Selected quad for the lemma: reason_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
reason_n	angle_n	equal_a	side_n	2,221	5	9.5367	5	false

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A03380	The schoole of skil containing two bookes: the first, of the sphere, of heauen, of the starres, of their orbes, and of the earth, &c. The second, of the sphericall elements, of the celestiall circles, and of their vses, &c. Orderly set forth according to art, with apt figures and proportions in their proper places, by Tho. Hill.	Hill, Thomas, b. ca. 1528.; Jaggard, William, 1569-1623.	1599 (1599)	STC 13502; ESTC S104125	144,541	253

longitude multiplied by 9. miles doeth produce 18. miles which multiplied againe in it selfe doe produce 324. that is the first quadrate The difference of the latitude being one degrée doth make containe 15. miles which also wrought againe in themselues doe offer the later quadrate which containeth 225. Now of these two quadrates conioyned the roote is of 23. which number is almost the distance of Viteberge in germaine miles from Brunsweeke Another THe longitude of Danske hath 39. degrées the latitude of 62. degrées The longitude of Noriberge hath 28. degrées the latitude is of 49. degrées The difference of the longitudes is of 11. degrées The difference of the latitudes is of 5. degrées The middle latitude is of 51. degrées The miles answering to one degrée in latitude are 9. The difference of the longitudes that is multiplied with the 9. doth yéeld 99. miles which againe multiplied in themselues do produce the first quadrate which containeth 9801. The difference of the latitudes being 5. degrées multiplied by the 15. doeth then produce 75. miles which wrought againe in themselues do offer the later quadrate which containeth 5625. The increase now of the two quadrates comprehendeth 15426. The root containeth 124. And so many are the miles almost betwéene Danske and Noriberge Another THe longitude of Ierusalem hath 66. degrées the latitude is of 31. degrées The longitude of Nazareth hath 67. degrées the latitude is of 32. degrées The difference of the longitudes is 1. degrée The difference of the latitudes is the like one degree The miles assigned to 1. degree in the Parallel of the lesser latitude are 12. The first quadrate doth containe 144. The miles answering to one degree of the difference of the latitude are 15. The later quadrate comprehendeth 225. The increase of the quadrates containeth 369. The root containeth 16. miles Now the distance in a maner is so much betweene Ierusalem and Nazareth And thus by other examples may young practisioners excercise without labour tediousnes and paine to finde the spaces of places giuen by the degrees of the longitudes and latitudes A demonstration of the third rule THe demonstration of this working or instruction is taken out of the last proposition of the first book of Euclide where hee doeth teach and demonstrate that in the tryangle right cornered the quadrate which by the line or side drawne and stretching to maketh a right angle that is equall in the two squares which are caused by the sides containing the right angle Which that you may easier conceaue and vnderstand in the page going before is placed an apte figure to this matter by which a reason not onely of the third but also of the rules of the first second may be practised and declared Also there is repeated those thinges which afore were declared of the Theoricke of the longitudes and latitudes that the yonger practisers may the readier and easier conceaue the rules hereafter taught The line E. F. doeth represent the Equinoctiall on earth lying vnder the celestial Equinoctial cyrcle The line B. C. doth represent the Parallell that is the cyrcle equ●ich ●unt to the Equinoctiall cyrcle drawne ouer the head or Zenith of the city C. The line A. D. doth represent the Parallel yea equidistant to that Equinoctiall drawne by the Zenith of the cities A. and D. The line A. B. E. doeth represent the meridian of the proper city or place A. The line D. CF. doth represent the meridian of the cities C. and D. The declaration of the first rule THe two Cities C. and D. agrée in longitude in that they are vnder one meridian that is they bee distant by like spaces from the West But they haue not alike latitude for that the City C. is nearer to the Equinoctiall than the City D. by thrée degrées To haue therefore the distance or that space betwéene you shall easily finde the same by the degrées of the meridian The declaration of the second rule THe two Cities A. and D. agrée in the latitude or they haue one like eleuation of the Pole in that they are vnder one Parallel and the Zenith of both is by fiue degrées distant from the Equinoctiall But the longitude of them is not alike that is they be not equally distant from the West for the city A. is more Westerly then the citty D. by foure degrées So that the distance is to bee gathered and learned by those degrées betwéene in that Parallell The declaration of the third rule THe two Cities A. an● ● be distant by vnlike spaces aswell from the 〈◊〉 from the Equinoctiall For they