflj thence towards P keeping the thread fiiff upon ,;te your Pen, draw a line from ^ by S to P, the ictme line fo drawn fhall be half an ElUffts, and in like itliaed manner you may draw the other half from P by Dto A. In which becaule the whole thred is lotbcit equal to the Diameter^?, therefore the two Lines made by thred in drawing of the ElUp/is, ijti j mult in every point ofthe faid ElUpfis be alfoe- im qual to the lame Diameter They that de- lire a demonflration thereof geometrically, may ^ £j COnfuIt Apollomas Pergaus, Claudius Mydorgius or Others, in their treatifes of Conical Scdions, this is fuihcient for our prefent purpofe, and ji, from the equality of thefe two Lines with the ; Diameter, a brief Method of calculation of iq PC.; 'S * to Bffroitomp. 339 Planets place m an Ellipjis, is thus Demonftra- ted by Dr. Ward now Bifhop of Salifiury. 6. In this Ell/pfis H denotes the place of the Suns Center, to which the true motion of the Planet is referred, A/" the other Fochs whereun- to the equal or middle motion is numbred, ji the y^phelioiwhtrethe Planet is fartheft diltant from the Sun and floweft in motion, P the P?- rihdion where the Planet is neareft the Sun and floweft in motion. In the points A and P the Line of the rnean and true motion do convene, and therefore in either of thefe places the Pla- net is from P insequality, but jnali other pomts the mean and true motion differ, and in D and C is the grcateft elliptick iEquation. 8. Now flippofe the Planet in 5, the lineof the middle motion according to this Figure is MB, the line of the true motion HE. The mean Anomaly A MB, The Eliptick ajquati- on or Profihapharefis MB H, which in this l^am- pie fubtradted from AMB, the rcmainer AHB is the true Anomaly. And here note that in the right lined Triangle MBH, the fide MHis al- ways the fame, being the diftance of the Eociy the other two fides MB and HB are together equal to AP. Now then if you continue the fide MB till be equal to BH and draw the line HE, in the right lined Triangle MEHy we have given ME=AD and MHviith the Angle EMH, to find the Angles MEH and MHE which in this cafe are equal, becaufe E B = B H Contradion, and therefore the double of BE Hot BHE^MBHy which is the Angle required. And that which yet remaineth to be done, is the Un fntroDuctfoit the finding the place of the Jfheliony the true Excentricity or diltance o fthe umbilique points, and the ftating of the Planets middle motion. 0 ^ C H A P. X. Of the finding of the Suns Apogeon, quantity of Excentricity and middle motion. THe place of the Suns Jpogaon and quantity of Excentricity may from the obfervatU ons of our countrey man Mr. Edward Wright be obtained in this manner, in the years 1596, and 1497, the Suns entrance intoT and and into the midft of«. SI. "K and ss were as in the Table following exprelfed. 1^96 1597 D. H. M. D. H. M. r. o 25'03-S4 L 15 28.09.56 1 15 12.19.15 I £3. o January. 25.00.07 ^ f 24.05.54 March. 9.18.43 I 10.00.37 } j4fril. 24.21.47 K I jMy. 28.01.43 September.^ 12.13.48 Okober. 27.15.23 J L 27.21.50 J H'. 15 And hence the Suns continuance in the Nor- thern Semicircle from T to£; in the year 1596 being Leap year, was thus found. From ;JiII XII ( to ^(Itonomp. d. h. ; From the I. of fa^uaryl 2^6.13.4^. to © Entrance S j ^ t From the i. of J«« to © Entrance T 6 9.18.41 Their difference, 186.19.05 In the year 1597 from the 1 of Januarry to the time of the © Entrance into 255.19.15 To the © entrance into r« 69.09.37 Their difference is 186.18.38 And the difference of the Suns continuance in thefe Arks in the year 1596 and 159715 27'. and therefore the mean time of his continuance in thofe Arks is days 186.hours 18. minuteS conds 30. And by confequence his continuance in the Southern Semicircle that is fromsss to T is 178 days. 11 hours, 8 minutes and 30 fe- conds. In like manner in the year 1596 between his entrance into « 15. and IE 15, there are days 185.17.36 And in the year 1597 therearedays 185.17.56 And to find the middle motion anfwering to days 186. hours 18. Minutes 51. ftconds 30 I fay. As 365 days, 6 hours, the length of the Julian,year is to 360, the degrees in a Circle. So is 186 days, 18 hours, 51'. 30'' to 184 degrees. 03'. 56". In like manner the mean motion anfwering to / 3^2 lln 3ffitrotucti'ott *^0 185 days, i^h* 46' is 183 degrees, 02'. 09," Apparent motion from T to ^ 180. 00. 00 Middle motion 184- 03- 56 Their Sum 364. 03. 56 Half Sum is the Arch. SME 182. 01. 58 In 1596 fronr 15 » to 15 si there are days 185, hours01, minutes 36. In 1597. days 135. hours 4. 02'. And the mean motion anfwering thereunto i?., I82<1' 30'. 36'', Apparent motion from 15 IS to 15 De. 180. Middle motion 185. 17. 56. 181.04.53 Half Sum is j83.32.26 From 15 ss to 15 a- Days. 185. 04''- 02' Apparent motion Middle motion Half Sum 180. 182. 30. 36 181. 15. 18 Now then in from PGC. 181.32.26 deduft NK D 180, the Rcmainer is DC+A^P. 1.32.26. Thereforeor iVF. 46. i3,wbofe Sine is And from XPG. 181. 15. 18 dedud TNK 180, the Remainer is- K G -\-TX i. 15. 18. Therefore KG or TX 37. 39 , Vyhofe Sine is HP. m i 0^' ii CI.| :eKilf; feiii itiiiii, i?c. i8i.5|,r 'SMiii \]M to3firottotnr. Now then to find the ApOgS OH. 363 ASH^46'. 13" So if ^37'. 39" ToTan£. HAR. 39'1-10'. 04" C AM. 45 5.12851105 15. 03948202 9.91097097 AfogMn 95. 49. 56. Hence to find the excentricity AKj AstheSine39.10.04 9.80043756 ToRad. So//.R. 37.39 15.03948202 To 1733.99 5.23904446 Or thus, in the Triangle ARy we have given Ay. and R yi As Ay. 37* 39 ToRad. 80^^7.46.13 ToTetng.RAy. 50.49.56 P AS. 45. 5.03948202 15. .12851105 zo. 08902903 [{fell Afogaon 95 deg. 49'. 56". as before. Then for the Excentricity R A. AstheSineof 50.49.56 9.88945938 Is to .Ry. 46'. 13" 5. Z2851105 ^oii Raditii. T o R A. 1734.01 5.23905167 And this agreeth with the excentricity, ufed by 3^4 jnttoDucttoti by Mr. Street in his J-firon. Carolina, Pag. 23. But Mr. Wing as well by obfervation in former ages, as our own, in his Afiron. Infiaur. Pag. 39. doth £nd it to be 1788 or 1791. The work by both obfervations as followeth. 2. And firft in the time of Ptolemy, Am Chrifii 139 by comparing many obfervations to- gether, he fets down for the meafure neareft truth, the interval between the vernal EqukiOx and the Tropick of Cancer to be days 93. hours 23- and minutes 03. And from the Vernal to the Autumnal Equinox, days 186. hours 13. and minutes 5. D. The apparent motion from r to $ 90.36.00 Middle motion for 93''' 23''- 3'. is 92.36-42 The half Sum is G P 91.18.21 Apparent motion from r to 180.00.00 Middle motion for i86<^' i3''- 5'. is 183.52.05 The half Sum is G £/C 181.56.oi The half of GEK is GE. 90.58.01 And GF lefs GE is 00.20.20 WhofeSum is AC 59146. Again from GEK 181. 56. 02. dedu(!^ the Semicircle FED 180. the remainer is the fumm DK and FG. 1. 56. 2. and therefore DK=FG. 58'. 01". whofe fign is BC. 168755. L is the place of the Aohehon, and AB the Excentricity. Now then in the Triangle ABC. in the Fig. 6 v\ e have given the twolides AC and BC. To find the Angle SAC and the Hypoteaufe AB^ ; for to ^fltonomp. 365 For which the proportions are. As the fide jiC $91^6 4-771 ^25 38 b to the Radius. io.oooooooo So is the fide 5C 168755 5.22725^5^5 ToTang. 70.41. 10. 10.45533127 Secondly for AB. As the Sine of jB^C. 7c. 41. io« 9.97484352 Is to the fide 168755. 5.22725665 So is the Radius. lo.oooooooo TotheHypot. AB. 1788. 10. 5.25241313 Therefore the Aphelion at that time was in H ■ID. 41. lo. And the excentricity. 1788. ' 3. Again ArmoChrifii 1652 the Suns place by obfervation was found to be as followeth, April. 24. hours. 10. OBober. 27. hours. 7. 10' January. 24, hours. 11.20' July. 27. hours. 16.30, Hence it appeareth that the Sun is running through one Semicircle of the Ecliptick, that is from » 15 to te 15.18 5 days 21 hours and 10'. And through the other Semicircle from ks i 5 to 51 15, days 184. hours 5. therefore the Suns mean motion, according to the praftice in the laft example, from tS 15 to 1? 15 is 181. 30.26. and from ss 15 to SI 15. 181. 16. 30. Now then in Fig. 7. if wefubtradl the femi- circle of the Orb KMH. 180. from WPF 36. 26. the remainer is the ivmofKW^diHy ; • 1-36 I 3^6 Jntrotjucti'ott 1.56. 26. the Siiieofhalf thereof 48'. i3"iseA qual to^C. 140252. Again the mean motion of the Sun in his Orb fromisj 15 to^5 15 is the Arch SKP. 181. i5. 30. whofe excefs above the Semicircle being hi- f^ed is 38. 15. whofe Sine C5; in 345. now then in the Triangle ABC to find the Angle BAC, the proportion is. As the fide ACj 140252 5.14^9090^ Is to the Radius. lo.oooooooo So is the Side CB 111345 5.04667072 To Tang. BACt 38. 36. 21, 9.89966166 Which being dedu(fted out of the Angle. 69 yiSl.45 it leaveth the Angle 69 AL6. 33. 39. the place of the 0 Aphelion fought, and this is the quantity whkh we retain. And for the excentrieity B G. As the Sum of 5>4C. 38.26.21 9.79356702 Is to the 10.00000000 So is the fide BC wi345 5.04667072 To the Hy pot. .45. 179103 5.25310370 Sothen Anno Chrifii. 16^2. ApheL 9^-33*39 Anno Chrifi. 139. the Aphelion 70.41.10 Their differen(e is 25.52.29 And the difference of time is 1513 julian years. Hence to find the motion of the Aphelion for 2. years, fay I, if 1513 years give 2 5.52.2 9,what (hail one year give, and the anfwer is oo'^- 01' 01", 'Ifc: to ^mtonornp. ^67 01". 33'"- 4:}.*. that is in Decimal num- bers. 0.00475. ®44+7' °55S* And the motion for. 1651 years. 7. 84298. 4208862, which being deducted from the place of the aphelion J^no Chrifli. 1652-"—26.82245. 3703703. The remainer, viz.. 18. 97946. 9494841 is the place thereof in the beginning of the Chriftian z^ra, which being reduced is, 68 deg. I9. min. 33. fee. 56. thirds. 4. The Earths middle vnoxXony AphdiondRd Excentricity being thus found, we will now (hew how the lame may be ftated to any particular time defired, and this mull be done by help of the Sun or Earths place taken by oblervatipn. In the 178 year then from the.death of der,Mechir the 27at It hours P.M. Hippar- CHS found in the Metidiah of AkxarJ. that the Sun entered T o. the which Vernal Equinox! happened in the Meridian of London according to Mr. Wings computation at 9 hours 14', and the Suns Aphelion then may thus be found. The motion of the Aphelion for one year, was before found to be. 0.00475.04447.05 5 5. there-' fore the motion thereof for one day is o.ocooi. 501491722. The Chriftian began in the 4713 year compleat of the Julian Period, iii which there hre days 1721423. ^hQzy£raA- kxandri began November the twelfth, in the year 4390 of the Julian Period,, in which there are 1603397 days. And from the death afy^- lexander to the 27 of Mechir 178, there are days 64781, therefore from the beginning of the Julian Period, to the 178 year .79560710 The double whercpf is the Angle MB H i.'5i.io.which being Subtrai3)cdfrom 360there- mainer 3581 08; 50. is the eftimate middle mo- tion of the Sun, from which fubtrading the A- fhelion before found, 65. 4p.' the remain'- er292. i8.' 57. is the mean Anomaly by which the ablblute^quation may. be found according to the former operation. ' ' Co. ari-'^ 4.69127^4^ X.ME—MH. 196424 i-;i 5;2932i855 Tang, i Anorn. 56.. 09. 28; r 10.17359517 Tang. IX. 55. 12' 18. I 10.15808715 Differ. 00. 57. .10. : ' n ' O •aO *W"-- Doubled i. 54.20, whicfa addedto the mid- die motion before fOund gives the © true Jilaoe y. 00. 3'. 10", which exceeds the obfervation b'i 10". therefore ■ I dedudt the lame ftom the middle motion before found, and the remainer 358. 05* 50. istfte middle motion at the time of the obfervatioa of Hipparchu^ m which if you add the middle motion of the Sun for 5 3 245 days, or for 323 iEgyptian years 131 days, 28c. 48. 08' the~Summ', rejecting the whole Circles, is 278* 51. 48 the Suns mean Longitude in the beginning of the Chriilian ^ya. 6. But one obfervation is not fufficient, where- B b 2 by ^ytoftatc the middle motion for any delired Eppcha, we will therefore examine the lame by .another obfervation made by AlbMegnius at A- i-a^a in the year ofChx'& S^Zy March: hours 22. 21. but in the Meridian of London at 18 hours. 58'. ' The motion of the A^heiion for 881 years, «74, days.is 3/ 80^06865 3737, which being ad- d^d 'tp the plade thereof inthebe^nning of the Ghriftian t/^r-^Tthe place at the time of the obfer- yationwill be-found to be 22.785538148578, that is reduced, Deg. 82. oi'- 40". And hence the,Equation according to the former operati- ens is Qeg. 2. 01'. 16" which being dedu(^ed from a whole Circle, the remainer 357''' 58'. 44" is the eftimate middle motion at that time, from which deducting the y^/^Mw deg. '82. 01. 40. the remainer 275. 57. 04 is the meanano- irijUy, and the /Equation anfwaing thereto is deg. 2.^02'., 18" which being added, to the middle motion before found, gives the0 place T. 00. 01'. 02" which exceeds the obfervadon 01'. 02". therefore dedudl the fame from the middle motioii before fbund^ the remainer 3 57; 57!j 22" is .the middle motion of the ^0 at the time of the obfervation, from which deducing the middIe;fn<>tion'for 881 years, 74 days, 18 hours, 58 minutes, pviz.. 8o^- :o6'. 10". the re- rnainer 277 deg. 51'. 22".