be vnder diuers meridians and Parallels The city A. is nearer to the West than the city C. by foure degrées and it is further distant from the Equinoctiall than C by thrée degrées Wherefore by those degrées in which it is nearer to the West and furthest distaunt from the Equinoctiall must the distance of the two cities A. and C. be sought For that the space betwéene the meridiane A. B. passing by the Zenith of the City A. and meridiane C. D. stretching by the Zenith of the city C. containeth foure degrées yet those degrées are not in the great cyrcle in that those two Parallels doe not deuide the earth into two iust halues but into vnequall halues so that of necessity it must follow that the degrées of diuers Parallels haue vnequall spaces Wherfore in the third rule are not the miles answering to the degrées of the lesser eleuation taken except the difference of the latitudes bee small nor the miles taken answering to the degrées of the greater eleuation but the miles are taken answering to the degrées of the middle latitude for that it lacketh in one part may be restored in the other Of the same may the distance in miles be sought according to the longitude After this in that the space betwéene the Parallel A. C. passing by the Zenith of the city A. and the parallel B. C. reaching by the zenith of the city C. containeth thrée degrées and these are the degrées of the meridian that is of the great Cyrcle where to one degrée doe alwaies and euery where fiftéene Germaine miles answere So that the distance of those Cities are easily found according to their latitude And in the same by that multiplication of the miles and degrées the adding of the product by the increase and extraction of the root that the distance of the Cities may necessarily and surely be gathered is thus demonstrated That in euery tryangle right cornered the square which is made by the side is drawne against a right angle and is equall to the two squares which are made by the sides containing a right angle As the quadrate which is made by the drawing of the line A. C. into it selfe that is equall to the squares which are caused by the drawing of the line A. B. into it selfe and B. C. into it selfe which by Arithmeticall practise may more readier and better bee

subtracted the right ascention of y e west part from the right ascention of the mid heauen or noonestead and the remainer or rest as afore taught was distributed into thrée equall parts After that in the ende of the first portion from the noonstead towardes the West the auncients constituted or placed the bound of the ninth house with the circle comming from the poles of y e world and in the bound of the second portion was the beginning of the eight house formed These attained the degrées and partes of the degrées of the Zodiack answering to ech arkes of the Equatoure were sought in the Tables of the right sphere but the houses standing vnder were defined and made like to their opposites And séeing this maner of forming the houses is vnperfect therefore shal here no further be taught of the same In which a e. is the verticall circle crossing a d e c. at right angles f g b. the equatour d g c. the horison d. and c. be the points in which the distinguishers of the houses concurre and méet which also do make equall distinctiōs in the verticall circle and thereby be the houses noted and diuided But the later Astronomers moued by the authority of the incomparable Mathematician Regiomontanus inur̄ted and deuised another order of the houses more agréeing to reason than the former For they deuided the quarters of the equatour comprehended betweene the horison and noonstead into thrée equall spaces and by each section they imagined great circles ioyning in the sections of the Meridian and horison as the former Although all these are plainer and more euidently taught and known in the materiall Sphere yet we thought good to speak somwhat as our possibility serueth in plaine forme Wherefore grant that a f c. is the Meridian a. the Top n. the Northerly pole k. the Southerly pole b. and c. the points of the sections of the horison and Meridian where the distinguishers of the houses concurre and méet which also are imagined by the equall distinctions of the equatour e i l. as to the eie sufficiently appeareth that b i c. is the horison circle d. the easterly point or rising of the equatour from which the first house taketh his beginning The Circle of position AL these Circles being set down the Astronomers notwithstanding do write of another Circle whose vse and office serueth to great purpose for the Art of directing searching other more secret matters in Astronomy and is thereof called the circle of Position which passeth at al times by the former sections of the meridian and Horizone and by the Center of the star or of any other purposed point in heauen like to the soresaid cyrcles whether that star be aboue the earth or vnder the earth That this may clearly appeare marke and consider this figure here expressed where the letter c. representeth the top pointe d. the Northerly Pole e. the opposite pole a g b f. the cyrcle of the position passing by the sections of the horizon and meridian b c d e. the meridian a b. the Horizone g f. the Centers of the stars of which the one is in g. aboue the earth and the other vnder the earth in the point f. And many other cyrcles besides all these which hetherto haue bene described may bee inuented and imagined in the sphere for the necessity of the workings The difinitions names and offices of the foure lesser Circles THe Parallels are lesser cyrcles which from either of the greater circles drawn thwartly on the sphere doe equally difand bee distant from the Equatoure or Zodiacke toward their poles so y ● they doe not deuide the Sphere into equall halfe Spheres but into vnequall portions For séeing the sphere from the middle streacheth or draweth by litle and litle straighter and narower toward the furthest aud highest toppes euen so must the parallels which are distant from the middle and greatest and that by equall spaces on each side agréeing drawe of necessity narrower and so much the narower as they nearer approach vnto the poles As writeth Theodosius in the sixte proposition of his first Booke of the sphere And the same Author in the 14. proposition of his first Book of the sphere and in the sixt of his second Booke writeth that all the parallels haue the same poles agréeing with the greater cyrcles vnto which the parallels are And certaine of the Paralels are applied vnto the plain of the Equatoure others vnto the plaine of the eccliptick These doe as well the fixed starres as the planets placed without the ecclipticke and drawne about the Exe-trée stretched b● the poles of the ecclipticke and Center of the worlde discribe yet do all their centers consist in the Exe-trée of the Zodiack and the middle cyrcle of them and the greatest is the ecclipticke These also doe the same stars and the verticiall or toppe points of each places or any other applied vnto the plaine of the equatour drawne as it were by the first mouer about the Exetrée and poles of the world define And the Centers of these be in the Exe-trée of the worlde or equatoure but the middle and greatest of these is the equatour It is manifest by that afore taught that the sun in euery day doth gaine toward the East against the dayly motion one degrée of the Zodiack and of this hapneth that he in each day through the thwartnesse of the Zodiack describeth a certaine newe cyrcle in heauen and in the nexte day another and so forth by order as the like may be compared by a small corde winded close about a Nun or top beginning from the foote vpward euen so the sun beginning to turne againe at the first degrée of Capricorne doth euery day after change a new Parallel vntill hée become backe vnto the first degrée of Cancer and by and by after returned from Cancer he in the like order goeth vnto the Capricorne so that in the next day following the Sun riseth not with the same Parallel aboue the Horizone that hee did in the morning before nor shall not run the nexte morrow in that Parallel that he did in this day And each of these Parallelles euen as the greater cyrcles containe 360. degrées which bée so much lesser then the degrées of the greater cyrcles and occupy or comprehend somuch the lesser space in heauen as answereth to the vpper face of the earth as by how much the more frō the compasse and largenesse of the greatest cyrcle they lacke by reason of the distance And although they yéeld and be lesse in the quantity yet vnto the degrées of the greatest cyrcles be they agréeable and like as writeth Theodosius in the 14. proposition of his second booke of the sphere These lesser cyrcles do offer and teach sundry vtilities First the Parallels of which on this side and beyond the Equatour are 182 that the sun yearly by his dayly motion describeth and doe expresse the causes of the continuall equallity of

depressed So that the cause of the diuersity of this appearance is onely the swelling of the earth To be briefe the beginnings and spaces of the dayes and nights and that in diuers places of the earth do vary and yet following in a maner one order But this variety could not happen if the earth were not Sphericall and all about equally rounde herein excluding both vallies and the toppes of hilles which applied vnto the body of the earth cause no inequalitie or diuersity at all For the swelling of the earth causeth that the stars be not séene togither in all countries but drawne about by little and little by a certaine succession and order that they so appeare sooner to them in the East part then to them in the West through the swelling as yet not aboue caried which swelling being high betwéene both is a let and cause of the later appearing of them to the west and by that meanes also kéepeth and hideth the stars the longer from their sight So that by these it euidently appeareth that the onely cause is the swelling of the earth If the earth were fashioned with a déepe hollownesse and compassed round about with a light inclosure then should the stars risen be soonest séene to them in the West partes and much later appeare to them in the East For that the higher