^ isthc 0 niean Lon- gitude in the beginning of the- Ghriftian zALra. ■. , : J a. - ■ : By the firft obfervation it is dcgk 48'' By the fecond 277.51.12 Their difference is j 1.00.36 to l^llronomr* ?7i" He that defires the fame to this or any other, Epocha, to more exaftnefs^ muft take the pains^ to compare the Colledion thereof from fun- dry Obfervations, with one another, this is lamcient to ihew how it is to be found. Here therefore I will only add the meafures fet'down by fome of our own Nation, and leave it to the Readers choice to make ule of that whidr pleafeth him belt. Tlx ® mean Longitude in the beginning of the Chriftian TEra according to. Vincent Wing h 9. 8^- 00". 31" Tho. Street is 9- 1' S'S' John Flamfied is 9- 7- 54* 39 By our firft Computation 9. 8. 51. 48 Byourfecond 9. 7. 51. 12 In the Enfting Tables of the © mean Lon- gitude, we have made ufe of that meafure gi- yen by Mr. FUmfied, a little pains will fit the Tables to any other meafure., n,', . io"P- CHAP. XI. Of the quantity of the Trofical and Sydereal Tear, '"pHe year Natural or Tropical ( fo called , X from the Greek word TgeTrtj, ( which lig- nifiesto turn ) becaufe the year doth ftill turn r'lH or return into it felf) is that part of time in which the 0 doth finifh his courfe in the Zodiack B b 3 by f 3iittrolntcttoti by coming to the feme point: front whence it j began- : . . , 2. Tfiat we ra^ determine the true quantU- ty thereof, wemuftErft find the time of the 0 j Ingres into the £quinoftral Points, about which ' * there is no finall difference amongft Aftrono- iners, and therefore an abfoluteexadnefs is not to be expeded, it is well that we are arrived fo near the Truth as we are. Leaving it therefore to the fcrutiny of after Ages, to make and com- pare fundry Obfervationsof the © entrance in- to the /Equinodial Points, it (hall fuffice to (hew here how the quantity of the Tropical year maybe determined, from tliele fcdlowingobfer- vations. 3. j^lbategniuSf Anno Chrifii 882 oblerved the 0 entrance into the Autumnal Equinox at ArnEiam Syria toht Sept. 19. i hour 15'in the Aforning. But according to Mr. Wings cor-® redion in his Aftron. Infiaur. Page 44, it was ^t I hour 43' in the Morning, and therefore ac- cording to the © middle motion, the mean time of this Autumnal iEquinox was Sfpr. 16. ii''' 14'. 25". that is at London at 8''* 54'. 25". 4. Again by lundry obiervations made in the year 1650. the fecond from Biflextile as that of Alhategnius was, the true time of the © in- \ ijds ' Ifmo fkid Vriil Mr,; Imcotf " 'feS® .t. ip' ml! {(dm ^.TTi iiSok :.N( lOD grefs into ^ was found to be Sept. 12. i4''* 40 Ao'. "^1 a and therefore his ingrefs according to his mid die motion was Sfft. 10. n''-02. ■ 5, Now the interval of the(e two oblertar^ tions is the time of 768 years, in which fpace by fubtrading the leifer from the greater, 1.^2; find an anticipation of 5 days, 9 hours, 52'. 25".'|'c' 5 y/hich divided by 768 giyeth in the quotient '^ to JEHrottomp. 375 lo'. 55". 39"' which being fubtra(!!led for 365 days, 6 hours, the quantity of the Julian year, the true quantity of the Tropical year will be 365 days, 5 hours, 49'. 04". 21 Others from other obfervations have found it Ibme- what lefs, our worthy countryman Mr. Edxvard takes it to be 5 hours. 48'. Mr. John Flamfied) 5^- 29'. Mr. Tho. Street 5^- 49'. 01". taking therefore the Tropical year to confift of 365 days, 5 hours, 49 Minutes, the Suns mean motion for one day is o deg. 59'. 8". 19"'. 43'v 47V. 21^'-29'"- 23""'orinde- cimal Numbers, the whole Circle being divi- ded into loo degrees, the® daily motion is o. 27379. 08048. 11873. 6. The Sydereal or Starry year is found from the Solar by adding the Annual Motion of the eighth Orb or pr?V • > : ' • i' ; / . » • ( ■4:, (v >5 'J ksi '^of 374 J^ttoDucti'oti «8. 16'. and therefore the motion in one ye^ 50", which being divided by 365 days, d hours, the quotientis the motion thereof in a days time, 00'. 8"'. I2'*'* 48*- 47"* 18*"* 30*"'* i3«* and in decimal Numbers, the motion for a year is 00385. 80246. 91358. The motion for a day. 00001. 05626. 95938. 9. Now the time in which the Sun moveth 50", is 20'. 17". 28"', thereforethelength of the fydereal year is 365 days, 6 hours, 9'. 17". 28"'. And the Suns mean motion for a day 59'. 8"'. 19"'. 4a'"-47^- 21"'' 29^"- 23"". converted into time is 00. 03'. 56", 33"'. 18'"-55''-9^'- ^jvii. ^.yviii. wMch being added to the ^quinofti- at day, 24 hours, giveththe mean fblar day, 24 hours. 3.'56". 33"'. 18'"- 55. 9. 23. 57. 10. And the daily motion of the fixed Stars, being converted into time is 32'"- 51^- 15^'* 9*". 14"". 24'*- and therefore the ^quinodial day being 24 hours, the lydereal day is 24hours, 00'. 00". 00"'. 32'^- 51. 15. 9. 14. 24. . 11. Hence to find the pr^cefiion of the ^qui- jioftial Points, or longitude of any fixed Scar, you mull add or fubtra(fl the motion thereof, from the time of the obfervation, to the time given, to or from the place given by oblervation, and you have your defire. Example. The place of the firft Star in A- ries found by Tycho'm the year 1601 current, was in T. 27'^- 37'. 00. and I would know the place thereof in the beginning of the Chriftian ty£ra. ft to Ifiic kpl Her, is I Qir 12. ffithtli ftiai], h} kind to 1x2 Ik Si kof kk to Sdtonomp* 22<1-13'.2o'' The motion of the fixed Stars for i 6qo years, Which being dedufted from ? tile place found by obferv. j The remainer. is the place thereof in the beginning of the Chriftian iy£ra. 27. 37. 00 5.23I.40 12. Having thus found the ©middle moti- on, the motion of the u4phelion and fixed Stars, with their places, in the beginning of the Chri- ftian r^ra-, we will now fet down the num- bers here exhibited Mra Chrifii. Mr. Wing from the like obfervations, takes the © motion to be as followeth. The © mean Longitude Place of Aphelion The Anom^y 9.8.00.51 2.8.20.05 c6.29.40.28 The which in decimal Numbers are The © mean Longitude Place of the Aphelion The Anomaly 77.22460.864.19 18.98171.29629 58.24289.56790 The mean motions for one year. The ® mean Longitude 99.933^4-37S^3-34 The Aphelion 00.00475.04447.05 The © mean Anomaly 99.92889.33116.29 The ® mean motions for one day. 37^ aiti 3[titroDU(tton The © mean Longitude 00.27379.08048.11 The Afhelion 00.00001.30149.17 The mean Anomaly 00.27377.77898.94 And according to thefemeafures are theTa- bles made fliewing the © mean Longitude and Anomaly, for Years, Months, Days and Hours. CHAP. XII. Tlie Sms meAtt motions othcrvaife fiated. SOme there are in our prefentage, that will not allow the Aphelion to have any motion, or alteration, but what proceeds from the mo- tion of the fixed Stars, the which as hath been fliewed, domove5Gfecondsinayear, and hence the place of the firft Star in Aries, in the begin- ningofthe Chrillian iy£ra was found to be r. 5.23''-40. Now then, if from the place of the Aphelion Anno Chrifii. 1652 aswaslhewed in the tenth Chapter^ deg. 96. 33'. 39. we deduftthemo- tion of the fcced Stars for that time. 28. 19. 12^.' theremainer 68. 14.271$the conftant place of the Aphelion J but Mr. Street in his Afirono- mia Carolina Page 23, makes the conftant place of the Aphelion to be 68'i' 20.00, and the ©ex- centricity 1732. And from the obfervation of Tycho 1590 March the eleventh, in the Meridian of V- ranihiirgy but reduced to the Meridian of London. March the tenth, hour 23. 2'. He determines to siafonomr* 377 determines the Earths mean Anomaly thus. The place of the Sun obferved r. o. 3 3.19 The prasceflion of Equinox 0.27.27.22 The Earths Sydereal Longitude 5-o3.05.57 The place of the Afhelion SubtraA 8.08.20.00 The Earths true Anomaly 8.24.45.57 Equation Subtradl 1.58.47 Theremainer isthe ?' 2,. ^imate M. Anom. 5 9.2.47.10 vfyxrfwowanfwcringthereto add. 1.58.27 The Earths true Anomaly 8.24.45.37 The place of the Aphelion 8.08.20.00 PrEceffion of the iEquinoX ca7.27.22 Place of the Sun r.00.32.59 But the place by obfervation r.00.33.15 The difference is 001.001.20 Which being added ? 2 , ^ tothemeanAnom. The mean Anomaly is 8.22.47.30 The abfolute Equation 1.58.27 The true Anomaly 8,24.45.57 Agreeing with obfervation. And fb the mean Anomaly ^ra ChriJH is 23. 19. 56. But Mr. Flamfied according to whofe rfieafure the enfuing Tables are compo- fed, takes the mean Anomaly i^raChrifli. to be 6. 24. 07. 091. The place of the Aphelion to be 8, 08. 23. 50. And fb the PrsEcellion of the iEquinox and Aphelion in the beginning of the 3(tt Unttobucttoit ^cChriftian B, 13. 47. 30. in decimal Numbers. cy£ra Chrifii. lltifS The Suns mean Anomaly 56.6^976.35185 jiKan The Suns and Prsc. ^q. 20.49768.51851 ;je3isf< The © mean motions for one Tedf- ilfuht The © mean Longitude 99-95364.37563.34 ® The PrasceflionofiEquin. .00385.80246.p1 Jm The © mean Anom^. 9p-p2978.57316.43 Day, Horn "The © mean Motions for one Day. The © mean Longitude po.27379.08048.11 The Prascefiion of^qui. 00.00001.056pp.30 The® mean Anom. 00.27378.02348.S1 ijtkl ■ CHAP. XIII, t/ovo to Calculate the Suns true place by either of the Tables of middle motion. . 'as T 7T TRite out the £pocha next before the V V given time, and feverally under that Lj' let the motions belonging to the years, months and days compleat, to the hours, fcruples, cur- rent every one under his like f only remember that in the BilTextile years after the end of fre- bruary the days muft be increafed by an unite) then adding all together, the fum (hall be the© mean motion for the time given. Example to siaronomp. JExamfU. Let the given time be Anno Chrifti 1672. f r- , iir»4r723.hours II. 34'. 54''. by the Tables of the © mean Longitude and Anomaly, the num- bersareasfollowcth. y. H ' .o:!4Si:' The Epeeha 1660 c Years 11 iiSi' January If Day. 23 f Hours 11 ' 34 -V 54 M. Longitud.M. AnoraaL- 80.67440. 53-79815 99.81766. 5'9'7<5S26 0848751. 0848711 06.29718. 06.29688 00.12548. 0C.12548 ■ 00.00646. 00.00646 00.0G017. 00.00017 kms its^ 95.408S6. 68.47951 : By the Tabks jof the Suns j mean Anomaly afld prxceiTion, of the Equinox, the numbers are thefe. Anomaly. Pr«ce.iEqui. 53.767^1. 26.90200 The Efocha 1660 sa Years?" M January Days 23 I. Hours 11 34' 54" 99.77520. 8.48718. 6.29694. c. 12 548. .00646. .00035. 00.04243 00032 00024 26.94499 68.45882 95.40381 Ei)|, ® mean Anomaly 68.45882 There Hti 3ntcotiiet(ott There is no great difference between the0 0^, mean Longitude and Anomaly found by the Ta-, bles of mean Longitude and Anomaly, and that found by the Tables of mean Arramaly and Pre- cefljon of the/Equinox. ^ The method of finding the Elliptical ^uation is the fame in both, we will inilance in the latter only, in wt^the® mean Anomaly is Degrees 68. 45882. And the kjyf precelfion of ^quin. deg. 26. 94499. But bocaufe there is no Canon of Sines andTitiF . ^ gents as yet publilhed, fuitable to this divifion ^ of the Circle jnto an 100 deg. or parts :We mull . firft convert the ® mean Anomaly, and prec. of'® qf the ^uin, given, into the degrees and parts the common Circle: And this may be done either into degrees and decimal parts of a degree, or into deg. and minutes: if it were required to ^ bedpne into degrees and minutes, the Table here exhibited for that purpofe will fer ve the turn,but if ;it be required to.be done into degrees and de- ciraal parts, Ijudgethefollowing method tO'be more convenient. Multiply the degrees and parts given by 3 6,the Prc^d, if you cut off one figure more towards ,, t^'tight hand: than there are.parts in the nunl- 7 bef given, IhpU be the degrees and parts of the common Circle. icaialp Anomalj.68v4.5,582.Pr£c.iEquinox. 26.9440s 36 ■oithe • 41075292 20537646 16166994 8983497= Anom. 246.451752 PrxvTEq.97.001964 And to ^Stotiomp* 38 , And if you multiply the parts of thefe Pro- duds, you will convert them into minutes, [j Other wife tlius.Multiply the degrees and parts loi^ given by 6 continually, the lecond Produd:,if you |U-cut off one figure more towards the right hand than are parts in the number given, lhali be the . degrees and parts of the common Circle. The ■ "1 third Produd of the parts only lhail give minutes, the fourth leconds, and lb forward as far as you ...apleafe. Example. fit it ®Mean Anom.68.45S82 6 Pf«c./Eq.a6.94499 6 p'tiii f ikk (iini|l)i <■ 41075292 246.431752 6 16166994 97.001964 6 27.10512 6 o. 11784 6 6.3072 7.0704 iGIS Miii iSllSlijff Aid thus the mean Anomu is deg. 246.451742 0t27'.o6.ThePrec. £q. 97.00196^ oroo'.oj". Hence to find the Elli[^icjil yEqiiation in degrees and decimal parts: InFig. S.wehavegiveninthe right lined plain Triangle £ AfHy the fides A/£, and M//,and the Angle £ AfH, 66.451742. the ex'cefs of the mean Anomaly a]bove a Se^circle, to find the Angle M£ H. 4' The 382 mn 3[nt(ot>u(tton The fide ioocoo The fide 34^3 Zctu. 203468 Co, dr. 4.69150389 me Hi 0.917042 the double whereof is the Angle MB H. 1.834084 or EIliptick\^qua- tion fought, which being added to the mean A- ; nomaly and prajceilion of the TEquinox, becaufe ' the Anomaly is more than a Semicircle, the fame is the Suns true place. The® mean Anomaly 246.451742 The Prajceifion of the iEquinoX 9 7.001964 Elliptick iEquation 1.834084 The Suns true place. 345.287790 But becaufe the Elliptick Equation thus found doth not ib exadly agree to obfervation as is de- fired, BdUaldus in Chap, 3.ofhis'3ookentitulcd jifirommia Philolatcd fundament a clarius explicata, Printed at 1657. (hews how to corted the feme by an Angle applied to the Focm of middle motion, fubtended by the part of the ordinate line, intercepted between the EUipfis and the Cir- cle circumfcribing it. This Mr. Street maketh uleof in his AfironomiaCarolinayTind this 1 thought not amiis to add here. In Fig. 9. let ABCPDF befiippoled anE/-^ lipfis, and the Circle defcribed upon the extremes of the tranfVerfe Diameter, and the Or- dinates KN and OB extended to G and M\n Xcru. i9<5S32 :fiZangle.5 6.774129 f J Xangle.5 5.857087 10.18374097 10.16867813 5.29343327 thsj to Hflronomp; 3 9? thqPeripheryoftheCircle: thenbytheiiofthe firft of Afollomus. XN.GX'-'.OB tmg. OEB.OMtmg. OEM. And the Angle bEM—OEB=BEM=-ETYy the variation to be deduded from the Elliptick Equation ETH, the Remainer is theabfolute iEquation T T S in the firft Quadrant. In the fecond and third Quadrants, the variati- on or difference between the mean and correded Anomaly, muft be added to theEIliptick iEqua- tion, to find the true and abfolute/Equation. For XN.XG.Ql^.tmg.QEj^. the comf.m.j4nom. QR.t.QER. and the Angle f^ER—ECO is the va- nation, and ECO-\~ECH=OCH is the abfolute Equation fought in the fecond Quadrant. Again, XN.XG: :a D,rat2g.aED. a b,tang, aEB. hndaEB—aED~DEf thcvariation=£fC> and EfO~\^EfH—OfH the abfolute iEquation fought b the third Quadrant. Laftly, in the fourth Quadrant of meanAno.' maly it is. XN.XG: :ch.tang.eEH.eg.ta»g^tFg. and hEg is the variation; And EFH~EFy=TFy the abfolute Equation fought in the fourth Quadrant. And to find XN the conjugate Semi-diameter, in the right angled Triangle ENX, we have gi- yen, EN=JX and£T the fembdiftanceofthe umbilick points. And Mr. in Chap. 19. of his Arithm.Logar. hath (hewed,that the half Sum of theLogarith. of the fumand difference of the Hypotenule, and the given leg. fliall be the Loga- ritli. of the other leg. Jt C c Now 3^4 3[nttoDttction Now then El1 kio EiK;,;. ^ Kofoli, i 'moiiol ckc to 3Iftroitomr. ' vation is in £3 25. 6'. 54. in the fecond T 14- .50. 9. in the thirds 4. 20. 20. Now then in Fig. 10. let the Circle iSHDCFE denote the Moons Equant T the Center of the Earth, the Semidiameters TD, TE and TF the apparent places of the Moon, in the firft, fecond and third obfervations, Cthe Center of the Exeentrick, CD, CE and CF the Lines of middle motion. 2^- 20' iUk, fflCB ibki rsaaJl lOKSli From the firfl: obfervation 2 to the lecond there are j" The true motion ofthe Moon is deg. 169.43.15" The motion of the ^jpo^ 50. 00. 14 mainer is the Arch FE 3 And deduding the Angle DCFp from the Angle Z)CE, the re-> ^6. 55. 45 mainer is the Angle FCE j Suppofe ID. cocooGoo the Logarithm of DC, continue EC to FT, and with the other fight Lines compleat the Diagram. 1. In the Triangle DCH we have given the Angle DCH 26. 13. 15. the complement of DCF 93. 4(5. 45 to a Semicircle. The Angle DHC 50. 02. 57. The half of the Arch DF and the fide CD looooco. To find CH. As the Sine of DHC $0. 02. 57 9.88456640 To the Side DC, fo the Sine 7 o . of HZ)C43. 48. j- TotheSideCy^ ' 9*95507557 2 In the* Triangle F/CE we have given CH as before, the Angle CHE 25. 00. 07. The half of the Arch F£, the Angle F/C£ 133. 04. 17 the complement of FCE, and by confequence the Angle CEH 21. 55. 36 To find the Side CE. • ■ to Sdtonomr* 38p As the Sine of CEi/21. 55. 3^ 9.57219707 TotheSideC// ^9*95507557 So is the Sine of CHE 25.00.07 9.62 597986 To the Sine CE 19.58105543 10.00885836 3. In the Triangle JDCE, we have given DC. CE and the Angle DCE 140. 42. 28. whofe complement 39. 17. 32 is the Suram of the An- gles, to find the Angle CED and DE, As the greater Side CE Is to the lefler Side DC So is the Radius To the tang, of 44. 24. 54 10.00885836 lO.COOOOOOO 10.00000000 19.99114x64 Which fubtrafled from 45. 2 the remainer is the half. Difference of the acute angles 35*16. jis the Radius. To the tang, of the com. 35.16 8.01109962 Is to the tang, of the J Z. 19.38.46 9.55265735 To the tang, of J X. 00. 12. 35 7*56375697 , Their Sum 19. 51. 21. istheangle CDE. j Their difference 19. 26. 11. istheangle C£Z>. C c 4 B9^ 3tn JntroDuction As the Sine of CE-D. 19. 26. 11. 9.52216126 Is to the Sine of 1^^.040.42.28. 9 .80159290 0,2^' So is the Side EC; lo.ooocoooo iiistliej To the Side EE. , 10.27943164 4. In the Ifofceles Triangle ETEwe havegi- ven the Side EE, the angle ETE 150. 06. 08 whofe complement 29. 53. 52 istheSummof iitheSi the other two angles, the half whereof is the 474i.( angleTEE 14. 56. 56 whichbeing fiibtrafted itotiie from the angle CEE. 19. 51. 21 the remainer oislki is the angle CET 4.54.25. iOtteS As the Sine of ETE? q, 150.06.08 Co. ar.^ 374 Is to the Sine of EET. 14.56.56 9.41154778 tbeliril So is the Side EE 10.27943164 ruetiK^ To the Side ET ^ 9.99335424 5. In the Triangle CET we have given EC. ET and the angle CET, to find CTD and CT. ; pfee ilie.tqi As the Side DT 9.99 3 3 5424 He j [ Is to the Side EC xo.oocooooo :j)iinf[ SoistheE.'jT lo.ocoooooo pijfjj To the tang, of 26. i3 10.00664576 Herere Pedu6t 45. Radius. , mis, Is to ^Qvonomr- 3pt Is to the Sine of the remainer? o. 26. 18. 5 So is the tang, of the i Z angle 87. 32. 57 To thetang. ^ X angle 10.08.04 9.25223668 Their Summ 97.41.01 is the angle CT!D } 7.88368672 11.36854996 As the Sine of CTD. 9 7.41.01. Co. Is to the Side DC So is the Sine of CDT* 4. To the Side CT 54. 25 The place oftheMoonin? the firft Oblervation 3 The true Anomaly CTD fub. The place of the Apogaon a place in the firft: Obfervation TheiEquation CDT Add. The in that time, deg. ^ 9-4* The true motion of the (t 119.43.54 Their Summ is in Fig. 11. ? The angle ^ 165.03.35 i Therefore in the oblique angled Spherical TriangleDP5 we have given BP. 89. 13. 45 the complement of the Moons Latitude in the firfl: oblervation 2. PD 89. 27. 50 the comple- ment of the Moons Latitude in the fecondobfer- vation,.and the angle jDP5 165. 03. 35,whofc complemenr to a Semicircle is Z)PF 14. 56. 25. The angle PBD is required. I. Proportion. stkcS ;iieCoi As the Cotangent of PD 89.27.50 9.97114485 Is to the Radius lo-oooooooo So is theCofine of DPP 14.56.25 9.98506483 hefa&i To thetang. ofPF 89.26.42 12.0119x998 ftSobt BP 89.13.45 rord TheirZ is FPP 178.40.27. whole complement Is the ArchFD 1.19.3 3. 2. Proportion. As the Sineof FP 89.26.42.0. <«r. 0.00002037 Is to the Cotang.of DPF 14.56.25 10.57376158 So is the Sine of FD 1.19.33. 8.36418419 To the Cotang.ofFCD 85.02.56 8.93796614 TGD—PBD inquired. And to Sftconomp. gps And in the right angled Spherical Triangle right angled at we have given uiB 046'. 15" the Latitude in the firft obfervation, and the Angle jiB^—PBD 85. 02. 56. to find ASl the Longitude of the Moon from the afcending Node. AstheCot. of 85.02.56 8.93796614 \stO Radins 10.00000000 Sois the Sine of ^5 o.46'.i5" 8.12882290 To the tang, of 8.49.17 9.19085676 2. To fiTid the Angle AfiB. Asthe tang, of 0.46.15 8.12886212 \% to l\iQ Radius lo.oooocooo So is the Sine A^ 8.49.17 9.18569718 TotheCotang. of AS\>B 5-041 11.05682506 The angle of the € orbite with the Ecliptick The firft obferved place of the ([ T. 19.14.18 A^ Subtract 8.49.17 There r efts the true place of the ^n^ deg. 78.82862.6543 20 TheMoonsmean Anomaly,^ The Moons Node Retro- ? , o ^ grade,deg. f 74-69845-d790i® The to siaronomp. The I mean nfdtion for one year. « The Moons mean Lonei-? tude,des. ^ 3S-9400I-4489M The Moons A^ogaon^ deg. 11.29551.125365 The Moons mean Anoma- \ ly,deg. 5 ^'^^4450.322566 The Moons Node Retro- ^ „ grade-, deg. i 05.36900.781604 The ([ mean motion for one day. The Moons mean Longi--i . tude.deg j-o?.66oio.962375 The Moons deg. 00.03094.660620 I The Moons mean Anoma-7 Iy,deg. j-03.62916.302253 i The Moons Node Retro-T grade, deg. S o=-0'47o-9«i94S I And according tothefe meafuresare theTa- I bles made (hewing the Moons mean Longitude, Apogaon, Anomaly, and Node retrograde for Vears, Months, Days and Hours. And hence to compute the Moons true place in j her Orbit, I (hall make u(e of the Method, which Mr. Horron in h.is Pofihumas works laltly pub- lifhed by Mr. Flamfied, in which from the Ra- iolfhin Tables he fets down thefe Dimenli- iOns. Tho 3n 3ntrotuction ifooi The Moons mean Semidiameter dcg. oo. 15'.3o" Her mean diftance in Semid. of "> the Earth Deg. / ' The half whereof deg.^. 5 3.41.*) nj he adds 45 the whole U I W Whofe Artificial cotangent is .9.91000022 V f And the double thereof makes'^ „ • , this ftanding Numb. j" 9-82000044 ,fc C Greateft 6685.44 7 The Moons < Mean 5523.69 S Excentricity cLeafl: 4361.945 And her greateft variation 00. 36'. 27"." ® Thefe things preraifcd his directions for com- ^ puting the Moons place, are as followeth. \ IDf2 CHAP. XVTI. iiKte How to Calculate the Moons true place in her Orbit. mi TO the given time find the true place of the^'^ Sun, or his Longitude from the Vernal ^tquinox, as hath been already ftiewed. 2. From the Tables of the Moons mean moti-;^G ons, write out the Epocha next before the gi- *ili] Yen time, andfeverally under that fetthemoti-jf-^i ons, belonging to the years, months and days 111 compleat, and to the hours and fcruples cur-. rent, every one under his like (only remember "ti that in the Biflextile years, after the end of Fe- h hruary^ the days muft be increafed by one Unite) l^io then adding them all together, the Summ lhall be •: a the to 3pp the Moons mean motions for the time given; But in her Node Retragrade you mult leave out the Radix or firfl; number, and the Summ of the remainer being deduded from tlie Radix, fhail be the mean place of her Node required. 3. Dedud the Moons Jfogaon from the © true' place, the reft is the annual Augment, the tan- gent of vvhofe Complement 180 or 3 60,being ad- ded to the artificial Number given 9.82000044. the Summ fliali be the tangent of an Arch, which being deduded from the faid Complement, gi- Vtth the Apogaon iEquation to be added to the mean Apogaon, in the firft and third (Quadrants of the annual Augment, and Subtraded in the fecond and fourth, their Summ or difference is tie true Apogaon. 4. The true Apagaof/ being Deduded from the >1 mean Longitude gives the Moons mean Anomaly. ' . 5. Double the annual Augment, and to the Coiine thereof add the Logaritlim. of, 1161.7 >• the difference between the Moons mean and ex- tream Excentricity, viz.. 3. 06511268, the Summ ffiall be the Logarithm of a number which being; added to the mean Excentricity, if the double annual Augment be in the firft or fourth quadrants-, or Subtraded from it, ifin the fecond or third quadrants; the Summ or difference lhall be the Moons true Excentricity. , 6. The Moons true Excentricity being takerf for a natural Sine, the Arch anfvvering thereto l^alibethe 1 greateft Phyfical .Equation. 7. To the half of the Moons greateft PhyllcM Equation add 45 deg. the cotagent of the Summ is the artificial Logarithm of tlie Exccntrick. D d To 400 %i\ 3Jntcot)uttioii To the double whereof if you add the tang, of half the mean Anomaly, the Summ fhall be the tangent of an Arch, which being added to half the mean Anomaly, ihali give the Excentrick A- nomaly. 8. To the Logarithm of the Excentrick, add the tangent of half the Excentrick Anomaly, the Summ lhall be the tangent of an Arch, whofe double {hall be the Goequated Anomaly, and the difference between this and the mean Anomaly is the terreftrial Equation, which being added to 54 the Moons mean Longitude, if the mean Anoma- ly be in the firft Semicircle, or Subtradtedfrom it, if in the latter, the Summ or difference lhall be the place of the Moon firft Equated. 9. From the place of the Moon firft Equated, Dedudt the true place of the Sun, and double M the remainer, and to the Sine of the double add 3 of the Sine of the greateft variation o. 36. zj,vtz. S. 02541571, the Summ lliall be the Sine of the iL true variation, atthattime, which being added' to the Moons place firft Equated, when her fingle 03 diftance from the Sun isin the firft or third qua- drants, or Subtraded when in the fecond or fourth, the Summ or difference lhall be the Moons i,\r true place in her Orbit. iCo Example. ilj] tia Letthe given time be Jmo Chrift-i 1672. Feb. ila 23. h. 11. 34'.54" at which time the Suns true ■ ( place ism>£ 15. 29133 and the Moons middle motions are as follovveth. j, « Long. to ^firoitomp; 401 16^0. C Longitude ^ A^og£on fi> Retrograde 13.3^650. 41.78372. 55-85177 II. 02.6603:^ 24.31246. 59.08945 January. 13.46339. 00.95934 .45599 D. 23 84.18252. ,.71177 .33832 H. II 1.67755. .01418 .00674 34' .08641. .00072 .00054 54 .00228. .0C012 .Qooor I Longitude 15.43897. 67.78229 59.89082 95.96094 Thefe Numbers reduced to the Degrees and Parts of the common Circle are for the d meaii Longitude. The 1 Afogaon. The © true place is The d Apogaon fubtra£t. The Annual Augment. The Complement whereof is TheTang.ofdeg.7?8.72462 The Handing Number. TheTang.ofdeg.7 3.20288 I n t r« ..-1 1 -r _ . 4 55. 580292 244. 015956 345.2913?' 244. 01595 101. 27538 78. 72462 10.70033391 9.82000044 10.* 3203343 5 ^ —"O- / J Their difference. 5.5 2174 tiicApogMn Equatioi Mean Apogaon 244.01595 Their difference 23 8.49421 is the true Apogaon, D d 2 Secotfdly: 4^2 M JntroBuctfoti Secondly. ttJtni The J mean Longitude. 55;. 58029 The true Apog£on fubtradt. 238.49421 Reils the d mean ^ww.colreft. 177.08608 j, Or thus. ci.2:s ^275 n The d mean Anomaly in the Tables for the time propounded, will be found to be 67.78221, which converted into the deg. and parts of jjgjjjj the common Circle is i?!