inclosure to the hollownesse as a wal built about should be a let and hinderance to the sight of the beholders in such sort that those starres arising it shoulde hinder their sight If the earth were formed with places standing in sharp piller forme or in right line vp then should the stars appeare set and be hidden alike to those places and no differences of dayes should be caused but that they shoulde haue one like day and the sun also appearing to that fide which they shewed so that whiles the Sun runneth and compasseth about the backe parts they should be without light of the sun and should remaine al the time in shadow and darknesse And if it should haue a Cubicke for me then should they sée the sun sixe houres and loose or be without light and sight of the sun other eightéene houres If in round piller-wise as if the howndes were playne vnto both the Poles and the hollow partes should decline vnto the East and West then should no stars continually appeare to them dwelling in the hollow but that certaine stars should arise vp and set in the West and other certain stars néere to both the Poles should alwaies be hid To conclude if the whole earth were framed with an equal playnesse throughout then should the stars appeare at one moment to all countries and setting againe should hide the like out of sight and by that meanes shoulde the dayes begin and end alike and no differences shoulde bée obserued To all such arguments seing experience onely doth repugne or contrary them It is therefore manifest that the earth from the West towarde the East riseth vp into an equall swelling If the earth also were plaine from the East vnto the West then shoulde the starres arise so soone to them in the West as to those of the East which is a manifest error Also if the earth wéere playne from the North vnto the South and like from the South vnto the North then the starres which were to some of a continuall appearance should alwaies séene the fame and like which way or into what quarter soeuer a man goeth which also is vntrue But the cause which maketh the earth séene plaine is through the ouer great quantity which causeth it so to appeare to euery mans sight But that the earth is round according to latitude the diuers eleuations of the Pole and stars eyther alwaies in sight or continually hidden doth euidently declare For from the Equatour in going forth easilie towardes the North and that the Pole Articke be higher raysed and the stars néere to the Pole raysed vp then are the Stares right against like depressed and as they were out of sight and so much the more as they go further from the Equatoure nor the Northerly stars neuer set but continually drawne about in sight with heauen But the contrarie happeneth by going from the saide Cyrcle or Equatoure vnto the contrary part So that there is no greater cause of this diuersitie than the swelling of the earth which if the same shoulde bee plaine the starres opposite or right against according to latitude about the Poles shoulde offer and appeare togither to all countries which the swelling of the earth hindreth to be séene An instrument by which the round nesse of the Earth according to latitude may be proued and all those may easily be shewed which are taught of the dayes Artificiall That the Water hath a like swelling and runneth round THis by two reasons is prooued the first is most certaine by a mark or marks standing on the sea banke like as a tower stéeple or such like erected of purpose so that a shippe sayling into the déepe and carried so far off that no more of the sides or bottom can be descerned sauing the top of the mast which only appeareth to the sight Or thus that a marke stoode on the sea banke and a ship passing forth of the hauen sayling so far into the sea that the eie of the beholder being néere the foote of the mast cannot decerne the marke the ship in the meane time staying or standing still so that his eie being in the top of the mast shall perfectly sée that marke but the others eie being néere the foote of the mast shoulde rather better sée the marke than he which is in the top of the mast as may more euidently appeare by lynes drawne from either place vnto the mark so that the manifest cause of this appeareth to bee none other then the swelling of the water But here are all other impediments excluded that may otherwise hinder as mists foggs and such like vapours ascending Also a like reason of the impediments of this aboue written is for that the water ariseth into a swelling which hindreth the sight of the bottom or sides of the ship that being in a high place doeth not hinder the sight of the same as the top of the mast which either excéedeth or is equall with the swelling of the water For men sayling on the mayne sea sée nothing round about but the Sky and the Sea but comming nearer the banks do by litle and litle descry and sée either high hilles or cliffes as if they were rising forth of the water Also to those that dwell on a high ground the sun first ariseth and last setteth And to this agréeth that out of the higher places both more and further may bée séene into the sea then in vallies or lower places By all these therefore it is euident that the vpper face of the water swelleth as by the example following more plainly shall appeare but an other example of the same shall