* 5^434 To which the Ayoo^n Equation? ^2174, of being Jtdded i ^ ^ie Their Sum is the mean Anem.QontCi. 177.08608 And hence it appears that working by the mean Anomaly inftead of the mean Longitude, the true Apog£on Equation muft be added to the mean Anomaly , in the fccond and fourth Qua- drants of the d Annual Augment, and fub- tracted from it in the firlt and third. a&fej 2geiit( Tnirdly. . * jaooia The Annual Augment, 101.27558 being dou- bled is deg. 202. 55076, the Cofine of whole excefs above 180 , that is the Cofme ofi(fiil)[ 22.54076 is 9.96545577 5ml The Logarithm of 1161.75 5. 06511268.^ The Logarithm of 1072. 92 3.03056845 " The d meanExcentr,5523.69 Their difference 4450. 77 is the d true Ex- . ... - ccntficiay . toHllronomr. 403 > centricity. \^/hich taken as a natural Sine, the Arch anfwering thereunto Deg. 2.55094 is the d greateftPhyfical Equation. r ,, • * FoHrthly. -To the half of the Phyfical Equation, deg. 01. 27547 add 45 degrees, the Sum is deg. 46.27547 , the Cotangent whereof; viz.. 9.98080957 is the Logarithm of the Excen- trick , the double of which Logarithm is 9.9^161914 Tangent i Anomaly correded? 88.54304 J"-59455229 Tang, of deg. 88.40849 _ 11.55620143 Their Sum deg. 176.9515 3 is the excentrick A- nomaly. Fifthly. The Logarithm of the Excen-7 « trick is f 900SC957 Tang.jexcent.Anom. 88.475765 11. 57505878 Tangent of deg. 88.407268 11.55586835 Thedortlewhereof.76.8,4S36};.^J^=~=^^^^^ M. Anomaly corred. 177.086080 Their difference 0.271544 Equati- ' -'^^3 on loughtto The Remainer S5-3o8748}fcJ^^„£= Dd Sfxfhly. 404 M 3IntroD(icti'on Sixthly. From the place of the , ^ ^ ^t.£r,Chrifii. }77-45>739370i'5 Which being dednded from-> the (L mean longitude, the >56.8114797531 remainer is the Moons mean-' diftance from the Sun, in the beginning of the Chriftian iy£ra. In like manner the Moons mean diftance from the Sun in a year or a day is thus found. From the ^ mean Longitude. 35.940014489J Moons diftance from the ©. 36.00637073 31 * © Anomaly for a year. Prxcefhon of the Equinox. Their Sum fubtrad. 99.9297857316 0038580246 £?9.93 3 6437562 Moons 4o8 Hn 3fittroliuctioit Moons dillaace from the Sun in a days time. © mean Anomaly. _ 27378.02348 Prxceflion of the ^Equinox. i .0569^ Their Sum fubtrad. _ 27379.08047 From the d mean Longitude. o3.<56oio.95287 d Daily motion from the ©. 03.38631.88240 And accqrding to thefe meafures are the Ta- bles made,(hewing the Moons mean motion from the Sun, by which the mean conjundbion of the © and Moon may be thus computed. To the given year and Month gather the mid- die motions of the Moon from the Sun, and take the complement thereof to a whole Circle,from which fubtraiting continually the neareft leiler middle motions, the day, hour, and minute en- fuing thereto is the mean time of the Conjun- dion. sq™ Example, Chrifii 1676. I would know the time of the mean Conjundion or New Moon in OStober. o Epocha 1660 32.697283 .jpjgjj Years Compl. 15. 50.254463 Septemb. Compl. 24.465038 | I. day for Leap-year. 03.386318 , Their Sum is the Moons motions q , ' ° ftomthe©. j-io.3o3!o2 Complement to a whole Circle. 89.196898 est Days 26 Subtrad. 88.044289 Honrs S.rubftiaft. ' 1.15^^09® I-I 20772 Minute? to liftronomp. Minutes lo Subtract. The Remainer giveth 8". , 4op 0.013837 0.023516 —— .00321 Therefore the mean Conjunction in O^eber, 1676. was the 26 day, 10 min. 8 feconds after 8 at night. ' And to find the mean oppofition. To the com- plemenf of the middle motion, add a femicircle, and then fubtraCt the neareft lefler middle moti- ons as before, the day, hour, and minute enfuing thereto, fliall be the mean oppofition required. Example, JnmChrifii, 16-] 6. I define toknow the mean oppofition in November, Epocha 1660 Years Compl. 15 Oblober Compl. I day for Leap-year. The € mean motion from the ©. Complement to a whole Circle. To which add a Semicircle. The Sum is Days 10 fubtraCt. Hours 2. Minutes 32, The Remainer. giveth 9 feconds. 32.697283 50.254463 29.440922 03.386318 15.778986 84.221014 50. 34.221014 33.863188 .357S26 .282193 .075633 .075251 .000382 Therefore ^ ]^nttot)uct{o(t to ^Stonom^ Therefore the Full Moon or mean Oppofition of the Sun and Moon was November tne icth. Hours 2,32' c$>". The like may be done for a- ny other. And here I fhould proceed to fhew the manner of finding the true Conjundlion or Oppofition of the Sun and Moon, but there being no decimal Canon yet extant, fuitable to the Tables of mid- die motions here exhibited, Ichule rather to re- fer my Reader to Mr. Street'^ J^ronomia Carolina^ for inftruftions in that particular, and what elfc fliall be found wanting in this Subject. 4ii A N INTRODUCTION T O Geography, O R, The Fourth Part of COSMOGRAPHY. CHAP. I. Of the N^ature and Divifionof Geographj. Geography is a science concern- ing the meafure and diftindion of the Earthly Globe, as it is a Spheri- cal Body compofed of Earth and Wa- ter, for that both thefe do together make but one Globe. v 2. And ,p 412 3lntroliucti'on 2. And hence the parts of Geography are two^ the one concerns the Earthy part, and the other the Water, 3. The Earthy part of this Globe is common- ly divided into Continents and Iflands. 4. A Continent is a great quantity of l.and not feparated by any Sea from the reft of the World, as the whole Continent of Jfiay and Africa, or the Continents of France, Spain, and Germany. 5. An Ifland is a part of Earth environed round about with feme Sea or other-, as the Iflc of Eri- tain with the Ocean, the Hleof Sicily with the Me- diterranean, and therefore in Latine it is called Infda , becaufe it is fcituate in Salo , in the Sea. 6. Both thefe are lubdivided into Peninfula > IfihmHS, Vromontormm. 7. Peninfala, cjuafi fene infula, is a tracH; of land which being almoft encompafled round by water, is joyned to the main land by fome little part of Earth. 8. Jfihmus is that narrow neck of Land which joyneth the Peninfula to the Continent. 9. Promontorium is a high mountain which Ihooteth it felf into the Sea, the outmofl: end whereof is called a Cape or Foreland, as the Cape of Good Hope in Africkz I o. The Watry part of this Globe may be alfo diftinguilhed by diverfe Names, as Seas, Rivers, Ponds, Lakes, and luchlike. 11. And this Terreftrial Globe may be meafu- red either in whole, or in any particular pjurt. 12. The meafiire of this Earthly Globe in whole, is either in relpedt of its circumference, ro its bulk and thicknefs. 13. For rjSft lifeia, ereDK); to 415 13. For the meafuring of the Earths circumfe- rence, it is fuppofcd to be compafled with a great Circle, and this Circle in imitation of Aitrono- mers, is divided into 360 degrees or parts, and each degree is fuppofed to be equal to 15 com- mon German miles,or <5 0 miles with us in 'Englmdf and hence the circumference of the Earth is found, by multiplying 360 by 15, to be 5400 German miles, or multiplying 360 by r5o, the circumfe- rence is2i6ooEngliih miles. 14. The circumference of the Earth being thus obtained, the Diameter may be found by rhe common proportion between the Circumference and the Diameter of a Circle, the which accord- Ing to Archimedes is as 22 to 7, or according to VanCulen as 1 to 3. 14159. and to bring an U- nite in the firft place. As the circumference 3.14159. is to i theDi. ameter, fo is 1 the circumference to 318308 the Diameter, which being multiplied by 5400, the Earths Diameter will be found to be 17x1? Ger- man miles and 8632 parts, but being multiplied by 21600, the Diameter will be 6875 Ehglilh miles, and parts 4528. 15. The meaiure of the Earth being thus found inrefped; of its whole circumference andDiame- ter, that which is next to be confidered, is the diftindtion of it into convenient fpaces. 16. And this is either Primary or Secon- dary. 17. The Primary diflindtion of the Earthly Globe into convenient fpaces, is by Circles con- fidered abfolutely in themfelves, dividing the Globe into feveral parts without any reference to one another. 'i8. And ,u'' ^ f 414 2fn ItitcoDuctioit 18. And thefe as in the Celeftial Globe, are 0) ^ either great or fmall. trk i 19. The Great Circles are fuch, as divide the whole,Globe into, two Equal Parts or Spaces, idell and are cither more or kfs Principal. ' jwith 20. The principal great Circles are fuch as i.toi have a principal or chief ufe in dividing the jittlie Globe into its Parts, and arc either fixed or iDegre moveable. .:oftt 21. There is but one fixed or immutable ilOc great Circle in the Tcrreltrial Globe, keeping one and the fame place, and without any xVIulti- plication,by reafon of the variety of places upon eft Par ^ the Earth, and this, is called the t^qiunor. df toi 22. The z/£qH:itor is a great Circle going round about the Terreftrial Globe, from Eafl to i Eirt] Weft. The ufe is to ihew the Latitude or Di- Tfw fiance of any Town or Place from thence to- ijitcde wards the or South Pole , and muft be (koK meafured by.theT^grees in the Meridian. 23. The mutaMe great Circles are two, the iiolfj Meridian and the Horizon. . xrwjfj 24. The Meridian Circle, is a great Circle tieiii drawn from Pole to Pole, or through the North ,':ovcro or South Points of every place •, fo that there are j; fj as many Meridians as there are diftinft places upon the Earth 3 but the chief arid firftMeridi- an was by the Greeks,and by the Ancients before them placed in the Fortunate inands,as they were termed of old, from an opinion of fome fingu- lar blelTing imagined by t e Ancients upon the Genmi of thofe Paits; They are now by the Spa- niards called the llles3 whereof and Plir:y out of Juha the African King, findeth to be but fix: but the late Diftovcrers meet with ftvcff, to dpcogjapftp. 415 Qven, \h2X. 'lSf Lancerottaf FortevemHra,Teniriffa^ Gamer At Fierro, Patma, and the Gran Canary ^ which givethname to the reft; for the Scituation of thefe Iflands, they lie not as Ptolemy placed them within one Degree of Longitude, or little lefs, but more fcattering, and lifted up a little above the Tropick of Cancer^ about the thirti- eth Degree of the Northern Latitude, in that part of the Weftern otherwife called the At- lantick Ocean, which Jieth upon the coaft of Africkj, and are therefore by Geographers recko- ned among the African Ifles. This was the far- theft Part of the Earth difcovered towards the Weft to thofe of about Ptolemy\ time : there- fore the great Meridian was fixed there, in the Ifle Hera •, or Jmonea, as then it was called ; now Tenerijf^: and from this Meridian all the Longitudes in the Greek Geography are taken. Our own Geographers, the later efptcially have affefted to tranfplant this great Meridian out of the Canary Ifles into the Jiz.ores, or o- therwife termed the Flemifh Ifles, becaufe feme of them have been famoufly poflefled, and firft difcovered by them. They are now in number nine: Tercera, S. Michael, S.Mary, S. George ^ Gratiofa^ Pico,Fay all, Corvo, F lores, they are fitu- ate in the fame Atlantick Ocean, but North Weft of the Canaries, and bending more upon the Spa- nifh Coaft, under the 3 0 Degree of Latitude or thereabout. Through thefe Ifles the late Geo- graphers will have the great Meridian to pafs, fome»of them in the Ifles Corvo, and Floras, the moft Weftern •, as Johnfon in his lefler Globe of the year 1602. others in S. Michael and S. Mary, the more Eaftcrn of the Az.ores, but Stevmm a E e Dutcif 416;^ 3ltl 3InttoT>ucUon Dutch Geographer inclines much to the bring- ^ ing back the great Meridian to the Fortunate lilands, more particularly to the Peak a Moun- tain fo called from the Iharpnefs in the top, in the Ifle Tmenff, which is believed to be the high- dl Mountain in the World ^ therefore the fame !-■ Johnfon in his greateft Globe of the year 1616, hath drawn the great Meridian in that place, and ^ it were to be wiflied, that this might be made the - ^ common and unchangeable pradlice. 25. The Horizon is a great Circle, defigning fo great a Part of the Earth, as a quick fight can ^ difcern in an open field •, it is twofold Rational and Senfible. 3^' ^ 16. The Rational Horizon is that which is Mtr fuppofed to pals through the Center of the j; A Earth, and is reprefented by the wooden Circle »Uiie in the Frame, as well of^e Celeftial, as the Ter- Line: reftrial Globe, this Rational Horizon belongcth «n, more to Aftrouomy than Geography. 27. The Senfible Horizon is that before de- Atf fined, the ufe of it is to difcern the divers rifings Idingt and fettings of the Stars, in divers places of the Earth, and why the days are fometiraes longer, and fometimeslhorter. 28. The great but lefs principal Circle upon the Terreftrial Globe is the Zodiack, in which the Sun doth always move. This Circle is de- , fcribed upon Globes and Maps for ornament fake, and to difcover under what part of the Zo- Uyi diack the feveral Nations lie. Iti 29. The leifer Circles are thofe which do not m divide the Terreftial Globe into two equal, but stto into two unequal Parts, and thefe by a general :(to name are called Parallels, or Circles ^quidiftant - from to 417 from the Equinodial •, of which as many may be drawn, as there can Meridians, namely 180 if but to each degree, but they are ufually dfawn to every ten Degrees in each Quadrant frem the t/Equator to the Poles. 30. Thefe Parallels are not df the fame Magni- tude, but are lefs and lefs as they are nearer and I nearer to each Pole: and their ufe is to diitin- guifh the Zones, Climates and Latitudes of all Countries,w ith the length of the Day and Night in any Part of the World. 31. Again, a Parallel is either named or un- named. 32. An unnamed Parallel is that which is drawn with finall black Circular Lines. 33. A named Parallel is that which is drawn upon the Globe with a more full ruddy andcircu- lar Line; fuch as are the Tropicks of Canser and Capricorn , with the Ardick and Antardick Cir- ties,of which having fpoken before in the general defcription of the Globe, there is no need of adding more concerning them now. CHAP. II. Of the Difl^inElionor Dimenfon of the Earthly Glohe by Zories and Climates. HAving fliewed the primary diftindion of the Globe into convenient fpaces by Cir- cles confidered .abfolutely in therafelves, we come now to confider the fecondary Dimenfion or 4i- ftindioii of convenient fpaces in the Globe, by the fa pe Circles compared with one another, £e 2 and ■418 Hn 5ntroliuctton and by the fpaces contained between thofe Cir- cles. 2. This fecundary Dimenfion or Diltindion of the terreftial Globe into Parts, is either a Zone or a Clime. 3. A Zone is a fpace of the Terreftial Globe included either between two of the leller nomi- nated Circles, or betw^een one and either Pole. They are in Number five, one over hot, two over cold, and two temperate. 4. The over hot or Torrid Zone, is between the two Tropicks, continually fcorched with the prefence of the Sun. 5. The two over cold or Frigid Zones, are fcituated between the two polar Circles and the very Poles,continually wanting the neighbour- hood of the Sun. 6. The two temperate Zones, are one of them betweerhtbe Tropick oiCancer and the Ar- ilick Circles and the other between the Tropick of Capricorn and the Antardick Circle , enjoy- ning an indiflerency between Heat and Cold •, fo that the parts next the Torrid Zone are the hotter, and the parts next the Frigid Zone are the Colder. 7. The Inhabitants of thefe Zowfv, in refped of thediverfity oftheir noon Shadows are divi- ded into three kinds, AmphifcU, HeterofcH and Penfcii. Thofe that inhabit between the two Tropicks are called AmphifcH, becaufe that their noon Shadows aredivcrfly caft, fbmetimes towards the South as when the Sun is more North- ward than their vertical point, and fbmetimes towards the North, as when the Sun declines Southward from the Zenith. Thofe to(!5eo8iapb^ 419 Thofe that live between the Tropick of Cmr m and the Ardick Circle or between theTro- pick of CapricoVn and the Antardick Circle, are ailed Heterofcii^hscanih theShado ws at noon are call one only way, and that either North or South. They that inhabit Northward of the Tropick of Cancer have their Shadows always towards the North, and they that inhabit South- ward of the Tropick oi Capricorn, have their noon Shadows always towards the South. Thofe that inhabit between the Poles and the Ardick or Antardick Circles are called PerifcH, becaufe that their Gnomons do call their Sha- dows circulary, and the reafon hereof is, for that the Sun is carried round about above their Hori- ixm in his whole diurnal revolution. 8. The nextfecundary Dimenfionor diftindi- on of the earthy Globe into convenient parts or fpaces, is by Climes. 9. And a Clime or Climate is a Ipace of Earth conteined between three Paralells, the middle- moll whereof divideth it into two equal parts, ferving for the fettingout the length and Ihort-' nefs of the days in every Country. 10. Thefe Climates and the Parallels by which they are conteined are none of them of equal quantity, for the firft Clime as alfothe Pa- rallel beginning at the Equator is larger than the fecond, and the fecond is likewile greater than the third. 11. The Antients reckoned but feven Cli- mates at the firft, to which Number there were afterward added two more, fo that in die firft of thefe Numbers were comprehended fourteen parallels, but in the latter eighteen. E e 3 12. Ptolemy '420 Untroljaction 12. Ptolemy accounted the Paralells 38 each way from the Equator, that is 38 towards the 0 North, and as many towards the South, 24 of ] which he reckoned by the difference of one quar- ter of an hour, 4 by the difference of half an ^ hour, 4 by an whole hours difference, and 6 by p Months difference, but now the parallels being reckoned by the difference of a quarter of an hour, the Climates are 24 in Number till you come to the Latitude of 66 degrees 31 Minutes, to which are afterwards added 6 Climates more unto the Pole it felf, where the Artificial day is 6 Months in length. 13. The diflances of all both Climates and Parallels, together with their Latitudes from the iEquator, and difference of the quantity of the longeft days, are here fully expreft in the Table following. to J Table of the Climates belonging to the three forts of Inhabitants. ERabitantsbc- Iqpging to the feveral Climes Bea of the Clime I • .ii y ul'! '.Jl .'■.S! '1^ '42 s 3ltt 3fnttot)ttctioti «it'. r r 111, !. I r a iilii -'m'- [ ■ ■■ >i'' ''i": ■ii". ' 1'.'.''' =11.; ,■ ( fi'm sm mp': •}*: r* ' i-Viij! iS't'? Hetsrofcli Clinics Paralells 1 Length of the Days Poles Ele- vation Breadth of the Clime 10 20 21 17. 00 17. IS 54- 29 55- 34 2. 17 II 22 *3 17. 30 17. 45 S6. 37 57- 34 2. 0 ii »4 25 13. 00 18. ly $8. 26 Sp. 14 I. 40 >3 25 27 18. 30 18. 45 79. 79 5o. 40 I. 25 H 28 2p 19. CO 19. IS 5i. 18 5i. S3 I. 13 ij 30 3* 19. 30 19. 4S 52. 2S 1^2. S4 I. 0 16 32 33 20. 00 20. 15 53. 22^ 53. 45 0. 52 17 34 3? 30. 30 20. 4S 54. o5 54. 30 0. 44 i8 3^ 37 21, 00 21. IS 54. 49 5s. 06 0. 35 38 39 21, 30 21. 4S 55. 21 <^7- 37 ®. 25 V/ 20 40 41 22. 'CO 22. IS 67. 47 '^7- 57 0. 22 21 42 43' 22. 30 22. 45 55. 00 55. 141 0. 17 to i. The Duke- - ' dom to (!5cogjap^p. 455 dom of Holfiein ; containing Wag^evUnd^ Bit- marflj.) Starmaria^ and Holfiein, eipecially fb call- ed. 2. The Kingdom of Denmarhj, comprc- bending both Juitlmds, part of Scandta, and the Hemodes, or Baltick Iflands. 3. The Kingdom of Norway confifting of Norway it (elf, and the Iflands of the Northern Ocean. 29. Swethland is bounded on the Eaft with Mafcovy, on the Weft with the DoferineHilh, which divide it from Norway ; on the -North with the great frozen Ocean Ipoken of before \ on the South with Denmatkj, Liefland, and the Baltick Sea. 30. It isfituatc under the fame Parallels and Degrees with Norway , that is, from the firft ParSlel of the 12 Clime, where the Pole is ele- vated 58 degrees ^6 minutes, as far as to the 71 degree of Latitude, by which account the long- eft day in the Southern Point is but 18 hours, whereas on the fartheft North of all the Coun- trey, they have no Night for almoft three whole Moneths together. 31. The whole Kingdom is divided into two Parts, the one lying on the Eaft, the other on the Weft of the Bay or Gulf of being a large and Ipacious branch of the BaltickSea, extending from the moft Southerly Point of Gothland, as far as to Lapland on the North. According to which Divifion we have the Provinces of i. Goth- land. 2. Sweden lying on the Weft fide of the Gulph. 3. Lapland ihutting it , up upon the North, 4. Bodiaor Bodden. 5. Finlandonthe Eaft fide thereof. 6. The Iflands, where it mingleth with the reft of the Baltkk^St^s. 32. ^njfta is bounded on the Eaft by Tartary, F f 3 on 43 5 KntroDuction on the Weft; with Livonia and Fmland^^rova which it is divided by great mountains and the River Foln y on the North by the frozen Ocean, and fome part of La^Undy and on the South by Li- tuania a Province of the Kingdom oi Poland, and the Grim Tartar inhabiting on the Banks of PaUu Maous y and the Euxine Sea. It ftandeth partly in Unrobe and partly in Afia, the River Pands or Don running through it, the common boundary of thofe great and noted parts of the world. 35. It is fcituate North within the Artick Cir- clefo far, that the longeft day in Summer will be full fix months, whereas the longeft day in the fouthern parts is but \6 hours and an half. 34. It is divided into the Provinces of covy Ipecially lb called. 2. SmkuftOy 3. Mofaij^, 4. Plefco, 5. Novagrod the great, 6. Corelia , •^.Biartnia, 8. Petzjora, ^^.Condora, \o. Obdora, II. jHgriay 12. Severiay Pertnia, 14. Rozjtny x^.WiathkAy \6.CafaUy Aflracan , i3. Novo- gordia inferiour, 10. Phe Mordnits or Morduay 20. WorotimeyZi, Pnba, ii.Wolodomir, Zi.Duinay 24. the Ruflian Iflands. 3 5. Poland is boundedontheEaft VJithRujfia, and the Grim-Tartar,from whom it is parted by the Kivcv Boryfihenes-, on the Weft with Germa- nyy on the North with theSaltick Sea and fome part of Rajfia, on the South with the Carpathian Mountains, which divide it from LLangaryyTran- ftlvania, and Moldavia, It is of figure round in compafs 2600 miles, fcituate under the 8 and 12 Climates, fo that the longeft day in the fouthern parts is but 16 hours, and about 18 hours to (I5eo8?dp^^p. 4^7 hours in the parts moft North. 3 6. The feveral Provinces of which this King- dom doth condil, are i. Livonia^ i. Samogitiay ^.LitHOftiay ^.Volkimaj ^.PoMd, 6. Rttffia mgra^ 7. Majfoviay 8. Podlajfia, 9. Prulfta, 10. Pomerel- lia, II. ipecialiy fo called. 37. is bounded on the Eaft with 7V4«- ftlvania and Walachia, on the Weft with Sterna^ Atfiriaand Moraviayon the North with the C«r- pathian mountains which divide it from Po/W,and on the South with Sclav onlay and fome part of Dacia; it extendeth in length from Prejhurg a- long the Danovo to the borders of Tranfdvania, for the Ipace of 300 Englilh miles, and 190 of the fame miles in breadth. 38. Itlieth in the Northern temperate Zone, betwixt the middle parallels of the 7 and 9 Cli- mates, fo that the longeft Summers day in the Southern parts is but 15 hours and an half, and not above 16 hours in the parts moft North. 40. This Country is commonly divided into the upper Hungary and the lower, the upper ly- ing on the North of the River Danow, the lower lying on the South of that River, comprehend- ing all Pannonia inferior and part of Superior, and is now polfelled by the King of Hungary and the Great Turk, who is Lord of the moft part by Arms and Conqueft. 04- Sclavonia is bounded on xthe Eaft with ServiOy Macedonia and Epirusy from which it is parted by the River Drlnus, and a line drawn from thence unto the Adriatick, on the Weft withCarnlola in Germany, and Ifiria in the Seig- F f 4 niorv 458 JnttoDuction niory of Fenke, trom which laft it is divided by the River Arjia •, on the North with Hungary^ on the South with the Adriatick Sea. 41. It is fcituacc in the Northern temperate Zone, between the middle Parallels of the fixth and feventh Climates, fo that the longefl; day in Summer is about 15 hours and an half. 42. This Country as it came at lafi; to be di- vided? between the Kings of Hungary and the State of Fmke\ isdiftinguilhcdinto i.Windtfch- land, 2. Croatia, ^.Bofnia, 4. Dalmatia, ^.Libar- pia or Cant^do di Zara, and 6. The Sclavonian I- flands. 43. Dacia is bounded on the Eafl; with the Euxine Sea and feme part of Thrace ■, on the Weft mt\\ Hungary Sclavonia-, on the North with Podolia, and fome other members of the Realm of Poland, on the South with the reft of Thrace and Macedonia. 44. It lieth on both fides of the Danow front- ing all along the upper and the lower Hungary, and fome part of Sclavonia; extended from the 7 Climate to the to-, lb that the longeft Summers day in the moft northern parts thereof, is near 17 hours, and in the moft fouthern 15 hours 3 quarters. 45. The feveral Provinces comprehended tin- der the name of Dacia,nr^ i.TranfUvania, 2.M0.I- davia, ■^.Walachia, 'j.Rafcia, ^.Servia, 6. Buka- ria, the firft four in old Dacia, on the North fide of the Danoxvthe two laft in new Dacta, on the South thereof. 45. Greece in the prcfent Latitude and ex- tent thereof, is bounded on the Eaft with the Propontick^ to (!5cosjap^p. 4^p Propontick, Hellcfpont , and iEgean Seas, on the Weft with the Adriatick ^ on the North with Mount H-impu which parteth it from Bulgaria , Servia. and fome part of IllyricHtn \ and on the South with theSea/o»/<««; fo that it is in a man- ner a Veninfula or Demi-Illand,environed on three fides by the Sea, on the fourth only united to the reft of Europe. 46. It is fcituate in the northern temperate Zone, under the fifth and fixth Climates, the longeft day being 15 hours. 47. In this Country formerly fo famous for learning and government, the leveral Provin- Ces are i.PelopomiefHs, Z.Achaia., ^.Epirus, y.Al' hania^ 5. Macedon, 6. Thrace, 7. The Iflands of the Propontick •, 8. Tigean, and 9. The Ionian Seas, and 10. finally the Ifle of G-efc. And thus I have given you a brief deftription of thofe Countries which are comprehended in the Continent of Europe-, the Wands in this part of the world are many; I will mention only fome few, Thefe two in the Britilh and Northern O- cean, known by the names of Great Britain and Ireland are the moft famous, to which may be ad- ■ ded Greenland. In the hiediterranaen Sea you have the Iflands, of Sicilia, Sardinia, Corjica arid Crete , which is now called Candia the greater and the lefs: As for the otlier Iflands belonging to this part of the world, the Reader may exped a more particular defcription from tht m who have or ihall write more birgely of fiik jbjedl: This we deem fufficient for our prefentpurpofe. Let this then fuffice for the ^ fcription of the Jirftparc of the World called En, ope. CHAP. 440 3^n 3ntroDuctiott j,Tfe CHAP. IV. 