through the difficultie of measuring And this whole compasse is not onely ment of the earth but of the earth and water ioyntly togither both which are saide to make one Sphere Also Eratostenes gathereth the compasse of all the earthly Orbe by the proportion of the perticular or the degree of the celestiall Cyrcle vnto the like space on earth For he affirmeth that to one degrée of the celestiall Equatour answere 700. furlongs or 15. Germayne myles but Ptolomie attributeth to a degrée 500. furlongs Which is thus to be vnderstoode that a Cyrcle be imagined on earth directly vnder the Equinoctiall or Merydian lyne deuiding the earth into twoe halfes and that this Cyrcle be likewise deuided into 360. parts or degrées as the celestiall Cyrcles are And ech of these parts doth like vnto the celestial parts containe 700. furlonges or 15. Germaine myles This nowe being tryed and found what the whole Summe eyther of the furlongs or myles of the whole cyrcumference of the earth which contayneth 360. parts or degrées you shall easily finde and knowe the same by this maner Multiply the whole compasse of the earth that is the 368. degrées by the 700. furlongs or fiftéene Germayne myles and the whole compasse shal either appeare to be 252000. furlongs or 5400. Germayne myles This whole compasse of the earth deuide by 22. and the number comming thereof shall bee the 22. part of the compasse of it that is 11454 12 22. furlongs or 254 ●0 22. Germayne myles And abate this 22. part from the whole Summe of the circumference and the number in furlongs shall remaine and be 240545 10 22. and in GErmayne miles 5154 1● 22. And if any of these sums be deuided a part by 3. it shal be found in furlongs to be 80181. a halfe and a third part or 3 2. 10 66. And in Germaine myles 1718 4 22. that is the dyameter of the earth aswell in the furlonges as Germayne miles And Archimedes by sundry labours and witty inuentions and by Geometrical practise hath found that the like proportion is of the Circumference of the whole Cyrcle vn to the diameter of the same as is 22. vnto 7. that is the diameter thrice with a seauenth part and a halfe But whensoeuer any man will by the cyrcumference of the Cyrcle gather and finde his diameter worke the numbers thus as this example teacheth First set down 22. at the left hand toward the right hand 7. and the cyrcumference betwéen those two numbers 22. 5400. 7. After multiply the first by the second that is 7. by 5400. the number increased which is 47800. deuide by the thirde that is 22. and you shall finde in the quotient 1718 4 22. Germayne myles Or thus in furlongs the number being set downe alike 22. 252000. 7. then multiplie the first by the second as 7. by 25200. and the increase shall be 1764000. after the increased number deuide by the third as by 22. and the diameter shall be 80181 18 22. If any couet to finde the vpper face of the earth by the dyameter and cyrcumference known worke one into the other and you shal haue that you séeke But if you desire to knowe the thicknesse of the earth then ioyne the superficiall solydenes of the Sphere vnto the sixt part of the diameter and you shall obtaine your desire THE SECOND PART OF THE SPHERICALL Elements of the Celestiall Circles with the vses of the same Circles What is the Summe of this Second Part. WHereas in the first part were only teh rudiments of the Sphere handeled and taught which are also written and contained in diuers Phy●●●e bookes as of the World and the many parts thereof that is of the Ethereall and Elementarie Region And also of the parts motion and forme● of ●he Etheriall Region as Heauen and the for●●●e 〈◊〉 and quantitie of the Earth Here in this second parte shall fully bee ●et●●● th● and largely handled the manifold vses of the Cyrcle of which the materiall Sphere is framed and made Further this second part is deuided into thrée partes the first teacheth the deuision of the Cyrcles in that the auncient Astronomers for a playner instruction deuided heauen into sundry Cyrcles and of these some in greater and other some in lesser Cyrcles In the second part are the definitions descriptions and vtilities of all the Cyrcles taught In the third and last part are the places of the Zones learnedly described and the vtilities of them So that this second part doeth especially intreate of the Cyrcles séeing the principall poynte of the Sphere is of the celestiall appearances which by reason of the celestiall Cyrcles or of the first moouer are caused as may appeare of the ascentions and descentions of the signes by which the whole knowledge aswell of the naturall as artificiall day is learned Wherefore in that this instruction of the ascentions of the signes consisteth in the Cyrcles which the auncient Astronomers imagined to bée in the first mouer therefore is this second part of the celestiall Cyrcles aptely placed and necessarily before taught That the Sphere of the worlde is either right or thwart THe roundnesse of the earth as is afore taught both altereth the standing of the Poles and the whole Sphere of the worlde in diuers partes of the earth For to them which dwell vnder the Equatour either Pole falleth to the playnesse of the Horizōt But to others dwelling without the Equatoure the one Pole is raysed and the other depressed hid through which diuersitie of the standinges of them are these differences caused that the risings and settings of the signes are altered the spaces betwéene the dayes and nights varied whose causes ought diligently to be sought Therefore is the right Sphere distinguished from the thwart Sphere of the worlde In this maner as here you may be holde by these figures following That is called the right Sphere in which either Pole resteth and standeth on the plaine of the Horizont and the Equatoure which there doeth exactly possesse the middle place betwéene the Poles and doeth with the Horrizont make a right sphericall angle of which it is so named a right Sphere For they haue such a standing vpon the Sphere of the worlde as that neyther of the Poles is eleuated aboue the Horizont to them which dwell vnder the Equatoure The thwart declined or bending Sphere is that in which either of the Poles of the world eleuated is séene aboue the Horizont and the other iust somuch set and hidde beneath the Horizont and also that the Equatoure frameth and maketh with the Horizont thwart and vnequall angles And that is called a blunte angle which séeth the Pole eleuated and that a sharpe angle declining vnto the contrary They which dwell on this side and beyonde the Equatoure haue such a Sphere But the same forme and condicion of the thwart Sphere is not euery where nor the positure of it the same reason but that the thwartnesse of the Sphere

that C. B. the ark of the distance of places which reacheth out right is a quarter of the greatest cyrcle Wherefore if the degrées bee multiplied by 15. and the minutes deuided by 4. the distance then shal be knowne As for example Nubarta of Taprobone hath the longitude 121. degrées and 20. minutes but no latitude the city Pyse of the Tuscanes in Italie hath the longitude 31. degrées and 20. minutes almost the latitude of 42. degrées and 11. minutes then the angle of the difference of longitude is right for the difference is of 90. degrées or a whole quadrant These then multiplied by 15. do procreate or bring forth the distance to be of 13 50. Germaine miles Essina a Mart-towne or principal ●itty of Aethiope vnder the gouernment of Aegipt hath the longitude of 70. degrées and 3. minutes but it hath no latitude The Ile of Tyrus hath the longitude of 67. degrées and no minutes the latitude of 33. degrées and 20. minutes The difference of longitude betweene the one and the other is of 3. degrées and 3. minutes The complement of the difference of longitude is of 86. degrées and 57. minutes of the latitude of the place not standing vnder the equatour the complement is 56. degrées and 40. minutes The royall citty Colipolis of Inde aboue the riuer Ganges hath the longitude of 164. degrées and 20. minutes but no latitude knowne The longitude of Tyrus is of 67. degrées and no minutes the latitude hath 33 degrées 20. minutes The difference of longitude greater then the quadrant is of 97. degrées and 20. minutes The quadrant being abated there remaineth 7. degrées and 20. minutes The complement of the latitude of Tyrus is of 56. degrées and 40. minutes If of two places giuen either standeth without the Equatoure toward some one of the opposite quarters and the other vnder the equatoure then is the reason of the standing considered and the angle of the difference of longitude For the one differeth either by like spaces from each bound and is nearer to the Pole the other to the Equatoure The same appeareth by the compared latitudes which like toppes of either place containe the same Parallel the vnlike being distant and the Parallell by a space seperated toward each place doe argue peculiar and proper tops But the angle of the difference of longitude either it is right blunt or sharpe This of the placing and diuersitie of the angles doeth much varie or alter the reason methode of the searching of these If two places giuen haue equall arks of the latitudes and from the middle or halfe of the equatoure bee alike distant and how much so euer the angle of the difference of longitude be as here vnder the difference of longitude is in the first of the example taught yet are the arkes of the latitudes agréeing and equally founde so that in this example appeareth no difference but in the only longitudes of the places offered As for example The longitude of Danske is of 39. degrées and twoe scruples or minutes the latitude of the same hath 54 degrées and 48. minutes The longitude of Lubecke is of 28. degrées and 20. scruples the latitude hath 54. degrées and 48. scruples The difference of longitude consisteth of 10. degrées and 42. minutes The halfe difference is of 5. degrées and 21. scruples The distance on earth betwéene Danske and Lubecke is of 92 Germaine miles and a halfe