'soiJte Of Afia. frf. j J(k!l0i Asia is bound on the Weft with the Medite- ranean and ^gsean Seas, the Hellefpoaty iWdiei Propontis, Thracian Bofphoms and the Euxine i®, ^ Sea, the Palus Maotis^ the Rivers Tanais and D«- jtaufeit' t»4, a Line being drawn from the firft of the i>k ^ two faid Rivers unto the other, by all which it is tiioftli( parted from Europe-^ on the North it hath the main Scy thick Ocean; but on theEaftthelndi- lolldiv an Ocean, and Mare del Eur by which it isfepa- p[m rated from America; on the South the Mediter- liecaii ranean, or that part of it, which is called the iltlieiil Carpathian Sea, wafhing the (hoars of Anatoliay lasmadi and the main Southern Ocean, pafTing along the . Indian, Perfian and Arabian Coafts: and finally ^ Tliii on the fbuth-weft, the red Sea or Bay of Ara- hia, by which it is parted from Affrick. Envi- roned on all fides with the Sea, or fome Sea like ;|ePcri Rivers, except a narrow IfihmHs in the fbuth- weft, which Joyns it to Africk^y and the fpace of r,,^ in ground ( whatfoevcr it be ) between Dnina and j Tanaisy on the North-weft which unites it to ^ Earope. ^ ^ 2. It is fituatedEaft and Weft, from the 52 5^ to the 169 degree of Longitude; and North and South from the 82 degree of Latitude to the very Equator;, fome of the Klands only ly- ing on the South of that Circle; (b that the longeft fumniers day in the fouthern parts, is but twelve hours, but in the moft northern parts |^jj, hereof almoft four whole Months together. 3- This to (!5eo8}ap^p. 441 3. This Country hath heretofore been had in Ipecial honour*, i. For the creation of Man, who had his firft making in this part of the World. 2. Becaufe in this part of it flood the Garden of Eden^ which he had for the firft place of his habitation. 3. Becaufe hereflou- rifhed the four firft great Monarchies of the Af^ fyrians, Babylonians, Medes and Perfians. 4. ifecaufeitwastheSceneof almoft all the memo- rable Aftions which are recorded by the pen- men of the Scriptures. 5. Becaufe our Saviour Chrift was borne here, and here wrought his moft divine Miracles, and accomplilhed the great work of our Redemption. 6. And final- ly, becaufe from hence all Nations of the World had their firft beginning, on the difperfion which was made by the Sons of Noah after their vain attempt at Babel. 4. This part of the World for the better un- derftandingofthe Greek and the Roman Stories and the eftate of the Affyrian, Babylonian and the Perfian Monarchies, to which the holy Scrip- tures do fo much relate, we fhall confider as di- vided into the Regions of i. Anatolia or Aftn minor. 2. Cyprus. 3. Syria. 4. Arabia. 5. Chat- dea. 6. Ajfyria. 7. Mefopotamia. 8. Turcoma' ttia. 9. Media. 10. Perfia. ii. Tartaria. 12. China. 13. India, and 14. the Orientallflands. Anatolia or Afia minor. Anatolia or Afia minory is bounded on the Eaft with the River Euphrates^ by which it is par- ted from the greater AJia-y on the Weft with the Thracian Boffhorusj Propontis, Hell?fpont,^nd the 442 ||iu3lntroBuct(oii the^gean Sea, by which it is parted from £«- rope-, onthe North with Pomns Ettxhrns, tailed alfothe black Sea, swdMare Maggiore, and on the South by the Rhodian, Lydian and Pam- philian Seas, fcveral parts of the Mediterranean. So that it isaDemi-Illand or PeninfHk environed on all fides with water,' excepting a fmall Ifihmus or Neck of Land extending from the head, of Euphrates to the Euxine Sea, by which it is joy- ned to the reft of Afia. It reachethfrom the 51 to the 72 degree,of Longitude, and from the 36 to the 45 degree of Latitude, and lyeth almoft in the fame poll- tion with Italy, extending from the middle Pa- rallel of the fourth Clime, to tiie middle Parallel of the llxtli, lb that the longeft fiiramers day in the Southern. Parts, is about 14 hours and a half-, and one hour longer in thofe parts which lie moil; towards the North. The Provinces into which it was divided be- fore the Roman Conqueftwere i.Bithynia. 2. Pontus. Paphlagonia. a^.(jalatia. ^, Cappadocta. 6. Armenia Major Minor. 7. Phrygia minor.-, 8. Phrygia major. 9. AEy/ia the greater and the Ids. 10. Afia fpecially fo called, comprehending (yEolis and Ionia. 11. Lydia. 12. Caria. 13. Ly- cia, 14. Lycaoi'ia. 15. P-iJidia. 16. Pamphylia.i i-l.-Ifanria. 18, Cilicia. 19. The Province of the Afian Hies, whereof the moft principal are I. Tencdos. 2. Chios. 3. Samos: 4. Choos. 5. lea- ria. 6. Lefbos. 7. Patmos. 8. Ciaros. 9 Carpa- thos. ig>. phodes. Cyprus to dpeogjap^p. . 445 Cyprus. Cyprfis is fituated in the Syrian and Cilician Seas, extended in length from Eaft to Weft 200 miles, in breadth 60 the whole compafs reck- oned 550, diftant about 60 miles from the rocky Shores of Olida in ^/la minor, and about one hundred from the main Land of It is fituated under the fourth Climate, fo that the longeft day in Summer is no more than 14 hours and a half. Divided by Ptolemy into the 4 provinces of I. Paphia. 2. Amathafia. 3. Lepathin. Sala-- mine. Syria^ Syria is bounded on the Eaft with the River Euphrates by which it is parted from Afefophta- mia-, on the Weft with the Mediterranean Sea 3 on the North with Cilida and Armenia minor, parted from the laft by mount Taunts-, and on the South with Palefline, and fome parts of A- rabia. The length hereof from Mount Taurus to the Edge of Arabia, is faid to be 525 Miles the breadth from the Mediterranean to the Ri- ver Euphrates 470 Miles, drawing foraewhac near unto a Square. The whole Country was antiently divided in- to thelefix parts. i.Phcenida. i. Palefiine. "^.Sy- mlpccially fo called. 4: Comagena, '^.Palmyrene. and Ctelofyria, or Syria Cava. Arabia 444 3|[ntrotuction Arabia. Ar^ia hath on the Eafl: Chaldaa and the Bay or Gulf of Perfiu\ on the Well Palefiine, fome part of and the whole courfe of the red Sea, on the North the River Euphrates Viith "fi Ibme parts of Syria and Palefiine^ and on the South the main fouthern Ocean. It is in circuit 3i about 4000 Miles, but of fo unequal and hetc- -'ami regeneous Compofition, that no general Cha- liiyt rafter can be given of it, and therefore we mult vd look upon it as it ftands divided into Arabia De- tii. fcrtOy 2. Arabia Pttraa. 3. Arabia Felix and itii 4. The Arabick^lHsnds. Chaldea. fm Chaldea is bounded on the Eaft with Suftana litte a Province of Perfta; on the Weft with Arabia deferta-y on %ht North With Mfopotamia'y and on iori: the South with the Perfian Bay and the reft of ytt, Deferta. ictli 'rrn Aflyria. ii,, Affyria is bounded on the Eaft with Media, from which it is parted by the Mountain called CMthras; on the Weft with Me/opetamia, from jif which it is divided by the River Tygris •, on the ^ j South withS/ A View 'of the more Notable Epchg. Epochie. Years of thejulian Period. Months The Julian Period Creation of the World of the Olympiades The building of Rome Epochte of Nabonajfer The beginning of Melons Cyrck. Thebeginningof the periods of calippns The Death of Alexander the great ufra of the caldees The t/£ra of Dionyfins 1 755 3938 4961 3dd7 4281 4384 4390 4403 4429 Jan. I Jan. I July 8 Ap. 2* Feb. 2^ June 2^ June 28 -Va. i2 0[l. if Mar.2^ The beginning of the Chriftiant/fra falls in the 4713 year of the Julian Period. Years of Chrift Month The Dioclefian i/£ra The Turkifh t/fra or Hegyra The Perfian t^ra from lefdagird The e/frd from the Perfian Sultan 284 622 532 1079 Aug. 29 July 16 June 16 Mar. 14 Days in the Year of Julian Accompt o£.g)^t and Perfian Accompt 1 0 0 0 355 2, 0 I 0 0 0 1^1 0 c 0 2 0 0 0 730 5 0 0 2 0 0 0 730 0 0 0 3 0 c 0 1095 7 5 0 3 0 0 0 1097 0 0 0 4 0 0 0 i4!5i 0 0 0 4 0 0 0 1460 0 0 0 5 0 0 0 1826 z 5 0 5 0 0 0 1825 0 0 0 6 0 0 0 2191 5 0 0 6 0 0 0 2190 0 0 0 7 0 0 0 2576 7 5 0 7 0 0 0 0 0 0 8 0 0 0 z^zz 0 0 0 8 D 0 0 2920 0 0 0 9 0 0 0 3287 z 0 9 0 0 0 3285 0 0 0 10 0 0 0 36J1 5 0 0 10 0 0 0 0 0 0 I 4^o Days in Tulian Days inAigyptian Days in Perfian Months Months Months Comon BilTex rhoth 30 Pharvadln 30 Jamnry 3' 30 paophl do Arepehad do Ftbrtmry $9 60 Athyr 90 Chortat 90 Mirch 90 9^ Chteae 120 Ttrmn 120 ■uipriL 120 121 tybi lyo Mertat 150 iMty 151 172 Michir i8o Sachriur 180 7me 181 182 Phamenoth 210 Machermi 2IO July 212 213 Pharmulbi 240 Apenina. '245 Augii^ 243 244 Pachon 270 ir hil^ J September 273' 274 Payny 300 Aderma 275 oUtber 304 30 J Pphephi 330 Dima 305 November 334 1 335 Me fori 330 Pecbmam 335 December 365 ' 7,66 Epagomeni 3<^5 Aphmder 3'^5 Days in Turkifh or Arabical Years Days in Turkifh Months I 354 21 7442 Muharraii 30 2 709 22 7795 Sapber 59 3 •1063 23 8150 Rabie i. 89 4 •1417 24 8505 Rible 2. 118 5 •1772 25 88S9 Ghmidi i. 148 6 •2125 26 9213 Glmidi 2. 177 7 • 2480 27 9568 Regeb 207 8 •2835 28 9922 Sahjihen 236 9 •3189 29 10275 ^Kamaddan 266 10 •3543 30 0 10631 0 Scheval 295 II *3898 do 0 21262 ° Dul^idati 325 12 •4252 90 0 31893 0 Dulhajati' j 13 •4507 120 0 42524 0 Djilbitts- > "354 14 •4951 lyo 0 53155 0 che true J 15 •5315 180 0 63786^ 0 In anno A-'] 16 •5670 210 0 74417 0 bundanti J '355 17 •6024 240 0 0J048 0 i8 *6378 270 0' 95679 0 19 .6733 300 0 ' 106310, 0 20 •7087 tcljc 3fun'an Calendar, a^c. 4^1 '■4 I '•»•< - i».! 1 .•» ! ■> ;■ 4^2 May Jmi ;i.| 41191^ 8 P 6\i6 5 7 8 9 I!! II ^3 14 16 17 18 19 20 l_ I 211 22I ;:^3 H G 13 A B A ijB C D iiE F G 9 A B I7,C .26,5 ^7! , 2814 F 1:1 h II B C 19 D E 16 G 5 A B 14C 10 E F JG 18 A Mark Evang. Phil.&j3c. .1 E F 12 G I A B 9 C D 17 E 6 F G 14 A 3 B C 11 D E 19 F 8 G 16 A y B '— 13 D 2 E F 10 G A 18 B 7 C p M;E S.Bamaby{ !a !b D E i7|f 6 G A l_ 14'B 3 f ID iilE F SJohn bJ Pct.Ap. '4^4 tiD^e HuU'au Caicnljat, &c. All Saints All Souls A P.Confpir. iijij D IZi4;E 15i :F 1411 2 G Luke Evaog. December iS. Sceph, |S. John Innocents! 'Sylvefter October November Is. Andrew is.Thomas F !G Ciiri. Nat. i|i6'A a J B 915C 42 D 6 loF z ;? 8 18A 9 to tEbe (!5?ie3o?ian Calentar, 46 5 Junuary I 2 XXIX XXVHl 4 XXVll 5 XXVI 6 25.XXV 7 XXIV 8 XXIII 9 XXII 10 XXI 11 XX 12 XIX »3 XVIII 14 XVII 15 XVI 16 XV 17 XIV 18 XIII 19 XII 20 XI 2J X 22 IX 23 VIII 24 VII 2y VI 26 V 27 IV 28 III 29 II I Jl February March A XXIX D p B XXVIII E XXIX E c xxvii F xxviii F D 2^XXVI G XXVII G E xxv.xxiv A xxvi A F XXIII B 2?.XXV B G XXII c XXIV c A XXI D XXIII D B XX E XXII E C XIX F XXI F D XVIII G XX G. E XVII A XIX F K XVI B xviii E G XV C XVII c A xiv XVI . D B XIII E XV E C XII F XIV F D xt G XIII G E x A XII A F IX E XI B G VIII C X C A VII D IX 0 B VI K VIII E C V F VII F D IV G VI G E III A V A F II B IV B G I C III C A li D B I £ C F H h i 4<5^ tlD^e (!5jC80?ian Calentjac, &c. April May Jm I XXIX 2 XXVIIl 2 XXVII J 4 2?.XXVI 5 xxv.xxiv 6 XXIII 7 XXII 8 XXI 9 XX lO XIX II XVIII 12 XVII I? XVI 14 XV M XIV 16 XIII 17 Xll 18 XI 19 X 20 IX 21 VIII 22 VII 23 VI 24 V 25 1'^ 26 i III 27 11 28 I 2^1^ 30 XXIX 3» 1 G XX VIII B XXVII E A XXVII c 2^XXV1 F B XXVI D XXV.XXIV G c 25.XXV E XXIII A D XXIV F xxn B E XXIII G XXI C F XXII A XX D G XXI B XIX E A XX C XVIII F B XIX D XVII G C XVIII E XVI A D XVII F XV B E XVI G XIV C F XV A XIII D G XIV B XII E A XIII C XI F B XII D X G C XI E IX A D X F VIII B E IX G VII F VIII A VI D G VII £ V E A VI C IV F B V D III G C IV E II A D III F I B E II G * C F I A XXIX D G * £ XXVIII E A XXIX G XXVII F XXVIII D tiDlbe d^jcgojfan CalenDat- , 467 ply Augu(l Siptemher, I t 3 4 5 6 7 8 S 10 11 17 13 14 i'S- 16 17 18 19 20 21 22 2? 24 25 26 27 28 29 3® I XXVI 25.XXV XXIV XXIII XXII G A B C D E F G A B C D E F G A ' B C D E F G A B C D E F G A B XXV. XXIV XXIII XXII XXI XX c D E F G A B c D E F G A B C D E F G A B C D E F G A B C D E XXIII XXII XXI XX XIX F G A B c D E F G A B C D E F G A B C D E F G A B C D E F G XXI XX XIX XVIII XVII XIX XVIII XVII XVI XV XVIII XVII XVI XV XIV XVI XV XIV XIII XII XIV XIII XII XI X XIII XII XI X IX XI X IX VIU VII IX VIII VII VI V VIU VII VI V IV VI V IV III II IV III II I III II I XXIX I XXIX XXVIII XXVII 2f.XXVI XXIX XXVIII XXVII XXVI 2S.XXV XXIV XXVIII XXVII 24.XXVI XXV. XXIV XXIII nh 3 4<58 d^^egojtian CalcnDar. il It, ;i tl i I i H' h 'i I i itM,- !■: 'sa t ■i ': # ;il l< 'f ;' .i; \ Odoher Ns7jember Vicmbir I XXII A XXI D XX F 2 XXI B XX E xix: G 3 XX c XIX F XVIII A 4 XIX D XVIII G XVII B 5 XVIII E XVII A XVI c <5 XVII F XVI B XV D 7 XVI G XV C XIV E 8 XV A XIV D XIII F 9 XIV B XIII E XII G 10 XIII C XII F XI A 11 XII D XI G X B 13 XI E X A IX C 13 X F IX B VIII D 14 IX G VIII C VII E 15 VIII A VII D VI F i6 Vil B VI E V Q 17 VI C V F IV A i8 V D IV G III B 19 IV E III A II C 20 III F II B I D 21 11 G I C X E 22 I A * D XXIX F 23 * B XXIX E XXVIII . G 24 XXIX C XXVIII F XXVII A 25 XXVUI D XXVII G XXVI 25 XXVII E 2 5.XX VI A 25.XXY c 27 XXVI F XXV. XXIV B XXIV D 28 2f.XXV G XXIII C XXIil E 29 XXIV A XXIl D XXII F 5<= XXIII B XXI E XXI G 3' XXII ! C XX A r. < 4^9 A Table {hewing the Dominical ^tter, Gol- den Number andEpaft, according to the Ju- lian account for ever, and in the Gregorian^ till the Year 1700. 1672 1675 1674 1675 1676 1^77 1678 1679 1680 1681 :68i 1683 1684 1685 1(586 1687 1688 1689 1690 1691 GF E D C BA G F E W B A G 13 »4 15 16, FE D C B AG F E D 1692 21 I693'22 169423 169^ 24 CB A G F 1696 25 1697 26 1698 27 1699 ;?3 ED C B A CB A G F ED C B A ~gY E D C BA G F E DC B A G FE D C B AG F E D Year 1672 1673 1674 167J 1^76 1677 1678 1679 1680 x68i i68a 1683 1684 i^8f 1686 1687 1688 1689 G Julian Gregor. N Epaft Epaft 1 11 I 2 22 12 3 3 23 4 14 4 S 25 15 6 6 26 7 I7 7 8 28 18 9 9 29 10 20 10 11 I 21 12 12 2 13 23 »3 14 4 24 15 15 7 16 26 16 17 7 17 18 18 8 19 29 19 The anticipation of the Gregorian Calendar. ' From ^OBober 1582D lo From 24 Ffi*. 1700D.11 From 24 Feb, 1800D.12 From 24 Feb. 1900D.13 From 24 Ffi. 2100D.14 From 24 Feb. 2200D.17! From 24 Feb. 2520D.16 Hh 4 4 i '47Q T abula EpaQiamtn Expanfa. Ill IV VI VII VIII IP I'N jM 4H ''I 7,E pC |o'B XXIX XXVIII XXVII XXVI 12 U Mf i^r 16 q l?|P" 18 n 19 Hi) 20,1 ■ XXV XXIV XXIII XXII XXI VI V IV III II XX XIX XVIII XVII XVI XV XIV XIII XII XI X IX VIII Vii VI 2iik "''I 25 h h'8 ^ I ' 26c |V ^'/Id jiv 28'c III 2pb' in [I XI X IX VIII VII XXII XXI XX XIX xvni XVII XVI XV XIV XIII "I II I XXIX XIV XIII XII XI X XXVIlIIX XXVII yni XXVI |vii XXV i"VI XXIV iV XXV XXIV XXIII XXII XXI XX XIX XVIII XVII XVI I * XXIX XXVIIlIX XXVII VIII XII XI X XXVI XXV XXIV XXIII XXII XXI XX XIX XVIII XVII XXIII XXII "I XXI " XX il, XIX F XV XIV XIII XII XI VII iVI V IV III XVIII XVII XVI XV IXIV Ixxix X XXVIII ,IX XXVII VIII XXVI XXV II I XXIX x XXVIIlIX XIII XII XI XVI [XXVII VIII XV XXVI VII ixiv XXV |VI XIII XXIV ;V XII XXIII IV I I • i - XXIV XXIII XXII XXI XX XTX XVIII XVII XVI VII VI V lY III II I * XXIX >XVIII XXVII XV , j'XXVI Tabula Epadarum Expanfa. 471 IX X XI XII XIII XIV XV VI V IV III II XVII XVI XV XIV XIII XXVIII XXVII XXVI XXV XXIV IX VIII VII VI V XX XIX XVIII XVII XVI r * XXIX XXVIII XXVII XII XI X IX VIII I XXIX XXVIII XXVII XII XI X IX VIII XXIII XXII XXI XX XiX IV III II I XV XIV XIII XII XI XXVI 25 XXIV XXIII XXII VII VI V IV III II I XXIX XXVUI XXVI XXV XXIV XXIII XXII VII VI V IV III xvin XVII XVI XV XIV xxrx XXVIII XXVII j XXVI XXV X IX VIII VII VI XXI XX XIX XVIII XVII XXI XX XIX XVIII XVII 1 II I < XXIX XXVIII XIII XII XI X IX XXIV XXIII XXII XXI XX V IV III II I XVI XV XIV XIII XII XXVI XXVII XXIV XXIII 1—^— iXVI XV XIV XIII IXII 1 XXVII XXVI XV XXIV XXIII VIII VII VI V IV XIX XVIII XVII XVI XV * !xi XXIX X XXVIltlX XXVII VllI XXVI VII 1 XXII XXI XX XIX XVIII 1 :xi ix IX jVIII Ivil XXII XXI XX XIX XVIU' i 'lb II t •XIX XIV XIII XII XI X 1 25 IVI XXIV V XXIII IV XXII III XXI II ) XVII XVI XV XIV XIII ■r-'TTylX 427 Tabula Epadarum Expanfa. M H G F E D C B A 11 t t r q p n m 1 1 k i h g f e d c b \3. 1 XVI XVII ■ XVIII XIX I II XXIII XXII XXI XX XIX IV III II I * XV XIV XIII XII XI XXVI 2f 1 XXIV 1 XXIII , xxir ^ VTII VII VI y IV XIX XVIII XVII XVI XV XVIII XVII XVI XV XIV XXIX XXVIII XXVII XXVI 2T X IX VIII VII VI XXI j XX XIX XVIII XVII III II I * XXIX XIV XIII XII XI X XIII XII XI X IX XXIV XXIII XX n XXI XX V IV III II I XVI XV XIV XIII XII xxvin XXVII XXVI 2^ XXIV IX VIII VII VI V VIII VII VI V IV XIX XVIII XVII XVI ixv * IXI xXK X XXVIII IX XXVII "VIII XXVI VII XXIII XXII XXI XX XIX IV III II I * III jXIV II 'xill I jxn ^ XI XXIX X 25 VI XXtV V XXIII IV XXII III XXI II XVIII XVII XVI XV XIV XXIX XXVIII XXVII XXVI XXVIIIIX XXVII VIII XXVI " VII 2< VI XXIV V XX I y V '-ic XVIII xxrx XVII xxvin XVI jXXVII XIII XII XI X IX XXIV XXIII XXII XXI XX J Tabula iEquationis Epadarum. 473 N| Anni Clirifti. [P 520 P 580 Biff, [a 800 BiiT. (C lb iioo BifT. !c i40o'Ei(r. [petraftis decern di- ebus. H84! 1600 Biff. 1700 iSooj ipoo 2000 Biff. I CC 2100 2200I 2300! 2409 Biff. 2500! 2600 27C0! 2800 Biff. 29001 3600 i3icoj 3200 3300 13400 3500 Biff Anni Chrifti. q 3600 Biff. C p 3700 n 3800 n 390c 400c Biff. c m 4100 1 420c 1 430c Biff. CC 1 44CC k 4500 k 4600 c i 47OC i 4800 Biff. i 4900 c h 5 000 g 5100 h 5200 Biff. c 5300 5400 f i__ 550c f 5600 Biff. e 570c 'e 00 0 0 c 5900 6coo Biff. jd 6100 1 c 6200 1 i 6300 1 1 r 6400 Biff- !c lb 1 6500 \ 474 A Table fhcwing the Dominical Letter both in the Julian and the Gregorian account for ever. Cy. © 9 10 11 12 13 15 16 17 18 19 20 21 22 23 24-I 25 0.6 27 28 Anni Chr. C B D C A : E G i A E D C B A 1 G F E D C B A G F E D C B A G F E D C B A G F E D 1^82 1600 F E D C B A G F E D C B A G F E D C B A G F E D C B A G F E [700 2 Jog 2600 E D C B A G F E D C . B A G F E b C B A G F E D C B A G F • E D C B A G F 1800 27C0 2800 F E D C B A G F E D B G F E D C B~A G F E D C B A G E D F~E G F A G E F D E C D B A G F E D C? B A G ipoo 2000 2^00 C B A G F E D C B A 2100 3000 A G F E D C B A G F E D C B A G F E D C B A G F E D C B A G F E D C B 2iOO 5100 3290 475 1 Afci. Corp 1 LXX (Vfh. Eall. Pent. Chri- Adv. 1 m. XXIII fan. Feb. IMay, May Nov. c XXII d 18 4 2Z 30 10 21 i9 XXI c 19 7 23 Ma. 1 11 22 30 I? XX t 20 6 24 2 12 23 D«. I 2 XIX g ZI 7 25 3 13 14 2 XVIII a 22 8 26 4 14 2f 3 lO XVII b 24 9 27 5 If 26 No. 2 J XVI c 24 10 28 6 16 17 28 i8 XV d J I 29 7 17 23 29 7 XIV e 26 iz 30 8 18 29 XIII t 27 13 SI 9 19 30 1 XII g i8 14 Ap. I lo 20 31 z 4 XI a 29 If 2 11 21 Jait.i 3 X b 40 16 3 12 22 2 A'i>.27 12 IX c SI 17 4 13 -3 3 28 I VIII d Fib.t 18 f 14 24 4 29 jvii e 2 19 6 15 2f f 30 givi t s 20 7 16 26 5 1 V g 4 21 8 17 27 7 2 17 IV a 2Z 9 18 23 8 3 6 III b d 23 10 19 29 9 No. 27 n 7 24 11 20 30 10 28 T4 I Id 8 12 21 31 II 29 J * c 9 26 13 22 yun.1 12 30 XXIX t 10 27 14 23 Z 13 Dec. 1 II XXVIII g 11 28 If 24 3 14 z XXVII a 12 ^2.1 2f 4 If s 19 2XXVI h IS z 17 26 f 16 No. 27 3 XXV.XXtV.C 14 3 18 27 6 »7 z3 4 19 28 7 18 29 e 16 f ao 29 8 19 30 17 6 zt 30 9 20 X»oo. I 'g 18 7 22 3« 10 21 2 « 'a.' 19 8 23 /««.! 11 22 3 b 20 9 24 2 12 23 No. 17 c 21 10 25 ? , I? H 28 ■ 'if! r ' X 7 If Jh ) 4 '.■ 'ill A >.• !-.i ji ' I'' w 'f I'll!" 47^ A Table to convert Sexagenary Degrees and Minutes into Decimals and the contrary. 1 00 37 10 73 20 109 30 145 40 181 5® 2 38 74 110 146 182 ? 39 75 I II 147 183 4 01 40 11 76 21 112 3> 148 41 184 51 5 4> 77 113 149 185 6 12 78 114 150 186 7 43 79 "5 I5» 187 8 02 44 12 80 22 116 32 152 42 188 V- 9 45 81 "7 153 189 lo 46 82 118 154 190 11 03 47 13 83 23 up 33 155 43 191 53 12 i! 84 120 156 191 13 49 85 121 157 193 14 50 86 122 158 194 If 04 5^ 14 87 24 123 35 159 44 195 54 16 52 83 124 i6o 196 17 53 89 125 161 197 18 21 54 £5 90 25 126 11 162 11 198 19 55 91 127 163 199 20 56 92 128 164 200 21 57 16 93 129 165 201 22 06 58 94 26 130 36 166 46 202 5 Minutes Seconds Thirds I 00462962 00C07716 00000128 2 OO92<;92 5 15432 257 01388889 23148- 385 4 01851851 30864 515 5 02314814 0 < ) 0 OJ CO 00 0 00000643 6 02777778 46296 771 7 03240740 54012 poo 8 03703703 61728 1028 9 04166667 69444 1157 lo 04629629 00077160 00001286 11 05092592 084876 1414 12 05555553 092592 1543 13 06018518 100308 1671 14 06481480 108024 1800 15 06944444 00115740 1929 16 07409407 123456 2057 17 07870370 131172 2186 18 08333333 138889 2314 19 08796296 146604 2443 20 09259259 00154320 2572 21 00722222 162036 2700 22 -10185185 169752 2829 23 10648148 177468 2957 24 111 111 II 185184 3086 25 11574074 0019290Q 3215 26 12037037 200616 3 343 27 12500000 208332 3472 28 12962962 216048 3600 29 13425926 223764 3729 30 13888889 00231481 00003858 51 32 33 3 + 35 36 37. 38 39 40 41 42 43 44 "il 46 47 48 49 5£ 51 52 53 ■54 55 5<5 57 53 59 60 'Minutes 44351852 14814814 15277777 15747C40 16203703 16666666 I y 12,^62p 17592592 18055555 18518518 18981481 49444414 1990.7407 •50370370 2O83-333 3 .21296296 2147^9259 22p2222 22085185 55148148 2^1 I 1 1 I 24074074 24537037 25ODOCGO v^-re-1 - ^ ^ ' 25625926 ,26588888 26S51852 2^114 Seconds 00239670 246913 • 254629 262345 270061 C0277777 285493 293209 300925 308640 00316356 3 240--2 531788 339504 ^bL^^o 00354936 362652 370370 378084 385802 00393518 401234 40F950 416666 4^4382 00432098 439814 447530 45525'6 ).2962 Thirds 00C03986 4115 4243 4372 4581 CO004629 - 475S 4886 501.5 5.144 00C05272 5401 5-529 ^5658 57^7 OGO05915 604.1, 6172 6301 643 0 00006558 6687 6815 6944 . . ^^073 OOC0720I 733a 7458 .- 7587 0^007716- \ i "'5^ 1 ^ J 44; i'-' .V'-'. t'' i •• 1 .4' ,8" ■ 4^0 A Table Converting Hours and Minutes intoDe- grees and Minutes of the ^y£qHiitor, and into Hours. 1104.16666667 208.33333333 3|I2.S ^ 16,16666667 5 20.83333333 625.0 7,2^.16666667 833-353333 i3 ?i3 -5 10 4i'66666667 lij45-33 333333 1 ijso. 13 15 16 54.16666667 S8.3333333 5 62.5 66.66606667 i7j-0.83 3 33 333 i8j75-oo 1^,79.16660667 20^83.33333333 21 87.5 22 91.66666667 23 95.83333333 ,2 41100.00000000 Minutes I 0.06944444 2 [ 0.13888888 h 0.20833333 , 4 0.27777777 ! 5. 0.34722212 6 0.41666666 1 0.48611111 8 0-55555J35 9 0.623 ' 10 0.69444444 II 0.76388888 12 0.83333333 13 0.90277777 14 0.97222222 IT 1.04166666 16 l.iiiiiiii 17 1.18035555 18 1.25 1.31944444 20 1.38888888 21 145833333 22 1.52777777 23 1.59721222 1.66666666 il 1.73611111 26 1.80555555 i? 1-875 28' ■1.94444444 ,29 ■2.01388888 ho ^•98333333 4<3I The Decimal parts of a Day and the coatrary. Seconds Minutes Seconds OOI1574O 3,1 2.15277777 .03587963: .00231481 3'2 2.22222222 .03703704 .00347222 33 2.29166666 .0381944.4 .00462962 34 2.36111111 .03935185 .00578703.. ; 35 2.43055555 ■.O4O5C926 .006^4444 36 2.5 .04166666 .00810184 37 2.56944444 .04282467 .00925925 38 2.63888888 .04398148 .01041660 39 2.70833333 .04513888 •.01157405 40 2-77777777 .04629629 .61273148 41 2.84722222 •04745370 .01388888 42 2.91 666666 .0486 i 111 .01504630 43 2.98611111 .04976852 .01626371 44 3-05555555 .05092592 .01736111 45 3.125 .0520835 3 .0185 1853 46 3.19444444 .05324074 .04967593 .47 3.26388888 .05439814 .02083333 .48. 3-333 33333 •05555555 .0219^074 49 3.40277777 .05671296 .02314816 50 3.47222222 .05787037 .02430555 SI 3.54166666 .05902777 .02546295 52 3.6111 nil .06018518 .026(^2037 53 3.68055555 .06134259 .02777777 54 3-75 .0625 .92893 51S 55 3.81944444 .06365741 .03069259 56 3.88888888 .06481481 .0.3125000 = 57 3.9583333.3 ,06597222 .03240741 58 4.02777777 .06712963 .03356482 59 4.0972 2222 .06828704 , .03472222 60 4.16666666 .06944444 I i 2 482 A Catalogue of fome of the mod eminent Git-' tesand Towns in England^ eland in is (hewed the difference of their Meridian from London, with the hightofthe Pole. Canterbury Carlile chejler Coventry Carmarthen Chicbefter Colchejler Darby Dublin in Irelard Durefme Dartmouth Eely Grantham Glocefler Halefax Hartford Hereford Huntington ma Lancafter Leiceflet ^483 i , ^ . .. . . . i ' ' ' : ' ^ 1 1 • . ', - ' Names of the Citties Differ. Merid. Hight Pole Lincoln. Middle of the Ifle of Mm Nottingham Newar!^ Newcafile 0 1 s o 17 s 04 s 03 s q 6 s 53.12 54.22 53.03 53.02 54.53 N. Lufingham Norwich 03 s 04 a 04 £ 0 ^ s 03 s 52.41 52.44 52. i3 5>. 54 52.44 Northampton Oxford Ol^nham Peterborough Richmond Rochefler Rofs St. Michaels Mount inCornwal Stafford 02 s 06 s 03 a 0 10 s 0 23 s 08 s 02 s oils 0 27 s 52. 35 54. 26 51. 28 52.07 50.38 52.55 52.41 52.43 53. 28 Stamford Shrewsbury fCredah in Ireland ■Uppingham in Rutland [warwicl;^ 03 s 06 s 52.40 52.25 tfinchefier u'aurford in Ireland \mrcefler Yarmouth in Suffolk 0 5 s 0 27 s op s 06 a 50.10 52. 22 52. 20 52-45 Yor^ 04 s. 54. 00 ! London .0 00 51. 3? 1 ■ <0.9593447022 01.8258497658 ,42.7851^4^580 --0-3.713-f9i-f4|o - ^04!6 7^9.^753.'^ 2 ^ 62i4l99427l4f; ^5.943 541.^^44.: 89.4066397741 99.209^286451 12.6733271348 05.66I3 3'.57242, '06.5606805144 07.5200253066 08.4484234926 09.4077682848 Nov. Uec: 22.4766159^58 ■35:94061448^ 10.3 361664708 1 I.'2955 U 2'^'6" 495 Th? Moons mean Anomaly and Node Retrograde 1 28 29 JO 98.8229599749 02.4830696036 06.1431792323 09.8032888610 13*4633984897 CO.8355583674 00.866 504973 6 00.89745 1 579^ 00.9283981860 00.9593447922 X The Moons mean Motions in Days. "Days t Mean Anomaly C Node Bctrograd-e 1 2 3 4 5 03.6291630225 07.2583260450 10 8874890675 14.51 66520900 18.1458151 125 CO.O147O951P4 00.0294192208 0O044128832 2 00.05883 844 i 6 CO.O73 5480520 6 7 8 P lO 21.7749781350 25.4041411575 29.0353041800 32.6624672025 36,29 I 6302250 00.0882576624 00.1029672728 00.1176768832 00.1323864936 CO. 1470961049 11 J2 13 H 1 1 i6 I? 18 19 20 21 22 ^3 24 25 26 27 28 29 30 31 39.9207932475 43.549956.1700 47.1-91192925 50.8082823 150 54.4374453375 00.1618057144. 00.1765153248 C0.19I2249352 00.2059345456 00.2206441560 58.0666083600 61.6957713825 65.3249344050 68.9540974275 72.,5832604500 00.23 53537664 00.2 5006 3 3 76 8 00.2647729872 00.2794825976 00.2941922080 76.2124234725 79.8415864950^ 83.4707495-175 87,0999125400 90.7290755625 00.308901 8184 CC.3236114288 00.3383210392 00.3530306496 00,3677402600 94.3582385850 97.9874016075 01.6165646300 05.2457276525 08.8748906750 12.5040536975 00.3824498704 00.3971594808 00.411869O912 CO.4265787O16 00 4412883120 00.4559979224 Kk- 5 500 The Moons mean Motions in Hours. Hours , Mean Longitude | C ApogjEon 1 2 3 4 5 00.152504.5678 1 00.3050091357 00.4575137035 00.6100182713 00.7625228301 00.0012894419 00.0025788838 00.0038683257 00.0041577676 00.00644.72095 6 7 " 8 9 lO 00.91 5O274O7I 01.06753 19749 01.2200365427 01.372541 I 105 01.5250456786 OO.Q077366515 00.0090260934 00.0103 155353 00.0116049772 00.0128944192 11 12 13 14 15 01.6775502464 01.8300548143 01.9825593821 02.1 350639499 02.2875685177 OC.Ol 418386 1 I 00.0154733031 00.0167627450 00.0180521869 00.0193416288 16 17 18 19 20 O2.44OO730S55 02.5925776533 02.7450822211 1 02.8975867891 03.050091 3 569 00.0206310707 00.0219205126 00.0232099545 00.0244993964 00.0257888384 21 22 23 24 03.2025959250 03.3551004928 03 50-6050607 03.6601096285 00,0270782803 00.0283677222 00.0296571642 00.0309466061 i ! mmtrnmrnuMi 50t 1 2 3 4 5 7 8 P lO 11 12 13 H 15 i6 I? 18 19 20 21 22 23 24 The Moons nacan Motions in Hours. Hours C Mean Anomaly 00.15121512^9 00.3024302518 00,4536453778 00.6048605037 00.7560736296 00.9072907556 01.0585058815 01.2097210074 01.3609361333 01.51 21 512593 01.6633663852 01.814581 5 I 12 01.9657966371 02.1 1701 1-630 02.2682268889 02.419^420148 02.5706571407 02.7218722666 02.8730873926 03.9243025185 03.17551764+5 63.32.67327704 03-477947^964 03.6291630223 C Nocfe Retrograde 00.0006 1 2 9004 00.0012258008 00.0018387013 60.00245 16017 CO.0039545021 00.0036774026 00.0042903030 00.0049032034 00.0055 16 1038 00,0061290043 00.0067419047 OO.OO7354S052 00.0079677056 CO.OO8 5 806060 CO.OO9 193 5064. 