The great citie Alexandria vnder the Turke after Ptolomie hath the longitude of 122. degrées the latitude of the same is of 41. degrées That famous Toletum or Toledo of Spaine hath a longitude to the same of 10. degrées the latitude of the same is of 41. degrees The difference of longitude betwéene the one and the other is of 102. degrées The halfe difference hath 51. degrées The complement of the equall latitudes of either is of 49. degrées The whole distance betwéen both appeareth to containe 1077. Germaine miles and a halfe If of two places giuen the one bee further distant from the equatour then the other and the greatnesse of the complements of either latitude differing as that the arkes of the latitudes be vnequall so that the diuersity of the angle included with the arks of the complements shal varie the methode or reason of the search for that the one giueth and formeth a right angle another a sharpe another a blunt angle yet to these the angle of the difference of longitude is right The example of two places differing alike both in the longitude and latitude here appeareth The citty Tacola which at this day is called Malchaia or Magna a place of much resort of Marchants This from the West hath the longitude of 160. degrées and 30. minutes of latitude from the equatour it is 4. degrées and 15. scruples distant The other city and place in the countrey of Pontus named Trapezus being a head city of Cappadocia and was the auncient seat of the Emperours This hath the longitude of 70. degrées and 30. minutes and the latitude of the same is of 43. degrées and 5. scruples The difference of longitude betwéene the one and the other is of 90. degrées The arke of distance betwéene both places is of 87. degrées and 6. minutes to which 1306. a halfe Germaine miles answere If the vnequall arkes of the latitudes and angle of the difference of longitude be lesser then the right it canseth a diuers reason of the search by which the arke of the complement of the greater latitude doth varie thrée waies as it is greater or lesser and as with the arke by the second in quisition surely knowne and beeing ioyned forme either more or lesse a quarter of the cyrcle Or thus that the angle which the vnequall complements of the vnequall latitudes include be sharpe that is and if the arks of the latitudes of either place be vnequall and the difference of longitude bee lesser then the quadrant As in this example more plainer appeareth of twoe places beeing of sundry longitudes That worthie citty Trapezus of Cappadocia whose longitude is of 70. degrées and 30. minutes the latitude 43. degrées and 5. minutes The longitude of that well knowne city of Rome hath 39. degrées and 8. scruples the latitude 41. degrées and 8. minutes The difference of longitude betwéene the one and the other is of 33. degrées and 22. minutes Another example not vnlike the former and not much varying from the former as the longitude of Ierusalem which is of 66. degrées and no minutes the latitude of 31. degrées and 40. scruples The longitude of Viteberge being of 30. degrées and 30. minutes the latitude 51. degrées and 50. scruples The difference betwéene the one and the other of longitude is of 35. degrées and 50. scruples If in places vnequally distant from the equatoure the angle of the difference of longitude shalbe blunt by which the difference of longitude shall appeare greater than the quadrant Or thus that

the angle of the difference of longitude be blunt séeing the places are further distant then a whole quarter and thereby causeth a diuers reason and way of serch from the former which semblably the diuers quantity of the complement of the greater latitude doeth thrée manner of waies varie as in the same arke which perfectly knowne by the second is either greater or lesser The example of this appeareth of these two places the noble city Antiochia in Syria which was after caled Seleucia hath the longitude of 106. degrées and no minutes the latitude is of 40. degrées and 40. scruples The other of Toletum whose longitude is of 7. degrées and 4. scruples the latitude hath 37. degrées and 50. minutes The difference of longitude is of 98. degrées and 56. scruples which deducted from the halfe cyrcle or 180. degrées the difference that remaineth vnto the halfe cyrcle is of 81. degrées and 4. minutes The like example not much varying from the former of these two places as the noble city of Portugale named Lysebone whose longitude is of 4. degrées and 18. scruples the latitude hath 39. degrées and 38. scruples The other named Calecute although the latitude differeth hath the longitude of 112. degrées and no minutes the latitude is of 5. degrées and no minutes The difference of longitude containeth 107. degrées and 42. scruples more then the quadrant The same deducted from the halfe cyrcle doth expresse the difference remaining vnto the halfe cyrcle to bee of 72. degrées and 18. minutes The complement of the greater latitude is of 50. degrées and 22. scruples The complement of the lesser latitude is of 85. degrées and no scruples Another example of two places distant from the Equatour of