00.0098064068 00.0104193072 00.01 10722076 00.01 1 645 108 I 00.0122580085 OO.O1287Q9O9O ^00.01 34838094 OO.OI4O967O99 00.0147096 103 K k 4 ) The Moons mean Motions in Minutes of an Hour 'M. 1 2 3 4 5 C M.Long. .0025414 .0050828 .0076242 .0101656 .01 27070 C Apog' ,0000214 .0000429 .0000643 .0000859 .0001074 <; M. Au. .0025202 .0050405 .0075607 .0100810 .0126012 S% Retrog. .0000102 .0000204 .0000306 .0000408 .0000510 .0000612 .COOO714 .0000816 .0000918 .0001021 .0001123 .0001225 .0001327 .0001429 .00015 31 .oco 1633 .0001735 .0001837 .0001939 .0002041 .0002143 I .0002245 .0002347 .00024.42 .0002544 .0002642 .0002744 .0002 846 .0092948 .0003064 .0152484 .0177898 .02033 12 .0228726 .0254141 .0279555 .0304969 .0330383 .0355797 .038121 I .0406624 .0432038 .0457452 .0482867 .0508284 .0533696 .0559110 .0584524 .0609938 .0635352 26 j.0660766 27 .0686180 28 .0711594 29;.0737008 3 o'.o 762422 .0001288 .0001502 .0001716 .0001930 .0002 149 .0002363 .0002577 .0002791 .0003004 .0003218 .0151214 .0176416' .0201618 j .0226820 .0252025 .0277227 .0302429 .0327631 ,0352833 .0378035 .0003432 .0003646 .0003860 .0004079 .0004298 .0004512 .0004726 .0004940 .0005154 .0005368 .0403237 .0428439 .0453641 .0478843 •0504045 .0529247 .0554449 .057965 X .0604853 .0630055 .0005582 .0005795 .0006008 .0006222 .0006437 .0655257 ,0680459 .0705661 .0730863 .0756075 503 The Moons mean Motions in Seconds. c M.Long. 1 C Apog. C M. An. Rctrog. I OCOO423 0000003 0000420 0000002 2 0000847 0000007 0000840 0000003 3 COO1270 0000010 0001260 0000005 4 0001693 000001 3 0001680 0000006 5 0002116 0000016 0002100 OOOOOOq 6 COO2539 0000019 0002520 0000010 7 0002969 0000022 0002940 0000012 8 0003392 0000025 000 3 3 60 0000013 9 000381 5 0000028 0003780 0000015 lO 0004275 0000035 000 L200 0000017 11 0004658 0000038 0004620 00000I9 12 0005078 000004I 000 5O4O 0000020 13 OOO55O4 0000044 0005460 0000022 H 0005930 0000047 0005880 0000023 IS OCO6357 0000050 0006300 0000025 i6 0006 784 000005 3 0006720 0000027 I? 0007207 0000056 0007140 0000028 i8 0007630 0000059 0007560 0000029 19 OCO805O 0000062 000-980 000003 1 20 0008470 000006 5 0008400 000003 3 21 0008893 0000068 0008820 0000035 22 0009316 0000071 0009240 0000036 2.3 OCO9736 0000074 0009660 0000038 24 OOIOI56 0000077 0010080 0000039 25 0 0 0 CO 0000080 00I0^00 0000041 26 0011C08 0000083 0010920 0000043 27 0011434 0000086 0011340 0000044 28 0011860 0000089 0011760 0000047 29 0012287 0000092 0012180 0000049 30 0012714 0000095 0012600 0000051 5o4 ^ oj tht <^qu, of (_ Nedt and inclin.ofhtr MOiLini. © Sig. 0. gf. 6 1 I fe I 7 2 & 8 0 Sig.0.6 S» 1*7 3. z« 8 > e/i Incr. Incr. Incr. ET Red. Red. Red. • o •00000 00000 09444 01166 09471 01166 30 1 •00388 00055 09638 01194 09277 01138 29 2 •007 fo 001II 09805 01222 090ff oiiri 28 •01138 00166 09972 01250 08833 01083 ^7 4 •01J27 00221 lOI II 01277 08611 Ioioff 26 5 .01888 00250 10250 01277 08388 01017 if 6 .02277 00305 10388 0 1 277 08138 01000 7 .02638 00333 10500 oi3of 07861 00972 23 8 .03000 00361 10583 01305 07611 00944 22 9 •03361 00416 10666 oi3of 07305 00916 21 lo .03722 00472 10750 01333 07017 >o 1 00 0 0 1 20 1 I •04083 00517 10805 01333 0671a 00833 19 12 .04444 oof5f 10861 0,333 06416 00805 18 I? .04777 00611 10888 01333 061H 00777 I7 14 .05111 00638 10916 01361 0 00 0 va 00722- 16 If •Of 444 00666 10916 01361 05472 00666 If 16 .05777 00722 10916 01361 Of 138 00638 14 I7 .06111 00777 10862 01361 04805 00611 13 18 .06416 00805 10861 01361 04444 005 ff 12 19 •06722 00833 10805 01333 04111 00517 II 10 .07000 00861 10750 01333 037fo 00472 10 21 .073of 00916 10694 oi3of 03388 00416 9 22 •07583 00944 io6i I 01305 03027 00361 8 25 •07888 00972 10500 01305 02092 00333 7 24 •08111 01000 10388 01277 02611 00305 6 2f .08361 01027 10277 01277 02222 00250 f 26 •08583 OI055 10138 01277 OIf27 00222 4 27 •08823 01083 10000i OIZ5O 01138 00166 3 28 .090 ff 01 111 09833 ; 0I2Z1 00750 00 HI 2 29 •09250 01138 09638 01194 00388 00055 I 30 .09444 oil 66 09472 01166 00000 00006 0 ir.f 1 10.4 9- 3 507 A Table (hewing the mean Motion of the Moon! from the Sun in Years and Months.. Mra C a © in Years C a © in Years Chr. 1600 1610 1640 1660 56.811475)7531 21.5206732464 58.5795367034 '95.6384101604 32.6^72836174 I Z J 4 •) 36.0063707331 72.0127414661 08.0191121993 47.4117836215 83.4181543546 1680 1700 1720 1740 1760 69.7561560744 06.8150305314 43.8739O39884 80.9027774454 17.9916509024 6 7 8 9 10 19.4245250877 55.4308958208 94.8235672430 40.8298379761 76.8362087092 Motion of the Moon from the Sun in Months. II 1 ^ 13 14 15 02.8426794423 42.2353508645 78.2417215976 14.2480923307 50.2.544630638 Feb. Aiar- April 04.9758835440 99.7928106 l60 04.7686941600 06.3582588800 16 17 18 19 20 89.6471344860 25.6535052191 61.6598759512, 97.6662466853 37.0589181075 May June \ Ahjit. 1 1.3341424240 12.9237071440 17.8995906880 22.8754742320 40 60 80 100 200 74.1178362150 11.1767543225 48.2356724300 85.294590537^ 70.5891810750 Sept. Olio. Nov. Dec. 24.4650389520 29.4409224960 3 1.0304872160 36.0063707331 ^00 400 500 600 7OO 55.8837716125 41.1783621500 26.4729526875 11.7675432250 97.0621337625 50S A Table fhewitig the mean Motion of the Moon from the Sun in Days and Hours. C i in Days. I 03.3863 188240 1 2 06.7726376480 2 ■5 10.1589564720 3 4 13-5452752960 4 5 16.93 1 '(941200 5 6 20.3 179129440 6 7 23.70423 I 7680 7 8 27.0905505920 8 9 30.4768694160 9 10 3 3-863 1882400 10 11 37.2495070640 11 12 47.6358258880 12 44.0221447720 13 '4 47.4084635360 14 15 50.-947823600 15 16 54.181lOI1840 16 ^7 57.5674200080 17 18 60.9537388320 18 19 64.3400576560 19 20 67.7263764800 20 21 71.1126953040 21 22 74.4990141280 22 ^3 77-^853329520 23 24; 81.3716517760 24 84.6579-06000 26 88.0442894240 2.7 91 4306082480 28 94.8169270720 29 98.2032458960 30 01.5895647200 31 04.9758835440 a C? in Hours. oo.i^icp66ij6 00.2821932352 OD.4232898530 00.5643864706 00.70 54830882 00.8465797060 00.9876763236 01.1287729412 01.2698695588 01.4109661766 01.55^20627942 01.6931594120 01.8342560296 01.9753526472 02.1164492648 02.2575458824 02.3986425000 02.5397391 176 02.6808357354 02.8219323520 02.9630289708 03.1041255884 03.2452222062 03-3.863 188240 $09 [a tabic fhewing the mean motion of the moon from the sun in minutes. (s) in minutes. c i 0 in minutes. i co.00235 16 102 00.0728999183 2 00,0047032205 32 00,0752515088 3 00.00 7054830s 33 00,077603 1390 4 00.0094064411 34 00.0799547492 5 00.0117^80^ i 3 3s 00,0823063 59j. 6 00.0141096617 36 od.9846579696 - 7 do.01646 12719 37 00.0870095798 • 8 00,0188128822 3^ co.08936119oo 9 00.0211644924 39 00.0917128002 lo 00.0235161029 4-c 00.0940644104 11 00.02586771 3 i 41 00.0964160206 12 00.0262193233 4; 00,0997676308 13 00.030570933 5 43 00.1011192410 00.032922543 7 4-1 00.1034708512 - is 00.0352741539/ 45 00.1058224614 i6 00.0376257644 46 00.1081740716 i? 00.0399773746 47 oo.i 105256818 ih 00.0423289848 00.1 128772920 19 00.0446805950 49 co.i x52289022 20 00.0470322052 5° oo.i 175805124 21 00.0493838154 51 go.1199321226 22 00.05 i 7354^5<5 52 00.1222837328 23 00.0540870358 s3 00.1246353430 24 00.0564386460 s4 00.1269869532 2s 00.0587902562 ss 00.1293385634 26 00.0611418664 56 00.1316901736 27 00.0634934766 s7 00.1340417838 28 00.0658450868 s8 00.1363933940 29 00.0681966970 s9 co.i 387450050 30 00.0705483080 60! 00.1410966152 ^r® A Tabic fhewing the mean Motion of the Moon from the Sun in Seconds. a («) in Seconds » a 0 in Seconds I 00.0000 3 p 1935 31 CO.OOI2149985 2 00.0000783870 32 oo.col 2541920 3 00.0001175805 33 00.OG1293 3855 4 00.0201567740 34 00.0013325790 5 00.0001959675 3 5 00.0013717725 6 00.0002351610 36 00.C014109660 7 00.0002743545 37 00.0014501595 8 00.00031 35480 38 00.0014893 530 9 00.000352741 5 39 00.0015285465 lO 60.00039193 50 40 00.00 T 5677400 11 CO.00043 11285 41 00.0016069335 12 00.0004703220 42 00.001646 1270 13 CO.0005 995155 43 OC.OO168532O5 H 00.0005487090 4 + 00.0017245140 IS 00.0005879025 45 00.091 7'^3 "O75 i6 00.0006270960 46 00.0018029010 17 00.0006662895 47 00.0018420945 i8 00.0007954830 48. 00.0018812880 ^9 GO.OCO7446765 49 OO.CO19204815 20 00.COO7838700 ■ 50 00.0019 5 96750 2l 00.0008230635 51 GO 0019988685 22 00.0008622570 52 00.0020380620 23 00.0009014505 S3 00.0020772555 24 CO.OOO9406440 54 CO.0021 164490 2^ 00.0009798375 55 C0.002I 556425 26 00.00101903 10 56 00.0021948360 27 00.0010582245 57 CO.OO22340295 28 00.0010974180 5« 00.0022732230 29 00.0011366115 59 C0.002 3 124165 30 00.0011758050 60 00.002 35 16 TOO A Catalogue of fbme of the moft notable fixed Stars according to the obfervations of Tycho Brahe^ and by him rectified to the beginning of the Year of Mans Re- demption, idoi. The Names of the Stars firjl Star of Aries. the bright Star in the top'f of the head of Aries. J the South Rye o/Taurus. the North Eye of Taurus. The bright Star of the Pleiades. t})e higher head of Gemini* the lower head of Gemini. the bright foot of (jemini. In the South Arm of Cancer. the bright Star in the nec^ of Leo. the heart of Leo. In the extream of the tail a/Leo. In Virgo's wing; Vindemiatrix. tAirgins Spil^e. South BaUance. Vorth Balldnce. Tbe highejl in the Forehead of Scorpio. The Scorpions heart. Former of the j in the head »/Sagittarius. Northern in the former horn of Capricorn. The left Shoulder 0/ Aquarius. In the mouth of the South Fifh. the Polar Star or laft Star in the~\ ail of the leffer Bear. J L 1 Longit. 07.671 "V b 9.57-N 3 ^ 5.91. S I r. 5.3I.S 1 b 2. 6. S 9 $ 4.1 I.N S S I0.2.N 2 "6.98.N 2 6.48. S 2 7.8. S 9 8.47.N 2 0.26.N I Si 12.18.N I Si id.i J.N 9 m 1.59. S I n: 0.26. N 2 m8.3J.N 2 t i.oj. M 9 yt 4.27. S I y; 1.24. N 4 i!:i;7.22. N 9 X 8.42.N 9 9-4- N 5 '05.400 X d5.02.N2 00.J83 ol.ldp 00.801 od.dao 04.078 04.921 01.069 02298 06.662 od.745 04.458 01.217 05.074 02.649 09.899 07.988 01.171 02.209 07.861 ,04.949 09.620 Latit; 7.8. N 4 t Longit. The Names of tlie Stars. The lalistar in the till of the great Bear, .oj.888 Ihe Tongue of the Dragon. p $.2 59 Arfturus inthe skjrt ofhisGarment. pj.iSi The bright Star oj the North Crovnn. pi.84$ The Head of Eeicaks. ,02.921 The bright Star of the Har^. 02.699 The Head oj Mcdufa. 05*727 The bright Star in the Goats left Shoulder. 04.$ i d Tne middle of the Serpents Necl^. p4*j83 Tl)e bright Star in the eagles Shoulder, 07.264 The bright Star in the Dolphins Tail. 02.570 The mouth of Pcgafus. 07.324 The head of Andromeda. 02.440 in the top oj the Triangle. 00.366 In the Snout of the ryhale. 01.643 Tje bright Star in the nhales Tail. 07481 Bright Shoulder of Orion. 06.444 Middlemofl in the belt of Orion. 04,972 T'elajlin the tail of the Hare. 07.324 Toe great Dogs mouth Sirius. 02.386 Tje lejfer Dog Procyon. ,05.^41 In the top of the Ships Stern. 01.636 Brightefi in Hydra's Heart* 06.044 54.25.N 76.17.N 31.2. N 44.23.N 37.23.N 61.47.N 22.22.N 22.50.N 25.35-N 29.2 i.N 29. 8.N 22. 7.N 25.42.N 16.49.N 7.50. S 20.47. S 16.06 s 24.33. s j[ 38.26. s ^138.30. s s I5-57* S St!43.i8. S (r, 22.24. ^ HI m A m J vCf « X H yf ta Y X II n: The Contents. THE CONTENTS OF THE Firft part, CONTAINING The Praffical Geometry or the Art of Surveying. CHapter I. Of the Defmtiojt dnd Dimfton of Geometry* Chap. 2* Of Figures in the Ceneral^mere particularly of a Circle and the AffeElions thereof. Chap. 3. Of Triangles. Chap. 4. Of Quadrangular and Multangular Figures. Chap. 5. Solid Bodies. L I 2 Chap. The Contents. Chap. 6. Of the meafutring of Lines both Kight and Circular. Chap. 7. Of the meafuring of a Cir-^ fle^ Chap. 8. Of the meafuring of plain Triangles. Chap.p. Of the meafuring of Heights and Defiances. Chap, 10. Of the taking of Difian- ces. Chap. 11. Hoia? to tal^e the Plot of a Field at one Station^ &c. Chap. 12. Hon> to taJ{e the Plot of a Wood^ Parh^^ or other Champian Plane, &c. Chap. 1The Plot of a Field being taken by an Infrument, how to compute the Content thereof in A- cres, Roodsy and Perches, Chap. 14. How to take the Plot of momttainous and unet/en Ground, ' &c. Chap.. 15. Ta reduce Statutemeafure into The Contents. into CuJiomaryj and the contrary. Chap. 16. Of the meafuring of folid Bodies. Tables. Page. A Table of Squares. pp A Table for the Gauging of Wine Vef- fels. 114, A Tablefor the Gauging of Beer and AleVeJfels. 120 A Table (hewing the third part of the Areas of Circles^ in Foot meafure and Decimal parts of a Foot. 132 A Table fhewing the third part of the Area of any Circle inFoot meafarcy not exceeding 10 f. circumf. 136 A Table for the fpeedy finding of the length or Circumference anfwering to any Arch in Degrees and Deci- mal parts. 151 A Common Di