which the one is distant from the middle of the Equatour into the North and the other into the South as this example further instructeth the one beeing the Ile of Thilen which in Ptholomies ●●me was the vttermost bond of the earth knowne Northward that hath the longitude of 33. degrées the latitude Northerly of 63. degrees The other called the Ile of S. Thomas hath the longitude of 27 degrées and 20. minutes the latitude Southerly of 16. degrées The difference of longitude is of 5● degrées ●nd 40. minutes The complement of the latitude Northerly is of 26. degrées A third example of the difference of other two places as Bas●a of Taprobane which Ptholomie affirmeth to bee in longitude 126. degrées and in latitude toward the South 6. degrées and 30. scruples The other named Stocholma in the Realme of Suecia hath the longitude of 42. degrées and 38. scruples and the latitude of 60. degrées 30. scruples The complement of the latitude Boreal is 29. degrées and 30. minutes The common way of measuring of places with their spaces by the rules of longitudes and latitudes HEre before I haue somewhat written of sundry habitable places on the earth whose sundry points differ betwéene the one and the other either in the onely longitude or in the onely latitude or in the longitude and latitude both together Those places which do differ in the onely longitude be distant by equal spaces from the equatoure toward either of the Poles of the worlde the verticiall pointes of those places ended by the same Parallell ioyning next the same space betwéene yet each haue their owne proper meridians being not distant by a like space from the Westerly bounde The distance of these is alwaies gathered and noted in the same Parallel which commonly belongeth to either place standing or hanging right ouer the tops of them Those places which doe differ in the onely latitude are standing ●●der the same meridiane but they haue diuers Parallels ●●d each proper and those continually distant vnequally either towarde one pole from the middle of the Equatour if either place declineth vnto one and the same quarter or otherwise from the middle of the equatoure seuered and distant into the contrary quarters by equall or vnequall spaces If that one of the places looke into the South and the other into the North the distance of these is alwaies accompted in the common meridian Those places which do differ both in the longitude and latitude togither or both decline towarde one Pole of the world or seperated and distant from the midst of the equatoure towarde the opposite Poles as the one looking into y ● North and the other into the south or els by equal Parallels distant from the equatour of which two onely are in the Sphere If they bee reduced and applied vnto one great cyrcle per 3. secundi Theodosij or els bee vnder by vnequall Parallels and by an vnequall space The difference of the longitude of those which either bee towarde them or toward the Poles equally distant is alway gathered in the middle Parallell betwéen either of the bonds by arithmeticall proportion as afore taught But in those places which haue equall Parallels and equally dastant vnto the opposite quarters the difference of longitude is imagined noted in eith●● of the equall Parallels Therefore the arke hath the distance of the places standing by the next space drawne ouerthwart by the pointes of those places which with the arks of the differēce of either both of the longitude and latitude doth forme and make a sphericall tryangle right cornered alwaies in the vpper face of the Globe If that two meridianes méete and ende at the poles of the worlde and beeing cut by the ouerthwart cyrcumferences of the Parallels doe make with the included arkes of them right cornered tryangles through the foure right lesser angles but the angles beeing not right the arke of the distance of the places doth deuide them into two right cornered tryangles One of those tryangles is vsed in the cōmon accompt for the right cornered because in places not farre distant from the equatoure the angles contained betwéene the mutuall sections of the meridians and Parallelles doe not so much varie from the right angles but in places far distant from the equatour they varie very much Now the rules for the diuers standing of places shall be taught in an easie and common maner If places doe differ in the onely longitude TO the searching and knowing of this like as in the former are the longitudes and latitudes of places giuen required by which they being founde séeing in the latitude there is no diuersity the difference of longitude is onely to be considered by deducting the lesser longitude out of the greater and then howemany miles by proportion of the Parallell vnder which the places stand or lie to the equatoure answere to one degrée of the same The same doth that rule set forth in the fourme of a table here following declare beeing drawne and made vnto this vse by the learned in which the miles that answere to one degrée of each Parallell are there founde and noted vnto one degrée of the distance of the Parallell from the equatour If to the whole